LMFDB - Embedded newform 6845.2.a.g.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6845,2,Mod(1,6845)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6845, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6845.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6845 = 5 \cdot 37^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6845.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-1,6,-5,2,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$54.6576001836$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.368464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 6x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.78948$ of defining polynomial
Character	$\chi$	$=$	6845.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.66356 q^{2} -0.671838 q^{3} +5.09455 q^{4} -1.00000 q^{5} -1.78948 q^{6} -0.305070 q^{7} +8.24252 q^{8} -2.54863 q^{9} -2.66356 q^{10} +1.85927 q^{11} -3.42271 q^{12} -4.78120 q^{13} -0.812573 q^{14} +0.671838 q^{15} +11.7653 q^{16} +2.54592 q^{17} -6.78844 q^{18} -8.13969 q^{19} -5.09455 q^{20} +0.204958 q^{21} +4.95226 q^{22} -6.67080 q^{23} -5.53764 q^{24} +1.00000 q^{25} -12.7350 q^{26} +3.72778 q^{27} -1.55420 q^{28} -0.374855 q^{29} +1.78948 q^{30} +4.02477 q^{31} +14.8527 q^{32} -1.24913 q^{33} +6.78120 q^{34} +0.305070 q^{35} -12.9841 q^{36} -21.6806 q^{38} +3.21219 q^{39} -8.24252 q^{40} +3.06337 q^{41} +0.545917 q^{42} -11.5328 q^{43} +9.47212 q^{44} +2.54863 q^{45} -17.7681 q^{46} +4.13587 q^{47} -7.90441 q^{48} -6.90693 q^{49} +2.66356 q^{50} -1.71044 q^{51} -24.3581 q^{52} -2.51559 q^{53} +9.92917 q^{54} -1.85927 q^{55} -2.51455 q^{56} +5.46855 q^{57} -0.998448 q^{58} +2.62947 q^{59} +3.42271 q^{60} +2.92264 q^{61} +10.7202 q^{62} +0.777512 q^{63} +16.0303 q^{64} +4.78120 q^{65} -3.32712 q^{66} -8.57520 q^{67} +12.9703 q^{68} +4.48169 q^{69} +0.812573 q^{70} -14.7798 q^{71} -21.0072 q^{72} -8.32984 q^{73} -0.671838 q^{75} -41.4681 q^{76} -0.567206 q^{77} +8.55587 q^{78} -8.41618 q^{79} -11.7653 q^{80} +5.14143 q^{81} +8.15947 q^{82} +10.8444 q^{83} +1.04417 q^{84} -2.54592 q^{85} -30.7182 q^{86} +0.251842 q^{87} +15.3250 q^{88} -17.7053 q^{89} +6.78844 q^{90} +1.45860 q^{91} -33.9847 q^{92} -2.70399 q^{93} +11.0161 q^{94} +8.13969 q^{95} -9.97858 q^{96} +13.6612 q^{97} -18.3970 q^{98} -4.73859 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 6 q^{4} - 5 q^{5} + 2 q^{6} + 7 q^{7} + 6 q^{8} + 2 q^{9} + 7 q^{11} - 2 q^{13} - 4 q^{14} + q^{15} + 8 q^{16} + 8 q^{17} + 6 q^{18} - 14 q^{19} - 6 q^{20} - 9 q^{21} - 2 q^{22} - 2 q^{23}+ \cdots + 18 q^{99}+O(q^{100}) $ 5 * q - q^3 + 6 * q^4 - 5 * q^5 + 2 * q^6 + 7 * q^7 + 6 * q^8 + 2 * q^9 + 7 * q^11 - 2 * q^13 - 4 * q^14 + q^15 + 8 * q^16 + 8 * q^17 + 6 * q^18 - 14 * q^19 - 6 * q^20 - 9 * q^21 - 2 * q^22 - 2 * q^23 - 12 * q^24 + 5 * q^25 - 20 * q^26 - 7 * q^27 - 14 * q^28 - 2 * q^29 - 2 * q^30 - 8 * q^31 + 22 * q^32 - 21 * q^33 + 12 * q^34 - 7 * q^35 - 36 * q^36 - 32 * q^38 - 12 * q^39 - 6 * q^40 - 9 * q^41 - 2 * q^42 - 14 * q^43 + 18 * q^44 - 2 * q^45 - 28 * q^46 - 5 * q^47 - 30 * q^48 + 2 * q^49 - 10 * q^51 - 20 * q^52 - 15 * q^53 + 12 * q^54 - 7 * q^55 + 14 * q^56 - 4 * q^57 - 12 * q^59 - 12 * q^61 + 10 * q^62 + 24 * q^63 + 12 * q^64 + 2 * q^65 + 10 * q^66 - 2 * q^67 + 4 * q^68 + 30 * q^69 + 4 * q^70 + 13 * q^71 - 26 * q^72 - 5 * q^73 - q^75 - 18 * q^76 - 7 * q^77 + 10 * q^78 - 36 * q^79 - 8 * q^80 + 21 * q^81 + 2 * q^82 - 9 * q^83 + 30 * q^84 - 8 * q^85 - 8 * q^86 + 6 * q^87 - 8 * q^88 + 16 * q^89 - 6 * q^90 + 4 * q^91 - 12 * q^92 + 16 * q^93 - 20 * q^94 + 14 * q^95 - 30 * q^98 + 18 * 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 $

$2$	2.66356	1.88342	0.941711	−	0.336424i	$-0.109218\pi$
$2$	2.66356	1.88342	0.941711	+	0.336424i	$0.109218\pi$
$3$	−0.671838	−0.387886	−0.193943	−	0.981013i	$-0.562128\pi$
$3$	−0.671838	−0.387886	−0.193943	+	0.981013i	$0.562128\pi$
$4$	5.09455	2.54728
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.78948	−0.730552
$7$	−0.305070	−0.115306	−0.0576528	−	0.998337i	$-0.518362\pi$
$7$	−0.305070	−0.115306	−0.0576528	+	0.998337i	$0.518362\pi$
$8$	8.24252	2.91417
$9$	−2.54863	−0.849545
$10$	−2.66356	−0.842292
$11$	1.85927	0.560590	0.280295	−	0.959914i	$-0.409568\pi$
$11$	1.85927	0.560590	0.280295	+	0.959914i	$0.409568\pi$
$12$	−3.42271	−0.988052
$13$	−4.78120	−1.32607	−0.663033	−	0.748590i	$-0.730731\pi$
$13$	−4.78120	−1.32607	−0.663033	+	0.748590i	$0.730731\pi$
$14$	−0.812573	−0.217169
$15$	0.671838	0.173468
$16$	11.7653	2.94134
$17$	2.54592	0.617476	0.308738	−	0.951147i	$-0.400093\pi$
$17$	2.54592	0.617476	0.308738	+	0.951147i	$0.400093\pi$
$18$	−6.78844	−1.60005
$19$	−8.13969	−1.86737	−0.933687	−	0.358091i	$-0.883428\pi$
$19$	−8.13969	−1.86737	−0.933687	+	0.358091i	$0.883428\pi$
$20$	−5.09455	−1.13918
$21$	0.204958	0.0447254
$22$	4.95226	1.05583
$23$	−6.67080	−1.39096	−0.695479	−	0.718547i	$-0.744808\pi$
$23$	−6.67080	−1.39096	−0.695479	+	0.718547i	$0.744808\pi$
$24$	−5.53764	−1.13037
$25$	1.00000	0.200000
$26$	−12.7350	−2.49754
$27$	3.72778	0.717412
$28$	−1.55420	−0.293715
$29$	−0.374855	−0.0696088	−0.0348044	−	0.999394i	$-0.511081\pi$
$29$	−0.374855	−0.0696088	−0.0348044	+	0.999394i	$0.511081\pi$
$30$	1.78948	0.326713
$31$	4.02477	0.722869	0.361435	−	0.932397i	$-0.382287\pi$
$31$	4.02477	0.722869	0.361435	+	0.932397i	$0.382287\pi$
$32$	14.8527	2.62560
$33$	−1.24913	−0.217445
$34$	6.78120	1.16297
$35$	0.305070	0.0515663
$36$	−12.9841	−2.16402
$37$	0	0
$37$	0	0
$38$	−21.6806	−3.51705
$39$	3.21219	0.514363
$40$	−8.24252	−1.30326
$41$	3.06337	0.478418	0.239209	−	0.970968i	$-0.423112\pi$
$41$	3.06337	0.478418	0.239209	+	0.970968i	$0.423112\pi$
$42$	0.545917	0.0842368
$43$	−11.5328	−1.75873	−0.879366	−	0.476146i	$-0.842033\pi$
$43$	−11.5328	−1.75873	−0.879366	+	0.476146i	$0.842033\pi$
$44$	9.47212	1.42798
$45$	2.54863	0.379928
$46$	−17.7681	−2.61976
$47$	4.13587	0.603279	0.301640	−	0.953422i	$-0.402466\pi$
$47$	4.13587	0.603279	0.301640	+	0.953422i	$0.402466\pi$
$48$	−7.90441	−1.14090
$49$	−6.90693	−0.986705
$50$	2.66356	0.376684
$51$	−1.71044	−0.239510
$52$	−24.3581	−3.37786
$53$	−2.51559	−0.345543	−0.172771	−	0.984962i	$-0.555272\pi$
$53$	−2.51559	−0.345543	−0.172771	+	0.984962i	$0.555272\pi$
$54$	9.92917	1.35119
$55$	−1.85927	−0.250703
$56$	−2.51455	−0.336020
$57$	5.46855	0.724328
$58$	−0.998448	−0.131103
$59$	2.62947	0.342328	0.171164	−	0.985243i	$-0.445247\pi$
$59$	2.62947	0.342328	0.171164	+	0.985243i	$0.445247\pi$
$60$	3.42271	0.441870
$61$	2.92264	0.374205	0.187103	−	0.982340i	$-0.440090\pi$
$61$	2.92264	0.374205	0.187103	+	0.982340i	$0.440090\pi$
$62$	10.7202	1.36147
$63$	0.777512	0.0979573
$64$	16.0303	2.00378
$65$	4.78120	0.593035
$66$	−3.32712	−0.409540
$67$	−8.57520	−1.04763	−0.523814	−	0.851833i	$-0.675491\pi$
$67$	−8.57520	−1.04763	−0.523814	+	0.851833i	$0.675491\pi$
$68$	12.9703	1.57288
$69$	4.48169	0.539533
$70$	0.812573	0.0971210
$71$	−14.7798	−1.75404	−0.877021	−	0.480452i	$-0.840473\pi$
$71$	−14.7798	−1.75404	−0.877021	+	0.480452i	$0.840473\pi$
$72$	−21.0072	−2.47572
$73$	−8.32984	−0.974934	−0.487467	−	0.873142i	$-0.662079\pi$
$73$	−8.32984	−0.974934	−0.487467	+	0.873142i	$0.662079\pi$
$74$	0	0
$75$	−0.671838	−0.0775772
$76$	−41.4681	−4.75671
$77$	−0.567206	−0.0646392
$78$	8.55587	0.968761
$79$	−8.41618	−0.946894	−0.473447	−	0.880822i	$-0.656990\pi$
$79$	−8.41618	−0.946894	−0.473447	+	0.880822i	$0.656990\pi$
$80$	−11.7653	−1.31541
$81$	5.14143	0.571271
$82$	8.15947	0.901063
$83$	10.8444	1.19033	0.595163	−	0.803605i	$-0.297088\pi$
$83$	10.8444	1.19033	0.595163	+	0.803605i	$0.297088\pi$
$84$	1.04417	0.113928
$85$	−2.54592	−0.276143
$86$	−30.7182	−3.31243
$87$	0.251842	0.0270003
$88$	15.3250	1.63365
$89$	−17.7053	−1.87676	−0.938380	−	0.345605i	$-0.887674\pi$
$89$	−17.7053	−1.87676	−0.938380	+	0.345605i	$0.887674\pi$
$90$	6.78844	0.715564
$91$	1.45860	0.152903
$92$	−33.9847	−3.54315
$93$	−2.70399	−0.280391
$94$	11.0161	1.13623
$95$	8.13969	0.835115
$96$	−9.97858	−1.01843
$97$	13.6612	1.38709	0.693544	−	0.720415i	$-0.256048\pi$
$97$	13.6612	1.38709	0.693544	+	0.720415i	$0.256048\pi$
$98$	−18.3970	−1.85838
$99$	−4.73859	−0.476246
$100$	5.09455	0.509455
$101$	2.57981	0.256701	0.128350	−	0.991729i	$-0.459032\pi$
$101$	2.57981	0.256701	0.128350	+	0.991729i	$0.459032\pi$
$102$	−4.55587	−0.451098
$103$	8.31211	0.819017	0.409509	−	0.912306i	$-0.365700\pi$
$103$	8.31211	0.819017	0.409509	+	0.912306i	$0.365700\pi$
$104$	−39.4092	−3.86439
$105$	−0.204958	−0.0200018
$106$	−6.70042	−0.650803
$107$	−10.2766	−0.993477	−0.496739	−	0.867900i	$-0.665469\pi$
$107$	−10.2766	−0.993477	−0.496739	+	0.867900i	$0.665469\pi$
$108$	18.9914	1.82745
$109$	19.7515	1.89185	0.945926	−	0.324384i	$-0.105157\pi$
$109$	19.7515	1.89185	0.945926	+	0.324384i	$0.105157\pi$
$110$	−4.95226	−0.472180
$111$	0	0
$112$	−3.58926	−0.339153
$113$	0.193052	0.0181608	0.00908039	−	0.999959i	$-0.497110\pi$
$113$	0.193052	0.0181608	0.00908039	+	0.999959i	$0.497110\pi$
$114$	14.5658	1.36421
$115$	6.67080	0.622055
$116$	−1.90972	−0.177313
$117$	12.1855	1.12655
$118$	7.00376	0.644748
$119$	−0.776683	−0.0711984
$120$	5.53764	0.505515
$121$	−7.54313	−0.685739
$122$	7.78462	0.704786
$123$	−2.05809	−0.185572
$124$	20.5044	1.84135
$125$	−1.00000	−0.0894427
$126$	2.07095	0.184495
$127$	3.21589	0.285364	0.142682	−	0.989769i	$-0.454427\pi$
$127$	3.21589	0.285364	0.142682	+	0.989769i	$0.454427\pi$
$128$	12.9922	1.14836
$129$	7.74816	0.682187
$130$	12.7350	1.11694
$131$	−12.9029	−1.12733	−0.563664	−	0.826004i	$-0.690609\pi$
$131$	−12.9029	−1.12733	−0.563664	+	0.826004i	$0.690609\pi$
$132$	−6.36373	−0.553892
$133$	2.48318	0.215319
$134$	−22.8406	−1.97312
$135$	−3.72778	−0.320836
$136$	20.9848	1.79943
$137$	−6.77934	−0.579198	−0.289599	−	0.957148i	$-0.593522\pi$
$137$	−6.77934	−0.579198	−0.289599	+	0.957148i	$0.593522\pi$
$138$	11.9373	1.01617
$139$	−6.06609	−0.514519	−0.257259	−	0.966342i	$-0.582819\pi$
$139$	−6.06609	−0.514519	−0.257259	+	0.966342i	$0.582819\pi$
$140$	1.55420	0.131353
$141$	−2.77864	−0.234003
$142$	−39.3669	−3.30360
$143$	−8.88952	−0.743379
$144$	−29.9856	−2.49880
$145$	0.374855	0.0311300
$146$	−22.1870	−1.83621
$147$	4.64034	0.382729
$148$	0	0
$149$	−10.2822	−0.842348	−0.421174	−	0.906980i	$-0.638382\pi$
$149$	−10.2822	−0.842348	−0.421174	+	0.906980i	$0.638382\pi$
$150$	−1.78948	−0.146110
$151$	11.1876	0.910430	0.455215	−	0.890382i	$-0.349562\pi$
$151$	11.1876	0.910430	0.455215	+	0.890382i	$0.349562\pi$
$152$	−67.0916	−5.44185
$153$	−6.48861	−0.524573
$154$	−1.51079	−0.121743
$155$	−4.02477	−0.323277
$156$	16.3647	1.31022
$157$	16.6402	1.32804	0.664018	−	0.747716i	$-0.268850\pi$
$157$	16.6402	1.32804	0.664018	+	0.747716i	$0.268850\pi$
$158$	−22.4170	−1.78340
$159$	1.69007	0.134031
$160$	−14.8527	−1.17421
$161$	2.03506	0.160385
$162$	13.6945	1.07594
$163$	11.1117	0.870339	0.435169	−	0.900349i	$-0.356689\pi$
$163$	11.1117	0.870339	0.435169	+	0.900349i	$0.356689\pi$
$164$	15.6065	1.21866
$165$	1.24913	0.0972443
$166$	28.8847	2.24188
$167$	−2.24836	−0.173983	−0.0869915	−	0.996209i	$-0.527725\pi$
$167$	−2.24836	−0.173983	−0.0869915	+	0.996209i	$0.527725\pi$
$168$	1.68937	0.130338
$169$	9.85990	0.758454
$170$	−6.78120	−0.520094
$171$	20.7451	1.58642
$172$	−58.7543	−4.47998
$173$	−10.3774	−0.788981	−0.394490	−	0.918900i	$-0.629079\pi$
$173$	−10.3774	−0.788981	−0.394490	+	0.918900i	$0.629079\pi$
$174$	0.670796	0.0508529
$175$	−0.305070	−0.0230611
$176$	21.8749	1.64888
$177$	−1.76658	−0.132784
$178$	−47.1592	−3.53473
$179$	9.67455	0.723110	0.361555	−	0.932351i	$-0.382246\pi$
$179$	9.67455	0.723110	0.361555	+	0.932351i	$0.382246\pi$
$180$	12.9841	0.967781
$181$	22.1236	1.64443	0.822216	−	0.569175i	$-0.192737\pi$
$181$	22.1236	1.64443	0.822216	+	0.569175i	$0.192737\pi$
$182$	3.88507	0.287981
$183$	−1.96354	−0.145149
$184$	−54.9842	−4.05349
$185$	0	0
$186$	−7.20224	−0.528094
$187$	4.73354	0.346150
$188$	21.0704	1.53672
$189$	−1.13724	−0.0827217
$190$	21.6806	1.57287
$191$	−13.4291	−0.971695	−0.485848	−	0.874043i	$-0.661489\pi$
$191$	−13.4291	−0.971695	−0.485848	+	0.874043i	$0.661489\pi$
$192$	−10.7697	−0.777239
$193$	−5.54030	−0.398799	−0.199400	−	0.979918i	$-0.563899\pi$
$193$	−5.54030	−0.398799	−0.199400	+	0.979918i	$0.563899\pi$
$194$	36.3875	2.61247
$195$	−3.21219	−0.230030
$196$	−35.1877	−2.51341
$197$	9.64388	0.687098	0.343549	−	0.939135i	$-0.388371\pi$
$197$	9.64388	0.687098	0.343549	+	0.939135i	$0.388371\pi$
$198$	−12.6215	−0.896972
$199$	−9.72036	−0.689058	−0.344529	−	0.938776i	$-0.611961\pi$
$199$	−9.72036	−0.689058	−0.344529	+	0.938776i	$0.611961\pi$
$200$	8.24252	0.582834
$201$	5.76115	0.406360
$202$	6.87149	0.483476
$203$	0.114357	0.00802629
$204$	−8.71394	−0.610098
$205$	−3.06337	−0.213955
$206$	22.1398	1.54255
$207$	17.0014	1.18168
$208$	−56.2525	−3.90041
$209$	−15.1338	−1.04683
$210$	−0.545917	−0.0376719
$211$	18.2671	1.25756	0.628779	−	0.777584i	$-0.283555\pi$
$211$	18.2671	1.25756	0.628779	+	0.777584i	$0.283555\pi$
$212$	−12.8158	−0.880193
$213$	9.92964	0.680368
$214$	−27.3724	−1.87114
$215$	11.5328	0.786529
$216$	30.7263	2.09066
$217$	−1.22784	−0.0833509
$218$	52.6093	3.56315
$219$	5.59630	0.378163
$220$	−9.47212	−0.638610
$221$	−12.1725	−0.818814
$222$	0	0
$223$	−3.90014	−0.261173	−0.130586	−	0.991437i	$-0.541686\pi$
$223$	−3.90014	−0.261173	−0.130586	+	0.991437i	$0.541686\pi$
$224$	−4.53110	−0.302747
$225$	−2.54863	−0.169909
$226$	0.514205	0.0342044
$227$	23.0180	1.52776	0.763878	−	0.645360i	$-0.223293\pi$
$227$	23.0180	1.52776	0.763878	+	0.645360i	$0.223293\pi$
$228$	27.8598	1.84506
$229$	−24.9325	−1.64758	−0.823792	−	0.566893i	$-0.808145\pi$
$229$	−24.9325	−1.64758	−0.823792	+	0.566893i	$0.808145\pi$
$230$	17.7681	1.17159
$231$	0.381071	0.0250726
$232$	−3.08975	−0.202852
$233$	27.1703	1.77999	0.889993	−	0.455974i	$-0.150709\pi$
$233$	27.1703	1.77999	0.889993	+	0.455974i	$0.150709\pi$
$234$	32.4569	2.12177
$235$	−4.13587	−0.269795
$236$	13.3960	0.872004
$237$	5.65431	0.367287
$238$	−2.06874	−0.134097
$239$	−11.4668	−0.741726	−0.370863	−	0.928688i	$-0.620938\pi$
$239$	−11.4668	−0.741726	−0.370863	+	0.928688i	$0.620938\pi$
$240$	7.90441	0.510227
$241$	13.6581	0.879797	0.439898	−	0.898048i	$-0.355014\pi$
$241$	13.6581	0.879797	0.439898	+	0.898048i	$0.355014\pi$
$242$	−20.0916	−1.29154
$243$	−14.6376	−0.939000
$244$	14.8895	0.953204
$245$	6.90693	0.441268
$246$	−5.48184	−0.349510
$247$	38.9175	2.47626
$248$	33.1742	2.10657
$249$	−7.28567	−0.461710
$250$	−2.66356	−0.168458
$251$	6.65785	0.420240	0.210120	−	0.977676i	$-0.432615\pi$
$251$	6.65785	0.420240	0.210120	+	0.977676i	$0.432615\pi$
$252$	3.96107	0.249524
$253$	−12.4028	−0.779756
$254$	8.56571	0.537461
$255$	1.71044	0.107112
$256$	2.54507	0.159067
$257$	11.5885	0.722869	0.361434	−	0.932398i	$-0.382287\pi$
$257$	11.5885	0.722869	0.361434	+	0.932398i	$0.382287\pi$
$258$	20.6377	1.28485
$259$	0	0
$260$	24.3581	1.51062
$261$	0.955368	0.0591358
$262$	−34.3675	−2.12323
$263$	−17.1403	−1.05692	−0.528459	−	0.848959i	$-0.677230\pi$
$263$	−17.1403	−1.05692	−0.528459	+	0.848959i	$0.677230\pi$
$264$	−10.2959	−0.633671
$265$	2.51559	0.154531
$266$	6.61409	0.405536
$267$	11.8951	0.727969
$268$	−43.6868	−2.66860
$269$	−2.00497	−0.122245	−0.0611225	−	0.998130i	$-0.519468\pi$
$269$	−2.00497	−0.122245	−0.0611225	+	0.998130i	$0.519468\pi$
$270$	−9.92917	−0.604270
$271$	5.82824	0.354040	0.177020	−	0.984207i	$-0.443354\pi$
$271$	5.82824	0.354040	0.177020	+	0.984207i	$0.443354\pi$
$272$	29.9536	1.81620
$273$	−0.979944	−0.0593089
$274$	−18.0572	−1.09087
$275$	1.85927	0.112118
$276$	22.8322	1.37434
$277$	−21.8323	−1.31177	−0.655887	−	0.754859i	$-0.727705\pi$
$277$	−21.8323	−1.31177	−0.655887	+	0.754859i	$0.727705\pi$
$278$	−16.1574	−0.969056
$279$	−10.2577	−0.614110
$280$	2.51455	0.150273
$281$	−3.56449	−0.212640	−0.106320	−	0.994332i	$-0.533907\pi$
$281$	−3.56449	−0.212640	−0.106320	+	0.994332i	$0.533907\pi$
$282$	−7.40107	−0.440727
$283$	14.0437	0.834812	0.417406	−	0.908720i	$-0.362939\pi$
$283$	14.0437	0.834812	0.417406	+	0.908720i	$0.362939\pi$
$284$	−75.2965	−4.46803
$285$	−5.46855	−0.323929
$286$	−23.6778	−1.40010
$287$	−0.934543	−0.0551643
$288$	−37.8540	−2.23057
$289$	−10.5183	−0.618724
$290$	0.998448	0.0586309
$291$	−9.17813	−0.538031
$292$	−42.4368	−2.48342
$293$	−23.9496	−1.39915	−0.699577	−	0.714558i	$-0.746628\pi$
$293$	−23.9496	−1.39915	−0.699577	+	0.714558i	$0.746628\pi$
$294$	12.3598	0.720839
$295$	−2.62947	−0.153094
$296$	0	0
$297$	6.93094	0.402174
$298$	−27.3872	−1.58650
$299$	31.8944	1.84450
$300$	−3.42271	−0.197610
$301$	3.51831	0.202792
$302$	29.7987	1.71472
$303$	−1.73322	−0.0995707
$304$	−95.7663	−5.49257
$305$	−2.92264	−0.167350
$306$	−17.2828	−0.987992
$307$	−3.40442	−0.194301	−0.0971503	−	0.995270i	$-0.530973\pi$
$307$	−3.40442	−0.194301	−0.0971503	+	0.995270i	$0.530973\pi$
$308$	−2.88966	−0.164654
$309$	−5.58439	−0.317685
$310$	−10.7202	−0.608867
$311$	−1.85014	−0.104912	−0.0524558	−	0.998623i	$-0.516705\pi$
$311$	−1.85014	−0.104912	−0.0524558	+	0.998623i	$0.516705\pi$
$312$	26.4766	1.49894
$313$	24.8106	1.40238	0.701189	−	0.712975i	$-0.252653\pi$
$313$	24.8106	1.40238	0.701189	+	0.712975i	$0.252653\pi$
$314$	44.3223	2.50125
$315$	−0.777512	−0.0438078
$316$	−42.8766	−2.41200
$317$	−29.0459	−1.63138	−0.815690	−	0.578489i	$-0.803642\pi$
$317$	−29.0459	−1.63138	−0.815690	+	0.578489i	$0.803642\pi$
$318$	4.50160	0.252437
$319$	−0.696955	−0.0390220
$320$	−16.0303	−0.896119
$321$	6.90422	0.385356
$322$	5.42051	0.302073
$323$	−20.7230	−1.15306
$324$	26.1933	1.45518
$325$	−4.78120	−0.265213
$326$	29.5968	1.63921
$327$	−13.2698	−0.733822
$328$	25.2499	1.39419
$329$	−1.26173	−0.0695615
$330$	3.32712	0.183152
$331$	5.75347	0.316239	0.158120	−	0.987420i	$-0.449457\pi$
$331$	5.75347	0.316239	0.158120	+	0.987420i	$0.449457\pi$
$332$	55.2473	3.03209
$333$	0	0
$334$	−5.98863	−0.327683
$335$	8.57520	0.468513
$336$	2.41140	0.131553
$337$	−1.93856	−0.105600	−0.0528001	−	0.998605i	$-0.516815\pi$
$337$	−1.93856	−0.105600	−0.0528001	+	0.998605i	$0.516815\pi$
$338$	26.2624	1.42849
$339$	−0.129700	−0.00704431
$340$	−12.9703	−0.703414
$341$	7.48311	0.405233
$342$	55.2558	2.98789
$343$	4.24259	0.229078
$344$	−95.0592	−5.12525
$345$	−4.48169	−0.241286
$346$	−27.6409	−1.48598
$347$	−21.1544	−1.13563	−0.567815	−	0.823156i	$-0.692211\pi$
$347$	−21.1544	−1.13563	−0.567815	+	0.823156i	$0.692211\pi$
$348$	1.28302	0.0687771
$349$	−16.4132	−0.878577	−0.439289	−	0.898346i	$-0.644769\pi$
$349$	−16.4132	−0.878577	−0.439289	+	0.898346i	$0.644769\pi$
$350$	−0.812573	−0.0434338
$351$	−17.8233	−0.951337
$352$	27.6150	1.47189
$353$	0.557505	0.0296730	0.0148365	−	0.999890i	$-0.495277\pi$
$353$	0.557505	0.0296730	0.0148365	+	0.999890i	$0.495277\pi$
$354$	−4.70539	−0.250089
$355$	14.7798	0.784431
$356$	−90.2007	−4.78063
$357$	0.521805	0.0276169
$358$	25.7688	1.36192
$359$	2.92356	0.154299	0.0771497	−	0.997020i	$-0.475418\pi$
$359$	2.92356	0.154299	0.0771497	+	0.997020i	$0.475418\pi$
$360$	21.0072	1.10717
$361$	47.2546	2.48708
$362$	58.9275	3.09716
$363$	5.06776	0.265989
$364$	7.43092	0.389486
$365$	8.32984	0.436004
$366$	−5.23000	−0.273377
$367$	9.11824	0.475968	0.237984	−	0.971269i	$-0.423513\pi$
$367$	9.11824	0.475968	0.237984	+	0.971269i	$0.423513\pi$
$368$	−78.4842	−4.09127
$369$	−7.80741	−0.406438
$370$	0	0
$371$	0.767431	0.0398430
$372$	−13.7756	−0.714233
$373$	14.0591	0.727953	0.363977	−	0.931408i	$-0.381419\pi$
$373$	14.0591	0.727953	0.363977	+	0.931408i	$0.381419\pi$
$374$	12.6081	0.651947
$375$	0.671838	0.0346936
$376$	34.0900	1.75806
$377$	1.79226	0.0923059
$378$	−3.02909	−0.155800
$379$	0.614911	0.0315858	0.0157929	−	0.999875i	$-0.494973\pi$
$379$	0.614911	0.0315858	0.0157929	+	0.999875i	$0.494973\pi$
$380$	41.4681	2.12727
$381$	−2.16056	−0.110689
$382$	−35.7692	−1.83011
$383$	−1.29749	−0.0662988	−0.0331494	−	0.999450i	$-0.510554\pi$
$383$	−1.29749	−0.0662988	−0.0331494	+	0.999450i	$0.510554\pi$
$384$	−8.72867	−0.445433
$385$	0.567206	0.0289075
$386$	−14.7569	−0.751107
$387$	29.3928	1.49412
$388$	69.5978	3.53329
$389$	−31.3099	−1.58747	−0.793737	−	0.608261i	$-0.791867\pi$
$389$	−31.3099	−1.58747	−0.793737	+	0.608261i	$0.791867\pi$
$390$	−8.55587	−0.433243
$391$	−16.9833	−0.858882
$392$	−56.9305	−2.87543
$393$	8.66863	0.437274
$394$	25.6871	1.29410
$395$	8.41618	0.423464
$396$	−24.1410	−1.21313
$397$	−6.53409	−0.327937	−0.163968	−	0.986466i	$-0.552430\pi$
$397$	−6.53409	−0.327937	−0.163968	+	0.986466i	$0.552430\pi$
$398$	−25.8907	−1.29779
$399$	−1.66829	−0.0835191
$400$	11.7653	0.588267
$401$	9.19104	0.458978	0.229489	−	0.973311i	$-0.426294\pi$
$401$	9.19104	0.458978	0.229489	+	0.973311i	$0.426294\pi$
$402$	15.3452	0.765347
$403$	−19.2432	−0.958573
$404$	13.1430	0.653888
$405$	−5.14143	−0.255480
$406$	0.304597	0.0151169
$407$	0	0
$408$	−14.0984	−0.697973
$409$	34.8027	1.72088	0.860442	−	0.509549i	$-0.170188\pi$
$409$	34.8027	1.72088	0.860442	+	0.509549i	$0.170188\pi$
$410$	−8.15947	−0.402968
$411$	4.55462	0.224663
$412$	42.3465	2.08626
$413$	−0.802174	−0.0394724
$414$	45.2843	2.22560
$415$	−10.8444	−0.532330
$416$	−71.0136	−3.48173
$417$	4.07543	0.199575
$418$	−40.3099	−1.97162
$419$	−13.4200	−0.655608	−0.327804	−	0.944746i	$-0.606309\pi$
$419$	−13.4200	−0.655608	−0.327804	+	0.944746i	$0.606309\pi$
$420$	−1.04417	−0.0509502
$421$	−21.6891	−1.05706	−0.528532	−	0.848913i	$-0.677257\pi$
$421$	−21.6891	−1.05706	−0.528532	+	0.848913i	$0.677257\pi$
$422$	48.6555	2.36851
$423$	−10.5408	−0.512512
$424$	−20.7348	−1.00697
$425$	2.54592	0.123495
$426$	26.4482	1.28142
$427$	−0.891609	−0.0431480
$428$	−52.3547	−2.53066
$429$	5.97232	0.288346
$430$	30.7182	1.48137
$431$	21.5555	1.03829	0.519145	−	0.854686i	$-0.326250\pi$
$431$	21.5555	1.03829	0.519145	+	0.854686i	$0.326250\pi$
$432$	43.8587	2.11015
$433$	−7.21533	−0.346747	−0.173373	−	0.984856i	$-0.555467\pi$
$433$	−7.21533	−0.346747	−0.173373	+	0.984856i	$0.555467\pi$
$434$	−3.27041	−0.156985
$435$	−0.251842	−0.0120749
$436$	100.625	4.81907
$437$	54.2982	2.59744
$438$	14.9061	0.712240
$439$	−11.6800	−0.557455	−0.278728	−	0.960370i	$-0.589913\pi$
$439$	−11.6800	−0.557455	−0.278728	+	0.960370i	$0.589913\pi$
$440$	−15.3250	−0.730592
$441$	17.6032	0.838250
$442$	−32.4223	−1.54217
$443$	−2.79428	−0.132760	−0.0663802	−	0.997794i	$-0.521145\pi$
$443$	−2.79428	−0.132760	−0.0663802	+	0.997794i	$0.521145\pi$
$444$	0	0
$445$	17.7053	0.839313
$446$	−10.3883	−0.491898
$447$	6.90795	0.326735
$448$	−4.89035	−0.231047
$449$	27.3557	1.29099	0.645497	−	0.763763i	$-0.276650\pi$
$449$	27.3557	1.29099	0.645497	+	0.763763i	$0.276650\pi$
$450$	−6.78844	−0.320010
$451$	5.69562	0.268196
$452$	0.983512	0.0462605
$453$	−7.51622	−0.353143
$454$	61.3097	2.87741
$455$	−1.45860	−0.0683803
$456$	45.0747	2.11082
$457$	−32.6562	−1.52759	−0.763796	−	0.645458i	$-0.776667\pi$
$457$	−32.6562	−1.52759	−0.763796	+	0.645458i	$0.776667\pi$
$458$	−66.4091	−3.10309
$459$	9.49063	0.442985
$460$	33.9847	1.58455
$461$	−16.8322	−0.783955	−0.391977	−	0.919975i	$-0.628209\pi$
$461$	−16.8322	−0.783955	−0.391977	+	0.919975i	$0.628209\pi$
$462$	1.01500	0.0472223
$463$	−11.0460	−0.513353	−0.256676	−	0.966497i	$-0.582627\pi$
$463$	−11.0460	−0.513353	−0.256676	+	0.966497i	$0.582627\pi$
$464$	−4.41030	−0.204743
$465$	2.70399	0.125395
$466$	72.3697	3.35246
$467$	3.59552	0.166381	0.0831904	−	0.996534i	$-0.473489\pi$
$467$	3.59552	0.166381	0.0831904	+	0.996534i	$0.473489\pi$
$468$	62.0798	2.86964
$469$	2.61604	0.120797
$470$	−11.0161	−0.508137
$471$	−11.1796	−0.515127
$472$	21.6735	0.997603
$473$	−21.4425	−0.985927
$474$	15.0606	0.691756
$475$	−8.13969	−0.373475
$476$	−3.95685	−0.181362
$477$	6.41132	0.293554
$478$	−30.5425	−1.39698
$479$	13.3049	0.607915	0.303958	−	0.952686i	$-0.401692\pi$
$479$	13.3049	0.607915	0.303958	+	0.952686i	$0.401692\pi$
$480$	9.97858	0.455458
$481$	0	0
$482$	36.3792	1.65703
$483$	−1.36723	−0.0622112
$484$	−38.4289	−1.74677
$485$	−13.6612	−0.620324
$486$	−38.9880	−1.76853
$487$	−14.6228	−0.662624	−0.331312	−	0.943521i	$-0.607491\pi$
$487$	−14.6228	−0.662624	−0.331312	+	0.943521i	$0.607491\pi$
$488$	24.0899	1.09050
$489$	−7.46529	−0.337592
$490$	18.3970	0.831093
$491$	−16.3252	−0.736745	−0.368373	−	0.929678i	$-0.620085\pi$
$491$	−16.3252	−0.736745	−0.368373	+	0.929678i	$0.620085\pi$
$492$	−10.4850	−0.472702
$493$	−0.954350	−0.0429817
$494$	103.659	4.66384
$495$	4.73859	0.212984
$496$	47.3528	2.12620
$497$	4.50888	0.202251
$498$	−19.4058	−0.869595
$499$	5.52070	0.247140	0.123570	−	0.992336i	$-0.460566\pi$
$499$	5.52070	0.247140	0.123570	+	0.992336i	$0.460566\pi$
$500$	−5.09455	−0.227835
$501$	1.51053	0.0674856
$502$	17.7336	0.791488
$503$	8.54778	0.381127	0.190563	−	0.981675i	$-0.438969\pi$
$503$	8.54778	0.381127	0.190563	+	0.981675i	$0.438969\pi$
$504$	6.40866	0.285464
$505$	−2.57981	−0.114800
$506$	−33.0355	−1.46861
$507$	−6.62425	−0.294193
$508$	16.3835	0.726901
$509$	28.9252	1.28209	0.641044	−	0.767505i	$-0.278502\pi$
$509$	28.9252	1.28209	0.641044	+	0.767505i	$0.278502\pi$
$510$	4.55587	0.201737
$511$	2.54118	0.112415
$512$	−19.2055	−0.848772
$513$	−30.3430	−1.33968
$514$	30.8666	1.36147
$515$	−8.31211	−0.366276
$516$	39.4734	1.73772
$517$	7.68969	0.338192
$518$	0	0
$519$	6.97195	0.306034
$520$	39.4092	1.72821
$521$	−7.22238	−0.316418	−0.158209	−	0.987406i	$-0.550572\pi$
$521$	−7.22238	−0.316418	−0.158209	+	0.987406i	$0.550572\pi$
$522$	2.54468	0.111378
$523$	−17.6241	−0.770647	−0.385324	−	0.922782i	$-0.625910\pi$
$523$	−17.6241	−0.770647	−0.385324	+	0.922782i	$0.625910\pi$
$524$	−65.7343	−2.87161
$525$	0.204958	0.00894509
$526$	−45.6543	−1.99062
$527$	10.2467	0.446354
$528$	−14.6964	−0.639578
$529$	21.4995	0.934761
$530$	6.70042	0.291048
$531$	−6.70156	−0.290823
$532$	12.6507	0.548476
$533$	−14.6466	−0.634415
$534$	31.6833	1.37107
$535$	10.2766	0.444297
$536$	−70.6813	−3.05297
$537$	−6.49973	−0.280484
$538$	−5.34035	−0.230239
$539$	−12.8418	−0.553136
$540$	−18.9914	−0.817259
$541$	−7.75499	−0.333413	−0.166707	−	0.986007i	$-0.553313\pi$
$541$	−7.75499	−0.333413	−0.166707	+	0.986007i	$0.553313\pi$
$542$	15.5239	0.666807
$543$	−14.8635	−0.637852
$544$	37.8136	1.62125
$545$	−19.7515	−0.846062
$546$	−2.61014	−0.111704
$547$	13.7097	0.586186	0.293093	−	0.956084i	$-0.405315\pi$
$547$	13.7097	0.586186	0.293093	+	0.956084i	$0.405315\pi$
$548$	−34.5377	−1.47538
$549$	−7.44873	−0.317904
$550$	4.95226	0.211165
$551$	3.05120	0.129986
$552$	36.9405	1.57229
$553$	2.56752	0.109182
$554$	−58.1516	−2.47062
$555$	0	0
$556$	−30.9040	−1.31062
$557$	18.5503	0.786001	0.393000	−	0.919538i	$-0.371437\pi$
$557$	18.5503	0.786001	0.393000	+	0.919538i	$0.371437\pi$
$558$	−27.3219	−1.15663
$559$	55.1405	2.33220
$560$	3.58926	0.151674
$561$	−3.18017	−0.134267
$562$	−9.49423	−0.400490
$563$	42.1208	1.77518	0.887590	−	0.460635i	$-0.152378\pi$
$563$	42.1208	1.77518	0.887590	+	0.460635i	$0.152378\pi$
$564$	−14.1559	−0.596071
$565$	−0.193052	−0.00812175
$566$	37.4063	1.57230
$567$	−1.56850	−0.0658707
$568$	−121.823	−5.11158
$569$	−2.03727	−0.0854066	−0.0427033	−	0.999088i	$-0.513597\pi$
$569$	−2.03727	−0.0854066	−0.0427033	+	0.999088i	$0.513597\pi$
$570$	−14.5658	−0.610095
$571$	−8.16633	−0.341750	−0.170875	−	0.985293i	$-0.554659\pi$
$571$	−8.16633	−0.341750	−0.170875	+	0.985293i	$0.554659\pi$
$572$	−45.2881	−1.89359
$573$	9.02218	0.376907
$574$	−2.48921	−0.103898
$575$	−6.67080	−0.278191
$576$	−40.8553	−1.70230
$577$	−9.12193	−0.379751	−0.189875	−	0.981808i	$-0.560808\pi$
$577$	−9.12193	−0.379751	−0.189875	+	0.981808i	$0.560808\pi$
$578$	−28.0161	−1.16532
$579$	3.72218	0.154689
$580$	1.90972	0.0792967
$581$	−3.30830	−0.137251
$582$	−24.4465	−1.01334
$583$	−4.67715	−0.193708
$584$	−68.6589	−2.84112
$585$	−12.1855	−0.503810
$586$	−63.7913	−2.63519
$587$	−10.4884	−0.432902	−0.216451	−	0.976293i	$-0.569448\pi$
$587$	−10.4884	−0.432902	−0.216451	+	0.976293i	$0.569448\pi$
$588$	23.6404	0.974916
$589$	−32.7604	−1.34987
$590$	−7.00376	−0.288340
$591$	−6.47913	−0.266516
$592$	0	0
$593$	−5.23762	−0.215083	−0.107542	−	0.994201i	$-0.534298\pi$
$593$	−5.23762	−0.215083	−0.107542	+	0.994201i	$0.534298\pi$
$594$	18.4610	0.757463
$595$	0.776683	0.0318409
$596$	−52.3830	−2.14569
$597$	6.53050	0.267276
$598$	84.9527	3.47397
$599$	19.5578	0.799108	0.399554	−	0.916710i	$-0.369165\pi$
$599$	19.5578	0.799108	0.399554	+	0.916710i	$0.369165\pi$
$600$	−5.53764	−0.226073
$601$	11.3565	0.463242	0.231621	−	0.972806i	$-0.425597\pi$
$601$	11.3565	0.463242	0.231621	+	0.972806i	$0.425597\pi$
$602$	9.37122	0.381942
$603$	21.8550	0.890006
$604$	56.9955	2.31912
$605$	7.54313	0.306672
$606$	−4.61652	−0.187534
$607$	27.7177	1.12503	0.562514	−	0.826788i	$-0.309834\pi$
$607$	27.7177	1.12503	0.562514	+	0.826788i	$0.309834\pi$
$608$	−120.896	−4.90298
$609$	−0.0768294	−0.00311328
$610$	−7.78462	−0.315190
$611$	−19.7744	−0.799989
$612$	−33.0566	−1.33623
$613$	17.1015	0.690725	0.345362	−	0.938469i	$-0.387756\pi$
$613$	17.1015	0.690725	0.345362	+	0.938469i	$0.387756\pi$
$614$	−9.06788	−0.365950
$615$	2.05809	0.0829902
$616$	−4.67521	−0.188370
$617$	−21.3110	−0.857950	−0.428975	−	0.903316i	$-0.641125\pi$
$617$	−21.3110	−0.857950	−0.428975	+	0.903316i	$0.641125\pi$
$618$	−14.8744	−0.598335
$619$	12.0242	0.483292	0.241646	−	0.970364i	$-0.422313\pi$
$619$	12.0242	0.483292	0.241646	+	0.970364i	$0.422313\pi$
$620$	−20.5044	−0.823476
$621$	−24.8673	−0.997889
$622$	−4.92795	−0.197593
$623$	5.40137	0.216401
$624$	37.7926	1.51291
$625$	1.00000	0.0400000
$626$	66.0845	2.64127
$627$	10.1675	0.406051
$628$	84.7746	3.38287
$629$	0	0
$630$	−2.07095	−0.0825086
$631$	33.0174	1.31440	0.657201	−	0.753715i	$-0.271740\pi$
$631$	33.0174	1.31440	0.657201	+	0.753715i	$0.271740\pi$
$632$	−69.3705	−2.75941
$633$	−12.2725	−0.487789
$634$	−77.3655	−3.07258
$635$	−3.21589	−0.127619
$636$	8.61014	0.341414
$637$	33.0234	1.30844
$638$	−1.85638	−0.0734948
$639$	37.6683	1.49014
$640$	−12.9922	−0.513563
$641$	14.2676	0.563535	0.281767	−	0.959483i	$-0.409079\pi$
$641$	14.2676	0.563535	0.281767	+	0.959483i	$0.409079\pi$
$642$	18.3898	0.725787
$643$	−22.5157	−0.887932	−0.443966	−	0.896044i	$-0.646429\pi$
$643$	−22.5157	−0.887932	−0.443966	+	0.896044i	$0.646429\pi$
$644$	10.3677	0.408545
$645$	−7.74816	−0.305083
$646$	−55.1969	−2.17169
$647$	41.2531	1.62183	0.810913	−	0.585167i	$-0.198971\pi$
$647$	41.2531	1.62183	0.810913	+	0.585167i	$0.198971\pi$
$648$	42.3784	1.66478
$649$	4.88889	0.191906
$650$	−12.7350	−0.499509
$651$	0.824907	0.0323306
$652$	56.6093	2.21699
$653$	13.4063	0.524628	0.262314	−	0.964983i	$-0.415514\pi$
$653$	13.4063	0.524628	0.262314	+	0.964983i	$0.415514\pi$
$654$	−35.3449	−1.38210
$655$	12.9029	0.504156
$656$	36.0416	1.40719
$657$	21.2297	0.828250
$658$	−3.36070	−0.131014
$659$	−35.6948	−1.39047	−0.695236	−	0.718781i	$-0.744700\pi$
$659$	−35.6948	−1.39047	−0.695236	+	0.718781i	$0.744700\pi$
$660$	6.36373	0.247708
$661$	33.2215	1.29217	0.646084	−	0.763266i	$-0.276405\pi$
$661$	33.2215	1.29217	0.646084	+	0.763266i	$0.276405\pi$
$662$	15.3247	0.595612
$663$	8.17798	0.317606
$664$	89.3851	3.46881
$665$	−2.48318	−0.0962935
$666$	0	0
$667$	2.50058	0.0968229
$668$	−11.4544	−0.443183
$669$	2.62026	0.101305
$670$	22.8406	0.882408
$671$	5.43396	0.209776
$672$	3.04417	0.117431
$673$	34.4820	1.32918	0.664592	−	0.747206i	$-0.268605\pi$
$673$	34.4820	1.32918	0.664592	+	0.747206i	$0.268605\pi$
$674$	−5.16348	−0.198890
$675$	3.72778	0.143482
$676$	50.2317	1.93199
$677$	13.6122	0.523158	0.261579	−	0.965182i	$-0.415757\pi$
$677$	13.6122	0.523158	0.261579	+	0.965182i	$0.415757\pi$
$678$	−0.345462	−0.0132674
$679$	−4.16763	−0.159939
$680$	−20.9848	−0.804729
$681$	−15.4643	−0.592595
$682$	19.9317	0.763225
$683$	22.3789	0.856304	0.428152	−	0.903707i	$-0.359165\pi$
$683$	22.3789	0.856304	0.428152	+	0.903707i	$0.359165\pi$
$684$	105.687	4.04104
$685$	6.77934	0.259025
$686$	11.3004	0.431451
$687$	16.7506	0.639074
$688$	−135.687	−5.17302
$689$	12.0275	0.458213
$690$	−11.9373	−0.454444
$691$	−24.7590	−0.941877	−0.470938	−	0.882166i	$-0.656085\pi$
$691$	−24.7590	−0.941877	−0.470938	+	0.882166i	$0.656085\pi$
$692$	−52.8683	−2.00975
$693$	1.44560	0.0549139
$694$	−56.3461	−2.13887
$695$	6.06609	0.230100
$696$	2.07581	0.0786834
$697$	7.79909	0.295412
$698$	−43.7175	−1.65473
$699$	−18.2540	−0.690431
$700$	−1.55420	−0.0587431
$701$	−4.21211	−0.159089	−0.0795446	−	0.996831i	$-0.525347\pi$
$701$	−4.21211	−0.159089	−0.0795446	+	0.996831i	$0.525347\pi$
$702$	−47.4734	−1.79177
$703$	0	0
$704$	29.8045	1.12330
$705$	2.77864	0.104650
$706$	1.48495	0.0558868
$707$	−0.787024	−0.0295991
$708$	−8.99993	−0.338238
$709$	−27.8603	−1.04632	−0.523158	−	0.852236i	$-0.675246\pi$
$709$	−27.8603	−1.04632	−0.523158	+	0.852236i	$0.675246\pi$
$710$	39.3669	1.47741
$711$	21.4498	0.804429
$712$	−145.936	−5.46920
$713$	−26.8484	−1.00548
$714$	1.38986	0.0520142
$715$	8.88952	0.332449
$716$	49.2875	1.84196
$717$	7.70384	0.287705
$718$	7.78707	0.290611
$719$	−16.3154	−0.608460	−0.304230	−	0.952599i	$-0.598399\pi$
$719$	−16.3154	−0.608460	−0.304230	+	0.952599i	$0.598399\pi$
$720$	29.9856	1.11750
$721$	−2.53578	−0.0944373
$722$	125.865	4.68423
$723$	−9.17604	−0.341261
$724$	112.710	4.18882
$725$	−0.374855	−0.0139218
$726$	13.4983	0.500968
$727$	−33.2581	−1.23348	−0.616738	−	0.787168i	$-0.711546\pi$
$727$	−33.2581	−1.23348	−0.616738	+	0.787168i	$0.711546\pi$
$728$	12.0226	0.445586
$729$	−5.59024	−0.207046
$730$	22.1870	0.821178
$731$	−29.3615	−1.08597
$732$	−10.0033	−0.369734
$733$	−2.17580	−0.0803648	−0.0401824	−	0.999192i	$-0.512794\pi$
$733$	−2.17580	−0.0803648	−0.0401824	+	0.999192i	$0.512794\pi$
$734$	24.2870	0.896448
$735$	−4.64034	−0.171162
$736$	−99.0791	−3.65210
$737$	−15.9436	−0.587289
$738$	−20.7955	−0.765493
$739$	−39.0865	−1.43782	−0.718910	−	0.695103i	$-0.755359\pi$
$739$	−39.0865	−1.43782	−0.718910	+	0.695103i	$0.755359\pi$
$740$	0	0
$741$	−26.1463	−0.960507
$742$	2.04410	0.0750412
$743$	32.4833	1.19170	0.595848	−	0.803097i	$-0.296816\pi$
$743$	32.4833	1.19170	0.595848	+	0.803097i	$0.296816\pi$
$744$	−22.2877	−0.817107
$745$	10.2822	0.376710
$746$	37.4473	1.37104
$747$	−27.6384	−1.01123
$748$	24.1152	0.881740
$749$	3.13509	0.114554
$750$	1.78948	0.0653426
$751$	33.0898	1.20747	0.603733	−	0.797187i	$-0.293679\pi$
$751$	33.0898	1.20747	0.603733	+	0.797187i	$0.293679\pi$
$752$	48.6600	1.77445
$753$	−4.47299	−0.163005
$754$	4.77378	0.173851
$755$	−11.1876	−0.407157
$756$	−5.79370	−0.210715
$757$	−14.4706	−0.525942	−0.262971	−	0.964804i	$-0.584702\pi$
$757$	−14.4706	−0.525942	−0.262971	+	0.964804i	$0.584702\pi$
$758$	1.63785	0.0594894
$759$	8.33266	0.302456
$760$	67.0916	2.43367
$761$	−0.396943	−0.0143892	−0.00719459	−	0.999974i	$-0.502290\pi$
$761$	−0.396943	−0.0143892	−0.00719459	+	0.999974i	$0.502290\pi$
$762$	−5.75477	−0.208473
$763$	−6.02559	−0.218141
$764$	−68.4152	−2.47518
$765$	6.48861	0.234596
$766$	−3.45595	−0.124868
$767$	−12.5720	−0.453950
$768$	−1.70987	−0.0616997
$769$	40.3582	1.45535	0.727676	−	0.685921i	$-0.240600\pi$
$769$	40.3582	1.45535	0.727676	+	0.685921i	$0.240600\pi$
$770$	1.51079	0.0544450
$771$	−7.78557	−0.280390
$772$	−28.2253	−1.01585
$773$	34.5410	1.24236	0.621178	−	0.783670i	$-0.286655\pi$
$773$	34.5410	1.24236	0.621178	+	0.783670i	$0.286655\pi$
$774$	78.2895	2.81406
$775$	4.02477	0.144574
$776$	112.603	4.04221
$777$	0	0
$778$	−83.3957	−2.98988
$779$	−24.9349	−0.893386
$780$	−16.3647	−0.585950
$781$	−27.4796	−0.983298
$782$	−45.2360	−1.61764
$783$	−1.39738	−0.0499382
$784$	−81.2625	−2.90223
$785$	−16.6402	−0.593916
$786$	23.0894	0.823572
$787$	−22.2006	−0.791365	−0.395682	−	0.918387i	$-0.629492\pi$
$787$	−22.2006	−0.791365	−0.395682	+	0.918387i	$0.629492\pi$
$788$	49.1313	1.75023
$789$	11.5155	0.409963
$790$	22.4170	0.797561
$791$	−0.0588943	−0.00209404
$792$	−39.0579	−1.38786
$793$	−13.9737	−0.496221
$794$	−17.4039	−0.617643
$795$	−1.69007	−0.0599406
$796$	−49.5208	−1.75522
$797$	−21.1930	−0.750695	−0.375348	−	0.926884i	$-0.622477\pi$
$797$	−21.1930	−0.750695	−0.375348	+	0.926884i	$0.622477\pi$
$798$	−4.44360	−0.157302
$799$	10.5296	0.372510
$800$	14.8527	0.525121
$801$	45.1244	1.59439
$802$	24.4809	0.864450
$803$	−15.4874	−0.546538
$804$	29.3505	1.03511
$805$	−2.03506	−0.0717265
$806$	−51.2555	−1.80540
$807$	1.34701	0.0474171
$808$	21.2642	0.748071
$809$	−9.51326	−0.334468	−0.167234	−	0.985917i	$-0.553484\pi$
$809$	−9.51326	−0.334468	−0.167234	+	0.985917i	$0.553484\pi$
$810$	−13.6945	−0.481176
$811$	−31.4978	−1.10604	−0.553019	−	0.833169i	$-0.686524\pi$
$811$	−31.4978	−1.10604	−0.553019	+	0.833169i	$0.686524\pi$
$812$	0.582598	0.0204452
$813$	−3.91563	−0.137327
$814$	0	0
$815$	−11.1117	−0.389227
$816$	−20.1240	−0.704480
$817$	93.8733	3.28421
$818$	92.6991	3.24115
$819$	−3.71744	−0.129898
$820$	−15.6065	−0.545003
$821$	24.2852	0.847558	0.423779	−	0.905766i	$-0.360703\pi$
$821$	24.2852	0.847558	0.423779	+	0.905766i	$0.360703\pi$
$822$	12.1315	0.423134
$823$	−23.7422	−0.827602	−0.413801	−	0.910367i	$-0.635799\pi$
$823$	−23.7422	−0.827602	−0.413801	+	0.910367i	$0.635799\pi$
$824$	68.5128	2.38676
$825$	−1.24913	−0.0434890
$826$	−2.13664	−0.0743431
$827$	30.7256	1.06844	0.534218	−	0.845347i	$-0.320606\pi$
$827$	30.7256	1.06844	0.534218	+	0.845347i	$0.320606\pi$
$828$	86.6146	3.01006
$829$	19.9615	0.693290	0.346645	−	0.937996i	$-0.387321\pi$
$829$	19.9615	0.693290	0.346645	+	0.937996i	$0.387321\pi$
$830$	−28.8847	−1.00260
$831$	14.6678	0.508819
$832$	−76.6439	−2.65715
$833$	−17.5845	−0.609266
$834$	10.8551	0.375883
$835$	2.24836	0.0778076
$836$	−77.1002	−2.66656
$837$	15.0035	0.518595
$838$	−35.7449	−1.23479
$839$	27.5692	0.951796	0.475898	−	0.879501i	$-0.342123\pi$
$839$	27.5692	0.951796	0.475898	+	0.879501i	$0.342123\pi$
$840$	−1.68937	−0.0582887
$841$	−28.8595	−0.995155
$842$	−57.7703	−1.99090
$843$	2.39476	0.0824799
$844$	93.0626	3.20335
$845$	−9.85990	−0.339191
$846$	−28.0761	−0.965277
$847$	2.30118	0.0790696
$848$	−29.5968	−1.01636
$849$	−9.43510	−0.323812
$850$	6.78120	0.232593
$851$	0	0
$852$	50.5871	1.73308
$853$	2.05731	0.0704408	0.0352204	−	0.999380i	$-0.488787\pi$
$853$	2.05731	0.0704408	0.0352204	+	0.999380i	$0.488787\pi$
$854$	−2.37485	−0.0812659
$855$	−20.7451	−0.709467
$856$	−84.7052	−2.89516
$857$	2.81604	0.0961942	0.0480971	−	0.998843i	$-0.484684\pi$
$857$	2.81604	0.0961942	0.0480971	+	0.998843i	$0.484684\pi$
$858$	15.9076	0.543078
$859$	9.11129	0.310873	0.155437	−	0.987846i	$-0.450322\pi$
$859$	9.11129	0.310873	0.155437	+	0.987846i	$0.450322\pi$
$860$	58.7543	2.00351
$861$	0.627862	0.0213975
$862$	57.4143	1.95554
$863$	−9.48116	−0.322743	−0.161371	−	0.986894i	$-0.551592\pi$
$863$	−9.48116	−0.322743	−0.161371	+	0.986894i	$0.551592\pi$
$864$	55.3675	1.88364
$865$	10.3774	0.352843
$866$	−19.2185	−0.653070
$867$	7.06660	0.239994
$868$	−6.25527	−0.212318
$869$	−15.6479	−0.530819
$870$	−0.670796	−0.0227421
$871$	40.9998	1.38922
$872$	162.802	5.51318
$873$	−34.8175	−1.17839
$874$	144.627	4.89207
$875$	0.305070	0.0103133
$876$	28.5106	0.963285
$877$	5.35815	0.180932	0.0904659	−	0.995900i	$-0.471164\pi$
$877$	5.35815	0.180932	0.0904659	+	0.995900i	$0.471164\pi$
$878$	−31.1103	−1.04992
$879$	16.0903	0.542712
$880$	−21.8749	−0.737403
$881$	−46.0547	−1.55162	−0.775812	−	0.630964i	$-0.782659\pi$
$881$	−46.0547	−1.55162	−0.775812	+	0.630964i	$0.782659\pi$
$882$	46.8873	1.57878
$883$	40.0651	1.34830	0.674150	−	0.738595i	$-0.264510\pi$
$883$	40.0651	1.34830	0.674150	+	0.738595i	$0.264510\pi$
$884$	−62.0137	−2.08574
$885$	1.76658	0.0593829
$886$	−7.44274	−0.250044
$887$	27.6759	0.929266	0.464633	−	0.885503i	$-0.346186\pi$
$887$	27.6759	0.929266	0.464633	+	0.885503i	$0.346186\pi$
$888$	0	0
$889$	−0.981072	−0.0329041
$890$	47.1592	1.58078
$891$	9.55929	0.320248
$892$	−19.8695	−0.665279
$893$	−33.6647	−1.12655
$894$	18.3997	0.615379
$895$	−9.67455	−0.323385
$896$	−3.96354	−0.132413
$897$	−21.4279	−0.715456
$898$	72.8635	2.43149
$899$	−1.50870	−0.0503181
$900$	−12.9841	−0.432805
$901$	−6.40448	−0.213364
$902$	15.1706	0.505127
$903$	−2.36373	−0.0786601
$904$	1.59123	0.0529236
$905$	−22.1236	−0.735413
$906$	−20.0199	−0.665117
$907$	−15.4980	−0.514603	−0.257302	−	0.966331i	$-0.582833\pi$
$907$	−15.4980	−0.514603	−0.257302	+	0.966331i	$0.582833\pi$
$908$	117.266	3.89162
$909$	−6.57500	−0.218079
$910$	−3.88507	−0.128789
$911$	32.7886	1.08633	0.543167	−	0.839625i	$-0.317225\pi$
$911$	32.7886	1.08633	0.543167	+	0.839625i	$0.317225\pi$
$912$	64.3394	2.13049
$913$	20.1626	0.667284
$914$	−86.9817	−2.87710
$915$	1.96354	0.0649126
$916$	−127.020	−4.19685
$917$	3.93628	0.129987
$918$	25.2789	0.834326
$919$	0.940187	0.0310139	0.0155070	−	0.999880i	$-0.495064\pi$
$919$	0.940187	0.0310139	0.0155070	+	0.999880i	$0.495064\pi$
$920$	54.9842	1.81277
$921$	2.28722	0.0753665
$922$	−44.8336	−1.47652
$923$	70.6653	2.32598
$924$	1.94138	0.0638669
$925$	0	0
$926$	−29.4218	−0.966859
$927$	−21.1845	−0.695791
$928$	−5.56759	−0.182765
$929$	−7.74285	−0.254035	−0.127017	−	0.991900i	$-0.540540\pi$
$929$	−7.74285	−0.254035	−0.127017	+	0.991900i	$0.540540\pi$
$930$	7.20224	0.236171
$931$	56.2203	1.84255
$932$	138.421	4.53411
$933$	1.24299	0.0406937
$934$	9.57688	0.313365
$935$	−4.73354	−0.154803
$936$	100.440	3.28297
$937$	−27.8977	−0.911377	−0.455689	−	0.890139i	$-0.650607\pi$
$937$	−27.8977	−0.911377	−0.455689	+	0.890139i	$0.650607\pi$
$938$	6.96797	0.227512
$939$	−16.6687	−0.543963
$940$	−21.0704	−0.687241
$941$	−49.2272	−1.60476	−0.802380	−	0.596813i	$-0.796433\pi$
$941$	−49.2272	−1.60476	−0.802380	+	0.596813i	$0.796433\pi$
$942$	−29.7774	−0.970200
$943$	−20.4351	−0.665459
$944$	30.9367	1.00690
$945$	1.13724	0.0369943
$946$	−57.1134	−1.85692
$947$	−37.8698	−1.23060	−0.615302	−	0.788291i	$-0.710966\pi$
$947$	−37.8698	−1.23060	−0.615302	+	0.788291i	$0.710966\pi$
$948$	28.8062	0.935581
$949$	39.8266	1.29283
$950$	−21.6806	−0.703410
$951$	19.5141	0.632789
$952$	−6.40183	−0.207484
$953$	33.5987	1.08837	0.544185	−	0.838965i	$-0.316839\pi$
$953$	33.5987	1.08837	0.544185	+	0.838965i	$0.316839\pi$
$954$	17.0769	0.552886
$955$	13.4291	0.434555
$956$	−58.4183	−1.88938
$957$	0.468241	0.0151361
$958$	35.4383	1.14496
$959$	2.06817	0.0667848
$960$	10.7697	0.347592
$961$	−14.8013	−0.477460
$962$	0	0
$963$	26.1913	0.844003
$964$	69.5820	2.24109
$965$	5.54030	0.178348
$966$	−3.64170	−0.117170
$967$	−32.5035	−1.04524	−0.522622	−	0.852565i	$-0.675046\pi$
$967$	−32.5035	−1.04524	−0.522622	+	0.852565i	$0.675046\pi$
$968$	−62.1744	−1.99836
$969$	13.9225	0.447255
$970$	−36.3875	−1.16833
$971$	−30.1776	−0.968445	−0.484223	−	0.874945i	$-0.660898\pi$
$971$	−30.1776	−0.968445	−0.484223	+	0.874945i	$0.660898\pi$
$972$	−74.5718	−2.39189
$973$	1.85058	0.0593269
$974$	−38.9488	−1.24800
$975$	3.21219	0.102873
$976$	34.3858	1.10066
$977$	41.4992	1.32768	0.663839	−	0.747876i	$-0.268926\pi$
$977$	41.4992	1.32768	0.663839	+	0.747876i	$0.268926\pi$
$978$	−19.8842	−0.635828
$979$	−32.9189	−1.05209
$980$	35.1877	1.12403
$981$	−50.3394	−1.60721
$982$	−43.4831	−1.38760
$983$	−21.6389	−0.690173	−0.345086	−	0.938571i	$-0.612150\pi$
$983$	−21.6389	−0.690173	−0.345086	+	0.938571i	$0.612150\pi$
$984$	−16.9638	−0.540788
$985$	−9.64388	−0.307280
$986$	−2.54197	−0.0809527
$987$	0.847679	0.0269819
$988$	198.267	6.30772
$989$	76.9328	2.44632
$990$	12.6215	0.401138
$991$	−19.4143	−0.616714	−0.308357	−	0.951271i	$-0.599779\pi$
$991$	−19.4143	−0.616714	−0.308357	+	0.951271i	$0.599779\pi$
$992$	59.7785	1.89797
$993$	−3.86540	−0.122665
$994$	12.0097	0.380924
$995$	9.72036	0.308156
$996$	−37.1172	−1.17610
$997$	10.2171	0.323578	0.161789	−	0.986825i	$-0.448274\pi$
$997$	10.2171	0.323578	0.161789	+	0.986825i	$0.448274\pi$
$998$	14.7047	0.465470
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6845.2.a.g.1.5		5
37.36	even	2		185.2.a.d.1.1	✓	5
111.110	odd	2		1665.2.a.q.1.5		5
148.147	odd	2		2960.2.a.ba.1.4		5
185.73	odd	4		925.2.b.g.149.10		10
185.147	odd	4		925.2.b.g.149.1		10
185.184	even	2		925.2.a.h.1.5		5
259.258	odd	2		9065.2.a.j.1.1		5
555.554	odd	2		8325.2.a.cc.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.a.d.1.1	✓	5	37.36	even	2
925.2.a.h.1.5		5	185.184	even	2
925.2.b.g.149.1		10	185.147	odd	4
925.2.b.g.149.10		10	185.73	odd	4
1665.2.a.q.1.5		5	111.110	odd	2
2960.2.a.ba.1.4		5	148.147	odd	2
6845.2.a.g.1.5		5	1.1	even	1	trivial
8325.2.a.cc.1.1		5	555.554	odd	2
9065.2.a.j.1.1		5	259.258	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:	\( N \)	\(=\)	\( 6845 = 5 \cdot 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6845.a (trivial)

\(f(q)\)	\(=\)	\(q+2.66356 q^{2} -0.671838 q^{3} +5.09455 q^{4} -1.00000 q^{5} -1.78948 q^{6} -0.305070 q^{7} +8.24252 q^{8} -2.54863 q^{9} -2.66356 q^{10} +1.85927 q^{11} -3.42271 q^{12} -4.78120 q^{13} -0.812573 q^{14} +0.671838 q^{15} +11.7653 q^{16} +2.54592 q^{17} -6.78844 q^{18} -8.13969 q^{19} -5.09455 q^{20} +0.204958 q^{21} +4.95226 q^{22} -6.67080 q^{23} -5.53764 q^{24} +1.00000 q^{25} -12.7350 q^{26} +3.72778 q^{27} -1.55420 q^{28} -0.374855 q^{29} +1.78948 q^{30} +4.02477 q^{31} +14.8527 q^{32} -1.24913 q^{33} +6.78120 q^{34} +0.305070 q^{35} -12.9841 q^{36} -21.6806 q^{38} +3.21219 q^{39} -8.24252 q^{40} +3.06337 q^{41} +0.545917 q^{42} -11.5328 q^{43} +9.47212 q^{44} +2.54863 q^{45} -17.7681 q^{46} +4.13587 q^{47} -7.90441 q^{48} -6.90693 q^{49} +2.66356 q^{50} -1.71044 q^{51} -24.3581 q^{52} -2.51559 q^{53} +9.92917 q^{54} -1.85927 q^{55} -2.51455 q^{56} +5.46855 q^{57} -0.998448 q^{58} +2.62947 q^{59} +3.42271 q^{60} +2.92264 q^{61} +10.7202 q^{62} +0.777512 q^{63} +16.0303 q^{64} +4.78120 q^{65} -3.32712 q^{66} -8.57520 q^{67} +12.9703 q^{68} +4.48169 q^{69} +0.812573 q^{70} -14.7798 q^{71} -21.0072 q^{72} -8.32984 q^{73} -0.671838 q^{75} -41.4681 q^{76} -0.567206 q^{77} +8.55587 q^{78} -8.41618 q^{79} -11.7653 q^{80} +5.14143 q^{81} +8.15947 q^{82} +10.8444 q^{83} +1.04417 q^{84} -2.54592 q^{85} -30.7182 q^{86} +0.251842 q^{87} +15.3250 q^{88} -17.7053 q^{89} +6.78844 q^{90} +1.45860 q^{91} -33.9847 q^{92} -2.70399 q^{93} +11.0161 q^{94} +8.13969 q^{95} -9.97858 q^{96} +13.6612 q^{97} -18.3970 q^{98} -4.73859 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 6 q^{4} - 5 q^{5} + 2 q^{6} + 7 q^{7} + 6 q^{8} + 2 q^{9} + 7 q^{11} - 2 q^{13} - 4 q^{14} + q^{15} + 8 q^{16} + 8 q^{17} + 6 q^{18} - 14 q^{19} - 6 q^{20} - 9 q^{21} - 2 q^{22} - 2 q^{23}+ \cdots + 18 q^{99}+O(q^{100}) \) 5 * q - q^3 + 6 * q^4 - 5 * q^5 + 2 * q^6 + 7 * q^7 + 6 * q^8 + 2 * q^9 + 7 * q^11 - 2 * q^13 - 4 * q^14 + q^15 + 8 * q^16 + 8 * q^17 + 6 * q^18 - 14 * q^19 - 6 * q^20 - 9 * q^21 - 2 * q^22 - 2 * q^23 - 12 * q^24 + 5 * q^25 - 20 * q^26 - 7 * q^27 - 14 * q^28 - 2 * q^29 - 2 * q^30 - 8 * q^31 + 22 * q^32 - 21 * q^33 + 12 * q^34 - 7 * q^35 - 36 * q^36 - 32 * q^38 - 12 * q^39 - 6 * q^40 - 9 * q^41 - 2 * q^42 - 14 * q^43 + 18 * q^44 - 2 * q^45 - 28 * q^46 - 5 * q^47 - 30 * q^48 + 2 * q^49 - 10 * q^51 - 20 * q^52 - 15 * q^53 + 12 * q^54 - 7 * q^55 + 14 * q^56 - 4 * q^57 - 12 * q^59 - 12 * q^61 + 10 * q^62 + 24 * q^63 + 12 * q^64 + 2 * q^65 + 10 * q^66 - 2 * q^67 + 4 * q^68 + 30 * q^69 + 4 * q^70 + 13 * q^71 - 26 * q^72 - 5 * q^73 - q^75 - 18 * q^76 - 7 * q^77 + 10 * q^78 - 36 * q^79 - 8 * q^80 + 21 * q^81 + 2 * q^82 - 9 * q^83 + 30 * q^84 - 8 * q^85 - 8 * q^86 + 6 * q^87 - 8 * q^88 + 16 * q^89 - 6 * q^90 + 4 * q^91 - 12 * q^92 + 16 * q^93 - 20 * q^94 + 14 * q^95 - 30 * q^98 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more