LMFDB - Embedded newform 7200.2.k.q.3601.5

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:	$8$
Coefficient field:	8.0.212336640000.29
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 2x^{4} + 16 $ x^8 + 2*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 1800)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3601.5
Root			$-0.622597 - 1.26979i$ of defining polynomial
Character	$\chi$	$=$	7200.3601
Dual form			7200.2.k.q.3601.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.44949 q^{7} +O(q^{10})$	$q+1.44949 q^{7} -1.14152i q^{11} -3.87298i q^{13} -3.05009 q^{17} -0.710706i q^{19} -5.54048 q^{23} +6.22069i q^{29} -3.44949 q^{31} -6.32456i q^{37} -8.03087 q^{41} +7.03526i q^{43} +4.98078 q^{47} -4.89898 q^{49} +3.93765i q^{53} +10.1583i q^{59} +2.45157i q^{61} +13.3598i q^{67} +11.6407 q^{71} -2.89898 q^{73} -1.65462i q^{77} +6.89898 q^{79} +12.4414i q^{83} -16.0617 q^{89} -5.61385i q^{91} -13.6969 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7}+O(q^{10}) $ 8 * q - 8 * q^7	$ 8 q - 8 q^{7} - 8 q^{31} + 16 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 - 8 * q^31 + 16 * q^73 + 16 * q^79 + 8 * q^97

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
$7$	1.44949	0.547856	0.273928	−	0.961750i	$-0.411677\pi$
$7$	1.44949	0.547856	0.273928	+	0.961750i	$0.411677\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.14152i	− 0.344180i	−0.985081	−	0.172090i	$-0.944948\pi$
$11$	− 1.14152i	− 0.344180i	0.985081	−	0.172090i	$-0.0550521\pi$
$12$	0	0
$13$	− 3.87298i	− 1.07417i	−0.843527	−	0.537086i	$-0.819525\pi$
$13$	− 3.87298i	− 1.07417i	0.843527	−	0.537086i	$-0.180475\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.05009	−0.739756	−0.369878	−	0.929080i	$-0.620601\pi$
$17$	−3.05009	−0.739756	−0.369878	+	0.929080i	$0.620601\pi$
$18$	0	0
$19$	− 0.710706i	− 0.163047i	−0.996671	−	0.0815235i	$-0.974021\pi$
$19$	− 0.710706i	− 0.163047i	0.996671	−	0.0815235i	$-0.0259786\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.54048	−1.15527	−0.577635	−	0.816295i	$-0.696024\pi$
$23$	−5.54048	−1.15527	−0.577635	+	0.816295i	$0.696024\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.22069i	1.15515i	0.816337	+	0.577576i	$0.196001\pi$
$29$	6.22069i	1.15515i	−0.816337	+	0.577576i	$0.803999\pi$
$30$	0	0
$31$	−3.44949	−0.619547	−0.309773	−	0.950810i	$-0.600253\pi$
$31$	−3.44949	−0.619547	−0.309773	+	0.950810i	$0.600253\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.32456i	− 1.03975i	−0.854242	−	0.519875i	$-0.825978\pi$
$37$	− 6.32456i	− 1.03975i	0.854242	−	0.519875i	$-0.174022\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.03087	−1.25421	−0.627106	−	0.778934i	$-0.715761\pi$
$41$	−8.03087	−1.25421	−0.627106	+	0.778934i	$0.715761\pi$
$42$	0	0
$43$	7.03526i	1.07287i	0.843943	+	0.536434i	$0.180229\pi$
$43$	7.03526i	1.07287i	−0.843943	+	0.536434i	$0.819771\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.98078	0.726521	0.363261	−	0.931688i	$-0.381663\pi$
$47$	4.98078	0.726521	0.363261	+	0.931688i	$0.381663\pi$
$48$	0	0
$49$	−4.89898	−0.699854
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.93765i	0.540878i	0.962737	+	0.270439i	$0.0871689\pi$
$53$	3.93765i	0.540878i	−0.962737	+	0.270439i	$0.912831\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.1583i	1.32250i	0.750164	+	0.661251i	$0.229974\pi$
$59$	10.1583i	1.32250i	−0.750164	+	0.661251i	$0.770026\pi$
$60$	0	0
$61$	2.45157i	0.313892i	0.987607	+	0.156946i	$0.0501648\pi$
$61$	2.45157i	0.313892i	−0.987607	+	0.156946i	$0.949835\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.3598i	1.63216i	0.577938	+	0.816081i	$0.303858\pi$
$67$	13.3598i	1.63216i	−0.577938	+	0.816081i	$0.696142\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.6407	1.38149	0.690746	−	0.723097i	$-0.257282\pi$
$71$	11.6407	1.38149	0.690746	+	0.723097i	$0.257282\pi$
$72$	0	0
$73$	−2.89898	−0.339300	−0.169650	−	0.985504i	$-0.554264\pi$
$73$	−2.89898	−0.339300	−0.169650	+	0.985504i	$0.554264\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.65462i	− 0.188561i
$78$	0	0
$79$	6.89898	0.776196	0.388098	−	0.921618i	$-0.373132\pi$
$79$	6.89898	0.776196	0.388098	+	0.921618i	$0.373132\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.4414i	1.36562i	0.730597	+	0.682809i	$0.239242\pi$
$83$	12.4414i	1.36562i	−0.730597	+	0.682809i	$0.760758\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.0617	−1.70254	−0.851271	−	0.524727i	$-0.824167\pi$
$89$	−16.0617	−1.70254	−0.851271	+	0.524727i	$0.824167\pi$
$90$	0	0
$91$	− 5.61385i	− 0.588491i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.6969	−1.39071	−0.695357	−	0.718665i	$-0.744754\pi$
$97$	−13.6969	−1.39071	−0.695357	+	0.718665i	$0.744754\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.50372i	0.846152i	0.906094	+	0.423076i	$0.139050\pi$
$101$	8.50372i	0.846152i	−0.906094	+	0.423076i	$0.860950\pi$
$102$	0	0
$103$	2.89898	0.285645	0.142822	−	0.989748i	$-0.454382\pi$
$103$	2.89898	0.285645	0.142822	+	0.989748i	$0.454382\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11.2999i	− 1.09240i	−0.837655	−	0.546199i	$-0.816074\pi$
$107$	− 11.2999i	− 1.09240i	0.837655	−	0.546199i	$-0.183926\pi$
$108$	0	0
$109$	− 8.77613i	− 0.840601i	−0.907385	−	0.420300i	$-0.861925\pi$
$109$	− 8.77613i	− 0.840601i	0.907385	−	0.420300i	$-0.138075\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.10018	0.573857	0.286929	−	0.957952i	$-0.407366\pi$
$113$	6.10018	0.573857	0.286929	+	0.957952i	$0.407366\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.42108	−0.405279
$120$	0	0
$121$	9.69694	0.881540
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.7980	1.40184	0.700921	−	0.713239i	$-0.252773\pi$
$127$	15.7980	1.40184	0.700921	+	0.713239i	$0.252773\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.01682i	0.787803i	0.919153	+	0.393902i	$0.128875\pi$
$131$	9.01682i	0.787803i	−0.919153	+	0.393902i	$0.871125\pi$
$132$	0	0
$133$	− 1.03016i	− 0.0893263i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.98078	0.425537	0.212768	−	0.977103i	$-0.431752\pi$
$137$	4.98078	0.425537	0.212768	+	0.977103i	$0.431752\pi$
$138$	0	0
$139$	21.8165i	1.85045i	0.379418	+	0.925225i	$0.376124\pi$
$139$	21.8165i	1.85045i	−0.379418	+	0.925225i	$0.623876\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.42108	−0.369709
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.0336i	1.47737i	0.674049	+	0.738687i	$0.264554\pi$
$149$	18.0336i	1.47737i	−0.674049	+	0.738687i	$0.735446\pi$
$150$	0	0
$151$	−2.34847	−0.191116	−0.0955579	−	0.995424i	$-0.530463\pi$
$151$	−2.34847	−0.191116	−0.0955579	+	0.995424i	$0.530463\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 16.5221i	− 1.31861i	−0.751877	−	0.659303i	$-0.770851\pi$
$157$	− 16.5221i	− 1.31861i	0.751877	−	0.659303i	$-0.229149\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.03087	−0.632921
$162$	0	0
$163$	− 8.45667i	− 0.662378i	−0.943564	−	0.331189i	$-0.892550\pi$
$163$	− 8.45667i	− 0.662378i	0.943564	−	0.331189i	$-0.107450\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.6214	1.28621	0.643103	−	0.765780i	$-0.277647\pi$
$167$	16.6214	1.28621	0.643103	+	0.765780i	$0.277647\pi$
$168$	0	0
$169$	−2.00000	−0.153846
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.3790i	1.24527i	0.782511	+	0.622637i	$0.213939\pi$
$173$	16.3790i	1.24527i	−0.782511	+	0.622637i	$0.786061\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.0336i	1.34790i	0.738778	+	0.673949i	$0.235403\pi$
$179$	18.0336i	1.34790i	−0.738778	+	0.673949i	$0.764597\pi$
$180$	0	0
$181$	17.9435i	1.33373i	0.745178	+	0.666865i	$0.232364\pi$
$181$	17.9435i	1.33373i	−0.745178	+	0.666865i	$0.767636\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.48173i	0.254610i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.6407	−0.842289	−0.421145	−	0.906994i	$-0.638371\pi$
$191$	−11.6407	−0.842289	−0.421145	+	0.906994i	$0.638371\pi$
$192$	0	0
$193$	−2.10102	−0.151235	−0.0756174	−	0.997137i	$-0.524093\pi$
$193$	−2.10102	−0.151235	−0.0756174	+	0.997137i	$0.524093\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 27.1658i	− 1.93548i	−0.251948	−	0.967741i	$-0.581071\pi$
$197$	− 27.1658i	− 1.93548i	0.251948	−	0.967741i	$-0.418929\pi$
$198$	0	0
$199$	3.24745	0.230206	0.115103	−	0.993354i	$-0.463280\pi$
$199$	3.24745	0.230206	0.115103	+	0.993354i	$0.463280\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.01682i	0.632857i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.811283	−0.0561176
$210$	0	0
$211$	− 21.1058i	− 1.45298i	−0.687176	−	0.726491i	$-0.741150\pi$
$211$	− 21.1058i	− 1.45298i	0.687176	−	0.726491i	$-0.258850\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.00000	−0.339422
$218$	0	0
$219$	0	0
$220$	0	0
$221$	11.8130i	0.794625i
$222$	0	0
$223$	14.3485	0.960845	0.480422	−	0.877037i	$-0.340483\pi$
$223$	14.3485	0.960845	0.480422	+	0.877037i	$0.340483\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 21.4582i	− 1.42423i	−0.702063	−	0.712115i	$-0.747737\pi$
$227$	− 21.4582i	− 1.42423i	0.702063	−	0.712115i	$-0.252263\pi$
$228$	0	0
$229$	− 17.9435i	− 1.18574i	−0.805298	−	0.592870i	$-0.797995\pi$
$229$	− 17.9435i	− 1.18574i	0.805298	−	0.592870i	$-0.202005\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.9924	−1.17872	−0.589362	−	0.807869i	$-0.700621\pi$
$233$	−17.9924	−1.17872	−0.589362	+	0.807869i	$0.700621\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.4828	−1.32492	−0.662462	−	0.749096i	$-0.730488\pi$
$239$	−20.4828	−1.32492	−0.662462	+	0.749096i	$0.730488\pi$
$240$	0	0
$241$	−5.69694	−0.366972	−0.183486	−	0.983022i	$-0.558738\pi$
$241$	−5.69694	−0.366972	−0.183486	+	0.983022i	$0.558738\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.75255	−0.175141
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.4414i	0.785292i	0.919690	+	0.392646i	$0.128440\pi$
$251$	12.4414i	0.785292i	−0.919690	+	0.392646i	$0.871560\pi$
$252$	0	0
$253$	6.32456i	0.397621i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.0810	0.691212	0.345606	−	0.938380i	$-0.387673\pi$
$257$	11.0810	0.691212	0.345606	+	0.938380i	$0.387673\pi$
$258$	0	0
$259$	− 9.16738i	− 0.569633i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.5213	−0.648769	−0.324384	−	0.945925i	$-0.605157\pi$
$263$	−10.5213	−0.648769	−0.324384	+	0.945925i	$0.605157\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	28.8204i	1.75721i	0.477549	+	0.878605i	$0.341525\pi$
$269$	28.8204i	1.75721i	−0.477549	+	0.878605i	$0.658475\pi$
$270$	0	0
$271$	−14.8990	−0.905049	−0.452524	−	0.891752i	$-0.649476\pi$
$271$	−14.8990	−0.905049	−0.452524	+	0.891752i	$0.649476\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.03016i	− 0.0618964i	−0.999521	−	0.0309482i	$-0.990147\pi$
$277$	− 1.03016i	− 0.0618964i	0.999521	−	0.0309482i	$-0.00985268\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.03087	0.479082	0.239541	−	0.970886i	$-0.423003\pi$
$281$	8.03087	0.479082	0.239541	+	0.970886i	$0.423003\pi$
$282$	0	0
$283$	7.03526i	0.418203i	0.977894	+	0.209101i	$0.0670539\pi$
$283$	7.03526i	0.418203i	−0.977894	+	0.209101i	$0.932946\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.6407	−0.687127
$288$	0	0
$289$	−7.69694	−0.452761
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.87530i	0.460080i	0.973181	+	0.230040i	$0.0738857\pi$
$293$	7.87530i	0.460080i	−0.973181	+	0.230040i	$0.926114\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	21.4582i	1.24096i
$300$	0	0
$301$	10.1975i	0.587776i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.45667i	0.482648i	0.970445	+	0.241324i	$0.0775816\pi$
$307$	8.45667i	0.482648i	−0.970445	+	0.241324i	$0.922418\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−32.1235	−1.82156	−0.910778	−	0.412897i	$-0.864517\pi$
$311$	−32.1235	−1.82156	−0.910778	+	0.412897i	$0.864517\pi$
$312$	0	0
$313$	−16.5959	−0.938057	−0.469028	−	0.883183i	$-0.655396\pi$
$313$	−16.5959	−0.938057	−0.469028	+	0.883183i	$0.655396\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 18.0336i	− 1.01287i	−0.862278	−	0.506435i	$-0.830963\pi$
$317$	− 18.0336i	− 1.01287i	0.862278	−	0.506435i	$-0.169037\pi$
$318$	0	0
$319$	7.10102	0.397581
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.16772i	0.120615i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.21959	0.398029
$330$	0	0
$331$	12.6491i	0.695258i	0.937632	+	0.347629i	$0.113013\pi$
$331$	12.6491i	0.695258i	−0.937632	+	0.347629i	$0.886987\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−0.797959	−0.0434676	−0.0217338	−	0.999764i	$-0.506919\pi$
$337$	−0.797959	−0.0434676	−0.0217338	+	0.999764i	$0.506919\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.93765i	0.213236i
$342$	0	0
$343$	−17.2474	−0.931275
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7.87530i	− 0.422768i	−0.977403	−	0.211384i	$-0.932203\pi$
$347$	− 7.87530i	− 0.422768i	0.977403	−	0.211384i	$-0.0677971\pi$
$348$	0	0
$349$	12.6491i	0.677091i	0.940950	+	0.338546i	$0.109935\pi$
$349$	12.6491i	0.677091i	−0.940950	+	0.338546i	$0.890065\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−30.1928	−1.60700	−0.803500	−	0.595304i	$-0.797031\pi$
$353$	−30.1928	−1.60700	−0.803500	+	0.595304i	$0.797031\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.42108	−0.233336	−0.116668	−	0.993171i	$-0.537221\pi$
$359$	−4.42108	−0.233336	−0.116668	+	0.993171i	$0.537221\pi$
$360$	0	0
$361$	18.4949	0.973416
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.5505	0.968329	0.484164	−	0.874977i	$-0.339124\pi$
$367$	18.5505	0.968329	0.484164	+	0.874977i	$0.339124\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.70759i	0.296323i
$372$	0	0
$373$	− 13.0404i	− 0.675204i	−0.941289	−	0.337602i	$-0.890384\pi$
$373$	− 13.0404i	− 0.675204i	0.941289	−	0.337602i	$-0.109616\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0926	1.24083
$378$	0	0
$379$	5.61385i	0.288364i	0.989551	+	0.144182i	$0.0460551\pi$
$379$	5.61385i	0.288364i	−0.989551	+	0.144182i	$0.953945\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.10018	0.311705	0.155852	−	0.987780i	$-0.450188\pi$
$383$	6.10018	0.311705	0.155852	+	0.987780i	$0.450188\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9.52992i	0.483186i	0.970378	+	0.241593i	$0.0776699\pi$
$389$	9.52992i	0.483186i	−0.970378	+	0.241593i	$0.922330\pi$
$390$	0	0
$391$	16.8990	0.854618
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.1975i	0.511800i	0.966703	+	0.255900i	$0.0823717\pi$
$397$	10.1975i	0.511800i	−0.966703	+	0.255900i	$0.917628\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.2505	0.761572	0.380786	−	0.924663i	$-0.375654\pi$
$401$	15.2505	0.761572	0.380786	+	0.924663i	$0.375654\pi$
$402$	0	0
$403$	13.3598i	0.665500i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.21959	−0.357862
$408$	0	0
$409$	1.89898	0.0938985	0.0469492	−	0.998897i	$-0.485050\pi$
$409$	1.89898	0.0938985	0.0469492	+	0.998897i	$0.485050\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	14.7244i	0.724541i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	38.4656i	1.87917i	0.342317	+	0.939585i	$0.388788\pi$
$419$	38.4656i	1.87917i	−0.342317	+	0.939585i	$0.611212\pi$
$420$	0	0
$421$	− 37.3084i	− 1.81830i	−0.416467	−	0.909151i	$-0.636732\pi$
$421$	− 37.3084i	− 1.81830i	0.416467	−	0.909151i	$-0.363268\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.55353i	0.171967i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.42108	−0.212956	−0.106478	−	0.994315i	$-0.533957\pi$
$431$	−4.42108	−0.212956	−0.106478	+	0.994315i	$0.533957\pi$
$432$	0	0
$433$	26.5959	1.27812	0.639059	−	0.769158i	$-0.279324\pi$
$433$	26.5959	1.27812	0.639059	+	0.769158i	$0.279324\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.93765i	0.188363i
$438$	0	0
$439$	−27.0454	−1.29081	−0.645403	−	0.763842i	$-0.723311\pi$
$439$	−27.0454	−1.29081	−0.645403	+	0.763842i	$0.723311\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	35.0411i	1.66485i	0.554136	+	0.832426i	$0.313049\pi$
$443$	35.0411i	1.66485i	−0.554136	+	0.832426i	$0.686951\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−39.3431	−1.85671	−0.928357	−	0.371689i	$-0.878779\pi$
$449$	−39.3431	−1.85671	−0.928357	+	0.371689i	$0.878779\pi$
$450$	0	0
$451$	9.16738i	0.431675i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.69694	0.406826	0.203413	−	0.979093i	$-0.434797\pi$
$457$	8.69694	0.406826	0.203413	+	0.979093i	$0.434797\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 10.7868i	− 0.502389i	−0.967937	−	0.251195i	$-0.919177\pi$
$461$	− 10.7868i	− 0.502389i	0.967937	−	0.251195i	$-0.0808234\pi$
$462$	0	0
$463$	−2.89898	−0.134727	−0.0673635	−	0.997728i	$-0.521459\pi$
$463$	−2.89898	−0.134727	−0.0673635	+	0.997728i	$0.521459\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4.45075i	− 0.205956i	−0.994684	−	0.102978i	$-0.967163\pi$
$467$	− 4.45075i	− 0.205956i	0.994684	−	0.102978i	$-0.0328372\pi$
$468$	0	0
$469$	19.3649i	0.894189i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.03087	0.369260
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.0617	−0.733880	−0.366940	−	0.930245i	$-0.619595\pi$
$479$	−16.0617	−0.733880	−0.366940	+	0.930245i	$0.619595\pi$
$480$	0	0
$481$	−24.4949	−1.11687
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−43.0454	−1.95057	−0.975287	−	0.220943i	$-0.929087\pi$
$487$	−43.0454	−1.95057	−0.975287	+	0.220943i	$0.929087\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	22.5997i	1.01991i	0.860201	+	0.509955i	$0.170338\pi$
$491$	22.5997i	1.01991i	−0.860201	+	0.509955i	$0.829662\pi$
$492$	0	0
$493$	− 18.9737i	− 0.854531i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.8730	0.756859
$498$	0	0
$499$	− 11.9384i	− 0.534436i	−0.963636	−	0.267218i	$-0.913896\pi$
$499$	− 11.9384i	− 0.534436i	0.963636	−	0.267218i	$-0.0861044\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	28.8218	1.28510	0.642551	−	0.766243i	$-0.277876\pi$
$503$	28.8218	1.28510	0.642551	+	0.766243i	$0.277876\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18.0336i	0.799327i	0.916662	+	0.399664i	$0.130873\pi$
$509$	18.0336i	0.799327i	−0.916662	+	0.399664i	$0.869127\pi$
$510$	0	0
$511$	−4.20204	−0.185887
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.68565i	− 0.250054i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.2505	0.668135	0.334067	−	0.942549i	$-0.391579\pi$
$521$	15.2505	0.668135	0.334067	+	0.942549i	$0.391579\pi$
$522$	0	0
$523$	− 10.5170i	− 0.459876i	−0.973205	−	0.229938i	$-0.926148\pi$
$523$	− 10.5170i	− 0.459876i	0.973205	−	0.229938i	$-0.0738523\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.5213	0.458313
$528$	0	0
$529$	7.69694	0.334649
$530$	0	0
$531$	0	0
$532$	0	0
$533$	31.1034i	1.34724i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.59227i	0.240876i
$540$	0	0
$541$	− 16.5221i	− 0.710340i	−0.934802	−	0.355170i	$-0.884423\pi$
$541$	− 16.5221i	− 0.710340i	0.934802	−	0.355170i	$-0.115577\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 21.8165i	− 0.932806i	−0.884572	−	0.466403i	$-0.845550\pi$
$547$	− 21.8165i	− 0.932806i	0.884572	−	0.466403i	$-0.154450\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.42108	0.188344
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	27.2474	1.15244
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 2.16772i	− 0.0913584i	−0.998956	−	0.0456792i	$-0.985455\pi$
$563$	− 2.16772i	− 0.0913584i	0.998956	−	0.0456792i	$-0.0145452\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.03087	0.336672	0.168336	−	0.985730i	$-0.446161\pi$
$569$	8.03087	0.336672	0.168336	+	0.985730i	$0.446161\pi$
$570$	0	0
$571$	14.7812i	0.618575i	0.950969	+	0.309288i	$0.100091\pi$
$571$	14.7812i	0.618575i	−0.950969	+	0.309288i	$0.899909\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.6969	0.570211	0.285106	−	0.958496i	$-0.407971\pi$
$577$	13.6969	0.570211	0.285106	+	0.958496i	$0.407971\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	18.0336i	0.748162i
$582$	0	0
$583$	4.49490	0.186160
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.42455i	0.141346i	0.997500	+	0.0706732i	$0.0225147\pi$
$587$	3.42455i	0.141346i	−0.997500	+	0.0706732i	$0.977485\pi$
$588$	0	0
$589$	2.45157i	0.101015i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.91147	−0.283820	−0.141910	−	0.989880i	$-0.545324\pi$
$593$	−6.91147	−0.283820	−0.141910	+	0.989880i	$0.545324\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−39.3431	−1.60751	−0.803757	−	0.594957i	$-0.797169\pi$
$599$	−39.3431	−1.60751	−0.803757	+	0.594957i	$0.797169\pi$
$600$	0	0
$601$	−9.89898	−0.403788	−0.201894	−	0.979407i	$-0.564710\pi$
$601$	−9.89898	−0.403788	−0.201894	+	0.979407i	$0.564710\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11.3031	0.458777	0.229389	−	0.973335i	$-0.426327\pi$
$607$	11.3031	0.458777	0.229389	+	0.973335i	$0.426327\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 19.2905i	− 0.780409i
$612$	0	0
$613$	31.6228i	1.27723i	0.769526	+	0.638616i	$0.220493\pi$
$613$	31.6228i	1.27723i	−0.769526	+	0.638616i	$0.779507\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−36.2930	−1.46110	−0.730550	−	0.682859i	$-0.760736\pi$
$617$	−36.2930	−1.46110	−0.730550	+	0.682859i	$0.760736\pi$
$618$	0	0
$619$	− 44.9826i	− 1.80800i	−0.427529	−	0.904002i	$-0.640616\pi$
$619$	− 44.9826i	− 1.80800i	0.427529	−	0.904002i	$-0.359384\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−23.2813	−0.932747
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	19.2905i	0.769162i
$630$	0	0
$631$	23.4495	0.933509	0.466755	−	0.884387i	$-0.345423\pi$
$631$	23.4495	0.933509	0.466755	+	0.884387i	$0.345423\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.9737i	0.751764i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.21959	−0.285157	−0.142578	−	0.989784i	$-0.545539\pi$
$641$	−7.21959	−0.285157	−0.142578	+	0.989784i	$0.545539\pi$
$642$	0	0
$643$	− 3.48173i	− 0.137306i	−0.997641	−	0.0686531i	$-0.978130\pi$
$643$	− 3.48173i	− 0.137306i	0.997641	−	0.0686531i	$-0.0218702\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.2004	−0.479646	−0.239823	−	0.970817i	$-0.577089\pi$
$647$	−12.2004	−0.479646	−0.239823	+	0.970817i	$0.577089\pi$
$648$	0	0
$649$	11.5959	0.455180
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 12.4414i	− 0.486869i	−0.969917	−	0.243434i	$-0.921726\pi$
$653$	− 12.4414i	− 0.486869i	0.969917	−	0.243434i	$-0.0782740\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 15.8659i	− 0.618049i	−0.951054	−	0.309024i	$-0.899998\pi$
$659$	− 15.8659i	− 0.618049i	0.951054	−	0.309024i	$-0.100002\pi$
$660$	0	0
$661$	24.6593i	0.959136i	0.877505	+	0.479568i	$0.159207\pi$
$661$	24.6593i	0.959136i	−0.877505	+	0.479568i	$0.840793\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 34.4656i	− 1.33451i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.79851	0.108035
$672$	0	0
$673$	−17.1010	−0.659196	−0.329598	−	0.944121i	$-0.606913\pi$
$673$	−17.1010	−0.659196	−0.329598	+	0.944121i	$0.606913\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 9.52992i	− 0.366265i	−0.983088	−	0.183132i	$-0.941376\pi$
$677$	− 9.52992i	− 0.366265i	0.983088	−	0.183132i	$-0.0586237\pi$
$678$	0	0
$679$	−19.8536	−0.761910
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 6.84910i	− 0.262074i	−0.991378	−	0.131037i	$-0.958169\pi$
$683$	− 6.84910i	− 0.262074i	0.991378	−	0.131037i	$-0.0418306\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	15.2505	0.580996
$690$	0	0
$691$	37.9473i	1.44358i	0.692110	+	0.721792i	$0.256681\pi$
$691$	37.9473i	1.44358i	−0.692110	+	0.721792i	$0.743319\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	24.4949	0.927810
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.91145i	0.109964i	0.998487	+	0.0549820i	$0.0175101\pi$
$701$	2.91145i	0.109964i	−0.998487	+	0.0549820i	$0.982490\pi$
$702$	0	0
$703$	−4.49490	−0.169528
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.3261i	0.463569i
$708$	0	0
$709$	24.2681i	0.911406i	0.890132	+	0.455703i	$0.150612\pi$
$709$	24.2681i	0.911406i	−0.890132	+	0.455703i	$0.849388\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	19.1118	0.715744
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	43.7642	1.63213	0.816064	−	0.577962i	$-0.196152\pi$
$719$	43.7642	1.63213	0.816064	+	0.577962i	$0.196152\pi$
$720$	0	0
$721$	4.20204	0.156492
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−38.5505	−1.42976	−0.714880	−	0.699248i	$-0.753519\pi$
$727$	−38.5505	−1.42976	−0.714880	+	0.699248i	$0.753519\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 21.4582i	− 0.793660i
$732$	0	0
$733$	30.9839i	1.14442i	0.820109	+	0.572208i	$0.193913\pi$
$733$	30.9839i	1.14442i	−0.820109	+	0.572208i	$0.806087\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15.2505	0.561758
$738$	0	0
$739$	30.9839i	1.13976i	0.821728	+	0.569880i	$0.193010\pi$
$739$	30.9839i	1.13976i	−0.821728	+	0.569880i	$0.806990\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−33.2429	−1.21956	−0.609782	−	0.792569i	$-0.708743\pi$
$743$	−33.2429	−1.21956	−0.609782	+	0.792569i	$0.708743\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 16.3790i	− 0.598477i
$750$	0	0
$751$	−16.2020	−0.591221	−0.295610	−	0.955309i	$-0.595523\pi$
$751$	−16.2020	−0.591221	−0.295610	+	0.955309i	$0.595523\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	50.9877i	1.85318i	0.376074	+	0.926590i	$0.377274\pi$
$757$	50.9877i	1.85318i	−0.376074	+	0.926590i	$0.622726\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−32.1235	−1.16448	−0.582238	−	0.813019i	$-0.697823\pi$
$761$	−32.1235	−1.16448	−0.582238	+	0.813019i	$0.697823\pi$
$762$	0	0
$763$	− 12.7209i	− 0.460528i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	39.3431	1.42060
$768$	0	0
$769$	24.7980	0.894237	0.447119	−	0.894475i	$-0.352450\pi$
$769$	24.7980	0.894237	0.447119	+	0.894475i	$0.352450\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 18.6621i	− 0.671228i	−0.942000	−	0.335614i	$-0.891056\pi$
$773$	− 18.6621i	− 0.671228i	0.942000	−	0.335614i	$-0.108944\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.70759i	0.204495i
$780$	0	0
$781$	− 13.2880i	− 0.475483i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 33.7549i	− 1.20323i	−0.798785	−	0.601616i	$-0.794524\pi$
$787$	− 33.7549i	− 1.20323i	0.798785	−	0.601616i	$-0.205476\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.84215	0.314391
$792$	0	0
$793$	9.49490	0.337174
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.59227i	0.198088i	0.995083	+	0.0990442i	$0.0315785\pi$
$797$	5.59227i	0.198088i	−0.995083	+	0.0990442i	$0.968421\pi$
$798$	0	0
$799$	−15.1918	−0.537449
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.30923i	0.116780i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.03087	−0.282350	−0.141175	−	0.989985i	$-0.545088\pi$
$809$	−8.03087	−0.282350	−0.141175	+	0.989985i	$0.545088\pi$
$810$	0	0
$811$	− 45.7651i	− 1.60703i	−0.595285	−	0.803515i	$-0.702961\pi$
$811$	− 45.7651i	− 1.60703i	0.595285	−	0.803515i	$-0.297039\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.00000	0.174928
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 28.1920i	− 0.983907i	−0.870622	−	0.491953i	$-0.836283\pi$
$821$	− 28.1920i	− 0.983907i	0.870622	−	0.491953i	$-0.163717\pi$
$822$	0	0
$823$	−28.8434	−1.00542	−0.502708	−	0.864456i	$-0.667663\pi$
$823$	−28.8434	−1.00542	−0.502708	+	0.864456i	$0.667663\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 13.5829i	− 0.472323i	−0.971714	−	0.236162i	$-0.924111\pi$
$827$	− 13.5829i	− 0.472323i	0.971714	−	0.236162i	$-0.0758895\pi$
$828$	0	0
$829$	− 24.6593i	− 0.856453i	−0.903671	−	0.428227i	$-0.859138\pi$
$829$	− 24.6593i	− 0.856453i	0.903671	−	0.428227i	$-0.140862\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	14.9423	0.517721
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	27.7024	0.956393	0.478197	−	0.878253i	$-0.341291\pi$
$839$	27.7024	0.956393	0.478197	+	0.878253i	$0.341291\pi$
$840$	0	0
$841$	−9.69694	−0.334377
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14.0556	0.482957
$848$	0	0
$849$	0	0
$850$	0	0
$851$	35.0411i	1.20119i
$852$	0	0
$853$	− 17.9435i	− 0.614374i	−0.951649	−	0.307187i	$-0.900612\pi$
$853$	− 17.9435i	− 0.614374i	0.951649	−	0.307187i	$-0.0993877\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.0425	−0.718799	−0.359399	−	0.933184i	$-0.617018\pi$
$857$	−21.0425	−0.718799	−0.359399	+	0.933184i	$0.617018\pi$
$858$	0	0
$859$	− 18.3348i	− 0.625574i	−0.949823	−	0.312787i	$-0.898737\pi$
$859$	− 18.3348i	− 0.625574i	0.949823	−	0.312787i	$-0.101263\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6.10018	−0.207653	−0.103826	−	0.994595i	$-0.533109\pi$
$863$	−6.10018	−0.207653	−0.103826	+	0.994595i	$0.533109\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 7.87530i	− 0.267151i
$870$	0	0
$871$	51.7423	1.75322
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	33.4354i	1.12903i	0.825421	+	0.564517i	$0.190938\pi$
$877$	33.4354i	1.12903i	−0.825421	+	0.564517i	$0.809062\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−39.3431	−1.32550	−0.662751	−	0.748840i	$-0.730611\pi$
$881$	−39.3431	−1.32550	−0.662751	+	0.748840i	$0.730611\pi$
$882$	0	0
$883$	− 38.6580i	− 1.30095i	−0.759529	−	0.650473i	$-0.774571\pi$
$883$	− 38.6580i	− 1.30095i	0.759529	−	0.650473i	$-0.225429\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.8229	−0.464129	−0.232064	−	0.972700i	$-0.574548\pi$
$887$	−13.8229	−0.464129	−0.232064	+	0.972700i	$0.574548\pi$
$888$	0	0
$889$	22.8990	0.768007
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 3.53987i	− 0.118457i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 21.4582i	− 0.715671i
$900$	0	0
$901$	− 12.0102i	− 0.400118i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5.68565i	0.188789i	0.995535	+	0.0943944i	$0.0300915\pi$
$907$	5.68565i	0.188789i	−0.995535	+	0.0943944i	$0.969909\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	48.1852	1.59645	0.798224	−	0.602361i	$-0.205773\pi$
$911$	48.1852	1.59645	0.798224	+	0.602361i	$0.205773\pi$
$912$	0	0
$913$	14.2020	0.470019
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.0698i	0.431602i
$918$	0	0
$919$	−48.6413	−1.60453	−0.802265	−	0.596969i	$-0.796372\pi$
$919$	−48.6413	−1.60453	−0.802265	+	0.596969i	$0.796372\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 45.0841i	− 1.48396i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0.811283	0.0266173	0.0133087	−	0.999911i	$-0.495764\pi$
$929$	0.811283	0.0266173	0.0133087	+	0.999911i	$0.495764\pi$
$930$	0	0
$931$	3.48173i	0.114109i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.7980	0.679440	0.339720	−	0.940527i	$-0.389668\pi$
$937$	20.7980	0.679440	0.339720	+	0.940527i	$0.389668\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	41.8902i	1.36558i	0.730614	+	0.682791i	$0.239234\pi$
$941$	41.8902i	1.36558i	−0.730614	+	0.682791i	$0.760766\pi$
$942$	0	0
$943$	44.4949	1.44895
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 21.4582i	− 0.697298i	−0.937253	−	0.348649i	$-0.886641\pi$
$947$	− 21.4582i	− 0.697298i	0.937253	−	0.348649i	$-0.113359\pi$
$948$	0	0
$949$	11.2277i	0.364467i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−22.1619	−0.717895	−0.358948	−	0.933358i	$-0.616864\pi$
$953$	−22.1619	−0.717895	−0.358948	+	0.933358i	$0.616864\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.21959	0.233133
$960$	0	0
$961$	−19.1010	−0.616162
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−60.2929	−1.93889	−0.969444	−	0.245314i	$-0.921109\pi$
$967$	−60.2929	−1.93889	−0.969444	+	0.245314i	$0.921109\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 31.7318i	− 1.01832i	−0.860671	−	0.509162i	$-0.829956\pi$
$971$	− 31.7318i	− 1.01832i	0.860671	−	0.509162i	$-0.170044\pi$
$972$	0	0
$973$	31.6228i	1.01378i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−53.1660	−1.70093	−0.850466	−	0.526031i	$-0.823680\pi$
$977$	−53.1660	−1.70093	−0.850466	+	0.526031i	$0.823680\pi$
$978$	0	0
$979$	18.3348i	0.585981i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.11940	0.0357035	0.0178517	−	0.999841i	$-0.494317\pi$
$983$	1.11940	0.0357035	0.0178517	+	0.999841i	$0.494317\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 38.9787i	− 1.23945i
$990$	0	0
$991$	−29.2474	−0.929076	−0.464538	−	0.885553i	$-0.653780\pi$
$991$	−29.2474	−0.929076	−0.464538	+	0.885553i	$0.653780\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 12.6491i	− 0.400601i	−0.979734	−	0.200301i	$-0.935808\pi$
$997$	− 12.6491i	− 0.400601i	0.979734	−	0.200301i	$-0.0641919\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.k.q.3601.5		8
3.2	odd	2	inner	7200.2.k.q.3601.7		8
4.3	odd	2		1800.2.k.s.901.3	yes	8
5.2	odd	4		7200.2.d.u.2449.9		16
5.3	odd	4		7200.2.d.u.2449.6		16
5.4	even	2		7200.2.k.t.3601.2		8
8.3	odd	2		1800.2.k.s.901.4	yes	8
8.5	even	2	inner	7200.2.k.q.3601.8		8
12.11	even	2		1800.2.k.s.901.6	yes	8
15.2	even	4		7200.2.d.u.2449.11		16
15.8	even	4		7200.2.d.u.2449.8		16
15.14	odd	2		7200.2.k.t.3601.4		8
20.3	even	4		1800.2.d.u.1549.4		16
20.7	even	4		1800.2.d.u.1549.13		16
20.19	odd	2		1800.2.k.r.901.6	yes	8
24.5	odd	2	inner	7200.2.k.q.3601.6		8
24.11	even	2		1800.2.k.s.901.5	yes	8
40.3	even	4		1800.2.d.u.1549.16		16
40.13	odd	4		7200.2.d.u.2449.7		16
40.19	odd	2		1800.2.k.r.901.5	yes	8
40.27	even	4		1800.2.d.u.1549.1		16
40.29	even	2		7200.2.k.t.3601.3		8
40.37	odd	4		7200.2.d.u.2449.12		16
60.23	odd	4		1800.2.d.u.1549.14		16
60.47	odd	4		1800.2.d.u.1549.3		16
60.59	even	2		1800.2.k.r.901.3	✓	8
120.29	odd	2		7200.2.k.t.3601.1		8
120.53	even	4		7200.2.d.u.2449.5		16
120.59	even	2		1800.2.k.r.901.4	yes	8
120.77	even	4		7200.2.d.u.2449.10		16
120.83	odd	4		1800.2.d.u.1549.2		16
120.107	odd	4		1800.2.d.u.1549.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1800.2.d.u.1549.1		16	40.27	even	4
1800.2.d.u.1549.2		16	120.83	odd	4
1800.2.d.u.1549.3		16	60.47	odd	4
1800.2.d.u.1549.4		16	20.3	even	4
1800.2.d.u.1549.13		16	20.7	even	4
1800.2.d.u.1549.14		16	60.23	odd	4
1800.2.d.u.1549.15		16	120.107	odd	4
1800.2.d.u.1549.16		16	40.3	even	4
1800.2.k.r.901.3	✓	8	60.59	even	2
1800.2.k.r.901.4	yes	8	120.59	even	2
1800.2.k.r.901.5	yes	8	40.19	odd	2
1800.2.k.r.901.6	yes	8	20.19	odd	2
1800.2.k.s.901.3	yes	8	4.3	odd	2
1800.2.k.s.901.4	yes	8	8.3	odd	2
1800.2.k.s.901.5	yes	8	24.11	even	2
1800.2.k.s.901.6	yes	8	12.11	even	2
7200.2.d.u.2449.5		16	120.53	even	4
7200.2.d.u.2449.6		16	5.3	odd	4
7200.2.d.u.2449.7		16	40.13	odd	4
7200.2.d.u.2449.8		16	15.8	even	4
7200.2.d.u.2449.9		16	5.2	odd	4
7200.2.d.u.2449.10		16	120.77	even	4
7200.2.d.u.2449.11		16	15.2	even	4
7200.2.d.u.2449.12		16	40.37	odd	4
7200.2.k.q.3601.5		8	1.1	even	1	trivial
7200.2.k.q.3601.6		8	24.5	odd	2	inner
7200.2.k.q.3601.7		8	3.2	odd	2	inner
7200.2.k.q.3601.8		8	8.5	even	2	inner
7200.2.k.t.3601.1		8	120.29	odd	2
7200.2.k.t.3601.2		8	5.4	even	2
7200.2.k.t.3601.3		8	40.29	even	2
7200.2.k.t.3601.4		8	15.14	odd	2

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)

\(f(q)\)	\(=\)	\(q+1.44949 q^{7} +O(q^{10})\)	\(q+1.44949 q^{7} -1.14152i q^{11} -3.87298i q^{13} -3.05009 q^{17} -0.710706i q^{19} -5.54048 q^{23} +6.22069i q^{29} -3.44949 q^{31} -6.32456i q^{37} -8.03087 q^{41} +7.03526i q^{43} +4.98078 q^{47} -4.89898 q^{49} +3.93765i q^{53} +10.1583i q^{59} +2.45157i q^{61} +13.3598i q^{67} +11.6407 q^{71} -2.89898 q^{73} -1.65462i q^{77} +6.89898 q^{79} +12.4414i q^{83} -16.0617 q^{89} -5.61385i q^{91} -13.6969 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7}+O(q^{10}) \) 8 * q - 8 * q^7	\( 8 q - 8 q^{7} - 8 q^{31} + 16 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 - 8 * q^31 + 16 * q^73 + 16 * q^79 + 8 * q^97

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