LMFDB - Embedded newform 1369.2.a.o.1.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1369,2,Mod(1,1369)] 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("1369.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1369, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1369 = 37^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1369.a (trivial)

Newform invariants

comment:select newform

sage:traces = [27,9,-1,25,17,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$10.9315200367$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	1369.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.526565 q^{2} -2.50816 q^{3} -1.72273 q^{4} -0.0684348 q^{5} -1.32071 q^{6} -3.53356 q^{7} -1.96026 q^{8} +3.29085 q^{9} -0.0360353 q^{10} -3.28404 q^{11} +4.32088 q^{12} -4.46058 q^{13} -1.86065 q^{14} +0.171645 q^{15} +2.41326 q^{16} -6.41920 q^{17} +1.73285 q^{18} -1.03070 q^{19} +0.117895 q^{20} +8.86271 q^{21} -1.72926 q^{22} +4.56098 q^{23} +4.91664 q^{24} -4.99532 q^{25} -2.34878 q^{26} -0.729508 q^{27} +6.08736 q^{28} -6.88401 q^{29} +0.0903823 q^{30} -3.11573 q^{31} +5.19125 q^{32} +8.23689 q^{33} -3.38012 q^{34} +0.241818 q^{35} -5.66925 q^{36} -0.542728 q^{38} +11.1878 q^{39} +0.134150 q^{40} +0.132134 q^{41} +4.66679 q^{42} +1.05842 q^{43} +5.65751 q^{44} -0.225209 q^{45} +2.40165 q^{46} +6.28952 q^{47} -6.05283 q^{48} +5.48602 q^{49} -2.63036 q^{50} +16.1004 q^{51} +7.68437 q^{52} -10.3751 q^{53} -0.384133 q^{54} +0.224743 q^{55} +6.92668 q^{56} +2.58515 q^{57} -3.62488 q^{58} -0.295332 q^{59} -0.295698 q^{60} +0.197725 q^{61} -1.64063 q^{62} -11.6284 q^{63} -2.09298 q^{64} +0.305259 q^{65} +4.33726 q^{66} -8.35060 q^{67} +11.0585 q^{68} -11.4396 q^{69} +0.127333 q^{70} +5.41508 q^{71} -6.45092 q^{72} +10.3221 q^{73} +12.5290 q^{75} +1.77561 q^{76} +11.6043 q^{77} +5.89112 q^{78} +13.9537 q^{79} -0.165151 q^{80} -8.04284 q^{81} +0.0695773 q^{82} +5.50740 q^{83} -15.2681 q^{84} +0.439296 q^{85} +0.557325 q^{86} +17.2662 q^{87} +6.43757 q^{88} +4.93951 q^{89} -0.118587 q^{90} +15.7617 q^{91} -7.85733 q^{92} +7.81474 q^{93} +3.31184 q^{94} +0.0705354 q^{95} -13.0205 q^{96} +5.81769 q^{97} +2.88874 q^{98} -10.8073 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 9 q^{2} - q^{3} + 25 q^{4} + 17 q^{5} + 10 q^{6} - 3 q^{7} + 21 q^{8} + 20 q^{9} - 11 q^{10} - 5 q^{11} - 10 q^{12} + 15 q^{13} + 25 q^{14} + 23 q^{15} + 13 q^{16} + 26 q^{17} + 15 q^{18} + 27 q^{19}+ \cdots + 35 q^{99}+O(q^{100}) $ 27 * q + 9 * q^2 - q^3 + 25 * q^4 + 17 * q^5 + 10 * q^6 - 3 * q^7 + 21 * q^8 + 20 * q^9 - 11 * q^10 - 5 * q^11 - 10 * q^12 + 15 * q^13 + 25 * q^14 + 23 * q^15 + 13 * q^16 + 26 * q^17 + 15 * q^18 + 27 * q^19 + 38 * q^20 - 17 * q^21 - 4 * q^22 + 41 * q^23 + 27 * q^24 + 16 * q^25 - 23 * q^26 - 13 * q^27 + 48 * q^28 + 42 * q^29 + 46 * q^30 + 45 * q^31 + 31 * q^32 - 33 * q^33 + 49 * q^34 + 51 * q^35 - 25 * q^36 - 11 * q^38 + 47 * q^39 + 31 * q^40 + 7 * q^41 - 7 * q^42 + 11 * q^43 - 34 * q^44 + 24 * q^45 - 15 * q^46 - 21 * q^47 + 6 * q^48 + 6 * q^49 + 19 * q^50 + 37 * q^51 + 6 * q^52 - 7 * q^53 + 7 * q^54 + 35 * q^55 + 54 * q^56 - 3 * q^57 + 37 * q^58 + 74 * q^59 - 14 * q^60 + 26 * q^61 - 33 * q^62 - 50 * q^63 - 27 * q^64 - 6 * q^65 + 19 * q^66 + 2 * q^67 + 57 * q^68 + 28 * q^69 + 6 * q^70 + 5 * q^71 - 12 * q^72 - 23 * q^73 + 34 * q^75 + 27 * q^76 - 11 * q^77 + 17 * q^78 + 58 * q^79 + 36 * q^80 - 37 * q^81 - 34 * q^82 + 56 * q^83 - 40 * q^84 - 16 * q^85 - 45 * q^86 + 66 * q^87 - 33 * q^88 + 64 * q^89 - 34 * q^90 + 17 * q^91 + 34 * q^92 - 30 * q^93 - 5 * q^94 - 22 * q^95 + 19 * q^96 - 17 * q^97 - 6 * q^98 + 35 * q^99

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 $

$2$	0.526565	0.372337	0.186169	−	0.982518i	$-0.440393\pi$
$2$	0.526565	0.372337	0.186169	+	0.982518i	$0.440393\pi$
$3$	−2.50816	−1.44809	−0.724043	−	0.689755i	$-0.757718\pi$
$3$	−2.50816	−1.44809	−0.724043	+	0.689755i	$0.757718\pi$
$4$	−1.72273	−0.861365
$5$	−0.0684348	−0.0306050	−0.0153025	−	0.999883i	$-0.504871\pi$
$5$	−0.0684348	−0.0306050	−0.0153025	+	0.999883i	$0.504871\pi$
$6$	−1.32071	−0.539176
$7$	−3.53356	−1.33556	−0.667779	−	0.744359i	$-0.732755\pi$
$7$	−3.53356	−1.33556	−0.667779	+	0.744359i	$0.732755\pi$
$8$	−1.96026	−0.693056
$9$	3.29085	1.09695
$10$	−0.0360353	−0.0113954
$11$	−3.28404	−0.990176	−0.495088	−	0.868843i	$-0.664864\pi$
$11$	−3.28404	−0.990176	−0.495088	+	0.868843i	$0.664864\pi$
$12$	4.32088	1.24733
$13$	−4.46058	−1.23714	−0.618571	−	0.785729i	$-0.712288\pi$
$13$	−4.46058	−1.23714	−0.618571	+	0.785729i	$0.712288\pi$
$14$	−1.86065	−0.497278
$15$	0.171645	0.0443186
$16$	2.41326	0.603314
$17$	−6.41920	−1.55688	−0.778442	−	0.627716i	$-0.783990\pi$
$17$	−6.41920	−1.55688	−0.778442	+	0.627716i	$0.783990\pi$
$18$	1.73285	0.408436
$19$	−1.03070	−0.236458	−0.118229	−	0.992986i	$-0.537722\pi$
$19$	−1.03070	−0.236458	−0.118229	+	0.992986i	$0.537722\pi$
$20$	0.117895	0.0263620
$21$	8.86271	1.93400
$22$	−1.72926	−0.368679
$23$	4.56098	0.951029	0.475515	−	0.879708i	$-0.342262\pi$
$23$	4.56098	0.951029	0.475515	+	0.879708i	$0.342262\pi$
$24$	4.91664	1.00360
$25$	−4.99532	−0.999063
$26$	−2.34878	−0.460634
$27$	−0.729508	−0.140394
$28$	6.08736	1.15040
$29$	−6.88401	−1.27833	−0.639164	−	0.769070i	$-0.720720\pi$
$29$	−6.88401	−1.27833	−0.639164	+	0.769070i	$0.720720\pi$
$30$	0.0903823	0.0165015
$31$	−3.11573	−0.559601	−0.279801	−	0.960058i	$-0.590268\pi$
$31$	−3.11573	−0.559601	−0.279801	+	0.960058i	$0.590268\pi$
$32$	5.19125	0.917692
$33$	8.23689	1.43386
$34$	−3.38012	−0.579686
$35$	0.241818	0.0408747
$36$	−5.66925	−0.944875
$37$	0	0
$37$	0	0
$38$	−0.542728	−0.0880421
$39$	11.1878	1.79149
$40$	0.134150	0.0212109
$41$	0.132134	0.0206359	0.0103180	−	0.999947i	$-0.496716\pi$
$41$	0.132134	0.0206359	0.0103180	+	0.999947i	$0.496716\pi$
$42$	4.66679	0.720102
$43$	1.05842	0.161407	0.0807035	−	0.996738i	$-0.474283\pi$
$43$	1.05842	0.161407	0.0807035	+	0.996738i	$0.474283\pi$
$44$	5.65751	0.852902
$45$	−0.225209	−0.0335722
$46$	2.40165	0.354104
$47$	6.28952	0.917421	0.458711	−	0.888586i	$-0.348311\pi$
$47$	6.28952	0.917421	0.458711	+	0.888586i	$0.348311\pi$
$48$	−6.05283	−0.873650
$49$	5.48602	0.783716
$50$	−2.63036	−0.371989
$51$	16.1004	2.25450
$52$	7.68437	1.06563
$53$	−10.3751	−1.42513	−0.712564	−	0.701607i	$-0.752466\pi$
$53$	−10.3751	−1.42513	−0.712564	+	0.701607i	$0.752466\pi$
$54$	−0.384133	−0.0522739
$55$	0.224743	0.0303043
$56$	6.92668	0.925617
$57$	2.58515	0.342411
$58$	−3.62488	−0.475970
$59$	−0.295332	−0.0384489	−0.0192245	−	0.999815i	$-0.506120\pi$
$59$	−0.295332	−0.0384489	−0.0192245	+	0.999815i	$0.506120\pi$
$60$	−0.295698	−0.0381745
$61$	0.197725	0.0253161	0.0126580	−	0.999920i	$-0.495971\pi$
$61$	0.197725	0.0253161	0.0126580	+	0.999920i	$0.495971\pi$
$62$	−1.64063	−0.208360
$63$	−11.6284	−1.46504
$64$	−2.09298	−0.261623
$65$	0.305259	0.0378627
$66$	4.33726	0.533879
$67$	−8.35060	−1.02019	−0.510094	−	0.860119i	$-0.670389\pi$
$67$	−8.35060	−1.02019	−0.510094	+	0.860119i	$0.670389\pi$
$68$	11.0585	1.34105
$69$	−11.4396	−1.37717
$70$	0.127333	0.0152192
$71$	5.41508	0.642652	0.321326	−	0.946969i	$-0.395872\pi$
$71$	5.41508	0.642652	0.321326	+	0.946969i	$0.395872\pi$
$72$	−6.45092	−0.760249
$73$	10.3221	1.20811	0.604057	−	0.796941i	$-0.293550\pi$
$73$	10.3221	1.20811	0.604057	+	0.796941i	$0.293550\pi$
$74$	0	0
$75$	12.5290	1.44673
$76$	1.77561	0.203676
$77$	11.6043	1.32244
$78$	5.89112	0.667038
$79$	13.9537	1.56991	0.784956	−	0.619551i	$-0.212685\pi$
$79$	13.9537	1.56991	0.784956	+	0.619551i	$0.212685\pi$
$80$	−0.165151	−0.0184644
$81$	−8.04284	−0.893649
$82$	0.0695773	0.00768353
$83$	5.50740	0.604516	0.302258	−	0.953226i	$-0.402260\pi$
$83$	5.50740	0.604516	0.302258	+	0.953226i	$0.402260\pi$
$84$	−15.2681	−1.66588
$85$	0.439296	0.0476484
$86$	0.557325	0.0600979
$87$	17.2662	1.85113
$88$	6.43757	0.686247
$89$	4.93951	0.523587	0.261794	−	0.965124i	$-0.415686\pi$
$89$	4.93951	0.523587	0.261794	+	0.965124i	$0.415686\pi$
$90$	−0.118587	−0.0125002
$91$	15.7617	1.65228
$92$	−7.85733	−0.819183
$93$	7.81474	0.810350
$94$	3.31184	0.341590
$95$	0.0705354	0.00723678
$96$	−13.0205	−1.32890
$97$	5.81769	0.590697	0.295348	−	0.955390i	$-0.404564\pi$
$97$	5.81769	0.590697	0.295348	+	0.955390i	$0.404564\pi$
$98$	2.88874	0.291807
$99$	−10.8073	−1.08617
$100$	8.60558	0.860558
$101$	−7.73145	−0.769308	−0.384654	−	0.923061i	$-0.625679\pi$
$101$	−7.73145	−0.769308	−0.384654	+	0.923061i	$0.625679\pi$
$102$	8.47788	0.839435
$103$	−9.36515	−0.922776	−0.461388	−	0.887199i	$-0.652648\pi$
$103$	−9.36515	−0.922776	−0.461388	+	0.887199i	$0.652648\pi$
$104$	8.74389	0.857409
$105$	−0.606518	−0.0591901
$106$	−5.46316	−0.530629
$107$	−8.71420	−0.842433	−0.421217	−	0.906960i	$-0.638397\pi$
$107$	−8.71420	−0.842433	−0.421217	+	0.906960i	$0.638397\pi$
$108$	1.25675	0.120930
$109$	2.09957	0.201102	0.100551	−	0.994932i	$-0.467939\pi$
$109$	2.09957	0.201102	0.100551	+	0.994932i	$0.467939\pi$
$110$	0.118342	0.0112834
$111$	0	0
$112$	−8.52738	−0.805761
$113$	−16.9125	−1.59100	−0.795499	−	0.605955i	$-0.792791\pi$
$113$	−16.9125	−1.59100	−0.795499	+	0.605955i	$0.792791\pi$
$114$	1.36125	0.127492
$115$	−0.312129	−0.0291062
$116$	11.8593	1.10111
$117$	−14.6791	−1.35708
$118$	−0.155511	−0.0143160
$119$	22.6826	2.07931
$120$	−0.336469	−0.0307153
$121$	−0.215076	−0.0195524
$122$	0.104115	0.00942612
$123$	−0.331414	−0.0298826
$124$	5.36756	0.482021
$125$	0.684027	0.0611813
$126$	−6.12311	−0.545490
$127$	−6.57555	−0.583486	−0.291743	−	0.956497i	$-0.594235\pi$
$127$	−6.57555	−0.583486	−0.291743	+	0.956497i	$0.594235\pi$
$128$	−11.4846	−1.01510
$129$	−2.65468	−0.233731
$130$	0.160738	0.0140977
$131$	9.80674	0.856819	0.428409	−	0.903585i	$-0.359074\pi$
$131$	9.80674	0.856819	0.428409	+	0.903585i	$0.359074\pi$
$132$	−14.1899	−1.23508
$133$	3.64202	0.315803
$134$	−4.39713	−0.379854
$135$	0.0499237	0.00429675
$136$	12.5833	1.07901
$137$	−17.7097	−1.51305	−0.756523	−	0.653967i	$-0.773103\pi$
$137$	−17.7097	−1.51305	−0.756523	+	0.653967i	$0.773103\pi$
$138$	−6.02371	−0.512773
$139$	−5.25256	−0.445516	−0.222758	−	0.974874i	$-0.571506\pi$
$139$	−5.25256	−0.445516	−0.222758	+	0.974874i	$0.571506\pi$
$140$	−0.416587	−0.0352080
$141$	−15.7751	−1.32850
$142$	2.85139	0.239283
$143$	14.6487	1.22499
$144$	7.94168	0.661806
$145$	0.471106	0.0391232
$146$	5.43527	0.449826
$147$	−13.7598	−1.13489
$148$	0	0
$149$	−5.44333	−0.445935	−0.222967	−	0.974826i	$-0.571574\pi$
$149$	−5.44333	−0.445935	−0.222967	+	0.974826i	$0.571574\pi$
$150$	6.59735	0.538671
$151$	0.818729	0.0666272	0.0333136	−	0.999445i	$-0.489394\pi$
$151$	0.818729	0.0666272	0.0333136	+	0.999445i	$0.489394\pi$
$152$	2.02043	0.163878
$153$	−21.1247	−1.70783
$154$	6.11044	0.492393
$155$	0.213224	0.0171266
$156$	−19.2736	−1.54312
$157$	−4.29533	−0.342805	−0.171402	−	0.985201i	$-0.554830\pi$
$157$	−4.29533	−0.342805	−0.171402	+	0.985201i	$0.554830\pi$
$158$	7.34752	0.584537
$159$	26.0224	2.06371
$160$	−0.355262	−0.0280859
$161$	−16.1165	−1.27016
$162$	−4.23508	−0.332739
$163$	1.00287	0.0785509	0.0392754	−	0.999228i	$-0.487495\pi$
$163$	1.00287	0.0785509	0.0392754	+	0.999228i	$0.487495\pi$
$164$	−0.227632	−0.0177751
$165$	−0.563690	−0.0438832
$166$	2.90000	0.225084
$167$	−14.3325	−1.10908	−0.554540	−	0.832157i	$-0.687106\pi$
$167$	−14.3325	−1.10908	−0.554540	+	0.832157i	$0.687106\pi$
$168$	−17.3732	−1.34037
$169$	6.89677	0.530521
$170$	0.231318	0.0177413
$171$	−3.39187	−0.259383
$172$	−1.82337	−0.139030
$173$	8.04485	0.611638	0.305819	−	0.952090i	$-0.401070\pi$
$173$	8.04485	0.611638	0.305819	+	0.952090i	$0.401070\pi$
$174$	9.09176	0.689245
$175$	17.6512	1.33431
$176$	−7.92523	−0.597387
$177$	0.740739	0.0556774
$178$	2.60097	0.194951
$179$	−6.30681	−0.471393	−0.235697	−	0.971827i	$-0.575737\pi$
$179$	−6.30681	−0.471393	−0.235697	+	0.971827i	$0.575737\pi$
$180$	0.387974	0.0289179
$181$	0.866005	0.0643696	0.0321848	−	0.999482i	$-0.489753\pi$
$181$	0.866005	0.0643696	0.0321848	+	0.999482i	$0.489753\pi$
$182$	8.29956	0.615204
$183$	−0.495925	−0.0366598
$184$	−8.94069	−0.659116
$185$	0	0
$186$	4.11496	0.301724
$187$	21.0809	1.54159
$188$	−10.8351	−0.790234
$189$	2.57776	0.187504
$190$	0.0371415	0.00269452
$191$	9.29054	0.672240	0.336120	−	0.941819i	$-0.390885\pi$
$191$	9.29054	0.672240	0.336120	+	0.941819i	$0.390885\pi$
$192$	5.24953	0.378852
$193$	−2.92124	−0.210276	−0.105138	−	0.994458i	$-0.533528\pi$
$193$	−2.92124	−0.210276	−0.105138	+	0.994458i	$0.533528\pi$
$194$	3.06339	0.219938
$195$	−0.765637	−0.0548284
$196$	−9.45092	−0.675066
$197$	−8.07471	−0.575300	−0.287650	−	0.957736i	$-0.592874\pi$
$197$	−8.07471	−0.575300	−0.287650	+	0.957736i	$0.592874\pi$
$198$	−5.69074	−0.404423
$199$	−26.7947	−1.89943	−0.949713	−	0.313123i	$-0.898625\pi$
$199$	−26.7947	−1.89943	−0.949713	+	0.313123i	$0.898625\pi$
$200$	9.79211	0.692407
$201$	20.9446	1.47732
$202$	−4.07111	−0.286442
$203$	24.3250	1.70728
$204$	−27.7366	−1.94195
$205$	−0.00904259	−0.000631562 0
$206$	−4.93136	−0.343584
$207$	15.0095	1.04323
$208$	−10.7645	−0.746386
$209$	3.38485	0.234135
$210$	−0.319371	−0.0220387
$211$	23.8048	1.63879	0.819395	−	0.573230i	$-0.194310\pi$
$211$	23.8048	1.63879	0.819395	+	0.573230i	$0.194310\pi$
$212$	17.8735	1.22756
$213$	−13.5819	−0.930615
$214$	−4.58859	−0.313669
$215$	−0.0724325	−0.00493986
$216$	1.43002	0.0973008
$217$	11.0096	0.747380
$218$	1.10556	0.0748779
$219$	−25.8895	−1.74945
$220$	−0.387171	−0.0261030
$221$	28.6334	1.92609
$222$	0	0
$223$	−14.5194	−0.972288	−0.486144	−	0.873879i	$-0.661597\pi$
$223$	−14.5194	−0.972288	−0.486144	+	0.873879i	$0.661597\pi$
$224$	−18.3436	−1.22563
$225$	−16.4389	−1.09592
$226$	−8.90555	−0.592388
$227$	−25.8042	−1.71269	−0.856344	−	0.516406i	$-0.827270\pi$
$227$	−25.8042	−1.71269	−0.856344	+	0.516406i	$0.827270\pi$
$228$	−4.45351	−0.294941
$229$	−12.9045	−0.852751	−0.426376	−	0.904546i	$-0.640210\pi$
$229$	−12.9045	−0.852751	−0.426376	+	0.904546i	$0.640210\pi$
$230$	−0.164356	−0.0108373
$231$	−29.1055	−1.91500
$232$	13.4944	0.885953
$233$	−25.7678	−1.68811	−0.844054	−	0.536258i	$-0.819837\pi$
$233$	−25.7678	−1.68811	−0.844054	+	0.536258i	$0.819837\pi$
$234$	−7.72951	−0.505294
$235$	−0.430422	−0.0280776
$236$	0.508777	0.0331186
$237$	−34.9980	−2.27337
$238$	11.9439	0.774205
$239$	−4.23509	−0.273945	−0.136973	−	0.990575i	$-0.543737\pi$
$239$	−4.23509	−0.273945	−0.136973	+	0.990575i	$0.543737\pi$
$240$	0.414224	0.0267380
$241$	26.7537	1.72335	0.861677	−	0.507457i	$-0.169414\pi$
$241$	26.7537	1.72335	0.861677	+	0.507457i	$0.169414\pi$
$242$	−0.113252	−0.00728009
$243$	22.3612	1.43447
$244$	−0.340627	−0.0218064
$245$	−0.375434	−0.0239856
$246$	−0.174511	−0.0111264
$247$	4.59750	0.292532
$248$	6.10763	0.387835
$249$	−13.8134	−0.875390
$250$	0.360185	0.0227801
$251$	20.3984	1.28753	0.643766	−	0.765222i	$-0.277371\pi$
$251$	20.3984	1.28753	0.643766	+	0.765222i	$0.277371\pi$
$252$	20.0326	1.26194
$253$	−14.9784	−0.941686
$254$	−3.46245	−0.217254
$255$	−1.10182	−0.0689989
$256$	−1.86141	−0.116338
$257$	−3.30196	−0.205971	−0.102985	−	0.994683i	$-0.532840\pi$
$257$	−3.30196	−0.205971	−0.102985	+	0.994683i	$0.532840\pi$
$258$	−1.39786	−0.0870269
$259$	0	0
$260$	−0.525878	−0.0326136
$261$	−22.6543	−1.40226
$262$	5.16388	0.319026
$263$	−9.32130	−0.574776	−0.287388	−	0.957814i	$-0.592787\pi$
$263$	−9.32130	−0.574776	−0.287388	+	0.957814i	$0.592787\pi$
$264$	−16.1464	−0.993744
$265$	0.710017	0.0436160
$266$	1.91776	0.117585
$267$	−12.3891	−0.758199
$268$	14.3858	0.878754
$269$	−0.204606	−0.0124751	−0.00623754	−	0.999981i	$-0.501985\pi$
$269$	−0.204606	−0.0124751	−0.00623754	+	0.999981i	$0.501985\pi$
$270$	0.0262881	0.00159984
$271$	−1.48951	−0.0904811	−0.0452406	−	0.998976i	$-0.514405\pi$
$271$	−1.48951	−0.0904811	−0.0452406	+	0.998976i	$0.514405\pi$
$272$	−15.4912	−0.939291
$273$	−39.5328	−2.39264
$274$	−9.32533	−0.563364
$275$	16.4048	0.989248
$276$	19.7074	1.18625
$277$	−24.2853	−1.45916	−0.729582	−	0.683893i	$-0.760285\pi$
$277$	−24.2853	−1.45916	−0.729582	+	0.683893i	$0.760285\pi$
$278$	−2.76581	−0.165882
$279$	−10.2534	−0.613855
$280$	−0.474026	−0.0283285
$281$	22.0161	1.31337	0.656684	−	0.754166i	$-0.271959\pi$
$281$	22.0161	1.31337	0.656684	+	0.754166i	$0.271959\pi$
$282$	−8.30662	−0.494652
$283$	−15.7565	−0.936627	−0.468313	−	0.883562i	$-0.655138\pi$
$283$	−15.7565	−0.936627	−0.468313	+	0.883562i	$0.655138\pi$
$284$	−9.32872	−0.553558
$285$	−0.176914	−0.0104795
$286$	7.71350	0.456109
$287$	−0.466905	−0.0275605
$288$	17.0837	1.00666
$289$	24.2061	1.42389
$290$	0.248068	0.0145670
$291$	−14.5917	−0.855379
$292$	−17.7822	−1.04063
$293$	12.6623	0.739741	0.369871	−	0.929083i	$-0.379402\pi$
$293$	12.6623	0.739741	0.369871	+	0.929083i	$0.379402\pi$
$294$	−7.24542	−0.422561
$295$	0.0202110	0.00117673
$296$	0	0
$297$	2.39573	0.139015
$298$	−2.86626	−0.166038
$299$	−20.3446	−1.17656
$300$	−21.5842	−1.24616
$301$	−3.73998	−0.215569
$302$	0.431114	0.0248078
$303$	19.3917	1.11402
$304$	−2.48733	−0.142658
$305$	−0.0135313	−0.000774798 0
$306$	−11.1235	−0.635888
$307$	33.4953	1.91168	0.955839	−	0.293891i	$-0.0949502\pi$
$307$	33.4953	1.91168	0.955839	+	0.293891i	$0.0949502\pi$
$308$	−19.9911	−1.13910
$309$	23.4893	1.33626
$310$	0.112276	0.00637686
$311$	27.4894	1.55878	0.779389	−	0.626540i	$-0.215529\pi$
$311$	27.4894	1.55878	0.779389	+	0.626540i	$0.215529\pi$
$312$	−21.9310	−1.24160
$313$	−14.7595	−0.834254	−0.417127	−	0.908848i	$-0.636963\pi$
$313$	−14.7595	−0.834254	−0.417127	+	0.908848i	$0.636963\pi$
$314$	−2.26177	−0.127639
$315$	0.795788	0.0448376
$316$	−24.0384	−1.35227
$317$	−23.4424	−1.31666	−0.658328	−	0.752731i	$-0.728736\pi$
$317$	−23.4424	−1.31666	−0.658328	+	0.752731i	$0.728736\pi$
$318$	13.7025	0.768396
$319$	22.6074	1.26577
$320$	0.143233	0.00800696
$321$	21.8566	1.21992
$322$	−8.48636	−0.472926
$323$	6.61624	0.368137
$324$	13.8556	0.769758
$325$	22.2820	1.23598
$326$	0.528076	0.0292474
$327$	−5.26605	−0.291213
$328$	−0.259018	−0.0143019
$329$	−22.2244	−1.22527
$330$	−0.296819	−0.0163394
$331$	7.78059	0.427660	0.213830	−	0.976871i	$-0.431406\pi$
$331$	7.78059	0.427660	0.213830	+	0.976871i	$0.431406\pi$
$332$	−9.48776	−0.520709
$333$	0	0
$334$	−7.54697	−0.412952
$335$	0.571471	0.0312228
$336$	21.3880	1.16681
$337$	2.12301	0.115648	0.0578238	−	0.998327i	$-0.481584\pi$
$337$	2.12301	0.115648	0.0578238	+	0.998327i	$0.481584\pi$
$338$	3.63160	0.197533
$339$	42.4193	2.30390
$340$	−0.756789	−0.0410427
$341$	10.2322	0.554103
$342$	−1.78604	−0.0965779
$343$	5.34975	0.288859
$344$	−2.07477	−0.111864
$345$	0.782870	0.0421483
$346$	4.23613	0.227736
$347$	0.922491	0.0495219	0.0247610	−	0.999693i	$-0.492118\pi$
$347$	0.922491	0.0495219	0.0247610	+	0.999693i	$0.492118\pi$
$348$	−29.7450	−1.59450
$349$	−11.1602	−0.597394	−0.298697	−	0.954348i	$-0.596552\pi$
$349$	−11.1602	−0.597394	−0.298697	+	0.954348i	$0.596552\pi$
$350$	9.29451	0.496813
$351$	3.25403	0.173687
$352$	−17.0483	−0.908676
$353$	32.1241	1.70979	0.854895	−	0.518800i	$-0.173621\pi$
$353$	32.1241	1.70979	0.854895	+	0.518800i	$0.173621\pi$
$354$	0.390047	0.0207308
$355$	−0.370580	−0.0196683
$356$	−8.50944	−0.450999
$357$	−56.8915	−3.01102
$358$	−3.32095	−0.175517
$359$	14.1001	0.744177	0.372089	−	0.928197i	$-0.378642\pi$
$359$	14.1001	0.744177	0.372089	+	0.928197i	$0.378642\pi$
$360$	0.441467	0.0232674
$361$	−17.9377	−0.944088
$362$	0.456008	0.0239672
$363$	0.539445	0.0283135
$364$	−27.1532	−1.42321
$365$	−0.706393	−0.0369743
$366$	−0.261137	−0.0136498
$367$	−10.4345	−0.544679	−0.272339	−	0.962201i	$-0.587797\pi$
$367$	−10.4345	−0.544679	−0.272339	+	0.962201i	$0.587797\pi$
$368$	11.0068	0.573769
$369$	0.434835	0.0226366
$370$	0	0
$371$	36.6610	1.90334
$372$	−13.4627	−0.698007
$373$	16.2539	0.841597	0.420799	−	0.907154i	$-0.361750\pi$
$373$	16.2539	0.841597	0.420799	+	0.907154i	$0.361750\pi$
$374$	11.1005	0.573991
$375$	−1.71565	−0.0885957
$376$	−12.3291	−0.635824
$377$	30.7067	1.58147
$378$	1.35736	0.0698149
$379$	−19.0074	−0.976344	−0.488172	−	0.872748i	$-0.662336\pi$
$379$	−19.0074	−0.976344	−0.488172	+	0.872748i	$0.662336\pi$
$380$	−0.121513	−0.00623351
$381$	16.4925	0.844937
$382$	4.89207	0.250300
$383$	−1.37881	−0.0704538	−0.0352269	−	0.999379i	$-0.511215\pi$
$383$	−1.37881	−0.0704538	−0.0352269	+	0.999379i	$0.511215\pi$
$384$	28.8052	1.46996
$385$	−0.794140	−0.0404731
$386$	−1.53822	−0.0782935
$387$	3.48310	0.177056
$388$	−10.0223	−0.508805
$389$	20.5060	1.03969	0.519847	−	0.854260i	$-0.325989\pi$
$389$	20.5060	1.03969	0.519847	+	0.854260i	$0.325989\pi$
$390$	−0.403157	−0.0204147
$391$	−29.2778	−1.48064
$392$	−10.7540	−0.543159
$393$	−24.5968	−1.24075
$394$	−4.25186	−0.214206
$395$	−0.954917	−0.0480471
$396$	18.6181	0.935592
$397$	−35.8997	−1.80176	−0.900878	−	0.434072i	$-0.857076\pi$
$397$	−35.8997	−1.80176	−0.900878	+	0.434072i	$0.857076\pi$
$398$	−14.1091	−0.707227
$399$	−9.13476	−0.457310
$400$	−12.0550	−0.602749
$401$	39.0228	1.94870	0.974352	−	0.225029i	$-0.0722478\pi$
$401$	39.0228	1.94870	0.974352	+	0.225029i	$0.0722478\pi$
$402$	11.0287	0.550061
$403$	13.8980	0.692306
$404$	13.3192	0.662655
$405$	0.550410	0.0273501
$406$	12.8087	0.635685
$407$	0	0
$408$	−31.5609	−1.56250
$409$	−12.0635	−0.596502	−0.298251	−	0.954488i	$-0.596403\pi$
$409$	−12.0635	−0.596502	−0.298251	+	0.954488i	$0.596403\pi$
$410$	−0.00476151	−0.000235154 0
$411$	44.4188	2.19102
$412$	16.1336	0.794846
$413$	1.04357	0.0513508
$414$	7.90348	0.388435
$415$	−0.376898	−0.0185012
$416$	−23.1560	−1.13532
$417$	13.1743	0.645146
$418$	1.78234	0.0871771
$419$	−26.7346	−1.30607	−0.653036	−	0.757327i	$-0.726505\pi$
$419$	−26.7346	−1.30607	−0.653036	+	0.757327i	$0.726505\pi$
$420$	1.04487	0.0509843
$421$	28.5640	1.39212	0.696062	−	0.717981i	$-0.254934\pi$
$421$	28.5640	1.39212	0.696062	+	0.717981i	$0.254934\pi$
$422$	12.5348	0.610183
$423$	20.6979	1.00637
$424$	20.3379	0.987693
$425$	32.0659	1.55543
$426$	−7.15173	−0.346503
$427$	−0.698672	−0.0338111
$428$	15.0122	0.725642
$429$	−36.7413	−1.77389
$430$	−0.0381404	−0.00183929
$431$	36.5545	1.76077	0.880384	−	0.474262i	$-0.157285\pi$
$431$	36.5545	1.76077	0.880384	+	0.474262i	$0.157285\pi$
$432$	−1.76049	−0.0847016
$433$	12.9039	0.620124	0.310062	−	0.950716i	$-0.399650\pi$
$433$	12.9039	0.620124	0.310062	+	0.950716i	$0.399650\pi$
$434$	5.79726	0.278278
$435$	−1.18161	−0.0566537
$436$	−3.61699	−0.173222
$437$	−4.70098	−0.224878
$438$	−13.6325	−0.651387
$439$	1.75636	0.0838264	0.0419132	−	0.999121i	$-0.486655\pi$
$439$	1.75636	0.0838264	0.0419132	+	0.999121i	$0.486655\pi$
$440$	−0.440553	−0.0210026
$441$	18.0537	0.859699
$442$	15.0773	0.717155
$443$	−4.25988	−0.202393	−0.101197	−	0.994866i	$-0.532267\pi$
$443$	−4.25988	−0.202393	−0.101197	+	0.994866i	$0.532267\pi$
$444$	0	0
$445$	−0.338034	−0.0160244
$446$	−7.64538	−0.362019
$447$	13.6527	0.645752
$448$	7.39568	0.349413
$449$	−5.66630	−0.267409	−0.133705	−	0.991021i	$-0.542687\pi$
$449$	−5.66630	−0.267409	−0.133705	+	0.991021i	$0.542687\pi$
$450$	−8.65612	−0.408054
$451$	−0.433935	−0.0204332
$452$	29.1357	1.37043
$453$	−2.05350	−0.0964819
$454$	−13.5876	−0.637698
$455$	−1.07865	−0.0505678
$456$	−5.06755	−0.237310
$457$	6.02235	0.281714	0.140857	−	0.990030i	$-0.455014\pi$
$457$	6.02235	0.281714	0.140857	+	0.990030i	$0.455014\pi$
$458$	−6.79504	−0.317511
$459$	4.68286	0.218577
$460$	0.537715	0.0250711
$461$	31.9058	1.48600	0.743002	−	0.669289i	$-0.233401\pi$
$461$	31.9058	1.48600	0.743002	+	0.669289i	$0.233401\pi$
$462$	−15.3259	−0.713027
$463$	−1.96921	−0.0915169	−0.0457584	−	0.998953i	$-0.514570\pi$
$463$	−1.96921	−0.0915169	−0.0457584	+	0.998953i	$0.514570\pi$
$464$	−16.6129	−0.771234
$465$	−0.534800	−0.0248007
$466$	−13.5684	−0.628546
$467$	23.6540	1.09458	0.547289	−	0.836944i	$-0.315660\pi$
$467$	23.6540	1.09458	0.547289	+	0.836944i	$0.315660\pi$
$468$	25.2882	1.16895
$469$	29.5073	1.36252
$470$	−0.226645	−0.0104544
$471$	10.7734	0.496410
$472$	0.578927	0.0266473
$473$	−3.47588	−0.159821
$474$	−18.4287	−0.846460
$475$	5.14865	0.236236
$476$	−39.0760	−1.79104
$477$	−34.1429	−1.56330
$478$	−2.23005	−0.102000
$479$	16.8900	0.771724	0.385862	−	0.922557i	$-0.373904\pi$
$479$	16.8900	0.771724	0.385862	+	0.922557i	$0.373904\pi$
$480$	0.891053	0.0406708
$481$	0	0
$482$	14.0875	0.641669
$483$	40.4226	1.83929
$484$	0.370518	0.0168417
$485$	−0.398132	−0.0180782
$486$	11.7746	0.534108
$487$	21.8897	0.991917	0.495959	−	0.868346i	$-0.334817\pi$
$487$	21.8897	0.991917	0.495959	+	0.868346i	$0.334817\pi$
$488$	−0.387592	−0.0175455
$489$	−2.51536	−0.113748
$490$	−0.197690	−0.00893074
$491$	−29.8896	−1.34890	−0.674449	−	0.738322i	$-0.735619\pi$
$491$	−29.8896	−1.34890	−0.674449	+	0.738322i	$0.735619\pi$
$492$	0.570937	0.0257398
$493$	44.1898	1.99021
$494$	2.42088	0.108921
$495$	0.739595	0.0332423
$496$	−7.51905	−0.337615
$497$	−19.1345	−0.858299
$498$	−7.27366	−0.325941
$499$	−19.1355	−0.856623	−0.428312	−	0.903631i	$-0.640891\pi$
$499$	−19.1355	−0.856623	−0.428312	+	0.903631i	$0.640891\pi$
$500$	−1.17839	−0.0526994
$501$	35.9481	1.60604
$502$	10.7411	0.479397
$503$	−2.93324	−0.130787	−0.0653935	−	0.997860i	$-0.520830\pi$
$503$	−2.93324	−0.130787	−0.0653935	+	0.997860i	$0.520830\pi$
$504$	22.7947	1.01536
$505$	0.529100	0.0235447
$506$	−7.88711	−0.350625
$507$	−17.2982	−0.768240
$508$	11.3279	0.502594
$509$	17.8704	0.792093	0.396047	−	0.918230i	$-0.370382\pi$
$509$	17.8704	0.792093	0.396047	+	0.918230i	$0.370382\pi$
$510$	−0.580182	−0.0256909
$511$	−36.4738	−1.61351
$512$	21.9890	0.971787
$513$	0.751901	0.0331972
$514$	−1.73870	−0.0766907
$515$	0.640902	0.0282415
$516$	4.57329	0.201328
$517$	−20.6551	−0.908408
$518$	0	0
$519$	−20.1777	−0.885705
$520$	−0.598386	−0.0262410
$521$	−23.8776	−1.04610	−0.523048	−	0.852304i	$-0.675205\pi$
$521$	−23.8776	−1.04610	−0.523048	+	0.852304i	$0.675205\pi$
$522$	−11.9289	−0.522115
$523$	−20.4872	−0.895842	−0.447921	−	0.894073i	$-0.647835\pi$
$523$	−20.4872	−0.895842	−0.447921	+	0.894073i	$0.647835\pi$
$524$	−16.8944	−0.738034
$525$	−44.2721	−1.93219
$526$	−4.90827	−0.214011
$527$	20.0005	0.871235
$528$	19.8777	0.865067
$529$	−2.19750	−0.0955434
$530$	0.373870	0.0162399
$531$	−0.971894	−0.0421766
$532$	−6.27422	−0.272022
$533$	−0.589396	−0.0255296
$534$	−6.52365	−0.282306
$535$	0.596354	0.0257826
$536$	16.3693	0.707047
$537$	15.8185	0.682618
$538$	−0.107739	−0.00464494
$539$	−18.0163	−0.776017
$540$	−0.0860051	−0.00370107
$541$	−41.5639	−1.78697	−0.893485	−	0.449092i	$-0.851747\pi$
$541$	−41.5639	−1.78697	−0.893485	+	0.449092i	$0.851747\pi$
$542$	−0.784322	−0.0336895
$543$	−2.17208	−0.0932127
$544$	−33.3237	−1.42874
$545$	−0.143684	−0.00615473
$546$	−20.8166	−0.890868
$547$	−30.7561	−1.31504	−0.657518	−	0.753439i	$-0.728394\pi$
$547$	−30.7561	−1.31504	−0.657518	+	0.753439i	$0.728394\pi$
$548$	30.5091	1.30328
$549$	0.650684	0.0277705
$550$	8.63820	0.368334
$551$	7.09532	0.302271
$552$	22.4247	0.954457
$553$	−49.3061	−2.09671
$554$	−12.7878	−0.543302
$555$	0	0
$556$	9.04874	0.383752
$557$	−6.28603	−0.266348	−0.133174	−	0.991093i	$-0.542517\pi$
$557$	−6.28603	−0.266348	−0.133174	+	0.991093i	$0.542517\pi$
$558$	−5.39908	−0.228561
$559$	−4.72115	−0.199684
$560$	0.583569	0.0246603
$561$	−52.8743	−2.23235
$562$	11.5929	0.489016
$563$	3.90668	0.164647	0.0823234	−	0.996606i	$-0.473766\pi$
$563$	3.90668	0.164647	0.0823234	+	0.996606i	$0.473766\pi$
$564$	27.1763	1.14433
$565$	1.15741	0.0486924
$566$	−8.29682	−0.348741
$567$	28.4198	1.19352
$568$	−10.6150	−0.445394
$569$	−9.18872	−0.385211	−0.192606	−	0.981276i	$-0.561694\pi$
$569$	−9.18872	−0.385211	−0.192606	+	0.981276i	$0.561694\pi$
$570$	−0.0931566	−0.00390190
$571$	−5.00637	−0.209510	−0.104755	−	0.994498i	$-0.533406\pi$
$571$	−5.00637	−0.209510	−0.104755	+	0.994498i	$0.533406\pi$
$572$	−25.2358	−1.05516
$573$	−23.3021	−0.973461
$574$	−0.245855	−0.0102618
$575$	−22.7835	−0.950138
$576$	−6.88770	−0.286988
$577$	23.7048	0.986843	0.493422	−	0.869790i	$-0.335746\pi$
$577$	23.7048	0.986843	0.493422	+	0.869790i	$0.335746\pi$
$578$	12.7461	0.530167
$579$	7.32694	0.304497
$580$	−0.811588	−0.0336993
$581$	−19.4607	−0.807366
$582$	−7.68346	−0.318490
$583$	34.0722	1.41113
$584$	−20.2340	−0.837291
$585$	1.00456	0.0415335
$586$	6.66754	0.275433
$587$	34.9252	1.44152	0.720760	−	0.693185i	$-0.243793\pi$
$587$	34.9252	1.44152	0.720760	+	0.693185i	$0.243793\pi$
$588$	23.7044	0.977553
$589$	3.21137	0.132322
$590$	0.0106424	0.000438140 0
$591$	20.2527	0.833083
$592$	0	0
$593$	30.3683	1.24708	0.623538	−	0.781793i	$-0.285695\pi$
$593$	30.3683	1.24708	0.623538	+	0.781793i	$0.285695\pi$
$594$	1.26151	0.0517603
$595$	−1.55228	−0.0636372
$596$	9.37738	0.384113
$597$	67.2053	2.75053
$598$	−10.7127	−0.438077
$599$	−2.63334	−0.107595	−0.0537977	−	0.998552i	$-0.517133\pi$
$599$	−2.63334	−0.107595	−0.0537977	+	0.998552i	$0.517133\pi$
$600$	−24.5602	−1.00266
$601$	28.8263	1.17585	0.587924	−	0.808916i	$-0.299946\pi$
$601$	28.8263	1.17585	0.587924	+	0.808916i	$0.299946\pi$
$602$	−1.96934	−0.0802643
$603$	−27.4806	−1.11910
$604$	−1.41045	−0.0573903
$605$	0.0147187	0.000598400 0
$606$	10.2110	0.414793
$607$	−17.9531	−0.728695	−0.364347	−	0.931263i	$-0.618708\pi$
$607$	−17.9531	−0.728695	−0.364347	+	0.931263i	$0.618708\pi$
$608$	−5.35060	−0.216995
$609$	−61.0110	−2.47229
$610$	−0.00712508	−0.000288486 0
$611$	−28.0549	−1.13498
$612$	36.3921	1.47106
$613$	37.0598	1.49683	0.748416	−	0.663230i	$-0.230815\pi$
$613$	37.0598	1.49683	0.748416	+	0.663230i	$0.230815\pi$
$614$	17.6374	0.711789
$615$	0.0226802	0.000914556 0
$616$	−22.7475	−0.916523
$617$	−27.7006	−1.11519	−0.557593	−	0.830115i	$-0.688275\pi$
$617$	−27.7006	−1.11519	−0.557593	+	0.830115i	$0.688275\pi$
$618$	12.3686	0.497539
$619$	−27.1079	−1.08956	−0.544779	−	0.838580i	$-0.683386\pi$
$619$	−27.1079	−1.08956	−0.544779	+	0.838580i	$0.683386\pi$
$620$	−0.367328	−0.0147522
$621$	−3.32727	−0.133519
$622$	14.4749	0.580392
$623$	−17.4540	−0.699281
$624$	26.9991	1.08083
$625$	24.9298	0.997191
$626$	−7.77181	−0.310624
$627$	−8.48973	−0.339047
$628$	7.39969	0.295280
$629$	0	0
$630$	0.419034	0.0166947
$631$	18.4257	0.733516	0.366758	−	0.930316i	$-0.380468\pi$
$631$	18.4257	0.733516	0.366758	+	0.930316i	$0.380468\pi$
$632$	−27.3528	−1.08804
$633$	−59.7062	−2.37311
$634$	−12.3439	−0.490240
$635$	0.449996	0.0178576
$636$	−44.8295	−1.77760
$637$	−24.4708	−0.969569
$638$	11.9042	0.471293
$639$	17.8202	0.704958
$640$	0.785946	0.0310672
$641$	−23.3565	−0.922527	−0.461263	−	0.887263i	$-0.652604\pi$
$641$	−23.3565	−0.922527	−0.461263	+	0.887263i	$0.652604\pi$
$642$	11.5089	0.454220
$643$	−12.8226	−0.505672	−0.252836	−	0.967509i	$-0.581363\pi$
$643$	−12.8226	−0.505672	−0.252836	+	0.967509i	$0.581363\pi$
$644$	27.7643	1.09407
$645$	0.181672	0.00715334
$646$	3.48388	0.137071
$647$	24.5104	0.963604	0.481802	−	0.876280i	$-0.339982\pi$
$647$	24.5104	0.963604	0.481802	+	0.876280i	$0.339982\pi$
$648$	15.7660	0.619349
$649$	0.969882	0.0380712
$650$	11.7329	0.460203
$651$	−27.6138	−1.08227
$652$	−1.72768	−0.0676610
$653$	35.6842	1.39643	0.698215	−	0.715888i	$-0.253978\pi$
$653$	35.6842	1.39643	0.698215	+	0.715888i	$0.253978\pi$
$654$	−2.77292	−0.108430
$655$	−0.671122	−0.0262229
$656$	0.318874	0.0124500
$657$	33.9686	1.32524
$658$	−11.7026	−0.456214
$659$	19.4230	0.756612	0.378306	−	0.925681i	$-0.376507\pi$
$659$	19.4230	0.756612	0.378306	+	0.925681i	$0.376507\pi$
$660$	0.971085	0.0377994
$661$	−34.1803	−1.32946	−0.664731	−	0.747083i	$-0.731454\pi$
$661$	−34.1803	−1.32946	−0.664731	+	0.747083i	$0.731454\pi$
$662$	4.09698	0.159234
$663$	−71.8170	−2.78914
$664$	−10.7959	−0.418963
$665$	−0.249241	−0.00966514
$666$	0	0
$667$	−31.3978	−1.21573
$668$	24.6910	0.955322
$669$	36.4168	1.40796
$670$	0.300917	0.0116254
$671$	−0.649337	−0.0250674
$672$	46.0086	1.77482
$673$	−26.5978	−1.02527	−0.512634	−	0.858607i	$-0.671330\pi$
$673$	−26.5978	−1.02527	−0.512634	+	0.858607i	$0.671330\pi$
$674$	1.11790	0.0430599
$675$	3.64412	0.140262
$676$	−11.8813	−0.456972
$677$	10.0461	0.386105	0.193052	−	0.981188i	$-0.438161\pi$
$677$	10.0461	0.386105	0.193052	+	0.981188i	$0.438161\pi$
$678$	22.3365	0.857829
$679$	−20.5571	−0.788910
$680$	−0.861134	−0.0330230
$681$	64.7211	2.48012
$682$	5.38790	0.206313
$683$	−32.5300	−1.24473	−0.622364	−	0.782728i	$-0.713828\pi$
$683$	−32.5300	−1.24473	−0.622364	+	0.782728i	$0.713828\pi$
$684$	5.84327	0.223423
$685$	1.21196	0.0463067
$686$	2.81699	0.107553
$687$	32.3664	1.23486
$688$	2.55423	0.0973792
$689$	46.2789	1.76309
$690$	0.412231	0.0156934
$691$	−45.2007	−1.71952	−0.859759	−	0.510700i	$-0.829386\pi$
$691$	−45.2007	−1.71952	−0.859759	+	0.510700i	$0.829386\pi$
$692$	−13.8591	−0.526844
$693$	38.1882	1.45065
$694$	0.485751	0.0184389
$695$	0.359458	0.0136350
$696$	−33.8462	−1.28294
$697$	−0.848198	−0.0321278
$698$	−5.87659	−0.222432
$699$	64.6298	2.44452
$700$	−30.4083	−1.14933
$701$	−6.75355	−0.255078	−0.127539	−	0.991834i	$-0.540708\pi$
$701$	−6.75355	−0.255078	−0.127539	+	0.991834i	$0.540708\pi$
$702$	1.71346	0.0646703
$703$	0	0
$704$	6.87344	0.259053
$705$	1.07957	0.0406588
$706$	16.9154	0.636619
$707$	27.3195	1.02746
$708$	−1.27609	−0.0479585
$709$	−37.9151	−1.42393	−0.711966	−	0.702214i	$-0.752195\pi$
$709$	−37.9151	−1.42393	−0.711966	+	0.702214i	$0.752195\pi$
$710$	−0.195134	−0.00732326
$711$	45.9195	1.72212
$712$	−9.68271	−0.362875
$713$	−14.2108	−0.532197
$714$	−29.9571	−1.12112
$715$	−1.00248	−0.0374907
$716$	10.8649	0.406042
$717$	10.6223	0.396696
$718$	7.42464	0.277085
$719$	26.0171	0.970274	0.485137	−	0.874438i	$-0.338770\pi$
$719$	26.0171	0.970274	0.485137	+	0.874438i	$0.338770\pi$
$720$	−0.543487	−0.0202546
$721$	33.0923	1.23242
$722$	−9.44534	−0.351519
$723$	−67.1024	−2.49556
$724$	−1.49189	−0.0554457
$725$	34.3878	1.27713
$726$	0.284053	0.0105422
$727$	52.6165	1.95144	0.975718	−	0.219029i	$-0.0702890\pi$
$727$	52.6165	1.95144	0.975718	+	0.219029i	$0.0702890\pi$
$728$	−30.8970	−1.14512
$729$	−31.9570	−1.18359
$730$	−0.371962	−0.0137669
$731$	−6.79419	−0.251292
$732$	0.854345	0.0315775
$733$	−7.94458	−0.293440	−0.146720	−	0.989178i	$-0.546872\pi$
$733$	−7.94458	−0.293440	−0.146720	+	0.989178i	$0.546872\pi$
$734$	−5.49446	−0.202804
$735$	0.941648	0.0347332
$736$	23.6772	0.872752
$737$	27.4237	1.01017
$738$	0.228969	0.00842846
$739$	−4.31152	−0.158602	−0.0793010	−	0.996851i	$-0.525269\pi$
$739$	−4.31152	−0.158602	−0.0793010	+	0.996851i	$0.525269\pi$
$740$	0	0
$741$	−11.5313	−0.423611
$742$	19.3044	0.708686
$743$	−10.8365	−0.397553	−0.198777	−	0.980045i	$-0.563697\pi$
$743$	−10.8365	−0.397553	−0.198777	+	0.980045i	$0.563697\pi$
$744$	−15.3189	−0.561618
$745$	0.372513	0.0136478
$746$	8.55875	0.313358
$747$	18.1241	0.663124
$748$	−36.3167	−1.32787
$749$	30.7921	1.12512
$750$	−0.903400	−0.0329875
$751$	−44.6876	−1.63067	−0.815336	−	0.578988i	$-0.803448\pi$
$751$	−44.6876	−1.63067	−0.815336	+	0.578988i	$0.803448\pi$
$752$	15.1782	0.553493
$753$	−51.1623	−1.86446
$754$	16.1691	0.588842
$755$	−0.0560295	−0.00203912
$756$	−4.44078	−0.161510
$757$	18.0245	0.655112	0.327556	−	0.944832i	$-0.393775\pi$
$757$	18.0245	0.655112	0.327556	+	0.944832i	$0.393775\pi$
$758$	−10.0086	−0.363529
$759$	37.5683	1.36364
$760$	−0.138268	−0.00501549
$761$	3.36170	0.121862	0.0609308	−	0.998142i	$-0.480593\pi$
$761$	3.36170	0.121862	0.0609308	+	0.998142i	$0.480593\pi$
$762$	8.68437	0.314602
$763$	−7.41894	−0.268584
$764$	−16.0051	−0.579044
$765$	1.44566	0.0522680
$766$	−0.726032	−0.0262326
$767$	1.31735	0.0475668
$768$	4.66872	0.168468
$769$	46.0044	1.65896	0.829480	−	0.558536i	$-0.188637\pi$
$769$	46.0044	1.65896	0.829480	+	0.558536i	$0.188637\pi$
$770$	−0.418166	−0.0150697
$771$	8.28185	0.298263
$772$	5.03251	0.181124
$773$	19.0909	0.686653	0.343327	−	0.939216i	$-0.388446\pi$
$773$	19.0909	0.686653	0.343327	+	0.939216i	$0.388446\pi$
$774$	1.83408	0.0659245
$775$	15.5640	0.559077
$776$	−11.4042	−0.409386
$777$	0	0
$778$	10.7977	0.387117
$779$	−0.136190	−0.00487953
$780$	1.31899	0.0472273
$781$	−17.7833	−0.636338
$782$	−15.4167	−0.551299
$783$	5.02194	0.179470
$784$	13.2392	0.472827
$785$	0.293950	0.0104915
$786$	−12.9518	−0.461977
$787$	22.1092	0.788109	0.394054	−	0.919087i	$-0.371072\pi$
$787$	22.1092	0.788109	0.394054	+	0.919087i	$0.371072\pi$
$788$	13.9106	0.495543
$789$	23.3793	0.832325
$790$	−0.502826	−0.0178897
$791$	59.7614	2.12487
$792$	21.1851	0.752780
$793$	−0.881968	−0.0313196
$794$	−18.9035	−0.670861
$795$	−1.78083	−0.0631597
$796$	46.1600	1.63610
$797$	−4.77394	−0.169102	−0.0845508	−	0.996419i	$-0.526946\pi$
$797$	−4.77394	−0.169102	−0.0845508	+	0.996419i	$0.526946\pi$
$798$	−4.81004	−0.170274
$799$	−40.3737	−1.42832
$800$	−25.9319	−0.916833
$801$	16.2552	0.574349
$802$	20.5480	0.725575
$803$	−33.8983	−1.19625
$804$	−36.0819	−1.27251
$805$	1.10293	0.0388731
$806$	7.31817	0.257772
$807$	0.513185	0.0180650
$808$	15.1556	0.533174
$809$	24.1639	0.849558	0.424779	−	0.905297i	$-0.360352\pi$
$809$	24.1639	0.849558	0.424779	+	0.905297i	$0.360352\pi$
$810$	0.289826	0.0101835
$811$	20.4560	0.718307	0.359154	−	0.933278i	$-0.383065\pi$
$811$	20.4560	0.718307	0.359154	+	0.933278i	$0.383065\pi$
$812$	−41.9055	−1.47059
$813$	3.73592	0.131024
$814$	0	0
$815$	−0.0686312	−0.00240405
$816$	38.8543	1.36017
$817$	−1.09091	−0.0381660
$818$	−6.35221	−0.222100
$819$	51.8695	1.81247
$820$	0.0155779	0.000544005 0
$821$	8.27234	0.288707	0.144353	−	0.989526i	$-0.453890\pi$
$821$	8.27234	0.288707	0.144353	+	0.989526i	$0.453890\pi$
$822$	23.3894	0.815799
$823$	−35.3049	−1.23065	−0.615326	−	0.788273i	$-0.710976\pi$
$823$	−35.3049	−1.23065	−0.615326	+	0.788273i	$0.710976\pi$
$824$	18.3581	0.639535
$825$	−41.1459	−1.43252
$826$	0.549508	0.0191198
$827$	21.9832	0.764430	0.382215	−	0.924073i	$-0.375161\pi$
$827$	21.9832	0.764430	0.382215	+	0.924073i	$0.375161\pi$
$828$	−25.8573	−0.898604
$829$	−49.5466	−1.72082	−0.860412	−	0.509598i	$-0.829794\pi$
$829$	−49.5466	−1.72082	−0.860412	+	0.509598i	$0.829794\pi$
$830$	−0.198461	−0.00688868
$831$	60.9114	2.11299
$832$	9.33592	0.323665
$833$	−35.2158	−1.22016
$834$	6.93710	0.240212
$835$	0.980839	0.0339433
$836$	−5.83118	−0.201675
$837$	2.27295	0.0785646
$838$	−14.0775	−0.486299
$839$	−41.0989	−1.41889	−0.709447	−	0.704759i	$-0.751055\pi$
$839$	−41.0989	−1.41889	−0.709447	+	0.704759i	$0.751055\pi$
$840$	1.18893	0.0410220
$841$	18.3896	0.634124
$842$	15.0408	0.518340
$843$	−55.2197	−1.90187
$844$	−41.0092	−1.41160
$845$	−0.471979	−0.0162366
$846$	10.8988	0.374708
$847$	0.759984	0.0261134
$848$	−25.0378	−0.859800
$849$	39.5198	1.35632
$850$	16.8848	0.579143
$851$	0	0
$852$	23.3979	0.801599
$853$	26.4660	0.906179	0.453089	−	0.891465i	$-0.350322\pi$
$853$	26.4660	0.906179	0.453089	+	0.891465i	$0.350322\pi$
$854$	−0.367896	−0.0125891
$855$	0.232122	0.00793840
$856$	17.0821	0.583853
$857$	−2.73858	−0.0935481	−0.0467741	−	0.998905i	$-0.514894\pi$
$857$	−2.73858	−0.0935481	−0.0467741	+	0.998905i	$0.514894\pi$
$858$	−19.3467	−0.660485
$859$	−6.12536	−0.208995	−0.104497	−	0.994525i	$-0.533323\pi$
$859$	−6.12536	−0.208995	−0.104497	+	0.994525i	$0.533323\pi$
$860$	0.124782	0.00425502
$861$	1.17107	0.0399100
$862$	19.2483	0.655600
$863$	−22.6823	−0.772116	−0.386058	−	0.922474i	$-0.626164\pi$
$863$	−22.6823	−0.772116	−0.386058	+	0.922474i	$0.626164\pi$
$864$	−3.78706	−0.128838
$865$	−0.550547	−0.0187192
$866$	6.79476	0.230895
$867$	−60.7128	−2.06191
$868$	−18.9666	−0.643767
$869$	−45.8245	−1.55449
$870$	−0.622193	−0.0210943
$871$	37.2485	1.26212
$872$	−4.11570	−0.139375
$873$	19.1452	0.647966
$874$	−2.47537	−0.0837306
$875$	−2.41705	−0.0817111
$876$	44.6007	1.50692
$877$	4.44443	0.150078	0.0750389	−	0.997181i	$-0.476092\pi$
$877$	4.44443	0.150078	0.0750389	+	0.997181i	$0.476092\pi$
$878$	0.924836	0.0312117
$879$	−31.7591	−1.07121
$880$	0.542362	0.0182830
$881$	46.7564	1.57526	0.787632	−	0.616146i	$-0.211307\pi$
$881$	46.7564	1.57526	0.787632	+	0.616146i	$0.211307\pi$
$882$	9.50643	0.320098
$883$	−23.0255	−0.774871	−0.387435	−	0.921897i	$-0.626639\pi$
$883$	−23.0255	−0.774871	−0.387435	+	0.921897i	$0.626639\pi$
$884$	−49.3275	−1.65906
$885$	−0.0506923	−0.00170400
$886$	−2.24310	−0.0753585
$887$	−34.5820	−1.16115	−0.580575	−	0.814207i	$-0.697172\pi$
$887$	−34.5820	−1.16115	−0.580575	+	0.814207i	$0.697172\pi$
$888$	0	0
$889$	23.2351	0.779279
$890$	−0.177997	−0.00596647
$891$	26.4130	0.884869
$892$	25.0129	0.837495
$893$	−6.48258	−0.216931
$894$	7.18904	0.240438
$895$	0.431605	0.0144270
$896$	40.5815	1.35573
$897$	51.0275	1.70376
$898$	−2.98367	−0.0995665
$899$	21.4487	0.715354
$900$	28.3197	0.943990
$901$	66.5998	2.21876
$902$	−0.228495	−0.00760805
$903$	9.38045	0.312162
$904$	33.1530	1.10265
$905$	−0.0592649	−0.00197003
$906$	−1.08130	−0.0359238
$907$	−26.9725	−0.895607	−0.447803	−	0.894132i	$-0.647794\pi$
$907$	−26.9725	−0.895607	−0.447803	+	0.894132i	$0.647794\pi$
$908$	44.4537	1.47525
$909$	−25.4431	−0.843894
$910$	−0.567978	−0.0188283
$911$	9.96573	0.330179	0.165090	−	0.986279i	$-0.447209\pi$
$911$	9.96573	0.330179	0.165090	+	0.986279i	$0.447209\pi$
$912$	6.23862	0.206581
$913$	−18.0865	−0.598577
$914$	3.17116	0.104893
$915$	0.0339385	0.00112197
$916$	22.2309	0.734530
$917$	−34.6527	−1.14433
$918$	2.46583	0.0813844
$919$	13.3269	0.439613	0.219806	−	0.975543i	$-0.429457\pi$
$919$	13.3269	0.439613	0.219806	+	0.975543i	$0.429457\pi$
$920$	0.611854	0.0201722
$921$	−84.0115	−2.76827
$922$	16.8005	0.553295
$923$	−24.1544	−0.795052
$924$	50.1409	1.64952
$925$	0	0
$926$	−1.03692	−0.0340752
$927$	−30.8193	−1.01224
$928$	−35.7366	−1.17311
$929$	52.1412	1.71070	0.855348	−	0.518054i	$-0.173343\pi$
$929$	52.1412	1.71070	0.855348	+	0.518054i	$0.173343\pi$
$930$	−0.281607	−0.00923424
$931$	−5.65441	−0.185316
$932$	44.3910	1.45408
$933$	−68.9476	−2.25724
$934$	12.4554	0.407552
$935$	−1.44267	−0.0471803
$936$	28.7749	0.940536
$937$	−9.76511	−0.319012	−0.159506	−	0.987197i	$-0.550990\pi$
$937$	−9.76511	−0.319012	−0.159506	+	0.987197i	$0.550990\pi$
$938$	15.5375	0.507318
$939$	37.0190	1.20807
$940$	0.741501	0.0241851
$941$	−43.3925	−1.41455	−0.707277	−	0.706937i	$-0.750077\pi$
$941$	−43.3925	−1.41455	−0.707277	+	0.706937i	$0.750077\pi$
$942$	5.67287	0.184832
$943$	0.602662	0.0196254
$944$	−0.712712	−0.0231968
$945$	−0.176408	−0.00573856
$946$	−1.83028	−0.0595075
$947$	−47.3912	−1.54001	−0.770004	−	0.638040i	$-0.779746\pi$
$947$	−47.3912	−1.54001	−0.770004	+	0.638040i	$0.779746\pi$
$948$	60.2922	1.95820
$949$	−46.0427	−1.49461
$950$	2.71110	0.0879596
$951$	58.7972	1.90663
$952$	−44.4637	−1.44108
$953$	37.2753	1.20747	0.603733	−	0.797186i	$-0.293679\pi$
$953$	37.2753	1.20747	0.603733	+	0.797186i	$0.293679\pi$
$954$	−17.9784	−0.582074
$955$	−0.635796	−0.0205739
$956$	7.29591	0.235967
$957$	−56.7028	−1.83294
$958$	8.89368	0.287342
$959$	62.5784	2.02076
$960$	−0.359251	−0.0115948
$961$	−21.2922	−0.686847
$962$	0	0
$963$	−28.6772	−0.924108
$964$	−46.0893	−1.48444
$965$	0.199915	0.00643548
$966$	21.2851	0.684838
$967$	24.5302	0.788837	0.394418	−	0.918931i	$-0.370946\pi$
$967$	24.5302	0.788837	0.394418	+	0.918931i	$0.370946\pi$
$968$	0.421605	0.0135509
$969$	−16.5946	−0.533095
$970$	−0.209642	−0.00673121
$971$	−2.85194	−0.0915232	−0.0457616	−	0.998952i	$-0.514571\pi$
$971$	−2.85194	−0.0915232	−0.0457616	+	0.998952i	$0.514571\pi$
$972$	−38.5224	−1.23561
$973$	18.5602	0.595013
$974$	11.5263	0.369328
$975$	−55.8868	−1.78981
$976$	0.477161	0.0152735
$977$	−40.4966	−1.29560	−0.647800	−	0.761811i	$-0.724311\pi$
$977$	−40.4966	−1.29560	−0.647800	+	0.761811i	$0.724311\pi$
$978$	−1.32450	−0.0423528
$979$	−16.2216	−0.518443
$980$	0.646772	0.0206604
$981$	6.90938	0.220599
$982$	−15.7388	−0.502245
$983$	19.9376	0.635911	0.317955	−	0.948106i	$-0.397004\pi$
$983$	19.9376	0.635911	0.317955	+	0.948106i	$0.397004\pi$
$984$	0.649657	0.0207103
$985$	0.552591	0.0176070
$986$	23.2688	0.741030
$987$	55.7423	1.77430
$988$	−7.92025	−0.251977
$989$	4.82742	0.153503
$990$	0.389445	0.0123774
$991$	31.8979	1.01327	0.506635	−	0.862161i	$-0.330889\pi$
$991$	31.8979	1.01327	0.506635	+	0.862161i	$0.330889\pi$
$992$	−16.1745	−0.513542
$993$	−19.5149	−0.619288
$994$	−10.0755	−0.319577
$995$	1.83369	0.0581318
$996$	23.7968	0.754031
$997$	22.1782	0.702391	0.351196	−	0.936302i	$-0.385775\pi$
$997$	22.1782	0.702391	0.351196	+	0.936302i	$0.385775\pi$
$998$	−10.0761	−0.318953
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1369.2.a.o.1.14	yes	27
37.6	odd	4		1369.2.b.h.1368.23		54
37.31	odd	4		1369.2.b.h.1368.32		54
37.36	even	2		1369.2.a.n.1.14	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1369.2.a.n.1.14	✓	27	37.36	even	2
1369.2.a.o.1.14	yes	27	1.1	even	1	trivial
1369.2.b.h.1368.23		54	37.6	odd	4
1369.2.b.h.1368.32		54	37.31	odd	4

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:	\( N \)	\(=\)	\( 1369 = 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1369.a (trivial)

\(f(q)\)	\(=\)	\(q+0.526565 q^{2} -2.50816 q^{3} -1.72273 q^{4} -0.0684348 q^{5} -1.32071 q^{6} -3.53356 q^{7} -1.96026 q^{8} +3.29085 q^{9} -0.0360353 q^{10} -3.28404 q^{11} +4.32088 q^{12} -4.46058 q^{13} -1.86065 q^{14} +0.171645 q^{15} +2.41326 q^{16} -6.41920 q^{17} +1.73285 q^{18} -1.03070 q^{19} +0.117895 q^{20} +8.86271 q^{21} -1.72926 q^{22} +4.56098 q^{23} +4.91664 q^{24} -4.99532 q^{25} -2.34878 q^{26} -0.729508 q^{27} +6.08736 q^{28} -6.88401 q^{29} +0.0903823 q^{30} -3.11573 q^{31} +5.19125 q^{32} +8.23689 q^{33} -3.38012 q^{34} +0.241818 q^{35} -5.66925 q^{36} -0.542728 q^{38} +11.1878 q^{39} +0.134150 q^{40} +0.132134 q^{41} +4.66679 q^{42} +1.05842 q^{43} +5.65751 q^{44} -0.225209 q^{45} +2.40165 q^{46} +6.28952 q^{47} -6.05283 q^{48} +5.48602 q^{49} -2.63036 q^{50} +16.1004 q^{51} +7.68437 q^{52} -10.3751 q^{53} -0.384133 q^{54} +0.224743 q^{55} +6.92668 q^{56} +2.58515 q^{57} -3.62488 q^{58} -0.295332 q^{59} -0.295698 q^{60} +0.197725 q^{61} -1.64063 q^{62} -11.6284 q^{63} -2.09298 q^{64} +0.305259 q^{65} +4.33726 q^{66} -8.35060 q^{67} +11.0585 q^{68} -11.4396 q^{69} +0.127333 q^{70} +5.41508 q^{71} -6.45092 q^{72} +10.3221 q^{73} +12.5290 q^{75} +1.77561 q^{76} +11.6043 q^{77} +5.89112 q^{78} +13.9537 q^{79} -0.165151 q^{80} -8.04284 q^{81} +0.0695773 q^{82} +5.50740 q^{83} -15.2681 q^{84} +0.439296 q^{85} +0.557325 q^{86} +17.2662 q^{87} +6.43757 q^{88} +4.93951 q^{89} -0.118587 q^{90} +15.7617 q^{91} -7.85733 q^{92} +7.81474 q^{93} +3.31184 q^{94} +0.0705354 q^{95} -13.0205 q^{96} +5.81769 q^{97} +2.88874 q^{98} -10.8073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 9 q^{2} - q^{3} + 25 q^{4} + 17 q^{5} + 10 q^{6} - 3 q^{7} + 21 q^{8} + 20 q^{9} - 11 q^{10} - 5 q^{11} - 10 q^{12} + 15 q^{13} + 25 q^{14} + 23 q^{15} + 13 q^{16} + 26 q^{17} + 15 q^{18} + 27 q^{19}+ \cdots + 35 q^{99}+O(q^{100}) \) 27 * q + 9 * q^2 - q^3 + 25 * q^4 + 17 * q^5 + 10 * q^6 - 3 * q^7 + 21 * q^8 + 20 * q^9 - 11 * q^10 - 5 * q^11 - 10 * q^12 + 15 * q^13 + 25 * q^14 + 23 * q^15 + 13 * q^16 + 26 * q^17 + 15 * q^18 + 27 * q^19 + 38 * q^20 - 17 * q^21 - 4 * q^22 + 41 * q^23 + 27 * q^24 + 16 * q^25 - 23 * q^26 - 13 * q^27 + 48 * q^28 + 42 * q^29 + 46 * q^30 + 45 * q^31 + 31 * q^32 - 33 * q^33 + 49 * q^34 + 51 * q^35 - 25 * q^36 - 11 * q^38 + 47 * q^39 + 31 * q^40 + 7 * q^41 - 7 * q^42 + 11 * q^43 - 34 * q^44 + 24 * q^45 - 15 * q^46 - 21 * q^47 + 6 * q^48 + 6 * q^49 + 19 * q^50 + 37 * q^51 + 6 * q^52 - 7 * q^53 + 7 * q^54 + 35 * q^55 + 54 * q^56 - 3 * q^57 + 37 * q^58 + 74 * q^59 - 14 * q^60 + 26 * q^61 - 33 * q^62 - 50 * q^63 - 27 * q^64 - 6 * q^65 + 19 * q^66 + 2 * q^67 + 57 * q^68 + 28 * q^69 + 6 * q^70 + 5 * q^71 - 12 * q^72 - 23 * q^73 + 34 * q^75 + 27 * q^76 - 11 * q^77 + 17 * q^78 + 58 * q^79 + 36 * q^80 - 37 * q^81 - 34 * q^82 + 56 * q^83 - 40 * q^84 - 16 * q^85 - 45 * q^86 + 66 * q^87 - 33 * q^88 + 64 * q^89 - 34 * q^90 + 17 * q^91 + 34 * q^92 - 30 * q^93 - 5 * q^94 - 22 * q^95 + 19 * q^96 - 17 * q^97 - 6 * q^98 + 35 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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