LMFDB - Embedded newform 592.3.z.a.47.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,3,Mod(47,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("592.47");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	592.z (of order $6$, degree $2$, 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:	$16.1308316501$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			47.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	592.47
Dual form			592.3.z.a.63.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.96410 + 3.40192i) q^{5} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})$	$q+(1.96410 + 3.40192i) q^{5} +(-4.50000 + 7.79423i) q^{9} +(-5.00000 - 8.66025i) q^{13} +(-14.4282 + 24.9904i) q^{17} +(4.78461 - 8.28719i) q^{25} -55.6410 q^{29} +(27.8923 + 24.3109i) q^{37} +(-30.1410 - 52.2058i) q^{41} -35.3538 q^{45} +(-24.5000 + 42.4352i) q^{49} +(-45.0000 + 77.9423i) q^{53} +(46.4615 + 80.4737i) q^{61} +(19.6410 - 34.0192i) q^{65} -110.000 q^{73} +(-40.5000 - 70.1481i) q^{81} -113.354 q^{85} +(-88.7820 + 153.775i) q^{89} +59.7077 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{5} - 18 q^{9}+O(q^{10}) $ 4 * q - 6 * q^5 - 18 * q^9	$ 4 q - 6 q^{5} - 18 q^{9} - 20 q^{13} - 30 q^{17} - 64 q^{25} - 84 q^{29} + 70 q^{37} + 18 q^{41} + 108 q^{45} - 98 q^{49} - 180 q^{53} - 22 q^{61} - 60 q^{65} - 440 q^{73} - 162 q^{81} - 204 q^{85} - 78 q^{89} - 260 q^{97}+O(q^{100}) $ 4 * q - 6 * q^5 - 18 * q^9 - 20 * q^13 - 30 * q^17 - 64 * q^25 - 84 * q^29 + 70 * q^37 + 18 * q^41 + 108 * q^45 - 98 * q^49 - 180 * q^53 - 22 * q^61 - 60 * q^65 - 440 * q^73 - 162 * q^81 - 204 * q^85 - 78 * q^89 - 260 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/592\mathbb{Z}\right)^\times$.

$n$	$113$	$149$	$223$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$-1$

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		−0.500000	−	0.866025i	$-0.666667\pi$
$3$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$4$		0			0
$5$	1.96410	+	3.40192i	0.392820	+	0.680385i	0.992820	−	0.119615i	$-0.0381661\pi$
$5$	1.96410	+	3.40192i	0.392820	+	0.680385i	−0.600000	+	0.800000i	$0.704833\pi$
$6$		0			0
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$		0			0
$9$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−5.00000	−	8.66025i	−0.384615	−	0.666173i	0.607100	−	0.794625i	$-0.292333\pi$
$13$	−5.00000	−	8.66025i	−0.384615	−	0.666173i	−0.991716	+	0.128452i	$0.958999\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−14.4282	+	24.9904i	−0.848718	+	1.47002i	0.0336351	+	0.999434i	$0.489292\pi$
$17$	−14.4282	+	24.9904i	−0.848718	+	1.47002i	−0.882353	+	0.470588i	$0.844042\pi$
$18$		0			0
$19$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	4.78461	−	8.28719i	0.191384	−	0.331487i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−55.6410			−1.91866			−0.959328	−	0.282294i	$-0.908905\pi$
$29$	−55.6410			−1.91866			−0.959328	+	0.282294i	$0.908905\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	27.8923	+	24.3109i	0.753846	+	0.657051i
$37$	27.8923	+	24.3109i	0.753846	+	0.657051i
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−30.1410	−	52.2058i	−0.735147	−	1.27331i	−0.954659	−	0.297702i	$-0.903780\pi$
$41$	−30.1410	−	52.2058i	−0.735147	−	1.27331i	0.219512	−	0.975610i	$-0.429553\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−35.3538			−0.785641
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−24.5000	+	42.4352i	−0.500000	+	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−45.0000	+	77.9423i	−0.849057	+	1.47061i	0.0329946	+	0.999456i	$0.489496\pi$
$53$	−45.0000	+	77.9423i	−0.849057	+	1.47061i	−0.882051	+	0.471154i	$0.843838\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	46.4615	+	80.4737i	0.761664	+	1.31924i	0.941992	+	0.335635i	$0.108951\pi$
$61$	46.4615	+	80.4737i	0.761664	+	1.31924i	−0.180328	+	0.983607i	$0.557716\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	19.6410	−	34.0192i	0.302169	−	0.523373i
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$71$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$72$		0			0
$73$	−110.000			−1.50685			−0.753425	−	0.657534i	$-0.771599\pi$
$73$	−110.000			−1.50685			−0.753425	+	0.657534i	$0.771599\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	−40.5000	−	70.1481i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$	−113.354			−1.33357
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−88.7820	+	153.775i	−0.997551	+	1.72781i	−0.438202	+	0.898876i	$0.644385\pi$
$89$	−88.7820	+	153.775i	−0.997551	+	1.72781i	−0.559349	+	0.828932i	$0.688949\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	59.7077			0.615543			0.307771	−	0.951460i	$-0.400417\pi$
$97$	59.7077			0.615543			0.307771	+	0.951460i	$0.400417\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	133.641			1.32318			0.661589	−	0.749866i	$-0.269882\pi$
$101$	133.641			1.32318			0.661589	+	0.749866i	$0.269882\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$	6.46152	−	11.1917i	0.0592800	−	0.102676i	−0.834862	−	0.550459i	$-0.814453\pi$
$109$	6.46152	−	11.1917i	0.0592800	−	0.102676i	0.894142	+	0.447783i	$0.147786\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	15.0000	−	25.9808i	0.132743	−	0.229918i	−0.791990	−	0.610534i	$-0.790955\pi$
$113$	15.0000	−	25.9808i	0.132743	−	0.229918i	0.924733	+	0.380616i	$0.124288\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	90.0000			0.769231
$118$		0			0
$119$		0			0
$120$		0			0
$121$	121.000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	135.795			1.08636
$126$		0			0
$127$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$127$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−257.420			−1.87898			−0.939491	−	0.342574i	$-0.888701\pi$
$137$	−257.420			−1.87898			−0.939491	+	0.342574i	$0.888701\pi$
$138$		0			0
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−109.285	−	189.286i	−0.753687	−	1.30542i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	293.487			1.96971			0.984856	−	0.173374i	$-0.0554669\pi$
$149$	293.487			1.96971			0.984856	+	0.173374i	$0.0554669\pi$
$150$		0			0
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$	−129.854	−	224.913i	−0.848718	−	1.47002i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	156.815	−	271.612i	0.998824	−	1.73001i	0.457423	−	0.889249i	$-0.348773\pi$
$157$	156.815	−	271.612i	0.998824	−	1.73001i	0.541401	−	0.840764i	$-0.317894\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	34.5000	−	59.7558i	0.204142	−	0.353584i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	37.4667	−	64.8942i	0.216570	−	0.375111i	−0.737187	−	0.675689i	$-0.763846\pi$
$173$	37.4667	−	64.8942i	0.216570	−	0.375111i	0.953757	+	0.300578i	$0.0971796\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	146.385	+	253.546i	0.808755	+	1.40080i	0.913727	+	0.406329i	$0.133191\pi$
$181$	146.385	+	253.546i	0.808755	+	1.40080i	−0.104972	+	0.994475i	$0.533475\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−27.9205	+	142.637i	−0.150921	+	0.771008i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	385.985			1.99992			0.999960	−	0.00895123i	$-0.00284930\pi$
$193$	385.985			1.99992			0.999960	+	0.00895123i	$0.00284930\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−73.2513	+	126.875i	−0.371834	+	0.644035i	−0.989848	−	0.142132i	$-0.954604\pi$
$197$	−73.2513	+	126.875i	−0.371834	+	0.644035i	0.618014	+	0.786167i	$0.287938\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	118.400	−	205.075i	0.577561	−	1.00037i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	288.564			1.30572
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	43.0615	+	74.5847i	0.191384	+	0.331487i
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$	58.5385	+	101.392i	0.255627	+	0.442758i	0.965066	−	0.262009i	$-0.0843849\pi$
$229$	58.5385	+	101.392i	0.255627	+	0.442758i	−0.709439	+	0.704767i	$0.751052\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	255.267			1.09556			0.547782	−	0.836621i	$-0.315472\pi$
$233$	255.267			1.09556			0.547782	+	0.836621i	$0.315472\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	−209.000	−	361.999i	−0.867220	−	1.50207i	−0.864826	−	0.502072i	$-0.832571\pi$
$241$	−209.000	−	361.999i	−0.867220	−	1.50207i	−0.00239399	−	0.999997i	$-0.500762\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−192.482			−0.785641
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−99.7872	+	172.836i	−0.388277	+	0.672515i	−0.992218	−	0.124514i	$-0.960263\pi$
$257$	−99.7872	+	172.836i	−0.388277	+	0.672515i	0.603941	+	0.797029i	$0.293596\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	250.385	−	433.679i	0.959328	−	1.66160i
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$	−353.538			−1.33411
$266$		0			0
$267$		0			0
$268$		0			0
$269$	138.000			0.513011			0.256506	−	0.966543i	$-0.417429\pi$
$269$	138.000			0.513011			0.256506	+	0.966543i	$0.417429\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	160.738	+	278.407i	0.580283	+	1.00508i	0.995445	+	0.0953324i	$0.0303914\pi$
$277$	160.738	+	278.407i	0.580283	+	1.00508i	−0.415162	+	0.909747i	$0.636275\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	23.0641	−	39.9481i	0.0820785	−	0.142164i	−0.822064	−	0.569395i	$-0.807178\pi$
$281$	23.0641	−	39.9481i	0.0820785	−	0.142164i	0.904143	+	0.427231i	$0.140511\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−271.846	−	470.851i	−0.940644	−	1.62924i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	83.6103	+	144.817i	0.285359	+	0.494257i	0.972696	−	0.232082i	$-0.0745537\pi$
$293$	83.6103	+	144.817i	0.285359	+	0.494257i	−0.687337	+	0.726339i	$0.741220\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−182.510	+	316.117i	−0.598394	+	1.03645i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	282.700	−	489.651i	0.903195	−	1.56438i	0.0798722	−	0.996805i	$-0.474549\pi$
$313$	282.700	−	489.651i	0.903195	−	1.56438i	0.823322	−	0.567574i	$-0.192118\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−304.236	+	526.952i	−0.959734	+	1.66231i	−0.236593	+	0.971609i	$0.576031\pi$
$317$	−304.236	+	526.952i	−0.959734	+	1.66231i	−0.723141	+	0.690700i	$0.757303\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−95.6922			−0.294438
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$	−315.000	+	108.000i	−0.945946	+	0.324324i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−336.915	−	583.554i	−0.999749	−	1.73162i	−0.519288	−	0.854599i	$-0.673803\pi$
$337$	−336.915	−	583.554i	−0.999749	−	1.73162i	−0.480461	−	0.877016i	$-0.659531\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	−305.385	+	528.942i	−0.875027	+	1.51559i	−0.0182939	+	0.999833i	$0.505823\pi$
$349$	−305.385	+	528.942i	−0.875027	+	1.51559i	−0.856734	+	0.515759i	$0.827510\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−123.059	+	213.144i	−0.348609	+	0.603808i	−0.986003	−	0.166730i	$-0.946679\pi$
$353$	−123.059	+	213.144i	−0.348609	+	0.603808i	0.637394	+	0.770538i	$0.280012\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−180.500	+	312.635i	−0.500000	+	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−216.051	−	374.212i	−0.591921	−	1.02524i
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		0			0
$369$	542.538			1.47029
$370$		0			0
$371$		0			0
$372$		0			0
$373$	80.7384	+	139.843i	0.216457	+	0.374914i	0.953722	−	0.300689i	$-0.0972166\pi$
$373$	80.7384	+	139.843i	0.216457	+	0.374914i	−0.737265	+	0.675603i	$0.763883\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	278.205	+	481.865i	0.737945	+	1.27816i
$378$		0			0
$379$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$379$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	388.949	+	673.679i	0.999868	+	1.73182i	0.514007	+	0.857786i	$0.328161\pi$
$389$	388.949	+	673.679i	0.999868	+	1.73182i	0.485861	+	0.874036i	$0.338506\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	69.9076			0.176090			0.0880448	−	0.996117i	$-0.471938\pi$
$397$	69.9076			0.176090			0.0880448	+	0.996117i	$0.471938\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−798.000			−1.99002			−0.995012	−	0.0997506i	$-0.968195\pi$
$401$	−798.000			−1.99002			−0.995012	+	0.0997506i	$0.968195\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	159.092	−	275.556i	0.392820	−	0.680385i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−299.423	+	518.616i	−0.732086	+	1.26801i	0.223905	+	0.974611i	$0.428120\pi$
$409$	−299.423	+	518.616i	−0.732086	+	1.26801i	−0.955990	+	0.293399i	$0.905214\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	−756.461			−1.79682			−0.898410	−	0.439157i	$-0.855277\pi$
$421$	−756.461			−1.79682			−0.898410	+	0.439157i	$0.855277\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	138.067	+	239.138i	0.324863	+	0.562679i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$	−851.677			−1.96692			−0.983460	−	0.181123i	$-0.942027\pi$
$433$	−851.677			−1.96692			−0.983460	+	0.181123i	$0.942027\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	−220.500	−	381.917i	−0.500000	−	0.866025i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−697.508			−1.56743
$446$		0			0
$447$		0			0
$448$		0			0
$449$	351.000	+	607.950i	0.781737	+	1.35401i	0.930929	+	0.365200i	$0.118999\pi$
$449$	351.000	+	607.950i	0.781737	+	1.35401i	−0.149192	+	0.988808i	$0.547667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	67.0077	+	116.061i	0.146625	+	0.253962i	0.929978	−	0.367615i	$-0.119826\pi$
$457$	67.0077	+	116.061i	0.146625	+	0.253962i	−0.783353	+	0.621577i	$0.786492\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−261.000	+	452.065i	−0.566161	+	0.980619i	0.430780	+	0.902457i	$0.358238\pi$
$461$	−261.000	+	452.065i	−0.566161	+	0.980619i	−0.996941	+	0.0781619i	$0.975095\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−405.000	−	701.481i	−0.849057	−	1.47061i
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	71.0770	−	363.109i	0.147769	−	0.754904i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	117.272	+	203.121i	0.241798	+	0.418806i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	802.800	−	1390.49i	1.62840	−	2.82047i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$503$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$504$		0			0
$505$	262.485	+	454.637i	0.519771	+	0.900270i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−38.9744	+	67.5057i	−0.0765706	+	0.132624i	−0.901768	−	0.432220i	$-0.857730\pi$
$509$	−38.9744	+	67.5057i	−0.0765706	+	0.132624i	0.825198	+	0.564844i	$0.191064\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	279.000	+	483.242i	0.535509	+	0.927528i	0.999139	+	0.0414992i	$0.0132134\pi$
$521$	279.000	+	483.242i	0.535509	+	0.927528i	−0.463630	+	0.886029i	$0.653453\pi$
$522$		0			0
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	529.000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−301.410	+	522.058i	−0.565497	+	0.979470i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	386.461			0.714346			0.357173	−	0.934038i	$-0.383741\pi$
$541$	386.461			0.714346			0.357173	+	0.934038i	$0.383741\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	50.7644			0.0931456
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	−836.307			−1.52333
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	543.226	−	940.894i	0.975270	−	1.68922i	0.296230	−	0.955117i	$-0.404271\pi$
$557$	543.226	−	940.894i	0.975270	−	1.68922i	0.679040	−	0.734101i	$-0.262396\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$	117.846			0.208577
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−669.666			−1.17692			−0.588459	−	0.808527i	$-0.700265\pi$
$569$	−669.666			−1.17692			−0.588459	+	0.808527i	$0.700265\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	575.000	−	995.929i	0.996534	−	1.72605i	0.426223	−	0.904618i	$-0.359844\pi$
$577$	575.000	−	995.929i	0.996534	−	1.72605i	0.570311	−	0.821429i	$-0.306823\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	176.769	+	306.173i	0.302169	+	0.523373i
$586$		0			0
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	172.395			0.290716			0.145358	−	0.989379i	$-0.453567\pi$
$593$	172.395			0.290716			0.145358	+	0.989379i	$0.453567\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	−67.6539	+	117.180i	−0.112569	+	0.194975i	−0.916805	−	0.399334i	$-0.869241\pi$
$601$	−67.6539	+	117.180i	−0.112569	+	0.194975i	0.804236	+	0.594309i	$0.202575\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	237.656	+	411.633i	0.392820	+	0.680385i
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	512.508	−	887.689i	0.836065	−	1.44811i	−0.0570962	−	0.998369i	$-0.518184\pi$
$613$	512.508	−	887.689i	0.836065	−	1.44811i	0.893161	−	0.449738i	$-0.148482\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−105.000	+	181.865i	−0.170178	+	0.294757i	−0.938482	−	0.345328i	$-0.887768\pi$
$617$	−105.000	+	181.865i	−0.170178	+	0.294757i	0.768304	+	0.640085i	$0.221101\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	147.100	+	254.784i	0.235360	+	0.407655i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1009.97	+	346.277i	−1.60568	+	0.550520i
$630$		0			0
$631$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$631$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	490.000			0.769231
$638$		0			0
$639$		0			0
$640$		0			0
$641$	131.295	−	227.409i	0.204828	−	0.354773i	−0.745250	−	0.666785i	$-0.767670\pi$
$641$	131.295	−	227.409i	0.204828	−	0.354773i	0.950078	+	0.312012i	$0.101003\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	337.867	+	585.202i	0.517407	+	0.896175i	0.999796	+	0.0202175i	$0.00643585\pi$
$653$	337.867	+	585.202i	0.517407	+	0.896175i	−0.482389	+	0.875957i	$0.660231\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	495.000	−	857.365i	0.753425	−	1.30497i
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	554.308	+	960.089i	0.838589	+	1.45248i	0.891074	+	0.453858i	$0.149953\pi$
$661$	554.308	+	960.089i	0.838589	+	1.45248i	−0.0524847	+	0.998622i	$0.516714\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−385.000	−	666.840i	−0.572065	−	0.990846i	−0.996354	−	0.0853191i	$-0.972809\pi$
$673$	−385.000	−	666.840i	−0.572065	−	0.990846i	0.424288	−	0.905527i	$-0.360524\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	584.933			0.864008			0.432004	−	0.901872i	$-0.357807\pi$
$677$	584.933			0.864008			0.432004	+	0.901872i	$0.357807\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$	−505.600	−	875.725i	−0.738102	−	1.27843i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	900.000			1.30624
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1739.52			2.49573
$698$		0			0
$699$		0			0
$700$		0			0
$701$	651.000	−	1127.57i	0.928673	−	1.60851i	0.143129	−	0.989704i	$-0.454284\pi$
$701$	651.000	−	1127.57i	0.928673	−	1.60851i	0.785544	−	0.618805i	$-0.212383\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−518.000			−0.730606			−0.365303	−	0.930889i	$-0.619035\pi$
$709$	−518.000			−0.730606			−0.365303	+	0.930889i	$0.619035\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−266.221	+	461.108i	−0.367201	+	0.636010i
$726$		0			0
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	729.000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−725.000	−	1255.74i	−0.989086	−	1.71315i	−0.622143	−	0.782904i	$-0.713738\pi$
$733$	−725.000	−	1255.74i	−0.989086	−	1.71315i	−0.366943	−	0.930243i	$-0.619596\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$	576.439	+	998.421i	0.773743	+	1.34016i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−702.800	+	1217.29i	−0.928401	+	1.60804i	−0.142404	+	0.989809i	$0.545483\pi$
$757$	−702.800	+	1217.29i	−0.928401	+	1.60804i	−0.785997	+	0.618230i	$0.787850\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−677.679	+	1173.77i	−0.890512	+	1.54241i	−0.0512484	+	0.998686i	$0.516320\pi$
$761$	−677.679	+	1173.77i	−0.890512	+	1.54241i	−0.839263	+	0.543725i	$0.817013\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	510.092	−	883.506i	0.666787	−	1.15491i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	962.000			1.25098			0.625488	−	0.780234i	$-0.284900\pi$
$769$	962.000			1.25098			0.625488	+	0.780234i	$0.284900\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−745.287	−	1290.87i	−0.964149	−	1.66995i	−0.711885	−	0.702296i	$-0.752158\pi$
$773$	−745.287	−	1290.87i	−0.964149	−	1.66995i	−0.252264	−	0.967658i	$-0.581175\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1232.01			1.56943
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	464.615	−	804.737i	0.585896	−	1.01480i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	555.000	−	961.288i	0.696361	−	1.20613i	−0.273358	−	0.961912i	$-0.588134\pi$
$797$	555.000	−	961.288i	0.696361	−	1.20613i	0.969720	−	0.244221i	$-0.0785322\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−799.038	−	1383.97i	−0.997551	−	1.72781i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	759.000	+	1314.63i	0.938195	+	1.62500i	0.768835	+	0.639448i	$0.220837\pi$
$809$	759.000	+	1314.63i	0.938195	+	1.62500i	0.169361	+	0.985554i	$0.445830\pi$
$810$		0			0
$811$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$811$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−429.000	−	743.050i	−0.522533	−	0.905055i	−0.999656	−	0.0262179i	$-0.991654\pi$
$821$	−429.000	−	743.050i	−0.522533	−	0.905055i	0.477123	−	0.878837i	$-0.341680\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$827$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$828$		0			0
$829$	−629.000	+	1089.46i	−0.758745	+	1.31419i	0.184745	+	0.982786i	$0.440854\pi$
$829$	−629.000	+	1089.46i	−0.758745	+	1.31419i	−0.943491	+	0.331399i	$0.892479\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−706.982	−	1224.53i	−0.848718	−	1.47002i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	2254.92			2.68124
$842$		0			0
$843$		0			0
$844$		0			0
$845$	271.046			0.320765
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−614.569	+	1064.46i	−0.720480	+	1.24791i	0.240328	+	0.970692i	$0.422745\pi$
$853$	−614.569	+	1064.46i	−0.720480	+	1.24791i	−0.960808	+	0.277215i	$0.910588\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1226.84			−1.43155			−0.715774	−	0.698333i	$-0.753926\pi$
$857$	−1226.84			−1.43155			−0.715774	+	0.698333i	$0.753926\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$	294.353			0.340293
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−268.684	+	465.375i	−0.307771	+	0.533076i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1407.75			−1.60519			−0.802596	−	0.596523i	$-0.796549\pi$
$877$	−1407.75			−1.60519			−0.802596	+	0.596523i	$0.796549\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−508.320	−	880.437i	−0.576981	−	0.999361i	−0.995823	−	0.0913015i	$-0.970897\pi$
$881$	−508.320	−	880.437i	−0.576981	−	0.999361i	0.418842	−	0.908059i	$-0.362436\pi$
$882$		0			0
$883$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$883$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−1298.54	−	2249.13i	−1.44122	−	2.49626i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−575.028	+	995.978i	−0.635390	+	1.10053i
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$		0			0
$909$	−601.385	+	1041.63i	−0.661589	+	1.14591i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	334.923	−	114.831i	0.362079	−	0.124141i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	861.243	+	1491.72i	0.927065	+	1.60572i	0.788206	+	0.615411i	$0.211010\pi$
$929$	861.243	+	1491.72i	0.927065	+	1.60572i	0.138859	+	0.990312i	$0.455657\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−897.315	+	1554.20i	−0.957647	+	1.65869i	−0.229456	+	0.973319i	$0.573695\pi$
$937$	−897.315	+	1554.20i	−0.957647	+	1.65869i	−0.728191	+	0.685374i	$0.759639\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−131.795	+	228.275i	−0.140058	+	0.242588i	−0.927518	−	0.373778i	$-0.878062\pi$
$941$	−131.795	+	228.275i	−0.140058	+	0.242588i	0.787460	+	0.616366i	$0.211396\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$	550.000	+	952.628i	0.579557	+	1.00382i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	615.000	−	1065.21i	0.645331	−	1.11775i	−0.338895	−	0.940824i	$-0.610053\pi$
$953$	615.000	−	1065.21i	0.645331	−	1.11775i	0.984225	−	0.176921i	$-0.0566137\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	961.000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	758.113	+	1313.09i	0.785609	+	1.36071i
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−945.000	+	1636.79i	−0.967247	+	1.67532i	−0.263793	+	0.964579i	$0.584974\pi$
$977$	−945.000	+	1636.79i	−0.967247	+	1.67532i	−0.703454	+	0.710741i	$0.748360\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	58.1537	+	100.725i	0.0592800	+	0.102676i
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$	−575.492			−0.584256
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−925.000	+	1602.15i	−0.927783	+	1.60697i	−0.140761	+	0.990044i	$0.544955\pi$
$997$	−925.000	+	1602.15i	−0.927783	+	1.60697i	−0.787023	+	0.616924i	$0.788378\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.3.z.a.47.2	✓	4
4.3	odd	2	CM	592.3.z.a.47.2	✓	4
37.26	even	3	inner	592.3.z.a.63.2	yes	4
148.63	odd	6	inner	592.3.z.a.63.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
592.3.z.a.47.2	✓	4	1.1	even	1	trivial
592.3.z.a.47.2	✓	4	4.3	odd	2	CM
592.3.z.a.63.2	yes	4	37.26	even	3	inner
592.3.z.a.63.2	yes	4	148.63	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	592.z (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.96410 + 3.40192i) q^{5} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)	\(q+(1.96410 + 3.40192i) q^{5} +(-4.50000 + 7.79423i) q^{9} +(-5.00000 - 8.66025i) q^{13} +(-14.4282 + 24.9904i) q^{17} +(4.78461 - 8.28719i) q^{25} -55.6410 q^{29} +(27.8923 + 24.3109i) q^{37} +(-30.1410 - 52.2058i) q^{41} -35.3538 q^{45} +(-24.5000 + 42.4352i) q^{49} +(-45.0000 + 77.9423i) q^{53} +(46.4615 + 80.4737i) q^{61} +(19.6410 - 34.0192i) q^{65} -110.000 q^{73} +(-40.5000 - 70.1481i) q^{81} -113.354 q^{85} +(-88.7820 + 153.775i) q^{89} +59.7077 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{5} - 18 q^{9}+O(q^{10}) \) 4 * q - 6 * q^5 - 18 * q^9	\( 4 q - 6 q^{5} - 18 q^{9} - 20 q^{13} - 30 q^{17} - 64 q^{25} - 84 q^{29} + 70 q^{37} + 18 q^{41} + 108 q^{45} - 98 q^{49} - 180 q^{53} - 22 q^{61} - 60 q^{65} - 440 q^{73} - 162 q^{81} - 204 q^{85} - 78 q^{89} - 260 q^{97}+O(q^{100}) \) 4 * q - 6 * q^5 - 18 * q^9 - 20 * q^13 - 30 * q^17 - 64 * q^25 - 84 * q^29 + 70 * q^37 + 18 * q^41 + 108 * q^45 - 98 * q^49 - 180 * q^53 - 22 * q^61 - 60 * q^65 - 440 * q^73 - 162 * q^81 - 204 * q^85 - 78 * q^89 - 260 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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