LMFDB - Embedded newform 1764.3.bk.g.1745.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(557,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("1764.557");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.bk (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:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	16.0.329365073333488765586374656.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 199x^{12} + 24960x^{8} - 2913559x^{4} + 214358881 $ x^16 - 199x^12 + 24960x^8 - 2913559*x^4 + 214358881
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1745.6
Root			$1.33156 + 3.03759i$ of defining polynomial
Character	$\chi$	$=$	1764.1745
Dual form			1764.3.bk.g.557.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.08083 + 1.77872i) q^{5} +O(q^{10})$	$q+(3.08083 + 1.77872i) q^{5} +(8.03119 - 4.63681i) q^{11} +4.24264 q^{13} +(2.11532 - 1.22128i) q^{17} +(10.3749 - 17.9698i) q^{19} +(0.682720 + 0.394169i) q^{23} +(-6.17232 - 10.6908i) q^{25} +7.69694i q^{29} +(11.7891 + 20.4193i) q^{31} +(11.6723 - 20.2170i) q^{37} -51.5574i q^{41} -49.3446 q^{43} +(-13.6574 - 7.88512i) q^{47} +(65.1929 - 37.6391i) q^{53} +32.9903 q^{55} +(13.6574 - 7.88512i) q^{59} +(-4.01093 + 6.94714i) q^{61} +(13.0709 + 7.54647i) q^{65} +(43.6723 + 75.6427i) q^{67} +40.0614i q^{71} +(17.2023 + 29.7952i) q^{73} +(-4.34463 + 7.52512i) q^{79} -115.115i q^{83} +8.68926 q^{85} +(-81.6200 - 47.1234i) q^{89} +(63.9266 - 36.9080i) q^{95} +154.173 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 216 q^{25} - 128 q^{37} - 160 q^{43} + 384 q^{67} + 560 q^{79} - 1120 q^{85}+O(q^{100}) $ 16 * q + 216 * q^25 - 128 * q^37 - 160 * q^43 + 384 * q^67 + 560 * q^79 - 1120 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	3.08083	+	1.77872i	0.616166	+	0.355744i	0.775375	−	0.631501i	$-0.217561\pi$
$5$	3.08083	+	1.77872i	0.616166	+	0.355744i	−0.159209	+	0.987245i	$0.550894\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	8.03119	−	4.63681i	0.730108	−	0.421528i	−0.0883536	−	0.996089i	$-0.528161\pi$
$11$	8.03119	−	4.63681i	0.730108	−	0.421528i	0.818462	+	0.574561i	$0.194827\pi$
$12$		0			0
$13$	4.24264			0.326357			0.163178	−	0.986597i	$-0.447825\pi$
$13$	4.24264			0.326357			0.163178	+	0.986597i	$0.447825\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.11532	−	1.22128i	0.124431	−	0.0718400i	−0.436493	−	0.899708i	$-0.643780\pi$
$17$	2.11532	−	1.22128i	0.124431	−	0.0718400i	0.560923	+	0.827868i	$0.310446\pi$
$18$		0			0
$19$	10.3749	−	17.9698i	0.546047	−	0.945781i	−0.452493	−	0.891768i	$-0.649465\pi$
$19$	10.3749	−	17.9698i	0.546047	−	0.945781i	0.998540	−	0.0540134i	$-0.0172014\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.682720	+	0.394169i	0.0296835	+	0.0171378i	0.514768	−	0.857329i	$-0.327878\pi$
$23$	0.682720	+	0.394169i	0.0296835	+	0.0171378i	−0.485085	+	0.874467i	$0.661211\pi$
$24$		0			0
$25$	−6.17232	−	10.6908i	−0.246893	−	0.427631i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			7.69694i			0.265412i	0.991155	+	0.132706i	$0.0423666\pi$
$29$			7.69694i			0.265412i	−0.991155	+	0.132706i	$0.957633\pi$
$30$		0			0
$31$	11.7891	+	20.4193i	0.380294	+	0.658688i	0.991104	−	0.133089i	$-0.0424895\pi$
$31$	11.7891	+	20.4193i	0.380294	+	0.658688i	−0.610810	+	0.791777i	$0.709156\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	11.6723	−	20.2170i	0.315468	−	0.546407i	−0.664069	−	0.747671i	$-0.731172\pi$
$37$	11.6723	−	20.2170i	0.315468	−	0.546407i	0.979537	+	0.201265i	$0.0645052\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	51.5574i		−	1.25750i	−0.777608	−	0.628749i	$-0.783567\pi$
$41$		−	51.5574i		−	1.25750i	0.777608	−	0.628749i	$-0.216433\pi$
$42$		0			0
$43$	−49.3446			−1.14755			−0.573775	−	0.819013i	$-0.694522\pi$
$43$	−49.3446			−1.14755			−0.573775	+	0.819013i	$0.694522\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−13.6574	−	7.88512i	−0.290584	−	0.167769i	0.347621	−	0.937635i	$-0.386989\pi$
$47$	−13.6574	−	7.88512i	−0.290584	−	0.167769i	−0.638205	+	0.769866i	$0.720323\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	65.1929	−	37.6391i	1.23005	−	0.710172i	0.263013	−	0.964792i	$-0.415284\pi$
$53$	65.1929	−	37.6391i	1.23005	−	0.710172i	0.967041	+	0.254620i	$0.0819505\pi$
$54$		0			0
$55$	32.9903			0.599824
$56$		0			0
$57$		0			0
$58$		0			0
$59$	13.6574	−	7.88512i	0.231482	−	0.133646i	−0.379774	−	0.925079i	$-0.623998\pi$
$59$	13.6574	−	7.88512i	0.231482	−	0.133646i	0.611256	+	0.791433i	$0.290665\pi$
$60$		0			0
$61$	−4.01093	+	6.94714i	−0.0657530	+	0.113888i	−0.897028	−	0.441974i	$-0.854278\pi$
$61$	−4.01093	+	6.94714i	−0.0657530	+	0.113888i	0.831275	+	0.555862i	$0.187612\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	13.0709	+	7.54647i	0.201090	+	0.116099i
$66$		0			0
$67$	43.6723	+	75.6427i	0.651826	+	1.12900i	0.982680	+	0.185313i	$0.0593299\pi$
$67$	43.6723	+	75.6427i	0.651826	+	1.12900i	−0.330854	+	0.943682i	$0.607337\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			40.0614i			0.564245i	0.959378	+	0.282123i	$0.0910385\pi$
$71$			40.0614i			0.564245i	−0.959378	+	0.282123i	$0.908962\pi$
$72$		0			0
$73$	17.2023	+	29.7952i	0.235648	+	0.408154i	0.959461	−	0.281843i	$-0.0909456\pi$
$73$	17.2023	+	29.7952i	0.235648	+	0.408154i	−0.723813	+	0.689996i	$0.757612\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.34463	+	7.52512i	−0.0549953	+	0.0952547i	−0.892212	−	0.451616i	$-0.850848\pi$
$79$	−4.34463	+	7.52512i	−0.0549953	+	0.0952547i	0.837217	+	0.546871i	$0.184181\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	115.115i		−	1.38693i	−0.720492	−	0.693463i	$-0.756084\pi$
$83$		−	115.115i		−	1.38693i	0.720492	−	0.693463i	$-0.243916\pi$
$84$		0			0
$85$	8.68926			0.102227
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−81.6200	−	47.1234i	−0.917079	−	0.529476i	−0.0343770	−	0.999409i	$-0.510945\pi$
$89$	−81.6200	−	47.1234i	−0.917079	−	0.529476i	−0.882702	+	0.469933i	$0.844278\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	63.9266	−	36.9080i	0.672912	−	0.388506i
$96$		0			0
$97$	154.173			1.58941			0.794707	−	0.606993i	$-0.207624\pi$
$97$	154.173			1.58941			0.794707	+	0.606993i	$0.207624\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	165.724	−	95.6808i	1.64083	−	0.947335i	0.660293	−	0.751008i	$-0.270432\pi$
$101$	165.724	−	95.6808i	1.64083	−	0.947335i	0.980539	−	0.196326i	$-0.0629012\pi$
$102$		0			0
$103$	40.5368	−	70.2118i	0.393561	−	0.681668i	−0.599355	−	0.800483i	$-0.704576\pi$
$103$	40.5368	−	70.2118i	0.393561	−	0.681668i	0.992916	+	0.118815i	$0.0379096\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−143.879	−	83.0684i	−1.34466	−	0.776340i	−0.357173	−	0.934038i	$-0.616259\pi$
$107$	−143.879	−	83.0684i	−1.34466	−	0.776340i	−0.987487	+	0.157698i	$0.949593\pi$
$108$		0			0
$109$	64.3446	+	111.448i	0.590318	+	1.02246i	0.994189	+	0.107645i	$0.0343309\pi$
$109$	64.3446	+	111.448i	0.590318	+	1.02246i	−0.403872	+	0.914816i	$0.632336\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			188.138i			1.66494i	0.554069	+	0.832471i	$0.313074\pi$
$113$			188.138i			1.66494i	−0.554069	+	0.832471i	$0.686926\pi$
$114$		0			0
$115$	1.40223	+	2.42873i	0.0121933	+	0.0211194i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−17.5000	+	30.3109i	−0.144628	+	0.250503i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	132.851i		−	1.06281i
$126$		0			0
$127$	−40.6554			−0.320121			−0.160061	−	0.987107i	$-0.551169\pi$
$127$	−40.6554			−0.320121			−0.160061	+	0.987107i	$0.551169\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−83.1384	−	48.0000i	−0.634645	−	0.366412i	0.147904	−	0.989002i	$-0.452747\pi$
$131$	−83.1384	−	48.0000i	−0.634645	−	0.366412i	−0.782549	+	0.622590i	$0.786081\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	100.830	−	58.2145i	0.735988	−	0.424923i	−0.0846205	−	0.996413i	$-0.526968\pi$
$137$	100.830	−	58.2145i	0.735988	−	0.424923i	0.820609	+	0.571490i	$0.193634\pi$
$138$		0			0
$139$	99.9218			0.718862			0.359431	−	0.933172i	$-0.382971\pi$
$139$	99.9218			0.718862			0.359431	+	0.933172i	$0.382971\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	34.0735	−	19.6723i	0.238276	−	0.137569i
$144$		0			0
$145$	−13.6907	+	23.7130i	−0.0944186	+	0.163538i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	94.0655	+	54.3087i	0.631312	+	0.364488i	0.781260	−	0.624206i	$-0.214577\pi$
$149$	94.0655	+	54.3087i	0.631312	+	0.364488i	−0.149948	+	0.988694i	$0.547911\pi$
$150$		0			0
$151$	112.017	+	194.019i	0.741834	+	1.28489i	0.951659	+	0.307155i	$0.0993771\pi$
$151$	112.017	+	194.019i	0.741834	+	1.28489i	−0.209825	+	0.977739i	$0.567290\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			83.8780i			0.541149i
$156$		0			0
$157$	52.5576	+	91.0325i	0.334762	+	0.579825i	0.983439	−	0.181239i	$-0.0580108\pi$
$157$	52.5576	+	91.0325i	0.334762	+	0.579825i	−0.648677	+	0.761064i	$0.724677\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	10.9831	−	19.0232i	0.0673807	−	0.116707i	−0.830367	−	0.557217i	$-0.811869\pi$
$163$	10.9831	−	19.0232i	0.0673807	−	0.116707i	0.897748	+	0.440510i	$0.145202\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	311.838i		−	1.86729i	−0.358195	−	0.933647i	$-0.616608\pi$
$167$		−	311.838i		−	1.86729i	0.358195	−	0.933647i	$-0.383392\pi$
$168$		0			0
$169$	−151.000			−0.893491
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−56.0079	−	32.3362i	−0.323745	−	0.186914i	0.329316	−	0.944220i	$-0.393182\pi$
$173$	−56.0079	−	32.3362i	−0.323745	−	0.186914i	−0.653061	+	0.757306i	$0.726515\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	289.483	−	167.133i	1.61722	−	0.933703i	0.629587	−	0.776930i	$-0.283224\pi$
$179$	289.483	−	167.133i	1.61722	−	0.933703i	0.987635	−	0.156773i	$-0.0501091\pi$
$180$		0			0
$181$	229.590			1.26845			0.634226	−	0.773147i	$-0.281319\pi$
$181$	229.590			1.26845			0.634226	+	0.773147i	$0.281319\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	71.9209	−	41.5235i	0.388762	−	0.224452i
$186$		0			0
$187$	11.3257	−	19.6167i	0.0605652	−	0.104902i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−65.4535	−	37.7896i	−0.342688	−	0.197851i	0.318772	−	0.947832i	$-0.396730\pi$
$191$	−65.4535	−	37.7896i	−0.342688	−	0.197851i	−0.661460	+	0.749980i	$0.730063\pi$
$192$		0			0
$193$	−4.68926	−	8.12204i	−0.0242967	−	0.0420831i	0.853621	−	0.520894i	$-0.174401\pi$
$193$	−4.68926	−	8.12204i	−0.0242967	−	0.0420831i	−0.877918	+	0.478811i	$0.841068\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	38.5566i		−	0.195719i	−0.995200	−	0.0978594i	$-0.968800\pi$
$197$		−	38.5566i		−	0.195719i	0.995200	−	0.0978594i	$-0.0311996\pi$
$198$		0			0
$199$	32.5509	+	56.3798i	0.163572	+	0.283315i	0.936147	−	0.351608i	$-0.114365\pi$
$199$	32.5509	+	56.3798i	0.163572	+	0.283315i	−0.772575	+	0.634923i	$0.781032\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	91.7062	−	158.840i	0.447347	−	0.774828i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	192.426i		−	0.920697i
$210$		0			0
$211$	−176.068			−0.834444			−0.417222	−	0.908804i	$-0.636996\pi$
$211$	−176.068			−0.834444			−0.417222	+	0.908804i	$0.636996\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−152.023	−	87.7702i	−0.707082	−	0.408234i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.97454	−	5.18146i	0.0406088	−	0.0234455i
$222$		0			0
$223$	−146.199			−0.655602			−0.327801	−	0.944747i	$-0.606308\pi$
$223$	−146.199			−0.655602			−0.327801	+	0.944747i	$0.606308\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	21.3815	−	12.3446i	0.0941918	−	0.0543816i	−0.452164	−	0.891935i	$-0.649348\pi$
$227$	21.3815	−	12.3446i	0.0941918	−	0.0543816i	0.546356	+	0.837553i	$0.316015\pi$
$228$		0			0
$229$	104.919	−	181.726i	0.458164	−	0.793563i	−0.540700	−	0.841215i	$-0.681841\pi$
$229$	104.919	−	181.726i	0.458164	−	0.793563i	0.998864	+	0.0476526i	$0.0151741\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	34.6942	+	20.0307i	0.148902	+	0.0859687i	0.572600	−	0.819835i	$-0.305935\pi$
$233$	34.6942	+	20.0307i	0.148902	+	0.0859687i	−0.423698	+	0.905804i	$0.639268\pi$
$234$		0			0
$235$	−28.0508	−	48.5855i	−0.119365	−	0.206747i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			229.662i			0.960928i	0.877014	+	0.480464i	$0.159532\pi$
$239$			229.662i			0.960928i	−0.877014	+	0.480464i	$0.840468\pi$
$240$		0			0
$241$	190.711	+	330.321i	0.791332	+	1.37063i	0.925142	+	0.379621i	$0.123946\pi$
$241$	190.711	+	330.321i	0.791332	+	1.37063i	−0.133810	+	0.991007i	$0.542721\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	44.0169	−	76.2396i	0.178206	−	0.308662i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			72.2636i			0.287903i	0.989585	+	0.143951i	$0.0459809\pi$
$251$			72.2636i			0.287903i	−0.989585	+	0.143951i	$0.954019\pi$
$252$		0			0
$253$	7.31074			0.0288962
$254$		0			0
$255$		0			0
$256$		0			0
$257$	192.899	+	111.370i	0.750578	+	0.433347i	0.825903	−	0.563813i	$-0.190666\pi$
$257$	192.899	+	111.370i	0.750578	+	0.433347i	−0.0753246	+	0.997159i	$0.523999\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	278.882	−	161.013i	1.06039	−	0.612215i	0.134848	−	0.990866i	$-0.456945\pi$
$263$	278.882	−	161.013i	1.06039	−	0.612215i	0.925540	+	0.378651i	$0.123612\pi$
$264$		0			0
$265$	267.798			1.01056
$266$		0			0
$267$		0			0
$268$		0			0
$269$	266.382	−	153.796i	0.990267	−	0.571731i	0.0849131	−	0.996388i	$-0.472939\pi$
$269$	266.382	−	153.796i	0.990267	−	0.571731i	0.905354	+	0.424657i	$0.139605\pi$
$270$		0			0
$271$	181.055	−	313.597i	0.668101	−	1.15718i	−0.310334	−	0.950628i	$-0.600441\pi$
$271$	181.055	−	313.597i	0.668101	−	1.15718i	0.978435	−	0.206557i	$-0.0662258\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−99.1421	−	57.2397i	−0.360517	−	0.208144i
$276$		0			0
$277$	−14.3446	−	24.8456i	−0.0517857	−	0.0896954i	0.838971	−	0.544177i	$-0.183158\pi$
$277$	−14.3446	−	24.8456i	−0.0517857	−	0.0896954i	−0.890756	+	0.454481i	$0.849825\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	89.7693i		−	0.319464i	−0.987160	−	0.159732i	$-0.948937\pi$
$281$		−	89.7693i		−	0.319464i	0.987160	−	0.159732i	$-0.0510629\pi$
$282$		0			0
$283$	−150.869	−	261.313i	−0.533107	−	0.923369i	−0.999252	−	0.0386609i	$-0.987691\pi$
$283$	−150.869	−	261.313i	−0.533107	−	0.923369i	0.466145	−	0.884708i	$-0.345643\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−141.517	+	245.115i	−0.489678	+	0.848147i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	235.855i		−	0.804966i	−0.915428	−	0.402483i	$-0.868147\pi$
$293$		−	235.855i		−	0.804966i	0.915428	−	0.402483i	$-0.131853\pi$
$294$		0			0
$295$	56.1017			0.190175
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.89654	+	1.67232i	0.00968741	+	0.00559303i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−24.7140	+	14.2686i	−0.0810296	+	0.0467824i
$306$		0			0
$307$	68.8810			0.224368			0.112184	−	0.993687i	$-0.464215\pi$
$307$	68.8810			0.224368			0.112184	+	0.993687i	$0.464215\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−432.931	+	249.953i	−1.39206	+	0.803707i	−0.993543	−	0.113453i	$-0.963809\pi$
$311$	−432.931	+	249.953i	−1.39206	+	0.803707i	−0.398518	+	0.917160i	$0.630475\pi$
$312$		0			0
$313$	−144.006	+	249.426i	−0.460083	+	0.796888i	−0.998965	−	0.0454940i	$-0.985514\pi$
$313$	−144.006	+	249.426i	−0.460083	+	0.796888i	0.538881	+	0.842382i	$0.318847\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	142.972	+	82.5451i	0.451017	+	0.260395i	0.708259	−	0.705952i	$-0.249481\pi$
$317$	142.972	+	82.5451i	0.451017	+	0.260395i	−0.257243	+	0.966347i	$0.582814\pi$
$318$		0			0
$319$	35.6893	+	61.8156i	0.111879	+	0.193779i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	50.6826i		−	0.156912i
$324$		0			0
$325$	−26.1869	−	45.3571i	−0.0805751	−	0.139560i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	132.655	−	229.766i	0.400772	−	0.694157i	−0.593048	−	0.805167i	$-0.702075\pi$
$331$	132.655	−	229.766i	0.400772	−	0.694157i	0.993819	+	0.111011i	$0.0354088\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			310.723i			0.927532i
$336$		0			0
$337$	220.655			0.654764			0.327382	−	0.944892i	$-0.393834\pi$
$337$	220.655			0.654764			0.327382	+	0.944892i	$0.393834\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	189.361	+	109.328i	0.555311	+	0.320609i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−162.151	+	93.6177i	−0.467293	+	0.269792i	−0.715106	−	0.699016i	$-0.753622\pi$
$347$	−162.151	+	93.6177i	−0.467293	+	0.269792i	0.247813	+	0.968808i	$0.420288\pi$
$348$		0			0
$349$	−318.270			−0.911948			−0.455974	−	0.889993i	$-0.650709\pi$
$349$	−318.270			−0.911948			−0.455974	+	0.889993i	$0.650709\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	146.502	−	84.5829i	0.415019	−	0.239611i	−0.277925	−	0.960603i	$-0.589647\pi$
$353$	146.502	−	84.5829i	0.415019	−	0.239611i	0.692944	+	0.720991i	$0.256313\pi$
$354$		0			0
$355$	−71.2580	+	123.422i	−0.200727	+	0.347669i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−96.2128	−	55.5485i	−0.268002	−	0.154731i	0.359977	−	0.932961i	$-0.382784\pi$
$359$	−96.2128	−	55.5485i	−0.268002	−	0.154731i	−0.627979	+	0.778230i	$0.716118\pi$
$360$		0			0
$361$	−34.7768	−	60.2353i	−0.0963347	−	0.166857i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			122.392i			0.335321i
$366$		0			0
$367$	−40.5248	−	70.1910i	−0.110422	−	0.191256i	0.805519	−	0.592571i	$-0.201887\pi$
$367$	−40.5248	−	70.1910i	−0.110422	−	0.191256i	−0.915940	+	0.401314i	$0.868554\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−219.723	+	380.572i	−0.589070	+	1.02030i	0.405284	+	0.914191i	$0.367172\pi$
$373$	−219.723	+	380.572i	−0.589070	+	1.02030i	−0.994355	+	0.106109i	$0.966161\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			32.6554i			0.0866190i
$378$		0			0
$379$	−81.3785			−0.214719			−0.107360	−	0.994220i	$-0.534240\pi$
$379$	−81.3785			−0.214719			−0.107360	+	0.994220i	$0.534240\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−399.226	−	230.493i	−1.04237	−	0.601810i	−0.121864	−	0.992547i	$-0.538887\pi$
$383$	−399.226	−	230.493i	−1.04237	−	0.601810i	−0.920503	+	0.390736i	$0.872220\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−480.866	+	277.628i	−1.23616	+	0.713697i	−0.968307	−	0.249764i	$-0.919647\pi$
$389$	−480.866	+	277.628i	−1.23616	+	0.713697i	−0.267852	+	0.963460i	$0.586314\pi$
$390$		0			0
$391$	1.92556			0.00492471
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−26.7702	+	15.4558i	−0.0677726	+	0.0391285i
$396$		0			0
$397$	−152.064	+	263.382i	−0.383033	+	0.663432i	−0.991494	−	0.130152i	$-0.958454\pi$
$397$	−152.064	+	263.382i	−0.383033	+	0.663432i	0.608462	+	0.793583i	$0.291787\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	213.465	+	123.244i	0.532333	+	0.307342i	0.741966	−	0.670438i	$-0.233894\pi$
$401$	213.465	+	123.244i	0.532333	+	0.307342i	−0.209633	+	0.977780i	$0.567227\pi$
$402$		0			0
$403$	50.0169	+	86.6319i	0.124112	+	0.214967i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	216.489i		−	0.531915i
$408$		0			0
$409$	7.30278	+	12.6488i	0.0178552	+	0.0309261i	0.874815	−	0.484457i	$-0.160983\pi$
$409$	7.30278	+	12.6488i	0.0178552	+	0.0309261i	−0.856960	+	0.515383i	$0.827650\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	204.757	−	354.650i	0.493390	−	0.854577i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			619.804i			1.47925i	0.673021	+	0.739623i	$0.264996\pi$
$419$			619.804i			1.47925i	−0.673021	+	0.739623i	$0.735004\pi$
$420$		0			0
$421$	−307.379			−0.730115			−0.365058	−	0.930985i	$-0.618951\pi$
$421$	−307.379			−0.730115			−0.365058	+	0.930985i	$0.618951\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−26.1128	−	15.0763i	−0.0614420	−	0.0354736i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−452.837	+	261.446i	−1.05067	+	0.606603i	−0.922836	−	0.385193i	$-0.874135\pi$
$431$	−452.837	+	261.446i	−1.05067	+	0.606603i	−0.127831	+	0.991796i	$0.540801\pi$
$432$		0			0
$433$	−547.444			−1.26431			−0.632153	−	0.774844i	$-0.717829\pi$
$433$	−547.444			−1.26431			−0.632153	+	0.774844i	$0.717829\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	14.1663	−	8.17891i	0.0324171	−	0.0187161i
$438$		0			0
$439$	−258.386	+	447.537i	−0.588578	+	1.01945i	0.405841	+	0.913944i	$0.366979\pi$
$439$	−258.386	+	447.537i	−0.588578	+	1.01945i	−0.994419	+	0.105503i	$0.966355\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−474.560	−	273.987i	−1.07124	−	0.618481i	−0.142721	−	0.989763i	$-0.545585\pi$
$443$	−474.560	−	273.987i	−1.07124	−	0.618481i	−0.928520	+	0.371282i	$0.878918\pi$
$444$		0			0
$445$	−167.638	−	290.358i	−0.376716	−	0.652490i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	575.681i		−	1.28214i	−0.767482	−	0.641070i	$-0.778491\pi$
$449$		−	575.681i		−	1.28214i	0.767482	−	0.641070i	$-0.221509\pi$
$450$		0			0
$451$	−239.062	−	414.068i	−0.530071	−	0.918110i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−203.412	+	352.321i	−0.445104	+	0.770942i	−0.998059	−	0.0622683i	$-0.980167\pi$
$457$	−203.412	+	352.321i	−0.445104	+	0.770942i	0.552956	+	0.833211i	$0.313500\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			209.524i			0.454498i	0.973837	+	0.227249i	$0.0729731\pi$
$461$			209.524i			0.454498i	−0.973837	+	0.227249i	$0.927027\pi$
$462$		0			0
$463$	870.169			1.87942			0.939708	−	0.341978i	$-0.111097\pi$
$463$	870.169			1.87942			0.939708	+	0.341978i	$0.111097\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	201.088	+	116.098i	0.430594	+	0.248604i	0.699600	−	0.714535i	$-0.253362\pi$
$467$	201.088	+	116.098i	0.430594	+	0.248604i	−0.269006	+	0.963139i	$0.586695\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−396.296	+	228.802i	−0.837835	+	0.483724i
$474$		0			0
$475$	−256.148			−0.539260
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−770.453	+	444.821i	−1.60846	+	0.928645i	−0.618746	+	0.785591i	$0.712359\pi$
$479$	−770.453	+	444.821i	−1.60846	+	0.928645i	−0.989715	+	0.143054i	$0.954308\pi$
$480$		0			0
$481$	49.5214	−	85.7737i	0.102955	−	0.178324i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	474.982	+	274.231i	0.979344	+	0.565425i
$486$		0			0
$487$	32.7062	+	56.6488i	0.0671585	+	0.116322i	0.897649	−	0.440710i	$-0.145273\pi$
$487$	32.7062	+	56.6488i	0.0671585	+	0.116322i	−0.830491	+	0.557032i	$0.811940\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	82.4878i		−	0.168000i	−0.996466	−	0.0839998i	$-0.973230\pi$
$491$		−	82.4878i		−	0.168000i	0.996466	−	0.0839998i	$-0.0267695\pi$
$492$		0			0
$493$	9.40013	+	16.2815i	0.0190672	+	0.0330254i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	33.3277	−	57.7252i	0.0667889	−	0.115682i	−0.830697	−	0.556724i	$-0.812058\pi$
$499$	33.3277	−	57.7252i	0.0667889	−	0.115682i	0.897486	+	0.441043i	$0.145391\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			592.885i			1.17870i	0.807879	+	0.589349i	$0.200616\pi$
$503$			592.885i			1.17870i	−0.807879	+	0.589349i	$0.799384\pi$
$504$		0			0
$505$	680.757			1.34803
$506$		0			0
$507$		0			0
$508$		0			0
$509$	673.701	+	388.961i	1.32358	+	0.764168i	0.984297	−	0.176518i	$-0.0564833\pi$
$509$	673.701	+	388.961i	1.32358	+	0.764168i	0.339280	+	0.940685i	$0.389817\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	249.774	−	144.207i	0.484998	−	0.280014i
$516$		0			0
$517$	−146.247			−0.282877
$518$		0			0
$519$		0			0
$520$		0			0
$521$	375.133	−	216.583i	0.720024	−	0.415706i	−0.0947375	−	0.995502i	$-0.530201\pi$
$521$	375.133	−	216.583i	0.720024	−	0.415706i	0.814762	+	0.579796i	$0.196868\pi$
$522$		0			0
$523$	138.617	−	240.092i	0.265042	−	0.459066i	−0.702533	−	0.711651i	$-0.747948\pi$
$523$	138.617	−	240.092i	0.265042	−	0.459066i	0.967575	+	0.252585i	$0.0812809\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	49.8755	+	28.7956i	0.0946404	+	0.0546406i
$528$		0			0
$529$	−264.189	−	457.589i	−0.499413	−	0.865008i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	218.740i		−	0.410393i
$534$		0			0
$535$	−295.511	−	511.840i	−0.552356	−	0.956709i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−360.379	+	624.194i	−0.666134	+	1.15378i	0.312842	+	0.949805i	$0.398719\pi$
$541$	−360.379	+	624.194i	−0.666134	+	1.15378i	−0.978977	+	0.203973i	$0.934615\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			457.804i			0.840008i
$546$		0			0
$547$	−744.825			−1.36165			−0.680827	−	0.732444i	$-0.738379\pi$
$547$	−744.825			−1.36165			−0.680827	+	0.732444i	$0.738379\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	138.313	+	79.8550i	0.251022	+	0.144927i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−620.101	+	358.016i	−1.11329	+	0.642757i	−0.939679	−	0.342057i	$-0.888876\pi$
$557$	−620.101	+	358.016i	−1.11329	+	0.642757i	−0.173609	+	0.984815i	$0.555543\pi$
$558$		0			0
$559$	−209.352			−0.374511
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−650.256	+	375.426i	−1.15498	+	0.666831i	−0.950097	−	0.311955i	$-0.899016\pi$
$563$	−650.256	+	375.426i	−1.15498	+	0.666831i	−0.204888	+	0.978786i	$0.565683\pi$
$564$		0			0
$565$	−334.645	+	579.623i	−0.592293	+	1.02588i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	147.006	+	84.8741i	0.258359	+	0.149164i	0.623586	−	0.781755i	$-0.285675\pi$
$569$	147.006	+	84.8741i	0.258359	+	0.149164i	−0.365227	+	0.930919i	$0.619009\pi$
$570$		0			0
$571$	−313.757	−	543.443i	−0.549487	−	0.951739i	−0.998310	−	0.0581185i	$-0.981490\pi$
$571$	−313.757	−	543.443i	−0.549487	−	0.951739i	0.448823	−	0.893621i	$-0.351843\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	9.73173i		−	0.0169247i
$576$		0			0
$577$	82.3160	+	142.576i	0.142662	+	0.247098i	0.928498	−	0.371337i	$-0.121100\pi$
$577$	82.3160	+	142.576i	0.142662	+	0.247098i	−0.785836	+	0.618435i	$0.787767\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	349.051	−	604.574i	0.598715	−	1.03700i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1040.76i			1.77301i	0.462719	+	0.886505i	$0.346874\pi$
$587$			1040.76i			1.77301i	−0.462719	+	0.886505i	$0.653126\pi$
$588$		0			0
$589$	489.243			0.830633
$590$		0			0
$591$		0			0
$592$		0			0
$593$	543.797	+	313.961i	0.917027	+	0.529446i	0.882685	−	0.469964i	$-0.155733\pi$
$593$	543.797	+	313.961i	0.917027	+	0.529446i	0.0343417	+	0.999410i	$0.489067\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−228.485	+	131.916i	−0.381445	+	0.220227i	−0.678447	−	0.734650i	$-0.737347\pi$
$599$	−228.485	+	131.916i	−0.381445	+	0.220227i	0.297002	+	0.954877i	$0.404013\pi$
$600$		0			0
$601$	−502.094			−0.835431			−0.417715	−	0.908578i	$-0.637169\pi$
$601$	−502.094			−0.835431			−0.417715	+	0.908578i	$0.637169\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−107.829	+	62.2552i	−0.178230	+	0.102901i
$606$		0			0
$607$	499.801	−	865.680i	0.823395	−	1.42616i	−0.0797451	−	0.996815i	$-0.525411\pi$
$607$	499.801	−	865.680i	0.823395	−	1.42616i	0.903140	−	0.429346i	$-0.141256\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−57.9436	−	33.4537i	−0.0948340	−	0.0547524i
$612$		0			0
$613$	−460.085	−	796.890i	−0.750546	−	1.29998i	−0.947558	−	0.319583i	$-0.896457\pi$
$613$	−460.085	−	796.890i	−0.750546	−	1.29998i	0.197012	−	0.980401i	$-0.436876\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	520.713i		−	0.843943i	−0.906609	−	0.421972i	$-0.861338\pi$
$617$		−	520.713i		−	0.843943i	0.906609	−	0.421972i	$-0.138662\pi$
$618$		0			0
$619$	−119.305	−	206.643i	−0.192739	−	0.333833i	0.753418	−	0.657542i	$-0.228404\pi$
$619$	−119.305	−	206.643i	−0.192739	−	0.333833i	−0.946157	+	0.323708i	$0.895070\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	81.9972	−	142.023i	0.131195	−	0.227237i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	57.0207i		−	0.0906529i
$630$		0			0
$631$	534.034			0.846329			0.423165	−	0.906053i	$-0.360919\pi$
$631$	534.034			0.846329			0.423165	+	0.906053i	$0.360919\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−125.252	−	72.3145i	−0.197248	−	0.113881i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−868.646	+	501.513i	−1.35514	+	0.782392i	−0.988964	−	0.148153i	$-0.952667\pi$
$641$	−868.646	+	501.513i	−1.35514	+	0.782392i	−0.366178	+	0.930545i	$0.619334\pi$
$642$		0			0
$643$	−744.923			−1.15851			−0.579256	−	0.815146i	$-0.696657\pi$
$643$	−744.923			−1.15851			−0.579256	+	0.815146i	$0.696657\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−772.015	+	445.723i	−1.19322	+	0.688908i	−0.959036	−	0.283284i	$-0.908576\pi$
$647$	−772.015	+	445.723i	−1.19322	+	0.688908i	−0.234187	+	0.972192i	$0.575243\pi$
$648$		0			0
$649$	73.1236	−	126.654i	0.112671	−	0.195152i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	70.5924	+	40.7565i	0.108105	+	0.0624143i	0.553078	−	0.833130i	$-0.313453\pi$
$653$	70.5924	+	40.7565i	0.108105	+	0.0624143i	−0.444973	+	0.895544i	$0.646787\pi$
$654$		0			0
$655$	−170.757	−	295.760i	−0.260698	−	0.451542i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	185.430i		−	0.281380i	−0.990054	−	0.140690i	$-0.955068\pi$
$659$		−	185.430i		−	0.281380i	0.990054	−	0.140690i	$-0.0449322\pi$
$660$		0			0
$661$	272.783	+	472.475i	0.412683	+	0.714788i	0.995182	−	0.0980435i	$-0.0312584\pi$
$661$	272.783	+	472.475i	0.412683	+	0.714788i	−0.582499	+	0.812831i	$0.697925\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−3.03389	+	5.25486i	−0.00454857	+	0.00787835i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			74.3917i			0.110867i
$672$		0			0
$673$	−994.136			−1.47717			−0.738585	−	0.674160i	$-0.764506\pi$
$673$	−994.136			−1.47717			−0.738585	+	0.674160i	$0.764506\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	181.821	+	104.975i	0.268569	+	0.155058i	0.628237	−	0.778022i	$-0.283777\pi$
$677$	181.821	+	104.975i	0.268569	+	0.155058i	−0.359668	+	0.933080i	$0.617110\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	355.619	−	205.317i	0.520672	−	0.300610i	−0.216538	−	0.976274i	$-0.569476\pi$
$683$	355.619	−	205.317i	0.520672	−	0.300610i	0.737209	+	0.675664i	$0.236143\pi$
$684$		0			0
$685$	414.189			0.604655
$686$		0			0
$687$		0			0
$688$		0			0
$689$	276.590	−	159.689i	0.401437	−	0.231770i
$690$		0			0
$691$	−662.339	+	1147.21i	−0.958523	+	1.66021i	−0.232431	+	0.972613i	$0.574668\pi$
$691$	−662.339	+	1147.21i	−0.958523	+	1.66021i	−0.726092	+	0.687597i	$0.758665\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	307.842	+	177.733i	0.442938	+	0.255731i
$696$		0			0
$697$	−62.9661	−	109.060i	−0.0903387	−	0.156471i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	580.339i		−	0.827873i	−0.910306	−	0.413936i	$-0.864154\pi$
$701$		−	580.339i		−	0.827873i	0.910306	−	0.413936i	$-0.135846\pi$
$702$		0			0
$703$	−242.198	−	419.499i	−0.344521	−	0.596727i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	163.689	−	283.518i	0.230873	−	0.399885i	−0.727192	−	0.686434i	$-0.759175\pi$
$709$	163.689	−	283.518i	0.230873	−	0.399885i	0.958065	+	0.286550i	$0.0925083\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			18.5876i			0.0260695i
$714$		0			0
$715$	139.966			0.195757
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−339.821	−	196.196i	−0.472630	−	0.272873i	0.244710	−	0.969596i	$-0.421307\pi$
$719$	−339.821	−	196.196i	−0.472630	−	0.272873i	−0.717340	+	0.696723i	$0.754641\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	82.2862	−	47.5080i	0.113498	−	0.0655282i
$726$		0			0
$727$	−1340.99			−1.84455			−0.922274	−	0.386537i	$-0.873671\pi$
$727$	−1340.99			−1.84455			−0.922274	+	0.386537i	$0.873671\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−104.380	+	60.2636i	−0.142790	+	0.0824400i
$732$		0			0
$733$	65.2016	−	112.933i	0.0889518	−	0.154069i	−0.818117	−	0.575052i	$-0.804982\pi$
$733$	65.2016	−	112.933i	0.0889518	−	0.154069i	0.907068	+	0.420983i	$0.138315\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	701.481	+	405.000i	0.951806	+	0.549526i
$738$		0			0
$739$	101.017	+	174.966i	0.136694	+	0.236761i	0.926243	−	0.376926i	$-0.123019\pi$
$739$	101.017	+	174.966i	0.136694	+	0.236761i	−0.789549	+	0.613687i	$0.789686\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	821.453i		−	1.10559i	−0.833317	−	0.552795i	$-0.813561\pi$
$743$		−	821.453i		−	1.10559i	0.833317	−	0.552795i	$-0.186439\pi$
$744$		0			0
$745$	193.200	+	334.632i	0.259329	+	0.449171i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−402.068	+	696.402i	−0.535377	+	0.927299i	0.463768	+	0.885956i	$0.346497\pi$
$751$	−402.068	+	696.402i	−0.535377	+	0.927299i	−0.999145	+	0.0413429i	$0.986836\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			796.987i			1.05561i
$756$		0			0
$757$	518.000			0.684280			0.342140	−	0.939649i	$-0.388848\pi$
$757$	518.000			0.684280			0.342140	+	0.939649i	$0.388848\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	583.576	+	336.927i	0.766854	+	0.442743i	0.831751	−	0.555149i	$-0.187339\pi$
$761$	583.576	+	336.927i	0.766854	+	0.442743i	−0.0648975	+	0.997892i	$0.520672\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	57.9436	−	33.4537i	0.0755457	−	0.0436164i
$768$		0			0
$769$	653.023			0.849185			0.424592	−	0.905385i	$-0.360417\pi$
$769$	653.023			0.849185			0.424592	+	0.905385i	$0.360417\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−399.499	+	230.651i	−0.516816	+	0.298384i	−0.735631	−	0.677383i	$-0.763114\pi$
$773$	−399.499	+	230.651i	−0.516816	+	0.298384i	0.218815	+	0.975766i	$0.429781\pi$
$774$		0			0
$775$	145.532	−	252.069i	0.187783	−	0.325250i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−926.479	−	534.903i	−1.18932	−	0.686653i
$780$		0			0
$781$	185.757	+	321.741i	0.237845	+	0.411960i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			373.941i			0.476358i
$786$		0			0
$787$	−396.563	−	686.867i	−0.503892	−	0.872767i	−0.999990	−	0.00450000i	$-0.998568\pi$
$787$	−396.563	−	686.867i	−0.503892	−	0.872767i	0.496098	−	0.868267i	$-0.334766\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−17.0169	+	29.4742i	−0.0214589	+	0.0371680i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			435.599i			0.546548i	0.961936	+	0.273274i	$0.0881066\pi$
$797$			435.599i			0.546548i	−0.961936	+	0.273274i	$0.911893\pi$
$798$		0			0
$799$	−38.5198			−0.0482100
$800$		0			0
$801$		0			0
$802$		0			0
$803$	276.309	+	159.527i	0.344096	+	0.198664i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−499.721	+	288.514i	−0.617702	+	0.356631i	−0.775974	−	0.630765i	$-0.782741\pi$
$809$	−499.721	+	288.514i	−0.617702	+	0.356631i	0.158272	+	0.987396i	$0.449408\pi$
$810$		0			0
$811$	38.5274			0.0475060			0.0237530	−	0.999718i	$-0.492438\pi$
$811$	38.5274			0.0475060			0.0237530	+	0.999718i	$0.492438\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	67.6739	−	39.0715i	0.0830354	−	0.0479405i
$816$		0			0
$817$	−511.945	+	886.715i	−0.626616	+	1.08533i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	394.707	+	227.884i	0.480764	+	0.277569i	0.720735	−	0.693211i	$-0.243805\pi$
$821$	394.707	+	227.884i	0.480764	+	0.277569i	−0.239971	+	0.970780i	$0.577138\pi$
$822$		0			0
$823$	199.791	+	346.048i	0.242759	+	0.420472i	0.961499	−	0.274808i	$-0.0886141\pi$
$823$	199.791	+	346.048i	0.242759	+	0.420472i	−0.718740	+	0.695279i	$0.755281\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1182.94i			1.43040i	0.698922	+	0.715198i	$0.253664\pi$
$827$			1182.94i			1.43040i	−0.698922	+	0.715198i	$0.746336\pi$
$828$		0			0
$829$	81.7807	+	141.648i	0.0986498	+	0.170866i	0.911126	−	0.412128i	$-0.135214\pi$
$829$	81.7807	+	141.648i	0.0986498	+	0.170866i	−0.812476	+	0.582994i	$0.801881\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	554.672	−	960.721i	0.664278	−	1.15056i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	980.535i		−	1.16869i	−0.811504	−	0.584347i	$-0.801351\pi$
$839$		−	980.535i		−	1.16869i	0.811504	−	0.584347i	$-0.198649\pi$
$840$		0			0
$841$	781.757			0.929557
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−465.206	−	268.587i	−0.550539	−	0.317854i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	15.9378	−	9.20172i	0.0187284	−	0.0108128i
$852$		0			0
$853$	184.455			0.216243			0.108121	−	0.994138i	$-0.465517\pi$
$853$	184.455			0.216243			0.108121	+	0.994138i	$0.465517\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	992.826	−	573.208i	1.15849	−	0.668854i	0.207548	−	0.978225i	$-0.433452\pi$
$857$	992.826	−	573.208i	1.15849	−	0.668854i	0.950942	+	0.309371i	$0.100118\pi$
$858$		0			0
$859$	445.537	−	771.693i	0.518670	−	0.898362i	−0.481095	−	0.876668i	$-0.659761\pi$
$859$	445.537	−	771.693i	0.518670	−	0.898362i	0.999765	−	0.0216937i	$-0.00690585\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1239.05	−	715.366i	−1.43575	−	0.828929i	−0.438197	−	0.898879i	$-0.644383\pi$
$863$	−1239.05	−	715.366i	−1.43575	−	0.828929i	−0.997550	+	0.0699503i	$0.977716\pi$
$864$		0			0
$865$	−115.034	−	199.245i	−0.132987	−	0.230341i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			80.5809i			0.0927283i
$870$		0			0
$871$	185.286	+	320.925i	0.212728	+	0.368455i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−707.429	+	1225.30i	−0.806647	+	1.39715i	0.108527	+	0.994094i	$0.465387\pi$
$877$	−707.429	+	1225.30i	−0.806647	+	1.39715i	−0.915174	+	0.403060i	$0.867947\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	407.260i		−	0.462270i	−0.972922	−	0.231135i	$-0.925756\pi$
$881$		−	407.260i		−	0.462270i	0.972922	−	0.231135i	$-0.0742439\pi$
$882$		0			0
$883$	−386.723			−0.437965			−0.218983	−	0.975729i	$-0.570274\pi$
$883$	−386.723			−0.437965			−0.218983	+	0.975729i	$0.570274\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−999.171	−	576.872i	−1.12646	−	0.650363i	−0.183419	−	0.983035i	$-0.558717\pi$
$887$	−999.171	−	576.872i	−1.12646	−	0.650363i	−0.943043	+	0.332672i	$0.892050\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−283.389	+	163.615i	−0.317345	+	0.183219i
$894$		0			0
$895$	1189.13			1.32864
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−157.166	+	90.7401i	−0.174824	+	0.100934i
$900$		0			0
$901$	91.9359	−	159.238i	0.102038	−	0.176734i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	707.328	+	408.376i	0.781578	+	0.451244i
$906$		0			0
$907$	284.085	+	492.049i	0.313214	+	0.542502i	0.979056	−	0.203590i	$-0.0652611\pi$
$907$	284.085	+	492.049i	0.313214	+	0.542502i	−0.665843	+	0.746092i	$0.731928\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			78.5035i			0.0861729i	0.999071	+	0.0430864i	$0.0137191\pi$
$911$			78.5035i			0.0861729i	−0.999071	+	0.0430864i	$0.986281\pi$
$912$		0			0
$913$	−533.766	−	924.509i	−0.584628	−	1.01261i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−271.463	+	470.188i	−0.295390	+	0.511630i	−0.975076	−	0.221873i	$-0.928783\pi$
$919$	−271.463	+	470.188i	−0.295390	+	0.511630i	0.679686	+	0.733504i	$0.262116\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			169.966i			0.184145i
$924$		0			0
$925$	−288.181			−0.311547
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−158.773	−	91.6676i	−0.170907	−	0.0986734i	0.412106	−	0.911136i	$-0.364793\pi$
$929$	−158.773	−	91.6676i	−0.170907	−	0.0986734i	−0.583014	+	0.812462i	$0.698127\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	69.7851	−	40.2905i	0.0746365	−	0.0430914i
$936$		0			0
$937$	532.184			0.567966			0.283983	−	0.958829i	$-0.408344\pi$
$937$	532.184			0.567966			0.283983	+	0.958829i	$0.408344\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1442.91	−	833.063i	1.53338	−	0.885295i	0.534174	−	0.845375i	$-0.320623\pi$
$941$	1442.91	−	833.063i	1.53338	−	0.885295i	0.999203	−	0.0399209i	$-0.0127106\pi$
$942$		0			0
$943$	20.3223	−	35.1993i	0.0215507	−	0.0373269i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−466.690	−	269.444i	−0.492809	−	0.284523i	0.232930	−	0.972493i	$-0.425169\pi$
$947$	−466.690	−	269.444i	−0.492809	−	0.284523i	−0.725739	+	0.687970i	$0.758502\pi$
$948$		0			0
$949$	72.9831	+	126.410i	0.0769052	+	0.133204i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1047.62i			1.09928i	0.835400	+	0.549642i	$0.185236\pi$
$953$			1047.62i			1.09928i	−0.835400	+	0.549642i	$0.814764\pi$
$954$		0			0
$955$	−134.434	−	232.847i	−0.140769	−	0.243819i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	202.534	−	350.799i	0.210753	−	0.365035i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	33.3635i		−	0.0345736i
$966$		0			0
$967$	347.209			0.359058			0.179529	−	0.983753i	$-0.442543\pi$
$967$	347.209			0.359058			0.179529	+	0.983753i	$0.442543\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	388.746	+	224.443i	0.400356	+	0.231146i	0.686638	−	0.727000i	$-0.259086\pi$
$971$	388.746	+	224.443i	0.400356	+	0.231146i	−0.286281	+	0.958146i	$0.592419\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	818.821	−	472.747i	0.838098	−	0.483876i	−0.0185194	−	0.999829i	$-0.505895\pi$
$977$	818.821	−	472.747i	0.838098	−	0.483876i	0.856617	+	0.515953i	$0.172562\pi$
$978$		0			0
$979$	−874.008			−0.892756
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−934.658	+	539.625i	−0.950822	+	0.548958i	−0.893336	−	0.449389i	$-0.851642\pi$
$983$	−934.658	+	539.625i	−0.950822	+	0.548958i	−0.0574860	+	0.998346i	$0.518308\pi$
$984$		0			0
$985$	68.5814	−	118.786i	0.0696258	−	0.120595i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−33.6886	−	19.4501i	−0.0340633	−	0.0196664i
$990$		0			0
$991$	393.429	+	681.440i	0.397002	+	0.687628i	0.993355	−	0.115095i	$-0.0367171\pi$
$991$	393.429	+	681.440i	0.397002	+	0.687628i	−0.596352	+	0.802723i	$0.703384\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			231.595i			0.232759i
$996$		0			0
$997$	937.999	+	1624.66i	0.940822	+	1.62955i	0.763909	+	0.645324i	$0.223278\pi$
$997$	937.999	+	1624.66i	0.940822	+	1.62955i	0.176913	+	0.984227i	$0.443389\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.bk.g.1745.6		16
3.2	odd	2	inner	1764.3.bk.g.1745.3		16
7.2	even	3		1764.3.c.h.197.4	yes	8
7.3	odd	6	inner	1764.3.bk.g.557.5		16
7.4	even	3	inner	1764.3.bk.g.557.3		16
7.5	odd	6		1764.3.c.h.197.6	yes	8
7.6	odd	2	inner	1764.3.bk.g.1745.4		16
21.2	odd	6		1764.3.c.h.197.5	yes	8
21.5	even	6		1764.3.c.h.197.3	✓	8
21.11	odd	6	inner	1764.3.bk.g.557.6		16
21.17	even	6	inner	1764.3.bk.g.557.4		16
21.20	even	2	inner	1764.3.bk.g.1745.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.3.c.h.197.3	✓	8	21.5	even	6
1764.3.c.h.197.4	yes	8	7.2	even	3
1764.3.c.h.197.5	yes	8	21.2	odd	6
1764.3.c.h.197.6	yes	8	7.5	odd	6
1764.3.bk.g.557.3		16	7.4	even	3	inner
1764.3.bk.g.557.4		16	21.17	even	6	inner
1764.3.bk.g.557.5		16	7.3	odd	6	inner
1764.3.bk.g.557.6		16	21.11	odd	6	inner
1764.3.bk.g.1745.3		16	3.2	odd	2	inner
1764.3.bk.g.1745.4		16	7.6	odd	2	inner
1764.3.bk.g.1745.5		16	21.20	even	2	inner
1764.3.bk.g.1745.6		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.bk (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.08083 + 1.77872i) q^{5} +O(q^{10})\)	\(q+(3.08083 + 1.77872i) q^{5} +(8.03119 - 4.63681i) q^{11} +4.24264 q^{13} +(2.11532 - 1.22128i) q^{17} +(10.3749 - 17.9698i) q^{19} +(0.682720 + 0.394169i) q^{23} +(-6.17232 - 10.6908i) q^{25} +7.69694i q^{29} +(11.7891 + 20.4193i) q^{31} +(11.6723 - 20.2170i) q^{37} -51.5574i q^{41} -49.3446 q^{43} +(-13.6574 - 7.88512i) q^{47} +(65.1929 - 37.6391i) q^{53} +32.9903 q^{55} +(13.6574 - 7.88512i) q^{59} +(-4.01093 + 6.94714i) q^{61} +(13.0709 + 7.54647i) q^{65} +(43.6723 + 75.6427i) q^{67} +40.0614i q^{71} +(17.2023 + 29.7952i) q^{73} +(-4.34463 + 7.52512i) q^{79} -115.115i q^{83} +8.68926 q^{85} +(-81.6200 - 47.1234i) q^{89} +(63.9266 - 36.9080i) q^{95} +154.173 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 216 q^{25} - 128 q^{37} - 160 q^{43} + 384 q^{67} + 560 q^{79} - 1120 q^{85}+O(q^{100}) \) 16 * q + 216 * q^25 - 128 * q^37 - 160 * q^43 + 384 * q^67 + 560 * q^79 - 1120 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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