LMFDB - Embedded newform 63.6.c.b.62.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,6,Mod(62,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("63.62");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	63.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$10.1041806482$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 456x^{6} - 612x^{5} + 66744x^{4} + 32004x^{3} + 3729464x^{2} + 2503804x + 66026577 $ x^8 - 4x^7 + 456x^6 - 612x^5 + 66744x^4 + 32004x^3 + 3729464x^2 + 2503804*x + 66026577
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			62.4
Root			$1.82288 + 12.8433i$ of defining polynomial
Character	$\chi$	$=$	63.62
Dual form			63.6.c.b.62.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.500983i q^{2} +31.7490 q^{4} +63.0754 q^{5} +(-53.6863 + 118.003i) q^{7} -31.9372i q^{8} +O(q^{10})$	\(q-0.500983i q^{2} +31.7490 q^{4} +63.0754 q^{5} +(-53.6863 + 118.003i) q^{7} -31.9372i q^{8} -31.5997i q^{10} -215.278i q^{11} +204.407i q^{13} +(59.1177 + 26.8959i) q^{14} +999.969 q^{16} +1947.42 q^{17} +2313.66i q^{19} +2002.58 q^{20} -107.851 q^{22} -3832.04i q^{23} +853.510 q^{25} +102.404 q^{26} +(-1704.49 + 3746.49i) q^{28} +2573.71i q^{29} -5950.54i q^{31} -1522.96i q^{32} -975.626i q^{34} +(-3386.28 + 7443.11i) q^{35} +4061.99 q^{37} +1159.10 q^{38} -2014.45i q^{40} -7655.38 q^{41} -5679.07 q^{43} -6834.88i q^{44} -1919.79 q^{46} -16249.2 q^{47} +(-11042.6 - 12670.3i) q^{49} -427.594i q^{50} +6489.72i q^{52} +24622.1i q^{53} -13578.8i q^{55} +(3768.69 + 1714.59i) q^{56} +1289.39 q^{58} -36472.9 q^{59} -19761.4i q^{61} -2981.12 q^{62} +31236.0 q^{64} +12893.1i q^{65} -21372.2 q^{67} +61828.8 q^{68} +(3728.87 + 1696.47i) q^{70} -76408.9i q^{71} +44656.7i q^{73} -2034.99i q^{74} +73456.4i q^{76} +(25403.6 + 11557.5i) q^{77} -38937.7 q^{79} +63073.4 q^{80} +3835.22i q^{82} +37932.3 q^{83} +122835. q^{85} +2845.12i q^{86} -6875.39 q^{88} -42411.2 q^{89} +(-24120.7 - 10973.8i) q^{91} -121663. i q^{92} +8140.59i q^{94} +145935. i q^{95} -100733. i q^{97} +(-6347.62 + 5532.14i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 112 q^{7}+O(q^{10}) $ 8 * q - 112 * q^7	$ 8 q - 112 q^{7} - 128 q^{16} + 5360 q^{22} + 9368 q^{25} - 10080 q^{28} - 34304 q^{37} + 32416 q^{43} + 71888 q^{46} - 106120 q^{49} - 115792 q^{58} + 253952 q^{64} + 152608 q^{67} - 260736 q^{70} - 94592 q^{79} + 475200 q^{85} - 26048 q^{88} - 501312 q^{91}+O(q^{100}) $ 8 * q - 112 * q^7 - 128 * q^16 + 5360 * q^22 + 9368 * q^25 - 10080 * q^28 - 34304 * q^37 + 32416 * q^43 + 71888 * q^46 - 106120 * q^49 - 115792 * q^58 + 253952 * q^64 + 152608 * q^67 - 260736 * q^70 - 94592 * q^79 + 475200 * q^85 - 26048 * q^88 - 501312 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$.

$n$	$10$	$29$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	0.500983i		−	0.0885622i	−0.999019	−	0.0442811i	$-0.985900\pi$
$2$		−	0.500983i		−	0.0885622i	0.999019	−	0.0442811i	$-0.0140997\pi$
$3$		0			0
$3$		0			0
$4$	31.7490			0.992157
$5$	63.0754			1.12833			0.564164	−	0.825663i	$-0.309199\pi$
$5$	63.0754			1.12833			0.564164	+	0.825663i	$0.309199\pi$
$6$		0			0
$7$	−53.6863	+	118.003i	−0.414112	+	0.910226i
$7$	−53.6863	+	118.003i	−0.414112	+	0.910226i
$8$		−	31.9372i		−	0.176430i
$9$		0			0
$10$		−	31.5997i		−	0.0999271i
$11$		−	215.278i		−	0.536437i	−0.963358	−	0.268219i	$-0.913565\pi$
$11$		−	215.278i		−	0.536437i	0.963358	−	0.268219i	$-0.0864349\pi$
$12$		0			0
$13$			204.407i			0.335457i	0.985833	+	0.167729i	$0.0536432\pi$
$13$			204.407i			0.335457i	−0.985833	+	0.167729i	$0.946357\pi$
$14$	59.1177	+	26.8959i	0.0806116	+	0.0366747i
$15$		0			0
$16$	999.969			0.976532
$17$	1947.42			1.63432			0.817162	−	0.576409i	$-0.195546\pi$
$17$	1947.42			1.63432			0.817162	+	0.576409i	$0.195546\pi$
$18$		0			0
$19$			2313.66i			1.47033i	0.677887	+	0.735166i	$0.262896\pi$
$19$			2313.66i			1.47033i	−0.677887	+	0.735166i	$0.737104\pi$
$20$	2002.58			1.11948
$21$		0			0
$22$	−107.851			−0.0475080
$23$		−	3832.04i		−	1.51046i	−0.655458	−	0.755232i	$-0.727524\pi$
$23$		−	3832.04i		−	1.51046i	0.655458	−	0.755232i	$-0.272476\pi$
$24$		0			0
$25$	853.510			0.273123
$26$	102.404			0.0297088
$27$		0			0
$28$	−1704.49	+	3746.49i	−0.410864	+	0.903087i
$29$			2573.71i			0.568283i	0.958782	+	0.284142i	$0.0917086\pi$
$29$			2573.71i			0.568283i	−0.958782	+	0.284142i	$0.908291\pi$
$30$		0			0
$31$		−	5950.54i		−	1.11212i	−0.831142	−	0.556061i	$-0.812312\pi$
$31$		−	5950.54i		−	1.11212i	0.831142	−	0.556061i	$-0.187688\pi$
$32$		−	1522.96i		−	0.262914i
$33$		0			0
$34$		−	975.626i		−	0.144739i
$35$	−3386.28	+	7443.11i	−0.467254	+	1.02703i
$36$		0			0
$37$	4061.99			0.487792			0.243896	−	0.969801i	$-0.421574\pi$
$37$	4061.99			0.487792			0.243896	+	0.969801i	$0.421574\pi$
$38$	1159.10			0.130216
$39$		0			0
$40$		−	2014.45i		−	0.199071i
$41$	−7655.38			−0.711225			−0.355612	−	0.934634i	$-0.615728\pi$
$41$	−7655.38			−0.711225			−0.355612	+	0.934634i	$0.615728\pi$
$42$		0			0
$43$	−5679.07			−0.468389			−0.234194	−	0.972190i	$-0.575245\pi$
$43$	−5679.07			−0.468389			−0.234194	+	0.972190i	$0.575245\pi$
$44$		−	6834.88i		−	0.532230i
$45$		0			0
$46$	−1919.79			−0.133770
$47$	−16249.2			−1.07297			−0.536485	−	0.843910i	$-0.680248\pi$
$47$	−16249.2			−1.07297			−0.536485	+	0.843910i	$0.680248\pi$
$48$		0			0
$49$	−11042.6	−	12670.3i	−0.657022	−	0.753871i
$50$		−	427.594i		−	0.0241884i
$51$		0			0
$52$			6489.72i			0.332826i
$53$			24622.1i			1.20403i	0.798486	+	0.602013i	$0.205635\pi$
$53$			24622.1i			1.20403i	−0.798486	+	0.602013i	$0.794365\pi$
$54$		0			0
$55$		−	13578.8i		−	0.605277i
$56$	3768.69	+	1714.59i	0.160591	+	0.0730617i
$57$		0			0
$58$	1289.39			0.0503284
$59$	−36472.9			−1.36408			−0.682041	−	0.731314i	$-0.738908\pi$
$59$	−36472.9			−1.36408			−0.682041	+	0.731314i	$0.738908\pi$
$60$		0			0
$61$		−	19761.4i		−	0.679974i	−0.940430	−	0.339987i	$-0.889577\pi$
$61$		−	19761.4i		−	0.679974i	0.940430	−	0.339987i	$-0.110423\pi$
$62$	−2981.12			−0.0984919
$63$		0			0
$64$	31236.0			0.953248
$65$			12893.1i			0.378506i
$66$		0			0
$67$	−21372.2			−0.581652			−0.290826	−	0.956776i	$-0.593930\pi$
$67$	−21372.2			−0.581652			−0.290826	+	0.956776i	$0.593930\pi$
$68$	61828.8			1.62150
$69$		0			0
$70$	3728.87	+	1696.47i	0.0909563	+	0.0413811i
$71$		−	76408.9i		−	1.79886i	−0.437061	−	0.899432i	$-0.643981\pi$
$71$		−	76408.9i		−	1.79886i	0.437061	−	0.899432i	$-0.356019\pi$
$72$		0			0
$73$			44656.7i			0.980797i	0.871498	+	0.490399i	$0.163149\pi$
$73$			44656.7i			0.980797i	−0.871498	+	0.490399i	$0.836851\pi$
$74$		−	2034.99i		−	0.0431999i
$75$		0			0
$76$			73456.4i			1.45880i
$77$	25403.6	+	11557.5i	0.488279	+	0.222145i
$78$		0			0
$79$	−38937.7			−0.701943			−0.350972	−	0.936386i	$-0.614149\pi$
$79$	−38937.7			−0.701943			−0.350972	+	0.936386i	$0.614149\pi$
$80$	63073.4			1.10185
$81$		0			0
$82$			3835.22i			0.0629876i
$83$	37932.3			0.604386			0.302193	−	0.953247i	$-0.402281\pi$
$83$	37932.3			0.604386			0.302193	+	0.953247i	$0.402281\pi$
$84$		0			0
$85$	122835.			1.84405
$86$			2845.12i			0.0414815i
$87$		0			0
$88$	−6875.39			−0.0946434
$89$	−42411.2			−0.567552			−0.283776	−	0.958891i	$-0.591587\pi$
$89$	−42411.2			−0.567552			−0.283776	+	0.958891i	$0.591587\pi$
$90$		0			0
$91$	−24120.7	−	10973.8i	−0.305342	−	0.138917i
$92$		−	121663.i		−	1.49862i
$93$		0			0
$94$			8140.59i			0.0950246i
$95$			145935.i			1.65902i
$96$		0			0
$97$		−	100733.i		−	1.08704i	−0.839397	−	0.543519i	$-0.817092\pi$
$97$		−	100733.i		−	1.08704i	0.839397	−	0.543519i	$-0.182908\pi$
$98$	−6347.62	+	5532.14i	−0.0667645	+	0.0581873i
$99$		0			0
$100$	27098.1			0.270981
$101$	−9349.75			−0.0912004			−0.0456002	−	0.998960i	$-0.514520\pi$
$101$	−9349.75			−0.0912004			−0.0456002	+	0.998960i	$0.514520\pi$
$102$		0			0
$103$			120974.i			1.12357i	0.827284	+	0.561783i	$0.189885\pi$
$103$			120974.i			1.12357i	−0.827284	+	0.561783i	$0.810115\pi$
$104$	6528.18			0.0591846
$105$		0			0
$106$	12335.3			0.106631
$107$		−	19336.0i		−	0.163270i	−0.996662	−	0.0816350i	$-0.973986\pi$
$107$		−	19336.0i		−	0.163270i	0.996662	−	0.0816350i	$-0.0260142\pi$
$108$		0			0
$109$	−162541.			−1.31038			−0.655188	−	0.755466i	$-0.727411\pi$
$109$	−162541.			−1.31038			−0.655188	+	0.755466i	$0.727411\pi$
$110$	−6802.74			−0.0536046
$111$		0			0
$112$	−53684.6	+	118000.i	−0.404394	+	0.888864i
$113$		−	194330.i		−	1.43167i	−0.698268	−	0.715836i	$-0.746046\pi$
$113$		−	194330.i		−	1.43167i	0.698268	−	0.715836i	$-0.253954\pi$
$114$		0			0
$115$		−	241707.i		−	1.70430i
$116$			81712.8i			0.563826i
$117$		0			0
$118$			18272.3i			0.120806i
$119$	−104550.	+	229802.i	−0.676793	+	1.48760i
$120$		0			0
$121$	114706.			0.712235
$122$	−9900.11			−0.0602200
$123$		0			0
$124$		−	188924.i		−	1.10340i
$125$	−143275.			−0.820155
$126$		0			0
$127$	142459.			0.783756			0.391878	−	0.920017i	$-0.371826\pi$
$127$	142459.			0.783756			0.391878	+	0.920017i	$0.371826\pi$
$128$		−	64383.4i		−	0.347335i
$129$		0			0
$130$	6459.20			0.0335213
$131$	−198723.			−1.01174			−0.505870	−	0.862609i	$-0.668829\pi$
$131$	−198723.			−1.01174			−0.505870	+	0.862609i	$0.668829\pi$
$132$		0			0
$133$	−273019.	−	124212.i	−1.33833	−	0.608882i
$134$			10707.1i			0.0515124i
$135$		0			0
$136$		−	62195.2i		−	0.288343i
$137$			216399.i			0.985039i	0.870301	+	0.492519i	$0.163924\pi$
$137$			216399.i			0.985039i	−0.870301	+	0.492519i	$0.836076\pi$
$138$		0			0
$139$		−	34409.9i		−	0.151059i	−0.997144	−	0.0755294i	$-0.975935\pi$
$139$		−	34409.9i		−	0.151059i	0.997144	−	0.0755294i	$-0.0240647\pi$
$140$	−107511.	+	236311.i	−0.463590	+	1.01898i
$141$		0			0
$142$	−38279.6			−0.159311
$143$	44004.4			0.179952
$144$		0			0
$145$			162338.i			0.641210i
$146$	22372.3			0.0868616
$147$		0			0
$148$	128964.			0.483966
$149$		−	424636.i		−	1.56694i	−0.621432	−	0.783468i	$-0.713449\pi$
$149$		−	424636.i		−	1.56694i	0.621432	−	0.783468i	$-0.286551\pi$
$150$		0			0
$151$	−275806.			−0.984376			−0.492188	−	0.870489i	$-0.663803\pi$
$151$	−275806.			−0.984376			−0.492188	+	0.870489i	$0.663803\pi$
$152$	73891.7			0.259410
$153$		0			0
$154$	5790.11	−	12726.8i	0.0196737	−	0.0432430i
$155$		−	375333.i		−	1.25484i
$156$		0			0
$157$			515757.i			1.66992i	0.550311	+	0.834960i	$0.314509\pi$
$157$			515757.i			1.66992i	−0.550311	+	0.834960i	$0.685491\pi$
$158$			19507.1i			0.0621656i
$159$		0			0
$160$		−	96061.2i		−	0.296653i
$161$	452193.	+	205728.i	1.37486	+	0.625501i
$162$		0			0
$163$	513291.			1.51319			0.756597	−	0.653882i	$-0.226861\pi$
$163$	513291.			1.51319			0.756597	+	0.653882i	$0.226861\pi$
$164$	−243051.			−0.705647
$165$		0			0
$166$		−	19003.5i		−	0.0535257i
$167$	388861.			1.07896			0.539478	−	0.842000i	$-0.318622\pi$
$167$	388861.			1.07896			0.539478	+	0.842000i	$0.318622\pi$
$168$		0			0
$169$	329511.			0.887468
$170$		−	61538.0i		−	0.163313i
$171$		0			0
$172$	−180305.			−0.464715
$173$	611437.			1.55323			0.776616	−	0.629974i	$-0.216934\pi$
$173$	611437.			1.55323			0.776616	+	0.629974i	$0.216934\pi$
$174$		0			0
$175$	−45821.8	+	100717.i	−0.113104	+	0.248604i
$176$		−	215272.i		−	0.523848i
$177$		0			0
$178$			21247.3i			0.0502636i
$179$			255217.i			0.595357i	0.954666	+	0.297678i	$0.0962123\pi$
$179$			255217.i			0.595357i	−0.954666	+	0.297678i	$0.903788\pi$
$180$		0			0
$181$			604856.i			1.37232i	0.727451	+	0.686160i	$0.240705\pi$
$181$			604856.i			1.37232i	−0.727451	+	0.686160i	$0.759295\pi$
$182$	−5497.71	+	12084.1i	−0.0123028	+	0.0270417i
$183$		0			0
$184$	−122385.			−0.266491
$185$	256212.			0.550389
$186$		0			0
$187$		−	419238.i		−	0.876712i
$188$	−515897.			−1.06455
$189$		0			0
$190$	73111.0			0.146926
$191$			129543.i			0.256938i	0.991714	+	0.128469i	$0.0410063\pi$
$191$			129543.i			0.256938i	−0.991714	+	0.128469i	$0.958994\pi$
$192$		0			0
$193$	176709.			0.341480			0.170740	−	0.985316i	$-0.445384\pi$
$193$	176709.			0.341480			0.170740	+	0.985316i	$0.445384\pi$
$194$	−50465.8			−0.0962704
$195$		0			0
$196$	−350591.	−	402270.i	−0.651869	−	0.747959i
$197$		−	814108.i		−	1.49457i	−0.664504	−	0.747285i	$-0.731357\pi$
$197$		−	814108.i		−	1.49457i	0.664504	−	0.747285i	$-0.268643\pi$
$198$		0			0
$199$			312577.i			0.559531i	0.960068	+	0.279765i	$0.0902567\pi$
$199$			312577.i			0.559531i	−0.960068	+	0.279765i	$0.909743\pi$
$200$		−	27258.7i		−	0.0481870i
$201$		0			0
$202$			4684.07i			0.00807690i
$203$	−303706.	−	138173.i	−0.517266	−	0.235333i
$204$		0			0
$205$	−482866.			−0.802495
$206$	60605.9			0.0995055
$207$		0			0
$208$			204400.i			0.327585i
$209$	498081.			0.788740
$210$		0			0
$211$	248493.			0.384245			0.192122	−	0.981371i	$-0.438463\pi$
$211$	248493.			0.384245			0.192122	+	0.981371i	$0.438463\pi$
$212$			781729.i			1.19458i
$213$		0			0
$214$	−9686.99			−0.0144595
$215$	−358210.			−0.528496
$216$		0			0
$217$	702183.	+	319462.i	1.01228	+	0.460543i
$218$			81430.2i			0.116050i
$219$		0			0
$220$		−	431113.i		−	0.600529i
$221$			398067.i			0.548246i
$222$		0			0
$223$		−	947790.i		−	1.27629i	−0.769915	−	0.638146i	$-0.779702\pi$
$223$		−	947790.i		−	1.27629i	0.769915	−	0.638146i	$-0.220298\pi$
$224$	179714.	+	81761.9i	0.239311	+	0.108876i
$225$		0			0
$226$	−97356.1			−0.126792
$227$	−553144.			−0.712482			−0.356241	−	0.934394i	$-0.615942\pi$
$227$	−553144.			−0.712482			−0.356241	+	0.934394i	$0.615942\pi$
$228$		0			0
$229$			456083.i			0.574718i	0.957823	+	0.287359i	$0.0927773\pi$
$229$			456083.i			0.574718i	−0.957823	+	0.287359i	$0.907223\pi$
$230$	−121091.			−0.150936
$231$		0			0
$232$	82197.1			0.100262
$233$			201829.i			0.243553i	0.992558	+	0.121777i	$0.0388592\pi$
$233$			201829.i			0.243553i	−0.992558	+	0.121777i	$0.961141\pi$
$234$		0			0
$235$	−1.02493e6			−1.21066
$236$	−1.15798e6			−1.35338
$237$		0			0
$238$	115127.	+	52377.7i	0.131745	+	0.0599383i
$239$			727594.i			0.823937i	0.911198	+	0.411969i	$0.135159\pi$
$239$			727594.i			0.823937i	−0.911198	+	0.411969i	$0.864841\pi$
$240$		0			0
$241$		−	1.49179e6i		−	1.65449i	−0.561842	−	0.827245i	$-0.689907\pi$
$241$		−	1.49179e6i		−	1.65449i	0.561842	−	0.827245i	$-0.310093\pi$
$242$		−	57465.9i		−	0.0630771i
$243$		0			0
$244$		−	627404.i		−	0.674641i
$245$	−696515.	−	799186.i	−0.741336	−	0.850614i
$246$		0			0
$247$	−472928.			−0.493233
$248$	−190044.			−0.196211
$249$		0			0
$250$			71778.5i			0.0726347i
$251$	−1.70164e6			−1.70484			−0.852418	−	0.522861i	$-0.824865\pi$
$251$	−1.70164e6			−1.70484			−0.852418	+	0.522861i	$0.824865\pi$
$252$		0			0
$253$	−824955.			−0.810268
$254$		−	71369.6i		−	0.0694111i
$255$		0			0
$256$	967298.			0.922487
$257$	596737.			0.563573			0.281787	−	0.959477i	$-0.409073\pi$
$257$	596737.			0.563573			0.281787	+	0.959477i	$0.409073\pi$
$258$		0			0
$259$	−218073.	+	479328.i	−0.202001	+	0.444001i
$260$			409342.i			0.375537i
$261$		0			0
$262$			99556.8i			0.0896020i
$263$		−	134437.i		−	0.119848i	−0.998203	−	0.0599240i	$-0.980914\pi$
$263$		−	134437.i		−	0.119848i	0.998203	−	0.0599240i	$-0.0190858\pi$
$264$		0			0
$265$			1.55305e6i			1.35854i
$266$	−62228.0	+	136778.i	−0.0539239	+	0.118526i
$267$		0			0
$268$	−678548.			−0.577090
$269$	2.01762e6			1.70004			0.850018	−	0.526754i	$-0.176591\pi$
$269$	2.01762e6			1.70004			0.850018	+	0.526754i	$0.176591\pi$
$270$		0			0
$271$		−	932314.i		−	0.771150i	−0.922676	−	0.385575i	$-0.874003\pi$
$271$		−	932314.i		−	0.771150i	0.922676	−	0.385575i	$-0.125997\pi$
$272$	1.94736e6			1.59597
$273$		0			0
$274$	108412.			0.0872372
$275$		−	183742.i		−	0.146513i
$276$		0			0
$277$	1.82936e6			1.43251			0.716257	−	0.697836i	$-0.245854\pi$
$277$	1.82936e6			1.43251			0.716257	+	0.697836i	$0.245854\pi$
$278$	−17238.8			−0.0133781
$279$		0			0
$280$	237712.	+	108148.i	0.181199	+	0.0824376i
$281$			1.87677e6i			1.41790i	0.705261	+	0.708948i	$0.250830\pi$
$281$			1.87677e6i			1.41790i	−0.705261	+	0.708948i	$0.749170\pi$
$282$		0			0
$283$			439799.i			0.326429i	0.986591	+	0.163214i	$0.0521862\pi$
$283$			439799.i			0.326429i	−0.986591	+	0.163214i	$0.947814\pi$
$284$		−	2.42591e6i		−	1.78475i
$285$		0			0
$286$		−	22045.5i		−	0.0159369i
$287$	410989.	−	903360.i	0.294527	−	0.647375i
$288$		0			0
$289$	2.37260e6			1.67101
$290$	81328.6			0.0567869
$291$		0			0
$292$			1.41781e6i			0.973105i
$293$	644698.			0.438720			0.219360	−	0.975644i	$-0.429603\pi$
$293$	644698.			0.438720			0.219360	+	0.975644i	$0.429603\pi$
$294$		0			0
$295$	−2.30055e6			−1.53913
$296$		−	129729.i		−	0.0860610i
$297$		0			0
$298$	−212736.			−0.138771
$299$	783295.			0.506696
$300$		0			0
$301$	304888.	−	670150.i	0.193965	−	0.426339i
$302$			138174.i			0.0871785i
$303$		0			0
$304$			2.31359e6i			1.43583i
$305$		−	1.24646e6i		−	0.767233i
$306$		0			0
$307$		−	567995.i		−	0.343953i	−0.985101	−	0.171976i	$-0.944985\pi$
$307$		−	567995.i		−	0.343953i	0.985101	−	0.171976i	$-0.0550153\pi$
$308$	806538.	+	366939.i	0.484449	+	0.220403i
$309$		0			0
$310$	−188035.			−0.111131
$311$	−2.81751e6			−1.65182			−0.825912	−	0.563799i	$-0.809339\pi$
$311$	−2.81751e6			−1.65182			−0.825912	+	0.563799i	$0.809339\pi$
$312$		0			0
$313$			174130.i			0.100465i	0.998738	+	0.0502324i	$0.0159962\pi$
$313$			174130.i			0.100465i	−0.998738	+	0.0502324i	$0.984004\pi$
$314$	258385.			0.147892
$315$		0			0
$316$	−1.23623e6			−0.696438
$317$		−	1.78930e6i		−	1.00008i	−0.866002	−	0.500040i	$-0.833319\pi$
$317$		−	1.78930e6i		−	1.00008i	0.866002	−	0.500040i	$-0.166681\pi$
$318$		0			0
$319$	554064.			0.304848
$320$	1.97023e6			1.07558
$321$		0			0
$322$	103066.	−	226541.i	0.0553958	−	0.121761i
$323$			4.50567e6i			2.40300i
$324$		0			0
$325$			174463.i			0.0916211i
$326$		−	257150.i		−	0.134012i
$327$		0			0
$328$			244491.i			0.125481i
$329$	872360.	−	1.91746e6i	0.444330	−	0.976646i
$330$		0			0
$331$	−697884.			−0.350117			−0.175058	−	0.984558i	$-0.556011\pi$
$331$	−697884.			−0.350117			−0.175058	+	0.984558i	$0.556011\pi$
$332$	1.20431e6			0.599645
$333$		0			0
$334$		−	194813.i		−	0.0955546i
$335$	−1.34806e6			−0.656294
$336$		0			0
$337$	−1.40112e6			−0.672048			−0.336024	−	0.941853i	$-0.609082\pi$
$337$	−1.40112e6			−0.672048			−0.336024	+	0.941853i	$0.609082\pi$
$338$		−	165079.i		−	0.0785961i
$339$		0			0
$340$	3.89988e6			1.82959
$341$	−1.28102e6			−0.596583
$342$		0			0
$343$	2.08797e6	−	622838.i	0.958274	−	0.285851i
$344$			181374.i			0.0826377i
$345$		0			0
$346$		−	306320.i		−	0.137558i
$347$			363601.i			0.162107i	0.996710	+	0.0810534i	$0.0258284\pi$
$347$			363601.i			0.162107i	−0.996710	+	0.0810534i	$0.974172\pi$
$348$		0			0
$349$			1.95507e6i			0.859211i	0.903017	+	0.429606i	$0.141347\pi$
$349$			1.95507e6i			0.859211i	−0.903017	+	0.429606i	$0.858653\pi$
$350$	50457.5	+	22955.9i	0.0220169	+	0.0100167i
$351$		0			0
$352$	−327860.			−0.141037
$353$	−465278.			−0.198736			−0.0993678	−	0.995051i	$-0.531682\pi$
$353$	−465278.			−0.198736			−0.0993678	+	0.995051i	$0.531682\pi$
$354$		0			0
$355$		−	4.81953e6i		−	2.02971i
$356$	−1.34651e6			−0.563100
$357$		0			0
$358$	127860.			0.0527261
$359$			1.35714e6i			0.555760i	0.960616	+	0.277880i	$0.0896317\pi$
$359$			1.35714e6i			0.555760i	−0.960616	+	0.277880i	$0.910368\pi$
$360$		0			0
$361$	−2.87692e6			−1.16187
$362$	303023.			0.121536
$363$		0			0
$364$	−765808.	−	348409.i	−0.302947	−	0.137827i
$365$			2.81674e6i			1.10666i
$366$		0			0
$367$			922112.i			0.357370i	0.983906	+	0.178685i	$0.0571843\pi$
$367$			922112.i			0.357370i	−0.983906	+	0.178685i	$0.942816\pi$
$368$		−	3.83192e6i		−	1.47502i
$369$		0			0
$370$		−	128358.i		−	0.0487437i
$371$	−2.90549e6	−	1.32187e6i	−1.09594	−	0.498602i
$372$		0			0
$373$	−2.81126e6			−1.04623			−0.523117	−	0.852261i	$-0.675231\pi$
$373$	−2.81126e6			−1.04623			−0.523117	+	0.852261i	$0.675231\pi$
$374$	−210031.			−0.0776435
$375$		0			0
$376$			518954.i			0.189304i
$377$	−526084.			−0.190635
$378$		0			0
$379$	−2.75476e6			−0.985113			−0.492556	−	0.870281i	$-0.663937\pi$
$379$	−2.75476e6			−0.985113			−0.492556	+	0.870281i	$0.663937\pi$
$380$			4.63329e6i			1.64600i
$381$		0			0
$382$	64898.6			0.0227550
$383$	−3.77768e6			−1.31592			−0.657958	−	0.753055i	$-0.728579\pi$
$383$	−3.77768e6			−1.31592			−0.657958	+	0.753055i	$0.728579\pi$
$384$		0			0
$385$	1.60234e6	+	728994.i	0.550938	+	0.250653i
$386$		−	88528.4i		−	0.0302423i
$387$		0			0
$388$		−	3.19819e6i		−	1.07851i
$389$			338761.i			0.113506i	0.998388	+	0.0567531i	$0.0180748\pi$
$389$			338761.i			0.113506i	−0.998388	+	0.0567531i	$0.981925\pi$
$390$		0			0
$391$		−	7.46260e6i		−	2.46859i
$392$	−404654.	+	352669.i	−0.133005	+	0.115918i
$393$		0			0
$394$	−407854.			−0.132362
$395$	−2.45601e6			−0.792022
$396$		0			0
$397$			4.60034e6i			1.46492i	0.680811	+	0.732459i	$0.261628\pi$
$397$			4.60034e6i			1.46492i	−0.680811	+	0.732459i	$0.738372\pi$
$398$	156596.			0.0495533
$399$		0			0
$400$	853483.			0.266713
$401$			2.00825e6i			0.623672i	0.950136	+	0.311836i	$0.100944\pi$
$401$			2.00825e6i			0.623672i	−0.950136	+	0.311836i	$0.899056\pi$
$402$		0			0
$403$	1.21633e6			0.373069
$404$	−296845.			−0.0904851
$405$		0			0
$406$	−69222.3	+	152152.i	−0.0208416	+	0.0458102i
$407$		−	874459.i		−	0.261670i
$408$		0			0
$409$		−	3.06590e6i		−	0.906253i	−0.891446	−	0.453126i	$-0.850309\pi$
$409$		−	3.06590e6i		−	0.906253i	0.891446	−	0.453126i	$-0.149691\pi$
$410$			241908.i			0.0710707i
$411$		0			0
$412$			3.84080e6i			1.11475i
$413$	1.95810e6	−	4.30393e6i	0.564883	−	1.24162i
$414$		0			0
$415$	2.39260e6			0.681945
$416$	311303.			0.0881962
$417$		0			0
$418$		−	249530.i		−	0.0698525i
$419$	3.68397e6			1.02513			0.512567	−	0.858647i	$-0.328695\pi$
$419$	3.68397e6			1.02513			0.512567	+	0.858647i	$0.328695\pi$
$420$		0			0
$421$	1.79276e6			0.492966			0.246483	−	0.969147i	$-0.420725\pi$
$421$	1.79276e6			0.492966			0.246483	+	0.969147i	$0.420725\pi$
$422$		−	124491.i		−	0.0340295i
$423$		0			0
$424$	786362.			0.212426
$425$	1.66214e6			0.446372
$426$		0			0
$427$	2.33191e6	+	1.06091e6i	0.618930	+	0.281586i
$428$		−	613898.i		−	0.161989i
$429$		0			0
$430$			179457.i			0.0468047i
$431$			4.74965e6i			1.23160i	0.787904	+	0.615798i	$0.211166\pi$
$431$			4.74965e6i			1.23160i	−0.787904	+	0.615798i	$0.788834\pi$
$432$		0			0
$433$			2.89216e6i			0.741315i	0.928770	+	0.370657i	$0.120868\pi$
$433$			2.89216e6i			0.741315i	−0.928770	+	0.370657i	$0.879132\pi$
$434$	160045.	−	351782.i	0.0407867	−	0.0896498i
$435$		0			0
$436$	−5.16051e6			−1.30010
$437$	8.86603e6			2.22088
$438$		0			0
$439$			6.49637e6i			1.60883i	0.594070	+	0.804413i	$0.297520\pi$
$439$			6.49637e6i			1.60883i	−0.594070	+	0.804413i	$0.702480\pi$
$440$	−433668.			−0.106789
$441$		0			0
$442$	199425.			0.0485538
$443$		−	2.64142e6i		−	0.639481i	−0.947505	−	0.319740i	$-0.896404\pi$
$443$		−	2.64142e6i		−	0.639481i	0.947505	−	0.319740i	$-0.103596\pi$
$444$		0			0
$445$	−2.67510e6			−0.640384
$446$	−474827.			−0.113031
$447$		0			0
$448$	−1.67695e6	+	3.68595e6i	−0.394752	+	0.867671i
$449$		−	4.15159e6i		−	0.971848i	−0.874001	−	0.485924i	$-0.838483\pi$
$449$		−	4.15159e6i		−	0.971848i	0.874001	−	0.485924i	$-0.161517\pi$
$450$		0			0
$451$			1.64804e6i			0.381527i
$452$		−	6.16978e6i		−	1.42044i
$453$		0			0
$454$			277116.i			0.0630989i
$455$	−1.52142e6	−	692180.i	−0.344526	−	0.156744i
$456$		0			0
$457$	5.17647e6			1.15943			0.579713	−	0.814821i	$-0.303165\pi$
$457$	5.17647e6			1.15943			0.579713	+	0.814821i	$0.303165\pi$
$458$	228490.			0.0508983
$459$		0			0
$460$		−	7.67397e6i		−	1.69093i
$461$	−972966.			−0.213229			−0.106614	−	0.994300i	$-0.534001\pi$
$461$	−972966.			−0.213229			−0.106614	+	0.994300i	$0.534001\pi$
$462$		0			0
$463$	738585.			0.160121			0.0800604	−	0.996790i	$-0.474489\pi$
$463$	738585.			0.160121			0.0800604	+	0.996790i	$0.474489\pi$
$464$			2.57363e6i			0.554947i
$465$		0			0
$466$	101113.			0.0215696
$467$	4.80458e6			1.01944			0.509721	−	0.860340i	$-0.329749\pi$
$467$	4.80458e6			1.01944			0.509721	+	0.860340i	$0.329749\pi$
$468$		0			0
$469$	1.14740e6	−	2.52200e6i	0.240869	−	0.529435i
$470$			513471.i			0.107219i
$471$		0			0
$472$			1.16484e6i			0.240665i
$473$			1.22258e6i			0.251261i
$474$		0			0
$475$			1.97473e6i			0.401582i
$476$	−3.31936e6	+	7.29600e6i	−0.671485	+	1.47594i
$477$		0			0
$478$	364512.			0.0729697
$479$	478272.			0.0952436			0.0476218	−	0.998865i	$-0.484836\pi$
$479$	478272.			0.0952436			0.0476218	+	0.998865i	$0.484836\pi$
$480$		0			0
$481$			830299.i			0.163633i
$482$	−747360.			−0.146525
$483$		0			0
$484$	3.64181e6			0.706649
$485$		−	6.35381e6i		−	1.22653i
$486$		0			0
$487$	−1.95384e6			−0.373308			−0.186654	−	0.982426i	$-0.559764\pi$
$487$	−1.95384e6			−0.373308			−0.186654	+	0.982426i	$0.559764\pi$
$488$	−631122.			−0.119968
$489$		0			0
$490$	−400379.	+	348942.i	−0.0753322	+	0.0656543i
$491$		−	2.30071e6i		−	0.430684i	−0.976539	−	0.215342i	$-0.930913\pi$
$491$		−	2.30071e6i		−	0.430684i	0.976539	−	0.215342i	$-0.0690866\pi$
$492$		0			0
$493$			5.01210e6i			0.928758i
$494$			236929.i			0.0436818i
$495$		0			0
$496$		−	5.95035e6i		−	1.08602i
$497$	9.01651e6	+	4.10211e6i	1.63737	+	0.744931i
$498$		0			0
$499$	371032.			0.0667053			0.0333526	−	0.999444i	$-0.489382\pi$
$499$	371032.			0.0667053			0.0333526	+	0.999444i	$0.489382\pi$
$500$	−4.54885e6			−0.813723
$501$		0			0
$502$			852491.i			0.150984i
$503$	8.06148e6			1.42067			0.710337	−	0.703861i	$-0.248542\pi$
$503$	8.06148e6			1.42067			0.710337	+	0.703861i	$0.248542\pi$
$504$		0			0
$505$	−589740.			−0.102904
$506$			413289.i			0.0717591i
$507$		0			0
$508$	4.52294e6			0.777609
$509$	−847086.			−0.144922			−0.0724608	−	0.997371i	$-0.523085\pi$
$509$	−847086.			−0.144922			−0.0724608	+	0.997371i	$0.523085\pi$
$510$		0			0
$511$	−5.26964e6	−	2.39745e6i	−0.892747	−	0.406160i
$512$		−	2.54487e6i		−	0.429033i
$513$		0			0
$514$		−	298955.i		−	0.0499113i
$515$			7.63048e6i			1.26775i
$516$		0			0
$517$			3.49811e6i			0.575581i
$518$	240136.	+	109251.i	0.0393217	+	0.0178896i
$519$		0			0
$520$	411768.			0.0667797
$521$	−1.07784e7			−1.73964			−0.869821	−	0.493367i	$-0.835766\pi$
$521$	−1.07784e7			−1.73964			−0.869821	+	0.493367i	$0.835766\pi$
$522$		0			0
$523$		−	166657.i		−	0.0266421i	−0.999911	−	0.0133210i	$-0.995760\pi$
$523$		−	166657.i		−	0.0266421i	0.999911	−	0.0133210i	$-0.00424034\pi$
$524$	−6.30925e6			−1.00381
$525$		0			0
$526$	−67350.9			−0.0106140
$527$		−	1.15882e7i		−	1.81757i
$528$		0			0
$529$	−8.24817e6			−1.28150
$530$	778053.			0.120315
$531$		0			0
$532$	−8.66810e6	−	3.94360e6i	−1.32784	−	0.604107i
$533$		−	1.56481e6i		−	0.238586i
$534$		0			0
$535$		−	1.21962e6i		−	0.184222i
$536$			682570.i			0.102621i
$537$		0			0
$538$		−	1.01079e6i		−	0.150559i
$539$	−2.72765e6	+	2.37723e6i	−0.404405	+	0.352451i
$540$		0			0
$541$	7.47615e6			1.09821			0.549104	−	0.835754i	$-0.314969\pi$
$541$	7.47615e6			1.09821			0.549104	+	0.835754i	$0.314969\pi$
$542$	−467074.			−0.0682947
$543$		0			0
$544$		−	2.96584e6i		−	0.429686i
$545$	−1.02523e7			−1.47853
$546$		0			0
$547$	4.21433e6			0.602227			0.301113	−	0.953588i	$-0.402642\pi$
$547$	4.21433e6			0.602227			0.301113	+	0.953588i	$0.402642\pi$
$548$			6.87045e6i			0.977313i
$549$		0			0
$550$	−92051.8			−0.0129755
$551$	−5.95469e6			−0.835565
$552$		0			0
$553$	2.09042e6	−	4.59477e6i	0.290683	−	0.638927i
$554$		−	916477.i		−	0.126867i
$555$		0			0
$556$		−	1.09248e6i		−	0.149874i
$557$			1.16271e7i			1.58794i	0.607955	+	0.793971i	$0.291990\pi$
$557$			1.16271e7i			1.58794i	−0.607955	+	0.793971i	$0.708010\pi$
$558$		0			0
$559$		−	1.16084e6i		−	0.157124i
$560$	−3.38618e6	+	7.44288e6i	−0.456289	+	1.00293i
$561$		0			0
$562$	940228.			0.125572
$563$	1.83029e6			0.243360			0.121680	−	0.992569i	$-0.461172\pi$
$563$	1.83029e6			0.243360			0.121680	+	0.992569i	$0.461172\pi$
$564$		0			0
$565$		−	1.22574e7i		−	1.61540i
$566$	220332.			0.0289092
$567$		0			0
$568$	−2.44029e6			−0.317373
$569$		−	3.47393e6i		−	0.449822i	−0.974379	−	0.224911i	$-0.927791\pi$
$569$		−	3.47393e6i		−	0.449822i	0.974379	−	0.224911i	$-0.0722091\pi$
$570$		0			0
$571$	−1.24919e7			−1.60339			−0.801695	−	0.597733i	$-0.796068\pi$
$571$	−1.24919e7			−1.60339			−0.801695	+	0.597733i	$0.796068\pi$
$572$	1.39710e6			0.178540
$573$		0			0
$574$	−452568.	−	205898.i	−0.0573330	−	0.0260839i
$575$		−	3.27068e6i		−	0.412543i
$576$		0			0
$577$			1.79598e6i			0.224575i	0.993676	+	0.112288i	$0.0358178\pi$
$577$			1.79598e6i			0.224575i	−0.993676	+	0.112288i	$0.964182\pi$
$578$		−	1.18863e6i		−	0.147988i
$579$		0			0
$580$			5.15407e6i			0.636180i
$581$	−2.03645e6	+	4.47614e6i	−0.250284	+	0.550128i
$582$		0			0
$583$	5.30062e6			0.645885
$584$	1.42621e6			0.173042
$585$		0			0
$586$		−	322983.i		−	0.0388540i
$587$	3.72811e6			0.446574			0.223287	−	0.974753i	$-0.428321\pi$
$587$	3.72811e6			0.446574			0.223287	+	0.974753i	$0.428321\pi$
$588$		0			0
$589$	1.37675e7			1.63519
$590$			1.15254e6i			0.136309i
$591$		0			0
$592$	4.06186e6			0.476344
$593$	515696.			0.0602222			0.0301111	−	0.999547i	$-0.490414\pi$
$593$	515696.			0.0602222			0.0301111	+	0.999547i	$0.490414\pi$
$594$		0			0
$595$	−6.59453e6	+	1.44949e7i	−0.763645	+	1.67850i
$596$		−	1.34818e7i		−	1.55465i
$597$		0			0
$598$		−	392418.i		−	0.0448741i
$599$			6.53346e6i			0.744006i	0.928232	+	0.372003i	$0.121329\pi$
$599$			6.53346e6i			0.744006i	−0.928232	+	0.372003i	$0.878671\pi$
$600$		0			0
$601$		−	8.04609e6i		−	0.908655i	−0.890835	−	0.454327i	$-0.849880\pi$
$601$		−	8.04609e6i		−	0.908655i	0.890835	−	0.454327i	$-0.150120\pi$
$602$	−335734.	−	152744.i	−0.0377575	−	0.0171780i
$603$		0			0
$604$	−8.75656e6			−0.976655
$605$	7.23514e6			0.803635
$606$		0			0
$607$			1.03860e7i			1.14413i	0.820208	+	0.572066i	$0.193858\pi$
$607$			1.03860e7i			1.14413i	−0.820208	+	0.572066i	$0.806142\pi$
$608$	3.52360e6			0.386570
$609$		0			0
$610$	−624454.			−0.0679478
$611$		−	3.32145e6i		−	0.359936i
$612$		0			0
$613$	5.16271e6			0.554915			0.277457	−	0.960738i	$-0.410508\pi$
$613$	5.16271e6			0.554915			0.277457	+	0.960738i	$0.410508\pi$
$614$	−284556.			−0.0304612
$615$		0			0
$616$	369114.	−	811319.i	0.0391930	−	0.0861469i
$617$		−	1.19447e7i		−	1.26318i	−0.775304	−	0.631588i	$-0.782403\pi$
$617$		−	1.19447e7i		−	1.26318i	0.775304	−	0.631588i	$-0.217597\pi$
$618$		0			0
$619$		−	3.37357e6i		−	0.353886i	−0.984221	−	0.176943i	$-0.943379\pi$
$619$		−	3.37357e6i		−	0.353886i	0.984221	−	0.176943i	$-0.0566208\pi$
$620$		−	1.19164e7i		−	1.24500i
$621$		0			0
$622$			1.41152e6i			0.146289i
$623$	2.27690e6	−	5.00466e6i	0.235030	−	0.516600i
$624$		0			0
$625$	−1.17044e7			−1.19853
$626$	87236.4			0.00889738
$627$		0			0
$628$			1.63748e7i			1.65682i
$629$	7.91041e6			0.797210
$630$		0			0
$631$	1.20832e7			1.20811			0.604057	−	0.796941i	$-0.293550\pi$
$631$	1.20832e7			1.20811			0.604057	+	0.796941i	$0.293550\pi$
$632$			1.24356e6i			0.123844i
$633$		0			0
$634$	−896408.			−0.0885692
$635$	8.98567e6			0.884333
$636$		0			0
$637$	2.58990e6	−	2.25718e6i	0.252892	−	0.220403i
$638$		−	277577.i		−	0.0269980i
$639$		0			0
$640$		−	4.06101e6i		−	0.391908i
$641$			2.05311e7i			1.97363i	0.161847	+	0.986816i	$0.448255\pi$
$641$			2.05311e7i			1.97363i	−0.161847	+	0.986816i	$0.551745\pi$
$642$		0			0
$643$			616862.i			0.0588384i	0.999567	+	0.0294192i	$0.00936577\pi$
$643$			616862.i			0.0588384i	−0.999567	+	0.0294192i	$0.990634\pi$
$644$	1.43567e7	+	6.53166e6i	1.36408	+	0.620595i
$645$		0			0
$646$	2.25727e6			0.212815
$647$	5.99062e6			0.562615			0.281308	−	0.959618i	$-0.409232\pi$
$647$	5.99062e6			0.562615			0.281308	+	0.959618i	$0.409232\pi$
$648$		0			0
$649$			7.85183e6i			0.731744i
$650$	87403.2			0.00811417
$651$		0			0
$652$	1.62965e7			1.50132
$653$			6.26768e6i			0.575206i	0.957750	+	0.287603i	$0.0928584\pi$
$653$			6.26768e6i			0.575206i	−0.957750	+	0.287603i	$0.907142\pi$
$654$		0			0
$655$	−1.25345e7			−1.14158
$656$	−7.65514e6			−0.694534
$657$		0			0
$658$	−960616.	−	437038.i	−0.0864938	−	0.0393509i
$659$		−	1.04856e7i		−	0.940542i	−0.882522	−	0.470271i	$-0.844156\pi$
$659$		−	1.04856e7i		−	0.940542i	0.882522	−	0.470271i	$-0.155844\pi$
$660$		0			0
$661$		−	7.74885e6i		−	0.689816i	−0.938636	−	0.344908i	$-0.887910\pi$
$661$		−	7.74885e6i		−	0.689816i	0.938636	−	0.344908i	$-0.112090\pi$
$662$			349628.i			0.0310071i
$663$		0			0
$664$		−	1.21145e6i		−	0.106632i
$665$	−1.72208e7	−	7.83471e6i	−1.51008	−	0.687019i
$666$		0			0
$667$	9.86256e6			0.858371
$668$	1.23460e7			1.07049
$669$		0			0
$670$			675357.i			0.0581228i
$671$	−4.25419e6			−0.364763
$672$		0			0
$673$	−1.68044e7			−1.43017			−0.715083	−	0.699039i	$-0.753611\pi$
$673$	−1.68044e7			−1.43017			−0.715083	+	0.699039i	$0.753611\pi$
$674$			701937.i			0.0595180i
$675$		0			0
$676$	1.04616e7			0.880508
$677$	−3.65107e6			−0.306160			−0.153080	−	0.988214i	$-0.548919\pi$
$677$	−3.65107e6			−0.306160			−0.153080	+	0.988214i	$0.548919\pi$
$678$		0			0
$679$	1.18869e7	+	5.40800e6i	0.989449	+	0.450155i
$680$		−	3.92299e6i		−	0.325346i
$681$		0			0
$682$			641771.i			0.0528347i
$683$			855959.i			0.0702104i	0.999384	+	0.0351052i	$0.0111766\pi$
$683$			855959.i			0.0702104i	−0.999384	+	0.0351052i	$0.988823\pi$
$684$		0			0
$685$			1.36494e7i			1.11145i
$686$	−312031.	−	1.04604e6i	−0.0253156	−	0.0848668i
$687$		0			0
$688$	−5.67889e6			−0.457396
$689$	−5.03294e6			−0.403900
$690$		0			0
$691$		−	2.19345e7i		−	1.74756i	−0.486321	−	0.873780i	$-0.661661\pi$
$691$		−	2.19345e7i		−	1.74756i	0.486321	−	0.873780i	$-0.338339\pi$
$692$	1.94125e7			1.54105
$693$		0			0
$694$	182158.			0.0143565
$695$		−	2.17042e6i		−	0.170444i
$696$		0			0
$697$	−1.49083e7			−1.16237
$698$	979460.			0.0760936
$699$		0			0
$700$	−1.45480e6	+	3.19767e6i	−0.112217	+	0.246654i
$701$		−	1.46197e6i		−	0.112368i	−0.998420	−	0.0561841i	$-0.982107\pi$
$701$		−	1.46197e6i		−	0.112368i	0.998420	−	0.0561841i	$-0.0178934\pi$
$702$		0			0
$703$			9.39806e6i			0.717216i
$704$		−	6.72444e6i		−	0.511357i
$705$		0			0
$706$			233096.i			0.0176004i
$707$	501953.	−	1.10330e6i	0.0377672	−	0.0830129i
$708$		0			0
$709$	304247.			0.0227306			0.0113653	−	0.999935i	$-0.496382\pi$
$709$	304247.			0.0227306			0.0113653	+	0.999935i	$0.496382\pi$
$710$	−2.41450e6			−0.179755
$711$		0			0
$712$			1.35449e6i			0.100133i
$713$	−2.28027e7			−1.67982
$714$		0			0
$715$	2.77560e6			0.203044
$716$			8.10289e6i			0.590687i
$717$		0			0
$718$	679902.			0.0492193
$719$	5.58661e6			0.403020			0.201510	−	0.979486i	$-0.435415\pi$
$719$	5.58661e6			0.403020			0.201510	+	0.979486i	$0.435415\pi$
$720$		0			0
$721$	−1.42753e7	−	6.49464e6i	−1.02270	−	0.465283i
$722$			1.44129e6i			0.102898i
$723$		0			0
$724$			1.92036e7i			1.36156i
$725$			2.19669e6i			0.155211i
$726$		0			0
$727$		−	8.31858e6i		−	0.583732i	−0.956459	−	0.291866i	$-0.905724\pi$
$727$		−	8.31858e6i		−	0.583732i	0.956459	−	0.291866i	$-0.0942761\pi$
$728$	−350474.	+	770347.i	−0.0245091	+	0.0538714i
$729$		0			0
$730$	1.41114e6			0.0980083
$731$	−1.10596e7			−0.765498
$732$		0			0
$733$		−	9.66621e6i		−	0.664502i	−0.943191	−	0.332251i	$-0.892192\pi$
$733$		−	9.66621e6i		−	0.664502i	0.943191	−	0.332251i	$-0.107808\pi$
$734$	461963.			0.0316495
$735$		0			0
$736$	−5.83603e6			−0.397121
$737$			4.60098e6i			0.312020i
$738$		0			0
$739$	2.23123e7			1.50291			0.751456	−	0.659783i	$-0.229352\pi$
$739$	2.23123e7			1.50291			0.751456	+	0.659783i	$0.229352\pi$
$740$	8.13447e6			0.546072
$741$		0			0
$742$	−662235.	+	1.45560e6i	−0.0441573	+	0.0970585i
$743$			2.76395e7i			1.83678i	0.395671	+	0.918392i	$0.370512\pi$
$743$			2.76395e7i			1.83678i	−0.395671	+	0.918392i	$0.629488\pi$
$744$		0			0
$745$		−	2.67841e7i		−	1.76802i
$746$			1.40839e6i			0.0926568i
$747$		0			0
$748$		−	1.33104e7i		−	0.869835i
$749$	2.28171e6	+	1.03808e6i	0.148612	+	0.0676121i
$750$		0			0
$751$	−8.21189e6			−0.531304			−0.265652	−	0.964069i	$-0.585587\pi$
$751$	−8.21189e6			−0.531304			−0.265652	+	0.964069i	$0.585587\pi$
$752$	−1.62487e7			−1.04779
$753$		0			0
$754$			263559.i			0.0168830i
$755$	−1.73966e7			−1.11070
$756$		0			0
$757$	−1.00914e7			−0.640044			−0.320022	−	0.947410i	$-0.603690\pi$
$757$	−1.00914e7			−0.640044			−0.320022	+	0.947410i	$0.603690\pi$
$758$			1.38009e6i			0.0872437i
$759$		0			0
$760$	4.66075e6			0.292700
$761$	−2.03226e7			−1.27209			−0.636046	−	0.771651i	$-0.719431\pi$
$761$	−2.03226e7			−1.27209			−0.636046	+	0.771651i	$0.719431\pi$
$762$		0			0
$763$	8.72620e6	−	1.91803e7i	0.542643	−	1.19274i
$764$			4.11285e6i			0.254923i
$765$		0			0
$766$			1.89255e6i			0.116540i
$767$		−	7.45532e6i		−	0.457591i
$768$		0			0
$769$		−	6.02918e6i		−	0.367657i	−0.982958	−	0.183828i	$-0.941151\pi$
$769$		−	6.02918e6i		−	0.367657i	0.982958	−	0.183828i	$-0.0588490\pi$
$770$	365214.	−	802746.i	0.0221983	−	0.0487923i
$771$		0			0
$772$	5.61034e6			0.338802
$773$	−2.29807e7			−1.38330			−0.691648	−	0.722235i	$-0.743115\pi$
$773$	−2.29807e7			−1.38330			−0.691648	+	0.722235i	$0.743115\pi$
$774$		0			0
$775$		−	5.07884e6i		−	0.303746i
$776$	−3.21714e6			−0.191786
$777$		0			0
$778$	169714.			0.0100524
$779$		−	1.77119e7i		−	1.04574i
$780$		0			0
$781$	−1.64492e7			−0.964977
$782$	−3.73864e6			−0.218623
$783$		0			0
$784$	−1.10422e7	−	1.26699e7i	−0.641603	−	0.736179i
$785$			3.25316e7i			1.88422i
$786$		0			0
$787$			2.09363e7i			1.20494i	0.798143	+	0.602468i	$0.205816\pi$
$787$			2.09363e7i			1.20494i	−0.798143	+	0.602468i	$0.794184\pi$
$788$		−	2.58471e7i		−	1.48285i
$789$		0			0
$790$			1.23042e6i			0.0701432i
$791$	2.29316e7	+	1.04329e7i	1.30315	+	0.592873i
$792$		0			0
$793$	4.03936e6			0.228102
$794$	2.30469e6			0.129736
$795$		0			0
$796$			9.92401e6i			0.555142i
$797$	1.00607e7			0.561024			0.280512	−	0.959851i	$-0.409496\pi$
$797$	1.00607e7			0.561024			0.280512	+	0.959851i	$0.409496\pi$
$798$		0			0
$799$	−3.16441e7			−1.75358
$800$		−	1.29986e6i		−	0.0718078i
$801$		0			0
$802$	1.00610e6			0.0552337
$803$	9.61362e6			0.526136
$804$		0			0
$805$	2.85223e7	+	1.29764e7i	1.55130	+	0.705771i
$806$		−	609362.i		−	0.0330398i
$807$		0			0
$808$			298605.i			0.0160905i
$809$		−	7.35336e6i		−	0.395016i	−0.980301	−	0.197508i	$-0.936715\pi$
$809$		−	7.35336e6i		−	0.395016i	0.980301	−	0.197508i	$-0.0632848\pi$
$810$		0			0
$811$			2.94645e7i			1.57307i	0.617547	+	0.786534i	$0.288126\pi$
$811$			2.94645e7i			1.57307i	−0.617547	+	0.786534i	$0.711874\pi$
$812$	−9.64238e6	−	4.38686e6i	−0.513209	−	0.233487i
$813$		0			0
$814$	−438089.			−0.0231740
$815$	3.23760e7			1.70738
$816$		0			0
$817$		−	1.31394e7i		−	0.688686i
$818$	−1.53596e6			−0.0802597
$819$		0			0
$820$	−1.53305e7			−0.796200
$821$		−	1.09524e7i		−	0.567088i	−0.958959	−	0.283544i	$-0.908490\pi$
$821$		−	1.09524e7i		−	0.567088i	0.958959	−	0.283544i	$-0.0915101\pi$
$822$		0			0
$823$	1.12004e7			0.576413			0.288207	−	0.957568i	$-0.406941\pi$
$823$	1.12004e7			0.576413			0.288207	+	0.957568i	$0.406941\pi$
$824$	3.86357e6			0.198231
$825$		0			0
$826$	−2.15620e6	−	980973.i	−0.109961	−	0.0500273i
$827$			1.16800e7i			0.593853i	0.954900	+	0.296927i	$0.0959617\pi$
$827$			1.16800e7i			0.593853i	−0.954900	+	0.296927i	$0.904038\pi$
$828$		0			0
$829$			6.68871e6i			0.338031i	0.985613	+	0.169015i	$0.0540588\pi$
$829$			6.68871e6i			0.338031i	−0.985613	+	0.169015i	$0.945941\pi$
$830$		−	1.19865e6i		−	0.0603945i
$831$		0			0
$832$			6.38486e6i			0.319774i
$833$	−2.15046e7	−	2.46745e7i	−1.07379	−	1.23207i
$834$		0			0
$835$	2.45276e7			1.21742
$836$	1.58136e7			0.782554
$837$		0			0
$838$		−	1.84561e6i		−	0.0907881i
$839$	2.99936e7			1.47104			0.735519	−	0.677505i	$-0.236939\pi$
$839$	2.99936e7			1.47104			0.735519	+	0.677505i	$0.236939\pi$
$840$		0			0
$841$	1.38872e7			0.677054
$842$		−	898142.i		−	0.0436581i
$843$		0			0
$844$	7.88940e6			0.381231
$845$	2.07840e7			1.00136
$846$		0			0
$847$	−6.15815e6	+	1.35357e7i	−0.294945	+	0.648295i
$848$			2.46214e7i			1.17577i
$849$		0			0
$850$		−	832707.i		−	0.0395316i
$851$		−	1.55657e7i		−	0.736792i
$852$		0			0
$853$		−	1.80757e7i		−	0.850595i	−0.905054	−	0.425297i	$-0.860169\pi$
$853$		−	1.80757e7i		−	0.850595i	0.905054	−	0.425297i	$-0.139831\pi$
$854$	531500.	−	1.16825e6i	0.0249378	−	0.0548138i
$855$		0			0
$856$	−617536.			−0.0288057
$857$	−8.65519e6			−0.402554			−0.201277	−	0.979534i	$-0.564509\pi$
$857$	−8.65519e6			−0.402554			−0.201277	+	0.979534i	$0.564509\pi$
$858$		0			0
$859$			2.15926e7i			0.998440i	0.866475	+	0.499220i	$0.166380\pi$
$859$			2.15926e7i			0.998440i	−0.866475	+	0.499220i	$0.833620\pi$
$860$	−1.13728e7			−0.524351
$861$		0			0
$862$	2.37950e6			0.109073
$863$		−	1.45980e6i		−	0.0667215i	−0.999443	−	0.0333608i	$-0.989379\pi$
$863$		−	1.45980e6i		−	0.0667215i	0.999443	−	0.0333608i	$-0.0106210\pi$
$864$		0			0
$865$	3.85667e7			1.75256
$866$	1.44892e6			0.0656524
$867$		0			0
$868$	2.22936e7	+	1.01426e7i	1.00434	+	0.456931i
$869$			8.38244e6i			0.376548i
$870$		0			0
$871$		−	4.36863e6i		−	0.195119i
$872$			5.19109e6i			0.231189i
$873$		0			0
$874$		−	4.44173e6i		−	0.196686i
$875$	7.69191e6	−	1.69070e7i	0.339636	−	0.746526i
$876$		0			0
$877$	−3.46821e7			−1.52267			−0.761336	−	0.648358i	$-0.775456\pi$
$877$	−3.46821e7			−1.52267			−0.761336	+	0.648358i	$0.775456\pi$
$878$	3.25457e6			0.142481
$879$		0			0
$880$		−	1.35784e7i		−	0.591072i
$881$	7.75832e6			0.336766			0.168383	−	0.985722i	$-0.446145\pi$
$881$	7.75832e6			0.336766			0.168383	+	0.985722i	$0.446145\pi$
$882$		0			0
$883$	−4.09308e7			−1.76664			−0.883320	−	0.468770i	$-0.844697\pi$
$883$	−4.09308e7			−1.76664			−0.883320	+	0.468770i	$0.844697\pi$
$884$			1.26382e7i			0.543946i
$885$		0			0
$886$	−1.32331e6			−0.0566338
$887$	−3.70958e6			−0.158313			−0.0791564	−	0.996862i	$-0.525223\pi$
$887$	−3.70958e6			−0.158313			−0.0791564	+	0.996862i	$0.525223\pi$
$888$		0			0
$889$	−7.64810e6	+	1.68106e7i	−0.324563	+	0.713395i
$890$			1.34018e6i			0.0567138i
$891$		0			0
$892$		−	3.00914e7i		−	1.26628i
$893$		−	3.75951e7i		−	1.57762i
$894$		0			0
$895$			1.60979e7i			0.671757i
$896$	7.59745e6	+	3.45650e6i	0.316153	+	0.143836i
$897$		0			0
$898$	−2.07988e6			−0.0860690
$899$	1.53150e7			0.632000
$900$		0			0
$901$			4.79497e7i			1.96777i
$902$	825639.			0.0337889
$903$		0			0
$904$	−6.20635e6			−0.252590
$905$			3.81515e7i			1.54843i
$906$		0			0
$907$	−3.93881e6			−0.158981			−0.0794907	−	0.996836i	$-0.525329\pi$
$907$	−3.93881e6			−0.158981			−0.0794907	+	0.996836i	$0.525329\pi$
$908$	−1.75618e7			−0.706894
$909$		0			0
$910$	−346771.	+	762208.i	−0.0138816	+	0.0305119i
$911$		−	1.26943e7i		−	0.506771i	−0.967365	−	0.253386i	$-0.918456\pi$
$911$		−	1.26943e7i		−	0.506771i	0.967365	−	0.253386i	$-0.0815441\pi$
$912$		0			0
$913$		−	8.16601e6i		−	0.324215i
$914$		−	2.59332e6i		−	0.102681i
$915$		0			0
$916$			1.44802e7i			0.570210i
$917$	1.06687e7	−	2.34499e7i	0.418974	−	0.920913i
$918$		0			0
$919$	2.42488e7			0.947112			0.473556	−	0.880764i	$-0.342970\pi$
$919$	2.42488e7			0.947112			0.473556	+	0.880764i	$0.342970\pi$
$920$	−7.71946e6			−0.300689
$921$		0			0
$922$			487440.i			0.0188840i
$923$	1.56185e7			0.603442
$924$		0			0
$925$	3.46695e6			0.133227
$926$		−	370019.i		−	0.0141807i
$927$		0			0
$928$	3.91965e6			0.149409
$929$	−2.02477e7			−0.769726			−0.384863	−	0.922974i	$-0.625751\pi$
$929$	−2.02477e7			−0.769726			−0.384863	+	0.922974i	$0.625751\pi$
$930$		0			0
$931$	2.93148e7	−	2.55487e7i	1.10844	−	0.966040i
$932$			6.40788e6i			0.241643i
$933$		0			0
$934$		−	2.40701e6i		−	0.0902841i
$935$		−	2.64436e7i		−	0.989218i
$936$		0			0
$937$		−	5.18764e7i		−	1.93028i	−0.261732	−	0.965141i	$-0.584294\pi$
$937$		−	5.18764e7i		−	1.93028i	0.261732	−	0.965141i	$-0.415706\pi$
$938$	−1.26348e6	−	574826.i	−0.0468879	−	0.0213319i
$939$		0			0
$940$	−3.25404e7			−1.20117
$941$	2.42851e7			0.894058			0.447029	−	0.894519i	$-0.352482\pi$
$941$	2.42851e7			0.894058			0.447029	+	0.894519i	$0.352482\pi$
$942$		0			0
$943$			2.93357e7i			1.07428i
$944$	−3.64718e7			−1.33207
$945$		0			0
$946$	612493.			0.0222522
$947$			2.07717e7i			0.752656i	0.926487	+	0.376328i	$0.122813\pi$
$947$			2.07717e7i			0.752656i	−0.926487	+	0.376328i	$0.877187\pi$
$948$		0			0
$949$	−9.12813e6			−0.329016
$950$	989307.			0.0355649
$951$		0			0
$952$	7.33924e6	+	3.33903e6i	0.262457	+	0.119406i
$953$			3.07837e7i			1.09797i	0.835834	+	0.548983i	$0.184985\pi$
$953$			3.07837e7i			1.09797i	−0.835834	+	0.548983i	$0.815015\pi$
$954$		0			0
$955$			8.17095e6i			0.289911i
$956$			2.31004e7i			0.817475i
$957$		0			0
$958$		−	239606.i		−	0.00843498i
$959$	−2.55358e7	−	1.16176e7i	−0.896608	−	0.407917i
$960$		0			0
$961$	−6.77977e6			−0.236814
$962$	415966.			0.0144917
$963$		0			0
$964$		−	4.73627e7i		−	1.64151i
$965$	1.11460e7			0.385302
$966$		0			0
$967$	2.32637e7			0.800042			0.400021	−	0.916506i	$-0.369003\pi$
$967$	2.32637e7			0.800042			0.400021	+	0.916506i	$0.369003\pi$
$968$		−	3.66339e6i		−	0.125659i
$969$		0			0
$970$	−3.18315e6			−0.108625
$971$	1.28300e6			0.0436696			0.0218348	−	0.999762i	$-0.493049\pi$
$971$	1.28300e6			0.0436696			0.0218348	+	0.999762i	$0.493049\pi$
$972$		0			0
$973$	4.06048e6	+	1.84734e6i	0.137498	+	0.0625553i
$974$			978843.i			0.0330610i
$975$		0			0
$976$		−	1.97607e7i		−	0.664016i
$977$			3.67144e7i			1.23055i	0.788311	+	0.615276i	$0.210956\pi$
$977$			3.67144e7i			1.23055i	−0.788311	+	0.615276i	$0.789044\pi$
$978$		0			0
$979$			9.13021e6i			0.304456i
$980$	−2.21137e7	−	2.53734e7i	−0.735522	−	0.843942i
$981$		0			0
$982$	−1.15262e6			−0.0381423
$983$	−2.29357e7			−0.757056			−0.378528	−	0.925590i	$-0.623570\pi$
$983$	−2.29357e7			−0.757056			−0.378528	+	0.925590i	$0.623570\pi$
$984$		0			0
$985$		−	5.13502e7i		−	1.68636i
$986$	2.51098e6			0.0822529
$987$		0			0
$988$	−1.50150e7			−0.489365
$989$			2.17624e7i			0.707484i
$990$		0			0
$991$	4.60249e6			0.148871			0.0744353	−	0.997226i	$-0.476285\pi$
$991$	4.60249e6			0.148871			0.0744353	+	0.997226i	$0.476285\pi$
$992$	−9.06242e6			−0.292392
$993$		0			0
$994$	2.05509e6	−	4.51712e6i	0.0659727	−	0.145009i
$995$			1.97159e7i			0.631334i
$996$		0			0
$997$			3.89131e7i			1.23982i	0.784674	+	0.619908i	$0.212830\pi$
$997$			3.89131e7i			1.23982i	−0.784674	+	0.619908i	$0.787170\pi$
$998$		−	185881.i		−	0.00590756i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.6.c.b.62.4	yes	8
3.2	odd	2	inner	63.6.c.b.62.5	yes	8
4.3	odd	2		1008.6.k.b.881.5		8
7.6	odd	2	inner	63.6.c.b.62.3	✓	8
12.11	even	2		1008.6.k.b.881.3		8
21.20	even	2	inner	63.6.c.b.62.6	yes	8
28.27	even	2		1008.6.k.b.881.4		8
84.83	odd	2		1008.6.k.b.881.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.c.b.62.3	✓	8	7.6	odd	2	inner
63.6.c.b.62.4	yes	8	1.1	even	1	trivial
63.6.c.b.62.5	yes	8	3.2	odd	2	inner
63.6.c.b.62.6	yes	8	21.20	even	2	inner
1008.6.k.b.881.3		8	12.11	even	2
1008.6.k.b.881.4		8	28.27	even	2
1008.6.k.b.881.5		8	4.3	odd	2
1008.6.k.b.881.6		8	84.83	odd	2

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:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.500983i q^{2} +31.7490 q^{4} +63.0754 q^{5} +(-53.6863 + 118.003i) q^{7} -31.9372i q^{8} +O(q^{10})\)	\(q-0.500983i q^{2} +31.7490 q^{4} +63.0754 q^{5} +(-53.6863 + 118.003i) q^{7} -31.9372i q^{8} -31.5997i q^{10} -215.278i q^{11} +204.407i q^{13} +(59.1177 + 26.8959i) q^{14} +999.969 q^{16} +1947.42 q^{17} +2313.66i q^{19} +2002.58 q^{20} -107.851 q^{22} -3832.04i q^{23} +853.510 q^{25} +102.404 q^{26} +(-1704.49 + 3746.49i) q^{28} +2573.71i q^{29} -5950.54i q^{31} -1522.96i q^{32} -975.626i q^{34} +(-3386.28 + 7443.11i) q^{35} +4061.99 q^{37} +1159.10 q^{38} -2014.45i q^{40} -7655.38 q^{41} -5679.07 q^{43} -6834.88i q^{44} -1919.79 q^{46} -16249.2 q^{47} +(-11042.6 - 12670.3i) q^{49} -427.594i q^{50} +6489.72i q^{52} +24622.1i q^{53} -13578.8i q^{55} +(3768.69 + 1714.59i) q^{56} +1289.39 q^{58} -36472.9 q^{59} -19761.4i q^{61} -2981.12 q^{62} +31236.0 q^{64} +12893.1i q^{65} -21372.2 q^{67} +61828.8 q^{68} +(3728.87 + 1696.47i) q^{70} -76408.9i q^{71} +44656.7i q^{73} -2034.99i q^{74} +73456.4i q^{76} +(25403.6 + 11557.5i) q^{77} -38937.7 q^{79} +63073.4 q^{80} +3835.22i q^{82} +37932.3 q^{83} +122835. q^{85} +2845.12i q^{86} -6875.39 q^{88} -42411.2 q^{89} +(-24120.7 - 10973.8i) q^{91} -121663. i q^{92} +8140.59i q^{94} +145935. i q^{95} -100733. i q^{97} +(-6347.62 + 5532.14i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 112 q^{7}+O(q^{10}) \) 8 * q - 112 * q^7	\( 8 q - 112 q^{7} - 128 q^{16} + 5360 q^{22} + 9368 q^{25} - 10080 q^{28} - 34304 q^{37} + 32416 q^{43} + 71888 q^{46} - 106120 q^{49} - 115792 q^{58} + 253952 q^{64} + 152608 q^{67} - 260736 q^{70} - 94592 q^{79} + 475200 q^{85} - 26048 q^{88} - 501312 q^{91}+O(q^{100}) \) 8 * q - 112 * q^7 - 128 * q^16 + 5360 * q^22 + 9368 * q^25 - 10080 * q^28 - 34304 * q^37 + 32416 * q^43 + 71888 * q^46 - 106120 * q^49 - 115792 * q^58 + 253952 * q^64 + 152608 * q^67 - 260736 * q^70 - 94592 * q^79 + 475200 * q^85 - 26048 * q^88 - 501312 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more