LMFDB - Embedded newform 576.4.d.e.289.3

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^{10} $
Twist minimal:	no (minimal twist has level 64)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+13.8564i q^{5} -27.7128 q^{7} +O(q^{10})$	$q+13.8564i q^{5} -27.7128 q^{7} +42.0000i q^{11} -41.5692i q^{13} +6.00000 q^{17} +94.0000i q^{19} -138.564 q^{23} -67.0000 q^{25} -235.559i q^{29} +110.851 q^{31} -384.000i q^{35} -13.8564i q^{37} +54.0000 q^{41} -442.000i q^{43} -55.4256 q^{47} +425.000 q^{49} +69.2820i q^{53} -581.969 q^{55} +138.000i q^{59} -429.549i q^{61} +576.000 q^{65} -178.000i q^{67} -859.097 q^{71} +434.000 q^{73} -1163.94i q^{77} +166.277 q^{79} -270.000i q^{83} +83.1384i q^{85} +1182.00 q^{89} +1152.00i q^{91} -1302.50 q^{95} -1238.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 24 q^{17} - 268 q^{25} + 216 q^{41} + 1700 q^{49} + 2304 q^{65} + 1736 q^{73} + 4728 q^{89} - 4952 q^{97}+O(q^{100}) $ 4 * q + 24 * q^17 - 268 * q^25 + 216 * q^41 + 1700 * q^49 + 2304 * q^65 + 1736 * q^73 + 4728 * q^89 - 4952 * 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	$0.212737\pi$
$5$	13.8564i	1.23935i	−0.784857	+	0.619677i	$0.787263\pi$
$6$	0	0
$7$	−27.7128	−1.49635	−0.748176	−	0.663501i	$-0.769070\pi$
$7$	−27.7128	−1.49635	−0.748176	+	0.663501i	$0.769070\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	42.0000i	1.15123i	0.817723	+	0.575613i	$0.195236\pi$
$11$	42.0000i	1.15123i	−0.817723	+	0.575613i	$0.804764\pi$
$12$	0	0
$13$	− 41.5692i	− 0.886864i	−0.896308	−	0.443432i	$-0.853761\pi$
$13$	− 41.5692i	− 0.886864i	0.896308	−	0.443432i	$-0.146239\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000	0.0856008	0.0428004	−	0.999084i	$-0.486372\pi$
$17$	6.00000	0.0856008	0.0428004	+	0.999084i	$0.486372\pi$
$18$	0	0
$19$	94.0000i	1.13500i	0.823372	+	0.567502i	$0.192090\pi$
$19$	94.0000i	1.13500i	−0.823372	+	0.567502i	$0.807910\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−138.564	−1.25620	−0.628100	−	0.778133i	$-0.716167\pi$
$23$	−138.564	−1.25620	−0.628100	+	0.778133i	$0.716167\pi$
$24$	0	0
$25$	−67.0000	−0.536000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 235.559i	− 1.50835i	−0.656673	−	0.754176i	$-0.728037\pi$
$29$	− 235.559i	− 1.50835i	0.656673	−	0.754176i	$-0.271963\pi$
$30$	0	0
$31$	110.851	0.642241	0.321121	−	0.947038i	$-0.395941\pi$
$31$	110.851	0.642241	0.321121	+	0.947038i	$0.395941\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 384.000i	− 1.85451i
$36$	0	0
$37$	− 13.8564i	− 0.0615670i	−0.999526	−	0.0307835i	$-0.990200\pi$
$37$	− 13.8564i	− 0.0615670i	0.999526	−	0.0307835i	$-0.00980024\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	54.0000	0.205692	0.102846	−	0.994697i	$-0.467205\pi$
$41$	54.0000	0.205692	0.102846	+	0.994697i	$0.467205\pi$
$42$	0	0
$43$	− 442.000i	− 1.56754i	−0.621049	−	0.783772i	$-0.713293\pi$
$43$	− 442.000i	− 1.56754i	0.621049	−	0.783772i	$-0.286707\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−55.4256	−0.172014	−0.0860070	−	0.996295i	$-0.527411\pi$
$47$	−55.4256	−0.172014	−0.0860070	+	0.996295i	$0.527411\pi$
$48$	0	0
$49$	425.000	1.23907
$50$	0	0
$51$	0	0
$52$	0	0
$53$	69.2820i	0.179559i	0.995962	+	0.0897794i	$0.0286162\pi$
$53$	69.2820i	0.179559i	−0.995962	+	0.0897794i	$0.971384\pi$
$54$	0	0
$55$	−581.969	−1.42678
$56$	0	0
$57$	0	0
$58$	0	0
$59$	138.000i	0.304510i	0.988341	+	0.152255i	$0.0486534\pi$
$59$	138.000i	0.304510i	−0.988341	+	0.152255i	$0.951347\pi$
$60$	0	0
$61$	− 429.549i	− 0.901608i	−0.892623	−	0.450804i	$-0.851137\pi$
$61$	− 429.549i	− 0.901608i	0.892623	−	0.450804i	$-0.148863\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	576.000	1.09914
$66$	0	0
$67$	− 178.000i	− 0.324570i	−0.986744	−	0.162285i	$-0.948114\pi$
$67$	− 178.000i	− 0.324570i	0.986744	−	0.162285i	$-0.0518863\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−859.097	−1.43600	−0.718001	−	0.696043i	$-0.754942\pi$
$71$	−859.097	−1.43600	−0.718001	+	0.696043i	$0.754942\pi$
$72$	0	0
$73$	434.000	0.695834	0.347917	−	0.937525i	$-0.386889\pi$
$73$	434.000	0.695834	0.347917	+	0.937525i	$0.386889\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1163.94i	− 1.72264i
$78$	0	0
$79$	166.277	0.236805	0.118403	−	0.992966i	$-0.462223\pi$
$79$	166.277	0.236805	0.118403	+	0.992966i	$0.462223\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 270.000i	− 0.357064i	−0.983934	−	0.178532i	$-0.942865\pi$
$83$	− 270.000i	− 0.357064i	0.983934	−	0.178532i	$-0.0571349\pi$
$84$	0	0
$85$	83.1384i	0.106090i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1182.00	1.40777	0.703886	−	0.710313i	$-0.251446\pi$
$89$	1182.00	1.40777	0.703886	+	0.710313i	$0.251446\pi$
$90$	0	0
$91$	1152.00i	1.32706i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1302.50	−1.40667
$96$	0	0
$97$	−1238.00	−1.29587	−0.647937	−	0.761694i	$-0.724368\pi$
$97$	−1238.00	−1.29587	−0.647937	+	0.761694i	$0.724368\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1759.76i	− 1.73369i	−0.498575	−	0.866847i	$-0.666143\pi$
$101$	− 1759.76i	− 1.73369i	0.498575	−	0.866847i	$-0.333857\pi$
$102$	0	0
$103$	−1025.37	−0.980904	−0.490452	−	0.871468i	$-0.663168\pi$
$103$	−1025.37	−0.980904	−0.490452	+	0.871468i	$0.663168\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1302.00i	− 1.17635i	−0.808735	−	0.588173i	$-0.799847\pi$
$107$	− 1302.00i	− 1.17635i	0.808735	−	0.588173i	$-0.200153\pi$
$108$	0	0
$109$	− 263.272i	− 0.231347i	−0.993287	−	0.115674i	$-0.963097\pi$
$109$	− 263.272i	− 0.231347i	0.993287	−	0.115674i	$-0.0369027\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1650.00	−1.37362	−0.686809	−	0.726837i	$-0.740989\pi$
$113$	−1650.00	−1.37362	−0.686809	+	0.726837i	$0.740989\pi$
$114$	0	0
$115$	− 1920.00i	− 1.55688i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−166.277	−0.128089
$120$	0	0
$121$	−433.000	−0.325319
$122$	0	0
$123$	0	0
$124$	0	0
$125$	803.672i	0.575061i
$126$	0	0
$127$	−1108.51	−0.774524	−0.387262	−	0.921970i	$-0.626579\pi$
$127$	−1108.51	−0.774524	−0.387262	+	0.921970i	$0.626579\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	210.000i	0.140059i	0.997545	+	0.0700297i	$0.0223094\pi$
$131$	210.000i	0.140059i	−0.997545	+	0.0700297i	$0.977691\pi$
$132$	0	0
$133$	− 2605.00i	− 1.69836i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−522.000	−0.325529	−0.162764	−	0.986665i	$-0.552041\pi$
$137$	−522.000	−0.325529	−0.162764	+	0.986665i	$0.552041\pi$
$138$	0	0
$139$	550.000i	0.335614i	0.985820	+	0.167807i	$0.0536686\pi$
$139$	550.000i	0.335614i	−0.985820	+	0.167807i	$0.946331\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1745.91	1.02098
$144$	0	0
$145$	3264.00	1.86938
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 374.123i	− 0.205700i	−0.994697	−	0.102850i	$-0.967204\pi$
$149$	− 374.123i	− 0.205700i	0.994697	−	0.102850i	$-0.0327962\pi$
$150$	0	0
$151$	1468.78	0.791573	0.395787	−	0.918343i	$-0.370472\pi$
$151$	1468.78	0.791573	0.395787	+	0.918343i	$0.370472\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1536.00i	0.795964i
$156$	0	0
$157$	1676.63i	0.852288i	0.904655	+	0.426144i	$0.140128\pi$
$157$	1676.63i	0.852288i	−0.904655	+	0.426144i	$0.859872\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3840.00	1.87972
$162$	0	0
$163$	− 578.000i	− 0.277745i	−0.990310	−	0.138873i	$-0.955652\pi$
$163$	− 578.000i	− 0.277745i	0.990310	−	0.138873i	$-0.0443478\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1191.65	−0.552172	−0.276086	−	0.961133i	$-0.589037\pi$
$167$	−1191.65	−0.552172	−0.276086	+	0.961133i	$0.589037\pi$
$168$	0	0
$169$	469.000	0.213473
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2591.15i	1.13874i	0.822083	+	0.569368i	$0.192812\pi$
$173$	2591.15i	1.13874i	−0.822083	+	0.569368i	$0.807188\pi$
$174$	0	0
$175$	1856.76	0.802044
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 3918.00i	− 1.63601i	−0.575214	−	0.818003i	$-0.695081\pi$
$179$	− 3918.00i	− 1.63601i	0.575214	−	0.818003i	$-0.304919\pi$
$180$	0	0
$181$	3588.81i	1.47378i	0.676013	+	0.736890i	$0.263706\pi$
$181$	3588.81i	1.47378i	−0.676013	+	0.736890i	$0.736294\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	192.000	0.0763034
$186$	0	0
$187$	252.000i	0.0985458i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3103.84	−1.17584	−0.587920	−	0.808919i	$-0.700053\pi$
$191$	−3103.84	−1.17584	−0.587920	+	0.808919i	$0.700053\pi$
$192$	0	0
$193$	−1846.00	−0.688487	−0.344243	−	0.938880i	$-0.611865\pi$
$193$	−1846.00	−0.688487	−0.344243	+	0.938880i	$0.611865\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	457.261i	0.165373i	0.996576	+	0.0826866i	$0.0263501\pi$
$197$	457.261i	0.165373i	−0.996576	+	0.0826866i	$0.973650\pi$
$198$	0	0
$199$	2411.01	0.858856	0.429428	−	0.903101i	$-0.358715\pi$
$199$	2411.01	0.858856	0.429428	+	0.903101i	$0.358715\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6528.00i	2.25702i
$204$	0	0
$205$	748.246i	0.254926i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3948.00	−1.30665
$210$	0	0
$211$	2990.00i	0.975545i	0.872971	+	0.487773i	$0.162191\pi$
$211$	2990.00i	0.975545i	−0.872971	+	0.487773i	$0.837809\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6124.53	1.94274
$216$	0	0
$217$	−3072.00	−0.961018
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 249.415i	− 0.0759162i
$222$	0	0
$223$	−5431.71	−1.63110	−0.815548	−	0.578690i	$-0.803564\pi$
$223$	−5431.71	−1.63110	−0.815548	+	0.578690i	$0.803564\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3294.00i	− 0.963130i	−0.876410	−	0.481565i	$-0.840069\pi$
$227$	− 3294.00i	− 0.963130i	0.876410	−	0.481565i	$-0.159931\pi$
$228$	0	0
$229$	− 1898.33i	− 0.547795i	−0.961759	−	0.273897i	$-0.911687\pi$
$229$	− 1898.33i	− 0.547795i	0.961759	−	0.273897i	$-0.0883129\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4818.00	−1.35467	−0.677334	−	0.735676i	$-0.736865\pi$
$233$	−4818.00	−1.35467	−0.677334	+	0.735676i	$0.736865\pi$
$234$	0	0
$235$	− 768.000i	− 0.213186i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5708.84	1.54508	0.772540	−	0.634966i	$-0.218986\pi$
$239$	5708.84	1.54508	0.772540	+	0.634966i	$0.218986\pi$
$240$	0	0
$241$	−550.000	−0.147007	−0.0735033	−	0.997295i	$-0.523418\pi$
$241$	−550.000	−0.147007	−0.0735033	+	0.997295i	$0.523418\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5888.97i	1.53564i
$246$	0	0
$247$	3907.51	1.00659
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6666.00i	1.67631i	0.545431	+	0.838156i	$0.316366\pi$
$251$	6666.00i	1.67631i	−0.545431	+	0.838156i	$0.683634\pi$
$252$	0	0
$253$	− 5819.69i	− 1.44617i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4962.00	−1.20436	−0.602181	−	0.798360i	$-0.705701\pi$
$257$	−4962.00	−1.20436	−0.602181	+	0.798360i	$0.705701\pi$
$258$	0	0
$259$	384.000i	0.0921259i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5570.28	1.30600	0.653000	−	0.757358i	$-0.273510\pi$
$263$	5570.28	1.30600	0.653000	+	0.757358i	$0.273510\pi$
$264$	0	0
$265$	−960.000	−0.222537
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1039.23i	0.235550i	0.993040	+	0.117775i	$0.0375762\pi$
$269$	1039.23i	0.235550i	−0.993040	+	0.117775i	$0.962424\pi$
$270$	0	0
$271$	1053.09	0.236053	0.118027	−	0.993010i	$-0.462343\pi$
$271$	1053.09	0.236053	0.118027	+	0.993010i	$0.462343\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2814.00i	− 0.617057i
$276$	0	0
$277$	4919.02i	1.06699i	0.845804	+	0.533494i	$0.179121\pi$
$277$	4919.02i	1.06699i	−0.845804	+	0.533494i	$0.820879\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3102.00	0.658540	0.329270	−	0.944236i	$-0.393197\pi$
$281$	3102.00	0.658540	0.329270	+	0.944236i	$0.393197\pi$
$282$	0	0
$283$	4102.00i	0.861620i	0.902443	+	0.430810i	$0.141772\pi$
$283$	4102.00i	0.861620i	−0.902443	+	0.430810i	$0.858228\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1496.49	−0.307788
$288$	0	0
$289$	−4877.00	−0.992673
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3893.65i	0.776346i	0.921586	+	0.388173i	$0.126894\pi$
$293$	3893.65i	0.776346i	−0.921586	+	0.388173i	$0.873106\pi$
$294$	0	0
$295$	−1912.18	−0.377395
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5760.00i	1.11408i
$300$	0	0
$301$	12249.1i	2.34560i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5952.00	1.11741
$306$	0	0
$307$	− 4738.00i	− 0.880821i	−0.897797	−	0.440410i	$-0.854833\pi$
$307$	− 4738.00i	− 0.880821i	0.897797	−	0.440410i	$-0.145167\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4683.47	−0.853939	−0.426969	−	0.904266i	$-0.640419\pi$
$311$	−4683.47	−0.853939	−0.426969	+	0.904266i	$0.640419\pi$
$312$	0	0
$313$	9466.00	1.70942	0.854712	−	0.519102i	$-0.173733\pi$
$313$	9466.00	1.70942	0.854712	+	0.519102i	$0.173733\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4115.35i	− 0.729152i	−0.931174	−	0.364576i	$-0.881214\pi$
$317$	− 4115.35i	− 0.729152i	0.931174	−	0.364576i	$-0.118786\pi$
$318$	0	0
$319$	9893.47	1.73645
$320$	0	0
$321$	0	0
$322$	0	0
$323$	564.000i	0.0971573i
$324$	0	0
$325$	2785.14i	0.475359i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1536.00	0.257393
$330$	0	0
$331$	8374.00i	1.39056i	0.718737	+	0.695282i	$0.244721\pi$
$331$	8374.00i	1.39056i	−0.718737	+	0.695282i	$0.755279\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2466.44	0.402257
$336$	0	0
$337$	−3566.00	−0.576417	−0.288208	−	0.957568i	$-0.593060\pi$
$337$	−3566.00	−0.576417	−0.288208	+	0.957568i	$0.593060\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4655.75i	0.739364i
$342$	0	0
$343$	−2272.45	−0.357728
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3894.00i	− 0.602423i	−0.953557	−	0.301212i	$-0.902609\pi$
$347$	− 3894.00i	− 0.602423i	0.953557	−	0.301212i	$-0.0973911\pi$
$348$	0	0
$349$	− 5085.30i	− 0.779971i	−0.920821	−	0.389986i	$-0.872480\pi$
$349$	− 5085.30i	− 0.779971i	0.920821	−	0.389986i	$-0.127520\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1698.00	−0.256021	−0.128011	−	0.991773i	$-0.540859\pi$
$353$	−1698.00	−0.256021	−0.128011	+	0.991773i	$0.540859\pi$
$354$	0	0
$355$	− 11904.0i	− 1.77971i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2521.87	−0.370749	−0.185375	−	0.982668i	$-0.559350\pi$
$359$	−2521.87	−0.370749	−0.185375	+	0.982668i	$0.559350\pi$
$360$	0	0
$361$	−1977.00	−0.288234
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6013.68i	0.862385i
$366$	0	0
$367$	−3270.11	−0.465118	−0.232559	−	0.972582i	$-0.574710\pi$
$367$	−3270.11	−0.465118	−0.232559	+	0.972582i	$0.574710\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1920.00i	− 0.268683i
$372$	0	0
$373$	374.123i	0.0519339i	0.999663	+	0.0259670i	$0.00826647\pi$
$373$	374.123i	0.0519339i	−0.999663	+	0.0259670i	$0.991734\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9792.00	−1.33770
$378$	0	0
$379$	− 106.000i	− 0.0143664i	−0.999974	−	0.00718319i	$-0.997714\pi$
$379$	− 106.000i	− 0.0143664i	0.999974	−	0.00718319i	$-0.00228650\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12858.7	1.71554	0.857769	−	0.514035i	$-0.171850\pi$
$383$	12858.7	1.71554	0.857769	+	0.514035i	$0.171850\pi$
$384$	0	0
$385$	16128.0	2.13496
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 5417.85i	− 0.706160i	−0.935593	−	0.353080i	$-0.885134\pi$
$389$	− 5417.85i	− 0.706160i	0.935593	−	0.353080i	$-0.114866\pi$
$390$	0	0
$391$	−831.384	−0.107532
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2304.00i	0.293486i
$396$	0	0
$397$	− 14119.7i	− 1.78500i	−0.451044	−	0.892502i	$-0.648948\pi$
$397$	− 14119.7i	− 1.78500i	0.451044	−	0.892502i	$-0.351052\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6918.00	0.861517	0.430759	−	0.902467i	$-0.358246\pi$
$401$	6918.00	0.861517	0.430759	+	0.902467i	$0.358246\pi$
$402$	0	0
$403$	− 4608.00i	− 0.569580i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	581.969	0.0708775
$408$	0	0
$409$	−7270.00	−0.878920	−0.439460	−	0.898262i	$-0.644830\pi$
$409$	−7270.00	−0.878920	−0.439460	+	0.898262i	$0.644830\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 3824.37i	− 0.455653i
$414$	0	0
$415$	3741.23	0.442530
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 10734.0i	− 1.25153i	−0.780012	−	0.625764i	$-0.784787\pi$
$419$	− 10734.0i	− 1.25153i	0.780012	−	0.625764i	$-0.215213\pi$
$420$	0	0
$421$	− 15643.9i	− 1.81101i	−0.424333	−	0.905506i	$-0.639491\pi$
$421$	− 15643.9i	− 1.81101i	0.424333	−	0.905506i	$-0.360509\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−402.000	−0.0458820
$426$	0	0
$427$	11904.0i	1.34912i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16239.7	−1.81494	−0.907470	−	0.420117i	$-0.861989\pi$
$431$	−16239.7	−1.81494	−0.907470	+	0.420117i	$0.861989\pi$
$432$	0	0
$433$	−10982.0	−1.21885	−0.609424	−	0.792844i	$-0.708599\pi$
$433$	−10982.0	−1.21885	−0.609424	+	0.792844i	$0.708599\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 13025.0i	− 1.42579i
$438$	0	0
$439$	−17043.4	−1.85293	−0.926465	−	0.376381i	$-0.877168\pi$
$439$	−17043.4	−1.85293	−0.926465	+	0.376381i	$0.877168\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2682.00i	0.287643i	0.989604	+	0.143821i	$0.0459390\pi$
$443$	2682.00i	0.287643i	−0.989604	+	0.143821i	$0.954061\pi$
$444$	0	0
$445$	16378.3i	1.74473i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14346.0	−1.50786	−0.753931	−	0.656954i	$-0.771844\pi$
$449$	−14346.0	−1.50786	−0.753931	+	0.656954i	$0.771844\pi$
$450$	0	0
$451$	2268.00i	0.236798i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−15962.6	−1.64470
$456$	0	0
$457$	−7894.00	−0.808021	−0.404011	−	0.914754i	$-0.632384\pi$
$457$	−7894.00	−0.808021	−0.404011	+	0.914754i	$0.632384\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 3616.52i	− 0.365376i	−0.983171	−	0.182688i	$-0.941520\pi$
$461$	− 3616.52i	− 0.365376i	0.983171	−	0.182688i	$-0.0584798\pi$
$462$	0	0
$463$	6817.35	0.684296	0.342148	−	0.939646i	$-0.388846\pi$
$463$	6817.35	0.684296	0.342148	+	0.939646i	$0.388846\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4734.00i	− 0.469086i	−0.972106	−	0.234543i	$-0.924641\pi$
$467$	− 4734.00i	− 0.469086i	0.972106	−	0.234543i	$-0.0753594\pi$
$468$	0	0
$469$	4932.88i	0.485670i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	18564.0	1.80460
$474$	0	0
$475$	− 6298.00i	− 0.608362i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12304.5	−1.17371	−0.586854	−	0.809693i	$-0.699634\pi$
$479$	−12304.5	−1.17371	−0.586854	+	0.809693i	$0.699634\pi$
$480$	0	0
$481$	−576.000	−0.0546015
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 17154.2i	− 1.60605i
$486$	0	0
$487$	−3907.51	−0.363585	−0.181793	−	0.983337i	$-0.558190\pi$
$487$	−3907.51	−0.363585	−0.181793	+	0.983337i	$0.558190\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6618.00i	0.608281i	0.952627	+	0.304141i	$0.0983693\pi$
$491$	6618.00i	0.608281i	−0.952627	+	0.304141i	$0.901631\pi$
$492$	0	0
$493$	− 1413.35i	− 0.129116i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	23808.0	2.14876
$498$	0	0
$499$	17006.0i	1.52564i	0.646612	+	0.762819i	$0.276185\pi$
$499$	17006.0i	1.52564i	−0.646612	+	0.762819i	$0.723815\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11777.9	−1.04404	−0.522021	−	0.852933i	$-0.674822\pi$
$503$	−11777.9	−1.04404	−0.522021	+	0.852933i	$0.674822\pi$
$504$	0	0
$505$	24384.0	2.14866
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9297.65i	0.809648i	0.914395	+	0.404824i	$0.132667\pi$
$509$	9297.65i	0.809648i	−0.914395	+	0.404824i	$0.867333\pi$
$510$	0	0
$511$	−12027.4	−1.04121
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 14208.0i	− 1.21569i
$516$	0	0
$517$	− 2327.88i	− 0.198027i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	246.000	0.0206861	0.0103430	−	0.999947i	$-0.496708\pi$
$521$	246.000	0.0206861	0.0103430	+	0.999947i	$0.496708\pi$
$522$	0	0
$523$	− 14218.0i	− 1.18874i	−0.804193	−	0.594369i	$-0.797402\pi$
$523$	− 14218.0i	− 1.18874i	0.804193	−	0.594369i	$-0.202598\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	665.108	0.0549764
$528$	0	0
$529$	7033.00	0.578039
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2244.74i	− 0.182421i
$534$	0	0
$535$	18041.0	1.45791
$536$	0	0
$537$	0	0
$538$	0	0
$539$	17850.0i	1.42645i
$540$	0	0
$541$	− 10073.6i	− 0.800551i	−0.916395	−	0.400276i	$-0.868914\pi$
$541$	− 10073.6i	− 0.800551i	0.916395	−	0.400276i	$-0.131086\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3648.00	0.286721
$546$	0	0
$547$	− 21458.0i	− 1.67729i	−0.544678	−	0.838645i	$-0.683348\pi$
$547$	− 21458.0i	− 1.67729i	0.544678	−	0.838645i	$-0.316652\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	22142.5	1.71199
$552$	0	0
$553$	−4608.00	−0.354344
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 10932.7i	− 0.831658i	−0.909443	−	0.415829i	$-0.863491\pi$
$557$	− 10932.7i	− 0.831658i	0.909443	−	0.415829i	$-0.136509\pi$
$558$	0	0
$559$	−18373.6	−1.39020
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 22206.0i	− 1.66229i	−0.556053	−	0.831147i	$-0.687685\pi$
$563$	− 22206.0i	− 1.66229i	0.556053	−	0.831147i	$-0.312315\pi$
$564$	0	0
$565$	− 22863.1i	− 1.70240i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2202.00	−0.162237	−0.0811183	−	0.996704i	$-0.525849\pi$
$569$	−2202.00	−0.162237	−0.0811183	+	0.996704i	$0.525849\pi$
$570$	0	0
$571$	− 5690.00i	− 0.417021i	−0.978020	−	0.208511i	$-0.933138\pi$
$571$	− 5690.00i	− 0.417021i	0.978020	−	0.208511i	$-0.0668616\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	9283.79	0.673323
$576$	0	0
$577$	18466.0	1.33232	0.666161	−	0.745808i	$-0.267936\pi$
$577$	18466.0	1.33232	0.666161	+	0.745808i	$0.267936\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7482.46i	0.534294i
$582$	0	0
$583$	−2909.85	−0.206713
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 4950.00i	− 0.348055i	−0.984741	−	0.174028i	$-0.944322\pi$
$587$	− 4950.00i	− 0.348055i	0.984741	−	0.174028i	$-0.0556782\pi$
$588$	0	0
$589$	10420.0i	0.728946i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6834.00	−0.473253	−0.236626	−	0.971601i	$-0.576042\pi$
$593$	−6834.00	−0.473253	−0.236626	+	0.971601i	$0.576042\pi$
$594$	0	0
$595$	− 2304.00i	− 0.158748i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1468.78	−0.100188	−0.0500941	−	0.998745i	$-0.515952\pi$
$599$	−1468.78	−0.100188	−0.0500941	+	0.998745i	$0.515952\pi$
$600$	0	0
$601$	−2014.00	−0.136693	−0.0683467	−	0.997662i	$-0.521772\pi$
$601$	−2014.00	−0.136693	−0.0683467	+	0.997662i	$0.521772\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 5999.82i	− 0.403186i
$606$	0	0
$607$	4544.90	0.303908	0.151954	−	0.988388i	$-0.451444\pi$
$607$	4544.90	0.303908	0.151954	+	0.988388i	$0.451444\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2304.00i	0.152553i
$612$	0	0
$613$	28918.3i	1.90538i	0.303939	+	0.952692i	$0.401698\pi$
$613$	28918.3i	1.90538i	−0.303939	+	0.952692i	$0.598302\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11730.0	−0.765368	−0.382684	−	0.923879i	$-0.625000\pi$
$617$	−11730.0	−0.765368	−0.382684	+	0.923879i	$0.625000\pi$
$618$	0	0
$619$	− 18314.0i	− 1.18918i	−0.804029	−	0.594590i	$-0.797315\pi$
$619$	− 18314.0i	− 1.18918i	0.804029	−	0.594590i	$-0.202685\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−32756.5	−2.10652
$624$	0	0
$625$	−19511.0	−1.24870
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 83.1384i	− 0.00527019i
$630$	0	0
$631$	−4628.04	−0.291980	−0.145990	−	0.989286i	$-0.546637\pi$
$631$	−4628.04	−0.291980	−0.145990	+	0.989286i	$0.546637\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 15360.0i	− 0.959910i
$636$	0	0
$637$	− 17666.9i	− 1.09888i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18570.0	−1.14426	−0.572130	−	0.820163i	$-0.693883\pi$
$641$	−18570.0	−1.14426	−0.572130	+	0.820163i	$0.693883\pi$
$642$	0	0
$643$	27022.0i	1.65730i	0.559767	+	0.828650i	$0.310890\pi$
$643$	27022.0i	1.65730i	−0.559767	+	0.828650i	$0.689110\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2854.42	−0.173445	−0.0867224	−	0.996233i	$-0.527639\pi$
$647$	−2854.42	−0.173445	−0.0867224	+	0.996233i	$0.527639\pi$
$648$	0	0
$649$	−5796.00	−0.350559
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 28225.5i	− 1.69150i	−0.533580	−	0.845750i	$-0.679154\pi$
$653$	− 28225.5i	− 1.69150i	0.533580	−	0.845750i	$-0.320846\pi$
$654$	0	0
$655$	−2909.85	−0.173583
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8562.00i	0.506113i	0.967452	+	0.253056i	$0.0814358\pi$
$659$	8562.00i	0.506113i	−0.967452	+	0.253056i	$0.918564\pi$
$660$	0	0
$661$	16115.0i	0.948262i	0.880454	+	0.474131i	$0.157238\pi$
$661$	16115.0i	0.948262i	−0.880454	+	0.474131i	$0.842762\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	36096.0	2.10488
$666$	0	0
$667$	32640.0i	1.89479i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	18041.0	1.03795
$672$	0	0
$673$	−25430.0	−1.45654	−0.728272	−	0.685288i	$-0.759676\pi$
$673$	−25430.0	−1.45654	−0.728272	+	0.685288i	$0.759676\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15976.4i	0.906978i	0.891262	+	0.453489i	$0.149821\pi$
$677$	15976.4i	0.906978i	−0.891262	+	0.453489i	$0.850179\pi$
$678$	0	0
$679$	34308.5	1.93908
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 8358.00i	− 0.468243i	−0.972207	−	0.234122i	$-0.924779\pi$
$683$	− 8358.00i	− 0.468243i	0.972207	−	0.234122i	$-0.0752214\pi$
$684$	0	0
$685$	− 7233.04i	− 0.403446i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2880.00	0.159244
$690$	0	0
$691$	− 22898.0i	− 1.26061i	−0.776348	−	0.630305i	$-0.782930\pi$
$691$	− 22898.0i	− 1.26061i	0.776348	−	0.630305i	$-0.217070\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7621.02	−0.415945
$696$	0	0
$697$	324.000	0.0176074
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8854.24i	0.477062i	0.971135	+	0.238531i	$0.0766658\pi$
$701$	8854.24i	0.477062i	−0.971135	+	0.238531i	$0.923334\pi$
$702$	0	0
$703$	1302.50	0.0698788
$704$	0	0
$705$	0	0
$706$	0	0
$707$	48768.0i	2.59421i
$708$	0	0
$709$	− 14757.1i	− 0.781683i	−0.920458	−	0.390842i	$-0.872184\pi$
$709$	− 14757.1i	− 0.781683i	0.920458	−	0.390842i	$-0.127816\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−15360.0	−0.806783
$714$	0	0
$715$	24192.0i	1.26536i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	13135.9	0.681343	0.340671	−	0.940182i	$-0.389346\pi$
$719$	13135.9	0.681343	0.340671	+	0.940182i	$0.389346\pi$
$720$	0	0
$721$	28416.0	1.46778
$722$	0	0
$723$	0	0
$724$	0	0
$725$	15782.4i	0.808476i
$726$	0	0
$727$	−16710.8	−0.852504	−0.426252	−	0.904605i	$-0.640166\pi$
$727$	−16710.8	−0.852504	−0.426252	+	0.904605i	$0.640166\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 2652.00i	− 0.134183i
$732$	0	0
$733$	31495.6i	1.58706i	0.608529	+	0.793531i	$0.291760\pi$
$733$	31495.6i	1.58706i	−0.608529	+	0.793531i	$0.708240\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7476.00	0.373653
$738$	0	0
$739$	14750.0i	0.734219i	0.930178	+	0.367109i	$0.119652\pi$
$739$	14750.0i	0.734219i	−0.930178	+	0.367109i	$0.880348\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2133.89	0.105363	0.0526815	−	0.998611i	$-0.483223\pi$
$743$	2133.89	0.105363	0.0526815	+	0.998611i	$0.483223\pi$
$744$	0	0
$745$	5184.00	0.254936
$746$	0	0
$747$	0	0
$748$	0	0
$749$	36082.1i	1.76023i
$750$	0	0
$751$	−5265.43	−0.255843	−0.127922	−	0.991784i	$-0.540831\pi$
$751$	−5265.43	−0.255843	−0.127922	+	0.991784i	$0.540831\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	20352.0i	0.981040i
$756$	0	0
$757$	− 27006.1i	− 1.29664i	−0.761369	−	0.648319i	$-0.775472\pi$
$757$	− 27006.1i	− 1.29664i	0.761369	−	0.648319i	$-0.224528\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20538.0	−0.978321	−0.489160	−	0.872194i	$-0.662697\pi$
$761$	−20538.0	−0.978321	−0.489160	+	0.872194i	$0.662697\pi$
$762$	0	0
$763$	7296.00i	0.346177i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5736.55	0.270059
$768$	0	0
$769$	−10678.0	−0.500726	−0.250363	−	0.968152i	$-0.580550\pi$
$769$	−10678.0	−0.500726	−0.250363	+	0.968152i	$0.580550\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	27837.5i	1.29527i	0.761949	+	0.647636i	$0.224242\pi$
$773$	27837.5i	1.29527i	−0.761949	+	0.647636i	$0.775758\pi$
$774$	0	0
$775$	−7427.03	−0.344241
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5076.00i	0.233462i
$780$	0	0
$781$	− 36082.1i	− 1.65316i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−23232.0	−1.05629
$786$	0	0
$787$	− 12818.0i	− 0.580575i	−0.956940	−	0.290287i	$-0.906249\pi$
$787$	− 12818.0i	− 0.580575i	0.956940	−	0.290287i	$-0.0937509\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	45726.1	2.05542
$792$	0	0
$793$	−17856.0	−0.799603
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6748.07i	0.299911i	0.988693	+	0.149955i	$0.0479130\pi$
$797$	6748.07i	0.299911i	−0.988693	+	0.149955i	$0.952087\pi$
$798$	0	0
$799$	−332.554	−0.0147245
$800$	0	0
$801$	0	0
$802$	0	0
$803$	18228.0i	0.801061i
$804$	0	0
$805$	53208.6i	2.32964i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−29130.0	−1.26595	−0.632977	−	0.774171i	$-0.718167\pi$
$809$	−29130.0	−1.26595	−0.632977	+	0.774171i	$0.718167\pi$
$810$	0	0
$811$	− 28618.0i	− 1.23910i	−0.784955	−	0.619552i	$-0.787314\pi$
$811$	− 28618.0i	− 1.23910i	0.784955	−	0.619552i	$-0.212686\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8009.00	0.344225
$816$	0	0
$817$	41548.0	1.77917
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 12124.4i	− 0.515399i	−0.966225	−	0.257700i	$-0.917035\pi$
$821$	− 12124.4i	− 0.515399i	0.966225	−	0.257700i	$-0.0829645\pi$
$822$	0	0
$823$	42816.3	1.81347	0.906733	−	0.421706i	$-0.138568\pi$
$823$	42816.3	1.81347	0.906733	+	0.421706i	$0.138568\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 41574.0i	− 1.74809i	−0.485846	−	0.874045i	$-0.661488\pi$
$827$	− 41574.0i	− 1.74809i	0.485846	−	0.874045i	$-0.338512\pi$
$828$	0	0
$829$	22405.8i	0.938704i	0.883011	+	0.469352i	$0.155513\pi$
$829$	22405.8i	0.938704i	−0.883011	+	0.469352i	$0.844487\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2550.00	0.106065
$834$	0	0
$835$	− 16512.0i	− 0.684337i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31287.8	1.28745	0.643727	−	0.765255i	$-0.277387\pi$
$839$	31287.8	1.28745	0.643727	+	0.765255i	$0.277387\pi$
$840$	0	0
$841$	−31099.0	−1.27512
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6498.65i	0.264569i
$846$	0	0
$847$	11999.6	0.486792
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1920.00i	0.0773405i
$852$	0	0
$853$	4253.92i	0.170752i	0.996349	+	0.0853759i	$0.0272091\pi$
$853$	4253.92i	0.170752i	−0.996349	+	0.0853759i	$0.972791\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41862.0	1.66859	0.834293	−	0.551321i	$-0.185876\pi$
$857$	41862.0	1.66859	0.834293	+	0.551321i	$0.185876\pi$
$858$	0	0
$859$	− 10762.0i	− 0.427468i	−0.976892	−	0.213734i	$-0.931437\pi$
$859$	− 10762.0i	− 0.427468i	0.976892	−	0.213734i	$-0.0685625\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1884.47	0.0743316	0.0371658	−	0.999309i	$-0.488167\pi$
$863$	1884.47	0.0743316	0.0371658	+	0.999309i	$0.488167\pi$
$864$	0	0
$865$	−35904.0	−1.41130
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6983.63i	0.272616i
$870$	0	0
$871$	−7399.32	−0.287849
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 22272.0i	− 0.860493i
$876$	0	0
$877$	37647.9i	1.44958i	0.688972	+	0.724788i	$0.258062\pi$
$877$	37647.9i	1.44958i	−0.688972	+	0.724788i	$0.741938\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4722.00	−0.180577	−0.0902884	−	0.995916i	$-0.528779\pi$
$881$	−4722.00	−0.180577	−0.0902884	+	0.995916i	$0.528779\pi$
$882$	0	0
$883$	31390.0i	1.19633i	0.801374	+	0.598164i	$0.204103\pi$
$883$	31390.0i	1.19633i	−0.801374	+	0.598164i	$0.795897\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39490.8	−1.49489	−0.747446	−	0.664322i	$-0.768720\pi$
$887$	−39490.8	−1.49489	−0.747446	+	0.664322i	$0.768720\pi$
$888$	0	0
$889$	30720.0	1.15896
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 5210.01i	− 0.195237i
$894$	0	0
$895$	54289.4	2.02759
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 26112.0i	− 0.968725i
$900$	0	0
$901$	415.692i	0.0153704i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−49728.0	−1.82654
$906$	0	0
$907$	− 5242.00i	− 0.191905i	−0.995386	−	0.0959525i	$-0.969410\pi$
$907$	− 5242.00i	− 0.191905i	0.995386	−	0.0959525i	$-0.0305897\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	28211.6	1.02601	0.513004	−	0.858386i	$-0.328533\pi$
$911$	28211.6	1.02601	0.513004	+	0.858386i	$0.328533\pi$
$912$	0	0
$913$	11340.0	0.411062
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5819.69i	− 0.209578i
$918$	0	0
$919$	1912.18	0.0686367	0.0343184	−	0.999411i	$-0.489074\pi$
$919$	1912.18	0.0686367	0.0343184	+	0.999411i	$0.489074\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	35712.0i	1.27354i
$924$	0	0
$925$	928.379i	0.0329999i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	20118.0	0.710495	0.355248	−	0.934772i	$-0.384397\pi$
$929$	20118.0	0.710495	0.355248	+	0.934772i	$0.384397\pi$
$930$	0	0
$931$	39950.0i	1.40635i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3491.81	−0.122133
$936$	0	0
$937$	−3950.00	−0.137717	−0.0688585	−	0.997626i	$-0.521936\pi$
$937$	−3950.00	−0.137717	−0.0688585	+	0.997626i	$0.521936\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11570.1i	0.400823i	0.979712	+	0.200412i	$0.0642279\pi$
$941$	11570.1i	0.400823i	−0.979712	+	0.200412i	$0.935772\pi$
$942$	0	0
$943$	−7482.46	−0.258391
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46146.0i	1.58347i	0.610866	+	0.791734i	$0.290821\pi$
$947$	46146.0i	1.58347i	−0.610866	+	0.791734i	$0.709179\pi$
$948$	0	0
$949$	− 18041.0i	− 0.617110i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	47718.0	1.62197	0.810985	−	0.585067i	$-0.198932\pi$
$953$	47718.0	1.62197	0.810985	+	0.585067i	$0.198932\pi$
$954$	0	0
$955$	− 43008.0i	− 1.45728i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14466.1	0.487106
$960$	0	0
$961$	−17503.0	−0.587526
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 25578.9i	− 0.853280i
$966$	0	0
$967$	−34391.6	−1.14370	−0.571851	−	0.820358i	$-0.693774\pi$
$967$	−34391.6	−1.14370	−0.571851	+	0.820358i	$0.693774\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 2118.00i	− 0.0699999i	−0.999387	−	0.0349999i	$-0.988857\pi$
$971$	− 2118.00i	− 0.0699999i	0.999387	−	0.0349999i	$-0.0111431\pi$
$972$	0	0
$973$	− 15242.0i	− 0.502197i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−8826.00	−0.289016	−0.144508	−	0.989504i	$-0.546160\pi$
$977$	−8826.00	−0.289016	−0.144508	+	0.989504i	$0.546160\pi$
$978$	0	0
$979$	49644.0i	1.62066i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	23361.9	0.758015	0.379008	−	0.925394i	$-0.376265\pi$
$983$	23361.9	0.758015	0.379008	+	0.925394i	$0.376265\pi$
$984$	0	0
$985$	−6336.00	−0.204956
$986$	0	0
$987$	0	0
$988$	0	0
$989$	61245.3i	1.96915i
$990$	0	0
$991$	−16738.5	−0.536546	−0.268273	−	0.963343i	$-0.586453\pi$
$991$	−16738.5	−0.536546	−0.268273	+	0.963343i	$0.586453\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	33408.0i	1.06443i
$996$	0	0
$997$	− 13094.3i	− 0.415949i	−0.978134	−	0.207974i	$-0.933313\pi$
$997$	− 13094.3i	− 0.415949i	0.978134	−	0.207974i	$-0.0666870\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.4.d.e.289.3		4
3.2	odd	2		64.4.b.b.33.1	✓	4
4.3	odd	2	inner	576.4.d.e.289.4		4
8.3	odd	2	inner	576.4.d.e.289.2		4
8.5	even	2	inner	576.4.d.e.289.1		4
12.11	even	2		64.4.b.b.33.3	yes	4
16.3	odd	4		2304.4.a.ba.1.2		2
16.5	even	4		2304.4.a.ba.1.1		2
16.11	odd	4		2304.4.a.bi.1.1		2
16.13	even	4		2304.4.a.bi.1.2		2
24.5	odd	2		64.4.b.b.33.4	yes	4
24.11	even	2		64.4.b.b.33.2	yes	4
48.5	odd	4		256.4.a.i.1.2		2
48.11	even	4		256.4.a.m.1.2		2
48.29	odd	4		256.4.a.m.1.1		2
48.35	even	4		256.4.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.4.b.b.33.1	✓	4	3.2	odd	2
64.4.b.b.33.2	yes	4	24.11	even	2
64.4.b.b.33.3	yes	4	12.11	even	2
64.4.b.b.33.4	yes	4	24.5	odd	2
256.4.a.i.1.1		2	48.35	even	4
256.4.a.i.1.2		2	48.5	odd	4
256.4.a.m.1.1		2	48.29	odd	4
256.4.a.m.1.2		2	48.11	even	4
576.4.d.e.289.1		4	8.5	even	2	inner
576.4.d.e.289.2		4	8.3	odd	2	inner
576.4.d.e.289.3		4	1.1	even	1	trivial
576.4.d.e.289.4		4	4.3	odd	2	inner
2304.4.a.ba.1.1		2	16.5	even	4
2304.4.a.ba.1.2		2	16.3	odd	4
2304.4.a.bi.1.1		2	16.11	odd	4
2304.4.a.bi.1.2		2	16.13	even	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)

\(f(q)\)	\(=\)	\(q+13.8564i q^{5} -27.7128 q^{7} +O(q^{10})\)	\(q+13.8564i q^{5} -27.7128 q^{7} +42.0000i q^{11} -41.5692i q^{13} +6.00000 q^{17} +94.0000i q^{19} -138.564 q^{23} -67.0000 q^{25} -235.559i q^{29} +110.851 q^{31} -384.000i q^{35} -13.8564i q^{37} +54.0000 q^{41} -442.000i q^{43} -55.4256 q^{47} +425.000 q^{49} +69.2820i q^{53} -581.969 q^{55} +138.000i q^{59} -429.549i q^{61} +576.000 q^{65} -178.000i q^{67} -859.097 q^{71} +434.000 q^{73} -1163.94i q^{77} +166.277 q^{79} -270.000i q^{83} +83.1384i q^{85} +1182.00 q^{89} +1152.00i q^{91} -1302.50 q^{95} -1238.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 24 q^{17} - 268 q^{25} + 216 q^{41} + 1700 q^{49} + 2304 q^{65} + 1736 q^{73} + 4728 q^{89} - 4952 q^{97}+O(q^{100}) \) 4 * q + 24 * q^17 - 268 * q^25 + 216 * q^41 + 1700 * q^49 + 2304 * q^65 + 1736 * q^73 + 4728 * q^89 - 4952 * q^97

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