LMFDB - Embedded newform 576.4.d.b.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:	$\Q(\zeta_{12})$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	576.289
Dual form			576.4.d.b.289.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-13.8564i q^{5} -3.46410 q^{7} +O(q^{10})$	$q-13.8564i q^{5} -3.46410 q^{7} -48.0000i q^{11} +20.7846i q^{13} -96.0000 q^{17} -40.0000i q^{19} +110.851 q^{23} -67.0000 q^{25} -13.8564i q^{29} -204.382 q^{31} +48.0000i q^{35} +297.913i q^{37} +288.000 q^{41} -152.000i q^{43} -554.256 q^{47} -331.000 q^{49} +180.133i q^{53} -665.108 q^{55} +480.000i q^{59} +755.174i q^{61} +288.000 q^{65} -848.000i q^{67} +886.810 q^{71} -538.000 q^{73} +166.277i q^{77} -1008.05 q^{79} -432.000i q^{83} +1330.22i q^{85} -1344.00 q^{89} -72.0000i q^{91} -554.256 q^{95} -590.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 384 q^{17} - 268 q^{25} + 1152 q^{41} - 1324 q^{49} + 1152 q^{65} - 2152 q^{73} - 5376 q^{89} - 2360 q^{97}+O(q^{100}) $ 4 * q - 384 * q^17 - 268 * q^25 + 1152 * q^41 - 1324 * q^49 + 1152 * q^65 - 2152 * q^73 - 5376 * q^89 - 2360 * 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
$5$	− 13.8564i	− 1.23935i	−0.784857	−	0.619677i	$-0.787263\pi$
$5$	− 13.8564i	− 1.23935i	0.784857	−	0.619677i	$-0.212737\pi$
$6$	0	0
$7$	−3.46410	−0.187044	−0.0935220	−	0.995617i	$-0.529813\pi$
$7$	−3.46410	−0.187044	−0.0935220	+	0.995617i	$0.529813\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 48.0000i	− 1.31569i	−0.753155	−	0.657843i	$-0.771469\pi$
$11$	− 48.0000i	− 1.31569i	0.753155	−	0.657843i	$-0.228531\pi$
$12$	0	0
$13$	20.7846i	0.443432i	0.975111	+	0.221716i	$0.0711658\pi$
$13$	20.7846i	0.443432i	−0.975111	+	0.221716i	$0.928834\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−96.0000	−1.36961	−0.684806	−	0.728725i	$-0.740113\pi$
$17$	−96.0000	−1.36961	−0.684806	+	0.728725i	$0.740113\pi$
$18$	0	0
$19$	− 40.0000i	− 0.482980i	−0.970403	−	0.241490i	$-0.922364\pi$
$19$	− 40.0000i	− 0.482980i	0.970403	−	0.241490i	$-0.0776362\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	110.851	1.00496	0.502480	−	0.864589i	$-0.332421\pi$
$23$	110.851	1.00496	0.502480	+	0.864589i	$0.332421\pi$
$24$	0	0
$25$	−67.0000	−0.536000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 13.8564i	− 0.0887266i	−0.999015	−	0.0443633i	$-0.985874\pi$
$29$	− 13.8564i	− 0.0887266i	0.999015	−	0.0443633i	$-0.0141259\pi$
$30$	0	0
$31$	−204.382	−1.18413	−0.592066	−	0.805890i	$-0.701688\pi$
$31$	−204.382	−1.18413	−0.592066	+	0.805890i	$0.701688\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	48.0000i	0.231814i
$36$	0	0
$37$	297.913i	1.32369i	0.749640	+	0.661845i	$0.230226\pi$
$37$	297.913i	1.32369i	−0.749640	+	0.661845i	$0.769774\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	288.000	1.09703	0.548513	−	0.836142i	$-0.315194\pi$
$41$	288.000	1.09703	0.548513	+	0.836142i	$0.315194\pi$
$42$	0	0
$43$	− 152.000i	− 0.539065i	−0.962991	−	0.269532i	$-0.913131\pi$
$43$	− 152.000i	− 0.539065i	0.962991	−	0.269532i	$-0.0868691\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−554.256	−1.72014	−0.860070	−	0.510176i	$-0.829580\pi$
$47$	−554.256	−1.72014	−0.860070	+	0.510176i	$0.829580\pi$
$48$	0	0
$49$	−331.000	−0.965015
$50$	0	0
$51$	0	0
$52$	0	0
$53$	180.133i	0.466853i	0.972374	+	0.233427i	$0.0749938\pi$
$53$	180.133i	0.466853i	−0.972374	+	0.233427i	$0.925006\pi$
$54$	0	0
$55$	−665.108	−1.63060
$56$	0	0
$57$	0	0
$58$	0	0
$59$	480.000i	1.05916i	0.848259	+	0.529582i	$0.177651\pi$
$59$	480.000i	1.05916i	−0.848259	+	0.529582i	$0.822349\pi$
$60$	0	0
$61$	755.174i	1.58508i	0.609817	+	0.792542i	$0.291243\pi$
$61$	755.174i	1.58508i	−0.609817	+	0.792542i	$0.708757\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	288.000	0.549569
$66$	0	0
$67$	− 848.000i	− 1.54626i	−0.634245	−	0.773132i	$-0.718689\pi$
$67$	− 848.000i	− 1.54626i	0.634245	−	0.773132i	$-0.281311\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	886.810	1.48232	0.741162	−	0.671326i	$-0.234275\pi$
$71$	886.810	1.48232	0.741162	+	0.671326i	$0.234275\pi$
$72$	0	0
$73$	−538.000	−0.862577	−0.431289	−	0.902214i	$-0.641941\pi$
$73$	−538.000	−0.862577	−0.431289	+	0.902214i	$0.641941\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	166.277i	0.246091i
$78$	0	0
$79$	−1008.05	−1.43563	−0.717816	−	0.696233i	$-0.754858\pi$
$79$	−1008.05	−1.43563	−0.717816	+	0.696233i	$0.754858\pi$
$80$	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
$85$	1330.22i	1.69744i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1344.00	−1.60072	−0.800358	−	0.599522i	$-0.795357\pi$
$89$	−1344.00	−1.60072	−0.800358	+	0.599522i	$0.795357\pi$
$90$	0	0
$91$	− 72.0000i	− 0.0829412i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−554.256	−0.598584
$96$	0	0
$97$	−590.000	−0.617582	−0.308791	−	0.951130i	$-0.599924\pi$
$97$	−590.000	−0.617582	−0.308791	+	0.951130i	$0.599924\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 983.805i	− 0.969230i	−0.874728	−	0.484615i	$-0.838960\pi$
$101$	− 983.805i	− 0.969230i	0.874728	−	0.484615i	$-0.161040\pi$
$102$	0	0
$103$	−1437.60	−1.37525	−0.687627	−	0.726064i	$-0.741348\pi$
$103$	−1437.60	−1.37525	−0.687627	+	0.726064i	$0.741348\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1824.00i	− 1.64797i	−0.566612	−	0.823985i	$-0.691746\pi$
$107$	− 1824.00i	− 1.64797i	0.566612	−	0.823985i	$-0.308254\pi$
$108$	0	0
$109$	1046.16i	0.919301i	0.888100	+	0.459651i	$0.152025\pi$
$109$	1046.16i	0.919301i	−0.888100	+	0.459651i	$0.847975\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	192.000	0.159839	0.0799196	−	0.996801i	$-0.474534\pi$
$113$	192.000	0.159839	0.0799196	+	0.996801i	$0.474534\pi$
$114$	0	0
$115$	− 1536.00i	− 1.24550i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	332.554	0.256178
$120$	0	0
$121$	−973.000	−0.731029
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 803.672i	− 0.575061i
$126$	0	0
$127$	−793.279	−0.554269	−0.277134	−	0.960831i	$-0.589385\pi$
$127$	−793.279	−0.554269	−0.277134	+	0.960831i	$0.589385\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1344.00i	0.896380i	0.893938	+	0.448190i	$0.147931\pi$
$131$	1344.00i	0.896380i	−0.893938	+	0.448190i	$0.852069\pi$
$132$	0	0
$133$	138.564i	0.0903386i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	864.000	0.538807	0.269403	−	0.963027i	$-0.413174\pi$
$137$	864.000	0.538807	0.269403	+	0.963027i	$0.413174\pi$
$138$	0	0
$139$	1232.00i	0.751776i	0.926665	+	0.375888i	$0.122662\pi$
$139$	1232.00i	0.751776i	−0.926665	+	0.375888i	$0.877338\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	997.661	0.583417
$144$	0	0
$145$	−192.000	−0.109964
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2868.28i	1.57704i	0.615012	+	0.788518i	$0.289151\pi$
$149$	2868.28i	1.57704i	−0.615012	+	0.788518i	$0.710849\pi$
$150$	0	0
$151$	744.782	0.401387	0.200694	−	0.979654i	$-0.435680\pi$
$151$	744.782	0.401387	0.200694	+	0.979654i	$0.435680\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2832.00i	1.46756i
$156$	0	0
$157$	616.610i	0.313445i	0.987643	+	0.156722i	$0.0500928\pi$
$157$	616.610i	0.313445i	−0.987643	+	0.156722i	$0.949907\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−384.000	−0.187972
$162$	0	0
$163$	− 1960.00i	− 0.941835i	−0.882177	−	0.470917i	$-0.843923\pi$
$163$	− 1960.00i	− 0.941835i	0.882177	−	0.470917i	$-0.156077\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3436.39	−1.59231	−0.796155	−	0.605093i	$-0.793136\pi$
$167$	−3436.39	−1.59231	−0.796155	+	0.605093i	$0.793136\pi$
$168$	0	0
$169$	1765.00	0.803368
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1898.33i	0.834261i	0.908847	+	0.417131i	$0.136964\pi$
$173$	1898.33i	0.834261i	−0.908847	+	0.417131i	$0.863036\pi$
$174$	0	0
$175$	232.095	0.100256
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1056.00i	− 0.440945i	−0.975393	−	0.220472i	$-0.929240\pi$
$179$	− 1056.00i	− 0.440945i	0.975393	−	0.220472i	$-0.0707599\pi$
$180$	0	0
$181$	− 1960.68i	− 0.805173i	−0.915382	−	0.402586i	$-0.868111\pi$
$181$	− 1960.68i	− 0.805173i	0.915382	−	0.402586i	$-0.131889\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4128.00	1.64052
$186$	0	0
$187$	4608.00i	1.80198i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1884.47	0.713903	0.356952	−	0.934123i	$-0.383816\pi$
$191$	1884.47	0.713903	0.356952	+	0.934123i	$0.383816\pi$
$192$	0	0
$193$	962.000	0.358789	0.179394	−	0.983777i	$-0.442586\pi$
$193$	962.000	0.358789	0.179394	+	0.983777i	$0.442586\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4697.32i	− 1.69883i	−0.527722	−	0.849417i	$-0.676954\pi$
$197$	− 4697.32i	− 1.69883i	0.527722	−	0.849417i	$-0.323046\pi$
$198$	0	0
$199$	3107.30	1.10689	0.553444	−	0.832887i	$-0.313313\pi$
$199$	3107.30	1.10689	0.553444	+	0.832887i	$0.313313\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	48.0000i	0.0165958i
$204$	0	0
$205$	− 3990.65i	− 1.35960i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1920.00	−0.635451
$210$	0	0
$211$	− 3152.00i	− 1.02840i	−0.857670	−	0.514201i	$-0.828089\pi$
$211$	− 3152.00i	− 1.02840i	0.857670	−	0.514201i	$-0.171911\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2106.17	−0.668092
$216$	0	0
$217$	708.000	0.221485
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1995.32i	− 0.607330i
$222$	0	0
$223$	5525.24	1.65918	0.829591	−	0.558372i	$-0.188574\pi$
$223$	5525.24	1.65918	0.829591	+	0.558372i	$0.188574\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2448.00i	− 0.715769i	−0.933766	−	0.357884i	$-0.883498\pi$
$227$	− 2448.00i	− 0.715769i	0.933766	−	0.357884i	$-0.116502\pi$
$228$	0	0
$229$	− 2210.10i	− 0.637761i	−0.947795	−	0.318881i	$-0.896693\pi$
$229$	− 2210.10i	− 0.637761i	0.947795	−	0.318881i	$-0.103307\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5568.00	−1.56554	−0.782772	−	0.622308i	$-0.786195\pi$
$233$	−5568.00	−1.56554	−0.782772	+	0.622308i	$0.786195\pi$
$234$	0	0
$235$	7680.00i	2.13186i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5210.01	1.41007	0.705037	−	0.709171i	$-0.250931\pi$
$239$	5210.01	1.41007	0.705037	+	0.709171i	$0.250931\pi$
$240$	0	0
$241$	2798.00	0.747863	0.373932	−	0.927456i	$-0.378009\pi$
$241$	2798.00	0.747863	0.373932	+	0.927456i	$0.378009\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4586.47i	1.19600i
$246$	0	0
$247$	831.384	0.214169
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 4656.00i	− 1.17085i	−0.810725	−	0.585427i	$-0.800927\pi$
$251$	− 4656.00i	− 1.17085i	0.810725	−	0.585427i	$-0.199073\pi$
$252$	0	0
$253$	− 5320.86i	− 1.32221i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1536.00	−0.372813	−0.186407	−	0.982473i	$-0.559684\pi$
$257$	−1536.00	−0.372813	−0.186407	+	0.982473i	$0.559684\pi$
$258$	0	0
$259$	− 1032.00i	− 0.247588i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1330.22	0.311881	0.155940	−	0.987766i	$-0.450159\pi$
$263$	1330.22	0.311881	0.155940	+	0.987766i	$0.450159\pi$
$264$	0	0
$265$	2496.00	0.578596
$266$	0	0
$267$	0	0
$268$	0	0
$269$	706.677i	0.160174i	0.996788	+	0.0800871i	$0.0255198\pi$
$269$	706.677i	0.160174i	−0.996788	+	0.0800871i	$0.974480\pi$
$270$	0	0
$271$	4777.00	1.07078	0.535391	−	0.844604i	$-0.320164\pi$
$271$	4777.00	1.07078	0.535391	+	0.844604i	$0.320164\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3216.00i	0.705208i
$276$	0	0
$277$	− 5868.19i	− 1.27287i	−0.771330	−	0.636435i	$-0.780408\pi$
$277$	− 5868.19i	− 1.27287i	0.771330	−	0.636435i	$-0.219592\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1536.00	−0.326086	−0.163043	−	0.986619i	$-0.552131\pi$
$281$	−1536.00	−0.326086	−0.163043	+	0.986619i	$0.552131\pi$
$282$	0	0
$283$	6752.00i	1.41825i	0.705083	+	0.709125i	$0.250910\pi$
$283$	6752.00i	1.41825i	−0.705083	+	0.709125i	$0.749090\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−997.661	−0.205192
$288$	0	0
$289$	4303.00	0.875840
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 5140.73i	− 1.02500i	−0.858688	−	0.512499i	$-0.828720\pi$
$293$	− 5140.73i	− 1.02500i	0.858688	−	0.512499i	$-0.171280\pi$
$294$	0	0
$295$	6651.08	1.31268
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2304.00i	0.445631i
$300$	0	0
$301$	526.543i	0.100829i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10464.0	1.96448
$306$	0	0
$307$	− 6224.00i	− 1.15708i	−0.815655	−	0.578538i	$-0.803623\pi$
$307$	− 6224.00i	− 1.15708i	0.815655	−	0.578538i	$-0.196377\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2438.73	−0.444655	−0.222327	−	0.974972i	$-0.571365\pi$
$311$	−2438.73	−0.444655	−0.222327	+	0.974972i	$0.571365\pi$
$312$	0	0
$313$	934.000	0.168667	0.0843335	−	0.996438i	$-0.473124\pi$
$313$	934.000	0.168667	0.0843335	+	0.996438i	$0.473124\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2618.86i	− 0.464006i	−0.972715	−	0.232003i	$-0.925472\pi$
$317$	− 2618.86i	− 0.464006i	0.972715	−	0.232003i	$-0.0745279\pi$
$318$	0	0
$319$	−665.108	−0.116736
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3840.00i	0.661496i
$324$	0	0
$325$	− 1392.57i	− 0.237679i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1920.00	0.321742
$330$	0	0
$331$	9824.00i	1.63135i	0.578512	+	0.815674i	$0.303633\pi$
$331$	9824.00i	1.63135i	−0.578512	+	0.815674i	$0.696367\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11750.2	−1.91637
$336$	0	0
$337$	−4862.00	−0.785905	−0.392953	−	0.919559i	$-0.628546\pi$
$337$	−4862.00	−0.785905	−0.392953	+	0.919559i	$0.628546\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9810.34i	1.55795i
$342$	0	0
$343$	2334.80	0.367544
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 528.000i	− 0.0816845i	−0.999166	−	0.0408423i	$-0.986996\pi$
$347$	− 528.000i	− 0.0816845i	0.999166	−	0.0408423i	$-0.0130041\pi$
$348$	0	0
$349$	− 6644.15i	− 1.01906i	−0.860452	−	0.509532i	$-0.829819\pi$
$349$	− 6644.15i	− 1.01906i	0.860452	−	0.509532i	$-0.170181\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2112.00	0.318443	0.159222	−	0.987243i	$-0.449102\pi$
$353$	2112.00	0.318443	0.159222	+	0.987243i	$0.449102\pi$
$354$	0	0
$355$	− 12288.0i	− 1.83712i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6207.67	0.912614	0.456307	−	0.889823i	$-0.349172\pi$
$359$	6207.67	0.912614	0.456307	+	0.889823i	$0.349172\pi$
$360$	0	0
$361$	5259.00	0.766730
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7454.75i	1.06904i
$366$	0	0
$367$	−65.8179	−0.00936149	−0.00468075	−	0.999989i	$-0.501490\pi$
$367$	−65.8179	−0.00936149	−0.00468075	+	0.999989i	$0.501490\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 624.000i	− 0.0873220i
$372$	0	0
$373$	8168.35i	1.13389i	0.823755	+	0.566945i	$0.191875\pi$
$373$	8168.35i	1.13389i	−0.823755	+	0.566945i	$0.808125\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	288.000	0.0393442
$378$	0	0
$379$	9448.00i	1.28050i	0.768165	+	0.640252i	$0.221170\pi$
$379$	9448.00i	1.28050i	−0.768165	+	0.640252i	$0.778830\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1884.47	0.251415	0.125708	−	0.992067i	$-0.459880\pi$
$383$	1884.47	0.251415	0.125708	+	0.992067i	$0.459880\pi$
$384$	0	0
$385$	2304.00	0.304994
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 9048.23i	− 1.17934i	−0.807644	−	0.589670i	$-0.799258\pi$
$389$	− 9048.23i	− 1.17934i	0.807644	−	0.589670i	$-0.200742\pi$
$390$	0	0
$391$	−10641.7	−1.37641
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13968.0i	1.77926i
$396$	0	0
$397$	6270.02i	0.792654i	0.918110	+	0.396327i	$0.129715\pi$
$397$	6270.02i	0.792654i	−0.918110	+	0.396327i	$0.870285\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15648.0	−1.94869	−0.974344	−	0.225064i	$-0.927741\pi$
$401$	−15648.0	−1.94869	−0.974344	+	0.225064i	$0.927741\pi$
$402$	0	0
$403$	− 4248.00i	− 0.525082i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14299.8	1.74156
$408$	0	0
$409$	−13642.0	−1.64928	−0.824638	−	0.565662i	$-0.808621\pi$
$409$	−13642.0	−1.64928	−0.824638	+	0.565662i	$0.808621\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1662.77i	− 0.198110i
$414$	0	0
$415$	−5985.97	−0.708047
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 16080.0i	− 1.87484i	−0.348196	−	0.937422i	$-0.613206\pi$
$419$	− 16080.0i	− 1.87484i	0.348196	−	0.937422i	$-0.386794\pi$
$420$	0	0
$421$	− 1489.56i	− 0.172439i	−0.996276	−	0.0862196i	$-0.972521\pi$
$421$	− 1489.56i	− 0.172439i	0.996276	−	0.0862196i	$-0.0274787\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6432.00	0.734113
$426$	0	0
$427$	− 2616.00i	− 0.296480i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6761.93	−0.755709	−0.377854	−	0.925865i	$-0.623338\pi$
$431$	−6761.93	−0.755709	−0.377854	+	0.925865i	$0.623338\pi$
$432$	0	0
$433$	−5474.00	−0.607537	−0.303769	−	0.952746i	$-0.598245\pi$
$433$	−5474.00	−0.607537	−0.303769	+	0.952746i	$0.598245\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4434.05i	− 0.485376i
$438$	0	0
$439$	1860.22	0.202240	0.101120	−	0.994874i	$-0.467757\pi$
$439$	1860.22	0.202240	0.101120	+	0.994874i	$0.467757\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 17424.0i	− 1.86871i	−0.356342	−	0.934356i	$-0.615976\pi$
$443$	− 17424.0i	− 1.86871i	0.356342	−	0.934356i	$-0.384024\pi$
$444$	0	0
$445$	18623.0i	1.98385i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3168.00	−0.332978	−0.166489	−	0.986043i	$-0.553243\pi$
$449$	−3168.00	−0.332978	−0.166489	+	0.986043i	$0.553243\pi$
$450$	0	0
$451$	− 13824.0i	− 1.44334i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−997.661	−0.102794
$456$	0	0
$457$	3878.00	0.396948	0.198474	−	0.980106i	$-0.436401\pi$
$457$	3878.00	0.396948	0.198474	+	0.980106i	$0.436401\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 9851.90i	− 0.995334i	−0.867368	−	0.497667i	$-0.834190\pi$
$461$	− 9851.90i	− 0.995334i	0.867368	−	0.497667i	$-0.165810\pi$
$462$	0	0
$463$	−16451.0	−1.65128	−0.825641	−	0.564196i	$-0.809186\pi$
$463$	−16451.0	−1.65128	−0.825641	+	0.564196i	$0.809186\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3888.00i	0.385257i	0.981272	+	0.192629i	$0.0617013\pi$
$467$	3888.00i	0.385257i	−0.981272	+	0.192629i	$0.938299\pi$
$468$	0	0
$469$	2937.56i	0.289219i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7296.00	−0.709240
$474$	0	0
$475$	2680.00i	0.258878i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9311.51	−0.888212	−0.444106	−	0.895974i	$-0.646479\pi$
$479$	−9311.51	−0.888212	−0.444106	+	0.895974i	$0.646479\pi$
$480$	0	0
$481$	−6192.00	−0.586967
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8175.28i	0.765403i
$486$	0	0
$487$	−12086.3	−1.12460	−0.562300	−	0.826933i	$-0.690083\pi$
$487$	−12086.3	−1.12460	−0.562300	+	0.826933i	$0.690083\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 7872.00i	− 0.723541i	−0.932267	−	0.361770i	$-0.882172\pi$
$491$	− 7872.00i	− 0.723541i	0.932267	−	0.361770i	$-0.117828\pi$
$492$	0	0
$493$	1330.22i	0.121521i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3072.00	−0.277260
$498$	0	0
$499$	− 16736.0i	− 1.50142i	−0.660635	−	0.750708i	$-0.729713\pi$
$499$	− 16736.0i	− 1.50142i	0.660635	−	0.750708i	$-0.270287\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1551.92	−0.137568	−0.0687839	−	0.997632i	$-0.521912\pi$
$503$	−1551.92	−0.137568	−0.0687839	+	0.997632i	$0.521912\pi$
$504$	0	0
$505$	−13632.0	−1.20122
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 1565.77i	− 0.136349i	−0.997673	−	0.0681746i	$-0.978283\pi$
$509$	− 1565.77i	− 0.136349i	0.997673	−	0.0681746i	$-0.0217175\pi$
$510$	0	0
$511$	1863.69	0.161340
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19920.0i	1.70443i
$516$	0	0
$517$	26604.3i	2.26316i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12192.0	1.02522	0.512612	−	0.858621i	$-0.328678\pi$
$521$	12192.0	1.02522	0.512612	+	0.858621i	$0.328678\pi$
$522$	0	0
$523$	3688.00i	0.308346i	0.988044	+	0.154173i	$0.0492713\pi$
$523$	3688.00i	0.308346i	−0.988044	+	0.154173i	$0.950729\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19620.7	1.62180
$528$	0	0
$529$	121.000	0.00994493
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5985.97i	0.486456i
$534$	0	0
$535$	−25274.1	−2.04242
$536$	0	0
$537$	0	0
$538$	0	0
$539$	15888.0i	1.26966i
$540$	0	0
$541$	7447.82i	0.591879i	0.955207	+	0.295940i	$0.0956327\pi$
$541$	7447.82i	0.591879i	−0.955207	+	0.295940i	$0.904367\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	14496.0	1.13934
$546$	0	0
$547$	− 952.000i	− 0.0744142i	−0.999308	−	0.0372071i	$-0.988154\pi$
$547$	− 952.000i	− 0.0744142i	0.999308	−	0.0372071i	$-0.0118461\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−554.256	−0.0428532
$552$	0	0
$553$	3492.00	0.268526
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13925.7i	1.05934i	0.848205	+	0.529668i	$0.177684\pi$
$557$	13925.7i	1.05934i	−0.848205	+	0.529668i	$0.822316\pi$
$558$	0	0
$559$	3159.26	0.239038
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3696.00i	0.276675i	0.990385	+	0.138337i	$0.0441758\pi$
$563$	3696.00i	0.276675i	−0.990385	+	0.138337i	$0.955824\pi$
$564$	0	0
$565$	− 2660.43i	− 0.198098i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6240.00	−0.459744	−0.229872	−	0.973221i	$-0.573831\pi$
$569$	−6240.00	−0.459744	−0.229872	+	0.973221i	$0.573831\pi$
$570$	0	0
$571$	− 7216.00i	− 0.528862i	−0.964405	−	0.264431i	$-0.914816\pi$
$571$	− 7216.00i	− 0.528862i	0.964405	−	0.264431i	$-0.0851841\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7427.03	−0.538659
$576$	0	0
$577$	−4754.00	−0.343001	−0.171501	−	0.985184i	$-0.554862\pi$
$577$	−4754.00	−0.343001	−0.171501	+	0.985184i	$0.554862\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1496.49i	0.106859i
$582$	0	0
$583$	8646.40	0.614232
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1152.00i	− 0.0810019i	−0.999179	−	0.0405010i	$-0.987105\pi$
$587$	− 1152.00i	− 0.0810019i	0.999179	−	0.0405010i	$-0.0128954\pi$
$588$	0	0
$589$	8175.28i	0.571913i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17472.0	1.20993	0.604965	−	0.796252i	$-0.293187\pi$
$593$	17472.0	1.20993	0.604965	+	0.796252i	$0.293187\pi$
$594$	0	0
$595$	− 4608.00i	− 0.317495i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27158.6	−1.85254	−0.926268	−	0.376867i	$-0.877002\pi$
$599$	−27158.6	−1.85254	−0.926268	+	0.376867i	$0.877002\pi$
$600$	0	0
$601$	−9142.00	−0.620482	−0.310241	−	0.950658i	$-0.600410\pi$
$601$	−9142.00	−0.620482	−0.310241	+	0.950658i	$0.600410\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	13482.3i	0.906005i
$606$	0	0
$607$	−28894.1	−1.93208	−0.966041	−	0.258388i	$-0.916809\pi$
$607$	−28894.1	−1.93208	−0.966041	+	0.258388i	$0.916809\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 11520.0i	− 0.762765i
$612$	0	0
$613$	− 9179.87i	− 0.604847i	−0.953174	−	0.302424i	$-0.902204\pi$
$613$	− 9179.87i	− 0.604847i	0.953174	−	0.302424i	$-0.0977957\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14208.0	−0.927054	−0.463527	−	0.886083i	$-0.653416\pi$
$617$	−14208.0	−0.927054	−0.463527	+	0.886083i	$0.653416\pi$
$618$	0	0
$619$	3680.00i	0.238953i	0.992837	+	0.119476i	$0.0381216\pi$
$619$	3680.00i	0.238953i	−0.992837	+	0.119476i	$0.961878\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4655.75	0.299404
$624$	0	0
$625$	−19511.0	−1.24870
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 28599.6i	− 1.81294i
$630$	0	0
$631$	8836.92	0.557516	0.278758	−	0.960361i	$-0.410077\pi$
$631$	8836.92	0.557516	0.278758	+	0.960361i	$0.410077\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10992.0i	0.686936i
$636$	0	0
$637$	− 6879.71i	− 0.427918i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2400.00	−0.147885	−0.0739425	−	0.997263i	$-0.523558\pi$
$641$	−2400.00	−0.147885	−0.0739425	+	0.997263i	$0.523558\pi$
$642$	0	0
$643$	− 184.000i	− 0.0112850i	−0.999984	−	0.00564250i	$-0.998204\pi$
$643$	− 184.000i	− 0.0112850i	0.999984	−	0.00564250i	$-0.00179607\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12082.8	−0.734194	−0.367097	−	0.930183i	$-0.619648\pi$
$647$	−12082.8	−0.734194	−0.367097	+	0.930183i	$0.619648\pi$
$648$	0	0
$649$	23040.0	1.39353
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 26147.0i	− 1.56694i	−0.621429	−	0.783471i	$-0.713447\pi$
$653$	− 26147.0i	− 1.56694i	0.621429	−	0.783471i	$-0.286553\pi$
$654$	0	0
$655$	18623.0	1.11093
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 18816.0i	− 1.11224i	−0.831101	−	0.556121i	$-0.812289\pi$
$659$	− 18816.0i	− 1.11224i	0.831101	−	0.556121i	$-0.187711\pi$
$660$	0	0
$661$	16676.2i	0.981284i	0.871361	+	0.490642i	$0.163238\pi$
$661$	16676.2i	0.981284i	−0.871361	+	0.490642i	$0.836762\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1920.00	0.111962
$666$	0	0
$667$	− 1536.00i	− 0.0891667i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	36248.4	2.08547
$672$	0	0
$673$	29326.0	1.67969	0.839847	−	0.542823i	$-0.182645\pi$
$673$	29326.0	1.67969	0.839847	+	0.542823i	$0.182645\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21934.7i	1.24523i	0.782530	+	0.622613i	$0.213929\pi$
$677$	21934.7i	1.24523i	−0.782530	+	0.622613i	$0.786071\pi$
$678$	0	0
$679$	2043.82	0.115515
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 21552.0i	− 1.20741i	−0.797206	−	0.603707i	$-0.793690\pi$
$683$	− 21552.0i	− 1.20741i	0.797206	−	0.603707i	$-0.206310\pi$
$684$	0	0
$685$	− 11971.9i	− 0.667772i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3744.00	−0.207017
$690$	0	0
$691$	26840.0i	1.47763i	0.673909	+	0.738815i	$0.264614\pi$
$691$	26840.0i	1.47763i	−0.673909	+	0.738815i	$0.735386\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17071.1	0.931717
$696$	0	0
$697$	−27648.0	−1.50250
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 5611.84i	− 0.302363i	−0.988506	−	0.151181i	$-0.951692\pi$
$701$	− 5611.84i	− 0.302363i	0.988506	−	0.151181i	$-0.0483078\pi$
$702$	0	0
$703$	11916.5	0.639317
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3408.00i	0.181289i
$708$	0	0
$709$	− 32278.5i	− 1.70979i	−0.518797	−	0.854897i	$-0.673620\pi$
$709$	− 32278.5i	− 1.70979i	0.518797	−	0.854897i	$-0.326380\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−22656.0	−1.19001
$714$	0	0
$715$	− 13824.0i	− 0.723061i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12637.0	0.655469	0.327734	−	0.944770i	$-0.393715\pi$
$719$	12637.0	0.655469	0.327734	+	0.944770i	$0.393715\pi$
$720$	0	0
$721$	4980.00	0.257233
$722$	0	0
$723$	0	0
$724$	0	0
$725$	928.379i	0.0475574i
$726$	0	0
$727$	20420.9	1.04177	0.520886	−	0.853626i	$-0.325602\pi$
$727$	20420.9	1.04177	0.520886	+	0.853626i	$0.325602\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14592.0i	0.738310i
$732$	0	0
$733$	− 616.610i	− 0.0310710i	−0.999879	−	0.0155355i	$-0.995055\pi$
$733$	− 616.610i	− 0.0310710i	0.999879	−	0.0155355i	$-0.00494530\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−40704.0	−2.03440
$738$	0	0
$739$	25264.0i	1.25758i	0.777575	+	0.628790i	$0.216449\pi$
$739$	25264.0i	1.25758i	−0.777575	+	0.628790i	$0.783551\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−18068.8	−0.892165	−0.446082	−	0.894992i	$-0.647181\pi$
$743$	−18068.8	−0.892165	−0.446082	+	0.894992i	$0.647181\pi$
$744$	0	0
$745$	39744.0	1.95451
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6318.52i	0.308243i
$750$	0	0
$751$	−1562.31	−0.0759114	−0.0379557	−	0.999279i	$-0.512085\pi$
$751$	−1562.31	−0.0759114	−0.0379557	+	0.999279i	$0.512085\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 10320.0i	− 0.497461i
$756$	0	0
$757$	− 7115.26i	− 0.341623i	−0.985304	−	0.170812i	$-0.945361\pi$
$757$	− 7115.26i	− 0.341623i	0.985304	−	0.170812i	$-0.0546389\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	26208.0	1.24841	0.624205	−	0.781261i	$-0.285423\pi$
$761$	26208.0	1.24841	0.624205	+	0.781261i	$0.285423\pi$
$762$	0	0
$763$	− 3624.00i	− 0.171950i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9976.61	−0.469667
$768$	0	0
$769$	12866.0	0.603329	0.301664	−	0.953414i	$-0.402458\pi$
$769$	12866.0	0.603329	0.301664	+	0.953414i	$0.402458\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	12318.3i	0.573170i	0.958055	+	0.286585i	$0.0925200\pi$
$773$	12318.3i	0.573170i	−0.958055	+	0.286585i	$0.907480\pi$
$774$	0	0
$775$	13693.6	0.634695
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 11520.0i	− 0.529842i
$780$	0	0
$781$	− 42566.9i	− 1.95027i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8544.00	0.388469
$786$	0	0
$787$	− 1816.00i	− 0.0822534i	−0.999154	−	0.0411267i	$-0.986905\pi$
$787$	− 1816.00i	− 0.0822534i	0.999154	−	0.0411267i	$-0.0130947\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−665.108	−0.0298970
$792$	0	0
$793$	−15696.0	−0.702877
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16697.0i	0.742079i	0.928617	+	0.371040i	$0.120999\pi$
$797$	16697.0i	0.742079i	−0.928617	+	0.371040i	$0.879001\pi$
$798$	0	0
$799$	53208.6	2.35593
$800$	0	0
$801$	0	0
$802$	0	0
$803$	25824.0i	1.13488i
$804$	0	0
$805$	5320.86i	0.232964i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24288.0	1.05553	0.527763	−	0.849392i	$-0.323031\pi$
$809$	24288.0	1.05553	0.527763	+	0.849392i	$0.323031\pi$
$810$	0	0
$811$	8152.00i	0.352966i	0.984304	+	0.176483i	$0.0564721\pi$
$811$	8152.00i	0.352966i	−0.984304	+	0.176483i	$0.943528\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−27158.6	−1.16727
$816$	0	0
$817$	−6080.00	−0.260358
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 44991.8i	− 1.91257i	−0.292430	−	0.956287i	$-0.594464\pi$
$821$	− 44991.8i	− 1.91257i	0.292430	−	0.956287i	$-0.405536\pi$
$822$	0	0
$823$	7285.01	0.308553	0.154277	−	0.988028i	$-0.450695\pi$
$823$	7285.01	0.308553	0.154277	+	0.988028i	$0.450695\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3936.00i	− 0.165500i	−0.996570	−	0.0827498i	$-0.973630\pi$
$827$	− 3936.00i	− 0.165500i	0.996570	−	0.0827498i	$-0.0263702\pi$
$828$	0	0
$829$	− 28163.1i	− 1.17991i	−0.807436	−	0.589956i	$-0.799145\pi$
$829$	− 28163.1i	− 1.17991i	0.807436	−	0.589956i	$-0.200855\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	31776.0	1.32170
$834$	0	0
$835$	47616.0i	1.97344i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	11085.1	0.456139	0.228070	−	0.973645i	$-0.426759\pi$
$839$	11085.1	0.456139	0.228070	+	0.973645i	$0.426759\pi$
$840$	0	0
$841$	24197.0	0.992128
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 24456.6i	− 0.995658i
$846$	0	0
$847$	3370.57	0.136735
$848$	0	0
$849$	0	0
$850$	0	0
$851$	33024.0i	1.33026i
$852$	0	0
$853$	− 44818.5i	− 1.79901i	−0.436908	−	0.899506i	$-0.643926\pi$
$853$	− 44818.5i	− 1.79901i	0.436908	−	0.899506i	$-0.356074\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39840.0	1.58799	0.793996	−	0.607923i	$-0.207997\pi$
$857$	39840.0	1.58799	0.793996	+	0.607923i	$0.207997\pi$
$858$	0	0
$859$	− 11432.0i	− 0.454080i	−0.973885	−	0.227040i	$-0.927095\pi$
$859$	− 11432.0i	− 0.454080i	0.973885	−	0.227040i	$-0.0729048\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4101.50	−0.161780	−0.0808902	−	0.996723i	$-0.525776\pi$
$863$	−4101.50	−0.161780	−0.0808902	+	0.996723i	$0.525776\pi$
$864$	0	0
$865$	26304.0	1.03395
$866$	0	0
$867$	0	0
$868$	0	0
$869$	48386.6i	1.88884i
$870$	0	0
$871$	17625.3	0.685663
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2784.00i	0.107562i
$876$	0	0
$877$	35839.6i	1.37995i	0.723833	+	0.689976i	$0.242379\pi$
$877$	35839.6i	1.37995i	−0.723833	+	0.689976i	$0.757621\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−33024.0	−1.26289	−0.631445	−	0.775420i	$-0.717538\pi$
$881$	−33024.0	−1.26289	−0.631445	+	0.775420i	$0.717538\pi$
$882$	0	0
$883$	36056.0i	1.37416i	0.726583	+	0.687079i	$0.241107\pi$
$883$	36056.0i	1.37416i	−0.726583	+	0.687079i	$0.758893\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11306.8	−0.428011	−0.214006	−	0.976832i	$-0.568651\pi$
$887$	−11306.8	−0.428011	−0.214006	+	0.976832i	$0.568651\pi$
$888$	0	0
$889$	2748.00	0.103673
$890$	0	0
$891$	0	0
$892$	0	0
$893$	22170.3i	0.830794i
$894$	0	0
$895$	−14632.4	−0.546487
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2832.00i	0.105064i
$900$	0	0
$901$	− 17292.8i	− 0.639408i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−27168.0	−0.997895
$906$	0	0
$907$	− 33800.0i	− 1.23739i	−0.785632	−	0.618694i	$-0.787662\pi$
$907$	− 33800.0i	− 1.23739i	0.785632	−	0.618694i	$-0.212338\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−31149.2	−1.13284	−0.566421	−	0.824116i	$-0.691672\pi$
$911$	−31149.2	−1.13284	−0.566421	+	0.824116i	$0.691672\pi$
$912$	0	0
$913$	−20736.0	−0.751655
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4655.75i	− 0.167662i
$918$	0	0
$919$	21751.1	0.780743	0.390371	−	0.920658i	$-0.372347\pi$
$919$	21751.1	0.780743	0.390371	+	0.920658i	$0.372347\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	18432.0i	0.657310i
$924$	0	0
$925$	− 19960.2i	− 0.709498i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9312.00	0.328866	0.164433	−	0.986388i	$-0.447421\pi$
$929$	9312.00	0.328866	0.164433	+	0.986388i	$0.447421\pi$
$930$	0	0
$931$	13240.0i	0.466083i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	63850.3	2.23329
$936$	0	0
$937$	37846.0	1.31950	0.659752	−	0.751484i	$-0.270661\pi$
$937$	37846.0	1.31950	0.659752	+	0.751484i	$0.270661\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	28835.2i	0.998937i	0.866332	+	0.499469i	$0.166471\pi$
$941$	28835.2i	0.998937i	−0.866332	+	0.499469i	$0.833529\pi$
$942$	0	0
$943$	31925.2	1.10247
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 34656.0i	− 1.18920i	−0.804023	−	0.594598i	$-0.797311\pi$
$947$	− 34656.0i	− 1.18920i	0.804023	−	0.594598i	$-0.202689\pi$
$948$	0	0
$949$	− 11182.1i	− 0.382494i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−52704.0	−1.79145	−0.895724	−	0.444610i	$-0.853342\pi$
$953$	−52704.0	−1.79145	−0.895724	+	0.444610i	$0.853342\pi$
$954$	0	0
$955$	− 26112.0i	− 0.884780i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2992.98	−0.100780
$960$	0	0
$961$	11981.0	0.402168
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 13329.9i	− 0.444667i
$966$	0	0
$967$	−33418.2	−1.11133	−0.555665	−	0.831406i	$-0.687536\pi$
$967$	−33418.2	−1.11133	−0.555665	+	0.831406i	$0.687536\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 6384.00i	− 0.210991i	−0.994420	−	0.105496i	$-0.966357\pi$
$971$	− 6384.00i	− 0.210991i	0.994420	−	0.105496i	$-0.0336429\pi$
$972$	0	0
$973$	− 4267.77i	− 0.140615i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31200.0	1.02167	0.510837	−	0.859677i	$-0.329335\pi$
$977$	31200.0	1.02167	0.510837	+	0.859677i	$0.329335\pi$
$978$	0	0
$979$	64512.0i	2.10604i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	24609.0	0.798479	0.399239	−	0.916847i	$-0.369274\pi$
$983$	24609.0	0.798479	0.399239	+	0.916847i	$0.369274\pi$
$984$	0	0
$985$	−65088.0	−2.10546
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 16849.4i	− 0.541739i
$990$	0	0
$991$	13153.2	0.421620	0.210810	−	0.977527i	$-0.432390\pi$
$991$	13153.2	0.421620	0.210810	+	0.977527i	$0.432390\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 43056.0i	− 1.37183i
$996$	0	0
$997$	30615.7i	0.972527i	0.873812	+	0.486264i	$0.161641\pi$
$997$	30615.7i	0.972527i	−0.873812	+	0.486264i	$0.838359\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.4.d.b.289.1	✓	4
3.2	odd	2		576.4.d.h.289.3	yes	4
4.3	odd	2	inner	576.4.d.b.289.2	yes	4
8.3	odd	2	inner	576.4.d.b.289.4	yes	4
8.5	even	2	inner	576.4.d.b.289.3	yes	4
12.11	even	2		576.4.d.h.289.4	yes	4
16.3	odd	4		2304.4.a.bj.1.1		2
16.5	even	4		2304.4.a.bj.1.2		2
16.11	odd	4		2304.4.a.w.1.2		2
16.13	even	4		2304.4.a.w.1.1		2
24.5	odd	2		576.4.d.h.289.1	yes	4
24.11	even	2		576.4.d.h.289.2	yes	4
48.5	odd	4		2304.4.a.z.1.1		2
48.11	even	4		2304.4.a.bm.1.1		2
48.29	odd	4		2304.4.a.bm.1.2		2
48.35	even	4		2304.4.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.4.d.b.289.1	✓	4	1.1	even	1	trivial
576.4.d.b.289.2	yes	4	4.3	odd	2	inner
576.4.d.b.289.3	yes	4	8.5	even	2	inner
576.4.d.b.289.4	yes	4	8.3	odd	2	inner
576.4.d.h.289.1	yes	4	24.5	odd	2
576.4.d.h.289.2	yes	4	24.11	even	2
576.4.d.h.289.3	yes	4	3.2	odd	2
576.4.d.h.289.4	yes	4	12.11	even	2
2304.4.a.w.1.1		2	16.13	even	4
2304.4.a.w.1.2		2	16.11	odd	4
2304.4.a.z.1.1		2	48.5	odd	4
2304.4.a.z.1.2		2	48.35	even	4
2304.4.a.bj.1.1		2	16.3	odd	4
2304.4.a.bj.1.2		2	16.5	even	4
2304.4.a.bm.1.1		2	48.11	even	4
2304.4.a.bm.1.2		2	48.29	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)

\(f(q)\)	\(=\)	\(q-13.8564i q^{5} -3.46410 q^{7} +O(q^{10})\)	\(q-13.8564i q^{5} -3.46410 q^{7} -48.0000i q^{11} +20.7846i q^{13} -96.0000 q^{17} -40.0000i q^{19} +110.851 q^{23} -67.0000 q^{25} -13.8564i q^{29} -204.382 q^{31} +48.0000i q^{35} +297.913i q^{37} +288.000 q^{41} -152.000i q^{43} -554.256 q^{47} -331.000 q^{49} +180.133i q^{53} -665.108 q^{55} +480.000i q^{59} +755.174i q^{61} +288.000 q^{65} -848.000i q^{67} +886.810 q^{71} -538.000 q^{73} +166.277i q^{77} -1008.05 q^{79} -432.000i q^{83} +1330.22i q^{85} -1344.00 q^{89} -72.0000i q^{91} -554.256 q^{95} -590.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 384 q^{17} - 268 q^{25} + 1152 q^{41} - 1324 q^{49} + 1152 q^{65} - 2152 q^{73} - 5376 q^{89} - 2360 q^{97}+O(q^{100}) \) 4 * q - 384 * q^17 - 268 * q^25 + 1152 * q^41 - 1324 * q^49 + 1152 * q^65 - 2152 * q^73 - 5376 * q^89 - 2360 * q^97

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