LMFDB - Embedded newform 418.2.a.g.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [418,2,Mod(1,418)] 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("418.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(418, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 418 = 2 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	418.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$3.33774680449$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	418.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -2.14510 q^{3} +1.00000 q^{4} -1.60147 q^{5} +2.14510 q^{6} +2.89167 q^{7} -1.00000 q^{8} +1.60147 q^{9} +1.60147 q^{10} -1.00000 q^{11} -2.14510 q^{12} +4.89167 q^{13} -2.89167 q^{14} +3.43531 q^{15} +1.00000 q^{16} -5.74657 q^{17} -1.60147 q^{18} -1.00000 q^{19} -1.60147 q^{20} -6.20293 q^{21} +1.00000 q^{22} -7.74657 q^{23} +2.14510 q^{24} -2.43531 q^{25} -4.89167 q^{26} +3.00000 q^{27} +2.89167 q^{28} +5.34803 q^{29} -3.43531 q^{30} -7.43531 q^{31} -1.00000 q^{32} +2.14510 q^{33} +5.74657 q^{34} -4.63091 q^{35} +1.60147 q^{36} -7.49314 q^{37} +1.00000 q^{38} -10.4931 q^{39} +1.60147 q^{40} -2.05783 q^{41} +6.20293 q^{42} -10.6887 q^{43} -1.00000 q^{44} -2.56469 q^{45} +7.74657 q^{46} -7.08727 q^{47} -2.14510 q^{48} +1.36176 q^{49} +2.43531 q^{50} +12.3270 q^{51} +4.89167 q^{52} +8.32698 q^{53} -3.00000 q^{54} +1.60147 q^{55} -2.89167 q^{56} +2.14510 q^{57} -5.34803 q^{58} -2.54364 q^{59} +3.43531 q^{60} +13.4931 q^{61} +7.43531 q^{62} +4.63091 q^{63} +1.00000 q^{64} -7.83384 q^{65} -2.14510 q^{66} -10.3848 q^{67} -5.74657 q^{68} +16.6172 q^{69} +4.63091 q^{70} +14.9789 q^{71} -1.60147 q^{72} -3.74657 q^{73} +7.49314 q^{74} +5.22399 q^{75} -1.00000 q^{76} -2.89167 q^{77} +10.4931 q^{78} -8.69607 q^{79} -1.60147 q^{80} -11.2397 q^{81} +2.05783 q^{82} -6.10833 q^{83} -6.20293 q^{84} +9.20293 q^{85} +10.6887 q^{86} -11.4721 q^{87} +1.00000 q^{88} +3.20293 q^{89} +2.56469 q^{90} +14.1451 q^{91} -7.74657 q^{92} +15.9495 q^{93} +7.08727 q^{94} +1.60147 q^{95} +2.14510 q^{96} -6.98627 q^{97} -1.36176 q^{98} -1.60147 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{10} - 3 q^{11} + 6 q^{14} - 9 q^{15} + 3 q^{16} - 9 q^{17} - 3 q^{18} - 3 q^{19} - 3 q^{20} - 15 q^{21} + 3 q^{22} - 15 q^{23} + 12 q^{25}+ \cdots - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 - 6 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^10 - 3 * q^11 + 6 * q^14 - 9 * q^15 + 3 * q^16 - 9 * q^17 - 3 * q^18 - 3 * q^19 - 3 * q^20 - 15 * q^21 + 3 * q^22 - 15 * q^23 + 12 * q^25 + 9 * q^27 - 6 * q^28 + 6 * q^29 + 9 * q^30 - 3 * q^31 - 3 * q^32 + 9 * q^34 + 3 * q^36 - 6 * q^37 + 3 * q^38 - 15 * q^39 + 3 * q^40 - 9 * q^41 + 15 * q^42 - 21 * q^43 - 3 * q^44 - 27 * q^45 + 15 * q^46 - 12 * q^47 + 27 * q^49 - 12 * q^50 + 3 * q^51 - 9 * q^53 - 9 * q^54 + 3 * q^55 + 6 * q^56 - 6 * q^58 - 3 * q^59 - 9 * q^60 + 24 * q^61 + 3 * q^62 + 3 * q^64 - 6 * q^65 - 9 * q^68 + 3 * q^69 + 21 * q^71 - 3 * q^72 - 3 * q^73 + 6 * q^74 + 36 * q^75 - 3 * q^76 + 6 * q^77 + 15 * q^78 - 6 * q^79 - 3 * q^80 - 9 * q^81 + 9 * q^82 - 33 * q^83 - 15 * q^84 + 24 * q^85 + 21 * q^86 + 6 * q^87 + 3 * q^88 + 6 * q^89 + 27 * q^90 + 36 * q^91 - 15 * q^92 + 36 * q^93 + 12 * q^94 + 3 * q^95 + 12 * q^97 - 27 * q^98 - 3 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.14510	−1.23848	−0.619238	−	0.785204i	$-0.712558\pi$
$3$	−2.14510	−1.23848	−0.619238	+	0.785204i	$0.712558\pi$
$4$	1.00000	0.500000
$5$	−1.60147	−0.716197	−0.358099	−	0.933684i	$-0.616575\pi$
$5$	−1.60147	−0.716197	−0.358099	+	0.933684i	$0.616575\pi$
$6$	2.14510	0.875735
$7$	2.89167	1.09295	0.546474	−	0.837476i	$-0.315970\pi$
$7$	2.89167	1.09295	0.546474	+	0.837476i	$0.315970\pi$
$8$	−1.00000	−0.353553
$9$	1.60147	0.533822
$10$	1.60147	0.506428
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.14510	−0.619238
$13$	4.89167	1.35671	0.678353	−	0.734736i	$-0.262694\pi$
$13$	4.89167	1.35671	0.678353	+	0.734736i	$0.262694\pi$
$14$	−2.89167	−0.772832
$15$	3.43531	0.886993
$16$	1.00000	0.250000
$17$	−5.74657	−1.39375	−0.696874	−	0.717194i	$-0.745426\pi$
$17$	−5.74657	−1.39375	−0.696874	+	0.717194i	$0.745426\pi$
$18$	−1.60147	−0.377469
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.60147	−0.358099
$21$	−6.20293	−1.35359
$22$	1.00000	0.213201
$23$	−7.74657	−1.61527	−0.807636	−	0.589682i	$-0.799253\pi$
$23$	−7.74657	−1.61527	−0.807636	+	0.589682i	$0.799253\pi$
$24$	2.14510	0.437867
$25$	−2.43531	−0.487062
$26$	−4.89167	−0.959336
$27$	3.00000	0.577350
$28$	2.89167	0.546474
$29$	5.34803	0.993105	0.496552	−	0.868007i	$-0.334599\pi$
$29$	5.34803	0.993105	0.496552	+	0.868007i	$0.334599\pi$
$30$	−3.43531	−0.627199
$31$	−7.43531	−1.33542	−0.667710	−	0.744421i	$-0.732726\pi$
$31$	−7.43531	−1.33542	−0.667710	+	0.744421i	$0.732726\pi$
$32$	−1.00000	−0.176777
$33$	2.14510	0.373414
$34$	5.74657	0.985528
$35$	−4.63091	−0.782767
$36$	1.60147	0.266911
$37$	−7.49314	−1.23186	−0.615932	−	0.787799i	$-0.711220\pi$
$37$	−7.49314	−1.23186	−0.615932	+	0.787799i	$0.711220\pi$
$38$	1.00000	0.162221
$39$	−10.4931	−1.68025
$40$	1.60147	0.253214
$41$	−2.05783	−0.321379	−0.160689	−	0.987005i	$-0.551372\pi$
$41$	−2.05783	−0.321379	−0.160689	+	0.987005i	$0.551372\pi$
$42$	6.20293	0.957133
$43$	−10.6887	−1.63002	−0.815009	−	0.579449i	$-0.803268\pi$
$43$	−10.6887	−1.63002	−0.815009	+	0.579449i	$0.803268\pi$
$44$	−1.00000	−0.150756
$45$	−2.56469	−0.382322
$46$	7.74657	1.14217
$47$	−7.08727	−1.03379	−0.516893	−	0.856050i	$-0.672911\pi$
$47$	−7.08727	−1.03379	−0.516893	+	0.856050i	$0.672911\pi$
$48$	−2.14510	−0.309619
$49$	1.36176	0.194537
$50$	2.43531	0.344405
$51$	12.3270	1.72612
$52$	4.89167	0.678353
$53$	8.32698	1.14380	0.571899	−	0.820324i	$-0.306207\pi$
$53$	8.32698	1.14380	0.571899	+	0.820324i	$0.306207\pi$
$54$	−3.00000	−0.408248
$55$	1.60147	0.215942
$56$	−2.89167	−0.386416
$57$	2.14510	0.284126
$58$	−5.34803	−0.702231
$59$	−2.54364	−0.331153	−0.165577	−	0.986197i	$-0.552949\pi$
$59$	−2.54364	−0.331153	−0.165577	+	0.986197i	$0.552949\pi$
$60$	3.43531	0.443496
$61$	13.4931	1.72762	0.863810	−	0.503818i	$-0.168072\pi$
$61$	13.4931	1.72762	0.863810	+	0.503818i	$0.168072\pi$
$62$	7.43531	0.944285
$63$	4.63091	0.583440
$64$	1.00000	0.125000
$65$	−7.83384	−0.971669
$66$	−2.14510	−0.264044
$67$	−10.3848	−1.26871	−0.634353	−	0.773043i	$-0.718733\pi$
$67$	−10.3848	−1.26871	−0.634353	+	0.773043i	$0.718733\pi$
$68$	−5.74657	−0.696874
$69$	16.6172	2.00047
$70$	4.63091	0.553500
$71$	14.9789	1.77767	0.888837	−	0.458224i	$-0.151514\pi$
$71$	14.9789	1.77767	0.888837	+	0.458224i	$0.151514\pi$
$72$	−1.60147	−0.188735
$73$	−3.74657	−0.438503	−0.219251	−	0.975668i	$-0.570361\pi$
$73$	−3.74657	−0.438503	−0.219251	+	0.975668i	$0.570361\pi$
$74$	7.49314	0.871059
$75$	5.22399	0.603214
$76$	−1.00000	−0.114708
$77$	−2.89167	−0.329536
$78$	10.4931	1.18811
$79$	−8.69607	−0.978384	−0.489192	−	0.872176i	$-0.662708\pi$
$79$	−8.69607	−0.978384	−0.489192	+	0.872176i	$0.662708\pi$
$80$	−1.60147	−0.179049
$81$	−11.2397	−1.24886
$82$	2.05783	0.227249
$83$	−6.10833	−0.670476	−0.335238	−	0.942133i	$-0.608817\pi$
$83$	−6.10833	−0.670476	−0.335238	+	0.942133i	$0.608817\pi$
$84$	−6.20293	−0.676795
$85$	9.20293	0.998198
$86$	10.6887	1.15260
$87$	−11.4721	−1.22994
$88$	1.00000	0.106600
$89$	3.20293	0.339510	0.169755	−	0.985486i	$-0.445702\pi$
$89$	3.20293	0.339510	0.169755	+	0.985486i	$0.445702\pi$
$90$	2.56469	0.270342
$91$	14.1451	1.48281
$92$	−7.74657	−0.807636
$93$	15.9495	1.65389
$94$	7.08727	0.730997
$95$	1.60147	0.164307
$96$	2.14510	0.218934
$97$	−6.98627	−0.709349	−0.354674	−	0.934990i	$-0.615408\pi$
$97$	−6.98627	−0.709349	−0.354674	+	0.934990i	$0.615408\pi$
$98$	−1.36176	−0.137559
$99$	−1.60147	−0.160953
$100$	−2.43531	−0.243531
$101$	−1.20293	−0.119696	−0.0598481	−	0.998207i	$-0.519062\pi$
$101$	−1.20293	−0.119696	−0.0598481	+	0.998207i	$0.519062\pi$
$102$	−12.3270	−1.22055
$103$	16.8779	1.66303	0.831517	−	0.555500i	$-0.187473\pi$
$103$	16.8779	1.66303	0.831517	+	0.555500i	$0.187473\pi$
$104$	−4.89167	−0.479668
$105$	9.93378	0.969438
$106$	−8.32698	−0.808788
$107$	−8.03677	−0.776944	−0.388472	−	0.921460i	$-0.626997\pi$
$107$	−8.03677	−0.776944	−0.388472	+	0.921460i	$0.626997\pi$
$108$	3.00000	0.288675
$109$	11.5299	1.10437	0.552183	−	0.833723i	$-0.313795\pi$
$109$	11.5299	1.10437	0.552183	+	0.833723i	$0.313795\pi$
$110$	−1.60147	−0.152694
$111$	16.0735	1.52563
$112$	2.89167	0.273237
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	−2.14510	−0.200907
$115$	12.4059	1.15685
$116$	5.34803	0.496552
$117$	7.83384	0.724239
$118$	2.54364	0.234161
$119$	−16.6172	−1.52329
$120$	−3.43531	−0.313599
$121$	1.00000	0.0909091
$122$	−13.4931	−1.22161
$123$	4.41425	0.398020
$124$	−7.43531	−0.667710
$125$	11.9074	1.06503
$126$	−4.63091	−0.412554
$127$	−21.2765	−1.88798	−0.943991	−	0.329971i	$-0.892961\pi$
$127$	−21.2765	−1.88798	−0.943991	+	0.329971i	$0.892961\pi$
$128$	−1.00000	−0.0883883
$129$	22.9284	2.01874
$130$	7.83384	0.687073
$131$	−3.43531	−0.300144	−0.150072	−	0.988675i	$-0.547951\pi$
$131$	−3.43531	−0.300144	−0.150072	+	0.988675i	$0.547951\pi$
$132$	2.14510	0.186707
$133$	−2.89167	−0.250740
$134$	10.3848	0.897111
$135$	−4.80440	−0.413497
$136$	5.74657	0.492764
$137$	13.5877	1.16088	0.580439	−	0.814303i	$-0.302881\pi$
$137$	13.5877	1.16088	0.580439	+	0.814303i	$0.302881\pi$
$138$	−16.6172	−1.41455
$139$	3.19560	0.271048	0.135524	−	0.990774i	$-0.456728\pi$
$139$	3.19560	0.271048	0.135524	+	0.990774i	$0.456728\pi$
$140$	−4.63091	−0.391383
$141$	15.2029	1.28032
$142$	−14.9789	−1.25701
$143$	−4.89167	−0.409062
$144$	1.60147	0.133455
$145$	−8.56469	−0.711259
$146$	3.74657	0.310068
$147$	−2.92112	−0.240930
$148$	−7.49314	−0.615932
$149$	−12.0735	−0.989104	−0.494552	−	0.869148i	$-0.664668\pi$
$149$	−12.0735	−0.989104	−0.494552	+	0.869148i	$0.664668\pi$
$150$	−5.22399	−0.426537
$151$	−4.91273	−0.399792	−0.199896	−	0.979817i	$-0.564060\pi$
$151$	−4.91273	−0.399792	−0.199896	+	0.979817i	$0.564060\pi$
$152$	1.00000	0.0811107
$153$	−9.20293	−0.744013
$154$	2.89167	0.233017
$155$	11.9074	0.956425
$156$	−10.4931	−0.840123
$157$	8.00733	0.639054	0.319527	−	0.947577i	$-0.396476\pi$
$157$	8.00733	0.639054	0.319527	+	0.947577i	$0.396476\pi$
$158$	8.69607	0.691822
$159$	−17.8622	−1.41657
$160$	1.60147	0.126607
$161$	−22.4005	−1.76541
$162$	11.2397	0.883075
$163$	1.88434	0.147593	0.0737966	−	0.997273i	$-0.476488\pi$
$163$	1.88434	0.147593	0.0737966	+	0.997273i	$0.476488\pi$
$164$	−2.05783	−0.160689
$165$	−3.43531	−0.267438
$166$	6.10833	0.474098
$167$	−4.98627	−0.385849	−0.192925	−	0.981214i	$-0.561797\pi$
$167$	−4.98627	−0.385849	−0.192925	+	0.981214i	$0.561797\pi$
$168$	6.20293	0.478567
$169$	10.9284	0.840650
$170$	−9.20293	−0.705833
$171$	−1.60147	−0.122467
$172$	−10.6887	−0.815009
$173$	6.51419	0.495265	0.247632	−	0.968854i	$-0.420347\pi$
$173$	6.51419	0.495265	0.247632	+	0.968854i	$0.420347\pi$
$174$	11.4721	0.869696
$175$	−7.04211	−0.532333
$176$	−1.00000	−0.0753778
$177$	5.45636	0.410125
$178$	−3.20293	−0.240070
$179$	−5.55096	−0.414899	−0.207449	−	0.978246i	$-0.566516\pi$
$179$	−5.55096	−0.414899	−0.207449	+	0.978246i	$0.566516\pi$
$180$	−2.56469	−0.191161
$181$	1.88434	0.140062	0.0700311	−	0.997545i	$-0.477690\pi$
$181$	1.88434	0.140062	0.0700311	+	0.997545i	$0.477690\pi$
$182$	−14.1451	−1.04850
$183$	−28.9442	−2.13961
$184$	7.74657	0.571085
$185$	12.0000	0.882258
$186$	−15.9495	−1.16947
$187$	5.74657	0.420231
$188$	−7.08727	−0.516893
$189$	8.67501	0.631014
$190$	−1.60147	−0.116183
$191$	2.44264	0.176743	0.0883715	−	0.996088i	$-0.471834\pi$
$191$	2.44264	0.176743	0.0883715	+	0.996088i	$0.471834\pi$
$192$	−2.14510	−0.154809
$193$	−9.67501	−0.696423	−0.348211	−	0.937416i	$-0.613211\pi$
$193$	−9.67501	−0.696423	−0.348211	+	0.937416i	$0.613211\pi$
$194$	6.98627	0.501585
$195$	16.8044	1.20339
$196$	1.36176	0.0972686
$197$	−17.4931	−1.24633	−0.623167	−	0.782089i	$-0.714154\pi$
$197$	−17.4931	−1.24633	−0.623167	+	0.782089i	$0.714154\pi$
$198$	1.60147	0.113811
$199$	−25.2397	−1.78920	−0.894598	−	0.446873i	$-0.852538\pi$
$199$	−25.2397	−1.78920	−0.894598	+	0.446873i	$0.852538\pi$
$200$	2.43531	0.172202
$201$	22.2765	1.57126
$202$	1.20293	0.0846379
$203$	15.4648	1.08541
$204$	12.3270	0.863061
$205$	3.29554	0.230171
$206$	−16.8779	−1.17594
$207$	−12.4059	−0.862267
$208$	4.89167	0.339176
$209$	1.00000	0.0691714
$210$	−9.93378	−0.685496
$211$	−14.8338	−1.02120	−0.510602	−	0.859817i	$-0.670577\pi$
$211$	−14.8338	−1.02120	−0.510602	+	0.859817i	$0.670577\pi$
$212$	8.32698	0.571899
$213$	−32.1314	−2.20161
$214$	8.03677	0.549383
$215$	17.1176	1.16741
$216$	−3.00000	−0.204124
$217$	−21.5005	−1.45955
$218$	−11.5299	−0.780904
$219$	8.03677	0.543075
$220$	1.60147	0.107971
$221$	−28.1103	−1.89090
$222$	−16.0735	−1.07879
$223$	11.4931	0.769637	0.384819	−	0.922992i	$-0.374264\pi$
$223$	11.4931	0.769637	0.384819	+	0.922992i	$0.374264\pi$
$224$	−2.89167	−0.193208
$225$	−3.90006	−0.260004
$226$	8.00000	0.532152
$227$	8.73284	0.579619	0.289810	−	0.957084i	$-0.406408\pi$
$227$	8.73284	0.579619	0.289810	+	0.957084i	$0.406408\pi$
$228$	2.14510	0.142063
$229$	7.84117	0.518159	0.259080	−	0.965856i	$-0.416581\pi$
$229$	7.84117	0.518159	0.259080	+	0.965856i	$0.416581\pi$
$230$	−12.4059	−0.818018
$231$	6.20293	0.408123
$232$	−5.34803	−0.351116
$233$	−12.5069	−0.819352	−0.409676	−	0.912231i	$-0.634358\pi$
$233$	−12.5069	−0.819352	−0.409676	+	0.912231i	$0.634358\pi$
$234$	−7.83384	−0.512114
$235$	11.3500	0.740394
$236$	−2.54364	−0.165577
$237$	18.6540	1.21170
$238$	16.6172	1.07713
$239$	24.6245	1.59283	0.796414	−	0.604752i	$-0.206728\pi$
$239$	24.6245	1.59283	0.796414	+	0.604752i	$0.206728\pi$
$240$	3.43531	0.221748
$241$	16.0809	1.03586	0.517930	−	0.855423i	$-0.326703\pi$
$241$	16.0809	1.03586	0.517930	+	0.855423i	$0.326703\pi$
$242$	−1.00000	−0.0642824
$243$	15.1103	0.969328
$244$	13.4931	0.863810
$245$	−2.18081	−0.139327
$246$	−4.41425	−0.281443
$247$	−4.89167	−0.311250
$248$	7.43531	0.472143
$249$	13.1030	0.830368
$250$	−11.9074	−0.753089
$251$	13.0873	0.826061	0.413031	−	0.910717i	$-0.364470\pi$
$251$	13.0873	0.826061	0.413031	+	0.910717i	$0.364470\pi$
$252$	4.63091	0.291720
$253$	7.74657	0.487023
$254$	21.2765	1.33500
$255$	−19.7412	−1.23624
$256$	1.00000	0.0625000
$257$	0.723522	0.0451320	0.0225660	−	0.999745i	$-0.492816\pi$
$257$	0.723522	0.0451320	0.0225660	+	0.999745i	$0.492816\pi$
$258$	−22.9284	−1.42746
$259$	−21.6677	−1.34636
$260$	−7.83384	−0.485834
$261$	8.56469	0.530141
$262$	3.43531	0.212234
$263$	13.7760	0.849465	0.424733	−	0.905319i	$-0.360368\pi$
$263$	13.7760	0.849465	0.424733	+	0.905319i	$0.360368\pi$
$264$	−2.14510	−0.132022
$265$	−13.3354	−0.819185
$266$	2.89167	0.177300
$267$	−6.87062	−0.420475
$268$	−10.3848	−0.634353
$269$	−19.9579	−1.21685	−0.608427	−	0.793610i	$-0.708199\pi$
$269$	−19.9579	−1.21685	−0.608427	+	0.793610i	$0.708199\pi$
$270$	4.80440	0.292386
$271$	28.7486	1.74635	0.873175	−	0.487406i	$-0.162057\pi$
$271$	28.7486	1.74635	0.873175	+	0.487406i	$0.162057\pi$
$272$	−5.74657	−0.348437
$273$	−30.3427	−1.83642
$274$	−13.5877	−0.820865
$275$	2.43531	0.146855
$276$	16.6172	1.00024
$277$	3.08727	0.185496	0.0927482	−	0.995690i	$-0.470435\pi$
$277$	3.08727	0.185496	0.0927482	+	0.995690i	$0.470435\pi$
$278$	−3.19560	−0.191660
$279$	−11.9074	−0.712877
$280$	4.63091	0.276750
$281$	9.09460	0.542538	0.271269	−	0.962504i	$-0.412557\pi$
$281$	9.09460	0.542538	0.271269	+	0.962504i	$0.412557\pi$
$282$	−15.2029	−0.905321
$283$	13.1451	0.781395	0.390698	−	0.920519i	$-0.372234\pi$
$283$	13.1451	0.781395	0.390698	+	0.920519i	$0.372234\pi$
$284$	14.9789	0.888837
$285$	−3.43531	−0.203490
$286$	4.89167	0.289251
$287$	−5.95056	−0.351251
$288$	−1.60147	−0.0943673
$289$	16.0230	0.942532
$290$	8.56469	0.502936
$291$	14.9863	0.878511
$292$	−3.74657	−0.219251
$293$	3.10833	0.181591	0.0907953	−	0.995870i	$-0.471059\pi$
$293$	3.10833	0.181591	0.0907953	+	0.995870i	$0.471059\pi$
$294$	2.92112	0.170363
$295$	4.07355	0.237171
$296$	7.49314	0.435530
$297$	−3.00000	−0.174078
$298$	12.0735	0.699402
$299$	−37.8937	−2.19145
$300$	5.22399	0.301607
$301$	−30.9083	−1.78153
$302$	4.91273	0.282696
$303$	2.58041	0.148241
$304$	−1.00000	−0.0573539
$305$	−21.6088	−1.23732
$306$	9.20293	0.526097
$307$	2.18920	0.124944	0.0624722	−	0.998047i	$-0.480102\pi$
$307$	2.18920	0.124944	0.0624722	+	0.998047i	$0.480102\pi$
$308$	−2.89167	−0.164768
$309$	−36.2049	−2.05963
$310$	−11.9074	−0.676294
$311$	3.92112	0.222346	0.111173	−	0.993801i	$-0.464539\pi$
$311$	3.92112	0.222346	0.111173	+	0.993801i	$0.464539\pi$
$312$	10.4931	0.594057
$313$	−21.7118	−1.22722	−0.613611	−	0.789608i	$-0.710284\pi$
$313$	−21.7118	−1.22722	−0.613611	+	0.789608i	$0.710284\pi$
$314$	−8.00733	−0.451880
$315$	−7.41625	−0.417858
$316$	−8.69607	−0.489192
$317$	6.90739	0.387958	0.193979	−	0.981006i	$-0.437861\pi$
$317$	6.90739	0.387958	0.193979	+	0.981006i	$0.437861\pi$
$318$	17.8622	1.00166
$319$	−5.34803	−0.299432
$320$	−1.60147	−0.0895246
$321$	17.2397	0.962226
$322$	22.4005	1.24833
$323$	5.74657	0.319748
$324$	−11.2397	−0.624428
$325$	−11.9127	−0.660799
$326$	−1.88434	−0.104364
$327$	−24.7328	−1.36773
$328$	2.05783	0.113625
$329$	−20.4941	−1.12987
$330$	3.43531	0.189107
$331$	32.1030	1.76454	0.882270	−	0.470744i	$-0.156014\pi$
$331$	32.1030	1.76454	0.882270	+	0.470744i	$0.156014\pi$
$332$	−6.10833	−0.335238
$333$	−12.0000	−0.657596
$334$	4.98627	0.272837
$335$	16.6309	0.908644
$336$	−6.20293	−0.338398
$337$	−1.55096	−0.0844864	−0.0422432	−	0.999107i	$-0.513450\pi$
$337$	−1.55096	−0.0844864	−0.0422432	+	0.999107i	$0.513450\pi$
$338$	−10.9284	−0.594429
$339$	17.1608	0.932048
$340$	9.20293	0.499099
$341$	7.43531	0.402645
$342$	1.60147	0.0865973
$343$	−16.3039	−0.880330
$344$	10.6887	0.576298
$345$	−26.6118	−1.43273
$346$	−6.51419	−0.350205
$347$	−0.506864	−0.0272099	−0.0136049	−	0.999907i	$-0.504331\pi$
$347$	−0.506864	−0.0272099	−0.0136049	+	0.999907i	$0.504331\pi$
$348$	−11.4721	−0.614968
$349$	10.8117	0.578738	0.289369	−	0.957218i	$-0.406554\pi$
$349$	10.8117	0.578738	0.289369	+	0.957218i	$0.406554\pi$
$350$	7.04211	0.376417
$351$	14.6750	0.783294
$352$	1.00000	0.0533002
$353$	27.0966	1.44221	0.721103	−	0.692828i	$-0.243635\pi$
$353$	27.0966	1.44221	0.721103	+	0.692828i	$0.243635\pi$
$354$	−5.45636	−0.290002
$355$	−23.9883	−1.27316
$356$	3.20293	0.169755
$357$	35.6456	1.88656
$358$	5.55096	0.293378
$359$	−20.7486	−1.09507	−0.547534	−	0.836784i	$-0.684433\pi$
$359$	−20.7486	−1.09507	−0.547534	+	0.836784i	$0.684433\pi$
$360$	2.56469	0.135171
$361$	1.00000	0.0526316
$362$	−1.88434	−0.0990389
$363$	−2.14510	−0.112589
$364$	14.1451	0.741405
$365$	6.00000	0.314054
$366$	28.9442	1.51294
$367$	25.6402	1.33841	0.669205	−	0.743078i	$-0.266635\pi$
$367$	25.6402	1.33841	0.669205	+	0.743078i	$0.266635\pi$
$368$	−7.74657	−0.403818
$369$	−3.29554	−0.171559
$370$	−12.0000	−0.623850
$371$	24.0789	1.25011
$372$	15.9495	0.826943
$373$	25.2049	1.30506	0.652531	−	0.757762i	$-0.273707\pi$
$373$	25.2049	1.30506	0.652531	+	0.757762i	$0.273707\pi$
$374$	−5.74657	−0.297148
$375$	−25.5426	−1.31901
$376$	7.08727	0.365498
$377$	26.1608	1.34735
$378$	−8.67501	−0.446195
$379$	3.47208	0.178349	0.0891744	−	0.996016i	$-0.471577\pi$
$379$	3.47208	0.178349	0.0891744	+	0.996016i	$0.471577\pi$
$380$	1.60147	0.0821534
$381$	45.6402	2.33822
$382$	−2.44264	−0.124976
$383$	23.1176	1.18126	0.590628	−	0.806944i	$-0.298880\pi$
$383$	23.1176	1.18126	0.590628	+	0.806944i	$0.298880\pi$
$384$	2.14510	0.109467
$385$	4.63091	0.236013
$386$	9.67501	0.492445
$387$	−17.1176	−0.870139
$388$	−6.98627	−0.354674
$389$	−26.7623	−1.35690	−0.678451	−	0.734646i	$-0.737348\pi$
$389$	−26.7623	−1.35690	−0.678451	+	0.734646i	$0.737348\pi$
$390$	−16.8044	−0.850924
$391$	44.5162	2.25128
$392$	−1.36176	−0.0687793
$393$	7.36909	0.371721
$394$	17.4931	0.881291
$395$	13.9265	0.700716
$396$	−1.60147	−0.0804767
$397$	−31.1030	−1.56101	−0.780507	−	0.625147i	$-0.785039\pi$
$397$	−31.1030	−1.56101	−0.780507	+	0.625147i	$0.785039\pi$
$398$	25.2397	1.26515
$399$	6.20293	0.310535
$400$	−2.43531	−0.121765
$401$	29.3500	1.46567	0.732835	−	0.680406i	$-0.238197\pi$
$401$	29.3500	1.46567	0.732835	+	0.680406i	$0.238197\pi$
$402$	−22.2765	−1.11105
$403$	−36.3711	−1.81177
$404$	−1.20293	−0.0598481
$405$	18.0000	0.894427
$406$	−15.4648	−0.767503
$407$	7.49314	0.371421
$408$	−12.3270	−0.610276
$409$	8.58774	0.424636	0.212318	−	0.977201i	$-0.431899\pi$
$409$	8.58774	0.424636	0.212318	+	0.977201i	$0.431899\pi$
$410$	−3.29554	−0.162755
$411$	−29.1471	−1.43772
$412$	16.8779	0.831517
$413$	−7.35536	−0.361934
$414$	12.4059	0.609715
$415$	9.78228	0.480193
$416$	−4.89167	−0.239834
$417$	−6.85490	−0.335686
$418$	−1.00000	−0.0489116
$419$	7.59414	0.370998	0.185499	−	0.982644i	$-0.440610\pi$
$419$	7.59414	0.370998	0.185499	+	0.982644i	$0.440610\pi$
$420$	9.93378	0.484719
$421$	8.76030	0.426951	0.213475	−	0.976948i	$-0.431522\pi$
$421$	8.76030	0.426951	0.213475	+	0.976948i	$0.431522\pi$
$422$	14.8338	0.722100
$423$	−11.3500	−0.551857
$424$	−8.32698	−0.404394
$425$	13.9947	0.678841
$426$	32.1314	1.55677
$427$	39.0177	1.88820
$428$	−8.03677	−0.388472
$429$	10.4931	0.506613
$430$	−17.1176	−0.825486
$431$	−33.6677	−1.62172	−0.810858	−	0.585243i	$-0.800999\pi$
$431$	−33.6677	−1.62172	−0.810858	+	0.585243i	$0.800999\pi$
$432$	3.00000	0.144338
$433$	30.5383	1.46758	0.733789	−	0.679378i	$-0.237750\pi$
$433$	30.5383	1.46758	0.733789	+	0.679378i	$0.237750\pi$
$434$	21.5005	1.03206
$435$	18.3721	0.880877
$436$	11.5299	0.552183
$437$	7.74657	0.370569
$438$	−8.03677	−0.384012
$439$	−12.5069	−0.596920	−0.298460	−	0.954422i	$-0.596473\pi$
$439$	−12.5069	−0.596920	−0.298460	+	0.954422i	$0.596473\pi$
$440$	−1.60147	−0.0763469
$441$	2.18081	0.103848
$442$	28.1103	1.33707
$443$	−17.6677	−0.839417	−0.419709	−	0.907659i	$-0.637868\pi$
$443$	−17.6677	−0.839417	−0.419709	+	0.907659i	$0.637868\pi$
$444$	16.0735	0.762817
$445$	−5.12938	−0.243156
$446$	−11.4931	−0.544216
$447$	25.8990	1.22498
$448$	2.89167	0.136619
$449$	−17.7098	−0.835777	−0.417888	−	0.908498i	$-0.637230\pi$
$449$	−17.7098	−0.835777	−0.417888	+	0.908498i	$0.637230\pi$
$450$	3.90006	0.183851
$451$	2.05783	0.0968994
$452$	−8.00000	−0.376288
$453$	10.5383	0.495133
$454$	−8.73284	−0.409853
$455$	−22.6529	−1.06198
$456$	−2.14510	−0.100454
$457$	−24.8064	−1.16039	−0.580197	−	0.814476i	$-0.697024\pi$
$457$	−24.8064	−1.16039	−0.580197	+	0.814476i	$0.697024\pi$
$458$	−7.84117	−0.366394
$459$	−17.2397	−0.804681
$460$	12.4059	0.578426
$461$	−2.98627	−0.139085	−0.0695423	−	0.997579i	$-0.522154\pi$
$461$	−2.98627	−0.139085	−0.0695423	+	0.997579i	$0.522154\pi$
$462$	−6.20293	−0.288586
$463$	−39.5667	−1.83882	−0.919410	−	0.393301i	$-0.871333\pi$
$463$	−39.5667	−1.83882	−0.919410	+	0.393301i	$0.871333\pi$
$464$	5.34803	0.248276
$465$	−25.5426	−1.18451
$466$	12.5069	0.579369
$467$	4.46475	0.206604	0.103302	−	0.994650i	$-0.467059\pi$
$467$	4.46475	0.206604	0.103302	+	0.994650i	$0.467059\pi$
$468$	7.83384	0.362119
$469$	−30.0294	−1.38663
$470$	−11.3500	−0.523538
$471$	−17.1765	−0.791453
$472$	2.54364	0.117080
$473$	10.6887	0.491469
$474$	−18.6540	−0.856805
$475$	2.43531	0.111740
$476$	−16.6172	−0.761647
$477$	13.3354	0.610585
$478$	−24.6245	−1.12630
$479$	−20.8863	−0.954321	−0.477161	−	0.878816i	$-0.658334\pi$
$479$	−20.8863	−0.954321	−0.477161	+	0.878816i	$0.658334\pi$
$480$	−3.43531	−0.156800
$481$	−36.6540	−1.67128
$482$	−16.0809	−0.732464
$483$	48.0514	2.18642
$484$	1.00000	0.0454545
$485$	11.1883	0.508033
$486$	−15.1103	−0.685418
$487$	15.2838	0.692575	0.346288	−	0.938128i	$-0.387442\pi$
$487$	15.2838	0.692575	0.346288	+	0.938128i	$0.387442\pi$
$488$	−13.4931	−0.610806
$489$	−4.04211	−0.182791
$490$	2.18081	0.0985191
$491$	3.31965	0.149814	0.0749069	−	0.997191i	$-0.476134\pi$
$491$	3.31965	0.149814	0.0749069	+	0.997191i	$0.476134\pi$
$492$	4.41425	0.199010
$493$	−30.7328	−1.38414
$494$	4.89167	0.220087
$495$	2.56469	0.115274
$496$	−7.43531	−0.333855
$497$	43.3142	1.94291
$498$	−13.1030	−0.587159
$499$	−2.07355	−0.0928247	−0.0464124	−	0.998922i	$-0.514779\pi$
$499$	−2.07355	−0.0928247	−0.0464124	+	0.998922i	$0.514779\pi$
$500$	11.9074	0.532515
$501$	10.6961	0.477865
$502$	−13.0873	−0.584114
$503$	−1.11672	−0.0497921	−0.0248960	−	0.999690i	$-0.507925\pi$
$503$	−1.11672	−0.0497921	−0.0248960	+	0.999690i	$0.507925\pi$
$504$	−4.63091	−0.206277
$505$	1.92645	0.0857260
$506$	−7.74657	−0.344377
$507$	−23.4426	−1.04112
$508$	−21.2765	−0.943991
$509$	−38.2481	−1.69532	−0.847659	−	0.530542i	$-0.821988\pi$
$509$	−38.2481	−1.69532	−0.847659	+	0.530542i	$0.821988\pi$
$510$	19.7412	0.874156
$511$	−10.8338	−0.479261
$512$	−1.00000	−0.0441942
$513$	−3.00000	−0.132453
$514$	−0.723522	−0.0319132
$515$	−27.0294	−1.19106
$516$	22.9284	1.00937
$517$	7.08727	0.311698
$518$	21.6677	0.952023
$519$	−13.9736	−0.613373
$520$	7.83384	0.343537
$521$	10.8853	0.476892	0.238446	−	0.971156i	$-0.423362\pi$
$521$	10.8853	0.476892	0.238446	+	0.971156i	$0.423362\pi$
$522$	−8.56469	−0.374866
$523$	−15.7191	−0.687349	−0.343674	−	0.939089i	$-0.611672\pi$
$523$	−15.7191	−0.687349	−0.343674	+	0.939089i	$0.611672\pi$
$524$	−3.43531	−0.150072
$525$	15.1060	0.659282
$526$	−13.7760	−0.600663
$527$	42.7275	1.86124
$528$	2.14510	0.0933536
$529$	37.0093	1.60910
$530$	13.3354	0.579251
$531$	−4.07355	−0.176777
$532$	−2.89167	−0.125370
$533$	−10.0662	−0.436016
$534$	6.87062	0.297321
$535$	12.8706	0.556445
$536$	10.3848	0.448555
$537$	11.9074	0.513842
$538$	19.9579	0.860446
$539$	−1.36176	−0.0586552
$540$	−4.80440	−0.206748
$541$	−26.4373	−1.13663	−0.568314	−	0.822812i	$-0.692404\pi$
$541$	−26.4373	−1.13663	−0.568314	+	0.822812i	$0.692404\pi$
$542$	−28.7486	−1.23486
$543$	−4.04211	−0.173464
$544$	5.74657	0.246382
$545$	−18.4648	−0.790943
$546$	30.3427	1.29855
$547$	−24.9588	−1.06716	−0.533581	−	0.845749i	$-0.679154\pi$
$547$	−24.9588	−1.06716	−0.533581	+	0.845749i	$0.679154\pi$
$548$	13.5877	0.580439
$549$	21.6088	0.922241
$550$	−2.43531	−0.103842
$551$	−5.34803	−0.227834
$552$	−16.6172	−0.707274
$553$	−25.1462	−1.06932
$554$	−3.08727	−0.131166
$555$	−25.7412	−1.09265
$556$	3.19560	0.135524
$557$	8.30486	0.351888	0.175944	−	0.984400i	$-0.443702\pi$
$557$	8.30486	0.351888	0.175944	+	0.984400i	$0.443702\pi$
$558$	11.9074	0.504080
$559$	−52.2858	−2.21145
$560$	−4.63091	−0.195692
$561$	−12.3270	−0.520445
$562$	−9.09460	−0.383633
$563$	41.6402	1.75493	0.877463	−	0.479644i	$-0.159234\pi$
$563$	41.6402	1.75493	0.877463	+	0.479644i	$0.159234\pi$
$564$	15.2029	0.640159
$565$	12.8117	0.538993
$566$	−13.1451	−0.552530
$567$	−32.5015	−1.36494
$568$	−14.9789	−0.628503
$569$	−14.9138	−0.625219	−0.312609	−	0.949882i	$-0.601203\pi$
$569$	−14.9138	−0.625219	−0.312609	+	0.949882i	$0.601203\pi$
$570$	3.43531	0.143889
$571$	−23.0020	−0.962603	−0.481302	−	0.876555i	$-0.659836\pi$
$571$	−23.0020	−0.962603	−0.481302	+	0.876555i	$0.659836\pi$
$572$	−4.89167	−0.204531
$573$	−5.23970	−0.218892
$574$	5.95056	0.248372
$575$	18.8653	0.786737
$576$	1.60147	0.0667277
$577$	−33.6613	−1.40134	−0.700669	−	0.713487i	$-0.747115\pi$
$577$	−33.6613	−1.40134	−0.700669	+	0.713487i	$0.747115\pi$
$578$	−16.0230	−0.666471
$579$	20.7539	0.862502
$580$	−8.56469	−0.355629
$581$	−17.6633	−0.732796
$582$	−14.9863	−0.621201
$583$	−8.32698	−0.344868
$584$	3.74657	0.155034
$585$	−12.5456	−0.518698
$586$	−3.10833	−0.128404
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	−2.92112	−0.120465
$589$	7.43531	0.306367
$590$	−4.07355	−0.167705
$591$	37.5246	1.54355
$592$	−7.49314	−0.307966
$593$	−30.9588	−1.27133	−0.635663	−	0.771967i	$-0.719273\pi$
$593$	−30.9588	−1.27133	−0.635663	+	0.771967i	$0.719273\pi$
$594$	3.00000	0.123091
$595$	26.6118	1.09098
$596$	−12.0735	−0.494552
$597$	54.1418	2.21587
$598$	37.8937	1.54959
$599$	3.93378	0.160730	0.0803650	−	0.996766i	$-0.474391\pi$
$599$	3.93378	0.160730	0.0803650	+	0.996766i	$0.474391\pi$
$600$	−5.22399	−0.213268
$601$	23.1451	0.944108	0.472054	−	0.881570i	$-0.343513\pi$
$601$	23.1451	0.944108	0.472054	+	0.881570i	$0.343513\pi$
$602$	30.9083	1.25973
$603$	−16.6309	−0.677263
$604$	−4.91273	−0.199896
$605$	−1.60147	−0.0651088
$606$	−2.58041	−0.104822
$607$	−1.78334	−0.0723836	−0.0361918	−	0.999345i	$-0.511523\pi$
$607$	−1.78334	−0.0723836	−0.0361918	+	0.999345i	$0.511523\pi$
$608$	1.00000	0.0405554
$609$	−33.1735	−1.34426
$610$	21.6088	0.874914
$611$	−34.6686	−1.40254
$612$	−9.20293	−0.372006
$613$	−33.0598	−1.33527	−0.667637	−	0.744487i	$-0.732694\pi$
$613$	−33.0598	−1.33527	−0.667637	+	0.744487i	$0.732694\pi$
$614$	−2.18920	−0.0883491
$615$	−7.06927	−0.285061
$616$	2.89167	0.116509
$617$	−21.8412	−0.879292	−0.439646	−	0.898171i	$-0.644896\pi$
$617$	−21.8412	−0.879292	−0.439646	+	0.898171i	$0.644896\pi$
$618$	36.2049	1.45638
$619$	−46.2206	−1.85776	−0.928882	−	0.370375i	$-0.879229\pi$
$619$	−46.2206	−1.85776	−0.928882	+	0.370375i	$0.879229\pi$
$620$	11.9074	0.478212
$621$	−23.2397	−0.932577
$622$	−3.92112	−0.157222
$623$	9.26182	0.371067
$624$	−10.4931	−0.420062
$625$	−6.89273	−0.275709
$626$	21.7118	0.867778
$627$	−2.14510	−0.0856671
$628$	8.00733	0.319527
$629$	43.0598	1.71691
$630$	7.41625	0.295470
$631$	0.317659	0.0126458	0.00632291	−	0.999980i	$-0.497987\pi$
$631$	0.317659	0.0126458	0.00632291	+	0.999980i	$0.497987\pi$
$632$	8.69607	0.345911
$633$	31.8201	1.26474
$634$	−6.90739	−0.274327
$635$	34.0735	1.35217
$636$	−17.8622	−0.708283
$637$	6.66129	0.263930
$638$	5.34803	0.211731
$639$	23.9883	0.948961
$640$	1.60147	0.0633035
$641$	−41.5246	−1.64012	−0.820061	−	0.572276i	$-0.806061\pi$
$641$	−41.5246	−1.64012	−0.820061	+	0.572276i	$0.806061\pi$
$642$	−17.2397	−0.680397
$643$	7.26182	0.286378	0.143189	−	0.989695i	$-0.454264\pi$
$643$	7.26182	0.286378	0.143189	+	0.989695i	$0.454264\pi$
$644$	−22.4005	−0.882704
$645$	−36.7191	−1.44581
$646$	−5.74657	−0.226096
$647$	29.0819	1.14333	0.571664	−	0.820487i	$-0.306298\pi$
$647$	29.0819	1.14333	0.571664	+	0.820487i	$0.306298\pi$
$648$	11.2397	0.441537
$649$	2.54364	0.0998465
$650$	11.9127	0.467256
$651$	46.1207	1.80761
$652$	1.88434	0.0737966
$653$	12.6529	0.495146	0.247573	−	0.968869i	$-0.420367\pi$
$653$	12.6529	0.495146	0.247573	+	0.968869i	$0.420367\pi$
$654$	24.7328	0.967131
$655$	5.50153	0.214962
$656$	−2.05783	−0.0803447
$657$	−6.00000	−0.234082
$658$	20.4941	0.798942
$659$	49.4878	1.92777	0.963886	−	0.266317i	$-0.0858068\pi$
$659$	49.4878	1.92777	0.963886	+	0.266317i	$0.0858068\pi$
$660$	−3.43531	−0.133719
$661$	4.94950	0.192513	0.0962566	−	0.995357i	$-0.469313\pi$
$661$	4.94950	0.192513	0.0962566	+	0.995357i	$0.469313\pi$
$662$	−32.1030	−1.24772
$663$	60.2995	2.34184
$664$	6.10833	0.237049
$665$	4.63091	0.179579
$666$	12.0000	0.464991
$667$	−41.4289	−1.60413
$668$	−4.98627	−0.192925
$669$	−24.6540	−0.953177
$670$	−16.6309	−0.642508
$671$	−13.4931	−0.520897
$672$	6.20293	0.239283
$673$	21.9001	0.844185	0.422093	−	0.906553i	$-0.361296\pi$
$673$	21.9001	0.844185	0.422093	+	0.906553i	$0.361296\pi$
$674$	1.55096	0.0597409
$675$	−7.30592	−0.281205
$676$	10.9284	0.420325
$677$	26.4353	1.01599	0.507996	−	0.861360i	$-0.330387\pi$
$677$	26.4353	1.01599	0.507996	+	0.861360i	$0.330387\pi$
$678$	−17.1608	−0.659057
$679$	−20.2020	−0.775282
$680$	−9.20293	−0.352916
$681$	−18.7328	−0.717844
$682$	−7.43531	−0.284713
$683$	9.82545	0.375960	0.187980	−	0.982173i	$-0.439806\pi$
$683$	9.82545	0.375960	0.187980	+	0.982173i	$0.439806\pi$
$684$	−1.60147	−0.0612336
$685$	−21.7603	−0.831418
$686$	16.3039	0.622487
$687$	−16.8201	−0.641727
$688$	−10.6887	−0.407504
$689$	40.7328	1.55180
$690$	26.6118	1.01310
$691$	−19.8255	−0.754196	−0.377098	−	0.926173i	$-0.623078\pi$
$691$	−19.8255	−0.754196	−0.377098	+	0.926173i	$0.623078\pi$
$692$	6.51419	0.247632
$693$	−4.63091	−0.175914
$694$	0.506864	0.0192403
$695$	−5.11765	−0.194123
$696$	11.4721	0.434848
$697$	11.8255	0.447921
$698$	−10.8117	−0.409230
$699$	26.8285	1.01475
$700$	−7.04211	−0.266167
$701$	−10.7550	−0.406209	−0.203105	−	0.979157i	$-0.565103\pi$
$701$	−10.7550	−0.406209	−0.203105	+	0.979157i	$0.565103\pi$
$702$	−14.6750	−0.553873
$703$	7.49314	0.282609
$704$	−1.00000	−0.0376889
$705$	−24.3470	−0.916960
$706$	−27.0966	−1.01979
$707$	−3.47848	−0.130822
$708$	5.45636	0.205063
$709$	−1.82651	−0.0685962	−0.0342981	−	0.999412i	$-0.510920\pi$
$709$	−1.82651	−0.0685962	−0.0342981	+	0.999412i	$0.510920\pi$
$710$	23.9883	0.900264
$711$	−13.9265	−0.522283
$712$	−3.20293	−0.120035
$713$	57.5981	2.15707
$714$	−35.6456	−1.33400
$715$	7.83384	0.292969
$716$	−5.55096	−0.207449
$717$	−52.8221	−1.97268
$718$	20.7486	0.774329
$719$	−4.65929	−0.173762	−0.0868812	−	0.996219i	$-0.527690\pi$
$719$	−4.65929	−0.173762	−0.0868812	+	0.996219i	$0.527690\pi$
$720$	−2.56469	−0.0955804
$721$	48.8055	1.81761
$722$	−1.00000	−0.0372161
$723$	−34.4951	−1.28289
$724$	1.88434	0.0700311
$725$	−13.0241	−0.483703
$726$	2.14510	0.0796122
$727$	19.9358	0.739377	0.369688	−	0.929156i	$-0.379464\pi$
$727$	19.9358	0.739377	0.369688	+	0.929156i	$0.379464\pi$
$728$	−14.1451	−0.524252
$729$	1.30592	0.0483676
$730$	−6.00000	−0.222070
$731$	61.4236	2.27183
$732$	−28.9442	−1.06981
$733$	13.7687	0.508558	0.254279	−	0.967131i	$-0.418162\pi$
$733$	13.7687	0.508558	0.254279	+	0.967131i	$0.418162\pi$
$734$	−25.6402	−0.946398
$735$	4.67807	0.172553
$736$	7.74657	0.285542
$737$	10.3848	0.382529
$738$	3.29554	0.121311
$739$	22.4867	0.827188	0.413594	−	0.910461i	$-0.364273\pi$
$739$	22.4867	0.827188	0.413594	+	0.910461i	$0.364273\pi$
$740$	12.0000	0.441129
$741$	10.4931	0.385475
$742$	−24.0789	−0.883964
$743$	−50.6540	−1.85831	−0.929157	−	0.369686i	$-0.879465\pi$
$743$	−50.6540	−1.85831	−0.929157	+	0.369686i	$0.879465\pi$
$744$	−15.9495	−0.584737
$745$	19.3354	0.708393
$746$	−25.2049	−0.922818
$747$	−9.78228	−0.357915
$748$	5.74657	0.210115
$749$	−23.2397	−0.849160
$750$	25.5426	0.932683
$751$	−38.9127	−1.41995	−0.709973	−	0.704229i	$-0.751293\pi$
$751$	−38.9127	−1.41995	−0.709973	+	0.704229i	$0.751293\pi$
$752$	−7.08727	−0.258446
$753$	−28.0735	−1.02306
$754$	−26.1608	−0.952721
$755$	7.86756	0.286330
$756$	8.67501	0.315507
$757$	−20.5877	−0.748274	−0.374137	−	0.927373i	$-0.622061\pi$
$757$	−20.5877	−0.748274	−0.374137	+	0.927373i	$0.622061\pi$
$758$	−3.47208	−0.126112
$759$	−16.6172	−0.603166
$760$	−1.60147	−0.0580913
$761$	10.6025	0.384341	0.192171	−	0.981362i	$-0.438447\pi$
$761$	10.6025	0.384341	0.192171	+	0.981362i	$0.438447\pi$
$762$	−45.6402	−1.65337
$763$	33.3407	1.20701
$764$	2.44264	0.0883715
$765$	14.7382	0.532860
$766$	−23.1176	−0.835275
$767$	−12.4426	−0.449278
$768$	−2.14510	−0.0774047
$769$	−8.77495	−0.316433	−0.158216	−	0.987404i	$-0.550574\pi$
$769$	−8.77495	−0.316433	−0.158216	+	0.987404i	$0.550574\pi$
$770$	−4.63091	−0.166886
$771$	−1.55203	−0.0558949
$772$	−9.67501	−0.348211
$773$	−46.8653	−1.68563	−0.842813	−	0.538206i	$-0.819102\pi$
$773$	−46.8653	−1.68563	−0.842813	+	0.538206i	$0.819102\pi$
$774$	17.1176	0.615281
$775$	18.1073	0.650432
$776$	6.98627	0.250793
$777$	46.4794	1.66744
$778$	26.7623	0.959474
$779$	2.05783	0.0737294
$780$	16.8044	0.601694
$781$	−14.9789	−0.535989
$782$	−44.5162	−1.59190
$783$	16.0441	0.573369
$784$	1.36176	0.0486343
$785$	−12.8235	−0.457689
$786$	−7.36909	−0.262847
$787$	−27.4143	−0.977213	−0.488606	−	0.872504i	$-0.662495\pi$
$787$	−27.4143	−0.977213	−0.488606	+	0.872504i	$0.662495\pi$
$788$	−17.4931	−0.623167
$789$	−29.5510	−1.05204
$790$	−13.9265	−0.495481
$791$	−23.1334	−0.822528
$792$	1.60147	0.0569056
$793$	66.0040	2.34387
$794$	31.1030	1.10380
$795$	28.6057	1.01454
$796$	−25.2397	−0.894598
$797$	−9.16616	−0.324682	−0.162341	−	0.986735i	$-0.551904\pi$
$797$	−9.16616	−0.324682	−0.162341	+	0.986735i	$0.551904\pi$
$798$	−6.20293	−0.219581
$799$	40.7275	1.44084
$800$	2.43531	0.0861011
$801$	5.12938	0.181238
$802$	−29.3500	−1.03639
$803$	3.74657	0.132214
$804$	22.2765	0.785631
$805$	35.8737	1.26438
$806$	36.3711	1.28112
$807$	42.8117	1.50704
$808$	1.20293	0.0423190
$809$	48.0829	1.69050	0.845252	−	0.534368i	$-0.179450\pi$
$809$	48.0829	1.69050	0.845252	+	0.534368i	$0.179450\pi$
$810$	−18.0000	−0.632456
$811$	−2.67395	−0.0938951	−0.0469475	−	0.998897i	$-0.514949\pi$
$811$	−2.67395	−0.0938951	−0.0469475	+	0.998897i	$0.514949\pi$
$812$	15.4648	0.542706
$813$	−61.6686	−2.16281
$814$	−7.49314	−0.262634
$815$	−3.01771	−0.105706
$816$	12.3270	0.431531
$817$	10.6887	0.373952
$818$	−8.58774	−0.300263
$819$	22.6529	0.791556
$820$	3.29554	0.115085
$821$	29.5078	1.02983	0.514915	−	0.857242i	$-0.327824\pi$
$821$	29.5078	1.02983	0.514915	+	0.857242i	$0.327824\pi$
$822$	29.1471	1.01662
$823$	−2.39654	−0.0835382	−0.0417691	−	0.999127i	$-0.513299\pi$
$823$	−2.39654	−0.0835382	−0.0417691	+	0.999127i	$0.513299\pi$
$824$	−16.8779	−0.587971
$825$	−5.22399	−0.181876
$826$	7.35536	0.255926
$827$	7.78868	0.270839	0.135419	−	0.990788i	$-0.456762\pi$
$827$	7.78868	0.270839	0.135419	+	0.990788i	$0.456762\pi$
$828$	−12.4059	−0.431134
$829$	13.2818	0.461296	0.230648	−	0.973037i	$-0.425915\pi$
$829$	13.2818	0.461296	0.230648	+	0.973037i	$0.425915\pi$
$830$	−9.78228	−0.339548
$831$	−6.62252	−0.229733
$832$	4.89167	0.169588
$833$	−7.82545	−0.271136
$834$	6.85490	0.237366
$835$	7.98534	0.276344
$836$	1.00000	0.0345857
$837$	−22.3059	−0.771006
$838$	−7.59414	−0.262335
$839$	−33.9063	−1.17058	−0.585288	−	0.810825i	$-0.699019\pi$
$839$	−33.9063	−1.17058	−0.585288	+	0.810825i	$0.699019\pi$
$840$	−9.93378	−0.342748
$841$	−0.398534	−0.0137426
$842$	−8.76030	−0.301900
$843$	−19.5089	−0.671921
$844$	−14.8338	−0.510602
$845$	−17.5015	−0.602071
$846$	11.3500	0.390222
$847$	2.89167	0.0993590
$848$	8.32698	0.285950
$849$	−28.1976	−0.967739
$850$	−13.9947	−0.480013
$851$	58.0461	1.98979
$852$	−32.1314	−1.10080
$853$	40.0735	1.37209	0.686046	−	0.727558i	$-0.259345\pi$
$853$	40.0735	1.37209	0.686046	+	0.727558i	$0.259345\pi$
$854$	−39.0177	−1.33516
$855$	2.56469	0.0877106
$856$	8.03677	0.274691
$857$	0.665693	0.0227396	0.0113698	−	0.999935i	$-0.496381\pi$
$857$	0.665693	0.0227396	0.0113698	+	0.999935i	$0.496381\pi$
$858$	−10.4931	−0.358230
$859$	13.8990	0.474228	0.237114	−	0.971482i	$-0.423799\pi$
$859$	13.8990	0.474228	0.237114	+	0.971482i	$0.423799\pi$
$860$	17.1176	0.583707
$861$	12.7646	0.435015
$862$	33.6677	1.14673
$863$	−40.3941	−1.37503	−0.687516	−	0.726169i	$-0.741299\pi$
$863$	−40.3941	−1.37503	−0.687516	+	0.726169i	$0.741299\pi$
$864$	−3.00000	−0.102062
$865$	−10.4323	−0.354707
$866$	−30.5383	−1.03773
$867$	−34.3711	−1.16730
$868$	−21.5005	−0.729773
$869$	8.69607	0.294994
$870$	−18.3721	−0.622874
$871$	−50.7991	−1.72126
$872$	−11.5299	−0.390452
$873$	−11.1883	−0.378666
$874$	−7.74657	−0.262032
$875$	34.4323	1.16402
$876$	8.03677	0.271537
$877$	15.6108	0.527139	0.263569	−	0.964640i	$-0.415100\pi$
$877$	15.6108	0.527139	0.263569	+	0.964640i	$0.415100\pi$
$878$	12.5069	0.422086
$879$	−6.66769	−0.224895
$880$	1.60147	0.0539854
$881$	19.9843	0.673288	0.336644	−	0.941632i	$-0.390708\pi$
$881$	19.9843	0.673288	0.336644	+	0.941632i	$0.390708\pi$
$882$	−2.18081	−0.0734318
$883$	20.2167	0.680345	0.340172	−	0.940363i	$-0.389515\pi$
$883$	20.2167	0.680345	0.340172	+	0.940363i	$0.389515\pi$
$884$	−28.1103	−0.945452
$885$	−8.73818	−0.293731
$886$	17.6677	0.593557
$887$	20.1471	0.676473	0.338237	−	0.941061i	$-0.390170\pi$
$887$	20.1471	0.676473	0.338237	+	0.941061i	$0.390170\pi$
$888$	−16.0735	−0.539393
$889$	−61.5246	−2.06347
$890$	5.12938	0.171937
$891$	11.2397	0.376544
$892$	11.4931	0.384819
$893$	7.08727	0.237167
$894$	−25.8990	−0.866192
$895$	8.88968	0.297149
$896$	−2.89167	−0.0966039
$897$	81.2858	2.71405
$898$	17.7098	0.590984
$899$	−39.7643	−1.32621
$900$	−3.90006	−0.130002
$901$	−47.8516	−1.59417
$902$	−2.05783	−0.0685182
$903$	66.3015	2.20638
$904$	8.00000	0.266076
$905$	−3.01771	−0.100312
$906$	−10.5383	−0.350112
$907$	4.57109	0.151781	0.0758903	−	0.997116i	$-0.475820\pi$
$907$	4.57109	0.151781	0.0758903	+	0.997116i	$0.475820\pi$
$908$	8.73284	0.289810
$909$	−1.92645	−0.0638964
$910$	22.6529	0.750936
$911$	51.6402	1.71092	0.855459	−	0.517871i	$-0.173275\pi$
$911$	51.6402	1.71092	0.855459	+	0.517871i	$0.173275\pi$
$912$	2.14510	0.0710314
$913$	6.10833	0.202156
$914$	24.8064	0.820522
$915$	46.3531	1.53239
$916$	7.84117	0.259080
$917$	−9.93378	−0.328042
$918$	17.2397	0.568995
$919$	−48.8412	−1.61112	−0.805561	−	0.592513i	$-0.798136\pi$
$919$	−48.8412	−1.61112	−0.805561	+	0.592513i	$0.798136\pi$
$920$	−12.4059	−0.409009
$921$	−4.69607	−0.154741
$922$	2.98627	0.0983477
$923$	73.2721	2.41178
$924$	6.20293	0.204061
$925$	18.2481	0.599994
$926$	39.5667	1.30024
$927$	27.0294	0.887763
$928$	−5.34803	−0.175558
$929$	−57.8819	−1.89904	−0.949522	−	0.313700i	$-0.898431\pi$
$929$	−57.8819	−1.89904	−0.949522	+	0.313700i	$0.898431\pi$
$930$	25.5426	0.837574
$931$	−1.36176	−0.0446299
$932$	−12.5069	−0.409676
$933$	−8.41120	−0.275370
$934$	−4.46475	−0.146091
$935$	−9.20293	−0.300968
$936$	−7.83384	−0.256057
$937$	−57.6496	−1.88333	−0.941664	−	0.336553i	$-0.890739\pi$
$937$	−57.6496	−1.88333	−0.941664	+	0.336553i	$0.890739\pi$
$938$	30.0294	0.980496
$939$	46.5740	1.51989
$940$	11.3500	0.370197
$941$	−15.0966	−0.492135	−0.246067	−	0.969253i	$-0.579138\pi$
$941$	−15.0966	−0.492135	−0.246067	+	0.969253i	$0.579138\pi$
$942$	17.1765	0.559642
$943$	15.9411	0.519114
$944$	−2.54364	−0.0827883
$945$	−13.8927	−0.451931
$946$	−10.6887	−0.347521
$947$	33.3775	1.08462	0.542311	−	0.840178i	$-0.317549\pi$
$947$	33.3775	1.08462	0.542311	+	0.840178i	$0.317549\pi$
$948$	18.6540	0.605852
$949$	−18.3270	−0.594919
$950$	−2.43531	−0.0790118
$951$	−14.8171	−0.480476
$952$	16.6172	0.538566
$953$	−12.4794	−0.404248	−0.202124	−	0.979360i	$-0.564784\pi$
$953$	−12.4794	−0.404248	−0.202124	+	0.979360i	$0.564784\pi$
$954$	−13.3354	−0.431749
$955$	−3.91180	−0.126583
$956$	24.6245	0.796414
$957$	11.4721	0.370840
$958$	20.8863	0.674807
$959$	39.2913	1.26878
$960$	3.43531	0.110874
$961$	24.2838	0.783349
$962$	36.6540	1.18177
$963$	−12.8706	−0.414750
$964$	16.0809	0.517930
$965$	15.4942	0.498776
$966$	−48.0514	−1.54603
$967$	−12.5804	−0.404559	−0.202279	−	0.979328i	$-0.564835\pi$
$967$	−12.5804	−0.404559	−0.202279	+	0.979328i	$0.564835\pi$
$968$	−1.00000	−0.0321412
$969$	−12.3270	−0.396000
$970$	−11.1883	−0.359234
$971$	−1.79067	−0.0574653	−0.0287327	−	0.999587i	$-0.509147\pi$
$971$	−1.79067	−0.0574653	−0.0287327	+	0.999587i	$0.509147\pi$
$972$	15.1103	0.484664
$973$	9.24063	0.296241
$974$	−15.2838	−0.489725
$975$	25.5540	0.818384
$976$	13.4931	0.431905
$977$	6.40586	0.204942	0.102471	−	0.994736i	$-0.467325\pi$
$977$	6.40586	0.204942	0.102471	+	0.994736i	$0.467325\pi$
$978$	4.04211	0.129252
$979$	−3.20293	−0.102366
$980$	−2.18081	−0.0696635
$981$	18.4648	0.589534
$982$	−3.31965	−0.105934
$983$	−20.5647	−0.655912	−0.327956	−	0.944693i	$-0.606360\pi$
$983$	−20.5647	−0.655912	−0.327956	+	0.944693i	$0.606360\pi$
$984$	−4.41425	−0.140721
$985$	28.0147	0.892621
$986$	30.7328	0.978733
$987$	43.9619	1.39932
$988$	−4.89167	−0.155625
$989$	82.8011	2.63292
$990$	−2.56469	−0.0815113
$991$	−29.0166	−0.921744	−0.460872	−	0.887467i	$-0.652463\pi$
$991$	−29.0166	−0.921744	−0.460872	+	0.887467i	$0.652463\pi$
$992$	7.43531	0.236071
$993$	−68.8642	−2.18534
$994$	−43.3142	−1.37384
$995$	40.4205	1.28142
$996$	13.1030	0.415184
$997$	42.3638	1.34167	0.670837	−	0.741605i	$-0.265935\pi$
$997$	42.3638	1.34167	0.670837	+	0.741605i	$0.265935\pi$
$998$	2.07355	0.0656370
$999$	−22.4794	−0.711217

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	418.2.a.g.1.1	✓	3
3.2	odd	2		3762.2.a.bg.1.2		3
4.3	odd	2		3344.2.a.q.1.3		3
11.10	odd	2		4598.2.a.bo.1.1		3
19.18	odd	2		7942.2.a.bi.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.g.1.1	✓	3	1.1	even	1	trivial
3344.2.a.q.1.3		3	4.3	odd	2
3762.2.a.bg.1.2		3	3.2	odd	2
4598.2.a.bo.1.1		3	11.10	odd	2
7942.2.a.bi.1.3		3	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 418 = 2 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	418.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.14510 q^{3} +1.00000 q^{4} -1.60147 q^{5} +2.14510 q^{6} +2.89167 q^{7} -1.00000 q^{8} +1.60147 q^{9} +1.60147 q^{10} -1.00000 q^{11} -2.14510 q^{12} +4.89167 q^{13} -2.89167 q^{14} +3.43531 q^{15} +1.00000 q^{16} -5.74657 q^{17} -1.60147 q^{18} -1.00000 q^{19} -1.60147 q^{20} -6.20293 q^{21} +1.00000 q^{22} -7.74657 q^{23} +2.14510 q^{24} -2.43531 q^{25} -4.89167 q^{26} +3.00000 q^{27} +2.89167 q^{28} +5.34803 q^{29} -3.43531 q^{30} -7.43531 q^{31} -1.00000 q^{32} +2.14510 q^{33} +5.74657 q^{34} -4.63091 q^{35} +1.60147 q^{36} -7.49314 q^{37} +1.00000 q^{38} -10.4931 q^{39} +1.60147 q^{40} -2.05783 q^{41} +6.20293 q^{42} -10.6887 q^{43} -1.00000 q^{44} -2.56469 q^{45} +7.74657 q^{46} -7.08727 q^{47} -2.14510 q^{48} +1.36176 q^{49} +2.43531 q^{50} +12.3270 q^{51} +4.89167 q^{52} +8.32698 q^{53} -3.00000 q^{54} +1.60147 q^{55} -2.89167 q^{56} +2.14510 q^{57} -5.34803 q^{58} -2.54364 q^{59} +3.43531 q^{60} +13.4931 q^{61} +7.43531 q^{62} +4.63091 q^{63} +1.00000 q^{64} -7.83384 q^{65} -2.14510 q^{66} -10.3848 q^{67} -5.74657 q^{68} +16.6172 q^{69} +4.63091 q^{70} +14.9789 q^{71} -1.60147 q^{72} -3.74657 q^{73} +7.49314 q^{74} +5.22399 q^{75} -1.00000 q^{76} -2.89167 q^{77} +10.4931 q^{78} -8.69607 q^{79} -1.60147 q^{80} -11.2397 q^{81} +2.05783 q^{82} -6.10833 q^{83} -6.20293 q^{84} +9.20293 q^{85} +10.6887 q^{86} -11.4721 q^{87} +1.00000 q^{88} +3.20293 q^{89} +2.56469 q^{90} +14.1451 q^{91} -7.74657 q^{92} +15.9495 q^{93} +7.08727 q^{94} +1.60147 q^{95} +2.14510 q^{96} -6.98627 q^{97} -1.36176 q^{98} -1.60147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{10} - 3 q^{11} + 6 q^{14} - 9 q^{15} + 3 q^{16} - 9 q^{17} - 3 q^{18} - 3 q^{19} - 3 q^{20} - 15 q^{21} + 3 q^{22} - 15 q^{23} + 12 q^{25}+ \cdots - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 - 6 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^10 - 3 * q^11 + 6 * q^14 - 9 * q^15 + 3 * q^16 - 9 * q^17 - 3 * q^18 - 3 * q^19 - 3 * q^20 - 15 * q^21 + 3 * q^22 - 15 * q^23 + 12 * q^25 + 9 * q^27 - 6 * q^28 + 6 * q^29 + 9 * q^30 - 3 * q^31 - 3 * q^32 + 9 * q^34 + 3 * q^36 - 6 * q^37 + 3 * q^38 - 15 * q^39 + 3 * q^40 - 9 * q^41 + 15 * q^42 - 21 * q^43 - 3 * q^44 - 27 * q^45 + 15 * q^46 - 12 * q^47 + 27 * q^49 - 12 * q^50 + 3 * q^51 - 9 * q^53 - 9 * q^54 + 3 * q^55 + 6 * q^56 - 6 * q^58 - 3 * q^59 - 9 * q^60 + 24 * q^61 + 3 * q^62 + 3 * q^64 - 6 * q^65 - 9 * q^68 + 3 * q^69 + 21 * q^71 - 3 * q^72 - 3 * q^73 + 6 * q^74 + 36 * q^75 - 3 * q^76 + 6 * q^77 + 15 * q^78 - 6 * q^79 - 3 * q^80 - 9 * q^81 + 9 * q^82 - 33 * q^83 - 15 * q^84 + 24 * q^85 + 21 * q^86 + 6 * q^87 + 3 * q^88 + 6 * q^89 + 27 * q^90 + 36 * q^91 - 15 * q^92 + 36 * q^93 + 12 * q^94 + 3 * q^95 + 12 * q^97 - 27 * q^98 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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