LMFDB - Embedded newform 32.18.b.a.17.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,18,Mod(17,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("32.17");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	32.b (of order $2$, degree $1$, not 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:	$58.6310679503$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 83403052 x^{14} - 583821224 x^{13} + \cdots + 21\!\cdots\!00 $ x^16 - 8x^15 + 83403052x^14 - 583821224x^13 + 2709232900692534x^12 - 16255389814479656x^11 + 44023334982114936772524x^10 - 220116525902848632203992x^9 + 375637035261256983364255923921x^8 - 1502546820346051325629336482512x^7 + 1580449203685347736142710540597573024x^6 - 4741342352143096486478474620456939520x^5 + 2671768262327077521053591460599154325518592x^4 - 5343528622418654441129876221219442847805440x^3 + 1662520959198696999402497079485716800591945908224x^2 - 1662518287435176013550209293272621607067374059520*x + 210991858777295491315579293406297919761107107446784000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{240}\cdot 3^{14}\cdot 7 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.7
Root			$0.500000 + 1062.13i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.18.b.a.17.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4248.51i q^{3} -663971. i q^{5} +1.66742e7 q^{7} +1.11090e8 q^{9} +O(q^{10})$	$q-4248.51i q^{3} -663971. i q^{5} +1.66742e7 q^{7} +1.11090e8 q^{9} +1.05479e9i q^{11} +9.19407e7i q^{13} -2.82089e9 q^{15} -1.98326e10 q^{17} +8.44846e10i q^{19} -7.08405e10i q^{21} +2.72263e10 q^{23} +3.22082e11 q^{25} -1.02062e12i q^{27} +3.75013e12i q^{29} -5.36921e12 q^{31} +4.48129e12 q^{33} -1.10712e13i q^{35} +1.92294e13i q^{37} +3.90612e11 q^{39} +5.42400e13 q^{41} -3.96416e13i q^{43} -7.37607e13i q^{45} -4.48999e13 q^{47} +4.53979e13 q^{49} +8.42591e13i q^{51} +7.81803e14i q^{53} +7.00349e14 q^{55} +3.58934e14 q^{57} +1.07092e15i q^{59} -1.92548e15i q^{61} +1.85234e15 q^{63} +6.10460e13 q^{65} +3.68678e15i q^{67} -1.15671e14i q^{69} +1.02421e16 q^{71} +1.25847e16 q^{73} -1.36837e15i q^{75} +1.75877e16i q^{77} -3.24102e15 q^{79} +1.00101e16 q^{81} -4.28316e15i q^{83} +1.31683e16i q^{85} +1.59325e16 q^{87} +6.95255e16 q^{89} +1.53304e15i q^{91} +2.28112e16i q^{93} +5.60953e16 q^{95} -7.74223e16 q^{97} +1.17177e17i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 11529600 q^{7} - 602654096 q^{9}+O(q^{10}) $ 16 * q - 11529600 * q^7 - 602654096 * q^9	$ 16 q - 11529600 q^{7} - 602654096 q^{9} + 9993282176 q^{15} - 7489125600 q^{17} - 746845345920 q^{23} - 1809682431664 q^{25} + 318979758592 q^{31} + 5633526177600 q^{33} + 18457706051456 q^{39} + 7482251536032 q^{41} + 376698804821760 q^{47} + 127691292101520 q^{49} - 22\!\cdots\!52 q^{55}+ \cdots + 95\!\cdots\!40 q^{97}+O(q^{100}) $ 16 * q - 11529600 * q^7 - 602654096 * q^9 + 9993282176 * q^15 - 7489125600 * q^17 - 746845345920 * q^23 - 1809682431664 * q^25 + 318979758592 * q^31 + 5633526177600 * q^33 + 18457706051456 * q^39 + 7482251536032 * q^41 + 376698804821760 * q^47 + 127691292101520 * q^49 - 2209036687713152 * q^55 - 190521298294720 * q^57 + 8131096607338880 * q^63 + 2385987975356160 * q^65 - 9025926285576576 * q^71 + 11332002046118560 * q^73 + 45299671392008448 * q^79 + 20101901999290832 * q^81 - 25965768920837760 * q^87 - 69879174608766048 * q^89 - 93790444358203776 * q^95 + 95593398602180640 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$.

$n$	$5$	$31$
$\chi(n)$	$-1$	$1$

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	0
$2$	0	0
$3$	− 4248.51i	− 0.373858i	−0.982373	−	0.186929i	$-0.940147\pi$
$3$	− 4248.51i	− 0.373858i	0.982373	−	0.186929i	$-0.0598534\pi$
$4$	0	0
$5$	− 663971.i	− 0.760158i	−0.924954	−	0.380079i	$-0.875897\pi$
$5$	− 663971.i	− 0.760158i	0.924954	−	0.380079i	$-0.124103\pi$
$6$	0	0
$7$	1.66742e7	1.09323	0.546615	−	0.837384i	$-0.315916\pi$
$7$	1.66742e7	1.09323	0.546615	+	0.837384i	$0.315916\pi$
$8$	0	0
$9$	1.11090e8	0.860230
$10$	0	0
$11$	1.05479e9i	1.48364i	0.670600	+	0.741819i	$0.266037\pi$
$11$	1.05479e9i	1.48364i	−0.670600	+	0.741819i	$0.733963\pi$
$12$	0	0
$13$	9.19407e7i	0.0312600i	0.999878	+	0.0156300i	$0.00497539\pi$
$13$	9.19407e7i	0.0312600i	−0.999878	+	0.0156300i	$0.995025\pi$
$14$	0	0
$15$	−2.82089e9	−0.284191
$16$	0	0
$17$	−1.98326e10	−0.689547	−0.344773	−	0.938686i	$-0.612044\pi$
$17$	−1.98326e10	−0.689547	−0.344773	+	0.938686i	$0.612044\pi$
$18$	0	0
$19$	8.44846e10i	1.14123i	0.821219	+	0.570613i	$0.193294\pi$
$19$	8.44846e10i	1.14123i	−0.821219	+	0.570613i	$0.806706\pi$
$20$	0	0
$21$	− 7.08405e10i	− 0.408712i
$22$	0	0
$23$	2.72263e10	0.0724941	0.0362470	−	0.999343i	$-0.488460\pi$
$23$	2.72263e10	0.0724941	0.0362470	+	0.999343i	$0.488460\pi$
$24$	0	0
$25$	3.22082e11	0.422160
$26$	0	0
$27$	− 1.02062e12i	− 0.695462i
$28$	0	0
$29$	3.75013e12i	1.39208i	0.718005	+	0.696038i	$0.245056\pi$
$29$	3.75013e12i	1.39208i	−0.718005	+	0.696038i	$0.754944\pi$
$30$	0	0
$31$	−5.36921e12	−1.13067	−0.565336	−	0.824861i	$-0.691253\pi$
$31$	−5.36921e12	−1.13067	−0.565336	+	0.824861i	$0.691253\pi$
$32$	0	0
$33$	4.48129e12	0.554669
$34$	0	0
$35$	− 1.10712e13i	− 0.831027i
$36$	0	0
$37$	1.92294e13i	0.900015i	0.893025	+	0.450008i	$0.148579\pi$
$37$	1.92294e13i	0.900015i	−0.893025	+	0.450008i	$0.851421\pi$
$38$	0	0
$39$	3.90612e11	0.0116868
$40$	0	0
$41$	5.42400e13	1.06086	0.530429	−	0.847730i	$-0.322031\pi$
$41$	5.42400e13	1.06086	0.530429	+	0.847730i	$0.322031\pi$
$42$	0	0
$43$	− 3.96416e13i	− 0.517213i	−0.965983	−	0.258607i	$-0.916737\pi$
$43$	− 3.96416e13i	− 0.517213i	0.965983	−	0.258607i	$-0.0832634\pi$
$44$	0	0
$45$	− 7.37607e13i	− 0.653911i
$46$	0	0
$47$	−4.48999e13	−0.275051	−0.137526	−	0.990498i	$-0.543915\pi$
$47$	−4.48999e13	−0.275051	−0.137526	+	0.990498i	$0.543915\pi$
$48$	0	0
$49$	4.53979e13	0.195150
$50$	0	0
$51$	8.42591e13i	0.257792i
$52$	0	0
$53$	7.81803e14i	1.72485i	0.506181	+	0.862427i	$0.331057\pi$
$53$	7.81803e14i	1.72485i	−0.506181	+	0.862427i	$0.668943\pi$
$54$	0	0
$55$	7.00349e14	1.12780
$56$	0	0
$57$	3.58934e14	0.426656
$58$	0	0
$59$	1.07092e15i	0.949541i	0.880110	+	0.474770i	$0.157469\pi$
$59$	1.07092e15i	0.949541i	−0.880110	+	0.474770i	$0.842531\pi$
$60$	0	0
$61$	− 1.92548e15i	− 1.28599i	−0.765872	−	0.642993i	$-0.777692\pi$
$61$	− 1.92548e15i	− 1.28599i	0.765872	−	0.642993i	$-0.222308\pi$
$62$	0	0
$63$	1.85234e15	0.940429
$64$	0	0
$65$	6.10460e13	0.0237626
$66$	0	0
$67$	3.68678e15i	1.10920i	0.832116	+	0.554602i	$0.187129\pi$
$67$	3.68678e15i	1.10920i	−0.832116	+	0.554602i	$0.812871\pi$
$68$	0	0
$69$	− 1.15671e14i	− 0.0271025i
$70$	0	0
$71$	1.02421e16	1.88232	0.941162	−	0.337957i	$-0.109736\pi$
$71$	1.02421e16	1.88232	0.941162	+	0.337957i	$0.109736\pi$
$72$	0	0
$73$	1.25847e16	1.82641	0.913204	−	0.407502i	$-0.133600\pi$
$73$	1.25847e16	1.82641	0.913204	+	0.407502i	$0.133600\pi$
$74$	0	0
$75$	− 1.36837e15i	− 0.157828i
$76$	0	0
$77$	1.75877e16i	1.62196i
$78$	0	0
$79$	−3.24102e15	−0.240354	−0.120177	−	0.992752i	$-0.538346\pi$
$79$	−3.24102e15	−0.240354	−0.120177	+	0.992752i	$0.538346\pi$
$80$	0	0
$81$	1.00101e16	0.600227
$82$	0	0
$83$	− 4.28316e15i	− 0.208737i	−0.994539	−	0.104369i	$-0.966718\pi$
$83$	− 4.28316e15i	− 0.208737i	0.994539	−	0.104369i	$-0.0332822\pi$
$84$	0	0
$85$	1.31683e16i	0.524165i
$86$	0	0
$87$	1.59325e16	0.520439
$88$	0	0
$89$	6.95255e16	1.87210	0.936050	−	0.351868i	$-0.114453\pi$
$89$	6.95255e16	1.87210	0.936050	+	0.351868i	$0.114453\pi$
$90$	0	0
$91$	1.53304e15i	0.0341744i
$92$	0	0
$93$	2.28112e16i	0.422710i
$94$	0	0
$95$	5.60953e16	0.867513
$96$	0	0
$97$	−7.74223e16	−1.00301	−0.501507	−	0.865154i	$-0.667221\pi$
$97$	−7.74223e16	−1.00301	−0.501507	+	0.865154i	$0.667221\pi$
$98$	0	0
$99$	1.17177e17i	1.27627i
$100$	0	0
$101$	− 8.38662e16i	− 0.770646i	−0.922782	−	0.385323i	$-0.874090\pi$
$101$	− 8.38662e16i	− 0.770646i	0.922782	−	0.385323i	$-0.125910\pi$
$102$	0	0
$103$	−8.10218e16	−0.630210	−0.315105	−	0.949057i	$-0.602040\pi$
$103$	−8.10218e16	−0.630210	−0.315105	+	0.949057i	$0.602040\pi$
$104$	0	0
$105$	−4.70360e16	−0.310686
$106$	0	0
$107$	− 1.60731e17i	− 0.904351i	−0.891929	−	0.452175i	$-0.850648\pi$
$107$	− 1.60731e17i	− 0.904351i	0.891929	−	0.452175i	$-0.149352\pi$
$108$	0	0
$109$	− 1.05482e17i	− 0.507053i	−0.967328	−	0.253526i	$-0.918410\pi$
$109$	− 1.05482e17i	− 0.507053i	0.967328	−	0.253526i	$-0.0815905\pi$
$110$	0	0
$111$	8.16962e16	0.336478
$112$	0	0
$113$	−1.30124e17	−0.460458	−0.230229	−	0.973136i	$-0.573948\pi$
$113$	−1.30124e17	−0.460458	−0.230229	+	0.973136i	$0.573948\pi$
$114$	0	0
$115$	− 1.80775e16i	− 0.0551069i
$116$	0	0
$117$	1.02137e16i	0.0268908i
$118$	0	0
$119$	−3.30692e17	−0.753833
$120$	0	0
$121$	−6.07133e17	−1.20118
$122$	0	0
$123$	− 2.30439e17i	− 0.396610i
$124$	0	0
$125$	− 7.20423e17i	− 1.08107i
$126$	0	0
$127$	−1.12618e18	−1.47664	−0.738322	−	0.674448i	$-0.764381\pi$
$127$	−1.12618e18	−1.47664	−0.738322	+	0.674448i	$0.764381\pi$
$128$	0	0
$129$	−1.68418e17	−0.193364
$130$	0	0
$131$	3.73512e17i	0.376269i	0.982143	+	0.188134i	$0.0602440\pi$
$131$	3.73512e17i	0.376269i	−0.982143	+	0.188134i	$0.939756\pi$
$132$	0	0
$133$	1.40871e18i	1.24762i
$134$	0	0
$135$	−6.77664e17	−0.528661
$136$	0	0
$137$	−3.64355e17	−0.250842	−0.125421	−	0.992104i	$-0.540028\pi$
$137$	−3.64355e17	−0.250842	−0.125421	+	0.992104i	$0.540028\pi$
$138$	0	0
$139$	− 2.51151e18i	− 1.52866i	−0.644828	−	0.764328i	$-0.723071\pi$
$139$	− 2.51151e18i	− 1.52866i	0.644828	−	0.764328i	$-0.276929\pi$
$140$	0	0
$141$	1.90758e17i	0.102830i
$142$	0	0
$143$	−9.69781e16	−0.0463786
$144$	0	0
$145$	2.48998e18	1.05820
$146$	0	0
$147$	− 1.92874e17i	− 0.0729585i
$148$	0	0
$149$	− 2.83942e18i	− 0.957517i	−0.877947	−	0.478758i	$-0.841087\pi$
$149$	− 2.83942e18i	− 0.957517i	0.877947	−	0.478758i	$-0.158913\pi$
$150$	0	0
$151$	2.74195e18	0.825574	0.412787	−	0.910828i	$-0.364555\pi$
$151$	2.74195e18	0.825574	0.412787	+	0.910828i	$0.364555\pi$
$152$	0	0
$153$	−2.20321e18	−0.593169
$154$	0	0
$155$	3.56500e18i	0.859489i
$156$	0	0
$157$	3.26134e18i	0.705096i	0.935794	+	0.352548i	$0.114685\pi$
$157$	3.26134e18i	0.705096i	−0.935794	+	0.352548i	$0.885315\pi$
$158$	0	0
$159$	3.32150e18	0.644850
$160$	0	0
$161$	4.53977e17	0.0792526
$162$	0	0
$163$	7.80461e18i	1.22675i	0.789791	+	0.613376i	$0.210189\pi$
$163$	7.80461e18i	1.22675i	−0.789791	+	0.613376i	$0.789811\pi$
$164$	0	0
$165$	− 2.97544e18i	− 0.421636i
$166$	0	0
$167$	8.54007e18	1.09237	0.546187	−	0.837663i	$-0.316079\pi$
$167$	8.54007e18	1.09237	0.546187	+	0.837663i	$0.316079\pi$
$168$	0	0
$169$	8.64196e18	0.999023
$170$	0	0
$171$	9.38542e18i	0.981718i
$172$	0	0
$173$	− 2.07391e18i	− 0.196516i	−0.995161	−	0.0982581i	$-0.968673\pi$
$173$	− 2.07391e18i	− 0.196516i	0.995161	−	0.0982581i	$-0.0313271\pi$
$174$	0	0
$175$	5.37046e18	0.461517
$176$	0	0
$177$	4.54981e18	0.354993
$178$	0	0
$179$	− 1.67289e19i	− 1.18636i	−0.805071	−	0.593179i	$-0.797873\pi$
$179$	− 1.67289e19i	− 1.18636i	0.805071	−	0.593179i	$-0.202127\pi$
$180$	0	0
$181$	3.02297e19i	1.95059i	0.220907	+	0.975295i	$0.429098\pi$
$181$	3.02297e19i	1.95059i	−0.220907	+	0.975295i	$0.570902\pi$
$182$	0	0
$183$	−8.18045e18	−0.480776
$184$	0	0
$185$	1.27677e19	0.684154
$186$	0	0
$187$	− 2.09192e19i	− 1.02304i
$188$	0	0
$189$	− 1.70180e19i	− 0.760299i
$190$	0	0
$191$	−1.22397e18	−0.0500019	−0.0250009	−	0.999687i	$-0.507959\pi$
$191$	−1.22397e18	−0.0500019	−0.0250009	+	0.999687i	$0.507959\pi$
$192$	0	0
$193$	−6.70836e18	−0.250830	−0.125415	−	0.992104i	$-0.540026\pi$
$193$	−6.70836e18	−0.250830	−0.125415	+	0.992104i	$0.540026\pi$
$194$	0	0
$195$	− 2.59355e17i	− 0.00888382i
$196$	0	0
$197$	1.35900e19i	0.426832i	0.976961	+	0.213416i	$0.0684590\pi$
$197$	1.35900e19i	0.426832i	−0.976961	+	0.213416i	$0.931541\pi$
$198$	0	0
$199$	−2.94272e19	−0.848198	−0.424099	−	0.905616i	$-0.639409\pi$
$199$	−2.94272e19	−0.848198	−0.424099	+	0.905616i	$0.639409\pi$
$200$	0	0
$201$	1.56633e19	0.414685
$202$	0	0
$203$	6.25303e19i	1.52186i
$204$	0	0
$205$	− 3.60138e19i	− 0.806419i
$206$	0	0
$207$	3.02458e18	0.0623616
$208$	0	0
$209$	−8.91134e19	−1.69317
$210$	0	0
$211$	− 7.34029e19i	− 1.28621i	−0.765777	−	0.643106i	$-0.777645\pi$
$211$	− 7.34029e19i	− 1.28621i	0.765777	−	0.643106i	$-0.222355\pi$
$212$	0	0
$213$	− 4.35138e19i	− 0.703721i
$214$	0	0
$215$	−2.63209e19	−0.393164
$216$	0	0
$217$	−8.95273e19	−1.23608
$218$	0	0
$219$	− 5.34662e19i	− 0.682817i
$220$	0	0
$221$	− 1.82342e18i	− 0.0215553i
$222$	0	0
$223$	−7.73683e19	−0.847172	−0.423586	−	0.905856i	$-0.639229\pi$
$223$	−7.73683e19	−0.847172	−0.423586	+	0.905856i	$0.639229\pi$
$224$	0	0
$225$	3.57802e19	0.363155
$226$	0	0
$227$	1.21923e20i	1.14780i	0.818924	+	0.573902i	$0.194571\pi$
$227$	1.21923e20i	1.14780i	−0.818924	+	0.573902i	$0.805429\pi$
$228$	0	0
$229$	− 1.42038e19i	− 0.124109i	−0.998073	−	0.0620543i	$-0.980235\pi$
$229$	− 1.42038e19i	− 0.124109i	0.998073	−	0.0620543i	$-0.0197652\pi$
$230$	0	0
$231$	7.47218e19	0.606381
$232$	0	0
$233$	−7.02873e19	−0.530092	−0.265046	−	0.964236i	$-0.585387\pi$
$233$	−7.02873e19	−0.530092	−0.265046	+	0.964236i	$0.585387\pi$
$234$	0	0
$235$	2.98122e19i	0.209082i
$236$	0	0
$237$	1.37695e19i	0.0898581i
$238$	0	0
$239$	−4.12244e19	−0.250480	−0.125240	−	0.992126i	$-0.539970\pi$
$239$	−4.12244e19	−0.250480	−0.125240	+	0.992126i	$0.539970\pi$
$240$	0	0
$241$	9.34265e19	0.528841	0.264420	−	0.964408i	$-0.414819\pi$
$241$	9.34265e19	0.528841	0.264420	+	0.964408i	$0.414819\pi$
$242$	0	0
$243$	− 1.74331e20i	− 0.919861i
$244$	0	0
$245$	− 3.01429e19i	− 0.148345i
$246$	0	0
$247$	−7.76758e18	−0.0356748
$248$	0	0
$249$	−1.81971e19	−0.0780381
$250$	0	0
$251$	− 8.23914e19i	− 0.330107i	−0.986285	−	0.165054i	$-0.947220\pi$
$251$	− 8.23914e19i	− 0.330107i	0.986285	−	0.165054i	$-0.0527797\pi$
$252$	0	0
$253$	2.87180e19i	0.107555i
$254$	0	0
$255$	5.59456e19	0.195963
$256$	0	0
$257$	3.96731e20	1.30036	0.650182	−	0.759779i	$-0.274693\pi$
$257$	3.96731e20	1.30036	0.650182	+	0.759779i	$0.274693\pi$
$258$	0	0
$259$	3.20634e20i	0.983923i
$260$	0	0
$261$	4.16603e20i	1.19751i
$262$	0	0
$263$	3.01079e20	0.811066	0.405533	−	0.914080i	$-0.367086\pi$
$263$	3.01079e20	0.811066	0.405533	+	0.914080i	$0.367086\pi$
$264$	0	0
$265$	5.19094e20	1.31116
$266$	0	0
$267$	− 2.95380e20i	− 0.699899i
$268$	0	0
$269$	7.99834e19i	0.177871i	0.996037	+	0.0889356i	$0.0283465\pi$
$269$	7.99834e19i	0.177871i	−0.996037	+	0.0889356i	$0.971653\pi$
$270$	0	0
$271$	3.82151e20	0.797986	0.398993	−	0.916954i	$-0.369360\pi$
$271$	3.82151e20	0.797986	0.398993	+	0.916954i	$0.369360\pi$
$272$	0	0
$273$	6.51313e18	0.0127764
$274$	0	0
$275$	3.39729e20i	0.626332i
$276$	0	0
$277$	3.87625e20i	0.671945i	0.941872	+	0.335972i	$0.109065\pi$
$277$	3.87625e20i	0.671945i	−0.941872	+	0.335972i	$0.890935\pi$
$278$	0	0
$279$	−5.96467e20	−0.972638
$280$	0	0
$281$	−5.68349e20	−0.872190	−0.436095	−	0.899901i	$-0.643639\pi$
$281$	−5.68349e20	−0.872190	−0.436095	+	0.899901i	$0.643639\pi$
$282$	0	0
$283$	4.50897e20i	0.651467i	0.945462	+	0.325734i	$0.105611\pi$
$283$	4.50897e20i	0.651467i	−0.945462	+	0.325734i	$0.894389\pi$
$284$	0	0
$285$	− 2.38322e20i	− 0.324326i
$286$	0	0
$287$	9.04408e20	1.15976
$288$	0	0
$289$	−4.33908e20	−0.524525
$290$	0	0
$291$	3.28930e20i	0.374984i
$292$	0	0
$293$	− 1.51002e21i	− 1.62408i	−0.583602	−	0.812040i	$-0.698357\pi$
$293$	− 1.51002e21i	− 1.62408i	0.583602	−	0.812040i	$-0.301643\pi$
$294$	0	0
$295$	7.11057e20	0.721801
$296$	0	0
$297$	1.07654e21	1.03181
$298$	0	0
$299$	2.50321e18i	0.00226617i
$300$	0	0
$301$	− 6.60992e20i	− 0.565433i
$302$	0	0
$303$	−3.56307e20	−0.288112
$304$	0	0
$305$	−1.27847e21	−0.977553
$306$	0	0
$307$	− 7.15201e20i	− 0.517311i	−0.965970	−	0.258655i	$-0.916721\pi$
$307$	− 7.15201e20i	− 0.517311i	0.965970	−	0.258655i	$-0.0832794\pi$
$308$	0	0
$309$	3.44222e20i	0.235609i
$310$	0	0
$311$	−1.55199e21	−1.00560	−0.502801	−	0.864402i	$-0.667697\pi$
$311$	−1.55199e21	−1.00560	−0.502801	+	0.864402i	$0.667697\pi$
$312$	0	0
$313$	−7.03438e20	−0.431617	−0.215809	−	0.976436i	$-0.569239\pi$
$313$	−7.03438e20	−0.431617	−0.215809	+	0.976436i	$0.569239\pi$
$314$	0	0
$315$	− 1.22990e21i	− 0.714875i
$316$	0	0
$317$	2.05646e21i	1.13270i	0.824163	+	0.566352i	$0.191646\pi$
$317$	2.05646e21i	1.13270i	−0.824163	+	0.566352i	$0.808354\pi$
$318$	0	0
$319$	−3.95559e21	−2.06534
$320$	0	0
$321$	−6.82867e20	−0.338099
$322$	0	0
$323$	− 1.67555e21i	− 0.786929i
$324$	0	0
$325$	2.96125e19i	0.0131967i
$326$	0	0
$327$	−4.48142e20	−0.189566
$328$	0	0
$329$	−7.48669e20	−0.300694
$330$	0	0
$331$	− 7.03264e20i	− 0.268275i	−0.990963	−	0.134138i	$-0.957174\pi$
$331$	− 7.03264e20i	− 0.268275i	0.990963	−	0.134138i	$-0.0428264\pi$
$332$	0	0
$333$	2.13619e21i	0.774221i
$334$	0	0
$335$	2.44791e21	0.843170
$336$	0	0
$337$	8.84366e20	0.289586	0.144793	−	0.989462i	$-0.453748\pi$
$337$	8.84366e20	0.289586	0.144793	+	0.989462i	$0.453748\pi$
$338$	0	0
$339$	5.52833e20i	0.172146i
$340$	0	0
$341$	− 5.66339e21i	− 1.67751i
$342$	0	0
$343$	−3.12195e21	−0.879885
$344$	0	0
$345$	−7.68024e19	−0.0206022
$346$	0	0
$347$	5.48605e21i	1.40107i	0.713620	+	0.700533i	$0.247055\pi$
$347$	5.48605e21i	1.40107i	−0.713620	+	0.700533i	$0.752945\pi$
$348$	0	0
$349$	2.09596e19i	0.00509762i	0.999997	+	0.00254881i	$0.000811312\pi$
$349$	2.09596e19i	0.00509762i	−0.999997	+	0.00254881i	$0.999189\pi$
$350$	0	0
$351$	9.38368e19	0.0217402
$352$	0	0
$353$	−8.34131e20	−0.184141	−0.0920703	−	0.995753i	$-0.529348\pi$
$353$	−8.34131e20	−0.184141	−0.0920703	+	0.995753i	$0.529348\pi$
$354$	0	0
$355$	− 6.80047e21i	− 1.43086i
$356$	0	0
$357$	1.40495e21i	0.281826i
$358$	0	0
$359$	7.98650e21	1.52775	0.763877	−	0.645361i	$-0.223293\pi$
$359$	7.98650e21	1.52775	0.763877	+	0.645361i	$0.223293\pi$
$360$	0	0
$361$	−1.65726e21	−0.302398
$362$	0	0
$363$	2.57941e21i	0.449071i
$364$	0	0
$365$	− 8.35586e21i	− 1.38836i
$366$	0	0
$367$	4.95976e21	0.786682	0.393341	−	0.919393i	$-0.371319\pi$
$367$	4.95976e21	0.786682	0.393341	+	0.919393i	$0.371319\pi$
$368$	0	0
$369$	6.02554e21	0.912581
$370$	0	0
$371$	1.30359e22i	1.88566i
$372$	0	0
$373$	1.32602e22i	1.83243i	0.400692	+	0.916213i	$0.368770\pi$
$373$	1.32602e22i	1.83243i	−0.400692	+	0.916213i	$0.631230\pi$
$374$	0	0
$375$	−3.06073e21	−0.404165
$376$	0	0
$377$	−3.44790e20	−0.0435164
$378$	0	0
$379$	− 4.66066e21i	− 0.562360i	−0.959655	−	0.281180i	$-0.909274\pi$
$379$	− 4.66066e21i	− 0.562360i	0.959655	−	0.281180i	$-0.0907258\pi$
$380$	0	0
$381$	4.78459e21i	0.552055i
$382$	0	0
$383$	9.41823e21	1.03939	0.519696	−	0.854351i	$-0.326045\pi$
$383$	9.41823e21	1.03939	0.519696	+	0.854351i	$0.326045\pi$
$384$	0	0
$385$	1.16778e22	1.23294
$386$	0	0
$387$	− 4.40380e21i	− 0.444923i
$388$	0	0
$389$	− 2.84056e21i	− 0.274683i	−0.990524	−	0.137342i	$-0.956144\pi$
$389$	− 2.84056e21i	− 0.274683i	0.990524	−	0.137342i	$-0.0438558\pi$
$390$	0	0
$391$	−5.39969e20	−0.0499881
$392$	0	0
$393$	1.58687e21	0.140671
$394$	0	0
$395$	2.15194e21i	0.182707i
$396$	0	0
$397$	6.92645e20i	0.0563367i	0.999603	+	0.0281683i	$0.00896745\pi$
$397$	6.92645e20i	0.0563367i	−0.999603	+	0.0281683i	$0.991033\pi$
$398$	0	0
$399$	5.98493e21	0.466433
$400$	0	0
$401$	−4.47868e21	−0.334521	−0.167260	−	0.985913i	$-0.553492\pi$
$401$	−4.47868e21	−0.334521	−0.167260	+	0.985913i	$0.553492\pi$
$402$	0	0
$403$	− 4.93650e20i	− 0.0353448i
$404$	0	0
$405$	− 6.64641e21i	− 0.456267i
$406$	0	0
$407$	−2.02829e22	−1.33530
$408$	0	0
$409$	1.38245e22	0.872972	0.436486	−	0.899711i	$-0.356223\pi$
$409$	1.38245e22	0.872972	0.436486	+	0.899711i	$0.356223\pi$
$410$	0	0
$411$	1.54797e21i	0.0937791i
$412$	0	0
$413$	1.78567e22i	1.03807i
$414$	0	0
$415$	−2.84389e21	−0.158673
$416$	0	0
$417$	−1.06702e22	−0.571500
$418$	0	0
$419$	4.06502e21i	0.209047i	0.994522	+	0.104523i	$0.0333317\pi$
$419$	4.06502e21i	0.209047i	−0.994522	+	0.104523i	$0.966668\pi$
$420$	0	0
$421$	− 8.16114e21i	− 0.403045i	−0.979484	−	0.201522i	$-0.935411\pi$
$421$	− 8.16114e21i	− 0.403045i	0.979484	−	0.201522i	$-0.0645888\pi$
$422$	0	0
$423$	−4.98794e21	−0.236607
$424$	0	0
$425$	−6.38773e21	−0.291099
$426$	0	0
$427$	− 3.21059e22i	− 1.40588i
$428$	0	0
$429$	4.12013e20i	0.0173390i
$430$	0	0
$431$	9.79174e21	0.396098	0.198049	−	0.980192i	$-0.436539\pi$
$431$	9.79174e21	0.396098	0.198049	+	0.980192i	$0.436539\pi$
$432$	0	0
$433$	−2.60425e22	−1.01283	−0.506414	−	0.862290i	$-0.669029\pi$
$433$	−2.60425e22	−1.01283	−0.506414	+	0.862290i	$0.669029\pi$
$434$	0	0
$435$	− 1.05787e22i	− 0.395616i
$436$	0	0
$437$	2.30020e21i	0.0827321i
$438$	0	0
$439$	−2.68066e22	−0.927455	−0.463727	−	0.885978i	$-0.653488\pi$
$439$	−2.68066e22	−0.927455	−0.463727	+	0.885978i	$0.653488\pi$
$440$	0	0
$441$	5.04327e21	0.167874
$442$	0	0
$443$	− 6.73832e21i	− 0.215834i	−0.994160	−	0.107917i	$-0.965582\pi$
$443$	− 6.73832e21i	− 0.215834i	0.994160	−	0.107917i	$-0.0344181\pi$
$444$	0	0
$445$	− 4.61629e22i	− 1.42309i
$446$	0	0
$447$	−1.20633e22	−0.357975
$448$	0	0
$449$	−3.12876e22	−0.893878	−0.446939	−	0.894564i	$-0.647486\pi$
$449$	−3.12876e22	−0.893878	−0.446939	+	0.894564i	$0.647486\pi$
$450$	0	0
$451$	5.72118e22i	1.57393i
$452$	0	0
$453$	− 1.16492e22i	− 0.308647i
$454$	0	0
$455$	1.01789e21	0.0259779
$456$	0	0
$457$	−5.48338e22	−1.34822	−0.674110	−	0.738631i	$-0.735473\pi$
$457$	−5.48338e22	−1.34822	−0.674110	+	0.738631i	$0.735473\pi$
$458$	0	0
$459$	2.02416e22i	0.479553i
$460$	0	0
$461$	− 6.78571e22i	− 1.54931i	−0.632386	−	0.774653i	$-0.717924\pi$
$461$	− 6.78571e22i	− 1.54931i	0.632386	−	0.774653i	$-0.282076\pi$
$462$	0	0
$463$	1.64300e21	0.0361576	0.0180788	−	0.999837i	$-0.494245\pi$
$463$	1.64300e21	0.0361576	0.0180788	+	0.999837i	$0.494245\pi$
$464$	0	0
$465$	1.51460e22	0.321327
$466$	0	0
$467$	− 4.77673e22i	− 0.977096i	−0.872537	−	0.488548i	$-0.837527\pi$
$467$	− 4.77673e22i	− 0.977096i	0.872537	−	0.488548i	$-0.162473\pi$
$468$	0	0
$469$	6.14740e22i	1.21261i
$470$	0	0
$471$	1.38558e22	0.263606
$472$	0	0
$473$	4.18136e22	0.767357
$474$	0	0
$475$	2.72110e22i	0.481780i
$476$	0	0
$477$	8.68507e22i	1.48377i
$478$	0	0
$479$	5.48658e22	0.904585	0.452293	−	0.891870i	$-0.350606\pi$
$479$	5.48658e22	0.904585	0.452293	+	0.891870i	$0.350606\pi$
$480$	0	0
$481$	−1.76796e21	−0.0281345
$482$	0	0
$483$	− 1.92873e21i	− 0.0296292i
$484$	0	0
$485$	5.14062e22i	0.762448i
$486$	0	0
$487$	−1.26074e23	−1.80564	−0.902818	−	0.430022i	$-0.858506\pi$
$487$	−1.26074e23	−1.80564	−0.902818	+	0.430022i	$0.858506\pi$
$488$	0	0
$489$	3.31580e22	0.458630
$490$	0	0
$491$	− 1.93871e22i	− 0.259013i	−0.991579	−	0.129506i	$-0.958661\pi$
$491$	− 1.93871e22i	− 0.259013i	0.991579	−	0.129506i	$-0.0413393\pi$
$492$	0	0
$493$	− 7.43748e22i	− 0.959902i
$494$	0	0
$495$	7.78020e22	0.970167
$496$	0	0
$497$	1.70779e23	2.05781
$498$	0	0
$499$	− 8.94608e22i	− 1.04178i	−0.853622	−	0.520892i	$-0.825599\pi$
$499$	− 8.94608e22i	− 1.04178i	0.853622	−	0.520892i	$-0.174401\pi$
$500$	0	0
$501$	− 3.62826e22i	− 0.408393i
$502$	0	0
$503$	−3.84163e22	−0.418011	−0.209005	−	0.977914i	$-0.567023\pi$
$503$	−3.84163e22	−0.418011	−0.209005	+	0.977914i	$0.567023\pi$
$504$	0	0
$505$	−5.56847e22	−0.585813
$506$	0	0
$507$	− 3.67155e22i	− 0.373492i
$508$	0	0
$509$	1.79276e23i	1.76369i	0.471541	+	0.881844i	$0.343698\pi$
$509$	1.79276e23i	1.76369i	−0.471541	+	0.881844i	$0.656302\pi$
$510$	0	0
$511$	2.09839e23	1.99668
$512$	0	0
$513$	8.62269e22	0.793679
$514$	0	0
$515$	5.37961e22i	0.479059i
$516$	0	0
$517$	− 4.73599e22i	− 0.408076i
$518$	0	0
$519$	−8.81104e21	−0.0734691
$520$	0	0
$521$	6.78107e22	0.547240	0.273620	−	0.961838i	$-0.411779\pi$
$521$	6.78107e22	0.547240	0.273620	+	0.961838i	$0.411779\pi$
$522$	0	0
$523$	2.09501e23i	1.63652i	0.574845	+	0.818262i	$0.305062\pi$
$523$	2.09501e23i	1.63652i	−0.574845	+	0.818262i	$0.694938\pi$
$524$	0	0
$525$	− 2.28165e22i	− 0.172542i
$526$	0	0
$527$	1.06485e23	0.779651
$528$	0	0
$529$	−1.40309e23	−0.994745
$530$	0	0
$531$	1.18968e23i	0.816824i
$532$	0	0
$533$	4.98687e21i	0.0331624i
$534$	0	0
$535$	−1.06721e23	−0.687450
$536$	0	0
$537$	−7.10728e22	−0.443529
$538$	0	0
$539$	4.78852e22i	0.289532i
$540$	0	0
$541$	− 1.24185e23i	− 0.727600i	−0.931477	−	0.363800i	$-0.881479\pi$
$541$	− 1.24185e23i	− 0.727600i	0.931477	−	0.363800i	$-0.118521\pi$
$542$	0	0
$543$	1.28431e23	0.729243
$544$	0	0
$545$	−7.00370e22	−0.385440
$546$	0	0
$547$	− 1.22112e23i	− 0.651429i	−0.945468	−	0.325714i	$-0.894395\pi$
$547$	− 1.22112e23i	− 0.651429i	0.945468	−	0.325714i	$-0.105605\pi$
$548$	0	0
$549$	− 2.13903e23i	− 1.10624i
$550$	0	0
$551$	−3.16828e23	−1.58867
$552$	0	0
$553$	−5.40413e22	−0.262762
$554$	0	0
$555$	− 5.42439e22i	− 0.255776i
$556$	0	0
$557$	2.07273e23i	0.947924i	0.880545	+	0.473962i	$0.157177\pi$
$557$	2.07273e23i	0.947924i	−0.880545	+	0.473962i	$0.842823\pi$
$558$	0	0
$559$	3.64468e21	0.0161681
$560$	0	0
$561$	−8.88756e22	−0.382471
$562$	0	0
$563$	− 1.43541e23i	− 0.599314i	−0.954047	−	0.299657i	$-0.903128\pi$
$563$	− 1.43541e23i	− 0.599314i	0.954047	−	0.299657i	$-0.0968722\pi$
$564$	0	0
$565$	8.63985e22i	0.350021i
$566$	0	0
$567$	1.66910e23	0.656185
$568$	0	0
$569$	1.29010e23	0.492231	0.246116	−	0.969240i	$-0.420846\pi$
$569$	1.29010e23	0.492231	0.246116	+	0.969240i	$0.420846\pi$
$570$	0	0
$571$	− 2.64243e23i	− 0.978579i	−0.872121	−	0.489290i	$-0.837256\pi$
$571$	− 2.64243e23i	− 0.978579i	0.872121	−	0.489290i	$-0.162744\pi$
$572$	0	0
$573$	5.20005e21i	0.0186936i
$574$	0	0
$575$	8.76912e21	0.0306041
$576$	0	0
$577$	−9.70397e22	−0.328818	−0.164409	−	0.986392i	$-0.552572\pi$
$577$	−9.70397e22	−0.328818	−0.164409	+	0.986392i	$0.552572\pi$
$578$	0	0
$579$	2.85005e22i	0.0937746i
$580$	0	0
$581$	− 7.14182e22i	− 0.228198i
$582$	0	0
$583$	−8.24637e23	−2.55906
$584$	0	0
$585$	6.78161e21	0.0204413
$586$	0	0
$587$	− 3.02349e23i	− 0.885288i	−0.896697	−	0.442644i	$-0.854041\pi$
$587$	− 3.02349e23i	− 0.885288i	0.896697	−	0.442644i	$-0.145959\pi$
$588$	0	0
$589$	− 4.53616e23i	− 1.29035i
$590$	0	0
$591$	5.77374e22	0.159574
$592$	0	0
$593$	−3.18008e23	−0.854030	−0.427015	−	0.904245i	$-0.640435\pi$
$593$	−3.18008e23	−0.854030	−0.427015	+	0.904245i	$0.640435\pi$
$594$	0	0
$595$	2.19570e23i	0.573032i
$596$	0	0
$597$	1.25022e23i	0.317105i
$598$	0	0
$599$	−3.24737e23	−0.800578	−0.400289	−	0.916389i	$-0.631090\pi$
$599$	−3.24737e23	−0.800578	−0.400289	+	0.916389i	$0.631090\pi$
$600$	0	0
$601$	−3.78448e23	−0.906929	−0.453464	−	0.891274i	$-0.649812\pi$
$601$	−3.78448e23	−0.906929	−0.453464	+	0.891274i	$0.649812\pi$
$602$	0	0
$603$	4.09565e23i	0.954171i
$604$	0	0
$605$	4.03118e23i	0.913087i
$606$	0	0
$607$	2.72859e23	0.600945	0.300472	−	0.953791i	$-0.402856\pi$
$607$	2.72859e23	0.600945	0.300472	+	0.953791i	$0.402856\pi$
$608$	0	0
$609$	2.65661e23	0.568959
$610$	0	0
$611$	− 4.12813e21i	− 0.00859811i
$612$	0	0
$613$	2.23929e23i	0.453625i	0.973938	+	0.226813i	$0.0728305\pi$
$613$	2.23929e23i	0.453625i	−0.973938	+	0.226813i	$0.927169\pi$
$614$	0	0
$615$	−1.53005e23	−0.301486
$616$	0	0
$617$	9.37688e23	1.79736	0.898679	−	0.438608i	$-0.144528\pi$
$617$	9.37688e23	1.79736	0.898679	+	0.438608i	$0.144528\pi$
$618$	0	0
$619$	− 8.46873e23i	− 1.57924i	−0.613597	−	0.789619i	$-0.710278\pi$
$619$	− 8.46873e23i	− 1.57924i	0.613597	−	0.789619i	$-0.289722\pi$
$620$	0	0
$621$	− 2.77878e22i	− 0.0504168i
$622$	0	0
$623$	1.15928e24	2.04663
$624$	0	0
$625$	−2.32610e23	−0.399622
$626$	0	0
$627$	3.78600e23i	0.633003i
$628$	0	0
$629$	− 3.81368e23i	− 0.620603i
$630$	0	0
$631$	−5.75474e23	−0.911541	−0.455771	−	0.890097i	$-0.650636\pi$
$631$	−5.75474e23	−0.911541	−0.455771	+	0.890097i	$0.650636\pi$
$632$	0	0
$633$	−3.11853e23	−0.480860
$634$	0	0
$635$	7.47750e23i	1.12248i
$636$	0	0
$637$	4.17392e21i	0.00610041i
$638$	0	0
$639$	1.13780e24	1.61923
$640$	0	0
$641$	4.70111e23	0.651489	0.325745	−	0.945458i	$-0.394385\pi$
$641$	4.70111e23	0.651489	0.325745	+	0.945458i	$0.394385\pi$
$642$	0	0
$643$	− 1.44971e23i	− 0.195653i	−0.995203	−	0.0978266i	$-0.968811\pi$
$643$	− 1.44971e23i	− 0.195653i	0.995203	−	0.0978266i	$-0.0311891\pi$
$644$	0	0
$645$	1.11825e23i	0.146987i
$646$	0	0
$647$	−1.43169e23	−0.183300	−0.0916502	−	0.995791i	$-0.529214\pi$
$647$	−1.43169e23	−0.183300	−0.0916502	+	0.995791i	$0.529214\pi$
$648$	0	0
$649$	−1.12959e24	−1.40877
$650$	0	0
$651$	3.80358e23i	0.462119i
$652$	0	0
$653$	3.23785e23i	0.383260i	0.981467	+	0.191630i	$0.0613774\pi$
$653$	3.23785e23i	0.383260i	−0.981467	+	0.191630i	$0.938623\pi$
$654$	0	0
$655$	2.48001e23	0.286024
$656$	0	0
$657$	1.39804e24	1.57113
$658$	0	0
$659$	8.58094e23i	0.939742i	0.882735	+	0.469871i	$0.155700\pi$
$659$	8.58094e23i	0.939742i	−0.882735	+	0.469871i	$0.844300\pi$
$660$	0	0
$661$	− 7.30451e23i	− 0.779612i	−0.920897	−	0.389806i	$-0.872542\pi$
$661$	− 7.30451e23i	− 0.779612i	0.920897	−	0.389806i	$-0.127458\pi$
$662$	0	0
$663$	−7.74684e21	−0.00805860
$664$	0	0
$665$	9.35343e23	0.948390
$666$	0	0
$667$	1.02102e23i	0.100917i
$668$	0	0
$669$	3.28700e23i	0.316722i
$670$	0	0
$671$	2.03098e24	1.90794
$672$	0	0
$673$	−1.25481e23	−0.114934	−0.0574671	−	0.998347i	$-0.518302\pi$
$673$	−1.25481e23	−0.114934	−0.0574671	+	0.998347i	$0.518302\pi$
$674$	0	0
$675$	− 3.28724e23i	− 0.293596i
$676$	0	0
$677$	3.68442e22i	0.0320897i	0.999871	+	0.0160448i	$0.00510745\pi$
$677$	3.68442e22i	0.0320897i	−0.999871	+	0.0160448i	$0.994893\pi$
$678$	0	0
$679$	−1.29095e24	−1.09652
$680$	0	0
$681$	5.17993e23	0.429115
$682$	0	0
$683$	− 3.46565e23i	− 0.280033i	−0.990149	−	0.140016i	$-0.955285\pi$
$683$	− 3.46565e23i	− 0.280033i	0.990149	−	0.140016i	$-0.0447155\pi$
$684$	0	0
$685$	2.41921e23i	0.190679i
$686$	0	0
$687$	−6.03449e22	−0.0463990
$688$	0	0
$689$	−7.18795e22	−0.0539190
$690$	0	0
$691$	− 1.64173e24i	− 1.20154i	−0.799421	−	0.600771i	$-0.794860\pi$
$691$	− 1.64173e24i	− 1.20154i	0.799421	−	0.600771i	$-0.205140\pi$
$692$	0	0
$693$	1.95383e24i	1.39526i
$694$	0	0
$695$	−1.66757e24	−1.16202
$696$	0	0
$697$	−1.07572e24	−0.731511
$698$	0	0
$699$	2.98617e23i	0.198179i
$700$	0	0
$701$	− 1.08179e24i	− 0.700709i	−0.936617	−	0.350355i	$-0.886061\pi$
$701$	− 1.08179e24i	− 0.700709i	0.936617	−	0.350355i	$-0.113939\pi$
$702$	0	0
$703$	−1.62458e24	−1.02712
$704$	0	0
$705$	1.26658e23	0.0781670
$706$	0	0
$707$	− 1.39840e24i	− 0.842493i
$708$	0	0
$709$	1.54173e23i	0.0906808i	0.998972	+	0.0453404i	$0.0144372\pi$
$709$	1.54173e23i	0.0906808i	−0.998972	+	0.0453404i	$0.985563\pi$
$710$	0	0
$711$	−3.60045e23	−0.206760
$712$	0	0
$713$	−1.46184e23	−0.0819669
$714$	0	0
$715$	6.43906e22i	0.0352550i
$716$	0	0
$717$	1.75143e23i	0.0936438i
$718$	0	0
$719$	1.87407e24	0.978565	0.489283	−	0.872125i	$-0.337259\pi$
$719$	1.87407e24	0.978565	0.489283	+	0.872125i	$0.337259\pi$
$720$	0	0
$721$	−1.35097e24	−0.688964
$722$	0	0
$723$	− 3.96924e23i	− 0.197711i
$724$	0	0
$725$	1.20785e24i	0.587679i
$726$	0	0
$727$	−3.49121e24	−1.65933	−0.829667	−	0.558259i	$-0.811469\pi$
$727$	−3.49121e24	−1.65933	−0.829667	+	0.558259i	$0.811469\pi$
$728$	0	0
$729$	5.52055e23	0.256329
$730$	0	0
$731$	7.86197e23i	0.356643i
$732$	0	0
$733$	− 4.17586e24i	− 1.85081i	−0.378979	−	0.925405i	$-0.623725\pi$
$733$	− 4.17586e24i	− 1.85081i	0.378979	−	0.925405i	$-0.376275\pi$
$734$	0	0
$735$	−1.28063e23	−0.0554600
$736$	0	0
$737$	−3.88877e24	−1.64566
$738$	0	0
$739$	3.77789e24i	1.56233i	0.624327	+	0.781163i	$0.285373\pi$
$739$	3.77789e24i	1.56233i	−0.624327	+	0.781163i	$0.714627\pi$
$740$	0	0
$741$	3.30007e22i	0.0133373i
$742$	0	0
$743$	2.48774e24	0.982651	0.491326	−	0.870976i	$-0.336513\pi$
$743$	2.48774e24	0.982651	0.491326	+	0.870976i	$0.336513\pi$
$744$	0	0
$745$	−1.88529e24	−0.727864
$746$	0	0
$747$	− 4.75817e23i	− 0.179562i
$748$	0	0
$749$	− 2.68005e24i	− 0.988663i
$750$	0	0
$751$	−1.46920e22	−0.00529836	−0.00264918	−	0.999996i	$-0.500843\pi$
$751$	−1.46920e22	−0.00529836	−0.00264918	+	0.999996i	$0.500843\pi$
$752$	0	0
$753$	−3.50041e23	−0.123413
$754$	0	0
$755$	− 1.82058e24i	− 0.627567i
$756$	0	0
$757$	− 4.58159e24i	− 1.54419i	−0.635506	−	0.772096i	$-0.719208\pi$
$757$	− 4.58159e24i	− 1.54419i	0.635506	−	0.772096i	$-0.280792\pi$
$758$	0	0
$759$	1.22009e23	0.0402102
$760$	0	0
$761$	2.36360e24	0.761736	0.380868	−	0.924629i	$-0.375625\pi$
$761$	2.36360e24	0.761736	0.380868	+	0.924629i	$0.375625\pi$
$762$	0	0
$763$	− 1.75883e24i	− 0.554325i
$764$	0	0
$765$	1.46287e24i	0.450902i
$766$	0	0
$767$	−9.84609e22	−0.0296827
$768$	0	0
$769$	−9.13423e23	−0.269338	−0.134669	−	0.990891i	$-0.542997\pi$
$769$	−9.13423e23	−0.269338	−0.134669	+	0.990891i	$0.542997\pi$
$770$	0	0
$771$	− 1.68552e24i	− 0.486151i
$772$	0	0
$773$	1.76234e24i	0.497238i	0.968601	+	0.248619i	$0.0799766\pi$
$773$	1.76234e24i	0.497238i	−0.968601	+	0.248619i	$0.920023\pi$
$774$	0	0
$775$	−1.72933e24	−0.477324
$776$	0	0
$777$	1.36222e24	0.367847
$778$	0	0
$779$	4.58244e24i	1.21068i
$780$	0	0
$781$	1.08033e25i	2.79268i
$782$	0	0
$783$	3.82747e24	0.968136
$784$	0	0
$785$	2.16543e24	0.535985
$786$	0	0
$787$	− 3.12907e24i	− 0.757933i	−0.925410	−	0.378966i	$-0.876280\pi$
$787$	− 3.12907e24i	− 0.757933i	0.925410	−	0.378966i	$-0.123720\pi$
$788$	0	0
$789$	− 1.27914e24i	− 0.303223i
$790$	0	0
$791$	−2.16971e24	−0.503387
$792$	0	0
$793$	1.77030e23	0.0402000
$794$	0	0
$795$	− 2.20538e24i	− 0.490188i
$796$	0	0
$797$	1.77982e24i	0.387241i	0.981077	+	0.193620i	$0.0620230\pi$
$797$	1.77982e24i	0.387241i	−0.981077	+	0.193620i	$0.937977\pi$
$798$	0	0
$799$	8.90481e23	0.189661
$800$	0	0
$801$	7.72360e24	1.61044
$802$	0	0
$803$	1.32742e25i	2.70973i
$804$	0	0
$805$	− 3.01427e23i	− 0.0602445i
$806$	0	0
$807$	3.39811e23	0.0664985
$808$	0	0
$809$	2.82672e23	0.0541653	0.0270826	−	0.999633i	$-0.491378\pi$
$809$	2.82672e23	0.0541653	0.0270826	+	0.999633i	$0.491378\pi$
$810$	0	0
$811$	− 9.40519e24i	− 1.76478i	−0.470519	−	0.882390i	$-0.655933\pi$
$811$	− 9.40519e24i	− 1.76478i	0.470519	−	0.882390i	$-0.344067\pi$
$812$	0	0
$813$	− 1.62357e24i	− 0.298333i
$814$	0	0
$815$	5.18203e24	0.932525
$816$	0	0
$817$	3.34911e24	0.590258
$818$	0	0
$819$	1.70306e23i	0.0293979i
$820$	0	0
$821$	2.44785e24i	0.413875i	0.978354	+	0.206937i	$0.0663496\pi$
$821$	2.44785e24i	0.413875i	−0.978354	+	0.206937i	$0.933650\pi$
$822$	0	0
$823$	−2.44893e24	−0.405581	−0.202791	−	0.979222i	$-0.565001\pi$
$823$	−2.44893e24	−0.405581	−0.202791	+	0.979222i	$0.565001\pi$
$824$	0	0
$825$	1.44334e24	0.234159
$826$	0	0
$827$	3.39629e24i	0.539769i	0.962893	+	0.269885i	$0.0869855\pi$
$827$	3.39629e24i	0.539769i	−0.962893	+	0.269885i	$0.913014\pi$
$828$	0	0
$829$	− 3.76176e24i	− 0.585703i	−0.956158	−	0.292851i	$-0.905396\pi$
$829$	− 3.76176e24i	− 0.585703i	0.956158	−	0.292851i	$-0.0946041\pi$
$830$	0	0
$831$	1.64683e24	0.251212
$832$	0	0
$833$	−9.00359e23	−0.134565
$834$	0	0
$835$	− 5.67036e24i	− 0.830377i
$836$	0	0
$837$	5.47994e24i	0.786338i
$838$	0	0
$839$	1.26128e25	1.77352	0.886758	−	0.462234i	$-0.152952\pi$
$839$	1.26128e25	1.77352	0.886758	+	0.462234i	$0.152952\pi$
$840$	0	0
$841$	−6.80632e24	−0.937877
$842$	0	0
$843$	2.41464e24i	0.326075i
$844$	0	0
$845$	− 5.73801e24i	− 0.759415i
$846$	0	0
$847$	−1.01234e25	−1.31317
$848$	0	0
$849$	1.91564e24	0.243556
$850$	0	0
$851$	5.23545e23i	0.0652458i
$852$	0	0
$853$	− 3.35040e24i	− 0.409289i	−0.978836	−	0.204644i	$-0.934396\pi$
$853$	− 3.35040e24i	− 0.409289i	0.978836	−	0.204644i	$-0.0656038\pi$
$854$	0	0
$855$	6.23164e24	0.746261
$856$	0	0
$857$	7.08522e24	0.831795	0.415897	−	0.909412i	$-0.363468\pi$
$857$	7.08522e24	0.831795	0.415897	+	0.909412i	$0.363468\pi$
$858$	0	0
$859$	8.34929e24i	0.960966i	0.877004	+	0.480483i	$0.159539\pi$
$859$	8.34929e24i	0.960966i	−0.877004	+	0.480483i	$0.840461\pi$
$860$	0	0
$861$	− 3.84239e24i	− 0.433585i
$862$	0	0
$863$	−6.81718e24	−0.754246	−0.377123	−	0.926163i	$-0.623087\pi$
$863$	−6.81718e24	−0.754246	−0.377123	+	0.926163i	$0.623087\pi$
$864$	0	0
$865$	−1.37702e24	−0.149383
$866$	0	0
$867$	1.84347e24i	0.196098i
$868$	0	0
$869$	− 3.41859e24i	− 0.356598i
$870$	0	0
$871$	−3.38965e23	−0.0346738
$872$	0	0
$873$	−8.60087e24	−0.862822
$874$	0	0
$875$	− 1.20125e25i	− 1.18185i
$876$	0	0
$877$	1.56397e25i	1.50914i	0.656217	+	0.754572i	$0.272156\pi$
$877$	1.56397e25i	1.50914i	−0.656217	+	0.754572i	$0.727844\pi$
$878$	0	0
$879$	−6.41533e24	−0.607175
$880$	0	0
$881$	−1.62905e25	−1.51230	−0.756152	−	0.654396i	$-0.772923\pi$
$881$	−1.62905e25	−1.51230	−0.756152	+	0.654396i	$0.772923\pi$
$882$	0	0
$883$	4.63346e24i	0.421929i	0.977494	+	0.210964i	$0.0676604\pi$
$883$	4.63346e24i	0.421929i	−0.977494	+	0.210964i	$0.932340\pi$
$884$	0	0
$885$	− 3.02094e24i	− 0.269851i
$886$	0	0
$887$	1.40637e24	0.123239	0.0616197	−	0.998100i	$-0.480373\pi$
$887$	1.40637e24	0.123239	0.0616197	+	0.998100i	$0.480373\pi$
$888$	0	0
$889$	−1.87781e25	−1.61431
$890$	0	0
$891$	1.05585e25i	0.890519i
$892$	0	0
$893$	− 3.79335e24i	− 0.313896i
$894$	0	0
$895$	−1.11075e25	−0.901820
$896$	0	0
$897$	1.06349e22	0.000847224 0
$898$	0	0
$899$	− 2.01352e25i	− 1.57398i
$900$	0	0
$901$	− 1.55052e25i	− 1.18937i
$902$	0	0
$903$	−2.80823e24	−0.211392
$904$	0	0
$905$	2.00716e25	1.48276
$906$	0	0
$907$	− 8.93640e24i	− 0.647889i	−0.946076	−	0.323945i	$-0.894991\pi$
$907$	− 8.93640e24i	− 0.647889i	0.946076	−	0.323945i	$-0.105009\pi$
$908$	0	0
$909$	− 9.31672e24i	− 0.662933i
$910$	0	0
$911$	−1.71271e25	−1.19613	−0.598064	−	0.801448i	$-0.704063\pi$
$911$	−1.71271e25	−1.19613	−0.598064	+	0.801448i	$0.704063\pi$
$912$	0	0
$913$	4.51783e24	0.309691
$914$	0	0
$915$	5.43158e24i	0.365466i
$916$	0	0
$917$	6.22800e24i	0.411348i
$918$	0	0
$919$	1.92318e25	1.24692	0.623459	−	0.781856i	$-0.285727\pi$
$919$	1.92318e25	1.24692	0.623459	+	0.781856i	$0.285727\pi$
$920$	0	0
$921$	−3.03854e24	−0.193401
$922$	0	0
$923$	9.41669e23i	0.0588415i
$924$	0	0
$925$	6.19343e24i	0.379950i
$926$	0	0
$927$	−9.00073e24	−0.542126
$928$	0	0
$929$	−1.01792e25	−0.601979	−0.300989	−	0.953627i	$-0.597317\pi$
$929$	−1.01792e25	−0.601979	−0.300989	+	0.953627i	$0.597317\pi$
$930$	0	0
$931$	3.83543e24i	0.222711i
$932$	0	0
$933$	6.59366e24i	0.375952i
$934$	0	0
$935$	−1.38897e25	−0.777670
$936$	0	0
$937$	2.57324e25	1.41480	0.707399	−	0.706815i	$-0.249869\pi$
$937$	2.57324e25	1.41480	0.707399	+	0.706815i	$0.249869\pi$
$938$	0	0
$939$	2.98857e24i	0.161364i
$940$	0	0
$941$	1.64698e25i	0.873327i	0.899625	+	0.436663i	$0.143840\pi$
$941$	1.64698e25i	0.873327i	−0.899625	+	0.436663i	$0.856160\pi$
$942$	0	0
$943$	1.47676e24	0.0769058
$944$	0	0
$945$	−1.12995e25	−0.577947
$946$	0	0
$947$	− 3.53295e25i	− 1.77485i	−0.460949	−	0.887427i	$-0.652491\pi$
$947$	− 3.53295e25i	− 1.77485i	0.460949	−	0.887427i	$-0.347509\pi$
$948$	0	0
$949$	1.15705e24i	0.0570936i
$950$	0	0
$951$	8.73690e24	0.423470
$952$	0	0
$953$	−4.30929e24	−0.205171	−0.102585	−	0.994724i	$-0.532712\pi$
$953$	−4.30929e24	−0.205171	−0.102585	+	0.994724i	$0.532712\pi$
$954$	0	0
$955$	8.12679e23i	0.0380093i
$956$	0	0
$957$	1.68054e25i	0.772142i
$958$	0	0
$959$	−6.07532e24	−0.274228
$960$	0	0
$961$	6.27834e24	0.278417
$962$	0	0
$963$	− 1.78556e25i	− 0.777950i
$964$	0	0
$965$	4.45415e24i	0.190670i
$966$	0	0
$967$	2.86774e25	1.20619	0.603094	−	0.797670i	$-0.293934\pi$
$967$	2.86774e25	1.20619	0.603094	+	0.797670i	$0.293934\pi$
$968$	0	0
$969$	−7.11859e24	−0.294200
$970$	0	0
$971$	3.16457e25i	1.28514i	0.766225	+	0.642572i	$0.222133\pi$
$971$	3.16457e25i	1.28514i	−0.766225	+	0.642572i	$0.777867\pi$
$972$	0	0
$973$	− 4.18775e25i	− 1.67117i
$974$	0	0
$975$	1.25809e23	0.00493370
$976$	0	0
$977$	3.66143e25	1.41107	0.705533	−	0.708677i	$-0.250708\pi$
$977$	3.66143e25	1.41107	0.705533	+	0.708677i	$0.250708\pi$
$978$	0	0
$979$	7.33347e25i	2.77752i
$980$	0	0
$981$	− 1.17180e25i	− 0.436182i
$982$	0	0
$983$	4.80762e25	1.75884	0.879418	−	0.476051i	$-0.157932\pi$
$983$	4.80762e25	1.75884	0.879418	+	0.476051i	$0.157932\pi$
$984$	0	0
$985$	9.02337e24	0.324460
$986$	0	0
$987$	3.18073e24i	0.112417i
$988$	0	0
$989$	− 1.07930e24i	− 0.0374949i
$990$	0	0
$991$	2.82526e25	0.964789	0.482395	−	0.875954i	$-0.339767\pi$
$991$	2.82526e25	0.964789	0.482395	+	0.875954i	$0.339767\pi$
$992$	0	0
$993$	−2.98783e24	−0.100297
$994$	0	0
$995$	1.95388e25i	0.644765i
$996$	0	0
$997$	2.70363e25i	0.877078i	0.898712	+	0.438539i	$0.144504\pi$
$997$	2.70363e25i	0.877078i	−0.898712	+	0.438539i	$0.855496\pi$
$998$	0	0
$999$	1.96259e25	0.625926

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.18.b.a.17.7		16
4.3	odd	2		8.18.b.a.5.13	✓	16
8.3	odd	2		8.18.b.a.5.14	yes	16
8.5	even	2	inner	32.18.b.a.17.10		16
12.11	even	2		72.18.d.b.37.4		16
24.11	even	2		72.18.d.b.37.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.18.b.a.5.13	✓	16	4.3	odd	2
8.18.b.a.5.14	yes	16	8.3	odd	2
32.18.b.a.17.7		16	1.1	even	1	trivial
32.18.b.a.17.10		16	8.5	even	2	inner
72.18.d.b.37.3		16	24.11	even	2
72.18.d.b.37.4		16	12.11	even	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:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	32.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4248.51i q^{3} -663971. i q^{5} +1.66742e7 q^{7} +1.11090e8 q^{9} +O(q^{10})\)	\(q-4248.51i q^{3} -663971. i q^{5} +1.66742e7 q^{7} +1.11090e8 q^{9} +1.05479e9i q^{11} +9.19407e7i q^{13} -2.82089e9 q^{15} -1.98326e10 q^{17} +8.44846e10i q^{19} -7.08405e10i q^{21} +2.72263e10 q^{23} +3.22082e11 q^{25} -1.02062e12i q^{27} +3.75013e12i q^{29} -5.36921e12 q^{31} +4.48129e12 q^{33} -1.10712e13i q^{35} +1.92294e13i q^{37} +3.90612e11 q^{39} +5.42400e13 q^{41} -3.96416e13i q^{43} -7.37607e13i q^{45} -4.48999e13 q^{47} +4.53979e13 q^{49} +8.42591e13i q^{51} +7.81803e14i q^{53} +7.00349e14 q^{55} +3.58934e14 q^{57} +1.07092e15i q^{59} -1.92548e15i q^{61} +1.85234e15 q^{63} +6.10460e13 q^{65} +3.68678e15i q^{67} -1.15671e14i q^{69} +1.02421e16 q^{71} +1.25847e16 q^{73} -1.36837e15i q^{75} +1.75877e16i q^{77} -3.24102e15 q^{79} +1.00101e16 q^{81} -4.28316e15i q^{83} +1.31683e16i q^{85} +1.59325e16 q^{87} +6.95255e16 q^{89} +1.53304e15i q^{91} +2.28112e16i q^{93} +5.60953e16 q^{95} -7.74223e16 q^{97} +1.17177e17i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 11529600 q^{7} - 602654096 q^{9}+O(q^{10}) \) 16 * q - 11529600 * q^7 - 602654096 * q^9	\( 16 q - 11529600 q^{7} - 602654096 q^{9} + 9993282176 q^{15} - 7489125600 q^{17} - 746845345920 q^{23} - 1809682431664 q^{25} + 318979758592 q^{31} + 5633526177600 q^{33} + 18457706051456 q^{39} + 7482251536032 q^{41} + 376698804821760 q^{47} + 127691292101520 q^{49} - 22\!\cdots\!52 q^{55}+ \cdots + 95\!\cdots\!40 q^{97}+O(q^{100}) \) 16 * q - 11529600 * q^7 - 602654096 * q^9 + 9993282176 * q^15 - 7489125600 * q^17 - 746845345920 * q^23 - 1809682431664 * q^25 + 318979758592 * q^31 + 5633526177600 * q^33 + 18457706051456 * q^39 + 7482251536032 * q^41 + 376698804821760 * q^47 + 127691292101520 * q^49 - 2209036687713152 * q^55 - 190521298294720 * q^57 + 8131096607338880 * q^63 + 2385987975356160 * q^65 - 9025926285576576 * q^71 + 11332002046118560 * q^73 + 45299671392008448 * q^79 + 20101901999290832 * q^81 - 25965768920837760 * q^87 - 69879174608766048 * q^89 - 93790444358203776 * q^95 + 95593398602180640 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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