LMFDB - Embedded newform 201.4.j.a.125.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [201,4,Mod(5,201)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(201, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 5]))

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

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

chi := DirichletCharacter("201.5");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 201 = 3 \cdot 67 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	201.j (of order $22$, degree $10$, 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:	$11.8593839112$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{22}]$

Embedding invariants

Embedding label			125.2
Root			$-0.995472 + 0.0950560i$ of defining polynomial
Character	$\chi$	$=$	201.125
Dual form			201.4.j.a.119.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.46393 - 4.98567i) q^{3} +(-3.32332 - 7.27706i) q^{4} +(-16.8377 + 26.2000i) q^{7} +(-22.7138 - 14.5973i) q^{9} +O(q^{10})$	$q+(1.46393 - 4.98567i) q^{3} +(-3.32332 - 7.27706i) q^{4} +(-16.8377 + 26.2000i) q^{7} +(-22.7138 - 14.5973i) q^{9} +(-41.1461 + 5.91592i) q^{12} +(-14.4618 + 2.07929i) q^{13} +(-41.9111 + 48.3680i) q^{16} +(8.25593 - 5.30577i) q^{19} +(105.975 + 122.302i) q^{21} +(17.7894 + 123.728i) q^{25} +(-106.029 + 91.8744i) q^{27} +(246.616 + 35.4580i) q^{28} +(129.358 + 18.5989i) q^{31} +(-30.7400 + 213.801i) q^{36} -416.379 q^{37} +(-10.8043 + 75.1458i) q^{39} +(-439.942 - 200.915i) q^{43} +(179.792 + 279.762i) q^{48} +(-260.444 - 570.293i) q^{49} +(63.1924 + 98.3292i) q^{52} +(-14.3667 - 48.9286i) q^{57} +(-639.474 + 554.108i) q^{61} +(764.899 - 349.317i) q^{63} +(491.260 + 144.247i) q^{64} +(261.407 - 482.109i) q^{67} +(-325.165 - 375.261i) q^{73} +(642.908 + 92.4362i) q^{75} +(-66.0475 - 42.4461i) q^{76} +(-1019.81 + 146.626i) q^{79} +(302.838 + 663.122i) q^{81} +(537.809 - 1177.64i) q^{84} +(189.026 - 413.910i) q^{91} +(282.098 - 617.709i) q^{93} -941.513i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 16 q^{4} + 54 q^{9}+O(q^{10}) $ 20 * q + 16 * q^4 + 54 * q^9	$ 20 q + 16 q^{4} + 54 q^{9} - 128 q^{16} + 112 q^{19} + 324 q^{21} + 250 q^{25} - 432 q^{36} + 220 q^{37} - 648 q^{39} + 1258 q^{49} - 6160 q^{52} + 8910 q^{57} + 5940 q^{63} + 1024 q^{64} - 880 q^{67} - 10710 q^{73} - 896 q^{76} - 9724 q^{79} - 1458 q^{81} + 11664 q^{84} - 3888 q^{91} - 1620 q^{93}+O(q^{100}) $ 20 * q + 16 * q^4 + 54 * q^9 - 128 * q^16 + 112 * q^19 + 324 * q^21 + 250 * q^25 - 432 * q^36 + 220 * q^37 - 648 * q^39 + 1258 * q^49 - 6160 * q^52 + 8910 * q^57 + 5940 * q^63 + 1024 * q^64 - 880 * q^67 - 10710 * q^73 - 896 * q^76 - 9724 * q^79 - 1458 * q^81 + 11664 * q^84 - 3888 * q^91 - 1620 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$.

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{15}{22}\right)$

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		0.540641	−	0.841254i	$-0.318182\pi$
$2$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$3$	1.46393	−	4.98567i	0.281733	−	0.959493i
$3$	1.46393	−	4.98567i	0.281733	−	0.959493i
$4$	−3.32332	−	7.27706i	−0.415415	−	0.909632i
$5$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$5$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$6$		0			0
$7$	−16.8377	+	26.2000i	−0.909151	+	1.41467i	0.000821854		1.00000i	$0.499738\pi$
$7$	−16.8377	+	26.2000i	−0.909151	+	1.41467i	−0.909973	+	0.414667i	$0.863898\pi$
$8$		0			0
$9$	−22.7138	−	14.5973i	−0.841254	−	0.540641i
$10$		0			0
$11$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$11$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$12$	−41.1461	+	5.91592i	−0.989821	+	0.142315i
$13$	−14.4618	+	2.07929i	−0.308537	+	0.0443610i	−0.294844	−	0.955545i	$-0.595268\pi$
$13$	−14.4618	+	2.07929i	−0.308537	+	0.0443610i	−0.0136929	+	0.999906i	$0.504359\pi$
$14$		0			0
$15$		0			0
$16$	−41.9111	+	48.3680i	−0.654861	+	0.755750i
$17$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$17$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$18$		0			0
$19$	8.25593	−	5.30577i	0.0996864	−	0.0640645i	−0.489848	−	0.871808i	$-0.662948\pi$
$19$	8.25593	−	5.30577i	0.0996864	−	0.0640645i	0.589534	+	0.807743i	$0.299311\pi$
$20$		0			0
$21$	105.975	+	122.302i	1.10123	+	1.27088i
$22$		0			0
$23$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$23$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$24$		0			0
$25$	17.7894	+	123.728i	0.142315	+	0.989821i
$26$		0			0
$27$	−106.029	+	91.8744i	−0.755750	+	0.654861i
$28$	246.616	+	35.4580i	1.66450	+	0.239319i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	129.358	+	18.5989i	0.749464	+	0.107757i	0.506454	−	0.862267i	$-0.330956\pi$
$31$	129.358	+	18.5989i	0.749464	+	0.107757i	0.243010	+	0.970024i	$0.421865\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−30.7400	+	213.801i	−0.142315	+	0.989821i
$37$	−416.379			−1.85006			−0.925032	−	0.379889i	$-0.875962\pi$
$37$	−416.379			−1.85006			−0.925032	+	0.379889i	$0.875962\pi$
$38$		0			0
$39$	−10.8043	+	75.1458i	−0.0443610	+	0.308537i
$40$		0			0
$41$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$41$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$42$		0			0
$43$	−439.942	−	200.915i	−1.56025	−	0.712540i	−0.566485	−	0.824072i	$-0.691697\pi$
$43$	−439.942	−	200.915i	−1.56025	−	0.712540i	−0.993761	+	0.111532i	$0.964424\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$47$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$48$	179.792	+	279.762i	0.540641	+	0.841254i
$49$	−260.444	−	570.293i	−0.759312	−	1.66266i
$50$		0			0
$51$		0			0
$52$	63.1924	+	98.3292i	0.168523	+	0.262227i
$53$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$53$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−14.3667	−	48.9286i	−0.0333846	−	0.113697i
$58$		0			0
$59$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$59$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$60$		0			0
$61$	−639.474	+	554.108i	−1.34223	+	1.16305i	−0.370003	+	0.929031i	$0.620643\pi$
$61$	−639.474	+	554.108i	−1.34223	+	1.16305i	−0.972231	+	0.234022i	$0.924811\pi$
$62$		0			0
$63$	764.899	−	349.317i	1.52965	−	0.698569i
$64$	491.260	+	144.247i	0.959493	+	0.281733i
$65$		0			0
$66$		0			0
$67$	261.407	−	482.109i	0.476656	−	0.879090i
$67$	261.407	−	482.109i	0.476656	−	0.879090i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$71$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$72$		0			0
$73$	−325.165	−	375.261i	−0.521338	−	0.601657i	0.432627	−	0.901573i	$-0.357587\pi$
$73$	−325.165	−	375.261i	−0.521338	−	0.601657i	−0.953966	+	0.299916i	$0.903041\pi$
$74$		0			0
$75$	642.908	+	92.4362i	0.989821	+	0.142315i
$76$	−66.0475	−	42.4461i	−0.0996864	−	0.0640645i
$77$		0			0
$78$		0			0
$79$	−1019.81	+	146.626i	−1.45237	+	0.208820i	−0.822892	−	0.568198i	$-0.807641\pi$
$79$	−1019.81	+	146.626i	−1.45237	+	0.208820i	−0.629480	+	0.777017i	$0.716732\pi$
$80$		0			0
$81$	302.838	+	663.122i	0.415415	+	0.909632i
$82$		0			0
$83$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$83$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$84$	537.809	−	1177.64i	0.698569	−	1.52965i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$89$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$90$		0			0
$91$	189.026	−	413.910i	0.217751	−	0.476808i
$92$		0			0
$93$	282.098	−	617.709i	0.314540	−	0.688747i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	941.513i		−	0.985527i	−0.870163	−	0.492764i	$-0.835987\pi$
$97$		−	941.513i		−	0.985527i	0.870163	−	0.492764i	$-0.164013\pi$
$98$		0			0
$99$		0			0
$100$	841.254	−	540.641i	0.841254	−	0.540641i
$101$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$101$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$102$		0			0
$103$	138.736	−	964.927i	0.132719	−	0.923078i	−0.809271	−	0.587436i	$-0.800137\pi$
$103$	138.736	−	964.927i	0.132719	−	0.923078i	0.941989	−	0.335643i	$-0.108953\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$107$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$108$	1020.94	+	466.249i	0.909632	+	0.415415i
$109$	1897.86	−	272.870i	1.66772	−	0.239782i	0.757182	−	0.653204i	$-0.226576\pi$
$109$	1897.86	−	272.870i	1.66772	−	0.239782i	0.910539	+	0.413422i	$0.135667\pi$
$110$		0			0
$111$	−609.548	+	2075.93i	−0.521223	+	1.77512i
$112$	−561.554	−	1912.48i	−0.473766	−	1.61350i
$113$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$113$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	358.835	+	163.875i	0.283541	+	0.129489i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	189.421	+	1317.45i	0.142315	+	0.989821i
$122$		0			0
$123$		0			0
$124$	−294.553	−	1003.16i	−0.213320	−	0.726500i
$125$		0			0
$126$		0			0
$127$	−2316.93	−	1489.00i	−1.61885	−	1.04037i	−0.956729	−	0.290981i	$-0.906018\pi$
$127$	−2316.93	−	1489.00i	−1.61885	−	1.04037i	−0.662125	−	0.749393i	$-0.730345\pi$
$128$		0			0
$129$	−1645.74	+	1899.28i	−1.12325	+	1.29630i
$130$		0			0
$131$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$131$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$132$		0			0
$133$			305.642i			0.199267i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$137$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$138$		0			0
$139$	−2017.97	−	1748.58i	−1.23138	−	1.06700i	−0.995456	−	0.0952257i	$-0.969643\pi$
$139$	−2017.97	−	1748.58i	−1.23138	−	1.06700i	−0.235925	−	0.971771i	$-0.575812\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1658.00	−	486.834i	0.959493	−	0.281733i
$145$		0			0
$146$		0			0
$147$	−3224.56	+	463.622i	−1.80923	+	0.260129i
$148$	1383.76	+	3030.02i	0.768544	+	1.68288i
$149$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$149$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$150$		0			0
$151$	−1503.26	+	3291.69i	−0.810157	+	1.77400i	−0.203426	+	0.979090i	$0.565208\pi$
$151$	−1503.26	+	3291.69i	−0.810157	+	1.77400i	−0.606732	+	0.794907i	$0.707520\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	582.746	−	171.110i	0.299084	−	0.0878188i
$157$	2687.14	+	789.015i	1.36597	+	0.401085i	0.880863	−	0.473371i	$-0.156963\pi$
$157$	2687.14	+	789.015i	1.36597	+	0.401085i	0.485106	+	0.874456i	$0.338781\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	771.272			0.370618			0.185309	−	0.982680i	$-0.440671\pi$
$163$	771.272			0.370618			0.185309	+	0.982680i	$0.440671\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$167$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$168$		0			0
$169$	−1903.19	+	558.826i	−0.866266	+	0.254359i
$170$		0			0
$171$	−264.974			−0.118497
$172$			3869.19i			1.71525i
$173$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$173$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$174$		0			0
$175$	−3541.20	−	1617.21i	−1.52965	−	0.698569i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$179$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$180$		0			0
$181$	4110.58	+	1206.98i	1.68805	+	0.495656i	0.978019	−	0.208515i	$-0.0668629\pi$
$181$	4110.58	+	1206.98i	1.68805	+	0.495656i	0.710031	+	0.704171i	$0.248681\pi$
$182$		0			0
$183$	1826.46	+	3999.38i	0.737790	+	1.61553i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−621.828	−	4324.91i	−0.239319	−	1.66450i
$190$		0			0
$191$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$191$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$192$	1438.34	−	2238.10i	0.540641	−	0.841254i
$193$	540.678	−	3760.50i	0.201652	−	1.40252i	−0.597730	−	0.801698i	$-0.703930\pi$
$193$	540.678	−	3760.50i	0.201652	−	1.40252i	0.799382	−	0.600823i	$-0.205160\pi$
$194$		0			0
$195$		0			0
$196$	−3284.51	+	3790.53i	−1.19698	+	1.38139i
$197$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$197$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$198$		0			0
$199$	3380.56	+	2172.56i	1.20423	+	0.773911i	0.979683	−	0.200552i	$-0.0642736\pi$
$199$	3380.56	+	2172.56i	1.20423	+	0.773911i	0.224547	+	0.974463i	$0.427910\pi$
$200$		0			0
$201$	−2020.96	−	2009.06i	−0.709191	−	0.705016i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	505.539	−	786.634i	0.168523	−	0.262227i
$209$		0			0
$210$		0			0
$211$	−4581.38	+	1345.21i	−1.49476	+	0.438902i	−0.924058	−	0.382253i	$-0.875148\pi$
$211$	−4581.38	+	1345.21i	−1.49476	+	0.438902i	−0.570705	+	0.821155i	$0.693330\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2665.38	+	3076.02i	−0.833816	+	0.962275i
$218$		0			0
$219$	−2346.94	+	1071.81i	−0.724163	+	0.330714i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1388.43	+	407.679i	−0.416933	+	0.122422i	−0.483469	−	0.875362i	$-0.660623\pi$
$223$	−1388.43	+	407.679i	−0.416933	+	0.122422i	0.0665362	+	0.997784i	$0.478805\pi$
$224$		0			0
$225$	1402.03	−	3070.01i	0.415415	−	0.909632i
$226$		0			0
$227$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$227$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$228$	−308.311	+	267.153i	−0.0895544	+	0.0775993i
$229$	5122.15	+	736.454i	1.47808	+	0.212516i	0.833713	−	0.552197i	$-0.186210\pi$
$229$	5122.15	+	736.454i	1.47808	+	0.212516i	0.644370	+	0.764714i	$0.277120\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$233$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−761.892	+	5299.08i	−0.208820	+	1.45237i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	120.746	+	139.348i	0.0322736	+	0.0372457i	0.771657	−	0.636039i	$-0.219428\pi$
$241$	120.746	+	139.348i	0.0322736	+	0.0372457i	−0.739383	+	0.673285i	$0.764883\pi$
$242$		0			0
$243$	3749.44	−	539.088i	0.989821	−	0.142315i
$244$	6157.45	+	2812.01i	1.61553	+	0.737790i
$245$		0			0
$246$		0			0
$247$	−108.363	+	93.8975i	−0.0279150	+	0.0241885i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$251$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$252$	−5084.01	−	4405.32i	−1.27088	−	1.10123i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−582.922	−	4054.31i	−0.142315	−	0.989821i
$257$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$257$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$258$		0			0
$259$	7010.88	−	10909.1i	1.68199	−	2.61722i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$263$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−4377.08	−	300.070i	−0.997658	−	0.0683944i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	2229.60	−	7593.32i	0.499773	−	1.70207i	−0.193243	−	0.981151i	$-0.561901\pi$
$271$	2229.60	−	7593.32i	0.499773	−	1.70207i	0.693017	−	0.720921i	$-0.256281\pi$
$272$		0			0
$273$	−1786.90	−	1548.36i	−0.396147	−	0.343263i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	7754.47	+	4983.49i	1.68202	+	1.08097i	0.853956	+	0.520344i	$0.174196\pi$
$277$	7754.47	+	4983.49i	1.68202	+	1.08097i	0.828068	+	0.560627i	$0.189440\pi$
$278$		0			0
$279$	−2666.72	−	2310.73i	−0.572231	−	0.495841i
$280$		0			0
$281$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$281$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$282$		0			0
$283$	−7980.99	+	5129.07i	−1.67640	+	1.07736i	−0.792168	+	0.610304i	$0.791047\pi$
$283$	−7980.99	+	5129.07i	−1.67640	+	1.07736i	−0.884230	+	0.467052i	$0.845316\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−3217.33	−	3713.00i	−0.654861	−	0.755750i
$290$		0			0
$291$	−4694.07	−	1378.30i	−0.945607	−	0.277655i
$292$	−1650.16	+	3613.36i	−0.330714	+	0.724163i
$293$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$293$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−1463.93	−	4985.67i	−0.281733	−	0.959493i
$301$	12671.6	−	8143.54i	2.42651	−	1.55942i
$302$		0			0
$303$		0			0
$304$	−89.3860	+	621.693i	−0.0168639	+	0.117291i
$305$		0			0
$306$		0			0
$307$	−423.199	+	2943.41i	−0.0786750	+	0.547197i	0.911920	+	0.410369i	$0.134600\pi$
$307$	−423.199	+	2943.41i	−0.0786750	+	0.547197i	−0.990595	+	0.136828i	$0.956309\pi$
$308$		0			0
$309$	−4607.71	−	2104.27i	−0.848296	−	0.387404i
$310$		0			0
$311$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$311$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$312$		0			0
$313$	−401.527	−	1367.47i	−0.0725100	−	0.246946i	0.915260	−	0.402863i	$-0.131985\pi$
$313$	−401.527	−	1367.47i	−0.0725100	−	0.246946i	−0.987770	+	0.155917i	$0.950167\pi$
$314$		0			0
$315$		0			0
$316$	4456.16	+	6933.91i	0.793286	+	1.23438i
$317$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$317$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	3819.15	−	4407.53i	0.654861	−	0.755750i
$325$	−514.532	−	1752.34i	−0.0878188	−	0.299084i
$326$		0			0
$327$	1417.88	−	9861.55i	0.239782	−	1.66772i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	10730.4	−	4900.39i	1.78186	−	0.813746i	0.807119	−	0.590389i	$-0.201026\pi$
$331$	10730.4	−	4900.39i	1.78186	−	0.813746i	0.974737	−	0.223357i	$-0.0717017\pi$
$332$		0			0
$333$	9457.58	+	6078.02i	1.55637	+	1.00022i
$334$		0			0
$335$		0			0
$336$	−10357.1			−1.68162
$337$	−1765.07	+	2746.50i	−0.285310	+	0.443951i	−0.954095	−	0.299505i	$-0.903178\pi$
$337$	−1765.07	+	2746.50i	−0.285310	+	0.443951i	0.668785	+	0.743456i	$0.266815\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	8753.28	+	1258.53i	1.37794	+	0.198118i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$347$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$348$		0			0
$349$	5353.92	+	11723.5i	0.821172	+	1.79812i	0.548881	+	0.835900i	$0.315054\pi$
$349$	5353.92	+	11723.5i	0.821172	+	1.79812i	0.272290	+	0.962215i	$0.412219\pi$
$350$		0			0
$351$	1342.33	−	1549.14i	0.204127	−	0.235575i
$352$		0			0
$353$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$353$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$359$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$360$		0			0
$361$	−2809.32	+	6151.56i	−0.409582	+	0.896859i
$362$		0			0
$363$	6845.68	+	984.261i	0.989821	+	0.142315i
$364$	−3640.24			−0.524177
$365$		0			0
$366$		0			0
$367$	−3391.68	−	11551.0i	−0.482409	−	1.64294i	−0.737005	−	0.675887i	$-0.763761\pi$
$367$	−3391.68	−	11551.0i	−0.482409	−	1.64294i	0.254596	−	0.967048i	$-0.418058\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−5432.61			−0.757171
$373$		−	434.724i		−	0.0603463i	−0.999545	−	0.0301731i	$-0.990394\pi$
$373$		−	434.724i		−	0.0603463i	0.999545	−	0.0301731i	$-0.00960586\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1537.36	+	5235.76i	−0.208361	+	0.709612i	0.787301	+	0.616569i	$0.211478\pi$
$379$	−1537.36	+	5235.76i	−0.208361	+	0.709612i	−0.995662	+	0.0930434i	$0.970340\pi$
$380$		0			0
$381$	−10815.5	+	9371.67i	−1.45432	+	1.26017i
$382$		0			0
$383$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$383$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	7059.96	+	10985.5i	0.927334	+	1.44296i
$388$	−6851.44	+	3128.95i	−0.896467	+	0.409403i
$389$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$389$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−9073.47	+	10471.3i	−1.14706	+	1.32378i	−0.208762	+	0.977967i	$0.566943\pi$
$397$	−9073.47	+	10471.3i	−1.14706	+	1.32378i	−0.938302	+	0.345816i	$0.887602\pi$
$398$		0			0
$399$	1523.83	+	447.438i	0.191196	+	0.0561401i
$400$	−6730.03	−	4325.13i	−0.841254	−	0.540641i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−1909.42			−0.236018
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−8446.71	+	13143.3i	−1.02118	+	1.58899i	−0.234029	+	0.972230i	$0.575191\pi$
$409$	−8446.71	+	13143.3i	−1.02118	+	1.58899i	−0.787152	+	0.616759i	$0.788445\pi$
$410$		0			0
$411$		0			0
$412$	−7482.89	+	2197.17i	−0.894795	+	0.262736i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−11672.0	+	7501.14i	−1.37070	+	0.880893i
$418$		0			0
$419$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$419$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$420$		0			0
$421$	3869.06	−	2486.49i	0.447901	−	0.287849i	−0.297181	−	0.954821i	$-0.596046\pi$
$421$	3869.06	−	2486.49i	0.447901	−	0.287849i	0.745082	+	0.666972i	$0.232410\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3750.33	−	26084.1i	−0.425038	−	2.95620i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		−	8978.95i		−	1.00000i
$433$	5049.64	+	726.029i	0.560439	+	0.0805790i	0.416712	−	0.909038i	$-0.363182\pi$
$433$	5049.64	+	726.029i	0.560439	+	0.0805790i	0.143727	+	0.989617i	$0.454091\pi$
$434$		0			0
$435$		0			0
$436$	−8292.88	−	12904.0i	−0.910910	−	1.41740i
$437$		0			0
$438$		0			0
$439$	3584.38			0.389688			0.194844	−	0.980834i	$-0.437580\pi$
$439$	3584.38			0.389688			0.194844	+	0.980834i	$0.437580\pi$
$440$		0			0
$441$	−2409.05	+	16755.3i	−0.260129	+	1.80923i
$442$		0			0
$443$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$443$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$444$	17132.4	−	2463.27i	1.83123	−	0.263292i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−12051.0	+	10442.2i	−1.27088	+	1.10123i
$449$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$449$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	14210.6	+	12313.6i	1.47389	+	1.27713i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−379.854	−	2641.94i	−0.0388814	−	0.270426i	0.961102	−	0.276194i	$-0.0890732\pi$
$457$	−379.854	−	2641.94i	−0.0388814	−	0.270426i	−0.999983	+	0.00576765i	$0.998164\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$461$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$462$		0			0
$463$	−6555.63	+	5680.49i	−0.658026	+	0.570183i	−0.918560	−	0.395282i	$-0.870647\pi$
$463$	−6555.63	+	5680.49i	−0.658026	+	0.570183i	0.260534	+	0.965465i	$0.416101\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$467$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$468$		−	3155.87i		−	0.311710i
$469$	8229.76	+	14966.5i	0.810267	+	1.47354i
$470$		0			0
$471$	7867.54	−	12242.1i	0.769676	−	1.19764i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	803.338	+	927.101i	0.0775993	+	0.0895544i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$479$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$480$		0			0
$481$	6021.60	−	865.775i	0.570814	−	0.0820706i
$482$		0			0
$483$		0			0
$484$	8957.67	−	5756.74i	0.841254	−	0.540641i
$485$		0			0
$486$		0			0
$487$	−15396.6	+	7031.37i	−1.43262	+	0.654254i	−0.972351	−	0.233526i	$-0.924974\pi$
$487$	−15396.6	+	7031.37i	−1.43262	+	0.654254i	−0.460267	+	0.887781i	$0.652246\pi$
$488$		0			0
$489$	1129.08	−	3845.31i	0.104415	−	0.355605i
$490$		0			0
$491$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$491$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−6321.12	+	5477.28i	−0.572231	+	0.495841i
$497$		0			0
$498$		0			0
$499$		−	17311.3i		−	1.55303i	−0.630099	−	0.776515i	$-0.716986\pi$
$499$		−	17311.3i		−	1.55303i	0.630099	−	0.776515i	$-0.283014\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$503$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			10306.7i			0.902837i
$508$	−3135.64	+	21808.9i	−0.273861	+	1.90475i
$509$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$509$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$510$		0			0
$511$	15306.9	−	2200.80i	1.32512	−	0.190523i
$512$		0			0
$513$	−387.902	+	1321.07i	−0.0333846	+	0.113697i
$514$		0			0
$515$		0			0
$516$	19290.5	+	5664.20i	1.64577	+	0.483242i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$521$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$522$		0			0
$523$	−3031.86	−	21087.1i	−0.253488	−	1.76304i	−0.576927	−	0.816796i	$-0.695748\pi$
$523$	−3031.86	−	21087.1i	−0.253488	−	1.76304i	0.323439	−	0.946249i	$-0.395161\pi$
$524$		0			0
$525$	−13246.9	+	15287.8i	−1.10123	+	1.27088i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	10235.5	+	6577.98i	0.841254	+	0.540641i
$530$		0			0
$531$		0			0
$532$	2224.18	−	1015.75i	0.181260	−	0.0827787i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−19015.9	−	16477.4i	−1.51119	−	1.30946i	−0.768739	−	0.639562i	$-0.779116\pi$
$541$	−19015.9	−	16477.4i	−1.51119	−	1.30946i	−0.742455	−	0.669895i	$-0.766339\pi$
$542$		0			0
$543$	12035.2	−	18727.1i	0.951157	−	1.48003i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−9143.78	−	7923.13i	−0.714735	−	0.619321i	0.219653	−	0.975578i	$-0.429507\pi$
$547$	−9143.78	−	7923.13i	−0.714735	−	0.619321i	−0.934388	+	0.356257i	$0.884053\pi$
$548$		0			0
$549$	22613.4	−	3251.31i	1.75795	−	0.252755i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	13329.6	−	29187.8i	1.02502	−	2.24447i
$554$		0			0
$555$		0			0
$556$	−6018.16	+	20496.0i	−0.459041	+	1.56335i
$557$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$557$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$558$		0			0
$559$	6780.12	+	1990.82i	0.513003	+	0.150631i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$563$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−22472.9	−	3231.11i	−1.66450	−	0.239319i
$568$		0			0
$569$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$569$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$570$		0			0
$571$	25296.3	−	7427.66i	1.85397	−	0.544374i	0.854270	−	0.519829i	$-0.174004\pi$
$571$	25296.3	−	7427.66i	1.85397	−	0.544374i	0.999699	−	0.0245456i	$-0.00781390\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−9052.79	−	10447.5i	−0.654861	−	0.755750i
$577$	−24877.7	−	11361.3i	−1.79493	−	0.819716i	−0.965183	−	0.261575i	$-0.915758\pi$
$577$	−24877.7	−	11361.3i	−1.79493	−	0.819716i	−0.829745	−	0.558142i	$-0.811514\pi$
$578$		0			0
$579$	−17957.1	−	8200.73i	−1.28890	−	0.588620i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$587$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$588$	14090.1	+	21924.6i	0.988204	+	1.53768i
$589$	1166.65	−	532.792i	0.0816147	−	0.0372722i
$590$		0			0
$591$		0			0
$592$	17450.9	−	20139.4i	1.21153	−	1.39818i
$593$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$593$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	15780.5	−	13673.9i	1.08183	−	0.937414i
$598$		0			0
$599$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$599$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$600$		0			0
$601$	−14160.4	−	9100.36i	−0.961091	−	0.617656i	−0.0367916	−	0.999323i	$-0.511714\pi$
$601$	−14160.4	−	9100.36i	−0.961091	−	0.617656i	−0.924300	+	0.381667i	$0.875350\pi$
$602$		0			0
$603$	−12975.1	+	7134.72i	−0.876260	+	0.481838i
$604$	28949.6			1.95024
$605$		0			0
$606$		0			0
$607$	−12025.5	−	26332.2i	−0.804120	−	1.76078i	−0.630852	−	0.775903i	$-0.717294\pi$
$607$	−12025.5	−	26332.2i	−0.804120	−	1.76078i	−0.173269	−	0.984875i	$-0.555433\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	14782.7	−	4340.61i	0.974012	−	0.285996i	0.244262	−	0.969709i	$-0.421454\pi$
$613$	14782.7	−	4340.61i	0.974012	−	0.285996i	0.729750	+	0.683714i	$0.239636\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$617$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$618$		0			0
$619$	−19069.0	+	22006.8i	−1.23820	+	1.42896i	−0.372777	+	0.927921i	$0.621594\pi$
$619$	−19069.0	+	22006.8i	−1.23820	+	1.42896i	−0.865424	+	0.501040i	$0.832951\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−3181.83	−	3672.02i	−0.204127	−	0.235575i
$625$	−14992.1	+	4402.07i	−0.959493	+	0.281733i
$626$		0			0
$627$		0			0
$628$	−3188.52	−	22176.6i	−0.202605	−	1.40915i
$629$		0			0
$630$		0			0
$631$	−29526.1	−	4245.21i	−1.86278	−	0.267827i	−0.883217	−	0.468964i	$-0.844628\pi$
$631$	−29526.1	−	4245.21i	−1.86278	−	0.267827i	−0.979563	+	0.201136i	$0.935537\pi$
$632$		0			0
$633$			24810.5i			1.55787i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	4952.30	+	7705.92i	0.308033	+	0.479309i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	−14192.9	−	16379.5i	−0.870473	−	1.00458i	−0.999916	−	0.0129929i	$-0.995864\pi$
$643$	−14192.9	−	16379.5i	−0.870473	−	1.00458i	0.129442	−	0.991587i	$-0.458681\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$647$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	11434.1	+	17791.8i	0.688383	+	1.07114i
$652$	−2563.18	−	5612.59i	−0.153960	−	0.337126i
$653$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$653$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1907.96	+	13270.1i	0.113298	+	0.788003i
$658$		0			0
$659$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$659$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$660$		0			0
$661$	16164.5	−	25152.4i	0.951174	−	1.48006i	0.0755037	−	0.997146i	$-0.475944\pi$
$661$	16164.5	−	25152.4i	0.951174	−	1.48006i	0.875670	−	0.482910i	$-0.160420\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			7519.06i			0.434534i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−450.082	+	1532.84i	−0.0257792	+	0.0877957i	−0.971378	−	0.237540i	$-0.923659\pi$
$673$	−450.082	+	1532.84i	−0.0257792	+	0.0877957i	0.945599	+	0.325335i	$0.105477\pi$
$674$		0			0
$675$	−13253.6	−	11484.3i	−0.755750	−	0.654861i
$676$	10391.5	+	11992.4i	0.591232	+	0.682319i
$677$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$677$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$678$		0			0
$679$	24667.6	+	15852.9i	1.39419	+	0.895994i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$683$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$684$	880.593	+	1928.23i	0.0492256	+	0.107789i
$685$		0			0
$686$		0			0
$687$	11170.2	−	24459.2i	0.620332	−	1.35834i
$688$	28156.3	−	12858.6i	1.56025	−	0.712540i
$689$		0			0
$690$		0			0
$691$	22616.0	+	26100.2i	1.24508	+	1.43690i	0.857031	+	0.515264i	$0.172306\pi$
$691$	22616.0	+	26100.2i	1.24508	+	1.43690i	0.388051	+	0.921638i	$0.373148\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$			31144.0i			1.68162i
$701$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$701$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$702$		0			0
$703$	−3437.60	+	2209.21i	−0.184426	+	0.118523i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−5373.25	+	37371.8i	−0.284621	+	1.97959i	−0.134802	+	0.990872i	$0.543040\pi$
$709$	−5373.25	+	37371.8i	−0.284621	+	1.97959i	−0.149819	+	0.988713i	$0.547869\pi$
$710$		0			0
$711$	25304.1	+	11556.0i	1.33471	+	0.609541i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$719$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$720$		0			0
$721$	22945.1	+	19882.0i	1.18519	+	1.02697i
$722$		0			0
$723$	871.508	−	398.004i	0.0448295	−	0.0204730i
$724$	−4877.55	−	33924.1i	−0.250377	−	1.74141i
$725$		0			0
$726$		0			0
$727$	7174.04	+	24432.6i	0.365984	+	1.24643i	0.912534	+	0.409001i	$0.134123\pi$
$727$	7174.04	+	24432.6i	0.365984	+	1.24643i	−0.546549	+	0.837427i	$0.684059\pi$
$728$		0			0
$729$	2801.18	−	19482.7i	0.142315	−	0.989821i
$730$		0			0
$731$		0			0
$732$	23033.8	−	26582.4i	1.16305	−	1.34223i
$733$	−35505.5	+	16214.8i	−1.78912	+	0.817065i	−0.819257	+	0.573427i	$0.805614\pi$
$733$	−35505.5	+	16214.8i	−1.78912	+	0.817065i	−0.969866	+	0.243638i	$0.921659\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−21066.9	+	32780.7i	−1.04866	+	1.63174i	−0.319235	+	0.947675i	$0.603426\pi$
$739$	−21066.9	+	32780.7i	−1.04866	+	1.63174i	−0.729421	+	0.684065i	$0.760210\pi$
$740$		0			0
$741$	309.506	+	677.724i	0.0153441	+	0.0335989i
$742$		0			0
$743$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$743$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−16829.5	−	36851.4i	−0.817730	−	1.79058i	−0.569757	−	0.821813i	$-0.692963\pi$
$751$	−16829.5	−	36851.4i	−0.817730	−	1.79058i	−0.247973	−	0.968767i	$-0.579764\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−29406.1	+	18898.1i	−1.41467	+	0.909151i
$757$	−9554.51	+	32539.7i	−0.458738	+	1.56232i	0.327782	+	0.944753i	$0.393699\pi$
$757$	−9554.51	+	32539.7i	−0.458738	+	1.56232i	−0.786520	+	0.617565i	$0.788119\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$761$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$762$		0			0
$763$	−24806.4	+	54318.3i	−1.17700	+	2.57727i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−21066.8	−	3028.95i	−0.989821	−	0.142315i
$769$	−415.589	−	1415.36i	−0.0194883	−	0.0663711i	0.949175	−	0.314750i	$-0.101921\pi$
$769$	−415.589	−	1415.36i	−0.0194883	−	0.0663711i	−0.968663	+	0.248378i	$0.920102\pi$
$770$		0			0
$771$		0			0
$772$	−29162.2	+	8562.79i	−1.35955	+	0.399199i
$773$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$773$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$774$		0			0
$775$			16336.0i			0.757171i
$776$		0			0
$777$	−44126.0	−	50924.1i	−2.03734	−	2.35121i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	38499.4	+	11304.4i	1.75380	+	0.514962i
$785$		0			0
$786$		0			0
$787$	−40129.6	−	18326.6i	−1.81762	−	0.830078i	−0.926476	−	0.376354i	$-0.877178\pi$
$787$	−40129.6	−	18326.6i	−1.81762	−	0.830078i	−0.891142	−	0.453724i	$-0.850095\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	8095.80	−	9343.05i	0.362535	−	0.418388i
$794$		0			0
$795$		0			0
$796$	4575.12	−	31820.6i	0.203719	−	1.41690i
$797$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$797$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−7903.76	+	21383.4i	−0.346697	+	0.937977i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$809$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$810$		0			0
$811$	−22926.4	+	35674.1i	−0.992668	+	1.54462i	−0.162802	+	0.986659i	$0.552053\pi$
$811$	−22926.4	+	35674.1i	−0.992668	+	1.54462i	−0.829866	+	0.557963i	$0.811583\pi$
$812$		0			0
$813$	−34593.8	−	22232.1i	−1.49232	−	0.959058i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−4698.14	+	675.491i	−0.201184	+	0.0289259i
$818$		0			0
$819$	−10335.5	+	6642.21i	−0.440966	+	0.283392i
$820$		0			0
$821$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$821$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$822$		0			0
$823$	29393.2	−	18889.8i	1.24494	−	0.800071i	0.258786	−	0.965935i	$-0.416677\pi$
$823$	29393.2	−	18889.8i	1.24494	−	0.800071i	0.986149	+	0.165863i	$0.0530411\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$827$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$828$		0			0
$829$	−4762.51	−	33124.0i	−0.199528	−	1.38775i	−0.805657	−	0.592383i	$-0.798187\pi$
$829$	−4762.51	−	33124.0i	−0.199528	−	1.38775i	0.606128	−	0.795367i	$-0.292722\pi$
$830$		0			0
$831$	36198.0	−	31365.8i	1.51107	−	1.30935i
$832$	−7404.45	−	1064.60i	−0.308537	−	0.0443610i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−15424.4	+	9912.67i	−0.636972	+	0.409357i
$838$		0			0
$839$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$839$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$840$		0			0
$841$	24389.0			1.00000
$842$		0			0
$843$		0			0
$844$	25014.6	+	28868.4i	1.02019	+	1.17736i
$845$		0			0
$846$		0			0
$847$	−37706.7	−	17220.1i	−1.52965	−	0.698569i
$848$		0			0
$849$	13888.3	+	47299.2i	0.561419	+	1.91202i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−4393.00	−	9619.33i	−0.176335	−	0.386119i	0.800741	−	0.599010i	$-0.204439\pi$
$853$	−4393.00	−	9619.33i	−0.176335	−	0.386119i	−0.977076	+	0.212891i	$0.931712\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$857$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$858$		0			0
$859$	6189.61	+	43049.7i	0.245852	+	1.70994i	0.621699	+	0.783256i	$0.286443\pi$
$859$	6189.61	+	43049.7i	0.245852	+	1.70994i	−0.375847	+	0.926682i	$0.622648\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$863$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−23221.7	+	10605.0i	−0.909632	+	0.415415i
$868$	31242.3	+	9173.56i	1.22170	+	0.358722i
$869$		0			0
$870$		0			0
$871$	−2777.97	+	7515.71i	−0.108069	+	0.292377i
$872$		0			0
$873$	−13743.6	+	21385.4i	−0.532816	+	0.829078i
$874$		0			0
$875$		0			0
$876$	15599.3	+	13516.9i	0.601657	+	0.521338i
$877$	32418.3	+	37412.7i	1.24822	+	1.44052i	0.852979	+	0.521946i	$0.174794\pi$
$877$	32418.3	+	37412.7i	1.24822	+	1.44052i	0.395242	+	0.918577i	$0.370661\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$881$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$882$		0			0
$883$	51572.9	−	7415.06i	1.96553	−	0.282601i	0.965806	−	0.259266i	$-0.0834806\pi$
$883$	51572.9	−	7415.06i	1.96553	−	0.282601i	0.999728	−	0.0233351i	$-0.00742847\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$887$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$888$		0			0
$889$	78023.7	−	35632.2i	2.94357	−	1.34428i
$890$		0			0
$891$		0			0
$892$	7580.89	+	8748.82i	0.284560	+	0.328399i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−27000.0			−1.00000
$901$		0			0
$902$		0			0
$903$	−22050.7	−	75097.9i	−0.812628	−	2.76756i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	7535.49	−	52410.5i	0.275868	−	1.91870i	−0.105772	−	0.994390i	$-0.533731\pi$
$907$	7535.49	−	52410.5i	0.275868	−	1.91870i	0.381640	−	0.924311i	$-0.375360\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$911$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$912$	2968.70	+	1355.76i	0.107789	+	0.0492256i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−11663.3	−	39721.6i	−0.420707	−	1.43279i
$917$		0			0
$918$		0			0
$919$	18576.8	+	28906.0i	0.666803	+	1.03756i	0.995649	+	0.0931816i	$0.0297037\pi$
$919$	18576.8	+	28906.0i	0.666803	+	1.03756i	−0.328847	+	0.944383i	$0.606660\pi$
$920$		0			0
$921$	14055.4	+	6418.87i	0.502866	+	0.229651i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−7407.12	−	51517.7i	−0.263292	−	1.83123i
$926$		0			0
$927$	−17236.5	+	19892.0i	−0.610704	+	0.704790i
$928$		0			0
$929$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$929$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$930$		0			0
$931$	−5176.05	−	3326.44i	−0.182211	−	0.117100i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			56503.1i			1.96998i	0.172597	+	0.984992i	$0.444784\pi$
$937$			56503.1i			1.96998i	−0.172597	+	0.984992i	$0.555216\pi$
$938$		0			0
$939$	−7405.59			−0.257372
$940$		0			0
$941$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$941$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$947$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$948$	41093.7	−	12066.2i	1.40787	−	0.413388i
$949$	5482.75	+	4750.83i	0.187542	+	0.162506i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$953$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−12196.7	−	3581.27i	−0.409409	−	0.120213i
$962$		0			0
$963$		0			0
$964$	612.768	−	1341.77i	0.0204730	−	0.0448295i
$965$		0			0
$966$		0			0
$967$	−57401.0			−1.90889			−0.954443	−	0.298394i	$-0.903549\pi$
$967$	−57401.0			−1.90889			−0.954443	+	0.298394i	$0.903549\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$971$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$972$	−16383.6	−	25493.3i	−0.540641	−	0.841254i
$973$	79790.8	−	23428.7i	2.62896	−	0.771931i
$974$		0			0
$975$	−9489.81			−0.311710
$976$		−	54153.3i		−	1.77603i
$977$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$977$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−47090.8	−	21505.6i	−1.53261	−	0.699921i
$982$		0			0
$983$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$983$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1043.42	+	476.516i	0.0335989	+	0.0153441i
$989$		0			0
$990$		0			0
$991$	56717.3	−	25901.9i	1.81805	−	0.830274i	0.893829	−	0.448407i	$-0.148008\pi$
$991$	56717.3	−	25901.9i	1.81805	−	0.830274i	0.924217	−	0.381867i	$-0.124719\pi$
$992$		0			0
$993$	−8723.30	−	60671.9i	−0.278777	−	1.93894i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1705.11	+	11859.3i	−0.0541639	+	0.376718i	0.944652	+	0.328073i	$0.106399\pi$
$997$	−1705.11	+	11859.3i	−0.0541639	+	0.376718i	−0.998816	+	0.0486447i	$0.984510\pi$
$998$		0			0
$999$	44148.2	−	38254.6i	1.39818	−	1.21153i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.a.125.2	yes	20
3.2	odd	2	CM	201.4.j.a.125.2	yes	20
67.52	odd	22	inner	201.4.j.a.119.2	✓	20
201.119	even	22	inner	201.4.j.a.119.2	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.a.119.2	✓	20	67.52	odd	22	inner
201.4.j.a.119.2	✓	20	201.119	even	22	inner
201.4.j.a.125.2	yes	20	1.1	even	1	trivial
201.4.j.a.125.2	yes	20	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.j (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(1.46393 - 4.98567i) q^{3} +(-3.32332 - 7.27706i) q^{4} +(-16.8377 + 26.2000i) q^{7} +(-22.7138 - 14.5973i) q^{9} +O(q^{10})\)	\(q+(1.46393 - 4.98567i) q^{3} +(-3.32332 - 7.27706i) q^{4} +(-16.8377 + 26.2000i) q^{7} +(-22.7138 - 14.5973i) q^{9} +(-41.1461 + 5.91592i) q^{12} +(-14.4618 + 2.07929i) q^{13} +(-41.9111 + 48.3680i) q^{16} +(8.25593 - 5.30577i) q^{19} +(105.975 + 122.302i) q^{21} +(17.7894 + 123.728i) q^{25} +(-106.029 + 91.8744i) q^{27} +(246.616 + 35.4580i) q^{28} +(129.358 + 18.5989i) q^{31} +(-30.7400 + 213.801i) q^{36} -416.379 q^{37} +(-10.8043 + 75.1458i) q^{39} +(-439.942 - 200.915i) q^{43} +(179.792 + 279.762i) q^{48} +(-260.444 - 570.293i) q^{49} +(63.1924 + 98.3292i) q^{52} +(-14.3667 - 48.9286i) q^{57} +(-639.474 + 554.108i) q^{61} +(764.899 - 349.317i) q^{63} +(491.260 + 144.247i) q^{64} +(261.407 - 482.109i) q^{67} +(-325.165 - 375.261i) q^{73} +(642.908 + 92.4362i) q^{75} +(-66.0475 - 42.4461i) q^{76} +(-1019.81 + 146.626i) q^{79} +(302.838 + 663.122i) q^{81} +(537.809 - 1177.64i) q^{84} +(189.026 - 413.910i) q^{91} +(282.098 - 617.709i) q^{93} -941.513i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{4} + 54 q^{9}+O(q^{10}) \) 20 * q + 16 * q^4 + 54 * q^9	\( 20 q + 16 q^{4} + 54 q^{9} - 128 q^{16} + 112 q^{19} + 324 q^{21} + 250 q^{25} - 432 q^{36} + 220 q^{37} - 648 q^{39} + 1258 q^{49} - 6160 q^{52} + 8910 q^{57} + 5940 q^{63} + 1024 q^{64} - 880 q^{67} - 10710 q^{73} - 896 q^{76} - 9724 q^{79} - 1458 q^{81} + 11664 q^{84} - 3888 q^{91} - 1620 q^{93}+O(q^{100}) \) 20 * q + 16 * q^4 + 54 * q^9 - 128 * q^16 + 112 * q^19 + 324 * q^21 + 250 * q^25 - 432 * q^36 + 220 * q^37 - 648 * q^39 + 1258 * q^49 - 6160 * q^52 + 8910 * q^57 + 5940 * q^63 + 1024 * q^64 - 880 * q^67 - 10710 * q^73 - 896 * q^76 - 9724 * q^79 - 1458 * q^81 + 11664 * q^84 - 3888 * q^91 - 1620 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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