LMFDB - Embedded newform 3654.2.a.bd.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3654,2,Mod(1,3654)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3654, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3654.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3654 = 2 \cdot 3^{2} \cdot 7 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3654.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3,-2,0,3,3,0,-2,3,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.1773368986$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1304.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 11x - 2 $ x^3 - 11*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1218)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.182370$ of defining polynomial
Character	$\chi$	$=$	3654.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +2.89219 q^{5} +1.00000 q^{7} +1.00000 q^{8} +2.89219 q^{10} +1.18237 q^{11} +3.70982 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.89219 q^{17} +3.18237 q^{19} +2.89219 q^{20} +1.18237 q^{22} -1.18237 q^{23} +3.36474 q^{25} +3.70982 q^{26} +1.00000 q^{28} -1.00000 q^{29} -0.892186 q^{31} +1.00000 q^{32} +2.89219 q^{34} +2.89219 q^{35} -4.96674 q^{37} +3.18237 q^{38} +2.89219 q^{40} -11.0413 q^{41} +5.63526 q^{43} +1.18237 q^{44} -1.18237 q^{46} -6.43929 q^{47} +1.00000 q^{49} +3.36474 q^{50} +3.70982 q^{52} -12.7511 q^{53} +3.41963 q^{55} +1.00000 q^{56} -1.00000 q^{58} +6.43929 q^{59} -0.364739 q^{61} -0.892186 q^{62} +1.00000 q^{64} +10.7295 q^{65} +3.18237 q^{67} +2.89219 q^{68} +2.89219 q^{70} -14.1491 q^{71} +16.5884 q^{73} -4.96674 q^{74} +3.18237 q^{76} +1.18237 q^{77} +0.729478 q^{79} +2.89219 q^{80} -11.0413 q^{82} +6.43929 q^{83} +8.36474 q^{85} +5.63526 q^{86} +1.18237 q^{88} +5.25693 q^{89} +3.70982 q^{91} -1.18237 q^{92} -6.43929 q^{94} +9.20400 q^{95} -2.07456 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{7} + 3 q^{8} - 2 q^{10} + 3 q^{11} + q^{13} + 3 q^{14} + 3 q^{16} - 2 q^{17} + 9 q^{19} - 2 q^{20} + 3 q^{22} - 3 q^{23} + 9 q^{25} + q^{26} + 3 q^{28} - 3 q^{29}+ \cdots + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^7 + 3 * q^8 - 2 * q^10 + 3 * q^11 + q^13 + 3 * q^14 + 3 * q^16 - 2 * q^17 + 9 * q^19 - 2 * q^20 + 3 * q^22 - 3 * q^23 + 9 * q^25 + q^26 + 3 * q^28 - 3 * q^29 + 8 * q^31 + 3 * q^32 - 2 * q^34 - 2 * q^35 + 7 * q^37 + 9 * q^38 - 2 * q^40 + 18 * q^43 + 3 * q^44 - 3 * q^46 - 7 * q^47 + 3 * q^49 + 9 * q^50 + q^52 + 5 * q^53 - 10 * q^55 + 3 * q^56 - 3 * q^58 + 7 * q^59 + 8 * q^62 + 3 * q^64 + 30 * q^65 + 9 * q^67 - 2 * q^68 - 2 * q^70 - 20 * q^71 + 15 * q^73 + 7 * q^74 + 9 * q^76 + 3 * q^77 - 2 * q^80 + 7 * q^83 + 24 * q^85 + 18 * q^86 + 3 * q^88 + 4 * q^89 + q^91 - 3 * q^92 - 7 * q^94 - 14 * q^95 + 5 * q^97 + 3 * 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
$5$	2.89219	1.29342	0.646712	−	0.762734i	$-0.276143\pi$
$5$	2.89219	1.29342	0.646712	+	0.762734i	$0.276143\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	2.89219	0.914589
$11$	1.18237	0.356498	0.178249	−	0.983985i	$-0.442957\pi$
$11$	1.18237	0.356498	0.178249	+	0.983985i	$0.442957\pi$
$12$	0	0
$13$	3.70982	1.02892	0.514459	−	0.857515i	$-0.327993\pi$
$13$	3.70982	1.02892	0.514459	+	0.857515i	$0.327993\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.89219	0.701458	0.350729	−	0.936477i	$-0.385934\pi$
$17$	2.89219	0.701458	0.350729	+	0.936477i	$0.385934\pi$
$18$	0	0
$19$	3.18237	0.730086	0.365043	−	0.930991i	$-0.381054\pi$
$19$	3.18237	0.730086	0.365043	+	0.930991i	$0.381054\pi$
$20$	2.89219	0.646712
$21$	0	0
$22$	1.18237	0.252082
$23$	−1.18237	−0.246541	−0.123271	−	0.992373i	$-0.539338\pi$
$23$	−1.18237	−0.246541	−0.123271	+	0.992373i	$0.539338\pi$
$24$	0	0
$25$	3.36474	0.672948
$26$	3.70982	0.727555
$27$	0	0
$28$	1.00000	0.188982
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−0.892186	−0.160241	−0.0801207	−	0.996785i	$-0.525531\pi$
$31$	−0.892186	−0.160241	−0.0801207	+	0.996785i	$0.525531\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.89219	0.496006
$35$	2.89219	0.488869
$36$	0	0
$37$	−4.96674	−0.816527	−0.408264	−	0.912864i	$-0.633866\pi$
$37$	−4.96674	−0.816527	−0.408264	+	0.912864i	$0.633866\pi$
$38$	3.18237	0.516249
$39$	0	0
$40$	2.89219	0.457295
$41$	−11.0413	−1.72436	−0.862180	−	0.506601i	$-0.830902\pi$
$41$	−11.0413	−1.72436	−0.862180	+	0.506601i	$0.830902\pi$
$42$	0	0
$43$	5.63526	0.859369	0.429685	−	0.902979i	$-0.358625\pi$
$43$	5.63526	0.859369	0.429685	+	0.902979i	$0.358625\pi$
$44$	1.18237	0.178249
$45$	0	0
$46$	−1.18237	−0.174331
$47$	−6.43929	−0.939268	−0.469634	−	0.882861i	$-0.655614\pi$
$47$	−6.43929	−0.939268	−0.469634	+	0.882861i	$0.655614\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.36474	0.475846
$51$	0	0
$52$	3.70982	0.514459
$53$	−12.7511	−1.75150	−0.875750	−	0.482765i	$-0.839633\pi$
$53$	−12.7511	−1.75150	−0.875750	+	0.482765i	$0.839633\pi$
$54$	0	0
$55$	3.41963	0.461103
$56$	1.00000	0.133631
$57$	0	0
$58$	−1.00000	−0.131306
$59$	6.43929	0.838325	0.419162	−	0.907911i	$-0.362324\pi$
$59$	6.43929	0.838325	0.419162	+	0.907911i	$0.362324\pi$
$60$	0	0
$61$	−0.364739	−0.0467001	−0.0233500	−	0.999727i	$-0.507433\pi$
$61$	−0.364739	−0.0467001	−0.0233500	+	0.999727i	$0.507433\pi$
$62$	−0.892186	−0.113308
$63$	0	0
$64$	1.00000	0.125000
$65$	10.7295	1.33083
$66$	0	0
$67$	3.18237	0.388788	0.194394	−	0.980923i	$-0.437726\pi$
$67$	3.18237	0.388788	0.194394	+	0.980923i	$0.437726\pi$
$68$	2.89219	0.350729
$69$	0	0
$70$	2.89219	0.345682
$71$	−14.1491	−1.67919	−0.839595	−	0.543212i	$-0.817208\pi$
$71$	−14.1491	−1.67919	−0.839595	+	0.543212i	$0.817208\pi$
$72$	0	0
$73$	16.5884	1.94153	0.970763	−	0.240040i	$-0.0771606\pi$
$73$	16.5884	1.94153	0.970763	+	0.240040i	$0.0771606\pi$
$74$	−4.96674	−0.577372
$75$	0	0
$76$	3.18237	0.365043
$77$	1.18237	0.134744
$78$	0	0
$79$	0.729478	0.0820727	0.0410364	−	0.999158i	$-0.486934\pi$
$79$	0.729478	0.0820727	0.0410364	+	0.999158i	$0.486934\pi$
$80$	2.89219	0.323356
$81$	0	0
$82$	−11.0413	−1.21931
$83$	6.43929	0.706804	0.353402	−	0.935471i	$-0.385025\pi$
$83$	6.43929	0.706804	0.353402	+	0.935471i	$0.385025\pi$
$84$	0	0
$85$	8.36474	0.907283
$86$	5.63526	0.607666
$87$	0	0
$88$	1.18237	0.126041
$89$	5.25693	0.557233	0.278616	−	0.960402i	$-0.410124\pi$
$89$	5.25693	0.557233	0.278616	+	0.960402i	$0.410124\pi$
$90$	0	0
$91$	3.70982	0.388894
$92$	−1.18237	−0.123271
$93$	0	0
$94$	−6.43929	−0.664163
$95$	9.20400	0.944311
$96$	0	0
$97$	−2.07456	−0.210639	−0.105320	−	0.994438i	$-0.533587\pi$
$97$	−2.07456	−0.210639	−0.105320	+	0.994438i	$0.533587\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	3.36474	0.336474
$101$	−17.5687	−1.74816	−0.874078	−	0.485786i	$-0.838533\pi$
$101$	−17.5687	−1.74816	−0.874078	+	0.485786i	$0.838533\pi$
$102$	0	0
$103$	−1.54711	−0.152441	−0.0762206	−	0.997091i	$-0.524285\pi$
$103$	−1.54711	−0.152441	−0.0762206	+	0.997091i	$0.524285\pi$
$104$	3.70982	0.363777
$105$	0	0
$106$	−12.7511	−1.23850
$107$	15.4196	1.49067	0.745336	−	0.666689i	$-0.232289\pi$
$107$	15.4196	1.49067	0.745336	+	0.666689i	$0.232289\pi$
$108$	0	0
$109$	−10.8786	−1.04198	−0.520990	−	0.853563i	$-0.674437\pi$
$109$	−10.8786	−1.04198	−0.520990	+	0.853563i	$0.674437\pi$
$110$	3.41963	0.326049
$111$	0	0
$112$	1.00000	0.0944911
$113$	3.54711	0.333684	0.166842	−	0.985984i	$-0.446643\pi$
$113$	3.54711	0.333684	0.166842	+	0.985984i	$0.446643\pi$
$114$	0	0
$115$	−3.41963	−0.318882
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	6.43929	0.592785
$119$	2.89219	0.265126
$120$	0	0
$121$	−9.60200	−0.872909
$122$	−0.364739	−0.0330219
$123$	0	0
$124$	−0.892186	−0.0801207
$125$	−4.72948	−0.423017
$126$	0	0
$127$	10.4529	0.927544	0.463772	−	0.885955i	$-0.346496\pi$
$127$	10.4529	0.927544	0.463772	+	0.885955i	$0.346496\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	10.7295	0.941037
$131$	−3.41963	−0.298775	−0.149387	−	0.988779i	$-0.547730\pi$
$131$	−3.41963	−0.298775	−0.149387	+	0.988779i	$0.547730\pi$
$132$	0	0
$133$	3.18237	0.275946
$134$	3.18237	0.274915
$135$	0	0
$136$	2.89219	0.248003
$137$	−4.60200	−0.393176	−0.196588	−	0.980486i	$-0.562986\pi$
$137$	−4.60200	−0.393176	−0.196588	+	0.980486i	$0.562986\pi$
$138$	0	0
$139$	10.6766	0.905574	0.452787	−	0.891619i	$-0.350430\pi$
$139$	10.6766	0.905574	0.452787	+	0.891619i	$0.350430\pi$
$140$	2.89219	0.244434
$141$	0	0
$142$	−14.1491	−1.18737
$143$	4.38637	0.366807
$144$	0	0
$145$	−2.89219	−0.240183
$146$	16.5884	1.37287
$147$	0	0
$148$	−4.96674	−0.408264
$149$	−0.751113	−0.0615336	−0.0307668	−	0.999527i	$-0.509795\pi$
$149$	−0.751113	−0.0615336	−0.0307668	+	0.999527i	$0.509795\pi$
$150$	0	0
$151$	−3.78437	−0.307968	−0.153984	−	0.988073i	$-0.549210\pi$
$151$	−3.78437	−0.307968	−0.153984	+	0.988073i	$0.549210\pi$
$152$	3.18237	0.258124
$153$	0	0
$154$	1.18237	0.0952781
$155$	−2.58037	−0.207260
$156$	0	0
$157$	7.78437	0.621261	0.310630	−	0.950531i	$-0.399460\pi$
$157$	7.78437	0.621261	0.310630	+	0.950531i	$0.399460\pi$
$158$	0.729478	0.0580342
$159$	0	0
$160$	2.89219	0.228647
$161$	−1.18237	−0.0931838
$162$	0	0
$163$	−8.51385	−0.666856	−0.333428	−	0.942776i	$-0.608205\pi$
$163$	−8.51385	−0.666856	−0.333428	+	0.942776i	$0.608205\pi$
$164$	−11.0413	−0.862180
$165$	0	0
$166$	6.43929	0.499786
$167$	−16.2982	−1.26119	−0.630597	−	0.776110i	$-0.717190\pi$
$167$	−16.2982	−1.26119	−0.630597	+	0.776110i	$0.717190\pi$
$168$	0	0
$169$	0.762737	0.0586721
$170$	8.36474	0.641546
$171$	0	0
$172$	5.63526	0.429685
$173$	−14.4609	−1.09944	−0.549722	−	0.835348i	$-0.685266\pi$
$173$	−14.4609	−1.09944	−0.549722	+	0.835348i	$0.685266\pi$
$174$	0	0
$175$	3.36474	0.254350
$176$	1.18237	0.0891245
$177$	0	0
$178$	5.25693	0.394023
$179$	9.20400	0.687940	0.343970	−	0.938981i	$-0.388228\pi$
$179$	9.20400	0.687940	0.343970	+	0.938981i	$0.388228\pi$
$180$	0	0
$181$	9.40604	0.699145	0.349573	−	0.936909i	$-0.386327\pi$
$181$	9.40604	0.699145	0.349573	+	0.936909i	$0.386327\pi$
$182$	3.70982	0.274990
$183$	0	0
$184$	−1.18237	−0.0871654
$185$	−14.3647	−1.05612
$186$	0	0
$187$	3.41963	0.250068
$188$	−6.43929	−0.469634
$189$	0	0
$190$	9.20400	0.667729
$191$	20.1491	1.45794	0.728969	−	0.684546i	$-0.240000\pi$
$191$	20.1491	1.45794	0.728969	+	0.684546i	$0.240000\pi$
$192$	0	0
$193$	−9.56874	−0.688773	−0.344387	−	0.938828i	$-0.611913\pi$
$193$	−9.56874	−0.688773	−0.344387	+	0.938828i	$0.611913\pi$
$194$	−2.07456	−0.148944
$195$	0	0
$196$	1.00000	0.0714286
$197$	16.0826	1.14584	0.572919	−	0.819612i	$-0.305811\pi$
$197$	16.0826	1.14584	0.572919	+	0.819612i	$0.305811\pi$
$198$	0	0
$199$	−1.54711	−0.109672	−0.0548358	−	0.998495i	$-0.517464\pi$
$199$	−1.54711	−0.109672	−0.0548358	+	0.998495i	$0.517464\pi$
$200$	3.36474	0.237923
$201$	0	0
$202$	−17.5687	−1.23613
$203$	−1.00000	−0.0701862
$204$	0	0
$205$	−31.9335	−2.23033
$206$	−1.54711	−0.107792
$207$	0	0
$208$	3.70982	0.257229
$209$	3.76274	0.260274
$210$	0	0
$211$	26.6630	1.83555	0.917777	−	0.397096i	$-0.129982\pi$
$211$	26.6630	1.83555	0.917777	+	0.397096i	$0.129982\pi$
$212$	−12.7511	−0.875750
$213$	0	0
$214$	15.4196	1.05406
$215$	16.2982	1.11153
$216$	0	0
$217$	−0.892186	−0.0605655
$218$	−10.8786	−0.736791
$219$	0	0
$220$	3.41963	0.230552
$221$	10.7295	0.721743
$222$	0	0
$223$	−20.3864	−1.36517	−0.682586	−	0.730805i	$-0.739145\pi$
$223$	−20.3864	−1.36517	−0.682586	+	0.730805i	$0.739145\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	3.54711	0.235950
$227$	13.6217	0.904102	0.452051	−	0.891992i	$-0.350693\pi$
$227$	13.6217	0.904102	0.452051	+	0.891992i	$0.350693\pi$
$228$	0	0
$229$	0.513850	0.0339562	0.0169781	−	0.999856i	$-0.494595\pi$
$229$	0.513850	0.0339562	0.0169781	+	0.999856i	$0.494595\pi$
$230$	−3.41963	−0.225484
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	−12.6630	−0.829578	−0.414789	−	0.909918i	$-0.636145\pi$
$233$	−12.6630	−0.829578	−0.414789	+	0.909918i	$0.636145\pi$
$234$	0	0
$235$	−18.6236	−1.21487
$236$	6.43929	0.419162
$237$	0	0
$238$	2.89219	0.187473
$239$	−12.5355	−0.810853	−0.405427	−	0.914128i	$-0.632877\pi$
$239$	−12.5355	−0.810853	−0.405427	+	0.914128i	$0.632877\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−9.60200	−0.617240
$243$	0	0
$244$	−0.364739	−0.0233500
$245$	2.89219	0.184775
$246$	0	0
$247$	11.8060	0.751198
$248$	−0.892186	−0.0566539
$249$	0	0
$250$	−4.72948	−0.299118
$251$	19.7179	1.24458	0.622290	−	0.782787i	$-0.286202\pi$
$251$	19.7179	1.24458	0.622290	+	0.782787i	$0.286202\pi$
$252$	0	0
$253$	−1.39800	−0.0878914
$254$	10.4529	0.655873
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.24333	−0.576583	−0.288291	−	0.957543i	$-0.593087\pi$
$257$	−9.24333	−0.576583	−0.288291	+	0.957543i	$0.593087\pi$
$258$	0	0
$259$	−4.96674	−0.308618
$260$	10.7295	0.665414
$261$	0	0
$262$	−3.41963	−0.211266
$263$	5.78437	0.356680	0.178340	−	0.983969i	$-0.442927\pi$
$263$	5.78437	0.356680	0.178340	+	0.983969i	$0.442927\pi$
$264$	0	0
$265$	−36.8786	−2.26543
$266$	3.18237	0.195124
$267$	0	0
$268$	3.18237	0.194394
$269$	8.90022	0.542656	0.271328	−	0.962487i	$-0.412537\pi$
$269$	8.90022	0.542656	0.271328	+	0.962487i	$0.412537\pi$
$270$	0	0
$271$	3.83729	0.233099	0.116549	−	0.993185i	$-0.462817\pi$
$271$	3.83729	0.233099	0.116549	+	0.993185i	$0.462817\pi$
$272$	2.89219	0.175365
$273$	0	0
$274$	−4.60200	−0.278017
$275$	3.97837	0.239904
$276$	0	0
$277$	−7.63526	−0.458758	−0.229379	−	0.973337i	$-0.573670\pi$
$277$	−7.63526	−0.458758	−0.229379	+	0.973337i	$0.573670\pi$
$278$	10.6766	0.640337
$279$	0	0
$280$	2.89219	0.172841
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	5.94708	0.353517	0.176759	−	0.984254i	$-0.443439\pi$
$283$	5.94708	0.353517	0.176759	+	0.984254i	$0.443439\pi$
$284$	−14.1491	−0.839595
$285$	0	0
$286$	4.38637	0.259372
$287$	−11.0413	−0.651747
$288$	0	0
$289$	−8.63526	−0.507957
$290$	−2.89219	−0.169835
$291$	0	0
$292$	16.5884	0.970763
$293$	15.1159	0.883078	0.441539	−	0.897242i	$-0.354433\pi$
$293$	15.1159	0.883078	0.441539	+	0.897242i	$0.354433\pi$
$294$	0	0
$295$	18.6236	1.08431
$296$	−4.96674	−0.288686
$297$	0	0
$298$	−0.751113	−0.0435108
$299$	−4.38637	−0.253671
$300$	0	0
$301$	5.63526	0.324811
$302$	−3.78437	−0.217766
$303$	0	0
$304$	3.18237	0.182521
$305$	−1.05489	−0.0604030
$306$	0	0
$307$	−12.6685	−0.723031	−0.361515	−	0.932366i	$-0.617740\pi$
$307$	−12.6685	−0.723031	−0.361515	+	0.932366i	$0.617740\pi$
$308$	1.18237	0.0673718
$309$	0	0
$310$	−2.58037	−0.146555
$311$	−20.4609	−1.16023	−0.580116	−	0.814534i	$-0.696993\pi$
$311$	−20.4609	−1.16023	−0.580116	+	0.814534i	$0.696993\pi$
$312$	0	0
$313$	−9.56874	−0.540857	−0.270429	−	0.962740i	$-0.587165\pi$
$313$	−9.56874	−0.540857	−0.270429	+	0.962740i	$0.587165\pi$
$314$	7.78437	0.439298
$315$	0	0
$316$	0.729478	0.0410364
$317$	29.5687	1.66075	0.830373	−	0.557208i	$-0.188127\pi$
$317$	29.5687	1.66075	0.830373	+	0.557208i	$0.188127\pi$
$318$	0	0
$319$	−1.18237	−0.0662000
$320$	2.89219	0.161678
$321$	0	0
$322$	−1.18237	−0.0658909
$323$	9.20400	0.512125
$324$	0	0
$325$	12.4826	0.692408
$326$	−8.51385	−0.471539
$327$	0	0
$328$	−11.0413	−0.609654
$329$	−6.43929	−0.355010
$330$	0	0
$331$	−2.06652	−0.113586	−0.0567930	−	0.998386i	$-0.518088\pi$
$331$	−2.06652	−0.113586	−0.0567930	+	0.998386i	$0.518088\pi$
$332$	6.43929	0.353402
$333$	0	0
$334$	−16.2982	−0.891799
$335$	9.20400	0.502868
$336$	0	0
$337$	9.71785	0.529365	0.264683	−	0.964336i	$-0.414733\pi$
$337$	9.71785	0.529365	0.264683	+	0.964336i	$0.414733\pi$
$338$	0.762737	0.0414874
$339$	0	0
$340$	8.36474	0.453642
$341$	−1.05489	−0.0571257
$342$	0	0
$343$	1.00000	0.0539949
$344$	5.63526	0.303833
$345$	0	0
$346$	−14.4609	−0.777424
$347$	−7.09422	−0.380838	−0.190419	−	0.981703i	$-0.560985\pi$
$347$	−7.09422	−0.380838	−0.190419	+	0.981703i	$0.560985\pi$
$348$	0	0
$349$	−30.4609	−1.63054	−0.815268	−	0.579084i	$-0.803410\pi$
$349$	−30.4609	−1.63054	−0.815268	+	0.579084i	$0.803410\pi$
$350$	3.36474	0.179853
$351$	0	0
$352$	1.18237	0.0630205
$353$	5.56874	0.296394	0.148197	−	0.988958i	$-0.452653\pi$
$353$	5.56874	0.296394	0.148197	+	0.988958i	$0.452653\pi$
$354$	0	0
$355$	−40.9219	−2.17191
$356$	5.25693	0.278616
$357$	0	0
$358$	9.20400	0.486447
$359$	11.5687	0.610575	0.305287	−	0.952260i	$-0.401247\pi$
$359$	11.5687	0.610575	0.305287	+	0.952260i	$0.401247\pi$
$360$	0	0
$361$	−8.87252	−0.466975
$362$	9.40604	0.494370
$363$	0	0
$364$	3.70982	0.194447
$365$	47.9768	2.51122
$366$	0	0
$367$	−12.4609	−0.650455	−0.325228	−	0.945636i	$-0.605441\pi$
$367$	−12.4609	−0.650455	−0.325228	+	0.945636i	$0.605441\pi$
$368$	−1.18237	−0.0616353
$369$	0	0
$370$	−14.3647	−0.746787
$371$	−12.7511	−0.662005
$372$	0	0
$373$	8.83927	0.457680	0.228840	−	0.973464i	$-0.426507\pi$
$373$	8.83927	0.457680	0.228840	+	0.973464i	$0.426507\pi$
$374$	3.41963	0.176825
$375$	0	0
$376$	−6.43929	−0.332081
$377$	−3.70982	−0.191065
$378$	0	0
$379$	7.56874	0.388780	0.194390	−	0.980924i	$-0.437727\pi$
$379$	7.56874	0.388780	0.194390	+	0.980924i	$0.437727\pi$
$380$	9.20400	0.472155
$381$	0	0
$382$	20.1491	1.03092
$383$	−23.1375	−1.18227	−0.591135	−	0.806573i	$-0.701320\pi$
$383$	−23.1375	−1.18227	−0.591135	+	0.806573i	$0.701320\pi$
$384$	0	0
$385$	3.41963	0.174281
$386$	−9.56874	−0.487036
$387$	0	0
$388$	−2.07456	−0.105320
$389$	2.54104	0.128836	0.0644180	−	0.997923i	$-0.479481\pi$
$389$	2.54104	0.128836	0.0644180	+	0.997923i	$0.479481\pi$
$390$	0	0
$391$	−3.41963	−0.172938
$392$	1.00000	0.0505076
$393$	0	0
$394$	16.0826	0.810229
$395$	2.10979	0.106155
$396$	0	0
$397$	−18.5491	−0.930952	−0.465476	−	0.885061i	$-0.654117\pi$
$397$	−18.5491	−0.930952	−0.465476	+	0.885061i	$0.654117\pi$
$398$	−1.54711	−0.0775496
$399$	0	0
$400$	3.36474	0.168237
$401$	22.7295	1.13506	0.567528	−	0.823354i	$-0.307900\pi$
$401$	22.7295	1.13506	0.567528	+	0.823354i	$0.307900\pi$
$402$	0	0
$403$	−3.30985	−0.164875
$404$	−17.5687	−0.874078
$405$	0	0
$406$	−1.00000	−0.0496292
$407$	−5.87252	−0.291090
$408$	0	0
$409$	−17.4060	−0.860673	−0.430337	−	0.902669i	$-0.641605\pi$
$409$	−17.4060	−0.860673	−0.430337	+	0.902669i	$0.641605\pi$
$410$	−31.9335	−1.57708
$411$	0	0
$412$	−1.54711	−0.0762206
$413$	6.43929	0.316857
$414$	0	0
$415$	18.6236	0.914198
$416$	3.70982	0.181889
$417$	0	0
$418$	3.76274	0.184041
$419$	−10.3783	−0.507015	−0.253507	−	0.967333i	$-0.581584\pi$
$419$	−10.3783	−0.507015	−0.253507	+	0.967333i	$0.581584\pi$
$420$	0	0
$421$	−22.6630	−1.10453	−0.552263	−	0.833670i	$-0.686235\pi$
$421$	−22.6630	−1.10453	−0.552263	+	0.833670i	$0.686235\pi$
$422$	26.6630	1.29793
$423$	0	0
$424$	−12.7511	−0.619249
$425$	9.73145	0.472045
$426$	0	0
$427$	−0.364739	−0.0176510
$428$	15.4196	0.745336
$429$	0	0
$430$	16.2982	0.785970
$431$	−5.87252	−0.282870	−0.141435	−	0.989948i	$-0.545172\pi$
$431$	−5.87252	−0.282870	−0.141435	+	0.989948i	$0.545172\pi$
$432$	0	0
$433$	38.7592	1.86265	0.931323	−	0.364194i	$-0.118656\pi$
$433$	38.7592	1.86265	0.931323	+	0.364194i	$0.118656\pi$
$434$	−0.892186	−0.0428263
$435$	0	0
$436$	−10.8786	−0.520990
$437$	−3.76274	−0.179996
$438$	0	0
$439$	27.6297	1.31869	0.659347	−	0.751839i	$-0.270833\pi$
$439$	27.6297	1.31869	0.659347	+	0.751839i	$0.270833\pi$
$440$	3.41963	0.163025
$441$	0	0
$442$	10.7295	0.510349
$443$	−25.1824	−1.19645	−0.598225	−	0.801328i	$-0.704127\pi$
$443$	−25.1824	−1.19645	−0.598225	+	0.801328i	$0.704127\pi$
$444$	0	0
$445$	15.2040	0.720739
$446$	−20.3864	−0.965323
$447$	0	0
$448$	1.00000	0.0472456
$449$	−8.27659	−0.390596	−0.195298	−	0.980744i	$-0.562567\pi$
$449$	−8.27659	−0.390596	−0.195298	+	0.980744i	$0.562567\pi$
$450$	0	0
$451$	−13.0549	−0.614731
$452$	3.54711	0.166842
$453$	0	0
$454$	13.6217	0.639296
$455$	10.7295	0.503006
$456$	0	0
$457$	−7.93904	−0.371373	−0.185686	−	0.982609i	$-0.559451\pi$
$457$	−7.93904	−0.371373	−0.185686	+	0.982609i	$0.559451\pi$
$458$	0.513850	0.0240106
$459$	0	0
$460$	−3.41963	−0.159441
$461$	9.50778	0.442822	0.221411	−	0.975181i	$-0.428934\pi$
$461$	9.50778	0.442822	0.221411	+	0.975181i	$0.428934\pi$
$462$	0	0
$463$	24.6902	1.14745	0.573724	−	0.819048i	$-0.305498\pi$
$463$	24.6902	1.14745	0.573724	+	0.819048i	$0.305498\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	−12.6630	−0.586600
$467$	−22.5139	−1.04182	−0.520908	−	0.853613i	$-0.674407\pi$
$467$	−22.5139	−1.04182	−0.520908	+	0.853613i	$0.674407\pi$
$468$	0	0
$469$	3.18237	0.146948
$470$	−18.6236	−0.859044
$471$	0	0
$472$	6.43929	0.296393
$473$	6.66296	0.306363
$474$	0	0
$475$	10.7078	0.491310
$476$	2.89219	0.132563
$477$	0	0
$478$	−12.5355	−0.573360
$479$	−40.6982	−1.85955	−0.929774	−	0.368131i	$-0.879998\pi$
$479$	−40.6982	−1.85955	−0.929774	+	0.368131i	$0.879998\pi$
$480$	0	0
$481$	−18.4257	−0.840140
$482$	−10.0000	−0.455488
$483$	0	0
$484$	−9.60200	−0.436455
$485$	−6.00000	−0.272446
$486$	0	0
$487$	31.1375	1.41097	0.705487	−	0.708723i	$-0.250728\pi$
$487$	31.1375	1.41097	0.705487	+	0.708723i	$0.250728\pi$
$488$	−0.364739	−0.0165110
$489$	0	0
$490$	2.89219	0.130656
$491$	38.9002	1.75554	0.877771	−	0.479080i	$-0.159030\pi$
$491$	38.9002	1.75554	0.877771	+	0.479080i	$0.159030\pi$
$492$	0	0
$493$	−2.89219	−0.130258
$494$	11.8060	0.531177
$495$	0	0
$496$	−0.892186	−0.0400603
$497$	−14.1491	−0.634674
$498$	0	0
$499$	−23.8060	−1.06570	−0.532852	−	0.846209i	$-0.678880\pi$
$499$	−23.8060	−1.06570	−0.532852	+	0.846209i	$0.678880\pi$
$500$	−4.72948	−0.211509
$501$	0	0
$502$	19.7179	0.880051
$503$	−22.2902	−0.993870	−0.496935	−	0.867788i	$-0.665541\pi$
$503$	−22.2902	−0.993870	−0.496935	+	0.867788i	$0.665541\pi$
$504$	0	0
$505$	−50.8121	−2.26111
$506$	−1.39800	−0.0621486
$507$	0	0
$508$	10.4529	0.463772
$509$	0.351143	0.0155641	0.00778206	−	0.999970i	$-0.497523\pi$
$509$	0.351143	0.0155641	0.00778206	+	0.999970i	$0.497523\pi$
$510$	0	0
$511$	16.5884	0.733828
$512$	1.00000	0.0441942
$513$	0	0
$514$	−9.24333	−0.407706
$515$	−4.47453	−0.197171
$516$	0	0
$517$	−7.61363	−0.334847
$518$	−4.96674	−0.218226
$519$	0	0
$520$	10.7295	0.470519
$521$	−1.35867	−0.0595246	−0.0297623	−	0.999557i	$-0.509475\pi$
$521$	−1.35867	−0.0595246	−0.0297623	+	0.999557i	$0.509475\pi$
$522$	0	0
$523$	29.7708	1.30179	0.650893	−	0.759170i	$-0.274395\pi$
$523$	29.7708	1.30179	0.650893	+	0.759170i	$0.274395\pi$
$524$	−3.41963	−0.149387
$525$	0	0
$526$	5.78437	0.252211
$527$	−2.58037	−0.112403
$528$	0	0
$529$	−21.6020	−0.939217
$530$	−36.8786	−1.60190
$531$	0	0
$532$	3.18237	0.137973
$533$	−40.9612	−1.77423
$534$	0	0
$535$	44.5964	1.92807
$536$	3.18237	0.137457
$537$	0	0
$538$	8.90022	0.383716
$539$	1.18237	0.0509283
$540$	0	0
$541$	−16.1275	−0.693374	−0.346687	−	0.937981i	$-0.612693\pi$
$541$	−16.1275	−0.693374	−0.346687	+	0.937981i	$0.612693\pi$
$542$	3.83729	0.164826
$543$	0	0
$544$	2.89219	0.124001
$545$	−31.4629	−1.34772
$546$	0	0
$547$	23.9551	1.02425	0.512123	−	0.858912i	$-0.328859\pi$
$547$	23.9551	1.02425	0.512123	+	0.858912i	$0.328859\pi$
$548$	−4.60200	−0.196588
$549$	0	0
$550$	3.97837	0.169638
$551$	−3.18237	−0.135574
$552$	0	0
$553$	0.729478	0.0310206
$554$	−7.63526	−0.324391
$555$	0	0
$556$	10.6766	0.452787
$557$	−3.97837	−0.168569	−0.0842844	−	0.996442i	$-0.526860\pi$
$557$	−3.97837	−0.168569	−0.0842844	+	0.996442i	$0.526860\pi$
$558$	0	0
$559$	20.9058	0.884220
$560$	2.89219	0.122217
$561$	0	0
$562$	24.0000	1.01238
$563$	7.71785	0.325269	0.162634	−	0.986686i	$-0.448001\pi$
$563$	7.71785	0.325269	0.162634	+	0.986686i	$0.448001\pi$
$564$	0	0
$565$	10.2589	0.431595
$566$	5.94708	0.249974
$567$	0	0
$568$	−14.1491	−0.593684
$569$	30.5355	1.28011	0.640057	−	0.768327i	$-0.278911\pi$
$569$	30.5355	1.28011	0.640057	+	0.768327i	$0.278911\pi$
$570$	0	0
$571$	−23.8060	−0.996250	−0.498125	−	0.867105i	$-0.665978\pi$
$571$	−23.8060	−0.996250	−0.498125	+	0.867105i	$0.665978\pi$
$572$	4.38637	0.183404
$573$	0	0
$574$	−11.0413	−0.460855
$575$	−3.97837	−0.165909
$576$	0	0
$577$	−17.9415	−0.746915	−0.373458	−	0.927647i	$-0.621828\pi$
$577$	−17.9415	−0.746915	−0.373458	+	0.927647i	$0.621828\pi$
$578$	−8.63526	−0.359179
$579$	0	0
$580$	−2.89219	−0.120091
$581$	6.43929	0.267147
$582$	0	0
$583$	−15.0765	−0.624406
$584$	16.5884	0.686433
$585$	0	0
$586$	15.1159	0.624430
$587$	−4.16271	−0.171813	−0.0859067	−	0.996303i	$-0.527379\pi$
$587$	−4.16271	−0.171813	−0.0859067	+	0.996303i	$0.527379\pi$
$588$	0	0
$589$	−2.83927	−0.116990
$590$	18.6236	0.766723
$591$	0	0
$592$	−4.96674	−0.204132
$593$	−37.3748	−1.53480	−0.767399	−	0.641170i	$-0.778449\pi$
$593$	−37.3748	−1.53480	−0.767399	+	0.641170i	$0.778449\pi$
$594$	0	0
$595$	8.36474	0.342921
$596$	−0.751113	−0.0307668
$597$	0	0
$598$	−4.38637	−0.179372
$599$	−34.0826	−1.39258	−0.696289	−	0.717762i	$-0.745167\pi$
$599$	−34.0826	−1.39258	−0.696289	+	0.717762i	$0.745167\pi$
$600$	0	0
$601$	3.27856	0.133735	0.0668676	−	0.997762i	$-0.478699\pi$
$601$	3.27856	0.133735	0.0668676	+	0.997762i	$0.478699\pi$
$602$	5.63526	0.229676
$603$	0	0
$604$	−3.78437	−0.153984
$605$	−27.7708	−1.12904
$606$	0	0
$607$	33.3667	1.35431	0.677157	−	0.735839i	$-0.263212\pi$
$607$	33.3667	1.35431	0.677157	+	0.735839i	$0.263212\pi$
$608$	3.18237	0.129062
$609$	0	0
$610$	−1.05489	−0.0427114
$611$	−23.8886	−0.966429
$612$	0	0
$613$	2.87859	0.116265	0.0581326	−	0.998309i	$-0.481485\pi$
$613$	2.87859	0.116265	0.0581326	+	0.998309i	$0.481485\pi$
$614$	−12.6685	−0.511260
$615$	0	0
$616$	1.18237	0.0476390
$617$	−12.8393	−0.516889	−0.258445	−	0.966026i	$-0.583210\pi$
$617$	−12.8393	−0.516889	−0.258445	+	0.966026i	$0.583210\pi$
$618$	0	0
$619$	−13.5471	−0.544504	−0.272252	−	0.962226i	$-0.587768\pi$
$619$	−13.5471	−0.544504	−0.272252	+	0.962226i	$0.587768\pi$
$620$	−2.58037	−0.103630
$621$	0	0
$622$	−20.4609	−0.820409
$623$	5.25693	0.210614
$624$	0	0
$625$	−30.5022	−1.22009
$626$	−9.56874	−0.382444
$627$	0	0
$628$	7.78437	0.310630
$629$	−14.3647	−0.572760
$630$	0	0
$631$	24.2982	0.967297	0.483648	−	0.875262i	$-0.339311\pi$
$631$	24.2982	0.967297	0.483648	+	0.875262i	$0.339311\pi$
$632$	0.729478	0.0290171
$633$	0	0
$634$	29.5687	1.17432
$635$	30.2317	1.19971
$636$	0	0
$637$	3.70982	0.146988
$638$	−1.18237	−0.0468105
$639$	0	0
$640$	2.89219	0.114324
$641$	−21.4357	−0.846660	−0.423330	−	0.905976i	$-0.639139\pi$
$641$	−21.4357	−0.846660	−0.423330	+	0.905976i	$0.639139\pi$
$642$	0	0
$643$	−4.75915	−0.187683	−0.0938413	−	0.995587i	$-0.529915\pi$
$643$	−4.75915	−0.187683	−0.0938413	+	0.995587i	$0.529915\pi$
$644$	−1.18237	−0.0465919
$645$	0	0
$646$	9.20400	0.362127
$647$	−24.4473	−0.961124	−0.480562	−	0.876961i	$-0.659567\pi$
$647$	−24.4473	−0.961124	−0.480562	+	0.876961i	$0.659567\pi$
$648$	0	0
$649$	7.61363	0.298861
$650$	12.4826	0.489606
$651$	0	0
$652$	−8.51385	−0.333428
$653$	−48.6630	−1.90433	−0.952164	−	0.305586i	$-0.901148\pi$
$653$	−48.6630	−1.90433	−0.952164	+	0.305586i	$0.901148\pi$
$654$	0	0
$655$	−9.89021	−0.386443
$656$	−11.0413	−0.431090
$657$	0	0
$658$	−6.43929	−0.251030
$659$	49.7179	1.93673	0.968366	−	0.249533i	$-0.0802771\pi$
$659$	49.7179	1.93673	0.968366	+	0.249533i	$0.0802771\pi$
$660$	0	0
$661$	−23.2786	−0.905431	−0.452716	−	0.891655i	$-0.649545\pi$
$661$	−23.2786	−0.905431	−0.452716	+	0.891655i	$0.649545\pi$
$662$	−2.06652	−0.0803175
$663$	0	0
$664$	6.43929	0.249893
$665$	9.20400	0.356916
$666$	0	0
$667$	1.18237	0.0457815
$668$	−16.2982	−0.630597
$669$	0	0
$670$	9.20400	0.355582
$671$	−0.431256	−0.0166485
$672$	0	0
$673$	−5.14305	−0.198250	−0.0991249	−	0.995075i	$-0.531604\pi$
$673$	−5.14305	−0.198250	−0.0991249	+	0.995075i	$0.531604\pi$
$674$	9.71785	0.374318
$675$	0	0
$676$	0.762737	0.0293360
$677$	−3.72341	−0.143102	−0.0715512	−	0.997437i	$-0.522795\pi$
$677$	−3.72341	−0.143102	−0.0715512	+	0.997437i	$0.522795\pi$
$678$	0	0
$679$	−2.07456	−0.0796141
$680$	8.36474	0.320773
$681$	0	0
$682$	−1.05489	−0.0403940
$683$	9.20400	0.352181	0.176091	−	0.984374i	$-0.443655\pi$
$683$	9.20400	0.352181	0.176091	+	0.984374i	$0.443655\pi$
$684$	0	0
$685$	−13.3098	−0.508543
$686$	1.00000	0.0381802
$687$	0	0
$688$	5.63526	0.214842
$689$	−47.3043	−1.80215
$690$	0	0
$691$	−41.8140	−1.59068	−0.795341	−	0.606163i	$-0.792708\pi$
$691$	−41.8140	−1.59068	−0.795341	+	0.606163i	$0.792708\pi$
$692$	−14.4609	−0.549722
$693$	0	0
$694$	−7.09422	−0.269293
$695$	30.8786	1.17129
$696$	0	0
$697$	−31.9335	−1.20957
$698$	−30.4609	−1.15296
$699$	0	0
$700$	3.36474	0.127175
$701$	31.7572	1.19945	0.599726	−	0.800205i	$-0.295276\pi$
$701$	31.7572	1.19945	0.599726	+	0.800205i	$0.295276\pi$
$702$	0	0
$703$	−15.8060	−0.596135
$704$	1.18237	0.0445622
$705$	0	0
$706$	5.56874	0.209582
$707$	−17.5687	−0.660741
$708$	0	0
$709$	−27.3531	−1.02727	−0.513634	−	0.858009i	$-0.671701\pi$
$709$	−27.3531	−1.02727	−0.513634	+	0.858009i	$0.671701\pi$
$710$	−40.9219	−1.53577
$711$	0	0
$712$	5.25693	0.197012
$713$	1.05489	0.0395061
$714$	0	0
$715$	12.6862	0.474437
$716$	9.20400	0.343970
$717$	0	0
$718$	11.5687	0.431742
$719$	−18.2156	−0.679328	−0.339664	−	0.940547i	$-0.610313\pi$
$719$	−18.2156	−0.679328	−0.339664	+	0.940547i	$0.610313\pi$
$720$	0	0
$721$	−1.54711	−0.0576173
$722$	−8.87252	−0.330201
$723$	0	0
$724$	9.40604	0.349573
$725$	−3.36474	−0.124963
$726$	0	0
$727$	−53.2065	−1.97332	−0.986660	−	0.162797i	$-0.947949\pi$
$727$	−53.2065	−1.97332	−0.986660	+	0.162797i	$0.947949\pi$
$728$	3.70982	0.137495
$729$	0	0
$730$	47.9768	1.77570
$731$	16.2982	0.602812
$732$	0	0
$733$	40.3647	1.49091	0.745453	−	0.666558i	$-0.232233\pi$
$733$	40.3647	1.49091	0.745453	+	0.666558i	$0.232233\pi$
$734$	−12.4609	−0.459941
$735$	0	0
$736$	−1.18237	−0.0435827
$737$	3.76274	0.138602
$738$	0	0
$739$	−15.5687	−0.572705	−0.286353	−	0.958124i	$-0.592443\pi$
$739$	−15.5687	−0.572705	−0.286353	+	0.958124i	$0.592443\pi$
$740$	−14.3647	−0.528058
$741$	0	0
$742$	−12.7511	−0.468108
$743$	1.48615	0.0545216	0.0272608	−	0.999628i	$-0.491322\pi$
$743$	1.48615	0.0545216	0.0272608	+	0.999628i	$0.491322\pi$
$744$	0	0
$745$	−2.17236	−0.0795891
$746$	8.83927	0.323628
$747$	0	0
$748$	3.41963	0.125034
$749$	15.4196	0.563421
$750$	0	0
$751$	15.8060	0.576769	0.288385	−	0.957515i	$-0.406882\pi$
$751$	15.8060	0.576769	0.288385	+	0.957515i	$0.406882\pi$
$752$	−6.43929	−0.234817
$753$	0	0
$754$	−3.70982	−0.135104
$755$	−10.9451	−0.398333
$756$	0	0
$757$	−17.5022	−0.636129	−0.318065	−	0.948069i	$-0.603033\pi$
$757$	−17.5022	−0.636129	−0.318065	+	0.948069i	$0.603033\pi$
$758$	7.56874	0.274909
$759$	0	0
$760$	9.20400	0.333864
$761$	45.0765	1.63402	0.817011	−	0.576621i	$-0.195629\pi$
$761$	45.0765	1.63402	0.817011	+	0.576621i	$0.195629\pi$
$762$	0	0
$763$	−10.8786	−0.393831
$764$	20.1491	0.728969
$765$	0	0
$766$	−23.1375	−0.835991
$767$	23.8886	0.862567
$768$	0	0
$769$	43.9199	1.58379	0.791896	−	0.610656i	$-0.209094\pi$
$769$	43.9199	1.58379	0.791896	+	0.610656i	$0.209094\pi$
$770$	3.41963	0.123235
$771$	0	0
$772$	−9.56874	−0.344387
$773$	−26.3254	−0.946859	−0.473430	−	0.880832i	$-0.656984\pi$
$773$	−26.3254	−0.946859	−0.473430	+	0.880832i	$0.656984\pi$
$774$	0	0
$775$	−3.00197	−0.107834
$776$	−2.07456	−0.0744722
$777$	0	0
$778$	2.54104	0.0911008
$779$	−35.1375	−1.25893
$780$	0	0
$781$	−16.7295	−0.598628
$782$	−3.41963	−0.122286
$783$	0	0
$784$	1.00000	0.0357143
$785$	22.5139	0.803554
$786$	0	0
$787$	−31.5551	−1.12482	−0.562410	−	0.826859i	$-0.690126\pi$
$787$	−31.5551	−1.12482	−0.562410	+	0.826859i	$0.690126\pi$
$788$	16.0826	0.572919
$789$	0	0
$790$	2.10979	0.0750628
$791$	3.54711	0.126121
$792$	0	0
$793$	−1.35312	−0.0480505
$794$	−18.5491	−0.658282
$795$	0	0
$796$	−1.54711	−0.0548358
$797$	−11.7844	−0.417424	−0.208712	−	0.977977i	$-0.566927\pi$
$797$	−11.7844	−0.417424	−0.208712	+	0.977977i	$0.566927\pi$
$798$	0	0
$799$	−18.6236	−0.658857
$800$	3.36474	0.118961
$801$	0	0
$802$	22.7295	0.802606
$803$	19.6136	0.692150
$804$	0	0
$805$	−3.41963	−0.120526
$806$	−3.30985	−0.116584
$807$	0	0
$808$	−17.5687	−0.618066
$809$	−53.7611	−1.89014	−0.945070	−	0.326867i	$-0.894007\pi$
$809$	−53.7611	−1.89014	−0.945070	+	0.326867i	$0.894007\pi$
$810$	0	0
$811$	15.2297	0.534788	0.267394	−	0.963587i	$-0.413837\pi$
$811$	15.2297	0.534788	0.267394	+	0.963587i	$0.413837\pi$
$812$	−1.00000	−0.0350931
$813$	0	0
$814$	−5.87252	−0.205832
$815$	−24.6236	−0.862528
$816$	0	0
$817$	17.9335	0.627413
$818$	−17.4060	−0.608588
$819$	0	0
$820$	−31.9335	−1.11517
$821$	4.15467	0.144999	0.0724995	−	0.997368i	$-0.476902\pi$
$821$	4.15467	0.144999	0.0724995	+	0.997368i	$0.476902\pi$
$822$	0	0
$823$	3.35867	0.117076	0.0585380	−	0.998285i	$-0.481356\pi$
$823$	3.35867	0.117076	0.0585380	+	0.998285i	$0.481356\pi$
$824$	−1.54711	−0.0538961
$825$	0	0
$826$	6.43929	0.224052
$827$	17.0493	0.592863	0.296432	−	0.955054i	$-0.404203\pi$
$827$	17.0493	0.592863	0.296432	+	0.955054i	$0.404203\pi$
$828$	0	0
$829$	10.5964	0.368030	0.184015	−	0.982923i	$-0.441091\pi$
$829$	10.5964	0.368030	0.184015	+	0.982923i	$0.441091\pi$
$830$	18.6236	0.646436
$831$	0	0
$832$	3.70982	0.128615
$833$	2.89219	0.100208
$834$	0	0
$835$	−47.1375	−1.63126
$836$	3.76274	0.130137
$837$	0	0
$838$	−10.3783	−0.358514
$839$	−49.8140	−1.71977	−0.859886	−	0.510486i	$-0.829465\pi$
$839$	−49.8140	−1.71977	−0.859886	+	0.510486i	$0.829465\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−22.6630	−0.781017
$843$	0	0
$844$	26.6630	0.917777
$845$	2.20598	0.0758879
$846$	0	0
$847$	−9.60200	−0.329929
$848$	−12.7511	−0.437875
$849$	0	0
$850$	9.73145	0.333786
$851$	5.87252	0.201308
$852$	0	0
$853$	−31.2040	−1.06840	−0.534202	−	0.845357i	$-0.679388\pi$
$853$	−31.2040	−1.06840	−0.534202	+	0.845357i	$0.679388\pi$
$854$	−0.364739	−0.0124811
$855$	0	0
$856$	15.4196	0.527032
$857$	44.6453	1.52505	0.762527	−	0.646957i	$-0.223959\pi$
$857$	44.6453	1.52505	0.762527	+	0.646957i	$0.223959\pi$
$858$	0	0
$859$	20.7118	0.706677	0.353339	−	0.935496i	$-0.385046\pi$
$859$	20.7118	0.706677	0.353339	+	0.935496i	$0.385046\pi$
$860$	16.2982	0.555765
$861$	0	0
$862$	−5.87252	−0.200019
$863$	12.5748	0.428051	0.214026	−	0.976828i	$-0.431342\pi$
$863$	12.5748	0.428051	0.214026	+	0.976828i	$0.431342\pi$
$864$	0	0
$865$	−41.8237	−1.42205
$866$	38.7592	1.31709
$867$	0	0
$868$	−0.892186	−0.0302828
$869$	0.862513	0.0292587
$870$	0	0
$871$	11.8060	0.400031
$872$	−10.8786	−0.368396
$873$	0	0
$874$	−3.76274	−0.127276
$875$	−4.72948	−0.159886
$876$	0	0
$877$	47.0438	1.58856	0.794278	−	0.607555i	$-0.207850\pi$
$877$	47.0438	1.58856	0.794278	+	0.607555i	$0.207850\pi$
$878$	27.6297	0.932457
$879$	0	0
$880$	3.41963	0.115276
$881$	−44.4216	−1.49660	−0.748301	−	0.663359i	$-0.769130\pi$
$881$	−44.4216	−1.49660	−0.748301	+	0.663359i	$0.769130\pi$
$882$	0	0
$883$	−6.27659	−0.211224	−0.105612	−	0.994407i	$-0.533680\pi$
$883$	−6.27659	−0.211224	−0.105612	+	0.994407i	$0.533680\pi$
$884$	10.7295	0.360871
$885$	0	0
$886$	−25.1824	−0.846018
$887$	10.0352	0.336950	0.168475	−	0.985706i	$-0.446116\pi$
$887$	10.0352	0.336950	0.168475	+	0.985706i	$0.446116\pi$
$888$	0	0
$889$	10.4529	0.350579
$890$	15.2040	0.509639
$891$	0	0
$892$	−20.3864	−0.682586
$893$	−20.4922	−0.685746
$894$	0	0
$895$	26.6197	0.889798
$896$	1.00000	0.0334077
$897$	0	0
$898$	−8.27659	−0.276193
$899$	0.892186	0.0297561
$900$	0	0
$901$	−36.8786	−1.22860
$902$	−13.0549	−0.434680
$903$	0	0
$904$	3.54711	0.117975
$905$	27.2040	0.904292
$906$	0	0
$907$	38.1924	1.26816	0.634079	−	0.773269i	$-0.281380\pi$
$907$	38.1924	1.26816	0.634079	+	0.773269i	$0.281380\pi$
$908$	13.6217	0.452051
$909$	0	0
$910$	10.7295	0.355679
$911$	−25.3259	−0.839085	−0.419543	−	0.907736i	$-0.637810\pi$
$911$	−25.3259	−0.839085	−0.419543	+	0.907736i	$0.637810\pi$
$912$	0	0
$913$	7.61363	0.251974
$914$	−7.93904	−0.262600
$915$	0	0
$916$	0.513850	0.0169781
$917$	−3.41963	−0.112926
$918$	0	0
$919$	15.0549	0.496615	0.248308	−	0.968681i	$-0.420126\pi$
$919$	15.0549	0.496615	0.248308	+	0.968681i	$0.420126\pi$
$920$	−3.41963	−0.112742
$921$	0	0
$922$	9.50778	0.313122
$923$	−52.4906	−1.72775
$924$	0	0
$925$	−16.7118	−0.549480
$926$	24.6902	0.811369
$927$	0	0
$928$	−1.00000	−0.0328266
$929$	−1.27052	−0.0416845	−0.0208422	−	0.999783i	$-0.506635\pi$
$929$	−1.27052	−0.0416845	−0.0208422	+	0.999783i	$0.506635\pi$
$930$	0	0
$931$	3.18237	0.104298
$932$	−12.6630	−0.414789
$933$	0	0
$934$	−22.5139	−0.736676
$935$	9.89021	0.323445
$936$	0	0
$937$	−43.0277	−1.40565	−0.702827	−	0.711361i	$-0.748079\pi$
$937$	−43.0277	−1.40565	−0.702827	+	0.711361i	$0.748079\pi$
$938$	3.18237	0.103908
$939$	0	0
$940$	−18.6236	−0.607436
$941$	−15.3234	−0.499530	−0.249765	−	0.968306i	$-0.580353\pi$
$941$	−15.3234	−0.499530	−0.249765	+	0.968306i	$0.580353\pi$
$942$	0	0
$943$	13.0549	0.425126
$944$	6.43929	0.209581
$945$	0	0
$946$	6.66296	0.216632
$947$	7.27052	0.236260	0.118130	−	0.992998i	$-0.462310\pi$
$947$	7.27052	0.236260	0.118130	+	0.992998i	$0.462310\pi$
$948$	0	0
$949$	61.5399	1.99767
$950$	10.7078	0.347408
$951$	0	0
$952$	2.89219	0.0937363
$953$	42.2317	1.36802	0.684010	−	0.729473i	$-0.260235\pi$
$953$	42.2317	1.36802	0.684010	+	0.729473i	$0.260235\pi$
$954$	0	0
$955$	58.2750	1.88573
$956$	−12.5355	−0.405427
$957$	0	0
$958$	−40.6982	−1.31490
$959$	−4.60200	−0.148606
$960$	0	0
$961$	−30.2040	−0.974323
$962$	−18.4257	−0.594068
$963$	0	0
$964$	−10.0000	−0.322078
$965$	−27.6746	−0.890876
$966$	0	0
$967$	−13.7395	−0.441832	−0.220916	−	0.975293i	$-0.570905\pi$
$967$	−13.7395	−0.441832	−0.220916	+	0.975293i	$0.570905\pi$
$968$	−9.60200	−0.308620
$969$	0	0
$970$	−6.00000	−0.192648
$971$	13.9174	0.446631	0.223315	−	0.974746i	$-0.428312\pi$
$971$	13.9174	0.446631	0.223315	+	0.974746i	$0.428312\pi$
$972$	0	0
$973$	10.6766	0.342275
$974$	31.1375	0.997709
$975$	0	0
$976$	−0.364739	−0.0116750
$977$	7.52547	0.240761	0.120381	−	0.992728i	$-0.461589\pi$
$977$	7.52547	0.240761	0.120381	+	0.992728i	$0.461589\pi$
$978$	0	0
$979$	6.21563	0.198652
$980$	2.89219	0.0923875
$981$	0	0
$982$	38.9002	1.24136
$983$	−42.9748	−1.37068	−0.685341	−	0.728222i	$-0.740347\pi$
$983$	−42.9748	−1.37068	−0.685341	+	0.728222i	$0.740347\pi$
$984$	0	0
$985$	46.5139	1.48205
$986$	−2.89219	−0.0921060
$987$	0	0
$988$	11.8060	0.375599
$989$	−6.66296	−0.211870
$990$	0	0
$991$	14.6630	0.465784	0.232892	−	0.972503i	$-0.425181\pi$
$991$	14.6630	0.465784	0.232892	+	0.972503i	$0.425181\pi$
$992$	−0.892186	−0.0283269
$993$	0	0
$994$	−14.1491	−0.448783
$995$	−4.47453	−0.141852
$996$	0	0
$997$	23.0277	0.729295	0.364647	−	0.931146i	$-0.381190\pi$
$997$	23.0277	0.729295	0.364647	+	0.931146i	$0.381190\pi$
$998$	−23.8060	−0.753566
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3654.2.a.bd.1.3		3
3.2	odd	2		1218.2.a.q.1.1	✓	3
12.11	even	2		9744.2.a.bk.1.1		3
21.20	even	2		8526.2.a.bm.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1218.2.a.q.1.1	✓	3	3.2	odd	2
3654.2.a.bd.1.3		3	1.1	even	1	trivial
8526.2.a.bm.1.3		3	21.20	even	2
9744.2.a.bk.1.1		3	12.11	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:	\( N \)	\(=\)	\( 3654 = 2 \cdot 3^{2} \cdot 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3654.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.89219 q^{5} +1.00000 q^{7} +1.00000 q^{8} +2.89219 q^{10} +1.18237 q^{11} +3.70982 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.89219 q^{17} +3.18237 q^{19} +2.89219 q^{20} +1.18237 q^{22} -1.18237 q^{23} +3.36474 q^{25} +3.70982 q^{26} +1.00000 q^{28} -1.00000 q^{29} -0.892186 q^{31} +1.00000 q^{32} +2.89219 q^{34} +2.89219 q^{35} -4.96674 q^{37} +3.18237 q^{38} +2.89219 q^{40} -11.0413 q^{41} +5.63526 q^{43} +1.18237 q^{44} -1.18237 q^{46} -6.43929 q^{47} +1.00000 q^{49} +3.36474 q^{50} +3.70982 q^{52} -12.7511 q^{53} +3.41963 q^{55} +1.00000 q^{56} -1.00000 q^{58} +6.43929 q^{59} -0.364739 q^{61} -0.892186 q^{62} +1.00000 q^{64} +10.7295 q^{65} +3.18237 q^{67} +2.89219 q^{68} +2.89219 q^{70} -14.1491 q^{71} +16.5884 q^{73} -4.96674 q^{74} +3.18237 q^{76} +1.18237 q^{77} +0.729478 q^{79} +2.89219 q^{80} -11.0413 q^{82} +6.43929 q^{83} +8.36474 q^{85} +5.63526 q^{86} +1.18237 q^{88} +5.25693 q^{89} +3.70982 q^{91} -1.18237 q^{92} -6.43929 q^{94} +9.20400 q^{95} -2.07456 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{7} + 3 q^{8} - 2 q^{10} + 3 q^{11} + q^{13} + 3 q^{14} + 3 q^{16} - 2 q^{17} + 9 q^{19} - 2 q^{20} + 3 q^{22} - 3 q^{23} + 9 q^{25} + q^{26} + 3 q^{28} - 3 q^{29}+ \cdots + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^7 + 3 * q^8 - 2 * q^10 + 3 * q^11 + q^13 + 3 * q^14 + 3 * q^16 - 2 * q^17 + 9 * q^19 - 2 * q^20 + 3 * q^22 - 3 * q^23 + 9 * q^25 + q^26 + 3 * q^28 - 3 * q^29 + 8 * q^31 + 3 * q^32 - 2 * q^34 - 2 * q^35 + 7 * q^37 + 9 * q^38 - 2 * q^40 + 18 * q^43 + 3 * q^44 - 3 * q^46 - 7 * q^47 + 3 * q^49 + 9 * q^50 + q^52 + 5 * q^53 - 10 * q^55 + 3 * q^56 - 3 * q^58 + 7 * q^59 + 8 * q^62 + 3 * q^64 + 30 * q^65 + 9 * q^67 - 2 * q^68 - 2 * q^70 - 20 * q^71 + 15 * q^73 + 7 * q^74 + 9 * q^76 + 3 * q^77 - 2 * q^80 + 7 * q^83 + 24 * q^85 + 18 * q^86 + 3 * q^88 + 4 * q^89 + q^91 - 3 * q^92 - 7 * q^94 - 14 * q^95 + 5 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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