LMFDB - Embedded newform 315.10.a.l.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,10,Mod(1,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$162.236288392$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 3018x^{4} + 3368x^{3} + 2066979x^{2} - 6329061x - 14714266 $ x^6 - 3x^5 - 3018x^4 + 3368x^3 + 2066979x^2 - 6329061*x - 14714266
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{4}\cdot 5 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$29.3435$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+26.3435 q^{2} +181.982 q^{4} -625.000 q^{5} -2401.00 q^{7} -8693.85 q^{8} -16464.7 q^{10} +2245.43 q^{11} -123379. q^{13} -63250.8 q^{14} -322201. q^{16} +18794.1 q^{17} +404776. q^{19} -113739. q^{20} +59152.4 q^{22} -1.75527e6 q^{23} +390625. q^{25} -3.25024e6 q^{26} -436938. q^{28} -3.94483e6 q^{29} +8.99428e6 q^{31} -4.03667e6 q^{32} +495103. q^{34} +1.50062e6 q^{35} -1.04609e7 q^{37} +1.06632e7 q^{38} +5.43366e6 q^{40} -3.02704e6 q^{41} +1.60767e7 q^{43} +408626. q^{44} -4.62401e7 q^{46} +3.97557e7 q^{47} +5.76480e6 q^{49} +1.02904e7 q^{50} -2.24528e7 q^{52} -4.08073e7 q^{53} -1.40339e6 q^{55} +2.08739e7 q^{56} -1.03921e8 q^{58} -1.60585e8 q^{59} +1.49289e8 q^{61} +2.36941e8 q^{62} +5.86269e7 q^{64} +7.71120e7 q^{65} +5.12683e7 q^{67} +3.42018e6 q^{68} +3.95318e7 q^{70} +3.35362e8 q^{71} +1.35293e8 q^{73} -2.75578e8 q^{74} +7.36618e7 q^{76} -5.39127e6 q^{77} +1.65365e8 q^{79} +2.01376e8 q^{80} -7.97429e7 q^{82} +1.61436e8 q^{83} -1.17463e7 q^{85} +4.23516e8 q^{86} -1.95214e7 q^{88} -1.00313e9 q^{89} +2.96233e8 q^{91} -3.19428e8 q^{92} +1.04731e9 q^{94} -2.52985e8 q^{95} -2.89165e8 q^{97} +1.51865e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 15 q^{2} + 3009 q^{4} - 3750 q^{5} - 14406 q^{7} - 22041 q^{8} + 9375 q^{10} + 47796 q^{11} + 102168 q^{13} + 36015 q^{14} + 2371065 q^{16} + 38472 q^{17} + 361056 q^{19} - 1880625 q^{20} + 2068680 q^{22}+ \cdots - 86472015 q^{98}+O(q^{100}) $ 6 * q - 15 * q^2 + 3009 * q^4 - 3750 * q^5 - 14406 * q^7 - 22041 * q^8 + 9375 * q^10 + 47796 * q^11 + 102168 * q^13 + 36015 * q^14 + 2371065 * q^16 + 38472 * q^17 + 361056 * q^19 - 1880625 * q^20 + 2068680 * q^22 - 697032 * q^23 + 2343750 * q^25 - 10847622 * q^26 - 7224609 * q^28 - 16028184 * q^29 + 1362912 * q^31 + 1148823 * q^32 - 9216294 * q^34 + 9003750 * q^35 - 3912924 * q^37 - 39877140 * q^38 + 13775625 * q^40 - 22452756 * q^41 - 29998992 * q^43 + 94378728 * q^44 + 49589520 * q^46 + 121271508 * q^47 + 34588806 * q^49 - 5859375 * q^50 - 44528550 * q^52 + 20308596 * q^53 - 29872500 * q^55 + 52920441 * q^56 + 256759494 * q^58 - 120280392 * q^59 - 87693540 * q^61 + 344802528 * q^62 + 640647105 * q^64 - 63855000 * q^65 + 495050664 * q^67 + 141443706 * q^68 - 22509375 * q^70 - 253762512 * q^71 + 187195308 * q^73 + 418806054 * q^74 - 584719476 * q^76 - 114758196 * q^77 + 831079500 * q^79 - 1481915625 * q^80 + 2286848814 * q^82 + 767650536 * q^83 - 24045000 * q^85 - 566199948 * q^86 + 2952495444 * q^88 - 582579684 * q^89 - 245305368 * q^91 + 4259382384 * q^92 - 1115946660 * q^94 - 225660000 * q^95 - 1184506872 * q^97 - 86472015 * q^98

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 $

$2$	26.3435	1.16423	0.582115	−	0.813106i	$-0.302225\pi$
$2$	26.3435	1.16423	0.582115	+	0.813106i	$0.302225\pi$
$3$	0	0
$3$	0	0
$4$	181.982	0.355433
$5$	−625.000	−0.447214
$5$	−625.000	−0.447214
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	−8693.85	−0.750425
$9$	0	0
$10$	−16464.7	−0.520660
$11$	2245.43	0.0462415	0.0231207	−	0.999733i	$-0.492640\pi$
$11$	2245.43	0.0462415	0.0231207	+	0.999733i	$0.492640\pi$
$12$	0	0
$13$	−123379.	−1.19811	−0.599055	−	0.800708i	$-0.704457\pi$
$13$	−123379.	−1.19811	−0.599055	+	0.800708i	$0.704457\pi$
$14$	−63250.8	−0.440038
$15$	0	0
$16$	−322201.	−1.22910
$17$	18794.1	0.0545760	0.0272880	−	0.999628i	$-0.491313\pi$
$17$	18794.1	0.0545760	0.0272880	+	0.999628i	$0.491313\pi$
$18$	0	0
$19$	404776.	0.712564	0.356282	−	0.934379i	$-0.384044\pi$
$19$	404776.	0.712564	0.356282	+	0.934379i	$0.384044\pi$
$20$	−113739.	−0.158954
$21$	0	0
$22$	59152.4	0.0538357
$23$	−1.75527e6	−1.30789	−0.653943	−	0.756544i	$-0.726886\pi$
$23$	−1.75527e6	−1.30789	−0.653943	+	0.756544i	$0.726886\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−3.25024e6	−1.39488
$27$	0	0
$28$	−436938.	−0.134341
$29$	−3.94483e6	−1.03571	−0.517854	−	0.855469i	$-0.673269\pi$
$29$	−3.94483e6	−1.03571	−0.517854	+	0.855469i	$0.673269\pi$
$30$	0	0
$31$	8.99428e6	1.74920	0.874599	−	0.484847i	$-0.161125\pi$
$31$	8.99428e6	1.74920	0.874599	+	0.484847i	$0.161125\pi$
$32$	−4.03667e6	−0.680532
$33$	0	0
$34$	495103.	0.0635390
$35$	1.50062e6	0.169031
$36$	0	0
$37$	−1.04609e7	−0.917618	−0.458809	−	0.888535i	$-0.651724\pi$
$37$	−1.04609e7	−0.917618	−0.458809	+	0.888535i	$0.651724\pi$
$38$	1.06632e7	0.829589
$39$	0	0
$40$	5.43366e6	0.335600
$41$	−3.02704e6	−0.167298	−0.0836489	−	0.996495i	$-0.526657\pi$
$41$	−3.02704e6	−0.167298	−0.0836489	+	0.996495i	$0.526657\pi$
$42$	0	0
$43$	1.60767e7	0.717114	0.358557	−	0.933508i	$-0.383269\pi$
$43$	1.60767e7	0.717114	0.358557	+	0.933508i	$0.383269\pi$
$44$	408626.	0.0164357
$45$	0	0
$46$	−4.62401e7	−1.52268
$47$	3.97557e7	1.18839	0.594196	−	0.804321i	$-0.297471\pi$
$47$	3.97557e7	1.18839	0.594196	+	0.804321i	$0.297471\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	1.02904e7	0.232846
$51$	0	0
$52$	−2.24528e7	−0.425848
$53$	−4.08073e7	−0.710389	−0.355195	−	0.934792i	$-0.615585\pi$
$53$	−4.08073e7	−0.710389	−0.355195	+	0.934792i	$0.615585\pi$
$54$	0	0
$55$	−1.40339e6	−0.0206798
$56$	2.08739e7	0.283634
$57$	0	0
$58$	−1.03921e8	−1.20580
$59$	−1.60585e8	−1.72532	−0.862660	−	0.505784i	$-0.831203\pi$
$59$	−1.60585e8	−1.72532	−0.862660	+	0.505784i	$0.831203\pi$
$60$	0	0
$61$	1.49289e8	1.38052	0.690260	−	0.723561i	$-0.257496\pi$
$61$	1.49289e8	1.38052	0.690260	+	0.723561i	$0.257496\pi$
$62$	2.36941e8	2.03647
$63$	0	0
$64$	5.86269e7	0.436805
$65$	7.71120e7	0.535811
$66$	0	0
$67$	5.12683e7	0.310822	0.155411	−	0.987850i	$-0.450330\pi$
$67$	5.12683e7	0.310822	0.155411	+	0.987850i	$0.450330\pi$
$68$	3.42018e6	0.0193981
$69$	0	0
$70$	3.95318e7	0.196791
$71$	3.35362e8	1.56622	0.783108	−	0.621886i	$-0.213633\pi$
$71$	3.35362e8	1.56622	0.783108	+	0.621886i	$0.213633\pi$
$72$	0	0
$73$	1.35293e8	0.557601	0.278801	−	0.960349i	$-0.410063\pi$
$73$	1.35293e8	0.557601	0.278801	+	0.960349i	$0.410063\pi$
$74$	−2.75578e8	−1.06832
$75$	0	0
$76$	7.36618e7	0.253269
$77$	−5.39127e6	−0.0174776
$78$	0	0
$79$	1.65365e8	0.477665	0.238832	−	0.971061i	$-0.423235\pi$
$79$	1.65365e8	0.477665	0.238832	+	0.971061i	$0.423235\pi$
$80$	2.01376e8	0.549670
$81$	0	0
$82$	−7.97429e7	−0.194773
$83$	1.61436e8	0.373379	0.186690	−	0.982419i	$-0.440224\pi$
$83$	1.61436e8	0.373379	0.186690	+	0.982419i	$0.440224\pi$
$84$	0	0
$85$	−1.17463e7	−0.0244071
$86$	4.23516e8	0.834886
$87$	0	0
$88$	−1.95214e7	−0.0347007
$89$	−1.00313e9	−1.69473	−0.847366	−	0.531009i	$-0.821813\pi$
$89$	−1.00313e9	−1.69473	−0.847366	+	0.531009i	$0.821813\pi$
$90$	0	0
$91$	2.96233e8	0.452843
$92$	−3.19428e8	−0.464866
$93$	0	0
$94$	1.04731e9	1.38356
$95$	−2.52985e8	−0.318668
$96$	0	0
$97$	−2.89165e8	−0.331644	−0.165822	−	0.986156i	$-0.553028\pi$
$97$	−2.89165e8	−0.331644	−0.165822	+	0.986156i	$0.553028\pi$
$98$	1.51865e8	0.166319
$99$	0	0
$100$	7.10866e7	0.0710866
$101$	−1.57210e8	−0.150326	−0.0751629	−	0.997171i	$-0.523948\pi$
$101$	−1.57210e8	−0.150326	−0.0751629	+	0.997171i	$0.523948\pi$
$102$	0	0
$103$	5.15072e8	0.450921	0.225461	−	0.974252i	$-0.427611\pi$
$103$	5.15072e8	0.450921	0.225461	+	0.974252i	$0.427611\pi$
$104$	1.07264e9	0.899092
$105$	0	0
$106$	−1.07501e9	−0.827057
$107$	2.36823e9	1.74662	0.873308	−	0.487168i	$-0.161970\pi$
$107$	2.36823e9	1.74662	0.873308	+	0.487168i	$0.161970\pi$
$108$	0	0
$109$	2.25821e8	0.153231	0.0766154	−	0.997061i	$-0.475589\pi$
$109$	2.25821e8	0.153231	0.0766154	+	0.997061i	$0.475589\pi$
$110$	−3.69703e7	−0.0240761
$111$	0	0
$112$	7.73605e8	0.464556
$113$	2.93829e9	1.69528	0.847640	−	0.530572i	$-0.178023\pi$
$113$	2.93829e9	1.69528	0.847640	+	0.530572i	$0.178023\pi$
$114$	0	0
$115$	1.09705e9	0.584904
$116$	−7.17886e8	−0.368125
$117$	0	0
$118$	−4.23036e9	−2.00867
$119$	−4.51246e7	−0.0206278
$120$	0	0
$121$	−2.35291e9	−0.997862
$122$	3.93279e9	1.60724
$123$	0	0
$124$	1.63679e9	0.621723
$125$	−2.44141e8	−0.0894427
$126$	0	0
$127$	3.08644e9	1.05279	0.526393	−	0.850241i	$-0.323544\pi$
$127$	3.08644e9	1.05279	0.526393	+	0.850241i	$0.323544\pi$
$128$	3.61122e9	1.18907
$129$	0	0
$130$	2.03140e9	0.623808
$131$	2.40858e9	0.714561	0.357281	−	0.933997i	$-0.383704\pi$
$131$	2.40858e9	0.714561	0.357281	+	0.933997i	$0.383704\pi$
$132$	0	0
$133$	−9.71868e8	−0.269324
$134$	1.35059e9	0.361869
$135$	0	0
$136$	−1.63393e8	−0.0409551
$137$	−5.47458e9	−1.32772	−0.663862	−	0.747855i	$-0.731084\pi$
$137$	−5.47458e9	−1.32772	−0.663862	+	0.747855i	$0.731084\pi$
$138$	0	0
$139$	9.08897e8	0.206513	0.103257	−	0.994655i	$-0.467074\pi$
$139$	9.08897e8	0.206513	0.103257	+	0.994655i	$0.467074\pi$
$140$	2.73086e8	0.0600791
$141$	0	0
$142$	8.83462e9	1.82344
$143$	−2.77039e8	−0.0554024
$144$	0	0
$145$	2.46552e9	0.463182
$146$	3.56411e9	0.649176
$147$	0	0
$148$	−1.90370e9	−0.326152
$149$	−6.00080e9	−0.997403	−0.498702	−	0.866774i	$-0.666190\pi$
$149$	−6.00080e9	−0.997403	−0.498702	+	0.866774i	$0.666190\pi$
$150$	0	0
$151$	5.35390e9	0.838058	0.419029	−	0.907973i	$-0.362371\pi$
$151$	5.35390e9	0.838058	0.419029	+	0.907973i	$0.362371\pi$
$152$	−3.51906e9	−0.534725
$153$	0	0
$154$	−1.42025e8	−0.0203480
$155$	−5.62143e9	−0.782265
$156$	0	0
$157$	−3.05256e8	−0.0400973	−0.0200487	−	0.999799i	$-0.506382\pi$
$157$	−3.05256e8	−0.0400973	−0.0200487	+	0.999799i	$0.506382\pi$
$158$	4.35631e9	0.556112
$159$	0	0
$160$	2.52292e9	0.304343
$161$	4.21441e9	0.494334
$162$	0	0
$163$	9.53729e9	1.05823	0.529116	−	0.848550i	$-0.322524\pi$
$163$	9.53729e9	1.05823	0.529116	+	0.848550i	$0.322524\pi$
$164$	−5.50865e8	−0.0594632
$165$	0	0
$166$	4.25281e9	0.434700
$167$	7.59729e9	0.755848	0.377924	−	0.925837i	$-0.376638\pi$
$167$	7.59729e9	0.755848	0.377924	+	0.925837i	$0.376638\pi$
$168$	0	0
$169$	4.61792e9	0.435468
$170$	−3.09439e8	−0.0284155
$171$	0	0
$172$	2.92566e9	0.254886
$173$	1.61106e10	1.36742	0.683712	−	0.729751i	$-0.260364\pi$
$173$	1.61106e10	1.36742	0.683712	+	0.729751i	$0.260364\pi$
$174$	0	0
$175$	−9.37891e8	−0.0755929
$176$	−7.23479e8	−0.0568354
$177$	0	0
$178$	−2.64259e10	−1.97306
$179$	−5.90583e9	−0.429974	−0.214987	−	0.976617i	$-0.568971\pi$
$179$	−5.90583e9	−0.429974	−0.214987	+	0.976617i	$0.568971\pi$
$180$	0	0
$181$	−4.57036e9	−0.316517	−0.158258	−	0.987398i	$-0.550588\pi$
$181$	−4.57036e9	−0.316517	−0.158258	+	0.987398i	$0.550588\pi$
$182$	7.80383e9	0.527214
$183$	0	0
$184$	1.52601e10	0.981469
$185$	6.53807e9	0.410371
$186$	0	0
$187$	4.22007e7	0.00252367
$188$	7.23481e9	0.422393
$189$	0	0
$190$	−6.66452e9	−0.371003
$191$	−1.08620e10	−0.590555	−0.295277	−	0.955412i	$-0.595412\pi$
$191$	−1.08620e10	−0.590555	−0.295277	+	0.955412i	$0.595412\pi$
$192$	0	0
$193$	3.53101e10	1.83185	0.915926	−	0.401346i	$-0.131458\pi$
$193$	3.53101e10	1.83185	0.915926	+	0.401346i	$0.131458\pi$
$194$	−7.61762e9	−0.386110
$195$	0	0
$196$	1.04909e9	0.0507761
$197$	1.34688e10	0.637136	0.318568	−	0.947900i	$-0.396798\pi$
$197$	1.34688e10	0.637136	0.318568	+	0.947900i	$0.396798\pi$
$198$	0	0
$199$	−2.38956e10	−1.08014	−0.540068	−	0.841621i	$-0.681601\pi$
$199$	−2.38956e10	−1.08014	−0.540068	+	0.841621i	$0.681601\pi$
$200$	−3.39603e9	−0.150085
$201$	0	0
$202$	−4.14146e9	−0.175014
$203$	9.47153e9	0.391461
$204$	0	0
$205$	1.89190e9	0.0748179
$206$	1.35688e10	0.524976
$207$	0	0
$208$	3.97529e10	1.47260
$209$	9.08895e8	0.0329500
$210$	0	0
$211$	−2.71767e10	−0.943898	−0.471949	−	0.881626i	$-0.656449\pi$
$211$	−2.71767e10	−0.943898	−0.471949	+	0.881626i	$0.656449\pi$
$212$	−7.42618e9	−0.252496
$213$	0	0
$214$	6.23876e10	2.03346
$215$	−1.00479e10	−0.320703
$216$	0	0
$217$	−2.15953e10	−0.661135
$218$	5.94893e9	0.178396
$219$	0	0
$220$	−2.55391e8	−0.00735029
$221$	−2.31880e9	−0.0653880
$222$	0	0
$223$	−2.65338e8	−0.00718502	−0.00359251	−	0.999994i	$-0.501144\pi$
$223$	−2.65338e8	−0.00718502	−0.00359251	+	0.999994i	$0.501144\pi$
$224$	9.69204e9	0.257217
$225$	0	0
$226$	7.74049e10	1.97370
$227$	2.92609e10	0.731428	0.365714	−	0.930727i	$-0.380825\pi$
$227$	2.92609e10	0.731428	0.365714	+	0.930727i	$0.380825\pi$
$228$	0	0
$229$	−3.93335e10	−0.945156	−0.472578	−	0.881289i	$-0.656676\pi$
$229$	−3.93335e10	−0.945156	−0.472578	+	0.881289i	$0.656676\pi$
$230$	2.89001e10	0.680963
$231$	0	0
$232$	3.42957e10	0.777220
$233$	−4.02983e10	−0.895746	−0.447873	−	0.894097i	$-0.647818\pi$
$233$	−4.02983e10	−0.895746	−0.447873	+	0.894097i	$0.647818\pi$
$234$	0	0
$235$	−2.48473e10	−0.531465
$236$	−2.92234e10	−0.613236
$237$	0	0
$238$	−1.18874e9	−0.0240155
$239$	−3.54020e10	−0.701839	−0.350919	−	0.936406i	$-0.614131\pi$
$239$	−3.54020e10	−0.701839	−0.350919	+	0.936406i	$0.614131\pi$
$240$	0	0
$241$	3.42729e9	0.0654446	0.0327223	−	0.999464i	$-0.489582\pi$
$241$	3.42729e9	0.0654446	0.0327223	+	0.999464i	$0.489582\pi$
$242$	−6.19838e10	−1.16174
$243$	0	0
$244$	2.71678e10	0.490683
$245$	−3.60300e9	−0.0638877
$246$	0	0
$247$	−4.99410e10	−0.853730
$248$	−7.81949e10	−1.31264
$249$	0	0
$250$	−6.43153e9	−0.104132
$251$	6.49927e10	1.03355	0.516776	−	0.856120i	$-0.327132\pi$
$251$	6.49927e10	1.03355	0.516776	+	0.856120i	$0.327132\pi$
$252$	0	0
$253$	−3.94134e9	−0.0604785
$254$	8.13076e10	1.22569
$255$	0	0
$256$	6.51152e10	0.947551
$257$	−7.71755e10	−1.10352	−0.551760	−	0.834003i	$-0.686044\pi$
$257$	−7.71755e10	−1.10352	−0.551760	+	0.834003i	$0.686044\pi$
$258$	0	0
$259$	2.51167e10	0.346827
$260$	1.40330e10	0.190445
$261$	0	0
$262$	6.34504e10	0.831914
$263$	−5.32154e10	−0.685862	−0.342931	−	0.939361i	$-0.611420\pi$
$263$	−5.32154e10	−0.685862	−0.342931	+	0.939361i	$0.611420\pi$
$264$	0	0
$265$	2.55046e10	0.317696
$266$	−2.56024e10	−0.313555
$267$	0	0
$268$	9.32989e9	0.110476
$269$	−2.15319e10	−0.250725	−0.125362	−	0.992111i	$-0.540009\pi$
$269$	−2.15319e10	−0.250725	−0.125362	+	0.992111i	$0.540009\pi$
$270$	0	0
$271$	4.81053e10	0.541790	0.270895	−	0.962609i	$-0.412680\pi$
$271$	4.81053e10	0.541790	0.270895	+	0.962609i	$0.412680\pi$
$272$	−6.05548e9	−0.0670793
$273$	0	0
$274$	−1.44220e11	−1.54578
$275$	8.77119e8	0.00924829
$276$	0	0
$277$	6.60074e10	0.673649	0.336824	−	0.941568i	$-0.390647\pi$
$277$	6.60074e10	0.673649	0.336824	+	0.941568i	$0.390647\pi$
$278$	2.39436e10	0.240429
$279$	0	0
$280$	−1.30462e10	−0.126845
$281$	−7.31129e10	−0.699545	−0.349772	−	0.936835i	$-0.613741\pi$
$281$	−7.31129e10	−0.699545	−0.349772	+	0.936835i	$0.613741\pi$
$282$	0	0
$283$	1.32293e11	1.22602	0.613012	−	0.790073i	$-0.289958\pi$
$283$	1.32293e11	1.22602	0.613012	+	0.790073i	$0.289958\pi$
$284$	6.10298e10	0.556685
$285$	0	0
$286$	−7.29818e9	−0.0645011
$287$	7.26792e9	0.0632327
$288$	0	0
$289$	−1.18235e11	−0.997021
$290$	6.49504e10	0.539251
$291$	0	0
$292$	2.46209e10	0.198190
$293$	4.00575e10	0.317526	0.158763	−	0.987317i	$-0.449249\pi$
$293$	4.00575e10	0.317526	0.158763	+	0.987317i	$0.449249\pi$
$294$	0	0
$295$	1.00365e11	0.771586
$296$	9.09456e10	0.688603
$297$	0	0
$298$	−1.58082e11	−1.16121
$299$	2.16564e11	1.56699
$300$	0	0
$301$	−3.86001e10	−0.271044
$302$	1.41041e11	0.975693
$303$	0	0
$304$	−1.30419e11	−0.875812
$305$	−9.33055e10	−0.617388
$306$	0	0
$307$	3.61839e10	0.232484	0.116242	−	0.993221i	$-0.462915\pi$
$307$	3.61839e10	0.232484	0.116242	+	0.993221i	$0.462915\pi$
$308$	−9.81112e8	−0.00621212
$309$	0	0
$310$	−1.48088e11	−0.910737
$311$	−1.58042e11	−0.957968	−0.478984	−	0.877824i	$-0.658995\pi$
$311$	−1.58042e11	−0.957968	−0.478984	+	0.877824i	$0.658995\pi$
$312$	0	0
$313$	1.78114e11	1.04894	0.524468	−	0.851430i	$-0.324264\pi$
$313$	1.78114e11	1.04894	0.524468	+	0.851430i	$0.324264\pi$
$314$	−8.04151e9	−0.0466825
$315$	0	0
$316$	3.00935e10	0.169778
$317$	−3.74923e10	−0.208533	−0.104267	−	0.994549i	$-0.533249\pi$
$317$	−3.74923e10	−0.208533	−0.104267	+	0.994549i	$0.533249\pi$
$318$	0	0
$319$	−8.85782e9	−0.0478926
$320$	−3.66418e10	−0.195345
$321$	0	0
$322$	1.11023e11	0.575519
$323$	7.60740e9	0.0388888
$324$	0	0
$325$	−4.81950e10	−0.239622
$326$	2.51246e11	1.23203
$327$	0	0
$328$	2.63166e10	0.125544
$329$	−9.54535e10	−0.449170
$330$	0	0
$331$	1.08393e11	0.496335	0.248167	−	0.968717i	$-0.420172\pi$
$331$	1.08393e11	0.496335	0.248167	+	0.968717i	$0.420172\pi$
$332$	2.93785e10	0.132711
$333$	0	0
$334$	2.00139e11	0.879981
$335$	−3.20427e10	−0.139004
$336$	0	0
$337$	3.53012e11	1.49092	0.745462	−	0.666548i	$-0.232229\pi$
$337$	3.53012e11	1.49092	0.745462	+	0.666548i	$0.232229\pi$
$338$	1.21652e11	0.506986
$339$	0	0
$340$	−2.13761e9	−0.00867509
$341$	2.01960e10	0.0808855
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	−1.39768e11	−0.538140
$345$	0	0
$346$	4.24409e11	1.59200
$347$	4.58221e11	1.69665	0.848324	−	0.529477i	$-0.177612\pi$
$347$	4.58221e11	1.69665	0.848324	+	0.529477i	$0.177612\pi$
$348$	0	0
$349$	4.77589e11	1.72322	0.861608	−	0.507575i	$-0.169458\pi$
$349$	4.77589e11	1.72322	0.861608	+	0.507575i	$0.169458\pi$
$350$	−2.47074e10	−0.0880076
$351$	0	0
$352$	−9.06404e9	−0.0314688
$353$	−4.13087e11	−1.41597	−0.707986	−	0.706226i	$-0.750396\pi$
$353$	−4.13087e11	−1.41597	−0.707986	+	0.706226i	$0.750396\pi$
$354$	0	0
$355$	−2.09601e11	−0.700433
$356$	−1.82551e11	−0.602364
$357$	0	0
$358$	−1.55581e11	−0.500589
$359$	1.83554e11	0.583230	0.291615	−	0.956536i	$-0.405807\pi$
$359$	1.83554e11	0.583230	0.291615	+	0.956536i	$0.405807\pi$
$360$	0	0
$361$	−1.58844e11	−0.492253
$362$	−1.20399e11	−0.368498
$363$	0	0
$364$	5.39091e10	0.160955
$365$	−8.45584e10	−0.249367
$366$	0	0
$367$	−1.52885e11	−0.439915	−0.219957	−	0.975509i	$-0.570592\pi$
$367$	−1.52885e11	−0.439915	−0.219957	+	0.975509i	$0.570592\pi$
$368$	5.65552e11	1.60752
$369$	0	0
$370$	1.72236e11	0.477767
$371$	9.79783e10	0.268502
$372$	0	0
$373$	3.94586e11	1.05548	0.527742	−	0.849405i	$-0.323039\pi$
$373$	3.94586e11	1.05548	0.527742	+	0.849405i	$0.323039\pi$
$374$	1.11172e9	0.00293814
$375$	0	0
$376$	−3.45630e11	−0.891798
$377$	4.86710e11	1.24089
$378$	0	0
$379$	7.26412e11	1.80845	0.904225	−	0.427057i	$-0.140450\pi$
$379$	7.26412e11	1.80845	0.904225	+	0.427057i	$0.140450\pi$
$380$	−4.60387e10	−0.113265
$381$	0	0
$382$	−2.86144e11	−0.687542
$383$	5.73936e11	1.36292	0.681458	−	0.731857i	$-0.261346\pi$
$383$	5.73936e11	1.36292	0.681458	+	0.731857i	$0.261346\pi$
$384$	0	0
$385$	3.36954e9	0.00781623
$386$	9.30192e11	2.13270
$387$	0	0
$388$	−5.26227e10	−0.117877
$389$	−2.27345e11	−0.503400	−0.251700	−	0.967805i	$-0.580990\pi$
$389$	−2.27345e11	−0.503400	−0.251700	+	0.967805i	$0.580990\pi$
$390$	0	0
$391$	−3.29888e10	−0.0713791
$392$	−5.01183e10	−0.107204
$393$	0	0
$394$	3.54817e11	0.741773
$395$	−1.03353e11	−0.213618
$396$	0	0
$397$	−9.03630e11	−1.82572	−0.912858	−	0.408277i	$-0.866130\pi$
$397$	−9.03630e11	−1.82572	−0.912858	+	0.408277i	$0.866130\pi$
$398$	−6.29493e11	−1.25753
$399$	0	0
$400$	−1.25860e11	−0.245820
$401$	−3.47203e11	−0.670555	−0.335277	−	0.942119i	$-0.608830\pi$
$401$	−3.47203e11	−0.670555	−0.335277	+	0.942119i	$0.608830\pi$
$402$	0	0
$403$	−1.10971e12	−2.09573
$404$	−2.86093e10	−0.0534307
$405$	0	0
$406$	2.49514e11	0.455750
$407$	−2.34892e10	−0.0424320
$408$	0	0
$409$	−1.06690e12	−1.88524	−0.942621	−	0.333866i	$-0.891647\pi$
$409$	−1.06690e12	−1.88524	−0.942621	+	0.333866i	$0.891647\pi$
$410$	4.98393e10	0.0871053
$411$	0	0
$412$	9.37337e10	0.160272
$413$	3.85563e11	0.652110
$414$	0	0
$415$	−1.00898e11	−0.166980
$416$	4.98041e11	0.815352
$417$	0	0
$418$	2.39435e10	0.0383614
$419$	−8.89018e11	−1.40912	−0.704559	−	0.709645i	$-0.748855\pi$
$419$	−8.89018e11	−1.40912	−0.704559	+	0.709645i	$0.748855\pi$
$420$	0	0
$421$	9.18431e11	1.42488	0.712438	−	0.701735i	$-0.247591\pi$
$421$	9.18431e11	1.42488	0.712438	+	0.701735i	$0.247591\pi$
$422$	−7.15929e11	−1.09891
$423$	0	0
$424$	3.54773e11	0.533094
$425$	7.34144e9	0.0109152
$426$	0	0
$427$	−3.58442e11	−0.521788
$428$	4.30975e11	0.620805
$429$	0	0
$430$	−2.64698e11	−0.373372
$431$	7.15695e11	0.999034	0.499517	−	0.866304i	$-0.333511\pi$
$431$	7.15695e11	0.999034	0.499517	+	0.866304i	$0.333511\pi$
$432$	0	0
$433$	−1.30709e12	−1.78694	−0.893472	−	0.449119i	$-0.851738\pi$
$433$	−1.30709e12	−1.78694	−0.893472	+	0.449119i	$0.851738\pi$
$434$	−5.68896e11	−0.769713
$435$	0	0
$436$	4.10954e10	0.0544633
$437$	−7.10493e11	−0.931952
$438$	0	0
$439$	−8.08851e11	−1.03939	−0.519695	−	0.854352i	$-0.673954\pi$
$439$	−8.08851e11	−1.03939	−0.519695	+	0.854352i	$0.673954\pi$
$440$	1.22009e10	0.0155186
$441$	0	0
$442$	−6.10854e10	−0.0761267
$443$	−2.49522e11	−0.307816	−0.153908	−	0.988085i	$-0.549186\pi$
$443$	−2.49522e11	−0.307816	−0.153908	+	0.988085i	$0.549186\pi$
$444$	0	0
$445$	6.26955e11	0.757907
$446$	−6.98995e9	−0.00836503
$447$	0	0
$448$	−1.40763e11	−0.165097
$449$	−1.12524e12	−1.30658	−0.653291	−	0.757107i	$-0.726612\pi$
$449$	−1.12524e12	−1.30658	−0.653291	+	0.757107i	$0.726612\pi$
$450$	0	0
$451$	−6.79699e9	−0.00773610
$452$	5.34715e11	0.602558
$453$	0	0
$454$	7.70835e11	0.851550
$455$	−1.85146e11	−0.202518
$456$	0	0
$457$	7.92565e11	0.849987	0.424993	−	0.905196i	$-0.360276\pi$
$457$	7.92565e11	0.849987	0.424993	+	0.905196i	$0.360276\pi$
$458$	−1.03618e12	−1.10038
$459$	0	0
$460$	1.99642e11	0.207894
$461$	−8.54573e11	−0.881241	−0.440621	−	0.897693i	$-0.645242\pi$
$461$	−8.54573e11	−0.881241	−0.440621	+	0.897693i	$0.645242\pi$
$462$	0	0
$463$	−1.55057e12	−1.56811	−0.784056	−	0.620690i	$-0.786852\pi$
$463$	−1.55057e12	−1.56811	−0.784056	+	0.620690i	$0.786852\pi$
$464$	1.27103e12	1.27299
$465$	0	0
$466$	−1.06160e12	−1.04285
$467$	1.54154e11	0.149979	0.0749893	−	0.997184i	$-0.476108\pi$
$467$	1.54154e11	0.149979	0.0749893	+	0.997184i	$0.476108\pi$
$468$	0	0
$469$	−1.23095e11	−0.117480
$470$	−6.54566e11	−0.618747
$471$	0	0
$472$	1.39610e12	1.29472
$473$	3.60990e10	0.0331604
$474$	0	0
$475$	1.58116e11	0.142513
$476$	−8.21186e9	−0.00733179
$477$	0	0
$478$	−9.32614e11	−0.817102
$479$	1.96197e12	1.70287	0.851437	−	0.524457i	$-0.175732\pi$
$479$	1.96197e12	1.70287	0.851437	+	0.524457i	$0.175732\pi$
$480$	0	0
$481$	1.29066e12	1.09941
$482$	9.02868e10	0.0761926
$483$	0	0
$484$	−4.28186e11	−0.354673
$485$	1.80728e11	0.148316
$486$	0	0
$487$	8.97598e11	0.723106	0.361553	−	0.932352i	$-0.382247\pi$
$487$	8.97598e11	0.723106	0.361553	+	0.932352i	$0.382247\pi$
$488$	−1.29789e12	−1.03598
$489$	0	0
$490$	−9.49158e10	−0.0743800
$491$	−4.80068e11	−0.372766	−0.186383	−	0.982477i	$-0.559676\pi$
$491$	−4.80068e11	−0.372766	−0.186383	+	0.982477i	$0.559676\pi$
$492$	0	0
$493$	−7.41395e10	−0.0565247
$494$	−1.31562e12	−0.993939
$495$	0	0
$496$	−2.89797e12	−2.14994
$497$	−8.05205e11	−0.591974
$498$	0	0
$499$	−1.81750e12	−1.31227	−0.656133	−	0.754646i	$-0.727809\pi$
$499$	−1.81750e12	−1.31227	−0.656133	+	0.754646i	$0.727809\pi$
$500$	−4.44291e10	−0.0317909
$501$	0	0
$502$	1.71214e12	1.20329
$503$	−9.78467e11	−0.681538	−0.340769	−	0.940147i	$-0.610687\pi$
$503$	−9.78467e11	−0.681538	−0.340769	+	0.940147i	$0.610687\pi$
$504$	0	0
$505$	9.82561e10	0.0672277
$506$	−1.03829e11	−0.0704109
$507$	0	0
$508$	5.61675e11	0.374195
$509$	−9.49400e11	−0.626930	−0.313465	−	0.949600i	$-0.601490\pi$
$509$	−9.49400e11	−0.626930	−0.313465	+	0.949600i	$0.601490\pi$
$510$	0	0
$511$	−3.24839e11	−0.210753
$512$	−1.33578e11	−0.0859056
$513$	0	0
$514$	−2.03308e12	−1.28475
$515$	−3.21920e11	−0.201658
$516$	0	0
$517$	8.92685e10	0.0549529
$518$	6.61662e11	0.403787
$519$	0	0
$520$	−6.70400e11	−0.402086
$521$	−3.20965e12	−1.90848	−0.954240	−	0.299042i	$-0.903333\pi$
$521$	−3.20965e12	−1.90848	−0.954240	+	0.299042i	$0.903333\pi$
$522$	0	0
$523$	−2.27961e12	−1.33230	−0.666152	−	0.745816i	$-0.732060\pi$
$523$	−2.27961e12	−1.33230	−0.666152	+	0.745816i	$0.732060\pi$
$524$	4.38317e11	0.253979
$525$	0	0
$526$	−1.40188e12	−0.798501
$527$	1.69039e11	0.0954642
$528$	0	0
$529$	1.27983e12	0.710564
$530$	6.71880e11	0.369871
$531$	0	0
$532$	−1.76862e11	−0.0957265
$533$	3.73473e11	0.200441
$534$	0	0
$535$	−1.48015e12	−0.781111
$536$	−4.45719e11	−0.233249
$537$	0	0
$538$	−5.67226e11	−0.291901
$539$	1.29444e10	0.00660592
$540$	0	0
$541$	2.51926e12	1.26440	0.632201	−	0.774804i	$-0.282152\pi$
$541$	2.51926e12	1.26440	0.632201	+	0.774804i	$0.282152\pi$
$542$	1.26726e12	0.630769
$543$	0	0
$544$	−7.58656e10	−0.0371407
$545$	−1.41138e11	−0.0685269
$546$	0	0
$547$	−2.15157e12	−1.02757	−0.513787	−	0.857918i	$-0.671758\pi$
$547$	−2.15157e12	−1.02757	−0.513787	+	0.857918i	$0.671758\pi$
$548$	−9.96272e11	−0.471917
$549$	0	0
$550$	2.31064e10	0.0107671
$551$	−1.59677e12	−0.738008
$552$	0	0
$553$	−3.97043e11	−0.180540
$554$	1.73887e12	0.784282
$555$	0	0
$556$	1.65403e11	0.0734016
$557$	2.27895e12	1.00320	0.501600	−	0.865100i	$-0.332745\pi$
$557$	2.27895e12	1.00320	0.501600	+	0.865100i	$0.332745\pi$
$558$	0	0
$559$	−1.98353e12	−0.859181
$560$	−4.83503e11	−0.207756
$561$	0	0
$562$	−1.92605e12	−0.814431
$563$	−3.82145e12	−1.60303	−0.801513	−	0.597978i	$-0.795971\pi$
$563$	−3.82145e12	−1.60303	−0.801513	+	0.597978i	$0.795971\pi$
$564$	0	0
$565$	−1.83643e12	−0.758152
$566$	3.48508e12	1.42738
$567$	0	0
$568$	−2.91559e12	−1.17533
$569$	2.94845e12	1.17920	0.589601	−	0.807695i	$-0.299285\pi$
$569$	2.94845e12	1.17920	0.589601	+	0.807695i	$0.299285\pi$
$570$	0	0
$571$	2.99642e12	1.17962	0.589808	−	0.807544i	$-0.299203\pi$
$571$	2.99642e12	1.17962	0.589808	+	0.807544i	$0.299203\pi$
$572$	−5.04160e10	−0.0196918
$573$	0	0
$574$	1.91463e11	0.0736174
$575$	−6.85654e11	−0.261577
$576$	0	0
$577$	−1.87283e12	−0.703407	−0.351703	−	0.936111i	$-0.614397\pi$
$577$	−1.87283e12	−0.703407	−0.351703	+	0.936111i	$0.614397\pi$
$578$	−3.11472e12	−1.16076
$579$	0	0
$580$	4.48679e11	0.164630
$581$	−3.87609e11	−0.141124
$582$	0	0
$583$	−9.16297e10	−0.0328494
$584$	−1.17622e12	−0.418438
$585$	0	0
$586$	1.05525e12	0.369673
$587$	1.94572e12	0.676410	0.338205	−	0.941073i	$-0.390180\pi$
$587$	1.94572e12	0.676410	0.338205	+	0.941073i	$0.390180\pi$
$588$	0	0
$589$	3.64067e12	1.24642
$590$	2.64398e12	0.898305
$591$	0	0
$592$	3.37052e12	1.12784
$593$	−8.61235e11	−0.286006	−0.143003	−	0.989722i	$-0.545676\pi$
$593$	−8.61235e11	−0.286006	−0.143003	+	0.989722i	$0.545676\pi$
$594$	0	0
$595$	2.82029e10	0.00922502
$596$	−1.09204e12	−0.354510
$597$	0	0
$598$	5.70507e12	1.82434
$599$	−2.81546e12	−0.893570	−0.446785	−	0.894641i	$-0.647431\pi$
$599$	−2.81546e12	−0.893570	−0.446785	+	0.894641i	$0.647431\pi$
$600$	0	0
$601$	4.79484e12	1.49913	0.749565	−	0.661931i	$-0.230263\pi$
$601$	4.79484e12	1.49913	0.749565	+	0.661931i	$0.230263\pi$
$602$	−1.01686e12	−0.315557
$603$	0	0
$604$	9.74312e11	0.297873
$605$	1.47057e12	0.446257
$606$	0	0
$607$	7.43279e11	0.222230	0.111115	−	0.993808i	$-0.464558\pi$
$607$	7.43279e11	0.222230	0.111115	+	0.993808i	$0.464558\pi$
$608$	−1.63395e12	−0.484922
$609$	0	0
$610$	−2.45800e12	−0.718782
$611$	−4.90503e12	−1.42382
$612$	0	0
$613$	6.45388e11	0.184607	0.0923036	−	0.995731i	$-0.470577\pi$
$613$	6.45388e11	0.184607	0.0923036	+	0.995731i	$0.470577\pi$
$614$	9.53213e11	0.270665
$615$	0	0
$616$	4.68708e10	0.0131156
$617$	−4.66543e12	−1.29601	−0.648005	−	0.761636i	$-0.724396\pi$
$617$	−4.66543e12	−1.29601	−0.648005	+	0.761636i	$0.724396\pi$
$618$	0	0
$619$	−2.49665e12	−0.683519	−0.341759	−	0.939787i	$-0.611023\pi$
$619$	−2.49665e12	−0.683519	−0.341759	+	0.939787i	$0.611023\pi$
$620$	−1.02300e12	−0.278043
$621$	0	0
$622$	−4.16338e12	−1.11530
$623$	2.40851e12	0.640549
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	4.69215e12	1.22120
$627$	0	0
$628$	−5.55509e10	−0.0142519
$629$	−1.96604e11	−0.0500799
$630$	0	0
$631$	2.92442e12	0.734357	0.367179	−	0.930150i	$-0.380324\pi$
$631$	2.92442e12	0.734357	0.367179	+	0.930150i	$0.380324\pi$
$632$	−1.43766e12	−0.358451
$633$	0	0
$634$	−9.87678e11	−0.242781
$635$	−1.92902e12	−0.470821
$636$	0	0
$637$	−7.11256e11	−0.171159
$638$	−2.33346e11	−0.0557581
$639$	0	0
$640$	−2.25701e12	−0.531770
$641$	4.23039e12	0.989736	0.494868	−	0.868968i	$-0.335216\pi$
$641$	4.23039e12	0.989736	0.494868	+	0.868968i	$0.335216\pi$
$642$	0	0
$643$	1.95718e12	0.451524	0.225762	−	0.974183i	$-0.427513\pi$
$643$	1.95718e12	0.451524	0.225762	+	0.974183i	$0.427513\pi$
$644$	7.66946e11	0.175703
$645$	0	0
$646$	2.00406e11	0.0452756
$647$	4.71751e12	1.05838	0.529192	−	0.848502i	$-0.322495\pi$
$647$	4.71751e12	1.05838	0.529192	+	0.848502i	$0.322495\pi$
$648$	0	0
$649$	−3.60580e11	−0.0797813
$650$	−1.26963e12	−0.278975
$651$	0	0
$652$	1.73561e12	0.376130
$653$	−5.50754e12	−1.18536	−0.592678	−	0.805440i	$-0.701929\pi$
$653$	−5.50754e12	−1.18536	−0.592678	+	0.805440i	$0.701929\pi$
$654$	0	0
$655$	−1.50536e12	−0.319562
$656$	9.75315e11	0.205626
$657$	0	0
$658$	−2.51458e12	−0.522937
$659$	4.05044e12	0.836599	0.418300	−	0.908309i	$-0.362626\pi$
$659$	4.05044e12	0.836599	0.418300	+	0.908309i	$0.362626\pi$
$660$	0	0
$661$	−2.16897e12	−0.441923	−0.220962	−	0.975283i	$-0.570920\pi$
$661$	−2.16897e12	−0.441923	−0.220962	+	0.975283i	$0.570920\pi$
$662$	2.85545e12	0.577848
$663$	0	0
$664$	−1.40350e12	−0.280193
$665$	6.07417e11	0.120445
$666$	0	0
$667$	6.92425e12	1.35459
$668$	1.38257e12	0.268653
$669$	0	0
$670$	−8.44117e11	−0.161833
$671$	3.35217e11	0.0638373
$672$	0	0
$673$	−7.79666e12	−1.46501	−0.732505	−	0.680761i	$-0.761649\pi$
$673$	−7.79666e12	−1.46501	−0.732505	+	0.680761i	$0.761649\pi$
$674$	9.29959e12	1.73578
$675$	0	0
$676$	8.40378e11	0.154780
$677$	3.98612e12	0.729293	0.364646	−	0.931146i	$-0.381190\pi$
$677$	3.98612e12	0.729293	0.364646	+	0.931146i	$0.381190\pi$
$678$	0	0
$679$	6.94284e11	0.125350
$680$	1.02121e11	0.0183157
$681$	0	0
$682$	5.32034e11	0.0941693
$683$	7.20461e12	1.26683	0.633413	−	0.773814i	$-0.281653\pi$
$683$	7.20461e12	1.26683	0.633413	+	0.773814i	$0.281653\pi$
$684$	0	0
$685$	3.42161e12	0.593776
$686$	−3.64628e11	−0.0628625
$687$	0	0
$688$	−5.17992e12	−0.881405
$689$	5.03477e12	0.851125
$690$	0	0
$691$	7.66976e12	1.27977	0.639883	−	0.768472i	$-0.278983\pi$
$691$	7.66976e12	1.27977	0.639883	+	0.768472i	$0.278983\pi$
$692$	2.93183e12	0.486028
$693$	0	0
$694$	1.20711e13	1.97529
$695$	−5.68061e11	−0.0923555
$696$	0	0
$697$	−5.68904e10	−0.00913044
$698$	1.25814e13	2.00622
$699$	0	0
$700$	−1.70679e11	−0.0268682
$701$	6.76464e11	0.105807	0.0529034	−	0.998600i	$-0.483152\pi$
$701$	6.76464e11	0.105807	0.0529034	+	0.998600i	$0.483152\pi$
$702$	0	0
$703$	−4.23433e12	−0.653861
$704$	1.31642e11	0.0201985
$705$	0	0
$706$	−1.08822e13	−1.64852
$707$	3.77461e11	0.0568178
$708$	0	0
$709$	1.19764e13	1.77999	0.889995	−	0.455970i	$-0.150708\pi$
$709$	1.19764e13	1.77999	0.889995	+	0.455970i	$0.150708\pi$
$710$	−5.52164e12	−0.815465
$711$	0	0
$712$	8.72104e12	1.27177
$713$	−1.57874e13	−2.28775
$714$	0	0
$715$	1.73149e11	0.0247767
$716$	−1.07475e12	−0.152827
$717$	0	0
$718$	4.83547e12	0.679014
$719$	9.56848e12	1.33525	0.667626	−	0.744497i	$-0.267311\pi$
$719$	9.56848e12	1.33525	0.667626	+	0.744497i	$0.267311\pi$
$720$	0	0
$721$	−1.23669e12	−0.170432
$722$	−4.18451e12	−0.573096
$723$	0	0
$724$	−8.31721e11	−0.112500
$725$	−1.54095e12	−0.207141
$726$	0	0
$727$	1.05787e13	1.40451	0.702257	−	0.711924i	$-0.252176\pi$
$727$	1.05787e13	1.40451	0.702257	+	0.711924i	$0.252176\pi$
$728$	−2.57541e12	−0.339825
$729$	0	0
$730$	−2.22757e12	−0.290320
$731$	3.02147e11	0.0391372
$732$	0	0
$733$	−5.59263e12	−0.715563	−0.357782	−	0.933805i	$-0.616467\pi$
$733$	−5.59263e12	−0.715563	−0.357782	+	0.933805i	$0.616467\pi$
$734$	−4.02754e12	−0.512162
$735$	0	0
$736$	7.08546e12	0.890057
$737$	1.15119e11	0.0143729
$738$	0	0
$739$	−7.75052e11	−0.0955941	−0.0477970	−	0.998857i	$-0.515220\pi$
$739$	−7.75052e11	−0.0955941	−0.0477970	+	0.998857i	$0.515220\pi$
$740$	1.18981e12	0.145859
$741$	0	0
$742$	2.58110e12	0.312598
$743$	−2.61033e12	−0.314228	−0.157114	−	0.987580i	$-0.550219\pi$
$743$	−2.61033e12	−0.314228	−0.157114	+	0.987580i	$0.550219\pi$
$744$	0	0
$745$	3.75050e12	0.446052
$746$	1.03948e13	1.22883
$747$	0	0
$748$	7.67976e9	0.000896996 0
$749$	−5.68613e12	−0.660159
$750$	0	0
$751$	1.00388e13	1.15160	0.575801	−	0.817590i	$-0.304690\pi$
$751$	1.00388e13	1.15160	0.575801	+	0.817590i	$0.304690\pi$
$752$	−1.28093e13	−1.46065
$753$	0	0
$754$	1.28217e13	1.44468
$755$	−3.34619e12	−0.374791
$756$	0	0
$757$	−1.40463e12	−0.155464	−0.0777320	−	0.996974i	$-0.524768\pi$
$757$	−1.40463e12	−0.155464	−0.0777320	+	0.996974i	$0.524768\pi$
$758$	1.91363e13	2.10545
$759$	0	0
$760$	2.19941e12	0.239136
$761$	1.42436e13	1.53954	0.769769	−	0.638323i	$-0.220372\pi$
$761$	1.42436e13	1.53954	0.769769	+	0.638323i	$0.220372\pi$
$762$	0	0
$763$	−5.42197e11	−0.0579158
$764$	−1.97669e12	−0.209903
$765$	0	0
$766$	1.51195e13	1.58675
$767$	1.98128e13	2.06712
$768$	0	0
$769$	−1.29105e12	−0.133129	−0.0665647	−	0.997782i	$-0.521204\pi$
$769$	−1.29105e12	−0.133129	−0.0665647	+	0.997782i	$0.521204\pi$
$770$	8.87656e10	0.00909990
$771$	0	0
$772$	6.42578e12	0.651101
$773$	8.58124e12	0.864455	0.432227	−	0.901765i	$-0.357728\pi$
$773$	8.58124e12	0.864455	0.432227	+	0.901765i	$0.357728\pi$
$774$	0	0
$775$	3.51339e12	0.349840
$776$	2.51395e12	0.248874
$777$	0	0
$778$	−5.98908e12	−0.586073
$779$	−1.22527e12	−0.119210
$780$	0	0
$781$	7.53031e11	0.0724241
$782$	−8.69041e11	−0.0831017
$783$	0	0
$784$	−1.85743e12	−0.175586
$785$	1.90785e11	0.0179321
$786$	0	0
$787$	−3.12167e11	−0.0290068	−0.0145034	−	0.999895i	$-0.504617\pi$
$787$	−3.12167e11	−0.0290068	−0.0145034	+	0.999895i	$0.504617\pi$
$788$	2.45108e12	0.226459
$789$	0	0
$790$	−2.72269e12	−0.248701
$791$	−7.05483e12	−0.640756
$792$	0	0
$793$	−1.84191e13	−1.65402
$794$	−2.38048e13	−2.12555
$795$	0	0
$796$	−4.34855e12	−0.383916
$797$	8.61341e12	0.756159	0.378079	−	0.925773i	$-0.376585\pi$
$797$	8.61341e12	0.756159	0.378079	+	0.925773i	$0.376585\pi$
$798$	0	0
$799$	7.47173e11	0.0648576
$800$	−1.57682e12	−0.136106
$801$	0	0
$802$	−9.14656e12	−0.780680
$803$	3.03791e11	0.0257843
$804$	0	0
$805$	−2.63401e12	−0.221073
$806$	−2.92336e13	−2.43992
$807$	0	0
$808$	1.36676e12	0.112808
$809$	2.01016e13	1.64992	0.824958	−	0.565193i	$-0.191198\pi$
$809$	2.01016e13	1.64992	0.824958	+	0.565193i	$0.191198\pi$
$810$	0	0
$811$	−1.62455e13	−1.31868	−0.659340	−	0.751845i	$-0.729164\pi$
$811$	−1.62455e13	−1.31868	−0.659340	+	0.751845i	$0.729164\pi$
$812$	1.72365e12	0.139138
$813$	0	0
$814$	−6.18789e11	−0.0494006
$815$	−5.96081e12	−0.473256
$816$	0	0
$817$	6.50745e12	0.510989
$818$	−2.81058e13	−2.19486
$819$	0	0
$820$	3.44291e11	0.0265927
$821$	8.24254e11	0.0633165	0.0316583	−	0.999499i	$-0.489921\pi$
$821$	8.24254e11	0.0633165	0.0316583	+	0.999499i	$0.489921\pi$
$822$	0	0
$823$	2.59567e13	1.97220	0.986099	−	0.166159i	$-0.0531364\pi$
$823$	2.59567e13	1.97220	0.986099	+	0.166159i	$0.0531364\pi$
$824$	−4.47796e12	−0.338382
$825$	0	0
$826$	1.01571e13	0.759206
$827$	1.31498e13	0.977560	0.488780	−	0.872407i	$-0.337442\pi$
$827$	1.31498e13	0.977560	0.488780	+	0.872407i	$0.337442\pi$
$828$	0	0
$829$	1.78218e13	1.31056	0.655279	−	0.755387i	$-0.272551\pi$
$829$	1.78218e13	1.31056	0.655279	+	0.755387i	$0.272551\pi$
$830$	−2.65800e12	−0.194404
$831$	0	0
$832$	−7.23334e12	−0.523340
$833$	1.08344e11	0.00779657
$834$	0	0
$835$	−4.74830e12	−0.338025
$836$	1.65402e11	0.0117115
$837$	0	0
$838$	−2.34199e13	−1.64054
$839$	−1.79273e13	−1.24906	−0.624532	−	0.780999i	$-0.714710\pi$
$839$	−1.79273e13	−1.24906	−0.624532	+	0.780999i	$0.714710\pi$
$840$	0	0
$841$	1.05452e12	0.0726898
$842$	2.41947e13	1.65888
$843$	0	0
$844$	−4.94565e12	−0.335492
$845$	−2.88620e12	−0.194747
$846$	0	0
$847$	5.64933e12	0.377156
$848$	1.31482e13	0.873140
$849$	0	0
$850$	1.93400e11	0.0127078
$851$	1.83618e13	1.20014
$852$	0	0
$853$	1.65652e13	1.07134	0.535668	−	0.844429i	$-0.320060\pi$
$853$	1.65652e13	1.07134	0.535668	+	0.844429i	$0.320060\pi$
$854$	−9.44264e12	−0.607481
$855$	0	0
$856$	−2.05891e13	−1.31070
$857$	1.00056e13	0.633619	0.316810	−	0.948489i	$-0.397388\pi$
$857$	1.00056e13	0.633619	0.316810	+	0.948489i	$0.397388\pi$
$858$	0	0
$859$	4.82691e12	0.302482	0.151241	−	0.988497i	$-0.451673\pi$
$859$	4.82691e12	0.302482	0.151241	+	0.988497i	$0.451673\pi$
$860$	−1.82854e12	−0.113988
$861$	0	0
$862$	1.88539e13	1.16311
$863$	−2.92326e12	−0.179398	−0.0896992	−	0.995969i	$-0.528591\pi$
$863$	−2.92326e12	−0.179398	−0.0896992	+	0.995969i	$0.528591\pi$
$864$	0	0
$865$	−1.00691e13	−0.611531
$866$	−3.44334e13	−2.08042
$867$	0	0
$868$	−3.92994e12	−0.234989
$869$	3.71316e11	0.0220879
$870$	0	0
$871$	−6.32544e12	−0.372399
$872$	−1.96326e12	−0.114988
$873$	0	0
$874$	−1.87169e13	−1.08501
$875$	5.86182e11	0.0338062
$876$	0	0
$877$	−9.67239e12	−0.552123	−0.276061	−	0.961140i	$-0.589029\pi$
$877$	−9.67239e12	−0.552123	−0.276061	+	0.961140i	$0.589029\pi$
$878$	−2.13080e13	−1.21009
$879$	0	0
$880$	4.52174e11	0.0254176
$881$	−1.87103e12	−0.104638	−0.0523191	−	0.998630i	$-0.516661\pi$
$881$	−1.87103e12	−0.104638	−0.0523191	+	0.998630i	$0.516661\pi$
$882$	0	0
$883$	8.57451e12	0.474664	0.237332	−	0.971429i	$-0.423727\pi$
$883$	8.57451e12	0.474664	0.237332	+	0.971429i	$0.423727\pi$
$884$	−4.21979e11	−0.0232411
$885$	0	0
$886$	−6.57328e12	−0.358369
$887$	1.80022e13	0.976494	0.488247	−	0.872705i	$-0.337636\pi$
$887$	1.80022e13	0.976494	0.488247	+	0.872705i	$0.337636\pi$
$888$	0	0
$889$	−7.41053e12	−0.397916
$890$	1.65162e13	0.882379
$891$	0	0
$892$	−4.82867e10	−0.00255379
$893$	1.60922e13	0.846804
$894$	0	0
$895$	3.69115e12	0.192290
$896$	−8.67053e12	−0.449427
$897$	0	0
$898$	−2.96428e13	−1.52116
$899$	−3.54809e13	−1.81166
$900$	0	0
$901$	−7.66937e11	−0.0387702
$902$	−1.79057e11	−0.00900660
$903$	0	0
$904$	−2.55450e13	−1.27218
$905$	2.85647e12	0.141551
$906$	0	0
$907$	2.53756e13	1.24504	0.622520	−	0.782604i	$-0.286109\pi$
$907$	2.53756e13	1.24504	0.622520	+	0.782604i	$0.286109\pi$
$908$	5.32495e12	0.259973
$909$	0	0
$910$	−4.87740e12	−0.235777
$911$	−1.84038e13	−0.885268	−0.442634	−	0.896702i	$-0.645956\pi$
$911$	−1.84038e13	−0.885268	−0.442634	+	0.896702i	$0.645956\pi$
$912$	0	0
$913$	3.62494e11	0.0172656
$914$	2.08790e13	0.989581
$915$	0	0
$916$	−7.15798e12	−0.335939
$917$	−5.78299e12	−0.270079
$918$	0	0
$919$	7.61943e12	0.352373	0.176186	−	0.984357i	$-0.443624\pi$
$919$	7.61943e12	0.352373	0.176186	+	0.984357i	$0.443624\pi$
$920$	−9.53755e12	−0.438926
$921$	0	0
$922$	−2.25125e13	−1.02597
$923$	−4.13767e13	−1.87650
$924$	0	0
$925$	−4.08630e12	−0.183524
$926$	−4.08475e13	−1.82564
$927$	0	0
$928$	1.59240e13	0.704832
$929$	3.02391e12	0.133198	0.0665991	−	0.997780i	$-0.478785\pi$
$929$	3.02391e12	0.133198	0.0665991	+	0.997780i	$0.478785\pi$
$930$	0	0
$931$	2.33345e12	0.101795
$932$	−7.33355e12	−0.318378
$933$	0	0
$934$	4.06097e12	0.174610
$935$	−2.63755e10	−0.00112862
$936$	0	0
$937$	−1.00516e13	−0.425999	−0.212999	−	0.977052i	$-0.568323\pi$
$937$	−1.00516e13	−0.425999	−0.212999	+	0.977052i	$0.568323\pi$
$938$	−3.24276e12	−0.136774
$939$	0	0
$940$	−4.52176e12	−0.188900
$941$	1.52744e13	0.635053	0.317526	−	0.948249i	$-0.397148\pi$
$941$	1.52744e13	0.635053	0.317526	+	0.948249i	$0.397148\pi$
$942$	0	0
$943$	5.31328e12	0.218806
$944$	5.17405e13	2.12059
$945$	0	0
$946$	9.50974e11	0.0386063
$947$	−4.29774e13	−1.73646	−0.868230	−	0.496162i	$-0.834742\pi$
$947$	−4.29774e13	−1.73646	−0.868230	+	0.496162i	$0.834742\pi$
$948$	0	0
$949$	−1.66924e13	−0.668068
$950$	4.16533e12	0.165918
$951$	0	0
$952$	3.92307e11	0.0154796
$953$	−1.01121e13	−0.397120	−0.198560	−	0.980089i	$-0.563627\pi$
$953$	−1.01121e13	−0.397120	−0.198560	+	0.980089i	$0.563627\pi$
$954$	0	0
$955$	6.78876e12	0.264104
$956$	−6.44251e12	−0.249457
$957$	0	0
$958$	5.16852e13	1.98254
$959$	1.31445e13	0.501833
$960$	0	0
$961$	5.44575e13	2.05969
$962$	3.40005e13	1.27996
$963$	0	0
$964$	6.23703e11	0.0232612
$965$	−2.20688e13	−0.819229
$966$	0	0
$967$	4.14688e13	1.52511	0.762556	−	0.646922i	$-0.223944\pi$
$967$	4.14688e13	1.52511	0.762556	+	0.646922i	$0.223944\pi$
$968$	2.04558e13	0.748820
$969$	0	0
$970$	4.76101e12	0.172674
$971$	−3.67205e12	−0.132563	−0.0662814	−	0.997801i	$-0.521114\pi$
$971$	−3.67205e12	−0.132563	−0.0662814	+	0.997801i	$0.521114\pi$
$972$	0	0
$973$	−2.18226e12	−0.0780547
$974$	2.36459e13	0.841862
$975$	0	0
$976$	−4.81010e13	−1.69680
$977$	−8.95608e10	−0.00314480	−0.00157240	−	0.999999i	$-0.500501\pi$
$977$	−8.95608e10	−0.00314480	−0.00157240	+	0.999999i	$0.500501\pi$
$978$	0	0
$979$	−2.25245e12	−0.0783669
$980$	−6.55680e11	−0.0227078
$981$	0	0
$982$	−1.26467e13	−0.433985
$983$	4.67974e13	1.59857	0.799284	−	0.600953i	$-0.205212\pi$
$983$	4.67974e13	1.59857	0.799284	+	0.600953i	$0.205212\pi$
$984$	0	0
$985$	−8.41802e12	−0.284936
$986$	−1.95310e12	−0.0658078
$987$	0	0
$988$	−9.08834e12	−0.303444
$989$	−2.82190e13	−0.937902
$990$	0	0
$991$	−1.94121e12	−0.0639353	−0.0319677	−	0.999489i	$-0.510177\pi$
$991$	−1.94121e12	−0.0639353	−0.0319677	+	0.999489i	$0.510177\pi$
$992$	−3.63070e13	−1.19038
$993$	0	0
$994$	−2.12119e13	−0.689194
$995$	1.49347e13	0.483051
$996$	0	0
$997$	2.62275e13	0.840677	0.420338	−	0.907367i	$-0.361911\pi$
$997$	2.62275e13	0.840677	0.420338	+	0.907367i	$0.361911\pi$
$998$	−4.78793e13	−1.52778
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.l.1.5		6
3.2	odd	2		35.10.a.e.1.2	✓	6
15.2	even	4		175.10.b.g.99.4		12
15.8	even	4		175.10.b.g.99.9		12
15.14	odd	2		175.10.a.g.1.5		6
21.20	even	2		245.10.a.g.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.10.a.e.1.2	✓	6	3.2	odd	2
175.10.a.g.1.5		6	15.14	odd	2
175.10.b.g.99.4		12	15.2	even	4
175.10.b.g.99.9		12	15.8	even	4
245.10.a.g.1.2		6	21.20	even	2
315.10.a.l.1.5		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+26.3435 q^{2} +181.982 q^{4} -625.000 q^{5} -2401.00 q^{7} -8693.85 q^{8} -16464.7 q^{10} +2245.43 q^{11} -123379. q^{13} -63250.8 q^{14} -322201. q^{16} +18794.1 q^{17} +404776. q^{19} -113739. q^{20} +59152.4 q^{22} -1.75527e6 q^{23} +390625. q^{25} -3.25024e6 q^{26} -436938. q^{28} -3.94483e6 q^{29} +8.99428e6 q^{31} -4.03667e6 q^{32} +495103. q^{34} +1.50062e6 q^{35} -1.04609e7 q^{37} +1.06632e7 q^{38} +5.43366e6 q^{40} -3.02704e6 q^{41} +1.60767e7 q^{43} +408626. q^{44} -4.62401e7 q^{46} +3.97557e7 q^{47} +5.76480e6 q^{49} +1.02904e7 q^{50} -2.24528e7 q^{52} -4.08073e7 q^{53} -1.40339e6 q^{55} +2.08739e7 q^{56} -1.03921e8 q^{58} -1.60585e8 q^{59} +1.49289e8 q^{61} +2.36941e8 q^{62} +5.86269e7 q^{64} +7.71120e7 q^{65} +5.12683e7 q^{67} +3.42018e6 q^{68} +3.95318e7 q^{70} +3.35362e8 q^{71} +1.35293e8 q^{73} -2.75578e8 q^{74} +7.36618e7 q^{76} -5.39127e6 q^{77} +1.65365e8 q^{79} +2.01376e8 q^{80} -7.97429e7 q^{82} +1.61436e8 q^{83} -1.17463e7 q^{85} +4.23516e8 q^{86} -1.95214e7 q^{88} -1.00313e9 q^{89} +2.96233e8 q^{91} -3.19428e8 q^{92} +1.04731e9 q^{94} -2.52985e8 q^{95} -2.89165e8 q^{97} +1.51865e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 15 q^{2} + 3009 q^{4} - 3750 q^{5} - 14406 q^{7} - 22041 q^{8} + 9375 q^{10} + 47796 q^{11} + 102168 q^{13} + 36015 q^{14} + 2371065 q^{16} + 38472 q^{17} + 361056 q^{19} - 1880625 q^{20} + 2068680 q^{22}+ \cdots - 86472015 q^{98}+O(q^{100}) \) 6 * q - 15 * q^2 + 3009 * q^4 - 3750 * q^5 - 14406 * q^7 - 22041 * q^8 + 9375 * q^10 + 47796 * q^11 + 102168 * q^13 + 36015 * q^14 + 2371065 * q^16 + 38472 * q^17 + 361056 * q^19 - 1880625 * q^20 + 2068680 * q^22 - 697032 * q^23 + 2343750 * q^25 - 10847622 * q^26 - 7224609 * q^28 - 16028184 * q^29 + 1362912 * q^31 + 1148823 * q^32 - 9216294 * q^34 + 9003750 * q^35 - 3912924 * q^37 - 39877140 * q^38 + 13775625 * q^40 - 22452756 * q^41 - 29998992 * q^43 + 94378728 * q^44 + 49589520 * q^46 + 121271508 * q^47 + 34588806 * q^49 - 5859375 * q^50 - 44528550 * q^52 + 20308596 * q^53 - 29872500 * q^55 + 52920441 * q^56 + 256759494 * q^58 - 120280392 * q^59 - 87693540 * q^61 + 344802528 * q^62 + 640647105 * q^64 - 63855000 * q^65 + 495050664 * q^67 + 141443706 * q^68 - 22509375 * q^70 - 253762512 * q^71 + 187195308 * q^73 + 418806054 * q^74 - 584719476 * q^76 - 114758196 * q^77 + 831079500 * q^79 - 1481915625 * q^80 + 2286848814 * q^82 + 767650536 * q^83 - 24045000 * q^85 - 566199948 * q^86 + 2952495444 * q^88 - 582579684 * q^89 - 245305368 * q^91 + 4259382384 * q^92 - 1115946660 * q^94 - 225660000 * q^95 - 1184506872 * q^97 - 86472015 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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