LMFDB - Embedded newform 7200.2.k.l.3601.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(3601,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7200.3601");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{53}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3601.1
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	7200.3601
Dual form			7200.2.k.l.3601.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44949 q^{7} +O(q^{10})$	$q-2.44949 q^{7} -3.46410i q^{11} +4.89898 q^{17} +3.46410i q^{19} +2.44949 q^{23} -4.00000 q^{31} +8.48528i q^{37} -4.24264i q^{43} +7.34847 q^{47} -1.00000 q^{49} -5.65685i q^{53} +10.3923i q^{59} +3.46410i q^{61} -4.24264i q^{67} -12.0000 q^{71} +4.89898 q^{73} +8.48528i q^{77} -4.00000 q^{79} -9.89949i q^{83} +6.00000 q^{89} +4.89898 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 16 q^{31} - 4 q^{49} - 48 q^{71} - 16 q^{79} + 24 q^{89}+O(q^{100}) $ 4 * q - 16 * q^31 - 4 * q^49 - 48 * q^71 - 16 * q^79 + 24 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$6401$	$6751$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.46410i	− 1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	− 3.46410i	− 1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.89898	1.18818	0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898	1.18818	0.594089	+	0.804400i	$0.297513\pi$
$18$	0	0
$19$	3.46410i	0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$	3.46410i	0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.44949	0.510754	0.255377	−	0.966842i	$-0.417800\pi$
$23$	2.44949	0.510754	0.255377	+	0.966842i	$0.417800\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.48528i	1.39497i	0.716599	+	0.697486i	$0.245698\pi$
$37$	8.48528i	1.39497i	−0.716599	+	0.697486i	$0.754302\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	− 4.24264i	− 0.646997i	−0.946229	−	0.323498i	$-0.895141\pi$
$43$	− 4.24264i	− 0.646997i	0.946229	−	0.323498i	$-0.104859\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.34847	1.07188	0.535942	−	0.844255i	$-0.319956\pi$
$47$	7.34847	1.07188	0.535942	+	0.844255i	$0.319956\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 5.65685i	− 0.777029i	−0.921443	−	0.388514i	$-0.872988\pi$
$53$	− 5.65685i	− 0.777029i	0.921443	−	0.388514i	$-0.127012\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.3923i	1.35296i	0.736460	+	0.676481i	$0.236496\pi$
$59$	10.3923i	1.35296i	−0.736460	+	0.676481i	$0.763504\pi$
$60$	0	0
$61$	3.46410i	0.443533i	0.975100	+	0.221766i	$0.0711822\pi$
$61$	3.46410i	0.443533i	−0.975100	+	0.221766i	$0.928818\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.24264i	− 0.518321i	−0.965834	−	0.259161i	$-0.916554\pi$
$67$	− 4.24264i	− 0.518321i	0.965834	−	0.259161i	$-0.0834459\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	4.89898	0.573382	0.286691	−	0.958023i	$-0.407445\pi$
$73$	4.89898	0.573382	0.286691	+	0.958023i	$0.407445\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.48528i	0.966988i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 9.89949i	− 1.08661i	−0.839535	−	0.543305i	$-0.817173\pi$
$83$	− 9.89949i	− 1.08661i	0.839535	−	0.543305i	$-0.182827\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.89898	0.497416	0.248708	−	0.968579i	$-0.419994\pi$
$97$	4.89898	0.497416	0.248708	+	0.968579i	$0.419994\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 13.8564i	− 1.37876i	−0.724398	−	0.689382i	$-0.757882\pi$
$101$	− 13.8564i	− 1.37876i	0.724398	−	0.689382i	$-0.242118\pi$
$102$	0	0
$103$	7.34847	0.724066	0.362033	−	0.932165i	$-0.382083\pi$
$103$	7.34847	0.724066	0.362033	+	0.932165i	$0.382083\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.41421i	− 0.136717i	−0.997661	−	0.0683586i	$-0.978224\pi$
$107$	− 1.41421i	− 0.136717i	0.997661	−	0.0683586i	$-0.0217762\pi$
$108$	0	0
$109$	3.46410i	0.331801i	0.986143	+	0.165900i	$0.0530530\pi$
$109$	3.46410i	0.331801i	−0.986143	+	0.165900i	$0.946947\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	17.1464	1.52150	0.760750	−	0.649045i	$-0.224831\pi$
$127$	17.1464	1.52150	0.760750	+	0.649045i	$0.224831\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.46410i	0.302660i	0.988483	+	0.151330i	$0.0483556\pi$
$131$	3.46410i	0.302660i	−0.988483	+	0.151330i	$0.951644\pi$
$132$	0	0
$133$	− 8.48528i	− 0.735767i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.79796	−0.837096	−0.418548	−	0.908195i	$-0.637461\pi$
$137$	−9.79796	−0.837096	−0.418548	+	0.908195i	$0.637461\pi$
$138$	0	0
$139$	10.3923i	0.881464i	0.897639	+	0.440732i	$0.145281\pi$
$139$	10.3923i	0.881464i	−0.897639	+	0.440732i	$0.854719\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 17.3205i	− 1.41895i	−0.704730	−	0.709476i	$-0.748932\pi$
$149$	− 17.3205i	− 1.41895i	0.704730	−	0.709476i	$-0.251068\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 8.48528i	− 0.677199i	−0.940931	−	0.338600i	$-0.890047\pi$
$157$	− 8.48528i	− 0.677199i	0.940931	−	0.338600i	$-0.109953\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	− 21.2132i	− 1.66155i	−0.556611	−	0.830773i	$-0.687899\pi$
$163$	− 21.2132i	− 1.66155i	0.556611	−	0.830773i	$-0.312101\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.2474	−0.947736	−0.473868	−	0.880596i	$-0.657143\pi$
$167$	−12.2474	−0.947736	−0.473868	+	0.880596i	$0.657143\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2.82843i	− 0.215041i	−0.994203	−	0.107521i	$-0.965709\pi$
$173$	− 2.82843i	− 0.215041i	0.994203	−	0.107521i	$-0.0342912\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.46410i	0.258919i	0.991585	+	0.129460i	$0.0413242\pi$
$179$	3.46410i	0.258919i	−0.991585	+	0.129460i	$0.958676\pi$
$180$	0	0
$181$	− 13.8564i	− 1.02994i	−0.857209	−	0.514969i	$-0.827803\pi$
$181$	− 13.8564i	− 1.02994i	0.857209	−	0.514969i	$-0.172197\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 16.9706i	− 1.24101i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	24.4949	1.76318	0.881591	−	0.472015i	$-0.156473\pi$
$193$	24.4949	1.76318	0.881591	+	0.472015i	$0.156473\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.65685i	− 0.403034i	−0.979485	−	0.201517i	$-0.935413\pi$
$197$	− 5.65685i	− 0.403034i	0.979485	−	0.201517i	$-0.0645872\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	24.2487i	1.66935i	0.550743	+	0.834675i	$0.314345\pi$
$211$	24.2487i	1.66935i	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.79796	0.665129
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	22.0454	1.47627	0.738135	−	0.674653i	$-0.235707\pi$
$223$	22.0454	1.47627	0.738135	+	0.674653i	$0.235707\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.41421i	0.0938647i	0.998898	+	0.0469323i	$0.0149445\pi$
$227$	1.41421i	0.0938647i	−0.998898	+	0.0469323i	$0.985055\pi$
$228$	0	0
$229$	− 27.7128i	− 1.83131i	−0.401960	−	0.915657i	$-0.631671\pi$
$229$	− 27.7128i	− 1.83131i	0.401960	−	0.915657i	$-0.368329\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.6969	0.962828	0.481414	−	0.876493i	$-0.340123\pi$
$233$	14.6969	0.962828	0.481414	+	0.876493i	$0.340123\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.3923i	0.655956i	0.944685	+	0.327978i	$0.106367\pi$
$251$	10.3923i	0.655956i	−0.944685	+	0.327978i	$0.893633\pi$
$252$	0	0
$253$	− 8.48528i	− 0.533465i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.79796	0.611180	0.305590	−	0.952163i	$-0.401146\pi$
$257$	9.79796	0.611180	0.305590	+	0.952163i	$0.401146\pi$
$258$	0	0
$259$	− 20.7846i	− 1.29149i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.34847	0.453126	0.226563	−	0.973997i	$-0.427251\pi$
$263$	7.34847	0.453126	0.226563	+	0.973997i	$0.427251\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 10.3923i	− 0.633630i	−0.948487	−	0.316815i	$-0.897387\pi$
$269$	− 10.3923i	− 0.633630i	0.948487	−	0.316815i	$-0.102613\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 25.4558i	− 1.52949i	−0.644331	−	0.764747i	$-0.722864\pi$
$277$	− 25.4558i	− 1.52949i	0.644331	−	0.764747i	$-0.277136\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	21.2132i	1.26099i	0.776192	+	0.630497i	$0.217149\pi$
$283$	21.2132i	1.26099i	−0.776192	+	0.630497i	$0.782851\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	7.00000	0.411765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.7990i	1.15667i	0.815800	+	0.578335i	$0.196297\pi$
$293$	19.7990i	1.15667i	−0.815800	+	0.578335i	$0.803703\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	10.3923i	0.599002i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	29.6985i	1.69498i	0.530810	+	0.847491i	$0.321888\pi$
$307$	29.6985i	1.69498i	−0.530810	+	0.847491i	$0.678112\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	9.79796	0.553813	0.276907	−	0.960897i	$-0.410691\pi$
$313$	9.79796	0.553813	0.276907	+	0.960897i	$0.410691\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 28.2843i	− 1.58860i	−0.607524	−	0.794301i	$-0.707837\pi$
$317$	− 28.2843i	− 1.58860i	0.607524	−	0.794301i	$-0.292163\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	16.9706i	0.944267i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−18.0000	−0.992372
$330$	0	0
$331$	− 31.1769i	− 1.71364i	−0.515617	−	0.856819i	$-0.672437\pi$
$331$	− 31.1769i	− 1.71364i	0.515617	−	0.856819i	$-0.327563\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	13.8564i	0.750366i
$342$	0	0
$343$	19.5959	1.05808
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 15.5563i	− 0.835109i	−0.908652	−	0.417554i	$-0.862887\pi$
$347$	− 15.5563i	− 0.835109i	0.908652	−	0.417554i	$-0.137113\pi$
$348$	0	0
$349$	13.8564i	0.741716i	0.928689	+	0.370858i	$0.120936\pi$
$349$	13.8564i	0.741716i	−0.928689	+	0.370858i	$0.879064\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	29.3939	1.56448	0.782239	−	0.622978i	$-0.214078\pi$
$353$	29.3939	1.56448	0.782239	+	0.622978i	$0.214078\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	7.00000	0.368421
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−12.2474	−0.639312	−0.319656	−	0.947534i	$-0.603567\pi$
$367$	−12.2474	−0.639312	−0.319656	+	0.947534i	$0.603567\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13.8564i	0.719389i
$372$	0	0
$373$	− 8.48528i	− 0.439351i	−0.975573	−	0.219676i	$-0.929500\pi$
$373$	− 8.48528i	− 0.439351i	0.975573	−	0.219676i	$-0.0704999\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	24.2487i	1.24557i	0.782392	+	0.622786i	$0.213999\pi$
$379$	24.2487i	1.24557i	−0.782392	+	0.622786i	$0.786001\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	26.9444	1.37679	0.688397	−	0.725334i	$-0.258315\pi$
$383$	26.9444	1.37679	0.688397	+	0.725334i	$0.258315\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 3.46410i	− 0.175637i	−0.996136	−	0.0878185i	$-0.972010\pi$
$389$	− 3.46410i	− 0.175637i	0.996136	−	0.0878185i	$-0.0279895\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	16.9706i	0.851728i	0.904787	+	0.425864i	$0.140030\pi$
$397$	16.9706i	0.851728i	−0.904787	+	0.425864i	$0.859970\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	29.3939	1.45700
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 25.4558i	− 1.25260i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 10.3923i	− 0.507697i	−0.967244	−	0.253849i	$-0.918303\pi$
$419$	− 10.3923i	− 0.507697i	0.967244	−	0.253849i	$-0.0816965\pi$
$420$	0	0
$421$	24.2487i	1.18181i	0.806741	+	0.590905i	$0.201229\pi$
$421$	24.2487i	1.18181i	−0.806741	+	0.590905i	$0.798771\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 8.48528i	− 0.410632i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−4.89898	−0.235430	−0.117715	−	0.993047i	$-0.537557\pi$
$433$	−4.89898	−0.235430	−0.117715	+	0.993047i	$0.537557\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.48528i	0.405906i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.41421i	0.0671913i	0.999436	+	0.0335957i	$0.0106958\pi$
$443$	1.41421i	0.0671913i	−0.999436	+	0.0335957i	$0.989304\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.5959	−0.916658	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	−19.5959	−0.916658	−0.458329	+	0.888783i	$0.651552\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13.8564i	0.645357i	0.946509	+	0.322679i	$0.104583\pi$
$461$	13.8564i	0.645357i	−0.946509	+	0.322679i	$0.895417\pi$
$462$	0	0
$463$	17.1464	0.796862	0.398431	−	0.917198i	$-0.369555\pi$
$463$	17.1464	0.796862	0.398431	+	0.917198i	$0.369555\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.07107i	0.327210i	0.986526	+	0.163605i	$0.0523123\pi$
$467$	7.07107i	0.327210i	−0.986526	+	0.163605i	$0.947688\pi$
$468$	0	0
$469$	10.3923i	0.479872i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−14.6969	−0.675766
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.34847	0.332991	0.166495	−	0.986042i	$-0.446755\pi$
$487$	7.34847	0.332991	0.166495	+	0.986042i	$0.446755\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 24.2487i	− 1.09433i	−0.837025	−	0.547165i	$-0.815707\pi$
$491$	− 24.2487i	− 1.09433i	0.837025	−	0.547165i	$-0.184293\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	29.3939	1.31850
$498$	0	0
$499$	− 17.3205i	− 0.775372i	−0.921791	−	0.387686i	$-0.873274\pi$
$499$	− 17.3205i	− 0.775372i	0.921791	−	0.387686i	$-0.126726\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.2474	0.546087	0.273043	−	0.962002i	$-0.411970\pi$
$503$	12.2474	0.546087	0.273043	+	0.962002i	$0.411970\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	27.7128i	1.22835i	0.789170	+	0.614174i	$0.210511\pi$
$509$	27.7128i	1.22835i	−0.789170	+	0.614174i	$0.789489\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 25.4558i	− 1.11955i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	− 12.7279i	− 0.556553i	−0.960501	−	0.278277i	$-0.910237\pi$
$523$	− 12.7279i	− 0.556553i	0.960501	−	0.278277i	$-0.0897632\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19.5959	−0.853612
$528$	0	0
$529$	−17.0000	−0.739130
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.46410i	0.149209i
$540$	0	0
$541$	− 41.5692i	− 1.78720i	−0.448864	−	0.893600i	$-0.648171\pi$
$541$	− 41.5692i	− 1.78720i	0.448864	−	0.893600i	$-0.351829\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.24264i	0.181402i	0.995878	+	0.0907011i	$0.0289108\pi$
$547$	4.24264i	0.181402i	−0.995878	+	0.0907011i	$0.971089\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	9.79796	0.416652
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14.1421i	− 0.599222i	−0.954062	−	0.299611i	$-0.903143\pi$
$557$	− 14.1421i	− 0.599222i	0.954062	−	0.299611i	$-0.0968568\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 41.0122i	− 1.72846i	−0.503099	−	0.864229i	$-0.667807\pi$
$563$	− 41.0122i	− 1.72846i	0.503099	−	0.864229i	$-0.332193\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	3.46410i	0.144968i	0.997370	+	0.0724841i	$0.0230926\pi$
$571$	3.46410i	0.144968i	−0.997370	+	0.0724841i	$0.976907\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	29.3939	1.22368	0.611842	−	0.790980i	$-0.290429\pi$
$577$	29.3939	1.22368	0.611842	+	0.790980i	$0.290429\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	24.2487i	1.00601i
$582$	0	0
$583$	−19.5959	−0.811580
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.89949i	− 0.408596i	−0.978909	−	0.204298i	$-0.934509\pi$
$587$	− 9.89949i	− 0.408596i	0.978909	−	0.204298i	$-0.0654911\pi$
$588$	0	0
$589$	− 13.8564i	− 0.570943i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.79796	−0.402354	−0.201177	−	0.979555i	$-0.564477\pi$
$593$	−9.79796	−0.402354	−0.201177	+	0.979555i	$0.564477\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7.34847	−0.298265	−0.149133	−	0.988817i	$-0.547648\pi$
$607$	−7.34847	−0.298265	−0.149133	+	0.988817i	$0.547648\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 33.9411i	− 1.37087i	−0.728134	−	0.685435i	$-0.759612\pi$
$613$	− 33.9411i	− 1.37087i	0.728134	−	0.685435i	$-0.240388\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−34.2929	−1.38058	−0.690289	−	0.723534i	$-0.742517\pi$
$617$	−34.2929	−1.38058	−0.690289	+	0.723534i	$0.742517\pi$
$618$	0	0
$619$	− 10.3923i	− 0.417702i	−0.977947	−	0.208851i	$-0.933028\pi$
$619$	− 10.3923i	− 0.417702i	0.977947	−	0.208851i	$-0.0669724\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−14.6969	−0.588820
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	41.5692i	1.65747i
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	− 29.6985i	− 1.17119i	−0.810602	−	0.585597i	$-0.800860\pi$
$643$	− 29.6985i	− 1.17119i	0.810602	−	0.585597i	$-0.199140\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.2474	0.481497	0.240748	−	0.970588i	$-0.422607\pi$
$647$	12.2474	0.481497	0.240748	+	0.970588i	$0.422607\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 11.3137i	− 0.442740i	−0.975190	−	0.221370i	$-0.928947\pi$
$653$	− 11.3137i	− 0.442740i	0.975190	−	0.221370i	$-0.0710528\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 24.2487i	− 0.944596i	−0.881439	−	0.472298i	$-0.843425\pi$
$659$	− 24.2487i	− 0.944596i	0.881439	−	0.472298i	$-0.156575\pi$
$660$	0	0
$661$	− 10.3923i	− 0.404214i	−0.979363	−	0.202107i	$-0.935221\pi$
$661$	− 10.3923i	− 0.404214i	0.979363	−	0.202107i	$-0.0647788\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	34.2929	1.32189	0.660946	−	0.750433i	$-0.270155\pi$
$673$	34.2929	1.32189	0.660946	+	0.750433i	$0.270155\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.6274i	0.869642i	0.900517	+	0.434821i	$0.143188\pi$
$677$	22.6274i	0.869642i	−0.900517	+	0.434821i	$0.856812\pi$
$678$	0	0
$679$	−12.0000	−0.460518
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.5563i	0.595247i	0.954683	+	0.297624i	$0.0961940\pi$
$683$	15.5563i	0.595247i	−0.954683	+	0.297624i	$0.903806\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 3.46410i	− 0.131781i	−0.997827	−	0.0658903i	$-0.979011\pi$
$691$	− 3.46410i	− 0.131781i	0.997827	−	0.0658903i	$-0.0209887\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 24.2487i	− 0.915861i	−0.888988	−	0.457931i	$-0.848591\pi$
$701$	− 24.2487i	− 0.915861i	0.888988	−	0.457931i	$-0.151409\pi$
$702$	0	0
$703$	−29.3939	−1.10861
$704$	0	0
$705$	0	0
$706$	0	0
$707$	33.9411i	1.27649i
$708$	0	0
$709$	41.5692i	1.56116i	0.625053	+	0.780582i	$0.285077\pi$
$709$	41.5692i	1.56116i	−0.625053	+	0.780582i	$0.714923\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9.79796	−0.366936
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	46.5403	1.72608	0.863042	−	0.505132i	$-0.168556\pi$
$727$	46.5403	1.72608	0.863042	+	0.505132i	$0.168556\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 20.7846i	− 0.768747i
$732$	0	0
$733$	42.4264i	1.56706i	0.621357	+	0.783528i	$0.286582\pi$
$733$	42.4264i	1.56706i	−0.621357	+	0.783528i	$0.713418\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14.6969	−0.541369
$738$	0	0
$739$	− 3.46410i	− 0.127429i	−0.997968	−	0.0637145i	$-0.979705\pi$
$739$	− 3.46410i	− 0.127429i	0.997968	−	0.0637145i	$-0.0202947\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−51.4393	−1.88712	−0.943562	−	0.331195i	$-0.892548\pi$
$743$	−51.4393	−1.88712	−0.943562	+	0.331195i	$0.892548\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.46410i	0.126576i
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.4558i	0.925208i	0.886565	+	0.462604i	$0.153085\pi$
$757$	25.4558i	0.925208i	−0.886565	+	0.462604i	$0.846915\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	− 8.48528i	− 0.307188i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 28.2843i	− 1.01731i	−0.860969	−	0.508657i	$-0.830142\pi$
$773$	− 28.2843i	− 1.01731i	0.860969	−	0.508657i	$-0.169858\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	41.5692i	1.48746i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 21.2132i	− 0.756169i	−0.925771	−	0.378085i	$-0.876583\pi$
$787$	− 21.2132i	− 0.756169i	0.925771	−	0.378085i	$-0.123417\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.5980i	1.40263i	0.712850	+	0.701316i	$0.247404\pi$
$797$	39.5980i	1.40263i	−0.712850	+	0.701316i	$0.752596\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 16.9706i	− 0.598878i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	− 10.3923i	− 0.364923i	−0.983213	−	0.182462i	$-0.941593\pi$
$811$	− 10.3923i	− 0.364923i	0.983213	−	0.182462i	$-0.0584065\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	14.6969	0.514181
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 31.1769i	− 1.08808i	−0.839059	−	0.544041i	$-0.816894\pi$
$821$	− 31.1769i	− 1.08808i	0.839059	−	0.544041i	$-0.183106\pi$
$822$	0	0
$823$	26.9444	0.939222	0.469611	−	0.882873i	$-0.344394\pi$
$823$	26.9444	0.939222	0.469611	+	0.882873i	$0.344394\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.3553i	1.22943i	0.788751	+	0.614713i	$0.210728\pi$
$827$	35.3553i	1.22943i	−0.788751	+	0.614713i	$0.789272\pi$
$828$	0	0
$829$	10.3923i	0.360940i	0.983581	+	0.180470i	$0.0577618\pi$
$829$	10.3923i	0.360940i	−0.983581	+	0.180470i	$0.942238\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.89898	−0.169740
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.44949	0.0841655
$848$	0	0
$849$	0	0
$850$	0	0
$851$	20.7846i	0.712487i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.1918	1.33877	0.669384	−	0.742917i	$-0.266558\pi$
$857$	39.1918	1.33877	0.669384	+	0.742917i	$0.266558\pi$
$858$	0	0
$859$	3.46410i	0.118194i	0.998252	+	0.0590968i	$0.0188221\pi$
$859$	3.46410i	0.118194i	−0.998252	+	0.0590968i	$0.981178\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.44949	0.0833816	0.0416908	−	0.999131i	$-0.486726\pi$
$863$	2.44949	0.0833816	0.0416908	+	0.999131i	$0.486726\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	13.8564i	0.470046i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.48528i	0.286528i	0.989685	+	0.143264i	$0.0457597\pi$
$877$	8.48528i	0.286528i	−0.989685	+	0.143264i	$0.954240\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	− 4.24264i	− 0.142776i	−0.997449	−	0.0713881i	$-0.977257\pi$
$883$	− 4.24264i	− 0.142776i	0.997449	−	0.0713881i	$-0.0227429\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−26.9444	−0.904704	−0.452352	−	0.891839i	$-0.649415\pi$
$887$	−26.9444	−0.904704	−0.452352	+	0.891839i	$0.649415\pi$
$888$	0	0
$889$	−42.0000	−1.40863
$890$	0	0
$891$	0	0
$892$	0	0
$893$	25.4558i	0.851847i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 27.7128i	− 0.923248i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 4.24264i	− 0.140875i	−0.997516	−	0.0704373i	$-0.977561\pi$
$907$	− 4.24264i	− 0.140875i	0.997516	−	0.0704373i	$-0.0224395\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−34.2929	−1.13493
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8.48528i	− 0.280209i
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	− 3.46410i	− 0.113531i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.89898	0.160043	0.0800213	−	0.996793i	$-0.474501\pi$
$937$	4.89898	0.160043	0.0800213	+	0.996793i	$0.474501\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 13.8564i	− 0.451706i	−0.974161	−	0.225853i	$-0.927483\pi$
$941$	− 13.8564i	− 0.451706i	0.974161	−	0.225853i	$-0.0725169\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 41.0122i	− 1.33272i	−0.745631	−	0.666359i	$-0.767852\pi$
$947$	− 41.0122i	− 1.33272i	0.745631	−	0.666359i	$-0.232148\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−17.1464	−0.551392	−0.275696	−	0.961245i	$-0.588908\pi$
$967$	−17.1464	−0.551392	−0.275696	+	0.961245i	$0.588908\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.46410i	0.111168i	0.998454	+	0.0555842i	$0.0177021\pi$
$971$	3.46410i	0.111168i	−0.998454	+	0.0555842i	$0.982298\pi$
$972$	0	0
$973$	− 25.4558i	− 0.816077i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−44.0908	−1.41059	−0.705295	−	0.708914i	$-0.749185\pi$
$977$	−44.0908	−1.41059	−0.705295	+	0.708914i	$0.749185\pi$
$978$	0	0
$979$	− 20.7846i	− 0.664279i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	56.3383	1.79691	0.898456	−	0.439064i	$-0.144690\pi$
$983$	56.3383	1.79691	0.898456	+	0.439064i	$0.144690\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 10.3923i	− 0.330456i
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	50.9117i	1.61239i	0.591650	+	0.806195i	$0.298477\pi$
$997$	50.9117i	1.61239i	−0.591650	+	0.806195i	$0.701523\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.k.l.3601.1		4
3.2	odd	2		800.2.d.f.401.3		4
4.3	odd	2		1800.2.k.m.901.4		4
5.2	odd	4		1440.2.d.c.1009.1		4
5.3	odd	4		1440.2.d.c.1009.3		4
5.4	even	2	inner	7200.2.k.l.3601.3		4
8.3	odd	2		1800.2.k.m.901.3		4
8.5	even	2	inner	7200.2.k.l.3601.2		4
12.11	even	2		200.2.d.e.101.1		4
15.2	even	4		160.2.f.a.49.4		4
15.8	even	4		160.2.f.a.49.2		4
15.14	odd	2		800.2.d.f.401.2		4
20.3	even	4		360.2.d.b.109.3		4
20.7	even	4		360.2.d.b.109.2		4
20.19	odd	2		1800.2.k.m.901.1		4
24.5	odd	2		800.2.d.f.401.1		4
24.11	even	2		200.2.d.e.101.2		4
40.3	even	4		360.2.d.b.109.1		4
40.13	odd	4		1440.2.d.c.1009.2		4
40.19	odd	2		1800.2.k.m.901.2		4
40.27	even	4		360.2.d.b.109.4		4
40.29	even	2	inner	7200.2.k.l.3601.4		4
40.37	odd	4		1440.2.d.c.1009.4		4
48.5	odd	4		6400.2.a.cm.1.4		4
48.11	even	4		6400.2.a.co.1.1		4
48.29	odd	4		6400.2.a.cm.1.2		4
48.35	even	4		6400.2.a.co.1.3		4
60.23	odd	4		40.2.f.a.29.2	yes	4
60.47	odd	4		40.2.f.a.29.3	yes	4
60.59	even	2		200.2.d.e.101.4		4
120.29	odd	2		800.2.d.f.401.4		4
120.53	even	4		160.2.f.a.49.3		4
120.59	even	2		200.2.d.e.101.3		4
120.77	even	4		160.2.f.a.49.1		4
120.83	odd	4		40.2.f.a.29.4	yes	4
120.107	odd	4		40.2.f.a.29.1	✓	4
240.29	odd	4		6400.2.a.cm.1.3		4
240.53	even	4		1280.2.c.k.769.3		4
240.59	even	4		6400.2.a.co.1.4		4
240.77	even	4		1280.2.c.k.769.4		4
240.83	odd	4		1280.2.c.i.769.4		4
240.107	odd	4		1280.2.c.i.769.3		4
240.149	odd	4		6400.2.a.cm.1.1		4
240.173	even	4		1280.2.c.k.769.2		4
240.179	even	4		6400.2.a.co.1.2		4
240.197	even	4		1280.2.c.k.769.1		4
240.203	odd	4		1280.2.c.i.769.1		4
240.227	odd	4		1280.2.c.i.769.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.f.a.29.1	✓	4	120.107	odd	4
40.2.f.a.29.2	yes	4	60.23	odd	4
40.2.f.a.29.3	yes	4	60.47	odd	4
40.2.f.a.29.4	yes	4	120.83	odd	4
160.2.f.a.49.1		4	120.77	even	4
160.2.f.a.49.2		4	15.8	even	4
160.2.f.a.49.3		4	120.53	even	4
160.2.f.a.49.4		4	15.2	even	4
200.2.d.e.101.1		4	12.11	even	2
200.2.d.e.101.2		4	24.11	even	2
200.2.d.e.101.3		4	120.59	even	2
200.2.d.e.101.4		4	60.59	even	2
360.2.d.b.109.1		4	40.3	even	4
360.2.d.b.109.2		4	20.7	even	4
360.2.d.b.109.3		4	20.3	even	4
360.2.d.b.109.4		4	40.27	even	4
800.2.d.f.401.1		4	24.5	odd	2
800.2.d.f.401.2		4	15.14	odd	2
800.2.d.f.401.3		4	3.2	odd	2
800.2.d.f.401.4		4	120.29	odd	2
1280.2.c.i.769.1		4	240.203	odd	4
1280.2.c.i.769.2		4	240.227	odd	4
1280.2.c.i.769.3		4	240.107	odd	4
1280.2.c.i.769.4		4	240.83	odd	4
1280.2.c.k.769.1		4	240.197	even	4
1280.2.c.k.769.2		4	240.173	even	4
1280.2.c.k.769.3		4	240.53	even	4
1280.2.c.k.769.4		4	240.77	even	4
1440.2.d.c.1009.1		4	5.2	odd	4
1440.2.d.c.1009.2		4	40.13	odd	4
1440.2.d.c.1009.3		4	5.3	odd	4
1440.2.d.c.1009.4		4	40.37	odd	4
1800.2.k.m.901.1		4	20.19	odd	2
1800.2.k.m.901.2		4	40.19	odd	2
1800.2.k.m.901.3		4	8.3	odd	2
1800.2.k.m.901.4		4	4.3	odd	2
6400.2.a.cm.1.1		4	240.149	odd	4
6400.2.a.cm.1.2		4	48.29	odd	4
6400.2.a.cm.1.3		4	240.29	odd	4
6400.2.a.cm.1.4		4	48.5	odd	4
6400.2.a.co.1.1		4	48.11	even	4
6400.2.a.co.1.2		4	240.179	even	4
6400.2.a.co.1.3		4	48.35	even	4
6400.2.a.co.1.4		4	240.59	even	4
7200.2.k.l.3601.1		4	1.1	even	1	trivial
7200.2.k.l.3601.2		4	8.5	even	2	inner
7200.2.k.l.3601.3		4	5.4	even	2	inner
7200.2.k.l.3601.4		4	40.29	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.44949 q^{7} +O(q^{10})\)	\(q-2.44949 q^{7} -3.46410i q^{11} +4.89898 q^{17} +3.46410i q^{19} +2.44949 q^{23} -4.00000 q^{31} +8.48528i q^{37} -4.24264i q^{43} +7.34847 q^{47} -1.00000 q^{49} -5.65685i q^{53} +10.3923i q^{59} +3.46410i q^{61} -4.24264i q^{67} -12.0000 q^{71} +4.89898 q^{73} +8.48528i q^{77} -4.00000 q^{79} -9.89949i q^{83} +6.00000 q^{89} +4.89898 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 16 q^{31} - 4 q^{49} - 48 q^{71} - 16 q^{79} + 24 q^{89}+O(q^{100}) \) 4 * q - 16 * q^31 - 4 * q^49 - 48 * q^71 - 16 * q^79 + 24 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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