LMFDB - Embedded newform 387.8.a.a.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,8,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("387.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	387.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:	$120.893004862$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 825 x^{8} + 431 x^{7} + 229838 x^{6} - 1804 x^{5} - 25242488 x^{4} - 2085744 x^{3} + \cdots - 5193030528 $ x^10 - x^9 - 825x^8 + 431x^7 + 229838x^6 - 1804x^5 - 25242488x^4 - 2085744x^3 + 933060896x^2 - 813446208x - 5193030528
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 129)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$16.8413$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.8413 q^{2} +155.630 q^{4} +309.730 q^{5} -729.159 q^{7} +465.334 q^{8} +O(q^{10})$	\(q+16.8413 q^{2} +155.630 q^{4} +309.730 q^{5} -729.159 q^{7} +465.334 q^{8} +5216.26 q^{10} -2838.34 q^{11} +2230.95 q^{13} -12280.0 q^{14} -12083.9 q^{16} -4513.16 q^{17} +1100.91 q^{19} +48203.4 q^{20} -47801.4 q^{22} -24101.1 q^{23} +17807.5 q^{25} +37572.1 q^{26} -113479. q^{28} -174466. q^{29} -30188.2 q^{31} -263071. q^{32} -76007.6 q^{34} -225842. q^{35} +175314. q^{37} +18540.8 q^{38} +144128. q^{40} -206811. q^{41} -79507.0 q^{43} -441732. q^{44} -405894. q^{46} +393333. q^{47} -291871. q^{49} +299903. q^{50} +347203. q^{52} -123642. q^{53} -879119. q^{55} -339302. q^{56} -2.93824e6 q^{58} -341123. q^{59} -2.31627e6 q^{61} -508410. q^{62} -2.88373e6 q^{64} +690991. q^{65} -1.43482e6 q^{67} -702385. q^{68} -3.80348e6 q^{70} +369372. q^{71} -2.24497e6 q^{73} +2.95251e6 q^{74} +171335. q^{76} +2.06960e6 q^{77} -4.59668e6 q^{79} -3.74273e6 q^{80} -3.48297e6 q^{82} +5.41678e6 q^{83} -1.39786e6 q^{85} -1.33900e6 q^{86} -1.32078e6 q^{88} +253544. q^{89} -1.62671e6 q^{91} -3.75086e6 q^{92} +6.62425e6 q^{94} +340984. q^{95} +1.14080e7 q^{97} -4.91549e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{2} + 371 q^{4} + 122 q^{5} - 2052 q^{7} + 927 q^{8}+O(q^{10}) $ 10 * q + q^2 + 371 * q^4 + 122 * q^5 - 2052 * q^7 + 927 * q^8	\( 10 q + q^{2} + 371 q^{4} + 122 q^{5} - 2052 q^{7} + 927 q^{8} - 10032 q^{10} + 8888 q^{11} - 16432 q^{13} + 28408 q^{14} - 26669 q^{16} + 48122 q^{17} - 56146 q^{19} + 88940 q^{20} - 100626 q^{22} + 236336 q^{23} - 135016 q^{25} + 166748 q^{26} - 259060 q^{28} + 248818 q^{29} - 430970 q^{31} - 69493 q^{32} + 445522 q^{34} - 298982 q^{35} - 261254 q^{37} - 257662 q^{38} - 671432 q^{40} + 126814 q^{41} - 795070 q^{43} + 620022 q^{44} - 809038 q^{46} - 627080 q^{47} - 1256116 q^{49} + 83117 q^{50} - 3674204 q^{52} + 1612384 q^{53} - 4732974 q^{55} + 4301484 q^{56} - 7268516 q^{58} + 3442492 q^{59} - 5217214 q^{61} + 2500324 q^{62} - 4657369 q^{64} + 1224166 q^{65} - 6810926 q^{67} + 3563486 q^{68} - 2745858 q^{70} + 13935120 q^{71} - 13743720 q^{73} + 1752692 q^{74} - 15817594 q^{76} + 7685750 q^{77} - 9007608 q^{79} + 13641024 q^{80} - 6329026 q^{82} + 21779128 q^{83} - 13177392 q^{85} - 79507 q^{86} - 13214750 q^{88} + 1895364 q^{89} - 16439838 q^{91} - 9614510 q^{92} + 3404276 q^{94} + 16861514 q^{95} - 20434472 q^{97} - 35731457 q^{98}+O(q^{100}) \) 10 * q + q^2 + 371 * q^4 + 122 * q^5 - 2052 * q^7 + 927 * q^8 - 10032 * q^10 + 8888 * q^11 - 16432 * q^13 + 28408 * q^14 - 26669 * q^16 + 48122 * q^17 - 56146 * q^19 + 88940 * q^20 - 100626 * q^22 + 236336 * q^23 - 135016 * q^25 + 166748 * q^26 - 259060 * q^28 + 248818 * q^29 - 430970 * q^31 - 69493 * q^32 + 445522 * q^34 - 298982 * q^35 - 261254 * q^37 - 257662 * q^38 - 671432 * q^40 + 126814 * q^41 - 795070 * q^43 + 620022 * q^44 - 809038 * q^46 - 627080 * q^47 - 1256116 * q^49 + 83117 * q^50 - 3674204 * q^52 + 1612384 * q^53 - 4732974 * q^55 + 4301484 * q^56 - 7268516 * q^58 + 3442492 * q^59 - 5217214 * q^61 + 2500324 * q^62 - 4657369 * q^64 + 1224166 * q^65 - 6810926 * q^67 + 3563486 * q^68 - 2745858 * q^70 + 13935120 * q^71 - 13743720 * q^73 + 1752692 * q^74 - 15817594 * q^76 + 7685750 * q^77 - 9007608 * q^79 + 13641024 * q^80 - 6329026 * q^82 + 21779128 * q^83 - 13177392 * q^85 - 79507 * q^86 - 13214750 * q^88 + 1895364 * q^89 - 16439838 * q^91 - 9614510 * q^92 + 3404276 * q^94 + 16861514 * q^95 - 20434472 * q^97 - 35731457 * q^98

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 $

$2$	16.8413	1.48858	0.744289	−	0.667858i	$-0.232789\pi$
$2$	16.8413	1.48858	0.744289	+	0.667858i	$0.232789\pi$
$3$	0	0
$3$	0	0
$4$	155.630	1.21586
$5$	309.730	1.10812	0.554061	−	0.832476i	$-0.313077\pi$
$5$	309.730	1.10812	0.554061	+	0.832476i	$0.313077\pi$
$6$	0	0
$7$	−729.159	−0.803487	−0.401744	−	0.915752i	$-0.631596\pi$
$7$	−729.159	−0.803487	−0.401744	+	0.915752i	$0.631596\pi$
$8$	465.334	0.321329
$9$	0	0
$10$	5216.26	1.64953
$11$	−2838.34	−0.642969	−0.321485	−	0.946915i	$-0.604182\pi$
$11$	−2838.34	−0.642969	−0.321485	+	0.946915i	$0.604182\pi$
$12$	0	0
$13$	2230.95	0.281636	0.140818	−	0.990036i	$-0.455027\pi$
$13$	2230.95	0.281636	0.140818	+	0.990036i	$0.455027\pi$
$14$	−12280.0	−1.19605
$15$	0	0
$16$	−12083.9	−0.737540
$17$	−4513.16	−0.222797	−0.111399	−	0.993776i	$-0.535533\pi$
$17$	−4513.16	−0.222797	−0.111399	+	0.993776i	$0.535533\pi$
$18$	0	0
$19$	1100.91	0.0368225	0.0184113	−	0.999830i	$-0.494139\pi$
$19$	1100.91	0.0368225	0.0184113	+	0.999830i	$0.494139\pi$
$20$	48203.4	1.34733
$21$	0	0
$22$	−47801.4	−0.957109
$23$	−24101.1	−0.413037	−0.206518	−	0.978443i	$-0.566213\pi$
$23$	−24101.1	−0.413037	−0.206518	+	0.978443i	$0.566213\pi$
$24$	0	0
$25$	17807.5	0.227937
$26$	37572.1	0.419236
$27$	0	0
$28$	−113479.	−0.976930
$29$	−174466.	−1.32837	−0.664183	−	0.747570i	$-0.731220\pi$
$29$	−174466.	−1.32837	−0.664183	+	0.747570i	$0.731220\pi$
$30$	0	0
$31$	−30188.2	−0.182000	−0.0910000	−	0.995851i	$-0.529006\pi$
$31$	−30188.2	−0.182000	−0.0910000	+	0.995851i	$0.529006\pi$
$32$	−263071.	−1.41921
$33$	0	0
$34$	−76007.6	−0.331651
$35$	−225842.	−0.890363
$36$	0	0
$37$	175314.	0.568996	0.284498	−	0.958677i	$-0.408173\pi$
$37$	175314.	0.568996	0.284498	+	0.958677i	$0.408173\pi$
$38$	18540.8	0.0548132
$39$	0	0
$40$	144128.	0.356072
$41$	−206811.	−0.468630	−0.234315	−	0.972161i	$-0.575285\pi$
$41$	−206811.	−0.468630	−0.234315	+	0.972161i	$0.575285\pi$
$42$	0	0
$43$	−79507.0	−0.152499
$43$	−79507.0	−0.152499
$44$	−441732.	−0.781762
$45$	0	0
$46$	−405894.	−0.614837
$47$	393333.	0.552608	0.276304	−	0.961070i	$-0.410890\pi$
$47$	393333.	0.552608	0.276304	+	0.961070i	$0.410890\pi$
$48$	0	0
$49$	−291871.	−0.354408
$50$	299903.	0.339301
$51$	0	0
$52$	347203.	0.342430
$53$	−123642.	−0.114077	−0.0570387	−	0.998372i	$-0.518166\pi$
$53$	−123642.	−0.114077	−0.0570387	+	0.998372i	$0.518166\pi$
$54$	0	0
$55$	−879119.	−0.712489
$56$	−339302.	−0.258184
$57$	0	0
$58$	−2.93824e6	−1.97737
$59$	−341123.	−0.216237	−0.108118	−	0.994138i	$-0.534483\pi$
$59$	−341123.	−0.216237	−0.108118	+	0.994138i	$0.534483\pi$
$60$	0	0
$61$	−2.31627e6	−1.30657	−0.653287	−	0.757110i	$-0.726610\pi$
$61$	−2.31627e6	−1.30657	−0.653287	+	0.757110i	$0.726610\pi$
$62$	−508410.	−0.270921
$63$	0	0
$64$	−2.88373e6	−1.37507
$65$	690991.	0.312087
$66$	0	0
$67$	−1.43482e6	−0.582821	−0.291411	−	0.956598i	$-0.594125\pi$
$67$	−1.43482e6	−0.582821	−0.291411	+	0.956598i	$0.594125\pi$
$68$	−702385.	−0.270891
$69$	0	0
$70$	−3.80348e6	−1.32537
$71$	369372.	0.122479	0.0612393	−	0.998123i	$-0.480495\pi$
$71$	369372.	0.122479	0.0612393	+	0.998123i	$0.480495\pi$
$72$	0	0
$73$	−2.24497e6	−0.675431	−0.337715	−	0.941248i	$-0.609654\pi$
$73$	−2.24497e6	−0.675431	−0.337715	+	0.941248i	$0.609654\pi$
$74$	2.95251e6	0.846995
$75$	0	0
$76$	171335.	0.0447711
$77$	2.06960e6	0.516617
$78$	0	0
$79$	−4.59668e6	−1.04894	−0.524468	−	0.851430i	$-0.675736\pi$
$79$	−4.59668e6	−1.04894	−0.524468	+	0.851430i	$0.675736\pi$
$80$	−3.74273e6	−0.817285
$81$	0	0
$82$	−3.48297e6	−0.697592
$83$	5.41678e6	1.03984	0.519921	−	0.854214i	$-0.325961\pi$
$83$	5.41678e6	1.03984	0.519921	+	0.854214i	$0.325961\pi$
$84$	0	0
$85$	−1.39786e6	−0.246886
$86$	−1.33900e6	−0.227006
$87$	0	0
$88$	−1.32078e6	−0.206604
$89$	253544.	0.0381231	0.0190615	−	0.999818i	$-0.493932\pi$
$89$	253544.	0.0381231	0.0190615	+	0.999818i	$0.493932\pi$
$90$	0	0
$91$	−1.62671e6	−0.226291
$92$	−3.75086e6	−0.502196
$93$	0	0
$94$	6.62425e6	0.822600
$95$	340984.	0.0408039
$96$	0	0
$97$	1.14080e7	1.26914	0.634571	−	0.772865i	$-0.281177\pi$
$97$	1.14080e7	1.26914	0.634571	+	0.772865i	$0.281177\pi$
$98$	−4.91549e6	−0.527564
$99$	0	0
$100$	2.77140e6	0.277140
$101$	1.23368e7	1.19146	0.595729	−	0.803186i	$-0.296863\pi$
$101$	1.23368e7	1.19146	0.595729	+	0.803186i	$0.296863\pi$
$102$	0	0
$103$	−2.12203e6	−0.191347	−0.0956735	−	0.995413i	$-0.530500\pi$
$103$	−2.12203e6	−0.191347	−0.0956735	+	0.995413i	$0.530500\pi$
$104$	1.03813e6	0.0904976
$105$	0	0
$106$	−2.08229e6	−0.169813
$107$	1.74818e7	1.37957	0.689786	−	0.724014i	$-0.257705\pi$
$107$	1.74818e7	1.37957	0.689786	+	0.724014i	$0.257705\pi$
$108$	0	0
$109$	−1.60682e7	−1.18843	−0.594217	−	0.804305i	$-0.702538\pi$
$109$	−1.60682e7	−1.18843	−0.594217	+	0.804305i	$0.702538\pi$
$110$	−1.48055e7	−1.06059
$111$	0	0
$112$	8.81105e6	0.592604
$113$	3.60370e6	0.234949	0.117475	−	0.993076i	$-0.462520\pi$
$113$	3.60370e6	0.234949	0.117475	+	0.993076i	$0.462520\pi$
$114$	0	0
$115$	−7.46482e6	−0.457695
$116$	−2.71522e7	−1.61511
$117$	0	0
$118$	−5.74497e6	−0.321885
$119$	3.29081e6	0.179015
$120$	0	0
$121$	−1.14310e7	−0.586591
$122$	−3.90090e7	−1.94494
$123$	0	0
$124$	−4.69821e6	−0.221287
$125$	−1.86821e7	−0.855541
$126$	0	0
$127$	−4.20986e6	−0.182371	−0.0911853	−	0.995834i	$-0.529066\pi$
$127$	−4.20986e6	−0.182371	−0.0911853	+	0.995834i	$0.529066\pi$
$128$	−1.48928e7	−0.627685
$129$	0	0
$130$	1.16372e7	0.464565
$131$	653698.	0.0254055	0.0127027	−	0.999919i	$-0.495956\pi$
$131$	653698.	0.0254055	0.0127027	+	0.999919i	$0.495956\pi$
$132$	0	0
$133$	−802737.	−0.0295864
$134$	−2.41643e7	−0.867575
$135$	0	0
$136$	−2.10012e6	−0.0715911
$137$	−1.85216e7	−0.615400	−0.307700	−	0.951483i	$-0.599559\pi$
$137$	−1.85216e7	−0.615400	−0.307700	+	0.951483i	$0.599559\pi$
$138$	0	0
$139$	2.06114e7	0.650963	0.325481	−	0.945548i	$-0.394474\pi$
$139$	2.06114e7	0.650963	0.325481	+	0.945548i	$0.394474\pi$
$140$	−3.51479e7	−1.08256
$141$	0	0
$142$	6.22072e6	0.182319
$143$	−6.33219e6	−0.181083
$144$	0	0
$145$	−5.40373e7	−1.47199
$146$	−3.78083e7	−1.00543
$147$	0	0
$148$	2.72841e7	0.691821
$149$	−1.29742e6	−0.0321313	−0.0160657	−	0.999871i	$-0.505114\pi$
$149$	−1.29742e6	−0.0321313	−0.0160657	+	0.999871i	$0.505114\pi$
$150$	0	0
$151$	−4.95267e7	−1.17063	−0.585316	−	0.810805i	$-0.699029\pi$
$151$	−4.95267e7	−1.17063	−0.585316	+	0.810805i	$0.699029\pi$
$152$	512290.	0.0118321
$153$	0	0
$154$	3.48548e7	0.769025
$155$	−9.35019e6	−0.201678
$156$	0	0
$157$	2.22814e6	0.0459509	0.0229755	−	0.999736i	$-0.492686\pi$
$157$	2.22814e6	0.0459509	0.0229755	+	0.999736i	$0.492686\pi$
$158$	−7.74142e7	−1.56142
$159$	0	0
$160$	−8.14809e7	−1.57266
$161$	1.75735e7	0.331870
$162$	0	0
$163$	−5.20514e7	−0.941403	−0.470702	−	0.882292i	$-0.655999\pi$
$163$	−5.20514e7	−0.941403	−0.470702	+	0.882292i	$0.655999\pi$
$164$	−3.21861e7	−0.569790
$165$	0	0
$166$	9.12257e7	1.54789
$167$	−3.54782e6	−0.0589459	−0.0294730	−	0.999566i	$-0.509383\pi$
$167$	−3.54782e6	−0.0589459	−0.0294730	+	0.999566i	$0.509383\pi$
$168$	0	0
$169$	−5.77714e7	−0.920681
$170$	−2.35418e7	−0.367510
$171$	0	0
$172$	−1.23737e7	−0.185417
$173$	6.21175e7	0.912121	0.456061	−	0.889949i	$-0.349260\pi$
$173$	6.21175e7	0.912121	0.456061	+	0.889949i	$0.349260\pi$
$174$	0	0
$175$	−1.29845e7	−0.183144
$176$	3.42981e7	0.474216
$177$	0	0
$178$	4.27002e6	0.0567492
$179$	−7.15362e7	−0.932267	−0.466134	−	0.884714i	$-0.654353\pi$
$179$	−7.15362e7	−0.932267	−0.466134	+	0.884714i	$0.654353\pi$
$180$	0	0
$181$	−9.85015e7	−1.23472	−0.617359	−	0.786681i	$-0.711798\pi$
$181$	−9.85015e7	−1.23472	−0.617359	+	0.786681i	$0.711798\pi$
$182$	−2.73960e7	−0.336851
$183$	0	0
$184$	−1.12150e7	−0.132721
$185$	5.42998e7	0.630518
$186$	0	0
$187$	1.28099e7	0.143252
$188$	6.12145e7	0.671896
$189$	0	0
$190$	5.74263e6	0.0607397
$191$	−7.13441e7	−0.740868	−0.370434	−	0.928859i	$-0.620791\pi$
$191$	−7.13441e7	−0.740868	−0.370434	+	0.928859i	$0.620791\pi$
$192$	0	0
$193$	9.18006e7	0.919168	0.459584	−	0.888134i	$-0.347998\pi$
$193$	9.18006e7	0.919168	0.459584	+	0.888134i	$0.347998\pi$
$194$	1.92127e8	1.88922
$195$	0	0
$196$	−4.54240e7	−0.430912
$197$	−1.10625e8	−1.03091	−0.515455	−	0.856916i	$-0.672377\pi$
$197$	−1.10625e8	−1.03091	−0.515455	+	0.856916i	$0.672377\pi$
$198$	0	0
$199$	1.36445e8	1.22736	0.613679	−	0.789556i	$-0.289689\pi$
$199$	1.36445e8	1.22736	0.613679	+	0.789556i	$0.289689\pi$
$200$	8.28645e6	0.0732426
$201$	0	0
$202$	2.07769e8	1.77358
$203$	1.27213e8	1.06732
$204$	0	0
$205$	−6.40555e7	−0.519299
$206$	−3.57378e7	−0.284835
$207$	0	0
$208$	−2.69584e7	−0.207718
$209$	−3.12475e6	−0.0236757
$210$	0	0
$211$	3.65532e7	0.267878	0.133939	−	0.990990i	$-0.457237\pi$
$211$	3.65532e7	0.267878	0.133939	+	0.990990i	$0.457237\pi$
$212$	−1.92424e7	−0.138703
$213$	0	0
$214$	2.94418e8	2.05360
$215$	−2.46257e7	−0.168987
$216$	0	0
$217$	2.20120e7	0.146235
$218$	−2.70610e8	−1.76908
$219$	0	0
$220$	−1.36818e8	−0.866289
$221$	−1.00686e7	−0.0627475
$222$	0	0
$223$	1.19926e8	0.724181	0.362090	−	0.932143i	$-0.382063\pi$
$223$	1.19926e8	0.724181	0.362090	+	0.932143i	$0.382063\pi$
$224$	1.91820e8	1.14032
$225$	0	0
$226$	6.06911e7	0.349740
$227$	−1.46239e8	−0.829801	−0.414900	−	0.909867i	$-0.636184\pi$
$227$	−1.46239e8	−0.829801	−0.414900	+	0.909867i	$0.636184\pi$
$228$	0	0
$229$	−4.46554e7	−0.245725	−0.122863	−	0.992424i	$-0.539207\pi$
$229$	−4.46554e7	−0.245725	−0.122863	+	0.992424i	$0.539207\pi$
$230$	−1.25717e8	−0.681315
$231$	0	0
$232$	−8.11849e7	−0.426842
$233$	−1.24363e8	−0.644089	−0.322045	−	0.946724i	$-0.604370\pi$
$233$	−1.24363e8	−0.644089	−0.322045	+	0.946724i	$0.604370\pi$
$234$	0	0
$235$	1.21827e8	0.612358
$236$	−5.30892e7	−0.262914
$237$	0	0
$238$	5.54216e7	0.266477
$239$	1.64150e8	0.777763	0.388881	−	0.921288i	$-0.372862\pi$
$239$	1.64150e8	0.777763	0.388881	+	0.921288i	$0.372862\pi$
$240$	0	0
$241$	1.62362e8	0.747177	0.373589	−	0.927594i	$-0.378127\pi$
$241$	1.62362e8	0.747177	0.373589	+	0.927594i	$0.378127\pi$
$242$	−1.92513e8	−0.873186
$243$	0	0
$244$	−3.60482e8	−1.58861
$245$	−9.04010e7	−0.392728
$246$	0	0
$247$	2.45607e6	0.0103705
$248$	−1.40476e7	−0.0584819
$249$	0	0
$250$	−3.14632e8	−1.27354
$251$	9.66679e7	0.385855	0.192928	−	0.981213i	$-0.438202\pi$
$251$	9.66679e7	0.385855	0.192928	+	0.981213i	$0.438202\pi$
$252$	0	0
$253$	6.84070e7	0.265570
$254$	−7.08997e7	−0.271473
$255$	0	0
$256$	1.18303e8	0.440713
$257$	2.18566e8	0.803187	0.401594	−	0.915818i	$-0.368456\pi$
$257$	2.18566e8	0.803187	0.401594	+	0.915818i	$0.368456\pi$
$258$	0	0
$259$	−1.27831e8	−0.457181
$260$	1.07539e8	0.379455
$261$	0	0
$262$	1.10091e7	0.0378180
$263$	3.86760e8	1.31098	0.655490	−	0.755204i	$-0.272462\pi$
$263$	3.86760e8	1.31098	0.655490	+	0.755204i	$0.272462\pi$
$264$	0	0
$265$	−3.82956e7	−0.126412
$266$	−1.35192e7	−0.0440417
$267$	0	0
$268$	−2.23302e8	−0.708631
$269$	5.81229e8	1.82060	0.910299	−	0.413952i	$-0.135852\pi$
$269$	5.81229e8	1.82060	0.910299	+	0.413952i	$0.135852\pi$
$270$	0	0
$271$	1.79603e8	0.548177	0.274088	−	0.961705i	$-0.411624\pi$
$271$	1.79603e8	0.548177	0.274088	+	0.961705i	$0.411624\pi$
$272$	5.45364e7	0.164322
$273$	0	0
$274$	−3.11929e8	−0.916070
$275$	−5.05439e7	−0.146556
$276$	0	0
$277$	−3.14331e8	−0.888604	−0.444302	−	0.895877i	$-0.646548\pi$
$277$	−3.14331e8	−0.888604	−0.444302	+	0.895877i	$0.646548\pi$
$278$	3.47124e8	0.969009
$279$	0	0
$280$	−1.05092e8	−0.286099
$281$	1.64974e7	0.0443552	0.0221776	−	0.999754i	$-0.492940\pi$
$281$	1.64974e7	0.0443552	0.0221776	+	0.999754i	$0.492940\pi$
$282$	0	0
$283$	2.19485e8	0.575643	0.287821	−	0.957684i	$-0.407069\pi$
$283$	2.19485e8	0.575643	0.287821	+	0.957684i	$0.407069\pi$
$284$	5.74856e7	0.148917
$285$	0	0
$286$	−1.06642e8	−0.269556
$287$	1.50798e8	0.376538
$288$	0	0
$289$	−3.89970e8	−0.950361
$290$	−9.10060e8	−2.19117
$291$	0	0
$292$	−3.49386e8	−0.821231
$293$	1.26932e8	0.294806	0.147403	−	0.989077i	$-0.452909\pi$
$293$	1.26932e8	0.294806	0.147403	+	0.989077i	$0.452909\pi$
$294$	0	0
$295$	−1.05656e8	−0.239617
$296$	8.15793e7	0.182835
$297$	0	0
$298$	−2.18503e7	−0.0478300
$299$	−5.37682e7	−0.116326
$300$	0	0
$301$	5.79732e7	0.122531
$302$	−8.34096e8	−1.74258
$303$	0	0
$304$	−1.33032e7	−0.0271581
$305$	−7.17417e8	−1.44784
$306$	0	0
$307$	7.98217e8	1.57448	0.787239	−	0.616647i	$-0.211510\pi$
$307$	7.98217e8	1.57448	0.787239	+	0.616647i	$0.211510\pi$
$308$	3.22093e8	0.628136
$309$	0	0
$310$	−1.57470e8	−0.300214
$311$	−6.92412e7	−0.130528	−0.0652639	−	0.997868i	$-0.520789\pi$
$311$	−6.92412e7	−0.130528	−0.0652639	+	0.997868i	$0.520789\pi$
$312$	0	0
$313$	−3.20822e8	−0.591370	−0.295685	−	0.955285i	$-0.595548\pi$
$313$	−3.20822e8	−0.591370	−0.295685	+	0.955285i	$0.595548\pi$
$314$	3.75249e7	0.0684015
$315$	0	0
$316$	−7.15383e8	−1.27536
$317$	7.47487e8	1.31794	0.658971	−	0.752168i	$-0.270992\pi$
$317$	7.47487e8	1.31794	0.658971	+	0.752168i	$0.270992\pi$
$318$	0	0
$319$	4.95194e8	0.854098
$320$	−8.93178e8	−1.52375
$321$	0	0
$322$	2.95961e8	0.494014
$323$	−4.96857e6	−0.00820395
$324$	0	0
$325$	3.97277e7	0.0641950
$326$	−8.76615e8	−1.40135
$327$	0	0
$328$	−9.62361e7	−0.150584
$329$	−2.86802e8	−0.444014
$330$	0	0
$331$	2.50730e8	0.380021	0.190011	−	0.981782i	$-0.439148\pi$
$331$	2.50730e8	0.380021	0.190011	+	0.981782i	$0.439148\pi$
$332$	8.43015e8	1.26431
$333$	0	0
$334$	−5.97500e7	−0.0877456
$335$	−4.44406e8	−0.645838
$336$	0	0
$337$	7.43325e8	1.05797	0.528986	−	0.848631i	$-0.322572\pi$
$337$	7.43325e8	1.05797	0.528986	+	0.848631i	$0.322572\pi$
$338$	−9.72947e8	−1.37051
$339$	0	0
$340$	−2.17550e8	−0.300180
$341$	8.56845e7	0.117020
$342$	0	0
$343$	8.13314e8	1.08825
$344$	−3.69973e7	−0.0490022
$345$	0	0
$346$	1.04614e9	1.35776
$347$	−1.38858e8	−0.178409	−0.0892046	−	0.996013i	$-0.528433\pi$
$347$	−1.38858e8	−0.178409	−0.0892046	+	0.996013i	$0.528433\pi$
$348$	0	0
$349$	−1.20522e9	−1.51766	−0.758832	−	0.651286i	$-0.774230\pi$
$349$	−1.20522e9	−1.51766	−0.758832	+	0.651286i	$0.774230\pi$
$350$	−2.18677e8	−0.272624
$351$	0	0
$352$	7.46685e8	0.912511
$353$	6.60929e8	0.799729	0.399865	−	0.916574i	$-0.369057\pi$
$353$	6.60929e8	0.799729	0.399865	+	0.916574i	$0.369057\pi$
$354$	0	0
$355$	1.14406e8	0.135721
$356$	3.94592e7	0.0463525
$357$	0	0
$358$	−1.20477e9	−1.38775
$359$	−9.22665e7	−0.105248	−0.0526240	−	0.998614i	$-0.516758\pi$
$359$	−9.22665e7	−0.105248	−0.0526240	+	0.998614i	$0.516758\pi$
$360$	0	0
$361$	−8.92660e8	−0.998644
$362$	−1.65890e9	−1.83797
$363$	0	0
$364$	−2.53166e8	−0.275138
$365$	−6.95334e8	−0.748460
$366$	0	0
$367$	3.85551e8	0.407146	0.203573	−	0.979060i	$-0.434745\pi$
$367$	3.85551e8	0.407146	0.203573	+	0.979060i	$0.434745\pi$
$368$	2.91234e8	0.304631
$369$	0	0
$370$	9.14481e8	0.938575
$371$	9.01545e7	0.0916598
$372$	0	0
$373$	−1.32246e9	−1.31948	−0.659740	−	0.751494i	$-0.729333\pi$
$373$	−1.32246e9	−1.31948	−0.659740	+	0.751494i	$0.729333\pi$
$374$	2.15735e8	0.213241
$375$	0	0
$376$	1.83031e8	0.177569
$377$	−3.89224e8	−0.374115
$378$	0	0
$379$	−1.21768e9	−1.14894	−0.574470	−	0.818526i	$-0.694792\pi$
$379$	−1.21768e9	−1.14894	−0.574470	+	0.818526i	$0.694792\pi$
$380$	5.30675e7	0.0496119
$381$	0	0
$382$	−1.20153e9	−1.10284
$383$	1.61232e9	1.46641	0.733204	−	0.680009i	$-0.238024\pi$
$383$	1.61232e9	1.46641	0.733204	+	0.680009i	$0.238024\pi$
$384$	0	0
$385$	6.41017e8	0.572476
$386$	1.54604e9	1.36825
$387$	0	0
$388$	1.77544e9	1.54310
$389$	−2.98305e8	−0.256943	−0.128472	−	0.991713i	$-0.541007\pi$
$389$	−2.98305e8	−0.256943	−0.128472	+	0.991713i	$0.541007\pi$
$390$	0	0
$391$	1.08772e8	0.0920233
$392$	−1.35817e8	−0.113882
$393$	0	0
$394$	−1.86307e9	−1.53459
$395$	−1.42373e9	−1.16235
$396$	0	0
$397$	1.02892e9	0.825303	0.412651	−	0.910889i	$-0.364603\pi$
$397$	1.02892e9	0.825303	0.412651	+	0.910889i	$0.364603\pi$
$398$	2.29791e9	1.82702
$399$	0	0
$400$	−2.15184e8	−0.168112
$401$	−1.29394e9	−1.00210	−0.501048	−	0.865420i	$-0.667052\pi$
$401$	−1.29394e9	−1.00210	−0.501048	+	0.865420i	$0.667052\pi$
$402$	0	0
$403$	−6.73483e7	−0.0512577
$404$	1.91999e9	1.44865
$405$	0	0
$406$	2.14244e9	1.58880
$407$	−4.97599e8	−0.365847
$408$	0	0
$409$	1.90377e9	1.37589	0.687943	−	0.725764i	$-0.258514\pi$
$409$	1.90377e9	1.37589	0.687943	+	0.725764i	$0.258514\pi$
$410$	−1.07878e9	−0.773017
$411$	0	0
$412$	−3.30253e8	−0.232652
$413$	2.48733e8	0.173743
$414$	0	0
$415$	1.67774e9	1.15227
$416$	−5.86897e8	−0.399701
$417$	0	0
$418$	−5.26250e7	−0.0352432
$419$	9.60415e8	0.637837	0.318919	−	0.947782i	$-0.396680\pi$
$419$	9.60415e8	0.637837	0.318919	+	0.947782i	$0.396680\pi$
$420$	0	0
$421$	−1.02619e9	−0.670258	−0.335129	−	0.942172i	$-0.608780\pi$
$421$	−1.02619e9	−0.670258	−0.335129	+	0.942172i	$0.608780\pi$
$422$	6.15605e8	0.398757
$423$	0	0
$424$	−5.75347e7	−0.0366564
$425$	−8.03683e7	−0.0507836
$426$	0	0
$427$	1.68893e9	1.04982
$428$	2.72071e9	1.67737
$429$	0	0
$430$	−4.14729e8	−0.251551
$431$	9.95323e8	0.598816	0.299408	−	0.954125i	$-0.403211\pi$
$431$	9.95323e8	0.598816	0.299408	+	0.954125i	$0.403211\pi$
$432$	0	0
$433$	1.86480e9	1.10388	0.551942	−	0.833882i	$-0.313887\pi$
$433$	1.86480e9	1.10388	0.551942	+	0.833882i	$0.313887\pi$
$434$	3.70711e8	0.217682
$435$	0	0
$436$	−2.50070e9	−1.44497
$437$	−2.65330e7	−0.0152091
$438$	0	0
$439$	−3.07487e9	−1.73461	−0.867304	−	0.497778i	$-0.834149\pi$
$439$	−3.07487e9	−1.73461	−0.867304	+	0.497778i	$0.834149\pi$
$440$	−4.09084e8	−0.228943
$441$	0	0
$442$	−1.69569e8	−0.0934046
$443$	3.08044e9	1.68345	0.841723	−	0.539910i	$-0.181542\pi$
$443$	3.08044e9	1.68345	0.841723	+	0.539910i	$0.181542\pi$
$444$	0	0
$445$	7.85301e7	0.0422451
$446$	2.01972e9	1.07800
$447$	0	0
$448$	2.10270e9	1.10485
$449$	1.06308e9	0.554250	0.277125	−	0.960834i	$-0.410618\pi$
$449$	1.06308e9	0.554250	0.277125	+	0.960834i	$0.410618\pi$
$450$	0	0
$451$	5.87000e8	0.301314
$452$	5.60845e8	0.285666
$453$	0	0
$454$	−2.46287e9	−1.23522
$455$	−5.03842e8	−0.250758
$456$	0	0
$457$	−1.10208e9	−0.540138	−0.270069	−	0.962841i	$-0.587046\pi$
$457$	−1.10208e9	−0.540138	−0.270069	+	0.962841i	$0.587046\pi$
$458$	−7.52056e8	−0.365781
$459$	0	0
$460$	−1.16175e9	−0.556495
$461$	−2.33182e9	−1.10852	−0.554258	−	0.832345i	$-0.686998\pi$
$461$	−2.33182e9	−1.10852	−0.554258	+	0.832345i	$0.686998\pi$
$462$	0	0
$463$	−1.26190e9	−0.590871	−0.295436	−	0.955363i	$-0.595465\pi$
$463$	−1.26190e9	−0.590871	−0.295436	+	0.955363i	$0.595465\pi$
$464$	2.10822e9	0.979723
$465$	0	0
$466$	−2.09444e9	−0.958777
$467$	1.69316e9	0.769288	0.384644	−	0.923065i	$-0.374324\pi$
$467$	1.69316e9	0.769288	0.384644	+	0.923065i	$0.374324\pi$
$468$	0	0
$469$	1.04621e9	0.468289
$470$	2.05173e9	0.911542
$471$	0	0
$472$	−1.58736e8	−0.0694830
$473$	2.25668e8	0.0980519
$474$	0	0
$475$	1.96045e7	0.00839320
$476$	5.12150e8	0.217657
$477$	0	0
$478$	2.76450e9	1.15776
$479$	4.64777e9	1.93228	0.966141	−	0.258016i	$-0.0830688\pi$
$479$	4.64777e9	1.93228	0.966141	+	0.258016i	$0.0830688\pi$
$480$	0	0
$481$	3.91115e8	0.160250
$482$	2.73439e9	1.11223
$483$	0	0
$484$	−1.77901e9	−0.713214
$485$	3.53341e9	1.40637
$486$	0	0
$487$	−1.39826e9	−0.548574	−0.274287	−	0.961648i	$-0.588442\pi$
$487$	−1.39826e9	−0.548574	−0.274287	+	0.961648i	$0.588442\pi$
$488$	−1.07784e9	−0.419840
$489$	0	0
$490$	−1.52247e9	−0.584606
$491$	1.68533e9	0.642540	0.321270	−	0.946988i	$-0.395890\pi$
$491$	1.68533e9	0.642540	0.321270	+	0.946988i	$0.395890\pi$
$492$	0	0
$493$	7.87392e8	0.295956
$494$	4.13634e7	0.0154373
$495$	0	0
$496$	3.64790e8	0.134232
$497$	−2.69331e8	−0.0984099
$498$	0	0
$499$	2.27490e9	0.819617	0.409808	−	0.912172i	$-0.365596\pi$
$499$	2.27490e9	0.819617	0.409808	+	0.912172i	$0.365596\pi$
$500$	−2.90751e9	−1.04022
$501$	0	0
$502$	1.62802e9	0.574375
$503$	9.28864e8	0.325435	0.162717	−	0.986673i	$-0.447974\pi$
$503$	9.28864e8	0.325435	0.162717	+	0.986673i	$0.447974\pi$
$504$	0	0
$505$	3.82108e9	1.32028
$506$	1.15207e9	0.395321
$507$	0	0
$508$	−6.55183e8	−0.221738
$509$	2.77307e9	0.932072	0.466036	−	0.884766i	$-0.345682\pi$
$509$	2.77307e9	0.932072	0.466036	+	0.884766i	$0.345682\pi$
$510$	0	0
$511$	1.63694e9	0.542700
$512$	3.89866e9	1.28372
$513$	0	0
$514$	3.68094e9	1.19561
$515$	−6.57256e8	−0.212036
$516$	0	0
$517$	−1.11641e9	−0.355310
$518$	−2.15285e9	−0.680549
$519$	0	0
$520$	3.21541e8	0.100282
$521$	−2.24360e9	−0.695045	−0.347522	−	0.937672i	$-0.612977\pi$
$521$	−2.24360e9	−0.695045	−0.347522	+	0.937672i	$0.612977\pi$
$522$	0	0
$523$	−1.67822e9	−0.512971	−0.256486	−	0.966548i	$-0.582565\pi$
$523$	−1.67822e9	−0.512971	−0.256486	+	0.966548i	$0.582565\pi$
$524$	1.01735e8	0.0308896
$525$	0	0
$526$	6.51355e9	1.95150
$527$	1.36244e8	0.0405491
$528$	0	0
$529$	−2.82396e9	−0.829401
$530$	−6.44948e8	−0.188174
$531$	0	0
$532$	−1.24930e8	−0.0359730
$533$	−4.61384e8	−0.131983
$534$	0	0
$535$	5.41465e9	1.52873
$536$	−6.67670e8	−0.187277
$537$	0	0
$538$	9.78866e9	2.71010
$539$	8.28428e8	0.227874
$540$	0	0
$541$	−2.19865e9	−0.596989	−0.298495	−	0.954411i	$-0.596484\pi$
$541$	−2.19865e9	−0.596989	−0.298495	+	0.954411i	$0.596484\pi$
$542$	3.02475e9	0.816003
$543$	0	0
$544$	1.18728e9	0.316197
$545$	−4.97681e9	−1.31693
$546$	0	0
$547$	2.48961e9	0.650392	0.325196	−	0.945647i	$-0.394570\pi$
$547$	2.48961e9	0.650392	0.325196	+	0.945647i	$0.394570\pi$
$548$	−2.88253e9	−0.748242
$549$	0	0
$550$	−8.51226e8	−0.218160
$551$	−1.92071e8	−0.0489138
$552$	0	0
$553$	3.35171e9	0.842807
$554$	−5.29376e9	−1.32276
$555$	0	0
$556$	3.20777e9	0.791482
$557$	−6.66391e9	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$557$	−6.66391e9	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$558$	0	0
$559$	−1.77376e8	−0.0429490
$560$	2.72904e9	0.656678
$561$	0	0
$562$	2.77839e8	0.0660261
$563$	−5.07158e9	−1.19774	−0.598872	−	0.800845i	$-0.704384\pi$
$563$	−5.07158e9	−1.19774	−0.598872	+	0.800845i	$0.704384\pi$
$564$	0	0
$565$	1.11617e9	0.260353
$566$	3.69643e9	0.856889
$567$	0	0
$568$	1.71881e8	0.0393559
$569$	−4.00987e9	−0.912510	−0.456255	−	0.889849i	$-0.650809\pi$
$569$	−4.00987e9	−0.912510	−0.456255	+	0.889849i	$0.650809\pi$
$570$	0	0
$571$	4.51386e8	0.101466	0.0507331	−	0.998712i	$-0.483844\pi$
$571$	4.51386e8	0.101466	0.0507331	+	0.998712i	$0.483844\pi$
$572$	−9.85481e8	−0.220172
$573$	0	0
$574$	2.53964e9	0.560506
$575$	−4.29181e8	−0.0941462
$576$	0	0
$577$	3.50937e8	0.0760525	0.0380262	−	0.999277i	$-0.487893\pi$
$577$	3.50937e8	0.0760525	0.0380262	+	0.999277i	$0.487893\pi$
$578$	−6.56762e9	−1.41469
$579$	0	0
$580$	−8.40985e9	−1.78974
$581$	−3.94969e9	−0.835500
$582$	0	0
$583$	3.50938e8	0.0733483
$584$	−1.04466e9	−0.217035
$585$	0	0
$586$	2.13771e9	0.438841
$587$	−9.76991e7	−0.0199369	−0.00996844	−	0.999950i	$-0.503173\pi$
$587$	−9.76991e7	−0.0199369	−0.00996844	+	0.999950i	$0.503173\pi$
$588$	0	0
$589$	−3.32345e7	−0.00670170
$590$	−1.77939e9	−0.356688
$591$	0	0
$592$	−2.11846e9	−0.419658
$593$	−3.00939e9	−0.592635	−0.296317	−	0.955090i	$-0.595759\pi$
$593$	−3.00939e9	−0.592635	−0.296317	+	0.955090i	$0.595759\pi$
$594$	0	0
$595$	1.01926e9	0.198370
$596$	−2.01918e8	−0.0390673
$597$	0	0
$598$	−9.05528e8	−0.173160
$599$	−3.08282e9	−0.586077	−0.293039	−	0.956101i	$-0.594666\pi$
$599$	−3.08282e9	−0.586077	−0.293039	+	0.956101i	$0.594666\pi$
$600$	0	0
$601$	1.38784e9	0.260783	0.130391	−	0.991463i	$-0.458377\pi$
$601$	1.38784e9	0.260783	0.130391	+	0.991463i	$0.458377\pi$
$602$	9.76346e8	0.182396
$603$	0	0
$604$	−7.70787e9	−1.42333
$605$	−3.54052e9	−0.650015
$606$	0	0
$607$	1.39258e8	0.0252733	0.0126366	−	0.999920i	$-0.495978\pi$
$607$	1.39258e8	0.0252733	0.0126366	+	0.999920i	$0.495978\pi$
$608$	−2.89617e8	−0.0522591
$609$	0	0
$610$	−1.20822e10	−2.15523
$611$	8.77504e8	0.155634
$612$	0	0
$613$	2.95089e9	0.517417	0.258709	−	0.965955i	$-0.416703\pi$
$613$	2.95089e9	0.517417	0.258709	+	0.965955i	$0.416703\pi$
$614$	1.34430e10	2.34373
$615$	0	0
$616$	9.63055e8	0.166004
$617$	−8.86339e9	−1.51915	−0.759577	−	0.650418i	$-0.774594\pi$
$617$	−8.86339e9	−1.51915	−0.759577	+	0.650418i	$0.774594\pi$
$618$	0	0
$619$	−4.47199e9	−0.757851	−0.378926	−	0.925427i	$-0.623706\pi$
$619$	−4.47199e9	−0.757851	−0.378926	+	0.925427i	$0.623706\pi$
$620$	−1.45517e9	−0.245213
$621$	0	0
$622$	−1.16611e9	−0.194301
$623$	−1.84874e8	−0.0306314
$624$	0	0
$625$	−7.17762e9	−1.17598
$626$	−5.40307e9	−0.880300
$627$	0	0
$628$	3.46767e8	0.0558701
$629$	−7.91218e8	−0.126771
$630$	0	0
$631$	−1.05699e10	−1.67481	−0.837406	−	0.546581i	$-0.815929\pi$
$631$	−1.05699e10	−1.67481	−0.837406	+	0.546581i	$0.815929\pi$
$632$	−2.13899e9	−0.337053
$633$	0	0
$634$	1.25887e10	1.96186
$635$	−1.30392e9	−0.202089
$636$	0	0
$637$	−6.51148e8	−0.0998140
$638$	8.33972e9	1.27139
$639$	0	0
$640$	−4.61274e9	−0.695552
$641$	3.54117e9	0.531060	0.265530	−	0.964103i	$-0.414453\pi$
$641$	3.54117e9	0.531060	0.265530	+	0.964103i	$0.414453\pi$
$642$	0	0
$643$	8.30426e9	1.23186	0.615931	−	0.787800i	$-0.288780\pi$
$643$	8.30426e9	1.23186	0.615931	+	0.787800i	$0.288780\pi$
$644$	2.73497e9	0.403508
$645$	0	0
$646$	−8.36774e7	−0.0122122
$647$	−1.28816e10	−1.86984	−0.934921	−	0.354856i	$-0.884530\pi$
$647$	−1.28816e10	−1.86984	−0.934921	+	0.354856i	$0.884530\pi$
$648$	0	0
$649$	9.68224e8	0.139033
$650$	6.69067e8	0.0955593
$651$	0	0
$652$	−8.10078e9	−1.14462
$653$	−1.91300e9	−0.268855	−0.134428	−	0.990923i	$-0.542920\pi$
$653$	−1.91300e9	−0.268855	−0.134428	+	0.990923i	$0.542920\pi$
$654$	0	0
$655$	2.02470e8	0.0281524
$656$	2.49907e9	0.345633
$657$	0	0
$658$	−4.83013e9	−0.660949
$659$	4.65956e9	0.634229	0.317114	−	0.948387i	$-0.397286\pi$
$659$	4.65956e9	0.634229	0.317114	+	0.948387i	$0.397286\pi$
$660$	0	0
$661$	−5.95034e9	−0.801376	−0.400688	−	0.916214i	$-0.631229\pi$
$661$	−5.95034e9	−0.801376	−0.400688	+	0.916214i	$0.631229\pi$
$662$	4.22262e9	0.565691
$663$	0	0
$664$	2.52061e9	0.334131
$665$	−2.48631e8	−0.0327854
$666$	0	0
$667$	4.20481e9	0.548664
$668$	−5.52149e8	−0.0716702
$669$	0	0
$670$	−7.48440e9	−0.961380
$671$	6.57435e9	0.840087
$672$	0	0
$673$	−5.91845e9	−0.748437	−0.374219	−	0.927341i	$-0.622089\pi$
$673$	−5.91845e9	−0.748437	−0.374219	+	0.927341i	$0.622089\pi$
$674$	1.25186e10	1.57487
$675$	0	0
$676$	−8.99099e9	−1.11942
$677$	−2.50997e9	−0.310890	−0.155445	−	0.987845i	$-0.549681\pi$
$677$	−2.50997e9	−0.310890	−0.155445	+	0.987845i	$0.549681\pi$
$678$	0	0
$679$	−8.31828e9	−1.01974
$680$	−6.50471e8	−0.0793317
$681$	0	0
$682$	1.44304e9	0.174194
$683$	−1.42786e10	−1.71479	−0.857396	−	0.514657i	$-0.827919\pi$
$683$	−1.42786e10	−1.71479	−0.857396	+	0.514657i	$0.827919\pi$
$684$	0	0
$685$	−5.73670e9	−0.681939
$686$	1.36973e10	1.61994
$687$	0	0
$688$	9.60751e8	0.112474
$689$	−2.75838e8	−0.0321283
$690$	0	0
$691$	5.80665e9	0.669502	0.334751	−	0.942307i	$-0.391348\pi$
$691$	5.80665e9	0.669502	0.334751	+	0.942307i	$0.391348\pi$
$692$	9.66737e9	1.10901
$693$	0	0
$694$	−2.33855e9	−0.265576
$695$	6.38397e9	0.721347
$696$	0	0
$697$	9.33370e8	0.104409
$698$	−2.02974e10	−2.25916
$699$	0	0
$700$	−2.02079e9	−0.222678
$701$	4.91452e9	0.538850	0.269425	−	0.963021i	$-0.413166\pi$
$701$	4.91452e9	0.538850	0.269425	+	0.963021i	$0.413166\pi$
$702$	0	0
$703$	1.93004e8	0.0209519
$704$	8.18501e9	0.884128
$705$	0	0
$706$	1.11309e10	1.19046
$707$	−8.99550e9	−0.957321
$708$	0	0
$709$	−1.52802e10	−1.61015	−0.805076	−	0.593171i	$-0.797876\pi$
$709$	−1.52802e10	−1.61015	−0.805076	+	0.593171i	$0.797876\pi$
$710$	1.92674e9	0.202032
$711$	0	0
$712$	1.17983e8	0.0122500
$713$	7.27568e8	0.0751727
$714$	0	0
$715$	−1.96127e9	−0.200662
$716$	−1.11332e10	−1.13351
$717$	0	0
$718$	−1.55389e9	−0.156670
$719$	−3.10746e9	−0.311784	−0.155892	−	0.987774i	$-0.549825\pi$
$719$	−3.10746e9	−0.311784	−0.155892	+	0.987774i	$0.549825\pi$
$720$	0	0
$721$	1.54730e9	0.153745
$722$	−1.50336e10	−1.48656
$723$	0	0
$724$	−1.53298e10	−1.50125
$725$	−3.10681e9	−0.302783
$726$	0	0
$727$	−1.33342e10	−1.28705	−0.643525	−	0.765425i	$-0.722529\pi$
$727$	−1.33342e10	−1.28705	−0.643525	+	0.765425i	$0.722529\pi$
$728$	−7.56965e8	−0.0727136
$729$	0	0
$730$	−1.17104e10	−1.11414
$731$	3.58828e8	0.0339762
$732$	0	0
$733$	1.76386e10	1.65425	0.827123	−	0.562021i	$-0.189976\pi$
$733$	1.76386e10	1.65425	0.827123	+	0.562021i	$0.189976\pi$
$734$	6.49319e9	0.606069
$735$	0	0
$736$	6.34029e9	0.586188
$737$	4.07251e9	0.374736
$738$	0	0
$739$	−7.25403e9	−0.661187	−0.330593	−	0.943773i	$-0.607249\pi$
$739$	−7.25403e9	−0.661187	−0.330593	+	0.943773i	$0.607249\pi$
$740$	8.45071e9	0.766623
$741$	0	0
$742$	1.51832e9	0.136443
$743$	1.76263e10	1.57652	0.788262	−	0.615340i	$-0.210981\pi$
$743$	1.76263e10	1.57652	0.788262	+	0.615340i	$0.210981\pi$
$744$	0	0
$745$	−4.01850e8	−0.0356055
$746$	−2.22720e10	−1.96415
$747$	0	0
$748$	1.99361e9	0.174174
$749$	−1.27470e10	−1.10847
$750$	0	0
$751$	−4.14335e9	−0.356954	−0.178477	−	0.983944i	$-0.557117\pi$
$751$	−4.14335e9	−0.356954	−0.178477	+	0.983944i	$0.557117\pi$
$752$	−4.75298e9	−0.407571
$753$	0	0
$754$	−6.55505e9	−0.556899
$755$	−1.53399e10	−1.29720
$756$	0	0
$757$	−2.75256e9	−0.230622	−0.115311	−	0.993329i	$-0.536786\pi$
$757$	−2.75256e9	−0.230622	−0.115311	+	0.993329i	$0.536786\pi$
$758$	−2.05074e10	−1.71029
$759$	0	0
$760$	1.58671e8	0.0131115
$761$	−2.07471e10	−1.70652	−0.853260	−	0.521485i	$-0.825378\pi$
$761$	−2.07471e10	−1.70652	−0.853260	+	0.521485i	$0.825378\pi$
$762$	0	0
$763$	1.17163e10	0.954891
$764$	−1.11033e10	−0.900794
$765$	0	0
$766$	2.71536e10	2.18286
$767$	−7.61028e8	−0.0608999
$768$	0	0
$769$	5.50318e9	0.436387	0.218193	−	0.975906i	$-0.429984\pi$
$769$	5.50318e9	0.436387	0.218193	+	0.975906i	$0.429984\pi$
$770$	1.07956e10	0.852174
$771$	0	0
$772$	1.42870e10	1.11758
$773$	−2.11458e10	−1.64663	−0.823316	−	0.567583i	$-0.807879\pi$
$773$	−2.11458e10	−1.64663	−0.823316	+	0.567583i	$0.807879\pi$
$774$	0	0
$775$	−5.37578e8	−0.0414845
$776$	5.30855e9	0.407812
$777$	0	0
$778$	−5.02385e9	−0.382480
$779$	−2.27680e8	−0.0172561
$780$	0	0
$781$	−1.04840e9	−0.0787499
$782$	1.83186e9	0.136984
$783$	0	0
$784$	3.52692e9	0.261391
$785$	6.90123e8	0.0509193
$786$	0	0
$787$	−1.18964e10	−0.869970	−0.434985	−	0.900438i	$-0.643246\pi$
$787$	−1.18964e10	−0.869970	−0.434985	+	0.900438i	$0.643246\pi$
$788$	−1.72166e10	−1.25345
$789$	0	0
$790$	−2.39775e10	−1.73025
$791$	−2.62767e9	−0.188779
$792$	0	0
$793$	−5.16747e9	−0.367978
$794$	1.73283e10	1.22853
$795$	0	0
$796$	2.12350e10	1.49230
$797$	−2.29290e9	−0.160428	−0.0802140	−	0.996778i	$-0.525560\pi$
$797$	−2.29290e9	−0.160428	−0.0802140	+	0.996778i	$0.525560\pi$
$798$	0	0
$799$	−1.77517e9	−0.123119
$800$	−4.68465e9	−0.323491
$801$	0	0
$802$	−2.17917e10	−1.49170
$803$	6.37199e9	0.434281
$804$	0	0
$805$	5.44303e9	0.367752
$806$	−1.13424e9	−0.0763010
$807$	0	0
$808$	5.74074e9	0.382850
$809$	1.40938e10	0.935857	0.467928	−	0.883766i	$-0.345000\pi$
$809$	1.40938e10	0.935857	0.467928	+	0.883766i	$0.345000\pi$
$810$	0	0
$811$	−9.79033e9	−0.644502	−0.322251	−	0.946654i	$-0.604440\pi$
$811$	−9.79033e9	−0.644502	−0.322251	+	0.946654i	$0.604440\pi$
$812$	1.97983e10	1.29772
$813$	0	0
$814$	−8.38024e9	−0.544592
$815$	−1.61219e10	−1.04319
$816$	0	0
$817$	−8.75299e7	−0.00561538
$818$	3.20620e10	2.04811
$819$	0	0
$820$	−9.96899e9	−0.631397
$821$	2.38646e10	1.50506	0.752528	−	0.658560i	$-0.228834\pi$
$821$	2.38646e10	1.50506	0.752528	+	0.658560i	$0.228834\pi$
$822$	0	0
$823$	−1.89600e9	−0.118560	−0.0592801	−	0.998241i	$-0.518881\pi$
$823$	−1.89600e9	−0.118560	−0.0592801	+	0.998241i	$0.518881\pi$
$824$	−9.87453e8	−0.0614853
$825$	0	0
$826$	4.18899e9	0.258630
$827$	−6.38221e9	−0.392375	−0.196188	−	0.980566i	$-0.562856\pi$
$827$	−6.38221e9	−0.392375	−0.196188	+	0.980566i	$0.562856\pi$
$828$	0	0
$829$	−1.64567e10	−1.00323	−0.501616	−	0.865090i	$-0.667261\pi$
$829$	−1.64567e10	−1.00323	−0.501616	+	0.865090i	$0.667261\pi$
$830$	2.82553e10	1.71525
$831$	0	0
$832$	−6.43345e9	−0.387269
$833$	1.31726e9	0.0789611
$834$	0	0
$835$	−1.09887e9	−0.0653194
$836$	−4.86307e8	−0.0287865
$837$	0	0
$838$	1.61747e10	0.949470
$839$	2.41700e10	1.41289	0.706447	−	0.707766i	$-0.250297\pi$
$839$	2.41700e10	1.41289	0.706447	+	0.707766i	$0.250297\pi$
$840$	0	0
$841$	1.31885e10	0.764555
$842$	−1.72825e10	−0.997731
$843$	0	0
$844$	5.68880e9	0.325703
$845$	−1.78935e10	−1.02023
$846$	0	0
$847$	8.33501e9	0.471318
$848$	1.49407e9	0.0841367
$849$	0	0
$850$	−1.35351e9	−0.0755953
$851$	−4.22524e9	−0.235016
$852$	0	0
$853$	1.63789e10	0.903575	0.451787	−	0.892126i	$-0.350787\pi$
$853$	1.63789e10	0.903575	0.451787	+	0.892126i	$0.350787\pi$
$854$	2.84438e10	1.56273
$855$	0	0
$856$	8.13489e9	0.443296
$857$	6.73056e9	0.365274	0.182637	−	0.983180i	$-0.441537\pi$
$857$	6.73056e9	0.365274	0.182637	+	0.983180i	$0.441537\pi$
$858$	0	0
$859$	2.21189e10	1.19066	0.595328	−	0.803483i	$-0.297022\pi$
$859$	2.21189e10	1.19066	0.595328	+	0.803483i	$0.297022\pi$
$860$	−3.83251e9	−0.205465
$861$	0	0
$862$	1.67626e10	0.891385
$863$	1.61806e10	0.856955	0.428478	−	0.903552i	$-0.359050\pi$
$863$	1.61806e10	0.856955	0.428478	+	0.903552i	$0.359050\pi$
$864$	0	0
$865$	1.92396e10	1.01074
$866$	3.14056e10	1.64322
$867$	0	0
$868$	3.42574e9	0.177801
$869$	1.30469e10	0.674434
$870$	0	0
$871$	−3.20101e9	−0.164143
$872$	−7.47708e9	−0.381878
$873$	0	0
$874$	−4.46852e8	−0.0226399
$875$	1.36222e10	0.687416
$876$	0	0
$877$	−1.22556e9	−0.0613528	−0.0306764	−	0.999529i	$-0.509766\pi$
$877$	−1.22556e9	−0.0613528	−0.0306764	+	0.999529i	$0.509766\pi$
$878$	−5.17850e10	−2.58210
$879$	0	0
$880$	1.06231e10	0.525489
$881$	−1.19682e10	−0.589675	−0.294838	−	0.955547i	$-0.595266\pi$
$881$	−1.19682e10	−0.589675	−0.294838	+	0.955547i	$0.595266\pi$
$882$	0	0
$883$	−1.75931e10	−0.859963	−0.429982	−	0.902838i	$-0.641480\pi$
$883$	−1.75931e10	−0.859963	−0.429982	+	0.902838i	$0.641480\pi$
$884$	−1.56698e9	−0.0762924
$885$	0	0
$886$	5.18786e10	2.50594
$887$	−1.10368e10	−0.531017	−0.265509	−	0.964108i	$-0.585540\pi$
$887$	−1.10368e10	−0.531017	−0.265509	+	0.964108i	$0.585540\pi$
$888$	0	0
$889$	3.06966e9	0.146532
$890$	1.32255e9	0.0628851
$891$	0	0
$892$	1.86642e10	0.880504
$893$	4.33023e8	0.0203484
$894$	0	0
$895$	−2.21569e10	−1.03307
$896$	1.08592e10	0.504337
$897$	0	0
$898$	1.79037e10	0.825043
$899$	5.26682e9	0.241763
$900$	0	0
$901$	5.58015e8	0.0254161
$902$	9.88586e9	0.448530
$903$	0	0
$904$	1.67692e9	0.0754959
$905$	−3.05089e10	−1.36822
$906$	0	0
$907$	−3.54273e9	−0.157657	−0.0788284	−	0.996888i	$-0.525118\pi$
$907$	−3.54273e9	−0.157657	−0.0788284	+	0.996888i	$0.525118\pi$
$908$	−2.27593e10	−1.00892
$909$	0	0
$910$	−8.48537e9	−0.373272
$911$	1.87857e10	0.823216	0.411608	−	0.911361i	$-0.364967\pi$
$911$	1.87857e10	0.823216	0.411608	+	0.911361i	$0.364967\pi$
$912$	0	0
$913$	−1.53747e10	−0.668586
$914$	−1.85604e10	−0.804037
$915$	0	0
$916$	−6.94974e9	−0.298768
$917$	−4.76649e8	−0.0204130
$918$	0	0
$919$	−4.06136e10	−1.72611	−0.863053	−	0.505114i	$-0.831450\pi$
$919$	−4.06136e10	−1.72611	−0.863053	+	0.505114i	$0.831450\pi$
$920$	−3.47363e9	−0.147071
$921$	0	0
$922$	−3.92710e10	−1.65011
$923$	8.24049e8	0.0344943
$924$	0	0
$925$	3.12190e9	0.129695
$926$	−2.12521e10	−0.879558
$927$	0	0
$928$	4.58969e10	1.88524
$929$	−9.02179e9	−0.369179	−0.184590	−	0.982816i	$-0.559096\pi$
$929$	−9.02179e9	−0.369179	−0.184590	+	0.982816i	$0.559096\pi$
$930$	0	0
$931$	−3.21323e8	−0.0130502
$932$	−1.93547e10	−0.783124
$933$	0	0
$934$	2.85151e10	1.14514
$935$	3.96760e9	0.158740
$936$	0	0
$937$	−3.93104e10	−1.56106	−0.780530	−	0.625119i	$-0.785050\pi$
$937$	−3.93104e10	−1.56106	−0.780530	+	0.625119i	$0.785050\pi$
$938$	1.76196e10	0.697085
$939$	0	0
$940$	1.89600e10	0.744543
$941$	2.27417e10	0.889732	0.444866	−	0.895597i	$-0.353251\pi$
$941$	2.27417e10	0.889732	0.444866	+	0.895597i	$0.353251\pi$
$942$	0	0
$943$	4.98436e9	0.193561
$944$	4.12208e9	0.159483
$945$	0	0
$946$	3.80055e9	0.145958
$947$	2.55265e10	0.976710	0.488355	−	0.872645i	$-0.337597\pi$
$947$	2.55265e10	0.976710	0.488355	+	0.872645i	$0.337597\pi$
$948$	0	0
$949$	−5.00841e9	−0.190225
$950$	3.30165e8	0.0124939
$951$	0	0
$952$	1.53132e9	0.0575225
$953$	−2.78690e10	−1.04303	−0.521515	−	0.853242i	$-0.674633\pi$
$953$	−2.78690e10	−1.04303	−0.521515	+	0.853242i	$0.674633\pi$
$954$	0	0
$955$	−2.20974e10	−0.820973
$956$	2.55467e10	0.945653
$957$	0	0
$958$	7.82746e10	2.87635
$959$	1.35052e10	0.494466
$960$	0	0
$961$	−2.66013e10	−0.966876
$962$	6.58690e9	0.238544
$963$	0	0
$964$	2.52684e10	0.908465
$965$	2.84334e10	1.01855
$966$	0	0
$967$	−1.07060e10	−0.380745	−0.190372	−	0.981712i	$-0.560970\pi$
$967$	−1.07060e10	−0.380745	−0.190372	+	0.981712i	$0.560970\pi$
$968$	−5.31923e9	−0.188488
$969$	0	0
$970$	5.95074e10	2.09348
$971$	2.27220e10	0.796489	0.398245	−	0.917279i	$-0.369620\pi$
$971$	2.27220e10	0.796489	0.398245	+	0.917279i	$0.369620\pi$
$972$	0	0
$973$	−1.50290e10	−0.523040
$974$	−2.35485e10	−0.816595
$975$	0	0
$976$	2.79894e10	0.963651
$977$	−4.39223e10	−1.50679	−0.753397	−	0.657566i	$-0.771586\pi$
$977$	−4.39223e10	−1.50679	−0.753397	+	0.657566i	$0.771586\pi$
$978$	0	0
$979$	−7.19644e8	−0.0245120
$980$	−1.40692e10	−0.477504
$981$	0	0
$982$	2.83832e10	0.956470
$983$	1.14421e10	0.384210	0.192105	−	0.981374i	$-0.438469\pi$
$983$	1.14421e10	0.384210	0.192105	+	0.981374i	$0.438469\pi$
$984$	0	0
$985$	−3.42638e10	−1.14238
$986$	1.32607e10	0.440553
$987$	0	0
$988$	3.82239e8	0.0126091
$989$	1.91620e9	0.0629875
$990$	0	0
$991$	3.59878e10	1.17462	0.587310	−	0.809362i	$-0.300187\pi$
$991$	3.59878e10	1.17462	0.587310	+	0.809362i	$0.300187\pi$
$992$	7.94165e9	0.258297
$993$	0	0
$994$	−4.53589e9	−0.146491
$995$	4.22610e10	1.36006
$996$	0	0
$997$	1.24025e9	0.0396347	0.0198173	−	0.999804i	$-0.493692\pi$
$997$	1.24025e9	0.0396347	0.0198173	+	0.999804i	$0.493692\pi$
$998$	3.83124e10	1.22006
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.8.a.a.1.9		10
3.2	odd	2		129.8.a.a.1.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.8.a.a.1.2	✓	10	3.2	odd	2
387.8.a.a.1.9		10	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:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q+16.8413 q^{2} +155.630 q^{4} +309.730 q^{5} -729.159 q^{7} +465.334 q^{8} +O(q^{10})\)	\(q+16.8413 q^{2} +155.630 q^{4} +309.730 q^{5} -729.159 q^{7} +465.334 q^{8} +5216.26 q^{10} -2838.34 q^{11} +2230.95 q^{13} -12280.0 q^{14} -12083.9 q^{16} -4513.16 q^{17} +1100.91 q^{19} +48203.4 q^{20} -47801.4 q^{22} -24101.1 q^{23} +17807.5 q^{25} +37572.1 q^{26} -113479. q^{28} -174466. q^{29} -30188.2 q^{31} -263071. q^{32} -76007.6 q^{34} -225842. q^{35} +175314. q^{37} +18540.8 q^{38} +144128. q^{40} -206811. q^{41} -79507.0 q^{43} -441732. q^{44} -405894. q^{46} +393333. q^{47} -291871. q^{49} +299903. q^{50} +347203. q^{52} -123642. q^{53} -879119. q^{55} -339302. q^{56} -2.93824e6 q^{58} -341123. q^{59} -2.31627e6 q^{61} -508410. q^{62} -2.88373e6 q^{64} +690991. q^{65} -1.43482e6 q^{67} -702385. q^{68} -3.80348e6 q^{70} +369372. q^{71} -2.24497e6 q^{73} +2.95251e6 q^{74} +171335. q^{76} +2.06960e6 q^{77} -4.59668e6 q^{79} -3.74273e6 q^{80} -3.48297e6 q^{82} +5.41678e6 q^{83} -1.39786e6 q^{85} -1.33900e6 q^{86} -1.32078e6 q^{88} +253544. q^{89} -1.62671e6 q^{91} -3.75086e6 q^{92} +6.62425e6 q^{94} +340984. q^{95} +1.14080e7 q^{97} -4.91549e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{2} + 371 q^{4} + 122 q^{5} - 2052 q^{7} + 927 q^{8}+O(q^{10}) \) 10 * q + q^2 + 371 * q^4 + 122 * q^5 - 2052 * q^7 + 927 * q^8	\( 10 q + q^{2} + 371 q^{4} + 122 q^{5} - 2052 q^{7} + 927 q^{8} - 10032 q^{10} + 8888 q^{11} - 16432 q^{13} + 28408 q^{14} - 26669 q^{16} + 48122 q^{17} - 56146 q^{19} + 88940 q^{20} - 100626 q^{22} + 236336 q^{23} - 135016 q^{25} + 166748 q^{26} - 259060 q^{28} + 248818 q^{29} - 430970 q^{31} - 69493 q^{32} + 445522 q^{34} - 298982 q^{35} - 261254 q^{37} - 257662 q^{38} - 671432 q^{40} + 126814 q^{41} - 795070 q^{43} + 620022 q^{44} - 809038 q^{46} - 627080 q^{47} - 1256116 q^{49} + 83117 q^{50} - 3674204 q^{52} + 1612384 q^{53} - 4732974 q^{55} + 4301484 q^{56} - 7268516 q^{58} + 3442492 q^{59} - 5217214 q^{61} + 2500324 q^{62} - 4657369 q^{64} + 1224166 q^{65} - 6810926 q^{67} + 3563486 q^{68} - 2745858 q^{70} + 13935120 q^{71} - 13743720 q^{73} + 1752692 q^{74} - 15817594 q^{76} + 7685750 q^{77} - 9007608 q^{79} + 13641024 q^{80} - 6329026 q^{82} + 21779128 q^{83} - 13177392 q^{85} - 79507 q^{86} - 13214750 q^{88} + 1895364 q^{89} - 16439838 q^{91} - 9614510 q^{92} + 3404276 q^{94} + 16861514 q^{95} - 20434472 q^{97} - 35731457 q^{98}+O(q^{100}) \) 10 * q + q^2 + 371 * q^4 + 122 * q^5 - 2052 * q^7 + 927 * q^8 - 10032 * q^10 + 8888 * q^11 - 16432 * q^13 + 28408 * q^14 - 26669 * q^16 + 48122 * q^17 - 56146 * q^19 + 88940 * q^20 - 100626 * q^22 + 236336 * q^23 - 135016 * q^25 + 166748 * q^26 - 259060 * q^28 + 248818 * q^29 - 430970 * q^31 - 69493 * q^32 + 445522 * q^34 - 298982 * q^35 - 261254 * q^37 - 257662 * q^38 - 671432 * q^40 + 126814 * q^41 - 795070 * q^43 + 620022 * q^44 - 809038 * q^46 - 627080 * q^47 - 1256116 * q^49 + 83117 * q^50 - 3674204 * q^52 + 1612384 * q^53 - 4732974 * q^55 + 4301484 * q^56 - 7268516 * q^58 + 3442492 * q^59 - 5217214 * q^61 + 2500324 * q^62 - 4657369 * q^64 + 1224166 * q^65 - 6810926 * q^67 + 3563486 * q^68 - 2745858 * q^70 + 13935120 * q^71 - 13743720 * q^73 + 1752692 * q^74 - 15817594 * q^76 + 7685750 * q^77 - 9007608 * q^79 + 13641024 * q^80 - 6329026 * q^82 + 21779128 * q^83 - 13177392 * q^85 - 79507 * q^86 - 13214750 * q^88 + 1895364 * q^89 - 16439838 * q^91 - 9614510 * q^92 + 3404276 * q^94 + 16861514 * q^95 - 20434472 * q^97 - 35731457 * q^98

Introduction

L-functions

Modular forms

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Database

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