LMFDB - Embedded newform 8002.2.a.g.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$0$
Dimension:	$95$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.76392 q^{3} +1.00000 q^{4} -0.367238 q^{5} -2.76392 q^{6} +3.23477 q^{7} +1.00000 q^{8} +4.63923 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.76392 q^{3} +1.00000 q^{4} -0.367238 q^{5} -2.76392 q^{6} +3.23477 q^{7} +1.00000 q^{8} +4.63923 q^{9} -0.367238 q^{10} +3.53302 q^{11} -2.76392 q^{12} -1.18080 q^{13} +3.23477 q^{14} +1.01501 q^{15} +1.00000 q^{16} +2.10842 q^{17} +4.63923 q^{18} +4.57720 q^{19} -0.367238 q^{20} -8.94062 q^{21} +3.53302 q^{22} +9.27908 q^{23} -2.76392 q^{24} -4.86514 q^{25} -1.18080 q^{26} -4.53070 q^{27} +3.23477 q^{28} -3.36348 q^{29} +1.01501 q^{30} +5.80266 q^{31} +1.00000 q^{32} -9.76496 q^{33} +2.10842 q^{34} -1.18793 q^{35} +4.63923 q^{36} +5.02846 q^{37} +4.57720 q^{38} +3.26364 q^{39} -0.367238 q^{40} -1.65639 q^{41} -8.94062 q^{42} +4.98733 q^{43} +3.53302 q^{44} -1.70370 q^{45} +9.27908 q^{46} +12.2816 q^{47} -2.76392 q^{48} +3.46372 q^{49} -4.86514 q^{50} -5.82749 q^{51} -1.18080 q^{52} -6.09083 q^{53} -4.53070 q^{54} -1.29746 q^{55} +3.23477 q^{56} -12.6510 q^{57} -3.36348 q^{58} +5.16847 q^{59} +1.01501 q^{60} +4.51662 q^{61} +5.80266 q^{62} +15.0068 q^{63} +1.00000 q^{64} +0.433635 q^{65} -9.76496 q^{66} +1.52221 q^{67} +2.10842 q^{68} -25.6466 q^{69} -1.18793 q^{70} -9.36135 q^{71} +4.63923 q^{72} -4.90737 q^{73} +5.02846 q^{74} +13.4468 q^{75} +4.57720 q^{76} +11.4285 q^{77} +3.26364 q^{78} -0.0377434 q^{79} -0.367238 q^{80} -1.39522 q^{81} -1.65639 q^{82} -9.47484 q^{83} -8.94062 q^{84} -0.774291 q^{85} +4.98733 q^{86} +9.29638 q^{87} +3.53302 q^{88} -13.3012 q^{89} -1.70370 q^{90} -3.81962 q^{91} +9.27908 q^{92} -16.0381 q^{93} +12.2816 q^{94} -1.68092 q^{95} -2.76392 q^{96} -10.0972 q^{97} +3.46372 q^{98} +16.3905 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 95 q + 95 q^{2} + 24 q^{3} + 95 q^{4} + 36 q^{5} + 24 q^{6} + 21 q^{7} + 95 q^{8} + 121 q^{9}+O(q^{10}) $ 95 * q + 95 * q^2 + 24 * q^3 + 95 * q^4 + 36 * q^5 + 24 * q^6 + 21 * q^7 + 95 * q^8 + 121 * q^9	\( 95 q + 95 q^{2} + 24 q^{3} + 95 q^{4} + 36 q^{5} + 24 q^{6} + 21 q^{7} + 95 q^{8} + 121 q^{9} + 36 q^{10} + 40 q^{11} + 24 q^{12} + 52 q^{13} + 21 q^{14} + 15 q^{15} + 95 q^{16} + 84 q^{17} + 121 q^{18} + 37 q^{19} + 36 q^{20} + 36 q^{21} + 40 q^{22} + 37 q^{23} + 24 q^{24} + 133 q^{25} + 52 q^{26} + 93 q^{27} + 21 q^{28} + 66 q^{29} + 15 q^{30} + 10 q^{31} + 95 q^{32} + 63 q^{33} + 84 q^{34} + 55 q^{35} + 121 q^{36} + 49 q^{37} + 37 q^{38} + 14 q^{39} + 36 q^{40} + 98 q^{41} + 36 q^{42} + 37 q^{43} + 40 q^{44} + 97 q^{45} + 37 q^{46} + 91 q^{47} + 24 q^{48} + 170 q^{49} + 133 q^{50} + 22 q^{51} + 52 q^{52} + 70 q^{53} + 93 q^{54} - q^{55} + 21 q^{56} + 50 q^{57} + 66 q^{58} + 72 q^{59} + 15 q^{60} + 97 q^{61} + 10 q^{62} + 75 q^{63} + 95 q^{64} + 75 q^{65} + 63 q^{66} + 39 q^{67} + 84 q^{68} + 65 q^{69} + 55 q^{70} + 28 q^{71} + 121 q^{72} + 117 q^{73} + 49 q^{74} + 62 q^{75} + 37 q^{76} + 92 q^{77} + 14 q^{78} + q^{79} + 36 q^{80} + 155 q^{81} + 98 q^{82} + 117 q^{83} + 36 q^{84} + 81 q^{85} + 37 q^{86} + 46 q^{87} + 40 q^{88} + 90 q^{89} + 97 q^{90} + 65 q^{91} + 37 q^{92} + 36 q^{93} + 91 q^{94} + 38 q^{95} + 24 q^{96} + 111 q^{97} + 170 q^{98} + 97 q^{99}+O(q^{100}) \) 95 * q + 95 * q^2 + 24 * q^3 + 95 * q^4 + 36 * q^5 + 24 * q^6 + 21 * q^7 + 95 * q^8 + 121 * q^9 + 36 * q^10 + 40 * q^11 + 24 * q^12 + 52 * q^13 + 21 * q^14 + 15 * q^15 + 95 * q^16 + 84 * q^17 + 121 * q^18 + 37 * q^19 + 36 * q^20 + 36 * q^21 + 40 * q^22 + 37 * q^23 + 24 * q^24 + 133 * q^25 + 52 * q^26 + 93 * q^27 + 21 * q^28 + 66 * q^29 + 15 * q^30 + 10 * q^31 + 95 * q^32 + 63 * q^33 + 84 * q^34 + 55 * q^35 + 121 * q^36 + 49 * q^37 + 37 * q^38 + 14 * q^39 + 36 * q^40 + 98 * q^41 + 36 * q^42 + 37 * q^43 + 40 * q^44 + 97 * q^45 + 37 * q^46 + 91 * q^47 + 24 * q^48 + 170 * q^49 + 133 * q^50 + 22 * q^51 + 52 * q^52 + 70 * q^53 + 93 * q^54 - q^55 + 21 * q^56 + 50 * q^57 + 66 * q^58 + 72 * q^59 + 15 * q^60 + 97 * q^61 + 10 * q^62 + 75 * q^63 + 95 * q^64 + 75 * q^65 + 63 * q^66 + 39 * q^67 + 84 * q^68 + 65 * q^69 + 55 * q^70 + 28 * q^71 + 121 * q^72 + 117 * q^73 + 49 * q^74 + 62 * q^75 + 37 * q^76 + 92 * q^77 + 14 * q^78 + q^79 + 36 * q^80 + 155 * q^81 + 98 * q^82 + 117 * q^83 + 36 * q^84 + 81 * q^85 + 37 * q^86 + 46 * q^87 + 40 * q^88 + 90 * q^89 + 97 * q^90 + 65 * q^91 + 37 * q^92 + 36 * q^93 + 91 * q^94 + 38 * q^95 + 24 * q^96 + 111 * q^97 + 170 * q^98 + 97 * 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
$3$	−2.76392	−1.59575	−0.797874	−	0.602824i	$-0.794042\pi$
$3$	−2.76392	−1.59575	−0.797874	+	0.602824i	$0.794042\pi$
$4$	1.00000	0.500000
$5$	−0.367238	−0.164234	−0.0821169	−	0.996623i	$-0.526168\pi$
$5$	−0.367238	−0.164234	−0.0821169	+	0.996623i	$0.526168\pi$
$6$	−2.76392	−1.12836
$7$	3.23477	1.22263	0.611313	−	0.791389i	$-0.290641\pi$
$7$	3.23477	1.22263	0.611313	+	0.791389i	$0.290641\pi$
$8$	1.00000	0.353553
$9$	4.63923	1.54641
$10$	−0.367238	−0.116131
$11$	3.53302	1.06524	0.532622	−	0.846353i	$-0.321207\pi$
$11$	3.53302	1.06524	0.532622	+	0.846353i	$0.321207\pi$
$12$	−2.76392	−0.797874
$13$	−1.18080	−0.327495	−0.163748	−	0.986502i	$-0.552358\pi$
$13$	−1.18080	−0.327495	−0.163748	+	0.986502i	$0.552358\pi$
$14$	3.23477	0.864528
$15$	1.01501	0.262076
$16$	1.00000	0.250000
$17$	2.10842	0.511366	0.255683	−	0.966761i	$-0.417700\pi$
$17$	2.10842	0.511366	0.255683	+	0.966761i	$0.417700\pi$
$18$	4.63923	1.09348
$19$	4.57720	1.05008	0.525041	−	0.851077i	$-0.324050\pi$
$19$	4.57720	1.05008	0.525041	+	0.851077i	$0.324050\pi$
$20$	−0.367238	−0.0821169
$21$	−8.94062	−1.95100
$22$	3.53302	0.753242
$23$	9.27908	1.93482	0.967411	−	0.253210i	$-0.0814863\pi$
$23$	9.27908	1.93482	0.967411	+	0.253210i	$0.0814863\pi$
$24$	−2.76392	−0.564182
$25$	−4.86514	−0.973027
$26$	−1.18080	−0.231574
$27$	−4.53070	−0.871933
$28$	3.23477	0.611313
$29$	−3.36348	−0.624583	−0.312291	−	0.949986i	$-0.601097\pi$
$29$	−3.36348	−0.624583	−0.312291	+	0.949986i	$0.601097\pi$
$30$	1.01501	0.185315
$31$	5.80266	1.04219	0.521094	−	0.853499i	$-0.325524\pi$
$31$	5.80266	1.04219	0.521094	+	0.853499i	$0.325524\pi$
$32$	1.00000	0.176777
$33$	−9.76496	−1.69986
$34$	2.10842	0.361591
$35$	−1.18793	−0.200797
$36$	4.63923	0.773205
$37$	5.02846	0.826674	0.413337	−	0.910578i	$-0.364363\pi$
$37$	5.02846	0.826674	0.413337	+	0.910578i	$0.364363\pi$
$38$	4.57720	0.742519
$39$	3.26364	0.522600
$40$	−0.367238	−0.0580654
$41$	−1.65639	−0.258685	−0.129342	−	0.991600i	$-0.541287\pi$
$41$	−1.65639	−0.258685	−0.129342	+	0.991600i	$0.541287\pi$
$42$	−8.94062	−1.37957
$43$	4.98733	0.760560	0.380280	−	0.924871i	$-0.375828\pi$
$43$	4.98733	0.760560	0.380280	+	0.924871i	$0.375828\pi$
$44$	3.53302	0.532622
$45$	−1.70370	−0.253973
$46$	9.27908	1.36813
$47$	12.2816	1.79146	0.895729	−	0.444601i	$-0.146655\pi$
$47$	12.2816	1.79146	0.895729	+	0.444601i	$0.146655\pi$
$48$	−2.76392	−0.398937
$49$	3.46372	0.494817
$50$	−4.86514	−0.688034
$51$	−5.82749	−0.816012
$52$	−1.18080	−0.163748
$53$	−6.09083	−0.836640	−0.418320	−	0.908300i	$-0.637381\pi$
$53$	−6.09083	−0.836640	−0.418320	+	0.908300i	$0.637381\pi$
$54$	−4.53070	−0.616550
$55$	−1.29746	−0.174949
$56$	3.23477	0.432264
$57$	−12.6510	−1.67566
$58$	−3.36348	−0.441647
$59$	5.16847	0.672878	0.336439	−	0.941705i	$-0.390777\pi$
$59$	5.16847	0.672878	0.336439	+	0.941705i	$0.390777\pi$
$60$	1.01501	0.131038
$61$	4.51662	0.578294	0.289147	−	0.957285i	$-0.406628\pi$
$61$	4.51662	0.578294	0.289147	+	0.957285i	$0.406628\pi$
$62$	5.80266	0.736939
$63$	15.0068	1.89068
$64$	1.00000	0.125000
$65$	0.433635	0.0537858
$66$	−9.76496	−1.20198
$67$	1.52221	0.185967	0.0929837	−	0.995668i	$-0.470360\pi$
$67$	1.52221	0.185967	0.0929837	+	0.995668i	$0.470360\pi$
$68$	2.10842	0.255683
$69$	−25.6466	−3.08749
$70$	−1.18793	−0.141985
$71$	−9.36135	−1.11099	−0.555494	−	0.831521i	$-0.687471\pi$
$71$	−9.36135	−1.11099	−0.555494	+	0.831521i	$0.687471\pi$
$72$	4.63923	0.546739
$73$	−4.90737	−0.574364	−0.287182	−	0.957876i	$-0.592718\pi$
$73$	−4.90737	−0.574364	−0.287182	+	0.957876i	$0.592718\pi$
$74$	5.02846	0.584547
$75$	13.4468	1.55271
$76$	4.57720	0.525041
$77$	11.4285	1.30240
$78$	3.26364	0.369534
$79$	−0.0377434	−0.00424647	−0.00212323	−	0.999998i	$-0.500676\pi$
$79$	−0.0377434	−0.00424647	−0.00212323	+	0.999998i	$0.500676\pi$
$80$	−0.367238	−0.0410584
$81$	−1.39522	−0.155025
$82$	−1.65639	−0.182918
$83$	−9.47484	−1.04000	−0.519999	−	0.854167i	$-0.674068\pi$
$83$	−9.47484	−1.04000	−0.519999	+	0.854167i	$0.674068\pi$
$84$	−8.94062	−0.975502
$85$	−0.774291	−0.0839836
$86$	4.98733	0.537797
$87$	9.29638	0.996677
$88$	3.53302	0.376621
$89$	−13.3012	−1.40993	−0.704964	−	0.709243i	$-0.749037\pi$
$89$	−13.3012	−1.40993	−0.704964	+	0.709243i	$0.749037\pi$
$90$	−1.70370	−0.179586
$91$	−3.81962	−0.400405
$92$	9.27908	0.967411
$93$	−16.0381	−1.66307
$94$	12.2816	1.26675
$95$	−1.68092	−0.172459
$96$	−2.76392	−0.282091
$97$	−10.0972	−1.02521	−0.512605	−	0.858624i	$-0.671320\pi$
$97$	−10.0972	−1.02521	−0.512605	+	0.858624i	$0.671320\pi$
$98$	3.46372	0.349888
$99$	16.3905	1.64731
$100$	−4.86514	−0.486514
$101$	8.66433	0.862133	0.431067	−	0.902320i	$-0.358137\pi$
$101$	8.66433	0.862133	0.431067	+	0.902320i	$0.358137\pi$
$102$	−5.82749	−0.577007
$103$	5.07218	0.499777	0.249888	−	0.968275i	$-0.419606\pi$
$103$	5.07218	0.499777	0.249888	+	0.968275i	$0.419606\pi$
$104$	−1.18080	−0.115787
$105$	3.28334	0.320421
$106$	−6.09083	−0.591594
$107$	3.40721	0.329387	0.164694	−	0.986345i	$-0.447336\pi$
$107$	3.40721	0.329387	0.164694	+	0.986345i	$0.447336\pi$
$108$	−4.53070	−0.435967
$109$	11.9031	1.14011	0.570057	−	0.821605i	$-0.306921\pi$
$109$	11.9031	1.14011	0.570057	+	0.821605i	$0.306921\pi$
$110$	−1.29746	−0.123708
$111$	−13.8983	−1.31916
$112$	3.23477	0.305657
$113$	−6.04206	−0.568389	−0.284195	−	0.958767i	$-0.591726\pi$
$113$	−6.04206	−0.568389	−0.284195	+	0.958767i	$0.591726\pi$
$114$	−12.6510	−1.18487
$115$	−3.40763	−0.317763
$116$	−3.36348	−0.312291
$117$	−5.47801	−0.506442
$118$	5.16847	0.475796
$119$	6.82024	0.625210
$120$	1.01501	0.0926577
$121$	1.48220	0.134746
$122$	4.51662	0.408916
$123$	4.57812	0.412795
$124$	5.80266	0.521094
$125$	3.62285	0.324038
$126$	15.0068	1.33691
$127$	−16.8140	−1.49200	−0.746001	−	0.665945i	$-0.768029\pi$
$127$	−16.8140	−1.49200	−0.746001	+	0.665945i	$0.768029\pi$
$128$	1.00000	0.0883883
$129$	−13.7846	−1.21366
$130$	0.433635	0.0380323
$131$	21.3303	1.86363	0.931817	−	0.362929i	$-0.118223\pi$
$131$	21.3303	1.86363	0.931817	+	0.362929i	$0.118223\pi$
$132$	−9.76496	−0.849931
$133$	14.8062	1.28386
$134$	1.52221	0.131499
$135$	1.66384	0.143201
$136$	2.10842	0.180795
$137$	−13.8121	−1.18004	−0.590022	−	0.807387i	$-0.700881\pi$
$137$	−13.8121	−1.18004	−0.590022	+	0.807387i	$0.700881\pi$
$138$	−25.6466	−2.18318
$139$	−13.4322	−1.13930	−0.569651	−	0.821887i	$-0.692922\pi$
$139$	−13.4322	−1.13930	−0.569651	+	0.821887i	$0.692922\pi$
$140$	−1.18793	−0.100398
$141$	−33.9453	−2.85871
$142$	−9.36135	−0.785587
$143$	−4.17179	−0.348863
$144$	4.63923	0.386603
$145$	1.23520	0.102578
$146$	−4.90737	−0.406136
$147$	−9.57342	−0.789602
$148$	5.02846	0.413337
$149$	16.6628	1.36507	0.682535	−	0.730853i	$-0.260878\pi$
$149$	16.6628	1.36507	0.682535	+	0.730853i	$0.260878\pi$
$150$	13.4468	1.09793
$151$	−10.3801	−0.844717	−0.422358	−	0.906429i	$-0.638798\pi$
$151$	−10.3801	−0.844717	−0.422358	+	0.906429i	$0.638798\pi$
$152$	4.57720	0.371260
$153$	9.78144	0.790782
$154$	11.4285	0.920933
$155$	−2.13096	−0.171163
$156$	3.26364	0.261300
$157$	24.6567	1.96782	0.983910	−	0.178666i	$-0.0571782\pi$
$157$	24.6567	1.96782	0.983910	+	0.178666i	$0.0571782\pi$
$158$	−0.0377434	−0.00300271
$159$	16.8346	1.33507
$160$	−0.367238	−0.0290327
$161$	30.0157	2.36557
$162$	−1.39522	−0.109619
$163$	−21.1001	−1.65269	−0.826345	−	0.563164i	$-0.809584\pi$
$163$	−21.1001	−1.65269	−0.826345	+	0.563164i	$0.809584\pi$
$164$	−1.65639	−0.129342
$165$	3.58606	0.279175
$166$	−9.47484	−0.735390
$167$	−12.7282	−0.984934	−0.492467	−	0.870331i	$-0.663905\pi$
$167$	−12.7282	−0.984934	−0.492467	+	0.870331i	$0.663905\pi$
$168$	−8.94062	−0.689784
$169$	−11.6057	−0.892747
$170$	−0.774291	−0.0593854
$171$	21.2347	1.62386
$172$	4.98733	0.380280
$173$	5.27910	0.401363	0.200681	−	0.979657i	$-0.435684\pi$
$173$	5.27910	0.401363	0.200681	+	0.979657i	$0.435684\pi$
$174$	9.29638	0.704757
$175$	−15.7376	−1.18965
$176$	3.53302	0.266311
$177$	−14.2852	−1.07374
$178$	−13.3012	−0.996969
$179$	−7.91251	−0.591409	−0.295704	−	0.955279i	$-0.595554\pi$
$179$	−7.91251	−0.591409	−0.295704	+	0.955279i	$0.595554\pi$
$180$	−1.70370	−0.126986
$181$	8.10424	0.602383	0.301192	−	0.953564i	$-0.402616\pi$
$181$	8.10424	0.602383	0.301192	+	0.953564i	$0.402616\pi$
$182$	−3.81962	−0.283129
$183$	−12.4836	−0.922812
$184$	9.27908	0.684063
$185$	−1.84664	−0.135768
$186$	−16.0381	−1.17597
$187$	7.44907	0.544730
$188$	12.2816	0.895729
$189$	−14.6558	−1.06605
$190$	−1.68092	−0.121947
$191$	−15.7247	−1.13780	−0.568900	−	0.822407i	$-0.692631\pi$
$191$	−15.7247	−1.13780	−0.568900	+	0.822407i	$0.692631\pi$
$192$	−2.76392	−0.199468
$193$	−12.2094	−0.878850	−0.439425	−	0.898279i	$-0.644818\pi$
$193$	−12.2094	−0.878850	−0.439425	+	0.898279i	$0.644818\pi$
$194$	−10.0972	−0.724933
$195$	−1.19853	−0.0858286
$196$	3.46372	0.247408
$197$	−6.42042	−0.457436	−0.228718	−	0.973493i	$-0.573453\pi$
$197$	−6.42042	−0.457436	−0.228718	+	0.973493i	$0.573453\pi$
$198$	16.3905	1.16482
$199$	−9.46239	−0.670771	−0.335386	−	0.942081i	$-0.608867\pi$
$199$	−9.46239	−0.670771	−0.335386	+	0.942081i	$0.608867\pi$
$200$	−4.86514	−0.344017
$201$	−4.20726	−0.296757
$202$	8.66433	0.609620
$203$	−10.8801	−0.763632
$204$	−5.82749	−0.408006
$205$	0.608289	0.0424848
$206$	5.07218	0.353396
$207$	43.0478	2.99203
$208$	−1.18080	−0.0818738
$209$	16.1713	1.11859
$210$	3.28334	0.226572
$211$	−25.4508	−1.75210	−0.876051	−	0.482218i	$-0.839831\pi$
$211$	−25.4508	−1.75210	−0.876051	+	0.482218i	$0.839831\pi$
$212$	−6.09083	−0.418320
$213$	25.8740	1.77286
$214$	3.40721	0.232912
$215$	−1.83154	−0.124910
$216$	−4.53070	−0.308275
$217$	18.7703	1.27421
$218$	11.9031	0.806183
$219$	13.5635	0.916540
$220$	−1.29746	−0.0874745
$221$	−2.48962	−0.167470
$222$	−13.8983	−0.932790
$223$	18.7013	1.25233	0.626166	−	0.779690i	$-0.284623\pi$
$223$	18.7013	1.25233	0.626166	+	0.779690i	$0.284623\pi$
$224$	3.23477	0.216132
$225$	−22.5705	−1.50470
$226$	−6.04206	−0.401912
$227$	−24.8723	−1.65083	−0.825415	−	0.564527i	$-0.809059\pi$
$227$	−24.8723	−1.65083	−0.825415	+	0.564527i	$0.809059\pi$
$228$	−12.6510	−0.837832
$229$	12.7385	0.841781	0.420890	−	0.907112i	$-0.361718\pi$
$229$	12.7385	0.841781	0.420890	+	0.907112i	$0.361718\pi$
$230$	−3.40763	−0.224693
$231$	−31.5874	−2.07830
$232$	−3.36348	−0.220823
$233$	−10.6215	−0.695836	−0.347918	−	0.937525i	$-0.613111\pi$
$233$	−10.6215	−0.695836	−0.347918	+	0.937525i	$0.613111\pi$
$234$	−5.47801	−0.358109
$235$	−4.51027	−0.294218
$236$	5.16847	0.336439
$237$	0.104320	0.00677629
$238$	6.82024	0.442090
$239$	22.3928	1.44847	0.724234	−	0.689554i	$-0.242193\pi$
$239$	22.3928	1.44847	0.724234	+	0.689554i	$0.242193\pi$
$240$	1.01501	0.0655189
$241$	28.1660	1.81433	0.907165	−	0.420775i	$-0.138242\pi$
$241$	28.1660	1.81433	0.907165	+	0.420775i	$0.138242\pi$
$242$	1.48220	0.0952795
$243$	17.4484	1.11931
$244$	4.51662	0.289147
$245$	−1.27201	−0.0812656
$246$	4.57812	0.291890
$247$	−5.40476	−0.343897
$248$	5.80266	0.368469
$249$	26.1877	1.65958
$250$	3.62285	0.229129
$251$	24.3493	1.53691	0.768457	−	0.639901i	$-0.221025\pi$
$251$	24.3493	1.53691	0.768457	+	0.639901i	$0.221025\pi$
$252$	15.0068	0.945342
$253$	32.7832	2.06106
$254$	−16.8140	−1.05500
$255$	2.14007	0.134017
$256$	1.00000	0.0625000
$257$	−8.52032	−0.531483	−0.265742	−	0.964044i	$-0.585617\pi$
$257$	−8.52032	−0.531483	−0.265742	+	0.964044i	$0.585617\pi$
$258$	−13.7846	−0.858189
$259$	16.2659	1.01071
$260$	0.433635	0.0268929
$261$	−15.6040	−0.965862
$262$	21.3303	1.31779
$263$	7.48188	0.461352	0.230676	−	0.973031i	$-0.425906\pi$
$263$	7.48188	0.461352	0.230676	+	0.973031i	$0.425906\pi$
$264$	−9.76496	−0.600992
$265$	2.23678	0.137405
$266$	14.8062	0.907824
$267$	36.7635	2.24989
$268$	1.52221	0.0929837
$269$	−6.36107	−0.387841	−0.193920	−	0.981017i	$-0.562120\pi$
$269$	−6.36107	−0.387841	−0.193920	+	0.981017i	$0.562120\pi$
$270$	1.66384	0.101258
$271$	9.74884	0.592200	0.296100	−	0.955157i	$-0.404314\pi$
$271$	9.74884	0.592200	0.296100	+	0.955157i	$0.404314\pi$
$272$	2.10842	0.127842
$273$	10.5571	0.638945
$274$	−13.8121	−0.834417
$275$	−17.1886	−1.03651
$276$	−25.6466	−1.54374
$277$	23.1651	1.39186	0.695928	−	0.718112i	$-0.254993\pi$
$277$	23.1651	1.39186	0.695928	+	0.718112i	$0.254993\pi$
$278$	−13.4322	−0.805608
$279$	26.9199	1.61165
$280$	−1.18793	−0.0709923
$281$	14.7330	0.878898	0.439449	−	0.898268i	$-0.355174\pi$
$281$	14.7330	0.878898	0.439449	+	0.898268i	$0.355174\pi$
$282$	−33.9453	−2.02142
$283$	−17.7812	−1.05698	−0.528492	−	0.848938i	$-0.677243\pi$
$283$	−17.7812	−1.05698	−0.528492	+	0.848938i	$0.677243\pi$
$284$	−9.36135	−0.555494
$285$	4.64592	0.275201
$286$	−4.17179	−0.246683
$287$	−5.35804	−0.316275
$288$	4.63923	0.273369
$289$	−12.5546	−0.738505
$290$	1.23520	0.0725333
$291$	27.9077	1.63598
$292$	−4.90737	−0.287182
$293$	−3.03766	−0.177462	−0.0887311	−	0.996056i	$-0.528281\pi$
$293$	−3.03766	−0.177462	−0.0887311	+	0.996056i	$0.528281\pi$
$294$	−9.57342	−0.558333
$295$	−1.89806	−0.110509
$296$	5.02846	0.292274
$297$	−16.0070	−0.928822
$298$	16.6628	0.965250
$299$	−10.9568	−0.633645
$300$	13.4468	0.776353
$301$	16.1328	0.929881
$302$	−10.3801	−0.597305
$303$	−23.9475	−1.37575
$304$	4.57720	0.262520
$305$	−1.65868	−0.0949755
$306$	9.78144	0.559167
$307$	−13.0290	−0.743603	−0.371802	−	0.928312i	$-0.621260\pi$
$307$	−13.0290	−0.743603	−0.371802	+	0.928312i	$0.621260\pi$
$308$	11.4285	0.651198
$309$	−14.0191	−0.797518
$310$	−2.13096	−0.121030
$311$	14.6894	0.832957	0.416479	−	0.909146i	$-0.363264\pi$
$311$	14.6894	0.832957	0.416479	+	0.909146i	$0.363264\pi$
$312$	3.26364	0.184767
$313$	16.3404	0.923616	0.461808	−	0.886980i	$-0.347201\pi$
$313$	16.3404	0.923616	0.461808	+	0.886980i	$0.347201\pi$
$314$	24.6567	1.39146
$315$	−5.51108	−0.310514
$316$	−0.0377434	−0.00212323
$317$	26.1965	1.47134	0.735670	−	0.677340i	$-0.236868\pi$
$317$	26.1965	1.47134	0.735670	+	0.677340i	$0.236868\pi$
$318$	16.8346	0.944035
$319$	−11.8832	−0.665334
$320$	−0.367238	−0.0205292
$321$	−9.41724	−0.525619
$322$	30.0157	1.67271
$323$	9.65064	0.536976
$324$	−1.39522	−0.0775125
$325$	5.74476	0.318662
$326$	−21.1001	−1.16863
$327$	−32.8993	−1.81934
$328$	−1.65639	−0.0914588
$329$	39.7281	2.19028
$330$	3.58606	0.197406
$331$	5.33640	0.293315	0.146657	−	0.989187i	$-0.453149\pi$
$331$	5.33640	0.293315	0.146657	+	0.989187i	$0.453149\pi$
$332$	−9.47484	−0.519999
$333$	23.3282	1.27838
$334$	−12.7282	−0.696454
$335$	−0.559013	−0.0305421
$336$	−8.94062	−0.487751
$337$	3.37252	0.183713	0.0918565	−	0.995772i	$-0.470720\pi$
$337$	3.37252	0.183713	0.0918565	+	0.995772i	$0.470720\pi$
$338$	−11.6057	−0.631267
$339$	16.6998	0.907006
$340$	−0.774291	−0.0419918
$341$	20.5009	1.11019
$342$	21.2347	1.14824
$343$	−11.4391	−0.617651
$344$	4.98733	0.268899
$345$	9.41841	0.507070
$346$	5.27910	0.283806
$347$	28.7355	1.54260	0.771301	−	0.636470i	$-0.219606\pi$
$347$	28.7355	1.54260	0.771301	+	0.636470i	$0.219606\pi$
$348$	9.29638	0.498338
$349$	32.9196	1.76215	0.881073	−	0.472981i	$-0.156822\pi$
$349$	32.9196	1.76215	0.881073	+	0.472981i	$0.156822\pi$
$350$	−15.7376	−0.841209
$351$	5.34985	0.285554
$352$	3.53302	0.188310
$353$	14.3387	0.763170	0.381585	−	0.924334i	$-0.375378\pi$
$353$	14.3387	0.763170	0.381585	+	0.924334i	$0.375378\pi$
$354$	−14.2852	−0.759251
$355$	3.43784	0.182462
$356$	−13.3012	−0.704964
$357$	−18.8506	−0.997678
$358$	−7.91251	−0.418189
$359$	11.7620	0.620777	0.310388	−	0.950610i	$-0.399541\pi$
$359$	11.7620	0.620777	0.310388	+	0.950610i	$0.399541\pi$
$360$	−1.70370	−0.0897930
$361$	1.95074	0.102670
$362$	8.10424	0.425949
$363$	−4.09668	−0.215020
$364$	−3.81962	−0.200202
$365$	1.80217	0.0943299
$366$	−12.4836	−0.652527
$367$	34.4734	1.79950	0.899750	−	0.436406i	$-0.143749\pi$
$367$	34.4734	1.79950	0.899750	+	0.436406i	$0.143749\pi$
$368$	9.27908	0.483706
$369$	−7.68438	−0.400033
$370$	−1.84664	−0.0960024
$371$	−19.7024	−1.02290
$372$	−16.0381	−0.831535
$373$	−4.92017	−0.254757	−0.127378	−	0.991854i	$-0.540656\pi$
$373$	−4.92017	−0.254757	−0.127378	+	0.991854i	$0.540656\pi$
$374$	7.44907	0.385182
$375$	−10.0133	−0.517082
$376$	12.2816	0.633376
$377$	3.97160	0.204548
$378$	−14.6558	−0.753811
$379$	14.7975	0.760097	0.380048	−	0.924967i	$-0.375907\pi$
$379$	14.7975	0.760097	0.380048	+	0.924967i	$0.375907\pi$
$380$	−1.68092	−0.0862294
$381$	46.4725	2.38086
$382$	−15.7247	−0.804545
$383$	−20.2608	−1.03528	−0.517639	−	0.855599i	$-0.673189\pi$
$383$	−20.2608	−1.03528	−0.517639	+	0.855599i	$0.673189\pi$
$384$	−2.76392	−0.141045
$385$	−4.19697	−0.213897
$386$	−12.2094	−0.621441
$387$	23.1374	1.17614
$388$	−10.0972	−0.512605
$389$	4.70167	0.238384	0.119192	−	0.992871i	$-0.461970\pi$
$389$	4.70167	0.238384	0.119192	+	0.992871i	$0.461970\pi$
$390$	−1.19853	−0.0606900
$391$	19.5642	0.989403
$392$	3.46372	0.174944
$393$	−58.9550	−2.97389
$394$	−6.42042	−0.323456
$395$	0.0138608	0.000697413 0
$396$	16.3905	0.823653
$397$	−23.6257	−1.18574	−0.592870	−	0.805298i	$-0.702005\pi$
$397$	−23.6257	−1.18574	−0.592870	+	0.805298i	$0.702005\pi$
$398$	−9.46239	−0.474307
$399$	−40.9230	−2.04871
$400$	−4.86514	−0.243257
$401$	−23.5480	−1.17593	−0.587965	−	0.808887i	$-0.700071\pi$
$401$	−23.5480	−1.17593	−0.587965	+	0.808887i	$0.700071\pi$
$402$	−4.20726	−0.209839
$403$	−6.85179	−0.341312
$404$	8.66433	0.431067
$405$	0.512379	0.0254603
$406$	−10.8801	−0.539969
$407$	17.7656	0.880610
$408$	−5.82749	−0.288504
$409$	−4.20954	−0.208148	−0.104074	−	0.994570i	$-0.533188\pi$
$409$	−4.20954	−0.208148	−0.104074	+	0.994570i	$0.533188\pi$
$410$	0.608289	0.0300413
$411$	38.1754	1.88305
$412$	5.07218	0.249888
$413$	16.7188	0.822678
$414$	43.0478	2.11568
$415$	3.47952	0.170803
$416$	−1.18080	−0.0578935
$417$	37.1254	1.81804
$418$	16.1713	0.790965
$419$	−13.9493	−0.681467	−0.340734	−	0.940160i	$-0.610675\pi$
$419$	−13.9493	−0.681467	−0.340734	+	0.940160i	$0.610675\pi$
$420$	3.28334	0.160210
$421$	−36.0979	−1.75930	−0.879651	−	0.475619i	$-0.842224\pi$
$421$	−36.0979	−1.75930	−0.879651	+	0.475619i	$0.842224\pi$
$422$	−25.4508	−1.23892
$423$	56.9772	2.77033
$424$	−6.09083	−0.295797
$425$	−10.2577	−0.497573
$426$	25.8740	1.25360
$427$	14.6102	0.707038
$428$	3.40721	0.164694
$429$	11.5305	0.556697
$430$	−1.83154	−0.0883245
$431$	11.3684	0.547598	0.273799	−	0.961787i	$-0.411720\pi$
$431$	11.3684	0.547598	0.273799	+	0.961787i	$0.411720\pi$
$432$	−4.53070	−0.217983
$433$	−32.7616	−1.57442	−0.787211	−	0.616684i	$-0.788476\pi$
$433$	−32.7616	−1.57442	−0.787211	+	0.616684i	$0.788476\pi$
$434$	18.7703	0.901001
$435$	−3.41398	−0.163688
$436$	11.9031	0.570057
$437$	42.4722	2.03172
$438$	13.5635	0.648091
$439$	7.27546	0.347239	0.173619	−	0.984813i	$-0.444454\pi$
$439$	7.27546	0.347239	0.173619	+	0.984813i	$0.444454\pi$
$440$	−1.29746	−0.0618538
$441$	16.0690	0.765189
$442$	−2.48962	−0.118419
$443$	−30.9023	−1.46821	−0.734106	−	0.679035i	$-0.762399\pi$
$443$	−30.9023	−1.46821	−0.734106	+	0.679035i	$0.762399\pi$
$444$	−13.8983	−0.659582
$445$	4.88471	0.231558
$446$	18.7013	0.885532
$447$	−46.0546	−2.17831
$448$	3.23477	0.152828
$449$	−29.5435	−1.39424	−0.697122	−	0.716953i	$-0.745536\pi$
$449$	−29.5435	−1.39424	−0.697122	+	0.716953i	$0.745536\pi$
$450$	−22.5705	−1.06398
$451$	−5.85205	−0.275562
$452$	−6.04206	−0.284195
$453$	28.6896	1.34795
$454$	−24.8723	−1.16731
$455$	1.40271	0.0657600
$456$	−12.6510	−0.592437
$457$	−15.2311	−0.712479	−0.356240	−	0.934395i	$-0.615941\pi$
$457$	−15.2311	−0.712479	−0.356240	+	0.934395i	$0.615941\pi$
$458$	12.7385	0.595229
$459$	−9.55260	−0.445877
$460$	−3.40763	−0.158882
$461$	−36.7514	−1.71168	−0.855842	−	0.517237i	$-0.826960\pi$
$461$	−36.7514	−1.71168	−0.855842	+	0.517237i	$0.826960\pi$
$462$	−31.5874	−1.46958
$463$	30.5211	1.41843	0.709217	−	0.704990i	$-0.249048\pi$
$463$	30.5211	1.41843	0.709217	+	0.704990i	$0.249048\pi$
$464$	−3.36348	−0.156146
$465$	5.88979	0.273132
$466$	−10.6215	−0.492030
$467$	30.9609	1.43270	0.716349	−	0.697742i	$-0.245812\pi$
$467$	30.9609	1.43270	0.716349	+	0.697742i	$0.245812\pi$
$468$	−5.47801	−0.253221
$469$	4.92399	0.227369
$470$	−4.51027	−0.208043
$471$	−68.1491	−3.14014
$472$	5.16847	0.237898
$473$	17.6203	0.810183
$474$	0.104320	0.00479156
$475$	−22.2687	−1.02176
$476$	6.82024	0.312605
$477$	−28.2568	−1.29379
$478$	22.3928	1.02422
$479$	−27.8347	−1.27180	−0.635900	−	0.771772i	$-0.719371\pi$
$479$	−27.8347	−1.27180	−0.635900	+	0.771772i	$0.719371\pi$
$480$	1.01501	0.0463289
$481$	−5.93762	−0.270732
$482$	28.1660	1.28292
$483$	−82.9608	−3.77485
$484$	1.48220	0.0673728
$485$	3.70806	0.168374
$486$	17.4484	0.791475
$487$	6.50020	0.294552	0.147276	−	0.989095i	$-0.452949\pi$
$487$	6.50020	0.294552	0.147276	+	0.989095i	$0.452949\pi$
$488$	4.51662	0.204458
$489$	58.3190	2.63728
$490$	−1.27201	−0.0574634
$491$	32.0032	1.44428	0.722141	−	0.691745i	$-0.243158\pi$
$491$	32.0032	1.44428	0.722141	+	0.691745i	$0.243158\pi$
$492$	4.57812	0.206398
$493$	−7.09162	−0.319391
$494$	−5.40476	−0.243172
$495$	−6.01920	−0.270543
$496$	5.80266	0.260547
$497$	−30.2818	−1.35832
$498$	26.1877	1.17350
$499$	37.5620	1.68151	0.840754	−	0.541418i	$-0.182112\pi$
$499$	37.5620	1.68151	0.840754	+	0.541418i	$0.182112\pi$
$500$	3.62285	0.162019
$501$	35.1796	1.57171
$502$	24.3493	1.08676
$503$	−5.25896	−0.234485	−0.117243	−	0.993103i	$-0.537406\pi$
$503$	−5.25896	−0.234485	−0.117243	+	0.993103i	$0.537406\pi$
$504$	15.0068	0.668457
$505$	−3.18187	−0.141591
$506$	32.7832	1.45739
$507$	32.0772	1.42460
$508$	−16.8140	−0.746001
$509$	35.5105	1.57397	0.786987	−	0.616969i	$-0.211640\pi$
$509$	35.5105	1.57397	0.786987	+	0.616969i	$0.211640\pi$
$510$	2.14007	0.0947641
$511$	−15.8742	−0.702233
$512$	1.00000	0.0441942
$513$	−20.7379	−0.915601
$514$	−8.52032	−0.375815
$515$	−1.86270	−0.0820802
$516$	−13.7846	−0.606831
$517$	43.3911	1.90834
$518$	16.2659	0.714683
$519$	−14.5910	−0.640473
$520$	0.433635	0.0190162
$521$	−29.2293	−1.28056	−0.640279	−	0.768143i	$-0.721181\pi$
$521$	−29.2293	−1.28056	−0.640279	+	0.768143i	$0.721181\pi$
$522$	−15.6040	−0.682967
$523$	−6.82922	−0.298621	−0.149310	−	0.988790i	$-0.547705\pi$
$523$	−6.82922	−0.298621	−0.149310	+	0.988790i	$0.547705\pi$
$524$	21.3303	0.931817
$525$	43.4974	1.89838
$526$	7.48188	0.326225
$527$	12.2344	0.532940
$528$	−9.76496	−0.424965
$529$	63.1014	2.74354
$530$	2.23678	0.0971597
$531$	23.9777	1.04055
$532$	14.8062	0.641929
$533$	1.95587	0.0847180
$534$	36.7635	1.59091
$535$	−1.25126	−0.0540965
$536$	1.52221	0.0657494
$537$	21.8695	0.943739
$538$	−6.36107	−0.274245
$539$	12.2374	0.527101
$540$	1.66384	0.0716004
$541$	−26.0088	−1.11821	−0.559104	−	0.829098i	$-0.688855\pi$
$541$	−26.0088	−1.11821	−0.559104	+	0.829098i	$0.688855\pi$
$542$	9.74884	0.418748
$543$	−22.3994	−0.961252
$544$	2.10842	0.0903976
$545$	−4.37129	−0.187245
$546$	10.5571	0.451802
$547$	5.34793	0.228661	0.114331	−	0.993443i	$-0.463528\pi$
$547$	5.34793	0.228661	0.114331	+	0.993443i	$0.463528\pi$
$548$	−13.8121	−0.590022
$549$	20.9537	0.894281
$550$	−17.1886	−0.732925
$551$	−15.3953	−0.655863
$552$	−25.6466	−1.09159
$553$	−0.122091	−0.00519185
$554$	23.1651	0.984191
$555$	5.10396	0.216651
$556$	−13.4322	−0.569651
$557$	30.9296	1.31053	0.655265	−	0.755399i	$-0.272557\pi$
$557$	30.9296	1.31053	0.655265	+	0.755399i	$0.272557\pi$
$558$	26.9199	1.13961
$559$	−5.88904	−0.249080
$560$	−1.18793	−0.0501992
$561$	−20.5886	−0.869252
$562$	14.7330	0.621475
$563$	23.3636	0.984658	0.492329	−	0.870409i	$-0.336146\pi$
$563$	23.3636	0.984658	0.492329	+	0.870409i	$0.336146\pi$
$564$	−33.9453	−1.42936
$565$	2.21887	0.0933487
$566$	−17.7812	−0.747401
$567$	−4.51323	−0.189538
$568$	−9.36135	−0.392793
$569$	6.51200	0.272997	0.136499	−	0.990640i	$-0.456415\pi$
$569$	6.51200	0.272997	0.136499	+	0.990640i	$0.456415\pi$
$570$	4.64592	0.194596
$571$	−22.3762	−0.936416	−0.468208	−	0.883618i	$-0.655100\pi$
$571$	−22.3762	−0.936416	−0.468208	+	0.883618i	$0.655100\pi$
$572$	−4.17179	−0.174431
$573$	43.4617	1.81564
$574$	−5.35804	−0.223640
$575$	−45.1440	−1.88264
$576$	4.63923	0.193301
$577$	6.37774	0.265509	0.132754	−	0.991149i	$-0.457618\pi$
$577$	6.37774	0.265509	0.132754	+	0.991149i	$0.457618\pi$
$578$	−12.5546	−0.522202
$579$	33.7457	1.40242
$580$	1.23520	0.0512888
$581$	−30.6489	−1.27153
$582$	27.9077	1.15681
$583$	−21.5190	−0.891226
$584$	−4.90737	−0.203068
$585$	2.01173	0.0831749
$586$	−3.03766	−0.125485
$587$	22.9372	0.946719	0.473359	−	0.880869i	$-0.343041\pi$
$587$	22.9372	0.946719	0.473359	+	0.880869i	$0.343041\pi$
$588$	−9.57342	−0.394801
$589$	26.5599	1.09438
$590$	−1.89806	−0.0781418
$591$	17.7455	0.729953
$592$	5.02846	0.206669
$593$	−11.5153	−0.472878	−0.236439	−	0.971646i	$-0.575980\pi$
$593$	−11.5153	−0.472878	−0.236439	+	0.971646i	$0.575980\pi$
$594$	−16.0070	−0.656776
$595$	−2.50465	−0.102681
$596$	16.6628	0.682535
$597$	26.1533	1.07038
$598$	−10.9568	−0.448055
$599$	−23.1818	−0.947184	−0.473592	−	0.880744i	$-0.657043\pi$
$599$	−23.1818	−0.947184	−0.473592	+	0.880744i	$0.657043\pi$
$600$	13.4468	0.548964
$601$	−48.7485	−1.98849	−0.994245	−	0.107126i	$-0.965835\pi$
$601$	−48.7485	−1.98849	−0.994245	+	0.107126i	$0.965835\pi$
$602$	16.1328	0.657525
$603$	7.06188	0.287582
$604$	−10.3801	−0.422358
$605$	−0.544320	−0.0221298
$606$	−23.9475	−0.972800
$607$	46.3290	1.88044	0.940218	−	0.340573i	$-0.110621\pi$
$607$	46.3290	1.88044	0.940218	+	0.340573i	$0.110621\pi$
$608$	4.57720	0.185630
$609$	30.0716	1.21856
$610$	−1.65868	−0.0671578
$611$	−14.5021	−0.586694
$612$	9.78144	0.395391
$613$	3.62438	0.146387	0.0731937	−	0.997318i	$-0.476681\pi$
$613$	3.62438	0.146387	0.0731937	+	0.997318i	$0.476681\pi$
$614$	−13.0290	−0.525807
$615$	−1.68126	−0.0677950
$616$	11.4285	0.460467
$617$	−27.6746	−1.11414	−0.557068	−	0.830467i	$-0.688074\pi$
$617$	−27.6746	−1.11414	−0.557068	+	0.830467i	$0.688074\pi$
$618$	−14.0191	−0.563930
$619$	10.1398	0.407552	0.203776	−	0.979017i	$-0.434679\pi$
$619$	10.1398	0.407552	0.203776	+	0.979017i	$0.434679\pi$
$620$	−2.13096	−0.0855813
$621$	−42.0407	−1.68704
$622$	14.6894	0.588990
$623$	−43.0264	−1.72382
$624$	3.26364	0.130650
$625$	22.9952	0.919809
$626$	16.3404	0.653095
$627$	−44.6961	−1.78499
$628$	24.6567	0.983910
$629$	10.6021	0.422733
$630$	−5.51108	−0.219567
$631$	−22.7351	−0.905072	−0.452536	−	0.891746i	$-0.649481\pi$
$631$	−22.7351	−0.905072	−0.452536	+	0.891746i	$0.649481\pi$
$632$	−0.0377434	−0.00150135
$633$	70.3437	2.79591
$634$	26.1965	1.04039
$635$	6.17474	0.245037
$636$	16.8346	0.667533
$637$	−4.08996	−0.162050
$638$	−11.8832	−0.470462
$639$	−43.4295	−1.71804
$640$	−0.367238	−0.0145164
$641$	0.350813	0.0138563	0.00692815	−	0.999976i	$-0.497795\pi$
$641$	0.350813	0.0138563	0.00692815	+	0.999976i	$0.497795\pi$
$642$	−9.41724	−0.371669
$643$	10.8997	0.429843	0.214921	−	0.976631i	$-0.431050\pi$
$643$	10.8997	0.429843	0.214921	+	0.976631i	$0.431050\pi$
$644$	30.0157	1.18278
$645$	5.06221	0.199324
$646$	9.65064	0.379699
$647$	20.6344	0.811223	0.405612	−	0.914046i	$-0.367059\pi$
$647$	20.6344	0.811223	0.405612	+	0.914046i	$0.367059\pi$
$648$	−1.39522	−0.0548096
$649$	18.2603	0.716779
$650$	5.74476	0.225328
$651$	−51.8794	−2.03331
$652$	−21.1001	−0.826345
$653$	40.5755	1.58784	0.793922	−	0.608020i	$-0.208036\pi$
$653$	40.5755	1.58784	0.793922	+	0.608020i	$0.208036\pi$
$654$	−32.8993	−1.28646
$655$	−7.83328	−0.306072
$656$	−1.65639	−0.0646712
$657$	−22.7664	−0.888202
$658$	39.7281	1.54876
$659$	−28.9645	−1.12830	−0.564148	−	0.825674i	$-0.690795\pi$
$659$	−28.9645	−1.12830	−0.564148	+	0.825674i	$0.690795\pi$
$660$	3.58606	0.139587
$661$	−33.5960	−1.30673	−0.653367	−	0.757042i	$-0.726644\pi$
$661$	−33.5960	−1.30673	−0.653367	+	0.757042i	$0.726644\pi$
$662$	5.33640	0.207405
$663$	6.88111	0.267240
$664$	−9.47484	−0.367695
$665$	−5.43738	−0.210853
$666$	23.3282	0.903950
$667$	−31.2100	−1.20846
$668$	−12.7282	−0.492467
$669$	−51.6888	−1.99841
$670$	−0.559013	−0.0215966
$671$	15.9573	0.616025
$672$	−8.94062	−0.344892
$673$	−0.189875	−0.00731913	−0.00365957	−	0.999993i	$-0.501165\pi$
$673$	−0.189875	−0.00731913	−0.00365957	+	0.999993i	$0.501165\pi$
$674$	3.37252	0.129905
$675$	22.0425	0.848415
$676$	−11.6057	−0.446373
$677$	32.2102	1.23794	0.618970	−	0.785415i	$-0.287550\pi$
$677$	32.2102	1.23794	0.618970	+	0.785415i	$0.287550\pi$
$678$	16.6998	0.641350
$679$	−32.6619	−1.25345
$680$	−0.774291	−0.0296927
$681$	68.7448	2.63431
$682$	20.5009	0.785020
$683$	9.54716	0.365312	0.182656	−	0.983177i	$-0.441531\pi$
$683$	9.54716	0.365312	0.182656	+	0.983177i	$0.441531\pi$
$684$	21.2347	0.811928
$685$	5.07231	0.193803
$686$	−11.4391	−0.436745
$687$	−35.2080	−1.34327
$688$	4.98733	0.190140
$689$	7.19206	0.273996
$690$	9.41841	0.358553
$691$	44.3119	1.68570	0.842852	−	0.538145i	$-0.180875\pi$
$691$	44.3119	1.68570	0.842852	+	0.538145i	$0.180875\pi$
$692$	5.27910	0.200681
$693$	53.0194	2.01404
$694$	28.7355	1.09078
$695$	4.93280	0.187112
$696$	9.29638	0.352378
$697$	−3.49236	−0.132283
$698$	32.9196	1.24602
$699$	29.3568	1.11038
$700$	−15.7376	−0.594825
$701$	−33.8221	−1.27744	−0.638721	−	0.769438i	$-0.720536\pi$
$701$	−33.8221	−1.27744	−0.638721	+	0.769438i	$0.720536\pi$
$702$	5.34985	0.201917
$703$	23.0163	0.868075
$704$	3.53302	0.133156
$705$	12.4660	0.469497
$706$	14.3387	0.539643
$707$	28.0271	1.05407
$708$	−14.2852	−0.536871
$709$	46.2329	1.73631	0.868156	−	0.496292i	$-0.165305\pi$
$709$	46.2329	1.73631	0.868156	+	0.496292i	$0.165305\pi$
$710$	3.43784	0.129020
$711$	−0.175101	−0.00656678
$712$	−13.3012	−0.498485
$713$	53.8434	2.01645
$714$	−18.8506	−0.705465
$715$	1.53204	0.0572950
$716$	−7.91251	−0.295704
$717$	−61.8918	−2.31139
$718$	11.7620	0.438955
$719$	−24.9984	−0.932285	−0.466142	−	0.884710i	$-0.654357\pi$
$719$	−24.9984	−0.932285	−0.466142	+	0.884710i	$0.654357\pi$
$720$	−1.70370	−0.0634932
$721$	16.4073	0.611041
$722$	1.95074	0.0725989
$723$	−77.8484	−2.89521
$724$	8.10424	0.301192
$725$	16.3638	0.607736
$726$	−4.09668	−0.152042
$727$	32.3088	1.19827	0.599134	−	0.800649i	$-0.295512\pi$
$727$	32.3088	1.19827	0.599134	+	0.800649i	$0.295512\pi$
$728$	−3.81962	−0.141564
$729$	−44.0402	−1.63112
$730$	1.80217	0.0667013
$731$	10.5154	0.388925
$732$	−12.4836	−0.461406
$733$	1.56010	0.0576237	0.0288119	−	0.999585i	$-0.490828\pi$
$733$	1.56010	0.0576237	0.0288119	+	0.999585i	$0.490828\pi$
$734$	34.4734	1.27244
$735$	3.51572	0.129679
$736$	9.27908	0.342032
$737$	5.37799	0.198101
$738$	−7.68438	−0.282866
$739$	−36.4643	−1.34136	−0.670680	−	0.741746i	$-0.733998\pi$
$739$	−36.4643	−1.34136	−0.670680	+	0.741746i	$0.733998\pi$
$740$	−1.84664	−0.0678839
$741$	14.9383	0.548772
$742$	−19.7024	−0.723299
$743$	10.4424	0.383094	0.191547	−	0.981483i	$-0.438650\pi$
$743$	10.4424	0.383094	0.191547	+	0.981483i	$0.438650\pi$
$744$	−16.0381	−0.587984
$745$	−6.11921	−0.224191
$746$	−4.92017	−0.180140
$747$	−43.9560	−1.60826
$748$	7.44907	0.272365
$749$	11.0215	0.402718
$750$	−10.0133	−0.365632
$751$	16.8965	0.616563	0.308282	−	0.951295i	$-0.400246\pi$
$751$	16.8965	0.616563	0.308282	+	0.951295i	$0.400246\pi$
$752$	12.2816	0.447864
$753$	−67.2995	−2.45253
$754$	3.97160	0.144637
$755$	3.81195	0.138731
$756$	−14.6558	−0.533025
$757$	−23.9349	−0.869929	−0.434964	−	0.900448i	$-0.643239\pi$
$757$	−23.9349	−0.869929	−0.434964	+	0.900448i	$0.643239\pi$
$758$	14.7975	0.537470
$759$	−90.6099	−3.28893
$760$	−1.68092	−0.0609734
$761$	−13.0076	−0.471525	−0.235763	−	0.971811i	$-0.575759\pi$
$761$	−13.0076	−0.471525	−0.235763	+	0.971811i	$0.575759\pi$
$762$	46.4725	1.68352
$763$	38.5039	1.39394
$764$	−15.7247	−0.568900
$765$	−3.59211	−0.129873
$766$	−20.2608	−0.732052
$767$	−6.10294	−0.220364
$768$	−2.76392	−0.0997342
$769$	−24.5112	−0.883895	−0.441948	−	0.897041i	$-0.645712\pi$
$769$	−24.5112	−0.883895	−0.441948	+	0.897041i	$0.645712\pi$
$770$	−4.19697	−0.151248
$771$	23.5495	0.848113
$772$	−12.2094	−0.439425
$773$	4.00059	0.143891	0.0719456	−	0.997409i	$-0.477079\pi$
$773$	4.00059	0.143891	0.0719456	+	0.997409i	$0.477079\pi$
$774$	23.1374	0.831655
$775$	−28.2307	−1.01408
$776$	−10.0972	−0.362467
$777$	−44.9576	−1.61285
$778$	4.70167	0.168563
$779$	−7.58162	−0.271640
$780$	−1.19853	−0.0429143
$781$	−33.0738	−1.18347
$782$	19.5642	0.699614
$783$	15.2389	0.544595
$784$	3.46372	0.123704
$785$	−9.05488	−0.323182
$786$	−58.9550	−2.10286
$787$	−0.294818	−0.0105091	−0.00525457	−	0.999986i	$-0.501673\pi$
$787$	−0.294818	−0.0105091	−0.00525457	+	0.999986i	$0.501673\pi$
$788$	−6.42042	−0.228718
$789$	−20.6793	−0.736202
$790$	0.0138608	0.000493146 0
$791$	−19.5447	−0.694928
$792$	16.3905	0.582410
$793$	−5.33324	−0.189389
$794$	−23.6257	−0.838445
$795$	−6.18228	−0.219263
$796$	−9.46239	−0.335386
$797$	−9.69830	−0.343532	−0.171766	−	0.985138i	$-0.554947\pi$
$797$	−9.69830	−0.343532	−0.171766	+	0.985138i	$0.554947\pi$
$798$	−40.9230	−1.44866
$799$	25.8948	0.916091
$800$	−4.86514	−0.172009
$801$	−61.7075	−2.18033
$802$	−23.5480	−0.831507
$803$	−17.3378	−0.611838
$804$	−4.20726	−0.148379
$805$	−11.0229	−0.388506
$806$	−6.85179	−0.241344
$807$	17.5815	0.618896
$808$	8.66433	0.304810
$809$	−21.1072	−0.742090	−0.371045	−	0.928615i	$-0.621000\pi$
$809$	−21.1072	−0.742090	−0.371045	+	0.928615i	$0.621000\pi$
$810$	0.512379	0.0180032
$811$	−51.3780	−1.80413	−0.902063	−	0.431604i	$-0.857948\pi$
$811$	−51.3780	−1.80413	−0.902063	+	0.431604i	$0.857948\pi$
$812$	−10.8801	−0.381816
$813$	−26.9450	−0.945001
$814$	17.7656	0.622686
$815$	7.74877	0.271428
$816$	−5.82749	−0.204003
$817$	22.8280	0.798650
$818$	−4.20954	−0.147183
$819$	−17.7201	−0.619190
$820$	0.608289	0.0212424
$821$	0.642775	0.0224330	0.0112165	−	0.999937i	$-0.496430\pi$
$821$	0.642775	0.0224330	0.0112165	+	0.999937i	$0.496430\pi$
$822$	38.1754	1.33152
$823$	−4.39757	−0.153290	−0.0766448	−	0.997058i	$-0.524421\pi$
$823$	−4.39757	−0.153290	−0.0766448	+	0.997058i	$0.524421\pi$
$824$	5.07218	0.176698
$825$	47.5079	1.65401
$826$	16.7188	0.581721
$827$	12.1213	0.421499	0.210750	−	0.977540i	$-0.432409\pi$
$827$	12.1213	0.421499	0.210750	+	0.977540i	$0.432409\pi$
$828$	43.0478	1.49602
$829$	18.0947	0.628455	0.314227	−	0.949348i	$-0.398255\pi$
$829$	18.0947	0.628455	0.314227	+	0.949348i	$0.398255\pi$
$830$	3.47952	0.120776
$831$	−64.0264	−2.22105
$832$	−1.18080	−0.0409369
$833$	7.30296	0.253032
$834$	37.1254	1.28555
$835$	4.67426	0.161759
$836$	16.1713	0.559297
$837$	−26.2901	−0.908719
$838$	−13.9493	−0.481870
$839$	48.6539	1.67972	0.839859	−	0.542804i	$-0.182637\pi$
$839$	48.6539	1.67972	0.839859	+	0.542804i	$0.182637\pi$
$840$	3.28334	0.113286
$841$	−17.6870	−0.609896
$842$	−36.0979	−1.24401
$843$	−40.7208	−1.40250
$844$	−25.4508	−0.876051
$845$	4.26206	0.146619
$846$	56.9772	1.95892
$847$	4.79458	0.164744
$848$	−6.09083	−0.209160
$849$	49.1458	1.68668
$850$	−10.2577	−0.351837
$851$	46.6595	1.59947
$852$	25.8740	0.886428
$853$	−17.7721	−0.608507	−0.304253	−	0.952591i	$-0.598407\pi$
$853$	−17.7721	−0.608507	−0.304253	+	0.952591i	$0.598407\pi$
$854$	14.6102	0.499952
$855$	−7.79818	−0.266692
$856$	3.40721	0.116456
$857$	−0.0321627	−0.00109866	−0.000549328	−	1.00000i	$-0.500175\pi$
$857$	−0.0321627	−0.00109866	−0.000549328		1.00000i	$0.500175\pi$
$858$	11.5305	0.393644
$859$	−33.6131	−1.14686	−0.573432	−	0.819253i	$-0.694388\pi$
$859$	−33.6131	−1.14686	−0.573432	+	0.819253i	$0.694388\pi$
$860$	−1.83154	−0.0624548
$861$	14.8092	0.504695
$862$	11.3684	0.387210
$863$	19.6122	0.667607	0.333804	−	0.942643i	$-0.391668\pi$
$863$	19.6122	0.667607	0.333804	+	0.942643i	$0.391668\pi$
$864$	−4.53070	−0.154137
$865$	−1.93869	−0.0659173
$866$	−32.7616	−1.11328
$867$	34.6998	1.17847
$868$	18.7703	0.637104
$869$	−0.133348	−0.00452353
$870$	−3.41398	−0.115745
$871$	−1.79743	−0.0609035
$872$	11.9031	0.403091
$873$	−46.8430	−1.58540
$874$	42.4722	1.43664
$875$	11.7191	0.396177
$876$	13.5635	0.458270
$877$	15.8206	0.534224	0.267112	−	0.963665i	$-0.413931\pi$
$877$	15.8206	0.534224	0.267112	+	0.963665i	$0.413931\pi$
$878$	7.27546	0.245535
$879$	8.39585	0.283185
$880$	−1.29746	−0.0437373
$881$	37.4923	1.26315	0.631574	−	0.775316i	$-0.282409\pi$
$881$	37.4923	1.26315	0.631574	+	0.775316i	$0.282409\pi$
$882$	16.0690	0.541071
$883$	−12.8139	−0.431221	−0.215611	−	0.976479i	$-0.569174\pi$
$883$	−12.8139	−0.431221	−0.215611	+	0.976479i	$0.569174\pi$
$884$	−2.48962	−0.0837350
$885$	5.24607	0.176345
$886$	−30.9023	−1.03818
$887$	14.6607	0.492259	0.246130	−	0.969237i	$-0.420841\pi$
$887$	14.6607	0.492259	0.246130	+	0.969237i	$0.420841\pi$
$888$	−13.8983	−0.466395
$889$	−54.3894	−1.82416
$890$	4.88471	0.163736
$891$	−4.92935	−0.165139
$892$	18.7013	0.626166
$893$	56.2154	1.88118
$894$	−46.0546	−1.54030
$895$	2.90577	0.0971293
$896$	3.23477	0.108066
$897$	30.2835	1.01114
$898$	−29.5435	−0.985879
$899$	−19.5171	−0.650933
$900$	−22.5705	−0.752350
$901$	−12.8420	−0.427830
$902$	−5.85205	−0.194852
$903$	−44.5898	−1.48386
$904$	−6.04206	−0.200956
$905$	−2.97618	−0.0989317
$906$	28.6896	0.953148
$907$	−26.6199	−0.883900	−0.441950	−	0.897040i	$-0.645713\pi$
$907$	−26.6199	−0.883900	−0.441950	+	0.897040i	$0.645713\pi$
$908$	−24.8723	−0.825415
$909$	40.1958	1.33321
$910$	1.40271	0.0464993
$911$	−47.3649	−1.56927	−0.784635	−	0.619958i	$-0.787150\pi$
$911$	−47.3649	−1.56927	−0.784635	+	0.619958i	$0.787150\pi$
$912$	−12.6510	−0.418916
$913$	−33.4748	−1.10785
$914$	−15.2311	−0.503799
$915$	4.58444	0.151557
$916$	12.7385	0.420890
$917$	68.9984	2.27853
$918$	−9.55260	−0.315283
$919$	−1.19887	−0.0395472	−0.0197736	−	0.999804i	$-0.506295\pi$
$919$	−1.19887	−0.0395472	−0.0197736	+	0.999804i	$0.506295\pi$
$920$	−3.40763	−0.112346
$921$	36.0110	1.18660
$922$	−36.7514	−1.21034
$923$	11.0539	0.363843
$924$	−31.5874	−1.03915
$925$	−24.4642	−0.804377
$926$	30.5211	1.00298
$927$	23.5310	0.772860
$928$	−3.36348	−0.110412
$929$	21.5913	0.708388	0.354194	−	0.935172i	$-0.384755\pi$
$929$	21.5913	0.708388	0.354194	+	0.935172i	$0.384755\pi$
$930$	5.88979	0.193134
$931$	15.8541	0.519597
$932$	−10.6215	−0.347918
$933$	−40.6002	−1.32919
$934$	30.9609	1.01307
$935$	−2.73558	−0.0894631
$936$	−5.47801	−0.179054
$937$	−6.42677	−0.209953	−0.104977	−	0.994475i	$-0.533477\pi$
$937$	−6.42677	−0.209953	−0.104977	+	0.994475i	$0.533477\pi$
$938$	4.92399	0.160774
$939$	−45.1636	−1.47386
$940$	−4.51027	−0.147109
$941$	−57.2468	−1.86619	−0.933095	−	0.359629i	$-0.882903\pi$
$941$	−57.2468	−1.86619	−0.933095	+	0.359629i	$0.882903\pi$
$942$	−68.1491	−2.22042
$943$	−15.3698	−0.500509
$944$	5.16847	0.168219
$945$	5.38215	0.175081
$946$	17.6203	0.572886
$947$	−10.7582	−0.349595	−0.174798	−	0.984604i	$-0.555927\pi$
$947$	−10.7582	−0.349595	−0.174798	+	0.984604i	$0.555927\pi$
$948$	0.104320	0.00338815
$949$	5.79462	0.188101
$950$	−22.2687	−0.722492
$951$	−72.4048	−2.34789
$952$	6.82024	0.221045
$953$	7.85467	0.254438	0.127219	−	0.991875i	$-0.459395\pi$
$953$	7.85467	0.254438	0.127219	+	0.991875i	$0.459395\pi$
$954$	−28.2568	−0.914847
$955$	5.77470	0.186865
$956$	22.3928	0.724234
$957$	32.8443	1.06170
$958$	−27.8347	−0.899298
$959$	−44.6788	−1.44275
$960$	1.01501	0.0327595
$961$	2.67087	0.0861571
$962$	−5.93762	−0.191436
$963$	15.8068	0.509368
$964$	28.1660	0.907165
$965$	4.48375	0.144337
$966$	−82.9608	−2.66922
$967$	−15.9658	−0.513426	−0.256713	−	0.966488i	$-0.582639\pi$
$967$	−15.9658	−0.513426	−0.256713	+	0.966488i	$0.582639\pi$
$968$	1.48220	0.0476398
$969$	−26.6736	−0.856878
$970$	3.70806	0.119059
$971$	2.15847	0.0692685	0.0346342	−	0.999400i	$-0.488973\pi$
$971$	2.15847	0.0692685	0.0346342	+	0.999400i	$0.488973\pi$
$972$	17.4484	0.559657
$973$	−43.4499	−1.39294
$974$	6.50020	0.208280
$975$	−15.8780	−0.508504
$976$	4.51662	0.144574
$977$	53.9120	1.72480	0.862399	−	0.506230i	$-0.168961\pi$
$977$	53.9120	1.72480	0.862399	+	0.506230i	$0.168961\pi$
$978$	58.3190	1.86484
$979$	−46.9935	−1.50192
$980$	−1.27201	−0.0406328
$981$	55.2215	1.76309
$982$	32.0032	1.02126
$983$	27.9252	0.890674	0.445337	−	0.895363i	$-0.353084\pi$
$983$	27.9252	0.890674	0.445337	+	0.895363i	$0.353084\pi$
$984$	4.57812	0.145945
$985$	2.35782	0.0751264
$986$	−7.09162	−0.225843
$987$	−109.805	−3.49514
$988$	−5.40476	−0.171948
$989$	46.2778	1.47155
$990$	−6.01920	−0.191303
$991$	−0.147461	−0.00468427	−0.00234213	−	0.999997i	$-0.500746\pi$
$991$	−0.147461	−0.00468427	−0.00234213	+	0.999997i	$0.500746\pi$
$992$	5.80266	0.184235
$993$	−14.7494	−0.468057
$994$	−30.2818	−0.960479
$995$	3.47495	0.110163
$996$	26.1877	0.829788
$997$	−59.8197	−1.89451	−0.947255	−	0.320482i	$-0.896155\pi$
$997$	−59.8197	−1.89451	−0.947255	+	0.320482i	$0.896155\pi$
$998$	37.5620	1.18901
$999$	−22.7825	−0.720805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.g.1.8	✓	95

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.g.1.8	✓	95	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.76392 q^{3} +1.00000 q^{4} -0.367238 q^{5} -2.76392 q^{6} +3.23477 q^{7} +1.00000 q^{8} +4.63923 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.76392 q^{3} +1.00000 q^{4} -0.367238 q^{5} -2.76392 q^{6} +3.23477 q^{7} +1.00000 q^{8} +4.63923 q^{9} -0.367238 q^{10} +3.53302 q^{11} -2.76392 q^{12} -1.18080 q^{13} +3.23477 q^{14} +1.01501 q^{15} +1.00000 q^{16} +2.10842 q^{17} +4.63923 q^{18} +4.57720 q^{19} -0.367238 q^{20} -8.94062 q^{21} +3.53302 q^{22} +9.27908 q^{23} -2.76392 q^{24} -4.86514 q^{25} -1.18080 q^{26} -4.53070 q^{27} +3.23477 q^{28} -3.36348 q^{29} +1.01501 q^{30} +5.80266 q^{31} +1.00000 q^{32} -9.76496 q^{33} +2.10842 q^{34} -1.18793 q^{35} +4.63923 q^{36} +5.02846 q^{37} +4.57720 q^{38} +3.26364 q^{39} -0.367238 q^{40} -1.65639 q^{41} -8.94062 q^{42} +4.98733 q^{43} +3.53302 q^{44} -1.70370 q^{45} +9.27908 q^{46} +12.2816 q^{47} -2.76392 q^{48} +3.46372 q^{49} -4.86514 q^{50} -5.82749 q^{51} -1.18080 q^{52} -6.09083 q^{53} -4.53070 q^{54} -1.29746 q^{55} +3.23477 q^{56} -12.6510 q^{57} -3.36348 q^{58} +5.16847 q^{59} +1.01501 q^{60} +4.51662 q^{61} +5.80266 q^{62} +15.0068 q^{63} +1.00000 q^{64} +0.433635 q^{65} -9.76496 q^{66} +1.52221 q^{67} +2.10842 q^{68} -25.6466 q^{69} -1.18793 q^{70} -9.36135 q^{71} +4.63923 q^{72} -4.90737 q^{73} +5.02846 q^{74} +13.4468 q^{75} +4.57720 q^{76} +11.4285 q^{77} +3.26364 q^{78} -0.0377434 q^{79} -0.367238 q^{80} -1.39522 q^{81} -1.65639 q^{82} -9.47484 q^{83} -8.94062 q^{84} -0.774291 q^{85} +4.98733 q^{86} +9.29638 q^{87} +3.53302 q^{88} -13.3012 q^{89} -1.70370 q^{90} -3.81962 q^{91} +9.27908 q^{92} -16.0381 q^{93} +12.2816 q^{94} -1.68092 q^{95} -2.76392 q^{96} -10.0972 q^{97} +3.46372 q^{98} +16.3905 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 95 q + 95 q^{2} + 24 q^{3} + 95 q^{4} + 36 q^{5} + 24 q^{6} + 21 q^{7} + 95 q^{8} + 121 q^{9}+O(q^{10}) \) 95 * q + 95 * q^2 + 24 * q^3 + 95 * q^4 + 36 * q^5 + 24 * q^6 + 21 * q^7 + 95 * q^8 + 121 * q^9	\( 95 q + 95 q^{2} + 24 q^{3} + 95 q^{4} + 36 q^{5} + 24 q^{6} + 21 q^{7} + 95 q^{8} + 121 q^{9} + 36 q^{10} + 40 q^{11} + 24 q^{12} + 52 q^{13} + 21 q^{14} + 15 q^{15} + 95 q^{16} + 84 q^{17} + 121 q^{18} + 37 q^{19} + 36 q^{20} + 36 q^{21} + 40 q^{22} + 37 q^{23} + 24 q^{24} + 133 q^{25} + 52 q^{26} + 93 q^{27} + 21 q^{28} + 66 q^{29} + 15 q^{30} + 10 q^{31} + 95 q^{32} + 63 q^{33} + 84 q^{34} + 55 q^{35} + 121 q^{36} + 49 q^{37} + 37 q^{38} + 14 q^{39} + 36 q^{40} + 98 q^{41} + 36 q^{42} + 37 q^{43} + 40 q^{44} + 97 q^{45} + 37 q^{46} + 91 q^{47} + 24 q^{48} + 170 q^{49} + 133 q^{50} + 22 q^{51} + 52 q^{52} + 70 q^{53} + 93 q^{54} - q^{55} + 21 q^{56} + 50 q^{57} + 66 q^{58} + 72 q^{59} + 15 q^{60} + 97 q^{61} + 10 q^{62} + 75 q^{63} + 95 q^{64} + 75 q^{65} + 63 q^{66} + 39 q^{67} + 84 q^{68} + 65 q^{69} + 55 q^{70} + 28 q^{71} + 121 q^{72} + 117 q^{73} + 49 q^{74} + 62 q^{75} + 37 q^{76} + 92 q^{77} + 14 q^{78} + q^{79} + 36 q^{80} + 155 q^{81} + 98 q^{82} + 117 q^{83} + 36 q^{84} + 81 q^{85} + 37 q^{86} + 46 q^{87} + 40 q^{88} + 90 q^{89} + 97 q^{90} + 65 q^{91} + 37 q^{92} + 36 q^{93} + 91 q^{94} + 38 q^{95} + 24 q^{96} + 111 q^{97} + 170 q^{98} + 97 q^{99}+O(q^{100}) \) 95 * q + 95 * q^2 + 24 * q^3 + 95 * q^4 + 36 * q^5 + 24 * q^6 + 21 * q^7 + 95 * q^8 + 121 * q^9 + 36 * q^10 + 40 * q^11 + 24 * q^12 + 52 * q^13 + 21 * q^14 + 15 * q^15 + 95 * q^16 + 84 * q^17 + 121 * q^18 + 37 * q^19 + 36 * q^20 + 36 * q^21 + 40 * q^22 + 37 * q^23 + 24 * q^24 + 133 * q^25 + 52 * q^26 + 93 * q^27 + 21 * q^28 + 66 * q^29 + 15 * q^30 + 10 * q^31 + 95 * q^32 + 63 * q^33 + 84 * q^34 + 55 * q^35 + 121 * q^36 + 49 * q^37 + 37 * q^38 + 14 * q^39 + 36 * q^40 + 98 * q^41 + 36 * q^42 + 37 * q^43 + 40 * q^44 + 97 * q^45 + 37 * q^46 + 91 * q^47 + 24 * q^48 + 170 * q^49 + 133 * q^50 + 22 * q^51 + 52 * q^52 + 70 * q^53 + 93 * q^54 - q^55 + 21 * q^56 + 50 * q^57 + 66 * q^58 + 72 * q^59 + 15 * q^60 + 97 * q^61 + 10 * q^62 + 75 * q^63 + 95 * q^64 + 75 * q^65 + 63 * q^66 + 39 * q^67 + 84 * q^68 + 65 * q^69 + 55 * q^70 + 28 * q^71 + 121 * q^72 + 117 * q^73 + 49 * q^74 + 62 * q^75 + 37 * q^76 + 92 * q^77 + 14 * q^78 + q^79 + 36 * q^80 + 155 * q^81 + 98 * q^82 + 117 * q^83 + 36 * q^84 + 81 * q^85 + 37 * q^86 + 46 * q^87 + 40 * q^88 + 90 * q^89 + 97 * q^90 + 65 * q^91 + 37 * q^92 + 36 * q^93 + 91 * q^94 + 38 * q^95 + 24 * q^96 + 111 * q^97 + 170 * q^98 + 97 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more