LMFDB - Embedded newform 2034.2.a.r.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2034,2,Mod(1,2034)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2034.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2034, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2034 = 2 \cdot 3^{2} \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2034.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-4,0,4,-4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$16.2415717711$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 226)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.17557$ of defining polynomial
Character	$\chi$	$=$	2034.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.07768 q^{5} -3.23607 q^{7} -1.00000 q^{8} -2.07768 q^{10} -3.80423 q^{11} +0.884927 q^{13} +3.23607 q^{14} +1.00000 q^{16} +4.47214 q^{17} -3.67853 q^{19} +2.07768 q^{20} +3.80423 q^{22} +3.85224 q^{23} -0.683231 q^{25} -0.884927 q^{26} -3.23607 q^{28} +2.96261 q^{29} +8.27636 q^{31} -1.00000 q^{32} -4.47214 q^{34} -6.72353 q^{35} -11.1180 q^{37} +3.67853 q^{38} -2.07768 q^{40} -11.9596 q^{41} -0.659481 q^{43} -3.80423 q^{44} -3.85224 q^{46} -3.75621 q^{47} +3.47214 q^{49} +0.683231 q^{50} +0.884927 q^{52} -7.27044 q^{53} -7.90398 q^{55} +3.23607 q^{56} -2.96261 q^{58} -7.67853 q^{59} +13.0615 q^{61} -8.27636 q^{62} +1.00000 q^{64} +1.83860 q^{65} -6.79360 q^{67} +4.47214 q^{68} +6.72353 q^{70} -9.18806 q^{71} +3.24920 q^{73} +11.1180 q^{74} -3.67853 q^{76} +12.3107 q^{77} -16.6139 q^{79} +2.07768 q^{80} +11.9596 q^{82} -2.21702 q^{83} +9.29168 q^{85} +0.659481 q^{86} +3.80423 q^{88} -14.0000 q^{89} -2.86368 q^{91} +3.85224 q^{92} +3.75621 q^{94} -7.64282 q^{95} +11.7213 q^{97} -3.47214 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{7} - 4 q^{8} + 4 q^{10} + 4 q^{13} + 4 q^{14} + 4 q^{16} - 6 q^{19} - 4 q^{20} + 6 q^{23} + 4 q^{25} - 4 q^{26} - 4 q^{28} - 4 q^{32} + 4 q^{35} - 8 q^{37} + 6 q^{38}+ \cdots + 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 - 4 * q^7 - 4 * q^8 + 4 * q^10 + 4 * q^13 + 4 * q^14 + 4 * q^16 - 6 * q^19 - 4 * q^20 + 6 * q^23 + 4 * q^25 - 4 * q^26 - 4 * q^28 - 4 * q^32 + 4 * q^35 - 8 * q^37 + 6 * q^38 + 4 * q^40 - 8 * q^41 - 6 * q^43 - 6 * q^46 + 6 * q^47 - 4 * q^49 - 4 * q^50 + 4 * q^52 - 4 * q^53 - 20 * q^55 + 4 * q^56 - 22 * q^59 + 16 * q^61 + 4 * q^64 - 24 * q^65 - 18 * q^67 - 4 * q^70 - 22 * q^71 + 8 * q^74 - 6 * q^76 - 2 * q^79 - 4 * q^80 + 8 * q^82 - 12 * q^83 + 6 * q^86 - 56 * q^89 - 24 * q^91 + 6 * q^92 - 6 * q^94 + 16 * q^95 + 16 * q^97 + 4 * q^98

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	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.07768	0.929168	0.464584	−	0.885529i	$-0.346204\pi$
$5$	2.07768	0.929168	0.464584	+	0.885529i	$0.346204\pi$
$6$	0	0
$7$	−3.23607	−1.22312	−0.611559	−	0.791199i	$-0.709457\pi$
$7$	−3.23607	−1.22312	−0.611559	+	0.791199i	$0.709457\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.07768	−0.657021
$11$	−3.80423	−1.14702	−0.573509	−	0.819199i	$-0.694418\pi$
$11$	−3.80423	−1.14702	−0.573509	+	0.819199i	$0.694418\pi$
$12$	0	0
$13$	0.884927	0.245435	0.122717	−	0.992442i	$-0.460839\pi$
$13$	0.884927	0.245435	0.122717	+	0.992442i	$0.460839\pi$
$14$	3.23607	0.864876
$15$	0	0
$16$	1.00000	0.250000
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	−3.67853	−0.843913	−0.421956	−	0.906616i	$-0.638657\pi$
$19$	−3.67853	−0.843913	−0.421956	+	0.906616i	$0.638657\pi$
$20$	2.07768	0.464584
$21$	0	0
$22$	3.80423	0.811064
$23$	3.85224	0.803247	0.401623	−	0.915805i	$-0.368446\pi$
$23$	3.85224	0.803247	0.401623	+	0.915805i	$0.368446\pi$
$24$	0	0
$25$	−0.683231	−0.136646
$26$	−0.884927	−0.173548
$27$	0	0
$28$	−3.23607	−0.611559
$29$	2.96261	0.550143	0.275071	−	0.961424i	$-0.411298\pi$
$29$	2.96261	0.550143	0.275071	+	0.961424i	$0.411298\pi$
$30$	0	0
$31$	8.27636	1.48648	0.743239	−	0.669026i	$-0.233288\pi$
$31$	8.27636	1.48648	0.743239	+	0.669026i	$0.233288\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.47214	−0.766965
$35$	−6.72353	−1.13648
$36$	0	0
$37$	−11.1180	−1.82778	−0.913892	−	0.405957i	$-0.866938\pi$
$37$	−11.1180	−1.82778	−0.913892	+	0.405957i	$0.866938\pi$
$38$	3.67853	0.596737
$39$	0	0
$40$	−2.07768	−0.328511
$41$	−11.9596	−1.86777	−0.933887	−	0.357567i	$-0.883606\pi$
$41$	−11.9596	−1.86777	−0.933887	+	0.357567i	$0.883606\pi$
$42$	0	0
$43$	−0.659481	−0.100570	−0.0502849	−	0.998735i	$-0.516013\pi$
$43$	−0.659481	−0.100570	−0.0502849	+	0.998735i	$0.516013\pi$
$44$	−3.80423	−0.573509
$45$	0	0
$46$	−3.85224	−0.567981
$47$	−3.75621	−0.547900	−0.273950	−	0.961744i	$-0.588330\pi$
$47$	−3.75621	−0.547900	−0.273950	+	0.961744i	$0.588330\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0.683231	0.0966235
$51$	0	0
$52$	0.884927	0.122717
$53$	−7.27044	−0.998672	−0.499336	−	0.866408i	$-0.666423\pi$
$53$	−7.27044	−0.998672	−0.499336	+	0.866408i	$0.666423\pi$
$54$	0	0
$55$	−7.90398	−1.06577
$56$	3.23607	0.432438
$57$	0	0
$58$	−2.96261	−0.389010
$59$	−7.67853	−0.999660	−0.499830	−	0.866124i	$-0.666604\pi$
$59$	−7.67853	−0.999660	−0.499830	+	0.866124i	$0.666604\pi$
$60$	0	0
$61$	13.0615	1.67236	0.836179	−	0.548456i	$-0.184784\pi$
$61$	13.0615	1.67236	0.836179	+	0.548456i	$0.184784\pi$
$62$	−8.27636	−1.05110
$63$	0	0
$64$	1.00000	0.125000
$65$	1.83860	0.228050
$66$	0	0
$67$	−6.79360	−0.829971	−0.414985	−	0.909828i	$-0.636213\pi$
$67$	−6.79360	−0.829971	−0.414985	+	0.909828i	$0.636213\pi$
$68$	4.47214	0.542326
$69$	0	0
$70$	6.72353	0.803615
$71$	−9.18806	−1.09042	−0.545211	−	0.838299i	$-0.683550\pi$
$71$	−9.18806	−1.09042	−0.545211	+	0.838299i	$0.683550\pi$
$72$	0	0
$73$	3.24920	0.380290	0.190145	−	0.981756i	$-0.439104\pi$
$73$	3.24920	0.380290	0.190145	+	0.981756i	$0.439104\pi$
$74$	11.1180	1.29244
$75$	0	0
$76$	−3.67853	−0.421956
$77$	12.3107	1.40294
$78$	0	0
$79$	−16.6139	−1.86921	−0.934603	−	0.355693i	$-0.884245\pi$
$79$	−16.6139	−1.86921	−0.934603	+	0.355693i	$0.884245\pi$
$80$	2.07768	0.232292
$81$	0	0
$82$	11.9596	1.32072
$83$	−2.21702	−0.243349	−0.121675	−	0.992570i	$-0.538826\pi$
$83$	−2.21702	−0.243349	−0.121675	+	0.992570i	$0.538826\pi$
$84$	0	0
$85$	9.29168	1.00782
$86$	0.659481	0.0711136
$87$	0	0
$88$	3.80423	0.405532
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	−2.86368	−0.300196
$92$	3.85224	0.401623
$93$	0	0
$94$	3.75621	0.387424
$95$	−7.64282	−0.784137
$96$	0	0
$97$	11.7213	1.19012	0.595061	−	0.803681i	$-0.297128\pi$
$97$	11.7213	1.19012	0.595061	+	0.803681i	$0.297128\pi$
$98$	−3.47214	−0.350739
$99$	0	0
$100$	−0.683231	−0.0683231
$101$	9.13922	0.909386	0.454693	−	0.890648i	$-0.349749\pi$
$101$	9.13922	0.909386	0.454693	+	0.890648i	$0.349749\pi$
$102$	0	0
$103$	−14.6624	−1.44473	−0.722364	−	0.691513i	$-0.756944\pi$
$103$	−14.6624	−1.44473	−0.722364	+	0.691513i	$0.756944\pi$
$104$	−0.884927	−0.0867742
$105$	0	0
$106$	7.27044	0.706168
$107$	−0.659481	−0.0637544	−0.0318772	−	0.999492i	$-0.510149\pi$
$107$	−0.659481	−0.0637544	−0.0318772	+	0.999492i	$0.510149\pi$
$108$	0	0
$109$	9.74258	0.933170	0.466585	−	0.884476i	$-0.345484\pi$
$109$	9.74258	0.933170	0.466585	+	0.884476i	$0.345484\pi$
$110$	7.90398	0.753615
$111$	0	0
$112$	−3.23607	−0.305780
$113$	1.00000	0.0940721
$113$	1.00000	0.0940721
$114$	0	0
$115$	8.00373	0.746352
$116$	2.96261	0.275071
$117$	0	0
$118$	7.67853	0.706866
$119$	−14.4721	−1.32666
$120$	0	0
$121$	3.47214	0.315649
$122$	−13.0615	−1.18254
$123$	0	0
$124$	8.27636	0.743239
$125$	−11.8080	−1.05614
$126$	0	0
$127$	−1.85776	−0.164850	−0.0824249	−	0.996597i	$-0.526266\pi$
$127$	−1.85776	−0.164850	−0.0824249	+	0.996597i	$0.526266\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.83860	−0.161256
$131$	12.1018	1.05734	0.528671	−	0.848827i	$-0.322691\pi$
$131$	12.1018	1.05734	0.528671	+	0.848827i	$0.322691\pi$
$132$	0	0
$133$	11.9040	1.03221
$134$	6.79360	0.586878
$135$	0	0
$136$	−4.47214	−0.383482
$137$	−7.40456	−0.632615	−0.316307	−	0.948657i	$-0.602443\pi$
$137$	−7.40456	−0.632615	−0.316307	+	0.948657i	$0.602443\pi$
$138$	0	0
$139$	−21.4508	−1.81943	−0.909716	−	0.415232i	$-0.863701\pi$
$139$	−21.4508	−1.81943	−0.909716	+	0.415232i	$0.863701\pi$
$140$	−6.72353	−0.568242
$141$	0	0
$142$	9.18806	0.771045
$143$	−3.36646	−0.281518
$144$	0	0
$145$	6.15537	0.511175
$146$	−3.24920	−0.268905
$147$	0	0
$148$	−11.1180	−0.913892
$149$	19.4104	1.59016	0.795080	−	0.606505i	$-0.207429\pi$
$149$	19.4104	1.59016	0.795080	+	0.606505i	$0.207429\pi$
$150$	0	0
$151$	−9.28408	−0.755528	−0.377764	−	0.925902i	$-0.623307\pi$
$151$	−9.28408	−0.755528	−0.377764	+	0.925902i	$0.623307\pi$
$152$	3.67853	0.298368
$153$	0	0
$154$	−12.3107	−0.992027
$155$	17.1957	1.38119
$156$	0	0
$157$	20.3701	1.62571	0.812855	−	0.582467i	$-0.197912\pi$
$157$	20.3701	1.62571	0.812855	+	0.582467i	$0.197912\pi$
$158$	16.6139	1.32173
$159$	0	0
$160$	−2.07768	−0.164255
$161$	−12.4661	−0.982466
$162$	0	0
$163$	−8.05934	−0.631257	−0.315628	−	0.948883i	$-0.602215\pi$
$163$	−8.05934	−0.631257	−0.315628	+	0.948883i	$0.602215\pi$
$164$	−11.9596	−0.933887
$165$	0	0
$166$	2.21702	0.172074
$167$	12.2759	0.949934	0.474967	−	0.880004i	$-0.342460\pi$
$167$	12.2759	0.949934	0.474967	+	0.880004i	$0.342460\pi$
$168$	0	0
$169$	−12.2169	−0.939762
$170$	−9.29168	−0.712640
$171$	0	0
$172$	−0.659481	−0.0502849
$173$	−2.56816	−0.195253	−0.0976267	−	0.995223i	$-0.531125\pi$
$173$	−2.56816	−0.195253	−0.0976267	+	0.995223i	$0.531125\pi$
$174$	0	0
$175$	2.21098	0.167135
$176$	−3.80423	−0.286754
$177$	0	0
$178$	14.0000	1.04934
$179$	−11.7473	−0.878033	−0.439016	−	0.898479i	$-0.644673\pi$
$179$	−11.7473	−0.878033	−0.439016	+	0.898479i	$0.644673\pi$
$180$	0	0
$181$	−8.21181	−0.610379	−0.305189	−	0.952292i	$-0.598720\pi$
$181$	−8.21181	−0.610379	−0.305189	+	0.952292i	$0.598720\pi$
$182$	2.86368	0.212270
$183$	0	0
$184$	−3.85224	−0.283991
$185$	−23.0996	−1.69832
$186$	0	0
$187$	−17.0130	−1.24411
$188$	−3.75621	−0.273950
$189$	0	0
$190$	7.64282	0.554469
$191$	−2.75841	−0.199591	−0.0997957	−	0.995008i	$-0.531819\pi$
$191$	−2.75841	−0.199591	−0.0997957	+	0.995008i	$0.531819\pi$
$192$	0	0
$193$	−19.0996	−1.37482	−0.687411	−	0.726269i	$-0.741253\pi$
$193$	−19.0996	−1.37482	−0.687411	+	0.726269i	$0.741253\pi$
$194$	−11.7213	−0.841543
$195$	0	0
$196$	3.47214	0.248010
$197$	−10.7706	−0.767371	−0.383686	−	0.923464i	$-0.625345\pi$
$197$	−10.7706	−0.767371	−0.383686	+	0.923464i	$0.625345\pi$
$198$	0	0
$199$	15.0860	1.06942	0.534709	−	0.845036i	$-0.320421\pi$
$199$	15.0860	1.06942	0.534709	+	0.845036i	$0.320421\pi$
$200$	0.683231	0.0483117
$201$	0	0
$202$	−9.13922	−0.643033
$203$	−9.58721	−0.672890
$204$	0	0
$205$	−24.8482	−1.73548
$206$	14.6624	1.02158
$207$	0	0
$208$	0.884927	0.0613586
$209$	13.9940	0.967983
$210$	0	0
$211$	−4.43403	−0.305251	−0.152626	−	0.988284i	$-0.548773\pi$
$211$	−4.43403	−0.305251	−0.152626	+	0.988284i	$0.548773\pi$
$212$	−7.27044	−0.499336
$213$	0	0
$214$	0.659481	0.0450812
$215$	−1.37019	−0.0934463
$216$	0	0
$217$	−26.7829	−1.81814
$218$	−9.74258	−0.659851
$219$	0	0
$220$	−7.90398	−0.532886
$221$	3.95751	0.266211
$222$	0	0
$223$	1.52607	0.102193	0.0510966	−	0.998694i	$-0.483728\pi$
$223$	1.52607	0.102193	0.0510966	+	0.998694i	$0.483728\pi$
$224$	3.23607	0.216219
$225$	0	0
$226$	−1.00000	−0.0665190
$227$	−6.08059	−0.403583	−0.201791	−	0.979429i	$-0.564676\pi$
$227$	−6.08059	−0.403583	−0.201791	+	0.979429i	$0.564676\pi$
$228$	0	0
$229$	−2.17371	−0.143643	−0.0718213	−	0.997418i	$-0.522881\pi$
$229$	−2.17371	−0.143643	−0.0718213	+	0.997418i	$0.522881\pi$
$230$	−8.00373	−0.527750
$231$	0	0
$232$	−2.96261	−0.194505
$233$	6.58129	0.431154	0.215577	−	0.976487i	$-0.430837\pi$
$233$	6.58129	0.431154	0.215577	+	0.976487i	$0.430837\pi$
$234$	0	0
$235$	−7.80423	−0.509092
$236$	−7.67853	−0.499830
$237$	0	0
$238$	14.4721	0.938089
$239$	2.07247	0.134057	0.0670286	−	0.997751i	$-0.478648\pi$
$239$	2.07247	0.134057	0.0670286	+	0.997751i	$0.478648\pi$
$240$	0	0
$241$	−11.8423	−0.762831	−0.381416	−	0.924404i	$-0.624563\pi$
$241$	−11.8423	−0.762831	−0.381416	+	0.924404i	$0.624563\pi$
$242$	−3.47214	−0.223197
$243$	0	0
$244$	13.0615	0.836179
$245$	7.21400	0.460886
$246$	0	0
$247$	−3.25523	−0.207125
$248$	−8.27636	−0.525550
$249$	0	0
$250$	11.8080	0.746801
$251$	14.7735	0.932493	0.466247	−	0.884655i	$-0.345606\pi$
$251$	14.7735	0.932493	0.466247	+	0.884655i	$0.345606\pi$
$252$	0	0
$253$	−14.6548	−0.921338
$254$	1.85776	0.116566
$255$	0	0
$256$	1.00000	0.0625000
$257$	26.4779	1.65165	0.825824	−	0.563927i	$-0.190710\pi$
$257$	26.4779	1.65165	0.825824	+	0.563927i	$0.190710\pi$
$258$	0	0
$259$	35.9785	2.23560
$260$	1.83860	0.114025
$261$	0	0
$262$	−12.1018	−0.747654
$263$	−11.8156	−0.728579	−0.364289	−	0.931286i	$-0.618688\pi$
$263$	−11.8156	−0.728579	−0.364289	+	0.931286i	$0.618688\pi$
$264$	0	0
$265$	−15.1057	−0.927934
$266$	−11.9040	−0.729880
$267$	0	0
$268$	−6.79360	−0.414985
$269$	−31.2733	−1.90677	−0.953385	−	0.301757i	$-0.902427\pi$
$269$	−31.2733	−1.90677	−0.953385	+	0.301757i	$0.902427\pi$
$270$	0	0
$271$	−1.58900	−0.0965251	−0.0482626	−	0.998835i	$-0.515368\pi$
$271$	−1.58900	−0.0965251	−0.0482626	+	0.998835i	$0.515368\pi$
$272$	4.47214	0.271163
$273$	0	0
$274$	7.40456	0.447326
$275$	2.59917	0.156736
$276$	0	0
$277$	19.7757	1.18820	0.594102	−	0.804390i	$-0.297507\pi$
$277$	19.7757	1.18820	0.594102	+	0.804390i	$0.297507\pi$
$278$	21.4508	1.28653
$279$	0	0
$280$	6.72353	0.401807
$281$	−9.65094	−0.575727	−0.287863	−	0.957671i	$-0.592945\pi$
$281$	−9.65094	−0.575727	−0.287863	+	0.957671i	$0.592945\pi$
$282$	0	0
$283$	−8.32386	−0.494802	−0.247401	−	0.968913i	$-0.579577\pi$
$283$	−8.32386	−0.494802	−0.247401	+	0.968913i	$0.579577\pi$
$284$	−9.18806	−0.545211
$285$	0	0
$286$	3.36646	0.199063
$287$	38.7021	2.28451
$288$	0	0
$289$	3.00000	0.176471
$290$	−6.15537	−0.361456
$291$	0	0
$292$	3.24920	0.190145
$293$	1.92232	0.112303	0.0561515	−	0.998422i	$-0.482117\pi$
$293$	1.92232	0.112303	0.0561515	+	0.998422i	$0.482117\pi$
$294$	0	0
$295$	−15.9536	−0.928852
$296$	11.1180	0.646219
$297$	0	0
$298$	−19.4104	−1.12441
$299$	3.40895	0.197145
$300$	0	0
$301$	2.13412	0.123009
$302$	9.28408	0.534239
$303$	0	0
$304$	−3.67853	−0.210978
$305$	27.1377	1.55390
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	12.3107	0.701469
$309$	0	0
$310$	−17.1957	−0.976648
$311$	1.50870	0.0855505	0.0427753	−	0.999085i	$-0.486380\pi$
$311$	1.50870	0.0855505	0.0427753	+	0.999085i	$0.486380\pi$
$312$	0	0
$313$	34.1150	1.92829	0.964146	−	0.265373i	$-0.0854952\pi$
$313$	34.1150	1.92829	0.964146	+	0.265373i	$0.0854952\pi$
$314$	−20.3701	−1.14955
$315$	0	0
$316$	−16.6139	−0.934603
$317$	−16.8041	−0.943813	−0.471907	−	0.881649i	$-0.656434\pi$
$317$	−16.8041	−0.943813	−0.471907	+	0.881649i	$0.656434\pi$
$318$	0	0
$319$	−11.2704	−0.631024
$320$	2.07768	0.116146
$321$	0	0
$322$	12.4661	0.694709
$323$	−16.4509	−0.915352
$324$	0	0
$325$	−0.604610	−0.0335377
$326$	8.05934	0.446366
$327$	0	0
$328$	11.9596	0.660358
$329$	12.1554	0.670147
$330$	0	0
$331$	11.3108	0.621700	0.310850	−	0.950459i	$-0.399386\pi$
$331$	11.3108	0.621700	0.310850	+	0.950459i	$0.399386\pi$
$332$	−2.21702	−0.121675
$333$	0	0
$334$	−12.2759	−0.671705
$335$	−14.1150	−0.771183
$336$	0	0
$337$	31.8525	1.73512	0.867559	−	0.497335i	$-0.165688\pi$
$337$	31.8525	1.73512	0.867559	+	0.497335i	$0.165688\pi$
$338$	12.2169	0.664512
$339$	0	0
$340$	9.29168	0.503912
$341$	−31.4852	−1.70502
$342$	0	0
$343$	11.4164	0.616428
$344$	0.659481	0.0355568
$345$	0	0
$346$	2.56816	0.138065
$347$	23.4164	1.25706	0.628529	−	0.777786i	$-0.283657\pi$
$347$	23.4164	1.25706	0.628529	+	0.777786i	$0.283657\pi$
$348$	0	0
$349$	1.74907	0.0936256	0.0468128	−	0.998904i	$-0.485094\pi$
$349$	1.74907	0.0936256	0.0468128	+	0.998904i	$0.485094\pi$
$350$	−2.21098	−0.118182
$351$	0	0
$352$	3.80423	0.202766
$353$	29.9977	1.59662	0.798308	−	0.602249i	$-0.205729\pi$
$353$	29.9977	1.59662	0.798308	+	0.602249i	$0.205729\pi$
$354$	0	0
$355$	−19.0899	−1.01319
$356$	−14.0000	−0.741999
$357$	0	0
$358$	11.7473	0.620863
$359$	−12.4160	−0.655292	−0.327646	−	0.944801i	$-0.606255\pi$
$359$	−12.4160	−0.655292	−0.327646	+	0.944801i	$0.606255\pi$
$360$	0	0
$361$	−5.46841	−0.287811
$362$	8.21181	0.431603
$363$	0	0
$364$	−2.86368	−0.150098
$365$	6.75080	0.353353
$366$	0	0
$367$	−22.1305	−1.15520	−0.577602	−	0.816318i	$-0.696012\pi$
$367$	−22.1305	−1.15520	−0.577602	+	0.816318i	$0.696012\pi$
$368$	3.85224	0.200812
$369$	0	0
$370$	23.0996	1.20089
$371$	23.5276	1.22149
$372$	0	0
$373$	12.1525	0.629231	0.314615	−	0.949219i	$-0.398124\pi$
$373$	12.1525	0.629231	0.314615	+	0.949219i	$0.398124\pi$
$374$	17.0130	0.879722
$375$	0	0
$376$	3.75621	0.193712
$377$	2.62169	0.135024
$378$	0	0
$379$	−5.25634	−0.270000	−0.135000	−	0.990846i	$-0.543103\pi$
$379$	−5.25634	−0.270000	−0.135000	+	0.990846i	$0.543103\pi$
$380$	−7.64282	−0.392069
$381$	0	0
$382$	2.75841	0.141132
$383$	10.4803	0.535516	0.267758	−	0.963486i	$-0.413717\pi$
$383$	10.4803	0.535516	0.267758	+	0.963486i	$0.413717\pi$
$384$	0	0
$385$	25.5778	1.30357
$386$	19.0996	0.972146
$387$	0	0
$388$	11.7213	0.595061
$389$	−26.6927	−1.35337	−0.676686	−	0.736272i	$-0.736584\pi$
$389$	−26.6927	−1.35337	−0.676686	+	0.736272i	$0.736584\pi$
$390$	0	0
$391$	17.2277	0.871244
$392$	−3.47214	−0.175369
$393$	0	0
$394$	10.7706	0.542613
$395$	−34.5184	−1.73681
$396$	0	0
$397$	18.1795	0.912404	0.456202	−	0.889876i	$-0.349210\pi$
$397$	18.1795	0.912404	0.456202	+	0.889876i	$0.349210\pi$
$398$	−15.0860	−0.756193
$399$	0	0
$400$	−0.683231	−0.0341616
$401$	16.7791	0.837910	0.418955	−	0.908007i	$-0.362396\pi$
$401$	16.7791	0.837910	0.418955	+	0.908007i	$0.362396\pi$
$402$	0	0
$403$	7.32398	0.364833
$404$	9.13922	0.454693
$405$	0	0
$406$	9.58721	0.475805
$407$	42.2953	2.09650
$408$	0	0
$409$	−8.20389	−0.405656	−0.202828	−	0.979214i	$-0.565013\pi$
$409$	−8.20389	−0.405656	−0.202828	+	0.979214i	$0.565013\pi$
$410$	24.8482	1.22717
$411$	0	0
$412$	−14.6624	−0.722364
$413$	24.8482	1.22270
$414$	0	0
$415$	−4.60626	−0.226112
$416$	−0.884927	−0.0433871
$417$	0	0
$418$	−13.9940	−0.684467
$419$	−31.9630	−1.56150	−0.780748	−	0.624847i	$-0.785162\pi$
$419$	−31.9630	−1.56150	−0.780748	+	0.624847i	$0.785162\pi$
$420$	0	0
$421$	−19.0996	−0.930859	−0.465430	−	0.885085i	$-0.654100\pi$
$421$	−19.0996	−0.930859	−0.465430	+	0.885085i	$0.654100\pi$
$422$	4.43403	0.215845
$423$	0	0
$424$	7.27044	0.353084
$425$	−3.05550	−0.148214
$426$	0	0
$427$	−42.2680	−2.04549
$428$	−0.659481	−0.0318772
$429$	0	0
$430$	1.37019	0.0660765
$431$	26.7055	1.28636	0.643179	−	0.765716i	$-0.277615\pi$
$431$	26.7055	1.28636	0.643179	+	0.765716i	$0.277615\pi$
$432$	0	0
$433$	1.30353	0.0626435	0.0313218	−	0.999509i	$-0.490028\pi$
$433$	1.30353	0.0626435	0.0313218	+	0.999509i	$0.490028\pi$
$434$	26.7829	1.28562
$435$	0	0
$436$	9.74258	0.466585
$437$	−14.1706	−0.677871
$438$	0	0
$439$	−1.48179	−0.0707219	−0.0353609	−	0.999375i	$-0.511258\pi$
$439$	−1.48179	−0.0707219	−0.0353609	+	0.999375i	$0.511258\pi$
$440$	7.90398	0.376807
$441$	0	0
$442$	−3.95751	−0.188240
$443$	19.1282	0.908808	0.454404	−	0.890796i	$-0.349852\pi$
$443$	19.1282	0.908808	0.454404	+	0.890796i	$0.349852\pi$
$444$	0	0
$445$	−29.0876	−1.37888
$446$	−1.52607	−0.0722615
$447$	0	0
$448$	−3.23607	−0.152890
$449$	1.17442	0.0554242	0.0277121	−	0.999616i	$-0.491178\pi$
$449$	1.17442	0.0554242	0.0277121	+	0.999616i	$0.491178\pi$
$450$	0	0
$451$	45.4970	2.14237
$452$	1.00000	0.0470360
$453$	0	0
$454$	6.08059	0.285376
$455$	−5.94983	−0.278932
$456$	0	0
$457$	−17.1377	−0.801670	−0.400835	−	0.916150i	$-0.631280\pi$
$457$	−17.1377	−0.801670	−0.400835	+	0.916150i	$0.631280\pi$
$458$	2.17371	0.101571
$459$	0	0
$460$	8.00373	0.373176
$461$	28.8695	1.34459	0.672293	−	0.740285i	$-0.265309\pi$
$461$	28.8695	1.34459	0.672293	+	0.740285i	$0.265309\pi$
$462$	0	0
$463$	−22.1341	−1.02866	−0.514330	−	0.857592i	$-0.671959\pi$
$463$	−22.1341	−1.02866	−0.514330	+	0.857592i	$0.671959\pi$
$464$	2.96261	0.137536
$465$	0	0
$466$	−6.58129	−0.304872
$467$	4.28138	0.198118	0.0990592	−	0.995082i	$-0.468417\pi$
$467$	4.28138	0.198118	0.0990592	+	0.995082i	$0.468417\pi$
$468$	0	0
$469$	21.9846	1.01515
$470$	7.80423	0.359982
$471$	0	0
$472$	7.67853	0.353433
$473$	2.50881	0.115355
$474$	0	0
$475$	2.51329	0.115318
$476$	−14.4721	−0.663329
$477$	0	0
$478$	−2.07247	−0.0947928
$479$	8.44164	0.385708	0.192854	−	0.981227i	$-0.438226\pi$
$479$	8.44164	0.385708	0.192854	+	0.981227i	$0.438226\pi$
$480$	0	0
$481$	−9.83860	−0.448601
$482$	11.8423	0.539403
$483$	0	0
$484$	3.47214	0.157824
$485$	24.3532	1.10582
$486$	0	0
$487$	−7.41820	−0.336151	−0.168075	−	0.985774i	$-0.553755\pi$
$487$	−7.41820	−0.336151	−0.168075	+	0.985774i	$0.553755\pi$
$488$	−13.0615	−0.591268
$489$	0	0
$490$	−7.21400	−0.325895
$491$	−20.0392	−0.904357	−0.452178	−	0.891927i	$-0.649353\pi$
$491$	−20.0392	−0.904357	−0.452178	+	0.891927i	$0.649353\pi$
$492$	0	0
$493$	13.2492	0.596714
$494$	3.25523	0.146460
$495$	0	0
$496$	8.27636	0.371620
$497$	29.7332	1.33372
$498$	0	0
$499$	9.83891	0.440450	0.220225	−	0.975449i	$-0.429321\pi$
$499$	9.83891	0.440450	0.220225	+	0.975449i	$0.429321\pi$
$500$	−11.8080	−0.528068
$501$	0	0
$502$	−14.7735	−0.659372
$503$	27.0784	1.20737	0.603683	−	0.797224i	$-0.293699\pi$
$503$	27.0784	1.20737	0.603683	+	0.797224i	$0.293699\pi$
$504$	0	0
$505$	18.9884	0.844973
$506$	14.6548	0.651484
$507$	0	0
$508$	−1.85776	−0.0824249
$509$	−1.13193	−0.0501720	−0.0250860	−	0.999685i	$-0.507986\pi$
$509$	−1.13193	−0.0501720	−0.0250860	+	0.999685i	$0.507986\pi$
$510$	0	0
$511$	−10.5146	−0.465140
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−26.4779	−1.16789
$515$	−30.4638	−1.34240
$516$	0	0
$517$	14.2895	0.628451
$518$	−35.9785	−1.58081
$519$	0	0
$520$	−1.83860	−0.0806279
$521$	−10.7771	−0.472152	−0.236076	−	0.971735i	$-0.575861\pi$
$521$	−10.7771	−0.472152	−0.236076	+	0.971735i	$0.575861\pi$
$522$	0	0
$523$	34.2467	1.49750	0.748752	−	0.662851i	$-0.230654\pi$
$523$	34.2467	1.49750	0.748752	+	0.662851i	$0.230654\pi$
$524$	12.1018	0.528671
$525$	0	0
$526$	11.8156	0.515183
$527$	37.0130	1.61231
$528$	0	0
$529$	−8.16027	−0.354794
$530$	15.1057	0.656149
$531$	0	0
$532$	11.9040	0.516103
$533$	−10.5834	−0.458416
$534$	0	0
$535$	−1.37019	−0.0592386
$536$	6.79360	0.293439
$537$	0	0
$538$	31.2733	1.34829
$539$	−13.2088	−0.568943
$540$	0	0
$541$	15.9425	0.685423	0.342712	−	0.939441i	$-0.388655\pi$
$541$	15.9425	0.685423	0.342712	+	0.939441i	$0.388655\pi$
$542$	1.58900	0.0682536
$543$	0	0
$544$	−4.47214	−0.191741
$545$	20.2420	0.867072
$546$	0	0
$547$	29.8040	1.27433	0.637164	−	0.770729i	$-0.280108\pi$
$547$	29.8040	1.27433	0.637164	+	0.770729i	$0.280108\pi$
$548$	−7.40456	−0.316307
$549$	0	0
$550$	−2.59917	−0.110829
$551$	−10.8981	−0.464273
$552$	0	0
$553$	53.7636	2.28626
$554$	−19.7757	−0.840187
$555$	0	0
$556$	−21.4508	−0.909716
$557$	6.36865	0.269849	0.134924	−	0.990856i	$-0.456921\pi$
$557$	6.36865	0.269849	0.134924	+	0.990856i	$0.456921\pi$
$558$	0	0
$559$	−0.583592	−0.0246833
$560$	−6.72353	−0.284121
$561$	0	0
$562$	9.65094	0.407100
$563$	38.7021	1.63110	0.815549	−	0.578689i	$-0.196435\pi$
$563$	38.7021	1.63110	0.815549	+	0.578689i	$0.196435\pi$
$564$	0	0
$565$	2.07768	0.0874088
$566$	8.32386	0.349878
$567$	0	0
$568$	9.18806	0.385522
$569$	7.57177	0.317425	0.158713	−	0.987325i	$-0.449266\pi$
$569$	7.57177	0.317425	0.158713	+	0.987325i	$0.449266\pi$
$570$	0	0
$571$	−12.2313	−0.511862	−0.255931	−	0.966695i	$-0.582382\pi$
$571$	−12.2313	−0.511862	−0.255931	+	0.966695i	$0.582382\pi$
$572$	−3.36646	−0.140759
$573$	0	0
$574$	−38.7021	−1.61539
$575$	−2.63197	−0.109761
$576$	0	0
$577$	−0.0118438	−0.000493065 0	−0.000246533	−	1.00000i	$-0.500078\pi$
$577$	−0.0118438	−0.000493065 0	−0.000246533		1.00000i	$0.500078\pi$
$578$	−3.00000	−0.124784
$579$	0	0
$580$	6.15537	0.255588
$581$	7.17442	0.297645
$582$	0	0
$583$	27.6584	1.14549
$584$	−3.24920	−0.134453
$585$	0	0
$586$	−1.92232	−0.0794102
$587$	24.7354	1.02094	0.510469	−	0.859896i	$-0.329472\pi$
$587$	24.7354	1.02094	0.510469	+	0.859896i	$0.329472\pi$
$588$	0	0
$589$	−30.4449	−1.25446
$590$	15.9536	0.656798
$591$	0	0
$592$	−11.1180	−0.456946
$593$	−20.6656	−0.848635	−0.424317	−	0.905514i	$-0.639486\pi$
$593$	−20.6656	−0.848635	−0.424317	+	0.905514i	$0.639486\pi$
$594$	0	0
$595$	−30.0685	−1.23269
$596$	19.4104	0.795080
$597$	0	0
$598$	−3.40895	−0.139402
$599$	−45.7304	−1.86849	−0.934246	−	0.356629i	$-0.883926\pi$
$599$	−45.7304	−1.86849	−0.934246	+	0.356629i	$0.883926\pi$
$600$	0	0
$601$	36.5894	1.49251	0.746256	−	0.665659i	$-0.231849\pi$
$601$	36.5894	1.49251	0.746256	+	0.665659i	$0.231849\pi$
$602$	−2.13412	−0.0869804
$603$	0	0
$604$	−9.28408	−0.377764
$605$	7.21400	0.293291
$606$	0	0
$607$	−0.367085	−0.0148995	−0.00744976	−	0.999972i	$-0.502371\pi$
$607$	−0.367085	−0.0148995	−0.00744976	+	0.999972i	$0.502371\pi$
$608$	3.67853	0.149184
$609$	0	0
$610$	−27.1377	−1.09878
$611$	−3.32398	−0.134474
$612$	0	0
$613$	13.6955	0.553157	0.276579	−	0.960991i	$-0.410799\pi$
$613$	13.6955	0.553157	0.276579	+	0.960991i	$0.410799\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	−12.3107	−0.496014
$617$	−27.8862	−1.12266	−0.561329	−	0.827593i	$-0.689710\pi$
$617$	−27.8862	−1.12266	−0.561329	+	0.827593i	$0.689710\pi$
$618$	0	0
$619$	−18.1827	−0.730826	−0.365413	−	0.930846i	$-0.619072\pi$
$619$	−18.1827	−0.730826	−0.365413	+	0.930846i	$0.619072\pi$
$620$	17.1957	0.690594
$621$	0	0
$622$	−1.50870	−0.0604934
$623$	45.3050	1.81510
$624$	0	0
$625$	−21.1170	−0.844682
$626$	−34.1150	−1.36351
$627$	0	0
$628$	20.3701	0.812855
$629$	−49.7211	−1.98251
$630$	0	0
$631$	4.87732	0.194163	0.0970816	−	0.995276i	$-0.469049\pi$
$631$	4.87732	0.194163	0.0970816	+	0.995276i	$0.469049\pi$
$632$	16.6139	0.660864
$633$	0	0
$634$	16.8041	0.667377
$635$	−3.85984	−0.153173
$636$	0	0
$637$	3.07259	0.121740
$638$	11.2704	0.446201
$639$	0	0
$640$	−2.07768	−0.0821277
$641$	−14.2741	−0.563791	−0.281896	−	0.959445i	$-0.590963\pi$
$641$	−14.2741	−0.563791	−0.281896	+	0.959445i	$0.590963\pi$
$642$	0	0
$643$	24.6286	0.971258	0.485629	−	0.874165i	$-0.338591\pi$
$643$	24.6286	0.971258	0.485629	+	0.874165i	$0.338591\pi$
$644$	−12.4661	−0.491233
$645$	0	0
$646$	16.4509	0.647252
$647$	17.6193	0.692685	0.346343	−	0.938108i	$-0.387423\pi$
$647$	17.6193	0.692685	0.346343	+	0.938108i	$0.387423\pi$
$648$	0	0
$649$	29.2109	1.14663
$650$	0.604610	0.0237147
$651$	0	0
$652$	−8.05934	−0.315628
$653$	−4.86949	−0.190558	−0.0952790	−	0.995451i	$-0.530374\pi$
$653$	−4.86949	−0.190558	−0.0952790	+	0.995451i	$0.530374\pi$
$654$	0	0
$655$	25.1438	0.982449
$656$	−11.9596	−0.466944
$657$	0	0
$658$	−12.1554	−0.473866
$659$	−11.6937	−0.455523	−0.227762	−	0.973717i	$-0.573141\pi$
$659$	−11.6937	−0.455523	−0.227762	+	0.973717i	$0.573141\pi$
$660$	0	0
$661$	−16.5225	−0.642652	−0.321326	−	0.946969i	$-0.604129\pi$
$661$	−16.5225	−0.642652	−0.321326	+	0.946969i	$0.604129\pi$
$662$	−11.3108	−0.439609
$663$	0	0
$664$	2.21702	0.0860369
$665$	24.7327	0.959093
$666$	0	0
$667$	11.4127	0.441901
$668$	12.2759	0.474967
$669$	0	0
$670$	14.1150	0.545308
$671$	−49.6890	−1.91822
$672$	0	0
$673$	−32.8958	−1.26804	−0.634019	−	0.773317i	$-0.718596\pi$
$673$	−32.8958	−1.26804	−0.634019	+	0.773317i	$0.718596\pi$
$674$	−31.8525	−1.22691
$675$	0	0
$676$	−12.2169	−0.469881
$677$	−50.6977	−1.94847	−0.974235	−	0.225536i	$-0.927587\pi$
$677$	−50.6977	−1.94847	−0.974235	+	0.225536i	$0.927587\pi$
$678$	0	0
$679$	−37.9310	−1.45566
$680$	−9.29168	−0.356320
$681$	0	0
$682$	31.4852	1.20563
$683$	−36.6711	−1.40318	−0.701590	−	0.712581i	$-0.747526\pi$
$683$	−36.6711	−1.40318	−0.701590	+	0.712581i	$0.747526\pi$
$684$	0	0
$685$	−15.3843	−0.587805
$686$	−11.4164	−0.435880
$687$	0	0
$688$	−0.659481	−0.0251425
$689$	−6.43381	−0.245109
$690$	0	0
$691$	22.1662	0.843242	0.421621	−	0.906772i	$-0.361461\pi$
$691$	22.1662	0.843242	0.421621	+	0.906772i	$0.361461\pi$
$692$	−2.56816	−0.0976267
$693$	0	0
$694$	−23.4164	−0.888875
$695$	−44.5679	−1.69056
$696$	0	0
$697$	−53.4849	−2.02589
$698$	−1.74907	−0.0662033
$699$	0	0
$700$	2.21098	0.0835673
$701$	−5.92813	−0.223902	−0.111951	−	0.993714i	$-0.535710\pi$
$701$	−5.92813	−0.223902	−0.111951	+	0.993714i	$0.535710\pi$
$702$	0	0
$703$	40.8978	1.54249
$704$	−3.80423	−0.143377
$705$	0	0
$706$	−29.9977	−1.12898
$707$	−29.5751	−1.11229
$708$	0	0
$709$	0.723751	0.0271810	0.0135905	−	0.999908i	$-0.495674\pi$
$709$	0.723751	0.0271810	0.0135905	+	0.999908i	$0.495674\pi$
$710$	19.0899	0.716430
$711$	0	0
$712$	14.0000	0.524672
$713$	31.8825	1.19401
$714$	0	0
$715$	−6.99444	−0.261577
$716$	−11.7473	−0.439016
$717$	0	0
$718$	12.4160	0.463361
$719$	−28.0081	−1.04453	−0.522263	−	0.852784i	$-0.674912\pi$
$719$	−28.0081	−1.04453	−0.522263	+	0.852784i	$0.674912\pi$
$720$	0	0
$721$	47.4485	1.76707
$722$	5.46841	0.203513
$723$	0	0
$724$	−8.21181	−0.305189
$725$	−2.02415	−0.0751750
$726$	0	0
$727$	−32.6332	−1.21030	−0.605149	−	0.796112i	$-0.706886\pi$
$727$	−32.6332	−1.21030	−0.605149	+	0.796112i	$0.706886\pi$
$728$	2.86368	0.106135
$729$	0	0
$730$	−6.75080	−0.249858
$731$	−2.94929	−0.109083
$732$	0	0
$733$	4.78600	0.176775	0.0883875	−	0.996086i	$-0.471829\pi$
$733$	4.78600	0.176775	0.0883875	+	0.996086i	$0.471829\pi$
$734$	22.1305	0.816853
$735$	0	0
$736$	−3.85224	−0.141995
$737$	25.8444	0.951991
$738$	0	0
$739$	−20.0200	−0.736446	−0.368223	−	0.929737i	$-0.620034\pi$
$739$	−20.0200	−0.736446	−0.368223	+	0.929737i	$0.620034\pi$
$740$	−23.0996	−0.849160
$741$	0	0
$742$	−23.5276	−0.863727
$743$	33.0018	1.21072	0.605360	−	0.795952i	$-0.293029\pi$
$743$	33.0018	1.21072	0.605360	+	0.795952i	$0.293029\pi$
$744$	0	0
$745$	40.3286	1.47753
$746$	−12.1525	−0.444933
$747$	0	0
$748$	−17.0130	−0.622057
$749$	2.13412	0.0779792
$750$	0	0
$751$	25.6966	0.937684	0.468842	−	0.883282i	$-0.344671\pi$
$751$	25.6966	0.937684	0.468842	+	0.883282i	$0.344671\pi$
$752$	−3.75621	−0.136975
$753$	0	0
$754$	−2.62169	−0.0954765
$755$	−19.2894	−0.702012
$756$	0	0
$757$	33.5214	1.21835	0.609177	−	0.793034i	$-0.291500\pi$
$757$	33.5214	1.21835	0.609177	+	0.793034i	$0.291500\pi$
$758$	5.25634	0.190919
$759$	0	0
$760$	7.64282	0.277234
$761$	−24.4921	−0.887838	−0.443919	−	0.896067i	$-0.646412\pi$
$761$	−24.4921	−0.887838	−0.443919	+	0.896067i	$0.646412\pi$
$762$	0	0
$763$	−31.5276	−1.14138
$764$	−2.75841	−0.0997957
$765$	0	0
$766$	−10.4803	−0.378667
$767$	−6.79494	−0.245351
$768$	0	0
$769$	−27.0841	−0.976676	−0.488338	−	0.872654i	$-0.662397\pi$
$769$	−27.0841	−0.976676	−0.488338	+	0.872654i	$0.662397\pi$
$770$	−25.5778	−0.921760
$771$	0	0
$772$	−19.0996	−0.687411
$773$	0.835776	0.0300608	0.0150304	−	0.999887i	$-0.495216\pi$
$773$	0.835776	0.0300608	0.0150304	+	0.999887i	$0.495216\pi$
$774$	0	0
$775$	−5.65467	−0.203122
$776$	−11.7213	−0.420771
$777$	0	0
$778$	26.6927	0.956978
$779$	43.9937	1.57624
$780$	0	0
$781$	34.9534	1.25073
$782$	−17.2277	−0.616062
$783$	0	0
$784$	3.47214	0.124005
$785$	42.3226	1.51056
$786$	0	0
$787$	54.3473	1.93727	0.968636	−	0.248482i	$-0.0799318\pi$
$787$	54.3473	1.93727	0.968636	+	0.248482i	$0.0799318\pi$
$788$	−10.7706	−0.383686
$789$	0	0
$790$	34.5184	1.22811
$791$	−3.23607	−0.115061
$792$	0	0
$793$	11.5585	0.410455
$794$	−18.1795	−0.645167
$795$	0	0
$796$	15.0860	0.534709
$797$	38.5129	1.36420	0.682099	−	0.731260i	$-0.261067\pi$
$797$	38.5129	1.36420	0.682099	+	0.731260i	$0.261067\pi$
$798$	0	0
$799$	−16.7983	−0.594281
$800$	0.683231	0.0241559
$801$	0	0
$802$	−16.7791	−0.592492
$803$	−12.3607	−0.436199
$804$	0	0
$805$	−25.9006	−0.912877
$806$	−7.32398	−0.257976
$807$	0	0
$808$	−9.13922	−0.321517
$809$	28.5623	1.00420	0.502098	−	0.864811i	$-0.332562\pi$
$809$	28.5623	1.00420	0.502098	+	0.864811i	$0.332562\pi$
$810$	0	0
$811$	32.1352	1.12842	0.564210	−	0.825631i	$-0.309181\pi$
$811$	32.1352	1.12842	0.564210	+	0.825631i	$0.309181\pi$
$812$	−9.58721	−0.336445
$813$	0	0
$814$	−42.2953	−1.48245
$815$	−16.7448	−0.586544
$816$	0	0
$817$	2.42592	0.0848722
$818$	8.20389	0.286842
$819$	0	0
$820$	−24.8482	−0.867739
$821$	4.75661	0.166007	0.0830035	−	0.996549i	$-0.473549\pi$
$821$	4.75661	0.166007	0.0830035	+	0.996549i	$0.473549\pi$
$822$	0	0
$823$	20.4982	0.714521	0.357261	−	0.934005i	$-0.383711\pi$
$823$	20.4982	0.714521	0.357261	+	0.934005i	$0.383711\pi$
$824$	14.6624	0.510788
$825$	0	0
$826$	−24.8482	−0.864581
$827$	13.6928	0.476144	0.238072	−	0.971247i	$-0.423485\pi$
$827$	13.6928	0.476144	0.238072	+	0.971247i	$0.423485\pi$
$828$	0	0
$829$	3.44853	0.119772	0.0598862	−	0.998205i	$-0.480926\pi$
$829$	3.44853	0.119772	0.0598862	+	0.998205i	$0.480926\pi$
$830$	4.60626	0.159886
$831$	0	0
$832$	0.884927	0.0306793
$833$	15.5279	0.538009
$834$	0	0
$835$	25.5053	0.882649
$836$	13.9940	0.483991
$837$	0	0
$838$	31.9630	1.10414
$839$	29.1429	1.00612	0.503062	−	0.864250i	$-0.332207\pi$
$839$	29.1429	1.00612	0.503062	+	0.864250i	$0.332207\pi$
$840$	0	0
$841$	−20.2229	−0.697343
$842$	19.0996	0.658217
$843$	0	0
$844$	−4.43403	−0.152626
$845$	−25.3829	−0.873197
$846$	0	0
$847$	−11.2361	−0.386076
$848$	−7.27044	−0.249668
$849$	0	0
$850$	3.05550	0.104803
$851$	−42.8291	−1.46816
$852$	0	0
$853$	5.54472	0.189848	0.0949238	−	0.995485i	$-0.469739\pi$
$853$	5.54472	0.189848	0.0949238	+	0.995485i	$0.469739\pi$
$854$	42.2680	1.44638
$855$	0	0
$856$	0.659481	0.0225406
$857$	23.1317	0.790164	0.395082	−	0.918646i	$-0.370716\pi$
$857$	23.1317	0.790164	0.395082	+	0.918646i	$0.370716\pi$
$858$	0	0
$859$	34.6930	1.18371	0.591854	−	0.806045i	$-0.298396\pi$
$859$	34.6930	1.18371	0.591854	+	0.806045i	$0.298396\pi$
$860$	−1.37019	−0.0467232
$861$	0	0
$862$	−26.7055	−0.909593
$863$	0.406756	0.0138461	0.00692307	−	0.999976i	$-0.497796\pi$
$863$	0.406756	0.0138461	0.00692307	+	0.999976i	$0.497796\pi$
$864$	0	0
$865$	−5.33582	−0.181423
$866$	−1.30353	−0.0442957
$867$	0	0
$868$	−26.7829	−0.909070
$869$	63.2029	2.14401
$870$	0	0
$871$	−6.01184	−0.203704
$872$	−9.74258	−0.329925
$873$	0	0
$874$	14.1706	0.479327
$875$	38.2113	1.29178
$876$	0	0
$877$	−32.8758	−1.11014	−0.555068	−	0.831805i	$-0.687308\pi$
$877$	−32.8758	−1.11014	−0.555068	+	0.831805i	$0.687308\pi$
$878$	1.48179	0.0500079
$879$	0	0
$880$	−7.90398	−0.266443
$881$	−22.2241	−0.748749	−0.374375	−	0.927278i	$-0.622143\pi$
$881$	−22.2241	−0.748749	−0.374375	+	0.927278i	$0.622143\pi$
$882$	0	0
$883$	−32.0428	−1.07833	−0.539163	−	0.842201i	$-0.681259\pi$
$883$	−32.0428	−1.07833	−0.539163	+	0.842201i	$0.681259\pi$
$884$	3.95751	0.133106
$885$	0	0
$886$	−19.1282	−0.642625
$887$	−19.6255	−0.658958	−0.329479	−	0.944163i	$-0.606873\pi$
$887$	−19.6255	−0.658958	−0.329479	+	0.944163i	$0.606873\pi$
$888$	0	0
$889$	6.01184	0.201631
$890$	29.0876	0.975018
$891$	0	0
$892$	1.52607	0.0510966
$893$	13.8174	0.462380
$894$	0	0
$895$	−24.4071	−0.815840
$896$	3.23607	0.108109
$897$	0	0
$898$	−1.17442	−0.0391908
$899$	24.5196	0.817776
$900$	0	0
$901$	−32.5144	−1.08321
$902$	−45.4970	−1.51488
$903$	0	0
$904$	−1.00000	−0.0332595
$905$	−17.0615	−0.567145
$906$	0	0
$907$	−30.5406	−1.01408	−0.507041	−	0.861922i	$-0.669261\pi$
$907$	−30.5406	−1.01408	−0.507041	+	0.861922i	$0.669261\pi$
$908$	−6.08059	−0.201791
$909$	0	0
$910$	5.94983	0.197235
$911$	−20.7448	−0.687305	−0.343652	−	0.939097i	$-0.611664\pi$
$911$	−20.7448	−0.687305	−0.343652	+	0.939097i	$0.611664\pi$
$912$	0	0
$913$	8.43403	0.279126
$914$	17.1377	0.566866
$915$	0	0
$916$	−2.17371	−0.0718213
$917$	−39.1623	−1.29325
$918$	0	0
$919$	−12.1496	−0.400777	−0.200388	−	0.979717i	$-0.564220\pi$
$919$	−12.1496	−0.400777	−0.200388	+	0.979717i	$0.564220\pi$
$920$	−8.00373	−0.263875
$921$	0	0
$922$	−28.8695	−0.950766
$923$	−8.13076	−0.267627
$924$	0	0
$925$	7.59615	0.249760
$926$	22.1341	0.727372
$927$	0	0
$928$	−2.96261	−0.0972525
$929$	−8.16140	−0.267767	−0.133883	−	0.990997i	$-0.542745\pi$
$929$	−8.16140	−0.267767	−0.133883	+	0.990997i	$0.542745\pi$
$930$	0	0
$931$	−12.7724	−0.418597
$932$	6.58129	0.215577
$933$	0	0
$934$	−4.28138	−0.140091
$935$	−35.3477	−1.15599
$936$	0	0
$937$	12.4705	0.407393	0.203697	−	0.979034i	$-0.434704\pi$
$937$	12.4705	0.407393	0.203697	+	0.979034i	$0.434704\pi$
$938$	−21.9846	−0.717822
$939$	0	0
$940$	−7.80423	−0.254546
$941$	−24.0087	−0.782662	−0.391331	−	0.920250i	$-0.627985\pi$
$941$	−24.0087	−0.782662	−0.391331	+	0.920250i	$0.627985\pi$
$942$	0	0
$943$	−46.0712	−1.50028
$944$	−7.67853	−0.249915
$945$	0	0
$946$	−2.50881	−0.0815686
$947$	−39.6607	−1.28880	−0.644399	−	0.764689i	$-0.722892\pi$
$947$	−39.6607	−1.28880	−0.644399	+	0.764689i	$0.722892\pi$
$948$	0	0
$949$	2.87530	0.0933363
$950$	−2.51329	−0.0815418
$951$	0	0
$952$	14.4721	0.469045
$953$	6.37905	0.206638	0.103319	−	0.994648i	$-0.467054\pi$
$953$	6.37905	0.206638	0.103319	+	0.994648i	$0.467054\pi$
$954$	0	0
$955$	−5.73110	−0.185454
$956$	2.07247	0.0670286
$957$	0	0
$958$	−8.44164	−0.272737
$959$	23.9617	0.773763
$960$	0	0
$961$	37.4982	1.20962
$962$	9.83860	0.317209
$963$	0	0
$964$	−11.8423	−0.381416
$965$	−39.6830	−1.27744
$966$	0	0
$967$	17.8376	0.573618	0.286809	−	0.957988i	$-0.407406\pi$
$967$	17.8376	0.573618	0.286809	+	0.957988i	$0.407406\pi$
$968$	−3.47214	−0.111599
$969$	0	0
$970$	−24.3532	−0.781935
$971$	38.6667	1.24087	0.620437	−	0.784256i	$-0.286955\pi$
$971$	38.6667	1.24087	0.620437	+	0.784256i	$0.286955\pi$
$972$	0	0
$973$	69.4162	2.22538
$974$	7.41820	0.237695
$975$	0	0
$976$	13.0615	0.418090
$977$	−57.9071	−1.85261	−0.926306	−	0.376771i	$-0.877034\pi$
$977$	−57.9071	−1.85261	−0.926306	+	0.376771i	$0.877034\pi$
$978$	0	0
$979$	53.2592	1.70217
$980$	7.21400	0.230443
$981$	0	0
$982$	20.0392	0.639477
$983$	33.2788	1.06143	0.530715	−	0.847550i	$-0.321923\pi$
$983$	33.2788	1.06143	0.530715	+	0.847550i	$0.321923\pi$
$984$	0	0
$985$	−22.3778	−0.713017
$986$	−13.2492	−0.421940
$987$	0	0
$988$	−3.25523	−0.103563
$989$	−2.54048	−0.0807824
$990$	0	0
$991$	−20.1764	−0.640924	−0.320462	−	0.947261i	$-0.603838\pi$
$991$	−20.1764	−0.640924	−0.320462	+	0.947261i	$0.603838\pi$
$992$	−8.27636	−0.262775
$993$	0	0
$994$	−29.7332	−0.943079
$995$	31.3439	0.993670
$996$	0	0
$997$	−24.6412	−0.780396	−0.390198	−	0.920731i	$-0.627593\pi$
$997$	−24.6412	−0.780396	−0.390198	+	0.920731i	$0.627593\pi$
$998$	−9.83891	−0.311445
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2034.2.a.r.1.4		4
3.2	odd	2		226.2.a.d.1.4	✓	4
12.11	even	2		1808.2.a.j.1.1		4
15.14	odd	2		5650.2.a.o.1.1		4
24.5	odd	2		7232.2.a.u.1.1		4
24.11	even	2		7232.2.a.v.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
226.2.a.d.1.4	✓	4	3.2	odd	2
1808.2.a.j.1.1		4	12.11	even	2
2034.2.a.r.1.4		4	1.1	even	1	trivial
5650.2.a.o.1.1		4	15.14	odd	2
7232.2.a.u.1.1		4	24.5	odd	2
7232.2.a.v.1.4		4	24.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2034 = 2 \cdot 3^{2} \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2034.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.07768 q^{5} -3.23607 q^{7} -1.00000 q^{8} -2.07768 q^{10} -3.80423 q^{11} +0.884927 q^{13} +3.23607 q^{14} +1.00000 q^{16} +4.47214 q^{17} -3.67853 q^{19} +2.07768 q^{20} +3.80423 q^{22} +3.85224 q^{23} -0.683231 q^{25} -0.884927 q^{26} -3.23607 q^{28} +2.96261 q^{29} +8.27636 q^{31} -1.00000 q^{32} -4.47214 q^{34} -6.72353 q^{35} -11.1180 q^{37} +3.67853 q^{38} -2.07768 q^{40} -11.9596 q^{41} -0.659481 q^{43} -3.80423 q^{44} -3.85224 q^{46} -3.75621 q^{47} +3.47214 q^{49} +0.683231 q^{50} +0.884927 q^{52} -7.27044 q^{53} -7.90398 q^{55} +3.23607 q^{56} -2.96261 q^{58} -7.67853 q^{59} +13.0615 q^{61} -8.27636 q^{62} +1.00000 q^{64} +1.83860 q^{65} -6.79360 q^{67} +4.47214 q^{68} +6.72353 q^{70} -9.18806 q^{71} +3.24920 q^{73} +11.1180 q^{74} -3.67853 q^{76} +12.3107 q^{77} -16.6139 q^{79} +2.07768 q^{80} +11.9596 q^{82} -2.21702 q^{83} +9.29168 q^{85} +0.659481 q^{86} +3.80423 q^{88} -14.0000 q^{89} -2.86368 q^{91} +3.85224 q^{92} +3.75621 q^{94} -7.64282 q^{95} +11.7213 q^{97} -3.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{7} - 4 q^{8} + 4 q^{10} + 4 q^{13} + 4 q^{14} + 4 q^{16} - 6 q^{19} - 4 q^{20} + 6 q^{23} + 4 q^{25} - 4 q^{26} - 4 q^{28} - 4 q^{32} + 4 q^{35} - 8 q^{37} + 6 q^{38}+ \cdots + 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 - 4 * q^7 - 4 * q^8 + 4 * q^10 + 4 * q^13 + 4 * q^14 + 4 * q^16 - 6 * q^19 - 4 * q^20 + 6 * q^23 + 4 * q^25 - 4 * q^26 - 4 * q^28 - 4 * q^32 + 4 * q^35 - 8 * q^37 + 6 * q^38 + 4 * q^40 - 8 * q^41 - 6 * q^43 - 6 * q^46 + 6 * q^47 - 4 * q^49 - 4 * q^50 + 4 * q^52 - 4 * q^53 - 20 * q^55 + 4 * q^56 - 22 * q^59 + 16 * q^61 + 4 * q^64 - 24 * q^65 - 18 * q^67 - 4 * q^70 - 22 * q^71 + 8 * q^74 - 6 * q^76 - 2 * q^79 - 4 * q^80 + 8 * q^82 - 12 * q^83 + 6 * q^86 - 56 * q^89 - 24 * q^91 + 6 * q^92 - 6 * q^94 + 16 * q^95 + 16 * q^97 + 4 * q^98

Introduction

L-functions

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