LMFDB - Embedded newform 108.4.h.a.35.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,4,Mod(35,108)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("108.35");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	108.h (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.37220628062$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.553553856144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 11x^{6} + 96x^{4} + 704x^{2} + 4096 $ x^8 + 11x^6 + 96x^4 + 704*x^2 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			35.4
Root			$-0.807795 + 2.71062i$ of defining polynomial
Character	$\chi$	$=$	108.35
Dual form			108.4.h.a.71.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.75136 + 0.655739i) q^{2} +(7.14001 + 3.60836i) q^{4} +(-12.1949 + 7.04075i) q^{5} +(21.1499 + 12.2109i) q^{7} +(17.2786 + 14.6099i) q^{8} +O(q^{10})$	\(q+(2.75136 + 0.655739i) q^{2} +(7.14001 + 3.60836i) q^{4} +(-12.1949 + 7.04075i) q^{5} +(21.1499 + 12.2109i) q^{7} +(17.2786 + 14.6099i) q^{8} +(-38.1696 + 11.3750i) q^{10} +(-5.17194 + 8.95806i) q^{11} +(22.8051 + 39.4995i) q^{13} +(50.1839 + 47.4655i) q^{14} +(37.9595 + 51.5274i) q^{16} -18.5146i q^{17} -142.642i q^{19} +(-112.477 + 6.26737i) q^{20} +(-20.1040 + 21.2554i) q^{22} +(-31.4938 - 54.5488i) q^{23} +(36.6443 - 63.4697i) q^{25} +(36.8437 + 123.632i) q^{26} +(106.949 + 163.502i) q^{28} +(-28.4835 - 16.4449i) q^{29} +(72.4520 - 41.8302i) q^{31} +(70.6520 + 166.662i) q^{32} +(12.1408 - 50.9405i) q^{34} -343.896 q^{35} -141.220 q^{37} +(93.5357 - 392.459i) q^{38} +(-313.576 - 56.5121i) q^{40} +(356.247 - 205.679i) q^{41} +(-194.409 - 112.242i) q^{43} +(-69.2515 + 45.2984i) q^{44} +(-50.8811 - 170.735i) q^{46} +(190.623 - 330.169i) q^{47} +(126.713 + 219.473i) q^{49} +(142.441 - 150.599i) q^{50} +(20.3001 + 364.316i) q^{52} +383.851i q^{53} -145.657i q^{55} +(187.042 + 519.986i) q^{56} +(-67.5848 - 63.9237i) q^{58} +(-65.2637 - 113.040i) q^{59} +(312.840 - 541.855i) q^{61} +(226.772 - 67.5805i) q^{62} +(85.1024 + 504.878i) q^{64} +(-556.213 - 321.129i) q^{65} +(-314.510 + 181.583i) q^{67} +(66.8073 - 132.195i) q^{68} +(-946.182 - 225.506i) q^{70} +280.360 q^{71} -178.139 q^{73} +(-388.548 - 92.6037i) q^{74} +(514.702 - 1018.46i) q^{76} +(-218.772 + 126.308i) q^{77} +(595.842 + 344.010i) q^{79} +(-825.706 - 361.110i) q^{80} +(1115.04 - 332.293i) q^{82} +(-362.048 + 627.085i) q^{83} +(130.357 + 225.785i) q^{85} +(-461.288 - 436.300i) q^{86} +(-220.240 + 79.2216i) q^{88} -72.7080i q^{89} +1113.88i q^{91} +(-28.0344 - 503.120i) q^{92} +(740.979 - 783.416i) q^{94} +(1004.30 + 1739.51i) q^{95} +(-434.434 + 752.462i) q^{97} +(204.716 + 686.939i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{2} + 11 q^{4} - 66 q^{5}+O(q^{10}) $ 8 * q + 3 * q^2 + 11 * q^4 - 66 * q^5	$ 8 q + 3 q^{2} + 11 q^{4} - 66 q^{5} - 116 q^{10} + 214 q^{13} + 42 q^{14} + 71 q^{16} - 306 q^{20} + 207 q^{22} - 54 q^{25} + 540 q^{28} + 498 q^{29} - 327 q^{32} + 469 q^{34} - 1256 q^{37} - 1035 q^{38} - 602 q^{40} + 1272 q^{41} - 912 q^{46} - 154 q^{49} + 1329 q^{50} - 464 q^{52} + 1314 q^{56} - 830 q^{58} + 262 q^{61} - 550 q^{64} - 3282 q^{65} + 843 q^{68} - 480 q^{70} + 3940 q^{73} - 222 q^{74} + 105 q^{76} - 330 q^{77} + 4786 q^{82} - 472 q^{85} - 1209 q^{86} - 1425 q^{88} + 1308 q^{92} + 1356 q^{94} - 572 q^{97}+O(q^{100}) $ 8 * q + 3 * q^2 + 11 * q^4 - 66 * q^5 - 116 * q^10 + 214 * q^13 + 42 * q^14 + 71 * q^16 - 306 * q^20 + 207 * q^22 - 54 * q^25 + 540 * q^28 + 498 * q^29 - 327 * q^32 + 469 * q^34 - 1256 * q^37 - 1035 * q^38 - 602 * q^40 + 1272 * q^41 - 912 * q^46 - 154 * q^49 + 1329 * q^50 - 464 * q^52 + 1314 * q^56 - 830 * q^58 + 262 * q^61 - 550 * q^64 - 3282 * q^65 + 843 * q^68 - 480 * q^70 + 3940 * q^73 - 222 * q^74 + 105 * q^76 - 330 * q^77 + 4786 * q^82 - 472 * q^85 - 1209 * q^86 - 1425 * q^88 + 1308 * q^92 + 1356 * q^94 - 572 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	2.75136	+	0.655739i	0.972754	+	0.231839i
$2$	2.75136	+	0.655739i	0.972754	+	0.231839i
$3$		0			0
$3$		0			0
$4$	7.14001	+	3.60836i	0.892501	+	0.451044i
$5$	−12.1949	+	7.04075i	−1.09075	+	0.629744i	−0.933776	−	0.357859i	$-0.883507\pi$
$5$	−12.1949	+	7.04075i	−1.09075	+	0.629744i	−0.156972	+	0.987603i	$0.550173\pi$
$6$		0			0
$7$	21.1499	+	12.2109i	1.14199	+	0.659327i	0.946922	−	0.321463i	$-0.104175\pi$
$7$	21.1499	+	12.2109i	1.14199	+	0.659327i	0.195066	+	0.980790i	$0.437508\pi$
$8$	17.2786	+	14.6099i	0.763615	+	0.645672i
$9$		0			0
$10$	−38.1696	+	11.3750i	−1.20703	+	0.359708i
$11$	−5.17194	+	8.95806i	−0.141763	+	0.245541i	−0.928161	−	0.372180i	$-0.878611\pi$
$11$	−5.17194	+	8.95806i	−0.141763	+	0.245541i	0.786397	+	0.617721i	$0.211944\pi$
$12$		0			0
$13$	22.8051	+	39.4995i	0.486537	+	0.842708i	0.999880	−	0.0154760i	$-0.00492637\pi$
$13$	22.8051	+	39.4995i	0.486537	+	0.842708i	−0.513343	+	0.858184i	$0.671593\pi$
$14$	50.1839	+	47.4655i	0.958016	+	0.906120i
$15$		0			0
$16$	37.9595	+	51.5274i	0.593118	+	0.805116i
$17$		−	18.5146i		−	0.264144i	−0.991240	−	0.132072i	$-0.957837\pi$
$17$		−	18.5146i		−	0.264144i	0.991240	−	0.132072i	$-0.0421631\pi$
$18$		0			0
$19$		−	142.642i		−	1.72233i	−0.508327	−	0.861164i	$-0.669736\pi$
$19$		−	142.642i		−	1.72233i	0.508327	−	0.861164i	$-0.330264\pi$
$20$	−112.477	+	6.26737i	−1.25754	+	0.0700713i
$21$		0			0
$22$	−20.1040	+	21.2554i	−0.194827	+	0.205985i
$23$	−31.4938	−	54.5488i	−0.285518	−	0.494531i	0.687217	−	0.726452i	$-0.258832\pi$
$23$	−31.4938	−	54.5488i	−0.285518	−	0.494531i	−0.972735	+	0.231921i	$0.925499\pi$
$24$		0			0
$25$	36.6443	−	63.4697i	0.293154	−	0.507758i
$26$	36.8437	+	123.632i	0.277909	+	0.932546i
$27$		0			0
$28$	106.949	+	163.502i	0.721840	+	1.10354i
$29$	−28.4835	−	16.4449i	−0.182388	−	0.105302i	0.406026	−	0.913861i	$-0.366914\pi$
$29$	−28.4835	−	16.4449i	−0.182388	−	0.105302i	−0.588414	+	0.808560i	$0.700247\pi$
$30$		0			0
$31$	72.4520	−	41.8302i	0.419767	−	0.242352i	−0.275211	−	0.961384i	$-0.588748\pi$
$31$	72.4520	−	41.8302i	0.419767	−	0.242352i	0.694977	+	0.719031i	$0.255414\pi$
$32$	70.6520	+	166.662i	0.390301	+	0.920687i
$33$		0			0
$34$	12.1408	−	50.9405i	0.0612389	−	0.256948i
$35$	−343.896			−1.66083
$36$		0			0
$37$	−141.220			−0.627472			−0.313736	−	0.949510i	$-0.601581\pi$
$37$	−141.220			−0.627472			−0.313736	+	0.949510i	$0.601581\pi$
$38$	93.5357	−	392.459i	0.399303	−	1.67540i
$39$		0			0
$40$	−313.576	−	56.5121i	−1.23952	−	0.223384i
$41$	356.247	−	205.679i	1.35698	−	0.783456i	0.367769	−	0.929917i	$-0.380122\pi$
$41$	356.247	−	205.679i	1.35698	−	0.783456i	0.989216	+	0.146462i	$0.0467885\pi$
$42$		0			0
$43$	−194.409	−	112.242i	−0.689466	−	0.398063i	0.113946	−	0.993487i	$-0.463651\pi$
$43$	−194.409	−	112.242i	−0.689466	−	0.398063i	−0.803412	+	0.595424i	$0.796984\pi$
$44$	−69.2515	+	45.2984i	−0.237274	+	0.155204i
$45$		0			0
$46$	−50.8811	−	170.735i	−0.163087	−	0.547251i
$47$	190.623	−	330.169i	0.591601	−	1.02468i	−0.402416	−	0.915457i	$-0.631829\pi$
$47$	190.623	−	330.169i	0.591601	−	1.02468i	0.994017	−	0.109226i	$-0.0348373\pi$
$48$		0			0
$49$	126.713	+	219.473i	0.369424	+	0.639862i
$50$	142.441	−	150.599i	0.402885	−	0.425959i
$51$		0			0
$52$	20.3001	+	364.316i	0.0541368	+	0.971568i
$53$			383.851i			0.994831i	0.867512	+	0.497416i	$0.165718\pi$
$53$			383.851i			0.994831i	−0.867512	+	0.497416i	$0.834282\pi$
$54$		0			0
$55$		−	145.657i		−	0.357098i
$56$	187.042	+	519.986i	0.446330	+	1.24082i
$57$		0			0
$58$	−67.5848	−	63.9237i	−0.153006	−	0.144717i
$59$	−65.2637	−	113.040i	−0.144010	−	0.249433i	0.784993	−	0.619505i	$-0.212667\pi$
$59$	−65.2637	−	113.040i	−0.144010	−	0.249433i	−0.929003	+	0.370071i	$0.879333\pi$
$60$		0			0
$61$	312.840	−	541.855i	0.656641	−	1.13734i	−0.324839	−	0.945769i	$-0.605310\pi$
$61$	312.840	−	541.855i	0.656641	−	1.13734i	0.981480	−	0.191566i	$-0.0613566\pi$
$62$	226.772	−	67.5805i	0.464517	−	0.138431i
$63$		0			0
$64$	85.1024	+	504.878i	0.166216	+	0.986089i
$65$	−556.213	−	321.129i	−1.06138	−	0.612788i
$66$		0			0
$67$	−314.510	+	181.583i	−0.573486	+	0.331102i	−0.758540	−	0.651626i	$-0.774087\pi$
$67$	−314.510	+	181.583i	−0.573486	+	0.331102i	0.185054	+	0.982728i	$0.440754\pi$
$68$	66.8073	−	132.195i	0.119141	−	0.235749i
$69$		0			0
$70$	−946.182	−	225.506i	−1.61558	−	0.385044i
$71$	280.360			0.468629			0.234314	−	0.972161i	$-0.424716\pi$
$71$	280.360			0.468629			0.234314	+	0.972161i	$0.424716\pi$
$72$		0			0
$73$	−178.139			−0.285610			−0.142805	−	0.989751i	$-0.545612\pi$
$73$	−178.139			−0.285610			−0.142805	+	0.989751i	$0.545612\pi$
$74$	−388.548	−	92.6037i	−0.610376	−	0.145472i
$75$		0			0
$76$	514.702	−	1018.46i	0.776847	−	1.53718i
$77$	−218.772	+	126.308i	−0.323784	+	0.186937i
$78$		0			0
$79$	595.842	+	344.010i	0.848575	+	0.489925i	0.860170	−	0.510008i	$-0.170357\pi$
$79$	595.842	+	344.010i	0.848575	+	0.489925i	−0.0115945	+	0.999933i	$0.503691\pi$
$80$	−825.706	−	361.110i	−1.15396	−	0.504666i
$81$		0			0
$82$	1115.04	−	332.293i	1.50165	−	0.447508i
$83$	−362.048	+	627.085i	−0.478794	+	0.829295i	−0.999704	−	0.0243162i	$-0.992259\pi$
$83$	−362.048	+	627.085i	−0.478794	+	0.829295i	0.520911	+	0.853611i	$0.325592\pi$
$84$		0			0
$85$	130.357	+	225.785i	0.166343	+	0.288115i
$86$	−461.288	−	436.300i	−0.578394	−	0.547063i
$87$		0			0
$88$	−220.240	+	79.2216i	−0.266792	+	0.0959665i
$89$		−	72.7080i		−	0.0865958i	−0.999062	−	0.0432979i	$-0.986214\pi$
$89$		−	72.7080i		−	0.0865958i	0.999062	−	0.0432979i	$-0.0137865\pi$
$90$		0			0
$91$			1113.88i			1.28315i
$92$	−28.0344	−	503.120i	−0.0317695	−	0.570151i
$93$		0			0
$94$	740.979	−	783.416i	0.813044	−	0.859609i
$95$	1004.30	+	1739.51i	1.08463	+	1.87863i
$96$		0			0
$97$	−434.434	+	752.462i	−0.454743	+	0.787638i	−0.998673	−	0.0514923i	$-0.983602\pi$
$97$	−434.434	+	752.462i	−0.454743	+	0.787638i	0.543930	+	0.839130i	$0.316936\pi$
$98$	204.716	+	686.939i	0.211014	+	0.708075i
$99$		0			0
$100$	490.662	−	320.949i	0.490662	−	0.320949i
$101$	1035.10	+	597.613i	1.01976	+	0.588759i	0.914035	−	0.405636i	$-0.132950\pi$
$101$	1035.10	+	597.613i	1.01976	+	0.588759i	0.105726	+	0.994395i	$0.466283\pi$
$102$		0			0
$103$	24.8677	−	14.3574i	0.0237892	−	0.0137347i	−0.488058	−	0.872811i	$-0.662295\pi$
$103$	24.8677	−	14.3574i	0.0237892	−	0.0137347i	0.511847	+	0.859076i	$0.328961\pi$
$104$	−183.043	+	1015.68i	−0.172585	+	0.957648i
$105$		0			0
$106$	−251.706	+	1056.12i	−0.230640	+	0.967726i
$107$	1378.07			1.24508			0.622539	−	0.782588i	$-0.286101\pi$
$107$	1378.07			1.24508			0.622539	+	0.782588i	$0.286101\pi$
$108$		0			0
$109$	−2087.06			−1.83398			−0.916990	−	0.398910i	$-0.869389\pi$
$109$	−2087.06			−1.83398			−0.916990	+	0.398910i	$0.869389\pi$
$110$	95.5132	−	400.756i	0.0827893	−	0.347369i
$111$		0			0
$112$	173.645	+	1553.32i	0.146499	+	1.31049i
$113$	−929.382	+	536.579i	−0.773707	+	0.446700i	−0.834195	−	0.551469i	$-0.814068\pi$
$113$	−929.382	+	536.579i	−0.773707	+	0.446700i	0.0604883	+	0.998169i	$0.480734\pi$
$114$		0			0
$115$	768.129	+	443.480i	0.622856	+	0.359606i
$116$	−144.033	−	220.196i	−0.115286	−	0.176247i
$117$		0			0
$118$	−105.439	−	353.810i	−0.0822584	−	0.276025i
$119$	226.080	−	391.583i	0.174158	−	0.301650i
$120$		0			0
$121$	612.002	+	1060.02i	0.459806	+	0.796408i
$122$	1216.05	−	1285.70i	0.902429	−	0.954113i
$123$		0			0
$124$	668.247	−	37.2354i	0.483954	−	0.0269665i
$125$		−	728.175i		−	0.521039i
$126$		0			0
$127$		−	887.035i		−	0.619776i	−0.950773	−	0.309888i	$-0.899708\pi$
$127$		−	887.035i		−	0.619776i	0.950773	−	0.309888i	$-0.100292\pi$
$128$	−96.9205	+	1444.91i	−0.0669269	+	0.997758i
$129$		0			0
$130$	−1319.77	−	1248.27i	−0.890393	−	0.842161i
$131$	−1094.78	−	1896.22i	−0.730164	−	1.26468i	−0.956813	−	0.290704i	$-0.906110\pi$
$131$	−1094.78	−	1896.22i	−0.730164	−	1.26468i	0.226649	−	0.973976i	$-0.427223\pi$
$132$		0			0
$133$	1741.78	−	3016.86i	1.13558	−	1.96688i
$134$	−984.404	+	293.363i	−0.634623	+	0.189125i
$135$		0			0
$136$	270.497	−	319.907i	0.170551	−	0.201705i
$137$	409.672	+	236.524i	0.255479	+	0.147501i	0.622270	−	0.782802i	$-0.286210\pi$
$137$	409.672	+	236.524i	0.255479	+	0.147501i	−0.366791	+	0.930303i	$0.619544\pi$
$138$		0			0
$139$	−1524.58	+	880.219i	−0.930313	+	0.537116i	−0.886910	−	0.461941i	$-0.847153\pi$
$139$	−1524.58	+	880.219i	−0.930313	+	0.537116i	−0.0434022	+	0.999058i	$0.513820\pi$
$140$	−2455.42	−	1240.90i	−1.48229	−	0.749107i
$141$		0			0
$142$	771.373	+	183.843i	0.455860	+	0.108646i
$143$	−471.785			−0.275893
$144$		0			0
$145$	463.139			0.265252
$146$	−490.124	−	116.813i	−0.277829	−	0.0662156i
$147$		0			0
$148$	−1008.31	−	509.573i	−0.560020	−	0.283018i
$149$	−369.301	+	213.216i	−0.203049	+	0.117230i	−0.598077	−	0.801439i	$-0.704068\pi$
$149$	−369.301	+	213.216i	−0.203049	+	0.117230i	0.395028	+	0.918669i	$0.370735\pi$
$150$		0			0
$151$	−1450.98	−	837.723i	−0.781980	−	0.451477i	0.0551513	−	0.998478i	$-0.482436\pi$
$151$	−1450.98	−	837.723i	−0.781980	−	0.451477i	−0.837132	+	0.547001i	$0.815769\pi$
$152$	2083.98	−	2464.65i	1.11206	−	1.31520i
$153$		0			0
$154$	−684.747	+	204.062i	−0.358302	+	0.106778i
$155$	−589.032	+	1020.23i	−0.305240	+	0.528691i
$156$		0			0
$157$	651.331	+	1128.14i	0.331095	+	0.573473i	0.982727	−	0.185062i	$-0.0592486\pi$
$157$	651.331	+	1128.14i	0.331095	+	0.573473i	−0.651632	+	0.758535i	$0.725915\pi$
$158$	1413.80	+	1337.21i	0.711872	+	0.673310i
$159$		0			0
$160$	−2035.02	−	1534.99i	−1.00552	−	0.758448i
$161$		−	1538.27i		−	0.752998i
$162$		0			0
$163$		−	622.806i		−	0.299276i	−0.988741	−	0.149638i	$-0.952189\pi$
$163$		−	622.806i		−	0.299276i	0.988741	−	0.149638i	$-0.0478108\pi$
$164$	3285.77	−	183.087i	1.56448	−	0.0871748i
$165$		0			0
$166$	−1407.33	+	1487.93i	−0.658011	+	0.695697i
$167$	−1129.22	−	1955.86i	−0.523241	−	0.906281i	−0.999634	−	0.0270481i	$-0.991389\pi$
$167$	−1129.22	−	1955.86i	−0.523241	−	0.906281i	0.476393	−	0.879233i	$-0.341944\pi$
$168$		0			0
$169$	58.3579	−	101.079i	0.0265625	−	0.0460077i
$170$	210.603	+	706.696i	0.0950149	+	0.318830i
$171$		0			0
$172$	−983.071	−	1502.90i	−0.435805	−	0.666252i
$173$	−545.349	−	314.857i	−0.239665	−	0.138371i	0.375358	−	0.926880i	$-0.377520\pi$
$173$	−545.349	−	314.857i	−0.239665	−	0.138371i	−0.615023	+	0.788509i	$0.710853\pi$
$174$		0			0
$175$	1550.05	−	894.920i	0.669557	−	0.386569i
$176$	−657.910	+	73.5473i	−0.281772	+	0.0314991i
$177$		0			0
$178$	47.6775	−	200.046i	0.0200763	−	0.0842365i
$179$	−2858.75			−1.19370			−0.596852	−	0.802351i	$-0.703582\pi$
$179$	−2858.75			−1.19370			−0.596852	+	0.802351i	$0.703582\pi$
$180$		0			0
$181$	1006.85			0.413475			0.206737	−	0.978396i	$-0.433715\pi$
$181$	1006.85			0.413475			0.206737	+	0.978396i	$0.433715\pi$
$182$	−730.416	+	3064.70i	−0.297484	+	1.24819i
$183$		0			0
$184$	252.783	−	1402.65i	0.101279	−	0.561982i
$185$	1722.17	−	994.296i	0.684414	−	0.395147i
$186$		0			0
$187$	165.855	+	95.7565i	0.0648584	+	0.0374460i
$188$	2552.42	−	1669.58i	0.990183	−	0.647693i
$189$		0			0
$190$	1622.54	+	5444.57i	0.619535	+	2.07890i
$191$	569.125	−	985.754i	0.215604	−	0.373438i	−0.737855	−	0.674959i	$-0.764161\pi$
$191$	569.125	−	985.754i	0.215604	−	0.373438i	0.953459	+	0.301521i	$0.0974945\pi$
$192$		0			0
$193$	−2087.93	−	3616.40i	−0.778717	−	1.34878i	−0.932681	−	0.360702i	$-0.882537\pi$
$193$	−2087.93	−	3616.40i	−0.778717	−	1.34878i	0.153964	−	0.988076i	$-0.450796\pi$
$194$	−1688.70	+	1785.42i	−0.624958	+	0.660751i
$195$		0			0
$196$	112.794	+	2024.26i	0.0411057	+	0.737704i
$197$			962.700i			0.348170i	0.984731	+	0.174085i	$0.0556968\pi$
$197$			962.700i			0.348170i	−0.984731	+	0.174085i	$0.944303\pi$
$198$		0			0
$199$		−	4885.92i		−	1.74047i	−0.492637	−	0.870235i	$-0.663967\pi$
$199$		−	4885.92i		−	1.74047i	0.492637	−	0.870235i	$-0.336033\pi$
$200$	1560.45	−	561.302i	0.551702	−	0.198450i
$201$		0			0
$202$	2456.05	+	2323.00i	0.855480	+	0.809138i
$203$	−401.615	−	695.618i	−0.138856	−	0.240506i
$204$		0			0
$205$	−2896.27	+	5016.49i	−0.986752	+	1.70911i
$206$	77.8348	−	23.1957i	0.0263253	−	0.00784523i
$207$		0			0
$208$	−1169.64	+	2674.47i	−0.389903	+	0.891544i
$209$	1277.79	+	737.734i	0.422903	+	0.244163i
$210$		0			0
$211$	958.429	−	553.349i	0.312706	−	0.180541i	−0.335431	−	0.942065i	$-0.608882\pi$
$211$	958.429	−	553.349i	0.312706	−	0.180541i	0.648137	+	0.761524i	$0.275549\pi$
$212$	−1385.07	+	2740.70i	−0.448713	+	0.887888i
$213$		0			0
$214$	3791.58	+	903.657i	1.21116	+	0.288658i
$215$	3161.07			1.00271
$216$		0			0
$217$	2043.14			0.639158
$218$	−5742.25	−	1368.57i	−1.78401	−	0.425188i
$219$		0			0
$220$	525.583	−	1039.99i	0.161067	−	0.318711i
$221$	731.319	−	422.227i	0.222597	−	0.128516i
$222$		0			0
$223$	413.125	+	238.518i	0.124058	+	0.0716247i	0.560745	−	0.827989i	$-0.310515\pi$
$223$	413.125	+	238.518i	0.124058	+	0.0716247i	−0.436687	+	0.899614i	$0.643848\pi$
$224$	−540.814	+	4387.62i	−0.161315	+	1.30875i
$225$		0			0
$226$	−2908.92	+	866.892i	−0.856189	+	0.255154i
$227$	137.957	−	238.949i	0.0403372	−	0.0698661i	−0.845152	−	0.534526i	$-0.820490\pi$
$227$	137.957	−	238.949i	0.0403372	−	0.0698661i	0.885489	+	0.464660i	$0.153823\pi$
$228$		0			0
$229$	1627.20	+	2818.39i	0.469556	+	0.813294i	0.999394	−	0.0348045i	$-0.0110808\pi$
$229$	1627.20	+	2818.39i	0.469556	+	0.813294i	−0.529839	+	0.848098i	$0.677748\pi$
$230$	1822.60	+	1723.87i	0.522515	+	0.494210i
$231$		0			0
$232$	−251.897	−	700.286i	−0.0712838	−	0.198173i
$233$			3772.46i			1.06069i	0.847781	+	0.530347i	$0.177938\pi$
$233$			3772.46i			1.06069i	−0.847781	+	0.530347i	$0.822062\pi$
$234$		0			0
$235$			5368.52i			1.49023i
$236$	−58.0950	−	1042.60i	−0.0160240	−	0.287575i
$237$		0			0
$238$	878.806	−	929.137i	0.239347	−	0.253055i
$239$	2561.38	+	4436.45i	0.693231	+	1.20071i	0.970773	+	0.239998i	$0.0771468\pi$
$239$	2561.38	+	4436.45i	0.693231	+	1.20071i	−0.277542	+	0.960713i	$0.589520\pi$
$240$		0			0
$241$	335.826	−	581.668i	0.0897612	−	0.155471i	−0.817649	−	0.575717i	$-0.804723\pi$
$241$	335.826	−	581.668i	0.0897612	−	0.155471i	0.907410	+	0.420246i	$0.138056\pi$
$242$	988.745	+	3317.81i	0.262640	+	0.881310i
$243$		0			0
$244$	4188.89	−	2740.01i	1.09904	−	0.718899i
$245$	−3090.50	−	1784.30i	−0.805898	−	0.465285i
$246$		0			0
$247$	5634.28	−	3252.95i	1.45142	−	0.837977i
$248$	1863.01	+	335.747i	0.477020	+	0.0859676i
$249$		0			0
$250$	477.493	−	2003.47i	0.120797	−	0.506843i
$251$	−7151.14			−1.79831			−0.899156	−	0.437629i	$-0.855818\pi$
$251$	−7151.14			−1.79831			−0.899156	+	0.437629i	$0.855818\pi$
$252$		0			0
$253$	651.535			0.161904
$254$	581.664	−	2440.56i	0.143688	−	0.602890i
$255$		0			0
$256$	−1214.15	+	3911.91i	−0.296422	+	0.955057i
$257$	−1550.29	+	895.062i	−0.376282	+	0.217247i	−0.676200	−	0.736718i	$-0.736374\pi$
$257$	−1550.29	+	895.062i	−0.376282	+	0.217247i	0.299917	+	0.953965i	$0.403041\pi$
$258$		0			0
$259$	−2986.80	−	1724.43i	−0.716566	−	0.413709i
$260$	−2812.61	−	4299.88i	−0.670888	−	1.02564i
$261$		0			0
$262$	−1768.72	−	5935.07i	−0.417068	−	1.39950i
$263$	−804.847	+	1394.04i	−0.188703	+	0.326844i	−0.944818	−	0.327595i	$-0.893762\pi$
$263$	−804.847	+	1394.04i	−0.188703	+	0.326844i	0.756115	+	0.654439i	$0.227095\pi$
$264$		0			0
$265$	−2702.60	−	4681.04i	−0.626489	−	1.08511i
$266$	6770.56	−	7158.32i	1.56064	−	1.65002i
$267$		0			0
$268$	−2900.82	+	161.637i	−0.661179	+	0.0368416i
$269$		−	6781.33i		−	1.53704i	−0.639823	−	0.768522i	$-0.720992\pi$
$269$		−	6781.33i		−	1.53704i	0.639823	−	0.768522i	$-0.279008\pi$
$270$		0			0
$271$			4536.90i			1.01696i	0.861073	+	0.508482i	$0.169793\pi$
$271$			4536.90i			1.01696i	−0.861073	+	0.508482i	$0.830207\pi$
$272$	954.010	−	702.807i	0.212667	−	0.156669i
$273$		0			0
$274$	972.058	+	919.402i	0.214322	+	0.202712i
$275$	379.044	+	656.523i	0.0831171	+	0.143963i
$276$		0			0
$277$	−271.730	+	470.650i	−0.0589410	+	0.102089i	−0.893990	−	0.448086i	$-0.852106\pi$
$277$	−271.730	+	470.650i	−0.0589410	+	0.102089i	0.835049	+	0.550175i	$0.185439\pi$
$278$	−4771.88	+	1422.07i	−1.02949	+	0.306799i
$279$		0			0
$280$	−5942.05	−	5024.28i	−1.26823	−	1.07235i
$281$	−2191.32	−	1265.16i	−0.465207	−	0.268587i	0.249024	−	0.968497i	$-0.419890\pi$
$281$	−2191.32	−	1265.16i	−0.465207	−	0.268587i	−0.714231	+	0.699910i	$0.753223\pi$
$282$		0			0
$283$	−8037.89	+	4640.68i	−1.68835	+	0.974769i	−0.732566	+	0.680696i	$0.761677\pi$
$283$	−8037.89	+	4640.68i	−1.68835	+	0.974769i	−0.955783	+	0.294072i	$0.904989\pi$
$284$	2001.77	+	1011.64i	0.418252	+	0.211372i
$285$		0			0
$286$	−1298.05	−	309.368i	−0.268376	−	0.0639627i
$287$	10046.1			2.06621
$288$		0			0
$289$	4570.21			0.930228
$290$	1274.26	+	303.698i	0.258025	+	0.0614958i
$291$		0			0
$292$	−1271.91	−	642.788i	−0.254908	−	0.128823i
$293$	−3008.44	+	1736.92i	−0.599846	+	0.346322i	−0.768981	−	0.639271i	$-0.779236\pi$
$293$	−3008.44	+	1736.92i	−0.599846	+	0.346322i	0.169135	+	0.985593i	$0.445903\pi$
$294$		0			0
$295$	1591.77	+	919.011i	0.314158	+	0.181379i
$296$	−2440.09	−	2063.21i	−0.479147	−	0.405141i
$297$		0			0
$298$	−1155.89	+	344.469i	−0.224695	+	0.0669617i
$299$	1436.44	−	2487.98i	0.277830	−	0.481216i
$300$		0			0
$301$	−2741.15	−	4747.81i	−0.524908	−	0.909167i
$302$	−3442.85	−	3256.35i	−0.656005	−	0.620469i
$303$		0			0
$304$	7349.95	−	5414.61i	1.38667	−	1.02154i
$305$			8810.52i			1.65406i
$306$		0			0
$307$			2913.47i			0.541630i	0.962631	+	0.270815i	$0.0872932\pi$
$307$			2913.47i			0.541630i	−0.962631	+	0.270815i	$0.912707\pi$
$308$	−2017.80	+	112.434i	−0.373295	+	0.0208004i
$309$		0			0
$310$	−2289.65	+	2420.78i	−0.419494	+	0.443520i
$311$	−621.241	−	1076.02i	−0.113271	−	0.196192i	0.803816	−	0.594878i	$-0.202800\pi$
$311$	−621.241	−	1076.02i	−0.113271	−	0.196192i	−0.917087	+	0.398686i	$0.869466\pi$
$312$		0			0
$313$	−2351.28	+	4072.53i	−0.424607	+	0.735441i	−0.996384	−	0.0849687i	$-0.972921\pi$
$313$	−2351.28	+	4072.53i	−0.424607	+	0.735441i	0.571777	+	0.820409i	$0.306254\pi$
$314$	1052.29	+	3531.03i	0.189121	+	0.634609i
$315$		0			0
$316$	3013.01	+	4606.24i	0.536377	+	0.820004i
$317$	7137.68	+	4120.94i	1.26464	+	0.730142i	0.973969	−	0.226680i	$-0.0727872\pi$
$317$	7137.68	+	4120.94i	1.26464	+	0.730142i	0.290674	+	0.956822i	$0.406121\pi$
$318$		0			0
$319$	294.629	−	170.104i	0.0517118	−	0.0298558i
$320$	−4592.54	−	5557.77i	−0.802283	−	0.970902i
$321$		0			0
$322$	1008.70	−	4232.34i	0.174574	−	0.732482i
$323$	−2640.96			−0.454944
$324$		0			0
$325$	3342.70			0.570522
$326$	408.399	−	1713.57i	0.0693838	−	0.291122i
$327$		0			0
$328$	9160.40	+	1650.87i	1.54207	+	0.277909i
$329$	8063.33	−	4655.37i	1.35120	−	0.780117i
$330$		0			0
$331$	−1120.04	−	646.654i	−0.185991	−	0.107382i	0.404114	−	0.914709i	$-0.367580\pi$
$331$	−1120.04	−	646.654i	−0.185991	−	0.107382i	−0.590104	+	0.807327i	$0.700913\pi$
$332$	−4847.77	+	3171.00i	−0.801373	+	0.524190i
$333$		0			0
$334$	−1824.35	−	6121.75i	−0.298874	−	1.00290i
$335$	2556.96	−	4428.78i	0.417019	−	0.722298i
$336$		0			0
$337$	807.929	+	1399.37i	0.130596	+	0.226198i	0.923906	−	0.382619i	$-0.124978\pi$
$337$	807.929	+	1399.37i	0.130596	+	0.226198i	−0.793311	+	0.608817i	$0.791644\pi$
$338$	226.845	−	239.837i	0.0365052	−	0.0385959i
$339$		0			0
$340$	116.038	+	2082.48i	0.0185090	+	0.332171i
$341$			865.372i			0.137427i
$342$		0			0
$343$		−	2187.58i		−	0.344368i
$344$	−1719.27	−	4779.68i	−0.269468	−	0.749136i
$345$		0			0
$346$	−1293.99	−	1223.89i	−0.201056	−	0.190165i
$347$	1787.50	+	3096.05i	0.276537	+	0.478975i	0.970522	−	0.241014i	$-0.0774800\pi$
$347$	1787.50	+	3096.05i	0.276537	+	0.478975i	−0.693985	+	0.719989i	$0.744147\pi$
$348$		0			0
$349$	−1372.37	+	2377.02i	−0.210491	+	0.364582i	−0.951868	−	0.306507i	$-0.900840\pi$
$349$	−1372.37	+	2377.02i	−0.210491	+	0.364582i	0.741377	+	0.671089i	$0.234173\pi$
$350$	4851.58	−	1445.82i	0.740936	−	0.220807i
$351$		0			0
$352$	−1858.38	−	229.062i	−0.281397	−	0.0346848i
$353$	−7100.32	−	4099.37i	−1.07057	−	0.618095i	−0.142234	−	0.989833i	$-0.545429\pi$
$353$	−7100.32	−	4099.37i	−1.07057	−	0.618095i	−0.928338	+	0.371738i	$0.878762\pi$
$354$		0			0
$355$	−3418.97	+	1973.95i	−0.511156	+	0.295116i
$356$	262.356	−	519.136i	0.0390586	−	0.0772869i
$357$		0			0
$358$	−7865.47	−	1874.60i	−1.16118	−	0.276747i
$359$	4315.47			0.634434			0.317217	−	0.948353i	$-0.397252\pi$
$359$	4315.47			0.634434			0.317217	+	0.948353i	$0.397252\pi$
$360$		0			0
$361$	−13487.6			−1.96641
$362$	2770.22	+	660.234i	0.402209	+	0.0958595i
$363$		0			0
$364$	−4019.28	+	7953.13i	−0.578757	+	1.14521i
$365$	2172.39	−	1254.23i	0.311529	−	0.179861i
$366$		0			0
$367$	4930.13	+	2846.41i	0.701228	+	0.404854i	0.807805	−	0.589450i	$-0.200656\pi$
$367$	4930.13	+	2846.41i	0.701228	+	0.404854i	−0.106576	+	0.994305i	$0.533989\pi$
$368$	1615.27	−	3693.44i	0.228809	−	0.523190i
$369$		0			0
$370$	5390.32	−	1606.38i	0.757377	−	0.225707i
$371$	−4687.17	+	8118.42i	−0.655919	+	1.13609i
$372$		0			0
$373$	−3397.58	−	5884.78i	−0.471635	−	0.816896i	0.527838	−	0.849345i	$-0.323003\pi$
$373$	−3397.58	−	5884.78i	−0.471635	−	0.816896i	−0.999473	+	0.0324489i	$0.989669\pi$
$374$	393.536	+	372.219i	0.0544098	+	0.0514625i
$375$		0			0
$376$	8117.44	−	2919.89i	1.11336	−	0.400483i
$377$		−	1500.11i		−	0.204933i
$378$		0			0
$379$		−	606.464i		−	0.0821951i	−0.999155	−	0.0410976i	$-0.986915\pi$
$379$		−	606.464i		−	0.0821951i	0.999155	−	0.0410976i	$-0.0130854\pi$
$380$	893.988	+	16044.0i	0.120686	+	2.16589i
$381$		0			0
$382$	2212.27	−	2338.97i	0.296308	−	0.313278i
$383$	3045.35	+	5274.71i	0.406293	+	0.703721i	0.994471	−	0.105011i	$-0.0334878\pi$
$383$	3045.35	+	5274.71i	0.406293	+	0.703721i	−0.588178	+	0.808732i	$0.700154\pi$
$384$		0			0
$385$	1778.61	−	3080.64i	0.235445	−	0.407802i
$386$	−3373.24	−	11319.2i	−0.444801	−	1.49257i
$387$		0			0
$388$	−5817.01	+	3804.99i	−0.761119	+	0.497859i
$389$	−861.445	−	497.356i	−0.112280	−	0.0648250i	0.442808	−	0.896616i	$-0.353982\pi$
$389$	−861.445	−	497.356i	−0.112280	−	0.0648250i	−0.555088	+	0.831791i	$0.687316\pi$
$390$		0			0
$391$	−1009.95	+	583.096i	−0.130628	+	0.0754179i
$392$	−1017.05	+	5643.44i	−0.131043	+	0.727135i
$393$		0			0
$394$	−631.280	+	2648.74i	−0.0807194	+	0.338684i
$395$	−9688.34			−1.23411
$396$		0			0
$397$	6940.21			0.877378			0.438689	−	0.898639i	$-0.355443\pi$
$397$	6940.21			0.877378			0.438689	+	0.898639i	$0.355443\pi$
$398$	3203.89	−	13442.9i	0.403508	−	1.69305i
$399$		0			0
$400$	4661.43	−	521.098i	0.582679	−	0.0651373i
$401$	−5377.43	+	3104.66i	−0.669666	+	0.386632i	−0.795950	−	0.605362i	$-0.793028\pi$
$401$	−5377.43	+	3104.66i	−0.669666	+	0.386632i	0.126284	+	0.991994i	$0.459695\pi$
$402$		0			0
$403$	3304.55	+	1907.88i	0.408464	+	0.235827i
$404$	5234.20	+	8001.96i	0.644582	+	0.985426i
$405$		0			0
$406$	−648.846	−	2177.25i	−0.0793145	−	0.266146i
$407$	730.382	−	1265.06i	0.0889526	−	0.154070i
$408$		0			0
$409$	7046.57	+	12205.0i	0.851908	+	1.47555i	0.879484	+	0.475928i	$0.157888\pi$
$409$	7046.57	+	12205.0i	0.851908	+	1.47555i	−0.0275761	+	0.999620i	$0.508779\pi$
$410$	−11258.2	+	11903.0i	−1.35610	+	1.43377i
$411$		0			0
$412$	229.362	−	12.7803i	0.0274269	−	0.00152826i
$413$		−	3187.72i		−	0.379800i
$414$		0			0
$415$		−	10196.3i		−	1.20607i
$416$	−4971.86	+	6591.46i	−0.585974	+	0.776858i
$417$		0			0
$418$	3031.91	+	2867.67i	0.354774	+	0.335556i
$419$	−3019.55	−	5230.01i	−0.352064	−	0.609792i	0.634547	−	0.772884i	$-0.281187\pi$
$419$	−3019.55	−	5230.01i	−0.352064	−	0.609792i	−0.986611	+	0.163092i	$0.947853\pi$
$420$		0			0
$421$	211.164	−	365.746i	0.0244453	−	0.0423406i	−0.853544	−	0.521021i	$-0.825551\pi$
$421$	211.164	−	365.746i	0.0244453	−	0.0423406i	0.877989	+	0.478680i	$0.158885\pi$
$422$	2999.84	−	893.986i	0.346043	−	0.103125i
$423$		0			0
$424$	−5608.03	+	6632.43i	−0.642334	+	0.759668i
$425$	−1175.12	−	678.455i	−0.134121	−	0.0774350i
$426$		0			0
$427$	13233.1	−	7640.13i	1.49975	−	0.865882i
$428$	9839.46	+	4972.58i	1.11123	+	0.561586i
$429$		0			0
$430$	8697.25	+	2072.84i	0.975392	+	0.232468i
$431$	−5360.56			−0.599093			−0.299547	−	0.954082i	$-0.596835\pi$
$431$	−5360.56			−0.599093			−0.299547	+	0.954082i	$0.596835\pi$
$432$		0			0
$433$	15780.1			1.75137			0.875683	−	0.482886i	$-0.160412\pi$
$433$	15780.1			1.75137			0.875683	+	0.482886i	$0.160412\pi$
$434$	5621.42	+	1339.77i	0.621744	+	0.148182i
$435$		0			0
$436$	−14901.6	−	7530.84i	−1.63683	−	0.827206i
$437$	−7780.94	+	4492.33i	−0.851745	+	0.491755i
$438$		0			0
$439$	−8408.37	−	4854.57i	−0.914144	−	0.527782i	−0.0323821	−	0.999476i	$-0.510309\pi$
$439$	−8408.37	−	4854.57i	−0.914144	−	0.527782i	−0.881762	+	0.471694i	$0.843643\pi$
$440$	2128.04	−	2516.76i	0.230568	−	0.272686i
$441$		0			0
$442$	2289.00	−	682.146i	0.246327	−	0.0734081i
$443$	2862.60	−	4958.17i	0.307012	−	0.531760i	−0.670696	−	0.741733i	$-0.734004\pi$
$443$	2862.60	−	4958.17i	0.307012	−	0.531760i	0.977707	+	0.209973i	$0.0673376\pi$
$444$		0			0
$445$	511.918	+	886.669i	0.0545332	+	0.0944542i
$446$	980.251	+	927.151i	0.104072	+	0.0984347i
$447$		0			0
$448$	−4365.11	+	11717.3i	−0.460339	+	1.23569i
$449$			1247.80i			0.131152i	0.997848	+	0.0655760i	$0.0208885\pi$
$449$			1247.80i			0.131152i	−0.997848	+	0.0655760i	$0.979112\pi$
$450$		0			0
$451$			4255.04i			0.444261i
$452$	−8571.97	+	477.640i	−0.892016	+	0.0497042i
$453$		0			0
$454$	536.259	−	566.972i	0.0554359	−	0.0586108i
$455$	−7842.56	−	13583.7i	−0.808055	−	1.39959i
$456$		0			0
$457$	807.738	−	1399.04i	0.0826792	−	0.143205i	−0.821721	−	0.569890i	$-0.806986\pi$
$457$	807.738	−	1399.04i	0.0826792	−	0.143205i	0.904400	+	0.426686i	$0.140319\pi$
$458$	2628.88	+	8821.43i	0.268209	+	0.899996i
$459$		0			0
$460$	3884.22	+	5938.13i	0.393701	+	0.601885i
$461$	8888.65	+	5131.86i	0.898016	+	0.518470i	0.876556	−	0.481300i	$-0.159835\pi$
$461$	8888.65	+	5131.86i	0.898016	+	0.518470i	0.0214602	+	0.999770i	$0.493168\pi$
$462$		0			0
$463$	8462.41	−	4885.77i	0.849420	−	0.490413i	−0.0110353	−	0.999939i	$-0.503513\pi$
$463$	8462.41	−	4885.77i	0.849420	−	0.490413i	0.860455	+	0.509526i	$0.170179\pi$
$464$	−233.854	−	2091.92i	−0.0233975	−	0.209300i
$465$		0			0
$466$	−2473.75	+	10379.4i	−0.245910	+	1.03180i
$467$	18652.1			1.84821			0.924106	−	0.382136i	$-0.124811\pi$
$467$	18652.1			1.84821			0.924106	+	0.382136i	$0.124811\pi$
$468$		0			0
$469$	−8869.16			−0.873219
$470$	−3520.35	+	14770.8i	−0.345493	+	1.44963i
$471$		0			0
$472$	523.835	−	2906.67i	0.0510836	−	0.283454i
$473$	2010.94	−	1161.02i	0.195482	−	0.112862i
$474$		0			0
$475$	−9053.43	−	5227.00i	−0.874526	−	0.504908i
$476$	3027.19	−	1980.13i	0.291493	−	0.190670i
$477$		0			0
$478$	4138.15	+	13885.9i	0.395972	+	1.32871i
$479$	−5940.11	+	10288.6i	−0.566619	+	0.981413i	0.430278	+	0.902696i	$0.358415\pi$
$479$	−5940.11	+	10288.6i	−0.566619	+	0.981413i	−0.996897	+	0.0787164i	$0.974918\pi$
$480$		0			0
$481$	−3220.54	−	5578.13i	−0.305289	−	0.528776i
$482$	1305.40	−	1380.17i	0.123360	−	0.130425i
$483$		0			0
$484$	544.778	+	9776.87i	0.0511625	+	0.918188i
$485$		−	12235.0i		−	1.14549i
$486$		0			0
$487$			5781.59i			0.537965i	0.963145	+	0.268982i	$0.0866874\pi$
$487$			5781.59i			0.537965i	−0.963145	+	0.268982i	$0.913313\pi$
$488$	13321.9	−	4791.96i	1.23577	−	0.444512i
$489$		0			0
$490$	−7333.06	−	6935.83i	−0.676069	−	0.639447i
$491$	−5097.04	−	8828.33i	−0.468485	−	0.811440i	0.530866	−	0.847456i	$-0.321867\pi$
$491$	−5097.04	−	8828.33i	−0.468485	−	0.811440i	−0.999351	+	0.0360157i	$0.988533\pi$
$492$		0			0
$493$	−304.472	+	527.361i	−0.0278149	+	0.0481767i
$494$	17635.0	−	5255.44i	1.60615	−	0.478651i
$495$		0			0
$496$	4905.65	+	2145.41i	0.444093	+	0.194217i
$497$	5929.59	+	3423.45i	0.535168	+	0.308979i
$498$		0			0
$499$	−625.048	+	360.872i	−0.0560741	+	0.0323744i	−0.527775	−	0.849384i	$-0.676974\pi$
$499$	−625.048	+	360.872i	−0.0560741	+	0.0323744i	0.471701	+	0.881759i	$0.343640\pi$
$500$	2627.51	−	5199.18i	0.235012	−	0.465028i
$501$		0			0
$502$	−19675.4	−	4689.29i	−1.74932	−	0.416919i
$503$	−11611.0			−1.02925			−0.514623	−	0.857416i	$-0.672068\pi$
$503$	−11611.0			−1.02925			−0.514623	+	0.857416i	$0.672068\pi$
$504$		0			0
$505$	−16830.6			−1.48307
$506$	1792.61	+	427.237i	0.157493	+	0.0375356i
$507$		0			0
$508$	3200.74	−	6333.44i	0.279547	−	0.553151i
$509$	−10673.3	+	6162.22i	−0.929440	+	0.536612i	−0.886634	−	0.462471i	$-0.846963\pi$
$509$	−10673.3	+	6162.22i	−0.929440	+	0.536612i	−0.0428055	+	0.999083i	$0.513630\pi$
$510$		0			0
$511$	−3767.62	−	2175.24i	−0.326164	−	0.188311i
$512$	−5905.75	+	9966.94i	−0.509765	+	0.860313i
$513$		0			0
$514$	−4852.35	+	1446.05i	−0.416397	+	0.124091i
$515$	−202.173	+	350.175i	−0.0172987	+	0.0299622i
$516$		0			0
$517$	1971.78	+	3415.23i	0.167735	+	0.290525i
$518$	−7086.99	−	6703.09i	−0.601128	−	0.568565i
$519$		0			0
$520$	−4918.93	−	13674.9i	−0.414825	−	1.15324i
$521$		−	18831.0i		−	1.58350i	−0.610847	−	0.791749i	$-0.709171\pi$
$521$		−	18831.0i		−	1.58350i	0.610847	−	0.791749i	$-0.290829\pi$
$522$		0			0
$523$		−	14290.2i		−	1.19478i	−0.801953	−	0.597388i	$-0.796205\pi$
$523$		−	14290.2i		−	1.19478i	0.801953	−	0.597388i	$-0.203795\pi$
$524$	−974.527	−	17489.4i	−0.0812451	−	1.45807i
$525$		0			0
$526$	−3128.55	+	3307.73i	−0.259337	+	0.274190i
$527$	−774.470	−	1341.42i	−0.0640161	−	0.110879i
$528$		0			0
$529$	4099.78	−	7101.03i	0.336959	−	0.583631i
$530$	−4366.30	−	14651.5i	−0.357849	−	1.20079i
$531$		0			0
$532$	23322.3	−	15255.4i	1.90065	−	1.24325i
$533$	16248.5	+	9381.05i	1.32045	+	0.762361i
$534$		0			0
$535$	−16805.5	+	9702.67i	−1.35807	+	0.784080i
$536$	−8087.21	−	1457.46i	−0.651706	−	0.117449i
$537$		0			0
$538$	4446.78	−	18657.9i	0.356347	−	1.49517i
$539$	−2621.40			−0.209483
$540$		0			0
$541$	5015.04			0.398546			0.199273	−	0.979944i	$-0.436142\pi$
$541$	5015.04			0.398546			0.199273	+	0.979944i	$0.436142\pi$
$542$	−2975.03	+	12482.7i	−0.235772	+	0.989256i
$543$		0			0
$544$	3085.69	−	1308.10i	0.243194	−	0.103096i
$545$	25451.5	−	14694.4i	2.00041	−	1.15494i
$546$		0			0
$547$	20650.0	+	11922.3i	1.61413	+	0.931920i	0.988398	+	0.151885i	$0.0485342\pi$
$547$	20650.0	+	11922.3i	1.61413	+	0.931920i	0.625735	+	0.780036i	$0.284799\pi$
$548$	2071.60	+	3167.03i	0.161486	+	0.246877i
$549$		0			0
$550$	612.379	+	2054.89i	0.0474763	+	0.159310i
$551$	−2345.73	+	4062.93i	−0.181364	+	0.314132i
$552$		0			0
$553$	8401.34	+	14551.5i	0.646042	+	1.11898i
$554$	−1056.25	+	1116.75i	−0.0810033	+	0.0856426i
$555$		0			0
$556$	−14061.7	+	783.533i	−1.07257	+	0.0597647i
$557$			17758.0i			1.35086i	0.737423	+	0.675431i	$0.236042\pi$
$557$			17758.0i			1.35086i	−0.737423	+	0.675431i	$0.763958\pi$
$558$		0			0
$559$		−	10238.7i		−	0.774691i
$560$	−13054.1	−	17720.1i	−0.985067	−	1.33716i
$561$		0			0
$562$	−5199.50	−	4917.85i	−0.390263	−	0.369122i
$563$	5239.99	+	9075.93i	0.392254	+	0.679405i	0.992747	−	0.120226i	$-0.0383620\pi$
$563$	5239.99	+	9075.93i	0.392254	+	0.679405i	−0.600492	+	0.799631i	$0.705029\pi$
$564$		0			0
$565$	7555.84	−	13087.1i	0.562613	−	0.974474i
$566$	−25158.2	+	7497.43i	−1.86834	+	0.556785i
$567$		0			0
$568$	4844.24	+	4096.03i	0.357852	+	0.302580i
$569$	−16336.8	−	9432.07i	−1.20365	−	0.694926i	−0.242283	−	0.970206i	$-0.577896\pi$
$569$	−16336.8	−	9432.07i	−1.20365	−	0.694926i	−0.961364	+	0.275279i	$0.911230\pi$
$570$		0			0
$571$	−19039.1	+	10992.2i	−1.39538	+	0.805621i	−0.993904	−	0.110250i	$-0.964835\pi$
$571$	−19039.1	+	10992.2i	−1.39538	+	0.805621i	−0.401473	+	0.915871i	$0.631502\pi$
$572$	−3368.55	−	1702.37i	−0.246235	−	0.124440i
$573$		0			0
$574$	27640.5	+	6587.63i	2.00992	+	0.479029i
$575$	−4616.27			−0.334803
$576$		0			0
$577$	−14949.4			−1.07860			−0.539298	−	0.842115i	$-0.681311\pi$
$577$	−14949.4			−1.07860			−0.539298	+	0.842115i	$0.681311\pi$
$578$	12574.3	+	2996.87i	0.904883	+	0.215663i
$579$		0			0
$580$	3306.82	+	1671.17i	0.236738	+	0.119641i
$581$	−15314.5	+	8841.86i	−1.09355	+	0.631363i
$582$		0			0
$583$	−3438.56	−	1985.26i	−0.244272	−	0.141031i
$584$	−3077.99	−	2602.59i	−0.218096	−	0.184411i
$585$		0			0
$586$	−9416.29	+	2806.16i	−0.663794	+	0.197818i
$587$	12448.8	−	21562.0i	0.875329	−	1.51612i	0.0189181	−	0.999821i	$-0.493978\pi$
$587$	12448.8	−	21562.0i	0.875329	−	1.51612i	0.856411	−	0.516294i	$-0.172689\pi$
$588$		0			0
$589$	−5966.73	−	10334.7i	−0.417410	−	0.722976i
$590$	3776.92	+	3572.32i	0.263548	+	0.249271i
$591$		0			0
$592$	−5360.66	−	7276.71i	−0.372165	−	0.505188i
$593$			9675.27i			0.670010i	0.942216	+	0.335005i	$0.108738\pi$
$593$			9675.27i			0.670010i	−0.942216	+	0.335005i	$0.891262\pi$
$594$		0			0
$595$			6367.10i			0.438699i
$596$	−3406.17	+	189.796i	−0.234097	+	0.0130442i
$597$		0			0
$598$	5583.62	−	5903.41i	0.381825	−	0.403693i
$599$	9430.05	+	16333.3i	0.643241	+	1.11413i	0.984705	+	0.174232i	$0.0557441\pi$
$599$	9430.05	+	16333.3i	0.643241	+	1.11413i	−0.341463	+	0.939895i	$0.610923\pi$
$600$		0			0
$601$	6948.70	−	12035.5i	0.471619	−	0.816869i	−0.527853	−	0.849336i	$-0.677003\pi$
$601$	6948.70	−	12035.5i	0.471619	−	0.816869i	0.999473	+	0.0324666i	$0.0103363\pi$
$602$	−4428.58	−	14860.4i	−0.299826	−	1.00609i
$603$		0			0
$604$	−7337.21	−	11217.0i	−0.494283	−	0.755651i
$605$	−14926.7	−	8617.91i	−1.00307	−	0.579120i
$606$		0			0
$607$	−421.745	+	243.495i	−0.0282012	+	0.0162819i	−0.514034	−	0.857770i	$-0.671850\pi$
$607$	−421.745	+	243.495i	−0.0282012	+	0.0162819i	0.485833	+	0.874052i	$0.338516\pi$
$608$	23773.0	−	10077.9i	1.58573	−	0.672226i
$609$		0			0
$610$	−5777.40	+	24240.9i	−0.383476	+	1.60900i
$611$	17388.7			1.15134
$612$		0			0
$613$	−9927.03			−0.654077			−0.327038	−	0.945011i	$-0.606051\pi$
$613$	−9927.03			−0.654077			−0.327038	+	0.945011i	$0.606051\pi$
$614$	−1910.48	+	8016.01i	−0.125571	+	0.526873i
$615$		0			0
$616$	−5625.43	−	1013.80i	−0.367946	−	0.0663105i
$617$	−5109.14	+	2949.76i	−0.333365	+	0.192468i	−0.657334	−	0.753599i	$-0.728316\pi$
$617$	−5109.14	+	2949.76i	−0.333365	+	0.192468i	0.323969	+	0.946068i	$0.394983\pi$
$618$		0			0
$619$	3876.03	+	2237.83i	0.251682	+	0.145308i	0.620534	−	0.784180i	$-0.286916\pi$
$619$	3876.03	+	2237.83i	0.251682	+	0.145308i	−0.368852	+	0.929488i	$0.620249\pi$
$620$	−7887.06	+	5159.04i	−0.510890	+	0.334181i
$621$		0			0
$622$	−1003.67	−	3367.90i	−0.0647003	−	0.217107i
$623$	887.830	−	1537.77i	0.0570950	−	0.0988914i
$624$		0			0
$625$	9707.43	+	16813.8i	0.621275	+	1.07608i
$626$	−9139.73	+	9663.18i	−0.583542	+	0.616963i
$627$		0			0
$628$	579.787	+	10405.2i	0.0368408	+	0.661164i
$629$			2614.64i			0.165743i
$630$		0			0
$631$			24373.4i			1.53770i	0.639429	+	0.768851i	$0.279171\pi$
$631$			24373.4i			1.53770i	−0.639429	+	0.768851i	$0.720829\pi$
$632$	5269.40	+	14649.2i	0.331654	+	0.922016i
$633$		0			0
$634$	16936.1	+	16018.7i	1.06091	+	1.00344i
$635$	6245.39	+	10817.3i	0.390300	+	0.676020i
$636$		0			0
$637$	−5779.38	+	10010.2i	−0.359478	+	0.622633i
$638$	922.177	−	274.819i	0.0572247	−	0.0170536i
$639$		0			0
$640$	−8991.29	−	18302.9i	−0.555331	−	1.13045i
$641$	−9760.55	−	5635.25i	−0.601433	−	0.347237i	0.168172	−	0.985758i	$-0.446214\pi$
$641$	−9760.55	−	5635.25i	−0.601433	−	0.347237i	−0.769605	+	0.638520i	$0.779547\pi$
$642$		0			0
$643$	22870.6	−	13204.3i	1.40269	−	0.809842i	0.408019	−	0.912973i	$-0.366220\pi$
$643$	22870.6	−	13204.3i	1.40269	−	0.809842i	0.994668	+	0.103132i	$0.0328863\pi$
$644$	5550.63	−	10983.3i	0.339636	−	0.672052i
$645$		0			0
$646$	−7266.23	−	1731.78i	−0.442548	−	0.105474i
$647$	14186.7			0.862037			0.431018	−	0.902343i	$-0.358154\pi$
$647$	14186.7			0.862037			0.431018	+	0.902343i	$0.358154\pi$
$648$		0			0
$649$	1350.16			0.0816616
$650$	9196.99	+	2191.94i	0.554978	+	0.132269i
$651$		0			0
$652$	2247.31	−	4446.85i	0.134987	−	0.267104i
$653$	22369.2	−	12914.9i	1.34054	−	0.773963i	0.353656	−	0.935376i	$-0.384938\pi$
$653$	22369.2	−	12914.9i	1.34054	−	0.773963i	0.986887	+	0.161413i	$0.0516050\pi$
$654$		0			0
$655$	26701.6	+	15416.2i	1.59285	+	0.919632i
$656$	24121.1	+	10549.0i	1.43562	+	0.627848i
$657$		0			0
$658$	25237.9	−	7521.16i	1.49525	−	0.445601i
$659$	−16381.0	+	28372.7i	−0.968305	+	1.67715i	−0.267843	+	0.963463i	$0.586311\pi$
$659$	−16381.0	+	28372.7i	−0.968305	+	1.67715i	−0.700462	+	0.713690i	$0.747022\pi$
$660$		0			0
$661$	8946.09	+	15495.1i	0.526418	+	0.911783i	0.999526	+	0.0307787i	$0.00979870\pi$
$661$	8946.09	+	15495.1i	0.526418	+	0.911783i	−0.473108	+	0.881004i	$0.656868\pi$
$662$	−2657.60	−	2513.63i	−0.156028	−	0.147576i
$663$		0			0
$664$	−15417.3	+	5545.69i	−0.901067	+	0.324118i
$665$			49053.9i			2.86049i
$666$		0			0
$667$			2071.65i			0.120262i
$668$	−1005.18	−	18039.5i	−0.0582209	−	1.04486i
$669$		0			0
$670$	9939.24	−	10508.5i	0.573114	−	0.605937i
$671$	3235.98	+	5604.88i	0.186175	+	0.322465i
$672$		0			0
$673$	−7354.65	+	12738.6i	−0.421250	+	0.729626i	−0.996062	−	0.0886596i	$-0.971742\pi$
$673$	−7354.65	+	12738.6i	−0.421250	+	0.729626i	0.574812	+	0.818285i	$0.305075\pi$
$674$	1305.28	+	4379.98i	0.0745959	+	0.250312i
$675$		0			0
$676$	781.404	−	511.128i	0.0444586	−	0.0290810i
$677$	−3170.02	−	1830.21i	−0.179962	−	0.103901i	0.407313	−	0.913289i	$-0.366466\pi$
$677$	−3170.02	−	1830.21i	−0.179962	−	0.103901i	−0.587275	+	0.809388i	$0.699799\pi$
$678$		0			0
$679$	−18376.5	+	10609.7i	−1.03862	+	0.599649i
$680$	−1046.30	+	5805.75i	−0.0590056	+	0.327412i
$681$		0			0
$682$	−567.459	+	2380.95i	−0.0318609	+	0.133683i
$683$	−924.804			−0.0518106			−0.0259053	−	0.999664i	$-0.508247\pi$
$683$	−924.804			−0.0518106			−0.0259053	+	0.999664i	$0.508247\pi$
$684$		0			0
$685$	−6661.23			−0.371551
$686$	1434.48	−	6018.84i	0.0798380	−	0.334986i
$687$		0			0
$688$	−1596.13	−	14278.0i	−0.0884475	−	0.791198i
$689$	−15162.0	+	8753.76i	−0.838352	+	0.484023i
$690$		0			0
$691$	−5345.61	−	3086.29i	−0.294293	−	0.169910i	0.345583	−	0.938388i	$-0.387681\pi$
$691$	−5345.61	−	3086.29i	−0.294293	−	0.169910i	−0.639876	+	0.768478i	$0.721014\pi$
$692$	−2757.68	−	4215.90i	−0.151490	−	0.231596i
$693$		0			0
$694$	2887.87	+	9690.49i	0.157957	+	0.530037i
$695$	12394.8	−	21468.4i	0.676491	−	1.17172i
$696$		0			0
$697$	−3808.07	−	6595.77i	−0.206945	−	0.358440i
$698$	−5334.60	+	5640.13i	−0.289280	+	0.305848i
$699$		0			0
$700$	14296.5	−	796.619i	0.771940	−	0.0430134i
$701$		−	32474.6i		−	1.74971i	−0.484381	−	0.874857i	$-0.660955\pi$
$701$		−	32474.6i		−	1.74971i	0.484381	−	0.874857i	$-0.339045\pi$
$702$		0			0
$703$			20143.9i			1.08071i
$704$	−4962.87	−	1848.84i	−0.265689	−	0.0989786i
$705$		0			0
$706$	−16847.4	−	15934.8i	−0.898105	−	0.849455i
$707$	14594.8	+	25278.9i	0.776370	+	1.34471i
$708$		0			0
$709$	7238.22	−	12537.0i	0.383409	−	0.664084i	−0.608138	−	0.793831i	$-0.708083\pi$
$709$	7238.22	−	12537.0i	0.383409	−	0.664084i	0.991547	+	0.129747i	$0.0414166\pi$
$710$	−10701.2	+	3189.09i	−0.565648	+	0.168569i
$711$		0			0
$712$	1062.26	−	1256.29i	0.0559125	−	0.0661259i
$713$	−4563.58	−	2634.78i	−0.239702	−	0.138392i
$714$		0			0
$715$	5753.39	−	3321.72i	0.300930	−	0.173742i
$716$	−20411.5	−	10315.4i	−1.06538	−	0.538414i
$717$		0			0
$718$	11873.4	+	2829.83i	0.617149	+	0.147087i
$719$	−35929.2			−1.86361			−0.931803	−	0.362964i	$-0.881765\pi$
$719$	−35929.2			−1.86361			−0.931803	+	0.362964i	$0.881765\pi$
$720$		0			0
$721$	701.267			0.0362227
$722$	−37109.4	−	8844.38i	−1.91284	−	0.455891i
$723$		0			0
$724$	7188.95	+	3633.09i	0.369027	+	0.186495i
$725$	−2087.51	+	1205.23i	−0.106936	+	0.0617392i
$726$		0			0
$727$	−5939.53	−	3429.19i	−0.303006	−	0.174940i	0.340787	−	0.940141i	$-0.389307\pi$
$727$	−5939.53	−	3429.19i	−0.303006	−	0.174940i	−0.643792	+	0.765200i	$0.722640\pi$
$728$	−16273.7	+	19246.4i	−0.828493	+	0.979832i
$729$		0			0
$730$	6799.48	−	2026.32i	0.344740	−	0.102736i
$731$	−2078.12	+	3599.40i	−0.105146	+	0.182119i
$732$		0			0
$733$	−4546.20	−	7874.25i	−0.229083	−	0.396783i	0.728454	−	0.685095i	$-0.240239\pi$
$733$	−4546.20	−	7874.25i	−0.229083	−	0.396783i	−0.957537	+	0.288312i	$0.906906\pi$
$734$	11698.1	+	11064.4i	0.588262	+	0.556396i
$735$		0			0
$736$	6866.13	−	9102.81i	0.343871	−	0.455889i
$737$		−	3756.54i		−	0.187753i
$738$		0			0
$739$		−	31899.1i		−	1.58786i	−0.608011	−	0.793929i	$-0.708032\pi$
$739$		−	31899.1i		−	1.58786i	0.608011	−	0.793929i	$-0.291968\pi$
$740$	15884.1	−	885.080i	0.789069	−	0.0439678i
$741$		0			0
$742$	−18219.7	+	19263.2i	−0.901437	+	0.953064i
$743$	−14153.9	−	24515.4i	−0.698867	−	1.21047i	−0.968860	−	0.247610i	$-0.920355\pi$
$743$	−14153.9	−	24515.4i	−0.698867	−	1.21047i	0.269993	−	0.962862i	$-0.412979\pi$
$744$		0			0
$745$	3002.40	−	5200.30i	0.147650	−	0.255737i
$746$	−5489.10	−	18419.1i	−0.269397	−	0.903982i
$747$		0			0
$748$	838.684	+	1282.17i	0.0409964	+	0.0626747i
$749$	29146.1	+	16827.5i	1.42186	+	0.820914i
$750$		0			0
$751$	4714.22	−	2721.76i	0.229060	−	0.132248i	−0.381078	−	0.924543i	$-0.624447\pi$
$751$	4714.22	−	2721.76i	0.229060	−	0.132248i	0.610139	+	0.792295i	$0.291114\pi$
$752$	24248.7	−	2710.75i	1.17588	−	0.131451i
$753$		0			0
$754$	983.682	−	4127.35i	0.0475114	−	0.199349i
$755$	23592.8			1.13726
$756$		0			0
$757$	−2726.44			−0.130904			−0.0654519	−	0.997856i	$-0.520849\pi$
$757$	−2726.44			−0.130904			−0.0654519	+	0.997856i	$0.520849\pi$
$758$	397.682	−	1668.60i	0.0190560	−	0.0799556i
$759$		0			0
$760$	−8060.98	+	44729.0i	−0.384740	+	2.13486i
$761$	33778.4	−	19502.0i	1.60902	−	0.928971i	0.619436	−	0.785047i	$-0.287361\pi$
$761$	33778.4	−	19502.0i	1.60902	−	0.928971i	0.989589	−	0.143923i	$-0.0459718\pi$
$762$		0			0
$763$	−44141.1	−	25484.9i	−2.09438	−	1.20919i
$764$	7620.51	−	4984.69i	0.360864	−	0.236047i
$765$		0			0
$766$	4920.05	+	16509.6i	0.232074	+	0.778742i
$767$	2976.69	−	5155.77i	0.140133	−	0.242717i
$768$		0			0
$769$	−19193.7	−	33244.4i	−0.900054	−	1.55894i	−0.827422	−	0.561581i	$-0.810193\pi$
$769$	−19193.7	−	33244.4i	−0.900054	−	1.55894i	−0.0726320	−	0.997359i	$-0.523140\pi$
$770$	6913.69	−	7309.65i	0.323574	−	0.342106i
$771$		0			0
$772$	−1858.59	−	33355.1i	−0.0866476	−	1.55502i
$773$		−	24295.7i		−	1.13047i	−0.824929	−	0.565237i	$-0.808785\pi$
$773$		−	24295.7i		−	1.13047i	0.824929	−	0.565237i	$-0.191215\pi$
$774$		0			0
$775$		−	6131.35i		−	0.284186i
$776$	−18499.8	+	6654.48i	−0.855804	+	0.307837i
$777$		0			0
$778$	−2044.01	−	1933.29i	−0.0941921	−	0.0890897i
$779$	−29338.4	−	50815.6i	−1.34937	−	2.33717i
$780$		0			0
$781$	−1450.00	+	2511.48i	−0.0664344	+	0.115068i
$782$	−3161.10	+	942.044i	−0.144553	+	0.0430785i
$783$		0			0
$784$	−6498.90	+	14860.2i	−0.296051	+	0.676943i
$785$	−15885.9	−	9171.72i	−0.722282	−	0.417010i
$786$		0			0
$787$	17901.4	−	10335.4i	0.810823	−	0.468129i	−0.0364189	−	0.999337i	$-0.511595\pi$
$787$	17901.4	−	10335.4i	0.810823	−	0.468129i	0.847241	+	0.531208i	$0.178262\pi$
$788$	−3473.76	+	6873.69i	−0.157040	+	0.310742i
$789$		0			0
$790$	−26656.1	−	6353.02i	−1.20049	−	0.286114i
$791$	−26208.5			−1.17809
$792$		0			0
$793$	28537.4			1.27792
$794$	19095.0	+	4550.97i	0.853473	+	0.203410i
$795$		0			0
$796$	17630.1	−	34885.5i	0.785029	−	1.55337i
$797$	−18581.4	+	10728.0i	−0.825832	+	0.476794i	−0.852423	−	0.522852i	$-0.824868\pi$
$797$	−18581.4	+	10728.0i	−0.825832	+	0.476794i	0.0265914	+	0.999646i	$0.491535\pi$
$798$		0			0
$799$	−6112.96	−	3529.32i	−0.270664	−	0.156268i
$800$	13167.0	+	1622.95i	0.581905	+	0.0717250i
$801$		0			0
$802$	−16831.1	+	5015.86i	−0.741057	+	0.220843i
$803$	921.322	−	1595.78i	0.0404891	−	0.0701292i
$804$		0			0
$805$	10830.6	+	18759.1i	0.474196	+	0.821331i
$806$	7840.94	+	7416.20i	0.342662	+	0.324100i
$807$		0			0
$808$	9153.99	+	25448.6i	0.398560	+	1.10802i
$809$			2327.97i			0.101171i	0.998720	+	0.0505854i	$0.0161087\pi$
$809$			2327.97i			0.101171i	−0.998720	+	0.0505854i	$0.983891\pi$
$810$		0			0
$811$		−	39772.9i		−	1.72209i	−0.508527	−	0.861046i	$-0.669810\pi$
$811$		−	39772.9i		−	1.72209i	0.508527	−	0.861046i	$-0.330190\pi$
$812$	−357.501	−	6415.89i	−0.0154505	−	0.277283i
$813$		0			0
$814$	2839.10	−	3001.70i	0.122249	−	0.129250i
$815$	4385.02	+	7595.08i	0.188467	+	0.326435i
$816$		0			0
$817$	−16010.4	+	27730.8i	−0.685596	+	1.18749i
$818$	11384.4	+	38201.2i	0.486608	+	1.63285i
$819$		0			0
$820$	−38780.7	+	25367.0i	−1.65156	+	1.08031i
$821$	21847.6	+	12613.7i	0.928730	+	0.536202i	0.886410	−	0.462902i	$-0.153192\pi$
$821$	21847.6	+	12613.7i	0.928730	+	0.536202i	0.0423203	+	0.999104i	$0.486525\pi$
$822$		0			0
$823$	−17312.0	+	9995.08i	−0.733241	+	0.423337i	−0.819607	−	0.572926i	$-0.805808\pi$
$823$	−17312.0	+	9995.08i	−0.733241	+	0.423337i	0.0863655	+	0.996264i	$0.472475\pi$
$824$	639.440	+	115.239i	0.0270339	+	0.00487200i
$825$		0			0
$826$	2090.31	−	8770.57i	0.0880523	−	0.369452i
$827$	−2050.36			−0.0862126			−0.0431063	−	0.999070i	$-0.513725\pi$
$827$	−2050.36			−0.0862126			−0.0431063	+	0.999070i	$0.513725\pi$
$828$		0			0
$829$	29597.2			1.23999			0.619997	−	0.784604i	$-0.287134\pi$
$829$	29597.2			1.23999			0.619997	+	0.784604i	$0.287134\pi$
$830$	6686.14	−	28053.9i	0.279614	−	1.17321i
$831$		0			0
$832$	−18001.7	+	14875.3i	−0.750115	+	0.619841i
$833$	4063.45	−	2346.04i	0.169016	−	0.0975814i
$834$		0			0
$835$	27541.4	+	15901.0i	1.14145	+	0.659016i
$836$	6461.45	+	9878.15i	0.267313	+	0.408664i
$837$		0			0
$838$	−4878.36	−	16369.7i	−0.201098	−	0.674800i
$839$	279.354	−	483.856i	0.0114951	−	0.0199101i	−0.860221	−	0.509922i	$-0.829674\pi$
$839$	279.354	−	483.856i	0.0114951	−	0.0199101i	0.871716	+	0.490012i	$0.163008\pi$
$840$		0			0
$841$	−11653.6	−	20184.7i	−0.477823	−	0.827614i
$842$	820.822	−	867.832i	0.0335955	−	0.0355196i
$843$		0			0
$844$	8839.88	−	492.568i	0.360523	−	0.0200887i
$845$			1643.53i			0.0669103i
$846$		0			0
$847$			29892.4i			1.21265i
$848$	−19778.9	+	14570.8i	−0.800954	+	0.590052i
$849$		0			0
$850$	−2788.29	−	2637.25i	−0.112515	−	0.106420i
$851$	4447.56	+	7703.40i	0.179154	+	0.310305i
$852$		0			0
$853$	−8160.16	+	14133.8i	−0.327548	+	0.567330i	−0.982025	−	0.188752i	$-0.939556\pi$
$853$	−8160.16	+	14133.8i	−0.327548	+	0.567330i	0.654476	+	0.756082i	$0.272889\pi$
$854$	41419.0	−	12343.3i	1.65964	−	0.494590i
$855$		0			0
$856$	23811.2	+	20133.5i	0.950761	+	0.803912i
$857$	15524.2	+	8962.93i	0.618784	+	0.357255i	0.776396	−	0.630246i	$-0.217046\pi$
$857$	15524.2	+	8962.93i	0.618784	+	0.357255i	−0.157611	+	0.987501i	$0.550379\pi$
$858$		0			0
$859$	−4829.99	+	2788.60i	−0.191848	+	0.110763i	−0.592847	−	0.805315i	$-0.701996\pi$
$859$	−4829.99	+	2788.60i	−0.191848	+	0.110763i	0.401000	+	0.916078i	$0.368663\pi$
$860$	22570.1	+	11406.3i	0.894922	+	0.452268i
$861$		0			0
$862$	−14748.9	−	3515.13i	−0.582770	−	0.138893i
$863$	−9701.55			−0.382670			−0.191335	−	0.981525i	$-0.561282\pi$
$863$	−9701.55			−0.382670			−0.191335	+	0.981525i	$0.561282\pi$
$864$		0			0
$865$	8867.33			0.348553
$866$	43416.7	+	10347.6i	1.70365	+	0.406035i
$867$		0			0
$868$	14588.0	+	7372.37i	0.570449	+	0.288289i
$869$	−6163.31	+	3558.39i	−0.240594	+	0.138907i
$870$		0			0
$871$	−14344.9	−	8282.01i	−0.558045	−	0.322187i
$872$	−36061.5	−	30491.7i	−1.40045	−	1.18415i
$873$		0			0
$874$	−24354.0	+	7257.76i	−0.942547	+	0.280889i
$875$	8891.68	−	15400.8i	0.343535	−	0.595021i
$876$		0			0
$877$	1954.41	+	3385.13i	0.0752516	+	0.130340i	0.901196	−	0.433412i	$-0.142691\pi$
$877$	1954.41	+	3385.13i	0.0752516	+	0.130340i	−0.825944	+	0.563752i	$0.809357\pi$
$878$	−19951.1	−	18870.4i	−0.766878	−	0.725336i
$879$		0			0
$880$	7505.34	−	5529.08i	0.287506	−	0.211801i
$881$			21674.8i			0.828881i	0.910076	+	0.414440i	$0.136023\pi$
$881$			21674.8i			0.828881i	−0.910076	+	0.414440i	$0.863977\pi$
$882$		0			0
$883$			16447.3i			0.626837i	0.949615	+	0.313419i	$0.101474\pi$
$883$			16447.3i			0.626837i	−0.949615	+	0.313419i	$0.898526\pi$
$884$	6745.17	−	375.849i	0.256634	−	0.0142999i
$885$		0			0
$886$	11127.3	−	11764.6i	0.421929	−	0.446094i
$887$	−16014.8	−	27738.5i	−0.606229	−	1.05002i	−0.991856	−	0.127365i	$-0.959348\pi$
$887$	−16014.8	−	27738.5i	−0.606229	−	1.05002i	0.385627	−	0.922655i	$-0.373985\pi$
$888$		0			0
$889$	10831.5	−	18760.7i	0.408635	−	0.707777i
$890$	827.051	+	2775.23i	0.0311492	+	0.104524i
$891$		0			0
$892$	2089.06	+	3193.72i	0.0784157	+	0.119881i
$893$	−47095.9	−	27190.8i	−1.76484	−	1.01893i
$894$		0			0
$895$	34862.3	−	20127.8i	1.30203	−	0.751728i
$896$	−19693.5	+	29376.2i	−0.734278	+	1.09530i
$897$		0			0
$898$	−818.230	+	3433.15i	−0.0304061	+	0.127579i
$899$	−2751.58			−0.102080
$900$		0			0
$901$	7106.86			0.262779
$902$	−2790.20	+	11707.2i	−0.102997	+	0.432157i
$903$		0			0
$904$	−23897.8	−	4306.82i	−0.879236	−	0.158454i
$905$	−12278.5	+	7089.01i	−0.450997	+	0.260383i
$906$		0			0
$907$	15475.2	+	8934.59i	0.566532	+	0.327087i	0.755763	−	0.654845i	$-0.227266\pi$
$907$	15475.2	+	8934.59i	0.566532	+	0.327087i	−0.189231	+	0.981933i	$0.560600\pi$
$908$	1847.23	−	1208.30i	0.0675138	−	0.0441617i
$909$		0			0
$910$	−12670.4	−	42516.4i	−0.461559	−	1.54880i
$911$	−25249.8	+	43734.0i	−0.918292	+	1.59053i	−0.116283	+	0.993216i	$0.537098\pi$
$911$	−25249.8	+	43734.0i	−0.918292	+	1.59053i	−0.802009	+	0.597312i	$0.796235\pi$
$912$		0			0
$913$	−3744.97	−	6486.48i	−0.135751	−	0.235127i
$914$	3139.79	−	3319.61i	0.113627	−	0.120135i
$915$		0			0
$916$	1448.46	+	25994.8i	0.0522473	+	0.937656i
$917$		−	53473.1i		−	1.92567i
$918$		0			0
$919$		−	20444.0i		−	0.733826i	−0.930255	−	0.366913i	$-0.880415\pi$
$919$		−	20444.0i		−	0.733826i	0.930255	−	0.366913i	$-0.119585\pi$
$920$	6793.04	+	18885.0i	0.243435	+	0.676761i
$921$		0			0
$922$	21090.7	+	19948.3i	0.753348	+	0.712539i
$923$	6393.63	+	11074.1i	0.228005	+	0.394917i
$924$		0			0
$925$	−5174.91	+	8963.21i	−0.183946	+	0.318604i
$926$	26487.0	−	7893.41i	0.939973	−	0.280123i
$927$		0			0
$928$	728.336	−	5908.99i	0.0257638	−	0.209022i
$929$	−2982.13	−	1721.73i	−0.105318	−	0.0608054i	0.446416	−	0.894826i	$-0.352700\pi$
$929$	−2982.13	−	1721.73i	−0.105318	−	0.0608054i	−0.551734	+	0.834020i	$0.686034\pi$
$930$		0			0
$931$	31305.9	−	18074.5i	1.10205	−	0.636270i
$932$	−13612.4	+	26935.4i	−0.478420	+	0.946671i
$933$		0			0
$934$	51318.7	+	12230.9i	1.79786	+	0.428487i
$935$	−2696.79			−0.0943256
$936$		0			0
$937$	23658.2			0.824844			0.412422	−	0.910993i	$-0.364683\pi$
$937$	23658.2			0.824844			0.412422	+	0.910993i	$0.364683\pi$
$938$	−24402.3	−	5815.85i	−0.849427	−	0.202446i
$939$		0			0
$940$	−19371.5	+	38331.3i	−0.672159	+	1.33003i
$941$	−43980.0	+	25391.9i	−1.52360	+	0.879651i	−0.523990	+	0.851724i	$0.675557\pi$
$941$	−43980.0	+	25391.9i	−1.52360	+	0.879651i	−0.999610	+	0.0279268i	$0.991109\pi$
$942$		0			0
$943$	−22439.1	−	12955.2i	−0.774887	−	0.447381i
$944$	3347.28	−	7653.82i	0.115408	−	0.263888i
$945$		0			0
$946$	6294.15	−	1875.73i	0.216322	−	0.0644663i
$947$	16965.1	−	29384.4i	0.582145	−	1.00830i	−0.413080	−	0.910695i	$-0.635547\pi$
$947$	16965.1	−	29384.4i	0.582145	−	1.00830i	0.995225	−	0.0976094i	$-0.0311196\pi$
$948$		0			0
$949$	−4062.46	−	7036.40i	−0.138960	−	0.240686i
$950$	−21481.7	−	20318.1i	−0.733641	−	0.693900i
$951$		0			0
$952$	9627.34	−	3463.01i	0.327756	−	0.117896i
$953$			26312.9i			0.894394i	0.894435	+	0.447197i	$0.147578\pi$
$953$			26312.9i			0.894394i	−0.894435	+	0.447197i	$0.852422\pi$
$954$		0			0
$955$			16028.3i			0.543102i
$956$	2280.03	+	40918.7i	0.0771356	+	1.38431i
$957$		0			0
$958$	−23090.0	+	24412.4i	−0.778711	+	0.823309i
$959$	5776.35	+	10004.9i	0.194503	+	0.336888i
$960$		0			0
$961$	−11396.0	+	19738.4i	−0.382531	+	0.662562i
$962$	−5203.07	−	17459.3i	−0.174380	−	0.585146i
$963$		0			0
$964$	4496.66	−	2941.33i	0.150236	−	0.0982718i
$965$	50924.3	+	29401.2i	1.69877	+	0.980785i
$966$		0			0
$967$	−29997.5	+	17319.1i	−0.997575	+	0.575950i	−0.907530	−	0.419987i	$-0.862035\pi$
$967$	−29997.5	+	17319.1i	−0.997575	+	0.575950i	−0.0900454	+	0.995938i	$0.528701\pi$
$968$	−4912.19	+	27257.0i	−0.163103	+	0.905033i
$969$		0			0
$970$	8022.94	−	33662.8i	0.265568	−	1.11428i
$971$	44852.9			1.48239			0.741194	−	0.671291i	$-0.234260\pi$
$971$	44852.9			1.48239			0.741194	+	0.671291i	$0.234260\pi$
$972$		0			0
$973$	−42993.1			−1.41654
$974$	−3791.22	+	15907.3i	−0.124721	+	0.523308i
$975$		0			0
$976$	39795.7	−	4448.73i	1.30515	−	0.145902i
$977$	46581.9	−	26894.1i	1.52537	−	0.880673i	0.525823	−	0.850594i	$-0.323757\pi$
$977$	46581.9	−	26894.1i	1.52537	−	0.880673i	0.999548	−	0.0300794i	$-0.00957601\pi$
$978$		0			0
$979$	651.322	+	376.041i	0.0212629	+	0.0122761i
$980$	−15627.8	−	23891.6i	−0.509401	−	0.778763i
$981$		0			0
$982$	−8234.73	−	27632.3i	−0.267598	−	0.897945i
$983$	−16298.7	+	28230.1i	−0.528837	+	0.915972i	0.470598	+	0.882348i	$0.344038\pi$
$983$	−16298.7	+	28230.1i	−0.528837	+	0.915972i	−0.999435	+	0.0336243i	$0.989295\pi$
$984$		0			0
$985$	−6778.13	−	11740.1i	−0.219258	−	0.379766i
$986$	−1183.52	+	1251.31i	−0.0382263	+	0.0404156i
$987$		0			0
$988$	51966.6	−	2895.64i	1.67336	−	0.0932414i
$989$			14139.7i			0.454617i
$990$		0			0
$991$		−	6930.76i		−	0.222162i	−0.993811	−	0.111081i	$-0.964569\pi$
$991$		−	6930.76i		−	0.222162i	0.993811	−	0.111081i	$-0.0354313\pi$
$992$	12090.4	+	9119.63i	0.386966	+	0.291884i
$993$		0			0
$994$	14069.6	+	13307.4i	0.448954	+	0.424634i
$995$	34400.5	+	59583.4i	1.09605	+	1.89841i
$996$		0			0
$997$	−12127.0	+	21004.6i	−0.385222	+	0.667225i	−0.991800	−	0.127799i	$-0.959209\pi$
$997$	−12127.0	+	21004.6i	−0.385222	+	0.667225i	0.606578	+	0.795024i	$0.292542\pi$
$998$	−1956.37	+	583.021i	−0.0620520	+	0.0184922i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.h.a.35.4		8
3.2	odd	2		36.4.h.a.11.1	✓	8
4.3	odd	2	inner	108.4.h.a.35.3		8
9.2	odd	6		324.4.b.b.323.6		8
9.4	even	3		36.4.h.a.23.2	yes	8
9.5	odd	6	inner	108.4.h.a.71.3		8
9.7	even	3		324.4.b.b.323.3		8
12.11	even	2		36.4.h.a.11.2	yes	8
36.7	odd	6		324.4.b.b.323.5		8
36.11	even	6		324.4.b.b.323.4		8
36.23	even	6	inner	108.4.h.a.71.4		8
36.31	odd	6		36.4.h.a.23.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.4.h.a.11.1	✓	8	3.2	odd	2
36.4.h.a.11.2	yes	8	12.11	even	2
36.4.h.a.23.1	yes	8	36.31	odd	6
36.4.h.a.23.2	yes	8	9.4	even	3
108.4.h.a.35.3		8	4.3	odd	2	inner
108.4.h.a.35.4		8	1.1	even	1	trivial
108.4.h.a.71.3		8	9.5	odd	6	inner
108.4.h.a.71.4		8	36.23	even	6	inner
324.4.b.b.323.3		8	9.7	even	3
324.4.b.b.323.4		8	36.11	even	6
324.4.b.b.323.5		8	36.7	odd	6
324.4.b.b.323.6		8	9.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.h (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.75136 + 0.655739i) q^{2} +(7.14001 + 3.60836i) q^{4} +(-12.1949 + 7.04075i) q^{5} +(21.1499 + 12.2109i) q^{7} +(17.2786 + 14.6099i) q^{8} +O(q^{10})\)	\(q+(2.75136 + 0.655739i) q^{2} +(7.14001 + 3.60836i) q^{4} +(-12.1949 + 7.04075i) q^{5} +(21.1499 + 12.2109i) q^{7} +(17.2786 + 14.6099i) q^{8} +(-38.1696 + 11.3750i) q^{10} +(-5.17194 + 8.95806i) q^{11} +(22.8051 + 39.4995i) q^{13} +(50.1839 + 47.4655i) q^{14} +(37.9595 + 51.5274i) q^{16} -18.5146i q^{17} -142.642i q^{19} +(-112.477 + 6.26737i) q^{20} +(-20.1040 + 21.2554i) q^{22} +(-31.4938 - 54.5488i) q^{23} +(36.6443 - 63.4697i) q^{25} +(36.8437 + 123.632i) q^{26} +(106.949 + 163.502i) q^{28} +(-28.4835 - 16.4449i) q^{29} +(72.4520 - 41.8302i) q^{31} +(70.6520 + 166.662i) q^{32} +(12.1408 - 50.9405i) q^{34} -343.896 q^{35} -141.220 q^{37} +(93.5357 - 392.459i) q^{38} +(-313.576 - 56.5121i) q^{40} +(356.247 - 205.679i) q^{41} +(-194.409 - 112.242i) q^{43} +(-69.2515 + 45.2984i) q^{44} +(-50.8811 - 170.735i) q^{46} +(190.623 - 330.169i) q^{47} +(126.713 + 219.473i) q^{49} +(142.441 - 150.599i) q^{50} +(20.3001 + 364.316i) q^{52} +383.851i q^{53} -145.657i q^{55} +(187.042 + 519.986i) q^{56} +(-67.5848 - 63.9237i) q^{58} +(-65.2637 - 113.040i) q^{59} +(312.840 - 541.855i) q^{61} +(226.772 - 67.5805i) q^{62} +(85.1024 + 504.878i) q^{64} +(-556.213 - 321.129i) q^{65} +(-314.510 + 181.583i) q^{67} +(66.8073 - 132.195i) q^{68} +(-946.182 - 225.506i) q^{70} +280.360 q^{71} -178.139 q^{73} +(-388.548 - 92.6037i) q^{74} +(514.702 - 1018.46i) q^{76} +(-218.772 + 126.308i) q^{77} +(595.842 + 344.010i) q^{79} +(-825.706 - 361.110i) q^{80} +(1115.04 - 332.293i) q^{82} +(-362.048 + 627.085i) q^{83} +(130.357 + 225.785i) q^{85} +(-461.288 - 436.300i) q^{86} +(-220.240 + 79.2216i) q^{88} -72.7080i q^{89} +1113.88i q^{91} +(-28.0344 - 503.120i) q^{92} +(740.979 - 783.416i) q^{94} +(1004.30 + 1739.51i) q^{95} +(-434.434 + 752.462i) q^{97} +(204.716 + 686.939i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} + 11 q^{4} - 66 q^{5}+O(q^{10}) \) 8 * q + 3 * q^2 + 11 * q^4 - 66 * q^5	\( 8 q + 3 q^{2} + 11 q^{4} - 66 q^{5} - 116 q^{10} + 214 q^{13} + 42 q^{14} + 71 q^{16} - 306 q^{20} + 207 q^{22} - 54 q^{25} + 540 q^{28} + 498 q^{29} - 327 q^{32} + 469 q^{34} - 1256 q^{37} - 1035 q^{38} - 602 q^{40} + 1272 q^{41} - 912 q^{46} - 154 q^{49} + 1329 q^{50} - 464 q^{52} + 1314 q^{56} - 830 q^{58} + 262 q^{61} - 550 q^{64} - 3282 q^{65} + 843 q^{68} - 480 q^{70} + 3940 q^{73} - 222 q^{74} + 105 q^{76} - 330 q^{77} + 4786 q^{82} - 472 q^{85} - 1209 q^{86} - 1425 q^{88} + 1308 q^{92} + 1356 q^{94} - 572 q^{97}+O(q^{100}) \) 8 * q + 3 * q^2 + 11 * q^4 - 66 * q^5 - 116 * q^10 + 214 * q^13 + 42 * q^14 + 71 * q^16 - 306 * q^20 + 207 * q^22 - 54 * q^25 + 540 * q^28 + 498 * q^29 - 327 * q^32 + 469 * q^34 - 1256 * q^37 - 1035 * q^38 - 602 * q^40 + 1272 * q^41 - 912 * q^46 - 154 * q^49 + 1329 * q^50 - 464 * q^52 + 1314 * q^56 - 830 * q^58 + 262 * q^61 - 550 * q^64 - 3282 * q^65 + 843 * q^68 - 480 * q^70 + 3940 * q^73 - 222 * q^74 + 105 * q^76 - 330 * q^77 + 4786 * q^82 - 472 * q^85 - 1209 * q^86 - 1425 * q^88 + 1308 * q^92 + 1356 * q^94 - 572 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more