LMFDB - Embedded newform 32.18.b.a.17.5

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			17.5
Root			$0.500000 + 3370.46i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.18.b.a.17.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-13481.8i q^{3} +1.59197e6i q^{5} +1.66055e7 q^{7} -5.26193e7 q^{9} +O(q^{10})$	$q-13481.8i q^{3} +1.59197e6i q^{5} +1.66055e7 q^{7} -5.26193e7 q^{9} +6.79322e8i q^{11} +3.13276e9i q^{13} +2.14627e10 q^{15} -1.26774e10 q^{17} +5.87947e9i q^{19} -2.23873e11i q^{21} -5.43400e11 q^{23} -1.77143e12 q^{25} -1.03164e12i q^{27} -2.82120e12i q^{29} -8.42117e11 q^{31} +9.15850e12 q^{33} +2.64355e13i q^{35} -6.17985e12i q^{37} +4.22353e13 q^{39} -6.38887e13 q^{41} +5.41782e13i q^{43} -8.37684e13i q^{45} +3.89159e13 q^{47} +4.31135e13 q^{49} +1.70914e14i q^{51} +5.92171e14i q^{53} -1.08146e15 q^{55} +7.92660e13 q^{57} -7.11422e14i q^{59} +9.01392e14i q^{61} -8.73773e14 q^{63} -4.98726e15 q^{65} +8.96950e14i q^{67} +7.32602e15i q^{69} -8.49830e13 q^{71} +1.01538e16 q^{73} +2.38821e16i q^{75} +1.12805e16i q^{77} -3.99449e15 q^{79} -2.07037e16 q^{81} -2.19564e15i q^{83} -2.01820e16i q^{85} -3.80349e16 q^{87} -6.74113e16 q^{89} +5.20212e16i q^{91} +1.13533e16i q^{93} -9.35994e15 q^{95} -1.67025e15 q^{97} -3.57455e16i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 13481.8i	− 1.18636i	−0.805068	−	0.593182i	$-0.797871\pi$
$3$	− 13481.8i	− 1.18636i	0.805068	−	0.593182i	$-0.202129\pi$
$4$	0	0
$5$	1.59197e6i	1.82259i	0.411750	+	0.911297i	$0.364918\pi$
$5$	1.59197e6i	1.82259i	−0.411750	+	0.911297i	$0.635082\pi$
$6$	0	0
$7$	1.66055e7	1.08873	0.544364	−	0.838849i	$-0.316771\pi$
$7$	1.66055e7	1.08873	0.544364	+	0.838849i	$0.316771\pi$
$8$	0	0
$9$	−5.26193e7	−0.407459
$10$	0	0
$11$	6.79322e8i	0.955516i	0.878492	+	0.477758i	$0.158550\pi$
$11$	6.79322e8i	0.955516i	−0.878492	+	0.477758i	$0.841450\pi$
$12$	0	0
$13$	3.13276e9i	1.06514i	0.846385	+	0.532572i	$0.178775\pi$
$13$	3.13276e9i	1.06514i	−0.846385	+	0.532572i	$0.821225\pi$
$14$	0	0
$15$	2.14627e10	2.16226
$16$	0	0
$17$	−1.26774e10	−0.440771	−0.220385	−	0.975413i	$-0.570732\pi$
$17$	−1.26774e10	−0.440771	−0.220385	+	0.975413i	$0.570732\pi$
$18$	0	0
$19$	5.87947e9i	0.0794205i	0.999211	+	0.0397103i	$0.0126435\pi$
$19$	5.87947e9i	0.0794205i	−0.999211	+	0.0397103i	$0.987357\pi$
$20$	0	0
$21$	− 2.23873e11i	− 1.29163i
$22$	0	0
$23$	−5.43400e11	−1.44688	−0.723441	−	0.690387i	$-0.757440\pi$
$23$	−5.43400e11	−1.44688	−0.723441	+	0.690387i	$0.757440\pi$
$24$	0	0
$25$	−1.77143e12	−2.32185
$26$	0	0
$27$	− 1.03164e12i	− 0.702969i
$28$	0	0
$29$	− 2.82120e12i	− 1.04725i	−0.851949	−	0.523625i	$-0.824579\pi$
$29$	− 2.82120e12i	− 1.04725i	0.851949	−	0.523625i	$-0.175421\pi$
$30$	0	0
$31$	−8.42117e11	−0.177336	−0.0886682	−	0.996061i	$-0.528261\pi$
$31$	−8.42117e11	−0.177336	−0.0886682	+	0.996061i	$0.528261\pi$
$32$	0	0
$33$	9.15850e12	1.13359
$34$	0	0
$35$	2.64355e13i	1.98431i
$36$	0	0
$37$	− 6.17985e12i	− 0.289243i	−0.989487	−	0.144622i	$-0.953804\pi$
$37$	− 6.17985e12i	− 0.289243i	0.989487	−	0.144622i	$-0.0461965\pi$
$38$	0	0
$39$	4.22353e13	1.26365
$40$	0	0
$41$	−6.38887e13	−1.24957	−0.624786	−	0.780796i	$-0.714814\pi$
$41$	−6.38887e13	−1.24957	−0.624786	+	0.780796i	$0.714814\pi$
$42$	0	0
$43$	5.41782e13i	0.706875i	0.935458	+	0.353438i	$0.114987\pi$
$43$	5.41782e13i	0.706875i	−0.935458	+	0.353438i	$0.885013\pi$
$44$	0	0
$45$	− 8.37684e13i	− 0.742633i
$46$	0	0
$47$	3.89159e13	0.238394	0.119197	−	0.992871i	$-0.461968\pi$
$47$	3.89159e13	0.238394	0.119197	+	0.992871i	$0.461968\pi$
$48$	0	0
$49$	4.31135e13	0.185330
$50$	0	0
$51$	1.70914e14i	0.522915i
$52$	0	0
$53$	5.92171e14i	1.30648i	0.757152	+	0.653239i	$0.226590\pi$
$53$	5.92171e14i	1.30648i	−0.757152	+	0.653239i	$0.773410\pi$
$54$	0	0
$55$	−1.08146e15	−1.74152
$56$	0	0
$57$	7.92660e13	0.0942216
$58$	0	0
$59$	− 7.11422e14i	− 0.630791i	−0.948960	−	0.315395i	$-0.897863\pi$
$59$	− 7.11422e14i	− 0.630791i	0.948960	−	0.315395i	$-0.102137\pi$
$60$	0	0
$61$	9.01392e14i	0.602019i	0.953621	+	0.301009i	$0.0973236\pi$
$61$	9.01392e14i	0.602019i	−0.953621	+	0.301009i	$0.902676\pi$
$62$	0	0
$63$	−8.73773e14	−0.443612
$64$	0	0
$65$	−4.98726e15	−1.94133
$66$	0	0
$67$	8.96950e14i	0.269856i	0.990855	+	0.134928i	$0.0430804\pi$
$67$	8.96950e14i	0.269856i	−0.990855	+	0.134928i	$0.956920\pi$
$68$	0	0
$69$	7.32602e15i	1.71653i
$70$	0	0
$71$	−8.49830e13	−0.0156184	−0.00780919	−	0.999970i	$-0.502486\pi$
$71$	−8.49830e13	−0.0156184	−0.00780919	+	0.999970i	$0.502486\pi$
$72$	0	0
$73$	1.01538e16	1.47361	0.736806	−	0.676104i	$-0.236333\pi$
$73$	1.01538e16	1.47361	0.736806	+	0.676104i	$0.236333\pi$
$74$	0	0
$75$	2.38821e16i	2.75456i
$76$	0	0
$77$	1.12805e16i	1.04030i
$78$	0	0
$79$	−3.99449e15	−0.296232	−0.148116	−	0.988970i	$-0.547321\pi$
$79$	−3.99449e15	−0.296232	−0.148116	+	0.988970i	$0.547321\pi$
$80$	0	0
$81$	−2.07037e16	−1.24144
$82$	0	0
$83$	− 2.19564e15i	− 0.107003i	−0.998568	−	0.0535016i	$-0.982962\pi$
$83$	− 2.19564e15i	− 0.107003i	0.998568	−	0.0535016i	$-0.0170382\pi$
$84$	0	0
$85$	− 2.01820e16i	− 0.803346i
$86$	0	0
$87$	−3.80349e16	−1.24242
$88$	0	0
$89$	−6.74113e16	−1.81517	−0.907586	−	0.419866i	$-0.862077\pi$
$89$	−6.74113e16	−1.81517	−0.907586	+	0.419866i	$0.862077\pi$
$90$	0	0
$91$	5.20212e16i	1.15965i
$92$	0	0
$93$	1.13533e16i	0.210386i
$94$	0	0
$95$	−9.35994e15	−0.144751
$96$	0	0
$97$	−1.67025e15	−0.0216383	−0.0108191	−	0.999941i	$-0.503444\pi$
$97$	−1.67025e15	−0.0216383	−0.0108191	+	0.999941i	$0.503444\pi$
$98$	0	0
$99$	− 3.57455e16i	− 0.389334i
$100$	0	0
$101$	9.20491e16i	0.845840i	0.906167	+	0.422920i	$0.138995\pi$
$101$	9.20491e16i	0.845840i	−0.906167	+	0.422920i	$0.861005\pi$
$102$	0	0
$103$	1.51701e17	1.17998	0.589988	−	0.807412i	$-0.299132\pi$
$103$	1.51701e17	1.17998	0.589988	+	0.807412i	$0.299132\pi$
$104$	0	0
$105$	3.56399e17	2.35411
$106$	0	0
$107$	4.02158e15i	0.0226274i	0.999936	+	0.0113137i	$0.00360134\pi$
$107$	4.02158e15i	0.0226274i	−0.999936	+	0.0113137i	$0.996399\pi$
$108$	0	0
$109$	6.68975e16i	0.321577i	0.986989	+	0.160788i	$0.0514037\pi$
$109$	6.68975e16i	0.321577i	−0.986989	+	0.160788i	$0.948596\pi$
$110$	0	0
$111$	−8.33156e16	−0.343148
$112$	0	0
$113$	−1.74001e17	−0.615721	−0.307860	−	0.951432i	$-0.599613\pi$
$113$	−1.74001e17	−0.615721	−0.307860	+	0.951432i	$0.599613\pi$
$114$	0	0
$115$	− 8.65076e17i	− 2.63708i
$116$	0	0
$117$	− 1.64844e17i	− 0.434003i
$118$	0	0
$119$	−2.10514e17	−0.479880
$120$	0	0
$121$	4.39685e16	0.0869893
$122$	0	0
$123$	8.61336e17i	1.48245i
$124$	0	0
$125$	− 1.60549e18i	− 2.40919i
$126$	0	0
$127$	9.21764e17	1.20861	0.604307	−	0.796751i	$-0.293450\pi$
$127$	9.21764e17	1.20861	0.604307	+	0.796751i	$0.293450\pi$
$128$	0	0
$129$	7.30421e17	0.838611
$130$	0	0
$131$	− 1.81435e18i	− 1.82774i	−0.406003	−	0.913872i	$-0.633078\pi$
$131$	− 1.81435e18i	− 1.82774i	0.406003	−	0.913872i	$-0.366922\pi$
$132$	0	0
$133$	9.76318e16i	0.0864674i
$134$	0	0
$135$	1.64234e18	1.28123
$136$	0	0
$137$	−2.37631e18	−1.63598	−0.817989	−	0.575234i	$-0.804911\pi$
$137$	−2.37631e18	−1.63598	−0.817989	+	0.575234i	$0.804911\pi$
$138$	0	0
$139$	2.37238e18i	1.44397i	0.691908	+	0.721986i	$0.256770\pi$
$139$	2.37238e18i	1.44397i	−0.691908	+	0.721986i	$0.743230\pi$
$140$	0	0
$141$	− 5.24657e17i	− 0.282822i
$142$	0	0
$143$	−2.12815e18	−1.01776
$144$	0	0
$145$	4.49126e18	1.90871
$146$	0	0
$147$	− 5.81248e17i	− 0.219869i
$148$	0	0
$149$	4.84759e17i	0.163472i	0.996654	+	0.0817359i	$0.0260464\pi$
$149$	4.84759e17i	0.163472i	−0.996654	+	0.0817359i	$0.973954\pi$
$150$	0	0
$151$	−3.48930e18	−1.05059	−0.525296	−	0.850920i	$-0.676045\pi$
$151$	−3.48930e18	−1.05059	−0.525296	+	0.850920i	$0.676045\pi$
$152$	0	0
$153$	6.67074e17	0.179596
$154$	0	0
$155$	− 1.34063e18i	− 0.323212i
$156$	0	0
$157$	5.04719e18i	1.09120i	0.838047	+	0.545598i	$0.183697\pi$
$157$	5.04719e18i	1.09120i	−0.838047	+	0.545598i	$0.816303\pi$
$158$	0	0
$159$	7.98354e18	1.54996
$160$	0	0
$161$	−9.02345e18	−1.57526
$162$	0	0
$163$	4.62747e18i	0.727359i	0.931524	+	0.363680i	$0.118480\pi$
$163$	4.62747e18i	0.727359i	−0.931524	+	0.363680i	$0.881520\pi$
$164$	0	0
$165$	1.45801e19i	2.06607i
$166$	0	0
$167$	−6.00884e18	−0.768600	−0.384300	−	0.923208i	$-0.625557\pi$
$167$	−6.00884e18	−0.768600	−0.384300	+	0.923208i	$0.625557\pi$
$168$	0	0
$169$	−1.16377e18	−0.134533
$170$	0	0
$171$	− 3.09374e17i	− 0.0323606i
$172$	0	0
$173$	− 1.31182e19i	− 1.24304i	−0.783400	−	0.621518i	$-0.786516\pi$
$173$	− 1.31182e19i	− 1.24304i	0.783400	−	0.621518i	$-0.213484\pi$
$174$	0	0
$175$	−2.94155e19	−2.52786
$176$	0	0
$177$	−9.59127e18	−0.748348
$178$	0	0
$179$	1.01351e19i	0.718749i	0.933193	+	0.359375i	$0.117010\pi$
$179$	1.01351e19i	0.718749i	−0.933193	+	0.359375i	$0.882990\pi$
$180$	0	0
$181$	5.37640e18i	0.346916i	0.984841	+	0.173458i	$0.0554940\pi$
$181$	5.37640e18i	0.346916i	−0.984841	+	0.173458i	$0.944506\pi$
$182$	0	0
$183$	1.21524e19	0.714213
$184$	0	0
$185$	9.83814e18	0.527173
$186$	0	0
$187$	− 8.61201e18i	− 0.421164i
$188$	0	0
$189$	− 1.71309e19i	− 0.765343i
$190$	0	0
$191$	3.15290e19	1.28803	0.644016	−	0.765012i	$-0.277267\pi$
$191$	3.15290e19	1.28803	0.644016	+	0.765012i	$0.277267\pi$
$192$	0	0
$193$	5.88587e18	0.220077	0.110038	−	0.993927i	$-0.464903\pi$
$193$	5.88587e18	0.220077	0.110038	+	0.993927i	$0.464903\pi$
$194$	0	0
$195$	6.72373e19i	2.30312i
$196$	0	0
$197$	4.95627e19i	1.55665i	0.627859	+	0.778327i	$0.283931\pi$
$197$	4.95627e19i	1.55665i	−0.627859	+	0.778327i	$0.716069\pi$
$198$	0	0
$199$	2.52157e19	0.726810	0.363405	−	0.931631i	$-0.381614\pi$
$199$	2.52157e19	0.726810	0.363405	+	0.931631i	$0.381614\pi$
$200$	0	0
$201$	1.20925e19	0.320148
$202$	0	0
$203$	− 4.68475e19i	− 1.14017i
$204$	0	0
$205$	− 1.01709e20i	− 2.27746i
$206$	0	0
$207$	2.85933e19	0.589545
$208$	0	0
$209$	−3.99405e18	−0.0758876
$210$	0	0
$211$	− 9.05451e19i	− 1.58659i	−0.608839	−	0.793294i	$-0.708364\pi$
$211$	− 9.05451e19i	− 1.58659i	0.608839	−	0.793294i	$-0.291636\pi$
$212$	0	0
$213$	1.14573e18i	0.0185291i
$214$	0	0
$215$	−8.62501e19	−1.28835
$216$	0	0
$217$	−1.39838e19	−0.193071
$218$	0	0
$219$	− 1.36891e20i	− 1.74824i
$220$	0	0
$221$	− 3.97151e19i	− 0.469485i
$222$	0	0
$223$	−1.01974e20	−1.11660	−0.558302	−	0.829637i	$-0.688547\pi$
$223$	−1.01974e20	−1.11660	−0.558302	+	0.829637i	$0.688547\pi$
$224$	0	0
$225$	9.32115e19	0.946058
$226$	0	0
$227$	4.34756e18i	0.0409285i	0.999791	+	0.0204643i	$0.00651443\pi$
$227$	4.34756e18i	0.0409285i	−0.999791	+	0.0204643i	$0.993486\pi$
$228$	0	0
$229$	5.56510e19i	0.486263i	0.969993	+	0.243132i	$0.0781747\pi$
$229$	5.56510e19i	0.486263i	−0.969993	+	0.243132i	$0.921825\pi$
$230$	0	0
$231$	1.52082e20	1.23417
$232$	0	0
$233$	8.39273e19	0.632963	0.316481	−	0.948599i	$-0.397499\pi$
$233$	8.39273e19	0.632963	0.316481	+	0.948599i	$0.397499\pi$
$234$	0	0
$235$	6.19529e19i	0.434495i
$236$	0	0
$237$	5.38530e19i	0.351438i
$238$	0	0
$239$	2.51863e20	1.53032	0.765158	−	0.643842i	$-0.222661\pi$
$239$	2.51863e20	1.53032	0.765158	+	0.643842i	$0.222661\pi$
$240$	0	0
$241$	−5.48650e19	−0.310563	−0.155282	−	0.987870i	$-0.549629\pi$
$241$	−5.48650e19	−0.310563	−0.155282	+	0.987870i	$0.549629\pi$
$242$	0	0
$243$	1.45897e20i	0.769826i
$244$	0	0
$245$	6.86353e19i	0.337782i
$246$	0	0
$247$	−1.84190e19	−0.0845943
$248$	0	0
$249$	−2.96012e19	−0.126945
$250$	0	0
$251$	2.36559e20i	0.947792i	0.880581	+	0.473896i	$0.157153\pi$
$251$	2.36559e20i	0.947792i	−0.880581	+	0.473896i	$0.842847\pi$
$252$	0	0
$253$	− 3.69143e20i	− 1.38252i
$254$	0	0
$255$	−2.72090e20	−0.953061
$256$	0	0
$257$	−1.56566e20	−0.513177	−0.256588	−	0.966521i	$-0.582598\pi$
$257$	−1.56566e20	−0.513177	−0.256588	+	0.966521i	$0.582598\pi$
$258$	0	0
$259$	− 1.02620e20i	− 0.314907i
$260$	0	0
$261$	1.48450e20i	0.426712i
$262$	0	0
$263$	−4.99732e20	−1.34621	−0.673106	−	0.739546i	$-0.735040\pi$
$263$	−4.99732e20	−1.34621	−0.673106	+	0.739546i	$0.735040\pi$
$264$	0	0
$265$	−9.42718e20	−2.38118
$266$	0	0
$267$	9.08827e20i	2.15345i
$268$	0	0
$269$	4.92718e20i	1.09573i	0.836566	+	0.547865i	$0.184559\pi$
$269$	4.92718e20i	1.09573i	−0.836566	+	0.547865i	$0.815441\pi$
$270$	0	0
$271$	1.88474e20	0.393561	0.196780	−	0.980448i	$-0.436951\pi$
$271$	1.88474e20	0.393561	0.196780	+	0.980448i	$0.436951\pi$
$272$	0	0
$273$	7.01340e20	1.37577
$274$	0	0
$275$	− 1.20337e21i	− 2.21856i
$276$	0	0
$277$	4.10225e20i	0.711122i	0.934653	+	0.355561i	$0.115710\pi$
$277$	4.10225e20i	0.711122i	−0.934653	+	0.355561i	$0.884290\pi$
$278$	0	0
$279$	4.43116e19	0.0722574
$280$	0	0
$281$	5.23813e20	0.803845	0.401922	−	0.915674i	$-0.368342\pi$
$281$	5.23813e20	0.803845	0.401922	+	0.915674i	$0.368342\pi$
$282$	0	0
$283$	8.12727e20i	1.17425i	0.809497	+	0.587124i	$0.199740\pi$
$283$	8.12727e20i	1.17425i	−0.809497	+	0.587124i	$0.800260\pi$
$284$	0	0
$285$	1.26189e20i	0.171728i
$286$	0	0
$287$	−1.06091e21	−1.36045
$288$	0	0
$289$	−6.66525e20	−0.805721
$290$	0	0
$291$	2.25181e19i	0.0256709i
$292$	0	0
$293$	− 1.33596e20i	− 0.143687i	−0.997416	−	0.0718436i	$-0.977112\pi$
$293$	− 1.33596e20i	− 0.143687i	0.997416	−	0.0718436i	$-0.0228882\pi$
$294$	0	0
$295$	1.13256e21	1.14968
$296$	0	0
$297$	7.00816e20	0.671698
$298$	0	0
$299$	− 1.70234e21i	− 1.54114i
$300$	0	0
$301$	8.99659e20i	0.769596i
$302$	0	0
$303$	1.24099e21	1.00347
$304$	0	0
$305$	−1.43499e21	−1.09724
$306$	0	0
$307$	1.20452e21i	0.871243i	0.900130	+	0.435622i	$0.143471\pi$
$307$	1.20452e21i	0.871243i	−0.900130	+	0.435622i	$0.856529\pi$
$308$	0	0
$309$	− 2.04521e21i	− 1.39988i
$310$	0	0
$311$	−1.25790e21	−0.815050	−0.407525	−	0.913194i	$-0.633608\pi$
$311$	−1.25790e21	−0.815050	−0.407525	+	0.913194i	$0.633608\pi$
$312$	0	0
$313$	2.89626e21	1.77709	0.888546	−	0.458787i	$-0.151716\pi$
$313$	2.89626e21	1.77709	0.888546	+	0.458787i	$0.151716\pi$
$314$	0	0
$315$	− 1.39102e21i	− 0.808525i
$316$	0	0
$317$	5.55170e20i	0.305789i	0.988242	+	0.152895i	$0.0488595\pi$
$317$	5.55170e20i	0.305789i	−0.988242	+	0.152895i	$0.951140\pi$
$318$	0	0
$319$	1.91650e21	1.00066
$320$	0	0
$321$	5.42182e19	0.0268443
$322$	0	0
$323$	− 7.45362e19i	− 0.0350063i
$324$	0	0
$325$	− 5.54946e21i	− 2.47310i
$326$	0	0
$327$	9.01900e20	0.381507
$328$	0	0
$329$	6.46219e20	0.259546
$330$	0	0
$331$	− 8.74883e20i	− 0.333743i	−0.985979	−	0.166871i	$-0.946634\pi$
$331$	− 8.74883e20i	− 0.333743i	0.985979	−	0.166871i	$-0.0533665\pi$
$332$	0	0
$333$	3.25180e20i	0.117855i
$334$	0	0
$335$	−1.42792e21	−0.491839
$336$	0	0
$337$	1.38557e21	0.453706	0.226853	−	0.973929i	$-0.427156\pi$
$337$	1.38557e21	0.453706	0.226853	+	0.973929i	$0.427156\pi$
$338$	0	0
$339$	2.34584e21i	0.730469i
$340$	0	0
$341$	− 5.72069e20i	− 0.169448i
$342$	0	0
$343$	−3.14703e21	−0.886954
$344$	0	0
$345$	−1.16628e22	−3.12853
$346$	0	0
$347$	− 3.85582e21i	− 0.984727i	−0.870390	−	0.492364i	$-0.836133\pi$
$347$	− 3.85582e21i	− 0.984727i	0.870390	−	0.492364i	$-0.163867\pi$
$348$	0	0
$349$	− 1.47067e20i	− 0.0357684i	−0.999840	−	0.0178842i	$-0.994307\pi$
$349$	− 1.47067e20i	− 0.0357684i	0.999840	−	0.0178842i	$-0.00569302\pi$
$350$	0	0
$351$	3.23188e21	0.748764
$352$	0	0
$353$	5.92536e21	1.30807	0.654033	−	0.756466i	$-0.273076\pi$
$353$	5.92536e21	1.30807	0.654033	+	0.756466i	$0.273076\pi$
$354$	0	0
$355$	− 1.35290e20i	− 0.0284660i
$356$	0	0
$357$	2.83812e21i	0.569312i
$358$	0	0
$359$	7.67655e21	1.46846	0.734232	−	0.678899i	$-0.237543\pi$
$359$	7.67655e21	1.46846	0.734232	+	0.678899i	$0.237543\pi$
$360$	0	0
$361$	5.44582e21	0.993692
$362$	0	0
$363$	− 5.92775e20i	− 0.103201i
$364$	0	0
$365$	1.61645e22i	2.68580i
$366$	0	0
$367$	−1.69701e21	−0.269168	−0.134584	−	0.990902i	$-0.542970\pi$
$367$	−1.69701e21	−0.269168	−0.134584	+	0.990902i	$0.542970\pi$
$368$	0	0
$369$	3.36178e21	0.509150
$370$	0	0
$371$	9.83331e21i	1.42240i
$372$	0	0
$373$	2.45745e21i	0.339594i	0.985479	+	0.169797i	$0.0543113\pi$
$373$	2.45745e21i	0.339594i	−0.985479	+	0.169797i	$0.945689\pi$
$374$	0	0
$375$	−2.16449e22	−2.85818
$376$	0	0
$377$	8.83814e21	1.11547
$378$	0	0
$379$	5.05987e21i	0.610529i	0.952268	+	0.305264i	$0.0987448\pi$
$379$	5.05987e21i	0.610529i	−0.952268	+	0.305264i	$0.901255\pi$
$380$	0	0
$381$	− 1.24271e22i	− 1.43386i
$382$	0	0
$383$	−2.16009e21	−0.238387	−0.119194	−	0.992871i	$-0.538031\pi$
$383$	−2.16009e21	−0.238387	−0.119194	+	0.992871i	$0.538031\pi$
$384$	0	0
$385$	−1.79582e22	−1.89604
$386$	0	0
$387$	− 2.85082e21i	− 0.288023i
$388$	0	0
$389$	1.57073e22i	1.51891i	0.650562	+	0.759453i	$0.274533\pi$
$389$	1.57073e22i	1.51891i	−0.650562	+	0.759453i	$0.725467\pi$
$390$	0	0
$391$	6.88887e21	0.637743
$392$	0	0
$393$	−2.44607e22	−2.16837
$394$	0	0
$395$	− 6.35911e21i	− 0.539910i
$396$	0	0
$397$	− 1.27818e21i	− 0.103961i	−0.998648	−	0.0519807i	$-0.983447\pi$
$397$	− 1.27818e21i	− 0.103961i	0.998648	−	0.0519807i	$-0.0165534\pi$
$398$	0	0
$399$	1.31625e21	0.102582
$400$	0	0
$401$	−1.13639e22	−0.848793	−0.424396	−	0.905477i	$-0.639514\pi$
$401$	−1.13639e22	−0.848793	−0.424396	+	0.905477i	$0.639514\pi$
$402$	0	0
$403$	− 2.63815e21i	− 0.188889i
$404$	0	0
$405$	− 3.29596e22i	− 2.26263i
$406$	0	0
$407$	4.19811e21	0.276377
$408$	0	0
$409$	1.19704e22	0.755891	0.377945	−	0.925828i	$-0.376631\pi$
$409$	1.19704e22	0.755891	0.377945	+	0.925828i	$0.376631\pi$
$410$	0	0
$411$	3.20369e22i	1.94087i
$412$	0	0
$413$	− 1.18136e22i	− 0.686760i
$414$	0	0
$415$	3.49539e21	0.195023
$416$	0	0
$417$	3.19840e22	1.71308
$418$	0	0
$419$	− 4.18757e21i	− 0.215349i	−0.994186	−	0.107674i	$-0.965660\pi$
$419$	− 4.18757e21i	− 0.215349i	0.994186	−	0.107674i	$-0.0343404\pi$
$420$	0	0
$421$	2.74877e22i	1.35750i	0.734368	+	0.678751i	$0.237479\pi$
$421$	2.74877e22i	1.35750i	−0.734368	+	0.678751i	$0.762521\pi$
$422$	0	0
$423$	−2.04773e21	−0.0971357
$424$	0	0
$425$	2.24571e22	1.02340
$426$	0	0
$427$	1.49681e22i	0.655435i
$428$	0	0
$429$	2.86914e22i	1.20744i
$430$	0	0
$431$	4.37127e22	1.76828	0.884139	−	0.467224i	$-0.154746\pi$
$431$	4.37127e22	1.76828	0.884139	+	0.467224i	$0.154746\pi$
$432$	0	0
$433$	6.06049e21	0.235701	0.117850	−	0.993031i	$-0.462400\pi$
$433$	6.06049e21	0.235701	0.117850	+	0.993031i	$0.462400\pi$
$434$	0	0
$435$	− 6.05504e22i	− 2.26443i
$436$	0	0
$437$	− 3.19490e21i	− 0.114912i
$438$	0	0
$439$	−1.64448e22	−0.568959	−0.284480	−	0.958682i	$-0.591821\pi$
$439$	−1.64448e22	−0.568959	−0.284480	+	0.958682i	$0.591821\pi$
$440$	0	0
$441$	−2.26860e21	−0.0755145
$442$	0	0
$443$	− 3.31744e22i	− 1.06260i	−0.847182	−	0.531302i	$-0.821703\pi$
$443$	− 3.31744e22i	− 1.06260i	0.847182	−	0.531302i	$-0.178297\pi$
$444$	0	0
$445$	− 1.07317e23i	− 3.30832i
$446$	0	0
$447$	6.53544e21	0.193937
$448$	0	0
$449$	−2.57130e22	−0.734615	−0.367307	−	0.930100i	$-0.619720\pi$
$449$	−2.57130e22	−0.734615	−0.367307	+	0.930100i	$0.619720\pi$
$450$	0	0
$451$	− 4.34010e22i	− 1.19399i
$452$	0	0
$453$	4.70421e22i	1.24638i
$454$	0	0
$455$	−8.28161e22	−2.11358
$456$	0	0
$457$	2.71668e22	0.667960	0.333980	−	0.942580i	$-0.391608\pi$
$457$	2.71668e22	0.667960	0.333980	+	0.942580i	$0.391608\pi$
$458$	0	0
$459$	1.30785e22i	0.309848i
$460$	0	0
$461$	1.14852e22i	0.262230i	0.991367	+	0.131115i	$0.0418557\pi$
$461$	1.14852e22i	0.262230i	−0.991367	+	0.131115i	$0.958144\pi$
$462$	0	0
$463$	4.81356e22	1.05932	0.529661	−	0.848209i	$-0.322319\pi$
$463$	4.81356e22	1.05932	0.529661	+	0.848209i	$0.322319\pi$
$464$	0	0
$465$	−1.80741e22	−0.383447
$466$	0	0
$467$	1.45571e22i	0.297770i	0.988855	+	0.148885i	$0.0475684\pi$
$467$	1.45571e22i	0.297770i	−0.988855	+	0.148885i	$0.952432\pi$
$468$	0	0
$469$	1.48943e22i	0.293800i
$470$	0	0
$471$	6.80453e22	1.29455
$472$	0	0
$473$	−3.68045e22	−0.675431
$474$	0	0
$475$	− 1.04151e22i	− 0.184402i
$476$	0	0
$477$	− 3.11596e22i	− 0.532336i
$478$	0	0
$479$	−4.36285e22	−0.719314	−0.359657	−	0.933085i	$-0.617106\pi$
$479$	−4.36285e22	−0.719314	−0.359657	+	0.933085i	$0.617106\pi$
$480$	0	0
$481$	1.93600e22	0.308086
$482$	0	0
$483$	1.21652e23i	1.86883i
$484$	0	0
$485$	− 2.65899e21i	− 0.0394378i
$486$	0	0
$487$	−1.77302e22	−0.253932	−0.126966	−	0.991907i	$-0.540524\pi$
$487$	−1.77302e22	−0.253932	−0.126966	+	0.991907i	$0.540524\pi$
$488$	0	0
$489$	6.23867e22	0.862913
$490$	0	0
$491$	3.92177e20i	0.00523949i	0.999997	+	0.00261975i	$0.000833892\pi$
$491$	3.92177e20i	0.00523949i	−0.999997	+	0.00261975i	$0.999166\pi$
$492$	0	0
$493$	3.57653e22i	0.461598i
$494$	0	0
$495$	5.69057e22	0.709597
$496$	0	0
$497$	−1.41119e21	−0.0170042
$498$	0	0
$499$	3.85769e22i	0.449234i	0.974447	+	0.224617i	$0.0721130\pi$
$499$	3.85769e22i	0.449234i	−0.974447	+	0.224617i	$0.927887\pi$
$500$	0	0
$501$	8.10101e22i	0.911840i
$502$	0	0
$503$	−1.70508e23	−1.85531	−0.927656	−	0.373435i	$-0.878180\pi$
$503$	−1.70508e23	−1.85531	−0.927656	+	0.373435i	$0.878180\pi$
$504$	0	0
$505$	−1.46539e23	−1.54162
$506$	0	0
$507$	1.56897e22i	0.159605i
$508$	0	0
$509$	− 1.15058e23i	− 1.13192i	−0.824434	−	0.565959i	$-0.808506\pi$
$509$	− 1.15058e23i	− 1.13192i	0.824434	−	0.565959i	$-0.191494\pi$
$510$	0	0
$511$	1.68609e23	1.60436
$512$	0	0
$513$	6.06550e21	0.0558302
$514$	0	0
$515$	2.41504e23i	2.15062i
$516$	0	0
$517$	2.64364e22i	0.227789i
$518$	0	0
$519$	−1.76858e23	−1.47469
$520$	0	0
$521$	−1.62883e23	−1.31448	−0.657241	−	0.753680i	$-0.728277\pi$
$521$	−1.62883e23	−1.31448	−0.657241	+	0.753680i	$0.728277\pi$
$522$	0	0
$523$	− 2.36470e23i	− 1.84719i	−0.383366	−	0.923596i	$-0.625235\pi$
$523$	− 2.36470e23i	− 1.84719i	0.383366	−	0.923596i	$-0.374765\pi$
$524$	0	0
$525$	3.96575e23i	2.99897i
$526$	0	0
$527$	1.06758e22	0.0781648
$528$	0	0
$529$	1.54233e23	1.09346
$530$	0	0
$531$	3.74346e22i	0.257022i
$532$	0	0
$533$	− 2.00148e23i	− 1.33098i
$534$	0	0
$535$	−6.40224e21	−0.0412406
$536$	0	0
$537$	1.36640e23	0.852698
$538$	0	0
$539$	2.92879e22i	0.177086i
$540$	0	0
$541$	− 1.05868e23i	− 0.620281i	−0.950691	−	0.310141i	$-0.899624\pi$
$541$	− 1.05868e23i	− 0.620281i	0.950691	−	0.310141i	$-0.100376\pi$
$542$	0	0
$543$	7.24837e22	0.411568
$544$	0	0
$545$	−1.06499e23	−0.586104
$546$	0	0
$547$	− 2.89118e23i	− 1.54235i	−0.636625	−	0.771174i	$-0.719670\pi$
$547$	− 2.89118e23i	− 1.54235i	0.636625	−	0.771174i	$-0.280330\pi$
$548$	0	0
$549$	− 4.74307e22i	− 0.245298i
$550$	0	0
$551$	1.65872e22	0.0831732
$552$	0	0
$553$	−6.63307e22	−0.322516
$554$	0	0
$555$	− 1.32636e23i	− 0.625419i
$556$	0	0
$557$	3.41183e23i	1.56034i	0.625570	+	0.780168i	$0.284866\pi$
$557$	3.41183e23i	1.56034i	−0.625570	+	0.780168i	$0.715134\pi$
$558$	0	0
$559$	−1.69727e23	−0.752925
$560$	0	0
$561$	−1.16106e23	−0.499653
$562$	0	0
$563$	− 1.44889e23i	− 0.604941i	−0.953159	−	0.302471i	$-0.902189\pi$
$563$	− 1.44889e23i	− 0.604941i	0.953159	−	0.302471i	$-0.0978114\pi$
$564$	0	0
$565$	− 2.77004e23i	− 1.12221i
$566$	0	0
$567$	−3.43795e23	−1.35159
$568$	0	0
$569$	4.55076e23	1.73632	0.868160	−	0.496285i	$-0.165303\pi$
$569$	4.55076e23	1.73632	0.868160	+	0.496285i	$0.165303\pi$
$570$	0	0
$571$	2.17154e23i	0.804193i	0.915597	+	0.402097i	$0.131718\pi$
$571$	2.17154e23i	0.804193i	−0.915597	+	0.402097i	$0.868282\pi$
$572$	0	0
$573$	− 4.25068e23i	− 1.52807i
$574$	0	0
$575$	9.62594e23	3.35944
$576$	0	0
$577$	2.28537e23	0.774394	0.387197	−	0.921997i	$-0.373443\pi$
$577$	2.28537e23	0.774394	0.387197	+	0.921997i	$0.373443\pi$
$578$	0	0
$579$	− 7.93523e22i	− 0.261091i
$580$	0	0
$581$	− 3.64597e22i	− 0.116497i
$582$	0	0
$583$	−4.02275e23	−1.24836
$584$	0	0
$585$	2.62426e23	0.791011
$586$	0	0
$587$	5.49823e23i	1.60990i	0.593342	+	0.804951i	$0.297808\pi$
$587$	5.49823e23i	1.60990i	−0.593342	+	0.804951i	$0.702192\pi$
$588$	0	0
$589$	− 4.95120e21i	− 0.0140842i
$590$	0	0
$591$	6.68196e23	1.84676
$592$	0	0
$593$	−4.92814e23	−1.32348	−0.661741	−	0.749732i	$-0.730182\pi$
$593$	−4.92814e23	−1.32348	−0.661741	+	0.749732i	$0.730182\pi$
$594$	0	0
$595$	− 3.35133e23i	− 0.874626i
$596$	0	0
$597$	− 3.39954e23i	− 0.862261i
$598$	0	0
$599$	4.19943e23	1.03529	0.517646	−	0.855595i	$-0.326809\pi$
$599$	4.19943e23	1.03529	0.517646	+	0.855595i	$0.326809\pi$
$600$	0	0
$601$	2.16713e23	0.519341	0.259671	−	0.965697i	$-0.416386\pi$
$601$	2.16713e23	0.519341	0.259671	+	0.965697i	$0.416386\pi$
$602$	0	0
$603$	− 4.71969e22i	− 0.109955i
$604$	0	0
$605$	6.99965e22i	0.158546i
$606$	0	0
$607$	4.33551e23	0.954852	0.477426	−	0.878672i	$-0.341570\pi$
$607$	4.33551e23	0.954852	0.477426	+	0.878672i	$0.341570\pi$
$608$	0	0
$609$	−6.31590e23	−1.35266
$610$	0	0
$611$	1.21914e23i	0.253924i
$612$	0	0
$613$	− 2.71131e23i	− 0.549244i	−0.961552	−	0.274622i	$-0.911447\pi$
$613$	− 2.71131e23i	− 0.549244i	0.961552	−	0.274622i	$-0.0885528\pi$
$614$	0	0
$615$	−1.37122e24	−2.70190
$616$	0	0
$617$	−5.91051e23	−1.13292	−0.566462	−	0.824088i	$-0.691688\pi$
$617$	−5.91051e23	−1.13292	−0.566462	+	0.824088i	$0.691688\pi$
$618$	0	0
$619$	− 9.44262e23i	− 1.76085i	−0.474187	−	0.880424i	$-0.657258\pi$
$619$	− 9.44262e23i	− 1.76085i	0.474187	−	0.880424i	$-0.342742\pi$
$620$	0	0
$621$	5.60593e23i	1.01711i
$622$	0	0
$623$	−1.11940e24	−1.97623
$624$	0	0
$625$	1.20439e24	2.06913
$626$	0	0
$627$	5.38471e22i	0.0900303i
$628$	0	0
$629$	7.83442e22i	0.127490i
$630$	0	0
$631$	−3.35285e23	−0.531086	−0.265543	−	0.964099i	$-0.585551\pi$
$631$	−3.35285e23	−0.531086	−0.265543	+	0.964099i	$0.585551\pi$
$632$	0	0
$633$	−1.22071e24	−1.88227
$634$	0	0
$635$	1.46742e24i	2.20281i
$636$	0	0
$637$	1.35064e23i	0.197403i
$638$	0	0
$639$	4.47175e21	0.00636385
$640$	0	0
$641$	1.02554e24	1.42122	0.710609	−	0.703587i	$-0.248419\pi$
$641$	1.02554e24	1.42122	0.710609	+	0.703587i	$0.248419\pi$
$642$	0	0
$643$	− 1.00641e24i	− 1.35826i	−0.734017	−	0.679131i	$-0.762357\pi$
$643$	− 1.00641e24i	− 1.35826i	0.734017	−	0.679131i	$-0.237643\pi$
$644$	0	0
$645$	1.16281e24i	1.52845i
$646$	0	0
$647$	−3.80957e23	−0.487742	−0.243871	−	0.969808i	$-0.578417\pi$
$647$	−3.80957e23	−0.487742	−0.243871	+	0.969808i	$0.578417\pi$
$648$	0	0
$649$	4.83285e23	0.602731
$650$	0	0
$651$	1.88527e23i	0.229053i
$652$	0	0
$653$	4.03993e23i	0.478202i	0.970995	+	0.239101i	$0.0768527\pi$
$653$	4.03993e23i	0.478202i	−0.970995	+	0.239101i	$0.923147\pi$
$654$	0	0
$655$	2.88839e24	3.33123
$656$	0	0
$657$	−5.34285e23	−0.600437
$658$	0	0
$659$	3.09111e23i	0.338523i	0.985571	+	0.169262i	$0.0541383\pi$
$659$	3.09111e23i	0.338523i	−0.985571	+	0.169262i	$0.945862\pi$
$660$	0	0
$661$	4.95209e23i	0.528537i	0.964449	+	0.264269i	$0.0851306\pi$
$661$	4.95209e23i	0.528537i	−0.964449	+	0.264269i	$0.914869\pi$
$662$	0	0
$663$	−5.35432e23	−0.556980
$664$	0	0
$665$	−1.55427e23	−0.157595
$666$	0	0
$667$	1.53304e24i	1.51525i
$668$	0	0
$669$	1.37480e24i	1.32470i
$670$	0	0
$671$	−6.12336e23	−0.575239
$672$	0	0
$673$	−7.41478e23	−0.679157	−0.339579	−	0.940578i	$-0.610284\pi$
$673$	−7.41478e23	−0.679157	−0.339579	+	0.940578i	$0.610284\pi$
$674$	0	0
$675$	1.82748e24i	1.63219i
$676$	0	0
$677$	1.13904e22i	0.00992051i	0.999988	+	0.00496025i	$0.00157890\pi$
$677$	1.13904e22i	0.00992051i	−0.999988	+	0.00496025i	$0.998421\pi$
$678$	0	0
$679$	−2.77355e22	−0.0235582
$680$	0	0
$681$	5.86131e22	0.0485561
$682$	0	0
$683$	− 2.41181e24i	− 1.94880i	−0.224828	−	0.974398i	$-0.572182\pi$
$683$	− 2.41181e24i	− 1.94880i	0.224828	−	0.974398i	$-0.427818\pi$
$684$	0	0
$685$	− 3.78301e24i	− 2.98172i
$686$	0	0
$687$	7.50277e23	0.576885
$688$	0	0
$689$	−1.85513e24	−1.39159
$690$	0	0
$691$	3.17702e23i	0.232518i	0.993219	+	0.116259i	$0.0370902\pi$
$691$	3.17702e23i	0.232518i	−0.993219	+	0.116259i	$0.962910\pi$
$692$	0	0
$693$	− 5.93573e23i	− 0.423879i
$694$	0	0
$695$	−3.77676e24	−2.63177
$696$	0	0
$697$	8.09940e23	0.550775
$698$	0	0
$699$	− 1.13149e24i	− 0.750924i
$700$	0	0
$701$	− 3.50256e23i	− 0.226873i	−0.993545	−	0.113436i	$-0.963814\pi$
$701$	− 3.50256e23i	− 0.226873i	0.993545	−	0.113436i	$-0.0361858\pi$
$702$	0	0
$703$	3.63342e22	0.0229718
$704$	0	0
$705$	8.35238e23	0.515469
$706$	0	0
$707$	1.52853e24i	0.920890i
$708$	0	0
$709$	− 3.69006e23i	− 0.217041i	−0.994094	−	0.108520i	$-0.965389\pi$
$709$	− 3.69006e23i	− 0.217041i	0.994094	−	0.108520i	$-0.0346112\pi$
$710$	0	0
$711$	2.10188e23	0.120702
$712$	0	0
$713$	4.57606e23	0.256585
$714$	0	0
$715$	− 3.38796e24i	− 1.85497i
$716$	0	0
$717$	− 3.39557e24i	− 1.81551i
$718$	0	0
$719$	3.63965e24	1.90048	0.950242	−	0.311512i	$-0.100835\pi$
$719$	3.63965e24	1.90048	0.950242	+	0.311512i	$0.100835\pi$
$720$	0	0
$721$	2.51909e24	1.28467
$722$	0	0
$723$	7.39680e23i	0.368441i
$724$	0	0
$725$	4.99755e24i	2.43156i
$726$	0	0
$727$	2.92373e24	1.38962	0.694808	−	0.719196i	$-0.255490\pi$
$727$	2.92373e24	1.38962	0.694808	+	0.719196i	$0.255490\pi$
$728$	0	0
$729$	−7.06719e23	−0.328143
$730$	0	0
$731$	− 6.86837e23i	− 0.311570i
$732$	0	0
$733$	8.20457e22i	0.0363640i	0.999835	+	0.0181820i	$0.00578783\pi$
$733$	8.20457e22i	0.0363640i	−0.999835	+	0.0181820i	$0.994212\pi$
$734$	0	0
$735$	9.25329e23	0.400732
$736$	0	0
$737$	−6.09318e23	−0.257852
$738$	0	0
$739$	3.42591e22i	0.0141677i	0.999975	+	0.00708383i	$0.00225487\pi$
$739$	3.42591e22i	0.0141677i	−0.999975	+	0.00708383i	$0.997745\pi$
$740$	0	0
$741$	2.48321e23i	0.100360i
$742$	0	0
$743$	−1.12234e24	−0.443324	−0.221662	−	0.975124i	$-0.571148\pi$
$743$	−1.12234e24	−0.443324	−0.221662	+	0.975124i	$0.571148\pi$
$744$	0	0
$745$	−7.71722e23	−0.297943
$746$	0	0
$747$	1.15533e23i	0.0435994i
$748$	0	0
$749$	6.67805e22i	0.0246351i
$750$	0	0
$751$	−5.53313e22	−0.0199540	−0.00997702	−	0.999950i	$-0.503176\pi$
$751$	−5.53313e22	−0.0199540	−0.00997702	+	0.999950i	$0.503176\pi$
$752$	0	0
$753$	3.18925e24	1.12443
$754$	0	0
$755$	− 5.55486e24i	− 1.91480i
$756$	0	0
$757$	− 4.11071e24i	− 1.38548i	−0.721185	−	0.692742i	$-0.756402\pi$
$757$	− 4.11071e24i	− 1.38548i	0.721185	−	0.692742i	$-0.243598\pi$
$758$	0	0
$759$	−4.97673e24	−1.64017
$760$	0	0
$761$	5.58495e24	1.79991	0.899954	−	0.435986i	$-0.143600\pi$
$761$	5.58495e24	1.79991	0.899954	+	0.435986i	$0.143600\pi$
$762$	0	0
$763$	1.11087e24i	0.350110i
$764$	0	0
$765$	1.06196e24i	0.327331i
$766$	0	0
$767$	2.22872e24	0.671884
$768$	0	0
$769$	−3.49045e24	−1.02922	−0.514609	−	0.857425i	$-0.672063\pi$
$769$	−3.49045e24	−1.02922	−0.514609	+	0.857425i	$0.672063\pi$
$770$	0	0
$771$	2.11080e24i	0.608814i
$772$	0	0
$773$	5.74685e23i	0.162145i	0.996708	+	0.0810726i	$0.0258346\pi$
$773$	5.74685e23i	0.162145i	−0.996708	+	0.0810726i	$0.974165\pi$
$774$	0	0
$775$	1.49175e24	0.411748
$776$	0	0
$777$	−1.38350e24	−0.373595
$778$	0	0
$779$	− 3.75632e23i	− 0.0992417i
$780$	0	0
$781$	− 5.77308e22i	− 0.0149236i
$782$	0	0
$783$	−2.91046e24	−0.736185
$784$	0	0
$785$	−8.03497e24	−1.98881
$786$	0	0
$787$	− 5.04324e24i	− 1.22159i	−0.791790	−	0.610794i	$-0.790850\pi$
$787$	− 5.04324e24i	− 1.22159i	0.791790	−	0.610794i	$-0.209150\pi$
$788$	0	0
$789$	6.73730e24i	1.59710i
$790$	0	0
$791$	−2.88937e24	−0.670353
$792$	0	0
$793$	−2.82384e24	−0.641237
$794$	0	0
$795$	1.27096e25i	2.82494i
$796$	0	0
$797$	5.78429e24i	1.25850i	0.777201	+	0.629252i	$0.216639\pi$
$797$	5.78429e24i	1.25850i	−0.777201	+	0.629252i	$0.783361\pi$
$798$	0	0
$799$	−4.93350e23	−0.105077
$800$	0	0
$801$	3.54714e24	0.739608
$802$	0	0
$803$	6.89768e24i	1.40806i
$804$	0	0
$805$	− 1.43651e25i	− 2.87106i
$806$	0	0
$807$	6.64273e24	1.29994
$808$	0	0
$809$	5.81148e24	1.11359	0.556793	−	0.830651i	$-0.312031\pi$
$809$	5.81148e24	1.11359	0.556793	+	0.830651i	$0.312031\pi$
$810$	0	0
$811$	− 3.15834e23i	− 0.0592627i	−0.999561	−	0.0296314i	$-0.990567\pi$
$811$	− 3.15834e23i	− 0.0592627i	0.999561	−	0.0296314i	$-0.00943334\pi$
$812$	0	0
$813$	− 2.54097e24i	− 0.466906i
$814$	0	0
$815$	−7.36679e24	−1.32568
$816$	0	0
$817$	−3.18539e23	−0.0561404
$818$	0	0
$819$	− 2.73732e24i	− 0.472511i
$820$	0	0
$821$	3.64320e24i	0.615980i	0.951390	+	0.307990i	$0.0996563\pi$
$821$	3.64320e24i	0.615980i	−0.951390	+	0.307990i	$0.900344\pi$
$822$	0	0
$823$	−2.26743e24	−0.375522	−0.187761	−	0.982215i	$-0.560123\pi$
$823$	−2.26743e24	−0.375522	−0.187761	+	0.982215i	$0.560123\pi$
$824$	0	0
$825$	−1.62236e25	−2.63202
$826$	0	0
$827$	1.05054e25i	1.66962i	0.550539	+	0.834809i	$0.314422\pi$
$827$	1.05054e25i	1.66962i	−0.550539	+	0.834809i	$0.685578\pi$
$828$	0	0
$829$	8.95945e24i	1.39498i	0.716595	+	0.697489i	$0.245700\pi$
$829$	8.95945e24i	1.39498i	−0.716595	+	0.697489i	$0.754300\pi$
$830$	0	0
$831$	5.53058e24	0.843649
$832$	0	0
$833$	−5.46565e23	−0.0816882
$834$	0	0
$835$	− 9.56589e24i	− 1.40085i
$836$	0	0
$837$	8.68762e23i	0.124662i
$838$	0	0
$839$	7.86032e22	0.0110526	0.00552629	−	0.999985i	$-0.498241\pi$
$839$	7.86032e22	0.0110526	0.00552629	+	0.999985i	$0.498241\pi$
$840$	0	0
$841$	−7.02012e23	−0.0967339
$842$	0	0
$843$	− 7.06195e24i	− 0.953652i
$844$	0	0
$845$	− 1.85268e24i	− 0.245199i
$846$	0	0
$847$	7.30120e23	0.0947077
$848$	0	0
$849$	1.09570e25	1.39309
$850$	0	0
$851$	3.35813e24i	0.418501i
$852$	0	0
$853$	4.09750e24i	0.500555i	0.968174	+	0.250277i	$0.0805218\pi$
$853$	4.09750e24i	0.500555i	−0.968174	+	0.250277i	$0.919478\pi$
$854$	0	0
$855$	4.92514e23	0.0589803
$856$	0	0
$857$	2.30079e24	0.270110	0.135055	−	0.990838i	$-0.456879\pi$
$857$	2.30079e24	0.270110	0.135055	+	0.990838i	$0.456879\pi$
$858$	0	0
$859$	6.99290e24i	0.804851i	0.915453	+	0.402426i	$0.131833\pi$
$859$	6.99290e24i	0.804851i	−0.915453	+	0.402426i	$0.868167\pi$
$860$	0	0
$861$	1.43030e25i	1.61398i
$862$	0	0
$863$	−3.06286e24	−0.338871	−0.169436	−	0.985541i	$-0.554194\pi$
$863$	−3.06286e24	−0.338871	−0.169436	+	0.985541i	$0.554194\pi$
$864$	0	0
$865$	2.08839e25	2.26555
$866$	0	0
$867$	8.98597e24i	0.955878i
$868$	0	0
$869$	− 2.71355e24i	− 0.283054i
$870$	0	0
$871$	−2.80993e24	−0.287436
$872$	0	0
$873$	8.78876e22	0.00881671
$874$	0	0
$875$	− 2.66600e25i	− 2.62296i
$876$	0	0
$877$	− 1.21076e25i	− 1.16832i	−0.811637	−	0.584162i	$-0.801423\pi$
$877$	− 1.21076e25i	− 1.16832i	0.811637	−	0.584162i	$-0.198577\pi$
$878$	0	0
$879$	−1.80112e24	−0.170465
$880$	0	0
$881$	−1.51486e25	−1.40629	−0.703147	−	0.711045i	$-0.748222\pi$
$881$	−1.51486e25	−1.40629	−0.703147	+	0.711045i	$0.748222\pi$
$882$	0	0
$883$	− 1.55372e25i	− 1.41484i	−0.706795	−	0.707418i	$-0.749860\pi$
$883$	− 1.55372e25i	− 1.41484i	0.706795	−	0.707418i	$-0.250140\pi$
$884$	0	0
$885$	− 1.52690e25i	− 1.36393i
$886$	0	0
$887$	1.56055e25	1.36750	0.683750	−	0.729716i	$-0.260348\pi$
$887$	1.56055e25	1.36750	0.683750	+	0.729716i	$0.260348\pi$
$888$	0	0
$889$	1.53064e25	1.31585
$890$	0	0
$891$	− 1.40645e25i	− 1.18621i
$892$	0	0
$893$	2.28805e23i	0.0189334i
$894$	0	0
$895$	−1.61348e25	−1.30999
$896$	0	0
$897$	−2.29507e25	−1.82835
$898$	0	0
$899$	2.37578e24i	0.185716i
$900$	0	0
$901$	− 7.50716e24i	− 0.575857i
$902$	0	0
$903$	1.21290e25	0.913020
$904$	0	0
$905$	−8.55907e24	−0.632286
$906$	0	0
$907$	1.36420e25i	0.989048i	0.869164	+	0.494524i	$0.164658\pi$
$907$	1.36420e25i	0.989048i	−0.869164	+	0.494524i	$0.835342\pi$
$908$	0	0
$909$	− 4.84356e24i	− 0.344645i
$910$	0	0
$911$	−3.50650e24	−0.244888	−0.122444	−	0.992475i	$-0.539073\pi$
$911$	−3.50650e24	−0.244888	−0.122444	+	0.992475i	$0.539073\pi$
$912$	0	0
$913$	1.49155e24	0.102243
$914$	0	0
$915$	1.93463e25i	1.30172i
$916$	0	0
$917$	− 3.01283e25i	− 1.98992i
$918$	0	0
$919$	1.32969e25	0.862123	0.431061	−	0.902323i	$-0.358139\pi$
$919$	1.32969e25	0.862123	0.431061	+	0.902323i	$0.358139\pi$
$920$	0	0
$921$	1.62392e25	1.03361
$922$	0	0
$923$	− 2.66231e23i	− 0.0166358i
$924$	0	0
$925$	1.09472e25i	0.671579i
$926$	0	0
$927$	−7.98243e24	−0.480792
$928$	0	0
$929$	−7.82134e24	−0.462538	−0.231269	−	0.972890i	$-0.574288\pi$
$929$	−7.82134e24	−0.462538	−0.231269	+	0.972890i	$0.574288\pi$
$930$	0	0
$931$	2.53484e23i	0.0147190i
$932$	0	0
$933$	1.69588e25i	0.966946i
$934$	0	0
$935$	1.37101e25	0.767610
$936$	0	0
$937$	−2.58422e23	−0.0142083	−0.00710417	−	0.999975i	$-0.502261\pi$
$937$	−2.58422e23	−0.0142083	−0.00710417	+	0.999975i	$0.502261\pi$
$938$	0	0
$939$	− 3.90468e25i	− 2.10828i
$940$	0	0
$941$	− 2.37973e24i	− 0.126187i	−0.998008	−	0.0630936i	$-0.979903\pi$
$941$	− 2.37973e24i	− 0.126187i	0.998008	−	0.0630936i	$-0.0200967\pi$
$942$	0	0
$943$	3.47171e25	1.80798
$944$	0	0
$945$	2.72719e25	1.39491
$946$	0	0
$947$	3.48762e25i	1.75208i	0.482237	+	0.876041i	$0.339824\pi$
$947$	3.48762e25i	1.75208i	−0.482237	+	0.876041i	$0.660176\pi$
$948$	0	0
$949$	3.18093e25i	1.56961i
$950$	0	0
$951$	7.48470e24	0.362777
$952$	0	0
$953$	2.89815e25	1.37985	0.689924	−	0.723882i	$-0.257644\pi$
$953$	2.89815e25	1.37985	0.689924	+	0.723882i	$0.257644\pi$
$954$	0	0
$955$	5.01932e25i	2.34756i
$956$	0	0
$957$	− 2.58379e25i	− 1.18715i
$958$	0	0
$959$	−3.94598e25	−1.78114
$960$	0	0
$961$	−2.18410e25	−0.968552
$962$	0	0
$963$	− 2.11613e23i	− 0.00921974i
$964$	0	0
$965$	9.37014e24i	0.401110i
$966$	0	0
$967$	−2.14679e25	−0.902953	−0.451476	−	0.892283i	$-0.649103\pi$
$967$	−2.14679e25	−0.902953	−0.451476	+	0.892283i	$0.649103\pi$
$968$	0	0
$969$	−1.00488e24	−0.0415302
$970$	0	0
$971$	− 2.38828e25i	− 0.969887i	−0.874546	−	0.484943i	$-0.838840\pi$
$971$	− 2.38828e25i	− 0.969887i	0.874546	−	0.484943i	$-0.161160\pi$
$972$	0	0
$973$	3.93947e25i	1.57209i
$974$	0	0
$975$	−7.48169e25	−2.93400
$976$	0	0
$977$	−6.22623e24	−0.239950	−0.119975	−	0.992777i	$-0.538282\pi$
$977$	−6.22623e24	−0.239950	−0.119975	+	0.992777i	$0.538282\pi$
$978$	0	0
$979$	− 4.57940e25i	− 1.73443i
$980$	0	0
$981$	− 3.52010e24i	− 0.131029i
$982$	0	0
$983$	4.13991e25	1.51456	0.757278	−	0.653092i	$-0.226529\pi$
$983$	4.13991e25	1.51456	0.757278	+	0.653092i	$0.226529\pi$
$984$	0	0
$985$	−7.89024e25	−2.83715
$986$	0	0
$987$	− 8.71221e24i	− 0.307916i
$988$	0	0
$989$	− 2.94404e25i	− 1.02276i
$990$	0	0
$991$	−1.74916e25	−0.597315	−0.298657	−	0.954360i	$-0.596539\pi$
$991$	−1.74916e25	−0.597315	−0.298657	+	0.954360i	$0.596539\pi$
$992$	0	0
$993$	−1.17950e25	−0.395940
$994$	0	0
$995$	4.01427e25i	1.32468i
$996$	0	0
$997$	4.25273e25i	1.37962i	0.723991	+	0.689810i	$0.242306\pi$
$997$	4.25273e25i	1.37962i	−0.723991	+	0.689810i	$0.757694\pi$
$998$	0	0
$999$	−6.37538e24	−0.203329

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.18.b.a.17.5		16
4.3	odd	2		8.18.b.a.5.5	✓	16
8.3	odd	2		8.18.b.a.5.6	yes	16
8.5	even	2	inner	32.18.b.a.17.12		16
12.11	even	2		72.18.d.b.37.12		16
24.11	even	2		72.18.d.b.37.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.18.b.a.5.5	✓	16	4.3	odd	2
8.18.b.a.5.6	yes	16	8.3	odd	2
32.18.b.a.17.5		16	1.1	even	1	trivial
32.18.b.a.17.12		16	8.5	even	2	inner
72.18.d.b.37.11		16	24.11	even	2
72.18.d.b.37.12		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)

\(f(q)\)	\(=\)	\(q-13481.8i q^{3} +1.59197e6i q^{5} +1.66055e7 q^{7} -5.26193e7 q^{9} +O(q^{10})\)	\(q-13481.8i q^{3} +1.59197e6i q^{5} +1.66055e7 q^{7} -5.26193e7 q^{9} +6.79322e8i q^{11} +3.13276e9i q^{13} +2.14627e10 q^{15} -1.26774e10 q^{17} +5.87947e9i q^{19} -2.23873e11i q^{21} -5.43400e11 q^{23} -1.77143e12 q^{25} -1.03164e12i q^{27} -2.82120e12i q^{29} -8.42117e11 q^{31} +9.15850e12 q^{33} +2.64355e13i q^{35} -6.17985e12i q^{37} +4.22353e13 q^{39} -6.38887e13 q^{41} +5.41782e13i q^{43} -8.37684e13i q^{45} +3.89159e13 q^{47} +4.31135e13 q^{49} +1.70914e14i q^{51} +5.92171e14i q^{53} -1.08146e15 q^{55} +7.92660e13 q^{57} -7.11422e14i q^{59} +9.01392e14i q^{61} -8.73773e14 q^{63} -4.98726e15 q^{65} +8.96950e14i q^{67} +7.32602e15i q^{69} -8.49830e13 q^{71} +1.01538e16 q^{73} +2.38821e16i q^{75} +1.12805e16i q^{77} -3.99449e15 q^{79} -2.07037e16 q^{81} -2.19564e15i q^{83} -2.01820e16i q^{85} -3.80349e16 q^{87} -6.74113e16 q^{89} +5.20212e16i q^{91} +1.13533e16i q^{93} -9.35994e15 q^{95} -1.67025e15 q^{97} -3.57455e16i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 11529600 q^{7} - 602654096 q^{9}+O(q^{10}) \) 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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