LMFDB - Embedded newform 6048.2.j.d.5615.24

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			5615.24
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.d.5615.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381416 q^{5} +1.00000i q^{7} +O(q^{10})$	$q-0.381416 q^{5} +1.00000i q^{7} -2.13458i q^{11} +6.49722i q^{13} +6.97095i q^{17} -6.19245 q^{19} -1.82497 q^{23} -4.85452 q^{25} -0.694888 q^{29} -1.67769i q^{31} -0.381416i q^{35} -8.06690i q^{37} -1.31125i q^{41} -7.67301 q^{43} +6.82360 q^{47} -1.00000 q^{49} +0.954792 q^{53} +0.814162i q^{55} -12.8165i q^{59} -12.4829i q^{61} -2.47815i q^{65} -0.634394 q^{67} +5.79187 q^{71} +8.13772 q^{73} +2.13458 q^{77} -14.1354i q^{79} +3.36133i q^{83} -2.65883i q^{85} +15.9304i q^{89} -6.49722 q^{91} +2.36190 q^{95} +10.8796 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.381416	−0.170575	−0.0852873	−	0.996356i	$-0.527181\pi$
$5$	−0.381416	−0.170575	−0.0852873	+	0.996356i	$0.527181\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.13458i	− 0.643599i	−0.946808	−	0.321799i	$-0.895712\pi$
$11$	− 2.13458i	− 0.643599i	0.946808	−	0.321799i	$-0.104288\pi$
$12$	0	0
$13$	6.49722i	1.80201i	0.433813	+	0.901003i	$0.357168\pi$
$13$	6.49722i	1.80201i	−0.433813	+	0.901003i	$0.642832\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.97095i	1.69070i	0.534211	+	0.845351i	$0.320609\pi$
$17$	6.97095i	1.69070i	−0.534211	+	0.845351i	$0.679391\pi$
$18$	0	0
$19$	−6.19245	−1.42065	−0.710323	−	0.703876i	$-0.751451\pi$
$19$	−6.19245	−1.42065	−0.710323	+	0.703876i	$0.751451\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.82497	−0.380533	−0.190266	−	0.981733i	$-0.560935\pi$
$23$	−1.82497	−0.380533	−0.190266	+	0.981733i	$0.560935\pi$
$24$	0	0
$25$	−4.85452	−0.970904
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.694888	−0.129038	−0.0645188	−	0.997916i	$-0.520551\pi$
$29$	−0.694888	−0.129038	−0.0645188	+	0.997916i	$0.520551\pi$
$30$	0	0
$31$	− 1.67769i	− 0.301322i	−0.988585	−	0.150661i	$-0.951860\pi$
$31$	− 1.67769i	− 0.301322i	0.988585	−	0.150661i	$-0.0481402\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 0.381416i	− 0.0644711i
$36$	0	0
$37$	− 8.06690i	− 1.32619i	−0.748535	−	0.663095i	$-0.769242\pi$
$37$	− 8.06690i	− 1.32619i	0.748535	−	0.663095i	$-0.230758\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1.31125i	− 0.204783i	−0.994744	−	0.102391i	$-0.967351\pi$
$41$	− 1.31125i	− 0.204783i	0.994744	−	0.102391i	$-0.0326494\pi$
$42$	0	0
$43$	−7.67301	−1.17012	−0.585061	−	0.810989i	$-0.698930\pi$
$43$	−7.67301	−1.17012	−0.585061	+	0.810989i	$0.698930\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.82360	0.995325	0.497662	−	0.867371i	$-0.334192\pi$
$47$	6.82360	0.995325	0.497662	+	0.867371i	$0.334192\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.954792	0.131151	0.0655754	−	0.997848i	$-0.479112\pi$
$53$	0.954792	0.131151	0.0655754	+	0.997848i	$0.479112\pi$
$54$	0	0
$55$	0.814162i	0.109782i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 12.8165i	− 1.66856i	−0.551341	−	0.834280i	$-0.685884\pi$
$59$	− 12.8165i	− 1.66856i	0.551341	−	0.834280i	$-0.314116\pi$
$60$	0	0
$61$	− 12.4829i	− 1.59827i	−0.601149	−	0.799137i	$-0.705290\pi$
$61$	− 12.4829i	− 1.59827i	0.601149	−	0.799137i	$-0.294710\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2.47815i	− 0.307376i
$66$	0	0
$67$	−0.634394	−0.0775036	−0.0387518	−	0.999249i	$-0.512338\pi$
$67$	−0.634394	−0.0775036	−0.0387518	+	0.999249i	$0.512338\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.79187	0.687368	0.343684	−	0.939085i	$-0.388325\pi$
$71$	5.79187	0.687368	0.343684	+	0.939085i	$0.388325\pi$
$72$	0	0
$73$	8.13772	0.952448	0.476224	−	0.879324i	$-0.342005\pi$
$73$	8.13772	0.952448	0.476224	+	0.879324i	$0.342005\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.13458	0.243258
$78$	0	0
$79$	− 14.1354i	− 1.59036i	−0.606377	−	0.795178i	$-0.707378\pi$
$79$	− 14.1354i	− 1.59036i	0.606377	−	0.795178i	$-0.292622\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.36133i	0.368954i	0.982837	+	0.184477i	$0.0590591\pi$
$83$	3.36133i	0.368954i	−0.982837	+	0.184477i	$0.940941\pi$
$84$	0	0
$85$	− 2.65883i	− 0.288391i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.9304i	1.68862i	0.535858	+	0.844308i	$0.319988\pi$
$89$	15.9304i	1.68862i	−0.535858	+	0.844308i	$0.680012\pi$
$90$	0	0
$91$	−6.49722	−0.681094
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.36190	0.242326
$96$	0	0
$97$	10.8796	1.10465	0.552326	−	0.833628i	$-0.313740\pi$
$97$	10.8796	1.10465	0.552326	+	0.833628i	$0.313740\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.1758	−1.11203	−0.556016	−	0.831171i	$-0.687671\pi$
$101$	−11.1758	−1.11203	−0.556016	+	0.831171i	$0.687671\pi$
$102$	0	0
$103$	− 4.65366i	− 0.458539i	−0.973363	−	0.229270i	$-0.926366\pi$
$103$	− 4.65366i	− 0.458539i	0.973363	−	0.229270i	$-0.0736337\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 17.4916i	− 1.69098i	−0.533993	−	0.845489i	$-0.679309\pi$
$107$	− 17.4916i	− 1.69098i	0.533993	−	0.845489i	$-0.320691\pi$
$108$	0	0
$109$	− 1.55555i	− 0.148994i	−0.997221	−	0.0744972i	$-0.976265\pi$
$109$	− 1.55555i	− 0.148994i	0.997221	−	0.0744972i	$-0.0237352\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.7024i	1.00679i	0.864055	+	0.503397i	$0.167917\pi$
$113$	10.7024i	1.00679i	−0.864055	+	0.503397i	$0.832083\pi$
$114$	0	0
$115$	0.696073	0.0649092
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.97095	−0.639026
$120$	0	0
$121$	6.44358	0.585780
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.75867	0.336186
$126$	0	0
$127$	− 19.8483i	− 1.76125i	−0.473814	−	0.880625i	$-0.657123\pi$
$127$	− 19.8483i	− 1.76125i	0.473814	−	0.880625i	$-0.342877\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.8474i	1.47197i	0.677000	+	0.735983i	$0.263280\pi$
$131$	16.8474i	1.47197i	−0.677000	+	0.735983i	$0.736720\pi$
$132$	0	0
$133$	− 6.19245i	− 0.536954i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.76935i	− 0.749216i	−0.927183	−	0.374608i	$-0.877777\pi$
$137$	− 8.76935i	− 0.749216i	0.927183	−	0.374608i	$-0.122223\pi$
$138$	0	0
$139$	0.920814	0.0781024	0.0390512	−	0.999237i	$-0.487566\pi$
$139$	0.920814	0.0781024	0.0390512	+	0.999237i	$0.487566\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.8688	1.15977
$144$	0	0
$145$	0.265042	0.0220105
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.7098	−1.20507	−0.602537	−	0.798091i	$-0.705843\pi$
$149$	−14.7098	−1.20507	−0.602537	+	0.798091i	$0.705843\pi$
$150$	0	0
$151$	15.1404i	1.23211i	0.787704	+	0.616054i	$0.211270\pi$
$151$	15.1404i	1.23211i	−0.787704	+	0.616054i	$0.788730\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.639898i	0.0513978i
$156$	0	0
$157$	− 5.91486i	− 0.472057i	−0.971746	−	0.236029i	$-0.924154\pi$
$157$	− 5.91486i	− 0.472057i	0.971746	−	0.236029i	$-0.0758460\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.82497i	− 0.143828i
$162$	0	0
$163$	−0.494679	−0.0387462	−0.0193731	−	0.999812i	$-0.506167\pi$
$163$	−0.494679	−0.0387462	−0.0193731	+	0.999812i	$0.506167\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.8984	−1.30764	−0.653820	−	0.756650i	$-0.726835\pi$
$167$	−16.8984	−1.30764	−0.653820	+	0.756650i	$0.726835\pi$
$168$	0	0
$169$	−29.2139	−2.24723
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.5249	1.02828	0.514141	−	0.857705i	$-0.328111\pi$
$173$	13.5249	1.02828	0.514141	+	0.857705i	$0.328111\pi$
$174$	0	0
$175$	− 4.85452i	− 0.366967i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1.44357i	− 0.107897i	−0.998544	−	0.0539487i	$-0.982819\pi$
$179$	− 1.44357i	− 0.107897i	0.998544	−	0.0539487i	$-0.0171807\pi$
$180$	0	0
$181$	6.40425i	0.476024i	0.971262	+	0.238012i	$0.0764958\pi$
$181$	6.40425i	0.476024i	−0.971262	+	0.238012i	$0.923504\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.07685i	0.226214i
$186$	0	0
$187$	14.8800	1.08813
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.60609	0.188570	0.0942850	−	0.995545i	$-0.469944\pi$
$191$	2.60609	0.188570	0.0942850	+	0.995545i	$0.469944\pi$
$192$	0	0
$193$	−11.5433	−0.830904	−0.415452	−	0.909615i	$-0.636377\pi$
$193$	−11.5433	−0.830904	−0.415452	+	0.909615i	$0.636377\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.6718	−1.11657	−0.558285	−	0.829650i	$-0.688540\pi$
$197$	−15.6718	−1.11657	−0.558285	+	0.829650i	$0.688540\pi$
$198$	0	0
$199$	2.80803i	0.199056i	0.995035	+	0.0995278i	$0.0317332\pi$
$199$	2.80803i	0.199056i	−0.995035	+	0.0995278i	$0.968267\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 0.694888i	− 0.0487716i
$204$	0	0
$205$	0.500132i	0.0349307i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	13.2183i	0.914327i
$210$	0	0
$211$	−22.7609	−1.56692	−0.783462	−	0.621440i	$-0.786548\pi$
$211$	−22.7609	−1.56692	−0.783462	+	0.621440i	$0.786548\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.92661	0.199593
$216$	0	0
$217$	1.67769	0.113889
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−45.2918	−3.04666
$222$	0	0
$223$	− 7.34083i	− 0.491578i	−0.969323	−	0.245789i	$-0.920953\pi$
$223$	− 7.34083i	− 0.491578i	0.969323	−	0.245789i	$-0.0790471\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.96015i	0.395589i	0.980244	+	0.197794i	$0.0633778\pi$
$227$	5.96015i	0.395589i	−0.980244	+	0.197794i	$0.936622\pi$
$228$	0	0
$229$	− 20.0338i	− 1.32387i	−0.749561	−	0.661935i	$-0.769736\pi$
$229$	− 20.0338i	− 1.32387i	0.749561	−	0.661935i	$-0.230264\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 19.0205i	− 1.24608i	−0.782192	−	0.623038i	$-0.785898\pi$
$233$	− 19.0205i	− 1.24608i	0.782192	−	0.623038i	$-0.214102\pi$
$234$	0	0
$235$	−2.60263	−0.169777
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.44700	0.611076	0.305538	−	0.952180i	$-0.401164\pi$
$239$	9.44700	0.611076	0.305538	+	0.952180i	$0.401164\pi$
$240$	0	0
$241$	−16.0144	−1.03158	−0.515789	−	0.856716i	$-0.672501\pi$
$241$	−16.0144	−1.03158	−0.515789	+	0.856716i	$0.672501\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.381416	0.0243678
$246$	0	0
$247$	− 40.2338i	− 2.56001i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 17.8027i	− 1.12370i	−0.827240	−	0.561849i	$-0.810090\pi$
$251$	− 17.8027i	− 1.12370i	0.827240	−	0.561849i	$-0.189910\pi$
$252$	0	0
$253$	3.89554i	0.244910i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 11.2136i	− 0.699486i	−0.936846	−	0.349743i	$-0.886269\pi$
$257$	− 11.2136i	− 0.699486i	0.936846	−	0.349743i	$-0.113731\pi$
$258$	0	0
$259$	8.06690	0.501253
$260$	0	0
$261$	0	0
$262$	0	0
$263$	30.8071	1.89965	0.949825	−	0.312782i	$-0.101261\pi$
$263$	30.8071	1.89965	0.949825	+	0.312782i	$0.101261\pi$
$264$	0	0
$265$	−0.364173	−0.0223710
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.9698	1.64438	0.822188	−	0.569216i	$-0.192753\pi$
$269$	26.9698	1.64438	0.822188	+	0.569216i	$0.192753\pi$
$270$	0	0
$271$	− 4.91487i	− 0.298557i	−0.988795	−	0.149279i	$-0.952305\pi$
$271$	− 4.91487i	− 0.298557i	0.988795	−	0.149279i	$-0.0476951\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.3623i	0.624873i
$276$	0	0
$277$	− 17.3250i	− 1.04096i	−0.853874	−	0.520480i	$-0.825753\pi$
$277$	− 17.3250i	− 1.04096i	0.853874	−	0.520480i	$-0.174247\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.5958i	0.870712i	0.900258	+	0.435356i	$0.143377\pi$
$281$	14.5958i	0.870712i	−0.900258	+	0.435356i	$0.856623\pi$
$282$	0	0
$283$	−17.2643	−1.02625	−0.513127	−	0.858313i	$-0.671513\pi$
$283$	−17.2643	−1.02625	−0.513127	+	0.858313i	$0.671513\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.31125	0.0774006
$288$	0	0
$289$	−31.5941	−1.85848
$290$	0	0
$291$	0	0
$292$	0	0
$293$	24.2206	1.41498	0.707491	−	0.706722i	$-0.249827\pi$
$293$	24.2206	1.41498	0.707491	+	0.706722i	$0.249827\pi$
$294$	0	0
$295$	4.88840i	0.284614i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 11.8572i	− 0.685722i
$300$	0	0
$301$	− 7.67301i	− 0.442265i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.76119i	0.272625i
$306$	0	0
$307$	0.805392	0.0459662	0.0229831	−	0.999736i	$-0.492684\pi$
$307$	0.805392	0.0459662	0.0229831	+	0.999736i	$0.492684\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.5723	−0.996432	−0.498216	−	0.867053i	$-0.666011\pi$
$311$	−17.5723	−0.996432	−0.498216	+	0.867053i	$0.666011\pi$
$312$	0	0
$313$	−12.6581	−0.715477	−0.357739	−	0.933822i	$-0.616452\pi$
$313$	−12.6581	−0.715477	−0.357739	+	0.933822i	$0.616452\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.9610	1.23345	0.616725	−	0.787178i	$-0.288459\pi$
$317$	21.9610	1.23345	0.616725	+	0.787178i	$0.288459\pi$
$318$	0	0
$319$	1.48329i	0.0830484i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 43.1673i	− 2.40189i
$324$	0	0
$325$	− 31.5409i	− 1.74958i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.82360i	0.376197i
$330$	0	0
$331$	3.55541	0.195423	0.0977115	−	0.995215i	$-0.468848\pi$
$331$	3.55541	0.195423	0.0977115	+	0.995215i	$0.468848\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.241968	0.0132201
$336$	0	0
$337$	−3.54995	−0.193378	−0.0966890	−	0.995315i	$-0.530825\pi$
$337$	−3.54995	−0.193378	−0.0966890	+	0.995315i	$0.530825\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.58116	−0.193930
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.7798i	1.27657i	0.769801	+	0.638284i	$0.220356\pi$
$347$	23.7798i	1.27657i	−0.769801	+	0.638284i	$0.779644\pi$
$348$	0	0
$349$	− 13.9749i	− 0.748058i	−0.927417	−	0.374029i	$-0.877976\pi$
$349$	− 13.9749i	− 0.748058i	0.927417	−	0.374029i	$-0.122024\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 10.3186i	− 0.549203i	−0.961558	−	0.274602i	$-0.911454\pi$
$353$	− 10.3186i	− 0.549203i	0.961558	−	0.274602i	$-0.0885460\pi$
$354$	0	0
$355$	−2.20911	−0.117247
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.36947	−0.388945	−0.194473	−	0.980908i	$-0.562300\pi$
$359$	−7.36947	−0.388945	−0.194473	+	0.980908i	$0.562300\pi$
$360$	0	0
$361$	19.3465	1.01824
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.10386	−0.162463
$366$	0	0
$367$	15.2286i	0.794928i	0.917618	+	0.397464i	$0.130110\pi$
$367$	15.2286i	0.794928i	−0.917618	+	0.397464i	$0.869890\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.954792i	0.0495703i
$372$	0	0
$373$	24.7361i	1.28079i	0.768047	+	0.640394i	$0.221229\pi$
$373$	24.7361i	1.28079i	−0.768047	+	0.640394i	$0.778771\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.51485i	− 0.232526i
$378$	0	0
$379$	2.09123	0.107419	0.0537096	−	0.998557i	$-0.482895\pi$
$379$	2.09123	0.107419	0.0537096	+	0.998557i	$0.482895\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−21.9617	−1.12219	−0.561094	−	0.827752i	$-0.689619\pi$
$383$	−21.9617	−1.12219	−0.561094	+	0.827752i	$0.689619\pi$
$384$	0	0
$385$	−0.814162	−0.0414935
$386$	0	0
$387$	0	0
$388$	0	0
$389$	19.7874	1.00326	0.501631	−	0.865082i	$-0.332734\pi$
$389$	19.7874	1.00326	0.501631	+	0.865082i	$0.332734\pi$
$390$	0	0
$391$	− 12.7218i	− 0.643367i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.39147i	0.271274i
$396$	0	0
$397$	− 2.02424i	− 0.101593i	−0.998709	−	0.0507967i	$-0.983824\pi$
$397$	− 2.02424i	− 0.101593i	0.998709	−	0.0507967i	$-0.0161761\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.5494i	1.12606i	0.826436	+	0.563031i	$0.190365\pi$
$401$	22.5494i	1.12606i	−0.826436	+	0.563031i	$0.809635\pi$
$402$	0	0
$403$	10.9003	0.542984
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−17.2194	−0.853535
$408$	0	0
$409$	−20.6609	−1.02161	−0.510807	−	0.859695i	$-0.670653\pi$
$409$	−20.6609	−1.02161	−0.510807	+	0.859695i	$0.670653\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	12.8165	0.630656
$414$	0	0
$415$	− 1.28207i	− 0.0629341i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 12.3671i	− 0.604171i	−0.953281	−	0.302086i	$-0.902317\pi$
$419$	− 12.3671i	− 0.604171i	0.953281	−	0.302086i	$-0.0976829\pi$
$420$	0	0
$421$	− 3.03328i	− 0.147833i	−0.997264	−	0.0739165i	$-0.976450\pi$
$421$	− 3.03328i	− 0.147833i	0.997264	−	0.0739165i	$-0.0235498\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 33.8406i	− 1.64151i
$426$	0	0
$427$	12.4829	0.604091
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.8066	0.761378	0.380689	−	0.924703i	$-0.375687\pi$
$431$	15.8066	0.761378	0.380689	+	0.924703i	$0.375687\pi$
$432$	0	0
$433$	−3.30149	−0.158659	−0.0793297	−	0.996848i	$-0.525278\pi$
$433$	−3.30149	−0.158659	−0.0793297	+	0.996848i	$0.525278\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11.3010	0.540602
$438$	0	0
$439$	− 10.8506i	− 0.517871i	−0.965895	−	0.258936i	$-0.916628\pi$
$439$	− 10.8506i	− 0.517871i	0.965895	−	0.258936i	$-0.0833717\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.1061i	0.575177i	0.957754	+	0.287589i	$0.0928536\pi$
$443$	12.1061i	0.575177i	−0.957754	+	0.287589i	$0.907146\pi$
$444$	0	0
$445$	− 6.07610i	− 0.288035i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.10578i	0.0521848i	0.999660	+	0.0260924i	$0.00830640\pi$
$449$	1.10578i	0.0521848i	−0.999660	+	0.0260924i	$0.991694\pi$
$450$	0	0
$451$	−2.79896	−0.131798
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.47815	0.116177
$456$	0	0
$457$	−2.60091	−0.121665	−0.0608327	−	0.998148i	$-0.519376\pi$
$457$	−2.60091	−0.121665	−0.0608327	+	0.998148i	$0.519376\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−25.7622	−1.19987	−0.599933	−	0.800050i	$-0.704806\pi$
$461$	−25.7622	−1.19987	−0.599933	+	0.800050i	$0.704806\pi$
$462$	0	0
$463$	9.73328i	0.452344i	0.974087	+	0.226172i	$0.0726211\pi$
$463$	9.73328i	0.452344i	−0.974087	+	0.226172i	$0.927379\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.70336i	0.402743i	0.979515	+	0.201372i	$0.0645399\pi$
$467$	8.70336i	0.402743i	−0.979515	+	0.201372i	$0.935460\pi$
$468$	0	0
$469$	− 0.634394i	− 0.0292936i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	16.3786i	0.753090i
$474$	0	0
$475$	30.0614	1.37931
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−13.7878	−0.629983	−0.314991	−	0.949095i	$-0.602002\pi$
$479$	−13.7878	−0.629983	−0.314991	+	0.949095i	$0.602002\pi$
$480$	0	0
$481$	52.4125	2.38980
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.14964	−0.188426
$486$	0	0
$487$	− 28.1191i	− 1.27420i	−0.770783	−	0.637098i	$-0.780135\pi$
$487$	− 28.1191i	− 1.27420i	0.770783	−	0.637098i	$-0.219865\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	22.2021i	1.00197i	0.865457	+	0.500983i	$0.167028\pi$
$491$	22.2021i	1.00197i	−0.865457	+	0.500983i	$0.832972\pi$
$492$	0	0
$493$	− 4.84403i	− 0.218164i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.79187i	0.259801i
$498$	0	0
$499$	−4.17365	−0.186838	−0.0934192	−	0.995627i	$-0.529780\pi$
$499$	−4.17365	−0.186838	−0.0934192	+	0.995627i	$0.529780\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−15.7948	−0.704256	−0.352128	−	0.935952i	$-0.614542\pi$
$503$	−15.7948	−0.704256	−0.352128	+	0.935952i	$0.614542\pi$
$504$	0	0
$505$	4.26263	0.189684
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−36.0454	−1.59769	−0.798843	−	0.601540i	$-0.794554\pi$
$509$	−36.0454	−1.59769	−0.798843	+	0.601540i	$0.794554\pi$
$510$	0	0
$511$	8.13772i	0.359992i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.77498i	0.0782151i
$516$	0	0
$517$	− 14.5655i	− 0.640590i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.7116i	0.688336i	0.938908	+	0.344168i	$0.111839\pi$
$521$	15.7116i	0.688336i	−0.938908	+	0.344168i	$0.888161\pi$
$522$	0	0
$523$	−5.96881	−0.260998	−0.130499	−	0.991448i	$-0.541658\pi$
$523$	−5.96881	−0.260998	−0.130499	+	0.991448i	$0.541658\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.6951	0.509446
$528$	0	0
$529$	−19.6695	−0.855195
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.51948	0.369020
$534$	0	0
$535$	6.67158i	0.288438i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.13458i	0.0919427i
$540$	0	0
$541$	24.3780i	1.04809i	0.851690	+	0.524047i	$0.175578\pi$
$541$	24.3780i	1.04809i	−0.851690	+	0.524047i	$0.824422\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.593311i	0.0254147i
$546$	0	0
$547$	−4.69571	−0.200774	−0.100387	−	0.994948i	$-0.532008\pi$
$547$	−4.69571	−0.200774	−0.100387	+	0.994948i	$0.532008\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.30306	0.183317
$552$	0	0
$553$	14.1354	0.601098
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−23.4437	−0.993340	−0.496670	−	0.867939i	$-0.665444\pi$
$557$	−23.4437	−0.993340	−0.496670	+	0.867939i	$0.665444\pi$
$558$	0	0
$559$	− 49.8533i	− 2.10857i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.1216i	0.510865i	0.966827	+	0.255433i	$0.0822179\pi$
$563$	12.1216i	0.510865i	−0.966827	+	0.255433i	$0.917782\pi$
$564$	0	0
$565$	− 4.08205i	− 0.171733i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 13.9834i	− 0.586215i	−0.956079	−	0.293108i	$-0.905311\pi$
$569$	− 13.9834i	− 0.586215i	0.956079	−	0.293108i	$-0.0946894\pi$
$570$	0	0
$571$	−30.9600	−1.29564	−0.647819	−	0.761795i	$-0.724319\pi$
$571$	−30.9600	−1.29564	−0.647819	+	0.761795i	$0.724319\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.85936	0.369461
$576$	0	0
$577$	−25.9858	−1.08180	−0.540901	−	0.841087i	$-0.681916\pi$
$577$	−25.9858	−1.08180	−0.540901	+	0.841087i	$0.681916\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.36133	−0.139451
$582$	0	0
$583$	− 2.03808i	− 0.0844085i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.2153i	0.751828i	0.926655	+	0.375914i	$0.122671\pi$
$587$	18.2153i	0.751828i	−0.926655	+	0.375914i	$0.877329\pi$
$588$	0	0
$589$	10.3890i	0.428072i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 37.4912i	− 1.53958i	−0.638297	−	0.769790i	$-0.720361\pi$
$593$	− 37.4912i	− 1.53958i	0.638297	−	0.769790i	$-0.279639\pi$
$594$	0	0
$595$	2.65883	0.109001
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−38.3963	−1.56883	−0.784416	−	0.620235i	$-0.787037\pi$
$599$	−38.3963	−1.56883	−0.784416	+	0.620235i	$0.787037\pi$
$600$	0	0
$601$	−25.6236	−1.04521	−0.522604	−	0.852576i	$-0.675039\pi$
$601$	−25.6236	−1.04521	−0.522604	+	0.852576i	$0.675039\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.45769	−0.0999192
$606$	0	0
$607$	24.6445i	1.00029i	0.865942	+	0.500145i	$0.166720\pi$
$607$	24.6445i	1.00029i	−0.865942	+	0.500145i	$0.833280\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	44.3345i	1.79358i
$612$	0	0
$613$	− 11.3317i	− 0.457685i	−0.973463	−	0.228842i	$-0.926506\pi$
$613$	− 11.3317i	− 0.457685i	0.973463	−	0.228842i	$-0.0734941\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 34.6800i	− 1.39616i	−0.716018	−	0.698082i	$-0.754037\pi$
$617$	− 34.6800i	− 1.39616i	0.716018	−	0.698082i	$-0.245963\pi$
$618$	0	0
$619$	−24.6276	−0.989866	−0.494933	−	0.868931i	$-0.664807\pi$
$619$	−24.6276	−0.989866	−0.494933	+	0.868931i	$0.664807\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−15.9304	−0.638237
$624$	0	0
$625$	22.8390	0.913560
$626$	0	0
$627$	0	0
$628$	0	0
$629$	56.2340	2.24219
$630$	0	0
$631$	10.8695i	0.432708i	0.976315	+	0.216354i	$0.0694166\pi$
$631$	10.8695i	0.432708i	−0.976315	+	0.216354i	$0.930583\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.57045i	0.300424i
$636$	0	0
$637$	− 6.49722i	− 0.257429i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 38.7871i	− 1.53200i	−0.642842	−	0.765999i	$-0.722245\pi$
$641$	− 38.7871i	− 1.53200i	0.642842	−	0.765999i	$-0.277755\pi$
$642$	0	0
$643$	−15.8941	−0.626801	−0.313400	−	0.949621i	$-0.601468\pi$
$643$	−15.8941	−0.626801	−0.313400	+	0.949621i	$0.601468\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.857423	−0.0337088	−0.0168544	−	0.999858i	$-0.505365\pi$
$647$	−0.857423	−0.0337088	−0.0168544	+	0.999858i	$0.505365\pi$
$648$	0	0
$649$	−27.3577	−1.07388
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.35844	−0.209692	−0.104846	−	0.994488i	$-0.533435\pi$
$653$	−5.35844	−0.209692	−0.104846	+	0.994488i	$0.533435\pi$
$654$	0	0
$655$	− 6.42588i	− 0.251080i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.84388i	0.149736i	0.997193	+	0.0748681i	$0.0238536\pi$
$659$	3.84388i	0.149736i	−0.997193	+	0.0748681i	$0.976146\pi$
$660$	0	0
$661$	38.4840i	1.49685i	0.663218	+	0.748426i	$0.269190\pi$
$661$	38.4840i	1.49685i	−0.663218	+	0.748426i	$0.730810\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.36190i	0.0915907i
$666$	0	0
$667$	1.26815	0.0491030
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−26.6457	−1.02865
$672$	0	0
$673$	−19.3543	−0.746055	−0.373027	−	0.927820i	$-0.621680\pi$
$673$	−19.3543	−0.746055	−0.373027	+	0.927820i	$0.621680\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16.5044	−0.634315	−0.317157	−	0.948373i	$-0.602728\pi$
$677$	−16.5044	−0.634315	−0.317157	+	0.948373i	$0.602728\pi$
$678$	0	0
$679$	10.8796i	0.417520i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	45.3881i	1.73673i	0.495929	+	0.868363i	$0.334828\pi$
$683$	45.3881i	1.73673i	−0.495929	+	0.868363i	$0.665172\pi$
$684$	0	0
$685$	3.34477i	0.127797i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.20350i	0.236335i
$690$	0	0
$691$	21.0358	0.800239	0.400119	−	0.916463i	$-0.368969\pi$
$691$	21.0358	0.800239	0.400119	+	0.916463i	$0.368969\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.351213	−0.0133223
$696$	0	0
$697$	9.14065	0.346227
$698$	0	0
$699$	0	0
$700$	0	0
$701$	44.0611	1.66417	0.832083	−	0.554652i	$-0.187148\pi$
$701$	44.0611	1.66417	0.832083	+	0.554652i	$0.187148\pi$
$702$	0	0
$703$	49.9539i	1.88405i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 11.1758i	− 0.420309i
$708$	0	0
$709$	14.8620i	0.558152i	0.960269	+	0.279076i	$0.0900282\pi$
$709$	14.8620i	0.558152i	−0.960269	+	0.279076i	$0.909972\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.06173i	0.114663i
$714$	0	0
$715$	−5.28979	−0.197827
$716$	0	0
$717$	0	0
$718$	0	0
$719$	52.1512	1.94491	0.972455	−	0.233090i	$-0.0748838\pi$
$719$	52.1512	1.94491	0.972455	+	0.233090i	$0.0748838\pi$
$720$	0	0
$721$	4.65366	0.173311
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.37335	0.125283
$726$	0	0
$727$	− 32.4943i	− 1.20515i	−0.798063	−	0.602574i	$-0.794142\pi$
$727$	− 32.4943i	− 1.20515i	0.798063	−	0.602574i	$-0.205858\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 53.4881i	− 1.97833i
$732$	0	0
$733$	− 29.8758i	− 1.10349i	−0.834014	−	0.551744i	$-0.813963\pi$
$733$	− 29.8758i	− 1.10349i	0.834014	−	0.551744i	$-0.186037\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.35416i	0.0498812i
$738$	0	0
$739$	−37.0819	−1.36408	−0.682040	−	0.731315i	$-0.738907\pi$
$739$	−37.0819	−1.36408	−0.682040	+	0.731315i	$0.738907\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−42.7019	−1.56658	−0.783290	−	0.621657i	$-0.786460\pi$
$743$	−42.7019	−1.56658	−0.783290	+	0.621657i	$0.786460\pi$
$744$	0	0
$745$	5.61055	0.205555
$746$	0	0
$747$	0	0
$748$	0	0
$749$	17.4916	0.639130
$750$	0	0
$751$	− 1.04931i	− 0.0382897i	−0.999817	−	0.0191449i	$-0.993906\pi$
$751$	− 1.04931i	− 0.0382897i	0.999817	−	0.0191449i	$-0.00609437\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 5.77479i	− 0.210166i
$756$	0	0
$757$	42.8516i	1.55747i	0.627354	+	0.778734i	$0.284138\pi$
$757$	42.8516i	1.55747i	−0.627354	+	0.778734i	$0.715862\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	16.6457i	0.603408i	0.953402	+	0.301704i	$0.0975554\pi$
$761$	16.6457i	0.603408i	−0.953402	+	0.301704i	$0.902445\pi$
$762$	0	0
$763$	1.55555	0.0563146
$764$	0	0
$765$	0	0
$766$	0	0
$767$	83.2714	3.00675
$768$	0	0
$769$	23.0100	0.829763	0.414881	−	0.909876i	$-0.363823\pi$
$769$	23.0100	0.829763	0.414881	+	0.909876i	$0.363823\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	40.5801	1.45956	0.729782	−	0.683680i	$-0.239622\pi$
$773$	40.5801	1.45956	0.729782	+	0.683680i	$0.239622\pi$
$774$	0	0
$775$	8.14438i	0.292555i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.11985i	0.290924i
$780$	0	0
$781$	− 12.3632i	− 0.442389i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.25602i	0.0805210i
$786$	0	0
$787$	−14.8449	−0.529162	−0.264581	−	0.964363i	$-0.585234\pi$
$787$	−14.8449	−0.529162	−0.264581	+	0.964363i	$0.585234\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−10.7024	−0.380532
$792$	0	0
$793$	81.1043	2.88010
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.99972	−0.0708337	−0.0354169	−	0.999373i	$-0.511276\pi$
$797$	−1.99972	−0.0708337	−0.0354169	+	0.999373i	$0.511276\pi$
$798$	0	0
$799$	47.5670i	1.68280i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 17.3706i	− 0.612995i
$804$	0	0
$805$	0.696073i	0.0245334i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 53.9508i	− 1.89681i	−0.317065	−	0.948404i	$-0.602697\pi$
$809$	− 53.9508i	− 1.89681i	0.317065	−	0.948404i	$-0.397303\pi$
$810$	0	0
$811$	15.4358	0.542025	0.271012	−	0.962576i	$-0.412642\pi$
$811$	15.4358	0.542025	0.271012	+	0.962576i	$0.412642\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.188679	0.00660912
$816$	0	0
$817$	47.5148	1.66233
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.2354	0.531720	0.265860	−	0.964012i	$-0.414344\pi$
$821$	15.2354	0.531720	0.265860	+	0.964012i	$0.414344\pi$
$822$	0	0
$823$	− 22.8119i	− 0.795173i	−0.917565	−	0.397586i	$-0.869848\pi$
$823$	− 22.8119i	− 0.795173i	0.917565	−	0.397586i	$-0.130152\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.8221i	1.24565i	0.782359	+	0.622827i	$0.214016\pi$
$827$	35.8221i	1.24565i	−0.782359	+	0.622827i	$0.785984\pi$
$828$	0	0
$829$	− 26.6357i	− 0.925094i	−0.886595	−	0.462547i	$-0.846936\pi$
$829$	− 26.6357i	− 0.925094i	0.886595	−	0.462547i	$-0.153064\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 6.97095i	− 0.241529i
$834$	0	0
$835$	6.44534	0.223050
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0.200994	0.00693908	0.00346954	−	0.999994i	$-0.498896\pi$
$839$	0.200994	0.00693908	0.00346954	+	0.999994i	$0.498896\pi$
$840$	0	0
$841$	−28.5171	−0.983349
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11.1427	0.383319
$846$	0	0
$847$	6.44358i	0.221404i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14.7219i	0.504659i
$852$	0	0
$853$	− 23.4171i	− 0.801787i	−0.916125	−	0.400893i	$-0.868700\pi$
$853$	− 23.4171i	− 0.801787i	0.916125	−	0.400893i	$-0.131300\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.65764i	0.159102i	0.996831	+	0.0795509i	$0.0253486\pi$
$857$	4.65764i	0.159102i	−0.996831	+	0.0795509i	$0.974651\pi$
$858$	0	0
$859$	−1.56210	−0.0532980	−0.0266490	−	0.999645i	$-0.508484\pi$
$859$	−1.56210	−0.0532980	−0.0266490	+	0.999645i	$0.508484\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	54.8529	1.86721	0.933607	−	0.358300i	$-0.116643\pi$
$863$	54.8529	1.86721	0.933607	+	0.358300i	$0.116643\pi$
$864$	0	0
$865$	−5.15863	−0.175399
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−30.1731	−1.02355
$870$	0	0
$871$	− 4.12180i	− 0.139662i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.75867i	0.127066i
$876$	0	0
$877$	2.53416i	0.0855724i	0.999084	+	0.0427862i	$0.0136234\pi$
$877$	2.53416i	0.0855724i	−0.999084	+	0.0427862i	$0.986377\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 21.8861i	− 0.737361i	−0.929556	−	0.368680i	$-0.879810\pi$
$881$	− 21.8861i	− 0.737361i	0.929556	−	0.368680i	$-0.120190\pi$
$882$	0	0
$883$	−49.1420	−1.65376	−0.826880	−	0.562378i	$-0.809887\pi$
$883$	−49.1420	−1.65376	−0.826880	+	0.562378i	$0.809887\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−21.2007	−0.711849	−0.355924	−	0.934515i	$-0.615834\pi$
$887$	−21.2007	−0.711849	−0.355924	+	0.934515i	$0.615834\pi$
$888$	0	0
$889$	19.8483	0.665690
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−42.2549	−1.41400
$894$	0	0
$895$	0.550600i	0.0184045i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.16581i	0.0388818i
$900$	0	0
$901$	6.65580i	0.221737i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 2.44268i	− 0.0811976i
$906$	0	0
$907$	−15.8245	−0.525443	−0.262722	−	0.964872i	$-0.584620\pi$
$907$	−15.8245	−0.525443	−0.262722	+	0.964872i	$0.584620\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−34.8223	−1.15372	−0.576858	−	0.816845i	$-0.695721\pi$
$911$	−34.8223	−1.15372	−0.576858	+	0.816845i	$0.695721\pi$
$912$	0	0
$913$	7.17502	0.237458
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−16.8474	−0.556351
$918$	0	0
$919$	7.23491i	0.238658i	0.992855	+	0.119329i	$0.0380743\pi$
$919$	7.23491i	0.238658i	−0.992855	+	0.119329i	$0.961926\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	37.6311i	1.23864i
$924$	0	0
$925$	39.1610i	1.28760i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	16.7468i	0.549445i	0.961524	+	0.274722i	$0.0885859\pi$
$929$	16.7468i	0.549445i	−0.961524	+	0.274722i	$0.911414\pi$
$930$	0	0
$931$	6.19245	0.202950
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.67548	−0.185608
$936$	0	0
$937$	48.8065	1.59444	0.797220	−	0.603689i	$-0.206303\pi$
$937$	48.8065	1.59444	0.797220	+	0.603689i	$0.206303\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−19.4343	−0.633540	−0.316770	−	0.948502i	$-0.602598\pi$
$941$	−19.4343	−0.633540	−0.316770	+	0.948502i	$0.602598\pi$
$942$	0	0
$943$	2.39299i	0.0779265i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 31.5298i	− 1.02458i	−0.858812	−	0.512290i	$-0.828797\pi$
$947$	− 31.5298i	− 1.02458i	0.858812	−	0.512290i	$-0.171203\pi$
$948$	0	0
$949$	52.8726i	1.71632i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 37.0260i	− 1.19939i	−0.800228	−	0.599696i	$-0.795288\pi$
$953$	− 37.0260i	− 1.19939i	0.800228	−	0.599696i	$-0.204712\pi$
$954$	0	0
$955$	−0.994004	−0.0321652
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.76935	0.283177
$960$	0	0
$961$	28.1854	0.909205
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.40280	0.141731
$966$	0	0
$967$	29.1641i	0.937854i	0.883237	+	0.468927i	$0.155359\pi$
$967$	29.1641i	0.937854i	−0.883237	+	0.468927i	$0.844641\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 10.8284i	− 0.347500i	−0.984790	−	0.173750i	$-0.944412\pi$
$971$	− 10.8284i	− 0.347500i	0.984790	−	0.173750i	$-0.0555885\pi$
$972$	0	0
$973$	0.920814i	0.0295199i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.12012i	− 0.0678285i	−0.999425	−	0.0339142i	$-0.989203\pi$
$977$	− 2.12012i	− 0.0678285i	0.999425	−	0.0339142i	$-0.0107973\pi$
$978$	0	0
$979$	34.0046	1.08679
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−38.6019	−1.23121	−0.615605	−	0.788055i	$-0.711088\pi$
$983$	−38.6019	−1.23121	−0.615605	+	0.788055i	$0.711088\pi$
$984$	0	0
$985$	5.97748	0.190458
$986$	0	0
$987$	0	0
$988$	0	0
$989$	14.0030	0.445270
$990$	0	0
$991$	61.3104i	1.94759i	0.227431	+	0.973794i	$0.426967\pi$
$991$	61.3104i	1.94759i	−0.227431	+	0.973794i	$0.573033\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.07103i	− 0.0339538i
$996$	0	0
$997$	11.0697i	0.350582i	0.984517	+	0.175291i	$0.0560866\pi$
$997$	11.0697i	0.350582i	−0.984517	+	0.175291i	$0.943913\pi$
$998$	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.5615.24		48
3.2	odd	2	inner	6048.2.j.d.5615.25		48
4.3	odd	2		1512.2.j.d.323.31	yes	48
8.3	odd	2	inner	6048.2.j.d.5615.26		48
8.5	even	2		1512.2.j.d.323.17	✓	48
12.11	even	2		1512.2.j.d.323.18	yes	48
24.5	odd	2		1512.2.j.d.323.32	yes	48
24.11	even	2	inner	6048.2.j.d.5615.23		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.d.323.17	✓	48	8.5	even	2
1512.2.j.d.323.18	yes	48	12.11	even	2
1512.2.j.d.323.31	yes	48	4.3	odd	2
1512.2.j.d.323.32	yes	48	24.5	odd	2
6048.2.j.d.5615.23		48	24.11	even	2	inner
6048.2.j.d.5615.24		48	1.1	even	1	trivial
6048.2.j.d.5615.25		48	3.2	odd	2	inner
6048.2.j.d.5615.26		48	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)

\(f(q)\)	\(=\)	\(q-0.381416 q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q-0.381416 q^{5} +1.00000i q^{7} -2.13458i q^{11} +6.49722i q^{13} +6.97095i q^{17} -6.19245 q^{19} -1.82497 q^{23} -4.85452 q^{25} -0.694888 q^{29} -1.67769i q^{31} -0.381416i q^{35} -8.06690i q^{37} -1.31125i q^{41} -7.67301 q^{43} +6.82360 q^{47} -1.00000 q^{49} +0.954792 q^{53} +0.814162i q^{55} -12.8165i q^{59} -12.4829i q^{61} -2.47815i q^{65} -0.634394 q^{67} +5.79187 q^{71} +8.13772 q^{73} +2.13458 q^{77} -14.1354i q^{79} +3.36133i q^{83} -2.65883i q^{85} +15.9304i q^{89} -6.49722 q^{91} +2.36190 q^{95} +10.8796 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more