LMFDB - Embedded newform 471.2.a.e.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [471,2,Mod(1,471)] 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("471.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(471, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 471 = 3 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	471.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.76095393520$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 20 x^{10} + 17 x^{9} + 149 x^{8} - 106 x^{7} - 500 x^{6} + 294 x^{5} + 711 x^{4} + \cdots - 15 $ x^12 - x^11 - 20x^10 + 17x^9 + 149x^8 - 106x^7 - 500x^6 + 294x^5 + 711x^4 - 349x^3 - 290x^2 + 173x - 15
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.576969$ of defining polynomial
Character	$\chi$	$=$	471.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.576969 q^{2} +1.00000 q^{3} -1.66711 q^{4} -4.25420 q^{5} -0.576969 q^{6} -0.240996 q^{7} +2.11581 q^{8} +1.00000 q^{9} +2.45454 q^{10} +3.20587 q^{11} -1.66711 q^{12} +1.47920 q^{13} +0.139047 q^{14} -4.25420 q^{15} +2.11346 q^{16} +1.69018 q^{17} -0.576969 q^{18} +6.68851 q^{19} +7.09221 q^{20} -0.240996 q^{21} -1.84968 q^{22} -4.99066 q^{23} +2.11581 q^{24} +13.0982 q^{25} -0.853451 q^{26} +1.00000 q^{27} +0.401766 q^{28} +0.612819 q^{29} +2.45454 q^{30} +8.86820 q^{31} -5.45101 q^{32} +3.20587 q^{33} -0.975179 q^{34} +1.02524 q^{35} -1.66711 q^{36} -8.56562 q^{37} -3.85906 q^{38} +1.47920 q^{39} -9.00107 q^{40} +0.629979 q^{41} +0.139047 q^{42} +4.51147 q^{43} -5.34452 q^{44} -4.25420 q^{45} +2.87946 q^{46} +0.799383 q^{47} +2.11346 q^{48} -6.94192 q^{49} -7.55727 q^{50} +1.69018 q^{51} -2.46598 q^{52} +1.27255 q^{53} -0.576969 q^{54} -13.6384 q^{55} -0.509900 q^{56} +6.68851 q^{57} -0.353578 q^{58} -5.28525 q^{59} +7.09221 q^{60} -3.62349 q^{61} -5.11667 q^{62} -0.240996 q^{63} -1.08186 q^{64} -6.29281 q^{65} -1.84968 q^{66} +15.0012 q^{67} -2.81771 q^{68} -4.99066 q^{69} -0.591534 q^{70} +14.8150 q^{71} +2.11581 q^{72} +8.80799 q^{73} +4.94209 q^{74} +13.0982 q^{75} -11.1505 q^{76} -0.772600 q^{77} -0.853451 q^{78} +16.5912 q^{79} -8.99109 q^{80} +1.00000 q^{81} -0.363478 q^{82} +2.31889 q^{83} +0.401766 q^{84} -7.19036 q^{85} -2.60297 q^{86} +0.612819 q^{87} +6.78299 q^{88} -3.62082 q^{89} +2.45454 q^{90} -0.356480 q^{91} +8.31997 q^{92} +8.86820 q^{93} -0.461219 q^{94} -28.4543 q^{95} -5.45101 q^{96} -2.28428 q^{97} +4.00527 q^{98} +3.20587 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{2} + 12 q^{3} + 17 q^{4} + 4 q^{5} - q^{6} + 8 q^{7} - 6 q^{8} + 12 q^{9} + 9 q^{10} - q^{11} + 17 q^{12} + 15 q^{13} - 7 q^{14} + 4 q^{15} + 19 q^{16} + 10 q^{17} - q^{18} + 14 q^{19} - 11 q^{20}+ \cdots - q^{99}+O(q^{100}) $ 12 * q - q^2 + 12 * q^3 + 17 * q^4 + 4 * q^5 - q^6 + 8 * q^7 - 6 * q^8 + 12 * q^9 + 9 * q^10 - q^11 + 17 * q^12 + 15 * q^13 - 7 * q^14 + 4 * q^15 + 19 * q^16 + 10 * q^17 - q^18 + 14 * q^19 - 11 * q^20 + 8 * q^21 - 9 * q^22 - 8 * q^23 - 6 * q^24 + 22 * q^25 - 17 * q^26 + 12 * q^27 + 7 * q^28 + 5 * q^29 + 9 * q^30 + 5 * q^31 - 33 * q^32 - q^33 - 13 * q^34 - 14 * q^35 + 17 * q^36 + 4 * q^37 - 32 * q^38 + 15 * q^39 - 4 * q^40 + 18 * q^41 - 7 * q^42 + 28 * q^43 - 20 * q^44 + 4 * q^45 - 15 * q^46 - 3 * q^47 + 19 * q^48 + 48 * q^49 - 2 * q^50 + 10 * q^51 + 26 * q^52 - 18 * q^53 - q^54 - 5 * q^55 - 10 * q^56 + 14 * q^57 - 6 * q^58 - 11 * q^60 + 20 * q^61 + q^62 + 8 * q^63 - 14 * q^64 - 11 * q^65 - 9 * q^66 + 27 * q^67 - 21 * q^68 - 8 * q^69 - 29 * q^70 + 7 * q^71 - 6 * q^72 + 15 * q^73 - 14 * q^74 + 22 * q^75 + 30 * q^76 - 23 * q^77 - 17 * q^78 + 4 * q^79 - 31 * q^80 + 12 * q^81 - 31 * q^82 - 13 * q^83 + 7 * q^84 + 8 * q^85 - 26 * q^86 + 5 * q^87 - 11 * q^88 - 9 * q^89 + 9 * q^90 - 31 * q^91 - 21 * q^92 + 5 * q^93 - 4 * q^94 - 44 * q^95 - 33 * q^96 - 16 * q^97 - 54 * q^98 - q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.576969	−0.407978	−0.203989	−	0.978973i	$-0.565391\pi$
$2$	−0.576969	−0.407978	−0.203989	+	0.978973i	$0.565391\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.66711	−0.833554
$5$	−4.25420	−1.90254	−0.951269	−	0.308364i	$-0.900219\pi$
$5$	−4.25420	−1.90254	−0.951269	+	0.308364i	$0.900219\pi$
$6$	−0.576969	−0.235546
$7$	−0.240996	−0.0910878	−0.0455439	−	0.998962i	$-0.514502\pi$
$7$	−0.240996	−0.0910878	−0.0455439	+	0.998962i	$0.514502\pi$
$8$	2.11581	0.748050
$9$	1.00000	0.333333
$10$	2.45454	0.776194
$11$	3.20587	0.966605	0.483303	−	0.875453i	$-0.339437\pi$
$11$	3.20587	0.966605	0.483303	+	0.875453i	$0.339437\pi$
$12$	−1.66711	−0.481252
$13$	1.47920	0.410256	0.205128	−	0.978735i	$-0.434239\pi$
$13$	1.47920	0.410256	0.205128	+	0.978735i	$0.434239\pi$
$14$	0.139047	0.0371619
$15$	−4.25420	−1.09843
$16$	2.11346	0.528365
$17$	1.69018	0.409928	0.204964	−	0.978769i	$-0.434292\pi$
$17$	1.69018	0.409928	0.204964	+	0.978769i	$0.434292\pi$
$18$	−0.576969	−0.135993
$19$	6.68851	1.53445	0.767225	−	0.641378i	$-0.221637\pi$
$19$	6.68851	1.53445	0.767225	+	0.641378i	$0.221637\pi$
$20$	7.09221	1.58587
$21$	−0.240996	−0.0525896
$22$	−1.84968	−0.394354
$23$	−4.99066	−1.04063	−0.520313	−	0.853976i	$-0.674185\pi$
$23$	−4.99066	−1.04063	−0.520313	+	0.853976i	$0.674185\pi$
$24$	2.11581	0.431887
$25$	13.0982	2.61965
$26$	−0.853451	−0.167375
$27$	1.00000	0.192450
$28$	0.401766	0.0759266
$29$	0.612819	0.113798	0.0568989	−	0.998380i	$-0.481879\pi$
$29$	0.612819	0.113798	0.0568989	+	0.998380i	$0.481879\pi$
$30$	2.45454	0.448136
$31$	8.86820	1.59278	0.796388	−	0.604787i	$-0.206742\pi$
$31$	8.86820	1.59278	0.796388	+	0.604787i	$0.206742\pi$
$32$	−5.45101	−0.963612
$33$	3.20587	0.558070
$34$	−0.975179	−0.167242
$35$	1.02524	0.173298
$36$	−1.66711	−0.277851
$37$	−8.56562	−1.40818	−0.704090	−	0.710111i	$-0.748645\pi$
$37$	−8.56562	−1.40818	−0.704090	+	0.710111i	$0.748645\pi$
$38$	−3.85906	−0.626023
$39$	1.47920	0.236861
$40$	−9.00107	−1.42319
$41$	0.629979	0.0983862	0.0491931	−	0.998789i	$-0.484335\pi$
$41$	0.629979	0.0983862	0.0491931	+	0.998789i	$0.484335\pi$
$42$	0.139047	0.0214554
$43$	4.51147	0.687992	0.343996	−	0.938971i	$-0.388219\pi$
$43$	4.51147	0.687992	0.343996	+	0.938971i	$0.388219\pi$
$44$	−5.34452	−0.805717
$45$	−4.25420	−0.634179
$46$	2.87946	0.424553
$47$	0.799383	0.116602	0.0583010	−	0.998299i	$-0.481432\pi$
$47$	0.799383	0.116602	0.0583010	+	0.998299i	$0.481432\pi$
$48$	2.11346	0.305052
$49$	−6.94192	−0.991703
$50$	−7.55727	−1.06876
$51$	1.69018	0.236672
$52$	−2.46598	−0.341970
$53$	1.27255	0.174798	0.0873989	−	0.996173i	$-0.472145\pi$
$53$	1.27255	0.174798	0.0873989	+	0.996173i	$0.472145\pi$
$54$	−0.576969	−0.0785155
$55$	−13.6384	−1.83900
$56$	−0.509900	−0.0681383
$57$	6.68851	0.885915
$58$	−0.353578	−0.0464270
$59$	−5.28525	−0.688081	−0.344041	−	0.938955i	$-0.611796\pi$
$59$	−5.28525	−0.688081	−0.344041	+	0.938955i	$0.611796\pi$
$60$	7.09221	0.915600
$61$	−3.62349	−0.463941	−0.231970	−	0.972723i	$-0.574517\pi$
$61$	−3.62349	−0.463941	−0.231970	+	0.972723i	$0.574517\pi$
$62$	−5.11667	−0.649818
$63$	−0.240996	−0.0303626
$64$	−1.08186	−0.135232
$65$	−6.29281	−0.780527
$66$	−1.84968	−0.227680
$67$	15.0012	1.83269	0.916346	−	0.400386i	$-0.131124\pi$
$67$	15.0012	1.83269	0.916346	+	0.400386i	$0.131124\pi$
$68$	−2.81771	−0.341697
$69$	−4.99066	−0.600805
$70$	−0.591534	−0.0707018
$71$	14.8150	1.75822	0.879110	−	0.476619i	$-0.158138\pi$
$71$	14.8150	1.75822	0.879110	+	0.476619i	$0.158138\pi$
$72$	2.11581	0.249350
$73$	8.80799	1.03090	0.515448	−	0.856921i	$-0.327625\pi$
$73$	8.80799	1.03090	0.515448	+	0.856921i	$0.327625\pi$
$74$	4.94209	0.574507
$75$	13.0982	1.51245
$76$	−11.1505	−1.27905
$77$	−0.772600	−0.0880460
$78$	−0.853451	−0.0966343
$79$	16.5912	1.86666	0.933328	−	0.359024i	$-0.116890\pi$
$79$	16.5912	1.86666	0.933328	+	0.359024i	$0.116890\pi$
$80$	−8.99109	−1.00523
$81$	1.00000	0.111111
$82$	−0.363478	−0.0401395
$83$	2.31889	0.254531	0.127266	−	0.991869i	$-0.459380\pi$
$83$	2.31889	0.254531	0.127266	+	0.991869i	$0.459380\pi$
$84$	0.401766	0.0438362
$85$	−7.19036	−0.779904
$86$	−2.60297	−0.280686
$87$	0.612819	0.0657011
$88$	6.78299	0.723069
$89$	−3.62082	−0.383806	−0.191903	−	0.981414i	$-0.561466\pi$
$89$	−3.62082	−0.383806	−0.191903	+	0.981414i	$0.561466\pi$
$90$	2.45454	0.258731
$91$	−0.356480	−0.0373693
$92$	8.31997	0.867417
$93$	8.86820	0.919589
$94$	−0.461219	−0.0475711
$95$	−28.4543	−2.91935
$96$	−5.45101	−0.556342
$97$	−2.28428	−0.231934	−0.115967	−	0.993253i	$-0.536997\pi$
$97$	−2.28428	−0.231934	−0.115967	+	0.993253i	$0.536997\pi$
$98$	4.00527	0.404593
$99$	3.20587	0.322202
$100$	−21.8362	−2.18362
$101$	2.38427	0.237244	0.118622	−	0.992939i	$-0.462152\pi$
$101$	2.38427	0.237244	0.118622	+	0.992939i	$0.462152\pi$
$102$	−0.975179	−0.0965571
$103$	1.12954	0.111297	0.0556485	−	0.998450i	$-0.482277\pi$
$103$	1.12954	0.111297	0.0556485	+	0.998450i	$0.482277\pi$
$104$	3.12970	0.306892
$105$	1.02524	0.100054
$106$	−0.734220	−0.0713138
$107$	−16.8343	−1.62744	−0.813719	−	0.581259i	$-0.802560\pi$
$107$	−16.8343	−1.62744	−0.813719	+	0.581259i	$0.802560\pi$
$108$	−1.66711	−0.160417
$109$	−6.08263	−0.582611	−0.291305	−	0.956630i	$-0.594090\pi$
$109$	−6.08263	−0.582611	−0.291305	+	0.956630i	$0.594090\pi$
$110$	7.86893	0.750273
$111$	−8.56562	−0.813013
$112$	−0.509335	−0.0481276
$113$	18.7474	1.76361	0.881803	−	0.471617i	$-0.156330\pi$
$113$	18.7474	1.76361	0.881803	+	0.471617i	$0.156330\pi$
$114$	−3.85906	−0.361434
$115$	21.2313	1.97983
$116$	−1.02164	−0.0948565
$117$	1.47920	0.136752
$118$	3.04943	0.280722
$119$	−0.407326	−0.0373395
$120$	−9.00107	−0.821681
$121$	−0.722420	−0.0656745
$122$	2.09064	0.189278
$123$	0.629979	0.0568033
$124$	−14.7842	−1.32766
$125$	−34.4515	−3.08144
$126$	0.139047	0.0123873
$127$	−7.51319	−0.666688	−0.333344	−	0.942805i	$-0.608177\pi$
$127$	−7.51319	−0.666688	−0.333344	+	0.942805i	$0.608177\pi$
$128$	11.5262	1.01878
$129$	4.51147	0.397212
$130$	3.63075	0.318438
$131$	8.66820	0.757344	0.378672	−	0.925531i	$-0.376381\pi$
$131$	8.66820	0.757344	0.378672	+	0.925531i	$0.376381\pi$
$132$	−5.34452	−0.465181
$133$	−1.61190	−0.139770
$134$	−8.65524	−0.747699
$135$	−4.25420	−0.366143
$136$	3.57609	0.306647
$137$	−7.15811	−0.611558	−0.305779	−	0.952102i	$-0.598917\pi$
$137$	−7.15811	−0.611558	−0.305779	+	0.952102i	$0.598917\pi$
$138$	2.87946	0.245116
$139$	1.34089	0.113733	0.0568664	−	0.998382i	$-0.481889\pi$
$139$	1.34089	0.113733	0.0568664	+	0.998382i	$0.481889\pi$
$140$	−1.70919	−0.144453
$141$	0.799383	0.0673202
$142$	−8.54780	−0.717316
$143$	4.74211	0.396555
$144$	2.11346	0.176122
$145$	−2.60706	−0.216504
$146$	−5.08193	−0.420584
$147$	−6.94192	−0.572560
$148$	14.2798	1.17379
$149$	−4.91050	−0.402284	−0.201142	−	0.979562i	$-0.564465\pi$
$149$	−4.91050	−0.402284	−0.201142	+	0.979562i	$0.564465\pi$
$150$	−7.55727	−0.617049
$151$	−9.69294	−0.788800	−0.394400	−	0.918939i	$-0.629048\pi$
$151$	−9.69294	−0.788800	−0.394400	+	0.918939i	$0.629048\pi$
$152$	14.1516	1.14785
$153$	1.69018	0.136643
$154$	0.445766	0.0359209
$155$	−37.7271	−3.03031
$156$	−2.46598	−0.197437
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−9.57261	−0.761556
$159$	1.27255	0.100920
$160$	23.1897	1.83331
$161$	1.20273	0.0947883
$162$	−0.576969	−0.0453309
$163$	2.99256	0.234395	0.117198	−	0.993109i	$-0.462609\pi$
$163$	2.99256	0.234395	0.117198	+	0.993109i	$0.462609\pi$
$164$	−1.05024	−0.0820102
$165$	−13.6384	−1.06175
$166$	−1.33793	−0.103843
$167$	−16.8997	−1.30773	−0.653867	−	0.756609i	$-0.726855\pi$
$167$	−16.8997	−1.30773	−0.653867	+	0.756609i	$0.726855\pi$
$168$	−0.509900	−0.0393397
$169$	−10.8120	−0.831690
$170$	4.14861	0.318184
$171$	6.68851	0.511483
$172$	−7.52110	−0.573478
$173$	17.3705	1.32065	0.660327	−	0.750978i	$-0.270418\pi$
$173$	17.3705	1.32065	0.660327	+	0.750978i	$0.270418\pi$
$174$	−0.353578	−0.0268046
$175$	−3.15662	−0.238618
$176$	6.77547	0.510720
$177$	−5.28525	−0.397264
$178$	2.08910	0.156584
$179$	18.8011	1.40526	0.702629	−	0.711556i	$-0.252009\pi$
$179$	18.8011	1.40526	0.702629	+	0.711556i	$0.252009\pi$
$180$	7.09221	0.528622
$181$	18.4966	1.37484	0.687421	−	0.726259i	$-0.258743\pi$
$181$	18.4966	1.37484	0.687421	+	0.726259i	$0.258743\pi$
$182$	0.205678	0.0152459
$183$	−3.62349	−0.267856
$184$	−10.5593	−0.778440
$185$	36.4399	2.67911
$186$	−5.11667	−0.375173
$187$	5.41848	0.396239
$188$	−1.33266	−0.0971940
$189$	−0.240996	−0.0175299
$190$	16.4172	1.19103
$191$	7.15009	0.517362	0.258681	−	0.965963i	$-0.416712\pi$
$191$	7.15009	0.517362	0.258681	+	0.965963i	$0.416712\pi$
$192$	−1.08186	−0.0780763
$193$	−25.8197	−1.85854	−0.929270	−	0.369401i	$-0.879563\pi$
$193$	−25.8197	−1.85854	−0.929270	+	0.369401i	$0.879563\pi$
$194$	1.31796	0.0946240
$195$	−6.29281	−0.450637
$196$	11.5729	0.826638
$197$	14.9728	1.06677	0.533383	−	0.845874i	$-0.320920\pi$
$197$	14.9728	1.06677	0.533383	+	0.845874i	$0.320920\pi$
$198$	−1.84968	−0.131451
$199$	2.43370	0.172521	0.0862603	−	0.996273i	$-0.472508\pi$
$199$	2.43370	0.172521	0.0862603	+	0.996273i	$0.472508\pi$
$200$	27.7133	1.95963
$201$	15.0012	1.05811
$202$	−1.37565	−0.0967905
$203$	−0.147687	−0.0103656
$204$	−2.81771	−0.197279
$205$	−2.68006	−0.187183
$206$	−0.651710	−0.0454068
$207$	−4.99066	−0.346875
$208$	3.12623	0.216765
$209$	21.4425	1.48321
$210$	−0.591534	−0.0408197
$211$	−17.3444	−1.19404	−0.597018	−	0.802228i	$-0.703648\pi$
$211$	−17.3444	−1.19404	−0.597018	+	0.802228i	$0.703648\pi$
$212$	−2.12147	−0.145703
$213$	14.8150	1.01511
$214$	9.71289	0.663959
$215$	−19.1927	−1.30893
$216$	2.11581	0.143962
$217$	−2.13720	−0.145082
$218$	3.50949	0.237693
$219$	8.80799	0.595189
$220$	22.7367	1.53291
$221$	2.50011	0.168175
$222$	4.94209	0.331692
$223$	−16.1554	−1.08185	−0.540923	−	0.841072i	$-0.681925\pi$
$223$	−16.1554	−1.08185	−0.540923	+	0.841072i	$0.681925\pi$
$224$	1.31367	0.0877733
$225$	13.0982	0.873216
$226$	−10.8167	−0.719514
$227$	−7.76601	−0.515448	−0.257724	−	0.966219i	$-0.582973\pi$
$227$	−7.76601	−0.515448	−0.257724	+	0.966219i	$0.582973\pi$
$228$	−11.1505	−0.738458
$229$	−9.78178	−0.646399	−0.323199	−	0.946331i	$-0.604758\pi$
$229$	−9.78178	−0.646399	−0.323199	+	0.946331i	$0.604758\pi$
$230$	−12.2498	−0.807727
$231$	−0.772600	−0.0508334
$232$	1.29661	0.0851264
$233$	11.8482	0.776201	0.388100	−	0.921617i	$-0.373131\pi$
$233$	11.8482	0.776201	0.388100	+	0.921617i	$0.373131\pi$
$234$	−0.853451	−0.0557918
$235$	−3.40074	−0.221840
$236$	8.81108	0.573553
$237$	16.5912	1.07771
$238$	0.235014	0.0152337
$239$	2.59130	0.167617	0.0838085	−	0.996482i	$-0.473292\pi$
$239$	2.59130	0.167617	0.0838085	+	0.996482i	$0.473292\pi$
$240$	−8.99109	−0.580372
$241$	18.1978	1.17223	0.586113	−	0.810229i	$-0.300657\pi$
$241$	18.1978	1.17223	0.586113	+	0.810229i	$0.300657\pi$
$242$	0.416814	0.0267938
$243$	1.00000	0.0641500
$244$	6.04075	0.386720
$245$	29.5323	1.88675
$246$	−0.363478	−0.0231745
$247$	9.89364	0.629517
$248$	18.7634	1.19148
$249$	2.31889	0.146954
$250$	19.8775	1.25716
$251$	−16.3733	−1.03347	−0.516737	−	0.856144i	$-0.672853\pi$
$251$	−16.3733	−1.03347	−0.516737	+	0.856144i	$0.672853\pi$
$252$	0.401766	0.0253089
$253$	−15.9994	−1.00587
$254$	4.33488	0.271994
$255$	−7.19036	−0.450278
$256$	−4.48655	−0.280410
$257$	3.97345	0.247857	0.123929	−	0.992291i	$-0.460451\pi$
$257$	3.97345	0.247857	0.123929	+	0.992291i	$0.460451\pi$
$258$	−2.60297	−0.162054
$259$	2.06428	0.128268
$260$	10.4908	0.650611
$261$	0.612819	0.0379326
$262$	−5.00128	−0.308980
$263$	11.0005	0.678318	0.339159	−	0.940729i	$-0.389857\pi$
$263$	11.0005	0.678318	0.339159	+	0.940729i	$0.389857\pi$
$264$	6.78299	0.417464
$265$	−5.41368	−0.332559
$266$	0.930018	0.0570231
$267$	−3.62082	−0.221590
$268$	−25.0087	−1.52765
$269$	19.7930	1.20680	0.603399	−	0.797439i	$-0.293813\pi$
$269$	19.7930	1.20680	0.603399	+	0.797439i	$0.293813\pi$
$270$	2.45454	0.149379
$271$	11.6000	0.704650	0.352325	−	0.935878i	$-0.385391\pi$
$271$	11.6000	0.704650	0.352325	+	0.935878i	$0.385391\pi$
$272$	3.57212	0.216592
$273$	−0.356480	−0.0215752
$274$	4.13000	0.249503
$275$	41.9912	2.53216
$276$	8.31997	0.500803
$277$	−19.0592	−1.14515	−0.572577	−	0.819851i	$-0.694056\pi$
$277$	−19.0592	−1.14515	−0.572577	+	0.819851i	$0.694056\pi$
$278$	−0.773651	−0.0464005
$279$	8.86820	0.530925
$280$	2.16922	0.129636
$281$	−30.7195	−1.83257	−0.916287	−	0.400523i	$-0.868828\pi$
$281$	−30.7195	−1.83257	−0.916287	+	0.400523i	$0.868828\pi$
$282$	−0.461219	−0.0274652
$283$	−10.1688	−0.604470	−0.302235	−	0.953233i	$-0.597733\pi$
$283$	−10.1688	−0.604470	−0.302235	+	0.953233i	$0.597733\pi$
$284$	−24.6982	−1.46557
$285$	−28.4543	−1.68549
$286$	−2.73605	−0.161786
$287$	−0.151822	−0.00896179
$288$	−5.45101	−0.321204
$289$	−14.1433	−0.831959
$290$	1.50419	0.0883291
$291$	−2.28428	−0.133907
$292$	−14.6839	−0.859308
$293$	−18.0760	−1.05601	−0.528005	−	0.849241i	$-0.677060\pi$
$293$	−18.0760	−1.05601	−0.528005	+	0.849241i	$0.677060\pi$
$294$	4.00527	0.233592
$295$	22.4845	1.30910
$296$	−18.1232	−1.05339
$297$	3.20587	0.186023
$298$	2.83321	0.164123
$299$	−7.38218	−0.426922
$300$	−21.8362	−1.26071
$301$	−1.08724	−0.0626677
$302$	5.59252	0.321813
$303$	2.38427	0.136973
$304$	14.1359	0.810750
$305$	15.4151	0.882665
$306$	−0.975179	−0.0557473
$307$	−28.8983	−1.64931	−0.824657	−	0.565633i	$-0.808632\pi$
$307$	−28.8983	−1.64931	−0.824657	+	0.565633i	$0.808632\pi$
$308$	1.28801	0.0733910
$309$	1.12954	0.0642574
$310$	21.7674	1.23630
$311$	18.8117	1.06671	0.533357	−	0.845890i	$-0.320930\pi$
$311$	18.8117	1.06671	0.533357	+	0.845890i	$0.320930\pi$
$312$	3.12970	0.177184
$313$	8.39372	0.474441	0.237220	−	0.971456i	$-0.423764\pi$
$313$	8.39372	0.474441	0.237220	+	0.971456i	$0.423764\pi$
$314$	0.576969	0.0325602
$315$	1.02524	0.0577660
$316$	−27.6593	−1.55596
$317$	−12.8553	−0.722024	−0.361012	−	0.932561i	$-0.617569\pi$
$317$	−12.8553	−0.722024	−0.361012	+	0.932561i	$0.617569\pi$
$318$	−0.734220	−0.0411730
$319$	1.96462	0.109997
$320$	4.60244	0.257284
$321$	−16.8343	−0.939601
$322$	−0.693937	−0.0386716
$323$	11.3048	0.629014
$324$	−1.66711	−0.0926171
$325$	19.3749	1.07473
$326$	−1.72661	−0.0956282
$327$	−6.08263	−0.336370
$328$	1.33291	0.0735978
$329$	−0.192648	−0.0106210
$330$	7.86893	0.433170
$331$	−7.05067	−0.387540	−0.193770	−	0.981047i	$-0.562072\pi$
$331$	−7.05067	−0.387540	−0.193770	+	0.981047i	$0.562072\pi$
$332$	−3.86584	−0.212165
$333$	−8.56562	−0.469393
$334$	9.75057	0.533528
$335$	−63.8183	−3.48677
$336$	−0.509335	−0.0277865
$337$	16.9975	0.925912	0.462956	−	0.886381i	$-0.346789\pi$
$337$	16.9975	0.925912	0.462956	+	0.886381i	$0.346789\pi$
$338$	6.23817	0.339312
$339$	18.7474	1.01822
$340$	11.9871	0.650091
$341$	28.4303	1.53958
$342$	−3.85906	−0.208674
$343$	3.35994	0.181420
$344$	9.54539	0.514653
$345$	21.2313	1.14305
$346$	−10.0222	−0.538799
$347$	14.9794	0.804136	0.402068	−	0.915610i	$-0.368291\pi$
$347$	14.9794	0.804136	0.402068	+	0.915610i	$0.368291\pi$
$348$	−1.02164	−0.0547654
$349$	4.72651	0.253004	0.126502	−	0.991966i	$-0.459625\pi$
$349$	4.72651	0.253004	0.126502	+	0.991966i	$0.459625\pi$
$350$	1.82127	0.0973510
$351$	1.47920	0.0789537
$352$	−17.4752	−0.931432
$353$	20.6345	1.09826	0.549131	−	0.835736i	$-0.314959\pi$
$353$	20.6345	1.09826	0.549131	+	0.835736i	$0.314959\pi$
$354$	3.04943	0.162075
$355$	−63.0261	−3.34508
$356$	6.03629	0.319923
$357$	−0.407326	−0.0215580
$358$	−10.8476	−0.573315
$359$	−4.83155	−0.255000	−0.127500	−	0.991839i	$-0.540695\pi$
$359$	−4.83155	−0.255000	−0.127500	+	0.991839i	$0.540695\pi$
$360$	−9.00107	−0.474398
$361$	25.7362	1.35454
$362$	−10.6720	−0.560906
$363$	−0.722420	−0.0379172
$364$	0.594291	0.0311493
$365$	−37.4710	−1.96132
$366$	2.09064	0.109280
$367$	30.3909	1.58639	0.793197	−	0.608965i	$-0.208415\pi$
$367$	30.3909	1.58639	0.793197	+	0.608965i	$0.208415\pi$
$368$	−10.5476	−0.549830
$369$	0.629979	0.0327954
$370$	−21.0247	−1.09302
$371$	−0.306679	−0.0159220
$372$	−14.7842	−0.766527
$373$	−4.47974	−0.231952	−0.115976	−	0.993252i	$-0.537000\pi$
$373$	−4.47974	−0.231952	−0.115976	+	0.993252i	$0.537000\pi$
$374$	−3.12629	−0.161657
$375$	−34.4515	−1.77907
$376$	1.69134	0.0872241
$377$	0.906481	0.0466862
$378$	0.139047	0.00715181
$379$	−28.2337	−1.45027	−0.725134	−	0.688608i	$-0.758222\pi$
$379$	−28.2337	−1.45027	−0.725134	+	0.688608i	$0.758222\pi$
$380$	47.4364	2.43343
$381$	−7.51319	−0.384913
$382$	−4.12538	−0.211073
$383$	−9.11508	−0.465759	−0.232879	−	0.972506i	$-0.574815\pi$
$383$	−9.11508	−0.465759	−0.232879	+	0.972506i	$0.574815\pi$
$384$	11.5262	0.588195
$385$	3.28680	0.167511
$386$	14.8971	0.758244
$387$	4.51147	0.229331
$388$	3.80814	0.193329
$389$	−23.4006	−1.18646	−0.593230	−	0.805033i	$-0.702147\pi$
$389$	−23.4006	−1.18646	−0.593230	+	0.805033i	$0.702147\pi$
$390$	3.63075	0.183850
$391$	−8.43511	−0.426582
$392$	−14.6878	−0.741844
$393$	8.66820	0.437253
$394$	−8.63883	−0.435218
$395$	−70.5824	−3.55138
$396$	−5.34452	−0.268572
$397$	33.9934	1.70608	0.853041	−	0.521844i	$-0.174756\pi$
$397$	33.9934	1.70608	0.853041	+	0.521844i	$0.174756\pi$
$398$	−1.40417	−0.0703847
$399$	−1.61190	−0.0806961
$400$	27.6826	1.38413
$401$	30.8413	1.54014	0.770072	−	0.637957i	$-0.220220\pi$
$401$	30.8413	1.54014	0.770072	+	0.637957i	$0.220220\pi$
$402$	−8.65524	−0.431684
$403$	13.1178	0.653445
$404$	−3.97484	−0.197756
$405$	−4.25420	−0.211393
$406$	0.0852107	0.00422894
$407$	−27.4602	−1.36115
$408$	3.57609	0.177043
$409$	33.1129	1.63733	0.818664	−	0.574272i	$-0.194715\pi$
$409$	33.1129	1.63733	0.818664	+	0.574272i	$0.194715\pi$
$410$	1.54631	0.0763668
$411$	−7.15811	−0.353083
$412$	−1.88307	−0.0927720
$413$	1.27372	0.0626758
$414$	2.87946	0.141518
$415$	−9.86503	−0.484255
$416$	−8.06313	−0.395327
$417$	1.34089	0.0656637
$418$	−12.3716	−0.605117
$419$	−8.76457	−0.428177	−0.214089	−	0.976814i	$-0.568678\pi$
$419$	−8.76457	−0.428177	−0.214089	+	0.976814i	$0.568678\pi$
$420$	−1.70919	−0.0834001
$421$	−0.895206	−0.0436297	−0.0218148	−	0.999762i	$-0.506944\pi$
$421$	−0.895206	−0.0436297	−0.0218148	+	0.999762i	$0.506944\pi$
$422$	10.0072	0.487141
$423$	0.799383	0.0388673
$424$	2.69246	0.130758
$425$	22.1383	1.07387
$426$	−8.54780	−0.414142
$427$	0.873247	0.0422594
$428$	28.0647	1.35656
$429$	4.74211	0.228951
$430$	11.0736	0.534015
$431$	21.7119	1.04582	0.522912	−	0.852387i	$-0.324846\pi$
$431$	21.7119	1.04582	0.522912	+	0.852387i	$0.324846\pi$
$432$	2.11346	0.101684
$433$	2.27586	0.109371	0.0546855	−	0.998504i	$-0.482584\pi$
$433$	2.27586	0.109371	0.0546855	+	0.998504i	$0.482584\pi$
$434$	1.23310	0.0591905
$435$	−2.60706	−0.124999
$436$	10.1404	0.485637
$437$	−33.3801	−1.59679
$438$	−5.08193	−0.242824
$439$	31.6739	1.51171	0.755856	−	0.654738i	$-0.227221\pi$
$439$	31.6739	1.51171	0.755856	+	0.654738i	$0.227221\pi$
$440$	−28.8562	−1.37567
$441$	−6.94192	−0.330568
$442$	−1.44248	−0.0686119
$443$	−24.8064	−1.17859	−0.589295	−	0.807918i	$-0.700594\pi$
$443$	−24.8064	−1.17859	−0.589295	+	0.807918i	$0.700594\pi$
$444$	14.2798	0.677690
$445$	15.4037	0.730205
$446$	9.32117	0.441370
$447$	−4.91050	−0.232259
$448$	0.260723	0.0123180
$449$	10.6775	0.503904	0.251952	−	0.967740i	$-0.418927\pi$
$449$	10.6775	0.503904	0.251952	+	0.967740i	$0.418927\pi$
$450$	−7.55727	−0.356253
$451$	2.01963	0.0951006
$452$	−31.2539	−1.47006
$453$	−9.69294	−0.455414
$454$	4.48075	0.210292
$455$	1.51654	0.0710965
$456$	14.1516	0.662709
$457$	−16.4735	−0.770596	−0.385298	−	0.922792i	$-0.625901\pi$
$457$	−16.4735	−0.770596	−0.385298	+	0.922792i	$0.625901\pi$
$458$	5.64378	0.263717
$459$	1.69018	0.0788907
$460$	−35.3948	−1.65029
$461$	−2.69572	−0.125552	−0.0627760	−	0.998028i	$-0.519995\pi$
$461$	−2.69572	−0.125552	−0.0627760	+	0.998028i	$0.519995\pi$
$462$	0.445766	0.0207389
$463$	−27.5090	−1.27845	−0.639227	−	0.769018i	$-0.720745\pi$
$463$	−27.5090	−1.27845	−0.639227	+	0.769018i	$0.720745\pi$
$464$	1.29517	0.0601267
$465$	−37.7271	−1.74955
$466$	−6.83603	−0.316673
$467$	−15.5576	−0.719920	−0.359960	−	0.932968i	$-0.617210\pi$
$467$	−15.5576	−0.719920	−0.359960	+	0.932968i	$0.617210\pi$
$468$	−2.46598	−0.113990
$469$	−3.61523	−0.166936
$470$	1.96212	0.0905058
$471$	−1.00000	−0.0460776
$472$	−11.1826	−0.514719
$473$	14.4632	0.665017
$474$	−9.57261	−0.439684
$475$	87.6077	4.01972
$476$	0.679055	0.0311245
$477$	1.27255	0.0582660
$478$	−1.49510	−0.0683841
$479$	−33.8351	−1.54597	−0.772983	−	0.634427i	$-0.781236\pi$
$479$	−33.8351	−1.54597	−0.772983	+	0.634427i	$0.781236\pi$
$480$	23.1897	1.05846
$481$	−12.6702	−0.577713
$482$	−10.4996	−0.478243
$483$	1.20273	0.0547261
$484$	1.20435	0.0547433
$485$	9.71780	0.441262
$486$	−0.576969	−0.0261718
$487$	−31.2170	−1.41458	−0.707289	−	0.706925i	$-0.750082\pi$
$487$	−31.2170	−1.41458	−0.707289	+	0.706925i	$0.750082\pi$
$488$	−7.66661	−0.347051
$489$	2.99256	0.135328
$490$	−17.0392	−0.769754
$491$	4.46169	0.201353	0.100677	−	0.994919i	$-0.467899\pi$
$491$	4.46169	0.201353	0.100677	+	0.994919i	$0.467899\pi$
$492$	−1.05024	−0.0473486
$493$	1.03577	0.0466489
$494$	−5.70832	−0.256829
$495$	−13.6384	−0.613001
$496$	18.7426	0.841567
$497$	−3.57036	−0.160152
$498$	−1.33793	−0.0599540
$499$	−9.66645	−0.432730	−0.216365	−	0.976313i	$-0.569420\pi$
$499$	−9.66645	−0.432730	−0.216365	+	0.976313i	$0.569420\pi$
$500$	57.4344	2.56854
$501$	−16.8997	−0.755021
$502$	9.44688	0.421635
$503$	36.1078	1.60997	0.804984	−	0.593297i	$-0.202174\pi$
$503$	36.1078	1.60997	0.804984	+	0.593297i	$0.202174\pi$
$504$	−0.509900	−0.0227128
$505$	−10.1432	−0.451366
$506$	9.23115	0.410375
$507$	−10.8120	−0.480177
$508$	12.5253	0.555720
$509$	42.2719	1.87367	0.936834	−	0.349775i	$-0.113742\pi$
$509$	42.2719	1.87367	0.936834	+	0.349775i	$0.113742\pi$
$510$	4.14861	0.183704
$511$	−2.12269	−0.0939022
$512$	−20.4638	−0.904383
$513$	6.68851	0.295305
$514$	−2.29256	−0.101120
$515$	−4.80530	−0.211747
$516$	−7.52110	−0.331098
$517$	2.56271	0.112708
$518$	−1.19102	−0.0523306
$519$	17.3705	0.762480
$520$	−13.3144	−0.583873
$521$	23.9877	1.05092	0.525461	−	0.850818i	$-0.323893\pi$
$521$	23.9877	1.05092	0.525461	+	0.850818i	$0.323893\pi$
$522$	−0.353578	−0.0154757
$523$	−40.4440	−1.76849	−0.884247	−	0.467020i	$-0.845328\pi$
$523$	−40.4440	−1.76849	−0.884247	+	0.467020i	$0.845328\pi$
$524$	−14.4508	−0.631287
$525$	−3.15662	−0.137766
$526$	−6.34693	−0.276739
$527$	14.9888	0.652923
$528$	6.77547	0.294865
$529$	1.90673	0.0829012
$530$	3.12352	0.135677
$531$	−5.28525	−0.229360
$532$	2.68722	0.116506
$533$	0.931864	0.0403635
$534$	2.08910	0.0904041
$535$	71.6167	3.09626
$536$	31.7397	1.37095
$537$	18.8011	0.811326
$538$	−11.4199	−0.492348
$539$	−22.2549	−0.958585
$540$	7.09221	0.305200
$541$	−39.7124	−1.70737	−0.853685	−	0.520790i	$-0.825637\pi$
$541$	−39.7124	−1.70737	−0.853685	+	0.520790i	$0.825637\pi$
$542$	−6.69283	−0.287482
$543$	18.4966	0.793765
$544$	−9.21318	−0.395012
$545$	25.8767	1.10844
$546$	0.205678	0.00880221
$547$	−43.1604	−1.84540	−0.922702	−	0.385513i	$-0.874024\pi$
$547$	−43.1604	−1.84540	−0.922702	+	0.385513i	$0.874024\pi$
$548$	11.9333	0.509767
$549$	−3.62349	−0.154647
$550$	−24.2276	−1.03307
$551$	4.09885	0.174617
$552$	−10.5593	−0.449433
$553$	−3.99841	−0.170030
$554$	10.9965	0.467198
$555$	36.4399	1.54679
$556$	−2.23541	−0.0948024
$557$	−34.3713	−1.45636	−0.728180	−	0.685386i	$-0.759633\pi$
$557$	−34.3713	−1.45636	−0.728180	+	0.685386i	$0.759633\pi$
$558$	−5.11667	−0.216606
$559$	6.67335	0.282253
$560$	2.16681	0.0915646
$561$	5.41848	0.228769
$562$	17.7242	0.747650
$563$	−25.4878	−1.07418	−0.537092	−	0.843524i	$-0.680477\pi$
$563$	−25.4878	−1.07418	−0.537092	+	0.843524i	$0.680477\pi$
$564$	−1.33266	−0.0561150
$565$	−79.7552	−3.35533
$566$	5.86706	0.246611
$567$	−0.240996	−0.0101209
$568$	31.3457	1.31524
$569$	−43.1003	−1.80686	−0.903428	−	0.428739i	$-0.858958\pi$
$569$	−43.1003	−1.80686	−0.903428	+	0.428739i	$0.858958\pi$
$570$	16.4172	0.687642
$571$	−31.5157	−1.31889	−0.659445	−	0.751753i	$-0.729209\pi$
$571$	−31.5157	−1.31889	−0.659445	+	0.751753i	$0.729209\pi$
$572$	−7.90561	−0.330550
$573$	7.15009	0.298699
$574$	0.0875967	0.00365622
$575$	−65.3689	−2.72607
$576$	−1.08186	−0.0450774
$577$	4.21286	0.175384	0.0876919	−	0.996148i	$-0.472051\pi$
$577$	4.21286	0.175384	0.0876919	+	0.996148i	$0.472051\pi$
$578$	8.16024	0.339421
$579$	−25.8197	−1.07303
$580$	4.34624	0.180468
$581$	−0.558843	−0.0231847
$582$	1.31796	0.0546312
$583$	4.07962	0.168961
$584$	18.6360	0.771163
$585$	−6.29281	−0.260176
$586$	10.4293	0.430829
$587$	27.1265	1.11963	0.559816	−	0.828617i	$-0.310872\pi$
$587$	27.1265	1.11963	0.559816	+	0.828617i	$0.310872\pi$
$588$	11.5729	0.477259
$589$	59.3151	2.44403
$590$	−12.9729	−0.534085
$591$	14.9728	0.615898
$592$	−18.1031	−0.744033
$593$	−24.0131	−0.986098	−0.493049	−	0.870001i	$-0.664118\pi$
$593$	−24.0131	−0.986098	−0.493049	+	0.870001i	$0.664118\pi$
$594$	−1.84968	−0.0758935
$595$	1.73285	0.0710397
$596$	8.18634	0.335325
$597$	2.43370	0.0996048
$598$	4.25929	0.174175
$599$	35.1778	1.43733	0.718663	−	0.695359i	$-0.244754\pi$
$599$	35.1778	1.43733	0.718663	+	0.695359i	$0.244754\pi$
$600$	27.7133	1.13139
$601$	27.3109	1.11403	0.557017	−	0.830501i	$-0.311946\pi$
$601$	27.3109	1.11403	0.557017	+	0.830501i	$0.311946\pi$
$602$	0.627306	0.0255671
$603$	15.0012	0.610898
$604$	16.1592	0.657507
$605$	3.07332	0.124948
$606$	−1.37565	−0.0558820
$607$	8.86125	0.359667	0.179834	−	0.983697i	$-0.442444\pi$
$607$	8.86125	0.359667	0.179834	+	0.983697i	$0.442444\pi$
$608$	−36.4592	−1.47861
$609$	−0.147687	−0.00598457
$610$	−8.89402	−0.360108
$611$	1.18245	0.0478366
$612$	−2.81771	−0.113899
$613$	−9.92598	−0.400907	−0.200453	−	0.979703i	$-0.564242\pi$
$613$	−9.92598	−0.400907	−0.200453	+	0.979703i	$0.564242\pi$
$614$	16.6734	0.672885
$615$	−2.68006	−0.108070
$616$	−1.63467	−0.0658628
$617$	−23.0094	−0.926325	−0.463163	−	0.886273i	$-0.653285\pi$
$617$	−23.0094	−0.926325	−0.463163	+	0.886273i	$0.653285\pi$
$618$	−0.651710	−0.0262156
$619$	0.404899	0.0162743	0.00813713	−	0.999967i	$-0.497410\pi$
$619$	0.404899	0.0162743	0.00813713	+	0.999967i	$0.497410\pi$
$620$	62.8951	2.52593
$621$	−4.99066	−0.200268
$622$	−10.8538	−0.435197
$623$	0.872601	0.0349600
$624$	3.12623	0.125149
$625$	81.0726	3.24290
$626$	−4.84291	−0.193562
$627$	21.4425	0.856330
$628$	1.66711	0.0665248
$629$	−14.4774	−0.577252
$630$	−0.591534	−0.0235673
$631$	2.09708	0.0834833	0.0417416	−	0.999128i	$-0.486709\pi$
$631$	2.09708	0.0834833	0.0417416	+	0.999128i	$0.486709\pi$
$632$	35.1038	1.39635
$633$	−17.3444	−0.689377
$634$	7.41709	0.294570
$635$	31.9626	1.26840
$636$	−2.12147	−0.0841219
$637$	−10.2685	−0.406852
$638$	−1.13352	−0.0448766
$639$	14.8150	0.586073
$640$	−49.0349	−1.93827
$641$	19.5858	0.773592	0.386796	−	0.922165i	$-0.373582\pi$
$641$	19.5858	0.773592	0.386796	+	0.922165i	$0.373582\pi$
$642$	9.71289	0.383337
$643$	−4.98027	−0.196403	−0.0982014	−	0.995167i	$-0.531309\pi$
$643$	−4.98027	−0.196403	−0.0982014	+	0.995167i	$0.531309\pi$
$644$	−2.00508	−0.0790111
$645$	−19.1927	−0.755711
$646$	−6.52250	−0.256624
$647$	−38.1011	−1.49791	−0.748954	−	0.662622i	$-0.769444\pi$
$647$	−38.1011	−1.49791	−0.748954	+	0.662622i	$0.769444\pi$
$648$	2.11581	0.0831167
$649$	−16.9438	−0.665103
$650$	−11.1787	−0.438465
$651$	−2.13720	−0.0837634
$652$	−4.98891	−0.195381
$653$	13.3209	0.521289	0.260644	−	0.965435i	$-0.416065\pi$
$653$	13.3209	0.521289	0.260644	+	0.965435i	$0.416065\pi$
$654$	3.50949	0.137232
$655$	−36.8763	−1.44087
$656$	1.33144	0.0519838
$657$	8.80799	0.343632
$658$	0.111152	0.00433315
$659$	−9.48278	−0.369397	−0.184698	−	0.982795i	$-0.559131\pi$
$659$	−9.48278	−0.369397	−0.184698	+	0.982795i	$0.559131\pi$
$660$	22.7367	0.885024
$661$	−42.1299	−1.63866	−0.819332	−	0.573320i	$-0.805655\pi$
$661$	−42.1299	−1.63866	−0.819332	+	0.573320i	$0.805655\pi$
$662$	4.06801	0.158108
$663$	2.50011	0.0970961
$664$	4.90632	0.190402
$665$	6.85736	0.265917
$666$	4.94209	0.191502
$667$	−3.05838	−0.118421
$668$	28.1735	1.09007
$669$	−16.1554	−0.624605
$670$	36.8212	1.42253
$671$	−11.6164	−0.448448
$672$	1.31367	0.0506760
$673$	30.6715	1.18230	0.591149	−	0.806562i	$-0.298674\pi$
$673$	30.6715	1.18230	0.591149	+	0.806562i	$0.298674\pi$
$674$	−9.80702	−0.377752
$675$	13.0982	0.504151
$676$	18.0247	0.693258
$677$	−26.7592	−1.02844	−0.514220	−	0.857658i	$-0.671918\pi$
$677$	−26.7592	−1.02844	−0.514220	+	0.857658i	$0.671918\pi$
$678$	−10.8167	−0.415411
$679$	0.550502	0.0211263
$680$	−15.2134	−0.583407
$681$	−7.76601	−0.297594
$682$	−16.4034	−0.628117
$683$	13.7122	0.524682	0.262341	−	0.964975i	$-0.415506\pi$
$683$	13.7122	0.524682	0.262341	+	0.964975i	$0.415506\pi$
$684$	−11.1505	−0.426349
$685$	30.4520	1.16351
$686$	−1.93858	−0.0740154
$687$	−9.78178	−0.373198
$688$	9.53480	0.363511
$689$	1.88235	0.0717118
$690$	−12.2498	−0.466342
$691$	−1.24745	−0.0474552	−0.0237276	−	0.999718i	$-0.507553\pi$
$691$	−1.24745	−0.0474552	−0.0237276	+	0.999718i	$0.507553\pi$
$692$	−28.9585	−1.10084
$693$	−0.772600	−0.0293487
$694$	−8.64265	−0.328070
$695$	−5.70442	−0.216381
$696$	1.29661	0.0491478
$697$	1.06478	0.0403313
$698$	−2.72705	−0.103220
$699$	11.8482	0.448140
$700$	5.26242	0.198901
$701$	−25.5232	−0.963997	−0.481999	−	0.876172i	$-0.660089\pi$
$701$	−25.5232	−0.963997	−0.481999	+	0.876172i	$0.660089\pi$
$702$	−0.853451	−0.0322114
$703$	−57.2913	−2.16078
$704$	−3.46829	−0.130716
$705$	−3.40074	−0.128079
$706$	−11.9054	−0.448067
$707$	−0.574600	−0.0216101
$708$	8.81108	0.331141
$709$	13.4476	0.505035	0.252518	−	0.967592i	$-0.418741\pi$
$709$	13.4476	0.505035	0.252518	+	0.967592i	$0.418741\pi$
$710$	36.3641	1.36472
$711$	16.5912	0.622219
$712$	−7.66094	−0.287106
$713$	−44.2582	−1.65748
$714$	0.235014	0.00879518
$715$	−20.1739	−0.754461
$716$	−31.3434	−1.17136
$717$	2.59130	0.0967737
$718$	2.78766	0.104034
$719$	22.9506	0.855912	0.427956	−	0.903800i	$-0.359234\pi$
$719$	22.9506	0.855912	0.427956	+	0.903800i	$0.359234\pi$
$720$	−8.99109	−0.335078
$721$	−0.272215	−0.0101378
$722$	−14.8490	−0.552622
$723$	18.1978	0.676785
$724$	−30.8358	−1.14600
$725$	8.02685	0.298110
$726$	0.416814	0.0154694
$727$	6.69101	0.248156	0.124078	−	0.992272i	$-0.460403\pi$
$727$	6.69101	0.248156	0.124078	+	0.992272i	$0.460403\pi$
$728$	−0.754243	−0.0279541
$729$	1.00000	0.0370370
$730$	21.6196	0.800176
$731$	7.62518	0.282027
$732$	6.04075	0.223273
$733$	−1.84344	−0.0680891	−0.0340446	−	0.999420i	$-0.510839\pi$
$733$	−1.84344	−0.0680891	−0.0340446	+	0.999420i	$0.510839\pi$
$734$	−17.5346	−0.647215
$735$	29.5323	1.08932
$736$	27.2042	1.00276
$737$	48.0920	1.77149
$738$	−0.363478	−0.0133798
$739$	43.9268	1.61588	0.807938	−	0.589268i	$-0.200584\pi$
$739$	43.9268	1.61588	0.807938	+	0.589268i	$0.200584\pi$
$740$	−60.7492	−2.23318
$741$	9.89364	0.363452
$742$	0.176944	0.00649582
$743$	−21.7989	−0.799722	−0.399861	−	0.916576i	$-0.630942\pi$
$743$	−21.7989	−0.799722	−0.399861	+	0.916576i	$0.630942\pi$
$744$	18.7634	0.687899
$745$	20.8903	0.765360
$746$	2.58467	0.0946316
$747$	2.31889	0.0848438
$748$	−9.03319	−0.330286
$749$	4.05701	0.148240
$750$	19.8775	0.725822
$751$	12.3609	0.451054	0.225527	−	0.974237i	$-0.427590\pi$
$751$	12.3609	0.451054	0.225527	+	0.974237i	$0.427590\pi$
$752$	1.68946	0.0616084
$753$	−16.3733	−0.596676
$754$	−0.523011	−0.0190469
$755$	41.2357	1.50072
$756$	0.401766	0.0146121
$757$	11.3778	0.413532	0.206766	−	0.978390i	$-0.433706\pi$
$757$	11.3778	0.413532	0.206766	+	0.978390i	$0.433706\pi$
$758$	16.2900	0.591678
$759$	−15.9994	−0.580742
$760$	−60.2038	−2.18382
$761$	4.16612	0.151022	0.0755109	−	0.997145i	$-0.475941\pi$
$761$	4.16612	0.151022	0.0755109	+	0.997145i	$0.475941\pi$
$762$	4.33488	0.157036
$763$	1.46589	0.0530687
$764$	−11.9200	−0.431249
$765$	−7.19036	−0.259968
$766$	5.25911	0.190020
$767$	−7.81793	−0.282289
$768$	−4.48655	−0.161895
$769$	26.0257	0.938512	0.469256	−	0.883062i	$-0.344522\pi$
$769$	26.0257	0.938512	0.469256	+	0.883062i	$0.344522\pi$
$770$	−1.89638	−0.0683408
$771$	3.97345	0.143100
$772$	43.0441	1.54919
$773$	−27.3737	−0.984564	−0.492282	−	0.870436i	$-0.663837\pi$
$773$	−27.3737	−0.984564	−0.492282	+	0.870436i	$0.663837\pi$
$774$	−2.60297	−0.0935620
$775$	116.158	4.17251
$776$	−4.83310	−0.173498
$777$	2.06428	0.0740556
$778$	13.5014	0.484050
$779$	4.21362	0.150969
$780$	10.4908	0.375630
$781$	47.4950	1.69950
$782$	4.86679	0.174036
$783$	0.612819	0.0219004
$784$	−14.6715	−0.523981
$785$	4.25420	0.151839
$786$	−5.00128	−0.178390
$787$	24.2951	0.866026	0.433013	−	0.901388i	$-0.357450\pi$
$787$	24.2951	0.866026	0.433013	+	0.901388i	$0.357450\pi$
$788$	−24.9612	−0.889207
$789$	11.0005	0.391627
$790$	40.7238	1.44889
$791$	−4.51804	−0.160643
$792$	6.78299	0.241023
$793$	−5.35986	−0.190334
$794$	−19.6132	−0.696045
$795$	−5.41368	−0.192003
$796$	−4.05724	−0.143805
$797$	−16.5668	−0.586827	−0.293414	−	0.955986i	$-0.594791\pi$
$797$	−16.5668	−0.586827	−0.293414	+	0.955986i	$0.594791\pi$
$798$	0.930018	0.0329223
$799$	1.35110	0.0477984
$800$	−71.3986	−2.52432
$801$	−3.62082	−0.127935
$802$	−17.7945	−0.628345
$803$	28.2372	0.996470
$804$	−25.0087	−0.881988
$805$	−5.11665	−0.180338
$806$	−7.56857	−0.266592
$807$	19.7930	0.696746
$808$	5.04466	0.177471
$809$	38.5972	1.35701	0.678504	−	0.734597i	$-0.262629\pi$
$809$	38.5972	1.35701	0.678504	+	0.734597i	$0.262629\pi$
$810$	2.45454	0.0862438
$811$	31.4546	1.10452	0.552260	−	0.833672i	$-0.313766\pi$
$811$	31.4546	1.10452	0.552260	+	0.833672i	$0.313766\pi$
$812$	0.246210	0.00864027
$813$	11.6000	0.406830
$814$	15.8437	0.555321
$815$	−12.7309	−0.445945
$816$	3.57212	0.125049
$817$	30.1750	1.05569
$818$	−19.1051	−0.667995
$819$	−0.356480	−0.0124564
$820$	4.46795	0.156027
$821$	−12.2153	−0.426318	−0.213159	−	0.977018i	$-0.568375\pi$
$821$	−12.2153	−0.426318	−0.213159	+	0.977018i	$0.568375\pi$
$822$	4.13000	0.144050
$823$	−18.3462	−0.639508	−0.319754	−	0.947501i	$-0.603600\pi$
$823$	−18.3462	−0.639508	−0.319754	+	0.947501i	$0.603600\pi$
$824$	2.38989	0.0832558
$825$	41.9912	1.46195
$826$	−0.734899	−0.0255704
$827$	18.1586	0.631437	0.315719	−	0.948853i	$-0.397754\pi$
$827$	18.1586	0.631437	0.315719	+	0.948853i	$0.397754\pi$
$828$	8.31997	0.289139
$829$	0.950272	0.0330043	0.0165022	−	0.999864i	$-0.494747\pi$
$829$	0.950272	0.0330043	0.0165022	+	0.999864i	$0.494747\pi$
$830$	5.69181	0.197566
$831$	−19.0592	−0.661155
$832$	−1.60028	−0.0554798
$833$	−11.7331	−0.406527
$834$	−0.773651	−0.0267894
$835$	71.8946	2.48801
$836$	−35.7469	−1.23633
$837$	8.86820	0.306530
$838$	5.05688	0.174687
$839$	−16.6297	−0.574121	−0.287060	−	0.957912i	$-0.592678\pi$
$839$	−16.6297	−0.574121	−0.287060	+	0.957912i	$0.592678\pi$
$840$	2.16922	0.0748452
$841$	−28.6245	−0.987050
$842$	0.516506	0.0178000
$843$	−30.7195	−1.05804
$844$	28.9149	0.995293
$845$	45.9963	1.58232
$846$	−0.461219	−0.0158570
$847$	0.174100	0.00598215
$848$	2.68948	0.0923571
$849$	−10.1688	−0.348991
$850$	−12.7731	−0.438115
$851$	42.7481	1.46539
$852$	−24.6982	−0.846147
$853$	−0.570344	−0.0195282	−0.00976411	−	0.999952i	$-0.503108\pi$
$853$	−0.570344	−0.0195282	−0.00976411	+	0.999952i	$0.503108\pi$
$854$	−0.503836	−0.0172409
$855$	−28.4543	−0.973116
$856$	−35.6182	−1.21741
$857$	4.52093	0.154432	0.0772160	−	0.997014i	$-0.475397\pi$
$857$	4.52093	0.154432	0.0772160	+	0.997014i	$0.475397\pi$
$858$	−2.73605	−0.0934072
$859$	−1.03793	−0.0354136	−0.0177068	−	0.999843i	$-0.505637\pi$
$859$	−1.03793	−0.0354136	−0.0177068	+	0.999843i	$0.505637\pi$
$860$	31.9963	1.09106
$861$	−0.151822	−0.00517409
$862$	−12.5271	−0.426674
$863$	−24.3909	−0.830275	−0.415137	−	0.909759i	$-0.636266\pi$
$863$	−24.3909	−0.830275	−0.415137	+	0.909759i	$0.636266\pi$
$864$	−5.45101	−0.185447
$865$	−73.8976	−2.51259
$866$	−1.31310	−0.0446210
$867$	−14.1433	−0.480332
$868$	3.56294	0.120934
$869$	53.1892	1.80432
$870$	1.50419	0.0509968
$871$	22.1898	0.751873
$872$	−12.8697	−0.435822
$873$	−2.28428	−0.0773112
$874$	19.2593	0.651455
$875$	8.30267	0.280682
$876$	−14.6839	−0.496122
$877$	−10.0555	−0.339552	−0.169776	−	0.985483i	$-0.554304\pi$
$877$	−10.0555	−0.339552	−0.169776	+	0.985483i	$0.554304\pi$
$878$	−18.2748	−0.616746
$879$	−18.0760	−0.609688
$880$	−28.8242	−0.971665
$881$	−26.9703	−0.908653	−0.454327	−	0.890835i	$-0.650120\pi$
$881$	−26.9703	−0.908653	−0.454327	+	0.890835i	$0.650120\pi$
$882$	4.00527	0.134864
$883$	−7.84527	−0.264014	−0.132007	−	0.991249i	$-0.542142\pi$
$883$	−7.84527	−0.264014	−0.132007	+	0.991249i	$0.542142\pi$
$884$	−4.16795	−0.140183
$885$	22.4845	0.755809
$886$	14.3125	0.480839
$887$	−21.1025	−0.708553	−0.354276	−	0.935141i	$-0.615273\pi$
$887$	−21.1025	−0.708553	−0.354276	+	0.935141i	$0.615273\pi$
$888$	−18.1232	−0.608174
$889$	1.81065	0.0607272
$890$	−8.88744	−0.297908
$891$	3.20587	0.107401
$892$	26.9328	0.901777
$893$	5.34668	0.178920
$894$	2.83321	0.0947566
$895$	−79.9836	−2.67356
$896$	−2.77777	−0.0927988
$897$	−7.38218	−0.246484
$898$	−6.16061	−0.205582
$899$	5.43460	0.181254
$900$	−21.8362	−0.727872
$901$	2.15083	0.0716546
$902$	−1.16526	−0.0387990
$903$	−1.08724	−0.0361812
$904$	39.6659	1.31927
$905$	−78.6883	−2.61569
$906$	5.59252	0.185799
$907$	−0.113586	−0.00377157	−0.00188579	−	0.999998i	$-0.500600\pi$
$907$	−0.113586	−0.00377157	−0.00188579	+	0.999998i	$0.500600\pi$
$908$	12.9468	0.429654
$909$	2.38427	0.0790814
$910$	−0.874996	−0.0290058
$911$	26.3479	0.872945	0.436472	−	0.899718i	$-0.356228\pi$
$911$	26.3479	0.872945	0.436472	+	0.899718i	$0.356228\pi$
$912$	14.1359	0.468087
$913$	7.43405	0.246031
$914$	9.50467	0.314386
$915$	15.4151	0.509607
$916$	16.3073	0.538808
$917$	−2.08900	−0.0689848
$918$	−0.975179	−0.0321857
$919$	−23.1980	−0.765232	−0.382616	−	0.923907i	$-0.624977\pi$
$919$	−23.1980	−0.765232	−0.382616	+	0.923907i	$0.624977\pi$
$920$	44.9213	1.48101
$921$	−28.8983	−0.952232
$922$	1.55534	0.0512225
$923$	21.9144	0.721320
$924$	1.28801	0.0423723
$925$	−112.194	−3.68893
$926$	15.8719	0.521581
$927$	1.12954	0.0370990
$928$	−3.34049	−0.109657
$929$	−53.8074	−1.76536	−0.882682	−	0.469971i	$-0.844264\pi$
$929$	−53.8074	−1.76536	−0.882682	+	0.469971i	$0.844264\pi$
$930$	21.7674	0.713780
$931$	−46.4311	−1.52172
$932$	−19.7522	−0.647005
$933$	18.8117	0.615868
$934$	8.97626	0.293712
$935$	−23.0513	−0.753859
$936$	3.12970	0.102297
$937$	22.8247	0.745649	0.372825	−	0.927902i	$-0.378389\pi$
$937$	22.8247	0.745649	0.372825	+	0.927902i	$0.378389\pi$
$938$	2.08588	0.0681063
$939$	8.39372	0.273919
$940$	5.66939	0.184915
$941$	32.0113	1.04354	0.521769	−	0.853087i	$-0.325272\pi$
$941$	32.0113	1.04354	0.521769	+	0.853087i	$0.325272\pi$
$942$	0.576969	0.0187987
$943$	−3.14401	−0.102383
$944$	−11.1702	−0.363558
$945$	1.02524	0.0333512
$946$	−8.34479	−0.271312
$947$	11.1494	0.362307	0.181153	−	0.983455i	$-0.442017\pi$
$947$	11.1494	0.362307	0.181153	+	0.983455i	$0.442017\pi$
$948$	−27.6593	−0.898333
$949$	13.0288	0.422931
$950$	−50.5469	−1.63996
$951$	−12.8553	−0.416861
$952$	−0.861822	−0.0279318
$953$	46.6378	1.51075	0.755374	−	0.655294i	$-0.227455\pi$
$953$	46.6378	1.51075	0.755374	+	0.655294i	$0.227455\pi$
$954$	−0.734220	−0.0237713
$955$	−30.4179	−0.984301
$956$	−4.31997	−0.139718
$957$	1.96462	0.0635071
$958$	19.5218	0.630721
$959$	1.72507	0.0557055
$960$	4.60244	0.148543
$961$	47.6449	1.53693
$962$	7.31033	0.235695
$963$	−16.8343	−0.542479
$964$	−30.3378	−0.977113
$965$	109.842	3.53594
$966$	−0.693937	−0.0223271
$967$	31.4584	1.01163	0.505817	−	0.862641i	$-0.331191\pi$
$967$	31.4584	1.01163	0.505817	+	0.862641i	$0.331191\pi$
$968$	−1.52850	−0.0491279
$969$	11.3048	0.363162
$970$	−5.60687	−0.180026
$971$	10.0664	0.323045	0.161523	−	0.986869i	$-0.448360\pi$
$971$	10.0664	0.323045	0.161523	+	0.986869i	$0.448360\pi$
$972$	−1.66711	−0.0534725
$973$	−0.323149	−0.0103597
$974$	18.0112	0.577117
$975$	19.3749	0.620493
$976$	−7.65811	−0.245130
$977$	−46.0718	−1.47397	−0.736983	−	0.675911i	$-0.763750\pi$
$977$	−46.0718	−1.47397	−0.736983	+	0.675911i	$0.763750\pi$
$978$	−1.72661	−0.0552110
$979$	−11.6079	−0.370989
$980$	−49.2336	−1.57271
$981$	−6.08263	−0.194204
$982$	−2.57426	−0.0821478
$983$	−28.2419	−0.900776	−0.450388	−	0.892833i	$-0.648714\pi$
$983$	−28.2419	−0.900776	−0.450388	+	0.892833i	$0.648714\pi$
$984$	1.33291	0.0424917
$985$	−63.6973	−2.02956
$986$	−0.597609	−0.0190317
$987$	−0.192648	−0.00613205
$988$	−16.4938	−0.524736
$989$	−22.5152	−0.715942
$990$	7.86893	0.250091
$991$	−35.3372	−1.12252	−0.561262	−	0.827638i	$-0.689684\pi$
$991$	−35.3372	−1.12252	−0.561262	+	0.827638i	$0.689684\pi$
$992$	−48.3407	−1.53482
$993$	−7.05067	−0.223746
$994$	2.05998	0.0653387
$995$	−10.3535	−0.328227
$996$	−3.86584	−0.122494
$997$	38.7805	1.22819	0.614096	−	0.789231i	$-0.289521\pi$
$997$	38.7805	1.22819	0.614096	+	0.789231i	$0.289521\pi$
$998$	5.57724	0.176544
$999$	−8.56562	−0.271004

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.a.e.1.5	✓	12
3.2	odd	2		1413.2.a.h.1.8		12
4.3	odd	2		7536.2.a.bm.1.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.a.e.1.5	✓	12	1.1	even	1	trivial
1413.2.a.h.1.8		12	3.2	odd	2
7536.2.a.bm.1.1		12	4.3	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:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	471.a (trivial)

\(f(q)\)	\(=\)	\(q-0.576969 q^{2} +1.00000 q^{3} -1.66711 q^{4} -4.25420 q^{5} -0.576969 q^{6} -0.240996 q^{7} +2.11581 q^{8} +1.00000 q^{9} +2.45454 q^{10} +3.20587 q^{11} -1.66711 q^{12} +1.47920 q^{13} +0.139047 q^{14} -4.25420 q^{15} +2.11346 q^{16} +1.69018 q^{17} -0.576969 q^{18} +6.68851 q^{19} +7.09221 q^{20} -0.240996 q^{21} -1.84968 q^{22} -4.99066 q^{23} +2.11581 q^{24} +13.0982 q^{25} -0.853451 q^{26} +1.00000 q^{27} +0.401766 q^{28} +0.612819 q^{29} +2.45454 q^{30} +8.86820 q^{31} -5.45101 q^{32} +3.20587 q^{33} -0.975179 q^{34} +1.02524 q^{35} -1.66711 q^{36} -8.56562 q^{37} -3.85906 q^{38} +1.47920 q^{39} -9.00107 q^{40} +0.629979 q^{41} +0.139047 q^{42} +4.51147 q^{43} -5.34452 q^{44} -4.25420 q^{45} +2.87946 q^{46} +0.799383 q^{47} +2.11346 q^{48} -6.94192 q^{49} -7.55727 q^{50} +1.69018 q^{51} -2.46598 q^{52} +1.27255 q^{53} -0.576969 q^{54} -13.6384 q^{55} -0.509900 q^{56} +6.68851 q^{57} -0.353578 q^{58} -5.28525 q^{59} +7.09221 q^{60} -3.62349 q^{61} -5.11667 q^{62} -0.240996 q^{63} -1.08186 q^{64} -6.29281 q^{65} -1.84968 q^{66} +15.0012 q^{67} -2.81771 q^{68} -4.99066 q^{69} -0.591534 q^{70} +14.8150 q^{71} +2.11581 q^{72} +8.80799 q^{73} +4.94209 q^{74} +13.0982 q^{75} -11.1505 q^{76} -0.772600 q^{77} -0.853451 q^{78} +16.5912 q^{79} -8.99109 q^{80} +1.00000 q^{81} -0.363478 q^{82} +2.31889 q^{83} +0.401766 q^{84} -7.19036 q^{85} -2.60297 q^{86} +0.612819 q^{87} +6.78299 q^{88} -3.62082 q^{89} +2.45454 q^{90} -0.356480 q^{91} +8.31997 q^{92} +8.86820 q^{93} -0.461219 q^{94} -28.4543 q^{95} -5.45101 q^{96} -2.28428 q^{97} +4.00527 q^{98} +3.20587 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + 12 q^{3} + 17 q^{4} + 4 q^{5} - q^{6} + 8 q^{7} - 6 q^{8} + 12 q^{9} + 9 q^{10} - q^{11} + 17 q^{12} + 15 q^{13} - 7 q^{14} + 4 q^{15} + 19 q^{16} + 10 q^{17} - q^{18} + 14 q^{19} - 11 q^{20}+ \cdots - q^{99}+O(q^{100}) \) 12 * q - q^2 + 12 * q^3 + 17 * q^4 + 4 * q^5 - q^6 + 8 * q^7 - 6 * q^8 + 12 * q^9 + 9 * q^10 - q^11 + 17 * q^12 + 15 * q^13 - 7 * q^14 + 4 * q^15 + 19 * q^16 + 10 * q^17 - q^18 + 14 * q^19 - 11 * q^20 + 8 * q^21 - 9 * q^22 - 8 * q^23 - 6 * q^24 + 22 * q^25 - 17 * q^26 + 12 * q^27 + 7 * q^28 + 5 * q^29 + 9 * q^30 + 5 * q^31 - 33 * q^32 - q^33 - 13 * q^34 - 14 * q^35 + 17 * q^36 + 4 * q^37 - 32 * q^38 + 15 * q^39 - 4 * q^40 + 18 * q^41 - 7 * q^42 + 28 * q^43 - 20 * q^44 + 4 * q^45 - 15 * q^46 - 3 * q^47 + 19 * q^48 + 48 * q^49 - 2 * q^50 + 10 * q^51 + 26 * q^52 - 18 * q^53 - q^54 - 5 * q^55 - 10 * q^56 + 14 * q^57 - 6 * q^58 - 11 * q^60 + 20 * q^61 + q^62 + 8 * q^63 - 14 * q^64 - 11 * q^65 - 9 * q^66 + 27 * q^67 - 21 * q^68 - 8 * q^69 - 29 * q^70 + 7 * q^71 - 6 * q^72 + 15 * q^73 - 14 * q^74 + 22 * q^75 + 30 * q^76 - 23 * q^77 - 17 * q^78 + 4 * q^79 - 31 * q^80 + 12 * q^81 - 31 * q^82 - 13 * q^83 + 7 * q^84 + 8 * q^85 - 26 * q^86 + 5 * q^87 - 11 * q^88 - 9 * q^89 + 9 * q^90 - 31 * q^91 - 21 * q^92 + 5 * q^93 - 4 * q^94 - 44 * q^95 - 33 * q^96 - 16 * q^97 - 54 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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