LMFDB - Embedded newform 6048.2.h.c.2591.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(2591,6048)]

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, 0, 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("6048.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.h (of order $2$, degree $1$, 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:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.6
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.c.2591.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.317837i q^{5} +1.00000i q^{7} +O(q^{10})$	$q+0.317837i q^{5} +1.00000i q^{7} -1.41421 q^{11} -4.44949 q^{13} +5.97469i q^{17} +2.44949i q^{19} +5.65685 q^{23} +4.89898 q^{25} -5.51399i q^{29} -6.44949i q^{31} -0.317837 q^{35} +9.00000 q^{37} -7.24604i q^{41} +6.34847i q^{43} +10.8530 q^{47} -1.00000 q^{49} +11.9494i q^{53} -0.449490i q^{55} -5.83183 q^{59} -10.4495 q^{61} -1.41421i q^{65} +5.79796i q^{67} -5.65685 q^{71} -9.79796 q^{73} -1.41421i q^{77} +14.3485i q^{79} -8.02458 q^{83} -1.89898 q^{85} -15.4135i q^{89} -4.44949i q^{91} -0.778539 q^{95} -14.2474 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{13} + 72 q^{37} - 8 q^{49} - 64 q^{61} + 24 q^{85} - 16 q^{97}+O(q^{100}) $ 8 * q - 16 * q^13 + 72 * q^37 - 8 * q^49 - 64 * q^61 + 24 * q^85 - 16 * 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
$5$	0.317837i	0.142141i	0.997471	+	0.0710706i	$0.0226416\pi$
$5$	0.317837i	0.142141i	−0.997471	+	0.0710706i	$0.977358\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	−4.44949	−1.23407	−0.617033	−	0.786937i	$-0.711666\pi$
$13$	−4.44949	−1.23407	−0.617033	+	0.786937i	$0.711666\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.97469i	1.44908i	0.689235	+	0.724538i	$0.257947\pi$
$17$	5.97469i	1.44908i	−0.689235	+	0.724538i	$0.742053\pi$
$18$	0	0
$19$	2.44949i	0.561951i	0.959715	+	0.280976i	$0.0906580\pi$
$19$	2.44949i	0.561951i	−0.959715	+	0.280976i	$0.909342\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.65685	1.17954	0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685	1.17954	0.589768	+	0.807573i	$0.299219\pi$
$24$	0	0
$25$	4.89898	0.979796
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 5.51399i	− 1.02392i	−0.859009	−	0.511961i	$-0.828919\pi$
$29$	− 5.51399i	− 1.02392i	0.859009	−	0.511961i	$-0.171081\pi$
$30$	0	0
$31$	− 6.44949i	− 1.15836i	−0.815199	−	0.579181i	$-0.803372\pi$
$31$	− 6.44949i	− 1.15836i	0.815199	−	0.579181i	$-0.196628\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.317837	−0.0537243
$36$	0	0
$37$	9.00000	1.47959	0.739795	−	0.672832i	$-0.234922\pi$
$37$	9.00000	1.47959	0.739795	+	0.672832i	$0.234922\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 7.24604i	− 1.13164i	−0.824528	−	0.565821i	$-0.808559\pi$
$41$	− 7.24604i	− 1.13164i	0.824528	−	0.565821i	$-0.191441\pi$
$42$	0	0
$43$	6.34847i	0.968132i	0.875031	+	0.484066i	$0.160841\pi$
$43$	6.34847i	0.968132i	−0.875031	+	0.484066i	$0.839159\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.8530	1.58307	0.791537	−	0.611121i	$-0.209281\pi$
$47$	10.8530	1.58307	0.791537	+	0.611121i	$0.209281\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.9494i	1.64137i	0.571378	+	0.820687i	$0.306409\pi$
$53$	11.9494i	1.64137i	−0.571378	+	0.820687i	$0.693591\pi$
$54$	0	0
$55$	− 0.449490i	− 0.0606092i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.83183	−0.759239	−0.379620	−	0.925143i	$-0.623945\pi$
$59$	−5.83183	−0.759239	−0.379620	+	0.925143i	$0.623945\pi$
$60$	0	0
$61$	−10.4495	−1.33792	−0.668960	−	0.743298i	$-0.733260\pi$
$61$	−10.4495	−1.33792	−0.668960	+	0.743298i	$0.733260\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.41421i	− 0.175412i
$66$	0	0
$67$	5.79796i	0.708333i	0.935182	+	0.354167i	$0.115235\pi$
$67$	5.79796i	0.708333i	−0.935182	+	0.354167i	$0.884765\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	−9.79796	−1.14676	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	−9.79796	−1.14676	−0.573382	+	0.819288i	$0.694369\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.41421i	− 0.161165i
$78$	0	0
$79$	14.3485i	1.61433i	0.590327	+	0.807164i	$0.298999\pi$
$79$	14.3485i	1.61433i	−0.590327	+	0.807164i	$0.701001\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.02458	−0.880812	−0.440406	−	0.897799i	$-0.645165\pi$
$83$	−8.02458	−0.880812	−0.440406	+	0.897799i	$0.645165\pi$
$84$	0	0
$85$	−1.89898	−0.205973
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 15.4135i	− 1.63383i	−0.576761	−	0.816913i	$-0.695684\pi$
$89$	− 15.4135i	− 1.63383i	0.576761	−	0.816913i	$-0.304316\pi$
$90$	0	0
$91$	− 4.44949i	− 0.466433i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.778539	−0.0798764
$96$	0	0
$97$	−14.2474	−1.44661	−0.723305	−	0.690529i	$-0.757378\pi$
$97$	−14.2474	−1.44661	−0.723305	+	0.690529i	$0.757378\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 5.65685i	− 0.562878i	−0.959579	−	0.281439i	$-0.909188\pi$
$101$	− 5.65685i	− 0.562878i	0.959579	−	0.281439i	$-0.0908117\pi$
$102$	0	0
$103$	13.7980i	1.35955i	0.733419	+	0.679777i	$0.237923\pi$
$103$	13.7980i	1.35955i	−0.733419	+	0.679777i	$0.762077\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.8564	−1.33955	−0.669775	−	0.742564i	$-0.733609\pi$
$107$	−13.8564	−1.33955	−0.669775	+	0.742564i	$0.733609\pi$
$108$	0	0
$109$	−18.7980	−1.80052	−0.900259	−	0.435355i	$-0.856623\pi$
$109$	−18.7980	−1.80052	−0.900259	+	0.435355i	$0.856623\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.60697i	0.339315i	0.985503	+	0.169657i	$0.0542661\pi$
$113$	3.60697i	0.339315i	−0.985503	+	0.169657i	$0.945734\pi$
$114$	0	0
$115$	1.79796i	0.167661i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.97469	−0.547699
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.14626i	0.281410i
$126$	0	0
$127$	17.4495i	1.54839i	0.632946	+	0.774196i	$0.281845\pi$
$127$	17.4495i	1.54839i	−0.632946	+	0.774196i	$0.718155\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.7778	1.29114	0.645572	−	0.763700i	$-0.276619\pi$
$131$	14.7778	1.29114	0.645572	+	0.763700i	$0.276619\pi$
$132$	0	0
$133$	−2.44949	−0.212398
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.2706i	1.30466i	0.757936	+	0.652329i	$0.226208\pi$
$137$	15.2706i	1.30466i	−0.757936	+	0.652329i	$0.773792\pi$
$138$	0	0
$139$	− 4.44949i	− 0.377401i	−0.982035	−	0.188700i	$-0.939572\pi$
$139$	− 4.44949i	− 0.377401i	0.982035	−	0.188700i	$-0.0604275\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.29253	0.526208
$144$	0	0
$145$	1.75255	0.145541
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.32124i	0.272086i	0.990703	+	0.136043i	$0.0434386\pi$
$149$	3.32124i	0.272086i	−0.990703	+	0.136043i	$0.956561\pi$
$150$	0	0
$151$	− 6.34847i	− 0.516631i	−0.966061	−	0.258316i	$-0.916833\pi$
$151$	− 6.34847i	− 0.516631i	0.966061	−	0.258316i	$-0.0831674\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.04989	0.164651
$156$	0	0
$157$	14.2474	1.13707	0.568535	−	0.822659i	$-0.307510\pi$
$157$	14.2474	1.13707	0.568535	+	0.822659i	$0.307510\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.65685i	0.445823i
$162$	0	0
$163$	5.44949i	0.426837i	0.976961	+	0.213418i	$0.0684598\pi$
$163$	5.44949i	0.426837i	−0.976961	+	0.213418i	$0.931540\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.0458	1.00951	0.504756	−	0.863262i	$-0.331583\pi$
$167$	13.0458	1.00951	0.504756	+	0.863262i	$0.331583\pi$
$168$	0	0
$169$	6.79796	0.522920
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.7778i	1.12354i	0.827295	+	0.561768i	$0.189879\pi$
$173$	14.7778i	1.12354i	−0.827295	+	0.561768i	$0.810121\pi$
$174$	0	0
$175$	4.89898i	0.370328i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.68556	0.200728	0.100364	−	0.994951i	$-0.467999\pi$
$179$	2.68556	0.200728	0.100364	+	0.994951i	$0.467999\pi$
$180$	0	0
$181$	−15.7980	−1.17425	−0.587127	−	0.809495i	$-0.699741\pi$
$181$	−15.7980	−1.17425	−0.587127	+	0.809495i	$0.699741\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.86054i	0.210311i
$186$	0	0
$187$	− 8.44949i	− 0.617888i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.7778	−1.06928	−0.534642	−	0.845078i	$-0.679554\pi$
$191$	−14.7778	−1.06928	−0.534642	+	0.845078i	$0.679554\pi$
$192$	0	0
$193$	−18.5959	−1.33856	−0.669282	−	0.743009i	$-0.733398\pi$
$193$	−18.5959	−1.33856	−0.669282	+	0.743009i	$0.733398\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.89949i	0.705310i	0.935753	+	0.352655i	$0.114721\pi$
$197$	9.89949i	0.705310i	−0.935753	+	0.352655i	$0.885279\pi$
$198$	0	0
$199$	16.6969i	1.18361i	0.806079	+	0.591807i	$0.201585\pi$
$199$	16.6969i	1.18361i	−0.806079	+	0.591807i	$0.798415\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.51399	0.387006
$204$	0	0
$205$	2.30306	0.160853
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 3.46410i	− 0.239617i
$210$	0	0
$211$	21.5959i	1.48672i	0.668889	+	0.743362i	$0.266770\pi$
$211$	21.5959i	1.48672i	−0.668889	+	0.743362i	$0.733230\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.01778	−0.137611
$216$	0	0
$217$	6.44949	0.437820
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 26.5843i	− 1.78826i
$222$	0	0
$223$	− 3.75255i	− 0.251289i	−0.992075	−	0.125645i	$-0.959900\pi$
$223$	− 3.75255i	− 0.251289i	0.992075	−	0.125645i	$-0.0400999\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.0280	−0.731953	−0.365976	−	0.930624i	$-0.619265\pi$
$227$	−11.0280	−0.731953	−0.365976	+	0.930624i	$0.619265\pi$
$228$	0	0
$229$	4.20204	0.277679	0.138839	−	0.990315i	$-0.455663\pi$
$229$	4.20204	0.277679	0.138839	+	0.990315i	$0.455663\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.07107i	0.463241i	0.972806	+	0.231621i	$0.0744028\pi$
$233$	7.07107i	0.463241i	−0.972806	+	0.231621i	$0.925597\pi$
$234$	0	0
$235$	3.44949i	0.225020i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.1988	1.43592	0.717961	−	0.696083i	$-0.245075\pi$
$239$	22.1988	1.43592	0.717961	+	0.696083i	$0.245075\pi$
$240$	0	0
$241$	−5.34847	−0.344525	−0.172263	−	0.985051i	$-0.555108\pi$
$241$	−5.34847	−0.344525	−0.172263	+	0.985051i	$0.555108\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 0.317837i	− 0.0203059i
$246$	0	0
$247$	− 10.8990i	− 0.693485i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3.28913	−0.207608	−0.103804	−	0.994598i	$-0.533101\pi$
$251$	−3.28913	−0.207608	−0.103804	+	0.994598i	$0.533101\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.1421i	− 0.882162i	−0.897467	−	0.441081i	$-0.854595\pi$
$257$	− 14.1421i	− 0.882162i	0.897467	−	0.441081i	$-0.145405\pi$
$258$	0	0
$259$	9.00000i	0.559233i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.9063	0.980824	0.490412	−	0.871491i	$-0.336846\pi$
$263$	15.9063	0.980824	0.490412	+	0.871491i	$0.336846\pi$
$264$	0	0
$265$	−3.79796	−0.233307
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 19.4812i	− 1.18779i	−0.804544	−	0.593893i	$-0.797590\pi$
$269$	− 19.4812i	− 1.18779i	0.804544	−	0.593893i	$-0.202410\pi$
$270$	0	0
$271$	− 20.6969i	− 1.25725i	−0.777709	−	0.628625i	$-0.783618\pi$
$271$	− 20.6969i	− 1.25725i	0.777709	−	0.628625i	$-0.216382\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.92820	−0.417786
$276$	0	0
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 18.2419i	− 1.08822i	−0.839013	−	0.544111i	$-0.816867\pi$
$281$	− 18.2419i	− 1.08822i	0.839013	−	0.544111i	$-0.183133\pi$
$282$	0	0
$283$	17.3485i	1.03126i	0.856812	+	0.515630i	$0.172442\pi$
$283$	17.3485i	1.03126i	−0.856812	+	0.515630i	$0.827558\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.24604	0.427720
$288$	0	0
$289$	−18.6969	−1.09982
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 8.16744i	− 0.477147i	−0.971124	−	0.238574i	$-0.923320\pi$
$293$	− 8.16744i	− 0.477147i	0.971124	−	0.238574i	$-0.0766798\pi$
$294$	0	0
$295$	− 1.85357i	− 0.107919i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−25.1701	−1.45563
$300$	0	0
$301$	−6.34847	−0.365920
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 3.32124i	− 0.190173i
$306$	0	0
$307$	− 21.7980i	− 1.24408i	−0.782987	−	0.622038i	$-0.786305\pi$
$307$	− 21.7980i	− 1.24408i	0.782987	−	0.622038i	$-0.213695\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.3595	1.38130	0.690649	−	0.723190i	$-0.257325\pi$
$311$	24.3595	1.38130	0.690649	+	0.723190i	$0.257325\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.27135i	− 0.0714061i	−0.999362	−	0.0357030i	$-0.988633\pi$
$317$	− 1.27135i	− 0.0714061i	0.999362	−	0.0357030i	$-0.0113670\pi$
$318$	0	0
$319$	7.79796i	0.436602i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−14.6349	−0.814310
$324$	0	0
$325$	−21.7980	−1.20913
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10.8530i	0.598346i
$330$	0	0
$331$	6.14643i	0.337838i	0.985630	+	0.168919i	$0.0540277\pi$
$331$	6.14643i	0.337838i	−0.985630	+	0.168919i	$0.945972\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.84281	−0.100683
$336$	0	0
$337$	18.5959	1.01298	0.506492	−	0.862245i	$-0.330942\pi$
$337$	18.5959	1.01298	0.506492	+	0.862245i	$0.330942\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9.12096i	0.493927i
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.7627	−1.59775	−0.798873	−	0.601500i	$-0.794570\pi$
$347$	−29.7627	−1.59775	−0.798873	+	0.601500i	$0.794570\pi$
$348$	0	0
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.51739i	0.453335i	0.973972	+	0.226667i	$0.0727831\pi$
$353$	8.51739i	0.453335i	−0.973972	+	0.226667i	$0.927217\pi$
$354$	0	0
$355$	− 1.79796i	− 0.0954258i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.3923	0.548485	0.274242	−	0.961661i	$-0.411573\pi$
$359$	10.3923	0.548485	0.274242	+	0.961661i	$0.411573\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3.11416i	− 0.163002i
$366$	0	0
$367$	26.8990i	1.40412i	0.712120	+	0.702058i	$0.247735\pi$
$367$	26.8990i	1.40412i	−0.712120	+	0.702058i	$0.752265\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.9494	−0.620381
$372$	0	0
$373$	−21.8990	−1.13389	−0.566943	−	0.823757i	$-0.691874\pi$
$373$	−21.8990	−1.13389	−0.566943	+	0.823757i	$0.691874\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.5344i	1.26359i
$378$	0	0
$379$	33.0454i	1.69743i	0.528852	+	0.848714i	$0.322623\pi$
$379$	33.0454i	1.69743i	−0.528852	+	0.848714i	$0.677377\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.36773	−0.120985	−0.0604926	−	0.998169i	$-0.519267\pi$
$383$	−2.36773	−0.120985	−0.0604926	+	0.998169i	$0.519267\pi$
$384$	0	0
$385$	0.449490	0.0229081
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.16404i	0.261827i	0.991394	+	0.130914i	$0.0417911\pi$
$389$	5.16404i	0.261827i	−0.991394	+	0.130914i	$0.958209\pi$
$390$	0	0
$391$	33.7980i	1.70924i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.56048	−0.229463
$396$	0	0
$397$	2.20204	0.110517	0.0552586	−	0.998472i	$-0.482402\pi$
$397$	2.20204	0.110517	0.0552586	+	0.998472i	$0.482402\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 37.1195i	− 1.85366i	−0.375482	−	0.926830i	$-0.622523\pi$
$401$	− 37.1195i	− 1.85366i	0.375482	−	0.926830i	$-0.377477\pi$
$402$	0	0
$403$	28.6969i	1.42950i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.7279	−0.630900
$408$	0	0
$409$	−13.7980	−0.682265	−0.341133	−	0.940015i	$-0.610811\pi$
$409$	−13.7980	−0.682265	−0.341133	+	0.940015i	$0.610811\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 5.83183i	− 0.286965i
$414$	0	0
$415$	− 2.55051i	− 0.125200i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−8.02458	−0.392026	−0.196013	−	0.980601i	$-0.562800\pi$
$419$	−8.02458	−0.392026	−0.196013	+	0.980601i	$0.562800\pi$
$420$	0	0
$421$	23.1010	1.12587	0.562937	−	0.826500i	$-0.309671\pi$
$421$	23.1010	1.12587	0.562937	+	0.826500i	$0.309671\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	29.2699i	1.41980i
$426$	0	0
$427$	− 10.4495i	− 0.505686i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.92820	−0.333720	−0.166860	−	0.985981i	$-0.553363\pi$
$431$	−6.92820	−0.333720	−0.166860	+	0.985981i	$0.553363\pi$
$432$	0	0
$433$	21.5505	1.03565	0.517826	−	0.855486i	$-0.326742\pi$
$433$	21.5505	1.03565	0.517826	+	0.855486i	$0.326742\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13.8564i	0.662842i
$438$	0	0
$439$	14.8990i	0.711089i	0.934659	+	0.355545i	$0.115705\pi$
$439$	14.8990i	0.711089i	−0.934659	+	0.355545i	$0.884295\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	38.8194	1.84437	0.922184	−	0.386752i	$-0.126403\pi$
$443$	38.8194	1.84437	0.922184	+	0.386752i	$0.126403\pi$
$444$	0	0
$445$	4.89898	0.232234
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 1.12848i	− 0.0532565i	−0.999645	−	0.0266282i	$-0.991523\pi$
$449$	− 1.12848i	− 0.0532565i	0.999645	−	0.0266282i	$-0.00847703\pi$
$450$	0	0
$451$	10.2474i	0.482534i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.41421	0.0662994
$456$	0	0
$457$	−27.5959	−1.29088	−0.645441	−	0.763810i	$-0.723327\pi$
$457$	−27.5959	−1.29088	−0.645441	+	0.763810i	$0.723327\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.26042i	0.291577i	0.989316	+	0.145788i	$0.0465719\pi$
$461$	6.26042i	0.291577i	−0.989316	+	0.145788i	$0.953428\pi$
$462$	0	0
$463$	− 6.75255i	− 0.313818i	−0.987613	−	0.156909i	$-0.949847\pi$
$463$	− 6.75255i	− 0.313818i	0.987613	−	0.156909i	$-0.0501529\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.1981	−1.67505	−0.837524	−	0.546401i	$-0.815998\pi$
$467$	−36.1981	−1.67505	−0.837524	+	0.546401i	$0.815998\pi$
$468$	0	0
$469$	−5.79796	−0.267725
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 8.97809i	− 0.412813i
$474$	0	0
$475$	12.0000i	0.550598i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.9951	1.14206	0.571029	−	0.820930i	$-0.306545\pi$
$479$	24.9951	1.14206	0.571029	+	0.820930i	$0.306545\pi$
$480$	0	0
$481$	−40.0454	−1.82591
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 4.52837i	− 0.205623i
$486$	0	0
$487$	19.5959i	0.887976i	0.896033	+	0.443988i	$0.146437\pi$
$487$	19.5959i	0.887976i	−0.896033	+	0.443988i	$0.853563\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.32124	−0.149885	−0.0749427	−	0.997188i	$-0.523877\pi$
$491$	−3.32124	−0.149885	−0.0749427	+	0.997188i	$0.523877\pi$
$492$	0	0
$493$	32.9444	1.48374
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.65685i	− 0.253745i
$498$	0	0
$499$	0.752551i	0.0336888i	0.999858	+	0.0168444i	$0.00536200\pi$
$499$	0.752551i	0.0336888i	−0.999858	+	0.0168444i	$0.994638\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.7600	0.568942	0.284471	−	0.958685i	$-0.408182\pi$
$503$	12.7600	0.568942	0.284471	+	0.958685i	$0.408182\pi$
$504$	0	0
$505$	1.79796	0.0800081
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 12.2672i	− 0.543735i	−0.962335	−	0.271867i	$-0.912359\pi$
$509$	− 12.2672i	− 0.543735i	0.962335	−	0.271867i	$-0.0876412\pi$
$510$	0	0
$511$	− 9.79796i	− 0.433436i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.38551	−0.193248
$516$	0	0
$517$	−15.3485	−0.675025
$518$	0	0
$519$	0	0
$520$	0	0
$521$	25.4880i	1.11665i	0.829623	+	0.558324i	$0.188555\pi$
$521$	25.4880i	1.11665i	−0.829623	+	0.558324i	$0.811445\pi$
$522$	0	0
$523$	32.0454i	1.40125i	0.713531	+	0.700624i	$0.247095\pi$
$523$	32.0454i	1.40125i	−0.713531	+	0.700624i	$0.752905\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	38.5337	1.67855
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	32.2412i	1.39652i
$534$	0	0
$535$	− 4.40408i	− 0.190405i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.41421	0.0609145
$540$	0	0
$541$	−29.0000	−1.24681	−0.623404	−	0.781900i	$-0.714251\pi$
$541$	−29.0000	−1.24681	−0.623404	+	0.781900i	$0.714251\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 5.97469i	− 0.255928i
$546$	0	0
$547$	− 34.1464i	− 1.46000i	−0.683449	−	0.729998i	$-0.739521\pi$
$547$	− 34.1464i	− 1.46000i	0.683449	−	0.729998i	$-0.260479\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13.5065	0.575395
$552$	0	0
$553$	−14.3485	−0.610159
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 27.7128i	− 1.17423i	−0.809504	−	0.587115i	$-0.800264\pi$
$557$	− 27.7128i	− 1.17423i	0.809504	−	0.587115i	$-0.199736\pi$
$558$	0	0
$559$	− 28.2474i	− 1.19474i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	46.5262	1.96084	0.980422	−	0.196906i	$-0.0630893\pi$
$563$	46.5262	1.96084	0.980422	+	0.196906i	$0.0630893\pi$
$564$	0	0
$565$	−1.14643	−0.0482306
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.2419i	0.764741i	0.924009	+	0.382370i	$0.124892\pi$
$569$	18.2419i	0.764741i	−0.924009	+	0.382370i	$0.875108\pi$
$570$	0	0
$571$	0.348469i	0.0145830i	0.999973	+	0.00729149i	$0.00232097\pi$
$571$	0.348469i	0.0145830i	−0.999973	+	0.00729149i	$0.997679\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	27.7128	1.15570
$576$	0	0
$577$	37.5959	1.56514	0.782569	−	0.622564i	$-0.213909\pi$
$577$	37.5959	1.56514	0.782569	+	0.622564i	$0.213909\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 8.02458i	− 0.332916i
$582$	0	0
$583$	− 16.8990i	− 0.699884i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−30.5412	−1.26057	−0.630286	−	0.776363i	$-0.717062\pi$
$587$	−30.5412	−1.26057	−0.630286	+	0.776363i	$0.717062\pi$
$588$	0	0
$589$	15.7980	0.650944
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.8523i	1.02056i	0.860008	+	0.510280i	$0.170458\pi$
$593$	24.8523i	1.02056i	−0.860008	+	0.510280i	$0.829542\pi$
$594$	0	0
$595$	− 1.89898i	− 0.0778506i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26.2986	1.07453	0.537266	−	0.843413i	$-0.319457\pi$
$599$	26.2986	1.07453	0.537266	+	0.843413i	$0.319457\pi$
$600$	0	0
$601$	−26.7423	−1.09084	−0.545422	−	0.838162i	$-0.683630\pi$
$601$	−26.7423	−1.09084	−0.545422	+	0.838162i	$0.683630\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 2.86054i	− 0.116297i
$606$	0	0
$607$	− 26.2474i	− 1.06535i	−0.846320	−	0.532676i	$-0.821187\pi$
$607$	− 26.2474i	− 1.06535i	0.846320	−	0.532676i	$-0.178813\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−48.2903	−1.95362
$612$	0	0
$613$	17.3939	0.702532	0.351266	−	0.936276i	$-0.385751\pi$
$613$	17.3939	0.702532	0.351266	+	0.936276i	$0.385751\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 18.2419i	− 0.734392i	−0.930144	−	0.367196i	$-0.880318\pi$
$617$	− 18.2419i	− 0.734392i	0.930144	−	0.367196i	$-0.119682\pi$
$618$	0	0
$619$	− 15.8434i	− 0.636799i	−0.947957	−	0.318399i	$-0.896855\pi$
$619$	− 15.8434i	− 0.636799i	0.947957	−	0.318399i	$-0.103145\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	15.4135	0.617528
$624$	0	0
$625$	23.4949	0.939796
$626$	0	0
$627$	0	0
$628$	0	0
$629$	53.7722i	2.14404i
$630$	0	0
$631$	39.7423i	1.58212i	0.611740	+	0.791059i	$0.290470\pi$
$631$	39.7423i	1.58212i	−0.611740	+	0.791059i	$0.709530\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.54610	−0.220090
$636$	0	0
$637$	4.44949	0.176295
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.16404i	0.203967i	0.994786	+	0.101984i	$0.0325190\pi$
$641$	5.16404i	0.203967i	−0.994786	+	0.101984i	$0.967481\pi$
$642$	0	0
$643$	− 10.4949i	− 0.413878i	−0.978354	−	0.206939i	$-0.933650\pi$
$643$	− 10.4949i	− 0.413878i	0.978354	−	0.206939i	$-0.0663502\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−49.3546	−1.94033	−0.970165	−	0.242446i	$-0.922050\pi$
$647$	−49.3546	−1.94033	−0.970165	+	0.242446i	$0.922050\pi$
$648$	0	0
$649$	8.24745	0.323741
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 13.3636i	− 0.522958i	−0.965209	−	0.261479i	$-0.915790\pi$
$653$	− 13.3636i	− 0.522958i	0.965209	−	0.261479i	$-0.0842102\pi$
$654$	0	0
$655$	4.69694i	0.183525i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−33.0197	−1.28627	−0.643133	−	0.765755i	$-0.722366\pi$
$659$	−33.0197	−1.28627	−0.643133	+	0.765755i	$0.722366\pi$
$660$	0	0
$661$	−2.04541	−0.0795571	−0.0397786	−	0.999209i	$-0.512665\pi$
$661$	−2.04541	−0.0795571	−0.0397786	+	0.999209i	$0.512665\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 0.778539i	− 0.0301905i
$666$	0	0
$667$	− 31.1918i	− 1.20775i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	14.7778	0.570491
$672$	0	0
$673$	−3.10102	−0.119536	−0.0597678	−	0.998212i	$-0.519036\pi$
$673$	−3.10102	−0.119536	−0.0597678	+	0.998212i	$0.519036\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 27.9985i	− 1.07607i	−0.842922	−	0.538036i	$-0.819167\pi$
$677$	− 27.9985i	− 1.07607i	0.842922	−	0.538036i	$-0.180833\pi$
$678$	0	0
$679$	− 14.2474i	− 0.546767i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−26.2344	−1.00383	−0.501915	−	0.864917i	$-0.667371\pi$
$683$	−26.2344	−1.00383	−0.501915	+	0.864917i	$0.667371\pi$
$684$	0	0
$685$	−4.85357	−0.185445
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 53.1687i	− 2.02556i
$690$	0	0
$691$	− 12.9444i	− 0.492428i	−0.969216	−	0.246214i	$-0.920813\pi$
$691$	− 12.9444i	− 0.492428i	0.969216	−	0.246214i	$-0.0791866\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.41421	0.0536442
$696$	0	0
$697$	43.2929	1.63983
$698$	0	0
$699$	0	0
$700$	0	0
$701$	50.1331i	1.89350i	0.321965	+	0.946751i	$0.395657\pi$
$701$	50.1331i	1.89350i	−0.321965	+	0.946751i	$0.604343\pi$
$702$	0	0
$703$	22.0454i	0.831458i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.65685	0.212748
$708$	0	0
$709$	−6.39388	−0.240127	−0.120064	−	0.992766i	$-0.538310\pi$
$709$	−6.39388	−0.240127	−0.120064	+	0.992766i	$0.538310\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 36.4838i	− 1.36633i
$714$	0	0
$715$	2.00000i	0.0747958i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.38211	0.0515438	0.0257719	−	0.999668i	$-0.491796\pi$
$719$	1.38211	0.0515438	0.0257719	+	0.999668i	$0.491796\pi$
$720$	0	0
$721$	−13.7980	−0.513863
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 27.0129i	− 1.00323i
$726$	0	0
$727$	− 18.0454i	− 0.669267i	−0.942348	−	0.334634i	$-0.891387\pi$
$727$	− 18.0454i	− 0.669267i	0.942348	−	0.334634i	$-0.108613\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−37.9301	−1.40290
$732$	0	0
$733$	−21.5959	−0.797663	−0.398832	−	0.917024i	$-0.630584\pi$
$733$	−21.5959	−0.797663	−0.398832	+	0.917024i	$0.630584\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.19955i	− 0.302034i
$738$	0	0
$739$	− 20.0000i	− 0.735712i	−0.929883	−	0.367856i	$-0.880092\pi$
$739$	− 20.0000i	− 0.735712i	0.929883	−	0.367856i	$-0.119908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.76416	−0.0647207	−0.0323604	−	0.999476i	$-0.510302\pi$
$743$	−1.76416	−0.0647207	−0.0323604	+	0.999476i	$0.510302\pi$
$744$	0	0
$745$	−1.05561	−0.0386747
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 13.8564i	− 0.506302i
$750$	0	0
$751$	6.20204i	0.226316i	0.993577	+	0.113158i	$0.0360966\pi$
$751$	6.20204i	0.226316i	−0.993577	+	0.113158i	$0.963903\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.01778	0.0734345
$756$	0	0
$757$	−4.30306	−0.156397	−0.0781987	−	0.996938i	$-0.524917\pi$
$757$	−4.30306	−0.156397	−0.0781987	+	0.996938i	$0.524917\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.05329i	0.183182i	0.995797	+	0.0915908i	$0.0291952\pi$
$761$	5.05329i	0.183182i	−0.995797	+	0.0915908i	$0.970805\pi$
$762$	0	0
$763$	− 18.7980i	− 0.680532i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25.9487	0.936952
$768$	0	0
$769$	16.2020	0.584261	0.292130	−	0.956379i	$-0.405636\pi$
$769$	16.2020	0.584261	0.292130	+	0.956379i	$0.405636\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 29.8735i	− 1.07447i	−0.843431	−	0.537237i	$-0.819468\pi$
$773$	− 29.8735i	− 1.07447i	0.843431	−	0.537237i	$-0.180532\pi$
$774$	0	0
$775$	− 31.5959i	− 1.13496i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	17.7491	0.635928
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.52837i	0.161624i
$786$	0	0
$787$	21.3939i	0.762609i	0.924449	+	0.381305i	$0.124525\pi$
$787$	21.3939i	0.762609i	−0.924449	+	0.381305i	$0.875475\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.60697	−0.128249
$792$	0	0
$793$	46.4949	1.65108
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0.985620i	0.0349124i	0.999848	+	0.0174562i	$0.00555677\pi$
$797$	0.985620i	0.0349124i	−0.999848	+	0.0174562i	$0.994443\pi$
$798$	0	0
$799$	64.8434i	2.29399i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13.8564	0.488982
$804$	0	0
$805$	−1.79796	−0.0633697
$806$	0	0
$807$	0	0
$808$	0	0
$809$	42.7764i	1.50394i	0.659199	+	0.751968i	$0.270895\pi$
$809$	42.7764i	1.50394i	−0.659199	+	0.751968i	$0.729105\pi$
$810$	0	0
$811$	28.0454i	0.984807i	0.870367	+	0.492404i	$0.163882\pi$
$811$	28.0454i	0.984807i	−0.870367	+	0.492404i	$0.836118\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.73205	−0.0606711
$816$	0	0
$817$	−15.5505	−0.544043
$818$	0	0
$819$	0	0
$820$	0	0
$821$	47.2261i	1.64820i	0.566443	+	0.824101i	$0.308319\pi$
$821$	47.2261i	1.64820i	−0.566443	+	0.824101i	$0.691681\pi$
$822$	0	0
$823$	− 20.8434i	− 0.726554i	−0.931681	−	0.363277i	$-0.881658\pi$
$823$	− 20.8434i	− 0.726554i	0.931681	−	0.363277i	$-0.118342\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.68556	−0.0933862	−0.0466931	−	0.998909i	$-0.514868\pi$
$827$	−2.68556	−0.0933862	−0.0466931	+	0.998909i	$0.514868\pi$
$828$	0	0
$829$	−7.95459	−0.276274	−0.138137	−	0.990413i	$-0.544111\pi$
$829$	−7.95459	−0.276274	−0.138137	+	0.990413i	$0.544111\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 5.97469i	− 0.207011i
$834$	0	0
$835$	4.14643i	0.143493i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.09638	−0.0378511	−0.0189256	−	0.999821i	$-0.506025\pi$
$839$	−1.09638	−0.0378511	−0.0189256	+	0.999821i	$0.506025\pi$
$840$	0	0
$841$	−1.40408	−0.0484166
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.16064i	0.0743284i
$846$	0	0
$847$	− 9.00000i	− 0.309244i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	50.9117	1.74523
$852$	0	0
$853$	43.7980	1.49961	0.749807	−	0.661657i	$-0.230146\pi$
$853$	43.7980	1.49961	0.749807	+	0.661657i	$0.230146\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.6021i	0.977029i	0.872556	+	0.488515i	$0.162461\pi$
$857$	28.6021i	0.977029i	−0.872556	+	0.488515i	$0.837539\pi$
$858$	0	0
$859$	− 38.8990i	− 1.32722i	−0.748080	−	0.663608i	$-0.769024\pi$
$859$	− 38.8990i	− 1.32722i	0.748080	−	0.663608i	$-0.230976\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13.5065	−0.459765	−0.229883	−	0.973218i	$-0.573834\pi$
$863$	−13.5065	−0.459765	−0.229883	+	0.973218i	$0.573834\pi$
$864$	0	0
$865$	−4.69694	−0.159701
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 20.2918i	− 0.688352i
$870$	0	0
$871$	− 25.7980i	− 0.874130i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.14626	−0.106363
$876$	0	0
$877$	−0.595918	−0.0201227	−0.0100614	−	0.999949i	$-0.503203\pi$
$877$	−0.595918	−0.0201227	−0.0100614	+	0.999949i	$0.503203\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 48.4974i	− 1.63392i	−0.576695	−	0.816960i	$-0.695658\pi$
$881$	− 48.4974i	− 1.63392i	0.576695	−	0.816960i	$-0.304342\pi$
$882$	0	0
$883$	− 22.3485i	− 0.752086i	−0.926602	−	0.376043i	$-0.877285\pi$
$883$	− 22.3485i	− 0.752086i	0.926602	−	0.376043i	$-0.122715\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.2030	0.376158	0.188079	−	0.982154i	$-0.439774\pi$
$887$	11.2030	0.376158	0.188079	+	0.982154i	$0.439774\pi$
$888$	0	0
$889$	−17.4495	−0.585237
$890$	0	0
$891$	0	0
$892$	0	0
$893$	26.5843i	0.889611i
$894$	0	0
$895$	0.853572i	0.0285318i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−35.5624	−1.18607
$900$	0	0
$901$	−71.3939	−2.37847
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 5.02118i	− 0.166910i
$906$	0	0
$907$	− 1.24745i	− 0.0414209i	−0.999786	−	0.0207104i	$-0.993407\pi$
$907$	− 1.24745i	− 0.0414209i	0.999786	−	0.0207104i	$-0.00659281\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−21.4989	−0.712291	−0.356146	−	0.934430i	$-0.615909\pi$
$911$	−21.4989	−0.712291	−0.356146	+	0.934430i	$0.615909\pi$
$912$	0	0
$913$	11.3485	0.375580
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.7778i	0.488006i
$918$	0	0
$919$	− 34.8434i	− 1.14938i	−0.818372	−	0.574688i	$-0.805123\pi$
$919$	− 34.8434i	− 1.14938i	0.818372	−	0.574688i	$-0.194877\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	25.1701	0.828484
$924$	0	0
$925$	44.0908	1.44970
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 0.317837i	− 0.0104279i	−0.999986	−	0.00521395i	$-0.998340\pi$
$929$	− 0.317837i	− 0.0104279i	0.999986	−	0.00521395i	$-0.00165966\pi$
$930$	0	0
$931$	− 2.44949i	− 0.0802788i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.68556	0.0878273
$936$	0	0
$937$	−18.7423	−0.612286	−0.306143	−	0.951986i	$-0.599039\pi$
$937$	−18.7423	−0.612286	−0.306143	+	0.951986i	$0.599039\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 27.7449i	− 0.904459i	−0.891902	−	0.452229i	$-0.850629\pi$
$941$	− 27.7449i	− 0.904459i	0.891902	−	0.452229i	$-0.149371\pi$
$942$	0	0
$943$	− 40.9898i	− 1.33481i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.84281	−0.0598832	−0.0299416	−	0.999552i	$-0.509532\pi$
$947$	−1.84281	−0.0598832	−0.0299416	+	0.999552i	$0.509532\pi$
$948$	0	0
$949$	43.5959	1.41518
$950$	0	0
$951$	0	0
$952$	0	0
$953$	37.7552i	1.22301i	0.791241	+	0.611505i	$0.209436\pi$
$953$	37.7552i	1.22301i	−0.791241	+	0.611505i	$0.790564\pi$
$954$	0	0
$955$	− 4.69694i	− 0.151989i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−15.2706	−0.493114
$960$	0	0
$961$	−10.5959	−0.341804
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 5.91048i	− 0.190265i
$966$	0	0
$967$	3.10102i	0.0997221i	0.998756	+	0.0498610i	$0.0158778\pi$
$967$	3.10102i	0.0997221i	−0.998756	+	0.0498610i	$0.984122\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33.5446	1.07650	0.538249	−	0.842786i	$-0.319086\pi$
$971$	33.5446	1.07650	0.538249	+	0.842786i	$0.319086\pi$
$972$	0	0
$973$	4.44949	0.142644
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.17837i	0.101685i	0.998707	+	0.0508426i	$0.0161907\pi$
$977$	3.17837i	0.101685i	−0.998707	+	0.0508426i	$0.983809\pi$
$978$	0	0
$979$	21.7980i	0.696666i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	13.0458	0.416095	0.208048	−	0.978119i	$-0.433289\pi$
$983$	13.0458	0.416095	0.208048	+	0.978119i	$0.433289\pi$
$984$	0	0
$985$	−3.14643	−0.100254
$986$	0	0
$987$	0	0
$988$	0	0
$989$	35.9124i	1.14195i
$990$	0	0
$991$	22.3485i	0.709923i	0.934881	+	0.354961i	$0.115506\pi$
$991$	22.3485i	0.709923i	−0.934881	+	0.354961i	$0.884494\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.30691	−0.168240
$996$	0	0
$997$	21.8434	0.691786	0.345893	−	0.938274i	$-0.387576\pi$
$997$	21.8434	0.691786	0.345893	+	0.938274i	$0.387576\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.h.c.2591.6	yes	8
3.2	odd	2	inner	6048.2.h.c.2591.4	yes	8
4.3	odd	2	inner	6048.2.h.c.2591.5	yes	8
12.11	even	2	inner	6048.2.h.c.2591.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.c.2591.3	✓	8	12.11	even	2	inner
6048.2.h.c.2591.4	yes	8	3.2	odd	2	inner
6048.2.h.c.2591.5	yes	8	4.3	odd	2	inner
6048.2.h.c.2591.6	yes	8	1.1	even	1	trivial

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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.317837i q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+0.317837i q^{5} +1.00000i q^{7} -1.41421 q^{11} -4.44949 q^{13} +5.97469i q^{17} +2.44949i q^{19} +5.65685 q^{23} +4.89898 q^{25} -5.51399i q^{29} -6.44949i q^{31} -0.317837 q^{35} +9.00000 q^{37} -7.24604i q^{41} +6.34847i q^{43} +10.8530 q^{47} -1.00000 q^{49} +11.9494i q^{53} -0.449490i q^{55} -5.83183 q^{59} -10.4495 q^{61} -1.41421i q^{65} +5.79796i q^{67} -5.65685 q^{71} -9.79796 q^{73} -1.41421i q^{77} +14.3485i q^{79} -8.02458 q^{83} -1.89898 q^{85} -15.4135i q^{89} -4.44949i q^{91} -0.778539 q^{95} -14.2474 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{13} + 72 q^{37} - 8 q^{49} - 64 q^{61} + 24 q^{85} - 16 q^{97}+O(q^{100}) \) 8 * q - 16 * q^13 + 72 * q^37 - 8 * q^49 - 64 * q^61 + 24 * q^85 - 16 * q^97

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