LMFDB - Embedded newform 441.8.a.r.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,8,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	441.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:	$137.761796238$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 306x^{2} - 228x + 10152 $ x^4 - x^3 - 306x^2 - 228x + 10152
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3\cdot 7 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.62848$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.62848 q^{2} -96.3202 q^{4} +502.942 q^{5} -1262.58 q^{8} +O(q^{10})$	$q+5.62848 q^{2} -96.3202 q^{4} +502.942 q^{5} -1262.58 q^{8} +2830.80 q^{10} +6201.78 q^{11} +9719.32 q^{13} +5222.58 q^{16} -13909.4 q^{17} +11828.6 q^{19} -48443.5 q^{20} +34906.6 q^{22} +2837.44 q^{23} +174826. q^{25} +54705.0 q^{26} +48836.2 q^{29} -126.044 q^{31} +191006. q^{32} -78288.8 q^{34} -367454. q^{37} +66576.8 q^{38} -635005. q^{40} -133287. q^{41} +812254. q^{43} -597357. q^{44} +15970.5 q^{46} -1.10481e6 q^{47} +984003. q^{50} -936168. q^{52} +772331. q^{53} +3.11914e6 q^{55} +274874. q^{58} +712666. q^{59} -1.25322e6 q^{61} -709.434 q^{62} +406580. q^{64} +4.88826e6 q^{65} -750217. q^{67} +1.33976e6 q^{68} +1.13668e6 q^{71} +4.17125e6 q^{73} -2.06821e6 q^{74} -1.13933e6 q^{76} -6.43960e6 q^{79} +2.62666e6 q^{80} -750202. q^{82} -1.83464e6 q^{83} -6.99563e6 q^{85} +4.57175e6 q^{86} -7.83025e6 q^{88} +1.40425e6 q^{89} -273303. q^{92} -6.21838e6 q^{94} +5.94908e6 q^{95} +6.70215e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 101 q^{4} + 196 q^{5} + 1347 q^{8}+O(q^{10}) $ 4 * q + q^2 + 101 * q^4 + 196 * q^5 + 1347 * q^8	$ 4 q + q^{2} + 101 q^{4} + 196 q^{5} + 1347 q^{8} + 5185 q^{10} + 5210 q^{11} + 3794 q^{13} - 21055 q^{16} - 11436 q^{17} + 69158 q^{19} - 87991 q^{20} + 57263 q^{22} + 146220 q^{23} + 6230 q^{25} + 107696 q^{26} + 70664 q^{29} + 288618 q^{31} - 2653 q^{32} + 495996 q^{34} - 448902 q^{37} + 528944 q^{38} - 767361 q^{40} - 663316 q^{41} + 554 q^{43} + 686635 q^{44} - 1064964 q^{46} + 762180 q^{47} + 525428 q^{50} - 3423848 q^{52} + 2761920 q^{53} + 1965056 q^{55} - 4451395 q^{58} + 3410898 q^{59} - 300892 q^{61} - 2066175 q^{62} - 2916551 q^{64} + 8019032 q^{65} - 4222478 q^{67} + 5770500 q^{68} + 380964 q^{71} + 451674 q^{73} + 10091220 q^{74} + 14747656 q^{76} - 12154822 q^{79} + 7870085 q^{80} - 8814904 q^{82} - 12087978 q^{83} - 7375500 q^{85} + 32881826 q^{86} - 14439051 q^{88} + 4955752 q^{89} - 206892 q^{92} - 5960646 q^{94} - 8460784 q^{95} + 22840614 q^{97}+O(q^{100}) $ 4 * q + q^2 + 101 * q^4 + 196 * q^5 + 1347 * q^8 + 5185 * q^10 + 5210 * q^11 + 3794 * q^13 - 21055 * q^16 - 11436 * q^17 + 69158 * q^19 - 87991 * q^20 + 57263 * q^22 + 146220 * q^23 + 6230 * q^25 + 107696 * q^26 + 70664 * q^29 + 288618 * q^31 - 2653 * q^32 + 495996 * q^34 - 448902 * q^37 + 528944 * q^38 - 767361 * q^40 - 663316 * q^41 + 554 * q^43 + 686635 * q^44 - 1064964 * q^46 + 762180 * q^47 + 525428 * q^50 - 3423848 * q^52 + 2761920 * q^53 + 1965056 * q^55 - 4451395 * q^58 + 3410898 * q^59 - 300892 * q^61 - 2066175 * q^62 - 2916551 * q^64 + 8019032 * q^65 - 4222478 * q^67 + 5770500 * q^68 + 380964 * q^71 + 451674 * q^73 + 10091220 * q^74 + 14747656 * q^76 - 12154822 * q^79 + 7870085 * q^80 - 8814904 * q^82 - 12087978 * q^83 - 7375500 * q^85 + 32881826 * q^86 - 14439051 * q^88 + 4955752 * q^89 - 206892 * q^92 - 5960646 * q^94 - 8460784 * q^95 + 22840614 * q^97

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 $

$2$	5.62848	0.497492	0.248746	−	0.968569i	$-0.419982\pi$
$2$	5.62848	0.497492	0.248746	+	0.968569i	$0.419982\pi$
$3$	0	0
$3$	0	0
$4$	−96.3202	−0.752502
$5$	502.942	1.79938	0.899690	−	0.436529i	$-0.143792\pi$
$5$	502.942	1.79938	0.899690	+	0.436529i	$0.143792\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1262.58	−0.871855
$9$	0	0
$10$	2830.80	0.895177
$11$	6201.78	1.40489	0.702444	−	0.711739i	$-0.252092\pi$
$11$	6201.78	1.40489	0.702444	+	0.711739i	$0.252092\pi$
$12$	0	0
$13$	9719.32	1.22697	0.613485	−	0.789706i	$-0.289767\pi$
$13$	9719.32	1.22697	0.613485	+	0.789706i	$0.289767\pi$
$14$	0	0
$15$	0	0
$16$	5222.58	0.318761
$17$	−13909.4	−0.686654	−0.343327	−	0.939216i	$-0.611554\pi$
$17$	−13909.4	−0.686654	−0.343327	+	0.939216i	$0.611554\pi$
$18$	0	0
$19$	11828.6	0.395635	0.197817	−	0.980239i	$-0.436615\pi$
$19$	11828.6	0.395635	0.197817	+	0.980239i	$0.436615\pi$
$20$	−48443.5	−1.35404
$21$	0	0
$22$	34906.6	0.698921
$23$	2837.44	0.0486273	0.0243136	−	0.999704i	$-0.492260\pi$
$23$	2837.44	0.0486273	0.0243136	+	0.999704i	$0.492260\pi$
$24$	0	0
$25$	174826.	2.23777
$26$	54705.0	0.610408
$27$	0	0
$28$	0	0
$29$	48836.2	0.371834	0.185917	−	0.982565i	$-0.440474\pi$
$29$	48836.2	0.371834	0.185917	+	0.982565i	$0.440474\pi$
$30$	0	0
$31$	−126.044	−0.000759898 0	−0.000379949	−	1.00000i	$-0.500121\pi$
$31$	−126.044	−0.000759898 0	−0.000379949		1.00000i	$0.500121\pi$
$32$	191006.	1.03044
$33$	0	0
$34$	−78288.8	−0.341604
$35$	0	0
$36$	0	0
$37$	−367454.	−1.19261	−0.596303	−	0.802759i	$-0.703364\pi$
$37$	−367454.	−1.19261	−0.596303	+	0.802759i	$0.703364\pi$
$38$	66576.8	0.196825
$39$	0	0
$40$	−635005.	−1.56880
$41$	−133287.	−0.302026	−0.151013	−	0.988532i	$-0.548253\pi$
$41$	−133287.	−0.302026	−0.151013	+	0.988532i	$0.548253\pi$
$42$	0	0
$43$	812254.	1.55795	0.778973	−	0.627058i	$-0.215741\pi$
$43$	812254.	1.55795	0.778973	+	0.627058i	$0.215741\pi$
$44$	−597357.	−1.05718
$45$	0	0
$46$	15970.5	0.0241917
$47$	−1.10481e6	−1.55219	−0.776093	−	0.630619i	$-0.782801\pi$
$47$	−1.10481e6	−1.55219	−0.776093	+	0.630619i	$0.782801\pi$
$48$	0	0
$49$	0	0
$50$	984003.	1.11327
$51$	0	0
$52$	−936168.	−0.923298
$53$	772331.	0.712587	0.356293	−	0.934374i	$-0.384040\pi$
$53$	772331.	0.712587	0.356293	+	0.934374i	$0.384040\pi$
$54$	0	0
$55$	3.11914e6	2.52793
$56$	0	0
$57$	0	0
$58$	274874.	0.184984
$59$	712666.	0.451756	0.225878	−	0.974156i	$-0.427475\pi$
$59$	712666.	0.451756	0.225878	+	0.974156i	$0.427475\pi$
$60$	0	0
$61$	−1.25322e6	−0.706924	−0.353462	−	0.935449i	$-0.614996\pi$
$61$	−1.25322e6	−0.706924	−0.353462	+	0.935449i	$0.614996\pi$
$62$	−709.434	−0.000378043 0
$63$	0	0
$64$	406580.	0.193873
$65$	4.88826e6	2.20779
$66$	0	0
$67$	−750217.	−0.304737	−0.152369	−	0.988324i	$-0.548690\pi$
$67$	−750217.	−0.304737	−0.152369	+	0.988324i	$0.548690\pi$
$68$	1.33976e6	0.516708
$69$	0	0
$70$	0	0
$71$	1.13668e6	0.376908	0.188454	−	0.982082i	$-0.439652\pi$
$71$	1.13668e6	0.376908	0.188454	+	0.982082i	$0.439652\pi$
$72$	0	0
$73$	4.17125e6	1.25498	0.627489	−	0.778625i	$-0.284083\pi$
$73$	4.17125e6	1.25498	0.627489	+	0.778625i	$0.284083\pi$
$74$	−2.06821e6	−0.593312
$75$	0	0
$76$	−1.13933e6	−0.297716
$77$	0	0
$78$	0	0
$79$	−6.43960e6	−1.46948	−0.734741	−	0.678348i	$-0.762696\pi$
$79$	−6.43960e6	−1.46948	−0.734741	+	0.678348i	$0.762696\pi$
$80$	2.62666e6	0.573572
$81$	0	0
$82$	−750202.	−0.150255
$83$	−1.83464e6	−0.352191	−0.176095	−	0.984373i	$-0.556347\pi$
$83$	−1.83464e6	−0.352191	−0.176095	+	0.984373i	$0.556347\pi$
$84$	0	0
$85$	−6.99563e6	−1.23555
$86$	4.57175e6	0.775065
$87$	0	0
$88$	−7.83025e6	−1.22486
$89$	1.40425e6	0.211145	0.105572	−	0.994412i	$-0.466333\pi$
$89$	1.40425e6	0.211145	0.105572	+	0.994412i	$0.466333\pi$
$90$	0	0
$91$	0	0
$92$	−273303.	−0.0365921
$93$	0	0
$94$	−6.21838e6	−0.772200
$95$	5.94908e6	0.711897
$96$	0	0
$97$	6.70215e6	0.745612	0.372806	−	0.927909i	$-0.378396\pi$
$97$	6.70215e6	0.745612	0.372806	+	0.927909i	$0.378396\pi$
$98$	0	0
$99$	0	0
$100$	−1.68393e7	−1.68393
$101$	383427.	0.0370303	0.0185152	−	0.999829i	$-0.494106\pi$
$101$	383427.	0.0370303	0.0185152	+	0.999829i	$0.494106\pi$
$102$	0	0
$103$	1.50402e7	1.35620	0.678099	−	0.734970i	$-0.262804\pi$
$103$	1.50402e7	1.35620	0.678099	+	0.734970i	$0.262804\pi$
$104$	−1.22714e7	−1.06974
$105$	0	0
$106$	4.34705e6	0.354506
$107$	−7.31846e6	−0.577532	−0.288766	−	0.957400i	$-0.593245\pi$
$107$	−7.31846e6	−0.577532	−0.288766	+	0.957400i	$0.593245\pi$
$108$	0	0
$109$	−1.28634e7	−0.951402	−0.475701	−	0.879607i	$-0.657806\pi$
$109$	−1.28634e7	−0.951402	−0.475701	+	0.879607i	$0.657806\pi$
$110$	1.75560e7	1.25762
$111$	0	0
$112$	0	0
$113$	900842.	0.0587319	0.0293660	−	0.999569i	$-0.490651\pi$
$113$	900842.	0.0587319	0.0293660	+	0.999569i	$0.490651\pi$
$114$	0	0
$115$	1.42707e6	0.0874990
$116$	−4.70392e6	−0.279806
$117$	0	0
$118$	4.01122e6	0.224745
$119$	0	0
$120$	0	0
$121$	1.89749e7	0.973712
$122$	−7.05372e6	−0.351689
$123$	0	0
$124$	12140.6	0.000571825 0
$125$	4.86349e7	2.22722
$126$	0	0
$127$	2.48188e6	0.107515	0.0537573	−	0.998554i	$-0.482880\pi$
$127$	2.48188e6	0.107515	0.0537573	+	0.998554i	$0.482880\pi$
$128$	−2.21603e7	−0.933986
$129$	0	0
$130$	2.75134e7	1.09836
$131$	2.80747e7	1.09110	0.545552	−	0.838077i	$-0.316320\pi$
$131$	2.80747e7	1.09110	0.545552	+	0.838077i	$0.316320\pi$
$132$	0	0
$133$	0	0
$134$	−4.22258e6	−0.151604
$135$	0	0
$136$	1.75618e7	0.598663
$137$	−2.19765e7	−0.730191	−0.365095	−	0.930970i	$-0.618964\pi$
$137$	−2.19765e7	−0.730191	−0.365095	+	0.930970i	$0.618964\pi$
$138$	0	0
$139$	−7.55942e6	−0.238746	−0.119373	−	0.992849i	$-0.538088\pi$
$139$	−7.55942e6	−0.238746	−0.119373	+	0.992849i	$0.538088\pi$
$140$	0	0
$141$	0	0
$142$	6.39779e6	0.187508
$143$	6.02771e7	1.72376
$144$	0	0
$145$	2.45618e7	0.669071
$146$	2.34778e7	0.624341
$147$	0	0
$148$	3.53933e7	0.897439
$149$	4.71800e7	1.16844	0.584220	−	0.811595i	$-0.301401\pi$
$149$	4.71800e7	1.16844	0.584220	+	0.811595i	$0.301401\pi$
$150$	0	0
$151$	5.33371e7	1.26070	0.630348	−	0.776313i	$-0.282912\pi$
$151$	5.33371e7	1.26070	0.630348	+	0.776313i	$0.282912\pi$
$152$	−1.49345e7	−0.344936
$153$	0	0
$154$	0	0
$155$	−63392.7	−0.00136735
$156$	0	0
$157$	7.09602e7	1.46341	0.731705	−	0.681622i	$-0.238725\pi$
$157$	7.09602e7	1.46341	0.731705	+	0.681622i	$0.238725\pi$
$158$	−3.62451e7	−0.731055
$159$	0	0
$160$	9.60648e7	1.85415
$161$	0	0
$162$	0	0
$163$	9.83380e7	1.77854	0.889272	−	0.457379i	$-0.151212\pi$
$163$	9.83380e7	1.77854	0.889272	+	0.457379i	$0.151212\pi$
$164$	1.28382e7	0.227275
$165$	0	0
$166$	−1.03262e7	−0.175212
$167$	7.36791e7	1.22416	0.612078	−	0.790797i	$-0.290334\pi$
$167$	7.36791e7	1.22416	0.612078	+	0.790797i	$0.290334\pi$
$168$	0	0
$169$	3.17167e7	0.505458
$170$	−3.93747e7	−0.614676
$171$	0	0
$172$	−7.82365e7	−1.17236
$173$	−1.11345e8	−1.63497	−0.817484	−	0.575952i	$-0.804632\pi$
$173$	−1.11345e8	−1.63497	−0.817484	+	0.575952i	$0.804632\pi$
$174$	0	0
$175$	0	0
$176$	3.23893e7	0.447824
$177$	0	0
$178$	7.90380e6	0.105043
$179$	−4.73180e6	−0.0616652	−0.0308326	−	0.999525i	$-0.509816\pi$
$179$	−4.73180e6	−0.0616652	−0.0308326	+	0.999525i	$0.509816\pi$
$180$	0	0
$181$	−1.00313e8	−1.25742	−0.628710	−	0.777640i	$-0.716417\pi$
$181$	−1.00313e8	−1.25742	−0.628710	+	0.777640i	$0.716417\pi$
$182$	0	0
$183$	0	0
$184$	−3.58250e6	−0.0423960
$185$	−1.84808e8	−2.14595
$186$	0	0
$187$	−8.62631e7	−0.964672
$188$	1.06415e8	1.16802
$189$	0	0
$190$	3.34843e7	0.354163
$191$	6.47581e7	0.672477	0.336238	−	0.941777i	$-0.390845\pi$
$191$	6.47581e7	0.672477	0.336238	+	0.941777i	$0.390845\pi$
$192$	0	0
$193$	7.66151e7	0.767121	0.383560	−	0.923516i	$-0.374698\pi$
$193$	7.66151e7	0.767121	0.383560	+	0.923516i	$0.374698\pi$
$194$	3.77229e7	0.370936
$195$	0	0
$196$	0	0
$197$	1.47076e8	1.37060	0.685301	−	0.728260i	$-0.259671\pi$
$197$	1.47076e8	1.37060	0.685301	+	0.728260i	$0.259671\pi$
$198$	0	0
$199$	294064.	0.00264518	0.00132259	−	0.999999i	$-0.499579\pi$
$199$	294064.	0.00264518	0.00132259	+	0.999999i	$0.499579\pi$
$200$	−2.20732e8	−1.95101
$201$	0	0
$202$	2.15811e6	0.0184223
$203$	0	0
$204$	0	0
$205$	−6.70355e7	−0.543459
$206$	8.46534e7	0.674698
$207$	0	0
$208$	5.07599e7	0.391111
$209$	7.33581e7	0.555823
$210$	0	0
$211$	−1.05229e8	−0.771163	−0.385582	−	0.922674i	$-0.625999\pi$
$211$	−1.05229e8	−0.771163	−0.385582	+	0.922674i	$0.625999\pi$
$212$	−7.43911e7	−0.536223
$213$	0	0
$214$	−4.11918e7	−0.287318
$215$	4.08517e8	2.80334
$216$	0	0
$217$	0	0
$218$	−7.24015e7	−0.473315
$219$	0	0
$220$	−3.00436e8	−1.90227
$221$	−1.35190e8	−0.842504
$222$	0	0
$223$	−3.68334e6	−0.0222420	−0.0111210	−	0.999938i	$-0.503540\pi$
$223$	−3.68334e6	−0.0222420	−0.0111210	+	0.999938i	$0.503540\pi$
$224$	0	0
$225$	0	0
$226$	5.07037e6	0.0292186
$227$	−7.72184e6	−0.0438157	−0.0219079	−	0.999760i	$-0.506974\pi$
$227$	−7.72184e6	−0.0438157	−0.0219079	+	0.999760i	$0.506974\pi$
$228$	0	0
$229$	2.95099e8	1.62384	0.811920	−	0.583769i	$-0.198423\pi$
$229$	2.95099e8	1.62384	0.811920	+	0.583769i	$0.198423\pi$
$230$	8.03223e6	0.0435300
$231$	0	0
$232$	−6.16597e7	−0.324185
$233$	9.97914e7	0.516830	0.258415	−	0.966034i	$-0.416800\pi$
$233$	9.97914e7	0.516830	0.258415	+	0.966034i	$0.416800\pi$
$234$	0	0
$235$	−5.55654e8	−2.79297
$236$	−6.86442e7	−0.339947
$237$	0	0
$238$	0	0
$239$	1.79058e8	0.848402	0.424201	−	0.905568i	$-0.360555\pi$
$239$	1.79058e8	0.848402	0.424201	+	0.905568i	$0.360555\pi$
$240$	0	0
$241$	7.10799e7	0.327105	0.163553	−	0.986535i	$-0.447705\pi$
$241$	7.10799e7	0.327105	0.163553	+	0.986535i	$0.447705\pi$
$242$	1.06800e8	0.484414
$243$	0	0
$244$	1.20710e8	0.531962
$245$	0	0
$246$	0	0
$247$	1.14966e8	0.485432
$248$	159140.	0.000662521 0
$249$	0	0
$250$	2.73740e8	1.10802
$251$	−5.85579e7	−0.233737	−0.116868	−	0.993147i	$-0.537286\pi$
$251$	−5.85579e7	−0.233737	−0.116868	+	0.993147i	$0.537286\pi$
$252$	0	0
$253$	1.75972e7	0.0683159
$254$	1.39692e7	0.0534876
$255$	0	0
$256$	−1.76771e8	−0.658523
$257$	2.78488e8	1.02339	0.511694	−	0.859168i	$-0.329018\pi$
$257$	2.78488e8	1.02339	0.511694	+	0.859168i	$0.329018\pi$
$258$	0	0
$259$	0	0
$260$	−4.70838e8	−1.66136
$261$	0	0
$262$	1.58018e8	0.542815
$263$	−2.48760e7	−0.0843210	−0.0421605	−	0.999111i	$-0.513424\pi$
$263$	−2.48760e7	−0.0843210	−0.0421605	+	0.999111i	$0.513424\pi$
$264$	0	0
$265$	3.88438e8	1.28221
$266$	0	0
$267$	0	0
$268$	7.22611e7	0.229315
$269$	4.54978e8	1.42514	0.712569	−	0.701602i	$-0.247531\pi$
$269$	4.54978e8	1.42514	0.712569	+	0.701602i	$0.247531\pi$
$270$	0	0
$271$	−2.74573e8	−0.838041	−0.419020	−	0.907977i	$-0.637626\pi$
$271$	−2.74573e8	−0.838041	−0.419020	+	0.907977i	$0.637626\pi$
$272$	−7.26430e7	−0.218878
$273$	0	0
$274$	−1.23694e8	−0.363264
$275$	1.08423e9	3.14382
$276$	0	0
$277$	4.08622e7	0.115516	0.0577581	−	0.998331i	$-0.481605\pi$
$277$	4.08622e7	0.115516	0.0577581	+	0.998331i	$0.481605\pi$
$278$	−4.25480e7	−0.118774
$279$	0	0
$280$	0	0
$281$	4.08644e8	1.09868	0.549342	−	0.835598i	$-0.314878\pi$
$281$	4.08644e8	1.09868	0.549342	+	0.835598i	$0.314878\pi$
$282$	0	0
$283$	−6.60171e8	−1.73143	−0.865713	−	0.500541i	$-0.833134\pi$
$283$	−6.60171e8	−1.73143	−0.865713	+	0.500541i	$0.833134\pi$
$284$	−1.09486e8	−0.283624
$285$	0	0
$286$	3.39268e8	0.857555
$287$	0	0
$288$	0	0
$289$	−2.16867e8	−0.528507
$290$	1.38245e8	0.332857
$291$	0	0
$292$	−4.01776e8	−0.944374
$293$	6.57223e8	1.52643	0.763213	−	0.646147i	$-0.223621\pi$
$293$	6.57223e8	1.52643	0.763213	+	0.646147i	$0.223621\pi$
$294$	0	0
$295$	3.58430e8	0.812881
$296$	4.63941e8	1.03978
$297$	0	0
$298$	2.65552e8	0.581289
$299$	2.75780e7	0.0596643
$300$	0	0
$301$	0	0
$302$	3.00207e8	0.627186
$303$	0	0
$304$	6.17756e7	0.126113
$305$	−6.30297e8	−1.27203
$306$	0	0
$307$	−2.21654e8	−0.437211	−0.218606	−	0.975813i	$-0.570151\pi$
$307$	−2.21654e8	−0.437211	−0.218606	+	0.975813i	$0.570151\pi$
$308$	0	0
$309$	0	0
$310$	−356804.	−0.000680243 0
$311$	−4.45985e8	−0.840734	−0.420367	−	0.907354i	$-0.638099\pi$
$311$	−4.45985e8	−0.840734	−0.420367	+	0.907354i	$0.638099\pi$
$312$	0	0
$313$	−5.38605e8	−0.992808	−0.496404	−	0.868092i	$-0.665347\pi$
$313$	−5.38605e8	−0.992808	−0.496404	+	0.868092i	$0.665347\pi$
$314$	3.99398e8	0.728034
$315$	0	0
$316$	6.20264e8	1.10579
$317$	−1.08023e9	−1.90461	−0.952307	−	0.305141i	$-0.901297\pi$
$317$	−1.08023e9	−1.90461	−0.952307	+	0.305141i	$0.901297\pi$
$318$	0	0
$319$	3.02871e8	0.522385
$320$	2.04486e8	0.348851
$321$	0	0
$322$	0	0
$323$	−1.64528e8	−0.271664
$324$	0	0
$325$	1.69919e9	2.74568
$326$	5.53493e8	0.884811
$327$	0	0
$328$	1.68285e8	0.263323
$329$	0	0
$330$	0	0
$331$	−7.72112e7	−0.117026	−0.0585130	−	0.998287i	$-0.518636\pi$
$331$	−7.72112e7	−0.117026	−0.0585130	+	0.998287i	$0.518636\pi$
$332$	1.76713e8	0.265024
$333$	0	0
$334$	4.14701e8	0.609008
$335$	−3.77316e8	−0.548338
$336$	0	0
$337$	−1.16026e9	−1.65139	−0.825697	−	0.564114i	$-0.809218\pi$
$337$	−1.16026e9	−1.65139	−0.825697	+	0.564114i	$0.809218\pi$
$338$	1.78517e8	0.251461
$339$	0	0
$340$	6.73821e8	0.929754
$341$	−781695.	−0.00106757
$342$	0	0
$343$	0	0
$344$	−1.02554e9	−1.35830
$345$	0	0
$346$	−6.26702e8	−0.813383
$347$	8.57808e8	1.10214	0.551070	−	0.834459i	$-0.314220\pi$
$347$	8.57808e8	1.10214	0.551070	+	0.834459i	$0.314220\pi$
$348$	0	0
$349$	2.41142e7	0.0303657	0.0151829	−	0.999885i	$-0.495167\pi$
$349$	2.41142e7	0.0303657	0.0151829	+	0.999885i	$0.495167\pi$
$350$	0	0
$351$	0	0
$352$	1.18457e9	1.44765
$353$	6.39421e8	0.773705	0.386853	−	0.922142i	$-0.373562\pi$
$353$	6.39421e8	0.773705	0.386853	+	0.922142i	$0.373562\pi$
$354$	0	0
$355$	5.71685e8	0.678200
$356$	−1.35258e8	−0.158887
$357$	0	0
$358$	−2.66328e7	−0.0306779
$359$	−1.04860e9	−1.19614	−0.598068	−	0.801445i	$-0.704065\pi$
$359$	−1.04860e9	−1.19614	−0.598068	+	0.801445i	$0.704065\pi$
$360$	0	0
$361$	−7.53957e8	−0.843473
$362$	−5.64607e8	−0.625557
$363$	0	0
$364$	0	0
$365$	2.09790e9	2.25818
$366$	0	0
$367$	−9.08871e7	−0.0959779	−0.0479889	−	0.998848i	$-0.515281\pi$
$367$	−9.08871e7	−0.0959779	−0.0479889	+	0.998848i	$0.515281\pi$
$368$	1.48188e7	0.0155005
$369$	0	0
$370$	−1.04019e9	−1.06759
$371$	0	0
$372$	0	0
$373$	−1.25097e9	−1.24814	−0.624072	−	0.781367i	$-0.714523\pi$
$373$	−1.25097e9	−1.24814	−0.624072	+	0.781367i	$0.714523\pi$
$374$	−4.85530e8	−0.479916
$375$	0	0
$376$	1.39491e9	1.35328
$377$	4.74655e8	0.456229
$378$	0	0
$379$	7.56067e8	0.713384	0.356692	−	0.934222i	$-0.383905\pi$
$379$	7.56067e8	0.713384	0.356692	+	0.934222i	$0.383905\pi$
$380$	−5.73017e8	−0.535704
$381$	0	0
$382$	3.64490e8	0.334552
$383$	5.73863e8	0.521931	0.260965	−	0.965348i	$-0.415959\pi$
$383$	5.73863e8	0.521931	0.260965	+	0.965348i	$0.415959\pi$
$384$	0	0
$385$	0	0
$386$	4.31226e8	0.381636
$387$	0	0
$388$	−6.45552e8	−0.561074
$389$	−2.11971e9	−1.82580	−0.912899	−	0.408185i	$-0.866162\pi$
$389$	−2.11971e9	−1.82580	−0.912899	+	0.408185i	$0.866162\pi$
$390$	0	0
$391$	−3.94672e7	−0.0333901
$392$	0	0
$393$	0	0
$394$	8.27817e8	0.681863
$395$	−3.23875e9	−2.64416
$396$	0	0
$397$	−1.38549e9	−1.11131	−0.555655	−	0.831413i	$-0.687532\pi$
$397$	−1.38549e9	−1.11131	−0.555655	+	0.831413i	$0.687532\pi$
$398$	1.65513e6	0.00131596
$399$	0	0
$400$	9.13042e8	0.713314
$401$	−4.40008e8	−0.340765	−0.170382	−	0.985378i	$-0.554500\pi$
$401$	−4.40008e8	−0.340765	−0.170382	+	0.985378i	$0.554500\pi$
$402$	0	0
$403$	−1.22506e6	−0.000932372 0
$404$	−3.69318e7	−0.0278654
$405$	0	0
$406$	0	0
$407$	−2.27887e9	−1.67548
$408$	0	0
$409$	7.21512e7	0.0521449	0.0260725	−	0.999660i	$-0.491700\pi$
$409$	7.21512e7	0.0521449	0.0260725	+	0.999660i	$0.491700\pi$
$410$	−3.77308e8	−0.270366
$411$	0	0
$412$	−1.44868e9	−1.02054
$413$	0	0
$414$	0	0
$415$	−9.22719e8	−0.633725
$416$	1.85645e9	1.26432
$417$	0	0
$418$	4.12895e8	0.276517
$419$	1.18463e9	0.786745	0.393373	−	0.919379i	$-0.371308\pi$
$419$	1.18463e9	0.786745	0.393373	+	0.919379i	$0.371308\pi$
$420$	0	0
$421$	−2.28422e8	−0.149194	−0.0745969	−	0.997214i	$-0.523767\pi$
$421$	−2.28422e8	−0.149194	−0.0745969	+	0.997214i	$0.523767\pi$
$422$	−5.92278e8	−0.383647
$423$	0	0
$424$	−9.75130e8	−0.621273
$425$	−2.43172e9	−1.53657
$426$	0	0
$427$	0	0
$428$	7.04916e8	0.434594
$429$	0	0
$430$	2.29933e9	1.39464
$431$	5.32136e8	0.320149	0.160074	−	0.987105i	$-0.448827\pi$
$431$	5.32136e8	0.320149	0.160074	+	0.987105i	$0.448827\pi$
$432$	0	0
$433$	3.55195e8	0.210261	0.105131	−	0.994458i	$-0.466474\pi$
$433$	3.55195e8	0.210261	0.105131	+	0.994458i	$0.466474\pi$
$434$	0	0
$435$	0	0
$436$	1.23901e9	0.715932
$437$	3.35629e7	0.0192386
$438$	0	0
$439$	5.73211e8	0.323362	0.161681	−	0.986843i	$-0.448309\pi$
$439$	5.73211e8	0.323362	0.161681	+	0.986843i	$0.448309\pi$
$440$	−3.93816e9	−2.20399
$441$	0	0
$442$	−7.60914e8	−0.419139
$443$	−1.60908e9	−0.879355	−0.439678	−	0.898156i	$-0.644907\pi$
$443$	−1.60908e9	−0.879355	−0.439678	+	0.898156i	$0.644907\pi$
$444$	0	0
$445$	7.06257e8	0.379929
$446$	−2.07316e7	−0.0110652
$447$	0	0
$448$	0	0
$449$	−2.60121e9	−1.35616	−0.678082	−	0.734986i	$-0.737189\pi$
$449$	−2.60121e9	−1.35616	−0.678082	+	0.734986i	$0.737189\pi$
$450$	0	0
$451$	−8.26615e8	−0.424312
$452$	−8.67693e7	−0.0441959
$453$	0	0
$454$	−4.34622e7	−0.0217980
$455$	0	0
$456$	0	0
$457$	−2.79230e9	−1.36853	−0.684267	−	0.729232i	$-0.739878\pi$
$457$	−2.79230e9	−1.36853	−0.684267	+	0.729232i	$0.739878\pi$
$458$	1.66096e9	0.807847
$459$	0	0
$460$	−1.37456e8	−0.0658431
$461$	−2.44371e9	−1.16171	−0.580854	−	0.814008i	$-0.697281\pi$
$461$	−2.44371e9	−1.16171	−0.580854	+	0.814008i	$0.697281\pi$
$462$	0	0
$463$	−5.78623e8	−0.270933	−0.135467	−	0.990782i	$-0.543253\pi$
$463$	−5.78623e8	−0.270933	−0.135467	+	0.990782i	$0.543253\pi$
$464$	2.55051e8	0.118526
$465$	0	0
$466$	5.61674e8	0.257118
$467$	−3.87321e9	−1.75979	−0.879897	−	0.475164i	$-0.842389\pi$
$467$	−3.87321e9	−1.75979	−0.879897	+	0.475164i	$0.842389\pi$
$468$	0	0
$469$	0	0
$470$	−3.12748e9	−1.38948
$471$	0	0
$472$	−8.99799e8	−0.393866
$473$	5.03742e9	2.18874
$474$	0	0
$475$	2.06794e9	0.885339
$476$	0	0
$477$	0	0
$478$	1.00783e9	0.422073
$479$	2.00235e9	0.832464	0.416232	−	0.909258i	$-0.363350\pi$
$479$	2.00235e9	0.832464	0.416232	+	0.909258i	$0.363350\pi$
$480$	0	0
$481$	−3.57141e9	−1.46329
$482$	4.00072e8	0.162732
$483$	0	0
$484$	−1.82767e9	−0.732720
$485$	3.37079e9	1.34164
$486$	0	0
$487$	4.33695e9	1.70150	0.850751	−	0.525568i	$-0.176147\pi$
$487$	4.33695e9	1.70150	0.850751	+	0.525568i	$0.176147\pi$
$488$	1.58229e9	0.616336
$489$	0	0
$490$	0	0
$491$	4.62835e9	1.76458	0.882289	−	0.470709i	$-0.156002\pi$
$491$	4.62835e9	1.76458	0.882289	+	0.470709i	$0.156002\pi$
$492$	0	0
$493$	−6.79283e8	−0.255321
$494$	6.47081e8	0.241499
$495$	0	0
$496$	−658273.	−0.000242226 0
$497$	0	0
$498$	0	0
$499$	2.51281e9	0.905333	0.452666	−	0.891680i	$-0.350473\pi$
$499$	2.51281e9	0.905333	0.452666	+	0.891680i	$0.350473\pi$
$500$	−4.68452e9	−1.67599
$501$	0	0
$502$	−3.29592e8	−0.116282
$503$	1.12749e9	0.395025	0.197512	−	0.980300i	$-0.436714\pi$
$503$	1.12749e9	0.395025	0.197512	+	0.980300i	$0.436714\pi$
$504$	0	0
$505$	1.92841e8	0.0666316
$506$	9.90455e7	0.0339866
$507$	0	0
$508$	−2.39055e8	−0.0809050
$509$	2.64670e9	0.889596	0.444798	−	0.895631i	$-0.353275\pi$
$509$	2.64670e9	0.889596	0.444798	+	0.895631i	$0.353275\pi$
$510$	0	0
$511$	0	0
$512$	1.84157e9	0.606376
$513$	0	0
$514$	1.56746e9	0.509127
$515$	7.56435e9	2.44032
$516$	0	0
$517$	−6.85177e9	−2.18065
$518$	0	0
$519$	0	0
$520$	−6.17182e9	−1.92487
$521$	4.99112e9	1.54620	0.773101	−	0.634283i	$-0.218705\pi$
$521$	4.99112e9	1.54620	0.773101	+	0.634283i	$0.218705\pi$
$522$	0	0
$523$	−1.95839e9	−0.598609	−0.299304	−	0.954158i	$-0.596755\pi$
$523$	−1.95839e9	−0.598609	−0.299304	+	0.954158i	$0.596755\pi$
$524$	−2.70416e9	−0.821057
$525$	0	0
$526$	−1.40014e8	−0.0419490
$527$	1.75319e6	0.000521786 0
$528$	0	0
$529$	−3.39677e9	−0.997635
$530$	2.18631e9	0.637891
$531$	0	0
$532$	0	0
$533$	−1.29546e9	−0.370577
$534$	0	0
$535$	−3.68076e9	−1.03920
$536$	9.47210e8	0.265687
$537$	0	0
$538$	2.56083e9	0.708995
$539$	0	0
$540$	0	0
$541$	−4.04832e9	−1.09922	−0.549610	−	0.835422i	$-0.685223\pi$
$541$	−4.04832e9	−1.09922	−0.549610	+	0.835422i	$0.685223\pi$
$542$	−1.54543e9	−0.416918
$543$	0	0
$544$	−2.65678e9	−0.707553
$545$	−6.46956e9	−1.71193
$546$	0	0
$547$	3.15001e9	0.822917	0.411458	−	0.911429i	$-0.365020\pi$
$547$	3.15001e9	0.822917	0.411458	+	0.911429i	$0.365020\pi$
$548$	2.11678e9	0.549470
$549$	0	0
$550$	6.10257e9	1.56402
$551$	5.77662e8	0.147110
$552$	0	0
$553$	0	0
$554$	2.29992e8	0.0574683
$555$	0	0
$556$	7.28125e8	0.179657
$557$	−2.56999e8	−0.0630141	−0.0315071	−	0.999504i	$-0.510031\pi$
$557$	−2.56999e8	−0.0630141	−0.0315071	+	0.999504i	$0.510031\pi$
$558$	0	0
$559$	7.89456e9	1.91155
$560$	0	0
$561$	0	0
$562$	2.30004e9	0.546586
$563$	−4.66156e9	−1.10091	−0.550455	−	0.834865i	$-0.685546\pi$
$563$	−4.66156e9	−1.10091	−0.550455	+	0.834865i	$0.685546\pi$
$564$	0	0
$565$	4.53071e8	0.105681
$566$	−3.71576e9	−0.861370
$567$	0	0
$568$	−1.43515e9	−0.328609
$569$	−2.82647e9	−0.643208	−0.321604	−	0.946874i	$-0.604222\pi$
$569$	−2.82647e9	−0.643208	−0.321604	+	0.946874i	$0.604222\pi$
$570$	0	0
$571$	−4.37915e9	−0.984382	−0.492191	−	0.870487i	$-0.663804\pi$
$571$	−4.37915e9	−0.984382	−0.492191	+	0.870487i	$0.663804\pi$
$572$	−5.80590e9	−1.29713
$573$	0	0
$574$	0	0
$575$	4.96058e8	0.108817
$576$	0	0
$577$	−2.54135e9	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$577$	−2.54135e9	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$578$	−1.22063e9	−0.262928
$579$	0	0
$580$	−2.36580e9	−0.503477
$581$	0	0
$582$	0	0
$583$	4.78982e9	1.00111
$584$	−5.26654e9	−1.09416
$585$	0	0
$586$	3.69916e9	0.759385
$587$	−1.61818e9	−0.330213	−0.165106	−	0.986276i	$-0.552797\pi$
$587$	−1.61818e9	−0.330213	−0.165106	+	0.986276i	$0.552797\pi$
$588$	0	0
$589$	−1.49092e6	−0.000300642 0
$590$	2.01741e9	0.404402
$591$	0	0
$592$	−1.91906e9	−0.380156
$593$	−2.27497e9	−0.448006	−0.224003	−	0.974588i	$-0.571913\pi$
$593$	−2.27497e9	−0.448006	−0.224003	+	0.974588i	$0.571913\pi$
$594$	0	0
$595$	0	0
$596$	−4.54439e9	−0.879253
$597$	0	0
$598$	1.55222e8	0.0296825
$599$	−5.23474e9	−0.995179	−0.497589	−	0.867413i	$-0.665781\pi$
$599$	−5.23474e9	−0.995179	−0.497589	+	0.867413i	$0.665781\pi$
$600$	0	0
$601$	−5.20968e9	−0.978926	−0.489463	−	0.872024i	$-0.662807\pi$
$601$	−5.20968e9	−0.978926	−0.489463	+	0.872024i	$0.662807\pi$
$602$	0	0
$603$	0	0
$604$	−5.13745e9	−0.948676
$605$	9.54327e9	1.75208
$606$	0	0
$607$	6.08182e9	1.10376	0.551878	−	0.833925i	$-0.313911\pi$
$607$	6.08182e9	1.10376	0.551878	+	0.833925i	$0.313911\pi$
$608$	2.25932e9	0.407676
$609$	0	0
$610$	−3.54761e9	−0.632822
$611$	−1.07380e10	−1.90449
$612$	0	0
$613$	−5.39577e9	−0.946110	−0.473055	−	0.881033i	$-0.656849\pi$
$613$	−5.39577e9	−0.946110	−0.473055	+	0.881033i	$0.656849\pi$
$614$	−1.24757e9	−0.217509
$615$	0	0
$616$	0	0
$617$	−9.38467e9	−1.60850	−0.804249	−	0.594292i	$-0.797432\pi$
$617$	−9.38467e9	−1.60850	−0.804249	+	0.594292i	$0.797432\pi$
$618$	0	0
$619$	2.06676e9	0.350246	0.175123	−	0.984547i	$-0.443968\pi$
$619$	2.06676e9	0.350246	0.175123	+	0.984547i	$0.443968\pi$
$620$	6.10600e6	0.00102893
$621$	0	0
$622$	−2.51021e9	−0.418258
$623$	0	0
$624$	0	0
$625$	1.08023e10	1.76984
$626$	−3.03153e9	−0.493914
$627$	0	0
$628$	−6.83490e9	−1.10122
$629$	5.11107e9	0.818907
$630$	0	0
$631$	−2.45121e9	−0.388399	−0.194200	−	0.980962i	$-0.562211\pi$
$631$	−2.45121e9	−0.388399	−0.194200	+	0.980962i	$0.562211\pi$
$632$	8.13052e9	1.28118
$633$	0	0
$634$	−6.08003e9	−0.947530
$635$	1.24824e9	0.193460
$636$	0	0
$637$	0	0
$638$	1.70470e9	0.259882
$639$	0	0
$640$	−1.11453e10	−1.68060
$641$	−7.26818e9	−1.08999	−0.544995	−	0.838439i	$-0.683469\pi$
$641$	−7.26818e9	−1.08999	−0.544995	+	0.838439i	$0.683469\pi$
$642$	0	0
$643$	−8.55284e9	−1.26874	−0.634369	−	0.773030i	$-0.718740\pi$
$643$	−8.55284e9	−1.26874	−0.634369	+	0.773030i	$0.718740\pi$
$644$	0	0
$645$	0	0
$646$	−9.26044e8	−0.135151
$647$	−6.60890e8	−0.0959322	−0.0479661	−	0.998849i	$-0.515274\pi$
$647$	−6.60890e8	−0.0959322	−0.0479661	+	0.998849i	$0.515274\pi$
$648$	0	0
$649$	4.41980e9	0.634667
$650$	9.56384e9	1.36595
$651$	0	0
$652$	−9.47194e9	−1.33836
$653$	1.72844e9	0.242917	0.121458	−	0.992597i	$-0.461243\pi$
$653$	1.72844e9	0.242917	0.121458	+	0.992597i	$0.461243\pi$
$654$	0	0
$655$	1.41200e10	1.96331
$656$	−6.96101e8	−0.0962740
$657$	0	0
$658$	0	0
$659$	4.19192e9	0.570576	0.285288	−	0.958442i	$-0.407911\pi$
$659$	4.19192e9	0.570576	0.285288	+	0.958442i	$0.407911\pi$
$660$	0	0
$661$	7.57756e9	1.02053	0.510263	−	0.860018i	$-0.329548\pi$
$661$	7.57756e9	1.02053	0.510263	+	0.860018i	$0.329548\pi$
$662$	−4.34581e8	−0.0582195
$663$	0	0
$664$	2.31639e9	0.307059
$665$	0	0
$666$	0	0
$667$	1.38570e8	0.0180813
$668$	−7.09679e9	−0.921180
$669$	0	0
$670$	−2.12371e9	−0.272794
$671$	−7.77219e9	−0.993149
$672$	0	0
$673$	5.50701e9	0.696407	0.348203	−	0.937419i	$-0.386792\pi$
$673$	5.50701e9	0.696407	0.348203	+	0.937419i	$0.386792\pi$
$674$	−6.53050e9	−0.821555
$675$	0	0
$676$	−3.05496e9	−0.380358
$677$	−9.73307e9	−1.20556	−0.602781	−	0.797907i	$-0.705941\pi$
$677$	−9.73307e9	−1.20556	−0.602781	+	0.797907i	$0.705941\pi$
$678$	0	0
$679$	0	0
$680$	8.83255e9	1.07722
$681$	0	0
$682$	−4.39975e6	−0.000531108 0
$683$	−9.68080e9	−1.16262	−0.581311	−	0.813681i	$-0.697460\pi$
$683$	−9.68080e9	−1.16262	−0.581311	+	0.813681i	$0.697460\pi$
$684$	0	0
$685$	−1.10529e10	−1.31389
$686$	0	0
$687$	0	0
$688$	4.24206e9	0.496612
$689$	7.50653e9	0.874323
$690$	0	0
$691$	5.53837e9	0.638571	0.319285	−	0.947659i	$-0.396557\pi$
$691$	5.53837e9	0.638571	0.319285	+	0.947659i	$0.396557\pi$
$692$	1.07248e10	1.23032
$693$	0	0
$694$	4.82815e9	0.548306
$695$	−3.80195e9	−0.429595
$696$	0	0
$697$	1.85394e9	0.207387
$698$	1.35726e8	0.0151067
$699$	0	0
$700$	0	0
$701$	1.27820e10	1.40148	0.700738	−	0.713419i	$-0.252854\pi$
$701$	1.27820e10	1.40148	0.700738	+	0.713419i	$0.252854\pi$
$702$	0	0
$703$	−4.34645e9	−0.471836
$704$	2.52152e9	0.272369
$705$	0	0
$706$	3.59897e9	0.384912
$707$	0	0
$708$	0	0
$709$	3.75099e9	0.395261	0.197630	−	0.980277i	$-0.436675\pi$
$709$	3.75099e9	0.395261	0.197630	+	0.980277i	$0.436675\pi$
$710$	3.21772e9	0.337399
$711$	0	0
$712$	−1.77298e9	−0.184087
$713$	−357642.	−3.69518e−5 0
$714$	0	0
$715$	3.03159e10	3.10169
$716$	4.55768e8	0.0464032
$717$	0	0
$718$	−5.90204e9	−0.595068
$719$	1.55760e10	1.56281	0.781403	−	0.624027i	$-0.214504\pi$
$719$	1.55760e10	1.56281	0.781403	+	0.624027i	$0.214504\pi$
$720$	0	0
$721$	0	0
$722$	−4.24363e9	−0.419621
$723$	0	0
$724$	9.66214e9	0.946212
$725$	8.53783e9	0.832079
$726$	0	0
$727$	2.04219e10	1.97118	0.985589	−	0.169158i	$-0.0541047\pi$
$727$	2.04219e10	1.97118	0.985589	+	0.169158i	$0.0541047\pi$
$728$	0	0
$729$	0	0
$730$	1.18080e10	1.12343
$731$	−1.12980e10	−1.06977
$732$	0	0
$733$	−2.26923e9	−0.212821	−0.106411	−	0.994322i	$-0.533936\pi$
$733$	−2.26923e9	−0.212821	−0.106411	+	0.994322i	$0.533936\pi$
$734$	−5.11556e8	−0.0477482
$735$	0	0
$736$	5.41968e8	0.0501073
$737$	−4.65268e9	−0.428122
$738$	0	0
$739$	1.84606e10	1.68264	0.841319	−	0.540539i	$-0.181780\pi$
$739$	1.84606e10	1.68264	0.841319	+	0.540539i	$0.181780\pi$
$740$	1.78008e10	1.61483
$741$	0	0
$742$	0	0
$743$	−1.77069e10	−1.58373	−0.791866	−	0.610695i	$-0.790890\pi$
$743$	−1.77069e10	−1.58373	−0.791866	+	0.610695i	$0.790890\pi$
$744$	0	0
$745$	2.37288e10	2.10247
$746$	−7.04103e9	−0.620941
$747$	0	0
$748$	8.30888e9	0.725917
$749$	0	0
$750$	0	0
$751$	−1.62678e10	−1.40148	−0.700742	−	0.713415i	$-0.747148\pi$
$751$	−1.62678e10	−1.40148	−0.700742	+	0.713415i	$0.747148\pi$
$752$	−5.76994e9	−0.494776
$753$	0	0
$754$	2.67158e9	0.226970
$755$	2.68255e10	2.26847
$756$	0	0
$757$	−2.16032e10	−1.81002	−0.905008	−	0.425394i	$-0.860135\pi$
$757$	−2.16032e10	−1.81002	−0.905008	+	0.425394i	$0.860135\pi$
$758$	4.25551e9	0.354902
$759$	0	0
$760$	−7.51120e9	−0.620671
$761$	−5.57029e9	−0.458175	−0.229088	−	0.973406i	$-0.573574\pi$
$761$	−5.57029e9	−0.458175	−0.229088	+	0.973406i	$0.573574\pi$
$762$	0	0
$763$	0	0
$764$	−6.23752e9	−0.506040
$765$	0	0
$766$	3.22998e9	0.259656
$767$	6.92663e9	0.554291
$768$	0	0
$769$	2.06958e10	1.64112	0.820558	−	0.571563i	$-0.193663\pi$
$769$	2.06958e10	1.64112	0.820558	+	0.571563i	$0.193663\pi$
$770$	0	0
$771$	0	0
$772$	−7.37958e9	−0.577260
$773$	1.62275e10	1.26364	0.631819	−	0.775116i	$-0.282309\pi$
$773$	1.62275e10	1.26364	0.631819	+	0.775116i	$0.282309\pi$
$774$	0	0
$775$	−2.20357e7	−0.00170048
$776$	−8.46200e9	−0.650066
$777$	0	0
$778$	−1.19307e10	−0.908320
$779$	−1.57659e9	−0.119492
$780$	0	0
$781$	7.04945e9	0.529513
$782$	−2.22140e8	−0.0166113
$783$	0	0
$784$	0	0
$785$	3.56889e10	2.63323
$786$	0	0
$787$	−8.20146e9	−0.599763	−0.299882	−	0.953976i	$-0.596947\pi$
$787$	−8.20146e9	−0.599763	−0.299882	+	0.953976i	$0.596947\pi$
$788$	−1.41664e10	−1.03138
$789$	0	0
$790$	−1.82292e10	−1.31545
$791$	0	0
$792$	0	0
$793$	−1.21804e10	−0.867375
$794$	−7.79817e9	−0.552868
$795$	0	0
$796$	−2.83243e7	−0.00199051
$797$	1.97600e10	1.38255	0.691276	−	0.722590i	$-0.257049\pi$
$797$	1.97600e10	1.38255	0.691276	+	0.722590i	$0.257049\pi$
$798$	0	0
$799$	1.53672e10	1.06581
$800$	3.33927e10	2.30588
$801$	0	0
$802$	−2.47657e9	−0.169528
$803$	2.58692e10	1.76310
$804$	0	0
$805$	0	0
$806$	−6.89522e6	−0.000463848 0
$807$	0	0
$808$	−4.84107e8	−0.0322851
$809$	1.16791e9	0.0775515	0.0387757	−	0.999248i	$-0.487654\pi$
$809$	1.16791e9	0.0775515	0.0387757	+	0.999248i	$0.487654\pi$
$810$	0	0
$811$	1.21328e10	0.798711	0.399356	−	0.916796i	$-0.369234\pi$
$811$	1.21328e10	0.798711	0.399356	+	0.916796i	$0.369234\pi$
$812$	0	0
$813$	0	0
$814$	−1.28266e10	−0.833537
$815$	4.94583e10	3.20028
$816$	0	0
$817$	9.60780e9	0.616377
$818$	4.06102e8	0.0259417
$819$	0	0
$820$	6.45688e9	0.408954
$821$	−1.90939e8	−0.0120419	−0.00602093	−	0.999982i	$-0.501917\pi$
$821$	−1.90939e8	−0.0120419	−0.00602093	+	0.999982i	$0.501917\pi$
$822$	0	0
$823$	−1.14186e10	−0.714023	−0.357012	−	0.934100i	$-0.616204\pi$
$823$	−1.14186e10	−0.714023	−0.357012	+	0.934100i	$0.616204\pi$
$824$	−1.89895e10	−1.18241
$825$	0	0
$826$	0	0
$827$	1.33839e9	0.0822834	0.0411417	−	0.999153i	$-0.486900\pi$
$827$	1.33839e9	0.0822834	0.0411417	+	0.999153i	$0.486900\pi$
$828$	0	0
$829$	2.02537e8	0.0123471	0.00617353	−	0.999981i	$-0.498035\pi$
$829$	2.02537e8	0.0123471	0.00617353	+	0.999981i	$0.498035\pi$
$830$	−5.19350e9	−0.315273
$831$	0	0
$832$	3.95169e9	0.237876
$833$	0	0
$834$	0	0
$835$	3.70563e10	2.20272
$836$	−7.06587e9	−0.418258
$837$	0	0
$838$	6.66767e9	0.391399
$839$	2.24187e10	1.31052	0.655259	−	0.755405i	$-0.272560\pi$
$839$	2.24187e10	1.31052	0.655259	+	0.755405i	$0.272560\pi$
$840$	0	0
$841$	−1.48649e10	−0.861739
$842$	−1.28567e9	−0.0742227
$843$	0	0
$844$	1.01357e10	0.580302
$845$	1.59517e10	0.909511
$846$	0	0
$847$	0	0
$848$	4.03356e9	0.227145
$849$	0	0
$850$	−1.36869e10	−0.764432
$851$	−1.04263e9	−0.0579932
$852$	0	0
$853$	1.43556e10	0.791952	0.395976	−	0.918261i	$-0.370406\pi$
$853$	1.43556e10	0.791952	0.395976	+	0.918261i	$0.370406\pi$
$854$	0	0
$855$	0	0
$856$	9.24015e9	0.503525
$857$	−8.58283e9	−0.465798	−0.232899	−	0.972501i	$-0.574821\pi$
$857$	−8.58283e9	−0.465798	−0.232899	+	0.972501i	$0.574821\pi$
$858$	0	0
$859$	−1.00819e10	−0.542707	−0.271354	−	0.962480i	$-0.587471\pi$
$859$	−1.00819e10	−0.542707	−0.271354	+	0.962480i	$0.587471\pi$
$860$	−3.93484e10	−2.10952
$861$	0	0
$862$	2.99511e9	0.159271
$863$	3.40980e10	1.80589	0.902946	−	0.429754i	$-0.141400\pi$
$863$	3.40980e10	1.80589	0.902946	+	0.429754i	$0.141400\pi$
$864$	0	0
$865$	−5.60000e10	−2.94193
$866$	1.99921e9	0.104603
$867$	0	0
$868$	0	0
$869$	−3.99370e10	−2.06446
$870$	0	0
$871$	−7.29160e9	−0.373903
$872$	1.62411e10	0.829485
$873$	0	0
$874$	1.88908e8	0.00957107
$875$	0	0
$876$	0	0
$877$	−2.77136e10	−1.38737	−0.693687	−	0.720276i	$-0.744015\pi$
$877$	−2.77136e10	−1.38737	−0.693687	+	0.720276i	$0.744015\pi$
$878$	3.22630e9	0.160870
$879$	0	0
$880$	1.62899e10	0.805805
$881$	−1.61765e10	−0.797020	−0.398510	−	0.917164i	$-0.630473\pi$
$881$	−1.61765e10	−0.797020	−0.398510	+	0.917164i	$0.630473\pi$
$882$	0	0
$883$	2.67909e10	1.30956	0.654780	−	0.755820i	$-0.272761\pi$
$883$	2.67909e10	1.30956	0.654780	+	0.755820i	$0.272761\pi$
$884$	1.30215e10	0.633986
$885$	0	0
$886$	−9.05667e9	−0.437472
$887$	3.49412e9	0.168115	0.0840573	−	0.996461i	$-0.473212\pi$
$887$	3.49412e9	0.168115	0.0840573	+	0.996461i	$0.473212\pi$
$888$	0	0
$889$	0	0
$890$	3.97515e9	0.189012
$891$	0	0
$892$	3.54780e8	0.0167372
$893$	−1.30683e10	−0.614099
$894$	0	0
$895$	−2.37982e9	−0.110959
$896$	0	0
$897$	0	0
$898$	−1.46408e10	−0.674681
$899$	−6.15550e6	−0.000282556 0
$900$	0	0
$901$	−1.07427e10	−0.489300
$902$	−4.65259e9	−0.211092
$903$	0	0
$904$	−1.13739e9	−0.0512057
$905$	−5.04514e10	−2.26258
$906$	0	0
$907$	−2.80537e10	−1.24843	−0.624215	−	0.781253i	$-0.714581\pi$
$907$	−2.80537e10	−1.24843	−0.624215	+	0.781253i	$0.714581\pi$
$908$	7.43769e8	0.0329714
$909$	0	0
$910$	0	0
$911$	2.81753e10	1.23468	0.617341	−	0.786696i	$-0.288210\pi$
$911$	2.81753e10	1.23468	0.617341	+	0.786696i	$0.288210\pi$
$912$	0	0
$913$	−1.13780e10	−0.494789
$914$	−1.57164e10	−0.680834
$915$	0	0
$916$	−2.84240e10	−1.22194
$917$	0	0
$918$	0	0
$919$	−2.83314e10	−1.20410	−0.602052	−	0.798457i	$-0.705650\pi$
$919$	−2.83314e10	−1.20410	−0.602052	+	0.798457i	$0.705650\pi$
$920$	−1.80179e9	−0.0762864
$921$	0	0
$922$	−1.37544e10	−0.577940
$923$	1.10478e10	0.462455
$924$	0	0
$925$	−6.42404e10	−2.66878
$926$	−3.25677e9	−0.134787
$927$	0	0
$928$	9.32799e9	0.383151
$929$	−4.02771e10	−1.64817	−0.824086	−	0.566464i	$-0.808311\pi$
$929$	−4.02771e10	−1.64817	−0.824086	+	0.566464i	$0.808311\pi$
$930$	0	0
$931$	0	0
$932$	−9.61193e9	−0.388915
$933$	0	0
$934$	−2.18003e10	−0.875483
$935$	−4.33853e10	−1.73581
$936$	0	0
$937$	2.09269e10	0.831029	0.415514	−	0.909587i	$-0.363602\pi$
$937$	2.09269e10	0.831029	0.415514	+	0.909587i	$0.363602\pi$
$938$	0	0
$939$	0	0
$940$	5.35207e10	2.10172
$941$	−1.74169e9	−0.0681410	−0.0340705	−	0.999419i	$-0.510847\pi$
$941$	−1.74169e9	−0.0681410	−0.0340705	+	0.999419i	$0.510847\pi$
$942$	0	0
$943$	−3.78194e8	−0.0146867
$944$	3.72196e9	0.144002
$945$	0	0
$946$	2.83530e10	1.08888
$947$	2.58951e10	0.990816	0.495408	−	0.868660i	$-0.335018\pi$
$947$	2.58951e10	0.990816	0.495408	+	0.868660i	$0.335018\pi$
$948$	0	0
$949$	4.05417e10	1.53982
$950$	1.16393e10	0.440449
$951$	0	0
$952$	0	0
$953$	3.87113e9	0.144881	0.0724407	−	0.997373i	$-0.476921\pi$
$953$	3.87113e9	0.144881	0.0724407	+	0.997373i	$0.476921\pi$
$954$	0	0
$955$	3.25696e10	1.21004
$956$	−1.72469e10	−0.638424
$957$	0	0
$958$	1.12702e10	0.414144
$959$	0	0
$960$	0	0
$961$	−2.75126e10	−0.999999
$962$	−2.01016e10	−0.727976
$963$	0	0
$964$	−6.84643e9	−0.246147
$965$	3.85329e10	1.38034
$966$	0	0
$967$	−1.34763e9	−0.0479266	−0.0239633	−	0.999713i	$-0.507628\pi$
$967$	−1.34763e9	−0.0479266	−0.0239633	+	0.999713i	$0.507628\pi$
$968$	−2.39573e10	−0.848936
$969$	0	0
$970$	1.89724e10	0.667454
$971$	5.24022e10	1.83689	0.918443	−	0.395553i	$-0.129447\pi$
$971$	5.24022e10	1.83689	0.918443	+	0.395553i	$0.129447\pi$
$972$	0	0
$973$	0	0
$974$	2.44104e10	0.846484
$975$	0	0
$976$	−6.54504e9	−0.225340
$977$	5.10191e10	1.75026	0.875128	−	0.483891i	$-0.160777\pi$
$977$	5.10191e10	1.75026	0.875128	+	0.483891i	$0.160777\pi$
$978$	0	0
$979$	8.70886e9	0.296635
$980$	0	0
$981$	0	0
$982$	2.60505e10	0.877863
$983$	−4.39614e10	−1.47616	−0.738082	−	0.674711i	$-0.764268\pi$
$983$	−4.39614e10	−1.47616	−0.738082	+	0.674711i	$0.764268\pi$
$984$	0	0
$985$	7.39709e10	2.46623
$986$	−3.82333e9	−0.127020
$987$	0	0
$988$	−1.10735e10	−0.365289
$989$	2.30473e9	0.0757586
$990$	0	0
$991$	1.21671e10	0.397127	0.198563	−	0.980088i	$-0.436372\pi$
$991$	1.21671e10	0.397127	0.198563	+	0.980088i	$0.436372\pi$
$992$	−2.40750e7	−0.000783026 0
$993$	0	0
$994$	0	0
$995$	1.47897e8	0.00475969
$996$	0	0
$997$	4.00545e9	0.128003	0.0640013	−	0.997950i	$-0.479614\pi$
$997$	4.00545e9	0.128003	0.0640013	+	0.997950i	$0.479614\pi$
$998$	1.41433e10	0.450396
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.8.a.r.1.3		4
3.2	odd	2		147.8.a.h.1.2		4
7.3	odd	6		63.8.e.c.37.2		8
7.5	odd	6		63.8.e.c.46.2		8
7.6	odd	2		441.8.a.q.1.3		4
21.2	odd	6		147.8.e.k.67.3		8
21.5	even	6		21.8.e.a.4.3	✓	8
21.11	odd	6		147.8.e.k.79.3		8
21.17	even	6		21.8.e.a.16.3	yes	8
21.20	even	2		147.8.a.i.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.8.e.a.4.3	✓	8	21.5	even	6
21.8.e.a.16.3	yes	8	21.17	even	6
63.8.e.c.37.2		8	7.3	odd	6
63.8.e.c.46.2		8	7.5	odd	6
147.8.a.h.1.2		4	3.2	odd	2
147.8.a.i.1.2		4	21.20	even	2
147.8.e.k.67.3		8	21.2	odd	6
147.8.e.k.79.3		8	21.11	odd	6
441.8.a.q.1.3		4	7.6	odd	2
441.8.a.r.1.3		4	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+5.62848 q^{2} -96.3202 q^{4} +502.942 q^{5} -1262.58 q^{8} +O(q^{10})\)	\(q+5.62848 q^{2} -96.3202 q^{4} +502.942 q^{5} -1262.58 q^{8} +2830.80 q^{10} +6201.78 q^{11} +9719.32 q^{13} +5222.58 q^{16} -13909.4 q^{17} +11828.6 q^{19} -48443.5 q^{20} +34906.6 q^{22} +2837.44 q^{23} +174826. q^{25} +54705.0 q^{26} +48836.2 q^{29} -126.044 q^{31} +191006. q^{32} -78288.8 q^{34} -367454. q^{37} +66576.8 q^{38} -635005. q^{40} -133287. q^{41} +812254. q^{43} -597357. q^{44} +15970.5 q^{46} -1.10481e6 q^{47} +984003. q^{50} -936168. q^{52} +772331. q^{53} +3.11914e6 q^{55} +274874. q^{58} +712666. q^{59} -1.25322e6 q^{61} -709.434 q^{62} +406580. q^{64} +4.88826e6 q^{65} -750217. q^{67} +1.33976e6 q^{68} +1.13668e6 q^{71} +4.17125e6 q^{73} -2.06821e6 q^{74} -1.13933e6 q^{76} -6.43960e6 q^{79} +2.62666e6 q^{80} -750202. q^{82} -1.83464e6 q^{83} -6.99563e6 q^{85} +4.57175e6 q^{86} -7.83025e6 q^{88} +1.40425e6 q^{89} -273303. q^{92} -6.21838e6 q^{94} +5.94908e6 q^{95} +6.70215e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 101 q^{4} + 196 q^{5} + 1347 q^{8}+O(q^{10}) \) 4 * q + q^2 + 101 * q^4 + 196 * q^5 + 1347 * q^8	\( 4 q + q^{2} + 101 q^{4} + 196 q^{5} + 1347 q^{8} + 5185 q^{10} + 5210 q^{11} + 3794 q^{13} - 21055 q^{16} - 11436 q^{17} + 69158 q^{19} - 87991 q^{20} + 57263 q^{22} + 146220 q^{23} + 6230 q^{25} + 107696 q^{26} + 70664 q^{29} + 288618 q^{31} - 2653 q^{32} + 495996 q^{34} - 448902 q^{37} + 528944 q^{38} - 767361 q^{40} - 663316 q^{41} + 554 q^{43} + 686635 q^{44} - 1064964 q^{46} + 762180 q^{47} + 525428 q^{50} - 3423848 q^{52} + 2761920 q^{53} + 1965056 q^{55} - 4451395 q^{58} + 3410898 q^{59} - 300892 q^{61} - 2066175 q^{62} - 2916551 q^{64} + 8019032 q^{65} - 4222478 q^{67} + 5770500 q^{68} + 380964 q^{71} + 451674 q^{73} + 10091220 q^{74} + 14747656 q^{76} - 12154822 q^{79} + 7870085 q^{80} - 8814904 q^{82} - 12087978 q^{83} - 7375500 q^{85} + 32881826 q^{86} - 14439051 q^{88} + 4955752 q^{89} - 206892 q^{92} - 5960646 q^{94} - 8460784 q^{95} + 22840614 q^{97}+O(q^{100}) \) 4 * q + q^2 + 101 * q^4 + 196 * q^5 + 1347 * q^8 + 5185 * q^10 + 5210 * q^11 + 3794 * q^13 - 21055 * q^16 - 11436 * q^17 + 69158 * q^19 - 87991 * q^20 + 57263 * q^22 + 146220 * q^23 + 6230 * q^25 + 107696 * q^26 + 70664 * q^29 + 288618 * q^31 - 2653 * q^32 + 495996 * q^34 - 448902 * q^37 + 528944 * q^38 - 767361 * q^40 - 663316 * q^41 + 554 * q^43 + 686635 * q^44 - 1064964 * q^46 + 762180 * q^47 + 525428 * q^50 - 3423848 * q^52 + 2761920 * q^53 + 1965056 * q^55 - 4451395 * q^58 + 3410898 * q^59 - 300892 * q^61 - 2066175 * q^62 - 2916551 * q^64 + 8019032 * q^65 - 4222478 * q^67 + 5770500 * q^68 + 380964 * q^71 + 451674 * q^73 + 10091220 * q^74 + 14747656 * q^76 - 12154822 * q^79 + 7870085 * q^80 - 8814904 * q^82 - 12087978 * q^83 - 7375500 * q^85 + 32881826 * q^86 - 14439051 * q^88 + 4955752 * q^89 - 206892 * q^92 - 5960646 * q^94 - 8460784 * q^95 + 22840614 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more