LMFDB - Embedded newform 4030.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4030,2,Mod(1,4030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$32.1797120146$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 12x^{6} + 40x^{5} + 24x^{4} - 118x^{3} + 25x^{2} + 30x + 4 $ x^8 - 3x^7 - 12x^6 + 40x^5 + 24x^4 - 118x^3 + 25x^2 + 30*x + 4
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			1.2
Root			$-1.95716$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.95716 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.95716 q^{6} -0.791637 q^{7} +1.00000 q^{8} +0.830474 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.95716 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.95716 q^{6} -0.791637 q^{7} +1.00000 q^{8} +0.830474 q^{9} +1.00000 q^{10} -2.46820 q^{11} -1.95716 q^{12} -1.00000 q^{13} -0.791637 q^{14} -1.95716 q^{15} +1.00000 q^{16} -2.10480 q^{17} +0.830474 q^{18} -0.660431 q^{19} +1.00000 q^{20} +1.54936 q^{21} -2.46820 q^{22} -1.45447 q^{23} -1.95716 q^{24} +1.00000 q^{25} -1.00000 q^{26} +4.24611 q^{27} -0.791637 q^{28} +5.06648 q^{29} -1.95716 q^{30} +1.00000 q^{31} +1.00000 q^{32} +4.83066 q^{33} -2.10480 q^{34} -0.791637 q^{35} +0.830474 q^{36} -1.69915 q^{37} -0.660431 q^{38} +1.95716 q^{39} +1.00000 q^{40} -6.79754 q^{41} +1.54936 q^{42} +11.9319 q^{43} -2.46820 q^{44} +0.830474 q^{45} -1.45447 q^{46} +11.4037 q^{47} -1.95716 q^{48} -6.37331 q^{49} +1.00000 q^{50} +4.11942 q^{51} -1.00000 q^{52} -3.50482 q^{53} +4.24611 q^{54} -2.46820 q^{55} -0.791637 q^{56} +1.29257 q^{57} +5.06648 q^{58} +4.12863 q^{59} -1.95716 q^{60} +10.4297 q^{61} +1.00000 q^{62} -0.657434 q^{63} +1.00000 q^{64} -1.00000 q^{65} +4.83066 q^{66} -7.91415 q^{67} -2.10480 q^{68} +2.84663 q^{69} -0.791637 q^{70} +7.99420 q^{71} +0.830474 q^{72} +2.84536 q^{73} -1.69915 q^{74} -1.95716 q^{75} -0.660431 q^{76} +1.95392 q^{77} +1.95716 q^{78} +7.48547 q^{79} +1.00000 q^{80} -10.8017 q^{81} -6.79754 q^{82} +7.55256 q^{83} +1.54936 q^{84} -2.10480 q^{85} +11.9319 q^{86} -9.91590 q^{87} -2.46820 q^{88} +9.87407 q^{89} +0.830474 q^{90} +0.791637 q^{91} -1.45447 q^{92} -1.95716 q^{93} +11.4037 q^{94} -0.660431 q^{95} -1.95716 q^{96} +6.94216 q^{97} -6.37331 q^{98} -2.04978 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9	\( 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} + 8 q^{10} + 10 q^{11} + 3 q^{12} - 8 q^{13} + 7 q^{14} + 3 q^{15} + 8 q^{16} + 11 q^{17} + 9 q^{18} + 2 q^{19} + 8 q^{20} + 5 q^{21} + 10 q^{22} + 12 q^{23} + 3 q^{24} + 8 q^{25} - 8 q^{26} - 3 q^{27} + 7 q^{28} + 9 q^{29} + 3 q^{30} + 8 q^{31} + 8 q^{32} + 6 q^{33} + 11 q^{34} + 7 q^{35} + 9 q^{36} + 19 q^{37} + 2 q^{38} - 3 q^{39} + 8 q^{40} + 6 q^{41} + 5 q^{42} + 21 q^{43} + 10 q^{44} + 9 q^{45} + 12 q^{46} + q^{47} + 3 q^{48} - 5 q^{49} + 8 q^{50} + 17 q^{51} - 8 q^{52} + 18 q^{53} - 3 q^{54} + 10 q^{55} + 7 q^{56} - 11 q^{57} + 9 q^{58} - 4 q^{59} + 3 q^{60} + 10 q^{61} + 8 q^{62} + 22 q^{63} + 8 q^{64} - 8 q^{65} + 6 q^{66} + 8 q^{67} + 11 q^{68} - 26 q^{69} + 7 q^{70} + 18 q^{71} + 9 q^{72} + 9 q^{73} + 19 q^{74} + 3 q^{75} + 2 q^{76} + 13 q^{77} - 3 q^{78} + 14 q^{79} + 8 q^{80} + 6 q^{82} + 3 q^{83} + 5 q^{84} + 11 q^{85} + 21 q^{86} - 21 q^{87} + 10 q^{88} - 15 q^{89} + 9 q^{90} - 7 q^{91} + 12 q^{92} + 3 q^{93} + q^{94} + 2 q^{95} + 3 q^{96} + 12 q^{97} - 5 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9 + 8 * q^10 + 10 * q^11 + 3 * q^12 - 8 * q^13 + 7 * q^14 + 3 * q^15 + 8 * q^16 + 11 * q^17 + 9 * q^18 + 2 * q^19 + 8 * q^20 + 5 * q^21 + 10 * q^22 + 12 * q^23 + 3 * q^24 + 8 * q^25 - 8 * q^26 - 3 * q^27 + 7 * q^28 + 9 * q^29 + 3 * q^30 + 8 * q^31 + 8 * q^32 + 6 * q^33 + 11 * q^34 + 7 * q^35 + 9 * q^36 + 19 * q^37 + 2 * q^38 - 3 * q^39 + 8 * q^40 + 6 * q^41 + 5 * q^42 + 21 * q^43 + 10 * q^44 + 9 * q^45 + 12 * q^46 + q^47 + 3 * q^48 - 5 * q^49 + 8 * q^50 + 17 * q^51 - 8 * q^52 + 18 * q^53 - 3 * q^54 + 10 * q^55 + 7 * q^56 - 11 * q^57 + 9 * q^58 - 4 * q^59 + 3 * q^60 + 10 * q^61 + 8 * q^62 + 22 * q^63 + 8 * q^64 - 8 * q^65 + 6 * q^66 + 8 * q^67 + 11 * q^68 - 26 * q^69 + 7 * q^70 + 18 * q^71 + 9 * q^72 + 9 * q^73 + 19 * q^74 + 3 * q^75 + 2 * q^76 + 13 * q^77 - 3 * q^78 + 14 * q^79 + 8 * q^80 + 6 * q^82 + 3 * q^83 + 5 * q^84 + 11 * q^85 + 21 * q^86 - 21 * q^87 + 10 * q^88 - 15 * q^89 + 9 * q^90 - 7 * q^91 + 12 * q^92 + 3 * q^93 + q^94 + 2 * q^95 + 3 * q^96 + 12 * q^97 - 5 * q^98 + 11 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.95716	−1.12997	−0.564983	−	0.825102i	$-0.691117\pi$
$3$	−1.95716	−1.12997	−0.564983	+	0.825102i	$0.691117\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.95716	−0.799007
$7$	−0.791637	−0.299211	−0.149605	−	0.988746i	$-0.547800\pi$
$7$	−0.791637	−0.299211	−0.149605	+	0.988746i	$0.547800\pi$
$8$	1.00000	0.353553
$9$	0.830474	0.276825
$10$	1.00000	0.316228
$11$	−2.46820	−0.744190	−0.372095	−	0.928195i	$-0.621360\pi$
$11$	−2.46820	−0.744190	−0.372095	+	0.928195i	$0.621360\pi$
$12$	−1.95716	−0.564983
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.791637	−0.211574
$15$	−1.95716	−0.505336
$16$	1.00000	0.250000
$17$	−2.10480	−0.510488	−0.255244	−	0.966877i	$-0.582156\pi$
$17$	−2.10480	−0.510488	−0.255244	+	0.966877i	$0.582156\pi$
$18$	0.830474	0.195745
$19$	−0.660431	−0.151513	−0.0757567	−	0.997126i	$-0.524137\pi$
$19$	−0.660431	−0.151513	−0.0757567	+	0.997126i	$0.524137\pi$
$20$	1.00000	0.223607
$21$	1.54936	0.338098
$22$	−2.46820	−0.526222
$23$	−1.45447	−0.303278	−0.151639	−	0.988436i	$-0.548455\pi$
$23$	−1.45447	−0.303278	−0.151639	+	0.988436i	$0.548455\pi$
$24$	−1.95716	−0.399504
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	4.24611	0.817164
$28$	−0.791637	−0.149605
$29$	5.06648	0.940821	0.470411	−	0.882448i	$-0.344106\pi$
$29$	5.06648	0.940821	0.470411	+	0.882448i	$0.344106\pi$
$30$	−1.95716	−0.357327
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	4.83066	0.840910
$34$	−2.10480	−0.360970
$35$	−0.791637	−0.133811
$36$	0.830474	0.138412
$37$	−1.69915	−0.279339	−0.139670	−	0.990198i	$-0.544604\pi$
$37$	−1.69915	−0.279339	−0.139670	+	0.990198i	$0.544604\pi$
$38$	−0.660431	−0.107136
$39$	1.95716	0.313396
$40$	1.00000	0.158114
$41$	−6.79754	−1.06160	−0.530799	−	0.847498i	$-0.678108\pi$
$41$	−6.79754	−1.06160	−0.530799	+	0.847498i	$0.678108\pi$
$42$	1.54936	0.239071
$43$	11.9319	1.81959	0.909797	−	0.415053i	$-0.136237\pi$
$43$	11.9319	1.81959	0.909797	+	0.415053i	$0.136237\pi$
$44$	−2.46820	−0.372095
$45$	0.830474	0.123800
$46$	−1.45447	−0.214450
$47$	11.4037	1.66339	0.831697	−	0.555230i	$-0.187370\pi$
$47$	11.4037	1.66339	0.831697	+	0.555230i	$0.187370\pi$
$48$	−1.95716	−0.282492
$49$	−6.37331	−0.910473
$50$	1.00000	0.141421
$51$	4.11942	0.576835
$52$	−1.00000	−0.138675
$53$	−3.50482	−0.481424	−0.240712	−	0.970597i	$-0.577381\pi$
$53$	−3.50482	−0.481424	−0.240712	+	0.970597i	$0.577381\pi$
$54$	4.24611	0.577822
$55$	−2.46820	−0.332812
$56$	−0.791637	−0.105787
$57$	1.29257	0.171205
$58$	5.06648	0.665261
$59$	4.12863	0.537502	0.268751	−	0.963210i	$-0.413389\pi$
$59$	4.12863	0.537502	0.268751	+	0.963210i	$0.413389\pi$
$60$	−1.95716	−0.252668
$61$	10.4297	1.33539	0.667693	−	0.744436i	$-0.267282\pi$
$61$	10.4297	1.33539	0.667693	+	0.744436i	$0.267282\pi$
$62$	1.00000	0.127000
$63$	−0.657434	−0.0828289
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	4.83066	0.594613
$67$	−7.91415	−0.966867	−0.483434	−	0.875381i	$-0.660611\pi$
$67$	−7.91415	−0.966867	−0.483434	+	0.875381i	$0.660611\pi$
$68$	−2.10480	−0.255244
$69$	2.84663	0.342694
$70$	−0.791637	−0.0946187
$71$	7.99420	0.948737	0.474369	−	0.880326i	$-0.342676\pi$
$71$	7.99420	0.948737	0.474369	+	0.880326i	$0.342676\pi$
$72$	0.830474	0.0978723
$73$	2.84536	0.333025	0.166512	−	0.986039i	$-0.446749\pi$
$73$	2.84536	0.333025	0.166512	+	0.986039i	$0.446749\pi$
$74$	−1.69915	−0.197523
$75$	−1.95716	−0.225993
$76$	−0.660431	−0.0757567
$77$	1.95392	0.222669
$78$	1.95716	0.221605
$79$	7.48547	0.842181	0.421091	−	0.907019i	$-0.361647\pi$
$79$	7.48547	0.842181	0.421091	+	0.907019i	$0.361647\pi$
$80$	1.00000	0.111803
$81$	−10.8017	−1.20019
$82$	−6.79754	−0.750663
$83$	7.55256	0.829001	0.414501	−	0.910049i	$-0.363956\pi$
$83$	7.55256	0.829001	0.414501	+	0.910049i	$0.363956\pi$
$84$	1.54936	0.169049
$85$	−2.10480	−0.228297
$86$	11.9319	1.28665
$87$	−9.91590	−1.06310
$88$	−2.46820	−0.263111
$89$	9.87407	1.04665	0.523325	−	0.852133i	$-0.324692\pi$
$89$	9.87407	1.04665	0.523325	+	0.852133i	$0.324692\pi$
$90$	0.830474	0.0875397
$91$	0.791637	0.0829861
$92$	−1.45447	−0.151639
$93$	−1.95716	−0.202948
$94$	11.4037	1.17620
$95$	−0.660431	−0.0677588
$96$	−1.95716	−0.199752
$97$	6.94216	0.704869	0.352435	−	0.935836i	$-0.385354\pi$
$97$	6.94216	0.704869	0.352435	+	0.935836i	$0.385354\pi$
$98$	−6.37331	−0.643802
$99$	−2.04978	−0.206010
$100$	1.00000	0.100000
$101$	−6.12827	−0.609786	−0.304893	−	0.952387i	$-0.598621\pi$
$101$	−6.12827	−0.609786	−0.304893	+	0.952387i	$0.598621\pi$
$102$	4.11942	0.407884
$103$	9.39365	0.925584	0.462792	−	0.886467i	$-0.346848\pi$
$103$	9.39365	0.925584	0.462792	+	0.886467i	$0.346848\pi$
$104$	−1.00000	−0.0980581
$105$	1.54936	0.151202
$106$	−3.50482	−0.340418
$107$	0.333370	0.0322281	0.0161141	−	0.999870i	$-0.494871\pi$
$107$	0.333370	0.0322281	0.0161141	+	0.999870i	$0.494871\pi$
$108$	4.24611	0.408582
$109$	−17.8447	−1.70922	−0.854608	−	0.519273i	$-0.826203\pi$
$109$	−17.8447	−1.70922	−0.854608	+	0.519273i	$0.826203\pi$
$110$	−2.46820	−0.235334
$111$	3.32551	0.315644
$112$	−0.791637	−0.0748026
$113$	10.3377	0.972490	0.486245	−	0.873823i	$-0.338366\pi$
$113$	10.3377	0.972490	0.486245	+	0.873823i	$0.338366\pi$
$114$	1.29257	0.121060
$115$	−1.45447	−0.135630
$116$	5.06648	0.470411
$117$	−0.830474	−0.0767774
$118$	4.12863	0.380071
$119$	1.66623	0.152743
$120$	−1.95716	−0.178663
$121$	−4.90799	−0.446181
$122$	10.4297	0.944261
$123$	13.3039	1.19957
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	−0.657434	−0.0585689
$127$	9.52741	0.845421	0.422710	−	0.906265i	$-0.361079\pi$
$127$	9.52741	0.845421	0.422710	+	0.906265i	$0.361079\pi$
$128$	1.00000	0.0883883
$129$	−23.3526	−2.05608
$130$	−1.00000	−0.0877058
$131$	−10.5187	−0.919026	−0.459513	−	0.888171i	$-0.651976\pi$
$131$	−10.5187	−0.919026	−0.459513	+	0.888171i	$0.651976\pi$
$132$	4.83066	0.420455
$133$	0.522822	0.0453344
$134$	−7.91415	−0.683678
$135$	4.24611	0.365447
$136$	−2.10480	−0.180485
$137$	7.42172	0.634080	0.317040	−	0.948412i	$-0.397311\pi$
$137$	7.42172	0.634080	0.317040	+	0.948412i	$0.397311\pi$
$138$	2.84663	0.242322
$139$	−7.72123	−0.654906	−0.327453	−	0.944867i	$-0.606190\pi$
$139$	−7.72123	−0.654906	−0.327453	+	0.944867i	$0.606190\pi$
$140$	−0.791637	−0.0669055
$141$	−22.3188	−1.87958
$142$	7.99420	0.670859
$143$	2.46820	0.206401
$144$	0.830474	0.0692062
$145$	5.06648	0.420748
$146$	2.84536	0.235484
$147$	12.4736	1.02880
$148$	−1.69915	−0.139670
$149$	3.17851	0.260394	0.130197	−	0.991488i	$-0.458439\pi$
$149$	3.17851	0.260394	0.130197	+	0.991488i	$0.458439\pi$
$150$	−1.95716	−0.159801
$151$	0.739176	0.0601533	0.0300766	−	0.999548i	$-0.490425\pi$
$151$	0.739176	0.0601533	0.0300766	+	0.999548i	$0.490425\pi$
$152$	−0.660431	−0.0535680
$153$	−1.74798	−0.141316
$154$	1.95392	0.157451
$155$	1.00000	0.0803219
$156$	1.95716	0.156698
$157$	−20.9472	−1.67177	−0.835883	−	0.548907i	$-0.815044\pi$
$157$	−20.9472	−1.67177	−0.835883	+	0.548907i	$0.815044\pi$
$158$	7.48547	0.595512
$159$	6.85949	0.543993
$160$	1.00000	0.0790569
$161$	1.15141	0.0907441
$162$	−10.8017	−0.848664
$163$	14.5155	1.13695	0.568473	−	0.822702i	$-0.307534\pi$
$163$	14.5155	1.13695	0.568473	+	0.822702i	$0.307534\pi$
$164$	−6.79754	−0.530799
$165$	4.83066	0.376066
$166$	7.55256	0.586192
$167$	4.20752	0.325588	0.162794	−	0.986660i	$-0.447949\pi$
$167$	4.20752	0.325588	0.162794	+	0.986660i	$0.447949\pi$
$168$	1.54936	0.119536
$169$	1.00000	0.0769231
$170$	−2.10480	−0.161431
$171$	−0.548471	−0.0419426
$172$	11.9319	0.909797
$173$	−13.2435	−1.00689	−0.503443	−	0.864029i	$-0.667934\pi$
$173$	−13.2435	−1.00689	−0.503443	+	0.864029i	$0.667934\pi$
$174$	−9.91590	−0.751723
$175$	−0.791637	−0.0598421
$176$	−2.46820	−0.186047
$177$	−8.08039	−0.607359
$178$	9.87407	0.740093
$179$	20.5455	1.53564	0.767820	−	0.640665i	$-0.221341\pi$
$179$	20.5455	1.53564	0.767820	+	0.640665i	$0.221341\pi$
$180$	0.830474	0.0618999
$181$	22.5596	1.67684	0.838422	−	0.545022i	$-0.183479\pi$
$181$	22.5596	1.67684	0.838422	+	0.545022i	$0.183479\pi$
$182$	0.791637	0.0586800
$183$	−20.4126	−1.50894
$184$	−1.45447	−0.107225
$185$	−1.69915	−0.124924
$186$	−1.95716	−0.143506
$187$	5.19506	0.379900
$188$	11.4037	0.831697
$189$	−3.36138	−0.244504
$190$	−0.660431	−0.0479127
$191$	0.283465	0.0205108	0.0102554	−	0.999947i	$-0.496736\pi$
$191$	0.283465	0.0205108	0.0102554	+	0.999947i	$0.496736\pi$
$192$	−1.95716	−0.141246
$193$	24.1413	1.73773	0.868865	−	0.495048i	$-0.164850\pi$
$193$	24.1413	1.73773	0.868865	+	0.495048i	$0.164850\pi$
$194$	6.94216	0.498418
$195$	1.95716	0.140155
$196$	−6.37331	−0.455237
$197$	−8.09922	−0.577045	−0.288523	−	0.957473i	$-0.593164\pi$
$197$	−8.09922	−0.577045	−0.288523	+	0.957473i	$0.593164\pi$
$198$	−2.04978	−0.145671
$199$	−14.6584	−1.03911	−0.519554	−	0.854438i	$-0.673902\pi$
$199$	−14.6584	−1.03911	−0.519554	+	0.854438i	$0.673902\pi$
$200$	1.00000	0.0707107
$201$	15.4893	1.09253
$202$	−6.12827	−0.431184
$203$	−4.01081	−0.281504
$204$	4.11942	0.288417
$205$	−6.79754	−0.474761
$206$	9.39365	0.654487
$207$	−1.20790	−0.0839549
$208$	−1.00000	−0.0693375
$209$	1.63008	0.112755
$210$	1.54936	0.106916
$211$	18.7956	1.29394	0.646970	−	0.762515i	$-0.276036\pi$
$211$	18.7956	1.29394	0.646970	+	0.762515i	$0.276036\pi$
$212$	−3.50482	−0.240712
$213$	−15.6459	−1.07204
$214$	0.333370	0.0227887
$215$	11.9319	0.813747
$216$	4.24611	0.288911
$217$	−0.791637	−0.0537398
$218$	−17.8447	−1.20860
$219$	−5.56883	−0.376307
$220$	−2.46820	−0.166406
$221$	2.10480	0.141584
$222$	3.32551	0.223194
$223$	11.8015	0.790284	0.395142	−	0.918620i	$-0.370695\pi$
$223$	11.8015	0.790284	0.395142	+	0.918620i	$0.370695\pi$
$224$	−0.791637	−0.0528935
$225$	0.830474	0.0553649
$226$	10.3377	0.687654
$227$	12.8072	0.850047	0.425023	−	0.905182i	$-0.360266\pi$
$227$	12.8072	0.850047	0.425023	+	0.905182i	$0.360266\pi$
$228$	1.29257	0.0856025
$229$	−9.07282	−0.599549	−0.299774	−	0.954010i	$-0.596911\pi$
$229$	−9.07282	−0.599549	−0.299774	+	0.954010i	$0.596911\pi$
$230$	−1.45447	−0.0959050
$231$	−3.82413	−0.251609
$232$	5.06648	0.332630
$233$	15.6742	1.02685	0.513425	−	0.858134i	$-0.328376\pi$
$233$	15.6742	1.02685	0.513425	+	0.858134i	$0.328376\pi$
$234$	−0.830474	−0.0542898
$235$	11.4037	0.743892
$236$	4.12863	0.268751
$237$	−14.6503	−0.951637
$238$	1.66623	0.108006
$239$	9.47079	0.612614	0.306307	−	0.951933i	$-0.400907\pi$
$239$	9.47079	0.612614	0.306307	+	0.951933i	$0.400907\pi$
$240$	−1.95716	−0.126334
$241$	−12.8737	−0.829268	−0.414634	−	0.909988i	$-0.636090\pi$
$241$	−12.8737	−0.829268	−0.414634	+	0.909988i	$0.636090\pi$
$242$	−4.90799	−0.315498
$243$	8.40239	0.539014
$244$	10.4297	0.667693
$245$	−6.37331	−0.407176
$246$	13.3039	0.848224
$247$	0.660431	0.0420222
$248$	1.00000	0.0635001
$249$	−14.7816	−0.936744
$250$	1.00000	0.0632456
$251$	23.9922	1.51438	0.757188	−	0.653197i	$-0.226573\pi$
$251$	23.9922	1.51438	0.757188	+	0.653197i	$0.226573\pi$
$252$	−0.657434	−0.0414144
$253$	3.58993	0.225697
$254$	9.52741	0.597803
$255$	4.11942	0.257968
$256$	1.00000	0.0625000
$257$	−12.1413	−0.757355	−0.378677	−	0.925529i	$-0.623621\pi$
$257$	−12.1413	−0.757355	−0.378677	+	0.925529i	$0.623621\pi$
$258$	−23.3526	−1.45387
$259$	1.34511	0.0835812
$260$	−1.00000	−0.0620174
$261$	4.20758	0.260442
$262$	−10.5187	−0.649850
$263$	7.68986	0.474177	0.237089	−	0.971488i	$-0.423807\pi$
$263$	7.68986	0.474177	0.237089	+	0.971488i	$0.423807\pi$
$264$	4.83066	0.297307
$265$	−3.50482	−0.215299
$266$	0.522822	0.0320562
$267$	−19.3251	−1.18268
$268$	−7.91415	−0.483434
$269$	1.61533	0.0984885	0.0492442	−	0.998787i	$-0.484319\pi$
$269$	1.61533	0.0984885	0.0492442	+	0.998787i	$0.484319\pi$
$270$	4.24611	0.258410
$271$	30.0037	1.82260	0.911299	−	0.411746i	$-0.135081\pi$
$271$	30.0037	1.82260	0.911299	+	0.411746i	$0.135081\pi$
$272$	−2.10480	−0.127622
$273$	−1.54936	−0.0937715
$274$	7.42172	0.448362
$275$	−2.46820	−0.148838
$276$	2.84663	0.171347
$277$	30.4894	1.83193	0.915967	−	0.401255i	$-0.131426\pi$
$277$	30.4894	1.83193	0.915967	+	0.401255i	$0.131426\pi$
$278$	−7.72123	−0.463089
$279$	0.830474	0.0497192
$280$	−0.791637	−0.0473093
$281$	2.36080	0.140834	0.0704168	−	0.997518i	$-0.477567\pi$
$281$	2.36080	0.140834	0.0704168	+	0.997518i	$0.477567\pi$
$282$	−22.3188	−1.32906
$283$	−17.2265	−1.02401	−0.512004	−	0.858983i	$-0.671097\pi$
$283$	−17.2265	−1.02401	−0.512004	+	0.858983i	$0.671097\pi$
$284$	7.99420	0.474369
$285$	1.29257	0.0765652
$286$	2.46820	0.145948
$287$	5.38118	0.317641
$288$	0.830474	0.0489362
$289$	−12.5698	−0.739402
$290$	5.06648	0.297514
$291$	−13.5869	−0.796479
$292$	2.84536	0.166512
$293$	15.4159	0.900604	0.450302	−	0.892876i	$-0.351316\pi$
$293$	15.4159	0.900604	0.450302	+	0.892876i	$0.351316\pi$
$294$	12.4736	0.727474
$295$	4.12863	0.240378
$296$	−1.69915	−0.0987613
$297$	−10.4802	−0.608125
$298$	3.17851	0.184126
$299$	1.45447	0.0841143
$300$	−1.95716	−0.112997
$301$	−9.44571	−0.544442
$302$	0.739176	0.0425348
$303$	11.9940	0.689037
$304$	−0.660431	−0.0378783
$305$	10.4297	0.597203
$306$	−1.74798	−0.0999253
$307$	−10.9588	−0.625452	−0.312726	−	0.949843i	$-0.601242\pi$
$307$	−10.9588	−0.625452	−0.312726	+	0.949843i	$0.601242\pi$
$308$	1.95392	0.111335
$309$	−18.3849	−1.04588
$310$	1.00000	0.0567962
$311$	−4.03705	−0.228920	−0.114460	−	0.993428i	$-0.536514\pi$
$311$	−4.03705	−0.228920	−0.114460	+	0.993428i	$0.536514\pi$
$312$	1.95716	0.110802
$313$	10.4680	0.591688	0.295844	−	0.955236i	$-0.404399\pi$
$313$	10.4680	0.591688	0.295844	+	0.955236i	$0.404399\pi$
$314$	−20.9472	−1.18212
$315$	−0.657434	−0.0370422
$316$	7.48547	0.421091
$317$	20.2623	1.13804	0.569022	−	0.822322i	$-0.307322\pi$
$317$	20.2623	1.13804	0.569022	+	0.822322i	$0.307322\pi$
$318$	6.85949	0.384661
$319$	−12.5051	−0.700150
$320$	1.00000	0.0559017
$321$	−0.652459	−0.0364167
$322$	1.15141	0.0641658
$323$	1.39007	0.0773457
$324$	−10.8017	−0.600096
$325$	−1.00000	−0.0554700
$326$	14.5155	0.803942
$327$	34.9250	1.93136
$328$	−6.79754	−0.375332
$329$	−9.02755	−0.497705
$330$	4.83066	0.265919
$331$	−8.17536	−0.449359	−0.224679	−	0.974433i	$-0.572133\pi$
$331$	−8.17536	−0.449359	−0.224679	+	0.974433i	$0.572133\pi$
$332$	7.55256	0.414501
$333$	−1.41110	−0.0773280
$334$	4.20752	0.230225
$335$	−7.91415	−0.432396
$336$	1.54936	0.0845245
$337$	5.70136	0.310573	0.155287	−	0.987869i	$-0.450370\pi$
$337$	5.70136	0.310573	0.155287	+	0.987869i	$0.450370\pi$
$338$	1.00000	0.0543928
$339$	−20.2325	−1.09888
$340$	−2.10480	−0.114149
$341$	−2.46820	−0.133660
$342$	−0.548471	−0.0296579
$343$	10.5868	0.571634
$344$	11.9319	0.643324
$345$	2.84663	0.153258
$346$	−13.2435	−0.711976
$347$	−14.7378	−0.791169	−0.395585	−	0.918430i	$-0.629458\pi$
$347$	−14.7378	−0.791169	−0.395585	+	0.918430i	$0.629458\pi$
$348$	−9.91590	−0.531548
$349$	17.1437	0.917679	0.458840	−	0.888519i	$-0.348265\pi$
$349$	17.1437	0.917679	0.458840	+	0.888519i	$0.348265\pi$
$350$	−0.791637	−0.0423148
$351$	−4.24611	−0.226641
$352$	−2.46820	−0.131555
$353$	20.2132	1.07584	0.537920	−	0.842996i	$-0.319210\pi$
$353$	20.2132	1.07584	0.537920	+	0.842996i	$0.319210\pi$
$354$	−8.08039	−0.429468
$355$	7.99420	0.424288
$356$	9.87407	0.523325
$357$	−3.26109	−0.172595
$358$	20.5455	1.08586
$359$	−26.7825	−1.41353	−0.706764	−	0.707449i	$-0.749846\pi$
$359$	−26.7825	−1.41353	−0.706764	+	0.707449i	$0.749846\pi$
$360$	0.830474	0.0437698
$361$	−18.5638	−0.977044
$362$	22.5596	1.18571
$363$	9.60573	0.504170
$364$	0.791637	0.0414930
$365$	2.84536	0.148933
$366$	−20.4126	−1.06698
$367$	−18.1619	−0.948046	−0.474023	−	0.880512i	$-0.657199\pi$
$367$	−18.1619	−0.948046	−0.474023	+	0.880512i	$0.657199\pi$
$368$	−1.45447	−0.0758196
$369$	−5.64518	−0.293876
$370$	−1.69915	−0.0883348
$371$	2.77454	0.144047
$372$	−1.95716	−0.101474
$373$	21.9089	1.13440	0.567200	−	0.823580i	$-0.308027\pi$
$373$	21.9089	1.13440	0.567200	+	0.823580i	$0.308027\pi$
$374$	5.19506	0.268630
$375$	−1.95716	−0.101067
$376$	11.4037	0.588099
$377$	−5.06648	−0.260937
$378$	−3.36138	−0.172891
$379$	0.912540	0.0468741	0.0234370	−	0.999725i	$-0.492539\pi$
$379$	0.912540	0.0468741	0.0234370	+	0.999725i	$0.492539\pi$
$380$	−0.660431	−0.0338794
$381$	−18.6467	−0.955297
$382$	0.283465	0.0145033
$383$	−26.3086	−1.34431	−0.672153	−	0.740412i	$-0.734630\pi$
$383$	−26.3086	−1.34431	−0.672153	+	0.740412i	$0.734630\pi$
$384$	−1.95716	−0.0998759
$385$	1.95392	0.0995808
$386$	24.1413	1.22876
$387$	9.90912	0.503709
$388$	6.94216	0.352435
$389$	9.14918	0.463882	0.231941	−	0.972730i	$-0.425492\pi$
$389$	9.14918	0.463882	0.231941	+	0.972730i	$0.425492\pi$
$390$	1.95716	0.0991046
$391$	3.06137	0.154820
$392$	−6.37331	−0.321901
$393$	20.5868	1.03847
$394$	−8.09922	−0.408033
$395$	7.48547	0.376635
$396$	−2.04978	−0.103005
$397$	−10.8659	−0.545343	−0.272671	−	0.962107i	$-0.587907\pi$
$397$	−10.8659	−0.545343	−0.272671	+	0.962107i	$0.587907\pi$
$398$	−14.6584	−0.734760
$399$	−1.02325	−0.0512263
$400$	1.00000	0.0500000
$401$	−20.8460	−1.04100	−0.520499	−	0.853862i	$-0.674254\pi$
$401$	−20.8460	−1.04100	−0.520499	+	0.853862i	$0.674254\pi$
$402$	15.4893	0.772534
$403$	−1.00000	−0.0498135
$404$	−6.12827	−0.304893
$405$	−10.8017	−0.536743
$406$	−4.01081	−0.199053
$407$	4.19385	0.207881
$408$	4.11942	0.203942
$409$	−15.1281	−0.748038	−0.374019	−	0.927421i	$-0.622020\pi$
$409$	−15.1281	−0.748038	−0.374019	+	0.927421i	$0.622020\pi$
$410$	−6.79754	−0.335707
$411$	−14.5255	−0.716489
$412$	9.39365	0.462792
$413$	−3.26838	−0.160826
$414$	−1.20790	−0.0593651
$415$	7.55256	0.370741
$416$	−1.00000	−0.0490290
$417$	15.1117	0.740022
$418$	1.63008	0.0797296
$419$	−38.4968	−1.88069	−0.940346	−	0.340221i	$-0.889498\pi$
$419$	−38.4968	−1.88069	−0.940346	+	0.340221i	$0.889498\pi$
$420$	1.54936	0.0756010
$421$	0.698826	0.0340587	0.0170293	−	0.999855i	$-0.494579\pi$
$421$	0.698826	0.0340587	0.0170293	+	0.999855i	$0.494579\pi$
$422$	18.7956	0.914954
$423$	9.47044	0.460469
$424$	−3.50482	−0.170209
$425$	−2.10480	−0.102098
$426$	−15.6459	−0.758048
$427$	−8.25654	−0.399562
$428$	0.333370	0.0161141
$429$	−4.83066	−0.233226
$430$	11.9319	0.575406
$431$	31.3528	1.51021	0.755105	−	0.655604i	$-0.227586\pi$
$431$	31.3528	1.51021	0.755105	+	0.655604i	$0.227586\pi$
$432$	4.24611	0.204291
$433$	−3.78243	−0.181772	−0.0908860	−	0.995861i	$-0.528970\pi$
$433$	−3.78243	−0.181772	−0.0908860	+	0.995861i	$0.528970\pi$
$434$	−0.791637	−0.0379998
$435$	−9.91590	−0.475431
$436$	−17.8447	−0.854608
$437$	0.960579	0.0459507
$438$	−5.56883	−0.266089
$439$	−26.3071	−1.25557	−0.627785	−	0.778387i	$-0.716038\pi$
$439$	−26.3071	−1.25557	−0.627785	+	0.778387i	$0.716038\pi$
$440$	−2.46820	−0.117667
$441$	−5.29287	−0.252041
$442$	2.10480	0.100115
$443$	−8.36096	−0.397241	−0.198621	−	0.980076i	$-0.563646\pi$
$443$	−8.36096	−0.397241	−0.198621	+	0.980076i	$0.563646\pi$
$444$	3.32551	0.157822
$445$	9.87407	0.468076
$446$	11.8015	0.558815
$447$	−6.22085	−0.294236
$448$	−0.791637	−0.0374013
$449$	−17.9260	−0.845982	−0.422991	−	0.906134i	$-0.639020\pi$
$449$	−17.9260	−0.845982	−0.422991	+	0.906134i	$0.639020\pi$
$450$	0.830474	0.0391489
$451$	16.7777	0.790030
$452$	10.3377	0.486245
$453$	−1.44668	−0.0679712
$454$	12.8072	0.601074
$455$	0.791637	0.0371125
$456$	1.29257	0.0605301
$457$	−6.72627	−0.314641	−0.157321	−	0.987548i	$-0.550286\pi$
$457$	−6.72627	−0.314641	−0.157321	+	0.987548i	$0.550286\pi$
$458$	−9.07282	−0.423945
$459$	−8.93720	−0.417153
$460$	−1.45447	−0.0678151
$461$	29.0254	1.35185	0.675923	−	0.736972i	$-0.263745\pi$
$461$	29.0254	1.35185	0.675923	+	0.736972i	$0.263745\pi$
$462$	−3.82413	−0.177915
$463$	−31.3944	−1.45902	−0.729510	−	0.683970i	$-0.760252\pi$
$463$	−31.3944	−1.45902	−0.729510	+	0.683970i	$0.760252\pi$
$464$	5.06648	0.235205
$465$	−1.95716	−0.0907611
$466$	15.6742	0.726093
$467$	19.8767	0.919785	0.459893	−	0.887975i	$-0.347888\pi$
$467$	19.8767	0.919785	0.459893	+	0.887975i	$0.347888\pi$
$468$	−0.830474	−0.0383887
$469$	6.26513	0.289297
$470$	11.4037	0.526011
$471$	40.9970	1.88904
$472$	4.12863	0.190036
$473$	−29.4502	−1.35412
$474$	−14.6503	−0.672909
$475$	−0.660431	−0.0303027
$476$	1.66623	0.0763717
$477$	−2.91066	−0.133270
$478$	9.47079	0.433184
$479$	−14.3497	−0.655654	−0.327827	−	0.944738i	$-0.606316\pi$
$479$	−14.3497	−0.655654	−0.327827	+	0.944738i	$0.606316\pi$
$480$	−1.95716	−0.0893317
$481$	1.69915	0.0774747
$482$	−12.8737	−0.586381
$483$	−2.25350	−0.102538
$484$	−4.90799	−0.223091
$485$	6.94216	0.315227
$486$	8.40239	0.381140
$487$	9.40110	0.426004	0.213002	−	0.977052i	$-0.431676\pi$
$487$	9.40110	0.426004	0.213002	+	0.977052i	$0.431676\pi$
$488$	10.4297	0.472130
$489$	−28.4092	−1.28471
$490$	−6.37331	−0.287917
$491$	14.2427	0.642763	0.321381	−	0.946950i	$-0.395853\pi$
$491$	14.2427	0.642763	0.321381	+	0.946950i	$0.395853\pi$
$492$	13.3039	0.599785
$493$	−10.6639	−0.480278
$494$	0.660431	0.0297142
$495$	−2.04978	−0.0921305
$496$	1.00000	0.0449013
$497$	−6.32850	−0.283872
$498$	−14.7816	−0.662378
$499$	−7.26576	−0.325260	−0.162630	−	0.986687i	$-0.551998\pi$
$499$	−7.26576	−0.325260	−0.162630	+	0.986687i	$0.551998\pi$
$500$	1.00000	0.0447214
$501$	−8.23479	−0.367903
$502$	23.9922	1.07083
$503$	−32.1385	−1.43299	−0.716494	−	0.697594i	$-0.754254\pi$
$503$	−32.1385	−1.43299	−0.716494	+	0.697594i	$0.754254\pi$
$504$	−0.657434	−0.0292844
$505$	−6.12827	−0.272704
$506$	3.58993	0.159592
$507$	−1.95716	−0.0869205
$508$	9.52741	0.422710
$509$	−33.9677	−1.50559	−0.752797	−	0.658253i	$-0.771296\pi$
$509$	−33.9677	−1.50559	−0.752797	+	0.658253i	$0.771296\pi$
$510$	4.11942	0.182411
$511$	−2.25249	−0.0996445
$512$	1.00000	0.0441942
$513$	−2.80426	−0.123811
$514$	−12.1413	−0.535531
$515$	9.39365	0.413934
$516$	−23.3526	−1.02804
$517$	−28.1465	−1.23788
$518$	1.34511	0.0591008
$519$	25.9197	1.13775
$520$	−1.00000	−0.0438529
$521$	−4.43043	−0.194101	−0.0970504	−	0.995279i	$-0.530941\pi$
$521$	−4.43043	−0.194101	−0.0970504	+	0.995279i	$0.530941\pi$
$522$	4.20758	0.184161
$523$	1.04777	0.0458157	0.0229079	−	0.999738i	$-0.492708\pi$
$523$	1.04777	0.0458157	0.0229079	+	0.999738i	$0.492708\pi$
$524$	−10.5187	−0.459513
$525$	1.54936	0.0676196
$526$	7.68986	0.335294
$527$	−2.10480	−0.0916864
$528$	4.83066	0.210227
$529$	−20.8845	−0.908022
$530$	−3.50482	−0.152240
$531$	3.42872	0.148794
$532$	0.522822	0.0226672
$533$	6.79754	0.294434
$534$	−19.3251	−0.836280
$535$	0.333370	0.0144128
$536$	−7.91415	−0.341839
$537$	−40.2108	−1.73522
$538$	1.61533	0.0696419
$539$	15.7306	0.677565
$540$	4.24611	0.182723
$541$	−40.1068	−1.72433	−0.862164	−	0.506629i	$-0.830891\pi$
$541$	−40.1068	−1.72433	−0.862164	+	0.506629i	$0.830891\pi$
$542$	30.0037	1.28877
$543$	−44.1528	−1.89478
$544$	−2.10480	−0.0902424
$545$	−17.8447	−0.764385
$546$	−1.54936	−0.0663065
$547$	41.8430	1.78908	0.894539	−	0.446989i	$-0.147504\pi$
$547$	41.8430	1.78908	0.894539	+	0.446989i	$0.147504\pi$
$548$	7.42172	0.317040
$549$	8.66160	0.369668
$550$	−2.46820	−0.105244
$551$	−3.34606	−0.142547
$552$	2.84663	0.121161
$553$	−5.92577	−0.251989
$554$	30.4894	1.29537
$555$	3.32551	0.141160
$556$	−7.72123	−0.327453
$557$	−18.5350	−0.785354	−0.392677	−	0.919677i	$-0.628451\pi$
$557$	−18.5350	−0.785354	−0.392677	+	0.919677i	$0.628451\pi$
$558$	0.830474	0.0351568
$559$	−11.9319	−0.504665
$560$	−0.791637	−0.0334528
$561$	−10.1676	−0.429275
$562$	2.36080	0.0995844
$563$	11.7626	0.495734	0.247867	−	0.968794i	$-0.420270\pi$
$563$	11.7626	0.495734	0.247867	+	0.968794i	$0.420270\pi$
$564$	−22.3188	−0.939790
$565$	10.3377	0.434911
$566$	−17.2265	−0.724083
$567$	8.55105	0.359110
$568$	7.99420	0.335429
$569$	−45.4504	−1.90538	−0.952690	−	0.303943i	$-0.901697\pi$
$569$	−45.4504	−1.90538	−0.952690	+	0.303943i	$0.901697\pi$
$570$	1.29257	0.0541398
$571$	13.8859	0.581105	0.290553	−	0.956859i	$-0.406161\pi$
$571$	13.8859	0.581105	0.290553	+	0.956859i	$0.406161\pi$
$572$	2.46820	0.103201
$573$	−0.554787	−0.0231766
$574$	5.38118	0.224606
$575$	−1.45447	−0.0606557
$576$	0.830474	0.0346031
$577$	32.0983	1.33627	0.668134	−	0.744041i	$-0.267093\pi$
$577$	32.0983	1.33627	0.668134	+	0.744041i	$0.267093\pi$
$578$	−12.5698	−0.522836
$579$	−47.2484	−1.96358
$580$	5.06648	0.210374
$581$	−5.97888	−0.248046
$582$	−13.5869	−0.563195
$583$	8.65059	0.358271
$584$	2.84536	0.117742
$585$	−0.830474	−0.0343359
$586$	15.4159	0.636823
$587$	42.6794	1.76157	0.880783	−	0.473520i	$-0.157017\pi$
$587$	42.6794	1.76157	0.880783	+	0.473520i	$0.157017\pi$
$588$	12.4736	0.514402
$589$	−0.660431	−0.0272126
$590$	4.12863	0.169973
$591$	15.8515	0.652042
$592$	−1.69915	−0.0698348
$593$	21.6838	0.890445	0.445223	−	0.895420i	$-0.353125\pi$
$593$	21.6838	0.890445	0.445223	+	0.895420i	$0.353125\pi$
$594$	−10.4802	−0.430009
$595$	1.66623	0.0683089
$596$	3.17851	0.130197
$597$	28.6889	1.17416
$598$	1.45447	0.0594778
$599$	−7.48644	−0.305888	−0.152944	−	0.988235i	$-0.548875\pi$
$599$	−7.48644	−0.305888	−0.152944	+	0.988235i	$0.548875\pi$
$600$	−1.95716	−0.0799007
$601$	−19.1181	−0.779842	−0.389921	−	0.920848i	$-0.627498\pi$
$601$	−19.1181	−0.779842	−0.389921	+	0.920848i	$0.627498\pi$
$602$	−9.44571	−0.384979
$603$	−6.57249	−0.267653
$604$	0.739176	0.0300766
$605$	−4.90799	−0.199538
$606$	11.9940	0.487223
$607$	−45.8203	−1.85979	−0.929894	−	0.367828i	$-0.880101\pi$
$607$	−45.8203	−1.85979	−0.929894	+	0.367828i	$0.880101\pi$
$608$	−0.660431	−0.0267840
$609$	7.84979	0.318090
$610$	10.4297	0.422286
$611$	−11.4037	−0.461342
$612$	−1.74798	−0.0706579
$613$	−3.29113	−0.132928	−0.0664638	−	0.997789i	$-0.521172\pi$
$613$	−3.29113	−0.132928	−0.0664638	+	0.997789i	$0.521172\pi$
$614$	−10.9588	−0.442262
$615$	13.3039	0.536464
$616$	1.95392	0.0787256
$617$	30.5674	1.23060	0.615299	−	0.788294i	$-0.289035\pi$
$617$	30.5674	1.23060	0.615299	+	0.788294i	$0.289035\pi$
$618$	−18.3849	−0.739548
$619$	−8.30994	−0.334005	−0.167002	−	0.985957i	$-0.553409\pi$
$619$	−8.30994	−0.334005	−0.167002	+	0.985957i	$0.553409\pi$
$620$	1.00000	0.0401610
$621$	−6.17585	−0.247828
$622$	−4.03705	−0.161871
$623$	−7.81668	−0.313168
$624$	1.95716	0.0783491
$625$	1.00000	0.0400000
$626$	10.4680	0.418387
$627$	−3.19032	−0.127409
$628$	−20.9472	−0.835883
$629$	3.57637	0.142599
$630$	−0.657434	−0.0261928
$631$	17.0035	0.676897	0.338449	−	0.940985i	$-0.390098\pi$
$631$	17.0035	0.676897	0.338449	+	0.940985i	$0.390098\pi$
$632$	7.48547	0.297756
$633$	−36.7859	−1.46211
$634$	20.2623	0.804719
$635$	9.52741	0.378084
$636$	6.85949	0.271996
$637$	6.37331	0.252520
$638$	−12.5051	−0.495080
$639$	6.63898	0.262634
$640$	1.00000	0.0395285
$641$	34.6455	1.36842	0.684208	−	0.729287i	$-0.260148\pi$
$641$	34.6455	1.36842	0.684208	+	0.729287i	$0.260148\pi$
$642$	−0.652459	−0.0257505
$643$	42.0800	1.65947	0.829737	−	0.558155i	$-0.188491\pi$
$643$	42.0800	1.65947	0.829737	+	0.558155i	$0.188491\pi$
$644$	1.15141	0.0453720
$645$	−23.3526	−0.919507
$646$	1.39007	0.0546917
$647$	−35.1652	−1.38249	−0.691244	−	0.722621i	$-0.742937\pi$
$647$	−35.1652	−1.38249	−0.691244	+	0.722621i	$0.742937\pi$
$648$	−10.8017	−0.424332
$649$	−10.1903	−0.400003
$650$	−1.00000	−0.0392232
$651$	1.54936	0.0607242
$652$	14.5155	0.568473
$653$	12.5765	0.492158	0.246079	−	0.969250i	$-0.420858\pi$
$653$	12.5765	0.492158	0.246079	+	0.969250i	$0.420858\pi$
$654$	34.9250	1.36568
$655$	−10.5187	−0.411001
$656$	−6.79754	−0.265399
$657$	2.36300	0.0921894
$658$	−9.02755	−0.351931
$659$	32.3488	1.26013	0.630066	−	0.776542i	$-0.283028\pi$
$659$	32.3488	1.26013	0.630066	+	0.776542i	$0.283028\pi$
$660$	4.83066	0.188033
$661$	9.27674	0.360823	0.180412	−	0.983591i	$-0.442257\pi$
$661$	9.27674	0.360823	0.180412	+	0.983591i	$0.442257\pi$
$662$	−8.17536	−0.317744
$663$	−4.11942	−0.159985
$664$	7.55256	0.293096
$665$	0.522822	0.0202742
$666$	−1.41110	−0.0546791
$667$	−7.36905	−0.285331
$668$	4.20752	0.162794
$669$	−23.0973	−0.892994
$670$	−7.91415	−0.305750
$671$	−25.7426	−0.993781
$672$	1.54936	0.0597678
$673$	−6.30492	−0.243037	−0.121518	−	0.992589i	$-0.538776\pi$
$673$	−6.30492	−0.243037	−0.121518	+	0.992589i	$0.538776\pi$
$674$	5.70136	0.219608
$675$	4.24611	0.163433
$676$	1.00000	0.0384615
$677$	−34.2102	−1.31481	−0.657403	−	0.753540i	$-0.728345\pi$
$677$	−34.2102	−1.31481	−0.657403	+	0.753540i	$0.728345\pi$
$678$	−20.2325	−0.777026
$679$	−5.49567	−0.210904
$680$	−2.10480	−0.0807153
$681$	−25.0658	−0.960525
$682$	−2.46820	−0.0945122
$683$	29.9619	1.14646	0.573231	−	0.819394i	$-0.305690\pi$
$683$	29.9619	1.14646	0.573231	+	0.819394i	$0.305690\pi$
$684$	−0.548471	−0.0209713
$685$	7.42172	0.283569
$686$	10.5868	0.404206
$687$	17.7570	0.677470
$688$	11.9319	0.454899
$689$	3.50482	0.133523
$690$	2.84663	0.108370
$691$	5.18647	0.197303	0.0986514	−	0.995122i	$-0.468547\pi$
$691$	5.18647	0.197303	0.0986514	+	0.995122i	$0.468547\pi$
$692$	−13.2435	−0.503443
$693$	1.62268	0.0616404
$694$	−14.7378	−0.559441
$695$	−7.72123	−0.292883
$696$	−9.91590	−0.375861
$697$	14.3074	0.541933
$698$	17.1437	0.648897
$699$	−30.6769	−1.16031
$700$	−0.791637	−0.0299211
$701$	22.2960	0.842107	0.421054	−	0.907036i	$-0.361660\pi$
$701$	22.2960	0.842107	0.421054	+	0.907036i	$0.361660\pi$
$702$	−4.24611	−0.160259
$703$	1.12217	0.0423236
$704$	−2.46820	−0.0930237
$705$	−22.3188	−0.840574
$706$	20.2132	0.760734
$707$	4.85136	0.182454
$708$	−8.08039	−0.303680
$709$	−6.67159	−0.250557	−0.125278	−	0.992122i	$-0.539982\pi$
$709$	−6.67159	−0.250557	−0.125278	+	0.992122i	$0.539982\pi$
$710$	7.99420	0.300017
$711$	6.21649	0.233137
$712$	9.87407	0.370046
$713$	−1.45447	−0.0544704
$714$	−3.26109	−0.122043
$715$	2.46820	0.0923054
$716$	20.5455	0.767820
$717$	−18.5358	−0.692234
$718$	−26.7825	−0.999516
$719$	25.6634	0.957085	0.478543	−	0.878064i	$-0.341165\pi$
$719$	25.6634	0.957085	0.478543	+	0.878064i	$0.341165\pi$
$720$	0.830474	0.0309499
$721$	−7.43636	−0.276945
$722$	−18.5638	−0.690874
$723$	25.1959	0.937045
$724$	22.5596	0.838422
$725$	5.06648	0.188164
$726$	9.60573	0.356502
$727$	17.7787	0.659376	0.329688	−	0.944090i	$-0.393056\pi$
$727$	17.7787	0.659376	0.329688	+	0.944090i	$0.393056\pi$
$728$	0.791637	0.0293400
$729$	15.9604	0.591125
$730$	2.84536	0.105312
$731$	−25.1142	−0.928881
$732$	−20.4126	−0.754471
$733$	−3.13833	−0.115917	−0.0579584	−	0.998319i	$-0.518459\pi$
$733$	−3.13833	−0.115917	−0.0579584	+	0.998319i	$0.518459\pi$
$734$	−18.1619	−0.670370
$735$	12.4736	0.460095
$736$	−1.45447	−0.0536125
$737$	19.5337	0.719533
$738$	−5.64518	−0.207802
$739$	4.27436	0.157235	0.0786173	−	0.996905i	$-0.474949\pi$
$739$	4.27436	0.157235	0.0786173	+	0.996905i	$0.474949\pi$
$740$	−1.69915	−0.0624621
$741$	−1.29257	−0.0474837
$742$	2.77454	0.101857
$743$	−23.2796	−0.854045	−0.427023	−	0.904241i	$-0.640438\pi$
$743$	−23.2796	−0.854045	−0.427023	+	0.904241i	$0.640438\pi$
$744$	−1.95716	−0.0717530
$745$	3.17851	0.116452
$746$	21.9089	0.802141
$747$	6.27221	0.229488
$748$	5.19506	0.189950
$749$	−0.263908	−0.00964299
$750$	−1.95716	−0.0714654
$751$	−19.5486	−0.713339	−0.356670	−	0.934231i	$-0.616088\pi$
$751$	−19.5486	−0.713339	−0.356670	+	0.934231i	$0.616088\pi$
$752$	11.4037	0.415848
$753$	−46.9566	−1.71119
$754$	−5.06648	−0.184510
$755$	0.739176	0.0269014
$756$	−3.36138	−0.122252
$757$	32.4199	1.17832	0.589160	−	0.808016i	$-0.299459\pi$
$757$	32.4199	1.17832	0.589160	+	0.808016i	$0.299459\pi$
$758$	0.912540	0.0331450
$759$	−7.02606	−0.255030
$760$	−0.660431	−0.0239564
$761$	−37.8771	−1.37304	−0.686522	−	0.727109i	$-0.740863\pi$
$761$	−37.8771	−1.37304	−0.686522	+	0.727109i	$0.740863\pi$
$762$	−18.6467	−0.675497
$763$	14.1266	0.511416
$764$	0.283465	0.0102554
$765$	−1.74798	−0.0631983
$766$	−26.3086	−0.950568
$767$	−4.12863	−0.149076
$768$	−1.95716	−0.0706229
$769$	−21.0855	−0.760362	−0.380181	−	0.924912i	$-0.624138\pi$
$769$	−21.0855	−0.760362	−0.380181	+	0.924912i	$0.624138\pi$
$770$	1.95392	0.0704143
$771$	23.7625	0.855786
$772$	24.1413	0.868865
$773$	−29.5436	−1.06261	−0.531304	−	0.847181i	$-0.678298\pi$
$773$	−29.5436	−1.06261	−0.531304	+	0.847181i	$0.678298\pi$
$774$	9.90912	0.356176
$775$	1.00000	0.0359211
$776$	6.94216	0.249209
$777$	−2.63260	−0.0944440
$778$	9.14918	0.328014
$779$	4.48931	0.160846
$780$	1.95716	0.0700776
$781$	−19.7313	−0.706041
$782$	3.06137	0.109474
$783$	21.5128	0.768805
$784$	−6.37331	−0.227618
$785$	−20.9472	−0.747637
$786$	20.5868	0.734308
$787$	32.3859	1.15443	0.577216	−	0.816591i	$-0.304139\pi$
$787$	32.3859	1.15443	0.577216	+	0.816591i	$0.304139\pi$
$788$	−8.09922	−0.288523
$789$	−15.0503	−0.535805
$790$	7.48547	0.266321
$791$	−8.18371	−0.290979
$792$	−2.04978	−0.0728356
$793$	−10.4297	−0.370370
$794$	−10.8659	−0.385616
$795$	6.85949	0.243281
$796$	−14.6584	−0.519554
$797$	−21.4882	−0.761153	−0.380576	−	0.924750i	$-0.624274\pi$
$797$	−21.4882	−0.761153	−0.380576	+	0.924750i	$0.624274\pi$
$798$	−1.02325	−0.0362225
$799$	−24.0024	−0.849143
$800$	1.00000	0.0353553
$801$	8.20016	0.289738
$802$	−20.8460	−0.736097
$803$	−7.02292	−0.247833
$804$	15.4893	0.546264
$805$	1.15141	0.0405820
$806$	−1.00000	−0.0352235
$807$	−3.16146	−0.111289
$808$	−6.12827	−0.215592
$809$	−25.5865	−0.899572	−0.449786	−	0.893136i	$-0.648500\pi$
$809$	−25.5865	−0.899572	−0.449786	+	0.893136i	$0.648500\pi$
$810$	−10.8017	−0.379534
$811$	−49.1469	−1.72578	−0.862891	−	0.505390i	$-0.831349\pi$
$811$	−49.1469	−1.72578	−0.862891	+	0.505390i	$0.831349\pi$
$812$	−4.01081	−0.140752
$813$	−58.7221	−2.05947
$814$	4.19385	0.146994
$815$	14.5155	0.508457
$816$	4.11942	0.144209
$817$	−7.88018	−0.275693
$818$	−15.1281	−0.528943
$819$	0.657434	0.0229726
$820$	−6.79754	−0.237380
$821$	14.6876	0.512600	0.256300	−	0.966597i	$-0.417497\pi$
$821$	14.6876	0.512600	0.256300	+	0.966597i	$0.417497\pi$
$822$	−14.5255	−0.506634
$823$	5.57595	0.194365	0.0971827	−	0.995267i	$-0.469017\pi$
$823$	5.57595	0.194365	0.0971827	+	0.995267i	$0.469017\pi$
$824$	9.39365	0.327243
$825$	4.83066	0.168182
$826$	−3.26838	−0.113721
$827$	−26.1182	−0.908218	−0.454109	−	0.890946i	$-0.650042\pi$
$827$	−26.1182	−0.908218	−0.454109	+	0.890946i	$0.650042\pi$
$828$	−1.20790	−0.0419775
$829$	15.0663	0.523273	0.261636	−	0.965167i	$-0.415738\pi$
$829$	15.0663	0.523273	0.261636	+	0.965167i	$0.415738\pi$
$830$	7.55256	0.262153
$831$	−59.6727	−2.07002
$832$	−1.00000	−0.0346688
$833$	13.4145	0.464786
$834$	15.1117	0.523275
$835$	4.20752	0.145607
$836$	1.63008	0.0563773
$837$	4.24611	0.146767
$838$	−38.4968	−1.32985
$839$	−42.1563	−1.45540	−0.727699	−	0.685897i	$-0.759410\pi$
$839$	−42.1563	−1.45540	−0.727699	+	0.685897i	$0.759410\pi$
$840$	1.54936	0.0534580
$841$	−3.33082	−0.114856
$842$	0.698826	0.0240831
$843$	−4.62047	−0.159137
$844$	18.7956	0.646970
$845$	1.00000	0.0344010
$846$	9.47044	0.325600
$847$	3.88535	0.133502
$848$	−3.50482	−0.120356
$849$	33.7150	1.15709
$850$	−2.10480	−0.0721939
$851$	2.47137	0.0847175
$852$	−15.6459	−0.536021
$853$	−30.1726	−1.03309	−0.516545	−	0.856260i	$-0.672782\pi$
$853$	−30.1726	−1.03309	−0.516545	+	0.856260i	$0.672782\pi$
$854$	−8.25654	−0.282533
$855$	−0.548471	−0.0187573
$856$	0.333370	0.0113944
$857$	−35.3057	−1.20602	−0.603010	−	0.797734i	$-0.706032\pi$
$857$	−35.3057	−1.20602	−0.603010	+	0.797734i	$0.706032\pi$
$858$	−4.83066	−0.164916
$859$	−2.77183	−0.0945738	−0.0472869	−	0.998881i	$-0.515057\pi$
$859$	−2.77183	−0.0945738	−0.0472869	+	0.998881i	$0.515057\pi$
$860$	11.9319	0.406874
$861$	−10.5318	−0.358924
$862$	31.3528	1.06788
$863$	−13.9706	−0.475563	−0.237782	−	0.971319i	$-0.576420\pi$
$863$	−13.9706	−0.475563	−0.237782	+	0.971319i	$0.576420\pi$
$864$	4.24611	0.144456
$865$	−13.2435	−0.450293
$866$	−3.78243	−0.128532
$867$	24.6012	0.835499
$868$	−0.791637	−0.0268699
$869$	−18.4756	−0.626743
$870$	−9.91590	−0.336181
$871$	7.91415	0.268161
$872$	−17.8447	−0.604299
$873$	5.76528	0.195125
$874$	0.960579	0.0324921
$875$	−0.791637	−0.0267622
$876$	−5.56883	−0.188153
$877$	16.8805	0.570015	0.285008	−	0.958525i	$-0.408004\pi$
$877$	16.8805	0.570015	0.285008	+	0.958525i	$0.408004\pi$
$878$	−26.3071	−0.887822
$879$	−30.1713	−1.01765
$880$	−2.46820	−0.0832030
$881$	−24.2823	−0.818090	−0.409045	−	0.912514i	$-0.634138\pi$
$881$	−24.2823	−0.818090	−0.409045	+	0.912514i	$0.634138\pi$
$882$	−5.29287	−0.178220
$883$	−10.4072	−0.350229	−0.175115	−	0.984548i	$-0.556030\pi$
$883$	−10.4072	−0.350229	−0.175115	+	0.984548i	$0.556030\pi$
$884$	2.10480	0.0707920
$885$	−8.08039	−0.271619
$886$	−8.36096	−0.280892
$887$	40.0077	1.34333	0.671663	−	0.740857i	$-0.265580\pi$
$887$	40.0077	1.34333	0.671663	+	0.740857i	$0.265580\pi$
$888$	3.32551	0.111597
$889$	−7.54225	−0.252959
$890$	9.87407	0.330980
$891$	26.6608	0.893171
$892$	11.8015	0.395142
$893$	−7.53133	−0.252026
$894$	−6.22085	−0.208057
$895$	20.5455	0.686760
$896$	−0.791637	−0.0264467
$897$	−2.84663	−0.0950463
$898$	−17.9260	−0.598200
$899$	5.06648	0.168976
$900$	0.830474	0.0276825
$901$	7.37693	0.245761
$902$	16.7777	0.558636
$903$	18.4868	0.615201
$904$	10.3377	0.343827
$905$	22.5596	0.749907
$906$	−1.44668	−0.0480629
$907$	−50.8508	−1.68847	−0.844237	−	0.535970i	$-0.819946\pi$
$907$	−50.8508	−1.68847	−0.844237	+	0.535970i	$0.819946\pi$
$908$	12.8072	0.425023
$909$	−5.08937	−0.168804
$910$	0.791637	0.0262425
$911$	14.4487	0.478708	0.239354	−	0.970932i	$-0.423064\pi$
$911$	14.4487	0.478708	0.239354	+	0.970932i	$0.423064\pi$
$912$	1.29257	0.0428012
$913$	−18.6412	−0.616934
$914$	−6.72627	−0.222485
$915$	−20.4126	−0.674820
$916$	−9.07282	−0.299774
$917$	8.32702	0.274982
$918$	−8.93720	−0.294971
$919$	−10.4034	−0.343177	−0.171588	−	0.985169i	$-0.554890\pi$
$919$	−10.4034	−0.343177	−0.171588	+	0.985169i	$0.554890\pi$
$920$	−1.45447	−0.0479525
$921$	21.4481	0.706740
$922$	29.0254	0.955900
$923$	−7.99420	−0.263132
$924$	−3.82413	−0.125805
$925$	−1.69915	−0.0558678
$926$	−31.3944	−1.03168
$927$	7.80119	0.256225
$928$	5.06648	0.166315
$929$	14.6391	0.480292	0.240146	−	0.970737i	$-0.422805\pi$
$929$	14.6391	0.480292	0.240146	+	0.970737i	$0.422805\pi$
$930$	−1.95716	−0.0641778
$931$	4.20913	0.137949
$932$	15.6742	0.513425
$933$	7.90115	0.258672
$934$	19.8767	0.650386
$935$	5.19506	0.169897
$936$	−0.830474	−0.0271449
$937$	−38.7845	−1.26703	−0.633517	−	0.773728i	$-0.718389\pi$
$937$	−38.7845	−1.26703	−0.633517	+	0.773728i	$0.718389\pi$
$938$	6.26513	0.204564
$939$	−20.4876	−0.668588
$940$	11.4037	0.371946
$941$	11.3948	0.371461	0.185730	−	0.982601i	$-0.440535\pi$
$941$	11.3948	0.371461	0.185730	+	0.982601i	$0.440535\pi$
$942$	40.9970	1.33575
$943$	9.88684	0.321960
$944$	4.12863	0.134375
$945$	−3.36138	−0.109346
$946$	−29.4502	−0.957510
$947$	−34.5911	−1.12406	−0.562030	−	0.827117i	$-0.689979\pi$
$947$	−34.5911	−1.12406	−0.562030	+	0.827117i	$0.689979\pi$
$948$	−14.6503	−0.475818
$949$	−2.84536	−0.0923644
$950$	−0.660431	−0.0214272
$951$	−39.6566	−1.28595
$952$	1.66623	0.0540030
$953$	33.2879	1.07830	0.539150	−	0.842210i	$-0.318746\pi$
$953$	33.2879	1.07830	0.539150	+	0.842210i	$0.318746\pi$
$954$	−2.91066	−0.0942361
$955$	0.283465	0.00917272
$956$	9.47079	0.306307
$957$	24.4744	0.791146
$958$	−14.3497	−0.463618
$959$	−5.87530	−0.189723
$960$	−1.95716	−0.0631671
$961$	1.00000	0.0322581
$962$	1.69915	0.0547829
$963$	0.276855	0.00892154
$964$	−12.8737	−0.414634
$965$	24.1413	0.777137
$966$	−2.25350	−0.0725052
$967$	22.5666	0.725693	0.362846	−	0.931849i	$-0.381805\pi$
$967$	22.5666	0.725693	0.362846	+	0.931849i	$0.381805\pi$
$968$	−4.90799	−0.157749
$969$	−2.72060	−0.0873981
$970$	6.94216	0.222899
$971$	38.0569	1.22130	0.610652	−	0.791899i	$-0.290907\pi$
$971$	38.0569	1.22130	0.610652	+	0.791899i	$0.290907\pi$
$972$	8.40239	0.269507
$973$	6.11241	0.195955
$974$	9.40110	0.301231
$975$	1.95716	0.0626793
$976$	10.4297	0.333847
$977$	−26.0747	−0.834204	−0.417102	−	0.908860i	$-0.636954\pi$
$977$	−26.0747	−0.834204	−0.417102	+	0.908860i	$0.636954\pi$
$978$	−28.4092	−0.908427
$979$	−24.3712	−0.778906
$980$	−6.37331	−0.203588
$981$	−14.8196	−0.473153
$982$	14.2427	0.454502
$983$	33.5096	1.06879	0.534395	−	0.845235i	$-0.320539\pi$
$983$	33.5096	1.06879	0.534395	+	0.845235i	$0.320539\pi$
$984$	13.3039	0.424112
$985$	−8.09922	−0.258062
$986$	−10.6639	−0.339608
$987$	17.6684	0.562390
$988$	0.660431	0.0210111
$989$	−17.3546	−0.551844
$990$	−2.04978	−0.0651461
$991$	−43.8747	−1.39373	−0.696863	−	0.717204i	$-0.745421\pi$
$991$	−43.8747	−1.39373	−0.696863	+	0.717204i	$0.745421\pi$
$992$	1.00000	0.0317500
$993$	16.0005	0.507760
$994$	−6.32850	−0.200728
$995$	−14.6584	−0.464703
$996$	−14.7816	−0.468372
$997$	29.3566	0.929733	0.464867	−	0.885381i	$-0.346102\pi$
$997$	29.3566	0.929733	0.464867	+	0.885381i	$0.346102\pi$
$998$	−7.26576	−0.229994
$999$	−7.21479	−0.228266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.o.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.o.1.2	✓	8	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:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.95716 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.95716 q^{6} -0.791637 q^{7} +1.00000 q^{8} +0.830474 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.95716 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.95716 q^{6} -0.791637 q^{7} +1.00000 q^{8} +0.830474 q^{9} +1.00000 q^{10} -2.46820 q^{11} -1.95716 q^{12} -1.00000 q^{13} -0.791637 q^{14} -1.95716 q^{15} +1.00000 q^{16} -2.10480 q^{17} +0.830474 q^{18} -0.660431 q^{19} +1.00000 q^{20} +1.54936 q^{21} -2.46820 q^{22} -1.45447 q^{23} -1.95716 q^{24} +1.00000 q^{25} -1.00000 q^{26} +4.24611 q^{27} -0.791637 q^{28} +5.06648 q^{29} -1.95716 q^{30} +1.00000 q^{31} +1.00000 q^{32} +4.83066 q^{33} -2.10480 q^{34} -0.791637 q^{35} +0.830474 q^{36} -1.69915 q^{37} -0.660431 q^{38} +1.95716 q^{39} +1.00000 q^{40} -6.79754 q^{41} +1.54936 q^{42} +11.9319 q^{43} -2.46820 q^{44} +0.830474 q^{45} -1.45447 q^{46} +11.4037 q^{47} -1.95716 q^{48} -6.37331 q^{49} +1.00000 q^{50} +4.11942 q^{51} -1.00000 q^{52} -3.50482 q^{53} +4.24611 q^{54} -2.46820 q^{55} -0.791637 q^{56} +1.29257 q^{57} +5.06648 q^{58} +4.12863 q^{59} -1.95716 q^{60} +10.4297 q^{61} +1.00000 q^{62} -0.657434 q^{63} +1.00000 q^{64} -1.00000 q^{65} +4.83066 q^{66} -7.91415 q^{67} -2.10480 q^{68} +2.84663 q^{69} -0.791637 q^{70} +7.99420 q^{71} +0.830474 q^{72} +2.84536 q^{73} -1.69915 q^{74} -1.95716 q^{75} -0.660431 q^{76} +1.95392 q^{77} +1.95716 q^{78} +7.48547 q^{79} +1.00000 q^{80} -10.8017 q^{81} -6.79754 q^{82} +7.55256 q^{83} +1.54936 q^{84} -2.10480 q^{85} +11.9319 q^{86} -9.91590 q^{87} -2.46820 q^{88} +9.87407 q^{89} +0.830474 q^{90} +0.791637 q^{91} -1.45447 q^{92} -1.95716 q^{93} +11.4037 q^{94} -0.660431 q^{95} -1.95716 q^{96} +6.94216 q^{97} -6.37331 q^{98} -2.04978 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9	\( 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} + 8 q^{10} + 10 q^{11} + 3 q^{12} - 8 q^{13} + 7 q^{14} + 3 q^{15} + 8 q^{16} + 11 q^{17} + 9 q^{18} + 2 q^{19} + 8 q^{20} + 5 q^{21} + 10 q^{22} + 12 q^{23} + 3 q^{24} + 8 q^{25} - 8 q^{26} - 3 q^{27} + 7 q^{28} + 9 q^{29} + 3 q^{30} + 8 q^{31} + 8 q^{32} + 6 q^{33} + 11 q^{34} + 7 q^{35} + 9 q^{36} + 19 q^{37} + 2 q^{38} - 3 q^{39} + 8 q^{40} + 6 q^{41} + 5 q^{42} + 21 q^{43} + 10 q^{44} + 9 q^{45} + 12 q^{46} + q^{47} + 3 q^{48} - 5 q^{49} + 8 q^{50} + 17 q^{51} - 8 q^{52} + 18 q^{53} - 3 q^{54} + 10 q^{55} + 7 q^{56} - 11 q^{57} + 9 q^{58} - 4 q^{59} + 3 q^{60} + 10 q^{61} + 8 q^{62} + 22 q^{63} + 8 q^{64} - 8 q^{65} + 6 q^{66} + 8 q^{67} + 11 q^{68} - 26 q^{69} + 7 q^{70} + 18 q^{71} + 9 q^{72} + 9 q^{73} + 19 q^{74} + 3 q^{75} + 2 q^{76} + 13 q^{77} - 3 q^{78} + 14 q^{79} + 8 q^{80} + 6 q^{82} + 3 q^{83} + 5 q^{84} + 11 q^{85} + 21 q^{86} - 21 q^{87} + 10 q^{88} - 15 q^{89} + 9 q^{90} - 7 q^{91} + 12 q^{92} + 3 q^{93} + q^{94} + 2 q^{95} + 3 q^{96} + 12 q^{97} - 5 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9 + 8 * q^10 + 10 * q^11 + 3 * q^12 - 8 * q^13 + 7 * q^14 + 3 * q^15 + 8 * q^16 + 11 * q^17 + 9 * q^18 + 2 * q^19 + 8 * q^20 + 5 * q^21 + 10 * q^22 + 12 * q^23 + 3 * q^24 + 8 * q^25 - 8 * q^26 - 3 * q^27 + 7 * q^28 + 9 * q^29 + 3 * q^30 + 8 * q^31 + 8 * q^32 + 6 * q^33 + 11 * q^34 + 7 * q^35 + 9 * q^36 + 19 * q^37 + 2 * q^38 - 3 * q^39 + 8 * q^40 + 6 * q^41 + 5 * q^42 + 21 * q^43 + 10 * q^44 + 9 * q^45 + 12 * q^46 + q^47 + 3 * q^48 - 5 * q^49 + 8 * q^50 + 17 * q^51 - 8 * q^52 + 18 * q^53 - 3 * q^54 + 10 * q^55 + 7 * q^56 - 11 * q^57 + 9 * q^58 - 4 * q^59 + 3 * q^60 + 10 * q^61 + 8 * q^62 + 22 * q^63 + 8 * q^64 - 8 * q^65 + 6 * q^66 + 8 * q^67 + 11 * q^68 - 26 * q^69 + 7 * q^70 + 18 * q^71 + 9 * q^72 + 9 * q^73 + 19 * q^74 + 3 * q^75 + 2 * q^76 + 13 * q^77 - 3 * q^78 + 14 * q^79 + 8 * q^80 + 6 * q^82 + 3 * q^83 + 5 * q^84 + 11 * q^85 + 21 * q^86 - 21 * q^87 + 10 * q^88 - 15 * q^89 + 9 * q^90 - 7 * q^91 + 12 * q^92 + 3 * q^93 + q^94 + 2 * q^95 + 3 * q^96 + 12 * q^97 - 5 * q^98 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more