LMFDB - Embedded newform 6048.2.j.c.5615.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(5615,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6048.5615");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.j (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:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5615.3
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.c.5615.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.47003 q^{5} +1.00000i q^{7} +O(q^{10})$	$q-3.47003 q^{5} +1.00000i q^{7} +5.11891i q^{11} +0.268602i q^{13} -1.96616i q^{17} -0.909926 q^{19} +5.86455 q^{23} +7.04110 q^{25} +7.19959 q^{29} -4.57491i q^{31} -3.47003i q^{35} -6.98312i q^{37} +8.34605i q^{41} +9.25237 q^{43} -6.49316 q^{47} -1.00000 q^{49} -4.54325 q^{53} -17.7628i q^{55} +3.50524i q^{59} -1.96254i q^{61} -0.932057i q^{65} +11.0280 q^{67} +1.20092 q^{71} -9.01697 q^{73} -5.11891 q^{77} -13.5570i q^{79} -4.96335i q^{83} +6.82263i q^{85} +18.2032i q^{89} -0.268602 q^{91} +3.15747 q^{95} +3.39139 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 64 q^{19} - 16 q^{25} + 48 q^{43} - 32 q^{49} - 16 q^{67} - 16 q^{73} - 16 q^{91} + 64 q^{97}+O(q^{100}) $ 32 * q + 64 * q^19 - 16 * q^25 + 48 * q^43 - 32 * q^49 - 16 * q^67 - 16 * q^73 - 16 * q^91 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$5$	−3.47003	−1.55184	−0.775922	−	0.630829i	$-0.782715\pi$
$5$	−3.47003	−1.55184	−0.775922	+	0.630829i	$0.782715\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.11891i	1.54341i	0.635981	+	0.771705i	$0.280596\pi$
$11$	5.11891i	1.54341i	−0.635981	+	0.771705i	$0.719404\pi$
$12$	0	0
$13$	0.268602i	0.0744968i	0.999306	+	0.0372484i	$0.0118593\pi$
$13$	0.268602i	0.0744968i	−0.999306	+	0.0372484i	$0.988141\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.96616i	− 0.476864i	−0.971159	−	0.238432i	$-0.923367\pi$
$17$	− 1.96616i	− 0.476864i	0.971159	−	0.238432i	$-0.0766334\pi$
$18$	0	0
$19$	−0.909926	−0.208751	−0.104376	−	0.994538i	$-0.533284\pi$
$19$	−0.909926	−0.208751	−0.104376	+	0.994538i	$0.533284\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.86455	1.22284	0.611421	−	0.791305i	$-0.290598\pi$
$23$	5.86455	1.22284	0.611421	+	0.791305i	$0.290598\pi$
$24$	0	0
$25$	7.04110	1.40822
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.19959	1.33693	0.668466	−	0.743743i	$-0.266951\pi$
$29$	7.19959	1.33693	0.668466	+	0.743743i	$0.266951\pi$
$30$	0	0
$31$	− 4.57491i	− 0.821678i	−0.911708	−	0.410839i	$-0.865236\pi$
$31$	− 4.57491i	− 0.821678i	0.911708	−	0.410839i	$-0.134764\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 3.47003i	− 0.586542i
$36$	0	0
$37$	− 6.98312i	− 1.14802i	−0.818849	−	0.574009i	$-0.805388\pi$
$37$	− 6.98312i	− 1.14802i	0.818849	−	0.574009i	$-0.194612\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.34605i	1.30343i	0.758462	+	0.651717i	$0.225951\pi$
$41$	8.34605i	1.30343i	−0.758462	+	0.651717i	$0.774049\pi$
$42$	0	0
$43$	9.25237	1.41097	0.705487	−	0.708723i	$-0.250728\pi$
$43$	9.25237	1.41097	0.705487	+	0.708723i	$0.250728\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.49316	−0.947125	−0.473562	−	0.880760i	$-0.657032\pi$
$47$	−6.49316	−0.947125	−0.473562	+	0.880760i	$0.657032\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.54325	−0.624063	−0.312032	−	0.950072i	$-0.601010\pi$
$53$	−4.54325	−0.624063	−0.312032	+	0.950072i	$0.601010\pi$
$54$	0	0
$55$	− 17.7628i	− 2.39513i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.50524i	0.456344i	0.973621	+	0.228172i	$0.0732748\pi$
$59$	3.50524i	0.456344i	−0.973621	+	0.228172i	$0.926725\pi$
$60$	0	0
$61$	− 1.96254i	− 0.251278i	−0.992076	−	0.125639i	$-0.959902\pi$
$61$	− 1.96254i	− 0.251278i	0.992076	−	0.125639i	$-0.0400980\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 0.932057i	− 0.115607i
$66$	0	0
$67$	11.0280	1.34728	0.673641	−	0.739059i	$-0.264729\pi$
$67$	11.0280	1.34728	0.673641	+	0.739059i	$0.264729\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.20092	0.142523	0.0712616	−	0.997458i	$-0.477297\pi$
$71$	1.20092	0.142523	0.0712616	+	0.997458i	$0.477297\pi$
$72$	0	0
$73$	−9.01697	−1.05536	−0.527679	−	0.849444i	$-0.676937\pi$
$73$	−9.01697	−1.05536	−0.527679	+	0.849444i	$0.676937\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.11891	−0.583354
$78$	0	0
$79$	− 13.5570i	− 1.52528i	−0.646825	−	0.762638i	$-0.723904\pi$
$79$	− 13.5570i	− 1.52528i	0.646825	−	0.762638i	$-0.276096\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4.96335i	− 0.544799i	−0.962184	−	0.272399i	$-0.912183\pi$
$83$	− 4.96335i	− 0.544799i	0.962184	−	0.272399i	$-0.0878172\pi$
$84$	0	0
$85$	6.82263i	0.740019i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	18.2032i	1.92954i	0.263100	+	0.964769i	$0.415255\pi$
$89$	18.2032i	1.92954i	−0.263100	+	0.964769i	$0.584745\pi$
$90$	0	0
$91$	−0.268602	−0.0281572
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.15747	0.323949
$96$	0	0
$97$	3.39139	0.344344	0.172172	−	0.985067i	$-0.444922\pi$
$97$	3.39139	0.344344	0.172172	+	0.985067i	$0.444922\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.18671	−0.814608	−0.407304	−	0.913293i	$-0.633531\pi$
$101$	−8.18671	−0.814608	−0.407304	+	0.913293i	$0.633531\pi$
$102$	0	0
$103$	16.5624i	1.63194i	0.578093	+	0.815971i	$0.303797\pi$
$103$	16.5624i	1.63194i	−0.578093	+	0.815971i	$0.696203\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.1480i	1.46441i	0.681083	+	0.732206i	$0.261509\pi$
$107$	15.1480i	1.46441i	−0.681083	+	0.732206i	$0.738491\pi$
$108$	0	0
$109$	7.70038i	0.737562i	0.929516	+	0.368781i	$0.120225\pi$
$109$	7.70038i	0.737562i	−0.929516	+	0.368781i	$0.879775\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.5822i	1.37178i	0.727706	+	0.685889i	$0.240587\pi$
$113$	14.5822i	1.37178i	−0.727706	+	0.685889i	$0.759413\pi$
$114$	0	0
$115$	−20.3501	−1.89766
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.96616	0.180238
$120$	0	0
$121$	−15.2033	−1.38211
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.08266	−0.633492
$126$	0	0
$127$	− 13.7782i	− 1.22262i	−0.791392	−	0.611309i	$-0.790643\pi$
$127$	− 13.7782i	− 1.22262i	0.791392	−	0.611309i	$-0.209357\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 11.4226i	− 0.997997i	−0.866603	−	0.498999i	$-0.833701\pi$
$131$	− 11.4226i	− 0.997997i	0.866603	−	0.498999i	$-0.166299\pi$
$132$	0	0
$133$	− 0.909926i	− 0.0789006i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.89343i	− 0.503510i	−0.967791	−	0.251755i	$-0.918992\pi$
$137$	− 5.89343i	− 0.503510i	0.967791	−	0.251755i	$-0.0810077\pi$
$138$	0	0
$139$	−3.67309	−0.311547	−0.155774	−	0.987793i	$-0.549787\pi$
$139$	−3.67309	−0.311547	−0.155774	+	0.987793i	$0.549787\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.37495	−0.114979
$144$	0	0
$145$	−24.9828	−2.07471
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−23.0240	−1.88620	−0.943102	−	0.332504i	$-0.892106\pi$
$149$	−23.0240	−1.88620	−0.943102	+	0.332504i	$0.892106\pi$
$150$	0	0
$151$	− 1.33860i	− 0.108934i	−0.998516	−	0.0544669i	$-0.982654\pi$
$151$	− 1.33860i	− 0.108934i	0.998516	−	0.0544669i	$-0.0173459\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	15.8751i	1.27512i
$156$	0	0
$157$	21.7839i	1.73854i	0.494333	+	0.869272i	$0.335412\pi$
$157$	21.7839i	1.73854i	−0.494333	+	0.869272i	$0.664588\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.86455i	0.462191i
$162$	0	0
$163$	20.2710	1.58775	0.793875	−	0.608082i	$-0.208061\pi$
$163$	20.2710	1.58775	0.793875	+	0.608082i	$0.208061\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.57632	0.508891	0.254445	−	0.967087i	$-0.418107\pi$
$167$	6.57632	0.508891	0.254445	+	0.967087i	$0.418107\pi$
$168$	0	0
$169$	12.9279	0.994450
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.3399	1.01421	0.507105	−	0.861884i	$-0.330716\pi$
$173$	13.3399	1.01421	0.507105	+	0.861884i	$0.330716\pi$
$174$	0	0
$175$	7.04110i	0.532257i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.82571i	0.734408i	0.930140	+	0.367204i	$0.119685\pi$
$179$	9.82571i	0.734408i	−0.930140	+	0.367204i	$0.880315\pi$
$180$	0	0
$181$	4.88753i	0.363287i	0.983364	+	0.181643i	$0.0581417\pi$
$181$	4.88753i	0.363287i	−0.983364	+	0.181643i	$0.941858\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	24.2316i	1.78154i
$186$	0	0
$187$	10.0646	0.735997
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.6129	0.840281	0.420141	−	0.907459i	$-0.361981\pi$
$191$	11.6129	0.840281	0.420141	+	0.907459i	$0.361981\pi$
$192$	0	0
$193$	−12.8166	−0.922562	−0.461281	−	0.887254i	$-0.652610\pi$
$193$	−12.8166	−0.922562	−0.461281	+	0.887254i	$0.652610\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.69147	−0.334253	−0.167127	−	0.985935i	$-0.553449\pi$
$197$	−4.69147	−0.334253	−0.167127	+	0.985935i	$0.553449\pi$
$198$	0	0
$199$	− 15.3555i	− 1.08852i	−0.838915	−	0.544262i	$-0.816810\pi$
$199$	− 15.3555i	− 1.08852i	0.838915	−	0.544262i	$-0.183190\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.19959i	0.505312i
$204$	0	0
$205$	− 28.9610i	− 2.02273i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 4.65783i	− 0.322189i
$210$	0	0
$211$	−5.09850	−0.350995	−0.175497	−	0.984480i	$-0.556153\pi$
$211$	−5.09850	−0.350995	−0.175497	+	0.984480i	$0.556153\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−32.1060	−2.18961
$216$	0	0
$217$	4.57491	0.310565
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.528115	0.0355249
$222$	0	0
$223$	29.5118i	1.97625i	0.153636	+	0.988127i	$0.450902\pi$
$223$	29.5118i	1.97625i	−0.153636	+	0.988127i	$0.549098\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 14.4769i	− 0.960868i	−0.877031	−	0.480434i	$-0.840479\pi$
$227$	− 14.4769i	− 0.960868i	0.877031	−	0.480434i	$-0.159521\pi$
$228$	0	0
$229$	11.9120i	0.787168i	0.919289	+	0.393584i	$0.128765\pi$
$229$	11.9120i	0.787168i	−0.919289	+	0.393584i	$0.871235\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 13.2938i	− 0.870904i	−0.900212	−	0.435452i	$-0.856589\pi$
$233$	− 13.2938i	− 0.870904i	0.900212	−	0.435452i	$-0.143411\pi$
$234$	0	0
$235$	22.5315	1.46979
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.7744	−0.761624	−0.380812	−	0.924652i	$-0.624356\pi$
$239$	−11.7744	−0.761624	−0.380812	+	0.924652i	$0.624356\pi$
$240$	0	0
$241$	−23.3009	−1.50094	−0.750471	−	0.660903i	$-0.770173\pi$
$241$	−23.3009	−1.50094	−0.750471	+	0.660903i	$0.770173\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.47003	0.221692
$246$	0	0
$247$	− 0.244408i	− 0.0155513i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.5562i	1.17126i	0.810579	+	0.585629i	$0.199153\pi$
$251$	18.5562i	1.17126i	−0.810579	+	0.585629i	$0.800847\pi$
$252$	0	0
$253$	30.0201i	1.88735i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.7179i	0.793318i	0.917966	+	0.396659i	$0.129831\pi$
$257$	12.7179i	0.793318i	−0.917966	+	0.396659i	$0.870169\pi$
$258$	0	0
$259$	6.98312	0.433910
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.04929	−0.249690	−0.124845	−	0.992176i	$-0.539843\pi$
$263$	−4.04929	−0.249690	−0.124845	+	0.992176i	$0.539843\pi$
$264$	0	0
$265$	15.7652	0.968449
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.6641	1.38185	0.690926	−	0.722925i	$-0.257203\pi$
$269$	22.6641	1.38185	0.690926	+	0.722925i	$0.257203\pi$
$270$	0	0
$271$	0.0619260i	0.00376174i	0.999998	+	0.00188087i	$0.000598699\pi$
$271$	0.0619260i	0.00376174i	−0.999998	+	0.00188087i	$0.999401\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	36.0427i	2.17346i
$276$	0	0
$277$	24.9177i	1.49716i	0.663045	+	0.748580i	$0.269264\pi$
$277$	24.9177i	1.49716i	−0.663045	+	0.748580i	$0.730736\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 8.26290i	− 0.492923i	−0.969153	−	0.246462i	$-0.920732\pi$
$281$	− 8.26290i	− 0.492923i	0.969153	−	0.246462i	$-0.0792679\pi$
$282$	0	0
$283$	−23.4764	−1.39553	−0.697763	−	0.716329i	$-0.745821\pi$
$283$	−23.4764	−1.39553	−0.697763	+	0.716329i	$0.745821\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.34605	−0.492652
$288$	0	0
$289$	13.1342	0.772601
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.18958	−0.361599	−0.180800	−	0.983520i	$-0.557869\pi$
$293$	−6.18958	−0.361599	−0.180800	+	0.983520i	$0.557869\pi$
$294$	0	0
$295$	− 12.1633i	− 0.708174i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.57523i	0.0910979i
$300$	0	0
$301$	9.25237i	0.533298i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.81007i	0.389944i
$306$	0	0
$307$	25.4140	1.45045	0.725226	−	0.688511i	$-0.241735\pi$
$307$	25.4140	1.45045	0.725226	+	0.688511i	$0.241735\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7.55273	−0.428276	−0.214138	−	0.976803i	$-0.568694\pi$
$311$	−7.55273	−0.428276	−0.214138	+	0.976803i	$0.568694\pi$
$312$	0	0
$313$	2.49790	0.141190	0.0705948	−	0.997505i	$-0.477510\pi$
$313$	2.49790	0.141190	0.0705948	+	0.997505i	$0.477510\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.9643	−0.615815	−0.307908	−	0.951416i	$-0.599629\pi$
$317$	−10.9643	−0.615815	−0.307908	+	0.951416i	$0.599629\pi$
$318$	0	0
$319$	36.8541i	2.06343i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.78906i	0.0995460i
$324$	0	0
$325$	1.89125i	0.104908i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 6.49316i	− 0.357980i
$330$	0	0
$331$	−35.6043	−1.95699	−0.978494	−	0.206275i	$-0.933866\pi$
$331$	−35.6043	−1.95699	−0.978494	+	0.206275i	$0.933866\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−38.2674	−2.09077
$336$	0	0
$337$	18.4308	1.00399	0.501995	−	0.864871i	$-0.332600\pi$
$337$	18.4308	1.00399	0.501995	+	0.864871i	$0.332600\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	23.4185	1.26819
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.2141i	1.35356i	0.736184	+	0.676782i	$0.236626\pi$
$347$	25.2141i	1.35356i	−0.736184	+	0.676782i	$0.763374\pi$
$348$	0	0
$349$	4.82396i	0.258221i	0.991630	+	0.129110i	$0.0412121\pi$
$349$	4.82396i	0.258221i	−0.991630	+	0.129110i	$0.958788\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.2660i	1.13187i	0.824448	+	0.565937i	$0.191486\pi$
$353$	21.2660i	1.13187i	−0.824448	+	0.565937i	$0.808514\pi$
$354$	0	0
$355$	−4.16723	−0.221174
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.9785	−1.31832	−0.659158	−	0.752004i	$-0.729087\pi$
$359$	−24.9785	−1.31832	−0.659158	+	0.752004i	$0.729087\pi$
$360$	0	0
$361$	−18.1720	−0.956423
$362$	0	0
$363$	0	0
$364$	0	0
$365$	31.2892	1.63775
$366$	0	0
$367$	13.8851i	0.724795i	0.932024	+	0.362397i	$0.118042\pi$
$367$	13.8851i	0.724795i	−0.932024	+	0.362397i	$0.881958\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 4.54325i	− 0.235874i
$372$	0	0
$373$	0.513970i	0.0266123i	0.999911	+	0.0133062i	$0.00423561\pi$
$373$	0.513970i	0.0266123i	−0.999911	+	0.0133062i	$0.995764\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.93383i	0.0995972i
$378$	0	0
$379$	7.97711	0.409757	0.204878	−	0.978787i	$-0.434320\pi$
$379$	7.97711	0.409757	0.204878	+	0.978787i	$0.434320\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8.03401	−0.410519	−0.205259	−	0.978708i	$-0.565804\pi$
$383$	−8.03401	−0.410519	−0.205259	+	0.978708i	$0.565804\pi$
$384$	0	0
$385$	17.7628	0.905275
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.60733	0.335005	0.167503	−	0.985872i	$-0.446430\pi$
$389$	6.60733	0.335005	0.167503	+	0.985872i	$0.446430\pi$
$390$	0	0
$391$	− 11.5306i	− 0.583130i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	47.0430i	2.36699i
$396$	0	0
$397$	− 17.4642i	− 0.876502i	−0.898853	−	0.438251i	$-0.855598\pi$
$397$	− 17.4642i	− 0.876502i	0.898853	−	0.438251i	$-0.144402\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 18.6133i	− 0.929503i	−0.885441	−	0.464751i	$-0.846144\pi$
$401$	− 18.6133i	− 0.929503i	0.885441	−	0.464751i	$-0.153856\pi$
$402$	0	0
$403$	1.22883	0.0612124
$404$	0	0
$405$	0	0
$406$	0	0
$407$	35.7460	1.77186
$408$	0	0
$409$	3.72844	0.184359	0.0921797	−	0.995742i	$-0.470617\pi$
$409$	3.72844	0.184359	0.0921797	+	0.995742i	$0.470617\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.50524	−0.172482
$414$	0	0
$415$	17.2230i	0.845443i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	30.1591i	1.47337i	0.676237	+	0.736684i	$0.263609\pi$
$419$	30.1591i	1.47337i	−0.676237	+	0.736684i	$0.736391\pi$
$420$	0	0
$421$	− 4.17416i	− 0.203436i	−0.994813	−	0.101718i	$-0.967566\pi$
$421$	− 4.17416i	− 0.203436i	0.994813	−	0.101718i	$-0.0324340\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 13.8439i	− 0.671529i
$426$	0	0
$427$	1.96254	0.0949740
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.3147	0.978524	0.489262	−	0.872137i	$-0.337266\pi$
$431$	20.3147	0.978524	0.489262	+	0.872137i	$0.337266\pi$
$432$	0	0
$433$	37.5740	1.80569	0.902847	−	0.429963i	$-0.141473\pi$
$433$	37.5740	1.80569	0.902847	+	0.429963i	$0.141473\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.33630	−0.255270
$438$	0	0
$439$	12.2219i	0.583317i	0.956522	+	0.291659i	$0.0942072\pi$
$439$	12.2219i	0.583317i	−0.956522	+	0.291659i	$0.905793\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 2.13132i	− 0.101262i	−0.998717	−	0.0506311i	$-0.983877\pi$
$443$	− 2.13132i	− 0.101262i	0.998717	−	0.0506311i	$-0.0161233\pi$
$444$	0	0
$445$	− 63.1657i	− 2.99434i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	24.6873i	1.16506i	0.812807	+	0.582532i	$0.197938\pi$
$449$	24.6873i	1.16506i	−0.812807	+	0.582532i	$0.802062\pi$
$450$	0	0
$451$	−42.7227	−2.01173
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.932057	0.0436955
$456$	0	0
$457$	−33.2814	−1.55684	−0.778419	−	0.627745i	$-0.783978\pi$
$457$	−33.2814	−1.55684	−0.778419	+	0.627745i	$0.783978\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15.2279	−0.709233	−0.354617	−	0.935012i	$-0.615389\pi$
$461$	−15.2279	−0.709233	−0.354617	+	0.935012i	$0.615389\pi$
$462$	0	0
$463$	3.21883i	0.149592i	0.997199	+	0.0747959i	$0.0238305\pi$
$463$	3.21883i	0.149592i	−0.997199	+	0.0747959i	$0.976169\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 22.5827i	− 1.04500i	−0.852638	−	0.522502i	$-0.824999\pi$
$467$	− 22.5827i	− 1.04500i	0.852638	−	0.522502i	$-0.175001\pi$
$468$	0	0
$469$	11.0280i	0.509225i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	47.3621i	2.17771i
$474$	0	0
$475$	−6.40688	−0.293968
$476$	0	0
$477$	0	0
$478$	0	0
$479$	28.0140	1.27999	0.639995	−	0.768379i	$-0.278936\pi$
$479$	28.0140	1.27999	0.639995	+	0.768379i	$0.278936\pi$
$480$	0	0
$481$	1.87568	0.0855237
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11.7682	−0.534368
$486$	0	0
$487$	− 2.55321i	− 0.115697i	−0.998325	−	0.0578485i	$-0.981576\pi$
$487$	− 2.55321i	− 0.115697i	0.998325	−	0.0578485i	$-0.0184240\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 0.803432i	− 0.0362584i	−0.999836	−	0.0181292i	$-0.994229\pi$
$491$	− 0.803432i	− 0.0362584i	0.999836	−	0.0181292i	$-0.00577102\pi$
$492$	0	0
$493$	− 14.1556i	− 0.637534i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.20092i	0.0538687i
$498$	0	0
$499$	−32.6029	−1.45950	−0.729752	−	0.683712i	$-0.760365\pi$
$499$	−32.6029	−1.45950	−0.729752	+	0.683712i	$0.760365\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11.6962	−0.521508	−0.260754	−	0.965405i	$-0.583971\pi$
$503$	−11.6962	−0.521508	−0.260754	+	0.965405i	$0.583971\pi$
$504$	0	0
$505$	28.4081	1.26414
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−40.0087	−1.77335	−0.886676	−	0.462391i	$-0.846992\pi$
$509$	−40.0087	−1.77335	−0.886676	+	0.462391i	$0.846992\pi$
$510$	0	0
$511$	− 9.01697i	− 0.398887i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 57.4720i	− 2.53252i
$516$	0	0
$517$	− 33.2379i	− 1.46180i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 5.13277i	− 0.224871i	−0.993659	−	0.112435i	$-0.964135\pi$
$521$	− 5.13277i	− 0.224871i	0.993659	−	0.112435i	$-0.0358651\pi$
$522$	0	0
$523$	−35.3652	−1.54641	−0.773207	−	0.634154i	$-0.781348\pi$
$523$	−35.3652	−1.54641	−0.773207	+	0.634154i	$0.781348\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.99500	−0.391829
$528$	0	0
$529$	11.3929	0.495344
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.24177	−0.0971018
$534$	0	0
$535$	− 52.5640i	− 2.27254i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 5.11891i	− 0.220487i
$540$	0	0
$541$	30.7038i	1.32006i	0.751240	+	0.660029i	$0.229456\pi$
$541$	30.7038i	1.32006i	−0.751240	+	0.660029i	$0.770544\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 26.7205i	− 1.14458i
$546$	0	0
$547$	2.21526	0.0947175	0.0473588	−	0.998878i	$-0.484920\pi$
$547$	2.21526	0.0947175	0.0473588	+	0.998878i	$0.484920\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.55110	−0.279086
$552$	0	0
$553$	13.5570	0.576500
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.88158	−0.376325	−0.188162	−	0.982138i	$-0.560253\pi$
$557$	−8.88158	−0.376325	−0.188162	+	0.982138i	$0.560253\pi$
$558$	0	0
$559$	2.48521i	0.105113i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 28.9398i	− 1.21967i	−0.792529	−	0.609834i	$-0.791236\pi$
$563$	− 28.9398i	− 1.21967i	0.792529	−	0.609834i	$-0.208764\pi$
$564$	0	0
$565$	− 50.6007i	− 2.12879i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 30.8816i	− 1.29463i	−0.762224	−	0.647313i	$-0.775893\pi$
$569$	− 30.8816i	− 1.29463i	0.762224	−	0.647313i	$-0.224107\pi$
$570$	0	0
$571$	31.5945	1.32219	0.661094	−	0.750303i	$-0.270092\pi$
$571$	31.5945	1.32219	0.661094	+	0.750303i	$0.270092\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	41.2928	1.72203
$576$	0	0
$577$	−37.6007	−1.56534	−0.782669	−	0.622438i	$-0.786142\pi$
$577$	−37.6007	−1.56534	−0.782669	+	0.622438i	$0.786142\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.96335	0.205915
$582$	0	0
$583$	− 23.2565i	− 0.963186i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 22.0321i	− 0.909361i	−0.890655	−	0.454680i	$-0.849754\pi$
$587$	− 22.0321i	− 0.909361i	0.890655	−	0.454680i	$-0.150246\pi$
$588$	0	0
$589$	4.16283i	0.171526i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17.8392i	0.732569i	0.930503	+	0.366284i	$0.119370\pi$
$593$	17.8392i	0.732569i	−0.930503	+	0.366284i	$0.880630\pi$
$594$	0	0
$595$	−6.82263	−0.279701
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−19.5363	−0.798230	−0.399115	−	0.916901i	$-0.630683\pi$
$599$	−19.5363	−0.798230	−0.399115	+	0.916901i	$0.630683\pi$
$600$	0	0
$601$	−20.1240	−0.820877	−0.410438	−	0.911888i	$-0.634624\pi$
$601$	−20.1240	−0.820877	−0.410438	+	0.911888i	$0.634624\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	52.7557	2.14483
$606$	0	0
$607$	− 15.3972i	− 0.624951i	−0.949926	−	0.312476i	$-0.898842\pi$
$607$	− 15.3972i	− 0.624951i	0.949926	−	0.312476i	$-0.101158\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1.74408i	− 0.0705578i
$612$	0	0
$613$	− 37.9629i	− 1.53331i	−0.642061	−	0.766653i	$-0.721921\pi$
$613$	− 37.9629i	− 1.53331i	0.642061	−	0.766653i	$-0.278079\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	43.0532i	1.73326i	0.498953	+	0.866629i	$0.333718\pi$
$617$	43.0532i	1.73326i	−0.498953	+	0.866629i	$0.666282\pi$
$618$	0	0
$619$	6.75463	0.271491	0.135746	−	0.990744i	$-0.456657\pi$
$619$	6.75463	0.271491	0.135746	+	0.990744i	$0.456657\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−18.2032	−0.729297
$624$	0	0
$625$	−10.6285	−0.425138
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.7299	−0.547448
$630$	0	0
$631$	1.96735i	0.0783189i	0.999233	+	0.0391594i	$0.0124680\pi$
$631$	1.96735i	0.0783189i	−0.999233	+	0.0391594i	$0.987532\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	47.8108i	1.89731i
$636$	0	0
$637$	− 0.268602i	− 0.0106424i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30.2998i	1.19677i	0.801209	+	0.598385i	$0.204191\pi$
$641$	30.2998i	1.19677i	−0.801209	+	0.598385i	$0.795809\pi$
$642$	0	0
$643$	−3.85901	−0.152185	−0.0760923	−	0.997101i	$-0.524244\pi$
$643$	−3.85901	−0.152185	−0.0760923	+	0.997101i	$0.524244\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.4004	1.31310	0.656552	−	0.754281i	$-0.272014\pi$
$647$	33.4004	1.31310	0.656552	+	0.754281i	$0.272014\pi$
$648$	0	0
$649$	−17.9430	−0.704326
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−26.9899	−1.05620	−0.528098	−	0.849184i	$-0.677095\pi$
$653$	−26.9899	−1.05620	−0.528098	+	0.849184i	$0.677095\pi$
$654$	0	0
$655$	39.6367i	1.54874i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	13.1888i	0.513763i	0.966443	+	0.256881i	$0.0826950\pi$
$659$	13.1888i	0.513763i	−0.966443	+	0.256881i	$0.917305\pi$
$660$	0	0
$661$	− 33.5767i	− 1.30598i	−0.757366	−	0.652990i	$-0.773514\pi$
$661$	− 33.5767i	− 1.30598i	0.757366	−	0.652990i	$-0.226486\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.15747i	0.122441i
$666$	0	0
$667$	42.2224	1.63486
$668$	0	0
$669$	0	0
$670$	0	0
$671$	10.0461	0.387824
$672$	0	0
$673$	13.5269	0.521424	0.260712	−	0.965417i	$-0.416043\pi$
$673$	13.5269	0.521424	0.260712	+	0.965417i	$0.416043\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25.3388	−0.973850	−0.486925	−	0.873444i	$-0.661882\pi$
$677$	−25.3388	−0.973850	−0.486925	+	0.873444i	$0.661882\pi$
$678$	0	0
$679$	3.39139i	0.130150i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	4.74712i	0.181644i	0.995867	+	0.0908218i	$0.0289494\pi$
$683$	4.74712i	0.181644i	−0.995867	+	0.0908218i	$0.971051\pi$
$684$	0	0
$685$	20.4504i	0.781369i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1.22033i	− 0.0464908i
$690$	0	0
$691$	46.8908	1.78381	0.891904	−	0.452224i	$-0.149369\pi$
$691$	46.8908	1.78381	0.891904	+	0.452224i	$0.149369\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.7457	0.483472
$696$	0	0
$697$	16.4097	0.621561
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11.8394	−0.447166	−0.223583	−	0.974685i	$-0.571775\pi$
$701$	−11.8394	−0.447166	−0.223583	+	0.974685i	$0.571775\pi$
$702$	0	0
$703$	6.35412i	0.239650i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 8.18671i	− 0.307893i
$708$	0	0
$709$	− 3.55487i	− 0.133506i	−0.997770	−	0.0667530i	$-0.978736\pi$
$709$	− 3.55487i	− 0.133506i	0.997770	−	0.0667530i	$-0.0212639\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 26.8298i	− 1.00478i
$714$	0	0
$715$	4.77112	0.178430
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0.174626	0.00651246	0.00325623	−	0.999995i	$-0.498964\pi$
$719$	0.174626	0.00651246	0.00325623	+	0.999995i	$0.498964\pi$
$720$	0	0
$721$	−16.5624	−0.616816
$722$	0	0
$723$	0	0
$724$	0	0
$725$	50.6930	1.88269
$726$	0	0
$727$	8.37928i	0.310770i	0.987854	+	0.155385i	$0.0496619\pi$
$727$	8.37928i	0.310770i	−0.987854	+	0.155385i	$0.950338\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 18.1917i	− 0.672843i
$732$	0	0
$733$	− 15.6312i	− 0.577351i	−0.957427	−	0.288676i	$-0.906785\pi$
$733$	− 15.6312i	− 0.577351i	0.957427	−	0.288676i	$-0.0932149\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	56.4512i	2.07941i
$738$	0	0
$739$	25.4382	0.935760	0.467880	−	0.883792i	$-0.345018\pi$
$739$	25.4382	0.935760	0.467880	+	0.883792i	$0.345018\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.0542	1.21264	0.606320	−	0.795220i	$-0.292645\pi$
$743$	33.0542	1.21264	0.606320	+	0.795220i	$0.292645\pi$
$744$	0	0
$745$	79.8941	2.92709
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−15.1480	−0.553496
$750$	0	0
$751$	40.4075i	1.47449i	0.675625	+	0.737246i	$0.263874\pi$
$751$	40.4075i	1.47449i	−0.675625	+	0.737246i	$0.736126\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.64498i	0.169048i
$756$	0	0
$757$	30.0219i	1.09116i	0.838057	+	0.545582i	$0.183691\pi$
$757$	30.0219i	1.09116i	−0.838057	+	0.545582i	$0.816309\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	13.9651i	0.506234i	0.967436	+	0.253117i	$0.0814557\pi$
$761$	13.9651i	0.506234i	−0.967436	+	0.253117i	$0.918544\pi$
$762$	0	0
$763$	−7.70038	−0.278772
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.941516	−0.0339962
$768$	0	0
$769$	17.9999	0.649094	0.324547	−	0.945869i	$-0.394788\pi$
$769$	17.9999	0.649094	0.324547	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−27.2134	−0.978798	−0.489399	−	0.872060i	$-0.662784\pi$
$773$	−27.2134	−0.978798	−0.489399	+	0.872060i	$0.662784\pi$
$774$	0	0
$775$	− 32.2124i	− 1.15710i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 7.59429i	− 0.272094i
$780$	0	0
$781$	6.14741i	0.219972i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 75.5908i	− 2.69795i
$786$	0	0
$787$	−3.45794	−0.123262	−0.0616312	−	0.998099i	$-0.519630\pi$
$787$	−3.45794	−0.123262	−0.0616312	+	0.998099i	$0.519630\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−14.5822	−0.518484
$792$	0	0
$793$	0.527143	0.0187194
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−44.2671	−1.56802	−0.784010	−	0.620748i	$-0.786829\pi$
$797$	−44.2671	−1.56802	−0.784010	+	0.620748i	$0.786829\pi$
$798$	0	0
$799$	12.7666i	0.451650i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 46.1571i	− 1.62885i
$804$	0	0
$805$	− 20.3501i	− 0.717248i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 14.4191i	− 0.506949i	−0.967342	−	0.253475i	$-0.918427\pi$
$809$	− 14.4191i	− 0.506949i	0.967342	−	0.253475i	$-0.0815734\pi$
$810$	0	0
$811$	−24.3314	−0.854389	−0.427195	−	0.904160i	$-0.640498\pi$
$811$	−24.3314	−0.854389	−0.427195	+	0.904160i	$0.640498\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−70.3410	−2.46394
$816$	0	0
$817$	−8.41897	−0.294543
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.63626	0.126906	0.0634532	−	0.997985i	$-0.479789\pi$
$821$	3.63626	0.126906	0.0634532	+	0.997985i	$0.479789\pi$
$822$	0	0
$823$	0.408592i	0.0142426i	0.999975	+	0.00712132i	$0.00226681\pi$
$823$	0.408592i	0.0142426i	−0.999975	+	0.00712132i	$0.997733\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.3528i	1.36843i	0.729280	+	0.684215i	$0.239855\pi$
$827$	39.3528i	1.36843i	−0.729280	+	0.684215i	$0.760145\pi$
$828$	0	0
$829$	− 37.5641i	− 1.30465i	−0.757938	−	0.652327i	$-0.773793\pi$
$829$	− 37.5641i	− 1.30465i	0.757938	−	0.652327i	$-0.226207\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.96616i	0.0681234i
$834$	0	0
$835$	−22.8200	−0.789719
$836$	0	0
$837$	0	0
$838$	0	0
$839$	35.7770	1.23516	0.617579	−	0.786509i	$-0.288113\pi$
$839$	35.7770	1.23516	0.617579	+	0.786509i	$0.288113\pi$
$840$	0	0
$841$	22.8342	0.787385
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−44.8600	−1.54323
$846$	0	0
$847$	− 15.2033i	− 0.522390i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 40.9528i	− 1.40384i
$852$	0	0
$853$	52.3988i	1.79410i	0.441929	+	0.897050i	$0.354294\pi$
$853$	52.3988i	1.79410i	−0.441929	+	0.897050i	$0.645706\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 19.1325i	− 0.653553i	−0.945102	−	0.326777i	$-0.894037\pi$
$857$	− 19.1325i	− 0.653553i	0.945102	−	0.326777i	$-0.105963\pi$
$858$	0	0
$859$	12.1998	0.416253	0.208127	−	0.978102i	$-0.433263\pi$
$859$	12.1998	0.416253	0.208127	+	0.978102i	$0.433263\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.3575	1.06742	0.533711	−	0.845667i	$-0.320797\pi$
$863$	31.3575	1.06742	0.533711	+	0.845667i	$0.320797\pi$
$864$	0	0
$865$	−46.2897	−1.57390
$866$	0	0
$867$	0	0
$868$	0	0
$869$	69.3968	2.35413
$870$	0	0
$871$	2.96214i	0.100368i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 7.08266i	− 0.239438i
$876$	0	0
$877$	4.64956i	0.157004i	0.996914	+	0.0785022i	$0.0250138\pi$
$877$	4.64956i	0.157004i	−0.996914	+	0.0785022i	$0.974986\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 43.3414i	− 1.46021i	−0.683336	−	0.730104i	$-0.739472\pi$
$881$	− 43.3414i	− 1.46021i	0.683336	−	0.730104i	$-0.260528\pi$
$882$	0	0
$883$	−0.420653	−0.0141561	−0.00707805	−	0.999975i	$-0.502253\pi$
$883$	−0.420653	−0.0141561	−0.00707805	+	0.999975i	$0.502253\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.5102	−0.487204	−0.243602	−	0.969875i	$-0.578329\pi$
$887$	−14.5102	−0.487204	−0.243602	+	0.969875i	$0.578329\pi$
$888$	0	0
$889$	13.7782	0.462106
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.90830	0.197714
$894$	0	0
$895$	− 34.0955i	− 1.13969i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 32.9375i	− 1.09853i
$900$	0	0
$901$	8.93276i	0.297593i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 16.9599i	− 0.563765i
$906$	0	0
$907$	30.1284	1.00040	0.500198	−	0.865911i	$-0.333260\pi$
$907$	30.1284	1.00040	0.500198	+	0.865911i	$0.333260\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	15.1934	0.503379	0.251689	−	0.967808i	$-0.419014\pi$
$911$	15.1934	0.503379	0.251689	+	0.967808i	$0.419014\pi$
$912$	0	0
$913$	25.4070	0.840848
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.4226	0.377207
$918$	0	0
$919$	38.7985i	1.27984i	0.768440	+	0.639922i	$0.221033\pi$
$919$	38.7985i	1.27984i	−0.768440	+	0.639922i	$0.778967\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0.322570i	0.0106175i
$924$	0	0
$925$	− 49.1688i	− 1.61666i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 43.7700i	− 1.43605i	−0.696019	−	0.718023i	$-0.745047\pi$
$929$	− 43.7700i	− 1.43605i	0.696019	−	0.718023i	$-0.254953\pi$
$930$	0	0
$931$	0.909926	0.0298216
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−34.9245	−1.14215
$936$	0	0
$937$	17.7605	0.580209	0.290104	−	0.956995i	$-0.406310\pi$
$937$	17.7605	0.580209	0.290104	+	0.956995i	$0.406310\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32.6364	1.06392	0.531958	−	0.846771i	$-0.321456\pi$
$941$	32.6364	1.06392	0.531958	+	0.846771i	$0.321456\pi$
$942$	0	0
$943$	48.9458i	1.59390i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.9843i	0.486925i	0.969910	+	0.243462i	$0.0782832\pi$
$947$	14.9843i	0.486925i	−0.969910	+	0.243462i	$0.921717\pi$
$948$	0	0
$949$	− 2.42198i	− 0.0786208i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 23.1053i	− 0.748453i	−0.927337	−	0.374226i	$-0.877908\pi$
$953$	− 23.1053i	− 0.748453i	0.927337	−	0.374226i	$-0.122092\pi$
$954$	0	0
$955$	−40.2971	−1.30398
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.89343	0.190309
$960$	0	0
$961$	10.0702	0.324846
$962$	0	0
$963$	0	0
$964$	0	0
$965$	44.4741	1.43167
$966$	0	0
$967$	25.1604i	0.809103i	0.914515	+	0.404552i	$0.132572\pi$
$967$	25.1604i	0.809103i	−0.914515	+	0.404552i	$0.867428\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 24.0973i	− 0.773318i	−0.922223	−	0.386659i	$-0.873629\pi$
$971$	− 24.0973i	− 0.773318i	0.922223	−	0.386659i	$-0.126371\pi$
$972$	0	0
$973$	− 3.67309i	− 0.117754i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 30.9215i	− 0.989267i	−0.869102	−	0.494634i	$-0.835302\pi$
$977$	− 30.9215i	− 0.989267i	0.869102	−	0.494634i	$-0.164698\pi$
$978$	0	0
$979$	−93.1807	−2.97807
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1.67927	−0.0535602	−0.0267801	−	0.999641i	$-0.508525\pi$
$983$	−1.67927	−0.0535602	−0.0267801	+	0.999641i	$0.508525\pi$
$984$	0	0
$985$	16.2795	0.518709
$986$	0	0
$987$	0	0
$988$	0	0
$989$	54.2610	1.72540
$990$	0	0
$991$	7.63921i	0.242667i	0.992612	+	0.121334i	$0.0387171\pi$
$991$	7.63921i	0.242667i	−0.992612	+	0.121334i	$0.961283\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	53.2841i	1.68922i
$996$	0	0
$997$	57.4039i	1.81800i	0.416798	+	0.908999i	$0.363152\pi$
$997$	57.4039i	1.81800i	−0.416798	+	0.908999i	$0.636848\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.j.c.5615.3		32
3.2	odd	2	inner	6048.2.j.c.5615.30		32
4.3	odd	2		1512.2.j.c.323.4	yes	32
8.3	odd	2	inner	6048.2.j.c.5615.29		32
8.5	even	2		1512.2.j.c.323.30	yes	32
12.11	even	2		1512.2.j.c.323.29	yes	32
24.5	odd	2		1512.2.j.c.323.3	✓	32
24.11	even	2	inner	6048.2.j.c.5615.4		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.c.323.3	✓	32	24.5	odd	2
1512.2.j.c.323.4	yes	32	4.3	odd	2
1512.2.j.c.323.29	yes	32	12.11	even	2
1512.2.j.c.323.30	yes	32	8.5	even	2
6048.2.j.c.5615.3		32	1.1	even	1	trivial
6048.2.j.c.5615.4		32	24.11	even	2	inner
6048.2.j.c.5615.29		32	8.3	odd	2	inner
6048.2.j.c.5615.30		32	3.2	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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.47003 q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q-3.47003 q^{5} +1.00000i q^{7} +5.11891i q^{11} +0.268602i q^{13} -1.96616i q^{17} -0.909926 q^{19} +5.86455 q^{23} +7.04110 q^{25} +7.19959 q^{29} -4.57491i q^{31} -3.47003i q^{35} -6.98312i q^{37} +8.34605i q^{41} +9.25237 q^{43} -6.49316 q^{47} -1.00000 q^{49} -4.54325 q^{53} -17.7628i q^{55} +3.50524i q^{59} -1.96254i q^{61} -0.932057i q^{65} +11.0280 q^{67} +1.20092 q^{71} -9.01697 q^{73} -5.11891 q^{77} -13.5570i q^{79} -4.96335i q^{83} +6.82263i q^{85} +18.2032i q^{89} -0.268602 q^{91} +3.15747 q^{95} +3.39139 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 64 q^{19} - 16 q^{25} + 48 q^{43} - 32 q^{49} - 16 q^{67} - 16 q^{73} - 16 q^{91} + 64 q^{97}+O(q^{100}) \) 32 * q + 64 * q^19 - 16 * q^25 + 48 * q^43 - 32 * q^49 - 16 * q^67 - 16 * q^73 - 16 * q^91 + 64 * q^97

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