LMFDB - Embedded newform 864.3.q.a.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(449,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("864.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	864.q (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:	$23.5422948407$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			449.4
Character	$\chi$	$=$	864.449
Dual form			864.3.q.a.737.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.41309 - 3.12525i) q^{5} +(3.74855 + 6.49268i) q^{7} +(6.17688 - 3.56622i) q^{11} +(-0.888802 + 1.53945i) q^{13} -14.7791i q^{17} -19.9460 q^{19} +(20.8716 + 12.0502i) q^{23} +(7.03438 + 12.1839i) q^{25} +(-40.1279 + 23.1679i) q^{29} +(14.1547 - 24.5167i) q^{31} -46.8606i q^{35} +63.0770 q^{37} +(28.0185 + 16.1765i) q^{41} +(-38.8176 - 67.2341i) q^{43} +(38.4719 - 22.2118i) q^{47} +(-3.60324 + 6.24099i) q^{49} -42.2846i q^{53} -44.5813 q^{55} +(-93.8917 - 54.2084i) q^{59} +(-25.3858 - 43.9695i) q^{61} +(9.62233 - 5.55546i) q^{65} +(56.9484 - 98.6376i) q^{67} -85.2129i q^{71} -94.5357 q^{73} +(46.3086 + 26.7363i) q^{77} +(-35.6059 - 61.6712i) q^{79} +(94.9037 - 54.7927i) q^{83} +(-46.1885 + 80.0009i) q^{85} +29.6936i q^{89} -13.3269 q^{91} +(107.970 + 62.3363i) q^{95} +(-62.4793 - 108.217i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 60 q^{25} - 72 q^{29} - 252 q^{41} - 36 q^{49} - 96 q^{61} + 288 q^{65} + 24 q^{73} + 720 q^{77} + 96 q^{85} - 132 q^{97}+O(q^{100}) $ 24 * q + 60 * q^25 - 72 * q^29 - 252 * q^41 - 36 * q^49 - 96 * q^61 + 288 * q^65 + 24 * q^73 + 720 * q^77 + 96 * q^85 - 132 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−5.41309	−	3.12525i	−1.08262	−	0.625050i	−0.151017	−	0.988531i	$-0.548255\pi$
$5$	−5.41309	−	3.12525i	−1.08262	−	0.625050i	−0.931602	+	0.363481i	$0.881588\pi$
$6$		0			0
$7$	3.74855	+	6.49268i	0.535507	+	0.927525i	0.999139	+	0.0414972i	$0.0132128\pi$
$7$	3.74855	+	6.49268i	0.535507	+	0.927525i	−0.463632	+	0.886028i	$0.653454\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	6.17688	−	3.56622i	0.561534	−	0.324202i	−0.192227	−	0.981351i	$-0.561571\pi$
$11$	6.17688	−	3.56622i	0.561534	−	0.324202i	0.753761	+	0.657149i	$0.228238\pi$
$12$		0			0
$13$	−0.888802	+	1.53945i	−0.0683694	+	0.118419i	−0.898184	−	0.439620i	$-0.855113\pi$
$13$	−0.888802	+	1.53945i	−0.0683694	+	0.118419i	0.829814	+	0.558040i	$0.188446\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	14.7791i		−	0.869362i	−0.900585	−	0.434681i	$-0.856861\pi$
$17$		−	14.7791i		−	0.869362i	0.900585	−	0.434681i	$-0.143139\pi$
$18$		0			0
$19$	−19.9460			−1.04979			−0.524896	−	0.851167i	$-0.675896\pi$
$19$	−19.9460			−1.04979			−0.524896	+	0.851167i	$0.675896\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	20.8716	+	12.0502i	0.907462	+	0.523923i	0.879614	−	0.475689i	$-0.157801\pi$
$23$	20.8716	+	12.0502i	0.907462	+	0.523923i	0.0278482	+	0.999612i	$0.491134\pi$
$24$		0			0
$25$	7.03438	+	12.1839i	0.281375	+	0.487356i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−40.1279	+	23.1679i	−1.38372	+	0.798891i	−0.992598	−	0.121447i	$-0.961246\pi$
$29$	−40.1279	+	23.1679i	−1.38372	+	0.798891i	−0.391123	+	0.920339i	$0.627913\pi$
$30$		0			0
$31$	14.1547	−	24.5167i	0.456603	−	0.790860i	−0.542176	−	0.840265i	$-0.682399\pi$
$31$	14.1547	−	24.5167i	0.456603	−	0.790860i	0.998779	+	0.0494054i	$0.0157326\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	46.8606i		−	1.33887i
$36$		0			0
$37$	63.0770			1.70478			0.852391	−	0.522904i	$-0.175151\pi$
$37$	63.0770			1.70478			0.852391	+	0.522904i	$0.175151\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	28.0185	+	16.1765i	0.683378	+	0.394549i	0.801127	−	0.598495i	$-0.204234\pi$
$41$	28.0185	+	16.1765i	0.683378	+	0.394549i	−0.117748	+	0.993043i	$0.537568\pi$
$42$		0			0
$43$	−38.8176	−	67.2341i	−0.902736	−	1.56358i	−0.823928	−	0.566694i	$-0.808222\pi$
$43$	−38.8176	−	67.2341i	−0.902736	−	1.56358i	−0.0788077	−	0.996890i	$-0.525111\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	38.4719	−	22.2118i	0.818551	−	0.472591i	−0.0313654	−	0.999508i	$-0.509986\pi$
$47$	38.4719	−	22.2118i	0.818551	−	0.472591i	0.849917	+	0.526917i	$0.176652\pi$
$48$		0			0
$49$	−3.60324	+	6.24099i	−0.0735354	+	0.127367i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	42.2846i		−	0.797823i	−0.916990	−	0.398911i	$-0.869388\pi$
$53$		−	42.2846i		−	0.797823i	0.916990	−	0.398911i	$-0.130612\pi$
$54$		0			0
$55$	−44.5813			−0.810570
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−93.8917	−	54.2084i	−1.59138	−	0.918786i	−0.993070	−	0.117524i	$-0.962504\pi$
$59$	−93.8917	−	54.2084i	−1.59138	−	0.918786i	−0.598314	−	0.801262i	$-0.704162\pi$
$60$		0			0
$61$	−25.3858	−	43.9695i	−0.416161	−	0.720812i	0.579389	−	0.815051i	$-0.303291\pi$
$61$	−25.3858	−	43.9695i	−0.416161	−	0.720812i	−0.995550	+	0.0942396i	$0.969958\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	9.62233	−	5.55546i	0.148036	−	0.0854686i
$66$		0			0
$67$	56.9484	−	98.6376i	0.849977	−	1.47220i	−0.0312514	−	0.999512i	$-0.509949\pi$
$67$	56.9484	−	98.6376i	0.849977	−	1.47220i	0.881228	−	0.472691i	$-0.156717\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	85.2129i		−	1.20018i	−0.799932	−	0.600091i	$-0.795131\pi$
$71$		−	85.2129i		−	1.20018i	0.799932	−	0.600091i	$-0.204869\pi$
$72$		0			0
$73$	−94.5357			−1.29501			−0.647505	−	0.762061i	$-0.724188\pi$
$73$	−94.5357			−1.29501			−0.647505	+	0.762061i	$0.724188\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	46.3086	+	26.7363i	0.601411	+	0.347225i
$78$		0			0
$79$	−35.6059	−	61.6712i	−0.450707	−	0.780648i	0.547723	−	0.836660i	$-0.315495\pi$
$79$	−35.6059	−	61.6712i	−0.450707	−	0.780648i	−0.998430	+	0.0560122i	$0.982161\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	94.9037	−	54.7927i	1.14342	−	0.660153i	0.196144	−	0.980575i	$-0.437158\pi$
$83$	94.9037	−	54.7927i	1.14342	−	0.660153i	0.947275	+	0.320422i	$0.103825\pi$
$84$		0			0
$85$	−46.1885	+	80.0009i	−0.543395	+	0.941187i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			29.6936i			0.333635i	0.985988	+	0.166818i	$0.0533491\pi$
$89$			29.6936i			0.333635i	−0.985988	+	0.166818i	$0.946651\pi$
$90$		0			0
$91$	−13.3269			−0.146449
$92$		0			0
$93$		0			0
$94$		0			0
$95$	107.970	+	62.3363i	1.13652	+	0.656172i
$96$		0			0
$97$	−62.4793	−	108.217i	−0.644116	−	1.11564i	−0.984505	−	0.175357i	$-0.943892\pi$
$97$	−62.4793	−	108.217i	−0.644116	−	1.11564i	0.340389	−	0.940285i	$-0.389441\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	34.0989	−	19.6870i	0.337613	−	0.194921i	−0.321603	−	0.946875i	$-0.604222\pi$
$101$	34.0989	−	19.6870i	0.337613	−	0.194921i	0.659216	+	0.751954i	$0.270888\pi$
$102$		0			0
$103$	−74.7230	+	129.424i	−0.725466	+	1.25654i	0.233316	+	0.972401i	$0.425042\pi$
$103$	−74.7230	+	129.424i	−0.725466	+	1.25654i	−0.958782	+	0.284143i	$0.908291\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	0.420211i		−	0.00392721i	−0.999998	−	0.00196360i	$-0.999375\pi$
$107$		−	0.420211i		−	0.00392721i	0.999998	−	0.00196360i	$-0.000625035\pi$
$108$		0			0
$109$	64.4616			0.591391			0.295695	−	0.955282i	$-0.404449\pi$
$109$	64.4616			0.591391			0.295695	+	0.955282i	$0.404449\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−22.4718	−	12.9741i	−0.198865	−	0.114815i	0.397261	−	0.917706i	$-0.369961\pi$
$113$	−22.4718	−	12.9741i	−0.198865	−	0.114815i	−0.596126	+	0.802891i	$0.703294\pi$
$114$		0			0
$115$	−75.3200	−	130.458i	−0.654957	−	1.13442i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	95.9562	−	55.4004i	0.806355	−	0.465549i
$120$		0			0
$121$	−35.0641	+	60.7329i	−0.289786	+	0.501925i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			68.3258i			0.546606i
$126$		0			0
$127$	−12.0916			−0.0952097			−0.0476049	−	0.998866i	$-0.515159\pi$
$127$	−12.0916			−0.0952097			−0.0476049	+	0.998866i	$0.515159\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	45.4518	+	26.2416i	0.346960	+	0.200318i	0.663346	−	0.748313i	$-0.269136\pi$
$131$	45.4518	+	26.2416i	0.346960	+	0.200318i	−0.316385	+	0.948631i	$0.602469\pi$
$132$		0			0
$133$	−74.7687	−	129.503i	−0.562171	−	0.973708i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	121.224	−	69.9888i	0.884848	−	0.510867i	0.0125942	−	0.999921i	$-0.495991\pi$
$137$	121.224	−	69.9888i	0.884848	−	0.510867i	0.872254	+	0.489053i	$0.162658\pi$
$138$		0			0
$139$	19.9656	−	34.5814i	0.143637	−	0.248787i	−0.785226	−	0.619209i	$-0.787453\pi$
$139$	19.9656	−	34.5814i	0.143637	−	0.248787i	0.928864	+	0.370422i	$0.120787\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			12.6787i			0.0886619i
$144$		0			0
$145$	289.621			1.99739
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.26226	+	1.88347i	0.0218944	+	0.0126407i	0.510907	−	0.859636i	$-0.329310\pi$
$149$	3.26226	+	1.88347i	0.0218944	+	0.0126407i	−0.489013	+	0.872277i	$0.662643\pi$
$150$		0			0
$151$	74.8795	+	129.695i	0.495891	+	0.858908i	0.999989	−	0.00473848i	$-0.00150831\pi$
$151$	74.8795	+	129.695i	0.495891	+	0.858908i	−0.504098	+	0.863646i	$0.668175\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−153.241	+	88.4739i	−0.988654	+	0.570800i
$156$		0			0
$157$	−58.9507	+	102.106i	−0.375482	+	0.650354i	−0.990399	−	0.138238i	$-0.955856\pi$
$157$	−58.9507	+	102.106i	−0.375482	+	0.650354i	0.614917	+	0.788592i	$0.289189\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			180.684i			1.12226i
$162$		0			0
$163$	111.960			0.686874			0.343437	−	0.939176i	$-0.388409\pi$
$163$	111.960			0.686874			0.343437	+	0.939176i	$0.388409\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.33677	−	0.771783i	−0.00800460	−	0.00462146i	0.495992	−	0.868327i	$-0.334804\pi$
$167$	−1.33677	−	0.771783i	−0.00800460	−	0.00462146i	−0.503997	+	0.863705i	$0.668138\pi$
$168$		0			0
$169$	82.9201	+	143.622i	0.490651	+	0.849833i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−46.0581	+	26.5917i	−0.266232	+	0.153709i	−0.627174	−	0.778879i	$-0.715789\pi$
$173$	−46.0581	+	26.5917i	−0.266232	+	0.153709i	0.360942	+	0.932588i	$0.382455\pi$
$174$		0			0
$175$	−52.7374	+	91.3439i	−0.301357	+	0.521965i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	239.444i		−	1.33768i	−0.743407	−	0.668839i	$-0.766792\pi$
$179$		−	239.444i		−	1.33768i	0.743407	−	0.668839i	$-0.233208\pi$
$180$		0			0
$181$	61.1557			0.337877			0.168938	−	0.985627i	$-0.445966\pi$
$181$	61.1557			0.337877			0.168938	+	0.985627i	$0.445966\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−341.441	−	197.131i	−1.84563	−	1.06557i
$186$		0			0
$187$	−52.7057	−	91.2890i	−0.281849	−	0.488176i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−28.9063	+	16.6891i	−0.151342	+	0.0873773i	−0.573759	−	0.819024i	$-0.694515\pi$
$191$	−28.9063	+	16.6891i	−0.151342	+	0.0873773i	0.422417	+	0.906402i	$0.361182\pi$
$192$		0			0
$193$	21.2796	−	36.8573i	0.110257	−	0.190971i	−0.805617	−	0.592437i	$-0.798166\pi$
$193$	21.2796	−	36.8573i	0.110257	−	0.190971i	0.915874	+	0.401466i	$0.131499\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	203.372i		−	1.03235i	−0.856484	−	0.516173i	$-0.827356\pi$
$197$		−	203.372i		−	1.03235i	0.856484	−	0.516173i	$-0.172644\pi$
$198$		0			0
$199$	128.160			0.644022			0.322011	−	0.946736i	$-0.395641\pi$
$199$	128.160			0.644022			0.322011	+	0.946736i	$0.395641\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−300.843	−	173.692i	−1.48198	−	0.855624i
$204$		0			0
$205$	−101.111	−	175.130i	−0.493225	−	0.854291i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−123.204	+	71.1320i	−0.589494	+	0.340344i
$210$		0			0
$211$	97.2675	−	168.472i	0.460983	−	0.798446i	−0.538027	−	0.842928i	$-0.680830\pi$
$211$	97.2675	−	168.472i	0.460983	−	0.798446i	0.999010	+	0.0444812i	$0.0141635\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			485.259i			2.25702i
$216$		0			0
$217$	212.238			0.978057
$218$		0			0
$219$		0			0
$220$		0			0
$221$	22.7518	+	13.1357i	0.102949	+	0.0594377i
$222$		0			0
$223$	95.1454	+	164.797i	0.426661	+	0.738998i	0.996574	−	0.0827067i	$-0.0263565\pi$
$223$	95.1454	+	164.797i	0.426661	+	0.738998i	−0.569913	+	0.821705i	$0.693023\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−320.930	+	185.289i	−1.41379	+	0.816252i	−0.995743	−	0.0921738i	$-0.970618\pi$
$227$	−320.930	+	185.289i	−1.41379	+	0.816252i	−0.418047	+	0.908426i	$0.637285\pi$
$228$		0			0
$229$	−203.685	+	352.793i	−0.889455	+	1.54058i	−0.0489338	+	0.998802i	$0.515582\pi$
$229$	−203.685	+	352.793i	−0.889455	+	1.54058i	−0.840521	+	0.541779i	$0.817751\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	21.5267i		−	0.0923895i	−0.998932	−	0.0461947i	$-0.985291\pi$
$233$		−	21.5267i		−	0.0923895i	0.998932	−	0.0461947i	$-0.0147095\pi$
$234$		0			0
$235$	−277.669			−1.18157
$236$		0			0
$237$		0			0
$238$		0			0
$239$	172.020	+	99.3155i	0.719747	+	0.415546i	0.814660	−	0.579939i	$-0.196924\pi$
$239$	172.020	+	99.3155i	0.719747	+	0.415546i	−0.0949125	+	0.995486i	$0.530257\pi$
$240$		0			0
$241$	−138.940	−	240.652i	−0.576516	−	0.998556i	−0.995875	−	0.0907351i	$-0.971078\pi$
$241$	−138.940	−	240.652i	−0.576516	−	0.998556i	0.419359	−	0.907821i	$-0.362255\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	39.0093	−	22.5220i	0.159222	−	0.0919267i
$246$		0			0
$247$	17.7281	−	30.7059i	0.0717736	−	0.124315i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			128.325i			0.511253i	0.966776	+	0.255627i	$0.0822817\pi$
$251$			128.325i			0.511253i	−0.966776	+	0.255627i	$0.917718\pi$
$252$		0			0
$253$	171.895			0.679428
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−62.3584	−	36.0026i	−0.242640	−	0.140088i	0.373750	−	0.927530i	$-0.378072\pi$
$257$	−62.3584	−	36.0026i	−0.242640	−	0.140088i	−0.616389	+	0.787442i	$0.711405\pi$
$258$		0			0
$259$	236.447	+	409.538i	0.912923	+	1.58123i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	113.783	−	65.6926i	0.432634	−	0.249782i	−0.267834	−	0.963465i	$-0.586308\pi$
$263$	113.783	−	65.6926i	0.432634	−	0.249782i	0.700468	+	0.713683i	$0.252975\pi$
$264$		0			0
$265$	−132.150	+	228.890i	−0.498679	+	0.863737i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	38.8284i		−	0.144344i	−0.997392	−	0.0721718i	$-0.977007\pi$
$269$		−	38.8284i		−	0.144344i	0.997392	−	0.0721718i	$-0.0229930\pi$
$270$		0			0
$271$	−0.802525			−0.00296135			−0.00148067	−	0.999999i	$-0.500471\pi$
$271$	−0.802525			−0.00296135			−0.00148067	+	0.999999i	$0.500471\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	86.9009	+	50.1723i	0.316003	+	0.182445i
$276$		0			0
$277$	61.1932	+	105.990i	0.220914	+	0.382635i	0.955086	−	0.296329i	$-0.0957626\pi$
$277$	61.1932	+	105.990i	0.220914	+	0.382635i	−0.734172	+	0.678964i	$0.762429\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	254.139	−	146.727i	0.904408	−	0.522160i	0.0257801	−	0.999668i	$-0.491793\pi$
$281$	254.139	−	146.727i	0.904408	−	0.522160i	0.878628	+	0.477508i	$0.158460\pi$
$282$		0			0
$283$	−90.6191	+	156.957i	−0.320209	+	0.554618i	−0.980531	−	0.196364i	$-0.937086\pi$
$283$	−90.6191	+	156.957i	−0.320209	+	0.554618i	0.660322	+	0.750983i	$0.270420\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			242.554i			0.845134i
$288$		0			0
$289$	70.5768			0.244210
$290$		0			0
$291$		0			0
$292$		0			0
$293$	290.545	+	167.746i	0.991622	+	0.572513i	0.905759	−	0.423794i	$-0.139302\pi$
$293$	290.545	+	167.746i	0.991622	+	0.572513i	0.0858631	+	0.996307i	$0.472635\pi$
$294$		0			0
$295$	338.829	+	586.870i	1.14857	+	1.98939i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−37.1015	+	21.4205i	−0.124085	+	0.0716406i
$300$		0			0
$301$	291.020	−	504.061i	0.966843	−	1.67462i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			317.348i			1.04049i
$306$		0			0
$307$	289.210			0.942053			0.471027	−	0.882119i	$-0.343884\pi$
$307$	289.210			0.942053			0.471027	+	0.882119i	$0.343884\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	23.6987	+	13.6824i	0.0762016	+	0.0439950i	0.537617	−	0.843189i	$-0.319325\pi$
$311$	23.6987	+	13.6824i	0.0762016	+	0.0439950i	−0.461415	+	0.887184i	$0.652658\pi$
$312$		0			0
$313$	−293.572	−	508.481i	−0.937928	−	1.62454i	−0.769327	−	0.638855i	$-0.779408\pi$
$313$	−293.572	−	508.481i	−0.937928	−	1.62454i	−0.168601	−	0.985684i	$-0.553925\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−326.081	+	188.263i	−1.02865	+	0.593889i	−0.916596	−	0.399814i	$-0.869075\pi$
$317$	−326.081	+	188.263i	−1.02865	+	0.593889i	−0.112049	+	0.993703i	$0.535741\pi$
$318$		0			0
$319$	−165.243	+	286.210i	−0.518004	+	0.897210i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			294.785i			0.912648i
$324$		0			0
$325$	−25.0087			−0.0769497
$326$		0			0
$327$		0			0
$328$		0			0
$329$	288.428	+	166.524i	0.876680	+	0.506151i
$330$		0			0
$331$	1.48388	+	2.57015i	0.00448301	+	0.00776480i	0.868258	−	0.496113i	$-0.165240\pi$
$331$	1.48388	+	2.57015i	0.00448301	+	0.00776480i	−0.863775	+	0.503877i	$0.831906\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−616.534	+	355.956i	−1.84040	+	1.06256i
$336$		0			0
$337$	−329.235	+	570.251i	−0.976958	+	1.69214i	−0.303641	+	0.952786i	$0.598202\pi$
$337$	−329.235	+	570.251i	−0.976958	+	1.69214i	−0.673317	+	0.739354i	$0.735131\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	201.915i		−	0.592126i
$342$		0			0
$343$	313.330			0.913499
$344$		0			0
$345$		0			0
$346$		0			0
$347$	255.949	+	147.772i	0.737606	+	0.425857i	0.821198	−	0.570643i	$-0.193306\pi$
$347$	255.949	+	147.772i	0.737606	+	0.425857i	−0.0835920	+	0.996500i	$0.526639\pi$
$348$		0			0
$349$	106.379	+	184.254i	0.304811	+	0.527949i	0.977219	−	0.212232i	$-0.0680733\pi$
$349$	106.379	+	184.254i	0.304811	+	0.527949i	−0.672408	+	0.740181i	$0.734740\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	124.287	−	71.7570i	0.352087	−	0.203278i	−0.313517	−	0.949583i	$-0.601507\pi$
$353$	124.287	−	71.7570i	0.352087	−	0.203278i	0.665604	+	0.746305i	$0.268174\pi$
$354$		0			0
$355$	−266.312	+	461.265i	−0.750173	+	1.29934i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	270.973i		−	0.754799i	−0.926051	−	0.377400i	$-0.876818\pi$
$359$		−	270.973i		−	0.754799i	0.926051	−	0.377400i	$-0.123182\pi$
$360$		0			0
$361$	36.8443			0.102062
$362$		0			0
$363$		0			0
$364$		0			0
$365$	511.731	+	295.448i	1.40200	+	0.809446i
$366$		0			0
$367$	−200.855	−	347.892i	−0.547290	−	0.947934i	−0.998459	−	0.0554953i	$-0.982326\pi$
$367$	−200.855	−	347.892i	−0.547290	−	0.947934i	0.451169	−	0.892438i	$-0.351007\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	274.540	−	158.506i	0.740001	−	0.427240i
$372$		0			0
$373$	−146.890	+	254.422i	−0.393808	+	0.682095i	−0.992948	−	0.118549i	$-0.962176\pi$
$373$	−146.890	+	254.422i	−0.393808	+	0.682095i	0.599140	+	0.800644i	$0.295509\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	82.3665i		−	0.218479i
$378$		0			0
$379$	280.802			0.740903			0.370452	−	0.928852i	$-0.379203\pi$
$379$	280.802			0.740903			0.370452	+	0.928852i	$0.379203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−546.665	−	315.617i	−1.42732	−	0.824066i	−0.430416	−	0.902631i	$-0.641633\pi$
$383$	−546.665	−	315.617i	−1.42732	−	0.824066i	−0.996909	+	0.0785646i	$0.974966\pi$
$384$		0			0
$385$	−167.115	−	289.452i	−0.434066	−	0.751824i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−378.049	+	218.267i	−0.971849	+	0.561097i	−0.899799	−	0.436304i	$-0.856287\pi$
$389$	−378.049	+	218.267i	−0.971849	+	0.561097i	−0.0720497	+	0.997401i	$0.522954\pi$
$390$		0			0
$391$	178.092	−	308.465i	0.455479	−	0.788913i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			445.109i			1.12686i
$396$		0			0
$397$	−405.382			−1.02111			−0.510557	−	0.859844i	$-0.670561\pi$
$397$	−405.382			−1.02111			−0.510557	+	0.859844i	$0.670561\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	66.9595	+	38.6591i	0.166981	+	0.0964068i	0.581162	−	0.813788i	$-0.302598\pi$
$401$	66.9595	+	38.6591i	0.166981	+	0.0964068i	−0.414180	+	0.910195i	$0.635932\pi$
$402$		0			0
$403$	25.1614	+	43.5809i	0.0624353	+	0.108141i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	389.619	−	224.946i	0.957294	−	0.552694i
$408$		0			0
$409$	85.3306	−	147.797i	0.208632	−	0.361362i	−0.742652	−	0.669678i	$-0.766432\pi$
$409$	85.3306	−	147.797i	0.208632	−	0.361362i	0.951284	+	0.308316i	$0.0997655\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	812.811i		−	1.96807i
$414$		0			0
$415$	−684.964			−1.65051
$416$		0			0
$417$		0			0
$418$		0			0
$419$	123.529	+	71.3195i	0.294819	+	0.170214i	0.640113	−	0.768281i	$-0.278888\pi$
$419$	123.529	+	71.3195i	0.294819	+	0.170214i	−0.345294	+	0.938494i	$0.612221\pi$
$420$		0			0
$421$	−307.819	−	533.158i	−0.731162	−	1.26641i	−0.956387	−	0.292102i	$-0.905645\pi$
$421$	−307.819	−	533.158i	−0.731162	−	1.26641i	0.225226	−	0.974307i	$-0.427688\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	180.068	−	103.962i	0.423689	−	0.244617i
$426$		0			0
$427$	190.320	−	329.644i	0.445714	−	0.772000i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	405.961i		−	0.941905i	−0.882159	−	0.470953i	$-0.843910\pi$
$431$		−	405.961i		−	0.941905i	0.882159	−	0.470953i	$-0.156090\pi$
$432$		0			0
$433$	50.1302			0.115774			0.0578870	−	0.998323i	$-0.481564\pi$
$433$	50.1302			0.115774			0.0578870	+	0.998323i	$0.481564\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−416.306	−	240.354i	−0.952646	−	0.550010i
$438$		0			0
$439$	−363.552	−	629.690i	−0.828136	−	1.43437i	−0.899499	−	0.436924i	$-0.856068\pi$
$439$	−363.552	−	629.690i	−0.828136	−	1.43437i	0.0713624	−	0.997450i	$-0.477265\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−324.072	+	187.103i	−0.731540	+	0.422355i	−0.818985	−	0.573815i	$-0.805463\pi$
$443$	−324.072	+	187.103i	−0.731540	+	0.422355i	0.0874454	+	0.996169i	$0.472130\pi$
$444$		0			0
$445$	92.7998	−	160.734i	0.208539	−	0.361200i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	225.368i		−	0.501934i	−0.967996	−	0.250967i	$-0.919251\pi$
$449$		−	225.368i		−	0.501934i	0.967996	−	0.250967i	$-0.0807486\pi$
$450$		0			0
$451$	230.756			0.511654
$452$		0			0
$453$		0			0
$454$		0			0
$455$	72.1396	+	41.6498i	0.158549	+	0.0915380i
$456$		0			0
$457$	390.263	+	675.955i	0.853966	+	1.47911i	0.877601	+	0.479392i	$0.159143\pi$
$457$	390.263	+	675.955i	0.853966	+	1.47911i	−0.0236344	+	0.999721i	$0.507524\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−474.085	+	273.713i	−1.02838	+	0.593738i	−0.916521	−	0.399986i	$-0.869015\pi$
$461$	−474.085	+	273.713i	−1.02838	+	0.593738i	−0.111863	+	0.993724i	$0.535682\pi$
$462$		0			0
$463$	174.664	−	302.526i	0.377243	−	0.653404i	−0.613417	−	0.789759i	$-0.710205\pi$
$463$	174.664	−	302.526i	0.377243	−	0.653404i	0.990660	+	0.136355i	$0.0435388\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	499.358i		−	1.06929i	−0.845077	−	0.534645i	$-0.820445\pi$
$467$		−	499.358i		−	1.06929i	0.845077	−	0.534645i	$-0.179555\pi$
$468$		0			0
$469$	853.896			1.82067
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−479.543	−	276.865i	−1.01383	−	0.585337i
$474$		0			0
$475$	−140.308	−	243.020i	−0.295385	−	0.511622i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−571.640	+	330.036i	−1.19340	+	0.689011i	−0.959076	−	0.283147i	$-0.908621\pi$
$479$	−571.640	+	330.036i	−1.19340	+	0.689011i	−0.234326	+	0.972158i	$0.575288\pi$
$480$		0			0
$481$	−56.0629	+	97.1038i	−0.116555	+	0.201879i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			781.053i			1.61042i
$486$		0			0
$487$	217.861			0.447352			0.223676	−	0.974664i	$-0.428194\pi$
$487$	217.861			0.447352			0.223676	+	0.974664i	$0.428194\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	132.712	+	76.6214i	0.270289	+	0.156052i	0.629019	−	0.777390i	$-0.283457\pi$
$491$	132.712	+	76.6214i	0.270289	+	0.156052i	−0.358730	+	0.933441i	$0.616790\pi$
$492$		0			0
$493$	342.401	+	593.056i	0.694526	+	1.20295i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	553.260	−	319.425i	1.11320	−	0.642706i
$498$		0			0
$499$	−153.693	+	266.204i	−0.308002	+	0.533476i	−0.977925	−	0.208955i	$-0.932994\pi$
$499$	−153.693	+	266.204i	−0.308002	+	0.533476i	0.669923	+	0.742431i	$0.266327\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	797.972i		−	1.58643i	−0.608945	−	0.793213i	$-0.708407\pi$
$503$		−	797.972i		−	1.58643i	0.608945	−	0.793213i	$-0.291593\pi$
$504$		0			0
$505$	−246.107			−0.487341
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−260.405	−	150.345i	−0.511602	−	0.295373i	0.221890	−	0.975072i	$-0.428777\pi$
$509$	−260.405	−	150.345i	−0.511602	−	0.295373i	−0.733492	+	0.679698i	$0.762111\pi$
$510$		0			0
$511$	−354.372	−	613.790i	−0.693487	−	1.20115i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	808.965	−	467.056i	1.57081	−	0.906905i
$516$		0			0
$517$	158.424	−	274.399i	0.306430	−	0.530752i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			928.990i			1.78309i	0.452932	+	0.891545i	$0.350378\pi$
$521$			928.990i			1.78309i	−0.452932	+	0.891545i	$0.649622\pi$
$522$		0			0
$523$	−519.086			−0.992516			−0.496258	−	0.868175i	$-0.665293\pi$
$523$	−519.086			−0.992516			−0.496258	+	0.868175i	$0.665293\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−362.335	−	209.194i	−0.687543	−	0.396953i
$528$		0			0
$529$	25.9164	+	44.8885i	0.0489913	+	0.0848554i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−49.8058	+	28.7554i	−0.0934443	+	0.0539501i
$534$		0			0
$535$	−1.31326	+	2.27464i	−0.00245470	+	0.00425167i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			51.3997i			0.0953613i
$540$		0			0
$541$	−109.959			−0.203251			−0.101625	−	0.994823i	$-0.532404\pi$
$541$	−109.959			−0.203251			−0.101625	+	0.994823i	$0.532404\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−348.937	−	201.459i	−0.640251	−	0.369649i
$546$		0			0
$547$	489.119	+	847.180i	0.894185	+	1.54877i	0.834810	+	0.550539i	$0.185578\pi$
$547$	489.119	+	847.180i	0.894185	+	1.54877i	0.0593757	+	0.998236i	$0.481089\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	800.392	−	462.107i	1.45262	−	0.838669i
$552$		0			0
$553$	266.941	−	462.355i	0.482714	−	0.836084i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	322.002i		−	0.578100i	−0.957314	−	0.289050i	$-0.906661\pi$
$557$		−	322.002i		−	0.578100i	0.957314	−	0.289050i	$-0.0933394\pi$
$558$		0			0
$559$	138.005			0.246878
$560$		0			0
$561$		0			0
$562$		0			0
$563$	174.603	+	100.807i	0.310129	+	0.179053i	0.646984	−	0.762503i	$-0.276030\pi$
$563$	174.603	+	100.807i	0.310129	+	0.179053i	−0.336855	+	0.941556i	$0.609363\pi$
$564$		0			0
$565$	81.0945	+	140.460i	0.143530	+	0.248601i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	123.786	−	71.4677i	0.217549	−	0.125602i	−0.387266	−	0.921968i	$-0.626580\pi$
$569$	123.786	−	71.4677i	0.217549	−	0.125602i	0.604815	+	0.796366i	$0.293247\pi$
$570$		0			0
$571$	−45.7205	+	79.1902i	−0.0800709	+	0.138687i	−0.903280	−	0.429051i	$-0.858848\pi$
$571$	−45.7205	+	79.1902i	−0.0800709	+	0.138687i	0.823209	+	0.567738i	$0.192181\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			339.064i			0.589676i
$576$		0			0
$577$	404.739			0.701455			0.350727	−	0.936478i	$-0.385934\pi$
$577$	404.739			0.701455			0.350727	+	0.936478i	$0.385934\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	711.503	+	410.786i	1.22462	+	0.707033i
$582$		0			0
$583$	−150.796	−	261.187i	−0.258656	−	0.448005i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	702.845	−	405.788i	1.19735	−	0.691291i	0.237387	−	0.971415i	$-0.423709\pi$
$587$	702.845	−	405.788i	1.19735	−	0.691291i	0.959964	+	0.280125i	$0.0903758\pi$
$588$		0			0
$589$	−282.330	+	489.010i	−0.479338	+	0.830238i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			95.3026i			0.160713i	0.996766	+	0.0803563i	$0.0256058\pi$
$593$			95.3026i			0.160713i	−0.996766	+	0.0803563i	$0.974394\pi$
$594$		0			0
$595$	−692.560			−1.16397
$596$		0			0
$597$		0			0
$598$		0			0
$599$	381.733	+	220.393i	0.637283	+	0.367936i	0.783567	−	0.621307i	$-0.213398\pi$
$599$	381.733	+	220.393i	0.637283	+	0.367936i	−0.146284	+	0.989243i	$0.546731\pi$
$600$		0			0
$601$	90.4814	+	156.718i	0.150551	+	0.260763i	0.931430	−	0.363920i	$-0.118562\pi$
$601$	90.4814	+	156.718i	0.150551	+	0.260763i	−0.780879	+	0.624682i	$0.785228\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	379.611	−	219.168i	0.627456	−	0.362262i
$606$		0			0
$607$	277.996	−	481.504i	0.457984	−	0.793252i	−0.540870	−	0.841106i	$-0.681905\pi$
$607$	277.996	−	481.504i	0.457984	−	0.793252i	0.998854	+	0.0478542i	$0.0152383\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			78.9674i			0.129243i
$612$		0			0
$613$	−335.352			−0.547067			−0.273534	−	0.961862i	$-0.588192\pi$
$613$	−335.352			−0.547067			−0.273534	+	0.961862i	$0.588192\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1020.64	−	589.268i	−1.65420	−	0.955053i	−0.975317	−	0.220808i	$-0.929131\pi$
$617$	−1020.64	−	589.268i	−1.65420	−	0.955053i	−0.678884	−	0.734246i	$-0.737536\pi$
$618$		0			0
$619$	−449.292	−	778.197i	−0.725835	−	1.25718i	−0.958629	−	0.284658i	$-0.908120\pi$
$619$	−449.292	−	778.197i	−0.725835	−	1.25718i	0.232794	−	0.972526i	$-0.425213\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−192.791	+	111.308i	−0.309455	+	0.178664i
$624$		0			0
$625$	389.395	−	674.451i	0.623031	−	1.07912i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	932.224i		−	1.48207i
$630$		0			0
$631$	431.017			0.683070			0.341535	−	0.939869i	$-0.389053\pi$
$631$	431.017			0.683070			0.341535	+	0.939869i	$0.389053\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	65.4531	+	37.7894i	0.103076	+	0.0595108i
$636$		0			0
$637$	−6.40513	−	11.0940i	−0.0100551	−	0.0174160i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−775.987	+	448.016i	−1.21059	+	0.698933i	−0.962887	−	0.269903i	$-0.913008\pi$
$641$	−775.987	+	448.016i	−1.21059	+	0.698933i	−0.247701	+	0.968837i	$0.579675\pi$
$642$		0			0
$643$	−110.854	+	192.004i	−0.172401	+	0.298607i	−0.939259	−	0.343210i	$-0.888486\pi$
$643$	−110.854	+	192.004i	−0.172401	+	0.298607i	0.766858	+	0.641817i	$0.221819\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	536.838i		−	0.829734i	−0.909882	−	0.414867i	$-0.863828\pi$
$647$		−	536.838i		−	0.829734i	0.909882	−	0.414867i	$-0.136172\pi$
$648$		0			0
$649$	−773.276			−1.19149
$650$		0			0
$651$		0			0
$652$		0			0
$653$	483.774	+	279.307i	0.740848	+	0.427729i	0.822378	−	0.568942i	$-0.192647\pi$
$653$	483.774	+	279.307i	0.740848	+	0.427729i	−0.0815296	+	0.996671i	$0.525981\pi$
$654$		0			0
$655$	−164.023	−	284.096i	−0.250417	−	0.433735i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	933.150	−	538.755i	1.41601	−	0.817534i	0.420064	−	0.907494i	$-0.362007\pi$
$659$	933.150	−	538.755i	1.41601	−	0.817534i	0.995945	+	0.0899606i	$0.0286741\pi$
$660$		0			0
$661$	147.880	−	256.135i	0.223721	−	0.387496i	−0.732214	−	0.681075i	$-0.761513\pi$
$661$	147.880	−	256.135i	0.223721	−	0.387496i	0.955935	+	0.293579i	$0.0948463\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			934.683i			1.40554i
$666$		0			0
$667$	−1116.71			−1.67423
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−313.610	−	181.063i	−0.467377	−	0.269840i
$672$		0			0
$673$	283.724	+	491.424i	0.421580	+	0.730199i	0.996094	−	0.0882963i	$-0.0281422\pi$
$673$	283.724	+	491.424i	0.421580	+	0.730199i	−0.574514	+	0.818495i	$0.694809\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	813.905	−	469.908i	1.20222	−	0.694104i	0.241174	−	0.970482i	$-0.422467\pi$
$677$	813.905	−	469.908i	1.20222	−	0.694104i	0.961049	+	0.276378i	$0.0891341\pi$
$678$		0			0
$679$	468.413	−	811.315i	0.689857	−	1.19487i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	14.2541i		−	0.0208699i	−0.999946	−	0.0104349i	$-0.996678\pi$
$683$		−	14.2541i		−	0.0208699i	0.999946	−	0.0104349i	$-0.00332161\pi$
$684$		0			0
$685$	−874.930			−1.27727
$686$		0			0
$687$		0			0
$688$		0			0
$689$	65.0950	+	37.5826i	0.0944775	+	0.0545466i
$690$		0			0
$691$	−83.1353	−	143.995i	−0.120312	−	0.208386i	0.799579	−	0.600561i	$-0.205056\pi$
$691$	−83.1353	−	143.995i	−0.120312	−	0.208386i	−0.919891	+	0.392175i	$0.871723\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−216.151	+	124.795i	−0.311009	+	0.179561i
$696$		0			0
$697$	239.075	−	414.090i	0.343006	−	0.594103i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			506.174i			0.722074i	0.932551	+	0.361037i	$0.117577\pi$
$701$			506.174i			0.722074i	−0.932551	+	0.361037i	$0.882423\pi$
$702$		0			0
$703$	−1258.14			−1.78967
$704$		0			0
$705$		0			0
$706$		0			0
$707$	255.643	+	147.595i	0.361588	+	0.208763i
$708$		0			0
$709$	−259.993	−	450.321i	−0.366704	−	0.635150i	0.622344	−	0.782744i	$-0.286181\pi$
$709$	−259.993	−	450.321i	−0.366704	−	0.635150i	−0.989048	+	0.147594i	$0.952847\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	590.863	−	341.135i	0.828700	−	0.478450i
$714$		0			0
$715$	39.6240	−	68.6307i	0.0554181	−	0.0959870i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	467.630i		−	0.650390i	−0.945647	−	0.325195i	$-0.894570\pi$
$719$		−	467.630i		−	0.650390i	0.945647	−	0.325195i	$-0.105430\pi$
$720$		0			0
$721$	−1120.41			−1.55397
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−564.549	−	325.943i	−0.778689	−	0.449576i
$726$		0			0
$727$	262.371	+	454.439i	0.360895	+	0.625088i	0.988108	−	0.153759i	$-0.0491380\pi$
$727$	262.371	+	454.439i	0.360895	+	0.625088i	−0.627214	+	0.778847i	$0.715805\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−993.663	+	573.692i	−1.35932	+	0.784804i
$732$		0			0
$733$	−329.348	+	570.447i	−0.449315	+	0.778236i	−0.998342	−	0.0575688i	$-0.981665\pi$
$733$	−329.348	+	570.447i	−0.449315	+	0.778236i	0.549027	+	0.835805i	$0.314998\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	812.363i		−	1.10226i
$738$		0			0
$739$	1209.27			1.63637			0.818183	−	0.574957i	$-0.194981\pi$
$739$	1209.27			1.63637			0.818183	+	0.574957i	$0.194981\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−339.650	−	196.097i	−0.457134	−	0.263926i	0.253704	−	0.967282i	$-0.418351\pi$
$743$	−339.650	−	196.097i	−0.457134	−	0.263926i	−0.710838	+	0.703355i	$0.751684\pi$
$744$		0			0
$745$	−11.7726	−	20.3908i	−0.0158022	−	0.0273701i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.72829	−	1.57518i	0.00364258	−	0.00210305i
$750$		0			0
$751$	558.493	−	967.339i	0.743666	−	1.28807i	−0.207149	−	0.978309i	$-0.566419\pi$
$751$	558.493	−	967.339i	0.743666	−	1.28807i	0.950815	−	0.309758i	$-0.100248\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	936.069i		−	1.23983i
$756$		0			0
$757$	−623.892			−0.824164			−0.412082	−	0.911147i	$-0.635198\pi$
$757$	−623.892			−0.824164			−0.412082	+	0.911147i	$0.635198\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	832.456	+	480.619i	1.09390	+	0.631562i	0.934611	−	0.355671i	$-0.115748\pi$
$761$	832.456	+	480.619i	1.09390	+	0.631562i	0.159286	+	0.987233i	$0.449081\pi$
$762$		0			0
$763$	241.638	+	418.528i	0.316694	+	0.548530i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	166.902	−	96.3610i	0.217604	−	0.125634i
$768$		0			0
$769$	−109.653	+	189.924i	−0.142592	+	0.246976i	−0.928472	−	0.371403i	$-0.878877\pi$
$769$	−109.653	+	189.924i	−0.142592	+	0.246976i	0.785880	+	0.618379i	$0.212210\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			761.394i			0.984986i	0.870316	+	0.492493i	$0.163914\pi$
$773$			761.394i			0.984986i	−0.870316	+	0.492493i	$0.836086\pi$
$774$		0			0
$775$	398.278			0.513907
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−558.858	−	322.657i	−0.717405	−	0.414194i
$780$		0			0
$781$	−303.888	−	526.349i	−0.389101	−	0.673943i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	638.211	−	368.471i	0.813007	−	0.469390i
$786$		0			0
$787$	263.023	−	455.569i	0.334210	−	0.578868i	−0.649123	−	0.760684i	$-0.724864\pi$
$787$	263.023	−	455.569i	0.334210	−	0.578868i	0.983333	+	0.181815i	$0.0581972\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	194.536i		−	0.245937i
$792$		0			0
$793$	90.2518			0.113811
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1230.12	+	710.212i	1.54344	+	0.891107i	0.998618	+	0.0525560i	$0.0167368\pi$
$797$	1230.12	+	710.212i	1.54344	+	0.891107i	0.544824	+	0.838551i	$0.316597\pi$
$798$		0			0
$799$	−328.271	−	568.582i	−0.410852	−	0.711617i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−583.935	+	337.135i	−0.727192	+	0.419845i
$804$		0			0
$805$	564.681	−	978.057i	0.701468	−	1.21498i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			749.612i			0.926590i	0.886204	+	0.463295i	$0.153333\pi$
$809$			749.612i			0.926590i	−0.886204	+	0.463295i	$0.846667\pi$
$810$		0			0
$811$	238.446			0.294015			0.147008	−	0.989135i	$-0.453036\pi$
$811$	238.446			0.294015			0.147008	+	0.989135i	$0.453036\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−606.052	−	349.905i	−0.743623	−	0.429331i
$816$		0			0
$817$	774.258	+	1341.05i	0.947684	+	1.64144i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−937.937	+	541.518i	−1.14243	+	0.659583i	−0.947032	−	0.321140i	$-0.895934\pi$
$821$	−937.937	+	541.518i	−1.14243	+	0.659583i	−0.195400	+	0.980724i	$0.562601\pi$
$822$		0			0
$823$	−787.881	+	1364.65i	−0.957328	+	1.65814i	−0.228379	+	0.973572i	$0.573343\pi$
$823$	−787.881	+	1364.65i	−0.957328	+	1.65814i	−0.728949	+	0.684568i	$0.759991\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			124.799i			0.150906i	0.997149	+	0.0754529i	$0.0240403\pi$
$827$			124.799i			0.150906i	−0.997149	+	0.0754529i	$0.975960\pi$
$828$		0			0
$829$	426.486			0.514458			0.257229	−	0.966350i	$-0.417190\pi$
$829$	426.486			0.514458			0.257229	+	0.966350i	$0.417190\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	92.2365	+	53.2528i	0.110728	+	0.0639289i
$834$		0			0
$835$	4.82403	+	8.35547i	0.00577728	+	0.0100065i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−430.710	+	248.671i	−0.513362	+	0.296389i	−0.734214	−	0.678918i	$-0.762449\pi$
$839$	−430.710	+	248.671i	−0.513362	+	0.296389i	0.220853	+	0.975307i	$0.429116\pi$
$840$		0			0
$841$	652.999	−	1131.03i	0.776455	−	1.34486i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	1036.58i		−	1.22673i
$846$		0			0
$847$	−525.759			−0.620730
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1316.52	+	760.092i	1.54703	+	0.893176i
$852$		0			0
$853$	767.481	+	1329.32i	0.899743	+	1.55840i	0.827823	+	0.560990i	$0.189579\pi$
$853$	767.481	+	1329.32i	0.899743	+	1.55840i	0.0719203	+	0.997410i	$0.477087\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1055.08	+	609.149i	−1.23113	+	0.710792i	−0.967265	−	0.253768i	$-0.918330\pi$
$857$	−1055.08	+	609.149i	−1.23113	+	0.710792i	−0.263863	+	0.964560i	$0.584997\pi$
$858$		0			0
$859$	−289.720	+	501.811i	−0.337276	+	0.584180i	−0.983919	−	0.178613i	$-0.942839\pi$
$859$	−289.720	+	501.811i	−0.337276	+	0.584180i	0.646643	+	0.762793i	$0.276172\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1361.09i			1.57716i	0.614934	+	0.788579i	$0.289183\pi$
$863$			1361.09i			1.57716i	−0.614934	+	0.788579i	$0.710817\pi$
$864$		0			0
$865$	332.422			0.384303
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−439.866	−	253.957i	−0.506175	−	0.292240i
$870$		0			0
$871$	101.232	+	175.339i	0.116225	+	0.201307i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−443.617	+	256.122i	−0.506991	+	0.292711i
$876$		0			0
$877$	150.947	−	261.448i	0.172118	−	0.298117i	−0.767042	−	0.641597i	$-0.778272\pi$
$877$	150.947	−	261.448i	0.172118	−	0.298117i	0.939160	+	0.343480i	$0.111606\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1170.94i		−	1.32911i	−0.747240	−	0.664554i	$-0.768622\pi$
$881$		−	1170.94i		−	1.32911i	0.747240	−	0.664554i	$-0.231378\pi$
$882$		0			0
$883$	1339.68			1.51719			0.758597	−	0.651560i	$-0.225885\pi$
$883$	1339.68			1.51719			0.758597	+	0.651560i	$0.225885\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−14.4375	−	8.33547i	−0.0162767	−	0.00939738i	0.491840	−	0.870686i	$-0.336325\pi$
$887$	−14.4375	−	8.33547i	−0.0162767	−	0.00939738i	−0.508116	+	0.861288i	$0.669658\pi$
$888$		0			0
$889$	−45.3261	−	78.5071i	−0.0509855	−	0.0883094i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−767.362	+	443.037i	−0.859308	+	0.496122i
$894$		0			0
$895$	−748.323	+	1296.13i	−0.836115	+	1.44819i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1311.74i			1.45911i
$900$		0			0
$901$	−624.930			−0.693596
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−331.042	−	191.127i	−0.365792	−	0.211190i
$906$		0			0
$907$	225.668	+	390.868i	0.248807	+	0.430946i	0.963195	−	0.268803i	$-0.0866282\pi$
$907$	225.668	+	390.868i	0.248807	+	0.430946i	−0.714388	+	0.699750i	$0.753295\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1243.11	−	717.711i	1.36456	−	0.787827i	0.374331	−	0.927295i	$-0.377873\pi$
$911$	1243.11	−	717.711i	1.36456	−	0.787827i	0.990227	+	0.139468i	$0.0445392\pi$
$912$		0			0
$913$	390.806	−	676.895i	0.428046	−	0.741397i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			393.472i			0.429086i
$918$		0			0
$919$	380.633			0.414181			0.207091	−	0.978322i	$-0.433600\pi$
$919$	380.633			0.414181			0.207091	+	0.978322i	$0.433600\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	131.181	+	75.7374i	0.142125	+	0.0820557i
$924$		0			0
$925$	443.707	+	768.523i	0.479683	+	0.830836i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	856.635	−	494.578i	0.922104	−	0.532377i	0.0377985	−	0.999285i	$-0.487965\pi$
$929$	856.635	−	494.578i	0.922104	−	0.532377i	0.884306	+	0.466908i	$0.154632\pi$
$930$		0			0
$931$	71.8703	−	124.483i	0.0771969	−	0.133709i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			658.874i			0.704678i
$936$		0			0
$937$	−549.202			−0.586128			−0.293064	−	0.956093i	$-0.594675\pi$
$937$	−549.202			−0.586128			−0.293064	+	0.956093i	$0.594675\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−405.479	−	234.103i	−0.430902	−	0.248781i	0.268829	−	0.963188i	$-0.413363\pi$
$941$	−405.479	−	234.103i	−0.430902	−	0.248781i	−0.699731	+	0.714407i	$0.746697\pi$
$942$		0			0
$943$	389.861	+	675.259i	0.413427	+	0.716076i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−260.869	+	150.613i	−0.275469	+	0.159042i	−0.631370	−	0.775482i	$-0.717507\pi$
$947$	−260.869	+	150.613i	−0.275469	+	0.159042i	0.355902	+	0.934523i	$0.384174\pi$
$948$		0			0
$949$	84.0235	−	145.533i	0.0885390	−	0.153354i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	319.384i		−	0.335135i	−0.985861	−	0.167567i	$-0.946409\pi$
$953$		−	319.384i		−	0.335135i	0.985861	−	0.167567i	$-0.0535912\pi$
$954$		0			0
$955$	208.630			0.218461
$956$		0			0
$957$		0			0
$958$		0			0
$959$	908.829	+	524.713i	0.947685	+	0.547146i
$960$		0			0
$961$	79.7892	+	138.199i	0.0830272	+	0.143807i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−230.377	+	133.008i	−0.238732	+	0.137832i
$966$		0			0
$967$	−302.086	+	523.229i	−0.312395	+	0.541085i	−0.978880	−	0.204434i	$-0.934465\pi$
$967$	−302.086	+	523.229i	−0.312395	+	0.541085i	0.666485	+	0.745518i	$0.267798\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			465.068i			0.478958i	0.970902	+	0.239479i	$0.0769766\pi$
$971$			465.068i			0.478958i	−0.970902	+	0.239479i	$0.923023\pi$
$972$		0			0
$973$	299.368			0.307675
$974$		0			0
$975$		0			0
$976$		0			0
$977$	332.512	+	191.976i	0.340340	+	0.196495i	0.660422	−	0.750894i	$-0.270377\pi$
$977$	332.512	+	191.976i	0.340340	+	0.196495i	−0.320083	+	0.947390i	$0.603711\pi$
$978$		0			0
$979$	105.894	+	183.413i	0.108165	+	0.187348i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−586.197	+	338.441i	−0.596335	+	0.344294i	−0.767598	−	0.640931i	$-0.778548\pi$
$983$	−586.197	+	338.441i	−0.596335	+	0.344294i	0.171264	+	0.985225i	$0.445215\pi$
$984$		0			0
$985$	−635.589	+	1100.87i	−0.645268	+	1.11764i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1871.05i		−	1.89186i
$990$		0			0
$991$	1097.79			1.10776			0.553878	−	0.832598i	$-0.313147\pi$
$991$	1097.79			1.10776			0.553878	+	0.832598i	$0.313147\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−693.744	−	400.533i	−0.697230	−	0.402546i
$996$		0			0
$997$	−90.7127	−	157.119i	−0.0909857	−	0.157592i	0.816940	−	0.576722i	$-0.195668\pi$
$997$	−90.7127	−	157.119i	−0.0909857	−	0.157592i	−0.907926	+	0.419130i	$0.862335\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.3.q.a.449.4		24
3.2	odd	2		288.3.q.b.257.12	yes	24
4.3	odd	2	inner	864.3.q.a.449.3		24
8.3	odd	2		1728.3.q.k.449.9		24
8.5	even	2		1728.3.q.k.449.10		24
9.2	odd	6	inner	864.3.q.a.737.4		24
9.4	even	3		2592.3.e.i.161.6		24
9.5	odd	6		2592.3.e.i.161.5		24
9.7	even	3		288.3.q.b.65.12	yes	24
12.11	even	2		288.3.q.b.257.1	yes	24
24.5	odd	2		576.3.q.l.257.1		24
24.11	even	2		576.3.q.l.257.12		24
36.7	odd	6		288.3.q.b.65.1	✓	24
36.11	even	6	inner	864.3.q.a.737.3		24
36.23	even	6		2592.3.e.i.161.19		24
36.31	odd	6		2592.3.e.i.161.20		24
72.11	even	6		1728.3.q.k.1601.9		24
72.29	odd	6		1728.3.q.k.1601.10		24
72.43	odd	6		576.3.q.l.65.12		24
72.61	even	6		576.3.q.l.65.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.q.b.65.1	✓	24	36.7	odd	6
288.3.q.b.65.12	yes	24	9.7	even	3
288.3.q.b.257.1	yes	24	12.11	even	2
288.3.q.b.257.12	yes	24	3.2	odd	2
576.3.q.l.65.1		24	72.61	even	6
576.3.q.l.65.12		24	72.43	odd	6
576.3.q.l.257.1		24	24.5	odd	2
576.3.q.l.257.12		24	24.11	even	2
864.3.q.a.449.3		24	4.3	odd	2	inner
864.3.q.a.449.4		24	1.1	even	1	trivial
864.3.q.a.737.3		24	36.11	even	6	inner
864.3.q.a.737.4		24	9.2	odd	6	inner
1728.3.q.k.449.9		24	8.3	odd	2
1728.3.q.k.449.10		24	8.5	even	2
1728.3.q.k.1601.9		24	72.11	even	6
1728.3.q.k.1601.10		24	72.29	odd	6
2592.3.e.i.161.5		24	9.5	odd	6
2592.3.e.i.161.6		24	9.4	even	3
2592.3.e.i.161.19		24	36.23	even	6
2592.3.e.i.161.20		24	36.31	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	864.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-5.41309 - 3.12525i) q^{5} +(3.74855 + 6.49268i) q^{7} +(6.17688 - 3.56622i) q^{11} +(-0.888802 + 1.53945i) q^{13} -14.7791i q^{17} -19.9460 q^{19} +(20.8716 + 12.0502i) q^{23} +(7.03438 + 12.1839i) q^{25} +(-40.1279 + 23.1679i) q^{29} +(14.1547 - 24.5167i) q^{31} -46.8606i q^{35} +63.0770 q^{37} +(28.0185 + 16.1765i) q^{41} +(-38.8176 - 67.2341i) q^{43} +(38.4719 - 22.2118i) q^{47} +(-3.60324 + 6.24099i) q^{49} -42.2846i q^{53} -44.5813 q^{55} +(-93.8917 - 54.2084i) q^{59} +(-25.3858 - 43.9695i) q^{61} +(9.62233 - 5.55546i) q^{65} +(56.9484 - 98.6376i) q^{67} -85.2129i q^{71} -94.5357 q^{73} +(46.3086 + 26.7363i) q^{77} +(-35.6059 - 61.6712i) q^{79} +(94.9037 - 54.7927i) q^{83} +(-46.1885 + 80.0009i) q^{85} +29.6936i q^{89} -13.3269 q^{91} +(107.970 + 62.3363i) q^{95} +(-62.4793 - 108.217i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 60 q^{25} - 72 q^{29} - 252 q^{41} - 36 q^{49} - 96 q^{61} + 288 q^{65} + 24 q^{73} + 720 q^{77} + 96 q^{85} - 132 q^{97}+O(q^{100}) \) 24 * q + 60 * q^25 - 72 * q^29 - 252 * q^41 - 36 * q^49 - 96 * q^61 + 288 * q^65 + 24 * q^73 + 720 * q^77 + 96 * q^85 - 132 * q^97

Introduction

L-functions

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