LMFDB - Embedded newform 403.2.a.e.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [403,2,Mod(1,403)] 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("403.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(403, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 403 = 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	403.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.21797120146$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 10x^{5} + 37x^{4} - 33x^{3} - 36x^{2} + 33x - 4 $ x^8 - x^7 - 11x^6 + 10x^5 + 37x^4 - 33x^3 - 36x^2 + 33x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.53950$ of defining polynomial
Character	$\chi$	$=$	403.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.53950 q^{2} +0.734778 q^{3} +0.370072 q^{4} +3.72666 q^{5} -1.13119 q^{6} +3.41650 q^{7} +2.50928 q^{8} -2.46010 q^{9} -5.73720 q^{10} +1.47104 q^{11} +0.271920 q^{12} +1.00000 q^{13} -5.25971 q^{14} +2.73826 q^{15} -4.60319 q^{16} -5.48536 q^{17} +3.78734 q^{18} +1.38091 q^{19} +1.37913 q^{20} +2.51036 q^{21} -2.26467 q^{22} -2.95176 q^{23} +1.84376 q^{24} +8.88797 q^{25} -1.53950 q^{26} -4.01196 q^{27} +1.26435 q^{28} +9.19281 q^{29} -4.21557 q^{30} -1.00000 q^{31} +2.06807 q^{32} +1.08089 q^{33} +8.44473 q^{34} +12.7321 q^{35} -0.910415 q^{36} -6.83535 q^{37} -2.12591 q^{38} +0.734778 q^{39} +9.35123 q^{40} +2.21323 q^{41} -3.86472 q^{42} -0.260364 q^{43} +0.544391 q^{44} -9.16795 q^{45} +4.54424 q^{46} +9.41355 q^{47} -3.38232 q^{48} +4.67245 q^{49} -13.6831 q^{50} -4.03052 q^{51} +0.370072 q^{52} +3.28201 q^{53} +6.17643 q^{54} +5.48207 q^{55} +8.57295 q^{56} +1.01466 q^{57} -14.1524 q^{58} +4.52700 q^{59} +1.01335 q^{60} -13.2347 q^{61} +1.53950 q^{62} -8.40493 q^{63} +6.02258 q^{64} +3.72666 q^{65} -1.66403 q^{66} -11.3154 q^{67} -2.02998 q^{68} -2.16889 q^{69} -19.6011 q^{70} -7.02292 q^{71} -6.17309 q^{72} +14.4428 q^{73} +10.5231 q^{74} +6.53068 q^{75} +0.511034 q^{76} +5.02581 q^{77} -1.13119 q^{78} -8.44263 q^{79} -17.1545 q^{80} +4.43241 q^{81} -3.40727 q^{82} -2.77489 q^{83} +0.929015 q^{84} -20.4421 q^{85} +0.400831 q^{86} +6.75467 q^{87} +3.69126 q^{88} +1.47710 q^{89} +14.1141 q^{90} +3.41650 q^{91} -1.09236 q^{92} -0.734778 q^{93} -14.4922 q^{94} +5.14616 q^{95} +1.51957 q^{96} +2.61380 q^{97} -7.19325 q^{98} -3.61891 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 7 q^{3} + 7 q^{4} + 11 q^{5} - 2 q^{7} + 9 q^{9} - 2 q^{10} - 2 q^{11} + 19 q^{12} + 8 q^{13} + 3 q^{14} + 6 q^{15} - 7 q^{16} + 7 q^{17} - 12 q^{18} - 5 q^{19} + 6 q^{20} + 8 q^{21} - 4 q^{22}+ \cdots - 8 q^{99}+O(q^{100}) $ 8 * q - q^2 + 7 * q^3 + 7 * q^4 + 11 * q^5 - 2 * q^7 + 9 * q^9 - 2 * q^10 - 2 * q^11 + 19 * q^12 + 8 * q^13 + 3 * q^14 + 6 * q^15 - 7 * q^16 + 7 * q^17 - 12 * q^18 - 5 * q^19 + 6 * q^20 + 8 * q^21 - 4 * q^22 + 14 * q^23 - 17 * q^24 + 17 * q^25 - q^26 + 7 * q^27 - 9 * q^28 + 12 * q^29 - 9 * q^30 - 8 * q^31 - 21 * q^32 + 10 * q^33 - 12 * q^34 - q^35 + 11 * q^36 + 2 * q^37 + 24 * q^38 + 7 * q^39 - 19 * q^40 + 13 * q^41 - 27 * q^42 - 5 * q^43 + 22 * q^44 + 19 * q^45 + 17 * q^46 + 23 * q^47 + 3 * q^48 + 26 * q^49 - 26 * q^50 + 18 * q^51 + 7 * q^52 + 25 * q^53 - 36 * q^54 - 17 * q^55 + 8 * q^56 - 35 * q^57 - 29 * q^58 - 5 * q^59 + 71 * q^60 - 9 * q^61 + q^62 - 37 * q^63 - 14 * q^64 + 11 * q^65 - 41 * q^66 + 22 * q^67 - 6 * q^68 - 7 * q^69 - 29 * q^70 - 17 * q^71 - 34 * q^72 - 27 * q^73 + 14 * q^74 - 33 * q^75 - 36 * q^76 + 31 * q^77 - 23 * q^79 + 9 * q^80 - 12 * q^81 + 18 * q^82 - 25 * q^83 - 62 * q^84 + 13 * q^85 + 11 * q^86 + 26 * q^87 + 5 * q^88 + 2 * q^89 - 14 * q^90 - 2 * q^91 - 20 * q^92 - 7 * q^93 - 38 * q^94 + 3 * q^95 - 52 * q^96 - 15 * q^97 + 39 * q^98 - 8 * 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 $

$2$	−1.53950	−1.08859	−0.544297	−	0.838893i	$-0.683203\pi$
$2$	−1.53950	−1.08859	−0.544297	+	0.838893i	$0.683203\pi$
$3$	0.734778	0.424224	0.212112	−	0.977245i	$-0.431966\pi$
$3$	0.734778	0.424224	0.212112	+	0.977245i	$0.431966\pi$
$4$	0.370072	0.185036
$5$	3.72666	1.66661	0.833306	−	0.552812i	$-0.186445\pi$
$5$	3.72666	1.66661	0.833306	+	0.552812i	$0.186445\pi$
$6$	−1.13119	−0.461808
$7$	3.41650	1.29131	0.645657	−	0.763627i	$-0.276584\pi$
$7$	3.41650	1.29131	0.645657	+	0.763627i	$0.276584\pi$
$8$	2.50928	0.887165
$9$	−2.46010	−0.820034
$10$	−5.73720	−1.81426
$11$	1.47104	0.443536	0.221768	−	0.975099i	$-0.428817\pi$
$11$	1.47104	0.443536	0.221768	+	0.975099i	$0.428817\pi$
$12$	0.271920	0.0784967
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−5.25971	−1.40572
$15$	2.73826	0.707017
$16$	−4.60319	−1.15080
$17$	−5.48536	−1.33040	−0.665198	−	0.746667i	$-0.731653\pi$
$17$	−5.48536	−1.33040	−0.665198	+	0.746667i	$0.731653\pi$
$18$	3.78734	0.892684
$19$	1.38091	0.316802	0.158401	−	0.987375i	$-0.449366\pi$
$19$	1.38091	0.316802	0.158401	+	0.987375i	$0.449366\pi$
$20$	1.37913	0.308383
$21$	2.51036	0.547807
$22$	−2.26467	−0.482830
$23$	−2.95176	−0.615484	−0.307742	−	0.951470i	$-0.599573\pi$
$23$	−2.95176	−0.615484	−0.307742	+	0.951470i	$0.599573\pi$
$24$	1.84376	0.376357
$25$	8.88797	1.77759
$26$	−1.53950	−0.301922
$27$	−4.01196	−0.772102
$28$	1.26435	0.238940
$29$	9.19281	1.70706	0.853531	−	0.521042i	$-0.174457\pi$
$29$	9.19281	1.70706	0.853531	+	0.521042i	$0.174457\pi$
$30$	−4.21557	−0.769654
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	2.06807	0.365586
$33$	1.08089	0.188158
$34$	8.44473	1.44826
$35$	12.7321	2.15212
$36$	−0.910415	−0.151736
$37$	−6.83535	−1.12373	−0.561863	−	0.827231i	$-0.689915\pi$
$37$	−6.83535	−1.12373	−0.561863	+	0.827231i	$0.689915\pi$
$38$	−2.12591	−0.344868
$39$	0.734778	0.117659
$40$	9.35123	1.47856
$41$	2.21323	0.345648	0.172824	−	0.984953i	$-0.444711\pi$
$41$	2.21323	0.345648	0.172824	+	0.984953i	$0.444711\pi$
$42$	−3.86472	−0.596339
$43$	−0.260364	−0.0397051	−0.0198525	−	0.999803i	$-0.506320\pi$
$43$	−0.260364	−0.0397051	−0.0198525	+	0.999803i	$0.506320\pi$
$44$	0.544391	0.0820700
$45$	−9.16795	−1.36668
$46$	4.54424	0.670012
$47$	9.41355	1.37311	0.686554	−	0.727079i	$-0.259123\pi$
$47$	9.41355	1.37311	0.686554	+	0.727079i	$0.259123\pi$
$48$	−3.38232	−0.488196
$49$	4.67245	0.667493
$50$	−13.6831	−1.93508
$51$	−4.03052	−0.564386
$52$	0.370072	0.0513197
$53$	3.28201	0.450818	0.225409	−	0.974264i	$-0.427628\pi$
$53$	3.28201	0.450818	0.225409	+	0.974264i	$0.427628\pi$
$54$	6.17643	0.840505
$55$	5.48207	0.739202
$56$	8.57295	1.14561
$57$	1.01466	0.134395
$58$	−14.1524	−1.85830
$59$	4.52700	0.589366	0.294683	−	0.955595i	$-0.404786\pi$
$59$	4.52700	0.589366	0.294683	+	0.955595i	$0.404786\pi$
$60$	1.01335	0.130823
$61$	−13.2347	−1.69453	−0.847263	−	0.531174i	$-0.821751\pi$
$61$	−13.2347	−1.69453	−0.847263	+	0.531174i	$0.821751\pi$
$62$	1.53950	0.195517
$63$	−8.40493	−1.05892
$64$	6.02258	0.752823
$65$	3.72666	0.462235
$66$	−1.66403	−0.204828
$67$	−11.3154	−1.38240	−0.691200	−	0.722664i	$-0.742917\pi$
$67$	−11.3154	−1.38240	−0.691200	+	0.722664i	$0.742917\pi$
$68$	−2.02998	−0.246171
$69$	−2.16889	−0.261103
$70$	−19.6011	−2.34278
$71$	−7.02292	−0.833467	−0.416734	−	0.909029i	$-0.636825\pi$
$71$	−7.02292	−0.833467	−0.416734	+	0.909029i	$0.636825\pi$
$72$	−6.17309	−0.727505
$73$	14.4428	1.69040	0.845200	−	0.534451i	$-0.179481\pi$
$73$	14.4428	1.69040	0.845200	+	0.534451i	$0.179481\pi$
$74$	10.5231	1.22328
$75$	6.53068	0.754098
$76$	0.511034	0.0586197
$77$	5.02581	0.572744
$78$	−1.13119	−0.128082
$79$	−8.44263	−0.949870	−0.474935	−	0.880021i	$-0.657529\pi$
$79$	−8.44263	−0.949870	−0.474935	+	0.880021i	$0.657529\pi$
$80$	−17.1545	−1.91793
$81$	4.43241	0.492490
$82$	−3.40727	−0.376270
$83$	−2.77489	−0.304584	−0.152292	−	0.988336i	$-0.548665\pi$
$83$	−2.77489	−0.304584	−0.152292	+	0.988336i	$0.548665\pi$
$84$	0.929015	0.101364
$85$	−20.4421	−2.21725
$86$	0.400831	0.0432227
$87$	6.75467	0.724177
$88$	3.69126	0.393489
$89$	1.47710	0.156572	0.0782860	−	0.996931i	$-0.475055\pi$
$89$	1.47710	0.156572	0.0782860	+	0.996931i	$0.475055\pi$
$90$	14.1141	1.48776
$91$	3.41650	0.358146
$92$	−1.09236	−0.113887
$93$	−0.734778	−0.0761929
$94$	−14.4922	−1.49476
$95$	5.14616	0.527985
$96$	1.51957	0.155090
$97$	2.61380	0.265391	0.132695	−	0.991157i	$-0.457637\pi$
$97$	2.61380	0.265391	0.132695	+	0.991157i	$0.457637\pi$
$98$	−7.19325	−0.726628
$99$	−3.61891	−0.363714
$100$	3.28919	0.328919
$101$	−0.124571	−0.0123952	−0.00619761	−	0.999981i	$-0.501973\pi$
$101$	−0.124571	−0.0123952	−0.00619761	+	0.999981i	$0.501973\pi$
$102$	6.20500	0.614387
$103$	−11.6410	−1.14702	−0.573512	−	0.819197i	$-0.694419\pi$
$103$	−11.6410	−1.14702	−0.573512	+	0.819197i	$0.694419\pi$
$104$	2.50928	0.246055
$105$	9.35527	0.912981
$106$	−5.05266	−0.490758
$107$	1.92303	0.185906	0.0929530	−	0.995670i	$-0.470369\pi$
$107$	1.92303	0.185906	0.0929530	+	0.995670i	$0.470369\pi$
$108$	−1.48471	−0.142867
$109$	−16.2199	−1.55358	−0.776792	−	0.629757i	$-0.783154\pi$
$109$	−16.2199	−1.55358	−0.776792	+	0.629757i	$0.783154\pi$
$110$	−8.43966	−0.804690
$111$	−5.02246	−0.476711
$112$	−15.7268	−1.48604
$113$	17.8881	1.68277	0.841386	−	0.540434i	$-0.181740\pi$
$113$	17.8881	1.68277	0.841386	+	0.540434i	$0.181740\pi$
$114$	−1.56207	−0.146301
$115$	−11.0002	−1.02577
$116$	3.40200	0.315868
$117$	−2.46010	−0.227437
$118$	−6.96934	−0.641579
$119$	−18.7407	−1.71796
$120$	6.87107	0.627240
$121$	−8.83604	−0.803276
$122$	20.3748	1.84465
$123$	1.62623	0.146632
$124$	−0.370072	−0.0332334
$125$	14.4891	1.29595
$126$	12.9394	1.15274
$127$	−13.6365	−1.21004	−0.605020	−	0.796210i	$-0.706835\pi$
$127$	−13.6365	−1.21004	−0.605020	+	0.796210i	$0.706835\pi$
$128$	−13.4079	−1.18510
$129$	−0.191309	−0.0168438
$130$	−5.73720	−0.503186
$131$	−18.1970	−1.58988	−0.794939	−	0.606690i	$-0.792497\pi$
$131$	−18.1970	−1.58988	−0.794939	+	0.606690i	$0.792497\pi$
$132$	0.400006	0.0348161
$133$	4.71786	0.409090
$134$	17.4201	1.50487
$135$	−14.9512	−1.28679
$136$	−13.7643	−1.18028
$137$	−8.55181	−0.730630	−0.365315	−	0.930884i	$-0.619039\pi$
$137$	−8.55181	−0.730630	−0.365315	+	0.930884i	$0.619039\pi$
$138$	3.33901	0.284235
$139$	−14.5561	−1.23464	−0.617318	−	0.786714i	$-0.711781\pi$
$139$	−14.5561	−1.23464	−0.617318	+	0.786714i	$0.711781\pi$
$140$	4.71180	0.398219
$141$	6.91687	0.582505
$142$	10.8118	0.907307
$143$	1.47104	0.123015
$144$	11.3243	0.943693
$145$	34.2584	2.84501
$146$	−22.2347	−1.84016
$147$	3.43321	0.283166
$148$	−2.52957	−0.207930
$149$	19.5389	1.60069	0.800343	−	0.599543i	$-0.204651\pi$
$149$	19.5389	1.60069	0.800343	+	0.599543i	$0.204651\pi$
$150$	−10.0540	−0.820906
$151$	11.9104	0.969253	0.484626	−	0.874721i	$-0.338956\pi$
$151$	11.9104	0.969253	0.484626	+	0.874721i	$0.338956\pi$
$152$	3.46508	0.281055
$153$	13.4945	1.09097
$154$	−7.73725	−0.623485
$155$	−3.72666	−0.299332
$156$	0.271920	0.0217711
$157$	15.3927	1.22847	0.614234	−	0.789124i	$-0.289465\pi$
$157$	15.3927	1.22847	0.614234	+	0.789124i	$0.289465\pi$
$158$	12.9975	1.03402
$159$	2.41154	0.191248
$160$	7.70698	0.609290
$161$	−10.0847	−0.794784
$162$	−6.82371	−0.536121
$163$	12.4659	0.976402	0.488201	−	0.872731i	$-0.337653\pi$
$163$	12.4659	0.976402	0.488201	+	0.872731i	$0.337653\pi$
$164$	0.819053	0.0639573
$165$	4.02810	0.313587
$166$	4.27195	0.331568
$167$	8.17898	0.632909	0.316454	−	0.948608i	$-0.397508\pi$
$167$	8.17898	0.632909	0.316454	+	0.948608i	$0.397508\pi$
$168$	6.29921	0.485995
$169$	1.00000	0.0769231
$170$	31.4706	2.41369
$171$	−3.39717	−0.259788
$172$	−0.0963532	−0.00734687
$173$	6.37604	0.484762	0.242381	−	0.970181i	$-0.422072\pi$
$173$	6.37604	0.484762	0.242381	+	0.970181i	$0.422072\pi$
$174$	−10.3988	−0.788334
$175$	30.3657	2.29543
$176$	−6.77148	−0.510420
$177$	3.32634	0.250023
$178$	−2.27400	−0.170443
$179$	−0.254712	−0.0190381	−0.00951904	−	0.999955i	$-0.503030\pi$
$179$	−0.254712	−0.0190381	−0.00951904	+	0.999955i	$0.503030\pi$
$180$	−3.39280	−0.252885
$181$	−0.510144	−0.0379187	−0.0189593	−	0.999820i	$-0.506035\pi$
$181$	−0.510144	−0.0379187	−0.0189593	+	0.999820i	$0.506035\pi$
$182$	−5.25971	−0.389876
$183$	−9.72454	−0.718858
$184$	−7.40679	−0.546036
$185$	−25.4730	−1.87281
$186$	1.13119	0.0829431
$187$	−8.06919	−0.590078
$188$	3.48369	0.254074
$189$	−13.7068	−0.997027
$190$	−7.92253	−0.574761
$191$	−4.79721	−0.347114	−0.173557	−	0.984824i	$-0.555526\pi$
$191$	−4.79721	−0.347114	−0.173557	+	0.984824i	$0.555526\pi$
$192$	4.42526	0.319365
$193$	16.7506	1.20574	0.602868	−	0.797841i	$-0.294025\pi$
$193$	16.7506	1.20574	0.602868	+	0.797841i	$0.294025\pi$
$194$	−4.02395	−0.288903
$195$	2.73826	0.196091
$196$	1.72914	0.123510
$197$	−18.3808	−1.30958	−0.654790	−	0.755811i	$-0.727243\pi$
$197$	−18.3808	−1.30958	−0.654790	+	0.755811i	$0.727243\pi$
$198$	5.57133	0.395937
$199$	−14.6113	−1.03577	−0.517884	−	0.855451i	$-0.673280\pi$
$199$	−14.6113	−1.03577	−0.517884	+	0.855451i	$0.673280\pi$
$200$	22.3024	1.57702
$201$	−8.31432	−0.586447
$202$	0.191777	0.0134934
$203$	31.4072	2.20435
$204$	−1.49158	−0.104432
$205$	8.24793	0.576061
$206$	17.9214	1.24864
$207$	7.26163	0.504718
$208$	−4.60319	−0.319174
$209$	2.03137	0.140513
$210$	−14.4025	−0.993865
$211$	11.4759	0.790032	0.395016	−	0.918674i	$-0.370739\pi$
$211$	11.4759	0.790032	0.395016	+	0.918674i	$0.370739\pi$
$212$	1.21458	0.0834176
$213$	−5.16028	−0.353577
$214$	−2.96051	−0.202376
$215$	−0.970286	−0.0661729
$216$	−10.0671	−0.684982
$217$	−3.41650	−0.231927
$218$	24.9706	1.69122
$219$	10.6122	0.717108
$220$	2.02876	0.136779
$221$	−5.48536	−0.368985
$222$	7.73210	0.518945
$223$	6.57040	0.439986	0.219993	−	0.975501i	$-0.429397\pi$
$223$	6.57040	0.439986	0.219993	+	0.975501i	$0.429397\pi$
$224$	7.06555	0.472087
$225$	−21.8653	−1.45769
$226$	−27.5388	−1.83186
$227$	−24.0202	−1.59428	−0.797138	−	0.603797i	$-0.793654\pi$
$227$	−24.0202	−1.59428	−0.797138	+	0.603797i	$0.793654\pi$
$228$	0.375497	0.0248679
$229$	−9.14500	−0.604318	−0.302159	−	0.953257i	$-0.597707\pi$
$229$	−9.14500	−0.604318	−0.302159	+	0.953257i	$0.597707\pi$
$230$	16.9348	1.11665
$231$	3.69285	0.242972
$232$	23.0673	1.51445
$233$	13.5108	0.885121	0.442560	−	0.896739i	$-0.354070\pi$
$233$	13.5108	0.885121	0.442560	+	0.896739i	$0.354070\pi$
$234$	3.78734	0.247586
$235$	35.0811	2.28844
$236$	1.67532	0.109054
$237$	−6.20346	−0.402958
$238$	28.8514	1.87016
$239$	−12.1027	−0.782856	−0.391428	−	0.920209i	$-0.628019\pi$
$239$	−12.1027	−0.782856	−0.391428	+	0.920209i	$0.628019\pi$
$240$	−12.6047	−0.813633
$241$	3.08477	0.198708	0.0993538	−	0.995052i	$-0.468322\pi$
$241$	3.08477	0.198708	0.0993538	+	0.995052i	$0.468322\pi$
$242$	13.6031	0.874441
$243$	15.2927	0.981028
$244$	−4.89778	−0.313548
$245$	17.4126	1.11245
$246$	−2.50359	−0.159623
$247$	1.38091	0.0878649
$248$	−2.50928	−0.159339
$249$	−2.03893	−0.129212
$250$	−22.3060	−1.41076
$251$	11.1443	0.703425	0.351712	−	0.936108i	$-0.385600\pi$
$251$	11.1443	0.703425	0.351712	+	0.936108i	$0.385600\pi$
$252$	−3.11043	−0.195939
$253$	−4.34216	−0.272989
$254$	20.9934	1.31724
$255$	−15.0204	−0.940612
$256$	8.59639	0.537274
$257$	−12.2076	−0.761492	−0.380746	−	0.924680i	$-0.624333\pi$
$257$	−12.2076	−0.761492	−0.380746	+	0.924680i	$0.624333\pi$
$258$	0.294521	0.0183361
$259$	−23.3530	−1.45108
$260$	1.37913	0.0855301
$261$	−22.6153	−1.39985
$262$	28.0143	1.73073
$263$	−16.8932	−1.04168	−0.520841	−	0.853654i	$-0.674381\pi$
$263$	−16.8932	−1.04168	−0.520841	+	0.853654i	$0.674381\pi$
$264$	2.71225	0.166928
$265$	12.2309	0.751339
$266$	−7.26316	−0.445333
$267$	1.08534	0.0664216
$268$	−4.18752	−0.255794
$269$	10.2679	0.626044	0.313022	−	0.949746i	$-0.398659\pi$
$269$	10.2679	0.626044	0.313022	+	0.949746i	$0.398659\pi$
$270$	23.0174	1.40080
$271$	−28.8588	−1.75305	−0.876524	−	0.481359i	$-0.840143\pi$
$271$	−28.8588	−1.75305	−0.876524	+	0.481359i	$0.840143\pi$
$272$	25.2502	1.53102
$273$	2.51036	0.151934
$274$	13.1655	0.795359
$275$	13.0746	0.788426
$276$	−0.802644	−0.0483135
$277$	−3.85467	−0.231604	−0.115802	−	0.993272i	$-0.536944\pi$
$277$	−3.85467	−0.231604	−0.115802	+	0.993272i	$0.536944\pi$
$278$	22.4092	1.34402
$279$	2.46010	0.147282
$280$	31.9484	1.90928
$281$	−22.2948	−1.33000	−0.664998	−	0.746846i	$-0.731567\pi$
$281$	−22.2948	−1.33000	−0.664998	+	0.746846i	$0.731567\pi$
$282$	−10.6485	−0.634111
$283$	−23.0516	−1.37027	−0.685136	−	0.728415i	$-0.740257\pi$
$283$	−23.0516	−1.37027	−0.685136	+	0.728415i	$0.740257\pi$
$284$	−2.59899	−0.154221
$285$	3.78128	0.223984
$286$	−2.26467	−0.133913
$287$	7.56148	0.446340
$288$	−5.08766	−0.299793
$289$	13.0892	0.769952
$290$	−52.7410	−3.09706
$291$	1.92056	0.112585
$292$	5.34487	0.312785
$293$	−7.07428	−0.413284	−0.206642	−	0.978417i	$-0.566254\pi$
$293$	−7.07428	−0.413284	−0.206642	+	0.978417i	$0.566254\pi$
$294$	−5.28544	−0.308253
$295$	16.8706	0.982243
$296$	−17.1518	−0.996929
$297$	−5.90176	−0.342455
$298$	−30.0801	−1.74250
$299$	−2.95176	−0.170705
$300$	2.41682	0.139535
$301$	−0.889531	−0.0512717
$302$	−18.3361	−1.05512
$303$	−0.0915316	−0.00525835
$304$	−6.35657	−0.364574
$305$	−49.3211	−2.82412
$306$	−20.7749	−1.18762
$307$	−20.1033	−1.14735	−0.573677	−	0.819081i	$-0.694484\pi$
$307$	−20.1033	−1.14735	−0.573677	+	0.819081i	$0.694484\pi$
$308$	1.85991	0.105978
$309$	−8.55356	−0.486595
$310$	5.73720	0.325851
$311$	17.2545	0.978410	0.489205	−	0.872169i	$-0.337287\pi$
$311$	17.2545	0.978410	0.489205	+	0.872169i	$0.337287\pi$
$312$	1.84376	0.104383
$313$	−6.66924	−0.376968	−0.188484	−	0.982076i	$-0.560357\pi$
$313$	−6.66924	−0.376968	−0.188484	+	0.982076i	$0.560357\pi$
$314$	−23.6971	−1.33730
$315$	−31.3223	−1.76481
$316$	−3.12438	−0.175760
$317$	−7.58760	−0.426162	−0.213081	−	0.977035i	$-0.568350\pi$
$317$	−7.58760	−0.426162	−0.213081	+	0.977035i	$0.568350\pi$
$318$	−3.71258	−0.208191
$319$	13.5230	0.757143
$320$	22.4441	1.25466
$321$	1.41300	0.0788658
$322$	15.5254	0.865196
$323$	−7.57477	−0.421471
$324$	1.64031	0.0911283
$325$	8.88797	0.493016
$326$	−19.1912	−1.06290
$327$	−11.9180	−0.659068
$328$	5.55360	0.306647
$329$	32.1614	1.77311
$330$	−6.20127	−0.341369
$331$	−35.1493	−1.93198	−0.965991	−	0.258574i	$-0.916747\pi$
$331$	−35.1493	−1.93198	−0.965991	+	0.258574i	$0.916747\pi$
$332$	−1.02691	−0.0563589
$333$	16.8157	0.921493
$334$	−12.5916	−0.688980
$335$	−42.1687	−2.30392
$336$	−11.5557	−0.630414
$337$	7.77952	0.423777	0.211889	−	0.977294i	$-0.432039\pi$
$337$	7.77952	0.423777	0.211889	+	0.977294i	$0.432039\pi$
$338$	−1.53950	−0.0837380
$339$	13.1438	0.713872
$340$	−7.56503	−0.410271
$341$	−1.47104	−0.0796614
$342$	5.22995	0.282804
$343$	−7.95207	−0.429371
$344$	−0.653325	−0.0352249
$345$	−8.08269	−0.435158
$346$	−9.81594	−0.527708
$347$	12.1489	0.652185	0.326092	−	0.945338i	$-0.394268\pi$
$347$	12.1489	0.652185	0.326092	+	0.945338i	$0.394268\pi$
$348$	2.49971	0.133999
$349$	−4.59026	−0.245711	−0.122855	−	0.992425i	$-0.539205\pi$
$349$	−4.59026	−0.245711	−0.122855	+	0.992425i	$0.539205\pi$
$350$	−46.7481	−2.49879
$351$	−4.01196	−0.214143
$352$	3.04221	0.162151
$353$	7.33123	0.390202	0.195101	−	0.980783i	$-0.437497\pi$
$353$	7.33123	0.390202	0.195101	+	0.980783i	$0.437497\pi$
$354$	−5.12091	−0.272173
$355$	−26.1720	−1.38907
$356$	0.546632	0.0289714
$357$	−13.7703	−0.728799
$358$	0.392130	0.0207247
$359$	34.8079	1.83709	0.918544	−	0.395318i	$-0.129366\pi$
$359$	34.8079	1.83709	0.918544	+	0.395318i	$0.129366\pi$
$360$	−23.0050	−1.21247
$361$	−17.0931	−0.899637
$362$	0.785368	0.0412780
$363$	−6.49252	−0.340769
$364$	1.26435	0.0662699
$365$	53.8233	2.81724
$366$	14.9710	0.782545
$367$	−8.38232	−0.437554	−0.218777	−	0.975775i	$-0.570207\pi$
$367$	−8.38232	−0.437554	−0.218777	+	0.975775i	$0.570207\pi$
$368$	13.5875	0.708298
$369$	−5.44476	−0.283443
$370$	39.2158	2.03873
$371$	11.2130	0.582148
$372$	−0.271920	−0.0140984
$373$	10.9817	0.568610	0.284305	−	0.958734i	$-0.408237\pi$
$373$	10.9817	0.568610	0.284305	+	0.958734i	$0.408237\pi$
$374$	12.4226	0.642355
$375$	10.6463	0.549771
$376$	23.6212	1.21817
$377$	9.19281	0.473454
$378$	21.1017	1.08536
$379$	34.6619	1.78046	0.890232	−	0.455507i	$-0.150542\pi$
$379$	34.6619	1.78046	0.890232	+	0.455507i	$0.150542\pi$
$380$	1.90445	0.0976962
$381$	−10.0198	−0.513328
$382$	7.38533	0.377866
$383$	18.3416	0.937213	0.468607	−	0.883407i	$-0.344756\pi$
$383$	18.3416	0.937213	0.468607	+	0.883407i	$0.344756\pi$
$384$	−9.85184	−0.502750
$385$	18.7295	0.954542
$386$	−25.7876	−1.31256
$387$	0.640521	0.0325595
$388$	0.967293	0.0491069
$389$	−5.31429	−0.269445	−0.134722	−	0.990883i	$-0.543014\pi$
$389$	−5.31429	−0.269445	−0.134722	+	0.990883i	$0.543014\pi$
$390$	−4.21557	−0.213464
$391$	16.1915	0.818838
$392$	11.7245	0.592176
$393$	−13.3707	−0.674464
$394$	28.2973	1.42560
$395$	−31.4628	−1.58306
$396$	−1.33926	−0.0673002
$397$	27.1145	1.36084	0.680418	−	0.732824i	$-0.261798\pi$
$397$	27.1145	1.36084	0.680418	+	0.732824i	$0.261798\pi$
$398$	22.4942	1.12753
$399$	3.46658	0.173546
$400$	−40.9130	−2.04565
$401$	2.84976	0.142310	0.0711551	−	0.997465i	$-0.477331\pi$
$401$	2.84976	0.142310	0.0711551	+	0.997465i	$0.477331\pi$
$402$	12.7999	0.638402
$403$	−1.00000	−0.0498135
$404$	−0.0461000	−0.00229356
$405$	16.5181	0.820789
$406$	−48.3515	−2.39965
$407$	−10.0551	−0.498412
$408$	−10.1137	−0.500703
$409$	−12.8347	−0.634634	−0.317317	−	0.948319i	$-0.602782\pi$
$409$	−12.8347	−0.634634	−0.317317	+	0.948319i	$0.602782\pi$
$410$	−12.6977	−0.627096
$411$	−6.28367	−0.309951
$412$	−4.30801	−0.212241
$413$	15.4665	0.761056
$414$	−11.1793	−0.549433
$415$	−10.3411	−0.507623
$416$	2.06807	0.101395
$417$	−10.6955	−0.523762
$418$	−3.12730	−0.152961
$419$	33.0742	1.61578	0.807889	−	0.589334i	$-0.200610\pi$
$419$	33.0742	1.61578	0.807889	+	0.589334i	$0.200610\pi$
$420$	3.46212	0.168934
$421$	38.3512	1.86912	0.934561	−	0.355803i	$-0.115793\pi$
$421$	38.3512	1.86912	0.934561	+	0.355803i	$0.115793\pi$
$422$	−17.6672	−0.860024
$423$	−23.1583	−1.12600
$424$	8.23548	0.399950
$425$	−48.7537	−2.36490
$426$	7.94428	0.384901
$427$	−45.2162	−2.18817
$428$	0.711658	0.0343993
$429$	1.08089	0.0521858
$430$	1.49376	0.0720354
$431$	−14.7506	−0.710513	−0.355257	−	0.934769i	$-0.615607\pi$
$431$	−14.7506	−0.710513	−0.355257	+	0.934769i	$0.615607\pi$
$432$	18.4678	0.888533
$433$	40.2610	1.93482	0.967409	−	0.253217i	$-0.0814887\pi$
$433$	40.2610	1.93482	0.967409	+	0.253217i	$0.0814887\pi$
$434$	5.25971	0.252474
$435$	25.1723	1.20692
$436$	−6.00253	−0.287469
$437$	−4.07610	−0.194986
$438$	−16.3376	−0.780639
$439$	−16.1611	−0.771327	−0.385663	−	0.922640i	$-0.626027\pi$
$439$	−16.1611	−0.771327	−0.385663	+	0.922640i	$0.626027\pi$
$440$	13.7560	0.655793
$441$	−11.4947	−0.547367
$442$	8.44473	0.401675
$443$	−23.4720	−1.11519	−0.557595	−	0.830113i	$-0.688276\pi$
$443$	−23.4720	−1.11519	−0.557595	+	0.830113i	$0.688276\pi$
$444$	−1.85867	−0.0882087
$445$	5.50463	0.260945
$446$	−10.1151	−0.478966
$447$	14.3567	0.679049
$448$	20.5761	0.972131
$449$	25.1699	1.18784	0.593919	−	0.804524i	$-0.297580\pi$
$449$	25.1699	1.18784	0.593919	+	0.804524i	$0.297580\pi$
$450$	33.6617	1.58683
$451$	3.25575	0.153307
$452$	6.61989	0.311373
$453$	8.75148	0.411180
$454$	36.9792	1.73552
$455$	12.7321	0.596890
$456$	2.54606	0.119230
$457$	37.2699	1.74341	0.871706	−	0.490029i	$-0.163014\pi$
$457$	37.2699	1.74341	0.871706	+	0.490029i	$0.163014\pi$
$458$	14.0788	0.657857
$459$	22.0070	1.02720
$460$	−4.07086	−0.189805
$461$	6.44702	0.300267	0.150134	−	0.988666i	$-0.452030\pi$
$461$	6.44702	0.300267	0.150134	+	0.988666i	$0.452030\pi$
$462$	−5.68516	−0.264497
$463$	5.87146	0.272870	0.136435	−	0.990649i	$-0.456436\pi$
$463$	5.87146	0.272870	0.136435	+	0.990649i	$0.456436\pi$
$464$	−42.3163	−1.96448
$465$	−2.73826	−0.126984
$466$	−20.7999	−0.963537
$467$	−19.4720	−0.901058	−0.450529	−	0.892762i	$-0.648765\pi$
$467$	−19.4720	−0.901058	−0.450529	+	0.892762i	$0.648765\pi$
$468$	−0.910415	−0.0420839
$469$	−38.6591	−1.78511
$470$	−54.0074	−2.49118
$471$	11.3102	0.521146
$472$	11.3595	0.522864
$473$	−0.383006	−0.0176106
$474$	9.55025	0.438657
$475$	12.2734	0.563144
$476$	−6.93541	−0.317884
$477$	−8.07407	−0.369686
$478$	18.6321	0.852212
$479$	−10.4707	−0.478417	−0.239208	−	0.970968i	$-0.576888\pi$
$479$	−10.4707	−0.478417	−0.239208	+	0.970968i	$0.576888\pi$
$480$	5.66292	0.258476
$481$	−6.83535	−0.311665
$482$	−4.74902	−0.216312
$483$	−7.40999	−0.337166
$484$	−3.26997	−0.148635
$485$	9.74072	0.442303
$486$	−23.5432	−1.06794
$487$	38.7006	1.75369	0.876846	−	0.480771i	$-0.159643\pi$
$487$	38.7006	1.75369	0.876846	+	0.480771i	$0.159643\pi$
$488$	−33.2095	−1.50332
$489$	9.15964	0.414213
$490$	−26.8068	−1.21101
$491$	24.8165	1.11995	0.559977	−	0.828508i	$-0.310810\pi$
$491$	24.8165	1.11995	0.559977	+	0.828508i	$0.310810\pi$
$492$	0.601821	0.0271322
$493$	−50.4259	−2.27107
$494$	−2.12591	−0.0956492
$495$	−13.4864	−0.606170
$496$	4.60319	0.206689
$497$	−23.9938	−1.07627
$498$	3.13893	0.140659
$499$	−26.3612	−1.18009	−0.590046	−	0.807370i	$-0.700890\pi$
$499$	−26.3612	−1.18009	−0.590046	+	0.807370i	$0.700890\pi$
$500$	5.36201	0.239797
$501$	6.00973	0.268495
$502$	−17.1568	−0.765744
$503$	−0.120043	−0.00535245	−0.00267622	−	0.999996i	$-0.500852\pi$
$503$	−0.120043	−0.00535245	−0.00267622	+	0.999996i	$0.500852\pi$
$504$	−21.0903	−0.939438
$505$	−0.464231	−0.0206580
$506$	6.68477	0.297174
$507$	0.734778	0.0326326
$508$	−5.04647	−0.223901
$509$	−15.0497	−0.667066	−0.333533	−	0.942738i	$-0.608241\pi$
$509$	−15.0497	−0.667066	−0.333533	+	0.942738i	$0.608241\pi$
$510$	23.1239	1.02394
$511$	49.3437	2.18284
$512$	13.5817	0.600231
$513$	−5.54014	−0.244603
$514$	18.7937	0.828955
$515$	−43.3821	−1.91164
$516$	−0.0707982	−0.00311672
$517$	13.8477	0.609022
$518$	35.9520	1.57964
$519$	4.68497	0.205647
$520$	9.35123	0.410078
$521$	−0.00259590	−0.000113729 0	−5.68643e−5	−	1.00000i	$-0.500018\pi$
$521$	−0.00259590	−0.000113729 0	−5.68643e−5		1.00000i	$0.500018\pi$
$522$	34.8163	1.52387
$523$	21.5268	0.941302	0.470651	−	0.882320i	$-0.344019\pi$
$523$	21.5268	0.941302	0.470651	+	0.882320i	$0.344019\pi$
$524$	−6.73419	−0.294184
$525$	22.3120	0.973777
$526$	26.0072	1.13397
$527$	5.48536	0.238946
$528$	−4.97553	−0.216532
$529$	−14.2871	−0.621179
$530$	−18.8295	−0.817903
$531$	−11.1369	−0.483300
$532$	1.74595	0.0756964
$533$	2.21323	0.0958655
$534$	−1.67088	−0.0723061
$535$	7.16646	0.309833
$536$	−28.3936	−1.22642
$537$	−0.187157	−0.00807641
$538$	−15.8075	−0.681508
$539$	6.87337	0.296057
$540$	−5.53302	−0.238103
$541$	−40.8707	−1.75717	−0.878583	−	0.477589i	$-0.841511\pi$
$541$	−40.8707	−1.75717	−0.878583	+	0.477589i	$0.841511\pi$
$542$	44.4282	1.90836
$543$	−0.374842	−0.0160860
$544$	−11.3441	−0.486374
$545$	−60.4460	−2.58922
$546$	−3.86472	−0.165395
$547$	27.4617	1.17418	0.587089	−	0.809523i	$-0.300274\pi$
$547$	27.4617	1.17418	0.587089	+	0.809523i	$0.300274\pi$
$548$	−3.16478	−0.135193
$549$	32.5586	1.38957
$550$	−20.1283	−0.858276
$551$	12.6944	0.540800
$552$	−5.44234	−0.231642
$553$	−28.8442	−1.22658
$554$	5.93427	0.252123
$555$	−18.7170	−0.794492
$556$	−5.38682	−0.228452
$557$	−21.1644	−0.896766	−0.448383	−	0.893842i	$-0.648000\pi$
$557$	−21.1644	−0.896766	−0.448383	+	0.893842i	$0.648000\pi$
$558$	−3.78734	−0.160331
$559$	−0.260364	−0.0110122
$560$	−58.6083	−2.47665
$561$	−5.92906	−0.250325
$562$	34.3229	1.44782
$563$	18.8491	0.794395	0.397198	−	0.917733i	$-0.369983\pi$
$563$	18.8491	0.794395	0.397198	+	0.917733i	$0.369983\pi$
$564$	2.55974	0.107784
$565$	66.6629	2.80453
$566$	35.4880	1.49167
$567$	15.1433	0.635959
$568$	−17.6225	−0.739423
$569$	18.9303	0.793601	0.396800	−	0.917905i	$-0.370121\pi$
$569$	18.9303	0.793601	0.396800	+	0.917905i	$0.370121\pi$
$570$	−5.82130	−0.243827
$571$	33.0754	1.38416	0.692082	−	0.721819i	$-0.256694\pi$
$571$	33.0754	1.38416	0.692082	+	0.721819i	$0.256694\pi$
$572$	0.544391	0.0227621
$573$	−3.52488	−0.147254
$574$	−11.6409	−0.485883
$575$	−26.2351	−1.09408
$576$	−14.8162	−0.617340
$577$	28.8553	1.20126	0.600631	−	0.799527i	$-0.294916\pi$
$577$	28.8553	1.20126	0.600631	+	0.799527i	$0.294916\pi$
$578$	−20.1509	−0.838165
$579$	12.3080	0.511502
$580$	12.6781	0.526429
$581$	−9.48040	−0.393313
$582$	−2.95671	−0.122560
$583$	4.82797	0.199954
$584$	36.2410	1.49966
$585$	−9.16795	−0.379048
$586$	10.8909	0.449898
$587$	−0.164259	−0.00677968	−0.00338984	−	0.999994i	$-0.501079\pi$
$587$	−0.164259	−0.00677968	−0.00338984	+	0.999994i	$0.501079\pi$
$588$	1.27053	0.0523960
$589$	−1.38091	−0.0568992
$590$	−25.9723	−1.06926
$591$	−13.5058	−0.555555
$592$	31.4644	1.29318
$593$	−33.6772	−1.38296	−0.691479	−	0.722397i	$-0.743040\pi$
$593$	−33.6772	−1.38296	−0.691479	+	0.722397i	$0.743040\pi$
$594$	9.08578	0.372794
$595$	−69.8402	−2.86317
$596$	7.23078	0.296184
$597$	−10.7361	−0.439398
$598$	4.54424	0.185828
$599$	0.0438430	0.00179138	0.000895689	−	1.00000i	$-0.499715\pi$
$599$	0.0438430	0.00179138	0.000895689		1.00000i	$0.499715\pi$
$600$	16.3873	0.669009
$601$	−2.17488	−0.0887154	−0.0443577	−	0.999016i	$-0.514124\pi$
$601$	−2.17488	−0.0887154	−0.0443577	+	0.999016i	$0.514124\pi$
$602$	1.36944	0.0558141
$603$	27.8371	1.13361
$604$	4.40770	0.179347
$605$	−32.9289	−1.33875
$606$	0.140913	0.00572421
$607$	27.5341	1.11758	0.558788	−	0.829311i	$-0.311267\pi$
$607$	27.5341	1.11758	0.558788	+	0.829311i	$0.311267\pi$
$608$	2.85581	0.115818
$609$	23.0773	0.935140
$610$	75.9300	3.07431
$611$	9.41355	0.380832
$612$	4.99395	0.201869
$613$	33.4199	1.34982	0.674908	−	0.737902i	$-0.264183\pi$
$613$	33.4199	1.34982	0.674908	+	0.737902i	$0.264183\pi$
$614$	30.9491	1.24900
$615$	6.06040	0.244379
$616$	12.6112	0.508118
$617$	35.0273	1.41015	0.705073	−	0.709135i	$-0.250914\pi$
$617$	35.0273	1.41015	0.705073	+	0.709135i	$0.250914\pi$
$618$	13.1682	0.529704
$619$	−36.1461	−1.45283	−0.726417	−	0.687254i	$-0.758816\pi$
$619$	−36.1461	−1.45283	−0.726417	+	0.687254i	$0.758816\pi$
$620$	−1.37913	−0.0553872
$621$	11.8423	0.475217
$622$	−26.5633	−1.06509
$623$	5.04650	0.202184
$624$	−3.38232	−0.135401
$625$	9.55612	0.382245
$626$	10.2673	0.410364
$627$	1.49260	0.0596089
$628$	5.69639	0.227311
$629$	37.4944	1.49500
$630$	48.2208	1.92116
$631$	−22.2778	−0.886864	−0.443432	−	0.896308i	$-0.646239\pi$
$631$	−22.2778	−0.886864	−0.443432	+	0.896308i	$0.646239\pi$
$632$	−21.1849	−0.842691
$633$	8.43222	0.335151
$634$	11.6811	0.463917
$635$	−50.8184	−2.01667
$636$	0.892445	0.0353877
$637$	4.67245	0.185129
$638$	−20.8187	−0.824221
$639$	17.2771	0.683471
$640$	−49.9667	−1.97511
$641$	20.8924	0.825198	0.412599	−	0.910913i	$-0.364621\pi$
$641$	20.8924	0.825198	0.412599	+	0.910913i	$0.364621\pi$
$642$	−2.17531	−0.0858528
$643$	−22.8406	−0.900746	−0.450373	−	0.892841i	$-0.648709\pi$
$643$	−22.8406	−0.900746	−0.450373	+	0.892841i	$0.648709\pi$
$644$	−3.73205	−0.147064
$645$	−0.712944	−0.0280721
$646$	11.6614	0.458811
$647$	11.1432	0.438084	0.219042	−	0.975715i	$-0.429707\pi$
$647$	11.1432	0.438084	0.219042	+	0.975715i	$0.429707\pi$
$648$	11.1222	0.436920
$649$	6.65941	0.261405
$650$	−13.6831	−0.536694
$651$	−2.51036	−0.0983890
$652$	4.61327	0.180669
$653$	15.5676	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$653$	15.5676	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$654$	18.3478	0.717457
$655$	−67.8139	−2.64971
$656$	−10.1879	−0.397771
$657$	−35.5307	−1.38618
$658$	−49.5126	−1.93020
$659$	−1.39049	−0.0541659	−0.0270829	−	0.999633i	$-0.508622\pi$
$659$	−1.39049	−0.0541659	−0.0270829	+	0.999633i	$0.508622\pi$
$660$	1.49069	0.0580249
$661$	16.6187	0.646392	0.323196	−	0.946332i	$-0.395243\pi$
$661$	16.6187	0.646392	0.323196	+	0.946332i	$0.395243\pi$
$662$	54.1125	2.10314
$663$	−4.03052	−0.156532
$664$	−6.96297	−0.270216
$665$	17.5818	0.681795
$666$	−25.8878	−1.00313
$667$	−27.1350	−1.05067
$668$	3.02681	0.117111
$669$	4.82778	0.186653
$670$	64.9189	2.50803
$671$	−19.4687	−0.751583
$672$	5.19161	0.200271
$673$	−0.981936	−0.0378508	−0.0189254	−	0.999821i	$-0.506025\pi$
$673$	−0.981936	−0.0378508	−0.0189254	+	0.999821i	$0.506025\pi$
$674$	−11.9766	−0.461321
$675$	−35.6582	−1.37248
$676$	0.370072	0.0142335
$677$	−6.63506	−0.255006	−0.127503	−	0.991838i	$-0.540696\pi$
$677$	−6.63506	−0.255006	−0.127503	+	0.991838i	$0.540696\pi$
$678$	−20.2349	−0.777117
$679$	8.93003	0.342703
$680$	−51.2949	−1.96707
$681$	−17.6495	−0.676331
$682$	2.26467	0.0867188
$683$	27.3380	1.04606	0.523030	−	0.852315i	$-0.324802\pi$
$683$	27.3380	1.04606	0.523030	+	0.852315i	$0.324802\pi$
$684$	−1.25720	−0.0480701
$685$	−31.8696	−1.21768
$686$	12.2422	0.467411
$687$	−6.71954	−0.256366
$688$	1.19850	0.0456925
$689$	3.28201	0.125035
$690$	12.4433	0.473710
$691$	7.25442	0.275971	0.137986	−	0.990434i	$-0.455937\pi$
$691$	7.25442	0.275971	0.137986	+	0.990434i	$0.455937\pi$
$692$	2.35959	0.0896983
$693$	−12.3640	−0.469670
$694$	−18.7032	−0.709964
$695$	−54.2457	−2.05766
$696$	16.9494	0.642464
$697$	−12.1403	−0.459848
$698$	7.06672	0.267479
$699$	9.92742	0.375489
$700$	11.2375	0.424737
$701$	19.0032	0.717741	0.358871	−	0.933387i	$-0.383162\pi$
$701$	19.0032	0.717741	0.358871	+	0.933387i	$0.383162\pi$
$702$	6.17643	0.233114
$703$	−9.43898	−0.355998
$704$	8.85947	0.333904
$705$	25.7768	0.970810
$706$	−11.2865	−0.424771
$707$	−0.425595	−0.0160061
$708$	1.23098	0.0462632
$709$	3.76793	0.141508	0.0707539	−	0.997494i	$-0.477460\pi$
$709$	3.76793	0.141508	0.0707539	+	0.997494i	$0.477460\pi$
$710$	40.2919	1.51213
$711$	20.7697	0.778926
$712$	3.70645	0.138905
$713$	2.95176	0.110544
$714$	21.1994	0.793366
$715$	5.48207	0.205018
$716$	−0.0942618	−0.00352273
$717$	−8.89276	−0.332106
$718$	−53.5868	−1.99984
$719$	−43.4267	−1.61954	−0.809772	−	0.586745i	$-0.800409\pi$
$719$	−43.4267	−1.61954	−0.809772	+	0.586745i	$0.800409\pi$
$720$	42.2018	1.57277
$721$	−39.7715	−1.48117
$722$	26.3149	0.979339
$723$	2.26662	0.0842965
$724$	−0.188790	−0.00701632
$725$	81.7054	3.03446
$726$	9.99526	0.370959
$727$	−16.3904	−0.607886	−0.303943	−	0.952690i	$-0.598303\pi$
$727$	−16.3904	−0.607886	−0.303943	+	0.952690i	$0.598303\pi$
$728$	8.57295	0.317735
$729$	−2.06048	−0.0763141
$730$	−82.8611	−3.06683
$731$	1.42819	0.0528234
$732$	−3.59878	−0.133015
$733$	14.4296	0.532970	0.266485	−	0.963839i	$-0.414138\pi$
$733$	14.4296	0.532970	0.266485	+	0.963839i	$0.414138\pi$
$734$	12.9046	0.476318
$735$	12.7944	0.471928
$736$	−6.10444	−0.225013
$737$	−16.6455	−0.613143
$738$	8.38223	0.308554
$739$	13.8985	0.511264	0.255632	−	0.966774i	$-0.417716\pi$
$739$	13.8985	0.511264	0.255632	+	0.966774i	$0.417716\pi$
$740$	−9.42684	−0.346538
$741$	1.01466	0.0372744
$742$	−17.2624	−0.633723
$743$	51.0664	1.87344	0.936722	−	0.350075i	$-0.113844\pi$
$743$	51.0664	1.87344	0.936722	+	0.350075i	$0.113844\pi$
$744$	−1.84376	−0.0675956
$745$	72.8146	2.66772
$746$	−16.9063	−0.618985
$747$	6.82651	0.249769
$748$	−2.98618	−0.109186
$749$	6.57001	0.240063
$750$	−16.3900	−0.598478
$751$	−10.1330	−0.369759	−0.184879	−	0.982761i	$-0.559189\pi$
$751$	−10.1330	−0.369759	−0.184879	+	0.982761i	$0.559189\pi$
$752$	−43.3324	−1.58017
$753$	8.18861	0.298410
$754$	−14.1524	−0.515399
$755$	44.3859	1.61537
$756$	−5.07252	−0.184486
$757$	48.8571	1.77574	0.887871	−	0.460092i	$-0.152183\pi$
$757$	48.8571	1.77574	0.887871	+	0.460092i	$0.152183\pi$
$758$	−53.3622	−1.93820
$759$	−3.19052	−0.115809
$760$	12.9132	0.468410
$761$	−40.2479	−1.45899	−0.729493	−	0.683988i	$-0.760244\pi$
$761$	−40.2479	−1.45899	−0.729493	+	0.683988i	$0.760244\pi$
$762$	15.4255	0.558805
$763$	−55.4152	−2.00617
$764$	−1.77531	−0.0642286
$765$	50.2895	1.81822
$766$	−28.2370	−1.02024
$767$	4.52700	0.163461
$768$	6.31643	0.227925
$769$	−20.0697	−0.723733	−0.361867	−	0.932230i	$-0.617860\pi$
$769$	−20.0697	−0.723733	−0.361867	+	0.932230i	$0.617860\pi$
$770$	−28.8341	−1.03911
$771$	−8.96990	−0.323043
$772$	6.19893	0.223104
$773$	29.0237	1.04391	0.521955	−	0.852973i	$-0.325203\pi$
$773$	29.0237	1.04391	0.521955	+	0.852973i	$0.325203\pi$
$774$	−0.986084	−0.0354441
$775$	−8.88797	−0.319265
$776$	6.55875	0.235445
$777$	−17.1592	−0.615584
$778$	8.18136	0.293316
$779$	3.05626	0.109502
$780$	1.01335	0.0362839
$781$	−10.3310	−0.369672
$782$	−24.9268	−0.891381
$783$	−36.8812	−1.31803
$784$	−21.5082	−0.768149
$785$	57.3632	2.04738
$786$	20.5843	0.734217
$787$	−2.27351	−0.0810418	−0.0405209	−	0.999179i	$-0.512902\pi$
$787$	−2.27351	−0.0810418	−0.0405209	+	0.999179i	$0.512902\pi$
$788$	−6.80222	−0.242319
$789$	−12.4128	−0.441906
$790$	48.4371	1.72331
$791$	61.1147	2.17299
$792$	−9.08086	−0.322674
$793$	−13.2347	−0.469977
$794$	−41.7428	−1.48140
$795$	8.98700	0.318736
$796$	−5.40724	−0.191654
$797$	37.0434	1.31214	0.656072	−	0.754698i	$-0.272217\pi$
$797$	37.0434	1.31214	0.656072	+	0.754698i	$0.272217\pi$
$798$	−5.33681	−0.188921
$799$	−51.6367	−1.82678
$800$	18.3809	0.649864
$801$	−3.63381	−0.128394
$802$	−4.38722	−0.154918
$803$	21.2459	0.749752
$804$	−3.07690	−0.108514
$805$	−37.5821	−1.32460
$806$	1.53950	0.0542267
$807$	7.54461	0.265583
$808$	−0.312582	−0.0109966
$809$	−53.3492	−1.87566	−0.937829	−	0.347097i	$-0.887168\pi$
$809$	−53.3492	−1.87566	−0.937829	+	0.347097i	$0.887168\pi$
$810$	−25.4296	−0.893506
$811$	23.9870	0.842296	0.421148	−	0.906992i	$-0.361627\pi$
$811$	23.9870	0.842296	0.421148	+	0.906992i	$0.361627\pi$
$812$	11.6229	0.407885
$813$	−21.2048	−0.743685
$814$	15.4798	0.542568
$815$	46.4560	1.62728
$816$	18.5533	0.649494
$817$	−0.359538	−0.0125786
$818$	19.7590	0.690859
$819$	−8.40493	−0.293692
$820$	3.05233	0.106592
$821$	1.35027	0.0471248	0.0235624	−	0.999722i	$-0.492499\pi$
$821$	1.35027	0.0471248	0.0235624	+	0.999722i	$0.492499\pi$
$822$	9.67374	0.337410
$823$	10.3871	0.362073	0.181037	−	0.983476i	$-0.442055\pi$
$823$	10.3871	0.362073	0.181037	+	0.983476i	$0.442055\pi$
$824$	−29.2106	−1.01760
$825$	9.60690	0.334469
$826$	−23.8107	−0.828481
$827$	−21.5358	−0.748874	−0.374437	−	0.927252i	$-0.622164\pi$
$827$	−21.5358	−0.748874	−0.374437	+	0.927252i	$0.622164\pi$
$828$	2.68732	0.0933910
$829$	−8.29139	−0.287972	−0.143986	−	0.989580i	$-0.545992\pi$
$829$	−8.29139	−0.287972	−0.143986	+	0.989580i	$0.545992\pi$
$830$	15.9201	0.552595
$831$	−2.83232	−0.0982522
$832$	6.02258	0.208795
$833$	−25.6301	−0.888029
$834$	16.4658	0.570164
$835$	30.4803	1.05481
$836$	0.751753	0.0259999
$837$	4.01196	0.138674
$838$	−50.9178	−1.75893
$839$	41.8699	1.44551	0.722754	−	0.691105i	$-0.242876\pi$
$839$	41.8699	1.44551	0.722754	+	0.691105i	$0.242876\pi$
$840$	23.4750	0.809964
$841$	55.5078	1.91406
$842$	−59.0418	−2.03471
$843$	−16.3817	−0.564216
$844$	4.24690	0.146184
$845$	3.72666	0.128201
$846$	35.6523	1.22575
$847$	−30.1883	−1.03728
$848$	−15.1077	−0.518801
$849$	−16.9378	−0.581303
$850$	75.0565	2.57442
$851$	20.1763	0.691635
$852$	−1.90968	−0.0654244
$853$	52.0452	1.78200	0.890998	−	0.454008i	$-0.150006\pi$
$853$	52.0452	1.78200	0.890998	+	0.454008i	$0.150006\pi$
$854$	69.6105	2.38202
$855$	−12.6601	−0.432966
$856$	4.82541	0.164929
$857$	30.2757	1.03420	0.517099	−	0.855926i	$-0.327012\pi$
$857$	30.2757	1.03420	0.517099	+	0.855926i	$0.327012\pi$
$858$	−1.66403	−0.0568091
$859$	17.0032	0.580142	0.290071	−	0.957005i	$-0.406321\pi$
$859$	17.0032	0.580142	0.290071	+	0.957005i	$0.406321\pi$
$860$	−0.359075	−0.0122444
$861$	5.55600	0.189348
$862$	22.7087	0.773460
$863$	−26.8426	−0.913731	−0.456866	−	0.889536i	$-0.651028\pi$
$863$	−26.8426	−0.913731	−0.456866	+	0.889536i	$0.651028\pi$
$864$	−8.29701	−0.282270
$865$	23.7613	0.807909
$866$	−61.9819	−2.10623
$867$	9.61764	0.326632
$868$	−1.26435	−0.0429148
$869$	−12.4195	−0.421301
$870$	−38.7529	−1.31385
$871$	−11.3154	−0.383409
$872$	−40.7003	−1.37828
$873$	−6.43021	−0.217630
$874$	6.27517	0.212261
$875$	49.5020	1.67347
$876$	3.92729	0.132691
$877$	−10.1497	−0.342731	−0.171366	−	0.985207i	$-0.554818\pi$
$877$	−10.1497	−0.342731	−0.171366	+	0.985207i	$0.554818\pi$
$878$	24.8801	0.839661
$879$	−5.19802	−0.175325
$880$	−25.2350	−0.850671
$881$	−6.43243	−0.216714	−0.108357	−	0.994112i	$-0.534559\pi$
$881$	−6.43243	−0.216714	−0.108357	+	0.994112i	$0.534559\pi$
$882$	17.6961	0.595860
$883$	−23.9973	−0.807573	−0.403786	−	0.914853i	$-0.632306\pi$
$883$	−23.9973	−0.807573	−0.403786	+	0.914853i	$0.632306\pi$
$884$	−2.02998	−0.0682755
$885$	12.3961	0.416691
$886$	36.1353	1.21399
$887$	29.1558	0.978955	0.489478	−	0.872016i	$-0.337187\pi$
$887$	29.1558	0.978955	0.489478	+	0.872016i	$0.337187\pi$
$888$	−12.6028	−0.422921
$889$	−46.5889	−1.56254
$890$	−8.47440	−0.284063
$891$	6.52026	0.218437
$892$	2.43152	0.0814133
$893$	12.9992	0.435003
$894$	−22.1022	−0.739209
$895$	−0.949225	−0.0317291
$896$	−45.8081	−1.53034
$897$	−2.16889	−0.0724170
$898$	−38.7491	−1.29307
$899$	−9.19281	−0.306597
$900$	−8.09173	−0.269724
$901$	−18.0030	−0.599767
$902$	−5.01223	−0.166889
$903$	−0.653608	−0.0217507
$904$	44.8863	1.49290
$905$	−1.90113	−0.0631957
$906$	−13.4729	−0.447608
$907$	5.88291	0.195339	0.0976695	−	0.995219i	$-0.468861\pi$
$907$	5.88291	0.195339	0.0976695	+	0.995219i	$0.468861\pi$
$908$	−8.88920	−0.294999
$909$	0.306456	0.0101645
$910$	−19.6011	−0.649771
$911$	−53.6501	−1.77751	−0.888754	−	0.458385i	$-0.848428\pi$
$911$	−53.6501	−1.77751	−0.888754	+	0.458385i	$0.848428\pi$
$912$	−4.67067	−0.154661
$913$	−4.08198	−0.135094
$914$	−57.3771	−1.89787
$915$	−36.2400	−1.19806
$916$	−3.38431	−0.111821
$917$	−62.1699	−2.05303
$918$	−33.8799	−1.11820
$919$	−12.6374	−0.416869	−0.208435	−	0.978036i	$-0.566837\pi$
$919$	−12.6374	−0.416869	−0.208435	+	0.978036i	$0.566837\pi$
$920$	−27.6026	−0.910030
$921$	−14.7714	−0.486735
$922$	−9.92521	−0.326869
$923$	−7.02292	−0.231162
$924$	1.36662	0.0449585
$925$	−60.7524	−1.99753
$926$	−9.03914	−0.297045
$927$	28.6381	0.940598
$928$	19.0114	0.624078
$929$	31.7973	1.04323	0.521617	−	0.853180i	$-0.325329\pi$
$929$	31.7973	1.04323	0.521617	+	0.853180i	$0.325329\pi$
$930$	4.21557	0.138234
$931$	6.45221	0.211463
$932$	4.99996	0.163779
$933$	12.6782	0.415065
$934$	29.9772	0.980886
$935$	−30.0711	−0.983430
$936$	−6.17309	−0.201774
$937$	−51.9324	−1.69656	−0.848278	−	0.529551i	$-0.822360\pi$
$937$	−51.9324	−1.69656	−0.848278	+	0.529551i	$0.822360\pi$
$938$	59.5158	1.94326
$939$	−4.90040	−0.159919
$940$	12.9825	0.423443
$941$	29.1612	0.950627	0.475313	−	0.879817i	$-0.342335\pi$
$941$	29.1612	0.950627	0.475313	+	0.879817i	$0.342335\pi$
$942$	−17.4121	−0.567316
$943$	−6.53291	−0.212741
$944$	−20.8387	−0.678240
$945$	−51.0807	−1.66166
$946$	0.589639	0.0191708
$947$	−30.6809	−0.996996	−0.498498	−	0.866891i	$-0.666115\pi$
$947$	−30.6809	−0.996996	−0.498498	+	0.866891i	$0.666115\pi$
$948$	−2.29572	−0.0745617
$949$	14.4428	0.468832
$950$	−18.8950	−0.613035
$951$	−5.57520	−0.180788
$952$	−47.0257	−1.52411
$953$	−51.8968	−1.68110	−0.840551	−	0.541732i	$-0.817769\pi$
$953$	−51.8968	−1.68110	−0.840551	+	0.541732i	$0.817769\pi$
$954$	12.4301	0.402438
$955$	−17.8776	−0.578504
$956$	−4.47885	−0.144856
$957$	9.93640	0.321198
$958$	16.1196	0.520802
$959$	−29.2172	−0.943473
$960$	16.4914	0.532258
$961$	1.00000	0.0322581
$962$	10.5231	0.339277
$963$	−4.73084	−0.152449
$964$	1.14159	0.0367680
$965$	62.4238	2.00949
$966$	11.4077	0.367037
$967$	51.3375	1.65090	0.825451	−	0.564474i	$-0.190921\pi$
$967$	51.3375	1.65090	0.825451	+	0.564474i	$0.190921\pi$
$968$	−22.1721	−0.712638
$969$	−5.56577	−0.178798
$970$	−14.9959	−0.481489
$971$	27.1645	0.871749	0.435874	−	0.900008i	$-0.356439\pi$
$971$	27.1645	0.871749	0.435874	+	0.900008i	$0.356439\pi$
$972$	5.65940	0.181525
$973$	−49.7310	−1.59430
$974$	−59.5797	−1.90906
$975$	6.53068	0.209149
$976$	60.9217	1.95006
$977$	10.8180	0.346100	0.173050	−	0.984913i	$-0.444638\pi$
$977$	10.8180	0.346100	0.173050	+	0.984913i	$0.444638\pi$
$978$	−14.1013	−0.450910
$979$	2.17287	0.0694452
$980$	6.44392	0.205843
$981$	39.9026	1.27399
$982$	−38.2051	−1.21917
$983$	4.39111	0.140055	0.0700274	−	0.997545i	$-0.477691\pi$
$983$	4.39111	0.140055	0.0700274	+	0.997545i	$0.477691\pi$
$984$	4.08066	0.130087
$985$	−68.4990	−2.18256
$986$	77.6308	2.47227
$987$	23.6315	0.752197
$988$	0.511034	0.0162582
$989$	0.768531	0.0244378
$990$	20.7624	0.659873
$991$	27.9776	0.888737	0.444368	−	0.895844i	$-0.353428\pi$
$991$	27.9776	0.888737	0.444368	+	0.895844i	$0.353428\pi$
$992$	−2.06807	−0.0656612
$993$	−25.8269	−0.819593
$994$	36.9385	1.17162
$995$	−54.4513	−1.72622
$996$	−0.754549	−0.0239088
$997$	−39.4392	−1.24905	−0.624526	−	0.781004i	$-0.714708\pi$
$997$	−39.4392	−1.24905	−0.624526	+	0.781004i	$0.714708\pi$
$998$	40.5832	1.28464
$999$	27.4232	0.867630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.a.e.1.3	✓	8
3.2	odd	2		3627.2.a.p.1.6		8
4.3	odd	2		6448.2.a.bd.1.5		8
13.12	even	2		5239.2.a.i.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.a.e.1.3	✓	8	1.1	even	1	trivial
3627.2.a.p.1.6		8	3.2	odd	2
5239.2.a.i.1.6		8	13.12	even	2
6448.2.a.bd.1.5		8	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:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.a (trivial)

\(f(q)\)	\(=\)	\(q-1.53950 q^{2} +0.734778 q^{3} +0.370072 q^{4} +3.72666 q^{5} -1.13119 q^{6} +3.41650 q^{7} +2.50928 q^{8} -2.46010 q^{9} -5.73720 q^{10} +1.47104 q^{11} +0.271920 q^{12} +1.00000 q^{13} -5.25971 q^{14} +2.73826 q^{15} -4.60319 q^{16} -5.48536 q^{17} +3.78734 q^{18} +1.38091 q^{19} +1.37913 q^{20} +2.51036 q^{21} -2.26467 q^{22} -2.95176 q^{23} +1.84376 q^{24} +8.88797 q^{25} -1.53950 q^{26} -4.01196 q^{27} +1.26435 q^{28} +9.19281 q^{29} -4.21557 q^{30} -1.00000 q^{31} +2.06807 q^{32} +1.08089 q^{33} +8.44473 q^{34} +12.7321 q^{35} -0.910415 q^{36} -6.83535 q^{37} -2.12591 q^{38} +0.734778 q^{39} +9.35123 q^{40} +2.21323 q^{41} -3.86472 q^{42} -0.260364 q^{43} +0.544391 q^{44} -9.16795 q^{45} +4.54424 q^{46} +9.41355 q^{47} -3.38232 q^{48} +4.67245 q^{49} -13.6831 q^{50} -4.03052 q^{51} +0.370072 q^{52} +3.28201 q^{53} +6.17643 q^{54} +5.48207 q^{55} +8.57295 q^{56} +1.01466 q^{57} -14.1524 q^{58} +4.52700 q^{59} +1.01335 q^{60} -13.2347 q^{61} +1.53950 q^{62} -8.40493 q^{63} +6.02258 q^{64} +3.72666 q^{65} -1.66403 q^{66} -11.3154 q^{67} -2.02998 q^{68} -2.16889 q^{69} -19.6011 q^{70} -7.02292 q^{71} -6.17309 q^{72} +14.4428 q^{73} +10.5231 q^{74} +6.53068 q^{75} +0.511034 q^{76} +5.02581 q^{77} -1.13119 q^{78} -8.44263 q^{79} -17.1545 q^{80} +4.43241 q^{81} -3.40727 q^{82} -2.77489 q^{83} +0.929015 q^{84} -20.4421 q^{85} +0.400831 q^{86} +6.75467 q^{87} +3.69126 q^{88} +1.47710 q^{89} +14.1141 q^{90} +3.41650 q^{91} -1.09236 q^{92} -0.734778 q^{93} -14.4922 q^{94} +5.14616 q^{95} +1.51957 q^{96} +2.61380 q^{97} -7.19325 q^{98} -3.61891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{3} + 7 q^{4} + 11 q^{5} - 2 q^{7} + 9 q^{9} - 2 q^{10} - 2 q^{11} + 19 q^{12} + 8 q^{13} + 3 q^{14} + 6 q^{15} - 7 q^{16} + 7 q^{17} - 12 q^{18} - 5 q^{19} + 6 q^{20} + 8 q^{21} - 4 q^{22}+ \cdots - 8 q^{99}+O(q^{100}) \) 8 * q - q^2 + 7 * q^3 + 7 * q^4 + 11 * q^5 - 2 * q^7 + 9 * q^9 - 2 * q^10 - 2 * q^11 + 19 * q^12 + 8 * q^13 + 3 * q^14 + 6 * q^15 - 7 * q^16 + 7 * q^17 - 12 * q^18 - 5 * q^19 + 6 * q^20 + 8 * q^21 - 4 * q^22 + 14 * q^23 - 17 * q^24 + 17 * q^25 - q^26 + 7 * q^27 - 9 * q^28 + 12 * q^29 - 9 * q^30 - 8 * q^31 - 21 * q^32 + 10 * q^33 - 12 * q^34 - q^35 + 11 * q^36 + 2 * q^37 + 24 * q^38 + 7 * q^39 - 19 * q^40 + 13 * q^41 - 27 * q^42 - 5 * q^43 + 22 * q^44 + 19 * q^45 + 17 * q^46 + 23 * q^47 + 3 * q^48 + 26 * q^49 - 26 * q^50 + 18 * q^51 + 7 * q^52 + 25 * q^53 - 36 * q^54 - 17 * q^55 + 8 * q^56 - 35 * q^57 - 29 * q^58 - 5 * q^59 + 71 * q^60 - 9 * q^61 + q^62 - 37 * q^63 - 14 * q^64 + 11 * q^65 - 41 * q^66 + 22 * q^67 - 6 * q^68 - 7 * q^69 - 29 * q^70 - 17 * q^71 - 34 * q^72 - 27 * q^73 + 14 * q^74 - 33 * q^75 - 36 * q^76 + 31 * q^77 - 23 * q^79 + 9 * q^80 - 12 * q^81 + 18 * q^82 - 25 * q^83 - 62 * q^84 + 13 * q^85 + 11 * q^86 + 26 * q^87 + 5 * q^88 + 2 * q^89 - 14 * q^90 - 2 * q^91 - 20 * q^92 - 7 * q^93 - 38 * q^94 + 3 * q^95 - 52 * q^96 - 15 * q^97 + 39 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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