LMFDB - Embedded newform 7200.2.a.ct.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(1,7200)]

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, 0, 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("7200.1");

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.a (trivial)

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:	yes
Analytic conductor:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.93185$ of defining polynomial
Character	$\chi$	$=$	7200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{7} +O(q^{10})$	$q-2.82843 q^{7} -4.89898 q^{11} -4.89898 q^{13} -3.46410 q^{17} -6.92820 q^{19} +4.00000 q^{23} -8.48528 q^{29} +6.92820 q^{31} -4.89898 q^{37} -5.65685 q^{41} +11.3137 q^{43} +4.00000 q^{47} +1.00000 q^{49} -3.46410 q^{53} -4.89898 q^{59} -6.00000 q^{61} -5.65685 q^{67} +9.79796 q^{71} +13.8564 q^{77} -6.92820 q^{79} +16.0000 q^{83} +13.8564 q^{91} +9.79796 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 16 q^{23} + 16 q^{47} + 4 q^{49} - 24 q^{61} + 64 q^{83}+O(q^{100}) $ 4 * q + 16 * q^23 + 16 * q^47 + 4 * q^49 - 24 * q^61 + 64 * q^83

Display 100 coefficients 10 coefficients

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
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.89898	−1.47710	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898	−1.47710	−0.738549	+	0.674200i	$0.764489\pi$
$12$	0	0
$13$	−4.89898	−1.35873	−0.679366	−	0.733799i	$-0.737745\pi$
$13$	−4.89898	−1.35873	−0.679366	+	0.733799i	$0.737745\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	−6.92820	−1.58944	−0.794719	−	0.606977i	$-0.792382\pi$
$19$	−6.92820	−1.58944	−0.794719	+	0.606977i	$0.792382\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.48528	−1.57568	−0.787839	−	0.615882i	$-0.788800\pi$
$29$	−8.48528	−1.57568	−0.787839	+	0.615882i	$0.788800\pi$
$30$	0	0
$31$	6.92820	1.24434	0.622171	−	0.782881i	$-0.286251\pi$
$31$	6.92820	1.24434	0.622171	+	0.782881i	$0.286251\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.89898	−0.805387	−0.402694	−	0.915335i	$-0.631926\pi$
$37$	−4.89898	−0.805387	−0.402694	+	0.915335i	$0.631926\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	0	0
$43$	11.3137	1.72532	0.862662	−	0.505781i	$-0.168795\pi$
$43$	11.3137	1.72532	0.862662	+	0.505781i	$0.168795\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.46410	−0.475831	−0.237915	−	0.971286i	$-0.576464\pi$
$53$	−3.46410	−0.475831	−0.237915	+	0.971286i	$0.576464\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.89898	−0.637793	−0.318896	−	0.947790i	$-0.603312\pi$
$59$	−4.89898	−0.637793	−0.318896	+	0.947790i	$0.603312\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.79796	1.16280	0.581402	−	0.813617i	$-0.302504\pi$
$71$	9.79796	1.16280	0.581402	+	0.813617i	$0.302504\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13.8564	1.57908
$78$	0	0
$79$	−6.92820	−0.779484	−0.389742	−	0.920924i	$-0.627436\pi$
$79$	−6.92820	−0.779484	−0.389742	+	0.920924i	$0.627436\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	13.8564	1.45255
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.79796	0.994832	0.497416	−	0.867512i	$-0.334282\pi$
$97$	9.79796	0.994832	0.497416	+	0.867512i	$0.334282\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.48528	0.844317	0.422159	−	0.906522i	$-0.361273\pi$
$101$	8.48528	0.844317	0.422159	+	0.906522i	$0.361273\pi$
$102$	0	0
$103$	−2.82843	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	17.3205	1.62938	0.814688	−	0.579899i	$-0.196908\pi$
$113$	17.3205	1.62938	0.814688	+	0.579899i	$0.196908\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	9.79796	0.898177
$120$	0	0
$121$	13.0000	1.18182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.1421	−1.25491	−0.627456	−	0.778652i	$-0.715904\pi$
$127$	−14.1421	−1.25491	−0.627456	+	0.778652i	$0.715904\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.89898	0.428026	0.214013	−	0.976831i	$-0.431347\pi$
$131$	4.89898	0.428026	0.214013	+	0.976831i	$0.431347\pi$
$132$	0	0
$133$	19.5959	1.69918
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.3923	−0.887875	−0.443937	−	0.896058i	$-0.646419\pi$
$137$	−10.3923	−0.887875	−0.443937	+	0.896058i	$0.646419\pi$
$138$	0	0
$139$	6.92820	0.587643	0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820	0.587643	0.293821	+	0.955860i	$0.405073\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	24.0000	2.00698
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.82843	0.231714	0.115857	−	0.993266i	$-0.463039\pi$
$149$	2.82843	0.231714	0.115857	+	0.993266i	$0.463039\pi$
$150$	0	0
$151$	−6.92820	−0.563809	−0.281905	−	0.959442i	$-0.590966\pi$
$151$	−6.92820	−0.563809	−0.281905	+	0.959442i	$0.590966\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.89898	0.390981	0.195491	−	0.980706i	$-0.437370\pi$
$157$	4.89898	0.390981	0.195491	+	0.980706i	$0.437370\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−11.3137	−0.891645
$162$	0	0
$163$	5.65685	0.443079	0.221540	−	0.975151i	$-0.428892\pi$
$163$	5.65685	0.443079	0.221540	+	0.975151i	$0.428892\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	0	0
$169$	11.0000	0.846154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−24.2487	−1.84360	−0.921798	−	0.387671i	$-0.873280\pi$
$173$	−24.2487	−1.84360	−0.921798	+	0.387671i	$0.873280\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.6969	1.09850	0.549250	−	0.835658i	$-0.314913\pi$
$179$	14.6969	1.09850	0.549250	+	0.835658i	$0.314913\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.9706	1.24101
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.79796	−0.708955	−0.354478	−	0.935064i	$-0.615341\pi$
$191$	−9.79796	−0.708955	−0.354478	+	0.935064i	$0.615341\pi$
$192$	0	0
$193$	9.79796	0.705273	0.352636	−	0.935760i	$-0.385285\pi$
$193$	9.79796	0.705273	0.352636	+	0.935760i	$0.385285\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.3923	−0.740421	−0.370211	−	0.928948i	$-0.620714\pi$
$197$	−10.3923	−0.740421	−0.370211	+	0.928948i	$0.620714\pi$
$198$	0	0
$199$	−6.92820	−0.491127	−0.245564	−	0.969380i	$-0.578973\pi$
$199$	−6.92820	−0.491127	−0.245564	+	0.969380i	$0.578973\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	24.0000	1.68447
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	33.9411	2.34776
$210$	0	0
$211$	−20.7846	−1.43087	−0.715436	−	0.698679i	$-0.753772\pi$
$211$	−20.7846	−1.43087	−0.715436	+	0.698679i	$0.753772\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−19.5959	−1.33026
$218$	0	0
$219$	0	0
$220$	0	0
$221$	16.9706	1.14156
$222$	0	0
$223$	−14.1421	−0.947027	−0.473514	−	0.880786i	$-0.657015\pi$
$223$	−14.1421	−0.947027	−0.473514	+	0.880786i	$0.657015\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.0000	1.06196	0.530979	−	0.847385i	$-0.321824\pi$
$227$	16.0000	1.06196	0.530979	+	0.847385i	$0.321824\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.46410	−0.226941	−0.113470	−	0.993541i	$-0.536197\pi$
$233$	−3.46410	−0.226941	−0.113470	+	0.993541i	$0.536197\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−29.3939	−1.90133	−0.950666	−	0.310217i	$-0.899598\pi$
$239$	−29.3939	−1.90133	−0.950666	+	0.310217i	$0.899598\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	33.9411	2.15962
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.89898	0.309221	0.154610	−	0.987976i	$-0.450588\pi$
$251$	4.89898	0.309221	0.154610	+	0.987976i	$0.450588\pi$
$252$	0	0
$253$	−19.5959	−1.23198
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.3923	−0.648254	−0.324127	−	0.946014i	$-0.605071\pi$
$257$	−10.3923	−0.648254	−0.324127	+	0.946014i	$0.605071\pi$
$258$	0	0
$259$	13.8564	0.860995
$260$	0	0
$261$	0	0
$262$	0	0
$263$	28.0000	1.72655	0.863277	−	0.504730i	$-0.168408\pi$
$263$	28.0000	1.72655	0.863277	+	0.504730i	$0.168408\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.82843	−0.172452	−0.0862261	−	0.996276i	$-0.527481\pi$
$269$	−2.82843	−0.172452	−0.0862261	+	0.996276i	$0.527481\pi$
$270$	0	0
$271$	−6.92820	−0.420858	−0.210429	−	0.977609i	$-0.567486\pi$
$271$	−6.92820	−0.420858	−0.210429	+	0.977609i	$0.567486\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.4949	1.47176	0.735878	−	0.677114i	$-0.236770\pi$
$277$	24.4949	1.47176	0.735878	+	0.677114i	$0.236770\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.6274	1.34984	0.674919	−	0.737892i	$-0.264178\pi$
$281$	22.6274	1.34984	0.674919	+	0.737892i	$0.264178\pi$
$282$	0	0
$283$	−28.2843	−1.68133	−0.840663	−	0.541559i	$-0.817834\pi$
$283$	−28.2843	−1.68133	−0.840663	+	0.541559i	$0.817834\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.0000	0.944450
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.3205	1.01187	0.505937	−	0.862570i	$-0.331147\pi$
$293$	17.3205	1.01187	0.505937	+	0.862570i	$0.331147\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−19.5959	−1.13326
$300$	0	0
$301$	−32.0000	−1.84445
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−11.3137	−0.645707	−0.322854	−	0.946449i	$-0.604642\pi$
$307$	−11.3137	−0.645707	−0.322854	+	0.946449i	$0.604642\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−29.3939	−1.66677	−0.833387	−	0.552690i	$-0.813601\pi$
$311$	−29.3939	−1.66677	−0.833387	+	0.552690i	$0.813601\pi$
$312$	0	0
$313$	−29.3939	−1.66144	−0.830720	−	0.556690i	$-0.812071\pi$
$313$	−29.3939	−1.66144	−0.830720	+	0.556690i	$0.812071\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.2487	1.36194	0.680972	−	0.732310i	$-0.261558\pi$
$317$	24.2487	1.36194	0.680972	+	0.732310i	$0.261558\pi$
$318$	0	0
$319$	41.5692	2.32743
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.0000	1.33540
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−11.3137	−0.623745
$330$	0	0
$331$	−20.7846	−1.14243	−0.571213	−	0.820802i	$-0.693527\pi$
$331$	−20.7846	−1.14243	−0.571213	+	0.820802i	$0.693527\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−19.5959	−1.06746	−0.533729	−	0.845656i	$-0.679210\pi$
$337$	−19.5959	−1.06746	−0.533729	+	0.845656i	$0.679210\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−33.9411	−1.83801
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.3923	0.553127	0.276563	−	0.960996i	$-0.410804\pi$
$353$	10.3923	0.553127	0.276563	+	0.960996i	$0.410804\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.5959	1.03423	0.517116	−	0.855915i	$-0.327005\pi$
$359$	19.5959	1.03423	0.517116	+	0.855915i	$0.327005\pi$
$360$	0	0
$361$	29.0000	1.52632
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−2.82843	−0.147643	−0.0738213	−	0.997271i	$-0.523519\pi$
$367$	−2.82843	−0.147643	−0.0738213	+	0.997271i	$0.523519\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.79796	0.508685
$372$	0	0
$373$	−14.6969	−0.760979	−0.380489	−	0.924785i	$-0.624244\pi$
$373$	−14.6969	−0.760979	−0.380489	+	0.924785i	$0.624244\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	41.5692	2.14092
$378$	0	0
$379$	−6.92820	−0.355878	−0.177939	−	0.984042i	$-0.556943\pi$
$379$	−6.92820	−0.355878	−0.177939	+	0.984042i	$0.556943\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14.1421	0.717035	0.358517	−	0.933523i	$-0.383282\pi$
$389$	14.1421	0.717035	0.358517	+	0.933523i	$0.383282\pi$
$390$	0	0
$391$	−13.8564	−0.700749
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−24.4949	−1.22936	−0.614682	−	0.788775i	$-0.710716\pi$
$397$	−24.4949	−1.22936	−0.614682	+	0.788775i	$0.710716\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.9706	−0.847469	−0.423735	−	0.905786i	$-0.639281\pi$
$401$	−16.9706	−0.847469	−0.423735	+	0.905786i	$0.639281\pi$
$402$	0	0
$403$	−33.9411	−1.69073
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	13.8564	0.681829
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24.4949	1.19665	0.598327	−	0.801252i	$-0.295832\pi$
$419$	24.4949	1.19665	0.598327	+	0.801252i	$0.295832\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	16.9706	0.821263
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	9.79796	0.470860	0.235430	−	0.971891i	$-0.424350\pi$
$433$	9.79796	0.470860	0.235430	+	0.971891i	$0.424350\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−27.7128	−1.32568
$438$	0	0
$439$	20.7846	0.991995	0.495998	−	0.868324i	$-0.334802\pi$
$439$	20.7846	0.991995	0.495998	+	0.868324i	$0.334802\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	32.0000	1.52037	0.760183	−	0.649709i	$-0.225109\pi$
$443$	32.0000	1.52037	0.760183	+	0.649709i	$0.225109\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−39.5980	−1.86874	−0.934372	−	0.356299i	$-0.884039\pi$
$449$	−39.5980	−1.86874	−0.934372	+	0.356299i	$0.884039\pi$
$450$	0	0
$451$	27.7128	1.30495
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	29.3939	1.37499	0.687494	−	0.726190i	$-0.258711\pi$
$457$	29.3939	1.37499	0.687494	+	0.726190i	$0.258711\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	25.4558	1.18560	0.592798	−	0.805351i	$-0.298023\pi$
$461$	25.4558	1.18560	0.592798	+	0.805351i	$0.298023\pi$
$462$	0	0
$463$	2.82843	0.131448	0.0657241	−	0.997838i	$-0.479064\pi$
$463$	2.82843	0.131448	0.0657241	+	0.997838i	$0.479064\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−55.4256	−2.54847
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9.79796	0.447680	0.223840	−	0.974626i	$-0.428141\pi$
$479$	9.79796	0.447680	0.223840	+	0.974626i	$0.428141\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.82843	0.128168	0.0640841	−	0.997944i	$-0.479587\pi$
$487$	2.82843	0.128168	0.0640841	+	0.997944i	$0.479587\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.89898	0.221088	0.110544	−	0.993871i	$-0.464741\pi$
$491$	4.89898	0.221088	0.110544	+	0.993871i	$0.464741\pi$
$492$	0	0
$493$	29.3939	1.32383
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−27.7128	−1.24309
$498$	0	0
$499$	6.92820	0.310149	0.155074	−	0.987903i	$-0.450438\pi$
$499$	6.92820	0.310149	0.155074	+	0.987903i	$0.450438\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.0000	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8.48528	−0.376103	−0.188052	−	0.982159i	$-0.560217\pi$
$509$	−8.48528	−0.376103	−0.188052	+	0.982159i	$0.560217\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−19.5959	−0.861827
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16.9706	0.743494	0.371747	−	0.928334i	$-0.378759\pi$
$521$	16.9706	0.743494	0.371747	+	0.928334i	$0.378759\pi$
$522$	0	0
$523$	28.2843	1.23678	0.618392	−	0.785869i	$-0.287784\pi$
$523$	28.2843	1.23678	0.618392	+	0.785869i	$0.287784\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24.0000	−1.04546
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	27.7128	1.20038
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.89898	−0.211014
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−11.3137	−0.483739	−0.241870	−	0.970309i	$-0.577761\pi$
$547$	−11.3137	−0.483739	−0.241870	+	0.970309i	$0.577761\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	58.7878	2.50444
$552$	0	0
$553$	19.5959	0.833303
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.1769	−1.32101	−0.660504	−	0.750822i	$-0.729657\pi$
$557$	−31.1769	−1.32101	−0.660504	+	0.750822i	$0.729657\pi$
$558$	0	0
$559$	−55.4256	−2.34425
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.00000	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$563$	−8.00000	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−39.5980	−1.66003	−0.830017	−	0.557738i	$-0.811669\pi$
$569$	−39.5980	−1.66003	−0.830017	+	0.557738i	$0.811669\pi$
$570$	0	0
$571$	−20.7846	−0.869809	−0.434904	−	0.900477i	$-0.643218\pi$
$571$	−20.7846	−0.869809	−0.434904	+	0.900477i	$0.643218\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	39.1918	1.63158	0.815789	−	0.578350i	$-0.196303\pi$
$577$	39.1918	1.63158	0.815789	+	0.578350i	$0.196303\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−45.2548	−1.87749
$582$	0	0
$583$	16.9706	0.702849
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.0000	1.65098	0.825488	−	0.564419i	$-0.190900\pi$
$587$	40.0000	1.65098	0.825488	+	0.564419i	$0.190900\pi$
$588$	0	0
$589$	−48.0000	−1.97781
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−10.3923	−0.426761	−0.213380	−	0.976969i	$-0.568447\pi$
$593$	−10.3923	−0.426761	−0.213380	+	0.976969i	$0.568447\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−19.5959	−0.800668	−0.400334	−	0.916369i	$-0.631106\pi$
$599$	−19.5959	−0.800668	−0.400334	+	0.916369i	$0.631106\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.82843	0.114802	0.0574012	−	0.998351i	$-0.481719\pi$
$607$	2.82843	0.114802	0.0574012	+	0.998351i	$0.481719\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−19.5959	−0.792766
$612$	0	0
$613$	−4.89898	−0.197868	−0.0989340	−	0.995094i	$-0.531543\pi$
$613$	−4.89898	−0.197868	−0.0989340	+	0.995094i	$0.531543\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.2487	0.976216	0.488108	−	0.872783i	$-0.337687\pi$
$617$	24.2487	0.976216	0.488108	+	0.872783i	$0.337687\pi$
$618$	0	0
$619$	−20.7846	−0.835404	−0.417702	−	0.908584i	$-0.637164\pi$
$619$	−20.7846	−0.835404	−0.417702	+	0.908584i	$0.637164\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.9706	0.676661
$630$	0	0
$631$	48.4974	1.93065	0.965326	−	0.261048i	$-0.0840679\pi$
$631$	48.4974	1.93065	0.965326	+	0.261048i	$0.0840679\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.89898	−0.194105
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5.65685	−0.223432	−0.111716	−	0.993740i	$-0.535635\pi$
$641$	−5.65685	−0.223432	−0.111716	+	0.993740i	$0.535635\pi$
$642$	0	0
$643$	−5.65685	−0.223085	−0.111542	−	0.993760i	$-0.535579\pi$
$643$	−5.65685	−0.223085	−0.111542	+	0.993760i	$0.535579\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.00000	−0.157256	−0.0786281	−	0.996904i	$-0.525054\pi$
$647$	−4.00000	−0.157256	−0.0786281	+	0.996904i	$0.525054\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−17.3205	−0.677804	−0.338902	−	0.940822i	$-0.610055\pi$
$653$	−17.3205	−0.677804	−0.338902	+	0.940822i	$0.610055\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−34.2929	−1.33586	−0.667930	−	0.744224i	$-0.732819\pi$
$659$	−34.2929	−1.33586	−0.667930	+	0.744224i	$0.732819\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−33.9411	−1.31421
$668$	0	0
$669$	0	0
$670$	0	0
$671$	29.3939	1.13474
$672$	0	0
$673$	9.79796	0.377684	0.188842	−	0.982008i	$-0.439527\pi$
$673$	9.79796	0.377684	0.188842	+	0.982008i	$0.439527\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.46410	−0.133136	−0.0665681	−	0.997782i	$-0.521205\pi$
$677$	−3.46410	−0.133136	−0.0665681	+	0.997782i	$0.521205\pi$
$678$	0	0
$679$	−27.7128	−1.06352
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.00000	0.306111	0.153056	−	0.988218i	$-0.451089\pi$
$683$	8.00000	0.306111	0.153056	+	0.988218i	$0.451089\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	16.9706	0.646527
$690$	0	0
$691$	34.6410	1.31781	0.658903	−	0.752228i	$-0.271021\pi$
$691$	34.6410	1.31781	0.658903	+	0.752228i	$0.271021\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	19.5959	0.742248
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−31.1127	−1.17511	−0.587555	−	0.809184i	$-0.699909\pi$
$701$	−31.1127	−1.17511	−0.587555	+	0.809184i	$0.699909\pi$
$702$	0	0
$703$	33.9411	1.28011
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−24.0000	−0.902613
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	27.7128	1.03785
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	19.5959	0.730804	0.365402	−	0.930850i	$-0.380931\pi$
$719$	19.5959	0.730804	0.365402	+	0.930850i	$0.380931\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−31.1127	−1.15391	−0.576953	−	0.816777i	$-0.695758\pi$
$727$	−31.1127	−1.15391	−0.576953	+	0.816777i	$0.695758\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−39.1918	−1.44956
$732$	0	0
$733$	24.4949	0.904740	0.452370	−	0.891830i	$-0.350579\pi$
$733$	24.4949	0.904740	0.452370	+	0.891830i	$0.350579\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	27.7128	1.02081
$738$	0	0
$739$	−20.7846	−0.764574	−0.382287	−	0.924044i	$-0.624863\pi$
$739$	−20.7846	−0.764574	−0.382287	+	0.924044i	$0.624863\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−22.6274	−0.826788
$750$	0	0
$751$	−20.7846	−0.758441	−0.379221	−	0.925306i	$-0.623808\pi$
$751$	−20.7846	−0.758441	−0.379221	+	0.925306i	$0.623808\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.89898	0.178056	0.0890282	−	0.996029i	$-0.471624\pi$
$757$	4.89898	0.178056	0.0890282	+	0.996029i	$0.471624\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	50.9117	1.84555	0.922774	−	0.385342i	$-0.125917\pi$
$761$	50.9117	1.84555	0.922774	+	0.385342i	$0.125917\pi$
$762$	0	0
$763$	16.9706	0.614376
$764$	0	0
$765$	0	0
$766$	0	0
$767$	24.0000	0.866590
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	38.1051	1.37055	0.685273	−	0.728286i	$-0.259683\pi$
$773$	38.1051	1.37055	0.685273	+	0.728286i	$0.259683\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	39.1918	1.40419
$780$	0	0
$781$	−48.0000	−1.71758
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−28.2843	−1.00823	−0.504113	−	0.863638i	$-0.668180\pi$
$787$	−28.2843	−1.00823	−0.504113	+	0.863638i	$0.668180\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−48.9898	−1.74188
$792$	0	0
$793$	29.3939	1.04381
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.46410	0.122705	0.0613524	−	0.998116i	$-0.480459\pi$
$797$	3.46410	0.122705	0.0613524	+	0.998116i	$0.480459\pi$
$798$	0	0
$799$	−13.8564	−0.490204
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16.9706	0.596653	0.298327	−	0.954464i	$-0.403572\pi$
$809$	16.9706	0.596653	0.298327	+	0.954464i	$0.403572\pi$
$810$	0	0
$811$	34.6410	1.21641	0.608205	−	0.793780i	$-0.291890\pi$
$811$	34.6410	1.21641	0.608205	+	0.793780i	$0.291890\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−78.3837	−2.74230
$818$	0	0
$819$	0	0
$820$	0	0
$821$	36.7696	1.28327	0.641633	−	0.767012i	$-0.278257\pi$
$821$	36.7696	1.28327	0.641633	+	0.767012i	$0.278257\pi$
$822$	0	0
$823$	−19.7990	−0.690149	−0.345075	−	0.938575i	$-0.612146\pi$
$823$	−19.7990	−0.690149	−0.345075	+	0.938575i	$0.612146\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.0000	−0.556375	−0.278187	−	0.960527i	$-0.589734\pi$
$827$	−16.0000	−0.556375	−0.278187	+	0.960527i	$0.589734\pi$
$828$	0	0
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.46410	−0.120024
$834$	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
$841$	43.0000	1.48276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−36.7696	−1.26342
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−19.5959	−0.671739
$852$	0	0
$853$	−44.0908	−1.50964	−0.754820	−	0.655932i	$-0.772276\pi$
$853$	−44.0908	−1.50964	−0.754820	+	0.655932i	$0.772276\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38.1051	1.30165	0.650823	−	0.759229i	$-0.274424\pi$
$857$	38.1051	1.30165	0.650823	+	0.759229i	$0.274424\pi$
$858$	0	0
$859$	20.7846	0.709162	0.354581	−	0.935025i	$-0.384624\pi$
$859$	20.7846	0.709162	0.354581	+	0.935025i	$0.384624\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−28.0000	−0.953131	−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000	−0.953131	−0.476566	+	0.879139i	$0.658119\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	33.9411	1.15137
$870$	0	0
$871$	27.7128	0.939013
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	44.0908	1.48884	0.744421	−	0.667711i	$-0.232726\pi$
$877$	44.0908	1.48884	0.744421	+	0.667711i	$0.232726\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	11.3137	0.381169	0.190584	−	0.981671i	$-0.438962\pi$
$881$	11.3137	0.381169	0.190584	+	0.981671i	$0.438962\pi$
$882$	0	0
$883$	39.5980	1.33258	0.666289	−	0.745694i	$-0.267882\pi$
$883$	39.5980	1.33258	0.666289	+	0.745694i	$0.267882\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	0	0
$889$	40.0000	1.34156
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−27.7128	−0.927374
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−58.7878	−1.96068
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	45.2548	1.50266	0.751331	−	0.659925i	$-0.229412\pi$
$907$	45.2548	1.50266	0.751331	+	0.659925i	$0.229412\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−39.1918	−1.29848	−0.649242	−	0.760582i	$-0.724914\pi$
$911$	−39.1918	−1.29848	−0.649242	+	0.760582i	$0.724914\pi$
$912$	0	0
$913$	−78.3837	−2.59412
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−13.8564	−0.457579
$918$	0	0
$919$	6.92820	0.228540	0.114270	−	0.993450i	$-0.463547\pi$
$919$	6.92820	0.228540	0.114270	+	0.993450i	$0.463547\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−48.0000	−1.57994
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	39.5980	1.29917	0.649584	−	0.760290i	$-0.274943\pi$
$929$	39.5980	1.29917	0.649584	+	0.760290i	$0.274943\pi$
$930$	0	0
$931$	−6.92820	−0.227063
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−19.5959	−0.640171	−0.320085	−	0.947389i	$-0.603712\pi$
$937$	−19.5959	−0.640171	−0.320085	+	0.947389i	$0.603712\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8.48528	−0.276612	−0.138306	−	0.990390i	$-0.544166\pi$
$941$	−8.48528	−0.276612	−0.138306	+	0.990390i	$0.544166\pi$
$942$	0	0
$943$	−22.6274	−0.736850
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−38.1051	−1.23435	−0.617173	−	0.786828i	$-0.711722\pi$
$953$	−38.1051	−1.23435	−0.617173	+	0.786828i	$0.711722\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	29.3939	0.949178
$960$	0	0
$961$	17.0000	0.548387
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	31.1127	1.00052	0.500258	−	0.865876i	$-0.333238\pi$
$967$	31.1127	1.00052	0.500258	+	0.865876i	$0.333238\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24.4949	0.786079	0.393039	−	0.919522i	$-0.371424\pi$
$971$	24.4949	0.786079	0.393039	+	0.919522i	$0.371424\pi$
$972$	0	0
$973$	−19.5959	−0.628216
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−31.1769	−0.997438	−0.498719	−	0.866764i	$-0.666196\pi$
$977$	−31.1769	−0.997438	−0.498719	+	0.866764i	$0.666196\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	45.2548	1.43902
$990$	0	0
$991$	6.92820	0.220082	0.110041	−	0.993927i	$-0.464902\pi$
$991$	6.92820	0.220082	0.110041	+	0.993927i	$0.464902\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−14.6969	−0.465457	−0.232728	−	0.972542i	$-0.574765\pi$
$997$	−14.6969	−0.465457	−0.232728	+	0.972542i	$0.574765\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.a.ct.1.1		4
3.2	odd	2		7200.2.a.cs.1.2		4
4.3	odd	2		7200.2.a.cs.1.4		4
5.2	odd	4		1440.2.f.j.289.3	yes	8
5.3	odd	4		1440.2.f.j.289.2	yes	8
5.4	even	2		7200.2.a.cs.1.3		4
12.11	even	2	inner	7200.2.a.ct.1.3		4
15.2	even	4		1440.2.f.j.289.5	yes	8
15.8	even	4		1440.2.f.j.289.8	yes	8
15.14	odd	2	inner	7200.2.a.ct.1.4		4
20.3	even	4		1440.2.f.j.289.1	✓	8
20.7	even	4		1440.2.f.j.289.4	yes	8
20.19	odd	2	inner	7200.2.a.ct.1.2		4
40.3	even	4		2880.2.f.x.1729.7		8
40.13	odd	4		2880.2.f.x.1729.8		8
40.27	even	4		2880.2.f.x.1729.6		8
40.37	odd	4		2880.2.f.x.1729.5		8
60.23	odd	4		1440.2.f.j.289.7	yes	8
60.47	odd	4		1440.2.f.j.289.6	yes	8
60.59	even	2		7200.2.a.cs.1.1		4
120.53	even	4		2880.2.f.x.1729.2		8
120.77	even	4		2880.2.f.x.1729.3		8
120.83	odd	4		2880.2.f.x.1729.1		8
120.107	odd	4		2880.2.f.x.1729.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.f.j.289.1	✓	8	20.3	even	4
1440.2.f.j.289.2	yes	8	5.3	odd	4
1440.2.f.j.289.3	yes	8	5.2	odd	4
1440.2.f.j.289.4	yes	8	20.7	even	4
1440.2.f.j.289.5	yes	8	15.2	even	4
1440.2.f.j.289.6	yes	8	60.47	odd	4
1440.2.f.j.289.7	yes	8	60.23	odd	4
1440.2.f.j.289.8	yes	8	15.8	even	4
2880.2.f.x.1729.1		8	120.83	odd	4
2880.2.f.x.1729.2		8	120.53	even	4
2880.2.f.x.1729.3		8	120.77	even	4
2880.2.f.x.1729.4		8	120.107	odd	4
2880.2.f.x.1729.5		8	40.37	odd	4
2880.2.f.x.1729.6		8	40.27	even	4
2880.2.f.x.1729.7		8	40.3	even	4
2880.2.f.x.1729.8		8	40.13	odd	4
7200.2.a.cs.1.1		4	60.59	even	2
7200.2.a.cs.1.2		4	3.2	odd	2
7200.2.a.cs.1.3		4	5.4	even	2
7200.2.a.cs.1.4		4	4.3	odd	2
7200.2.a.ct.1.1		4	1.1	even	1	trivial
7200.2.a.ct.1.2		4	20.19	odd	2	inner
7200.2.a.ct.1.3		4	12.11	even	2	inner
7200.2.a.ct.1.4		4	15.14	odd	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.a (trivial)

\(f(q)\)	\(=\)	\(q-2.82843 q^{7} +O(q^{10})\)	\(q-2.82843 q^{7} -4.89898 q^{11} -4.89898 q^{13} -3.46410 q^{17} -6.92820 q^{19} +4.00000 q^{23} -8.48528 q^{29} +6.92820 q^{31} -4.89898 q^{37} -5.65685 q^{41} +11.3137 q^{43} +4.00000 q^{47} +1.00000 q^{49} -3.46410 q^{53} -4.89898 q^{59} -6.00000 q^{61} -5.65685 q^{67} +9.79796 q^{71} +13.8564 q^{77} -6.92820 q^{79} +16.0000 q^{83} +13.8564 q^{91} +9.79796 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 16 q^{23} + 16 q^{47} + 4 q^{49} - 24 q^{61} + 64 q^{83}+O(q^{100}) \) 4 * q + 16 * q^23 + 16 * q^47 + 4 * q^49 - 24 * q^61 + 64 * q^83

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