LMFDB - Embedded newform 378.4.k.d.269.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(215,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

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

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

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.k (of order $6$, degree $2$, 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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} + 2 x^{18} - 32 x^{17} - 669 x^{16} + 1752 x^{15} - 1654 x^{14} + 13878 x^{13} + \cdots + 2458624 $ x^20 - 2x^19 + 2x^18 - 32x^17 - 669x^16 + 1752x^15 - 1654x^14 + 13878x^13 + 494085x^12 - 780096x^11 + 556152x^10 - 2898760x^9 - 1242580x^8 + 9104368x^7 - 4109728x^6 + 12152000x^5 + 28375312x^4 - 23774016x^3 + 7683200x^2 - 6146560*x + 2458624
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{18}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.8
Root			$-0.131056 - 0.489109i$ of defining polynomial
Character	$\chi$	$=$	378.269
Dual form			378.4.k.d.215.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-0.523601 - 0.906903i) q^{5} +(9.66492 + 15.7984i) q^{7} -8.00000i q^{8} +O(q^{10})$	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-0.523601 - 0.906903i) q^{5} +(9.66492 + 15.7984i) q^{7} -8.00000i q^{8} +(-1.81381 - 1.04720i) q^{10} +(-53.5189 - 30.8992i) q^{11} -70.7290i q^{13} +(32.5385 + 17.6987i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(7.82902 - 13.5603i) q^{17} +(-2.91867 + 1.68510i) q^{19} -4.18881 q^{20} -123.597 q^{22} +(141.553 - 81.7258i) q^{23} +(61.9517 - 107.303i) q^{25} +(-70.7290 - 122.506i) q^{26} +(74.0571 - 1.88348i) q^{28} -39.3634i q^{29} +(200.448 + 115.729i) q^{31} +(-27.7128 - 16.0000i) q^{32} -31.3161i q^{34} +(9.26706 - 17.0372i) q^{35} +(-115.930 - 200.797i) q^{37} +(-3.37019 + 5.83734i) q^{38} +(-7.25523 + 4.18881i) q^{40} -404.216 q^{41} -65.6847 q^{43} +(-214.076 + 123.597i) q^{44} +(163.452 - 283.107i) q^{46} +(-298.112 - 516.346i) q^{47} +(-156.179 + 305.381i) q^{49} -247.807i q^{50} +(-245.012 - 141.458i) q^{52} +(-190.400 - 109.928i) q^{53} +64.7153i q^{55} +(126.387 - 77.3194i) q^{56} +(-39.3634 - 68.1794i) q^{58} +(248.471 - 430.364i) q^{59} +(313.494 - 180.996i) q^{61} +462.916 q^{62} -64.0000 q^{64} +(-64.1443 + 37.0338i) q^{65} +(-365.704 + 633.418i) q^{67} +(-31.3161 - 54.2410i) q^{68} +(-0.986194 - 38.7764i) q^{70} +141.660i q^{71} +(688.060 + 397.252i) q^{73} +(-401.593 - 231.860i) q^{74} +13.4808i q^{76} +(-29.0990 - 1144.15i) q^{77} +(530.911 + 919.564i) q^{79} +(-8.37762 + 14.5105i) q^{80} +(-700.122 + 404.216i) q^{82} +101.325 q^{83} -16.3971 q^{85} +(-113.769 + 65.6847i) q^{86} +(-247.193 + 428.151i) q^{88} +(333.470 + 577.586i) q^{89} +(1117.40 - 683.590i) q^{91} -653.807i q^{92} +(-1032.69 - 596.224i) q^{94} +(3.05644 + 1.76464i) q^{95} -588.490i q^{97} +(34.8713 + 685.113i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 40 q^{4} + 4 q^{7}+O(q^{10}) $ 20 * q + 40 * q^4 + 4 * q^7	$ 20 q + 40 q^{4} + 4 q^{7} - 24 q^{10} - 160 q^{16} + 408 q^{19} + 240 q^{22} - 646 q^{25} + 56 q^{28} - 102 q^{31} + 194 q^{37} - 96 q^{40} - 2332 q^{43} - 624 q^{46} + 2840 q^{49} - 648 q^{52} + 96 q^{58} + 1878 q^{61} - 1280 q^{64} - 386 q^{67} + 3672 q^{70} + 1788 q^{73} + 814 q^{79} - 672 q^{82} - 4560 q^{85} + 480 q^{88} + 2724 q^{91} - 1536 q^{94}+O(q^{100}) $ 20 * q + 40 * q^4 + 4 * q^7 - 24 * q^10 - 160 * q^16 + 408 * q^19 + 240 * q^22 - 646 * q^25 + 56 * q^28 - 102 * q^31 + 194 * q^37 - 96 * q^40 - 2332 * q^43 - 624 * q^46 + 2840 * q^49 - 648 * q^52 + 96 * q^58 + 1878 * q^61 - 1280 * q^64 - 386 * q^67 + 3672 * q^70 + 1788 * q^73 + 814 * q^79 - 672 * q^82 - 4560 * q^85 + 480 * q^88 + 2724 * q^91 - 1536 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	1.73205	−	1.00000i	0.612372	−	0.353553i
$2$	1.73205	−	1.00000i	0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	2.00000	−	3.46410i	0.250000	−	0.433013i
$5$	−0.523601	−	0.906903i	−0.0468323	−	0.0811159i	0.841659	−	0.540009i	$-0.181579\pi$
$5$	−0.523601	−	0.906903i	−0.0468323	−	0.0811159i	−0.888491	+	0.458893i	$0.848246\pi$
$6$		0			0
$7$	9.66492	+	15.7984i	0.521857	+	0.853033i
$7$	9.66492	+	15.7984i	0.521857	+	0.853033i
$8$		−	8.00000i		−	0.353553i
$9$		0			0
$10$	−1.81381	−	1.04720i	−0.0573576	−	0.0331154i
$11$	−53.5189	−	30.8992i	−1.46696	−	0.846950i	−0.467644	−	0.883917i	$-0.654897\pi$
$11$	−53.5189	−	30.8992i	−1.46696	−	0.846950i	−0.999316	+	0.0369671i	$0.988230\pi$
$12$		0			0
$13$		−	70.7290i		−	1.50898i	−0.656314	−	0.754488i	$-0.727885\pi$
$13$		−	70.7290i		−	1.50898i	0.656314	−	0.754488i	$-0.272115\pi$
$14$	32.5385	+	17.6987i	0.621163	+	0.337870i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	7.82902	−	13.5603i	0.111695	−	0.193461i	−0.804759	−	0.593602i	$-0.797705\pi$
$17$	7.82902	−	13.5603i	0.111695	−	0.193461i	0.916454	+	0.400141i	$0.131039\pi$
$18$		0			0
$19$	−2.91867	+	1.68510i	−0.0352415	+	0.0203467i	−0.517517	−	0.855673i	$-0.673144\pi$
$19$	−2.91867	+	1.68510i	−0.0352415	+	0.0203467i	0.482276	+	0.876019i	$0.339810\pi$
$20$	−4.18881			−0.0468323
$21$		0			0
$22$	−123.597			−1.19777
$23$	141.553	−	81.7258i	1.28330	−	0.740914i	0.305850	−	0.952080i	$-0.401059\pi$
$23$	141.553	−	81.7258i	1.28330	−	0.740914i	0.977450	+	0.211166i	$0.0677260\pi$
$24$		0			0
$25$	61.9517	−	107.303i	0.495613	−	0.858428i
$26$	−70.7290	−	122.506i	−0.533504	−	0.924055i
$27$		0			0
$28$	74.0571	−	1.88348i	0.499838	−	0.0127123i
$29$		−	39.3634i		−	0.252055i	−0.992027	−	0.126028i	$-0.959777\pi$
$29$		−	39.3634i		−	0.252055i	0.992027	−	0.126028i	$-0.0402228\pi$
$30$		0			0
$31$	200.448	+	115.729i	1.16134	+	0.670501i	0.951625	−	0.307261i	$-0.0994125\pi$
$31$	200.448	+	115.729i	1.16134	+	0.670501i	0.209717	+	0.977762i	$0.432746\pi$
$32$	−27.7128	−	16.0000i	−0.153093	−	0.0883883i
$33$		0			0
$34$		−	31.3161i		−	0.157961i
$35$	9.26706	−	17.0372i	0.0447548	−	0.0822804i
$36$		0			0
$37$	−115.930	−	200.797i	−0.515102	−	0.892183i	−0.999846	−	0.0175269i	$-0.994421\pi$
$37$	−115.930	−	200.797i	−0.515102	−	0.892183i	0.484744	−	0.874656i	$-0.338913\pi$
$38$	−3.37019	+	5.83734i	−0.0143873	+	0.0249195i
$39$		0			0
$40$	−7.25523	+	4.18881i	−0.0286788	+	0.0165577i
$41$	−404.216			−1.53970			−0.769852	−	0.638222i	$-0.779670\pi$
$41$	−404.216			−1.53970			−0.769852	+	0.638222i	$0.779670\pi$
$42$		0			0
$43$	−65.6847			−0.232949			−0.116475	−	0.993194i	$-0.537159\pi$
$43$	−65.6847			−0.232949			−0.116475	+	0.993194i	$0.537159\pi$
$44$	−214.076	+	123.597i	−0.733480	+	0.423475i
$45$		0			0
$46$	163.452	−	283.107i	0.523905	−	0.907430i
$47$	−298.112	−	516.346i	−0.925194	−	1.60248i	−0.791248	−	0.611495i	$-0.790568\pi$
$47$	−298.112	−	516.346i	−0.925194	−	1.60248i	−0.133946	−	0.990989i	$-0.542765\pi$
$48$		0			0
$49$	−156.179	+	305.381i	−0.455331	+	0.890322i
$50$		−	247.807i		−	0.700903i
$51$		0			0
$52$	−245.012	−	141.458i	−0.653406	−	0.377244i
$53$	−190.400	−	109.928i	−0.493461	−	0.284900i	0.232548	−	0.972585i	$-0.425294\pi$
$53$	−190.400	−	109.928i	−0.493461	−	0.284900i	−0.726009	+	0.687685i	$0.758627\pi$
$54$		0			0
$55$			64.7153i			0.158658i
$56$	126.387	−	77.3194i	0.301593	−	0.184504i
$57$		0			0
$58$	−39.3634	−	68.1794i	−0.0891150	−	0.154352i
$59$	248.471	−	430.364i	0.548273	−	0.949637i	−0.450120	−	0.892968i	$-0.648619\pi$
$59$	248.471	−	430.364i	0.548273	−	0.949637i	0.998393	−	0.0566690i	$-0.0180480\pi$
$60$		0			0
$61$	313.494	−	180.996i	0.658014	−	0.379904i	−0.133506	−	0.991048i	$-0.542624\pi$
$61$	313.494	−	180.996i	0.658014	−	0.379904i	0.791520	+	0.611144i	$0.209290\pi$
$62$	462.916			0.948232
$63$		0			0
$64$	−64.0000			−0.125000
$65$	−64.1443	+	37.0338i	−0.122402	+	0.0706688i
$66$		0			0
$67$	−365.704	+	633.418i	−0.666834	+	1.15499i	0.311950	+	0.950098i	$0.399018\pi$
$67$	−365.704	+	633.418i	−0.666834	+	1.15499i	−0.978785	+	0.204892i	$0.934316\pi$
$68$	−31.3161	−	54.2410i	−0.0558475	−	0.0967307i
$69$		0			0
$70$	−0.986194	−	38.7764i	−0.00168390	−	0.0662095i
$71$			141.660i			0.236788i	0.992967	+	0.118394i	$0.0377747\pi$
$71$			141.660i			0.236788i	−0.992967	+	0.118394i	$0.962225\pi$
$72$		0			0
$73$	688.060	+	397.252i	1.10317	+	0.636915i	0.937052	−	0.349191i	$-0.113544\pi$
$73$	688.060	+	397.252i	1.10317	+	0.636915i	0.166117	+	0.986106i	$0.446877\pi$
$74$	−401.593	−	231.860i	−0.630868	−	0.364232i
$75$		0			0
$76$			13.4808i			0.0203467i
$77$	−29.0990	−	1144.15i	−0.0430668	−	1.69335i
$78$		0			0
$79$	530.911	+	919.564i	0.756103	+	1.30961i	0.944824	+	0.327578i	$0.106232\pi$
$79$	530.911	+	919.564i	0.756103	+	1.30961i	−0.188722	+	0.982031i	$0.560434\pi$
$80$	−8.37762	+	14.5105i	−0.0117081	+	0.0202790i
$81$		0			0
$82$	−700.122	+	404.216i	−0.942873	+	0.544368i
$83$	101.325			0.133998			0.0669989	−	0.997753i	$-0.478658\pi$
$83$	101.325			0.133998			0.0669989	+	0.997753i	$0.478658\pi$
$84$		0			0
$85$	−16.3971			−0.0209237
$86$	−113.769	+	65.6847i	−0.142652	+	0.0823601i
$87$		0			0
$88$	−247.193	+	428.151i	−0.299442	+	0.518649i
$89$	333.470	+	577.586i	0.397165	+	0.687910i	0.993375	−	0.114918i	$-0.0366606\pi$
$89$	333.470	+	577.586i	0.397165	+	0.687910i	−0.596210	+	0.802829i	$0.703327\pi$
$90$		0			0
$91$	1117.40	−	683.590i	1.28721	−	0.787469i
$92$		−	653.807i		−	0.740914i
$93$		0			0
$94$	−1032.69	−	596.224i	−1.13313	−	0.654211i
$95$	3.05644	+	1.76464i	0.00330088	+	0.00190577i
$96$		0			0
$97$		−	588.490i		−	0.616001i	−0.951386	−	0.308000i	$-0.900340\pi$
$97$		−	588.490i		−	0.616001i	0.951386	−	0.308000i	$-0.0996598\pi$
$98$	34.8713	+	685.113i	0.0359442	+	0.706193i
$99$		0			0
$100$	−247.807	−	429.214i	−0.247807	−	0.429214i
$101$	900.098	−	1559.01i	0.886763	−	1.53592i	0.0430833	−	0.999071i	$-0.486282\pi$
$101$	900.098	−	1559.01i	0.886763	−	1.53592i	0.843680	−	0.536847i	$-0.180385\pi$
$102$		0			0
$103$	−318.888	+	184.110i	−0.305058	+	0.176125i	−0.644713	−	0.764425i	$-0.723023\pi$
$103$	−318.888	+	184.110i	−0.305058	+	0.176125i	0.339655	+	0.940550i	$0.389690\pi$
$104$	−565.832			−0.533504
$105$		0			0
$106$	−439.710			−0.402910
$107$	−1204.40	+	695.360i	−1.08816	+	0.628252i	−0.933087	−	0.359650i	$-0.882896\pi$
$107$	−1204.40	+	695.360i	−1.08816	+	0.628252i	−0.155077	+	0.987902i	$0.549563\pi$
$108$		0			0
$109$	−779.495	+	1350.12i	−0.684973	+	1.18641i	0.288472	+	0.957488i	$0.406853\pi$
$109$	−779.495	+	1350.12i	−0.684973	+	1.18641i	−0.973445	+	0.228920i	$0.926481\pi$
$110$	64.7153	+	112.090i	0.0560942	+	0.0971580i
$111$		0			0
$112$	141.590	−	260.308i	0.119455	−	0.219614i
$113$			1429.08i			1.18971i	0.803834	+	0.594854i	$0.202790\pi$
$113$			1429.08i			1.18971i	−0.803834	+	0.594854i	$0.797210\pi$
$114$		0			0
$115$	−148.235	−	85.5835i	−0.120200	−	0.0693974i
$116$	−136.359	−	78.7268i	−0.109143	−	0.0630138i
$117$		0			0
$118$		−	993.883i		−	0.775376i
$119$	289.897	−	7.37291i	0.223318	−	0.00567961i
$120$		0			0
$121$	1244.02	+	2154.70i	0.934648	+	1.61886i
$122$	361.992	−	626.989i	0.268633	−	0.465286i
$123$		0			0
$124$	801.794	−	462.916i	0.580671	−	0.335251i
$125$	−260.652			−0.186507
$126$		0			0
$127$	1027.45			0.717883			0.358942	−	0.933360i	$-0.383138\pi$
$127$	1027.45			0.717883			0.358942	+	0.933360i	$0.383138\pi$
$128$	−110.851	+	64.0000i	−0.0765466	+	0.0441942i
$129$		0			0
$130$	−74.0675	+	128.289i	−0.0499704	+	0.0865513i
$131$	−56.5352	−	97.9218i	−0.0377061	−	0.0653089i	0.846556	−	0.532299i	$-0.178672\pi$
$131$	−56.5352	−	97.9218i	−0.0377061	−	0.0653089i	−0.884263	+	0.466990i	$0.845338\pi$
$132$		0			0
$133$	−54.8305	−	29.8240i	−0.0357474	−	0.0194441i
$134$			1462.82i			0.943046i
$135$		0			0
$136$	−108.482	−	62.6321i	−0.0683990	−	0.0394902i
$137$	968.042	+	558.899i	0.603689	+	0.348540i	0.770492	−	0.637450i	$-0.220011\pi$
$137$	968.042	+	558.899i	0.603689	+	0.348540i	−0.166802	+	0.985990i	$0.553344\pi$
$138$		0			0
$139$			16.9155i			0.0103219i	0.999987	+	0.00516097i	$0.00164280\pi$
$139$			16.9155i			0.0103219i	−0.999987	+	0.00516097i	$0.998357\pi$
$140$	−40.4845	−	66.1764i	−0.0244397	−	0.0399495i
$141$		0			0
$142$	141.660	+	245.363i	0.0837174	+	0.145003i
$143$	−2185.47	+	3785.34i	−1.27803	+	2.21361i
$144$		0			0
$145$	−35.6988	+	20.6107i	−0.0204457	+	0.0118043i
$146$	1589.01			0.900734
$147$		0			0
$148$	−927.440			−0.515102
$149$	−869.817	+	502.189i	−0.478243	+	0.276114i	−0.719684	−	0.694302i	$-0.755713\pi$
$149$	−869.817	+	502.189i	−0.478243	+	0.276114i	0.241441	+	0.970415i	$0.422380\pi$
$150$		0			0
$151$	1358.27	−	2352.59i	0.732014	−	1.26789i	−0.224007	−	0.974588i	$-0.571914\pi$
$151$	1358.27	−	2352.59i	0.732014	−	1.26789i	0.956021	−	0.293298i	$-0.0947528\pi$
$152$	13.4808	+	23.3494i	0.00719365	+	0.0124598i
$153$		0			0
$154$	−1194.55	−	1952.63i	−0.625063	−	1.02174i
$155$		−	242.383i		−	0.125604i
$156$		0			0
$157$	−297.129	−	171.548i	−0.151041	−	0.0872038i	0.422575	−	0.906328i	$-0.361126\pi$
$157$	−297.129	−	171.548i	−0.151041	−	0.0872038i	−0.573616	+	0.819124i	$0.694460\pi$
$158$	1839.13	+	1061.82i	0.926033	+	0.534645i
$159$		0			0
$160$			33.5105i			0.0165577i
$161$	2659.24	+	1446.44i	1.30172	+	0.708047i
$162$		0			0
$163$	−599.580	−	1038.50i	−0.288115	−	0.499030i	0.685245	−	0.728313i	$-0.259695\pi$
$163$	−599.580	−	1038.50i	−0.288115	−	0.499030i	−0.973360	+	0.229283i	$0.926362\pi$
$164$	−808.431	+	1400.24i	−0.384926	+	0.666712i
$165$		0			0
$166$	175.499	−	101.325i	0.0820566	−	0.0473754i
$167$	2792.01			1.29373			0.646863	−	0.762606i	$-0.276080\pi$
$167$	2792.01			1.29373			0.646863	+	0.762606i	$0.276080\pi$
$168$		0			0
$169$	−2805.59			−1.27701
$170$	−28.4007	+	16.3971i	−0.0128131	+	0.00739766i
$171$		0			0
$172$	−131.369	+	227.539i	−0.0582374	+	0.100870i
$173$	−1369.93	−	2372.79i	−0.602045	−	1.04277i	−0.992511	−	0.122156i	$-0.961019\pi$
$173$	−1369.93	−	2372.79i	−0.602045	−	1.04277i	0.390466	−	0.920618i	$-0.372314\pi$
$174$		0			0
$175$	2293.98	−	58.3425i	0.990907	−	0.0252016i
$176$			988.773i			0.423475i
$177$		0			0
$178$	1155.17	+	666.939i	0.486426	+	0.280838i
$179$	2995.63	+	1729.53i	1.25086	+	0.722185i	0.971280	−	0.237937i	$-0.0764713\pi$
$179$	2995.63	+	1729.53i	1.25086	+	0.722185i	0.279580	+	0.960122i	$0.409805\pi$
$180$		0			0
$181$		−	1112.90i		−	0.457025i	−0.973541	−	0.228513i	$-0.926614\pi$
$181$		−	1112.90i		−	0.457025i	0.973541	−	0.228513i	$-0.0733862\pi$
$182$	1251.81	−	2301.42i	0.509837	−	0.937321i
$183$		0			0
$184$	−653.807	−	1132.43i	−0.261953	−	0.453715i
$185$	−121.402	+	210.275i	−0.0482468	+	0.0835659i
$186$		0			0
$187$	−838.001	+	483.820i	−0.327704	+	0.189200i
$188$	−2384.90			−0.925194
$189$		0			0
$190$	7.05854			0.00269516
$191$	−193.052	+	111.459i	−0.0731348	+	0.0422244i	−0.536121	−	0.844141i	$-0.680111\pi$
$191$	−193.052	+	111.459i	−0.0731348	+	0.0422244i	0.462987	+	0.886365i	$0.346778\pi$
$192$		0			0
$193$	1297.49	−	2247.32i	0.483914	−	0.838164i	−0.515915	−	0.856640i	$-0.672548\pi$
$193$	1297.49	−	2247.32i	0.483914	−	0.838164i	0.999829	+	0.0184760i	$0.00588142\pi$
$194$	−588.490	−	1019.29i	−0.217789	−	0.377222i
$195$		0			0
$196$	745.512	+	1151.78i	0.271688	+	0.419745i
$197$			5097.29i			1.84349i	0.387801	+	0.921743i	$0.373235\pi$
$197$			5097.29i			1.84349i	−0.387801	+	0.921743i	$0.626765\pi$
$198$		0			0
$199$	3897.14	+	2250.02i	1.38825	+	0.801504i	0.993118	−	0.117121i	$-0.0373667\pi$
$199$	3897.14	+	2250.02i	1.38825	+	0.801504i	0.395129	+	0.918626i	$0.370700\pi$
$200$	−858.428	−	495.613i	−0.303500	−	0.175226i
$201$		0			0
$202$		−	3600.39i		−	1.25407i
$203$	621.878	−	380.444i	0.215011	−	0.131537i
$204$		0			0
$205$	211.648	+	366.585i	0.0721079	+	0.124895i
$206$	−368.220	+	637.775i	−0.124539	+	0.215708i
$207$		0			0
$208$	−980.049	+	565.832i	−0.326703	+	0.188622i
$209$	208.272			0.0689306
$210$		0			0
$211$	−953.941			−0.311242			−0.155621	−	0.987817i	$-0.549738\pi$
$211$	−953.941			−0.311242			−0.155621	+	0.987817i	$0.549738\pi$
$212$	−761.600	+	439.710i	−0.246731	+	0.142450i
$213$		0			0
$214$	−1390.72	+	2408.80i	−0.444241	+	0.769449i
$215$	34.3926	+	59.5697i	0.0109096	+	0.0188959i
$216$		0			0
$217$	108.987	+	4285.28i	0.0340945	+	1.34057i
$218$			3117.98i			0.968698i
$219$		0			0
$220$	224.180	+	129.431i	0.0687011	+	0.0396646i
$221$	−959.103	−	553.738i	−0.291929	−	0.168545i
$222$		0			0
$223$		−	2395.19i		−	0.719256i	−0.933096	−	0.359628i	$-0.882904\pi$
$223$		−	2395.19i		−	0.719256i	0.933096	−	0.359628i	$-0.117096\pi$
$224$	−15.0679	−	592.457i	−0.00449448	−	0.176720i
$225$		0			0
$226$	1429.08	+	2475.25i	0.420625	+	0.728544i
$227$	219.518	−	380.217i	0.0641848	−	0.111171i	−0.832147	−	0.554555i	$-0.812889\pi$
$227$	219.518	−	380.217i	0.0641848	−	0.111171i	0.896332	+	0.443383i	$0.146222\pi$
$228$		0			0
$229$	−1366.19	+	788.770i	−0.394237	+	0.227613i	−0.683995	−	0.729487i	$-0.739759\pi$
$229$	−1366.19	+	788.770i	−0.394237	+	0.227613i	0.289757	+	0.957100i	$0.406425\pi$
$230$	−342.334			−0.0981427
$231$		0			0
$232$	−314.907			−0.0891150
$233$	−3158.10	+	1823.33i	−0.887956	+	0.512662i	−0.873273	−	0.487230i	$-0.838007\pi$
$233$	−3158.10	+	1823.33i	−0.887956	+	0.512662i	−0.0146828	+	0.999892i	$0.504674\pi$
$234$		0			0
$235$	−312.184	+	540.718i	−0.0866580	+	0.150096i
$236$	−993.883	−	1721.46i	−0.274137	−	0.474819i
$237$		0			0
$238$	494.744	−	302.667i	0.134746	−	0.0824328i
$239$		−	6644.29i		−	1.79826i	−0.437684	−	0.899129i	$-0.644201\pi$
$239$		−	6644.29i		−	1.79826i	0.437684	−	0.899129i	$-0.355799\pi$
$240$		0			0
$241$	−1434.78	−	828.371i	−0.383495	−	0.221411i	0.295843	−	0.955237i	$-0.404400\pi$
$241$	−1434.78	−	828.371i	−0.383495	−	0.221411i	−0.679338	+	0.733826i	$0.737733\pi$
$242$	4309.40	+	2488.03i	1.14471	+	0.660896i
$243$		0			0
$244$		−	1447.97i		−	0.379904i
$245$	358.726	−	18.2587i	0.0935435	−	0.00476124i
$246$		0			0
$247$	119.185	+	206.435i	0.0307027	+	0.0531786i
$248$	925.832	−	1603.59i	0.237058	−	0.410596i
$249$		0			0
$250$	−451.463	+	260.652i	−0.114212	+	0.0659403i
$251$	1861.69			0.468162			0.234081	−	0.972217i	$-0.424792\pi$
$251$	1861.69			0.468162			0.234081	+	0.972217i	$0.424792\pi$
$252$		0			0
$253$	−10101.0			−2.51007
$254$	1779.59	−	1027.45i	0.439612	−	0.253810i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	−69.9982	−	121.241i	−0.0169898	−	0.0294271i	0.857406	−	0.514641i	$-0.172075\pi$
$257$	−69.9982	−	121.241i	−0.0169898	−	0.0294271i	−0.874395	+	0.485214i	$0.838742\pi$
$258$		0			0
$259$	2051.81	−	3772.19i	0.492252	−	0.904991i
$260$			296.270i			0.0706688i
$261$		0			0
$262$	−195.844	−	113.070i	−0.0461804	−	0.0266622i
$263$	4180.51	+	2413.62i	0.980156	+	0.565894i	0.902317	−	0.431072i	$-0.141865\pi$
$263$	4180.51	+	2413.62i	0.980156	+	0.565894i	0.0778391	+	0.996966i	$0.475198\pi$
$264$		0			0
$265$			230.233i			0.0533701i
$266$	−124.793	+	3.17385i	−0.0287653	+	0.000731583i
$267$		0			0
$268$	1462.82	+	2533.67i	0.333417	+	0.577495i
$269$	−49.6315	+	85.9642i	−0.0112494	+	0.0194845i	−0.871595	−	0.490226i	$-0.836914\pi$
$269$	−49.6315	+	85.9642i	−0.0112494	+	0.0194845i	0.860346	+	0.509711i	$0.170248\pi$
$270$		0			0
$271$	1956.90	−	1129.82i	0.438646	−	0.253252i	−0.264377	−	0.964419i	$-0.585166\pi$
$271$	1956.90	−	1129.82i	0.438646	−	0.253252i	0.703023	+	0.711167i	$0.251833\pi$
$272$	−250.529			−0.0558475
$273$		0			0
$274$	2235.60			0.492910
$275$	−6631.18	+	3828.51i	−1.45409	+	0.839520i
$276$		0			0
$277$	−961.503	+	1665.37i	−0.208560	+	0.361237i	−0.951261	−	0.308386i	$-0.900211\pi$
$277$	−961.503	+	1665.37i	−0.208560	+	0.361237i	0.742701	+	0.669623i	$0.233544\pi$
$278$	16.9155	+	29.2984i	0.00364936	+	0.00632087i
$279$		0			0
$280$	−136.298	−	74.1364i	−0.0290905	−	0.0158232i
$281$		−	136.085i		−	0.0288901i	−0.999896	−	0.0144451i	$-0.995402\pi$
$281$		−	136.085i		−	0.0288901i	0.999896	−	0.0144451i	$-0.00459817\pi$
$282$		0			0
$283$	5340.40	+	3083.28i	1.12174	+	0.647639i	0.941846	−	0.336046i	$-0.109090\pi$
$283$	5340.40	+	3083.28i	1.12174	+	0.647639i	0.179899	+	0.983685i	$0.442423\pi$
$284$	490.726	+	283.321i	0.102532	+	0.0591971i
$285$		0			0
$286$			8741.86i			1.80740i
$287$	−3906.71	−	6385.96i	−0.803505	−	1.31342i
$288$		0			0
$289$	2333.91	+	4042.46i	0.475048	+	0.822808i
$290$	−41.2214	+	71.3976i	−0.00834692	+	0.0144573i
$291$		0			0
$292$	2752.24	−	1589.01i	0.551584	−	0.318457i
$293$	4227.34			0.842880			0.421440	−	0.906856i	$-0.361525\pi$
$293$	4227.34			0.842880			0.421440	+	0.906856i	$0.361525\pi$
$294$		0			0
$295$	−520.398			−0.102708
$296$	−1606.37	+	927.440i	−0.315434	+	0.182116i
$297$		0			0
$298$	−1004.38	+	1739.63i	−0.195242	+	0.338169i
$299$	−5780.38	−	10011.9i	−1.11802	−	1.93647i
$300$		0			0
$301$	−634.838	−	1037.71i	−0.121566	−	0.198714i
$302$		−	5433.06i		−	1.03522i
$303$		0			0
$304$	46.6987	+	26.9615i	0.00881038	+	0.00508668i
$305$	−328.292	−	189.539i	−0.0616326	−	0.0355836i
$306$		0			0
$307$		−	2990.10i		−	0.555876i	−0.960599	−	0.277938i	$-0.910349\pi$
$307$		−	2990.10i		−	0.555876i	0.960599	−	0.277938i	$-0.0896510\pi$
$308$	−4021.65	−	2187.50i	−0.744010	−	0.404690i
$309$		0			0
$310$	−242.383	−	419.820i	−0.0444079	−	0.0769167i
$311$	−2149.84	+	3723.64i	−0.391982	+	0.678932i	−0.992711	−	0.120520i	$-0.961544\pi$
$311$	−2149.84	+	3723.64i	−0.391982	+	0.678932i	0.600729	+	0.799453i	$0.294877\pi$
$312$		0			0
$313$	−4209.01	+	2430.07i	−0.760087	+	0.438836i	−0.829327	−	0.558764i	$-0.811276\pi$
$313$	−4209.01	+	2430.07i	−0.760087	+	0.438836i	0.0692399	+	0.997600i	$0.477943\pi$
$314$	−686.191			−0.123325
$315$		0			0
$316$	4247.28			0.756103
$317$	−5259.05	+	3036.31i	−0.931791	+	0.537970i	−0.887377	−	0.461044i	$-0.847475\pi$
$317$	−5259.05	+	3036.31i	−0.931791	+	0.537970i	−0.0444133	+	0.999013i	$0.514142\pi$
$318$		0			0
$319$	−1216.30	+	2106.69i	−0.213478	+	0.369755i
$320$	33.5105	+	58.0418i	0.00585404	+	0.0101395i
$321$		0			0
$322$	6052.38	−	153.929i	1.04747	−	0.0266402i
$323$			52.7706i			0.00909051i
$324$		0			0
$325$	−7589.46	−	4381.78i	−1.29535	−	0.747869i
$326$	−2077.01	−	1199.16i	−0.352867	−	0.203728i
$327$		0			0
$328$			3233.73i			0.544368i
$329$	5276.20	−	9700.13i	0.884153	−	1.62549i
$330$		0			0
$331$	−4840.39	−	8383.80i	−0.803782	−	1.39219i	−0.917110	−	0.398633i	$-0.869485\pi$
$331$	−4840.39	−	8383.80i	−0.803782	−	1.39219i	0.113329	−	0.993558i	$-0.463849\pi$
$332$	202.649	−	350.999i	0.0334995	−	0.0580228i
$333$		0			0
$334$	4835.91	−	2792.01i	0.792243	−	0.457402i
$335$	765.932			0.124917
$336$		0			0
$337$	977.429			0.157994			0.0789970	−	0.996875i	$-0.474828\pi$
$337$	977.429			0.157994			0.0789970	+	0.996875i	$0.474828\pi$
$338$	−4859.42	+	2805.59i	−0.782005	+	0.451491i
$339$		0			0
$340$	−32.7942	+	56.8013i	−0.00523093	+	0.00906024i
$341$	−7151.86	−	12387.4i	−1.13576	−	1.96720i
$342$		0			0
$343$	−6333.98	+	484.108i	−0.997092	+	0.0762082i
$344$			525.478i			0.0823601i
$345$		0			0
$346$	−4745.58	−	2739.86i	−0.737352	−	0.425710i
$347$	8689.17	+	5016.69i	1.34426	+	0.776110i	0.987430	−	0.158059i	$-0.0505235\pi$
$347$	8689.17	+	5016.69i	1.34426	+	0.776110i	0.356832	+	0.934169i	$0.383857\pi$
$348$		0			0
$349$		−	3525.19i		−	0.540685i	−0.962764	−	0.270342i	$-0.912863\pi$
$349$		−	3525.19i		−	0.540685i	0.962764	−	0.270342i	$-0.0871369\pi$
$350$	3914.95	−	2395.03i	0.597894	−	0.365771i
$351$		0			0
$352$	988.773	+	1712.61i	0.149721	+	0.259324i
$353$	1986.78	−	3441.20i	0.299563	−	0.518858i	−0.676473	−	0.736467i	$-0.736493\pi$
$353$	1986.78	−	3441.20i	0.299563	−	0.518858i	0.976036	+	0.217609i	$0.0698259\pi$
$354$		0			0
$355$	128.472	−	74.1734i	0.0192073	−	0.0110893i
$356$	2667.76			0.397165
$357$		0			0
$358$	6918.12			1.02132
$359$	4177.52	−	2411.89i	0.614154	−	0.354582i	−0.160436	−	0.987046i	$-0.551290\pi$
$359$	4177.52	−	2411.89i	0.614154	−	0.354582i	0.774589	+	0.632465i	$0.217957\pi$
$360$		0			0
$361$	−3423.82	+	5930.23i	−0.499172	+	0.864591i
$362$	−1112.90	−	1927.61i	−0.161583	−	0.279870i
$363$		0			0
$364$	−133.217	−	5237.98i	−0.0191826	−	0.754244i
$365$		−	832.005i		−	0.119313i
$366$		0			0
$367$	−2543.41	−	1468.44i	−0.361757	−	0.208861i	0.308094	−	0.951356i	$-0.400309\pi$
$367$	−2543.41	−	1468.44i	−0.361757	−	0.208861i	−0.669851	+	0.742495i	$0.733642\pi$
$368$	−2264.85	−	1307.61i	−0.320825	−	0.185228i
$369$		0			0
$370$			485.608i			0.0682313i
$371$	−103.523	−	4070.46i	−0.0144870	−	0.569616i
$372$		0			0
$373$	−2676.89	−	4636.51i	−0.371592	−	0.643617i	0.618218	−	0.786007i	$-0.287855\pi$
$373$	−2676.89	−	4636.51i	−0.371592	−	0.643617i	−0.989811	+	0.142389i	$0.954521\pi$
$374$	−967.640	+	1676.00i	−0.133785	+	0.231722i
$375$		0			0
$376$	−4130.76	+	2384.90i	−0.566564	+	0.327106i
$377$	−2784.13			−0.380345
$378$		0			0
$379$	8850.81			1.19957			0.599783	−	0.800163i	$-0.295254\pi$
$379$	8850.81			1.19957			0.599783	+	0.800163i	$0.295254\pi$
$380$	12.2258	−	7.05854i	0.00165044	−	0.000952883i
$381$		0			0
$382$	−222.917	+	386.104i	−0.0298572	+	0.0517141i
$383$	−3747.48	−	6490.83i	−0.499967	−	0.865968i	0.500033	−	0.866006i	$-0.333321\pi$
$383$	−3747.48	−	6490.83i	−0.499967	−	0.865968i	−1.00000		3.80396e-5i	$0.999988\pi$
$384$		0			0
$385$	−1022.40	+	625.469i	−0.135341	+	0.0827970i
$386$		−	5189.96i		−	0.684358i
$387$		0			0
$388$	−2038.59	−	1176.98i	−0.266736	−	0.154000i
$389$	2925.04	+	1688.77i	0.381248	+	0.220114i	0.678361	−	0.734729i	$-0.262690\pi$
$389$	2925.04	+	1688.77i	0.381248	+	0.220114i	−0.297113	+	0.954842i	$0.596024\pi$
$390$		0			0
$391$		−	2559.33i		−	0.331026i
$392$	2443.04	+	1249.43i	0.314776	+	0.160984i
$393$		0			0
$394$	5097.29	+	8828.76i	0.651771	+	1.12890i
$395$	555.971	−	962.969i	0.0708200	−	0.122664i
$396$		0			0
$397$	6584.28	−	3801.43i	0.832381	−	0.480575i	−0.0222861	−	0.999752i	$-0.507094\pi$
$397$	6584.28	−	3801.43i	0.832381	−	0.480575i	0.854667	+	0.519176i	$0.173761\pi$
$398$	9000.06			1.13350
$399$		0			0
$400$	−1982.45			−0.247807
$401$	−8254.51	+	4765.75i	−1.02796	+	0.593491i	−0.916399	−	0.400267i	$-0.868918\pi$
$401$	−8254.51	+	4765.75i	−1.02796	+	0.593491i	−0.111558	+	0.993758i	$0.535584\pi$
$402$		0			0
$403$	8185.39	−	14177.5i	1.01177	−	1.75244i
$404$	−3600.39	−	6236.06i	−0.443381	−	0.767959i
$405$		0			0
$406$	696.681	−	1280.83i	0.0851618	−	0.156567i
$407$			14328.6i			1.74506i
$408$		0			0
$409$	−2296.96	−	1326.15i	−0.277695	−	0.160327i	0.354684	−	0.934986i	$-0.384588\pi$
$409$	−2296.96	−	1326.15i	−0.277695	−	0.160327i	−0.632380	+	0.774659i	$0.717922\pi$
$410$	733.169	+	423.295i	0.0883138	+	0.0509880i
$411$		0			0
$412$			1472.88i			0.176125i
$413$	9200.51	−	233.995i	1.09619	−	0.0278793i
$414$		0			0
$415$	−53.0537	−	91.8916i	−0.00627543	−	0.0108694i
$416$	−1131.66	+	1960.10i	−0.133376	+	0.231014i
$417$		0			0
$418$	360.738	−	208.272i	0.0422112	−	0.0243706i
$419$	−12792.8			−1.49157			−0.745786	−	0.666185i	$-0.767926\pi$
$419$	−12792.8			−1.49157			−0.745786	+	0.666185i	$0.767926\pi$
$420$		0			0
$421$	−7693.22			−0.890605			−0.445302	−	0.895380i	$-0.646904\pi$
$421$	−7693.22			−0.890605			−0.445302	+	0.895380i	$0.646904\pi$
$422$	−1652.27	+	953.941i	−0.190596	+	0.110041i
$423$		0			0
$424$	−879.420	+	1523.20i	−0.100727	+	0.174465i
$425$	−970.042	−	1680.16i	−0.110715	−	0.191764i
$426$		0			0
$427$	5889.35	+	3203.39i	0.667460	+	0.363052i
$428$			5562.88i			0.628252i
$429$		0			0
$430$	119.139	+	68.7852i	0.0133614	+	0.00771422i
$431$	2195.93	+	1267.82i	0.245415	+	0.141691i	0.617663	−	0.786443i	$-0.288080\pi$
$431$	2195.93	+	1267.82i	0.245415	+	0.141691i	−0.372248	+	0.928133i	$0.621413\pi$
$432$		0			0
$433$		−	9611.55i		−	1.06675i	−0.845880	−	0.533374i	$-0.820924\pi$
$433$		−	9611.55i		−	1.06675i	0.845880	−	0.533374i	$-0.179076\pi$
$434$	4474.05	+	7313.33i	0.494841	+	0.808873i
$435$		0			0
$436$	3117.98	+	5400.50i	0.342486	+	0.593204i
$437$	−275.432	+	477.062i	−0.0301503	+	0.0522219i
$438$		0			0
$439$	−396.640	+	229.000i	−0.0431221	+	0.0248965i	−0.521406	−	0.853309i	$-0.674592\pi$
$439$	−396.640	+	229.000i	−0.0431221	+	0.0248965i	0.478284	+	0.878205i	$0.341259\pi$
$440$	517.723			0.0560942
$441$		0			0
$442$	−2214.95			−0.238359
$443$	2446.18	−	1412.30i	0.262351	−	0.151468i	−0.363056	−	0.931767i	$-0.618267\pi$
$443$	2446.18	−	1412.30i	0.262351	−	0.151468i	0.625406	+	0.780299i	$0.284933\pi$
$444$		0			0
$445$	349.210	−	604.849i	0.0372003	−	0.0644328i
$446$	−2395.19	−	4148.60i	−0.254295	−	0.440452i
$447$		0			0
$448$	−618.555	−	1011.10i	−0.0652321	−	0.106629i
$449$		−	6366.85i		−	0.669199i	−0.942360	−	0.334599i	$-0.891399\pi$
$449$		−	6366.85i		−	0.669199i	0.942360	−	0.334599i	$-0.108601\pi$
$450$		0			0
$451$	21633.2	+	12489.9i	2.25869	+	1.30405i
$452$	4950.49	+	2858.17i	0.515158	+	0.297427i
$453$		0			0
$454$		−	878.074i		−	0.0907710i
$455$	−1205.02	−	655.449i	−0.124159	−	0.0675339i
$456$		0			0
$457$	−5963.62	−	10329.3i	−0.610430	−	1.05730i	−0.991168	−	0.132612i	$-0.957663\pi$
$457$	−5963.62	−	10329.3i	−0.610430	−	1.05730i	0.380738	−	0.924683i	$-0.375670\pi$
$458$	−1577.54	+	2732.38i	−0.160947	+	0.278768i
$459$		0			0
$460$	−592.940	+	342.334i	−0.0600999	+	0.0346987i
$461$	7106.81			0.717998			0.358999	−	0.933338i	$-0.383118\pi$
$461$	7106.81			0.717998			0.358999	+	0.933338i	$0.383118\pi$
$462$		0			0
$463$	−7835.44			−0.786487			−0.393244	−	0.919434i	$-0.628647\pi$
$463$	−7835.44			−0.786487			−0.393244	+	0.919434i	$0.628647\pi$
$464$	−545.435	+	314.907i	−0.0545715	+	0.0315069i
$465$		0			0
$466$	−3646.66	+	6316.19i	−0.362507	+	0.627880i
$467$	−1124.62	−	1947.90i	−0.111437	−	0.193015i	0.804913	−	0.593393i	$-0.202212\pi$
$467$	−1124.62	−	1947.90i	−0.111437	−	0.193015i	−0.916350	+	0.400378i	$0.868879\pi$
$468$		0			0
$469$	−13541.5	+	344.399i	−1.33324	+	0.0339080i
$470$			1248.73i			0.122553i
$471$		0			0
$472$	−3442.91	−	1987.77i	−0.335747	−	0.193844i
$473$	3515.38	+	2029.60i	0.341728	+	0.197297i
$474$		0			0
$475$			417.578i			0.0403364i
$476$	554.254	−	1018.98i	0.0533701	−	0.0981194i
$477$		0			0
$478$	−6644.29	−	11508.3i	−0.635780	−	1.10120i
$479$	−7779.25	+	13474.1i	−0.742052	+	1.28527i	0.209507	+	0.977807i	$0.432814\pi$
$479$	−7779.25	+	13474.1i	−0.742052	+	1.28527i	−0.951559	+	0.307465i	$0.900519\pi$
$480$		0			0
$481$	−14202.1	+	8199.61i	−1.34628	+	0.777276i
$482$	−3313.48			−0.313122
$483$		0			0
$484$	9952.14			0.934648
$485$	−533.703	+	308.134i	−0.0499675	+	0.0288487i
$486$		0			0
$487$	5927.28	−	10266.3i	0.551521	−	0.955262i	−0.446645	−	0.894711i	$-0.647381\pi$
$487$	5927.28	−	10266.3i	0.551521	−	0.955262i	0.998165	−	0.0605502i	$-0.0192855\pi$
$488$	−1447.97	−	2507.95i	−0.134316	−	0.232643i
$489$		0			0
$490$	603.073	−	390.351i	0.0556001	−	0.0359883i
$491$			16825.9i			1.54653i	0.634086	+	0.773263i	$0.281377\pi$
$491$			16825.9i			1.54653i	−0.634086	+	0.773263i	$0.718623\pi$
$492$		0			0
$493$	−533.778	−	308.177i	−0.0487630	−	0.0281533i
$494$	412.869	+	238.370i	0.0376030	+	0.0217101i
$495$		0			0
$496$		−	3703.33i		−	0.335251i
$497$	−2238.00	+	1369.14i	−0.201988	+	0.123570i
$498$		0			0
$499$	3027.11	+	5243.11i	0.271567	+	0.470368i	0.969263	−	0.246026i	$-0.0791248\pi$
$499$	3027.11	+	5243.11i	0.271567	+	0.470368i	−0.697696	+	0.716394i	$0.745791\pi$
$500$	−521.304	+	902.925i	−0.0466269	+	0.0807601i
$501$		0			0
$502$	3224.54	−	1861.69i	0.286689	−	0.165520i
$503$	4048.76			0.358897			0.179449	−	0.983767i	$-0.442569\pi$
$503$	4048.76			0.358897			0.179449	+	0.983767i	$0.442569\pi$
$504$		0			0
$505$	−1885.17			−0.166117
$506$	−17495.5	+	10101.0i	−1.53710	+	0.887443i
$507$		0			0
$508$	2054.89	−	3559.18i	0.179471	−	0.310853i
$509$	8200.52	+	14203.7i	0.714109	+	1.23687i	0.963302	+	0.268419i	$0.0865012\pi$
$509$	8200.52	+	14203.7i	0.714109	+	1.23687i	−0.249193	+	0.968454i	$0.580165\pi$
$510$		0			0
$511$	374.108	+	14709.6i	0.0323866	+	1.27342i
$512$			512.000i			0.0441942i
$513$		0			0
$514$	−242.481	−	139.996i	−0.0208081	−	0.0120136i
$515$	333.940	+	192.800i	0.0285731	+	0.0164967i
$516$		0			0
$517$			36845.7i			3.13437i
$518$	−218.352	−	8585.44i	−0.0185209	−	0.728229i
$519$		0			0
$520$	296.270	+	513.155i	0.0249852	+	0.0432756i
$521$	−203.991	+	353.323i	−0.0171536	+	0.0297109i	−0.874475	−	0.485071i	$-0.838794\pi$
$521$	−203.991	+	353.323i	−0.0171536	+	0.0297109i	0.857321	+	0.514782i	$0.172127\pi$
$522$		0			0
$523$	−1188.25	+	686.039i	−0.0993474	+	0.0573583i	−0.548851	−	0.835920i	$-0.684934\pi$
$523$	−1188.25	+	686.039i	−0.0993474	+	0.0573583i	0.449503	+	0.893279i	$0.351601\pi$
$524$	−452.281			−0.0377061
$525$		0			0
$526$	9654.47			0.800294
$527$	3138.63	−	1812.09i	0.259432	−	0.149783i
$528$		0			0
$529$	7274.73	−	12600.2i	0.597906	−	1.03560i
$530$	230.233	+	398.775i	0.0188692	+	0.0326824i
$531$		0			0
$532$	−212.974	+	130.291i	−0.0173564	+	0.0106181i
$533$			28589.8i			2.32338i
$534$		0			0
$535$	1261.25	+	728.182i	0.101922	+	0.0588450i
$536$	5067.35	+	2925.63i	0.408351	+	0.235761i
$537$		0			0
$538$			198.526i			0.0159090i
$539$	17794.5	−	11517.8i	1.42201	−	0.920425i
$540$		0			0
$541$	−4060.87	−	7033.63i	−0.322718	−	0.558963i	0.658330	−	0.752729i	$-0.271263\pi$
$541$	−4060.87	−	7033.63i	−0.322718	−	0.558963i	−0.981048	+	0.193766i	$0.937930\pi$
$542$	2259.63	−	3913.80i	0.179077	−	0.310170i
$543$		0			0
$544$	−433.928	+	250.529i	−0.0341995	+	0.0197451i
$545$	1632.58			0.128315
$546$		0			0
$547$	885.602			0.0692241			0.0346121	−	0.999401i	$-0.488980\pi$
$547$	885.602			0.0692241			0.0346121	+	0.999401i	$0.488980\pi$
$548$	3872.17	−	2235.60i	0.301845	−	0.174270i
$549$		0			0
$550$	−7657.02	+	13262.4i	−0.593630	+	1.02820i
$551$	66.3311	+	114.889i	0.00512849	+	0.00888281i
$552$		0			0
$553$	−9396.43	+	17275.0i	−0.722562	+	1.32841i
$554$			3846.01i			0.294948i
$555$		0			0
$556$	58.5968	+	33.8309i	0.00446953	+	0.00258049i
$557$	9805.15	+	5661.00i	0.745884	+	0.430636i	0.824205	−	0.566292i	$-0.191622\pi$
$557$	9805.15	+	5661.00i	0.745884	+	0.430636i	−0.0783207	+	0.996928i	$0.524956\pi$
$558$		0			0
$559$			4645.81i			0.351515i
$560$	−310.211	+	7.88955i	−0.0234086	+	0.000595347i
$561$		0			0
$562$	−136.085	−	235.706i	−0.0102142	−	0.0176915i
$563$	3068.25	−	5314.37i	0.229683	−	0.397823i	−0.728031	−	0.685544i	$-0.759564\pi$
$563$	3068.25	−	5314.37i	0.229683	−	0.397823i	0.957714	+	0.287721i	$0.0928977\pi$
$564$		0			0
$565$	1296.04	−	748.270i	0.0965042	−	0.0557167i
$566$	12333.1			0.915900
$567$		0			0
$568$	1133.28			0.0837174
$569$	−16176.2	+	9339.34i	−1.19181	+	0.688094i	−0.958718	−	0.284360i	$-0.908219\pi$
$569$	−16176.2	+	9339.34i	−1.19181	+	0.688094i	−0.233096	+	0.972454i	$0.574886\pi$
$570$		0			0
$571$	11480.5	−	19884.8i	0.841409	−	1.45736i	−0.0472940	−	0.998881i	$-0.515060\pi$
$571$	11480.5	−	19884.8i	0.841409	−	1.45736i	0.888703	−	0.458483i	$-0.151607\pi$
$572$	8741.86	+	15141.4i	0.639014	+	1.10680i
$573$		0			0
$574$	−13152.6	−	7154.09i	−0.956408	−	0.520220i
$575$		−	20252.2i		−	1.46883i
$576$		0			0
$577$	15917.7	+	9190.11i	1.14847	+	0.663067i	0.948513	−	0.316739i	$-0.102588\pi$
$577$	15917.7	+	9190.11i	1.14847	+	0.663067i	0.199952	+	0.979806i	$0.435921\pi$
$578$	8084.91	+	4667.83i	0.581813	+	0.335910i
$579$		0			0
$580$			164.886i			0.0118043i
$581$	979.295	+	1600.77i	0.0699277	+	0.114305i
$582$		0			0
$583$	6793.34	+	11766.4i	0.482592	+	0.835874i
$584$	3178.01	−	5504.48i	0.225183	−	0.390029i
$585$		0			0
$586$	7321.97	−	4227.34i	0.516156	−	0.298003i
$587$	24372.5			1.71374			0.856868	−	0.515536i	$-0.172407\pi$
$587$	24372.5			1.71374			0.856868	+	0.515536i	$0.172407\pi$
$588$		0			0
$589$	−780.058			−0.0545700
$590$	−901.356	+	520.398i	−0.0628953	+	0.0363126i
$591$		0			0
$592$	−1854.88	+	3212.75i	−0.128775	+	0.223046i
$593$	−6530.78	−	11311.6i	−0.452255	−	0.783328i	0.546271	−	0.837609i	$-0.316047\pi$
$593$	−6530.78	−	11311.6i	−0.452255	−	0.783328i	−0.998526	+	0.0542803i	$0.982714\pi$
$594$		0			0
$595$	−158.477	−	259.048i	−0.0109192	−	0.0178486i
$596$			4017.51i			0.276114i
$597$		0			0
$598$	−20023.8	−	11560.8i	−1.36929	−	0.790560i
$599$	18806.2	+	10857.8i	1.28280	+	0.740628i	0.977360	−	0.211582i	$-0.0678616\pi$
$599$	18806.2	+	10857.8i	1.28280	+	0.740628i	0.305445	+	0.952210i	$0.401195\pi$
$600$		0			0
$601$			9329.14i			0.633184i	0.948562	+	0.316592i	$0.102539\pi$
$601$			9329.14i			0.633184i	−0.948562	+	0.316592i	$0.897461\pi$
$602$	−2137.28	−	1162.53i	−0.144700	−	0.0787066i
$603$		0			0
$604$	−5433.06	−	9410.34i	−0.366007	−	0.633943i
$605$	1302.74	−	2256.41i	0.0875434	−	0.151630i
$606$		0			0
$607$	10975.9	−	6336.94i	0.733934	−	0.423737i	−0.0859257	−	0.996302i	$-0.527385\pi$
$607$	10975.9	−	6336.94i	0.733934	−	0.423737i	0.819860	+	0.572565i	$0.194051\pi$
$608$	107.846			0.00719365
$609$		0			0
$610$	−758.158			−0.0503228
$611$	−36520.6	+	21085.2i	−2.41811	+	1.39610i
$612$		0			0
$613$	−5945.64	+	10298.1i	−0.391749	+	0.678529i	−0.992680	−	0.120772i	$-0.961463\pi$
$613$	−5945.64	+	10298.1i	−0.391749	+	0.678529i	0.600932	+	0.799301i	$0.294796\pi$
$614$	−2990.10	−	5179.00i	−0.196532	−	0.340403i
$615$		0			0
$616$	−9153.21	+	232.792i	−0.598690	+	0.0152264i
$617$			1855.59i			0.121075i	0.998166	+	0.0605375i	$0.0192815\pi$
$617$			1855.59i			0.121075i	−0.998166	+	0.0605375i	$0.980719\pi$
$618$		0			0
$619$	−18981.8	−	10959.1i	−1.23254	−	0.711607i	−0.264981	−	0.964254i	$-0.585366\pi$
$619$	−18981.8	−	10959.1i	−1.23254	−	0.711607i	−0.967559	+	0.252647i	$0.918699\pi$
$620$	−839.640	−	484.766i	−0.0543883	−	0.0314011i
$621$		0			0
$622$			8599.37i			0.554346i
$623$	−5901.98	+	10850.6i	−0.379547	+	0.697786i
$624$		0			0
$625$	−7607.48	−	13176.5i	−0.486879	−	0.843299i
$626$	−4860.14	+	8418.02i	−0.310304	+	0.537463i
$627$		0			0
$628$	−1188.52	+	686.191i	−0.0755207	+	0.0436019i
$629$	−3630.47			−0.230137
$630$		0			0
$631$	26713.2			1.68532			0.842660	−	0.538446i	$-0.180988\pi$
$631$	26713.2			1.68532			0.842660	+	0.538446i	$0.180988\pi$
$632$	7356.51	−	4247.28i	0.463016	−	0.267323i
$633$		0			0
$634$	−6072.63	+	10518.1i	−0.380402	+	0.658876i
$635$	−537.972	−	931.795i	−0.0336201	−	0.0582318i
$636$		0			0
$637$	21599.2	+	11046.3i	1.34347	+	0.687084i
$638$			4865.18i			0.301904i
$639$		0			0
$640$	116.084	+	67.0209i	0.00716970	+	0.00413943i
$641$	−10903.6	−	6295.18i	−0.671865	−	0.387901i	0.124918	−	0.992167i	$-0.460133\pi$
$641$	−10903.6	−	6295.18i	−0.671865	−	0.387901i	−0.796783	+	0.604266i	$0.793467\pi$
$642$		0			0
$643$		−	16074.3i		−	0.985863i	−0.870068	−	0.492931i	$-0.835925\pi$
$643$		−	16074.3i		−	0.985863i	0.870068	−	0.492931i	$-0.164075\pi$
$644$	10329.1	−	6318.99i	0.632024	−	0.386651i
$645$		0			0
$646$	52.7706	+	91.4013i	0.00321398	+	0.00556678i
$647$	12167.9	−	21075.4i	0.739364	−	1.28062i	−0.213418	−	0.976961i	$-0.568460\pi$
$647$	12167.9	−	21075.4i	0.739364	−	1.28062i	0.952782	−	0.303655i	$-0.0982070\pi$
$648$		0			0
$649$	−26595.8	+	15355.1i	−1.60859	+	0.928720i
$650$	−17527.1			−1.05765
$651$		0			0
$652$	−4796.64			−0.288115
$653$	−7900.38	+	4561.29i	−0.473454	+	0.273349i	−0.717685	−	0.696368i	$-0.754798\pi$
$653$	−7900.38	+	4561.29i	−0.473454	+	0.273349i	0.244230	+	0.969717i	$0.421465\pi$
$654$		0			0
$655$	−59.2037	+	102.544i	−0.00353173	+	0.00611713i
$656$	3233.73	+	5600.98i	0.192463	+	0.333356i
$657$		0			0
$658$	−561.489	−	22077.3i	−0.0332662	−	1.30800i
$659$			11132.3i			0.658044i	0.944322	+	0.329022i	$0.106719\pi$
$659$			11132.3i			0.658044i	−0.944322	+	0.329022i	$0.893281\pi$
$660$		0			0
$661$	18775.1	+	10839.8i	1.10479	+	0.637852i	0.937475	−	0.348052i	$-0.113157\pi$
$661$	18775.1	+	10839.8i	1.10479	+	0.637852i	0.167316	+	0.985903i	$0.446490\pi$
$662$	−16767.6	−	9680.77i	−0.984427	−	0.568359i
$663$		0			0
$664$		−	810.597i		−	0.0473754i
$665$	1.66183	+	65.3419i	9.69068e−5	+	0.00381030i
$666$		0			0
$667$	−3217.01	−	5572.02i	−0.186751	−	0.323462i
$668$	5584.02	−	9671.81i	0.323432	−	0.560200i
$669$		0			0
$670$	1326.63	−	765.932i	0.0764960	−	0.0441650i
$671$	−22370.5			−1.28704
$672$		0			0
$673$	7578.24			0.434056			0.217028	−	0.976165i	$-0.430364\pi$
$673$	7578.24			0.434056			0.217028	+	0.976165i	$0.430364\pi$
$674$	1692.96	−	977.429i	0.0967512	−	0.0558593i
$675$		0			0
$676$	−5611.18	+	9718.84i	−0.319252	+	0.552961i
$677$	−6159.13	−	10667.9i	−0.349653	−	0.605616i	0.636535	−	0.771248i	$-0.280367\pi$
$677$	−6159.13	−	10667.9i	−0.349653	−	0.605616i	−0.986188	+	0.165632i	$0.947034\pi$
$678$		0			0
$679$	9297.20	−	5687.71i	0.525469	−	0.321464i
$680$			131.177i			0.00739766i
$681$		0			0
$682$	−24774.8	−	14303.7i	−1.39102	−	0.803105i
$683$	−12404.6	−	7161.79i	−0.694946	−	0.401227i	0.110516	−	0.993874i	$-0.464750\pi$
$683$	−12404.6	−	7161.79i	−0.694946	−	0.401227i	−0.805462	+	0.592647i	$0.798083\pi$
$684$		0			0
$685$		−	1170.56i		−	0.0652917i
$686$	−10486.7	+	7172.48i	−0.583648	+	0.399193i
$687$		0			0
$688$	525.478	+	910.154i	0.0291187	+	0.0504350i
$689$	−7775.06	+	13466.8i	−0.429907	+	0.744621i
$690$		0			0
$691$	19602.1	−	11317.3i	1.07916	−	0.623053i	0.148490	−	0.988914i	$-0.452559\pi$
$691$	19602.1	−	11317.3i	1.07916	−	0.623053i	0.930670	+	0.365861i	$0.119225\pi$
$692$	−10959.4			−0.602045
$693$		0			0
$694$	20066.8			1.09759
$695$	15.3407	−	8.85695i	0.000837274	−	0.000483400i
$696$		0			0
$697$	−3164.61	+	5481.27i	−0.171977	+	0.297874i
$698$	−3525.19	−	6105.80i	−0.191161	−	0.331100i
$699$		0			0
$700$	4385.86	−	8063.27i	0.236814	−	0.435376i
$701$		−	14459.7i		−	0.779080i	−0.921010	−	0.389540i	$-0.872634\pi$
$701$		−	14459.7i		−	0.779080i	0.921010	−	0.389540i	$-0.127366\pi$
$702$		0			0
$703$	676.723	+	390.706i	0.0363060	+	0.0209613i
$704$	3425.21	+	1977.55i	0.183370	+	0.105869i
$705$		0			0
$706$		−	7947.12i		−	0.423646i
$707$	33329.3	−	847.659i	1.77295	−	0.0450912i
$708$		0			0
$709$	5833.95	+	10104.7i	0.309025	+	0.535247i	0.978149	−	0.207904i	$-0.0666640\pi$
$709$	5833.95	+	10104.7i	0.309025	+	0.535247i	−0.669124	+	0.743150i	$0.733331\pi$
$710$	148.347	−	256.944i	0.00784135	−	0.0135816i
$711$		0			0
$712$	4620.69	−	2667.76i	0.243213	−	0.140419i
$713$	37832.2			1.98713
$714$		0			0
$715$	4577.25			0.239412
$716$	11982.5	−	6918.12i	0.625430	−	0.361092i
$717$		0			0
$718$	4823.79	−	8355.04i	0.250727	−	0.434272i
$719$	14170.9	+	24544.8i	0.735029	+	1.27311i	0.954710	+	0.297536i	$0.0961650\pi$
$719$	14170.9	+	24544.8i	0.735029	+	1.27311i	−0.219681	+	0.975572i	$0.570502\pi$
$720$		0			0
$721$	−5990.66	−	3258.50i	−0.309437	−	0.168312i
$722$			13695.3i			0.705936i
$723$		0			0
$724$	−3855.22	−	2225.81i	−0.197898	−	0.114256i
$725$	−4223.83	−	2438.63i	−0.216371	−	0.124922i
$726$		0			0
$727$			25532.4i			1.30254i	0.758847	+	0.651269i	$0.225763\pi$
$727$			25532.4i			1.30254i	−0.758847	+	0.651269i	$0.774237\pi$
$728$	−5468.72	−	8939.23i	−0.278412	−	0.455096i
$729$		0			0
$730$	−832.005	−	1441.08i	−0.0421834	−	0.0730638i
$731$	−514.247	+	890.702i	−0.0260193	+	0.0450667i
$732$		0			0
$733$	14203.4	−	8200.33i	0.715709	−	0.413215i	−0.0974626	−	0.995239i	$-0.531073\pi$
$733$	14203.4	−	8200.33i	0.715709	−	0.413215i	0.813171	+	0.582025i	$0.197739\pi$
$734$	−5873.75			−0.295373
$735$		0			0
$736$	−5230.45			−0.261953
$737$	39144.2	−	22599.9i	1.95644	−	1.12955i
$738$		0			0
$739$	6091.41	−	10550.6i	0.303215	−	0.525184i	−0.673647	−	0.739053i	$-0.735273\pi$
$739$	6091.41	−	10550.6i	0.303215	−	0.525184i	0.976862	+	0.213869i	$0.0686066\pi$
$740$	485.608	+	841.098i	0.0241234	+	0.0417830i
$741$		0			0
$742$	−4249.76	−	6946.71i	−0.210261	−	0.343695i
$743$		−	34275.9i		−	1.69241i	−0.532856	−	0.846206i	$-0.678881\pi$
$743$		−	34275.9i		−	1.69241i	0.532856	−	0.846206i	$-0.321119\pi$
$744$		0			0
$745$	910.874	+	525.893i	0.0447944	+	0.0258621i
$746$	−9273.01	−	5353.78i	−0.455106	−	0.262756i
$747$		0			0
$748$			3870.56i			0.189200i
$749$	−22626.0	−	12307.0i	−1.10379	−	0.600383i
$750$		0			0
$751$	2241.80	+	3882.91i	0.108927	+	0.188668i	0.915336	−	0.402691i	$-0.131925\pi$
$751$	2241.80	+	3882.91i	0.108927	+	0.188668i	−0.806409	+	0.591359i	$0.798592\pi$
$752$	−4769.80	+	8261.53i	−0.231299	+	0.400621i
$753$		0			0
$754$	−4822.26	+	2784.13i	−0.232913	+	0.134472i
$755$	−2844.76			−0.137128
$756$		0			0
$757$	−32462.7			−1.55862			−0.779312	−	0.626636i	$-0.784431\pi$
$757$	−32462.7			−1.55862			−0.779312	+	0.626636i	$0.784431\pi$
$758$	15330.1	−	8850.81i	0.734581	−	0.424111i
$759$		0			0
$760$	14.1171	−	24.4515i	0.000673790	−	0.00116704i
$761$	−13686.1	−	23705.0i	−0.651932	−	1.12918i	−0.982653	−	0.185451i	$-0.940625\pi$
$761$	−13686.1	−	23705.0i	−0.651932	−	1.12918i	0.330721	−	0.943728i	$-0.392708\pi$
$762$		0			0
$763$	−28863.6	+	734.083i	−1.36950	+	0.0348304i
$764$			891.669i			0.0422244i
$765$		0			0
$766$	−12981.7	−	7494.96i	−0.612332	−	0.353530i
$767$	−30439.2	−	17574.1i	−1.43298	−	0.827331i
$768$		0			0
$769$		−	13708.3i		−	0.642828i	−0.946939	−	0.321414i	$-0.895842\pi$
$769$		−	13708.3i		−	0.642828i	0.946939	−	0.321414i	$-0.104158\pi$
$770$	−1145.38	+	2105.74i	−0.0536059	+	0.0985528i
$771$		0			0
$772$	−5189.96	−	8989.28i	−0.241957	−	0.419082i
$773$	−12405.2	+	21486.4i	−0.577210	+	0.999757i	0.418588	+	0.908176i	$0.362525\pi$
$773$	−12405.2	+	21486.4i	−0.577210	+	0.999757i	−0.995798	+	0.0915806i	$0.970808\pi$
$774$		0			0
$775$	24836.2	−	14339.2i	1.15115	−	0.664619i
$776$	−4707.92			−0.217789
$777$		0			0
$778$	6755.10			0.311288
$779$	1179.77	−	681.142i	0.0542616	−	0.0313279i
$780$		0			0
$781$	4377.18	−	7581.50i	0.200548	−	0.347359i
$782$	−2559.33	−	4432.89i	−0.117035	−	0.202711i
$783$		0			0
$784$	5480.90	−	278.971i	0.249677	−	0.0127082i
$785$			359.290i			0.0163358i
$786$		0			0
$787$	606.028	+	349.890i	0.0274492	+	0.0158478i	0.513662	−	0.857993i	$-0.328289\pi$
$787$	606.028	+	349.890i	0.0274492	+	0.0158478i	−0.486213	+	0.873841i	$0.661622\pi$
$788$	17657.5	+	10194.6i	0.798253	+	0.460872i
$789$		0			0
$790$		−	2223.88i		−	0.100155i
$791$	−22577.2	+	13812.0i	−1.01486	+	0.620857i
$792$		0			0
$793$	−12801.7	−	22173.1i	−0.573267	−	0.992927i
$794$	7602.87	−	13168.6i	0.339818	−	0.588582i
$795$		0			0
$796$	15588.6	−	9000.06i	0.694123	−	0.400752i
$797$	−27109.3			−1.20484			−0.602422	−	0.798178i	$-0.705798\pi$
$797$	−27109.3			−1.20484			−0.602422	+	0.798178i	$0.705798\pi$
$798$		0			0
$799$	−9335.70			−0.413359
$800$	−3433.71	+	1982.45i	−0.151750	+	0.0876129i
$801$		0			0
$802$	−9531.49	+	16509.0i	−0.419662	+	0.726875i
$803$	−24549.5	−	42521.0i	−1.07887	−	1.86866i
$804$		0			0
$805$	−80.5975	−	3169.03i	−0.00352881	−	0.138750i
$806$		−	32741.6i		−	1.43086i
$807$		0			0
$808$	−12472.1	−	7200.78i	−0.543029	−	0.313518i
$809$	18559.7	+	10715.4i	0.806581	+	0.465680i	0.845767	−	0.533552i	$-0.179143\pi$
$809$	18559.7	+	10715.4i	0.806581	+	0.465680i	−0.0391863	+	0.999232i	$0.512477\pi$
$810$		0			0
$811$			4504.47i			0.195035i	0.995234	+	0.0975175i	$0.0310902\pi$
$811$			4504.47i			0.195035i	−0.995234	+	0.0975175i	$0.968910\pi$
$812$	−74.1403	−	2915.14i	−0.00320420	−	0.125987i
$813$		0			0
$814$	14328.6	+	24817.8i	0.616973	+	1.06863i
$815$	−627.881	+	1087.52i	−0.0269862	+	0.0467414i
$816$		0			0
$817$	191.712	−	110.685i	0.00820950	−	0.00473975i
$818$	−5304.60			−0.226737
$819$		0			0
$820$	1693.18			0.0721079
$821$	37115.4	−	21428.6i	1.57775	−	0.910917i	0.582582	−	0.812772i	$-0.302042\pi$
$821$	37115.4	−	21428.6i	1.57775	−	0.910917i	0.995172	−	0.0981448i	$-0.0312908\pi$
$822$		0			0
$823$	5329.30	−	9230.62i	0.225720	−	0.390959i	−0.730815	−	0.682576i	$-0.760860\pi$
$823$	5329.30	−	9230.62i	0.225720	−	0.390959i	0.956535	+	0.291617i	$0.0941931\pi$
$824$	1472.88	+	2551.10i	0.0622696	+	0.107854i
$825$		0			0
$826$	15701.8	−	9605.80i	0.661421	−	0.404635i
$827$			12138.7i			0.510405i	0.966888	+	0.255203i	$0.0821421\pi$
$827$			12138.7i			0.510405i	−0.966888	+	0.255203i	$0.917858\pi$
$828$		0			0
$829$	26860.1	+	15507.7i	1.12532	+	0.649704i	0.942754	−	0.333489i	$-0.108226\pi$
$829$	26860.1	+	15507.7i	1.12532	+	0.649704i	0.182567	+	0.983193i	$0.441559\pi$
$830$	−183.783	−	106.107i	−0.00768580	−	0.00443740i
$831$		0			0
$832$			4526.65i			0.188622i
$833$	2918.31	+	4508.65i	0.121385	+	0.187534i
$834$		0			0
$835$	−1461.90	−	2532.09i	−0.0605882	−	0.104942i
$836$	416.544	−	721.476i	0.0172326	−	0.0298478i
$837$		0			0
$838$	−22157.8	+	12792.8i	−0.913398	+	0.527350i
$839$	8304.65			0.341726			0.170863	−	0.985295i	$-0.445344\pi$
$839$	8304.65			0.341726			0.170863	+	0.985295i	$0.445344\pi$
$840$		0			0
$841$	22839.5			0.936468
$842$	−13325.0	+	7693.22i	−0.545382	+	0.314876i
$843$		0			0
$844$	−1907.88	+	3304.55i	−0.0778105	+	0.134772i
$845$	1469.01	+	2544.40i	0.0598052	+	0.103586i
$846$		0			0
$847$	−22017.5	+	40478.5i	−0.893187	+	1.64210i
$848$			3517.68i			0.142450i
$849$		0			0
$850$	−3360.32	−	1940.08i	−0.135598	−	0.0782874i
$851$	−32820.5	−	18949.0i	−1.32206	−	0.763292i
$852$		0			0
$853$			16024.0i			0.643202i	0.946875	+	0.321601i	$0.104221\pi$
$853$			16024.0i			0.643202i	−0.946875	+	0.321601i	$0.895779\pi$
$854$	13404.0	−	340.903i	0.537092	−	0.0136598i
$855$		0			0
$856$	5562.88	+	9635.19i	0.222121	+	0.384724i
$857$	13747.5	−	23811.4i	0.547966	−	0.949105i	−0.450447	−	0.892803i	$-0.648736\pi$
$857$	13747.5	−	23811.4i	0.547966	−	0.949105i	0.998414	−	0.0563025i	$-0.0179311\pi$
$858$		0			0
$859$	−42053.8	+	24279.8i	−1.67038	+	0.964394i	−0.702956	+	0.711233i	$0.748137\pi$
$859$	−42053.8	+	24279.8i	−1.67038	+	0.964394i	−0.967424	+	0.253161i	$0.918530\pi$
$860$	275.141			0.0109096
$861$		0			0
$862$	5071.27			0.200381
$863$	−7843.23	+	4528.29i	−0.309371	+	0.178615i	−0.646645	−	0.762791i	$-0.723828\pi$
$863$	−7843.23	+	4528.29i	−0.309371	+	0.178615i	0.337274	+	0.941406i	$0.390495\pi$
$864$		0			0
$865$	−1434.59	+	2484.79i	−0.0563903	+	0.0976709i
$866$	−9611.55	−	16647.7i	−0.377152	−	0.653247i
$867$		0			0
$868$	15062.6	+	8193.01i	0.589007	+	0.320379i
$869$		−	65618.8i		−	2.56152i
$870$		0			0
$871$	44801.0	+	25865.9i	1.74285	+	1.00624i
$872$	10801.0	+	6235.96i	0.419459	+	0.242175i
$873$		0			0
$874$			1101.73i			0.0426390i
$875$	−2519.18	−	4117.88i	−0.0973302	−	0.159097i
$876$		0			0
$877$	−24349.6	−	42174.8i	−0.937547	−	1.62388i	−0.770028	−	0.638010i	$-0.779758\pi$
$877$	−24349.6	−	42174.8i	−0.937547	−	1.62388i	−0.167518	−	0.985869i	$-0.553575\pi$
$878$	−458.000	+	793.280i	−0.0176045	+	0.0304919i
$879$		0			0
$880$	896.722	−	517.723i	0.0343506	−	0.0198323i
$881$	−47118.8			−1.80190			−0.900950	−	0.433923i	$-0.857129\pi$
$881$	−47118.8			−1.80190			−0.900950	+	0.433923i	$0.857129\pi$
$882$		0			0
$883$	−1664.93			−0.0634535			−0.0317268	−	0.999497i	$-0.510101\pi$
$883$	−1664.93			−0.0634535			−0.0317268	+	0.999497i	$0.510101\pi$
$884$	−3836.41	+	2214.95i	−0.145964	+	0.0842726i
$885$		0			0
$886$	2824.60	−	4892.36i	0.107104	−	0.185510i
$887$	14901.7	+	25810.6i	0.564094	+	0.977039i	0.997133	+	0.0756641i	$0.0241077\pi$
$887$	14901.7	+	25810.6i	0.564094	+	0.977039i	−0.433040	+	0.901375i	$0.642559\pi$
$888$		0			0
$889$	9930.20	+	16232.0i	0.374632	+	0.612378i
$890$		−	1396.84i		−	0.0526092i
$891$		0			0
$892$	−8297.19	−	4790.39i	−0.311447	−	0.179814i
$893$	1740.18	+	1004.70i	0.0652105	+	0.0376493i
$894$		0			0
$895$		−	3622.33i		−	0.135286i
$896$	−2082.47	−	1132.72i	−0.0776454	−	0.0422337i
$897$		0			0
$898$	−6366.85	−	11027.7i	−0.236598	−	0.409799i
$899$	4555.49	−	7890.33i	0.169003	−	0.292722i
$900$		0			0
$901$	−2981.29	+	1721.25i	−0.110234	+	0.0636439i
$902$	49959.7			1.84421
$903$		0			0
$904$	11432.7			0.420625
$905$	−1009.30	+	582.718i	−0.0370720	+	0.0214035i
$906$		0			0
$907$	3781.78	−	6550.24i	0.138448	−	0.239798i	−0.788462	−	0.615084i	$-0.789122\pi$
$907$	3781.78	−	6550.24i	0.138448	−	0.239798i	0.926909	+	0.375286i	$0.122455\pi$
$908$	−878.074	−	1520.87i	−0.0320924	−	0.0555857i
$909$		0			0
$910$	−2742.61	+	69.7525i	−0.0999085	+	0.00254096i
$911$			19685.0i			0.715908i	0.933739	+	0.357954i	$0.116525\pi$
$911$			19685.0i			0.715908i	−0.933739	+	0.357954i	$0.883475\pi$
$912$		0			0
$913$	−5422.78	−	3130.85i	−0.196570	−	0.113489i
$914$	−20658.6	−	11927.2i	−0.747621	−	0.431639i
$915$		0			0
$916$			6310.16i			0.227613i
$917$	1000.60	−	1839.57i	0.0360335	−	0.0662464i
$918$		0			0
$919$	21691.6	+	37570.9i	0.778606	+	1.34858i	0.932745	+	0.360536i	$0.117406\pi$
$919$	21691.6	+	37570.9i	0.778606	+	1.34858i	−0.154140	+	0.988049i	$0.549261\pi$
$920$	−684.668	+	1185.88i	−0.0245357	+	0.0424970i
$921$		0			0
$922$	12309.4	−	7106.81i	0.439682	−	0.253851i
$923$	10019.5			0.357308
$924$		0			0
$925$	−28728.2			−1.02117
$926$	−13571.4	+	7835.44i	−0.481623	+	0.278065i
$927$		0			0
$928$	−629.814	+	1090.87i	−0.0222787	+	0.0385879i
$929$	18264.4	+	31634.9i	0.645033	+	1.11723i	0.984294	+	0.176538i	$0.0564899\pi$
$929$	18264.4	+	31634.9i	0.645033	+	1.11723i	−0.339260	+	0.940693i	$0.610177\pi$
$930$		0			0
$931$	−58.7615	−	1154.48i	−0.00206856	−	0.0406408i
$932$			14586.6i			0.512662i
$933$		0			0
$934$	−3895.79	−	2249.24i	−0.136482	−	0.0787980i
$935$	877.556	+	506.657i	0.0306943	+	0.0177214i
$936$		0			0
$937$		−	52122.8i		−	1.81727i	−0.417595	−	0.908633i	$-0.637127\pi$
$937$		−	52122.8i		−	1.81727i	0.417595	−	0.908633i	$-0.362873\pi$
$938$	−23110.2	+	14138.0i	−0.804449	+	0.492135i
$939$		0			0
$940$	1248.73	+	2162.87i	0.0433290	+	0.0750480i
$941$	−2557.97	+	4430.53i	−0.0886157	+	0.153487i	−0.906926	−	0.421289i	$-0.861578\pi$
$941$	−2557.97	+	4430.53i	−0.0886157	+	0.153487i	0.818310	+	0.574776i	$0.194911\pi$
$942$		0			0
$943$	−57218.1	+	33034.9i	−1.97590	+	1.14079i
$944$	−7951.06			−0.274137
$945$		0			0
$946$	8118.41			0.279019
$947$	15364.2	−	8870.50i	0.527210	−	0.304385i	−0.212670	−	0.977124i	$-0.568216\pi$
$947$	15364.2	−	8870.50i	0.527210	−	0.304385i	0.739880	+	0.672739i	$0.234882\pi$
$948$		0			0
$949$	28097.2	−	48665.8i	0.961089	−	1.66466i
$950$	417.578	+	723.266i	0.0142611	+	0.0247009i
$951$		0			0
$952$	−58.9833	−	2319.18i	−0.00200805	−	0.0789548i
$953$			28803.1i			0.979037i	0.871993	+	0.489519i	$0.162827\pi$
$953$			28803.1i			0.979037i	−0.871993	+	0.489519i	$0.837173\pi$
$954$		0			0
$955$	202.164	+	116.720i	0.00685014	+	0.00395493i
$956$	−23016.5	−	13288.6i	−0.778668	−	0.449564i
$957$		0			0
$958$			31117.0i			1.04942i
$959$	526.339	+	20695.2i	0.0177230	+	0.696855i
$960$		0			0
$961$	11890.9	+	20595.6i	0.399144	+	0.691337i
$962$	−16399.2	+	28404.3i	−0.549617	+	0.951965i
$963$		0			0
$964$	−5739.12	+	3313.48i	−0.191748	+	0.110705i
$965$	−2717.47			−0.0906512
$966$		0			0
$967$	−51049.4			−1.69766			−0.848831	−	0.528665i	$-0.822693\pi$
$967$	−51049.4			−1.69766			−0.848831	+	0.528665i	$0.822693\pi$
$968$	17237.6	−	9952.14i	0.572353	−	0.330448i
$969$		0			0
$970$	−616.268	+	1067.41i	−0.0203991	+	0.0353323i
$971$	26201.9	+	45383.0i	0.865971	+	1.49991i	0.866080	+	0.499906i	$0.166632\pi$
$971$	26201.9	+	45383.0i	0.865971	+	1.49991i	−0.000108644		1.00000i	$0.500035\pi$
$972$		0			0
$973$	−267.237	+	163.487i	−0.00880496	+	0.00538657i
$974$		−	23709.1i		−	0.779968i
$975$		0			0
$976$	−5015.91	−	2895.94i	−0.164503	−	0.0949761i
$977$	36603.6	+	21133.1i	1.19862	+	0.692025i	0.960248	−	0.279148i	$-0.0900520\pi$
$977$	36603.6	+	21133.1i	1.19862	+	0.692025i	0.238375	+	0.971173i	$0.423385\pi$
$978$		0			0
$979$		−	41215.7i		−	1.34552i
$980$	654.202	−	1279.18i	0.0213242	−	0.0416958i
$981$		0			0
$982$	16825.9	+	29143.4i	0.546779	+	0.947050i
$983$	29518.7	−	51127.9i	0.957783	−	1.65893i	0.229915	−	0.973211i	$-0.426155\pi$
$983$	29518.7	−	51127.9i	0.957783	−	1.65893i	0.727868	−	0.685718i	$-0.240512\pi$
$984$		0			0
$985$	4622.75	−	2668.95i	0.149536	−	0.0863347i
$986$	−1232.71			−0.0398148
$987$		0			0
$988$	953.481			0.0307027
$989$	−9297.89	+	5368.14i	−0.298944	+	0.172595i
$990$		0			0
$991$	−10543.9	+	18262.5i	−0.337979	+	0.585397i	−0.984052	−	0.177878i	$-0.943077\pi$
$991$	−10543.9	+	18262.5i	−0.337979	+	0.585397i	0.646073	+	0.763275i	$0.276410\pi$
$992$	−3703.33	−	6414.35i	−0.118529	−	0.205298i
$993$		0			0
$994$	−2507.20	+	4609.42i	−0.0800037	+	0.147084i
$995$		−	4712.44i		−	0.150145i
$996$		0			0
$997$	46292.8	+	26727.1i	1.47052	+	0.849004i	0.999452	−	0.0330943i	$-0.0105362\pi$
$997$	46292.8	+	26727.1i	1.47052	+	0.849004i	0.471066	+	0.882098i	$0.343869\pi$
$998$	10486.2	+	6054.22i	0.332600	+	0.192027i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.k.d.269.8	yes	20
3.2	odd	2	inner	378.4.k.d.269.3	yes	20
7.5	odd	6	inner	378.4.k.d.215.3	✓	20
21.5	even	6	inner	378.4.k.d.215.8	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.k.d.215.3	✓	20	7.5	odd	6	inner
378.4.k.d.215.8	yes	20	21.5	even	6	inner
378.4.k.d.269.3	yes	20	3.2	odd	2	inner
378.4.k.d.269.8	yes	20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-0.523601 - 0.906903i) q^{5} +(9.66492 + 15.7984i) q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-0.523601 - 0.906903i) q^{5} +(9.66492 + 15.7984i) q^{7} -8.00000i q^{8} +(-1.81381 - 1.04720i) q^{10} +(-53.5189 - 30.8992i) q^{11} -70.7290i q^{13} +(32.5385 + 17.6987i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(7.82902 - 13.5603i) q^{17} +(-2.91867 + 1.68510i) q^{19} -4.18881 q^{20} -123.597 q^{22} +(141.553 - 81.7258i) q^{23} +(61.9517 - 107.303i) q^{25} +(-70.7290 - 122.506i) q^{26} +(74.0571 - 1.88348i) q^{28} -39.3634i q^{29} +(200.448 + 115.729i) q^{31} +(-27.7128 - 16.0000i) q^{32} -31.3161i q^{34} +(9.26706 - 17.0372i) q^{35} +(-115.930 - 200.797i) q^{37} +(-3.37019 + 5.83734i) q^{38} +(-7.25523 + 4.18881i) q^{40} -404.216 q^{41} -65.6847 q^{43} +(-214.076 + 123.597i) q^{44} +(163.452 - 283.107i) q^{46} +(-298.112 - 516.346i) q^{47} +(-156.179 + 305.381i) q^{49} -247.807i q^{50} +(-245.012 - 141.458i) q^{52} +(-190.400 - 109.928i) q^{53} +64.7153i q^{55} +(126.387 - 77.3194i) q^{56} +(-39.3634 - 68.1794i) q^{58} +(248.471 - 430.364i) q^{59} +(313.494 - 180.996i) q^{61} +462.916 q^{62} -64.0000 q^{64} +(-64.1443 + 37.0338i) q^{65} +(-365.704 + 633.418i) q^{67} +(-31.3161 - 54.2410i) q^{68} +(-0.986194 - 38.7764i) q^{70} +141.660i q^{71} +(688.060 + 397.252i) q^{73} +(-401.593 - 231.860i) q^{74} +13.4808i q^{76} +(-29.0990 - 1144.15i) q^{77} +(530.911 + 919.564i) q^{79} +(-8.37762 + 14.5105i) q^{80} +(-700.122 + 404.216i) q^{82} +101.325 q^{83} -16.3971 q^{85} +(-113.769 + 65.6847i) q^{86} +(-247.193 + 428.151i) q^{88} +(333.470 + 577.586i) q^{89} +(1117.40 - 683.590i) q^{91} -653.807i q^{92} +(-1032.69 - 596.224i) q^{94} +(3.05644 + 1.76464i) q^{95} -588.490i q^{97} +(34.8713 + 685.113i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 40 q^{4} + 4 q^{7}+O(q^{10}) \) 20 * q + 40 * q^4 + 4 * q^7	\( 20 q + 40 q^{4} + 4 q^{7} - 24 q^{10} - 160 q^{16} + 408 q^{19} + 240 q^{22} - 646 q^{25} + 56 q^{28} - 102 q^{31} + 194 q^{37} - 96 q^{40} - 2332 q^{43} - 624 q^{46} + 2840 q^{49} - 648 q^{52} + 96 q^{58} + 1878 q^{61} - 1280 q^{64} - 386 q^{67} + 3672 q^{70} + 1788 q^{73} + 814 q^{79} - 672 q^{82} - 4560 q^{85} + 480 q^{88} + 2724 q^{91} - 1536 q^{94}+O(q^{100}) \) 20 * q + 40 * q^4 + 4 * q^7 - 24 * q^10 - 160 * q^16 + 408 * q^19 + 240 * q^22 - 646 * q^25 + 56 * q^28 - 102 * q^31 + 194 * q^37 - 96 * q^40 - 2332 * q^43 - 624 * q^46 + 2840 * q^49 - 648 * q^52 + 96 * q^58 + 1878 * q^61 - 1280 * q^64 - 386 * q^67 + 3672 * q^70 + 1788 * q^73 + 814 * q^79 - 672 * q^82 - 4560 * q^85 + 480 * q^88 + 2724 * q^91 - 1536 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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