LMFDB - Embedded newform 378.4.f.b.253.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(127,378)]

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([4, 0]))

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

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

chi := DirichletCharacter("378.127");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.f (of order $3$, 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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 2x^{6} + 53x^{5} + 38x^{4} - 166x^{3} + 7x^{2} + 1543x + 2707 $ x^8 - 2x^7 - 2x^6 + 53x^5 + 38x^4 - 166x^3 + 7x^2 + 1543*x + 2707
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{6} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			253.3
Root			$2.52060 - 3.08349i$ of defining polynomial
Character	$\chi$	$=$	378.253
Dual form			378.4.f.b.127.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(2.98310 - 5.16688i) q^{5} +(3.50000 + 6.06218i) q^{7} -8.00000 q^{8} +O(q^{10})$	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(2.98310 - 5.16688i) q^{5} +(3.50000 + 6.06218i) q^{7} -8.00000 q^{8} +11.9324 q^{10} +(3.67432 + 6.36411i) q^{11} +(27.4070 - 47.4703i) q^{13} +(-7.00000 + 12.1244i) q^{14} +(-8.00000 - 13.8564i) q^{16} +53.3544 q^{17} +88.7964 q^{19} +(11.9324 + 20.6675i) q^{20} +(-7.34865 + 12.7282i) q^{22} +(-98.0698 + 169.862i) q^{23} +(44.7022 + 77.4265i) q^{25} +109.628 q^{26} -28.0000 q^{28} +(-48.8199 - 84.5585i) q^{29} +(82.9760 - 143.719i) q^{31} +(16.0000 - 27.7128i) q^{32} +(53.3544 + 92.4126i) q^{34} +41.7634 q^{35} +431.884 q^{37} +(88.7964 + 153.800i) q^{38} +(-23.8648 + 41.3351i) q^{40} +(47.5332 - 82.3299i) q^{41} +(187.423 + 324.627i) q^{43} -29.3946 q^{44} -392.279 q^{46} +(57.5647 + 99.7049i) q^{47} +(-24.5000 + 42.4352i) q^{49} +(-89.4044 + 154.853i) q^{50} +(109.628 + 189.881i) q^{52} +145.549 q^{53} +43.8435 q^{55} +(-28.0000 - 48.4974i) q^{56} +(97.6398 - 169.117i) q^{58} +(-202.457 + 350.666i) q^{59} +(288.425 + 499.567i) q^{61} +331.904 q^{62} +64.0000 q^{64} +(-163.516 - 283.218i) q^{65} +(21.8954 - 37.9239i) q^{67} +(-106.709 + 184.825i) q^{68} +(41.7634 + 72.3364i) q^{70} -196.885 q^{71} -510.969 q^{73} +(431.884 + 748.045i) q^{74} +(-177.593 + 307.600i) q^{76} +(-25.7203 + 44.5488i) q^{77} +(-196.883 - 341.011i) q^{79} -95.4592 q^{80} +190.133 q^{82} +(263.437 + 456.286i) q^{83} +(159.162 - 275.676i) q^{85} +(-374.847 + 649.254i) q^{86} +(-29.3946 - 50.9129i) q^{88} +589.818 q^{89} +383.698 q^{91} +(-392.279 - 679.448i) q^{92} +(-115.129 + 199.410i) q^{94} +(264.889 - 458.800i) q^{95} +(-468.232 - 811.002i) q^{97} -98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 - 16 * q^4 - q^5 + 28 * q^7 - 64 * q^8	$ 8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8} - 4 q^{10} - 5 q^{11} + 21 q^{13} - 56 q^{14} - 64 q^{16} + 46 q^{17} - 188 q^{19} - 4 q^{20} + 10 q^{22} - 374 q^{23} - 41 q^{25} + 84 q^{26} - 224 q^{28} - 271 q^{29} + 243 q^{31} + 128 q^{32} + 46 q^{34} - 14 q^{35} - 362 q^{37} - 188 q^{38} + 8 q^{40} + 213 q^{41} + 238 q^{43} + 40 q^{44} - 1496 q^{46} - 675 q^{47} - 196 q^{49} + 82 q^{50} + 84 q^{52} - 108 q^{53} - 2828 q^{55} - 224 q^{56} + 542 q^{58} - 202 q^{59} + 1212 q^{61} + 972 q^{62} + 512 q^{64} - 549 q^{65} - 139 q^{67} - 92 q^{68} - 14 q^{70} + 2590 q^{71} - 4000 q^{73} - 362 q^{74} + 376 q^{76} + 35 q^{77} + 1545 q^{79} + 32 q^{80} + 852 q^{82} + 142 q^{83} + 793 q^{85} - 476 q^{86} + 40 q^{88} + 264 q^{89} + 294 q^{91} - 1496 q^{92} + 1350 q^{94} - 1244 q^{95} + 638 q^{97} - 784 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 - 16 * q^4 - q^5 + 28 * q^7 - 64 * q^8 - 4 * q^10 - 5 * q^11 + 21 * q^13 - 56 * q^14 - 64 * q^16 + 46 * q^17 - 188 * q^19 - 4 * q^20 + 10 * q^22 - 374 * q^23 - 41 * q^25 + 84 * q^26 - 224 * q^28 - 271 * q^29 + 243 * q^31 + 128 * q^32 + 46 * q^34 - 14 * q^35 - 362 * q^37 - 188 * q^38 + 8 * q^40 + 213 * q^41 + 238 * q^43 + 40 * q^44 - 1496 * q^46 - 675 * q^47 - 196 * q^49 + 82 * q^50 + 84 * q^52 - 108 * q^53 - 2828 * q^55 - 224 * q^56 + 542 * q^58 - 202 * q^59 + 1212 * q^61 + 972 * q^62 + 512 * q^64 - 549 * q^65 - 139 * q^67 - 92 * q^68 - 14 * q^70 + 2590 * q^71 - 4000 * q^73 - 362 * q^74 + 376 * q^76 + 35 * q^77 + 1545 * q^79 + 32 * q^80 + 852 * q^82 + 142 * q^83 + 793 * q^85 - 476 * q^86 + 40 * q^88 + 264 * q^89 + 294 * q^91 - 1496 * q^92 + 1350 * q^94 - 1244 * q^95 + 638 * q^97 - 784 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	2.98310	−	5.16688i	0.266817	−	0.462140i	−0.701221	−	0.712944i	$-0.747362\pi$
$5$	2.98310	−	5.16688i	0.266817	−	0.462140i	0.968038	+	0.250804i	$0.0806948\pi$
$6$		0			0
$7$	3.50000	+	6.06218i	0.188982	+	0.327327i
$7$	3.50000	+	6.06218i	0.188982	+	0.327327i
$8$	−8.00000			−0.353553
$9$		0			0
$10$	11.9324			0.377336
$11$	3.67432	+	6.36411i	0.100714	+	0.174441i	0.911979	−	0.410237i	$-0.134554\pi$
$11$	3.67432	+	6.36411i	0.100714	+	0.174441i	−0.811265	+	0.584678i	$0.801221\pi$
$12$		0			0
$13$	27.4070	−	47.4703i	0.584718	−	1.01276i	−0.410192	−	0.911999i	$-0.634538\pi$
$13$	27.4070	−	47.4703i	0.584718	−	1.01276i	0.994910	−	0.100763i	$-0.0321283\pi$
$14$	−7.00000	+	12.1244i	−0.133631	+	0.231455i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	53.3544			0.761197			0.380598	−	0.924740i	$-0.375718\pi$
$17$	53.3544			0.761197			0.380598	+	0.924740i	$0.375718\pi$
$18$		0			0
$19$	88.7964			1.07217			0.536087	−	0.844163i	$-0.319902\pi$
$19$	88.7964			1.07217			0.536087	+	0.844163i	$0.319902\pi$
$20$	11.9324	+	20.6675i	0.133408	+	0.231070i
$21$		0			0
$22$	−7.34865	+	12.7282i	−0.0712153	+	0.123349i
$23$	−98.0698	+	169.862i	−0.889086	+	1.53994i	−0.0481279	+	0.998841i	$0.515326\pi$
$23$	−98.0698	+	169.862i	−0.889086	+	1.53994i	−0.840958	+	0.541101i	$0.818008\pi$
$24$		0			0
$25$	44.7022	+	77.4265i	0.357618	+	0.619412i
$26$	109.628			0.826916
$27$		0			0
$28$	−28.0000			−0.188982
$29$	−48.8199	−	84.5585i	−0.312608	−	0.541453i	0.666318	−	0.745667i	$-0.267869\pi$
$29$	−48.8199	−	84.5585i	−0.312608	−	0.541453i	−0.978926	+	0.204215i	$0.934536\pi$
$30$		0			0
$31$	82.9760	−	143.719i	0.480740	−	0.832666i	−0.519016	−	0.854765i	$-0.673701\pi$
$31$	82.9760	−	143.719i	0.480740	−	0.832666i	0.999756	+	0.0220988i	$0.00703483\pi$
$32$	16.0000	−	27.7128i	0.0883883	−	0.153093i
$33$		0			0
$34$	53.3544	+	92.4126i	0.269124	+	0.466136i
$35$	41.7634			0.201694
$36$		0			0
$37$	431.884			1.91895			0.959477	−	0.281789i	$-0.0909277\pi$
$37$	431.884			1.91895			0.959477	+	0.281789i	$0.0909277\pi$
$38$	88.7964	+	153.800i	0.379070	+	0.656569i
$39$		0			0
$40$	−23.8648	+	41.3351i	−0.0943339	+	0.163391i
$41$	47.5332	−	82.3299i	0.181059	−	0.313604i	−0.761182	−	0.648538i	$-0.775381\pi$
$41$	47.5332	−	82.3299i	0.181059	−	0.313604i	0.942242	+	0.334934i	$0.108714\pi$
$42$		0			0
$43$	187.423	+	324.627i	0.664693	+	1.15128i	0.979368	+	0.202083i	$0.0647711\pi$
$43$	187.423	+	324.627i	0.664693	+	1.15128i	−0.314675	+	0.949199i	$0.601896\pi$
$44$	−29.3946			−0.100714
$45$		0			0
$46$	−392.279			−1.25736
$47$	57.5647	+	99.7049i	0.178652	+	0.309435i	0.941419	−	0.337239i	$-0.109493\pi$
$47$	57.5647	+	99.7049i	0.178652	+	0.309435i	−0.762767	+	0.646674i	$0.776160\pi$
$48$		0			0
$49$	−24.5000	+	42.4352i	−0.0714286	+	0.123718i
$50$	−89.4044	+	154.853i	−0.252874	+	0.437990i
$51$		0			0
$52$	109.628	+	189.881i	0.292359	+	0.506381i
$53$	145.549			0.377220			0.188610	−	0.982052i	$-0.439602\pi$
$53$	145.549			0.377220			0.188610	+	0.982052i	$0.439602\pi$
$54$		0			0
$55$	43.8435			0.107488
$56$	−28.0000	−	48.4974i	−0.0668153	−	0.115728i
$57$		0			0
$58$	97.6398	−	169.117i	0.221047	−	0.382865i
$59$	−202.457	+	350.666i	−0.446741	+	0.773778i	−0.998172	−	0.0604428i	$-0.980749\pi$
$59$	−202.457	+	350.666i	−0.446741	+	0.773778i	0.551431	+	0.834221i	$0.314082\pi$
$60$		0			0
$61$	288.425	+	499.567i	0.605394	+	1.04857i	0.991989	+	0.126324i	$0.0403177\pi$
$61$	288.425	+	499.567i	0.605394	+	1.04857i	−0.386595	+	0.922250i	$0.626349\pi$
$62$	331.904			0.679869
$63$		0			0
$64$	64.0000			0.125000
$65$	−163.516	−	283.218i	−0.312025	−	0.540443i
$66$		0			0
$67$	21.8954	−	37.9239i	0.0399246	−	0.0691514i	−0.845373	−	0.534177i	$-0.820622\pi$
$67$	21.8954	−	37.9239i	0.0399246	−	0.0691514i	0.885297	+	0.465026i	$0.153955\pi$
$68$	−106.709	+	184.825i	−0.190299	+	0.329608i
$69$		0			0
$70$	41.7634	+	72.3364i	0.0713098	+	0.123512i
$71$	−196.885			−0.329098			−0.164549	−	0.986369i	$-0.552617\pi$
$71$	−196.885			−0.329098			−0.164549	+	0.986369i	$0.552617\pi$
$72$		0			0
$73$	−510.969			−0.819239			−0.409619	−	0.912257i	$-0.634338\pi$
$73$	−510.969			−0.819239			−0.409619	+	0.912257i	$0.634338\pi$
$74$	431.884	+	748.045i	0.678452	+	1.17511i
$75$		0			0
$76$	−177.593	+	307.600i	−0.268043	+	0.464265i
$77$	−25.7203	+	44.5488i	−0.0380662	+	0.0659326i
$78$		0			0
$79$	−196.883	−	341.011i	−0.280393	−	0.485655i	0.691088	−	0.722770i	$-0.257132\pi$
$79$	−196.883	−	341.011i	−0.280393	−	0.485655i	−0.971482	+	0.237115i	$0.923798\pi$
$80$	−95.4592			−0.133408
$81$		0			0
$82$	190.133			0.256057
$83$	263.437	+	456.286i	0.348385	+	0.603420i	0.985963	−	0.166966i	$-0.0533969\pi$
$83$	263.437	+	456.286i	0.348385	+	0.603420i	−0.637578	+	0.770386i	$0.720064\pi$
$84$		0			0
$85$	159.162	−	275.676i	0.203100	−	0.351780i
$86$	−374.847	+	649.254i	−0.470009	+	0.814080i
$87$		0			0
$88$	−29.3946	−	50.9129i	−0.0356076	−	0.0616743i
$89$	589.818			0.702479			0.351239	−	0.936286i	$-0.385760\pi$
$89$	589.818			0.702479			0.351239	+	0.936286i	$0.385760\pi$
$90$		0			0
$91$	383.698			0.442005
$92$	−392.279	−	679.448i	−0.444543	−	0.769971i
$93$		0			0
$94$	−115.129	+	199.410i	−0.126326	+	0.218804i
$95$	264.889	−	458.800i	0.286074	−	0.495494i
$96$		0			0
$97$	−468.232	−	811.002i	−0.490122	−	0.848915i	0.509814	−	0.860285i	$-0.329714\pi$
$97$	−468.232	−	811.002i	−0.490122	−	0.848915i	−0.999935	+	0.0113693i	$0.996381\pi$
$98$	−98.0000			−0.101015
$99$		0			0
$100$	−357.618			−0.357618
$101$	−213.573	−	369.920i	−0.210409	−	0.364439i	0.741433	−	0.671026i	$-0.234146\pi$
$101$	−213.573	−	369.920i	−0.210409	−	0.364439i	−0.951843	+	0.306587i	$0.900813\pi$
$102$		0			0
$103$	997.566	−	1727.84i	0.954302	−	1.65290i	0.218344	−	0.975872i	$-0.429934\pi$
$103$	997.566	−	1727.84i	0.954302	−	1.65290i	0.735958	−	0.677028i	$-0.236732\pi$
$104$	−219.256	+	379.763i	−0.206729	+	0.358065i
$105$		0			0
$106$	145.549	+	252.098i	0.133368	+	0.230999i
$107$	−1866.87			−1.68671			−0.843353	−	0.537361i	$-0.819421\pi$
$107$	−1866.87			−1.68671			−0.843353	+	0.537361i	$0.819421\pi$
$108$		0			0
$109$	−585.651			−0.514635			−0.257317	−	0.966327i	$-0.582839\pi$
$109$	−585.651			−0.514635			−0.257317	+	0.966327i	$0.582839\pi$
$110$	43.8435	+	75.9392i	0.0380029	+	0.0658229i
$111$		0			0
$112$	56.0000	−	96.9948i	0.0472456	−	0.0818317i
$113$	−370.541	+	641.795i	−0.308474	+	0.534292i	−0.978029	−	0.208470i	$-0.933152\pi$
$113$	−370.541	+	641.795i	−0.308474	+	0.534292i	0.669555	+	0.742763i	$0.266485\pi$
$114$		0			0
$115$	585.104	+	1013.43i	0.474446	+	0.821764i
$116$	390.559			0.312608
$117$		0			0
$118$	−809.830			−0.631787
$119$	186.740	+	323.444i	0.143853	+	0.249160i
$120$		0			0
$121$	638.499	−	1105.91i	0.479714	−	0.830888i
$122$	−576.850	+	999.133i	−0.428078	+	0.741453i
$123$		0			0
$124$	331.904	+	574.875i	0.240370	+	0.416333i
$125$	1279.18			0.915307
$126$		0			0
$127$	−72.9178			−0.0509481			−0.0254740	−	0.999675i	$-0.508110\pi$
$127$	−72.9178			−0.0509481			−0.0254740	+	0.999675i	$0.508110\pi$
$128$	64.0000	+	110.851i	0.0441942	+	0.0765466i
$129$		0			0
$130$	327.032	−	566.435i	0.220635	−	0.382151i
$131$	470.494	−	814.920i	0.313796	−	0.543510i	−0.665385	−	0.746500i	$-0.731733\pi$
$131$	470.494	−	814.920i	0.313796	−	0.543510i	0.979181	+	0.202990i	$0.0650659\pi$
$132$		0			0
$133$	310.787	+	538.299i	0.202622	+	0.350951i
$134$	87.5816			0.0564619
$135$		0			0
$136$	−426.835			−0.269124
$137$	−896.728	−	1553.18i	−0.559216	−	0.968591i	−0.997562	−	0.0697848i	$-0.977769\pi$
$137$	−896.728	−	1553.18i	−0.559216	−	0.968591i	0.438346	−	0.898807i	$-0.355565\pi$
$138$		0			0
$139$	−296.550	+	513.639i	−0.180957	+	0.313427i	−0.942207	−	0.335032i	$-0.891253\pi$
$139$	−296.550	+	513.639i	−0.180957	+	0.313427i	0.761250	+	0.648459i	$0.224586\pi$
$140$	−83.5268	+	144.673i	−0.0504236	+	0.0873363i
$141$		0			0
$142$	−196.885	−	341.015i	−0.116354	−	0.201531i
$143$	402.809			0.235556
$144$		0			0
$145$	−582.539			−0.333636
$146$	−510.969	−	885.025i	−0.289645	−	0.501679i
$147$		0			0
$148$	−863.768	+	1496.09i	−0.479738	+	0.830931i
$149$	1762.35	−	3052.49i	0.968978	−	1.67832i	0.270454	−	0.962733i	$-0.412826\pi$
$149$	1762.35	−	3052.49i	0.968978	−	1.67832i	0.698524	−	0.715586i	$-0.253840\pi$
$150$		0			0
$151$	−99.9885	−	173.185i	−0.0538871	−	0.0933352i	0.837824	−	0.545941i	$-0.183828\pi$
$151$	−99.9885	−	173.185i	−0.0538871	−	0.0933352i	−0.891711	+	0.452606i	$0.850494\pi$
$152$	−710.371			−0.379070
$153$		0			0
$154$	−102.881			−0.0538337
$155$	−495.052	−	857.455i	−0.256539	−	0.444338i
$156$		0			0
$157$	−385.823	+	668.266i	−0.196128	+	0.339703i	−0.947270	−	0.320438i	$-0.896170\pi$
$157$	−385.823	+	668.266i	−0.196128	+	0.339703i	0.751142	+	0.660141i	$0.229503\pi$
$158$	393.766	−	682.023i	0.198268	−	0.343410i
$159$		0			0
$160$	−95.4592	−	165.340i	−0.0471670	−	0.0816956i
$161$	−1372.98			−0.672086
$162$		0			0
$163$	−215.179			−0.103399			−0.0516996	−	0.998663i	$-0.516464\pi$
$163$	−215.179			−0.103399			−0.0516996	+	0.998663i	$0.516464\pi$
$164$	190.133	+	329.320i	0.0905297	+	0.156802i
$165$		0			0
$166$	−526.874	+	912.572i	−0.246345	+	0.426683i
$167$	185.964	−	322.099i	0.0861695	−	0.149250i	−0.819719	−	0.572765i	$-0.805871\pi$
$167$	185.964	−	322.099i	0.0861695	−	0.149250i	0.905889	+	0.423515i	$0.139204\pi$
$168$		0			0
$169$	−403.789	−	699.382i	−0.183791	−	0.318335i
$170$	636.647			0.287227
$171$		0			0
$172$	−1499.39			−0.664693
$173$	−51.3645	−	88.9659i	−0.0225732	−	0.0390980i	0.854518	−	0.519422i	$-0.173853\pi$
$173$	−51.3645	−	88.9659i	−0.0225732	−	0.0390980i	−0.877091	+	0.480324i	$0.840519\pi$
$174$		0			0
$175$	−312.916	+	541.986i	−0.135167	+	0.234116i
$176$	58.7892	−	101.826i	0.0251784	−	0.0436103i
$177$		0			0
$178$	589.818	+	1021.60i	0.248364	+	0.430179i
$179$	4299.18			1.79517			0.897585	−	0.440841i	$-0.145320\pi$
$179$	4299.18			1.79517			0.897585	+	0.440841i	$0.145320\pi$
$180$		0			0
$181$	−3032.50			−1.24533			−0.622663	−	0.782490i	$-0.713949\pi$
$181$	−3032.50			−1.24533			−0.622663	+	0.782490i	$0.713949\pi$
$182$	383.698	+	664.585i	0.156273	+	0.270672i
$183$		0			0
$184$	784.559	−	1358.90i	0.314339	−	0.544452i
$185$	1288.35	−	2231.49i	0.512009	−	0.886825i
$186$		0			0
$187$	196.041	+	339.554i	0.0766629	+	0.132784i
$188$	−460.517			−0.178652
$189$		0			0
$190$	1059.55			0.404569
$191$	−1246.15	−	2158.39i	−0.472084	−	0.817674i	0.527406	−	0.849614i	$-0.323165\pi$
$191$	−1246.15	−	2158.39i	−0.472084	−	0.817674i	−0.999490	+	0.0319400i	$0.989831\pi$
$192$		0			0
$193$	1076.98	−	1865.39i	0.401673	−	0.695718i	−0.592255	−	0.805751i	$-0.701762\pi$
$193$	1076.98	−	1865.39i	0.401673	−	0.695718i	0.993928	+	0.110033i	$0.0350955\pi$
$194$	936.465	−	1622.00i	0.346568	−	0.600274i
$195$		0			0
$196$	−98.0000	−	169.741i	−0.0357143	−	0.0618590i
$197$	−5115.11			−1.84993			−0.924966	−	0.380049i	$-0.875907\pi$
$197$	−5115.11			−1.84993			−0.924966	+	0.380049i	$0.875907\pi$
$198$		0			0
$199$	−2870.67			−1.02259			−0.511297	−	0.859404i	$-0.670835\pi$
$199$	−2870.67			−1.02259			−0.511297	+	0.859404i	$0.670835\pi$
$200$	−357.618	−	619.412i	−0.126437	−	0.218995i
$201$		0			0
$202$	427.146	−	739.839i	0.148782	−	0.257698i
$203$	341.739	−	591.910i	0.118155	−	0.204650i
$204$		0			0
$205$	−283.593	−	491.197i	−0.0966193	−	0.167350i
$206$	3990.26			1.34959
$207$		0			0
$208$	−877.024			−0.292359
$209$	326.267	+	565.110i	0.107982	+	0.187031i
$210$		0			0
$211$	1100.18	−	1905.57i	0.358956	−	0.621730i	−0.628831	−	0.777542i	$-0.716466\pi$
$211$	1100.18	−	1905.57i	0.358956	−	0.621730i	0.987787	+	0.155812i	$0.0497995\pi$
$212$	−291.098	+	504.196i	−0.0943051	+	0.163341i
$213$		0			0
$214$	−1866.87	−	3233.52i	−0.596340	−	1.03289i
$215$	2236.41			0.709405
$216$		0			0
$217$	1161.66			0.363405
$218$	−585.651	−	1014.38i	−0.181951	−	0.315148i
$219$		0			0
$220$	−87.6870	+	151.878i	−0.0268721	+	0.0465438i
$221$	1462.29	−	2532.75i	0.445086	−	0.770911i
$222$		0			0
$223$	1494.30	+	2588.20i	0.448724	+	0.777213i	0.998303	−	0.0582288i	$-0.0185453\pi$
$223$	1494.30	+	2588.20i	0.448724	+	0.777213i	−0.549579	+	0.835442i	$0.685212\pi$
$224$	224.000			0.0668153
$225$		0			0
$226$	−1482.16			−0.436248
$227$	871.858	+	1510.10i	0.254922	+	0.441538i	0.964874	−	0.262712i	$-0.0846169\pi$
$227$	871.858	+	1510.10i	0.254922	+	0.441538i	−0.709952	+	0.704250i	$0.751284\pi$
$228$		0			0
$229$	812.307	−	1406.96i	0.234405	−	0.406001i	−0.724695	−	0.689070i	$-0.758019\pi$
$229$	812.307	−	1406.96i	0.234405	−	0.406001i	0.959100	+	0.283069i	$0.0913525\pi$
$230$	−1170.21	+	2026.86i	−0.335484	+	0.581075i
$231$		0			0
$232$	390.559	+	676.468i	0.110524	+	0.191432i
$233$	−4815.42			−1.35394			−0.676972	−	0.736009i	$-0.736708\pi$
$233$	−4815.42			−1.35394			−0.676972	+	0.736009i	$0.736708\pi$
$234$		0			0
$235$	686.885			0.190670
$236$	−809.830	−	1402.67i	−0.223370	−	0.386889i
$237$		0			0
$238$	−373.481	+	646.888i	−0.101719	+	0.176183i
$239$	−1719.02	+	2977.42i	−0.465247	+	0.805831i	−0.999213	−	0.0396751i	$-0.987368\pi$
$239$	−1719.02	+	2977.42i	−0.465247	+	0.805831i	0.533966	+	0.845506i	$0.320701\pi$
$240$		0			0
$241$	2828.48	+	4899.07i	0.756010	+	1.30945i	0.944871	+	0.327444i	$0.106187\pi$
$241$	2828.48	+	4899.07i	0.756010	+	1.30945i	−0.188861	+	0.982004i	$0.560480\pi$
$242$	2553.99			0.678417
$243$		0			0
$244$	−2307.40			−0.605394
$245$	146.172	+	253.177i	0.0381167	+	0.0660200i
$246$		0			0
$247$	2433.64	−	4215.19i	0.626919	−	1.08586i
$248$	−663.808	+	1149.75i	−0.169967	+	0.294392i
$249$		0			0
$250$	1279.18	+	2215.61i	0.323610	+	0.560509i
$251$	−7375.50			−1.85473			−0.927365	−	0.374157i	$-0.877932\pi$
$251$	−7375.50			−1.85473			−0.927365	+	0.374157i	$0.877932\pi$
$252$		0			0
$253$	−1441.36			−0.358172
$254$	−72.9178	−	126.297i	−0.0180129	−	0.0311992i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	−129.494	+	224.291i	−0.0314305	+	0.0544392i	−0.881313	−	0.472533i	$-0.843340\pi$
$257$	−129.494	+	224.291i	−0.0314305	+	0.0544392i	0.849882	+	0.526973i	$0.176673\pi$
$258$		0			0
$259$	1511.59	+	2618.16i	0.362648	+	0.628125i
$260$	1308.13			0.312025
$261$		0			0
$262$	1881.98			0.443774
$263$	2069.48	+	3584.45i	0.485208	+	0.840405i	0.999856	−	0.0169970i	$-0.00541057\pi$
$263$	2069.48	+	3584.45i	0.485208	+	0.840405i	−0.514648	+	0.857402i	$0.672077\pi$
$264$		0			0
$265$	434.187	−	752.034i	0.100649	−	0.174329i
$266$	−621.575	+	1076.60i	−0.143275	+	0.248160i
$267$		0			0
$268$	87.5816	+	151.696i	0.0199623	+	0.0345757i
$269$	−5571.27			−1.26277			−0.631387	−	0.775468i	$-0.717514\pi$
$269$	−5571.27			−1.26277			−0.631387	+	0.775468i	$0.717514\pi$
$270$		0			0
$271$	−7479.10			−1.67647			−0.838234	−	0.545311i	$-0.816412\pi$
$271$	−7479.10			−1.67647			−0.838234	+	0.545311i	$0.816412\pi$
$272$	−426.835	−	739.301i	−0.0951496	−	0.164804i
$273$		0			0
$274$	1793.46	−	3106.36i	0.395426	−	0.684898i
$275$	−328.501	+	568.980i	−0.0720340	+	0.124766i
$276$		0			0
$277$	−2717.90	−	4707.54i	−0.589541	−	1.02112i	−0.994292	−	0.106688i	$-0.965975\pi$
$277$	−2717.90	−	4707.54i	−0.589541	−	1.02112i	0.404751	−	0.914427i	$-0.367358\pi$
$278$	−1186.20			−0.255912
$279$		0			0
$280$	−334.107			−0.0713098
$281$	3683.92	+	6380.74i	0.782079	+	1.35460i	0.930729	+	0.365711i	$0.119174\pi$
$281$	3683.92	+	6380.74i	0.782079	+	1.35460i	−0.148649	+	0.988890i	$0.547493\pi$
$282$		0			0
$283$	−2897.42	+	5018.47i	−0.608599	+	1.05412i	0.382872	+	0.923801i	$0.374935\pi$
$283$	−2897.42	+	5018.47i	−0.608599	+	1.05412i	−0.991472	+	0.130324i	$0.958398\pi$
$284$	393.771	−	682.031i	0.0822746	−	0.142504i
$285$		0			0
$286$	402.809	+	697.685i	0.0832818	+	0.144248i
$287$	665.465			0.136868
$288$		0			0
$289$	−2066.31			−0.420579
$290$	−582.539	−	1008.99i	−0.117958	−	0.204309i
$291$		0			0
$292$	1021.94	−	1770.05i	0.204810	−	0.354741i
$293$	−2282.20	+	3952.89i	−0.455044	+	0.788159i	−0.998691	−	0.0511553i	$-0.983710\pi$
$293$	−2282.20	+	3952.89i	−0.455044	+	0.788159i	0.543647	+	0.839314i	$0.317043\pi$
$294$		0			0
$295$	1207.90	+	2092.15i	0.238396	+	0.412914i
$296$	−3455.07			−0.678452
$297$		0			0
$298$	7049.42			1.37034
$299$	5375.60	+	9310.82i	1.03973	+	1.80086i
$300$		0			0
$301$	−1311.96	+	2272.39i	−0.251230	+	0.435144i
$302$	199.977	−	346.370i	0.0381039	−	0.0659980i
$303$		0			0
$304$	−710.371	−	1230.40i	−0.134022	−	0.232132i
$305$	3441.60			0.646117
$306$		0			0
$307$	9770.57			1.81640			0.908202	−	0.418533i	$-0.137455\pi$
$307$	9770.57			1.81640			0.908202	+	0.418533i	$0.137455\pi$
$308$	−102.881	−	178.195i	−0.0190331	−	0.0329663i
$309$		0			0
$310$	990.103	−	1714.91i	0.181400	−	0.314195i
$311$	−2917.06	+	5052.50i	−0.531870	+	0.921226i	0.467438	+	0.884026i	$0.345177\pi$
$311$	−2917.06	+	5052.50i	−0.531870	+	0.921226i	−0.999308	+	0.0371997i	$0.988156\pi$
$312$		0			0
$313$	−4032.66	−	6984.77i	−0.728241	−	1.26135i	−0.957626	−	0.288014i	$-0.907005\pi$
$313$	−4032.66	−	6984.77i	−0.728241	−	1.26135i	0.229385	−	0.973336i	$-0.426328\pi$
$314$	−1543.29			−0.277367
$315$		0			0
$316$	1575.06			0.280393
$317$	1216.50	+	2107.04i	0.215538	+	0.373323i	0.953439	−	0.301586i	$-0.0975161\pi$
$317$	1216.50	+	2107.04i	0.215538	+	0.373323i	−0.737901	+	0.674909i	$0.764183\pi$
$318$		0			0
$319$	358.760	−	621.391i	0.0629677	−	0.109063i
$320$	190.918	−	330.680i	0.0333521	−	0.0577675i
$321$		0			0
$322$	−1372.98	−	2378.07i	−0.237618	−	0.411567i
$323$	4737.68			0.816135
$324$		0			0
$325$	4900.62			0.836422
$326$	−215.179	−	372.700i	−0.0365572	−	0.0633189i
$327$		0			0
$328$	−380.265	+	658.639i	−0.0640142	+	0.110876i
$329$	−402.953	+	697.934i	−0.0675243	+	0.116956i
$330$		0			0
$331$	1455.34	+	2520.72i	0.241670	+	0.418584i	0.961190	−	0.275887i	$-0.0889717\pi$
$331$	1455.34	+	2520.72i	0.241670	+	0.418584i	−0.719520	+	0.694471i	$0.755638\pi$
$332$	−2107.49			−0.348385
$333$		0			0
$334$	743.855			0.121862
$335$	−130.632	−	226.262i	−0.0213051	−	0.0369015i
$336$		0			0
$337$	−2355.52	+	4079.88i	−0.380752	+	0.659482i	−0.991170	−	0.132598i	$-0.957668\pi$
$337$	−2355.52	+	4079.88i	−0.380752	+	0.659482i	0.610418	+	0.792080i	$0.291002\pi$
$338$	807.577	−	1398.76i	0.129960	−	0.225097i
$339$		0			0
$340$	636.647	+	1102.70i	0.101550	+	0.175890i
$341$	1219.52			0.193668
$342$		0			0
$343$	−343.000			−0.0539949
$344$	−1499.39	−	2597.02i	−0.235005	−	0.407040i
$345$		0			0
$346$	102.729	−	177.932i	0.0159617	−	0.0276465i
$347$	4201.39	−	7277.01i	0.649977	−	1.12579i	−0.333150	−	0.942874i	$-0.608112\pi$
$347$	4201.39	−	7277.01i	0.649977	−	1.12579i	0.983128	−	0.182920i	$-0.0585550\pi$
$348$		0			0
$349$	3161.78	+	5476.36i	0.484946	+	0.839951i	0.999850	−	0.0172969i	$-0.00550604\pi$
$349$	3161.78	+	5476.36i	0.484946	+	0.839951i	−0.514905	+	0.857247i	$0.672173\pi$
$350$	−1251.66			−0.191155
$351$		0			0
$352$	235.157			0.0356076
$353$	2692.85	+	4664.15i	0.406022	+	0.703251i	0.994440	−	0.105306i	$-0.0335821\pi$
$353$	2692.85	+	4664.15i	0.406022	+	0.703251i	−0.588417	+	0.808557i	$0.700249\pi$
$354$		0			0
$355$	−587.329	+	1017.28i	−0.0878090	+	0.152090i
$356$	−1179.64	+	2043.19i	−0.175620	+	0.304182i
$357$		0			0
$358$	4299.18	+	7446.39i	0.634689	+	1.09931i
$359$	11993.9			1.76327			0.881637	−	0.471927i	$-0.156442\pi$
$359$	11993.9			1.76327			0.881637	+	0.471927i	$0.156442\pi$
$360$		0			0
$361$	1025.80			0.149555
$362$	−3032.50	−	5252.44i	−0.440289	−	0.762603i
$363$		0			0
$364$	−767.396	+	1329.17i	−0.110501	+	0.191394i
$365$	−1524.27	+	2640.12i	−0.218587	+	0.378603i
$366$		0			0
$367$	2615.33	+	4529.89i	0.371987	+	0.644301i	0.989871	−	0.141968i	$-0.0453432\pi$
$367$	2615.33	+	4529.89i	0.371987	+	0.644301i	−0.617884	+	0.786269i	$0.712010\pi$
$368$	3138.23			0.444543
$369$		0			0
$370$	5153.41			0.724090
$371$	509.421	+	882.343i	0.0712880	+	0.123474i
$372$		0			0
$373$	−4334.34	+	7507.30i	−0.601672	+	1.04213i	0.390896	+	0.920435i	$0.372165\pi$
$373$	−4334.34	+	7507.30i	−0.601672	+	1.04213i	−0.992568	+	0.121692i	$0.961168\pi$
$374$	−392.083	+	679.107i	−0.0542089	+	0.0938925i
$375$		0			0
$376$	−460.517	−	797.639i	−0.0631632	−	0.109402i
$377$	−5352.03			−0.731150
$378$		0			0
$379$	−6465.25			−0.876247			−0.438123	−	0.898915i	$-0.644357\pi$
$379$	−6465.25			−0.876247			−0.438123	+	0.898915i	$0.644357\pi$
$380$	1059.55	+	1835.20i	0.143037	+	0.247747i
$381$		0			0
$382$	2492.29	−	4316.78i	0.333814	−	0.578183i
$383$	6827.24	−	11825.1i	0.910850	−	1.57764i	0.0979832	−	0.995188i	$-0.468761\pi$
$383$	6827.24	−	11825.1i	0.910850	−	1.57764i	0.812867	−	0.582450i	$-0.197906\pi$
$384$		0			0
$385$	153.452	+	265.787i	0.0203134	+	0.0351838i
$386$	4307.93			0.568051
$387$		0			0
$388$	3745.86			0.490122
$389$	−4249.25	−	7359.92i	−0.553844	−	0.959287i	−0.997992	−	0.0633332i	$-0.979827\pi$
$389$	−4249.25	−	7359.92i	−0.553844	−	0.959287i	0.444148	−	0.895953i	$-0.353506\pi$
$390$		0			0
$391$	−5232.46	+	9062.88i	−0.676769	+	1.17220i
$392$	196.000	−	339.482i	0.0252538	−	0.0437409i
$393$		0			0
$394$	−5115.11	−	8859.64i	−0.654050	−	1.13285i
$395$	−2349.29			−0.299254
$396$		0			0
$397$	12203.2			1.54272			0.771360	−	0.636399i	$-0.219577\pi$
$397$	12203.2			1.54272			0.771360	+	0.636399i	$0.219577\pi$
$398$	−2870.67	−	4972.14i	−0.361541	−	0.626208i
$399$		0			0
$400$	715.235	−	1238.82i	0.0894044	−	0.154853i
$401$	4882.89	−	8457.42i	0.608080	−	1.05323i	−0.383477	−	0.923551i	$-0.625273\pi$
$401$	4882.89	−	8457.42i	0.608080	−	1.05323i	0.991557	−	0.129675i	$-0.0413933\pi$
$402$		0			0
$403$	−4548.25	−	7877.80i	−0.562195	−	0.973750i
$404$	1708.59			0.210409
$405$		0			0
$406$	1366.96			0.167096
$407$	1586.88	+	2748.56i	0.193265	+	0.334744i
$408$		0			0
$409$	3897.13	−	6750.04i	0.471151	−	0.816058i	−0.528304	−	0.849055i	$-0.677172\pi$
$409$	3897.13	−	6750.04i	0.471151	−	0.816058i	0.999455	+	0.0329971i	$0.0105052\pi$
$410$	567.185	−	982.393i	0.0683202	−	0.118334i
$411$		0			0
$412$	3990.26	+	6911.34i	0.477151	+	0.826450i
$413$	−2834.40			−0.337704
$414$		0			0
$415$	3143.43			0.371820
$416$	−877.024	−	1519.05i	−0.103365	−	0.179033i
$417$		0			0
$418$	−652.533	+	1130.22i	−0.0763551	+	0.132251i
$419$	−6042.62	+	10466.1i	−0.704538	+	1.22030i	0.262320	+	0.964981i	$0.415512\pi$
$419$	−6042.62	+	10466.1i	−0.704538	+	1.22030i	−0.966858	+	0.255314i	$0.917821\pi$
$420$		0			0
$421$	−4925.64	−	8531.46i	−0.570217	−	0.987644i	−0.996543	−	0.0830748i	$-0.973526\pi$
$421$	−4925.64	−	8531.46i	−0.570217	−	0.987644i	0.426327	−	0.904569i	$-0.359807\pi$
$422$	4400.73			0.507640
$423$		0			0
$424$	−1164.39			−0.133368
$425$	2385.06	+	4131.05i	0.272218	+	0.471495i
$426$		0			0
$427$	−2018.97	+	3496.97i	−0.228817	+	0.396323i
$428$	3733.75	−	6467.04i	0.421676	−	0.730365i
$429$		0			0
$430$	2236.41	+	3873.58i	0.250813	+	0.434420i
$431$	268.348			0.0299904			0.0149952	−	0.999888i	$-0.495227\pi$
$431$	268.348			0.0299904			0.0149952	+	0.999888i	$0.495227\pi$
$432$		0			0
$433$	−2696.15			−0.299235			−0.149617	−	0.988744i	$-0.547804\pi$
$433$	−2696.15			−0.299235			−0.149617	+	0.988744i	$0.547804\pi$
$434$	1161.66	+	2012.06i	0.128483	+	0.222539i
$435$		0			0
$436$	1171.30	−	2028.76i	0.128659	−	0.222843i
$437$	−8708.25	+	15083.1i	−0.953254	+	1.65108i
$438$		0			0
$439$	4691.03	+	8125.10i	0.510001	+	0.883348i	0.999933	+	0.0115874i	$0.00368845\pi$
$439$	4691.03	+	8125.10i	0.510001	+	0.883348i	−0.489931	+	0.871761i	$0.662978\pi$
$440$	−350.748			−0.0380029
$441$		0			0
$442$	5849.14			0.629446
$443$	6343.60	+	10987.4i	0.680347	+	1.17839i	0.974875	+	0.222752i	$0.0715041\pi$
$443$	6343.60	+	10987.4i	0.680347	+	1.17839i	−0.294529	+	0.955643i	$0.595163\pi$
$444$		0			0
$445$	1759.49	−	3047.52i	0.187433	−	0.324644i
$446$	−2988.59	+	5176.39i	−0.317296	+	0.549572i
$447$		0			0
$448$	224.000	+	387.979i	0.0236228	+	0.0409159i
$449$	−8285.34			−0.870845			−0.435423	−	0.900226i	$-0.643401\pi$
$449$	−8285.34			−0.870845			−0.435423	+	0.900226i	$0.643401\pi$
$450$		0			0
$451$	698.609			0.0729406
$452$	−1482.16	−	2567.18i	−0.154237	−	0.267146i
$453$		0			0
$454$	−1743.72	+	3020.21i	−0.180257	+	0.312214i
$455$	1144.61	−	1982.52i	0.117934	−	0.204268i
$456$		0			0
$457$	−7970.71	−	13805.7i	−0.815873	−	1.41313i	−0.908699	−	0.417451i	$-0.862923\pi$
$457$	−7970.71	−	13805.7i	−0.815873	−	1.41313i	0.0928263	−	0.995682i	$-0.470410\pi$
$458$	3249.23			0.331499
$459$		0			0
$460$	−4680.84			−0.474446
$461$	−5159.87	−	8937.15i	−0.521299	−	0.902917i	−0.999693	−	0.0247714i	$-0.992114\pi$
$461$	−5159.87	−	8937.15i	−0.521299	−	0.902917i	0.478394	−	0.878145i	$-0.341219\pi$
$462$		0			0
$463$	−9235.81	+	15996.9i	−0.927050	+	1.60570i	−0.138820	+	0.990318i	$0.544331\pi$
$463$	−9235.81	+	15996.9i	−0.927050	+	1.60570i	−0.788230	+	0.615381i	$0.789002\pi$
$464$	−781.118	+	1352.94i	−0.0781520	+	0.135363i
$465$		0			0
$466$	−4815.42	−	8340.56i	−0.478691	−	0.829118i
$467$	1991.38			0.197324			0.0986619	−	0.995121i	$-0.468544\pi$
$467$	1991.38			0.197324			0.0986619	+	0.995121i	$0.468544\pi$
$468$		0			0
$469$	306.535			0.0301802
$470$	686.885	+	1189.72i	0.0674120	+	0.116761i
$471$		0			0
$472$	1619.66	−	2805.33i	0.157947	−	0.273572i
$473$	−1377.31	+	2385.57i	−0.133887	+	0.231900i
$474$		0			0
$475$	3969.40	+	6875.19i	0.383428	+	0.664117i
$476$	−1493.92			−0.143853
$477$		0			0
$478$	−6876.07			−0.657958
$479$	−3702.84	−	6413.51i	−0.353209	−	0.611777i	0.633600	−	0.773660i	$-0.281576\pi$
$479$	−3702.84	−	6413.51i	−0.353209	−	0.611777i	−0.986810	+	0.161884i	$0.948243\pi$
$480$		0			0
$481$	11836.6	−	20501.7i	1.12205	−	1.94344i
$482$	−5656.96	+	9798.14i	−0.534580	+	0.925919i
$483$		0			0
$484$	2553.99	+	4423.65i	0.239857	+	0.415444i
$485$	−5587.14			−0.523090
$486$		0			0
$487$	7064.90			0.657374			0.328687	−	0.944439i	$-0.393394\pi$
$487$	7064.90			0.657374			0.328687	+	0.944439i	$0.393394\pi$
$488$	−2307.40	−	3996.53i	−0.214039	−	0.370727i
$489$		0			0
$490$	−292.344	+	506.355i	−0.0269526	+	0.0466832i
$491$	−2706.72	+	4688.17i	−0.248783	+	0.430904i	−0.963188	−	0.268827i	$-0.913364\pi$
$491$	−2706.72	+	4688.17i	−0.248783	+	0.430904i	0.714406	+	0.699732i	$0.246697\pi$
$492$		0			0
$493$	−2604.76	−	4511.57i	−0.237956	−	0.412152i
$494$	9734.57			0.886598
$495$		0			0
$496$	−2655.23			−0.240370
$497$	−689.099	−	1193.55i	−0.0621938	−	0.107723i
$498$		0			0
$499$	−6682.03	+	11573.6i	−0.599456	+	1.03829i	0.393445	+	0.919348i	$0.371283\pi$
$499$	−6682.03	+	11573.6i	−0.599456	+	1.03829i	−0.992901	+	0.118941i	$0.962050\pi$
$500$	−2558.36	+	4431.21i	−0.228827	+	0.396339i
$501$		0			0
$502$	−7375.50	−	12774.7i	−0.655746	−	1.13579i
$503$	19112.4			1.69420			0.847098	−	0.531437i	$-0.178348\pi$
$503$	19112.4			1.69420			0.847098	+	0.531437i	$0.178348\pi$
$504$		0			0
$505$	−2548.44			−0.224563
$506$	−1441.36	−	2496.51i	−0.126633	−	0.219335i
$507$		0			0
$508$	145.836	−	252.595i	0.0127370	−	0.0220612i
$509$	9593.67	−	16616.7i	0.835426	−	1.44700i	−0.0582574	−	0.998302i	$-0.518554\pi$
$509$	9593.67	−	16616.7i	0.835426	−	1.44700i	0.893683	−	0.448698i	$-0.148112\pi$
$510$		0			0
$511$	−1788.39	−	3097.59i	−0.154822	−	0.268159i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	−517.977			−0.0444494
$515$	−5951.68	−	10308.6i	−0.509247	−	0.882042i
$516$		0			0
$517$	−423.022	+	732.696i	−0.0359855	+	0.0623287i
$518$	−3023.19	+	5236.31i	−0.256431	+	0.444151i
$519$		0			0
$520$	1308.13	+	2265.74i	0.110318	+	0.191076i
$521$	−17848.7			−1.50089			−0.750445	−	0.660933i	$-0.770161\pi$
$521$	−17848.7			−1.50089			−0.750445	+	0.660933i	$0.770161\pi$
$522$		0			0
$523$	−18283.8			−1.52867			−0.764334	−	0.644821i	$-0.776932\pi$
$523$	−18283.8			−1.52867			−0.764334	+	0.644821i	$0.776932\pi$
$524$	1881.98	+	3259.68i	0.156898	+	0.271755i
$525$		0			0
$526$	−4138.96	+	7168.89i	−0.343094	+	0.594256i
$527$	4427.14	−	7668.03i	0.365938	−	0.633823i
$528$		0			0
$529$	−13151.9	−	22779.7i	−1.08095	−	1.87226i
$530$	1736.75			0.142339
$531$		0			0
$532$	−2486.30			−0.202622
$533$	−2605.48	−	4512.83i	−0.211737	−	0.366740i
$534$		0			0
$535$	−5569.07	+	9645.91i	−0.450041	+	0.779494i
$536$	−175.163	+	303.391i	−0.0141155	+	0.0244487i
$537$		0			0
$538$	−5571.27	−	9649.72i	−0.446458	−	0.773288i
$539$	−360.084			−0.0287753
$540$		0			0
$541$	16801.9			1.33525			0.667623	−	0.744500i	$-0.267312\pi$
$541$	16801.9			1.33525			0.667623	+	0.744500i	$0.267312\pi$
$542$	−7479.10	−	12954.2i	−0.592721	−	1.02662i
$543$		0			0
$544$	853.671	−	1478.60i	0.0672809	−	0.116534i
$545$	−1747.06	+	3025.99i	−0.137313	+	0.237833i
$546$		0			0
$547$	−3217.95	−	5573.66i	−0.251535	−	0.435672i	0.712414	−	0.701760i	$-0.247602\pi$
$547$	−3217.95	−	5573.66i	−0.251535	−	0.435672i	−0.963949	+	0.266088i	$0.914269\pi$
$548$	7173.83			0.559216
$549$		0			0
$550$	−1314.00			−0.101871
$551$	−4335.03	−	7508.49i	−0.335170	−	0.580531i
$552$		0			0
$553$	1378.18	−	2387.08i	0.105979	−	0.183560i
$554$	5435.80	−	9415.09i	0.416869	−	0.722038i
$555$		0			0
$556$	−1186.20	−	2054.56i	−0.0904784	−	0.156713i
$557$	1258.90			0.0957657			0.0478829	−	0.998853i	$-0.484753\pi$
$557$	1258.90			0.0957657			0.0478829	+	0.998853i	$0.484753\pi$
$558$		0			0
$559$	20546.9			1.55463
$560$	−334.107	−	578.691i	−0.0252118	−	0.0436681i
$561$		0			0
$562$	−7367.84	+	12761.5i	−0.553013	+	0.957847i
$563$	1032.66	−	1788.63i	0.0773030	−	0.133893i	−0.824782	−	0.565450i	$-0.808702\pi$
$563$	1032.66	−	1788.63i	0.0773030	−	0.133893i	0.902085	+	0.431557i	$0.142036\pi$
$564$		0			0
$565$	2210.72	+	3829.08i	0.164612	+	0.285116i
$566$	−11589.7			−0.860689
$567$		0			0
$568$	1575.08			0.116354
$569$	61.9784	+	107.350i	0.00456638	+	0.00790920i	0.868300	−	0.496040i	$-0.165213\pi$
$569$	61.9784	+	107.350i	0.00456638	+	0.00790920i	−0.863733	+	0.503949i	$0.831880\pi$
$570$		0			0
$571$	612.046	−	1060.10i	0.0448570	−	0.0776946i	−0.842725	−	0.538344i	$-0.819050\pi$
$571$	612.046	−	1060.10i	0.0448570	−	0.0776946i	0.887582	+	0.460649i	$0.152383\pi$
$572$	−805.618	+	1395.37i	−0.0588891	+	0.101999i
$573$		0			0
$574$	665.465	+	1152.62i	0.0483902	+	0.0838142i
$575$	−17535.8			−1.27181
$576$		0			0
$577$	−22288.4			−1.60811			−0.804055	−	0.594555i	$-0.797328\pi$
$577$	−22288.4			−1.60811			−0.804055	+	0.594555i	$0.797328\pi$
$578$	−2066.31	−	3578.95i	−0.148697	−	0.257551i
$579$		0			0
$580$	1165.08	−	2017.97i	0.0834090	−	0.144469i
$581$	−1844.06	+	3194.00i	−0.131677	+	0.228071i
$582$		0			0
$583$	534.794	+	926.290i	0.0379912	+	0.0658028i
$584$	4087.75			0.289645
$585$		0			0
$586$	−9128.82			−0.643529
$587$	−8503.97	−	14729.3i	−0.597949	−	1.03568i	−0.993123	−	0.117073i	$-0.962649\pi$
$587$	−8503.97	−	14729.3i	−0.597949	−	1.03568i	0.395174	−	0.918606i	$-0.370684\pi$
$588$		0			0
$589$	7367.97	−	12761.7i	0.515436	−	0.892762i
$590$	−2415.80	+	4184.29i	−0.168571	+	0.291974i
$591$		0			0
$592$	−3455.07	−	5984.36i	−0.239869	−	0.415466i
$593$	−2043.54			−0.141515			−0.0707574	−	0.997494i	$-0.522542\pi$
$593$	−2043.54			−0.141515			−0.0707574	+	0.997494i	$0.522542\pi$
$594$		0			0
$595$	2228.26			0.153529
$596$	7049.42	+	12210.0i	0.484489	+	0.839160i
$597$		0			0
$598$	−10751.2	+	18621.6i	−0.735200	+	1.27340i
$599$	5409.54	−	9369.60i	0.368995	−	0.639118i	−0.620414	−	0.784275i	$-0.713035\pi$
$599$	5409.54	−	9369.60i	0.368995	−	0.639118i	0.989409	+	0.145157i	$0.0463687\pi$
$600$		0			0
$601$	2611.91	+	4523.95i	0.177274	+	0.307048i	0.940946	−	0.338557i	$-0.109939\pi$
$601$	2611.91	+	4523.95i	0.177274	+	0.307048i	−0.763672	+	0.645605i	$0.776605\pi$
$602$	−5247.86			−0.355294
$603$		0			0
$604$	799.908			0.0538871
$605$	−3809.41	−	6598.10i	−0.255991	−	0.443390i
$606$		0			0
$607$	4311.44	−	7467.63i	0.288296	−	0.499344i	−0.685107	−	0.728442i	$-0.740245\pi$
$607$	4311.44	−	7467.63i	0.288296	−	0.499344i	0.973403	+	0.229099i	$0.0735779\pi$
$608$	1420.74	−	2460.80i	0.0947676	−	0.164142i
$609$		0			0
$610$	3441.60	+	5961.03i	0.228437	+	0.395664i
$611$	6310.70			0.417846
$612$		0			0
$613$	−1749.52			−0.115273			−0.0576366	−	0.998338i	$-0.518356\pi$
$613$	−1749.52			−0.115273			−0.0576366	+	0.998338i	$0.518356\pi$
$614$	9770.57	+	16923.1i	0.642196	+	1.11232i
$615$		0			0
$616$	205.762	−	356.390i	0.0134584	−	0.0233107i
$617$	1166.01	−	2019.59i	0.0760809	−	0.131776i	−0.825475	−	0.564439i	$-0.809093\pi$
$617$	1166.01	−	2019.59i	0.0760809	−	0.131776i	0.901556	+	0.432663i	$0.142426\pi$
$618$		0			0
$619$	−8104.50	−	14037.4i	−0.526248	−	0.911488i	−0.999532	−	0.0305782i	$-0.990265\pi$
$619$	−8104.50	−	14037.4i	−0.526248	−	0.911488i	0.473285	−	0.880910i	$-0.343068\pi$
$620$	3960.41			0.256539
$621$		0			0
$622$	−11668.3			−0.752178
$623$	2064.36	+	3575.58i	0.132756	+	0.229940i
$624$		0			0
$625$	−1771.85	+	3068.94i	−0.113399	+	0.196412i
$626$	8065.32	−	13969.5i	0.514944	−	0.891909i
$627$		0			0
$628$	−1543.29	−	2673.06i	−0.0980639	−	0.169852i
$629$	23042.9			1.46070
$630$		0			0
$631$	−14574.4			−0.919488			−0.459744	−	0.888052i	$-0.652059\pi$
$631$	−14574.4			−0.919488			−0.459744	+	0.888052i	$0.652059\pi$
$632$	1575.06	+	2728.09i	0.0991340	+	0.171705i
$633$		0			0
$634$	−2433.01	+	4214.09i	−0.152409	+	0.263979i
$635$	−217.521	+	376.758i	−0.0135938	+	0.0235451i
$636$		0			0
$637$	1342.94	+	2326.05i	0.0835312	+	0.144680i
$638$	1435.04			0.0890498
$639$		0			0
$640$	763.674			0.0471670
$641$	685.232	+	1186.86i	0.0422232	+	0.0731327i	0.886365	−	0.462988i	$-0.153223\pi$
$641$	685.232	+	1186.86i	0.0422232	+	0.0731327i	−0.844142	+	0.536120i	$0.819889\pi$
$642$		0			0
$643$	1379.94	−	2390.12i	0.0846336	−	0.146590i	−0.820601	−	0.571501i	$-0.806361\pi$
$643$	1379.94	−	2390.12i	0.0846336	−	0.146590i	0.905235	+	0.424911i	$0.139695\pi$
$644$	2745.96	−	4756.13i	0.168021	−	0.291022i
$645$		0			0
$646$	4737.68	+	8205.90i	0.288547	+	0.499778i
$647$	3675.83			0.223357			0.111678	−	0.993744i	$-0.464377\pi$
$647$	3675.83			0.223357			0.111678	+	0.993744i	$0.464377\pi$
$648$		0			0
$649$	−2975.58			−0.179972
$650$	4900.62	+	8488.12i	0.295720	+	0.512202i
$651$		0			0
$652$	430.357	−	745.400i	0.0258498	−	0.0447732i
$653$	4565.83	−	7908.26i	0.273622	−	0.473926i	−0.696165	−	0.717882i	$-0.745112\pi$
$653$	4565.83	−	7908.26i	0.273622	−	0.473926i	0.969786	+	0.243955i	$0.0784451\pi$
$654$		0			0
$655$	−2807.06	−	4861.98i	−0.167452	−	0.290035i
$656$	−1521.06			−0.0905297
$657$		0			0
$658$	−1611.81			−0.0954938
$659$	3378.65	+	5851.99i	0.199717	+	0.345920i	0.948437	−	0.316967i	$-0.102664\pi$
$659$	3378.65	+	5851.99i	0.199717	+	0.345920i	−0.748720	+	0.662887i	$0.769331\pi$
$660$		0			0
$661$	−11202.9	+	19403.9i	−0.659215	+	1.14179i	0.321604	+	0.946874i	$0.395778\pi$
$661$	−11202.9	+	19403.9i	−0.659215	+	1.14179i	−0.980819	+	0.194920i	$0.937555\pi$
$662$	−2910.68	+	5041.44i	−0.170886	+	0.295983i
$663$		0			0
$664$	−2107.49	−	3650.29i	−0.123173	−	0.213341i
$665$	3708.44			0.216251
$666$		0			0
$667$	19151.0			1.11174
$668$	743.855	+	1288.40i	0.0430848	+	0.0746250i
$669$		0			0
$670$	261.265	−	452.524i	0.0150650	−	0.0260933i
$671$	−2119.53	+	3671.14i	−0.121943	+	0.211211i
$672$		0			0
$673$	7266.75	+	12586.4i	0.416215	+	0.720905i	0.995555	−	0.0941807i	$-0.0300232\pi$
$673$	7266.75	+	12586.4i	0.416215	+	0.720905i	−0.579340	+	0.815086i	$0.696690\pi$
$674$	−9422.09			−0.538465
$675$		0			0
$676$	3230.31			0.183791
$677$	−14651.4	−	25377.0i	−0.831758	−	1.44065i	−0.896643	−	0.442754i	$-0.854002\pi$
$677$	−14651.4	−	25377.0i	−0.831758	−	1.44065i	0.0648858	−	0.997893i	$-0.479332\pi$
$678$		0			0
$679$	3277.63	−	5677.02i	0.185249	−	0.320860i
$680$	−1273.29	+	2205.41i	−0.0718067	+	0.124373i
$681$		0			0
$682$	1219.52	+	2112.28i	0.0684721	+	0.118597i
$683$	15718.2			0.880584			0.440292	−	0.897855i	$-0.354875\pi$
$683$	15718.2			0.880584			0.440292	+	0.897855i	$0.354875\pi$
$684$		0			0
$685$	−10700.1			−0.596833
$686$	−343.000	−	594.093i	−0.0190901	−	0.0330650i
$687$		0			0
$688$	2998.78	−	5194.03i	0.166173	−	0.287821i
$689$	3989.06	−	6909.26i	0.220568	−	0.382034i
$690$		0			0
$691$	11677.9	+	20226.7i	0.642906	+	1.11355i	0.984781	+	0.173800i	$0.0556048\pi$
$691$	11677.9	+	20226.7i	0.642906	+	1.11355i	−0.341875	+	0.939745i	$0.611062\pi$
$692$	410.916			0.0225732
$693$		0			0
$694$	16805.5			0.919207
$695$	1769.27	+	3064.47i	0.0965646	+	0.167255i
$696$		0			0
$697$	2536.11	−	4392.66i	0.137822	−	0.238714i
$698$	−6323.55	+	10952.7i	−0.342908	+	0.593935i
$699$		0			0
$700$	−1251.66	−	2167.94i	−0.0675834	−	0.117058i
$701$	−34760.7			−1.87289			−0.936445	−	0.350815i	$-0.885904\pi$
$701$	−34760.7			−1.87289			−0.936445	+	0.350815i	$0.885904\pi$
$702$		0			0
$703$	38349.7			2.05745
$704$	235.157	+	407.303i	0.0125892	+	0.0218051i
$705$		0			0
$706$	−5385.70	+	9328.30i	−0.287101	+	0.497274i
$707$	1495.01	−	2589.44i	0.0795272	−	0.137745i
$708$		0			0
$709$	3237.66	+	5607.79i	0.171499	+	0.297045i	0.938944	−	0.344070i	$-0.111806\pi$
$709$	3237.66	+	5607.79i	0.171499	+	0.297045i	−0.767445	+	0.641115i	$0.778472\pi$
$710$	−2349.32			−0.124181
$711$		0			0
$712$	−4718.55			−0.248364
$713$	16274.9	+	28188.9i	0.854838	+	1.48062i
$714$		0			0
$715$	1201.62	−	2081.27i	0.0628504	−	0.108860i
$716$	−8598.35	+	14892.8i	−0.448793	+	0.777332i
$717$		0			0
$718$	11993.9	+	20774.1i	0.623412	+	1.07978i
$719$	7178.95			0.372364			0.186182	−	0.982515i	$-0.440389\pi$
$719$	7178.95			0.372364			0.186182	+	0.982515i	$0.440389\pi$
$720$		0			0
$721$	13965.9			0.721384
$722$	1025.80	+	1776.73i	0.0528757	+	0.0915833i
$723$		0			0
$724$	6065.00	−	10504.9i	0.311331	−	0.539242i
$725$	4364.71	−	7559.91i	0.223588	−	0.387266i
$726$		0			0
$727$	4077.92	+	7063.16i	0.208035	+	0.360328i	0.951096	−	0.308897i	$-0.0999598\pi$
$727$	4077.92	+	7063.16i	0.208035	+	0.360328i	−0.743060	+	0.669225i	$0.766626\pi$
$728$	−3069.59			−0.156273
$729$		0			0
$730$	−6097.09			−0.309128
$731$	9999.87	+	17320.3i	0.505962	+	0.876353i
$732$		0			0
$733$	7672.05	−	13288.4i	0.386594	−	0.669601i	−0.605395	−	0.795925i	$-0.706985\pi$
$733$	7672.05	−	13288.4i	0.386594	−	0.669601i	0.991989	+	0.126325i	$0.0403181\pi$
$734$	−5230.67	+	9059.78i	−0.263035	+	0.455590i
$735$		0			0
$736$	3138.23	+	5435.58i	0.157170	+	0.272226i
$737$	321.803			0.0160838
$738$		0			0
$739$	−10730.9			−0.534155			−0.267078	−	0.963675i	$-0.586058\pi$
$739$	−10730.9			−0.534155			−0.267078	+	0.963675i	$0.586058\pi$
$740$	5153.41	+	8925.97i	0.256004	+	0.443413i
$741$		0			0
$742$	−1018.84	+	1764.69i	−0.0504082	+	0.0873096i
$743$	−7772.66	+	13462.6i	−0.383783	+	0.664732i	−0.991600	−	0.129345i	$-0.958712\pi$
$743$	−7772.66	+	13462.6i	−0.383783	+	0.664732i	0.607816	+	0.794078i	$0.292046\pi$
$744$		0			0
$745$	−10514.6	−	18211.8i	−0.517079	−	0.895607i
$746$	−17337.4			−0.850893
$747$		0			0
$748$	−1568.33			−0.0766629
$749$	−6534.06	−	11317.3i	−0.318757	−	0.552104i
$750$		0			0
$751$	1264.26	−	2189.76i	0.0614292	−	0.106399i	−0.833675	−	0.552255i	$-0.813767\pi$
$751$	1264.26	−	2189.76i	0.0614292	−	0.106399i	0.895104	+	0.445856i	$0.147101\pi$
$752$	921.034	−	1595.28i	0.0446631	−	0.0773588i
$753$		0			0
$754$	−5352.03	−	9269.99i	−0.258501	−	0.447736i
$755$	−1193.10			−0.0575119
$756$		0			0
$757$	−890.832			−0.0427712			−0.0213856	−	0.999771i	$-0.506808\pi$
$757$	−890.832			−0.0427712			−0.0213856	+	0.999771i	$0.506808\pi$
$758$	−6465.25	−	11198.1i	−0.309800	−	0.536589i
$759$		0			0
$760$	−2119.11	+	3670.40i	−0.101142	+	0.175184i
$761$	−7254.29	+	12564.8i	−0.345556	+	0.598520i	−0.985455	−	0.169939i	$-0.945643\pi$
$761$	−7254.29	+	12564.8i	−0.345556	+	0.598520i	0.639899	+	0.768459i	$0.278976\pi$
$762$		0			0
$763$	−2049.78	−	3550.32i	−0.0972569	−	0.168454i
$764$	9969.18			0.472084
$765$		0			0
$766$	27308.9			1.28814
$767$	11097.5	+	19221.4i	0.522435	+	0.904884i
$768$		0			0
$769$	−3721.33	+	6445.54i	−0.174505	+	0.302252i	−0.939990	−	0.341202i	$-0.889166\pi$
$769$	−3721.33	+	6445.54i	−0.174505	+	0.302252i	0.765485	+	0.643454i	$0.222499\pi$
$770$	−306.905	+	531.574i	−0.0143637	+	0.0248787i
$771$		0			0
$772$	4307.93	+	7461.55i	0.200836	+	0.347859i
$773$	2614.02			0.121630			0.0608148	−	0.998149i	$-0.480630\pi$
$773$	2614.02			0.121630			0.0608148	+	0.998149i	$0.480630\pi$
$774$		0			0
$775$	14836.8			0.687684
$776$	3745.86	+	6488.02i	0.173284	+	0.300137i
$777$		0			0
$778$	8498.50	−	14719.8i	0.391627	−	0.678318i
$779$	4220.77	−	7310.60i	0.194127	−	0.336238i
$780$		0			0
$781$	−723.420	−	1253.00i	−0.0331447	−	0.0574083i
$782$	−20929.8			−0.957096
$783$		0			0
$784$	784.000			0.0357143
$785$	2301.90	+	3987.01i	0.104660	+	0.181277i
$786$		0			0
$787$	13238.7	−	22930.1i	0.599629	−	1.03859i	−0.393247	−	0.919433i	$-0.628648\pi$
$787$	13238.7	−	22930.1i	0.599629	−	1.03859i	0.992876	−	0.119155i	$-0.0380185\pi$
$788$	10230.2	−	17719.3i	0.462483	−	0.801044i
$789$		0			0
$790$	−2349.29	−	4069.08i	−0.105802	−	0.183255i
$791$	−5187.57			−0.233184
$792$		0			0
$793$	31619.5			1.41594
$794$	12203.2	+	21136.5i	0.545434	+	0.944719i
$795$		0			0
$796$	5741.33	−	9944.28i	0.255648	−	0.442796i
$797$	14542.1	−	25187.7i	0.646309	−	1.11944i	−0.337689	−	0.941258i	$-0.609645\pi$
$797$	14542.1	−	25187.7i	0.646309	−	1.11944i	0.983998	−	0.178182i	$-0.0570215\pi$
$798$		0			0
$799$	3071.33	+	5319.70i	0.135990	+	0.235541i
$800$	2860.94			0.126437
$801$		0			0
$802$	19531.6			0.859955
$803$	−1877.47	−	3251.87i	−0.0825085	−	0.142909i
$804$		0			0
$805$	−4095.73	+	7094.01i	−0.179324	+	0.310598i
$806$	9096.50	−	15755.6i	0.397532	−	0.688545i
$807$		0			0
$808$	1708.59	+	2959.36i	0.0743909	+	0.128849i
$809$	−8808.99			−0.382828			−0.191414	−	0.981509i	$-0.561307\pi$
$809$	−8808.99			−0.382828			−0.191414	+	0.981509i	$0.561307\pi$
$810$		0			0
$811$	−14856.3			−0.643251			−0.321625	−	0.946867i	$-0.604229\pi$
$811$	−14856.3			−0.643251			−0.321625	+	0.946867i	$0.604229\pi$
$812$	1366.96	+	2367.64i	0.0590773	+	0.102325i
$813$		0			0
$814$	−3173.76	+	5497.12i	−0.136659	+	0.236700i
$815$	−641.899	+	1111.80i	−0.0275887	+	0.0477850i
$816$		0			0
$817$	16642.5	+	28825.7i	0.712666	+	1.23437i
$818$	15588.5			0.666309
$819$		0			0
$820$	2268.74			0.0966193
$821$	12778.9	+	22133.7i	0.543223	+	0.940890i	0.998716	+	0.0506511i	$0.0161296\pi$
$821$	12778.9	+	22133.7i	0.543223	+	0.940890i	−0.455493	+	0.890239i	$0.650537\pi$
$822$		0			0
$823$	−3718.84	+	6441.23i	−0.157510	+	0.272815i	−0.933970	−	0.357351i	$-0.883680\pi$
$823$	−3718.84	+	6441.23i	−0.157510	+	0.272815i	0.776460	+	0.630166i	$0.217013\pi$
$824$	−7980.53	+	13822.7i	−0.337397	+	0.584388i
$825$		0			0
$826$	−2834.40	−	4909.33i	−0.119397	−	0.206801i
$827$	12890.0			0.541995			0.270998	−	0.962580i	$-0.412646\pi$
$827$	12890.0			0.541995			0.270998	+	0.962580i	$0.412646\pi$
$828$		0			0
$829$	−8592.44			−0.359985			−0.179993	−	0.983668i	$-0.557607\pi$
$829$	−8592.44			−0.359985			−0.179993	+	0.983668i	$0.557607\pi$
$830$	3143.43	+	5444.59i	0.131458	+	0.227692i
$831$		0			0
$832$	1754.05	−	3038.10i	0.0730898	−	0.126595i
$833$	−1307.18	+	2264.11i	−0.0543712	+	0.0941737i
$834$		0			0
$835$	−1109.50	−	1921.71i	−0.0459829	−	0.0796448i
$836$	−2610.13			−0.107982
$837$		0			0
$838$	−24170.5			−0.996367
$839$	−17776.0	−	30789.0i	−0.731462	−	1.26693i	−0.956258	−	0.292523i	$-0.905505\pi$
$839$	−17776.0	−	30789.0i	−0.731462	−	1.26693i	0.224797	−	0.974406i	$-0.427828\pi$
$840$		0			0
$841$	7427.74	−	12865.2i	0.304553	−	0.527501i
$842$	9851.29	−	17062.9i	0.403204	−	0.698370i
$843$		0			0
$844$	4400.73	+	7622.29i	0.179478	+	0.310865i
$845$	−4818.17			−0.196154
$846$		0			0
$847$	8938.98			0.362629
$848$	−1164.39	−	2016.78i	−0.0471526	−	0.0816706i
$849$		0			0
$850$	−4770.12	+	8262.09i	−0.192487	+	0.333397i
$851$	−42354.8	+	73360.6i	−1.70611	+	2.95508i
$852$		0			0
$853$	2104.95	+	3645.88i	0.0844926	+	0.146345i	0.905175	−	0.425039i	$-0.139740\pi$
$853$	2104.95	+	3645.88i	0.0844926	+	0.146345i	−0.820682	+	0.571385i	$0.806406\pi$
$854$	−8075.90			−0.323597
$855$		0			0
$856$	14935.0			0.596340
$857$	−1438.67	−	2491.85i	−0.0573443	−	0.0993232i	0.835928	−	0.548839i	$-0.184930\pi$
$857$	−1438.67	−	2491.85i	−0.0573443	−	0.0993232i	−0.893272	+	0.449516i	$0.851597\pi$
$858$		0			0
$859$	2494.94	−	4321.37i	0.0990993	−	0.171645i	−0.812213	−	0.583361i	$-0.801737\pi$
$859$	2494.94	−	4321.37i	0.0990993	−	0.171645i	0.911312	+	0.411716i	$0.135071\pi$
$860$	−4472.83	+	7747.16i	−0.177351	+	0.307181i
$861$		0			0
$862$	268.348	+	464.793i	0.0106032	+	0.0183653i
$863$	20313.7			0.801260			0.400630	−	0.916240i	$-0.368791\pi$
$863$	20313.7			0.801260			0.400630	+	0.916240i	$0.368791\pi$
$864$		0			0
$865$	−612.902			−0.0240917
$866$	−2696.15	−	4669.87i	−0.105795	−	0.183243i
$867$		0			0
$868$	−2323.33	+	4024.12i	−0.0908513	+	0.157359i
$869$	1446.82	−	2505.97i	0.0564788	−	0.0978242i
$870$		0			0
$871$	−1200.17	−	2078.76i	−0.0466893	−	0.0808682i
$872$	4685.21			0.181951
$873$		0			0
$874$	−34833.0			−1.34810
$875$	4477.13	+	7754.62i	0.172977	+	0.299604i
$876$		0			0
$877$	−1966.37	+	3405.85i	−0.0757120	+	0.131137i	−0.901396	−	0.432996i	$-0.857456\pi$
$877$	−1966.37	+	3405.85i	−0.0757120	+	0.131137i	0.825684	+	0.564133i	$0.190790\pi$
$878$	−9382.06	+	16250.2i	−0.360625	+	0.624622i
$879$		0			0
$880$	−350.748	−	607.513i	−0.0134360	−	0.0232719i
$881$	36905.8			1.41134			0.705668	−	0.708543i	$-0.250647\pi$
$881$	36905.8			1.41134			0.705668	+	0.708543i	$0.250647\pi$
$882$		0			0
$883$	−26603.9			−1.01392			−0.506960	−	0.861970i	$-0.669231\pi$
$883$	−26603.9			−1.01392			−0.506960	+	0.861970i	$0.669231\pi$
$884$	5849.14	+	10131.0i	0.222543	+	0.385456i
$885$		0			0
$886$	−12687.2	+	21974.9i	−0.481078	+	0.833251i
$887$	3983.32	−	6899.32i	0.150786	−	0.261168i	−0.780731	−	0.624867i	$-0.785153\pi$
$887$	3983.32	−	6899.32i	0.150786	−	0.261168i	0.931517	+	0.363699i	$0.118486\pi$
$888$		0			0
$889$	−255.212	−	442.041i	−0.00962828	−	0.0166767i
$890$	7037.95			0.265070
$891$		0			0
$892$	−11954.4			−0.448724
$893$	5111.53	+	8853.43i	0.191546	+	0.331768i
$894$		0			0
$895$	12824.9	−	22213.3i	0.478981	−	0.829620i
$896$	−448.000	+	775.959i	−0.0167038	+	0.0289319i
$897$		0			0
$898$	−8285.34	−	14350.6i	−0.307890	−	0.533282i
$899$	−16203.5			−0.601132
$900$		0			0
$901$	7765.68			0.287139
$902$	698.609	+	1210.03i	0.0257884	+	0.0446668i
$903$		0			0
$904$	2964.33	−	5134.36i	0.109062	−	0.188901i
$905$	−9046.25	+	15668.6i	−0.332274	+	0.575515i
$906$		0			0
$907$	−2900.71	−	5024.17i	−0.106192	−	0.183930i	0.808032	−	0.589138i	$-0.200533\pi$
$907$	−2900.71	−	5024.17i	−0.106192	−	0.183930i	−0.914225	+	0.405208i	$0.867199\pi$
$908$	−6974.87			−0.254922
$909$		0			0
$910$	4578.44			0.166784
$911$	−9314.42	−	16133.0i	−0.338749	−	0.586731i	0.645449	−	0.763804i	$-0.276670\pi$
$911$	−9314.42	−	16133.0i	−0.338749	−	0.586731i	−0.984198	+	0.177073i	$0.943337\pi$
$912$		0			0
$913$	−1935.90	+	3353.08i	−0.0701742	+	0.121545i
$914$	15941.4	−	27611.3i	0.576909	−	0.999236i
$915$		0			0
$916$	3249.23	+	5627.83i	0.117203	+	0.203001i
$917$	6586.92			0.237207
$918$		0			0
$919$	−36920.3			−1.32523			−0.662615	−	0.748960i	$-0.730554\pi$
$919$	−36920.3			−1.32523			−0.662615	+	0.748960i	$0.730554\pi$
$920$	−4680.84	−	8107.44i	−0.167742	−	0.290538i
$921$		0			0
$922$	10319.7	−	17874.3i	0.368614	−	0.638459i
$923$	−5396.04	+	9346.21i	−0.192430	+	0.333298i
$924$		0			0
$925$	19306.2	+	33439.3i	0.686252	+	1.18862i
$926$	−36943.2			−1.31105
$927$		0			0
$928$	−3124.47			−0.110524
$929$	11683.8	+	20236.9i	0.412628	+	0.714693i	0.995176	−	0.0981030i	$-0.0312775\pi$
$929$	11683.8	+	20236.9i	0.412628	+	0.714693i	−0.582548	+	0.812796i	$0.697944\pi$
$930$		0			0
$931$	−2175.51	+	3768.10i	−0.0765838	+	0.132647i
$932$	9630.85	−	16681.1i	0.338486	−	0.586275i
$933$		0			0
$934$	1991.38	+	3449.18i	0.0697645	+	0.120836i
$935$	2339.24			0.0818198
$936$		0			0
$937$	−17216.9			−0.600268			−0.300134	−	0.953897i	$-0.597031\pi$
$937$	−17216.9			−0.600268			−0.300134	+	0.953897i	$0.597031\pi$
$938$	306.535	+	530.935i	0.0106703	+	0.0184815i
$939$		0			0
$940$	−1373.77	+	2379.44i	−0.0476675	+	0.0825625i
$941$	−2597.04	+	4498.20i	−0.0899692	+	0.155831i	−0.907498	−	0.420057i	$-0.862010\pi$
$941$	−2597.04	+	4498.20i	−0.0899692	+	0.155831i	0.817529	+	0.575888i	$0.195344\pi$
$942$		0			0
$943$	9323.14	+	16148.2i	0.321955	+	0.557642i
$944$	6478.64			0.223370
$945$		0			0
$946$	−5509.24			−0.189345
$947$	−4610.28	−	7985.23i	−0.158198	−	0.274008i	0.776021	−	0.630707i	$-0.217235\pi$
$947$	−4610.28	−	7985.23i	−0.158198	−	0.274008i	−0.934219	+	0.356700i	$0.883902\pi$
$948$		0			0
$949$	−14004.1	+	24255.9i	−0.479024	+	0.829694i
$950$	−7938.79	+	13750.4i	−0.271125	+	0.469602i
$951$		0			0
$952$	−1493.92	−	2587.55i	−0.0508596	−	0.0880914i
$953$	−3344.36			−0.113677			−0.0568386	−	0.998383i	$-0.518102\pi$
$953$	−3344.36			−0.113677			−0.0568386	+	0.998383i	$0.518102\pi$
$954$		0			0
$955$	−14869.5			−0.503840
$956$	−6876.07	−	11909.7i	−0.232623	−	0.402915i
$957$		0			0
$958$	7405.69	−	12827.0i	0.249757	−	0.432591i
$959$	6277.10	−	10872.3i	0.211364	−	0.366093i
$960$		0			0
$961$	1125.46	+	1949.35i	0.0377785	+	0.0654342i
$962$	47346.6			1.58681
$963$		0			0
$964$	−22627.8			−0.756010
$965$	−6425.49	−	11129.3i	−0.214346	−	0.371258i
$966$		0			0
$967$	17167.6	−	29735.2i	0.570914	−	0.988851i	−0.425559	−	0.904931i	$-0.639922\pi$
$967$	17167.6	−	29735.2i	0.570914	−	0.988851i	0.996472	−	0.0839207i	$-0.0267442\pi$
$968$	−5107.99	+	8847.30i	−0.169604	+	0.293763i
$969$		0			0
$970$	−5587.14	−	9677.21i	−0.184940	−	0.320326i
$971$	−31841.4			−1.05236			−0.526179	−	0.850374i	$-0.676376\pi$
$971$	−31841.4			−1.05236			−0.526179	+	0.850374i	$0.676376\pi$
$972$		0			0
$973$	−4151.69			−0.136791
$974$	7064.90	+	12236.8i	0.232417	+	0.402558i
$975$		0			0
$976$	4614.80	−	7993.07i	0.151348	−	0.262143i
$977$	14123.0	−	24461.8i	0.462472	−	0.801025i	−0.536612	−	0.843829i	$-0.680296\pi$
$977$	14123.0	−	24461.8i	0.462472	−	0.801025i	0.999083	+	0.0428046i	$0.0136293\pi$
$978$		0			0
$979$	2167.18	+	3753.67i	0.0707492	+	0.122541i
$980$	−1169.38			−0.0381167
$981$		0			0
$982$	−10826.9			−0.351832
$983$	−4222.78	−	7314.07i	−0.137015	−	0.237317i	0.789350	−	0.613943i	$-0.210418\pi$
$983$	−4222.78	−	7314.07i	−0.137015	−	0.237317i	−0.926365	+	0.376626i	$0.877084\pi$
$984$		0			0
$985$	−15258.9	+	26429.2i	−0.493593	+	0.854928i
$986$	5209.51	−	9023.14i	0.168260	−	0.291436i
$987$		0			0
$988$	9734.57	+	16860.8i	0.313460	+	0.542928i
$989$	−73522.4			−2.36388
$990$		0			0
$991$	26998.8			0.865435			0.432718	−	0.901529i	$-0.357555\pi$
$991$	26998.8			0.865435			0.432718	+	0.901529i	$0.357555\pi$
$992$	−2655.23	−	4599.00i	−0.0849836	−	0.147196i
$993$		0			0
$994$	1378.20	−	2387.11i	0.0439776	−	0.0761715i
$995$	−8563.49	+	14832.4i	−0.272845	+	0.472581i
$996$		0			0
$997$	−18784.0	−	32534.8i	−0.596684	−	1.03349i	−0.993307	−	0.115506i	$-0.963151\pi$
$997$	−18784.0	−	32534.8i	−0.596684	−	1.03349i	0.396623	−	0.917982i	$-0.370182\pi$
$998$	−26728.1			−0.847759
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.f.b.253.3		8
3.2	odd	2		126.4.f.b.85.1	yes	8
9.2	odd	6		126.4.f.b.43.1	✓	8
9.4	even	3		1134.4.a.l.1.2		4
9.5	odd	6		1134.4.a.o.1.3		4
9.7	even	3	inner	378.4.f.b.127.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.4.f.b.43.1	✓	8	9.2	odd	6
126.4.f.b.85.1	yes	8	3.2	odd	2
378.4.f.b.127.3		8	9.7	even	3	inner
378.4.f.b.253.3		8	1.1	even	1	trivial
1134.4.a.l.1.2		4	9.4	even	3
1134.4.a.o.1.3		4	9.5	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.f (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(2.98310 - 5.16688i) q^{5} +(3.50000 + 6.06218i) q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(2.98310 - 5.16688i) q^{5} +(3.50000 + 6.06218i) q^{7} -8.00000 q^{8} +11.9324 q^{10} +(3.67432 + 6.36411i) q^{11} +(27.4070 - 47.4703i) q^{13} +(-7.00000 + 12.1244i) q^{14} +(-8.00000 - 13.8564i) q^{16} +53.3544 q^{17} +88.7964 q^{19} +(11.9324 + 20.6675i) q^{20} +(-7.34865 + 12.7282i) q^{22} +(-98.0698 + 169.862i) q^{23} +(44.7022 + 77.4265i) q^{25} +109.628 q^{26} -28.0000 q^{28} +(-48.8199 - 84.5585i) q^{29} +(82.9760 - 143.719i) q^{31} +(16.0000 - 27.7128i) q^{32} +(53.3544 + 92.4126i) q^{34} +41.7634 q^{35} +431.884 q^{37} +(88.7964 + 153.800i) q^{38} +(-23.8648 + 41.3351i) q^{40} +(47.5332 - 82.3299i) q^{41} +(187.423 + 324.627i) q^{43} -29.3946 q^{44} -392.279 q^{46} +(57.5647 + 99.7049i) q^{47} +(-24.5000 + 42.4352i) q^{49} +(-89.4044 + 154.853i) q^{50} +(109.628 + 189.881i) q^{52} +145.549 q^{53} +43.8435 q^{55} +(-28.0000 - 48.4974i) q^{56} +(97.6398 - 169.117i) q^{58} +(-202.457 + 350.666i) q^{59} +(288.425 + 499.567i) q^{61} +331.904 q^{62} +64.0000 q^{64} +(-163.516 - 283.218i) q^{65} +(21.8954 - 37.9239i) q^{67} +(-106.709 + 184.825i) q^{68} +(41.7634 + 72.3364i) q^{70} -196.885 q^{71} -510.969 q^{73} +(431.884 + 748.045i) q^{74} +(-177.593 + 307.600i) q^{76} +(-25.7203 + 44.5488i) q^{77} +(-196.883 - 341.011i) q^{79} -95.4592 q^{80} +190.133 q^{82} +(263.437 + 456.286i) q^{83} +(159.162 - 275.676i) q^{85} +(-374.847 + 649.254i) q^{86} +(-29.3946 - 50.9129i) q^{88} +589.818 q^{89} +383.698 q^{91} +(-392.279 - 679.448i) q^{92} +(-115.129 + 199.410i) q^{94} +(264.889 - 458.800i) q^{95} +(-468.232 - 811.002i) q^{97} -98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 - 16 * q^4 - q^5 + 28 * q^7 - 64 * q^8	\( 8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8} - 4 q^{10} - 5 q^{11} + 21 q^{13} - 56 q^{14} - 64 q^{16} + 46 q^{17} - 188 q^{19} - 4 q^{20} + 10 q^{22} - 374 q^{23} - 41 q^{25} + 84 q^{26} - 224 q^{28} - 271 q^{29} + 243 q^{31} + 128 q^{32} + 46 q^{34} - 14 q^{35} - 362 q^{37} - 188 q^{38} + 8 q^{40} + 213 q^{41} + 238 q^{43} + 40 q^{44} - 1496 q^{46} - 675 q^{47} - 196 q^{49} + 82 q^{50} + 84 q^{52} - 108 q^{53} - 2828 q^{55} - 224 q^{56} + 542 q^{58} - 202 q^{59} + 1212 q^{61} + 972 q^{62} + 512 q^{64} - 549 q^{65} - 139 q^{67} - 92 q^{68} - 14 q^{70} + 2590 q^{71} - 4000 q^{73} - 362 q^{74} + 376 q^{76} + 35 q^{77} + 1545 q^{79} + 32 q^{80} + 852 q^{82} + 142 q^{83} + 793 q^{85} - 476 q^{86} + 40 q^{88} + 264 q^{89} + 294 q^{91} - 1496 q^{92} + 1350 q^{94} - 1244 q^{95} + 638 q^{97} - 784 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 - 16 * q^4 - q^5 + 28 * q^7 - 64 * q^8 - 4 * q^10 - 5 * q^11 + 21 * q^13 - 56 * q^14 - 64 * q^16 + 46 * q^17 - 188 * q^19 - 4 * q^20 + 10 * q^22 - 374 * q^23 - 41 * q^25 + 84 * q^26 - 224 * q^28 - 271 * q^29 + 243 * q^31 + 128 * q^32 + 46 * q^34 - 14 * q^35 - 362 * q^37 - 188 * q^38 + 8 * q^40 + 213 * q^41 + 238 * q^43 + 40 * q^44 - 1496 * q^46 - 675 * q^47 - 196 * q^49 + 82 * q^50 + 84 * q^52 - 108 * q^53 - 2828 * q^55 - 224 * q^56 + 542 * q^58 - 202 * q^59 + 1212 * q^61 + 972 * q^62 + 512 * q^64 - 549 * q^65 - 139 * q^67 - 92 * q^68 - 14 * q^70 + 2590 * q^71 - 4000 * q^73 - 362 * q^74 + 376 * q^76 + 35 * q^77 + 1545 * q^79 + 32 * q^80 + 852 * q^82 + 142 * q^83 + 793 * q^85 - 476 * q^86 + 40 * q^88 + 264 * q^89 + 294 * q^91 - 1496 * q^92 + 1350 * q^94 - 1244 * q^95 + 638 * q^97 - 784 * q^98

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