LMFDB - Embedded newform 8007.2.a.i.1.32

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$0$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.32
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.195227 q^{2} +1.00000 q^{3} -1.96189 q^{4} -1.81098 q^{5} +0.195227 q^{6} +0.259125 q^{7} -0.773466 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.195227 q^{2} +1.00000 q^{3} -1.96189 q^{4} -1.81098 q^{5} +0.195227 q^{6} +0.259125 q^{7} -0.773466 q^{8} +1.00000 q^{9} -0.353552 q^{10} +3.82564 q^{11} -1.96189 q^{12} -2.75701 q^{13} +0.0505881 q^{14} -1.81098 q^{15} +3.77277 q^{16} +1.00000 q^{17} +0.195227 q^{18} +7.35313 q^{19} +3.55294 q^{20} +0.259125 q^{21} +0.746867 q^{22} +1.13310 q^{23} -0.773466 q^{24} -1.72035 q^{25} -0.538242 q^{26} +1.00000 q^{27} -0.508374 q^{28} +1.17177 q^{29} -0.353552 q^{30} -3.38321 q^{31} +2.28348 q^{32} +3.82564 q^{33} +0.195227 q^{34} -0.469271 q^{35} -1.96189 q^{36} +2.71892 q^{37} +1.43553 q^{38} -2.75701 q^{39} +1.40073 q^{40} +0.0121521 q^{41} +0.0505881 q^{42} +1.65097 q^{43} -7.50547 q^{44} -1.81098 q^{45} +0.221210 q^{46} -9.37297 q^{47} +3.77277 q^{48} -6.93285 q^{49} -0.335857 q^{50} +1.00000 q^{51} +5.40894 q^{52} +3.54711 q^{53} +0.195227 q^{54} -6.92816 q^{55} -0.200424 q^{56} +7.35313 q^{57} +0.228761 q^{58} +1.23417 q^{59} +3.55294 q^{60} -6.02802 q^{61} -0.660492 q^{62} +0.259125 q^{63} -7.09975 q^{64} +4.99289 q^{65} +0.746867 q^{66} +11.2463 q^{67} -1.96189 q^{68} +1.13310 q^{69} -0.0916142 q^{70} -15.9505 q^{71} -0.773466 q^{72} +6.73104 q^{73} +0.530805 q^{74} -1.72035 q^{75} -14.4260 q^{76} +0.991319 q^{77} -0.538242 q^{78} +1.25964 q^{79} -6.83242 q^{80} +1.00000 q^{81} +0.00237241 q^{82} -5.53845 q^{83} -0.508374 q^{84} -1.81098 q^{85} +0.322314 q^{86} +1.17177 q^{87} -2.95900 q^{88} +11.3301 q^{89} -0.353552 q^{90} -0.714410 q^{91} -2.22300 q^{92} -3.38321 q^{93} -1.82985 q^{94} -13.3164 q^{95} +2.28348 q^{96} -14.6090 q^{97} -1.35348 q^{98} +3.82564 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9}+O(q^{10}) $ 63 * q + 10 * q^2 + 63 * q^3 + 70 * q^4 + 19 * q^5 + 10 * q^6 + 11 * q^7 + 27 * q^8 + 63 * q^9	\( 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9} + 4 q^{10} + 23 q^{11} + 70 q^{12} + 10 q^{13} + 18 q^{14} + 19 q^{15} + 72 q^{16} + 63 q^{17} + 10 q^{18} + 6 q^{19} + 48 q^{20} + 11 q^{21} + 21 q^{22} + 44 q^{23} + 27 q^{24} + 110 q^{25} + 41 q^{26} + 63 q^{27} + 26 q^{28} + 35 q^{29} + 4 q^{30} + q^{31} + 54 q^{32} + 23 q^{33} + 10 q^{34} + 47 q^{35} + 70 q^{36} + 40 q^{37} + 38 q^{38} + 10 q^{39} - 10 q^{40} + 35 q^{41} + 18 q^{42} + 27 q^{43} + 46 q^{44} + 19 q^{45} + 8 q^{46} + 29 q^{47} + 72 q^{48} + 114 q^{49} + 27 q^{50} + 63 q^{51} - q^{52} + 75 q^{53} + 10 q^{54} + 5 q^{55} + 24 q^{56} + 6 q^{57} + 41 q^{58} + 105 q^{59} + 48 q^{60} + 5 q^{61} + 22 q^{62} + 11 q^{63} + 61 q^{64} + 49 q^{65} + 21 q^{66} + 4 q^{67} + 70 q^{68} + 44 q^{69} - 16 q^{70} + 16 q^{71} + 27 q^{72} + 39 q^{73} + 54 q^{74} + 110 q^{75} + 6 q^{76} + 88 q^{77} + 41 q^{78} + 16 q^{79} + 102 q^{80} + 63 q^{81} - 29 q^{82} + 73 q^{83} + 26 q^{84} + 19 q^{85} + 46 q^{86} + 35 q^{87} + 18 q^{88} + 88 q^{89} + 4 q^{90} - 15 q^{91} + 110 q^{92} + q^{93} - 8 q^{94} + 28 q^{95} + 54 q^{96} + 70 q^{97} + 33 q^{98} + 23 q^{99}+O(q^{100}) \) 63 * q + 10 * q^2 + 63 * q^3 + 70 * q^4 + 19 * q^5 + 10 * q^6 + 11 * q^7 + 27 * q^8 + 63 * q^9 + 4 * q^10 + 23 * q^11 + 70 * q^12 + 10 * q^13 + 18 * q^14 + 19 * q^15 + 72 * q^16 + 63 * q^17 + 10 * q^18 + 6 * q^19 + 48 * q^20 + 11 * q^21 + 21 * q^22 + 44 * q^23 + 27 * q^24 + 110 * q^25 + 41 * q^26 + 63 * q^27 + 26 * q^28 + 35 * q^29 + 4 * q^30 + q^31 + 54 * q^32 + 23 * q^33 + 10 * q^34 + 47 * q^35 + 70 * q^36 + 40 * q^37 + 38 * q^38 + 10 * q^39 - 10 * q^40 + 35 * q^41 + 18 * q^42 + 27 * q^43 + 46 * q^44 + 19 * q^45 + 8 * q^46 + 29 * q^47 + 72 * q^48 + 114 * q^49 + 27 * q^50 + 63 * q^51 - q^52 + 75 * q^53 + 10 * q^54 + 5 * q^55 + 24 * q^56 + 6 * q^57 + 41 * q^58 + 105 * q^59 + 48 * q^60 + 5 * q^61 + 22 * q^62 + 11 * q^63 + 61 * q^64 + 49 * q^65 + 21 * q^66 + 4 * q^67 + 70 * q^68 + 44 * q^69 - 16 * q^70 + 16 * q^71 + 27 * q^72 + 39 * q^73 + 54 * q^74 + 110 * q^75 + 6 * q^76 + 88 * q^77 + 41 * q^78 + 16 * q^79 + 102 * q^80 + 63 * q^81 - 29 * q^82 + 73 * q^83 + 26 * q^84 + 19 * q^85 + 46 * q^86 + 35 * q^87 + 18 * q^88 + 88 * q^89 + 4 * q^90 - 15 * q^91 + 110 * q^92 + q^93 - 8 * q^94 + 28 * q^95 + 54 * q^96 + 70 * q^97 + 33 * q^98 + 23 * 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 $

$2$	0.195227	0.138046	0.0690231	−	0.997615i	$-0.478012\pi$
$2$	0.195227	0.138046	0.0690231	+	0.997615i	$0.478012\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.96189	−0.980943
$5$	−1.81098	−0.809896	−0.404948	−	0.914340i	$-0.632710\pi$
$5$	−1.81098	−0.809896	−0.404948	+	0.914340i	$0.632710\pi$
$6$	0.195227	0.0797010
$7$	0.259125	0.0979401	0.0489700	−	0.998800i	$-0.484406\pi$
$7$	0.259125	0.0979401	0.0489700	+	0.998800i	$0.484406\pi$
$8$	−0.773466	−0.273462
$9$	1.00000	0.333333
$10$	−0.353552	−0.111803
$11$	3.82564	1.15347	0.576737	−	0.816930i	$-0.304326\pi$
$11$	3.82564	1.15347	0.576737	+	0.816930i	$0.304326\pi$
$12$	−1.96189	−0.566348
$13$	−2.75701	−0.764657	−0.382328	−	0.924027i	$-0.624878\pi$
$13$	−2.75701	−0.764657	−0.382328	+	0.924027i	$0.624878\pi$
$14$	0.0505881	0.0135202
$15$	−1.81098	−0.467593
$16$	3.77277	0.943193
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0.195227	0.0460154
$19$	7.35313	1.68692	0.843461	−	0.537190i	$-0.180514\pi$
$19$	7.35313	1.68692	0.843461	+	0.537190i	$0.180514\pi$
$20$	3.55294	0.794462
$21$	0.259125	0.0565457
$22$	0.746867	0.159232
$23$	1.13310	0.236267	0.118133	−	0.992998i	$-0.462309\pi$
$23$	1.13310	0.236267	0.118133	+	0.992998i	$0.462309\pi$
$24$	−0.773466	−0.157883
$25$	−1.72035	−0.344069
$26$	−0.538242	−0.105558
$27$	1.00000	0.192450
$28$	−0.508374	−0.0960737
$29$	1.17177	0.217593	0.108796	−	0.994064i	$-0.465300\pi$
$29$	1.17177	0.217593	0.108796	+	0.994064i	$0.465300\pi$
$30$	−0.353552	−0.0645494
$31$	−3.38321	−0.607642	−0.303821	−	0.952729i	$-0.598262\pi$
$31$	−3.38321	−0.607642	−0.303821	+	0.952729i	$0.598262\pi$
$32$	2.28348	0.403666
$33$	3.82564	0.665958
$34$	0.195227	0.0334811
$35$	−0.469271	−0.0793212
$36$	−1.96189	−0.326981
$37$	2.71892	0.446987	0.223494	−	0.974705i	$-0.428254\pi$
$37$	2.71892	0.446987	0.223494	+	0.974705i	$0.428254\pi$
$38$	1.43553	0.232873
$39$	−2.75701	−0.441475
$40$	1.40073	0.221475
$41$	0.0121521	0.00189783	0.000948917	−	1.00000i	$-0.499698\pi$
$41$	0.0121521	0.00189783	0.000948917		1.00000i	$0.499698\pi$
$42$	0.0505881	0.00780592
$43$	1.65097	0.251771	0.125886	−	0.992045i	$-0.459823\pi$
$43$	1.65097	0.251771	0.125886	+	0.992045i	$0.459823\pi$
$44$	−7.50547	−1.13149
$45$	−1.81098	−0.269965
$46$	0.221210	0.0326157
$47$	−9.37297	−1.36719	−0.683594	−	0.729862i	$-0.739584\pi$
$47$	−9.37297	−1.36719	−0.683594	+	0.729862i	$0.739584\pi$
$48$	3.77277	0.544553
$49$	−6.93285	−0.990408
$50$	−0.335857	−0.0474974
$51$	1.00000	0.140028
$52$	5.40894	0.750085
$53$	3.54711	0.487233	0.243617	−	0.969872i	$-0.421666\pi$
$53$	3.54711	0.487233	0.243617	+	0.969872i	$0.421666\pi$
$54$	0.195227	0.0265670
$55$	−6.92816	−0.934193
$56$	−0.200424	−0.0267828
$57$	7.35313	0.973945
$58$	0.228761	0.0300378
$59$	1.23417	0.160676	0.0803378	−	0.996768i	$-0.474400\pi$
$59$	1.23417	0.160676	0.0803378	+	0.996768i	$0.474400\pi$
$60$	3.55294	0.458683
$61$	−6.02802	−0.771808	−0.385904	−	0.922539i	$-0.626110\pi$
$61$	−6.02802	−0.771808	−0.385904	+	0.922539i	$0.626110\pi$
$62$	−0.660492	−0.0838826
$63$	0.259125	0.0326467
$64$	−7.09975	−0.887469
$65$	4.99289	0.619292
$66$	0.746867	0.0919329
$67$	11.2463	1.37395	0.686974	−	0.726682i	$-0.258939\pi$
$67$	11.2463	1.37395	0.686974	+	0.726682i	$0.258939\pi$
$68$	−1.96189	−0.237914
$69$	1.13310	0.136409
$70$	−0.0916142	−0.0109500
$71$	−15.9505	−1.89298	−0.946491	−	0.322731i	$-0.895399\pi$
$71$	−15.9505	−1.89298	−0.946491	+	0.322731i	$0.895399\pi$
$72$	−0.773466	−0.0911538
$73$	6.73104	0.787808	0.393904	−	0.919152i	$-0.371124\pi$
$73$	6.73104	0.787808	0.393904	+	0.919152i	$0.371124\pi$
$74$	0.530805	0.0617049
$75$	−1.72035	−0.198648
$76$	−14.4260	−1.65478
$77$	0.991319	0.112971
$78$	−0.538242	−0.0609439
$79$	1.25964	0.141720	0.0708601	−	0.997486i	$-0.477426\pi$
$79$	1.25964	0.141720	0.0708601	+	0.997486i	$0.477426\pi$
$80$	−6.83242	−0.763888
$81$	1.00000	0.111111
$82$	0.00237241	0.000261989 0
$83$	−5.53845	−0.607924	−0.303962	−	0.952684i	$-0.598310\pi$
$83$	−5.53845	−0.607924	−0.303962	+	0.952684i	$0.598310\pi$
$84$	−0.508374	−0.0554682
$85$	−1.81098	−0.196429
$86$	0.322314	0.0347560
$87$	1.17177	0.125627
$88$	−2.95900	−0.315431
$89$	11.3301	1.20099	0.600495	−	0.799629i	$-0.294970\pi$
$89$	11.3301	1.20099	0.600495	+	0.799629i	$0.294970\pi$
$90$	−0.353552	−0.0372676
$91$	−0.714410	−0.0748906
$92$	−2.22300	−0.231764
$93$	−3.38321	−0.350822
$94$	−1.82985	−0.188735
$95$	−13.3164	−1.36623
$96$	2.28348	0.233056
$97$	−14.6090	−1.48332	−0.741661	−	0.670774i	$-0.765962\pi$
$97$	−14.6090	−1.48332	−0.741661	+	0.670774i	$0.765962\pi$
$98$	−1.35348	−0.136722
$99$	3.82564	0.384491
$100$	3.37512	0.337512
$101$	6.34535	0.631386	0.315693	−	0.948861i	$-0.397763\pi$
$101$	6.34535	0.631386	0.315693	+	0.948861i	$0.397763\pi$
$102$	0.195227	0.0193303
$103$	7.91482	0.779870	0.389935	−	0.920842i	$-0.372497\pi$
$103$	7.91482	0.779870	0.389935	+	0.920842i	$0.372497\pi$
$104$	2.13245	0.209104
$105$	−0.469271	−0.0457961
$106$	0.692491	0.0672607
$107$	15.8680	1.53402	0.767011	−	0.641634i	$-0.221743\pi$
$107$	15.8680	1.53402	0.767011	+	0.641634i	$0.221743\pi$
$108$	−1.96189	−0.188783
$109$	3.03748	0.290938	0.145469	−	0.989363i	$-0.453531\pi$
$109$	3.03748	0.290938	0.145469	+	0.989363i	$0.453531\pi$
$110$	−1.35256	−0.128962
$111$	2.71892	0.258068
$112$	0.977620	0.0923764
$113$	−1.07822	−0.101430	−0.0507151	−	0.998713i	$-0.516150\pi$
$113$	−1.07822	−0.101430	−0.0507151	+	0.998713i	$0.516150\pi$
$114$	1.43553	0.134449
$115$	−2.05201	−0.191351
$116$	−2.29888	−0.213446
$117$	−2.75701	−0.254886
$118$	0.240943	0.0221806
$119$	0.259125	0.0237540
$120$	1.40073	0.127869
$121$	3.63551	0.330501
$122$	−1.17683	−0.106545
$123$	0.0121521	0.00109571
$124$	6.63747	0.596062
$125$	12.1704	1.08856
$126$	0.0505881	0.00450675
$127$	15.8271	1.40443	0.702214	−	0.711966i	$-0.252195\pi$
$127$	15.8271	1.40443	0.702214	+	0.711966i	$0.252195\pi$
$128$	−5.95302	−0.526177
$129$	1.65097	0.145360
$130$	0.974746	0.0854909
$131$	−0.898506	−0.0785028	−0.0392514	−	0.999229i	$-0.512497\pi$
$131$	−0.898506	−0.0785028	−0.0392514	+	0.999229i	$0.512497\pi$
$132$	−7.50547	−0.653267
$133$	1.90538	0.165217
$134$	2.19557	0.189668
$135$	−1.81098	−0.155864
$136$	−0.773466	−0.0663242
$137$	−5.83089	−0.498167	−0.249083	−	0.968482i	$-0.580129\pi$
$137$	−5.83089	−0.498167	−0.249083	+	0.968482i	$0.580129\pi$
$138$	0.221210	0.0188307
$139$	−2.33804	−0.198310	−0.0991548	−	0.995072i	$-0.531614\pi$
$139$	−2.33804	−0.198310	−0.0991548	+	0.995072i	$0.531614\pi$
$140$	0.920656	0.0778096
$141$	−9.37297	−0.789347
$142$	−3.11397	−0.261319
$143$	−10.5473	−0.882011
$144$	3.77277	0.314398
$145$	−2.12206	−0.176227
$146$	1.31408	0.108754
$147$	−6.93285	−0.571812
$148$	−5.33421	−0.438469
$149$	−1.78614	−0.146326	−0.0731632	−	0.997320i	$-0.523309\pi$
$149$	−1.78614	−0.146326	−0.0731632	+	0.997320i	$0.523309\pi$
$150$	−0.335857	−0.0274226
$151$	16.1250	1.31224	0.656118	−	0.754659i	$-0.272197\pi$
$151$	16.1250	1.31224	0.656118	+	0.754659i	$0.272197\pi$
$152$	−5.68739	−0.461308
$153$	1.00000	0.0808452
$154$	0.193532	0.0155952
$155$	6.12692	0.492126
$156$	5.40894	0.433062
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	0.245915	0.0195639
$159$	3.54711	0.281304
$160$	−4.13534	−0.326927
$161$	0.293613	0.0231400
$162$	0.195227	0.0153385
$163$	−15.5717	−1.21967	−0.609835	−	0.792529i	$-0.708764\pi$
$163$	−15.5717	−1.21967	−0.609835	+	0.792529i	$0.708764\pi$
$164$	−0.0238410	−0.00186167
$165$	−6.92816	−0.539357
$166$	−1.08125	−0.0839215
$167$	−14.8648	−1.15028	−0.575138	−	0.818056i	$-0.695052\pi$
$167$	−14.8648	−1.15028	−0.575138	+	0.818056i	$0.695052\pi$
$168$	−0.200424	−0.0154631
$169$	−5.39890	−0.415300
$170$	−0.353552	−0.0271162
$171$	7.35313	0.562308
$172$	−3.23902	−0.246973
$173$	24.0832	1.83101	0.915505	−	0.402306i	$-0.131791\pi$
$173$	24.0832	1.83101	0.915505	+	0.402306i	$0.131791\pi$
$174$	0.228761	0.0173423
$175$	−0.445785	−0.0336982
$176$	14.4333	1.08795
$177$	1.23417	0.0927661
$178$	2.21194	0.165792
$179$	20.5241	1.53405	0.767023	−	0.641620i	$-0.221737\pi$
$179$	20.5241	1.53405	0.767023	+	0.641620i	$0.221737\pi$
$180$	3.55294	0.264821
$181$	14.3862	1.06932	0.534659	−	0.845068i	$-0.320440\pi$
$181$	14.3862	1.06932	0.534659	+	0.845068i	$0.320440\pi$
$182$	−0.139472	−0.0103383
$183$	−6.02802	−0.445604
$184$	−0.876411	−0.0646099
$185$	−4.92391	−0.362013
$186$	−0.660492	−0.0484296
$187$	3.82564	0.279758
$188$	18.3887	1.34113
$189$	0.259125	0.0188486
$190$	−2.59971	−0.188603
$191$	11.7021	0.846733	0.423367	−	0.905958i	$-0.360848\pi$
$191$	11.7021	0.846733	0.423367	+	0.905958i	$0.360848\pi$
$192$	−7.09975	−0.512380
$193$	8.43235	0.606974	0.303487	−	0.952836i	$-0.401849\pi$
$193$	8.43235	0.606974	0.303487	+	0.952836i	$0.401849\pi$
$194$	−2.85207	−0.204767
$195$	4.99289	0.357548
$196$	13.6015	0.971534
$197$	−1.89134	−0.134752	−0.0673760	−	0.997728i	$-0.521463\pi$
$197$	−1.89134	−0.134752	−0.0673760	+	0.997728i	$0.521463\pi$
$198$	0.746867	0.0530775
$199$	19.5505	1.38590	0.692949	−	0.720987i	$-0.256311\pi$
$199$	19.5505	1.38590	0.692949	+	0.720987i	$0.256311\pi$
$200$	1.33063	0.0940897
$201$	11.2463	0.793250
$202$	1.23878	0.0871604
$203$	0.303636	0.0213110
$204$	−1.96189	−0.137360
$205$	−0.0220072	−0.00153705
$206$	1.54518	0.107658
$207$	1.13310	0.0787556
$208$	−10.4016	−0.721219
$209$	28.1304	1.94582
$210$	−0.0916142	−0.00632198
$211$	26.6958	1.83782	0.918909	−	0.394470i	$-0.129072\pi$
$211$	26.6958	1.83782	0.918909	+	0.394470i	$0.129072\pi$
$212$	−6.95903	−0.477948
$213$	−15.9505	−1.09291
$214$	3.09787	0.211766
$215$	−2.98988	−0.203908
$216$	−0.773466	−0.0526277
$217$	−0.876673	−0.0595125
$218$	0.592997	0.0401628
$219$	6.73104	0.454841
$220$	13.5923	0.916390
$221$	−2.75701	−0.185457
$222$	0.530805	0.0356253
$223$	1.14364	0.0765835	0.0382917	−	0.999267i	$-0.487808\pi$
$223$	1.14364	0.0765835	0.0382917	+	0.999267i	$0.487808\pi$
$224$	0.591706	0.0395350
$225$	−1.72035	−0.114690
$226$	−0.210497	−0.0140020
$227$	8.63700	0.573258	0.286629	−	0.958042i	$-0.407465\pi$
$227$	8.63700	0.573258	0.286629	+	0.958042i	$0.407465\pi$
$228$	−14.4260	−0.955385
$229$	−4.80834	−0.317744	−0.158872	−	0.987299i	$-0.550786\pi$
$229$	−4.80834	−0.317744	−0.158872	+	0.987299i	$0.550786\pi$
$230$	−0.400608	−0.0264153
$231$	0.991319	0.0652240
$232$	−0.906326	−0.0595032
$233$	4.16837	0.273079	0.136539	−	0.990635i	$-0.456402\pi$
$233$	4.16837	0.273079	0.136539	+	0.990635i	$0.456402\pi$
$234$	−0.538242	−0.0351860
$235$	16.9743	1.10728
$236$	−2.42131	−0.157614
$237$	1.25964	0.0818222
$238$	0.0505881	0.00327914
$239$	−0.437027	−0.0282689	−0.0141345	−	0.999900i	$-0.504499\pi$
$239$	−0.437027	−0.0282689	−0.0141345	+	0.999900i	$0.504499\pi$
$240$	−6.83242	−0.441031
$241$	13.7753	0.887342	0.443671	−	0.896190i	$-0.353676\pi$
$241$	13.7753	0.887342	0.443671	+	0.896190i	$0.353676\pi$
$242$	0.709748	0.0456243
$243$	1.00000	0.0641500
$244$	11.8263	0.757100
$245$	12.5553	0.802127
$246$	0.00237241	0.000151259 0
$247$	−20.2726	−1.28992
$248$	2.61679	0.166167
$249$	−5.53845	−0.350985
$250$	2.37599	0.150271
$251$	4.80678	0.303401	0.151701	−	0.988426i	$-0.451525\pi$
$251$	4.80678	0.303401	0.151701	+	0.988426i	$0.451525\pi$
$252$	−0.508374	−0.0320246
$253$	4.33481	0.272527
$254$	3.08987	0.193876
$255$	−1.81098	−0.113408
$256$	13.0373	0.814832
$257$	−14.8787	−0.928107	−0.464053	−	0.885807i	$-0.653605\pi$
$257$	−14.8787	−0.928107	−0.464053	+	0.885807i	$0.653605\pi$
$258$	0.322314	0.0200664
$259$	0.704540	0.0437780
$260$	−9.79549	−0.607490
$261$	1.17177	0.0725309
$262$	−0.175412	−0.0108370
$263$	11.8370	0.729902	0.364951	−	0.931027i	$-0.381086\pi$
$263$	11.8370	0.729902	0.364951	+	0.931027i	$0.381086\pi$
$264$	−2.95900	−0.182114
$265$	−6.42375	−0.394608
$266$	0.371981	0.0228076
$267$	11.3301	0.693392
$268$	−22.0639	−1.34777
$269$	−4.47200	−0.272663	−0.136331	−	0.990663i	$-0.543531\pi$
$269$	−4.47200	−0.272663	−0.136331	+	0.990663i	$0.543531\pi$
$270$	−0.353552	−0.0215165
$271$	−19.9956	−1.21465	−0.607323	−	0.794455i	$-0.707757\pi$
$271$	−19.9956	−1.21465	−0.607323	+	0.794455i	$0.707757\pi$
$272$	3.77277	0.228758
$273$	−0.714410	−0.0432381
$274$	−1.13835	−0.0687700
$275$	−6.58142	−0.396875
$276$	−2.22300	−0.133809
$277$	−19.4601	−1.16924	−0.584621	−	0.811307i	$-0.698757\pi$
$277$	−19.4601	−1.16924	−0.584621	+	0.811307i	$0.698757\pi$
$278$	−0.456447	−0.0273759
$279$	−3.38321	−0.202547
$280$	0.362965	0.0216913
$281$	3.88184	0.231571	0.115786	−	0.993274i	$-0.463061\pi$
$281$	3.88184	0.231571	0.115786	+	0.993274i	$0.463061\pi$
$282$	−1.82985	−0.108966
$283$	23.7421	1.41132	0.705662	−	0.708549i	$-0.250650\pi$
$283$	23.7421	1.41132	0.705662	+	0.708549i	$0.250650\pi$
$284$	31.2932	1.85691
$285$	−13.3164	−0.788794
$286$	−2.05912	−0.121758
$287$	0.00314891	0.000185874 0
$288$	2.28348	0.134555
$289$	1.00000	0.0588235
$290$	−0.414282	−0.0243275
$291$	−14.6090	−0.856397
$292$	−13.2055	−0.772795
$293$	2.04715	0.119596	0.0597980	−	0.998210i	$-0.480954\pi$
$293$	2.04715	0.119596	0.0597980	+	0.998210i	$0.480954\pi$
$294$	−1.35348	−0.0789364
$295$	−2.23506	−0.130130
$296$	−2.10299	−0.122234
$297$	3.82564	0.221986
$298$	−0.348703	−0.0201998
$299$	−3.12395	−0.180663
$300$	3.37512	0.194863
$301$	0.427809	0.0246585
$302$	3.14803	0.181149
$303$	6.34535	0.364531
$304$	27.7417	1.59109
$305$	10.9166	0.625084
$306$	0.195227	0.0111604
$307$	5.92042	0.337896	0.168948	−	0.985625i	$-0.445963\pi$
$307$	5.92042	0.337896	0.168948	+	0.985625i	$0.445963\pi$
$308$	−1.94486	−0.110818
$309$	7.91482	0.450258
$310$	1.19614	0.0679361
$311$	4.79451	0.271872	0.135936	−	0.990718i	$-0.456596\pi$
$311$	4.79451	0.271872	0.135936	+	0.990718i	$0.456596\pi$
$312$	2.13245	0.120726
$313$	−33.6435	−1.90164	−0.950821	−	0.309741i	$-0.899758\pi$
$313$	−33.6435	−1.90164	−0.950821	+	0.309741i	$0.899758\pi$
$314$	0.195227	0.0110173
$315$	−0.469271	−0.0264404
$316$	−2.47126	−0.139019
$317$	32.0544	1.80035	0.900176	−	0.435526i	$-0.143438\pi$
$317$	32.0544	1.80035	0.900176	+	0.435526i	$0.143438\pi$
$318$	0.692491	0.0388330
$319$	4.48278	0.250987
$320$	12.8575	0.718757
$321$	15.8680	0.885668
$322$	0.0573212	0.00319438
$323$	7.35313	0.409139
$324$	−1.96189	−0.108994
$325$	4.74301	0.263095
$326$	−3.04001	−0.168371
$327$	3.03748	0.167973
$328$	−0.00939921	−0.000518985 0
$329$	−2.42877	−0.133903
$330$	−1.35256	−0.0744561
$331$	0.895134	0.0492010	0.0246005	−	0.999697i	$-0.492169\pi$
$331$	0.895134	0.0492010	0.0246005	+	0.999697i	$0.492169\pi$
$332$	10.8658	0.596339
$333$	2.71892	0.148996
$334$	−2.90201	−0.158791
$335$	−20.3668	−1.11275
$336$	0.977620	0.0533335
$337$	0.998166	0.0543735	0.0271868	−	0.999630i	$-0.491345\pi$
$337$	0.998166	0.0543735	0.0271868	+	0.999630i	$0.491345\pi$
$338$	−1.05401	−0.0573305
$339$	−1.07822	−0.0585608
$340$	3.55294	0.192685
$341$	−12.9429	−0.700898
$342$	1.43553	0.0776244
$343$	−3.61035	−0.194941
$344$	−1.27697	−0.0688497
$345$	−2.05201	−0.110477
$346$	4.70168	0.252764
$347$	26.1432	1.40344	0.701721	−	0.712451i	$-0.252415\pi$
$347$	26.1432	1.40344	0.701721	+	0.712451i	$0.252415\pi$
$348$	−2.29888	−0.123233
$349$	−34.9836	−1.87263	−0.936313	−	0.351166i	$-0.885785\pi$
$349$	−34.9836	−1.87263	−0.936313	+	0.351166i	$0.885785\pi$
$350$	−0.0870291	−0.00465190
$351$	−2.75701	−0.147158
$352$	8.73576	0.465618
$353$	0.151673	0.00807275	0.00403638	−	0.999992i	$-0.498715\pi$
$353$	0.151673	0.00807275	0.00403638	+	0.999992i	$0.498715\pi$
$354$	0.240943	0.0128060
$355$	28.8861	1.53312
$356$	−22.2284	−1.17810
$357$	0.259125	0.0137144
$358$	4.00686	0.211769
$359$	−11.0641	−0.583942	−0.291971	−	0.956427i	$-0.594311\pi$
$359$	−11.0641	−0.583942	−0.291971	+	0.956427i	$0.594311\pi$
$360$	1.40073	0.0738251
$361$	35.0684	1.84571
$362$	2.80857	0.147615
$363$	3.63551	0.190815
$364$	1.40159	0.0734634
$365$	−12.1898	−0.638042
$366$	−1.17683	−0.0615139
$367$	−17.0163	−0.888242	−0.444121	−	0.895967i	$-0.646484\pi$
$367$	−17.0163	−0.888242	−0.444121	+	0.895967i	$0.646484\pi$
$368$	4.27491	0.222845
$369$	0.0121521	0.000632611 0
$370$	−0.961279	−0.0499745
$371$	0.919146	0.0477197
$372$	6.63747	0.344137
$373$	−2.61114	−0.135200	−0.0675999	−	0.997713i	$-0.521534\pi$
$373$	−2.61114	−0.135200	−0.0675999	+	0.997713i	$0.521534\pi$
$374$	0.746867	0.0386196
$375$	12.1704	0.628478
$376$	7.24967	0.373873
$377$	−3.23059	−0.166384
$378$	0.0505881	0.00260197
$379$	23.0385	1.18341	0.591705	−	0.806154i	$-0.298455\pi$
$379$	23.0385	1.18341	0.591705	+	0.806154i	$0.298455\pi$
$380$	26.1252	1.34020
$381$	15.8271	0.810847
$382$	2.28456	0.116888
$383$	22.7050	1.16017	0.580086	−	0.814555i	$-0.303019\pi$
$383$	22.7050	1.16017	0.580086	+	0.814555i	$0.303019\pi$
$384$	−5.95302	−0.303789
$385$	−1.79526	−0.0914949
$386$	1.64622	0.0837904
$387$	1.65097	0.0839237
$388$	28.6613	1.45506
$389$	−1.63077	−0.0826832	−0.0413416	−	0.999145i	$-0.513163\pi$
$389$	−1.63077	−0.0826832	−0.0413416	+	0.999145i	$0.513163\pi$
$390$	0.974746	0.0493582
$391$	1.13310	0.0573031
$392$	5.36233	0.270838
$393$	−0.898506	−0.0453236
$394$	−0.369239	−0.0186020
$395$	−2.28118	−0.114779
$396$	−7.50547	−0.377164
$397$	−24.6627	−1.23778	−0.618892	−	0.785476i	$-0.712418\pi$
$397$	−24.6627	−1.23778	−0.618892	+	0.785476i	$0.712418\pi$
$398$	3.81678	0.191318
$399$	1.90538	0.0953883
$400$	−6.49047	−0.324524
$401$	−28.1219	−1.40434	−0.702171	−	0.712009i	$-0.747786\pi$
$401$	−28.1219	−1.40434	−0.702171	+	0.712009i	$0.747786\pi$
$402$	2.19557	0.109505
$403$	9.32753	0.464637
$404$	−12.4489	−0.619354
$405$	−1.81098	−0.0899884
$406$	0.0592778	0.00294191
$407$	10.4016	0.515588
$408$	−0.773466	−0.0382923
$409$	−27.1371	−1.34184	−0.670921	−	0.741529i	$-0.734101\pi$
$409$	−27.1371	−1.34184	−0.670921	+	0.741529i	$0.734101\pi$
$410$	−0.00429639	−0.000212183 0
$411$	−5.83089	−0.287617
$412$	−15.5280	−0.765009
$413$	0.319805	0.0157366
$414$	0.221210	0.0108719
$415$	10.0300	0.492355
$416$	−6.29557	−0.308666
$417$	−2.33804	−0.114494
$418$	5.49180	0.268613
$419$	3.41203	0.166689	0.0833443	−	0.996521i	$-0.473440\pi$
$419$	3.41203	0.166689	0.0833443	+	0.996521i	$0.473440\pi$
$420$	0.920656	0.0449234
$421$	−3.34257	−0.162907	−0.0814535	−	0.996677i	$-0.525956\pi$
$421$	−3.34257	−0.162907	−0.0814535	+	0.996677i	$0.525956\pi$
$422$	5.21174	0.253704
$423$	−9.37297	−0.455730
$424$	−2.74357	−0.133240
$425$	−1.72035	−0.0834490
$426$	−3.11397	−0.150872
$427$	−1.56201	−0.0755910
$428$	−31.1313	−1.50479
$429$	−10.5473	−0.509229
$430$	−0.583705	−0.0281487
$431$	9.86545	0.475202	0.237601	−	0.971363i	$-0.423639\pi$
$431$	9.86545	0.475202	0.237601	+	0.971363i	$0.423639\pi$
$432$	3.77277	0.181518
$433$	41.4264	1.99083	0.995413	−	0.0956693i	$-0.0304991\pi$
$433$	41.4264	1.99083	0.995413	+	0.0956693i	$0.0304991\pi$
$434$	−0.171150	−0.00821547
$435$	−2.12206	−0.101745
$436$	−5.95919	−0.285393
$437$	8.33179	0.398564
$438$	1.31408	0.0627891
$439$	12.4094	0.592269	0.296135	−	0.955146i	$-0.404302\pi$
$439$	12.4094	0.592269	0.296135	+	0.955146i	$0.404302\pi$
$440$	5.35870	0.255466
$441$	−6.93285	−0.330136
$442$	−0.538242	−0.0256016
$443$	−4.86482	−0.231134	−0.115567	−	0.993300i	$-0.536869\pi$
$443$	−4.86482	−0.231134	−0.115567	+	0.993300i	$0.536869\pi$
$444$	−5.33421	−0.253150
$445$	−20.5186	−0.972676
$446$	0.223268	0.0105720
$447$	−1.78614	−0.0844816
$448$	−1.83972	−0.0869187
$449$	−30.9991	−1.46294	−0.731468	−	0.681876i	$-0.761164\pi$
$449$	−30.9991	−1.46294	−0.731468	+	0.681876i	$0.761164\pi$
$450$	−0.335857	−0.0158325
$451$	0.0464894	0.00218910
$452$	2.11534	0.0994973
$453$	16.1250	0.757620
$454$	1.68617	0.0791360
$455$	1.29378	0.0606535
$456$	−5.68739	−0.266337
$457$	36.5791	1.71110	0.855549	−	0.517723i	$-0.173220\pi$
$457$	36.5791	1.71110	0.855549	+	0.517723i	$0.173220\pi$
$458$	−0.938715	−0.0438633
$459$	1.00000	0.0466760
$460$	4.02582	0.187705
$461$	22.2892	1.03811	0.519055	−	0.854741i	$-0.326284\pi$
$461$	22.2892	1.03811	0.519055	+	0.854741i	$0.326284\pi$
$462$	0.193532	0.00900392
$463$	27.9273	1.29789	0.648947	−	0.760834i	$-0.275210\pi$
$463$	27.9273	1.29789	0.648947	+	0.760834i	$0.275210\pi$
$464$	4.42083	0.205232
$465$	6.12692	0.284129
$466$	0.813777	0.0376975
$467$	36.1612	1.67334	0.836671	−	0.547706i	$-0.184499\pi$
$467$	36.1612	1.67334	0.836671	+	0.547706i	$0.184499\pi$
$468$	5.40894	0.250028
$469$	2.91419	0.134565
$470$	3.31383	0.152856
$471$	1.00000	0.0460776
$472$	−0.954590	−0.0439386
$473$	6.31603	0.290411
$474$	0.245915	0.0112952
$475$	−12.6499	−0.580418
$476$	−0.508374	−0.0233013
$477$	3.54711	0.162411
$478$	−0.0853193	−0.00390241
$479$	9.05944	0.413936	0.206968	−	0.978348i	$-0.433640\pi$
$479$	9.05944	0.413936	0.206968	+	0.978348i	$0.433640\pi$
$480$	−4.13534	−0.188751
$481$	−7.49608	−0.341792
$482$	2.68930	0.122494
$483$	0.293613	0.0133599
$484$	−7.13245	−0.324202
$485$	26.4567	1.20134
$486$	0.195227	0.00885566
$487$	−7.37793	−0.334326	−0.167163	−	0.985929i	$-0.553461\pi$
$487$	−7.37793	−0.334326	−0.167163	+	0.985929i	$0.553461\pi$
$488$	4.66246	0.211060
$489$	−15.5717	−0.704177
$490$	2.45112	0.110730
$491$	36.6131	1.65233	0.826163	−	0.563431i	$-0.190519\pi$
$491$	36.6131	1.65233	0.826163	+	0.563431i	$0.190519\pi$
$492$	−0.0238410	−0.00107483
$493$	1.17177	0.0527740
$494$	−3.95776	−0.178068
$495$	−6.92816	−0.311398
$496$	−12.7641	−0.573123
$497$	−4.13319	−0.185399
$498$	−1.08125	−0.0484521
$499$	−27.4806	−1.23020	−0.615100	−	0.788449i	$-0.710884\pi$
$499$	−27.4806	−1.23020	−0.615100	+	0.788449i	$0.710884\pi$
$500$	−23.8770	−1.06781
$501$	−14.8648	−0.664112
$502$	0.938412	0.0418834
$503$	11.8114	0.526643	0.263321	−	0.964708i	$-0.415182\pi$
$503$	11.8114	0.526643	0.263321	+	0.964708i	$0.415182\pi$
$504$	−0.200424	−0.00892761
$505$	−11.4913	−0.511357
$506$	0.846271	0.0376213
$507$	−5.39890	−0.239774
$508$	−31.0510	−1.37766
$509$	−34.4557	−1.52722	−0.763612	−	0.645676i	$-0.776576\pi$
$509$	−34.4557	−1.52722	−0.763612	+	0.645676i	$0.776576\pi$
$510$	−0.353552	−0.0156555
$511$	1.74418	0.0771580
$512$	14.4513	0.638662
$513$	7.35313	0.324648
$514$	−2.90472	−0.128122
$515$	−14.3336	−0.631613
$516$	−3.23902	−0.142590
$517$	−35.8576	−1.57702
$518$	0.137545	0.00604338
$519$	24.0832	1.05713
$520$	−3.86183	−0.169353
$521$	12.2595	0.537098	0.268549	−	0.963266i	$-0.413456\pi$
$521$	12.2595	0.537098	0.268549	+	0.963266i	$0.413456\pi$
$522$	0.228761	0.0100126
$523$	−2.48526	−0.108673	−0.0543364	−	0.998523i	$-0.517304\pi$
$523$	−2.48526	−0.108673	−0.0543364	+	0.998523i	$0.517304\pi$
$524$	1.76277	0.0770068
$525$	−0.445785	−0.0194556
$526$	2.31090	0.100760
$527$	−3.38321	−0.147375
$528$	14.4333	0.628127
$529$	−21.7161	−0.944178
$530$	−1.25409	−0.0544741
$531$	1.23417	0.0535585
$532$	−3.73814	−0.162069
$533$	−0.0335034	−0.00145119
$534$	2.21194	0.0957200
$535$	−28.7367	−1.24240
$536$	−8.69859	−0.375722
$537$	20.5241	0.885682
$538$	−0.873054	−0.0376400
$539$	−26.5226	−1.14241
$540$	3.55294	0.152894
$541$	26.7357	1.14946	0.574729	−	0.818344i	$-0.305107\pi$
$541$	26.7357	1.14946	0.574729	+	0.818344i	$0.305107\pi$
$542$	−3.90368	−0.167677
$543$	14.3862	0.617372
$544$	2.28348	0.0979033
$545$	−5.50082	−0.235629
$546$	−0.139472	−0.00596885
$547$	34.5132	1.47568	0.737839	−	0.674977i	$-0.235846\pi$
$547$	34.5132	1.47568	0.737839	+	0.674977i	$0.235846\pi$
$548$	11.4395	0.488673
$549$	−6.02802	−0.257269
$550$	−1.28487	−0.0547870
$551$	8.61619	0.367062
$552$	−0.876411	−0.0373025
$553$	0.326403	0.0138801
$554$	−3.79912	−0.161409
$555$	−4.92391	−0.209008
$556$	4.58696	0.194531
$557$	13.9138	0.589546	0.294773	−	0.955567i	$-0.404756\pi$
$557$	13.9138	0.589546	0.294773	+	0.955567i	$0.404756\pi$
$558$	−0.660492	−0.0279609
$559$	−4.55175	−0.192518
$560$	−1.77045	−0.0748152
$561$	3.82564	0.161519
$562$	0.757839	0.0319675
$563$	32.6420	1.37570	0.687848	−	0.725855i	$-0.258556\pi$
$563$	32.6420	1.37570	0.687848	+	0.725855i	$0.258556\pi$
$564$	18.3887	0.774304
$565$	1.95263	0.0821479
$566$	4.63510	0.194828
$567$	0.259125	0.0108822
$568$	12.3372	0.517658
$569$	−19.3580	−0.811528	−0.405764	−	0.913978i	$-0.632995\pi$
$569$	−19.3580	−0.811528	−0.405764	+	0.913978i	$0.632995\pi$
$570$	−2.59971	−0.108890
$571$	14.9259	0.624630	0.312315	−	0.949979i	$-0.398895\pi$
$571$	14.9259	0.624630	0.312315	+	0.949979i	$0.398895\pi$
$572$	20.6926	0.865203
$573$	11.7021	0.488862
$574$	0.000614750 0	2.56592e−5 0
$575$	−1.94932	−0.0812921
$576$	−7.09975	−0.295823
$577$	12.8220	0.533786	0.266893	−	0.963726i	$-0.414003\pi$
$577$	12.8220	0.533786	0.266893	+	0.963726i	$0.414003\pi$
$578$	0.195227	0.00812036
$579$	8.43235	0.350436
$580$	4.16324	0.172869
$581$	−1.43515	−0.0595401
$582$	−2.85207	−0.118222
$583$	13.5700	0.562011
$584$	−5.20623	−0.215435
$585$	4.99289	0.206431
$586$	0.399659	0.0165098
$587$	35.8977	1.48166	0.740828	−	0.671695i	$-0.234433\pi$
$587$	35.8977	1.48166	0.740828	+	0.671695i	$0.234433\pi$
$588$	13.6015	0.560915
$589$	−24.8771	−1.02504
$590$	−0.436344	−0.0179640
$591$	−1.89134	−0.0777992
$592$	10.2579	0.421595
$593$	5.58440	0.229324	0.114662	−	0.993405i	$-0.463421\pi$
$593$	5.58440	0.229324	0.114662	+	0.993405i	$0.463421\pi$
$594$	0.746867	0.0306443
$595$	−0.469271	−0.0192382
$596$	3.50421	0.143538
$597$	19.5505	0.800148
$598$	−0.609879	−0.0249398
$599$	−19.7990	−0.808966	−0.404483	−	0.914546i	$-0.632549\pi$
$599$	−19.7990	−0.808966	−0.404483	+	0.914546i	$0.632549\pi$
$600$	1.33063	0.0543227
$601$	−19.1592	−0.781520	−0.390760	−	0.920493i	$-0.627788\pi$
$601$	−19.1592	−0.781520	−0.390760	+	0.920493i	$0.627788\pi$
$602$	0.0835197	0.00340401
$603$	11.2463	0.457983
$604$	−31.6355	−1.28723
$605$	−6.58384	−0.267671
$606$	1.23878	0.0503221
$607$	−25.4510	−1.03302	−0.516512	−	0.856280i	$-0.672770\pi$
$607$	−25.4510	−1.03302	−0.516512	+	0.856280i	$0.672770\pi$
$608$	16.7907	0.680953
$609$	0.303636	0.0123039
$610$	2.13122	0.0862904
$611$	25.8414	1.04543
$612$	−1.96189	−0.0793046
$613$	−9.01313	−0.364037	−0.182018	−	0.983295i	$-0.558263\pi$
$613$	−9.01313	−0.364037	−0.182018	+	0.983295i	$0.558263\pi$
$614$	1.15582	0.0466453
$615$	−0.0220072	−0.000887415 0
$616$	−0.766751	−0.0308933
$617$	−27.8484	−1.12114	−0.560568	−	0.828108i	$-0.689417\pi$
$617$	−27.8484	−1.12114	−0.560568	+	0.828108i	$0.689417\pi$
$618$	1.54518	0.0621564
$619$	6.74067	0.270930	0.135465	−	0.990782i	$-0.456747\pi$
$619$	6.74067	0.270930	0.135465	+	0.990782i	$0.456747\pi$
$620$	−12.0203	−0.482748
$621$	1.13310	0.0454696
$622$	0.936016	0.0375308
$623$	2.93592	0.117625
$624$	−10.4016	−0.416396
$625$	−13.4387	−0.537547
$626$	−6.56811	−0.262514
$627$	28.1304	1.12342
$628$	−1.96189	−0.0782878
$629$	2.71892	0.108410
$630$	−0.0916142	−0.00365000
$631$	−12.2182	−0.486397	−0.243199	−	0.969976i	$-0.578197\pi$
$631$	−12.2182	−0.486397	−0.243199	+	0.969976i	$0.578197\pi$
$632$	−0.974286	−0.0387550
$633$	26.6958	1.06106
$634$	6.25787	0.248532
$635$	−28.6626	−1.13744
$636$	−6.95903	−0.275944
$637$	19.1139	0.757322
$638$	0.875158	0.0346478
$639$	−15.9505	−0.630994
$640$	10.7808	0.426149
$641$	34.4938	1.36243	0.681213	−	0.732086i	$-0.261453\pi$
$641$	34.4938	1.36243	0.681213	+	0.732086i	$0.261453\pi$
$642$	3.09787	0.122263
$643$	−4.12449	−0.162654	−0.0813269	−	0.996687i	$-0.525916\pi$
$643$	−4.12449	−0.162654	−0.0813269	+	0.996687i	$0.525916\pi$
$644$	−0.576036	−0.0226990
$645$	−2.98988	−0.117726
$646$	1.43553	0.0564800
$647$	1.51738	0.0596544	0.0298272	−	0.999555i	$-0.490504\pi$
$647$	1.51738	0.0596544	0.0298272	+	0.999555i	$0.490504\pi$
$648$	−0.773466	−0.0303846
$649$	4.72150	0.185335
$650$	0.925962	0.0363192
$651$	−0.876673	−0.0343595
$652$	30.5499	1.19643
$653$	−36.5695	−1.43107	−0.715537	−	0.698575i	$-0.753818\pi$
$653$	−36.5695	−1.43107	−0.715537	+	0.698575i	$0.753818\pi$
$654$	0.592997	0.0231880
$655$	1.62718	0.0635791
$656$	0.0458470	0.00179002
$657$	6.73104	0.262603
$658$	−0.474161	−0.0184847
$659$	7.73406	0.301276	0.150638	−	0.988589i	$-0.451867\pi$
$659$	7.73406	0.301276	0.150638	+	0.988589i	$0.451867\pi$
$660$	13.5923	0.529078
$661$	−0.211829	−0.00823919	−0.00411960	−	0.999992i	$-0.501311\pi$
$661$	−0.211829	−0.00823919	−0.00411960	+	0.999992i	$0.501311\pi$
$662$	0.174754	0.00679201
$663$	−2.75701	−0.107073
$664$	4.28380	0.166244
$665$	−3.45061	−0.133809
$666$	0.530805	0.0205683
$667$	1.32773	0.0514099
$668$	29.1631	1.12836
$669$	1.14364	0.0442155
$670$	−3.97613	−0.153611
$671$	−23.0610	−0.890260
$672$	0.591706	0.0228256
$673$	−1.33482	−0.0514537	−0.0257269	−	0.999669i	$-0.508190\pi$
$673$	−1.33482	−0.0514537	−0.0257269	+	0.999669i	$0.508190\pi$
$674$	0.194869	0.00750606
$675$	−1.72035	−0.0662162
$676$	10.5920	0.407386
$677$	12.7339	0.489402	0.244701	−	0.969599i	$-0.421310\pi$
$677$	12.7339	0.489402	0.244701	+	0.969599i	$0.421310\pi$
$678$	−0.210497	−0.00808409
$679$	−3.78557	−0.145277
$680$	1.40073	0.0537156
$681$	8.63700	0.330970
$682$	−2.52680	−0.0967563
$683$	19.7264	0.754809	0.377405	−	0.926049i	$-0.376817\pi$
$683$	19.7264	0.754809	0.377405	+	0.926049i	$0.376817\pi$
$684$	−14.4260	−0.551592
$685$	10.5596	0.403463
$686$	−0.704837	−0.0269108
$687$	−4.80834	−0.183449
$688$	6.22875	0.237469
$689$	−9.77942	−0.372566
$690$	−0.400608	−0.0152509
$691$	34.3443	1.30652	0.653259	−	0.757135i	$-0.273401\pi$
$691$	34.3443	1.30652	0.653259	+	0.757135i	$0.273401\pi$
$692$	−47.2485	−1.79612
$693$	0.991319	0.0376571
$694$	5.10386	0.193740
$695$	4.23414	0.160610
$696$	−0.906326	−0.0343542
$697$	0.0121521	0.000460292 0
$698$	−6.82972	−0.258509
$699$	4.16837	0.157662
$700$	0.874579	0.0330560
$701$	17.1608	0.648155	0.324077	−	0.946031i	$-0.394946\pi$
$701$	17.1608	0.648155	0.324077	+	0.946031i	$0.394946\pi$
$702$	−0.538242	−0.0203146
$703$	19.9925	0.754033
$704$	−27.1611	−1.02367
$705$	16.9743	0.639288
$706$	0.0296107	0.00111441
$707$	1.64424	0.0618380
$708$	−2.42131	−0.0909982
$709$	2.19887	0.0825803	0.0412901	−	0.999147i	$-0.486853\pi$
$709$	2.19887	0.0825803	0.0412901	+	0.999147i	$0.486853\pi$
$710$	5.63935	0.211641
$711$	1.25964	0.0472400
$712$	−8.76346	−0.328424
$713$	−3.83349	−0.143566
$714$	0.0505881	0.00189321
$715$	19.1010	0.714337
$716$	−40.2660	−1.50481
$717$	−0.437027	−0.0163211
$718$	−2.16001	−0.0806109
$719$	44.1187	1.64535	0.822675	−	0.568512i	$-0.192481\pi$
$719$	44.1187	1.64535	0.822675	+	0.568512i	$0.192481\pi$
$720$	−6.83242	−0.254629
$721$	2.05093	0.0763806
$722$	6.84630	0.254793
$723$	13.7753	0.512307
$724$	−28.2241	−1.04894
$725$	−2.01585	−0.0748669
$726$	0.709748	0.0263412
$727$	−38.0525	−1.41129	−0.705644	−	0.708567i	$-0.749342\pi$
$727$	−38.0525	−1.41129	−0.705644	+	0.708567i	$0.749342\pi$
$728$	0.552572	0.0204797
$729$	1.00000	0.0370370
$730$	−2.37977	−0.0880792
$731$	1.65097	0.0610635
$732$	11.8263	0.437112
$733$	−40.3949	−1.49202	−0.746009	−	0.665936i	$-0.768032\pi$
$733$	−40.3949	−1.49202	−0.746009	+	0.665936i	$0.768032\pi$
$734$	−3.32203	−0.122618
$735$	12.5553	0.463108
$736$	2.58740	0.0953728
$737$	43.0241	1.58481
$738$	0.00237241	8.73295e−5 0
$739$	23.6484	0.869921	0.434960	−	0.900450i	$-0.356762\pi$
$739$	23.6484	0.869921	0.434960	+	0.900450i	$0.356762\pi$
$740$	9.66015	0.355114
$741$	−20.2726	−0.744734
$742$	0.179442	0.00658751
$743$	18.8675	0.692183	0.346091	−	0.938201i	$-0.387509\pi$
$743$	18.8675	0.692183	0.346091	+	0.938201i	$0.387509\pi$
$744$	2.61679	0.0959363
$745$	3.23467	0.118509
$746$	−0.509764	−0.0186638
$747$	−5.53845	−0.202641
$748$	−7.50547	−0.274427
$749$	4.11181	0.150242
$750$	2.37599	0.0867589
$751$	−22.3360	−0.815052	−0.407526	−	0.913194i	$-0.633608\pi$
$751$	−22.3360	−0.815052	−0.407526	+	0.913194i	$0.633608\pi$
$752$	−35.3621	−1.28952
$753$	4.80678	0.175169
$754$	−0.630697	−0.0229686
$755$	−29.2021	−1.06277
$756$	−0.508374	−0.0184894
$757$	−40.6229	−1.47646	−0.738232	−	0.674546i	$-0.764339\pi$
$757$	−40.6229	−1.47646	−0.738232	+	0.674546i	$0.764339\pi$
$758$	4.49774	0.163365
$759$	4.33481	0.157344
$760$	10.2998	0.373612
$761$	24.0050	0.870182	0.435091	−	0.900387i	$-0.356716\pi$
$761$	24.0050	0.870182	0.435091	+	0.900387i	$0.356716\pi$
$762$	3.08987	0.111934
$763$	0.787087	0.0284945
$764$	−22.9582	−0.830597
$765$	−1.81098	−0.0654762
$766$	4.43262	0.160157
$767$	−3.40262	−0.122862
$768$	13.0373	0.470443
$769$	6.11176	0.220396	0.110198	−	0.993910i	$-0.464852\pi$
$769$	6.11176	0.220396	0.110198	+	0.993910i	$0.464852\pi$
$770$	−0.350483	−0.0126305
$771$	−14.8787	−0.535843
$772$	−16.5433	−0.595407
$773$	−2.72466	−0.0979991	−0.0489996	−	0.998799i	$-0.515603\pi$
$773$	−2.72466	−0.0979991	−0.0489996	+	0.998799i	$0.515603\pi$
$774$	0.322314	0.0115853
$775$	5.82028	0.209071
$776$	11.2996	0.405632
$777$	0.704540	0.0252752
$778$	−0.318370	−0.0114141
$779$	0.0893557	0.00320150
$780$	−9.79549	−0.350735
$781$	−61.0210	−2.18350
$782$	0.221210	0.00791047
$783$	1.17177	0.0418757
$784$	−26.1561	−0.934146
$785$	−1.81098	−0.0646367
$786$	−0.175412	−0.00625675
$787$	−7.06739	−0.251925	−0.125963	−	0.992035i	$-0.540202\pi$
$787$	−7.06739	−0.251925	−0.125963	+	0.992035i	$0.540202\pi$
$788$	3.71059	0.132184
$789$	11.8370	0.421409
$790$	−0.445347	−0.0158447
$791$	−0.279393	−0.00993408
$792$	−2.95900	−0.105144
$793$	16.6193	0.590169
$794$	−4.81481	−0.170871
$795$	−6.42375	−0.227827
$796$	−38.3558	−1.35949
$797$	−22.6753	−0.803202	−0.401601	−	0.915815i	$-0.631546\pi$
$797$	−22.6753	−0.803202	−0.401601	+	0.915815i	$0.631546\pi$
$798$	0.371981	0.0131680
$799$	−9.37297	−0.331592
$800$	−3.92837	−0.138889
$801$	11.3301	0.400330
$802$	−5.49015	−0.193864
$803$	25.7505	0.908716
$804$	−22.0639	−0.778133
$805$	−0.531729	−0.0187410
$806$	1.82098	0.0641414
$807$	−4.47200	−0.157422
$808$	−4.90791	−0.172660
$809$	−17.9738	−0.631924	−0.315962	−	0.948772i	$-0.602327\pi$
$809$	−17.9738	−0.631924	−0.315962	+	0.948772i	$0.602327\pi$
$810$	−0.353552	−0.0124225
$811$	−36.0182	−1.26477	−0.632385	−	0.774654i	$-0.717924\pi$
$811$	−36.0182	−1.26477	−0.632385	+	0.774654i	$0.717924\pi$
$812$	−0.595699	−0.0209049
$813$	−19.9956	−0.701277
$814$	2.03067	0.0711749
$815$	28.2001	0.987805
$816$	3.77277	0.132073
$817$	12.1398	0.424718
$818$	−5.29788	−0.185236
$819$	−0.714410	−0.0249635
$820$	0.0431756	0.00150776
$821$	52.7557	1.84119	0.920594	−	0.390522i	$-0.127705\pi$
$821$	52.7557	1.84119	0.920594	+	0.390522i	$0.127705\pi$
$822$	−1.13835	−0.0397044
$823$	−23.1104	−0.805579	−0.402789	−	0.915293i	$-0.631959\pi$
$823$	−23.1104	−0.805579	−0.402789	+	0.915293i	$0.631959\pi$
$824$	−6.12184	−0.213265
$825$	−6.58142	−0.229136
$826$	0.0624345	0.00217237
$827$	−18.8162	−0.654302	−0.327151	−	0.944972i	$-0.606089\pi$
$827$	−18.8162	−0.654302	−0.327151	+	0.944972i	$0.606089\pi$
$828$	−2.22300	−0.0772548
$829$	1.81269	0.0629575	0.0314787	−	0.999504i	$-0.489978\pi$
$829$	1.81269	0.0629575	0.0314787	+	0.999504i	$0.489978\pi$
$830$	1.95813	0.0679677
$831$	−19.4601	−0.675062
$832$	19.5741	0.678609
$833$	−6.93285	−0.240209
$834$	−0.456447	−0.0158055
$835$	26.9200	0.931604
$836$	−55.1886	−1.90874
$837$	−3.38321	−0.116941
$838$	0.666119	0.0230107
$839$	−50.5071	−1.74370	−0.871849	−	0.489776i	$-0.837079\pi$
$839$	−50.5071	−1.74370	−0.871849	+	0.489776i	$0.837079\pi$
$840$	0.362965	0.0125235
$841$	−27.6269	−0.952653
$842$	−0.652559	−0.0224887
$843$	3.88184	0.133698
$844$	−52.3742	−1.80279
$845$	9.77731	0.336350
$846$	−1.82985	−0.0629117
$847$	0.942051	0.0323693
$848$	13.3824	0.459555
$849$	23.7421	0.814828
$850$	−0.335857	−0.0115198
$851$	3.08079	0.105608
$852$	31.2932	1.07209
$853$	33.8528	1.15910	0.579548	−	0.814938i	$-0.303229\pi$
$853$	33.8528	1.15910	0.579548	+	0.814938i	$0.303229\pi$
$854$	−0.304946	−0.0104350
$855$	−13.3164	−0.455410
$856$	−12.2734	−0.419496
$857$	−10.4086	−0.355552	−0.177776	−	0.984071i	$-0.556890\pi$
$857$	−10.4086	−0.355552	−0.177776	+	0.984071i	$0.556890\pi$
$858$	−2.05912	−0.0702971
$859$	−15.5580	−0.530832	−0.265416	−	0.964134i	$-0.585509\pi$
$859$	−15.5580	−0.530832	−0.265416	+	0.964134i	$0.585509\pi$
$860$	5.86581	0.200022
$861$	0.00314891	0.000107314 0
$862$	1.92600	0.0655998
$863$	−30.2707	−1.03043	−0.515214	−	0.857062i	$-0.672287\pi$
$863$	−30.2707	−1.03043	−0.515214	+	0.857062i	$0.672287\pi$
$864$	2.28348	0.0776855
$865$	−43.6142	−1.48293
$866$	8.08754	0.274826
$867$	1.00000	0.0339618
$868$	1.71993	0.0583784
$869$	4.81891	0.163470
$870$	−0.414282	−0.0140455
$871$	−31.0060	−1.05060
$872$	−2.34939	−0.0795603
$873$	−14.6090	−0.494441
$874$	1.62659	0.0550202
$875$	3.15366	0.106613
$876$	−13.2055	−0.446173
$877$	22.0789	0.745550	0.372775	−	0.927922i	$-0.378406\pi$
$877$	22.0789	0.745550	0.372775	+	0.927922i	$0.378406\pi$
$878$	2.42265	0.0817604
$879$	2.04715	0.0690488
$880$	−26.1384	−0.881124
$881$	−33.2656	−1.12075	−0.560373	−	0.828240i	$-0.689342\pi$
$881$	−33.2656	−1.12075	−0.560373	+	0.828240i	$0.689342\pi$
$882$	−1.35348	−0.0455740
$883$	5.33638	0.179584	0.0897918	−	0.995961i	$-0.471380\pi$
$883$	5.33638	0.179584	0.0897918	+	0.995961i	$0.471380\pi$
$884$	5.40894	0.181922
$885$	−2.23506	−0.0751308
$886$	−0.949742	−0.0319072
$887$	26.5313	0.890835	0.445417	−	0.895323i	$-0.353055\pi$
$887$	26.5313	0.890835	0.445417	+	0.895323i	$0.353055\pi$
$888$	−2.10299	−0.0705717
$889$	4.10120	0.137550
$890$	−4.00578	−0.134274
$891$	3.82564	0.128164
$892$	−2.24368	−0.0751240
$893$	−68.9206	−2.30634
$894$	−0.348703	−0.0116624
$895$	−37.1688	−1.24242
$896$	−1.54258	−0.0515338
$897$	−3.12395	−0.104306
$898$	−6.05184	−0.201953
$899$	−3.96435	−0.132218
$900$	3.37512	0.112504
$901$	3.54711	0.118171
$902$	0.00907597	0.000302197 0
$903$	0.427809	0.0142366
$904$	0.833965	0.0277373
$905$	−26.0532	−0.866037
$906$	3.14803	0.104586
$907$	8.37317	0.278027	0.139013	−	0.990291i	$-0.455607\pi$
$907$	8.37317	0.278027	0.139013	+	0.990291i	$0.455607\pi$
$908$	−16.9448	−0.562333
$909$	6.34535	0.210462
$910$	0.252581	0.00837298
$911$	−3.83602	−0.127093	−0.0635465	−	0.997979i	$-0.520241\pi$
$911$	−3.83602	−0.127093	−0.0635465	+	0.997979i	$0.520241\pi$
$912$	27.7417	0.918618
$913$	−21.1881	−0.701224
$914$	7.14121	0.236210
$915$	10.9166	0.360892
$916$	9.43341	0.311689
$917$	−0.232825	−0.00768857
$918$	0.195227	0.00644344
$919$	−56.5559	−1.86561	−0.932804	−	0.360384i	$-0.882646\pi$
$919$	−56.5559	−1.86561	−0.932804	+	0.360384i	$0.882646\pi$
$920$	1.58716	0.0523272
$921$	5.92042	0.195085
$922$	4.35144	0.143307
$923$	43.9758	1.44748
$924$	−1.94486	−0.0639810
$925$	−4.67748	−0.153795
$926$	5.45216	0.179169
$927$	7.91482	0.259957
$928$	2.67572	0.0878347
$929$	23.2962	0.764324	0.382162	−	0.924095i	$-0.375180\pi$
$929$	23.2962	0.764324	0.382162	+	0.924095i	$0.375180\pi$
$930$	1.19614	0.0392229
$931$	−50.9781	−1.67074
$932$	−8.17787	−0.267875
$933$	4.79451	0.156965
$934$	7.05964	0.230998
$935$	−6.92816	−0.226575
$936$	2.13245	0.0697014
$937$	−14.3884	−0.470048	−0.235024	−	0.971990i	$-0.575517\pi$
$937$	−14.3884	−0.470048	−0.235024	+	0.971990i	$0.575517\pi$
$938$	0.568927	0.0185761
$939$	−33.6435	−1.09791
$940$	−33.3016	−1.08618
$941$	24.1338	0.786740	0.393370	−	0.919380i	$-0.371309\pi$
$941$	24.1338	0.786740	0.393370	+	0.919380i	$0.371309\pi$
$942$	0.195227	0.00636083
$943$	0.0137695	0.000448395 0
$944$	4.65625	0.151548
$945$	−0.469271	−0.0152654
$946$	1.23306	0.0400901
$947$	34.8843	1.13359	0.566794	−	0.823860i	$-0.308184\pi$
$947$	34.8843	1.13359	0.566794	+	0.823860i	$0.308184\pi$
$948$	−2.47126	−0.0802629
$949$	−18.5575	−0.602403
$950$	−2.46960	−0.0801245
$951$	32.0544	1.03943
$952$	−0.200424	−0.00649579
$953$	−19.1761	−0.621175	−0.310587	−	0.950545i	$-0.600526\pi$
$953$	−19.1761	−0.621175	−0.310587	+	0.950545i	$0.600526\pi$
$954$	0.692491	0.0224202
$955$	−21.1923	−0.685766
$956$	0.857397	0.0277302
$957$	4.48278	0.144908
$958$	1.76864	0.0571423
$959$	−1.51093	−0.0487905
$960$	12.8575	0.414974
$961$	−19.5539	−0.630772
$962$	−1.46344	−0.0471830
$963$	15.8680	0.511341
$964$	−27.0255	−0.870433
$965$	−15.2708	−0.491585
$966$	0.0573212	0.00184428
$967$	−14.9082	−0.479416	−0.239708	−	0.970845i	$-0.577052\pi$
$967$	−14.9082	−0.479416	−0.239708	+	0.970845i	$0.577052\pi$
$968$	−2.81194	−0.0903792
$969$	7.35313	0.236216
$970$	5.16505	0.165840
$971$	−6.79874	−0.218182	−0.109091	−	0.994032i	$-0.534794\pi$
$971$	−6.79874	−0.218182	−0.109091	+	0.994032i	$0.534794\pi$
$972$	−1.96189	−0.0629275
$973$	−0.605844	−0.0194225
$974$	−1.44037	−0.0461524
$975$	4.74301	0.151898
$976$	−22.7423	−0.727964
$977$	−34.5384	−1.10498	−0.552491	−	0.833519i	$-0.686323\pi$
$977$	−34.5384	−1.10498	−0.552491	+	0.833519i	$0.686323\pi$
$978$	−3.04001	−0.0972088
$979$	43.3449	1.38531
$980$	−24.6320	−0.786841
$981$	3.03748	0.0969792
$982$	7.14786	0.228097
$983$	20.3299	0.648423	0.324212	−	0.945985i	$-0.394901\pi$
$983$	20.3299	0.648423	0.324212	+	0.945985i	$0.394901\pi$
$984$	−0.00939921	−0.000299636 0
$985$	3.42517	0.109135
$986$	0.228761	0.00728524
$987$	−2.42877	−0.0773087
$988$	39.7726	1.26534
$989$	1.87071	0.0594851
$990$	−1.35256	−0.0429872
$991$	−7.15915	−0.227418	−0.113709	−	0.993514i	$-0.536273\pi$
$991$	−7.15915	−0.227418	−0.113709	+	0.993514i	$0.536273\pi$
$992$	−7.72547	−0.245284
$993$	0.895134	0.0284062
$994$	−0.806908	−0.0255936
$995$	−35.4056	−1.12243
$996$	10.8658	0.344296
$997$	−39.0340	−1.23622	−0.618109	−	0.786092i	$-0.712101\pi$
$997$	−39.0340	−1.23622	−0.618109	+	0.786092i	$0.712101\pi$
$998$	−5.36495	−0.169824
$999$	2.71892	0.0860228

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.i.1.32	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.i.1.32	✓	63	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q+0.195227 q^{2} +1.00000 q^{3} -1.96189 q^{4} -1.81098 q^{5} +0.195227 q^{6} +0.259125 q^{7} -0.773466 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.195227 q^{2} +1.00000 q^{3} -1.96189 q^{4} -1.81098 q^{5} +0.195227 q^{6} +0.259125 q^{7} -0.773466 q^{8} +1.00000 q^{9} -0.353552 q^{10} +3.82564 q^{11} -1.96189 q^{12} -2.75701 q^{13} +0.0505881 q^{14} -1.81098 q^{15} +3.77277 q^{16} +1.00000 q^{17} +0.195227 q^{18} +7.35313 q^{19} +3.55294 q^{20} +0.259125 q^{21} +0.746867 q^{22} +1.13310 q^{23} -0.773466 q^{24} -1.72035 q^{25} -0.538242 q^{26} +1.00000 q^{27} -0.508374 q^{28} +1.17177 q^{29} -0.353552 q^{30} -3.38321 q^{31} +2.28348 q^{32} +3.82564 q^{33} +0.195227 q^{34} -0.469271 q^{35} -1.96189 q^{36} +2.71892 q^{37} +1.43553 q^{38} -2.75701 q^{39} +1.40073 q^{40} +0.0121521 q^{41} +0.0505881 q^{42} +1.65097 q^{43} -7.50547 q^{44} -1.81098 q^{45} +0.221210 q^{46} -9.37297 q^{47} +3.77277 q^{48} -6.93285 q^{49} -0.335857 q^{50} +1.00000 q^{51} +5.40894 q^{52} +3.54711 q^{53} +0.195227 q^{54} -6.92816 q^{55} -0.200424 q^{56} +7.35313 q^{57} +0.228761 q^{58} +1.23417 q^{59} +3.55294 q^{60} -6.02802 q^{61} -0.660492 q^{62} +0.259125 q^{63} -7.09975 q^{64} +4.99289 q^{65} +0.746867 q^{66} +11.2463 q^{67} -1.96189 q^{68} +1.13310 q^{69} -0.0916142 q^{70} -15.9505 q^{71} -0.773466 q^{72} +6.73104 q^{73} +0.530805 q^{74} -1.72035 q^{75} -14.4260 q^{76} +0.991319 q^{77} -0.538242 q^{78} +1.25964 q^{79} -6.83242 q^{80} +1.00000 q^{81} +0.00237241 q^{82} -5.53845 q^{83} -0.508374 q^{84} -1.81098 q^{85} +0.322314 q^{86} +1.17177 q^{87} -2.95900 q^{88} +11.3301 q^{89} -0.353552 q^{90} -0.714410 q^{91} -2.22300 q^{92} -3.38321 q^{93} -1.82985 q^{94} -13.3164 q^{95} +2.28348 q^{96} -14.6090 q^{97} -1.35348 q^{98} +3.82564 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9}+O(q^{10}) \) 63 * q + 10 * q^2 + 63 * q^3 + 70 * q^4 + 19 * q^5 + 10 * q^6 + 11 * q^7 + 27 * q^8 + 63 * q^9	\( 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9} + 4 q^{10} + 23 q^{11} + 70 q^{12} + 10 q^{13} + 18 q^{14} + 19 q^{15} + 72 q^{16} + 63 q^{17} + 10 q^{18} + 6 q^{19} + 48 q^{20} + 11 q^{21} + 21 q^{22} + 44 q^{23} + 27 q^{24} + 110 q^{25} + 41 q^{26} + 63 q^{27} + 26 q^{28} + 35 q^{29} + 4 q^{30} + q^{31} + 54 q^{32} + 23 q^{33} + 10 q^{34} + 47 q^{35} + 70 q^{36} + 40 q^{37} + 38 q^{38} + 10 q^{39} - 10 q^{40} + 35 q^{41} + 18 q^{42} + 27 q^{43} + 46 q^{44} + 19 q^{45} + 8 q^{46} + 29 q^{47} + 72 q^{48} + 114 q^{49} + 27 q^{50} + 63 q^{51} - q^{52} + 75 q^{53} + 10 q^{54} + 5 q^{55} + 24 q^{56} + 6 q^{57} + 41 q^{58} + 105 q^{59} + 48 q^{60} + 5 q^{61} + 22 q^{62} + 11 q^{63} + 61 q^{64} + 49 q^{65} + 21 q^{66} + 4 q^{67} + 70 q^{68} + 44 q^{69} - 16 q^{70} + 16 q^{71} + 27 q^{72} + 39 q^{73} + 54 q^{74} + 110 q^{75} + 6 q^{76} + 88 q^{77} + 41 q^{78} + 16 q^{79} + 102 q^{80} + 63 q^{81} - 29 q^{82} + 73 q^{83} + 26 q^{84} + 19 q^{85} + 46 q^{86} + 35 q^{87} + 18 q^{88} + 88 q^{89} + 4 q^{90} - 15 q^{91} + 110 q^{92} + q^{93} - 8 q^{94} + 28 q^{95} + 54 q^{96} + 70 q^{97} + 33 q^{98} + 23 q^{99}+O(q^{100}) \) 63 * q + 10 * q^2 + 63 * q^3 + 70 * q^4 + 19 * q^5 + 10 * q^6 + 11 * q^7 + 27 * q^8 + 63 * q^9 + 4 * q^10 + 23 * q^11 + 70 * q^12 + 10 * q^13 + 18 * q^14 + 19 * q^15 + 72 * q^16 + 63 * q^17 + 10 * q^18 + 6 * q^19 + 48 * q^20 + 11 * q^21 + 21 * q^22 + 44 * q^23 + 27 * q^24 + 110 * q^25 + 41 * q^26 + 63 * q^27 + 26 * q^28 + 35 * q^29 + 4 * q^30 + q^31 + 54 * q^32 + 23 * q^33 + 10 * q^34 + 47 * q^35 + 70 * q^36 + 40 * q^37 + 38 * q^38 + 10 * q^39 - 10 * q^40 + 35 * q^41 + 18 * q^42 + 27 * q^43 + 46 * q^44 + 19 * q^45 + 8 * q^46 + 29 * q^47 + 72 * q^48 + 114 * q^49 + 27 * q^50 + 63 * q^51 - q^52 + 75 * q^53 + 10 * q^54 + 5 * q^55 + 24 * q^56 + 6 * q^57 + 41 * q^58 + 105 * q^59 + 48 * q^60 + 5 * q^61 + 22 * q^62 + 11 * q^63 + 61 * q^64 + 49 * q^65 + 21 * q^66 + 4 * q^67 + 70 * q^68 + 44 * q^69 - 16 * q^70 + 16 * q^71 + 27 * q^72 + 39 * q^73 + 54 * q^74 + 110 * q^75 + 6 * q^76 + 88 * q^77 + 41 * q^78 + 16 * q^79 + 102 * q^80 + 63 * q^81 - 29 * q^82 + 73 * q^83 + 26 * q^84 + 19 * q^85 + 46 * q^86 + 35 * q^87 + 18 * q^88 + 88 * q^89 + 4 * q^90 - 15 * q^91 + 110 * q^92 + q^93 - 8 * q^94 + 28 * q^95 + 54 * q^96 + 70 * q^97 + 33 * q^98 + 23 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more