LMFDB - Embedded newform 6048.2.j.d.5615.13

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:	$48$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5615.13
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.d.5615.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.29653 q^{5} -1.00000i q^{7} +O(q^{10})$	$q-2.29653 q^{5} -1.00000i q^{7} +4.02419i q^{11} +1.82461i q^{13} -0.430472i q^{17} -5.01046 q^{19} -3.49479 q^{23} +0.274031 q^{25} -2.16016 q^{29} -2.10635i q^{31} +2.29653i q^{35} +2.19775i q^{37} -4.35032i q^{41} +12.0354 q^{43} +1.72531 q^{47} -1.00000 q^{49} -10.0938 q^{53} -9.24166i q^{55} +10.6175i q^{59} -8.44575i q^{61} -4.19026i q^{65} +12.6358 q^{67} +8.95532 q^{71} -7.16051 q^{73} +4.02419 q^{77} -15.2405i q^{79} +13.4507i q^{83} +0.988589i q^{85} +7.44987i q^{89} +1.82461 q^{91} +11.5067 q^{95} +1.68878 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 16 q^{19} + 48 q^{25} - 64 q^{43} - 48 q^{49} - 32 q^{67}+O(q^{100}) $ 48 * q - 16 * q^19 + 48 * q^25 - 64 * q^43 - 48 * q^49 - 32 * q^67

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
$5$	−2.29653	−1.02704	−0.513519	−	0.858078i	$-0.671658\pi$
$5$	−2.29653	−1.02704	−0.513519	+	0.858078i	$0.671658\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.02419i	1.21334i	0.794954	+	0.606670i	$0.207495\pi$
$11$	4.02419i	1.21334i	−0.794954	+	0.606670i	$0.792505\pi$
$12$	0	0
$13$	1.82461i	0.506055i	0.967459	+	0.253027i	$0.0814263\pi$
$13$	1.82461i	0.506055i	−0.967459	+	0.253027i	$0.918574\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.430472i	− 0.104405i	−0.998637	−	0.0522024i	$-0.983376\pi$
$17$	− 0.430472i	− 0.104405i	0.998637	−	0.0522024i	$-0.0166241\pi$
$18$	0	0
$19$	−5.01046	−1.14948	−0.574739	−	0.818336i	$-0.694897\pi$
$19$	−5.01046	−1.14948	−0.574739	+	0.818336i	$0.694897\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.49479	−0.728715	−0.364357	−	0.931259i	$-0.618711\pi$
$23$	−3.49479	−0.728715	−0.364357	+	0.931259i	$0.618711\pi$
$24$	0	0
$25$	0.274031	0.0548062
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.16016	−0.401131	−0.200565	−	0.979680i	$-0.564278\pi$
$29$	−2.16016	−0.401131	−0.200565	+	0.979680i	$0.564278\pi$
$30$	0	0
$31$	− 2.10635i	− 0.378311i	−0.981947	−	0.189156i	$-0.939425\pi$
$31$	− 2.10635i	− 0.378311i	0.981947	−	0.189156i	$-0.0605750\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.29653i	0.388184i
$36$	0	0
$37$	2.19775i	0.361309i	0.983547	+	0.180654i	$0.0578215\pi$
$37$	2.19775i	0.361309i	−0.983547	+	0.180654i	$0.942178\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 4.35032i	− 0.679405i	−0.940533	−	0.339703i	$-0.889674\pi$
$41$	− 4.35032i	− 0.679405i	0.940533	−	0.339703i	$-0.110326\pi$
$42$	0	0
$43$	12.0354	1.83538	0.917690	−	0.397298i	$-0.130052\pi$
$43$	12.0354	1.83538	0.917690	+	0.397298i	$0.130052\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.72531	0.251663	0.125831	−	0.992052i	$-0.459840\pi$
$47$	1.72531	0.251663	0.125831	+	0.992052i	$0.459840\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.0938	−1.38649	−0.693246	−	0.720701i	$-0.743820\pi$
$53$	−10.0938	−1.38649	−0.693246	+	0.720701i	$0.743820\pi$
$54$	0	0
$55$	− 9.24166i	− 1.24615i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.6175i	1.38228i	0.722721	+	0.691140i	$0.242891\pi$
$59$	10.6175i	1.38228i	−0.722721	+	0.691140i	$0.757109\pi$
$60$	0	0
$61$	− 8.44575i	− 1.08137i	−0.841226	−	0.540684i	$-0.818166\pi$
$61$	− 8.44575i	− 1.08137i	0.841226	−	0.540684i	$-0.181834\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 4.19026i	− 0.519737i
$66$	0	0
$67$	12.6358	1.54370	0.771852	−	0.635802i	$-0.219331\pi$
$67$	12.6358	1.54370	0.771852	+	0.635802i	$0.219331\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.95532	1.06280	0.531401	−	0.847121i	$-0.321666\pi$
$71$	8.95532	1.06280	0.531401	+	0.847121i	$0.321666\pi$
$72$	0	0
$73$	−7.16051	−0.838075	−0.419037	−	0.907969i	$-0.637632\pi$
$73$	−7.16051	−0.838075	−0.419037	+	0.907969i	$0.637632\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.02419	0.458599
$78$	0	0
$79$	− 15.2405i	− 1.71468i	−0.514747	−	0.857342i	$-0.672114\pi$
$79$	− 15.2405i	− 1.71468i	0.514747	−	0.857342i	$-0.327886\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.4507i	1.47640i	0.674581	+	0.738201i	$0.264324\pi$
$83$	13.4507i	1.47640i	−0.674581	+	0.738201i	$0.735676\pi$
$84$	0	0
$85$	0.988589i	0.107228i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.44987i	0.789684i	0.918749	+	0.394842i	$0.129201\pi$
$89$	7.44987i	0.789684i	−0.918749	+	0.394842i	$0.870799\pi$
$90$	0	0
$91$	1.82461	0.191271
$92$	0	0
$93$	0	0
$94$	0	0
$95$	11.5067	1.18056
$96$	0	0
$97$	1.68878	0.171469	0.0857347	−	0.996318i	$-0.472676\pi$
$97$	1.68878	0.171469	0.0857347	+	0.996318i	$0.472676\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.3668	−1.33005	−0.665024	−	0.746822i	$-0.731579\pi$
$101$	−13.3668	−1.33005	−0.665024	+	0.746822i	$0.731579\pi$
$102$	0	0
$103$	4.59723i	0.452978i	0.974014	+	0.226489i	$0.0727248\pi$
$103$	4.59723i	0.452978i	−0.974014	+	0.226489i	$0.927275\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 15.7048i	− 1.51824i	−0.650951	−	0.759120i	$-0.725630\pi$
$107$	− 15.7048i	− 1.51824i	0.650951	−	0.759120i	$-0.274370\pi$
$108$	0	0
$109$	− 16.0140i	− 1.53386i	−0.641728	−	0.766932i	$-0.721782\pi$
$109$	− 16.0140i	− 1.53386i	0.641728	−	0.766932i	$-0.278218\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.46858i	− 0.138152i	−0.997611	−	0.0690760i	$-0.977995\pi$
$113$	− 1.46858i	− 0.138152i	0.997611	−	0.0690760i	$-0.0220051\pi$
$114$	0	0
$115$	8.02588	0.748417
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.430472	−0.0394613
$120$	0	0
$121$	−5.19413	−0.472193
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.8533	0.970750
$126$	0	0
$127$	6.35551i	0.563960i	0.959420	+	0.281980i	$0.0909913\pi$
$127$	6.35551i	0.563960i	−0.959420	+	0.281980i	$0.909009\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.75532i	0.677585i	0.940861	+	0.338793i	$0.110018\pi$
$131$	7.75532i	0.677585i	−0.940861	+	0.338793i	$0.889982\pi$
$132$	0	0
$133$	5.01046i	0.434462i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.09831i	0.264706i	0.991203	+	0.132353i	$0.0422533\pi$
$137$	3.09831i	0.264706i	−0.991203	+	0.132353i	$0.957747\pi$
$138$	0	0
$139$	3.76422	0.319277	0.159638	−	0.987176i	$-0.448967\pi$
$139$	3.76422	0.319277	0.159638	+	0.987176i	$0.448967\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−7.34257	−0.614016
$144$	0	0
$145$	4.96085	0.411976
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.57894	0.538968	0.269484	−	0.963005i	$-0.413147\pi$
$149$	6.57894	0.538968	0.269484	+	0.963005i	$0.413147\pi$
$150$	0	0
$151$	− 1.08336i	− 0.0881627i	−0.999028	−	0.0440814i	$-0.985964\pi$
$151$	− 1.08336i	− 0.0881627i	0.999028	−	0.0440814i	$-0.0140361\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.83728i	0.388540i
$156$	0	0
$157$	− 16.6130i	− 1.32586i	−0.748680	−	0.662932i	$-0.769312\pi$
$157$	− 16.6130i	− 1.32586i	0.748680	−	0.662932i	$-0.230688\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.49479i	0.275428i
$162$	0	0
$163$	−11.4881	−0.899818	−0.449909	−	0.893074i	$-0.648544\pi$
$163$	−11.4881	−0.899818	−0.449909	+	0.893074i	$0.648544\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.4761	1.04281	0.521405	−	0.853309i	$-0.325408\pi$
$167$	13.4761	1.04281	0.521405	+	0.853309i	$0.325408\pi$
$168$	0	0
$169$	9.67081	0.743909
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.98344	−0.378884	−0.189442	−	0.981892i	$-0.560668\pi$
$173$	−4.98344	−0.378884	−0.189442	+	0.981892i	$0.560668\pi$
$174$	0	0
$175$	− 0.274031i	− 0.0207148i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 3.45929i	− 0.258559i	−0.991608	−	0.129280i	$-0.958733\pi$
$179$	− 3.45929i	− 0.258559i	0.991608	−	0.129280i	$-0.0412665\pi$
$180$	0	0
$181$	− 10.1409i	− 0.753768i	−0.926260	−	0.376884i	$-0.876996\pi$
$181$	− 10.1409i	− 0.753768i	0.926260	−	0.376884i	$-0.123004\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 5.04720i	− 0.371078i
$186$	0	0
$187$	1.73230	0.126678
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.9743	0.794074	0.397037	−	0.917803i	$-0.370038\pi$
$191$	10.9743	0.794074	0.397037	+	0.917803i	$0.370038\pi$
$192$	0	0
$193$	2.07226	0.149165	0.0745824	−	0.997215i	$-0.476238\pi$
$193$	2.07226	0.149165	0.0745824	+	0.997215i	$0.476238\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.69547	−0.334539	−0.167269	−	0.985911i	$-0.553495\pi$
$197$	−4.69547	−0.334539	−0.167269	+	0.985911i	$0.553495\pi$
$198$	0	0
$199$	− 16.7556i	− 1.18777i	−0.804549	−	0.593887i	$-0.797593\pi$
$199$	− 16.7556i	− 1.18777i	0.804549	−	0.593887i	$-0.202407\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.16016i	0.151613i
$204$	0	0
$205$	9.99061i	0.697775i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 20.1631i	− 1.39471i
$210$	0	0
$211$	−15.2955	−1.05299	−0.526494	−	0.850179i	$-0.676494\pi$
$211$	−15.2955	−1.05299	−0.526494	+	0.850179i	$0.676494\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−27.6396	−1.88500
$216$	0	0
$217$	−2.10635	−0.142988
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.785442	0.0528345
$222$	0	0
$223$	− 23.5657i	− 1.57808i	−0.614343	−	0.789039i	$-0.710579\pi$
$223$	− 23.5657i	− 1.57808i	0.614343	−	0.789039i	$-0.289421\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 16.6514i	− 1.10519i	−0.833450	−	0.552595i	$-0.813638\pi$
$227$	− 16.6514i	− 1.10519i	0.833450	−	0.552595i	$-0.186362\pi$
$228$	0	0
$229$	− 24.4291i	− 1.61432i	−0.590332	−	0.807161i	$-0.701003\pi$
$229$	− 24.4291i	− 1.61432i	0.590332	−	0.807161i	$-0.298997\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 18.3470i	− 1.20195i	−0.799268	−	0.600975i	$-0.794779\pi$
$233$	− 18.3470i	− 1.20195i	0.799268	−	0.600975i	$-0.205221\pi$
$234$	0	0
$235$	−3.96222	−0.258467
$236$	0	0
$237$	0	0
$238$	0	0
$239$	30.1173	1.94812	0.974062	−	0.226280i	$-0.0726563\pi$
$239$	30.1173	1.94812	0.974062	+	0.226280i	$0.0726563\pi$
$240$	0	0
$241$	−5.51878	−0.355496	−0.177748	−	0.984076i	$-0.556881\pi$
$241$	−5.51878	−0.355496	−0.177748	+	0.984076i	$0.556881\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.29653	0.146720
$246$	0	0
$247$	− 9.14212i	− 0.581699i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 18.2010i	− 1.14884i	−0.818561	−	0.574420i	$-0.805228\pi$
$251$	− 18.2010i	− 1.14884i	0.818561	−	0.574420i	$-0.194772\pi$
$252$	0	0
$253$	− 14.0637i	− 0.884178i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5.91116i	− 0.368728i	−0.982858	−	0.184364i	$-0.940977\pi$
$257$	− 5.91116i	− 0.368728i	0.982858	−	0.184364i	$-0.0590226\pi$
$258$	0	0
$259$	2.19775	0.136562
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.7132	0.968916	0.484458	−	0.874815i	$-0.339017\pi$
$263$	15.7132	0.968916	0.484458	+	0.874815i	$0.339017\pi$
$264$	0	0
$265$	23.1807	1.42398
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−26.9473	−1.64301	−0.821504	−	0.570202i	$-0.806865\pi$
$269$	−26.9473	−1.64301	−0.821504	+	0.570202i	$0.806865\pi$
$270$	0	0
$271$	5.68443i	0.345305i	0.984983	+	0.172652i	$0.0552337\pi$
$271$	5.68443i	0.345305i	−0.984983	+	0.172652i	$0.944766\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.10275i	0.0664985i
$276$	0	0
$277$	7.86390i	0.472496i	0.971693	+	0.236248i	$0.0759178\pi$
$277$	7.86390i	0.472496i	−0.971693	+	0.236248i	$0.924082\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0.186660i	0.0111352i	0.999985	+	0.00556760i	$0.00177223\pi$
$281$	0.186660i	0.0111352i	−0.999985	+	0.00556760i	$0.998228\pi$
$282$	0	0
$283$	12.4545	0.740343	0.370171	−	0.928963i	$-0.379299\pi$
$283$	12.4545	0.740343	0.370171	+	0.928963i	$0.379299\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.35032	−0.256791
$288$	0	0
$289$	16.8147	0.989100
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8.52169	0.497843	0.248921	−	0.968524i	$-0.419924\pi$
$293$	8.52169	0.497843	0.248921	+	0.968524i	$0.419924\pi$
$294$	0	0
$295$	− 24.3833i	− 1.41965i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 6.37662i	− 0.368770i
$300$	0	0
$301$	− 12.0354i	− 0.693708i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	19.3959i	1.11060i
$306$	0	0
$307$	−23.4380	−1.33768	−0.668838	−	0.743408i	$-0.733208\pi$
$307$	−23.4380	−1.33768	−0.668838	+	0.743408i	$0.733208\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.1365	−0.688199	−0.344099	−	0.938933i	$-0.611816\pi$
$311$	−12.1365	−0.688199	−0.344099	+	0.938933i	$0.611816\pi$
$312$	0	0
$313$	−7.54204	−0.426301	−0.213151	−	0.977019i	$-0.568372\pi$
$313$	−7.54204	−0.426301	−0.213151	+	0.977019i	$0.568372\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−29.7545	−1.67118	−0.835590	−	0.549354i	$-0.814874\pi$
$317$	−29.7545	−1.67118	−0.835590	+	0.549354i	$0.814874\pi$
$318$	0	0
$319$	− 8.69288i	− 0.486708i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.15686i	0.120011i
$324$	0	0
$325$	0.499998i	0.0277349i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1.72531i	− 0.0951195i
$330$	0	0
$331$	16.2670	0.894117	0.447058	−	0.894505i	$-0.352472\pi$
$331$	16.2670	0.894117	0.447058	+	0.894505i	$0.352472\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−29.0184	−1.58544
$336$	0	0
$337$	32.0056	1.74345	0.871727	−	0.489992i	$-0.163000\pi$
$337$	32.0056	1.74345	0.871727	+	0.489992i	$0.163000\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.47634	0.459020
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 15.3562i	− 0.824366i	−0.911101	−	0.412183i	$-0.864766\pi$
$347$	− 15.3562i	− 0.824366i	0.911101	−	0.412183i	$-0.135234\pi$
$348$	0	0
$349$	− 10.7044i	− 0.572996i	−0.958081	−	0.286498i	$-0.907509\pi$
$349$	− 10.7044i	− 0.572996i	0.958081	−	0.286498i	$-0.0924912\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 17.0572i	− 0.907866i	−0.891036	−	0.453933i	$-0.850021\pi$
$353$	− 17.0572i	− 0.907866i	0.891036	−	0.453933i	$-0.149979\pi$
$354$	0	0
$355$	−20.5661	−1.09154
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.2174	−0.908697	−0.454349	−	0.890824i	$-0.650128\pi$
$359$	−17.2174	−0.908697	−0.454349	+	0.890824i	$0.650128\pi$
$360$	0	0
$361$	6.10474	0.321302
$362$	0	0
$363$	0	0
$364$	0	0
$365$	16.4443	0.860734
$366$	0	0
$367$	25.6116i	1.33692i	0.743750	+	0.668458i	$0.233045\pi$
$367$	25.6116i	1.33692i	−0.743750	+	0.668458i	$0.766955\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.0938i	0.524045i
$372$	0	0
$373$	− 3.99889i	− 0.207054i	−0.994627	−	0.103527i	$-0.966987\pi$
$373$	− 3.99889i	− 0.207054i	0.994627	−	0.103527i	$-0.0330129\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.94143i	− 0.202994i
$378$	0	0
$379$	21.0603	1.08180	0.540898	−	0.841088i	$-0.318085\pi$
$379$	21.0603	1.08180	0.540898	+	0.841088i	$0.318085\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	27.1156	1.38554	0.692771	−	0.721158i	$-0.256390\pi$
$383$	27.1156	1.38554	0.692771	+	0.721158i	$0.256390\pi$
$384$	0	0
$385$	−9.24166	−0.470999
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8.59148	0.435605	0.217803	−	0.975993i	$-0.430111\pi$
$389$	8.59148	0.435605	0.217803	+	0.975993i	$0.430111\pi$
$390$	0	0
$391$	1.50441i	0.0760813i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	35.0001i	1.76105i
$396$	0	0
$397$	− 33.3145i	− 1.67201i	−0.548725	−	0.836003i	$-0.684887\pi$
$397$	− 33.3145i	− 1.67201i	0.548725	−	0.836003i	$-0.315113\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 5.42643i	− 0.270983i	−0.990779	−	0.135491i	$-0.956739\pi$
$401$	− 5.42643i	− 0.270983i	0.990779	−	0.135491i	$-0.0432613\pi$
$402$	0	0
$403$	3.84325	0.191446
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.84419	−0.438390
$408$	0	0
$409$	37.1175	1.83534	0.917672	−	0.397339i	$-0.130066\pi$
$409$	37.1175	1.83534	0.917672	+	0.397339i	$0.130066\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.6175	0.522452
$414$	0	0
$415$	− 30.8898i	− 1.51632i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	19.7739i	0.966016i	0.875616	+	0.483008i	$0.160456\pi$
$419$	19.7739i	0.966016i	−0.875616	+	0.483008i	$0.839544\pi$
$420$	0	0
$421$	19.4861i	0.949692i	0.880069	+	0.474846i	$0.157496\pi$
$421$	19.4861i	0.949692i	−0.880069	+	0.474846i	$0.842504\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 0.117963i	− 0.00572202i
$426$	0	0
$427$	−8.44575	−0.408718
$428$	0	0
$429$	0	0
$430$	0	0
$431$	33.0237	1.59069	0.795347	−	0.606154i	$-0.207288\pi$
$431$	33.0237	1.59069	0.795347	+	0.606154i	$0.207288\pi$
$432$	0	0
$433$	−33.2428	−1.59755	−0.798773	−	0.601632i	$-0.794517\pi$
$433$	−33.2428	−1.59755	−0.798773	+	0.601632i	$0.794517\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.5105	0.837642
$438$	0	0
$439$	9.28199i	0.443005i	0.975160	+	0.221503i	$0.0710962\pi$
$439$	9.28199i	0.443005i	−0.975160	+	0.221503i	$0.928904\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 2.61270i	− 0.124133i	−0.998072	−	0.0620667i	$-0.980231\pi$
$443$	− 2.61270i	− 0.124133i	0.998072	−	0.0620667i	$-0.0197691\pi$
$444$	0	0
$445$	− 17.1088i	− 0.811035i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 30.3318i	− 1.43145i	−0.698384	−	0.715723i	$-0.746097\pi$
$449$	− 30.3318i	− 1.43145i	0.698384	−	0.715723i	$-0.253903\pi$
$450$	0	0
$451$	17.5065	0.824349
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.19026	−0.196442
$456$	0	0
$457$	6.64077	0.310642	0.155321	−	0.987864i	$-0.450359\pi$
$457$	6.64077	0.310642	0.155321	+	0.987864i	$0.450359\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	35.1189	1.63565	0.817826	−	0.575465i	$-0.195179\pi$
$461$	35.1189	1.63565	0.817826	+	0.575465i	$0.195179\pi$
$462$	0	0
$463$	− 2.53401i	− 0.117765i	−0.998265	−	0.0588826i	$-0.981246\pi$
$463$	− 2.53401i	− 0.117765i	0.998265	−	0.0588826i	$-0.0187538\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 16.0379i	− 0.742145i	−0.928604	−	0.371072i	$-0.878990\pi$
$467$	− 16.0379i	− 0.742145i	0.928604	−	0.371072i	$-0.121010\pi$
$468$	0	0
$469$	− 12.6358i	− 0.583465i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	48.4327i	2.22694i
$474$	0	0
$475$	−1.37302	−0.0629985
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−29.2616	−1.33700	−0.668498	−	0.743714i	$-0.733062\pi$
$479$	−29.2616	−1.33700	−0.668498	+	0.743714i	$0.733062\pi$
$480$	0	0
$481$	−4.01004	−0.182842
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.87832	−0.176106
$486$	0	0
$487$	2.81399i	0.127514i	0.997965	+	0.0637570i	$0.0203083\pi$
$487$	2.81399i	0.127514i	−0.997965	+	0.0637570i	$0.979692\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19.2668i	0.869497i	0.900552	+	0.434748i	$0.143163\pi$
$491$	19.2668i	0.869497i	−0.900552	+	0.434748i	$0.856837\pi$
$492$	0	0
$493$	0.929886i	0.0418799i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.95532i	− 0.401701i
$498$	0	0
$499$	−39.4436	−1.76574	−0.882869	−	0.469620i	$-0.844391\pi$
$499$	−39.4436	−1.76574	−0.882869	+	0.469620i	$0.844391\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−33.3860	−1.48861	−0.744304	−	0.667841i	$-0.767219\pi$
$503$	−33.3860	−1.48861	−0.744304	+	0.667841i	$0.767219\pi$
$504$	0	0
$505$	30.6973	1.36601
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.16584	−0.406269	−0.203134	−	0.979151i	$-0.565113\pi$
$509$	−9.16584	−0.406269	−0.203134	+	0.979151i	$0.565113\pi$
$510$	0	0
$511$	7.16051i	0.316762i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 10.5577i	− 0.465226i
$516$	0	0
$517$	6.94299i	0.305352i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.14191i	0.0500280i	0.999687	+	0.0250140i	$0.00796304\pi$
$521$	1.14191i	0.0500280i	−0.999687	+	0.0250140i	$0.992037\pi$
$522$	0	0
$523$	19.5606	0.855323	0.427662	−	0.903939i	$-0.359337\pi$
$523$	19.5606	0.855323	0.427662	+	0.903939i	$0.359337\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.906723	−0.0394975
$528$	0	0
$529$	−10.7864	−0.468975
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.93762	0.343816
$534$	0	0
$535$	36.0665i	1.55929i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 4.02419i	− 0.173334i
$540$	0	0
$541$	35.9815i	1.54696i	0.633818	+	0.773482i	$0.281487\pi$
$541$	35.9815i	1.54696i	−0.633818	+	0.773482i	$0.718513\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	36.7766i	1.57534i
$546$	0	0
$547$	19.0634	0.815092	0.407546	−	0.913185i	$-0.366385\pi$
$547$	19.0634	0.815092	0.407546	+	0.913185i	$0.366385\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.8234	0.461091
$552$	0	0
$553$	−15.2405	−0.648090
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.4543	−0.781934	−0.390967	−	0.920405i	$-0.627859\pi$
$557$	−18.4543	−0.781934	−0.390967	+	0.920405i	$0.627859\pi$
$558$	0	0
$559$	21.9598i	0.928803i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 31.6916i	− 1.33564i	−0.744322	−	0.667821i	$-0.767227\pi$
$563$	− 31.6916i	− 1.33564i	0.744322	−	0.667821i	$-0.232773\pi$
$564$	0	0
$565$	3.37262i	0.141887i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	27.5168i	1.15357i	0.816897	+	0.576783i	$0.195692\pi$
$569$	27.5168i	1.15357i	−0.816897	+	0.576783i	$0.804308\pi$
$570$	0	0
$571$	27.7326	1.16057	0.580287	−	0.814412i	$-0.302940\pi$
$571$	27.7326	1.16057	0.580287	+	0.814412i	$0.302940\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.957681	−0.0399381
$576$	0	0
$577$	38.3530	1.59666	0.798328	−	0.602223i	$-0.205718\pi$
$577$	38.3530	1.59666	0.798328	+	0.602223i	$0.205718\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.4507	0.558027
$582$	0	0
$583$	− 40.6194i	− 1.68229i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.7550i	0.567731i	0.958864	+	0.283865i	$0.0916169\pi$
$587$	13.7550i	0.567731i	−0.958864	+	0.283865i	$0.908383\pi$
$588$	0	0
$589$	10.5538i	0.434861i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.5293i	1.29475i	0.762171	+	0.647376i	$0.224134\pi$
$593$	31.5293i	1.29475i	−0.762171	+	0.647376i	$0.775866\pi$
$594$	0	0
$595$	0.988589	0.0405282
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.6650	0.599196	0.299598	−	0.954066i	$-0.403147\pi$
$599$	14.6650	0.599196	0.299598	+	0.954066i	$0.403147\pi$
$600$	0	0
$601$	−40.8677	−1.66703	−0.833515	−	0.552497i	$-0.813675\pi$
$601$	−40.8677	−1.66703	−0.833515	+	0.552497i	$0.813675\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11.9284	0.484960
$606$	0	0
$607$	− 40.0141i	− 1.62412i	−0.583572	−	0.812061i	$-0.698345\pi$
$607$	− 40.0141i	− 1.62412i	0.583572	−	0.812061i	$-0.301655\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.14802i	0.127355i
$612$	0	0
$613$	3.46053i	0.139770i	0.997555	+	0.0698848i	$0.0222632\pi$
$613$	3.46053i	0.139770i	−0.997555	+	0.0698848i	$0.977737\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.6562i	1.47572i	0.674952	+	0.737861i	$0.264164\pi$
$617$	36.6562i	1.47572i	−0.674952	+	0.737861i	$0.735836\pi$
$618$	0	0
$619$	10.0770	0.405029	0.202514	−	0.979279i	$-0.435089\pi$
$619$	10.0770	0.405029	0.202514	+	0.979279i	$0.435089\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.44987	0.298473
$624$	0	0
$625$	−26.2951	−1.05180
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.946071	0.0377223
$630$	0	0
$631$	28.1634i	1.12117i	0.828098	+	0.560583i	$0.189423\pi$
$631$	28.1634i	1.12117i	−0.828098	+	0.560583i	$0.810577\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 14.5956i	− 0.579209i
$636$	0	0
$637$	− 1.82461i	− 0.0722935i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.3588i	0.488142i	0.969757	+	0.244071i	$0.0784830\pi$
$641$	12.3588i	0.488142i	−0.969757	+	0.244071i	$0.921517\pi$
$642$	0	0
$643$	−28.0875	−1.10766	−0.553831	−	0.832629i	$-0.686835\pi$
$643$	−28.0875	−1.10766	−0.553831	+	0.832629i	$0.686835\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−43.3350	−1.70368	−0.851838	−	0.523806i	$-0.824512\pi$
$647$	−43.3350	−1.70368	−0.851838	+	0.523806i	$0.824512\pi$
$648$	0	0
$649$	−42.7268	−1.67717
$650$	0	0
$651$	0	0
$652$	0	0
$653$	32.0509	1.25425	0.627125	−	0.778919i	$-0.284232\pi$
$653$	32.0509	1.25425	0.627125	+	0.778919i	$0.284232\pi$
$654$	0	0
$655$	− 17.8103i	− 0.695905i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 6.88401i	− 0.268163i	−0.990970	−	0.134081i	$-0.957192\pi$
$659$	− 6.88401i	− 0.268163i	0.990970	−	0.134081i	$-0.0428084\pi$
$660$	0	0
$661$	− 47.2652i	− 1.83840i	−0.393788	−	0.919201i	$-0.628836\pi$
$661$	− 47.2652i	− 1.83840i	0.393788	−	0.919201i	$-0.371164\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 11.5067i	− 0.446209i
$666$	0	0
$667$	7.54929	0.292310
$668$	0	0
$669$	0	0
$670$	0	0
$671$	33.9873	1.31207
$672$	0	0
$673$	6.89911	0.265941	0.132971	−	0.991120i	$-0.457548\pi$
$673$	6.89911	0.265941	0.132971	+	0.991120i	$0.457548\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−23.7479	−0.912707	−0.456353	−	0.889799i	$-0.650845\pi$
$677$	−23.7479	−0.912707	−0.456353	+	0.889799i	$0.650845\pi$
$678$	0	0
$679$	− 1.68878i	− 0.0648094i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 41.5546i	− 1.59004i	−0.606583	−	0.795020i	$-0.707460\pi$
$683$	− 41.5546i	− 1.59004i	0.606583	−	0.795020i	$-0.292540\pi$
$684$	0	0
$685$	− 7.11534i	− 0.271863i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 18.4172i	− 0.701641i
$690$	0	0
$691$	17.5146	0.666285	0.333142	−	0.942876i	$-0.391891\pi$
$691$	17.5146	0.666285	0.333142	+	0.942876i	$0.391891\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.64462	−0.327909
$696$	0	0
$697$	−1.87269	−0.0709331
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.39361	0.0526361	0.0263180	−	0.999654i	$-0.491622\pi$
$701$	1.39361	0.0526361	0.0263180	+	0.999654i	$0.491622\pi$
$702$	0	0
$703$	− 11.0118i	− 0.415317i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.3668i	0.502711i
$708$	0	0
$709$	− 42.1733i	− 1.58385i	−0.610616	−	0.791927i	$-0.709078\pi$
$709$	− 42.1733i	− 1.58385i	0.610616	−	0.791927i	$-0.290922\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.36125i	0.275681i
$714$	0	0
$715$	16.8624	0.630618
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−11.8108	−0.440468	−0.220234	−	0.975447i	$-0.570682\pi$
$719$	−11.8108	−0.440468	−0.220234	+	0.975447i	$0.570682\pi$
$720$	0	0
$721$	4.59723	0.171210
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.591949	−0.0219844
$726$	0	0
$727$	− 45.5504i	− 1.68937i	−0.535264	−	0.844685i	$-0.679788\pi$
$727$	− 45.5504i	− 1.68937i	0.535264	−	0.844685i	$-0.320212\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 5.18089i	− 0.191622i
$732$	0	0
$733$	38.4714i	1.42097i	0.703711	+	0.710487i	$0.251525\pi$
$733$	38.4714i	1.42097i	−0.703711	+	0.710487i	$0.748475\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	50.8487i	1.87304i
$738$	0	0
$739$	−12.1292	−0.446179	−0.223090	−	0.974798i	$-0.571614\pi$
$739$	−12.1292	−0.446179	−0.223090	+	0.974798i	$0.571614\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−46.7546	−1.71526	−0.857630	−	0.514267i	$-0.828064\pi$
$743$	−46.7546	−1.71526	−0.857630	+	0.514267i	$0.828064\pi$
$744$	0	0
$745$	−15.1087	−0.553540
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−15.7048	−0.573841
$750$	0	0
$751$	29.7196i	1.08448i	0.840222	+	0.542242i	$0.182424\pi$
$751$	29.7196i	1.08448i	−0.840222	+	0.542242i	$0.817576\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.48797i	0.0905464i
$756$	0	0
$757$	16.0932i	0.584919i	0.956278	+	0.292459i	$0.0944737\pi$
$757$	16.0932i	0.584919i	−0.956278	+	0.292459i	$0.905526\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.21482i	0.189037i	0.995523	+	0.0945186i	$0.0301312\pi$
$761$	5.21482i	0.189037i	−0.995523	+	0.0945186i	$0.969869\pi$
$762$	0	0
$763$	−16.0140	−0.579746
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−19.3727	−0.699509
$768$	0	0
$769$	17.7856	0.641367	0.320683	−	0.947186i	$-0.396087\pi$
$769$	17.7856	0.641367	0.320683	+	0.947186i	$0.396087\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−19.9688	−0.718228	−0.359114	−	0.933294i	$-0.616921\pi$
$773$	−19.9688	−0.718228	−0.359114	+	0.933294i	$0.616921\pi$
$774$	0	0
$775$	− 0.577204i	− 0.0207338i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	21.7971i	0.780962i
$780$	0	0
$781$	36.0379i	1.28954i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	38.1522i	1.36171i
$786$	0	0
$787$	32.1178	1.14487	0.572437	−	0.819948i	$-0.305998\pi$
$787$	32.1178	1.14487	0.572437	+	0.819948i	$0.305998\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.46858	−0.0522166
$792$	0	0
$793$	15.4102	0.547231
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.4964	0.371800	0.185900	−	0.982569i	$-0.440480\pi$
$797$	10.4964	0.371800	0.185900	+	0.982569i	$0.440480\pi$
$798$	0	0
$799$	− 0.742698i	− 0.0262748i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 28.8153i	− 1.01687i
$804$	0	0
$805$	− 8.02588i	− 0.282875i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 31.8333i	− 1.11920i	−0.828763	−	0.559599i	$-0.810955\pi$
$809$	− 31.8333i	− 1.11920i	0.828763	−	0.559599i	$-0.189045\pi$
$810$	0	0
$811$	1.39041	0.0488238	0.0244119	−	0.999702i	$-0.492229\pi$
$811$	1.39041	0.0488238	0.0244119	+	0.999702i	$0.492229\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	26.3827	0.924147
$816$	0	0
$817$	−60.3029	−2.10973
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18.4554	−0.644097	−0.322048	−	0.946723i	$-0.604371\pi$
$821$	−18.4554	−0.644097	−0.322048	+	0.946723i	$0.604371\pi$
$822$	0	0
$823$	41.1895i	1.43578i	0.696159	+	0.717888i	$0.254891\pi$
$823$	41.1895i	1.43578i	−0.696159	+	0.717888i	$0.745109\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.08227i	0.211501i	0.994393	+	0.105751i	$0.0337245\pi$
$827$	6.08227i	0.211501i	−0.994393	+	0.105751i	$0.966275\pi$
$828$	0	0
$829$	44.2474i	1.53677i	0.639985	+	0.768387i	$0.278940\pi$
$829$	44.2474i	1.53677i	−0.639985	+	0.768387i	$0.721060\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.430472i	0.0149150i
$834$	0	0
$835$	−30.9482	−1.07101
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−45.9488	−1.58633	−0.793164	−	0.609008i	$-0.791568\pi$
$839$	−45.9488	−1.58633	−0.793164	+	0.609008i	$0.791568\pi$
$840$	0	0
$841$	−24.3337	−0.839094
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−22.2093	−0.764022
$846$	0	0
$847$	5.19413i	0.178472i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 7.68070i	− 0.263291i
$852$	0	0
$853$	22.7862i	0.780185i	0.920776	+	0.390093i	$0.127557\pi$
$853$	22.7862i	0.780185i	−0.920776	+	0.390093i	$0.872443\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 28.8660i	− 0.986042i	−0.870017	−	0.493021i	$-0.835892\pi$
$857$	− 28.8660i	− 0.986042i	0.870017	−	0.493021i	$-0.164108\pi$
$858$	0	0
$859$	−53.1350	−1.81294	−0.906471	−	0.422268i	$-0.861234\pi$
$859$	−53.1350	−1.81294	−0.906471	+	0.422268i	$0.861234\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.2397	−0.927250	−0.463625	−	0.886032i	$-0.653451\pi$
$863$	−27.2397	−0.927250	−0.463625	+	0.886032i	$0.653451\pi$
$864$	0	0
$865$	11.4446	0.389128
$866$	0	0
$867$	0	0
$868$	0	0
$869$	61.3305	2.08050
$870$	0	0
$871$	23.0553i	0.781199i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 10.8533i	− 0.366909i
$876$	0	0
$877$	24.5161i	0.827850i	0.910311	+	0.413925i	$0.135842\pi$
$877$	24.5161i	0.827850i	−0.910311	+	0.413925i	$0.864158\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 18.5560i	− 0.625167i	−0.949890	−	0.312584i	$-0.898806\pi$
$881$	− 18.5560i	− 0.625167i	0.949890	−	0.312584i	$-0.101194\pi$
$882$	0	0
$883$	48.1013	1.61874	0.809369	−	0.587300i	$-0.199809\pi$
$883$	48.1013	1.61874	0.809369	+	0.587300i	$0.199809\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.36841	0.180254	0.0901268	−	0.995930i	$-0.471273\pi$
$887$	5.36841	0.180254	0.0901268	+	0.995930i	$0.471273\pi$
$888$	0	0
$889$	6.35551	0.213157
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.64461	−0.289281
$894$	0	0
$895$	7.94435i	0.265550i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.55004i	0.151752i
$900$	0	0
$901$	4.34510i	0.144756i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	23.2889i	0.774148i
$906$	0	0
$907$	−4.80217	−0.159453	−0.0797267	−	0.996817i	$-0.525405\pi$
$907$	−4.80217	−0.159453	−0.0797267	+	0.996817i	$0.525405\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.79257	−0.0925219	−0.0462609	−	0.998929i	$-0.514731\pi$
$911$	−2.79257	−0.0925219	−0.0462609	+	0.998929i	$0.514731\pi$
$912$	0	0
$913$	−54.1280	−1.79138
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.75532	0.256103
$918$	0	0
$919$	7.90433i	0.260740i	0.991465	+	0.130370i	$0.0416165\pi$
$919$	7.90433i	0.260740i	−0.991465	+	0.130370i	$0.958384\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	16.3399i	0.537836i
$924$	0	0
$925$	0.602253i	0.0198019i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 4.24927i	− 0.139414i	−0.997568	−	0.0697070i	$-0.977794\pi$
$929$	− 4.24927i	− 0.139414i	0.997568	−	0.0697070i	$-0.0222064\pi$
$930$	0	0
$931$	5.01046	0.164211
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.97827	−0.130103
$936$	0	0
$937$	1.70393	0.0556649	0.0278324	−	0.999613i	$-0.491140\pi$
$937$	1.70393	0.0556649	0.0278324	+	0.999613i	$0.491140\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	53.7297	1.75154	0.875769	−	0.482730i	$-0.160355\pi$
$941$	53.7297	1.75154	0.875769	+	0.482730i	$0.160355\pi$
$942$	0	0
$943$	15.2035i	0.495093i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 48.4816i	− 1.57544i	−0.616033	−	0.787720i	$-0.711261\pi$
$947$	− 48.4816i	− 1.57544i	0.616033	−	0.787720i	$-0.288739\pi$
$948$	0	0
$949$	− 13.0651i	− 0.424112i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 51.7471i	− 1.67625i	−0.545476	−	0.838126i	$-0.683651\pi$
$953$	− 51.7471i	− 1.67625i	0.545476	−	0.838126i	$-0.316349\pi$
$954$	0	0
$955$	−25.2028	−0.815544
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.09831	0.100050
$960$	0	0
$961$	26.5633	0.856881
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.75901	−0.153198
$966$	0	0
$967$	− 7.89472i	− 0.253877i	−0.991911	−	0.126939i	$-0.959485\pi$
$967$	− 7.89472i	− 0.253877i	0.991911	−	0.126939i	$-0.0405151\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 15.1077i	− 0.484829i	−0.970173	−	0.242414i	$-0.922061\pi$
$971$	− 15.1077i	− 0.484829i	0.970173	−	0.242414i	$-0.0779393\pi$
$972$	0	0
$973$	− 3.76422i	− 0.120675i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.3398i	0.362792i	0.983410	+	0.181396i	$0.0580616\pi$
$977$	11.3398i	0.362792i	−0.983410	+	0.181396i	$0.941938\pi$
$978$	0	0
$979$	−29.9797	−0.958155
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−34.3365	−1.09517	−0.547583	−	0.836751i	$-0.684452\pi$
$983$	−34.3365	−1.09517	−0.547583	+	0.836751i	$0.684452\pi$
$984$	0	0
$985$	10.7833	0.343584
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−42.0612	−1.33747
$990$	0	0
$991$	20.9504i	0.665510i	0.943013	+	0.332755i	$0.107978\pi$
$991$	20.9504i	0.665510i	−0.943013	+	0.332755i	$0.892022\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	38.4797i	1.21989i
$996$	0	0
$997$	− 8.15182i	− 0.258171i	−0.991633	−	0.129085i	$-0.958796\pi$
$997$	− 8.15182i	− 0.258171i	0.991633	−	0.129085i	$-0.0412041\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.j.d.5615.13		48
3.2	odd	2	inner	6048.2.j.d.5615.36		48
4.3	odd	2		1512.2.j.d.323.19	✓	48
8.3	odd	2	inner	6048.2.j.d.5615.35		48
8.5	even	2		1512.2.j.d.323.29	yes	48
12.11	even	2		1512.2.j.d.323.30	yes	48
24.5	odd	2		1512.2.j.d.323.20	yes	48
24.11	even	2	inner	6048.2.j.d.5615.14		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.d.323.19	✓	48	4.3	odd	2
1512.2.j.d.323.20	yes	48	24.5	odd	2
1512.2.j.d.323.29	yes	48	8.5	even	2
1512.2.j.d.323.30	yes	48	12.11	even	2
6048.2.j.d.5615.13		48	1.1	even	1	trivial
6048.2.j.d.5615.14		48	24.11	even	2	inner
6048.2.j.d.5615.35		48	8.3	odd	2	inner
6048.2.j.d.5615.36		48	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)

\(f(q)\)	\(=\)	\(q-2.29653 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-2.29653 q^{5} -1.00000i q^{7} +4.02419i q^{11} +1.82461i q^{13} -0.430472i q^{17} -5.01046 q^{19} -3.49479 q^{23} +0.274031 q^{25} -2.16016 q^{29} -2.10635i q^{31} +2.29653i q^{35} +2.19775i q^{37} -4.35032i q^{41} +12.0354 q^{43} +1.72531 q^{47} -1.00000 q^{49} -10.0938 q^{53} -9.24166i q^{55} +10.6175i q^{59} -8.44575i q^{61} -4.19026i q^{65} +12.6358 q^{67} +8.95532 q^{71} -7.16051 q^{73} +4.02419 q^{77} -15.2405i q^{79} +13.4507i q^{83} +0.988589i q^{85} +7.44987i q^{89} +1.82461 q^{91} +11.5067 q^{95} +1.68878 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 16 q^{19} + 48 q^{25} - 64 q^{43} - 48 q^{49} - 32 q^{67}+O(q^{100}) \) 48 * q - 16 * q^19 + 48 * q^25 - 64 * q^43 - 48 * q^49 - 32 * q^67

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