LMFDB - Embedded newform 6048.2.j.c.5615.6

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			5615.6
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.c.5615.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21305 q^{5} -1.00000i q^{7} +O(q^{10})$	$q-2.21305 q^{5} -1.00000i q^{7} +2.42373i q^{11} +1.11593i q^{13} -0.701326i q^{17} -0.938340 q^{19} +4.30502 q^{23} -0.102426 q^{25} +4.26130 q^{29} -7.02221i q^{31} +2.21305i q^{35} -3.12583i q^{37} -0.157922i q^{41} -7.08975 q^{43} -0.867970 q^{47} -1.00000 q^{49} +3.91444 q^{53} -5.36382i q^{55} +7.28820i q^{59} -4.57877i q^{61} -2.46961i q^{65} -15.1677 q^{67} -6.93451 q^{71} +7.02133 q^{73} +2.42373 q^{77} +6.65903i q^{79} +8.41084i q^{83} +1.55207i q^{85} +7.34335i q^{89} +1.11593 q^{91} +2.07659 q^{95} +7.99425 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.21305	−0.989704	−0.494852	−	0.868977i	$-0.664778\pi$
$5$	−2.21305	−0.989704	−0.494852	+	0.868977i	$0.664778\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.42373i	0.730781i	0.930854	+	0.365390i	$0.119065\pi$
$11$	2.42373i	0.730781i	−0.930854	+	0.365390i	$0.880935\pi$
$12$	0	0
$13$	1.11593i	0.309504i	0.987953	+	0.154752i	$0.0494579\pi$
$13$	1.11593i	0.309504i	−0.987953	+	0.154752i	$0.950542\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.701326i	− 0.170097i	−0.996377	−	0.0850483i	$-0.972896\pi$
$17$	− 0.701326i	− 0.170097i	0.996377	−	0.0850483i	$-0.0271044\pi$
$18$	0	0
$19$	−0.938340	−0.215270	−0.107635	−	0.994190i	$-0.534328\pi$
$19$	−0.938340	−0.215270	−0.107635	+	0.994190i	$0.534328\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.30502	0.897658	0.448829	−	0.893618i	$-0.351841\pi$
$23$	4.30502	0.897658	0.448829	+	0.893618i	$0.351841\pi$
$24$	0	0
$25$	−0.102426	−0.0204852
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.26130	0.791303	0.395652	−	0.918401i	$-0.370519\pi$
$29$	4.26130	0.791303	0.395652	+	0.918401i	$0.370519\pi$
$30$	0	0
$31$	− 7.02221i	− 1.26123i	−0.776098	−	0.630613i	$-0.782804\pi$
$31$	− 7.02221i	− 1.26123i	0.776098	−	0.630613i	$-0.217196\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.21305i	0.374073i
$36$	0	0
$37$	− 3.12583i	− 0.513883i	−0.966427	−	0.256941i	$-0.917285\pi$
$37$	− 3.12583i	− 0.513883i	0.966427	−	0.256941i	$-0.0827148\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 0.157922i	− 0.0246632i	−0.999924	−	0.0123316i	$-0.996075\pi$
$41$	− 0.157922i	− 0.0246632i	0.999924	−	0.0123316i	$-0.00392537\pi$
$42$	0	0
$43$	−7.08975	−1.08118	−0.540588	−	0.841287i	$-0.681798\pi$
$43$	−7.08975	−1.08118	−0.540588	+	0.841287i	$0.681798\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.867970	−0.126607	−0.0633033	−	0.997994i	$-0.520164\pi$
$47$	−0.867970	−0.126607	−0.0633033	+	0.997994i	$0.520164\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.91444	0.537689	0.268845	−	0.963184i	$-0.413358\pi$
$53$	3.91444	0.537689	0.268845	+	0.963184i	$0.413358\pi$
$54$	0	0
$55$	− 5.36382i	− 0.723257i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.28820i	0.948843i	0.880298	+	0.474421i	$0.157343\pi$
$59$	7.28820i	0.948843i	−0.880298	+	0.474421i	$0.842657\pi$
$60$	0	0
$61$	− 4.57877i	− 0.586251i	−0.956074	−	0.293126i	$-0.905305\pi$
$61$	− 4.57877i	− 0.586251i	0.956074	−	0.293126i	$-0.0946955\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2.46961i	− 0.306318i
$66$	0	0
$67$	−15.1677	−1.85303	−0.926513	−	0.376264i	$-0.877209\pi$
$67$	−15.1677	−1.85303	−0.926513	+	0.376264i	$0.877209\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.93451	−0.822975	−0.411487	−	0.911416i	$-0.634991\pi$
$71$	−6.93451	−0.822975	−0.411487	+	0.911416i	$0.634991\pi$
$72$	0	0
$73$	7.02133	0.821785	0.410893	−	0.911684i	$-0.365217\pi$
$73$	7.02133	0.821785	0.410893	+	0.911684i	$0.365217\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.42373	0.276209
$78$	0	0
$79$	6.65903i	0.749200i	0.927187	+	0.374600i	$0.122220\pi$
$79$	6.65903i	0.749200i	−0.927187	+	0.374600i	$0.877780\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.41084i	0.923210i	0.887086	+	0.461605i	$0.152726\pi$
$83$	8.41084i	0.923210i	−0.887086	+	0.461605i	$0.847274\pi$
$84$	0	0
$85$	1.55207i	0.168345i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.34335i	0.778393i	0.921155	+	0.389197i	$0.127247\pi$
$89$	7.34335i	0.778393i	−0.921155	+	0.389197i	$0.872753\pi$
$90$	0	0
$91$	1.11593	0.116982
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.07659	0.213054
$96$	0	0
$97$	7.99425	0.811693	0.405846	−	0.913941i	$-0.366977\pi$
$97$	7.99425	0.811693	0.405846	+	0.913941i	$0.366977\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.6519	1.55742	0.778709	−	0.627385i	$-0.215875\pi$
$101$	15.6519	1.55742	0.778709	+	0.627385i	$0.215875\pi$
$102$	0	0
$103$	− 4.90314i	− 0.483121i	−0.970386	−	0.241560i	$-0.922341\pi$
$103$	− 4.90314i	− 0.483121i	0.970386	−	0.241560i	$-0.0776592\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.1434i	1.46397i	0.681320	+	0.731986i	$0.261406\pi$
$107$	15.1434i	1.46397i	−0.681320	+	0.731986i	$0.738594\pi$
$108$	0	0
$109$	2.72939i	0.261428i	0.991420	+	0.130714i	$0.0417270\pi$
$109$	2.72939i	0.261428i	−0.991420	+	0.130714i	$0.958273\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.84331i	0.267476i	0.991017	+	0.133738i	$0.0426980\pi$
$113$	2.84331i	0.267476i	−0.991017	+	0.133738i	$0.957302\pi$
$114$	0	0
$115$	−9.52720	−0.888416
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.701326	−0.0642905
$120$	0	0
$121$	5.12555	0.465959
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.2919	1.00998
$126$	0	0
$127$	10.4243i	0.925006i	0.886618	+	0.462503i	$0.153049\pi$
$127$	10.4243i	0.925006i	−0.886618	+	0.462503i	$0.846951\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 21.3573i	− 1.86600i	−0.359881	−	0.932998i	$-0.617183\pi$
$131$	− 21.3573i	− 1.86600i	0.359881	−	0.932998i	$-0.382817\pi$
$132$	0	0
$133$	0.938340i	0.0813644i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.2577i	0.961812i	0.876772	+	0.480906i	$0.159692\pi$
$137$	11.2577i	0.961812i	−0.876772	+	0.480906i	$0.840308\pi$
$138$	0	0
$139$	−8.11586	−0.688379	−0.344189	−	0.938900i	$-0.611846\pi$
$139$	−8.11586	−0.688379	−0.344189	+	0.938900i	$0.611846\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.70472	−0.226180
$144$	0	0
$145$	−9.43045	−0.783156
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.90580	−0.401899	−0.200949	−	0.979602i	$-0.564403\pi$
$149$	−4.90580	−0.401899	−0.200949	+	0.979602i	$0.564403\pi$
$150$	0	0
$151$	18.9521i	1.54230i	0.636654	+	0.771150i	$0.280318\pi$
$151$	18.9521i	1.54230i	−0.636654	+	0.771150i	$0.719682\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	15.5405i	1.24824i
$156$	0	0
$157$	− 10.0907i	− 0.805326i	−0.915348	−	0.402663i	$-0.868085\pi$
$157$	− 10.0907i	− 0.805326i	0.915348	−	0.402663i	$-0.131915\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 4.30502i	− 0.339283i
$162$	0	0
$163$	19.9762	1.56466	0.782330	−	0.622864i	$-0.214031\pi$
$163$	19.9762	1.56466	0.782330	+	0.622864i	$0.214031\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.8623	1.07270	0.536348	−	0.843997i	$-0.319803\pi$
$167$	13.8623	1.07270	0.536348	+	0.843997i	$0.319803\pi$
$168$	0	0
$169$	11.7547	0.904207
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.4796	1.63306	0.816531	−	0.577301i	$-0.195894\pi$
$173$	21.4796	1.63306	0.816531	+	0.577301i	$0.195894\pi$
$174$	0	0
$175$	0.102426i	0.00774268i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.94291i	0.294707i	0.989084	+	0.147354i	$0.0470755\pi$
$179$	3.94291i	0.294707i	−0.989084	+	0.147354i	$0.952925\pi$
$180$	0	0
$181$	11.7849i	0.875962i	0.898984	+	0.437981i	$0.144306\pi$
$181$	11.7849i	0.875962i	−0.898984	+	0.437981i	$0.855694\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.91760i	0.508592i
$186$	0	0
$187$	1.69982	0.124303
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.6064	1.20160	0.600798	−	0.799401i	$-0.294850\pi$
$191$	16.6064	1.20160	0.600798	+	0.799401i	$0.294850\pi$
$192$	0	0
$193$	6.90416	0.496972	0.248486	−	0.968635i	$-0.420067\pi$
$193$	6.90416	0.496972	0.248486	+	0.968635i	$0.420067\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.98273	−0.639993	−0.319997	−	0.947419i	$-0.603682\pi$
$197$	−8.98273	−0.639993	−0.319997	+	0.947419i	$0.603682\pi$
$198$	0	0
$199$	− 11.4427i	− 0.811153i	−0.914061	−	0.405577i	$-0.867071\pi$
$199$	− 11.4427i	− 0.811153i	0.914061	−	0.405577i	$-0.132929\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 4.26130i	− 0.299085i
$204$	0	0
$205$	0.349488i	0.0244093i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 2.27428i	− 0.157315i
$210$	0	0
$211$	−23.8035	−1.63870	−0.819351	−	0.573292i	$-0.805666\pi$
$211$	−23.8035	−1.63870	−0.819351	+	0.573292i	$0.805666\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	15.6899	1.07004
$216$	0	0
$217$	−7.02221	−0.476698
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.782633	0.0526456
$222$	0	0
$223$	4.42841i	0.296549i	0.988946	+	0.148274i	$0.0473718\pi$
$223$	4.42841i	0.296549i	−0.988946	+	0.148274i	$0.952628\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.46751i	0.429264i	0.976695	+	0.214632i	$0.0688552\pi$
$227$	6.46751i	0.429264i	−0.976695	+	0.214632i	$0.931145\pi$
$228$	0	0
$229$	− 11.4715i	− 0.758056i	−0.925385	−	0.379028i	$-0.876258\pi$
$229$	− 11.4715i	− 0.758056i	0.925385	−	0.379028i	$-0.123742\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 19.8678i	− 1.30158i	−0.759256	−	0.650792i	$-0.774437\pi$
$233$	− 19.8678i	− 1.30158i	0.759256	−	0.650792i	$-0.225563\pi$
$234$	0	0
$235$	1.92086	0.125303
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.2210	0.661138	0.330569	−	0.943782i	$-0.392759\pi$
$239$	10.2210	0.661138	0.330569	+	0.943782i	$0.392759\pi$
$240$	0	0
$241$	28.1808	1.81529	0.907643	−	0.419744i	$-0.137880\pi$
$241$	28.1808	1.81529	0.907643	+	0.419744i	$0.137880\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.21305	0.141386
$246$	0	0
$247$	− 1.04712i	− 0.0666269i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.62338i	0.291825i	0.989297	+	0.145913i	$0.0466118\pi$
$251$	4.62338i	0.291825i	−0.989297	+	0.145913i	$0.953388\pi$
$252$	0	0
$253$	10.4342i	0.655991i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	29.6682i	1.85065i	0.379174	+	0.925326i	$0.376208\pi$
$257$	29.6682i	1.85065i	−0.379174	+	0.925326i	$0.623792\pi$
$258$	0	0
$259$	−3.12583	−0.194230
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.8635	0.978185	0.489093	−	0.872232i	$-0.337328\pi$
$263$	15.8635	0.978185	0.489093	+	0.872232i	$0.337328\pi$
$264$	0	0
$265$	−8.66283	−0.532153
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.13803	−0.191329	−0.0956644	−	0.995414i	$-0.530498\pi$
$269$	−3.13803	−0.191329	−0.0956644	+	0.995414i	$0.530498\pi$
$270$	0	0
$271$	22.7174i	1.37998i	0.723817	+	0.689992i	$0.242386\pi$
$271$	22.7174i	1.37998i	−0.723817	+	0.689992i	$0.757614\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 0.248252i	− 0.0149702i
$276$	0	0
$277$	− 24.8267i	− 1.49169i	−0.666119	−	0.745845i	$-0.732046\pi$
$277$	− 24.8267i	− 1.49169i	0.666119	−	0.745845i	$-0.267954\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	30.2207i	1.80282i	0.432970	+	0.901408i	$0.357465\pi$
$281$	30.2207i	1.80282i	−0.432970	+	0.901408i	$0.642535\pi$
$282$	0	0
$283$	14.9803	0.890488	0.445244	−	0.895409i	$-0.353117\pi$
$283$	14.9803	0.890488	0.445244	+	0.895409i	$0.353117\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.157922	−0.00932182
$288$	0	0
$289$	16.5081	0.971067
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7.58993	−0.443408	−0.221704	−	0.975114i	$-0.571162\pi$
$293$	−7.58993	−0.443408	−0.221704	+	0.975114i	$0.571162\pi$
$294$	0	0
$295$	− 16.1291i	− 0.939074i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.80411i	0.277829i
$300$	0	0
$301$	7.08975i	0.408646i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.1330i	0.580216i
$306$	0	0
$307$	18.1246	1.03443	0.517214	−	0.855856i	$-0.326969\pi$
$307$	18.1246	1.03443	0.517214	+	0.855856i	$0.326969\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.0291	−0.965631	−0.482816	−	0.875722i	$-0.660386\pi$
$311$	−17.0291	−0.965631	−0.482816	+	0.875722i	$0.660386\pi$
$312$	0	0
$313$	7.44270	0.420686	0.210343	−	0.977628i	$-0.432542\pi$
$313$	7.44270	0.420686	0.210343	+	0.977628i	$0.432542\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.40549	0.135106	0.0675528	−	0.997716i	$-0.478481\pi$
$317$	2.40549	0.135106	0.0675528	+	0.997716i	$0.478481\pi$
$318$	0	0
$319$	10.3282i	0.578269i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.658082i	0.0366167i
$324$	0	0
$325$	− 0.114301i	− 0.00634025i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.867970i	0.0478528i
$330$	0	0
$331$	−14.3776	−0.790265	−0.395133	−	0.918624i	$-0.629301\pi$
$331$	−14.3776	−0.790265	−0.395133	+	0.918624i	$0.629301\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	33.5667	1.83395
$336$	0	0
$337$	20.4430	1.11360	0.556800	−	0.830646i	$-0.312029\pi$
$337$	20.4430	1.11360	0.556800	+	0.830646i	$0.312029\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	17.0199	0.921679
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.62453i	0.355623i	0.984065	+	0.177812i	$0.0569018\pi$
$347$	6.62453i	0.355623i	−0.984065	+	0.177812i	$0.943098\pi$
$348$	0	0
$349$	33.7876i	1.80861i	0.426889	+	0.904304i	$0.359609\pi$
$349$	33.7876i	1.80861i	−0.426889	+	0.904304i	$0.640391\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.5976i	1.41565i	0.706390	+	0.707823i	$0.250323\pi$
$353$	26.5976i	1.41565i	−0.706390	+	0.707823i	$0.749677\pi$
$354$	0	0
$355$	15.3464	0.814501
$356$	0	0
$357$	0	0
$358$	0	0
$359$	29.7098	1.56802	0.784012	−	0.620746i	$-0.213170\pi$
$359$	29.7098	1.56802	0.784012	+	0.620746i	$0.213170\pi$
$360$	0	0
$361$	−18.1195	−0.953659
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−15.5385	−0.813324
$366$	0	0
$367$	− 15.6932i	− 0.819176i	−0.912271	−	0.409588i	$-0.865672\pi$
$367$	− 15.6932i	− 0.819176i	0.912271	−	0.409588i	$-0.134328\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 3.91444i	− 0.203227i
$372$	0	0
$373$	11.0269i	0.570951i	0.958386	+	0.285475i	$0.0921515\pi$
$373$	11.0269i	0.570951i	−0.958386	+	0.285475i	$0.907849\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.75533i	0.244912i
$378$	0	0
$379$	7.84736	0.403092	0.201546	−	0.979479i	$-0.435403\pi$
$379$	7.84736	0.403092	0.201546	+	0.979479i	$0.435403\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.26238	−0.166700	−0.0833500	−	0.996520i	$-0.526562\pi$
$383$	−3.26238	−0.166700	−0.0833500	+	0.996520i	$0.526562\pi$
$384$	0	0
$385$	−5.36382	−0.273365
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.15697	0.0586609	0.0293304	−	0.999570i	$-0.490662\pi$
$389$	1.15697	0.0586609	0.0293304	+	0.999570i	$0.490662\pi$
$390$	0	0
$391$	− 3.01922i	− 0.152688i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 14.7367i	− 0.741486i
$396$	0	0
$397$	− 2.41295i	− 0.121103i	−0.998165	−	0.0605513i	$-0.980714\pi$
$397$	− 2.41295i	− 0.121103i	0.998165	−	0.0605513i	$-0.0192859\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 7.48047i	− 0.373557i	−0.982402	−	0.186778i	$-0.940195\pi$
$401$	− 7.48047i	− 0.373557i	0.982402	−	0.186778i	$-0.0598047\pi$
$402$	0	0
$403$	7.83631	0.390355
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.57615	0.375536
$408$	0	0
$409$	−7.49823	−0.370764	−0.185382	−	0.982667i	$-0.559352\pi$
$409$	−7.49823	−0.370764	−0.185382	+	0.982667i	$0.559352\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.28820	0.358629
$414$	0	0
$415$	− 18.6136i	− 0.913705i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 26.7682i	− 1.30771i	−0.756619	−	0.653856i	$-0.773150\pi$
$419$	− 26.7682i	− 1.30771i	0.756619	−	0.653856i	$-0.226850\pi$
$420$	0	0
$421$	28.1616i	1.37251i	0.727359	+	0.686257i	$0.240747\pi$
$421$	28.1616i	1.37251i	−0.727359	+	0.686257i	$0.759253\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.0718340i	0.00348446i
$426$	0	0
$427$	−4.57877	−0.221582
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−29.9359	−1.44196	−0.720980	−	0.692955i	$-0.756308\pi$
$431$	−29.9359	−1.44196	−0.720980	+	0.692955i	$0.756308\pi$
$432$	0	0
$433$	−39.1106	−1.87954	−0.939768	−	0.341814i	$-0.888959\pi$
$433$	−39.1106	−1.87954	−0.939768	+	0.341814i	$0.888959\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.03957	−0.193239
$438$	0	0
$439$	− 31.4351i	− 1.50031i	−0.661260	−	0.750157i	$-0.729978\pi$
$439$	− 31.4351i	− 1.50031i	0.661260	−	0.750157i	$-0.270022\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.2344i	1.15141i	0.817656	+	0.575707i	$0.195273\pi$
$443$	24.2344i	1.15141i	−0.817656	+	0.575707i	$0.804727\pi$
$444$	0	0
$445$	− 16.2512i	− 0.770379i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	22.6066i	1.06687i	0.845841	+	0.533435i	$0.179099\pi$
$449$	22.6066i	1.06687i	−0.845841	+	0.533435i	$0.820901\pi$
$450$	0	0
$451$	0.382759	0.0180234
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.46961	−0.115777
$456$	0	0
$457$	−9.35271	−0.437502	−0.218751	−	0.975781i	$-0.570198\pi$
$457$	−9.35271	−0.437502	−0.218751	+	0.975781i	$0.570198\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11.5898	0.539789	0.269895	−	0.962890i	$-0.413011\pi$
$461$	11.5898	0.539789	0.269895	+	0.962890i	$0.413011\pi$
$462$	0	0
$463$	17.8299i	0.828625i	0.910135	+	0.414312i	$0.135978\pi$
$463$	17.8299i	0.828625i	−0.910135	+	0.414312i	$0.864022\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.50180i	− 0.162044i	−0.996712	−	0.0810221i	$-0.974182\pi$
$467$	− 3.50180i	− 0.162044i	0.996712	−	0.0810221i	$-0.0258184\pi$
$468$	0	0
$469$	15.1677i	0.700378i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 17.1836i	− 0.790103i
$474$	0	0
$475$	0.0961103	0.00440985
$476$	0	0
$477$	0	0
$478$	0	0
$479$	18.2987	0.836087	0.418043	−	0.908427i	$-0.362716\pi$
$479$	18.2987	0.836087	0.418043	+	0.908427i	$0.362716\pi$
$480$	0	0
$481$	3.48822	0.159049
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−17.6916	−0.803336
$486$	0	0
$487$	24.3559i	1.10367i	0.833953	+	0.551836i	$0.186072\pi$
$487$	24.3559i	1.10367i	−0.833953	+	0.551836i	$0.813928\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.8479i	1.16650i	0.812292	+	0.583250i	$0.198219\pi$
$491$	25.8479i	1.16650i	−0.812292	+	0.583250i	$0.801781\pi$
$492$	0	0
$493$	− 2.98856i	− 0.134598i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.93451i	0.311055i
$498$	0	0
$499$	13.5430	0.606270	0.303135	−	0.952948i	$-0.401967\pi$
$499$	13.5430	0.606270	0.303135	+	0.952948i	$0.401967\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17.6930	−0.788890	−0.394445	−	0.918920i	$-0.629063\pi$
$503$	−17.6930	−0.788890	−0.394445	+	0.918920i	$0.629063\pi$
$504$	0	0
$505$	−34.6383	−1.54138
$506$	0	0
$507$	0	0
$508$	0	0
$509$	12.0147	0.532544	0.266272	−	0.963898i	$-0.414208\pi$
$509$	12.0147	0.532544	0.266272	+	0.963898i	$0.414208\pi$
$510$	0	0
$511$	− 7.02133i	− 0.310606i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.8509i	0.478147i
$516$	0	0
$517$	− 2.10372i	− 0.0925216i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17.5217i	0.767638i	0.923408	+	0.383819i	$0.125391\pi$
$521$	17.5217i	0.767638i	−0.923408	+	0.383819i	$0.874609\pi$
$522$	0	0
$523$	5.61428	0.245495	0.122748	−	0.992438i	$-0.460829\pi$
$523$	5.61428	0.245495	0.122748	+	0.992438i	$0.460829\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.92486	−0.214530
$528$	0	0
$529$	−4.46684	−0.194210
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.176230	0.00763337
$534$	0	0
$535$	− 33.5131i	− 1.44890i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.42373i	− 0.104397i
$540$	0	0
$541$	− 34.1532i	− 1.46836i	−0.678954	−	0.734181i	$-0.737567\pi$
$541$	− 34.1532i	− 1.46836i	0.678954	−	0.734181i	$-0.262433\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 6.04027i	− 0.258737i
$546$	0	0
$547$	−21.6399	−0.925255	−0.462627	−	0.886553i	$-0.653093\pi$
$547$	−21.6399	−0.925255	−0.462627	+	0.886553i	$0.653093\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.99855	−0.170344
$552$	0	0
$553$	6.65903	0.283171
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.9554	1.09976	0.549882	−	0.835242i	$-0.314673\pi$
$557$	25.9554	1.09976	0.549882	+	0.835242i	$0.314673\pi$
$558$	0	0
$559$	− 7.91168i	− 0.334629i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13.9262i	0.586921i	0.955971	+	0.293460i	$0.0948069\pi$
$563$	13.9262i	0.586921i	−0.955971	+	0.293460i	$0.905193\pi$
$564$	0	0
$565$	− 6.29237i	− 0.264722i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.6130i	0.780296i	0.920752	+	0.390148i	$0.127576\pi$
$569$	18.6130i	0.780296i	−0.920752	+	0.390148i	$0.872424\pi$
$570$	0	0
$571$	−4.69112	−0.196317	−0.0981586	−	0.995171i	$-0.531295\pi$
$571$	−4.69112	−0.196317	−0.0981586	+	0.995171i	$0.531295\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.440945	−0.0183887
$576$	0	0
$577$	−9.27967	−0.386318	−0.193159	−	0.981167i	$-0.561873\pi$
$577$	−9.27967	−0.386318	−0.193159	+	0.981167i	$0.561873\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.41084	0.348940
$582$	0	0
$583$	9.48752i	0.392933i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.3528i	1.41789i	0.705264	+	0.708945i	$0.250829\pi$
$587$	34.3528i	1.41789i	−0.705264	+	0.708945i	$0.749171\pi$
$588$	0	0
$589$	6.58922i	0.271504i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 44.4288i	− 1.82447i	−0.409666	−	0.912236i	$-0.634355\pi$
$593$	− 44.4288i	− 1.82447i	0.409666	−	0.912236i	$-0.365645\pi$
$594$	0	0
$595$	1.55207	0.0636285
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−13.8604	−0.566322	−0.283161	−	0.959072i	$-0.591383\pi$
$599$	−13.8604	−0.566322	−0.283161	+	0.959072i	$0.591383\pi$
$600$	0	0
$601$	−6.51128	−0.265601	−0.132800	−	0.991143i	$-0.542397\pi$
$601$	−6.51128	−0.265601	−0.132800	+	0.991143i	$0.542397\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−11.3431	−0.461162
$606$	0	0
$607$	43.5994i	1.76964i	0.465929	+	0.884822i	$0.345720\pi$
$607$	43.5994i	1.76964i	−0.465929	+	0.884822i	$0.654280\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 0.968597i	− 0.0391852i
$612$	0	0
$613$	− 22.4072i	− 0.905017i	−0.891760	−	0.452508i	$-0.850529\pi$
$613$	− 22.4072i	− 0.905017i	0.891760	−	0.452508i	$-0.149471\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 30.5128i	− 1.22840i	−0.789151	−	0.614199i	$-0.789479\pi$
$617$	− 30.5128i	− 1.22840i	0.789151	−	0.614199i	$-0.210521\pi$
$618$	0	0
$619$	13.5504	0.544637	0.272319	−	0.962207i	$-0.412210\pi$
$619$	13.5504	0.544637	0.272319	+	0.962207i	$0.412210\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.34335	0.294205
$624$	0	0
$625$	−24.4774	−0.979095
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.19222	−0.0874097
$630$	0	0
$631$	− 26.5393i	− 1.05651i	−0.849085	−	0.528257i	$-0.822846\pi$
$631$	− 26.5393i	− 1.05651i	0.849085	−	0.528257i	$-0.177154\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 23.0694i	− 0.915482i
$636$	0	0
$637$	− 1.11593i	− 0.0442149i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	10.3318i	0.408083i	0.978962	+	0.204041i	$0.0654077\pi$
$641$	10.3318i	0.408083i	−0.978962	+	0.204041i	$0.934592\pi$
$642$	0	0
$643$	−13.5130	−0.532901	−0.266450	−	0.963849i	$-0.585851\pi$
$643$	−13.5130	−0.532901	−0.266450	+	0.963849i	$0.585851\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.31247	−0.326797	−0.163399	−	0.986560i	$-0.552246\pi$
$647$	−8.31247	−0.326797	−0.163399	+	0.986560i	$0.552246\pi$
$648$	0	0
$649$	−17.6646	−0.693396
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.56899	−0.100533	−0.0502663	−	0.998736i	$-0.516007\pi$
$653$	−2.56899	−0.100533	−0.0502663	+	0.998736i	$0.516007\pi$
$654$	0	0
$655$	47.2647i	1.84678i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.7747i	0.497632i	0.968551	+	0.248816i	$0.0800414\pi$
$659$	12.7747i	0.497632i	−0.968551	+	0.248816i	$0.919959\pi$
$660$	0	0
$661$	25.8624i	1.00593i	0.864307	+	0.502965i	$0.167758\pi$
$661$	25.8624i	1.00593i	−0.864307	+	0.502965i	$0.832242\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 2.07659i	− 0.0805267i
$666$	0	0
$667$	18.3450	0.710320
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11.0977	0.428421
$672$	0	0
$673$	−33.9446	−1.30847	−0.654235	−	0.756292i	$-0.727009\pi$
$673$	−33.9446	−1.30847	−0.654235	+	0.756292i	$0.727009\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.1280	1.31165	0.655823	−	0.754915i	$-0.272322\pi$
$677$	34.1280	1.31165	0.655823	+	0.754915i	$0.272322\pi$
$678$	0	0
$679$	− 7.99425i	− 0.306791i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 6.80951i	− 0.260559i	−0.991477	−	0.130279i	$-0.958413\pi$
$683$	− 6.80951i	− 0.260559i	0.991477	−	0.130279i	$-0.0415874\pi$
$684$	0	0
$685$	− 24.9139i	− 0.951910i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.36825i	0.166417i
$690$	0	0
$691$	5.06521	0.192690	0.0963448	−	0.995348i	$-0.469285\pi$
$691$	5.06521	0.192690	0.0963448	+	0.995348i	$0.469285\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.9608	0.681291
$696$	0	0
$697$	−0.110755	−0.00419513
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−44.7933	−1.69182	−0.845910	−	0.533326i	$-0.820942\pi$
$701$	−44.7933	−1.69182	−0.845910	+	0.533326i	$0.820942\pi$
$702$	0	0
$703$	2.93309i	0.110624i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 15.6519i	− 0.588649i
$708$	0	0
$709$	− 40.5597i	− 1.52325i	−0.648018	−	0.761625i	$-0.724402\pi$
$709$	− 40.5597i	− 1.52325i	0.648018	−	0.761625i	$-0.275598\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 30.2307i	− 1.13215i
$714$	0	0
$715$	5.98566	0.223851
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−13.2926	−0.495729	−0.247864	−	0.968795i	$-0.579729\pi$
$719$	−13.2926	−0.495729	−0.247864	+	0.968795i	$0.579729\pi$
$720$	0	0
$721$	−4.90314	−0.182603
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.436468	−0.0162100
$726$	0	0
$727$	− 28.6102i	− 1.06109i	−0.847656	−	0.530546i	$-0.821987\pi$
$727$	− 28.6102i	− 1.06109i	0.847656	−	0.530546i	$-0.178013\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.97222i	0.183904i
$732$	0	0
$733$	− 33.9007i	− 1.25215i	−0.779763	−	0.626075i	$-0.784660\pi$
$733$	− 33.9007i	− 1.25215i	0.779763	−	0.626075i	$-0.215340\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 36.7623i	− 1.35416i
$738$	0	0
$739$	11.1200	0.409055	0.204528	−	0.978861i	$-0.434434\pi$
$739$	11.1200	0.409055	0.204528	+	0.978861i	$0.434434\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−45.7857	−1.67971	−0.839857	−	0.542808i	$-0.817361\pi$
$743$	−45.7857	−1.67971	−0.839857	+	0.542808i	$0.817361\pi$
$744$	0	0
$745$	10.8568	0.397761
$746$	0	0
$747$	0	0
$748$	0	0
$749$	15.1434	0.553329
$750$	0	0
$751$	13.8013i	0.503616i	0.967777	+	0.251808i	$0.0810252\pi$
$751$	13.8013i	0.503616i	−0.967777	+	0.251808i	$0.918975\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 41.9418i	− 1.52642i
$756$	0	0
$757$	− 5.94985i	− 0.216251i	−0.994137	−	0.108126i	$-0.965515\pi$
$757$	− 5.94985i	− 0.216251i	0.994137	−	0.108126i	$-0.0344848\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 38.1237i	− 1.38198i	−0.722864	−	0.690991i	$-0.757175\pi$
$761$	− 38.1237i	− 1.38198i	0.722864	−	0.690991i	$-0.242825\pi$
$762$	0	0
$763$	2.72939	0.0988106
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.13314	−0.293671
$768$	0	0
$769$	15.5207	0.559689	0.279845	−	0.960045i	$-0.409717\pi$
$769$	15.5207	0.559689	0.279845	+	0.960045i	$0.409717\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	39.6525	1.42620	0.713101	−	0.701061i	$-0.247290\pi$
$773$	39.6525	1.42620	0.713101	+	0.701061i	$0.247290\pi$
$774$	0	0
$775$	0.719256i	0.0258365i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.148184i	0.00530925i
$780$	0	0
$781$	− 16.8073i	− 0.601414i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	22.3312i	0.797035i
$786$	0	0
$787$	50.2403	1.79087	0.895436	−	0.445189i	$-0.146864\pi$
$787$	50.2403	1.79087	0.895436	+	0.445189i	$0.146864\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.84331	0.101096
$792$	0	0
$793$	5.10960	0.181447
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.78519	0.0632345	0.0316173	−	0.999500i	$-0.489934\pi$
$797$	1.78519	0.0632345	0.0316173	+	0.999500i	$0.489934\pi$
$798$	0	0
$799$	0.608730i	0.0215353i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	17.0178i	0.600545i
$804$	0	0
$805$	9.52720i	0.335790i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	31.9043i	1.12169i	0.827919	+	0.560847i	$0.189524\pi$
$809$	31.9043i	1.12169i	−0.827919	+	0.560847i	$0.810476\pi$
$810$	0	0
$811$	12.5528	0.440789	0.220394	−	0.975411i	$-0.429266\pi$
$811$	12.5528	0.440789	0.220394	+	0.975411i	$0.429266\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−44.2084	−1.54855
$816$	0	0
$817$	6.65259	0.232745
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42.1349	1.47052	0.735260	−	0.677786i	$-0.237060\pi$
$821$	42.1349	1.47052	0.735260	+	0.677786i	$0.237060\pi$
$822$	0	0
$823$	1.50592i	0.0524931i	0.999656	+	0.0262465i	$0.00835549\pi$
$823$	1.50592i	0.0524931i	−0.999656	+	0.0262465i	$0.991645\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	40.6098i	1.41214i	0.708141	+	0.706071i	$0.249534\pi$
$827$	40.6098i	1.41214i	−0.708141	+	0.706071i	$0.750466\pi$
$828$	0	0
$829$	35.4547i	1.23139i	0.787984	+	0.615696i	$0.211125\pi$
$829$	35.4547i	1.23139i	−0.787984	+	0.615696i	$0.788875\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.701326i	0.0242995i
$834$	0	0
$835$	−30.6779	−1.06165
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.8372	1.09914	0.549571	−	0.835447i	$-0.314791\pi$
$839$	31.8372	1.09914	0.549571	+	0.835447i	$0.314791\pi$
$840$	0	0
$841$	−10.8413	−0.373839
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−26.0137	−0.894898
$846$	0	0
$847$	− 5.12555i	− 0.176116i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 13.4567i	− 0.461291i
$852$	0	0
$853$	30.0499i	1.02889i	0.857523	+	0.514445i	$0.172002\pi$
$853$	30.0499i	1.02889i	−0.857523	+	0.514445i	$0.827998\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.7594i	0.914085i	0.889445	+	0.457043i	$0.151091\pi$
$857$	26.7594i	0.914085i	−0.889445	+	0.457043i	$0.848909\pi$
$858$	0	0
$859$	3.98544	0.135981	0.0679907	−	0.997686i	$-0.478341\pi$
$859$	3.98544	0.135981	0.0679907	+	0.997686i	$0.478341\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33.1318	1.12782	0.563910	−	0.825836i	$-0.309296\pi$
$863$	33.1318	1.12782	0.563910	+	0.825836i	$0.309296\pi$
$864$	0	0
$865$	−47.5353	−1.61625
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−16.1397	−0.547501
$870$	0	0
$871$	− 16.9261i	− 0.573519i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 11.2919i	− 0.381736i
$876$	0	0
$877$	4.35210i	0.146960i	0.997297	+	0.0734800i	$0.0234105\pi$
$877$	4.35210i	0.146960i	−0.997297	+	0.0734800i	$0.976589\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 37.2945i	− 1.25648i	−0.778018	−	0.628242i	$-0.783775\pi$
$881$	− 37.2945i	− 1.25648i	0.778018	−	0.628242i	$-0.216225\pi$
$882$	0	0
$883$	44.7359	1.50548	0.752742	−	0.658315i	$-0.228731\pi$
$883$	44.7359	1.50548	0.752742	+	0.658315i	$0.228731\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.69357	−0.124018	−0.0620090	−	0.998076i	$-0.519751\pi$
$887$	−3.69357	−0.124018	−0.0620090	+	0.998076i	$0.519751\pi$
$888$	0	0
$889$	10.4243	0.349619
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.814451	0.0272546
$894$	0	0
$895$	− 8.72584i	− 0.291673i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 29.9237i	− 0.998012i
$900$	0	0
$901$	− 2.74530i	− 0.0914591i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 26.0804i	− 0.866943i
$906$	0	0
$907$	9.61629	0.319304	0.159652	−	0.987173i	$-0.448963\pi$
$907$	9.61629	0.319304	0.159652	+	0.987173i	$0.448963\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−53.1598	−1.76126	−0.880632	−	0.473802i	$-0.842881\pi$
$911$	−53.1598	−1.76126	−0.880632	+	0.473802i	$0.842881\pi$
$912$	0	0
$913$	−20.3856	−0.674664
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−21.3573	−0.705280
$918$	0	0
$919$	− 33.3183i	− 1.09907i	−0.835471	−	0.549534i	$-0.814805\pi$
$919$	− 33.3183i	− 1.09907i	0.835471	−	0.549534i	$-0.185195\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 7.73845i	− 0.254714i
$924$	0	0
$925$	0.320166i	0.0105270i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 13.5434i	− 0.444343i	−0.975008	−	0.222172i	$-0.928685\pi$
$929$	− 13.5434i	− 0.444343i	0.975008	−	0.222172i	$-0.0713146\pi$
$930$	0	0
$931$	0.938340	0.0307528
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.76178	−0.123024
$936$	0	0
$937$	42.3432	1.38329	0.691647	−	0.722236i	$-0.256886\pi$
$937$	42.3432	1.38329	0.691647	+	0.722236i	$0.256886\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−60.9773	−1.98780	−0.993902	−	0.110267i	$-0.964829\pi$
$941$	−60.9773	−1.98780	−0.993902	+	0.110267i	$0.964829\pi$
$942$	0	0
$943$	− 0.679855i	− 0.0221391i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 39.4879i	− 1.28318i	−0.767047	−	0.641591i	$-0.778275\pi$
$947$	− 39.4879i	− 1.28318i	0.767047	−	0.641591i	$-0.221725\pi$
$948$	0	0
$949$	7.83534i	0.254346i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	25.8210i	0.836425i	0.908349	+	0.418213i	$0.137343\pi$
$953$	25.8210i	0.836425i	−0.908349	+	0.418213i	$0.862657\pi$
$954$	0	0
$955$	−36.7507	−1.18922
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.2577	0.363531
$960$	0	0
$961$	−18.3114	−0.590690
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−15.2792	−0.491856
$966$	0	0
$967$	− 4.92813i	− 0.158478i	−0.996856	−	0.0792390i	$-0.974751\pi$
$967$	− 4.92813i	− 0.158478i	0.996856	−	0.0792390i	$-0.0252490\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 20.9043i	− 0.670850i	−0.942067	−	0.335425i	$-0.891120\pi$
$971$	− 20.9043i	− 0.670850i	0.942067	−	0.335425i	$-0.108880\pi$
$972$	0	0
$973$	8.11586i	0.260183i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 16.6995i	− 0.534266i	−0.963660	−	0.267133i	$-0.913924\pi$
$977$	− 16.6995i	− 0.534266i	0.963660	−	0.267133i	$-0.0860763\pi$
$978$	0	0
$979$	−17.7983	−0.568835
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−27.7932	−0.886465	−0.443232	−	0.896407i	$-0.646168\pi$
$983$	−27.7932	−0.886465	−0.443232	+	0.896407i	$0.646168\pi$
$984$	0	0
$985$	19.8792	0.633404
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−30.5215	−0.970526
$990$	0	0
$991$	38.3482i	1.21817i	0.793104	+	0.609086i	$0.208464\pi$
$991$	38.3482i	1.21817i	−0.793104	+	0.609086i	$0.791536\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25.3233i	0.802802i
$996$	0	0
$997$	− 27.7418i	− 0.878590i	−0.898343	−	0.439295i	$-0.855228\pi$
$997$	− 27.7418i	− 0.878590i	0.898343	−	0.439295i	$-0.144772\pi$
$998$	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.5615.6		32
3.2	odd	2	inner	6048.2.j.c.5615.27		32
4.3	odd	2		1512.2.j.c.323.11	✓	32
8.3	odd	2	inner	6048.2.j.c.5615.28		32
8.5	even	2		1512.2.j.c.323.21	yes	32
12.11	even	2		1512.2.j.c.323.22	yes	32
24.5	odd	2		1512.2.j.c.323.12	yes	32
24.11	even	2	inner	6048.2.j.c.5615.5		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.c.323.11	✓	32	4.3	odd	2
1512.2.j.c.323.12	yes	32	24.5	odd	2
1512.2.j.c.323.21	yes	32	8.5	even	2
1512.2.j.c.323.22	yes	32	12.11	even	2
6048.2.j.c.5615.5		32	24.11	even	2	inner
6048.2.j.c.5615.6		32	1.1	even	1	trivial
6048.2.j.c.5615.27		32	3.2	odd	2	inner
6048.2.j.c.5615.28		32	8.3	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)

\(f(q)\)	\(=\)	\(q-2.21305 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-2.21305 q^{5} -1.00000i q^{7} +2.42373i q^{11} +1.11593i q^{13} -0.701326i q^{17} -0.938340 q^{19} +4.30502 q^{23} -0.102426 q^{25} +4.26130 q^{29} -7.02221i q^{31} +2.21305i q^{35} -3.12583i q^{37} -0.157922i q^{41} -7.08975 q^{43} -0.867970 q^{47} -1.00000 q^{49} +3.91444 q^{53} -5.36382i q^{55} +7.28820i q^{59} -4.57877i q^{61} -2.46961i q^{65} -15.1677 q^{67} -6.93451 q^{71} +7.02133 q^{73} +2.42373 q^{77} +6.65903i q^{79} +8.41084i q^{83} +1.55207i q^{85} +7.34335i q^{89} +1.11593 q^{91} +2.07659 q^{95} +7.99425 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more