LMFDB - Embedded newform 6048.2.j.d.5615.15

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			5615.15
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.d.5615.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.27819 q^{5} -1.00000i q^{7} +O(q^{10})$	$q-1.27819 q^{5} -1.00000i q^{7} +2.84853i q^{11} +0.223683i q^{13} -0.397844i q^{17} +4.64862 q^{19} -7.36177 q^{23} -3.36622 q^{25} +10.6317 q^{29} -7.67558i q^{31} +1.27819i q^{35} -4.74489i q^{37} +1.97843i q^{41} -7.62888 q^{43} -11.0724 q^{47} -1.00000 q^{49} -0.295366 q^{53} -3.64098i q^{55} +7.25779i q^{59} +9.45592i q^{61} -0.285910i q^{65} +3.52321 q^{67} +8.37710 q^{71} +10.6704 q^{73} +2.84853 q^{77} -5.22135i q^{79} -9.11868i q^{83} +0.508521i q^{85} +8.94887i q^{89} +0.223683 q^{91} -5.94184 q^{95} +0.228078 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.27819	−0.571625	−0.285813	−	0.958285i	$-0.592264\pi$
$5$	−1.27819	−0.571625	−0.285813	+	0.958285i	$0.592264\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.84853i	0.858865i	0.903099	+	0.429433i	$0.141286\pi$
$11$	2.84853i	0.858865i	−0.903099	+	0.429433i	$0.858714\pi$
$12$	0	0
$13$	0.223683i	0.0620385i	0.999519	+	0.0310192i	$0.00987531\pi$
$13$	0.223683i	0.0620385i	−0.999519	+	0.0310192i	$0.990125\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.397844i	− 0.0964913i	−0.998836	−	0.0482456i	$-0.984637\pi$
$17$	− 0.397844i	− 0.0964913i	0.998836	−	0.0482456i	$-0.0153630\pi$
$18$	0	0
$19$	4.64862	1.06647	0.533234	−	0.845968i	$-0.320977\pi$
$19$	4.64862	1.06647	0.533234	+	0.845968i	$0.320977\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.36177	−1.53503	−0.767517	−	0.641028i	$-0.778508\pi$
$23$	−7.36177	−1.53503	−0.767517	+	0.641028i	$0.778508\pi$
$24$	0	0
$25$	−3.36622	−0.673244
$26$	0	0
$27$	0	0
$28$	0	0
$29$	10.6317	1.97425	0.987124	−	0.159957i	$-0.0511356\pi$
$29$	10.6317	1.97425	0.987124	+	0.159957i	$0.0511356\pi$
$30$	0	0
$31$	− 7.67558i	− 1.37857i	−0.724488	−	0.689287i	$-0.757924\pi$
$31$	− 7.67558i	− 1.37857i	0.724488	−	0.689287i	$-0.242076\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.27819i	0.216054i
$36$	0	0
$37$	− 4.74489i	− 0.780055i	−0.920803	−	0.390027i	$-0.872466\pi$
$37$	− 4.74489i	− 0.780055i	0.920803	−	0.390027i	$-0.127534\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.97843i	0.308979i	0.987994	+	0.154490i	$0.0493733\pi$
$41$	1.97843i	0.308979i	−0.987994	+	0.154490i	$0.950627\pi$
$42$	0	0
$43$	−7.62888	−1.16339	−0.581697	−	0.813406i	$-0.697611\pi$
$43$	−7.62888	−1.16339	−0.581697	+	0.813406i	$0.697611\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.0724	−1.61508	−0.807538	−	0.589815i	$-0.799201\pi$
$47$	−11.0724	−1.61508	−0.807538	+	0.589815i	$0.799201\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.295366	−0.0405717	−0.0202858	−	0.999794i	$-0.506458\pi$
$53$	−0.295366	−0.0405717	−0.0202858	+	0.999794i	$0.506458\pi$
$54$	0	0
$55$	− 3.64098i	− 0.490949i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.25779i	0.944884i	0.881362	+	0.472442i	$0.156627\pi$
$59$	7.25779i	0.944884i	−0.881362	+	0.472442i	$0.843373\pi$
$60$	0	0
$61$	9.45592i	1.21071i	0.795957	+	0.605354i	$0.206968\pi$
$61$	9.45592i	1.21071i	−0.795957	+	0.605354i	$0.793032\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 0.285910i	− 0.0354628i
$66$	0	0
$67$	3.52321	0.430428	0.215214	−	0.976567i	$-0.430955\pi$
$67$	3.52321	0.430428	0.215214	+	0.976567i	$0.430955\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.37710	0.994179	0.497089	−	0.867699i	$-0.334402\pi$
$71$	8.37710	0.994179	0.497089	+	0.867699i	$0.334402\pi$
$72$	0	0
$73$	10.6704	1.24887	0.624437	−	0.781076i	$-0.285329\pi$
$73$	10.6704	1.24887	0.624437	+	0.781076i	$0.285329\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.84853	0.324621
$78$	0	0
$79$	− 5.22135i	− 0.587448i	−0.955890	−	0.293724i	$-0.905105\pi$
$79$	− 5.22135i	− 0.587448i	0.955890	−	0.293724i	$-0.0948947\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 9.11868i	− 1.00091i	−0.865764	−	0.500453i	$-0.833167\pi$
$83$	− 9.11868i	− 1.00091i	0.865764	−	0.500453i	$-0.166833\pi$
$84$	0	0
$85$	0.508521i	0.0551569i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.94887i	0.948578i	0.880369	+	0.474289i	$0.157295\pi$
$89$	8.94887i	0.948578i	−0.880369	+	0.474289i	$0.842705\pi$
$90$	0	0
$91$	0.223683	0.0234483
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.94184	−0.609620
$96$	0	0
$97$	0.228078	0.0231578	0.0115789	−	0.999933i	$-0.496314\pi$
$97$	0.228078	0.0231578	0.0115789	+	0.999933i	$0.496314\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.36504	0.633346	0.316673	−	0.948535i	$-0.397434\pi$
$101$	6.36504	0.633346	0.316673	+	0.948535i	$0.397434\pi$
$102$	0	0
$103$	− 11.6557i	− 1.14847i	−0.818690	−	0.574236i	$-0.805299\pi$
$103$	− 11.6557i	− 1.14847i	0.818690	−	0.574236i	$-0.194701\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 9.01403i	− 0.871419i	−0.900087	−	0.435709i	$-0.856498\pi$
$107$	− 9.01403i	− 0.871419i	0.900087	−	0.435709i	$-0.143502\pi$
$108$	0	0
$109$	− 5.45206i	− 0.522213i	−0.965310	−	0.261106i	$-0.915913\pi$
$109$	− 5.45206i	− 0.522213i	0.965310	−	0.261106i	$-0.0840873\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 7.00229i	− 0.658720i	−0.944204	−	0.329360i	$-0.893167\pi$
$113$	− 7.00229i	− 0.658720i	0.944204	−	0.329360i	$-0.106833\pi$
$114$	0	0
$115$	9.40976	0.877465
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.397844	−0.0364703
$120$	0	0
$121$	2.88586	0.262351
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.6936	0.956469
$126$	0	0
$127$	− 9.48046i	− 0.841255i	−0.907234	−	0.420627i	$-0.861810\pi$
$127$	− 9.48046i	− 0.841255i	0.907234	−	0.420627i	$-0.138190\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.34169i	0.554076i	0.960859	+	0.277038i	$0.0893528\pi$
$131$	6.34169i	0.554076i	−0.960859	+	0.277038i	$0.910647\pi$
$132$	0	0
$133$	− 4.64862i	− 0.403087i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 11.3108i	− 0.966349i	−0.875524	−	0.483174i	$-0.839484\pi$
$137$	− 11.3108i	− 0.966349i	0.875524	−	0.483174i	$-0.160516\pi$
$138$	0	0
$139$	−21.7583	−1.84551	−0.922757	−	0.385382i	$-0.874070\pi$
$139$	−21.7583	−1.84551	−0.922757	+	0.385382i	$0.874070\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.637168	−0.0532827
$144$	0	0
$145$	−13.5893	−1.12853
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.42505	0.198668	0.0993340	−	0.995054i	$-0.468329\pi$
$149$	2.42505	0.198668	0.0993340	+	0.995054i	$0.468329\pi$
$150$	0	0
$151$	− 20.0398i	− 1.63082i	−0.578887	−	0.815408i	$-0.696513\pi$
$151$	− 20.0398i	− 1.63082i	0.578887	−	0.815408i	$-0.303487\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.81087i	0.788028i
$156$	0	0
$157$	− 15.5990i	− 1.24494i	−0.782644	−	0.622470i	$-0.786129\pi$
$157$	− 15.5990i	− 1.24494i	0.782644	−	0.622470i	$-0.213871\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.36177i	0.580189i
$162$	0	0
$163$	−12.8755	−1.00848	−0.504242	−	0.863562i	$-0.668228\pi$
$163$	−12.8755	−1.00848	−0.504242	+	0.863562i	$0.668228\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.55771	−0.120539	−0.0602697	−	0.998182i	$-0.519196\pi$
$167$	−1.55771	−0.120539	−0.0602697	+	0.998182i	$0.519196\pi$
$168$	0	0
$169$	12.9500	0.996151
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.60553	0.198095	0.0990474	−	0.995083i	$-0.468420\pi$
$173$	2.60553	0.198095	0.0990474	+	0.995083i	$0.468420\pi$
$174$	0	0
$175$	3.36622i	0.254462i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.9293i	1.41484i	0.706793	+	0.707420i	$0.250141\pi$
$179$	18.9293i	1.41484i	−0.706793	+	0.707420i	$0.749859\pi$
$180$	0	0
$181$	− 15.6813i	− 1.16558i	−0.812621	−	0.582792i	$-0.801960\pi$
$181$	− 15.6813i	− 1.16558i	0.812621	−	0.582792i	$-0.198040\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.06488i	0.445899i
$186$	0	0
$187$	1.13327	0.0828730
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.86888	−0.279943	−0.139971	−	0.990156i	$-0.544701\pi$
$191$	−3.86888	−0.279943	−0.139971	+	0.990156i	$0.544701\pi$
$192$	0	0
$193$	−12.1061	−0.871419	−0.435710	−	0.900087i	$-0.643503\pi$
$193$	−12.1061	−0.871419	−0.435710	+	0.900087i	$0.643503\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.20058	−0.370526	−0.185263	−	0.982689i	$-0.559314\pi$
$197$	−5.20058	−0.370526	−0.185263	+	0.982689i	$0.559314\pi$
$198$	0	0
$199$	15.5030i	1.09898i	0.835501	+	0.549489i	$0.185177\pi$
$199$	15.5030i	1.09898i	−0.835501	+	0.549489i	$0.814823\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 10.6317i	− 0.746196i
$204$	0	0
$205$	− 2.52882i	− 0.176620i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	13.2418i	0.915952i
$210$	0	0
$211$	9.04096	0.622406	0.311203	−	0.950344i	$-0.399268\pi$
$211$	9.04096	0.622406	0.311203	+	0.950344i	$0.399268\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.75118	0.665025
$216$	0	0
$217$	−7.67558	−0.521052
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.0889909	0.00598617
$222$	0	0
$223$	− 10.2327i	− 0.685231i	−0.939476	−	0.342616i	$-0.888687\pi$
$223$	− 10.2327i	− 0.685231i	0.939476	−	0.342616i	$-0.111313\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.9987i	− 0.730007i	−0.931006	−	0.365003i	$-0.881068\pi$
$227$	− 10.9987i	− 0.730007i	0.931006	−	0.365003i	$-0.118932\pi$
$228$	0	0
$229$	− 24.1988i	− 1.59910i	−0.600600	−	0.799550i	$-0.705072\pi$
$229$	− 24.1988i	− 1.59910i	0.600600	−	0.799550i	$-0.294928\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 21.5448i	− 1.41144i	−0.708489	−	0.705722i	$-0.750623\pi$
$233$	− 21.5448i	− 1.41144i	0.708489	−	0.705722i	$-0.249377\pi$
$234$	0	0
$235$	14.1527	0.923219
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.5708	−1.33061	−0.665307	−	0.746570i	$-0.731699\pi$
$239$	−20.5708	−1.33061	−0.665307	+	0.746570i	$0.731699\pi$
$240$	0	0
$241$	−11.0840	−0.713984	−0.356992	−	0.934107i	$-0.616198\pi$
$241$	−11.0840	−0.713984	−0.356992	+	0.934107i	$0.616198\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.27819	0.0816608
$246$	0	0
$247$	1.03982i	0.0661620i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 20.7016i	− 1.30667i	−0.757068	−	0.653336i	$-0.773369\pi$
$251$	− 20.7016i	− 1.30667i	0.757068	−	0.653336i	$-0.226631\pi$
$252$	0	0
$253$	− 20.9702i	− 1.31839i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 10.6516i	− 0.664427i	−0.943204	−	0.332213i	$-0.892205\pi$
$257$	− 10.6516i	− 0.664427i	0.943204	−	0.332213i	$-0.107795\pi$
$258$	0	0
$259$	−4.74489	−0.294833
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.36313	0.392367	0.196184	−	0.980567i	$-0.437145\pi$
$263$	6.36313	0.392367	0.196184	+	0.980567i	$0.437145\pi$
$264$	0	0
$265$	0.377535	0.0231918
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.3840	−0.877007	−0.438503	−	0.898730i	$-0.644491\pi$
$269$	−14.3840	−0.877007	−0.438503	+	0.898730i	$0.644491\pi$
$270$	0	0
$271$	− 12.9317i	− 0.785548i	−0.919635	−	0.392774i	$-0.871515\pi$
$271$	− 12.9317i	− 0.785548i	0.919635	−	0.392774i	$-0.128485\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 9.58880i	− 0.578226i
$276$	0	0
$277$	− 13.5242i	− 0.812593i	−0.913741	−	0.406297i	$-0.866820\pi$
$277$	− 13.5242i	− 0.812593i	0.913741	−	0.406297i	$-0.133180\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 20.7882i	− 1.24012i	−0.784554	−	0.620061i	$-0.787108\pi$
$281$	− 20.7882i	− 1.24012i	0.784554	−	0.620061i	$-0.212892\pi$
$282$	0	0
$283$	17.6119	1.04692	0.523459	−	0.852051i	$-0.324641\pi$
$283$	17.6119	1.04692	0.523459	+	0.852051i	$0.324641\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.97843	0.116783
$288$	0	0
$289$	16.8417	0.990689
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.42581	−0.141718	−0.0708588	−	0.997486i	$-0.522574\pi$
$293$	−2.42581	−0.141718	−0.0708588	+	0.997486i	$0.522574\pi$
$294$	0	0
$295$	− 9.27687i	− 0.540120i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 1.64670i	− 0.0952312i
$300$	0	0
$301$	7.62888i	0.439721i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 12.0865i	− 0.692071i
$306$	0	0
$307$	5.15871	0.294423	0.147212	−	0.989105i	$-0.452970\pi$
$307$	5.15871	0.294423	0.147212	+	0.989105i	$0.452970\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−31.1304	−1.76524	−0.882620	−	0.470086i	$-0.844223\pi$
$311$	−31.1304	−1.76524	−0.882620	+	0.470086i	$0.844223\pi$
$312$	0	0
$313$	−21.8080	−1.23266	−0.616330	−	0.787488i	$-0.711381\pi$
$313$	−21.8080	−1.23266	−0.616330	+	0.787488i	$0.711381\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.309869	−0.0174040	−0.00870199	−	0.999962i	$-0.502770\pi$
$317$	−0.309869	−0.0174040	−0.00870199	+	0.999962i	$0.502770\pi$
$318$	0	0
$319$	30.2846i	1.69561i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1.84943i	− 0.102905i
$324$	0	0
$325$	− 0.752966i	− 0.0417670i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11.0724i	0.610441i
$330$	0	0
$331$	−28.5893	−1.57141	−0.785706	−	0.618600i	$-0.787700\pi$
$331$	−28.5893	−1.57141	−0.785706	+	0.618600i	$0.787700\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.50334	−0.246044
$336$	0	0
$337$	26.7103	1.45500	0.727502	−	0.686106i	$-0.240681\pi$
$337$	26.7103	1.45500	0.727502	+	0.686106i	$0.240681\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	21.8641	1.18401
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.0302i	0.592132i	0.955168	+	0.296066i	$0.0956748\pi$
$347$	11.0302i	0.592132i	−0.955168	+	0.296066i	$0.904325\pi$
$348$	0	0
$349$	9.30132i	0.497888i	0.968518	+	0.248944i	$0.0800836\pi$
$349$	9.30132i	0.497888i	−0.968518	+	0.248944i	$0.919916\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.4585i	0.982446i	0.871034	+	0.491223i	$0.163450\pi$
$353$	18.4585i	0.982446i	−0.871034	+	0.491223i	$0.836550\pi$
$354$	0	0
$355$	−10.7076	−0.568298
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.70098	0.406442	0.203221	−	0.979133i	$-0.434859\pi$
$359$	7.70098	0.406442	0.203221	+	0.979133i	$0.434859\pi$
$360$	0	0
$361$	2.60971	0.137353
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−13.6388	−0.713888
$366$	0	0
$367$	− 36.0127i	− 1.87985i	−0.341386	−	0.939923i	$-0.610896\pi$
$367$	− 36.0127i	− 1.87985i	0.341386	−	0.939923i	$-0.389104\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.295366i	0.0153346i
$372$	0	0
$373$	− 32.8248i	− 1.69960i	−0.527103	−	0.849801i	$-0.676722\pi$
$373$	− 32.8248i	− 1.69960i	0.527103	−	0.849801i	$-0.323278\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.37812i	0.122479i
$378$	0	0
$379$	27.8200	1.42902	0.714509	−	0.699626i	$-0.246650\pi$
$379$	27.8200	1.42902	0.714509	+	0.699626i	$0.246650\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.9047	−0.608301	−0.304151	−	0.952624i	$-0.598373\pi$
$383$	−11.9047	−0.608301	−0.304151	+	0.952624i	$0.598373\pi$
$384$	0	0
$385$	−3.64098	−0.185561
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.8388	1.25938	0.629688	−	0.776848i	$-0.283183\pi$
$389$	24.8388	1.25938	0.629688	+	0.776848i	$0.283183\pi$
$390$	0	0
$391$	2.92883i	0.148117i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.67390i	0.335800i
$396$	0	0
$397$	− 4.49543i	− 0.225619i	−0.993617	−	0.112810i	$-0.964015\pi$
$397$	− 4.49543i	− 0.225619i	0.993617	−	0.112810i	$-0.0359850\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.8094i	0.739545i	0.929122	+	0.369772i	$0.120564\pi$
$401$	14.8094i	0.739545i	−0.929122	+	0.369772i	$0.879436\pi$
$402$	0	0
$403$	1.71689	0.0855246
$404$	0	0
$405$	0	0
$406$	0	0
$407$	13.5160	0.669962
$408$	0	0
$409$	−33.4088	−1.65196	−0.825980	−	0.563699i	$-0.809378\pi$
$409$	−33.4088	−1.65196	−0.825980	+	0.563699i	$0.809378\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.25779	0.357133
$414$	0	0
$415$	11.6554i	0.572143i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.31558i	0.308536i	0.988029	+	0.154268i	$0.0493020\pi$
$419$	6.31558i	0.308536i	−0.988029	+	0.154268i	$0.950698\pi$
$420$	0	0
$421$	10.6716i	0.520104i	0.965595	+	0.260052i	$0.0837397\pi$
$421$	10.6716i	0.520104i	−0.965595	+	0.260052i	$0.916260\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.33923i	0.0649622i
$426$	0	0
$427$	9.45592	0.457604
$428$	0	0
$429$	0	0
$430$	0	0
$431$	35.8686	1.72773	0.863865	−	0.503723i	$-0.168037\pi$
$431$	35.8686	1.72773	0.863865	+	0.503723i	$0.168037\pi$
$432$	0	0
$433$	4.43908	0.213328	0.106664	−	0.994295i	$-0.465983\pi$
$433$	4.43908	0.213328	0.106664	+	0.994295i	$0.465983\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−34.2221	−1.63706
$438$	0	0
$439$	− 18.7939i	− 0.896985i	−0.893787	−	0.448493i	$-0.851961\pi$
$439$	− 18.7939i	− 0.896985i	0.893787	−	0.448493i	$-0.148039\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	30.9630i	1.47109i	0.677473	+	0.735547i	$0.263075\pi$
$443$	30.9630i	1.47109i	−0.677473	+	0.735547i	$0.736925\pi$
$444$	0	0
$445$	− 11.4384i	− 0.542232i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	24.6744i	1.16446i	0.813025	+	0.582229i	$0.197819\pi$
$449$	24.6744i	1.16446i	−0.813025	+	0.582229i	$0.802181\pi$
$450$	0	0
$451$	−5.63563	−0.265372
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.285910	−0.0134037
$456$	0	0
$457$	13.3915	0.626429	0.313214	−	0.949682i	$-0.398594\pi$
$457$	13.3915	0.626429	0.313214	+	0.949682i	$0.398594\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.3835	0.576759	0.288379	−	0.957516i	$-0.406884\pi$
$461$	12.3835	0.576759	0.288379	+	0.957516i	$0.406884\pi$
$462$	0	0
$463$	− 3.34707i	− 0.155551i	−0.996971	−	0.0777757i	$-0.975218\pi$
$463$	− 3.34707i	− 0.155551i	0.996971	−	0.0777757i	$-0.0247818\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.05078i	− 0.141173i	−0.997506	−	0.0705866i	$-0.977513\pi$
$467$	− 3.05078i	− 0.141173i	0.997506	−	0.0705866i	$-0.0224871\pi$
$468$	0	0
$469$	− 3.52321i	− 0.162687i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 21.7311i	− 0.999198i
$474$	0	0
$475$	−15.6483	−0.717993
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.4934	−0.936368	−0.468184	−	0.883631i	$-0.655092\pi$
$479$	−20.4934	−0.936368	−0.468184	+	0.883631i	$0.655092\pi$
$480$	0	0
$481$	1.06135	0.0483934
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.291528	−0.0132376
$486$	0	0
$487$	− 21.9186i	− 0.993226i	−0.867972	−	0.496613i	$-0.834577\pi$
$487$	− 21.9186i	− 0.993226i	0.867972	−	0.496613i	$-0.165423\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 10.2880i	− 0.464291i	−0.972681	−	0.232146i	$-0.925425\pi$
$491$	− 10.2880i	− 0.464291i	0.972681	−	0.232146i	$-0.0745746\pi$
$492$	0	0
$493$	− 4.22974i	− 0.190498i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.37710i	− 0.375764i
$498$	0	0
$499$	1.87931	0.0841294	0.0420647	−	0.999115i	$-0.486606\pi$
$499$	1.87931	0.0841294	0.0420647	+	0.999115i	$0.486606\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14.5489	0.648702	0.324351	−	0.945937i	$-0.394854\pi$
$503$	14.5489	0.648702	0.324351	+	0.945937i	$0.394854\pi$
$504$	0	0
$505$	−8.13576	−0.362036
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−25.7122	−1.13967	−0.569837	−	0.821758i	$-0.692994\pi$
$509$	−25.7122	−1.13967	−0.569837	+	0.821758i	$0.692994\pi$
$510$	0	0
$511$	− 10.6704i	− 0.472030i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	14.8983i	0.656496i
$516$	0	0
$517$	− 31.5401i	− 1.38713i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 8.03293i	− 0.351929i	−0.984397	−	0.175965i	$-0.943696\pi$
$521$	− 8.03293i	− 0.351929i	0.984397	−	0.175965i	$-0.0563044\pi$
$522$	0	0
$523$	−17.2938	−0.756206	−0.378103	−	0.925764i	$-0.623423\pi$
$523$	−17.2938	−0.756206	−0.378103	+	0.925764i	$0.623423\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.05368	−0.133020
$528$	0	0
$529$	31.1956	1.35633
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.442542	−0.0191686
$534$	0	0
$535$	11.5217i	0.498125i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.84853i	− 0.122695i
$540$	0	0
$541$	− 6.58504i	− 0.283113i	−0.989930	−	0.141557i	$-0.954789\pi$
$541$	− 6.58504i	− 0.283113i	0.989930	−	0.141557i	$-0.0452107\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.96879i	0.298510i
$546$	0	0
$547$	33.4478	1.43012	0.715062	−	0.699061i	$-0.246398\pi$
$547$	33.4478	1.43012	0.715062	+	0.699061i	$0.246398\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	49.4226	2.10547
$552$	0	0
$553$	−5.22135	−0.222035
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.92223	0.250933	0.125467	−	0.992098i	$-0.459957\pi$
$557$	5.92223	0.250933	0.125467	+	0.992098i	$0.459957\pi$
$558$	0	0
$559$	− 1.70645i	− 0.0721751i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 23.1453i	− 0.975459i	−0.872995	−	0.487730i	$-0.837825\pi$
$563$	− 23.1453i	− 0.975459i	0.872995	−	0.487730i	$-0.162175\pi$
$564$	0	0
$565$	8.95028i	0.376541i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.41139i	0.0591686i	0.999562	+	0.0295843i	$0.00941835\pi$
$569$	1.41139i	0.0591686i	−0.999562	+	0.0295843i	$0.990582\pi$
$570$	0	0
$571$	6.82194	0.285489	0.142745	−	0.989760i	$-0.454407\pi$
$571$	6.82194	0.285489	0.142745	+	0.989760i	$0.454407\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	24.7813	1.03345
$576$	0	0
$577$	6.97587	0.290409	0.145205	−	0.989402i	$-0.453616\pi$
$577$	6.97587	0.290409	0.145205	+	0.989402i	$0.453616\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.11868	−0.378307
$582$	0	0
$583$	− 0.841360i	− 0.0348456i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 40.3481i	− 1.66534i	−0.553766	−	0.832672i	$-0.686810\pi$
$587$	− 40.3481i	− 1.66534i	0.553766	−	0.832672i	$-0.313190\pi$
$588$	0	0
$589$	− 35.6809i	− 1.47020i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 6.40544i	− 0.263040i	−0.991314	−	0.131520i	$-0.958014\pi$
$593$	− 6.40544i	− 0.263040i	0.991314	−	0.131520i	$-0.0419858\pi$
$594$	0	0
$595$	0.508521	0.0208473
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.80603	0.114651	0.0573257	−	0.998356i	$-0.481743\pi$
$599$	2.80603	0.114651	0.0573257	+	0.998356i	$0.481743\pi$
$600$	0	0
$601$	11.6165	0.473847	0.236924	−	0.971528i	$-0.423861\pi$
$601$	11.6165	0.473847	0.236924	+	0.971528i	$0.423861\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.68868	−0.149966
$606$	0	0
$607$	38.5388i	1.56424i	0.623126	+	0.782122i	$0.285862\pi$
$607$	38.5388i	1.56424i	−0.623126	+	0.782122i	$0.714138\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 2.47671i	− 0.100197i
$612$	0	0
$613$	15.8274i	0.639261i	0.947542	+	0.319631i	$0.103559\pi$
$613$	15.8274i	0.639261i	−0.947542	+	0.319631i	$0.896441\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23.9040i	0.962339i	0.876628	+	0.481170i	$0.159788\pi$
$617$	23.9040i	0.962339i	−0.876628	+	0.481170i	$0.840212\pi$
$618$	0	0
$619$	31.4193	1.26285	0.631424	−	0.775437i	$-0.282471\pi$
$619$	31.4193	1.26285	0.631424	+	0.775437i	$0.282471\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.94887	0.358529
$624$	0	0
$625$	3.16256	0.126502
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.88772	−0.0752685
$630$	0	0
$631$	6.94219i	0.276364i	0.990407	+	0.138182i	$0.0441260\pi$
$631$	6.94219i	0.276364i	−0.990407	+	0.138182i	$0.955874\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.1179i	0.480883i
$636$	0	0
$637$	− 0.223683i	− 0.00886264i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 31.1823i	− 1.23163i	−0.787892	−	0.615814i	$-0.788827\pi$
$641$	− 31.1823i	− 1.23163i	0.787892	−	0.615814i	$-0.211173\pi$
$642$	0	0
$643$	−22.6881	−0.894733	−0.447366	−	0.894351i	$-0.647638\pi$
$643$	−22.6881	−0.894733	−0.447366	+	0.894351i	$0.647638\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	37.7608	1.48453	0.742266	−	0.670106i	$-0.233751\pi$
$647$	37.7608	1.48453	0.742266	+	0.670106i	$0.233751\pi$
$648$	0	0
$649$	−20.6741	−0.811528
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−47.8848	−1.87388	−0.936939	−	0.349494i	$-0.886353\pi$
$653$	−47.8848	−1.87388	−0.936939	+	0.349494i	$0.886353\pi$
$654$	0	0
$655$	− 8.10591i	− 0.316724i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 7.00120i	− 0.272728i	−0.990659	−	0.136364i	$-0.956458\pi$
$659$	− 7.00120i	− 0.272728i	0.990659	−	0.136364i	$-0.0435417\pi$
$660$	0	0
$661$	41.8042i	1.62599i	0.582267	+	0.812997i	$0.302166\pi$
$661$	41.8042i	1.62599i	−0.582267	+	0.812997i	$0.697834\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.94184i	0.230415i
$666$	0	0
$667$	−78.2677	−3.03054
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−26.9355	−1.03983
$672$	0	0
$673$	22.2519	0.857747	0.428874	−	0.903364i	$-0.358911\pi$
$673$	22.2519	0.857747	0.428874	+	0.903364i	$0.358911\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−29.2219	−1.12309	−0.561544	−	0.827447i	$-0.689793\pi$
$677$	−29.2219	−1.12309	−0.561544	+	0.827447i	$0.689793\pi$
$678$	0	0
$679$	− 0.228078i	− 0.00875283i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	35.0080i	1.33954i	0.742567	+	0.669772i	$0.233608\pi$
$683$	35.0080i	1.33954i	−0.742567	+	0.669772i	$0.766392\pi$
$684$	0	0
$685$	14.4574i	0.552390i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 0.0660684i	− 0.00251700i
$690$	0	0
$691$	−1.33849	−0.0509185	−0.0254593	−	0.999676i	$-0.508105\pi$
$691$	−1.33849	−0.0509185	−0.0254593	+	0.999676i	$0.508105\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	27.8113	1.05494
$696$	0	0
$697$	0.787107	0.0298138
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23.0621	0.871044	0.435522	−	0.900178i	$-0.356564\pi$
$701$	23.0621	0.871044	0.435522	+	0.900178i	$0.356564\pi$
$702$	0	0
$703$	− 22.0572i	− 0.831903i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 6.36504i	− 0.239382i
$708$	0	0
$709$	46.8150i	1.75818i	0.476660	+	0.879088i	$0.341847\pi$
$709$	46.8150i	1.75818i	−0.476660	+	0.879088i	$0.658153\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	56.5058i	2.11616i
$714$	0	0
$715$	0.814424	0.0304577
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−34.8849	−1.30099	−0.650493	−	0.759512i	$-0.725438\pi$
$719$	−34.8849	−1.30099	−0.650493	+	0.759512i	$0.725438\pi$
$720$	0	0
$721$	−11.6557	−0.434082
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−35.7885	−1.32915
$726$	0	0
$727$	16.4491i	0.610063i	0.952342	+	0.305032i	$0.0986671\pi$
$727$	16.4491i	0.610063i	−0.952342	+	0.305032i	$0.901333\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.03510i	0.112257i
$732$	0	0
$733$	35.9855i	1.32915i	0.747220	+	0.664577i	$0.231388\pi$
$733$	35.9855i	1.32915i	−0.747220	+	0.664577i	$0.768612\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.0360i	0.369680i
$738$	0	0
$739$	5.44866	0.200432	0.100216	−	0.994966i	$-0.468047\pi$
$739$	5.44866	0.200432	0.100216	+	0.994966i	$0.468047\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.2115	1.14504	0.572520	−	0.819891i	$-0.305966\pi$
$743$	31.2115	1.14504	0.572520	+	0.819891i	$0.305966\pi$
$744$	0	0
$745$	−3.09969	−0.113564
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.01403	−0.329365
$750$	0	0
$751$	− 23.1812i	− 0.845894i	−0.906154	−	0.422947i	$-0.860996\pi$
$751$	− 23.1812i	− 0.845894i	0.906154	−	0.422947i	$-0.139004\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	25.6147i	0.932216i
$756$	0	0
$757$	39.3962i	1.43188i	0.698162	+	0.715940i	$0.254002\pi$
$757$	39.3962i	1.43188i	−0.698162	+	0.715940i	$0.745998\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 17.1356i	− 0.621166i	−0.950546	−	0.310583i	$-0.899476\pi$
$761$	− 17.1356i	− 0.621166i	0.950546	−	0.310583i	$-0.100524\pi$
$762$	0	0
$763$	−5.45206	−0.197378
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.62344	−0.0586192
$768$	0	0
$769$	−44.9084	−1.61944	−0.809718	−	0.586819i	$-0.800380\pi$
$769$	−44.9084	−1.61944	−0.809718	+	0.586819i	$0.800380\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	27.0273	0.972106	0.486053	−	0.873929i	$-0.338436\pi$
$773$	27.0273	0.972106	0.486053	+	0.873929i	$0.338436\pi$
$774$	0	0
$775$	25.8377i	0.928117i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.19699i	0.329516i
$780$	0	0
$781$	23.8624i	0.853866i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	19.9386i	0.711639i
$786$	0	0
$787$	34.1562	1.21754	0.608769	−	0.793348i	$-0.291664\pi$
$787$	34.1562	1.21754	0.608769	+	0.793348i	$0.291664\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.00229	−0.248973
$792$	0	0
$793$	−2.11513	−0.0751104
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−46.2712	−1.63901	−0.819505	−	0.573072i	$-0.805751\pi$
$797$	−46.2712	−1.63901	−0.819505	+	0.573072i	$0.805751\pi$
$798$	0	0
$799$	4.40509i	0.155841i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	30.3949i	1.07261i
$804$	0	0
$805$	− 9.40976i	− 0.331651i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 26.3429i	− 0.926167i	−0.886315	−	0.463084i	$-0.846743\pi$
$809$	− 26.3429i	− 0.926167i	0.886315	−	0.463084i	$-0.153257\pi$
$810$	0	0
$811$	−2.96401	−0.104080	−0.0520402	−	0.998645i	$-0.516572\pi$
$811$	−2.96401	−0.104080	−0.0520402	+	0.998645i	$0.516572\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	16.4573	0.576475
$816$	0	0
$817$	−35.4638	−1.24072
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.3996	−0.362949	−0.181475	−	0.983396i	$-0.558087\pi$
$821$	−10.3996	−0.362949	−0.181475	+	0.983396i	$0.558087\pi$
$822$	0	0
$823$	− 39.8598i	− 1.38942i	−0.719288	−	0.694712i	$-0.755532\pi$
$823$	− 39.8598i	− 1.38942i	0.719288	−	0.694712i	$-0.244468\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 27.5060i	− 0.956477i	−0.878230	−	0.478239i	$-0.841275\pi$
$827$	− 27.5060i	− 0.956477i	0.878230	−	0.478239i	$-0.158725\pi$
$828$	0	0
$829$	− 38.0086i	− 1.32009i	−0.751225	−	0.660047i	$-0.770536\pi$
$829$	− 38.0086i	− 1.32009i	0.751225	−	0.660047i	$-0.229464\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.397844i	0.0137845i
$834$	0	0
$835$	1.99106	0.0689034
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30.9037	1.06691	0.533457	−	0.845827i	$-0.320893\pi$
$839$	30.9037	1.06691	0.533457	+	0.845827i	$0.320893\pi$
$840$	0	0
$841$	84.0320	2.89765
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−16.5526	−0.569425
$846$	0	0
$847$	− 2.88586i	− 0.0991592i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	34.9308i	1.19741i
$852$	0	0
$853$	25.3302i	0.867290i	0.901084	+	0.433645i	$0.142773\pi$
$853$	25.3302i	0.867290i	−0.901084	+	0.433645i	$0.857227\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.1574i	1.33759i	0.743447	+	0.668795i	$0.233190\pi$
$857$	39.1574i	1.33759i	−0.743447	+	0.668795i	$0.766810\pi$
$858$	0	0
$859$	−11.2783	−0.384810	−0.192405	−	0.981316i	$-0.561629\pi$
$859$	−11.2783	−0.384810	−0.192405	+	0.981316i	$0.561629\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−15.6543	−0.532878	−0.266439	−	0.963852i	$-0.585847\pi$
$863$	−15.6543	−0.532878	−0.266439	+	0.963852i	$0.585847\pi$
$864$	0	0
$865$	−3.33037	−0.113236
$866$	0	0
$867$	0	0
$868$	0	0
$869$	14.8732	0.504539
$870$	0	0
$871$	0.788081i	0.0267031i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 10.6936i	− 0.361511i
$876$	0	0
$877$	14.6357i	0.494211i	0.968989	+	0.247106i	$0.0794795\pi$
$877$	14.6357i	0.494211i	−0.968989	+	0.247106i	$0.920521\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	21.9054i	0.738013i	0.929427	+	0.369007i	$0.120302\pi$
$881$	21.9054i	0.738013i	−0.929427	+	0.369007i	$0.879698\pi$
$882$	0	0
$883$	36.3852	1.22446	0.612230	−	0.790680i	$-0.290273\pi$
$883$	36.3852	1.22446	0.612230	+	0.790680i	$0.290273\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−10.3554	−0.347700	−0.173850	−	0.984772i	$-0.555621\pi$
$887$	−10.3554	−0.347700	−0.173850	+	0.984772i	$0.555621\pi$
$888$	0	0
$889$	−9.48046	−0.317964
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−51.4715	−1.72243
$894$	0	0
$895$	− 24.1953i	− 0.808759i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 81.6040i	− 2.72165i
$900$	0	0
$901$	0.117510i	0.00391481i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.0438i	0.666278i
$906$	0	0
$907$	−10.5615	−0.350689	−0.175344	−	0.984507i	$-0.556104\pi$
$907$	−10.5615	−0.350689	−0.175344	+	0.984507i	$0.556104\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−18.4732	−0.612044	−0.306022	−	0.952024i	$-0.598998\pi$
$911$	−18.4732	−0.612044	−0.306022	+	0.952024i	$0.598998\pi$
$912$	0	0
$913$	25.9749	0.859643
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.34169	0.209421
$918$	0	0
$919$	− 27.7252i	− 0.914569i	−0.889320	−	0.457285i	$-0.848822\pi$
$919$	− 27.7252i	− 0.914569i	0.889320	−	0.457285i	$-0.151178\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.87381i	0.0616773i
$924$	0	0
$925$	15.9723i	0.525167i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 57.6318i	− 1.89084i	−0.325858	−	0.945419i	$-0.605653\pi$
$929$	− 57.6318i	− 1.89084i	0.325858	−	0.945419i	$-0.394347\pi$
$930$	0	0
$931$	−4.64862	−0.152353
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.44854	−0.0473723
$936$	0	0
$937$	−44.9081	−1.46708	−0.733542	−	0.679644i	$-0.762134\pi$
$937$	−44.9081	−1.46708	−0.733542	+	0.679644i	$0.762134\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	25.0167	0.815521	0.407761	−	0.913089i	$-0.366310\pi$
$941$	25.0167	0.815521	0.407761	+	0.913089i	$0.366310\pi$
$942$	0	0
$943$	− 14.5648i	− 0.474294i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.7909i	1.29303i	0.762901	+	0.646516i	$0.223775\pi$
$947$	39.7909i	1.29303i	−0.762901	+	0.646516i	$0.776225\pi$
$948$	0	0
$949$	2.38678i	0.0774782i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12.4544i	0.403437i	0.979444	+	0.201718i	$0.0646526\pi$
$953$	12.4544i	0.403437i	−0.979444	+	0.201718i	$0.935347\pi$
$954$	0	0
$955$	4.94518	0.160022
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11.3108	−0.365246
$960$	0	0
$961$	−27.9145	−0.900466
$962$	0	0
$963$	0	0
$964$	0	0
$965$	15.4740	0.498125
$966$	0	0
$967$	6.97106i	0.224174i	0.993698	+	0.112087i	$0.0357536\pi$
$967$	6.97106i	0.224174i	−0.993698	+	0.112087i	$0.964246\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	9.18374i	0.294720i	0.989083	+	0.147360i	$0.0470776\pi$
$971$	9.18374i	0.294720i	−0.989083	+	0.147360i	$0.952922\pi$
$972$	0	0
$973$	21.7583i	0.697539i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 39.1316i	− 1.25193i	−0.779851	−	0.625965i	$-0.784705\pi$
$977$	− 39.1316i	− 1.25193i	0.779851	−	0.625965i	$-0.215295\pi$
$978$	0	0
$979$	−25.4912	−0.814701
$980$	0	0
$981$	0	0
$982$	0	0
$983$	50.4121	1.60790	0.803948	−	0.594700i	$-0.202729\pi$
$983$	50.4121	1.60790	0.803948	+	0.594700i	$0.202729\pi$
$984$	0	0
$985$	6.64735	0.211802
$986$	0	0
$987$	0	0
$988$	0	0
$989$	56.1620	1.78585
$990$	0	0
$991$	55.9071i	1.77595i	0.459894	+	0.887974i	$0.347887\pi$
$991$	55.9071i	1.77595i	−0.459894	+	0.887974i	$0.652113\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 19.8158i	− 0.628204i
$996$	0	0
$997$	11.6669i	0.369494i	0.982786	+	0.184747i	$0.0591466\pi$
$997$	11.6669i	0.369494i	−0.982786	+	0.184747i	$0.940853\pi$
$998$	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.5615.15		48
3.2	odd	2	inner	6048.2.j.d.5615.34		48
4.3	odd	2		1512.2.j.d.323.24	yes	48
8.3	odd	2	inner	6048.2.j.d.5615.33		48
8.5	even	2		1512.2.j.d.323.26	yes	48
12.11	even	2		1512.2.j.d.323.25	yes	48
24.5	odd	2		1512.2.j.d.323.23	✓	48
24.11	even	2	inner	6048.2.j.d.5615.16		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.d.323.23	✓	48	24.5	odd	2
1512.2.j.d.323.24	yes	48	4.3	odd	2
1512.2.j.d.323.25	yes	48	12.11	even	2
1512.2.j.d.323.26	yes	48	8.5	even	2
6048.2.j.d.5615.15		48	1.1	even	1	trivial
6048.2.j.d.5615.16		48	24.11	even	2	inner
6048.2.j.d.5615.33		48	8.3	odd	2	inner
6048.2.j.d.5615.34		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)

\(f(q)\)	\(=\)	\(q-1.27819 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-1.27819 q^{5} -1.00000i q^{7} +2.84853i q^{11} +0.223683i q^{13} -0.397844i q^{17} +4.64862 q^{19} -7.36177 q^{23} -3.36622 q^{25} +10.6317 q^{29} -7.67558i q^{31} +1.27819i q^{35} -4.74489i q^{37} +1.97843i q^{41} -7.62888 q^{43} -11.0724 q^{47} -1.00000 q^{49} -0.295366 q^{53} -3.64098i q^{55} +7.25779i q^{59} +9.45592i q^{61} -0.285910i q^{65} +3.52321 q^{67} +8.37710 q^{71} +10.6704 q^{73} +2.84853 q^{77} -5.22135i q^{79} -9.11868i q^{83} +0.508521i q^{85} +8.94887i q^{89} +0.223683 q^{91} -5.94184 q^{95} +0.228078 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 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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