LMFDB - Embedded newform 8006.2.a.c.1.6

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			1.6
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)

$f(q)$	$=$	$q-1.00000 q^{2} -3.11194 q^{3} +1.00000 q^{4} +0.282330 q^{5} +3.11194 q^{6} -3.77246 q^{7} -1.00000 q^{8} +6.68416 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.11194 q^{3} +1.00000 q^{4} +0.282330 q^{5} +3.11194 q^{6} -3.77246 q^{7} -1.00000 q^{8} +6.68416 q^{9} -0.282330 q^{10} -3.16321 q^{11} -3.11194 q^{12} +3.69999 q^{13} +3.77246 q^{14} -0.878595 q^{15} +1.00000 q^{16} -3.88814 q^{17} -6.68416 q^{18} -5.09605 q^{19} +0.282330 q^{20} +11.7397 q^{21} +3.16321 q^{22} -3.51118 q^{23} +3.11194 q^{24} -4.92029 q^{25} -3.69999 q^{26} -11.4649 q^{27} -3.77246 q^{28} +3.32528 q^{29} +0.878595 q^{30} +4.38907 q^{31} -1.00000 q^{32} +9.84371 q^{33} +3.88814 q^{34} -1.06508 q^{35} +6.68416 q^{36} +5.30109 q^{37} +5.09605 q^{38} -11.5142 q^{39} -0.282330 q^{40} -10.4068 q^{41} -11.7397 q^{42} -1.61158 q^{43} -3.16321 q^{44} +1.88714 q^{45} +3.51118 q^{46} -8.52577 q^{47} -3.11194 q^{48} +7.23148 q^{49} +4.92029 q^{50} +12.0996 q^{51} +3.69999 q^{52} -7.37335 q^{53} +11.4649 q^{54} -0.893070 q^{55} +3.77246 q^{56} +15.8586 q^{57} -3.32528 q^{58} -10.9408 q^{59} -0.878595 q^{60} -0.0726800 q^{61} -4.38907 q^{62} -25.2158 q^{63} +1.00000 q^{64} +1.04462 q^{65} -9.84371 q^{66} -0.00327689 q^{67} -3.88814 q^{68} +10.9266 q^{69} +1.06508 q^{70} +8.00400 q^{71} -6.68416 q^{72} +4.09852 q^{73} -5.30109 q^{74} +15.3116 q^{75} -5.09605 q^{76} +11.9331 q^{77} +11.5142 q^{78} +3.80393 q^{79} +0.282330 q^{80} +15.6256 q^{81} +10.4068 q^{82} -5.55535 q^{83} +11.7397 q^{84} -1.09774 q^{85} +1.61158 q^{86} -10.3481 q^{87} +3.16321 q^{88} +11.2802 q^{89} -1.88714 q^{90} -13.9581 q^{91} -3.51118 q^{92} -13.6585 q^{93} +8.52577 q^{94} -1.43877 q^{95} +3.11194 q^{96} -6.09044 q^{97} -7.23148 q^{98} -21.1434 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.11194	−1.79668	−0.898339	−	0.439302i	$-0.855226\pi$
$3$	−3.11194	−1.79668	−0.898339	+	0.439302i	$0.855226\pi$
$4$	1.00000	0.500000
$5$	0.282330	0.126262	0.0631310	−	0.998005i	$-0.479891\pi$
$5$	0.282330	0.126262	0.0631310	+	0.998005i	$0.479891\pi$
$6$	3.11194	1.27044
$7$	−3.77246	−1.42586	−0.712929	−	0.701237i	$-0.752632\pi$
$7$	−3.77246	−1.42586	−0.712929	+	0.701237i	$0.752632\pi$
$8$	−1.00000	−0.353553
$9$	6.68416	2.22805
$10$	−0.282330	−0.0892807
$11$	−3.16321	−0.953743	−0.476872	−	0.878973i	$-0.658229\pi$
$11$	−3.16321	−0.953743	−0.476872	+	0.878973i	$0.658229\pi$
$12$	−3.11194	−0.898339
$13$	3.69999	1.02619	0.513097	−	0.858331i	$-0.328498\pi$
$13$	3.69999	1.02619	0.513097	+	0.858331i	$0.328498\pi$
$14$	3.77246	1.00823
$15$	−0.878595	−0.226852
$16$	1.00000	0.250000
$17$	−3.88814	−0.943011	−0.471506	−	0.881863i	$-0.656289\pi$
$17$	−3.88814	−0.943011	−0.471506	+	0.881863i	$0.656289\pi$
$18$	−6.68416	−1.57547
$19$	−5.09605	−1.16911	−0.584557	−	0.811352i	$-0.698732\pi$
$19$	−5.09605	−1.16911	−0.584557	+	0.811352i	$0.698732\pi$
$20$	0.282330	0.0631310
$21$	11.7397	2.56181
$22$	3.16321	0.674398
$23$	−3.51118	−0.732131	−0.366065	−	0.930589i	$-0.619295\pi$
$23$	−3.51118	−0.732131	−0.366065	+	0.930589i	$0.619295\pi$
$24$	3.11194	0.635222
$25$	−4.92029	−0.984058
$26$	−3.69999	−0.725629
$27$	−11.4649	−2.20642
$28$	−3.77246	−0.712929
$29$	3.32528	0.617489	0.308744	−	0.951145i	$-0.400091\pi$
$29$	3.32528	0.617489	0.308744	+	0.951145i	$0.400091\pi$
$30$	0.878595	0.160409
$31$	4.38907	0.788301	0.394150	−	0.919046i	$-0.371039\pi$
$31$	4.38907	0.788301	0.394150	+	0.919046i	$0.371039\pi$
$32$	−1.00000	−0.176777
$33$	9.84371	1.71357
$34$	3.88814	0.666810
$35$	−1.06508	−0.180032
$36$	6.68416	1.11403
$37$	5.30109	0.871493	0.435747	−	0.900069i	$-0.356484\pi$
$37$	5.30109	0.871493	0.435747	+	0.900069i	$0.356484\pi$
$38$	5.09605	0.826689
$39$	−11.5142	−1.84374
$40$	−0.282330	−0.0446403
$41$	−10.4068	−1.62527	−0.812634	−	0.582775i	$-0.801967\pi$
$41$	−10.4068	−1.62527	−0.812634	+	0.582775i	$0.801967\pi$
$42$	−11.7397	−1.81147
$43$	−1.61158	−0.245764	−0.122882	−	0.992421i	$-0.539214\pi$
$43$	−1.61158	−0.245764	−0.122882	+	0.992421i	$0.539214\pi$
$44$	−3.16321	−0.476872
$45$	1.88714	0.281319
$46$	3.51118	0.517695
$47$	−8.52577	−1.24361	−0.621806	−	0.783172i	$-0.713601\pi$
$47$	−8.52577	−1.24361	−0.621806	+	0.783172i	$0.713601\pi$
$48$	−3.11194	−0.449170
$49$	7.23148	1.03307
$50$	4.92029	0.695834
$51$	12.0996	1.69429
$52$	3.69999	0.513097
$53$	−7.37335	−1.01281	−0.506404	−	0.862297i	$-0.669025\pi$
$53$	−7.37335	−1.01281	−0.506404	+	0.862297i	$0.669025\pi$
$54$	11.4649	1.56017
$55$	−0.893070	−0.120422
$56$	3.77246	0.504117
$57$	15.8586	2.10052
$58$	−3.32528	−0.436630
$59$	−10.9408	−1.42438	−0.712188	−	0.701989i	$-0.752296\pi$
$59$	−10.9408	−1.42438	−0.712188	+	0.701989i	$0.752296\pi$
$60$	−0.878595	−0.113426
$61$	−0.0726800	−0.00930572	−0.00465286	−	0.999989i	$-0.501481\pi$
$61$	−0.0726800	−0.00930572	−0.00465286	+	0.999989i	$0.501481\pi$
$62$	−4.38907	−0.557413
$63$	−25.2158	−3.17689
$64$	1.00000	0.125000
$65$	1.04462	0.129569
$66$	−9.84371	−1.21168
$67$	−0.00327689	−0.000400336 0	−0.000200168	−	1.00000i	$-0.500064\pi$
$67$	−0.00327689	−0.000400336 0	−0.000200168		1.00000i	$0.500064\pi$
$68$	−3.88814	−0.471506
$69$	10.9266	1.31540
$70$	1.06508	0.127302
$71$	8.00400	0.949900	0.474950	−	0.880013i	$-0.342466\pi$
$71$	8.00400	0.949900	0.474950	+	0.880013i	$0.342466\pi$
$72$	−6.68416	−0.787736
$73$	4.09852	0.479696	0.239848	−	0.970810i	$-0.422902\pi$
$73$	4.09852	0.479696	0.239848	+	0.970810i	$0.422902\pi$
$74$	−5.30109	−0.616239
$75$	15.3116	1.76804
$76$	−5.09605	−0.584557
$77$	11.9331	1.35990
$78$	11.5142	1.30372
$79$	3.80393	0.427976	0.213988	−	0.976836i	$-0.431355\pi$
$79$	3.80393	0.427976	0.213988	+	0.976836i	$0.431355\pi$
$80$	0.282330	0.0315655
$81$	15.6256	1.73617
$82$	10.4068	1.14924
$83$	−5.55535	−0.609779	−0.304889	−	0.952388i	$-0.598619\pi$
$83$	−5.55535	−0.609779	−0.304889	+	0.952388i	$0.598619\pi$
$84$	11.7397	1.28090
$85$	−1.09774	−0.119066
$86$	1.61158	0.173781
$87$	−10.3481	−1.10943
$88$	3.16321	0.337199
$89$	11.2802	1.19570	0.597850	−	0.801608i	$-0.296022\pi$
$89$	11.2802	1.19570	0.597850	+	0.801608i	$0.296022\pi$
$90$	−1.88714	−0.198922
$91$	−13.9581	−1.46321
$92$	−3.51118	−0.366065
$93$	−13.6585	−1.41632
$94$	8.52577	0.879366
$95$	−1.43877	−0.147615
$96$	3.11194	0.317611
$97$	−6.09044	−0.618390	−0.309195	−	0.950999i	$-0.600060\pi$
$97$	−6.09044	−0.618390	−0.309195	+	0.950999i	$0.600060\pi$
$98$	−7.23148	−0.730490
$99$	−21.1434	−2.12499
$100$	−4.92029	−0.492029
$101$	−7.65780	−0.761980	−0.380990	−	0.924579i	$-0.624417\pi$
$101$	−7.65780	−0.761980	−0.380990	+	0.924579i	$0.624417\pi$
$102$	−12.0996	−1.19804
$103$	−1.41633	−0.139555	−0.0697777	−	0.997563i	$-0.522229\pi$
$103$	−1.41633	−0.139555	−0.0697777	+	0.997563i	$0.522229\pi$
$104$	−3.69999	−0.362814
$105$	3.31447	0.323459
$106$	7.37335	0.716163
$107$	−9.36505	−0.905353	−0.452677	−	0.891675i	$-0.649531\pi$
$107$	−9.36505	−0.905353	−0.452677	+	0.891675i	$0.649531\pi$
$108$	−11.4649	−1.10321
$109$	−16.3622	−1.56722	−0.783608	−	0.621255i	$-0.786623\pi$
$109$	−16.3622	−1.56722	−0.783608	+	0.621255i	$0.786623\pi$
$110$	0.893070	0.0851509
$111$	−16.4967	−1.56579
$112$	−3.77246	−0.356464
$113$	−3.77776	−0.355382	−0.177691	−	0.984086i	$-0.556863\pi$
$113$	−3.77776	−0.355382	−0.177691	+	0.984086i	$0.556863\pi$
$114$	−15.8586	−1.48529
$115$	−0.991312	−0.0924403
$116$	3.32528	0.308744
$117$	24.7314	2.28642
$118$	10.9408	1.00719
$119$	14.6679	1.34460
$120$	0.878595	0.0802044
$121$	−0.994110	−0.0903736
$122$	0.0726800	0.00658014
$123$	32.3853	2.92008
$124$	4.38907	0.394150
$125$	−2.80080	−0.250511
$126$	25.2158	2.24640
$127$	−20.4627	−1.81577	−0.907884	−	0.419221i	$-0.862303\pi$
$127$	−20.4627	−1.81577	−0.907884	+	0.419221i	$0.862303\pi$
$128$	−1.00000	−0.0883883
$129$	5.01514	0.441559
$130$	−1.04462	−0.0916193
$131$	−4.49275	−0.392534	−0.196267	−	0.980551i	$-0.562882\pi$
$131$	−4.49275	−0.392534	−0.196267	+	0.980551i	$0.562882\pi$
$132$	9.84371	0.856785
$133$	19.2247	1.66699
$134$	0.00327689	0.000283081 0
$135$	−3.23689	−0.278587
$136$	3.88814	0.333405
$137$	−1.14700	−0.0979944	−0.0489972	−	0.998799i	$-0.515603\pi$
$137$	−1.14700	−0.0979944	−0.0489972	+	0.998799i	$0.515603\pi$
$138$	−10.9266	−0.930131
$139$	0.532834	0.0451944	0.0225972	−	0.999745i	$-0.492806\pi$
$139$	0.532834	0.0451944	0.0225972	+	0.999745i	$0.492806\pi$
$140$	−1.06508	−0.0900158
$141$	26.5317	2.23437
$142$	−8.00400	−0.671680
$143$	−11.7039	−0.978725
$144$	6.68416	0.557014
$145$	0.938827	0.0779653
$146$	−4.09852	−0.339196
$147$	−22.5039	−1.85609
$148$	5.30109	0.435747
$149$	−4.00920	−0.328447	−0.164223	−	0.986423i	$-0.552512\pi$
$149$	−4.00920	−0.328447	−0.164223	+	0.986423i	$0.552512\pi$
$150$	−15.3116	−1.25019
$151$	6.98721	0.568611	0.284306	−	0.958734i	$-0.408237\pi$
$151$	6.98721	0.568611	0.284306	+	0.958734i	$0.408237\pi$
$152$	5.09605	0.413345
$153$	−25.9889	−2.10108
$154$	−11.9331	−0.961596
$155$	1.23917	0.0995324
$156$	−11.5142	−0.921870
$157$	−10.9016	−0.870040	−0.435020	−	0.900421i	$-0.643259\pi$
$157$	−10.9016	−0.870040	−0.435020	+	0.900421i	$0.643259\pi$
$158$	−3.80393	−0.302625
$159$	22.9454	1.81969
$160$	−0.282330	−0.0223202
$161$	13.2458	1.04391
$162$	−15.6256	−1.22766
$163$	−0.741372	−0.0580687	−0.0290344	−	0.999578i	$-0.509243\pi$
$163$	−0.741372	−0.0580687	−0.0290344	+	0.999578i	$0.509243\pi$
$164$	−10.4068	−0.812634
$165$	2.77918	0.216359
$166$	5.55535	0.431179
$167$	−15.9444	−1.23382	−0.616909	−	0.787034i	$-0.711615\pi$
$167$	−15.9444	−1.23382	−0.616909	+	0.787034i	$0.711615\pi$
$168$	−11.7397	−0.905736
$169$	0.689956	0.0530735
$170$	1.09774	0.0841927
$171$	−34.0629	−2.60485
$172$	−1.61158	−0.122882
$173$	0.548112	0.0416722	0.0208361	−	0.999783i	$-0.493367\pi$
$173$	0.548112	0.0416722	0.0208361	+	0.999783i	$0.493367\pi$
$174$	10.3481	0.784484
$175$	18.5616	1.40313
$176$	−3.16321	−0.238436
$177$	34.0472	2.55915
$178$	−11.2802	−0.845488
$179$	−4.31099	−0.322219	−0.161109	−	0.986937i	$-0.551507\pi$
$179$	−4.31099	−0.322219	−0.161109	+	0.986937i	$0.551507\pi$
$180$	1.88714	0.140659
$181$	−8.93195	−0.663907	−0.331953	−	0.943296i	$-0.607708\pi$
$181$	−8.93195	−0.663907	−0.331953	+	0.943296i	$0.607708\pi$
$182$	13.9581	1.03464
$183$	0.226176	0.0167194
$184$	3.51118	0.258847
$185$	1.49666	0.110036
$186$	13.6585	1.00149
$187$	12.2990	0.899391
$188$	−8.52577	−0.621806
$189$	43.2509	3.14604
$190$	1.43877	0.104379
$191$	19.4085	1.40435	0.702173	−	0.712006i	$-0.252213\pi$
$191$	19.4085	1.40435	0.702173	+	0.712006i	$0.252213\pi$
$192$	−3.11194	−0.224585
$193$	−11.9915	−0.863166	−0.431583	−	0.902073i	$-0.642045\pi$
$193$	−11.9915	−0.863166	−0.431583	+	0.902073i	$0.642045\pi$
$194$	6.09044	0.437268
$195$	−3.25080	−0.232794
$196$	7.23148	0.516534
$197$	−21.5872	−1.53802	−0.769012	−	0.639235i	$-0.779251\pi$
$197$	−21.5872	−1.53802	−0.769012	+	0.639235i	$0.779251\pi$
$198$	21.1434	1.50260
$199$	−12.4673	−0.883780	−0.441890	−	0.897069i	$-0.645692\pi$
$199$	−12.4673	−0.883780	−0.441890	+	0.897069i	$0.645692\pi$
$200$	4.92029	0.347917
$201$	0.0101975	0.000719276 0
$202$	7.65780	0.538801
$203$	−12.5445	−0.880451
$204$	12.0996	0.847144
$205$	−2.93815	−0.205209
$206$	1.41633	0.0986806
$207$	−23.4693	−1.63123
$208$	3.69999	0.256548
$209$	16.1199	1.11504
$210$	−3.31447	−0.228720
$211$	13.5767	0.934660	0.467330	−	0.884083i	$-0.345216\pi$
$211$	13.5767	0.934660	0.467330	+	0.884083i	$0.345216\pi$
$212$	−7.37335	−0.506404
$213$	−24.9079	−1.70666
$214$	9.36505	0.640181
$215$	−0.454998	−0.0310306
$216$	11.4649	0.780087
$217$	−16.5576	−1.12400
$218$	16.3622	1.10819
$219$	−12.7543	−0.861859
$220$	−0.893070	−0.0602108
$221$	−14.3861	−0.967712
$222$	16.4967	1.10718
$223$	−16.0420	−1.07425	−0.537127	−	0.843501i	$-0.680490\pi$
$223$	−16.0420	−1.07425	−0.537127	+	0.843501i	$0.680490\pi$
$224$	3.77246	0.252058
$225$	−32.8880	−2.19253
$226$	3.77776	0.251293
$227$	−22.0138	−1.46111	−0.730554	−	0.682855i	$-0.760738\pi$
$227$	−22.0138	−1.46111	−0.730554	+	0.682855i	$0.760738\pi$
$228$	15.8586	1.05026
$229$	15.5101	1.02494	0.512469	−	0.858706i	$-0.328731\pi$
$229$	15.5101	1.02494	0.512469	+	0.858706i	$0.328731\pi$
$230$	0.991312	0.0653651
$231$	−37.1350	−2.44331
$232$	−3.32528	−0.218315
$233$	−18.2881	−1.19809	−0.599047	−	0.800714i	$-0.704454\pi$
$233$	−18.2881	−1.19809	−0.599047	+	0.800714i	$0.704454\pi$
$234$	−24.7314	−1.61674
$235$	−2.40708	−0.157021
$236$	−10.9408	−0.712188
$237$	−11.8376	−0.768935
$238$	−14.6679	−0.950776
$239$	14.1894	0.917839	0.458919	−	0.888478i	$-0.348237\pi$
$239$	14.1894	0.917839	0.458919	+	0.888478i	$0.348237\pi$
$240$	−0.878595	−0.0567130
$241$	2.90395	0.187060	0.0935298	−	0.995616i	$-0.470185\pi$
$241$	2.90395	0.187060	0.0935298	+	0.995616i	$0.470185\pi$
$242$	0.994110	0.0639038
$243$	−14.2311	−0.912925
$244$	−0.0726800	−0.00465286
$245$	2.04167	0.130437
$246$	−32.3853	−2.06481
$247$	−18.8554	−1.19974
$248$	−4.38907	−0.278706
$249$	17.2879	1.09558
$250$	2.80080	0.177138
$251$	−19.9334	−1.25818	−0.629091	−	0.777332i	$-0.716573\pi$
$251$	−19.9334	−1.25818	−0.629091	+	0.777332i	$0.716573\pi$
$252$	−25.2158	−1.58844
$253$	11.1066	0.698265
$254$	20.4627	1.28394
$255$	3.41610	0.213924
$256$	1.00000	0.0625000
$257$	7.70448	0.480592	0.240296	−	0.970700i	$-0.422755\pi$
$257$	7.70448	0.480592	0.240296	+	0.970700i	$0.422755\pi$
$258$	−5.01514	−0.312229
$259$	−19.9982	−1.24263
$260$	1.04462	0.0647846
$261$	22.2267	1.37580
$262$	4.49275	0.277563
$263$	8.96845	0.553018	0.276509	−	0.961011i	$-0.410822\pi$
$263$	8.96845	0.553018	0.276509	+	0.961011i	$0.410822\pi$
$264$	−9.84371	−0.605839
$265$	−2.08172	−0.127879
$266$	−19.2247	−1.17874
$267$	−35.1033	−2.14829
$268$	−0.00327689	−0.000200168 0
$269$	−31.8671	−1.94297	−0.971485	−	0.237103i	$-0.923802\pi$
$269$	−31.8671	−1.94297	−0.971485	+	0.237103i	$0.923802\pi$
$270$	3.23689	0.196991
$271$	−14.3524	−0.871847	−0.435923	−	0.899984i	$-0.643578\pi$
$271$	−14.3524	−0.871847	−0.435923	+	0.899984i	$0.643578\pi$
$272$	−3.88814	−0.235753
$273$	43.4367	2.62891
$274$	1.14700	0.0692925
$275$	15.5639	0.938539
$276$	10.9266	0.657702
$277$	14.7361	0.885407	0.442704	−	0.896668i	$-0.354019\pi$
$277$	14.7361	0.885407	0.442704	+	0.896668i	$0.354019\pi$
$278$	−0.532834	−0.0319572
$279$	29.3373	1.75638
$280$	1.06508	0.0636508
$281$	22.1019	1.31849	0.659246	−	0.751928i	$-0.270876\pi$
$281$	22.1019	1.31849	0.659246	+	0.751928i	$0.270876\pi$
$282$	−26.5317	−1.57994
$283$	27.3595	1.62636	0.813178	−	0.582015i	$-0.197736\pi$
$283$	27.3595	1.62636	0.813178	+	0.582015i	$0.197736\pi$
$284$	8.00400	0.474950
$285$	4.47737	0.265216
$286$	11.7039	0.692063
$287$	39.2592	2.31740
$288$	−6.68416	−0.393868
$289$	−1.88240	−0.110729
$290$	−0.938827	−0.0551298
$291$	18.9531	1.11105
$292$	4.09852	0.239848
$293$	12.5301	0.732015	0.366008	−	0.930612i	$-0.380724\pi$
$293$	12.5301	0.732015	0.366008	+	0.930612i	$0.380724\pi$
$294$	22.5039	1.31246
$295$	−3.08893	−0.179844
$296$	−5.30109	−0.308119
$297$	36.2659	2.10436
$298$	4.00920	0.232247
$299$	−12.9913	−0.751308
$300$	15.3116	0.884018
$301$	6.07963	0.350424
$302$	−6.98721	−0.402069
$303$	23.8306	1.36903
$304$	−5.09605	−0.292279
$305$	−0.0205198	−0.00117496
$306$	25.9889	1.48569
$307$	−2.01237	−0.114852	−0.0574261	−	0.998350i	$-0.518289\pi$
$307$	−2.01237	−0.114852	−0.0574261	+	0.998350i	$0.518289\pi$
$308$	11.9331	0.679951
$309$	4.40754	0.250736
$310$	−1.23917	−0.0703800
$311$	8.69566	0.493086	0.246543	−	0.969132i	$-0.420705\pi$
$311$	8.69566	0.493086	0.246543	+	0.969132i	$0.420705\pi$
$312$	11.5142	0.651861
$313$	−31.0829	−1.75691	−0.878454	−	0.477827i	$-0.841425\pi$
$313$	−31.0829	−1.75691	−0.878454	+	0.477827i	$0.841425\pi$
$314$	10.9016	0.615211
$315$	−7.11918	−0.401120
$316$	3.80393	0.213988
$317$	−22.8210	−1.28176	−0.640879	−	0.767642i	$-0.721430\pi$
$317$	−22.8210	−1.28176	−0.640879	+	0.767642i	$0.721430\pi$
$318$	−22.9454	−1.28671
$319$	−10.5185	−0.588926
$320$	0.282330	0.0157827
$321$	29.1435	1.62663
$322$	−13.2458	−0.738159
$323$	19.8141	1.10249
$324$	15.6256	0.868086
$325$	−18.2050	−1.00983
$326$	0.741372	0.0410608
$327$	50.9182	2.81578
$328$	10.4068	0.574619
$329$	32.1632	1.77321
$330$	−2.77918	−0.152989
$331$	−18.6697	−1.02618	−0.513090	−	0.858335i	$-0.671499\pi$
$331$	−18.6697	−1.02618	−0.513090	+	0.858335i	$0.671499\pi$
$332$	−5.55535	−0.304889
$333$	35.4333	1.94173
$334$	15.9444	0.872441
$335$	−0.000925167 0	−5.05473e−5 0
$336$	11.7397	0.640452
$337$	9.43269	0.513831	0.256916	−	0.966434i	$-0.417294\pi$
$337$	9.43269	0.513831	0.256916	+	0.966434i	$0.417294\pi$
$338$	−0.689956	−0.0375287
$339$	11.7562	0.638507
$340$	−1.09774	−0.0595332
$341$	−13.8836	−0.751837
$342$	34.0629	1.84191
$343$	−0.873260	−0.0471516
$344$	1.61158	0.0868906
$345$	3.08490	0.166085
$346$	−0.548112	−0.0294667
$347$	−32.2294	−1.73017	−0.865083	−	0.501629i	$-0.832734\pi$
$347$	−32.2294	−1.73017	−0.865083	+	0.501629i	$0.832734\pi$
$348$	−10.3481	−0.554714
$349$	−31.3719	−1.67930	−0.839649	−	0.543130i	$-0.817239\pi$
$349$	−31.3719	−1.67930	−0.839649	+	0.543130i	$0.817239\pi$
$350$	−18.5616	−0.992160
$351$	−42.4200	−2.26421
$352$	3.16321	0.168600
$353$	11.4706	0.610519	0.305259	−	0.952269i	$-0.401257\pi$
$353$	11.4706	0.610519	0.305259	+	0.952269i	$0.401257\pi$
$354$	−34.0472	−1.80959
$355$	2.25977	0.119936
$356$	11.2802	0.597850
$357$	−45.6455	−2.41581
$358$	4.31099	0.227843
$359$	2.65590	0.140173	0.0700866	−	0.997541i	$-0.477672\pi$
$359$	2.65590	0.140173	0.0700866	+	0.997541i	$0.477672\pi$
$360$	−1.88714	−0.0994611
$361$	6.96976	0.366830
$362$	8.93195	0.469453
$363$	3.09361	0.162372
$364$	−13.9581	−0.731603
$365$	1.15714	0.0605673
$366$	−0.226176	−0.0118224
$367$	−17.3458	−0.905443	−0.452722	−	0.891652i	$-0.649547\pi$
$367$	−17.3458	−0.905443	−0.452722	+	0.891652i	$0.649547\pi$
$368$	−3.51118	−0.183033
$369$	−69.5607	−3.62119
$370$	−1.49666	−0.0778075
$371$	27.8157	1.44412
$372$	−13.6585	−0.708162
$373$	1.16230	0.0601815	0.0300907	−	0.999547i	$-0.490420\pi$
$373$	1.16230	0.0601815	0.0300907	+	0.999547i	$0.490420\pi$
$374$	−12.2990	−0.635965
$375$	8.71591	0.450088
$376$	8.52577	0.439683
$377$	12.3035	0.633663
$378$	−43.2509	−2.22459
$379$	−5.10563	−0.262259	−0.131129	−	0.991365i	$-0.541860\pi$
$379$	−5.10563	−0.262259	−0.131129	+	0.991365i	$0.541860\pi$
$380$	−1.43877	−0.0738074
$381$	63.6786	3.26235
$382$	−19.4085	−0.993023
$383$	−35.7308	−1.82576	−0.912879	−	0.408229i	$-0.866146\pi$
$383$	−35.7308	−1.82576	−0.912879	+	0.408229i	$0.866146\pi$
$384$	3.11194	0.158805
$385$	3.36907	0.171704
$386$	11.9915	0.610351
$387$	−10.7721	−0.547575
$388$	−6.09044	−0.309195
$389$	16.8913	0.856421	0.428210	−	0.903679i	$-0.359144\pi$
$389$	16.8913	0.856421	0.428210	+	0.903679i	$0.359144\pi$
$390$	3.25080	0.164610
$391$	13.6519	0.690408
$392$	−7.23148	−0.365245
$393$	13.9812	0.705257
$394$	21.5872	1.08755
$395$	1.07397	0.0540371
$396$	−21.1434	−1.06250
$397$	32.1989	1.61602	0.808008	−	0.589171i	$-0.200546\pi$
$397$	32.1989	1.61602	0.808008	+	0.589171i	$0.200546\pi$
$398$	12.4673	0.624927
$399$	−59.8260	−2.99505
$400$	−4.92029	−0.246014
$401$	−15.6610	−0.782074	−0.391037	−	0.920375i	$-0.627884\pi$
$401$	−15.6610	−0.782074	−0.391037	+	0.920375i	$0.627884\pi$
$402$	−0.0101975	−0.000508605 0
$403$	16.2395	0.808949
$404$	−7.65780	−0.380990
$405$	4.41157	0.219213
$406$	12.5445	0.622573
$407$	−16.7684	−0.831181
$408$	−12.0996	−0.599021
$409$	−17.9319	−0.886673	−0.443337	−	0.896355i	$-0.646205\pi$
$409$	−17.9319	−0.886673	−0.443337	+	0.896355i	$0.646205\pi$
$410$	2.93815	0.145105
$411$	3.56938	0.176064
$412$	−1.41633	−0.0697777
$413$	41.2739	2.03096
$414$	23.4693	1.15345
$415$	−1.56844	−0.0769919
$416$	−3.69999	−0.181407
$417$	−1.65815	−0.0811998
$418$	−16.1199	−0.788449
$419$	−21.7346	−1.06181	−0.530903	−	0.847432i	$-0.678147\pi$
$419$	−21.7346	−1.06181	−0.530903	+	0.847432i	$0.678147\pi$
$420$	3.31447	0.161729
$421$	−6.90605	−0.336580	−0.168290	−	0.985737i	$-0.553825\pi$
$421$	−6.90605	−0.336580	−0.168290	+	0.985737i	$0.553825\pi$
$422$	−13.5767	−0.660904
$423$	−56.9876	−2.77083
$424$	7.37335	0.358081
$425$	19.1308	0.927978
$426$	24.9079	1.20679
$427$	0.274183	0.0132686
$428$	−9.36505	−0.452677
$429$	36.4217	1.75846
$430$	0.454998	0.0219420
$431$	31.3313	1.50917	0.754587	−	0.656200i	$-0.227837\pi$
$431$	31.3313	1.50917	0.754587	+	0.656200i	$0.227837\pi$
$432$	−11.4649	−0.551605
$433$	8.45111	0.406134	0.203067	−	0.979165i	$-0.434909\pi$
$433$	8.45111	0.406134	0.203067	+	0.979165i	$0.434909\pi$
$434$	16.5576	0.794791
$435$	−2.92157	−0.140079
$436$	−16.3622	−0.783608
$437$	17.8931	0.855945
$438$	12.7543	0.609426
$439$	37.0443	1.76803	0.884014	−	0.467460i	$-0.154831\pi$
$439$	37.0443	1.76803	0.884014	+	0.467460i	$0.154831\pi$
$440$	0.893070	0.0425754
$441$	48.3364	2.30173
$442$	14.3861	0.684276
$443$	6.88564	0.327146	0.163573	−	0.986531i	$-0.447698\pi$
$443$	6.88564	0.327146	0.163573	+	0.986531i	$0.447698\pi$
$444$	−16.4967	−0.782897
$445$	3.18475	0.150972
$446$	16.0420	0.759613
$447$	12.4764	0.590114
$448$	−3.77246	−0.178232
$449$	−11.4882	−0.542161	−0.271080	−	0.962557i	$-0.587381\pi$
$449$	−11.4882	−0.542161	−0.271080	+	0.962557i	$0.587381\pi$
$450$	32.8880	1.55036
$451$	32.9189	1.55009
$452$	−3.77776	−0.177691
$453$	−21.7438	−1.02161
$454$	22.0138	1.03316
$455$	−3.94079	−0.184747
$456$	−15.8586	−0.742647
$457$	17.3587	0.812007	0.406003	−	0.913872i	$-0.366922\pi$
$457$	17.3587	0.812007	0.406003	+	0.913872i	$0.366922\pi$
$458$	−15.5101	−0.724741
$459$	44.5771	2.08068
$460$	−0.991312	−0.0462201
$461$	34.8308	1.62223	0.811116	−	0.584885i	$-0.198860\pi$
$461$	34.8308	1.62223	0.811116	+	0.584885i	$0.198860\pi$
$462$	37.1350	1.72768
$463$	−40.6142	−1.88750	−0.943751	−	0.330657i	$-0.892730\pi$
$463$	−40.6142	−1.88750	−0.943751	+	0.330657i	$0.892730\pi$
$464$	3.32528	0.154372
$465$	−3.85622	−0.178828
$466$	18.2881	0.847181
$467$	13.6161	0.630077	0.315038	−	0.949079i	$-0.397983\pi$
$467$	13.6161	0.630077	0.315038	+	0.949079i	$0.397983\pi$
$468$	24.7314	1.14321
$469$	0.0123620	0.000570823 0
$470$	2.40708	0.111030
$471$	33.9250	1.56318
$472$	10.9408	0.503593
$473$	5.09777	0.234396
$474$	11.8376	0.543719
$475$	25.0741	1.15048
$476$	14.6679	0.672300
$477$	−49.2847	−2.25659
$478$	−14.1894	−0.649010
$479$	−12.4128	−0.567157	−0.283578	−	0.958949i	$-0.591522\pi$
$479$	−12.4128	−0.567157	−0.283578	+	0.958949i	$0.591522\pi$
$480$	0.878595	0.0401022
$481$	19.6140	0.894321
$482$	−2.90395	−0.132271
$483$	−41.2201	−1.87558
$484$	−0.994110	−0.0451868
$485$	−1.71952	−0.0780792
$486$	14.2311	0.645536
$487$	−6.24012	−0.282767	−0.141384	−	0.989955i	$-0.545155\pi$
$487$	−6.24012	−0.282767	−0.141384	+	0.989955i	$0.545155\pi$
$488$	0.0726800	0.00329007
$489$	2.30710	0.104331
$490$	−2.04167	−0.0922331
$491$	31.6616	1.42887	0.714433	−	0.699704i	$-0.246685\pi$
$491$	31.6616	1.42887	0.714433	+	0.699704i	$0.246685\pi$
$492$	32.3853	1.46004
$493$	−12.9291	−0.582299
$494$	18.8554	0.848343
$495$	−5.96943	−0.268306
$496$	4.38907	0.197075
$497$	−30.1948	−1.35442
$498$	−17.2879	−0.774690
$499$	17.1921	0.769622	0.384811	−	0.922995i	$-0.374267\pi$
$499$	17.1921	0.769622	0.384811	+	0.922995i	$0.374267\pi$
$500$	−2.80080	−0.125256
$501$	49.6181	2.21677
$502$	19.9334	0.889669
$503$	−3.32507	−0.148257	−0.0741287	−	0.997249i	$-0.523618\pi$
$503$	−3.32507	−0.148257	−0.0741287	+	0.997249i	$0.523618\pi$
$504$	25.2158	1.12320
$505$	−2.16203	−0.0962091
$506$	−11.1066	−0.493748
$507$	−2.14710	−0.0953561
$508$	−20.4627	−0.907884
$509$	20.3738	0.903055	0.451527	−	0.892257i	$-0.350879\pi$
$509$	20.3738	0.903055	0.451527	+	0.892257i	$0.350879\pi$
$510$	−3.41610	−0.151267
$511$	−15.4615	−0.683978
$512$	−1.00000	−0.0441942
$513$	58.4257	2.57956
$514$	−7.70448	−0.339830
$515$	−0.399874	−0.0176206
$516$	5.01514	0.220779
$517$	26.9688	1.18609
$518$	19.9982	0.878669
$519$	−1.70569	−0.0748715
$520$	−1.04462	−0.0458096
$521$	5.81618	0.254811	0.127406	−	0.991851i	$-0.459335\pi$
$521$	5.81618	0.254811	0.127406	+	0.991851i	$0.459335\pi$
$522$	−22.2267	−0.972836
$523$	−4.13739	−0.180915	−0.0904577	−	0.995900i	$-0.528833\pi$
$523$	−4.13739	−0.180915	−0.0904577	+	0.995900i	$0.528833\pi$
$524$	−4.49275	−0.196267
$525$	−57.7626	−2.52097
$526$	−8.96845	−0.391043
$527$	−17.0653	−0.743377
$528$	9.84371	0.428393
$529$	−10.6716	−0.463984
$530$	2.08172	0.0904241
$531$	−73.1303	−3.17359
$532$	19.2247	0.833496
$533$	−38.5051	−1.66784
$534$	35.1033	1.51907
$535$	−2.64404	−0.114312
$536$	0.00327689	0.000141540 0
$537$	13.4156	0.578924
$538$	31.8671	1.37389
$539$	−22.8747	−0.985283
$540$	−3.23689	−0.139293
$541$	32.3849	1.39234	0.696168	−	0.717879i	$-0.254887\pi$
$541$	32.3849	1.39234	0.696168	+	0.717879i	$0.254887\pi$
$542$	14.3524	0.616489
$543$	27.7957	1.19283
$544$	3.88814	0.166702
$545$	−4.61955	−0.197880
$546$	−43.4367	−1.85892
$547$	40.2077	1.71916	0.859580	−	0.511002i	$-0.170726\pi$
$547$	40.2077	1.71916	0.859580	+	0.511002i	$0.170726\pi$
$548$	−1.14700	−0.0489972
$549$	−0.485805	−0.0207337
$550$	−15.5639	−0.663647
$551$	−16.9458	−0.721915
$552$	−10.9266	−0.465066
$553$	−14.3502	−0.610232
$554$	−14.7361	−0.626077
$555$	−4.65751	−0.197700
$556$	0.532834	0.0225972
$557$	−18.7105	−0.792791	−0.396395	−	0.918080i	$-0.629739\pi$
$557$	−18.7105	−0.792791	−0.396395	+	0.918080i	$0.629739\pi$
$558$	−29.3373	−1.24195
$559$	−5.96284	−0.252201
$560$	−1.06508	−0.0450079
$561$	−38.2737	−1.61592
$562$	−22.1019	−0.932314
$563$	37.4568	1.57862	0.789308	−	0.613997i	$-0.210439\pi$
$563$	37.4568	1.57862	0.789308	+	0.613997i	$0.210439\pi$
$564$	26.5317	1.11718
$565$	−1.06658	−0.0448712
$566$	−27.3595	−1.15001
$567$	−58.9468	−2.47553
$568$	−8.00400	−0.335840
$569$	7.00570	0.293694	0.146847	−	0.989159i	$-0.453087\pi$
$569$	7.00570	0.293694	0.146847	+	0.989159i	$0.453087\pi$
$570$	−4.47737	−0.187536
$571$	−1.39061	−0.0581952	−0.0290976	−	0.999577i	$-0.509263\pi$
$571$	−1.39061	−0.0581952	−0.0290976	+	0.999577i	$0.509263\pi$
$572$	−11.7039	−0.489363
$573$	−60.3979	−2.52316
$574$	−39.2592	−1.63865
$575$	17.2760	0.720459
$576$	6.68416	0.278507
$577$	21.8005	0.907567	0.453783	−	0.891112i	$-0.350074\pi$
$577$	21.8005	0.907567	0.453783	+	0.891112i	$0.350074\pi$
$578$	1.88240	0.0782975
$579$	37.3168	1.55083
$580$	0.938827	0.0389827
$581$	20.9573	0.869457
$582$	−18.9531	−0.785630
$583$	23.3234	0.965958
$584$	−4.09852	−0.169598
$585$	6.98242	0.288687
$586$	−12.5301	−0.517613
$587$	8.93439	0.368762	0.184381	−	0.982855i	$-0.440972\pi$
$587$	8.93439	0.368762	0.184381	+	0.982855i	$0.440972\pi$
$588$	−22.5039	−0.928047
$589$	−22.3670	−0.921614
$590$	3.08893	0.127169
$591$	67.1780	2.76333
$592$	5.30109	0.217873
$593$	23.7677	0.976023	0.488012	−	0.872837i	$-0.337722\pi$
$593$	23.7677	0.976023	0.488012	+	0.872837i	$0.337722\pi$
$594$	−36.2659	−1.48801
$595$	4.14118	0.169772
$596$	−4.00920	−0.164223
$597$	38.7973	1.58787
$598$	12.9913	0.531255
$599$	−26.5521	−1.08489	−0.542445	−	0.840091i	$-0.682501\pi$
$599$	−26.5521	−1.08489	−0.542445	+	0.840091i	$0.682501\pi$
$600$	−15.3116	−0.625095
$601$	7.17329	0.292605	0.146302	−	0.989240i	$-0.453263\pi$
$601$	7.17329	0.292605	0.146302	+	0.989240i	$0.453263\pi$
$602$	−6.07963	−0.247787
$603$	−0.0219033	−0.000891971 0
$604$	6.98721	0.284306
$605$	−0.280667	−0.0114107
$606$	−23.8306	−0.968053
$607$	−28.2513	−1.14669	−0.573343	−	0.819315i	$-0.694354\pi$
$607$	−28.2513	−1.14669	−0.573343	+	0.819315i	$0.694354\pi$
$608$	5.09605	0.206672
$609$	39.0377	1.58189
$610$	0.0205198	0.000830821 0
$611$	−31.5453	−1.27619
$612$	−25.9889	−1.05054
$613$	−27.4831	−1.11003	−0.555017	−	0.831839i	$-0.687288\pi$
$613$	−27.4831	−1.11003	−0.555017	+	0.831839i	$0.687288\pi$
$614$	2.01237	0.0812128
$615$	9.14335	0.368696
$616$	−11.9331	−0.480798
$617$	−28.5611	−1.14983	−0.574913	−	0.818215i	$-0.694964\pi$
$617$	−28.5611	−1.14983	−0.574913	+	0.818215i	$0.694964\pi$
$618$	−4.40754	−0.177297
$619$	8.65279	0.347785	0.173893	−	0.984765i	$-0.444365\pi$
$619$	8.65279	0.347785	0.173893	+	0.984765i	$0.444365\pi$
$620$	1.23917	0.0497662
$621$	40.2553	1.61539
$622$	−8.69566	−0.348664
$623$	−42.5542	−1.70490
$624$	−11.5142	−0.460935
$625$	23.8107	0.952428
$626$	31.0829	1.24232
$627$	−50.1641	−2.00336
$628$	−10.9016	−0.435020
$629$	−20.6113	−0.821828
$630$	7.11918	0.283635
$631$	9.66457	0.384740	0.192370	−	0.981322i	$-0.438383\pi$
$631$	9.66457	0.384740	0.192370	+	0.981322i	$0.438383\pi$
$632$	−3.80393	−0.151312
$633$	−42.2499	−1.67928
$634$	22.8210	0.906339
$635$	−5.77723	−0.229262
$636$	22.9454	0.909845
$637$	26.7564	1.06013
$638$	10.5185	0.416433
$639$	53.5000	2.11643
$640$	−0.282330	−0.0111601
$641$	45.2798	1.78844	0.894222	−	0.447625i	$-0.147730\pi$
$641$	45.2798	1.78844	0.894222	+	0.447625i	$0.147730\pi$
$642$	−29.1435	−1.15020
$643$	25.1321	0.991113	0.495556	−	0.868576i	$-0.334964\pi$
$643$	25.1321	0.991113	0.495556	+	0.868576i	$0.334964\pi$
$644$	13.2458	0.521957
$645$	1.41593	0.0557521
$646$	−19.8141	−0.779577
$647$	45.9835	1.80780	0.903900	−	0.427744i	$-0.140692\pi$
$647$	45.9835	1.80780	0.903900	+	0.427744i	$0.140692\pi$
$648$	−15.6256	−0.613830
$649$	34.6082	1.35849
$650$	18.2050	0.714061
$651$	51.5263	2.01948
$652$	−0.741372	−0.0290344
$653$	−21.1273	−0.826777	−0.413388	−	0.910555i	$-0.635655\pi$
$653$	−21.1273	−0.826777	−0.413388	+	0.910555i	$0.635655\pi$
$654$	−50.9182	−1.99106
$655$	−1.26844	−0.0495621
$656$	−10.4068	−0.406317
$657$	27.3952	1.06879
$658$	−32.1632	−1.25385
$659$	14.4411	0.562543	0.281272	−	0.959628i	$-0.409244\pi$
$659$	14.4411	0.562543	0.281272	+	0.959628i	$0.409244\pi$
$660$	2.77918	0.108179
$661$	27.8043	1.08146	0.540731	−	0.841196i	$-0.318148\pi$
$661$	27.8043	1.08146	0.540731	+	0.841196i	$0.318148\pi$
$662$	18.6697	0.725619
$663$	44.7686	1.73867
$664$	5.55535	0.215589
$665$	5.42771	0.210478
$666$	−35.4333	−1.37301
$667$	−11.6756	−0.452082
$668$	−15.9444	−0.616909
$669$	49.9218	1.93009
$670$	0.000925167 0	3.57423e−5 0
$671$	0.229902	0.00887527
$672$	−11.7397	−0.452868
$673$	−36.5174	−1.40764	−0.703821	−	0.710377i	$-0.748524\pi$
$673$	−36.5174	−1.40764	−0.703821	+	0.710377i	$0.748524\pi$
$674$	−9.43269	−0.363334
$675$	56.4106	2.17124
$676$	0.689956	0.0265368
$677$	−4.11894	−0.158304	−0.0791518	−	0.996863i	$-0.525221\pi$
$677$	−4.11894	−0.158304	−0.0791518	+	0.996863i	$0.525221\pi$
$678$	−11.7562	−0.451493
$679$	22.9760	0.881736
$680$	1.09774	0.0420964
$681$	68.5056	2.62514
$682$	13.8836	0.531629
$683$	1.80061	0.0688982	0.0344491	−	0.999406i	$-0.489032\pi$
$683$	1.80061	0.0688982	0.0344491	+	0.999406i	$0.489032\pi$
$684$	−34.0629	−1.30243
$685$	−0.323832	−0.0123730
$686$	0.873260	0.0333412
$687$	−48.2666	−1.84148
$688$	−1.61158	−0.0614410
$689$	−27.2813	−1.03934
$690$	−3.08490	−0.117440
$691$	16.8821	0.642226	0.321113	−	0.947041i	$-0.395943\pi$
$691$	16.8821	0.642226	0.321113	+	0.947041i	$0.395943\pi$
$692$	0.548112	0.0208361
$693$	79.7627	3.02994
$694$	32.2294	1.22341
$695$	0.150435	0.00570633
$696$	10.3481	0.392242
$697$	40.4630	1.53265
$698$	31.3719	1.18744
$699$	56.9115	2.15259
$700$	18.5616	0.701563
$701$	−38.4082	−1.45066	−0.725329	−	0.688403i	$-0.758312\pi$
$701$	−38.4082	−1.45066	−0.725329	+	0.688403i	$0.758312\pi$
$702$	42.4200	1.60104
$703$	−27.0146	−1.01888
$704$	−3.16321	−0.119218
$705$	7.49070	0.282116
$706$	−11.4706	−0.431702
$707$	28.8888	1.08647
$708$	34.0472	1.27957
$709$	21.0831	0.791791	0.395895	−	0.918296i	$-0.370434\pi$
$709$	21.0831	0.791791	0.395895	+	0.918296i	$0.370434\pi$
$710$	−2.25977	−0.0848077
$711$	25.4261	0.953553
$712$	−11.2802	−0.422744
$713$	−15.4108	−0.577139
$714$	45.6455	1.70824
$715$	−3.30435	−0.123576
$716$	−4.31099	−0.161109
$717$	−44.1567	−1.64906
$718$	−2.65590	−0.0991174
$719$	−39.1888	−1.46150	−0.730748	−	0.682647i	$-0.760829\pi$
$719$	−39.1888	−1.46150	−0.730748	+	0.682647i	$0.760829\pi$
$720$	1.88714	0.0703296
$721$	5.34307	0.198986
$722$	−6.96976	−0.259388
$723$	−9.03690	−0.336086
$724$	−8.93195	−0.331953
$725$	−16.3613	−0.607644
$726$	−3.09361	−0.114815
$727$	−20.0876	−0.745006	−0.372503	−	0.928031i	$-0.621500\pi$
$727$	−20.0876	−0.745006	−0.372503	+	0.928031i	$0.621500\pi$
$728$	13.9581	0.517321
$729$	−2.59036	−0.0959393
$730$	−1.15714	−0.0428276
$731$	6.26605	0.231758
$732$	0.226176	0.00835970
$733$	13.2528	0.489504	0.244752	−	0.969586i	$-0.421294\pi$
$733$	13.2528	0.489504	0.244752	+	0.969586i	$0.421294\pi$
$734$	17.3458	0.640245
$735$	−6.35354	−0.234354
$736$	3.51118	0.129424
$737$	0.0103655	0.000381818 0
$738$	69.5607	2.56056
$739$	44.0551	1.62059	0.810296	−	0.586020i	$-0.199306\pi$
$739$	44.0551	1.62059	0.810296	+	0.586020i	$0.199306\pi$
$740$	1.49666	0.0550182
$741$	58.6768	2.15554
$742$	−27.8157	−1.02115
$743$	−15.8103	−0.580025	−0.290012	−	0.957023i	$-0.593659\pi$
$743$	−15.8103	−0.580025	−0.290012	+	0.957023i	$0.593659\pi$
$744$	13.6585	0.500746
$745$	−1.13192	−0.0414703
$746$	−1.16230	−0.0425547
$747$	−37.1329	−1.35862
$748$	12.2990	0.449695
$749$	35.3293	1.29090
$750$	−8.71591	−0.318260
$751$	14.9140	0.544220	0.272110	−	0.962266i	$-0.412279\pi$
$751$	14.9140	0.544220	0.272110	+	0.962266i	$0.412279\pi$
$752$	−8.52577	−0.310903
$753$	62.0314	2.26055
$754$	−12.3035	−0.448067
$755$	1.97270	0.0717940
$756$	43.2509	1.57302
$757$	28.1464	1.02300	0.511500	−	0.859283i	$-0.329090\pi$
$757$	28.1464	1.02300	0.511500	+	0.859283i	$0.329090\pi$
$758$	5.10563	0.185445
$759$	−34.5630	−1.25456
$760$	1.43877	0.0521897
$761$	15.8387	0.574154	0.287077	−	0.957908i	$-0.407316\pi$
$761$	15.8387	0.574154	0.287077	+	0.957908i	$0.407316\pi$
$762$	−63.6786	−2.30683
$763$	61.7259	2.23463
$764$	19.4085	0.702173
$765$	−7.33747	−0.265287
$766$	35.7308	1.29101
$767$	−40.4810	−1.46169
$768$	−3.11194	−0.112292
$769$	−6.14184	−0.221480	−0.110740	−	0.993849i	$-0.535322\pi$
$769$	−6.14184	−0.221480	−0.110740	+	0.993849i	$0.535322\pi$
$770$	−3.36907	−0.121413
$771$	−23.9759	−0.863470
$772$	−11.9915	−0.431583
$773$	−14.7294	−0.529779	−0.264889	−	0.964279i	$-0.585335\pi$
$773$	−14.7294	−0.529779	−0.264889	+	0.964279i	$0.585335\pi$
$774$	10.7721	0.387194
$775$	−21.5955	−0.775734
$776$	6.09044	0.218634
$777$	62.2330	2.23260
$778$	−16.8913	−0.605581
$779$	53.0336	1.90012
$780$	−3.25080	−0.116397
$781$	−25.3183	−0.905960
$782$	−13.6519	−0.488192
$783$	−38.1240	−1.36244
$784$	7.23148	0.258267
$785$	−3.07784	−0.109853
$786$	−13.9812	−0.498692
$787$	17.9792	0.640888	0.320444	−	0.947267i	$-0.396168\pi$
$787$	17.9792	0.640888	0.320444	+	0.947267i	$0.396168\pi$
$788$	−21.5872	−0.769012
$789$	−27.9093	−0.993596
$790$	−1.07397	−0.0382100
$791$	14.2515	0.506724
$792$	21.1434	0.751298
$793$	−0.268916	−0.00954947
$794$	−32.1989	−1.14270
$795$	6.47818	0.229758
$796$	−12.4673	−0.441890
$797$	−41.3323	−1.46406	−0.732032	−	0.681270i	$-0.761428\pi$
$797$	−41.3323	−1.46406	−0.732032	+	0.681270i	$0.761428\pi$
$798$	59.8260	2.11782
$799$	33.1493	1.17274
$800$	4.92029	0.173959
$801$	75.3988	2.66409
$802$	15.6610	0.553010
$803$	−12.9645	−0.457507
$804$	0.0101975	0.000359638 0
$805$	3.73969	0.131807
$806$	−16.2395	−0.572014
$807$	99.1684	3.49089
$808$	7.65780	0.269401
$809$	53.3135	1.87440	0.937202	−	0.348788i	$-0.113407\pi$
$809$	53.3135	1.87440	0.937202	+	0.348788i	$0.113407\pi$
$810$	−4.41157	−0.155007
$811$	12.4645	0.437686	0.218843	−	0.975760i	$-0.429772\pi$
$811$	12.4645	0.437686	0.218843	+	0.975760i	$0.429772\pi$
$812$	−12.5445	−0.440225
$813$	44.6638	1.56643
$814$	16.7684	0.587734
$815$	−0.209312	−0.00733187
$816$	12.0996	0.423572
$817$	8.21270	0.287326
$818$	17.9319	0.626973
$819$	−93.2982	−3.26010
$820$	−2.93815	−0.102605
$821$	37.3191	1.30244	0.651222	−	0.758887i	$-0.274256\pi$
$821$	37.3191	1.30244	0.651222	+	0.758887i	$0.274256\pi$
$822$	−3.56938	−0.124496
$823$	−0.201405	−0.00702055	−0.00351028	−	0.999994i	$-0.501117\pi$
$823$	−0.201405	−0.00702055	−0.00351028	+	0.999994i	$0.501117\pi$
$824$	1.41633	0.0493403
$825$	−48.4339	−1.68625
$826$	−41.2739	−1.43610
$827$	−3.61206	−0.125604	−0.0628019	−	0.998026i	$-0.520004\pi$
$827$	−3.61206	−0.125604	−0.0628019	+	0.998026i	$0.520004\pi$
$828$	−23.4693	−0.815614
$829$	−48.0844	−1.67004	−0.835021	−	0.550218i	$-0.814544\pi$
$829$	−48.0844	−1.67004	−0.835021	+	0.550218i	$0.814544\pi$
$830$	1.56844	0.0544415
$831$	−45.8579	−1.59079
$832$	3.69999	0.128274
$833$	−28.1170	−0.974196
$834$	1.65815	0.0574169
$835$	−4.50160	−0.155784
$836$	16.1199	0.557518
$837$	−50.3203	−1.73932
$838$	21.7346	0.750811
$839$	10.0106	0.345603	0.172801	−	0.984957i	$-0.444718\pi$
$839$	10.0106	0.345603	0.172801	+	0.984957i	$0.444718\pi$
$840$	−3.31447	−0.114360
$841$	−17.9425	−0.618708
$842$	6.90605	0.237998
$843$	−68.7799	−2.36891
$844$	13.5767	0.467330
$845$	0.194795	0.00670117
$846$	56.9876	1.95928
$847$	3.75024	0.128860
$848$	−7.37335	−0.253202
$849$	−85.1412	−2.92204
$850$	−19.1308	−0.656179
$851$	−18.6130	−0.638047
$852$	−24.9079	−0.853332
$853$	35.9302	1.23023	0.615114	−	0.788438i	$-0.289110\pi$
$853$	35.9302	1.23023	0.615114	+	0.788438i	$0.289110\pi$
$854$	−0.274183	−0.00938234
$855$	−9.61698	−0.328894
$856$	9.36505	0.320091
$857$	−7.48934	−0.255831	−0.127916	−	0.991785i	$-0.540829\pi$
$857$	−7.48934	−0.255831	−0.127916	+	0.991785i	$0.540829\pi$
$858$	−36.4217	−1.24342
$859$	29.4887	1.00614	0.503070	−	0.864246i	$-0.332204\pi$
$859$	29.4887	1.00614	0.503070	+	0.864246i	$0.332204\pi$
$860$	−0.454998	−0.0155153
$861$	−122.172	−4.16362
$862$	−31.3313	−1.06715
$863$	45.8617	1.56115	0.780576	−	0.625061i	$-0.214926\pi$
$863$	45.8617	1.56115	0.780576	+	0.625061i	$0.214926\pi$
$864$	11.4649	0.390044
$865$	0.154749	0.00526161
$866$	−8.45111	−0.287180
$867$	5.85792	0.198945
$868$	−16.5576	−0.562002
$869$	−12.0326	−0.408179
$870$	2.92157	0.0990505
$871$	−0.0121245	−0.000410823 0
$872$	16.3622	0.554095
$873$	−40.7095	−1.37781
$874$	−17.8931	−0.605245
$875$	10.5659	0.357193
$876$	−12.7543	−0.430930
$877$	13.5808	0.458591	0.229296	−	0.973357i	$-0.426358\pi$
$877$	13.5808	0.458591	0.229296	+	0.973357i	$0.426358\pi$
$878$	−37.0443	−1.25019
$879$	−38.9929	−1.31520
$880$	−0.893070	−0.0301054
$881$	−10.3050	−0.347183	−0.173591	−	0.984818i	$-0.555537\pi$
$881$	−10.3050	−0.347183	−0.173591	+	0.984818i	$0.555537\pi$
$882$	−48.3364	−1.62757
$883$	42.5835	1.43305	0.716524	−	0.697562i	$-0.245732\pi$
$883$	42.5835	1.43305	0.716524	+	0.697562i	$0.245732\pi$
$884$	−14.3861	−0.483856
$885$	9.61256	0.323123
$886$	−6.88564	−0.231327
$887$	−39.8940	−1.33951	−0.669754	−	0.742583i	$-0.733601\pi$
$887$	−39.8940	−1.33951	−0.669754	+	0.742583i	$0.733601\pi$
$888$	16.4967	0.553592
$889$	77.1947	2.58903
$890$	−3.18475	−0.106753
$891$	−49.4269	−1.65586
$892$	−16.0420	−0.537127
$893$	43.4478	1.45392
$894$	−12.4764	−0.417273
$895$	−1.21712	−0.0406840
$896$	3.77246	0.126029
$897$	40.4282	1.34986
$898$	11.4882	0.383366
$899$	14.5949	0.486767
$900$	−32.8880	−1.09627
$901$	28.6686	0.955089
$902$	−32.9189	−1.09608
$903$	−18.9194	−0.629600
$904$	3.77776	0.125646
$905$	−2.52176	−0.0838261
$906$	21.7438	0.722389
$907$	49.8659	1.65577	0.827885	−	0.560898i	$-0.189544\pi$
$907$	49.8659	1.65577	0.827885	+	0.560898i	$0.189544\pi$
$908$	−22.0138	−0.730554
$909$	−51.1860	−1.69773
$910$	3.94079	0.130636
$911$	−60.1061	−1.99140	−0.995702	−	0.0926120i	$-0.970478\pi$
$911$	−60.1061	−1.99140	−0.995702	+	0.0926120i	$0.970478\pi$
$912$	15.8586	0.525131
$913$	17.5727	0.581572
$914$	−17.3587	−0.574175
$915$	0.0638563	0.00211102
$916$	15.5101	0.512469
$917$	16.9488	0.559697
$918$	−44.5771	−1.47126
$919$	−3.94937	−0.130278	−0.0651388	−	0.997876i	$-0.520749\pi$
$919$	−3.94937	−0.130278	−0.0651388	+	0.997876i	$0.520749\pi$
$920$	0.991312	0.0326826
$921$	6.26239	0.206353
$922$	−34.8308	−1.14709
$923$	29.6147	0.974781
$924$	−37.1350	−1.22165
$925$	−26.0829	−0.857600
$926$	40.6142	1.33467
$927$	−9.46701	−0.310937
$928$	−3.32528	−0.109158
$929$	−12.6422	−0.414776	−0.207388	−	0.978259i	$-0.566496\pi$
$929$	−12.6422	−0.414776	−0.207388	+	0.978259i	$0.566496\pi$
$930$	3.85622	0.126450
$931$	−36.8520	−1.20778
$932$	−18.2881	−0.599047
$933$	−27.0604	−0.885917
$934$	−13.6161	−0.445532
$935$	3.47238	0.113559
$936$	−24.7314	−0.808370
$937$	38.3161	1.25173	0.625866	−	0.779930i	$-0.284746\pi$
$937$	38.3161	1.25173	0.625866	+	0.779930i	$0.284746\pi$
$938$	−0.0123620	−0.000403632 0
$939$	96.7280	3.15660
$940$	−2.40708	−0.0785104
$941$	−6.55470	−0.213677	−0.106839	−	0.994276i	$-0.534073\pi$
$941$	−6.55470	−0.213677	−0.106839	+	0.994276i	$0.534073\pi$
$942$	−33.9250	−1.10534
$943$	36.5401	1.18991
$944$	−10.9408	−0.356094
$945$	12.2110	0.397225
$946$	−5.09777	−0.165743
$947$	0.271701	0.00882911	0.00441455	−	0.999990i	$-0.498595\pi$
$947$	0.271701	0.00882911	0.00441455	+	0.999990i	$0.498595\pi$
$948$	−11.8376	−0.384467
$949$	15.1645	0.492261
$950$	−25.0741	−0.813510
$951$	71.0177	2.30291
$952$	−14.6679	−0.475388
$953$	32.1577	1.04169	0.520845	−	0.853651i	$-0.325617\pi$
$953$	32.1577	1.04169	0.520845	+	0.853651i	$0.325617\pi$
$954$	49.2847	1.59565
$955$	5.47959	0.177316
$956$	14.1894	0.458919
$957$	32.7331	1.05811
$958$	12.4128	0.401040
$959$	4.32700	0.139726
$960$	−0.878595	−0.0283565
$961$	−11.7360	−0.378582
$962$	−19.6140	−0.632380
$963$	−62.5975	−2.01718
$964$	2.90395	0.0935298
$965$	−3.38556	−0.108985
$966$	41.2201	1.32623
$967$	42.4708	1.36577	0.682885	−	0.730526i	$-0.260725\pi$
$967$	42.4708	1.36577	0.682885	+	0.730526i	$0.260725\pi$
$968$	0.994110	0.0319519
$969$	−61.6604	−1.98082
$970$	1.71952	0.0552103
$971$	10.3154	0.331037	0.165518	−	0.986207i	$-0.447070\pi$
$971$	10.3154	0.331037	0.165518	+	0.986207i	$0.447070\pi$
$972$	−14.2311	−0.456463
$973$	−2.01010	−0.0644407
$974$	6.24012	0.199946
$975$	56.6530	1.81435
$976$	−0.0726800	−0.00232643
$977$	−35.3676	−1.13151	−0.565755	−	0.824573i	$-0.691415\pi$
$977$	−35.3676	−1.13151	−0.565755	+	0.824573i	$0.691415\pi$
$978$	−2.30710	−0.0737730
$979$	−35.6817	−1.14039
$980$	2.04167	0.0652187
$981$	−109.368	−3.49184
$982$	−31.6616	−1.01036
$983$	−46.8945	−1.49570	−0.747851	−	0.663866i	$-0.768914\pi$
$983$	−46.8945	−1.49570	−0.747851	+	0.663866i	$0.768914\pi$
$984$	−32.3853	−1.03241
$985$	−6.09472	−0.194194
$986$	12.9291	0.411747
$987$	−100.090	−3.18589
$988$	−18.8554	−0.599869
$989$	5.65855	0.179931
$990$	5.96943	0.189721
$991$	−47.9449	−1.52302	−0.761510	−	0.648153i	$-0.775542\pi$
$991$	−47.9449	−1.52302	−0.761510	+	0.648153i	$0.775542\pi$
$992$	−4.38907	−0.139353
$993$	58.0990	1.84372
$994$	30.1948	0.957720
$995$	−3.51988	−0.111588
$996$	17.2879	0.547788
$997$	−40.2215	−1.27383	−0.636914	−	0.770935i	$-0.719790\pi$
$997$	−40.2215	−1.27383	−0.636914	+	0.770935i	$0.719790\pi$
$998$	−17.1921	−0.544205
$999$	−60.7764	−1.92288

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8006.2.a.c.1.6	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8006.2.a.c.1.6	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.11194 q^{3} +1.00000 q^{4} +0.282330 q^{5} +3.11194 q^{6} -3.77246 q^{7} -1.00000 q^{8} +6.68416 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.11194 q^{3} +1.00000 q^{4} +0.282330 q^{5} +3.11194 q^{6} -3.77246 q^{7} -1.00000 q^{8} +6.68416 q^{9} -0.282330 q^{10} -3.16321 q^{11} -3.11194 q^{12} +3.69999 q^{13} +3.77246 q^{14} -0.878595 q^{15} +1.00000 q^{16} -3.88814 q^{17} -6.68416 q^{18} -5.09605 q^{19} +0.282330 q^{20} +11.7397 q^{21} +3.16321 q^{22} -3.51118 q^{23} +3.11194 q^{24} -4.92029 q^{25} -3.69999 q^{26} -11.4649 q^{27} -3.77246 q^{28} +3.32528 q^{29} +0.878595 q^{30} +4.38907 q^{31} -1.00000 q^{32} +9.84371 q^{33} +3.88814 q^{34} -1.06508 q^{35} +6.68416 q^{36} +5.30109 q^{37} +5.09605 q^{38} -11.5142 q^{39} -0.282330 q^{40} -10.4068 q^{41} -11.7397 q^{42} -1.61158 q^{43} -3.16321 q^{44} +1.88714 q^{45} +3.51118 q^{46} -8.52577 q^{47} -3.11194 q^{48} +7.23148 q^{49} +4.92029 q^{50} +12.0996 q^{51} +3.69999 q^{52} -7.37335 q^{53} +11.4649 q^{54} -0.893070 q^{55} +3.77246 q^{56} +15.8586 q^{57} -3.32528 q^{58} -10.9408 q^{59} -0.878595 q^{60} -0.0726800 q^{61} -4.38907 q^{62} -25.2158 q^{63} +1.00000 q^{64} +1.04462 q^{65} -9.84371 q^{66} -0.00327689 q^{67} -3.88814 q^{68} +10.9266 q^{69} +1.06508 q^{70} +8.00400 q^{71} -6.68416 q^{72} +4.09852 q^{73} -5.30109 q^{74} +15.3116 q^{75} -5.09605 q^{76} +11.9331 q^{77} +11.5142 q^{78} +3.80393 q^{79} +0.282330 q^{80} +15.6256 q^{81} +10.4068 q^{82} -5.55535 q^{83} +11.7397 q^{84} -1.09774 q^{85} +1.61158 q^{86} -10.3481 q^{87} +3.16321 q^{88} +11.2802 q^{89} -1.88714 q^{90} -13.9581 q^{91} -3.51118 q^{92} -13.6585 q^{93} +8.52577 q^{94} -1.43877 q^{95} +3.11194 q^{96} -6.09044 q^{97} -7.23148 q^{98} -21.1434 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q - 92 q^{2} - 2 q^{3} + 92 q^{4} + 10 q^{5} + 2 q^{6} + 8 q^{7} - 92 q^{8} + 104 q^{9}+O(q^{10}) \) 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