LMFDB - Embedded newform 127.4.a.b.1.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [127,4,Mod(1,127)] 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("127.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(127, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 127 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	127.a (trivial)

Newform invariants

comment:select newform

sage:traces = [13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.49324257073$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 5 x^{12} - 55 x^{11} + 264 x^{10} + 1126 x^{9} - 5085 x^{8} - 10823 x^{7} + 44242 x^{6} + \cdots - 130048 $ x^13 - 5x^12 - 55x^11 + 264x^10 + 1126x^9 - 5085x^8 - 10823x^7 + 44242x^6 + 52047x^5 - 178872x^4 - 113160x^3 + 298560x^2 + 65136x - 130048
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$4.55672$ of defining polynomial
Character	$\chi$	$=$	127.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.55672 q^{2} -1.25597 q^{3} +4.65028 q^{4} -21.6593 q^{5} -4.46714 q^{6} +3.10281 q^{7} -11.9140 q^{8} -25.4225 q^{9} -77.0363 q^{10} +4.68401 q^{11} -5.84061 q^{12} +34.7041 q^{13} +11.0358 q^{14} +27.2035 q^{15} -79.5771 q^{16} +26.9343 q^{17} -90.4209 q^{18} -71.9403 q^{19} -100.722 q^{20} -3.89704 q^{21} +16.6597 q^{22} +36.4787 q^{23} +14.9637 q^{24} +344.127 q^{25} +123.433 q^{26} +65.8412 q^{27} +14.4289 q^{28} -207.690 q^{29} +96.7553 q^{30} -148.506 q^{31} -187.722 q^{32} -5.88298 q^{33} +95.7979 q^{34} -67.2048 q^{35} -118.222 q^{36} +221.460 q^{37} -255.872 q^{38} -43.5874 q^{39} +258.050 q^{40} -233.491 q^{41} -13.8607 q^{42} -67.9365 q^{43} +21.7819 q^{44} +550.635 q^{45} +129.744 q^{46} -365.440 q^{47} +99.9465 q^{48} -333.373 q^{49} +1223.96 q^{50} -33.8287 q^{51} +161.384 q^{52} +275.718 q^{53} +234.179 q^{54} -101.453 q^{55} -36.9670 q^{56} +90.3549 q^{57} -738.697 q^{58} -568.484 q^{59} +126.504 q^{60} +549.271 q^{61} -528.195 q^{62} -78.8813 q^{63} -31.0564 q^{64} -751.669 q^{65} -20.9241 q^{66} +567.770 q^{67} +125.252 q^{68} -45.8161 q^{69} -239.029 q^{70} -408.724 q^{71} +302.885 q^{72} -340.676 q^{73} +787.672 q^{74} -432.213 q^{75} -334.543 q^{76} +14.5336 q^{77} -155.028 q^{78} -231.048 q^{79} +1723.59 q^{80} +603.714 q^{81} -830.464 q^{82} +1124.47 q^{83} -18.1223 q^{84} -583.380 q^{85} -241.631 q^{86} +260.853 q^{87} -55.8054 q^{88} -394.186 q^{89} +1958.46 q^{90} +107.680 q^{91} +169.636 q^{92} +186.519 q^{93} -1299.77 q^{94} +1558.18 q^{95} +235.773 q^{96} -18.6833 q^{97} -1185.71 q^{98} -119.079 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 8 q^{2} - 16 q^{3} + 34 q^{4} - 46 q^{5} - 2 q^{6} - 26 q^{7} - 117 q^{8} + 3 q^{9} - 25 q^{10} - 53 q^{11} - 255 q^{12} - 75 q^{13} - 152 q^{14} - 62 q^{15} + 82 q^{16} - 479 q^{17} - 292 q^{18}+ \cdots + 1769 q^{99}+O(q^{100}) $ 13 * q - 8 * q^2 - 16 * q^3 + 34 * q^4 - 46 * q^5 - 2 * q^6 - 26 * q^7 - 117 * q^8 + 3 * q^9 - 25 * q^10 - 53 * q^11 - 255 * q^12 - 75 * q^13 - 152 * q^14 - 62 * q^15 + 82 * q^16 - 479 * q^17 - 292 * q^18 - 209 * q^19 - 533 * q^20 - 174 * q^21 - 407 * q^22 - 376 * q^23 + 108 * q^24 + 9 * q^25 - 622 * q^26 - 562 * q^27 - 258 * q^28 - 158 * q^29 - 287 * q^30 - 307 * q^31 - 609 * q^32 - 494 * q^33 + 948 * q^34 - 512 * q^35 + 1556 * q^36 - 171 * q^37 + 712 * q^38 - 154 * q^39 + 1897 * q^40 - 641 * q^41 + 1489 * q^42 + 530 * q^43 + 1328 * q^44 - 566 * q^45 + 2051 * q^46 - 555 * q^47 + 268 * q^48 + 357 * q^49 + 808 * q^50 + 1228 * q^51 + 2473 * q^52 - 1640 * q^53 + 2957 * q^54 + 540 * q^55 + 551 * q^56 + 968 * q^57 + 1328 * q^58 - 860 * q^59 + 4466 * q^60 + 191 * q^61 + 367 * q^62 + 120 * q^63 + 1915 * q^64 - 1584 * q^65 + 3416 * q^66 + 912 * q^67 - 1873 * q^68 - 2142 * q^69 + 2329 * q^70 + 115 * q^71 - 1864 * q^72 - 2563 * q^73 + 470 * q^74 + 44 * q^75 - 192 * q^76 - 3338 * q^77 + 4226 * q^78 + 169 * q^79 - 3194 * q^80 - 395 * q^81 + 54 * q^82 - 1688 * q^83 + 853 * q^84 + 480 * q^85 - 485 * q^86 - 1068 * q^87 - 1192 * q^88 - 2752 * q^89 + 4948 * q^90 - 398 * q^91 - 846 * q^92 - 804 * q^93 + 531 * q^94 - 1766 * q^95 + 3965 * q^96 - 2124 * q^97 - 1479 * q^98 + 1769 * 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 $

$2$	3.55672	1.25749	0.628746	−	0.777611i	$-0.283569\pi$
$2$	3.55672	1.25749	0.628746	+	0.777611i	$0.283569\pi$
$3$	−1.25597	−0.241712	−0.120856	−	0.992670i	$-0.538564\pi$
$3$	−1.25597	−0.241712	−0.120856	+	0.992670i	$0.538564\pi$
$4$	4.65028	0.581285
$5$	−21.6593	−1.93727	−0.968635	−	0.248487i	$-0.920067\pi$
$5$	−21.6593	−1.93727	−0.968635	+	0.248487i	$0.920067\pi$
$6$	−4.46714	−0.303950
$7$	3.10281	0.167536	0.0837680	−	0.996485i	$-0.473305\pi$
$7$	3.10281	0.167536	0.0837680	+	0.996485i	$0.473305\pi$
$8$	−11.9140	−0.526531
$9$	−25.4225	−0.941576
$10$	−77.0363	−2.43610
$11$	4.68401	0.128389	0.0641946	−	0.997937i	$-0.479552\pi$
$11$	4.68401	0.128389	0.0641946	+	0.997937i	$0.479552\pi$
$12$	−5.84061	−0.140503
$13$	34.7041	0.740400	0.370200	−	0.928952i	$-0.379289\pi$
$13$	34.7041	0.740400	0.370200	+	0.928952i	$0.379289\pi$
$14$	11.0358	0.210675
$15$	27.2035	0.468261
$16$	−79.5771	−1.24339
$17$	26.9343	0.384267	0.192133	−	0.981369i	$-0.438459\pi$
$17$	26.9343	0.384267	0.192133	+	0.981369i	$0.438459\pi$
$18$	−90.4209	−1.18402
$19$	−71.9403	−0.868645	−0.434322	−	0.900758i	$-0.643012\pi$
$19$	−71.9403	−0.868645	−0.434322	+	0.900758i	$0.643012\pi$
$20$	−100.722	−1.12611
$21$	−3.89704	−0.0404954
$22$	16.6597	0.161448
$23$	36.4787	0.330710	0.165355	−	0.986234i	$-0.447123\pi$
$23$	36.4787	0.330710	0.165355	+	0.986234i	$0.447123\pi$
$24$	14.9637	0.127269
$25$	344.127	2.75302
$26$	123.433	0.931046
$27$	65.8412	0.469301
$28$	14.4289	0.0973861
$29$	−207.690	−1.32990	−0.664951	−	0.746887i	$-0.731547\pi$
$29$	−207.690	−1.32990	−0.664951	+	0.746887i	$0.731547\pi$
$30$	96.7553	0.588834
$31$	−148.506	−0.860402	−0.430201	−	0.902733i	$-0.641557\pi$
$31$	−148.506	−0.860402	−0.430201	+	0.902733i	$0.641557\pi$
$32$	−187.722	−1.03702
$33$	−5.88298	−0.0310332
$34$	95.7979	0.483212
$35$	−67.2048	−0.324562
$36$	−118.222	−0.547323
$37$	221.460	0.983994	0.491997	−	0.870597i	$-0.336267\pi$
$37$	221.460	0.983994	0.491997	+	0.870597i	$0.336267\pi$
$38$	−255.872	−1.09231
$39$	−43.5874	−0.178963
$40$	258.050	1.02003
$41$	−233.491	−0.889396	−0.444698	−	0.895681i	$-0.646689\pi$
$41$	−233.491	−0.889396	−0.444698	+	0.895681i	$0.646689\pi$
$42$	−13.8607	−0.0509226
$43$	−67.9365	−0.240935	−0.120468	−	0.992717i	$-0.538439\pi$
$43$	−67.9365	−0.240935	−0.120468	+	0.992717i	$0.538439\pi$
$44$	21.7819	0.0746307
$45$	550.635	1.82409
$46$	129.744	0.415865
$47$	−365.440	−1.13415	−0.567073	−	0.823667i	$-0.691924\pi$
$47$	−365.440	−1.13415	−0.567073	+	0.823667i	$0.691924\pi$
$48$	99.9465	0.300542
$49$	−333.373	−0.971932
$50$	1223.96	3.46189
$51$	−33.8287	−0.0928817
$52$	161.384	0.430383
$53$	275.718	0.714580	0.357290	−	0.933994i	$-0.383701\pi$
$53$	275.718	0.714580	0.357290	+	0.933994i	$0.383701\pi$
$54$	234.179	0.590142
$55$	−101.453	−0.248725
$56$	−36.9670	−0.0882128
$57$	90.3549	0.209961
$58$	−738.697	−1.67234
$59$	−568.484	−1.25441	−0.627206	−	0.778853i	$-0.715802\pi$
$59$	−568.484	−1.25441	−0.627206	+	0.778853i	$0.715802\pi$
$60$	126.504	0.272193
$61$	549.271	1.15290	0.576450	−	0.817132i	$-0.304437\pi$
$61$	549.271	1.15290	0.576450	+	0.817132i	$0.304437\pi$
$62$	−528.195	−1.08195
$63$	−78.8813	−0.157748
$64$	−31.0564	−0.0606571
$65$	−751.669	−1.43435
$66$	−20.9241	−0.0390240
$67$	567.770	1.03529	0.517643	−	0.855597i	$-0.326809\pi$
$67$	567.770	1.03529	0.517643	+	0.855597i	$0.326809\pi$
$68$	125.252	0.223368
$69$	−45.8161	−0.0799364
$70$	−239.029	−0.408134
$71$	−408.724	−0.683192	−0.341596	−	0.939847i	$-0.610967\pi$
$71$	−408.724	−0.683192	−0.341596	+	0.939847i	$0.610967\pi$
$72$	302.885	0.495769
$73$	−340.676	−0.546207	−0.273103	−	0.961985i	$-0.588050\pi$
$73$	−340.676	−0.546207	−0.273103	+	0.961985i	$0.588050\pi$
$74$	787.672	1.23736
$75$	−432.213	−0.665436
$76$	−334.543	−0.504930
$77$	14.5336	0.0215098
$78$	−155.028	−0.225045
$79$	−231.048	−0.329050	−0.164525	−	0.986373i	$-0.552609\pi$
$79$	−231.048	−0.329050	−0.164525	+	0.986373i	$0.552609\pi$
$80$	1723.59	2.40879
$81$	603.714	0.828140
$82$	−830.464	−1.11841
$83$	1124.47	1.48707	0.743535	−	0.668698i	$-0.233148\pi$
$83$	1124.47	1.48707	0.743535	+	0.668698i	$0.233148\pi$
$84$	−18.1223	−0.0235393
$85$	−583.380	−0.744428
$86$	−241.631	−0.302974
$87$	260.853	0.321453
$88$	−55.8054	−0.0676009
$89$	−394.186	−0.469478	−0.234739	−	0.972058i	$-0.575424\pi$
$89$	−394.186	−0.469478	−0.234739	+	0.972058i	$0.575424\pi$
$90$	1958.46	2.29377
$91$	107.680	0.124044
$92$	169.636	0.192237
$93$	186.519	0.207969
$94$	−1299.77	−1.42618
$95$	1558.18	1.68280
$96$	235.773	0.250661
$97$	−18.6833	−0.0195567	−0.00977837	−	0.999952i	$-0.503113\pi$
$97$	−18.6833	−0.0195567	−0.00977837	+	0.999952i	$0.503113\pi$
$98$	−1185.71	−1.22220
$99$	−119.079	−0.120888
$100$	1600.29	1.60029
$101$	−462.387	−0.455537	−0.227769	−	0.973715i	$-0.573143\pi$
$101$	−462.387	−0.455537	−0.227769	+	0.973715i	$0.573143\pi$
$102$	−120.319	−0.116798
$103$	−2002.96	−1.91609	−0.958045	−	0.286618i	$-0.907469\pi$
$103$	−2002.96	−1.91609	−0.958045	+	0.286618i	$0.907469\pi$
$104$	−413.466	−0.389843
$105$	84.4072	0.0784505
$106$	980.651	0.898578
$107$	306.145	0.276600	0.138300	−	0.990390i	$-0.455836\pi$
$107$	306.145	0.276600	0.138300	+	0.990390i	$0.455836\pi$
$108$	306.180	0.272798
$109$	943.846	0.829395	0.414697	−	0.909959i	$-0.363887\pi$
$109$	943.846	0.829395	0.414697	+	0.909959i	$0.363887\pi$
$110$	−360.839	−0.312769
$111$	−278.147	−0.237843
$112$	−246.913	−0.208313
$113$	−1543.29	−1.28478	−0.642390	−	0.766378i	$-0.722057\pi$
$113$	−1543.29	−1.28478	−0.642390	+	0.766378i	$0.722057\pi$
$114$	321.367	0.264025
$115$	−790.104	−0.640674
$116$	−965.818	−0.773051
$117$	−882.267	−0.697142
$118$	−2021.94	−1.57741
$119$	83.5721	0.0643785
$120$	−324.103	−0.246554
$121$	−1309.06	−0.983516
$122$	1953.60	1.44976
$123$	293.258	0.214977
$124$	−690.594	−0.500139
$125$	−4746.15	−3.39607
$126$	−280.559	−0.198366
$127$	127.000	0.0887357
$127$	127.000	0.0887357
$128$	1391.31	0.960749
$129$	85.3263	0.0582369
$130$	−2673.48	−1.80369
$131$	859.195	0.573040	0.286520	−	0.958074i	$-0.407502\pi$
$131$	859.195	0.573040	0.286520	+	0.958074i	$0.407502\pi$
$132$	−27.3575	−0.0180391
$133$	−223.217	−0.145529
$134$	2019.40	1.30186
$135$	−1426.08	−0.909163
$136$	−320.896	−0.202328
$137$	478.637	0.298487	0.149243	−	0.988800i	$-0.452316\pi$
$137$	478.637	0.298487	0.149243	+	0.988800i	$0.452316\pi$
$138$	−162.955	−0.100519
$139$	−2638.87	−1.61026	−0.805129	−	0.593099i	$-0.797904\pi$
$139$	−2638.87	−1.61026	−0.805129	+	0.593099i	$0.797904\pi$
$140$	−312.521	−0.188663
$141$	458.982	0.274136
$142$	−1453.72	−0.859108
$143$	162.555	0.0950594
$144$	2023.05	1.17075
$145$	4498.44	2.57638
$146$	−1211.69	−0.686851
$147$	418.706	0.234927
$148$	1029.85	0.571981
$149$	−2916.90	−1.60377	−0.801885	−	0.597478i	$-0.796170\pi$
$149$	−2916.90	−1.60377	−0.801885	+	0.597478i	$0.796170\pi$
$150$	−1537.26	−0.836780
$151$	3618.46	1.95011	0.975054	−	0.221966i	$-0.0712474\pi$
$151$	3618.46	1.95011	0.975054	+	0.221966i	$0.0712474\pi$
$152$	857.100	0.457368
$153$	−684.739	−0.361816
$154$	51.6919	0.0270484
$155$	3216.54	1.66683
$156$	−202.693	−0.104029
$157$	−20.0791	−0.0102069	−0.00510347	−	0.999987i	$-0.501624\pi$
$157$	−20.0791	−0.0102069	−0.00510347	+	0.999987i	$0.501624\pi$
$158$	−821.774	−0.413777
$159$	−346.293	−0.172722
$160$	4065.92	2.00900
$161$	113.186	0.0554058
$162$	2147.24	1.04138
$163$	2962.91	1.42376	0.711880	−	0.702301i	$-0.247844\pi$
$163$	2962.91	1.42376	0.711880	+	0.702301i	$0.247844\pi$
$164$	−1085.80	−0.516992
$165$	127.421	0.0601196
$166$	3999.43	1.86998
$167$	167.860	0.0777809	0.0388905	−	0.999243i	$-0.487618\pi$
$167$	167.860	0.0777809	0.0388905	+	0.999243i	$0.487618\pi$
$168$	46.4294	0.0213221
$169$	−992.623	−0.451808
$170$	−2074.92	−0.936112
$171$	1828.91	0.817895
$172$	−315.924	−0.140052
$173$	−2507.56	−1.10200	−0.551000	−	0.834505i	$-0.685754\pi$
$173$	−2507.56	−1.10200	−0.551000	+	0.834505i	$0.685754\pi$
$174$	927.782	0.404224
$175$	1067.76	0.461229
$176$	−372.740	−0.159638
$177$	713.999	0.303206
$178$	−1402.01	−0.590365
$179$	−3301.08	−1.37840	−0.689202	−	0.724570i	$-0.742039\pi$
$179$	−3301.08	−1.37840	−0.689202	+	0.724570i	$0.742039\pi$
$180$	2560.61	1.06031
$181$	−555.448	−0.228100	−0.114050	−	0.993475i	$-0.536382\pi$
$181$	−555.448	−0.228100	−0.114050	+	0.993475i	$0.536382\pi$
$182$	382.989	0.155984
$183$	−689.868	−0.278669
$184$	−434.608	−0.174129
$185$	−4796.68	−1.90626
$186$	663.397	0.261519
$187$	126.161	0.0493357
$188$	−1699.40	−0.659262
$189$	204.293	0.0786248
$190$	5542.02	2.11611
$191$	4790.14	1.81467	0.907337	−	0.420404i	$-0.138112\pi$
$191$	4790.14	1.81467	0.907337	+	0.420404i	$0.138112\pi$
$192$	39.0060	0.0146615
$193$	2800.98	1.04466	0.522330	−	0.852744i	$-0.325063\pi$
$193$	2800.98	1.04466	0.522330	+	0.852744i	$0.325063\pi$
$194$	−66.4514	−0.0245924
$195$	944.074	0.346700
$196$	−1550.28	−0.564969
$197$	5277.12	1.90853	0.954263	−	0.298969i	$-0.0966427\pi$
$197$	5277.12	1.90853	0.954263	+	0.298969i	$0.0966427\pi$
$198$	−423.532	−0.152016
$199$	3819.16	1.36047	0.680233	−	0.732996i	$-0.261878\pi$
$199$	3819.16	1.36047	0.680233	+	0.732996i	$0.261878\pi$
$200$	−4099.94	−1.44955
$201$	−713.103	−0.250241
$202$	−1644.58	−0.572834
$203$	−644.424	−0.222806
$204$	−157.313	−0.0539907
$205$	5057.27	1.72300
$206$	−7123.97	−2.40947
$207$	−927.380	−0.311388
$208$	−2761.66	−0.920608
$209$	−336.969	−0.111525
$210$	300.213	0.0986508
$211$	3123.75	1.01919	0.509593	−	0.860416i	$-0.329796\pi$
$211$	3123.75	1.01919	0.509593	+	0.860416i	$0.329796\pi$
$212$	1282.16	0.415374
$213$	513.346	0.165135
$214$	1088.87	0.347822
$215$	1471.46	0.466757
$216$	−784.434	−0.247102
$217$	−460.786	−0.144148
$218$	3357.00	1.04296
$219$	427.879	0.132025
$220$	−471.783	−0.144580
$221$	934.732	0.284511
$222$	−989.292	−0.299085
$223$	2937.08	0.881980	0.440990	−	0.897512i	$-0.354627\pi$
$223$	2937.08	0.881980	0.440990	+	0.897512i	$0.354627\pi$
$224$	−582.464	−0.173739
$225$	−8748.58	−2.59217
$226$	−5489.04	−1.61560
$227$	742.599	0.217128	0.108564	−	0.994089i	$-0.465375\pi$
$227$	742.599	0.217128	0.108564	+	0.994089i	$0.465375\pi$
$228$	420.176	0.122047
$229$	1537.21	0.443589	0.221795	−	0.975093i	$-0.428809\pi$
$229$	1537.21	0.443589	0.221795	+	0.975093i	$0.428809\pi$
$230$	−2810.18	−0.805642
$231$	−18.2538	−0.00519917
$232$	2474.43	0.700234
$233$	−2758.38	−0.775569	−0.387784	−	0.921750i	$-0.626759\pi$
$233$	−2758.38	−0.775569	−0.387784	+	0.921750i	$0.626759\pi$
$234$	−3137.98	−0.876650
$235$	7915.19	2.19715
$236$	−2643.61	−0.729171
$237$	290.190	0.0795352
$238$	297.243	0.0809554
$239$	4991.06	1.35081	0.675407	−	0.737445i	$-0.263968\pi$
$239$	4991.06	1.35081	0.675407	+	0.737445i	$0.263968\pi$
$240$	−2164.78	−0.582232
$241$	−1898.98	−0.507570	−0.253785	−	0.967261i	$-0.581676\pi$
$241$	−1898.98	−0.507570	−0.253785	+	0.967261i	$0.581676\pi$
$242$	−4655.96	−1.23676
$243$	−2535.96	−0.669472
$244$	2554.26	0.670163
$245$	7220.63	1.88289
$246$	1043.04	0.270332
$247$	−2496.63	−0.643144
$248$	1769.31	0.453028
$249$	−1412.30	−0.359442
$250$	−16880.7	−4.27052
$251$	−7098.60	−1.78510	−0.892550	−	0.450949i	$-0.851086\pi$
$251$	−7098.60	−1.78510	−0.892550	+	0.450949i	$0.851086\pi$
$252$	−366.820	−0.0916963
$253$	170.866	0.0424596
$254$	451.704	0.111584
$255$	732.708	0.179937
$256$	5196.97	1.26879
$257$	3001.39	0.728489	0.364244	−	0.931303i	$-0.381327\pi$
$257$	3001.39	0.728489	0.364244	+	0.931303i	$0.381327\pi$
$258$	303.482	0.0732324
$259$	687.148	0.164854
$260$	−3495.47	−0.833768
$261$	5280.02	1.25220
$262$	3055.92	0.720592
$263$	−6894.12	−1.61639	−0.808193	−	0.588917i	$-0.799554\pi$
$263$	−6894.12	−1.61639	−0.808193	+	0.588917i	$0.799554\pi$
$264$	70.0900	0.0163399
$265$	−5971.86	−1.38433
$266$	−793.922	−0.183002
$267$	495.085	0.113478
$268$	2640.29	0.601796
$269$	35.6896	0.00808934	0.00404467	−	0.999992i	$-0.498713\pi$
$269$	35.6896	0.00808934	0.00404467	+	0.999992i	$0.498713\pi$
$270$	−5072.16	−1.14327
$271$	−8189.13	−1.83562	−0.917812	−	0.397015i	$-0.870046\pi$
$271$	−8189.13	−1.83562	−0.917812	+	0.397015i	$0.870046\pi$
$272$	−2143.36	−0.477794
$273$	−135.243	−0.0299828
$274$	1702.38	0.375345
$275$	1611.89	0.353458
$276$	−213.058	−0.0464658
$277$	1571.30	0.340831	0.170415	−	0.985372i	$-0.445489\pi$
$277$	1571.30	0.340831	0.170415	+	0.985372i	$0.445489\pi$
$278$	−9385.73	−2.02489
$279$	3775.40	0.810134
$280$	800.680	0.170892
$281$	−6413.57	−1.36157	−0.680786	−	0.732483i	$-0.738362\pi$
$281$	−6413.57	−1.36157	−0.680786	+	0.732483i	$0.738362\pi$
$282$	1632.47	0.344724
$283$	−4423.94	−0.929244	−0.464622	−	0.885509i	$-0.653810\pi$
$283$	−4423.94	−0.929244	−0.464622	+	0.885509i	$0.653810\pi$
$284$	−1900.68	−0.397129
$285$	−1957.03	−0.406752
$286$	578.161	0.119536
$287$	−724.479	−0.149006
$288$	4772.36	0.976437
$289$	−4187.54	−0.852339
$290$	15999.7	3.23977
$291$	23.4657	0.00472709
$292$	−1584.24	−0.317502
$293$	−1838.00	−0.366475	−0.183237	−	0.983069i	$-0.558658\pi$
$293$	−1838.00	−0.366475	−0.183237	+	0.983069i	$0.558658\pi$
$294$	1489.22	0.295419
$295$	12313.0	2.43014
$296$	−2638.48	−0.518103
$297$	308.401	0.0602533
$298$	−10374.6	−2.01673
$299$	1265.96	0.244857
$300$	−2009.91	−0.386808
$301$	−210.794	−0.0403653
$302$	12869.9	2.45225
$303$	580.745	0.110109
$304$	5724.81	1.08007
$305$	−11896.8	−2.23348
$306$	−2435.43	−0.454981
$307$	4992.98	0.928223	0.464112	−	0.885777i	$-0.346374\pi$
$307$	4992.98	0.928223	0.464112	+	0.885777i	$0.346374\pi$
$308$	67.5852	0.0125033
$309$	2515.66	0.463141
$310$	11440.4	2.09603
$311$	−329.661	−0.0601072	−0.0300536	−	0.999548i	$-0.509568\pi$
$311$	−329.661	−0.0601072	−0.0300536	+	0.999548i	$0.509568\pi$
$312$	519.301	0.0942296
$313$	3767.23	0.680308	0.340154	−	0.940370i	$-0.389521\pi$
$313$	3767.23	0.680308	0.340154	+	0.940370i	$0.389521\pi$
$314$	−71.4159	−0.0128351
$315$	1708.52	0.305600
$316$	−1074.44	−0.191272
$317$	5522.55	0.978478	0.489239	−	0.872150i	$-0.337275\pi$
$317$	5522.55	0.978478	0.489239	+	0.872150i	$0.337275\pi$
$318$	−1231.67	−0.217197
$319$	−972.824	−0.170745
$320$	672.662	0.117509
$321$	−384.509	−0.0668573
$322$	402.572	0.0696723
$323$	−1937.66	−0.333791
$324$	2807.44	0.481385
$325$	11942.6	2.03833
$326$	10538.2	1.79037
$327$	−1185.44	−0.200474
$328$	2781.83	0.468295
$329$	−1133.89	−0.190010
$330$	453.203	0.0755999
$331$	2509.49	0.416719	0.208359	−	0.978052i	$-0.433188\pi$
$331$	2509.49	0.416719	0.208359	+	0.978052i	$0.433188\pi$
$332$	5229.10	0.864411
$333$	−5630.07	−0.926505
$334$	597.032	0.0978088
$335$	−12297.5	−2.00563
$336$	310.115	0.0503517
$337$	−6782.08	−1.09627	−0.548136	−	0.836389i	$-0.684662\pi$
$337$	−6782.08	−1.09627	−0.548136	+	0.836389i	$0.684662\pi$
$338$	−3530.48	−0.568145
$339$	1938.32	0.310546
$340$	−2712.88	−0.432725
$341$	−695.604	−0.110466
$342$	6504.91	1.02850
$343$	−2098.65	−0.330369
$344$	809.398	0.126860
$345$	992.347	0.154858
$346$	−8918.68	−1.38576
$347$	259.660	0.0401708	0.0200854	−	0.999798i	$-0.493606\pi$
$347$	259.660	0.0401708	0.0200854	+	0.999798i	$0.493606\pi$
$348$	1213.04	0.186855
$349$	−3640.92	−0.558436	−0.279218	−	0.960228i	$-0.590075\pi$
$349$	−3640.92	−0.558436	−0.279218	+	0.960228i	$0.590075\pi$
$350$	3797.73	0.579992
$351$	2284.96	0.347471
$352$	−879.290	−0.133143
$353$	−4840.66	−0.729864	−0.364932	−	0.931034i	$-0.618908\pi$
$353$	−4840.66	−0.729864	−0.364932	+	0.931034i	$0.618908\pi$
$354$	2539.50	0.381279
$355$	8852.70	1.32353
$356$	−1833.07	−0.272901
$357$	−104.964	−0.0155610
$358$	−11741.0	−1.73333
$359$	−2758.30	−0.405509	−0.202754	−	0.979230i	$-0.564989\pi$
$359$	−2758.30	−0.405509	−0.202754	+	0.979230i	$0.564989\pi$
$360$	−6560.29	−0.960438
$361$	−1683.59	−0.245456
$362$	−1975.57	−0.286834
$363$	1644.14	0.237727
$364$	500.743	0.0721046
$365$	7378.82	1.05815
$366$	−2453.67	−0.350424
$367$	−1924.67	−0.273752	−0.136876	−	0.990588i	$-0.543706\pi$
$367$	−1924.67	−0.273752	−0.136876	+	0.990588i	$0.543706\pi$
$368$	−2902.87	−0.411202
$369$	5935.95	0.837434
$370$	−17060.4	−2.39711
$371$	855.499	0.119718
$372$	867.366	0.120889
$373$	10669.6	1.48110	0.740548	−	0.672003i	$-0.234566\pi$
$373$	10669.6	1.48110	0.740548	+	0.672003i	$0.234566\pi$
$374$	448.718	0.0620392
$375$	5961.02	0.820869
$376$	4353.86	0.597163
$377$	−7207.72	−0.984659
$378$	726.612	0.0988701
$379$	−13147.8	−1.78195	−0.890973	−	0.454055i	$-0.849977\pi$
$379$	−13147.8	−1.78195	−0.890973	+	0.454055i	$0.849977\pi$
$380$	7245.97	0.978186
$381$	−159.508	−0.0214484
$382$	17037.2	2.28194
$383$	−1516.27	−0.202292	−0.101146	−	0.994872i	$-0.532251\pi$
$383$	−1516.27	−0.202292	−0.101146	+	0.994872i	$0.532251\pi$
$384$	−1747.45	−0.232224
$385$	−314.788	−0.0416703
$386$	9962.33	1.31365
$387$	1727.12	0.226859
$388$	−86.8826	−0.0113680
$389$	−2833.57	−0.369325	−0.184663	−	0.982802i	$-0.559119\pi$
$389$	−2833.57	−0.369325	−0.184663	+	0.982802i	$0.559119\pi$
$390$	3357.81	0.435972
$391$	982.528	0.127081
$392$	3971.81	0.511752
$393$	−1079.12	−0.138510
$394$	18769.3	2.39995
$395$	5004.35	0.637459
$396$	−553.752	−0.0702705
$397$	8181.42	1.03429	0.517146	−	0.855897i	$-0.326994\pi$
$397$	8181.42	1.03429	0.517146	+	0.855897i	$0.326994\pi$
$398$	13583.7	1.71078
$399$	280.354	0.0351761
$400$	−27384.6	−3.42308
$401$	−3816.00	−0.475217	−0.237609	−	0.971361i	$-0.576364\pi$
$401$	−3816.00	−0.475217	−0.237609	+	0.971361i	$0.576364\pi$
$402$	−2536.31	−0.314676
$403$	−5153.77	−0.637042
$404$	−2150.23	−0.264797
$405$	−13076.0	−1.60433
$406$	−2292.04	−0.280177
$407$	1037.32	0.126334
$408$	403.036	0.0489051
$409$	4710.01	0.569425	0.284713	−	0.958613i	$-0.408102\pi$
$409$	4710.01	0.569425	0.284713	+	0.958613i	$0.408102\pi$
$410$	17987.3	2.16666
$411$	−601.153	−0.0721477
$412$	−9314.31	−1.11379
$413$	−1763.90	−0.210159
$414$	−3298.43	−0.391568
$415$	−24355.3	−2.88085
$416$	−6514.71	−0.767813
$417$	3314.34	0.389218
$418$	−1198.51	−0.140241
$419$	3426.47	0.399508	0.199754	−	0.979846i	$-0.435986\pi$
$419$	3426.47	0.399508	0.199754	+	0.979846i	$0.435986\pi$
$420$	392.517	0.0456021
$421$	−16395.3	−1.89800	−0.948998	−	0.315282i	$-0.897901\pi$
$421$	−16395.3	−1.89800	−0.948998	+	0.315282i	$0.897901\pi$
$422$	11110.3	1.28162
$423$	9290.41	1.06788
$424$	−3284.91	−0.376248
$425$	9268.83	1.05789
$426$	1825.83	0.207656
$427$	1704.28	0.193152
$428$	1423.66	0.160783
$429$	−204.164	−0.0229770
$430$	5233.58	0.586943
$431$	−1633.82	−0.182595	−0.0912973	−	0.995824i	$-0.529101\pi$
$431$	−1633.82	−0.182595	−0.0912973	+	0.995824i	$0.529101\pi$
$432$	−5239.45	−0.583526
$433$	−5080.57	−0.563872	−0.281936	−	0.959433i	$-0.590977\pi$
$433$	−5080.57	−0.563872	−0.281936	+	0.959433i	$0.590977\pi$
$434$	−1638.89	−0.181265
$435$	−5649.90	−0.622741
$436$	4389.14	0.482114
$437$	−2624.29	−0.287269
$438$	1521.85	0.166020
$439$	14265.8	1.55095	0.775476	−	0.631377i	$-0.217510\pi$
$439$	14265.8	1.55095	0.775476	+	0.631377i	$0.217510\pi$
$440$	1208.71	0.130961
$441$	8475.18	0.915147
$442$	3324.58	0.357770
$443$	5429.63	0.582324	0.291162	−	0.956674i	$-0.405958\pi$
$443$	5429.63	0.582324	0.291162	+	0.956674i	$0.405958\pi$
$444$	−1293.46	−0.138254
$445$	8537.80	0.909507
$446$	10446.4	1.10908
$447$	3663.54	0.387650
$448$	−96.3622	−0.0101622
$449$	11238.8	1.18127	0.590636	−	0.806938i	$-0.298877\pi$
$449$	11238.8	1.18127	0.590636	+	0.806938i	$0.298877\pi$
$450$	−31116.3	−3.25963
$451$	−1093.68	−0.114189
$452$	−7176.71	−0.746823
$453$	−4544.68	−0.471364
$454$	2641.22	0.273036
$455$	−2332.28	−0.240306
$456$	−1076.49	−0.110551
$457$	−15023.4	−1.53778	−0.768891	−	0.639379i	$-0.779191\pi$
$457$	−15023.4	−1.53778	−0.768891	+	0.639379i	$0.779191\pi$
$458$	5467.44	0.557809
$459$	1773.39	0.180337
$460$	−3674.20	−0.372414
$461$	−5963.18	−0.602457	−0.301229	−	0.953552i	$-0.597397\pi$
$461$	−5963.18	−0.602457	−0.301229	+	0.953552i	$0.597397\pi$
$462$	−64.9235	−0.00653791
$463$	−18727.4	−1.87978	−0.939890	−	0.341477i	$-0.889073\pi$
$463$	−18727.4	−1.87978	−0.939890	+	0.341477i	$0.889073\pi$
$464$	16527.4	1.65359
$465$	−4039.88	−0.402893
$466$	−9810.79	−0.975271
$467$	−10274.3	−1.01807	−0.509033	−	0.860747i	$-0.669997\pi$
$467$	−10274.3	−1.01807	−0.509033	+	0.860747i	$0.669997\pi$
$468$	−4102.79	−0.405238
$469$	1761.68	0.173448
$470$	28152.1	2.76290
$471$	25.2188	0.00246713
$472$	6772.94	0.660487
$473$	−318.215	−0.0309335
$474$	1032.12	0.100015
$475$	−24756.6	−2.39139
$476$	388.633	0.0374222
$477$	−7009.44	−0.672831
$478$	17751.8	1.69864
$479$	−8907.75	−0.849699	−0.424849	−	0.905264i	$-0.639673\pi$
$479$	−8907.75	−0.849699	−0.424849	+	0.905264i	$0.639673\pi$
$480$	−5106.68	−0.485598
$481$	7685.58	0.728549
$482$	−6754.16	−0.638265
$483$	−142.159	−0.0133922
$484$	−6087.49	−0.571703
$485$	404.668	0.0378867
$486$	−9019.70	−0.841856
$487$	14163.8	1.31791	0.658957	−	0.752180i	$-0.270998\pi$
$487$	14163.8	1.31791	0.658957	+	0.752180i	$0.270998\pi$
$488$	−6544.03	−0.607038
$489$	−3721.33	−0.344139
$490$	25681.8	2.36772
$491$	10620.0	0.976117	0.488058	−	0.872811i	$-0.337705\pi$
$491$	10620.0	0.976117	0.488058	+	0.872811i	$0.337705\pi$
$492$	1363.73	0.124963
$493$	−5594.00	−0.511037
$494$	−8879.81	−0.808748
$495$	2579.18	0.234193
$496$	11817.7	1.06982
$497$	−1268.19	−0.114459
$498$	−5023.17	−0.451995
$499$	60.1581	0.00539689	0.00269844	−	0.999996i	$-0.499141\pi$
$499$	60.1581	0.00539689	0.00269844	+	0.999996i	$0.499141\pi$
$500$	−22070.9	−1.97408
$501$	−210.827	−0.0188005
$502$	−25247.8	−2.24475
$503$	−2042.42	−0.181047	−0.0905237	−	0.995894i	$-0.528854\pi$
$503$	−2042.42	−0.181047	−0.0905237	+	0.995894i	$0.528854\pi$
$504$	939.794	0.0830591
$505$	10015.0	0.882499
$506$	607.724	0.0533926
$507$	1246.70	0.109207
$508$	590.585	0.0515807
$509$	507.623	0.0442043	0.0221022	−	0.999756i	$-0.492964\pi$
$509$	507.623	0.0442043	0.0221022	+	0.999756i	$0.492964\pi$
$510$	2606.04	0.226269
$511$	−1057.05	−0.0915093
$512$	7353.67	0.634745
$513$	−4736.64	−0.407656
$514$	10675.1	0.916069
$515$	43382.7	3.71198
$516$	396.791	0.0338522
$517$	−1711.72	−0.145612
$518$	2443.99	0.207303
$519$	3149.42	0.266366
$520$	8955.41	0.755232
$521$	−10385.4	−0.873304	−0.436652	−	0.899631i	$-0.643836\pi$
$521$	−10385.4	−0.873304	−0.436652	+	0.899631i	$0.643836\pi$
$522$	18779.6	1.57463
$523$	9516.81	0.795681	0.397840	−	0.917455i	$-0.369760\pi$
$523$	9516.81	0.795681	0.397840	+	0.917455i	$0.369760\pi$
$524$	3995.49	0.333099
$525$	−1341.08	−0.111484
$526$	−24520.5	−2.03259
$527$	−3999.91	−0.330624
$528$	468.150	0.0385864
$529$	−10836.3	−0.890631
$530$	−21240.3	−1.74079
$531$	14452.3	1.18112
$532$	−1038.02	−0.0845939
$533$	−8103.12	−0.658509
$534$	1760.88	0.142698
$535$	−6630.90	−0.535848
$536$	−6764.44	−0.545110
$537$	4146.06	0.333176
$538$	126.938	0.0101723
$539$	−1561.52	−0.124786
$540$	−6631.65	−0.528483
$541$	15484.2	1.23053	0.615267	−	0.788319i	$-0.289048\pi$
$541$	15484.2	1.23053	0.615267	+	0.788319i	$0.289048\pi$
$542$	−29126.5	−2.30828
$543$	697.626	0.0551344
$544$	−5056.15	−0.398494
$545$	−20443.1	−1.60676
$546$	−481.023	−0.0377031
$547$	−16557.4	−1.29423	−0.647115	−	0.762392i	$-0.724025\pi$
$547$	−16557.4	−1.29423	−0.647115	+	0.762392i	$0.724025\pi$
$548$	2225.79	0.173506
$549$	−13963.9	−1.08554
$550$	5733.06	0.444470
$551$	14941.3	1.15521
$552$	545.855	0.0420890
$553$	−716.898	−0.0551277
$554$	5588.67	0.428592
$555$	6024.48	0.460766
$556$	−12271.5	−0.936019
$557$	−20606.8	−1.56757	−0.783786	−	0.621031i	$-0.786714\pi$
$557$	−20606.8	−1.56757	−0.783786	+	0.621031i	$0.786714\pi$
$558$	13428.1	1.01874
$559$	−2357.68	−0.178389
$560$	5347.96	0.403559
$561$	−158.454	−0.0119250
$562$	−22811.3	−1.71216
$563$	−4874.74	−0.364913	−0.182456	−	0.983214i	$-0.558405\pi$
$563$	−4874.74	−0.364913	−0.182456	+	0.983214i	$0.558405\pi$
$564$	2134.39	0.159351
$565$	33426.6	2.48897
$566$	−15734.7	−1.16852
$567$	1873.21	0.138743
$568$	4869.56	0.359722
$569$	8937.52	0.658489	0.329245	−	0.944245i	$-0.393206\pi$
$569$	8937.52	0.658489	0.329245	+	0.944245i	$0.393206\pi$
$570$	−6960.61	−0.511487
$571$	−15728.0	−1.15270	−0.576352	−	0.817201i	$-0.695524\pi$
$571$	−15728.0	−1.15270	−0.576352	+	0.817201i	$0.695524\pi$
$572$	755.924	0.0552566
$573$	−6016.28	−0.438628
$574$	−2576.77	−0.187374
$575$	12553.3	0.910449
$576$	789.533	0.0571132
$577$	−16312.3	−1.17693	−0.588465	−	0.808522i	$-0.700268\pi$
$577$	−16312.3	−1.17693	−0.588465	+	0.808522i	$0.700268\pi$
$578$	−14893.9	−1.07181
$579$	−3517.95	−0.252506
$580$	20919.0	1.49761
$581$	3489.02	0.249138
$582$	83.4609	0.00594427
$583$	1291.46	0.0917444
$584$	4058.83	0.287595
$585$	19109.3	1.35055
$586$	−6537.26	−0.460839
$587$	−3925.07	−0.275988	−0.137994	−	0.990433i	$-0.544066\pi$
$587$	−3925.07	−0.275988	−0.137994	+	0.990433i	$0.544066\pi$
$588$	1947.10	0.136560
$589$	10683.6	0.747384
$590$	43793.9	3.05588
$591$	−6627.91	−0.461313
$592$	−17623.1	−1.22349
$593$	−5393.72	−0.373514	−0.186757	−	0.982406i	$-0.559798\pi$
$593$	−5393.72	−0.373514	−0.186757	+	0.982406i	$0.559798\pi$
$594$	1096.90	0.0757680
$595$	−1810.12	−0.124719
$596$	−13564.4	−0.932247
$597$	−4796.75	−0.328841
$598$	4502.67	0.307906
$599$	20576.9	1.40359	0.701795	−	0.712379i	$-0.252382\pi$
$599$	20576.9	1.40359	0.701795	+	0.712379i	$0.252382\pi$
$600$	5149.40	0.350373
$601$	−22195.4	−1.50644	−0.753218	−	0.657771i	$-0.771500\pi$
$601$	−22195.4	−1.50644	−0.753218	+	0.657771i	$0.771500\pi$
$602$	−749.736	−0.0507591
$603$	−14434.2	−0.974800
$604$	16826.9	1.13357
$605$	28353.4	1.90534
$606$	2065.55	0.138461
$607$	6045.19	0.404229	0.202114	−	0.979362i	$-0.435219\pi$
$607$	6045.19	0.404229	0.202114	+	0.979362i	$0.435219\pi$
$608$	13504.8	0.900806
$609$	809.377	0.0538549
$610$	−42313.8	−2.80858
$611$	−12682.3	−0.839722
$612$	−3184.23	−0.210318
$613$	3343.79	0.220317	0.110159	−	0.993914i	$-0.464864\pi$
$613$	3343.79	0.220317	0.110159	+	0.993914i	$0.464864\pi$
$614$	17758.7	1.16723
$615$	−6351.78	−0.416469
$616$	−173.154	−0.0113256
$617$	−15547.1	−1.01443	−0.507215	−	0.861820i	$-0.669325\pi$
$617$	−15547.1	−1.01443	−0.507215	+	0.861820i	$0.669325\pi$
$618$	8947.49	0.582396
$619$	22619.3	1.46873	0.734366	−	0.678754i	$-0.237480\pi$
$619$	22619.3	1.46873	0.734366	+	0.678754i	$0.237480\pi$
$620$	14957.8	0.968904
$621$	2401.80	0.155203
$622$	−1172.51	−0.0755843
$623$	−1223.08	−0.0786545
$624$	3468.56	0.222522
$625$	59782.5	3.82608
$626$	13399.0	0.855482
$627$	423.223	0.0269568
$628$	−93.3735	−0.00593314
$629$	5964.87	0.378116
$630$	6076.72	0.384289
$631$	26867.2	1.69504	0.847518	−	0.530767i	$-0.178096\pi$
$631$	26867.2	1.69504	0.847518	+	0.530767i	$0.178096\pi$
$632$	2752.72	0.173255
$633$	−3923.34	−0.246349
$634$	19642.2	1.23043
$635$	−2750.74	−0.171905
$636$	−1610.36	−0.100401
$637$	−11569.4	−0.719618
$638$	−3460.06	−0.214710
$639$	10390.8	0.643277
$640$	−30134.9	−1.86123
$641$	8374.16	0.516006	0.258003	−	0.966144i	$-0.416936\pi$
$641$	8374.16	0.516006	0.258003	+	0.966144i	$0.416936\pi$
$642$	−1367.59	−0.0840725
$643$	7862.35	0.482210	0.241105	−	0.970499i	$-0.422490\pi$
$643$	7862.35	0.482210	0.241105	+	0.970499i	$0.422490\pi$
$644$	526.348	0.0322065
$645$	−1848.11	−0.112821
$646$	−6891.74	−0.419740
$647$	17180.5	1.04395	0.521973	−	0.852962i	$-0.325196\pi$
$647$	17180.5	1.04395	0.521973	+	0.852962i	$0.325196\pi$
$648$	−7192.67	−0.436041
$649$	−2662.79	−0.161053
$650$	42476.6	2.56319
$651$	578.733	0.0348423
$652$	13778.4	0.827610
$653$	−23797.5	−1.42614	−0.713069	−	0.701094i	$-0.752695\pi$
$653$	−23797.5	−1.42614	−0.713069	+	0.701094i	$0.752695\pi$
$654$	−4216.29	−0.252095
$655$	−18609.6	−1.11013
$656$	18580.6	1.10587
$657$	8660.85	0.514295
$658$	−4032.93	−0.238936
$659$	2006.16	0.118587	0.0592936	−	0.998241i	$-0.481115\pi$
$659$	2006.16	0.118587	0.0592936	+	0.998241i	$0.481115\pi$
$660$	592.545	0.0349466
$661$	6108.34	0.359436	0.179718	−	0.983718i	$-0.442482\pi$
$661$	6108.34	0.359436	0.179718	+	0.983718i	$0.442482\pi$
$662$	8925.56	0.524020
$663$	−1174.00	−0.0687696
$664$	−13397.0	−0.782988
$665$	4834.74	0.281929
$666$	−20024.6	−1.16507
$667$	−7576.27	−0.439811
$668$	780.596	0.0452128
$669$	−3688.89	−0.213185
$670$	−43738.9	−2.52206
$671$	2572.79	0.148020
$672$	731.558	0.0419947
$673$	−27856.8	−1.59554	−0.797771	−	0.602961i	$-0.793987\pi$
$673$	−27856.8	−1.59554	−0.797771	+	0.602961i	$0.793987\pi$
$674$	−24122.0	−1.37855
$675$	22657.7	1.29199
$676$	−4615.97	−0.262629
$677$	10994.6	0.624164	0.312082	−	0.950055i	$-0.398974\pi$
$677$	10994.6	0.624164	0.312082	+	0.950055i	$0.398974\pi$
$678$	6894.07	0.390509
$679$	−57.9707	−0.00327646
$680$	6950.41	0.391965
$681$	−932.682	−0.0524823
$682$	−2474.07	−0.138911
$683$	6307.36	0.353359	0.176680	−	0.984268i	$-0.443464\pi$
$683$	6307.36	0.353359	0.176680	+	0.984268i	$0.443464\pi$
$684$	8504.92	0.475430
$685$	−10367.0	−0.578250
$686$	−7464.33	−0.415437
$687$	−1930.69	−0.107221
$688$	5406.20	0.299577
$689$	9568.54	0.529075
$690$	3529.50	0.194733
$691$	26464.0	1.45693	0.728464	−	0.685084i	$-0.240235\pi$
$691$	26464.0	1.45693	0.728464	+	0.685084i	$0.240235\pi$
$692$	−11660.8	−0.640576
$693$	−369.481	−0.0202531
$694$	923.537	0.0505144
$695$	57156.2	3.11951
$696$	−3107.81	−0.169255
$697$	−6288.94	−0.341765
$698$	−12949.8	−0.702228
$699$	3464.44	0.187464
$700$	4965.38	0.268105
$701$	3019.55	0.162692	0.0813459	−	0.996686i	$-0.474078\pi$
$701$	3019.55	0.162692	0.0813459	+	0.996686i	$0.474078\pi$
$702$	8126.97	0.436941
$703$	−15931.9	−0.854741
$704$	−145.469	−0.00778772
$705$	−9941.24	−0.531076
$706$	−17216.9	−0.917798
$707$	−1434.70	−0.0763189
$708$	3320.30	0.176249
$709$	31902.5	1.68988	0.844940	−	0.534862i	$-0.179636\pi$
$709$	31902.5	1.68988	0.844940	+	0.534862i	$0.179636\pi$
$710$	31486.6	1.66433
$711$	5873.83	0.309825
$712$	4696.34	0.247195
$713$	−5417.30	−0.284543
$714$	−373.328	−0.0195679
$715$	−3520.82	−0.184156
$716$	−15350.9	−0.801245
$717$	−6268.62	−0.326508
$718$	−9810.52	−0.509924
$719$	−28697.6	−1.48851	−0.744255	−	0.667896i	$-0.767195\pi$
$719$	−28697.6	−1.48851	−0.744255	+	0.667896i	$0.767195\pi$
$720$	−43818.0	−2.26806
$721$	−6214.79	−0.321014
$722$	−5988.05	−0.308659
$723$	2385.07	0.122685
$724$	−2582.99	−0.132591
$725$	−71471.9	−3.66124
$726$	5847.75	0.298940
$727$	15909.5	0.811623	0.405812	−	0.913957i	$-0.366989\pi$
$727$	15909.5	0.811623	0.405812	+	0.913957i	$0.366989\pi$
$728$	−1282.91	−0.0653128
$729$	−13115.2	−0.666321
$730$	26244.4	1.33062
$731$	−1829.82	−0.0925835
$732$	−3208.08	−0.161986
$733$	4339.64	0.218674	0.109337	−	0.994005i	$-0.465127\pi$
$733$	4339.64	0.218674	0.109337	+	0.994005i	$0.465127\pi$
$734$	−6845.52	−0.344241
$735$	−9068.90	−0.455117
$736$	−6847.83	−0.342954
$737$	2659.44	0.132920
$738$	21112.5	1.05307
$739$	−21253.4	−1.05794	−0.528970	−	0.848640i	$-0.677422\pi$
$739$	−21253.4	−1.05794	−0.528970	+	0.848640i	$0.677422\pi$
$740$	−22305.9	−1.10808
$741$	3135.69	0.155455
$742$	3042.77	0.150544
$743$	−12418.6	−0.613183	−0.306591	−	0.951841i	$-0.599189\pi$
$743$	−12418.6	−0.613183	−0.306591	+	0.951841i	$0.599189\pi$
$744$	−2222.20	−0.109502
$745$	63178.2	3.10694
$746$	37948.7	1.86247
$747$	−28586.9	−1.40019
$748$	586.682	0.0286781
$749$	949.910	0.0463404
$750$	21201.7	1.03224
$751$	−8975.98	−0.436136	−0.218068	−	0.975934i	$-0.569975\pi$
$751$	−8975.98	−0.436136	−0.218068	+	0.975934i	$0.569975\pi$
$752$	29080.7	1.41019
$753$	8915.64	0.431479
$754$	−25635.8	−1.23820
$755$	−78373.5	−3.77789
$756$	950.017	0.0457034
$757$	−21920.9	−1.05248	−0.526241	−	0.850336i	$-0.676399\pi$
$757$	−21920.9	−1.05248	−0.526241	+	0.850336i	$0.676399\pi$
$758$	−46763.1	−2.24078
$759$	−214.603	−0.0102630
$760$	−18564.2	−0.886046
$761$	−20443.5	−0.973817	−0.486908	−	0.873453i	$-0.661875\pi$
$761$	−20443.5	−0.973817	−0.486908	+	0.873453i	$0.661875\pi$
$762$	−567.327	−0.0269712
$763$	2928.57	0.138953
$764$	22275.5	1.05484
$765$	14831.0	0.700936
$766$	−5392.96	−0.254381
$767$	−19728.8	−0.928767
$768$	−6527.24	−0.306681
$769$	16001.1	0.750342	0.375171	−	0.926956i	$-0.377584\pi$
$769$	16001.1	0.750342	0.375171	+	0.926956i	$0.377584\pi$
$770$	−1119.61	−0.0524001
$771$	−3769.66	−0.176084
$772$	13025.4	0.607245
$773$	12671.2	0.589588	0.294794	−	0.955561i	$-0.404749\pi$
$773$	12671.2	0.589588	0.294794	+	0.955561i	$0.404749\pi$
$774$	6142.88	0.285273
$775$	−51104.9	−2.36870
$776$	222.594	0.0102972
$777$	−863.037	−0.0398472
$778$	−10078.2	−0.464423
$779$	16797.5	0.772569
$780$	4390.20	0.201531
$781$	−1914.47	−0.0877146
$782$	3494.58	0.159803
$783$	−13674.6	−0.624124
$784$	26528.8	1.20849
$785$	434.901	0.0197736
$786$	−3838.14	−0.174176
$787$	−19819.4	−0.897696	−0.448848	−	0.893608i	$-0.648165\pi$
$787$	−19819.4	−0.897696	−0.448848	+	0.893608i	$0.648165\pi$
$788$	24540.1	1.10940
$789$	8658.81	0.390699
$790$	17799.1	0.801599
$791$	−4788.52	−0.215247
$792$	1418.72	0.0636514
$793$	19062.0	0.853607
$794$	29099.1	1.30061
$795$	7500.48	0.334610
$796$	17760.1	0.790819
$797$	−5844.03	−0.259732	−0.129866	−	0.991532i	$-0.541455\pi$
$797$	−5844.03	−0.259732	−0.129866	+	0.991532i	$0.541455\pi$
$798$	997.142	0.0442336
$799$	−9842.88	−0.435815
$800$	−64600.1	−2.85495
$801$	10021.2	0.442049
$802$	−13572.5	−0.597582
$803$	−1595.73	−0.0701271
$804$	−3316.13	−0.145461
$805$	−2451.54	−0.107336
$806$	−18330.5	−0.801074
$807$	−44.8250	−0.00195529
$808$	5508.90	0.239854
$809$	1202.95	0.0522785	0.0261393	−	0.999658i	$-0.491679\pi$
$809$	1202.95	0.0522785	0.0261393	+	0.999658i	$0.491679\pi$
$810$	−46507.9	−2.01743
$811$	13021.4	0.563804	0.281902	−	0.959443i	$-0.409035\pi$
$811$	13021.4	0.563804	0.281902	+	0.959443i	$0.409035\pi$
$812$	−2996.75	−0.129514
$813$	10285.3	0.443692
$814$	3689.46	0.158864
$815$	−64174.7	−2.75821
$816$	2691.99	0.115488
$817$	4887.38	0.209287
$818$	16752.2	0.716048
$819$	−2737.51	−0.116796
$820$	23517.7	1.00155
$821$	41429.4	1.76114	0.880571	−	0.473915i	$-0.157160\pi$
$821$	41429.4	1.76114	0.880571	+	0.473915i	$0.157160\pi$
$822$	−2138.14	−0.0907251
$823$	−659.098	−0.0279158	−0.0139579	−	0.999903i	$-0.504443\pi$
$823$	−659.098	−0.0279158	−0.0139579	+	0.999903i	$0.504443\pi$
$824$	23863.3	1.00888
$825$	−2024.49	−0.0854348
$826$	−6273.70	−0.264273
$827$	−7705.47	−0.323997	−0.161999	−	0.986791i	$-0.551794\pi$
$827$	−7705.47	−0.323997	−0.161999	+	0.986791i	$0.551794\pi$
$828$	−4312.57	−0.181005
$829$	−17314.6	−0.725405	−0.362702	−	0.931905i	$-0.618146\pi$
$829$	−17314.6	−0.725405	−0.362702	+	0.931905i	$0.618146\pi$
$830$	−86625.1	−3.62265
$831$	−1973.50	−0.0823828
$832$	−1077.79	−0.0449105
$833$	−8979.17	−0.373481
$834$	11788.2	0.489439
$835$	−3635.74	−0.150683
$836$	−1567.00	−0.0648276
$837$	−9777.81	−0.403788
$838$	12187.0	0.502378
$839$	27022.8	1.11196	0.555978	−	0.831197i	$-0.312344\pi$
$839$	27022.8	1.11196	0.555978	+	0.831197i	$0.312344\pi$
$840$	−1005.63	−0.0413066
$841$	18746.3	0.768638
$842$	−58313.4	−2.38671
$843$	8055.26	0.329108
$844$	14526.3	0.592437
$845$	21499.6	0.875275
$846$	33043.4	1.34286
$847$	−4061.76	−0.164774
$848$	−21940.8	−0.888503
$849$	5556.34	0.224609
$850$	32966.7	1.33029
$851$	8078.56	0.325417
$852$	2387.20	0.0959907
$853$	31941.6	1.28213	0.641066	−	0.767486i	$-0.278493\pi$
$853$	31941.6	1.28213	0.641066	+	0.767486i	$0.278493\pi$
$854$	6061.66	0.242887
$855$	−39612.9	−1.58448
$856$	−3647.42	−0.145638
$857$	32460.4	1.29384	0.646922	−	0.762556i	$-0.276056\pi$
$857$	32460.4	1.29384	0.646922	+	0.762556i	$0.276056\pi$
$858$	−726.153	−0.0288933
$859$	−15544.1	−0.617412	−0.308706	−	0.951157i	$-0.599896\pi$
$859$	−15544.1	−0.617412	−0.308706	+	0.951157i	$0.599896\pi$
$860$	6842.70	0.271319
$861$	909.925	0.0360164
$862$	−5811.04	−0.229611
$863$	−38037.6	−1.50036	−0.750182	−	0.661231i	$-0.770034\pi$
$863$	−38037.6	−1.50036	−0.750182	+	0.661231i	$0.770034\pi$
$864$	−12359.8	−0.486677
$865$	54312.0	2.13487
$866$	−18070.2	−0.709064
$867$	5259.43	0.206020
$868$	−2142.78	−0.0837912
$869$	−1082.23	−0.0422465
$870$	−20095.1	−0.783091
$871$	19704.0	0.766526
$872$	−11245.0	−0.436702
$873$	474.977	0.0184141
$874$	−9333.86	−0.361239
$875$	−14726.4	−0.568963
$876$	1989.76	0.0767439
$877$	27706.4	1.06679	0.533396	−	0.845865i	$-0.320915\pi$
$877$	27706.4	1.06679	0.533396	+	0.845865i	$0.320915\pi$
$878$	50739.4	1.95031
$879$	2308.47	0.0885812
$880$	8073.30	0.309263
$881$	31693.3	1.21200	0.606002	−	0.795463i	$-0.292772\pi$
$881$	31693.3	1.21200	0.606002	+	0.795463i	$0.292772\pi$
$882$	30143.9	1.15079
$883$	23723.0	0.904126	0.452063	−	0.891986i	$-0.350688\pi$
$883$	23723.0	0.904126	0.452063	+	0.891986i	$0.350688\pi$
$884$	4346.77	0.165382
$885$	−15464.8	−0.587392
$886$	19311.7	0.732267
$887$	−19424.6	−0.735305	−0.367652	−	0.929963i	$-0.619838\pi$
$887$	−19424.6	−0.735305	−0.367652	+	0.929963i	$0.619838\pi$
$888$	3313.85	0.125232
$889$	394.057	0.0148664
$890$	30366.6	1.14370
$891$	2827.80	0.106324
$892$	13658.2	0.512681
$893$	26289.9	0.985170
$894$	13030.2	0.487467
$895$	71499.2	2.67034
$896$	4316.98	0.160960
$897$	−1590.01	−0.0591849
$898$	39973.2	1.48544
$899$	30843.3	1.14425
$900$	−40683.3	−1.50679
$901$	7426.27	0.274589
$902$	−3889.90	−0.143592
$903$	264.751	0.00975677
$904$	18386.8	0.676477
$905$	12030.6	0.441892
$906$	−16164.2	−0.592736
$907$	19267.5	0.705365	0.352683	−	0.935743i	$-0.385270\pi$
$907$	19267.5	0.705365	0.352683	+	0.935743i	$0.385270\pi$
$908$	3453.29	0.126213
$909$	11755.1	0.428923
$910$	−8295.29	−0.302183
$911$	1028.87	0.0374181	0.0187090	−	0.999825i	$-0.494044\pi$
$911$	1028.87	0.0374181	0.0187090	+	0.999825i	$0.494044\pi$
$912$	−7190.19	−0.261065
$913$	5267.03	0.190924
$914$	−53434.2	−1.93375
$915$	14942.1	0.539858
$916$	7148.47	0.257852
$917$	2665.92	0.0960047
$918$	6307.45	0.226772
$919$	4737.40	0.170046	0.0850231	−	0.996379i	$-0.472904\pi$
$919$	4737.40	0.170046	0.0850231	+	0.996379i	$0.472904\pi$
$920$	9413.32	0.337335
$921$	−6271.04	−0.224362
$922$	−21209.4	−0.757585
$923$	−14184.4	−0.505835
$924$	−84.8850	−0.00302220
$925$	76210.3	2.70895
$926$	−66608.3	−2.36381
$927$	50920.3	1.80414
$928$	38988.0	1.37914
$929$	−8146.55	−0.287707	−0.143853	−	0.989599i	$-0.545949\pi$
$929$	−8146.55	−0.287707	−0.143853	+	0.989599i	$0.545949\pi$
$930$	−14368.7	−0.506634
$931$	23982.9	0.844263
$932$	−12827.2	−0.450826
$933$	414.044	0.0145286
$934$	−36542.7	−1.28021
$935$	−2732.56	−0.0955766
$936$	10511.4	0.367067
$937$	−20190.7	−0.703951	−0.351976	−	0.936009i	$-0.614490\pi$
$937$	−20190.7	−0.703951	−0.351976	+	0.936009i	$0.614490\pi$
$938$	6265.82	0.218109
$939$	−4731.53	−0.164438
$940$	36807.8	1.27717
$941$	629.020	0.0217911	0.0108956	−	0.999941i	$-0.496532\pi$
$941$	629.020	0.0217911	0.0108956	+	0.999941i	$0.496532\pi$
$942$	89.6962	0.00310240
$943$	−8517.45	−0.294132
$944$	45238.4	1.55973
$945$	−4424.84	−0.152318
$946$	−1131.80	−0.0388986
$947$	7085.45	0.243132	0.121566	−	0.992583i	$-0.461208\pi$
$947$	7085.45	0.243132	0.121566	+	0.992583i	$0.461208\pi$
$948$	1349.46	0.0462326
$949$	−11822.9	−0.404411
$950$	−88052.4	−3.00716
$951$	−6936.16	−0.236509
$952$	−995.680	−0.0338973
$953$	17711.3	0.602020	0.301010	−	0.953621i	$-0.402676\pi$
$953$	17711.3	0.602020	0.301010	+	0.953621i	$0.402676\pi$
$954$	−24930.6	−0.846079
$955$	−103751.	−3.51551
$956$	23209.8	0.785208
$957$	1221.84	0.0412711
$958$	−31682.4	−1.06849
$959$	1485.12	0.0500073
$960$	−844.843	−0.0284033
$961$	−7736.96	−0.259708
$962$	27335.5	0.916144
$963$	−7782.99	−0.260439
$964$	−8830.80	−0.295043
$965$	−60667.5	−2.02379
$966$	−505.619	−0.0168406
$967$	−33449.0	−1.11236	−0.556178	−	0.831063i	$-0.687733\pi$
$967$	−33449.0	−1.11236	−0.556178	+	0.831063i	$0.687733\pi$
$968$	15596.2	0.517852
$969$	2433.65	0.0806812
$970$	1439.29	0.0476422
$971$	36271.6	1.19878	0.599388	−	0.800459i	$-0.295411\pi$
$971$	36271.6	1.19878	0.599388	+	0.800459i	$0.295411\pi$
$972$	−11792.9	−0.389154
$973$	−8187.91	−0.269776
$974$	50376.8	1.65727
$975$	−14999.6	−0.492689
$976$	−43709.4	−1.43351
$977$	3918.58	0.128318	0.0641589	−	0.997940i	$-0.479564\pi$
$977$	3918.58	0.128318	0.0641589	+	0.997940i	$0.479564\pi$
$978$	−13235.7	−0.432752
$979$	−1846.37	−0.0602760
$980$	33577.9	1.09450
$981$	−23995.0	−0.780938
$982$	37772.4	1.22746
$983$	−15639.7	−0.507456	−0.253728	−	0.967276i	$-0.581657\pi$
$983$	−15639.7	−0.507456	−0.253728	+	0.967276i	$0.581657\pi$
$984$	−3493.89	−0.113192
$985$	−114299.	−3.69733
$986$	−19896.3	−0.642624
$987$	1424.13	0.0459277
$988$	−11610.0	−0.373850
$989$	−2478.23	−0.0796797
$990$	9173.43	0.294496
$991$	25003.7	0.801481	0.400741	−	0.916192i	$-0.368753\pi$
$991$	25003.7	0.801481	0.400741	+	0.916192i	$0.368753\pi$
$992$	27877.8	0.892259
$993$	−3151.84	−0.100726
$994$	−4510.61	−0.143932
$995$	−82720.4	−2.63559
$996$	−6567.60	−0.208938
$997$	33551.2	1.06578	0.532888	−	0.846186i	$-0.321107\pi$
$997$	33551.2	1.06578	0.532888	+	0.846186i	$0.321107\pi$
$998$	213.966	0.00678654
$999$	14581.2	0.461790

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	127.4.a.b.1.12	✓	13
3.2	odd	2		1143.4.a.d.1.2		13
4.3	odd	2		2032.4.a.g.1.8		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
127.4.a.b.1.12	✓	13	1.1	even	1	trivial
1143.4.a.d.1.2		13	3.2	odd	2
2032.4.a.g.1.8		13	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:	\( N \)	\(=\)	\( 127 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	127.a (trivial)

\(f(q)\)	\(=\)	\(q+3.55672 q^{2} -1.25597 q^{3} +4.65028 q^{4} -21.6593 q^{5} -4.46714 q^{6} +3.10281 q^{7} -11.9140 q^{8} -25.4225 q^{9} -77.0363 q^{10} +4.68401 q^{11} -5.84061 q^{12} +34.7041 q^{13} +11.0358 q^{14} +27.2035 q^{15} -79.5771 q^{16} +26.9343 q^{17} -90.4209 q^{18} -71.9403 q^{19} -100.722 q^{20} -3.89704 q^{21} +16.6597 q^{22} +36.4787 q^{23} +14.9637 q^{24} +344.127 q^{25} +123.433 q^{26} +65.8412 q^{27} +14.4289 q^{28} -207.690 q^{29} +96.7553 q^{30} -148.506 q^{31} -187.722 q^{32} -5.88298 q^{33} +95.7979 q^{34} -67.2048 q^{35} -118.222 q^{36} +221.460 q^{37} -255.872 q^{38} -43.5874 q^{39} +258.050 q^{40} -233.491 q^{41} -13.8607 q^{42} -67.9365 q^{43} +21.7819 q^{44} +550.635 q^{45} +129.744 q^{46} -365.440 q^{47} +99.9465 q^{48} -333.373 q^{49} +1223.96 q^{50} -33.8287 q^{51} +161.384 q^{52} +275.718 q^{53} +234.179 q^{54} -101.453 q^{55} -36.9670 q^{56} +90.3549 q^{57} -738.697 q^{58} -568.484 q^{59} +126.504 q^{60} +549.271 q^{61} -528.195 q^{62} -78.8813 q^{63} -31.0564 q^{64} -751.669 q^{65} -20.9241 q^{66} +567.770 q^{67} +125.252 q^{68} -45.8161 q^{69} -239.029 q^{70} -408.724 q^{71} +302.885 q^{72} -340.676 q^{73} +787.672 q^{74} -432.213 q^{75} -334.543 q^{76} +14.5336 q^{77} -155.028 q^{78} -231.048 q^{79} +1723.59 q^{80} +603.714 q^{81} -830.464 q^{82} +1124.47 q^{83} -18.1223 q^{84} -583.380 q^{85} -241.631 q^{86} +260.853 q^{87} -55.8054 q^{88} -394.186 q^{89} +1958.46 q^{90} +107.680 q^{91} +169.636 q^{92} +186.519 q^{93} -1299.77 q^{94} +1558.18 q^{95} +235.773 q^{96} -18.6833 q^{97} -1185.71 q^{98} -119.079 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 8 q^{2} - 16 q^{3} + 34 q^{4} - 46 q^{5} - 2 q^{6} - 26 q^{7} - 117 q^{8} + 3 q^{9} - 25 q^{10} - 53 q^{11} - 255 q^{12} - 75 q^{13} - 152 q^{14} - 62 q^{15} + 82 q^{16} - 479 q^{17} - 292 q^{18}+ \cdots + 1769 q^{99}+O(q^{100}) \) 13 * q - 8 * q^2 - 16 * q^3 + 34 * q^4 - 46 * q^5 - 2 * q^6 - 26 * q^7 - 117 * q^8 + 3 * q^9 - 25 * q^10 - 53 * q^11 - 255 * q^12 - 75 * q^13 - 152 * q^14 - 62 * q^15 + 82 * q^16 - 479 * q^17 - 292 * q^18 - 209 * q^19 - 533 * q^20 - 174 * q^21 - 407 * q^22 - 376 * q^23 + 108 * q^24 + 9 * q^25 - 622 * q^26 - 562 * q^27 - 258 * q^28 - 158 * q^29 - 287 * q^30 - 307 * q^31 - 609 * q^32 - 494 * q^33 + 948 * q^34 - 512 * q^35 + 1556 * q^36 - 171 * q^37 + 712 * q^38 - 154 * q^39 + 1897 * q^40 - 641 * q^41 + 1489 * q^42 + 530 * q^43 + 1328 * q^44 - 566 * q^45 + 2051 * q^46 - 555 * q^47 + 268 * q^48 + 357 * q^49 + 808 * q^50 + 1228 * q^51 + 2473 * q^52 - 1640 * q^53 + 2957 * q^54 + 540 * q^55 + 551 * q^56 + 968 * q^57 + 1328 * q^58 - 860 * q^59 + 4466 * q^60 + 191 * q^61 + 367 * q^62 + 120 * q^63 + 1915 * q^64 - 1584 * q^65 + 3416 * q^66 + 912 * q^67 - 1873 * q^68 - 2142 * q^69 + 2329 * q^70 + 115 * q^71 - 1864 * q^72 - 2563 * q^73 + 470 * q^74 + 44 * q^75 - 192 * q^76 - 3338 * q^77 + 4226 * q^78 + 169 * q^79 - 3194 * q^80 - 395 * q^81 + 54 * q^82 - 1688 * q^83 + 853 * q^84 + 480 * q^85 - 485 * q^86 - 1068 * q^87 - 1192 * q^88 - 2752 * q^89 + 4948 * q^90 - 398 * q^91 - 846 * q^92 - 804 * q^93 + 531 * q^94 - 1766 * q^95 + 3965 * q^96 - 2124 * q^97 - 1479 * q^98 + 1769 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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