LMFDB - Embedded newform 538.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [538,2,Mod(1,538)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(538, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("538.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	538.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:	$4.29595162874$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 $ x^7 - x^6 - 15x^5 + 16x^4 + 49x^3 - 53x^2 - 44*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.26381$ of defining polynomial
Character	$\chi$	$=$	538.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -3.13760 q^{3} +1.00000 q^{4} +3.00565 q^{5} -3.13760 q^{6} -0.324393 q^{7} +1.00000 q^{8} +6.84453 q^{9} +3.00565 q^{10} -4.08405 q^{11} -3.13760 q^{12} -0.304135 q^{13} -0.324393 q^{14} -9.43051 q^{15} +1.00000 q^{16} +5.91934 q^{17} +6.84453 q^{18} +2.22348 q^{19} +3.00565 q^{20} +1.01781 q^{21} -4.08405 q^{22} +8.27520 q^{23} -3.13760 q^{24} +4.03391 q^{25} -0.304135 q^{26} -12.0626 q^{27} -0.324393 q^{28} -5.84659 q^{29} -9.43051 q^{30} +10.2994 q^{31} +1.00000 q^{32} +12.8141 q^{33} +5.91934 q^{34} -0.975010 q^{35} +6.84453 q^{36} +3.68773 q^{37} +2.22348 q^{38} +0.954253 q^{39} +3.00565 q^{40} +5.70620 q^{41} +1.01781 q^{42} +3.80314 q^{43} -4.08405 q^{44} +20.5722 q^{45} +8.27520 q^{46} -10.2865 q^{47} -3.13760 q^{48} -6.89477 q^{49} +4.03391 q^{50} -18.5725 q^{51} -0.304135 q^{52} +6.39465 q^{53} -12.0626 q^{54} -12.2752 q^{55} -0.324393 q^{56} -6.97638 q^{57} -5.84659 q^{58} -2.31530 q^{59} -9.43051 q^{60} -8.06087 q^{61} +10.2994 q^{62} -2.22032 q^{63} +1.00000 q^{64} -0.914122 q^{65} +12.8141 q^{66} +7.93283 q^{67} +5.91934 q^{68} -25.9643 q^{69} -0.975010 q^{70} +7.43426 q^{71} +6.84453 q^{72} -16.5934 q^{73} +3.68773 q^{74} -12.6568 q^{75} +2.22348 q^{76} +1.32484 q^{77} +0.954253 q^{78} -1.09880 q^{79} +3.00565 q^{80} +17.3140 q^{81} +5.70620 q^{82} -5.59412 q^{83} +1.01781 q^{84} +17.7914 q^{85} +3.80314 q^{86} +18.3442 q^{87} -4.08405 q^{88} -14.9648 q^{89} +20.5722 q^{90} +0.0986592 q^{91} +8.27520 q^{92} -32.3153 q^{93} -10.2865 q^{94} +6.68299 q^{95} -3.13760 q^{96} -2.53547 q^{97} -6.89477 q^{98} -27.9534 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9} + 7 q^{10} - 3 q^{11} + q^{12} - 9 q^{13} + 6 q^{14} + 8 q^{15} + 7 q^{16} + 8 q^{17} + 12 q^{18} - 11 q^{19} + 7 q^{20}+ \cdots - 37 q^{99}+O(q^{100}) $ 7 * q + 7 * q^2 + q^3 + 7 * q^4 + 7 * q^5 + q^6 + 6 * q^7 + 7 * q^8 + 12 * q^9 + 7 * q^10 - 3 * q^11 + q^12 - 9 * q^13 + 6 * q^14 + 8 * q^15 + 7 * q^16 + 8 * q^17 + 12 * q^18 - 11 * q^19 + 7 * q^20 - 6 * q^21 - 3 * q^22 + 12 * q^23 + q^24 + 22 * q^25 - 9 * q^26 - 14 * q^27 + 6 * q^28 - 5 * q^29 + 8 * q^30 + 14 * q^31 + 7 * q^32 - 4 * q^33 + 8 * q^34 - 4 * q^35 + 12 * q^36 + 13 * q^37 - 11 * q^38 - 18 * q^39 + 7 * q^40 + 12 * q^41 - 6 * q^42 - 11 * q^43 - 3 * q^44 + 3 * q^45 + 12 * q^46 + 2 * q^47 + q^48 + 15 * q^49 + 22 * q^50 - 26 * q^51 - 9 * q^52 + 19 * q^53 - 14 * q^54 - 40 * q^55 + 6 * q^56 - 12 * q^57 - 5 * q^58 - 9 * q^59 + 8 * q^60 - 3 * q^61 + 14 * q^62 - 26 * q^63 + 7 * q^64 - 10 * q^65 - 4 * q^66 - 33 * q^67 + 8 * q^68 - 64 * q^69 - 4 * q^70 + 28 * q^71 + 12 * q^72 - 14 * q^73 + 13 * q^74 - 45 * q^75 - 11 * q^76 + 10 * q^77 - 18 * q^78 + 2 * q^79 + 7 * q^80 + 15 * q^81 + 12 * q^82 - 7 * q^83 - 6 * q^84 - 16 * q^85 - 11 * q^86 + 16 * q^87 - 3 * q^88 + 18 * q^89 + 3 * q^90 - 26 * q^91 + 12 * q^92 + 6 * q^93 + 2 * q^94 - 34 * q^95 + q^96 - 4 * q^97 + 15 * q^98 - 37 * q^99

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.13760	−1.81149	−0.905747	−	0.423819i	$-0.860689\pi$
$3$	−3.13760	−1.81149	−0.905747	+	0.423819i	$0.860689\pi$
$4$	1.00000	0.500000
$5$	3.00565	1.34417	0.672083	−	0.740476i	$-0.265400\pi$
$5$	3.00565	1.34417	0.672083	+	0.740476i	$0.265400\pi$
$6$	−3.13760	−1.28092
$7$	−0.324393	−0.122609	−0.0613045	−	0.998119i	$-0.519526\pi$
$7$	−0.324393	−0.122609	−0.0613045	+	0.998119i	$0.519526\pi$
$8$	1.00000	0.353553
$9$	6.84453	2.28151
$10$	3.00565	0.950469
$11$	−4.08405	−1.23139	−0.615693	−	0.787986i	$-0.711124\pi$
$11$	−4.08405	−1.23139	−0.615693	+	0.787986i	$0.711124\pi$
$12$	−3.13760	−0.905747
$13$	−0.304135	−0.0843518	−0.0421759	−	0.999110i	$-0.513429\pi$
$13$	−0.304135	−0.0843518	−0.0421759	+	0.999110i	$0.513429\pi$
$14$	−0.324393	−0.0866976
$15$	−9.43051	−2.43495
$16$	1.00000	0.250000
$17$	5.91934	1.43565	0.717826	−	0.696223i	$-0.245137\pi$
$17$	5.91934	1.43565	0.717826	+	0.696223i	$0.245137\pi$
$18$	6.84453	1.61327
$19$	2.22348	0.510101	0.255050	−	0.966928i	$-0.417908\pi$
$19$	2.22348	0.510101	0.255050	+	0.966928i	$0.417908\pi$
$20$	3.00565	0.672083
$21$	1.01781	0.222105
$22$	−4.08405	−0.870722
$23$	8.27520	1.72550	0.862749	−	0.505632i	$-0.168741\pi$
$23$	8.27520	1.72550	0.862749	+	0.505632i	$0.168741\pi$
$24$	−3.13760	−0.640460
$25$	4.03391	0.806781
$26$	−0.304135	−0.0596458
$27$	−12.0626	−2.32145
$28$	−0.324393	−0.0613045
$29$	−5.84659	−1.08568	−0.542842	−	0.839835i	$-0.682652\pi$
$29$	−5.84659	−1.08568	−0.542842	+	0.839835i	$0.682652\pi$
$30$	−9.43051	−1.72177
$31$	10.2994	1.84982	0.924912	−	0.380181i	$-0.124138\pi$
$31$	10.2994	1.84982	0.924912	+	0.380181i	$0.124138\pi$
$32$	1.00000	0.176777
$33$	12.8141	2.23065
$34$	5.91934	1.01516
$35$	−0.975010	−0.164807
$36$	6.84453	1.14075
$37$	3.68773	0.606258	0.303129	−	0.952949i	$-0.401969\pi$
$37$	3.68773	0.606258	0.303129	+	0.952949i	$0.401969\pi$
$38$	2.22348	0.360696
$39$	0.954253	0.152803
$40$	3.00565	0.475234
$41$	5.70620	0.891159	0.445580	−	0.895242i	$-0.352998\pi$
$41$	5.70620	0.891159	0.445580	+	0.895242i	$0.352998\pi$
$42$	1.01781	0.157052
$43$	3.80314	0.579974	0.289987	−	0.957031i	$-0.406349\pi$
$43$	3.80314	0.579974	0.289987	+	0.957031i	$0.406349\pi$
$44$	−4.08405	−0.615693
$45$	20.5722	3.06673
$46$	8.27520	1.22011
$47$	−10.2865	−1.50044	−0.750219	−	0.661189i	$-0.770052\pi$
$47$	−10.2865	−1.50044	−0.750219	+	0.661189i	$0.770052\pi$
$48$	−3.13760	−0.452873
$49$	−6.89477	−0.984967
$50$	4.03391	0.570480
$51$	−18.5725	−2.60067
$52$	−0.304135	−0.0421759
$53$	6.39465	0.878373	0.439187	−	0.898396i	$-0.355267\pi$
$53$	6.39465	0.878373	0.439187	+	0.898396i	$0.355267\pi$
$54$	−12.0626	−1.64151
$55$	−12.2752	−1.65519
$56$	−0.324393	−0.0433488
$57$	−6.97638	−0.924044
$58$	−5.84659	−0.767694
$59$	−2.31530	−0.301426	−0.150713	−	0.988578i	$-0.548157\pi$
$59$	−2.31530	−0.301426	−0.150713	+	0.988578i	$0.548157\pi$
$60$	−9.43051	−1.21747
$61$	−8.06087	−1.03209	−0.516044	−	0.856562i	$-0.672596\pi$
$61$	−8.06087	−1.03209	−0.516044	+	0.856562i	$0.672596\pi$
$62$	10.2994	1.30802
$63$	−2.22032	−0.279733
$64$	1.00000	0.125000
$65$	−0.914122	−0.113383
$66$	12.8141	1.57731
$67$	7.93283	0.969150	0.484575	−	0.874750i	$-0.338974\pi$
$67$	7.93283	0.969150	0.484575	+	0.874750i	$0.338974\pi$
$68$	5.91934	0.717826
$69$	−25.9643	−3.12573
$70$	−0.975010	−0.116536
$71$	7.43426	0.882284	0.441142	−	0.897437i	$-0.354574\pi$
$71$	7.43426	0.882284	0.441142	+	0.897437i	$0.354574\pi$
$72$	6.84453	0.806635
$73$	−16.5934	−1.94211	−0.971055	−	0.238854i	$-0.923228\pi$
$73$	−16.5934	−1.94211	−0.971055	+	0.238854i	$0.923228\pi$
$74$	3.68773	0.428689
$75$	−12.6568	−1.46148
$76$	2.22348	0.255050
$77$	1.32484	0.150979
$78$	0.954253	0.108048
$79$	−1.09880	−0.123625	−0.0618123	−	0.998088i	$-0.519688\pi$
$79$	−1.09880	−0.123625	−0.0618123	+	0.998088i	$0.519688\pi$
$80$	3.00565	0.336041
$81$	17.3140	1.92378
$82$	5.70620	0.630145
$83$	−5.59412	−0.614035	−0.307017	−	0.951704i	$-0.599331\pi$
$83$	−5.59412	−0.614035	−0.307017	+	0.951704i	$0.599331\pi$
$84$	1.01781	0.111053
$85$	17.7914	1.92975
$86$	3.80314	0.410104
$87$	18.3442	1.96671
$88$	−4.08405	−0.435361
$89$	−14.9648	−1.58627	−0.793134	−	0.609047i	$-0.791552\pi$
$89$	−14.9648	−1.58627	−0.793134	+	0.609047i	$0.791552\pi$
$90$	20.5722	2.16850
$91$	0.0986592	0.0103423
$92$	8.27520	0.862749
$93$	−32.3153	−3.35094
$94$	−10.2865	−1.06097
$95$	6.68299	0.685660
$96$	−3.13760	−0.320230
$97$	−2.53547	−0.257438	−0.128719	−	0.991681i	$-0.541087\pi$
$97$	−2.53547	−0.257438	−0.128719	+	0.991681i	$0.541087\pi$
$98$	−6.89477	−0.696477
$99$	−27.9534	−2.80942
$100$	4.03391	0.403391
$101$	−13.1899	−1.31244	−0.656221	−	0.754568i	$-0.727846\pi$
$101$	−13.1899	−1.31244	−0.656221	+	0.754568i	$0.727846\pi$
$102$	−18.5725	−1.83895
$103$	6.16809	0.607760	0.303880	−	0.952710i	$-0.401718\pi$
$103$	6.16809	0.607760	0.303880	+	0.952710i	$0.401718\pi$
$104$	−0.304135	−0.0298229
$105$	3.05919	0.298546
$106$	6.39465	0.621104
$107$	−6.39157	−0.617897	−0.308948	−	0.951079i	$-0.599977\pi$
$107$	−6.39157	−0.617897	−0.308948	+	0.951079i	$0.599977\pi$
$108$	−12.0626	−1.16072
$109$	1.47459	0.141240	0.0706198	−	0.997503i	$-0.477502\pi$
$109$	1.47459	0.141240	0.0706198	+	0.997503i	$0.477502\pi$
$110$	−12.2752	−1.17039
$111$	−11.5706	−1.09823
$112$	−0.324393	−0.0306522
$113$	14.8028	1.39253	0.696266	−	0.717784i	$-0.254844\pi$
$113$	14.8028	1.39253	0.696266	+	0.717784i	$0.254844\pi$
$114$	−6.97638	−0.653398
$115$	24.8723	2.31936
$116$	−5.84659	−0.542842
$117$	−2.08166	−0.192449
$118$	−2.31530	−0.213141
$119$	−1.92019	−0.176024
$120$	−9.43051	−0.860884
$121$	5.67944	0.516313
$122$	−8.06087	−0.729797
$123$	−17.9038	−1.61433
$124$	10.2994	0.924912
$125$	−2.90374	−0.259718
$126$	−2.22032	−0.197801
$127$	−15.9776	−1.41778	−0.708889	−	0.705320i	$-0.750803\pi$
$127$	−15.9776	−1.41778	−0.708889	+	0.705320i	$0.750803\pi$
$128$	1.00000	0.0883883
$129$	−11.9327	−1.05062
$130$	−0.914122	−0.0801738
$131$	0.541102	0.0472763	0.0236381	−	0.999721i	$-0.492475\pi$
$131$	0.541102	0.0472763	0.0236381	+	0.999721i	$0.492475\pi$
$132$	12.8141	1.11532
$133$	−0.721280	−0.0625429
$134$	7.93283	0.685292
$135$	−36.2559	−3.12041
$136$	5.91934	0.507579
$137$	13.0516	1.11507	0.557535	−	0.830153i	$-0.311747\pi$
$137$	13.0516	1.11507	0.557535	+	0.830153i	$0.311747\pi$
$138$	−25.9643	−2.21022
$139$	−3.83630	−0.325391	−0.162695	−	0.986676i	$-0.552019\pi$
$139$	−3.83630	−0.325391	−0.162695	+	0.986676i	$0.552019\pi$
$140$	−0.975010	−0.0824034
$141$	32.2749	2.71804
$142$	7.43426	0.623869
$143$	1.24210	0.103870
$144$	6.84453	0.570377
$145$	−17.5728	−1.45934
$146$	−16.5934	−1.37328
$147$	21.6330	1.78426
$148$	3.68773	0.303129
$149$	−13.5271	−1.10818	−0.554091	−	0.832456i	$-0.686934\pi$
$149$	−13.5271	−1.10818	−0.554091	+	0.832456i	$0.686934\pi$
$150$	−12.6568	−1.03342
$151$	2.86134	0.232853	0.116426	−	0.993199i	$-0.462856\pi$
$151$	2.86134	0.232853	0.116426	+	0.993199i	$0.462856\pi$
$152$	2.22348	0.180348
$153$	40.5151	3.27545
$154$	1.32484	0.106758
$155$	30.9563	2.48647
$156$	0.954253	0.0764014
$157$	5.27923	0.421328	0.210664	−	0.977559i	$-0.432437\pi$
$157$	5.27923	0.421328	0.210664	+	0.977559i	$0.432437\pi$
$158$	−1.09880	−0.0874159
$159$	−20.0639	−1.59117
$160$	3.00565	0.237617
$161$	−2.68441	−0.211562
$162$	17.3140	1.36031
$163$	10.3269	0.808867	0.404433	−	0.914567i	$-0.367469\pi$
$163$	10.3269	0.808867	0.404433	+	0.914567i	$0.367469\pi$
$164$	5.70620	0.445580
$165$	38.5147	2.99836
$166$	−5.59412	−0.434188
$167$	−2.66027	−0.205858	−0.102929	−	0.994689i	$-0.532821\pi$
$167$	−2.66027	−0.205858	−0.102929	+	0.994689i	$0.532821\pi$
$168$	1.01781	0.0785261
$169$	−12.9075	−0.992885
$170$	17.7914	1.36454
$171$	15.2187	1.16380
$172$	3.80314	0.289987
$173$	19.5720	1.48803	0.744014	−	0.668164i	$-0.232919\pi$
$173$	19.5720	1.48803	0.744014	+	0.668164i	$0.232919\pi$
$174$	18.3442	1.39067
$175$	−1.30857	−0.0989186
$176$	−4.08405	−0.307847
$177$	7.26448	0.546032
$178$	−14.9648	−1.12166
$179$	12.2712	0.917194	0.458597	−	0.888644i	$-0.348352\pi$
$179$	12.2712	0.917194	0.458597	+	0.888644i	$0.348352\pi$
$180$	20.5722	1.53336
$181$	−12.9260	−0.960781	−0.480390	−	0.877055i	$-0.659505\pi$
$181$	−12.9260	−0.960781	−0.480390	+	0.877055i	$0.659505\pi$
$182$	0.0986592	0.00731310
$183$	25.2918	1.86962
$184$	8.27520	0.610056
$185$	11.0840	0.814912
$186$	−32.3153	−2.36948
$187$	−24.1749	−1.76784
$188$	−10.2865	−0.750219
$189$	3.91302	0.284630
$190$	6.68299	0.484835
$191$	1.87883	0.135947	0.0679736	−	0.997687i	$-0.478347\pi$
$191$	1.87883	0.135947	0.0679736	+	0.997687i	$0.478347\pi$
$192$	−3.13760	−0.226437
$193$	3.16511	0.227829	0.113915	−	0.993491i	$-0.463661\pi$
$193$	3.16511	0.227829	0.113915	+	0.993491i	$0.463661\pi$
$194$	−2.53547	−0.182036
$195$	2.86815	0.205392
$196$	−6.89477	−0.492484
$197$	14.9516	1.06526	0.532630	−	0.846348i	$-0.321204\pi$
$197$	14.9516	1.06526	0.532630	+	0.846348i	$0.321204\pi$
$198$	−27.9534	−1.98656
$199$	−5.49517	−0.389542	−0.194771	−	0.980849i	$-0.562396\pi$
$199$	−5.49517	−0.389542	−0.194771	+	0.980849i	$0.562396\pi$
$200$	4.03391	0.285240
$201$	−24.8900	−1.75561
$202$	−13.1899	−0.928037
$203$	1.89659	0.133115
$204$	−18.5725	−1.30034
$205$	17.1508	1.19787
$206$	6.16809	0.429751
$207$	56.6398	3.93674
$208$	−0.304135	−0.0210880
$209$	−9.08079	−0.628131
$210$	3.05919	0.211104
$211$	−3.88750	−0.267626	−0.133813	−	0.991007i	$-0.542722\pi$
$211$	−3.88750	−0.267626	−0.133813	+	0.991007i	$0.542722\pi$
$212$	6.39465	0.439187
$213$	−23.3257	−1.59825
$214$	−6.39157	−0.436919
$215$	11.4309	0.779581
$216$	−12.0626	−0.820755
$217$	−3.34105	−0.226805
$218$	1.47459	0.0998715
$219$	52.0634	3.51812
$220$	−12.2752	−0.827594
$221$	−1.80028	−0.121100
$222$	−11.5706	−0.776568
$223$	−14.0272	−0.939333	−0.469667	−	0.882844i	$-0.655626\pi$
$223$	−14.0272	−0.939333	−0.469667	+	0.882844i	$0.655626\pi$
$224$	−0.324393	−0.0216744
$225$	27.6102	1.84068
$226$	14.8028	0.984668
$227$	−11.3821	−0.755459	−0.377730	−	0.925916i	$-0.623295\pi$
$227$	−11.3821	−0.755459	−0.377730	+	0.925916i	$0.623295\pi$
$228$	−6.97638	−0.462022
$229$	−20.3763	−1.34650	−0.673251	−	0.739414i	$-0.735103\pi$
$229$	−20.3763	−1.34650	−0.673251	+	0.739414i	$0.735103\pi$
$230$	24.8723	1.64003
$231$	−4.15680	−0.273498
$232$	−5.84659	−0.383847
$233$	20.9537	1.37272	0.686360	−	0.727262i	$-0.259207\pi$
$233$	20.9537	1.37272	0.686360	+	0.727262i	$0.259207\pi$
$234$	−2.08166	−0.136082
$235$	−30.9175	−2.01684
$236$	−2.31530	−0.150713
$237$	3.44759	0.223945
$238$	−1.92019	−0.124468
$239$	−25.4814	−1.64825	−0.824126	−	0.566407i	$-0.808333\pi$
$239$	−25.4814	−1.64825	−0.824126	+	0.566407i	$0.808333\pi$
$240$	−9.43051	−0.608737
$241$	6.56813	0.423090	0.211545	−	0.977368i	$-0.432150\pi$
$241$	6.56813	0.423090	0.211545	+	0.977368i	$0.432150\pi$
$242$	5.67944	0.365088
$243$	−18.1366	−1.16346
$244$	−8.06087	−0.516044
$245$	−20.7232	−1.32396
$246$	−17.9038	−1.14150
$247$	−0.676237	−0.0430279
$248$	10.2994	0.654012
$249$	17.5521	1.11232
$250$	−2.90374	−0.183648
$251$	−13.0147	−0.821480	−0.410740	−	0.911753i	$-0.634730\pi$
$251$	−13.0147	−0.821480	−0.410740	+	0.911753i	$0.634730\pi$
$252$	−2.22032	−0.139867
$253$	−33.7963	−2.12476
$254$	−15.9776	−1.00252
$255$	−55.8224	−3.49574
$256$	1.00000	0.0625000
$257$	27.0712	1.68865	0.844327	−	0.535828i	$-0.180000\pi$
$257$	27.0712	1.68865	0.844327	+	0.535828i	$0.180000\pi$
$258$	−11.9327	−0.742900
$259$	−1.19627	−0.0743327
$260$	−0.914122	−0.0566914
$261$	−40.0171	−2.47700
$262$	0.541102	0.0334294
$263$	−14.4165	−0.888959	−0.444479	−	0.895789i	$-0.646611\pi$
$263$	−14.4165	−0.888959	−0.444479	+	0.895789i	$0.646611\pi$
$264$	12.8141	0.788653
$265$	19.2201	1.18068
$266$	−0.721280	−0.0442245
$267$	46.9536	2.87351
$268$	7.93283	0.484575
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	−36.2559	−2.20646
$271$	−9.86488	−0.599249	−0.299624	−	0.954057i	$-0.596861\pi$
$271$	−9.86488	−0.599249	−0.299624	+	0.954057i	$0.596861\pi$
$272$	5.91934	0.358913
$273$	−0.309553	−0.0187350
$274$	13.0516	0.788474
$275$	−16.4747	−0.993460
$276$	−25.9643	−1.56286
$277$	−27.4029	−1.64648	−0.823240	−	0.567693i	$-0.807836\pi$
$277$	−27.4029	−1.64648	−0.823240	+	0.567693i	$0.807836\pi$
$278$	−3.83630	−0.230086
$279$	70.4944	4.22039
$280$	−0.975010	−0.0582680
$281$	9.72279	0.580013	0.290007	−	0.957025i	$-0.406342\pi$
$281$	9.72279	0.580013	0.290007	+	0.957025i	$0.406342\pi$
$282$	32.2749	1.92194
$283$	−17.5086	−1.04078	−0.520390	−	0.853929i	$-0.674213\pi$
$283$	−17.5086	−1.04078	−0.520390	+	0.853929i	$0.674213\pi$
$284$	7.43426	0.441142
$285$	−20.9685	−1.24207
$286$	1.24210	0.0734470
$287$	−1.85105	−0.109264
$288$	6.84453	0.403318
$289$	18.0386	1.06110
$290$	−17.5728	−1.03191
$291$	7.95530	0.466348
$292$	−16.5934	−0.971055
$293$	5.81945	0.339976	0.169988	−	0.985446i	$-0.445627\pi$
$293$	5.81945	0.339976	0.169988	+	0.985446i	$0.445627\pi$
$294$	21.6330	1.26166
$295$	−6.95897	−0.405167
$296$	3.68773	0.214345
$297$	49.2642	2.85860
$298$	−13.5271	−0.783602
$299$	−2.51678	−0.145549
$300$	−12.6568	−0.730740
$301$	−1.23371	−0.0711100
$302$	2.86134	0.164652
$303$	41.3846	2.37748
$304$	2.22348	0.127525
$305$	−24.2281	−1.38730
$306$	40.5151	2.31609
$307$	27.8232	1.58795	0.793977	−	0.607947i	$-0.208007\pi$
$307$	27.8232	1.58795	0.793977	+	0.607947i	$0.208007\pi$
$308$	1.32484	0.0754895
$309$	−19.3530	−1.10095
$310$	30.9563	1.75820
$311$	18.2180	1.03305	0.516524	−	0.856273i	$-0.327226\pi$
$311$	18.2180	1.03305	0.516524	+	0.856273i	$0.327226\pi$
$312$	0.954253	0.0540240
$313$	−11.4600	−0.647756	−0.323878	−	0.946099i	$-0.604987\pi$
$313$	−11.4600	−0.647756	−0.323878	+	0.946099i	$0.604987\pi$
$314$	5.27923	0.297924
$315$	−6.67348	−0.376008
$316$	−1.09880	−0.0618123
$317$	13.8558	0.778221	0.389111	−	0.921191i	$-0.372782\pi$
$317$	13.8558	0.778221	0.389111	+	0.921191i	$0.372782\pi$
$318$	−20.0639	−1.12513
$319$	23.8777	1.33690
$320$	3.00565	0.168021
$321$	20.0542	1.11932
$322$	−2.68441	−0.149597
$323$	13.1615	0.732327
$324$	17.3140	0.961888
$325$	−1.22685	−0.0680535
$326$	10.3269	0.571955
$327$	−4.62666	−0.255855
$328$	5.70620	0.315072
$329$	3.33686	0.183967
$330$	38.5147	2.12016
$331$	−34.5901	−1.90125	−0.950623	−	0.310349i	$-0.899554\pi$
$331$	−34.5901	−1.90125	−0.950623	+	0.310349i	$0.899554\pi$
$332$	−5.59412	−0.307017
$333$	25.2407	1.38318
$334$	−2.66027	−0.145564
$335$	23.8433	1.30270
$336$	1.01781	0.0555263
$337$	−25.2373	−1.37476	−0.687381	−	0.726297i	$-0.741240\pi$
$337$	−25.2373	−1.37476	−0.687381	+	0.726297i	$0.741240\pi$
$338$	−12.9075	−0.702076
$339$	−46.4453	−2.52256
$340$	17.7914	0.964877
$341$	−42.0632	−2.27785
$342$	15.2187	0.822931
$343$	4.50736	0.243375
$344$	3.80314	0.205052
$345$	−78.0393	−4.20150
$346$	19.5720	1.05220
$347$	16.6544	0.894056	0.447028	−	0.894520i	$-0.352482\pi$
$347$	16.6544	0.894056	0.447028	+	0.894520i	$0.352482\pi$
$348$	18.3442	0.983355
$349$	−4.22147	−0.225970	−0.112985	−	0.993597i	$-0.536041\pi$
$349$	−4.22147	−0.225970	−0.112985	+	0.993597i	$0.536041\pi$
$350$	−1.30857	−0.0699460
$351$	3.66865	0.195818
$352$	−4.08405	−0.217680
$353$	13.4438	0.715544	0.357772	−	0.933809i	$-0.383537\pi$
$353$	13.4438	0.715544	0.357772	+	0.933809i	$0.383537\pi$
$354$	7.26448	0.386103
$355$	22.3447	1.18594
$356$	−14.9648	−0.793134
$357$	6.02479	0.318866
$358$	12.2712	0.648554
$359$	−17.9654	−0.948177	−0.474089	−	0.880477i	$-0.657222\pi$
$359$	−17.9654	−0.948177	−0.474089	+	0.880477i	$0.657222\pi$
$360$	20.5722	1.08425
$361$	−14.0561	−0.739797
$362$	−12.9260	−0.679374
$363$	−17.8198	−0.935297
$364$	0.0986592	0.00517115
$365$	−49.8739	−2.61052
$366$	25.2918	1.32202
$367$	−19.2339	−1.00400	−0.502000	−	0.864868i	$-0.667402\pi$
$367$	−19.2339	−1.00400	−0.502000	+	0.864868i	$0.667402\pi$
$368$	8.27520	0.431375
$369$	39.0563	2.03319
$370$	11.0840	0.576230
$371$	−2.07438	−0.107696
$372$	−32.3153	−1.67547
$373$	6.12369	0.317073	0.158536	−	0.987353i	$-0.449322\pi$
$373$	6.12369	0.317073	0.158536	+	0.987353i	$0.449322\pi$
$374$	−24.1749	−1.25005
$375$	9.11076	0.470478
$376$	−10.2865	−0.530485
$377$	1.77815	0.0915794
$378$	3.91302	0.201264
$379$	4.30176	0.220967	0.110483	−	0.993878i	$-0.464760\pi$
$379$	4.30176	0.220967	0.110483	+	0.993878i	$0.464760\pi$
$380$	6.68299	0.342830
$381$	50.1312	2.56830
$382$	1.87883	0.0961291
$383$	−30.1561	−1.54090	−0.770452	−	0.637498i	$-0.779969\pi$
$383$	−30.1561	−1.54090	−0.770452	+	0.637498i	$0.779969\pi$
$384$	−3.13760	−0.160115
$385$	3.98199	0.202941
$386$	3.16511	0.161100
$387$	26.0307	1.32322
$388$	−2.53547	−0.128719
$389$	16.5905	0.841172	0.420586	−	0.907253i	$-0.361825\pi$
$389$	16.5905	0.841172	0.420586	+	0.907253i	$0.361825\pi$
$390$	2.86815	0.145234
$391$	48.9837	2.47721
$392$	−6.89477	−0.348238
$393$	−1.69776	−0.0856406
$394$	14.9516	0.753253
$395$	−3.30260	−0.166172
$396$	−27.9534	−1.40471
$397$	−6.67285	−0.334901	−0.167450	−	0.985881i	$-0.553553\pi$
$397$	−6.67285	−0.334901	−0.167450	+	0.985881i	$0.553553\pi$
$398$	−5.49517	−0.275448
$399$	2.26309	0.113296
$400$	4.03391	0.201695
$401$	27.7436	1.38545	0.692725	−	0.721202i	$-0.256410\pi$
$401$	27.7436	1.38545	0.692725	+	0.721202i	$0.256410\pi$
$402$	−24.8900	−1.24140
$403$	−3.13240	−0.156036
$404$	−13.1899	−0.656221
$405$	52.0397	2.58587
$406$	1.89659	0.0941262
$407$	−15.0608	−0.746538
$408$	−18.5725	−0.919477
$409$	30.6637	1.51622	0.758111	−	0.652125i	$-0.226122\pi$
$409$	30.6637	1.51622	0.758111	+	0.652125i	$0.226122\pi$
$410$	17.1508	0.847019
$411$	−40.9506	−2.01994
$412$	6.16809	0.303880
$413$	0.751066	0.0369576
$414$	56.6398	2.78370
$415$	−16.8140	−0.825364
$416$	−0.304135	−0.0149114
$417$	12.0368	0.589443
$418$	−9.08079	−0.444156
$419$	7.47427	0.365142	0.182571	−	0.983193i	$-0.441558\pi$
$419$	7.47427	0.365142	0.182571	+	0.983193i	$0.441558\pi$
$420$	3.05919	0.149273
$421$	−36.8824	−1.79754	−0.898769	−	0.438422i	$-0.855538\pi$
$421$	−36.8824	−1.79754	−0.898769	+	0.438422i	$0.855538\pi$
$422$	−3.88750	−0.189240
$423$	−70.4062	−3.42326
$424$	6.39465	0.310552
$425$	23.8781	1.15826
$426$	−23.3257	−1.13013
$427$	2.61489	0.126543
$428$	−6.39157	−0.308948
$429$	−3.89722	−0.188159
$430$	11.4309	0.551247
$431$	37.8309	1.82225	0.911125	−	0.412130i	$-0.135215\pi$
$431$	37.8309	1.82225	0.911125	+	0.412130i	$0.135215\pi$
$432$	−12.0626	−0.580362
$433$	9.88884	0.475227	0.237614	−	0.971360i	$-0.423635\pi$
$433$	9.88884	0.475227	0.237614	+	0.971360i	$0.423635\pi$
$434$	−3.34105	−0.160375
$435$	55.1363	2.64358
$436$	1.47459	0.0706198
$437$	18.3997	0.880178
$438$	52.0634	2.48769
$439$	16.8597	0.804669	0.402334	−	0.915493i	$-0.368199\pi$
$439$	16.8597	0.804669	0.402334	+	0.915493i	$0.368199\pi$
$440$	−12.2752	−0.585197
$441$	−47.1914	−2.24721
$442$	−1.80028	−0.0856305
$443$	−12.7251	−0.604586	−0.302293	−	0.953215i	$-0.597752\pi$
$443$	−12.7251	−0.604586	−0.302293	+	0.953215i	$0.597752\pi$
$444$	−11.5706	−0.549117
$445$	−44.9790	−2.13221
$446$	−14.0272	−0.664209
$447$	42.4425	2.00746
$448$	−0.324393	−0.0153261
$449$	27.2026	1.28377	0.641885	−	0.766801i	$-0.278153\pi$
$449$	27.2026	1.28377	0.641885	+	0.766801i	$0.278153\pi$
$450$	27.6102	1.30156
$451$	−23.3044	−1.09736
$452$	14.8028	0.696266
$453$	−8.97774	−0.421811
$454$	−11.3821	−0.534190
$455$	0.296534	0.0139018
$456$	−6.97638	−0.326699
$457$	19.5630	0.915121	0.457560	−	0.889179i	$-0.348723\pi$
$457$	19.5630	0.915121	0.457560	+	0.889179i	$0.348723\pi$
$458$	−20.3763	−0.952121
$459$	−71.4026	−3.33279
$460$	24.8723	1.15968
$461$	30.5373	1.42226	0.711132	−	0.703058i	$-0.248183\pi$
$461$	30.5373	1.42226	0.711132	+	0.703058i	$0.248183\pi$
$462$	−4.15680	−0.193392
$463$	1.65808	0.0770576	0.0385288	−	0.999257i	$-0.487733\pi$
$463$	1.65808	0.0770576	0.0385288	+	0.999257i	$0.487733\pi$
$464$	−5.84659	−0.271421
$465$	−97.1285	−4.50422
$466$	20.9537	0.970660
$467$	19.9513	0.923237	0.461618	−	0.887079i	$-0.347269\pi$
$467$	19.9513	0.923237	0.461618	+	0.887079i	$0.347269\pi$
$468$	−2.08166	−0.0962247
$469$	−2.57335	−0.118826
$470$	−30.9175	−1.42612
$471$	−16.5641	−0.763233
$472$	−2.31530	−0.106570
$473$	−15.5322	−0.714172
$474$	3.44759	0.158353
$475$	8.96930	0.411540
$476$	−1.92019	−0.0880119
$477$	43.7684	2.00402
$478$	−25.4814	−1.16549
$479$	−1.41839	−0.0648078	−0.0324039	−	0.999475i	$-0.510316\pi$
$479$	−1.41839	−0.0648078	−0.0324039	+	0.999475i	$0.510316\pi$
$480$	−9.43051	−0.430442
$481$	−1.12157	−0.0511390
$482$	6.56813	0.299170
$483$	8.42262	0.383242
$484$	5.67944	0.258156
$485$	−7.62074	−0.346040
$486$	−18.1366	−0.822691
$487$	39.0907	1.77137	0.885685	−	0.464286i	$-0.153689\pi$
$487$	39.0907	1.77137	0.885685	+	0.464286i	$0.153689\pi$
$488$	−8.06087	−0.364898
$489$	−32.4017	−1.46526
$490$	−20.7232	−0.936180
$491$	4.88763	0.220575	0.110288	−	0.993900i	$-0.464823\pi$
$491$	4.88763	0.220575	0.110288	+	0.993900i	$0.464823\pi$
$492$	−17.9038	−0.807165
$493$	−34.6080	−1.55866
$494$	−0.676237	−0.0304253
$495$	−84.0179	−3.77633
$496$	10.2994	0.462456
$497$	−2.41162	−0.108176
$498$	17.5521	0.786529
$499$	−25.2729	−1.13137	−0.565685	−	0.824621i	$-0.691388\pi$
$499$	−25.2729	−1.13137	−0.565685	+	0.824621i	$0.691388\pi$
$500$	−2.90374	−0.129859
$501$	8.34687	0.372911
$502$	−13.0147	−0.580874
$503$	−2.63475	−0.117478	−0.0587388	−	0.998273i	$-0.518708\pi$
$503$	−2.63475	−0.117478	−0.0587388	+	0.998273i	$0.518708\pi$
$504$	−2.22032	−0.0989007
$505$	−39.6441	−1.76414
$506$	−33.7963	−1.50243
$507$	40.4986	1.79860
$508$	−15.9776	−0.708889
$509$	−11.4330	−0.506760	−0.253380	−	0.967367i	$-0.581542\pi$
$509$	−11.4330	−0.506760	−0.253380	+	0.967367i	$0.581542\pi$
$510$	−55.8224	−2.47186
$511$	5.38278	0.238120
$512$	1.00000	0.0441942
$513$	−26.8209	−1.18417
$514$	27.0712	1.19406
$515$	18.5391	0.816931
$516$	−11.9327	−0.525310
$517$	42.0105	1.84762
$518$	−1.19627	−0.0525612
$519$	−61.4090	−2.69555
$520$	−0.914122	−0.0400869
$521$	2.05705	0.0901210	0.0450605	−	0.998984i	$-0.485652\pi$
$521$	2.05705	0.0901210	0.0450605	+	0.998984i	$0.485652\pi$
$522$	−40.0171	−1.75150
$523$	7.68774	0.336161	0.168081	−	0.985773i	$-0.446243\pi$
$523$	7.68774	0.336161	0.168081	+	0.985773i	$0.446243\pi$
$524$	0.541102	0.0236381
$525$	4.10577	0.179190
$526$	−14.4165	−0.628589
$527$	60.9656	2.65570
$528$	12.8141	0.557662
$529$	45.4789	1.97734
$530$	19.2201	0.834866
$531$	−15.8471	−0.687707
$532$	−0.721280	−0.0312715
$533$	−1.73546	−0.0751709
$534$	46.9536	2.03188
$535$	−19.2108	−0.830555
$536$	7.93283	0.342646
$537$	−38.5022	−1.66149
$538$	−1.00000	−0.0431131
$539$	28.1586	1.21288
$540$	−36.2559	−1.56020
$541$	6.80481	0.292562	0.146281	−	0.989243i	$-0.453270\pi$
$541$	6.80481	0.292562	0.146281	+	0.989243i	$0.453270\pi$
$542$	−9.86488	−0.423733
$543$	40.5565	1.74045
$544$	5.91934	0.253790
$545$	4.43208	0.189850
$546$	−0.309553	−0.0132476
$547$	−26.6131	−1.13790	−0.568948	−	0.822374i	$-0.692649\pi$
$547$	−26.6131	−1.13790	−0.568948	+	0.822374i	$0.692649\pi$
$548$	13.0516	0.557535
$549$	−55.1729	−2.35472
$550$	−16.4747	−0.702482
$551$	−12.9998	−0.553808
$552$	−25.9643	−1.10511
$553$	0.356443	0.0151575
$554$	−27.4029	−1.16424
$555$	−34.7771	−1.47621
$556$	−3.83630	−0.162695
$557$	36.6320	1.55215	0.776073	−	0.630643i	$-0.217209\pi$
$557$	36.6320	1.55215	0.776073	+	0.630643i	$0.217209\pi$
$558$	70.4944	2.98427
$559$	−1.15667	−0.0489219
$560$	−0.975010	−0.0412017
$561$	75.8511	3.20243
$562$	9.72279	0.410131
$563$	3.66713	0.154551	0.0772755	−	0.997010i	$-0.475378\pi$
$563$	3.66713	0.154551	0.0772755	+	0.997010i	$0.475378\pi$
$564$	32.2749	1.35902
$565$	44.4920	1.87179
$566$	−17.5086	−0.735943
$567$	−5.61653	−0.235872
$568$	7.43426	0.311934
$569$	−18.5648	−0.778277	−0.389138	−	0.921179i	$-0.627227\pi$
$569$	−18.5648	−0.778277	−0.389138	+	0.921179i	$0.627227\pi$
$570$	−20.9685	−0.878275
$571$	−29.1207	−1.21866	−0.609332	−	0.792915i	$-0.708562\pi$
$571$	−29.1207	−1.21866	−0.609332	+	0.792915i	$0.708562\pi$
$572$	1.24210	0.0519349
$573$	−5.89500	−0.246267
$574$	−1.85105	−0.0772614
$575$	33.3814	1.39210
$576$	6.84453	0.285189
$577$	−40.7975	−1.69842	−0.849211	−	0.528053i	$-0.822922\pi$
$577$	−40.7975	−1.69842	−0.849211	+	0.528053i	$0.822922\pi$
$578$	18.0386	0.750308
$579$	−9.93084	−0.412711
$580$	−17.5728	−0.729670
$581$	1.81469	0.0752862
$582$	7.95530	0.329758
$583$	−26.1161	−1.08162
$584$	−16.5934	−0.686640
$585$	−6.25673	−0.258684
$586$	5.81945	0.240399
$587$	−23.6978	−0.978112	−0.489056	−	0.872252i	$-0.662659\pi$
$587$	−23.6978	−0.978112	−0.489056	+	0.872252i	$0.662659\pi$
$588$	21.6330	0.892131
$589$	22.9004	0.943597
$590$	−6.95897	−0.286496
$591$	−46.9123	−1.92971
$592$	3.68773	0.151565
$593$	−44.8423	−1.84145	−0.920726	−	0.390210i	$-0.872402\pi$
$593$	−44.8423	−1.84145	−0.920726	+	0.390210i	$0.872402\pi$
$594$	49.2642	2.02133
$595$	−5.77142	−0.236605
$596$	−13.5271	−0.554091
$597$	17.2416	0.705654
$598$	−2.51678	−0.102919
$599$	−14.3773	−0.587442	−0.293721	−	0.955891i	$-0.594894\pi$
$599$	−14.3773	−0.587442	−0.293721	+	0.955891i	$0.594894\pi$
$600$	−12.6568	−0.516711
$601$	10.7316	0.437753	0.218876	−	0.975753i	$-0.429761\pi$
$601$	10.7316	0.437753	0.218876	+	0.975753i	$0.429761\pi$
$602$	−1.23371	−0.0502824
$603$	54.2965	2.21112
$604$	2.86134	0.116426
$605$	17.0704	0.694010
$606$	41.3846	1.68113
$607$	−19.0640	−0.773785	−0.386892	−	0.922125i	$-0.626452\pi$
$607$	−19.0640	−0.773785	−0.386892	+	0.922125i	$0.626452\pi$
$608$	2.22348	0.0901739
$609$	−5.95074	−0.241136
$610$	−24.2281	−0.980968
$611$	3.12848	0.126565
$612$	40.5151	1.63773
$613$	−4.32440	−0.174661	−0.0873305	−	0.996179i	$-0.527834\pi$
$613$	−4.32440	−0.174661	−0.0873305	+	0.996179i	$0.527834\pi$
$614$	27.8232	1.12285
$615$	−53.8124	−2.16993
$616$	1.32484	0.0533791
$617$	−7.09359	−0.285577	−0.142789	−	0.989753i	$-0.545607\pi$
$617$	−7.09359	−0.285577	−0.142789	+	0.989753i	$0.545607\pi$
$618$	−19.3530	−0.778492
$619$	0.121741	0.00489317	0.00244658	−	0.999997i	$-0.499221\pi$
$619$	0.121741	0.00489317	0.00244658	+	0.999997i	$0.499221\pi$
$620$	30.9563	1.24323
$621$	−99.8203	−4.00565
$622$	18.2180	0.730475
$623$	4.85448	0.194491
$624$	0.954253	0.0382007
$625$	−28.8971	−1.15589
$626$	−11.4600	−0.458032
$627$	28.4919	1.13786
$628$	5.27923	0.210664
$629$	21.8289	0.870376
$630$	−6.67348	−0.265878
$631$	−11.4215	−0.454683	−0.227341	−	0.973815i	$-0.573003\pi$
$631$	−11.4215	−0.454683	−0.227341	+	0.973815i	$0.573003\pi$
$632$	−1.09880	−0.0437079
$633$	12.1974	0.484803
$634$	13.8558	0.550286
$635$	−48.0229	−1.90573
$636$	−20.0639	−0.795584
$637$	2.09694	0.0830838
$638$	23.8777	0.945329
$639$	50.8840	2.01294
$640$	3.00565	0.118809
$641$	−41.7220	−1.64792	−0.823959	−	0.566649i	$-0.808240\pi$
$641$	−41.7220	−1.64792	−0.823959	+	0.566649i	$0.808240\pi$
$642$	20.0542	0.791476
$643$	40.3144	1.58984	0.794922	−	0.606712i	$-0.207512\pi$
$643$	40.3144	1.58984	0.794922	+	0.606712i	$0.207512\pi$
$644$	−2.68441	−0.105781
$645$	−35.8656	−1.41221
$646$	13.1615	0.517833
$647$	38.3534	1.50783	0.753913	−	0.656974i	$-0.228164\pi$
$647$	38.3534	1.50783	0.753913	+	0.656974i	$0.228164\pi$
$648$	17.3140	0.680157
$649$	9.45579	0.371172
$650$	−1.22685	−0.0481211
$651$	10.4829	0.410856
$652$	10.3269	0.404433
$653$	−47.3581	−1.85327	−0.926634	−	0.375965i	$-0.877311\pi$
$653$	−47.3581	−1.85327	−0.926634	+	0.375965i	$0.877311\pi$
$654$	−4.62666	−0.180917
$655$	1.62636	0.0635471
$656$	5.70620	0.222790
$657$	−113.574	−4.43094
$658$	3.33686	0.130084
$659$	−37.2534	−1.45119	−0.725594	−	0.688123i	$-0.758435\pi$
$659$	−37.2534	−1.45119	−0.725594	+	0.688123i	$0.758435\pi$
$660$	38.5147	1.49918
$661$	23.8806	0.928848	0.464424	−	0.885613i	$-0.346261\pi$
$661$	23.8806	0.928848	0.464424	+	0.885613i	$0.346261\pi$
$662$	−34.5901	−1.34438
$663$	5.64855	0.219372
$664$	−5.59412	−0.217094
$665$	−2.16791	−0.0840680
$666$	25.2407	0.978059
$667$	−48.3817	−1.87335
$668$	−2.66027	−0.102929
$669$	44.0118	1.70160
$670$	23.8433	0.921146
$671$	32.9210	1.27090
$672$	1.01781	0.0392630
$673$	−23.1189	−0.891167	−0.445584	−	0.895240i	$-0.647004\pi$
$673$	−23.1189	−0.891167	−0.445584	+	0.895240i	$0.647004\pi$
$674$	−25.2373	−0.972103
$675$	−48.6593	−1.87290
$676$	−12.9075	−0.496442
$677$	−33.7962	−1.29889	−0.649446	−	0.760408i	$-0.724999\pi$
$677$	−33.7962	−1.29889	−0.649446	+	0.760408i	$0.724999\pi$
$678$	−46.4453	−1.78372
$679$	0.822489	0.0315642
$680$	17.7914	0.682271
$681$	35.7126	1.36851
$682$	−42.0632	−1.61068
$683$	22.1931	0.849194	0.424597	−	0.905382i	$-0.360416\pi$
$683$	22.1931	0.849194	0.424597	+	0.905382i	$0.360416\pi$
$684$	15.2187	0.581900
$685$	39.2284	1.49884
$686$	4.50736	0.172092
$687$	63.9326	2.43918
$688$	3.80314	0.144994
$689$	−1.94484	−0.0740924
$690$	−78.0393	−2.97091
$691$	37.3843	1.42217	0.711084	−	0.703107i	$-0.248205\pi$
$691$	37.3843	1.42217	0.711084	+	0.703107i	$0.248205\pi$
$692$	19.5720	0.744014
$693$	9.06787	0.344460
$694$	16.6544	0.632193
$695$	−11.5306	−0.437379
$696$	18.3442	0.695337
$697$	33.7770	1.27939
$698$	−4.22147	−0.159785
$699$	−65.7442	−2.48667
$700$	−1.30857	−0.0494593
$701$	9.26870	0.350074	0.175037	−	0.984562i	$-0.443995\pi$
$701$	9.26870	0.350074	0.175037	+	0.984562i	$0.443995\pi$
$702$	3.66865	0.138464
$703$	8.19957	0.309253
$704$	−4.08405	−0.153923
$705$	97.0069	3.65349
$706$	13.4438	0.505966
$707$	4.27870	0.160917
$708$	7.26448	0.273016
$709$	26.4983	0.995164	0.497582	−	0.867417i	$-0.334221\pi$
$709$	26.4983	0.995164	0.497582	+	0.867417i	$0.334221\pi$
$710$	22.3447	0.838583
$711$	−7.52077	−0.282051
$712$	−14.9648	−0.560830
$713$	85.2294	3.19187
$714$	6.02479	0.225472
$715$	3.73332	0.139618
$716$	12.2712	0.458597
$717$	79.9503	2.98580
$718$	−17.9654	−0.670462
$719$	25.9263	0.966887	0.483443	−	0.875376i	$-0.339386\pi$
$719$	25.9263	0.966887	0.483443	+	0.875376i	$0.339386\pi$
$720$	20.5722	0.766682
$721$	−2.00089	−0.0745169
$722$	−14.0561	−0.523116
$723$	−20.6082	−0.766425
$724$	−12.9260	−0.480390
$725$	−23.5846	−0.875909
$726$	−17.8198	−0.661355
$727$	−24.6376	−0.913756	−0.456878	−	0.889529i	$-0.651032\pi$
$727$	−24.6376	−0.913756	−0.456878	+	0.889529i	$0.651032\pi$
$728$	0.0986592	0.00365655
$729$	4.96333	0.183827
$730$	−49.8739	−1.84592
$731$	22.5121	0.832641
$732$	25.2918	0.934811
$733$	26.5503	0.980658	0.490329	−	0.871537i	$-0.336877\pi$
$733$	26.5503	0.980658	0.490329	+	0.871537i	$0.336877\pi$
$734$	−19.2339	−0.709935
$735$	65.0212	2.39834
$736$	8.27520	0.305028
$737$	−32.3981	−1.19340
$738$	39.0563	1.43768
$739$	−35.5016	−1.30595	−0.652974	−	0.757380i	$-0.726479\pi$
$739$	−35.5016	−1.30595	−0.652974	+	0.757380i	$0.726479\pi$
$740$	11.0840	0.407456
$741$	2.12176	0.0779448
$742$	−2.07438	−0.0761529
$743$	−35.0512	−1.28590	−0.642951	−	0.765907i	$-0.722290\pi$
$743$	−35.0512	−1.28590	−0.642951	+	0.765907i	$0.722290\pi$
$744$	−32.3153	−1.18474
$745$	−40.6576	−1.48958
$746$	6.12369	0.224204
$747$	−38.2891	−1.40093
$748$	−24.1749	−0.883921
$749$	2.07338	0.0757596
$750$	9.11076	0.332678
$751$	12.9887	0.473965	0.236983	−	0.971514i	$-0.423842\pi$
$751$	12.9887	0.473965	0.236983	+	0.971514i	$0.423842\pi$
$752$	−10.2865	−0.375110
$753$	40.8349	1.48811
$754$	1.77815	0.0647564
$755$	8.60018	0.312992
$756$	3.91302	0.142315
$757$	42.2051	1.53397	0.766985	−	0.641665i	$-0.221756\pi$
$757$	42.2051	1.53397	0.766985	+	0.641665i	$0.221756\pi$
$758$	4.30176	0.156247
$759$	106.039	3.84898
$760$	6.68299	0.242417
$761$	16.8709	0.611568	0.305784	−	0.952101i	$-0.401081\pi$
$761$	16.8709	0.611568	0.305784	+	0.952101i	$0.401081\pi$
$762$	50.1312	1.81606
$763$	−0.478345	−0.0173172
$764$	1.87883	0.0679736
$765$	121.774	4.40275
$766$	−30.1561	−1.08958
$767$	0.704163	0.0254259
$768$	−3.13760	−0.113218
$769$	−22.6543	−0.816933	−0.408467	−	0.912773i	$-0.633936\pi$
$769$	−22.6543	−0.816933	−0.408467	+	0.912773i	$0.633936\pi$
$770$	3.98199	0.143501
$771$	−84.9385	−3.05899
$772$	3.16511	0.113915
$773$	−28.6726	−1.03128	−0.515642	−	0.856804i	$-0.672446\pi$
$773$	−28.6726	−1.03128	−0.515642	+	0.856804i	$0.672446\pi$
$774$	26.0307	0.935655
$775$	41.5468	1.49240
$776$	−2.53547	−0.0910182
$777$	3.75342	0.134653
$778$	16.5905	0.594798
$779$	12.6876	0.454581
$780$	2.86815	0.102696
$781$	−30.3619	−1.08643
$782$	48.9837	1.75165
$783$	70.5250	2.52036
$784$	−6.89477	−0.246242
$785$	15.8675	0.566335
$786$	−1.69776	−0.0605571
$787$	−53.5435	−1.90862	−0.954310	−	0.298820i	$-0.903407\pi$
$787$	−53.5435	−1.90862	−0.954310	+	0.298820i	$0.903407\pi$
$788$	14.9516	0.532630
$789$	45.2332	1.61034
$790$	−3.30260	−0.117501
$791$	−4.80193	−0.170737
$792$	−27.9534	−0.993280
$793$	2.45159	0.0870586
$794$	−6.67285	−0.236811
$795$	−60.3049	−2.13879
$796$	−5.49517	−0.194771
$797$	−51.8420	−1.83634	−0.918169	−	0.396188i	$-0.870333\pi$
$797$	−51.8420	−1.83634	−0.918169	+	0.396188i	$0.870333\pi$
$798$	2.26309	0.0801124
$799$	−60.8893	−2.15411
$800$	4.03391	0.142620
$801$	−102.427	−3.61909
$802$	27.7436	0.979661
$803$	67.7682	2.39149
$804$	−24.8900	−0.877804
$805$	−8.06840	−0.284374
$806$	−3.13240	−0.110334
$807$	3.13760	0.110449
$808$	−13.1899	−0.464019
$809$	29.8709	1.05021	0.525103	−	0.851039i	$-0.324027\pi$
$809$	29.8709	1.05021	0.525103	+	0.851039i	$0.324027\pi$
$810$	52.0397	1.82849
$811$	26.0612	0.915132	0.457566	−	0.889176i	$-0.348721\pi$
$811$	26.0612	0.915132	0.457566	+	0.889176i	$0.348721\pi$
$812$	1.89659	0.0665573
$813$	30.9520	1.08554
$814$	−15.0608	−0.527882
$815$	31.0391	1.08725
$816$	−18.5725	−0.650168
$817$	8.45621	0.295845
$818$	30.6637	1.07213
$819$	0.675275	0.0235960
$820$	17.1508	0.598933
$821$	3.94591	0.137713	0.0688567	−	0.997627i	$-0.478065\pi$
$821$	3.94591	0.137713	0.0688567	+	0.997627i	$0.478065\pi$
$822$	−40.9506	−1.42832
$823$	18.6762	0.651010	0.325505	−	0.945540i	$-0.394466\pi$
$823$	18.6762	0.651010	0.325505	+	0.945540i	$0.394466\pi$
$824$	6.16809	0.214876
$825$	51.6909	1.79965
$826$	0.751066	0.0261329
$827$	13.4024	0.466046	0.233023	−	0.972471i	$-0.425138\pi$
$827$	13.4024	0.466046	0.233023	+	0.972471i	$0.425138\pi$
$828$	56.6398	1.96837
$829$	−14.8573	−0.516015	−0.258007	−	0.966143i	$-0.583066\pi$
$829$	−14.8573	−0.516015	−0.258007	+	0.966143i	$0.583066\pi$
$830$	−16.8140	−0.583621
$831$	85.9793	2.98259
$832$	−0.304135	−0.0105440
$833$	−40.8125	−1.41407
$834$	12.0368	0.416799
$835$	−7.99584	−0.276708
$836$	−9.08079	−0.314066
$837$	−124.237	−4.29427
$838$	7.47427	0.258194
$839$	−28.3441	−0.978548	−0.489274	−	0.872130i	$-0.662738\pi$
$839$	−28.3441	−0.978548	−0.489274	+	0.872130i	$0.662738\pi$
$840$	3.05919	0.105552
$841$	5.18258	0.178710
$842$	−36.8824	−1.27105
$843$	−30.5062	−1.05069
$844$	−3.88750	−0.133813
$845$	−38.7954	−1.33460
$846$	−70.4062	−2.42061
$847$	−1.84237	−0.0633046
$848$	6.39465	0.219593
$849$	54.9351	1.88537
$850$	23.8781	0.819011
$851$	30.5167	1.04610
$852$	−23.3257	−0.799126
$853$	−16.4185	−0.562159	−0.281080	−	0.959684i	$-0.590693\pi$
$853$	−16.4185	−0.562159	−0.281080	+	0.959684i	$0.590693\pi$
$854$	2.61489	0.0894796
$855$	45.7419	1.56434
$856$	−6.39157	−0.218459
$857$	32.4378	1.10806	0.554028	−	0.832498i	$-0.313090\pi$
$857$	32.4378	1.10806	0.554028	+	0.832498i	$0.313090\pi$
$858$	−3.89722	−0.133049
$859$	−50.8594	−1.73530	−0.867651	−	0.497174i	$-0.834371\pi$
$859$	−50.8594	−1.73530	−0.867651	+	0.497174i	$0.834371\pi$
$860$	11.4309	0.389791
$861$	5.80786	0.197931
$862$	37.8309	1.28853
$863$	6.34534	0.215998	0.107999	−	0.994151i	$-0.465556\pi$
$863$	6.34534	0.215998	0.107999	+	0.994151i	$0.465556\pi$
$864$	−12.0626	−0.410378
$865$	58.8264	2.00016
$866$	9.88884	0.336036
$867$	−56.5979	−1.92217
$868$	−3.34105	−0.113403
$869$	4.48755	0.152230
$870$	55.1363	1.86930
$871$	−2.41265	−0.0817496
$872$	1.47459	0.0499358
$873$	−17.3541	−0.587348
$874$	18.3997	0.622380
$875$	0.941951	0.0318438
$876$	52.0634	1.75906
$877$	12.6512	0.427199	0.213600	−	0.976921i	$-0.431481\pi$
$877$	12.6512	0.427199	0.213600	+	0.976921i	$0.431481\pi$
$878$	16.8597	0.568987
$879$	−18.2591	−0.615864
$880$	−12.2752	−0.413797
$881$	17.4300	0.587233	0.293616	−	0.955923i	$-0.405141\pi$
$881$	17.4300	0.587233	0.293616	+	0.955923i	$0.405141\pi$
$882$	−47.1914	−1.58902
$883$	29.6353	0.997307	0.498654	−	0.866801i	$-0.333828\pi$
$883$	29.6353	0.997307	0.498654	+	0.866801i	$0.333828\pi$
$884$	−1.80028	−0.0605499
$885$	21.8345	0.733957
$886$	−12.7251	−0.427507
$887$	−53.5405	−1.79772	−0.898858	−	0.438241i	$-0.855602\pi$
$887$	−53.5405	−1.79772	−0.898858	+	0.438241i	$0.855602\pi$
$888$	−11.5706	−0.388284
$889$	5.18300	0.173832
$890$	−44.9790	−1.50770
$891$	−70.7111	−2.36891
$892$	−14.0272	−0.469667
$893$	−22.8718	−0.765375
$894$	42.4425	1.41949
$895$	36.8829	1.23286
$896$	−0.324393	−0.0108372
$897$	7.89663	0.263661
$898$	27.2026	0.907762
$899$	−60.2162	−2.00832
$900$	27.6102	0.920339
$901$	37.8522	1.26104
$902$	−23.3044	−0.775952
$903$	3.87090	0.128815
$904$	14.8028	0.492334
$905$	−38.8509	−1.29145
$906$	−8.97774	−0.298265
$907$	−22.9439	−0.761841	−0.380921	−	0.924608i	$-0.624393\pi$
$907$	−22.9439	−0.761841	−0.380921	+	0.924608i	$0.624393\pi$
$908$	−11.3821	−0.377730
$909$	−90.2786	−2.99435
$910$	0.296534	0.00983002
$911$	25.6304	0.849173	0.424586	−	0.905387i	$-0.360419\pi$
$911$	25.6304	0.849173	0.424586	+	0.905387i	$0.360419\pi$
$912$	−6.97638	−0.231011
$913$	22.8467	0.756114
$914$	19.5630	0.647088
$915$	76.0181	2.51308
$916$	−20.3763	−0.673251
$917$	−0.175529	−0.00579649
$918$	−71.4026	−2.35664
$919$	37.3824	1.23313	0.616567	−	0.787303i	$-0.288523\pi$
$919$	37.3824	1.23313	0.616567	+	0.787303i	$0.288523\pi$
$920$	24.8723	0.820016
$921$	−87.2981	−2.87657
$922$	30.5373	1.00569
$923$	−2.26102	−0.0744223
$924$	−4.15680	−0.136749
$925$	14.8759	0.489118
$926$	1.65808	0.0544879
$927$	42.2177	1.38661
$928$	−5.84659	−0.191924
$929$	−3.64426	−0.119564	−0.0597822	−	0.998211i	$-0.519041\pi$
$929$	−3.64426	−0.119564	−0.0597822	+	0.998211i	$0.519041\pi$
$930$	−97.1285	−3.18497
$931$	−15.3304	−0.502432
$932$	20.9537	0.686360
$933$	−57.1608	−1.87136
$934$	19.9513	0.652827
$935$	−72.6611	−2.37627
$936$	−2.08166	−0.0680412
$937$	−46.0310	−1.50377	−0.751884	−	0.659296i	$-0.770854\pi$
$937$	−46.0310	−1.50377	−0.751884	+	0.659296i	$0.770854\pi$
$938$	−2.57335	−0.0840230
$939$	35.9568	1.17341
$940$	−30.9175	−1.00842
$941$	−31.8933	−1.03969	−0.519846	−	0.854260i	$-0.674011\pi$
$941$	−31.8933	−1.03969	−0.519846	+	0.854260i	$0.674011\pi$
$942$	−16.5641	−0.539688
$943$	47.2200	1.53769
$944$	−2.31530	−0.0753566
$945$	11.7611	0.382590
$946$	−15.5322	−0.504996
$947$	19.5834	0.636375	0.318188	−	0.948028i	$-0.396926\pi$
$947$	19.5834	0.636375	0.318188	+	0.948028i	$0.396926\pi$
$948$	3.44759	0.111973
$949$	5.04663	0.163821
$950$	8.96930	0.291002
$951$	−43.4741	−1.40974
$952$	−1.92019	−0.0622338
$953$	−51.3328	−1.66283	−0.831416	−	0.555650i	$-0.812470\pi$
$953$	−51.3328	−1.66283	−0.831416	+	0.555650i	$0.812470\pi$
$954$	43.7684	1.41705
$955$	5.64709	0.182735
$956$	−25.4814	−0.824126
$957$	−74.9188	−2.42178
$958$	−1.41839	−0.0458261
$959$	−4.23383	−0.136718
$960$	−9.43051	−0.304368
$961$	75.0773	2.42185
$962$	−1.12157	−0.0361607
$963$	−43.7473	−1.40974
$964$	6.56813	0.211545
$965$	9.51319	0.306240
$966$	8.42262	0.270993
$967$	−29.5177	−0.949227	−0.474613	−	0.880194i	$-0.657412\pi$
$967$	−29.5177	−0.949227	−0.474613	+	0.880194i	$0.657412\pi$
$968$	5.67944	0.182544
$969$	−41.2956	−1.32661
$970$	−7.62074	−0.244687
$971$	12.1128	0.388717	0.194359	−	0.980931i	$-0.437737\pi$
$971$	12.1128	0.388717	0.194359	+	0.980931i	$0.437737\pi$
$972$	−18.1366	−0.581731
$973$	1.24447	0.0398958
$974$	39.0907	1.25255
$975$	3.84937	0.123278
$976$	−8.06087	−0.258022
$977$	−2.25355	−0.0720975	−0.0360487	−	0.999350i	$-0.511477\pi$
$977$	−2.25355	−0.0720975	−0.0360487	+	0.999350i	$0.511477\pi$
$978$	−32.4017	−1.03609
$979$	61.1170	1.95331
$980$	−20.7232	−0.661979
$981$	10.0928	0.322240
$982$	4.88763	0.155970
$983$	−29.7683	−0.949462	−0.474731	−	0.880131i	$-0.657455\pi$
$983$	−29.7683	−0.949462	−0.474731	+	0.880131i	$0.657455\pi$
$984$	−17.9038	−0.570752
$985$	44.9393	1.43189
$986$	−34.6080	−1.10214
$987$	−10.4697	−0.333255
$988$	−0.676237	−0.0215140
$989$	31.4718	1.00074
$990$	−84.0179	−2.67027
$991$	29.6081	0.940532	0.470266	−	0.882525i	$-0.344158\pi$
$991$	29.6081	0.940532	0.470266	+	0.882525i	$0.344158\pi$
$992$	10.2994	0.327006
$993$	108.530	3.44409
$994$	−2.41162	−0.0764919
$995$	−16.5165	−0.523609
$996$	17.5521	0.556160
$997$	41.7985	1.32377	0.661885	−	0.749605i	$-0.269757\pi$
$997$	41.7985	1.32377	0.661885	+	0.749605i	$0.269757\pi$
$998$	−25.2729	−0.800000
$999$	−44.4835	−1.40740

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.2.a.e.1.1	✓	7
3.2	odd	2		4842.2.a.n.1.3		7
4.3	odd	2		4304.2.a.h.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.e.1.1	✓	7	1.1	even	1	trivial
4304.2.a.h.1.7		7	4.3	odd	2
4842.2.a.n.1.3		7	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.13760 q^{3} +1.00000 q^{4} +3.00565 q^{5} -3.13760 q^{6} -0.324393 q^{7} +1.00000 q^{8} +6.84453 q^{9} +3.00565 q^{10} -4.08405 q^{11} -3.13760 q^{12} -0.304135 q^{13} -0.324393 q^{14} -9.43051 q^{15} +1.00000 q^{16} +5.91934 q^{17} +6.84453 q^{18} +2.22348 q^{19} +3.00565 q^{20} +1.01781 q^{21} -4.08405 q^{22} +8.27520 q^{23} -3.13760 q^{24} +4.03391 q^{25} -0.304135 q^{26} -12.0626 q^{27} -0.324393 q^{28} -5.84659 q^{29} -9.43051 q^{30} +10.2994 q^{31} +1.00000 q^{32} +12.8141 q^{33} +5.91934 q^{34} -0.975010 q^{35} +6.84453 q^{36} +3.68773 q^{37} +2.22348 q^{38} +0.954253 q^{39} +3.00565 q^{40} +5.70620 q^{41} +1.01781 q^{42} +3.80314 q^{43} -4.08405 q^{44} +20.5722 q^{45} +8.27520 q^{46} -10.2865 q^{47} -3.13760 q^{48} -6.89477 q^{49} +4.03391 q^{50} -18.5725 q^{51} -0.304135 q^{52} +6.39465 q^{53} -12.0626 q^{54} -12.2752 q^{55} -0.324393 q^{56} -6.97638 q^{57} -5.84659 q^{58} -2.31530 q^{59} -9.43051 q^{60} -8.06087 q^{61} +10.2994 q^{62} -2.22032 q^{63} +1.00000 q^{64} -0.914122 q^{65} +12.8141 q^{66} +7.93283 q^{67} +5.91934 q^{68} -25.9643 q^{69} -0.975010 q^{70} +7.43426 q^{71} +6.84453 q^{72} -16.5934 q^{73} +3.68773 q^{74} -12.6568 q^{75} +2.22348 q^{76} +1.32484 q^{77} +0.954253 q^{78} -1.09880 q^{79} +3.00565 q^{80} +17.3140 q^{81} +5.70620 q^{82} -5.59412 q^{83} +1.01781 q^{84} +17.7914 q^{85} +3.80314 q^{86} +18.3442 q^{87} -4.08405 q^{88} -14.9648 q^{89} +20.5722 q^{90} +0.0986592 q^{91} +8.27520 q^{92} -32.3153 q^{93} -10.2865 q^{94} +6.68299 q^{95} -3.13760 q^{96} -2.53547 q^{97} -6.89477 q^{98} -27.9534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9} + 7 q^{10} - 3 q^{11} + q^{12} - 9 q^{13} + 6 q^{14} + 8 q^{15} + 7 q^{16} + 8 q^{17} + 12 q^{18} - 11 q^{19} + 7 q^{20}+ \cdots - 37 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 + q^3 + 7 * q^4 + 7 * q^5 + q^6 + 6 * q^7 + 7 * q^8 + 12 * q^9 + 7 * q^10 - 3 * q^11 + q^12 - 9 * q^13 + 6 * q^14 + 8 * q^15 + 7 * q^16 + 8 * q^17 + 12 * q^18 - 11 * q^19 + 7 * q^20 - 6 * q^21 - 3 * q^22 + 12 * q^23 + q^24 + 22 * q^25 - 9 * q^26 - 14 * q^27 + 6 * q^28 - 5 * q^29 + 8 * q^30 + 14 * q^31 + 7 * q^32 - 4 * q^33 + 8 * q^34 - 4 * q^35 + 12 * q^36 + 13 * q^37 - 11 * q^38 - 18 * q^39 + 7 * q^40 + 12 * q^41 - 6 * q^42 - 11 * q^43 - 3 * q^44 + 3 * q^45 + 12 * q^46 + 2 * q^47 + q^48 + 15 * q^49 + 22 * q^50 - 26 * q^51 - 9 * q^52 + 19 * q^53 - 14 * q^54 - 40 * q^55 + 6 * q^56 - 12 * q^57 - 5 * q^58 - 9 * q^59 + 8 * q^60 - 3 * q^61 + 14 * q^62 - 26 * q^63 + 7 * q^64 - 10 * q^65 - 4 * q^66 - 33 * q^67 + 8 * q^68 - 64 * q^69 - 4 * q^70 + 28 * q^71 + 12 * q^72 - 14 * q^73 + 13 * q^74 - 45 * q^75 - 11 * q^76 + 10 * q^77 - 18 * q^78 + 2 * q^79 + 7 * q^80 + 15 * q^81 + 12 * q^82 - 7 * q^83 - 6 * q^84 - 16 * q^85 - 11 * q^86 + 16 * q^87 - 3 * q^88 + 18 * q^89 + 3 * q^90 - 26 * q^91 + 12 * q^92 + 6 * q^93 + 2 * q^94 - 34 * q^95 + q^96 - 4 * q^97 + 15 * q^98 - 37 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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