LMFDB - Embedded newform 1369.2.a.o.1.19

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)

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")

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);

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			1.19
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)

$f(q)$	$=$	\(q+1.81380 q^{2} -2.67610 q^{3} +1.28987 q^{4} -2.99386 q^{5} -4.85391 q^{6} -2.99597 q^{7} -1.28804 q^{8} +4.16150 q^{9} -5.43027 q^{10} +0.160906 q^{11} -3.45181 q^{12} +0.718529 q^{13} -5.43409 q^{14} +8.01187 q^{15} -4.91598 q^{16} -1.94564 q^{17} +7.54813 q^{18} +5.77250 q^{19} -3.86169 q^{20} +8.01751 q^{21} +0.291852 q^{22} -4.84015 q^{23} +3.44691 q^{24} +3.96322 q^{25} +1.30327 q^{26} -3.10830 q^{27} -3.86441 q^{28} +1.54800 q^{29} +14.5319 q^{30} +7.02347 q^{31} -6.34052 q^{32} -0.430601 q^{33} -3.52900 q^{34} +8.96953 q^{35} +5.36779 q^{36} +10.4702 q^{38} -1.92285 q^{39} +3.85621 q^{40} +2.08799 q^{41} +14.5422 q^{42} -6.85957 q^{43} +0.207548 q^{44} -12.4590 q^{45} -8.77906 q^{46} +11.0776 q^{47} +13.1556 q^{48} +1.97584 q^{49} +7.18848 q^{50} +5.20672 q^{51} +0.926808 q^{52} +10.0761 q^{53} -5.63783 q^{54} -0.481731 q^{55} +3.85892 q^{56} -15.4478 q^{57} +2.80776 q^{58} +9.76150 q^{59} +10.3343 q^{60} -11.0862 q^{61} +12.7392 q^{62} -12.4677 q^{63} -1.66848 q^{64} -2.15118 q^{65} -0.781024 q^{66} -7.90412 q^{67} -2.50962 q^{68} +12.9527 q^{69} +16.2689 q^{70} +2.02386 q^{71} -5.36017 q^{72} -9.41664 q^{73} -10.6060 q^{75} +7.44576 q^{76} -0.482070 q^{77} -3.48767 q^{78} +1.38332 q^{79} +14.7178 q^{80} -4.16640 q^{81} +3.78719 q^{82} +0.956139 q^{83} +10.3415 q^{84} +5.82497 q^{85} -12.4419 q^{86} -4.14259 q^{87} -0.207253 q^{88} +11.9126 q^{89} -22.5981 q^{90} -2.15269 q^{91} -6.24315 q^{92} -18.7955 q^{93} +20.0925 q^{94} -17.2821 q^{95} +16.9679 q^{96} +8.94904 q^{97} +3.58378 q^{98} +0.669612 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.81380	1.28255	0.641275	−	0.767311i	$-0.278406\pi$
$2$	1.81380	1.28255	0.641275	+	0.767311i	$0.278406\pi$
$3$	−2.67610	−1.54505	−0.772523	−	0.634987i	$-0.781006\pi$
$3$	−2.67610	−1.54505	−0.772523	+	0.634987i	$0.781006\pi$
$4$	1.28987	0.644934
$5$	−2.99386	−1.33890	−0.669448	−	0.742859i	$-0.733469\pi$
$5$	−2.99386	−1.33890	−0.669448	+	0.742859i	$0.733469\pi$
$6$	−4.85391	−1.98160
$7$	−2.99597	−1.13237	−0.566185	−	0.824278i	$-0.691581\pi$
$7$	−2.99597	−1.13237	−0.566185	+	0.824278i	$0.691581\pi$
$8$	−1.28804	−0.455390
$9$	4.16150	1.38717
$10$	−5.43027	−1.71720
$11$	0.160906	0.0485151	0.0242575	−	0.999706i	$-0.492278\pi$
$11$	0.160906	0.0485151	0.0242575	+	0.999706i	$0.492278\pi$
$12$	−3.45181	−0.996453
$13$	0.718529	0.199284	0.0996421	−	0.995023i	$-0.468230\pi$
$13$	0.718529	0.199284	0.0996421	+	0.995023i	$0.468230\pi$
$14$	−5.43409	−1.45232
$15$	8.01187	2.06866
$16$	−4.91598	−1.22899
$17$	−1.94564	−0.471886	−0.235943	−	0.971767i	$-0.575818\pi$
$17$	−1.94564	−0.471886	−0.235943	+	0.971767i	$0.575818\pi$
$18$	7.54813	1.77911
$19$	5.77250	1.32430	0.662151	−	0.749370i	$-0.269644\pi$
$19$	5.77250	1.32430	0.662151	+	0.749370i	$0.269644\pi$
$20$	−3.86169	−0.863500
$21$	8.01751	1.74956
$22$	0.291852	0.0622230
$23$	−4.84015	−1.00924	−0.504620	−	0.863341i	$-0.668367\pi$
$23$	−4.84015	−1.00924	−0.504620	+	0.863341i	$0.668367\pi$
$24$	3.44691	0.703599
$25$	3.96322	0.792643
$26$	1.30327	0.255592
$27$	−3.10830	−0.598192
$28$	−3.86441	−0.730304
$29$	1.54800	0.287456	0.143728	−	0.989617i	$-0.454091\pi$
$29$	1.54800	0.287456	0.143728	+	0.989617i	$0.454091\pi$
$30$	14.5319	2.65315
$31$	7.02347	1.26145	0.630726	−	0.776006i	$-0.282757\pi$
$31$	7.02347	1.26145	0.630726	+	0.776006i	$0.282757\pi$
$32$	−6.34052	−1.12086
$33$	−0.430601	−0.0749580
$34$	−3.52900	−0.605218
$35$	8.96953	1.51613
$36$	5.36779	0.894632
$37$	0	0
$37$	0	0
$38$	10.4702	1.69848
$39$	−1.92285	−0.307903
$40$	3.85621	0.609720
$41$	2.08799	0.326089	0.163044	−	0.986619i	$-0.447869\pi$
$41$	2.08799	0.326089	0.163044	+	0.986619i	$0.447869\pi$
$42$	14.5422	2.24390
$43$	−6.85957	−1.04607	−0.523037	−	0.852310i	$-0.675201\pi$
$43$	−6.85957	−1.04607	−0.523037	+	0.852310i	$0.675201\pi$
$44$	0.207548	0.0312890
$45$	−12.4590	−1.85727
$46$	−8.77906	−1.29440
$47$	11.0776	1.61583	0.807916	−	0.589298i	$-0.200595\pi$
$47$	11.0776	1.61583	0.807916	+	0.589298i	$0.200595\pi$
$48$	13.1556	1.89885
$49$	1.97584	0.282263
$50$	7.18848	1.01660
$51$	5.20672	0.729086
$52$	0.926808	0.128525
$53$	10.0761	1.38406	0.692032	−	0.721867i	$-0.256716\pi$
$53$	10.0761	1.38406	0.692032	+	0.721867i	$0.256716\pi$
$54$	−5.63783	−0.767211
$55$	−0.481731	−0.0649566
$56$	3.85892	0.515670
$57$	−15.4478	−2.04611
$58$	2.80776	0.368676
$59$	9.76150	1.27084	0.635420	−	0.772167i	$-0.280827\pi$
$59$	9.76150	1.27084	0.635420	+	0.772167i	$0.280827\pi$
$60$	10.3343	1.33415
$61$	−11.0862	−1.41945	−0.709724	−	0.704480i	$-0.751180\pi$
$61$	−11.0862	−1.41945	−0.709724	+	0.704480i	$0.751180\pi$
$62$	12.7392	1.61787
$63$	−12.4677	−1.57079
$64$	−1.66848	−0.208560
$65$	−2.15118	−0.266821
$66$	−0.781024	−0.0961374
$67$	−7.90412	−0.965642	−0.482821	−	0.875719i	$-0.660388\pi$
$67$	−7.90412	−0.965642	−0.482821	+	0.875719i	$0.660388\pi$
$68$	−2.50962	−0.304336
$69$	12.9527	1.55932
$70$	16.2689	1.94451
$71$	2.02386	0.240188	0.120094	−	0.992763i	$-0.461680\pi$
$71$	2.02386	0.240188	0.120094	+	0.992763i	$0.461680\pi$
$72$	−5.36017	−0.631702
$73$	−9.41664	−1.10213	−0.551067	−	0.834461i	$-0.685779\pi$
$73$	−9.41664	−1.10213	−0.551067	+	0.834461i	$0.685779\pi$
$74$	0	0
$75$	−10.6060	−1.22467
$76$	7.44576	0.854087
$77$	−0.482070	−0.0549370
$78$	−3.48767	−0.394901
$79$	1.38332	0.155635	0.0778177	−	0.996968i	$-0.475205\pi$
$79$	1.38332	0.155635	0.0778177	+	0.996968i	$0.475205\pi$
$80$	14.7178	1.64550
$81$	−4.16640	−0.462933
$82$	3.78719	0.418225
$83$	0.956139	0.104950	0.0524749	−	0.998622i	$-0.483289\pi$
$83$	0.956139	0.104950	0.0524749	+	0.998622i	$0.483289\pi$
$84$	10.3415	1.12835
$85$	5.82497	0.631807
$86$	−12.4419	−1.34164
$87$	−4.14259	−0.444132
$88$	−0.207253	−0.0220933
$89$	11.9126	1.26273	0.631365	−	0.775486i	$-0.282495\pi$
$89$	11.9126	1.26273	0.631365	+	0.775486i	$0.282495\pi$
$90$	−22.5981	−2.38205
$91$	−2.15269	−0.225663
$92$	−6.24315	−0.650893
$93$	−18.7955	−1.94900
$94$	20.0925	2.07238
$95$	−17.2821	−1.77310
$96$	16.9679	1.73177
$97$	8.94904	0.908638	0.454319	−	0.890839i	$-0.349883\pi$
$97$	8.94904	0.908638	0.454319	+	0.890839i	$0.349883\pi$
$98$	3.58378	0.362016
$99$	0.669612	0.0672985
$100$	5.11202	0.511202
$101$	−8.31722	−0.827594	−0.413797	−	0.910369i	$-0.635798\pi$
$101$	−8.31722	−0.827594	−0.413797	+	0.910369i	$0.635798\pi$
$102$	9.44394	0.935090
$103$	12.3487	1.21676	0.608378	−	0.793648i	$-0.291821\pi$
$103$	12.3487	1.21676	0.608378	+	0.793648i	$0.291821\pi$
$104$	−0.925492	−0.0907520
$105$	−24.0033	−2.34249
$106$	18.2761	1.77513
$107$	−5.62939	−0.544214	−0.272107	−	0.962267i	$-0.587720\pi$
$107$	−5.62939	−0.544214	−0.272107	+	0.962267i	$0.587720\pi$
$108$	−4.00929	−0.385794
$109$	−5.63253	−0.539499	−0.269749	−	0.962931i	$-0.586941\pi$
$109$	−5.63253	−0.539499	−0.269749	+	0.962931i	$0.586941\pi$
$110$	−0.873764	−0.0833101
$111$	0	0
$112$	14.7281	1.39168
$113$	−10.3234	−0.971146	−0.485573	−	0.874196i	$-0.661389\pi$
$113$	−10.3234	−0.971146	−0.485573	+	0.874196i	$0.661389\pi$
$114$	−28.0192	−2.62424
$115$	14.4907	1.35127
$116$	1.99671	0.185390
$117$	2.99016	0.276440
$118$	17.7054	1.62991
$119$	5.82907	0.534350
$120$	−10.3196	−0.942045
$121$	−10.9741	−0.997646
$122$	−20.1082	−1.82051
$123$	−5.58766	−0.503822
$124$	9.05934	0.813553
$125$	3.10399	0.277629
$126$	−22.6140	−2.01461
$127$	−4.71502	−0.418391	−0.209195	−	0.977874i	$-0.567084\pi$
$127$	−4.71502	−0.418391	−0.209195	+	0.977874i	$0.567084\pi$
$128$	9.65476	0.853368
$129$	18.3569	1.61623
$130$	−3.90180	−0.342211
$131$	14.4660	1.26390	0.631952	−	0.775007i	$-0.282254\pi$
$131$	14.4660	1.26390	0.631952	+	0.775007i	$0.282254\pi$
$132$	−0.555418	−0.0483430
$133$	−17.2942	−1.49960
$134$	−14.3365	−1.23848
$135$	9.30582	0.800917
$136$	2.50605	0.214892
$137$	5.17046	0.441742	0.220871	−	0.975303i	$-0.429110\pi$
$137$	5.17046	0.441742	0.220871	+	0.975303i	$0.429110\pi$
$138$	23.4936	1.99991
$139$	−0.912550	−0.0774015	−0.0387008	−	0.999251i	$-0.512322\pi$
$139$	−0.912550	−0.0774015	−0.0387008	+	0.999251i	$0.512322\pi$
$140$	11.5695	0.977801
$141$	−29.6447	−2.49654
$142$	3.67087	0.308053
$143$	0.115616	0.00966828
$144$	−20.4579	−1.70482
$145$	−4.63449	−0.384873
$146$	−17.0799	−1.41354
$147$	−5.28755	−0.436109
$148$	0	0
$149$	7.41563	0.607512	0.303756	−	0.952750i	$-0.401759\pi$
$149$	7.41563	0.607512	0.303756	+	0.952750i	$0.401759\pi$
$150$	−19.2371	−1.57070
$151$	13.3117	1.08329	0.541644	−	0.840608i	$-0.317802\pi$
$151$	13.3117	1.08329	0.541644	+	0.840608i	$0.317802\pi$
$152$	−7.43519	−0.603074
$153$	−8.09678	−0.654586
$154$	−0.874379	−0.0704595
$155$	−21.0273	−1.68895
$156$	−2.48023	−0.198577
$157$	−11.9589	−0.954425	−0.477213	−	0.878788i	$-0.658353\pi$
$157$	−11.9589	−0.954425	−0.477213	+	0.878788i	$0.658353\pi$
$158$	2.50906	0.199610
$159$	−26.9647	−2.13844
$160$	18.9826	1.50071
$161$	14.5009	1.14283
$162$	−7.55701	−0.593735
$163$	−3.10717	−0.243373	−0.121686	−	0.992569i	$-0.538830\pi$
$163$	−3.10717	−0.243373	−0.121686	+	0.992569i	$0.538830\pi$
$164$	2.69323	0.210306
$165$	1.28916	0.100361
$166$	1.73424	0.134603
$167$	−4.89510	−0.378794	−0.189397	−	0.981901i	$-0.560653\pi$
$167$	−4.89510	−0.378794	−0.189397	+	0.981901i	$0.560653\pi$
$168$	−10.3269	−0.796734
$169$	−12.4837	−0.960286
$170$	10.5653	0.810324
$171$	24.0223	1.83703
$172$	−8.84793	−0.674649
$173$	−19.0784	−1.45050	−0.725251	−	0.688484i	$-0.758276\pi$
$173$	−19.0784	−1.45050	−0.725251	+	0.688484i	$0.758276\pi$
$174$	−7.51383	−0.569622
$175$	−11.8737	−0.897566
$176$	−0.791011	−0.0596247
$177$	−26.1227	−1.96351
$178$	21.6070	1.61951
$179$	15.4965	1.15826	0.579132	−	0.815234i	$-0.303392\pi$
$179$	15.4965	1.15826	0.579132	+	0.815234i	$0.303392\pi$
$180$	−16.0704	−1.19782
$181$	22.7039	1.68757	0.843783	−	0.536684i	$-0.180323\pi$
$181$	22.7039	1.68757	0.843783	+	0.536684i	$0.180323\pi$
$182$	−3.90455	−0.289425
$183$	29.6679	2.19311
$184$	6.23429	0.459598
$185$	0	0
$186$	−34.0912	−2.49969
$187$	−0.313065	−0.0228936
$188$	14.2886	1.04210
$189$	9.31237	0.677375
$190$	−31.3462	−2.27409
$191$	14.7727	1.06892	0.534459	−	0.845194i	$-0.320515\pi$
$191$	14.7727	1.06892	0.534459	+	0.845194i	$0.320515\pi$
$192$	4.46501	0.322234
$193$	26.3754	1.89854	0.949272	−	0.314455i	$-0.101822\pi$
$193$	26.3754	1.89854	0.949272	+	0.314455i	$0.101822\pi$
$194$	16.2318	1.16537
$195$	5.75676	0.412250
$196$	2.54857	0.182041
$197$	4.60161	0.327851	0.163925	−	0.986473i	$-0.447584\pi$
$197$	4.60161	0.327851	0.163925	+	0.986473i	$0.447584\pi$
$198$	1.21454	0.0863137
$199$	2.96417	0.210125	0.105062	−	0.994466i	$-0.466496\pi$
$199$	2.96417	0.210125	0.105062	+	0.994466i	$0.466496\pi$
$200$	−5.10477	−0.360962
$201$	21.1522	1.49196
$202$	−15.0858	−1.06143
$203$	−4.63775	−0.325506
$204$	6.71598	0.470213
$205$	−6.25115	−0.436599
$206$	22.3981	1.56055
$207$	−20.1423	−1.39999
$208$	−3.53227	−0.244919
$209$	0.928831	0.0642486
$210$	−43.5372	−3.00435
$211$	−21.6699	−1.49182	−0.745909	−	0.666048i	$-0.767984\pi$
$211$	−21.6699	−1.49182	−0.745909	+	0.666048i	$0.767984\pi$
$212$	12.9969	0.892630
$213$	−5.41605	−0.371101
$214$	−10.2106	−0.697982
$215$	20.5366	1.40058
$216$	4.00360	0.272411
$217$	−21.0421	−1.42843
$218$	−10.2163	−0.691934
$219$	25.1998	1.70285
$220$	−0.621370	−0.0418927
$221$	−1.39800	−0.0940395
$222$	0	0
$223$	12.6558	0.847494	0.423747	−	0.905781i	$-0.360715\pi$
$223$	12.6558	0.847494	0.423747	+	0.905781i	$0.360715\pi$
$224$	18.9960	1.26922
$225$	16.4929	1.09953
$226$	−18.7246	−1.24554
$227$	6.57096	0.436130	0.218065	−	0.975934i	$-0.430025\pi$
$227$	6.57096	0.436130	0.218065	+	0.975934i	$0.430025\pi$
$228$	−19.9256	−1.31960
$229$	24.4422	1.61519	0.807593	−	0.589741i	$-0.200770\pi$
$229$	24.4422	1.61519	0.807593	+	0.589741i	$0.200770\pi$
$230$	26.2833	1.73307
$231$	1.29007	0.0848802
$232$	−1.99388	−0.130904
$233$	17.1463	1.12329	0.561646	−	0.827378i	$-0.310168\pi$
$233$	17.1463	1.12329	0.561646	+	0.827378i	$0.310168\pi$
$234$	5.42355	0.354549
$235$	−33.1648	−2.16343
$236$	12.5910	0.819607
$237$	−3.70190	−0.240464
$238$	10.5728	0.685331
$239$	17.8579	1.15513	0.577566	−	0.816344i	$-0.304003\pi$
$239$	17.8579	1.15513	0.577566	+	0.816344i	$0.304003\pi$
$240$	−39.3862	−2.54237
$241$	6.75241	0.434961	0.217480	−	0.976065i	$-0.430216\pi$
$241$	6.75241	0.434961	0.217480	+	0.976065i	$0.430216\pi$
$242$	−19.9048	−1.27953
$243$	20.4746	1.31345
$244$	−14.2998	−0.915450
$245$	−5.91540	−0.377921
$246$	−10.1349	−0.646177
$247$	4.14771	0.263912
$248$	−9.04649	−0.574452
$249$	−2.55872	−0.162152
$250$	5.63002	0.356074
$251$	−9.16599	−0.578552	−0.289276	−	0.957246i	$-0.593415\pi$
$251$	−9.16599	−0.578552	−0.289276	+	0.957246i	$0.593415\pi$
$252$	−16.0817	−1.01305
$253$	−0.778810	−0.0489634
$254$	−8.55210	−0.536607
$255$	−15.5882	−0.976171
$256$	20.8487	1.30305
$257$	−7.19490	−0.448806	−0.224403	−	0.974496i	$-0.572043\pi$
$257$	−7.19490	−0.448806	−0.224403	+	0.974496i	$0.572043\pi$
$258$	33.2957	2.07290
$259$	0	0
$260$	−2.77473	−0.172082
$261$	6.44199	0.398749
$262$	26.2385	1.62102
$263$	8.99331	0.554551	0.277276	−	0.960790i	$-0.410568\pi$
$263$	8.99331	0.554551	0.277276	+	0.960790i	$0.410568\pi$
$264$	0.554630	0.0341351
$265$	−30.1666	−1.85312
$266$	−31.3683	−1.92331
$267$	−31.8792	−1.95098
$268$	−10.1953	−0.622775
$269$	−27.0278	−1.64791	−0.823956	−	0.566654i	$-0.808238\pi$
$269$	−27.0278	−1.64791	−0.823956	+	0.566654i	$0.808238\pi$
$270$	16.8789	1.02722
$271$	17.1483	1.04168	0.520841	−	0.853653i	$-0.325618\pi$
$271$	17.1483	1.04168	0.520841	+	0.853653i	$0.325618\pi$
$272$	9.56471	0.579946
$273$	5.76082	0.348660
$274$	9.37817	0.566556
$275$	0.637706	0.0384551
$276$	16.7073	1.00566
$277$	10.4875	0.630134	0.315067	−	0.949069i	$-0.397973\pi$
$277$	10.4875	0.630134	0.315067	+	0.949069i	$0.397973\pi$
$278$	−1.65518	−0.0992713
$279$	29.2282	1.74985
$280$	−11.5531	−0.690429
$281$	14.5354	0.867110	0.433555	−	0.901127i	$-0.357259\pi$
$281$	14.5354	0.867110	0.433555	+	0.901127i	$0.357259\pi$
$282$	−53.7696	−3.20193
$283$	17.0774	1.01515	0.507573	−	0.861609i	$-0.330543\pi$
$283$	17.0774	1.01515	0.507573	+	0.861609i	$0.330543\pi$
$284$	2.61051	0.154905
$285$	46.2485	2.73953
$286$	0.209704	0.0124000
$287$	−6.25555	−0.369253
$288$	−26.3861	−1.55482
$289$	−13.2145	−0.777323
$290$	−8.40604	−0.493619
$291$	−23.9485	−1.40389
$292$	−12.1462	−0.710804
$293$	−13.6089	−0.795042	−0.397521	−	0.917593i	$-0.630129\pi$
$293$	−13.6089	−0.795042	−0.397521	+	0.917593i	$0.630129\pi$
$294$	−9.59055	−0.559332
$295$	−29.2246	−1.70152
$296$	0	0
$297$	−0.500144	−0.0290213
$298$	13.4505	0.779165
$299$	−3.47779	−0.201126
$300$	−13.6803	−0.789831
$301$	20.5511	1.18454
$302$	24.1447	1.38937
$303$	22.2577	1.27867
$304$	−28.3775	−1.62756
$305$	33.1907	1.90049
$306$	−14.6859	−0.839539
$307$	−20.2910	−1.15807	−0.579035	−	0.815303i	$-0.696570\pi$
$307$	−20.2910	−1.15807	−0.579035	+	0.815303i	$0.696570\pi$
$308$	−0.621807	−0.0354307
$309$	−33.0464	−1.87994
$310$	−38.1393	−2.16617
$311$	5.91474	0.335394	0.167697	−	0.985839i	$-0.446367\pi$
$311$	5.91474	0.335394	0.167697	+	0.985839i	$0.446367\pi$
$312$	2.47671	0.140216
$313$	23.5054	1.32860	0.664301	−	0.747465i	$-0.268729\pi$
$313$	23.5054	1.32860	0.664301	+	0.747465i	$0.268729\pi$
$314$	−21.6911	−1.22410
$315$	37.3267	2.10312
$316$	1.78430	0.100375
$317$	−0.715392	−0.0401804	−0.0200902	−	0.999798i	$-0.506395\pi$
$317$	−0.715392	−0.0401804	−0.0200902	+	0.999798i	$0.506395\pi$
$318$	−48.9086	−2.74266
$319$	0.249082	0.0139459
$320$	4.99519	0.279240
$321$	15.0648	0.840836
$322$	26.3018	1.46574
$323$	−11.2312	−0.624920
$324$	−5.37411	−0.298561
$325$	2.84769	0.157961
$326$	−5.63579	−0.312138
$327$	15.0732	0.833550
$328$	−2.68941	−0.148498
$329$	−33.1881	−1.82972
$330$	2.33828	0.128718
$331$	−25.6185	−1.40812	−0.704061	−	0.710139i	$-0.748632\pi$
$331$	−25.6185	−1.40812	−0.704061	+	0.710139i	$0.748632\pi$
$332$	1.23329	0.0676857
$333$	0	0
$334$	−8.87873	−0.485823
$335$	23.6638	1.29289
$336$	−39.4139	−2.15020
$337$	−14.4040	−0.784636	−0.392318	−	0.919830i	$-0.628327\pi$
$337$	−14.4040	−0.784636	−0.392318	+	0.919830i	$0.628327\pi$
$338$	−22.6430	−1.23161
$339$	27.6265	1.50047
$340$	7.51344	0.407474
$341$	1.13012	0.0611994
$342$	43.5716	2.35608
$343$	15.0522	0.812744
$344$	8.83538	0.476372
$345$	−38.7786	−2.08777
$346$	−34.6044	−1.86034
$347$	28.0587	1.50627	0.753134	−	0.657867i	$-0.228541\pi$
$347$	28.0587	1.50627	0.753134	+	0.657867i	$0.228541\pi$
$348$	−5.34340	−0.286436
$349$	−33.1943	−1.77685	−0.888425	−	0.459022i	$-0.848200\pi$
$349$	−33.1943	−1.77685	−0.888425	+	0.459022i	$0.848200\pi$
$350$	−21.5365	−1.15117
$351$	−2.23340	−0.119210
$352$	−1.02023	−0.0543784
$353$	−11.6435	−0.619723	−0.309862	−	0.950782i	$-0.600283\pi$
$353$	−11.6435	−0.619723	−0.309862	+	0.950782i	$0.600283\pi$
$354$	−47.3814	−2.51829
$355$	−6.05916	−0.321587
$356$	15.3656	0.814377
$357$	−15.5992	−0.825596
$358$	28.1076	1.48553
$359$	29.0400	1.53267	0.766335	−	0.642441i	$-0.222078\pi$
$359$	29.0400	1.53267	0.766335	+	0.642441i	$0.222078\pi$
$360$	16.0476	0.845784
$361$	14.3217	0.753776
$362$	41.1803	2.16439
$363$	29.3678	1.54141
$364$	−2.77669	−0.145538
$365$	28.1921	1.47564
$366$	53.8116	2.81278
$367$	−30.9655	−1.61638	−0.808192	−	0.588919i	$-0.799554\pi$
$367$	−30.9655	−1.61638	−0.808192	+	0.588919i	$0.799554\pi$
$368$	23.7940	1.24035
$369$	8.68917	0.452340
$370$	0	0
$371$	−30.1878	−1.56727
$372$	−24.2437	−1.25698
$373$	0.834072	0.0431866	0.0215933	−	0.999767i	$-0.493126\pi$
$373$	0.834072	0.0431866	0.0215933	+	0.999767i	$0.493126\pi$
$374$	−0.567837	−0.0293622
$375$	−8.30659	−0.428950
$376$	−14.2683	−0.735834
$377$	1.11228	0.0572854
$378$	16.8908	0.868767
$379$	15.4128	0.791701	0.395850	−	0.918315i	$-0.370450\pi$
$379$	15.4128	0.791701	0.395850	+	0.918315i	$0.370450\pi$
$380$	−22.2916	−1.14353
$381$	12.6179	0.646433
$382$	26.7948	1.37094
$383$	4.42729	0.226224	0.113112	−	0.993582i	$-0.463918\pi$
$383$	4.42729	0.226224	0.113112	+	0.993582i	$0.463918\pi$
$384$	−25.8371	−1.31849
$385$	1.44325	0.0735550
$386$	47.8397	2.43498
$387$	−28.5461	−1.45108
$388$	11.5431	0.586011
$389$	−4.08832	−0.207286	−0.103643	−	0.994615i	$-0.533050\pi$
$389$	−4.08832	−0.207286	−0.103643	+	0.994615i	$0.533050\pi$
$390$	10.4416	0.528732
$391$	9.41717	0.476247
$392$	−2.54496	−0.128540
$393$	−38.7126	−1.95279
$394$	8.34639	0.420485
$395$	−4.14146	−0.208380
$396$	0.863711	0.0434031
$397$	−6.68820	−0.335671	−0.167836	−	0.985815i	$-0.553678\pi$
$397$	−6.68820	−0.335671	−0.167836	+	0.985815i	$0.553678\pi$
$398$	5.37641	0.269495
$399$	46.2811	2.31695
$400$	−19.4831	−0.974154
$401$	−22.5654	−1.12686	−0.563432	−	0.826163i	$-0.690519\pi$
$401$	−22.5654	−1.12686	−0.563432	+	0.826163i	$0.690519\pi$
$402$	38.3658	1.91351
$403$	5.04656	0.251387
$404$	−10.7281	−0.533743
$405$	12.4736	0.619820
$406$	−8.41195	−0.417478
$407$	0	0
$408$	−6.70645	−0.332019
$409$	−3.17318	−0.156904	−0.0784518	−	0.996918i	$-0.524998\pi$
$409$	−3.17318	−0.156904	−0.0784518	+	0.996918i	$0.524998\pi$
$410$	−11.3383	−0.559960
$411$	−13.8367	−0.682512
$412$	15.9282	0.784727
$413$	−29.2452	−1.43906
$414$	−36.5341	−1.79555
$415$	−2.86255	−0.140517
$416$	−4.55585	−0.223369
$417$	2.44207	0.119589
$418$	1.68471	0.0824020
$419$	2.30553	0.112633	0.0563163	−	0.998413i	$-0.482064\pi$
$419$	2.30553	0.112633	0.0563163	+	0.998413i	$0.482064\pi$
$420$	−30.9611	−1.51075
$421$	10.4729	0.510419	0.255210	−	0.966886i	$-0.417855\pi$
$421$	10.4729	0.510419	0.255210	+	0.966886i	$0.417855\pi$
$422$	−39.3048	−1.91333
$423$	46.0994	2.24143
$424$	−12.9784	−0.630289
$425$	−7.71098	−0.374038
$426$	−9.82362	−0.475956
$427$	33.2141	1.60734
$428$	−7.26117	−0.350982
$429$	−0.309399	−0.0149379
$430$	37.2493	1.79632
$431$	10.9329	0.526617	0.263309	−	0.964712i	$-0.415186\pi$
$431$	10.9329	0.526617	0.263309	+	0.964712i	$0.415186\pi$
$432$	15.2803	0.735175
$433$	4.42652	0.212725	0.106362	−	0.994327i	$-0.466080\pi$
$433$	4.42652	0.212725	0.106362	+	0.994327i	$0.466080\pi$
$434$	−38.1661	−1.83203
$435$	12.4024	0.594647
$436$	−7.26522	−0.347941
$437$	−27.9397	−1.33654
$438$	45.7075	2.18399
$439$	−13.1721	−0.628672	−0.314336	−	0.949312i	$-0.601782\pi$
$439$	−13.1721	−0.628672	−0.314336	+	0.949312i	$0.601782\pi$
$440$	0.620488	0.0295806
$441$	8.22247	0.391546
$442$	−2.53569	−0.120610
$443$	32.5541	1.54669	0.773346	−	0.633984i	$-0.218582\pi$
$443$	32.5541	1.54669	0.773346	+	0.633984i	$0.218582\pi$
$444$	0	0
$445$	−35.6646	−1.69066
$446$	22.9551	1.08695
$447$	−19.8450	−0.938634
$448$	4.99871	0.236167
$449$	3.93429	0.185671	0.0928354	−	0.995681i	$-0.470407\pi$
$449$	3.93429	0.185671	0.0928354	+	0.995681i	$0.470407\pi$
$450$	29.9149	1.41020
$451$	0.335970	0.0158202
$452$	−13.3158	−0.626325
$453$	−35.6233	−1.67373
$454$	11.9184	0.559359
$455$	6.44487	0.302140
$456$	19.8973	0.931777
$457$	40.9703	1.91651	0.958255	−	0.285915i	$-0.0922976\pi$
$457$	40.9703	1.91651	0.958255	+	0.285915i	$0.0922976\pi$
$458$	44.3332	2.07156
$459$	6.04762	0.282279
$460$	18.6911	0.871479
$461$	15.1267	0.704522	0.352261	−	0.935902i	$-0.385413\pi$
$461$	15.1267	0.704522	0.352261	+	0.935902i	$0.385413\pi$
$462$	2.33992	0.108863
$463$	1.56923	0.0729285	0.0364642	−	0.999335i	$-0.488390\pi$
$463$	1.56923	0.0729285	0.0364642	+	0.999335i	$0.488390\pi$
$464$	−7.60992	−0.353281
$465$	56.2711	2.60951
$466$	31.0999	1.44068
$467$	12.8427	0.594288	0.297144	−	0.954833i	$-0.403966\pi$
$467$	12.8427	0.594288	0.297144	+	0.954833i	$0.403966\pi$
$468$	3.85691	0.178286
$469$	23.6805	1.09346
$470$	−60.1542	−2.77471
$471$	32.0032	1.47463
$472$	−12.5732	−0.578727
$473$	−1.10375	−0.0507503
$474$	−6.71450	−0.308407
$475$	22.8777	1.04970
$476$	7.51873	0.344621
$477$	41.9319	1.91993
$478$	32.3907	1.48151
$479$	6.58986	0.301098	0.150549	−	0.988603i	$-0.451896\pi$
$479$	6.58986	0.301098	0.150549	+	0.988603i	$0.451896\pi$
$480$	−50.7994	−2.31867
$481$	0	0
$482$	12.2475	0.557859
$483$	−38.8059	−1.76573
$484$	−14.1551	−0.643416
$485$	−26.7922	−1.21657
$486$	37.1368	1.68456
$487$	−0.565847	−0.0256410	−0.0128205	−	0.999918i	$-0.504081\pi$
$487$	−0.565847	−0.0256410	−0.0128205	+	0.999918i	$0.504081\pi$
$488$	14.2795	0.646402
$489$	8.31511	0.376022
$490$	−10.7293	−0.484702
$491$	−0.958656	−0.0432636	−0.0216318	−	0.999766i	$-0.506886\pi$
$491$	−0.958656	−0.0432636	−0.0216318	+	0.999766i	$0.506886\pi$
$492$	−7.20734	−0.324932
$493$	−3.01184	−0.135646
$494$	7.52311	0.338481
$495$	−2.00473	−0.0901057
$496$	−34.5272	−1.55032
$497$	−6.06342	−0.271982
$498$	−4.64101	−0.207969
$499$	−3.49941	−0.156655	−0.0783276	−	0.996928i	$-0.524958\pi$
$499$	−3.49941	−0.156655	−0.0783276	+	0.996928i	$0.524958\pi$
$500$	4.00374	0.179053
$501$	13.0998	0.585255
$502$	−16.6253	−0.742022
$503$	36.9239	1.64635	0.823177	−	0.567785i	$-0.192199\pi$
$503$	36.9239	1.64635	0.823177	+	0.567785i	$0.192199\pi$
$504$	16.0589	0.715321
$505$	24.9006	1.10806
$506$	−1.41260	−0.0627979
$507$	33.4077	1.48369
$508$	−6.08176	−0.269834
$509$	3.16897	0.140462	0.0702311	−	0.997531i	$-0.477626\pi$
$509$	3.16897	0.140462	0.0702311	+	0.997531i	$0.477626\pi$
$510$	−28.2739	−1.25199
$511$	28.2120	1.24802
$512$	18.5059	0.817854
$513$	−17.9426	−0.792187
$514$	−13.0501	−0.575615
$515$	−36.9704	−1.62911
$516$	23.6779	1.04236
$517$	1.78245	0.0783922
$518$	0	0
$519$	51.0556	2.24109
$520$	2.77080	0.121507
$521$	37.9628	1.66318	0.831590	−	0.555390i	$-0.187431\pi$
$521$	37.9628	1.66318	0.831590	+	0.555390i	$0.187431\pi$
$522$	11.6845	0.511416
$523$	24.6627	1.07842	0.539212	−	0.842170i	$-0.318722\pi$
$523$	24.6627	1.07842	0.539212	+	0.842170i	$0.318722\pi$
$524$	18.6593	0.815135
$525$	31.7751	1.38678
$526$	16.3121	0.711240
$527$	−13.6651	−0.595262
$528$	2.11682	0.0921229
$529$	0.427023	0.0185662
$530$	−54.7161	−2.37672
$531$	40.6225	1.76287
$532$	−22.3073	−0.967143
$533$	1.50028	0.0649843
$534$	−57.8225	−2.50222
$535$	16.8536	0.728646
$536$	10.1808	0.439743
$537$	−41.4702	−1.78957
$538$	−49.0229	−2.11353
$539$	0.317925	0.0136940
$540$	12.0033	0.516539
$541$	12.4634	0.535843	0.267921	−	0.963441i	$-0.413663\pi$
$541$	12.4634	0.535843	0.267921	+	0.963441i	$0.413663\pi$
$542$	31.1035	1.33601
$543$	−60.7579	−2.60737
$544$	12.3364	0.528917
$545$	16.8630	0.722333
$546$	10.4490	0.447174
$547$	−20.9831	−0.897172	−0.448586	−	0.893740i	$-0.648072\pi$
$547$	−20.9831	−0.897172	−0.448586	+	0.893740i	$0.648072\pi$
$548$	6.66921	0.284894
$549$	−46.1354	−1.96901
$550$	1.15667	0.0493206
$551$	8.93581	0.380678
$552$	−16.6836	−0.710100
$553$	−4.14438	−0.176237
$554$	19.0223	0.808179
$555$	0	0
$556$	−1.17707	−0.0499189
$557$	−29.6106	−1.25464	−0.627321	−	0.778761i	$-0.715849\pi$
$557$	−29.6106	−1.25464	−0.627321	+	0.778761i	$0.715849\pi$
$558$	53.0140	2.24426
$559$	−4.92880	−0.208466
$560$	−44.0940	−1.86331
$561$	0.837793	0.0353717
$562$	26.3643	1.11211
$563$	−8.15606	−0.343737	−0.171868	−	0.985120i	$-0.554980\pi$
$563$	−8.15606	−0.343737	−0.171868	+	0.985120i	$0.554980\pi$
$564$	−38.2378	−1.61010
$565$	30.9069	1.30026
$566$	30.9750	1.30198
$567$	12.4824	0.524212
$568$	−2.60681	−0.109379
$569$	−14.2731	−0.598359	−0.299180	−	0.954197i	$-0.596713\pi$
$569$	−14.2731	−0.598359	−0.299180	+	0.954197i	$0.596713\pi$
$570$	83.8855	3.51358
$571$	−20.4929	−0.857600	−0.428800	−	0.903399i	$-0.641064\pi$
$571$	−20.4929	−0.857600	−0.428800	+	0.903399i	$0.641064\pi$
$572$	0.149129	0.00623540
$573$	−39.5333	−1.65153
$574$	−11.3463	−0.473586
$575$	−19.1825	−0.799967
$576$	−6.94338	−0.289307
$577$	−31.7274	−1.32083	−0.660415	−	0.750901i	$-0.729619\pi$
$577$	−31.7274	−1.32083	−0.660415	+	0.750901i	$0.729619\pi$
$578$	−23.9684	−0.996956
$579$	−70.5832	−2.93334
$580$	−5.97788	−0.248218
$581$	−2.86456	−0.118842
$582$	−43.4378	−1.80056
$583$	1.62131	0.0671480
$584$	12.1290	0.501901
$585$	−8.95213	−0.370125
$586$	−24.6839	−1.01968
$587$	−6.38709	−0.263624	−0.131812	−	0.991275i	$-0.542079\pi$
$587$	−6.38709	−0.263624	−0.131812	+	0.991275i	$0.542079\pi$
$588$	−6.82024	−0.281262
$589$	40.5429	1.67054
$590$	−53.0075	−2.18229
$591$	−12.3144	−0.506545
$592$	0	0
$593$	23.4355	0.962379	0.481190	−	0.876617i	$-0.340205\pi$
$593$	23.4355	0.962379	0.481190	+	0.876617i	$0.340205\pi$
$594$	−0.907162	−0.0372213
$595$	−17.4514	−0.715440
$596$	9.56518	0.391805
$597$	−7.93242	−0.324652
$598$	−6.30801	−0.257954
$599$	−16.1611	−0.660323	−0.330162	−	0.943924i	$-0.607103\pi$
$599$	−16.1611	−0.660323	−0.330162	+	0.943924i	$0.607103\pi$
$600$	13.6609	0.557702
$601$	11.2355	0.458304	0.229152	−	0.973391i	$-0.426405\pi$
$601$	11.2355	0.458304	0.229152	+	0.973391i	$0.426405\pi$
$602$	37.2755	1.51924
$603$	−32.8930	−1.33951
$604$	17.1703	0.698649
$605$	32.8550	1.33574
$606$	40.3710	1.63996
$607$	7.11264	0.288693	0.144347	−	0.989527i	$-0.453892\pi$
$607$	7.11264	0.288693	0.144347	+	0.989527i	$0.453892\pi$
$608$	−36.6006	−1.48435
$609$	12.4111	0.502923
$610$	60.2013	2.43748
$611$	7.95957	0.322010
$612$	−10.4438	−0.422164
$613$	8.66767	0.350084	0.175042	−	0.984561i	$-0.443994\pi$
$613$	8.66767	0.350084	0.175042	+	0.984561i	$0.443994\pi$
$614$	−36.8038	−1.48528
$615$	16.7287	0.674566
$616$	0.620925	0.0250178
$617$	−10.1412	−0.408271	−0.204135	−	0.978943i	$-0.565438\pi$
$617$	−10.1412	−0.408271	−0.204135	+	0.978943i	$0.565438\pi$
$618$	−59.9395	−2.41112
$619$	30.5369	1.22738	0.613691	−	0.789546i	$-0.289684\pi$
$619$	30.5369	1.22738	0.613691	+	0.789546i	$0.289684\pi$
$620$	−27.1224	−1.08926
$621$	15.0446	0.603720
$622$	10.7282	0.430160
$623$	−35.6897	−1.42988
$624$	9.45271	0.378411
$625$	−29.1090	−1.16436
$626$	42.6341	1.70400
$627$	−2.48564	−0.0992670
$628$	−15.4254	−0.615541
$629$	0	0
$630$	67.7032	2.69736
$631$	7.70802	0.306851	0.153426	−	0.988160i	$-0.450969\pi$
$631$	7.70802	0.306851	0.153426	+	0.988160i	$0.450969\pi$
$632$	−1.78177	−0.0708748
$633$	57.9908	2.30493
$634$	−1.29758	−0.0515334
$635$	14.1161	0.560182
$636$	−34.7810	−1.37915
$637$	1.41970	0.0562505
$638$	0.451785	0.0178864
$639$	8.42230	0.333181
$640$	−28.9050	−1.14257
$641$	−25.7413	−1.01672	−0.508360	−	0.861144i	$-0.669748\pi$
$641$	−25.7413	−1.01672	−0.508360	+	0.861144i	$0.669748\pi$
$642$	27.3245	1.07841
$643$	−32.4876	−1.28118	−0.640592	−	0.767881i	$-0.721311\pi$
$643$	−32.4876	−1.28118	−0.640592	+	0.767881i	$0.721311\pi$
$644$	18.7043	0.737052
$645$	−54.9580	−2.16397
$646$	−20.3711	−0.801491
$647$	27.6429	1.08676	0.543378	−	0.839488i	$-0.317145\pi$
$647$	27.6429	1.08676	0.543378	+	0.839488i	$0.317145\pi$
$648$	5.36648	0.210815
$649$	1.57069	0.0616548
$650$	5.16513	0.202593
$651$	56.3107	2.20699
$652$	−4.00784	−0.156959
$653$	−4.48121	−0.175363	−0.0876817	−	0.996149i	$-0.527946\pi$
$653$	−4.48121	−0.175363	−0.0876817	+	0.996149i	$0.527946\pi$
$654$	27.3398	1.06907
$655$	−43.3094	−1.69224
$656$	−10.2645	−0.400761
$657$	−39.1874	−1.52884
$658$	−60.1966	−2.34671
$659$	−19.0352	−0.741508	−0.370754	−	0.928731i	$-0.620901\pi$
$659$	−19.0352	−0.741508	−0.370754	+	0.928731i	$0.620901\pi$
$660$	1.66285	0.0647262
$661$	25.4148	0.988520	0.494260	−	0.869314i	$-0.335439\pi$
$661$	25.4148	0.988520	0.494260	+	0.869314i	$0.335439\pi$
$662$	−46.4669	−1.80599
$663$	3.74118	0.145295
$664$	−1.23154	−0.0477931
$665$	51.7766	2.00781
$666$	0	0
$667$	−7.49253	−0.290112
$668$	−6.31404	−0.244297
$669$	−33.8681	−1.30942
$670$	42.9215	1.65820
$671$	−1.78385	−0.0688646
$672$	−50.8352	−1.96101
$673$	−33.0227	−1.27293	−0.636467	−	0.771304i	$-0.719605\pi$
$673$	−33.0227	−1.27293	−0.636467	+	0.771304i	$0.719605\pi$
$674$	−26.1260	−1.00633
$675$	−12.3189	−0.474153
$676$	−16.1023	−0.619321
$677$	−19.5563	−0.751609	−0.375805	−	0.926699i	$-0.622634\pi$
$677$	−19.5563	−0.751609	−0.375805	+	0.926699i	$0.622634\pi$
$678$	50.1089	1.92442
$679$	−26.8111	−1.02891
$680$	−7.50278	−0.287719
$681$	−17.5845	−0.673841
$682$	2.04981	0.0784913
$683$	−42.0083	−1.60740	−0.803702	−	0.595032i	$-0.797139\pi$
$683$	−42.0083	−1.60740	−0.803702	+	0.595032i	$0.797139\pi$
$684$	30.9856	1.18476
$685$	−15.4796	−0.591447
$686$	27.3017	1.04238
$687$	−65.4097	−2.49554
$688$	33.7215	1.28562
$689$	7.24000	0.275822
$690$	−70.3367	−2.67767
$691$	−16.4203	−0.624658	−0.312329	−	0.949974i	$-0.601109\pi$
$691$	−16.4203	−0.624658	−0.312329	+	0.949974i	$0.601109\pi$
$692$	−24.6086	−0.935478
$693$	−2.00614	−0.0762069
$694$	50.8928	1.93186
$695$	2.73205	0.103633
$696$	5.33581	0.202253
$697$	−4.06247	−0.153877
$698$	−60.2078	−2.27890
$699$	−45.8852	−1.73554
$700$	−15.3155	−0.578871
$701$	8.26474	0.312155	0.156077	−	0.987745i	$-0.450115\pi$
$701$	8.26474	0.312155	0.156077	+	0.987745i	$0.450115\pi$
$702$	−4.05094	−0.152893
$703$	0	0
$704$	−0.268468	−0.0101183
$705$	88.7522	3.34260
$706$	−21.1190	−0.794826
$707$	24.9181	0.937143
$708$	−33.6949	−1.26633
$709$	−9.91812	−0.372483	−0.186241	−	0.982504i	$-0.559631\pi$
$709$	−9.91812	−0.372483	−0.186241	+	0.982504i	$0.559631\pi$
$710$	−10.9901	−0.412451
$711$	5.75668	0.215892
$712$	−15.3438	−0.575034
$713$	−33.9946	−1.27311
$714$	−28.2938	−1.05887
$715$	−0.346138	−0.0129448
$716$	19.9885	0.747003
$717$	−47.7895	−1.78473
$718$	52.6727	1.96573
$719$	−6.40048	−0.238698	−0.119349	−	0.992852i	$-0.538081\pi$
$719$	−6.40048	−0.238698	−0.119349	+	0.992852i	$0.538081\pi$
$720$	61.2480	2.28258
$721$	−36.9964	−1.37782
$722$	25.9768	0.966755
$723$	−18.0701	−0.672035
$724$	29.2850	1.08837
$725$	6.13504	0.227850
$726$	53.2673	1.97693
$727$	−23.1107	−0.857130	−0.428565	−	0.903511i	$-0.640981\pi$
$727$	−23.1107	−0.857130	−0.428565	+	0.903511i	$0.640981\pi$
$728$	2.77275	0.102765
$729$	−42.2928	−1.56640
$730$	51.1348	1.89259
$731$	13.3462	0.493628
$732$	38.2677	1.41441
$733$	50.6416	1.87049	0.935245	−	0.354000i	$-0.115179\pi$
$733$	50.6416	1.87049	0.935245	+	0.354000i	$0.115179\pi$
$734$	−56.1652	−2.07309
$735$	15.8302	0.583905
$736$	30.6891	1.13121
$737$	−1.27182	−0.0468482
$738$	15.7604	0.580148
$739$	1.34343	0.0494190	0.0247095	−	0.999695i	$-0.492134\pi$
$739$	1.34343	0.0494190	0.0247095	+	0.999695i	$0.492134\pi$
$740$	0	0
$741$	−11.0997	−0.407757
$742$	−54.7547	−2.01011
$743$	28.7002	1.05291	0.526454	−	0.850203i	$-0.323521\pi$
$743$	28.7002	1.05291	0.526454	+	0.850203i	$0.323521\pi$
$744$	24.2093	0.887556
$745$	−22.2014	−0.813396
$746$	1.51284	0.0553890
$747$	3.97897	0.145583
$748$	−0.403813	−0.0147649
$749$	16.8655	0.616252
$750$	−15.0665	−0.550150
$751$	28.4247	1.03723	0.518616	−	0.855007i	$-0.326447\pi$
$751$	28.4247	1.03723	0.518616	+	0.855007i	$0.326447\pi$
$752$	−54.4572	−1.98585
$753$	24.5291	0.893890
$754$	2.01745	0.0734713
$755$	−39.8533	−1.45041
$756$	12.0117	0.436862
$757$	14.3460	0.521413	0.260706	−	0.965418i	$-0.416045\pi$
$757$	14.3460	0.521413	0.260706	+	0.965418i	$0.416045\pi$
$758$	27.9557	1.01540
$759$	2.08417	0.0756506
$760$	22.2600	0.807453
$761$	−17.3036	−0.627254	−0.313627	−	0.949546i	$-0.601544\pi$
$761$	−17.3036	−0.627254	−0.313627	+	0.949546i	$0.601544\pi$
$762$	22.8863	0.829082
$763$	16.8749	0.610912
$764$	19.0549	0.689382
$765$	24.2406	0.876422
$766$	8.03021	0.290143
$767$	7.01392	0.253258
$768$	−55.7933	−2.01327
$769$	18.1048	0.652875	0.326438	−	0.945219i	$-0.394152\pi$
$769$	18.1048	0.652875	0.326438	+	0.945219i	$0.394152\pi$
$770$	2.61777	0.0943379
$771$	19.2543	0.693425
$772$	34.0208	1.22444
$773$	23.0226	0.828064	0.414032	−	0.910262i	$-0.364120\pi$
$773$	23.0226	0.828064	0.414032	+	0.910262i	$0.364120\pi$
$774$	−51.7769	−1.86108
$775$	27.8355	0.999881
$776$	−11.5267	−0.413785
$777$	0	0
$778$	−7.41539	−0.265854
$779$	12.0529	0.431840
$780$	7.42546	0.265874
$781$	0.325652	0.0116527
$782$	17.0809	0.610810
$783$	−4.81163	−0.171954
$784$	−9.71319	−0.346900
$785$	35.8033	1.27788
$786$	−70.2168	−2.50455
$787$	34.6914	1.23661	0.618307	−	0.785937i	$-0.287819\pi$
$787$	34.6914	1.23661	0.618307	+	0.785937i	$0.287819\pi$
$788$	5.93546	0.211442
$789$	−24.0670	−0.856807
$790$	−7.51179	−0.267257
$791$	30.9287	1.09970
$792$	−0.862485	−0.0306471
$793$	−7.96579	−0.282873
$794$	−12.1310	−0.430515
$795$	80.7287	2.86315
$796$	3.82339	0.135516
$797$	−10.9268	−0.387047	−0.193524	−	0.981096i	$-0.561992\pi$
$797$	−10.9268	−0.387047	−0.193524	+	0.981096i	$0.561992\pi$
$798$	83.9446	2.97161
$799$	−21.5530	−0.762489
$800$	−25.1288	−0.888439
$801$	49.5742	1.75162
$802$	−40.9291	−1.44526
$803$	−1.51520	−0.0534701
$804$	27.2835	0.962216
$805$	−43.4138	−1.53014
$806$	9.15345	0.322417
$807$	72.3290	2.54610
$808$	10.7129	0.376878
$809$	−9.36038	−0.329093	−0.164547	−	0.986369i	$-0.552616\pi$
$809$	−9.36038	−0.329093	−0.164547	+	0.986369i	$0.552616\pi$
$810$	22.6247	0.794950
$811$	−56.2663	−1.97578	−0.987888	−	0.155170i	$-0.950408\pi$
$811$	−56.2663	−1.97578	−0.987888	+	0.155170i	$0.950408\pi$
$812$	−5.98209	−0.209930
$813$	−45.8904	−1.60945
$814$	0	0
$815$	9.30246	0.325851
$816$	−25.5961	−0.896043
$817$	−39.5968	−1.38532
$818$	−5.75551	−0.201237
$819$	−8.95844	−0.313033
$820$	−8.06315	−0.281578
$821$	41.0592	1.43297	0.716487	−	0.697600i	$-0.245749\pi$
$821$	41.0592	1.43297	0.716487	+	0.697600i	$0.245749\pi$
$822$	−25.0969	−0.875356
$823$	8.34795	0.290991	0.145496	−	0.989359i	$-0.453522\pi$
$823$	8.34795	0.290991	0.145496	+	0.989359i	$0.453522\pi$
$824$	−15.9056	−0.554098
$825$	−1.70656	−0.0594149
$826$	−53.0449	−1.84567
$827$	−12.1894	−0.423868	−0.211934	−	0.977284i	$-0.567976\pi$
$827$	−12.1894	−0.423868	−0.211934	+	0.977284i	$0.567976\pi$
$828$	−25.9809	−0.902898
$829$	7.15697	0.248572	0.124286	−	0.992246i	$-0.460336\pi$
$829$	7.15697	0.248572	0.124286	+	0.992246i	$0.460336\pi$
$830$	−5.19209	−0.180220
$831$	−28.0657	−0.973587
$832$	−1.19885	−0.0415626
$833$	−3.84427	−0.133196
$834$	4.42943	0.153379
$835$	14.6553	0.507166
$836$	1.19807	0.0414361
$837$	−21.8310	−0.754590
$838$	4.18177	0.144457
$839$	15.8792	0.548210	0.274105	−	0.961700i	$-0.411618\pi$
$839$	15.8792	0.548210	0.274105	+	0.961700i	$0.411618\pi$
$840$	30.9172	1.06674
$841$	−26.6037	−0.917369
$842$	18.9958	0.654638
$843$	−38.8982	−1.33973
$844$	−27.9513	−0.962124
$845$	37.3745	1.28572
$846$	83.6151	2.87475
$847$	32.8781	1.12971
$848$	−49.5341	−1.70101
$849$	−45.7008	−1.56845
$850$	−13.9862	−0.479722
$851$	0	0
$852$	−6.98598	−0.239336
$853$	−25.1344	−0.860586	−0.430293	−	0.902689i	$-0.641590\pi$
$853$	−25.1344	−0.860586	−0.430293	+	0.902689i	$0.641590\pi$
$854$	60.2437	2.06150
$855$	−71.9194	−2.45959
$856$	7.25087	0.247830
$857$	40.8577	1.39567	0.697836	−	0.716258i	$-0.254147\pi$
$857$	40.8577	1.39567	0.697836	+	0.716258i	$0.254147\pi$
$858$	−0.561188	−0.0191586
$859$	1.92974	0.0658420	0.0329210	−	0.999458i	$-0.489519\pi$
$859$	1.92974	0.0658420	0.0329210	+	0.999458i	$0.489519\pi$
$860$	26.4895	0.903285
$861$	16.7405	0.570514
$862$	19.8300	0.675413
$863$	38.2352	1.30154	0.650771	−	0.759274i	$-0.274446\pi$
$863$	38.2352	1.30154	0.650771	+	0.759274i	$0.274446\pi$
$864$	19.7082	0.670487
$865$	57.1181	1.94207
$866$	8.02882	0.272830
$867$	35.3633	1.20100
$868$	−27.1415	−0.921243
$869$	0.222584	0.00755066
$870$	22.4954	0.762665
$871$	−5.67934	−0.192437
$872$	7.25491	0.245682
$873$	37.2415	1.26043
$874$	−50.6771	−1.71418
$875$	−9.29947	−0.314379
$876$	32.5045	1.09822
$877$	−18.1316	−0.612259	−0.306130	−	0.951990i	$-0.599034\pi$
$877$	−18.1316	−0.612259	−0.306130	+	0.951990i	$0.599034\pi$
$878$	−23.8916	−0.806303
$879$	36.4188	1.22838
$880$	2.36818	0.0798313
$881$	−8.13173	−0.273965	−0.136982	−	0.990573i	$-0.543740\pi$
$881$	−8.13173	−0.273965	−0.136982	+	0.990573i	$0.543740\pi$
$882$	14.9139	0.502177
$883$	−37.4877	−1.26156	−0.630780	−	0.775962i	$-0.717265\pi$
$883$	−37.4877	−1.26156	−0.630780	+	0.775962i	$0.717265\pi$
$884$	−1.80323	−0.0606492
$885$	78.2079	2.62893
$886$	59.0466	1.98371
$887$	8.50512	0.285574	0.142787	−	0.989753i	$-0.454394\pi$
$887$	8.50512	0.285574	0.142787	+	0.989753i	$0.454394\pi$
$888$	0	0
$889$	14.1261	0.473773
$890$	−64.6884	−2.16836
$891$	−0.670400	−0.0224592
$892$	16.3243	0.546578
$893$	63.9454	2.13985
$894$	−35.9948	−1.20385
$895$	−46.3944	−1.55079
$896$	−28.9254	−0.966329
$897$	9.30690	0.310748
$898$	7.13602	0.238132
$899$	10.8723	0.362612
$900$	21.2737	0.709123
$901$	−19.6045	−0.653121
$902$	0.609382	0.0202902
$903$	−54.9967	−1.83017
$904$	13.2970	0.442250
$905$	−67.9723	−2.25948
$906$	−64.6135	−2.14664
$907$	−18.8969	−0.627463	−0.313731	−	0.949512i	$-0.601579\pi$
$907$	−18.8969	−0.627463	−0.313731	+	0.949512i	$0.601579\pi$
$908$	8.47567	0.281275
$909$	−34.6121	−1.14801
$910$	11.6897	0.387510
$911$	4.72015	0.156386	0.0781928	−	0.996938i	$-0.475085\pi$
$911$	4.72015	0.156386	0.0781928	+	0.996938i	$0.475085\pi$
$912$	75.9409	2.51465
$913$	0.153849	0.00509165
$914$	74.3119	2.45802
$915$	−88.8216	−2.93635
$916$	31.5272	1.04169
$917$	−43.3399	−1.43121
$918$	10.9692	0.362037
$919$	22.3416	0.736981	0.368490	−	0.929632i	$-0.379875\pi$
$919$	22.3416	0.736981	0.368490	+	0.929632i	$0.379875\pi$
$920$	−18.6646	−0.615354
$921$	54.3008	1.78927
$922$	27.4369	0.903585
$923$	1.45420	0.0478656
$924$	1.66402	0.0547421
$925$	0	0
$926$	2.84628	0.0935344
$927$	51.3892	1.68784
$928$	−9.81511	−0.322197
$929$	33.2743	1.09169	0.545846	−	0.837885i	$-0.316208\pi$
$929$	33.2743	1.09169	0.545846	+	0.837885i	$0.316208\pi$
$930$	102.064	3.34683
$931$	11.4055	0.373802
$932$	22.1165	0.724448
$933$	−15.8284	−0.518200
$934$	23.2940	0.762204
$935$	0.937274	0.0306521
$936$	−3.85144	−0.125888
$937$	−11.6182	−0.379550	−0.189775	−	0.981828i	$-0.560776\pi$
$937$	−11.6182	−0.379550	−0.189775	+	0.981828i	$0.560776\pi$
$938$	42.9517	1.40242
$939$	−62.9027	−2.05275
$940$	−42.7782	−1.39527
$941$	23.5693	0.768339	0.384169	−	0.923263i	$-0.374488\pi$
$941$	23.5693	0.768339	0.384169	+	0.923263i	$0.374488\pi$
$942$	58.0474	1.89129
$943$	−10.1062	−0.329102
$944$	−47.9873	−1.56185
$945$	−27.8800	−0.906935
$946$	−2.00198	−0.0650898
$947$	−43.5842	−1.41630	−0.708149	−	0.706063i	$-0.750469\pi$
$947$	−43.5842	−1.41630	−0.708149	+	0.706063i	$0.750469\pi$
$948$	−4.77496	−0.155083
$949$	−6.76613	−0.219638
$950$	41.4955	1.34629
$951$	1.91446	0.0620806
$952$	−7.50806	−0.243338
$953$	−43.6803	−1.41494	−0.707471	−	0.706742i	$-0.750164\pi$
$953$	−43.6803	−1.41494	−0.707471	+	0.706742i	$0.750164\pi$
$954$	76.0560	2.46241
$955$	−44.2276	−1.43117
$956$	23.0343	0.744984
$957$	−0.666569	−0.0215471
$958$	11.9527	0.386174
$959$	−15.4905	−0.500216
$960$	−13.3676	−0.431438
$961$	18.3291	0.591260
$962$	0	0
$963$	−23.4267	−0.754916
$964$	8.70971	0.280521
$965$	−78.9644	−2.54195
$966$	−70.3862	−2.26464
$967$	4.45096	0.143133	0.0715667	−	0.997436i	$-0.477200\pi$
$967$	4.45096	0.143133	0.0715667	+	0.997436i	$0.477200\pi$
$968$	14.1351	0.454318
$969$	30.0558	0.965531
$970$	−48.5957	−1.56031
$971$	−14.3321	−0.459940	−0.229970	−	0.973198i	$-0.573863\pi$
$971$	−14.3321	−0.459940	−0.229970	+	0.973198i	$0.573863\pi$
$972$	26.4095	0.847086
$973$	2.73397	0.0876472
$974$	−1.02633	−0.0328858
$975$	−7.62069	−0.244057
$976$	54.4997	1.74449
$977$	45.4283	1.45338	0.726690	−	0.686966i	$-0.241058\pi$
$977$	45.4283	1.45338	0.726690	+	0.686966i	$0.241058\pi$
$978$	15.0819	0.482267
$979$	1.91681	0.0612614
$980$	−7.63008	−0.243734
$981$	−23.4398	−0.748375
$982$	−1.73881	−0.0554877
$983$	24.6902	0.787495	0.393747	−	0.919219i	$-0.371178\pi$
$983$	24.6902	0.787495	0.393747	+	0.919219i	$0.371178\pi$
$984$	7.19711	0.229436
$985$	−13.7766	−0.438958
$986$	−5.46287	−0.173973
$987$	88.8147	2.82700
$988$	5.35000	0.170206
$989$	33.2013	1.05574
$990$	−3.63617	−0.115565
$991$	48.3428	1.53566	0.767829	−	0.640654i	$-0.221337\pi$
$991$	48.3428	1.53566	0.767829	+	0.640654i	$0.221337\pi$
$992$	−44.5324	−1.41391
$993$	68.5577	2.17561
$994$	−10.9978	−0.348830
$995$	−8.87432	−0.281335
$996$	−3.30041	−0.104578
$997$	12.0140	0.380488	0.190244	−	0.981737i	$-0.439072\pi$
$997$	12.0140	0.380488	0.190244	+	0.981737i	$0.439072\pi$
$998$	−6.34723	−0.200918
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1369.2.a.o.1.19	yes	27
37.6	odd	4		1369.2.b.h.1368.12		54
37.31	odd	4		1369.2.b.h.1368.43		54
37.36	even	2		1369.2.a.n.1.9	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1369.2.a.n.1.9	✓	27	37.36	even	2
1369.2.a.o.1.19	yes	27	1.1	even	1	trivial
1369.2.b.h.1368.12		54	37.6	odd	4
1369.2.b.h.1368.43		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)

\(f(q)\)	\(=\)	\(q+1.81380 q^{2} -2.67610 q^{3} +1.28987 q^{4} -2.99386 q^{5} -4.85391 q^{6} -2.99597 q^{7} -1.28804 q^{8} +4.16150 q^{9} -5.43027 q^{10} +0.160906 q^{11} -3.45181 q^{12} +0.718529 q^{13} -5.43409 q^{14} +8.01187 q^{15} -4.91598 q^{16} -1.94564 q^{17} +7.54813 q^{18} +5.77250 q^{19} -3.86169 q^{20} +8.01751 q^{21} +0.291852 q^{22} -4.84015 q^{23} +3.44691 q^{24} +3.96322 q^{25} +1.30327 q^{26} -3.10830 q^{27} -3.86441 q^{28} +1.54800 q^{29} +14.5319 q^{30} +7.02347 q^{31} -6.34052 q^{32} -0.430601 q^{33} -3.52900 q^{34} +8.96953 q^{35} +5.36779 q^{36} +10.4702 q^{38} -1.92285 q^{39} +3.85621 q^{40} +2.08799 q^{41} +14.5422 q^{42} -6.85957 q^{43} +0.207548 q^{44} -12.4590 q^{45} -8.77906 q^{46} +11.0776 q^{47} +13.1556 q^{48} +1.97584 q^{49} +7.18848 q^{50} +5.20672 q^{51} +0.926808 q^{52} +10.0761 q^{53} -5.63783 q^{54} -0.481731 q^{55} +3.85892 q^{56} -15.4478 q^{57} +2.80776 q^{58} +9.76150 q^{59} +10.3343 q^{60} -11.0862 q^{61} +12.7392 q^{62} -12.4677 q^{63} -1.66848 q^{64} -2.15118 q^{65} -0.781024 q^{66} -7.90412 q^{67} -2.50962 q^{68} +12.9527 q^{69} +16.2689 q^{70} +2.02386 q^{71} -5.36017 q^{72} -9.41664 q^{73} -10.6060 q^{75} +7.44576 q^{76} -0.482070 q^{77} -3.48767 q^{78} +1.38332 q^{79} +14.7178 q^{80} -4.16640 q^{81} +3.78719 q^{82} +0.956139 q^{83} +10.3415 q^{84} +5.82497 q^{85} -12.4419 q^{86} -4.14259 q^{87} -0.207253 q^{88} +11.9126 q^{89} -22.5981 q^{90} -2.15269 q^{91} -6.24315 q^{92} -18.7955 q^{93} +20.0925 q^{94} -17.2821 q^{95} +16.9679 q^{96} +8.94904 q^{97} +3.58378 q^{98} +0.669612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 9 q^{2} - q^{3} + 25 q^{4} + 17 q^{5} + 10 q^{6} - 3 q^{7} + 21 q^{8} + 20 q^{9} - 11 q^{10} - 5 q^{11} - 10 q^{12} + 15 q^{13} + 25 q^{14} + 23 q^{15} + 13 q^{16} + 26 q^{17} + 15 q^{18} + 27 q^{19}+ \cdots + 35 q^{99}+O(q^{100}) \) 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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