LMFDB - Embedded newform 4641.2.a.ba.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4641,2,Mod(1,4641)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4641, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4641.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4641 = 3 \cdot 7 \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4641.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:	$37.0585715781$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - x^{16} - 29 x^{15} + 26 x^{14} + 339 x^{13} - 266 x^{12} - 2047 x^{11} + 1356 x^{10} + \cdots + 18 $ x^17 - x^16 - 29x^15 + 26x^14 + 339x^13 - 266x^12 - 2047x^11 + 1356x^10 + 6792x^9 - 3573x^8 - 12125x^7 + 4456x^6 + 10521x^5 - 1858x^4 - 3551x^3 + 34x^2 + 323*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$2.35965$ of defining polynomial
Character	$\chi$	$=$	4641.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.35965 q^{2} -1.00000 q^{3} +3.56795 q^{4} -4.35328 q^{5} -2.35965 q^{6} -1.00000 q^{7} +3.69982 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.35965 q^{2} -1.00000 q^{3} +3.56795 q^{4} -4.35328 q^{5} -2.35965 q^{6} -1.00000 q^{7} +3.69982 q^{8} +1.00000 q^{9} -10.2722 q^{10} -5.11664 q^{11} -3.56795 q^{12} +1.00000 q^{13} -2.35965 q^{14} +4.35328 q^{15} +1.59438 q^{16} -1.00000 q^{17} +2.35965 q^{18} +1.30183 q^{19} -15.5323 q^{20} +1.00000 q^{21} -12.0735 q^{22} -0.350825 q^{23} -3.69982 q^{24} +13.9511 q^{25} +2.35965 q^{26} -1.00000 q^{27} -3.56795 q^{28} +3.43190 q^{29} +10.2722 q^{30} +4.39928 q^{31} -3.63747 q^{32} +5.11664 q^{33} -2.35965 q^{34} +4.35328 q^{35} +3.56795 q^{36} +7.72759 q^{37} +3.07187 q^{38} -1.00000 q^{39} -16.1064 q^{40} -9.55566 q^{41} +2.35965 q^{42} +4.42629 q^{43} -18.2559 q^{44} -4.35328 q^{45} -0.827825 q^{46} -1.48330 q^{47} -1.59438 q^{48} +1.00000 q^{49} +32.9196 q^{50} +1.00000 q^{51} +3.56795 q^{52} +6.16066 q^{53} -2.35965 q^{54} +22.2742 q^{55} -3.69982 q^{56} -1.30183 q^{57} +8.09809 q^{58} -6.43215 q^{59} +15.5323 q^{60} -8.63428 q^{61} +10.3808 q^{62} -1.00000 q^{63} -11.7719 q^{64} -4.35328 q^{65} +12.0735 q^{66} +13.1877 q^{67} -3.56795 q^{68} +0.350825 q^{69} +10.2722 q^{70} +4.38996 q^{71} +3.69982 q^{72} +7.04459 q^{73} +18.2344 q^{74} -13.9511 q^{75} +4.64488 q^{76} +5.11664 q^{77} -2.35965 q^{78} +5.42343 q^{79} -6.94077 q^{80} +1.00000 q^{81} -22.5480 q^{82} -4.27184 q^{83} +3.56795 q^{84} +4.35328 q^{85} +10.4445 q^{86} -3.43190 q^{87} -18.9306 q^{88} +15.5066 q^{89} -10.2722 q^{90} -1.00000 q^{91} -1.25173 q^{92} -4.39928 q^{93} -3.50007 q^{94} -5.66725 q^{95} +3.63747 q^{96} -0.530529 q^{97} +2.35965 q^{98} -5.11664 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + q^{2} - 17 q^{3} + 25 q^{4} - 4 q^{5} - q^{6} - 17 q^{7} + 6 q^{8} + 17 q^{9}+O(q^{10}) $ 17 * q + q^2 - 17 * q^3 + 25 * q^4 - 4 * q^5 - q^6 - 17 * q^7 + 6 * q^8 + 17 * q^9	\( 17 q + q^{2} - 17 q^{3} + 25 q^{4} - 4 q^{5} - q^{6} - 17 q^{7} + 6 q^{8} + 17 q^{9} - q^{10} + 4 q^{11} - 25 q^{12} + 17 q^{13} - q^{14} + 4 q^{15} + 53 q^{16} - 17 q^{17} + q^{18} + 13 q^{19} - 5 q^{20} + 17 q^{21} + 4 q^{22} + 8 q^{23} - 6 q^{24} + 37 q^{25} + q^{26} - 17 q^{27} - 25 q^{28} - 3 q^{29} + q^{30} + 10 q^{31} + 18 q^{32} - 4 q^{33} - q^{34} + 4 q^{35} + 25 q^{36} - 25 q^{38} - 17 q^{39} - 12 q^{41} + q^{42} + 29 q^{43} + 12 q^{44} - 4 q^{45} + 19 q^{46} - 4 q^{47} - 53 q^{48} + 17 q^{49} + 9 q^{50} + 17 q^{51} + 25 q^{52} - 8 q^{53} - q^{54} + 27 q^{55} - 6 q^{56} - 13 q^{57} + 2 q^{58} + 25 q^{59} + 5 q^{60} - 5 q^{61} - 37 q^{62} - 17 q^{63} + 94 q^{64} - 4 q^{65} - 4 q^{66} + 15 q^{67} - 25 q^{68} - 8 q^{69} + q^{70} + 32 q^{71} + 6 q^{72} - 15 q^{73} + 16 q^{74} - 37 q^{75} + 3 q^{76} - 4 q^{77} - q^{78} + 27 q^{79} + 41 q^{80} + 17 q^{81} - 8 q^{82} - 24 q^{83} + 25 q^{84} + 4 q^{85} + 53 q^{86} + 3 q^{87} - 9 q^{88} - 8 q^{89} - q^{90} - 17 q^{91} + 14 q^{92} - 10 q^{93} + 51 q^{94} - 9 q^{95} - 18 q^{96} - 22 q^{97} + q^{98} + 4 q^{99}+O(q^{100}) \) 17 * q + q^2 - 17 * q^3 + 25 * q^4 - 4 * q^5 - q^6 - 17 * q^7 + 6 * q^8 + 17 * q^9 - q^10 + 4 * q^11 - 25 * q^12 + 17 * q^13 - q^14 + 4 * q^15 + 53 * q^16 - 17 * q^17 + q^18 + 13 * q^19 - 5 * q^20 + 17 * q^21 + 4 * q^22 + 8 * q^23 - 6 * q^24 + 37 * q^25 + q^26 - 17 * q^27 - 25 * q^28 - 3 * q^29 + q^30 + 10 * q^31 + 18 * q^32 - 4 * q^33 - q^34 + 4 * q^35 + 25 * q^36 - 25 * q^38 - 17 * q^39 - 12 * q^41 + q^42 + 29 * q^43 + 12 * q^44 - 4 * q^45 + 19 * q^46 - 4 * q^47 - 53 * q^48 + 17 * q^49 + 9 * q^50 + 17 * q^51 + 25 * q^52 - 8 * q^53 - q^54 + 27 * q^55 - 6 * q^56 - 13 * q^57 + 2 * q^58 + 25 * q^59 + 5 * q^60 - 5 * q^61 - 37 * q^62 - 17 * q^63 + 94 * q^64 - 4 * q^65 - 4 * q^66 + 15 * q^67 - 25 * q^68 - 8 * q^69 + q^70 + 32 * q^71 + 6 * q^72 - 15 * q^73 + 16 * q^74 - 37 * q^75 + 3 * q^76 - 4 * q^77 - q^78 + 27 * q^79 + 41 * q^80 + 17 * q^81 - 8 * q^82 - 24 * q^83 + 25 * q^84 + 4 * q^85 + 53 * q^86 + 3 * q^87 - 9 * q^88 - 8 * q^89 - q^90 - 17 * q^91 + 14 * q^92 - 10 * q^93 + 51 * q^94 - 9 * q^95 - 18 * q^96 - 22 * q^97 + q^98 + 4 * 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 $

$2$	2.35965	1.66853	0.834263	−	0.551367i	$-0.185894\pi$
$2$	2.35965	1.66853	0.834263	+	0.551367i	$0.185894\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.56795	1.78398
$5$	−4.35328	−1.94685	−0.973424	−	0.229012i	$-0.926450\pi$
$5$	−4.35328	−1.94685	−0.973424	+	0.229012i	$0.926450\pi$
$6$	−2.35965	−0.963323
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.69982	1.30808
$9$	1.00000	0.333333
$10$	−10.2722	−3.24836
$11$	−5.11664	−1.54273	−0.771363	−	0.636396i	$-0.780425\pi$
$11$	−5.11664	−1.54273	−0.771363	+	0.636396i	$0.780425\pi$
$12$	−3.56795	−1.02998
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.35965	−0.630643
$15$	4.35328	1.12401
$16$	1.59438	0.398594
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	2.35965	0.556175
$19$	1.30183	0.298661	0.149331	−	0.988787i	$-0.452288\pi$
$19$	1.30183	0.298661	0.149331	+	0.988787i	$0.452288\pi$
$20$	−15.5323	−3.47313
$21$	1.00000	0.218218
$22$	−12.0735	−2.57408
$23$	−0.350825	−0.0731521	−0.0365761	−	0.999331i	$-0.511645\pi$
$23$	−0.350825	−0.0731521	−0.0365761	+	0.999331i	$0.511645\pi$
$24$	−3.69982	−0.755222
$25$	13.9511	2.79021
$26$	2.35965	0.462766
$27$	−1.00000	−0.192450
$28$	−3.56795	−0.674279
$29$	3.43190	0.637288	0.318644	−	0.947874i	$-0.396773\pi$
$29$	3.43190	0.637288	0.318644	+	0.947874i	$0.396773\pi$
$30$	10.2722	1.87544
$31$	4.39928	0.790134	0.395067	−	0.918652i	$-0.370721\pi$
$31$	4.39928	0.790134	0.395067	+	0.918652i	$0.370721\pi$
$32$	−3.63747	−0.643019
$33$	5.11664	0.890693
$34$	−2.35965	−0.404677
$35$	4.35328	0.735839
$36$	3.56795	0.594659
$37$	7.72759	1.27041	0.635204	−	0.772344i	$-0.280916\pi$
$37$	7.72759	1.27041	0.635204	+	0.772344i	$0.280916\pi$
$38$	3.07187	0.498324
$39$	−1.00000	−0.160128
$40$	−16.1064	−2.54664
$41$	−9.55566	−1.49234	−0.746172	−	0.665754i	$-0.768110\pi$
$41$	−9.55566	−1.49234	−0.746172	+	0.665754i	$0.768110\pi$
$42$	2.35965	0.364102
$43$	4.42629	0.675003	0.337501	−	0.941325i	$-0.390418\pi$
$43$	4.42629	0.675003	0.337501	+	0.941325i	$0.390418\pi$
$44$	−18.2559	−2.75219
$45$	−4.35328	−0.648949
$46$	−0.827825	−0.122056
$47$	−1.48330	−0.216361	−0.108181	−	0.994131i	$-0.534503\pi$
$47$	−1.48330	−0.216361	−0.108181	+	0.994131i	$0.534503\pi$
$48$	−1.59438	−0.230128
$49$	1.00000	0.142857
$50$	32.9196	4.65554
$51$	1.00000	0.140028
$52$	3.56795	0.494786
$53$	6.16066	0.846231	0.423116	−	0.906076i	$-0.360936\pi$
$53$	6.16066	0.846231	0.423116	+	0.906076i	$0.360936\pi$
$54$	−2.35965	−0.321108
$55$	22.2742	3.00345
$56$	−3.69982	−0.494409
$57$	−1.30183	−0.172432
$58$	8.09809	1.06333
$59$	−6.43215	−0.837395	−0.418697	−	0.908126i	$-0.637513\pi$
$59$	−6.43215	−0.837395	−0.418697	+	0.908126i	$0.637513\pi$
$60$	15.5323	2.00521
$61$	−8.63428	−1.10551	−0.552753	−	0.833345i	$-0.686423\pi$
$61$	−8.63428	−1.10551	−0.552753	+	0.833345i	$0.686423\pi$
$62$	10.3808	1.31836
$63$	−1.00000	−0.125988
$64$	−11.7719	−1.47149
$65$	−4.35328	−0.539958
$66$	12.0735	1.48614
$67$	13.1877	1.61114	0.805568	−	0.592504i	$-0.201861\pi$
$67$	13.1877	1.61114	0.805568	+	0.592504i	$0.201861\pi$
$68$	−3.56795	−0.432678
$69$	0.350825	0.0422344
$70$	10.2722	1.22777
$71$	4.38996	0.520992	0.260496	−	0.965475i	$-0.416114\pi$
$71$	4.38996	0.520992	0.260496	+	0.965475i	$0.416114\pi$
$72$	3.69982	0.436028
$73$	7.04459	0.824507	0.412254	−	0.911069i	$-0.364742\pi$
$73$	7.04459	0.824507	0.412254	+	0.911069i	$0.364742\pi$
$74$	18.2344	2.11971
$75$	−13.9511	−1.61093
$76$	4.64488	0.532804
$77$	5.11664	0.583095
$78$	−2.35965	−0.267178
$79$	5.42343	0.610183	0.305092	−	0.952323i	$-0.401313\pi$
$79$	5.42343	0.610183	0.305092	+	0.952323i	$0.401313\pi$
$80$	−6.94077	−0.776001
$81$	1.00000	0.111111
$82$	−22.5480	−2.49001
$83$	−4.27184	−0.468895	−0.234448	−	0.972129i	$-0.575328\pi$
$83$	−4.27184	−0.468895	−0.234448	+	0.972129i	$0.575328\pi$
$84$	3.56795	0.389295
$85$	4.35328	0.472180
$86$	10.4445	1.12626
$87$	−3.43190	−0.367938
$88$	−18.9306	−2.01801
$89$	15.5066	1.64370	0.821850	−	0.569704i	$-0.192942\pi$
$89$	15.5066	1.64370	0.821850	+	0.569704i	$0.192942\pi$
$90$	−10.2722	−1.08279
$91$	−1.00000	−0.104828
$92$	−1.25173	−0.130502
$93$	−4.39928	−0.456184
$94$	−3.50007	−0.361005
$95$	−5.66725	−0.581448
$96$	3.63747	0.371247
$97$	−0.530529	−0.0538670	−0.0269335	−	0.999637i	$-0.508574\pi$
$97$	−0.530529	−0.0538670	−0.0269335	+	0.999637i	$0.508574\pi$
$98$	2.35965	0.238361
$99$	−5.11664	−0.514242
$100$	49.7767	4.97767
$101$	−5.28858	−0.526233	−0.263117	−	0.964764i	$-0.584750\pi$
$101$	−5.28858	−0.526233	−0.263117	+	0.964764i	$0.584750\pi$
$102$	2.35965	0.233640
$103$	17.4267	1.71710	0.858551	−	0.512728i	$-0.171365\pi$
$103$	17.4267	1.71710	0.858551	+	0.512728i	$0.171365\pi$
$104$	3.69982	0.362797
$105$	−4.35328	−0.424837
$106$	14.5370	1.41196
$107$	11.1630	1.07916	0.539582	−	0.841933i	$-0.318582\pi$
$107$	11.1630	1.07916	0.539582	+	0.841933i	$0.318582\pi$
$108$	−3.56795	−0.343326
$109$	3.38240	0.323975	0.161987	−	0.986793i	$-0.448210\pi$
$109$	3.38240	0.323975	0.161987	+	0.986793i	$0.448210\pi$
$110$	52.5593	5.01133
$111$	−7.72759	−0.733471
$112$	−1.59438	−0.150654
$113$	18.4283	1.73359	0.866796	−	0.498663i	$-0.166175\pi$
$113$	18.4283	1.73359	0.866796	+	0.498663i	$0.166175\pi$
$114$	−3.07187	−0.287707
$115$	1.52724	0.142416
$116$	12.2449	1.13691
$117$	1.00000	0.0924500
$118$	−15.1776	−1.39721
$119$	1.00000	0.0916698
$120$	16.1064	1.47030
$121$	15.1800	1.38000
$122$	−20.3739	−1.84456
$123$	9.55566	0.861605
$124$	15.6964	1.40958
$125$	−38.9665	−3.48527
$126$	−2.35965	−0.210214
$127$	−0.491581	−0.0436207	−0.0218104	−	0.999762i	$-0.506943\pi$
$127$	−0.491581	−0.0436207	−0.0218104	+	0.999762i	$0.506943\pi$
$128$	−20.5026	−1.81219
$129$	−4.42629	−0.389713
$130$	−10.2722	−0.900934
$131$	12.3515	1.07915	0.539577	−	0.841936i	$-0.318584\pi$
$131$	12.3515	1.07915	0.539577	+	0.841936i	$0.318584\pi$
$132$	18.2559	1.58897
$133$	−1.30183	−0.112883
$134$	31.1184	2.68822
$135$	4.35328	0.374671
$136$	−3.69982	−0.317257
$137$	−22.0768	−1.88615	−0.943073	−	0.332586i	$-0.892079\pi$
$137$	−22.0768	−1.88615	−0.943073	+	0.332586i	$0.892079\pi$
$138$	0.827825	0.0704692
$139$	−7.81561	−0.662911	−0.331456	−	0.943471i	$-0.607540\pi$
$139$	−7.81561	−0.662911	−0.331456	+	0.943471i	$0.607540\pi$
$140$	15.5323	1.31272
$141$	1.48330	0.124916
$142$	10.3588	0.869288
$143$	−5.11664	−0.427875
$144$	1.59438	0.132865
$145$	−14.9400	−1.24070
$146$	16.6228	1.37571
$147$	−1.00000	−0.0824786
$148$	27.5717	2.26638
$149$	0.0473596	0.00387985	0.00193992	−	0.999998i	$-0.499383\pi$
$149$	0.0473596	0.00387985	0.00193992	+	0.999998i	$0.499383\pi$
$150$	−32.9196	−2.68788
$151$	−18.1700	−1.47866	−0.739328	−	0.673346i	$-0.764857\pi$
$151$	−18.1700	−1.47866	−0.739328	+	0.673346i	$0.764857\pi$
$152$	4.81655	0.390674
$153$	−1.00000	−0.0808452
$154$	12.0735	0.972909
$155$	−19.1513	−1.53827
$156$	−3.56795	−0.285665
$157$	−20.3914	−1.62741	−0.813707	−	0.581275i	$-0.802554\pi$
$157$	−20.3914	−1.62741	−0.813707	+	0.581275i	$0.802554\pi$
$158$	12.7974	1.01811
$159$	−6.16066	−0.488572
$160$	15.8349	1.25186
$161$	0.350825	0.0276489
$162$	2.35965	0.185392
$163$	−12.0662	−0.945099	−0.472549	−	0.881304i	$-0.656666\pi$
$163$	−12.0662	−0.945099	−0.472549	+	0.881304i	$0.656666\pi$
$164$	−34.0941	−2.66230
$165$	−22.2742	−1.73404
$166$	−10.0800	−0.782364
$167$	23.0795	1.78595	0.892973	−	0.450111i	$-0.148615\pi$
$167$	23.0795	1.78595	0.892973	+	0.450111i	$0.148615\pi$
$168$	3.69982	0.285447
$169$	1.00000	0.0769231
$170$	10.2722	0.787844
$171$	1.30183	0.0995537
$172$	15.7928	1.20419
$173$	9.86511	0.750030	0.375015	−	0.927019i	$-0.377638\pi$
$173$	9.86511	0.750030	0.375015	+	0.927019i	$0.377638\pi$
$174$	−8.09809	−0.613914
$175$	−13.9511	−1.05460
$176$	−8.15785	−0.614921
$177$	6.43215	0.483470
$178$	36.5903	2.74256
$179$	−5.68671	−0.425045	−0.212522	−	0.977156i	$-0.568168\pi$
$179$	−5.68671	−0.425045	−0.212522	+	0.977156i	$0.568168\pi$
$180$	−15.5323	−1.15771
$181$	−8.81847	−0.655472	−0.327736	−	0.944769i	$-0.606286\pi$
$181$	−8.81847	−0.655472	−0.327736	+	0.944769i	$0.606286\pi$
$182$	−2.35965	−0.174909
$183$	8.63428	0.638264
$184$	−1.29799	−0.0956891
$185$	−33.6404	−2.47329
$186$	−10.3808	−0.761155
$187$	5.11664	0.374166
$188$	−5.29234	−0.385984
$189$	1.00000	0.0727393
$190$	−13.3727	−0.970160
$191$	13.9963	1.01273	0.506367	−	0.862318i	$-0.330988\pi$
$191$	13.9963	1.01273	0.506367	+	0.862318i	$0.330988\pi$
$192$	11.7719	0.849564
$193$	0.757935	0.0545574	0.0272787	−	0.999628i	$-0.491316\pi$
$193$	0.757935	0.0545574	0.0272787	+	0.999628i	$0.491316\pi$
$194$	−1.25186	−0.0898785
$195$	4.35328	0.311745
$196$	3.56795	0.254854
$197$	−10.4251	−0.742755	−0.371378	−	0.928482i	$-0.621114\pi$
$197$	−10.4251	−0.742755	−0.371378	+	0.928482i	$0.621114\pi$
$198$	−12.0735	−0.858025
$199$	−6.99330	−0.495742	−0.247871	−	0.968793i	$-0.579731\pi$
$199$	−6.99330	−0.495742	−0.247871	+	0.968793i	$0.579731\pi$
$200$	51.6164	3.64983
$201$	−13.1877	−0.930189
$202$	−12.4792	−0.878033
$203$	−3.43190	−0.240872
$204$	3.56795	0.249807
$205$	41.5985	2.90536
$206$	41.1209	2.86503
$207$	−0.350825	−0.0243840
$208$	1.59438	0.110550
$209$	−6.66102	−0.460752
$210$	−10.2722	−0.708851
$211$	−4.67740	−0.322005	−0.161003	−	0.986954i	$-0.551473\pi$
$211$	−4.67740	−0.322005	−0.161003	+	0.986954i	$0.551473\pi$
$212$	21.9809	1.50966
$213$	−4.38996	−0.300795
$214$	26.3407	1.80061
$215$	−19.2689	−1.31413
$216$	−3.69982	−0.251741
$217$	−4.39928	−0.298643
$218$	7.98127	0.540560
$219$	−7.04459	−0.476030
$220$	79.4732	5.35808
$221$	−1.00000	−0.0672673
$222$	−18.2344	−1.22381
$223$	14.4145	0.965267	0.482634	−	0.875822i	$-0.339680\pi$
$223$	14.4145	0.965267	0.482634	+	0.875822i	$0.339680\pi$
$224$	3.63747	0.243038
$225$	13.9511	0.930071
$226$	43.4844	2.89254
$227$	10.5575	0.700725	0.350363	−	0.936614i	$-0.386058\pi$
$227$	10.5575	0.700725	0.350363	+	0.936614i	$0.386058\pi$
$228$	−4.64488	−0.307615
$229$	−13.8061	−0.912332	−0.456166	−	0.889895i	$-0.650778\pi$
$229$	−13.8061	−0.912332	−0.456166	+	0.889895i	$0.650778\pi$
$230$	3.60376	0.237625
$231$	−5.11664	−0.336650
$232$	12.6974	0.833626
$233$	25.7253	1.68532	0.842659	−	0.538448i	$-0.180989\pi$
$233$	25.7253	1.68532	0.842659	+	0.538448i	$0.180989\pi$
$234$	2.35965	0.154255
$235$	6.45722	0.421223
$236$	−22.9496	−1.49389
$237$	−5.42343	−0.352290
$238$	2.35965	0.152953
$239$	−23.2127	−1.50150	−0.750752	−	0.660584i	$-0.770309\pi$
$239$	−23.2127	−1.50150	−0.750752	+	0.660584i	$0.770309\pi$
$240$	6.94077	0.448025
$241$	−0.174539	−0.0112430	−0.00562152	−	0.999984i	$-0.501789\pi$
$241$	−0.174539	−0.0112430	−0.00562152	+	0.999984i	$0.501789\pi$
$242$	35.8196	2.30257
$243$	−1.00000	−0.0641500
$244$	−30.8067	−1.97220
$245$	−4.35328	−0.278121
$246$	22.5480	1.43761
$247$	1.30183	0.0828337
$248$	16.2765	1.03356
$249$	4.27184	0.270717
$250$	−91.9474	−5.81526
$251$	9.85856	0.622267	0.311133	−	0.950366i	$-0.399291\pi$
$251$	9.85856	0.622267	0.311133	+	0.950366i	$0.399291\pi$
$252$	−3.56795	−0.224760
$253$	1.79505	0.112854
$254$	−1.15996	−0.0727823
$255$	−4.35328	−0.272613
$256$	−24.8353	−1.55220
$257$	−6.41871	−0.400388	−0.200194	−	0.979756i	$-0.564157\pi$
$257$	−6.41871	−0.400388	−0.200194	+	0.979756i	$0.564157\pi$
$258$	−10.4445	−0.650246
$259$	−7.72759	−0.480169
$260$	−15.5323	−0.963272
$261$	3.43190	0.212429
$262$	29.1452	1.80060
$263$	−5.81291	−0.358439	−0.179220	−	0.983809i	$-0.557357\pi$
$263$	−5.81291	−0.358439	−0.179220	+	0.983809i	$0.557357\pi$
$264$	18.9306	1.16510
$265$	−26.8191	−1.64748
$266$	−3.07187	−0.188349
$267$	−15.5066	−0.948991
$268$	47.0531	2.87423
$269$	1.40533	0.0856847	0.0428424	−	0.999082i	$-0.486359\pi$
$269$	1.40533	0.0856847	0.0428424	+	0.999082i	$0.486359\pi$
$270$	10.2722	0.625148
$271$	21.0432	1.27829	0.639143	−	0.769088i	$-0.279289\pi$
$271$	21.0432	1.27829	0.639143	+	0.769088i	$0.279289\pi$
$272$	−1.59438	−0.0966732
$273$	1.00000	0.0605228
$274$	−52.0935	−3.14708
$275$	−71.3826	−4.30453
$276$	1.25173	0.0753452
$277$	−19.1718	−1.15192	−0.575960	−	0.817478i	$-0.695372\pi$
$277$	−19.1718	−1.15192	−0.575960	+	0.817478i	$0.695372\pi$
$278$	−18.4421	−1.10608
$279$	4.39928	0.263378
$280$	16.1064	0.962539
$281$	−3.38732	−0.202071	−0.101035	−	0.994883i	$-0.532216\pi$
$281$	−3.38732	−0.202071	−0.101035	+	0.994883i	$0.532216\pi$
$282$	3.50007	0.208426
$283$	19.2888	1.14660	0.573300	−	0.819346i	$-0.305663\pi$
$283$	19.2888	1.14660	0.573300	+	0.819346i	$0.305663\pi$
$284$	15.6632	0.929437
$285$	5.66725	0.335699
$286$	−12.0735	−0.713920
$287$	9.55566	0.564053
$288$	−3.63747	−0.214340
$289$	1.00000	0.0588235
$290$	−35.2533	−2.07014
$291$	0.530529	0.0311001
$292$	25.1348	1.47090
$293$	1.59215	0.0930143	0.0465071	−	0.998918i	$-0.485191\pi$
$293$	1.59215	0.0930143	0.0465071	+	0.998918i	$0.485191\pi$
$294$	−2.35965	−0.137618
$295$	28.0010	1.63028
$296$	28.5907	1.66180
$297$	5.11664	0.296898
$298$	0.111752	0.00647363
$299$	−0.350825	−0.0202888
$300$	−49.7767	−2.87386
$301$	−4.42629	−0.255127
$302$	−42.8749	−2.46717
$303$	5.28858	0.303821
$304$	2.07561	0.119045
$305$	37.5875	2.15225
$306$	−2.35965	−0.134892
$307$	−1.40067	−0.0799403	−0.0399701	−	0.999201i	$-0.512726\pi$
$307$	−1.40067	−0.0799403	−0.0399701	+	0.999201i	$0.512726\pi$
$308$	18.2559	1.04023
$309$	−17.4267	−0.991370
$310$	−45.1904	−2.56664
$311$	15.0265	0.852077	0.426039	−	0.904705i	$-0.359909\pi$
$311$	15.0265	0.852077	0.426039	+	0.904705i	$0.359909\pi$
$312$	−3.69982	−0.209461
$313$	9.59980	0.542613	0.271306	−	0.962493i	$-0.412544\pi$
$313$	9.59980	0.542613	0.271306	+	0.962493i	$0.412544\pi$
$314$	−48.1167	−2.71538
$315$	4.35328	0.245280
$316$	19.3505	1.08855
$317$	−12.4445	−0.698954	−0.349477	−	0.936945i	$-0.613641\pi$
$317$	−12.4445	−0.698954	−0.349477	+	0.936945i	$0.613641\pi$
$318$	−14.5370	−0.815194
$319$	−17.5598	−0.983161
$320$	51.2464	2.86476
$321$	−11.1630	−0.623055
$322$	0.827825	0.0461329
$323$	−1.30183	−0.0724360
$324$	3.56795	0.198220
$325$	13.9511	0.773866
$326$	−28.4720	−1.57692
$327$	−3.38240	−0.187047
$328$	−35.3542	−1.95211
$329$	1.48330	0.0817769
$330$	−52.5593	−2.89329
$331$	1.55371	0.0853995	0.0426998	−	0.999088i	$-0.486404\pi$
$331$	1.55371	0.0853995	0.0426998	+	0.999088i	$0.486404\pi$
$332$	−15.2417	−0.836498
$333$	7.72759	0.423470
$334$	54.4596	2.97989
$335$	−57.4098	−3.13663
$336$	1.59438	0.0869803
$337$	13.5523	0.738240	0.369120	−	0.929382i	$-0.379659\pi$
$337$	13.5523	0.738240	0.369120	+	0.929382i	$0.379659\pi$
$338$	2.35965	0.128348
$339$	−18.4283	−1.00089
$340$	15.5323	0.842357
$341$	−22.5096	−1.21896
$342$	3.07187	0.166108
$343$	−1.00000	−0.0539949
$344$	16.3765	0.882959
$345$	−1.52724	−0.0822239
$346$	23.2782	1.25144
$347$	20.9720	1.12584	0.562919	−	0.826512i	$-0.309678\pi$
$347$	20.9720	1.12584	0.562919	+	0.826512i	$0.309678\pi$
$348$	−12.2449	−0.656393
$349$	16.5294	0.884799	0.442399	−	0.896818i	$-0.354127\pi$
$349$	16.5294	0.884799	0.442399	+	0.896818i	$0.354127\pi$
$350$	−32.9196	−1.75963
$351$	−1.00000	−0.0533761
$352$	18.6116	0.992003
$353$	−29.3791	−1.56369	−0.781847	−	0.623471i	$-0.785722\pi$
$353$	−29.3791	−1.56369	−0.781847	+	0.623471i	$0.785722\pi$
$354$	15.1776	0.806682
$355$	−19.1107	−1.01429
$356$	55.3269	2.93232
$357$	−1.00000	−0.0529256
$358$	−13.4186	−0.709197
$359$	33.0162	1.74253	0.871264	−	0.490815i	$-0.163301\pi$
$359$	33.0162	1.74253	0.871264	+	0.490815i	$0.163301\pi$
$360$	−16.1064	−0.848879
$361$	−17.3052	−0.910802
$362$	−20.8085	−1.09367
$363$	−15.1800	−0.796745
$364$	−3.56795	−0.187011
$365$	−30.6671	−1.60519
$366$	20.3739	1.06496
$367$	8.79250	0.458965	0.229482	−	0.973313i	$-0.426297\pi$
$367$	8.79250	0.458965	0.229482	+	0.973313i	$0.426297\pi$
$368$	−0.559347	−0.0291580
$369$	−9.55566	−0.497448
$370$	−79.3796	−4.12675
$371$	−6.16066	−0.319845
$372$	−15.6964	−0.813822
$373$	21.9401	1.13602	0.568009	−	0.823022i	$-0.307714\pi$
$373$	21.9401	1.13602	0.568009	+	0.823022i	$0.307714\pi$
$374$	12.0735	0.624305
$375$	38.9665	2.01222
$376$	−5.48794	−0.283019
$377$	3.43190	0.176752
$378$	2.35965	0.121367
$379$	18.6467	0.957816	0.478908	−	0.877865i	$-0.341033\pi$
$379$	18.6467	0.957816	0.478908	+	0.877865i	$0.341033\pi$
$380$	−20.2205	−1.03729
$381$	0.491581	0.0251844
$382$	33.0263	1.68977
$383$	11.9881	0.612563	0.306281	−	0.951941i	$-0.400915\pi$
$383$	11.9881	0.612563	0.306281	+	0.951941i	$0.400915\pi$
$384$	20.5026	1.04627
$385$	−22.2742	−1.13520
$386$	1.78846	0.0910303
$387$	4.42629	0.225001
$388$	−1.89290	−0.0960975
$389$	−32.8980	−1.66799	−0.833997	−	0.551769i	$-0.813953\pi$
$389$	−32.8980	−1.66799	−0.833997	+	0.551769i	$0.813953\pi$
$390$	10.2722	0.520154
$391$	0.350825	0.0177420
$392$	3.69982	0.186869
$393$	−12.3515	−0.623050
$394$	−24.5995	−1.23931
$395$	−23.6097	−1.18793
$396$	−18.2559	−0.917395
$397$	5.20258	0.261110	0.130555	−	0.991441i	$-0.458324\pi$
$397$	5.20258	0.261110	0.130555	+	0.991441i	$0.458324\pi$
$398$	−16.5017	−0.827157
$399$	1.30183	0.0651732
$400$	22.2432	1.11216
$401$	−23.3033	−1.16371	−0.581856	−	0.813292i	$-0.697673\pi$
$401$	−23.3033	−1.16371	−0.581856	+	0.813292i	$0.697673\pi$
$402$	−31.1184	−1.55204
$403$	4.39928	0.219144
$404$	−18.8694	−0.938787
$405$	−4.35328	−0.216316
$406$	−8.09809	−0.401901
$407$	−39.5393	−1.95989
$408$	3.69982	0.183168
$409$	15.4271	0.762821	0.381411	−	0.924406i	$-0.375438\pi$
$409$	15.4271	0.762821	0.381411	+	0.924406i	$0.375438\pi$
$410$	98.1579	4.84767
$411$	22.0768	1.08897
$412$	62.1776	3.06327
$413$	6.43215	0.316505
$414$	−0.827825	−0.0406854
$415$	18.5965	0.912867
$416$	−3.63747	−0.178341
$417$	7.81561	0.382732
$418$	−15.7177	−0.768777
$419$	0.902582	0.0440940	0.0220470	−	0.999757i	$-0.492982\pi$
$419$	0.902582	0.0440940	0.0220470	+	0.999757i	$0.492982\pi$
$420$	−15.5323	−0.757899
$421$	25.0418	1.22046	0.610230	−	0.792224i	$-0.291077\pi$
$421$	25.0418	1.22046	0.610230	+	0.792224i	$0.291077\pi$
$422$	−11.0370	−0.537274
$423$	−1.48330	−0.0721205
$424$	22.7933	1.10694
$425$	−13.9511	−0.676726
$426$	−10.3588	−0.501884
$427$	8.63428	0.417842
$428$	39.8289	1.92520
$429$	5.11664	0.247034
$430$	−45.4678	−2.19265
$431$	30.3620	1.46249	0.731243	−	0.682117i	$-0.238940\pi$
$431$	30.3620	1.46249	0.731243	+	0.682117i	$0.238940\pi$
$432$	−1.59438	−0.0767094
$433$	22.9457	1.10270	0.551350	−	0.834274i	$-0.314113\pi$
$433$	22.9457	1.10270	0.551350	+	0.834274i	$0.314113\pi$
$434$	−10.3808	−0.498293
$435$	14.9400	0.716320
$436$	12.0682	0.577963
$437$	−0.456716	−0.0218477
$438$	−16.6228	−0.794267
$439$	−28.5793	−1.36401	−0.682007	−	0.731345i	$-0.738893\pi$
$439$	−28.5793	−1.36401	−0.682007	+	0.731345i	$0.738893\pi$
$440$	82.4104	3.92876
$441$	1.00000	0.0476190
$442$	−2.35965	−0.112237
$443$	19.5348	0.928126	0.464063	−	0.885802i	$-0.346391\pi$
$443$	19.5348	0.928126	0.464063	+	0.885802i	$0.346391\pi$
$444$	−27.5717	−1.30849
$445$	−67.5048	−3.20003
$446$	34.0132	1.61057
$447$	−0.0473596	−0.00224003
$448$	11.7719	0.556170
$449$	28.3149	1.33626	0.668132	−	0.744043i	$-0.267094\pi$
$449$	28.3149	1.33626	0.668132	+	0.744043i	$0.267094\pi$
$450$	32.9196	1.55185
$451$	48.8929	2.30228
$452$	65.7514	3.09269
$453$	18.1700	0.853702
$454$	24.9120	1.16918
$455$	4.35328	0.204085
$456$	−4.81655	−0.225556
$457$	23.8617	1.11620	0.558101	−	0.829773i	$-0.311530\pi$
$457$	23.8617	1.11620	0.558101	+	0.829773i	$0.311530\pi$
$458$	−32.5776	−1.52225
$459$	1.00000	0.0466760
$460$	5.44913	0.254067
$461$	18.6224	0.867331	0.433665	−	0.901074i	$-0.357220\pi$
$461$	18.6224	0.867331	0.433665	+	0.901074i	$0.357220\pi$
$462$	−12.0735	−0.561710
$463$	−1.96658	−0.0913945	−0.0456973	−	0.998955i	$-0.514551\pi$
$463$	−1.96658	−0.0913945	−0.0456973	+	0.998955i	$0.514551\pi$
$464$	5.47174	0.254019
$465$	19.1513	0.888121
$466$	60.7026	2.81199
$467$	−8.50987	−0.393790	−0.196895	−	0.980425i	$-0.563086\pi$
$467$	−8.50987	−0.393790	−0.196895	+	0.980425i	$0.563086\pi$
$468$	3.56795	0.164929
$469$	−13.1877	−0.608952
$470$	15.2368	0.702821
$471$	20.3914	0.939588
$472$	−23.7978	−1.09538
$473$	−22.6477	−1.04134
$474$	−12.7974	−0.587804
$475$	18.1620	0.833328
$476$	3.56795	0.163537
$477$	6.16066	0.282077
$478$	−54.7738	−2.50530
$479$	13.5020	0.616923	0.308462	−	0.951237i	$-0.400186\pi$
$479$	13.5020	0.616923	0.308462	+	0.951237i	$0.400186\pi$
$480$	−15.8349	−0.722762
$481$	7.72759	0.352348
$482$	−0.411851	−0.0187593
$483$	−0.350825	−0.0159631
$484$	54.1616	2.46189
$485$	2.30954	0.104871
$486$	−2.35965	−0.107036
$487$	−17.4244	−0.789577	−0.394789	−	0.918772i	$-0.629182\pi$
$487$	−17.4244	−0.789577	−0.394789	+	0.918772i	$0.629182\pi$
$488$	−31.9453	−1.44609
$489$	12.0662	0.545653
$490$	−10.2722	−0.464052
$491$	39.3023	1.77369	0.886844	−	0.462069i	$-0.152893\pi$
$491$	39.3023	1.77369	0.886844	+	0.462069i	$0.152893\pi$
$492$	34.0941	1.53708
$493$	−3.43190	−0.154565
$494$	3.07187	0.138210
$495$	22.2742	1.00115
$496$	7.01411	0.314943
$497$	−4.38996	−0.196916
$498$	10.0800	0.451698
$499$	−34.1542	−1.52895	−0.764475	−	0.644654i	$-0.777002\pi$
$499$	−34.1542	−1.52895	−0.764475	+	0.644654i	$0.777002\pi$
$500$	−139.031	−6.21764
$501$	−23.0795	−1.03112
$502$	23.2628	1.03827
$503$	−15.8657	−0.707415	−0.353707	−	0.935356i	$-0.615079\pi$
$503$	−15.8657	−0.707415	−0.353707	+	0.935356i	$0.615079\pi$
$504$	−3.69982	−0.164803
$505$	23.0227	1.02450
$506$	4.23569	0.188299
$507$	−1.00000	−0.0444116
$508$	−1.75394	−0.0778183
$509$	16.4974	0.731237	0.365618	−	0.930765i	$-0.380858\pi$
$509$	16.4974	0.731237	0.365618	+	0.930765i	$0.380858\pi$
$510$	−10.2722	−0.454862
$511$	−7.04459	−0.311634
$512$	−17.5973	−0.777698
$513$	−1.30183	−0.0574774
$514$	−15.1459	−0.668058
$515$	−75.8633	−3.34294
$516$	−15.7928	−0.695238
$517$	7.58951	0.333786
$518$	−18.2344	−0.801175
$519$	−9.86511	−0.433030
$520$	−16.1064	−0.706310
$521$	30.5696	1.33928	0.669639	−	0.742686i	$-0.266449\pi$
$521$	30.5696	1.33928	0.669639	+	0.742686i	$0.266449\pi$
$522$	8.09809	0.354444
$523$	−35.9681	−1.57277	−0.786387	−	0.617735i	$-0.788051\pi$
$523$	−35.9681	−1.57277	−0.786387	+	0.617735i	$0.788051\pi$
$524$	44.0695	1.92518
$525$	13.9511	0.608874
$526$	−13.7164	−0.598065
$527$	−4.39928	−0.191636
$528$	8.15785	0.355025
$529$	−22.8769	−0.994649
$530$	−63.2836	−2.74887
$531$	−6.43215	−0.279132
$532$	−4.64488	−0.201381
$533$	−9.55566	−0.413902
$534$	−36.5903	−1.58342
$535$	−48.5955	−2.10097
$536$	48.7921	2.10750
$537$	5.68671	0.245400
$538$	3.31610	0.142967
$539$	−5.11664	−0.220389
$540$	15.5323	0.668404
$541$	−16.0233	−0.688895	−0.344448	−	0.938806i	$-0.611934\pi$
$541$	−16.0233	−0.688895	−0.344448	+	0.938806i	$0.611934\pi$
$542$	49.6547	2.13285
$543$	8.81847	0.378437
$544$	3.63747	0.155955
$545$	−14.7245	−0.630729
$546$	2.35965	0.100984
$547$	−30.5538	−1.30639	−0.653193	−	0.757192i	$-0.726571\pi$
$547$	−30.5538	−1.30639	−0.653193	+	0.757192i	$0.726571\pi$
$548$	−78.7688	−3.36484
$549$	−8.63428	−0.368502
$550$	−168.438	−7.18222
$551$	4.46776	0.190333
$552$	1.29799	0.0552461
$553$	−5.42343	−0.230628
$554$	−45.2387	−1.92201
$555$	33.6404	1.42796
$556$	−27.8857	−1.18262
$557$	−38.9470	−1.65024	−0.825120	−	0.564958i	$-0.808892\pi$
$557$	−38.9470	−1.65024	−0.825120	+	0.564958i	$0.808892\pi$
$558$	10.3808	0.439453
$559$	4.42629	0.187212
$560$	6.94077	0.293301
$561$	−5.11664	−0.216025
$562$	−7.99289	−0.337160
$563$	17.5927	0.741445	0.370723	−	0.928744i	$-0.379110\pi$
$563$	17.5927	0.741445	0.370723	+	0.928744i	$0.379110\pi$
$564$	5.29234	0.222848
$565$	−80.2238	−3.37504
$566$	45.5148	1.91313
$567$	−1.00000	−0.0419961
$568$	16.2420	0.681501
$569$	36.9806	1.55031	0.775154	−	0.631772i	$-0.217672\pi$
$569$	36.9806	1.55031	0.775154	+	0.631772i	$0.217672\pi$
$570$	13.3727	0.560122
$571$	9.97736	0.417539	0.208770	−	0.977965i	$-0.433054\pi$
$571$	9.97736	0.417539	0.208770	+	0.977965i	$0.433054\pi$
$572$	−18.2559	−0.763319
$573$	−13.9963	−0.584702
$574$	22.5480	0.941136
$575$	−4.89439	−0.204110
$576$	−11.7719	−0.490496
$577$	−3.77464	−0.157140	−0.0785702	−	0.996909i	$-0.525035\pi$
$577$	−3.77464	−0.157140	−0.0785702	+	0.996909i	$0.525035\pi$
$578$	2.35965	0.0981485
$579$	−0.757935	−0.0314987
$580$	−53.3053	−2.21338
$581$	4.27184	0.177226
$582$	1.25186	0.0518914
$583$	−31.5219	−1.30550
$584$	26.0637	1.07852
$585$	−4.35328	−0.179986
$586$	3.75691	0.155197
$587$	−7.32245	−0.302230	−0.151115	−	0.988516i	$-0.548286\pi$
$587$	−7.32245	−0.302230	−0.151115	+	0.988516i	$0.548286\pi$
$588$	−3.56795	−0.147140
$589$	5.72713	0.235982
$590$	66.0725	2.72016
$591$	10.4251	0.428830
$592$	12.3207	0.506377
$593$	−12.7384	−0.523102	−0.261551	−	0.965190i	$-0.584234\pi$
$593$	−12.7384	−0.523102	−0.261551	+	0.965190i	$0.584234\pi$
$594$	12.0735	0.495381
$595$	−4.35328	−0.178467
$596$	0.168977	0.00692156
$597$	6.99330	0.286217
$598$	−0.827825	−0.0338523
$599$	13.9379	0.569486	0.284743	−	0.958604i	$-0.408092\pi$
$599$	13.9379	0.569486	0.284743	+	0.958604i	$0.408092\pi$
$600$	−51.6164	−2.10723
$601$	−0.763927	−0.0311612	−0.0155806	−	0.999879i	$-0.504960\pi$
$601$	−0.763927	−0.0311612	−0.0155806	+	0.999879i	$0.504960\pi$
$602$	−10.4445	−0.425686
$603$	13.1877	0.537045
$604$	−64.8298	−2.63789
$605$	−66.0829	−2.68665
$606$	12.4792	0.506933
$607$	−9.57453	−0.388618	−0.194309	−	0.980940i	$-0.562246\pi$
$607$	−9.57453	−0.388618	−0.194309	+	0.980940i	$0.562246\pi$
$608$	−4.73538	−0.192045
$609$	3.43190	0.139068
$610$	88.6933	3.59109
$611$	−1.48330	−0.0600079
$612$	−3.56795	−0.144226
$613$	47.5038	1.91866	0.959331	−	0.282285i	$-0.0910924\pi$
$613$	47.5038	1.91866	0.959331	+	0.282285i	$0.0910924\pi$
$614$	−3.30508	−0.133382
$615$	−41.5985	−1.67741
$616$	18.9306	0.762737
$617$	−3.59228	−0.144620	−0.0723099	−	0.997382i	$-0.523037\pi$
$617$	−3.59228	−0.144620	−0.0723099	+	0.997382i	$0.523037\pi$
$618$	−41.1209	−1.65413
$619$	37.1799	1.49439	0.747194	−	0.664606i	$-0.231401\pi$
$619$	37.1799	1.49439	0.747194	+	0.664606i	$0.231401\pi$
$620$	−68.3310	−2.74424
$621$	0.350825	0.0140781
$622$	35.4574	1.42171
$623$	−15.5066	−0.621260
$624$	−1.59438	−0.0638261
$625$	99.8769	3.99508
$626$	22.6522	0.905363
$627$	6.66102	0.266015
$628$	−72.7557	−2.90327
$629$	−7.72759	−0.308119
$630$	10.2722	0.409255
$631$	−43.9969	−1.75149	−0.875745	−	0.482773i	$-0.839630\pi$
$631$	−43.9969	−1.75149	−0.875745	+	0.482773i	$0.839630\pi$
$632$	20.0657	0.798171
$633$	4.67740	0.185910
$634$	−29.3647	−1.16622
$635$	2.13999	0.0849229
$636$	−21.9809	−0.871600
$637$	1.00000	0.0396214
$638$	−41.4350	−1.64043
$639$	4.38996	0.173664
$640$	89.2538	3.52807
$641$	−9.36747	−0.369993	−0.184996	−	0.982739i	$-0.559227\pi$
$641$	−9.36747	−0.369993	−0.184996	+	0.982739i	$0.559227\pi$
$642$	−26.3407	−1.03958
$643$	−19.0992	−0.753199	−0.376599	−	0.926376i	$-0.622907\pi$
$643$	−19.0992	−0.753199	−0.376599	+	0.926376i	$0.622907\pi$
$644$	1.25173	0.0493250
$645$	19.2689	0.758711
$646$	−3.07187	−0.120861
$647$	−10.2233	−0.401919	−0.200960	−	0.979600i	$-0.564406\pi$
$647$	−10.2233	−0.401919	−0.200960	+	0.979600i	$0.564406\pi$
$648$	3.69982	0.145343
$649$	32.9110	1.29187
$650$	32.9196	1.29121
$651$	4.39928	0.172421
$652$	−43.0517	−1.68603
$653$	1.59165	0.0622859	0.0311430	−	0.999515i	$-0.490085\pi$
$653$	1.59165	0.0622859	0.0311430	+	0.999515i	$0.490085\pi$
$654$	−7.98127	−0.312092
$655$	−53.7695	−2.10095
$656$	−15.2353	−0.594839
$657$	7.04459	0.274836
$658$	3.50007	0.136447
$659$	23.6685	0.921993	0.460996	−	0.887402i	$-0.347492\pi$
$659$	23.6685	0.921993	0.460996	+	0.887402i	$0.347492\pi$
$660$	−79.4732	−3.09349
$661$	9.06903	0.352744	0.176372	−	0.984324i	$-0.443564\pi$
$661$	9.06903	0.352744	0.176372	+	0.984324i	$0.443564\pi$
$662$	3.66621	0.142491
$663$	1.00000	0.0388368
$664$	−15.8050	−0.613354
$665$	5.66725	0.219767
$666$	18.2344	0.706570
$667$	−1.20400	−0.0466190
$668$	82.3466	3.18608
$669$	−14.4145	−0.557297
$670$	−135.467	−5.23355
$671$	44.1785	1.70549
$672$	−3.63747	−0.140318
$673$	−44.8522	−1.72892	−0.864462	−	0.502698i	$-0.832341\pi$
$673$	−44.8522	−1.72892	−0.864462	+	0.502698i	$0.832341\pi$
$674$	31.9787	1.23177
$675$	−13.9511	−0.536977
$676$	3.56795	0.137229
$677$	6.09052	0.234078	0.117039	−	0.993127i	$-0.462660\pi$
$677$	6.09052	0.234078	0.117039	+	0.993127i	$0.462660\pi$
$678$	−43.4844	−1.67001
$679$	0.530529	0.0203598
$680$	16.1064	0.617650
$681$	−10.5575	−0.404564
$682$	−53.1147	−2.03387
$683$	−39.5456	−1.51317	−0.756585	−	0.653896i	$-0.773134\pi$
$683$	−39.5456	−1.51317	−0.756585	+	0.653896i	$0.773134\pi$
$684$	4.64488	0.177601
$685$	96.1064	3.67204
$686$	−2.35965	−0.0900919
$687$	13.8061	0.526735
$688$	7.05716	0.269052
$689$	6.16066	0.234702
$690$	−3.60376	−0.137193
$691$	−31.9003	−1.21354	−0.606771	−	0.794876i	$-0.707536\pi$
$691$	−31.9003	−1.21354	−0.606771	+	0.794876i	$0.707536\pi$
$692$	35.1982	1.33804
$693$	5.11664	0.194365
$694$	49.4867	1.87849
$695$	34.0235	1.29059
$696$	−12.6974	−0.481294
$697$	9.55566	0.361946
$698$	39.0036	1.47631
$699$	−25.7253	−0.973018
$700$	−49.7767	−1.88138
$701$	−28.1543	−1.06337	−0.531686	−	0.846941i	$-0.678441\pi$
$701$	−28.1543	−1.06337	−0.531686	+	0.846941i	$0.678441\pi$
$702$	−2.35965	−0.0890593
$703$	10.0600	0.379422
$704$	60.2326	2.27010
$705$	−6.45722	−0.243193
$706$	−69.3245	−2.60906
$707$	5.28858	0.198897
$708$	22.9496	0.862499
$709$	39.8214	1.49552	0.747762	−	0.663967i	$-0.231128\pi$
$709$	39.8214	1.49552	0.747762	+	0.663967i	$0.231128\pi$
$710$	−45.0946	−1.69237
$711$	5.42343	0.203394
$712$	57.3718	2.15010
$713$	−1.54338	−0.0578000
$714$	−2.35965	−0.0883077
$715$	22.2742	0.833007
$716$	−20.2899	−0.758269
$717$	23.2127	0.866894
$718$	77.9067	2.90745
$719$	−24.8876	−0.928151	−0.464076	−	0.885796i	$-0.653613\pi$
$719$	−24.8876	−0.928151	−0.464076	+	0.885796i	$0.653613\pi$
$720$	−6.94077	−0.258667
$721$	−17.4267	−0.649004
$722$	−40.8343	−1.51970
$723$	0.174539	0.00649117
$724$	−31.4639	−1.16935
$725$	47.8787	1.77817
$726$	−35.8196	−1.32939
$727$	−8.18696	−0.303637	−0.151819	−	0.988408i	$-0.548513\pi$
$727$	−8.18696	−0.303637	−0.151819	+	0.988408i	$0.548513\pi$
$728$	−3.69982	−0.137124
$729$	1.00000	0.0370370
$730$	−72.3637	−2.67830
$731$	−4.42629	−0.163712
$732$	30.8067	1.13865
$733$	0.410841	0.0151748	0.00758738	−	0.999971i	$-0.497585\pi$
$733$	0.410841	0.0151748	0.00758738	+	0.999971i	$0.497585\pi$
$734$	20.7472	0.765794
$735$	4.35328	0.160573
$736$	1.27612	0.0470383
$737$	−67.4768	−2.48554
$738$	−22.5480	−0.830004
$739$	−53.0106	−1.95002	−0.975012	−	0.222150i	$-0.928692\pi$
$739$	−53.0106	−1.95002	−0.975012	+	0.222150i	$0.928692\pi$
$740$	−120.027	−4.41229
$741$	−1.30183	−0.0478241
$742$	−14.5370	−0.533670
$743$	17.5846	0.645118	0.322559	−	0.946549i	$-0.395457\pi$
$743$	17.5846	0.645118	0.322559	+	0.946549i	$0.395457\pi$
$744$	−16.2765	−0.596727
$745$	−0.206170	−0.00755347
$746$	51.7711	1.89547
$747$	−4.27184	−0.156298
$748$	18.2559	0.667503
$749$	−11.1630	−0.407885
$750$	91.9474	3.35744
$751$	18.5091	0.675406	0.337703	−	0.941253i	$-0.390350\pi$
$751$	18.5091	0.675406	0.337703	+	0.941253i	$0.390350\pi$
$752$	−2.36494	−0.0862404
$753$	−9.85856	−0.359266
$754$	8.09809	0.294915
$755$	79.0992	2.87872
$756$	3.56795	0.129765
$757$	34.2332	1.24423	0.622113	−	0.782928i	$-0.286274\pi$
$757$	34.2332	1.24423	0.622113	+	0.782928i	$0.286274\pi$
$758$	43.9997	1.59814
$759$	−1.79505	−0.0651561
$760$	−20.9678	−0.760582
$761$	23.4084	0.848554	0.424277	−	0.905532i	$-0.360528\pi$
$761$	23.4084	0.848554	0.424277	+	0.905532i	$0.360528\pi$
$762$	1.15996	0.0420209
$763$	−3.38240	−0.122451
$764$	49.9379	1.80669
$765$	4.35328	0.157393
$766$	28.2877	1.02208
$767$	−6.43215	−0.232252
$768$	24.8353	0.896166
$769$	−23.2190	−0.837299	−0.418650	−	0.908148i	$-0.637496\pi$
$769$	−23.2190	−0.837299	−0.418650	+	0.908148i	$0.637496\pi$
$770$	−52.5593	−1.89411
$771$	6.41871	0.231164
$772$	2.70428	0.0973290
$773$	0.0280417	0.00100859	0.000504296	−	1.00000i	$-0.499839\pi$
$773$	0.0280417	0.00100859	0.000504296		1.00000i	$0.499839\pi$
$774$	10.4445	0.375420
$775$	61.3747	2.20464
$776$	−1.96286	−0.0704626
$777$	7.72759	0.277226
$778$	−77.6278	−2.78309
$779$	−12.4399	−0.445705
$780$	15.5323	0.556146
$781$	−22.4618	−0.803748
$782$	0.827825	0.0296030
$783$	−3.43190	−0.122646
$784$	1.59438	0.0569420
$785$	88.7697	3.16833
$786$	−29.1452	−1.03957
$787$	−24.1258	−0.859991	−0.429996	−	0.902831i	$-0.641485\pi$
$787$	−24.1258	−0.859991	−0.429996	+	0.902831i	$0.641485\pi$
$788$	−37.1961	−1.32506
$789$	5.81291	0.206945
$790$	−55.7107	−1.98210
$791$	−18.4283	−0.655236
$792$	−18.9306	−0.672671
$793$	−8.63428	−0.306612
$794$	12.2763	0.435669
$795$	26.8191	0.951175
$796$	−24.9517	−0.884391
$797$	44.0773	1.56130	0.780649	−	0.624969i	$-0.214889\pi$
$797$	44.0773	1.56130	0.780649	+	0.624969i	$0.214889\pi$
$798$	3.07187	0.108743
$799$	1.48330	0.0524754
$800$	−50.7465	−1.79416
$801$	15.5066	0.547900
$802$	−54.9877	−1.94168
$803$	−36.0447	−1.27199
$804$	−47.0531	−1.65944
$805$	−1.52724	−0.0538282
$806$	10.3808	0.365647
$807$	−1.40533	−0.0494701
$808$	−19.5668	−0.688357
$809$	−45.2068	−1.58939	−0.794693	−	0.607011i	$-0.792368\pi$
$809$	−45.2068	−1.58939	−0.794693	+	0.607011i	$0.792368\pi$
$810$	−10.2722	−0.360929
$811$	−32.3332	−1.13537	−0.567686	−	0.823245i	$-0.692161\pi$
$811$	−32.3332	−1.13537	−0.567686	+	0.823245i	$0.692161\pi$
$812$	−12.2449	−0.429710
$813$	−21.0432	−0.738019
$814$	−93.2990	−3.27013
$815$	52.5276	1.83996
$816$	1.59438	0.0558143
$817$	5.76229	0.201597
$818$	36.4026	1.27279
$819$	−1.00000	−0.0349428
$820$	148.421	5.18310
$821$	−4.31979	−0.150762	−0.0753809	−	0.997155i	$-0.524017\pi$
$821$	−4.31979	−0.150762	−0.0753809	+	0.997155i	$0.524017\pi$
$822$	52.0935	1.81697
$823$	3.86070	0.134576	0.0672878	−	0.997734i	$-0.478565\pi$
$823$	3.86070	0.134576	0.0672878	+	0.997734i	$0.478565\pi$
$824$	64.4756	2.24611
$825$	71.3826	2.48522
$826$	15.1776	0.528097
$827$	41.1683	1.43156	0.715781	−	0.698325i	$-0.246071\pi$
$827$	41.1683	1.43156	0.715781	+	0.698325i	$0.246071\pi$
$828$	−1.25173	−0.0435006
$829$	28.3550	0.984811	0.492405	−	0.870366i	$-0.336118\pi$
$829$	28.3550	0.984811	0.492405	+	0.870366i	$0.336118\pi$
$830$	43.8813	1.52314
$831$	19.1718	0.665062
$832$	−11.7719	−0.408117
$833$	−1.00000	−0.0346479
$834$	18.4421	0.638598
$835$	−100.472	−3.47696
$836$	−23.7662	−0.821971
$837$	−4.39928	−0.152061
$838$	2.12978	0.0735720
$839$	12.6336	0.436161	0.218080	−	0.975931i	$-0.430021\pi$
$839$	12.6336	0.436161	0.218080	+	0.975931i	$0.430021\pi$
$840$	−16.1064	−0.555722
$841$	−17.2221	−0.593864
$842$	59.0898	2.03637
$843$	3.38732	0.116666
$844$	−16.6887	−0.574450
$845$	−4.35328	−0.149757
$846$	−3.50007	−0.120335
$847$	−15.1800	−0.521592
$848$	9.82240	0.337303
$849$	−19.2888	−0.661989
$850$	−32.9196	−1.12913
$851$	−2.71104	−0.0929331
$852$	−15.6632	−0.536611
$853$	43.0263	1.47319	0.736597	−	0.676332i	$-0.236431\pi$
$853$	43.0263	1.47319	0.736597	+	0.676332i	$0.236431\pi$
$854$	20.3739	0.697180
$855$	−5.66725	−0.193816
$856$	41.3009	1.41164
$857$	−3.00587	−0.102679	−0.0513393	−	0.998681i	$-0.516349\pi$
$857$	−3.00587	−0.102679	−0.0513393	+	0.998681i	$0.516349\pi$
$858$	12.0735	0.412182
$859$	45.3955	1.54887	0.774437	−	0.632652i	$-0.218033\pi$
$859$	45.3955	1.54887	0.774437	+	0.632652i	$0.218033\pi$
$860$	−68.7504	−2.34437
$861$	−9.55566	−0.325656
$862$	71.6438	2.44020
$863$	−32.8994	−1.11991	−0.559955	−	0.828523i	$-0.689182\pi$
$863$	−32.8994	−1.11991	−0.559955	+	0.828523i	$0.689182\pi$
$864$	3.63747	0.123749
$865$	−42.9456	−1.46019
$866$	54.1438	1.83988
$867$	−1.00000	−0.0339618
$868$	−15.6964	−0.532771
$869$	−27.7497	−0.941346
$870$	35.2533	1.19520
$871$	13.1877	0.446849
$872$	12.5142	0.423786
$873$	−0.530529	−0.0179557
$874$	−1.07769	−0.0364534
$875$	38.9665	1.31731
$876$	−25.1348	−0.849225
$877$	56.1670	1.89662	0.948312	−	0.317338i	$-0.102789\pi$
$877$	56.1670	1.89662	0.948312	+	0.317338i	$0.102789\pi$
$878$	−67.4371	−2.27589
$879$	−1.59215	−0.0537018
$880$	35.5134	1.19716
$881$	52.5063	1.76898	0.884491	−	0.466557i	$-0.154506\pi$
$881$	52.5063	1.76898	0.884491	+	0.466557i	$0.154506\pi$
$882$	2.35965	0.0794536
$883$	35.7356	1.20260	0.601299	−	0.799024i	$-0.294650\pi$
$883$	35.7356	1.20260	0.601299	+	0.799024i	$0.294650\pi$
$884$	−3.56795	−0.120003
$885$	−28.0010	−0.941242
$886$	46.0953	1.54860
$887$	53.6536	1.80151	0.900756	−	0.434324i	$-0.143013\pi$
$887$	53.6536	1.80151	0.900756	+	0.434324i	$0.143013\pi$
$888$	−28.5907	−0.959441
$889$	0.491581	0.0164871
$890$	−159.288	−5.33934
$891$	−5.11664	−0.171414
$892$	51.4303	1.72201
$893$	−1.93101	−0.0646188
$894$	−0.111752	−0.00373755
$895$	24.7558	0.827497
$896$	20.5026	0.684945
$897$	0.350825	0.0117137
$898$	66.8133	2.22959
$899$	15.0979	0.503543
$900$	49.7767	1.65922
$901$	−6.16066	−0.205241
$902$	115.370	3.84141
$903$	4.42629	0.147298
$904$	68.1815	2.26768
$905$	38.3893	1.27610
$906$	42.8749	1.42442
$907$	−47.2444	−1.56872	−0.784361	−	0.620304i	$-0.787009\pi$
$907$	−47.2444	−1.56872	−0.784361	+	0.620304i	$0.787009\pi$
$908$	37.6686	1.25008
$909$	−5.28858	−0.175411
$910$	10.2722	0.340521
$911$	16.2104	0.537074	0.268537	−	0.963269i	$-0.413460\pi$
$911$	16.2104	0.537074	0.268537	+	0.963269i	$0.413460\pi$
$912$	−2.07561	−0.0687304
$913$	21.8575	0.723377
$914$	56.3052	1.86241
$915$	−37.5875	−1.24260
$916$	−49.2595	−1.62758
$917$	−12.3515	−0.407882
$918$	2.35965	0.0778801
$919$	−9.00406	−0.297017	−0.148508	−	0.988911i	$-0.547447\pi$
$919$	−9.00406	−0.297017	−0.148508	+	0.988911i	$0.547447\pi$
$920$	5.65052	0.186292
$921$	1.40067	0.0461535
$922$	43.9423	1.44716
$923$	4.38996	0.144497
$924$	−18.2559	−0.600576
$925$	107.808	3.54471
$926$	−4.64043	−0.152494
$927$	17.4267	0.572368
$928$	−12.4834	−0.409789
$929$	35.5591	1.16666	0.583328	−	0.812237i	$-0.301750\pi$
$929$	35.5591	1.16666	0.583328	+	0.812237i	$0.301750\pi$
$930$	45.1904	1.48185
$931$	1.30183	0.0426659
$932$	91.7865	3.00657
$933$	−15.0265	−0.491947
$934$	−20.0803	−0.657048
$935$	−22.2742	−0.728444
$936$	3.69982	0.120932
$937$	21.2755	0.695041	0.347520	−	0.937672i	$-0.387024\pi$
$937$	21.2755	0.695041	0.347520	+	0.937672i	$0.387024\pi$
$938$	−31.1184	−1.01605
$939$	−9.59980	−0.313278
$940$	23.0391	0.751451
$941$	−18.3951	−0.599662	−0.299831	−	0.953992i	$-0.596930\pi$
$941$	−18.3951	−0.599662	−0.299831	+	0.953992i	$0.596930\pi$
$942$	48.1167	1.56773
$943$	3.35237	0.109168
$944$	−10.2553	−0.333780
$945$	−4.35328	−0.141612
$946$	−53.4407	−1.73751
$947$	21.0455	0.683887	0.341943	−	0.939721i	$-0.388915\pi$
$947$	21.0455	0.683887	0.341943	+	0.939721i	$0.388915\pi$
$948$	−19.3505	−0.628476
$949$	7.04459	0.228677
$950$	42.8559	1.39043
$951$	12.4445	0.403542
$952$	3.69982	0.119912
$953$	7.45040	0.241342	0.120671	−	0.992693i	$-0.461495\pi$
$953$	7.45040	0.241342	0.120671	+	0.992693i	$0.461495\pi$
$954$	14.5370	0.470653
$955$	−60.9296	−1.97164
$956$	−82.8218	−2.67865
$957$	17.5598	0.567628
$958$	31.8601	1.02935
$959$	22.0768	0.712896
$960$	−51.2464	−1.65397
$961$	−11.6463	−0.375688
$962$	18.2344	0.587901
$963$	11.1630	0.359721
$964$	−0.622747	−0.0200573
$965$	−3.29951	−0.106215
$966$	−0.827825	−0.0266348
$967$	−39.6310	−1.27445	−0.637224	−	0.770679i	$-0.719917\pi$
$967$	−39.6310	−1.27445	−0.637224	+	0.770679i	$0.719917\pi$
$968$	56.1633	1.80516
$969$	1.30183	0.0418209
$970$	5.44971	0.174980
$971$	−35.4617	−1.13802	−0.569010	−	0.822331i	$-0.692673\pi$
$971$	−35.4617	−1.13802	−0.569010	+	0.822331i	$0.692673\pi$
$972$	−3.56795	−0.114442
$973$	7.81561	0.250557
$974$	−41.1156	−1.31743
$975$	−13.9511	−0.446792
$976$	−13.7663	−0.440648
$977$	−5.82024	−0.186206	−0.0931029	−	0.995656i	$-0.529679\pi$
$977$	−5.82024	−0.186206	−0.0931029	+	0.995656i	$0.529679\pi$
$978$	28.4720	0.910436
$979$	−79.3419	−2.53578
$980$	−15.5323	−0.496161
$981$	3.38240	0.107992
$982$	92.7397	2.95944
$983$	15.3485	0.489540	0.244770	−	0.969581i	$-0.421288\pi$
$983$	15.3485	0.489540	0.244770	+	0.969581i	$0.421288\pi$
$984$	35.3542	1.12705
$985$	45.3833	1.44603
$986$	−8.09809	−0.257896
$987$	−1.48330	−0.0472139
$988$	4.64488	0.147773
$989$	−1.55285	−0.0493779
$990$	52.5593	1.67044
$991$	35.1063	1.11519	0.557594	−	0.830114i	$-0.311724\pi$
$991$	35.1063	1.11519	0.557594	+	0.830114i	$0.311724\pi$
$992$	−16.0022	−0.508072
$993$	−1.55371	−0.0493055
$994$	−10.3588	−0.328560
$995$	30.4438	0.965133
$996$	15.2417	0.482952
$997$	−50.3771	−1.59546	−0.797729	−	0.603016i	$-0.793966\pi$
$997$	−50.3771	−1.59546	−0.797729	+	0.603016i	$0.793966\pi$
$998$	−80.5919	−2.55109
$999$	−7.72759	−0.244490

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4641.2.a.ba.1.15	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4641.2.a.ba.1.15	✓	17	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:	\( N \)	\(=\)	\( 4641 = 3 \cdot 7 \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4641.a (trivial)

\(f(q)\)	\(=\)	\(q+2.35965 q^{2} -1.00000 q^{3} +3.56795 q^{4} -4.35328 q^{5} -2.35965 q^{6} -1.00000 q^{7} +3.69982 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.35965 q^{2} -1.00000 q^{3} +3.56795 q^{4} -4.35328 q^{5} -2.35965 q^{6} -1.00000 q^{7} +3.69982 q^{8} +1.00000 q^{9} -10.2722 q^{10} -5.11664 q^{11} -3.56795 q^{12} +1.00000 q^{13} -2.35965 q^{14} +4.35328 q^{15} +1.59438 q^{16} -1.00000 q^{17} +2.35965 q^{18} +1.30183 q^{19} -15.5323 q^{20} +1.00000 q^{21} -12.0735 q^{22} -0.350825 q^{23} -3.69982 q^{24} +13.9511 q^{25} +2.35965 q^{26} -1.00000 q^{27} -3.56795 q^{28} +3.43190 q^{29} +10.2722 q^{30} +4.39928 q^{31} -3.63747 q^{32} +5.11664 q^{33} -2.35965 q^{34} +4.35328 q^{35} +3.56795 q^{36} +7.72759 q^{37} +3.07187 q^{38} -1.00000 q^{39} -16.1064 q^{40} -9.55566 q^{41} +2.35965 q^{42} +4.42629 q^{43} -18.2559 q^{44} -4.35328 q^{45} -0.827825 q^{46} -1.48330 q^{47} -1.59438 q^{48} +1.00000 q^{49} +32.9196 q^{50} +1.00000 q^{51} +3.56795 q^{52} +6.16066 q^{53} -2.35965 q^{54} +22.2742 q^{55} -3.69982 q^{56} -1.30183 q^{57} +8.09809 q^{58} -6.43215 q^{59} +15.5323 q^{60} -8.63428 q^{61} +10.3808 q^{62} -1.00000 q^{63} -11.7719 q^{64} -4.35328 q^{65} +12.0735 q^{66} +13.1877 q^{67} -3.56795 q^{68} +0.350825 q^{69} +10.2722 q^{70} +4.38996 q^{71} +3.69982 q^{72} +7.04459 q^{73} +18.2344 q^{74} -13.9511 q^{75} +4.64488 q^{76} +5.11664 q^{77} -2.35965 q^{78} +5.42343 q^{79} -6.94077 q^{80} +1.00000 q^{81} -22.5480 q^{82} -4.27184 q^{83} +3.56795 q^{84} +4.35328 q^{85} +10.4445 q^{86} -3.43190 q^{87} -18.9306 q^{88} +15.5066 q^{89} -10.2722 q^{90} -1.00000 q^{91} -1.25173 q^{92} -4.39928 q^{93} -3.50007 q^{94} -5.66725 q^{95} +3.63747 q^{96} -0.530529 q^{97} +2.35965 q^{98} -5.11664 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + q^{2} - 17 q^{3} + 25 q^{4} - 4 q^{5} - q^{6} - 17 q^{7} + 6 q^{8} + 17 q^{9}+O(q^{10}) \) 17 * q + q^2 - 17 * q^3 + 25 * q^4 - 4 * q^5 - q^6 - 17 * q^7 + 6 * q^8 + 17 * q^9	\( 17 q + q^{2} - 17 q^{3} + 25 q^{4} - 4 q^{5} - q^{6} - 17 q^{7} + 6 q^{8} + 17 q^{9} - q^{10} + 4 q^{11} - 25 q^{12} + 17 q^{13} - q^{14} + 4 q^{15} + 53 q^{16} - 17 q^{17} + q^{18} + 13 q^{19} - 5 q^{20} + 17 q^{21} + 4 q^{22} + 8 q^{23} - 6 q^{24} + 37 q^{25} + q^{26} - 17 q^{27} - 25 q^{28} - 3 q^{29} + q^{30} + 10 q^{31} + 18 q^{32} - 4 q^{33} - q^{34} + 4 q^{35} + 25 q^{36} - 25 q^{38} - 17 q^{39} - 12 q^{41} + q^{42} + 29 q^{43} + 12 q^{44} - 4 q^{45} + 19 q^{46} - 4 q^{47} - 53 q^{48} + 17 q^{49} + 9 q^{50} + 17 q^{51} + 25 q^{52} - 8 q^{53} - q^{54} + 27 q^{55} - 6 q^{56} - 13 q^{57} + 2 q^{58} + 25 q^{59} + 5 q^{60} - 5 q^{61} - 37 q^{62} - 17 q^{63} + 94 q^{64} - 4 q^{65} - 4 q^{66} + 15 q^{67} - 25 q^{68} - 8 q^{69} + q^{70} + 32 q^{71} + 6 q^{72} - 15 q^{73} + 16 q^{74} - 37 q^{75} + 3 q^{76} - 4 q^{77} - q^{78} + 27 q^{79} + 41 q^{80} + 17 q^{81} - 8 q^{82} - 24 q^{83} + 25 q^{84} + 4 q^{85} + 53 q^{86} + 3 q^{87} - 9 q^{88} - 8 q^{89} - q^{90} - 17 q^{91} + 14 q^{92} - 10 q^{93} + 51 q^{94} - 9 q^{95} - 18 q^{96} - 22 q^{97} + q^{98} + 4 q^{99}+O(q^{100}) \) 17 * q + q^2 - 17 * q^3 + 25 * q^4 - 4 * q^5 - q^6 - 17 * q^7 + 6 * q^8 + 17 * q^9 - q^10 + 4 * q^11 - 25 * q^12 + 17 * q^13 - q^14 + 4 * q^15 + 53 * q^16 - 17 * q^17 + q^18 + 13 * q^19 - 5 * q^20 + 17 * q^21 + 4 * q^22 + 8 * q^23 - 6 * q^24 + 37 * q^25 + q^26 - 17 * q^27 - 25 * q^28 - 3 * q^29 + q^30 + 10 * q^31 + 18 * q^32 - 4 * q^33 - q^34 + 4 * q^35 + 25 * q^36 - 25 * q^38 - 17 * q^39 - 12 * q^41 + q^42 + 29 * q^43 + 12 * q^44 - 4 * q^45 + 19 * q^46 - 4 * q^47 - 53 * q^48 + 17 * q^49 + 9 * q^50 + 17 * q^51 + 25 * q^52 - 8 * q^53 - q^54 + 27 * q^55 - 6 * q^56 - 13 * q^57 + 2 * q^58 + 25 * q^59 + 5 * q^60 - 5 * q^61 - 37 * q^62 - 17 * q^63 + 94 * q^64 - 4 * q^65 - 4 * q^66 + 15 * q^67 - 25 * q^68 - 8 * q^69 + q^70 + 32 * q^71 + 6 * q^72 - 15 * q^73 + 16 * q^74 - 37 * q^75 + 3 * q^76 - 4 * q^77 - q^78 + 27 * q^79 + 41 * q^80 + 17 * q^81 - 8 * q^82 - 24 * q^83 + 25 * q^84 + 4 * q^85 + 53 * q^86 + 3 * q^87 - 9 * q^88 - 8 * q^89 - q^90 - 17 * q^91 + 14 * q^92 - 10 * q^93 + 51 * q^94 - 9 * q^95 - 18 * q^96 - 22 * q^97 + q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more