LMFDB - Embedded newform 2052.3.bl.a.145.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(145,2052)]

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, 2, 5]))

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

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

chi := DirichletCharacter("2052.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.bl (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			145.10
Character	$\chi$	$=$	2052.145
Dual form			2052.3.bl.a.1585.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.76547 q^{5} +(3.51868 + 6.09454i) q^{7} +O(q^{10})$	$q-4.76547 q^{5} +(3.51868 + 6.09454i) q^{7} +(-2.56395 - 4.44089i) q^{11} +(15.6455 - 9.03291i) q^{13} +(0.508862 + 0.881376i) q^{17} +(-0.399427 + 18.9958i) q^{19} +(-11.7073 - 20.2777i) q^{23} -2.29026 q^{25} -45.8197i q^{29} +(2.83600 + 1.63736i) q^{31} +(-16.7682 - 29.0434i) q^{35} +57.4081i q^{37} +56.4600i q^{41} +(39.7226 - 68.8016i) q^{43} -66.2077 q^{47} +(-0.262246 + 0.454223i) q^{49} +(36.1806 + 20.8889i) q^{53} +(12.2184 + 21.1629i) q^{55} +16.6241i q^{59} -78.6497 q^{61} +(-74.5581 + 43.0461i) q^{65} +(104.061 - 60.0797i) q^{67} +(40.9176 - 23.6238i) q^{71} +(-47.5240 - 82.3140i) q^{73} +(18.0434 - 31.2521i) q^{77} +(101.882 + 58.8214i) q^{79} +(-13.8505 - 23.9898i) q^{83} +(-2.42497 - 4.20017i) q^{85} +(-24.4672 - 14.1261i) q^{89} +(110.103 + 63.5679i) q^{91} +(1.90346 - 90.5240i) q^{95} +(87.8655 + 50.7292i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - q^{7}+O(q^{10}) $ 80 * q - q^7	$ 80 q - q^{7} + 6 q^{11} - 15 q^{13} + 21 q^{17} - 20 q^{19} - 24 q^{23} + 400 q^{25} + 24 q^{31} + 54 q^{35} + 76 q^{43} - 24 q^{47} - 267 q^{49} + 36 q^{53} + 14 q^{61} - 288 q^{65} - 21 q^{67} + 81 q^{71} + 55 q^{73} - 30 q^{77} - 51 q^{79} + 93 q^{83} - 216 q^{89} + 96 q^{91} + 432 q^{95} + 90 q^{97}+O(q^{100}) $ 80 * q - q^7 + 6 * q^11 - 15 * q^13 + 21 * q^17 - 20 * q^19 - 24 * q^23 + 400 * q^25 + 24 * q^31 + 54 * q^35 + 76 * q^43 - 24 * q^47 - 267 * q^49 + 36 * q^53 + 14 * q^61 - 288 * q^65 - 21 * q^67 + 81 * q^71 + 55 * q^73 - 30 * q^77 - 51 * q^79 + 93 * q^83 - 216 * q^89 + 96 * q^91 + 432 * q^95 + 90 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	−4.76547			−0.953095			−0.476547	−	0.879149i	$-0.658112\pi$
$5$	−4.76547			−0.953095			−0.476547	+	0.879149i	$0.658112\pi$
$6$		0			0
$7$	3.51868	+	6.09454i	0.502669	+	0.870648i	0.999995	+	0.00308448i	$0.000981821\pi$
$7$	3.51868	+	6.09454i	0.502669	+	0.870648i	−0.497326	+	0.867564i	$0.665685\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.56395	−	4.44089i	−0.233086	−	0.403717i	0.725629	−	0.688086i	$-0.241549\pi$
$11$	−2.56395	−	4.44089i	−0.233086	−	0.403717i	−0.958715	+	0.284370i	$0.908216\pi$
$12$		0			0
$13$	15.6455	−	9.03291i	1.20350	−	0.694840i	0.242166	−	0.970235i	$-0.422142\pi$
$13$	15.6455	−	9.03291i	1.20350	−	0.694840i	0.961331	+	0.275395i	$0.0888087\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.508862	+	0.881376i	0.0299331	+	0.0518456i	0.880604	−	0.473853i	$-0.157137\pi$
$17$	0.508862	+	0.881376i	0.0299331	+	0.0518456i	−0.850671	+	0.525699i	$0.823804\pi$
$18$		0			0
$19$	−0.399427	+	18.9958i	−0.0210225	+	0.999779i
$19$	−0.399427	+	18.9958i	−0.0210225	+	0.999779i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−11.7073	−	20.2777i	−0.509014	−	0.881638i	−0.999946	−	0.0104398i	$-0.996677\pi$
$23$	−11.7073	−	20.2777i	−0.509014	−	0.881638i	0.490932	−	0.871198i	$-0.336656\pi$
$24$		0			0
$25$	−2.29026			−0.0916104
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	45.8197i		−	1.57999i	−0.613114	−	0.789995i	$-0.710083\pi$
$29$		−	45.8197i		−	1.57999i	0.613114	−	0.789995i	$-0.289917\pi$
$30$		0			0
$31$	2.83600	+	1.63736i	0.0914837	+	0.0528181i	0.545044	−	0.838407i	$-0.316513\pi$
$31$	2.83600	+	1.63736i	0.0914837	+	0.0528181i	−0.453560	+	0.891226i	$0.649846\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−16.7682	−	29.0434i	−0.479091	−	0.829810i
$36$		0			0
$37$			57.4081i			1.55157i	0.630997	+	0.775786i	$0.282646\pi$
$37$			57.4081i			1.55157i	−0.630997	+	0.775786i	$0.717354\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			56.4600i			1.37707i	0.725202	+	0.688536i	$0.241746\pi$
$41$			56.4600i			1.37707i	−0.725202	+	0.688536i	$0.758254\pi$
$42$		0			0
$43$	39.7226	−	68.8016i	0.923782	−	1.60004i	0.130273	−	0.991478i	$-0.458415\pi$
$43$	39.7226	−	68.8016i	0.923782	−	1.60004i	0.793509	−	0.608559i	$-0.208252\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−66.2077			−1.40867			−0.704337	−	0.709865i	$-0.748756\pi$
$47$	−66.2077			−1.40867			−0.704337	+	0.709865i	$0.748756\pi$
$48$		0			0
$49$	−0.262246	+	0.454223i	−0.00535196	+	0.00926986i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	36.1806	+	20.8889i	0.682653	+	0.394130i	0.800854	−	0.598860i	$-0.204379\pi$
$53$	36.1806	+	20.8889i	0.682653	+	0.394130i	−0.118201	+	0.992990i	$0.537713\pi$
$54$		0			0
$55$	12.2184	+	21.1629i	0.222153	+	0.384780i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			16.6241i			0.281764i	0.990026	+	0.140882i	$0.0449938\pi$
$59$			16.6241i			0.281764i	−0.990026	+	0.140882i	$0.955006\pi$
$60$		0			0
$61$	−78.6497			−1.28934			−0.644669	−	0.764461i	$-0.723005\pi$
$61$	−78.6497			−1.28934			−0.644669	+	0.764461i	$0.723005\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−74.5581	+	43.0461i	−1.14705	+	0.662248i
$66$		0			0
$67$	104.061	−	60.0797i	1.55315	−	0.896711i	0.555267	−	0.831672i	$-0.312616\pi$
$67$	104.061	−	60.0797i	1.55315	−	0.896711i	0.997883	−	0.0650392i	$-0.0207173\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	40.9176	−	23.6238i	0.576304	−	0.332729i	−0.183359	−	0.983046i	$-0.558697\pi$
$71$	40.9176	−	23.6238i	0.576304	−	0.332729i	0.759663	+	0.650317i	$0.225364\pi$
$72$		0			0
$73$	−47.5240	−	82.3140i	−0.651014	−	1.12759i	−0.982877	−	0.184262i	$-0.941010\pi$
$73$	−47.5240	−	82.3140i	−0.651014	−	1.12759i	0.331863	−	0.943328i	$-0.392323\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	18.0434	−	31.2521i	0.234330	−	0.405872i
$78$		0			0
$79$	101.882	+	58.8214i	1.28964	+	0.744575i	0.978590	−	0.205819i	$-0.0659857\pi$
$79$	101.882	+	58.8214i	1.28964	+	0.744575i	0.311051	+	0.950393i	$0.399319\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−13.8505	−	23.9898i	−0.166874	−	0.289034i	0.770445	−	0.637506i	$-0.220034\pi$
$83$	−13.8505	−	23.9898i	−0.166874	−	0.289034i	−0.937319	+	0.348472i	$0.886701\pi$
$84$		0			0
$85$	−2.42497	−	4.20017i	−0.0285291	−	0.0494138i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−24.4672	−	14.1261i	−0.274912	−	0.158721i	0.356206	−	0.934408i	$-0.384070\pi$
$89$	−24.4672	−	14.1261i	−0.274912	−	0.158721i	−0.631118	+	0.775687i	$0.717404\pi$
$90$		0			0
$91$	110.103	+	63.5679i	1.20992	+	0.698548i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.90346	−	90.5240i	0.0200364	−	0.952884i
$96$		0			0
$97$	87.8655	+	50.7292i	0.905830	+	0.522981i	0.879087	−	0.476661i	$-0.158153\pi$
$97$	87.8655	+	50.7292i	0.905830	+	0.522981i	0.0267428	+	0.999642i	$0.491486\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	165.307			1.63670			0.818349	−	0.574721i	$-0.194890\pi$
$101$	165.307			1.63670			0.818349	+	0.574721i	$0.194890\pi$
$102$		0			0
$103$	−51.6210	−	29.8034i	−0.501174	−	0.289353i	0.228024	−	0.973656i	$-0.426774\pi$
$103$	−51.6210	−	29.8034i	−0.501174	−	0.289353i	−0.729198	+	0.684302i	$0.760107\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			112.820i			1.05439i	0.849744	+	0.527195i	$0.176756\pi$
$107$			112.820i			1.05439i	−0.849744	+	0.527195i	$0.823244\pi$
$108$		0			0
$109$	77.6418	−	44.8265i	0.712310	−	0.411252i	−0.0996059	−	0.995027i	$-0.531758\pi$
$109$	77.6418	−	44.8265i	0.712310	−	0.411252i	0.811916	+	0.583775i	$0.198425\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−44.8713	−	25.9065i	−0.397091	−	0.229261i	0.288137	−	0.957589i	$-0.406964\pi$
$113$	−44.8713	−	25.9065i	−0.397091	−	0.229261i	−0.685228	+	0.728328i	$0.740298\pi$
$114$		0			0
$115$	55.7909	+	96.6327i	0.485138	+	0.840284i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.58105	+	6.20256i	−0.0300929	+	0.0521224i
$120$		0			0
$121$	47.3524	−	82.0167i	0.391342	−	0.677824i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	130.051			1.04041
$126$		0			0
$127$	128.735	+	74.3249i	1.01366	+	0.585236i	0.912261	−	0.409610i	$-0.134335\pi$
$127$	128.735	+	74.3249i	1.01366	+	0.585236i	0.101398	+	0.994846i	$0.467669\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	111.844			0.853770			0.426885	−	0.904306i	$-0.359611\pi$
$131$	111.844			0.853770			0.426885	+	0.904306i	$0.359611\pi$
$132$		0			0
$133$	−117.176	+	64.4059i	−0.881023	+	0.484255i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	235.683			1.72032			0.860158	−	0.510028i	$-0.170365\pi$
$137$	235.683			1.72032			0.860158	+	0.510028i	$0.170365\pi$
$138$		0			0
$139$	−7.37284	−	12.7701i	−0.0530420	−	0.0918715i	0.838285	−	0.545232i	$-0.183558\pi$
$139$	−7.37284	−	12.7701i	−0.0530420	−	0.0918715i	−0.891327	+	0.453360i	$0.850225\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−80.2283	−	46.3198i	−0.561037	−	0.323915i
$144$		0			0
$145$			218.353i			1.50588i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	148.528			0.996834			0.498417	−	0.866937i	$-0.333915\pi$
$149$	148.528			0.996834			0.498417	+	0.866937i	$0.333915\pi$
$150$		0			0
$151$	−7.73412	+	4.46529i	−0.0512193	+	0.0295715i	−0.525391	−	0.850861i	$-0.676081\pi$
$151$	−7.73412	+	4.46529i	−0.0512193	+	0.0295715i	0.474172	+	0.880432i	$0.342748\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−13.5149	−	7.80281i	−0.0871926	−	0.0503407i
$156$		0			0
$157$	−20.0125			−0.127468			−0.0637342	−	0.997967i	$-0.520301\pi$
$157$	−20.0125			−0.127468			−0.0637342	+	0.997967i	$0.520301\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	82.3887	−	142.701i	0.511731	−	0.886344i
$162$		0			0
$163$	28.7108			0.176140			0.0880701	−	0.996114i	$-0.471930\pi$
$163$	28.7108			0.176140			0.0880701	+	0.996114i	$0.471930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	160.430	−	92.6241i	0.960656	−	0.554635i	0.0642812	−	0.997932i	$-0.479525\pi$
$167$	160.430	−	92.6241i	0.960656	−	0.554635i	0.896375	+	0.443297i	$0.146191\pi$
$168$		0			0
$169$	78.6871	−	136.290i	0.465604	−	0.806450i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−244.864	−	141.372i	−1.41540	−	0.817180i	−0.419507	−	0.907752i	$-0.637797\pi$
$173$	−244.864	−	141.372i	−1.41540	−	0.817180i	−0.995890	+	0.0905721i	$0.971130\pi$
$174$		0			0
$175$	−8.05869	−	13.9581i	−0.0460497	−	0.0797604i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	11.5013i		−	0.0642530i	−0.999484	−	0.0321265i	$-0.989772\pi$
$179$		−	11.5013i		−	0.0642530i	0.999484	−	0.0321265i	$-0.0102279\pi$
$180$		0			0
$181$	70.0041	+	40.4169i	0.386763	+	0.223298i	0.680757	−	0.732510i	$-0.261651\pi$
$181$	70.0041	+	40.4169i	0.386763	+	0.223298i	−0.293994	+	0.955807i	$0.594984\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	273.577i		−	1.47879i
$186$		0			0
$187$	2.60939	−	4.51960i	0.0139540	−	0.0241690i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.8496	+	22.2562i	0.0672756	+	0.116525i	0.897701	−	0.440605i	$-0.145236\pi$
$191$	12.8496	+	22.2562i	0.0672756	+	0.116525i	−0.830426	+	0.557130i	$0.811903\pi$
$192$		0			0
$193$		−	265.176i		−	1.37397i	−0.726671	−	0.686986i	$-0.758934\pi$
$193$		−	265.176i		−	1.37397i	0.726671	−	0.686986i	$-0.241066\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−333.924			−1.69505			−0.847523	−	0.530759i	$-0.821907\pi$
$197$	−333.924			−1.69505			−0.847523	+	0.530759i	$0.821907\pi$
$198$		0			0
$199$	44.0469	−	76.2915i	0.221341	−	0.383374i	−0.733874	−	0.679285i	$-0.762290\pi$
$199$	44.0469	−	76.2915i	0.221341	−	0.383374i	0.955215	+	0.295911i	$0.0956232\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	279.250	−	161.225i	1.37561	−	0.794212i
$204$		0			0
$205$		−	269.059i		−	1.31248i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	85.3823	−	46.9304i	0.408528	−	0.224547i
$210$		0			0
$211$		−	165.596i		−	0.784814i	−0.919792	−	0.392407i	$-0.871643\pi$
$211$		−	165.596i		−	0.784814i	0.919792	−	0.392407i	$-0.128357\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−189.297	+	327.872i	−0.880452	+	1.52499i
$216$		0			0
$217$			23.0454i			0.106200i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	15.9228	+	9.19302i	0.0720488	+	0.0415974i
$222$		0			0
$223$	10.4225	+	6.01742i	0.0467376	+	0.0269840i	0.523187	−	0.852218i	$-0.324743\pi$
$223$	10.4225	+	6.01742i	0.0467376	+	0.0269840i	−0.476449	+	0.879202i	$0.658076\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	350.478	−	202.349i	1.54396	−	0.891404i	0.545374	−	0.838193i	$-0.316388\pi$
$227$	350.478	−	202.349i	1.54396	−	0.891404i	0.998583	−	0.0532114i	$-0.0169457\pi$
$228$		0			0
$229$	−124.232	+	215.175i	−0.542496	+	0.939630i	0.456264	+	0.889844i	$0.349187\pi$
$229$	−124.232	+	215.175i	−0.542496	+	0.939630i	−0.998760	+	0.0497858i	$0.984146\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	112.559	+	194.958i	0.483086	+	0.836729i	0.999811	−	0.0194221i	$-0.00618264\pi$
$233$	112.559	+	194.958i	0.483086	+	0.836729i	−0.516726	+	0.856151i	$0.672849\pi$
$234$		0			0
$235$	315.511			1.34260
$236$		0			0
$237$		0			0
$238$		0			0
$239$	228.602	−	395.950i	0.956494	−	1.65670i	0.225582	−	0.974224i	$-0.427572\pi$
$239$	228.602	−	395.950i	0.956494	−	1.65670i	0.730912	−	0.682472i	$-0.239095\pi$
$240$		0			0
$241$			369.002i			1.53113i	0.643359	+	0.765564i	$0.277540\pi$
$241$			369.002i			1.53113i	−0.643359	+	0.765564i	$0.722460\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.24973	−	2.16459i	0.00510092	−	0.00883506i
$246$		0			0
$247$	165.338	+	300.806i	0.669386	+	1.21784i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−50.8316	+	88.0429i	−0.202516	+	0.350769i	−0.949339	−	0.314255i	$-0.898245\pi$
$251$	−50.8316	+	88.0429i	−0.202516	+	0.350769i	0.746822	+	0.665024i	$0.231579\pi$
$252$		0			0
$253$	−60.0339	+	103.982i	−0.237288	+	0.410995i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	54.1646	−	31.2720i	0.210757	−	0.121681i	−0.390906	−	0.920431i	$-0.627838\pi$
$257$	54.1646	−	31.2720i	0.210757	−	0.121681i	0.601663	+	0.798750i	$0.294505\pi$
$258$		0			0
$259$	−349.876	+	202.001i	−1.35087	+	0.779926i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	94.5566	−	163.777i	0.359531	−	0.622726i	−0.628352	−	0.777929i	$-0.716270\pi$
$263$	94.5566	−	163.777i	0.359531	−	0.622726i	0.987882	+	0.155204i	$0.0496034\pi$
$264$		0			0
$265$	−172.418	−	99.5455i	−0.650633	−	0.375643i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	109.722	−	63.3477i	0.407887	−	0.235493i	−0.281995	−	0.959416i	$-0.590996\pi$
$269$	109.722	−	63.3477i	0.407887	−	0.235493i	0.689881	+	0.723923i	$0.257663\pi$
$270$		0			0
$271$	−142.679	−	247.128i	−0.526492	−	0.911911i	−0.999524	−	0.0308656i	$-0.990174\pi$
$271$	−142.679	−	247.128i	−0.526492	−	0.911911i	0.473031	−	0.881046i	$-0.343160\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.87210	+	10.1708i	0.0213531	+	0.0369846i
$276$		0			0
$277$	119.422	+	206.845i	0.431127	+	0.746735i	0.996971	−	0.0777783i	$-0.0247826\pi$
$277$	119.422	+	206.845i	0.431127	+	0.746735i	−0.565843	+	0.824513i	$0.691449\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			158.413i			0.563748i	0.959451	+	0.281874i	$0.0909560\pi$
$281$			158.413i			0.563748i	−0.959451	+	0.281874i	$0.909044\pi$
$282$		0			0
$283$	27.7159			0.0979360			0.0489680	−	0.998800i	$-0.484407\pi$
$283$	27.7159			0.0979360			0.0489680	+	0.998800i	$0.484407\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−344.097	+	198.665i	−1.19895	+	0.692212i
$288$		0			0
$289$	143.982	−	249.384i	0.498208	−	0.862922i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	296.143	+	170.978i	1.01073	+	0.583543i	0.911404	−	0.411512i	$-0.134999\pi$
$293$	296.143	+	170.978i	1.01073	+	0.583543i	0.0993218	+	0.995055i	$0.468333\pi$
$294$		0			0
$295$		−	79.2216i		−	0.268548i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−366.333	−	211.502i	−1.22519	−	0.707366i
$300$		0			0
$301$	559.085			1.85743
$302$		0			0
$303$		0			0
$304$		0			0
$305$	374.803			1.22886
$306$		0			0
$307$	−246.917	+	142.558i	−0.804292	+	0.464358i	−0.844970	−	0.534814i	$-0.820382\pi$
$307$	−246.917	+	142.558i	−0.804292	+	0.464358i	0.0406781	+	0.999172i	$0.487048\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	158.490	−	274.513i	0.509615	−	0.882678i	−0.490323	−	0.871541i	$-0.663121\pi$
$311$	158.490	−	274.513i	0.509615	−	0.882678i	0.999938	−	0.0111378i	$-0.00354534\pi$
$312$		0			0
$313$	−47.0329			−0.150265			−0.0751325	−	0.997174i	$-0.523938\pi$
$313$	−47.0329			−0.150265			−0.0751325	+	0.997174i	$0.523938\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			518.079i			1.63432i	0.576413	+	0.817159i	$0.304452\pi$
$317$			518.079i			1.63432i	−0.576413	+	0.817159i	$0.695548\pi$
$318$		0			0
$319$	−203.480	+	117.479i	−0.637868	+	0.368274i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−16.9457	+	9.31420i	−0.0524634	+	0.0288365i
$324$		0			0
$325$	−35.8322	+	20.6877i	−0.110253	+	0.0636545i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−232.964	−	403.505i	−0.708097	−	1.22646i
$330$		0			0
$331$	1.17840	−	0.680349i	0.00356012	−	0.00205544i	−0.498219	−	0.867051i	$-0.666012\pi$
$331$	1.17840	−	0.680349i	0.00356012	−	0.00205544i	0.501779	+	0.864996i	$0.332679\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−495.900	+	286.308i	−1.48030	+	0.854651i
$336$		0			0
$337$			344.768i			1.02305i	0.859268	+	0.511525i	$0.170919\pi$
$337$			344.768i			1.02305i	−0.859268	+	0.511525i	$0.829081\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	16.7924i		−	0.0492447i
$342$		0			0
$343$	341.140			0.994577
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−251.526			−0.724860			−0.362430	−	0.932011i	$-0.618053\pi$
$347$	−251.526			−0.724860			−0.362430	+	0.932011i	$0.618053\pi$
$348$		0			0
$349$	−91.0768	−	157.750i	−0.260965	−	0.452005i	0.705534	−	0.708676i	$-0.250707\pi$
$349$	−91.0768	−	157.750i	−0.260965	−	0.452005i	−0.966499	+	0.256672i	$0.917374\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	187.453	+	324.678i	0.531028	+	0.919768i	0.999344	+	0.0362067i	$0.0115275\pi$
$353$	187.453	+	324.678i	0.531028	+	0.919768i	−0.468316	+	0.883561i	$0.655139\pi$
$354$		0			0
$355$	−194.992	+	112.578i	−0.549272	+	0.317122i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	15.9655	+	27.6530i	0.0444720	+	0.0770278i	0.887405	−	0.460991i	$-0.152506\pi$
$359$	15.9655	+	27.6530i	0.0444720	+	0.0770278i	−0.842933	+	0.538019i	$0.819173\pi$
$360$		0			0
$361$	−360.681	−	15.1749i	−0.999116	−	0.0420356i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	226.475	+	392.265i	0.620478	+	1.07470i
$366$		0			0
$367$	−1.26055			−0.00343474			−0.00171737	−	0.999999i	$-0.500547\pi$
$367$	−1.26055			−0.00343474			−0.00171737	+	0.999999i	$0.500547\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			294.006i			0.792468i
$372$		0			0
$373$	−119.825	−	69.1813i	−0.321248	−	0.185473i	0.330701	−	0.943736i	$-0.392715\pi$
$373$	−119.825	−	69.1813i	−0.321248	−	0.185473i	−0.651949	+	0.758263i	$0.726048\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−413.885	−	716.871i	−1.09784	−	1.90151i
$378$		0			0
$379$			282.758i			0.746064i	0.927818	+	0.373032i	$0.121682\pi$
$379$			282.758i			0.746064i	−0.927818	+	0.373032i	$0.878318\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	438.518i		−	1.14496i	−0.819920	−	0.572478i	$-0.805982\pi$
$383$		−	438.518i		−	1.14496i	0.819920	−	0.572478i	$-0.194018\pi$
$384$		0			0
$385$	−85.9855	+	148.931i	−0.223339	+	0.386834i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	718.244			1.84639			0.923193	−	0.384336i	$-0.125570\pi$
$389$	718.244			1.84639			0.923193	+	0.384336i	$0.125570\pi$
$390$		0			0
$391$	11.9148	−	20.6371i	0.0304727	−	0.0527803i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−485.514	−	280.312i	−1.22915	−	0.709650i
$396$		0			0
$397$	−127.044	−	220.047i	−0.320011	−	0.554275i	0.660479	−	0.750844i	$-0.270353\pi$
$397$	−127.044	−	220.047i	−0.320011	−	0.554275i	−0.980490	+	0.196569i	$0.937020\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			494.191i			1.23240i	0.787591	+	0.616199i	$0.211328\pi$
$401$			494.191i			1.23240i	−0.787591	+	0.616199i	$0.788672\pi$
$402$		0			0
$403$	59.1606			0.146801
$404$		0			0
$405$		0			0
$406$		0			0
$407$	254.943	−	147.191i	0.626395	−	0.361650i
$408$		0			0
$409$	602.747	−	347.996i	1.47371	−	0.850846i	0.474147	−	0.880446i	$-0.342756\pi$
$409$	602.747	−	347.996i	1.47371	−	0.850846i	0.999562	+	0.0295991i	$0.00942308\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−101.316	+	58.4949i	−0.245317	+	0.141634i
$414$		0			0
$415$	66.0042	+	114.323i	0.159046	+	0.275476i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−189.788	+	328.722i	−0.452955	+	0.784540i	−0.998568	−	0.0534967i	$-0.982963\pi$
$419$	−189.788	+	328.722i	−0.452955	+	0.784540i	0.545613	+	0.838037i	$0.316297\pi$
$420$		0			0
$421$	453.099	+	261.597i	1.07624	+	0.621370i	0.929881	−	0.367861i	$-0.119910\pi$
$421$	453.099	+	261.597i	1.07624	+	0.621370i	0.146363	+	0.989231i	$0.453243\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.16543	−	2.01858i	−0.00274218	−	0.00474960i
$426$		0			0
$427$	−276.743	−	479.333i	−0.648110	−	1.12256i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−222.148	−	128.257i	−0.515426	−	0.297581i	0.219635	−	0.975582i	$-0.429513\pi$
$431$	−222.148	−	128.257i	−0.515426	−	0.297581i	−0.735061	+	0.678001i	$0.762847\pi$
$432$		0			0
$433$	−234.232	−	135.234i	−0.540951	−	0.312318i	0.204513	−	0.978864i	$-0.434439\pi$
$433$	−234.232	−	135.234i	−0.540951	−	0.312318i	−0.745464	+	0.666545i	$0.767772\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	389.867	−	214.290i	0.892144	−	0.490367i
$438$		0			0
$439$	−550.597	−	317.887i	−1.25421	−	0.724117i	−0.282266	−	0.959336i	$-0.591086\pi$
$439$	−550.597	−	317.887i	−1.25421	−	0.724117i	−0.971942	+	0.235219i	$0.924419\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−48.8213			−0.110206			−0.0551031	−	0.998481i	$-0.517549\pi$
$443$	−48.8213			−0.110206			−0.0551031	+	0.998481i	$0.517549\pi$
$444$		0			0
$445$	116.598	+	67.3177i	0.262017	+	0.151276i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			11.4958i			0.0256031i	0.999918	+	0.0128015i	$0.00407497\pi$
$449$			11.4958i			0.0256031i	−0.999918	+	0.0128015i	$0.995925\pi$
$450$		0			0
$451$	250.732	−	144.760i	0.555947	−	0.320976i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−524.692	−	302.931i	−1.15317	−	0.665783i
$456$		0			0
$457$	−168.104	−	291.165i	−0.367842	−	0.637122i	0.621386	−	0.783505i	$-0.286570\pi$
$457$	−168.104	−	291.165i	−0.367842	−	0.637122i	−0.989228	+	0.146383i	$0.953237\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−195.834	+	339.194i	−0.424803	+	0.735780i	−0.996402	−	0.0847531i	$-0.972990\pi$
$461$	−195.834	+	339.194i	−0.424803	+	0.735780i	0.571599	+	0.820533i	$0.306323\pi$
$462$		0			0
$463$	217.097	−	376.022i	0.468891	−	0.812143i	−0.530477	−	0.847700i	$-0.677987\pi$
$463$	217.097	−	376.022i	0.468891	−	0.812143i	0.999368	+	0.0355563i	$0.0113203\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	176.659			0.378286			0.189143	−	0.981950i	$-0.439429\pi$
$467$	176.659			0.378286			0.189143	+	0.981950i	$0.439429\pi$
$468$		0			0
$469$	732.315	+	422.802i	1.56144	+	0.901498i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−407.387			−0.861283
$474$		0			0
$475$	0.914790	−	43.5053i	0.00192587	−	0.0915901i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−264.903			−0.553032			−0.276516	−	0.961009i	$-0.589180\pi$
$479$	−264.903			−0.553032			−0.276516	+	0.961009i	$0.589180\pi$
$480$		0			0
$481$	518.563	+	898.177i	1.07809	+	1.86731i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−418.721	−	241.749i	−0.863342	−	0.498451i
$486$		0			0
$487$		−	104.744i		−	0.215081i	−0.994201	−	0.107540i	$-0.965703\pi$
$487$		−	104.744i		−	0.215081i	0.994201	−	0.107540i	$-0.0342975\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−571.058			−1.16305			−0.581525	−	0.813529i	$-0.697544\pi$
$491$	−571.058			−1.16305			−0.581525	+	0.813529i	$0.697544\pi$
$492$		0			0
$493$	40.3844	−	23.3159i	0.0819155	−	0.0472940i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	287.952	+	166.249i	0.579380	+	0.334505i
$498$		0			0
$499$	−176.290			−0.353286			−0.176643	−	0.984275i	$-0.556524\pi$
$499$	−176.290			−0.353286			−0.176643	+	0.984275i	$0.556524\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−277.472	+	480.596i	−0.551635	+	0.955459i	0.446522	+	0.894773i	$0.352662\pi$
$503$	−277.472	+	480.596i	−0.551635	+	0.955459i	−0.998157	+	0.0606867i	$0.980671\pi$
$504$		0			0
$505$	−787.764			−1.55993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	618.884	−	357.313i	1.21588	−	0.701990i	0.251848	−	0.967767i	$-0.418962\pi$
$509$	618.884	−	357.313i	1.21588	−	0.701990i	0.964035	+	0.265776i	$0.0856283\pi$
$510$		0			0
$511$	334.444	−	579.274i	0.654489	−	1.13361i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	245.998	+	142.027i	0.477667	+	0.275781i
$516$		0			0
$517$	169.753	+	294.021i	0.328342	+	0.568706i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			327.567i			0.628728i	0.949303	+	0.314364i	$0.101791\pi$
$521$			327.567i			0.628728i	−0.949303	+	0.314364i	$0.898209\pi$
$522$		0			0
$523$	229.651	+	132.589i	0.439103	+	0.253516i	0.703217	−	0.710975i	$-0.251746\pi$
$523$	229.651	+	132.589i	0.439103	+	0.253516i	−0.264114	+	0.964492i	$0.585079\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			3.33277i			0.00632404i
$528$		0			0
$529$	−9.62261	+	16.6669i	−0.0181902	+	0.0315063i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	509.998	+	883.343i	0.956845	+	1.65730i
$534$		0			0
$535$		−	537.640i		−	1.00493i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.68954			0.00498987
$540$		0			0
$541$	317.733	−	550.330i	0.587307	−	1.01725i	−0.407277	−	0.913305i	$-0.633521\pi$
$541$	317.733	−	550.330i	0.587307	−	1.01725i	0.994584	−	0.103941i	$-0.0331452\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−370.000	+	213.619i	−0.678899	+	0.391962i
$546$		0			0
$547$		−	94.4294i		−	0.172631i	−0.996268	−	0.0863157i	$-0.972491\pi$
$547$		−	94.4294i		−	0.172631i	0.996268	−	0.0863157i	$-0.0275094\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	870.382	+	18.3016i	1.57964	+	0.0332153i
$552$		0			0
$553$			827.895i			1.49710i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−118.599	+	205.419i	−0.212925	+	0.368796i	−0.952629	−	0.304136i	$-0.901632\pi$
$557$	−118.599	+	205.419i	−0.212925	+	0.368796i	0.739704	+	0.672932i	$0.234966\pi$
$558$		0			0
$559$		−	1435.24i		−	2.56752i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−591.014	−	341.222i	−1.04976	−	0.606078i	−0.127176	−	0.991880i	$-0.540591\pi$
$563$	−591.014	−	341.222i	−1.04976	−	0.606078i	−0.922582	+	0.385802i	$0.873925\pi$
$564$		0			0
$565$	213.833	+	123.457i	0.378466	+	0.218507i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−60.1817	+	34.7459i	−0.105768	+	0.0610649i	−0.551951	−	0.833877i	$-0.686116\pi$
$569$	−60.1817	+	34.7459i	−0.105768	+	0.0610649i	0.446183	+	0.894942i	$0.352783\pi$
$570$		0			0
$571$	88.0919	−	152.580i	0.154277	−	0.267215i	−0.778519	−	0.627621i	$-0.784029\pi$
$571$	88.0919	−	152.580i	0.154277	−	0.267215i	0.932795	+	0.360406i	$0.117362\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	26.8128	+	46.4411i	0.0466309	+	0.0807672i
$576$		0			0
$577$	−215.231			−0.373018			−0.186509	−	0.982453i	$-0.559717\pi$
$577$	−215.231			−0.373018			−0.186509	+	0.982453i	$0.559717\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	97.4711	−	168.825i	0.167764	−	0.290576i
$582$		0			0
$583$		−	214.232i		−	0.367465i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	65.1043	−	112.764i	0.110910	−	0.192102i	−0.805227	−	0.592966i	$-0.797957\pi$
$587$	65.1043	−	112.764i	0.110910	−	0.192102i	0.916137	+	0.400864i	$0.131290\pi$
$588$		0			0
$589$	−32.2358	+	53.2180i	−0.0547297	+	0.0903531i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	388.044	−	672.113i	0.654375	−	1.13341i	−0.327675	−	0.944790i	$-0.606265\pi$
$593$	388.044	−	672.113i	0.654375	−	1.13341i	0.982050	−	0.188620i	$-0.0604016\pi$
$594$		0			0
$595$	17.0654	−	29.5581i	0.0286813	−	0.0496775i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−224.477	+	129.602i	−0.374753	+	0.216363i	−0.675533	−	0.737330i	$-0.736086\pi$
$599$	−224.477	+	129.602i	−0.374753	+	0.216363i	0.300780	+	0.953693i	$0.402753\pi$
$600$		0			0
$601$	−111.340	+	64.2824i	−0.185259	+	0.106959i	−0.589761	−	0.807578i	$-0.700778\pi$
$601$	−111.340	+	64.2824i	−0.185259	+	0.106959i	0.404502	+	0.914537i	$0.367445\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−225.656	+	390.848i	−0.372986	+	0.646030i
$606$		0			0
$607$	−428.107	−	247.168i	−0.705284	−	0.407196i	0.104028	−	0.994574i	$-0.466827\pi$
$607$	−428.107	−	247.168i	−0.705284	−	0.407196i	−0.809312	+	0.587378i	$0.800160\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1035.85	+	598.049i	−1.69534	+	0.978803i
$612$		0			0
$613$	−27.2301	−	47.1640i	−0.0444211	−	0.0769396i	0.842960	−	0.537976i	$-0.180811\pi$
$613$	−27.2301	−	47.1640i	−0.0444211	−	0.0769396i	−0.887381	+	0.461037i	$0.847478\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−84.1434	−	145.741i	−0.136375	−	0.236208i	0.789747	−	0.613433i	$-0.210212\pi$
$617$	−84.1434	−	145.741i	−0.136375	−	0.236208i	−0.926122	+	0.377224i	$0.876879\pi$
$618$		0			0
$619$	−226.776	−	392.788i	−0.366359	−	0.634553i	0.622634	−	0.782513i	$-0.286062\pi$
$619$	−226.776	−	392.788i	−0.366359	−	0.634553i	−0.988993	+	0.147961i	$0.952729\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	198.821i		−	0.319136i
$624$		0			0
$625$	−562.498			−0.899997
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−50.5981	+	29.2128i	−0.0804422	+	0.0464433i
$630$		0			0
$631$	−394.174	+	682.730i	−0.624682	+	1.08198i	0.363920	+	0.931430i	$0.381438\pi$
$631$	−394.174	+	682.730i	−0.624682	+	1.08198i	−0.988602	+	0.150551i	$0.951895\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−613.481	−	354.194i	−0.966112	−	0.557785i
$636$		0			0
$637$			9.47538i			0.0148750i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−456.096	−	263.327i	−0.711539	−	0.410807i	0.100092	−	0.994978i	$-0.468086\pi$
$641$	−456.096	−	263.327i	−0.711539	−	0.410807i	−0.811631	+	0.584171i	$0.801420\pi$
$642$		0			0
$643$	−1263.20			−1.96455			−0.982273	−	0.187456i	$-0.939976\pi$
$643$	−1263.20			−1.96455			−0.982273	+	0.187456i	$0.939976\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−498.316			−0.770195			−0.385098	−	0.922876i	$-0.625832\pi$
$647$	−498.316			−0.770195			−0.385098	+	0.922876i	$0.625832\pi$
$648$		0			0
$649$	73.8256	−	42.6233i	0.113753	−	0.0656753i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−406.675	+	704.382i	−0.622779	+	1.07869i	0.366186	+	0.930542i	$0.380663\pi$
$653$	−406.675	+	704.382i	−0.622779	+	1.07869i	−0.988966	+	0.148144i	$0.952670\pi$
$654$		0			0
$655$	−532.989			−0.813724
$656$		0			0
$657$		0			0
$658$		0			0
$659$			778.385i			1.18116i	0.806979	+	0.590580i	$0.201101\pi$
$659$			778.385i			1.18116i	−0.806979	+	0.590580i	$0.798899\pi$
$660$		0			0
$661$	7.66220	−	4.42377i	0.0115918	−	0.00669255i	−0.494193	−	0.869352i	$-0.664536\pi$
$661$	7.66220	−	4.42377i	0.0115918	−	0.00669255i	0.505785	+	0.862660i	$0.331203\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	558.399	−	306.924i	0.839698	−	0.461541i
$666$		0			0
$667$	−929.117	+	536.426i	−1.39298	+	0.804237i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	201.654	+	349.274i	0.300527	+	0.520528i
$672$		0			0
$673$	281.617	−	162.592i	0.418451	−	0.241593i	−0.275964	−	0.961168i	$-0.588997\pi$
$673$	281.617	−	162.592i	0.418451	−	0.241593i	0.694414	+	0.719576i	$0.255664\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−109.957	+	63.4837i	−0.162418	+	0.0937721i	−0.579006	−	0.815323i	$-0.696559\pi$
$677$	−109.957	+	63.4837i	−0.162418	+	0.0937721i	0.416588	+	0.909095i	$0.363226\pi$
$678$		0			0
$679$			713.999i			1.05155i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1318.05i			1.92979i	0.262639	+	0.964894i	$0.415407\pi$
$683$			1318.05i			1.92979i	−0.262639	+	0.964894i	$0.584593\pi$
$684$		0			0
$685$	−1123.14			−1.63962
$686$		0			0
$687$		0			0
$688$		0			0
$689$	754.751			1.09543
$690$		0			0
$691$	−7.62894	−	13.2137i	−0.0110404	−	0.0191226i	0.860452	−	0.509531i	$-0.170181\pi$
$691$	−7.62894	−	13.2137i	−0.0110404	−	0.0191226i	−0.871493	+	0.490408i	$0.836848\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	35.1351	+	60.8557i	0.0505541	+	0.0875622i
$696$		0			0
$697$	−49.7624	+	28.7304i	−0.0713952	+	0.0412200i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−195.166	−	338.037i	−0.278411	−	0.482221i	0.692579	−	0.721342i	$-0.256474\pi$
$701$	−195.166	−	338.037i	−0.278411	−	0.482221i	−0.970990	+	0.239120i	$0.923141\pi$
$702$		0			0
$703$	−1090.51	−	22.9303i	−1.55123	−	0.0326178i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	581.661	+	1007.47i	0.822717	+	1.42499i
$708$		0			0
$709$	625.820			0.882680			0.441340	−	0.897340i	$-0.354503\pi$
$709$	625.820			0.882680			0.441340	+	0.897340i	$0.354503\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	76.6765i		−	0.107541i
$714$		0			0
$715$	382.326	+	220.736i	0.534721	+	0.308721i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−464.868	−	805.175i	−0.646548	−	1.11985i	−0.983942	−	0.178490i	$-0.942879\pi$
$719$	−464.868	−	805.175i	−0.646548	−	1.11985i	0.337394	−	0.941364i	$-0.390455\pi$
$720$		0			0
$721$		−	419.474i		−	0.581795i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			104.939i			0.144743i
$726$		0			0
$727$	−400.376	+	693.471i	−0.550723	+	0.953881i	0.447499	+	0.894284i	$0.352315\pi$
$727$	−400.376	+	693.471i	−0.550723	+	0.953881i	−0.998223	+	0.0595965i	$0.981019\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	80.8534			0.110607
$732$		0			0
$733$	−453.750	+	785.917i	−0.619031	+	1.07219i	0.370632	+	0.928780i	$0.379141\pi$
$733$	−453.750	+	785.917i	−0.619031	+	1.07219i	−0.989663	+	0.143413i	$0.954192\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−533.614	−	308.082i	−0.724035	−	0.418022i
$738$		0			0
$739$	469.579	+	813.334i	0.635424	+	1.10059i	0.986425	+	0.164212i	$0.0525082\pi$
$739$	469.579	+	813.334i	0.635424	+	1.10059i	−0.351001	+	0.936375i	$0.614158\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1170.32i		−	1.57513i	−0.616234	−	0.787563i	$-0.711342\pi$
$743$		−	1170.32i		−	1.57513i	0.616234	−	0.787563i	$-0.288658\pi$
$744$		0			0
$745$	−707.807			−0.950077
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−687.584	+	396.977i	−0.918003	+	0.530009i
$750$		0			0
$751$	636.483	−	367.474i	0.847514	−	0.489313i	−0.0122971	−	0.999924i	$-0.503914\pi$
$751$	636.483	−	367.474i	0.847514	−	0.489313i	0.859811	+	0.510612i	$0.170581\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	36.8567	−	21.2792i	0.0488169	−	0.0281844i
$756$		0			0
$757$	−159.609	−	276.450i	−0.210843	−	0.365192i	0.741135	−	0.671356i	$-0.234288\pi$
$757$	−159.609	−	276.450i	−0.210843	−	0.365192i	−0.951979	+	0.306164i	$0.900954\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−248.077	+	429.681i	−0.325988	+	0.564627i	−0.981712	−	0.190373i	$-0.939030\pi$
$761$	−248.077	+	429.681i	−0.325988	+	0.564627i	0.655724	+	0.755001i	$0.272363\pi$
$762$		0			0
$763$	546.393	+	315.460i	0.716112	+	0.413447i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	150.164	+	260.092i	0.195781	+	0.339102i
$768$		0			0
$769$	−219.421	−	380.048i	−0.285333	−	0.494211i	0.687357	−	0.726320i	$-0.258771\pi$
$769$	−219.421	−	380.048i	−0.285333	−	0.494211i	−0.972690	+	0.232109i	$0.925438\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	494.966	+	285.769i	0.640319	+	0.369688i	0.784737	−	0.619828i	$-0.212798\pi$
$773$	494.966	+	285.769i	0.640319	+	0.369688i	−0.144418	+	0.989517i	$0.546131\pi$
$774$		0			0
$775$	−6.49516	−	3.74998i	−0.00838086	−	0.00483869i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1072.50	−	22.5516i	−1.37677	−	0.0289494i
$780$		0			0
$781$	−209.821	−	121.140i	−0.268657	−	0.155109i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	95.3692			0.121489
$786$		0			0
$787$	−648.532	−	374.430i	−0.824056	−	0.475769i	0.0277571	−	0.999615i	$-0.491164\pi$
$787$	−648.532	−	374.430i	−0.824056	−	0.475769i	−0.851813	+	0.523846i	$0.824497\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	364.627i		−	0.460969i
$792$		0			0
$793$	−1230.51	+	710.436i	−1.55172	+	0.895884i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	853.076	+	492.523i	1.07036	+	0.617972i	0.928279	−	0.371884i	$-0.121288\pi$
$797$	853.076	+	492.523i	1.07036	+	0.617972i	0.142079	+	0.989855i	$0.454621\pi$
$798$		0			0
$799$	−33.6906	−	58.3539i	−0.0421660	−	0.0730336i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−243.698	+	422.098i	−0.303485	+	0.525651i
$804$		0			0
$805$	−392.621	+	680.040i	−0.487728	+	0.844770i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1601.15			−1.97917			−0.989587	−	0.143939i	$-0.954023\pi$
$809$	−1601.15			−1.97917			−0.989587	+	0.143939i	$0.954023\pi$
$810$		0			0
$811$	−715.560	−	413.129i	−0.882318	−	0.509406i	−0.0108958	−	0.999941i	$-0.503468\pi$
$811$	−715.560	−	413.129i	−0.882318	−	0.509406i	−0.871422	+	0.490534i	$0.836802\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−136.821			−0.167878
$816$		0			0
$817$	1291.08	+	782.044i	1.58026	+	0.957214i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	175.615			0.213904			0.106952	−	0.994264i	$-0.465891\pi$
$821$	175.615			0.213904			0.106952	+	0.994264i	$0.465891\pi$
$822$		0			0
$823$	−407.940	−	706.572i	−0.495674	−	0.858533i	0.504313	−	0.863521i	$-0.331746\pi$
$823$	−407.940	−	706.572i	−0.495674	−	0.858533i	−0.999988	+	0.00498796i	$0.998412\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	982.422	+	567.202i	1.18793	+	0.685854i	0.957837	−	0.287313i	$-0.0927620\pi$
$827$	982.422	+	567.202i	1.18793	+	0.685854i	0.230098	+	0.973167i	$0.426095\pi$
$828$		0			0
$829$		−	764.117i		−	0.921734i	−0.887469	−	0.460867i	$-0.847539\pi$
$829$		−	764.117i		−	0.921734i	0.887469	−	0.460867i	$-0.152461\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.533788			−0.000640802
$834$		0			0
$835$	−764.523	+	441.398i	−0.915596	+	0.528620i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	105.221	+	60.7491i	0.125412	+	0.0724066i	0.561394	−	0.827549i	$-0.310265\pi$
$839$	105.221	+	60.7491i	0.125412	+	0.0724066i	−0.435982	+	0.899956i	$0.643599\pi$
$840$		0			0
$841$	−1258.44			−1.49637
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−374.981	+	649.487i	−0.443765	+	0.768623i
$846$		0			0
$847$	666.472			0.786861
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1164.10	−	672.095i	1.36792	−	0.789771i
$852$		0			0
$853$	34.8614	−	60.3817i	0.0408691	−	0.0707874i	−0.844867	−	0.534976i	$-0.820321\pi$
$853$	34.8614	−	60.3817i	0.0408691	−	0.0707874i	0.885736	+	0.464189i	$0.153654\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	301.084	+	173.831i	0.351324	+	0.202837i	0.665268	−	0.746605i	$-0.268317\pi$
$857$	301.084	+	173.831i	0.351324	+	0.202837i	−0.313945	+	0.949441i	$0.601651\pi$
$858$		0			0
$859$	−367.065	−	635.775i	−0.427317	−	0.740134i	0.569317	−	0.822118i	$-0.307208\pi$
$859$	−367.065	−	635.775i	−0.427317	−	0.740134i	−0.996634	+	0.0819838i	$0.973874\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1462.37i		−	1.69452i	−0.531181	−	0.847258i	$-0.678252\pi$
$863$		−	1462.37i		−	1.69452i	0.531181	−	0.847258i	$-0.321748\pi$
$864$		0			0
$865$	1166.89	+	673.705i	1.34901	+	0.778850i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	603.260i		−	0.694200i
$870$		0			0
$871$	1085.39	−	1879.95i	1.24614	−	2.15838i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	457.608	+	792.601i	0.522981	+	0.905829i
$876$		0			0
$877$			1267.52i			1.44529i	0.691220	+	0.722644i	$0.257074\pi$
$877$			1267.52i			1.44529i	−0.691220	+	0.722644i	$0.742926\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1356.03			1.53919			0.769596	−	0.638531i	$-0.220458\pi$
$881$	1356.03			1.53919			0.769596	+	0.638531i	$0.220458\pi$
$882$		0			0
$883$	276.740	−	479.328i	0.313409	−	0.542841i	−0.665689	−	0.746229i	$-0.731862\pi$
$883$	276.740	−	479.328i	0.313409	−	0.542841i	0.979098	+	0.203389i	$0.0651956\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	714.123	−	412.299i	0.805099	−	0.464824i	−0.0401519	−	0.999194i	$-0.512784\pi$
$887$	714.123	−	412.299i	0.805099	−	0.464824i	0.845251	+	0.534369i	$0.179451\pi$
$888$		0			0
$889$			1046.10i			1.17672i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	26.4451	−	1257.67i	0.0296138	−	1.40836i
$894$		0			0
$895$			54.8091i			0.0612392i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	75.0235	−	129.944i	0.0834521	−	0.144543i
$900$		0			0
$901$			42.5183i			0.0471901i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−333.603	−	192.606i	−0.368622	−	0.212824i
$906$		0			0
$907$	−729.211	−	421.010i	−0.803981	−	0.464179i	0.0408804	−	0.999164i	$-0.486984\pi$
$907$	−729.211	−	421.010i	−0.803981	−	0.464179i	−0.844861	+	0.534985i	$0.820317\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	316.343	−	182.641i	0.347248	−	0.200484i	−0.316224	−	0.948684i	$-0.602415\pi$
$911$	316.343	−	182.641i	0.347248	−	0.200484i	0.663473	+	0.748200i	$0.269082\pi$
$912$		0			0
$913$	−71.0239	+	123.017i	−0.0777918	+	0.134739i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	393.543	+	681.637i	0.429164	+	0.743334i
$918$		0			0
$919$	−1460.11			−1.58880			−0.794400	−	0.607395i	$-0.792215\pi$
$919$	−1460.11			−1.58880			−0.794400	+	0.607395i	$0.792215\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	426.783	−	739.210i	0.462387	−	0.800877i
$924$		0			0
$925$		−	131.479i		−	0.142140i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	120.980	−	209.544i	0.130226	−	0.225559i	−0.793537	−	0.608522i	$-0.791763\pi$
$929$	120.980	−	209.544i	0.130226	−	0.225559i	0.923764	+	0.382963i	$0.125096\pi$
$930$		0			0
$931$	−8.52359	−	5.16300i	−0.00915530	−	0.00554565i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12.4350	+	21.5380i	−0.0132995	+	0.0230353i
$936$		0			0
$937$	−402.399	+	696.975i	−0.429454	+	0.743837i	−0.996825	−	0.0796259i	$-0.974627\pi$
$937$	−402.399	+	696.975i	−0.429454	+	0.743837i	0.567370	+	0.823463i	$0.307961\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−623.096	+	359.745i	−0.662164	+	0.382300i	−0.793101	−	0.609090i	$-0.791535\pi$
$941$	−623.096	+	359.745i	−0.662164	+	0.382300i	0.130937	+	0.991391i	$0.458201\pi$
$942$		0			0
$943$	1144.88	−	660.995i	1.21408	−	0.700949i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	206.107	−	356.989i	0.217643	−	0.376968i	−0.736444	−	0.676498i	$-0.763497\pi$
$947$	206.107	−	356.989i	0.217643	−	0.376968i	0.954087	+	0.299530i	$0.0968300\pi$
$948$		0			0
$949$	−1487.07	−	858.561i	−1.56699	−	0.904701i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1176.65	−	679.339i	1.23468	−	0.712843i	0.266678	−	0.963786i	$-0.414074\pi$
$953$	1176.65	−	679.339i	1.23468	−	0.712843i	0.968002	+	0.250943i	$0.0807406\pi$
$954$		0			0
$955$	−61.2346	−	106.061i	−0.0641200	−	0.111059i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	829.295	+	1436.38i	0.864749	+	1.49779i
$960$		0			0
$961$	−475.138	−	822.963i	−0.494420	−	0.856361i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			1263.69i			1.30952i
$966$		0			0
$967$	896.753			0.927356			0.463678	−	0.886004i	$-0.346529\pi$
$967$	896.753			0.927356			0.463678	+	0.886004i	$0.346529\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−952.730	+	550.059i	−0.981184	+	0.566487i	−0.902627	−	0.430423i	$-0.858364\pi$
$971$	−952.730	+	550.059i	−0.981184	+	0.566487i	−0.0785567	+	0.996910i	$0.525031\pi$
$972$		0			0
$973$	51.8854	−	89.8681i	0.0533251	−	0.0923619i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.98302	−	1.14490i	−0.00202970	−	0.00117185i	0.498985	−	0.866611i	$-0.333706\pi$
$977$	−1.98302	−	1.14490i	−0.00202970	−	0.00117185i	−0.501015	+	0.865439i	$0.667040\pi$
$978$		0			0
$979$			144.875i			0.147982i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−568.653	−	328.312i	−0.578487	−	0.333990i	0.182045	−	0.983290i	$-0.441728\pi$
$983$	−568.653	−	328.312i	−0.578487	−	0.333990i	−0.760532	+	0.649301i	$0.775062\pi$
$984$		0			0
$985$	1591.31			1.61554
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1860.18			−1.88087
$990$		0			0
$991$	1396.56	−	806.305i	1.40924	−	0.813628i	0.413930	−	0.910309i	$-0.364156\pi$
$991$	1396.56	−	806.305i	1.40924	−	0.813628i	0.995315	+	0.0966810i	$0.0308227\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−209.904	+	363.565i	−0.210959	+	0.365392i
$996$		0			0
$997$	101.509			0.101814			0.0509072	−	0.998703i	$-0.483789\pi$
$997$	101.509			0.101814			0.0509072	+	0.998703i	$0.483789\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.bl.a.145.10		80
3.2	odd	2		684.3.bl.a.373.19	yes	80
9.2	odd	6		684.3.s.a.601.32	yes	80
9.7	even	3		2052.3.s.a.829.31		80
19.8	odd	6		2052.3.s.a.901.31		80
57.8	even	6		684.3.s.a.445.32	✓	80
171.65	even	6		684.3.bl.a.673.19	yes	80
171.160	odd	6	inner	2052.3.bl.a.1585.10		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.s.a.445.32	✓	80	57.8	even	6
684.3.s.a.601.32	yes	80	9.2	odd	6
684.3.bl.a.373.19	yes	80	3.2	odd	2
684.3.bl.a.673.19	yes	80	171.65	even	6
2052.3.s.a.829.31		80	9.7	even	3
2052.3.s.a.901.31		80	19.8	odd	6
2052.3.bl.a.145.10		80	1.1	even	1	trivial
2052.3.bl.a.1585.10		80	171.160	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.bl (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-4.76547 q^{5} +(3.51868 + 6.09454i) q^{7} +O(q^{10})\)	\(q-4.76547 q^{5} +(3.51868 + 6.09454i) q^{7} +(-2.56395 - 4.44089i) q^{11} +(15.6455 - 9.03291i) q^{13} +(0.508862 + 0.881376i) q^{17} +(-0.399427 + 18.9958i) q^{19} +(-11.7073 - 20.2777i) q^{23} -2.29026 q^{25} -45.8197i q^{29} +(2.83600 + 1.63736i) q^{31} +(-16.7682 - 29.0434i) q^{35} +57.4081i q^{37} +56.4600i q^{41} +(39.7226 - 68.8016i) q^{43} -66.2077 q^{47} +(-0.262246 + 0.454223i) q^{49} +(36.1806 + 20.8889i) q^{53} +(12.2184 + 21.1629i) q^{55} +16.6241i q^{59} -78.6497 q^{61} +(-74.5581 + 43.0461i) q^{65} +(104.061 - 60.0797i) q^{67} +(40.9176 - 23.6238i) q^{71} +(-47.5240 - 82.3140i) q^{73} +(18.0434 - 31.2521i) q^{77} +(101.882 + 58.8214i) q^{79} +(-13.8505 - 23.9898i) q^{83} +(-2.42497 - 4.20017i) q^{85} +(-24.4672 - 14.1261i) q^{89} +(110.103 + 63.5679i) q^{91} +(1.90346 - 90.5240i) q^{95} +(87.8655 + 50.7292i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - q^{7}+O(q^{10}) \) 80 * q - q^7	\( 80 q - q^{7} + 6 q^{11} - 15 q^{13} + 21 q^{17} - 20 q^{19} - 24 q^{23} + 400 q^{25} + 24 q^{31} + 54 q^{35} + 76 q^{43} - 24 q^{47} - 267 q^{49} + 36 q^{53} + 14 q^{61} - 288 q^{65} - 21 q^{67} + 81 q^{71} + 55 q^{73} - 30 q^{77} - 51 q^{79} + 93 q^{83} - 216 q^{89} + 96 q^{91} + 432 q^{95} + 90 q^{97}+O(q^{100}) \) 80 * q - q^7 + 6 * q^11 - 15 * q^13 + 21 * q^17 - 20 * q^19 - 24 * q^23 + 400 * q^25 + 24 * q^31 + 54 * q^35 + 76 * q^43 - 24 * q^47 - 267 * q^49 + 36 * q^53 + 14 * q^61 - 288 * q^65 - 21 * q^67 + 81 * q^71 + 55 * q^73 - 30 * q^77 - 51 * q^79 + 93 * q^83 - 216 * q^89 + 96 * q^91 + 432 * q^95 + 90 * q^97

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