LMFDB - Embedded newform 3645.2.a.l.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3645,2,Mod(1,3645)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3645, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3645.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3645 = 3^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3645.a (trivial)

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:	yes
Analytic conductor:	$29.1054715368$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	3645.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.86636 q^{2} +1.48328 q^{4} +1.00000 q^{5} +3.47631 q^{7} -0.964379 q^{8} +O(q^{10})$	\(q+1.86636 q^{2} +1.48328 q^{4} +1.00000 q^{5} +3.47631 q^{7} -0.964379 q^{8} +1.86636 q^{10} +3.42573 q^{11} -2.99117 q^{13} +6.48802 q^{14} -4.76644 q^{16} +1.83001 q^{17} -5.66300 q^{19} +1.48328 q^{20} +6.39363 q^{22} +8.50448 q^{23} +1.00000 q^{25} -5.58258 q^{26} +5.15634 q^{28} +3.54878 q^{29} +3.62762 q^{31} -6.96711 q^{32} +3.41545 q^{34} +3.47631 q^{35} +8.25062 q^{37} -10.5692 q^{38} -0.964379 q^{40} +4.56318 q^{41} +6.81253 q^{43} +5.08133 q^{44} +15.8724 q^{46} -4.31388 q^{47} +5.08470 q^{49} +1.86636 q^{50} -4.43675 q^{52} -6.55503 q^{53} +3.42573 q^{55} -3.35248 q^{56} +6.62328 q^{58} +3.26003 q^{59} +3.39466 q^{61} +6.77043 q^{62} -3.47022 q^{64} -2.99117 q^{65} -4.16593 q^{67} +2.71443 q^{68} +6.48802 q^{70} -8.70813 q^{71} -7.94401 q^{73} +15.3986 q^{74} -8.39983 q^{76} +11.9089 q^{77} +14.6559 q^{79} -4.76644 q^{80} +8.51652 q^{82} -14.4374 q^{83} +1.83001 q^{85} +12.7146 q^{86} -3.30370 q^{88} -3.57607 q^{89} -10.3982 q^{91} +12.6145 q^{92} -8.05123 q^{94} -5.66300 q^{95} -3.66068 q^{97} +9.48986 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 3 q^{2} + 27 q^{4} + 21 q^{5} + 12 q^{7} + 9 q^{8}+O(q^{10}) $ 21 * q + 3 * q^2 + 27 * q^4 + 21 * q^5 + 12 * q^7 + 9 * q^8	$ 21 q + 3 q^{2} + 27 q^{4} + 21 q^{5} + 12 q^{7} + 9 q^{8} + 3 q^{10} + 12 q^{13} + 39 q^{16} + 12 q^{17} + 24 q^{19} + 27 q^{20} + 18 q^{22} + 18 q^{23} + 21 q^{25} - 9 q^{26} + 30 q^{28} - 3 q^{29} + 24 q^{31} + 6 q^{32} + 18 q^{34} + 12 q^{35} + 24 q^{37} + 18 q^{38} + 9 q^{40} - 9 q^{41} + 42 q^{43} - 12 q^{44} + 30 q^{46} + 21 q^{47} + 39 q^{49} + 3 q^{50} + 36 q^{52} + 18 q^{53} - 30 q^{56} + 30 q^{58} - 6 q^{59} + 33 q^{61} + 18 q^{62} + 63 q^{64} + 12 q^{65} + 63 q^{67} + 36 q^{68} - 12 q^{71} + 39 q^{73} - 21 q^{74} + 48 q^{76} + 9 q^{77} + 48 q^{79} + 39 q^{80} + 24 q^{82} + 33 q^{83} + 12 q^{85} - 69 q^{86} + 54 q^{88} - 9 q^{89} + 69 q^{91} + 21 q^{92} + 30 q^{94} + 24 q^{95} + 30 q^{97} + 15 q^{98}+O(q^{100}) $ 21 * q + 3 * q^2 + 27 * q^4 + 21 * q^5 + 12 * q^7 + 9 * q^8 + 3 * q^10 + 12 * q^13 + 39 * q^16 + 12 * q^17 + 24 * q^19 + 27 * q^20 + 18 * q^22 + 18 * q^23 + 21 * q^25 - 9 * q^26 + 30 * q^28 - 3 * q^29 + 24 * q^31 + 6 * q^32 + 18 * q^34 + 12 * q^35 + 24 * q^37 + 18 * q^38 + 9 * q^40 - 9 * q^41 + 42 * q^43 - 12 * q^44 + 30 * q^46 + 21 * q^47 + 39 * q^49 + 3 * q^50 + 36 * q^52 + 18 * q^53 - 30 * q^56 + 30 * q^58 - 6 * q^59 + 33 * q^61 + 18 * q^62 + 63 * q^64 + 12 * q^65 + 63 * q^67 + 36 * q^68 - 12 * q^71 + 39 * q^73 - 21 * q^74 + 48 * q^76 + 9 * q^77 + 48 * q^79 + 39 * q^80 + 24 * q^82 + 33 * q^83 + 12 * q^85 - 69 * q^86 + 54 * q^88 - 9 * q^89 + 69 * q^91 + 21 * q^92 + 30 * q^94 + 24 * q^95 + 30 * q^97 + 15 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.86636	1.31971	0.659856	−	0.751392i	$-0.270617\pi$
$2$	1.86636	1.31971	0.659856	+	0.751392i	$0.270617\pi$
$3$	0	0
$3$	0	0
$4$	1.48328	0.741641
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.47631	1.31392	0.656960	−	0.753925i	$-0.271842\pi$
$7$	3.47631	1.31392	0.656960	+	0.753925i	$0.271842\pi$
$8$	−0.964379	−0.340960
$9$	0	0
$10$	1.86636	0.590193
$11$	3.42573	1.03290	0.516449	−	0.856318i	$-0.327254\pi$
$11$	3.42573	1.03290	0.516449	+	0.856318i	$0.327254\pi$
$12$	0	0
$13$	−2.99117	−0.829601	−0.414800	−	0.909912i	$-0.636149\pi$
$13$	−2.99117	−0.829601	−0.414800	+	0.909912i	$0.636149\pi$
$14$	6.48802	1.73400
$15$	0	0
$16$	−4.76644	−1.19161
$17$	1.83001	0.443843	0.221922	−	0.975064i	$-0.428767\pi$
$17$	1.83001	0.443843	0.221922	+	0.975064i	$0.428767\pi$
$18$	0	0
$19$	−5.66300	−1.29918	−0.649591	−	0.760284i	$-0.725060\pi$
$19$	−5.66300	−1.29918	−0.649591	+	0.760284i	$0.725060\pi$
$20$	1.48328	0.331672
$21$	0	0
$22$	6.39363	1.36313
$23$	8.50448	1.77331	0.886653	−	0.462435i	$-0.153024\pi$
$23$	8.50448	1.77331	0.886653	+	0.462435i	$0.153024\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−5.58258	−1.09483
$27$	0	0
$28$	5.15634	0.974457
$29$	3.54878	0.658992	0.329496	−	0.944157i	$-0.393121\pi$
$29$	3.54878	0.658992	0.329496	+	0.944157i	$0.393121\pi$
$30$	0	0
$31$	3.62762	0.651540	0.325770	−	0.945449i	$-0.394377\pi$
$31$	3.62762	0.651540	0.325770	+	0.945449i	$0.394377\pi$
$32$	−6.96711	−1.23162
$33$	0	0
$34$	3.41545	0.585746
$35$	3.47631	0.587603
$36$	0	0
$37$	8.25062	1.35639	0.678197	−	0.734880i	$-0.262762\pi$
$37$	8.25062	1.35639	0.678197	+	0.734880i	$0.262762\pi$
$38$	−10.5692	−1.71455
$39$	0	0
$40$	−0.964379	−0.152482
$41$	4.56318	0.712649	0.356325	−	0.934362i	$-0.384030\pi$
$41$	4.56318	0.712649	0.356325	+	0.934362i	$0.384030\pi$
$42$	0	0
$43$	6.81253	1.03890	0.519450	−	0.854501i	$-0.326137\pi$
$43$	6.81253	1.03890	0.519450	+	0.854501i	$0.326137\pi$
$44$	5.08133	0.766039
$45$	0	0
$46$	15.8724	2.34025
$47$	−4.31388	−0.629244	−0.314622	−	0.949217i	$-0.601878\pi$
$47$	−4.31388	−0.629244	−0.314622	+	0.949217i	$0.601878\pi$
$48$	0	0
$49$	5.08470	0.726386
$50$	1.86636	0.263942
$51$	0	0
$52$	−4.43675	−0.615266
$53$	−6.55503	−0.900403	−0.450201	−	0.892927i	$-0.648648\pi$
$53$	−6.55503	−0.900403	−0.450201	+	0.892927i	$0.648648\pi$
$54$	0	0
$55$	3.42573	0.461926
$56$	−3.35248	−0.447994
$57$	0	0
$58$	6.62328	0.869679
$59$	3.26003	0.424419	0.212210	−	0.977224i	$-0.431934\pi$
$59$	3.26003	0.424419	0.212210	+	0.977224i	$0.431934\pi$
$60$	0	0
$61$	3.39466	0.434642	0.217321	−	0.976100i	$-0.430268\pi$
$61$	3.39466	0.434642	0.217321	+	0.976100i	$0.430268\pi$
$62$	6.77043	0.859845
$63$	0	0
$64$	−3.47022	−0.433778
$65$	−2.99117	−0.371009
$66$	0	0
$67$	−4.16593	−0.508949	−0.254475	−	0.967079i	$-0.581903\pi$
$67$	−4.16593	−0.508949	−0.254475	+	0.967079i	$0.581903\pi$
$68$	2.71443	0.329172
$69$	0	0
$70$	6.48802	0.775467
$71$	−8.70813	−1.03346	−0.516732	−	0.856147i	$-0.672852\pi$
$71$	−8.70813	−1.03346	−0.516732	+	0.856147i	$0.672852\pi$
$72$	0	0
$73$	−7.94401	−0.929776	−0.464888	−	0.885370i	$-0.653905\pi$
$73$	−7.94401	−0.929776	−0.464888	+	0.885370i	$0.653905\pi$
$74$	15.3986	1.79005
$75$	0	0
$76$	−8.39983	−0.963527
$77$	11.9089	1.35714
$78$	0	0
$79$	14.6559	1.64892	0.824460	−	0.565921i	$-0.191479\pi$
$79$	14.6559	1.64892	0.824460	+	0.565921i	$0.191479\pi$
$80$	−4.76644	−0.532904
$81$	0	0
$82$	8.51652	0.940492
$83$	−14.4374	−1.58471	−0.792353	−	0.610062i	$-0.791144\pi$
$83$	−14.4374	−1.58471	−0.792353	+	0.610062i	$0.791144\pi$
$84$	0	0
$85$	1.83001	0.198493
$86$	12.7146	1.37105
$87$	0	0
$88$	−3.30370	−0.352176
$89$	−3.57607	−0.379063	−0.189531	−	0.981875i	$-0.560697\pi$
$89$	−3.57607	−0.379063	−0.189531	+	0.981875i	$0.560697\pi$
$90$	0	0
$91$	−10.3982	−1.09003
$92$	12.6145	1.31516
$93$	0	0
$94$	−8.05123	−0.830421
$95$	−5.66300	−0.581012
$96$	0	0
$97$	−3.66068	−0.371686	−0.185843	−	0.982579i	$-0.559502\pi$
$97$	−3.66068	−0.371686	−0.185843	+	0.982579i	$0.559502\pi$
$98$	9.48986	0.958620
$99$	0	0
$100$	1.48328	0.148328
$101$	0.391827	0.0389883	0.0194941	−	0.999810i	$-0.493794\pi$
$101$	0.391827	0.0389883	0.0194941	+	0.999810i	$0.493794\pi$
$102$	0	0
$103$	−16.4498	−1.62085	−0.810425	−	0.585842i	$-0.800764\pi$
$103$	−16.4498	−1.62085	−0.810425	+	0.585842i	$0.800764\pi$
$104$	2.88462	0.282860
$105$	0	0
$106$	−12.2340	−1.18827
$107$	12.6842	1.22623	0.613113	−	0.789995i	$-0.289917\pi$
$107$	12.6842	1.22623	0.613113	+	0.789995i	$0.289917\pi$
$108$	0	0
$109$	14.0464	1.34540	0.672701	−	0.739914i	$-0.265134\pi$
$109$	14.0464	1.34540	0.672701	+	0.739914i	$0.265134\pi$
$110$	6.39363	0.609609
$111$	0	0
$112$	−16.5696	−1.56568
$113$	7.45364	0.701180	0.350590	−	0.936529i	$-0.385981\pi$
$113$	7.45364	0.701180	0.350590	+	0.936529i	$0.385981\pi$
$114$	0	0
$115$	8.50448	0.793047
$116$	5.26384	0.488735
$117$	0	0
$118$	6.08437	0.560112
$119$	6.36168	0.583175
$120$	0	0
$121$	0.735642	0.0668765
$122$	6.33564	0.573602
$123$	0	0
$124$	5.38078	0.483209
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.8527	1.14050	0.570248	−	0.821473i	$-0.306847\pi$
$127$	12.8527	1.14050	0.570248	+	0.821473i	$0.306847\pi$
$128$	7.45755	0.659160
$129$	0	0
$130$	−5.58258	−0.489625
$131$	−17.5349	−1.53203	−0.766015	−	0.642823i	$-0.777763\pi$
$131$	−17.5349	−1.53203	−0.766015	+	0.642823i	$0.777763\pi$
$132$	0	0
$133$	−19.6863	−1.70702
$134$	−7.77510	−0.671667
$135$	0	0
$136$	−1.76483	−0.151333
$137$	−9.93917	−0.849160	−0.424580	−	0.905390i	$-0.639578\pi$
$137$	−9.93917	−0.849160	−0.424580	+	0.905390i	$0.639578\pi$
$138$	0	0
$139$	3.20540	0.271879	0.135939	−	0.990717i	$-0.456595\pi$
$139$	3.20540	0.271879	0.135939	+	0.990717i	$0.456595\pi$
$140$	5.15634	0.435790
$141$	0	0
$142$	−16.2525	−1.36388
$143$	−10.2469	−0.856892
$144$	0	0
$145$	3.54878	0.294710
$146$	−14.8263	−1.22704
$147$	0	0
$148$	12.2380	1.00596
$149$	1.00991	0.0827354	0.0413677	−	0.999144i	$-0.486828\pi$
$149$	1.00991	0.0827354	0.0413677	+	0.999144i	$0.486828\pi$
$150$	0	0
$151$	−11.8807	−0.966840	−0.483420	−	0.875388i	$-0.660606\pi$
$151$	−11.8807	−0.966840	−0.483420	+	0.875388i	$0.660606\pi$
$152$	5.46128	0.442968
$153$	0	0
$154$	22.2262	1.79104
$155$	3.62762	0.291377
$156$	0	0
$157$	19.5763	1.56236	0.781181	−	0.624305i	$-0.214618\pi$
$157$	19.5763	1.56236	0.781181	+	0.624305i	$0.214618\pi$
$158$	27.3531	2.17610
$159$	0	0
$160$	−6.96711	−0.550798
$161$	29.5642	2.32998
$162$	0	0
$163$	9.26458	0.725658	0.362829	−	0.931856i	$-0.381811\pi$
$163$	9.26458	0.725658	0.362829	+	0.931856i	$0.381811\pi$
$164$	6.76848	0.528530
$165$	0	0
$166$	−26.9453	−2.09136
$167$	10.1964	0.789023	0.394512	−	0.918891i	$-0.370914\pi$
$167$	10.1964	0.789023	0.394512	+	0.918891i	$0.370914\pi$
$168$	0	0
$169$	−4.05292	−0.311763
$170$	3.41545	0.261953
$171$	0	0
$172$	10.1049	0.770491
$173$	−11.3070	−0.859659	−0.429829	−	0.902910i	$-0.641426\pi$
$173$	−11.3070	−0.859659	−0.429829	+	0.902910i	$0.641426\pi$
$174$	0	0
$175$	3.47631	0.262784
$176$	−16.3285	−1.23081
$177$	0	0
$178$	−6.67422	−0.500254
$179$	−24.0632	−1.79857	−0.899285	−	0.437363i	$-0.855913\pi$
$179$	−24.0632	−1.79857	−0.899285	+	0.437363i	$0.855913\pi$
$180$	0	0
$181$	9.21372	0.684850	0.342425	−	0.939545i	$-0.388752\pi$
$181$	9.21372	0.684850	0.342425	+	0.939545i	$0.388752\pi$
$182$	−19.4068	−1.43852
$183$	0	0
$184$	−8.20154	−0.604626
$185$	8.25062	0.606598
$186$	0	0
$187$	6.26913	0.458445
$188$	−6.39870	−0.466673
$189$	0	0
$190$	−10.5692	−0.766768
$191$	−1.95990	−0.141813	−0.0709067	−	0.997483i	$-0.522589\pi$
$191$	−1.95990	−0.141813	−0.0709067	+	0.997483i	$0.522589\pi$
$192$	0	0
$193$	10.5582	0.759992	0.379996	−	0.924988i	$-0.375925\pi$
$193$	10.5582	0.759992	0.379996	+	0.924988i	$0.375925\pi$
$194$	−6.83213	−0.490518
$195$	0	0
$196$	7.54205	0.538718
$197$	−1.96925	−0.140303	−0.0701515	−	0.997536i	$-0.522348\pi$
$197$	−1.96925	−0.140303	−0.0701515	+	0.997536i	$0.522348\pi$
$198$	0	0
$199$	14.5404	1.03074	0.515369	−	0.856968i	$-0.327655\pi$
$199$	14.5404	1.03074	0.515369	+	0.856968i	$0.327655\pi$
$200$	−0.964379	−0.0681919
$201$	0	0
$202$	0.731289	0.0514533
$203$	12.3366	0.865862
$204$	0	0
$205$	4.56318	0.318706
$206$	−30.7012	−2.13906
$207$	0	0
$208$	14.2572	0.988560
$209$	−19.3999	−1.34192
$210$	0	0
$211$	12.0816	0.831735	0.415867	−	0.909425i	$-0.363478\pi$
$211$	12.0816	0.831735	0.415867	+	0.909425i	$0.363478\pi$
$212$	−9.72296	−0.667776
$213$	0	0
$214$	23.6732	1.61827
$215$	6.81253	0.464610
$216$	0	0
$217$	12.6107	0.856071
$218$	26.2156	1.77555
$219$	0	0
$220$	5.08133	0.342583
$221$	−5.47388	−0.368213
$222$	0	0
$223$	−20.8118	−1.39366	−0.696832	−	0.717234i	$-0.745408\pi$
$223$	−20.8118	−1.39366	−0.696832	+	0.717234i	$0.745408\pi$
$224$	−24.2198	−1.61825
$225$	0	0
$226$	13.9111	0.925356
$227$	8.52959	0.566129	0.283064	−	0.959101i	$-0.408649\pi$
$227$	8.52959	0.566129	0.283064	+	0.959101i	$0.408649\pi$
$228$	0	0
$229$	15.7824	1.04293	0.521465	−	0.853273i	$-0.325386\pi$
$229$	15.7824	1.04293	0.521465	+	0.853273i	$0.325386\pi$
$230$	15.8724	1.04659
$231$	0	0
$232$	−3.42237	−0.224689
$233$	18.1784	1.19090	0.595452	−	0.803391i	$-0.296973\pi$
$233$	18.1784	1.19090	0.595452	+	0.803391i	$0.296973\pi$
$234$	0	0
$235$	−4.31388	−0.281406
$236$	4.83554	0.314767
$237$	0	0
$238$	11.8732	0.769623
$239$	−12.3962	−0.801845	−0.400922	−	0.916112i	$-0.631310\pi$
$239$	−12.3962	−0.801845	−0.400922	+	0.916112i	$0.631310\pi$
$240$	0	0
$241$	−8.96913	−0.577752	−0.288876	−	0.957366i	$-0.593282\pi$
$241$	−8.96913	−0.577752	−0.288876	+	0.957366i	$0.593282\pi$
$242$	1.37297	0.0882578
$243$	0	0
$244$	5.03524	0.322348
$245$	5.08470	0.324850
$246$	0	0
$247$	16.9390	1.07780
$248$	−3.49840	−0.222149
$249$	0	0
$250$	1.86636	0.118039
$251$	11.7836	0.743776	0.371888	−	0.928278i	$-0.378711\pi$
$251$	11.7836	0.743776	0.371888	+	0.928278i	$0.378711\pi$
$252$	0	0
$253$	29.1341	1.83164
$254$	23.9878	1.50513
$255$	0	0
$256$	20.8589	1.30368
$257$	−10.2221	−0.637636	−0.318818	−	0.947816i	$-0.603286\pi$
$257$	−10.2221	−0.637636	−0.318818	+	0.947816i	$0.603286\pi$
$258$	0	0
$259$	28.6817	1.78219
$260$	−4.43675	−0.275155
$261$	0	0
$262$	−32.7263	−2.02184
$263$	−22.8909	−1.41151	−0.705757	−	0.708454i	$-0.749393\pi$
$263$	−22.8909	−1.41151	−0.705757	+	0.708454i	$0.749393\pi$
$264$	0	0
$265$	−6.55503	−0.402672
$266$	−36.7417	−2.25278
$267$	0	0
$268$	−6.17925	−0.377458
$269$	0.457689	0.0279058	0.0139529	−	0.999903i	$-0.495559\pi$
$269$	0.457689	0.0279058	0.0139529	+	0.999903i	$0.495559\pi$
$270$	0	0
$271$	−24.7898	−1.50588	−0.752938	−	0.658092i	$-0.771364\pi$
$271$	−24.7898	−1.50588	−0.752938	+	0.658092i	$0.771364\pi$
$272$	−8.72264	−0.528888
$273$	0	0
$274$	−18.5500	−1.12065
$275$	3.42573	0.206579
$276$	0	0
$277$	−6.15690	−0.369932	−0.184966	−	0.982745i	$-0.559218\pi$
$277$	−6.15690	−0.369932	−0.184966	+	0.982745i	$0.559218\pi$
$278$	5.98242	0.358802
$279$	0	0
$280$	−3.35248	−0.200349
$281$	−18.1653	−1.08365	−0.541824	−	0.840492i	$-0.682266\pi$
$281$	−18.1653	−1.08365	−0.541824	+	0.840492i	$0.682266\pi$
$282$	0	0
$283$	−7.03117	−0.417960	−0.208980	−	0.977920i	$-0.567014\pi$
$283$	−7.03117	−0.417960	−0.208980	+	0.977920i	$0.567014\pi$
$284$	−12.9166	−0.766460
$285$	0	0
$286$	−19.1244	−1.13085
$287$	15.8630	0.936364
$288$	0	0
$289$	−13.6511	−0.803003
$290$	6.62328	0.388932
$291$	0	0
$292$	−11.7832	−0.689560
$293$	−23.7515	−1.38758	−0.693790	−	0.720177i	$-0.744060\pi$
$293$	−23.7515	−1.38758	−0.693790	+	0.720177i	$0.744060\pi$
$294$	0	0
$295$	3.26003	0.189806
$296$	−7.95672	−0.462475
$297$	0	0
$298$	1.88486	0.109187
$299$	−25.4383	−1.47114
$300$	0	0
$301$	23.6824	1.36503
$302$	−22.1737	−1.27595
$303$	0	0
$304$	26.9924	1.54812
$305$	3.39466	0.194378
$306$	0	0
$307$	−18.6253	−1.06300	−0.531500	−	0.847058i	$-0.678372\pi$
$307$	−18.6253	−1.06300	−0.531500	+	0.847058i	$0.678372\pi$
$308$	17.6642	1.00651
$309$	0	0
$310$	6.77043	0.384534
$311$	1.93805	0.109897	0.0549485	−	0.998489i	$-0.482501\pi$
$311$	1.93805	0.109897	0.0549485	+	0.998489i	$0.482501\pi$
$312$	0	0
$313$	−30.2522	−1.70995	−0.854977	−	0.518667i	$-0.826429\pi$
$313$	−30.2522	−1.70995	−0.854977	+	0.518667i	$0.826429\pi$
$314$	36.5364	2.06187
$315$	0	0
$316$	21.7389	1.22291
$317$	2.93059	0.164598	0.0822992	−	0.996608i	$-0.473774\pi$
$317$	2.93059	0.164598	0.0822992	+	0.996608i	$0.473774\pi$
$318$	0	0
$319$	12.1572	0.680671
$320$	−3.47022	−0.193991
$321$	0	0
$322$	55.1772	3.07491
$323$	−10.3634	−0.576633
$324$	0	0
$325$	−2.99117	−0.165920
$326$	17.2910	0.957660
$327$	0	0
$328$	−4.40064	−0.242984
$329$	−14.9964	−0.826776
$330$	0	0
$331$	−10.8182	−0.594623	−0.297312	−	0.954780i	$-0.596090\pi$
$331$	−10.8182	−0.594623	−0.297312	+	0.954780i	$0.596090\pi$
$332$	−21.4147	−1.17528
$333$	0	0
$334$	19.0302	1.04128
$335$	−4.16593	−0.227609
$336$	0	0
$337$	−13.6817	−0.745289	−0.372645	−	0.927974i	$-0.621549\pi$
$337$	−13.6817	−0.745289	−0.372645	+	0.927974i	$0.621549\pi$
$338$	−7.56418	−0.411437
$339$	0	0
$340$	2.71443	0.147210
$341$	12.4273	0.672973
$342$	0	0
$343$	−6.65817	−0.359507
$344$	−6.56986	−0.354223
$345$	0	0
$346$	−21.1030	−1.13450
$347$	−17.9888	−0.965691	−0.482846	−	0.875705i	$-0.660397\pi$
$347$	−17.9888	−0.965691	−0.482846	+	0.875705i	$0.660397\pi$
$348$	0	0
$349$	3.74734	0.200591	0.100295	−	0.994958i	$-0.468021\pi$
$349$	3.74734	0.200591	0.100295	+	0.994958i	$0.468021\pi$
$350$	6.48802	0.346799
$351$	0	0
$352$	−23.8675	−1.27214
$353$	−0.326966	−0.0174026	−0.00870132	−	0.999962i	$-0.502770\pi$
$353$	−0.326966	−0.0174026	−0.00870132	+	0.999962i	$0.502770\pi$
$354$	0	0
$355$	−8.70813	−0.462179
$356$	−5.30432	−0.281128
$357$	0	0
$358$	−44.9105	−2.37360
$359$	−6.41154	−0.338388	−0.169194	−	0.985583i	$-0.554116\pi$
$359$	−6.41154	−0.338388	−0.169194	+	0.985583i	$0.554116\pi$
$360$	0	0
$361$	13.0696	0.687873
$362$	17.1961	0.903805
$363$	0	0
$364$	−15.4235	−0.808410
$365$	−7.94401	−0.415808
$366$	0	0
$367$	−8.15683	−0.425783	−0.212891	−	0.977076i	$-0.568288\pi$
$367$	−8.15683	−0.425783	−0.212891	+	0.977076i	$0.568288\pi$
$368$	−40.5361	−2.11309
$369$	0	0
$370$	15.3986	0.800534
$371$	−22.7873	−1.18306
$372$	0	0
$373$	0.837348	0.0433563	0.0216781	−	0.999765i	$-0.493099\pi$
$373$	0.837348	0.0433563	0.0216781	+	0.999765i	$0.493099\pi$
$374$	11.7004	0.605015
$375$	0	0
$376$	4.16021	0.214547
$377$	−10.6150	−0.546700
$378$	0	0
$379$	−7.38310	−0.379245	−0.189622	−	0.981857i	$-0.560726\pi$
$379$	−7.38310	−0.379245	−0.189622	+	0.981857i	$0.560726\pi$
$380$	−8.39983	−0.430902
$381$	0	0
$382$	−3.65787	−0.187153
$383$	17.1462	0.876130	0.438065	−	0.898943i	$-0.355664\pi$
$383$	17.1462	0.876130	0.438065	+	0.898943i	$0.355664\pi$
$384$	0	0
$385$	11.9089	0.606933
$386$	19.7053	1.00297
$387$	0	0
$388$	−5.42982	−0.275657
$389$	−6.49728	−0.329425	−0.164713	−	0.986342i	$-0.552670\pi$
$389$	−6.49728	−0.329425	−0.164713	+	0.986342i	$0.552670\pi$
$390$	0	0
$391$	15.5633	0.787070
$392$	−4.90358	−0.247668
$393$	0	0
$394$	−3.67531	−0.185160
$395$	14.6559	0.737419
$396$	0	0
$397$	7.21228	0.361974	0.180987	−	0.983485i	$-0.442071\pi$
$397$	7.21228	0.361974	0.180987	+	0.983485i	$0.442071\pi$
$398$	27.1375	1.36028
$399$	0	0
$400$	−4.76644	−0.238322
$401$	−10.6360	−0.531139	−0.265569	−	0.964092i	$-0.585560\pi$
$401$	−10.6360	−0.531139	−0.265569	+	0.964092i	$0.585560\pi$
$402$	0	0
$403$	−10.8508	−0.540518
$404$	0.581191	0.0289153
$405$	0	0
$406$	23.0245	1.14269
$407$	28.2644	1.40101
$408$	0	0
$409$	9.90790	0.489914	0.244957	−	0.969534i	$-0.421226\pi$
$409$	9.90790	0.489914	0.244957	+	0.969534i	$0.421226\pi$
$410$	8.51652	0.420601
$411$	0	0
$412$	−24.3998	−1.20209
$413$	11.3329	0.557653
$414$	0	0
$415$	−14.4374	−0.708702
$416$	20.8398	1.02175
$417$	0	0
$418$	−36.2072	−1.77095
$419$	−10.6489	−0.520234	−0.260117	−	0.965577i	$-0.583761\pi$
$419$	−10.6489	−0.520234	−0.260117	+	0.965577i	$0.583761\pi$
$420$	0	0
$421$	38.5898	1.88075	0.940376	−	0.340136i	$-0.110473\pi$
$421$	38.5898	1.88075	0.940376	+	0.340136i	$0.110473\pi$
$422$	22.5486	1.09765
$423$	0	0
$424$	6.32154	0.307001
$425$	1.83001	0.0887687
$426$	0	0
$427$	11.8009	0.571084
$428$	18.8142	0.909420
$429$	0	0
$430$	12.7146	0.613152
$431$	−24.1205	−1.16184	−0.580922	−	0.813960i	$-0.697308\pi$
$431$	−24.1205	−1.16184	−0.580922	+	0.813960i	$0.697308\pi$
$432$	0	0
$433$	−9.73516	−0.467842	−0.233921	−	0.972256i	$-0.575156\pi$
$433$	−9.73516	−0.467842	−0.233921	+	0.972256i	$0.575156\pi$
$434$	23.5361	1.12977
$435$	0	0
$436$	20.8348	0.997806
$437$	−48.1609	−2.30385
$438$	0	0
$439$	28.5848	1.36428	0.682139	−	0.731223i	$-0.261050\pi$
$439$	28.5848	1.36428	0.682139	+	0.731223i	$0.261050\pi$
$440$	−3.30370	−0.157498
$441$	0	0
$442$	−10.2162	−0.485935
$443$	29.3199	1.39303	0.696515	−	0.717543i	$-0.254733\pi$
$443$	29.3199	1.39303	0.696515	+	0.717543i	$0.254733\pi$
$444$	0	0
$445$	−3.57607	−0.169522
$446$	−38.8423	−1.83924
$447$	0	0
$448$	−12.0636	−0.569950
$449$	−27.2250	−1.28483	−0.642413	−	0.766359i	$-0.722067\pi$
$449$	−27.2250	−1.28483	−0.642413	+	0.766359i	$0.722067\pi$
$450$	0	0
$451$	15.6322	0.736093
$452$	11.0559	0.520024
$453$	0	0
$454$	15.9192	0.747127
$455$	−10.3982	−0.487476
$456$	0	0
$457$	−32.8154	−1.53504	−0.767520	−	0.641026i	$-0.778509\pi$
$457$	−32.8154	−1.53504	−0.767520	+	0.641026i	$0.778509\pi$
$458$	29.4556	1.37637
$459$	0	0
$460$	12.6145	0.588156
$461$	6.28230	0.292596	0.146298	−	0.989241i	$-0.453264\pi$
$461$	6.28230	0.292596	0.146298	+	0.989241i	$0.453264\pi$
$462$	0	0
$463$	−24.8804	−1.15629	−0.578145	−	0.815934i	$-0.696223\pi$
$463$	−24.8804	−1.15629	−0.578145	+	0.815934i	$0.696223\pi$
$464$	−16.9150	−0.785261
$465$	0	0
$466$	33.9273	1.57165
$467$	−30.2311	−1.39893	−0.699464	−	0.714668i	$-0.746578\pi$
$467$	−30.2311	−1.39893	−0.699464	+	0.714668i	$0.746578\pi$
$468$	0	0
$469$	−14.4820	−0.668719
$470$	−8.05123	−0.371376
$471$	0	0
$472$	−3.14390	−0.144710
$473$	23.3379	1.07308
$474$	0	0
$475$	−5.66300	−0.259836
$476$	9.43617	0.432506
$477$	0	0
$478$	−23.1357	−1.05820
$479$	−8.39900	−0.383760	−0.191880	−	0.981418i	$-0.561458\pi$
$479$	−8.39900	−0.383760	−0.191880	+	0.981418i	$0.561458\pi$
$480$	0	0
$481$	−24.6790	−1.12526
$482$	−16.7396	−0.762467
$483$	0	0
$484$	1.09116	0.0495984
$485$	−3.66068	−0.166223
$486$	0	0
$487$	25.5873	1.15947	0.579735	−	0.814805i	$-0.303156\pi$
$487$	25.5873	1.15947	0.579735	+	0.814805i	$0.303156\pi$
$488$	−3.27374	−0.148195
$489$	0	0
$490$	9.48986	0.428708
$491$	−2.37298	−0.107091	−0.0535456	−	0.998565i	$-0.517052\pi$
$491$	−2.37298	−0.107091	−0.0535456	+	0.998565i	$0.517052\pi$
$492$	0	0
$493$	6.49431	0.292489
$494$	31.6142	1.42239
$495$	0	0
$496$	−17.2908	−0.776381
$497$	−30.2721	−1.35789
$498$	0	0
$499$	1.46603	0.0656286	0.0328143	−	0.999461i	$-0.489553\pi$
$499$	1.46603	0.0656286	0.0328143	+	0.999461i	$0.489553\pi$
$500$	1.48328	0.0663344
$501$	0	0
$502$	21.9924	0.981570
$503$	−27.2110	−1.21328	−0.606640	−	0.794977i	$-0.707483\pi$
$503$	−27.2110	−1.21328	−0.606640	+	0.794977i	$0.707483\pi$
$504$	0	0
$505$	0.391827	0.0174361
$506$	54.3745	2.41724
$507$	0	0
$508$	19.0642	0.845839
$509$	−2.72382	−0.120731	−0.0603657	−	0.998176i	$-0.519227\pi$
$509$	−2.72382	−0.120731	−0.0603657	+	0.998176i	$0.519227\pi$
$510$	0	0
$511$	−27.6158	−1.22165
$512$	24.0150	1.06132
$513$	0	0
$514$	−19.0781	−0.841497
$515$	−16.4498	−0.724866
$516$	0	0
$517$	−14.7782	−0.649944
$518$	53.5302	2.35198
$519$	0	0
$520$	2.88462	0.126499
$521$	−4.45756	−0.195289	−0.0976447	−	0.995221i	$-0.531131\pi$
$521$	−4.45756	−0.195289	−0.0976447	+	0.995221i	$0.531131\pi$
$522$	0	0
$523$	−36.5047	−1.59624	−0.798119	−	0.602500i	$-0.794171\pi$
$523$	−36.5047	−1.59624	−0.798119	+	0.602500i	$0.794171\pi$
$524$	−26.0092	−1.13622
$525$	0	0
$526$	−42.7225	−1.86279
$527$	6.63859	0.289182
$528$	0	0
$529$	49.3262	2.14462
$530$	−12.2340	−0.531412
$531$	0	0
$532$	−29.2004	−1.26600
$533$	−13.6492	−0.591214
$534$	0	0
$535$	12.6842	0.548385
$536$	4.01754	0.173531
$537$	0	0
$538$	0.854211	0.0368276
$539$	17.4188	0.750282
$540$	0	0
$541$	−5.40959	−0.232577	−0.116288	−	0.993216i	$-0.537100\pi$
$541$	−5.40959	−0.232577	−0.116288	+	0.993216i	$0.537100\pi$
$542$	−46.2666	−1.98732
$543$	0	0
$544$	−12.7499	−0.546647
$545$	14.0464	0.601682
$546$	0	0
$547$	21.6469	0.925553	0.462776	−	0.886475i	$-0.346853\pi$
$547$	21.6469	0.925553	0.462776	+	0.886475i	$0.346853\pi$
$548$	−14.7426	−0.629772
$549$	0	0
$550$	6.39363	0.272625
$551$	−20.0967	−0.856150
$552$	0	0
$553$	50.9484	2.16655
$554$	−11.4910	−0.488204
$555$	0	0
$556$	4.75452	0.201636
$557$	−5.25820	−0.222797	−0.111399	−	0.993776i	$-0.535533\pi$
$557$	−5.25820	−0.222797	−0.111399	+	0.993776i	$0.535533\pi$
$558$	0	0
$559$	−20.3774	−0.861872
$560$	−16.5696	−0.700193
$561$	0	0
$562$	−33.9028	−1.43010
$563$	23.7804	1.00222	0.501112	−	0.865383i	$-0.332925\pi$
$563$	23.7804	1.00222	0.501112	+	0.865383i	$0.332925\pi$
$564$	0	0
$565$	7.45364	0.313577
$566$	−13.1227	−0.551587
$567$	0	0
$568$	8.39794	0.352370
$569$	−30.4284	−1.27562	−0.637812	−	0.770192i	$-0.720160\pi$
$569$	−30.4284	−1.27562	−0.637812	+	0.770192i	$0.720160\pi$
$570$	0	0
$571$	18.6866	0.782011	0.391005	−	0.920388i	$-0.372127\pi$
$571$	18.6866	0.782011	0.391005	+	0.920388i	$0.372127\pi$
$572$	−15.1991	−0.635506
$573$	0	0
$574$	29.6060	1.23573
$575$	8.50448	0.354661
$576$	0	0
$577$	18.4819	0.769410	0.384705	−	0.923040i	$-0.374303\pi$
$577$	18.4819	0.769410	0.384705	+	0.923040i	$0.374303\pi$
$578$	−25.4777	−1.05973
$579$	0	0
$580$	5.26384	0.218569
$581$	−50.1887	−2.08218
$582$	0	0
$583$	−22.4558	−0.930024
$584$	7.66104	0.317016
$585$	0	0
$586$	−44.3288	−1.83121
$587$	26.6430	1.09967	0.549836	−	0.835272i	$-0.314690\pi$
$587$	26.6430	1.09967	0.549836	+	0.835272i	$0.314690\pi$
$588$	0	0
$589$	−20.5432	−0.846468
$590$	6.08437	0.250490
$591$	0	0
$592$	−39.3261	−1.61629
$593$	−4.62357	−0.189867	−0.0949337	−	0.995484i	$-0.530264\pi$
$593$	−4.62357	−0.189867	−0.0949337	+	0.995484i	$0.530264\pi$
$594$	0	0
$595$	6.36168	0.260804
$596$	1.49799	0.0613600
$597$	0	0
$598$	−47.4769	−1.94148
$599$	7.76496	0.317268	0.158634	−	0.987337i	$-0.449291\pi$
$599$	7.76496	0.317268	0.158634	+	0.987337i	$0.449291\pi$
$600$	0	0
$601$	−14.9594	−0.610206	−0.305103	−	0.952319i	$-0.598691\pi$
$601$	−14.9594	−0.610206	−0.305103	+	0.952319i	$0.598691\pi$
$602$	44.1998	1.80145
$603$	0	0
$604$	−17.6225	−0.717049
$605$	0.735642	0.0299081
$606$	0	0
$607$	25.7325	1.04445	0.522224	−	0.852808i	$-0.325102\pi$
$607$	25.7325	1.04445	0.522224	+	0.852808i	$0.325102\pi$
$608$	39.4548	1.60010
$609$	0	0
$610$	6.33564	0.256523
$611$	12.9035	0.522021
$612$	0	0
$613$	29.1445	1.17713	0.588567	−	0.808448i	$-0.299692\pi$
$613$	29.1445	1.17713	0.588567	+	0.808448i	$0.299692\pi$
$614$	−34.7614	−1.40286
$615$	0	0
$616$	−11.4847	−0.462731
$617$	8.56104	0.344654	0.172327	−	0.985040i	$-0.444871\pi$
$617$	8.56104	0.344654	0.172327	+	0.985040i	$0.444871\pi$
$618$	0	0
$619$	21.7639	0.874764	0.437382	−	0.899276i	$-0.355906\pi$
$619$	21.7639	0.874764	0.437382	+	0.899276i	$0.355906\pi$
$620$	5.38078	0.216097
$621$	0	0
$622$	3.61710	0.145032
$623$	−12.4315	−0.498058
$624$	0	0
$625$	1.00000	0.0400000
$626$	−56.4613	−2.25665
$627$	0	0
$628$	29.0372	1.15871
$629$	15.0987	0.602026
$630$	0	0
$631$	−44.7461	−1.78131	−0.890657	−	0.454676i	$-0.849755\pi$
$631$	−44.7461	−1.78131	−0.890657	+	0.454676i	$0.849755\pi$
$632$	−14.1339	−0.562215
$633$	0	0
$634$	5.46952	0.217223
$635$	12.8527	0.510045
$636$	0	0
$637$	−15.2092	−0.602610
$638$	22.6896	0.898289
$639$	0	0
$640$	7.45755	0.294785
$641$	26.3418	1.04044	0.520219	−	0.854033i	$-0.325850\pi$
$641$	26.3418	1.04044	0.520219	+	0.854033i	$0.325850\pi$
$642$	0	0
$643$	36.3366	1.43297	0.716487	−	0.697600i	$-0.245749\pi$
$643$	36.3366	1.43297	0.716487	+	0.697600i	$0.245749\pi$
$644$	43.8520	1.72801
$645$	0	0
$646$	−19.3417	−0.760990
$647$	−39.1069	−1.53745	−0.768725	−	0.639580i	$-0.779108\pi$
$647$	−39.1069	−1.53745	−0.768725	+	0.639580i	$0.779108\pi$
$648$	0	0
$649$	11.1680	0.438382
$650$	−5.58258	−0.218967
$651$	0	0
$652$	13.7420	0.538178
$653$	−14.6250	−0.572319	−0.286159	−	0.958182i	$-0.592379\pi$
$653$	−14.6250	−0.572319	−0.286159	+	0.958182i	$0.592379\pi$
$654$	0	0
$655$	−17.5349	−0.685145
$656$	−21.7501	−0.849199
$657$	0	0
$658$	−27.9885	−1.09111
$659$	−20.8541	−0.812360	−0.406180	−	0.913793i	$-0.633139\pi$
$659$	−20.8541	−0.812360	−0.406180	+	0.913793i	$0.633139\pi$
$660$	0	0
$661$	17.6730	0.687401	0.343700	−	0.939079i	$-0.388320\pi$
$661$	17.6730	0.687401	0.343700	+	0.939079i	$0.388320\pi$
$662$	−20.1907	−0.784732
$663$	0	0
$664$	13.9231	0.540321
$665$	−19.6863	−0.763403
$666$	0	0
$667$	30.1805	1.16859
$668$	15.1242	0.585172
$669$	0	0
$670$	−7.77510	−0.300378
$671$	11.6292	0.448940
$672$	0	0
$673$	−23.5326	−0.907114	−0.453557	−	0.891227i	$-0.649845\pi$
$673$	−23.5326	−0.907114	−0.453557	+	0.891227i	$0.649845\pi$
$674$	−25.5349	−0.983567
$675$	0	0
$676$	−6.01162	−0.231216
$677$	38.2284	1.46924	0.734618	−	0.678481i	$-0.237361\pi$
$677$	38.2284	1.46924	0.734618	+	0.678481i	$0.237361\pi$
$678$	0	0
$679$	−12.7256	−0.488365
$680$	−1.76483	−0.0676780
$681$	0	0
$682$	23.1937	0.888131
$683$	5.55002	0.212366	0.106183	−	0.994347i	$-0.466137\pi$
$683$	5.55002	0.212366	0.106183	+	0.994347i	$0.466137\pi$
$684$	0	0
$685$	−9.93917	−0.379756
$686$	−12.4265	−0.474446
$687$	0	0
$688$	−32.4715	−1.23796
$689$	19.6072	0.746975
$690$	0	0
$691$	−30.4698	−1.15913	−0.579563	−	0.814927i	$-0.696777\pi$
$691$	−30.4698	−1.15913	−0.579563	+	0.814927i	$0.696777\pi$
$692$	−16.7715	−0.637558
$693$	0	0
$694$	−33.5736	−1.27443
$695$	3.20540	0.121588
$696$	0	0
$697$	8.35068	0.316305
$698$	6.99387	0.264722
$699$	0	0
$700$	5.15634	0.194891
$701$	−9.84780	−0.371946	−0.185973	−	0.982555i	$-0.559544\pi$
$701$	−9.84780	−0.371946	−0.185973	+	0.982555i	$0.559544\pi$
$702$	0	0
$703$	−46.7233	−1.76220
$704$	−11.8881	−0.448048
$705$	0	0
$706$	−0.610234	−0.0229665
$707$	1.36211	0.0512275
$708$	0	0
$709$	−42.7668	−1.60614	−0.803070	−	0.595885i	$-0.796801\pi$
$709$	−42.7668	−1.60614	−0.803070	+	0.595885i	$0.796801\pi$
$710$	−16.2525	−0.609944
$711$	0	0
$712$	3.44869	0.129245
$713$	30.8510	1.15538
$714$	0	0
$715$	−10.2469	−0.383214
$716$	−35.6926	−1.33389
$717$	0	0
$718$	−11.9662	−0.446575
$719$	−10.5740	−0.394342	−0.197171	−	0.980369i	$-0.563175\pi$
$719$	−10.5740	−0.394342	−0.197171	+	0.980369i	$0.563175\pi$
$720$	0	0
$721$	−57.1847	−2.12967
$722$	24.3925	0.907795
$723$	0	0
$724$	13.6665	0.507913
$725$	3.54878	0.131798
$726$	0	0
$727$	40.2638	1.49330	0.746650	−	0.665217i	$-0.231661\pi$
$727$	40.2638	1.49330	0.746650	+	0.665217i	$0.231661\pi$
$728$	10.0278	0.371656
$729$	0	0
$730$	−14.8263	−0.548748
$731$	12.4670	0.461109
$732$	0	0
$733$	44.7055	1.65124	0.825618	−	0.564229i	$-0.190827\pi$
$733$	44.7055	1.65124	0.825618	+	0.564229i	$0.190827\pi$
$734$	−15.2235	−0.561911
$735$	0	0
$736$	−59.2516	−2.18404
$737$	−14.2714	−0.525692
$738$	0	0
$739$	14.3039	0.526177	0.263088	−	0.964772i	$-0.415259\pi$
$739$	14.3039	0.526177	0.263088	+	0.964772i	$0.415259\pi$
$740$	12.2380	0.449878
$741$	0	0
$742$	−42.5292	−1.56130
$743$	21.9052	0.803625	0.401812	−	0.915722i	$-0.368380\pi$
$743$	21.9052	0.803625	0.401812	+	0.915722i	$0.368380\pi$
$744$	0	0
$745$	1.00991	0.0370004
$746$	1.56279	0.0572178
$747$	0	0
$748$	9.29890	0.340001
$749$	44.0941	1.61116
$750$	0	0
$751$	2.95673	0.107892	0.0539462	−	0.998544i	$-0.482820\pi$
$751$	2.95673	0.107892	0.0539462	+	0.998544i	$0.482820\pi$
$752$	20.5618	0.749813
$753$	0	0
$754$	−19.8113	−0.721487
$755$	−11.8807	−0.432384
$756$	0	0
$757$	31.6130	1.14899	0.574497	−	0.818507i	$-0.305198\pi$
$757$	31.6130	1.14899	0.574497	+	0.818507i	$0.305198\pi$
$758$	−13.7795	−0.500494
$759$	0	0
$760$	5.46128	0.198101
$761$	−16.9308	−0.613740	−0.306870	−	0.951751i	$-0.599282\pi$
$761$	−16.9308	−0.613740	−0.306870	+	0.951751i	$0.599282\pi$
$762$	0	0
$763$	48.8296	1.76775
$764$	−2.90708	−0.105175
$765$	0	0
$766$	32.0009	1.15624
$767$	−9.75129	−0.352099
$768$	0	0
$769$	−20.6348	−0.744108	−0.372054	−	0.928211i	$-0.621346\pi$
$769$	−20.6348	−0.744108	−0.372054	+	0.928211i	$0.621346\pi$
$770$	22.2262	0.800978
$771$	0	0
$772$	15.6607	0.563641
$773$	14.3231	0.515168	0.257584	−	0.966256i	$-0.417074\pi$
$773$	14.3231	0.515168	0.257584	+	0.966256i	$0.417074\pi$
$774$	0	0
$775$	3.62762	0.130308
$776$	3.53028	0.126730
$777$	0	0
$778$	−12.1262	−0.434747
$779$	−25.8413	−0.925861
$780$	0	0
$781$	−29.8317	−1.06746
$782$	29.0467	1.03871
$783$	0	0
$784$	−24.2359	−0.865568
$785$	19.5763	0.698709
$786$	0	0
$787$	30.7258	1.09526	0.547628	−	0.836722i	$-0.315531\pi$
$787$	30.7258	1.09526	0.547628	+	0.836722i	$0.315531\pi$
$788$	−2.92095	−0.104054
$789$	0	0
$790$	27.3531	0.973181
$791$	25.9111	0.921294
$792$	0	0
$793$	−10.1540	−0.360579
$794$	13.4607	0.477702
$795$	0	0
$796$	21.5675	0.764438
$797$	−30.9667	−1.09690	−0.548448	−	0.836185i	$-0.684781\pi$
$797$	−30.9667	−1.09690	−0.548448	+	0.836185i	$0.684781\pi$
$798$	0	0
$799$	−7.89445	−0.279286
$800$	−6.96711	−0.246325
$801$	0	0
$802$	−19.8506	−0.700950
$803$	−27.2140	−0.960363
$804$	0	0
$805$	29.5642	1.04200
$806$	−20.2515	−0.713328
$807$	0	0
$808$	−0.377870	−0.0132934
$809$	−2.58326	−0.0908226	−0.0454113	−	0.998968i	$-0.514460\pi$
$809$	−2.58326	−0.0908226	−0.0454113	+	0.998968i	$0.514460\pi$
$810$	0	0
$811$	−24.6638	−0.866064	−0.433032	−	0.901379i	$-0.642556\pi$
$811$	−24.6638	−0.866064	−0.433032	+	0.901379i	$0.642556\pi$
$812$	18.2987	0.642159
$813$	0	0
$814$	52.7514	1.84894
$815$	9.26458	0.324524
$816$	0	0
$817$	−38.5793	−1.34972
$818$	18.4917	0.646546
$819$	0	0
$820$	6.76848	0.236366
$821$	22.7169	0.792826	0.396413	−	0.918072i	$-0.370255\pi$
$821$	22.7169	0.792826	0.396413	+	0.918072i	$0.370255\pi$
$822$	0	0
$823$	26.7299	0.931745	0.465872	−	0.884852i	$-0.345741\pi$
$823$	26.7299	0.931745	0.465872	+	0.884852i	$0.345741\pi$
$824$	15.8639	0.552644
$825$	0	0
$826$	21.1511	0.735942
$827$	15.4490	0.537214	0.268607	−	0.963250i	$-0.413437\pi$
$827$	15.4490	0.537214	0.268607	+	0.963250i	$0.413437\pi$
$828$	0	0
$829$	−13.7566	−0.477788	−0.238894	−	0.971046i	$-0.576785\pi$
$829$	−13.7566	−0.477788	−0.238894	+	0.971046i	$0.576785\pi$
$830$	−26.9453	−0.935283
$831$	0	0
$832$	10.3800	0.359863
$833$	9.30507	0.322402
$834$	0	0
$835$	10.1964	0.352862
$836$	−28.7756	−0.995224
$837$	0	0
$838$	−19.8747	−0.686559
$839$	9.61015	0.331779	0.165890	−	0.986144i	$-0.446950\pi$
$839$	9.61015	0.331779	0.165890	+	0.986144i	$0.446950\pi$
$840$	0	0
$841$	−16.4062	−0.565730
$842$	72.0223	2.48205
$843$	0	0
$844$	17.9205	0.616849
$845$	−4.05292	−0.139425
$846$	0	0
$847$	2.55732	0.0878704
$848$	31.2442	1.07293
$849$	0	0
$850$	3.41545	0.117149
$851$	70.1672	2.40530
$852$	0	0
$853$	−20.5572	−0.703864	−0.351932	−	0.936025i	$-0.614475\pi$
$853$	−20.5572	−0.703864	−0.351932	+	0.936025i	$0.614475\pi$
$854$	22.0246	0.753667
$855$	0	0
$856$	−12.2324	−0.418094
$857$	−40.5054	−1.38364	−0.691820	−	0.722070i	$-0.743191\pi$
$857$	−40.5054	−1.38364	−0.691820	+	0.722070i	$0.743191\pi$
$858$	0	0
$859$	25.2768	0.862432	0.431216	−	0.902249i	$-0.358085\pi$
$859$	25.2768	0.862432	0.431216	+	0.902249i	$0.358085\pi$
$860$	10.1049	0.344574
$861$	0	0
$862$	−45.0174	−1.53330
$863$	47.4531	1.61532	0.807661	−	0.589648i	$-0.200733\pi$
$863$	47.4531	1.61532	0.807661	+	0.589648i	$0.200733\pi$
$864$	0	0
$865$	−11.3070	−0.384451
$866$	−18.1693	−0.617417
$867$	0	0
$868$	18.7052	0.634897
$869$	50.2072	1.70316
$870$	0	0
$871$	12.4610	0.422225
$872$	−13.5461	−0.458728
$873$	0	0
$874$	−89.8853	−3.04042
$875$	3.47631	0.117521
$876$	0	0
$877$	−3.38363	−0.114257	−0.0571285	−	0.998367i	$-0.518194\pi$
$877$	−3.38363	−0.114257	−0.0571285	+	0.998367i	$0.518194\pi$
$878$	53.3494	1.80045
$879$	0	0
$880$	−16.3285	−0.550435
$881$	23.8060	0.802045	0.401022	−	0.916068i	$-0.368655\pi$
$881$	23.8060	0.802045	0.401022	+	0.916068i	$0.368655\pi$
$882$	0	0
$883$	23.3657	0.786319	0.393159	−	0.919470i	$-0.371382\pi$
$883$	23.3657	0.786319	0.393159	+	0.919470i	$0.371382\pi$
$884$	−8.11930	−0.273082
$885$	0	0
$886$	54.7213	1.83840
$887$	30.8708	1.03654	0.518270	−	0.855217i	$-0.326576\pi$
$887$	30.8708	1.03654	0.518270	+	0.855217i	$0.326576\pi$
$888$	0	0
$889$	44.6800	1.49852
$890$	−6.67422	−0.223720
$891$	0	0
$892$	−30.8698	−1.03360
$893$	24.4295	0.817502
$894$	0	0
$895$	−24.0632	−0.804345
$896$	25.9247	0.866084
$897$	0	0
$898$	−50.8115	−1.69560
$899$	12.8736	0.429359
$900$	0	0
$901$	−11.9958	−0.399638
$902$	29.1753	0.971431
$903$	0	0
$904$	−7.18814	−0.239074
$905$	9.21372	0.306274
$906$	0	0
$907$	−11.2024	−0.371969	−0.185985	−	0.982553i	$-0.559547\pi$
$907$	−11.2024	−0.371969	−0.185985	+	0.982553i	$0.559547\pi$
$908$	12.6518	0.419864
$909$	0	0
$910$	−19.4068	−0.643328
$911$	−25.9949	−0.861250	−0.430625	−	0.902531i	$-0.641707\pi$
$911$	−25.9949	−0.861250	−0.430625	+	0.902531i	$0.641707\pi$
$912$	0	0
$913$	−49.4585	−1.63684
$914$	−61.2452	−2.02581
$915$	0	0
$916$	23.4097	0.773480
$917$	−60.9566	−2.01296
$918$	0	0
$919$	−9.14191	−0.301564	−0.150782	−	0.988567i	$-0.548179\pi$
$919$	−9.14191	−0.301564	−0.150782	+	0.988567i	$0.548179\pi$
$920$	−8.20154	−0.270397
$921$	0	0
$922$	11.7250	0.386143
$923$	26.0475	0.857363
$924$	0	0
$925$	8.25062	0.271279
$926$	−46.4357	−1.52597
$927$	0	0
$928$	−24.7247	−0.811629
$929$	−54.3613	−1.78354	−0.891768	−	0.452493i	$-0.850535\pi$
$929$	−54.3613	−1.78354	−0.891768	+	0.452493i	$0.850535\pi$
$930$	0	0
$931$	−28.7947	−0.943707
$932$	26.9636	0.883223
$933$	0	0
$934$	−56.4220	−1.84618
$935$	6.26913	0.205023
$936$	0	0
$937$	−1.02725	−0.0335587	−0.0167794	−	0.999859i	$-0.505341\pi$
$937$	−1.02725	−0.0335587	−0.0167794	+	0.999859i	$0.505341\pi$
$938$	−27.0286	−0.882516
$939$	0	0
$940$	−6.39870	−0.208703
$941$	51.4258	1.67643	0.838216	−	0.545338i	$-0.183599\pi$
$941$	51.4258	1.67643	0.838216	+	0.545338i	$0.183599\pi$
$942$	0	0
$943$	38.8075	1.26375
$944$	−15.5387	−0.505742
$945$	0	0
$946$	43.5568	1.41615
$947$	−27.2357	−0.885041	−0.442520	−	0.896759i	$-0.645916\pi$
$947$	−27.2357	−0.885041	−0.442520	+	0.896759i	$0.645916\pi$
$948$	0	0
$949$	23.7619	0.771343
$950$	−10.5692	−0.342909
$951$	0	0
$952$	−6.13508	−0.198839
$953$	−13.2086	−0.427870	−0.213935	−	0.976848i	$-0.568628\pi$
$953$	−13.2086	−0.427870	−0.213935	+	0.976848i	$0.568628\pi$
$954$	0	0
$955$	−1.95990	−0.0634209
$956$	−18.3871	−0.594681
$957$	0	0
$958$	−15.6755	−0.506453
$959$	−34.5516	−1.11573
$960$	0	0
$961$	−17.8404	−0.575496
$962$	−46.0597	−1.48503
$963$	0	0
$964$	−13.3037	−0.428485
$965$	10.5582	0.339879
$966$	0	0
$967$	−8.66434	−0.278626	−0.139313	−	0.990248i	$-0.544489\pi$
$967$	−8.66434	−0.278626	−0.139313	+	0.990248i	$0.544489\pi$
$968$	−0.709438	−0.0228022
$969$	0	0
$970$	−6.83213	−0.219366
$971$	27.5409	0.883831	0.441915	−	0.897057i	$-0.354299\pi$
$971$	27.5409	0.883831	0.441915	+	0.897057i	$0.354299\pi$
$972$	0	0
$973$	11.1430	0.357227
$974$	47.7549	1.53017
$975$	0	0
$976$	−16.1804	−0.517923
$977$	6.61583	0.211659	0.105830	−	0.994384i	$-0.466250\pi$
$977$	6.61583	0.211659	0.105830	+	0.994384i	$0.466250\pi$
$978$	0	0
$979$	−12.2507	−0.391533
$980$	7.54205	0.240922
$981$	0	0
$982$	−4.42883	−0.141330
$983$	37.3057	1.18987	0.594933	−	0.803775i	$-0.297179\pi$
$983$	37.3057	1.18987	0.594933	+	0.803775i	$0.297179\pi$
$984$	0	0
$985$	−1.96925	−0.0627454
$986$	12.1207	0.386001
$987$	0	0
$988$	25.1253	0.799342
$989$	57.9370	1.84229
$990$	0	0
$991$	−40.9731	−1.30155	−0.650777	−	0.759269i	$-0.725557\pi$
$991$	−40.9731	−1.30155	−0.650777	+	0.759269i	$0.725557\pi$
$992$	−25.2740	−0.802451
$993$	0	0
$994$	−56.4985	−1.79202
$995$	14.5404	0.460960
$996$	0	0
$997$	−23.1577	−0.733412	−0.366706	−	0.930337i	$-0.619515\pi$
$997$	−23.1577	−0.733412	−0.366706	+	0.930337i	$0.619515\pi$
$998$	2.73614	0.0866109
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3645.2.a.l.1.17		21
3.2	odd	2		3645.2.a.k.1.5		21
27.4	even	9		135.2.k.b.16.6	✓	42
27.7	even	9		135.2.k.b.76.6	yes	42
27.20	odd	18		405.2.k.b.361.2		42
27.23	odd	18		405.2.k.b.46.2		42
135.4	even	18		675.2.l.e.151.2		42
135.7	odd	36		675.2.u.d.49.12		84
135.34	even	18		675.2.l.e.76.2		42
135.58	odd	36		675.2.u.d.124.12		84
135.88	odd	36		675.2.u.d.49.3		84
135.112	odd	36		675.2.u.d.124.3		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.k.b.16.6	✓	42	27.4	even	9
135.2.k.b.76.6	yes	42	27.7	even	9
405.2.k.b.46.2		42	27.23	odd	18
405.2.k.b.361.2		42	27.20	odd	18
675.2.l.e.76.2		42	135.34	even	18
675.2.l.e.151.2		42	135.4	even	18
675.2.u.d.49.3		84	135.88	odd	36
675.2.u.d.49.12		84	135.7	odd	36
675.2.u.d.124.3		84	135.112	odd	36
675.2.u.d.124.12		84	135.58	odd	36
3645.2.a.k.1.5		21	3.2	odd	2
3645.2.a.l.1.17		21	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3645 = 3^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3645.a (trivial)

\(f(q)\)	\(=\)	\(q+1.86636 q^{2} +1.48328 q^{4} +1.00000 q^{5} +3.47631 q^{7} -0.964379 q^{8} +O(q^{10})\)	\(q+1.86636 q^{2} +1.48328 q^{4} +1.00000 q^{5} +3.47631 q^{7} -0.964379 q^{8} +1.86636 q^{10} +3.42573 q^{11} -2.99117 q^{13} +6.48802 q^{14} -4.76644 q^{16} +1.83001 q^{17} -5.66300 q^{19} +1.48328 q^{20} +6.39363 q^{22} +8.50448 q^{23} +1.00000 q^{25} -5.58258 q^{26} +5.15634 q^{28} +3.54878 q^{29} +3.62762 q^{31} -6.96711 q^{32} +3.41545 q^{34} +3.47631 q^{35} +8.25062 q^{37} -10.5692 q^{38} -0.964379 q^{40} +4.56318 q^{41} +6.81253 q^{43} +5.08133 q^{44} +15.8724 q^{46} -4.31388 q^{47} +5.08470 q^{49} +1.86636 q^{50} -4.43675 q^{52} -6.55503 q^{53} +3.42573 q^{55} -3.35248 q^{56} +6.62328 q^{58} +3.26003 q^{59} +3.39466 q^{61} +6.77043 q^{62} -3.47022 q^{64} -2.99117 q^{65} -4.16593 q^{67} +2.71443 q^{68} +6.48802 q^{70} -8.70813 q^{71} -7.94401 q^{73} +15.3986 q^{74} -8.39983 q^{76} +11.9089 q^{77} +14.6559 q^{79} -4.76644 q^{80} +8.51652 q^{82} -14.4374 q^{83} +1.83001 q^{85} +12.7146 q^{86} -3.30370 q^{88} -3.57607 q^{89} -10.3982 q^{91} +12.6145 q^{92} -8.05123 q^{94} -5.66300 q^{95} -3.66068 q^{97} +9.48986 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 3 q^{2} + 27 q^{4} + 21 q^{5} + 12 q^{7} + 9 q^{8}+O(q^{10}) \) 21 * q + 3 * q^2 + 27 * q^4 + 21 * q^5 + 12 * q^7 + 9 * q^8	\( 21 q + 3 q^{2} + 27 q^{4} + 21 q^{5} + 12 q^{7} + 9 q^{8} + 3 q^{10} + 12 q^{13} + 39 q^{16} + 12 q^{17} + 24 q^{19} + 27 q^{20} + 18 q^{22} + 18 q^{23} + 21 q^{25} - 9 q^{26} + 30 q^{28} - 3 q^{29} + 24 q^{31} + 6 q^{32} + 18 q^{34} + 12 q^{35} + 24 q^{37} + 18 q^{38} + 9 q^{40} - 9 q^{41} + 42 q^{43} - 12 q^{44} + 30 q^{46} + 21 q^{47} + 39 q^{49} + 3 q^{50} + 36 q^{52} + 18 q^{53} - 30 q^{56} + 30 q^{58} - 6 q^{59} + 33 q^{61} + 18 q^{62} + 63 q^{64} + 12 q^{65} + 63 q^{67} + 36 q^{68} - 12 q^{71} + 39 q^{73} - 21 q^{74} + 48 q^{76} + 9 q^{77} + 48 q^{79} + 39 q^{80} + 24 q^{82} + 33 q^{83} + 12 q^{85} - 69 q^{86} + 54 q^{88} - 9 q^{89} + 69 q^{91} + 21 q^{92} + 30 q^{94} + 24 q^{95} + 30 q^{97} + 15 q^{98}+O(q^{100}) \) 21 * q + 3 * q^2 + 27 * q^4 + 21 * q^5 + 12 * q^7 + 9 * q^8 + 3 * q^10 + 12 * q^13 + 39 * q^16 + 12 * q^17 + 24 * q^19 + 27 * q^20 + 18 * q^22 + 18 * q^23 + 21 * q^25 - 9 * q^26 + 30 * q^28 - 3 * q^29 + 24 * q^31 + 6 * q^32 + 18 * q^34 + 12 * q^35 + 24 * q^37 + 18 * q^38 + 9 * q^40 - 9 * q^41 + 42 * q^43 - 12 * q^44 + 30 * q^46 + 21 * q^47 + 39 * q^49 + 3 * q^50 + 36 * q^52 + 18 * q^53 - 30 * q^56 + 30 * q^58 - 6 * q^59 + 33 * q^61 + 18 * q^62 + 63 * q^64 + 12 * q^65 + 63 * q^67 + 36 * q^68 - 12 * q^71 + 39 * q^73 - 21 * q^74 + 48 * q^76 + 9 * q^77 + 48 * q^79 + 39 * q^80 + 24 * q^82 + 33 * q^83 + 12 * q^85 - 69 * q^86 + 54 * q^88 - 9 * q^89 + 69 * q^91 + 21 * q^92 + 30 * q^94 + 24 * q^95 + 30 * q^97 + 15 * q^98

Introduction

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