LMFDB - Embedded newform 576.4.d.f.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,4,Mod(289,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("576.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	576.d (of order $2$, degree $1$, 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:	$33.9851001633$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 9 $ x^4 - 5*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 192)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.1
Root			$-1.65831 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	576.289
Dual form			576.4.d.f.289.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.8997i q^{5} -19.8997 q^{7} +O(q^{10})$	$q-19.8997i q^{5} -19.8997 q^{7} +48.0000i q^{11} -79.5990i q^{13} +42.0000 q^{17} -92.0000i q^{19} -39.7995 q^{23} -271.000 q^{25} -19.8997i q^{29} -139.298 q^{31} +396.000i q^{35} +198.997i q^{37} +6.00000 q^{41} +92.0000i q^{43} -39.7995 q^{47} +53.0000 q^{49} +497.494i q^{53} +955.188 q^{55} +516.000i q^{59} +358.195i q^{61} -1584.00 q^{65} +524.000i q^{67} +994.987 q^{71} -430.000 q^{73} -955.188i q^{77} -1174.09 q^{79} -432.000i q^{83} -835.789i q^{85} -630.000 q^{89} +1584.00i q^{91} -1830.78 q^{95} +862.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 168 q^{17} - 1084 q^{25} + 24 q^{41} + 212 q^{49} - 6336 q^{65} - 1720 q^{73} - 2520 q^{89} + 3448 q^{97}+O(q^{100}) $ 4 * q + 168 * q^17 - 1084 * q^25 + 24 * q^41 + 212 * q^49 - 6336 * q^65 - 1720 * q^73 - 2520 * q^89 + 3448 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 19.8997i	− 1.77989i	−0.456070	−	0.889944i	$-0.650743\pi$
$5$	− 19.8997i	− 1.77989i	0.456070	−	0.889944i	$-0.349257\pi$
$6$	0	0
$7$	−19.8997	−1.07449	−0.537243	−	0.843428i	$-0.680534\pi$
$7$	−19.8997	−1.07449	−0.537243	+	0.843428i	$0.680534\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	48.0000i	1.31569i	0.753155	+	0.657843i	$0.228531\pi$
$11$	48.0000i	1.31569i	−0.753155	+	0.657843i	$0.771469\pi$
$12$	0	0
$13$	− 79.5990i	− 1.69821i	−0.528220	−	0.849107i	$-0.677141\pi$
$13$	− 79.5990i	− 1.69821i	0.528220	−	0.849107i	$-0.322859\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	42.0000	0.599206	0.299603	−	0.954064i	$-0.403146\pi$
$17$	42.0000	0.599206	0.299603	+	0.954064i	$0.403146\pi$
$18$	0	0
$19$	− 92.0000i	− 1.11086i	−0.831565	−	0.555428i	$-0.812555\pi$
$19$	− 92.0000i	− 1.11086i	0.831565	−	0.555428i	$-0.187445\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−39.7995	−0.360816	−0.180408	−	0.983592i	$-0.557742\pi$
$23$	−39.7995	−0.360816	−0.180408	+	0.983592i	$0.557742\pi$
$24$	0	0
$25$	−271.000	−2.16800
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 19.8997i	− 0.127424i	−0.997968	−	0.0637119i	$-0.979706\pi$
$29$	− 19.8997i	− 0.127424i	0.997968	−	0.0637119i	$-0.0202939\pi$
$30$	0	0
$31$	−139.298	−0.807055	−0.403527	−	0.914968i	$-0.632216\pi$
$31$	−139.298	−0.807055	−0.403527	+	0.914968i	$0.632216\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	396.000i	1.91246i
$36$	0	0
$37$	198.997i	0.884189i	0.896969	+	0.442094i	$0.145764\pi$
$37$	198.997i	0.884189i	−0.896969	+	0.442094i	$0.854236\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.0228547	0.0114273	−	0.999935i	$-0.496362\pi$
$41$	6.00000	0.0228547	0.0114273	+	0.999935i	$0.496362\pi$
$42$	0	0
$43$	92.0000i	0.326276i	0.986603	+	0.163138i	$0.0521616\pi$
$43$	92.0000i	0.326276i	−0.986603	+	0.163138i	$0.947838\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−39.7995	−0.123518	−0.0617591	−	0.998091i	$-0.519671\pi$
$47$	−39.7995	−0.123518	−0.0617591	+	0.998091i	$0.519671\pi$
$48$	0	0
$49$	53.0000	0.154519
$50$	0	0
$51$	0	0
$52$	0	0
$53$	497.494i	1.28936i	0.764453	+	0.644679i	$0.223009\pi$
$53$	497.494i	1.28936i	−0.764453	+	0.644679i	$0.776991\pi$
$54$	0	0
$55$	955.188	2.34177
$56$	0	0
$57$	0	0
$58$	0	0
$59$	516.000i	1.13860i	0.822129	+	0.569301i	$0.192786\pi$
$59$	516.000i	1.13860i	−0.822129	+	0.569301i	$0.807214\pi$
$60$	0	0
$61$	358.195i	0.751840i	0.926652	+	0.375920i	$0.122673\pi$
$61$	358.195i	0.751840i	−0.926652	+	0.375920i	$0.877327\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1584.00	−3.02263
$66$	0	0
$67$	524.000i	0.955474i	0.878503	+	0.477737i	$0.158543\pi$
$67$	524.000i	0.955474i	−0.878503	+	0.477737i	$0.841457\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	994.987	1.66314	0.831572	−	0.555416i	$-0.187441\pi$
$71$	994.987	1.66314	0.831572	+	0.555416i	$0.187441\pi$
$72$	0	0
$73$	−430.000	−0.689420	−0.344710	−	0.938709i	$-0.612023\pi$
$73$	−430.000	−0.689420	−0.344710	+	0.938709i	$0.612023\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 955.188i	− 1.41369i
$78$	0	0
$79$	−1174.09	−1.67209	−0.836044	−	0.548663i	$-0.815137\pi$
$79$	−1174.09	−1.67209	−0.836044	+	0.548663i	$0.815137\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 432.000i	− 0.571303i	−0.958334	−	0.285652i	$-0.907790\pi$
$83$	− 432.000i	− 0.571303i	0.958334	−	0.285652i	$-0.0922100\pi$
$84$	0	0
$85$	− 835.789i	− 1.06652i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−630.000	−0.750336	−0.375168	−	0.926957i	$-0.622415\pi$
$89$	−630.000	−0.750336	−0.375168	+	0.926957i	$0.622415\pi$
$90$	0	0
$91$	1584.00i	1.82471i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1830.78	−1.97720
$96$	0	0
$97$	862.000	0.902297	0.451149	−	0.892449i	$-0.351014\pi$
$97$	862.000	0.902297	0.451149	+	0.892449i	$0.351014\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 457.694i	− 0.450914i	−0.974253	−	0.225457i	$-0.927613\pi$
$101$	− 457.694i	− 0.450914i	0.974253	−	0.225457i	$-0.0723875\pi$
$102$	0	0
$103$	−537.293	−0.513991	−0.256996	−	0.966413i	$-0.582733\pi$
$103$	−537.293	−0.513991	−0.256996	+	0.966413i	$0.582733\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1380.00i	1.24682i	0.781896	+	0.623410i	$0.214253\pi$
$107$	1380.00i	1.24682i	−0.781896	+	0.623410i	$0.785747\pi$
$108$	0	0
$109$	− 1273.58i	− 1.11915i	−0.828780	−	0.559574i	$-0.810965\pi$
$109$	− 1273.58i	− 1.11915i	0.828780	−	0.559574i	$-0.189035\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2106.00	−1.75324	−0.876619	−	0.481186i	$-0.840206\pi$
$113$	−2106.00	−1.75324	−0.876619	+	0.481186i	$0.840206\pi$
$114$	0	0
$115$	792.000i	0.642212i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−835.789	−0.643838
$120$	0	0
$121$	−973.000	−0.731029
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2905.36i	2.07891i
$126$	0	0
$127$	139.298	0.0973285	0.0486643	−	0.998815i	$-0.484504\pi$
$127$	139.298	0.0973285	0.0486643	+	0.998815i	$0.484504\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	540.000i	0.360153i	0.983653	+	0.180076i	$0.0576345\pi$
$131$	540.000i	0.360153i	−0.983653	+	0.180076i	$0.942365\pi$
$132$	0	0
$133$	1830.78i	1.19360i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1446.00	0.901753	0.450876	−	0.892586i	$-0.351112\pi$
$137$	1446.00	0.901753	0.450876	+	0.892586i	$0.351112\pi$
$138$	0	0
$139$	− 1892.00i	− 1.15451i	−0.816563	−	0.577257i	$-0.804123\pi$
$139$	− 1892.00i	− 1.15451i	0.816563	−	0.577257i	$-0.195877\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3820.75	2.23432
$144$	0	0
$145$	−396.000	−0.226800
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2805.86i	− 1.54272i	−0.636399	−	0.771360i	$-0.719577\pi$
$149$	− 2805.86i	− 1.54272i	0.636399	−	0.771360i	$-0.280423\pi$
$150$	0	0
$151$	298.496	0.160869	0.0804347	−	0.996760i	$-0.474369\pi$
$151$	298.496	0.160869	0.0804347	+	0.996760i	$0.474369\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2772.00i	1.43647i
$156$	0	0
$157$	2348.17i	1.19366i	0.802368	+	0.596829i	$0.203573\pi$
$157$	2348.17i	1.19366i	−0.802368	+	0.596829i	$0.796427\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	792.000	0.387692
$162$	0	0
$163$	− 3188.00i	− 1.53192i	−0.642887	−	0.765961i	$-0.722263\pi$
$163$	− 3188.00i	− 1.53192i	0.642887	−	0.765961i	$-0.277737\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1989.97	−0.922089	−0.461045	−	0.887377i	$-0.652525\pi$
$167$	−1989.97	−0.922089	−0.461045	+	0.887377i	$0.652525\pi$
$168$	0	0
$169$	−4139.00	−1.88393
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 378.095i	− 0.166162i	−0.996543	−	0.0830811i	$-0.973524\pi$
$173$	− 378.095i	− 0.166162i	0.996543	−	0.0830811i	$-0.0264760\pi$
$174$	0	0
$175$	5392.83	2.32948
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 2844.00i	− 1.18754i	−0.804633	−	0.593772i	$-0.797638\pi$
$179$	− 2844.00i	− 1.18754i	0.804633	−	0.593772i	$-0.202362\pi$
$180$	0	0
$181$	− 2228.77i	− 0.915267i	−0.889141	−	0.457633i	$-0.848697\pi$
$181$	− 2228.77i	− 0.915267i	0.889141	−	0.457633i	$-0.151303\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3960.00	1.57376
$186$	0	0
$187$	2016.00i	0.788366i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	875.589	0.331704	0.165852	−	0.986151i	$-0.446963\pi$
$191$	875.589	0.331704	0.165852	+	0.986151i	$0.446963\pi$
$192$	0	0
$193$	2882.00	1.07488	0.537438	−	0.843304i	$-0.319392\pi$
$193$	2882.00	1.07488	0.537438	+	0.843304i	$0.319392\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4238.65i	1.53295i	0.642274	+	0.766475i	$0.277991\pi$
$197$	4238.65i	1.53295i	−0.642274	+	0.766475i	$0.722009\pi$
$198$	0	0
$199$	−3243.66	−1.15546	−0.577731	−	0.816227i	$-0.696062\pi$
$199$	−3243.66	−1.15546	−0.577731	+	0.816227i	$0.696062\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	396.000i	0.136915i
$204$	0	0
$205$	− 119.398i	− 0.0406788i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4416.00	1.46154
$210$	0	0
$211$	− 2356.00i	− 0.768691i	−0.923189	−	0.384345i	$-0.874427\pi$
$211$	− 2356.00i	− 0.768691i	0.923189	−	0.384345i	$-0.125573\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1830.78	0.580735
$216$	0	0
$217$	2772.00	0.867169
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 3343.16i	− 1.01758i
$222$	0	0
$223$	−1771.08	−0.531839	−0.265920	−	0.963995i	$-0.585676\pi$
$223$	−1771.08	−0.531839	−0.265920	+	0.963995i	$0.585676\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3864.00i	− 1.12979i	−0.825162	−	0.564896i	$-0.808916\pi$
$227$	− 3864.00i	− 1.12979i	0.825162	−	0.564896i	$-0.191084\pi$
$228$	0	0
$229$	557.193i	0.160788i	0.996763	+	0.0803938i	$0.0256178\pi$
$229$	557.193i	0.160788i	−0.996763	+	0.0803938i	$0.974382\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2814.00	−0.791207	−0.395604	−	0.918421i	$-0.629465\pi$
$233$	−2814.00	−0.791207	−0.395604	+	0.918421i	$0.629465\pi$
$234$	0	0
$235$	792.000i	0.219848i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4616.74	−1.24951	−0.624754	−	0.780822i	$-0.714801\pi$
$239$	−4616.74	−1.24951	−0.624754	+	0.780822i	$0.714801\pi$
$240$	0	0
$241$	362.000	0.0967571	0.0483786	−	0.998829i	$-0.484595\pi$
$241$	362.000	0.0967571	0.0483786	+	0.998829i	$0.484595\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1054.69i	− 0.275026i
$246$	0	0
$247$	−7323.11	−1.88647
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 5136.00i	− 1.29156i	−0.763524	−	0.645780i	$-0.776532\pi$
$251$	− 5136.00i	− 1.29156i	0.763524	−	0.645780i	$-0.223468\pi$
$252$	0	0
$253$	− 1910.38i	− 0.474721i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	174.000	0.0422328	0.0211164	−	0.999777i	$-0.493278\pi$
$257$	174.000	0.0422328	0.0211164	+	0.999777i	$0.493278\pi$
$258$	0	0
$259$	− 3960.00i	− 0.950048i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4775.94	−1.11976	−0.559880	−	0.828573i	$-0.689153\pi$
$263$	−4775.94	−1.11976	−0.559880	+	0.828573i	$0.689153\pi$
$264$	0	0
$265$	9900.00	2.29491
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 1134.29i	− 0.257095i	−0.991703	−	0.128548i	$-0.958968\pi$
$269$	− 1134.29i	− 0.257095i	0.991703	−	0.128548i	$-0.0410315\pi$
$270$	0	0
$271$	−179.098	−0.0401454	−0.0200727	−	0.999799i	$-0.506390\pi$
$271$	−179.098	−0.0401454	−0.0200727	+	0.999799i	$0.506390\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 13008.0i	− 2.85241i
$276$	0	0
$277$	− 6367.92i	− 1.38127i	−0.723205	−	0.690634i	$-0.757332\pi$
$277$	− 6367.92i	− 1.38127i	0.723205	−	0.690634i	$-0.242668\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−558.000	−0.118461	−0.0592304	−	0.998244i	$-0.518865\pi$
$281$	−558.000	−0.118461	−0.0592304	+	0.998244i	$0.518865\pi$
$282$	0	0
$283$	2068.00i	0.434381i	0.976129	+	0.217191i	$0.0696893\pi$
$283$	2068.00i	0.434381i	−0.976129	+	0.217191i	$0.930311\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−119.398	−0.0245570
$288$	0	0
$289$	−3149.00	−0.640953
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3363.06i	0.670553i	0.942120	+	0.335276i	$0.108830\pi$
$293$	3363.06i	0.670553i	−0.942120	+	0.335276i	$0.891170\pi$
$294$	0	0
$295$	10268.3	2.02658
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3168.00i	0.612743i
$300$	0	0
$301$	− 1830.78i	− 0.350579i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7128.00	1.33819
$306$	0	0
$307$	6860.00i	1.27531i	0.770321	+	0.637656i	$0.220096\pi$
$307$	6860.00i	1.27531i	−0.770321	+	0.637656i	$0.779904\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	10427.5	1.90125	0.950623	−	0.310348i	$-0.100446\pi$
$311$	10427.5	1.90125	0.950623	+	0.310348i	$0.100446\pi$
$312$	0	0
$313$	6262.00	1.13083	0.565414	−	0.824807i	$-0.308716\pi$
$313$	6262.00	1.13083	0.565414	+	0.824807i	$0.308716\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	776.090i	0.137507i	0.997634	+	0.0687533i	$0.0219021\pi$
$317$	776.090i	0.137507i	−0.997634	+	0.0687533i	$0.978098\pi$
$318$	0	0
$319$	955.188	0.167650
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 3864.00i	− 0.665631i
$324$	0	0
$325$	21571.3i	3.68173i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	792.000	0.132718
$330$	0	0
$331$	6460.00i	1.07273i	0.843986	+	0.536365i	$0.180203\pi$
$331$	6460.00i	1.07273i	−0.843986	+	0.536365i	$0.819797\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10427.5	1.70064
$336$	0	0
$337$	−74.0000	−0.0119615	−0.00598077	−	0.999982i	$-0.501904\pi$
$337$	−74.0000	−0.0119615	−0.00598077	+	0.999982i	$0.501904\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 6686.32i	− 1.06183i
$342$	0	0
$343$	5770.93	0.908457
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6888.00i	1.06561i	0.846238	+	0.532806i	$0.178862\pi$
$347$	6888.00i	1.06561i	−0.846238	+	0.532806i	$0.821138\pi$
$348$	0	0
$349$	676.591i	0.103774i	0.998653	+	0.0518870i	$0.0165236\pi$
$349$	676.591i	0.103774i	−0.998653	+	0.0518870i	$0.983476\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1830.00	0.275924	0.137962	−	0.990438i	$-0.455945\pi$
$353$	1830.00	0.275924	0.137962	+	0.990438i	$0.455945\pi$
$354$	0	0
$355$	− 19800.0i	− 2.96021i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6646.52	−0.977130	−0.488565	−	0.872527i	$-0.662479\pi$
$359$	−6646.52	−0.977130	−0.488565	+	0.872527i	$0.662479\pi$
$360$	0	0
$361$	−1605.00	−0.233999
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8556.89i	1.22709i
$366$	0	0
$367$	−3164.06	−0.450034	−0.225017	−	0.974355i	$-0.572244\pi$
$367$	−3164.06	−0.450034	−0.225017	+	0.974355i	$0.572244\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 9900.00i	− 1.38540i
$372$	0	0
$373$	− 9750.88i	− 1.35357i	−0.736181	−	0.676785i	$-0.763373\pi$
$373$	− 9750.88i	− 1.35357i	0.736181	−	0.676785i	$-0.236627\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1584.00	−0.216393
$378$	0	0
$379$	− 8332.00i	− 1.12925i	−0.825347	−	0.564625i	$-0.809021\pi$
$379$	− 8332.00i	− 1.12925i	0.825347	−	0.564625i	$-0.190979\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6765.91	−0.902669	−0.451334	−	0.892355i	$-0.649052\pi$
$383$	−6765.91	−0.902669	−0.451334	+	0.892355i	$0.649052\pi$
$384$	0	0
$385$	−19008.0	−2.51620
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5870.43i	0.765148i	0.923925	+	0.382574i	$0.124962\pi$
$389$	5870.43i	0.765148i	−0.923925	+	0.382574i	$0.875038\pi$
$390$	0	0
$391$	−1671.58	−0.216203
$392$	0	0
$393$	0	0
$394$	0	0
$395$	23364.0i	2.97613i
$396$	0	0
$397$	− 2666.57i	− 0.337106i	−0.985693	−	0.168553i	$-0.946091\pi$
$397$	− 2666.57i	− 0.337106i	0.985693	−	0.168553i	$-0.0539095\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2634.00	0.328019	0.164010	−	0.986459i	$-0.447557\pi$
$401$	2634.00	0.328019	0.164010	+	0.986459i	$0.447557\pi$
$402$	0	0
$403$	11088.0i	1.37055i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9551.88	−1.16331
$408$	0	0
$409$	506.000	0.0611738	0.0305869	−	0.999532i	$-0.490262\pi$
$409$	506.000	0.0611738	0.0305869	+	0.999532i	$0.490262\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 10268.3i	− 1.22341i
$414$	0	0
$415$	−8596.69	−1.01686
$416$	0	0
$417$	0	0
$418$	0	0
$419$	15480.0i	1.80489i	0.430809	+	0.902443i	$0.358228\pi$
$419$	15480.0i	1.80489i	−0.430809	+	0.902443i	$0.641772\pi$
$420$	0	0
$421$	− 10666.3i	− 1.23478i	−0.786658	−	0.617390i	$-0.788190\pi$
$421$	− 10666.3i	− 1.23478i	0.786658	−	0.617390i	$-0.211810\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−11382.0	−1.29908
$426$	0	0
$427$	− 7128.00i	− 0.807841i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17153.6	−1.91707	−0.958537	−	0.284968i	$-0.908017\pi$
$431$	−17153.6	−1.91707	−0.958537	+	0.284968i	$0.908017\pi$
$432$	0	0
$433$	2014.00	0.223526	0.111763	−	0.993735i	$-0.464350\pi$
$433$	2014.00	0.223526	0.111763	+	0.993735i	$0.464350\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3661.55i	0.400814i
$438$	0	0
$439$	−776.090	−0.0843753	−0.0421877	−	0.999110i	$-0.513433\pi$
$439$	−776.090	−0.0843753	−0.0421877	+	0.999110i	$0.513433\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 408.000i	− 0.0437577i	−0.999761	−	0.0218789i	$-0.993035\pi$
$443$	− 408.000i	− 0.0437577i	0.999761	−	0.0218789i	$-0.00696481\pi$
$444$	0	0
$445$	12536.8i	1.33551i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11658.0	1.22533	0.612667	−	0.790341i	$-0.290097\pi$
$449$	11658.0	1.22533	0.612667	+	0.790341i	$0.290097\pi$
$450$	0	0
$451$	288.000i	0.0300696i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	31521.2	3.24777
$456$	0	0
$457$	−16486.0	−1.68749	−0.843745	−	0.536745i	$-0.819654\pi$
$457$	−16486.0	−1.68749	−0.843745	+	0.536745i	$0.819654\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 4119.25i	− 0.416166i	−0.978111	−	0.208083i	$-0.933278\pi$
$461$	− 4119.25i	− 0.416166i	0.978111	−	0.208083i	$-0.0667224\pi$
$462$	0	0
$463$	−13472.1	−1.35227	−0.676137	−	0.736776i	$-0.736347\pi$
$463$	−13472.1	−1.35227	−0.676137	+	0.736776i	$0.736347\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4440.00i	− 0.439954i	−0.975505	−	0.219977i	$-0.929402\pi$
$467$	− 4440.00i	− 0.439954i	0.975505	−	0.219977i	$-0.0705983\pi$
$468$	0	0
$469$	− 10427.5i	− 1.02664i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4416.00	−0.429277
$474$	0	0
$475$	24932.0i	2.40833i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10626.5	1.01364	0.506822	−	0.862051i	$-0.330820\pi$
$479$	10626.5	1.01364	0.506822	+	0.862051i	$0.330820\pi$
$480$	0	0
$481$	15840.0	1.50154
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 17153.6i	− 1.60599i
$486$	0	0
$487$	1333.28	0.124059	0.0620296	−	0.998074i	$-0.480243\pi$
$487$	1333.28	0.124059	0.0620296	+	0.998074i	$0.480243\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7404.00i	0.680525i	0.940330	+	0.340263i	$0.110516\pi$
$491$	7404.00i	0.680525i	−0.940330	+	0.340263i	$0.889484\pi$
$492$	0	0
$493$	− 835.789i	− 0.0763531i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−19800.0	−1.78702
$498$	0	0
$499$	− 5740.00i	− 0.514945i	−0.966286	−	0.257473i	$-0.917110\pi$
$499$	− 5740.00i	− 0.514945i	0.966286	−	0.257473i	$-0.0828897\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16039.2	−1.42177	−0.710887	−	0.703306i	$-0.751706\pi$
$503$	−16039.2	−1.42177	−0.710887	+	0.703306i	$0.751706\pi$
$504$	0	0
$505$	−9108.00	−0.802576
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16377.5i	1.42617i	0.701078	+	0.713084i	$0.252702\pi$
$509$	16377.5i	1.42617i	−0.701078	+	0.713084i	$0.747298\pi$
$510$	0	0
$511$	8556.89	0.740772
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10692.0i	0.914846i
$516$	0	0
$517$	− 1910.38i	− 0.162511i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−22098.0	−1.85822	−0.929108	−	0.369807i	$-0.879424\pi$
$521$	−22098.0	−1.85822	−0.929108	+	0.369807i	$0.879424\pi$
$522$	0	0
$523$	16436.0i	1.37418i	0.726572	+	0.687090i	$0.241112\pi$
$523$	16436.0i	1.37418i	−0.726572	+	0.687090i	$0.758888\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5850.53	−0.483592
$528$	0	0
$529$	−10583.0	−0.869812
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 477.594i	− 0.0388122i
$534$	0	0
$535$	27461.7	2.21920
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2544.00i	0.203298i
$540$	0	0
$541$	− 11143.9i	− 0.885604i	−0.896619	−	0.442802i	$-0.853984\pi$
$541$	− 11143.9i	− 0.885604i	0.896619	−	0.442802i	$-0.146016\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−25344.0	−1.99196
$546$	0	0
$547$	− 1316.00i	− 0.102867i	−0.998676	−	0.0514334i	$-0.983621\pi$
$547$	− 1316.00i	− 0.102867i	0.998676	−	0.0514334i	$-0.0163790\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1830.78	−0.141549
$552$	0	0
$553$	23364.0	1.79663
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3402.86i	− 0.258858i	−0.991589	−	0.129429i	$-0.958686\pi$
$557$	− 3402.86i	− 0.258858i	0.991589	−	0.129429i	$-0.0413144\pi$
$558$	0	0
$559$	7323.11	0.554087
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 19560.0i	− 1.46422i	−0.681187	−	0.732110i	$-0.738536\pi$
$563$	− 19560.0i	− 1.46422i	0.681187	−	0.732110i	$-0.261464\pi$
$564$	0	0
$565$	41908.9i	3.12057i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20838.0	1.53528	0.767640	−	0.640881i	$-0.221431\pi$
$569$	20838.0	1.53528	0.767640	+	0.640881i	$0.221431\pi$
$570$	0	0
$571$	− 1100.00i	− 0.0806192i	−0.999187	−	0.0403096i	$-0.987166\pi$
$571$	− 1100.00i	− 0.0806192i	0.999187	−	0.0403096i	$-0.0128344\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	10785.7	0.782249
$576$	0	0
$577$	−1730.00	−0.124819	−0.0624097	−	0.998051i	$-0.519879\pi$
$577$	−1730.00	−0.124819	−0.0624097	+	0.998051i	$0.519879\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8596.69i	0.613857i
$582$	0	0
$583$	−23879.7	−1.69639
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7644.00i	0.537482i	0.963213	+	0.268741i	$0.0866075\pi$
$587$	7644.00i	0.537482i	−0.963213	+	0.268741i	$0.913393\pi$
$588$	0	0
$589$	12815.4i	0.896521i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26874.0	−1.86102	−0.930508	−	0.366271i	$-0.880634\pi$
$593$	−26874.0	−1.86102	−0.930508	+	0.366271i	$0.880634\pi$
$594$	0	0
$595$	16632.0i	1.14596i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9750.88	0.665125	0.332563	−	0.943081i	$-0.392087\pi$
$599$	9750.88	0.665125	0.332563	+	0.943081i	$0.392087\pi$
$600$	0	0
$601$	650.000	0.0441166	0.0220583	−	0.999757i	$-0.492978\pi$
$601$	650.000	0.0441166	0.0220583	+	0.999757i	$0.492978\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	19362.5i	1.30115i
$606$	0	0
$607$	16695.9	1.11642	0.558209	−	0.829701i	$-0.311489\pi$
$607$	16695.9	1.11642	0.558209	+	0.829701i	$0.311489\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3168.00i	0.209760i
$612$	0	0
$613$	− 7522.11i	− 0.495620i	−0.968809	−	0.247810i	$-0.920289\pi$
$613$	− 7522.11i	− 0.495620i	0.968809	−	0.247810i	$-0.0797108\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2526.00	−0.164818	−0.0824092	−	0.996599i	$-0.526261\pi$
$617$	−2526.00	−0.164818	−0.0824092	+	0.996599i	$0.526261\pi$
$618$	0	0
$619$	4372.00i	0.283886i	0.989875	+	0.141943i	$0.0453350\pi$
$619$	4372.00i	0.283886i	−0.989875	+	0.141943i	$0.954665\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12536.8	0.806225
$624$	0	0
$625$	23941.0	1.53222
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8357.89i	0.529811i
$630$	0	0
$631$	−23979.2	−1.51283	−0.756416	−	0.654091i	$-0.773051\pi$
$631$	−23979.2	−1.51283	−0.756416	+	0.654091i	$0.773051\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 2772.00i	− 0.173234i
$636$	0	0
$637$	− 4218.75i	− 0.262406i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1374.00	−0.0846642	−0.0423321	−	0.999104i	$-0.513479\pi$
$641$	−1374.00	−0.0846642	−0.0423321	+	0.999104i	$0.513479\pi$
$642$	0	0
$643$	− 4196.00i	− 0.257347i	−0.991687	−	0.128673i	$-0.958928\pi$
$643$	− 4196.00i	− 0.257347i	0.991687	−	0.128673i	$-0.0410719\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17392.4	−1.05682	−0.528412	−	0.848988i	$-0.677212\pi$
$647$	−17392.4	−1.05682	−0.528412	+	0.848988i	$0.677212\pi$
$648$	0	0
$649$	−24768.0	−1.49804
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 18685.9i	− 1.11981i	−0.828558	−	0.559904i	$-0.810838\pi$
$653$	− 18685.9i	− 1.11981i	0.828558	−	0.559904i	$-0.189162\pi$
$654$	0	0
$655$	10745.9	0.641032
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16380.0i	0.968246i	0.875000	+	0.484123i	$0.160861\pi$
$659$	16380.0i	0.968246i	−0.875000	+	0.484123i	$0.839139\pi$
$660$	0	0
$661$	− 5850.53i	− 0.344265i	−0.985074	−	0.172132i	$-0.944934\pi$
$661$	− 5850.53i	− 0.344265i	0.985074	−	0.172132i	$-0.0550657\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	36432.0	2.12447
$666$	0	0
$667$	792.000i	0.0459766i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−17193.4	−0.989185
$672$	0	0
$673$	−13250.0	−0.758915	−0.379458	−	0.925209i	$-0.623889\pi$
$673$	−13250.0	−0.758915	−0.379458	+	0.925209i	$0.623889\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 14347.7i	− 0.814516i	−0.913313	−	0.407258i	$-0.866485\pi$
$677$	− 14347.7i	− 0.814516i	0.913313	−	0.407258i	$-0.133515\pi$
$678$	0	0
$679$	−17153.6	−0.969505
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 28272.0i	− 1.58389i	−0.610591	−	0.791946i	$-0.709068\pi$
$683$	− 28272.0i	− 1.58389i	0.610591	−	0.791946i	$-0.290932\pi$
$684$	0	0
$685$	− 28775.0i	− 1.60502i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	39600.0	2.18961
$690$	0	0
$691$	16828.0i	0.926436i	0.886244	+	0.463218i	$0.153305\pi$
$691$	16828.0i	0.926436i	−0.886244	+	0.463218i	$0.846695\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−37650.3	−2.05490
$696$	0	0
$697$	252.000	0.0136947
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 18088.9i	− 0.974618i	−0.873230	−	0.487309i	$-0.837979\pi$
$701$	− 18088.9i	− 0.974618i	0.873230	−	0.487309i	$-0.162021\pi$
$702$	0	0
$703$	18307.8	0.982206
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9108.00i	0.484500i
$708$	0	0
$709$	− 36854.3i	− 1.95218i	−0.217373	−	0.976089i	$-0.569749\pi$
$709$	− 36854.3i	− 1.95218i	0.217373	−	0.976089i	$-0.430251\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5544.00	0.291198
$714$	0	0
$715$	− 76032.0i	− 3.97683i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21849.9	1.13333	0.566665	−	0.823948i	$-0.308233\pi$
$719$	21849.9	1.13333	0.566665	+	0.823948i	$0.308233\pi$
$720$	0	0
$721$	10692.0	0.552276
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5392.83i	0.276255i
$726$	0	0
$727$	−35282.3	−1.79993	−0.899963	−	0.435966i	$-0.856407\pi$
$727$	−35282.3	−1.79993	−0.899963	+	0.435966i	$0.856407\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3864.00i	0.195506i
$732$	0	0
$733$	− 8198.70i	− 0.413132i	−0.978433	−	0.206566i	$-0.933771\pi$
$733$	− 8198.70i	− 0.413132i	0.978433	−	0.206566i	$-0.0662288\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25152.0	−1.25710
$738$	0	0
$739$	− 35404.0i	− 1.76232i	−0.472815	−	0.881162i	$-0.656762\pi$
$739$	− 35404.0i	− 1.76232i	0.472815	−	0.881162i	$-0.343238\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9711.08	0.479495	0.239748	−	0.970835i	$-0.422935\pi$
$743$	9711.08	0.479495	0.239748	+	0.970835i	$0.422935\pi$
$744$	0	0
$745$	−55836.0	−2.74587
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 27461.7i	− 1.33969i
$750$	0	0
$751$	−21631.0	−1.05104	−0.525518	−	0.850783i	$-0.676128\pi$
$751$	−21631.0	−1.05104	−0.525518	+	0.850783i	$0.676128\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 5940.00i	− 0.286329i
$756$	0	0
$757$	26984.1i	1.29558i	0.761820	+	0.647789i	$0.224306\pi$
$757$	26984.1i	1.29558i	−0.761820	+	0.647789i	$0.775694\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14382.0	0.685082	0.342541	−	0.939503i	$-0.388713\pi$
$761$	14382.0	0.685082	0.342541	+	0.939503i	$0.388713\pi$
$762$	0	0
$763$	25344.0i	1.20251i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	41073.1	1.93359
$768$	0	0
$769$	32978.0	1.54645	0.773223	−	0.634134i	$-0.218643\pi$
$769$	32978.0	1.54645	0.773223	+	0.634134i	$0.218643\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28914.3i	1.34538i	0.739926	+	0.672688i	$0.234861\pi$
$773$	28914.3i	1.34538i	−0.739926	+	0.672688i	$0.765139\pi$
$774$	0	0
$775$	37749.8	1.74970
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 552.000i	− 0.0253883i
$780$	0	0
$781$	47759.4i	2.18818i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	46728.0	2.12458
$786$	0	0
$787$	− 24572.0i	− 1.11296i	−0.830862	−	0.556479i	$-0.812152\pi$
$787$	− 24572.0i	− 1.11296i	0.830862	−	0.556479i	$-0.187848\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	41908.9	1.88383
$792$	0	0
$793$	28512.0	1.27679
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19720.7i	− 0.876463i	−0.898862	−	0.438232i	$-0.855605\pi$
$797$	− 19720.7i	− 0.876463i	0.898862	−	0.438232i	$-0.144395\pi$
$798$	0	0
$799$	−1671.58	−0.0740128
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 20640.0i	− 0.907061i
$804$	0	0
$805$	− 15760.6i	− 0.690047i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	35286.0	1.53349	0.766743	−	0.641955i	$-0.221876\pi$
$809$	35286.0	1.53349	0.766743	+	0.641955i	$0.221876\pi$
$810$	0	0
$811$	− 556.000i	− 0.0240737i	−0.999928	−	0.0120369i	$-0.996168\pi$
$811$	− 556.000i	− 0.0240737i	0.999928	−	0.0120369i	$-0.00383155\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−63440.4	−2.72665
$816$	0	0
$817$	8464.00	0.362445
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33769.9i	1.43554i	0.696281	+	0.717769i	$0.254837\pi$
$821$	33769.9i	1.43554i	−0.696281	+	0.717769i	$0.745163\pi$
$822$	0	0
$823$	26247.8	1.11171	0.555856	−	0.831278i	$-0.312390\pi$
$823$	26247.8	1.11171	0.555856	+	0.831278i	$0.312390\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1116.00i	− 0.0469252i	−0.999725	−	0.0234626i	$-0.992531\pi$
$827$	− 1116.00i	− 0.0469252i	0.999725	−	0.0234626i	$-0.00746906\pi$
$828$	0	0
$829$	23242.9i	0.973775i	0.873465	+	0.486888i	$0.161868\pi$
$829$	23242.9i	0.973775i	−0.873465	+	0.486888i	$0.838132\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2226.00	0.0925886
$834$	0	0
$835$	39600.0i	1.64121i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−29650.6	−1.22009	−0.610044	−	0.792368i	$-0.708848\pi$
$839$	−29650.6	−1.22009	−0.610044	+	0.792368i	$0.708848\pi$
$840$	0	0
$841$	23993.0	0.983763
$842$	0	0
$843$	0	0
$844$	0	0
$845$	82365.1i	3.35319i
$846$	0	0
$847$	19362.5	0.785480
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 7920.00i	− 0.319029i
$852$	0	0
$853$	17074.0i	0.685348i	0.939454	+	0.342674i	$0.111333\pi$
$853$	17074.0i	0.685348i	−0.939454	+	0.342674i	$0.888667\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1206.00	0.0480702	0.0240351	−	0.999711i	$-0.492349\pi$
$857$	1206.00	0.0480702	0.0240351	+	0.999711i	$0.492349\pi$
$858$	0	0
$859$	− 34756.0i	− 1.38051i	−0.723565	−	0.690256i	$-0.757498\pi$
$859$	− 34756.0i	− 1.38051i	0.723565	−	0.690256i	$-0.242502\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1034.79	−0.0408164	−0.0204082	−	0.999792i	$-0.506497\pi$
$863$	−1034.79	−0.0408164	−0.0204082	+	0.999792i	$0.506497\pi$
$864$	0	0
$865$	−7524.00	−0.295750
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 56356.1i	− 2.19994i
$870$	0	0
$871$	41709.9	1.62260
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 57816.0i	− 2.23376i
$876$	0	0
$877$	− 19382.4i	− 0.746289i	−0.927773	−	0.373145i	$-0.878280\pi$
$877$	− 19382.4i	− 0.746289i	0.927773	−	0.373145i	$-0.121720\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5310.00	0.203063	0.101531	−	0.994832i	$-0.467626\pi$
$881$	5310.00	0.203063	0.101531	+	0.994832i	$0.467626\pi$
$882$	0	0
$883$	23308.0i	0.888309i	0.895950	+	0.444154i	$0.146496\pi$
$883$	23308.0i	0.888309i	−0.895950	+	0.444154i	$0.853504\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21969.3	0.831632	0.415816	−	0.909449i	$-0.363496\pi$
$887$	21969.3	0.831632	0.415816	+	0.909449i	$0.363496\pi$
$888$	0	0
$889$	−2772.00	−0.104578
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3661.55i	0.137211i
$894$	0	0
$895$	−56594.9	−2.11370
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2772.00i	0.102838i
$900$	0	0
$901$	20894.7i	0.772591i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−44352.0	−1.62907
$906$	0	0
$907$	27596.0i	1.01026i	0.863042	+	0.505132i	$0.168556\pi$
$907$	27596.0i	1.01026i	−0.863042	+	0.505132i	$0.831444\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−12099.0	−0.440021	−0.220011	−	0.975497i	$-0.570609\pi$
$911$	−12099.0	−0.440021	−0.220011	+	0.975497i	$0.570609\pi$
$912$	0	0
$913$	20736.0	0.751655
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 10745.9i	− 0.386979i
$918$	0	0
$919$	−14029.3	−0.503574	−0.251787	−	0.967783i	$-0.581018\pi$
$919$	−14029.3	−0.503574	−0.251787	+	0.967783i	$0.581018\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 79200.0i	− 2.82438i
$924$	0	0
$925$	− 53928.3i	− 1.91692i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−12438.0	−0.439265	−0.219633	−	0.975583i	$-0.570486\pi$
$929$	−12438.0	−0.439265	−0.219633	+	0.975583i	$0.570486\pi$
$930$	0	0
$931$	− 4876.00i	− 0.171648i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	40117.9	1.40320
$936$	0	0
$937$	8278.00	0.288613	0.144307	−	0.989533i	$-0.453905\pi$
$937$	8278.00	0.288613	0.144307	+	0.989533i	$0.453905\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5114.24i	0.177172i	0.996069	+	0.0885862i	$0.0282349\pi$
$941$	5114.24i	0.177172i	−0.996069	+	0.0885862i	$0.971765\pi$
$942$	0	0
$943$	−238.797	−0.00824634
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16980.0i	0.582657i	0.956623	+	0.291328i	$0.0940972\pi$
$947$	16980.0i	0.582657i	−0.956623	+	0.291328i	$0.905903\pi$
$948$	0	0
$949$	34227.6i	1.17078i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	31998.0	1.08764	0.543818	−	0.839203i	$-0.316978\pi$
$953$	31998.0	1.08764	0.543818	+	0.839203i	$0.316978\pi$
$954$	0	0
$955$	− 17424.0i	− 0.590395i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−28775.0	−0.968920
$960$	0	0
$961$	−10387.0	−0.348662
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 57351.1i	− 1.91316i
$966$	0	0
$967$	8059.40	0.268017	0.134009	−	0.990980i	$-0.457215\pi$
$967$	8059.40	0.268017	0.134009	+	0.990980i	$0.457215\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 29664.0i	− 0.980395i	−0.871612	−	0.490197i	$-0.836925\pi$
$971$	− 29664.0i	− 0.980395i	0.871612	−	0.490197i	$-0.163075\pi$
$972$	0	0
$973$	37650.3i	1.24051i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3354.00	0.109830	0.0549150	−	0.998491i	$-0.482511\pi$
$977$	3354.00	0.109830	0.0549150	+	0.998491i	$0.482511\pi$
$978$	0	0
$979$	− 30240.0i	− 0.987206i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	29610.8	0.960772	0.480386	−	0.877057i	$-0.340497\pi$
$983$	29610.8	0.960772	0.480386	+	0.877057i	$0.340497\pi$
$984$	0	0
$985$	84348.0	2.72848
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 3661.55i	− 0.117726i
$990$	0	0
$991$	−38506.0	−1.23429	−0.617146	−	0.786848i	$-0.711711\pi$
$991$	−38506.0	−1.23429	−0.617146	+	0.786848i	$0.711711\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	64548.0i	2.05659i
$996$	0	0
$997$	− 54326.3i	− 1.72571i	−0.505453	−	0.862854i	$-0.668674\pi$
$997$	− 54326.3i	− 1.72571i	0.505453	−	0.862854i	$-0.331326\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.4.d.f.289.1		4
3.2	odd	2		192.4.d.b.97.2	yes	4
4.3	odd	2	inner	576.4.d.f.289.2		4
8.3	odd	2	inner	576.4.d.f.289.4		4
8.5	even	2	inner	576.4.d.f.289.3		4
12.11	even	2		192.4.d.b.97.4	yes	4
16.3	odd	4		2304.4.a.y.1.1		2
16.5	even	4		2304.4.a.y.1.2		2
16.11	odd	4		2304.4.a.bl.1.2		2
16.13	even	4		2304.4.a.bl.1.1		2
24.5	odd	2		192.4.d.b.97.3	yes	4
24.11	even	2		192.4.d.b.97.1	✓	4
48.5	odd	4		768.4.a.i.1.1		2
48.11	even	4		768.4.a.l.1.1		2
48.29	odd	4		768.4.a.l.1.2		2
48.35	even	4		768.4.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.4.d.b.97.1	✓	4	24.11	even	2
192.4.d.b.97.2	yes	4	3.2	odd	2
192.4.d.b.97.3	yes	4	24.5	odd	2
192.4.d.b.97.4	yes	4	12.11	even	2
576.4.d.f.289.1		4	1.1	even	1	trivial
576.4.d.f.289.2		4	4.3	odd	2	inner
576.4.d.f.289.3		4	8.5	even	2	inner
576.4.d.f.289.4		4	8.3	odd	2	inner
768.4.a.i.1.1		2	48.5	odd	4
768.4.a.i.1.2		2	48.35	even	4
768.4.a.l.1.1		2	48.11	even	4
768.4.a.l.1.2		2	48.29	odd	4
2304.4.a.y.1.1		2	16.3	odd	4
2304.4.a.y.1.2		2	16.5	even	4
2304.4.a.bl.1.1		2	16.13	even	4
2304.4.a.bl.1.2		2	16.11	odd	4

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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	576.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-19.8997i q^{5} -19.8997 q^{7} +O(q^{10})\)	\(q-19.8997i q^{5} -19.8997 q^{7} +48.0000i q^{11} -79.5990i q^{13} +42.0000 q^{17} -92.0000i q^{19} -39.7995 q^{23} -271.000 q^{25} -19.8997i q^{29} -139.298 q^{31} +396.000i q^{35} +198.997i q^{37} +6.00000 q^{41} +92.0000i q^{43} -39.7995 q^{47} +53.0000 q^{49} +497.494i q^{53} +955.188 q^{55} +516.000i q^{59} +358.195i q^{61} -1584.00 q^{65} +524.000i q^{67} +994.987 q^{71} -430.000 q^{73} -955.188i q^{77} -1174.09 q^{79} -432.000i q^{83} -835.789i q^{85} -630.000 q^{89} +1584.00i q^{91} -1830.78 q^{95} +862.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 168 q^{17} - 1084 q^{25} + 24 q^{41} + 212 q^{49} - 6336 q^{65} - 1720 q^{73} - 2520 q^{89} + 3448 q^{97}+O(q^{100}) \) 4 * q + 168 * q^17 - 1084 * q^25 + 24 * q^41 + 212 * q^49 - 6336 * q^65 - 1720 * q^73 - 2520 * q^89 + 3448 * q^97

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