LMFDB - Embedded newform 32.16.b.a.17.2

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			17.2
Root			$0.500000 + 1589.49i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.16.b.a.17.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-6357.96i q^{3} -267219. i q^{5} -120583. q^{7} -2.60748e7 q^{9} +9.60941e7i q^{11} +1.64009e8i q^{13} -1.69897e9 q^{15} -1.25524e8 q^{17} +4.61207e9i q^{19} +7.66663e8i q^{21} -6.78047e9 q^{23} -4.08883e10 q^{25} +7.45526e10i q^{27} -7.66507e10i q^{29} -2.28582e10 q^{31} +6.10963e11 q^{33} +3.22221e10i q^{35} -8.61660e11i q^{37} +1.04276e12 q^{39} -3.52968e11 q^{41} +1.68998e12i q^{43} +6.96767e12i q^{45} -3.84783e12 q^{47} -4.73302e12 q^{49} +7.98078e11i q^{51} +9.03015e12i q^{53} +2.56782e13 q^{55} +2.93234e13 q^{57} +1.13978e13i q^{59} +3.96597e12i q^{61} +3.14418e12 q^{63} +4.38262e13 q^{65} +4.34443e13i q^{67} +4.31100e13i q^{69} +3.11414e13 q^{71} -8.34468e13 q^{73} +2.59966e14i q^{75} -1.15873e13i q^{77} -7.31635e12 q^{79} +9.98580e13 q^{81} -3.96179e14i q^{83} +3.35424e13i q^{85} -4.87342e14 q^{87} +9.54011e13 q^{89} -1.97767e13i q^{91} +1.45332e14i q^{93} +1.23243e15 q^{95} -1.16047e15 q^{97} -2.50563e15i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 6357.96i	− 1.67845i	−0.543784	−	0.839225i	$-0.683009\pi$
$3$	− 6357.96i	− 1.67845i	0.543784	−	0.839225i	$-0.316991\pi$
$4$	0	0
$5$	− 267219.i	− 1.52965i	−0.644238	−	0.764825i	$-0.722825\pi$
$5$	− 267219.i	− 1.52965i	0.644238	−	0.764825i	$-0.277175\pi$
$6$	0	0
$7$	−120583.	−0.0553415	−0.0276708	−	0.999617i	$-0.508809\pi$
$7$	−120583.	−0.0553415	−0.0276708	+	0.999617i	$0.508809\pi$
$8$	0	0
$9$	−2.60748e7	−1.81720
$10$	0	0
$11$	9.60941e7i	1.48680i	0.668849	+	0.743398i	$0.266787\pi$
$11$	9.60941e7i	1.48680i	−0.668849	+	0.743398i	$0.733213\pi$
$12$	0	0
$13$	1.64009e8i	0.724923i	0.931999	+	0.362461i	$0.118063\pi$
$13$	1.64009e8i	0.724923i	−0.931999	+	0.362461i	$0.881937\pi$
$14$	0	0
$15$	−1.69897e9	−2.56744
$16$	0	0
$17$	−1.25524e8	−0.0741926	−0.0370963	−	0.999312i	$-0.511811\pi$
$17$	−1.25524e8	−0.0741926	−0.0370963	+	0.999312i	$0.511811\pi$
$18$	0	0
$19$	4.61207e9i	1.18371i	0.806046	+	0.591853i	$0.201603\pi$
$19$	4.61207e9i	1.18371i	−0.806046	+	0.591853i	$0.798397\pi$
$20$	0	0
$21$	7.66663e8i	0.0928880i
$22$	0	0
$23$	−6.78047e9	−0.415242	−0.207621	−	0.978209i	$-0.566572\pi$
$23$	−6.78047e9	−0.415242	−0.207621	+	0.978209i	$0.566572\pi$
$24$	0	0
$25$	−4.08883e10	−1.33983
$26$	0	0
$27$	7.45526e10i	1.37162i
$28$	0	0
$29$	− 7.66507e10i	− 0.825146i	−0.910924	−	0.412573i	$-0.864630\pi$
$29$	− 7.66507e10i	− 0.825146i	0.910924	−	0.412573i	$-0.135370\pi$
$30$	0	0
$31$	−2.28582e10	−0.149221	−0.0746104	−	0.997213i	$-0.523771\pi$
$31$	−2.28582e10	−0.149221	−0.0746104	+	0.997213i	$0.523771\pi$
$32$	0	0
$33$	6.10963e11	2.49551
$34$	0	0
$35$	3.22221e10i	0.0846532i
$36$	0	0
$37$	− 8.61660e11i	− 1.49219i	−0.665841	−	0.746093i	$-0.731927\pi$
$37$	− 8.61660e11i	− 1.49219i	0.665841	−	0.746093i	$-0.268073\pi$
$38$	0	0
$39$	1.04276e12	1.21675
$40$	0	0
$41$	−3.52968e11	−0.283045	−0.141523	−	0.989935i	$-0.545200\pi$
$41$	−3.52968e11	−0.283045	−0.141523	+	0.989935i	$0.545200\pi$
$42$	0	0
$43$	1.68998e12i	0.948132i	0.880489	+	0.474066i	$0.157214\pi$
$43$	1.68998e12i	0.948132i	−0.880489	+	0.474066i	$0.842786\pi$
$44$	0	0
$45$	6.96767e12i	2.77967i
$46$	0	0
$47$	−3.84783e12	−1.10785	−0.553927	−	0.832565i	$-0.686871\pi$
$47$	−3.84783e12	−1.10785	−0.553927	+	0.832565i	$0.686871\pi$
$48$	0	0
$49$	−4.73302e12	−0.996937
$50$	0	0
$51$	7.98078e11i	0.124529i
$52$	0	0
$53$	9.03015e12i	1.05591i	0.849273	+	0.527954i	$0.177041\pi$
$53$	9.03015e12i	1.05591i	−0.849273	+	0.527954i	$0.822959\pi$
$54$	0	0
$55$	2.56782e13	2.27428
$56$	0	0
$57$	2.93234e13	1.98679
$58$	0	0
$59$	1.13978e13i	0.596254i	0.954526	+	0.298127i	$0.0963620\pi$
$59$	1.13978e13i	0.596254i	−0.954526	+	0.298127i	$0.903638\pi$
$60$	0	0
$61$	3.96597e12i	0.161575i	0.996731	+	0.0807877i	$0.0257436\pi$
$61$	3.96597e12i	0.161575i	−0.996731	+	0.0807877i	$0.974256\pi$
$62$	0	0
$63$	3.14418e12	0.100566
$64$	0	0
$65$	4.38262e13	1.10888
$66$	0	0
$67$	4.34443e13i	0.875733i	0.899040	+	0.437866i	$0.144266\pi$
$67$	4.34443e13i	0.875733i	−0.899040	+	0.437866i	$0.855734\pi$
$68$	0	0
$69$	4.31100e13i	0.696963i
$70$	0	0
$71$	3.11414e13	0.406350	0.203175	−	0.979142i	$-0.434874\pi$
$71$	3.11414e13	0.406350	0.203175	+	0.979142i	$0.434874\pi$
$72$	0	0
$73$	−8.34468e13	−0.884073	−0.442037	−	0.896997i	$-0.645744\pi$
$73$	−8.34468e13	−0.884073	−0.442037	+	0.896997i	$0.645744\pi$
$74$	0	0
$75$	2.59966e14i	2.24884i
$76$	0	0
$77$	− 1.15873e13i	− 0.0822816i
$78$	0	0
$79$	−7.31635e12	−0.0428638	−0.0214319	−	0.999770i	$-0.506823\pi$
$79$	−7.31635e12	−0.0428638	−0.0214319	+	0.999770i	$0.506823\pi$
$80$	0	0
$81$	9.98580e13	0.485004
$82$	0	0
$83$	− 3.96179e14i	− 1.60253i	−0.598312	−	0.801263i	$-0.704162\pi$
$83$	− 3.96179e14i	− 1.60253i	0.598312	−	0.801263i	$-0.295838\pi$
$84$	0	0
$85$	3.35424e13i	0.113489i
$86$	0	0
$87$	−4.87342e14	−1.38497
$88$	0	0
$89$	9.54011e13	0.228628	0.114314	−	0.993445i	$-0.463533\pi$
$89$	9.54011e13	0.228628	0.114314	+	0.993445i	$0.463533\pi$
$90$	0	0
$91$	− 1.97767e13i	− 0.0401183i
$92$	0	0
$93$	1.45332e14i	0.250460i
$94$	0	0
$95$	1.23243e15	1.81066
$96$	0	0
$97$	−1.16047e15	−1.45830	−0.729150	−	0.684354i	$-0.760084\pi$
$97$	−1.16047e15	−1.45830	−0.729150	+	0.684354i	$0.760084\pi$
$98$	0	0
$99$	− 2.50563e15i	− 2.70180i
$100$	0	0
$101$	− 2.07199e15i	− 1.92299i	−0.274818	−	0.961496i	$-0.588618\pi$
$101$	− 2.07199e15i	− 1.92299i	0.274818	−	0.961496i	$-0.411382\pi$
$102$	0	0
$103$	−4.51671e14	−0.361862	−0.180931	−	0.983496i	$-0.557911\pi$
$103$	−4.51671e14	−0.361862	−0.180931	+	0.983496i	$0.557911\pi$
$104$	0	0
$105$	2.04867e14	0.142086
$106$	0	0
$107$	1.58456e15i	0.953962i	0.878914	+	0.476981i	$0.158269\pi$
$107$	1.58456e15i	0.953962i	−0.878914	+	0.476981i	$0.841731\pi$
$108$	0	0
$109$	1.34658e15i	0.705562i	0.935706	+	0.352781i	$0.114764\pi$
$109$	1.34658e15i	0.705562i	−0.935706	+	0.352781i	$0.885236\pi$
$110$	0	0
$111$	−5.47840e15	−2.50456
$112$	0	0
$113$	1.52322e13	0.00609081	0.00304541	−	0.999995i	$-0.499031\pi$
$113$	1.52322e13	0.00609081	0.00304541	+	0.999995i	$0.499031\pi$
$114$	0	0
$115$	1.81187e15i	0.635175i
$116$	0	0
$117$	− 4.27649e15i	− 1.31733i
$118$	0	0
$119$	1.51361e13	0.00410593
$120$	0	0
$121$	−5.05683e15	−1.21056
$122$	0	0
$123$	2.24416e15i	0.475078i
$124$	0	0
$125$	2.77126e15i	0.519819i
$126$	0	0
$127$	7.11367e15	1.18458	0.592292	−	0.805724i	$-0.298223\pi$
$127$	7.11367e15	1.18458	0.592292	+	0.805724i	$0.298223\pi$
$128$	0	0
$129$	1.07448e16	1.59139
$130$	0	0
$131$	− 6.33055e15i	− 0.835424i	−0.908580	−	0.417712i	$-0.862832\pi$
$131$	− 6.33055e15i	− 0.835424i	0.908580	−	0.417712i	$-0.137168\pi$
$132$	0	0
$133$	− 5.56138e14i	− 0.0655081i
$134$	0	0
$135$	1.99219e16	2.09810
$136$	0	0
$137$	1.26171e16	1.19002	0.595011	−	0.803718i	$-0.297148\pi$
$137$	1.26171e16	1.19002	0.595011	+	0.803718i	$0.297148\pi$
$138$	0	0
$139$	− 3.53179e15i	− 0.298802i	−0.988777	−	0.149401i	$-0.952265\pi$
$139$	− 3.53179e15i	− 0.298802i	0.988777	−	0.149401i	$-0.0477345\pi$
$140$	0	0
$141$	2.44644e16i	1.85948i
$142$	0	0
$143$	−1.57603e16	−1.07781
$144$	0	0
$145$	−2.04825e16	−1.26219
$146$	0	0
$147$	3.00924e16i	1.67331i
$148$	0	0
$149$	5.94079e15i	0.298502i	0.988799	+	0.149251i	$0.0476862\pi$
$149$	5.94079e15i	0.298502i	−0.988799	+	0.149251i	$0.952314\pi$
$150$	0	0
$151$	−2.55527e16	−1.16174	−0.580871	−	0.813995i	$-0.697288\pi$
$151$	−2.55527e16	−1.16174	−0.580871	+	0.813995i	$0.697288\pi$
$152$	0	0
$153$	3.27301e15	0.134822
$154$	0	0
$155$	6.10815e15i	0.228256i
$156$	0	0
$157$	5.29272e16i	1.79652i	0.439466	+	0.898259i	$0.355168\pi$
$157$	5.29272e16i	1.79652i	−0.439466	+	0.898259i	$0.644832\pi$
$158$	0	0
$159$	5.74133e16	1.77229
$160$	0	0
$161$	8.17610e14	0.0229801
$162$	0	0
$163$	2.94167e16i	0.753679i	0.926279	+	0.376840i	$0.122989\pi$
$163$	2.94167e16i	0.753679i	−0.926279	+	0.376840i	$0.877011\pi$
$164$	0	0
$165$	− 1.63261e17i	− 3.81726i
$166$	0	0
$167$	−4.03066e16	−0.860999	−0.430500	−	0.902591i	$-0.641663\pi$
$167$	−4.03066e16	−0.860999	−0.430500	+	0.902591i	$0.641663\pi$
$168$	0	0
$169$	2.42871e16	0.474487
$170$	0	0
$171$	− 1.20259e17i	− 2.15102i
$172$	0	0
$173$	8.00780e16i	1.31271i	0.754454	+	0.656353i	$0.227902\pi$
$173$	8.00780e16i	1.31271i	−0.754454	+	0.656353i	$0.772098\pi$
$174$	0	0
$175$	4.93044e15	0.0741482
$176$	0	0
$177$	7.24669e16	1.00078
$178$	0	0
$179$	6.28967e16i	0.798418i	0.916860	+	0.399209i	$0.130715\pi$
$179$	6.28967e16i	0.798418i	−0.916860	+	0.399209i	$0.869285\pi$
$180$	0	0
$181$	− 1.40644e17i	− 1.64260i	−0.570496	−	0.821300i	$-0.693249\pi$
$181$	− 1.40644e17i	− 1.64260i	0.570496	−	0.821300i	$-0.306751\pi$
$182$	0	0
$183$	2.52155e16	0.271196
$184$	0	0
$185$	−2.30252e17	−2.28252
$186$	0	0
$187$	− 1.20621e16i	− 0.110309i
$188$	0	0
$189$	− 8.98978e15i	− 0.0759077i
$190$	0	0
$191$	−2.42747e17	−1.89411	−0.947053	−	0.321076i	$-0.895955\pi$
$191$	−2.42747e17	−1.89411	−0.947053	+	0.321076i	$0.895955\pi$
$192$	0	0
$193$	−8.44339e16	−0.609308	−0.304654	−	0.952463i	$-0.598541\pi$
$193$	−8.44339e16	−0.609308	−0.304654	+	0.952463i	$0.598541\pi$
$194$	0	0
$195$	− 2.78645e17i	− 1.86120i
$196$	0	0
$197$	1.91310e17i	1.18370i	0.806048	+	0.591850i	$0.201602\pi$
$197$	1.91310e17i	1.18370i	−0.806048	+	0.591850i	$0.798398\pi$
$198$	0	0
$199$	−1.18533e16	−0.0679895	−0.0339948	−	0.999422i	$-0.510823\pi$
$199$	−1.18533e16	−0.0679895	−0.0339948	+	0.999422i	$0.510823\pi$
$200$	0	0
$201$	2.76217e17	1.46987
$202$	0	0
$203$	9.24278e15i	0.0456649i
$204$	0	0
$205$	9.43197e16i	0.432960i
$206$	0	0
$207$	1.76799e17	0.754576
$208$	0	0
$209$	−4.43193e17	−1.75993
$210$	0	0
$211$	9.00677e16i	0.333005i	0.986041	+	0.166502i	$0.0532474\pi$
$211$	9.00677e16i	0.333005i	−0.986041	+	0.166502i	$0.946753\pi$
$212$	0	0
$213$	− 1.97996e17i	− 0.682038i
$214$	0	0
$215$	4.51595e17	1.45031
$216$	0	0
$217$	2.75631e15	0.00825811
$218$	0	0
$219$	5.30551e17i	1.48387i
$220$	0	0
$221$	− 2.05870e16i	− 0.0537839i
$222$	0	0
$223$	−4.15081e17	−1.01355	−0.506777	−	0.862077i	$-0.669163\pi$
$223$	−4.15081e17	−1.01355	−0.506777	+	0.862077i	$0.669163\pi$
$224$	0	0
$225$	1.06615e18	2.43473
$226$	0	0
$227$	− 3.49083e17i	− 0.745993i	−0.927833	−	0.372996i	$-0.878330\pi$
$227$	− 3.49083e17i	− 0.745993i	0.927833	−	0.372996i	$-0.121670\pi$
$228$	0	0
$229$	− 4.66508e15i	− 0.00933453i	−0.999989	−	0.00466727i	$-0.998514\pi$
$229$	− 4.66508e15i	− 0.00933453i	0.999989	−	0.00466727i	$-0.00148564\pi$
$230$	0	0
$231$	−7.36718e16	−0.138106
$232$	0	0
$233$	6.90379e17	1.21316	0.606580	−	0.795022i	$-0.292541\pi$
$233$	6.90379e17	1.21316	0.606580	+	0.795022i	$0.292541\pi$
$234$	0	0
$235$	1.02821e18i	1.69463i
$236$	0	0
$237$	4.65170e16i	0.0719448i
$238$	0	0
$239$	9.82227e17	1.42635	0.713177	−	0.700984i	$-0.247255\pi$
$239$	9.82227e17	1.42635	0.713177	+	0.700984i	$0.247255\pi$
$240$	0	0
$241$	−6.25947e16	−0.0853906	−0.0426953	−	0.999088i	$-0.513594\pi$
$241$	−6.25947e16	−0.0853906	−0.0426953	+	0.999088i	$0.513594\pi$
$242$	0	0
$243$	4.34855e17i	0.557567i
$244$	0	0
$245$	1.26475e18i	1.52497i
$246$	0	0
$247$	−7.56419e17	−0.858095
$248$	0	0
$249$	−2.51889e18	−2.68976
$250$	0	0
$251$	− 8.66947e17i	− 0.871846i	−0.899984	−	0.435923i	$-0.856422\pi$
$251$	− 8.66947e17i	− 0.871846i	0.899984	−	0.435923i	$-0.143578\pi$
$252$	0	0
$253$	− 6.51563e17i	− 0.617380i
$254$	0	0
$255$	2.13261e17	0.190485
$256$	0	0
$257$	−6.75670e17	−0.569162	−0.284581	−	0.958652i	$-0.591854\pi$
$257$	−6.75670e17	−0.569162	−0.284581	+	0.958652i	$0.591854\pi$
$258$	0	0
$259$	1.03902e17i	0.0825799i
$260$	0	0
$261$	1.99865e18i	1.49945i
$262$	0	0
$263$	4.36818e16	0.0309480	0.0154740	−	0.999880i	$-0.495074\pi$
$263$	4.36818e16	0.0309480	0.0154740	+	0.999880i	$0.495074\pi$
$264$	0	0
$265$	2.41303e18	1.61517
$266$	0	0
$267$	− 6.06557e17i	− 0.383740i
$268$	0	0
$269$	− 1.39181e18i	− 0.832604i	−0.909226	−	0.416302i	$-0.863326\pi$
$269$	− 1.39181e18i	− 0.832604i	0.909226	−	0.416302i	$-0.136674\pi$
$270$	0	0
$271$	−1.21335e18	−0.686617	−0.343309	−	0.939223i	$-0.611548\pi$
$271$	−1.21335e18	−0.686617	−0.343309	+	0.939223i	$0.611548\pi$
$272$	0	0
$273$	−1.25739e17	−0.0673366
$274$	0	0
$275$	− 3.92913e18i	− 1.99205i
$276$	0	0
$277$	− 1.97331e18i	− 0.947540i	−0.880649	−	0.473770i	$-0.842893\pi$
$277$	− 1.97331e18i	− 0.947540i	0.880649	−	0.473770i	$-0.157107\pi$
$278$	0	0
$279$	5.96023e17	0.271164
$280$	0	0
$281$	2.31539e18	0.998449	0.499225	−	0.866473i	$-0.333618\pi$
$281$	2.31539e18	0.998449	0.499225	+	0.866473i	$0.333618\pi$
$282$	0	0
$283$	− 2.72039e18i	− 1.11233i	−0.831073	−	0.556163i	$-0.812273\pi$
$283$	− 2.72039e18i	− 1.11233i	0.831073	−	0.556163i	$-0.187727\pi$
$284$	0	0
$285$	− 7.83575e18i	− 3.03910i
$286$	0	0
$287$	4.25620e16	0.0156642
$288$	0	0
$289$	−2.84667e18	−0.994495
$290$	0	0
$291$	7.37824e18i	2.44768i
$292$	0	0
$293$	5.75600e18i	1.81390i	0.421237	+	0.906951i	$0.361596\pi$
$293$	5.75600e18i	1.81390i	−0.421237	+	0.906951i	$0.638404\pi$
$294$	0	0
$295$	3.04571e18	0.912061
$296$	0	0
$297$	−7.16406e18	−2.03932
$298$	0	0
$299$	− 1.11206e18i	− 0.301018i
$300$	0	0
$301$	− 2.03783e17i	− 0.0524711i
$302$	0	0
$303$	−1.31736e19	−3.22765
$304$	0	0
$305$	1.05978e18	0.247154
$306$	0	0
$307$	2.54727e18i	0.565636i	0.959174	+	0.282818i	$0.0912692\pi$
$307$	2.54727e18i	0.565636i	−0.959174	+	0.282818i	$0.908731\pi$
$308$	0	0
$309$	2.87171e18i	0.607368i
$310$	0	0
$311$	5.78305e17	0.116534	0.0582671	−	0.998301i	$-0.481442\pi$
$311$	5.78305e17	0.116534	0.0582671	+	0.998301i	$0.481442\pi$
$312$	0	0
$313$	2.11174e18	0.405562	0.202781	−	0.979224i	$-0.435002\pi$
$313$	2.11174e18	0.405562	0.202781	+	0.979224i	$0.435002\pi$
$314$	0	0
$315$	− 8.40183e17i	− 0.153831i
$316$	0	0
$317$	− 6.00631e17i	− 0.104873i	−0.998624	−	0.0524364i	$-0.983301\pi$
$317$	− 6.00631e17i	− 0.104873i	0.998624	−	0.0524364i	$-0.0166987\pi$
$318$	0	0
$319$	7.36568e18	1.22682
$320$	0	0
$321$	1.00746e19	1.60118
$322$	0	0
$323$	− 5.78926e17i	− 0.0878222i
$324$	0	0
$325$	− 6.70604e18i	− 0.971272i
$326$	0	0
$327$	8.56153e18	1.18425
$328$	0	0
$329$	4.63984e17	0.0613103
$330$	0	0
$331$	− 3.00162e18i	− 0.379006i	−0.981880	−	0.189503i	$-0.939312\pi$
$331$	− 3.00162e18i	− 0.379006i	0.981880	−	0.189503i	$-0.0606877\pi$
$332$	0	0
$333$	2.24676e19i	2.71159i
$334$	0	0
$335$	1.16091e19	1.33956
$336$	0	0
$337$	6.11724e16	0.00675042	0.00337521	−	0.999994i	$-0.498926\pi$
$337$	6.11724e16	0.00675042	0.00337521	+	0.999994i	$0.498926\pi$
$338$	0	0
$339$	− 9.68460e16i	− 0.0102231i
$340$	0	0
$341$	− 2.19654e18i	− 0.221861i
$342$	0	0
$343$	1.14320e18	0.110514
$344$	0	0
$345$	1.15198e19	1.06611
$346$	0	0
$347$	8.50746e18i	0.753926i	0.926228	+	0.376963i	$0.123032\pi$
$347$	8.50746e18i	0.753926i	−0.926228	+	0.376963i	$0.876968\pi$
$348$	0	0
$349$	7.39145e17i	0.0627392i	0.999508	+	0.0313696i	$0.00998689\pi$
$349$	7.39145e17i	0.0627392i	−0.999508	+	0.0313696i	$0.990013\pi$
$350$	0	0
$351$	−1.22273e19	−0.994320
$352$	0	0
$353$	−2.03790e19	−1.58808	−0.794042	−	0.607863i	$-0.792027\pi$
$353$	−2.03790e19	−1.58808	−0.794042	+	0.607863i	$0.792027\pi$
$354$	0	0
$355$	− 8.32156e18i	− 0.621573i
$356$	0	0
$357$	− 9.62347e16i	− 0.00689160i
$358$	0	0
$359$	−6.59964e18	−0.453223	−0.226611	−	0.973985i	$-0.572765\pi$
$359$	−6.59964e18	−0.453223	−0.226611	+	0.973985i	$0.572765\pi$
$360$	0	0
$361$	−6.09005e18	−0.401159
$362$	0	0
$363$	3.21511e19i	2.03187i
$364$	0	0
$365$	2.22986e19i	1.35232i
$366$	0	0
$367$	−7.81363e18	−0.454839	−0.227419	−	0.973797i	$-0.573029\pi$
$367$	−7.81363e18	−0.454839	−0.227419	+	0.973797i	$0.573029\pi$
$368$	0	0
$369$	9.20355e18	0.514349
$370$	0	0
$371$	− 1.08888e18i	− 0.0584355i
$372$	0	0
$373$	− 3.65402e19i	− 1.88345i	−0.336382	−	0.941726i	$-0.609203\pi$
$373$	− 3.65402e19i	− 1.88345i	0.336382	−	0.941726i	$-0.390797\pi$
$374$	0	0
$375$	1.76196e19	0.872491
$376$	0	0
$377$	1.25714e19	0.598167
$378$	0	0
$379$	− 5.85785e18i	− 0.267882i	−0.990989	−	0.133941i	$-0.957237\pi$
$379$	− 5.85785e18i	− 0.267882i	0.990989	−	0.133941i	$-0.0427633\pi$
$380$	0	0
$381$	− 4.52284e19i	− 1.98826i
$382$	0	0
$383$	−3.82207e19	−1.61550	−0.807752	−	0.589522i	$-0.799316\pi$
$383$	−3.82207e19	−1.61550	−0.807752	+	0.589522i	$0.799316\pi$
$384$	0	0
$385$	−3.09635e18	−0.125862
$386$	0	0
$387$	− 4.40659e19i	− 1.72294i
$388$	0	0
$389$	− 9.25088e18i	− 0.347985i	−0.984747	−	0.173993i	$-0.944333\pi$
$389$	− 9.25088e18i	− 0.347985i	0.984747	−	0.173993i	$-0.0556669\pi$
$390$	0	0
$391$	8.51113e17	0.0308078
$392$	0	0
$393$	−4.02494e19	−1.40222
$394$	0	0
$395$	1.95507e18i	0.0655667i
$396$	0	0
$397$	5.20456e19i	1.68056i	0.542150	+	0.840282i	$0.317611\pi$
$397$	5.20456e19i	1.68056i	−0.542150	+	0.840282i	$0.682389\pi$
$398$	0	0
$399$	−3.53590e18	−0.109952
$400$	0	0
$401$	3.31013e19	0.991430	0.495715	−	0.868485i	$-0.334906\pi$
$401$	3.31013e19	0.991430	0.495715	+	0.868485i	$0.334906\pi$
$402$	0	0
$403$	− 3.74894e18i	− 0.108174i
$404$	0	0
$405$	− 2.66839e19i	− 0.741886i
$406$	0	0
$407$	8.28005e19	2.21858
$408$	0	0
$409$	−2.01464e19	−0.520323	−0.260162	−	0.965565i	$-0.583776\pi$
$409$	−2.01464e19	−0.520323	−0.260162	+	0.965565i	$0.583776\pi$
$410$	0	0
$411$	− 8.02190e19i	− 1.99739i
$412$	0	0
$413$	− 1.37438e18i	− 0.0329976i
$414$	0	0
$415$	−1.05866e20	−2.45131
$416$	0	0
$417$	−2.24550e19	−0.501524
$418$	0	0
$419$	6.19828e18i	0.133557i	0.997768	+	0.0667784i	$0.0212720\pi$
$419$	6.19828e18i	0.133557i	−0.997768	+	0.0667784i	$0.978728\pi$
$420$	0	0
$421$	− 1.87462e19i	− 0.389760i	−0.980827	−	0.194880i	$-0.937568\pi$
$421$	− 1.87462e19i	− 0.389760i	0.980827	−	0.194880i	$-0.0624317\pi$
$422$	0	0
$423$	1.00331e20	2.01319
$424$	0	0
$425$	5.13247e18	0.0994053
$426$	0	0
$427$	− 4.78228e17i	− 0.00894183i
$428$	0	0
$429$	1.00203e20i	1.80905i
$430$	0	0
$431$	−4.50478e19	−0.785405	−0.392702	−	0.919666i	$-0.628460\pi$
$431$	−4.50478e19	−0.785405	−0.392702	+	0.919666i	$0.628460\pi$
$432$	0	0
$433$	−8.20647e19	−1.38197	−0.690983	−	0.722871i	$-0.742822\pi$
$433$	−8.20647e19	−1.38197	−0.690983	+	0.722871i	$0.742822\pi$
$434$	0	0
$435$	1.30227e20i	2.11851i
$436$	0	0
$437$	− 3.12720e19i	− 0.491524i
$438$	0	0
$439$	−8.92387e19	−1.35541	−0.677703	−	0.735336i	$-0.737025\pi$
$439$	−8.92387e19	−1.35541	−0.677703	+	0.735336i	$0.737025\pi$
$440$	0	0
$441$	1.23412e20	1.81163
$442$	0	0
$443$	8.32102e19i	1.18072i	0.807138	+	0.590362i	$0.201015\pi$
$443$	8.32102e19i	1.18072i	−0.807138	+	0.590362i	$0.798985\pi$
$444$	0	0
$445$	− 2.54930e19i	− 0.349720i
$446$	0	0
$447$	3.77713e19	0.501020
$448$	0	0
$449$	6.70535e19	0.860150	0.430075	−	0.902793i	$-0.358487\pi$
$449$	6.70535e19	0.860150	0.430075	+	0.902793i	$0.358487\pi$
$450$	0	0
$451$	− 3.39181e19i	− 0.420831i
$452$	0	0
$453$	1.62463e20i	1.94993i
$454$	0	0
$455$	−5.28470e18	−0.0613670
$456$	0	0
$457$	2.49692e19	0.280565	0.140282	−	0.990112i	$-0.455199\pi$
$457$	2.49692e19	0.280565	0.140282	+	0.990112i	$0.455199\pi$
$458$	0	0
$459$	− 9.35815e18i	− 0.101764i
$460$	0	0
$461$	− 1.25078e20i	− 1.31650i	−0.752797	−	0.658252i	$-0.771296\pi$
$461$	− 1.25078e20i	− 1.31650i	0.752797	−	0.658252i	$-0.228704\pi$
$462$	0	0
$463$	2.00408e19	0.204201	0.102101	−	0.994774i	$-0.467444\pi$
$463$	2.00408e19	0.204201	0.102101	+	0.994774i	$0.467444\pi$
$464$	0	0
$465$	3.88354e19	0.383116
$466$	0	0
$467$	1.69751e20i	1.62157i	0.585344	+	0.810785i	$0.300960\pi$
$467$	1.69751e20i	1.62157i	−0.585344	+	0.810785i	$0.699040\pi$
$468$	0	0
$469$	− 5.23865e18i	− 0.0484644i
$470$	0	0
$471$	3.36509e20	3.01537
$472$	0	0
$473$	−1.62397e20	−1.40968
$474$	0	0
$475$	− 1.88580e20i	− 1.58596i
$476$	0	0
$477$	− 2.35459e20i	− 1.91879i
$478$	0	0
$479$	5.75801e19	0.454733	0.227366	−	0.973809i	$-0.426989\pi$
$479$	5.75801e19	0.454733	0.227366	+	0.973809i	$0.426989\pi$
$480$	0	0
$481$	1.41320e20	1.08172
$482$	0	0
$483$	− 5.19833e18i	− 0.0385710i
$484$	0	0
$485$	3.10100e20i	2.23069i
$486$	0	0
$487$	7.29502e19	0.508814	0.254407	−	0.967097i	$-0.418120\pi$
$487$	7.29502e19	0.508814	0.254407	+	0.967097i	$0.418120\pi$
$488$	0	0
$489$	1.87030e20	1.26501
$490$	0	0
$491$	1.69291e20i	1.11051i	0.831680	+	0.555256i	$0.187380\pi$
$491$	1.69291e20i	1.11051i	−0.831680	+	0.555256i	$0.812620\pi$
$492$	0	0
$493$	9.62151e18i	0.0612197i
$494$	0	0
$495$	−6.69552e20	−4.13281
$496$	0	0
$497$	−3.75512e18	−0.0224880
$498$	0	0
$499$	− 3.02072e20i	− 1.75532i	−0.479281	−	0.877661i	$-0.659103\pi$
$499$	− 3.02072e20i	− 1.75532i	0.479281	−	0.877661i	$-0.340897\pi$
$500$	0	0
$501$	2.56268e20i	1.44514i
$502$	0	0
$503$	−2.31647e20	−1.26785	−0.633925	−	0.773395i	$-0.718557\pi$
$503$	−2.31647e20	−1.26785	−0.633925	+	0.773395i	$0.718557\pi$
$504$	0	0
$505$	−5.53675e20	−2.94151
$506$	0	0
$507$	− 1.54416e20i	− 0.796403i
$508$	0	0
$509$	− 7.79841e19i	− 0.390502i	−0.980753	−	0.195251i	$-0.937448\pi$
$509$	− 7.79841e19i	− 0.390502i	0.980753	−	0.195251i	$-0.0625521\pi$
$510$	0	0
$511$	1.00623e19	0.0489260
$512$	0	0
$513$	−3.43842e20	−1.62360
$514$	0	0
$515$	1.20695e20i	0.553523i
$516$	0	0
$517$	− 3.69754e20i	− 1.64715i
$518$	0	0
$519$	5.09133e20	2.20331
$520$	0	0
$521$	3.41951e20	1.43774	0.718871	−	0.695144i	$-0.244659\pi$
$521$	3.41951e20	1.43774	0.718871	+	0.695144i	$0.244659\pi$
$522$	0	0
$523$	2.11396e20i	0.863642i	0.901959	+	0.431821i	$0.142129\pi$
$523$	2.11396e20i	0.863642i	−0.901959	+	0.431821i	$0.857871\pi$
$524$	0	0
$525$	− 3.13476e19i	− 0.124454i
$526$	0	0
$527$	2.86926e18	0.0110711
$528$	0	0
$529$	−2.20660e20	−0.827574
$530$	0	0
$531$	− 2.97195e20i	− 1.08351i
$532$	0	0
$533$	− 5.78898e19i	− 0.205186i
$534$	0	0
$535$	4.23425e20	1.45923
$536$	0	0
$537$	3.99895e20	1.34010
$538$	0	0
$539$	− 4.54815e20i	− 1.48224i
$540$	0	0
$541$	− 5.13131e20i	− 1.62648i	−0.581929	−	0.813240i	$-0.697702\pi$
$541$	− 5.13131e20i	− 1.62648i	0.581929	−	0.813240i	$-0.302298\pi$
$542$	0	0
$543$	−8.94209e20	−2.75702
$544$	0	0
$545$	3.59833e20	1.07926
$546$	0	0
$547$	− 6.07670e20i	− 1.77322i	−0.462516	−	0.886611i	$-0.653053\pi$
$547$	− 6.07670e20i	− 1.77322i	0.462516	−	0.886611i	$-0.346947\pi$
$548$	0	0
$549$	− 1.03412e20i	− 0.293614i
$550$	0	0
$551$	3.53518e20	0.976731
$552$	0	0
$553$	8.82228e17	0.00237215
$554$	0	0
$555$	1.46393e21i	3.83110i
$556$	0	0
$557$	− 1.16265e20i	− 0.296166i	−0.988975	−	0.148083i	$-0.952690\pi$
$557$	− 1.16265e20i	− 0.296166i	0.988975	−	0.148083i	$-0.0473102\pi$
$558$	0	0
$559$	−2.77172e20	−0.687322
$560$	0	0
$561$	−7.66905e19	−0.185149
$562$	0	0
$563$	4.19500e20i	0.986095i	0.870002	+	0.493048i	$0.164117\pi$
$563$	4.19500e20i	0.986095i	−0.870002	+	0.493048i	$0.835883\pi$
$564$	0	0
$565$	− 4.07034e18i	− 0.00931681i
$566$	0	0
$567$	−1.20412e19	−0.0268409
$568$	0	0
$569$	−2.54681e20	−0.552910	−0.276455	−	0.961027i	$-0.589160\pi$
$569$	−2.54681e20	−0.552910	−0.276455	+	0.961027i	$0.589160\pi$
$570$	0	0
$571$	− 5.64239e20i	− 1.19314i	−0.802560	−	0.596571i	$-0.796529\pi$
$571$	− 5.64239e20i	− 1.19314i	0.802560	−	0.596571i	$-0.203471\pi$
$572$	0	0
$573$	1.54338e21i	3.17916i
$574$	0	0
$575$	2.77242e20	0.556353
$576$	0	0
$577$	2.42327e20	0.473787	0.236893	−	0.971536i	$-0.423871\pi$
$577$	2.42327e20	0.473787	0.236893	+	0.971536i	$0.423871\pi$
$578$	0	0
$579$	5.36828e20i	1.02269i
$580$	0	0
$581$	4.77724e19i	0.0886863i
$582$	0	0
$583$	−8.67744e20	−1.56992
$584$	0	0
$585$	−1.14276e21	−2.01505
$586$	0	0
$587$	2.53984e20i	0.436536i	0.975889	+	0.218268i	$0.0700407\pi$
$587$	2.53984e20i	0.436536i	−0.975889	+	0.218268i	$0.929959\pi$
$588$	0	0
$589$	− 1.05424e20i	− 0.176634i
$590$	0	0
$591$	1.21634e21	1.98678
$592$	0	0
$593$	4.91791e20	0.783196	0.391598	−	0.920136i	$-0.371922\pi$
$593$	4.91791e20	0.783196	0.391598	+	0.920136i	$0.371922\pi$
$594$	0	0
$595$	− 4.04465e18i	− 0.00628063i
$596$	0	0
$597$	7.53630e19i	0.114117i
$598$	0	0
$599$	1.22676e21	1.81159	0.905793	−	0.423721i	$-0.139276\pi$
$599$	1.22676e21	1.81159	0.905793	+	0.423721i	$0.139276\pi$
$600$	0	0
$601$	8.81075e20	1.26898	0.634489	−	0.772932i	$-0.281210\pi$
$601$	8.81075e20	1.26898	0.634489	+	0.772932i	$0.281210\pi$
$602$	0	0
$603$	− 1.13280e21i	− 1.59138i
$604$	0	0
$605$	1.35128e21i	1.85174i
$606$	0	0
$607$	−8.63946e20	−1.15497	−0.577486	−	0.816401i	$-0.695966\pi$
$607$	−8.63946e20	−1.15497	−0.577486	+	0.816401i	$0.695966\pi$
$608$	0	0
$609$	5.87652e19	0.0766462
$610$	0	0
$611$	− 6.31078e20i	− 0.803108i
$612$	0	0
$613$	− 5.43647e20i	− 0.675092i	−0.941309	−	0.337546i	$-0.890403\pi$
$613$	− 5.43647e20i	− 0.675092i	0.941309	−	0.337546i	$-0.109597\pi$
$614$	0	0
$615$	5.99681e20	0.726703
$616$	0	0
$617$	9.53349e20	1.12749	0.563745	−	0.825949i	$-0.309360\pi$
$617$	9.53349e20	1.12749	0.563745	+	0.825949i	$0.309360\pi$
$618$	0	0
$619$	5.58681e20i	0.644888i	0.946589	+	0.322444i	$0.104504\pi$
$619$	5.58681e20i	0.644888i	−0.946589	+	0.322444i	$0.895496\pi$
$620$	0	0
$621$	− 5.05502e20i	− 0.569555i
$622$	0	0
$623$	−1.15038e19	−0.0126526
$624$	0	0
$625$	−5.07280e20	−0.544687
$626$	0	0
$627$	2.81780e21i	2.95395i
$628$	0	0
$629$	1.08159e20i	0.110709i
$630$	0	0
$631$	−3.62782e20	−0.362599	−0.181299	−	0.983428i	$-0.558030\pi$
$631$	−3.62782e20	−0.362599	−0.181299	+	0.983428i	$0.558030\pi$
$632$	0	0
$633$	5.72647e20	0.558932
$634$	0	0
$635$	− 1.90091e21i	− 1.81200i
$636$	0	0
$637$	− 7.76256e20i	− 0.722702i
$638$	0	0
$639$	−8.12004e20	−0.738417
$640$	0	0
$641$	−1.61218e21	−1.43212	−0.716059	−	0.698040i	$-0.754056\pi$
$641$	−1.61218e21	−1.43212	−0.716059	+	0.698040i	$0.754056\pi$
$642$	0	0
$643$	− 7.38606e20i	− 0.640959i	−0.947255	−	0.320480i	$-0.896156\pi$
$643$	− 7.38606e20i	− 0.640959i	0.947255	−	0.320480i	$-0.103844\pi$
$644$	0	0
$645$	− 2.87122e21i	− 2.43427i
$646$	0	0
$647$	1.36574e20	0.113132	0.0565659	−	0.998399i	$-0.481985\pi$
$647$	1.36574e20	0.113132	0.0565659	+	0.998399i	$0.481985\pi$
$648$	0	0
$649$	−1.09526e21	−0.886509
$650$	0	0
$651$	− 1.75245e19i	− 0.0138608i
$652$	0	0
$653$	1.25905e21i	0.973183i	0.873630	+	0.486592i	$0.161760\pi$
$653$	1.25905e21i	0.973183i	−0.873630	+	0.486592i	$0.838240\pi$
$654$	0	0
$655$	−1.69164e21	−1.27791
$656$	0	0
$657$	2.17586e21	1.60653
$658$	0	0
$659$	8.06495e20i	0.582051i	0.956715	+	0.291026i	$0.0939965\pi$
$659$	8.06495e20i	0.582051i	−0.956715	+	0.291026i	$0.906003\pi$
$660$	0	0
$661$	− 1.10006e21i	− 0.776081i	−0.921642	−	0.388041i	$-0.873152\pi$
$661$	− 1.10006e21i	− 0.776081i	0.921642	−	0.388041i	$-0.126848\pi$
$662$	0	0
$663$	−1.30892e20	−0.0902735
$664$	0	0
$665$	−1.48610e20	−0.100204
$666$	0	0
$667$	5.19728e20i	0.342635i
$668$	0	0
$669$	2.63907e21i	1.70120i
$670$	0	0
$671$	−3.81106e20	−0.240230
$672$	0	0
$673$	−2.15849e21	−1.33057	−0.665285	−	0.746589i	$-0.731690\pi$
$673$	−2.15849e21	−1.33057	−0.665285	+	0.746589i	$0.731690\pi$
$674$	0	0
$675$	− 3.04833e21i	− 1.83774i
$676$	0	0
$677$	− 3.56159e20i	− 0.210005i	−0.994472	−	0.105002i	$-0.966515\pi$
$677$	− 3.56159e20i	− 0.210005i	0.994472	−	0.105002i	$-0.0334850\pi$
$678$	0	0
$679$	1.39933e20	0.0807046
$680$	0	0
$681$	−2.21946e21	−1.25211
$682$	0	0
$683$	1.66633e21i	0.919614i	0.888019	+	0.459807i	$0.152081\pi$
$683$	1.66633e21i	0.919614i	−0.888019	+	0.459807i	$0.847919\pi$
$684$	0	0
$685$	− 3.37152e21i	− 1.82032i
$686$	0	0
$687$	−2.96604e19	−0.0156676
$688$	0	0
$689$	−1.48102e21	−0.765451
$690$	0	0
$691$	− 1.01961e21i	− 0.515640i	−0.966193	−	0.257820i	$-0.916996\pi$
$691$	− 1.01961e21i	− 0.515640i	0.966193	−	0.257820i	$-0.0830042\pi$
$692$	0	0
$693$	3.02137e20i	0.149522i
$694$	0	0
$695$	−9.43760e20	−0.457063
$696$	0	0
$697$	4.43060e19	0.0209999
$698$	0	0
$699$	− 4.38940e21i	− 2.03623i
$700$	0	0
$701$	2.21860e21i	1.00738i	0.863885	+	0.503690i	$0.168024\pi$
$701$	2.21860e21i	1.00738i	−0.863885	+	0.503690i	$0.831976\pi$
$702$	0	0
$703$	3.97404e21	1.76631
$704$	0	0
$705$	6.53734e21	2.84435
$706$	0	0
$707$	2.49847e20i	0.106421i
$708$	0	0
$709$	4.74711e21i	1.97962i	0.142383	+	0.989812i	$0.454524\pi$
$709$	4.74711e21i	1.97962i	−0.142383	+	0.989812i	$0.545476\pi$
$710$	0	0
$711$	1.90772e20	0.0778920
$712$	0	0
$713$	1.54989e20	0.0619627
$714$	0	0
$715$	4.21144e21i	1.64868i
$716$	0	0
$717$	− 6.24496e21i	− 2.39407i
$718$	0	0
$719$	2.72163e20	0.102179	0.0510896	−	0.998694i	$-0.483731\pi$
$719$	2.72163e20	0.102179	0.0510896	+	0.998694i	$0.483731\pi$
$720$	0	0
$721$	5.44639e19	0.0200260
$722$	0	0
$723$	3.97975e20i	0.143324i
$724$	0	0
$725$	3.13412e21i	1.10555i
$726$	0	0
$727$	−3.64179e21	−1.25837	−0.629184	−	0.777257i	$-0.716611\pi$
$727$	−3.64179e21	−1.25837	−0.629184	+	0.777257i	$0.716611\pi$
$728$	0	0
$729$	4.19764e21	1.42085
$730$	0	0
$731$	− 2.12134e20i	− 0.0703443i
$732$	0	0
$733$	− 2.24321e21i	− 0.728770i	−0.931248	−	0.364385i	$-0.881279\pi$
$733$	− 2.24321e21i	− 0.728770i	0.931248	−	0.364385i	$-0.118721\pi$
$734$	0	0
$735$	8.04125e21	2.55958
$736$	0	0
$737$	−4.17474e21	−1.30204
$738$	0	0
$739$	− 4.72145e21i	− 1.44292i	−0.692456	−	0.721460i	$-0.743471\pi$
$739$	− 4.72145e21i	− 1.44292i	0.692456	−	0.721460i	$-0.256529\pi$
$740$	0	0
$741$	4.80928e21i	1.44027i
$742$	0	0
$743$	−3.14778e21	−0.923822	−0.461911	−	0.886926i	$-0.652836\pi$
$743$	−3.14778e21	−0.923822	−0.461911	+	0.886926i	$0.652836\pi$
$744$	0	0
$745$	1.58749e21	0.456603
$746$	0	0
$747$	1.03303e22i	2.91210i
$748$	0	0
$749$	− 1.91071e20i	− 0.0527937i
$750$	0	0
$751$	1.50411e20	0.0407361	0.0203681	−	0.999793i	$-0.493516\pi$
$751$	1.50411e20	0.0407361	0.0203681	+	0.999793i	$0.493516\pi$
$752$	0	0
$753$	−5.51202e21	−1.46335
$754$	0	0
$755$	6.82817e21i	1.77706i
$756$	0	0
$757$	4.49997e21i	1.14813i	0.818811	+	0.574064i	$0.194634\pi$
$757$	4.49997e21i	1.14813i	−0.818811	+	0.574064i	$0.805366\pi$
$758$	0	0
$759$	−4.14261e21	−1.03624
$760$	0	0
$761$	−2.33535e21	−0.572754	−0.286377	−	0.958117i	$-0.592451\pi$
$761$	−2.33535e21	−0.572754	−0.286377	+	0.958117i	$0.592451\pi$
$762$	0	0
$763$	− 1.62375e20i	− 0.0390469i
$764$	0	0
$765$	− 8.74611e20i	− 0.206231i
$766$	0	0
$767$	−1.86934e21	−0.432238
$768$	0	0
$769$	1.43868e21	0.326224	0.163112	−	0.986608i	$-0.447847\pi$
$769$	1.43868e21	0.326224	0.163112	+	0.986608i	$0.447847\pi$
$770$	0	0
$771$	4.29588e21i	0.955310i
$772$	0	0
$773$	− 5.51890e21i	− 1.20367i	−0.798621	−	0.601834i	$-0.794437\pi$
$773$	− 5.51890e21i	− 1.20367i	0.798621	−	0.601834i	$-0.205563\pi$
$774$	0	0
$775$	9.34634e20	0.199930
$776$	0	0
$777$	6.60603e20	0.138606
$778$	0	0
$779$	− 1.62791e21i	− 0.335043i
$780$	0	0
$781$	2.99250e21i	0.604160i
$782$	0	0
$783$	5.71451e21	1.13179
$784$	0	0
$785$	1.41432e22	2.74804
$786$	0	0
$787$	− 6.07126e21i	− 1.15736i	−0.815555	−	0.578680i	$-0.803568\pi$
$787$	− 6.07126e21i	− 1.15736i	0.815555	−	0.578680i	$-0.196432\pi$
$788$	0	0
$789$	− 2.77727e20i	− 0.0519447i
$790$	0	0
$791$	−1.83675e18	−0.000337075 0
$792$	0	0
$793$	−6.50453e20	−0.117130
$794$	0	0
$795$	− 1.53419e22i	− 2.71098i
$796$	0	0
$797$	− 1.11112e22i	− 1.92674i	−0.268184	−	0.963368i	$-0.586423\pi$
$797$	− 1.11112e22i	− 1.92674i	0.268184	−	0.963368i	$-0.413577\pi$
$798$	0	0
$799$	4.82996e20	0.0821945
$800$	0	0
$801$	−2.48756e21	−0.415461
$802$	0	0
$803$	− 8.01874e21i	− 1.31444i
$804$	0	0
$805$	− 2.18481e20i	− 0.0351515i
$806$	0	0
$807$	−8.84909e21	−1.39748
$808$	0	0
$809$	4.64825e21	0.720568	0.360284	−	0.932843i	$-0.382680\pi$
$809$	4.64825e21	0.720568	0.360284	+	0.932843i	$0.382680\pi$
$810$	0	0
$811$	6.95327e21i	1.05811i	0.848586	+	0.529057i	$0.177454\pi$
$811$	6.95327e21i	1.05811i	−0.848586	+	0.529057i	$0.822546\pi$
$812$	0	0
$813$	7.71440e21i	1.15245i
$814$	0	0
$815$	7.86069e21	1.15287
$816$	0	0
$817$	−7.79431e21	−1.12231
$818$	0	0
$819$	5.15672e20i	0.0729028i
$820$	0	0
$821$	− 1.63651e21i	− 0.227167i	−0.993528	−	0.113584i	$-0.963767\pi$
$821$	− 1.63651e21i	− 0.227167i	0.993528	−	0.113584i	$-0.0362330\pi$
$822$	0	0
$823$	1.30316e22	1.77623	0.888116	−	0.459620i	$-0.152014\pi$
$823$	1.30316e22	1.77623	0.888116	+	0.459620i	$0.152014\pi$
$824$	0	0
$825$	−2.49812e22	−3.34356
$826$	0	0
$827$	− 5.43834e21i	− 0.714784i	−0.933954	−	0.357392i	$-0.883666\pi$
$827$	− 5.43834e21i	− 0.714784i	0.933954	−	0.357392i	$-0.116334\pi$
$828$	0	0
$829$	4.84909e21i	0.625894i	0.949771	+	0.312947i	$0.101316\pi$
$829$	4.84909e21i	0.625894i	−0.949771	+	0.312947i	$0.898684\pi$
$830$	0	0
$831$	−1.25462e22	−1.59040
$832$	0	0
$833$	5.94108e20	0.0739653
$834$	0	0
$835$	1.07707e22i	1.31703i
$836$	0	0
$837$	− 1.70414e21i	− 0.204675i
$838$	0	0
$839$	−9.02485e21	−1.06470	−0.532348	−	0.846526i	$-0.678690\pi$
$839$	−9.02485e21	−1.06470	−0.532348	+	0.846526i	$0.678690\pi$
$840$	0	0
$841$	2.75386e21	0.319133
$842$	0	0
$843$	− 1.47211e22i	− 1.67585i
$844$	0	0
$845$	− 6.48996e21i	− 0.725799i
$846$	0	0
$847$	6.09768e20	0.0669945
$848$	0	0
$849$	−1.72961e22	−1.86698
$850$	0	0
$851$	5.84246e21i	0.619618i
$852$	0	0
$853$	− 4.32653e21i	− 0.450840i	−0.974262	−	0.225420i	$-0.927625\pi$
$853$	− 4.32653e21i	− 0.450840i	0.974262	−	0.225420i	$-0.0723754\pi$
$854$	0	0
$855$	−3.21354e22	−3.29031
$856$	0	0
$857$	1.15996e22	1.16704	0.583521	−	0.812098i	$-0.301675\pi$
$857$	1.15996e22	1.16704	0.583521	+	0.812098i	$0.301675\pi$
$858$	0	0
$859$	2.37330e21i	0.234641i	0.993094	+	0.117321i	$0.0374305\pi$
$859$	2.37330e21i	0.234641i	−0.993094	+	0.117321i	$0.962569\pi$
$860$	0	0
$861$	− 2.70607e20i	− 0.0262915i
$862$	0	0
$863$	9.83961e21	0.939501	0.469750	−	0.882799i	$-0.344344\pi$
$863$	9.83961e21	0.939501	0.469750	+	0.882799i	$0.344344\pi$
$864$	0	0
$865$	2.13984e22	2.00798
$866$	0	0
$867$	1.80990e22i	1.66921i
$868$	0	0
$869$	− 7.03058e20i	− 0.0637298i
$870$	0	0
$871$	−7.12524e21	−0.634838
$872$	0	0
$873$	3.02591e22	2.65002
$874$	0	0
$875$	− 3.34167e20i	− 0.0287676i
$876$	0	0
$877$	1.13967e22i	0.964457i	0.876046	+	0.482228i	$0.160172\pi$
$877$	1.13967e22i	0.964457i	−0.876046	+	0.482228i	$0.839828\pi$
$878$	0	0
$879$	3.65964e22	3.04454
$880$	0	0
$881$	−1.32703e22	−1.08533	−0.542665	−	0.839949i	$-0.682585\pi$
$881$	−1.32703e22	−1.08533	−0.542665	+	0.839949i	$0.682585\pi$
$882$	0	0
$883$	− 1.60387e22i	− 1.28963i	−0.764340	−	0.644813i	$-0.776935\pi$
$883$	− 1.60387e22i	− 1.28963i	0.764340	−	0.644813i	$-0.223065\pi$
$884$	0	0
$885$	− 1.93645e22i	− 1.53085i
$886$	0	0
$887$	1.17227e22	0.911169	0.455584	−	0.890193i	$-0.349430\pi$
$887$	1.17227e22	0.911169	0.455584	+	0.890193i	$0.349430\pi$
$888$	0	0
$889$	−8.57788e20	−0.0655566
$890$	0	0
$891$	9.59577e21i	0.721102i
$892$	0	0
$893$	− 1.77465e22i	− 1.31137i
$894$	0	0
$895$	1.68072e22	1.22130
$896$	0	0
$897$	−7.07041e21	−0.505244
$898$	0	0
$899$	1.75210e21i	0.123129i
$900$	0	0
$901$	− 1.13350e21i	− 0.0783405i
$902$	0	0
$903$	−1.29565e21	−0.0880701
$904$	0	0
$905$	−3.75827e22	−2.51260
$906$	0	0
$907$	1.91784e22i	1.26112i	0.776139	+	0.630562i	$0.217175\pi$
$907$	1.91784e22i	1.26112i	−0.776139	+	0.630562i	$0.782825\pi$
$908$	0	0
$909$	5.40267e22i	3.49445i
$910$	0	0
$911$	−2.32929e22	−1.48196	−0.740978	−	0.671530i	$-0.765638\pi$
$911$	−2.32929e22	−1.48196	−0.740978	+	0.671530i	$0.765638\pi$
$912$	0	0
$913$	3.80704e22	2.38263
$914$	0	0
$915$	− 6.73805e21i	− 0.414835i
$916$	0	0
$917$	7.63357e20i	0.0462336i
$918$	0	0
$919$	−2.98532e21	−0.177879	−0.0889394	−	0.996037i	$-0.528348\pi$
$919$	−2.98532e21	−0.177879	−0.0889394	+	0.996037i	$0.528348\pi$
$920$	0	0
$921$	1.61954e22	0.949392
$922$	0	0
$923$	5.10746e21i	0.294572i
$924$	0	0
$925$	3.52318e22i	1.99927i
$926$	0	0
$927$	1.17772e22	0.657574
$928$	0	0
$929$	1.47039e21	0.0807821	0.0403910	−	0.999184i	$-0.487140\pi$
$929$	1.47039e21	0.0807821	0.0403910	+	0.999184i	$0.487140\pi$
$930$	0	0
$931$	− 2.18290e22i	− 1.18008i
$932$	0	0
$933$	− 3.67684e21i	− 0.195597i
$934$	0	0
$935$	−3.22323e21	−0.168735
$936$	0	0
$937$	−1.35569e22	−0.698417	−0.349208	−	0.937045i	$-0.613549\pi$
$937$	−1.35569e22	−0.698417	−0.349208	+	0.937045i	$0.613549\pi$
$938$	0	0
$939$	− 1.34264e22i	− 0.680716i
$940$	0	0
$941$	1.13116e22i	0.564418i	0.959353	+	0.282209i	$0.0910672\pi$
$941$	1.13116e22i	0.564418i	−0.959353	+	0.282209i	$0.908933\pi$
$942$	0	0
$943$	2.39329e21	0.117532
$944$	0	0
$945$	−2.40224e21	−0.116112
$946$	0	0
$947$	1.72242e22i	0.819433i	0.912213	+	0.409716i	$0.134372\pi$
$947$	1.72242e22i	0.819433i	−0.912213	+	0.409716i	$0.865628\pi$
$948$	0	0
$949$	− 1.36860e22i	− 0.640885i
$950$	0	0
$951$	−3.81879e21	−0.176024
$952$	0	0
$953$	−8.21718e21	−0.372843	−0.186421	−	0.982470i	$-0.559689\pi$
$953$	−8.21718e21	−0.372843	−0.186421	+	0.982470i	$0.559689\pi$
$954$	0	0
$955$	6.48667e22i	2.89732i
$956$	0	0
$957$	− 4.68307e22i	− 2.05916i
$958$	0	0
$959$	−1.52141e21	−0.0658576
$960$	0	0
$961$	−2.29428e22	−0.977733
$962$	0	0
$963$	− 4.13171e22i	− 1.73353i
$964$	0	0
$965$	2.25623e22i	0.932029i
$966$	0	0
$967$	1.45625e22	0.592296	0.296148	−	0.955142i	$-0.404298\pi$
$967$	1.45625e22	0.592296	0.296148	+	0.955142i	$0.404298\pi$
$968$	0	0
$969$	−3.68079e21	−0.147405
$970$	0	0
$971$	− 4.44965e22i	− 1.75462i	−0.479928	−	0.877308i	$-0.659337\pi$
$971$	− 4.44965e22i	− 1.75462i	0.479928	−	0.877308i	$-0.340663\pi$
$972$	0	0
$973$	4.25874e20i	0.0165362i
$974$	0	0
$975$	−4.26367e22	−1.63023
$976$	0	0
$977$	7.34006e21	0.276370	0.138185	−	0.990406i	$-0.455873\pi$
$977$	7.34006e21	0.276370	0.138185	+	0.990406i	$0.455873\pi$
$978$	0	0
$979$	9.16749e21i	0.339923i
$980$	0	0
$981$	− 3.51119e22i	− 1.28214i
$982$	0	0
$983$	−1.35957e22	−0.488934	−0.244467	−	0.969658i	$-0.578613\pi$
$983$	−1.35957e22	−0.488934	−0.244467	+	0.969658i	$0.578613\pi$
$984$	0	0
$985$	5.11217e22	1.81065
$986$	0	0
$987$	− 2.94999e21i	− 0.102906i
$988$	0	0
$989$	− 1.14589e22i	− 0.393704i
$990$	0	0
$991$	2.67627e21	0.0905686	0.0452843	−	0.998974i	$-0.485581\pi$
$991$	2.67627e21	0.0905686	0.0452843	+	0.998974i	$0.485581\pi$
$992$	0	0
$993$	−1.90842e22	−0.636143
$994$	0	0
$995$	3.16743e21i	0.104000i
$996$	0	0
$997$	1.79920e22i	0.581924i	0.956735	+	0.290962i	$0.0939754\pi$
$997$	1.79920e22i	0.581924i	−0.956735	+	0.290962i	$0.906025\pi$
$998$	0	0
$999$	6.42390e22	2.04672

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.16.b.a.17.2		14
4.3	odd	2		8.16.b.a.5.6	yes	14
8.3	odd	2		8.16.b.a.5.5	✓	14
8.5	even	2	inner	32.16.b.a.17.13		14
12.11	even	2		72.16.d.b.37.9		14
24.11	even	2		72.16.d.b.37.10		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.16.b.a.5.5	✓	14	8.3	odd	2
8.16.b.a.5.6	yes	14	4.3	odd	2
32.16.b.a.17.2		14	1.1	even	1	trivial
32.16.b.a.17.13		14	8.5	even	2	inner
72.16.d.b.37.9		14	12.11	even	2
72.16.d.b.37.10		14	24.11	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)

\(f(q)\)	\(=\)	\(q-6357.96i q^{3} -267219. i q^{5} -120583. q^{7} -2.60748e7 q^{9} +9.60941e7i q^{11} +1.64009e8i q^{13} -1.69897e9 q^{15} -1.25524e8 q^{17} +4.61207e9i q^{19} +7.66663e8i q^{21} -6.78047e9 q^{23} -4.08883e10 q^{25} +7.45526e10i q^{27} -7.66507e10i q^{29} -2.28582e10 q^{31} +6.10963e11 q^{33} +3.22221e10i q^{35} -8.61660e11i q^{37} +1.04276e12 q^{39} -3.52968e11 q^{41} +1.68998e12i q^{43} +6.96767e12i q^{45} -3.84783e12 q^{47} -4.73302e12 q^{49} +7.98078e11i q^{51} +9.03015e12i q^{53} +2.56782e13 q^{55} +2.93234e13 q^{57} +1.13978e13i q^{59} +3.96597e12i q^{61} +3.14418e12 q^{63} +4.38262e13 q^{65} +4.34443e13i q^{67} +4.31100e13i q^{69} +3.11414e13 q^{71} -8.34468e13 q^{73} +2.59966e14i q^{75} -1.15873e13i q^{77} -7.31635e12 q^{79} +9.98580e13 q^{81} -3.96179e14i q^{83} +3.35424e13i q^{85} -4.87342e14 q^{87} +9.54011e13 q^{89} -1.97767e13i q^{91} +1.45332e14i q^{93} +1.23243e15 q^{95} -1.16047e15 q^{97} -2.50563e15i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 1647088 q^{7} - 57395630 q^{9} - 712135312 q^{15} + 728554812 q^{17} + 35548816080 q^{23} - 75899954794 q^{25} + 105758138816 q^{31} - 150458001384 q^{33} + 2251546247120 q^{39} - 53229185940 q^{41}+ \cdots - 672574291859236 q^{97}+O(q^{100}) \) 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

Related objects

Downloads

Learn more