LMFDB - Embedded newform 2052.3.be.a.125.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(125,2052)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2052, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2052.125");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.be (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:	$55.9129502467$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$40$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			125.12
Character	$\chi$	$=$	2052.125
Dual form			2052.3.be.a.197.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.94117 + 2.27544i) q^{5} +(5.85150 + 10.1351i) q^{7} +O(q^{10})$	$q+(-3.94117 + 2.27544i) q^{5} +(5.85150 + 10.1351i) q^{7} +(-14.0645 + 8.12016i) q^{11} +7.26450 q^{13} +(-19.6143 - 11.3243i) q^{17} +(-13.5931 - 13.2751i) q^{19} +38.6626i q^{23} +(-2.14477 + 3.71485i) q^{25} +(-22.3737 - 12.9174i) q^{29} +(14.0349 - 24.3091i) q^{31} +(-46.1236 - 26.6295i) q^{35} -47.8191 q^{37} +(-7.23503 + 4.17715i) q^{41} +78.3646 q^{43} +(-14.2151 - 8.20711i) q^{47} +(-43.9802 + 76.1759i) q^{49} +(43.6381 - 25.1945i) q^{53} +(36.9538 - 64.0059i) q^{55} +(64.9066 - 37.4738i) q^{59} +(-31.1644 + 53.9783i) q^{61} +(-28.6306 + 16.5299i) q^{65} -57.1284 q^{67} +(-73.0839 - 42.1950i) q^{71} +(28.4345 - 49.2501i) q^{73} +(-164.597 - 95.0303i) q^{77} +119.049 q^{79} +(71.4114 - 41.2294i) q^{83} +103.071 q^{85} +(123.100 - 71.0719i) q^{89} +(42.5082 + 73.6264i) q^{91} +(83.7793 + 21.3894i) q^{95} +118.506 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + q^{7}+O(q^{10}) $ 80 * q + q^7	$ 80 q + q^{7} - 18 q^{11} + 10 q^{13} - 9 q^{17} + 20 q^{19} + 200 q^{25} + 27 q^{29} - 8 q^{31} + 22 q^{37} + 54 q^{41} + 88 q^{43} - 198 q^{47} - 267 q^{49} - 36 q^{53} - 171 q^{59} + 7 q^{61} + 144 q^{65} + 154 q^{67} - 135 q^{71} + 43 q^{73} - 216 q^{77} + 34 q^{79} + 171 q^{83} + 216 q^{89} + 122 q^{91} + 216 q^{95} + 16 q^{97}+O(q^{100}) $ 80 * q + q^7 - 18 * q^11 + 10 * q^13 - 9 * q^17 + 20 * q^19 + 200 * q^25 + 27 * q^29 - 8 * q^31 + 22 * q^37 + 54 * q^41 + 88 * q^43 - 198 * q^47 - 267 * q^49 - 36 * q^53 - 171 * q^59 + 7 * q^61 + 144 * q^65 + 154 * q^67 - 135 * q^71 + 43 * q^73 - 216 * q^77 + 34 * q^79 + 171 * q^83 + 216 * q^89 + 122 * q^91 + 216 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times$.

$n$	$325$	$1027$	$1217$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	−3.94117	+	2.27544i	−0.788235	+	0.455087i	−0.839341	−	0.543606i	$-0.817059\pi$
$5$	−3.94117	+	2.27544i	−0.788235	+	0.455087i	0.0511060	+	0.998693i	$0.483725\pi$
$6$		0			0
$7$	5.85150	+	10.1351i	0.835929	+	1.44787i	0.893272	+	0.449517i	$0.148404\pi$
$7$	5.85150	+	10.1351i	0.835929	+	1.44787i	−0.0573428	+	0.998355i	$0.518263\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−14.0645	+	8.12016i	−1.27859	+	0.738197i	−0.976590	−	0.215110i	$-0.930989\pi$
$11$	−14.0645	+	8.12016i	−1.27859	+	0.738197i	−0.302004	+	0.953307i	$0.597656\pi$
$12$		0			0
$13$	7.26450			0.558808			0.279404	−	0.960174i	$-0.409863\pi$
$13$	7.26450			0.558808			0.279404	+	0.960174i	$0.409863\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−19.6143	−	11.3243i	−1.15378	−	0.666137i	−0.203976	−	0.978976i	$-0.565387\pi$
$17$	−19.6143	−	11.3243i	−1.15378	−	0.666137i	−0.949806	+	0.312839i	$0.898720\pi$
$18$		0			0
$19$	−13.5931	−	13.2751i	−0.715424	−	0.698691i
$19$	−13.5931	−	13.2751i	−0.715424	−	0.698691i
$20$		0			0
$21$		0			0
$22$		0			0
$23$			38.6626i			1.68098i	0.541826	+	0.840491i	$0.317733\pi$
$23$			38.6626i			1.68098i	−0.541826	+	0.840491i	$0.682267\pi$
$24$		0			0
$25$	−2.14477	+	3.71485i	−0.0857908	+	0.148594i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−22.3737	−	12.9174i	−0.771506	−	0.445429i	0.0619056	−	0.998082i	$-0.480282\pi$
$29$	−22.3737	−	12.9174i	−0.771506	−	0.445429i	−0.833412	+	0.552653i	$0.813616\pi$
$30$		0			0
$31$	14.0349	−	24.3091i	0.452738	−	0.784166i	−0.545817	−	0.837905i	$-0.683780\pi$
$31$	14.0349	−	24.3091i	0.452738	−	0.784166i	0.998555	+	0.0537388i	$0.0171138\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−46.1236	−	26.6295i	−1.31782	−	0.760842i
$36$		0			0
$37$	−47.8191			−1.29241			−0.646204	−	0.763164i	$-0.723645\pi$
$37$	−47.8191			−1.29241			−0.646204	+	0.763164i	$0.723645\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.23503	+	4.17715i	−0.176464	+	0.101882i	−0.585630	−	0.810578i	$-0.699153\pi$
$41$	−7.23503	+	4.17715i	−0.176464	+	0.101882i	0.409166	+	0.912460i	$0.365820\pi$
$42$		0			0
$43$	78.3646			1.82243			0.911216	−	0.411929i	$-0.135145\pi$
$43$	78.3646			1.82243			0.911216	+	0.411929i	$0.135145\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−14.2151	−	8.20711i	−0.302449	−	0.174619i	0.341093	−	0.940029i	$-0.389203\pi$
$47$	−14.2151	−	8.20711i	−0.302449	−	0.174619i	−0.643543	+	0.765410i	$0.722536\pi$
$48$		0			0
$49$	−43.9802	+	76.1759i	−0.897555	+	1.55461i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	43.6381	−	25.1945i	0.823360	−	0.475367i	−0.0282138	−	0.999602i	$-0.508982\pi$
$53$	43.6381	−	25.1945i	0.823360	−	0.475367i	0.851574	+	0.524235i	$0.175649\pi$
$54$		0			0
$55$	36.9538	−	64.0059i	0.671888	−	1.16374i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	64.9066	−	37.4738i	1.10011	−	0.635150i	0.163861	−	0.986483i	$-0.447605\pi$
$59$	64.9066	−	37.4738i	1.10011	−	0.635150i	0.936250	+	0.351334i	$0.114272\pi$
$60$		0			0
$61$	−31.1644	+	53.9783i	−0.510891	+	0.884890i	0.489029	+	0.872267i	$0.337351\pi$
$61$	−31.1644	+	53.9783i	−0.510891	+	0.884890i	−0.999920	+	0.0126221i	$0.995982\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−28.6306	+	16.5299i	−0.440471	+	0.254306i
$66$		0			0
$67$	−57.1284			−0.852662			−0.426331	−	0.904567i	$-0.640194\pi$
$67$	−57.1284			−0.852662			−0.426331	+	0.904567i	$0.640194\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−73.0839	−	42.1950i	−1.02935	−	0.594296i	−0.112552	−	0.993646i	$-0.535902\pi$
$71$	−73.0839	−	42.1950i	−1.02935	−	0.594296i	−0.916799	+	0.399350i	$0.869236\pi$
$72$		0			0
$73$	28.4345	−	49.2501i	0.389514	−	0.674658i	−0.602870	−	0.797839i	$-0.705976\pi$
$73$	28.4345	−	49.2501i	0.389514	−	0.674658i	0.992384	+	0.123181i	$0.0393096\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−164.597	−	95.0303i	−2.13763	−	1.23416i
$78$		0			0
$79$	119.049			1.50694			0.753472	−	0.657480i	$-0.228378\pi$
$79$	119.049			1.50694			0.753472	+	0.657480i	$0.228378\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	71.4114	−	41.2294i	0.860379	−	0.496740i	−0.00376048	−	0.999993i	$-0.501197\pi$
$83$	71.4114	−	41.2294i	0.860379	−	0.496740i	0.864139	+	0.503253i	$0.167864\pi$
$84$		0			0
$85$	103.071			1.21260
$86$		0			0
$87$		0			0
$88$		0			0
$89$	123.100	−	71.0719i	1.38315	−	0.798560i	0.390616	−	0.920554i	$-0.372262\pi$
$89$	123.100	−	71.0719i	1.38315	−	0.798560i	0.992531	+	0.121993i	$0.0389286\pi$
$90$		0			0
$91$	42.5082	+	73.6264i	0.467123	+	0.809081i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	83.7793	+	21.3894i	0.881887	+	0.225152i
$96$		0			0
$97$	118.506			1.22171			0.610854	−	0.791743i	$-0.290826\pi$
$97$	118.506			1.22171			0.610854	+	0.791743i	$0.290826\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−78.0417	−	45.0574i	−0.772690	−	0.446113i	0.0611432	−	0.998129i	$-0.480525\pi$
$101$	−78.0417	−	45.0574i	−0.772690	−	0.446113i	−0.833833	+	0.552016i	$0.813859\pi$
$102$		0			0
$103$	−43.0937	+	74.6404i	−0.418385	+	0.724664i	−0.995777	−	0.0918028i	$-0.970737\pi$
$103$	−43.0937	+	74.6404i	−0.418385	+	0.724664i	0.577392	+	0.816467i	$0.304070\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			116.854i			1.09209i	0.837756	+	0.546045i	$0.183867\pi$
$107$			116.854i			1.09209i	−0.837756	+	0.546045i	$0.816133\pi$
$108$		0			0
$109$	−59.8897	+	103.732i	−0.549447	+	0.951670i	0.448865	+	0.893599i	$0.351828\pi$
$109$	−59.8897	+	103.732i	−0.549447	+	0.951670i	−0.998312	+	0.0580708i	$0.981505\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	121.648	+	70.2335i	1.07653	+	0.621535i	0.929958	−	0.367665i	$-0.119843\pi$
$113$	121.648	+	70.2335i	1.07653	+	0.621535i	0.146572	+	0.989200i	$0.453176\pi$
$114$		0			0
$115$	−87.9743	−	152.376i	−0.764994	−	1.32501i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	265.057i		−	2.22737i
$120$		0			0
$121$	71.3740	−	123.623i	0.589868	−	1.02168i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	133.293i		−	1.06634i
$126$		0			0
$127$	−35.1261	−	60.8401i	−0.276583	−	0.479056i	0.693950	−	0.720023i	$-0.255869\pi$
$127$	−35.1261	−	60.8401i	−0.276583	−	0.479056i	−0.970533	+	0.240967i	$0.922535\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.24670	+	1.29713i	−0.0171504	+	0.00990178i	−0.508551	−	0.861032i	$-0.669819\pi$
$131$	−2.24670	+	1.29713i	−0.0171504	+	0.00990178i	0.491400	+	0.870934i	$0.336485\pi$
$132$		0			0
$133$	55.0050	−	215.446i	0.413571	−	1.61990i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−128.331	−	74.0922i	−0.936726	−	0.540819i	−0.0477932	−	0.998857i	$-0.515219\pi$
$137$	−128.331	−	74.0922i	−0.936726	−	0.540819i	−0.888932	+	0.458038i	$0.848552\pi$
$138$		0			0
$139$	−133.439			−0.959992			−0.479996	−	0.877271i	$-0.659362\pi$
$139$	−133.439			−0.959992			−0.479996	+	0.877271i	$0.659362\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−102.172	+	58.9889i	−0.714488	+	0.412510i
$144$		0			0
$145$	117.571			0.810837
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−106.043	+	61.2239i	−0.711697	+	0.410899i	−0.811689	−	0.584090i	$-0.801452\pi$
$149$	−106.043	+	61.2239i	−0.711697	+	0.410899i	0.0999919	+	0.994988i	$0.468118\pi$
$150$		0			0
$151$	−36.8974	−	63.9082i	−0.244354	−	0.423233i	0.717596	−	0.696460i	$-0.245243\pi$
$151$	−36.8974	−	63.9082i	−0.244354	−	0.423233i	−0.961950	+	0.273226i	$0.911909\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			127.742i			0.824142i
$156$		0			0
$157$	−146.014	−	252.904i	−0.930027	−	1.61085i	−0.783270	−	0.621682i	$-0.786450\pi$
$157$	−146.014	−	252.904i	−0.930027	−	1.61085i	−0.146757	−	0.989173i	$-0.546884\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−391.849	+	226.234i	−2.43385	+	1.40518i
$162$		0			0
$163$	39.5636			0.242722			0.121361	−	0.992608i	$-0.461274\pi$
$163$	39.5636			0.242722			0.121361	+	0.992608i	$0.461274\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2.72800i			0.0163353i	0.999967	+	0.00816766i	$0.00259988\pi$
$167$			2.72800i			0.0163353i	−0.999967	+	0.00816766i	$0.997400\pi$
$168$		0			0
$169$	−116.227			−0.687734
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	119.559i		−	0.691092i	−0.938402	−	0.345546i	$-0.887694\pi$
$173$		−	119.559i		−	0.691092i	0.938402	−	0.345546i	$-0.112306\pi$
$174$		0			0
$175$	−50.2005			−0.286860
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	16.1873i		−	0.0904318i	−0.998977	−	0.0452159i	$-0.985602\pi$
$179$		−	16.1873i		−	0.0904318i	0.998977	−	0.0452159i	$-0.0143976\pi$
$180$		0			0
$181$	−171.692	−	297.380i	−0.948576	−	1.64298i	−0.748429	−	0.663215i	$-0.769191\pi$
$181$	−171.692	−	297.380i	−0.948576	−	1.64298i	−0.200147	−	0.979766i	$-0.564142\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	188.463	−	108.809i	1.01872	−	0.588159i
$186$		0			0
$187$	367.821			1.96696
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.9165	−	6.30263i	0.0571543	−	0.0329981i	−0.471151	−	0.882053i	$-0.656161\pi$
$191$	10.9165	−	6.30263i	0.0571543	−	0.0329981i	0.528305	+	0.849055i	$0.322828\pi$
$192$		0			0
$193$	−68.7102	−	119.009i	−0.356011	−	0.616629i	0.631279	−	0.775556i	$-0.282530\pi$
$193$	−68.7102	−	119.009i	−0.356011	−	0.616629i	−0.987290	+	0.158926i	$0.949197\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			191.963i			0.974429i	0.873282	+	0.487215i	$0.161987\pi$
$197$			191.963i			0.974429i	−0.873282	+	0.487215i	$0.838013\pi$
$198$		0			0
$199$	−64.5484	−	111.801i	−0.324364	−	0.561815i	0.657019	−	0.753874i	$-0.271817\pi$
$199$	−64.5484	−	111.801i	−0.324364	−	0.561815i	−0.981383	+	0.192059i	$0.938484\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	302.346i		−	1.48939i
$204$		0			0
$205$	19.0097	−	32.9257i	0.0927302	−	0.160613i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	298.976	+	76.3307i	1.43051	+	0.365218i
$210$		0			0
$211$	−78.4672	−	135.909i	−0.371882	−	0.644119i	0.617973	−	0.786200i	$-0.287954\pi$
$211$	−78.4672	−	135.909i	−0.371882	−	0.644119i	−0.989855	+	0.142080i	$0.954621\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−308.848	+	178.314i	−1.43650	+	0.829366i
$216$		0			0
$217$	328.501			1.51383
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−142.488	−	82.2655i	−0.644742	−	0.372242i
$222$		0			0
$223$	100.674			0.451453			0.225727	−	0.974191i	$-0.427524\pi$
$223$	100.674			0.451453			0.225727	+	0.974191i	$0.427524\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	27.5603	−	15.9119i	0.121411	−	0.0700967i	−0.438065	−	0.898943i	$-0.644336\pi$
$227$	27.5603	−	15.9119i	0.121411	−	0.0700967i	0.559476	+	0.828847i	$0.311003\pi$
$228$		0			0
$229$	−77.4200	+	134.095i	−0.338078	+	0.585569i	−0.984071	−	0.177774i	$-0.943110\pi$
$229$	−77.4200	+	134.095i	−0.338078	+	0.585569i	0.645993	+	0.763344i	$0.276444\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	334.241	+	192.974i	1.43451	+	0.828215i	0.997460	−	0.0712220i	$-0.0226899\pi$
$233$	334.241	+	192.974i	1.43451	+	0.828215i	0.437050	+	0.899437i	$0.356023\pi$
$234$		0			0
$235$	74.6990			0.317868
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−19.8128	−	11.4390i	−0.0828990	−	0.0478617i	0.457978	−	0.888964i	$-0.348574\pi$
$239$	−19.8128	−	11.4390i	−0.0828990	−	0.0478617i	−0.540876	+	0.841102i	$0.681907\pi$
$240$		0			0
$241$	−103.286	+	178.896i	−0.428571	+	0.742307i	−0.996747	−	0.0806002i	$-0.974316\pi$
$241$	−103.286	+	178.896i	−0.428571	+	0.742307i	0.568175	+	0.822908i	$0.307650\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	400.297i		−	1.63386i
$246$		0			0
$247$	−98.7467	−	96.4371i	−0.399784	−	0.390434i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−170.812	+	98.6182i	−0.680525	+	0.392901i	−0.800053	−	0.599930i	$-0.795195\pi$
$251$	−170.812	+	98.6182i	−0.680525	+	0.392901i	0.119528	+	0.992831i	$0.461862\pi$
$252$		0			0
$253$	−313.946	−	543.771i	−1.24089	−	2.14929i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	202.723i		−	0.788805i	−0.918938	−	0.394403i	$-0.870952\pi$
$257$		−	202.723i		−	0.788805i	0.918938	−	0.394403i	$-0.129048\pi$
$258$		0			0
$259$	−279.814	−	484.652i	−1.08036	−	1.87124i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			206.933i			0.786819i	0.919363	+	0.393409i	$0.128704\pi$
$263$			206.933i			0.786819i	−0.919363	+	0.393409i	$0.871296\pi$
$264$		0			0
$265$	−114.657	+	198.591i	−0.432667	+	0.749402i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	54.5399	+	31.4886i	0.202751	+	0.117058i	0.597938	−	0.801542i	$-0.295987\pi$
$269$	54.5399	+	31.4886i	0.202751	+	0.117058i	−0.395187	+	0.918601i	$0.629320\pi$
$270$		0			0
$271$	−101.148	+	175.193i	−0.373239	+	0.646469i	−0.990062	−	0.140633i	$-0.955086\pi$
$271$	−101.148	+	175.193i	−0.373239	+	0.646469i	0.616823	+	0.787102i	$0.288420\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	69.6635i		−	0.253322i
$276$		0			0
$277$	−48.8226	−	84.5632i	−0.176255	−	0.305282i	0.764340	−	0.644813i	$-0.223065\pi$
$277$	−48.8226	−	84.5632i	−0.176255	−	0.305282i	−0.940595	+	0.339531i	$0.889732\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−197.525	−	114.041i	−0.702935	−	0.405840i	0.105505	−	0.994419i	$-0.466354\pi$
$281$	−197.525	−	114.041i	−0.702935	−	0.405840i	−0.808440	+	0.588579i	$0.799687\pi$
$282$		0			0
$283$	−103.040	−	178.471i	−0.364099	−	0.630639i	0.624532	−	0.780999i	$-0.285290\pi$
$283$	−103.040	−	178.471i	−0.364099	−	0.630639i	−0.988631	+	0.150361i	$0.951957\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−84.6716	−	48.8852i	−0.295023	−	0.170332i
$288$		0			0
$289$	111.981	+	193.956i	0.387476	+	0.671128i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−218.824	−	126.338i	−0.746840	−	0.431189i	0.0777107	−	0.996976i	$-0.475239\pi$
$293$	−218.824	−	126.338i	−0.746840	−	0.431189i	−0.824551	+	0.565787i	$0.808572\pi$
$294$		0			0
$295$	−170.539	+	295.382i	−0.578097	+	1.00129i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			280.864i			0.939345i
$300$		0			0
$301$	458.551	+	794.233i	1.52342	+	2.63865i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	283.650i		−	0.930001i
$306$		0			0
$307$	261.675	−	453.234i	0.852361	−	1.47633i	−0.0267099	−	0.999643i	$-0.508503\pi$
$307$	261.675	−	453.234i	0.852361	−	1.47633i	0.879071	−	0.476690i	$-0.158164\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−332.891	−	192.195i	−1.07039	−	0.617989i	−0.142101	−	0.989852i	$-0.545386\pi$
$311$	−332.891	−	192.195i	−1.07039	−	0.617989i	−0.928287	+	0.371863i	$0.878719\pi$
$312$		0			0
$313$	−110.259	+	190.974i	−0.352265	+	0.610141i	−0.986646	−	0.162879i	$-0.947922\pi$
$313$	−110.259	+	190.974i	−0.352265	+	0.610141i	0.634381	+	0.773021i	$0.281255\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	166.527	+	96.1444i	0.525322	+	0.303295i	0.739109	−	0.673585i	$-0.235247\pi$
$317$	166.527	+	96.1444i	0.525322	+	0.303295i	−0.213787	+	0.976880i	$0.568580\pi$
$318$		0			0
$319$	419.567			1.31526
$320$		0			0
$321$		0			0
$322$		0			0
$323$	116.286	+	414.314i	0.360020	+	1.28271i
$324$		0			0
$325$	−15.5807	+	26.9865i	−0.0479405	+	0.0830354i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	192.096i		−	0.583877i
$330$		0			0
$331$	140.614	+	243.551i	0.424817	+	0.735804i	0.996403	−	0.0847374i	$-0.0270051\pi$
$331$	140.614	+	243.551i	0.424817	+	0.735804i	−0.571586	+	0.820542i	$0.693672\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	225.153	−	129.992i	0.672098	−	0.388036i
$336$		0			0
$337$	249.153	+	431.546i	0.739328	+	1.28055i	0.952798	+	0.303604i	$0.0981899\pi$
$337$	249.153	+	431.546i	0.739328	+	1.28055i	−0.213471	+	0.976949i	$0.568477\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			455.862i			1.33684i
$342$		0			0
$343$	−455.953			−1.32931
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−135.766	+	78.3843i	−0.391255	+	0.225891i	−0.682704	−	0.730695i	$-0.739196\pi$
$347$	−135.766	+	78.3843i	−0.391255	+	0.225891i	0.291449	+	0.956586i	$0.405863\pi$
$348$		0			0
$349$	−27.3145	−	47.3101i	−0.0782651	−	0.135559i	0.824236	−	0.566246i	$-0.191605\pi$
$349$	−27.3145	−	47.3101i	−0.0782651	−	0.135559i	−0.902501	+	0.430687i	$0.858271\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−22.4457	+	12.9590i	−0.0635854	+	0.0367111i	−0.531456	−	0.847086i	$-0.678355\pi$
$353$	−22.4457	+	12.9590i	−0.0635854	+	0.0367111i	0.467870	+	0.883797i	$0.345021\pi$
$354$		0			0
$355$	384.048			1.08183
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−551.226	−	318.250i	−1.53545	−	0.886491i	−0.999097	−	0.0424966i	$-0.986469\pi$
$359$	−551.226	−	318.250i	−1.53545	−	0.886491i	−0.536351	−	0.843995i	$-0.680198\pi$
$360$		0			0
$361$	8.54208	+	360.899i	0.0236623	+	0.999720i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			258.804i			0.709052i
$366$		0			0
$367$	−99.5425	+	172.413i	−0.271233	+	0.469789i	−0.969178	−	0.246362i	$-0.920765\pi$
$367$	−99.5425	+	172.413i	−0.271233	+	0.469789i	0.697945	+	0.716152i	$0.254098\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	510.697	+	294.851i	1.37654	+	0.794746i
$372$		0			0
$373$	225.834	−	391.156i	0.605453	−	1.04867i	−0.386527	−	0.922278i	$-0.626325\pi$
$373$	225.834	−	391.156i	0.605453	−	1.04867i	0.991980	−	0.126397i	$-0.0403412\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−162.533	−	93.8388i	−0.431123	−	0.248909i
$378$		0			0
$379$	−98.6773			−0.260362			−0.130181	−	0.991490i	$-0.541556\pi$
$379$	−98.6773			−0.260362			−0.130181	+	0.991490i	$0.541556\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	342.171	−	197.553i	0.893397	−	0.515803i	0.0183451	−	0.999832i	$-0.494160\pi$
$383$	342.171	−	197.553i	0.893397	−	0.515803i	0.875052	+	0.484029i	$0.160827\pi$
$384$		0			0
$385$	864.942			2.24660
$386$		0			0
$387$		0			0
$388$		0			0
$389$	217.495	+	125.571i	0.559114	+	0.322805i	0.752790	−	0.658261i	$-0.228708\pi$
$389$	217.495	+	125.571i	0.559114	+	0.322805i	−0.193676	+	0.981066i	$0.562041\pi$
$390$		0			0
$391$	437.828	−	758.340i	1.11976	−	1.93949i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−469.191	+	270.887i	−1.18782	+	0.685791i
$396$		0			0
$397$	157.630	−	273.022i	0.397052	−	0.687714i	−0.596309	−	0.802755i	$-0.703367\pi$
$397$	157.630	−	273.022i	0.397052	−	0.687714i	0.993361	+	0.115041i	$0.0367000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	241.283	−	139.305i	0.601704	−	0.347394i	−0.168008	−	0.985786i	$-0.553733\pi$
$401$	241.283	−	139.305i	0.601704	−	0.347394i	0.769712	+	0.638392i	$0.220400\pi$
$402$		0			0
$403$	101.956	−	176.594i	0.252994	−	0.438198i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	672.554	−	388.299i	1.65247	−	0.954052i
$408$		0			0
$409$	−200.542			−0.490323			−0.245161	−	0.969482i	$-0.578841\pi$
$409$	−200.542			−0.490323			−0.245161	+	0.969482i	$0.578841\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	759.602	+	438.557i	1.83923	+	1.06188i
$414$		0			0
$415$	−187.630	+	324.984i	−0.452120	+	0.783095i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−452.302	−	261.137i	−1.07948	−	0.623238i	−0.148723	−	0.988879i	$-0.547516\pi$
$419$	−452.302	−	261.137i	−1.07948	−	0.623238i	−0.930756	+	0.365641i	$0.880850\pi$
$420$		0			0
$421$	−258.641			−0.614349			−0.307174	−	0.951653i	$-0.599383\pi$
$421$	−258.641			−0.614349			−0.307174	+	0.951653i	$0.599383\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	84.1363	−	48.5761i	0.197968	−	0.114297i
$426$		0			0
$427$	−729.433			−1.70828
$428$		0			0
$429$		0			0
$430$		0			0
$431$	263.486	−	152.124i	0.611336	−	0.352955i	−0.162152	−	0.986766i	$-0.551843\pi$
$431$	263.486	−	152.124i	0.611336	−	0.352955i	0.773488	+	0.633811i	$0.218510\pi$
$432$		0			0
$433$	398.394	+	690.039i	0.920079	+	1.59362i	0.799290	+	0.600945i	$0.205209\pi$
$433$	398.394	+	690.039i	0.920079	+	1.59362i	0.120789	+	0.992678i	$0.461458\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	513.251	−	525.542i	1.17449	−	1.20261i
$438$		0			0
$439$	−503.675			−1.14732			−0.573662	−	0.819092i	$-0.694478\pi$
$439$	−503.675			−1.14732			−0.573662	+	0.819092i	$0.694478\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−292.376	−	168.803i	−0.659990	−	0.381045i	0.132283	−	0.991212i	$-0.457769\pi$
$443$	−292.376	−	168.803i	−0.659990	−	0.381045i	−0.792273	+	0.610167i	$0.791103\pi$
$444$		0			0
$445$	−323.439	+	560.213i	−0.726830	+	1.25891i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			771.592i			1.71847i	0.511584	+	0.859233i	$0.329059\pi$
$449$			771.592i			1.71847i	−0.511584	+	0.859233i	$0.670941\pi$
$450$		0			0
$451$	67.8382	−	117.499i	0.150417	−	0.260531i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−335.065	−	193.450i	−0.736406	−	0.425164i
$456$		0			0
$457$	174.821	+	302.800i	0.382542	+	0.662581i	0.991425	−	0.130678i	$-0.0417155\pi$
$457$	174.821	+	302.800i	0.382542	+	0.662581i	−0.608883	+	0.793260i	$0.708382\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			295.156i			0.640252i	0.947375	+	0.320126i	$0.103725\pi$
$461$			295.156i			0.640252i	−0.947375	+	0.320126i	$0.896275\pi$
$462$		0			0
$463$	78.9995	−	136.831i	0.170625	−	0.295532i	−0.768013	−	0.640434i	$-0.778755\pi$
$463$	78.9995	−	136.831i	0.170625	−	0.295532i	0.938639	+	0.344902i	$0.112088\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	666.947i		−	1.42815i	−0.700068	−	0.714076i	$-0.746847\pi$
$467$		−	666.947i		−	1.42815i	0.700068	−	0.714076i	$-0.253153\pi$
$468$		0			0
$469$	−334.287	−	579.002i	−0.712765	−	1.23455i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1102.16	+	636.333i	−2.33015	+	1.34531i
$474$		0			0
$475$	78.4690	−	22.0241i	0.165198	−	0.0463664i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	167.295	+	96.5879i	0.349259	+	0.201645i	0.664359	−	0.747414i	$-0.268705\pi$
$479$	167.295	+	96.5879i	0.349259	+	0.201645i	−0.315100	+	0.949059i	$0.602038\pi$
$480$		0			0
$481$	−347.382			−0.722208
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−467.051	+	269.652i	−0.962992	+	0.555984i
$486$		0			0
$487$	−550.037			−1.12944			−0.564720	−	0.825282i	$-0.691016\pi$
$487$	−550.037			−1.12944			−0.564720	+	0.825282i	$0.691016\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	220.915	−	127.546i	0.449930	−	0.259767i	−0.257871	−	0.966179i	$-0.583021\pi$
$491$	220.915	−	127.546i	0.449930	−	0.259767i	0.707801	+	0.706412i	$0.249688\pi$
$492$		0			0
$493$	292.563	+	506.733i	0.593433	+	1.02786i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	987.617i		−	1.98716i
$498$		0			0
$499$	−242.669	−	420.316i	−0.486311	−	0.842316i	0.513565	−	0.858051i	$-0.328325\pi$
$499$	−242.669	−	420.316i	−0.486311	−	0.842316i	−0.999876	+	0.0157350i	$0.994991\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−172.886	+	99.8159i	−0.343710	+	0.198441i	−0.661912	−	0.749582i	$-0.730255\pi$
$503$	−172.886	+	99.8159i	−0.343710	+	0.198441i	0.318201	+	0.948023i	$0.396921\pi$
$504$		0			0
$505$	410.101			0.812082
$506$		0			0
$507$		0			0
$508$		0			0
$509$			205.885i			0.404489i	0.979335	+	0.202244i	$0.0648235\pi$
$509$			205.885i			0.404489i	−0.979335	+	0.202244i	$0.935176\pi$
$510$		0			0
$511$	665.539			1.30242
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	392.228i		−	0.761607i
$516$		0			0
$517$	266.572			0.515613
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	570.510i		−	1.09503i	−0.836796	−	0.547514i	$-0.815574\pi$
$521$		−	570.510i		−	1.09503i	0.836796	−	0.547514i	$-0.184426\pi$
$522$		0			0
$523$	−466.633	−	808.231i	−0.892223	−	1.54538i	−0.837205	−	0.546890i	$-0.815812\pi$
$523$	−466.633	−	808.231i	−0.892223	−	1.54538i	−0.0550183	−	0.998485i	$-0.517522\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−550.569	+	317.871i	−1.04472	+	0.603171i
$528$		0			0
$529$	−965.795			−1.82570
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−52.5589	+	30.3449i	−0.0986095	+	0.0569322i
$534$		0			0
$535$	−265.893	−	460.540i	−0.496997	−	0.860823i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1428.50i		−	2.65029i
$540$		0			0
$541$	77.8507	+	134.841i	0.143901	+	0.249245i	0.928963	−	0.370174i	$-0.120702\pi$
$541$	77.8507	+	134.841i	0.143901	+	0.249245i	−0.785061	+	0.619418i	$0.787368\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	545.101i		−	1.00019i
$546$		0			0
$547$	−17.0612	+	29.5509i	−0.0311906	+	0.0540237i	−0.881199	−	0.472745i	$-0.843263\pi$
$547$	−17.0612	+	29.5509i	−0.0311906	+	0.0540237i	0.850009	+	0.526769i	$0.176597\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	132.646	+	472.601i	0.240736	+	0.857715i
$552$		0			0
$553$	696.613	+	1206.57i	1.25970	+	2.18186i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	34.3037	−	19.8053i	0.0615865	−	0.0355570i	−0.468891	−	0.883256i	$-0.655346\pi$
$557$	34.3037	−	19.8053i	0.0615865	−	0.0355570i	0.530477	+	0.847699i	$0.322013\pi$
$558$		0			0
$559$	569.279			1.01839
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−31.8404	−	18.3831i	−0.0565549	−	0.0326520i	0.471456	−	0.881890i	$-0.343729\pi$
$563$	−31.8404	−	18.3831i	−0.0565549	−	0.0326520i	−0.528011	+	0.849238i	$0.677062\pi$
$564$		0			0
$565$	−639.248			−1.13141
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.7340	−	7.92934i	0.0241371	−	0.0139356i	−0.487883	−	0.872909i	$-0.662231\pi$
$569$	13.7340	−	7.92934i	0.0241371	−	0.0139356i	0.512020	+	0.858974i	$0.328897\pi$
$570$		0			0
$571$	445.433	−	771.512i	0.780093	−	1.35116i	−0.151794	−	0.988412i	$-0.548505\pi$
$571$	445.433	−	771.512i	0.780093	−	1.35116i	0.931887	−	0.362748i	$-0.118161\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−143.626	−	82.9223i	−0.249784	−	0.144213i
$576$		0			0
$577$	420.790			0.729272			0.364636	−	0.931150i	$-0.381193\pi$
$577$	420.790			0.729272			0.364636	+	0.931150i	$0.381193\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	835.728	+	482.508i	1.43843	+	0.830478i
$582$		0			0
$583$	−409.166	+	708.696i	−0.701829	+	1.21560i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			648.942i			1.10552i	0.833339	+	0.552762i	$0.186426\pi$
$587$			648.942i			1.10552i	−0.833339	+	0.552762i	$0.813574\pi$
$588$		0			0
$589$	−513.484	+	144.120i	−0.871789	+	0.244687i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	104.519	−	60.3443i	0.176255	−	0.101761i	−0.409277	−	0.912410i	$-0.634219\pi$
$593$	104.519	−	60.3443i	0.176255	−	0.101761i	0.585532	+	0.810649i	$0.300886\pi$
$594$		0			0
$595$	603.121	+	1044.64i	1.01365	+	1.75569i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			482.506i			0.805520i	0.915306	+	0.402760i	$0.131949\pi$
$599$			482.506i			0.805520i	−0.915306	+	0.402760i	$0.868051\pi$
$600$		0			0
$601$	−191.870	−	332.329i	−0.319251	−	0.552960i	0.661081	−	0.750315i	$-0.270098\pi$
$601$	−191.870	−	332.329i	−0.319251	−	0.552960i	−0.980332	+	0.197355i	$0.936765\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			649.629i			1.07377i
$606$		0			0
$607$	−131.396	+	227.585i	−0.216468	+	0.374934i	−0.953726	−	0.300678i	$-0.902787\pi$
$607$	−131.396	+	227.585i	−0.216468	+	0.374934i	0.737258	+	0.675612i	$0.236120\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−103.266	−	59.6205i	−0.169011	−	0.0975786i
$612$		0			0
$613$	−416.617	+	721.601i	−0.679636	+	1.17716i	0.295455	+	0.955357i	$0.404529\pi$
$613$	−416.617	+	721.601i	−0.679636	+	1.17716i	−0.975091	+	0.221807i	$0.928805\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1190.60i			1.92965i	0.262888	+	0.964826i	$0.415325\pi$
$617$			1190.60i			1.92965i	−0.262888	+	0.964826i	$0.584675\pi$
$618$		0			0
$619$	−515.688	−	893.198i	−0.833099	−	1.44297i	−0.895569	−	0.444922i	$-0.853231\pi$
$619$	−515.688	−	893.198i	−0.833099	−	1.44297i	0.0624707	−	0.998047i	$-0.480102\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1440.64	+	831.755i	2.31243	+	1.33508i
$624$		0			0
$625$	249.681	+	432.460i	0.399489	+	0.691935i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	937.939	+	541.519i	1.49116	+	0.860921i
$630$		0			0
$631$	121.585	+	210.592i	0.192686	+	0.333743i	0.946140	−	0.323759i	$-0.104947\pi$
$631$	121.585	+	210.592i	0.192686	+	0.333743i	−0.753453	+	0.657502i	$0.771613\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	276.876	+	159.854i	0.436025	+	0.251739i
$636$		0			0
$637$	−319.494	+	553.380i	−0.501560	+	0.868728i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			580.450i			0.905538i	0.891628	+	0.452769i	$0.149564\pi$
$641$			580.450i			0.905538i	−0.891628	+	0.452769i	$0.850436\pi$
$642$		0			0
$643$	391.744	+	678.520i	0.609244	+	1.05524i	0.991365	+	0.131129i	$0.0418601\pi$
$643$	391.744	+	678.520i	0.609244	+	1.05524i	−0.382122	+	0.924112i	$0.624807\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			642.116i			0.992452i	0.868193	+	0.496226i	$0.165281\pi$
$647$			642.116i			0.992452i	−0.868193	+	0.496226i	$0.834719\pi$
$648$		0			0
$649$	−608.587	+	1054.10i	−0.937731	+	1.62420i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−546.629	−	315.597i	−0.837104	−	0.483302i	0.0191745	−	0.999816i	$-0.493896\pi$
$653$	−546.629	−	315.597i	−0.837104	−	0.483302i	−0.856279	+	0.516514i	$0.827230\pi$
$654$		0			0
$655$	5.90309	−	10.2245i	0.00901235	−	0.0156099i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−147.654	−	85.2482i	−0.224058	−	0.129360i	0.383770	−	0.923429i	$-0.374626\pi$
$659$	−147.654	−	85.2482i	−0.224058	−	0.129360i	−0.607828	+	0.794069i	$0.707959\pi$
$660$		0			0
$661$	−1189.45			−1.79948			−0.899738	−	0.436430i	$-0.856243\pi$
$661$	−1189.45			−1.79948			−0.899738	+	0.436430i	$0.856243\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	273.451	+	974.272i	0.411204	+	1.46507i
$666$		0			0
$667$	499.422	−	865.024i	0.748758	−	1.29689i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1012.24i		−	1.50855i
$672$		0			0
$673$	−50.1653	−	86.8888i	−0.0745398	−	0.129107i	0.826346	−	0.563162i	$-0.190415\pi$
$673$	−50.1653	−	86.8888i	−0.0745398	−	0.129107i	−0.900886	+	0.434056i	$0.857082\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−301.480	+	174.060i	−0.445318	+	0.257105i	−0.705851	−	0.708360i	$-0.749435\pi$
$677$	−301.480	+	174.060i	−0.445318	+	0.257105i	0.260533	+	0.965465i	$0.416102\pi$
$678$		0			0
$679$	693.436	+	1201.07i	1.02126	+	1.76888i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			50.5434i			0.0740021i	0.999315	+	0.0370010i	$0.0117805\pi$
$683$			50.5434i			0.0740021i	−0.999315	+	0.0370010i	$0.988220\pi$
$684$		0			0
$685$	674.368			0.984479
$686$		0			0
$687$		0			0
$688$		0			0
$689$	317.009	−	183.025i	0.460100	−	0.265639i
$690$		0			0
$691$	203.726	+	352.863i	0.294827	+	0.510656i	0.974945	−	0.222447i	$-0.0714045\pi$
$691$	203.726	+	352.863i	0.294827	+	0.510656i	−0.680117	+	0.733103i	$0.738071\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	525.906	−	303.632i	0.756699	−	0.436880i
$696$		0			0
$697$	189.214			0.271468
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−106.817	−	61.6711i	−0.152379	−	0.0879758i	0.421872	−	0.906655i	$-0.361373\pi$
$701$	−106.817	−	61.6711i	−0.152379	−	0.0879758i	−0.574251	+	0.818680i	$0.694706\pi$
$702$		0			0
$703$	650.008	+	634.805i	0.924620	+	0.902994i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	1054.61i		−	1.49167i
$708$		0			0
$709$	115.421	−	199.915i	0.162794	−	0.281967i	−0.773076	−	0.634314i	$-0.781283\pi$
$709$	115.421	−	199.915i	0.162794	−	0.281967i	0.935870	+	0.352347i	$0.114616\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	939.854	+	542.625i	1.31817	+	0.761045i
$714$		0			0
$715$	268.451	−	464.971i	0.375456	−	0.650309i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	524.661	+	302.913i	0.729710	+	0.421298i	0.818316	−	0.574769i	$-0.194908\pi$
$719$	524.661	+	302.913i	0.729710	+	0.421298i	−0.0886062	+	0.996067i	$0.528241\pi$
$720$		0			0
$721$	−1008.65			−1.39896
$722$		0			0
$723$		0			0
$724$		0			0
$725$	95.9727	−	55.4099i	0.132376	−	0.0764274i
$726$		0			0
$727$	555.764			0.764462			0.382231	−	0.924067i	$-0.375156\pi$
$727$	555.764			0.764462			0.382231	+	0.924067i	$0.375156\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1537.07	−	887.426i	−2.10269	−	1.21399i
$732$		0			0
$733$	−42.5759	+	73.7436i	−0.0580844	+	0.100605i	−0.893605	−	0.448853i	$-0.851833\pi$
$733$	−42.5759	+	73.7436i	−0.0580844	+	0.100605i	0.835521	+	0.549458i	$0.185166\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	803.484	−	463.892i	1.09021	−	0.629432i
$738$		0			0
$739$	−294.756	+	510.533i	−0.398859	+	0.690843i	−0.993585	−	0.113085i	$-0.963927\pi$
$739$	−294.756	+	510.533i	−0.398859	+	0.690843i	0.594727	+	0.803928i	$0.297260\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1217.64	+	703.004i	−1.63881	+	0.946169i	−0.657571	+	0.753393i	$0.728416\pi$
$743$	−1217.64	+	703.004i	−1.63881	+	0.946169i	−0.981243	+	0.192777i	$0.938251\pi$
$744$		0			0
$745$	278.622	−	482.588i	0.373990	−	0.647769i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1184.32	+	683.770i	−1.58121	+	0.912910i
$750$		0			0
$751$	9.97744			0.0132855			0.00664277	−	0.999978i	$-0.497886\pi$
$751$	9.97744			0.0132855			0.00664277	+	0.999978i	$0.497886\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	290.838	+	167.916i	0.385216	+	0.222405i
$756$		0			0
$757$	−546.784	+	947.058i	−0.722304	+	1.25107i	0.237770	+	0.971322i	$0.423584\pi$
$757$	−546.784	+	947.058i	−0.722304	+	1.25107i	−0.960074	+	0.279746i	$0.909750\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−864.824	−	499.306i	−1.13643	−	0.656118i	−0.190886	−	0.981612i	$-0.561136\pi$
$761$	−864.824	−	499.306i	−1.13643	−	0.656118i	−0.945544	+	0.325494i	$0.894470\pi$
$762$		0			0
$763$	−1401.78			−1.83719
$764$		0			0
$765$		0			0
$766$		0			0
$767$	471.514	−	272.229i	0.614751	−	0.354926i
$768$		0			0
$769$	27.7029			0.0360246			0.0180123	−	0.999838i	$-0.494266\pi$
$769$	27.7029			0.0360246			0.0180123	+	0.999838i	$0.494266\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	258.082	−	149.004i	0.333870	−	0.192760i	−0.323688	−	0.946164i	$-0.604923\pi$
$773$	258.082	−	149.004i	0.333870	−	0.192760i	0.657558	+	0.753404i	$0.271589\pi$
$774$		0			0
$775$	60.2032	+	104.275i	0.0776815	+	0.134548i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	153.798	+	39.2658i	0.197431	+	0.0504054i
$780$		0			0
$781$	1370.52			1.75483
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1150.94	+	664.493i	1.46616	+	0.846487i
$786$		0			0
$787$	200.548	−	347.359i	0.254825	−	0.441370i	−0.710023	−	0.704179i	$-0.751315\pi$
$787$	200.548	−	347.359i	0.254825	−	0.441370i	0.964848	+	0.262808i	$0.0846487\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1643.89i			2.07824i
$792$		0			0
$793$	−226.393	+	392.125i	−0.285490	+	0.494483i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	876.314	+	505.940i	1.09952	+	0.634806i	0.936094	−	0.351750i	$-0.114413\pi$
$797$	876.314	+	505.940i	1.09952	+	0.634806i	0.163422	+	0.986556i	$0.447747\pi$
$798$		0			0
$799$	185.880	+	321.953i	0.232641	+	0.402945i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			923.572i			1.15015i
$804$		0			0
$805$	1029.56	−	1783.26i	1.27896	−	2.21523i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1149.91i			1.42139i	0.703499	+	0.710696i	$0.251620\pi$
$809$			1149.91i			1.42139i	−0.703499	+	0.710696i	$0.748380\pi$
$810$		0			0
$811$	433.837	+	751.428i	0.534941	+	0.926545i	0.999166	+	0.0408280i	$0.0129996\pi$
$811$	433.837	+	751.428i	0.534941	+	0.926545i	−0.464225	+	0.885717i	$0.653667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−155.927	+	90.0245i	−0.191322	+	0.110460i
$816$		0			0
$817$	−1065.21	−	1040.30i	−1.30381	−	1.27332i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	157.983	+	91.2117i	0.192428	+	0.111098i	0.593119	−	0.805115i	$-0.297897\pi$
$821$	157.983	+	91.2117i	0.192428	+	0.111098i	−0.400691	+	0.916213i	$0.631230\pi$
$822$		0			0
$823$	−1614.01			−1.96113			−0.980566	−	0.196191i	$-0.937143\pi$
$823$	−1614.01			−1.96113			−0.980566	+	0.196191i	$0.937143\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1058.90	−	611.359i	1.28042	−	0.739249i	0.303492	−	0.952834i	$-0.401847\pi$
$827$	1058.90	−	611.359i	1.28042	−	0.739249i	0.976924	+	0.213585i	$0.0685140\pi$
$828$		0			0
$829$	−281.377			−0.339417			−0.169708	−	0.985494i	$-0.554283\pi$
$829$	−281.377			−0.339417			−0.169708	+	0.985494i	$0.554283\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1725.28	−	996.091i	2.07117	−	1.19579i
$834$		0			0
$835$	−6.20739	−	10.7515i	−0.00743400	−	0.0128761i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1271.47i			1.51546i	0.652567	+	0.757731i	$0.273692\pi$
$839$			1271.47i			1.51546i	−0.652567	+	0.757731i	$0.726308\pi$
$840$		0			0
$841$	−86.7792	−	150.306i	−0.103186	−	0.178723i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	458.071	−	264.467i	0.542096	−	0.312979i
$846$		0			0
$847$	1670.58			1.97235
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	1848.81i		−	2.17252i
$852$		0			0
$853$	−449.429			−0.526880			−0.263440	−	0.964676i	$-0.584857\pi$
$853$	−449.429			−0.526880			−0.263440	+	0.964676i	$0.584857\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	710.057i		−	0.828538i	−0.910154	−	0.414269i	$-0.864037\pi$
$857$		−	710.057i		−	0.828538i	0.910154	−	0.414269i	$-0.135963\pi$
$858$		0			0
$859$	−173.681			−0.202190			−0.101095	−	0.994877i	$-0.532235\pi$
$859$	−173.681			−0.202190			−0.101095	+	0.994877i	$0.532235\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	560.406i		−	0.649370i	−0.945822	−	0.324685i	$-0.894742\pi$
$863$		−	560.406i		−	0.649370i	0.945822	−	0.324685i	$-0.105258\pi$
$864$		0			0
$865$	272.049	+	471.202i	0.314507	+	0.544742i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1674.36	+	966.693i	−1.92677	+	1.11242i
$870$		0			0
$871$	−415.009			−0.476474
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1350.94	−	779.965i	1.54393	−	0.891388i
$876$		0			0
$877$	−686.285	−	1188.68i	−0.782537	−	1.35539i	−0.930460	−	0.366395i	$-0.880592\pi$
$877$	−686.285	−	1188.68i	−0.782537	−	1.35539i	0.147923	−	0.988999i	$-0.452741\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	105.391i		−	0.119627i	−0.998210	−	0.0598135i	$-0.980949\pi$
$881$		−	105.391i		−	0.119627i	0.998210	−	0.0598135i	$-0.0190506\pi$
$882$		0			0
$883$	−439.165	−	760.656i	−0.497356	−	0.861446i	0.502640	−	0.864496i	$-0.332362\pi$
$883$	−439.165	−	760.656i	−0.497356	−	0.861446i	−0.999995	+	0.00305054i	$0.999029\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1153.04i			1.29993i	0.759963	+	0.649966i	$0.225217\pi$
$887$			1153.04i			1.29993i	−0.759963	+	0.649966i	$0.774783\pi$
$888$		0			0
$889$	411.080	−	712.012i	0.462408	−	0.800914i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	84.2766	+	300.267i	0.0943746	+	0.336245i
$894$		0			0
$895$	36.8332	+	63.7969i	0.0411544	+	0.0712815i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−628.024	+	362.590i	−0.698581	+	0.403326i
$900$		0			0
$901$	−1141.24			−1.26664
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1353.34	+	781.350i	1.49540	+	0.863370i
$906$		0			0
$907$	1081.82			1.19274			0.596370	−	0.802709i	$-0.296609\pi$
$907$	1081.82			1.19274			0.596370	+	0.802709i	$0.296609\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	348.268	−	201.073i	0.382292	−	0.220716i	−0.296523	−	0.955026i	$-0.595827\pi$
$911$	348.268	−	201.073i	0.382292	−	0.220716i	0.678815	+	0.734309i	$0.262494\pi$
$912$		0			0
$913$	−669.579	+	1159.74i	−0.733383	+	1.27026i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−26.2931	−	15.1804i	−0.0286730	−	0.0165544i
$918$		0			0
$919$	−761.698			−0.828834			−0.414417	−	0.910087i	$-0.636014\pi$
$919$	−761.698			−0.828834			−0.414417	+	0.910087i	$0.636014\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−530.918	−	306.526i	−0.575209	−	0.332097i
$924$		0			0
$925$	102.561	−	177.641i	0.110877	−	0.192044i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1054.20i		−	1.13476i	−0.823455	−	0.567382i	$-0.807956\pi$
$929$		−	1054.20i		−	1.13476i	0.823455	−	0.567382i	$-0.192044\pi$
$930$		0			0
$931$	1609.07	−	451.621i	1.72832	−	0.485092i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1449.65	+	836.954i	−1.55043	+	0.895138i
$936$		0			0
$937$	−77.1282	−	133.590i	−0.0823140	−	0.142572i	0.821929	−	0.569589i	$-0.192898\pi$
$937$	−77.1282	−	133.590i	−0.0823140	−	0.142572i	−0.904244	+	0.427017i	$0.859564\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	351.356i		−	0.373386i	−0.982418	−	0.186693i	$-0.940223\pi$
$941$		−	351.356i		−	0.373386i	0.982418	−	0.186693i	$-0.0597769\pi$
$942$		0			0
$943$	−161.499	−	279.725i	−0.171261	−	0.296633i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			994.185i			1.04983i	0.851156	+	0.524913i	$0.175902\pi$
$947$			994.185i			1.04983i	−0.851156	+	0.524913i	$0.824098\pi$
$948$		0			0
$949$	206.563	−	357.777i	0.217663	−	0.377004i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−941.583	−	543.623i	−0.988020	−	0.570433i	−0.0833378	−	0.996521i	$-0.526558\pi$
$953$	−941.583	−	543.623i	−0.988020	−	0.570433i	−0.904682	+	0.426088i	$0.859891\pi$
$954$		0			0
$955$	−28.6825	+	49.6795i	−0.0300340	+	0.0520204i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	1734.20i		−	1.80834i
$960$		0			0
$961$	86.5438	+	149.898i	0.0900560	+	0.155981i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	541.597	+	312.691i	0.561241	+	0.324032i
$966$		0			0
$967$	−311.541	−	539.605i	−0.322173	−	0.558019i	0.658763	−	0.752350i	$-0.271080\pi$
$967$	−311.541	−	539.605i	−0.322173	−	0.558019i	−0.980936	+	0.194331i	$0.937747\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−614.569	−	354.822i	−0.632924	−	0.365419i	0.148960	−	0.988843i	$-0.452408\pi$
$971$	−614.569	−	354.822i	−0.632924	−	0.365419i	−0.781884	+	0.623424i	$0.785741\pi$
$972$		0			0
$973$	−780.818	−	1352.42i	−0.802485	−	1.38995i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−486.933	−	281.131i	−0.498396	−	0.287749i	0.229655	−	0.973272i	$-0.426240\pi$
$977$	−486.933	−	281.131i	−0.498396	−	0.287749i	−0.728051	+	0.685523i	$0.759574\pi$
$978$		0			0
$979$	−1154.23	+	1999.19i	−1.17899	+	2.04207i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	602.827i		−	0.613252i	−0.951830	−	0.306626i	$-0.900800\pi$
$983$		−	602.827i		−	0.613252i	0.951830	−	0.306626i	$-0.0992001\pi$
$984$		0			0
$985$	−436.799	−	756.558i	−0.443451	−	0.768079i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			3029.78i			3.06348i
$990$		0			0
$991$	−476.739	+	825.736i	−0.481068	+	0.833235i	−0.999764	−	0.0217240i	$-0.993085\pi$
$991$	−476.739	+	825.736i	−0.481068	+	0.833235i	0.518696	+	0.854959i	$0.326418\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	508.793	+	293.752i	0.511350	+	0.295228i
$996$		0			0
$997$	−199.801	+	346.066i	−0.200403	+	0.347108i	−0.948658	−	0.316303i	$-0.897558\pi$
$997$	−199.801	+	346.066i	−0.200403	+	0.347108i	0.748256	+	0.663411i	$0.230892\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.be.a.125.12		80
3.2	odd	2		684.3.be.a.581.15	yes	80
9.2	odd	6		2052.3.m.a.1493.29		80
9.7	even	3		684.3.m.a.353.11	✓	80
19.7	even	3		2052.3.m.a.881.12		80
57.26	odd	6		684.3.m.a.653.11	yes	80
171.7	even	3		684.3.be.a.425.15	yes	80
171.83	odd	6	inner	2052.3.be.a.197.12		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.11	✓	80	9.7	even	3
684.3.m.a.653.11	yes	80	57.26	odd	6
684.3.be.a.425.15	yes	80	171.7	even	3
684.3.be.a.581.15	yes	80	3.2	odd	2
2052.3.m.a.881.12		80	19.7	even	3
2052.3.m.a.1493.29		80	9.2	odd	6
2052.3.be.a.125.12		80	1.1	even	1	trivial
2052.3.be.a.197.12		80	171.83	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:	\( N \)	\(=\)	\( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2052.be (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.94117 + 2.27544i) q^{5} +(5.85150 + 10.1351i) q^{7} +O(q^{10})\)	\(q+(-3.94117 + 2.27544i) q^{5} +(5.85150 + 10.1351i) q^{7} +(-14.0645 + 8.12016i) q^{11} +7.26450 q^{13} +(-19.6143 - 11.3243i) q^{17} +(-13.5931 - 13.2751i) q^{19} +38.6626i q^{23} +(-2.14477 + 3.71485i) q^{25} +(-22.3737 - 12.9174i) q^{29} +(14.0349 - 24.3091i) q^{31} +(-46.1236 - 26.6295i) q^{35} -47.8191 q^{37} +(-7.23503 + 4.17715i) q^{41} +78.3646 q^{43} +(-14.2151 - 8.20711i) q^{47} +(-43.9802 + 76.1759i) q^{49} +(43.6381 - 25.1945i) q^{53} +(36.9538 - 64.0059i) q^{55} +(64.9066 - 37.4738i) q^{59} +(-31.1644 + 53.9783i) q^{61} +(-28.6306 + 16.5299i) q^{65} -57.1284 q^{67} +(-73.0839 - 42.1950i) q^{71} +(28.4345 - 49.2501i) q^{73} +(-164.597 - 95.0303i) q^{77} +119.049 q^{79} +(71.4114 - 41.2294i) q^{83} +103.071 q^{85} +(123.100 - 71.0719i) q^{89} +(42.5082 + 73.6264i) q^{91} +(83.7793 + 21.3894i) q^{95} +118.506 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 80 * q + q^7	\( 80 q + q^{7} - 18 q^{11} + 10 q^{13} - 9 q^{17} + 20 q^{19} + 200 q^{25} + 27 q^{29} - 8 q^{31} + 22 q^{37} + 54 q^{41} + 88 q^{43} - 198 q^{47} - 267 q^{49} - 36 q^{53} - 171 q^{59} + 7 q^{61} + 144 q^{65} + 154 q^{67} - 135 q^{71} + 43 q^{73} - 216 q^{77} + 34 q^{79} + 171 q^{83} + 216 q^{89} + 122 q^{91} + 216 q^{95} + 16 q^{97}+O(q^{100}) \) 80 * q + q^7 - 18 * q^11 + 10 * q^13 - 9 * q^17 + 20 * q^19 + 200 * q^25 + 27 * q^29 - 8 * q^31 + 22 * q^37 + 54 * q^41 + 88 * q^43 - 198 * q^47 - 267 * q^49 - 36 * q^53 - 171 * q^59 + 7 * q^61 + 144 * q^65 + 154 * q^67 - 135 * q^71 + 43 * q^73 - 216 * q^77 + 34 * q^79 + 171 * q^83 + 216 * q^89 + 122 * q^91 + 216 * q^95 + 16 * q^97

Introduction

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