LMFDB - Embedded newform 32.20.b.a.17.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,20,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, 20, names="a")

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

chi := DirichletCharacter("32.17");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 20 $
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:	$73.2213428980$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 9 x^{17} + 847482358 x^{16} - 6779858660 x^{15} + \cdots + 11\!\cdots\!76 $ x^18 - 9x^17 + 847482358x^16 - 6779858660x^15 + 288755694526787238x^14 - 2021289743039984830x^13 + 51121850635988052944521232x^12 - 306731077539161966630890980x^11 + 5053028465804290954773516593713297x^10 - 25265139517319958838954290038554593x^9 + 277525242785226225995541366728053703523138x^8 - 1110100819550071174104182779845774739997744x^7 + 7843581170028346695466748185748667347430921666496x^6 - 23530739624732277774733733894667178736900394654048x^5 + 88699644118206331836031766951870758907094189945148981184x^4 - 177399249018514584235839520819873862547126510378177038080x^3 + 27738447216461514660719783835924541644179956872586393406592768x^2 - 27738358516840927193180104500114450400133860395870028292288768*x + 1155804113673024778339834102840085457131269774629483366903990104576
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	multiple of $ 2^{306}\cdot 3^{16}\cdot 5^{4}\cdot 7 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.3
Root			$0.500000 + 12463.1i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.20.b.a.17.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-49852.5i q^{3} -3.24834e6i q^{5} -1.08353e8 q^{7} -1.32301e9 q^{9} +O(q^{10})$	$q-49852.5i q^{3} -3.24834e6i q^{5} -1.08353e8 q^{7} -1.32301e9 q^{9} +6.67212e9i q^{11} +4.23789e10i q^{13} -1.61938e11 q^{15} +9.60587e11 q^{17} -1.90113e11i q^{19} +5.40167e12i q^{21} +7.67154e12 q^{23} +8.52179e12 q^{25} +8.01381e12i q^{27} +1.13055e14i q^{29} +3.94702e13 q^{31} +3.32622e14 q^{33} +3.51967e14i q^{35} +6.09095e14i q^{37} +2.11270e15 q^{39} -1.93765e15 q^{41} -4.35407e15i q^{43} +4.29759e15i q^{45} -1.43084e16 q^{47} +3.41459e14 q^{49} -4.78877e16i q^{51} -3.20081e15i q^{53} +2.16733e16 q^{55} -9.47763e15 q^{57} +8.85551e16i q^{59} +4.76424e16i q^{61} +1.43352e17 q^{63} +1.37661e17 q^{65} +1.35781e17i q^{67} -3.82446e17i q^{69} +1.68191e17 q^{71} +7.51049e17 q^{73} -4.24833e17i q^{75} -7.22944e17i q^{77} -7.44514e17 q^{79} -1.13818e18 q^{81} -1.14882e18i q^{83} -3.12031e18i q^{85} +5.63609e18 q^{87} +4.58017e18 q^{89} -4.59188e18i q^{91} -1.96769e18i q^{93} -6.17553e17 q^{95} +5.33795e18 q^{97} -8.82730e18i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 80707216 q^{7} - 6198727826 q^{9}+O(q^{10}) $ 18 * q + 80707216 * q^7 - 6198727826 * q^9	$ 18 q + 80707216 q^{7} - 6198727826 q^{9} - 156097960432 q^{15} + 14121426692 q^{17} - 2177121583952 q^{23} - 44414474211734 q^{25} - 428505770260416 q^{31} - 185380269683736 q^{33} - 942830575043152 q^{39} + 12\!\cdots\!24 q^{41}+ \cdots + 82\!\cdots\!20 q^{97}+O(q^{100}) $ 18 * q + 80707216 * q^7 - 6198727826 * q^9 - 156097960432 * q^15 + 14121426692 * q^17 - 2177121583952 * q^23 - 44414474211734 * q^25 - 428505770260416 * q^31 - 185380269683736 * q^33 - 942830575043152 * q^39 + 1200260021141524 * q^41 + 1134160959992928 * q^47 + 31570092936329058 * q^49 + 51757047863478736 * q^55 - 7740744140670520 * q^57 - 430168668665078416 * q^63 - 184007586011668640 * q^65 + 1599477989199799056 * q^71 - 93004298989809068 * q^73 - 1813975801025264480 * q^79 + 1801762045144484330 * q^81 + 13124960835754102512 * q^87 + 4470509771045743540 * q^89 - 23809730986995679856 * q^95 + 8246429789464117220 * 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
$3$	− 49852.5i	− 1.46230i	−0.682219	−	0.731148i	$-0.738985\pi$
$3$	− 49852.5i	− 1.46230i	0.682219	−	0.731148i	$-0.261015\pi$
$4$	0	0
$5$	− 3.24834e6i	− 0.743783i	−0.928276	−	0.371891i	$-0.878709\pi$
$5$	− 3.24834e6i	− 0.743783i	0.928276	−	0.371891i	$-0.121291\pi$
$6$	0	0
$7$	−1.08353e8	−1.01487	−0.507434	−	0.861691i	$-0.669406\pi$
$7$	−1.08353e8	−1.01487	−0.507434	+	0.861691i	$0.669406\pi$
$8$	0	0
$9$	−1.32301e9	−1.13831
$10$	0	0
$11$	6.67212e9i	0.853166i	0.904448	+	0.426583i	$0.140283\pi$
$11$	6.67212e9i	0.853166i	−0.904448	+	0.426583i	$0.859717\pi$
$12$	0	0
$13$	4.23789e10i	1.10838i	0.832391	+	0.554189i	$0.186972\pi$
$13$	4.23789e10i	1.10838i	−0.832391	+	0.554189i	$0.813028\pi$
$14$	0	0
$15$	−1.61938e11	−1.08763
$16$	0	0
$17$	9.60587e11	1.96459	0.982295	−	0.187341i	$-0.0599869\pi$
$17$	9.60587e11	1.96459	0.982295	+	0.187341i	$0.0599869\pi$
$18$	0	0
$19$	− 1.90113e11i	− 0.135162i	−0.997714	−	0.0675809i	$-0.978472\pi$
$19$	− 1.90113e11i	− 0.135162i	0.997714	−	0.0675809i	$-0.0215281\pi$
$20$	0	0
$21$	5.40167e12i	1.48404i
$22$	0	0
$23$	7.67154e12	0.888113	0.444056	−	0.895999i	$-0.353539\pi$
$23$	7.67154e12	0.888113	0.444056	+	0.895999i	$0.353539\pi$
$24$	0	0
$25$	8.52179e12	0.446787
$26$	0	0
$27$	8.01381e12i	0.202247i
$28$	0	0
$29$	1.13055e14i	1.44714i	0.690252	+	0.723569i	$0.257499\pi$
$29$	1.13055e14i	1.44714i	−0.690252	+	0.723569i	$0.742501\pi$
$30$	0	0
$31$	3.94702e13	0.268123	0.134061	−	0.990973i	$-0.457198\pi$
$31$	3.94702e13	0.268123	0.134061	+	0.990973i	$0.457198\pi$
$32$	0	0
$33$	3.32622e14	1.24758
$34$	0	0
$35$	3.51967e14i	0.754841i
$36$	0	0
$37$	6.09095e14i	0.770492i	0.922814	+	0.385246i	$0.125883\pi$
$37$	6.09095e14i	0.770492i	−0.922814	+	0.385246i	$0.874117\pi$
$38$	0	0
$39$	2.11270e15	1.62078
$40$	0	0
$41$	−1.93765e15	−0.924333	−0.462167	−	0.886793i	$-0.652928\pi$
$41$	−1.93765e15	−0.924333	−0.462167	+	0.886793i	$0.652928\pi$
$42$	0	0
$43$	− 4.35407e15i	− 1.32113i	−0.750770	−	0.660564i	$-0.770317\pi$
$43$	− 4.35407e15i	− 1.32113i	0.750770	−	0.660564i	$-0.229683\pi$
$44$	0	0
$45$	4.29759e15i	0.846654i
$46$	0	0
$47$	−1.43084e16	−1.86493	−0.932464	−	0.361263i	$-0.882346\pi$
$47$	−1.43084e16	−1.86493	−0.932464	+	0.361263i	$0.882346\pi$
$48$	0	0
$49$	3.41459e14	0.0299554
$50$	0	0
$51$	− 4.78877e16i	− 2.87281i
$52$	0	0
$53$	− 3.20081e15i	− 0.133241i	−0.997778	−	0.0666207i	$-0.978778\pi$
$53$	− 3.20081e15i	− 0.133241i	0.997778	−	0.0666207i	$-0.0212218\pi$
$54$	0	0
$55$	2.16733e16	0.634570
$56$	0	0
$57$	−9.47763e15	−0.197646
$58$	0	0
$59$	8.85551e16i	1.33082i	0.746478	+	0.665411i	$0.231744\pi$
$59$	8.85551e16i	1.33082i	−0.746478	+	0.665411i	$0.768256\pi$
$60$	0	0
$61$	4.76424e16i	0.521627i	0.965389	+	0.260814i	$0.0839908\pi$
$61$	4.76424e16i	0.521627i	−0.965389	+	0.260814i	$0.916009\pi$
$62$	0	0
$63$	1.43352e17	1.15523
$64$	0	0
$65$	1.37661e17	0.824393
$66$	0	0
$67$	1.35781e17i	0.609717i	0.952398	+	0.304859i	$0.0986093\pi$
$67$	1.35781e17i	0.609717i	−0.952398	+	0.304859i	$0.901391\pi$
$68$	0	0
$69$	− 3.82446e17i	− 1.29868i
$70$	0	0
$71$	1.68191e17	0.435359	0.217679	−	0.976020i	$-0.430151\pi$
$71$	1.68191e17	0.435359	0.217679	+	0.976020i	$0.430151\pi$
$72$	0	0
$73$	7.51049e17	1.49314	0.746571	−	0.665306i	$-0.231699\pi$
$73$	7.51049e17	1.49314	0.746571	+	0.665306i	$0.231699\pi$
$74$	0	0
$75$	− 4.24833e17i	− 0.653335i
$76$	0	0
$77$	− 7.22944e17i	− 0.865850i
$78$	0	0
$79$	−7.44514e17	−0.698901	−0.349450	−	0.936955i	$-0.613632\pi$
$79$	−7.44514e17	−0.698901	−0.349450	+	0.936955i	$0.613632\pi$
$80$	0	0
$81$	−1.13818e18	−0.842563
$82$	0	0
$83$	− 1.14882e18i	− 0.674542i	−0.941408	−	0.337271i	$-0.890496\pi$
$83$	− 1.14882e18i	− 0.674542i	0.941408	−	0.337271i	$-0.109504\pi$
$84$	0	0
$85$	− 3.12031e18i	− 1.46123i
$86$	0	0
$87$	5.63609e18	2.11614
$88$	0	0
$89$	4.58017e18	1.38572	0.692861	−	0.721071i	$-0.256350\pi$
$89$	4.58017e18	1.38572	0.692861	+	0.721071i	$0.256350\pi$
$90$	0	0
$91$	− 4.59188e18i	− 1.12486i
$92$	0	0
$93$	− 1.96769e18i	− 0.392075i
$94$	0	0
$95$	−6.17553e17	−0.100531
$96$	0	0
$97$	5.33795e18	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$97$	5.33795e18	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$98$	0	0
$99$	− 8.82730e18i	− 0.971166i
$100$	0	0
$101$	1.38239e19i	1.25770i	0.777525	+	0.628852i	$0.216475\pi$
$101$	1.38239e19i	1.25770i	−0.777525	+	0.628852i	$0.783525\pi$
$102$	0	0
$103$	1.91639e19	1.44721	0.723604	−	0.690215i	$-0.242484\pi$
$103$	1.91639e19	1.44721	0.723604	+	0.690215i	$0.242484\pi$
$104$	0	0
$105$	1.75464e19	1.10380
$106$	0	0
$107$	− 7.06297e18i	− 0.371399i	−0.982607	−	0.185700i	$-0.940545\pi$
$107$	− 7.06297e18i	− 0.371399i	0.982607	−	0.185700i	$-0.0594552\pi$
$108$	0	0
$109$	− 9.28370e18i	− 0.409420i	−0.978823	−	0.204710i	$-0.934375\pi$
$109$	− 9.28370e18i	− 0.409420i	0.978823	−	0.204710i	$-0.0656252\pi$
$110$	0	0
$111$	3.03649e19	1.12669
$112$	0	0
$113$	−9.46581e17	−0.0296423	−0.0148212	−	0.999890i	$-0.504718\pi$
$113$	−9.46581e17	−0.0296423	−0.0148212	+	0.999890i	$0.504718\pi$
$114$	0	0
$115$	− 2.49197e19i	− 0.660563i
$116$	0	0
$117$	− 5.60678e19i	− 1.26168i
$118$	0	0
$119$	−1.04082e20	−1.99380
$120$	0	0
$121$	1.66419e19	0.272108
$122$	0	0
$123$	9.65967e19i	1.35165i
$124$	0	0
$125$	− 8.96388e19i	− 1.07610i
$126$	0	0
$127$	3.90334e19	0.402997	0.201498	−	0.979489i	$-0.435419\pi$
$127$	3.90334e19	0.402997	0.201498	+	0.979489i	$0.435419\pi$
$128$	0	0
$129$	−2.17061e20	−1.93188
$130$	0	0
$131$	− 1.83175e19i	− 0.140860i	−0.997517	−	0.0704301i	$-0.977563\pi$
$131$	− 1.83175e19i	− 0.140860i	0.997517	−	0.0704301i	$-0.0224372\pi$
$132$	0	0
$133$	2.05993e19i	0.137171i
$134$	0	0
$135$	2.60315e19	0.150428
$136$	0	0
$137$	1.21220e20	0.609159	0.304579	−	0.952487i	$-0.401484\pi$
$137$	1.21220e20	0.609159	0.304579	+	0.952487i	$0.401484\pi$
$138$	0	0
$139$	2.85008e20i	1.24801i	0.781422	+	0.624003i	$0.214495\pi$
$139$	2.85008e20i	1.24801i	−0.781422	+	0.624003i	$0.785505\pi$
$140$	0	0
$141$	7.13311e20i	2.72708i
$142$	0	0
$143$	−2.82757e20	−0.945631
$144$	0	0
$145$	3.67242e20	1.07636
$146$	0	0
$147$	− 1.70226e19i	− 0.0438037i
$148$	0	0
$149$	3.46011e20i	0.783106i	0.920155	+	0.391553i	$0.128062\pi$
$149$	3.46011e20i	0.783106i	−0.920155	+	0.391553i	$0.871938\pi$
$150$	0	0
$151$	−2.26945e20	−0.452522	−0.226261	−	0.974067i	$-0.572650\pi$
$151$	−2.26945e20	−0.452522	−0.226261	+	0.974067i	$0.572650\pi$
$152$	0	0
$153$	−1.27087e21	−2.23631
$154$	0	0
$155$	− 1.28213e20i	− 0.199425i
$156$	0	0
$157$	− 7.69546e19i	− 0.105971i	−0.998595	−	0.0529856i	$-0.983126\pi$
$157$	− 7.69546e19i	− 0.105971i	0.998595	−	0.0529856i	$-0.0168737\pi$
$158$	0	0
$159$	−1.59569e20	−0.194838
$160$	0	0
$161$	−8.31234e20	−0.901317
$162$	0	0
$163$	− 2.35119e20i	− 0.226728i	−0.993554	−	0.113364i	$-0.963837\pi$
$163$	− 2.35119e20i	− 0.226728i	0.993554	−	0.113364i	$-0.0361627\pi$
$164$	0	0
$165$	− 1.08047e21i	− 0.927929i
$166$	0	0
$167$	−3.06448e20	−0.234720	−0.117360	−	0.993089i	$-0.537443\pi$
$167$	−3.06448e20	−0.234720	−0.117360	+	0.993089i	$0.537443\pi$
$168$	0	0
$169$	−3.34054e20	−0.228503
$170$	0	0
$171$	2.51522e20i	0.153856i
$172$	0	0
$173$	4.05558e20i	0.222134i	0.993813	+	0.111067i	$0.0354268\pi$
$173$	4.05558e20i	0.222134i	−0.993813	+	0.111067i	$0.964573\pi$
$174$	0	0
$175$	−9.23361e20	−0.453430
$176$	0	0
$177$	4.41470e21	1.94605
$178$	0	0
$179$	− 5.58910e20i	− 0.221431i	−0.993852	−	0.110715i	$-0.964686\pi$
$179$	− 5.58910e20i	− 0.221431i	0.993852	−	0.110715i	$-0.0353142\pi$
$180$	0	0
$181$	− 5.71923e20i	− 0.203888i	−0.994790	−	0.101944i	$-0.967494\pi$
$181$	− 5.71923e20i	− 0.203888i	0.994790	−	0.101944i	$-0.0325062\pi$
$182$	0	0
$183$	2.37509e21	0.762773
$184$	0	0
$185$	1.97854e21	0.573079
$186$	0	0
$187$	6.40915e21i	1.67612i
$188$	0	0
$189$	− 8.68319e20i	− 0.205254i
$190$	0	0
$191$	5.07893e21	1.08631	0.543157	−	0.839631i	$-0.317229\pi$
$191$	5.07893e21	1.08631	0.543157	+	0.839631i	$0.317229\pi$
$192$	0	0
$193$	−2.57982e21	−0.499798	−0.249899	−	0.968272i	$-0.580397\pi$
$193$	−2.57982e21	−0.499798	−0.249899	+	0.968272i	$0.580397\pi$
$194$	0	0
$195$	− 6.86275e21i	− 1.20551i
$196$	0	0
$197$	− 1.06709e22i	− 1.70126i	−0.525764	−	0.850630i	$-0.676221\pi$
$197$	− 1.06709e22i	− 1.70126i	0.525764	−	0.850630i	$-0.323779\pi$
$198$	0	0
$199$	9.52516e21	1.37965	0.689823	−	0.723978i	$-0.257688\pi$
$199$	9.52516e21	1.37965	0.689823	+	0.723978i	$0.257688\pi$
$200$	0	0
$201$	6.76903e21	0.891587
$202$	0	0
$203$	− 1.22499e22i	− 1.46865i
$204$	0	0
$205$	6.29414e21i	0.687503i
$206$	0	0
$207$	−1.01495e22	−1.01095
$208$	0	0
$209$	1.26846e21	0.115315
$210$	0	0
$211$	4.84082e21i	0.402009i	0.979590	+	0.201004i	$0.0644205\pi$
$211$	4.84082e21i	0.402009i	−0.979590	+	0.201004i	$0.935579\pi$
$212$	0	0
$213$	− 8.38472e21i	− 0.636623i
$214$	0	0
$215$	−1.41435e22	−0.982632
$216$	0	0
$217$	−4.27671e21	−0.272109
$218$	0	0
$219$	− 3.74417e22i	− 2.18342i
$220$	0	0
$221$	4.07086e22i	2.17751i
$222$	0	0
$223$	3.44558e22	1.69187	0.845933	−	0.533289i	$-0.179044\pi$
$223$	3.44558e22	1.69187	0.845933	+	0.533289i	$0.179044\pi$
$224$	0	0
$225$	−1.12744e22	−0.508582
$226$	0	0
$227$	2.02926e22i	0.841575i	0.907159	+	0.420787i	$0.138246\pi$
$227$	2.02926e22i	0.841575i	−0.907159	+	0.420787i	$0.861754\pi$
$228$	0	0
$229$	− 4.45214e22i	− 1.69876i	−0.527783	−	0.849379i	$-0.676976\pi$
$229$	− 4.45214e22i	− 1.69876i	0.527783	−	0.849379i	$-0.323024\pi$
$230$	0	0
$231$	−3.60406e22	−1.26613
$232$	0	0
$233$	2.24432e22	0.726447	0.363224	−	0.931702i	$-0.381676\pi$
$233$	2.24432e22	0.726447	0.363224	+	0.931702i	$0.381676\pi$
$234$	0	0
$235$	4.64786e22i	1.38710i
$236$	0	0
$237$	3.71159e22i	1.02200i
$238$	0	0
$239$	9.79425e21	0.248995	0.124498	−	0.992220i	$-0.460268\pi$
$239$	9.79425e21	0.248995	0.124498	+	0.992220i	$0.460268\pi$
$240$	0	0
$241$	−2.87487e22	−0.675237	−0.337618	−	0.941283i	$-0.609621\pi$
$241$	−2.87487e22	−0.675237	−0.337618	+	0.941283i	$0.609621\pi$
$242$	0	0
$243$	6.60551e22i	1.43432i
$244$	0	0
$245$	− 1.10917e21i	− 0.0222803i
$246$	0	0
$247$	8.05681e21	0.149810
$248$	0	0
$249$	−5.72714e22	−0.986380
$250$	0	0
$251$	9.38618e22i	1.49826i	0.662421	+	0.749132i	$0.269529\pi$
$251$	9.38618e22i	1.49826i	−0.662421	+	0.749132i	$0.730471\pi$
$252$	0	0
$253$	5.11855e22i	0.757708i
$254$	0	0
$255$	−1.55555e23	−2.13675
$256$	0	0
$257$	3.38399e22	0.431583	0.215792	−	0.976439i	$-0.430767\pi$
$257$	3.38399e22	0.431583	0.215792	+	0.976439i	$0.430767\pi$
$258$	0	0
$259$	− 6.59972e22i	− 0.781948i
$260$	0	0
$261$	− 1.49573e23i	− 1.64729i
$262$	0	0
$263$	8.72821e22	0.894016	0.447008	−	0.894530i	$-0.352489\pi$
$263$	8.72821e22	0.894016	0.447008	+	0.894530i	$0.352489\pi$
$264$	0	0
$265$	−1.03973e22	−0.0991027
$266$	0	0
$267$	− 2.28333e23i	− 2.02634i
$268$	0	0
$269$	1.71472e23i	1.41758i	0.705422	+	0.708788i	$0.250758\pi$
$269$	1.71472e23i	1.41758i	−0.705422	+	0.708788i	$0.749242\pi$
$270$	0	0
$271$	−8.52402e22	−0.656804	−0.328402	−	0.944538i	$-0.606510\pi$
$271$	−8.52402e22	−0.656804	−0.328402	+	0.944538i	$0.606510\pi$
$272$	0	0
$273$	−2.28917e23	−1.64487
$274$	0	0
$275$	5.68585e22i	0.381184i
$276$	0	0
$277$	4.37653e22i	0.273888i	0.990579	+	0.136944i	$0.0437280\pi$
$277$	4.37653e22i	0.273888i	−0.990579	+	0.136944i	$0.956272\pi$
$278$	0	0
$279$	−5.22196e22	−0.305206
$280$	0	0
$281$	−3.82071e21	−0.0208658	−0.0104329	−	0.999946i	$-0.503321\pi$
$281$	−3.82071e21	−0.0208658	−0.0104329	+	0.999946i	$0.503321\pi$
$282$	0	0
$283$	5.37010e22i	0.274165i	0.990560	+	0.137082i	$0.0437725\pi$
$283$	5.37010e22i	0.274165i	−0.990560	+	0.137082i	$0.956227\pi$
$284$	0	0
$285$	3.07865e22i	0.147006i
$286$	0	0
$287$	2.09950e23	0.938075
$288$	0	0
$289$	6.83655e23	2.85961
$290$	0	0
$291$	− 2.66110e23i	− 1.04250i
$292$	0	0
$293$	− 2.33878e23i	− 0.858513i	−0.903183	−	0.429256i	$-0.858776\pi$
$293$	− 2.33878e23i	− 0.858513i	0.903183	−	0.429256i	$-0.141224\pi$
$294$	0	0
$295$	2.87657e23	0.989842
$296$	0	0
$297$	−5.34691e22	−0.172551
$298$	0	0
$299$	3.25112e23i	0.984365i
$300$	0	0
$301$	4.71776e23i	1.34077i
$302$	0	0
$303$	6.89158e23	1.83914
$304$	0	0
$305$	1.54759e23	0.387977
$306$	0	0
$307$	− 6.48358e22i	− 0.152757i	−0.997079	−	0.0763784i	$-0.975664\pi$
$307$	− 6.48358e22i	− 0.152757i	0.997079	−	0.0763784i	$-0.0243357\pi$
$308$	0	0
$309$	− 9.55370e23i	− 2.11625i
$310$	0	0
$311$	−3.48265e23	−0.725581	−0.362791	−	0.931871i	$-0.618176\pi$
$311$	−3.48265e23	−0.725581	−0.362791	+	0.931871i	$0.618176\pi$
$312$	0	0
$313$	−2.29940e23	−0.450757	−0.225378	−	0.974271i	$-0.572362\pi$
$313$	−2.29940e23	−0.450757	−0.225378	+	0.974271i	$0.572362\pi$
$314$	0	0
$315$	− 4.65656e23i	− 0.859241i
$316$	0	0
$317$	− 6.29056e23i	− 1.09301i	−0.837454	−	0.546507i	$-0.815957\pi$
$317$	− 6.29056e23i	− 1.09301i	0.837454	−	0.546507i	$-0.184043\pi$
$318$	0	0
$319$	−7.54319e23	−1.23465
$320$	0	0
$321$	−3.52107e23	−0.543096
$322$	0	0
$323$	− 1.82621e23i	− 0.265537i
$324$	0	0
$325$	3.61145e23i	0.495210i
$326$	0	0
$327$	−4.62816e23	−0.598693
$328$	0	0
$329$	1.55036e24	1.89265
$330$	0	0
$331$	− 8.20208e20i	0 0.000945275i	−1.00000		0.000472638i	$-0.999850\pi$
$331$	− 8.20208e20i	0 0.000945275i	1.00000		0.000472638i	$-0.000150445\pi$
$332$	0	0
$333$	− 8.05839e23i	− 0.877058i
$334$	0	0
$335$	4.41063e23	0.453497
$336$	0	0
$337$	−9.44584e23	−0.917818	−0.458909	−	0.888483i	$-0.651760\pi$
$337$	−9.44584e23	−0.917818	−0.458909	+	0.888483i	$0.651760\pi$
$338$	0	0
$339$	4.71894e22i	0.0433459i
$340$	0	0
$341$	2.63350e23i	0.228753i
$342$	0	0
$343$	1.19811e24	0.984466
$344$	0	0
$345$	−1.24231e24	−0.965938
$346$	0	0
$347$	2.51038e24i	1.84761i	0.382865	+	0.923804i	$0.374938\pi$
$347$	2.51038e24i	1.84761i	−0.382865	+	0.923804i	$0.625062\pi$
$348$	0	0
$349$	1.66169e24i	1.15800i	0.815329	+	0.578999i	$0.196556\pi$
$349$	1.66169e24i	1.15800i	−0.815329	+	0.578999i	$0.803444\pi$
$350$	0	0
$351$	−3.39617e23	−0.224167
$352$	0	0
$353$	−1.89780e24	−1.18684	−0.593418	−	0.804895i	$-0.702222\pi$
$353$	−1.89780e24	−1.18684	−0.593418	+	0.804895i	$0.702222\pi$
$354$	0	0
$355$	− 5.46340e23i	− 0.323812i
$356$	0	0
$357$	5.18877e24i	2.91552i
$358$	0	0
$359$	−1.73809e23	−0.0926138	−0.0463069	−	0.998927i	$-0.514745\pi$
$359$	−1.73809e23	−0.0926138	−0.0463069	+	0.998927i	$0.514745\pi$
$360$	0	0
$361$	1.94228e24	0.981731
$362$	0	0
$363$	− 8.29639e23i	− 0.397902i
$364$	0	0
$365$	− 2.43966e24i	− 1.11057i
$366$	0	0
$367$	−2.37308e24	−1.02562	−0.512808	−	0.858503i	$-0.671395\pi$
$367$	−2.37308e24	−1.02562	−0.512808	+	0.858503i	$0.671395\pi$
$368$	0	0
$369$	2.56353e24	1.05218
$370$	0	0
$371$	3.46817e23i	0.135222i
$372$	0	0
$373$	2.60620e24i	0.965547i	0.875745	+	0.482773i	$0.160371\pi$
$373$	2.60620e24i	0.965547i	−0.875745	+	0.482773i	$0.839629\pi$
$374$	0	0
$375$	−4.46872e24	−1.57357
$376$	0	0
$377$	−4.79116e24	−1.60398
$378$	0	0
$379$	− 3.73330e24i	− 1.18856i	−0.804258	−	0.594280i	$-0.797437\pi$
$379$	− 3.73330e24i	− 1.18856i	0.804258	−	0.594280i	$-0.202563\pi$
$380$	0	0
$381$	− 1.94592e24i	− 0.589300i
$382$	0	0
$383$	−5.20583e23	−0.150003	−0.0750017	−	0.997183i	$-0.523896\pi$
$383$	−5.20583e23	−0.150003	−0.0750017	+	0.997183i	$0.523896\pi$
$384$	0	0
$385$	−2.34837e24	−0.644004
$386$	0	0
$387$	5.76048e24i	1.50385i
$388$	0	0
$389$	− 3.58050e24i	− 0.890066i	−0.895514	−	0.445033i	$-0.853192\pi$
$389$	− 3.58050e24i	− 0.890066i	0.895514	−	0.445033i	$-0.146808\pi$
$390$	0	0
$391$	7.36918e24	1.74478
$392$	0	0
$393$	−9.13172e23	−0.205979
$394$	0	0
$395$	2.41843e24i	0.519830i
$396$	0	0
$397$	− 6.94274e24i	− 1.42240i	−0.702991	−	0.711199i	$-0.748153\pi$
$397$	− 6.94274e24i	− 1.42240i	0.702991	−	0.711199i	$-0.251847\pi$
$398$	0	0
$399$	1.02693e24	0.200585
$400$	0	0
$401$	9.52158e23	0.177353	0.0886763	−	0.996060i	$-0.471736\pi$
$401$	9.52158e23	0.177353	0.0886763	+	0.996060i	$0.471736\pi$
$402$	0	0
$403$	1.67271e24i	0.297181i
$404$	0	0
$405$	3.69718e24i	0.626683i
$406$	0	0
$407$	−4.06395e24	−0.657358
$408$	0	0
$409$	2.62892e24	0.405887	0.202944	−	0.979190i	$-0.434949\pi$
$409$	2.62892e24	0.405887	0.202944	+	0.979190i	$0.434949\pi$
$410$	0	0
$411$	− 6.04315e24i	− 0.890770i
$412$	0	0
$413$	− 9.59521e24i	− 1.35061i
$414$	0	0
$415$	−3.73174e24	−0.501713
$416$	0	0
$417$	1.42084e25	1.82495
$418$	0	0
$419$	1.31213e24i	0.161044i	0.996753	+	0.0805219i	$0.0256587\pi$
$419$	1.31213e24i	0.161044i	−0.996753	+	0.0805219i	$0.974341\pi$
$420$	0	0
$421$	5.96864e24i	0.700157i	0.936720	+	0.350079i	$0.113845\pi$
$421$	5.96864e24i	0.700157i	−0.936720	+	0.350079i	$0.886155\pi$
$422$	0	0
$423$	1.89302e25	2.12286
$424$	0	0
$425$	8.18592e24	0.877754
$426$	0	0
$427$	− 5.16219e24i	− 0.529382i
$428$	0	0
$429$	1.40962e25i	1.38279i
$430$	0	0
$431$	8.91123e24	0.836380	0.418190	−	0.908360i	$-0.362665\pi$
$431$	8.91123e24	0.836380	0.418190	+	0.908360i	$0.362665\pi$
$432$	0	0
$433$	−6.03283e24	−0.541859	−0.270929	−	0.962599i	$-0.587331\pi$
$433$	−6.03283e24	−0.541859	−0.270929	+	0.962599i	$0.587331\pi$
$434$	0	0
$435$	− 1.83079e25i	− 1.57395i
$436$	0	0
$437$	− 1.45846e24i	− 0.120039i
$438$	0	0
$439$	−1.30807e25	−1.03090	−0.515451	−	0.856919i	$-0.672376\pi$
$439$	−1.30807e25	−1.03090	−0.515451	+	0.856919i	$0.672376\pi$
$440$	0	0
$441$	−4.51754e23	−0.0340985
$442$	0	0
$443$	5.83057e24i	0.421576i	0.977532	+	0.210788i	$0.0676029\pi$
$443$	5.83057e24i	0.421576i	−0.977532	+	0.210788i	$0.932397\pi$
$444$	0	0
$445$	− 1.48779e25i	− 1.03068i
$446$	0	0
$447$	1.72495e25	1.14513
$448$	0	0
$449$	3.10018e24	0.197263	0.0986317	−	0.995124i	$-0.468553\pi$
$449$	3.10018e24	0.197263	0.0986317	+	0.995124i	$0.468553\pi$
$450$	0	0
$451$	− 1.29282e25i	− 0.788610i
$452$	0	0
$453$	1.13138e25i	0.661721i
$454$	0	0
$455$	−1.49160e25	−0.836649
$456$	0	0
$457$	−2.62842e25	−1.41414	−0.707068	−	0.707146i	$-0.749982\pi$
$457$	−2.62842e25	−1.41414	−0.707068	+	0.707146i	$0.749982\pi$
$458$	0	0
$459$	7.69796e24i	0.397333i
$460$	0	0
$461$	− 1.14359e25i	− 0.566384i	−0.959063	−	0.283192i	$-0.908607\pi$
$461$	− 1.14359e25i	− 0.566384i	0.959063	−	0.283192i	$-0.0913934\pi$
$462$	0	0
$463$	−1.33811e25	−0.636024	−0.318012	−	0.948087i	$-0.603015\pi$
$463$	−1.33811e25	−0.636024	−0.318012	+	0.948087i	$0.603015\pi$
$464$	0	0
$465$	−6.39172e24	−0.291618
$466$	0	0
$467$	4.46496e24i	0.195572i	0.995207	+	0.0977861i	$0.0311761\pi$
$467$	4.46496e24i	0.195572i	−0.995207	+	0.0977861i	$0.968824\pi$
$468$	0	0
$469$	− 1.47123e25i	− 0.618782i
$470$	0	0
$471$	−3.83638e24	−0.154961
$472$	0	0
$473$	2.90509e25	1.12714
$474$	0	0
$475$	− 1.62011e24i	− 0.0603885i
$476$	0	0
$477$	4.23471e24i	0.151670i
$478$	0	0
$479$	−3.95808e25	−1.36238	−0.681188	−	0.732108i	$-0.738537\pi$
$479$	−3.95808e25	−1.36238	−0.681188	+	0.732108i	$0.738537\pi$
$480$	0	0
$481$	−2.58128e25	−0.853998
$482$	0	0
$483$	4.14391e25i	1.31799i
$484$	0	0
$485$	− 1.73394e25i	− 0.530260i
$486$	0	0
$487$	4.66024e25	1.37051	0.685256	−	0.728302i	$-0.259690\pi$
$487$	4.66024e25	1.37051	0.685256	+	0.728302i	$0.259690\pi$
$488$	0	0
$489$	−1.17213e25	−0.331544
$490$	0	0
$491$	4.13355e25i	1.12473i	0.826889	+	0.562365i	$0.190109\pi$
$491$	4.13355e25i	1.12473i	−0.826889	+	0.562365i	$0.809891\pi$
$492$	0	0
$493$	1.08599e26i	2.84303i
$494$	0	0
$495$	−2.86740e25	−0.722336
$496$	0	0
$497$	−1.82239e25	−0.441831
$498$	0	0
$499$	4.26378e25i	0.995038i	0.867453	+	0.497519i	$0.165755\pi$
$499$	4.26378e25i	0.995038i	−0.867453	+	0.497519i	$0.834245\pi$
$500$	0	0
$501$	1.52772e25i	0.343231i
$502$	0	0
$503$	3.12731e25	0.676512	0.338256	−	0.941054i	$-0.390163\pi$
$503$	3.12731e25	0.676512	0.338256	+	0.941054i	$0.390163\pi$
$504$	0	0
$505$	4.49048e25	0.935458
$506$	0	0
$507$	1.66534e25i	0.334140i
$508$	0	0
$509$	− 9.07982e24i	− 0.175492i	−0.996143	−	0.0877462i	$-0.972034\pi$
$509$	− 9.07982e24i	− 0.175492i	0.996143	−	0.0877462i	$-0.0279665\pi$
$510$	0	0
$511$	−8.13784e25	−1.51534
$512$	0	0
$513$	1.52353e24	0.0273361
$514$	0	0
$515$	− 6.22509e25i	− 1.07641i
$516$	0	0
$517$	− 9.54676e25i	− 1.59109i
$518$	0	0
$519$	2.02181e25	0.324825
$520$	0	0
$521$	−6.28755e25	−0.973920	−0.486960	−	0.873424i	$-0.661894\pi$
$521$	−6.28755e25	−0.973920	−0.486960	+	0.873424i	$0.661894\pi$
$522$	0	0
$523$	− 2.85949e25i	− 0.427093i	−0.976933	−	0.213546i	$-0.931499\pi$
$523$	− 2.85949e25i	− 0.427093i	0.976933	−	0.213546i	$-0.0685014\pi$
$524$	0	0
$525$	4.60319e25i	0.663048i
$526$	0	0
$527$	3.79146e25	0.526751
$528$	0	0
$529$	−1.57629e25	−0.211256
$530$	0	0
$531$	− 1.17160e26i	− 1.51488i
$532$	0	0
$533$	− 8.21155e25i	− 1.02451i
$534$	0	0
$535$	−2.29429e25	−0.276240
$536$	0	0
$537$	−2.78631e25	−0.323798
$538$	0	0
$539$	2.27826e24i	0.0255570i
$540$	0	0
$541$	1.29199e26i	1.39921i	0.714528	+	0.699607i	$0.246642\pi$
$541$	1.29199e26i	1.39921i	−0.714528	+	0.699607i	$0.753358\pi$
$542$	0	0
$543$	−2.85118e25	−0.298144
$544$	0	0
$545$	−3.01566e25	−0.304520
$546$	0	0
$547$	− 4.90707e25i	− 0.478567i	−0.970950	−	0.239284i	$-0.923087\pi$
$547$	− 4.90707e25i	− 0.478567i	0.970950	−	0.239284i	$-0.0769126\pi$
$548$	0	0
$549$	− 6.30314e25i	− 0.593772i
$550$	0	0
$551$	2.14933e25	0.195598
$552$	0	0
$553$	8.06703e25	0.709291
$554$	0	0
$555$	− 9.86354e25i	− 0.838011i
$556$	0	0
$557$	1.47557e25i	0.121153i	0.998164	+	0.0605766i	$0.0192939\pi$
$557$	1.47557e25i	0.121153i	−0.998164	+	0.0605766i	$0.980706\pi$
$558$	0	0
$559$	1.84521e26	1.46431
$560$	0	0
$561$	3.19512e26	2.45098
$562$	0	0
$563$	1.71235e26i	1.26988i	0.772561	+	0.634940i	$0.218975\pi$
$563$	1.71235e26i	1.26988i	−0.772561	+	0.634940i	$0.781025\pi$
$564$	0	0
$565$	3.07481e24i	0.0220475i
$566$	0	0
$567$	1.23325e26	0.855089
$568$	0	0
$569$	5.00277e25	0.335463	0.167731	−	0.985833i	$-0.446356\pi$
$569$	5.00277e25	0.335463	0.167731	+	0.985833i	$0.446356\pi$
$570$	0	0
$571$	− 7.56372e25i	− 0.490561i	−0.969452	−	0.245280i	$-0.921120\pi$
$571$	− 7.56372e25i	− 0.490561i	0.969452	−	0.245280i	$-0.0788800\pi$
$572$	0	0
$573$	− 2.53197e26i	− 1.58851i
$574$	0	0
$575$	6.53753e25	0.396798
$576$	0	0
$577$	9.40149e25	0.552111	0.276055	−	0.961142i	$-0.410973\pi$
$577$	9.40149e25	0.552111	0.276055	+	0.961142i	$0.410973\pi$
$578$	0	0
$579$	1.28610e26i	0.730852i
$580$	0	0
$581$	1.24478e26i	0.684571i
$582$	0	0
$583$	2.13562e25	0.113677
$584$	0	0
$585$	−1.82127e26	−0.938413
$586$	0	0
$587$	4.89889e25i	0.244363i	0.992508	+	0.122181i	$0.0389890\pi$
$587$	4.89889e25i	0.244363i	−0.992508	+	0.122181i	$0.961011\pi$
$588$	0	0
$589$	− 7.50382e24i	− 0.0362399i
$590$	0	0
$591$	−5.31970e26	−2.48775
$592$	0	0
$593$	2.09589e25	0.0949182	0.0474591	−	0.998873i	$-0.484888\pi$
$593$	2.09589e25	0.0949182	0.0474591	+	0.998873i	$0.484888\pi$
$594$	0	0
$595$	3.38095e26i	1.48295i
$596$	0	0
$597$	− 4.74853e26i	− 2.01745i
$598$	0	0
$599$	4.37283e26	1.79973	0.899866	−	0.436167i	$-0.143664\pi$
$599$	4.37283e26	1.79973	0.899866	+	0.436167i	$0.143664\pi$
$600$	0	0
$601$	−1.76961e26	−0.705621	−0.352810	−	0.935695i	$-0.614774\pi$
$601$	−1.76961e26	−0.705621	−0.352810	+	0.935695i	$0.614774\pi$
$602$	0	0
$603$	− 1.79640e26i	− 0.694046i
$604$	0	0
$605$	− 5.40584e25i	− 0.202389i
$606$	0	0
$607$	−2.20019e26	−0.798302	−0.399151	−	0.916885i	$-0.630695\pi$
$607$	−2.20019e26	−0.798302	−0.399151	+	0.916885i	$0.630695\pi$
$608$	0	0
$609$	−6.10687e26	−2.14760
$610$	0	0
$611$	− 6.06376e26i	− 2.06705i
$612$	0	0
$613$	5.09110e26i	1.68243i	0.540701	+	0.841215i	$0.318159\pi$
$613$	5.09110e26i	1.68243i	−0.540701	+	0.841215i	$0.681841\pi$
$614$	0	0
$615$	3.13779e26	1.00533
$616$	0	0
$617$	4.19716e26	1.30391	0.651953	−	0.758260i	$-0.273950\pi$
$617$	4.19716e26	1.30391	0.651953	+	0.758260i	$0.273950\pi$
$618$	0	0
$619$	2.09098e26i	0.629926i	0.949104	+	0.314963i	$0.101992\pi$
$619$	2.09098e26i	0.629926i	−0.949104	+	0.314963i	$0.898008\pi$
$620$	0	0
$621$	6.14783e25i	0.179619i
$622$	0	0
$623$	−4.96274e26	−1.40632
$624$	0	0
$625$	−1.28637e26	−0.353594
$626$	0	0
$627$	− 6.32359e25i	− 0.168625i
$628$	0	0
$629$	5.85088e26i	1.51370i
$630$	0	0
$631$	−5.75808e26	−1.44544	−0.722718	−	0.691143i	$-0.757107\pi$
$631$	−5.75808e26	−1.44544	−0.722718	+	0.691143i	$0.757107\pi$
$632$	0	0
$633$	2.41327e26	0.587855
$634$	0	0
$635$	− 1.26794e26i	− 0.299742i
$636$	0	0
$637$	1.44707e25i	0.0332020i
$638$	0	0
$639$	−2.22518e26	−0.495573
$640$	0	0
$641$	2.10914e26	0.455990	0.227995	−	0.973662i	$-0.426783\pi$
$641$	2.10914e26	0.455990	0.227995	+	0.973662i	$0.426783\pi$
$642$	0	0
$643$	7.02503e26i	1.47450i	0.675622	+	0.737249i	$0.263875\pi$
$643$	7.02503e26i	1.47450i	−0.675622	+	0.737249i	$0.736125\pi$
$644$	0	0
$645$	7.05088e26i	1.43690i
$646$	0	0
$647$	1.43888e26	0.284730	0.142365	−	0.989814i	$-0.454529\pi$
$647$	1.43888e26	0.284730	0.142365	+	0.989814i	$0.454529\pi$
$648$	0	0
$649$	−5.90851e26	−1.13541
$650$	0	0
$651$	2.13205e26i	0.397904i
$652$	0	0
$653$	5.08869e26i	0.922425i	0.887290	+	0.461213i	$0.152585\pi$
$653$	5.08869e26i	0.922425i	−0.887290	+	0.461213i	$0.847415\pi$
$654$	0	0
$655$	−5.95013e25	−0.104769
$656$	0	0
$657$	−9.93647e26	−1.69966
$658$	0	0
$659$	− 3.48001e26i	− 0.578321i	−0.957281	−	0.289160i	$-0.906624\pi$
$659$	− 3.48001e26i	− 0.578321i	0.957281	−	0.289160i	$-0.0933761\pi$
$660$	0	0
$661$	− 5.73431e26i	− 0.925907i	−0.886383	−	0.462953i	$-0.846790\pi$
$661$	− 5.73431e26i	− 0.925907i	0.886383	−	0.462953i	$-0.153210\pi$
$662$	0	0
$663$	2.02943e27	3.18416
$664$	0	0
$665$	6.69136e25	0.102026
$666$	0	0
$667$	8.67308e26i	1.28522i
$668$	0	0
$669$	− 1.71771e27i	− 2.47401i
$670$	0	0
$671$	−3.17876e26	−0.445034
$672$	0	0
$673$	9.22067e26	1.25493	0.627465	−	0.778645i	$-0.284093\pi$
$673$	9.22067e26	1.25493	0.627465	+	0.778645i	$0.284093\pi$
$674$	0	0
$675$	6.82920e25i	0.0903616i
$676$	0	0
$677$	4.92154e26i	0.633153i	0.948567	+	0.316576i	$0.102533\pi$
$677$	4.92154e26i	0.633153i	−0.948567	+	0.316576i	$0.897467\pi$
$678$	0	0
$679$	−5.78382e26	−0.723522
$680$	0	0
$681$	1.01164e27	1.23063
$682$	0	0
$683$	4.82762e26i	0.571132i	0.958359	+	0.285566i	$0.0921815\pi$
$683$	4.82762e26i	0.571132i	−0.958359	+	0.285566i	$0.907818\pi$
$684$	0	0
$685$	− 3.93765e26i	− 0.453082i
$686$	0	0
$687$	−2.21950e27	−2.48409
$688$	0	0
$689$	1.35647e26	0.147682
$690$	0	0
$691$	− 9.84631e26i	− 1.04287i	−0.853290	−	0.521437i	$-0.825396\pi$
$691$	− 9.84631e26i	− 1.04287i	0.853290	−	0.521437i	$-0.174604\pi$
$692$	0	0
$693$	9.56463e26i	0.985604i
$694$	0	0
$695$	9.25802e26	0.928245
$696$	0	0
$697$	−1.86128e27	−1.81594
$698$	0	0
$699$	− 1.11885e27i	− 1.06228i
$700$	0	0
$701$	− 1.70700e27i	− 1.57729i	−0.614846	−	0.788647i	$-0.710782\pi$
$701$	− 1.70700e27i	− 1.57729i	0.614846	−	0.788647i	$-0.289218\pi$
$702$	0	0
$703$	1.15797e26	0.104141
$704$	0	0
$705$	2.31707e27	2.02835
$706$	0	0
$707$	− 1.49786e27i	− 1.27640i
$708$	0	0
$709$	− 1.25967e27i	− 1.04500i	−0.852638	−	0.522502i	$-0.824999\pi$
$709$	− 1.25967e27i	− 1.04500i	0.852638	−	0.522502i	$-0.175001\pi$
$710$	0	0
$711$	9.85001e26	0.795565
$712$	0	0
$713$	3.02797e26	0.238123
$714$	0	0
$715$	9.18492e26i	0.703344i
$716$	0	0
$717$	− 4.88268e26i	− 0.364105i
$718$	0	0
$719$	1.11025e27	0.806303	0.403151	−	0.915133i	$-0.367915\pi$
$719$	1.11025e27	0.806303	0.403151	+	0.915133i	$0.367915\pi$
$720$	0	0
$721$	−2.07647e27	−1.46872
$722$	0	0
$723$	1.43319e27i	0.987396i
$724$	0	0
$725$	9.63433e26i	0.646563i
$726$	0	0
$727$	1.66901e27	1.09114	0.545571	−	0.838065i	$-0.316313\pi$
$727$	1.66901e27	1.09114	0.545571	+	0.838065i	$0.316313\pi$
$728$	0	0
$729$	1.97016e27	1.25484
$730$	0	0
$731$	− 4.18246e27i	− 2.59547i
$732$	0	0
$733$	1.71730e27i	1.03839i	0.854657	+	0.519193i	$0.173767\pi$
$733$	1.71730e27i	1.03839i	−0.854657	+	0.519193i	$0.826233\pi$
$734$	0	0
$735$	−5.52951e25	−0.0325804
$736$	0	0
$737$	−9.05948e26	−0.520190
$738$	0	0
$739$	1.15277e27i	0.645088i	0.946554	+	0.322544i	$0.104538\pi$
$739$	1.15277e27i	0.645088i	−0.946554	+	0.322544i	$0.895462\pi$
$740$	0	0
$741$	− 4.01652e26i	− 0.219067i
$742$	0	0
$743$	1.30371e27	0.693084	0.346542	−	0.938034i	$-0.387356\pi$
$743$	1.30371e27	0.693084	0.346542	+	0.938034i	$0.387356\pi$
$744$	0	0
$745$	1.12396e27	0.582461
$746$	0	0
$747$	1.51990e27i	0.767837i
$748$	0	0
$749$	7.65293e26i	0.376921i
$750$	0	0
$751$	−3.01810e27	−1.44929	−0.724643	−	0.689125i	$-0.757995\pi$
$751$	−3.01810e27	−1.44929	−0.724643	+	0.689125i	$0.757995\pi$
$752$	0	0
$753$	4.67925e27	2.19091
$754$	0	0
$755$	7.37195e26i	0.336578i
$756$	0	0
$757$	1.21517e27i	0.541034i	0.962715	+	0.270517i	$0.0871947\pi$
$757$	1.21517e27i	0.541034i	−0.962715	+	0.270517i	$0.912805\pi$
$758$	0	0
$759$	2.55172e27	1.10799
$760$	0	0
$761$	−2.83984e27	−1.20265	−0.601326	−	0.799004i	$-0.705361\pi$
$761$	−2.83984e27	−1.20265	−0.601326	+	0.799004i	$0.705361\pi$
$762$	0	0
$763$	1.00592e27i	0.415507i
$764$	0	0
$765$	4.12821e27i	1.66333i
$766$	0	0
$767$	−3.75287e27	−1.47505
$768$	0	0
$769$	−2.44029e27	−0.935711	−0.467856	−	0.883805i	$-0.654973\pi$
$769$	−2.44029e27	−0.935711	−0.467856	+	0.883805i	$0.654973\pi$
$770$	0	0
$771$	− 1.68700e27i	− 0.631102i
$772$	0	0
$773$	− 4.44946e27i	− 1.62406i	−0.583616	−	0.812030i	$-0.698363\pi$
$773$	− 4.44946e27i	− 1.62406i	0.583616	−	0.812030i	$-0.301637\pi$
$774$	0	0
$775$	3.36357e26	0.119794
$776$	0	0
$777$	−3.29013e27	−1.14344
$778$	0	0
$779$	3.68373e26i	0.124934i
$780$	0	0
$781$	1.12219e27i	0.371433i
$782$	0	0
$783$	−9.06003e26	−0.292680
$784$	0	0
$785$	−2.49974e26	−0.0788195
$786$	0	0
$787$	− 1.79506e27i	− 0.552482i	−0.961088	−	0.276241i	$-0.910911\pi$
$787$	− 1.79506e27i	− 0.552482i	0.961088	−	0.276241i	$-0.0890887\pi$
$788$	0	0
$789$	− 4.35123e27i	− 1.30732i
$790$	0	0
$791$	1.02565e26	0.0300830
$792$	0	0
$793$	−2.01903e27	−0.578160
$794$	0	0
$795$	5.18332e26i	0.144917i
$796$	0	0
$797$	− 2.30489e27i	− 0.629211i	−0.949223	−	0.314605i	$-0.898128\pi$
$797$	− 2.30489e27i	− 0.629211i	0.949223	−	0.314605i	$-0.101872\pi$
$798$	0	0
$799$	−1.37445e28	−3.66382
$800$	0	0
$801$	−6.05961e27	−1.57738
$802$	0	0
$803$	5.01109e27i	1.27390i
$804$	0	0
$805$	2.70013e27i	0.670384i
$806$	0	0
$807$	8.54830e27	2.07291
$808$	0	0
$809$	7.21315e27	1.70850	0.854248	−	0.519866i	$-0.174018\pi$
$809$	7.21315e27	1.70850	0.854248	+	0.519866i	$0.174018\pi$
$810$	0	0
$811$	6.07611e27i	1.40581i	0.711282	+	0.702907i	$0.248115\pi$
$811$	6.07611e27i	1.40581i	−0.711282	+	0.702907i	$0.751885\pi$
$812$	0	0
$813$	4.24944e27i	0.960442i
$814$	0	0
$815$	−7.63746e26	−0.168637
$816$	0	0
$817$	−8.27767e26	−0.178566
$818$	0	0
$819$	6.07511e27i	1.28043i
$820$	0	0
$821$	− 7.09411e27i	− 1.46096i	−0.682935	−	0.730479i	$-0.739297\pi$
$821$	− 7.09411e27i	− 1.46096i	0.682935	−	0.730479i	$-0.260703\pi$
$822$	0	0
$823$	1.84261e27	0.370796	0.185398	−	0.982663i	$-0.440642\pi$
$823$	1.84261e27	0.370796	0.185398	+	0.982663i	$0.440642\pi$
$824$	0	0
$825$	2.83454e27	0.557403
$826$	0	0
$827$	− 7.94719e27i	− 1.52725i	−0.645657	−	0.763627i	$-0.723416\pi$
$827$	− 7.94719e27i	− 1.52725i	0.645657	−	0.763627i	$-0.276584\pi$
$828$	0	0
$829$	1.02585e28i	1.92670i	0.268247	+	0.963350i	$0.413556\pi$
$829$	1.02585e28i	1.92670i	−0.268247	+	0.963350i	$0.586444\pi$
$830$	0	0
$831$	2.18181e27	0.400505
$832$	0	0
$833$	3.28001e26	0.0588501
$834$	0	0
$835$	9.95448e26i	0.174581i
$836$	0	0
$837$	3.16307e26i	0.0542271i
$838$	0	0
$839$	4.15109e27	0.695702	0.347851	−	0.937550i	$-0.386912\pi$
$839$	4.15109e27	0.695702	0.347851	+	0.937550i	$0.386912\pi$
$840$	0	0
$841$	−6.67823e27	−1.09421
$842$	0	0
$843$	1.90472e26i	0.0305119i
$844$	0	0
$845$	1.08512e27i	0.169957i
$846$	0	0
$847$	−1.80319e27	−0.276153
$848$	0	0
$849$	2.67713e27	0.400910
$850$	0	0
$851$	4.67269e27i	0.684284i
$852$	0	0
$853$	1.07973e28i	1.54633i	0.634207	+	0.773163i	$0.281327\pi$
$853$	1.07973e28i	1.54633i	−0.634207	+	0.773163i	$0.718673\pi$
$854$	0	0
$855$	8.17029e26	0.114435
$856$	0	0
$857$	1.39277e28	1.90792	0.953960	−	0.299934i	$-0.0969648\pi$
$857$	1.39277e28	1.90792	0.953960	+	0.299934i	$0.0969648\pi$
$858$	0	0
$859$	− 1.18324e28i	− 1.58540i	−0.609613	−	0.792699i	$-0.708675\pi$
$859$	− 1.18324e28i	− 1.58540i	0.609613	−	0.792699i	$-0.291325\pi$
$860$	0	0
$861$	− 1.04665e28i	− 1.37174i
$862$	0	0
$863$	1.25518e28	1.60917	0.804586	−	0.593836i	$-0.202387\pi$
$863$	1.25518e28	1.60917	0.804586	+	0.593836i	$0.202387\pi$
$864$	0	0
$865$	1.31739e27	0.165219
$866$	0	0
$867$	− 3.40819e28i	− 4.18160i
$868$	0	0
$869$	− 4.96749e27i	− 0.596278i
$870$	0	0
$871$	−5.75426e27	−0.675798
$872$	0	0
$873$	−7.06217e27	−0.811526
$874$	0	0
$875$	9.71262e27i	1.09209i
$876$	0	0
$877$	− 7.63666e26i	− 0.0840248i	−0.999117	−	0.0420124i	$-0.986623\pi$
$877$	− 7.63666e26i	− 0.0840248i	0.999117	−	0.0420124i	$-0.0133769\pi$
$878$	0	0
$879$	−1.16594e28	−1.25540
$880$	0	0
$881$	−3.52443e27	−0.371379	−0.185690	−	0.982608i	$-0.559452\pi$
$881$	−3.52443e27	−0.371379	−0.185690	+	0.982608i	$0.559452\pi$
$882$	0	0
$883$	8.12815e27i	0.838233i	0.907932	+	0.419117i	$0.137660\pi$
$883$	8.12815e27i	0.838233i	−0.907932	+	0.419117i	$0.862340\pi$
$884$	0	0
$885$	− 1.43404e28i	− 1.44744i
$886$	0	0
$887$	−1.97455e27	−0.195072	−0.0975359	−	0.995232i	$-0.531096\pi$
$887$	−1.97455e27	−0.195072	−0.0975359	+	0.995232i	$0.531096\pi$
$888$	0	0
$889$	−4.22939e27	−0.408988
$890$	0	0
$891$	− 7.59406e27i	− 0.718846i
$892$	0	0
$893$	2.72022e27i	0.252067i
$894$	0	0
$895$	−1.81553e27	−0.164697
$896$	0	0
$897$	1.62076e28	1.43943
$898$	0	0
$899$	4.46232e27i	0.388010i
$900$	0	0
$901$	− 3.07466e27i	− 0.261765i
$902$	0	0
$903$	2.35192e28	1.96060
$904$	0	0
$905$	−1.85780e27	−0.151648
$906$	0	0
$907$	2.34497e28i	1.87443i	0.348758	+	0.937213i	$0.386604\pi$
$907$	2.34497e28i	1.87443i	−0.348758	+	0.937213i	$0.613396\pi$
$908$	0	0
$909$	− 1.82892e28i	− 1.43165i
$910$	0	0
$911$	4.33758e27	0.332524	0.166262	−	0.986082i	$-0.446830\pi$
$911$	4.33758e27	0.332524	0.166262	+	0.986082i	$0.446830\pi$
$912$	0	0
$913$	7.66505e27	0.575496
$914$	0	0
$915$	− 7.71510e27i	− 0.567337i
$916$	0	0
$917$	1.98475e27i	0.142954i
$918$	0	0
$919$	1.56252e28	1.10237	0.551187	−	0.834382i	$-0.314175\pi$
$919$	1.56252e28	1.10237	0.551187	+	0.834382i	$0.314175\pi$
$920$	0	0
$921$	−3.23223e27	−0.223376
$922$	0	0
$923$	7.12774e27i	0.482543i
$924$	0	0
$925$	5.19058e27i	0.344246i
$926$	0	0
$927$	−2.53541e28	−1.64737
$928$	0	0
$929$	4.07635e27	0.259491	0.129746	−	0.991547i	$-0.458584\pi$
$929$	4.07635e27	0.259491	0.129746	+	0.991547i	$0.458584\pi$
$930$	0	0
$931$	− 6.49159e25i	− 0.00404883i
$932$	0	0
$933$	1.73619e28i	1.06101i
$934$	0	0
$935$	2.08191e28	1.24667
$936$	0	0
$937$	−1.43374e28	−0.841285	−0.420642	−	0.907227i	$-0.638195\pi$
$937$	−1.43374e28	−0.841285	−0.420642	+	0.907227i	$0.638195\pi$
$938$	0	0
$939$	1.14631e28i	0.659140i
$940$	0	0
$941$	− 2.14704e28i	− 1.20987i	−0.796275	−	0.604935i	$-0.793199\pi$
$941$	− 2.14704e28i	− 1.20987i	0.796275	−	0.604935i	$-0.206801\pi$
$942$	0	0
$943$	−1.48648e28	−0.820912
$944$	0	0
$945$	−2.82059e27	−0.152665
$946$	0	0
$947$	6.72605e27i	0.356809i	0.983957	+	0.178404i	$0.0570935\pi$
$947$	6.72605e27i	0.356809i	−0.983957	+	0.178404i	$0.942907\pi$
$948$	0	0
$949$	3.18287e28i	1.65497i
$950$	0	0
$951$	−3.13600e28	−1.59831
$952$	0	0
$953$	2.63734e28	1.31760	0.658801	−	0.752317i	$-0.271064\pi$
$953$	2.63734e28	1.31760	0.658801	+	0.752317i	$0.271064\pi$
$954$	0	0
$955$	− 1.64981e28i	− 0.807981i
$956$	0	0
$957$	3.76047e28i	1.80542i
$958$	0	0
$959$	−1.31346e28	−0.618215
$960$	0	0
$961$	−2.01128e28	−0.928110
$962$	0	0
$963$	9.34439e27i	0.422767i
$964$	0	0
$965$	8.38011e27i	0.371741i
$966$	0	0
$967$	−1.20363e28	−0.523530	−0.261765	−	0.965132i	$-0.584305\pi$
$967$	−1.20363e28	−0.523530	−0.261765	+	0.965132i	$0.584305\pi$
$968$	0	0
$969$	−9.10409e27	−0.388294
$970$	0	0
$971$	− 6.80649e27i	− 0.284669i	−0.989819	−	0.142335i	$-0.954539\pi$
$971$	− 6.80649e27i	− 0.284669i	0.989819	−	0.142335i	$-0.0454609\pi$
$972$	0	0
$973$	− 3.08814e28i	− 1.26656i
$974$	0	0
$975$	1.80040e28	0.724143
$976$	0	0
$977$	−2.87955e28	−1.13586	−0.567932	−	0.823076i	$-0.692256\pi$
$977$	−2.87955e28	−1.13586	−0.567932	+	0.823076i	$0.692256\pi$
$978$	0	0
$979$	3.05594e28i	1.18225i
$980$	0	0
$981$	1.22824e28i	0.466046i
$982$	0	0
$983$	−1.22714e28	−0.456705	−0.228352	−	0.973579i	$-0.573334\pi$
$983$	−1.22714e28	−0.456705	−0.228352	+	0.973579i	$0.573334\pi$
$984$	0	0
$985$	−3.46626e28	−1.26537
$986$	0	0
$987$	− 7.72893e28i	− 2.76762i
$988$	0	0
$989$	− 3.34024e28i	− 1.17331i
$990$	0	0
$991$	2.34043e28	0.806484	0.403242	−	0.915093i	$-0.367883\pi$
$991$	2.34043e28	0.806484	0.403242	+	0.915093i	$0.367883\pi$
$992$	0	0
$993$	−4.08894e25	−0.00138227
$994$	0	0
$995$	− 3.09409e28i	− 1.02616i
$996$	0	0
$997$	4.36013e28i	1.41872i	0.704848	+	0.709359i	$0.251015\pi$
$997$	4.36013e28i	1.41872i	−0.704848	+	0.709359i	$0.748985\pi$
$998$	0	0
$999$	−4.88117e27	−0.155830

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.20.b.a.17.3		18
4.3	odd	2		8.20.b.a.5.5	✓	18
8.3	odd	2		8.20.b.a.5.6	yes	18
8.5	even	2	inner	32.20.b.a.17.16		18
12.11	even	2		72.20.d.b.37.14		18
24.11	even	2		72.20.d.b.37.13		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.20.b.a.5.5	✓	18	4.3	odd	2
8.20.b.a.5.6	yes	18	8.3	odd	2
32.20.b.a.17.3		18	1.1	even	1	trivial
32.20.b.a.17.16		18	8.5	even	2	inner
72.20.d.b.37.13		18	24.11	even	2
72.20.d.b.37.14		18	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:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	32.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-49852.5i q^{3} -3.24834e6i q^{5} -1.08353e8 q^{7} -1.32301e9 q^{9} +O(q^{10})\)	\(q-49852.5i q^{3} -3.24834e6i q^{5} -1.08353e8 q^{7} -1.32301e9 q^{9} +6.67212e9i q^{11} +4.23789e10i q^{13} -1.61938e11 q^{15} +9.60587e11 q^{17} -1.90113e11i q^{19} +5.40167e12i q^{21} +7.67154e12 q^{23} +8.52179e12 q^{25} +8.01381e12i q^{27} +1.13055e14i q^{29} +3.94702e13 q^{31} +3.32622e14 q^{33} +3.51967e14i q^{35} +6.09095e14i q^{37} +2.11270e15 q^{39} -1.93765e15 q^{41} -4.35407e15i q^{43} +4.29759e15i q^{45} -1.43084e16 q^{47} +3.41459e14 q^{49} -4.78877e16i q^{51} -3.20081e15i q^{53} +2.16733e16 q^{55} -9.47763e15 q^{57} +8.85551e16i q^{59} +4.76424e16i q^{61} +1.43352e17 q^{63} +1.37661e17 q^{65} +1.35781e17i q^{67} -3.82446e17i q^{69} +1.68191e17 q^{71} +7.51049e17 q^{73} -4.24833e17i q^{75} -7.22944e17i q^{77} -7.44514e17 q^{79} -1.13818e18 q^{81} -1.14882e18i q^{83} -3.12031e18i q^{85} +5.63609e18 q^{87} +4.58017e18 q^{89} -4.59188e18i q^{91} -1.96769e18i q^{93} -6.17553e17 q^{95} +5.33795e18 q^{97} -8.82730e18i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 80707216 q^{7} - 6198727826 q^{9}+O(q^{10}) \) 18 * q + 80707216 * q^7 - 6198727826 * q^9	\( 18 q + 80707216 q^{7} - 6198727826 q^{9} - 156097960432 q^{15} + 14121426692 q^{17} - 2177121583952 q^{23} - 44414474211734 q^{25} - 428505770260416 q^{31} - 185380269683736 q^{33} - 942830575043152 q^{39} + 12\!\cdots\!24 q^{41}+ \cdots + 82\!\cdots\!20 q^{97}+O(q^{100}) \) 18 * q + 80707216 * q^7 - 6198727826 * q^9 - 156097960432 * q^15 + 14121426692 * q^17 - 2177121583952 * q^23 - 44414474211734 * q^25 - 428505770260416 * q^31 - 185380269683736 * q^33 - 942830575043152 * q^39 + 1200260021141524 * q^41 + 1134160959992928 * q^47 + 31570092936329058 * q^49 + 51757047863478736 * q^55 - 7740744140670520 * q^57 - 430168668665078416 * q^63 - 184007586011668640 * q^65 + 1599477989199799056 * q^71 - 93004298989809068 * q^73 - 1813975801025264480 * q^79 + 1801762045144484330 * q^81 + 13124960835754102512 * q^87 + 4470509771045743540 * q^89 - 23809730986995679856 * q^95 + 8246429789464117220 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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