LMFDB - Embedded newform 8006.2.a.c.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8006,2,Mod(1,8006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8006, 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("8006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8006 = 2 \cdot 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8006.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:	$63.9282318582$
Analytic rank:	$0$
Dimension:	$92$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.07631 q^{3} +1.00000 q^{4} +0.619812 q^{5} +2.07631 q^{6} +2.12229 q^{7} -1.00000 q^{8} +1.31108 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.07631 q^{3} +1.00000 q^{4} +0.619812 q^{5} +2.07631 q^{6} +2.12229 q^{7} -1.00000 q^{8} +1.31108 q^{9} -0.619812 q^{10} +4.13318 q^{11} -2.07631 q^{12} -5.70819 q^{13} -2.12229 q^{14} -1.28692 q^{15} +1.00000 q^{16} +7.66383 q^{17} -1.31108 q^{18} +6.77264 q^{19} +0.619812 q^{20} -4.40654 q^{21} -4.13318 q^{22} +0.578738 q^{23} +2.07631 q^{24} -4.61583 q^{25} +5.70819 q^{26} +3.50673 q^{27} +2.12229 q^{28} -1.89626 q^{29} +1.28692 q^{30} +4.89895 q^{31} -1.00000 q^{32} -8.58177 q^{33} -7.66383 q^{34} +1.31542 q^{35} +1.31108 q^{36} +10.6592 q^{37} -6.77264 q^{38} +11.8520 q^{39} -0.619812 q^{40} -6.56786 q^{41} +4.40654 q^{42} -3.96066 q^{43} +4.13318 q^{44} +0.812621 q^{45} -0.578738 q^{46} +11.0720 q^{47} -2.07631 q^{48} -2.49588 q^{49} +4.61583 q^{50} -15.9125 q^{51} -5.70819 q^{52} +8.58880 q^{53} -3.50673 q^{54} +2.56179 q^{55} -2.12229 q^{56} -14.0621 q^{57} +1.89626 q^{58} +7.77859 q^{59} -1.28692 q^{60} -8.78585 q^{61} -4.89895 q^{62} +2.78249 q^{63} +1.00000 q^{64} -3.53801 q^{65} +8.58177 q^{66} +6.83033 q^{67} +7.66383 q^{68} -1.20164 q^{69} -1.31542 q^{70} +8.68286 q^{71} -1.31108 q^{72} -9.16675 q^{73} -10.6592 q^{74} +9.58392 q^{75} +6.77264 q^{76} +8.77180 q^{77} -11.8520 q^{78} +10.4431 q^{79} +0.619812 q^{80} -11.2143 q^{81} +6.56786 q^{82} -1.26996 q^{83} -4.40654 q^{84} +4.75013 q^{85} +3.96066 q^{86} +3.93723 q^{87} -4.13318 q^{88} -5.05939 q^{89} -0.812621 q^{90} -12.1144 q^{91} +0.578738 q^{92} -10.1718 q^{93} -11.0720 q^{94} +4.19776 q^{95} +2.07631 q^{96} +4.95816 q^{97} +2.49588 q^{98} +5.41891 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q - 92 q^{2} - 2 q^{3} + 92 q^{4} + 10 q^{5} + 2 q^{6} + 8 q^{7} - 92 q^{8} + 104 q^{9}+O(q^{10}) $ 92 * q - 92 * q^2 - 2 * q^3 + 92 * q^4 + 10 * q^5 + 2 * q^6 + 8 * q^7 - 92 * q^8 + 104 * q^9	\( 92 q - 92 q^{2} - 2 q^{3} + 92 q^{4} + 10 q^{5} + 2 q^{6} + 8 q^{7} - 92 q^{8} + 104 q^{9} - 10 q^{10} + 4 q^{11} - 2 q^{12} + 40 q^{13} - 8 q^{14} + 15 q^{15} + 92 q^{16} - 14 q^{17} - 104 q^{18} + 64 q^{19} + 10 q^{20} + 54 q^{21} - 4 q^{22} - 49 q^{23} + 2 q^{24} + 116 q^{25} - 40 q^{26} - 8 q^{27} + 8 q^{28} + 39 q^{29} - 15 q^{30} + 53 q^{31} - 92 q^{32} + q^{33} + 14 q^{34} - 22 q^{35} + 104 q^{36} + 58 q^{37} - 64 q^{38} + 58 q^{39} - 10 q^{40} + 27 q^{41} - 54 q^{42} + 40 q^{43} + 4 q^{44} + 43 q^{45} + 49 q^{46} - 28 q^{47} - 2 q^{48} + 148 q^{49} - 116 q^{50} + 48 q^{51} + 40 q^{52} + 32 q^{53} + 8 q^{54} + 36 q^{55} - 8 q^{56} + 48 q^{57} - 39 q^{58} + 8 q^{59} + 15 q^{60} + 99 q^{61} - 53 q^{62} + 92 q^{64} + 13 q^{65} - q^{66} + 48 q^{67} - 14 q^{68} + 63 q^{69} + 22 q^{70} - 13 q^{71} - 104 q^{72} + 49 q^{73} - 58 q^{74} + 16 q^{75} + 64 q^{76} + 41 q^{77} - 58 q^{78} + 143 q^{79} + 10 q^{80} + 124 q^{81} - 27 q^{82} - 24 q^{83} + 54 q^{84} + 121 q^{85} - 40 q^{86} + 5 q^{87} - 4 q^{88} + 25 q^{89} - 43 q^{90} + 67 q^{91} - 49 q^{92} + 43 q^{93} + 28 q^{94} - 38 q^{95} + 2 q^{96} + 74 q^{97} - 148 q^{98} + 86 q^{99}+O(q^{100}) \) 92 * q - 92 * q^2 - 2 * q^3 + 92 * q^4 + 10 * q^5 + 2 * q^6 + 8 * q^7 - 92 * q^8 + 104 * q^9 - 10 * q^10 + 4 * q^11 - 2 * q^12 + 40 * q^13 - 8 * q^14 + 15 * q^15 + 92 * q^16 - 14 * q^17 - 104 * q^18 + 64 * q^19 + 10 * q^20 + 54 * q^21 - 4 * q^22 - 49 * q^23 + 2 * q^24 + 116 * q^25 - 40 * q^26 - 8 * q^27 + 8 * q^28 + 39 * q^29 - 15 * q^30 + 53 * q^31 - 92 * q^32 + q^33 + 14 * q^34 - 22 * q^35 + 104 * q^36 + 58 * q^37 - 64 * q^38 + 58 * q^39 - 10 * q^40 + 27 * q^41 - 54 * q^42 + 40 * q^43 + 4 * q^44 + 43 * q^45 + 49 * q^46 - 28 * q^47 - 2 * q^48 + 148 * q^49 - 116 * q^50 + 48 * q^51 + 40 * q^52 + 32 * q^53 + 8 * q^54 + 36 * q^55 - 8 * q^56 + 48 * q^57 - 39 * q^58 + 8 * q^59 + 15 * q^60 + 99 * q^61 - 53 * q^62 + 92 * q^64 + 13 * q^65 - q^66 + 48 * q^67 - 14 * q^68 + 63 * q^69 + 22 * q^70 - 13 * q^71 - 104 * q^72 + 49 * q^73 - 58 * q^74 + 16 * q^75 + 64 * q^76 + 41 * q^77 - 58 * q^78 + 143 * q^79 + 10 * q^80 + 124 * q^81 - 27 * q^82 - 24 * q^83 + 54 * q^84 + 121 * q^85 - 40 * q^86 + 5 * q^87 - 4 * q^88 + 25 * q^89 - 43 * q^90 + 67 * q^91 - 49 * q^92 + 43 * q^93 + 28 * q^94 - 38 * q^95 + 2 * q^96 + 74 * q^97 - 148 * q^98 + 86 * 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
$3$	−2.07631	−1.19876	−0.599380	−	0.800465i	$-0.704586\pi$
$3$	−2.07631	−1.19876	−0.599380	+	0.800465i	$0.704586\pi$
$4$	1.00000	0.500000
$5$	0.619812	0.277188	0.138594	−	0.990349i	$-0.455742\pi$
$5$	0.619812	0.277188	0.138594	+	0.990349i	$0.455742\pi$
$6$	2.07631	0.847651
$7$	2.12229	0.802150	0.401075	−	0.916045i	$-0.368637\pi$
$7$	2.12229	0.802150	0.401075	+	0.916045i	$0.368637\pi$
$8$	−1.00000	−0.353553
$9$	1.31108	0.437026
$10$	−0.619812	−0.196002
$11$	4.13318	1.24620	0.623100	−	0.782142i	$-0.285873\pi$
$11$	4.13318	1.24620	0.623100	+	0.782142i	$0.285873\pi$
$12$	−2.07631	−0.599380
$13$	−5.70819	−1.58317	−0.791584	−	0.611060i	$-0.790743\pi$
$13$	−5.70819	−1.58317	−0.791584	+	0.611060i	$0.790743\pi$
$14$	−2.12229	−0.567206
$15$	−1.28692	−0.332282
$16$	1.00000	0.250000
$17$	7.66383	1.85875	0.929376	−	0.369135i	$-0.120346\pi$
$17$	7.66383	1.85875	0.929376	+	0.369135i	$0.120346\pi$
$18$	−1.31108	−0.309024
$19$	6.77264	1.55375	0.776875	−	0.629655i	$-0.216804\pi$
$19$	6.77264	1.55375	0.776875	+	0.629655i	$0.216804\pi$
$20$	0.619812	0.138594
$21$	−4.40654	−0.961586
$22$	−4.13318	−0.881196
$23$	0.578738	0.120675	0.0603376	−	0.998178i	$-0.480782\pi$
$23$	0.578738	0.120675	0.0603376	+	0.998178i	$0.480782\pi$
$24$	2.07631	0.423826
$25$	−4.61583	−0.923167
$26$	5.70819	1.11947
$27$	3.50673	0.674871
$28$	2.12229	0.401075
$29$	−1.89626	−0.352126	−0.176063	−	0.984379i	$-0.556336\pi$
$29$	−1.89626	−0.352126	−0.176063	+	0.984379i	$0.556336\pi$
$30$	1.28692	0.234959
$31$	4.89895	0.879877	0.439939	−	0.898028i	$-0.355000\pi$
$31$	4.89895	0.879877	0.439939	+	0.898028i	$0.355000\pi$
$32$	−1.00000	−0.176777
$33$	−8.58177	−1.49389
$34$	−7.66383	−1.31434
$35$	1.31542	0.222347
$36$	1.31108	0.218513
$37$	10.6592	1.75237	0.876184	−	0.481976i	$-0.160081\pi$
$37$	10.6592	1.75237	0.876184	+	0.481976i	$0.160081\pi$
$38$	−6.77264	−1.09867
$39$	11.8520	1.89784
$40$	−0.619812	−0.0980009
$41$	−6.56786	−1.02573	−0.512863	−	0.858470i	$-0.671415\pi$
$41$	−6.56786	−1.02573	−0.512863	+	0.858470i	$0.671415\pi$
$42$	4.40654	0.679944
$43$	−3.96066	−0.603995	−0.301998	−	0.953309i	$-0.597654\pi$
$43$	−3.96066	−0.603995	−0.301998	+	0.953309i	$0.597654\pi$
$44$	4.13318	0.623100
$45$	0.812621	0.121138
$46$	−0.578738	−0.0853303
$47$	11.0720	1.61502	0.807511	−	0.589852i	$-0.200814\pi$
$47$	11.0720	1.61502	0.807511	+	0.589852i	$0.200814\pi$
$48$	−2.07631	−0.299690
$49$	−2.49588	−0.356555
$50$	4.61583	0.652777
$51$	−15.9125	−2.22820
$52$	−5.70819	−0.791584
$53$	8.58880	1.17976	0.589881	−	0.807490i	$-0.299175\pi$
$53$	8.58880	1.17976	0.589881	+	0.807490i	$0.299175\pi$
$54$	−3.50673	−0.477206
$55$	2.56179	0.345432
$56$	−2.12229	−0.283603
$57$	−14.0621	−1.86257
$58$	1.89626	0.248991
$59$	7.77859	1.01269	0.506343	−	0.862332i	$-0.330997\pi$
$59$	7.77859	1.01269	0.506343	+	0.862332i	$0.330997\pi$
$60$	−1.28692	−0.166141
$61$	−8.78585	−1.12491	−0.562457	−	0.826827i	$-0.690144\pi$
$61$	−8.78585	−1.12491	−0.562457	+	0.826827i	$0.690144\pi$
$62$	−4.89895	−0.622167
$63$	2.78249	0.350560
$64$	1.00000	0.125000
$65$	−3.53801	−0.438836
$66$	8.58177	1.05634
$67$	6.83033	0.834458	0.417229	−	0.908801i	$-0.363001\pi$
$67$	6.83033	0.834458	0.417229	+	0.908801i	$0.363001\pi$
$68$	7.66383	0.929376
$69$	−1.20164	−0.144661
$70$	−1.31542	−0.157223
$71$	8.68286	1.03047	0.515233	−	0.857050i	$-0.327705\pi$
$71$	8.68286	1.03047	0.515233	+	0.857050i	$0.327705\pi$
$72$	−1.31108	−0.154512
$73$	−9.16675	−1.07289	−0.536443	−	0.843936i	$-0.680232\pi$
$73$	−9.16675	−1.07289	−0.536443	+	0.843936i	$0.680232\pi$
$74$	−10.6592	−1.23911
$75$	9.58392	1.10666
$76$	6.77264	0.776875
$77$	8.77180	0.999640
$78$	−11.8520	−1.34197
$79$	10.4431	1.17494	0.587469	−	0.809247i	$-0.300124\pi$
$79$	10.4431	1.17494	0.587469	+	0.809247i	$0.300124\pi$
$80$	0.619812	0.0692971
$81$	−11.2143	−1.24603
$82$	6.56786	0.725298
$83$	−1.26996	−0.139397	−0.0696983	−	0.997568i	$-0.522204\pi$
$83$	−1.26996	−0.139397	−0.0696983	+	0.997568i	$0.522204\pi$
$84$	−4.40654	−0.480793
$85$	4.75013	0.515224
$86$	3.96066	0.427089
$87$	3.93723	0.422115
$88$	−4.13318	−0.440598
$89$	−5.05939	−0.536294	−0.268147	−	0.963378i	$-0.586411\pi$
$89$	−5.05939	−0.536294	−0.268147	+	0.963378i	$0.586411\pi$
$90$	−0.812621	−0.0856578
$91$	−12.1144	−1.26994
$92$	0.578738	0.0603376
$93$	−10.1718	−1.05476
$94$	−11.0720	−1.14199
$95$	4.19776	0.430681
$96$	2.07631	0.211913
$97$	4.95816	0.503425	0.251712	−	0.967802i	$-0.419006\pi$
$97$	4.95816	0.503425	0.251712	+	0.967802i	$0.419006\pi$
$98$	2.49588	0.252122
$99$	5.41891	0.544621
$100$	−4.61583	−0.461583
$101$	13.0349	1.29702	0.648508	−	0.761208i	$-0.275393\pi$
$101$	13.0349	1.29702	0.648508	+	0.761208i	$0.275393\pi$
$102$	15.9125	1.57557
$103$	−10.7553	−1.05975	−0.529875	−	0.848076i	$-0.677761\pi$
$103$	−10.7553	−1.05975	−0.529875	+	0.848076i	$0.677761\pi$
$104$	5.70819	0.559735
$105$	−2.73123	−0.266540
$106$	−8.58880	−0.834218
$107$	−20.0664	−1.93990	−0.969948	−	0.243311i	$-0.921766\pi$
$107$	−20.0664	−1.93990	−0.969948	+	0.243311i	$0.921766\pi$
$108$	3.50673	0.337436
$109$	−0.173859	−0.0166527	−0.00832636	−	0.999965i	$-0.502650\pi$
$109$	−0.173859	−0.0166527	−0.00832636	+	0.999965i	$0.502650\pi$
$110$	−2.56179	−0.244257
$111$	−22.1319	−2.10067
$112$	2.12229	0.200538
$113$	18.1049	1.70316	0.851582	−	0.524222i	$-0.175644\pi$
$113$	18.1049	1.70316	0.851582	+	0.524222i	$0.175644\pi$
$114$	14.0621	1.31704
$115$	0.358709	0.0334498
$116$	−1.89626	−0.176063
$117$	−7.48388	−0.691885
$118$	−7.77859	−0.716078
$119$	16.2649	1.49100
$120$	1.28692	0.117480
$121$	6.08315	0.553013
$122$	8.78585	0.795434
$123$	13.6369	1.22960
$124$	4.89895	0.439939
$125$	−5.96001	−0.533079
$126$	−2.78249	−0.247884
$127$	1.94140	0.172271	0.0861357	−	0.996283i	$-0.472548\pi$
$127$	1.94140	0.172271	0.0861357	+	0.996283i	$0.472548\pi$
$128$	−1.00000	−0.0883883
$129$	8.22358	0.724046
$130$	3.53801	0.310304
$131$	7.37250	0.644138	0.322069	−	0.946716i	$-0.395622\pi$
$131$	7.37250	0.644138	0.322069	+	0.946716i	$0.395622\pi$
$132$	−8.58177	−0.746947
$133$	14.3735	1.24634
$134$	−6.83033	−0.590051
$135$	2.17352	0.187066
$136$	−7.66383	−0.657168
$137$	−1.73402	−0.148148	−0.0740738	−	0.997253i	$-0.523600\pi$
$137$	−1.73402	−0.148148	−0.0740738	+	0.997253i	$0.523600\pi$
$138$	1.20164	0.102291
$139$	19.2174	1.63000	0.815000	−	0.579460i	$-0.196737\pi$
$139$	19.2174	1.63000	0.815000	+	0.579460i	$0.196737\pi$
$140$	1.31542	0.111173
$141$	−22.9890	−1.93602
$142$	−8.68286	−0.728649
$143$	−23.5930	−1.97294
$144$	1.31108	0.109256
$145$	−1.17532	−0.0976053
$146$	9.16675	0.758645
$147$	5.18223	0.427423
$148$	10.6592	0.876184
$149$	−14.6899	−1.20344	−0.601721	−	0.798706i	$-0.705518\pi$
$149$	−14.6899	−1.20344	−0.601721	+	0.798706i	$0.705518\pi$
$150$	−9.58392	−0.782523
$151$	6.56625	0.534354	0.267177	−	0.963648i	$-0.413909\pi$
$151$	6.56625	0.534354	0.267177	+	0.963648i	$0.413909\pi$
$152$	−6.77264	−0.549333
$153$	10.0479	0.812322
$154$	−8.77180	−0.706852
$155$	3.03643	0.243892
$156$	11.8520	0.948920
$157$	−15.6820	−1.25156	−0.625781	−	0.779999i	$-0.715219\pi$
$157$	−15.6820	−1.25156	−0.625781	+	0.779999i	$0.715219\pi$
$158$	−10.4431	−0.830807
$159$	−17.8330	−1.41425
$160$	−0.619812	−0.0490004
$161$	1.22825	0.0967997
$162$	11.2143	0.881079
$163$	−6.72384	−0.526652	−0.263326	−	0.964707i	$-0.584819\pi$
$163$	−6.72384	−0.526652	−0.263326	+	0.964707i	$0.584819\pi$
$164$	−6.56786	−0.512863
$165$	−5.31908	−0.414090
$166$	1.26996	0.0985684
$167$	8.75081	0.677158	0.338579	−	0.940938i	$-0.390054\pi$
$167$	8.75081	0.677158	0.338579	+	0.940938i	$0.390054\pi$
$168$	4.40654	0.339972
$169$	19.5835	1.50642
$170$	−4.75013	−0.364319
$171$	8.87945	0.679028
$172$	−3.96066	−0.301998
$173$	−15.5173	−1.17976	−0.589878	−	0.807493i	$-0.700824\pi$
$173$	−15.5173	−1.17976	−0.589878	+	0.807493i	$0.700824\pi$
$174$	−3.93723	−0.298480
$175$	−9.79614	−0.740519
$176$	4.13318	0.311550
$177$	−16.1508	−1.21397
$178$	5.05939	0.379217
$179$	−21.6060	−1.61491	−0.807453	−	0.589932i	$-0.799155\pi$
$179$	−21.6060	−1.61491	−0.807453	+	0.589932i	$0.799155\pi$
$180$	0.812621	0.0605692
$181$	14.0455	1.04399	0.521997	−	0.852947i	$-0.325187\pi$
$181$	14.0455	1.04399	0.521997	+	0.852947i	$0.325187\pi$
$182$	12.1144	0.897983
$183$	18.2422	1.34850
$184$	−0.578738	−0.0426651
$185$	6.60673	0.485736
$186$	10.1718	0.745829
$187$	31.6759	2.31637
$188$	11.0720	0.807511
$189$	7.44231	0.541348
$190$	−4.19776	−0.304538
$191$	−23.6236	−1.70935	−0.854673	−	0.519166i	$-0.826243\pi$
$191$	−23.6236	−1.70935	−0.854673	+	0.519166i	$0.826243\pi$
$192$	−2.07631	−0.149845
$193$	4.91520	0.353804	0.176902	−	0.984229i	$-0.443392\pi$
$193$	4.91520	0.353804	0.176902	+	0.984229i	$0.443392\pi$
$194$	−4.95816	−0.355975
$195$	7.34601	0.526059
$196$	−2.49588	−0.178277
$197$	15.4192	1.09857	0.549286	−	0.835635i	$-0.314900\pi$
$197$	15.4192	1.09857	0.549286	+	0.835635i	$0.314900\pi$
$198$	−5.41891	−0.385105
$199$	−2.48034	−0.175827	−0.0879134	−	0.996128i	$-0.528020\pi$
$199$	−2.48034	−0.175827	−0.0879134	+	0.996128i	$0.528020\pi$
$200$	4.61583	0.326389
$201$	−14.1819	−1.00031
$202$	−13.0349	−0.917129
$203$	−4.02441	−0.282458
$204$	−15.9125	−1.11410
$205$	−4.07084	−0.284320
$206$	10.7553	0.749356
$207$	0.758770	0.0527382
$208$	−5.70819	−0.395792
$209$	27.9925	1.93628
$210$	2.73123	0.188473
$211$	5.41635	0.372877	0.186438	−	0.982467i	$-0.440306\pi$
$211$	5.41635	0.372877	0.186438	+	0.982467i	$0.440306\pi$
$212$	8.58880	0.589881
$213$	−18.0283	−1.23528
$214$	20.0664	1.37171
$215$	−2.45487	−0.167421
$216$	−3.50673	−0.238603
$217$	10.3970	0.705794
$218$	0.173859	0.0117752
$219$	19.0330	1.28613
$220$	2.56179	0.172716
$221$	−43.7466	−2.94272
$222$	22.1319	1.48540
$223$	−22.5204	−1.50808	−0.754039	−	0.656830i	$-0.771897\pi$
$223$	−22.5204	−1.50808	−0.754039	+	0.656830i	$0.771897\pi$
$224$	−2.12229	−0.141802
$225$	−6.05171	−0.403447
$226$	−18.1049	−1.20432
$227$	0.434007	0.0288060	0.0144030	−	0.999896i	$-0.495415\pi$
$227$	0.434007	0.0288060	0.0144030	+	0.999896i	$0.495415\pi$
$228$	−14.0621	−0.931286
$229$	4.25464	0.281154	0.140577	−	0.990070i	$-0.455104\pi$
$229$	4.25464	0.281154	0.140577	+	0.990070i	$0.455104\pi$
$230$	−0.358709	−0.0236526
$231$	−18.2130	−1.19833
$232$	1.89626	0.124495
$233$	−5.33330	−0.349396	−0.174698	−	0.984622i	$-0.555895\pi$
$233$	−5.33330	−0.349396	−0.174698	+	0.984622i	$0.555895\pi$
$234$	7.48388	0.489237
$235$	6.86258	0.447665
$236$	7.77859	0.506343
$237$	−21.6831	−1.40847
$238$	−16.2649	−1.05429
$239$	7.63898	0.494125	0.247062	−	0.969000i	$-0.420535\pi$
$239$	7.63898	0.494125	0.247062	+	0.969000i	$0.420535\pi$
$240$	−1.28692	−0.0830706
$241$	8.75487	0.563951	0.281975	−	0.959422i	$-0.409010\pi$
$241$	8.75487	0.563951	0.281975	+	0.959422i	$0.409010\pi$
$242$	−6.08315	−0.391039
$243$	12.7642	0.818825
$244$	−8.78585	−0.562457
$245$	−1.54698	−0.0988328
$246$	−13.6369	−0.869459
$247$	−38.6595	−2.45985
$248$	−4.89895	−0.311084
$249$	2.63684	0.167103
$250$	5.96001	0.376944
$251$	−4.32487	−0.272984	−0.136492	−	0.990641i	$-0.543583\pi$
$251$	−4.32487	−0.272984	−0.136492	+	0.990641i	$0.543583\pi$
$252$	2.78249	0.175280
$253$	2.39203	0.150385
$254$	−1.94140	−0.121814
$255$	−9.86276	−0.617630
$256$	1.00000	0.0625000
$257$	15.0188	0.936845	0.468423	−	0.883505i	$-0.344822\pi$
$257$	15.0188	0.936845	0.468423	+	0.883505i	$0.344822\pi$
$258$	−8.22358	−0.511978
$259$	22.6220	1.40566
$260$	−3.53801	−0.219418
$261$	−2.48614	−0.153888
$262$	−7.37250	−0.455474
$263$	−11.4598	−0.706643	−0.353322	−	0.935502i	$-0.614948\pi$
$263$	−11.4598	−0.706643	−0.353322	+	0.935502i	$0.614948\pi$
$264$	8.58177	0.528171
$265$	5.32344	0.327016
$266$	−14.3735	−0.881296
$267$	10.5049	0.642888
$268$	6.83033	0.417229
$269$	18.8723	1.15066	0.575332	−	0.817920i	$-0.304873\pi$
$269$	18.8723	1.15066	0.575332	+	0.817920i	$0.304873\pi$
$270$	−2.17352	−0.132276
$271$	−10.1495	−0.616541	−0.308270	−	0.951299i	$-0.599750\pi$
$271$	−10.1495	−0.616541	−0.308270	+	0.951299i	$0.599750\pi$
$272$	7.66383	0.464688
$273$	25.1534	1.52235
$274$	1.73402	0.104756
$275$	−19.0781	−1.15045
$276$	−1.20164	−0.0723303
$277$	−25.3443	−1.52279	−0.761396	−	0.648287i	$-0.775486\pi$
$277$	−25.3443	−1.52279	−0.761396	+	0.648287i	$0.775486\pi$
$278$	−19.2174	−1.15258
$279$	6.42290	0.384529
$280$	−1.31542	−0.0786115
$281$	−20.0125	−1.19384	−0.596922	−	0.802299i	$-0.703610\pi$
$281$	−20.0125	−1.19384	−0.596922	+	0.802299i	$0.703610\pi$
$282$	22.9890	1.36898
$283$	−20.8823	−1.24133	−0.620663	−	0.784078i	$-0.713136\pi$
$283$	−20.8823	−1.24133	−0.620663	+	0.784078i	$0.713136\pi$
$284$	8.68286	0.515233
$285$	−8.71587	−0.516284
$286$	23.5930	1.39508
$287$	−13.9389	−0.822787
$288$	−1.31108	−0.0772559
$289$	41.7343	2.45496
$290$	1.17532	0.0690174
$291$	−10.2947	−0.603485
$292$	−9.16675	−0.536443
$293$	5.14643	0.300658	0.150329	−	0.988636i	$-0.451967\pi$
$293$	5.14643	0.300658	0.150329	+	0.988636i	$0.451967\pi$
$294$	−5.18223	−0.302234
$295$	4.82127	0.280705
$296$	−10.6592	−0.619556
$297$	14.4939	0.841024
$298$	14.6899	0.850963
$299$	−3.30355	−0.191049
$300$	9.58392	0.553328
$301$	−8.40568	−0.484495
$302$	−6.56625	−0.377845
$303$	−27.0644	−1.55481
$304$	6.77264	0.388437
$305$	−5.44558	−0.311813
$306$	−10.0479	−0.574398
$307$	−10.0076	−0.571164	−0.285582	−	0.958354i	$-0.592187\pi$
$307$	−10.0076	−0.571164	−0.285582	+	0.958354i	$0.592187\pi$
$308$	8.77180	0.499820
$309$	22.3313	1.27039
$310$	−3.03643	−0.172457
$311$	2.18590	0.123951	0.0619754	−	0.998078i	$-0.480260\pi$
$311$	2.18590	0.123951	0.0619754	+	0.998078i	$0.480260\pi$
$312$	−11.8520	−0.670987
$313$	−22.5262	−1.27326	−0.636629	−	0.771170i	$-0.719672\pi$
$313$	−22.5262	−1.27326	−0.636629	+	0.771170i	$0.719672\pi$
$314$	15.6820	0.884987
$315$	1.72462	0.0971712
$316$	10.4431	0.587469
$317$	−13.5939	−0.763509	−0.381755	−	0.924264i	$-0.624680\pi$
$317$	−13.5939	−0.763509	−0.381755	+	0.924264i	$0.624680\pi$
$318$	17.8330	1.00003
$319$	−7.83757	−0.438820
$320$	0.619812	0.0346485
$321$	41.6642	2.32547
$322$	−1.22825	−0.0684477
$323$	51.9043	2.88803
$324$	−11.2143	−0.623017
$325$	26.3481	1.46153
$326$	6.72384	0.372399
$327$	0.360987	0.0199626
$328$	6.56786	0.362649
$329$	23.4981	1.29549
$330$	5.31908	0.292806
$331$	19.5256	1.07322	0.536612	−	0.843829i	$-0.319704\pi$
$331$	19.5256	1.07322	0.536612	+	0.843829i	$0.319704\pi$
$332$	−1.26996	−0.0696983
$333$	13.9751	0.765830
$334$	−8.75081	−0.478823
$335$	4.23352	0.231302
$336$	−4.40654	−0.240396
$337$	−7.57805	−0.412803	−0.206401	−	0.978467i	$-0.566175\pi$
$337$	−7.57805	−0.412803	−0.206401	+	0.978467i	$0.566175\pi$
$338$	−19.5835	−1.06520
$339$	−37.5914	−2.04168
$340$	4.75013	0.257612
$341$	20.2482	1.09650
$342$	−8.87945	−0.480146
$343$	−20.1530	−1.08816
$344$	3.96066	0.213545
$345$	−0.744792	−0.0400982
$346$	15.5173	0.834213
$347$	−6.01228	−0.322756	−0.161378	−	0.986893i	$-0.551594\pi$
$347$	−6.01228	−0.322756	−0.161378	+	0.986893i	$0.551594\pi$
$348$	3.93723	0.211057
$349$	23.5146	1.25871	0.629354	−	0.777119i	$-0.283320\pi$
$349$	23.5146	1.25871	0.629354	+	0.777119i	$0.283320\pi$
$350$	9.79614	0.523626
$351$	−20.0171	−1.06843
$352$	−4.13318	−0.220299
$353$	−19.1995	−1.02189	−0.510943	−	0.859615i	$-0.670704\pi$
$353$	−19.1995	−1.02189	−0.510943	+	0.859615i	$0.670704\pi$
$354$	16.1508	0.858405
$355$	5.38174	0.285633
$356$	−5.05939	−0.268147
$357$	−33.7710	−1.78735
$358$	21.6060	1.14191
$359$	15.8925	0.838772	0.419386	−	0.907808i	$-0.362245\pi$
$359$	15.8925	0.838772	0.419386	+	0.907808i	$0.362245\pi$
$360$	−0.812621	−0.0428289
$361$	26.8686	1.41414
$362$	−14.0455	−0.738215
$363$	−12.6305	−0.662930
$364$	−12.1144	−0.634970
$365$	−5.68166	−0.297392
$366$	−18.2422	−0.953534
$367$	−28.0729	−1.46539	−0.732696	−	0.680556i	$-0.761738\pi$
$367$	−28.0729	−1.46539	−0.732696	+	0.680556i	$0.761738\pi$
$368$	0.578738	0.0301688
$369$	−8.61096	−0.448269
$370$	−6.60673	−0.343467
$371$	18.2279	0.946347
$372$	−10.1718	−0.527381
$373$	3.37640	0.174823	0.0874117	−	0.996172i	$-0.472140\pi$
$373$	3.37640	0.174823	0.0874117	+	0.996172i	$0.472140\pi$
$374$	−31.6759	−1.63792
$375$	12.3748	0.639034
$376$	−11.0720	−0.570997
$377$	10.8242	0.557475
$378$	−7.44231	−0.382791
$379$	−2.73564	−0.140520	−0.0702602	−	0.997529i	$-0.522383\pi$
$379$	−2.73564	−0.140520	−0.0702602	+	0.997529i	$0.522383\pi$
$380$	4.19776	0.215341
$381$	−4.03096	−0.206512
$382$	23.6236	1.20869
$383$	−29.0148	−1.48259	−0.741293	−	0.671182i	$-0.765787\pi$
$383$	−29.0148	−1.48259	−0.741293	+	0.671182i	$0.765787\pi$
$384$	2.07631	0.105956
$385$	5.43687	0.277088
$386$	−4.91520	−0.250177
$387$	−5.19273	−0.263961
$388$	4.95816	0.251712
$389$	−23.8843	−1.21098	−0.605490	−	0.795853i	$-0.707023\pi$
$389$	−23.8843	−1.21098	−0.605490	+	0.795853i	$0.707023\pi$
$390$	−7.34601	−0.371980
$391$	4.43535	0.224305
$392$	2.49588	0.126061
$393$	−15.3076	−0.772167
$394$	−15.4192	−0.776807
$395$	6.47275	0.325679
$396$	5.41891	0.272311
$397$	−21.0189	−1.05491	−0.527455	−	0.849583i	$-0.676854\pi$
$397$	−21.0189	−1.05491	−0.527455	+	0.849583i	$0.676854\pi$
$398$	2.48034	0.124328
$399$	−29.8439	−1.49406
$400$	−4.61583	−0.230792
$401$	6.86026	0.342585	0.171293	−	0.985220i	$-0.445206\pi$
$401$	6.86026	0.342585	0.171293	+	0.985220i	$0.445206\pi$
$402$	14.1819	0.707329
$403$	−27.9642	−1.39299
$404$	13.0349	0.648508
$405$	−6.95076	−0.345386
$406$	4.02441	0.199728
$407$	44.0565	2.18380
$408$	15.9125	0.787786
$409$	20.7929	1.02814	0.514072	−	0.857747i	$-0.328136\pi$
$409$	20.7929	1.02814	0.514072	+	0.857747i	$0.328136\pi$
$410$	4.07084	0.201044
$411$	3.60037	0.177593
$412$	−10.7553	−0.529875
$413$	16.5084	0.812327
$414$	−0.758770	−0.0372915
$415$	−0.787139	−0.0386391
$416$	5.70819	0.279867
$417$	−39.9014	−1.95398
$418$	−27.9925	−1.36916
$419$	7.75666	0.378938	0.189469	−	0.981887i	$-0.439323\pi$
$419$	7.75666	0.378938	0.189469	+	0.981887i	$0.439323\pi$
$420$	−2.73123	−0.133270
$421$	31.0612	1.51383	0.756916	−	0.653512i	$-0.226705\pi$
$421$	31.0612	1.51383	0.756916	+	0.653512i	$0.226705\pi$
$422$	−5.41635	−0.263664
$423$	14.5163	0.705806
$424$	−8.58880	−0.417109
$425$	−35.3749	−1.71594
$426$	18.0283	0.873475
$427$	−18.6461	−0.902350
$428$	−20.0664	−0.969948
$429$	48.9864	2.36509
$430$	2.45487	0.118384
$431$	26.3775	1.27056	0.635280	−	0.772282i	$-0.280885\pi$
$431$	26.3775	1.27056	0.635280	+	0.772282i	$0.280885\pi$
$432$	3.50673	0.168718
$433$	−4.99465	−0.240028	−0.120014	−	0.992772i	$-0.538294\pi$
$433$	−4.99465	−0.240028	−0.120014	+	0.992772i	$0.538294\pi$
$434$	−10.3970	−0.499072
$435$	2.44034	0.117005
$436$	−0.173859	−0.00832636
$437$	3.91958	0.187499
$438$	−19.0330	−0.909434
$439$	12.9224	0.616751	0.308376	−	0.951265i	$-0.400215\pi$
$439$	12.9224	0.616751	0.308376	+	0.951265i	$0.400215\pi$
$440$	−2.56179	−0.122129
$441$	−3.27229	−0.155823
$442$	43.7466	2.08081
$443$	11.5909	0.550701	0.275350	−	0.961344i	$-0.411206\pi$
$443$	11.5909	0.550701	0.275350	+	0.961344i	$0.411206\pi$
$444$	−22.1319	−1.05033
$445$	−3.13587	−0.148655
$446$	22.5204	1.06637
$447$	30.5008	1.44264
$448$	2.12229	0.100269
$449$	−14.0300	−0.662118	−0.331059	−	0.943610i	$-0.607406\pi$
$449$	−14.0300	−0.662118	−0.331059	+	0.943610i	$0.607406\pi$
$450$	6.05171	0.285280
$451$	−27.1461	−1.27826
$452$	18.1049	0.851582
$453$	−13.6336	−0.640562
$454$	−0.434007	−0.0203689
$455$	−7.50868	−0.352012
$456$	14.0621	0.658519
$457$	17.4430	0.815947	0.407974	−	0.912994i	$-0.366236\pi$
$457$	17.4430	0.815947	0.407974	+	0.912994i	$0.366236\pi$
$458$	−4.25464	−0.198806
$459$	26.8750	1.25442
$460$	0.358709	0.0167249
$461$	−35.8735	−1.67080	−0.835399	−	0.549644i	$-0.814763\pi$
$461$	−35.8735	−1.67080	−0.835399	+	0.549644i	$0.814763\pi$
$462$	18.2130	0.847346
$463$	0.0122851	0.000570937 0	0.000285469	−	1.00000i	$-0.499909\pi$
$463$	0.0122851	0.000570937 0	0.000285469		1.00000i	$0.499909\pi$
$464$	−1.89626	−0.0880316
$465$	−6.30457	−0.292368
$466$	5.33330	0.247060
$467$	−23.0251	−1.06548	−0.532738	−	0.846280i	$-0.678837\pi$
$467$	−23.0251	−1.06548	−0.532738	+	0.846280i	$0.678837\pi$
$468$	−7.48388	−0.345943
$469$	14.4959	0.669361
$470$	−6.86258	−0.316547
$471$	32.5608	1.50032
$472$	−7.77859	−0.358039
$473$	−16.3701	−0.752699
$474$	21.6831	0.995938
$475$	−31.2614	−1.43437
$476$	16.2649	0.745499
$477$	11.2606	0.515586
$478$	−7.63898	−0.349399
$479$	13.3949	0.612030	0.306015	−	0.952027i	$-0.401004\pi$
$479$	13.3949	0.612030	0.306015	+	0.952027i	$0.401004\pi$
$480$	1.28692	0.0587398
$481$	−60.8450	−2.77429
$482$	−8.75487	−0.398773
$483$	−2.55023	−0.116040
$484$	6.08315	0.276507
$485$	3.07313	0.139543
$486$	−12.7642	−0.578997
$487$	21.0447	0.953627	0.476814	−	0.879004i	$-0.341792\pi$
$487$	21.0447	0.953627	0.476814	+	0.879004i	$0.341792\pi$
$488$	8.78585	0.397717
$489$	13.9608	0.631329
$490$	1.54698	0.0698853
$491$	7.87216	0.355266	0.177633	−	0.984097i	$-0.443156\pi$
$491$	7.87216	0.355266	0.177633	+	0.984097i	$0.443156\pi$
$492$	13.6369	0.614800
$493$	−14.5326	−0.654515
$494$	38.6595	1.73937
$495$	3.35871	0.150963
$496$	4.89895	0.219969
$497$	18.4275	0.826588
$498$	−2.63684	−0.118160
$499$	19.7552	0.884363	0.442181	−	0.896926i	$-0.354205\pi$
$499$	19.7552	0.884363	0.442181	+	0.896926i	$0.354205\pi$
$500$	−5.96001	−0.266540
$501$	−18.1694	−0.811750
$502$	4.32487	0.193029
$503$	12.7516	0.568565	0.284282	−	0.958741i	$-0.408245\pi$
$503$	12.7516	0.568565	0.284282	+	0.958741i	$0.408245\pi$
$504$	−2.78249	−0.123942
$505$	8.07916	0.359518
$506$	−2.39203	−0.106339
$507$	−40.6615	−1.80584
$508$	1.94140	0.0861357
$509$	27.2944	1.20980	0.604900	−	0.796301i	$-0.293213\pi$
$509$	27.2944	1.20980	0.604900	+	0.796301i	$0.293213\pi$
$510$	9.86276	0.436731
$511$	−19.4545	−0.860616
$512$	−1.00000	−0.0441942
$513$	23.7498	1.04858
$514$	−15.0188	−0.662450
$515$	−6.66626	−0.293750
$516$	8.22358	0.362023
$517$	45.7627	2.01264
$518$	−22.6220	−0.993954
$519$	32.2187	1.41424
$520$	3.53801	0.155152
$521$	17.5393	0.768412	0.384206	−	0.923247i	$-0.374475\pi$
$521$	17.5393	0.768412	0.384206	+	0.923247i	$0.374475\pi$
$522$	2.48614	0.108815
$523$	16.1436	0.705912	0.352956	−	0.935640i	$-0.385177\pi$
$523$	16.1436	0.705912	0.352956	+	0.935640i	$0.385177\pi$
$524$	7.37250	0.322069
$525$	20.3399	0.887704
$526$	11.4598	0.499672
$527$	37.5447	1.63547
$528$	−8.58177	−0.373474
$529$	−22.6651	−0.985437
$530$	−5.32344	−0.231236
$531$	10.1983	0.442570
$532$	14.3735	0.623170
$533$	37.4906	1.62390
$534$	−10.5049	−0.454591
$535$	−12.4374	−0.537717
$536$	−6.83033	−0.295025
$537$	44.8608	1.93589
$538$	−18.8723	−0.813642
$539$	−10.3159	−0.444338
$540$	2.17352	0.0935332
$541$	24.0293	1.03310	0.516550	−	0.856257i	$-0.327216\pi$
$541$	24.0293	1.03310	0.516550	+	0.856257i	$0.327216\pi$
$542$	10.1495	0.435960
$543$	−29.1629	−1.25150
$544$	−7.66383	−0.328584
$545$	−0.107760	−0.00461594
$546$	−25.1534	−1.07647
$547$	25.4221	1.08697	0.543485	−	0.839419i	$-0.317105\pi$
$547$	25.4221	1.08697	0.543485	+	0.839419i	$0.317105\pi$
$548$	−1.73402	−0.0740738
$549$	−11.5189	−0.491616
$550$	19.0781	0.813491
$551$	−12.8427	−0.547116
$552$	1.20164	0.0511453
$553$	22.1633	0.942477
$554$	25.3443	1.07678
$555$	−13.7176	−0.582281
$556$	19.2174	0.815000
$557$	11.9948	0.508237	0.254118	−	0.967173i	$-0.418215\pi$
$557$	11.9948	0.508237	0.254118	+	0.967173i	$0.418215\pi$
$558$	−6.42290	−0.271903
$559$	22.6082	0.956227
$560$	1.31542	0.0555867
$561$	−65.7692	−2.77678
$562$	20.0125	0.844175
$563$	47.4330	1.99906	0.999531	−	0.0306264i	$-0.00975021\pi$
$563$	47.4330	1.99906	0.999531	+	0.0306264i	$0.00975021\pi$
$564$	−22.9890	−0.968012
$565$	11.2216	0.472097
$566$	20.8823	0.877750
$567$	−23.8000	−0.999507
$568$	−8.68286	−0.364325
$569$	−18.4281	−0.772547	−0.386274	−	0.922384i	$-0.626238\pi$
$569$	−18.4281	−0.772547	−0.386274	+	0.922384i	$0.626238\pi$
$570$	8.71587	0.365068
$571$	37.9828	1.58953	0.794766	−	0.606917i	$-0.207594\pi$
$571$	37.9828	1.58953	0.794766	+	0.606917i	$0.207594\pi$
$572$	−23.5930	−0.986472
$573$	49.0501	2.04910
$574$	13.9389	0.581798
$575$	−2.67136	−0.111403
$576$	1.31108	0.0546282
$577$	16.0310	0.667377	0.333689	−	0.942683i	$-0.391707\pi$
$577$	16.0310	0.667377	0.333689	+	0.942683i	$0.391707\pi$
$578$	−41.7343	−1.73592
$579$	−10.2055	−0.424126
$580$	−1.17532	−0.0488026
$581$	−2.69523	−0.111817
$582$	10.2947	0.426729
$583$	35.4990	1.47022
$584$	9.16675	0.379323
$585$	−4.63860	−0.191783
$586$	−5.14643	−0.212597
$587$	−4.16180	−0.171776	−0.0858879	−	0.996305i	$-0.527373\pi$
$587$	−4.16180	−0.171776	−0.0858879	+	0.996305i	$0.527373\pi$
$588$	5.18223	0.213712
$589$	33.1788	1.36711
$590$	−4.82127	−0.198488
$591$	−32.0151	−1.31692
$592$	10.6592	0.438092
$593$	37.7687	1.55097	0.775487	−	0.631363i	$-0.217504\pi$
$593$	37.7687	1.55097	0.775487	+	0.631363i	$0.217504\pi$
$594$	−14.4939	−0.594694
$595$	10.0812	0.413287
$596$	−14.6899	−0.601721
$597$	5.14997	0.210774
$598$	3.30355	0.135092
$599$	5.88364	0.240399	0.120200	−	0.992750i	$-0.461647\pi$
$599$	5.88364	0.240399	0.120200	+	0.992750i	$0.461647\pi$
$600$	−9.58392	−0.391262
$601$	−0.137513	−0.00560926	−0.00280463	−	0.999996i	$-0.500893\pi$
$601$	−0.137513	−0.00560926	−0.00280463	+	0.999996i	$0.500893\pi$
$602$	8.40568	0.342590
$603$	8.95509	0.364679
$604$	6.56625	0.267177
$605$	3.77041	0.153289
$606$	27.0644	1.09942
$607$	21.2574	0.862812	0.431406	−	0.902158i	$-0.358018\pi$
$607$	21.2574	0.862812	0.431406	+	0.902158i	$0.358018\pi$
$608$	−6.77264	−0.274667
$609$	8.35594	0.338600
$610$	5.44558	0.220485
$611$	−63.2013	−2.55685
$612$	10.0479	0.406161
$613$	19.4566	0.785846	0.392923	−	0.919571i	$-0.371464\pi$
$613$	19.4566	0.785846	0.392923	+	0.919571i	$0.371464\pi$
$614$	10.0076	0.403874
$615$	8.45233	0.340831
$616$	−8.77180	−0.353426
$617$	−27.2055	−1.09525	−0.547627	−	0.836723i	$-0.684469\pi$
$617$	−27.2055	−1.09525	−0.547627	+	0.836723i	$0.684469\pi$
$618$	−22.3313	−0.898298
$619$	29.2063	1.17390	0.586950	−	0.809623i	$-0.300328\pi$
$619$	29.2063	1.17390	0.586950	+	0.809623i	$0.300328\pi$
$620$	3.03643	0.121946
$621$	2.02948	0.0814402
$622$	−2.18590	−0.0876464
$623$	−10.7375	−0.430189
$624$	11.8520	0.474460
$625$	19.3851	0.775403
$626$	22.5262	0.900329
$627$	−58.1212	−2.32114
$628$	−15.6820	−0.625781
$629$	81.6906	3.25722
$630$	−1.72462	−0.0687104
$631$	20.0740	0.799135	0.399567	−	0.916704i	$-0.369160\pi$
$631$	20.0740	0.799135	0.399567	+	0.916704i	$0.369160\pi$
$632$	−10.4431	−0.415403
$633$	−11.2460	−0.446990
$634$	13.5939	0.539883
$635$	1.20330	0.0477517
$636$	−17.8330	−0.707126
$637$	14.2470	0.564486
$638$	7.83757	0.310292
$639$	11.3839	0.450340
$640$	−0.619812	−0.0245002
$641$	28.3294	1.11894	0.559471	−	0.828850i	$-0.311004\pi$
$641$	28.3294	1.11894	0.559471	+	0.828850i	$0.311004\pi$
$642$	−41.6642	−1.64436
$643$	28.2093	1.11247	0.556233	−	0.831027i	$-0.312246\pi$
$643$	28.2093	1.11247	0.556233	+	0.831027i	$0.312246\pi$
$644$	1.22825	0.0483998
$645$	5.09707	0.200697
$646$	−51.9043	−2.04215
$647$	44.2970	1.74150	0.870748	−	0.491729i	$-0.163635\pi$
$647$	44.2970	1.74150	0.870748	+	0.491729i	$0.163635\pi$
$648$	11.2143	0.440540
$649$	32.1503	1.26201
$650$	−26.3481	−1.03346
$651$	−21.5874	−0.846077
$652$	−6.72384	−0.263326
$653$	−40.1826	−1.57247	−0.786233	−	0.617930i	$-0.787971\pi$
$653$	−40.1826	−1.57247	−0.786233	+	0.617930i	$0.787971\pi$
$654$	−0.360987	−0.0141157
$655$	4.56956	0.178548
$656$	−6.56786	−0.256432
$657$	−12.0183	−0.468879
$658$	−23.4981	−0.916050
$659$	42.1367	1.64141	0.820707	−	0.571349i	$-0.193580\pi$
$659$	42.1367	1.64141	0.820707	+	0.571349i	$0.193580\pi$
$660$	−5.31908	−0.207045
$661$	43.1279	1.67748	0.838740	−	0.544533i	$-0.183293\pi$
$661$	43.1279	1.67748	0.838740	+	0.544533i	$0.183293\pi$
$662$	−19.5256	−0.758884
$663$	90.8317	3.52761
$664$	1.26996	0.0492842
$665$	8.90887	0.345471
$666$	−13.9751	−0.541524
$667$	−1.09744	−0.0424929
$668$	8.75081	0.338579
$669$	46.7594	1.80782
$670$	−4.23352	−0.163555
$671$	−36.3135	−1.40187
$672$	4.40654	0.169986
$673$	13.2350	0.510170	0.255085	−	0.966919i	$-0.417896\pi$
$673$	13.2350	0.510170	0.255085	+	0.966919i	$0.417896\pi$
$674$	7.57805	0.291896
$675$	−16.1865	−0.623019
$676$	19.5835	0.753211
$677$	−5.90605	−0.226988	−0.113494	−	0.993539i	$-0.536204\pi$
$677$	−5.90605	−0.226988	−0.113494	+	0.993539i	$0.536204\pi$
$678$	37.5914	1.44369
$679$	10.5227	0.403822
$680$	−4.75013	−0.182159
$681$	−0.901134	−0.0345315
$682$	−20.2482	−0.775344
$683$	−28.9506	−1.10777	−0.553883	−	0.832595i	$-0.686854\pi$
$683$	−28.9506	−1.10777	−0.553883	+	0.832595i	$0.686854\pi$
$684$	8.87945	0.339514
$685$	−1.07477	−0.0410648
$686$	20.1530	0.769446
$687$	−8.83396	−0.337037
$688$	−3.96066	−0.150999
$689$	−49.0265	−1.86776
$690$	0.744792	0.0283537
$691$	22.0364	0.838305	0.419153	−	0.907916i	$-0.362327\pi$
$691$	22.0364	0.838305	0.419153	+	0.907916i	$0.362327\pi$
$692$	−15.5173	−0.589878
$693$	11.5005	0.436868
$694$	6.01228	0.228223
$695$	11.9112	0.451817
$696$	−3.93723	−0.149240
$697$	−50.3349	−1.90657
$698$	−23.5146	−0.890040
$699$	11.0736	0.418842
$700$	−9.79614	−0.370259
$701$	22.6020	0.853665	0.426833	−	0.904331i	$-0.359629\pi$
$701$	22.6020	0.853665	0.426833	+	0.904331i	$0.359629\pi$
$702$	20.0171	0.755497
$703$	72.1912	2.72274
$704$	4.13318	0.155775
$705$	−14.2489	−0.536643
$706$	19.1995	0.722583
$707$	27.6638	1.04040
$708$	−16.1508	−0.606984
$709$	−20.1724	−0.757592	−0.378796	−	0.925480i	$-0.623662\pi$
$709$	−20.1724	−0.757592	−0.378796	+	0.925480i	$0.623662\pi$
$710$	−5.38174	−0.201973
$711$	13.6917	0.513478
$712$	5.05939	0.189609
$713$	2.83521	0.106179
$714$	33.7710	1.26385
$715$	−14.6232	−0.546877
$716$	−21.6060	−0.807453
$717$	−15.8609	−0.592337
$718$	−15.8925	−0.593102
$719$	−36.1806	−1.34931	−0.674655	−	0.738133i	$-0.735708\pi$
$719$	−36.1806	−1.34931	−0.674655	+	0.738133i	$0.735708\pi$
$720$	0.812621	0.0302846
$721$	−22.8258	−0.850079
$722$	−26.8686	−0.999946
$723$	−18.1779	−0.676042
$724$	14.0455	0.521997
$725$	8.75281	0.325071
$726$	12.6305	0.468762
$727$	−51.1169	−1.89582	−0.947910	−	0.318538i	$-0.896808\pi$
$727$	−51.1169	−1.89582	−0.947910	+	0.318538i	$0.896808\pi$
$728$	12.1144	0.448991
$729$	7.14041	0.264460
$730$	5.68166	0.210288
$731$	−30.3538	−1.12268
$732$	18.2422	0.674251
$733$	−6.87248	−0.253841	−0.126921	−	0.991913i	$-0.540509\pi$
$733$	−6.87248	−0.253841	−0.126921	+	0.991913i	$0.540509\pi$
$734$	28.0729	1.03619
$735$	3.21201	0.118477
$736$	−0.578738	−0.0213326
$737$	28.2310	1.03990
$738$	8.61096	0.316974
$739$	−28.5148	−1.04893	−0.524467	−	0.851431i	$-0.675735\pi$
$739$	−28.5148	−1.04893	−0.524467	+	0.851431i	$0.675735\pi$
$740$	6.60673	0.242868
$741$	80.2693	2.94877
$742$	−18.2279	−0.669168
$743$	−41.2590	−1.51364	−0.756822	−	0.653621i	$-0.773249\pi$
$743$	−41.2590	−1.51364	−0.756822	+	0.653621i	$0.773249\pi$
$744$	10.1718	0.372914
$745$	−9.10497	−0.333580
$746$	−3.37640	−0.123619
$747$	−1.66502	−0.0609199
$748$	31.6759	1.15819
$749$	−42.5868	−1.55609
$750$	−12.3748	−0.451865
$751$	7.99212	0.291637	0.145818	−	0.989311i	$-0.453418\pi$
$751$	7.99212	0.291637	0.145818	+	0.989311i	$0.453418\pi$
$752$	11.0720	0.403756
$753$	8.97979	0.327242
$754$	−10.8242	−0.394194
$755$	4.06984	0.148117
$756$	7.44231	0.270674
$757$	−53.7877	−1.95495	−0.977473	−	0.211062i	$-0.932308\pi$
$757$	−53.7877	−1.95495	−0.977473	+	0.211062i	$0.932308\pi$
$758$	2.73564	0.0993629
$759$	−4.96660	−0.180276
$760$	−4.19776	−0.152269
$761$	14.1985	0.514696	0.257348	−	0.966319i	$-0.417151\pi$
$761$	14.1985	0.514696	0.257348	+	0.966319i	$0.417151\pi$
$762$	4.03096	0.146026
$763$	−0.368980	−0.0133580
$764$	−23.6236	−0.854673
$765$	6.22779	0.225166
$766$	29.0148	1.04835
$767$	−44.4017	−1.60325
$768$	−2.07631	−0.0749225
$769$	20.0040	0.721362	0.360681	−	0.932689i	$-0.382544\pi$
$769$	20.0040	0.721362	0.360681	+	0.932689i	$0.382544\pi$
$770$	−5.43687	−0.195931
$771$	−31.1837	−1.12305
$772$	4.91520	0.176902
$773$	−14.1270	−0.508113	−0.254057	−	0.967189i	$-0.581765\pi$
$773$	−14.1270	−0.508113	−0.254057	+	0.967189i	$0.581765\pi$
$774$	5.19273	0.186649
$775$	−22.6127	−0.812273
$776$	−4.95816	−0.177988
$777$	−46.9704	−1.68505
$778$	23.8843	0.856293
$779$	−44.4817	−1.59372
$780$	7.34601	0.263029
$781$	35.8878	1.28417
$782$	−4.43535	−0.158608
$783$	−6.64967	−0.237640
$784$	−2.49588	−0.0891386
$785$	−9.71990	−0.346918
$786$	15.3076	0.546005
$787$	12.7964	0.456142	0.228071	−	0.973644i	$-0.426758\pi$
$787$	12.7964	0.456142	0.228071	+	0.973644i	$0.426758\pi$
$788$	15.4192	0.549286
$789$	23.7942	0.847096
$790$	−6.47275	−0.230290
$791$	38.4238	1.36619
$792$	−5.41891	−0.192553
$793$	50.1514	1.78093
$794$	21.0189	0.745934
$795$	−11.0531	−0.392014
$796$	−2.48034	−0.0879134
$797$	22.9314	0.812272	0.406136	−	0.913813i	$-0.366876\pi$
$797$	22.9314	0.812272	0.406136	+	0.913813i	$0.366876\pi$
$798$	29.8439	1.05646
$799$	84.8542	3.00192
$800$	4.61583	0.163194
$801$	−6.63325	−0.234374
$802$	−6.86026	−0.242244
$803$	−37.8878	−1.33703
$804$	−14.1819	−0.500157
$805$	0.761284	0.0268318
$806$	27.9642	0.984995
$807$	−39.1848	−1.37937
$808$	−13.0349	−0.458565
$809$	44.2205	1.55471	0.777355	−	0.629062i	$-0.216561\pi$
$809$	44.2205	1.55471	0.777355	+	0.629062i	$0.216561\pi$
$810$	6.95076	0.244225
$811$	32.0932	1.12695	0.563473	−	0.826135i	$-0.309465\pi$
$811$	32.0932	1.12695	0.563473	+	0.826135i	$0.309465\pi$
$812$	−4.02441	−0.141229
$813$	21.0736	0.739084
$814$	−44.0565	−1.54418
$815$	−4.16752	−0.145982
$816$	−15.9125	−0.557049
$817$	−26.8241	−0.938458
$818$	−20.7929	−0.727008
$819$	−15.8830	−0.554996
$820$	−4.07084	−0.142160
$821$	−29.4680	−1.02844	−0.514219	−	0.857659i	$-0.671918\pi$
$821$	−29.4680	−1.02844	−0.514219	+	0.857659i	$0.671918\pi$
$822$	−3.60037	−0.125577
$823$	−21.8805	−0.762706	−0.381353	−	0.924429i	$-0.624542\pi$
$823$	−21.8805	−0.762706	−0.381353	+	0.924429i	$0.624542\pi$
$824$	10.7553	0.374678
$825$	39.6120	1.37911
$826$	−16.5084	−0.574402
$827$	14.7425	0.512647	0.256323	−	0.966591i	$-0.417489\pi$
$827$	14.7425	0.512647	0.256323	+	0.966591i	$0.417489\pi$
$828$	0.758770	0.0263691
$829$	40.0181	1.38989	0.694944	−	0.719064i	$-0.255429\pi$
$829$	40.0181	1.38989	0.694944	+	0.719064i	$0.255429\pi$
$830$	0.787139	0.0273220
$831$	52.6227	1.82546
$832$	−5.70819	−0.197896
$833$	−19.1280	−0.662746
$834$	39.9014	1.38167
$835$	5.42386	0.187700
$836$	27.9925	0.968141
$837$	17.1793	0.593804
$838$	−7.75666	−0.267949
$839$	−32.9992	−1.13926	−0.569630	−	0.821901i	$-0.692913\pi$
$839$	−32.9992	−1.13926	−0.569630	+	0.821901i	$0.692913\pi$
$840$	2.73123	0.0942363
$841$	−25.4042	−0.876007
$842$	−31.0612	−1.07044
$843$	41.5521	1.43113
$844$	5.41635	0.186438
$845$	12.1381	0.417563
$846$	−14.5163	−0.499080
$847$	12.9102	0.443600
$848$	8.58880	0.294941
$849$	43.3583	1.48805
$850$	35.3749	1.21335
$851$	6.16891	0.211468
$852$	−18.0283	−0.617640
$853$	−16.5929	−0.568131	−0.284066	−	0.958805i	$-0.591683\pi$
$853$	−16.5929	−0.568131	−0.284066	+	0.958805i	$0.591683\pi$
$854$	18.6461	0.638058
$855$	5.50359	0.188219
$856$	20.0664	0.685857
$857$	−42.9039	−1.46557	−0.732784	−	0.680461i	$-0.761780\pi$
$857$	−42.9039	−1.46557	−0.732784	+	0.680461i	$0.761780\pi$
$858$	−48.9864	−1.67237
$859$	25.4481	0.868277	0.434139	−	0.900846i	$-0.357053\pi$
$859$	25.4481	0.868277	0.434139	+	0.900846i	$0.357053\pi$
$860$	−2.45487	−0.0837103
$861$	28.9415	0.986324
$862$	−26.3775	−0.898421
$863$	4.25167	0.144728	0.0723642	−	0.997378i	$-0.476946\pi$
$863$	4.25167	0.144728	0.0723642	+	0.997378i	$0.476946\pi$
$864$	−3.50673	−0.119302
$865$	−9.61778	−0.327014
$866$	4.99465	0.169725
$867$	−86.6534	−2.94290
$868$	10.3970	0.352897
$869$	43.1631	1.46421
$870$	−2.44034	−0.0827353
$871$	−38.9889	−1.32109
$872$	0.173859	0.00588762
$873$	6.50053	0.220009
$874$	−3.91958	−0.132582
$875$	−12.6489	−0.427610
$876$	19.0330	0.643067
$877$	17.6501	0.596000	0.298000	−	0.954566i	$-0.403680\pi$
$877$	17.6501	0.596000	0.298000	+	0.954566i	$0.403680\pi$
$878$	−12.9224	−0.436109
$879$	−10.6856	−0.360417
$880$	2.56179	0.0863580
$881$	−14.7939	−0.498420	−0.249210	−	0.968449i	$-0.580171\pi$
$881$	−14.7939	−0.498420	−0.249210	+	0.968449i	$0.580171\pi$
$882$	3.27229	0.110184
$883$	37.2369	1.25312	0.626561	−	0.779372i	$-0.284462\pi$
$883$	37.2369	1.25312	0.626561	+	0.779372i	$0.284462\pi$
$884$	−43.7466	−1.47136
$885$	−10.0105	−0.336498
$886$	−11.5909	−0.389404
$887$	−9.29065	−0.311950	−0.155975	−	0.987761i	$-0.549852\pi$
$887$	−9.29065	−0.311950	−0.155975	+	0.987761i	$0.549852\pi$
$888$	22.1319	0.742699
$889$	4.12022	0.138188
$890$	3.13587	0.105115
$891$	−46.3507	−1.55281
$892$	−22.5204	−0.754039
$893$	74.9869	2.50934
$894$	−30.5008	−1.02010
$895$	−13.3916	−0.447633
$896$	−2.12229	−0.0709008
$897$	6.85920	0.229022
$898$	14.0300	0.468188
$899$	−9.28967	−0.309828
$900$	−6.05171	−0.201724
$901$	65.8231	2.19288
$902$	27.1461	0.903866
$903$	17.4528	0.580794
$904$	−18.1049	−0.602159
$905$	8.70557	0.289383
$906$	13.6336	0.452945
$907$	29.6996	0.986157	0.493079	−	0.869985i	$-0.335871\pi$
$907$	29.6996	0.986157	0.493079	+	0.869985i	$0.335871\pi$
$908$	0.434007	0.0144030
$909$	17.0897	0.566830
$910$	7.50868	0.248910
$911$	−23.9672	−0.794070	−0.397035	−	0.917804i	$-0.629961\pi$
$911$	−23.9672	−0.794070	−0.397035	+	0.917804i	$0.629961\pi$
$912$	−14.0621	−0.465643
$913$	−5.24899	−0.173716
$914$	−17.4430	−0.576962
$915$	11.3067	0.373789
$916$	4.25464	0.140577
$917$	15.6466	0.516696
$918$	−26.8750	−0.887007
$919$	−40.3326	−1.33045	−0.665225	−	0.746643i	$-0.731664\pi$
$919$	−40.3326	−1.33045	−0.665225	+	0.746643i	$0.731664\pi$
$920$	−0.358709	−0.0118263
$921$	20.7789	0.684688
$922$	35.8735	1.18143
$923$	−49.5634	−1.63140
$924$	−18.2130	−0.599164
$925$	−49.2013	−1.61773
$926$	−0.0122851	−0.000403714 0
$927$	−14.1010	−0.463138
$928$	1.89626	0.0622477
$929$	−38.7270	−1.27059	−0.635296	−	0.772269i	$-0.719122\pi$
$929$	−38.7270	−1.27059	−0.635296	+	0.772269i	$0.719122\pi$
$930$	6.30457	0.206735
$931$	−16.9037	−0.553997
$932$	−5.33330	−0.174698
$933$	−4.53860	−0.148587
$934$	23.0251	0.753405
$935$	19.6331	0.642072
$936$	7.48388	0.244618
$937$	44.9330	1.46790	0.733948	−	0.679206i	$-0.237676\pi$
$937$	44.9330	1.46790	0.733948	+	0.679206i	$0.237676\pi$
$938$	−14.4959	−0.473309
$939$	46.7715	1.52633
$940$	6.86258	0.223833
$941$	3.32227	0.108303	0.0541514	−	0.998533i	$-0.482755\pi$
$941$	3.32227	0.108303	0.0541514	+	0.998533i	$0.482755\pi$
$942$	−32.5608	−1.06089
$943$	−3.80107	−0.123780
$944$	7.77859	0.253172
$945$	4.61283	0.150055
$946$	16.3701	0.532238
$947$	10.6556	0.346261	0.173131	−	0.984899i	$-0.444612\pi$
$947$	10.6556	0.346261	0.173131	+	0.984899i	$0.444612\pi$
$948$	−21.6831	−0.704235
$949$	52.3256	1.69856
$950$	31.2614	1.01425
$951$	28.2252	0.915264
$952$	−16.2649	−0.527147
$953$	−28.4574	−0.921826	−0.460913	−	0.887445i	$-0.652478\pi$
$953$	−28.4574	−0.921826	−0.460913	+	0.887445i	$0.652478\pi$
$954$	−11.2606	−0.364575
$955$	−14.6422	−0.473811
$956$	7.63898	0.247062
$957$	16.2732	0.526039
$958$	−13.3949	−0.432771
$959$	−3.68010	−0.118837
$960$	−1.28692	−0.0415353
$961$	−7.00031	−0.225816
$962$	60.8450	1.96172
$963$	−26.3087	−0.847784
$964$	8.75487	0.281975
$965$	3.04650	0.0980702
$966$	2.55023	0.0820524
$967$	−17.2549	−0.554881	−0.277440	−	0.960743i	$-0.589486\pi$
$967$	−17.2549	−0.554881	−0.277440	+	0.960743i	$0.589486\pi$
$968$	−6.08315	−0.195520
$969$	−107.770	−3.46206
$970$	−3.07313	−0.0986721
$971$	38.9285	1.24928	0.624638	−	0.780915i	$-0.285247\pi$
$971$	38.9285	1.24928	0.624638	+	0.780915i	$0.285247\pi$
$972$	12.7642	0.409412
$973$	40.7850	1.30751
$974$	−21.0447	−0.674316
$975$	−54.7069	−1.75202
$976$	−8.78585	−0.281228
$977$	−17.6937	−0.566071	−0.283035	−	0.959109i	$-0.591341\pi$
$977$	−17.6937	−0.566071	−0.283035	+	0.959109i	$0.591341\pi$
$978$	−13.9608	−0.446417
$979$	−20.9114	−0.668330
$980$	−1.54698	−0.0494164
$981$	−0.227943	−0.00727766
$982$	−7.87216	−0.251211
$983$	−12.5997	−0.401868	−0.200934	−	0.979605i	$-0.564398\pi$
$983$	−12.5997	−0.401868	−0.200934	+	0.979605i	$0.564398\pi$
$984$	−13.6369	−0.434729
$985$	9.55699	0.304511
$986$	14.5326	0.462812
$987$	−48.7894	−1.55298
$988$	−38.6595	−1.22992
$989$	−2.29219	−0.0728873
$990$	−3.35871	−0.106747
$991$	14.4201	0.458068	0.229034	−	0.973418i	$-0.426443\pi$
$991$	14.4201	0.458068	0.229034	+	0.973418i	$0.426443\pi$
$992$	−4.89895	−0.155542
$993$	−40.5413	−1.28654
$994$	−18.4275	−0.584486
$995$	−1.53735	−0.0487372
$996$	2.63684	0.0835516
$997$	0.179891	0.00569720	0.00284860	−	0.999996i	$-0.499093\pi$
$997$	0.179891	0.00569720	0.00284860	+	0.999996i	$0.499093\pi$
$998$	−19.7552	−0.625339
$999$	37.3791	1.18262

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8006.2.a.c.1.18	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8006.2.a.c.1.18	✓	92	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:	\( N \)	\(=\)	\( 8006 = 2 \cdot 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.07631 q^{3} +1.00000 q^{4} +0.619812 q^{5} +2.07631 q^{6} +2.12229 q^{7} -1.00000 q^{8} +1.31108 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.07631 q^{3} +1.00000 q^{4} +0.619812 q^{5} +2.07631 q^{6} +2.12229 q^{7} -1.00000 q^{8} +1.31108 q^{9} -0.619812 q^{10} +4.13318 q^{11} -2.07631 q^{12} -5.70819 q^{13} -2.12229 q^{14} -1.28692 q^{15} +1.00000 q^{16} +7.66383 q^{17} -1.31108 q^{18} +6.77264 q^{19} +0.619812 q^{20} -4.40654 q^{21} -4.13318 q^{22} +0.578738 q^{23} +2.07631 q^{24} -4.61583 q^{25} +5.70819 q^{26} +3.50673 q^{27} +2.12229 q^{28} -1.89626 q^{29} +1.28692 q^{30} +4.89895 q^{31} -1.00000 q^{32} -8.58177 q^{33} -7.66383 q^{34} +1.31542 q^{35} +1.31108 q^{36} +10.6592 q^{37} -6.77264 q^{38} +11.8520 q^{39} -0.619812 q^{40} -6.56786 q^{41} +4.40654 q^{42} -3.96066 q^{43} +4.13318 q^{44} +0.812621 q^{45} -0.578738 q^{46} +11.0720 q^{47} -2.07631 q^{48} -2.49588 q^{49} +4.61583 q^{50} -15.9125 q^{51} -5.70819 q^{52} +8.58880 q^{53} -3.50673 q^{54} +2.56179 q^{55} -2.12229 q^{56} -14.0621 q^{57} +1.89626 q^{58} +7.77859 q^{59} -1.28692 q^{60} -8.78585 q^{61} -4.89895 q^{62} +2.78249 q^{63} +1.00000 q^{64} -3.53801 q^{65} +8.58177 q^{66} +6.83033 q^{67} +7.66383 q^{68} -1.20164 q^{69} -1.31542 q^{70} +8.68286 q^{71} -1.31108 q^{72} -9.16675 q^{73} -10.6592 q^{74} +9.58392 q^{75} +6.77264 q^{76} +8.77180 q^{77} -11.8520 q^{78} +10.4431 q^{79} +0.619812 q^{80} -11.2143 q^{81} +6.56786 q^{82} -1.26996 q^{83} -4.40654 q^{84} +4.75013 q^{85} +3.96066 q^{86} +3.93723 q^{87} -4.13318 q^{88} -5.05939 q^{89} -0.812621 q^{90} -12.1144 q^{91} +0.578738 q^{92} -10.1718 q^{93} -11.0720 q^{94} +4.19776 q^{95} +2.07631 q^{96} +4.95816 q^{97} +2.49588 q^{98} +5.41891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q - 92 q^{2} - 2 q^{3} + 92 q^{4} + 10 q^{5} + 2 q^{6} + 8 q^{7} - 92 q^{8} + 104 q^{9}+O(q^{10}) \) 92 * q - 92 * q^2 - 2 * q^3 + 92 * q^4 + 10 * q^5 + 2 * q^6 + 8 * q^7 - 92 * q^8 + 104 * q^9	\( 92 q - 92 q^{2} - 2 q^{3} + 92 q^{4} + 10 q^{5} + 2 q^{6} + 8 q^{7} - 92 q^{8} + 104 q^{9} - 10 q^{10} + 4 q^{11} - 2 q^{12} + 40 q^{13} - 8 q^{14} + 15 q^{15} + 92 q^{16} - 14 q^{17} - 104 q^{18} + 64 q^{19} + 10 q^{20} + 54 q^{21} - 4 q^{22} - 49 q^{23} + 2 q^{24} + 116 q^{25} - 40 q^{26} - 8 q^{27} + 8 q^{28} + 39 q^{29} - 15 q^{30} + 53 q^{31} - 92 q^{32} + q^{33} + 14 q^{34} - 22 q^{35} + 104 q^{36} + 58 q^{37} - 64 q^{38} + 58 q^{39} - 10 q^{40} + 27 q^{41} - 54 q^{42} + 40 q^{43} + 4 q^{44} + 43 q^{45} + 49 q^{46} - 28 q^{47} - 2 q^{48} + 148 q^{49} - 116 q^{50} + 48 q^{51} + 40 q^{52} + 32 q^{53} + 8 q^{54} + 36 q^{55} - 8 q^{56} + 48 q^{57} - 39 q^{58} + 8 q^{59} + 15 q^{60} + 99 q^{61} - 53 q^{62} + 92 q^{64} + 13 q^{65} - q^{66} + 48 q^{67} - 14 q^{68} + 63 q^{69} + 22 q^{70} - 13 q^{71} - 104 q^{72} + 49 q^{73} - 58 q^{74} + 16 q^{75} + 64 q^{76} + 41 q^{77} - 58 q^{78} + 143 q^{79} + 10 q^{80} + 124 q^{81} - 27 q^{82} - 24 q^{83} + 54 q^{84} + 121 q^{85} - 40 q^{86} + 5 q^{87} - 4 q^{88} + 25 q^{89} - 43 q^{90} + 67 q^{91} - 49 q^{92} + 43 q^{93} + 28 q^{94} - 38 q^{95} + 2 q^{96} + 74 q^{97} - 148 q^{98} + 86 q^{99}+O(q^{100}) \) 92 * q - 92 * q^2 - 2 * q^3 + 92 * q^4 + 10 * q^5 + 2 * q^6 + 8 * q^7 - 92 * q^8 + 104 * q^9 - 10 * q^10 + 4 * q^11 - 2 * q^12 + 40 * q^13 - 8 * q^14 + 15 * q^15 + 92 * q^16 - 14 * q^17 - 104 * q^18 + 64 * q^19 + 10 * q^20 + 54 * q^21 - 4 * q^22 - 49 * q^23 + 2 * q^24 + 116 * q^25 - 40 * q^26 - 8 * q^27 + 8 * q^28 + 39 * q^29 - 15 * q^30 + 53 * q^31 - 92 * q^32 + q^33 + 14 * q^34 - 22 * q^35 + 104 * q^36 + 58 * q^37 - 64 * q^38 + 58 * q^39 - 10 * q^40 + 27 * q^41 - 54 * q^42 + 40 * q^43 + 4 * q^44 + 43 * q^45 + 49 * q^46 - 28 * q^47 - 2 * q^48 + 148 * q^49 - 116 * q^50 + 48 * q^51 + 40 * q^52 + 32 * q^53 + 8 * q^54 + 36 * q^55 - 8 * q^56 + 48 * q^57 - 39 * q^58 + 8 * q^59 + 15 * q^60 + 99 * q^61 - 53 * q^62 + 92 * q^64 + 13 * q^65 - q^66 + 48 * q^67 - 14 * q^68 + 63 * q^69 + 22 * q^70 - 13 * q^71 - 104 * q^72 + 49 * q^73 - 58 * q^74 + 16 * q^75 + 64 * q^76 + 41 * q^77 - 58 * q^78 + 143 * q^79 + 10 * q^80 + 124 * q^81 - 27 * q^82 - 24 * q^83 + 54 * q^84 + 121 * q^85 - 40 * q^86 + 5 * q^87 - 4 * q^88 + 25 * q^89 - 43 * q^90 + 67 * q^91 - 49 * q^92 + 43 * q^93 + 28 * q^94 - 38 * q^95 + 2 * q^96 + 74 * q^97 - 148 * q^98 + 86 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more