LMFDB - Embedded newform 1764.4.k.bd.1549.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(361,1764)]

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, 0, 4]))

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

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

chi := DirichletCharacter("1764.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.k (of order $3$, 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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 27x^{6} + 10x^{5} + 446x^{4} + 62x^{3} + 3061x^{2} + 2142x + 14161 $ x^8 - 2x^7 + 27x^6 + 10x^5 + 446x^4 + 62x^3 + 3061x^2 + 2142*x + 14161
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{8}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1549.4
Root			$-1.25866 - 2.18006i$ of defining polynomial
Character	$\chi$	$=$	1764.1549
Dual form			1764.4.k.bd.361.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(9.57016 - 16.5760i) q^{5} +O(q^{10})$	$q+(9.57016 - 16.5760i) q^{5} +(20.2976 + 35.1564i) q^{11} -50.4776 q^{13} +(25.9597 + 44.9636i) q^{17} +(-16.5666 + 28.6941i) q^{19} +(31.4249 - 54.4296i) q^{23} +(-120.676 - 209.017i) q^{25} -129.921 q^{29} +(121.208 + 209.938i) q^{31} +(194.693 - 337.217i) q^{37} -470.110 q^{41} -125.003 q^{43} +(193.312 - 334.826i) q^{47} +(-305.718 - 529.519i) q^{53} +777.004 q^{55} +(-113.216 - 196.096i) q^{59} +(362.595 - 628.034i) q^{61} +(-483.079 + 836.718i) q^{65} +(-522.555 - 905.092i) q^{67} -169.839 q^{71} +(-190.866 - 330.589i) q^{73} +(580.560 - 1005.56i) q^{79} -808.448 q^{83} +993.756 q^{85} +(159.933 - 277.013i) q^{89} +(317.089 + 549.215i) q^{95} -1133.24 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 48 q^{17} - 192 q^{19} + 192 q^{23} - 324 q^{25} - 192 q^{29} - 48 q^{31} - 256 q^{37} - 2016 q^{41} - 224 q^{43} + 864 q^{47} - 648 q^{53} + 4704 q^{55} + 336 q^{59} - 960 q^{61} - 360 q^{65} - 720 q^{67} + 2688 q^{71} - 672 q^{73} + 1984 q^{79} - 6240 q^{83} + 1360 q^{85} + 2160 q^{89} - 3744 q^{95} + 4032 q^{97}+O(q^{100}) $ 8 * q + 48 * q^17 - 192 * q^19 + 192 * q^23 - 324 * q^25 - 192 * q^29 - 48 * q^31 - 256 * q^37 - 2016 * q^41 - 224 * q^43 + 864 * q^47 - 648 * q^53 + 4704 * q^55 + 336 * q^59 - 960 * q^61 - 360 * q^65 - 720 * q^67 + 2688 * q^71 - 672 * q^73 + 1984 * q^79 - 6240 * q^83 + 1360 * q^85 + 2160 * q^89 - 3744 * q^95 + 4032 * q^97

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
$5$	9.57016	−	16.5760i	0.855982	−	1.48260i	−0.0197499	−	0.999805i	$-0.506287\pi$
$5$	9.57016	−	16.5760i	0.855982	−	1.48260i	0.875731	−	0.482799i	$-0.160380\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	20.2976	+	35.1564i	0.556359	+	0.963641i	0.997796	+	0.0663495i	$0.0211352\pi$
$11$	20.2976	+	35.1564i	0.556359	+	0.963641i	−0.441438	+	0.897292i	$0.645531\pi$
$12$		0			0
$13$	−50.4776			−1.07692			−0.538460	−	0.842651i	$-0.680994\pi$
$13$	−50.4776			−1.07692			−0.538460	+	0.842651i	$0.680994\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	25.9597	+	44.9636i	0.370362	+	0.641486i	0.989621	−	0.143700i	$-0.0459002\pi$
$17$	25.9597	+	44.9636i	0.370362	+	0.641486i	−0.619259	+	0.785187i	$0.712567\pi$
$18$		0			0
$19$	−16.5666	+	28.6941i	−0.200033	+	0.346468i	−0.948539	−	0.316661i	$-0.897438\pi$
$19$	−16.5666	+	28.6941i	−0.200033	+	0.346468i	0.748506	+	0.663128i	$0.230772\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	31.4249	−	54.4296i	0.284894	−	0.493450i	−0.687690	−	0.726005i	$-0.741375\pi$
$23$	31.4249	−	54.4296i	0.284894	−	0.493450i	0.972583	+	0.232554i	$0.0747084\pi$
$24$		0			0
$25$	−120.676	−	209.017i	−0.965409	−	1.67214i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−129.921			−0.831924			−0.415962	−	0.909382i	$-0.636555\pi$
$29$	−129.921			−0.831924			−0.415962	+	0.909382i	$0.636555\pi$
$30$		0			0
$31$	121.208	+	209.938i	0.702242	+	1.21632i	0.967678	+	0.252191i	$0.0811510\pi$
$31$	121.208	+	209.938i	0.702242	+	1.21632i	−0.265435	+	0.964129i	$0.585516\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	194.693	−	337.217i	0.865061	−	1.49833i	−0.00192559	−	0.999998i	$-0.500613\pi$
$37$	194.693	−	337.217i	0.865061	−	1.49833i	0.866987	−	0.498331i	$-0.166054\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−470.110			−1.79071			−0.895353	−	0.445358i	$-0.853076\pi$
$41$	−470.110			−1.79071			−0.895353	+	0.445358i	$0.853076\pi$
$42$		0			0
$43$	−125.003			−0.443321			−0.221661	−	0.975124i	$-0.571148\pi$
$43$	−125.003			−0.443321			−0.221661	+	0.975124i	$0.571148\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	193.312	−	334.826i	0.599946	−	1.03914i	−0.392882	−	0.919589i	$-0.628522\pi$
$47$	193.312	−	334.826i	0.599946	−	1.03914i	0.992828	−	0.119548i	$-0.0381446\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−305.718	−	529.519i	−0.792332	−	1.37236i	−0.924520	−	0.381134i	$-0.875534\pi$
$53$	−305.718	−	529.519i	−0.792332	−	1.37236i	0.132188	−	0.991225i	$-0.457800\pi$
$54$		0			0
$55$	777.004			1.90493
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−113.216	−	196.096i	−0.249822	−	0.432704i	0.713654	−	0.700498i	$-0.247039\pi$
$59$	−113.216	−	196.096i	−0.249822	−	0.432704i	−0.963476	+	0.267794i	$0.913705\pi$
$60$		0			0
$61$	362.595	−	628.034i	0.761075	−	1.31822i	−0.181222	−	0.983442i	$-0.558005\pi$
$61$	362.595	−	628.034i	0.761075	−	1.31822i	0.942297	−	0.334779i	$-0.108662\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−483.079	+	836.718i	−0.921824	+	1.59665i
$66$		0			0
$67$	−522.555	−	905.092i	−0.952840	−	1.65037i	−0.739236	−	0.673446i	$-0.764813\pi$
$67$	−522.555	−	905.092i	−0.952840	−	1.65037i	−0.213603	−	0.976921i	$-0.568520\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−169.839			−0.283889			−0.141945	−	0.989875i	$-0.545336\pi$
$71$	−169.839			−0.283889			−0.141945	+	0.989875i	$0.545336\pi$
$72$		0			0
$73$	−190.866	−	330.589i	−0.306015	−	0.530034i	0.671472	−	0.741030i	$-0.265663\pi$
$73$	−190.866	−	330.589i	−0.306015	−	0.530034i	−0.977487	+	0.210996i	$0.932329\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	580.560	−	1005.56i	0.826812	−	1.43208i	−0.0737148	−	0.997279i	$-0.523485\pi$
$79$	580.560	−	1005.56i	0.826812	−	1.43208i	0.900527	−	0.434801i	$-0.143181\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−808.448			−1.06914			−0.534570	−	0.845124i	$-0.679527\pi$
$83$	−808.448			−1.06914			−0.534570	+	0.845124i	$0.679527\pi$
$84$		0			0
$85$	993.756			1.26809
$86$		0			0
$87$		0			0
$88$		0			0
$89$	159.933	−	277.013i	0.190482	−	0.329924i	−0.754928	−	0.655808i	$-0.772328\pi$
$89$	159.933	−	277.013i	0.190482	−	0.329924i	0.945410	+	0.325883i	$0.105662\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	317.089	+	549.215i	0.342449	+	0.593140i
$96$		0			0
$97$	−1133.24			−1.18622			−0.593110	−	0.805121i	$-0.702100\pi$
$97$	−1133.24			−1.18622			−0.593110	+	0.805121i	$0.702100\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	522.872	+	905.640i	0.515126	+	0.892224i	0.999846	+	0.0175545i	$0.00558806\pi$
$101$	522.872	+	905.640i	0.515126	+	0.892224i	−0.484720	+	0.874669i	$0.661079\pi$
$102$		0			0
$103$	−949.106	+	1643.90i	−0.907943	+	1.57260i	−0.0910261	+	0.995849i	$0.529015\pi$
$103$	−949.106	+	1643.90i	−0.907943	+	1.57260i	−0.816917	+	0.576755i	$0.804319\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−434.392	+	752.390i	−0.392470	+	0.679778i	−0.992775	−	0.119993i	$-0.961713\pi$
$107$	−434.392	+	752.390i	−0.392470	+	0.679778i	0.600305	+	0.799771i	$0.295046\pi$
$108$		0			0
$109$	−113.824	−	197.150i	−0.100022	−	0.173243i	0.811671	−	0.584114i	$-0.198558\pi$
$109$	−113.824	−	197.150i	−0.100022	−	0.173243i	−0.911694	+	0.410871i	$0.865225\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−581.462			−0.484065			−0.242032	−	0.970268i	$-0.577814\pi$
$113$	−581.462			−0.484065			−0.242032	+	0.970268i	$0.577814\pi$
$114$		0			0
$115$	−601.484	−	1041.80i	−0.487727	−	0.844769i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−158.482	+	274.498i	−0.119070	+	0.206235i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2227.02			−1.59353
$126$		0			0
$127$	1129.10			0.788908			0.394454	−	0.918916i	$-0.370934\pi$
$127$	1129.10			0.788908			0.394454	+	0.918916i	$0.370934\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−439.831	+	761.810i	−0.293345	+	0.508089i	−0.974599	−	0.223959i	$-0.928102\pi$
$131$	−439.831	+	761.810i	−0.293345	+	0.508089i	0.681253	+	0.732048i	$0.261435\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−161.650	−	279.986i	−0.100808	−	0.174604i	0.811210	−	0.584755i	$-0.198809\pi$
$137$	−161.650	−	279.986i	−0.100808	−	0.174604i	−0.912018	+	0.410151i	$0.865476\pi$
$138$		0			0
$139$	1710.64			1.04385			0.521923	−	0.852993i	$-0.325215\pi$
$139$	1710.64			1.04385			0.521923	+	0.852993i	$0.325215\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1024.57	−	1774.61i	−0.599154	−	1.03777i
$144$		0			0
$145$	−1243.37	+	2153.58i	−0.712111	+	1.23341i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−479.889	+	831.193i	−0.263853	+	0.457006i	−0.967262	−	0.253778i	$-0.918327\pi$
$149$	−479.889	+	831.193i	−0.263853	+	0.457006i	0.703410	+	0.710785i	$0.251660\pi$
$150$		0			0
$151$	809.422	+	1401.96i	0.436224	+	0.755563i	0.997395	−	0.0721377i	$-0.0229821\pi$
$151$	809.422	+	1401.96i	0.436224	+	0.755563i	−0.561170	+	0.827700i	$0.689649\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4639.90			2.40443
$156$		0			0
$157$	278.666	+	482.664i	0.141656	+	0.245355i	0.928120	−	0.372280i	$-0.121424\pi$
$157$	278.666	+	482.664i	0.141656	+	0.245355i	−0.786464	+	0.617636i	$0.788091\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	183.017	−	316.995i	0.0879447	−	0.152325i	−0.818697	−	0.574225i	$-0.805303\pi$
$163$	183.017	−	316.995i	0.0879447	−	0.152325i	0.906642	+	0.421900i	$0.138637\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2834.00			−1.31318			−0.656591	−	0.754247i	$-0.728002\pi$
$167$	−2834.00			−1.31318			−0.656591	+	0.754247i	$0.728002\pi$
$168$		0			0
$169$	350.990			0.159759
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1785.20	+	3092.05i	−0.784543	+	1.35887i	0.144729	+	0.989471i	$0.453769\pi$
$173$	−1785.20	+	3092.05i	−0.784543	+	1.35887i	−0.929272	+	0.369397i	$0.879564\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1838.69	−	3184.71i	−0.767767	−	1.32981i	−0.938771	−	0.344541i	$-0.888035\pi$
$179$	−1838.69	−	3184.71i	−0.767767	−	1.32981i	0.171004	−	0.985270i	$-0.445299\pi$
$180$		0			0
$181$	−1718.18			−0.705588			−0.352794	−	0.935701i	$-0.614768\pi$
$181$	−1718.18			−0.705588			−0.352794	+	0.935701i	$0.614768\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3726.48	−	6454.45i	−1.48095	−	2.56509i
$186$		0			0
$187$	−1053.84	+	1825.30i	−0.412108	+	0.713793i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	979.523	−	1696.58i	0.371077	−	0.642725i	−0.618654	−	0.785663i	$-0.712322\pi$
$191$	979.523	−	1696.58i	0.371077	−	0.642725i	0.989732	+	0.142939i	$0.0456551\pi$
$192$		0			0
$193$	1246.23	+	2158.53i	0.464795	+	0.805049i	0.999192	−	0.0401843i	$-0.0127945\pi$
$193$	1246.23	+	2158.53i	0.464795	+	0.805049i	−0.534397	+	0.845234i	$0.679461\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2174.45			−0.786410			−0.393205	−	0.919451i	$-0.628634\pi$
$197$	−2174.45			−0.786410			−0.393205	+	0.919451i	$0.628634\pi$
$198$		0			0
$199$	−1932.30	−	3346.85i	−0.688329	−	1.19222i	−0.972378	−	0.233411i	$-0.925011\pi$
$199$	−1932.30	−	3346.85i	−0.688329	−	1.19222i	0.284050	−	0.958810i	$-0.408322\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−4499.03	+	7792.55i	−1.53281	+	2.65491i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1345.04			−0.445161
$210$		0			0
$211$	−127.267			−0.0415233			−0.0207616	−	0.999784i	$-0.506609\pi$
$211$	−127.267			−0.0415233			−0.0207616	+	0.999784i	$0.506609\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1196.30	+	2072.06i	−0.379475	+	0.657270i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1310.39	−	2269.65i	−0.398851	−	0.690830i
$222$		0			0
$223$	−4071.36			−1.22259			−0.611297	−	0.791401i	$-0.709352\pi$
$223$	−4071.36			−1.22259			−0.611297	+	0.791401i	$0.709352\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2641.18	+	4574.66i	0.772252	+	1.33758i	0.936326	+	0.351132i	$0.114203\pi$
$227$	2641.18	+	4574.66i	0.772252	+	1.33758i	−0.164073	+	0.986448i	$0.552463\pi$
$228$		0			0
$229$	2290.09	−	3966.56i	0.660846	−	1.14462i	−0.319548	−	0.947570i	$-0.603531\pi$
$229$	2290.09	−	3966.56i	0.660846	−	1.14462i	0.980394	−	0.197048i	$-0.0631356\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1078.26	+	1867.60i	−0.303171	+	0.525108i	−0.976853	−	0.213914i	$-0.931379\pi$
$233$	−1078.26	+	1867.60i	−0.303171	+	0.525108i	0.673681	+	0.739022i	$0.264712\pi$
$234$		0			0
$235$	−3700.06	−	6408.69i	−1.02709	−	1.77896i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5755.76			1.55778			0.778889	−	0.627162i	$-0.215783\pi$
$239$	5755.76			1.55778			0.778889	+	0.627162i	$0.215783\pi$
$240$		0			0
$241$	1295.78	+	2244.35i	0.346342	+	0.599881i	0.985597	−	0.169114i	$-0.0540905\pi$
$241$	1295.78	+	2244.35i	0.346342	+	0.599881i	−0.639255	+	0.768995i	$0.720757\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	836.240	−	1448.41i	0.215420	−	0.373118i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3809.47			0.957974			0.478987	−	0.877822i	$-0.341004\pi$
$251$	3809.47			0.957974			0.478987	+	0.877822i	$0.341004\pi$
$252$		0			0
$253$	2551.40			0.634012
$254$		0			0
$255$		0			0
$256$		0			0
$257$	678.844	−	1175.79i	0.164767	−	0.285385i	−0.771806	−	0.635859i	$-0.780646\pi$
$257$	678.844	−	1175.79i	0.164767	−	0.285385i	0.936573	+	0.350474i	$0.113979\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2208.87	−	3825.87i	−0.517888	−	0.897009i	−0.999784	−	0.0207801i	$-0.993385\pi$
$263$	−2208.87	−	3825.87i	−0.517888	−	0.897009i	0.481896	−	0.876228i	$-0.339948\pi$
$264$		0			0
$265$	−11703.1			−2.71289
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1813.66	−	3141.35i	−0.411081	−	0.712013i	0.583927	−	0.811806i	$-0.301515\pi$
$269$	−1813.66	−	3141.35i	−0.411081	−	0.712013i	−0.995008	+	0.0997927i	$0.968182\pi$
$270$		0			0
$271$	2585.52	−	4478.24i	0.579553	−	1.00382i	−0.415978	−	0.909375i	$-0.636561\pi$
$271$	2585.52	−	4478.24i	0.579553	−	1.00382i	0.995531	−	0.0944403i	$-0.0301062\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4898.86	−	8485.07i	1.07423	−	1.86062i
$276$		0			0
$277$	−897.316	−	1554.20i	−0.194637	−	0.337121i	0.752144	−	0.658998i	$-0.229020\pi$
$277$	−897.316	−	1554.20i	−0.194637	−	0.337121i	−0.946781	+	0.321877i	$0.895686\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9118.63			1.93584			0.967921	−	0.251254i	$-0.0808430\pi$
$281$	9118.63			1.93584			0.967921	+	0.251254i	$0.0808430\pi$
$282$		0			0
$283$	−2902.36	−	5027.03i	−0.609637	−	1.05592i	−0.991300	−	0.131621i	$-0.957982\pi$
$283$	−2902.36	−	5027.03i	−0.609637	−	1.05592i	0.381663	−	0.924302i	$-0.375352\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1108.68	−	1920.30i	0.225663	−	0.390861i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1384.21			−0.275995			−0.137997	−	0.990433i	$-0.544067\pi$
$293$	−1384.21			−0.275995			−0.137997	+	0.990433i	$0.544067\pi$
$294$		0			0
$295$	−4333.99			−0.855372
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1586.26	+	2747.48i	−0.306808	+	0.531407i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6940.20	−	12020.8i	−1.30293	−	2.25675i
$306$		0			0
$307$	−337.112			−0.0626709			−0.0313355	−	0.999509i	$-0.509976\pi$
$307$	−337.112			−0.0626709			−0.0313355	+	0.999509i	$0.509976\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−982.347	−	1701.48i	−0.179112	−	0.310231i	0.762465	−	0.647030i	$-0.223989\pi$
$311$	−982.347	−	1701.48i	−0.179112	−	0.310231i	−0.941577	+	0.336799i	$0.890656\pi$
$312$		0			0
$313$	1798.49	−	3115.07i	0.324781	−	0.562538i	−0.656687	−	0.754163i	$-0.728043\pi$
$313$	1798.49	−	3115.07i	0.324781	−	0.562538i	0.981468	+	0.191626i	$0.0613760\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−107.725	+	186.585i	−0.0190865	+	0.0330589i	−0.875411	−	0.483379i	$-0.839409\pi$
$317$	−107.725	+	186.585i	−0.0190865	+	0.0330589i	0.856324	+	0.516438i	$0.172742\pi$
$318$		0			0
$319$	−2637.09	−	4567.57i	−0.462848	−	0.801676i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1720.25			−0.296339
$324$		0			0
$325$	6091.44	+	10550.7i	1.03967	+	1.80076i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	842.677	−	1459.56i	0.139933	−	0.242371i	−0.787538	−	0.616266i	$-0.788645\pi$
$331$	842.677	−	1459.56i	0.139933	−	0.242371i	0.927471	+	0.373895i	$0.121978\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−20003.7			−3.26245
$336$		0			0
$337$	−7497.87			−1.21197			−0.605987	−	0.795475i	$-0.707222\pi$
$337$	−7497.87			−1.21197			−0.605987	+	0.795475i	$0.707222\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4920.43	+	8522.44i	−0.781397	+	1.35342i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3019.42	−	5229.78i	−0.467120	−	0.809076i	0.532174	−	0.846635i	$-0.321375\pi$
$347$	−3019.42	−	5229.78i	−0.467120	−	0.809076i	−0.999294	+	0.0375588i	$0.988042\pi$
$348$		0			0
$349$	−1992.86			−0.305659			−0.152830	−	0.988253i	$-0.548839\pi$
$349$	−1992.86			−0.305659			−0.152830	+	0.988253i	$0.548839\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3168.04	+	5487.21i	0.477671	+	0.827350i	0.999672	−	0.0255944i	$-0.00814785\pi$
$353$	3168.04	+	5487.21i	0.477671	+	0.827350i	−0.522002	+	0.852944i	$0.674815\pi$
$354$		0			0
$355$	−1625.38	+	2815.25i	−0.243004	+	0.420895i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	452.255	−	783.328i	0.0664877	−	0.115160i	−0.830865	−	0.556474i	$-0.812154\pi$
$359$	452.255	−	783.328i	0.0664877	−	0.115160i	0.897353	+	0.441314i	$0.145487\pi$
$360$		0			0
$361$	2880.60	+	4989.34i	0.419974	+	0.727415i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7306.46			−1.04777
$366$		0			0
$367$	626.380	+	1084.92i	0.0890920	+	0.154312i	0.907128	−	0.420856i	$-0.138270\pi$
$367$	626.380	+	1084.92i	0.0890920	+	0.154312i	−0.818036	+	0.575168i	$0.804937\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1282.95	−	2222.14i	0.178094	−	0.308467i	−0.763134	−	0.646240i	$-0.776340\pi$
$373$	1282.95	−	2222.14i	0.178094	−	0.308467i	0.941228	+	0.337773i	$0.109674\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6558.12			0.895916
$378$		0			0
$379$	−3900.45			−0.528634			−0.264317	−	0.964436i	$-0.585147\pi$
$379$	−3900.45			−0.528634			−0.264317	+	0.964436i	$0.585147\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	834.557	−	1445.49i	0.111342	−	0.192849i	−0.804970	−	0.593316i	$-0.797819\pi$
$383$	834.557	−	1445.49i	0.111342	−	0.192849i	0.916311	+	0.400466i	$0.131152\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	594.871	+	1030.35i	0.0775351	+	0.134295i	0.902186	−	0.431347i	$-0.141962\pi$
$389$	594.871	+	1030.35i	0.0775351	+	0.134295i	−0.824651	+	0.565642i	$0.808628\pi$
$390$		0			0
$391$	3263.13			0.422056
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11112.1	−	19246.7i	−1.41547	−	2.45167i
$396$		0			0
$397$	−401.095	+	694.717i	−0.0507062	+	0.0878257i	−0.890264	−	0.455444i	$-0.849481\pi$
$397$	−401.095	+	694.717i	−0.0507062	+	0.0878257i	0.839558	+	0.543270i	$0.182814\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5266.11	+	9121.17i	−0.655803	+	1.13588i	0.325888	+	0.945408i	$0.394337\pi$
$401$	−5266.11	+	9121.17i	−0.655803	+	1.13588i	−0.981692	+	0.190476i	$0.938997\pi$
$402$		0			0
$403$	−6118.27	−	10597.1i	−0.756260	−	1.30988i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	15807.1			1.92514
$408$		0			0
$409$	−1835.76	−	3179.63i	−0.221937	−	0.384407i	0.733459	−	0.679734i	$-0.237905\pi$
$409$	−1835.76	−	3179.63i	−0.221937	−	0.384407i	−0.955396	+	0.295327i	$0.904571\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−7736.98	+	13400.8i	−0.915165	+	1.58511i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1668.29			−0.194514			−0.0972569	−	0.995259i	$-0.531007\pi$
$419$	−1668.29			−0.194514			−0.0972569	+	0.995259i	$0.531007\pi$
$420$		0			0
$421$	16043.7			1.85730			0.928649	−	0.370961i	$-0.120972\pi$
$421$	16043.7			1.85730			0.928649	+	0.370961i	$0.120972\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6265.44	−	10852.1i	0.715102	−	1.23859i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6493.34	+	11246.8i	0.725692	+	1.25694i	0.958689	+	0.284458i	$0.0918135\pi$
$431$	6493.34	+	11246.8i	0.725692	+	1.25694i	−0.232997	+	0.972478i	$0.574853\pi$
$432$		0			0
$433$	−943.959			−0.104766			−0.0523831	−	0.998627i	$-0.516682\pi$
$433$	−943.959			−0.104766			−0.0523831	+	0.998627i	$0.516682\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1041.21	+	1803.42i	0.113976	+	0.197413i
$438$		0			0
$439$	−3213.66	+	5566.22i	−0.349384	+	0.605151i	−0.986140	−	0.165914i	$-0.946943\pi$
$439$	−3213.66	+	5566.22i	−0.349384	+	0.605151i	0.636756	+	0.771065i	$0.280276\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3627.46	−	6282.94i	0.389042	−	0.673841i	−0.603279	−	0.797530i	$-0.706139\pi$
$443$	3627.46	−	6282.94i	0.389042	−	0.673841i	0.992321	+	0.123690i	$0.0394727\pi$
$444$		0			0
$445$	−3061.18	−	5302.11i	−0.326098	−	0.564818i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−17381.9			−1.82696			−0.913478	−	0.406887i	$-0.866614\pi$
$449$	−17381.9			−1.82696			−0.913478	+	0.406887i	$0.866614\pi$
$450$		0			0
$451$	−9542.09	−	16527.4i	−0.996274	−	1.72560i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6900.41	−	11951.9i	0.706319	−	1.22338i	−0.259895	−	0.965637i	$-0.583688\pi$
$457$	6900.41	−	11951.9i	0.706319	−	1.22338i	0.966214	−	0.257743i	$-0.0829787\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13.6989			0.00138400			0.000691998	−	1.00000i	$-0.499780\pi$
$461$	13.6989			0.00138400			0.000691998		1.00000i	$0.499780\pi$
$462$		0			0
$463$	13910.8			1.39631			0.698153	−	0.715949i	$-0.254006\pi$
$463$	13910.8			1.39631			0.698153	+	0.715949i	$0.254006\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3909.26	−	6771.04i	0.387364	−	0.670935i	−0.604730	−	0.796431i	$-0.706719\pi$
$467$	3909.26	−	6771.04i	0.387364	−	0.670935i	0.992094	+	0.125496i	$0.0400522\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2537.26	−	4394.67i	−0.246646	−	0.427203i
$474$		0			0
$475$	7996.75			0.772455
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9012.70	+	15610.5i	0.859709	+	1.48906i	0.872206	+	0.489138i	$0.162689\pi$
$479$	9012.70	+	15610.5i	0.859709	+	1.48906i	−0.0124969	+	0.999922i	$0.503978\pi$
$480$		0			0
$481$	−9827.62	+	17021.9i	−0.931602	+	1.61358i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10845.3	+	18784.6i	−1.01538	+	1.75869i
$486$		0			0
$487$	8715.70	+	15096.0i	0.810977	+	1.40465i	0.912181	+	0.409788i	$0.134397\pi$
$487$	8715.70	+	15096.0i	0.810977	+	1.40465i	−0.101204	+	0.994866i	$0.532269\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7107.17			−0.653243			−0.326621	−	0.945155i	$-0.605910\pi$
$491$	−7107.17			−0.653243			−0.326621	+	0.945155i	$0.605910\pi$
$492$		0			0
$493$	−3372.72	−	5841.73i	−0.308113	−	0.533668i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1749.80	+	3030.75i	−0.156978	+	0.271894i	−0.933777	−	0.357854i	$-0.883508\pi$
$499$	−1749.80	+	3030.75i	−0.156978	+	0.271894i	0.776800	+	0.629748i	$0.216842\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−9115.24			−0.808009			−0.404005	−	0.914757i	$-0.632382\pi$
$503$	−9115.24			−0.808009			−0.404005	+	0.914757i	$0.632382\pi$
$504$		0			0
$505$	20015.9			1.76375
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8024.85	+	13899.4i	−0.698811	+	1.21038i	0.270067	+	0.962841i	$0.412954\pi$
$509$	−8024.85	+	13899.4i	−0.698811	+	1.21038i	−0.968879	+	0.247536i	$0.920379\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	18166.2	+	31464.8i	1.55437	+	2.69224i
$516$		0			0
$517$	15695.1			1.33514
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2787.24	−	4827.64i	−0.234379	−	0.405956i	0.724713	−	0.689051i	$-0.241972\pi$
$521$	−2787.24	−	4827.64i	−0.234379	−	0.405956i	−0.959092	+	0.283095i	$0.908639\pi$
$522$		0			0
$523$	229.445	−	397.411i	0.0191834	−	0.0332267i	−0.856274	−	0.516521i	$-0.827227\pi$
$523$	229.445	−	397.411i	0.0191834	−	0.0332267i	0.875458	+	0.483295i	$0.160560\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6293.03	+	10899.8i	−0.520168	+	0.900958i
$528$		0			0
$529$	4108.45	+	7116.04i	0.337671	+	0.584864i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	23730.1			1.92845
$534$		0			0
$535$	8314.41	+	14401.0i	0.671894	+	1.16376i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	5486.33	−	9502.61i	0.436000	−	0.755174i	−0.561377	−	0.827560i	$-0.689728\pi$
$541$	5486.33	−	9502.61i	0.436000	−	0.755174i	0.997377	+	0.0723866i	$0.0230615\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4357.28			−0.342468
$546$		0			0
$547$	−19276.8			−1.50679			−0.753396	−	0.657567i	$-0.771586\pi$
$547$	−19276.8			−1.50679			−0.753396	+	0.657567i	$0.771586\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2152.35	−	3727.98i	0.166412	−	0.288235i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6095.71	−	10558.1i	−0.463705	−	0.803161i	0.535437	−	0.844575i	$-0.320147\pi$
$557$	−6095.71	−	10558.1i	−0.463705	−	0.803161i	−0.999142	+	0.0414146i	$0.986814\pi$
$558$		0			0
$559$	6309.87			0.477422
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1618.03	+	2802.51i	0.121122	+	0.209790i	0.920210	−	0.391424i	$-0.128017\pi$
$563$	1618.03	+	2802.51i	0.121122	+	0.209790i	−0.799088	+	0.601214i	$0.794684\pi$
$564$		0			0
$565$	−5564.68	+	9638.31i	−0.414350	+	0.717676i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6083.91	−	10537.6i	0.448244	−	0.776381i	−0.550028	−	0.835146i	$-0.685383\pi$
$569$	6083.91	−	10537.6i	0.448244	−	0.776381i	0.998272	+	0.0587651i	$0.0187163\pi$
$570$		0			0
$571$	12206.2	+	21141.7i	0.894592	+	1.54948i	0.834309	+	0.551297i	$0.185867\pi$
$571$	12206.2	+	21141.7i	0.894592	+	1.54948i	0.0602828	+	0.998181i	$0.480800\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−15169.0			−1.10016
$576$		0			0
$577$	−2173.42	−	3764.48i	−0.156812	−	0.271607i	0.776905	−	0.629618i	$-0.216788\pi$
$577$	−2173.42	−	3764.48i	−0.156812	−	0.271607i	−0.933718	+	0.358011i	$0.883455\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	12410.7	−	21495.9i	0.881641	−	1.52705i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8752.61			−0.615432			−0.307716	−	0.951478i	$-0.599565\pi$
$587$	−8752.61			−0.615432			−0.307716	+	0.951478i	$0.599565\pi$
$588$		0			0
$589$	−8031.97			−0.561887
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3894.39	+	6745.27i	−0.269685	+	0.467108i	−0.968780	−	0.247920i	$-0.920253\pi$
$593$	−3894.39	+	6745.27i	−0.269685	+	0.467108i	0.699095	+	0.715028i	$0.253586\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−572.008	−	990.746i	−0.0390177	−	0.0675806i	0.845857	−	0.533410i	$-0.179090\pi$
$599$	−572.008	−	990.746i	−0.0390177	−	0.0675806i	−0.884875	+	0.465829i	$0.845756\pi$
$600$		0			0
$601$	24673.4			1.67463			0.837314	−	0.546723i	$-0.184125\pi$
$601$	24673.4			1.67463			0.837314	+	0.546723i	$0.184125\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3033.39	+	5253.99i	0.203843	+	0.353066i
$606$		0			0
$607$	3162.81	−	5478.14i	0.211490	−	0.366311i	−0.740691	−	0.671846i	$-0.765502\pi$
$607$	3162.81	−	5478.14i	0.211490	−	0.366311i	0.952181	+	0.305534i	$0.0988351\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9757.93	+	16901.2i	−0.646094	+	1.11907i
$612$		0			0
$613$	−11488.1	−	19898.0i	−0.756935	−	1.31105i	−0.944407	−	0.328779i	$-0.893363\pi$
$613$	−11488.1	−	19898.0i	−0.756935	−	1.31105i	0.187472	−	0.982270i	$-0.439971\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8229.26			−0.536949			−0.268475	−	0.963287i	$-0.586520\pi$
$617$	−8229.26			−0.536949			−0.268475	+	0.963287i	$0.586520\pi$
$618$		0			0
$619$	−8426.40	−	14595.0i	−0.547150	−	0.947691i	−0.998468	−	0.0553284i	$-0.982379\pi$
$619$	−8426.40	−	14595.0i	−0.547150	−	0.947691i	0.451318	−	0.892363i	$-0.350954\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−6228.43	+	10788.0i	−0.398620	+	0.690429i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	20216.7			1.28154
$630$		0			0
$631$	−8408.46			−0.530484			−0.265242	−	0.964182i	$-0.585452\pi$
$631$	−8408.46			−0.530484			−0.265242	+	0.964182i	$0.585452\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10805.7	−	18715.9i	0.675290	−	1.16964i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4561.26	+	7900.33i	0.281059	+	0.486809i	0.971646	−	0.236441i	$-0.0759810\pi$
$641$	4561.26	+	7900.33i	0.281059	+	0.486809i	−0.690587	+	0.723250i	$0.742648\pi$
$642$		0			0
$643$	−19279.4			−1.18243			−0.591217	−	0.806513i	$-0.701352\pi$
$643$	−19279.4			−1.18243			−0.591217	+	0.806513i	$0.701352\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1088.93	−	1886.08i	−0.0661673	−	0.114605i	0.831044	−	0.556207i	$-0.187744\pi$
$647$	−1088.93	−	1886.08i	−0.0661673	−	0.114605i	−0.897211	+	0.441602i	$0.854410\pi$
$648$		0			0
$649$	4596.03	−	7960.55i	0.277981	−	0.481478i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3144.42	+	5446.30i	−0.188439	+	0.326386i	−0.944730	−	0.327849i	$-0.893676\pi$
$653$	−3144.42	+	5446.30i	−0.188439	+	0.326386i	0.756291	+	0.654236i	$0.227010\pi$
$654$		0			0
$655$	8418.51	+	14581.3i	0.502196	+	0.869829i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−20346.5			−1.20271			−0.601357	−	0.798980i	$-0.705373\pi$
$659$	−20346.5			−1.20271			−0.601357	+	0.798980i	$0.705373\pi$
$660$		0			0
$661$	13834.6	+	23962.2i	0.814074	+	1.41002i	0.909991	+	0.414628i	$0.136088\pi$
$661$	13834.6	+	23962.2i	0.814074	+	1.41002i	−0.0959168	+	0.995389i	$0.530578\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−4082.77	+	7071.57i	−0.237010	+	0.410513i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	29439.2			1.69372
$672$		0			0
$673$	11862.6			0.679450			0.339725	−	0.940525i	$-0.389666\pi$
$673$	11862.6			0.679450			0.339725	+	0.940525i	$0.389666\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6763.29	−	11714.4i	0.383950	−	0.665022i	−0.607673	−	0.794188i	$-0.707897\pi$
$677$	6763.29	−	11714.4i	0.383950	−	0.665022i	0.991623	+	0.129166i	$0.0412300\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−356.931	−	618.223i	−0.0199965	−	0.0346349i	0.855854	−	0.517217i	$-0.173032\pi$
$683$	−356.931	−	618.223i	−0.0199965	−	0.0346349i	−0.875850	+	0.482583i	$0.839699\pi$
$684$		0			0
$685$	−6188.06			−0.345159
$686$		0			0
$687$		0			0
$688$		0			0
$689$	15431.9	+	26728.9i	0.853279	+	1.47792i
$690$		0			0
$691$	7908.04	−	13697.1i	0.435364	−	0.754072i	−0.561962	−	0.827163i	$-0.689953\pi$
$691$	7908.04	−	13697.1i	0.435364	−	0.754072i	0.997325	+	0.0730915i	$0.0232865\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	16371.1	−	28355.6i	0.893512	−	1.54761i
$696$		0			0
$697$	−12203.9	−	21137.8i	−0.663210	−	1.14871i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−28556.9			−1.53863			−0.769314	−	0.638871i	$-0.779402\pi$
$701$	−28556.9			−1.53863			−0.769314	+	0.638871i	$0.779402\pi$
$702$		0			0
$703$	6450.77	+	11173.1i	0.346082	+	0.599431i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	967.358	−	1675.51i	0.0512411	−	0.0887521i	−0.839267	−	0.543719i	$-0.817016\pi$
$709$	967.358	−	1675.51i	0.0512411	−	0.0887521i	0.890508	+	0.454967i	$0.150349\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	15235.8			0.800258
$714$		0			0
$715$	−39221.3			−2.05146
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9333.33	−	16165.8i	0.484109	−	0.838502i	−0.515724	−	0.856755i	$-0.672477\pi$
$719$	9333.33	−	16165.8i	0.484109	−	0.838502i	0.999833	+	0.0182531i	$0.00581046\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	15678.4	+	27155.8i	0.803146	+	1.39109i
$726$		0			0
$727$	−25955.3			−1.32411			−0.662055	−	0.749455i	$-0.730315\pi$
$727$	−25955.3			−1.32411			−0.662055	+	0.749455i	$0.730315\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3245.05	−	5620.59i	−0.164190	−	0.284385i
$732$		0			0
$733$	1041.89	−	1804.61i	0.0525010	−	0.0909344i	−0.838581	−	0.544778i	$-0.816614\pi$
$733$	1041.89	−	1804.61i	0.0525010	−	0.0909344i	0.891082	+	0.453843i	$0.149947\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	21213.2	−	36742.3i	1.06024	−	1.83639i
$738$		0			0
$739$	−14428.7	−	24991.2i	−0.718223	−	1.24400i	−0.961703	−	0.274092i	$-0.911623\pi$
$739$	−14428.7	−	24991.2i	−0.718223	−	1.24400i	0.243481	−	0.969906i	$-0.421711\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	899.017			0.0443900			0.0221950	−	0.999754i	$-0.492935\pi$
$743$	899.017			0.0443900			0.0221950	+	0.999754i	$0.492935\pi$
$744$		0			0
$745$	9185.24	+	15909.3i	0.451706	+	0.782378i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	14504.7	−	25122.8i	0.704770	−	1.22070i	−0.262004	−	0.965067i	$-0.584383\pi$
$751$	14504.7	−	25122.8i	0.704770	−	1.22070i	0.966774	−	0.255631i	$-0.0822832\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	30985.2			1.49360
$756$		0			0
$757$	−10932.4			−0.524892			−0.262446	−	0.964947i	$-0.584529\pi$
$757$	−10932.4			−0.524892			−0.262446	+	0.964947i	$0.584529\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6606.40	+	11442.6i	−0.314693	+	0.545065i	−0.979372	−	0.202064i	$-0.935235\pi$
$761$	−6606.40	+	11442.6i	−0.314693	+	0.545065i	0.664679	+	0.747129i	$0.268568\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5714.89	+	9898.47i	0.269039	+	0.465988i
$768$		0			0
$769$	30129.7			1.41288			0.706440	−	0.707773i	$-0.250300\pi$
$769$	30129.7			1.41288			0.706440	+	0.707773i	$0.250300\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5920.37	+	10254.4i	0.275474	+	0.477134i	0.970254	−	0.242087i	$-0.0778320\pi$
$773$	5920.37	+	10254.4i	0.275474	+	0.477134i	−0.694781	+	0.719221i	$0.744499\pi$
$774$		0			0
$775$	29253.7	−	50668.9i	1.35590	−	2.34849i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7788.11	−	13489.4i	0.358200	−	0.620421i
$780$		0			0
$781$	−3447.31	−	5970.92i	−0.157944	−	0.273567i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	10667.5			0.485020
$786$		0			0
$787$	−3264.55	−	5654.36i	−0.147863	−	0.256107i	0.782574	−	0.622557i	$-0.213906\pi$
$787$	−3264.55	−	5654.36i	−0.147863	−	0.256107i	−0.930438	+	0.366450i	$0.880573\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−18303.0	+	31701.6i	−0.819618	+	1.41962i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	41987.6			1.86609			0.933046	−	0.359757i	$-0.117140\pi$
$797$	41987.6			1.86609			0.933046	+	0.359757i	$0.117140\pi$
$798$		0			0
$799$	20073.3			0.888790
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7748.21	−	13420.3i	0.340509	−	0.589778i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6817.06	−	11807.5i	−0.296261	−	0.513139i	0.679017	−	0.734123i	$-0.262406\pi$
$809$	−6817.06	−	11807.5i	−0.296261	−	0.513139i	−0.975277	+	0.220984i	$0.929073\pi$
$810$		0			0
$811$	20093.9			0.870025			0.435013	−	0.900424i	$-0.356744\pi$
$811$	20093.9			0.870025			0.435013	+	0.900424i	$0.356744\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3503.00	−	6067.38i	−0.150558	−	0.260774i
$816$		0			0
$817$	2070.87	−	3586.86i	0.0886790	−	0.153596i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3818.15	+	6613.23i	−0.162307	+	0.281124i	−0.935696	−	0.352808i	$-0.885227\pi$
$821$	−3818.15	+	6613.23i	−0.162307	+	0.281124i	0.773389	+	0.633932i	$0.218560\pi$
$822$		0			0
$823$	14394.2	+	24931.5i	0.609660	+	1.05596i	0.991296	+	0.131650i	$0.0420275\pi$
$823$	14394.2	+	24931.5i	0.609660	+	1.05596i	−0.381636	+	0.924313i	$0.624639\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20515.4			0.862623			0.431312	−	0.902203i	$-0.358051\pi$
$827$	20515.4			0.862623			0.431312	+	0.902203i	$0.358051\pi$
$828$		0			0
$829$	8132.21	+	14085.4i	0.340704	+	0.590116i	0.984564	−	0.175027i	$-0.0560014\pi$
$829$	8132.21	+	14085.4i	0.340704	+	0.590116i	−0.643860	+	0.765143i	$0.722668\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−27121.8	+	46976.4i	−1.12406	+	1.94693i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−13770.2			−0.566629			−0.283314	−	0.959027i	$-0.591434\pi$
$839$	−13770.2			−0.566629			−0.283314	+	0.959027i	$0.591434\pi$
$840$		0			0
$841$	−7509.45			−0.307903
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3359.03	−	5818.01i	0.136751	−	0.236859i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−12236.4	−	21194.1i	−0.492901	−	0.853729i
$852$		0			0
$853$	8031.48			0.322383			0.161191	−	0.986923i	$-0.448466\pi$
$853$	8031.48			0.322383			0.161191	+	0.986923i	$0.448466\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	21382.0	+	37034.7i	0.852270	+	1.47618i	0.879154	+	0.476537i	$0.158108\pi$
$857$	21382.0	+	37034.7i	0.852270	+	1.47618i	−0.0268841	+	0.999639i	$0.508558\pi$
$858$		0			0
$859$	4077.52	−	7062.48i	0.161960	−	0.280522i	−0.773612	−	0.633660i	$-0.781552\pi$
$859$	4077.52	−	7062.48i	0.161960	−	0.280522i	0.935571	+	0.353138i	$0.114885\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	20330.0	−	35212.6i	0.801901	−	1.38893i	−0.116462	−	0.993195i	$-0.537155\pi$
$863$	20330.0	−	35212.6i	0.801901	−	1.38893i	0.918363	−	0.395738i	$-0.129511\pi$
$864$		0			0
$865$	34169.2	+	59182.9i	1.34311	+	2.32633i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	47135.8			1.84002
$870$		0			0
$871$	26377.3	+	45686.9i	1.02613	+	1.77731i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	14130.3	−	24474.4i	0.544067	−	0.942351i	−0.454598	−	0.890697i	$-0.650217\pi$
$877$	14130.3	−	24474.4i	0.544067	−	0.942351i	0.998665	−	0.0516546i	$-0.0164495\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	15951.2			0.609998			0.304999	−	0.952353i	$-0.401344\pi$
$881$	15951.2			0.609998			0.304999	+	0.952353i	$0.401344\pi$
$882$		0			0
$883$	−1750.48			−0.0667137			−0.0333569	−	0.999444i	$-0.510620\pi$
$883$	−1750.48			−0.0667137			−0.0333569	+	0.999444i	$0.510620\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1003.51	+	1738.13i	−0.0379871	+	0.0657955i	−0.884394	−	0.466741i	$-0.845428\pi$
$887$	−1003.51	+	1738.13i	−0.0379871	+	0.0657955i	0.846407	+	0.532537i	$0.178761\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6405.03	+	11093.8i	0.240018	+	0.415724i
$894$		0			0
$895$	−70386.3			−2.62878
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−15747.4	−	27275.4i	−0.584212	−	1.01188i
$900$		0			0
$901$	15872.7	−	27492.3i	0.586900	−	1.01654i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−16443.3	+	28480.6i	−0.603970	+	1.04611i
$906$		0			0
$907$	6034.85	+	10452.7i	0.220930	+	0.382662i	0.955091	−	0.296314i	$-0.0957573\pi$
$907$	6034.85	+	10452.7i	0.220930	+	0.382662i	−0.734160	+	0.678976i	$0.762424\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−40167.1			−1.46081			−0.730404	−	0.683016i	$-0.760668\pi$
$911$	−40167.1			−1.46081			−0.730404	+	0.683016i	$0.760668\pi$
$912$		0			0
$913$	−16409.5	−	28422.1i	−0.594826	−	1.03027i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	6751.73	−	11694.3i	0.242349	−	0.419761i	−0.719034	−	0.694975i	$-0.755415\pi$
$919$	6751.73	−	11694.3i	0.242349	−	0.419761i	0.961383	+	0.275214i	$0.0887486\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	8573.05			0.305726
$924$		0			0
$925$	−93978.9			−3.34055
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−15501.0	+	26848.5i	−0.547438	+	0.948191i	0.451011	+	0.892518i	$0.351064\pi$
$929$	−15501.0	+	26848.5i	−0.547438	+	0.948191i	−0.998449	+	0.0556722i	$0.982270\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	20170.8	+	34936.9i	0.705515	+	1.22199i
$936$		0			0
$937$	34998.0			1.22021			0.610104	−	0.792322i	$-0.291128\pi$
$937$	34998.0			1.22021			0.610104	+	0.792322i	$0.291128\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5826.27	+	10091.4i	0.201839	+	0.349596i	0.949121	−	0.314911i	$-0.101975\pi$
$941$	5826.27	+	10091.4i	0.201839	+	0.349596i	−0.747282	+	0.664507i	$0.768641\pi$
$942$		0			0
$943$	−14773.2	+	25587.9i	−0.510160	+	0.883624i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1460.92	−	2530.39i	0.0501306	−	0.0868287i	−0.839871	−	0.542786i	$-0.817370\pi$
$947$	1460.92	−	2530.39i	0.0501306	−	0.0868287i	0.890002	+	0.455957i	$0.150703\pi$
$948$		0			0
$949$	9634.44	+	16687.3i	0.329554	+	0.570805i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−24262.5			−0.824702			−0.412351	−	0.911025i	$-0.635292\pi$
$953$	−24262.5			−0.824702			−0.412351	+	0.911025i	$0.635292\pi$
$954$		0			0
$955$	−18748.4	−	32473.2i	−0.635271	−	1.10032i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−14487.0	+	25092.3i	−0.486289	+	0.842277i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	47706.5			1.59143
$966$		0			0
$967$	10258.0			0.341131			0.170565	−	0.985346i	$-0.445441\pi$
$967$	10258.0			0.341131			0.170565	+	0.985346i	$0.445441\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−8317.85	+	14406.9i	−0.274905	+	0.476149i	−0.970111	−	0.242661i	$-0.921980\pi$
$971$	−8317.85	+	14406.9i	−0.274905	+	0.476149i	0.695206	+	0.718810i	$0.255313\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	6425.08	+	11128.6i	0.210395	+	0.364416i	0.951838	−	0.306600i	$-0.0991915\pi$
$977$	6425.08	+	11128.6i	0.210395	+	0.364416i	−0.741443	+	0.671016i	$0.765858\pi$
$978$		0			0
$979$	12985.0			0.423905
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−21136.4	−	36609.3i	−0.685805	−	1.18785i	−0.973183	−	0.230032i	$-0.926117\pi$
$983$	−21136.4	−	36609.3i	−0.685805	−	1.18785i	0.287378	−	0.957817i	$-0.407217\pi$
$984$		0			0
$985$	−20809.8	+	36043.6i	−0.673153	+	1.16593i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3928.22	+	6803.88i	−0.126299	+	0.218757i
$990$		0			0
$991$	5752.08	+	9962.90i	0.184380	+	0.319356i	0.943368	−	0.331749i	$-0.107639\pi$
$991$	5752.08	+	9962.90i	0.184380	+	0.319356i	−0.758987	+	0.651106i	$0.774305\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−73969.9			−2.35679
$996$		0			0
$997$	−8590.47	−	14879.1i	−0.272881	−	0.472645i	0.696717	−	0.717346i	$-0.254643\pi$
$997$	−8590.47	−	14879.1i	−0.272881	−	0.472645i	−0.969598	+	0.244702i	$0.921310\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.bd.1549.4		8
3.2	odd	2		588.4.i.k.373.1		8
7.2	even	3		1764.4.a.ba.1.1		4
7.3	odd	6		1764.4.k.bb.361.1		8
7.4	even	3	inner	1764.4.k.bd.361.4		8
7.5	odd	6		1764.4.a.bc.1.4		4
7.6	odd	2		1764.4.k.bb.1549.1		8
21.2	odd	6		588.4.a.k.1.4	yes	4
21.5	even	6		588.4.a.j.1.1	✓	4
21.11	odd	6		588.4.i.k.361.1		8
21.17	even	6		588.4.i.l.361.4		8
21.20	even	2		588.4.i.l.373.4		8
84.23	even	6		2352.4.a.cl.1.4		4
84.47	odd	6		2352.4.a.cq.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.4.a.j.1.1	✓	4	21.5	even	6
588.4.a.k.1.4	yes	4	21.2	odd	6
588.4.i.k.361.1		8	21.11	odd	6
588.4.i.k.373.1		8	3.2	odd	2
588.4.i.l.361.4		8	21.17	even	6
588.4.i.l.373.4		8	21.20	even	2
1764.4.a.ba.1.1		4	7.2	even	3
1764.4.a.bc.1.4		4	7.5	odd	6
1764.4.k.bb.361.1		8	7.3	odd	6
1764.4.k.bb.1549.1		8	7.6	odd	2
1764.4.k.bd.361.4		8	7.4	even	3	inner
1764.4.k.bd.1549.4		8	1.1	even	1	trivial
2352.4.a.cl.1.4		4	84.23	even	6
2352.4.a.cq.1.1		4	84.47	odd	6

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(9.57016 - 16.5760i) q^{5} +O(q^{10})\)	\(q+(9.57016 - 16.5760i) q^{5} +(20.2976 + 35.1564i) q^{11} -50.4776 q^{13} +(25.9597 + 44.9636i) q^{17} +(-16.5666 + 28.6941i) q^{19} +(31.4249 - 54.4296i) q^{23} +(-120.676 - 209.017i) q^{25} -129.921 q^{29} +(121.208 + 209.938i) q^{31} +(194.693 - 337.217i) q^{37} -470.110 q^{41} -125.003 q^{43} +(193.312 - 334.826i) q^{47} +(-305.718 - 529.519i) q^{53} +777.004 q^{55} +(-113.216 - 196.096i) q^{59} +(362.595 - 628.034i) q^{61} +(-483.079 + 836.718i) q^{65} +(-522.555 - 905.092i) q^{67} -169.839 q^{71} +(-190.866 - 330.589i) q^{73} +(580.560 - 1005.56i) q^{79} -808.448 q^{83} +993.756 q^{85} +(159.933 - 277.013i) q^{89} +(317.089 + 549.215i) q^{95} -1133.24 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 48 q^{17} - 192 q^{19} + 192 q^{23} - 324 q^{25} - 192 q^{29} - 48 q^{31} - 256 q^{37} - 2016 q^{41} - 224 q^{43} + 864 q^{47} - 648 q^{53} + 4704 q^{55} + 336 q^{59} - 960 q^{61} - 360 q^{65} - 720 q^{67} + 2688 q^{71} - 672 q^{73} + 1984 q^{79} - 6240 q^{83} + 1360 q^{85} + 2160 q^{89} - 3744 q^{95} + 4032 q^{97}+O(q^{100}) \) 8 * q + 48 * q^17 - 192 * q^19 + 192 * q^23 - 324 * q^25 - 192 * q^29 - 48 * q^31 - 256 * q^37 - 2016 * q^41 - 224 * q^43 + 864 * q^47 - 648 * q^53 + 4704 * q^55 + 336 * q^59 - 960 * q^61 - 360 * q^65 - 720 * q^67 + 2688 * q^71 - 672 * q^73 + 1984 * q^79 - 6240 * q^83 + 1360 * q^85 + 2160 * q^89 - 3744 * q^95 + 4032 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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