LMFDB - Embedded newform 32.16.b.a.17.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [32,16,Mod(17,32)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("32.17"); S:= CuspForms(chi, 16); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(32, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 16, names="a")

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	32.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$45.6619216320$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 7 x^{13} + 8071283 x^{12} - 48427607 x^{11} + 24279249501785 x^{10} - 121395803589361 x^{9} + \cdots + 29\!\cdots\!76 $ x^14 - 7x^13 + 8071283x^12 - 48427607x^11 + 24279249501785x^10 - 121395803589361x^9 + 33741121406602393257x^8 - 134963757252120740253x^7 + 22086556619849459465129634x^6 - 66259197486907857815169956x^5 + 6303156156087301477255966489160x^4 - 12606201880336248915418374398320x^3 + 622010974073555190418422676033290592x^2 - 622004670983658199381193153162550208*x + 29066150105813428852057441233286606976
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{182}\cdot 3^{6}\cdot 5^{4}\cdot 31^{2} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.5
Root			$0.500000 + 595.041i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.16.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)

$f(q)$	$=$	$q-2380.16i q^{3} -9943.39i q^{5} -1.24365e6 q^{7} +8.68373e6 q^{9} +6.55864e7i q^{11} -1.91929e8i q^{13} -2.36669e7 q^{15} +1.98739e9 q^{17} +8.13478e8i q^{19} +2.96009e9i q^{21} -8.30671e8 q^{23} +3.04187e10 q^{25} -5.48214e10i q^{27} +5.26318e10i q^{29} -2.31580e11 q^{31} +1.56106e11 q^{33} +1.23661e10i q^{35} -5.88641e11i q^{37} -4.56822e11 q^{39} +2.88417e11 q^{41} -1.77551e12i q^{43} -8.63457e10i q^{45} +6.45781e12 q^{47} -3.20090e12 q^{49} -4.73032e12i q^{51} -1.55187e13i q^{53} +6.52151e11 q^{55} +1.93621e12 q^{57} -1.28387e13i q^{59} -3.34293e13i q^{61} -1.07995e13 q^{63} -1.90843e12 q^{65} -8.71055e13i q^{67} +1.97713e12i q^{69} -1.73139e13 q^{71} +1.25837e14 q^{73} -7.24015e13i q^{75} -8.15665e13i q^{77} +4.05635e12 q^{79} -5.88183e12 q^{81} +2.09495e14i q^{83} -1.97614e13i q^{85} +1.25272e14 q^{87} -3.43643e14 q^{89} +2.38693e14i q^{91} +5.51198e14i q^{93} +8.08872e12 q^{95} -7.44295e14 q^{97} +5.69535e14i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 1647088 q^{7} - 57395630 q^{9} - 712135312 q^{15} + 728554812 q^{17} + 35548816080 q^{23} - 75899954794 q^{25} + 105758138816 q^{31} - 150458001384 q^{33} + 2251546247120 q^{39} - 53229185940 q^{41}+ \cdots - 672574291859236 q^{97}+O(q^{100}) $ 14 * q + 1647088 * q^7 - 57395630 * q^9 - 712135312 * q^15 + 728554812 * q^17 + 35548816080 * q^23 - 75899954794 * q^25 + 105758138816 * q^31 - 150458001384 * q^33 + 2251546247120 * q^39 - 53229185940 * q^41 - 12527998446432 * q^47 + 8427385380990 * q^49 + 30557833792176 * q^55 + 18277230892472 * q^57 - 36142362113776 * q^63 + 5437123965600 * q^65 + 173249927708016 * q^71 - 182057837882196 * q^73 + 294370273271392 * q^79 + 256903428263798 * q^81 - 1493385390675312 * q^87 + 314213951649228 * q^89 + 851301047679984 * q^95 - 672574291859236 * q^97

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
$3$	− 2380.16i	− 0.628344i	−0.949366	−	0.314172i	$-0.898273\pi$
$3$	− 2380.16i	− 0.628344i	0.949366	−	0.314172i	$-0.101727\pi$
$4$	0	0
$5$	− 9943.39i	− 0.0569193i	−0.999595	−	0.0284596i	$-0.990940\pi$
$5$	− 9943.39i	− 0.0569193i	0.999595	−	0.0284596i	$-0.00906021\pi$
$6$	0	0
$7$	−1.24365e6	−0.570772	−0.285386	−	0.958413i	$-0.592122\pi$
$7$	−1.24365e6	−0.570772	−0.285386	+	0.958413i	$0.592122\pi$
$8$	0	0
$9$	8.68373e6	0.605184
$10$	0	0
$11$	6.55864e7i	1.01477i	0.861719	+	0.507386i	$0.169388\pi$
$11$	6.55864e7i	1.01477i	−0.861719	+	0.507386i	$0.830612\pi$
$12$	0	0
$13$	− 1.91929e8i	− 0.848332i	−0.905585	−	0.424166i	$-0.860567\pi$
$13$	− 1.91929e8i	− 0.848332i	0.905585	−	0.424166i	$-0.139433\pi$
$14$	0	0
$15$	−2.36669e7	−0.0357649
$16$	0	0
$17$	1.98739e9	1.17467	0.587336	−	0.809343i	$-0.300177\pi$
$17$	1.98739e9	1.17467	0.587336	+	0.809343i	$0.300177\pi$
$18$	0	0
$19$	8.13478e8i	0.208782i	0.994536	+	0.104391i	$0.0332894\pi$
$19$	8.13478e8i	0.208782i	−0.994536	+	0.104391i	$0.966711\pi$
$20$	0	0
$21$	2.96009e9i	0.358641i
$22$	0	0
$23$	−8.30671e8	−0.0508710	−0.0254355	−	0.999676i	$-0.508097\pi$
$23$	−8.30671e8	−0.0508710	−0.0254355	+	0.999676i	$0.508097\pi$
$24$	0	0
$25$	3.04187e10	0.996760
$26$	0	0
$27$	− 5.48214e10i	− 1.00861i
$28$	0	0
$29$	5.26318e10i	0.566582i	0.959034	+	0.283291i	$0.0914262\pi$
$29$	5.26318e10i	0.566582i	−0.959034	+	0.283291i	$0.908574\pi$
$30$	0	0
$31$	−2.31580e11	−1.51178	−0.755889	−	0.654700i	$-0.772795\pi$
$31$	−2.31580e11	−1.51178	−0.755889	+	0.654700i	$0.772795\pi$
$32$	0	0
$33$	1.56106e11	0.637626
$34$	0	0
$35$	1.23661e10i	0.0324879i
$36$	0	0
$37$	− 5.88641e11i	− 1.01938i	−0.860357	−	0.509691i	$-0.829760\pi$
$37$	− 5.88641e11i	− 1.01938i	0.860357	−	0.509691i	$-0.170240\pi$
$38$	0	0
$39$	−4.56822e11	−0.533044
$40$	0	0
$41$	2.88417e11	0.231282	0.115641	−	0.993291i	$-0.463108\pi$
$41$	2.88417e11	0.231282	0.115641	+	0.993291i	$0.463108\pi$
$42$	0	0
$43$	− 1.77551e12i	− 0.996115i	−0.867144	−	0.498057i	$-0.834047\pi$
$43$	− 1.77551e12i	− 0.996115i	0.867144	−	0.498057i	$-0.165953\pi$
$44$	0	0
$45$	− 8.63457e10i	− 0.0344466i
$46$	0	0
$47$	6.45781e12	1.85931	0.929654	−	0.368434i	$-0.120106\pi$
$47$	6.45781e12	1.85931	0.929654	+	0.368434i	$0.120106\pi$
$48$	0	0
$49$	−3.20090e12	−0.674219
$50$	0	0
$51$	− 4.73032e12i	− 0.738098i
$52$	0	0
$53$	− 1.55187e13i	− 1.81463i	−0.420454	−	0.907314i	$-0.638129\pi$
$53$	− 1.55187e13i	− 1.81463i	0.420454	−	0.907314i	$-0.361871\pi$
$54$	0	0
$55$	6.52151e11	0.0577601
$56$	0	0
$57$	1.93621e12	0.131187
$58$	0	0
$59$	− 1.28387e13i	− 0.671633i	−0.941927	−	0.335816i	$-0.890988\pi$
$59$	− 1.28387e13i	− 0.671633i	0.941927	−	0.335816i	$-0.109012\pi$
$60$	0	0
$61$	− 3.34293e13i	− 1.36193i	−0.732317	−	0.680963i	$-0.761561\pi$
$61$	− 3.34293e13i	− 1.36193i	0.732317	−	0.680963i	$-0.238439\pi$
$62$	0	0
$63$	−1.07995e13	−0.345422
$64$	0	0
$65$	−1.90843e12	−0.0482864
$66$	0	0
$67$	− 8.71055e13i	− 1.75584i	−0.478808	−	0.877920i	$-0.658931\pi$
$67$	− 8.71055e13i	− 1.75584i	0.478808	−	0.877920i	$-0.341069\pi$
$68$	0	0
$69$	1.97713e12i	0.0319645i
$70$	0	0
$71$	−1.73139e13	−0.225922	−0.112961	−	0.993599i	$-0.536034\pi$
$71$	−1.73139e13	−0.225922	−0.112961	+	0.993599i	$0.536034\pi$
$72$	0	0
$73$	1.25837e14	1.33318	0.666589	−	0.745425i	$-0.267753\pi$
$73$	1.25837e14	1.33318	0.666589	+	0.745425i	$0.267753\pi$
$74$	0	0
$75$	− 7.24015e13i	− 0.626308i
$76$	0	0
$77$	− 8.15665e13i	− 0.579204i
$78$	0	0
$79$	4.05635e12	0.0237647	0.0118824	−	0.999929i	$-0.496218\pi$
$79$	4.05635e12	0.0237647	0.0118824	+	0.999929i	$0.496218\pi$
$80$	0	0
$81$	−5.88183e12	−0.0285677
$82$	0	0
$83$	2.09495e14i	0.847400i	0.905803	+	0.423700i	$0.139269\pi$
$83$	2.09495e14i	0.847400i	−0.905803	+	0.423700i	$0.860731\pi$
$84$	0	0
$85$	− 1.97614e13i	− 0.0668615i
$86$	0	0
$87$	1.25272e14	0.356008
$88$	0	0
$89$	−3.43643e14	−0.823536	−0.411768	−	0.911289i	$-0.635089\pi$
$89$	−3.43643e14	−0.823536	−0.411768	+	0.911289i	$0.635089\pi$
$90$	0	0
$91$	2.38693e14i	0.484204i
$92$	0	0
$93$	5.51198e14i	0.949916i
$94$	0	0
$95$	8.08872e12	0.0118837
$96$	0	0
$97$	−7.44295e14	−0.935313	−0.467657	−	0.883910i	$-0.654902\pi$
$97$	−7.44295e14	−0.935313	−0.467657	+	0.883910i	$0.654902\pi$
$98$	0	0
$99$	5.69535e14i	0.614124i
$100$	0	0
$101$	5.21858e14i	0.484331i	0.970235	+	0.242165i	$0.0778576\pi$
$101$	5.21858e14i	0.484331i	−0.970235	+	0.242165i	$0.922142\pi$
$102$	0	0
$103$	3.05095e14	0.244431	0.122215	−	0.992504i	$-0.461000\pi$
$103$	3.05095e14	0.244431	0.122215	+	0.992504i	$0.461000\pi$
$104$	0	0
$105$	2.94333e13	0.0204136
$106$	0	0
$107$	1.74238e15i	1.04898i	0.851418	+	0.524488i	$0.175743\pi$
$107$	1.74238e15i	1.04898i	−0.851418	+	0.524488i	$0.824257\pi$
$108$	0	0
$109$	2.64214e15i	1.38439i	0.721712	+	0.692194i	$0.243356\pi$
$109$	2.64214e15i	1.38439i	−0.721712	+	0.692194i	$0.756644\pi$
$110$	0	0
$111$	−1.40106e15	−0.640523
$112$	0	0
$113$	3.80929e15	1.52320	0.761598	−	0.648050i	$-0.224415\pi$
$113$	3.80929e15	1.52320	0.761598	+	0.648050i	$0.224415\pi$
$114$	0	0
$115$	8.25969e12i	0.00289554i
$116$	0	0
$117$	− 1.66666e15i	− 0.513397i
$118$	0	0
$119$	−2.47162e15	−0.670470
$120$	0	0
$121$	−1.24325e14	−0.0297625
$122$	0	0
$123$	− 6.86478e14i	− 0.145324i
$124$	0	0
$125$	− 6.05913e14i	− 0.113654i
$126$	0	0
$127$	3.11928e15	0.519430	0.259715	−	0.965685i	$-0.416371\pi$
$127$	3.11928e15	0.519430	0.259715	+	0.965685i	$0.416371\pi$
$128$	0	0
$129$	−4.22600e15	−0.625902
$130$	0	0
$131$	− 5.48452e15i	− 0.723775i	−0.932222	−	0.361887i	$-0.882132\pi$
$131$	− 5.48452e15i	− 0.723775i	0.932222	−	0.361887i	$-0.117868\pi$
$132$	0	0
$133$	− 1.01168e15i	− 0.119167i
$134$	0	0
$135$	−5.45111e14	−0.0574092
$136$	0	0
$137$	−5.52159e15	−0.520786	−0.260393	−	0.965503i	$-0.583852\pi$
$137$	−5.52159e15	−0.520786	−0.260393	+	0.965503i	$0.583852\pi$
$138$	0	0
$139$	− 1.40813e16i	− 1.19133i	−0.803233	−	0.595665i	$-0.796889\pi$
$139$	− 1.40813e16i	− 1.19133i	0.803233	−	0.595665i	$-0.203111\pi$
$140$	0	0
$141$	− 1.53706e16i	− 1.16828i
$142$	0	0
$143$	1.25879e16	0.860863
$144$	0	0
$145$	5.23338e14	0.0322494
$146$	0	0
$147$	7.61866e15i	0.423641i
$148$	0	0
$149$	− 3.53835e16i	− 1.77789i	−0.458018	−	0.888943i	$-0.651440\pi$
$149$	− 3.53835e16i	− 1.77789i	0.458018	−	0.888943i	$-0.348560\pi$
$150$	0	0
$151$	1.54452e16	0.702208	0.351104	−	0.936336i	$-0.385806\pi$
$151$	1.54452e16	0.702208	0.351104	+	0.936336i	$0.385806\pi$
$152$	0	0
$153$	1.72580e16	0.710893
$154$	0	0
$155$	2.30269e15i	0.0860493i
$156$	0	0
$157$	3.86897e16i	1.31325i	0.754216	+	0.656627i	$0.228017\pi$
$157$	3.86897e16i	1.31325i	−0.754216	+	0.656627i	$0.771983\pi$
$158$	0	0
$159$	−3.69371e16	−1.14021
$160$	0	0
$161$	1.03306e15	0.0290358
$162$	0	0
$163$	− 4.32097e16i	− 1.10707i	−0.832827	−	0.553534i	$-0.813279\pi$
$163$	− 4.32097e16i	− 1.10707i	0.832827	−	0.553534i	$-0.186721\pi$
$164$	0	0
$165$	− 1.55222e15i	− 0.0362932i
$166$	0	0
$167$	3.46562e15	0.0740299	0.0370149	−	0.999315i	$-0.488215\pi$
$167$	3.46562e15	0.0740299	0.0370149	+	0.999315i	$0.488215\pi$
$168$	0	0
$169$	1.43491e16	0.280333
$170$	0	0
$171$	7.06402e15i	0.126352i
$172$	0	0
$173$	7.03681e16i	1.15353i	0.816909	+	0.576766i	$0.195686\pi$
$173$	7.03681e16i	1.15353i	−0.816909	+	0.576766i	$0.804314\pi$
$174$	0	0
$175$	−3.78302e16	−0.568923
$176$	0	0
$177$	−3.05583e16	−0.422016
$178$	0	0
$179$	1.11332e17i	1.41326i	0.707584	+	0.706629i	$0.249785\pi$
$179$	1.11332e17i	1.41326i	−0.707584	+	0.706629i	$0.750215\pi$
$180$	0	0
$181$	− 1.29611e16i	− 0.151374i	−0.997132	−	0.0756871i	$-0.975885\pi$
$181$	− 1.29611e16i	− 0.151374i	0.997132	−	0.0756871i	$-0.0241150\pi$
$182$	0	0
$183$	−7.95672e16	−0.855758
$184$	0	0
$185$	−5.85308e15	−0.0580225
$186$	0	0
$187$	1.30346e17i	1.19202i
$188$	0	0
$189$	6.81786e16i	0.575685i
$190$	0	0
$191$	1.61349e17	1.25897	0.629485	−	0.777013i	$-0.283266\pi$
$191$	1.61349e17	1.25897	0.629485	+	0.777013i	$0.283266\pi$
$192$	0	0
$193$	3.53401e16	0.255028	0.127514	−	0.991837i	$-0.459300\pi$
$193$	3.53401e16	0.255028	0.127514	+	0.991837i	$0.459300\pi$
$194$	0	0
$195$	4.54236e15i	0.0303405i
$196$	0	0
$197$	5.78781e16i	0.358111i	0.983839	+	0.179056i	$0.0573042\pi$
$197$	5.78781e16i	0.358111i	−0.983839	+	0.179056i	$0.942696\pi$
$198$	0	0
$199$	1.30039e17	0.745894	0.372947	−	0.927853i	$-0.378347\pi$
$199$	1.30039e17	0.745894	0.372947	+	0.927853i	$0.378347\pi$
$200$	0	0
$201$	−2.07325e17	−1.10327
$202$	0	0
$203$	− 6.54554e16i	− 0.323389i
$204$	0	0
$205$	− 2.86784e15i	− 0.0131644i
$206$	0	0
$207$	−7.21333e15	−0.0307863
$208$	0	0
$209$	−5.33530e16	−0.211866
$210$	0	0
$211$	4.82411e16i	0.178361i	0.996016	+	0.0891803i	$0.0284247\pi$
$211$	4.82411e16i	0.178361i	−0.996016	+	0.0891803i	$0.971575\pi$
$212$	0	0
$213$	4.12100e16i	0.141957i
$214$	0	0
$215$	−1.76546e16	−0.0566981
$216$	0	0
$217$	2.88004e17	0.862881
$218$	0	0
$219$	− 2.99514e17i	− 0.837694i
$220$	0	0
$221$	− 3.81438e17i	− 0.996512i
$222$	0	0
$223$	−5.47989e17	−1.33809	−0.669045	−	0.743222i	$-0.733297\pi$
$223$	−5.47989e17	−1.33809	−0.669045	+	0.743222i	$0.733297\pi$
$224$	0	0
$225$	2.64148e17	0.603224
$226$	0	0
$227$	− 3.57860e17i	− 0.764750i	−0.924007	−	0.382375i	$-0.875106\pi$
$227$	− 3.57860e17i	− 0.764750i	0.924007	−	0.382375i	$-0.124894\pi$
$228$	0	0
$229$	− 5.19138e17i	− 1.03876i	−0.854542	−	0.519381i	$-0.826162\pi$
$229$	− 5.19138e17i	− 1.03876i	0.854542	−	0.519381i	$-0.173838\pi$
$230$	0	0
$231$	−1.94141e17	−0.363939
$232$	0	0
$233$	−2.40828e17	−0.423192	−0.211596	−	0.977357i	$-0.567866\pi$
$233$	−2.40828e17	−0.423192	−0.211596	+	0.977357i	$0.567866\pi$
$234$	0	0
$235$	− 6.42125e16i	− 0.105830i
$236$	0	0
$237$	− 9.65478e15i	− 0.0149324i
$238$	0	0
$239$	6.71814e17	0.975584	0.487792	−	0.872960i	$-0.337802\pi$
$239$	6.71814e17	0.975584	0.487792	+	0.872960i	$0.337802\pi$
$240$	0	0
$241$	5.56278e17	0.758864	0.379432	−	0.925220i	$-0.376119\pi$
$241$	5.56278e17	0.758864	0.379432	+	0.925220i	$0.376119\pi$
$242$	0	0
$243$	− 7.72628e17i	− 0.990657i
$244$	0	0
$245$	3.18278e16i	0.0383761i
$246$	0	0
$247$	1.56130e17	0.177117
$248$	0	0
$249$	4.98633e17	0.532458
$250$	0	0
$251$	1.15081e17i	0.115731i	0.998324	+	0.0578654i	$0.0184294\pi$
$251$	1.15081e17i	0.115731i	−0.998324	+	0.0578654i	$0.981571\pi$
$252$	0	0
$253$	− 5.44807e16i	− 0.0516225i
$254$	0	0
$255$	−4.70354e16	−0.0420120
$256$	0	0
$257$	−2.14601e18	−1.80773	−0.903864	−	0.427819i	$-0.859282\pi$
$257$	−2.14601e18	−1.80773	−0.903864	+	0.427819i	$0.859282\pi$
$258$	0	0
$259$	7.32063e17i	0.581835i
$260$	0	0
$261$	4.57040e17i	0.342887i
$262$	0	0
$263$	2.26162e18	1.60233	0.801164	−	0.598444i	$-0.204214\pi$
$263$	2.26162e18	1.60233	0.801164	+	0.598444i	$0.204214\pi$
$264$	0	0
$265$	−1.54309e17	−0.103287
$266$	0	0
$267$	8.17927e17i	0.517464i
$268$	0	0
$269$	− 9.71587e17i	− 0.581218i	−0.956842	−	0.290609i	$-0.906142\pi$
$269$	− 9.71587e17i	− 0.581218i	0.956842	−	0.290609i	$-0.0938580\pi$
$270$	0	0
$271$	−1.94678e18	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$271$	−1.94678e18	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$272$	0	0
$273$	5.68127e17	0.304246
$274$	0	0
$275$	1.99505e18i	1.01148i
$276$	0	0
$277$	3.11603e18i	1.49625i	0.663560	+	0.748123i	$0.269045\pi$
$277$	3.11603e18i	1.49625i	−0.663560	+	0.748123i	$0.730955\pi$
$278$	0	0
$279$	−2.01098e18	−0.914905
$280$	0	0
$281$	−1.24836e18	−0.538323	−0.269161	−	0.963095i	$-0.586747\pi$
$281$	−1.24836e18	−0.538323	−0.269161	+	0.963095i	$0.586747\pi$
$282$	0	0
$283$	1.29236e16i	0.00528429i	0.999997	+	0.00264215i	$0.000841022\pi$
$283$	1.29236e16i	0.00528429i	−0.999997	+	0.00264215i	$0.999159\pi$
$284$	0	0
$285$	− 1.92525e16i	− 0.00746707i
$286$	0	0
$287$	−3.58689e17	−0.132009
$288$	0	0
$289$	1.08731e18	0.379855
$290$	0	0
$291$	1.77154e18i	0.587698i
$292$	0	0
$293$	2.74547e18i	0.865186i	0.901589	+	0.432593i	$0.142401\pi$
$293$	2.74547e18i	0.865186i	−0.901589	+	0.432593i	$0.857599\pi$
$294$	0	0
$295$	−1.27660e17	−0.0382288
$296$	0	0
$297$	3.59554e18	1.02351
$298$	0	0
$299$	1.59430e17i	0.0431555i
$300$	0	0
$301$	2.20811e18i	0.568555i
$302$	0	0
$303$	1.24211e18	0.304326
$304$	0	0
$305$	−3.32401e17	−0.0775199
$306$	0	0
$307$	− 4.87988e18i	− 1.08361i	−0.840506	−	0.541803i	$-0.817742\pi$
$307$	− 4.87988e18i	− 1.08361i	0.840506	−	0.541803i	$-0.182258\pi$
$308$	0	0
$309$	− 7.26176e17i	− 0.153587i
$310$	0	0
$311$	5.97100e17	0.120322	0.0601609	−	0.998189i	$-0.480839\pi$
$311$	5.97100e17	0.120322	0.0601609	+	0.998189i	$0.480839\pi$
$312$	0	0
$313$	−1.48478e18	−0.285153	−0.142577	−	0.989784i	$-0.545539\pi$
$313$	−1.48478e18	−0.285153	−0.142577	+	0.989784i	$0.545539\pi$
$314$	0	0
$315$	1.07384e17i	0.0196612i
$316$	0	0
$317$	− 4.83677e18i	− 0.844521i	−0.906474	−	0.422261i	$-0.861237\pi$
$317$	− 4.83677e18i	− 0.844521i	0.906474	−	0.422261i	$-0.138763\pi$
$318$	0	0
$319$	−3.45193e18	−0.574952
$320$	0	0
$321$	4.14716e18	0.659118
$322$	0	0
$323$	1.61670e18i	0.245251i
$324$	0	0
$325$	− 5.83823e18i	− 0.845583i
$326$	0	0
$327$	6.28873e18	0.869871
$328$	0	0
$329$	−8.03125e18	−1.06124
$330$	0	0
$331$	1.17425e18i	0.148270i	0.997248	+	0.0741348i	$0.0236195\pi$
$331$	1.17425e18i	0.148270i	−0.997248	+	0.0741348i	$0.976380\pi$
$332$	0	0
$333$	− 5.11160e18i	− 0.616915i
$334$	0	0
$335$	−8.66124e17	−0.0999411
$336$	0	0
$337$	−6.61936e18	−0.730452	−0.365226	−	0.930919i	$-0.619008\pi$
$337$	−6.61936e18	−0.730452	−0.365226	+	0.930919i	$0.619008\pi$
$338$	0	0
$339$	− 9.06673e18i	− 0.957090i
$340$	0	0
$341$	− 1.51885e19i	− 1.53411i
$342$	0	0
$343$	9.88510e18	0.955598
$344$	0	0
$345$	1.96594e16	0.00181939
$346$	0	0
$347$	− 1.48938e19i	− 1.31988i	−0.751318	−	0.659941i	$-0.770581\pi$
$347$	− 1.48938e19i	− 1.31988i	0.751318	−	0.659941i	$-0.229419\pi$
$348$	0	0
$349$	1.42663e19i	1.21093i	0.795871	+	0.605466i	$0.207013\pi$
$349$	1.42663e19i	1.21093i	−0.795871	+	0.605466i	$0.792987\pi$
$350$	0	0
$351$	−1.05218e19	−0.855634
$352$	0	0
$353$	4.56762e18	0.355942	0.177971	−	0.984036i	$-0.443047\pi$
$353$	4.56762e18	0.355942	0.177971	+	0.984036i	$0.443047\pi$
$354$	0	0
$355$	1.72159e17i	0.0128593i
$356$	0	0
$357$	5.88286e18i	0.421286i
$358$	0	0
$359$	1.36834e19	0.939692	0.469846	−	0.882749i	$-0.344310\pi$
$359$	1.36834e19	0.939692	0.469846	+	0.882749i	$0.344310\pi$
$360$	0	0
$361$	1.45194e19	0.956410
$362$	0	0
$363$	2.95914e17i	0.0187011i
$364$	0	0
$365$	− 1.25125e18i	− 0.0758836i
$366$	0	0
$367$	−1.90250e19	−1.10747	−0.553733	−	0.832694i	$-0.686797\pi$
$367$	−1.90250e19	−1.10747	−0.553733	+	0.832694i	$0.686797\pi$
$368$	0	0
$369$	2.50453e18	0.139968
$370$	0	0
$371$	1.92999e19i	1.03574i
$372$	0	0
$373$	2.25771e18i	0.116373i	0.998306	+	0.0581865i	$0.0185318\pi$
$373$	2.25771e18i	0.116373i	−0.998306	+	0.0581865i	$0.981468\pi$
$374$	0	0
$375$	−1.44217e18	−0.0714138
$376$	0	0
$377$	1.01016e19	0.480650
$378$	0	0
$379$	2.46190e19i	1.12584i	0.826511	+	0.562920i	$0.190322\pi$
$379$	2.46190e19i	1.12584i	−0.826511	+	0.562920i	$0.809678\pi$
$380$	0	0
$381$	− 7.42440e18i	− 0.326380i
$382$	0	0
$383$	−1.64962e19	−0.697257	−0.348629	−	0.937261i	$-0.613353\pi$
$383$	−1.64962e19	−0.697257	−0.348629	+	0.937261i	$0.613353\pi$
$384$	0	0
$385$	−8.11047e17	−0.0329678
$386$	0	0
$387$	− 1.54180e19i	− 0.602833i
$388$	0	0
$389$	2.02343e19i	0.761143i	0.924751	+	0.380572i	$0.124273\pi$
$389$	2.02343e19i	0.761143i	−0.924751	+	0.380572i	$0.875727\pi$
$390$	0	0
$391$	−1.65087e18	−0.0597568
$392$	0	0
$393$	−1.30540e19	−0.454779
$394$	0	0
$395$	− 4.03339e16i	− 0.00135267i
$396$	0	0
$397$	4.94869e19i	1.59795i	0.601367	+	0.798973i	$0.294623\pi$
$397$	4.94869e19i	1.59795i	−0.601367	+	0.798973i	$0.705377\pi$
$398$	0	0
$399$	−2.40796e18	−0.0748779
$400$	0	0
$401$	−2.17764e19	−0.652234	−0.326117	−	0.945329i	$-0.605740\pi$
$401$	−2.17764e19	−0.652234	−0.326117	+	0.945329i	$0.605740\pi$
$402$	0	0
$403$	4.44469e19i	1.28249i
$404$	0	0
$405$	5.84853e16i	0.00162605i
$406$	0	0
$407$	3.86068e19	1.03444
$408$	0	0
$409$	−3.84378e19	−0.992735	−0.496368	−	0.868112i	$-0.665333\pi$
$409$	−3.84378e19	−0.992735	−0.496368	+	0.868112i	$0.665333\pi$
$410$	0	0
$411$	1.31423e19i	0.327233i
$412$	0	0
$413$	1.59669e19i	0.383349i
$414$	0	0
$415$	2.08309e18	0.0482334
$416$	0	0
$417$	−3.35158e19	−0.748565
$418$	0	0
$419$	− 4.27455e19i	− 0.921053i	−0.887646	−	0.460527i	$-0.847661\pi$
$419$	− 4.27455e19i	− 0.921053i	0.887646	−	0.460527i	$-0.152339\pi$
$420$	0	0
$421$	− 5.62283e19i	− 1.16907i	−0.811370	−	0.584533i	$-0.801278\pi$
$421$	− 5.62283e19i	− 1.16907i	0.811370	−	0.584533i	$-0.198722\pi$
$422$	0	0
$423$	5.60779e19	1.12522
$424$	0	0
$425$	6.04539e19	1.17087
$426$	0	0
$427$	4.15743e19i	0.777350i
$428$	0	0
$429$	− 2.99613e19i	− 0.540918i
$430$	0	0
$431$	−9.82364e19	−1.71275	−0.856373	−	0.516358i	$-0.827288\pi$
$431$	−9.82364e19	−1.71275	−0.856373	+	0.516358i	$0.827288\pi$
$432$	0	0
$433$	6.33031e19	1.06602	0.533010	−	0.846109i	$-0.321061\pi$
$433$	6.33031e19	1.06602	0.533010	+	0.846109i	$0.321061\pi$
$434$	0	0
$435$	− 1.24563e18i	− 0.0202637i
$436$	0	0
$437$	− 6.75733e17i	− 0.0106210i
$438$	0	0
$439$	9.36988e19	1.42315	0.711574	−	0.702611i	$-0.247982\pi$
$439$	9.36988e19	1.42315	0.711574	+	0.702611i	$0.247982\pi$
$440$	0	0
$441$	−2.77957e19	−0.408027
$442$	0	0
$443$	− 9.95376e19i	− 1.41241i	−0.708010	−	0.706203i	$-0.750407\pi$
$443$	− 9.95376e19i	− 1.41241i	0.708010	−	0.706203i	$-0.249593\pi$
$444$	0	0
$445$	3.41698e18i	0.0468751i
$446$	0	0
$447$	−8.42185e19	−1.11712
$448$	0	0
$449$	6.08712e19	0.780844	0.390422	−	0.920636i	$-0.372329\pi$
$449$	6.08712e19	0.780844	0.390422	+	0.920636i	$0.372329\pi$
$450$	0	0
$451$	1.89162e19i	0.234698i
$452$	0	0
$453$	− 3.67620e19i	− 0.441228i
$454$	0	0
$455$	2.37341e18	0.0275605
$456$	0	0
$457$	−5.20069e19	−0.584372	−0.292186	−	0.956362i	$-0.594383\pi$
$457$	−5.20069e19	−0.584372	−0.292186	+	0.956362i	$0.594383\pi$
$458$	0	0
$459$	− 1.08952e20i	− 1.18478i
$460$	0	0
$461$	− 9.63171e19i	− 1.01379i	−0.862009	−	0.506893i	$-0.830794\pi$
$461$	− 9.63171e19i	− 1.01379i	0.862009	−	0.506893i	$-0.169206\pi$
$462$	0	0
$463$	−7.66416e19	−0.780921	−0.390460	−	0.920620i	$-0.627684\pi$
$463$	−7.66416e19	−0.780921	−0.390460	+	0.920620i	$0.627684\pi$
$464$	0	0
$465$	5.48077e18	0.0540685
$466$	0	0
$467$	− 1.17176e20i	− 1.11934i	−0.828714	−	0.559672i	$-0.810927\pi$
$467$	− 1.17176e20i	− 1.11934i	0.828714	−	0.559672i	$-0.189073\pi$
$468$	0	0
$469$	1.08329e20i	1.00218i
$470$	0	0
$471$	9.20879e19	0.825174
$472$	0	0
$473$	1.16449e20	1.01083
$474$	0	0
$475$	2.47449e19i	0.208106i
$476$	0	0
$477$	− 1.34761e20i	− 1.09818i
$478$	0	0
$479$	−1.22565e20	−0.967944	−0.483972	−	0.875084i	$-0.660806\pi$
$479$	−1.22565e20	−0.967944	−0.483972	+	0.875084i	$0.660806\pi$
$480$	0	0
$481$	−1.12977e20	−0.864775
$482$	0	0
$483$	− 2.45886e18i	− 0.0182444i
$484$	0	0
$485$	7.40081e18i	0.0532373i
$486$	0	0
$487$	1.18086e20	0.823628	0.411814	−	0.911268i	$-0.364895\pi$
$487$	1.18086e20	0.823628	0.411814	+	0.911268i	$0.364895\pi$
$488$	0	0
$489$	−1.02846e20	−0.695619
$490$	0	0
$491$	1.58783e20i	1.04158i	0.853685	+	0.520789i	$0.174362\pi$
$491$	1.58783e20i	1.04158i	−0.853685	+	0.520789i	$0.825638\pi$
$492$	0	0
$493$	1.04600e20i	0.665548i
$494$	0	0
$495$	5.66310e18	0.0349555
$496$	0	0
$497$	2.15325e19	0.128950
$498$	0	0
$499$	2.07261e20i	1.20438i	0.798353	+	0.602190i	$0.205705\pi$
$499$	2.07261e20i	1.20438i	−0.798353	+	0.602190i	$0.794295\pi$
$500$	0	0
$501$	− 8.24873e18i	− 0.0465162i
$502$	0	0
$503$	2.98541e20	1.63397	0.816984	−	0.576661i	$-0.195644\pi$
$503$	2.98541e20	1.63397	0.816984	+	0.576661i	$0.195644\pi$
$504$	0	0
$505$	5.18903e18	0.0275677
$506$	0	0
$507$	− 3.41532e19i	− 0.176146i
$508$	0	0
$509$	− 1.35908e20i	− 0.680552i	−0.940326	−	0.340276i	$-0.889479\pi$
$509$	− 1.35908e20i	− 0.680552i	0.940326	−	0.340276i	$-0.110521\pi$
$510$	0	0
$511$	−1.56498e20	−0.760941
$512$	0	0
$513$	4.45960e19	0.210579
$514$	0	0
$515$	− 3.03368e18i	− 0.0139128i
$516$	0	0
$517$	4.23544e20i	1.88677i
$518$	0	0
$519$	1.67487e20	0.724815
$520$	0	0
$521$	−4.87015e19	−0.204767	−0.102383	−	0.994745i	$-0.532647\pi$
$521$	−4.87015e19	−0.204767	−0.102383	+	0.994745i	$0.532647\pi$
$522$	0	0
$523$	3.31732e20i	1.35527i	0.735400	+	0.677633i	$0.236994\pi$
$523$	3.31732e20i	1.35527i	−0.735400	+	0.677633i	$0.763006\pi$
$524$	0	0
$525$	9.00420e19i	0.357479i
$526$	0	0
$527$	−4.60240e20	−1.77584
$528$	0	0
$529$	−2.65945e20	−0.997412
$530$	0	0
$531$	− 1.11488e20i	− 0.406462i
$532$	0	0
$533$	− 5.53555e19i	− 0.196204i
$534$	0	0
$535$	1.73252e19	0.0597070
$536$	0	0
$537$	2.64988e20	0.888012
$538$	0	0
$539$	− 2.09935e20i	− 0.684179i
$540$	0	0
$541$	5.67118e20i	1.79760i	0.438356	+	0.898801i	$0.355561\pi$
$541$	5.67118e20i	1.79760i	−0.438356	+	0.898801i	$0.644439\pi$
$542$	0	0
$543$	−3.08495e19	−0.0951150
$544$	0	0
$545$	2.62719e19	0.0787983
$546$	0	0
$547$	2.33879e20i	0.682475i	0.939977	+	0.341238i	$0.110846\pi$
$547$	2.33879e20i	0.682475i	−0.939977	+	0.341238i	$0.889154\pi$
$548$	0	0
$549$	− 2.90291e20i	− 0.824217i
$550$	0	0
$551$	−4.28147e19	−0.118292
$552$	0	0
$553$	−5.04468e18	−0.0135642
$554$	0	0
$555$	1.39313e19i	0.0364581i
$556$	0	0
$557$	− 2.53453e20i	− 0.645630i	−0.946462	−	0.322815i	$-0.895371\pi$
$557$	− 2.53453e20i	− 0.645630i	0.946462	−	0.322815i	$-0.104629\pi$
$558$	0	0
$559$	−3.40772e20	−0.845036
$560$	0	0
$561$	3.10244e20	0.749001
$562$	0	0
$563$	− 1.13398e20i	− 0.266558i	−0.991079	−	0.133279i	$-0.957449\pi$
$563$	− 1.13398e20i	− 0.266558i	0.991079	−	0.133279i	$-0.0425507\pi$
$564$	0	0
$565$	− 3.78772e19i	− 0.0866992i
$566$	0	0
$567$	7.31493e18	0.0163056
$568$	0	0
$569$	−8.17119e19	−0.177396	−0.0886979	−	0.996059i	$-0.528271\pi$
$569$	−8.17119e19	−0.177396	−0.0886979	+	0.996059i	$0.528271\pi$
$570$	0	0
$571$	− 2.75197e20i	− 0.581933i	−0.956733	−	0.290967i	$-0.906023\pi$
$571$	− 2.75197e20i	− 0.581933i	0.956733	−	0.290967i	$-0.0939768\pi$
$572$	0	0
$573$	− 3.84036e20i	− 0.791066i
$574$	0	0
$575$	−2.52680e19	−0.0507062
$576$	0	0
$577$	3.55434e20	0.694929	0.347465	−	0.937693i	$-0.387043\pi$
$577$	3.55434e20	0.694929	0.347465	+	0.937693i	$0.387043\pi$
$578$	0	0
$579$	− 8.41153e19i	− 0.160245i
$580$	0	0
$581$	− 2.60539e20i	− 0.483672i
$582$	0	0
$583$	1.01782e21	1.84143
$584$	0	0
$585$	−1.65723e19	−0.0292222
$586$	0	0
$587$	− 1.74771e20i	− 0.300388i	−0.988657	−	0.150194i	$-0.952010\pi$
$587$	− 1.74771e20i	− 0.300388i	0.988657	−	0.150194i	$-0.0479899\pi$
$588$	0	0
$589$	− 1.88385e20i	− 0.315632i
$590$	0	0
$591$	1.37759e20	0.225017
$592$	0	0
$593$	−5.71741e20	−0.910519	−0.455260	−	0.890359i	$-0.650454\pi$
$593$	−5.71741e20	−0.910519	−0.455260	+	0.890359i	$0.650454\pi$
$594$	0	0
$595$	2.45763e19i	0.0381627i
$596$	0	0
$597$	− 3.09515e20i	− 0.468678i
$598$	0	0
$599$	1.14459e21	1.69024	0.845119	−	0.534578i	$-0.179529\pi$
$599$	1.14459e21	1.69024	0.845119	+	0.534578i	$0.179529\pi$
$600$	0	0
$601$	−7.89436e20	−1.13699	−0.568497	−	0.822685i	$-0.692475\pi$
$601$	−7.89436e20	−1.13699	−0.568497	+	0.822685i	$0.692475\pi$
$602$	0	0
$603$	− 7.56401e20i	− 1.06261i
$604$	0	0
$605$	1.23621e18i	0.00169406i
$606$	0	0
$607$	−1.43171e21	−1.91399	−0.956994	−	0.290108i	$-0.906309\pi$
$607$	−1.43171e21	−1.91399	−0.956994	+	0.290108i	$0.906309\pi$
$608$	0	0
$609$	−1.55795e20	−0.203200
$610$	0	0
$611$	− 1.23944e21i	− 1.57731i
$612$	0	0
$613$	2.16151e20i	0.268413i	0.990953	+	0.134207i	$0.0428486\pi$
$613$	2.16151e20i	0.268413i	−0.990953	+	0.134207i	$0.957151\pi$
$614$	0	0
$615$	−6.82592e18	−0.00827176
$616$	0	0
$617$	4.00200e20	0.473302	0.236651	−	0.971595i	$-0.423950\pi$
$617$	4.00200e20	0.473302	0.236651	+	0.971595i	$0.423950\pi$
$618$	0	0
$619$	1.50880e21i	1.74162i	0.491623	+	0.870808i	$0.336404\pi$
$619$	1.50880e21i	1.74162i	−0.491623	+	0.870808i	$0.663596\pi$
$620$	0	0
$621$	4.55386e19i	0.0513089i
$622$	0	0
$623$	4.27372e20	0.470052
$624$	0	0
$625$	9.22280e20	0.990291
$626$	0	0
$627$	1.26989e20i	0.133125i
$628$	0	0
$629$	− 1.16986e21i	− 1.19744i
$630$	0	0
$631$	−1.14570e21	−1.14512	−0.572558	−	0.819864i	$-0.694049\pi$
$631$	−1.14570e21	−1.14512	−0.572558	+	0.819864i	$0.694049\pi$
$632$	0	0
$633$	1.14822e20	0.112072
$634$	0	0
$635$	− 3.10162e19i	− 0.0295656i
$636$	0	0
$637$	6.14345e20i	0.571962i
$638$	0	0
$639$	−1.50350e20	−0.136724
$640$	0	0
$641$	−9.28279e20	−0.824600	−0.412300	−	0.911048i	$-0.635274\pi$
$641$	−9.28279e20	−0.824600	−0.412300	+	0.911048i	$0.635274\pi$
$642$	0	0
$643$	1.38224e21i	1.19950i	0.800188	+	0.599749i	$0.204733\pi$
$643$	1.38224e21i	1.19950i	−0.800188	+	0.599749i	$0.795267\pi$
$644$	0	0
$645$	4.20208e19i	0.0356259i
$646$	0	0
$647$	−4.33047e20	−0.358718	−0.179359	−	0.983784i	$-0.557402\pi$
$647$	−4.33047e20	−0.358718	−0.179359	+	0.983784i	$0.557402\pi$
$648$	0	0
$649$	8.42046e20	0.681554
$650$	0	0
$651$	− 6.85497e20i	− 0.542186i
$652$	0	0
$653$	1.49679e21i	1.15694i	0.815702	+	0.578472i	$0.196351\pi$
$653$	1.49679e21i	1.15694i	−0.815702	+	0.578472i	$0.803649\pi$
$654$	0	0
$655$	−5.45347e19	−0.0411967
$656$	0	0
$657$	1.09274e21	0.806819
$658$	0	0
$659$	1.32459e21i	0.955961i	0.878370	+	0.477981i	$0.158631\pi$
$659$	1.32459e21i	0.955961i	−0.878370	+	0.477981i	$0.841369\pi$
$660$	0	0
$661$	2.40747e21i	1.69844i	0.528042	+	0.849218i	$0.322926\pi$
$661$	2.40747e21i	1.69844i	−0.528042	+	0.849218i	$0.677074\pi$
$662$	0	0
$663$	−9.07886e20	−0.626152
$664$	0	0
$665$	−1.00595e19	−0.00678290
$666$	0	0
$667$	− 4.37197e19i	− 0.0288226i
$668$	0	0
$669$	1.30430e21i	0.840780i
$670$	0	0
$671$	2.19251e21	1.38205
$672$	0	0
$673$	2.34745e19	0.0144705	0.00723524	−	0.999974i	$-0.497697\pi$
$673$	2.34745e19	0.0144705	0.00723524	+	0.999974i	$0.497697\pi$
$674$	0	0
$675$	− 1.66760e21i	− 1.00534i
$676$	0	0
$677$	− 1.10436e21i	− 0.651171i	−0.945513	−	0.325586i	$-0.894439\pi$
$677$	− 1.10436e21i	− 0.651171i	0.945513	−	0.325586i	$-0.105561\pi$
$678$	0	0
$679$	9.25642e20	0.533850
$680$	0	0
$681$	−8.51766e20	−0.480526
$682$	0	0
$683$	1.38917e21i	0.766658i	0.923612	+	0.383329i	$0.125222\pi$
$683$	1.38917e21i	0.766658i	−0.923612	+	0.383329i	$0.874778\pi$
$684$	0	0
$685$	5.49033e19i	0.0296428i
$686$	0	0
$687$	−1.23563e21	−0.652700
$688$	0	0
$689$	−2.97850e21	−1.53941
$690$	0	0
$691$	2.53593e21i	1.28248i	0.767338	+	0.641242i	$0.221581\pi$
$691$	2.53593e21i	1.28248i	−0.767338	+	0.641242i	$0.778419\pi$
$692$	0	0
$693$	− 7.08301e20i	− 0.350525i
$694$	0	0
$695$	−1.40016e20	−0.0678096
$696$	0	0
$697$	5.73197e20	0.271680
$698$	0	0
$699$	5.73209e20i	0.265910i
$700$	0	0
$701$	− 2.17192e21i	− 0.986185i	−0.869977	−	0.493093i	$-0.835866\pi$
$701$	− 2.17192e21i	− 0.986185i	0.869977	−	0.493093i	$-0.164134\pi$
$702$	0	0
$703$	4.78846e20	0.212829
$704$	0	0
$705$	−1.52836e20	−0.0664979
$706$	0	0
$707$	− 6.49008e20i	− 0.276442i
$708$	0	0
$709$	5.09501e20i	0.212470i	0.994341	+	0.106235i	$0.0338797\pi$
$709$	5.09501e20i	0.212470i	−0.994341	+	0.106235i	$0.966120\pi$
$710$	0	0
$711$	3.52243e19	0.0143820
$712$	0	0
$713$	1.92367e20	0.0769057
$714$	0	0
$715$	− 1.25167e20i	− 0.0489997i
$716$	0	0
$717$	− 1.59903e21i	− 0.613002i
$718$	0	0
$719$	1.66156e21	0.623804	0.311902	−	0.950114i	$-0.399034\pi$
$719$	1.66156e21	0.623804	0.311902	+	0.950114i	$0.399034\pi$
$720$	0	0
$721$	−3.79431e20	−0.139514
$722$	0	0
$723$	− 1.32403e21i	− 0.476828i
$724$	0	0
$725$	1.60099e21i	0.564746i
$726$	0	0
$727$	−1.87981e21	−0.649539	−0.324769	−	0.945793i	$-0.605287\pi$
$727$	−1.87981e21	−0.649539	−0.324769	+	0.945793i	$0.605287\pi$
$728$	0	0
$729$	−1.92338e21	−0.651041
$730$	0	0
$731$	− 3.52863e21i	− 1.17011i
$732$	0	0
$733$	− 2.94480e21i	− 0.956701i	−0.878169	−	0.478351i	$-0.841235\pi$
$733$	− 2.94480e21i	− 0.956701i	0.878169	−	0.478351i	$-0.158765\pi$
$734$	0	0
$735$	7.57552e19	0.0241134
$736$	0	0
$737$	5.71294e21	1.78178
$738$	0	0
$739$	− 5.74402e21i	− 1.75543i	−0.479187	−	0.877713i	$-0.659068\pi$
$739$	− 5.74402e21i	− 1.75543i	0.479187	−	0.877713i	$-0.340932\pi$
$740$	0	0
$741$	− 3.71615e20i	− 0.111290i
$742$	0	0
$743$	−2.12554e21	−0.623810	−0.311905	−	0.950113i	$-0.600967\pi$
$743$	−2.12554e21	−0.623810	−0.311905	+	0.950113i	$0.600967\pi$
$744$	0	0
$745$	−3.51832e20	−0.101196
$746$	0	0
$747$	1.81920e21i	0.512833i
$748$	0	0
$749$	− 2.16692e21i	− 0.598726i
$750$	0	0
$751$	5.41638e21	1.46693	0.733465	−	0.679727i	$-0.237902\pi$
$751$	5.41638e21	1.46693	0.733465	+	0.679727i	$0.237902\pi$
$752$	0	0
$753$	2.73910e20	0.0727187
$754$	0	0
$755$	− 1.53577e20i	− 0.0399692i
$756$	0	0
$757$	1.53010e21i	0.390392i	0.980764	+	0.195196i	$0.0625342\pi$
$757$	1.53010e21i	0.390392i	−0.980764	+	0.195196i	$0.937466\pi$
$758$	0	0
$759$	−1.29673e20	−0.0324367
$760$	0	0
$761$	4.57526e21	1.12210	0.561049	−	0.827782i	$-0.310398\pi$
$761$	4.57526e21	1.12210	0.561049	+	0.827782i	$0.310398\pi$
$762$	0	0
$763$	− 3.28590e21i	− 0.790170i
$764$	0	0
$765$	− 1.71603e20i	− 0.0404635i
$766$	0	0
$767$	−2.46412e21	−0.569767
$768$	0	0
$769$	−1.36697e21	−0.309965	−0.154983	−	0.987917i	$-0.549532\pi$
$769$	−1.36697e21	−0.309965	−0.154983	+	0.987917i	$0.549532\pi$
$770$	0	0
$771$	5.10785e21i	1.13587i
$772$	0	0
$773$	7.87361e20i	0.171723i	0.996307	+	0.0858614i	$0.0273642\pi$
$773$	7.87361e20i	0.171723i	−0.996307	+	0.0858614i	$0.972636\pi$
$774$	0	0
$775$	−7.04436e21	−1.50688
$776$	0	0
$777$	1.74243e21	0.365592
$778$	0	0
$779$	2.34620e20i	0.0482875i
$780$	0	0
$781$	− 1.13556e21i	− 0.229259i
$782$	0	0
$783$	2.88535e21	0.571459
$784$	0	0
$785$	3.84707e20	0.0747494
$786$	0	0
$787$	− 3.99883e21i	− 0.762294i	−0.924514	−	0.381147i	$-0.875529\pi$
$787$	− 3.99883e21i	− 0.762294i	0.924514	−	0.381147i	$-0.124471\pi$
$788$	0	0
$789$	− 5.38302e21i	− 1.00681i
$790$	0	0
$791$	−4.73742e21	−0.869397
$792$	0	0
$793$	−6.41606e21	−1.15537
$794$	0	0
$795$	3.67280e20i	0.0648999i
$796$	0	0
$797$	5.29831e21i	0.918756i	0.888241	+	0.459378i	$0.151928\pi$
$797$	5.29831e21i	0.918756i	−0.888241	+	0.459378i	$0.848072\pi$
$798$	0	0
$799$	1.28342e22	2.18408
$800$	0	0
$801$	−2.98411e21	−0.498391
$802$	0	0
$803$	8.25322e21i	1.35287i
$804$	0	0
$805$	− 1.02722e19i	− 0.00165269i
$806$	0	0
$807$	−2.31253e21	−0.365205
$808$	0	0
$809$	−8.17463e21	−1.26723	−0.633613	−	0.773650i	$-0.718429\pi$
$809$	−8.17463e21	−1.26723	−0.633613	+	0.773650i	$0.718429\pi$
$810$	0	0
$811$	6.50807e21i	0.990367i	0.868788	+	0.495183i	$0.164899\pi$
$811$	6.50807e21i	0.990367i	−0.868788	+	0.495183i	$0.835101\pi$
$812$	0	0
$813$	4.63364e21i	0.692219i
$814$	0	0
$815$	−4.29650e20	−0.0630134
$816$	0	0
$817$	1.44434e21	0.207971
$818$	0	0
$819$	2.07274e21i	0.293033i
$820$	0	0
$821$	5.84596e21i	0.811488i	0.913987	+	0.405744i	$0.132988\pi$
$821$	5.84596e21i	0.811488i	−0.913987	+	0.405744i	$0.867012\pi$
$822$	0	0
$823$	−1.50787e21	−0.205525	−0.102762	−	0.994706i	$-0.532768\pi$
$823$	−1.50787e21	−0.205525	−0.102762	+	0.994706i	$0.532768\pi$
$824$	0	0
$825$	4.74855e21	0.635560
$826$	0	0
$827$	− 8.62150e20i	− 0.113316i	−0.998394	−	0.0566580i	$-0.981956\pi$
$827$	− 8.62150e20i	− 0.113316i	0.998394	−	0.0566580i	$-0.0180445\pi$
$828$	0	0
$829$	4.85501e20i	0.0626658i	0.999509	+	0.0313329i	$0.00997521\pi$
$829$	4.85501e20i	0.0626658i	−0.999509	+	0.0313329i	$0.990025\pi$
$830$	0	0
$831$	7.41665e21	0.940157
$832$	0	0
$833$	−6.36144e21	−0.791987
$834$	0	0
$835$	− 3.44600e19i	− 0.00421373i
$836$	0	0
$837$	1.26955e22i	1.52479i
$838$	0	0
$839$	−4.54768e21	−0.536507	−0.268254	−	0.963348i	$-0.586446\pi$
$839$	−4.54768e21	−0.536507	−0.268254	+	0.963348i	$0.586446\pi$
$840$	0	0
$841$	5.85909e21	0.678985
$842$	0	0
$843$	2.97130e21i	0.338252i
$844$	0	0
$845$	− 1.42679e20i	− 0.0159564i
$846$	0	0
$847$	1.54617e20	0.0169876
$848$	0	0
$849$	3.07604e19	0.00332035
$850$	0	0
$851$	4.88967e20i	0.0518571i
$852$	0	0
$853$	− 1.10589e22i	− 1.15238i	−0.817317	−	0.576188i	$-0.804540\pi$
$853$	− 1.10589e22i	− 1.15238i	0.817317	−	0.576188i	$-0.195460\pi$
$854$	0	0
$855$	7.02403e19	0.00719185
$856$	0	0
$857$	1.25141e22	1.25905	0.629526	−	0.776979i	$-0.283249\pi$
$857$	1.25141e22	1.25905	0.629526	+	0.776979i	$0.283249\pi$
$858$	0	0
$859$	8.74633e20i	0.0864723i	0.999065	+	0.0432362i	$0.0137668\pi$
$859$	8.74633e20i	0.0864723i	−0.999065	+	0.0432362i	$0.986233\pi$
$860$	0	0
$861$	8.53738e20i	0.0829471i
$862$	0	0
$863$	7.93718e20	0.0757854	0.0378927	−	0.999282i	$-0.487935\pi$
$863$	7.93718e20	0.0757854	0.0378927	+	0.999282i	$0.487935\pi$
$864$	0	0
$865$	6.99697e20	0.0656582
$866$	0	0
$867$	− 2.58797e21i	− 0.238680i
$868$	0	0
$869$	2.66042e20i	0.0241158i
$870$	0	0
$871$	−1.67181e22	−1.48953
$872$	0	0
$873$	−6.46326e21	−0.566037
$874$	0	0
$875$	7.53543e20i	0.0648706i
$876$	0	0
$877$	− 4.20486e21i	− 0.355840i	−0.984045	−	0.177920i	$-0.943063\pi$
$877$	− 4.20486e21i	− 0.355840i	0.984045	−	0.177920i	$-0.0569368\pi$
$878$	0	0
$879$	6.53466e21	0.543634
$880$	0	0
$881$	1.10473e22	0.903519	0.451760	−	0.892140i	$-0.350797\pi$
$881$	1.10473e22	0.903519	0.451760	+	0.892140i	$0.350797\pi$
$882$	0	0
$883$	− 4.28634e21i	− 0.344653i	−0.985040	−	0.172326i	$-0.944872\pi$
$883$	− 4.28634e21i	− 0.344653i	0.985040	−	0.172326i	$-0.0551284\pi$
$884$	0	0
$885$	3.03853e20i	0.0240208i
$886$	0	0
$887$	−1.74633e22	−1.35737	−0.678686	−	0.734428i	$-0.737450\pi$
$887$	−1.74633e22	−1.35737	−0.678686	+	0.734428i	$0.737450\pi$
$888$	0	0
$889$	−3.87929e21	−0.296476
$890$	0	0
$891$	− 3.85768e20i	− 0.0289897i
$892$	0	0
$893$	5.25328e21i	0.388191i
$894$	0	0
$895$	1.10702e21	0.0804416
$896$	0	0
$897$	3.79469e20	0.0271165
$898$	0	0
$899$	− 1.21885e22i	− 0.856546i
$900$	0	0
$901$	− 3.08418e22i	− 2.13159i
$902$	0	0
$903$	5.25566e21	0.357248
$904$	0	0
$905$	−1.28877e20	−0.00861611
$906$	0	0
$907$	− 2.71139e22i	− 1.78294i	−0.453077	−	0.891471i	$-0.649674\pi$
$907$	− 2.71139e22i	− 1.78294i	0.453077	−	0.891471i	$-0.350326\pi$
$908$	0	0
$909$	4.53167e21i	0.293109i
$910$	0	0
$911$	1.30962e22	0.833217	0.416609	−	0.909086i	$-0.363219\pi$
$911$	1.30962e22	0.833217	0.416609	+	0.909086i	$0.363219\pi$
$912$	0	0
$913$	−1.37400e22	−0.859918
$914$	0	0
$915$	7.91168e20i	0.0487091i
$916$	0	0
$917$	6.82081e21i	0.413110i
$918$	0	0
$919$	2.44676e22	1.45789	0.728946	−	0.684571i	$-0.240010\pi$
$919$	2.44676e22	1.45789	0.728946	+	0.684571i	$0.240010\pi$
$920$	0	0
$921$	−1.16149e22	−0.680877
$922$	0	0
$923$	3.32305e21i	0.191657i
$924$	0	0
$925$	− 1.79057e22i	− 1.01608i
$926$	0	0
$927$	2.64937e21	0.147926
$928$	0	0
$929$	1.99321e21	0.109506	0.0547528	−	0.998500i	$-0.482563\pi$
$929$	1.99321e21	0.109506	0.0547528	+	0.998500i	$0.482563\pi$
$930$	0	0
$931$	− 2.60386e21i	− 0.140765i
$932$	0	0
$933$	− 1.42119e21i	− 0.0756034i
$934$	0	0
$935$	1.29608e21	0.0678492
$936$	0	0
$937$	1.03535e22	0.533386	0.266693	−	0.963782i	$-0.414069\pi$
$937$	1.03535e22	0.533386	0.266693	+	0.963782i	$0.414069\pi$
$938$	0	0
$939$	3.53401e21i	0.179174i
$940$	0	0
$941$	8.05907e21i	0.402127i	0.979578	+	0.201063i	$0.0644397\pi$
$941$	8.05907e21i	0.402127i	−0.979578	+	0.201063i	$0.935560\pi$
$942$	0	0
$943$	−2.39579e20	−0.0117655
$944$	0	0
$945$	6.77927e20	0.0327676
$946$	0	0
$947$	1.49168e22i	0.709663i	0.934930	+	0.354832i	$0.115462\pi$
$947$	1.49168e22i	0.709663i	−0.934930	+	0.354832i	$0.884538\pi$
$948$	0	0
$949$	− 2.41519e22i	− 1.13098i
$950$	0	0
$951$	−1.15123e22	−0.530650
$952$	0	0
$953$	−1.13867e22	−0.516654	−0.258327	−	0.966058i	$-0.583171\pi$
$953$	−1.13867e22	−0.516654	−0.258327	+	0.966058i	$0.583171\pi$
$954$	0	0
$955$	− 1.60435e21i	− 0.0716596i
$956$	0	0
$957$	8.21615e21i	0.361267i
$958$	0	0
$959$	6.86692e21	0.297250
$960$	0	0
$961$	3.01640e22	1.28547
$962$	0	0
$963$	1.51304e22i	0.634824i
$964$	0	0
$965$	− 3.51401e20i	− 0.0145160i
$966$	0	0
$967$	1.30131e21	0.0529274	0.0264637	−	0.999650i	$-0.491575\pi$
$967$	1.30131e21	0.0529274	0.0264637	+	0.999650i	$0.491575\pi$
$968$	0	0
$969$	3.84801e21	0.154102
$970$	0	0
$971$	1.76428e22i	0.695703i	0.937550	+	0.347851i	$0.113089\pi$
$971$	1.76428e22i	0.695703i	−0.937550	+	0.347851i	$0.886911\pi$
$972$	0	0
$973$	1.75122e22i	0.679978i
$974$	0	0
$975$	−1.38959e22	−0.531317
$976$	0	0
$977$	−2.07355e22	−0.780740	−0.390370	−	0.920658i	$-0.627653\pi$
$977$	−2.07355e22	−0.780740	−0.390370	+	0.920658i	$0.627653\pi$
$978$	0	0
$979$	− 2.25383e22i	− 0.835702i
$980$	0	0
$981$	2.29437e22i	0.837810i
$982$	0	0
$983$	−1.68619e22	−0.606393	−0.303196	−	0.952928i	$-0.598054\pi$
$983$	−1.68619e22	−0.606393	−0.303196	+	0.952928i	$0.598054\pi$
$984$	0	0
$985$	5.75505e20	0.0203834
$986$	0	0
$987$	1.91157e22i	0.666824i
$988$	0	0
$989$	1.47486e21i	0.0506734i
$990$	0	0
$991$	−4.56273e21	−0.154409	−0.0772045	−	0.997015i	$-0.524599\pi$
$991$	−4.56273e21	−0.154409	−0.0772045	+	0.997015i	$0.524599\pi$
$992$	0	0
$993$	2.79492e21	0.0931643
$994$	0	0
$995$	− 1.29303e21i	− 0.0424557i
$996$	0	0
$997$	4.32033e22i	1.39734i	0.715442	+	0.698672i	$0.246225\pi$
$997$	4.32033e22i	1.39734i	−0.715442	+	0.698672i	$0.753775\pi$
$998$	0	0
$999$	−3.22701e22	−1.02816

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.16.b.a.17.5		14
4.3	odd	2		8.16.b.a.5.7	✓	14
8.3	odd	2		8.16.b.a.5.8	yes	14
8.5	even	2	inner	32.16.b.a.17.10		14
12.11	even	2		72.16.d.b.37.8		14
24.11	even	2		72.16.d.b.37.7		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.16.b.a.5.7	✓	14	4.3	odd	2
8.16.b.a.5.8	yes	14	8.3	odd	2
32.16.b.a.17.5		14	1.1	even	1	trivial
32.16.b.a.17.10		14	8.5	even	2	inner
72.16.d.b.37.7		14	24.11	even	2
72.16.d.b.37.8		14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	32.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2380.16i q^{3} -9943.39i q^{5} -1.24365e6 q^{7} +8.68373e6 q^{9} +6.55864e7i q^{11} -1.91929e8i q^{13} -2.36669e7 q^{15} +1.98739e9 q^{17} +8.13478e8i q^{19} +2.96009e9i q^{21} -8.30671e8 q^{23} +3.04187e10 q^{25} -5.48214e10i q^{27} +5.26318e10i q^{29} -2.31580e11 q^{31} +1.56106e11 q^{33} +1.23661e10i q^{35} -5.88641e11i q^{37} -4.56822e11 q^{39} +2.88417e11 q^{41} -1.77551e12i q^{43} -8.63457e10i q^{45} +6.45781e12 q^{47} -3.20090e12 q^{49} -4.73032e12i q^{51} -1.55187e13i q^{53} +6.52151e11 q^{55} +1.93621e12 q^{57} -1.28387e13i q^{59} -3.34293e13i q^{61} -1.07995e13 q^{63} -1.90843e12 q^{65} -8.71055e13i q^{67} +1.97713e12i q^{69} -1.73139e13 q^{71} +1.25837e14 q^{73} -7.24015e13i q^{75} -8.15665e13i q^{77} +4.05635e12 q^{79} -5.88183e12 q^{81} +2.09495e14i q^{83} -1.97614e13i q^{85} +1.25272e14 q^{87} -3.43643e14 q^{89} +2.38693e14i q^{91} +5.51198e14i q^{93} +8.08872e12 q^{95} -7.44295e14 q^{97} +5.69535e14i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 1647088 q^{7} - 57395630 q^{9} - 712135312 q^{15} + 728554812 q^{17} + 35548816080 q^{23} - 75899954794 q^{25} + 105758138816 q^{31} - 150458001384 q^{33} + 2251546247120 q^{39} - 53229185940 q^{41}+ \cdots - 672574291859236 q^{97}+O(q^{100}) \) 14 * q + 1647088 * q^7 - 57395630 * q^9 - 712135312 * q^15 + 728554812 * q^17 + 35548816080 * q^23 - 75899954794 * q^25 + 105758138816 * q^31 - 150458001384 * q^33 + 2251546247120 * q^39 - 53229185940 * q^41 - 12527998446432 * q^47 + 8427385380990 * q^49 + 30557833792176 * q^55 + 18277230892472 * q^57 - 36142362113776 * q^63 + 5437123965600 * q^65 + 173249927708016 * q^71 - 182057837882196 * q^73 + 294370273271392 * q^79 + 256903428263798 * q^81 - 1493385390675312 * q^87 + 314213951649228 * q^89 + 851301047679984 * q^95 - 672574291859236 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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