LMFDB - Embedded newform 1008.4.bt.d.17.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(17,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1008.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.bt (of order $6$, degree $2$, 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:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.10
Character	$\chi$	$=$	1008.17
Dual form			1008.4.bt.d.593.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.08296 - 3.60779i) q^{5} +(18.2790 + 2.97950i) q^{7} +O(q^{10})$	$q+(-2.08296 - 3.60779i) q^{5} +(18.2790 + 2.97950i) q^{7} +(-33.1649 - 19.1477i) q^{11} +49.6184i q^{13} +(7.65240 - 13.2543i) q^{17} +(-122.914 + 70.9647i) q^{19} +(136.779 - 78.9695i) q^{23} +(53.8226 - 93.2234i) q^{25} -204.644i q^{29} +(90.5273 + 52.2660i) q^{31} +(-27.3250 - 72.1531i) q^{35} +(-194.054 - 336.111i) q^{37} -325.463 q^{41} -191.352 q^{43} +(249.874 + 432.795i) q^{47} +(325.245 + 108.925i) q^{49} +(-37.8772 - 21.8684i) q^{53} +159.536i q^{55} +(-86.5601 + 149.927i) q^{59} +(208.382 - 120.309i) q^{61} +(179.013 - 103.353i) q^{65} +(-440.624 + 763.183i) q^{67} -1017.80i q^{71} +(-361.028 - 208.439i) q^{73} +(-549.170 - 448.817i) q^{77} +(-237.361 - 411.121i) q^{79} +652.093 q^{83} -63.7586 q^{85} +(-298.995 - 517.874i) q^{89} +(-147.838 + 906.976i) q^{91} +(512.052 + 295.633i) q^{95} -1771.43i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 24 q^{7}+O(q^{10}) $ 48 * q + 24 * q^7	$ 48 q + 24 q^{7} - 540 q^{19} - 924 q^{25} - 648 q^{31} - 132 q^{37} + 792 q^{43} + 672 q^{49} + 12 q^{67} + 2412 q^{73} - 1680 q^{79} + 480 q^{85} - 1404 q^{91}+O(q^{100}) $ 48 * q + 24 * q^7 - 540 * q^19 - 924 * q^25 - 648 * q^31 - 132 * q^37 + 792 * q^43 + 672 * q^49 + 12 * q^67 + 2412 * q^73 - 1680 * q^79 + 480 * q^85 - 1404 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.08296	−	3.60779i	−0.186306	−	0.322691i	0.757710	−	0.652591i	$-0.226318\pi$
$5$	−2.08296	−	3.60779i	−0.186306	−	0.322691i	−0.944016	+	0.329901i	$0.892985\pi$
$6$		0			0
$7$	18.2790	+	2.97950i	0.986974	+	0.160878i
$7$	18.2790	+	2.97950i	0.986974	+	0.160878i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−33.1649	−	19.1477i	−0.909053	−	0.524842i	−0.0289264	−	0.999582i	$-0.509209\pi$
$11$	−33.1649	−	19.1477i	−0.909053	−	0.524842i	−0.880126	+	0.474740i	$0.842542\pi$
$12$		0			0
$13$			49.6184i			1.05859i	0.848438	+	0.529295i	$0.177544\pi$
$13$			49.6184i			1.05859i	−0.848438	+	0.529295i	$0.822456\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.65240	−	13.2543i	0.109175	−	0.189097i	−0.806261	−	0.591560i	$-0.798512\pi$
$17$	7.65240	−	13.2543i	0.109175	−	0.189097i	0.915436	+	0.402463i	$0.131846\pi$
$18$		0			0
$19$	−122.914	+	70.9647i	−1.48413	+	0.856864i	−0.999837	−	0.0180413i	$-0.994257\pi$
$19$	−122.914	+	70.9647i	−1.48413	+	0.856864i	−0.484294	+	0.874905i	$0.660924\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	136.779	−	78.9695i	1.24002	−	0.715925i	0.270920	−	0.962602i	$-0.412672\pi$
$23$	136.779	−	78.9695i	1.24002	−	0.715925i	0.969098	+	0.246677i	$0.0793386\pi$
$24$		0			0
$25$	53.8226	−	93.2234i	0.430580	−	0.745787i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	204.644i		−	1.31039i	−0.755458	−	0.655197i	$-0.772586\pi$
$29$		−	204.644i		−	1.31039i	0.755458	−	0.655197i	$-0.227414\pi$
$30$		0			0
$31$	90.5273	+	52.2660i	0.524490	+	0.302814i	0.738770	−	0.673958i	$-0.235407\pi$
$31$	90.5273	+	52.2660i	0.524490	+	0.302814i	−0.214280	+	0.976772i	$0.568740\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−27.3250	−	72.1531i	−0.131965	−	0.348460i
$36$		0			0
$37$	−194.054	−	336.111i	−0.862224	−	1.49341i	−0.869778	−	0.493443i	$-0.835738\pi$
$37$	−194.054	−	336.111i	−0.862224	−	1.49341i	0.00755433	−	0.999971i	$-0.497595\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−325.463			−1.23973			−0.619864	−	0.784709i	$-0.712812\pi$
$41$	−325.463			−1.23973			−0.619864	+	0.784709i	$0.712812\pi$
$42$		0			0
$43$	−191.352			−0.678626			−0.339313	−	0.940674i	$-0.610195\pi$
$43$	−191.352			−0.678626			−0.339313	+	0.940674i	$0.610195\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	249.874	+	432.795i	0.775487	+	1.34318i	0.934520	+	0.355910i	$0.115829\pi$
$47$	249.874	+	432.795i	0.775487	+	1.34318i	−0.159033	+	0.987273i	$0.550838\pi$
$48$		0			0
$49$	325.245	+	108.925i	0.948237	+	0.317565i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−37.8772	−	21.8684i	−0.0981667	−	0.0566766i	0.450113	−	0.892972i	$-0.351384\pi$
$53$	−37.8772	−	21.8684i	−0.0981667	−	0.0566766i	−0.548280	+	0.836295i	$0.684717\pi$
$54$		0			0
$55$			159.536i			0.391124i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−86.5601	+	149.927i	−0.191003	+	0.330827i	−0.945583	−	0.325381i	$-0.894507\pi$
$59$	−86.5601	+	149.927i	−0.191003	+	0.330827i	0.754580	+	0.656208i	$0.227841\pi$
$60$		0			0
$61$	208.382	−	120.309i	0.437386	−	0.252525i	−0.265102	−	0.964220i	$-0.585406\pi$
$61$	208.382	−	120.309i	0.437386	−	0.252525i	0.702488	+	0.711695i	$0.252072\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	179.013	−	103.353i	0.341597	−	0.197221i
$66$		0			0
$67$	−440.624	+	763.183i	−0.803445	+	1.39161i	0.113891	+	0.993493i	$0.463669\pi$
$67$	−440.624	+	763.183i	−0.803445	+	1.39161i	−0.917336	+	0.398114i	$0.869665\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1017.80i		−	1.70128i	−0.525746	−	0.850642i	$-0.676214\pi$
$71$		−	1017.80i		−	1.70128i	0.525746	−	0.850642i	$-0.323786\pi$
$72$		0			0
$73$	−361.028	−	208.439i	−0.578837	−	0.334192i	0.181834	−	0.983329i	$-0.441797\pi$
$73$	−361.028	−	208.439i	−0.578837	−	0.334192i	−0.760671	+	0.649138i	$0.775130\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−549.170	−	448.817i	−0.812776	−	0.664252i
$78$		0			0
$79$	−237.361	−	411.121i	−0.338041	−	0.585504i	0.646024	−	0.763318i	$-0.276431\pi$
$79$	−237.361	−	411.121i	−0.338041	−	0.585504i	−0.984064	+	0.177814i	$0.943097\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	652.093			0.862368			0.431184	−	0.902264i	$-0.358096\pi$
$83$	652.093			0.862368			0.431184	+	0.902264i	$0.358096\pi$
$84$		0			0
$85$	−63.7586			−0.0813598
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−298.995	−	517.874i	−0.356105	−	0.616793i	0.631201	−	0.775619i	$-0.282562\pi$
$89$	−298.995	−	517.874i	−0.356105	−	0.616793i	−0.987307	+	0.158826i	$0.949229\pi$
$90$		0			0
$91$	−147.838	+	906.976i	−0.170304	+	1.04480i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	512.052	+	295.633i	0.553004	+	0.319277i
$96$		0			0
$97$		−	1771.43i		−	1.85424i	−0.374763	−	0.927121i	$-0.622276\pi$
$97$		−	1771.43i		−	1.85424i	0.374763	−	0.927121i	$-0.377724\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	106.303	−	184.123i	0.104728	−	0.181395i	−0.808899	−	0.587948i	$-0.799936\pi$
$101$	106.303	−	184.123i	0.104728	−	0.181395i	0.913627	+	0.406553i	$0.133269\pi$
$102$		0			0
$103$	−1148.55	+	663.115i	−1.09874	+	0.634356i	−0.935889	−	0.352295i	$-0.885401\pi$
$103$	−1148.55	+	663.115i	−1.09874	+	0.634356i	−0.162848	+	0.986651i	$0.552068\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−106.629	+	61.5621i	−0.0963382	+	0.0556209i	−0.547395	−	0.836874i	$-0.684380\pi$
$107$	−106.629	+	61.5621i	−0.0963382	+	0.0556209i	0.451057	+	0.892495i	$0.351047\pi$
$108$		0			0
$109$	8.14724	−	14.1114i	0.00715930	−	0.0124003i	−0.862424	−	0.506187i	$-0.831054\pi$
$109$	8.14724	−	14.1114i	0.00715930	−	0.0124003i	0.869583	+	0.493787i	$0.164388\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1537.79i		−	1.28020i	−0.768291	−	0.640101i	$-0.778892\pi$
$113$		−	1537.79i		−	1.28020i	0.768291	−	0.640101i	$-0.221108\pi$
$114$		0			0
$115$	−569.811	−	328.980i	−0.462045	−	0.266762i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	179.370	−	219.476i	0.138175	−	0.169070i
$120$		0			0
$121$	67.7716	+	117.384i	0.0509178	+	0.0881922i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−969.181			−0.693489
$126$		0			0
$127$	−1097.31			−0.766697			−0.383348	−	0.923604i	$-0.625229\pi$
$127$	−1097.31			−0.766697			−0.383348	+	0.923604i	$0.625229\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1417.08	−	2454.45i	−0.945119	−	1.63699i	−0.755513	−	0.655134i	$-0.772612\pi$
$131$	−1417.08	−	2454.45i	−0.945119	−	1.63699i	−0.189606	−	0.981860i	$-0.560721\pi$
$132$		0			0
$133$	−2458.19	+	930.941i	−1.60265	+	0.606939i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−980.159	−	565.895i	−0.611245	−	0.352903i	0.162207	−	0.986757i	$-0.448139\pi$
$137$	−980.159	−	565.895i	−0.611245	−	0.352903i	−0.773453	+	0.633854i	$0.781472\pi$
$138$		0			0
$139$		−	1088.19i		−	0.664020i	−0.943276	−	0.332010i	$-0.892273\pi$
$139$		−	1088.19i		−	0.664020i	0.943276	−	0.332010i	$-0.107727\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	950.080	−	1645.59i	0.555592	−	0.962314i
$144$		0			0
$145$	−738.313	+	426.265i	−0.422852	+	0.244134i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1485.18	−	857.470i	0.816583	−	0.471454i	−0.0326539	−	0.999467i	$-0.510396\pi$
$149$	1485.18	−	857.470i	0.816583	−	0.471454i	0.849237	+	0.528012i	$0.177063\pi$
$150$		0			0
$151$	−128.682	+	222.885i	−0.0693512	+	0.120120i	−0.898616	−	0.438736i	$-0.855426\pi$
$151$	−128.682	+	222.885i	−0.0693512	+	0.120120i	0.829265	+	0.558856i	$0.188760\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	435.472i		−	0.225664i
$156$		0			0
$157$	−644.785	−	372.267i	−0.327767	−	0.189237i	0.327082	−	0.944996i	$-0.393935\pi$
$157$	−644.785	−	372.267i	−0.327767	−	0.189237i	−0.654849	+	0.755759i	$0.727268\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2735.48	−	1035.95i	1.33904	−	0.507108i
$162$		0			0
$163$	−218.407	−	378.292i	−0.104951	−	0.181780i	0.808767	−	0.588129i	$-0.200135\pi$
$163$	−218.407	−	378.292i	−0.104951	−	0.181780i	−0.913718	+	0.406349i	$0.866802\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1110.76			−0.514688			−0.257344	−	0.966320i	$-0.582847\pi$
$167$	−1110.76			−0.514688			−0.257344	+	0.966320i	$0.582847\pi$
$168$		0			0
$169$	−264.987			−0.120613
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1967.21	−	3407.31i	−0.864533	−	1.49741i	−0.867510	−	0.497419i	$-0.834281\pi$
$173$	−1967.21	−	3407.31i	−0.864533	−	1.49741i	0.00297733	−	0.999996i	$-0.499052\pi$
$174$		0			0
$175$	1261.58	−	1543.67i	0.544953	−	0.666802i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−791.218	−	456.810i	−0.330382	−	0.190746i	0.325628	−	0.945498i	$-0.394424\pi$
$179$	−791.218	−	456.810i	−0.330382	−	0.190746i	−0.656011	+	0.754751i	$0.727757\pi$
$180$		0			0
$181$			2909.50i			1.19482i	0.801938	+	0.597408i	$0.203803\pi$
$181$			2909.50i			1.19482i	−0.801938	+	0.597408i	$0.796197\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−808.413	+	1400.21i	−0.321274	+	0.556463i
$186$		0			0
$187$	−507.581	+	293.052i	−0.198492	+	0.114599i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1342.95	+	775.351i	−0.508756	+	0.293730i	−0.732322	−	0.680959i	$-0.761563\pi$
$191$	−1342.95	+	775.351i	−0.508756	+	0.293730i	0.223566	+	0.974689i	$0.428230\pi$
$192$		0			0
$193$	−840.304	+	1455.45i	−0.313401	+	0.542827i	−0.979096	−	0.203397i	$-0.934802\pi$
$193$	−840.304	+	1455.45i	−0.313401	+	0.542827i	0.665695	+	0.746224i	$0.268135\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			3085.14i			1.11577i	0.829917	+	0.557887i	$0.188388\pi$
$197$			3085.14i			1.11577i	−0.829917	+	0.557887i	$0.811612\pi$
$198$		0			0
$199$	359.537	+	207.579i	0.128075	+	0.0739442i	0.562669	−	0.826682i	$-0.309775\pi$
$199$	359.537	+	207.579i	0.128075	+	0.0739442i	−0.434594	+	0.900627i	$0.643108\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	609.737	−	3740.69i	0.210813	−	1.29333i
$204$		0			0
$205$	677.927	+	1174.20i	0.230968	+	0.400049i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5435.25			1.79887
$210$		0			0
$211$	2166.75			0.706944			0.353472	−	0.935445i	$-0.385001\pi$
$211$	2166.75			0.706944			0.353472	+	0.935445i	$0.385001\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	398.579	+	690.359i	0.126432	+	0.218986i
$216$		0			0
$217$	1499.02	+	1225.10i	0.468942	+	0.383249i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	657.660	+	379.700i	0.200176	+	0.115572i
$222$		0			0
$223$			2089.66i			0.627505i	0.949505	+	0.313753i	$0.101586\pi$
$223$			2089.66i			0.627505i	−0.949505	+	0.313753i	$0.898414\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	137.016	−	237.319i	0.0400621	−	0.0693896i	−0.845299	−	0.534293i	$-0.820578\pi$
$227$	137.016	−	237.319i	0.0400621	−	0.0693896i	0.885361	+	0.464904i	$0.153911\pi$
$228$		0			0
$229$	1942.69	−	1121.61i	0.560595	−	0.323660i	−0.192789	−	0.981240i	$-0.561753\pi$
$229$	1942.69	−	1121.61i	0.560595	−	0.323660i	0.753384	+	0.657580i	$0.228420\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−522.653	+	301.754i	−0.146953	+	0.0848436i	−0.571674	−	0.820481i	$-0.693706\pi$
$233$	−522.653	+	301.754i	−0.146953	+	0.0848436i	0.424720	+	0.905325i	$0.360372\pi$
$234$		0			0
$235$	1040.96	−	1802.99i	0.288955	−	0.500485i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	1504.81i		−	0.407273i	−0.979047	−	0.203637i	$-0.934724\pi$
$239$		−	1504.81i		−	0.407273i	0.979047	−	0.203637i	$-0.0652761\pi$
$240$		0			0
$241$	−4456.97	−	2573.23i	−1.19128	−	0.687787i	−0.232685	−	0.972552i	$-0.574751\pi$
$241$	−4456.97	−	2573.23i	−1.19128	−	0.687787i	−0.958597	+	0.284765i	$0.908084\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−284.495	−	1400.30i	−0.0741866	−	0.365151i
$246$		0			0
$247$	−3521.15	−	6098.82i	−0.907068	−	1.57109i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2852.49			0.717321			0.358661	−	0.933468i	$-0.383234\pi$
$251$	2852.49			0.717321			0.358661	+	0.933468i	$0.383234\pi$
$252$		0			0
$253$	−6048.35			−1.50299
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1871.38	+	3241.33i	0.454216	+	0.786726i	0.998643	−	0.0520828i	$-0.0165860\pi$
$257$	1871.38	+	3241.33i	0.454216	+	0.786726i	−0.544426	+	0.838809i	$0.683253\pi$
$258$		0			0
$259$	−2545.67	−	6721.97i	−0.610735	−	1.61267i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6755.29	−	3900.17i	−1.58384	−	0.914428i	−0.994292	−	0.106691i	$-0.965974\pi$
$263$	−6755.29	−	3900.17i	−1.58384	−	0.914428i	−0.589543	−	0.807737i	$-0.700692\pi$
$264$		0			0
$265$			182.204i			0.0422366i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2102.33	−	3641.34i	0.476510	−	0.825340i	−0.523128	−	0.852254i	$-0.675235\pi$
$269$	2102.33	−	3641.34i	0.476510	−	0.825340i	0.999638	+	0.0269146i	$0.00856823\pi$
$270$		0			0
$271$	1823.57	−	1052.84i	0.408761	−	0.235998i	−0.281496	−	0.959562i	$-0.590831\pi$
$271$	1823.57	−	1052.84i	0.408761	−	0.235998i	0.690257	+	0.723564i	$0.257497\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3570.03	+	2061.16i	−0.782841	+	0.451973i
$276$		0			0
$277$	−3935.66	+	6816.76i	−0.853685	+	1.47863i	0.0241750	+	0.999708i	$0.492304\pi$
$277$	−3935.66	+	6816.76i	−0.853685	+	1.47863i	−0.877860	+	0.478918i	$0.841029\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	826.499i		−	0.175462i	−0.996144	−	0.0877310i	$-0.972038\pi$
$281$		−	826.499i		−	0.175462i	0.996144	−	0.0877310i	$-0.0279616\pi$
$282$		0			0
$283$	3946.80	+	2278.68i	0.829020	+	0.478635i	0.853517	−	0.521065i	$-0.174465\pi$
$283$	3946.80	+	2278.68i	0.829020	+	0.478635i	−0.0244968	+	0.999700i	$0.507798\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5949.15	−	969.718i	−1.22358	−	0.199445i
$288$		0			0
$289$	2339.38	+	4051.93i	0.476162	+	0.824736i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7373.68			−1.47022			−0.735111	−	0.677947i	$-0.762870\pi$
$293$	−7373.68			−1.47022			−0.735111	+	0.677947i	$0.762870\pi$
$294$		0			0
$295$	721.205			0.142340
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3918.34	+	6786.76i	0.757871	+	1.31267i
$300$		0			0
$301$	−3497.73	−	570.134i	−0.669787	−	0.109176i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−868.101	−	501.198i	−0.162975	−	0.0940936i
$306$		0			0
$307$			2406.09i			0.447306i	0.974669	+	0.223653i	$0.0717982\pi$
$307$			2406.09i			0.447306i	−0.974669	+	0.223653i	$0.928202\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1437.77	−	2490.28i	0.262149	−	0.454055i	−0.704664	−	0.709541i	$-0.748902\pi$
$311$	1437.77	−	2490.28i	0.262149	−	0.454055i	0.966813	+	0.255486i	$0.0822356\pi$
$312$		0			0
$313$	6586.58	−	3802.76i	1.18944	−	0.686725i	0.231263	−	0.972891i	$-0.425714\pi$
$313$	6586.58	−	3802.76i	1.18944	−	0.686725i	0.958180	+	0.286166i	$0.0923809\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2360.31	−	1362.73i	0.418197	−	0.241446i	−0.276109	−	0.961126i	$-0.589045\pi$
$317$	2360.31	−	1362.73i	0.418197	−	0.241446i	0.694306	+	0.719680i	$0.255712\pi$
$318$		0			0
$319$	−3918.47	+	6786.99i	−0.687750	+	1.19122i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			2172.20i			0.374193i
$324$		0			0
$325$	4625.60	+	2670.59i	0.789483	+	0.455808i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3277.94	+	8655.57i	0.549298	+	1.45045i
$330$		0			0
$331$	−2410.09	−	4174.40i	−0.400214	−	0.693190i	0.593538	−	0.804806i	$-0.297731\pi$
$331$	−2410.09	−	4174.40i	−0.400214	−	0.693190i	−0.993751	+	0.111616i	$0.964397\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3671.21			0.598745
$336$		0			0
$337$	−3069.99			−0.496241			−0.248121	−	0.968729i	$-0.579813\pi$
$337$	−3069.99			−0.496241			−0.248121	+	0.968729i	$0.579813\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2001.55	−	3466.79i	−0.317859	−	0.550549i
$342$		0			0
$343$	5620.62	+	2960.10i	0.884796	+	0.465979i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3507.33	+	2024.96i	0.542603	+	0.313272i	0.746133	−	0.665797i	$-0.231908\pi$
$347$	3507.33	+	2024.96i	0.542603	+	0.313272i	−0.203530	+	0.979069i	$0.565242\pi$
$348$		0			0
$349$		−	4056.76i		−	0.622216i	−0.950375	−	0.311108i	$-0.899300\pi$
$349$		−	4056.76i		−	0.622216i	0.950375	−	0.311108i	$-0.100700\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4848.88	+	8398.51i	−0.731105	+	1.26631i	0.225307	+	0.974288i	$0.427662\pi$
$353$	−4848.88	+	8398.51i	−0.731105	+	1.26631i	−0.956411	+	0.292023i	$0.905672\pi$
$354$		0			0
$355$	−3672.03	+	2120.04i	−0.548988	+	0.316959i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4481.02	−	2587.12i	0.658773	−	0.380343i	−0.133036	−	0.991111i	$-0.542473\pi$
$359$	4481.02	−	2587.12i	0.658773	−	0.380343i	0.791809	+	0.610769i	$0.209139\pi$
$360$		0			0
$361$	6642.47	−	11505.1i	0.968431	−	1.67737i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			1736.68i			0.249047i
$366$		0			0
$367$	8693.29	+	5019.07i	1.23647	+	0.713878i	0.968372	−	0.249512i	$-0.0802702\pi$
$367$	8693.29	+	5019.07i	1.23647	+	0.713878i	0.268102	+	0.963391i	$0.413604\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−627.201	−	512.588i	−0.0877700	−	0.0717312i
$372$		0			0
$373$	1372.10	+	2376.55i	0.190469	+	0.329901i	0.945406	−	0.325896i	$-0.105666\pi$
$373$	1372.10	+	2376.55i	0.190469	+	0.329901i	−0.754937	+	0.655797i	$0.772333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	10154.1			1.38717
$378$		0			0
$379$	−412.530			−0.0559109			−0.0279555	−	0.999609i	$-0.508900\pi$
$379$	−412.530			−0.0559109			−0.0279555	+	0.999609i	$0.508900\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2792.64	+	4837.00i	0.372578	+	0.645324i	0.989961	−	0.141339i	$-0.0451406\pi$
$383$	2792.64	+	4837.00i	0.372578	+	0.645324i	−0.617383	+	0.786662i	$0.711807\pi$
$384$		0			0
$385$	−475.337	+	2916.16i	−0.0629232	+	0.386029i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12438.4	−	7181.34i	−1.62122	−	0.936011i	−0.986595	−	0.163191i	$-0.947821\pi$
$389$	−12438.4	−	7181.34i	−1.62122	−	0.936011i	−0.634625	−	0.772820i	$-0.718845\pi$
$390$		0			0
$391$		−	2417.22i		−	0.312645i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−988.827	+	1712.70i	−0.125958	+	0.218165i
$396$		0			0
$397$	9573.43	−	5527.22i	1.21027	−	0.698749i	0.247451	−	0.968900i	$-0.420407\pi$
$397$	9573.43	−	5527.22i	1.21027	−	0.698749i	0.962818	+	0.270152i	$0.0870738\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	10385.2	−	5995.91i	1.29330	−	0.746686i	0.314061	−	0.949403i	$-0.398310\pi$
$401$	10385.2	−	5995.91i	1.29330	−	0.746686i	0.979237	+	0.202716i	$0.0649770\pi$
$402$		0			0
$403$	−2593.36	+	4491.82i	−0.320556	+	0.555220i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			14862.8i			1.81012i
$408$		0			0
$409$	−1478.37	−	853.536i	−0.178730	−	0.103190i	0.407966	−	0.912997i	$-0.366238\pi$
$409$	−1478.37	−	853.536i	−0.178730	−	0.103190i	−0.586696	+	0.809807i	$0.699572\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2028.94	+	2482.60i	−0.241738	+	0.295789i
$414$		0			0
$415$	−1358.28	−	2352.62i	−0.160664	−	0.278278i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2025.17			0.236124			0.118062	−	0.993006i	$-0.462332\pi$
$419$	2025.17			0.236124			0.118062	+	0.993006i	$0.462332\pi$
$420$		0			0
$421$	16388.2			1.89718			0.948592	−	0.316503i	$-0.102509\pi$
$421$	16388.2			1.89718			0.948592	+	0.316503i	$0.102509\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−823.744	−	1426.77i	−0.0940175	−	0.162843i
$426$		0			0
$427$	4167.47	−	1578.26i	0.472314	−	0.178870i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9721.77	+	5612.87i	1.08650	+	0.627290i	0.932642	−	0.360803i	$-0.117497\pi$
$431$	9721.77	+	5612.87i	1.08650	+	0.627290i	0.153857	+	0.988093i	$0.450831\pi$
$432$		0			0
$433$		−	15239.6i		−	1.69139i	−0.533670	−	0.845693i	$-0.679187\pi$
$433$		−	15239.6i		−	1.69139i	0.533670	−	0.845693i	$-0.320813\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−11208.1	+	19413.0i	−1.22690	+	2.12505i
$438$		0			0
$439$	−1650.32	+	952.810i	−0.179420	+	0.103588i	−0.587020	−	0.809572i	$-0.699699\pi$
$439$	−1650.32	+	952.810i	−0.179420	+	0.103588i	0.407600	+	0.913160i	$0.366366\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10032.5	+	5792.26i	−1.07598	+	0.621215i	−0.929808	−	0.368044i	$-0.880027\pi$
$443$	−10032.5	+	5792.26i	−1.07598	+	0.621215i	−0.146168	+	0.989260i	$0.546694\pi$
$444$		0			0
$445$	−1245.59	+	2157.42i	−0.132689	+	0.229824i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			16331.3i			1.71652i	0.513211	+	0.858262i	$0.328456\pi$
$449$			16331.3i			1.71652i	−0.513211	+	0.858262i	$0.671544\pi$
$450$		0			0
$451$	10793.9	+	6231.89i	1.12698	+	0.650661i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3580.12	−	1355.83i	0.368876	−	0.139697i
$456$		0			0
$457$	3320.32	+	5750.97i	0.339865	+	0.588663i	0.984407	−	0.175906i	$-0.0562853\pi$
$457$	3320.32	+	5750.97i	0.339865	+	0.588663i	−0.644542	+	0.764569i	$0.722952\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6825.82			−0.689610			−0.344805	−	0.938674i	$-0.612055\pi$
$461$	−6825.82			−0.689610			−0.344805	+	0.938674i	$0.612055\pi$
$462$		0			0
$463$	3630.09			0.364373			0.182186	−	0.983264i	$-0.441683\pi$
$463$	3630.09			0.364373			0.182186	+	0.983264i	$0.441683\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2775.83	+	4807.88i	0.275054	+	0.476407i	0.970149	−	0.242511i	$-0.0779711\pi$
$467$	2775.83	+	4807.88i	0.275054	+	0.476407i	−0.695095	+	0.718918i	$0.744638\pi$
$468$		0			0
$469$	−10328.1	+	12637.4i	−1.01686	+	1.24422i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6346.16	+	3663.96i	0.616907	+	0.356171i
$474$		0			0
$475$			15278.0i			1.47580i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8908.33	+	15429.7i	−0.849753	+	1.47182i	0.0316748	+	0.999498i	$0.489916\pi$
$479$	−8908.33	+	15429.7i	−0.849753	+	1.47182i	−0.881428	+	0.472318i	$0.843417\pi$
$480$		0			0
$481$	16677.3	−	9628.65i	1.58091	−	0.912741i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6390.95	+	3689.82i	−0.598346	+	0.345455i
$486$		0			0
$487$	532.969	−	923.130i	0.0495917	−	0.0858953i	−0.840164	−	0.542332i	$-0.817541\pi$
$487$	532.969	−	923.130i	0.0495917	−	0.0858953i	0.889756	+	0.456437i	$0.150875\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	8952.43i		−	0.822846i	−0.911444	−	0.411423i	$-0.865032\pi$
$491$		−	8952.43i		−	0.822846i	0.911444	−	0.411423i	$-0.134968\pi$
$492$		0			0
$493$	−2712.42	−	1566.02i	−0.247792	−	0.143063i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3032.55	−	18604.5i	0.273699	−	1.67912i
$498$		0			0
$499$	−725.335	−	1256.32i	−0.0650711	−	0.112706i	0.831654	−	0.555294i	$-0.187394\pi$
$499$	−725.335	−	1256.32i	−0.0650711	−	0.112706i	−0.896726	+	0.442587i	$0.854061\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10311.3			0.914032			0.457016	−	0.889458i	$-0.348918\pi$
$503$	10311.3			0.914032			0.457016	+	0.889458i	$0.348918\pi$
$504$		0			0
$505$	−885.701			−0.0780459
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4844.36	−	8390.67i	−0.421851	−	0.730668i	0.574269	−	0.818666i	$-0.305286\pi$
$509$	−4844.36	−	8390.67i	−0.421851	−	0.730668i	−0.996121	+	0.0879987i	$0.971953\pi$
$510$		0			0
$511$	−5978.19	−	4885.75i	−0.517533	−	0.422961i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4784.76	+	2762.48i	0.409402	+	0.236368i
$516$		0			0
$517$		−	19138.1i		−	1.62803i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9666.31	+	16742.5i	−0.812838	+	1.40788i	0.0980312	+	0.995183i	$0.468746\pi$
$521$	−9666.31	+	16742.5i	−0.812838	+	1.40788i	−0.910870	+	0.412694i	$0.864588\pi$
$522$		0			0
$523$	−19453.0	+	11231.2i	−1.62643	+	0.939018i	−0.641280	+	0.767307i	$0.721596\pi$
$523$	−19453.0	+	11231.2i	−1.62643	+	0.939018i	−0.985147	+	0.171711i	$0.945070\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1385.50	−	799.920i	0.114523	−	0.0661197i
$528$		0			0
$529$	6388.85	−	11065.8i	0.525097	−	0.909494i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	16149.0i		−	1.31236i
$534$		0			0
$535$	444.206	+	256.463i	0.0358967	+	0.0207250i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8701.05	−	9840.18i	−0.695326	−	0.786357i
$540$		0			0
$541$	10846.0	+	18785.8i	0.861934	+	1.49291i	0.870060	+	0.492946i	$0.164080\pi$
$541$	10846.0	+	18785.8i	0.861934	+	1.49291i	−0.00812568	+	0.999967i	$0.502587\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−67.8815			−0.00533527
$546$		0			0
$547$	22932.0			1.79251			0.896255	−	0.443538i	$-0.146277\pi$
$547$	22932.0			1.79251			0.896255	+	0.443538i	$0.146277\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14522.5	+	25153.7i	1.12283	+	1.94480i
$552$		0			0
$553$	−3113.79	−	8222.11i	−0.239443	−	0.632260i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4941.90	−	2853.21i	−0.375934	−	0.217045i	0.300114	−	0.953903i	$-0.402975\pi$
$557$	−4941.90	−	2853.21i	−0.375934	−	0.217045i	−0.676048	+	0.736858i	$0.736309\pi$
$558$		0			0
$559$		−	9494.59i		−	0.718387i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	491.280	−	850.922i	0.0367762	−	0.0636982i	−0.847051	−	0.531511i	$-0.821624\pi$
$563$	491.280	−	850.922i	0.0367762	−	0.0636982i	0.883828	+	0.467813i	$0.154958\pi$
$564$		0			0
$565$	−5548.02	+	3203.15i	−0.413109	+	0.238509i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	16337.6	−	9432.51i	1.20370	−	0.694958i	0.242326	−	0.970195i	$-0.422090\pi$
$569$	16337.6	−	9432.51i	1.20370	−	0.694958i	0.961377	+	0.275237i	$0.0887562\pi$
$570$		0			0
$571$	4441.37	−	7692.67i	0.325509	−	0.563797i	−0.656107	−	0.754668i	$-0.727798\pi$
$571$	4441.37	−	7692.67i	0.325509	−	0.563797i	0.981615	+	0.190871i	$0.0611312\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	17001.4i		−	1.23305i
$576$		0			0
$577$	3694.42	+	2132.98i	0.266553	+	0.153894i	0.627320	−	0.778762i	$-0.284152\pi$
$577$	3694.42	+	2132.98i	0.266553	+	0.153894i	−0.360767	+	0.932656i	$0.617485\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	11919.6	+	1942.91i	0.851135	+	0.138736i
$582$		0			0
$583$	837.461	+	1450.53i	0.0594925	+	0.103044i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	19210.8			1.35079			0.675397	−	0.737454i	$-0.263972\pi$
$587$	19210.8			1.35079			0.675397	+	0.737454i	$0.263972\pi$
$588$		0			0
$589$	−14836.2			−1.03788
$590$		0			0
$591$		0			0
$592$		0			0
$593$	10815.0	+	18732.1i	0.748934	+	1.29719i	0.948334	+	0.317274i	$0.102768\pi$
$593$	10815.0	+	18732.1i	0.748934	+	1.29719i	−0.199399	+	0.979918i	$0.563899\pi$
$594$		0			0
$595$	−1165.44	−	189.969i	−0.0803001	−	0.0130890i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−22218.7	−	12827.9i	−1.51558	−	0.875018i	−0.999833	−	0.0182745i	$-0.994183\pi$
$599$	−22218.7	−	12827.9i	−1.51558	−	0.875018i	−0.515743	−	0.856744i	$-0.672484\pi$
$600$		0			0
$601$			10757.1i			0.730099i	0.930988	+	0.365050i	$0.118948\pi$
$601$			10757.1i			0.730099i	−0.930988	+	0.365050i	$0.881052\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	282.331	−	489.012i	0.0189725	−	0.0328614i
$606$		0			0
$607$	−17158.2	+	9906.32i	−1.14733	+	0.662414i	−0.948236	−	0.317566i	$-0.897134\pi$
$607$	−17158.2	+	9906.32i	−1.14733	+	0.662414i	−0.199098	+	0.979980i	$0.563801\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−21474.6	+	12398.4i	−1.42188	+	0.820923i
$612$		0			0
$613$	−8947.03	+	15496.7i	−0.589506	+	1.02105i	0.404791	+	0.914409i	$0.367344\pi$
$613$	−8947.03	+	15496.7i	−0.589506	+	1.02105i	−0.994297	+	0.106645i	$0.965989\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			26131.8i			1.70507i	0.522673	+	0.852533i	$0.324935\pi$
$617$			26131.8i			1.70507i	−0.522673	+	0.852533i	$0.675065\pi$
$618$		0			0
$619$	12114.7	+	6994.42i	0.786640	+	0.454167i	0.838778	−	0.544473i	$-0.183270\pi$
$619$	12114.7	+	6994.42i	0.786640	+	0.454167i	−0.0521381	+	0.998640i	$0.516604\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3922.33	−	10357.1i	−0.252239	−	0.666048i
$624$		0			0
$625$	−4709.06	−	8156.32i	−0.301380	−	0.522005i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5939.91			−0.376534
$630$		0			0
$631$	1366.56			0.0862151			0.0431076	−	0.999070i	$-0.486274\pi$
$631$	1366.56			0.0862151			0.0431076	+	0.999070i	$0.486274\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2285.65	+	3958.86i	0.142840	+	0.247406i
$636$		0			0
$637$	−5404.67	+	16138.2i	−0.336171	+	1.00379i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2412.26	+	1392.72i	0.148641	+	0.0858177i	0.572476	−	0.819922i	$-0.305983\pi$
$641$	2412.26	+	1392.72i	0.148641	+	0.0858177i	−0.423835	+	0.905739i	$0.639316\pi$
$642$		0			0
$643$		−	29078.1i		−	1.78340i	−0.452623	−	0.891702i	$-0.649511\pi$
$643$		−	29078.1i		−	1.78340i	0.452623	−	0.891702i	$-0.350489\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1249.03	−	2163.38i	0.0758955	−	0.131455i	−0.825580	−	0.564285i	$-0.809152\pi$
$647$	1249.03	−	2163.38i	0.0758955	−	0.131455i	0.901475	+	0.432830i	$0.142485\pi$
$648$		0			0
$649$	5741.51	−	3314.86i	0.347263	−	0.200493i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11418.2	+	6592.32i	−0.684272	+	0.395065i	−0.801463	−	0.598045i	$-0.795945\pi$
$653$	−11418.2	+	6592.32i	−0.684272	+	0.395065i	0.117191	+	0.993109i	$0.462611\pi$
$654$		0			0
$655$	−5903.43	+	10225.0i	−0.352162	+	0.609962i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			13405.1i			0.792398i	0.918165	+	0.396199i	$0.129671\pi$
$659$			13405.1i			0.792398i	−0.918165	+	0.396199i	$0.870329\pi$
$660$		0			0
$661$	−6676.89	−	3854.91i	−0.392891	−	0.226836i	0.290521	−	0.956869i	$-0.406171\pi$
$661$	−6676.89	−	3854.91i	−0.392891	−	0.226836i	−0.683412	+	0.730033i	$0.739505\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8478.96	+	6929.54i	0.494436	+	0.404084i
$666$		0			0
$667$	−16160.6	−	27991.0i	−0.938144	−	1.62491i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9214.60			−0.530142
$672$		0			0
$673$	25045.3			1.43451			0.717255	−	0.696810i	$-0.245398\pi$
$673$	25045.3			1.43451			0.717255	+	0.696810i	$0.245398\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3354.60	−	5810.33i	−0.190440	−	0.329851i	0.754956	−	0.655775i	$-0.227658\pi$
$677$	−3354.60	−	5810.33i	−0.190440	−	0.329851i	−0.945396	+	0.325924i	$0.894325\pi$
$678$		0			0
$679$	5277.97	−	32380.0i	0.298306	−	1.83009i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−19735.9	−	11394.5i	−1.10567	−	0.638361i	−0.167968	−	0.985792i	$-0.553720\pi$
$683$	−19735.9	−	11394.5i	−1.10567	−	0.638361i	−0.937705	+	0.347432i	$0.887054\pi$
$684$		0			0
$685$			4714.94i			0.262991i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1085.08	−	1879.41i	0.0599973	−	0.103918i
$690$		0			0
$691$	17740.1	−	10242.2i	0.976650	−	0.563869i	0.0753929	−	0.997154i	$-0.475979\pi$
$691$	17740.1	−	10242.2i	0.976650	−	0.563869i	0.901257	+	0.433285i	$0.142646\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3925.95	+	2266.65i	−0.214273	+	0.123711i
$696$		0			0
$697$	−2490.58	+	4313.80i	−0.135348	+	0.234429i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	16624.7i		−	0.895731i	−0.894101	−	0.447866i	$-0.852184\pi$
$701$		−	16624.7i		−	0.895731i	0.894101	−	0.447866i	$-0.147816\pi$
$702$		0			0
$703$	47704.1	+	27541.9i	2.55931	+	1.47762i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2491.71	−	3048.85i	0.132547	−	0.162184i
$708$		0			0
$709$	2183.47	+	3781.89i	0.115659	+	0.200327i	0.918043	−	0.396481i	$-0.129769\pi$
$709$	2183.47	+	3781.89i	0.115659	+	0.200327i	−0.802384	+	0.596808i	$0.796435\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	16509.7			0.867170
$714$		0			0
$715$	−7915.92			−0.414040
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2729.14	−	4727.01i	−0.141557	−	0.245185i	0.786526	−	0.617557i	$-0.211878\pi$
$719$	−2729.14	−	4727.01i	−0.141557	−	0.245185i	−0.928083	+	0.372373i	$0.878544\pi$
$720$		0			0
$721$	−22970.1	+	8698.99i	−1.18648	+	0.449331i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−19077.6	−	11014.5i	−0.977275	−	0.564230i
$726$		0			0
$727$		−	765.750i		−	0.0390648i	−0.999809	−	0.0195324i	$-0.993782\pi$
$727$		−	765.750i		−	0.0390648i	0.999809	−	0.0195324i	$-0.00621775\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1464.30	+	2536.25i	−0.0740892	+	0.128326i
$732$		0			0
$733$	−12534.5	+	7236.81i	−0.631614	+	0.364662i	−0.781377	−	0.624059i	$-0.785482\pi$
$733$	−12534.5	+	7236.81i	−0.631614	+	0.364662i	0.149763	+	0.988722i	$0.452149\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	29226.5	−	16873.9i	1.46075	−	0.843363i
$738$		0			0
$739$	−17644.2	+	30560.6i	−0.878284	+	1.52123i	−0.0250604	+	0.999686i	$0.507978\pi$
$739$	−17644.2	+	30560.6i	−0.878284	+	1.52123i	−0.853223	+	0.521546i	$0.825356\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	9060.44i		−	0.447369i	−0.974662	−	0.223685i	$-0.928191\pi$
$743$		−	9060.44i		−	0.447369i	0.974662	−	0.223685i	$-0.0718086\pi$
$744$		0			0
$745$	−6187.15	−	3572.15i	−0.304268	−	0.175669i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2132.49	+	807.595i	−0.104031	+	0.0393977i
$750$		0			0
$751$	−14811.1	−	25653.7i	−0.719662	−	1.24649i	−0.961134	−	0.276084i	$-0.910963\pi$
$751$	−14811.1	−	25653.7i	−0.719662	−	1.24649i	0.241471	−	0.970408i	$-0.422370\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1072.16			0.0516820
$756$		0			0
$757$	−12661.2			−0.607900			−0.303950	−	0.952688i	$-0.598306\pi$
$757$	−12661.2			−0.607900			−0.303950	+	0.952688i	$0.598306\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−613.595	−	1062.78i	−0.0292284	−	0.0506250i	0.851041	−	0.525099i	$-0.175972\pi$
$761$	−613.595	−	1062.78i	−0.0292284	−	0.0506250i	−0.880270	+	0.474474i	$0.842638\pi$
$762$		0			0
$763$	190.969	−	233.668i	0.00906098	−	0.0110870i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7439.12	−	4294.98i	−0.350210	−	0.202194i
$768$		0			0
$769$		−	12271.0i		−	0.575428i	−0.957716	−	0.287714i	$-0.907105\pi$
$769$		−	12271.0i		−	0.575428i	0.957716	−	0.287714i	$-0.0928953\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−7280.79	+	12610.7i	−0.338773	+	0.586773i	−0.984202	−	0.177048i	$-0.943345\pi$
$773$	−7280.79	+	12610.7i	−0.338773	+	0.586773i	0.645429	+	0.763820i	$0.276679\pi$
$774$		0			0
$775$	9744.83	−	5626.18i	0.451670	−	0.260772i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	40004.1	−	23096.4i	1.83992	−	1.06228i
$780$		0			0
$781$	−19488.6	+	33755.3i	−0.892905	+	1.54656i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			3101.67i			0.141023i
$786$		0			0
$787$	−21263.2	−	12276.3i	−0.963090	−	0.556040i	−0.0659671	−	0.997822i	$-0.521013\pi$
$787$	−21263.2	−	12276.3i	−0.963090	−	0.556040i	−0.897123	+	0.441782i	$0.854347\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4581.84	−	28109.2i	0.205956	−	1.26353i
$792$		0			0
$793$	5969.55	+	10339.6i	0.267320	+	0.463012i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	20920.3			0.929781			0.464891	−	0.885368i	$-0.346094\pi$
$797$	20920.3			0.929781			0.464891	+	0.885368i	$0.346094\pi$
$798$		0			0
$799$	7648.55			0.338656
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7982.29	+	13825.7i	0.350795	+	0.607596i
$804$		0			0
$805$	−9435.38	−	7711.19i	−0.413110	−	0.337620i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−19985.0	−	11538.3i	−0.868522	−	0.501441i	−0.00166527	−	0.999999i	$-0.500530\pi$
$809$	−19985.0	−	11538.3i	−0.868522	−	0.501441i	−0.866857	+	0.498557i	$0.833863\pi$
$810$		0			0
$811$			5406.96i			0.234111i	0.993125	+	0.117055i	$0.0373455\pi$
$811$			5406.96i			0.234111i	−0.993125	+	0.117055i	$0.962654\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−909.867	+	1575.94i	−0.0391058	+	0.0677333i
$816$		0			0
$817$	23519.9	−	13579.2i	1.00717	−	0.581490i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	39007.4	−	22520.9i	1.65818	−	0.957353i	0.684631	−	0.728890i	$-0.259964\pi$
$821$	39007.4	−	22520.9i	1.65818	−	0.957353i	0.973553	−	0.228462i	$-0.0733698\pi$
$822$		0			0
$823$	−5506.70	+	9537.89i	−0.233234	+	0.403973i	−0.958758	−	0.284224i	$-0.908264\pi$
$823$	−5506.70	+	9537.89i	−0.233234	+	0.403973i	0.725524	+	0.688197i	$0.241597\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	19467.6i		−	0.818565i	−0.912408	−	0.409283i	$-0.865779\pi$
$827$		−	19467.6i		−	0.818565i	0.912408	−	0.409283i	$-0.134221\pi$
$828$		0			0
$829$	16023.1	+	9250.94i	0.671297	+	0.387574i	0.796568	−	0.604549i	$-0.206647\pi$
$829$	16023.1	+	9250.94i	0.671297	+	0.387574i	−0.125271	+	0.992123i	$0.539980\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3932.63	−	3477.38i	0.163575	−	0.144639i
$834$		0			0
$835$	2313.66	+	4007.37i	0.0958892	+	0.166085i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	9278.29			0.381790			0.190895	−	0.981610i	$-0.438861\pi$
$839$	9278.29			0.381790			0.190895	+	0.981610i	$0.438861\pi$
$840$		0			0
$841$	−17490.2			−0.717133
$842$		0			0
$843$		0			0
$844$		0			0
$845$	551.958	+	956.020i	0.0224709	+	0.0389208i
$846$		0			0
$847$	889.053	+	2347.59i	0.0360664	+	0.0952350i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−53085.1	−	30648.7i	−2.13835	−	1.23457i
$852$		0			0
$853$			4213.13i			0.169115i	0.996419	+	0.0845574i	$0.0269476\pi$
$853$			4213.13i			0.169115i	−0.996419	+	0.0845574i	$0.973052\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	11563.2	−	20028.0i	0.460898	−	0.798299i	−0.538108	−	0.842876i	$-0.680861\pi$
$857$	11563.2	−	20028.0i	0.460898	−	0.798299i	0.999006	+	0.0445770i	$0.0141940\pi$
$858$		0			0
$859$	13231.7	−	7639.30i	0.525563	−	0.303434i	−0.213645	−	0.976911i	$-0.568534\pi$
$859$	13231.7	−	7639.30i	0.525563	−	0.303434i	0.739208	+	0.673478i	$0.235200\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−14156.6	+	8173.34i	−0.558398	+	0.322391i	−0.752502	−	0.658590i	$-0.771153\pi$
$863$	−14156.6	+	8173.34i	−0.558398	+	0.322391i	0.194104	+	0.980981i	$0.437820\pi$
$864$		0			0
$865$	−8195.24	+	14194.6i	−0.322135	+	0.557953i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			18179.7i			0.709671i
$870$		0			0
$871$	−37868.0	−	21863.1i	−1.47314	−	0.850519i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−17715.7	−	2887.67i	−0.684456	−	0.111567i
$876$		0			0
$877$	10112.3	+	17515.0i	0.389359	+	0.674390i	0.992364	−	0.123348i	$-0.0393631\pi$
$877$	10112.3	+	17515.0i	0.389359	+	0.674390i	−0.603004	+	0.797738i	$0.706030\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1026.37			0.0392501			0.0196251	−	0.999807i	$-0.493753\pi$
$881$	1026.37			0.0392501			0.0196251	+	0.999807i	$0.493753\pi$
$882$		0			0
$883$	−4513.41			−0.172014			−0.0860069	−	0.996295i	$-0.527411\pi$
$883$	−4513.41			−0.172014			−0.0860069	+	0.996295i	$0.527411\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−20971.0	−	36322.9i	−0.793843	−	1.37498i	−0.923571	−	0.383427i	$-0.874744\pi$
$887$	−20971.0	−	36322.9i	−0.793843	−	1.37498i	0.129729	−	0.991550i	$-0.458589\pi$
$888$		0			0
$889$	−20057.7	−	3269.43i	−0.756710	−	0.123345i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−61426.3	−	35464.5i	−2.30185	−	1.32897i
$894$		0			0
$895$			3806.07i			0.142148i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	10695.9	−	18525.9i	0.396806	−	0.687289i
$900$		0			0
$901$	−579.703	+	334.692i	−0.0214348	+	0.0123754i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	10496.9	−	6060.37i	0.385556	−	0.222601i
$906$		0			0
$907$	9408.95	−	16296.8i	0.344453	−	0.596611i	−0.640801	−	0.767707i	$-0.721398\pi$
$907$	9408.95	−	16296.8i	0.344453	−	0.596611i	0.985254	+	0.171096i	$0.0547309\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			26132.8i			0.950405i	0.879877	+	0.475202i	$0.157625\pi$
$911$			26132.8i			0.950405i	−0.879877	+	0.475202i	$0.842375\pi$
$912$		0			0
$913$	−21626.6	−	12486.1i	−0.783938	−	0.452607i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−18589.7	−	49087.1i	−0.669452	−	1.76772i
$918$		0			0
$919$	10741.1	+	18604.2i	0.385547	+	0.667787i	0.991845	−	0.127450i	$-0.0406794\pi$
$919$	10741.1	+	18604.2i	0.385547	+	0.667787i	−0.606298	+	0.795238i	$0.707346\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	50501.8			1.80096
$924$		0			0
$925$	−41777.9			−1.48503
$926$		0			0
$927$		0			0
$928$		0			0
$929$	296.256	+	513.130i	0.0104627	+	0.0181219i	0.871209	−	0.490912i	$-0.163336\pi$
$929$	296.256	+	513.130i	0.0104627	+	0.0181219i	−0.860747	+	0.509034i	$0.830003\pi$
$930$		0			0
$931$	−47707.1	+	9692.50i	−1.67942	+	0.341202i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2114.54	+	1220.83i	0.0739604	+	0.0427010i
$936$		0			0
$937$		−	40362.0i		−	1.40722i	−0.710585	−	0.703612i	$-0.751569\pi$
$937$		−	40362.0i		−	1.40722i	0.710585	−	0.703612i	$-0.248431\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−21679.6	+	37550.1i	−0.751046	+	1.30085i	0.196271	+	0.980550i	$0.437117\pi$
$941$	−21679.6	+	37550.1i	−0.751046	+	1.30085i	−0.947316	+	0.320299i	$0.896216\pi$
$942$		0			0
$943$	−44516.6	+	25701.7i	−1.53729	+	0.887552i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1390.00	−	802.516i	0.0476968	−	0.0275378i	−0.475962	−	0.879466i	$-0.657900\pi$
$947$	1390.00	−	802.516i	0.0476968	−	0.0275378i	0.523659	+	0.851928i	$0.324567\pi$
$948$		0			0
$949$	10342.4	−	17913.6i	0.353772	−	0.612751i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	24091.7i		−	0.818895i	−0.912334	−	0.409447i	$-0.865722\pi$
$953$		−	24091.7i		−	0.818895i	0.912334	−	0.409447i	$-0.134278\pi$
$954$		0			0
$955$	5594.61	+	3230.05i	0.189568	+	0.109447i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−16230.3	−	13264.4i	−0.546509	−	0.446642i
$960$		0			0
$961$	−9432.03	−	16336.8i	−0.316607	−	0.548379i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	7001.28			0.233554
$966$		0			0
$967$	−27331.1			−0.908903			−0.454451	−	0.890772i	$-0.650165\pi$
$967$	−27331.1			−0.908903			−0.454451	+	0.890772i	$0.650165\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	14845.9	+	25713.8i	0.490656	+	0.849841i	0.999942	−	0.0107564i	$-0.00342394\pi$
$971$	14845.9	+	25713.8i	0.490656	+	0.849841i	−0.509286	+	0.860597i	$0.670091\pi$
$972$		0			0
$973$	3242.25	−	19891.0i	0.106826	−	0.655371i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	6699.29	+	3867.84i	0.219375	+	0.126656i	0.605661	−	0.795723i	$-0.292909\pi$
$977$	6699.29	+	3867.84i	0.219375	+	0.126656i	−0.386286	+	0.922379i	$0.626242\pi$
$978$		0			0
$979$			22900.3i			0.747596i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2020.18	−	3499.06i	0.0655482	−	0.113533i	−0.831389	−	0.555691i	$-0.812454\pi$
$983$	2020.18	−	3499.06i	0.0655482	−	0.113533i	0.896937	+	0.442158i	$0.145787\pi$
$984$		0			0
$985$	11130.6	−	6426.23i	0.360050	−	0.207875i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−26173.0	+	15111.0i	−0.841509	+	0.485845i
$990$		0			0
$991$	−6527.11	+	11305.3i	−0.209224	+	0.362386i	−0.951470	−	0.307741i	$-0.900427\pi$
$991$	−6527.11	+	11305.3i	−0.209224	+	0.362386i	0.742247	+	0.670127i	$0.233760\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	1729.51i		−	0.0551048i
$996$		0			0
$997$	5017.76	+	2897.00i	0.159392	+	0.0920251i	0.577574	−	0.816338i	$-0.303999\pi$
$997$	5017.76	+	2897.00i	0.159392	+	0.0920251i	−0.418182	+	0.908363i	$0.637333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.bt.d.17.10		48
3.2	odd	2	inner	1008.4.bt.d.17.15		48
4.3	odd	2		504.4.bl.a.17.10	✓	48
7.5	odd	6	inner	1008.4.bt.d.593.15		48
12.11	even	2		504.4.bl.a.17.15	yes	48
21.5	even	6	inner	1008.4.bt.d.593.10		48
28.19	even	6		504.4.bl.a.89.15	yes	48
84.47	odd	6		504.4.bl.a.89.10	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.bl.a.17.10	✓	48	4.3	odd	2
504.4.bl.a.17.15	yes	48	12.11	even	2
504.4.bl.a.89.10	yes	48	84.47	odd	6
504.4.bl.a.89.15	yes	48	28.19	even	6
1008.4.bt.d.17.10		48	1.1	even	1	trivial
1008.4.bt.d.17.15		48	3.2	odd	2	inner
1008.4.bt.d.593.10		48	21.5	even	6	inner
1008.4.bt.d.593.15		48	7.5	odd	6	inner

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.bt (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.08296 - 3.60779i) q^{5} +(18.2790 + 2.97950i) q^{7} +O(q^{10})\)	\(q+(-2.08296 - 3.60779i) q^{5} +(18.2790 + 2.97950i) q^{7} +(-33.1649 - 19.1477i) q^{11} +49.6184i q^{13} +(7.65240 - 13.2543i) q^{17} +(-122.914 + 70.9647i) q^{19} +(136.779 - 78.9695i) q^{23} +(53.8226 - 93.2234i) q^{25} -204.644i q^{29} +(90.5273 + 52.2660i) q^{31} +(-27.3250 - 72.1531i) q^{35} +(-194.054 - 336.111i) q^{37} -325.463 q^{41} -191.352 q^{43} +(249.874 + 432.795i) q^{47} +(325.245 + 108.925i) q^{49} +(-37.8772 - 21.8684i) q^{53} +159.536i q^{55} +(-86.5601 + 149.927i) q^{59} +(208.382 - 120.309i) q^{61} +(179.013 - 103.353i) q^{65} +(-440.624 + 763.183i) q^{67} -1017.80i q^{71} +(-361.028 - 208.439i) q^{73} +(-549.170 - 448.817i) q^{77} +(-237.361 - 411.121i) q^{79} +652.093 q^{83} -63.7586 q^{85} +(-298.995 - 517.874i) q^{89} +(-147.838 + 906.976i) q^{91} +(512.052 + 295.633i) q^{95} -1771.43i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 24 q^{7}+O(q^{10}) \) 48 * q + 24 * q^7	\( 48 q + 24 q^{7} - 540 q^{19} - 924 q^{25} - 648 q^{31} - 132 q^{37} + 792 q^{43} + 672 q^{49} + 12 q^{67} + 2412 q^{73} - 1680 q^{79} + 480 q^{85} - 1404 q^{91}+O(q^{100}) \) 48 * q + 24 * q^7 - 540 * q^19 - 924 * q^25 - 648 * q^31 - 132 * q^37 + 792 * q^43 + 672 * q^49 + 12 * q^67 + 2412 * q^73 - 1680 * q^79 + 480 * q^85 - 1404 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more