LMFDB - Embedded newform 6048.2.j.c.5615.31

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(5615,6048)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))

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

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

chi := DirichletCharacter("6048.5615");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5615.31
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.c.5615.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.61017 q^{5} -1.00000i q^{7} +O(q^{10})$	$q+3.61017 q^{5} -1.00000i q^{7} +2.00899i q^{11} +1.59265i q^{13} +7.79257i q^{17} +5.05480 q^{19} -7.14612 q^{23} +8.03333 q^{25} +4.94298 q^{29} +2.53252i q^{31} -3.61017i q^{35} +4.76592i q^{37} -0.836835i q^{41} -0.151434 q^{43} -0.795430 q^{47} -1.00000 q^{49} +1.05980 q^{53} +7.25280i q^{55} -8.34506i q^{59} +10.2367i q^{61} +5.74974i q^{65} -0.655357 q^{67} -11.3675 q^{71} -14.3835 q^{73} +2.00899 q^{77} -3.81177i q^{79} +14.9522i q^{83} +28.1325i q^{85} -0.0444167i q^{89} +1.59265 q^{91} +18.2487 q^{95} +9.87000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 64 q^{19} - 16 q^{25} + 48 q^{43} - 32 q^{49} - 16 q^{67} - 16 q^{73} - 16 q^{91} + 64 q^{97}+O(q^{100}) $ 32 * q + 64 * q^19 - 16 * q^25 + 48 * q^43 - 32 * q^49 - 16 * q^67 - 16 * q^73 - 16 * q^91 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.61017	1.61452	0.807259	−	0.590198i	$-0.200950\pi$
$5$	3.61017	1.61452	0.807259	+	0.590198i	$0.200950\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00899i	0.605734i	0.953033	+	0.302867i	$0.0979438\pi$
$11$	2.00899i	0.605734i	−0.953033	+	0.302867i	$0.902056\pi$
$12$	0	0
$13$	1.59265i	0.441722i	0.975305	+	0.220861i	$0.0708867\pi$
$13$	1.59265i	0.441722i	−0.975305	+	0.220861i	$0.929113\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.79257i	1.88998i	0.327106	+	0.944988i	$0.393926\pi$
$17$	7.79257i	1.88998i	−0.327106	+	0.944988i	$0.606074\pi$
$18$	0	0
$19$	5.05480	1.15965	0.579825	−	0.814741i	$-0.303121\pi$
$19$	5.05480	1.15965	0.579825	+	0.814741i	$0.303121\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.14612	−1.49007	−0.745034	−	0.667026i	$-0.767567\pi$
$23$	−7.14612	−1.49007	−0.745034	+	0.667026i	$0.767567\pi$
$24$	0	0
$25$	8.03333	1.60667
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.94298	0.917889	0.458945	−	0.888465i	$-0.348228\pi$
$29$	4.94298	0.917889	0.458945	+	0.888465i	$0.348228\pi$
$30$	0	0
$31$	2.53252i	0.454853i	0.973795	+	0.227427i	$0.0730312\pi$
$31$	2.53252i	0.454853i	−0.973795	+	0.227427i	$0.926969\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 3.61017i	− 0.610230i
$36$	0	0
$37$	4.76592i	0.783512i	0.920069	+	0.391756i	$0.128132\pi$
$37$	4.76592i	0.783512i	−0.920069	+	0.391756i	$0.871868\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 0.836835i	− 0.130692i	−0.997863	−	0.0653458i	$-0.979185\pi$
$41$	− 0.836835i	− 0.130692i	0.997863	−	0.0653458i	$-0.0208151\pi$
$42$	0	0
$43$	−0.151434	−0.0230934	−0.0115467	−	0.999933i	$-0.503676\pi$
$43$	−0.151434	−0.0230934	−0.0115467	+	0.999933i	$0.503676\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.795430	−0.116025	−0.0580127	−	0.998316i	$-0.518476\pi$
$47$	−0.795430	−0.116025	−0.0580127	+	0.998316i	$0.518476\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.05980	0.145575	0.0727873	−	0.997347i	$-0.476811\pi$
$53$	1.05980	0.145575	0.0727873	+	0.997347i	$0.476811\pi$
$54$	0	0
$55$	7.25280i	0.977968i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 8.34506i	− 1.08643i	−0.839592	−	0.543217i	$-0.817206\pi$
$59$	− 8.34506i	− 1.08643i	0.839592	−	0.543217i	$-0.182794\pi$
$60$	0	0
$61$	10.2367i	1.31067i	0.755336	+	0.655337i	$0.227474\pi$
$61$	10.2367i	1.31067i	−0.755336	+	0.655337i	$0.772526\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.74974i	0.713168i
$66$	0	0
$67$	−0.655357	−0.0800646	−0.0400323	−	0.999198i	$-0.512746\pi$
$67$	−0.655357	−0.0800646	−0.0400323	+	0.999198i	$0.512746\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.3675	−1.34907	−0.674536	−	0.738242i	$-0.735656\pi$
$71$	−11.3675	−1.34907	−0.674536	+	0.738242i	$0.735656\pi$
$72$	0	0
$73$	−14.3835	−1.68346	−0.841728	−	0.539901i	$-0.818462\pi$
$73$	−14.3835	−1.68346	−0.841728	+	0.539901i	$0.818462\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00899	0.228946
$78$	0	0
$79$	− 3.81177i	− 0.428858i	−0.976740	−	0.214429i	$-0.931211\pi$
$79$	− 3.81177i	− 0.428858i	0.976740	−	0.214429i	$-0.0687890\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.9522i	1.64121i	0.571494	+	0.820606i	$0.306364\pi$
$83$	14.9522i	1.64121i	−0.571494	+	0.820606i	$0.693636\pi$
$84$	0	0
$85$	28.1325i	3.05140i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 0.0444167i	− 0.00470816i	−0.999997	−	0.00235408i	$-0.999251\pi$
$89$	− 0.0444167i	− 0.00470816i	0.999997	−	0.00235408i	$-0.000749328\pi$
$90$	0	0
$91$	1.59265	0.166955
$92$	0	0
$93$	0	0
$94$	0	0
$95$	18.2487	1.87228
$96$	0	0
$97$	9.87000	1.00215	0.501073	−	0.865405i	$-0.332939\pi$
$97$	9.87000	1.00215	0.501073	+	0.865405i	$0.332939\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.43029	−0.440830	−0.220415	−	0.975406i	$-0.570741\pi$
$101$	−4.43029	−0.440830	−0.220415	+	0.975406i	$0.570741\pi$
$102$	0	0
$103$	11.2939i	1.11282i	0.830906	+	0.556412i	$0.187823\pi$
$103$	11.2939i	1.11282i	−0.830906	+	0.556412i	$0.812177\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.60018i	0.831411i	0.909499	+	0.415705i	$0.136465\pi$
$107$	8.60018i	0.831411i	−0.909499	+	0.415705i	$0.863535\pi$
$108$	0	0
$109$	− 19.9155i	− 1.90756i	−0.300512	−	0.953778i	$-0.597158\pi$
$109$	− 19.9155i	− 1.90756i	0.300512	−	0.953778i	$-0.402842\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.81372i	0.358764i	0.983779	+	0.179382i	$0.0574098\pi$
$113$	3.81372i	0.358764i	−0.983779	+	0.179382i	$0.942590\pi$
$114$	0	0
$115$	−25.7987	−2.40574
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.79257	0.714343
$120$	0	0
$121$	6.96395	0.633086
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.9508	0.979472
$126$	0	0
$127$	− 15.9943i	− 1.41926i	−0.704574	−	0.709630i	$-0.748862\pi$
$127$	− 15.9943i	− 1.41926i	0.704574	−	0.709630i	$-0.251138\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	19.8839i	1.73726i	0.495460	+	0.868631i	$0.335000\pi$
$131$	19.8839i	1.73726i	−0.495460	+	0.868631i	$0.665000\pi$
$132$	0	0
$133$	− 5.05480i	− 0.438307i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.93576i	0.763433i	0.924279	+	0.381717i	$0.124667\pi$
$137$	8.93576i	0.763433i	−0.924279	+	0.381717i	$0.875333\pi$
$138$	0	0
$139$	−1.20344	−0.102074	−0.0510372	−	0.998697i	$-0.516253\pi$
$139$	−1.20344	−0.102074	−0.0510372	+	0.998697i	$0.516253\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.19962	−0.267566
$144$	0	0
$145$	17.8450	1.48195
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.6791	−1.03871	−0.519355	−	0.854559i	$-0.673828\pi$
$149$	−12.6791	−1.03871	−0.519355	+	0.854559i	$0.673828\pi$
$150$	0	0
$151$	− 17.5094i	− 1.42489i	−0.701727	−	0.712446i	$-0.747588\pi$
$151$	− 17.5094i	− 1.42489i	0.701727	−	0.712446i	$-0.252412\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.14281i	0.734368i
$156$	0	0
$157$	15.2891i	1.22020i	0.792324	+	0.610100i	$0.208871\pi$
$157$	15.2891i	1.22020i	−0.792324	+	0.610100i	$0.791129\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.14612i	0.563193i
$162$	0	0
$163$	−17.5634	−1.37567	−0.687835	−	0.725867i	$-0.741439\pi$
$163$	−17.5634	−1.37567	−0.687835	+	0.725867i	$0.741439\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.20290	0.325230	0.162615	−	0.986690i	$-0.448007\pi$
$167$	4.20290	0.325230	0.162615	+	0.986690i	$0.448007\pi$
$168$	0	0
$169$	10.4635	0.804882
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.0074	1.36907	0.684537	−	0.728978i	$-0.260005\pi$
$173$	18.0074	1.36907	0.684537	+	0.728978i	$0.260005\pi$
$174$	0	0
$175$	− 8.03333i	− 0.607263i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0.886091i	0.0662295i	0.999452	+	0.0331148i	$0.0105427\pi$
$179$	0.886091i	0.0662295i	−0.999452	+	0.0331148i	$0.989457\pi$
$180$	0	0
$181$	− 8.47338i	− 0.629821i	−0.949121	−	0.314911i	$-0.898025\pi$
$181$	− 8.47338i	− 0.629821i	0.949121	−	0.314911i	$-0.101975\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	17.2058i	1.26499i
$186$	0	0
$187$	−15.6552	−1.14482
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.6102	1.49130	0.745652	−	0.666335i	$-0.232138\pi$
$191$	20.6102	1.49130	0.745652	+	0.666335i	$0.232138\pi$
$192$	0	0
$193$	18.2972	1.31706	0.658530	−	0.752555i	$-0.271179\pi$
$193$	18.2972	1.31706	0.658530	+	0.752555i	$0.271179\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.6580	0.759353	0.379677	−	0.925119i	$-0.376035\pi$
$197$	10.6580	0.759353	0.379677	+	0.925119i	$0.376035\pi$
$198$	0	0
$199$	− 26.1305i	− 1.85234i	−0.377108	−	0.926169i	$-0.623081\pi$
$199$	− 26.1305i	− 1.85234i	0.377108	−	0.926169i	$-0.376919\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 4.94298i	− 0.346929i
$204$	0	0
$205$	− 3.02112i	− 0.211004i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	10.1551i	0.702440i
$210$	0	0
$211$	21.1542	1.45632	0.728159	−	0.685409i	$-0.240376\pi$
$211$	21.1542	1.45632	0.728159	+	0.685409i	$0.240376\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.546701	−0.0372847
$216$	0	0
$217$	2.53252	0.171918
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−12.4108	−0.834844
$222$	0	0
$223$	6.04274i	0.404652i	0.979318	+	0.202326i	$0.0648500\pi$
$223$	6.04274i	0.404652i	−0.979318	+	0.202326i	$0.935150\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.0726i	1.33227i	0.745832	+	0.666134i	$0.232052\pi$
$227$	20.0726i	1.33227i	−0.745832	+	0.666134i	$0.767948\pi$
$228$	0	0
$229$	− 18.5176i	− 1.22368i	−0.790983	−	0.611838i	$-0.790430\pi$
$229$	− 18.5176i	− 1.22368i	0.790983	−	0.611838i	$-0.209570\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 17.1203i	− 1.12159i	−0.827956	−	0.560793i	$-0.810496\pi$
$233$	− 17.1203i	− 1.12159i	0.827956	−	0.560793i	$-0.189504\pi$
$234$	0	0
$235$	−2.87164	−0.187325
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.95612	0.255900	0.127950	−	0.991781i	$-0.459160\pi$
$239$	3.95612	0.255900	0.127950	+	0.991781i	$0.459160\pi$
$240$	0	0
$241$	−15.3726	−0.990235	−0.495117	−	0.868826i	$-0.664875\pi$
$241$	−15.3726	−0.990235	−0.495117	+	0.868826i	$0.664875\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.61017	−0.230645
$246$	0	0
$247$	8.05054i	0.512243i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 13.1732i	− 0.831482i	−0.909483	−	0.415741i	$-0.863522\pi$
$251$	− 13.1732i	− 0.831482i	0.909483	−	0.415741i	$-0.136478\pi$
$252$	0	0
$253$	− 14.3565i	− 0.902585i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 26.9072i	− 1.67843i	−0.543802	−	0.839213i	$-0.683016\pi$
$257$	− 26.9072i	− 1.67843i	0.543802	−	0.839213i	$-0.316984\pi$
$258$	0	0
$259$	4.76592	0.296140
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.15057	−0.502586	−0.251293	−	0.967911i	$-0.580856\pi$
$263$	−8.15057	−0.502586	−0.251293	+	0.967911i	$0.580856\pi$
$264$	0	0
$265$	3.82605	0.235033
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.84304	−0.234314	−0.117157	−	0.993113i	$-0.537378\pi$
$269$	−3.84304	−0.234314	−0.117157	+	0.993113i	$0.537378\pi$
$270$	0	0
$271$	12.8705i	0.781829i	0.920427	+	0.390914i	$0.127841\pi$
$271$	12.8705i	0.781829i	−0.920427	+	0.390914i	$0.872159\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	16.1389i	0.973212i
$276$	0	0
$277$	2.72420i	0.163682i	0.996645	+	0.0818408i	$0.0260799\pi$
$277$	2.72420i	0.163682i	−0.996645	+	0.0818408i	$0.973920\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.91234i	0.531666i	0.964019	+	0.265833i	$0.0856469\pi$
$281$	8.91234i	0.531666i	−0.964019	+	0.265833i	$0.914353\pi$
$282$	0	0
$283$	−4.90080	−0.291322	−0.145661	−	0.989335i	$-0.546531\pi$
$283$	−4.90080	−0.291322	−0.145661	+	0.989335i	$0.546531\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.836835	−0.0493968
$288$	0	0
$289$	−43.7241	−2.57201
$290$	0	0
$291$	0	0
$292$	0	0
$293$	26.1386	1.52704	0.763518	−	0.645786i	$-0.223470\pi$
$293$	26.1386	1.52704	0.763518	+	0.645786i	$0.223470\pi$
$294$	0	0
$295$	− 30.1271i	− 1.75407i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 11.3813i	− 0.658196i
$300$	0	0
$301$	0.151434i	0.00872849i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	36.9562i	2.11611i
$306$	0	0
$307$	23.4257	1.33697	0.668487	−	0.743724i	$-0.266942\pi$
$307$	23.4257	1.33697	0.668487	+	0.743724i	$0.266942\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.4333	0.875143	0.437571	−	0.899184i	$-0.355839\pi$
$311$	15.4333	0.875143	0.437571	+	0.899184i	$0.355839\pi$
$312$	0	0
$313$	−31.1894	−1.76293	−0.881464	−	0.472251i	$-0.843442\pi$
$313$	−31.1894	−1.76293	−0.881464	+	0.472251i	$0.843442\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	30.7471	1.72693	0.863466	−	0.504407i	$-0.168289\pi$
$317$	30.7471	1.72693	0.863466	+	0.504407i	$0.168289\pi$
$318$	0	0
$319$	9.93042i	0.555997i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	39.3899i	2.19171i
$324$	0	0
$325$	12.7943i	0.709700i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.795430i	0.0438535i
$330$	0	0
$331$	−1.00125	−0.0550339	−0.0275169	−	0.999621i	$-0.508760\pi$
$331$	−1.00125	−0.0550339	−0.0275169	+	0.999621i	$0.508760\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.36595	−0.129266
$336$	0	0
$337$	28.1818	1.53516	0.767580	−	0.640953i	$-0.221461\pi$
$337$	28.1818	1.53516	0.767580	+	0.640953i	$0.221461\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.08780	−0.275520
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 19.6496i	− 1.05485i	−0.849602	−	0.527424i	$-0.823158\pi$
$347$	− 19.6496i	− 1.05485i	0.849602	−	0.527424i	$-0.176842\pi$
$348$	0	0
$349$	11.2275i	0.600993i	0.953783	+	0.300497i	$0.0971524\pi$
$349$	11.2275i	0.600993i	−0.953783	+	0.300497i	$0.902848\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.2047i	1.71408i	0.515249	+	0.857041i	$0.327700\pi$
$353$	32.2047i	1.71408i	−0.515249	+	0.857041i	$0.672300\pi$
$354$	0	0
$355$	−41.0385	−2.17810
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.1918	1.06568	0.532842	−	0.846215i	$-0.321124\pi$
$359$	20.1918	1.06568	0.532842	+	0.846215i	$0.321124\pi$
$360$	0	0
$361$	6.55100	0.344790
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−51.9267	−2.71797
$366$	0	0
$367$	1.42641i	0.0744578i	0.999307	+	0.0372289i	$0.0118531\pi$
$367$	1.42641i	0.0744578i	−0.999307	+	0.0372289i	$0.988147\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.05980i	− 0.0550220i
$372$	0	0
$373$	− 9.80760i	− 0.507818i	−0.967228	−	0.253909i	$-0.918284\pi$
$373$	− 9.80760i	− 0.507818i	0.967228	−	0.253909i	$-0.0817165\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.87245i	0.405452i
$378$	0	0
$379$	17.5378	0.900857	0.450428	−	0.892813i	$-0.351271\pi$
$379$	17.5378	0.900857	0.450428	+	0.892813i	$0.351271\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8.32355	0.425313	0.212657	−	0.977127i	$-0.431788\pi$
$383$	8.32355	0.425313	0.212657	+	0.977127i	$0.431788\pi$
$384$	0	0
$385$	7.25280	0.369637
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−8.79829	−0.446091	−0.223045	−	0.974808i	$-0.571600\pi$
$389$	−8.79829	−0.446091	−0.223045	+	0.974808i	$0.571600\pi$
$390$	0	0
$391$	− 55.6866i	− 2.81619i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 13.7611i	− 0.692398i
$396$	0	0
$397$	− 31.5495i	− 1.58342i	−0.610896	−	0.791711i	$-0.709190\pi$
$397$	− 31.5495i	− 1.58342i	0.610896	−	0.791711i	$-0.290810\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 8.99955i	− 0.449416i	−0.974426	−	0.224708i	$-0.927857\pi$
$401$	− 8.99955i	− 0.449416i	0.974426	−	0.224708i	$-0.0721428\pi$
$402$	0	0
$403$	−4.03341	−0.200919
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.57470	−0.474600
$408$	0	0
$409$	−35.7330	−1.76688	−0.883441	−	0.468542i	$-0.844779\pi$
$409$	−35.7330	−1.76688	−0.883441	+	0.468542i	$0.844779\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8.34506	−0.410633
$414$	0	0
$415$	53.9798i	2.64976i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	35.1521i	1.71729i	0.512569	+	0.858646i	$0.328694\pi$
$419$	35.1521i	1.71729i	−0.512569	+	0.858646i	$0.671306\pi$
$420$	0	0
$421$	− 22.7156i	− 1.10709i	−0.832819	−	0.553546i	$-0.813274\pi$
$421$	− 22.7156i	− 1.10709i	0.832819	−	0.553546i	$-0.186726\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	62.6002i	3.03656i
$426$	0	0
$427$	10.2367	0.495388
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.00587	0.0484511	0.0242256	−	0.999707i	$-0.492288\pi$
$431$	1.00587	0.0484511	0.0242256	+	0.999707i	$0.492288\pi$
$432$	0	0
$433$	−33.8256	−1.62555	−0.812776	−	0.582576i	$-0.802045\pi$
$433$	−33.8256	−1.62555	−0.812776	+	0.582576i	$0.802045\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−36.1222	−1.72796
$438$	0	0
$439$	− 11.9514i	− 0.570411i	−0.958466	−	0.285206i	$-0.907938\pi$
$439$	− 11.9514i	− 0.570411i	0.958466	−	0.285206i	$-0.0920619\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.6849i	0.840234i	0.907470	+	0.420117i	$0.138011\pi$
$443$	17.6849i	0.840234i	−0.907470	+	0.420117i	$0.861989\pi$
$444$	0	0
$445$	− 0.160352i	− 0.00760141i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 33.4546i	− 1.57882i	−0.613867	−	0.789409i	$-0.710387\pi$
$449$	− 33.4546i	− 1.57882i	0.613867	−	0.789409i	$-0.289613\pi$
$450$	0	0
$451$	1.68120	0.0791644
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.74974	0.269552
$456$	0	0
$457$	34.7127	1.62379	0.811896	−	0.583802i	$-0.198436\pi$
$457$	34.7127	1.62379	0.811896	+	0.583802i	$0.198436\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.30870	0.433549	0.216775	−	0.976222i	$-0.430446\pi$
$461$	9.30870	0.433549	0.216775	+	0.976222i	$0.430446\pi$
$462$	0	0
$463$	1.56421i	0.0726950i	0.999339	+	0.0363475i	$0.0115723\pi$
$463$	1.56421i	0.0726950i	−0.999339	+	0.0363475i	$0.988428\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.47725i	0.114633i	0.998356	+	0.0573167i	$0.0182545\pi$
$467$	2.47725i	0.114633i	−0.998356	+	0.0573167i	$0.981746\pi$
$468$	0	0
$469$	0.655357i	0.0302616i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 0.304229i	− 0.0139885i
$474$	0	0
$475$	40.6069	1.86317
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.20750	0.192245	0.0961227	−	0.995369i	$-0.469356\pi$
$479$	4.20750	0.192245	0.0961227	+	0.995369i	$0.469356\pi$
$480$	0	0
$481$	−7.59045	−0.346095
$482$	0	0
$483$	0	0
$484$	0	0
$485$	35.6324	1.61798
$486$	0	0
$487$	29.7159i	1.34656i	0.739389	+	0.673278i	$0.235114\pi$
$487$	29.7159i	1.34656i	−0.739389	+	0.673278i	$0.764886\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 6.98629i	− 0.315287i	−0.987496	−	0.157643i	$-0.949610\pi$
$491$	− 6.98629i	− 0.315287i	0.987496	−	0.157643i	$-0.0503896\pi$
$492$	0	0
$493$	38.5185i	1.73479i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.3675i	0.509901i
$498$	0	0
$499$	−9.65859	−0.432378	−0.216189	−	0.976352i	$-0.569363\pi$
$499$	−9.65859	−0.432378	−0.216189	+	0.976352i	$0.569363\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	33.8339	1.50858	0.754290	−	0.656542i	$-0.227981\pi$
$503$	33.8339	1.50858	0.754290	+	0.656542i	$0.227981\pi$
$504$	0	0
$505$	−15.9941	−0.711728
$506$	0	0
$507$	0	0
$508$	0	0
$509$	12.8961	0.571611	0.285806	−	0.958288i	$-0.407739\pi$
$509$	12.8961	0.571611	0.285806	+	0.958288i	$0.407739\pi$
$510$	0	0
$511$	14.3835i	0.636287i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	40.7730i	1.79667i
$516$	0	0
$517$	− 1.59801i	− 0.0702805i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 15.4735i	− 0.677905i	−0.940803	−	0.338953i	$-0.889927\pi$
$521$	− 15.4735i	− 0.677905i	0.940803	−	0.338953i	$-0.110073\pi$
$522$	0	0
$523$	20.0632	0.877301	0.438651	−	0.898658i	$-0.355457\pi$
$523$	20.0632	0.877301	0.438651	+	0.898658i	$0.355457\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19.7348	−0.859661
$528$	0	0
$529$	28.0670	1.22031
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.33279	0.0577294
$534$	0	0
$535$	31.0481i	1.34233i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.00899i	− 0.0865334i
$540$	0	0
$541$	18.5169i	0.796103i	0.917363	+	0.398052i	$0.130313\pi$
$541$	18.5169i	0.796103i	−0.917363	+	0.398052i	$0.869687\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 71.8982i	− 3.07978i
$546$	0	0
$547$	24.8132	1.06094	0.530468	−	0.847705i	$-0.322016\pi$
$547$	24.8132	1.06094	0.530468	+	0.847705i	$0.322016\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	24.9858	1.06443
$552$	0	0
$553$	−3.81177	−0.162093
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.4177	−0.822756	−0.411378	−	0.911465i	$-0.634952\pi$
$557$	−19.4177	−0.822756	−0.411378	+	0.911465i	$0.634952\pi$
$558$	0	0
$559$	− 0.241181i	− 0.0102009i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 14.6696i	− 0.618252i	−0.951021	−	0.309126i	$-0.899964\pi$
$563$	− 14.6696i	− 0.618252i	0.951021	−	0.309126i	$-0.100036\pi$
$564$	0	0
$565$	13.7682i	0.579231i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 17.3133i	− 0.725809i	−0.931826	−	0.362905i	$-0.881785\pi$
$569$	− 17.3133i	− 0.725809i	0.931826	−	0.362905i	$-0.118215\pi$
$570$	0	0
$571$	−11.7082	−0.489975	−0.244987	−	0.969526i	$-0.578784\pi$
$571$	−11.7082	−0.489975	−0.244987	+	0.969526i	$0.578784\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−57.4071	−2.39404
$576$	0	0
$577$	4.79465	0.199604	0.0998020	−	0.995007i	$-0.468179\pi$
$577$	4.79465	0.199604	0.0998020	+	0.995007i	$0.468179\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	14.9522	0.620320
$582$	0	0
$583$	2.12913i	0.0881795i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	30.1643i	1.24501i	0.782615	+	0.622506i	$0.213886\pi$
$587$	30.1643i	1.24501i	−0.782615	+	0.622506i	$0.786114\pi$
$588$	0	0
$589$	12.8014i	0.527471i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.60974i	0.107169i	0.998563	+	0.0535847i	$0.0170647\pi$
$593$	2.60974i	0.107169i	−0.998563	+	0.0535847i	$0.982935\pi$
$594$	0	0
$595$	28.1325	1.15332
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27.6903	−1.13139	−0.565697	−	0.824613i	$-0.691393\pi$
$599$	−27.6903	−1.13139	−0.565697	+	0.824613i	$0.691393\pi$
$600$	0	0
$601$	16.3291	0.666079	0.333039	−	0.942913i	$-0.391926\pi$
$601$	16.3291	0.666079	0.333039	+	0.942913i	$0.391926\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	25.1410	1.02213
$606$	0	0
$607$	12.9385i	0.525159i	0.964910	+	0.262580i	$0.0845732\pi$
$607$	12.9385i	0.525159i	−0.964910	+	0.262580i	$0.915427\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1.26684i	− 0.0512509i
$612$	0	0
$613$	− 21.7992i	− 0.880463i	−0.897884	−	0.440231i	$-0.854896\pi$
$613$	− 21.7992i	− 0.880463i	0.897884	−	0.440231i	$-0.145104\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	41.8226i	1.68371i	0.539700	+	0.841857i	$0.318538\pi$
$617$	41.8226i	1.68371i	−0.539700	+	0.841857i	$0.681462\pi$
$618$	0	0
$619$	4.82763	0.194039	0.0970193	−	0.995282i	$-0.469069\pi$
$619$	4.82763	0.194039	0.0970193	+	0.995282i	$0.469069\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.0444167	−0.00177952
$624$	0	0
$625$	−0.632276	−0.0252911
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−37.1388	−1.48082
$630$	0	0
$631$	20.3929i	0.811830i	0.913911	+	0.405915i	$0.133047\pi$
$631$	20.3929i	0.811830i	−0.913911	+	0.405915i	$0.866953\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 57.7420i	− 2.29142i
$636$	0	0
$637$	− 1.59265i	− 0.0631032i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9.42755i	0.372366i	0.982515	+	0.186183i	$0.0596117\pi$
$641$	9.42755i	0.372366i	−0.982515	+	0.186183i	$0.940388\pi$
$642$	0	0
$643$	27.2077	1.07297	0.536484	−	0.843911i	$-0.319752\pi$
$643$	27.2077	1.07297	0.536484	+	0.843911i	$0.319752\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.24081	0.284666	0.142333	−	0.989819i	$-0.454540\pi$
$647$	7.24081	0.284666	0.142333	+	0.989819i	$0.454540\pi$
$648$	0	0
$649$	16.7652	0.658090
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30.0954	1.17772	0.588861	−	0.808234i	$-0.299576\pi$
$653$	30.0954	1.17772	0.588861	+	0.808234i	$0.299576\pi$
$654$	0	0
$655$	71.7842i	2.80484i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	41.9316i	1.63342i	0.577045	+	0.816712i	$0.304206\pi$
$659$	41.9316i	1.63342i	−0.577045	+	0.816712i	$0.695794\pi$
$660$	0	0
$661$	− 0.735053i	− 0.0285903i	−0.999898	−	0.0142951i	$-0.995450\pi$
$661$	− 0.735053i	− 0.0285903i	0.999898	−	0.0142951i	$-0.00455044\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 18.2487i	− 0.707654i
$666$	0	0
$667$	−35.3232	−1.36772
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−20.5654	−0.793920
$672$	0	0
$673$	−21.6142	−0.833165	−0.416582	−	0.909098i	$-0.636772\pi$
$673$	−21.6142	−0.833165	−0.416582	+	0.909098i	$0.636772\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−0.631879	−0.0242851	−0.0121425	−	0.999926i	$-0.503865\pi$
$677$	−0.631879	−0.0242851	−0.0121425	+	0.999926i	$0.503865\pi$
$678$	0	0
$679$	− 9.87000i	− 0.378776i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 22.4241i	− 0.858033i	−0.903297	−	0.429017i	$-0.858860\pi$
$683$	− 22.4241i	− 0.858033i	0.903297	−	0.429017i	$-0.141140\pi$
$684$	0	0
$685$	32.2596i	1.23258i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.68789i	0.0643035i
$690$	0	0
$691$	2.69349	0.102465	0.0512325	−	0.998687i	$-0.483685\pi$
$691$	2.69349	0.102465	0.0512325	+	0.998687i	$0.483685\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.34462	−0.164801
$696$	0	0
$697$	6.52109	0.247004
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−28.4555	−1.07475	−0.537375	−	0.843344i	$-0.680584\pi$
$701$	−28.4555	−1.07475	−0.537375	+	0.843344i	$0.680584\pi$
$702$	0	0
$703$	24.0908i	0.908601i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.43029i	0.166618i
$708$	0	0
$709$	− 13.4255i	− 0.504204i	−0.967701	−	0.252102i	$-0.918878\pi$
$709$	− 13.4255i	− 0.504204i	0.967701	−	0.252102i	$-0.0811218\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 18.0977i	− 0.677763i
$714$	0	0
$715$	−11.5512	−0.431990
$716$	0	0
$717$	0	0
$718$	0	0
$719$	49.4437	1.84394	0.921969	−	0.387264i	$-0.126580\pi$
$719$	49.4437	1.84394	0.921969	+	0.387264i	$0.126580\pi$
$720$	0	0
$721$	11.2939	0.420608
$722$	0	0
$723$	0	0
$724$	0	0
$725$	39.7086	1.47474
$726$	0	0
$727$	− 5.65644i	− 0.209786i	−0.994484	−	0.104893i	$-0.966550\pi$
$727$	− 5.65644i	− 0.209786i	0.994484	−	0.104893i	$-0.0334500\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1.18006i	− 0.0436460i
$732$	0	0
$733$	− 42.3705i	− 1.56499i	−0.622656	−	0.782495i	$-0.713946\pi$
$733$	− 42.3705i	− 1.56499i	0.622656	−	0.782495i	$-0.286054\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.31661i	− 0.0484979i
$738$	0	0
$739$	41.5942	1.53007	0.765034	−	0.643989i	$-0.222722\pi$
$739$	41.5942	1.53007	0.765034	+	0.643989i	$0.222722\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.5821	0.608338	0.304169	−	0.952618i	$-0.401621\pi$
$743$	16.5821	0.608338	0.304169	+	0.952618i	$0.401621\pi$
$744$	0	0
$745$	−45.7736	−1.67702
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.60018	0.314244
$750$	0	0
$751$	− 4.26395i	− 0.155594i	−0.996969	−	0.0777968i	$-0.975211\pi$
$751$	− 4.26395i	− 0.155594i	0.996969	−	0.0777968i	$-0.0247886\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 63.2118i	− 2.30051i
$756$	0	0
$757$	− 19.5706i	− 0.711305i	−0.934618	−	0.355653i	$-0.884259\pi$
$757$	− 19.5706i	− 0.711305i	0.934618	−	0.355653i	$-0.115741\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.1256i	0.802053i	0.916066	+	0.401027i	$0.131347\pi$
$761$	22.1256i	0.802053i	−0.916066	+	0.401027i	$0.868653\pi$
$762$	0	0
$763$	−19.9155	−0.720988
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.2908	0.479902
$768$	0	0
$769$	−16.1961	−0.584046	−0.292023	−	0.956411i	$-0.594328\pi$
$769$	−16.1961	−0.584046	−0.292023	+	0.956411i	$0.594328\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.34825	−0.0844608	−0.0422304	−	0.999108i	$-0.513446\pi$
$773$	−2.34825	−0.0844608	−0.0422304	+	0.999108i	$0.513446\pi$
$774$	0	0
$775$	20.3445i	0.730797i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 4.23003i	− 0.151557i
$780$	0	0
$781$	− 22.8372i	− 0.817179i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	55.1962i	1.97004i
$786$	0	0
$787$	−10.9012	−0.388586	−0.194293	−	0.980943i	$-0.562241\pi$
$787$	−10.9012	−0.388586	−0.194293	+	0.980943i	$0.562241\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.81372	0.135600
$792$	0	0
$793$	−16.3035	−0.578954
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.08600	0.180155	0.0900776	−	0.995935i	$-0.471288\pi$
$797$	5.08600	0.180155	0.0900776	+	0.995935i	$0.471288\pi$
$798$	0	0
$799$	− 6.19844i	− 0.219285i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 28.8963i	− 1.01973i
$804$	0	0
$805$	25.7987i	0.909285i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 26.8140i	− 0.942729i	−0.881938	−	0.471365i	$-0.843762\pi$
$809$	− 26.8140i	− 0.942729i	0.881938	−	0.471365i	$-0.156238\pi$
$810$	0	0
$811$	−38.8383	−1.36380	−0.681899	−	0.731446i	$-0.738846\pi$
$811$	−38.8383	−1.36380	−0.681899	+	0.731446i	$0.738846\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−63.4068	−2.22104
$816$	0	0
$817$	−0.765467	−0.0267803
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29.3421	−1.02405	−0.512023	−	0.858972i	$-0.671104\pi$
$821$	−29.3421	−1.02405	−0.512023	+	0.858972i	$0.671104\pi$
$822$	0	0
$823$	− 23.9770i	− 0.835784i	−0.908497	−	0.417892i	$-0.862769\pi$
$823$	− 23.9770i	− 0.835784i	0.908497	−	0.417892i	$-0.137231\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.92468i	0.171248i	0.996328	+	0.0856239i	$0.0272883\pi$
$827$	4.92468i	0.171248i	−0.996328	+	0.0856239i	$0.972712\pi$
$828$	0	0
$829$	8.00324i	0.277964i	0.990295	+	0.138982i	$0.0443830\pi$
$829$	8.00324i	0.277964i	−0.990295	+	0.138982i	$0.955617\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 7.79257i	− 0.269996i
$834$	0	0
$835$	15.1732	0.525089
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.4595	1.08610	0.543051	−	0.839700i	$-0.317269\pi$
$839$	31.4595	1.08610	0.543051	+	0.839700i	$0.317269\pi$
$840$	0	0
$841$	−4.56691	−0.157479
$842$	0	0
$843$	0	0
$844$	0	0
$845$	37.7749	1.29950
$846$	0	0
$847$	− 6.96395i	− 0.239284i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 34.0578i	− 1.16749i
$852$	0	0
$853$	1.31213i	0.0449263i	0.999748	+	0.0224632i	$0.00715085\pi$
$853$	1.31213i	0.0449263i	−0.999748	+	0.0224632i	$0.992849\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.2200i	− 0.827341i	−0.910427	−	0.413670i	$-0.864247\pi$
$857$	− 24.2200i	− 0.827341i	0.910427	−	0.413670i	$-0.135753\pi$
$858$	0	0
$859$	−8.49401	−0.289812	−0.144906	−	0.989445i	$-0.546288\pi$
$859$	−8.49401	−0.289812	−0.144906	+	0.989445i	$0.546288\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.8759	−1.05103	−0.525513	−	0.850785i	$-0.676127\pi$
$863$	−30.8759	−1.05103	−0.525513	+	0.850785i	$0.676127\pi$
$864$	0	0
$865$	65.0096	2.21039
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7.65782	0.259774
$870$	0	0
$871$	− 1.04376i	− 0.0353663i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 10.9508i	− 0.370206i
$876$	0	0
$877$	− 33.4689i	− 1.13016i	−0.825035	−	0.565082i	$-0.808845\pi$
$877$	− 33.4689i	− 1.13016i	0.825035	−	0.565082i	$-0.191155\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	33.7535i	1.13719i	0.822619	+	0.568593i	$0.192512\pi$
$881$	33.7535i	1.13719i	−0.822619	+	0.568593i	$0.807488\pi$
$882$	0	0
$883$	−1.59496	−0.0536746	−0.0268373	−	0.999640i	$-0.508544\pi$
$883$	−1.59496	−0.0536746	−0.0268373	+	0.999640i	$0.508544\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17.9639	−0.603167	−0.301584	−	0.953440i	$-0.597515\pi$
$887$	−17.9639	−0.603167	−0.301584	+	0.953440i	$0.597515\pi$
$888$	0	0
$889$	−15.9943	−0.536430
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.02074	−0.134549
$894$	0	0
$895$	3.19894i	0.106929i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.5182i	0.417505i
$900$	0	0
$901$	8.25855i	0.275132i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 30.5903i	− 1.01686i
$906$	0	0
$907$	−52.9653	−1.75868	−0.879342	−	0.476190i	$-0.842017\pi$
$907$	−52.9653	−1.75868	−0.879342	+	0.476190i	$0.842017\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−52.6435	−1.74416	−0.872079	−	0.489365i	$-0.837229\pi$
$911$	−52.6435	−1.74416	−0.872079	+	0.489365i	$0.837229\pi$
$912$	0	0
$913$	−30.0388	−0.994138
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.8839	0.656623
$918$	0	0
$919$	− 36.5794i	− 1.20664i	−0.797498	−	0.603322i	$-0.793843\pi$
$919$	− 36.5794i	− 1.20664i	0.797498	−	0.603322i	$-0.206157\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 18.1044i	− 0.595915i
$924$	0	0
$925$	38.2862i	1.25884i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 24.8413i	− 0.815018i	−0.913201	−	0.407509i	$-0.866398\pi$
$929$	− 24.8413i	− 0.815018i	0.913201	−	0.407509i	$-0.133602\pi$
$930$	0	0
$931$	−5.05480	−0.165664
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−56.5180	−1.84833
$936$	0	0
$937$	0.904448	0.0295470	0.0147735	−	0.999891i	$-0.495297\pi$
$937$	0.904448	0.0295470	0.0147735	+	0.999891i	$0.495297\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8.11337	−0.264488	−0.132244	−	0.991217i	$-0.542218\pi$
$941$	−8.11337	−0.264488	−0.132244	+	0.991217i	$0.542218\pi$
$942$	0	0
$943$	5.98012i	0.194740i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.2396i	1.11264i	0.830969	+	0.556319i	$0.187787\pi$
$947$	34.2396i	1.11264i	−0.830969	+	0.556319i	$0.812213\pi$
$948$	0	0
$949$	− 22.9078i	− 0.743620i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 5.47739i	− 0.177430i	−0.996057	−	0.0887151i	$-0.971724\pi$
$953$	− 5.47739i	− 0.177430i	0.996057	−	0.0887151i	$-0.0282761\pi$
$954$	0	0
$955$	74.4065	2.40774
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.93576	0.288551
$960$	0	0
$961$	24.5864	0.793109
$962$	0	0
$963$	0	0
$964$	0	0
$965$	66.0559	2.12641
$966$	0	0
$967$	− 45.0597i	− 1.44902i	−0.689262	−	0.724512i	$-0.742065\pi$
$967$	− 45.0597i	− 1.44902i	0.689262	−	0.724512i	$-0.257935\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 28.4013i	− 0.911442i	−0.890123	−	0.455721i	$-0.849382\pi$
$971$	− 28.4013i	− 0.911442i	0.890123	−	0.455721i	$-0.150618\pi$
$972$	0	0
$973$	1.20344i	0.0385805i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.06448i	0.130034i	0.997884	+	0.0650171i	$0.0207102\pi$
$977$	4.06448i	0.130034i	−0.997884	+	0.0650171i	$0.979290\pi$
$978$	0	0
$979$	0.0892328	0.00285189
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−13.0487	−0.416189	−0.208094	−	0.978109i	$-0.566726\pi$
$983$	−13.0487	−0.416189	−0.208094	+	0.978109i	$0.566726\pi$
$984$	0	0
$985$	38.4773	1.22599
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.08216	0.0344108
$990$	0	0
$991$	− 0.598647i	− 0.0190166i	−0.999955	−	0.00950832i	$-0.996973\pi$
$991$	− 0.598647i	− 0.0190166i	0.999955	−	0.00950832i	$-0.00302664\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 94.3354i	− 2.99063i
$996$	0	0
$997$	4.04553i	0.128123i	0.997946	+	0.0640616i	$0.0204054\pi$
$997$	4.04553i	0.128123i	−0.997946	+	0.0640616i	$0.979595\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.j.c.5615.31		32
3.2	odd	2	inner	6048.2.j.c.5615.2		32
4.3	odd	2		1512.2.j.c.323.5	✓	32
8.3	odd	2	inner	6048.2.j.c.5615.1		32
8.5	even	2		1512.2.j.c.323.27	yes	32
12.11	even	2		1512.2.j.c.323.28	yes	32
24.5	odd	2		1512.2.j.c.323.6	yes	32
24.11	even	2	inner	6048.2.j.c.5615.32		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.c.323.5	✓	32	4.3	odd	2
1512.2.j.c.323.6	yes	32	24.5	odd	2
1512.2.j.c.323.27	yes	32	8.5	even	2
1512.2.j.c.323.28	yes	32	12.11	even	2
6048.2.j.c.5615.1		32	8.3	odd	2	inner
6048.2.j.c.5615.2		32	3.2	odd	2	inner
6048.2.j.c.5615.31		32	1.1	even	1	trivial
6048.2.j.c.5615.32		32	24.11	even	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)

\(f(q)\)	\(=\)	\(q+3.61017 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+3.61017 q^{5} -1.00000i q^{7} +2.00899i q^{11} +1.59265i q^{13} +7.79257i q^{17} +5.05480 q^{19} -7.14612 q^{23} +8.03333 q^{25} +4.94298 q^{29} +2.53252i q^{31} -3.61017i q^{35} +4.76592i q^{37} -0.836835i q^{41} -0.151434 q^{43} -0.795430 q^{47} -1.00000 q^{49} +1.05980 q^{53} +7.25280i q^{55} -8.34506i q^{59} +10.2367i q^{61} +5.74974i q^{65} -0.655357 q^{67} -11.3675 q^{71} -14.3835 q^{73} +2.00899 q^{77} -3.81177i q^{79} +14.9522i q^{83} +28.1325i q^{85} -0.0444167i q^{89} +1.59265 q^{91} +18.2487 q^{95} +9.87000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 64 q^{19} - 16 q^{25} + 48 q^{43} - 32 q^{49} - 16 q^{67} - 16 q^{73} - 16 q^{91} + 64 q^{97}+O(q^{100}) \) 32 * q + 64 * q^19 - 16 * q^25 + 48 * q^43 - 32 * q^49 - 16 * q^67 - 16 * q^73 - 16 * q^91 + 64 * q^97

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