LMFDB - Embedded newform 6048.2.c.f.3025.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(3025,6048)]

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([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("6048.3025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.10
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.f.3025.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.25392i q^{5} -1.00000 q^{7} +O(q^{10})$	$q-1.25392i q^{5} -1.00000 q^{7} +1.55766i q^{11} +1.07281i q^{13} +0.0158095 q^{17} -2.35077i q^{19} -5.95129 q^{23} +3.42768 q^{25} -0.469737i q^{29} -1.69031 q^{31} +1.25392i q^{35} +4.59800i q^{37} +12.2190 q^{41} +1.97482i q^{43} -7.12571 q^{47} +1.00000 q^{49} -1.86735i q^{53} +1.95319 q^{55} +8.54291i q^{59} +3.92502i q^{61} +1.34521 q^{65} -12.7403i q^{67} +4.22205 q^{71} +6.51561 q^{73} -1.55766i q^{77} +6.15343 q^{79} +8.88604i q^{83} -0.0198239i q^{85} -0.240788 q^{89} -1.07281i q^{91} -2.94767 q^{95} +5.86131 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{7}+O(q^{10}) $ 24 * q - 24 * q^7	$ 24 q - 24 q^{7} - 24 q^{25} - 8 q^{31} + 24 q^{49} - 16 q^{55} - 8 q^{79}+O(q^{100}) $ 24 * q - 24 * q^7 - 24 * q^25 - 8 * q^31 + 24 * q^49 - 16 * q^55 - 8 * q^79

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
$5$	− 1.25392i	− 0.560770i	−0.959888	−	0.280385i	$-0.909538\pi$
$5$	− 1.25392i	− 0.560770i	0.959888	−	0.280385i	$-0.0904622\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.55766i	0.469653i	0.972037	+	0.234827i	$0.0754522\pi$
$11$	1.55766i	0.469653i	−0.972037	+	0.234827i	$0.924548\pi$
$12$	0	0
$13$	1.07281i	0.297543i	0.988872	+	0.148771i	$0.0475319\pi$
$13$	1.07281i	0.297543i	−0.988872	+	0.148771i	$0.952468\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.0158095	0.00383438	0.00191719	−	0.999998i	$-0.499390\pi$
$17$	0.0158095	0.00383438	0.00191719	+	0.999998i	$0.499390\pi$
$18$	0	0
$19$	− 2.35077i	− 0.539303i	−0.962958	−	0.269651i	$-0.913092\pi$
$19$	− 2.35077i	− 0.539303i	0.962958	−	0.269651i	$-0.0869085\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.95129	−1.24093	−0.620465	−	0.784235i	$-0.713056\pi$
$23$	−5.95129	−1.24093	−0.620465	+	0.784235i	$0.713056\pi$
$24$	0	0
$25$	3.42768	0.685537
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 0.469737i	− 0.0872279i	−0.999048	−	0.0436139i	$-0.986113\pi$
$29$	− 0.469737i	− 0.0872279i	0.999048	−	0.0436139i	$-0.0138872\pi$
$30$	0	0
$31$	−1.69031	−0.303589	−0.151795	−	0.988412i	$-0.548505\pi$
$31$	−1.69031	−0.303589	−0.151795	+	0.988412i	$0.548505\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.25392i	0.211951i
$36$	0	0
$37$	4.59800i	0.755906i	0.925825	+	0.377953i	$0.123372\pi$
$37$	4.59800i	0.755906i	−0.925825	+	0.377953i	$0.876628\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	12.2190	1.90829	0.954143	−	0.299350i	$-0.0967698\pi$
$41$	12.2190	1.90829	0.954143	+	0.299350i	$0.0967698\pi$
$42$	0	0
$43$	1.97482i	0.301158i	0.988598	+	0.150579i	$0.0481137\pi$
$43$	1.97482i	0.301158i	−0.988598	+	0.150579i	$0.951886\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.12571	−1.03939	−0.519696	−	0.854351i	$-0.673955\pi$
$47$	−7.12571	−1.03939	−0.519696	+	0.854351i	$0.673955\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.86735i	− 0.256500i	−0.991742	−	0.128250i	$-0.959064\pi$
$53$	− 1.86735i	− 0.256500i	0.991742	−	0.128250i	$-0.0409360\pi$
$54$	0	0
$55$	1.95319	0.263368
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.54291i	1.11219i	0.831118	+	0.556096i	$0.187701\pi$
$59$	8.54291i	1.11219i	−0.831118	+	0.556096i	$0.812299\pi$
$60$	0	0
$61$	3.92502i	0.502547i	0.967916	+	0.251274i	$0.0808494\pi$
$61$	3.92502i	0.502547i	−0.967916	+	0.251274i	$0.919151\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.34521	0.166853
$66$	0	0
$67$	− 12.7403i	− 1.55647i	−0.627972	−	0.778236i	$-0.716115\pi$
$67$	− 12.7403i	− 1.55647i	0.627972	−	0.778236i	$-0.283885\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.22205	0.501066	0.250533	−	0.968108i	$-0.419394\pi$
$71$	4.22205	0.501066	0.250533	+	0.968108i	$0.419394\pi$
$72$	0	0
$73$	6.51561	0.762595	0.381297	−	0.924452i	$-0.375477\pi$
$73$	6.51561	0.762595	0.381297	+	0.924452i	$0.375477\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.55766i	− 0.177512i
$78$	0	0
$79$	6.15343	0.692315	0.346158	−	0.938176i	$-0.387486\pi$
$79$	6.15343	0.692315	0.346158	+	0.938176i	$0.387486\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.88604i	0.975370i	0.873020	+	0.487685i	$0.162158\pi$
$83$	8.88604i	0.975370i	−0.873020	+	0.487685i	$0.837842\pi$
$84$	0	0
$85$	− 0.0198239i	− 0.00215021i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.240788	−0.0255235	−0.0127618	−	0.999919i	$-0.504062\pi$
$89$	−0.240788	−0.0255235	−0.0127618	+	0.999919i	$0.504062\pi$
$90$	0	0
$91$	− 1.07281i	− 0.112461i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.94767	−0.302425
$96$	0	0
$97$	5.86131	0.595126	0.297563	−	0.954702i	$-0.403826\pi$
$97$	5.86131	0.595126	0.297563	+	0.954702i	$0.403826\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.12870i	0.311317i	0.987811	+	0.155659i	$0.0497500\pi$
$101$	3.12870i	0.311317i	−0.987811	+	0.155659i	$0.950250\pi$
$102$	0	0
$103$	−2.72832	−0.268830	−0.134415	−	0.990925i	$-0.542915\pi$
$103$	−2.72832	−0.268830	−0.134415	+	0.990925i	$0.542915\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.05577i	− 0.198738i	−0.995051	−	0.0993692i	$-0.968318\pi$
$107$	− 2.05577i	− 0.198738i	0.995051	−	0.0993692i	$-0.0316825\pi$
$108$	0	0
$109$	13.8179i	1.32351i	0.749719	+	0.661756i	$0.230189\pi$
$109$	13.8179i	1.32351i	−0.749719	+	0.661756i	$0.769811\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.72061	−0.350005	−0.175003	−	0.984568i	$-0.555993\pi$
$113$	−3.72061	−0.350005	−0.175003	+	0.984568i	$0.555993\pi$
$114$	0	0
$115$	7.46244i	0.695876i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.0158095	−0.00144926
$120$	0	0
$121$	8.57368	0.779426
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 10.5676i	− 0.945199i
$126$	0	0
$127$	19.8627	1.76253	0.881264	−	0.472625i	$-0.156693\pi$
$127$	19.8627	1.76253	0.881264	+	0.472625i	$0.156693\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.24798i	0.633259i	0.948549	+	0.316629i	$0.102551\pi$
$131$	7.24798i	0.633259i	−0.948549	+	0.316629i	$0.897449\pi$
$132$	0	0
$133$	2.35077i	0.203837i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.1526	−1.12371	−0.561853	−	0.827237i	$-0.689911\pi$
$137$	−13.1526	−1.12371	−0.561853	+	0.827237i	$0.689911\pi$
$138$	0	0
$139$	5.29486i	0.449104i	0.974462	+	0.224552i	$0.0720919\pi$
$139$	5.29486i	0.449104i	−0.974462	+	0.224552i	$0.927908\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.67107	−0.139742
$144$	0	0
$145$	−0.589012	−0.0489148
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 6.59368i	− 0.540175i	−0.962836	−	0.270088i	$-0.912947\pi$
$149$	− 6.59368i	− 0.540175i	0.962836	−	0.270088i	$-0.0870526\pi$
$150$	0	0
$151$	9.51585	0.774389	0.387195	−	0.921998i	$-0.373444\pi$
$151$	9.51585	0.774389	0.387195	+	0.921998i	$0.373444\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.11952i	0.170244i
$156$	0	0
$157$	16.0109i	1.27781i	0.769285	+	0.638905i	$0.220612\pi$
$157$	16.0109i	1.27781i	−0.769285	+	0.638905i	$0.779388\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.95129	0.469027
$162$	0	0
$163$	12.5751i	0.984959i	0.870324	+	0.492480i	$0.163909\pi$
$163$	12.5751i	0.984959i	−0.870324	+	0.492480i	$0.836091\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.2841	−1.33748	−0.668741	−	0.743495i	$-0.733166\pi$
$167$	−17.2841	−1.33748	−0.668741	+	0.743495i	$0.733166\pi$
$168$	0	0
$169$	11.8491	0.911468
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.78719i	0.668078i	0.942559	+	0.334039i	$0.108412\pi$
$173$	8.78719i	0.668078i	−0.942559	+	0.334039i	$0.891588\pi$
$174$	0	0
$175$	−3.42768	−0.259109
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 11.6964i	− 0.874233i	−0.899405	−	0.437117i	$-0.856000\pi$
$179$	− 11.6964i	− 0.874233i	0.899405	−	0.437117i	$-0.144000\pi$
$180$	0	0
$181$	− 24.2624i	− 1.80341i	−0.432351	−	0.901705i	$-0.642316\pi$
$181$	− 24.2624i	− 1.80341i	0.432351	−	0.901705i	$-0.357684\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.76552	0.423890
$186$	0	0
$187$	0.0246260i	0.00180083i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.4850	0.758665	0.379333	−	0.925260i	$-0.376154\pi$
$191$	10.4850	0.758665	0.379333	+	0.925260i	$0.376154\pi$
$192$	0	0
$193$	12.7092	0.914832	0.457416	−	0.889253i	$-0.348775\pi$
$193$	12.7092	0.914832	0.457416	+	0.889253i	$0.348775\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 16.0117i	− 1.14079i	−0.821372	−	0.570394i	$-0.806791\pi$
$197$	− 16.0117i	− 1.14079i	0.821372	−	0.570394i	$-0.193209\pi$
$198$	0	0
$199$	11.1888	0.793156	0.396578	−	0.918001i	$-0.370198\pi$
$199$	11.1888	0.793156	0.396578	+	0.918001i	$0.370198\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.469737i	0.0329690i
$204$	0	0
$205$	− 15.3217i	− 1.07011i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.66170	0.253285
$210$	0	0
$211$	20.1657i	1.38827i	0.719847	+	0.694133i	$0.244212\pi$
$211$	20.1657i	1.38827i	−0.719847	+	0.694133i	$0.755788\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.47627	0.168880
$216$	0	0
$217$	1.69031	0.114746
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.0169606i	0.00114089i
$222$	0	0
$223$	11.1501	0.746663	0.373332	−	0.927698i	$-0.378215\pi$
$223$	11.1501	0.746663	0.373332	+	0.927698i	$0.378215\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.7414i	1.24391i	0.783053	+	0.621955i	$0.213662\pi$
$227$	18.7414i	1.24391i	−0.783053	+	0.621955i	$0.786338\pi$
$228$	0	0
$229$	13.5444i	0.895038i	0.894274	+	0.447519i	$0.147692\pi$
$229$	13.5444i	0.895038i	−0.894274	+	0.447519i	$0.852308\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.0070	0.721094	0.360547	−	0.932741i	$-0.382590\pi$
$233$	11.0070	0.721094	0.360547	+	0.932741i	$0.382590\pi$
$234$	0	0
$235$	8.93508i	0.582860i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.4366	0.998509	0.499255	−	0.866455i	$-0.333607\pi$
$239$	15.4366	0.998509	0.499255	+	0.866455i	$0.333607\pi$
$240$	0	0
$241$	16.0354	1.03293	0.516466	−	0.856308i	$-0.327247\pi$
$241$	16.0354	1.03293	0.516466	+	0.856308i	$0.327247\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.25392i	− 0.0801100i
$246$	0	0
$247$	2.52192	0.160466
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 11.4441i	− 0.722343i	−0.932499	−	0.361172i	$-0.882377\pi$
$251$	− 11.4441i	− 0.722343i	0.932499	−	0.361172i	$-0.117623\pi$
$252$	0	0
$253$	− 9.27010i	− 0.582806i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.7447	0.670236	0.335118	−	0.942176i	$-0.391224\pi$
$257$	10.7447	0.670236	0.335118	+	0.942176i	$0.391224\pi$
$258$	0	0
$259$	− 4.59800i	− 0.285706i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	14.6747	0.904878	0.452439	−	0.891795i	$-0.350554\pi$
$263$	14.6747	0.904878	0.452439	+	0.891795i	$0.350554\pi$
$264$	0	0
$265$	−2.34151	−0.143838
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 4.93110i	− 0.300655i	−0.988636	−	0.150327i	$-0.951967\pi$
$269$	− 4.93110i	− 0.300655i	0.988636	−	0.150327i	$-0.0480328\pi$
$270$	0	0
$271$	22.6605	1.37652	0.688262	−	0.725462i	$-0.258374\pi$
$271$	22.6605	1.37652	0.688262	+	0.725462i	$0.258374\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.33918i	0.321965i
$276$	0	0
$277$	5.99172i	0.360008i	0.983666	+	0.180004i	$0.0576110\pi$
$277$	5.99172i	0.360008i	−0.983666	+	0.180004i	$0.942389\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.92544	−0.114862	−0.0574309	−	0.998349i	$-0.518291\pi$
$281$	−1.92544	−0.114862	−0.0574309	+	0.998349i	$0.518291\pi$
$282$	0	0
$283$	− 28.2195i	− 1.67747i	−0.544537	−	0.838737i	$-0.683295\pi$
$283$	− 28.2195i	− 1.67747i	0.544537	−	0.838737i	$-0.316705\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.2190	−0.721265
$288$	0	0
$289$	−16.9998	−0.999985
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 29.0957i	− 1.69979i	−0.526955	−	0.849893i	$-0.676666\pi$
$293$	− 29.0957i	− 1.69979i	0.526955	−	0.849893i	$-0.323334\pi$
$294$	0	0
$295$	10.7121	0.623685
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 6.38458i	− 0.369230i
$300$	0	0
$301$	− 1.97482i	− 0.113827i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.92166	0.281814
$306$	0	0
$307$	15.8372i	0.903879i	0.892049	+	0.451940i	$0.149268\pi$
$307$	15.8372i	0.903879i	−0.892049	+	0.451940i	$0.850732\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−14.5620	−0.825737	−0.412869	−	0.910791i	$-0.635473\pi$
$311$	−14.5620	−0.825737	−0.412869	+	0.910791i	$0.635473\pi$
$312$	0	0
$313$	6.47077	0.365749	0.182875	−	0.983136i	$-0.441460\pi$
$313$	6.47077	0.365749	0.182875	+	0.983136i	$0.441460\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.9823i	0.672994i	0.941685	+	0.336497i	$0.109242\pi$
$317$	11.9823i	0.672994i	−0.941685	+	0.336497i	$0.890758\pi$
$318$	0	0
$319$	0.731691	0.0409669
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 0.0371646i	− 0.00206789i
$324$	0	0
$325$	3.67724i	0.203977i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.12571	0.392853
$330$	0	0
$331$	21.9422i	1.20605i	0.797720	+	0.603027i	$0.206039\pi$
$331$	21.9422i	1.20605i	−0.797720	+	0.603027i	$0.793961\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−15.9753	−0.872823
$336$	0	0
$337$	22.3150	1.21558	0.607788	−	0.794100i	$-0.292057\pi$
$337$	22.3150	1.21558	0.607788	+	0.794100i	$0.292057\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 2.63294i	− 0.142582i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 24.5806i	− 1.31956i	−0.751460	−	0.659779i	$-0.770650\pi$
$347$	− 24.5806i	− 1.31956i	0.751460	−	0.659779i	$-0.229350\pi$
$348$	0	0
$349$	12.5930i	0.674089i	0.941489	+	0.337045i	$0.109427\pi$
$349$	12.5930i	0.674089i	−0.941489	+	0.337045i	$0.890573\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−16.1946	−0.861953	−0.430977	−	0.902363i	$-0.641831\pi$
$353$	−16.1946	−0.861953	−0.430977	+	0.902363i	$0.641831\pi$
$354$	0	0
$355$	− 5.29412i	− 0.280983i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.1809	−0.537329	−0.268665	−	0.963234i	$-0.586582\pi$
$359$	−10.1809	−0.537329	−0.268665	+	0.963234i	$0.586582\pi$
$360$	0	0
$361$	13.4739	0.709152
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 8.17006i	− 0.427640i
$366$	0	0
$367$	12.3939	0.646955	0.323478	−	0.946236i	$-0.395148\pi$
$367$	12.3939	0.646955	0.323478	+	0.946236i	$0.395148\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.86735i	0.0969480i
$372$	0	0
$373$	− 7.57921i	− 0.392437i	−0.980560	−	0.196218i	$-0.937134\pi$
$373$	− 7.57921i	− 0.392437i	0.980560	−	0.196218i	$-0.0628661\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.503936	0.0259540
$378$	0	0
$379$	14.1179i	0.725187i	0.931947	+	0.362594i	$0.118109\pi$
$379$	14.1179i	0.725187i	−0.931947	+	0.362594i	$0.881891\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.28678	−0.0657512	−0.0328756	−	0.999459i	$-0.510467\pi$
$383$	−1.28678	−0.0657512	−0.0328756	+	0.999459i	$0.510467\pi$
$384$	0	0
$385$	−1.95319	−0.0995436
$386$	0	0
$387$	0	0
$388$	0	0
$389$	33.3310i	1.68995i	0.534806	+	0.844975i	$0.320385\pi$
$389$	33.3310i	1.68995i	−0.534806	+	0.844975i	$0.679615\pi$
$390$	0	0
$391$	−0.0940872	−0.00475819
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 7.71591i	− 0.388230i
$396$	0	0
$397$	− 20.7770i	− 1.04277i	−0.853322	−	0.521385i	$-0.825416\pi$
$397$	− 20.7770i	− 1.04277i	0.853322	−	0.521385i	$-0.174584\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7.38157	−0.368618	−0.184309	−	0.982868i	$-0.559005\pi$
$401$	−7.38157	−0.368618	−0.184309	+	0.982868i	$0.559005\pi$
$402$	0	0
$403$	− 1.81338i	− 0.0903308i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.16213	−0.355014
$408$	0	0
$409$	31.9846	1.58154	0.790769	−	0.612114i	$-0.209681\pi$
$409$	31.9846	1.58154	0.790769	+	0.612114i	$0.209681\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 8.54291i	− 0.420369i
$414$	0	0
$415$	11.1424	0.546958
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.6646i	0.716413i	0.933642	+	0.358207i	$0.116612\pi$
$419$	14.6646i	0.716413i	−0.933642	+	0.358207i	$0.883388\pi$
$420$	0	0
$421$	28.4958i	1.38880i	0.719589	+	0.694400i	$0.244330\pi$
$421$	28.4958i	1.38880i	−0.719589	+	0.694400i	$0.755670\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.0541901	0.00262861
$426$	0	0
$427$	− 3.92502i	− 0.189945i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	35.4443	1.70729	0.853645	−	0.520855i	$-0.174387\pi$
$431$	35.4443	1.70729	0.853645	+	0.520855i	$0.174387\pi$
$432$	0	0
$433$	−31.5907	−1.51815	−0.759077	−	0.651001i	$-0.774349\pi$
$433$	−31.5907	−1.51815	−0.759077	+	0.651001i	$0.774349\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13.9901i	0.669237i
$438$	0	0
$439$	−23.3732	−1.11554	−0.557771	−	0.829995i	$-0.688343\pi$
$439$	−23.3732	−1.11554	−0.557771	+	0.829995i	$0.688343\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.52932i	0.167683i	0.996479	+	0.0838416i	$0.0267190\pi$
$443$	3.52932i	0.167683i	−0.996479	+	0.0838416i	$0.973281\pi$
$444$	0	0
$445$	0.301929i	0.0143128i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.94601	0.139031	0.0695154	−	0.997581i	$-0.477855\pi$
$449$	2.94601	0.139031	0.0695154	+	0.997581i	$0.477855\pi$
$450$	0	0
$451$	19.0331i	0.896233i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.34521	−0.0630646
$456$	0	0
$457$	−22.1796	−1.03752	−0.518758	−	0.854921i	$-0.673605\pi$
$457$	−22.1796	−1.03752	−0.518758	+	0.854921i	$0.673605\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	23.4545i	1.09238i	0.837660	+	0.546192i	$0.183923\pi$
$461$	23.4545i	1.09238i	−0.837660	+	0.546192i	$0.816077\pi$
$462$	0	0
$463$	29.2354	1.35868	0.679342	−	0.733822i	$-0.262265\pi$
$463$	29.2354	1.35868	0.679342	+	0.733822i	$0.262265\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 5.43057i	− 0.251297i	−0.992075	−	0.125648i	$-0.959899\pi$
$467$	− 5.43057i	− 0.251297i	0.992075	−	0.125648i	$-0.0401011\pi$
$468$	0	0
$469$	12.7403i	0.588291i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.07611	−0.141440
$474$	0	0
$475$	− 8.05768i	− 0.369712i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.07525	−0.231894	−0.115947	−	0.993255i	$-0.536990\pi$
$479$	−5.07525	−0.231894	−0.115947	+	0.993255i	$0.536990\pi$
$480$	0	0
$481$	−4.93276	−0.224914
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 7.34962i	− 0.333729i
$486$	0	0
$487$	3.23366	0.146531	0.0732656	−	0.997312i	$-0.476658\pi$
$487$	3.23366	0.146531	0.0732656	+	0.997312i	$0.476658\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 32.6437i	− 1.47319i	−0.676335	−	0.736594i	$-0.736433\pi$
$491$	− 32.6437i	− 1.47319i	0.676335	−	0.736594i	$-0.263567\pi$
$492$	0	0
$493$	− 0.00742632i	0 0.000334465i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.22205	−0.189385
$498$	0	0
$499$	− 1.77339i	− 0.0793877i	−0.999212	−	0.0396939i	$-0.987362\pi$
$499$	− 1.77339i	− 0.0793877i	0.999212	−	0.0396939i	$-0.0126383\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	25.3666	1.13104	0.565519	−	0.824735i	$-0.308676\pi$
$503$	25.3666	1.13104	0.565519	+	0.824735i	$0.308676\pi$
$504$	0	0
$505$	3.92314	0.174578
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.3204i	0.634739i	0.948302	+	0.317369i	$0.102799\pi$
$509$	14.3204i	0.634739i	−0.948302	+	0.317369i	$0.897201\pi$
$510$	0	0
$511$	−6.51561	−0.288234
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.42110i	0.150752i
$516$	0	0
$517$	− 11.0995i	− 0.488154i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5.72827	−0.250960	−0.125480	−	0.992096i	$-0.540047\pi$
$521$	−5.72827	−0.250960	−0.125480	+	0.992096i	$0.540047\pi$
$522$	0	0
$523$	11.3779i	0.497519i	0.968565	+	0.248759i	$0.0800229\pi$
$523$	11.3779i	0.497519i	−0.968565	+	0.248759i	$0.919977\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.0267231	−0.00116408
$528$	0	0
$529$	12.4178	0.539905
$530$	0	0
$531$	0	0
$532$	0	0
$533$	13.1086i	0.567797i
$534$	0	0
$535$	−2.57777	−0.111447
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.55766i	0.0670933i
$540$	0	0
$541$	− 43.3953i	− 1.86571i	−0.360252	−	0.932855i	$-0.617309\pi$
$541$	− 43.3953i	− 1.86571i	0.360252	−	0.932855i	$-0.382691\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	17.3265	0.742187
$546$	0	0
$547$	− 18.5677i	− 0.793898i	−0.917841	−	0.396949i	$-0.870069\pi$
$547$	− 18.5677i	− 0.793898i	0.917841	−	0.396949i	$-0.129931\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.10424	−0.0470422
$552$	0	0
$553$	−6.15343	−0.261671
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.4689i	0.740182i	0.928996	+	0.370091i	$0.120673\pi$
$557$	17.4689i	0.740182i	−0.928996	+	0.370091i	$0.879327\pi$
$558$	0	0
$559$	−2.11860	−0.0896073
$560$	0	0
$561$	0	0
$562$	0	0
$563$	26.8370i	1.13105i	0.824733	+	0.565523i	$0.191325\pi$
$563$	26.8370i	1.13105i	−0.824733	+	0.565523i	$0.808675\pi$
$564$	0	0
$565$	4.66534i	0.196272i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−32.4375	−1.35985	−0.679926	−	0.733281i	$-0.737988\pi$
$569$	−32.4375	−1.35985	−0.679926	+	0.733281i	$0.737988\pi$
$570$	0	0
$571$	34.0452i	1.42475i	0.701800	+	0.712374i	$0.252380\pi$
$571$	34.0452i	1.42475i	−0.701800	+	0.712374i	$0.747620\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−20.3991	−0.850702
$576$	0	0
$577$	6.09534	0.253752	0.126876	−	0.991919i	$-0.459505\pi$
$577$	6.09534	0.253752	0.126876	+	0.991919i	$0.459505\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 8.88604i	− 0.368655i
$582$	0	0
$583$	2.90870	0.120466
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.7735i	0.444669i	0.974970	+	0.222334i	$0.0713677\pi$
$587$	10.7735i	0.444669i	−0.974970	+	0.222334i	$0.928632\pi$
$588$	0	0
$589$	3.97353i	0.163727i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−13.4720	−0.553230	−0.276615	−	0.960981i	$-0.589213\pi$
$593$	−13.4720	−0.553230	−0.276615	+	0.960981i	$0.589213\pi$
$594$	0	0
$595$	0.0198239i	0 0.000812701i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	37.4766	1.53125	0.765626	−	0.643285i	$-0.222429\pi$
$599$	37.4766	1.53125	0.765626	+	0.643285i	$0.222429\pi$
$600$	0	0
$601$	−3.99803	−0.163083	−0.0815415	−	0.996670i	$-0.525984\pi$
$601$	−3.99803	−0.163083	−0.0815415	+	0.996670i	$0.525984\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 10.7507i	− 0.437079i
$606$	0	0
$607$	−36.8295	−1.49486	−0.747432	−	0.664338i	$-0.768713\pi$
$607$	−36.8295	−1.49486	−0.747432	+	0.664338i	$0.768713\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 7.64451i	− 0.309264i
$612$	0	0
$613$	− 9.76469i	− 0.394392i	−0.980364	−	0.197196i	$-0.936816\pi$
$613$	− 9.76469i	− 0.394392i	0.980364	−	0.197196i	$-0.0631836\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14.4832	−0.583070	−0.291535	−	0.956560i	$-0.594166\pi$
$617$	−14.4832	−0.583070	−0.291535	+	0.956560i	$0.594166\pi$
$618$	0	0
$619$	− 6.72500i	− 0.270301i	−0.990825	−	0.135150i	$-0.956848\pi$
$619$	− 6.72500i	− 0.270301i	0.990825	−	0.135150i	$-0.0431518\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.240788	0.00964698
$624$	0	0
$625$	3.88743	0.155497
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.0726923i	0.00289843i
$630$	0	0
$631$	23.2251	0.924577	0.462288	−	0.886730i	$-0.347029\pi$
$631$	23.2251	0.924577	0.462288	+	0.886730i	$0.347029\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 24.9062i	− 0.988373i
$636$	0	0
$637$	1.07281i	0.0425061i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−30.3809	−1.19998	−0.599988	−	0.800009i	$-0.704828\pi$
$641$	−30.3809	−1.19998	−0.599988	+	0.800009i	$0.704828\pi$
$642$	0	0
$643$	5.74408i	0.226524i	0.993565	+	0.113262i	$0.0361300\pi$
$643$	5.74408i	0.226524i	−0.993565	+	0.113262i	$0.963870\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.19948	0.165099	0.0825494	−	0.996587i	$-0.473694\pi$
$647$	4.19948	0.165099	0.0825494	+	0.996587i	$0.473694\pi$
$648$	0	0
$649$	−13.3070	−0.522345
$650$	0	0
$651$	0	0
$652$	0	0
$653$	45.2069i	1.76908i	0.466460	+	0.884542i	$0.345529\pi$
$653$	45.2069i	1.76908i	−0.466460	+	0.884542i	$0.654471\pi$
$654$	0	0
$655$	9.08839	0.355113
$656$	0	0
$657$	0	0
$658$	0	0
$659$	47.0951i	1.83456i	0.398239	+	0.917282i	$0.369621\pi$
$659$	47.0951i	1.83456i	−0.398239	+	0.917282i	$0.630379\pi$
$660$	0	0
$661$	41.9805i	1.63285i	0.577449	+	0.816427i	$0.304048\pi$
$661$	41.9805i	1.63285i	−0.577449	+	0.816427i	$0.695952\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.94767	0.114306
$666$	0	0
$667$	2.79554i	0.108244i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.11386	−0.236023
$672$	0	0
$673$	30.0898	1.15987	0.579937	−	0.814661i	$-0.303077\pi$
$673$	30.0898	1.15987	0.579937	+	0.814661i	$0.303077\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	40.5494i	1.55844i	0.626749	+	0.779221i	$0.284385\pi$
$677$	40.5494i	1.55844i	−0.626749	+	0.779221i	$0.715615\pi$
$678$	0	0
$679$	−5.86131	−0.224936
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 50.1042i	− 1.91718i	−0.284786	−	0.958591i	$-0.591923\pi$
$683$	− 50.1042i	− 1.91718i	0.284786	−	0.958591i	$-0.408077\pi$
$684$	0	0
$685$	16.4924i	0.630141i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.00330	0.0763198
$690$	0	0
$691$	− 43.2174i	− 1.64407i	−0.569438	−	0.822034i	$-0.692839\pi$
$691$	− 43.2174i	− 1.64407i	0.569438	−	0.822034i	$-0.307161\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.63933	0.251844
$696$	0	0
$697$	0.193177	0.00731709
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 19.4337i	− 0.734002i	−0.930221	−	0.367001i	$-0.880385\pi$
$701$	− 19.4337i	− 0.734002i	0.930221	−	0.367001i	$-0.119615\pi$
$702$	0	0
$703$	10.8088	0.407662
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 3.12870i	− 0.117667i
$708$	0	0
$709$	− 24.0279i	− 0.902387i	−0.892426	−	0.451194i	$-0.850998\pi$
$709$	− 24.0279i	− 0.902387i	0.892426	−	0.451194i	$-0.149002\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.0595	0.376733
$714$	0	0
$715$	2.09539i	0.0783631i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	36.0898	1.34592	0.672961	−	0.739678i	$-0.265022\pi$
$719$	36.0898	1.34592	0.672961	+	0.739678i	$0.265022\pi$
$720$	0	0
$721$	2.72832	0.101608
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1.61011i	− 0.0597979i
$726$	0	0
$727$	−36.3230	−1.34714	−0.673572	−	0.739122i	$-0.735241\pi$
$727$	−36.3230	−1.34714	−0.673572	+	0.739122i	$0.735241\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0.0312211i	0.00115475i
$732$	0	0
$733$	11.9814i	0.442544i	0.975212	+	0.221272i	$0.0710209\pi$
$733$	11.9814i	0.442544i	−0.975212	+	0.221272i	$0.928979\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19.8451	0.731002
$738$	0	0
$739$	− 36.7377i	− 1.35142i	−0.737169	−	0.675708i	$-0.763838\pi$
$739$	− 36.7377i	− 1.35142i	0.737169	−	0.675708i	$-0.236162\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13.2966	0.487805	0.243903	−	0.969800i	$-0.421572\pi$
$743$	13.2966	0.487805	0.243903	+	0.969800i	$0.421572\pi$
$744$	0	0
$745$	−8.26795	−0.302914
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.05577i	0.0751161i
$750$	0	0
$751$	13.5953	0.496101	0.248051	−	0.968747i	$-0.420210\pi$
$751$	13.5953	0.496101	0.248051	+	0.968747i	$0.420210\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 11.9321i	− 0.434254i
$756$	0	0
$757$	8.04991i	0.292579i	0.989242	+	0.146290i	$0.0467331\pi$
$757$	8.04991i	0.292579i	−0.989242	+	0.146290i	$0.953267\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−53.5813	−1.94232	−0.971160	−	0.238429i	$-0.923368\pi$
$761$	−53.5813	−1.94232	−0.971160	+	0.238429i	$0.923368\pi$
$762$	0	0
$763$	− 13.8179i	− 0.500241i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.16489	−0.330925
$768$	0	0
$769$	3.40107	0.122646	0.0613229	−	0.998118i	$-0.480468\pi$
$769$	3.40107	0.122646	0.0613229	+	0.998118i	$0.480468\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 12.6537i	− 0.455123i	−0.973764	−	0.227561i	$-0.926925\pi$
$773$	− 12.6537i	− 0.455123i	0.973764	−	0.227561i	$-0.0730752\pi$
$774$	0	0
$775$	−5.79386	−0.208122
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 28.7240i	− 1.02914i
$780$	0	0
$781$	6.57654i	0.235327i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	20.0764	0.716558
$786$	0	0
$787$	− 37.3539i	− 1.33152i	−0.746165	−	0.665761i	$-0.768107\pi$
$787$	− 37.3539i	− 1.33152i	0.746165	−	0.665761i	$-0.231893\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.72061	0.132290
$792$	0	0
$793$	−4.21079	−0.149529
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 15.3333i	− 0.543134i	−0.962420	−	0.271567i	$-0.912458\pi$
$797$	− 15.3333i	− 0.543134i	0.962420	−	0.271567i	$-0.0875418\pi$
$798$	0	0
$799$	−0.112654	−0.00398542
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10.1491i	0.358155i
$804$	0	0
$805$	− 7.46244i	− 0.263016i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−48.1700	−1.69357	−0.846784	−	0.531938i	$-0.821464\pi$
$809$	−48.1700	−1.69357	−0.846784	+	0.531938i	$0.821464\pi$
$810$	0	0
$811$	− 21.4720i	− 0.753984i	−0.926216	−	0.376992i	$-0.876958\pi$
$811$	− 21.4720i	− 0.753984i	0.926216	−	0.376992i	$-0.123042\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	15.7682	0.552336
$816$	0	0
$817$	4.64235	0.162415
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.2783i	0.847318i	0.905822	+	0.423659i	$0.139254\pi$
$821$	24.2783i	0.847318i	−0.905822	+	0.423659i	$0.860746\pi$
$822$	0	0
$823$	−15.7002	−0.547275	−0.273637	−	0.961833i	$-0.588227\pi$
$823$	−15.7002	−0.547275	−0.273637	+	0.961833i	$0.588227\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.47290i	− 0.0512177i	−0.999672	−	0.0256089i	$-0.991848\pi$
$827$	− 1.47290i	− 0.0512177i	0.999672	−	0.0256089i	$-0.00815244\pi$
$828$	0	0
$829$	25.1759i	0.874395i	0.899365	+	0.437198i	$0.144029\pi$
$829$	25.1759i	0.874395i	−0.899365	+	0.437198i	$0.855971\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.0158095	0.000547768 0
$834$	0	0
$835$	21.6729i	0.750020i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14.4703	−0.499570	−0.249785	−	0.968301i	$-0.580360\pi$
$839$	−14.4703	−0.499570	−0.249785	+	0.968301i	$0.580360\pi$
$840$	0	0
$841$	28.7793	0.992391
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 14.8578i	− 0.511124i
$846$	0	0
$847$	−8.57368	−0.294595
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 27.3640i	− 0.938026i
$852$	0	0
$853$	0.552410i	0.0189142i	0.999955	+	0.00945708i	$0.00301033\pi$
$853$	0.552410i	0.0189142i	−0.999955	+	0.00945708i	$0.996990\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	47.3043	1.61588	0.807942	−	0.589262i	$-0.200581\pi$
$857$	47.3043	1.61588	0.807942	+	0.589262i	$0.200581\pi$
$858$	0	0
$859$	− 42.9031i	− 1.46384i	−0.681393	−	0.731918i	$-0.738625\pi$
$859$	− 42.9031i	− 1.46384i	0.681393	−	0.731918i	$-0.261375\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.39600	0.0815609	0.0407805	−	0.999168i	$-0.487016\pi$
$863$	2.39600	0.0815609	0.0407805	+	0.999168i	$0.487016\pi$
$864$	0	0
$865$	11.0184	0.374638
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.58498i	0.325148i
$870$	0	0
$871$	13.6678	0.463117
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.5676i	0.357252i
$876$	0	0
$877$	36.8060i	1.24285i	0.783473	+	0.621426i	$0.213446\pi$
$877$	36.8060i	1.24285i	−0.783473	+	0.621426i	$0.786554\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−45.1322	−1.52054	−0.760271	−	0.649606i	$-0.774934\pi$
$881$	−45.1322	−1.52054	−0.760271	+	0.649606i	$0.774934\pi$
$882$	0	0
$883$	38.0204i	1.27949i	0.768588	+	0.639744i	$0.220960\pi$
$883$	38.0204i	1.27949i	−0.768588	+	0.639744i	$0.779040\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30.4326	1.02183	0.510913	−	0.859633i	$-0.329308\pi$
$887$	30.4326	1.02183	0.510913	+	0.859633i	$0.329308\pi$
$888$	0	0
$889$	−19.8627	−0.666173
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16.7509i	0.560547i
$894$	0	0
$895$	−14.6664	−0.490244
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.794002i	0.0264814i
$900$	0	0
$901$	− 0.0295220i	0 0.000983519i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−30.4231	−1.01130
$906$	0	0
$907$	8.80211i	0.292269i	0.989265	+	0.146135i	$0.0466833\pi$
$907$	8.80211i	0.292269i	−0.989265	+	0.146135i	$0.953317\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−31.9516	−1.05860	−0.529302	−	0.848434i	$-0.677546\pi$
$911$	−31.9516	−1.05860	−0.529302	+	0.848434i	$0.677546\pi$
$912$	0	0
$913$	−13.8415	−0.458086
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 7.24798i	− 0.239349i
$918$	0	0
$919$	−55.5377	−1.83202	−0.916009	−	0.401157i	$-0.868608\pi$
$919$	−55.5377	−1.83202	−0.916009	+	0.401157i	$0.868608\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.52944i	0.149088i
$924$	0	0
$925$	15.7605i	0.518201i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−22.4706	−0.737235	−0.368617	−	0.929581i	$-0.620169\pi$
$929$	−22.4706	−0.737235	−0.368617	+	0.929581i	$0.620169\pi$
$930$	0	0
$931$	− 2.35077i	− 0.0770433i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.0308790	0.00100985
$936$	0	0
$937$	−47.5975	−1.55494	−0.777471	−	0.628919i	$-0.783498\pi$
$937$	−47.5975	−1.55494	−0.777471	+	0.628919i	$0.783498\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.03276i	0.164063i	0.996630	+	0.0820317i	$0.0261409\pi$
$941$	5.03276i	0.164063i	−0.996630	+	0.0820317i	$0.973859\pi$
$942$	0	0
$943$	−72.7188	−2.36805
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.9034i	0.614277i	0.951665	+	0.307138i	$0.0993714\pi$
$947$	18.9034i	0.614277i	−0.951665	+	0.307138i	$0.900629\pi$
$948$	0	0
$949$	6.98999i	0.226905i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	59.5108	1.92774	0.963872	−	0.266365i	$-0.0858225\pi$
$953$	59.5108	1.92774	0.963872	+	0.266365i	$0.0858225\pi$
$954$	0	0
$955$	− 13.1473i	− 0.425437i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13.1526	0.424721
$960$	0	0
$961$	−28.1428	−0.907834
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 15.9364i	− 0.513010i
$966$	0	0
$967$	26.3394	0.847018	0.423509	−	0.905892i	$-0.360798\pi$
$967$	26.3394	0.847018	0.423509	+	0.905892i	$0.360798\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 50.9352i	− 1.63459i	−0.576222	−	0.817293i	$-0.695474\pi$
$971$	− 50.9352i	− 1.63459i	0.576222	−	0.817293i	$-0.304526\pi$
$972$	0	0
$973$	− 5.29486i	− 0.169745i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23.5022	−0.751903	−0.375951	−	0.926639i	$-0.622684\pi$
$977$	−23.5022	−0.751903	−0.375951	+	0.926639i	$0.622684\pi$
$978$	0	0
$979$	− 0.375067i	− 0.0119872i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−41.3639	−1.31930	−0.659651	−	0.751572i	$-0.729296\pi$
$983$	−41.3639	−1.31930	−0.659651	+	0.751572i	$0.729296\pi$
$984$	0	0
$985$	−20.0774	−0.639720
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 11.7527i	− 0.373715i
$990$	0	0
$991$	8.06637	0.256237	0.128118	−	0.991759i	$-0.459106\pi$
$991$	8.06637	0.256237	0.128118	+	0.991759i	$0.459106\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 14.0299i	− 0.444778i
$996$	0	0
$997$	− 38.8503i	− 1.23040i	−0.788371	−	0.615200i	$-0.789075\pi$
$997$	− 38.8503i	− 1.23040i	0.788371	−	0.615200i	$-0.210925\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.f.3025.10		24
3.2	odd	2	inner	6048.2.c.f.3025.16		24
4.3	odd	2		1512.2.c.g.757.16	yes	24
8.3	odd	2		1512.2.c.g.757.15	yes	24
8.5	even	2	inner	6048.2.c.f.3025.15		24
12.11	even	2		1512.2.c.g.757.9	✓	24
24.5	odd	2	inner	6048.2.c.f.3025.9		24
24.11	even	2		1512.2.c.g.757.10	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.g.757.9	✓	24	12.11	even	2
1512.2.c.g.757.10	yes	24	24.11	even	2
1512.2.c.g.757.15	yes	24	8.3	odd	2
1512.2.c.g.757.16	yes	24	4.3	odd	2
6048.2.c.f.3025.9		24	24.5	odd	2	inner
6048.2.c.f.3025.10		24	1.1	even	1	trivial
6048.2.c.f.3025.15		24	8.5	even	2	inner
6048.2.c.f.3025.16		24	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.25392i q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-1.25392i q^{5} -1.00000 q^{7} +1.55766i q^{11} +1.07281i q^{13} +0.0158095 q^{17} -2.35077i q^{19} -5.95129 q^{23} +3.42768 q^{25} -0.469737i q^{29} -1.69031 q^{31} +1.25392i q^{35} +4.59800i q^{37} +12.2190 q^{41} +1.97482i q^{43} -7.12571 q^{47} +1.00000 q^{49} -1.86735i q^{53} +1.95319 q^{55} +8.54291i q^{59} +3.92502i q^{61} +1.34521 q^{65} -12.7403i q^{67} +4.22205 q^{71} +6.51561 q^{73} -1.55766i q^{77} +6.15343 q^{79} +8.88604i q^{83} -0.0198239i q^{85} -0.240788 q^{89} -1.07281i q^{91} -2.94767 q^{95} +5.86131 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{7}+O(q^{10}) \) 24 * q - 24 * q^7	\( 24 q - 24 q^{7} - 24 q^{25} - 8 q^{31} + 24 q^{49} - 16 q^{55} - 8 q^{79}+O(q^{100}) \) 24 * q - 24 * q^7 - 24 * q^25 - 8 * q^31 + 24 * q^49 - 16 * q^55 - 8 * q^79

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more