LMFDB - Embedded newform 2160.3.c.m.1889.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,3,Mod(1889,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2160.1889");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.c (of order $2$, degree $1$, 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:	$58.8557371018$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.31744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 14x^{2} + 31 $ x^4 + 14*x^2 + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.1
Root			$3.35300i$ of defining polynomial
Character	$\chi$	$=$	2160.1889
Dual form			2160.3.c.m.1889.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.58579 - 4.74186i) q^{5} -0.813575i q^{7} +15.3762i q^{11} +5.89243i q^{13} -12.8995 q^{17} -1.24264 q^{19} +4.79899 q^{23} +(-19.9706 - 15.0392i) q^{25} -42.6768i q^{29} -4.21320 q^{31} +(-3.85786 - 1.29016i) q^{35} -70.3144i q^{37} -7.04300i q^{41} +41.0496i q^{43} +79.7990 q^{47} +48.3381 q^{49} +63.7279 q^{53} +(72.9117 + 24.3833i) q^{55} -36.6448i q^{59} -82.9411 q^{61} +(27.9411 + 9.34414i) q^{65} -89.0027i q^{67} -69.6982i q^{71} -89.6188i q^{73} +12.5097 q^{77} -134.095 q^{79} -109.024 q^{83} +(-20.4558 + 61.1677i) q^{85} +137.514i q^{89} +4.79394 q^{91} +(-1.97056 + 5.89243i) q^{95} -90.6298i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{5} - 12 q^{17} + 12 q^{19} - 60 q^{23} - 12 q^{25} + 68 q^{31} - 72 q^{35} + 240 q^{47} - 180 q^{49} + 204 q^{53} + 88 q^{55} - 196 q^{61} - 24 q^{65} - 312 q^{77} - 180 q^{79} - 108 q^{83}+ \cdots + 60 q^{95}+O(q^{100}) $ 4 * q + 12 * q^5 - 12 * q^17 + 12 * q^19 - 60 * q^23 - 12 * q^25 + 68 * q^31 - 72 * q^35 + 240 * q^47 - 180 * q^49 + 204 * q^53 + 88 * q^55 - 196 * q^61 - 24 * q^65 - 312 * q^77 - 180 * q^79 - 108 * q^83 + 20 * q^85 - 456 * q^91 + 60 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.58579	−	4.74186i	0.317157	−	0.948373i
$5$	1.58579	−	4.74186i	0.317157	−	0.948373i
$6$		0			0
$7$		−	0.813575i		−	0.116225i	−0.998310	−	0.0581125i	$-0.981492\pi$
$7$		−	0.813575i		−	0.116225i	0.998310	−	0.0581125i	$-0.0185082\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			15.3762i			1.39783i	0.715203	+	0.698917i	$0.246334\pi$
$11$			15.3762i			1.39783i	−0.715203	+	0.698917i	$0.753666\pi$
$12$		0			0
$13$			5.89243i			0.453264i	0.973980	+	0.226632i	$0.0727715\pi$
$13$			5.89243i			0.453264i	−0.973980	+	0.226632i	$0.927229\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−12.8995			−0.758794			−0.379397	−	0.925234i	$-0.623869\pi$
$17$	−12.8995			−0.758794			−0.379397	+	0.925234i	$0.623869\pi$
$18$		0			0
$19$	−1.24264			−0.0654021			−0.0327011	−	0.999465i	$-0.510411\pi$
$19$	−1.24264			−0.0654021			−0.0327011	+	0.999465i	$0.510411\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.79899			0.208652			0.104326	−	0.994543i	$-0.466732\pi$
$23$	4.79899			0.208652			0.104326	+	0.994543i	$0.466732\pi$
$24$		0			0
$25$	−19.9706	−	15.0392i	−0.798823	−	0.601567i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	42.6768i		−	1.47161i	−0.677192	−	0.735807i	$-0.736803\pi$
$29$		−	42.6768i		−	1.47161i	0.677192	−	0.735807i	$-0.263197\pi$
$30$		0			0
$31$	−4.21320			−0.135910			−0.0679549	−	0.997688i	$-0.521647\pi$
$31$	−4.21320			−0.135910			−0.0679549	+	0.997688i	$0.521647\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.85786	−	1.29016i	−0.110225	−	0.0368616i
$36$		0			0
$37$		−	70.3144i		−	1.90039i	−0.311657	−	0.950195i	$-0.600884\pi$
$37$		−	70.3144i		−	1.90039i	0.311657	−	0.950195i	$-0.399116\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	7.04300i		−	0.171781i	−0.996305	−	0.0858903i	$-0.972627\pi$
$41$		−	7.04300i		−	0.171781i	0.996305	−	0.0858903i	$-0.0273735\pi$
$42$		0			0
$43$			41.0496i			0.954643i	0.878729	+	0.477321i	$0.158392\pi$
$43$			41.0496i			0.954643i	−0.878729	+	0.477321i	$0.841608\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	79.7990			1.69785			0.848925	−	0.528513i	$-0.177250\pi$
$47$	79.7990			1.69785			0.848925	+	0.528513i	$0.177250\pi$
$48$		0			0
$49$	48.3381			0.986492
$50$		0			0
$51$		0			0
$52$		0			0
$53$	63.7279			1.20241			0.601207	−	0.799093i	$-0.294687\pi$
$53$	63.7279			1.20241			0.601207	+	0.799093i	$0.294687\pi$
$54$		0			0
$55$	72.9117	+	24.3833i	1.32567	+	0.443333i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	36.6448i		−	0.621098i	−0.950557	−	0.310549i	$-0.899487\pi$
$59$		−	36.6448i		−	0.621098i	0.950557	−	0.310549i	$-0.100513\pi$
$60$		0			0
$61$	−82.9411			−1.35969			−0.679845	−	0.733356i	$-0.737953\pi$
$61$	−82.9411			−1.35969			−0.679845	+	0.733356i	$0.737953\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	27.9411	+	9.34414i	0.429863	+	0.143756i
$66$		0			0
$67$		−	89.0027i		−	1.32840i	−0.747556	−	0.664199i	$-0.768773\pi$
$67$		−	89.0027i		−	1.32840i	0.747556	−	0.664199i	$-0.231227\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	69.6982i		−	0.981665i	−0.871254	−	0.490833i	$-0.836693\pi$
$71$		−	69.6982i		−	0.981665i	0.871254	−	0.490833i	$-0.163307\pi$
$72$		0			0
$73$		−	89.6188i		−	1.22766i	−0.789440	−	0.613828i	$-0.789629\pi$
$73$		−	89.6188i		−	1.22766i	0.789440	−	0.613828i	$-0.210371\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.5097			0.162463
$78$		0			0
$79$	−134.095			−1.69741			−0.848705	−	0.528866i	$-0.822617\pi$
$79$	−134.095			−1.69741			−0.848705	+	0.528866i	$0.822617\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−109.024			−1.31355			−0.656773	−	0.754088i	$-0.728079\pi$
$83$	−109.024			−1.31355			−0.656773	+	0.754088i	$0.728079\pi$
$84$		0			0
$85$	−20.4558	+	61.1677i	−0.240657	+	0.719620i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			137.514i			1.54510i	0.634953	+	0.772551i	$0.281020\pi$
$89$			137.514i			1.54510i	−0.634953	+	0.772551i	$0.718980\pi$
$90$		0			0
$91$	4.79394			0.0526807
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.97056	+	5.89243i	−0.0207428	+	0.0620256i
$96$		0			0
$97$		−	90.6298i		−	0.934328i	−0.884171	−	0.467164i	$-0.845276\pi$
$97$		−	90.6298i		−	0.934328i	0.884171	−	0.467164i	$-0.154724\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			49.7198i			0.492275i	0.969235	+	0.246138i	$0.0791615\pi$
$101$			49.7198i			0.492275i	−0.969235	+	0.246138i	$0.920839\pi$
$102$		0			0
$103$		−	128.844i		−	1.25091i	−0.780259	−	0.625456i	$-0.784913\pi$
$103$		−	128.844i		−	1.25091i	0.780259	−	0.625456i	$-0.215087\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.42641			−0.0600599			−0.0300299	−	0.999549i	$-0.509560\pi$
$107$	−6.42641			−0.0600599			−0.0300299	+	0.999549i	$0.509560\pi$
$108$		0			0
$109$	108.279			0.993387			0.496694	−	0.867926i	$-0.334547\pi$
$109$	108.279			0.993387			0.496694	+	0.867926i	$0.334547\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−30.4264			−0.269260			−0.134630	−	0.990896i	$-0.542985\pi$
$113$	−30.4264			−0.269260			−0.134630	+	0.990896i	$0.542985\pi$
$114$		0			0
$115$	7.61017	−	22.7562i	0.0661754	−	0.197880i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			10.4947i			0.0881909i
$120$		0			0
$121$	−115.426			−0.953937
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−102.983	+	70.8488i	−0.823862	+	0.566790i
$126$		0			0
$127$		−	133.725i		−	1.05296i	−0.850189	−	0.526478i	$-0.823512\pi$
$127$		−	133.725i		−	1.05296i	0.850189	−	0.526478i	$-0.176488\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			61.7838i			0.471632i	0.971798	+	0.235816i	$0.0757763\pi$
$131$			61.7838i			0.471632i	−0.971798	+	0.235816i	$0.924224\pi$
$132$		0			0
$133$			1.01098i			0.00760137i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−163.919			−1.19649			−0.598244	−	0.801314i	$-0.704135\pi$
$137$	−163.919			−1.19649			−0.598244	+	0.801314i	$0.704135\pi$
$138$		0			0
$139$	−182.250			−1.31115			−0.655575	−	0.755130i	$-0.727574\pi$
$139$	−182.250			−1.31115			−0.655575	+	0.755130i	$0.727574\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−90.6030			−0.633588
$144$		0			0
$145$	−202.368	−	67.6763i	−1.39564	−	0.466733i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	189.535i		−	1.27205i	−0.771670	−	0.636023i	$-0.780578\pi$
$149$		−	189.535i		−	1.27205i	0.771670	−	0.636023i	$-0.219422\pi$
$150$		0			0
$151$	113.338			0.750583			0.375292	−	0.926907i	$-0.377543\pi$
$151$	113.338			0.750583			0.375292	+	0.926907i	$0.377543\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.68124	+	19.9784i	−0.0431048	+	0.128893i
$156$		0			0
$157$		−	113.189i		−	0.720946i	−0.932770	−	0.360473i	$-0.882615\pi$
$157$		−	113.189i		−	0.720946i	0.932770	−	0.360473i	$-0.117385\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	3.90434i		−	0.0242506i
$162$		0			0
$163$		−	152.414i		−	0.935053i	−0.883979	−	0.467527i	$-0.845145\pi$
$163$		−	152.414i		−	0.935053i	0.883979	−	0.467527i	$-0.154855\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−105.853			−0.633849			−0.316925	−	0.948451i	$-0.602650\pi$
$167$	−105.853			−0.633849			−0.316925	+	0.948451i	$0.602650\pi$
$168$		0			0
$169$	134.279			0.794552
$170$		0			0
$171$		0			0
$172$		0			0
$173$	110.012			0.635909			0.317954	−	0.948106i	$-0.397004\pi$
$173$	110.012			0.635909			0.317954	+	0.948106i	$0.397004\pi$
$174$		0			0
$175$	−12.2355	+	16.2476i	−0.0699171	+	0.0928432i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			118.407i			0.661492i	0.943720	+	0.330746i	$0.107300\pi$
$179$			118.407i			0.661492i	−0.943720	+	0.330746i	$0.892700\pi$
$180$		0			0
$181$	172.397			0.952469			0.476235	−	0.879318i	$-0.342001\pi$
$181$	172.397			0.952469			0.476235	+	0.879318i	$0.342001\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−333.421	−	111.504i	−1.80228	−	0.602722i
$186$		0			0
$187$		−	198.345i		−	1.06067i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			289.393i			1.51515i	0.652749	+	0.757574i	$0.273616\pi$
$191$			289.393i			1.51515i	−0.652749	+	0.757574i	$0.726384\pi$
$192$		0			0
$193$		−	127.833i		−	0.662347i	−0.943570	−	0.331173i	$-0.892555\pi$
$193$		−	127.833i		−	0.662347i	0.943570	−	0.331173i	$-0.107445\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	280.414			1.42342			0.711711	−	0.702472i	$-0.247920\pi$
$197$	280.414			1.42342			0.711711	+	0.702472i	$0.247920\pi$
$198$		0			0
$199$	−156.426			−0.786062			−0.393031	−	0.919525i	$-0.628574\pi$
$199$	−156.426			−0.786062			−0.393031	+	0.919525i	$0.628574\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−34.7208			−0.171038
$204$		0			0
$205$	−33.3970	−	11.1687i	−0.162912	−	0.0544815i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	19.1070i		−	0.0914213i
$210$		0			0
$211$	−22.5076			−0.106671			−0.0533355	−	0.998577i	$-0.516985\pi$
$211$	−22.5076			−0.106671			−0.0533355	+	0.998577i	$0.516985\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	194.652	+	65.0959i	0.905357	+	0.302772i
$216$		0			0
$217$			3.42776i			0.0157961i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	76.0094i		−	0.343934i
$222$		0			0
$223$		−	354.826i		−	1.59115i	−0.605856	−	0.795575i	$-0.707169\pi$
$223$		−	354.826i		−	1.59115i	0.605856	−	0.795575i	$-0.292831\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	415.098			1.82862			0.914312	−	0.405011i	$-0.132732\pi$
$227$	415.098			1.82862			0.914312	+	0.405011i	$0.132732\pi$
$228$		0			0
$229$	−220.220			−0.961661			−0.480830	−	0.876814i	$-0.659665\pi$
$229$	−220.220			−0.961661			−0.480830	+	0.876814i	$0.659665\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.0345			0.0430665			0.0215332	−	0.999768i	$-0.493145\pi$
$233$	10.0345			0.0430665			0.0215332	+	0.999768i	$0.493145\pi$
$234$		0			0
$235$	126.544	−	378.396i	0.538486	−	1.61020i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	114.363i		−	0.478507i	−0.970957	−	0.239253i	$-0.923097\pi$
$239$		−	114.363i		−	0.478507i	0.970957	−	0.239253i	$-0.0769027\pi$
$240$		0			0
$241$	−63.7208			−0.264402			−0.132201	−	0.991223i	$-0.542204\pi$
$241$	−63.7208			−0.264402			−0.132201	+	0.991223i	$0.542204\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	76.6539	−	229.213i	0.312873	−	0.935562i
$246$		0			0
$247$		−	7.32218i		−	0.0296444i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	229.631i		−	0.914866i	−0.889244	−	0.457433i	$-0.848769\pi$
$251$		−	229.631i		−	0.914866i	0.889244	−	0.457433i	$-0.151231\pi$
$252$		0			0
$253$			73.7901i			0.291660i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	388.227			1.51061			0.755306	−	0.655372i	$-0.227488\pi$
$257$	388.227			1.51061			0.755306	+	0.655372i	$0.227488\pi$
$258$		0			0
$259$	−57.2061			−0.220873
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−226.877			−0.862651			−0.431325	−	0.902196i	$-0.641954\pi$
$263$	−226.877			−0.862651			−0.431325	+	0.902196i	$0.641954\pi$
$264$		0			0
$265$	101.059	−	302.189i	0.381354	−	1.14034i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			226.772i			0.843019i	0.906824	+	0.421509i	$0.138500\pi$
$269$			226.772i			0.843019i	−0.906824	+	0.421509i	$0.861500\pi$
$270$		0			0
$271$	−322.889			−1.19147			−0.595737	−	0.803180i	$-0.703140\pi$
$271$	−322.889			−1.19147			−0.595737	+	0.803180i	$0.703140\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	231.245	−	307.071i	0.840890	−	1.11662i
$276$		0			0
$277$			200.564i			0.724058i	0.932167	+	0.362029i	$0.117916\pi$
$277$			200.564i			0.724058i	−0.932167	+	0.362029i	$0.882084\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			128.030i			0.455624i	0.973705	+	0.227812i	$0.0731572\pi$
$281$			128.030i			0.455624i	−0.973705	+	0.227812i	$0.926843\pi$
$282$		0			0
$283$			420.235i			1.48493i	0.669885	+	0.742465i	$0.266344\pi$
$283$			420.235i			1.48493i	−0.669885	+	0.742465i	$0.733656\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.73001			−0.0199652
$288$		0			0
$289$	−122.603			−0.424232
$290$		0			0
$291$		0			0
$292$		0			0
$293$	137.434			0.469056			0.234528	−	0.972109i	$-0.424645\pi$
$293$	137.434			0.469056			0.234528	+	0.972109i	$0.424645\pi$
$294$		0			0
$295$	−173.765	−	58.1108i	−0.589032	−	0.196986i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			28.2777i			0.0945744i
$300$		0			0
$301$	33.3970			0.110953
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−131.527	+	393.296i	−0.431236	+	1.28949i
$306$		0			0
$307$			117.849i			0.383872i	0.981407	+	0.191936i	$0.0614766\pi$
$307$			117.849i			0.383872i	−0.981407	+	0.191936i	$0.938523\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	499.326i		−	1.60555i	−0.596283	−	0.802774i	$-0.703356\pi$
$311$		−	499.326i		−	1.60555i	0.596283	−	0.802774i	$-0.296644\pi$
$312$		0			0
$313$		−	491.806i		−	1.57127i	−0.618693	−	0.785633i	$-0.712338\pi$
$313$		−	491.806i		−	1.57127i	0.618693	−	0.785633i	$-0.287662\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−241.399			−0.761511			−0.380756	−	0.924676i	$-0.624336\pi$
$317$	−241.399			−0.761511			−0.380756	+	0.924676i	$0.624336\pi$
$318$		0			0
$319$	656.205			2.05707
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.0294			0.0496267
$324$		0			0
$325$	88.6173	−	117.675i	0.272669	−	0.362078i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	64.9225i		−	0.197333i
$330$		0			0
$331$	87.8091			0.265284			0.132642	−	0.991164i	$-0.457654\pi$
$331$	87.8091			0.265284			0.132642	+	0.991164i	$0.457654\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−422.039	−	141.139i	−1.25982	−	0.421311i
$336$		0			0
$337$		−	452.138i		−	1.34166i	−0.741613	−	0.670828i	$-0.765939\pi$
$337$		−	452.138i		−	1.34166i	0.741613	−	0.670828i	$-0.234061\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	64.7829i		−	0.189979i
$342$		0			0
$343$		−	79.1919i		−	0.230880i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−189.411			−0.545854			−0.272927	−	0.962035i	$-0.587992\pi$
$347$	−189.411			−0.545854			−0.272927	+	0.962035i	$0.587992\pi$
$348$		0			0
$349$	−155.794			−0.446401			−0.223200	−	0.974773i	$-0.571650\pi$
$349$	−155.794			−0.446401			−0.223200	+	0.974773i	$0.571650\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	263.387			0.746138			0.373069	−	0.927804i	$-0.378305\pi$
$353$	263.387			0.746138			0.373069	+	0.927804i	$0.378305\pi$
$354$		0			0
$355$	−330.500	−	110.527i	−0.930985	−	0.311342i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			79.6346i			0.221823i	0.993830	+	0.110912i	$0.0353771\pi$
$359$			79.6346i			0.221823i	−0.993830	+	0.110912i	$0.964623\pi$
$360$		0			0
$361$	−359.456			−0.995723
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−424.960	−	142.116i	−1.16428	−	0.389360i
$366$		0			0
$367$			475.953i			1.29688i	0.761268	+	0.648438i	$0.224577\pi$
$367$			475.953i			1.29688i	−0.761268	+	0.648438i	$0.775423\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	51.8475i		−	0.139751i
$372$		0			0
$373$		−	61.7838i		−	0.165640i	−0.996565	−	0.0828201i	$-0.973607\pi$
$373$		−	61.7838i		−	0.165640i	0.996565	−	0.0828201i	$-0.0263927\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	251.470			0.667030
$378$		0			0
$379$	142.345			0.375581			0.187791	−	0.982209i	$-0.439867\pi$
$379$	142.345			0.375581			0.187791	+	0.982209i	$0.439867\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−357.520			−0.933472			−0.466736	−	0.884397i	$-0.654570\pi$
$383$	−357.520			−0.933472			−0.466736	+	0.884397i	$0.654570\pi$
$384$		0			0
$385$	19.8377	−	59.3192i	0.0515264	−	0.154076i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			411.079i			1.05676i	0.849009	+	0.528379i	$0.177200\pi$
$389$			411.079i			1.05676i	−0.849009	+	0.528379i	$0.822800\pi$
$390$		0			0
$391$	−61.9045			−0.158324
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−212.647	+	635.863i	−0.538346	+	1.60978i
$396$		0			0
$397$			457.462i			1.15230i	0.817345	+	0.576149i	$0.195445\pi$
$397$			457.462i			1.15230i	−0.817345	+	0.576149i	$0.804555\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	467.283i		−	1.16529i	−0.812725	−	0.582647i	$-0.802017\pi$
$401$		−	467.283i		−	1.16529i	0.812725	−	0.582647i	$-0.197983\pi$
$402$		0			0
$403$		−	24.8260i		−	0.0616030i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1081.17			2.65643
$408$		0			0
$409$	−448.574			−1.09676			−0.548378	−	0.836230i	$-0.684755\pi$
$409$	−448.574			−1.09676			−0.548378	+	0.836230i	$0.684755\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−29.8133			−0.0721871
$414$		0			0
$415$	−172.889	+	516.979i	−0.416601	+	1.24573i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	360.834i		−	0.861180i	−0.902548	−	0.430590i	$-0.858306\pi$
$419$		−	360.834i		−	0.861180i	0.902548	−	0.430590i	$-0.141694\pi$
$420$		0			0
$421$	357.059			0.848121			0.424060	−	0.905634i	$-0.360604\pi$
$421$	357.059			0.848121			0.424060	+	0.905634i	$0.360604\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	257.610	+	193.998i	0.606142	+	0.456465i
$426$		0			0
$427$			67.4789i			0.158030i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			655.947i			1.52192i	0.648800	+	0.760959i	$0.275271\pi$
$431$			655.947i			1.52192i	−0.648800	+	0.760959i	$0.724729\pi$
$432$		0			0
$433$		−	164.372i		−	0.379612i	−0.981822	−	0.189806i	$-0.939214\pi$
$433$		−	164.372i		−	0.379612i	0.981822	−	0.189806i	$-0.0607859\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.96342			−0.0136463
$438$		0			0
$439$	132.286			0.301336			0.150668	−	0.988584i	$-0.451858\pi$
$439$	132.286			0.301336			0.150668	+	0.988584i	$0.451858\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−455.485			−1.02818			−0.514092	−	0.857735i	$-0.671871\pi$
$443$	−455.485			−1.02818			−0.514092	+	0.857735i	$0.671871\pi$
$444$		0			0
$445$	652.073	+	218.068i	1.46533	+	0.490040i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	415.749i		−	0.925944i	−0.886373	−	0.462972i	$-0.846783\pi$
$449$		−	415.749i		−	0.925944i	0.886373	−	0.462972i	$-0.153217\pi$
$450$		0			0
$451$	108.294			0.240121
$452$		0			0
$453$		0			0
$454$		0			0
$455$	7.60216	−	22.7322i	0.0167081	−	0.0499609i
$456$		0			0
$457$		−	303.643i		−	0.664426i	−0.943204	−	0.332213i	$-0.892205\pi$
$457$		−	303.643i		−	0.664426i	0.943204	−	0.332213i	$-0.107795\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			428.790i			0.930130i	0.885277	+	0.465065i	$0.153969\pi$
$461$			428.790i			0.930130i	−0.885277	+	0.465065i	$0.846031\pi$
$462$		0			0
$463$		−	814.484i		−	1.75914i	−0.475765	−	0.879572i	$-0.657829\pi$
$463$		−	814.484i		−	1.75914i	0.475765	−	0.879572i	$-0.342171\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	607.118			1.30004			0.650019	−	0.759918i	$-0.274761\pi$
$467$	607.118			1.30004			0.650019	+	0.759918i	$0.274761\pi$
$468$		0			0
$469$	−72.4104			−0.154393
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−631.186			−1.33443
$474$		0			0
$475$	24.8162	+	18.6883i	0.0522447	+	0.0393438i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			343.889i			0.717931i	0.933351	+	0.358965i	$0.116870\pi$
$479$			343.889i			0.717931i	−0.933351	+	0.358965i	$0.883130\pi$
$480$		0			0
$481$	414.323			0.861378
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−429.754	−	143.720i	−0.886092	−	0.296329i
$486$		0			0
$487$			300.759i			0.617576i	0.951131	+	0.308788i	$0.0999233\pi$
$487$			300.759i			0.617576i	−0.951131	+	0.308788i	$0.900077\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			173.914i			0.354203i	0.984193	+	0.177101i	$0.0566720\pi$
$491$			173.914i			0.354203i	−0.984193	+	0.177101i	$0.943328\pi$
$492$		0			0
$493$			550.509i			1.11665i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−56.7048			−0.114094
$498$		0			0
$499$	761.610			1.52627			0.763136	−	0.646237i	$-0.223658\pi$
$499$	761.610			1.52627			0.763136	+	0.646237i	$0.223658\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	280.632			0.557917			0.278959	−	0.960303i	$-0.410011\pi$
$503$	280.632			0.557917			0.278959	+	0.960303i	$0.410011\pi$
$504$		0			0
$505$	235.765	+	78.8450i	0.466860	+	0.156129i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	633.176i		−	1.24396i	−0.783032	−	0.621981i	$-0.786328\pi$
$509$		−	633.176i		−	1.24396i	0.783032	−	0.621981i	$-0.213672\pi$
$510$		0			0
$511$	−72.9117			−0.142684
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−610.960	−	204.319i	−1.18633	−	0.396736i
$516$		0			0
$517$			1227.00i			2.37331i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			92.3966i			0.177345i	0.996061	+	0.0886723i	$0.0282624\pi$
$521$			92.3966i			0.177345i	−0.996061	+	0.0886723i	$0.971738\pi$
$522$		0			0
$523$			818.552i			1.56511i	0.622582	+	0.782554i	$0.286084\pi$
$523$			818.552i			1.56511i	−0.622582	+	0.782554i	$0.713916\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	54.3482			0.103128
$528$		0			0
$529$	−505.970			−0.956464
$530$		0			0
$531$		0			0
$532$		0			0
$533$	41.5004			0.0778620
$534$		0			0
$535$	−10.1909	+	30.4732i	−0.0190484	+	0.0569592i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			743.254i			1.37895i
$540$		0			0
$541$	−376.985			−0.696830			−0.348415	−	0.937340i	$-0.613280\pi$
$541$	−376.985			−0.696830			−0.348415	+	0.937340i	$0.613280\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	171.708	−	513.445i	0.315060	−	0.942102i
$546$		0			0
$547$		−	337.346i		−	0.616721i	−0.951270	−	0.308360i	$-0.900220\pi$
$547$		−	337.346i		−	0.616721i	0.951270	−	0.308360i	$-0.0997803\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			53.0319i			0.0962467i
$552$		0			0
$553$			109.097i			0.197282i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	303.338			0.544593			0.272296	−	0.962213i	$-0.412217\pi$
$557$	303.338			0.544593			0.272296	+	0.962213i	$0.412217\pi$
$558$		0			0
$559$	−241.882			−0.432705
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−234.093			−0.415796			−0.207898	−	0.978150i	$-0.566662\pi$
$563$	−234.093			−0.415796			−0.207898	+	0.978150i	$0.566662\pi$
$564$		0			0
$565$	−48.2498	+	144.278i	−0.0853978	+	0.255359i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	66.5257i		−	0.116917i	−0.998290	−	0.0584584i	$-0.981381\pi$
$569$		−	66.5257i		−	0.116917i	0.998290	−	0.0584584i	$-0.0186185\pi$
$570$		0			0
$571$	−1115.54			−1.95365			−0.976826	−	0.214035i	$-0.931339\pi$
$571$	−1115.54			−1.95365			−0.976826	+	0.214035i	$0.931339\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−95.8385	−	72.1728i	−0.166676	−	0.125518i
$576$		0			0
$577$		−	957.356i		−	1.65920i	−0.558361	−	0.829598i	$-0.688570\pi$
$577$		−	957.356i		−	1.65920i	0.558361	−	0.829598i	$-0.311430\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			88.6996i			0.152667i
$582$		0			0
$583$			979.891i			1.68077i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	79.8974			0.136111			0.0680557	−	0.997682i	$-0.478320\pi$
$587$	79.8974			0.136111			0.0680557	+	0.997682i	$0.478320\pi$
$588$		0			0
$589$	5.23550			0.00888879
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−213.831			−0.360593			−0.180296	−	0.983612i	$-0.557706\pi$
$593$	−213.831			−0.360593			−0.180296	+	0.983612i	$0.557706\pi$
$594$		0			0
$595$	49.7645	+	16.6424i	0.0836378	+	0.0279704i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	915.493i		−	1.52837i	−0.644998	−	0.764184i	$-0.723142\pi$
$599$		−	915.493i		−	1.52837i	0.644998	−	0.764184i	$-0.276858\pi$
$600$		0			0
$601$	479.426			0.797714			0.398857	−	0.917013i	$-0.369407\pi$
$601$	479.426			0.797714			0.398857	+	0.917013i	$0.369407\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−183.042	+	547.336i	−0.302548	+	0.904688i
$606$		0			0
$607$		−	219.893i		−	0.362261i	−0.983459	−	0.181131i	$-0.942024\pi$
$607$		−	219.893i		−	0.362261i	0.983459	−	0.181131i	$-0.0579757\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			470.210i			0.769575i
$612$		0			0
$613$			862.066i			1.40631i	0.711038	+	0.703154i	$0.248225\pi$
$613$			862.066i			1.40631i	−0.711038	+	0.703154i	$0.751775\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	695.423			1.12710			0.563552	−	0.826080i	$-0.309434\pi$
$617$	695.423			1.12710			0.563552	+	0.826080i	$0.309434\pi$
$618$		0			0
$619$	512.073			0.827259			0.413629	−	0.910445i	$-0.364261\pi$
$619$	512.073			0.827259			0.413629	+	0.910445i	$0.364261\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	111.878			0.179580
$624$		0			0
$625$	172.647	+	600.681i	0.276235	+	0.961090i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			907.020i			1.44200i
$630$		0			0
$631$	643.375			1.01961			0.509806	−	0.860290i	$-0.329717\pi$
$631$	643.375			1.01961			0.509806	+	0.860290i	$0.329717\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−634.108	−	212.060i	−0.998595	−	0.333953i
$636$		0			0
$637$			284.829i			0.447141i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			196.925i			0.307215i	0.988132	+	0.153608i	$0.0490892\pi$
$641$			196.925i			0.307215i	−0.988132	+	0.153608i	$0.950911\pi$
$642$		0			0
$643$			236.954i			0.368513i	0.982878	+	0.184256i	$0.0589877\pi$
$643$			236.954i			0.368513i	−0.982878	+	0.184256i	$0.941012\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−17.3818			−0.0268653			−0.0134326	−	0.999910i	$-0.504276\pi$
$647$	−17.3818			−0.0268653			−0.0134326	+	0.999910i	$0.504276\pi$
$648$		0			0
$649$	563.456			0.868191
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−785.704			−1.20322			−0.601611	−	0.798789i	$-0.705474\pi$
$653$	−785.704			−1.20322			−0.601611	+	0.798789i	$0.705474\pi$
$654$		0			0
$655$	292.971	+	97.9760i	0.447283	+	0.149582i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	498.767i		−	0.756855i	−0.925631	−	0.378427i	$-0.876465\pi$
$659$		−	498.767i		−	0.756855i	0.925631	−	0.378427i	$-0.123535\pi$
$660$		0			0
$661$	303.133			0.458597			0.229299	−	0.973356i	$-0.426357\pi$
$661$	303.133			0.458597			0.229299	+	0.973356i	$0.426357\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.79394	+	1.60320i	0.00720893	+	0.00241083i
$666$		0			0
$667$		−	204.805i		−	0.307055i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1275.32i		−	1.90062i
$672$		0			0
$673$			463.133i			0.688163i	0.938940	+	0.344081i	$0.111810\pi$
$673$			463.133i			0.688163i	−0.938940	+	0.344081i	$0.888190\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	311.064			0.459474			0.229737	−	0.973253i	$-0.426213\pi$
$677$	311.064			0.459474			0.229737	+	0.973253i	$0.426213\pi$
$678$		0			0
$679$	−73.7342			−0.108592
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−232.009			−0.339691			−0.169846	−	0.985471i	$-0.554327\pi$
$683$	−232.009			−0.339691			−0.169846	+	0.985471i	$0.554327\pi$
$684$		0			0
$685$	−259.940	+	777.281i	−0.379475	+	1.13472i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			375.513i			0.545011i
$690$		0			0
$691$	552.904			0.800150			0.400075	−	0.916482i	$-0.368984\pi$
$691$	552.904			0.800150			0.400075	+	0.916482i	$0.368984\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−289.009	+	864.204i	−0.415841	+	1.24346i
$696$		0			0
$697$			90.8512i			0.130346i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			250.274i			0.357024i	0.983938	+	0.178512i	$0.0571284\pi$
$701$			250.274i			0.357024i	−0.983938	+	0.178512i	$0.942872\pi$
$702$		0			0
$703$			87.3755i			0.124290i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	40.4508			0.0572147
$708$		0			0
$709$	1170.57			1.65102			0.825510	−	0.564388i	$-0.190888\pi$
$709$	1170.57			1.65102			0.825510	+	0.564388i	$0.190888\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−20.2191			−0.0283578
$714$		0			0
$715$	−143.677	+	429.627i	−0.200947	+	0.600877i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			168.127i			0.233834i	0.993142	+	0.116917i	$0.0373012\pi$
$719$			168.127i			0.233834i	−0.993142	+	0.116917i	$0.962699\pi$
$720$		0			0
$721$	−104.824			−0.145387
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−641.823	+	852.279i	−0.885274	+	1.17556i
$726$		0			0
$727$		−	382.835i		−	0.526595i	−0.964715	−	0.263298i	$-0.915190\pi$
$727$		−	382.835i		−	0.526595i	0.964715	−	0.263298i	$-0.0848101\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	529.520i		−	0.724377i
$732$		0			0
$733$			1403.08i			1.91416i	0.289819	+	0.957082i	$0.406405\pi$
$733$			1403.08i			1.91416i	−0.289819	+	0.957082i	$0.593595\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1368.52			1.85688
$738$		0			0
$739$	−899.125			−1.21668			−0.608339	−	0.793677i	$-0.708164\pi$
$739$	−899.125			−1.21668			−0.608339	+	0.793677i	$0.708164\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	316.587			0.426093			0.213046	−	0.977042i	$-0.431661\pi$
$743$	316.587			0.426093			0.213046	+	0.977042i	$0.431661\pi$
$744$		0			0
$745$	−898.749	−	300.562i	−1.20637	−	0.403439i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			5.22837i			0.00698046i
$750$		0			0
$751$	286.433			0.381402			0.190701	−	0.981648i	$-0.438924\pi$
$751$	286.433			0.381402			0.190701	+	0.981648i	$0.438924\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	179.730	−	537.434i	0.238053	−	0.711833i
$756$		0			0
$757$			783.221i			1.03464i	0.855793	+	0.517319i	$0.173070\pi$
$757$			783.221i			1.03464i	−0.855793	+	0.517319i	$0.826930\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1031.91i		−	1.35599i	−0.735064	−	0.677997i	$-0.762848\pi$
$761$		−	1031.91i		−	1.35599i	0.735064	−	0.677997i	$-0.237152\pi$
$762$		0			0
$763$		−	88.0933i		−	0.115456i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	215.927			0.281521
$768$		0			0
$769$	−831.250			−1.08095			−0.540475	−	0.841360i	$-0.681755\pi$
$769$	−831.250			−1.08095			−0.540475	+	0.841360i	$0.681755\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−395.360			−0.511462			−0.255731	−	0.966748i	$-0.582316\pi$
$773$	−395.360			−0.511462			−0.255731	+	0.966748i	$0.582316\pi$
$774$		0			0
$775$	84.1400	+	63.3631i	0.108568	+	0.0817588i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			8.75192i			0.0112348i
$780$		0			0
$781$	1071.69			1.37220
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−536.725	−	179.493i	−0.683726	−	0.228653i
$786$		0			0
$787$			573.929i			0.729262i	0.931152	+	0.364631i	$0.118805\pi$
$787$			573.929i			0.729262i	−0.931152	+	0.364631i	$0.881195\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			24.7542i			0.0312948i
$792$		0			0
$793$		−	488.725i		−	0.616299i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1408.39			1.76712			0.883560	−	0.468319i	$-0.155140\pi$
$797$	1408.39			1.76712			0.883560	+	0.468319i	$0.155140\pi$
$798$		0			0
$799$	−1029.37			−1.28832
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1377.99			1.71606
$804$		0			0
$805$	−18.5139	−	6.19145i	−0.0229986	−	0.00769124i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	105.645i		−	0.130587i	−0.997866	−	0.0652936i	$-0.979202\pi$
$809$		−	105.645i		−	0.130587i	0.997866	−	0.0652936i	$-0.0207984\pi$
$810$		0			0
$811$	1090.81			1.34502			0.672508	−	0.740090i	$-0.265217\pi$
$811$	1090.81			1.34502			0.672508	+	0.740090i	$0.265217\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−722.725	−	241.696i	−0.886779	−	0.296559i
$816$		0			0
$817$		−	51.0099i		−	0.0624357i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	851.302i		−	1.03691i	−0.855105	−	0.518454i	$-0.826508\pi$
$821$		−	851.302i		−	1.03691i	0.855105	−	0.518454i	$-0.173492\pi$
$822$		0			0
$823$			1120.12i			1.36103i	0.732736	+	0.680513i	$0.238243\pi$
$823$			1120.12i			1.36103i	−0.732736	+	0.680513i	$0.761757\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1214.63			1.46871			0.734357	−	0.678763i	$-0.237484\pi$
$827$	1214.63			1.46871			0.734357	+	0.678763i	$0.237484\pi$
$828$		0			0
$829$	1495.22			1.80364			0.901821	−	0.432111i	$-0.142231\pi$
$829$	1495.22			1.80364			0.901821	+	0.432111i	$0.142231\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−623.537			−0.748544
$834$		0			0
$835$	−167.860	+	501.940i	−0.201030	+	0.601125i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			463.865i			0.552879i	0.961031	+	0.276439i	$0.0891545\pi$
$839$			463.865i			0.552879i	−0.961031	+	0.276439i	$0.910846\pi$
$840$		0			0
$841$	−980.308			−1.16565
$842$		0			0
$843$		0			0
$844$		0			0
$845$	212.938	−	636.734i	0.251998	−	0.753531i
$846$		0			0
$847$			93.9081i			0.110871i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	337.438i		−	0.396519i
$852$		0			0
$853$			495.850i			0.581301i	0.956829	+	0.290651i	$0.0938717\pi$
$853$			495.850i			0.581301i	−0.956829	+	0.290651i	$0.906128\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1704.43			1.98884			0.994418	−	0.105511i	$-0.0336478\pi$
$857$	1704.43			1.98884			0.994418	+	0.105511i	$0.0336478\pi$
$858$		0			0
$859$	207.523			0.241586			0.120793	−	0.992678i	$-0.461456\pi$
$859$	207.523			0.241586			0.120793	+	0.992678i	$0.461456\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−117.426			−0.136068			−0.0680338	−	0.997683i	$-0.521673\pi$
$863$	−117.426			−0.136068			−0.0680338	+	0.997683i	$0.521673\pi$
$864$		0			0
$865$	174.456	−	521.663i	0.201683	−	0.603079i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	2061.87i		−	2.37270i
$870$		0			0
$871$	524.442			0.602115
$872$		0			0
$873$		0			0
$874$		0			0
$875$	57.6408	+	83.7842i	0.0658752	+	0.0957534i
$876$		0			0
$877$		−	908.637i		−	1.03607i	−0.855358	−	0.518037i	$-0.826663\pi$
$877$		−	908.637i		−	1.03607i	0.855358	−	0.518037i	$-0.173337\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1226.54i			1.39221i	0.717941	+	0.696104i	$0.245085\pi$
$881$			1226.54i			1.39221i	−0.717941	+	0.696104i	$0.754915\pi$
$882$		0			0
$883$		−	760.022i		−	0.860728i	−0.902656	−	0.430364i	$-0.858385\pi$
$883$		−	760.022i		−	0.860728i	0.902656	−	0.430364i	$-0.141615\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	597.542			0.673667			0.336833	−	0.941564i	$-0.390644\pi$
$887$	597.542			0.673667			0.336833	+	0.941564i	$0.390644\pi$
$888$		0			0
$889$	−108.796			−0.122380
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−99.1615			−0.111043
$894$		0			0
$895$	561.470	+	187.768i	0.627341	+	0.209797i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			179.806i			0.200007i
$900$		0			0
$901$	−822.058			−0.912384
$902$		0			0
$903$		0			0
$904$		0			0
$905$	273.385	−	817.483i	0.302083	−	0.903296i
$906$		0			0
$907$			1708.26i			1.88341i	0.336435	+	0.941707i	$0.390779\pi$
$907$			1708.26i			1.88341i	−0.336435	+	0.941707i	$0.609221\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1432.15i			1.57206i	0.618187	+	0.786031i	$0.287868\pi$
$911$			1432.15i			1.57206i	−0.618187	+	0.786031i	$0.712132\pi$
$912$		0			0
$913$		−	1676.38i		−	1.83612i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	50.2658			0.0548155
$918$		0			0
$919$	244.398			0.265939			0.132969	−	0.991120i	$-0.457549\pi$
$919$	244.398			0.265939			0.132969	+	0.991120i	$0.457549\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	410.692			0.444954
$924$		0			0
$925$	−1057.47	+	1404.22i	−1.14321	+	1.51807i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			376.003i			0.404740i	0.979309	+	0.202370i	$0.0648643\pi$
$929$			376.003i			0.404740i	−0.979309	+	0.202370i	$0.935136\pi$
$930$		0			0
$931$	−60.0669			−0.0645187
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−940.524	−	314.532i	−1.00591	−	0.336398i
$936$		0			0
$937$			412.076i			0.439782i	0.975524	+	0.219891i	$0.0705701\pi$
$937$			412.076i			0.439782i	−0.975524	+	0.219891i	$0.929430\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	334.579i		−	0.355556i	−0.984071	−	0.177778i	$-0.943109\pi$
$941$		−	334.579i		−	0.355556i	0.984071	−	0.177778i	$-0.0568910\pi$
$942$		0			0
$943$		−	33.7993i		−	0.0358423i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−611.132			−0.645335			−0.322667	−	0.946512i	$-0.604580\pi$
$947$	−611.132			−0.645335			−0.322667	+	0.946512i	$0.604580\pi$
$948$		0			0
$949$	528.073			0.556452
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1445.17			−1.51644			−0.758221	−	0.651997i	$-0.773931\pi$
$953$	−1445.17			−1.51644			−0.758221	+	0.651997i	$0.773931\pi$
$954$		0			0
$955$	1372.26	+	458.916i	1.43693	+	0.480540i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			133.360i			0.139062i
$960$		0			0
$961$	−943.249			−0.981529
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−606.167	−	202.716i	−0.628152	−	0.210068i
$966$		0			0
$967$			1183.07i			1.22344i	0.791073	+	0.611721i	$0.209523\pi$
$967$			1183.07i			1.22344i	−0.791073	+	0.611721i	$0.790477\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1444.94i			1.48810i	0.668124	+	0.744050i	$0.267097\pi$
$971$			1444.94i			1.48810i	−0.668124	+	0.744050i	$0.732903\pi$
$972$		0			0
$973$			148.274i			0.152388i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1246.51			−1.27586			−0.637929	−	0.770095i	$-0.720209\pi$
$977$	−1246.51			−1.27586			−0.637929	+	0.770095i	$0.720209\pi$
$978$		0			0
$979$	−2114.44			−2.15979
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−144.005			−0.146495			−0.0732477	−	0.997314i	$-0.523336\pi$
$983$	−144.005			−0.146495			−0.0732477	+	0.997314i	$0.523336\pi$
$984$		0			0
$985$	444.677	−	1329.69i	0.451449	−	1.34994i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			196.997i			0.199188i
$990$		0			0
$991$	−725.742			−0.732333			−0.366167	−	0.930549i	$-0.619330\pi$
$991$	−725.742			−0.732333			−0.366167	+	0.930549i	$0.619330\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−248.059	+	741.753i	−0.249305	+	0.745480i
$996$		0			0
$997$			881.323i			0.883975i	0.897021	+	0.441987i	$0.145726\pi$
$997$			881.323i			0.883975i	−0.897021	+	0.441987i	$0.854274\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.3.c.m.1889.1		4
3.2	odd	2		2160.3.c.g.1889.4		4
4.3	odd	2		270.3.b.d.269.1	yes	4
5.4	even	2		2160.3.c.g.1889.3		4
12.11	even	2		270.3.b.a.269.4	yes	4
15.14	odd	2	inner	2160.3.c.m.1889.2		4
20.3	even	4		1350.3.d.o.701.7		8
20.7	even	4		1350.3.d.o.701.2		8
20.19	odd	2		270.3.b.a.269.3	✓	4
36.7	odd	6		810.3.j.a.539.4		8
36.11	even	6		810.3.j.f.539.1		8
36.23	even	6		810.3.j.f.269.2		8
36.31	odd	6		810.3.j.a.269.3		8
60.23	odd	4		1350.3.d.o.701.3		8
60.47	odd	4		1350.3.d.o.701.6		8
60.59	even	2		270.3.b.d.269.2	yes	4
180.59	even	6		810.3.j.a.269.4		8
180.79	odd	6		810.3.j.f.539.2		8
180.119	even	6		810.3.j.a.539.3		8
180.139	odd	6		810.3.j.f.269.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.3.b.a.269.3	✓	4	20.19	odd	2
270.3.b.a.269.4	yes	4	12.11	even	2
270.3.b.d.269.1	yes	4	4.3	odd	2
270.3.b.d.269.2	yes	4	60.59	even	2
810.3.j.a.269.3		8	36.31	odd	6
810.3.j.a.269.4		8	180.59	even	6
810.3.j.a.539.3		8	180.119	even	6
810.3.j.a.539.4		8	36.7	odd	6
810.3.j.f.269.1		8	180.139	odd	6
810.3.j.f.269.2		8	36.23	even	6
810.3.j.f.539.1		8	36.11	even	6
810.3.j.f.539.2		8	180.79	odd	6
1350.3.d.o.701.2		8	20.7	even	4
1350.3.d.o.701.3		8	60.23	odd	4
1350.3.d.o.701.6		8	60.47	odd	4
1350.3.d.o.701.7		8	20.3	even	4
2160.3.c.g.1889.3		4	5.4	even	2
2160.3.c.g.1889.4		4	3.2	odd	2
2160.3.c.m.1889.1		4	1.1	even	1	trivial
2160.3.c.m.1889.2		4	15.14	odd	2	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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.58579 - 4.74186i) q^{5} -0.813575i q^{7} +15.3762i q^{11} +5.89243i q^{13} -12.8995 q^{17} -1.24264 q^{19} +4.79899 q^{23} +(-19.9706 - 15.0392i) q^{25} -42.6768i q^{29} -4.21320 q^{31} +(-3.85786 - 1.29016i) q^{35} -70.3144i q^{37} -7.04300i q^{41} +41.0496i q^{43} +79.7990 q^{47} +48.3381 q^{49} +63.7279 q^{53} +(72.9117 + 24.3833i) q^{55} -36.6448i q^{59} -82.9411 q^{61} +(27.9411 + 9.34414i) q^{65} -89.0027i q^{67} -69.6982i q^{71} -89.6188i q^{73} +12.5097 q^{77} -134.095 q^{79} -109.024 q^{83} +(-20.4558 + 61.1677i) q^{85} +137.514i q^{89} +4.79394 q^{91} +(-1.97056 + 5.89243i) q^{95} -90.6298i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{5} - 12 q^{17} + 12 q^{19} - 60 q^{23} - 12 q^{25} + 68 q^{31} - 72 q^{35} + 240 q^{47} - 180 q^{49} + 204 q^{53} + 88 q^{55} - 196 q^{61} - 24 q^{65} - 312 q^{77} - 180 q^{79} - 108 q^{83}+ \cdots + 60 q^{95}+O(q^{100}) \) 4 * q + 12 * q^5 - 12 * q^17 + 12 * q^19 - 60 * q^23 - 12 * q^25 + 68 * q^31 - 72 * q^35 + 240 * q^47 - 180 * q^49 + 204 * q^53 + 88 * q^55 - 196 * q^61 - 24 * q^65 - 312 * q^77 - 180 * q^79 - 108 * q^83 + 20 * q^85 - 456 * q^91 + 60 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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