LMFDB - Embedded newform 5184.2.c.i.5183.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(5183,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5184, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5184.5183");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.c (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:	$41.3944484078$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 2592)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5183.1
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	5184.5183
Dual form			5184.2.c.i.5183.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93185i q^{5} -0.732051i q^{7} +O(q^{10})$	$q-1.93185i q^{5} -0.732051i q^{7} +5.27792 q^{11} -4.46410 q^{13} -0.896575i q^{17} +1.26795i q^{19} -1.41421 q^{23} +1.26795 q^{25} -5.41662i q^{29} +7.46410i q^{31} -1.41421 q^{35} +7.73205 q^{37} -0.378937i q^{41} -8.73205i q^{43} +4.62158 q^{47} +6.46410 q^{49} -2.44949i q^{53} -10.1962i q^{55} +10.8332 q^{59} +1.19615 q^{61} +8.62398i q^{65} -13.1244i q^{67} -13.2827 q^{71} -13.7321 q^{73} -3.86370i q^{77} -16.5885i q^{79} +10.5558 q^{83} -1.73205 q^{85} +14.9372i q^{89} +3.26795i q^{91} +2.44949 q^{95} -6.39230 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{13} + 24 q^{25} + 48 q^{37} + 24 q^{49} - 32 q^{61} - 96 q^{73} + 32 q^{97}+O(q^{100}) $ 8 * q - 8 * q^13 + 24 * q^25 + 48 * q^37 + 24 * q^49 - 32 * q^61 - 96 * q^73 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 1.93185i	− 0.863950i	−0.901886	−	0.431975i	$-0.857817\pi$
$5$	− 1.93185i	− 0.863950i	0.901886	−	0.431975i	$-0.142183\pi$
$6$	0	0
$7$	− 0.732051i	− 0.276689i	−0.990384	−	0.138345i	$-0.955822\pi$
$7$	− 0.732051i	− 0.276689i	0.990384	−	0.138345i	$-0.0441781\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.27792	1.59135	0.795676	−	0.605723i	$-0.207116\pi$
$11$	5.27792	1.59135	0.795676	+	0.605723i	$0.207116\pi$
$12$	0	0
$13$	−4.46410	−1.23812	−0.619060	−	0.785344i	$-0.712486\pi$
$13$	−4.46410	−1.23812	−0.619060	+	0.785344i	$0.712486\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.896575i	− 0.217451i	−0.994072	−	0.108726i	$-0.965323\pi$
$17$	− 0.896575i	− 0.217451i	0.994072	−	0.108726i	$-0.0346770\pi$
$18$	0	0
$19$	1.26795i	0.290887i	0.989367	+	0.145444i	$0.0464610\pi$
$19$	1.26795i	0.290887i	−0.989367	+	0.145444i	$0.953539\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.41421	−0.294884	−0.147442	−	0.989071i	$-0.547104\pi$
$23$	−1.41421	−0.294884	−0.147442	+	0.989071i	$0.547104\pi$
$24$	0	0
$25$	1.26795	0.253590
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 5.41662i	− 1.00584i	−0.864333	−	0.502920i	$-0.832259\pi$
$29$	− 5.41662i	− 1.00584i	0.864333	−	0.502920i	$-0.167741\pi$
$30$	0	0
$31$	7.46410i	1.34059i	0.742094	+	0.670296i	$0.233833\pi$
$31$	7.46410i	1.34059i	−0.742094	+	0.670296i	$0.766167\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.41421	−0.239046
$36$	0	0
$37$	7.73205	1.27114	0.635571	−	0.772043i	$-0.280765\pi$
$37$	7.73205	1.27114	0.635571	+	0.772043i	$0.280765\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 0.378937i	− 0.0591801i	−0.999562	−	0.0295900i	$-0.990580\pi$
$41$	− 0.378937i	− 0.0591801i	0.999562	−	0.0295900i	$-0.00942018\pi$
$42$	0	0
$43$	− 8.73205i	− 1.33163i	−0.746119	−	0.665813i	$-0.768085\pi$
$43$	− 8.73205i	− 1.33163i	0.746119	−	0.665813i	$-0.231915\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.62158	0.674126	0.337063	−	0.941482i	$-0.390566\pi$
$47$	4.62158	0.674126	0.337063	+	0.941482i	$0.390566\pi$
$48$	0	0
$49$	6.46410	0.923443
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.44949i	− 0.336463i	−0.985747	−	0.168232i	$-0.946194\pi$
$53$	− 2.44949i	− 0.336463i	0.985747	−	0.168232i	$-0.0538057\pi$
$54$	0	0
$55$	− 10.1962i	− 1.37485i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.8332	1.41037	0.705184	−	0.709025i	$-0.250865\pi$
$59$	10.8332	1.41037	0.705184	+	0.709025i	$0.250865\pi$
$60$	0	0
$61$	1.19615	0.153152	0.0765758	−	0.997064i	$-0.475601\pi$
$61$	1.19615	0.153152	0.0765758	+	0.997064i	$0.475601\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8.62398i	1.06967i
$66$	0	0
$67$	− 13.1244i	− 1.60340i	−0.597730	−	0.801698i	$-0.703930\pi$
$67$	− 13.1244i	− 1.60340i	0.597730	−	0.801698i	$-0.296070\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.2827	−1.57637	−0.788185	−	0.615439i	$-0.788979\pi$
$71$	−13.2827	−1.57637	−0.788185	+	0.615439i	$0.788979\pi$
$72$	0	0
$73$	−13.7321	−1.60721	−0.803607	−	0.595160i	$-0.797089\pi$
$73$	−13.7321	−1.60721	−0.803607	+	0.595160i	$0.797089\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.86370i	− 0.440310i
$78$	0	0
$79$	− 16.5885i	− 1.86635i	−0.359426	−	0.933174i	$-0.617027\pi$
$79$	− 16.5885i	− 1.86635i	0.359426	−	0.933174i	$-0.382973\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.5558	1.15865	0.579327	−	0.815095i	$-0.303316\pi$
$83$	10.5558	1.15865	0.579327	+	0.815095i	$0.303316\pi$
$84$	0	0
$85$	−1.73205	−0.187867
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.9372i	1.58334i	0.610951	+	0.791669i	$0.290787\pi$
$89$	14.9372i	1.58334i	−0.610951	+	0.791669i	$0.709213\pi$
$90$	0	0
$91$	3.26795i	0.342574i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.44949	0.251312
$96$	0	0
$97$	−6.39230	−0.649040	−0.324520	−	0.945879i	$-0.605203\pi$
$97$	−6.39230	−0.649040	−0.324520	+	0.945879i	$0.605203\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.31319i	0.628186i	0.949392	+	0.314093i	$0.101700\pi$
$101$	6.31319i	0.628186i	−0.949392	+	0.314093i	$0.898300\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.79315	0.173350	0.0866752	−	0.996237i	$-0.472376\pi$
$107$	1.79315	0.173350	0.0866752	+	0.996237i	$0.472376\pi$
$108$	0	0
$109$	−3.92820	−0.376254	−0.188127	−	0.982145i	$-0.560242\pi$
$109$	−3.92820	−0.376254	−0.188127	+	0.982145i	$0.560242\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	20.2151i	1.90168i	0.309689	+	0.950838i	$0.399775\pi$
$113$	20.2151i	1.90168i	−0.309689	+	0.950838i	$0.600225\pi$
$114$	0	0
$115$	2.73205i	0.254765i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.656339	−0.0601665
$120$	0	0
$121$	16.8564	1.53240
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 12.1087i	− 1.08304i
$126$	0	0
$127$	− 21.1244i	− 1.87448i	−0.348680	−	0.937242i	$-0.613370\pi$
$127$	− 21.1244i	− 1.87448i	0.348680	−	0.937242i	$-0.386630\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−0.101536	−0.00887124	−0.00443562	−	0.999990i	$-0.501412\pi$
$131$	−0.101536	−0.00887124	−0.00443562	+	0.999990i	$0.501412\pi$
$132$	0	0
$133$	0.928203	0.0804854
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.86181i	− 0.415373i	−0.978195	−	0.207686i	$-0.933407\pi$
$137$	− 4.86181i	− 0.415373i	0.978195	−	0.207686i	$-0.0665934\pi$
$138$	0	0
$139$	− 12.3923i	− 1.05110i	−0.850762	−	0.525551i	$-0.823859\pi$
$139$	− 12.3923i	− 1.05110i	0.850762	−	0.525551i	$-0.176141\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−23.5612	−1.97028
$144$	0	0
$145$	−10.4641	−0.868996
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 1.83032i	− 0.149945i	−0.997186	−	0.0749727i	$-0.976113\pi$
$149$	− 1.83032i	− 0.149945i	0.997186	−	0.0749727i	$-0.0238869\pi$
$150$	0	0
$151$	1.07180i	0.0872216i	0.999049	+	0.0436108i	$0.0138862\pi$
$151$	1.07180i	0.0872216i	−0.999049	+	0.0436108i	$0.986114\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	14.4195	1.15821
$156$	0	0
$157$	−10.6603	−0.850781	−0.425390	−	0.905010i	$-0.639863\pi$
$157$	−10.6603	−0.850781	−0.425390	+	0.905010i	$0.639863\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.03528i	0.0815912i
$162$	0	0
$163$	1.60770i	0.125924i	0.998016	+	0.0629622i	$0.0200548\pi$
$163$	1.60770i	0.125924i	−0.998016	+	0.0629622i	$0.979945\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.6574	−0.824692	−0.412346	−	0.911027i	$-0.635291\pi$
$167$	−10.6574	−0.824692	−0.412346	+	0.911027i	$0.635291\pi$
$168$	0	0
$169$	6.92820	0.532939
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 11.0735i	− 0.841900i	−0.907084	−	0.420950i	$-0.861697\pi$
$173$	− 11.0735i	− 0.841900i	0.907084	−	0.420950i	$-0.138303\pi$
$174$	0	0
$175$	− 0.928203i	− 0.0701656i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−21.8695	−1.63461	−0.817303	−	0.576208i	$-0.804532\pi$
$179$	−21.8695	−1.63461	−0.817303	+	0.576208i	$0.804532\pi$
$180$	0	0
$181$	23.3205	1.73340	0.866700	−	0.498830i	$-0.166237\pi$
$181$	23.3205	1.73340	0.866700	+	0.498830i	$0.166237\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 14.9372i	− 1.09820i
$186$	0	0
$187$	− 4.73205i	− 0.346042i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.4553	1.48010	0.740048	−	0.672554i	$-0.234803\pi$
$191$	20.4553	1.48010	0.740048	+	0.672554i	$0.234803\pi$
$192$	0	0
$193$	−8.46410	−0.609259	−0.304630	−	0.952471i	$-0.598533\pi$
$193$	−8.46410	−0.609259	−0.304630	+	0.952471i	$0.598533\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.76079i	0.695428i	0.937601	+	0.347714i	$0.113042\pi$
$197$	9.76079i	0.695428i	−0.937601	+	0.347714i	$0.886958\pi$
$198$	0	0
$199$	− 8.00000i	− 0.567105i	−0.958957	−	0.283552i	$-0.908487\pi$
$199$	− 8.00000i	− 0.567105i	0.958957	−	0.283552i	$-0.0915130\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.96524	−0.278305
$204$	0	0
$205$	−0.732051	−0.0511286
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.69213i	0.462904i
$210$	0	0
$211$	− 19.1244i	− 1.31657i	−0.752767	−	0.658287i	$-0.771281\pi$
$211$	− 19.1244i	− 1.31657i	0.752767	−	0.658287i	$-0.228719\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−16.8690	−1.15046
$216$	0	0
$217$	5.46410	0.370927
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00240i	0.269231i
$222$	0	0
$223$	17.2679i	1.15635i	0.815914	+	0.578174i	$0.196234\pi$
$223$	17.2679i	1.15635i	−0.815914	+	0.578174i	$0.803766\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.3495	1.15153	0.575763	−	0.817616i	$-0.304705\pi$
$227$	17.3495	1.15153	0.575763	+	0.817616i	$0.304705\pi$
$228$	0	0
$229$	−11.0000	−0.726900	−0.363450	−	0.931614i	$-0.618401\pi$
$229$	−11.0000	−0.726900	−0.363450	+	0.931614i	$0.618401\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 18.8009i	− 1.23169i	−0.787869	−	0.615843i	$-0.788815\pi$
$233$	− 18.8009i	− 1.23169i	0.787869	−	0.615843i	$-0.211185\pi$
$234$	0	0
$235$	− 8.92820i	− 0.582412i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.0411	−1.23167	−0.615834	−	0.787876i	$-0.711181\pi$
$239$	−19.0411	−1.23167	−0.615834	+	0.787876i	$0.711181\pi$
$240$	0	0
$241$	−16.1244	−1.03866	−0.519331	−	0.854573i	$-0.673819\pi$
$241$	−16.1244	−1.03866	−0.519331	+	0.854573i	$0.673819\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 12.4877i	− 0.797809i
$246$	0	0
$247$	− 5.66025i	− 0.360153i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−23.8386	−1.50468	−0.752338	−	0.658777i	$-0.771074\pi$
$251$	−23.8386	−1.50468	−0.752338	+	0.658777i	$0.771074\pi$
$252$	0	0
$253$	−7.46410	−0.469264
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.795040i	0.0495932i	0.999693	+	0.0247966i	$0.00789381\pi$
$257$	0.795040i	0.0495932i	−0.999693	+	0.0247966i	$0.992106\pi$
$258$	0	0
$259$	− 5.66025i	− 0.351711i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.45001	0.459387	0.229694	−	0.973263i	$-0.426228\pi$
$263$	7.45001	0.459387	0.229694	+	0.973263i	$0.426228\pi$
$264$	0	0
$265$	−4.73205	−0.290688
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 22.7661i	− 1.38807i	−0.719939	−	0.694037i	$-0.755830\pi$
$269$	− 22.7661i	− 1.38807i	0.719939	−	0.694037i	$-0.244170\pi$
$270$	0	0
$271$	13.2679i	0.805971i	0.915206	+	0.402985i	$0.132027\pi$
$271$	13.2679i	0.805971i	−0.915206	+	0.402985i	$0.867973\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.69213	0.403551
$276$	0	0
$277$	−1.07180	−0.0643980	−0.0321990	−	0.999481i	$-0.510251\pi$
$277$	−1.07180	−0.0643980	−0.0321990	+	0.999481i	$0.510251\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.4524i	0.683193i	0.939847	+	0.341597i	$0.110968\pi$
$281$	11.4524i	0.683193i	−0.939847	+	0.341597i	$0.889032\pi$
$282$	0	0
$283$	− 19.3205i	− 1.14848i	−0.818685	−	0.574242i	$-0.805297\pi$
$283$	− 19.3205i	− 1.14848i	0.818685	−	0.574242i	$-0.194703\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.277401	−0.0163745
$288$	0	0
$289$	16.1962	0.952715
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 23.8014i	− 1.39049i	−0.718772	−	0.695246i	$-0.755295\pi$
$293$	− 23.8014i	− 1.39049i	0.718772	−	0.695246i	$-0.244705\pi$
$294$	0	0
$295$	− 20.9282i	− 1.21849i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.31319	0.365101
$300$	0	0
$301$	−6.39230	−0.368446
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2.31079i	− 0.132315i
$306$	0	0
$307$	− 18.0000i	− 1.02731i	−0.857996	−	0.513657i	$-0.828290\pi$
$307$	− 18.0000i	− 1.02731i	0.857996	−	0.513657i	$-0.171710\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.2175	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$311$	24.2175	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$312$	0	0
$313$	28.3205	1.60077	0.800385	−	0.599486i	$-0.204628\pi$
$313$	28.3205	1.60077	0.800385	+	0.599486i	$0.204628\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.31130i	0.410644i	0.978694	+	0.205322i	$0.0658241\pi$
$317$	7.31130i	0.410644i	−0.978694	+	0.205322i	$0.934176\pi$
$318$	0	0
$319$	− 28.5885i	− 1.60065i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.13681	0.0632539
$324$	0	0
$325$	−5.66025	−0.313974
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3.38323i	− 0.186524i
$330$	0	0
$331$	7.80385i	0.428938i	0.976731	+	0.214469i	$0.0688021\pi$
$331$	7.80385i	0.428938i	−0.976731	+	0.214469i	$0.931198\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−25.3543	−1.38525
$336$	0	0
$337$	−4.92820	−0.268456	−0.134228	−	0.990950i	$-0.542856\pi$
$337$	−4.92820	−0.268456	−0.134228	+	0.990950i	$0.542856\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	39.3949i	2.13335i
$342$	0	0
$343$	− 9.85641i	− 0.532196i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.7632	−0.738847	−0.369424	−	0.929261i	$-0.620445\pi$
$347$	−13.7632	−0.738847	−0.369424	+	0.929261i	$0.620445\pi$
$348$	0	0
$349$	2.14359	0.114744	0.0573720	−	0.998353i	$-0.481728\pi$
$349$	2.14359	0.114744	0.0573720	+	0.998353i	$0.481728\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 27.9797i	− 1.48921i	−0.667507	−	0.744604i	$-0.732639\pi$
$353$	− 27.9797i	− 1.48921i	0.667507	−	0.744604i	$-0.267361\pi$
$354$	0	0
$355$	25.6603i	1.36190i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.44949	−0.129279	−0.0646396	−	0.997909i	$-0.520590\pi$
$359$	−2.44949	−0.129279	−0.0646396	+	0.997909i	$0.520590\pi$
$360$	0	0
$361$	17.3923	0.915384
$362$	0	0
$363$	0	0
$364$	0	0
$365$	26.5283i	1.38855i
$366$	0	0
$367$	− 12.5359i	− 0.654369i	−0.944961	−	0.327184i	$-0.893900\pi$
$367$	− 12.5359i	− 0.654369i	0.944961	−	0.327184i	$-0.106100\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.79315	−0.0930958
$372$	0	0
$373$	32.2487	1.66977	0.834887	−	0.550421i	$-0.185533\pi$
$373$	32.2487	1.66977	0.834887	+	0.550421i	$0.185533\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.1803i	1.24535i
$378$	0	0
$379$	− 14.3923i	− 0.739283i	−0.929174	−	0.369642i	$-0.879480\pi$
$379$	− 14.3923i	− 0.739283i	0.929174	−	0.369642i	$-0.120520\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.2170	0.981942	0.490971	−	0.871176i	$-0.336642\pi$
$383$	19.2170	0.981942	0.490971	+	0.871176i	$0.336642\pi$
$384$	0	0
$385$	−7.46410	−0.380406
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.7317i	0.544119i	0.962280	+	0.272059i	$0.0877047\pi$
$389$	10.7317i	0.544119i	−0.962280	+	0.272059i	$0.912295\pi$
$390$	0	0
$391$	1.26795i	0.0641229i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−32.0464	−1.61243
$396$	0	0
$397$	−21.5885	−1.08349	−0.541747	−	0.840542i	$-0.682237\pi$
$397$	−21.5885	−1.08349	−0.541747	+	0.840542i	$0.682237\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.8009i	0.938871i	0.882967	+	0.469436i	$0.155543\pi$
$401$	18.8009i	0.938871i	−0.882967	+	0.469436i	$0.844457\pi$
$402$	0	0
$403$	− 33.3205i	− 1.65981i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	40.8091	2.02283
$408$	0	0
$409$	14.8038	0.732003	0.366002	−	0.930614i	$-0.380726\pi$
$409$	14.8038	0.732003	0.366002	+	0.930614i	$0.380726\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 7.93048i	− 0.390233i
$414$	0	0
$415$	− 20.3923i	− 1.00102i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.59008	0.0776804	0.0388402	−	0.999245i	$-0.487634\pi$
$419$	1.59008	0.0776804	0.0388402	+	0.999245i	$0.487634\pi$
$420$	0	0
$421$	−8.85641	−0.431635	−0.215817	−	0.976434i	$-0.569242\pi$
$421$	−8.85641	−0.431635	−0.215817	+	0.976434i	$0.569242\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.13681i	− 0.0551435i
$426$	0	0
$427$	− 0.875644i	− 0.0423754i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.48528	−0.408722	−0.204361	−	0.978896i	$-0.565512\pi$
$431$	−8.48528	−0.408722	−0.204361	+	0.978896i	$0.565512\pi$
$432$	0	0
$433$	−34.1769	−1.64244	−0.821219	−	0.570613i	$-0.806706\pi$
$433$	−34.1769	−1.64244	−0.821219	+	0.570613i	$0.806706\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.79315i	− 0.0857780i
$438$	0	0
$439$	− 7.46410i	− 0.356242i	−0.984009	−	0.178121i	$-0.942998\pi$
$439$	− 7.46410i	− 0.356242i	0.984009	−	0.178121i	$-0.0570019\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.0449	−0.809827	−0.404914	−	0.914355i	$-0.632698\pi$
$443$	−17.0449	−0.809827	−0.404914	+	0.914355i	$0.632698\pi$
$444$	0	0
$445$	28.8564	1.36792
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 23.0064i	− 1.08574i	−0.839818	−	0.542868i	$-0.817338\pi$
$449$	− 23.0064i	− 1.08574i	0.839818	−	0.542868i	$-0.182662\pi$
$450$	0	0
$451$	− 2.00000i	− 0.0941763i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.31319	0.295967
$456$	0	0
$457$	−2.80385	−0.131158	−0.0655792	−	0.997847i	$-0.520890\pi$
$457$	−2.80385	−0.131158	−0.0655792	+	0.997847i	$0.520890\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	35.6327i	1.65958i	0.558074	+	0.829791i	$0.311540\pi$
$461$	35.6327i	1.65958i	−0.558074	+	0.829791i	$0.688460\pi$
$462$	0	0
$463$	− 18.3923i	− 0.854763i	−0.904071	−	0.427381i	$-0.859436\pi$
$463$	− 18.3923i	− 0.854763i	0.904071	−	0.427381i	$-0.140564\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0716	0.558606	0.279303	−	0.960203i	$-0.409897\pi$
$467$	12.0716	0.558606	0.279303	+	0.960203i	$0.409897\pi$
$468$	0	0
$469$	−9.60770	−0.443642
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 46.0870i	− 2.11908i
$474$	0	0
$475$	1.60770i	0.0737661i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	31.2886	1.42961	0.714805	−	0.699323i	$-0.246515\pi$
$479$	31.2886	1.42961	0.714805	+	0.699323i	$0.246515\pi$
$480$	0	0
$481$	−34.5167	−1.57382
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.3490i	0.560739i
$486$	0	0
$487$	36.9282i	1.67338i	0.547679	+	0.836688i	$0.315511\pi$
$487$	36.9282i	1.67338i	−0.547679	+	0.836688i	$0.684489\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14.2437	0.642808	0.321404	−	0.946942i	$-0.395845\pi$
$491$	14.2437	0.642808	0.321404	+	0.946942i	$0.395845\pi$
$492$	0	0
$493$	−4.85641	−0.218722
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.72363i	0.436164i
$498$	0	0
$499$	36.4449i	1.63150i	0.578407	+	0.815748i	$0.303674\pi$
$499$	36.4449i	1.63150i	−0.578407	+	0.815748i	$0.696326\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	29.1165	1.29824	0.649120	−	0.760686i	$-0.275137\pi$
$503$	29.1165	1.29824	0.649120	+	0.760686i	$0.275137\pi$
$504$	0	0
$505$	12.1962	0.542722
$506$	0	0
$507$	0	0
$508$	0	0
$509$	36.9454i	1.63758i	0.574095	+	0.818788i	$0.305354\pi$
$509$	36.9454i	1.63758i	−0.574095	+	0.818788i	$0.694646\pi$
$510$	0	0
$511$	10.0526i	0.444699i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−7.72741	−0.340510
$516$	0	0
$517$	24.3923	1.07277
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.2789i	0.669383i	0.942328	+	0.334691i	$0.108632\pi$
$521$	15.2789i	0.669383i	−0.942328	+	0.334691i	$0.891368\pi$
$522$	0	0
$523$	− 16.1962i	− 0.708208i	−0.935206	−	0.354104i	$-0.884786\pi$
$523$	− 16.1962i	− 0.708208i	0.935206	−	0.354104i	$-0.115214\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.69213	0.291514
$528$	0	0
$529$	−21.0000	−0.913043
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.69161i	0.0732720i
$534$	0	0
$535$	− 3.46410i	− 0.149766i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	34.1170	1.46952
$540$	0	0
$541$	−7.00000	−0.300954	−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000	−0.300954	−0.150477	+	0.988614i	$0.548081\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.58871i	0.325064i
$546$	0	0
$547$	− 1.32051i	− 0.0564608i	−0.999601	−	0.0282304i	$-0.991013\pi$
$547$	− 1.32051i	− 0.0564608i	0.999601	−	0.0282304i	$-0.00898722\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.86800	0.292586
$552$	0	0
$553$	−12.1436	−0.516398
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.03339i	0.0861574i	0.999072	+	0.0430787i	$0.0137166\pi$
$557$	2.03339i	0.0861574i	−0.999072	+	0.0430787i	$0.986283\pi$
$558$	0	0
$559$	38.9808i	1.64871i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.55961	−0.360745	−0.180372	−	0.983598i	$-0.557730\pi$
$563$	−8.55961	−0.360745	−0.180372	+	0.983598i	$0.557730\pi$
$564$	0	0
$565$	39.0526	1.64295
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 7.03390i	− 0.294877i	−0.989071	−	0.147438i	$-0.952897\pi$
$569$	− 7.03390i	− 0.294877i	0.989071	−	0.147438i	$-0.0471028\pi$
$570$	0	0
$571$	39.8564i	1.66794i	0.551811	+	0.833969i	$0.313937\pi$
$571$	39.8564i	1.66794i	−0.551811	+	0.833969i	$0.686063\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.79315	−0.0747796
$576$	0	0
$577$	18.4641	0.768671	0.384335	−	0.923194i	$-0.374431\pi$
$577$	18.4641	0.768671	0.384335	+	0.923194i	$0.374431\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 7.72741i	− 0.320587i
$582$	0	0
$583$	− 12.9282i	− 0.535431i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.69161	−0.0698204	−0.0349102	−	0.999390i	$-0.511115\pi$
$587$	−1.69161	−0.0698204	−0.0349102	+	0.999390i	$0.511115\pi$
$588$	0	0
$589$	−9.46410	−0.389962
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 21.1488i	− 0.868478i	−0.900798	−	0.434239i	$-0.857017\pi$
$593$	− 21.1488i	− 0.868478i	0.900798	−	0.434239i	$-0.142983\pi$
$594$	0	0
$595$	1.26795i	0.0519808i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−14.4195	−0.589166	−0.294583	−	0.955626i	$-0.595181\pi$
$599$	−14.4195	−0.589166	−0.294583	+	0.955626i	$0.595181\pi$
$600$	0	0
$601$	3.78461	0.154377	0.0771887	−	0.997016i	$-0.475406\pi$
$601$	3.78461	0.154377	0.0771887	+	0.997016i	$0.475406\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 32.5641i	− 1.32392i
$606$	0	0
$607$	6.73205i	0.273246i	0.990623	+	0.136623i	$0.0436248\pi$
$607$	6.73205i	0.273246i	−0.990623	+	0.136623i	$0.956375\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−20.6312	−0.834649
$612$	0	0
$613$	−13.6077	−0.549610	−0.274805	−	0.961500i	$-0.588613\pi$
$613$	−13.6077	−0.549610	−0.274805	+	0.961500i	$0.588613\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9.28032i	− 0.373612i	−0.982397	−	0.186806i	$-0.940186\pi$
$617$	− 9.28032i	− 0.373612i	0.982397	−	0.186806i	$-0.0598135\pi$
$618$	0	0
$619$	4.92820i	0.198081i	0.995083	+	0.0990406i	$0.0315774\pi$
$619$	4.92820i	0.198081i	−0.995083	+	0.0990406i	$0.968423\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.9348	0.438092
$624$	0	0
$625$	−17.0526	−0.682102
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 6.93237i	− 0.276412i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−40.8091	−1.61946
$636$	0	0
$637$	−28.8564	−1.14333
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 37.0841i	− 1.46473i	−0.680910	−	0.732367i	$-0.738415\pi$
$641$	− 37.0841i	− 1.46473i	0.680910	−	0.732367i	$-0.261585\pi$
$642$	0	0
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.6660	0.655206	0.327603	−	0.944815i	$-0.393759\pi$
$647$	16.6660	0.655206	0.327603	+	0.944815i	$0.393759\pi$
$648$	0	0
$649$	57.1769	2.24439
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 11.1378i	− 0.435857i	−0.975965	−	0.217929i	$-0.930070\pi$
$653$	− 11.1378i	− 0.435857i	0.975965	−	0.217929i	$-0.0699300\pi$
$654$	0	0
$655$	0.196152i	0.00766431i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21.7680	0.847961	0.423981	−	0.905671i	$-0.360632\pi$
$659$	21.7680	0.847961	0.423981	+	0.905671i	$0.360632\pi$
$660$	0	0
$661$	8.51666	0.331260	0.165630	−	0.986188i	$-0.447034\pi$
$661$	8.51666	0.331260	0.165630	+	0.986188i	$0.447034\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.79315i	− 0.0695354i
$666$	0	0
$667$	7.66025i	0.296606i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.31319	0.243718
$672$	0	0
$673$	30.3205	1.16877	0.584385	−	0.811477i	$-0.301336\pi$
$673$	30.3205	1.16877	0.584385	+	0.811477i	$0.301336\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.8129i	1.49170i	0.666113	+	0.745850i	$0.267957\pi$
$677$	38.8129i	1.49170i	−0.666113	+	0.745850i	$0.732043\pi$
$678$	0	0
$679$	4.67949i	0.179582i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5.37945	−0.205839	−0.102920	−	0.994690i	$-0.532818\pi$
$683$	−5.37945	−0.205839	−0.102920	+	0.994690i	$0.532818\pi$
$684$	0	0
$685$	−9.39230	−0.358862
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10.9348i	0.416582i
$690$	0	0
$691$	2.58846i	0.0984696i	0.998787	+	0.0492348i	$0.0156783\pi$
$691$	2.58846i	0.0984696i	−0.998787	+	0.0492348i	$0.984322\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−23.9401	−0.908100
$696$	0	0
$697$	−0.339746	−0.0128688
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 8.06918i	− 0.304769i	−0.988321	−	0.152384i	$-0.951305\pi$
$701$	− 8.06918i	− 0.304769i	0.988321	−	0.152384i	$-0.0486952\pi$
$702$	0	0
$703$	9.80385i	0.369759i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.62158	0.173812
$708$	0	0
$709$	13.0000	0.488225	0.244113	−	0.969747i	$-0.421503\pi$
$709$	13.0000	0.488225	0.244113	+	0.969747i	$0.421503\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 10.5558i	− 0.395319i
$714$	0	0
$715$	45.5167i	1.70223i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	36.3906	1.35714	0.678570	−	0.734535i	$-0.262600\pi$
$719$	36.3906	1.35714	0.678570	+	0.734535i	$0.262600\pi$
$720$	0	0
$721$	−2.92820	−0.109052
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 6.86800i	− 0.255071i
$726$	0	0
$727$	25.9090i	0.960910i	0.877019	+	0.480455i	$0.159529\pi$
$727$	25.9090i	0.960910i	−0.877019	+	0.480455i	$0.840471\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7.82894	−0.289564
$732$	0	0
$733$	−39.3205	−1.45234	−0.726168	−	0.687517i	$-0.758701\pi$
$733$	−39.3205	−1.45234	−0.726168	+	0.687517i	$0.758701\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 69.2693i	− 2.55157i
$738$	0	0
$739$	− 32.0000i	− 1.17714i	−0.808447	−	0.588570i	$-0.799691\pi$
$739$	− 32.0000i	− 1.17714i	0.808447	−	0.588570i	$-0.200309\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.8000	1.09326	0.546628	−	0.837375i	$-0.315911\pi$
$743$	29.8000	1.09326	0.546628	+	0.837375i	$0.315911\pi$
$744$	0	0
$745$	−3.53590	−0.129545
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1.31268i	− 0.0479642i
$750$	0	0
$751$	− 9.41154i	− 0.343432i	−0.985146	−	0.171716i	$-0.945069\pi$
$751$	− 9.41154i	− 0.343432i	0.985146	−	0.171716i	$-0.0549312\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.07055	0.0753551
$756$	0	0
$757$	24.7846	0.900812	0.450406	−	0.892824i	$-0.351279\pi$
$757$	24.7846	0.900812	0.450406	+	0.892824i	$0.351279\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 2.10772i	− 0.0764047i	−0.999270	−	0.0382023i	$-0.987837\pi$
$761$	− 2.10772i	− 0.0764047i	0.999270	−	0.0382023i	$-0.0121631\pi$
$762$	0	0
$763$	2.87564i	0.104105i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−48.3607	−1.74620
$768$	0	0
$769$	−40.0333	−1.44364	−0.721819	−	0.692082i	$-0.756694\pi$
$769$	−40.0333	−1.44364	−0.721819	+	0.692082i	$0.756694\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	35.9745i	1.29391i	0.762527	+	0.646957i	$0.223959\pi$
$773$	35.9745i	1.29391i	−0.762527	+	0.646957i	$0.776041\pi$
$774$	0	0
$775$	9.46410i	0.339961i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.480473	0.0172147
$780$	0	0
$781$	−70.1051	−2.50856
$782$	0	0
$783$	0	0
$784$	0	0
$785$	20.5940i	0.735032i
$786$	0	0
$787$	− 24.7321i	− 0.881602i	−0.897605	−	0.440801i	$-0.854694\pi$
$787$	− 24.7321i	− 0.881602i	0.897605	−	0.440801i	$-0.145306\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	14.7985	0.526173
$792$	0	0
$793$	−5.33975	−0.189620
$794$	0	0
$795$	0	0
$796$	0	0
$797$	51.2263i	1.81453i	0.420563	+	0.907264i	$0.361833\pi$
$797$	51.2263i	1.81453i	−0.420563	+	0.907264i	$0.638167\pi$
$798$	0	0
$799$	− 4.14359i	− 0.146590i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−72.4766	−2.55764
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	0	0
$808$	0	0
$809$	40.3930i	1.42014i	0.704130	+	0.710071i	$0.251337\pi$
$809$	40.3930i	1.42014i	−0.704130	+	0.710071i	$0.748663\pi$
$810$	0	0
$811$	46.6410i	1.63779i	0.573945	+	0.818894i	$0.305412\pi$
$811$	46.6410i	1.63779i	−0.573945	+	0.818894i	$0.694588\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.10583	0.108792
$816$	0	0
$817$	11.0718	0.387353
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 47.1867i	− 1.64683i	−0.567442	−	0.823413i	$-0.692067\pi$
$821$	− 47.1867i	− 1.64683i	0.567442	−	0.823413i	$-0.307933\pi$
$822$	0	0
$823$	− 3.60770i	− 0.125756i	−0.998021	−	0.0628782i	$-0.979972\pi$
$823$	− 3.60770i	− 0.125756i	0.998021	−	0.0628782i	$-0.0200280\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.92996	−0.101885	−0.0509424	−	0.998702i	$-0.516222\pi$
$827$	−2.92996	−0.101885	−0.0509424	+	0.998702i	$0.516222\pi$
$828$	0	0
$829$	−15.7128	−0.545729	−0.272864	−	0.962053i	$-0.587971\pi$
$829$	−15.7128	−0.545729	−0.272864	+	0.962053i	$0.587971\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 5.79555i	− 0.200804i
$834$	0	0
$835$	20.5885i	0.712493i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−29.0149	−1.00171	−0.500853	−	0.865532i	$-0.666980\pi$
$839$	−29.0149	−1.00171	−0.500853	+	0.865532i	$0.666980\pi$
$840$	0	0
$841$	−0.339746	−0.0117154
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 13.3843i	− 0.460433i
$846$	0	0
$847$	− 12.3397i	− 0.423999i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10.9348	−0.374839
$852$	0	0
$853$	43.5692	1.49178	0.745891	−	0.666068i	$-0.232024\pi$
$853$	43.5692	1.49178	0.745891	+	0.666068i	$0.232024\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.0371647i	0.00126952i	1.00000		0.000634762i	$0.000202051\pi$
$857$	0.0371647i	0.00126952i	−1.00000		0.000634762i	$0.999798\pi$
$858$	0	0
$859$	− 36.5359i	− 1.24659i	−0.781987	−	0.623294i	$-0.785794\pi$
$859$	− 36.5359i	− 1.24659i	0.781987	−	0.623294i	$-0.214206\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.5355	1.31176	0.655882	−	0.754864i	$-0.272297\pi$
$863$	38.5355	1.31176	0.655882	+	0.754864i	$0.272297\pi$
$864$	0	0
$865$	−21.3923	−0.727360
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 87.5525i	− 2.97002i
$870$	0	0
$871$	58.5885i	1.98519i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−8.86422	−0.299665
$876$	0	0
$877$	−4.12436	−0.139270	−0.0696348	−	0.997573i	$-0.522183\pi$
$877$	−4.12436	−0.139270	−0.0696348	+	0.997573i	$0.522183\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 13.2084i	− 0.445002i	−0.974932	−	0.222501i	$-0.928578\pi$
$881$	− 13.2084i	− 0.445002i	0.974932	−	0.222501i	$-0.0714221\pi$
$882$	0	0
$883$	− 0.679492i	− 0.0228667i	−0.999935	−	0.0114334i	$-0.996361\pi$
$883$	− 0.679492i	− 0.0228667i	0.999935	−	0.0114334i	$-0.00363943\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.5964	−0.825867	−0.412934	−	0.910761i	$-0.635496\pi$
$887$	−24.5964	−0.825867	−0.412934	+	0.910761i	$0.635496\pi$
$888$	0	0
$889$	−15.4641	−0.518649
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.85993i	0.196095i
$894$	0	0
$895$	42.2487i	1.41222i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	40.4302	1.34842
$900$	0	0
$901$	−2.19615	−0.0731644
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 45.0518i	− 1.49757i
$906$	0	0
$907$	− 12.5359i	− 0.416248i	−0.978102	−	0.208124i	$-0.933264\pi$
$907$	− 12.5359i	− 0.416248i	0.978102	−	0.208124i	$-0.0667357\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10.7317	0.355557	0.177779	−	0.984071i	$-0.443109\pi$
$911$	10.7317	0.355557	0.177779	+	0.984071i	$0.443109\pi$
$912$	0	0
$913$	55.7128	1.84382
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.0743295i	0.00245458i
$918$	0	0
$919$	38.4449i	1.26818i	0.773260	+	0.634090i	$0.218625\pi$
$919$	38.4449i	1.26818i	−0.773260	+	0.634090i	$0.781375\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	59.2954	1.95173
$924$	0	0
$925$	9.80385	0.322349
$926$	0	0
$927$	0	0
$928$	0	0
$929$	40.0141i	1.31282i	0.754405	+	0.656410i	$0.227926\pi$
$929$	40.0141i	1.31282i	−0.754405	+	0.656410i	$0.772074\pi$
$930$	0	0
$931$	8.19615i	0.268618i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.14162	−0.298963
$936$	0	0
$937$	7.39230	0.241496	0.120748	−	0.992683i	$-0.461471\pi$
$937$	7.39230	0.241496	0.120748	+	0.992683i	$0.461471\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	48.1948i	1.57110i	0.618796	+	0.785552i	$0.287621\pi$
$941$	48.1948i	1.57110i	−0.618796	+	0.785552i	$0.712379\pi$
$942$	0	0
$943$	0.535898i	0.0174513i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−43.8406	−1.42463	−0.712314	−	0.701861i	$-0.752353\pi$
$947$	−43.8406	−1.42463	−0.712314	+	0.701861i	$0.752353\pi$
$948$	0	0
$949$	61.3013	1.98992
$950$	0	0
$951$	0	0
$952$	0	0
$953$	42.1862i	1.36654i	0.730164	+	0.683272i	$0.239444\pi$
$953$	42.1862i	1.36654i	−0.730164	+	0.683272i	$0.760556\pi$
$954$	0	0
$955$	− 39.5167i	− 1.27873i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.55910	−0.114929
$960$	0	0
$961$	−24.7128	−0.797188
$962$	0	0
$963$	0	0
$964$	0	0
$965$	16.3514i	0.526370i
$966$	0	0
$967$	− 38.3397i	− 1.23292i	−0.787385	−	0.616462i	$-0.788566\pi$
$967$	− 38.3397i	− 1.23292i	0.787385	−	0.616462i	$-0.211434\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	50.0523	1.60625	0.803127	−	0.595808i	$-0.203168\pi$
$971$	50.0523	1.60625	0.803127	+	0.595808i	$0.203168\pi$
$972$	0	0
$973$	−9.07180	−0.290828
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43.1571i	1.38072i	0.723467	+	0.690359i	$0.242547\pi$
$977$	43.1571i	1.38072i	−0.723467	+	0.690359i	$0.757453\pi$
$978$	0	0
$979$	78.8372i	2.51965i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−43.4617	−1.38621	−0.693106	−	0.720835i	$-0.743758\pi$
$983$	−43.4617	−1.38621	−0.693106	+	0.720835i	$0.743758\pi$
$984$	0	0
$985$	18.8564	0.600815
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.3490i	0.392675i
$990$	0	0
$991$	26.4449i	0.840049i	0.907513	+	0.420024i	$0.137978\pi$
$991$	26.4449i	0.840049i	−0.907513	+	0.420024i	$0.862022\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−15.4548	−0.489951
$996$	0	0
$997$	3.19615	0.101223	0.0506116	−	0.998718i	$-0.483883\pi$
$997$	3.19615	0.101223	0.0506116	+	0.998718i	$0.483883\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.c.i.5183.1		8
3.2	odd	2	inner	5184.2.c.i.5183.7		8
4.3	odd	2	inner	5184.2.c.i.5183.2		8
8.3	odd	2		2592.2.c.a.2591.8	yes	8
8.5	even	2		2592.2.c.a.2591.7	yes	8
12.11	even	2	inner	5184.2.c.i.5183.8		8
24.5	odd	2		2592.2.c.a.2591.1	✓	8
24.11	even	2		2592.2.c.a.2591.2	yes	8
72.5	odd	6		2592.2.s.f.1727.1		8
72.11	even	6		2592.2.s.f.863.4		8
72.13	even	6		2592.2.s.f.1727.4		8
72.29	odd	6		2592.2.s.b.863.4		8
72.43	odd	6		2592.2.s.f.863.1		8
72.59	even	6		2592.2.s.b.1727.1		8
72.61	even	6		2592.2.s.b.863.1		8
72.67	odd	6		2592.2.s.b.1727.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2592.2.c.a.2591.1	✓	8	24.5	odd	2
2592.2.c.a.2591.2	yes	8	24.11	even	2
2592.2.c.a.2591.7	yes	8	8.5	even	2
2592.2.c.a.2591.8	yes	8	8.3	odd	2
2592.2.s.b.863.1		8	72.61	even	6
2592.2.s.b.863.4		8	72.29	odd	6
2592.2.s.b.1727.1		8	72.59	even	6
2592.2.s.b.1727.4		8	72.67	odd	6
2592.2.s.f.863.1		8	72.43	odd	6
2592.2.s.f.863.4		8	72.11	even	6
2592.2.s.f.1727.1		8	72.5	odd	6
2592.2.s.f.1727.4		8	72.13	even	6
5184.2.c.i.5183.1		8	1.1	even	1	trivial
5184.2.c.i.5183.2		8	4.3	odd	2	inner
5184.2.c.i.5183.7		8	3.2	odd	2	inner
5184.2.c.i.5183.8		8	12.11	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:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.93185i q^{5} -0.732051i q^{7} +O(q^{10})\)	\(q-1.93185i q^{5} -0.732051i q^{7} +5.27792 q^{11} -4.46410 q^{13} -0.896575i q^{17} +1.26795i q^{19} -1.41421 q^{23} +1.26795 q^{25} -5.41662i q^{29} +7.46410i q^{31} -1.41421 q^{35} +7.73205 q^{37} -0.378937i q^{41} -8.73205i q^{43} +4.62158 q^{47} +6.46410 q^{49} -2.44949i q^{53} -10.1962i q^{55} +10.8332 q^{59} +1.19615 q^{61} +8.62398i q^{65} -13.1244i q^{67} -13.2827 q^{71} -13.7321 q^{73} -3.86370i q^{77} -16.5885i q^{79} +10.5558 q^{83} -1.73205 q^{85} +14.9372i q^{89} +3.26795i q^{91} +2.44949 q^{95} -6.39230 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{13} + 24 q^{25} + 48 q^{37} + 24 q^{49} - 32 q^{61} - 96 q^{73} + 32 q^{97}+O(q^{100}) \) 8 * q - 8 * q^13 + 24 * q^25 + 48 * q^37 + 24 * q^49 - 32 * q^61 - 96 * q^73 + 32 * q^97

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