LMFDB - Embedded newform 4031.2.a.c.1.44

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$32.1876970548$
Analytic rank:	$1$
Dimension:	$61$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.44
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.22202 q^{2} +2.56418 q^{3} -0.506669 q^{4} -0.815585 q^{5} +3.13347 q^{6} +1.64270 q^{7} -3.06320 q^{8} +3.57499 q^{9} +O(q^{10})$	\(q+1.22202 q^{2} +2.56418 q^{3} -0.506669 q^{4} -0.815585 q^{5} +3.13347 q^{6} +1.64270 q^{7} -3.06320 q^{8} +3.57499 q^{9} -0.996661 q^{10} -3.10611 q^{11} -1.29919 q^{12} -4.40903 q^{13} +2.00741 q^{14} -2.09130 q^{15} -2.72995 q^{16} -6.21598 q^{17} +4.36871 q^{18} +4.31329 q^{19} +0.413232 q^{20} +4.21216 q^{21} -3.79572 q^{22} +3.23122 q^{23} -7.85458 q^{24} -4.33482 q^{25} -5.38792 q^{26} +1.47439 q^{27} -0.832304 q^{28} -1.00000 q^{29} -2.55561 q^{30} -8.33892 q^{31} +2.79035 q^{32} -7.96460 q^{33} -7.59605 q^{34} -1.33976 q^{35} -1.81134 q^{36} -2.70989 q^{37} +5.27092 q^{38} -11.3055 q^{39} +2.49830 q^{40} +8.45534 q^{41} +5.14734 q^{42} -2.79729 q^{43} +1.57377 q^{44} -2.91571 q^{45} +3.94861 q^{46} -7.54263 q^{47} -7.00006 q^{48} -4.30155 q^{49} -5.29723 q^{50} -15.9389 q^{51} +2.23392 q^{52} -3.43806 q^{53} +1.80173 q^{54} +2.53330 q^{55} -5.03190 q^{56} +11.0600 q^{57} -1.22202 q^{58} +6.79534 q^{59} +1.05960 q^{60} -6.00294 q^{61} -10.1903 q^{62} +5.87263 q^{63} +8.86975 q^{64} +3.59594 q^{65} -9.73289 q^{66} +7.63281 q^{67} +3.14945 q^{68} +8.28541 q^{69} -1.63721 q^{70} +6.99398 q^{71} -10.9509 q^{72} +4.10515 q^{73} -3.31154 q^{74} -11.1152 q^{75} -2.18541 q^{76} -5.10239 q^{77} -13.8156 q^{78} +16.0432 q^{79} +2.22651 q^{80} -6.94440 q^{81} +10.3326 q^{82} -13.7960 q^{83} -2.13417 q^{84} +5.06967 q^{85} -3.41835 q^{86} -2.56418 q^{87} +9.51462 q^{88} -12.4847 q^{89} -3.56306 q^{90} -7.24270 q^{91} -1.63716 q^{92} -21.3825 q^{93} -9.21724 q^{94} -3.51785 q^{95} +7.15494 q^{96} -3.07842 q^{97} -5.25658 q^{98} -11.1043 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) $ 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.22202	0.864098	0.432049	−	0.901850i	$-0.357791\pi$
$2$	1.22202	0.864098	0.432049	+	0.901850i	$0.357791\pi$
$3$	2.56418	1.48043	0.740214	−	0.672372i	$-0.234724\pi$
$3$	2.56418	1.48043	0.740214	+	0.672372i	$0.234724\pi$
$4$	−0.506669	−0.253335
$5$	−0.815585	−0.364741	−0.182370	−	0.983230i	$-0.558377\pi$
$5$	−0.815585	−0.364741	−0.182370	+	0.983230i	$0.558377\pi$
$6$	3.13347	1.27923
$7$	1.64270	0.620881	0.310440	−	0.950593i	$-0.399524\pi$
$7$	1.64270	0.620881	0.310440	+	0.950593i	$0.399524\pi$
$8$	−3.06320	−1.08300
$9$	3.57499	1.19166
$10$	−0.996661	−0.315172
$11$	−3.10611	−0.936526	−0.468263	−	0.883589i	$-0.655120\pi$
$11$	−3.10611	−0.936526	−0.468263	+	0.883589i	$0.655120\pi$
$12$	−1.29919	−0.375044
$13$	−4.40903	−1.22285	−0.611423	−	0.791304i	$-0.709403\pi$
$13$	−4.40903	−1.22285	−0.611423	+	0.791304i	$0.709403\pi$
$14$	2.00741	0.536502
$15$	−2.09130	−0.539972
$16$	−2.72995	−0.682487
$17$	−6.21598	−1.50760	−0.753799	−	0.657105i	$-0.771781\pi$
$17$	−6.21598	−1.50760	−0.753799	+	0.657105i	$0.771781\pi$
$18$	4.36871	1.02971
$19$	4.31329	0.989535	0.494768	−	0.869025i	$-0.335253\pi$
$19$	4.31329	0.989535	0.494768	+	0.869025i	$0.335253\pi$
$20$	0.413232	0.0924015
$21$	4.21216	0.919169
$22$	−3.79572	−0.809251
$23$	3.23122	0.673756	0.336878	−	0.941548i	$-0.390629\pi$
$23$	3.23122	0.673756	0.336878	+	0.941548i	$0.390629\pi$
$24$	−7.85458	−1.60331
$25$	−4.33482	−0.866964
$26$	−5.38792	−1.05666
$27$	1.47439	0.283746
$28$	−0.832304	−0.157291
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−2.55561	−0.466589
$31$	−8.33892	−1.49771	−0.748857	−	0.662731i	$-0.769397\pi$
$31$	−8.33892	−1.49771	−0.748857	+	0.662731i	$0.769397\pi$
$32$	2.79035	0.493268
$33$	−7.96460	−1.38646
$34$	−7.59605	−1.30271
$35$	−1.33976	−0.226461
$36$	−1.81134	−0.301890
$37$	−2.70989	−0.445503	−0.222752	−	0.974875i	$-0.571504\pi$
$37$	−2.70989	−0.445503	−0.222752	+	0.974875i	$0.571504\pi$
$38$	5.27092	0.855056
$39$	−11.3055	−1.81033
$40$	2.49830	0.395016
$41$	8.45534	1.32050	0.660251	−	0.751045i	$-0.270450\pi$
$41$	8.45534	1.32050	0.660251	+	0.751045i	$0.270450\pi$
$42$	5.14734	0.794252
$43$	−2.79729	−0.426583	−0.213292	−	0.976989i	$-0.568418\pi$
$43$	−2.79729	−0.426583	−0.213292	+	0.976989i	$0.568418\pi$
$44$	1.57377	0.237255
$45$	−2.91571	−0.434649
$46$	3.94861	0.582191
$47$	−7.54263	−1.10021	−0.550103	−	0.835097i	$-0.685412\pi$
$47$	−7.54263	−1.10021	−0.550103	+	0.835097i	$0.685412\pi$
$48$	−7.00006	−1.01037
$49$	−4.30155	−0.614507
$50$	−5.29723	−0.749142
$51$	−15.9389	−2.23189
$52$	2.23392	0.309789
$53$	−3.43806	−0.472254	−0.236127	−	0.971722i	$-0.575878\pi$
$53$	−3.43806	−0.472254	−0.236127	+	0.971722i	$0.575878\pi$
$54$	1.80173	0.245184
$55$	2.53330	0.341589
$56$	−5.03190	−0.672416
$57$	11.0600	1.46494
$58$	−1.22202	−0.160459
$59$	6.79534	0.884677	0.442339	−	0.896848i	$-0.354149\pi$
$59$	6.79534	0.884677	0.442339	+	0.896848i	$0.354149\pi$
$60$	1.05960	0.136794
$61$	−6.00294	−0.768598	−0.384299	−	0.923209i	$-0.625557\pi$
$61$	−6.00294	−0.768598	−0.384299	+	0.923209i	$0.625557\pi$
$62$	−10.1903	−1.29417
$63$	5.87263	0.739881
$64$	8.86975	1.10872
$65$	3.59594	0.446022
$66$	−9.73289	−1.19804
$67$	7.63281	0.932496	0.466248	−	0.884654i	$-0.345605\pi$
$67$	7.63281	0.932496	0.466248	+	0.884654i	$0.345605\pi$
$68$	3.14945	0.381927
$69$	8.28541	0.997446
$70$	−1.63721	−0.195684
$71$	6.99398	0.830032	0.415016	−	0.909814i	$-0.363776\pi$
$71$	6.99398	0.830032	0.415016	+	0.909814i	$0.363776\pi$
$72$	−10.9509	−1.29058
$73$	4.10515	0.480471	0.240236	−	0.970715i	$-0.422775\pi$
$73$	4.10515	0.480471	0.240236	+	0.970715i	$0.422775\pi$
$74$	−3.31154	−0.384958
$75$	−11.1152	−1.28348
$76$	−2.18541	−0.250684
$77$	−5.10239	−0.581471
$78$	−13.8156	−1.56431
$79$	16.0432	1.80500	0.902501	−	0.430688i	$-0.141729\pi$
$79$	16.0432	1.80500	0.902501	+	0.430688i	$0.141729\pi$
$80$	2.22651	0.248931
$81$	−6.94440	−0.771600
$82$	10.3326	1.14104
$83$	−13.7960	−1.51430	−0.757151	−	0.653240i	$-0.773409\pi$
$83$	−13.7960	−1.51430	−0.757151	+	0.653240i	$0.773409\pi$
$84$	−2.13417	−0.232857
$85$	5.06967	0.549882
$86$	−3.41835	−0.368610
$87$	−2.56418	−0.274908
$88$	9.51462	1.01426
$89$	−12.4847	−1.32337	−0.661686	−	0.749781i	$-0.730159\pi$
$89$	−12.4847	−1.32337	−0.661686	+	0.749781i	$0.730159\pi$
$90$	−3.56306	−0.375579
$91$	−7.24270	−0.759241
$92$	−1.63716	−0.170686
$93$	−21.3825	−2.21726
$94$	−9.21724	−0.950686
$95$	−3.51785	−0.360924
$96$	7.15494	0.730248
$97$	−3.07842	−0.312567	−0.156283	−	0.987712i	$-0.549951\pi$
$97$	−3.07842	−0.312567	−0.156283	+	0.987712i	$0.549951\pi$
$98$	−5.25658	−0.530994
$99$	−11.1043	−1.11603
$100$	2.19632	0.219632
$101$	−1.65730	−0.164907	−0.0824536	−	0.996595i	$-0.526276\pi$
$101$	−1.65730	−0.164907	−0.0824536	+	0.996595i	$0.526276\pi$
$102$	−19.4776	−1.92857
$103$	7.87764	0.776207	0.388104	−	0.921616i	$-0.373130\pi$
$103$	7.87764	0.776207	0.388104	+	0.921616i	$0.373130\pi$
$104$	13.5057	1.32435
$105$	−3.43538	−0.335258
$106$	−4.20138	−0.408074
$107$	−12.8042	−1.23783	−0.618913	−	0.785459i	$-0.712427\pi$
$107$	−12.8042	−1.23783	−0.618913	+	0.785459i	$0.712427\pi$
$108$	−0.747026	−0.0718826
$109$	6.22592	0.596335	0.298168	−	0.954514i	$-0.403625\pi$
$109$	6.22592	0.596335	0.298168	+	0.954514i	$0.403625\pi$
$110$	3.09574	0.295167
$111$	−6.94863	−0.659535
$112$	−4.48447	−0.423743
$113$	12.4307	1.16939	0.584693	−	0.811255i	$-0.301215\pi$
$113$	12.4307	1.16939	0.584693	+	0.811255i	$0.301215\pi$
$114$	13.5156	1.26585
$115$	−2.63533	−0.245746
$116$	0.506669	0.0470431
$117$	−15.7623	−1.45722
$118$	8.30403	0.764448
$119$	−10.2110	−0.936038
$120$	6.40608	0.584792
$121$	−1.35210	−0.122918
$122$	−7.33571	−0.664144
$123$	21.6810	1.95491
$124$	4.22508	0.379423
$125$	7.61334	0.680958
$126$	7.17646	0.639330
$127$	−21.8710	−1.94074	−0.970370	−	0.241622i	$-0.922320\pi$
$127$	−21.8710	−1.94074	−0.970370	+	0.241622i	$0.922320\pi$
$128$	5.25831	0.464773
$129$	−7.17275	−0.631525
$130$	4.39431	0.385407
$131$	12.1984	1.06578	0.532888	−	0.846186i	$-0.321107\pi$
$131$	12.1984	1.06578	0.532888	+	0.846186i	$0.321107\pi$
$132$	4.03542	0.351238
$133$	7.08542	0.614383
$134$	9.32744	0.805768
$135$	−1.20249	−0.103494
$136$	19.0408	1.63273
$137$	−6.34500	−0.542090	−0.271045	−	0.962567i	$-0.587369\pi$
$137$	−6.34500	−0.542090	−0.271045	+	0.962567i	$0.587369\pi$
$138$	10.1249	0.861891
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	0.678815	0.0573703
$141$	−19.3406	−1.62877
$142$	8.54677	0.717229
$143$	13.6949	1.14523
$144$	−9.75955	−0.813295
$145$	0.815585	0.0677307
$146$	5.01657	0.415174
$147$	−11.0299	−0.909733
$148$	1.37302	0.112861
$149$	−11.5190	−0.943671	−0.471835	−	0.881687i	$-0.656408\pi$
$149$	−11.5190	−0.943671	−0.471835	+	0.881687i	$0.656408\pi$
$150$	−13.5830	−1.10905
$151$	−21.3044	−1.73373	−0.866865	−	0.498543i	$-0.833869\pi$
$151$	−21.3044	−1.73373	−0.866865	+	0.498543i	$0.833869\pi$
$152$	−13.2124	−1.07167
$153$	−22.2221	−1.79655
$154$	−6.23522	−0.502448
$155$	6.80110	0.546278
$156$	5.72817	0.458620
$157$	21.0244	1.67793	0.838964	−	0.544187i	$-0.183162\pi$
$157$	21.0244	1.67793	0.838964	+	0.544187i	$0.183162\pi$
$158$	19.6051	1.55970
$159$	−8.81579	−0.699138
$160$	−2.27577	−0.179915
$161$	5.30791	0.418322
$162$	−8.48619	−0.666738
$163$	−3.71791	−0.291209	−0.145605	−	0.989343i	$-0.546513\pi$
$163$	−3.71791	−0.291209	−0.145605	+	0.989343i	$0.546513\pi$
$164$	−4.28406	−0.334529
$165$	6.49581	0.505698
$166$	−16.8589	−1.30851
$167$	5.61312	0.434356	0.217178	−	0.976132i	$-0.430315\pi$
$167$	5.61312	0.434356	0.217178	+	0.976132i	$0.430315\pi$
$168$	−12.9027	−0.995463
$169$	6.43957	0.495352
$170$	6.19523	0.475152
$171$	15.4200	1.17919
$172$	1.41730	0.108068
$173$	8.56793	0.651408	0.325704	−	0.945472i	$-0.394399\pi$
$173$	8.56793	0.651408	0.325704	+	0.945472i	$0.394399\pi$
$174$	−3.13347	−0.237548
$175$	−7.12079	−0.538281
$176$	8.47951	0.639167
$177$	17.4244	1.30970
$178$	−15.2565	−1.14352
$179$	4.07113	0.304290	0.152145	−	0.988358i	$-0.451382\pi$
$179$	4.07113	0.304290	0.152145	+	0.988358i	$0.451382\pi$
$180$	1.47730	0.110112
$181$	0.884577	0.0657501	0.0328750	−	0.999459i	$-0.489534\pi$
$181$	0.884577	0.0657501	0.0328750	+	0.999459i	$0.489534\pi$
$182$	−8.85072	−0.656059
$183$	−15.3926	−1.13785
$184$	−9.89786	−0.729680
$185$	2.21015	0.162493
$186$	−26.1298	−1.91593
$187$	19.3075	1.41190
$188$	3.82162	0.278720
$189$	2.42197	0.176172
$190$	−4.29888	−0.311874
$191$	23.0208	1.66573	0.832863	−	0.553479i	$-0.186700\pi$
$191$	23.0208	1.66573	0.832863	+	0.553479i	$0.186700\pi$
$192$	22.7436	1.64138
$193$	−9.29232	−0.668876	−0.334438	−	0.942418i	$-0.608546\pi$
$193$	−9.29232	−0.668876	−0.334438	+	0.942418i	$0.608546\pi$
$194$	−3.76189	−0.270088
$195$	9.22063	0.660303
$196$	2.17946	0.155676
$197$	18.4780	1.31650	0.658250	−	0.752799i	$-0.271297\pi$
$197$	18.4780	1.31650	0.658250	+	0.752799i	$0.271297\pi$
$198$	−13.5697	−0.964355
$199$	19.9938	1.41732	0.708661	−	0.705549i	$-0.249300\pi$
$199$	19.9938	1.41732	0.708661	+	0.705549i	$0.249300\pi$
$200$	13.2784	0.938926
$201$	19.5719	1.38049
$202$	−2.02525	−0.142496
$203$	−1.64270	−0.115295
$204$	8.07574	0.565415
$205$	−6.89605	−0.481641
$206$	9.62663	0.670719
$207$	11.5516	0.802891
$208$	12.0364	0.834576
$209$	−13.3975	−0.926726
$210$	−4.19809	−0.289696
$211$	−20.9197	−1.44017	−0.720087	−	0.693884i	$-0.755898\pi$
$211$	−20.9197	−1.44017	−0.720087	+	0.693884i	$0.755898\pi$
$212$	1.74196	0.119638
$213$	17.9338	1.22880
$214$	−15.6470	−1.06960
$215$	2.28143	0.155592
$216$	−4.51633	−0.307298
$217$	−13.6983	−0.929902
$218$	7.60820	0.515292
$219$	10.5263	0.711303
$220$	−1.28354	−0.0865365
$221$	27.4065	1.84356
$222$	−8.49136	−0.569903
$223$	−18.5473	−1.24202	−0.621011	−	0.783802i	$-0.713278\pi$
$223$	−18.5473	−1.24202	−0.621011	+	0.783802i	$0.713278\pi$
$224$	4.58369	0.306261
$225$	−15.4970	−1.03313
$226$	15.1906	1.01046
$227$	−12.7237	−0.844504	−0.422252	−	0.906479i	$-0.638760\pi$
$227$	−12.7237	−0.844504	−0.422252	+	0.906479i	$0.638760\pi$
$228$	−5.60377	−0.371119
$229$	−6.98258	−0.461422	−0.230711	−	0.973022i	$-0.574105\pi$
$229$	−6.98258	−0.461422	−0.230711	+	0.973022i	$0.574105\pi$
$230$	−3.22043	−0.212349
$231$	−13.0834	−0.860826
$232$	3.06320	0.201109
$233$	17.8626	1.17022	0.585110	−	0.810954i	$-0.301051\pi$
$233$	17.8626	1.17022	0.585110	+	0.810954i	$0.301051\pi$
$234$	−19.2618	−1.25918
$235$	6.15166	0.401290
$236$	−3.44299	−0.224119
$237$	41.1376	2.67217
$238$	−12.4780	−0.808829
$239$	−0.615173	−0.0397922	−0.0198961	−	0.999802i	$-0.506334\pi$
$239$	−0.615173	−0.0397922	−0.0198961	+	0.999802i	$0.506334\pi$
$240$	5.70915	0.368524
$241$	26.1322	1.68333	0.841663	−	0.540003i	$-0.181577\pi$
$241$	26.1322	1.68333	0.841663	+	0.540003i	$0.181577\pi$
$242$	−1.65229	−0.106213
$243$	−22.2298	−1.42604
$244$	3.04151	0.194713
$245$	3.50828	0.224136
$246$	26.4946	1.68923
$247$	−19.0174	−1.21005
$248$	25.5438	1.62203
$249$	−35.3752	−2.24181
$250$	9.30365	0.588415
$251$	7.78324	0.491274	0.245637	−	0.969362i	$-0.421003\pi$
$251$	7.78324	0.491274	0.245637	+	0.969362i	$0.421003\pi$
$252$	−2.97548	−0.187438
$253$	−10.0365	−0.630990
$254$	−26.7268	−1.67699
$255$	12.9995	0.814061
$256$	−11.3137	−0.707109
$257$	−2.20377	−0.137467	−0.0687336	−	0.997635i	$-0.521896\pi$
$257$	−2.20377	−0.137467	−0.0687336	+	0.997635i	$0.521896\pi$
$258$	−8.76524	−0.545700
$259$	−4.45152	−0.276604
$260$	−1.82195	−0.112993
$261$	−3.57499	−0.221287
$262$	14.9066	0.920934
$263$	−3.07684	−0.189726	−0.0948632	−	0.995490i	$-0.530241\pi$
$263$	−3.07684	−0.189726	−0.0948632	+	0.995490i	$0.530241\pi$
$264$	24.3971	1.50154
$265$	2.80403	0.172250
$266$	8.65851	0.530887
$267$	−32.0129	−1.95916
$268$	−3.86731	−0.236234
$269$	−16.9896	−1.03588	−0.517938	−	0.855418i	$-0.673300\pi$
$269$	−16.9896	−1.03588	−0.517938	+	0.855418i	$0.673300\pi$
$270$	−1.46946	−0.0894286
$271$	6.62762	0.402599	0.201299	−	0.979530i	$-0.435484\pi$
$271$	6.62762	0.402599	0.201299	+	0.979530i	$0.435484\pi$
$272$	16.9693	1.02892
$273$	−18.5716	−1.12400
$274$	−7.75371	−0.468418
$275$	13.4644	0.811935
$276$	−4.19796	−0.252688
$277$	−7.98922	−0.480026	−0.240013	−	0.970770i	$-0.577152\pi$
$277$	−7.98922	−0.480026	−0.240013	+	0.970770i	$0.577152\pi$
$278$	−1.22202	−0.0732918
$279$	−29.8116	−1.78477
$280$	4.10395	0.245258
$281$	6.10477	0.364180	0.182090	−	0.983282i	$-0.441714\pi$
$281$	6.10477	0.364180	0.182090	+	0.983282i	$0.441714\pi$
$282$	−23.6346	−1.40742
$283$	−23.5742	−1.40134	−0.700669	−	0.713487i	$-0.747115\pi$
$283$	−23.5742	−1.40134	−0.700669	+	0.713487i	$0.747115\pi$
$284$	−3.54363	−0.210276
$285$	−9.02039	−0.534322
$286$	16.7355	0.989589
$287$	13.8896	0.819874
$288$	9.97548	0.587811
$289$	21.6385	1.27285
$290$	0.996661	0.0585259
$291$	−7.89362	−0.462732
$292$	−2.07995	−0.121720
$293$	14.8839	0.869527	0.434764	−	0.900545i	$-0.356832\pi$
$293$	14.8839	0.869527	0.434764	+	0.900545i	$0.356832\pi$
$294$	−13.4788	−0.786099
$295$	−5.54218	−0.322678
$296$	8.30093	0.482482
$297$	−4.57960	−0.265735
$298$	−14.0764	−0.815424
$299$	−14.2466	−0.823899
$300$	5.63175	0.325149
$301$	−4.59510	−0.264857
$302$	−26.0344	−1.49811
$303$	−4.24960	−0.244133
$304$	−11.7750	−0.675345
$305$	4.89591	0.280339
$306$	−27.1558	−1.55240
$307$	4.18406	0.238797	0.119398	−	0.992846i	$-0.461903\pi$
$307$	4.18406	0.238797	0.119398	+	0.992846i	$0.461903\pi$
$308$	2.58522	0.147307
$309$	20.1997	1.14912
$310$	8.31108	0.472038
$311$	−31.7838	−1.80229	−0.901146	−	0.433516i	$-0.857273\pi$
$311$	−31.7838	−1.80229	−0.901146	+	0.433516i	$0.857273\pi$
$312$	34.6311	1.96060
$313$	−32.6820	−1.84730	−0.923648	−	0.383241i	$-0.874808\pi$
$313$	−32.6820	−1.84730	−0.923648	+	0.383241i	$0.874808\pi$
$314$	25.6922	1.44989
$315$	−4.78963	−0.269865
$316$	−8.12860	−0.457270
$317$	−2.26046	−0.126960	−0.0634801	−	0.997983i	$-0.520220\pi$
$317$	−2.26046	−0.126960	−0.0634801	+	0.997983i	$0.520220\pi$
$318$	−10.7731	−0.604123
$319$	3.10611	0.173909
$320$	−7.23404	−0.404395
$321$	−32.8322	−1.83251
$322$	6.48637	0.361471
$323$	−26.8113	−1.49182
$324$	3.51852	0.195473
$325$	19.1124	1.06016
$326$	−4.54336	−0.251634
$327$	15.9644	0.882831
$328$	−25.9004	−1.43011
$329$	−12.3902	−0.683096
$330$	7.93801	0.436973
$331$	15.7568	0.866071	0.433035	−	0.901377i	$-0.357443\pi$
$331$	15.7568	0.866071	0.433035	+	0.901377i	$0.357443\pi$
$332$	6.98999	0.383625
$333$	−9.68784	−0.530890
$334$	6.85934	0.375326
$335$	−6.22521	−0.340119
$336$	−11.4990	−0.627320
$337$	−18.8953	−1.02929	−0.514646	−	0.857403i	$-0.672077\pi$
$337$	−18.8953	−1.02929	−0.514646	+	0.857403i	$0.672077\pi$
$338$	7.86928	0.428033
$339$	31.8746	1.73119
$340$	−2.56864	−0.139304
$341$	25.9016	1.40265
$342$	18.8435	1.01894
$343$	−18.5650	−1.00242
$344$	8.56866	0.461991
$345$	−6.75746	−0.363809
$346$	10.4702	0.562880
$347$	5.21808	0.280121	0.140061	−	0.990143i	$-0.455270\pi$
$347$	5.21808	0.280121	0.140061	+	0.990143i	$0.455270\pi$
$348$	1.29919	0.0696438
$349$	4.28258	0.229241	0.114621	−	0.993409i	$-0.463435\pi$
$349$	4.28258	0.229241	0.114621	+	0.993409i	$0.463435\pi$
$350$	−8.70174	−0.465128
$351$	−6.50061	−0.346977
$352$	−8.66712	−0.461959
$353$	12.8643	0.684696	0.342348	−	0.939573i	$-0.388778\pi$
$353$	12.8643	0.684696	0.342348	+	0.939573i	$0.388778\pi$
$354$	21.2930	1.13171
$355$	−5.70419	−0.302747
$356$	6.32560	0.335256
$357$	−26.1827	−1.38574
$358$	4.97500	0.262937
$359$	−19.1563	−1.01103	−0.505516	−	0.862817i	$-0.668698\pi$
$359$	−19.1563	−1.01103	−0.505516	+	0.862817i	$0.668698\pi$
$360$	8.93141	0.470726
$361$	−0.395571	−0.0208195
$362$	1.08097	0.0568145
$363$	−3.46703	−0.181972
$364$	3.66965	0.192342
$365$	−3.34810	−0.175248
$366$	−18.8101	−0.983217
$367$	11.9175	0.622090	0.311045	−	0.950395i	$-0.399321\pi$
$367$	11.9175	0.622090	0.311045	+	0.950395i	$0.399321\pi$
$368$	−8.82106	−0.459829
$369$	30.2278	1.57360
$370$	2.70084	0.140410
$371$	−5.64769	−0.293213
$372$	10.8338	0.561708
$373$	25.7166	1.33155	0.665776	−	0.746151i	$-0.268100\pi$
$373$	25.7166	1.33155	0.665776	+	0.746151i	$0.268100\pi$
$374$	23.5941	1.22002
$375$	19.5219	1.00811
$376$	23.1046	1.19153
$377$	4.40903	0.227077
$378$	2.95969	0.152230
$379$	−17.2057	−0.883799	−0.441899	−	0.897065i	$-0.645695\pi$
$379$	−17.2057	−0.883799	−0.441899	+	0.897065i	$0.645695\pi$
$380$	1.78239	0.0914346
$381$	−56.0812	−2.87313
$382$	28.1318	1.43935
$383$	25.9702	1.32702	0.663508	−	0.748170i	$-0.269067\pi$
$383$	25.9702	1.32702	0.663508	+	0.748170i	$0.269067\pi$
$384$	13.4832	0.688063
$385$	4.16143	0.212086
$386$	−11.3554	−0.577974
$387$	−10.0003	−0.508344
$388$	1.55974	0.0791839
$389$	−5.16129	−0.261688	−0.130844	−	0.991403i	$-0.541769\pi$
$389$	−5.16129	−0.261688	−0.130844	+	0.991403i	$0.541769\pi$
$390$	11.2678	0.570566
$391$	−20.0852	−1.01575
$392$	13.1765	0.665514
$393$	31.2787	1.57780
$394$	22.5804	1.13759
$395$	−13.0846	−0.658358
$396$	5.62622	0.282728
$397$	−32.0018	−1.60612	−0.803061	−	0.595896i	$-0.796797\pi$
$397$	−32.0018	−1.60612	−0.803061	+	0.595896i	$0.796797\pi$
$398$	24.4328	1.22471
$399$	18.1682	0.909550
$400$	11.8338	0.591692
$401$	−21.6125	−1.07928	−0.539639	−	0.841896i	$-0.681439\pi$
$401$	−21.6125	−1.07928	−0.539639	+	0.841896i	$0.681439\pi$
$402$	23.9172	1.19288
$403$	36.7666	1.83147
$404$	0.839702	0.0417767
$405$	5.66375	0.281434
$406$	−2.00741	−0.0996259
$407$	8.41720	0.417225
$408$	48.8239	2.41714
$409$	−14.2270	−0.703482	−0.351741	−	0.936097i	$-0.614410\pi$
$409$	−14.2270	−0.703482	−0.351741	+	0.936097i	$0.614410\pi$
$410$	−8.42711	−0.416185
$411$	−16.2697	−0.802524
$412$	−3.99136	−0.196640
$413$	11.1627	0.549279
$414$	14.1163	0.693776
$415$	11.2518	0.552328
$416$	−12.3027	−0.603191
$417$	−2.56418	−0.125568
$418$	−16.3720	−0.800782
$419$	22.2285	1.08593	0.542967	−	0.839754i	$-0.317301\pi$
$419$	22.2285	1.08593	0.542967	+	0.839754i	$0.317301\pi$
$420$	1.74060	0.0849326
$421$	11.9603	0.582907	0.291454	−	0.956585i	$-0.405861\pi$
$421$	11.9603	0.582907	0.291454	+	0.956585i	$0.405861\pi$
$422$	−25.5643	−1.24445
$423$	−26.9649	−1.31108
$424$	10.5315	0.511453
$425$	26.9452	1.30703
$426$	21.9154	1.06181
$427$	−9.86101	−0.477208
$428$	6.48749	0.313584
$429$	35.1162	1.69543
$430$	2.78795	0.134447
$431$	−24.2144	−1.16636	−0.583182	−	0.812341i	$-0.698193\pi$
$431$	−24.2144	−1.16636	−0.583182	+	0.812341i	$0.698193\pi$
$432$	−4.02499	−0.193653
$433$	18.4024	0.884361	0.442181	−	0.896926i	$-0.354205\pi$
$433$	18.4024	0.884361	0.442181	+	0.896926i	$0.354205\pi$
$434$	−16.7396	−0.803526
$435$	2.09130	0.100270
$436$	−3.15448	−0.151072
$437$	13.9372	0.666705
$438$	12.8634	0.614635
$439$	5.64319	0.269335	0.134667	−	0.990891i	$-0.457003\pi$
$439$	5.64319	0.269335	0.134667	+	0.990891i	$0.457003\pi$
$440$	−7.75998	−0.369943
$441$	−15.3780	−0.732287
$442$	33.4912	1.59302
$443$	31.5682	1.49985	0.749926	−	0.661522i	$-0.230089\pi$
$443$	31.5682	1.49985	0.749926	+	0.661522i	$0.230089\pi$
$444$	3.52066	0.167083
$445$	10.1823	0.482688
$446$	−22.6652	−1.07323
$447$	−29.5367	−1.39704
$448$	14.5703	0.688382
$449$	1.76256	0.0831803	0.0415902	−	0.999135i	$-0.486758\pi$
$449$	1.76256	0.0831803	0.0415902	+	0.999135i	$0.486758\pi$
$450$	−18.9376	−0.892726
$451$	−26.2632	−1.23669
$452$	−6.29828	−0.296246
$453$	−54.6283	−2.56666
$454$	−15.5486	−0.729734
$455$	5.90704	0.276926
$456$	−33.8790	−1.58653
$457$	23.9685	1.12120	0.560600	−	0.828087i	$-0.310571\pi$
$457$	23.9685	1.12120	0.560600	+	0.828087i	$0.310571\pi$
$458$	−8.53285	−0.398714
$459$	−9.16475	−0.427774
$460$	1.33524	0.0622561
$461$	−17.4182	−0.811248	−0.405624	−	0.914040i	$-0.632946\pi$
$461$	−17.4182	−0.811248	−0.405624	+	0.914040i	$0.632946\pi$
$462$	−15.9882	−0.743838
$463$	−5.02980	−0.233754	−0.116877	−	0.993146i	$-0.537288\pi$
$463$	−5.02980	−0.233754	−0.116877	+	0.993146i	$0.537288\pi$
$464$	2.72995	0.126735
$465$	17.4392	0.808724
$466$	21.8285	1.01119
$467$	30.5159	1.41211	0.706054	−	0.708158i	$-0.250474\pi$
$467$	30.5159	1.41211	0.706054	+	0.708158i	$0.250474\pi$
$468$	7.98626	0.369165
$469$	12.5384	0.578969
$470$	7.51745	0.346754
$471$	53.9102	2.48405
$472$	−20.8155	−0.958109
$473$	8.68869	0.399506
$474$	50.2709	2.30902
$475$	−18.6973	−0.857892
$476$	5.17359	0.237131
$477$	−12.2910	−0.562768
$478$	−0.751753	−0.0343844
$479$	−16.9569	−0.774780	−0.387390	−	0.921916i	$-0.626623\pi$
$479$	−16.9569	−0.774780	−0.387390	+	0.921916i	$0.626623\pi$
$480$	−5.83547	−0.266351
$481$	11.9480	0.544782
$482$	31.9341	1.45456
$483$	13.6104	0.619295
$484$	0.685069	0.0311395
$485$	2.51072	0.114006
$486$	−27.1653	−1.23224
$487$	29.7756	1.34926	0.674632	−	0.738155i	$-0.264303\pi$
$487$	29.7756	1.34926	0.674632	+	0.738155i	$0.264303\pi$
$488$	18.3882	0.832395
$489$	−9.53338	−0.431114
$490$	4.28719	0.193675
$491$	−19.2238	−0.867560	−0.433780	−	0.901019i	$-0.642821\pi$
$491$	−19.2238	−0.867560	−0.433780	+	0.901019i	$0.642821\pi$
$492$	−10.9851	−0.495246
$493$	6.21598	0.279954
$494$	−23.2396	−1.04560
$495$	9.05651	0.407060
$496$	22.7648	1.02217
$497$	11.4890	0.515351
$498$	−43.2292	−1.93715
$499$	−27.7575	−1.24260	−0.621298	−	0.783574i	$-0.713394\pi$
$499$	−27.7575	−1.24260	−0.621298	+	0.783574i	$0.713394\pi$
$500$	−3.85745	−0.172510
$501$	14.3930	0.643032
$502$	9.51127	0.424509
$503$	−13.3138	−0.593634	−0.296817	−	0.954934i	$-0.595925\pi$
$503$	−13.3138	−0.593634	−0.296817	+	0.954934i	$0.595925\pi$
$504$	−17.9890	−0.801295
$505$	1.35167	0.0601484
$506$	−12.2648	−0.545237
$507$	16.5122	0.733332
$508$	11.0814	0.491657
$509$	3.08356	0.136676	0.0683381	−	0.997662i	$-0.478230\pi$
$509$	3.08356	0.136676	0.0683381	+	0.997662i	$0.478230\pi$
$510$	15.8856	0.703428
$511$	6.74351	0.298315
$512$	−24.3422	−1.07579
$513$	6.35944	0.280776
$514$	−2.69304	−0.118785
$515$	−6.42489	−0.283114
$516$	3.63421	0.159987
$517$	23.4282	1.03037
$518$	−5.43985	−0.239013
$519$	21.9697	0.964362
$520$	−11.0151	−0.483043
$521$	−15.1165	−0.662266	−0.331133	−	0.943584i	$-0.607431\pi$
$521$	−15.1165	−0.662266	−0.331133	+	0.943584i	$0.607431\pi$
$522$	−4.36871	−0.191213
$523$	−29.0954	−1.27225	−0.636126	−	0.771586i	$-0.719464\pi$
$523$	−29.0954	−1.27225	−0.636126	+	0.771586i	$0.719464\pi$
$524$	−6.18053	−0.269998
$525$	−18.2590	−0.796886
$526$	−3.75996	−0.163942
$527$	51.8346	2.25795
$528$	21.7429	0.946240
$529$	−12.5592	−0.546053
$530$	3.42658	0.148841
$531$	24.2933	1.05424
$532$	−3.58996	−0.155645
$533$	−37.2799	−1.61477
$534$	−39.1203	−1.69290
$535$	10.4429	0.451486
$536$	−23.3808	−1.00990
$537$	10.4391	0.450480
$538$	−20.7617	−0.895099
$539$	13.3611	0.575502
$540$	0.609263	0.0262185
$541$	14.9490	0.642710	0.321355	−	0.946959i	$-0.395862\pi$
$541$	14.9490	0.642710	0.321355	+	0.946959i	$0.395862\pi$
$542$	8.09907	0.347885
$543$	2.26821	0.0973382
$544$	−17.3448	−0.743650
$545$	−5.07777	−0.217508
$546$	−22.6948	−0.971247
$547$	14.3411	0.613183	0.306592	−	0.951841i	$-0.400811\pi$
$547$	14.3411	0.613183	0.306592	+	0.951841i	$0.400811\pi$
$548$	3.21481	0.137330
$549$	−21.4605	−0.915911
$550$	16.4538	0.701591
$551$	−4.31329	−0.183752
$552$	−25.3799	−1.08024
$553$	26.3541	1.12069
$554$	−9.76298	−0.414789
$555$	5.66720	0.240559
$556$	0.506669	0.0214876
$557$	6.31143	0.267424	0.133712	−	0.991020i	$-0.457310\pi$
$557$	6.31143	0.267424	0.133712	+	0.991020i	$0.457310\pi$
$558$	−36.4303	−1.54222
$559$	12.3334	0.521645
$560$	3.65747	0.154556
$561$	49.5078	2.09022
$562$	7.46015	0.314687
$563$	34.2896	1.44513	0.722566	−	0.691302i	$-0.242962\pi$
$563$	34.2896	1.44513	0.722566	+	0.691302i	$0.242962\pi$
$564$	9.79930	0.412625
$565$	−10.1383	−0.426523
$566$	−28.8081	−1.21089
$567$	−11.4075	−0.479072
$568$	−21.4239	−0.898928
$569$	−11.3769	−0.476946	−0.238473	−	0.971149i	$-0.576647\pi$
$569$	−11.3769	−0.476946	−0.238473	+	0.971149i	$0.576647\pi$
$570$	−11.0231	−0.461706
$571$	−1.73907	−0.0727778	−0.0363889	−	0.999338i	$-0.511586\pi$
$571$	−1.73907	−0.0727778	−0.0363889	+	0.999338i	$0.511586\pi$
$572$	−6.93880	−0.290126
$573$	59.0293	2.46599
$574$	16.9733	0.708452
$575$	−14.0068	−0.584122
$576$	31.7093	1.32122
$577$	22.4614	0.935079	0.467540	−	0.883972i	$-0.345140\pi$
$577$	22.4614	0.935079	0.467540	+	0.883972i	$0.345140\pi$
$578$	26.4426	1.09987
$579$	−23.8271	−0.990222
$580$	−0.413232	−0.0171585
$581$	−22.6626	−0.940201
$582$	−9.64615	−0.399846
$583$	10.6790	0.442278
$584$	−12.5749	−0.520352
$585$	12.8555	0.531508
$586$	18.1884	0.751357
$587$	−6.73149	−0.277838	−0.138919	−	0.990304i	$-0.544363\pi$
$587$	−6.73149	−0.277838	−0.138919	+	0.990304i	$0.544363\pi$
$588$	5.58853	0.230467
$589$	−35.9682	−1.48204
$590$	−6.77265	−0.278825
$591$	47.3807	1.94898
$592$	7.39785	0.304050
$593$	30.8803	1.26810	0.634051	−	0.773291i	$-0.281391\pi$
$593$	30.8803	1.26810	0.634051	+	0.773291i	$0.281391\pi$
$594$	−5.59636	−0.229621
$595$	8.32792	0.341411
$596$	5.83631	0.239065
$597$	51.2676	2.09824
$598$	−17.4096	−0.711930
$599$	−5.93097	−0.242333	−0.121166	−	0.992632i	$-0.538663\pi$
$599$	−5.93097	−0.242333	−0.121166	+	0.992632i	$0.538663\pi$
$600$	34.0482	1.39001
$601$	24.3532	0.993388	0.496694	−	0.867926i	$-0.334547\pi$
$601$	24.3532	0.993388	0.496694	+	0.867926i	$0.334547\pi$
$602$	−5.61530	−0.228863
$603$	27.2872	1.11122
$604$	10.7943	0.439214
$605$	1.10275	0.0448333
$606$	−5.19309	−0.210955
$607$	26.7339	1.08510	0.542548	−	0.840025i	$-0.317460\pi$
$607$	26.7339	1.08510	0.542548	+	0.840025i	$0.317460\pi$
$608$	12.0356	0.488107
$609$	−4.21216	−0.170685
$610$	5.98290	0.242241
$611$	33.2557	1.34538
$612$	11.2593	0.455129
$613$	−12.6725	−0.511839	−0.255919	−	0.966698i	$-0.582378\pi$
$613$	−12.6725	−0.511839	−0.255919	+	0.966698i	$0.582378\pi$
$614$	5.11300	0.206344
$615$	−17.6827	−0.713035
$616$	15.6296	0.629735
$617$	34.4780	1.38803	0.694016	−	0.719959i	$-0.255839\pi$
$617$	34.4780	1.38803	0.694016	+	0.719959i	$0.255839\pi$
$618$	24.6844	0.992951
$619$	−27.5857	−1.10876	−0.554382	−	0.832263i	$-0.687045\pi$
$619$	−27.5857	−1.10876	−0.554382	+	0.832263i	$0.687045\pi$
$620$	−3.44591	−0.138391
$621$	4.76406	0.191175
$622$	−38.8404	−1.55736
$623$	−20.5085	−0.821656
$624$	30.8635	1.23553
$625$	15.4648	0.618591
$626$	−39.9380	−1.59625
$627$	−34.3536	−1.37195
$628$	−10.6524	−0.425077
$629$	16.8446	0.671639
$630$	−5.85302	−0.233190
$631$	−25.7556	−1.02532	−0.512658	−	0.858593i	$-0.671339\pi$
$631$	−25.7556	−1.02532	−0.512658	+	0.858593i	$0.671339\pi$
$632$	−49.1435	−1.95482
$633$	−53.6419	−2.13207
$634$	−2.76233	−0.109706
$635$	17.8377	0.707867
$636$	4.46669	0.177116
$637$	18.9657	0.751448
$638$	3.79572	0.150274
$639$	25.0034	0.989120
$640$	−4.28860	−0.169522
$641$	−27.4121	−1.08271	−0.541355	−	0.840794i	$-0.682089\pi$
$641$	−27.4121	−1.08271	−0.541355	+	0.840794i	$0.682089\pi$
$642$	−40.1215	−1.58347
$643$	−1.80944	−0.0713572	−0.0356786	−	0.999363i	$-0.511359\pi$
$643$	−1.80944	−0.0713572	−0.0356786	+	0.999363i	$0.511359\pi$
$644$	−2.68936	−0.105975
$645$	5.84999	0.230343
$646$	−32.7639	−1.28908
$647$	−13.2906	−0.522509	−0.261254	−	0.965270i	$-0.584136\pi$
$647$	−13.2906	−0.522509	−0.261254	+	0.965270i	$0.584136\pi$
$648$	21.2721	0.835646
$649$	−21.1070	−0.828524
$650$	23.3557	0.916085
$651$	−35.1249	−1.37665
$652$	1.88375	0.0737735
$653$	−22.1778	−0.867884	−0.433942	−	0.900941i	$-0.642878\pi$
$653$	−22.1778	−0.867884	−0.433942	+	0.900941i	$0.642878\pi$
$654$	19.5087	0.762852
$655$	−9.94880	−0.388732
$656$	−23.0826	−0.901226
$657$	14.6759	0.572561
$658$	−15.1411	−0.590262
$659$	−41.3265	−1.60985	−0.804926	−	0.593376i	$-0.797795\pi$
$659$	−41.3265	−1.60985	−0.804926	+	0.593376i	$0.797795\pi$
$660$	−3.29123	−0.128111
$661$	−46.2465	−1.79878	−0.899391	−	0.437146i	$-0.855989\pi$
$661$	−46.2465	−1.79878	−0.899391	+	0.437146i	$0.855989\pi$
$662$	19.2551	0.748370
$663$	70.2750	2.72926
$664$	42.2597	1.64000
$665$	−5.77876	−0.224091
$666$	−11.8387	−0.458741
$667$	−3.23122	−0.125113
$668$	−2.84399	−0.110037
$669$	−47.5586	−1.83872
$670$	−7.60732	−0.293896
$671$	18.6458	0.719813
$672$	11.7534	0.453397
$673$	−31.8702	−1.22850	−0.614252	−	0.789110i	$-0.710542\pi$
$673$	−31.8702	−1.22850	−0.614252	+	0.789110i	$0.710542\pi$
$674$	−23.0904	−0.889408
$675$	−6.39119	−0.245997
$676$	−3.26274	−0.125490
$677$	35.2624	1.35524	0.677622	−	0.735410i	$-0.263011\pi$
$677$	35.2624	1.35524	0.677622	+	0.735410i	$0.263011\pi$
$678$	38.9514	1.49592
$679$	−5.05691	−0.194067
$680$	−15.5294	−0.595525
$681$	−32.6259	−1.25023
$682$	31.6522	1.21203
$683$	−36.3887	−1.39238	−0.696188	−	0.717860i	$-0.745122\pi$
$683$	−36.3887	−1.39238	−0.696188	+	0.717860i	$0.745122\pi$
$684$	−7.81283	−0.298731
$685$	5.17489	0.197722
$686$	−22.6868	−0.866186
$687$	−17.9046	−0.683102
$688$	7.63646	0.291137
$689$	15.1585	0.577494
$690$	−8.25775	−0.314367
$691$	−34.4604	−1.31094	−0.655468	−	0.755223i	$-0.727529\pi$
$691$	−34.4604	−1.31094	−0.655468	+	0.755223i	$0.727529\pi$
$692$	−4.34111	−0.165024
$693$	−18.2410	−0.692919
$694$	6.37660	0.242052
$695$	0.815585	0.0309369
$696$	7.85458	0.297727
$697$	−52.5583	−1.99079
$698$	5.23339	0.198087
$699$	45.8029	1.73243
$700$	3.60789	0.136365
$701$	29.3851	1.10986	0.554930	−	0.831897i	$-0.312745\pi$
$701$	29.3851	1.10986	0.554930	+	0.831897i	$0.312745\pi$
$702$	−7.94387	−0.299822
$703$	−11.6885	−0.440841
$704$	−27.5504	−1.03834
$705$	15.7739	0.594081
$706$	15.7204	0.591644
$707$	−2.72243	−0.102388
$708$	−8.82843	−0.331793
$709$	−44.8301	−1.68363	−0.841814	−	0.539767i	$-0.818512\pi$
$709$	−44.8301	−1.68363	−0.841814	+	0.539767i	$0.818512\pi$
$710$	−6.97062	−0.261603
$711$	57.3544	2.15096
$712$	38.2430	1.43322
$713$	−26.9449	−1.00909
$714$	−31.9958	−1.19741
$715$	−11.1694	−0.417711
$716$	−2.06272	−0.0770873
$717$	−1.57741	−0.0589095
$718$	−23.4094	−0.873630
$719$	−0.0921431	−0.00343636	−0.00171818	−	0.999999i	$-0.500547\pi$
$719$	−0.0921431	−0.00343636	−0.00171818	+	0.999999i	$0.500547\pi$
$720$	7.95974	0.296642
$721$	12.9406	0.481932
$722$	−0.483395	−0.0179901
$723$	67.0077	2.49204
$724$	−0.448188	−0.0166568
$725$	4.33482	0.160991
$726$	−4.23677	−0.157241
$727$	−44.3808	−1.64599	−0.822997	−	0.568046i	$-0.807700\pi$
$727$	−44.3808	−1.64599	−0.822997	+	0.568046i	$0.807700\pi$
$728$	22.1858	0.822261
$729$	−36.1679	−1.33955
$730$	−4.09144	−0.151431
$731$	17.3879	0.643116
$732$	7.79896	0.288258
$733$	20.1530	0.744368	0.372184	−	0.928159i	$-0.378609\pi$
$733$	20.1530	0.744368	0.372184	+	0.928159i	$0.378609\pi$
$734$	14.5634	0.537546
$735$	8.99585	0.331817
$736$	9.01622	0.332342
$737$	−23.7083	−0.873307
$738$	36.9389	1.35974
$739$	−5.64234	−0.207557	−0.103778	−	0.994600i	$-0.533093\pi$
$739$	−5.64234	−0.207557	−0.103778	+	0.994600i	$0.533093\pi$
$740$	−1.11981	−0.0411652
$741$	−48.7640	−1.79139
$742$	−6.90158	−0.253365
$743$	24.4355	0.896453	0.448226	−	0.893920i	$-0.352056\pi$
$743$	24.4355	0.896453	0.448226	+	0.893920i	$0.352056\pi$
$744$	65.4987	2.40130
$745$	9.39470	0.344195
$746$	31.4261	1.15059
$747$	−49.3204	−1.80454
$748$	−9.78252	−0.357684
$749$	−21.0334	−0.768543
$750$	23.8562	0.871105
$751$	27.9110	1.01849	0.509243	−	0.860623i	$-0.329925\pi$
$751$	27.9110	1.01849	0.509243	+	0.860623i	$0.329925\pi$
$752$	20.5910	0.750876
$753$	19.9576	0.727295
$754$	5.38792	0.196217
$755$	17.3756	0.632362
$756$	−1.22714	−0.0446305
$757$	−7.88960	−0.286753	−0.143376	−	0.989668i	$-0.545796\pi$
$757$	−7.88960	−0.286753	−0.143376	+	0.989668i	$0.545796\pi$
$758$	−21.0257	−0.763689
$759$	−25.7354	−0.934135
$760$	10.7759	0.390882
$761$	10.6767	0.387029	0.193515	−	0.981097i	$-0.438011\pi$
$761$	10.6767	0.387029	0.193515	+	0.981097i	$0.438011\pi$
$762$	−68.5323	−2.48266
$763$	10.2273	0.370253
$764$	−11.6639	−0.421986
$765$	18.1240	0.655275
$766$	31.7361	1.14667
$767$	−29.9609	−1.08182
$768$	−29.0104	−1.04682
$769$	24.1775	0.871864	0.435932	−	0.899980i	$-0.356419\pi$
$769$	24.1775	0.871864	0.435932	+	0.899980i	$0.356419\pi$
$770$	5.08535	0.183263
$771$	−5.65084	−0.203510
$772$	4.70813	0.169449
$773$	−10.1485	−0.365017	−0.182509	−	0.983204i	$-0.558422\pi$
$773$	−10.1485	−0.365017	−0.182509	+	0.983204i	$0.558422\pi$
$774$	−12.2206	−0.439259
$775$	36.1477	1.29846
$776$	9.42982	0.338511
$777$	−11.4145	−0.409492
$778$	−6.30720	−0.226124
$779$	36.4703	1.30668
$780$	−4.67181	−0.167278
$781$	−21.7240	−0.777347
$782$	−24.5445	−0.877710
$783$	−1.47439	−0.0526902
$784$	11.7430	0.419393
$785$	−17.1472	−0.612009
$786$	38.2232	1.36338
$787$	−26.7996	−0.955302	−0.477651	−	0.878550i	$-0.658512\pi$
$787$	−26.7996	−0.955302	−0.477651	+	0.878550i	$0.658512\pi$
$788$	−9.36222	−0.333515
$789$	−7.88957	−0.280876
$790$	−15.9896	−0.568886
$791$	20.4199	0.726049
$792$	34.0147	1.20866
$793$	26.4672	0.939877
$794$	−39.1068	−1.38785
$795$	7.19003	0.255004
$796$	−10.1302	−0.359057
$797$	5.34781	0.189429	0.0947145	−	0.995504i	$-0.469806\pi$
$797$	5.34781	0.189429	0.0947145	+	0.995504i	$0.469806\pi$
$798$	22.2019	0.785940
$799$	46.8849	1.65867
$800$	−12.0957	−0.427646
$801$	−44.6326	−1.57702
$802$	−26.4109	−0.932602
$803$	−12.7510	−0.449974
$804$	−9.91646	−0.349727
$805$	−4.32905	−0.152579
$806$	44.9295	1.58257
$807$	−43.5644	−1.53354
$808$	5.07663	0.178595
$809$	−31.1029	−1.09352	−0.546760	−	0.837289i	$-0.684139\pi$
$809$	−31.1029	−1.09352	−0.546760	+	0.837289i	$0.684139\pi$
$810$	6.92121	0.243187
$811$	−56.1312	−1.97103	−0.985516	−	0.169582i	$-0.945758\pi$
$811$	−56.1312	−1.97103	−0.985516	+	0.169582i	$0.945758\pi$
$812$	0.832304	0.0292081
$813$	16.9944	0.596018
$814$	10.2860	0.360524
$815$	3.03228	0.106216
$816$	43.5123	1.52323
$817$	−12.0655	−0.422119
$818$	−17.3857	−0.607877
$819$	−25.8926	−0.904761
$820$	3.49402	0.122016
$821$	39.9036	1.39265	0.696323	−	0.717728i	$-0.254818\pi$
$821$	39.9036	1.39265	0.696323	+	0.717728i	$0.254818\pi$
$822$	−19.8819	−0.693459
$823$	12.8573	0.448177	0.224089	−	0.974569i	$-0.428059\pi$
$823$	12.8573	0.448177	0.224089	+	0.974569i	$0.428059\pi$
$824$	−24.1308	−0.840635
$825$	34.5251	1.20201
$826$	13.6410	0.474631
$827$	15.5391	0.540347	0.270174	−	0.962812i	$-0.412919\pi$
$827$	15.5391	0.540347	0.270174	+	0.962812i	$0.412919\pi$
$828$	−5.85284	−0.203400
$829$	24.6403	0.855794	0.427897	−	0.903827i	$-0.359255\pi$
$829$	24.6403	0.855794	0.427897	+	0.903827i	$0.359255\pi$
$830$	13.7499	0.477266
$831$	−20.4858	−0.710643
$832$	−39.1070	−1.35579
$833$	26.7384	0.926430
$834$	−3.13347	−0.108503
$835$	−4.57798	−0.158427
$836$	6.78812	0.234772
$837$	−12.2948	−0.424970
$838$	27.1636	0.938353
$839$	35.3999	1.22214	0.611070	−	0.791577i	$-0.290739\pi$
$839$	35.3999	1.22214	0.611070	+	0.791577i	$0.290739\pi$
$840$	10.5232	0.363086
$841$	1.00000	0.0344828
$842$	14.6157	0.503689
$843$	15.6537	0.539142
$844$	10.5994	0.364846
$845$	−5.25202	−0.180675
$846$	−32.9516	−1.13290
$847$	−2.22109	−0.0763176
$848$	9.38572	0.322307
$849$	−60.4482	−2.07458
$850$	32.9275	1.12940
$851$	−8.75624	−0.300160
$852$	−9.08650	−0.311298
$853$	20.5181	0.702526	0.351263	−	0.936277i	$-0.385752\pi$
$853$	20.5181	0.702526	0.351263	+	0.936277i	$0.385752\pi$
$854$	−12.0503	−0.412354
$855$	−12.5763	−0.430100
$856$	39.2217	1.34057
$857$	29.7782	1.01720	0.508602	−	0.861002i	$-0.330162\pi$
$857$	29.7782	1.01720	0.508602	+	0.861002i	$0.330162\pi$
$858$	42.9127	1.46501
$859$	12.6672	0.432199	0.216099	−	0.976371i	$-0.430667\pi$
$859$	12.6672	0.432199	0.216099	+	0.976371i	$0.430667\pi$
$860$	−1.15593	−0.0394169
$861$	35.6153	1.21376
$862$	−29.5904	−1.00785
$863$	44.2537	1.50641	0.753207	−	0.657784i	$-0.228506\pi$
$863$	44.2537	1.50641	0.753207	+	0.657784i	$0.228506\pi$
$864$	4.11405	0.139963
$865$	−6.98788	−0.237595
$866$	22.4880	0.764175
$867$	55.4848	1.88436
$868$	6.94052	0.235576
$869$	−49.8319	−1.69043
$870$	2.55561	0.0866434
$871$	−33.6533	−1.14030
$872$	−19.0712	−0.645833
$873$	−11.0053	−0.372474
$874$	17.0315	0.576099
$875$	12.5064	0.422794
$876$	−5.33336	−0.180198
$877$	10.0759	0.340240	0.170120	−	0.985423i	$-0.445584\pi$
$877$	10.0759	0.340240	0.170120	+	0.985423i	$0.445584\pi$
$878$	6.89609	0.232732
$879$	38.1649	1.28727
$880$	−6.91576	−0.233130
$881$	3.10037	0.104454	0.0522271	−	0.998635i	$-0.483368\pi$
$881$	3.10037	0.104454	0.0522271	+	0.998635i	$0.483368\pi$
$882$	−18.7922	−0.632767
$883$	−36.6780	−1.23431	−0.617156	−	0.786841i	$-0.711715\pi$
$883$	−36.6780	−1.23431	−0.617156	+	0.786841i	$0.711715\pi$
$884$	−13.8860	−0.467038
$885$	−14.2111	−0.477701
$886$	38.5770	1.29602
$887$	27.7503	0.931764	0.465882	−	0.884847i	$-0.345737\pi$
$887$	27.7503	0.931764	0.465882	+	0.884847i	$0.345737\pi$
$888$	21.2850	0.714279
$889$	−35.9275	−1.20497
$890$	12.4430	0.417090
$891$	21.5700	0.722624
$892$	9.39737	0.314647
$893$	−32.5335	−1.08869
$894$	−36.0944	−1.20718
$895$	−3.32035	−0.110987
$896$	8.63781	0.288569
$897$	−36.5307	−1.21972
$898$	2.15388	0.0718759
$899$	8.33892	0.278119
$900$	7.85183	0.261728
$901$	21.3709	0.711969
$902$	−32.0941	−1.06862
$903$	−11.7826	−0.392102
$904$	−38.0778	−1.26645
$905$	−0.721448	−0.0239817
$906$	−66.7568	−2.21785
$907$	33.5271	1.11325	0.556625	−	0.830764i	$-0.312096\pi$
$907$	33.5271	1.11325	0.556625	+	0.830764i	$0.312096\pi$
$908$	6.44672	0.213942
$909$	−5.92483	−0.196514
$910$	7.21852	0.239291
$911$	−32.3820	−1.07286	−0.536432	−	0.843944i	$-0.680228\pi$
$911$	−32.3820	−1.07286	−0.536432	+	0.843944i	$0.680228\pi$
$912$	−30.1933	−0.999799
$913$	42.8517	1.41818
$914$	29.2900	0.968827
$915$	12.5540	0.415022
$916$	3.53786	0.116894
$917$	20.0382	0.661719
$918$	−11.1995	−0.369639
$919$	1.21348	0.0400289	0.0200145	−	0.999800i	$-0.493629\pi$
$919$	1.21348	0.0400289	0.0200145	+	0.999800i	$0.493629\pi$
$920$	8.07255	0.266144
$921$	10.7287	0.353521
$922$	−21.2854	−0.700998
$923$	−30.8367	−1.01500
$924$	6.62897	0.218077
$925$	11.7469	0.386235
$926$	−6.14651	−0.201987
$927$	28.1625	0.924979
$928$	−2.79035	−0.0915977
$929$	39.2195	1.28675	0.643374	−	0.765552i	$-0.277534\pi$
$929$	39.2195	1.28675	0.643374	+	0.765552i	$0.277534\pi$
$930$	21.3111	0.698817
$931$	−18.5538	−0.608077
$932$	−9.05045	−0.296457
$933$	−81.4991	−2.66816
$934$	37.2910	1.22020
$935$	−15.7469	−0.514979
$936$	48.2829	1.57818
$937$	−11.3824	−0.371847	−0.185923	−	0.982564i	$-0.559528\pi$
$937$	−11.3824	−0.371847	−0.185923	+	0.982564i	$0.559528\pi$
$938$	15.3221	0.500286
$939$	−83.8024	−2.73479
$940$	−3.11686	−0.101661
$941$	51.0127	1.66297	0.831484	−	0.555549i	$-0.187492\pi$
$941$	51.0127	1.66297	0.831484	+	0.555549i	$0.187492\pi$
$942$	65.8793	2.14646
$943$	27.3211	0.889696
$944$	−18.5509	−0.603781
$945$	−1.97532	−0.0642572
$946$	10.6177	0.345213
$947$	4.19804	0.136418	0.0682089	−	0.997671i	$-0.478272\pi$
$947$	4.19804	0.136418	0.0682089	+	0.997671i	$0.478272\pi$
$948$	−20.8432	−0.676954
$949$	−18.0997	−0.587542
$950$	−22.8485	−0.741303
$951$	−5.79622	−0.187955
$952$	31.2782	1.01373
$953$	−51.3512	−1.66343	−0.831714	−	0.555205i	$-0.812640\pi$
$953$	−51.3512	−1.66343	−0.831714	+	0.555205i	$0.812640\pi$
$954$	−15.0199	−0.486287
$955$	−18.7754	−0.607558
$956$	0.311689	0.0100808
$957$	7.96460	0.257459
$958$	−20.7217	−0.669486
$959$	−10.4229	−0.336573
$960$	−18.5493	−0.598678
$961$	38.5376	1.24315
$962$	14.6007	0.470745
$963$	−45.7749	−1.47507
$964$	−13.2404	−0.426445
$965$	7.57868	0.243966
$966$	16.6322	0.535132
$967$	−40.7130	−1.30924	−0.654621	−	0.755958i	$-0.727172\pi$
$967$	−40.7130	−1.30924	−0.654621	+	0.755958i	$0.727172\pi$
$968$	4.14176	0.133121
$969$	−68.7489	−2.20853
$970$	3.06814	0.0985122
$971$	−19.6193	−0.629612	−0.314806	−	0.949156i	$-0.601939\pi$
$971$	−19.6193	−0.629612	−0.314806	+	0.949156i	$0.601939\pi$
$972$	11.2632	0.361266
$973$	−1.64270	−0.0526624
$974$	36.3864	1.16590
$975$	49.0075	1.56949
$976$	16.3877	0.524558
$977$	−27.3821	−0.876030	−0.438015	−	0.898968i	$-0.644318\pi$
$977$	−27.3821	−0.876030	−0.438015	+	0.898968i	$0.644318\pi$
$978$	−11.6500	−0.372525
$979$	38.7787	1.23937
$980$	−1.77754	−0.0567814
$981$	22.2576	0.710632
$982$	−23.4919	−0.749657
$983$	−9.05269	−0.288736	−0.144368	−	0.989524i	$-0.546115\pi$
$983$	−9.05269	−0.288736	−0.144368	+	0.989524i	$0.546115\pi$
$984$	−66.4131	−2.11717
$985$	−15.0704	−0.480182
$986$	7.59605	0.241908
$987$	−31.7708	−1.01127
$988$	9.63554	0.306547
$989$	−9.03867	−0.287413
$990$	11.0672	0.351740
$991$	−45.3127	−1.43940	−0.719702	−	0.694283i	$-0.755722\pi$
$991$	−45.3127	−1.43940	−0.719702	+	0.694283i	$0.755722\pi$
$992$	−23.2685	−0.738775
$993$	40.4031	1.28215
$994$	14.0398	0.445314
$995$	−16.3066	−0.516955
$996$	17.9236	0.567929
$997$	18.0606	0.571985	0.285993	−	0.958232i	$-0.407677\pi$
$997$	18.0606	0.571985	0.285993	+	0.958232i	$0.407677\pi$
$998$	−33.9202	−1.07373
$999$	−3.99542	−0.126409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.44	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.44	✓	61	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q+1.22202 q^{2} +2.56418 q^{3} -0.506669 q^{4} -0.815585 q^{5} +3.13347 q^{6} +1.64270 q^{7} -3.06320 q^{8} +3.57499 q^{9} +O(q^{10})\)	\(q+1.22202 q^{2} +2.56418 q^{3} -0.506669 q^{4} -0.815585 q^{5} +3.13347 q^{6} +1.64270 q^{7} -3.06320 q^{8} +3.57499 q^{9} -0.996661 q^{10} -3.10611 q^{11} -1.29919 q^{12} -4.40903 q^{13} +2.00741 q^{14} -2.09130 q^{15} -2.72995 q^{16} -6.21598 q^{17} +4.36871 q^{18} +4.31329 q^{19} +0.413232 q^{20} +4.21216 q^{21} -3.79572 q^{22} +3.23122 q^{23} -7.85458 q^{24} -4.33482 q^{25} -5.38792 q^{26} +1.47439 q^{27} -0.832304 q^{28} -1.00000 q^{29} -2.55561 q^{30} -8.33892 q^{31} +2.79035 q^{32} -7.96460 q^{33} -7.59605 q^{34} -1.33976 q^{35} -1.81134 q^{36} -2.70989 q^{37} +5.27092 q^{38} -11.3055 q^{39} +2.49830 q^{40} +8.45534 q^{41} +5.14734 q^{42} -2.79729 q^{43} +1.57377 q^{44} -2.91571 q^{45} +3.94861 q^{46} -7.54263 q^{47} -7.00006 q^{48} -4.30155 q^{49} -5.29723 q^{50} -15.9389 q^{51} +2.23392 q^{52} -3.43806 q^{53} +1.80173 q^{54} +2.53330 q^{55} -5.03190 q^{56} +11.0600 q^{57} -1.22202 q^{58} +6.79534 q^{59} +1.05960 q^{60} -6.00294 q^{61} -10.1903 q^{62} +5.87263 q^{63} +8.86975 q^{64} +3.59594 q^{65} -9.73289 q^{66} +7.63281 q^{67} +3.14945 q^{68} +8.28541 q^{69} -1.63721 q^{70} +6.99398 q^{71} -10.9509 q^{72} +4.10515 q^{73} -3.31154 q^{74} -11.1152 q^{75} -2.18541 q^{76} -5.10239 q^{77} -13.8156 q^{78} +16.0432 q^{79} +2.22651 q^{80} -6.94440 q^{81} +10.3326 q^{82} -13.7960 q^{83} -2.13417 q^{84} +5.06967 q^{85} -3.41835 q^{86} -2.56418 q^{87} +9.51462 q^{88} -12.4847 q^{89} -3.56306 q^{90} -7.24270 q^{91} -1.63716 q^{92} -21.3825 q^{93} -9.21724 q^{94} -3.51785 q^{95} +7.15494 q^{96} -3.07842 q^{97} -5.25658 q^{98} -11.1043 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more