LMFDB - Embedded newform 6048.2.c.f.3025.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6048.3025");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			3025.12
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.f.3025.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.940450i q^{5} -1.00000 q^{7} +O(q^{10})$	$q-0.940450i q^{5} -1.00000 q^{7} +5.98632i q^{11} +6.59017i q^{13} +2.64225 q^{17} -5.83430i q^{19} +2.88452 q^{23} +4.11555 q^{25} +3.09794i q^{29} -3.52412 q^{31} +0.940450i q^{35} +0.213560i q^{37} +1.63291 q^{41} +7.16716i q^{43} -9.32639 q^{47} +1.00000 q^{49} -7.51044i q^{53} +5.62983 q^{55} -11.9280i q^{59} -1.48304i q^{61} +6.19772 q^{65} +13.0555i q^{67} -1.54642 q^{71} -2.96871 q^{73} -5.98632i q^{77} -15.9515 q^{79} +8.74782i q^{83} -2.48490i q^{85} -7.50339 q^{89} -6.59017i q^{91} -5.48686 q^{95} +10.7529 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{7}+O(q^{10}) $ 24 * q - 24 * q^7	$ 24 q - 24 q^{7} - 24 q^{25} - 8 q^{31} + 24 q^{49} - 16 q^{55} - 8 q^{79}+O(q^{100}) $ 24 * q - 24 * q^7 - 24 * q^25 - 8 * q^31 + 24 * q^49 - 16 * q^55 - 8 * q^79

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 0.940450i	− 0.420582i	−0.977639	−	0.210291i	$-0.932559\pi$
$5$	− 0.940450i	− 0.420582i	0.977639	−	0.210291i	$-0.0674411\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.98632i	1.80494i	0.430749	+	0.902472i	$0.358249\pi$
$11$	5.98632i	1.80494i	−0.430749	+	0.902472i	$0.641751\pi$
$12$	0	0
$13$	6.59017i	1.82778i	0.405957	+	0.913892i	$0.366938\pi$
$13$	6.59017i	1.82778i	−0.405957	+	0.913892i	$0.633062\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.64225	0.640840	0.320420	−	0.947276i	$-0.396176\pi$
$17$	2.64225	0.640840	0.320420	+	0.947276i	$0.396176\pi$
$18$	0	0
$19$	− 5.83430i	− 1.33848i	−0.743047	−	0.669240i	$-0.766620\pi$
$19$	− 5.83430i	− 1.33848i	0.743047	−	0.669240i	$-0.233380\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.88452	0.601464	0.300732	−	0.953709i	$-0.402769\pi$
$23$	2.88452	0.601464	0.300732	+	0.953709i	$0.402769\pi$
$24$	0	0
$25$	4.11555	0.823111
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.09794i	0.575274i	0.957740	+	0.287637i	$0.0928696\pi$
$29$	3.09794i	0.575274i	−0.957740	+	0.287637i	$0.907130\pi$
$30$	0	0
$31$	−3.52412	−0.632950	−0.316475	−	0.948601i	$-0.602499\pi$
$31$	−3.52412	−0.632950	−0.316475	+	0.948601i	$0.602499\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.940450i	0.158965i
$36$	0	0
$37$	0.213560i	0.0351090i	0.999846	+	0.0175545i	$0.00558806\pi$
$37$	0.213560i	0.0351090i	−0.999846	+	0.0175545i	$0.994412\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.63291	0.255017	0.127509	−	0.991837i	$-0.459302\pi$
$41$	1.63291	0.255017	0.127509	+	0.991837i	$0.459302\pi$
$42$	0	0
$43$	7.16716i	1.09298i	0.837465	+	0.546491i	$0.184037\pi$
$43$	7.16716i	1.09298i	−0.837465	+	0.546491i	$0.815963\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.32639	−1.36039	−0.680197	−	0.733030i	$-0.738106\pi$
$47$	−9.32639	−1.36039	−0.680197	+	0.733030i	$0.738106\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.51044i	− 1.03164i	−0.856698	−	0.515819i	$-0.827488\pi$
$53$	− 7.51044i	− 1.03164i	0.856698	−	0.515819i	$-0.172512\pi$
$54$	0	0
$55$	5.62983	0.759126
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 11.9280i	− 1.55289i	−0.630186	−	0.776444i	$-0.717021\pi$
$59$	− 11.9280i	− 1.55289i	0.630186	−	0.776444i	$-0.282979\pi$
$60$	0	0
$61$	− 1.48304i	− 0.189884i	−0.995483	−	0.0949419i	$-0.969733\pi$
$61$	− 1.48304i	− 0.189884i	0.995483	−	0.0949419i	$-0.0302665\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.19772	0.768733
$66$	0	0
$67$	13.0555i	1.59498i	0.603329	+	0.797492i	$0.293840\pi$
$67$	13.0555i	1.59498i	−0.603329	+	0.797492i	$0.706160\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.54642	−0.183526	−0.0917632	−	0.995781i	$-0.529250\pi$
$71$	−1.54642	−0.183526	−0.0917632	+	0.995781i	$0.529250\pi$
$72$	0	0
$73$	−2.96871	−0.347461	−0.173731	−	0.984793i	$-0.555582\pi$
$73$	−2.96871	−0.347461	−0.173731	+	0.984793i	$0.555582\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.98632i	− 0.682204i
$78$	0	0
$79$	−15.9515	−1.79468	−0.897341	−	0.441338i	$-0.854504\pi$
$79$	−15.9515	−1.79468	−0.897341	+	0.441338i	$0.854504\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.74782i	0.960198i	0.877214	+	0.480099i	$0.159399\pi$
$83$	8.74782i	0.960198i	−0.877214	+	0.480099i	$0.840601\pi$
$84$	0	0
$85$	− 2.48490i	− 0.269526i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.50339	−0.795358	−0.397679	−	0.917525i	$-0.630184\pi$
$89$	−7.50339	−0.795358	−0.397679	+	0.917525i	$0.630184\pi$
$90$	0	0
$91$	− 6.59017i	− 0.690837i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.48686	−0.562940
$96$	0	0
$97$	10.7529	1.09179	0.545894	−	0.837854i	$-0.316190\pi$
$97$	10.7529	1.09179	0.545894	+	0.837854i	$0.316190\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 3.54920i	− 0.353159i	−0.984286	−	0.176579i	$-0.943497\pi$
$101$	− 3.54920i	− 0.353159i	0.984286	−	0.176579i	$-0.0565033\pi$
$102$	0	0
$103$	−15.3303	−1.51054	−0.755271	−	0.655413i	$-0.772495\pi$
$103$	−15.3303	−1.51054	−0.755271	+	0.655413i	$0.772495\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.41633i	0.233596i	0.993156	+	0.116798i	$0.0372629\pi$
$107$	2.41633i	0.233596i	−0.993156	+	0.116798i	$0.962737\pi$
$108$	0	0
$109$	6.86711i	0.657750i	0.944373	+	0.328875i	$0.106669\pi$
$109$	6.86711i	0.657750i	−0.944373	+	0.328875i	$0.893331\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.4539	1.45378	0.726891	−	0.686752i	$-0.240964\pi$
$113$	15.4539	1.45378	0.726891	+	0.686752i	$0.240964\pi$
$114$	0	0
$115$	− 2.71274i	− 0.252965i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.64225	−0.242215
$120$	0	0
$121$	−24.8360	−2.25782
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 8.57272i	− 0.766767i
$126$	0	0
$127$	−7.96760	−0.707010	−0.353505	−	0.935433i	$-0.615010\pi$
$127$	−7.96760	−0.707010	−0.353505	+	0.935433i	$0.615010\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.53780i	− 0.134358i	−0.997741	−	0.0671790i	$-0.978600\pi$
$131$	− 1.53780i	− 0.134358i	0.997741	−	0.0671790i	$-0.0213999\pi$
$132$	0	0
$133$	5.83430i	0.505898i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.34042	−0.541699	−0.270849	−	0.962622i	$-0.587305\pi$
$137$	−6.34042	−0.541699	−0.270849	+	0.962622i	$0.587305\pi$
$138$	0	0
$139$	8.13659i	0.690137i	0.938578	+	0.345068i	$0.112144\pi$
$139$	8.13659i	0.690137i	−0.938578	+	0.345068i	$0.887856\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−39.4509	−3.29905
$144$	0	0
$145$	2.91346	0.241950
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.2594i	1.25010i	0.780586	+	0.625048i	$0.214921\pi$
$149$	15.2594i	1.25010i	−0.780586	+	0.625048i	$0.785079\pi$
$150$	0	0
$151$	0.728441	0.0592797	0.0296398	−	0.999561i	$-0.490564\pi$
$151$	0.728441	0.0592797	0.0296398	+	0.999561i	$0.490564\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.31425i	0.266207i
$156$	0	0
$157$	12.1207i	0.967337i	0.875251	+	0.483669i	$0.160696\pi$
$157$	12.1207i	0.967337i	−0.875251	+	0.483669i	$0.839304\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.88452	−0.227332
$162$	0	0
$163$	− 5.45067i	− 0.426929i	−0.976951	−	0.213465i	$-0.931525\pi$
$163$	− 5.45067i	− 0.426929i	0.976951	−	0.213465i	$-0.0684748\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.96762	−0.152259	−0.0761297	−	0.997098i	$-0.524256\pi$
$167$	−1.96762	−0.152259	−0.0761297	+	0.997098i	$0.524256\pi$
$168$	0	0
$169$	−30.4303	−2.34079
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 4.48602i	− 0.341066i	−0.985352	−	0.170533i	$-0.945451\pi$
$173$	− 4.48602i	− 0.341066i	0.985352	−	0.170533i	$-0.0545489\pi$
$174$	0	0
$175$	−4.11555	−0.311107
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 24.8086i	− 1.85428i	−0.374716	−	0.927140i	$-0.622260\pi$
$179$	− 24.8086i	− 1.85428i	0.374716	−	0.927140i	$-0.377740\pi$
$180$	0	0
$181$	22.6114i	1.68070i	0.542048	+	0.840348i	$0.317649\pi$
$181$	22.6114i	1.68070i	−0.542048	+	0.840348i	$0.682351\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.200842	0.0147662
$186$	0	0
$187$	15.8174i	1.15668i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.3035	1.17968	0.589838	−	0.807521i	$-0.299191\pi$
$191$	16.3035	1.17968	0.589838	+	0.807521i	$0.299191\pi$
$192$	0	0
$193$	6.98388	0.502711	0.251355	−	0.967895i	$-0.419124\pi$
$193$	6.98388	0.502711	0.251355	+	0.967895i	$0.419124\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.5891i	0.968182i	0.875018	+	0.484091i	$0.160850\pi$
$197$	13.5891i	0.968182i	−0.875018	+	0.484091i	$0.839150\pi$
$198$	0	0
$199$	−20.2305	−1.43410	−0.717052	−	0.697020i	$-0.754509\pi$
$199$	−20.2305	−1.43410	−0.717052	+	0.697020i	$0.754509\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3.09794i	− 0.217433i
$204$	0	0
$205$	− 1.53567i	− 0.107256i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	34.9260	2.41588
$210$	0	0
$211$	19.1090i	1.31552i	0.753230	+	0.657758i	$0.228495\pi$
$211$	19.1090i	1.31552i	−0.753230	+	0.657758i	$0.771505\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.74035	0.459688
$216$	0	0
$217$	3.52412	0.239233
$218$	0	0
$219$	0	0
$220$	0	0
$221$	17.4129i	1.17132i
$222$	0	0
$223$	20.9241	1.40118	0.700592	−	0.713562i	$-0.252920\pi$
$223$	20.9241	1.40118	0.700592	+	0.713562i	$0.252920\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2.48329i	− 0.164821i	−0.996598	−	0.0824107i	$-0.973738\pi$
$227$	− 2.48329i	− 0.164821i	0.996598	−	0.0824107i	$-0.0262619\pi$
$228$	0	0
$229$	− 10.3155i	− 0.681669i	−0.940123	−	0.340835i	$-0.889290\pi$
$229$	− 10.3155i	− 0.681669i	0.940123	−	0.340835i	$-0.110710\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.5208	1.27885	0.639424	−	0.768855i	$-0.279173\pi$
$233$	19.5208	1.27885	0.639424	+	0.768855i	$0.279173\pi$
$234$	0	0
$235$	8.77100i	0.572157i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−17.8520	−1.15475	−0.577376	−	0.816478i	$-0.695923\pi$
$239$	−17.8520	−1.15475	−0.577376	+	0.816478i	$0.695923\pi$
$240$	0	0
$241$	6.72097	0.432935	0.216468	−	0.976290i	$-0.430546\pi$
$241$	6.72097	0.432935	0.216468	+	0.976290i	$0.430546\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 0.940450i	− 0.0600831i
$246$	0	0
$247$	38.4490	2.44645
$248$	0	0
$249$	0	0
$250$	0	0
$251$	28.1618i	1.77756i	0.458336	+	0.888779i	$0.348445\pi$
$251$	28.1618i	1.77756i	−0.458336	+	0.888779i	$0.651555\pi$
$252$	0	0
$253$	17.2677i	1.08561i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.9286	−1.55500	−0.777502	−	0.628880i	$-0.783514\pi$
$257$	−24.9286	−1.55500	−0.777502	+	0.628880i	$0.783514\pi$
$258$	0	0
$259$	− 0.213560i	− 0.0132700i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.0635	1.17550	0.587751	−	0.809042i	$-0.300013\pi$
$263$	19.0635	1.17550	0.587751	+	0.809042i	$0.300013\pi$
$264$	0	0
$265$	−7.06319	−0.433888
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 10.2410i	− 0.624406i	−0.950015	−	0.312203i	$-0.898933\pi$
$269$	− 10.2410i	− 0.624406i	0.950015	−	0.312203i	$-0.101067\pi$
$270$	0	0
$271$	4.45674	0.270728	0.135364	−	0.990796i	$-0.456780\pi$
$271$	4.45674	0.270728	0.135364	+	0.990796i	$0.456780\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	24.6370i	1.48567i
$276$	0	0
$277$	16.0596i	0.964929i	0.875915	+	0.482465i	$0.160258\pi$
$277$	16.0596i	0.964929i	−0.875915	+	0.482465i	$0.839742\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	31.3739	1.87161	0.935806	−	0.352516i	$-0.114674\pi$
$281$	31.3739	1.87161	0.935806	+	0.352516i	$0.114674\pi$
$282$	0	0
$283$	16.4627i	0.978605i	0.872114	+	0.489302i	$0.162749\pi$
$283$	16.4627i	0.978605i	−0.872114	+	0.489302i	$0.837251\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.63291	−0.0963874
$288$	0	0
$289$	−10.0185	−0.589324
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.78733i	0.338100i	0.985608	+	0.169050i	$0.0540699\pi$
$293$	5.78733i	0.338100i	−0.985608	+	0.169050i	$0.945930\pi$
$294$	0	0
$295$	−11.2177	−0.653117
$296$	0	0
$297$	0	0
$298$	0	0
$299$	19.0095i	1.09935i
$300$	0	0
$301$	− 7.16716i	− 0.413108i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.39472	−0.0798617
$306$	0	0
$307$	23.0312i	1.31446i	0.753691	+	0.657229i	$0.228272\pi$
$307$	23.0312i	1.31446i	−0.753691	+	0.657229i	$0.771728\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.7123	0.890964	0.445482	−	0.895291i	$-0.353032\pi$
$311$	15.7123	0.890964	0.445482	+	0.895291i	$0.353032\pi$
$312$	0	0
$313$	−6.53462	−0.369358	−0.184679	−	0.982799i	$-0.559125\pi$
$313$	−6.53462	−0.369358	−0.184679	+	0.982799i	$0.559125\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 19.9169i	− 1.11865i	−0.828949	−	0.559324i	$-0.811061\pi$
$317$	− 19.9169i	− 1.11865i	0.828949	−	0.559324i	$-0.188939\pi$
$318$	0	0
$319$	−18.5453	−1.03834
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 15.4157i	− 0.857751i
$324$	0	0
$325$	27.1222i	1.50447i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.32639	0.514180
$330$	0	0
$331$	21.3692i	1.17456i	0.809384	+	0.587280i	$0.199801\pi$
$331$	21.3692i	1.17456i	−0.809384	+	0.587280i	$0.800199\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.2780	0.670821
$336$	0	0
$337$	−9.56768	−0.521185	−0.260592	−	0.965449i	$-0.583918\pi$
$337$	−9.56768	−0.521185	−0.260592	+	0.965449i	$0.583918\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 21.0965i	− 1.14244i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 5.63471i	− 0.302487i	−0.988497	−	0.151243i	$-0.951672\pi$
$347$	− 5.63471i	− 0.302487i	0.988497	−	0.151243i	$-0.0483278\pi$
$348$	0	0
$349$	− 33.2219i	− 1.77833i	−0.457591	−	0.889163i	$-0.651287\pi$
$349$	− 33.2219i	− 1.77833i	0.457591	−	0.889163i	$-0.348713\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.20112	−0.330052	−0.165026	−	0.986289i	$-0.552771\pi$
$353$	−6.20112	−0.330052	−0.165026	+	0.986289i	$0.552771\pi$
$354$	0	0
$355$	1.45433i	0.0771879i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.76894	0.0933609	0.0466805	−	0.998910i	$-0.485136\pi$
$359$	1.76894	0.0933609	0.0466805	+	0.998910i	$0.485136\pi$
$360$	0	0
$361$	−15.0390	−0.791526
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.79192i	0.146136i
$366$	0	0
$367$	−9.62872	−0.502615	−0.251308	−	0.967907i	$-0.580861\pi$
$367$	−9.62872	−0.502615	−0.251308	+	0.967907i	$0.580861\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.51044i	0.389922i
$372$	0	0
$373$	− 19.6983i	− 1.01994i	−0.860192	−	0.509970i	$-0.829657\pi$
$373$	− 19.6983i	− 1.01994i	0.860192	−	0.509970i	$-0.170343\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−20.4160	−1.05148
$378$	0	0
$379$	− 16.3324i	− 0.838941i	−0.907769	−	0.419470i	$-0.862216\pi$
$379$	− 16.3324i	− 0.838941i	0.907769	−	0.419470i	$-0.137784\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	17.7037	0.904618	0.452309	−	0.891861i	$-0.350600\pi$
$383$	17.7037	0.904618	0.452309	+	0.891861i	$0.350600\pi$
$384$	0	0
$385$	−5.62983	−0.286923
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.4174i	1.23801i	0.785387	+	0.619005i	$0.212464\pi$
$389$	24.4174i	1.23801i	−0.785387	+	0.619005i	$0.787536\pi$
$390$	0	0
$391$	7.62162	0.385442
$392$	0	0
$393$	0	0
$394$	0	0
$395$	15.0016i	0.754811i
$396$	0	0
$397$	21.4996i	1.07903i	0.841975	+	0.539517i	$0.181393\pi$
$397$	21.4996i	1.07903i	−0.841975	+	0.539517i	$0.818607\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−29.0629	−1.45133	−0.725667	−	0.688046i	$-0.758469\pi$
$401$	−29.0629	−1.45133	−0.725667	+	0.688046i	$0.758469\pi$
$402$	0	0
$403$	− 23.2245i	− 1.15690i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.27844	−0.0633698
$408$	0	0
$409$	−5.96315	−0.294859	−0.147429	−	0.989073i	$-0.547100\pi$
$409$	−5.96315	−0.294859	−0.147429	+	0.989073i	$0.547100\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.9280i	0.586937i
$414$	0	0
$415$	8.22689	0.403842
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 28.8317i	− 1.40852i	−0.709941	−	0.704261i	$-0.751279\pi$
$419$	− 28.8317i	− 1.40852i	0.709941	−	0.704261i	$-0.248721\pi$
$420$	0	0
$421$	30.3324i	1.47831i	0.673534	+	0.739156i	$0.264776\pi$
$421$	30.3324i	1.47831i	−0.673534	+	0.739156i	$0.735224\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	10.8743	0.527482
$426$	0	0
$427$	1.48304i	0.0717693i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	32.3775	1.55957	0.779785	−	0.626048i	$-0.215329\pi$
$431$	32.3775	1.55957	0.779785	+	0.626048i	$0.215329\pi$
$432$	0	0
$433$	−30.9777	−1.48869	−0.744346	−	0.667794i	$-0.767239\pi$
$433$	−30.9777	−1.48869	−0.744346	+	0.667794i	$0.767239\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 16.8291i	− 0.805047i
$438$	0	0
$439$	−8.76381	−0.418274	−0.209137	−	0.977886i	$-0.567065\pi$
$439$	−8.76381	−0.418274	−0.209137	+	0.977886i	$0.567065\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.53477i	0.357988i	0.983850	+	0.178994i	$0.0572842\pi$
$443$	7.53477i	0.357988i	−0.983850	+	0.178994i	$0.942716\pi$
$444$	0	0
$445$	7.05656i	0.334513i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−32.5584	−1.53653	−0.768263	−	0.640134i	$-0.778879\pi$
$449$	−32.5584	−1.53653	−0.768263	+	0.640134i	$0.778879\pi$
$450$	0	0
$451$	9.77510i	0.460292i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.19772	−0.290554
$456$	0	0
$457$	−3.81122	−0.178281	−0.0891407	−	0.996019i	$-0.528412\pi$
$457$	−3.81122	−0.178281	−0.0891407	+	0.996019i	$0.528412\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.29754i	0.293306i	0.989188	+	0.146653i	$0.0468500\pi$
$461$	6.29754i	0.293306i	−0.989188	+	0.146653i	$0.953150\pi$
$462$	0	0
$463$	−12.8418	−0.596811	−0.298405	−	0.954439i	$-0.596455\pi$
$463$	−12.8418	−0.596811	−0.298405	+	0.954439i	$0.596455\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 34.5588i	− 1.59919i	−0.600538	−	0.799596i	$-0.705047\pi$
$467$	− 34.5588i	− 1.59919i	0.600538	−	0.799596i	$-0.294953\pi$
$468$	0	0
$469$	− 13.0555i	− 0.602847i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−42.9049	−1.97277
$474$	0	0
$475$	− 24.0114i	− 1.10172i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.5950	−0.758244	−0.379122	−	0.925347i	$-0.623774\pi$
$479$	−16.5950	−0.758244	−0.379122	+	0.925347i	$0.623774\pi$
$480$	0	0
$481$	−1.40739	−0.0641717
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 10.1125i	− 0.459186i
$486$	0	0
$487$	−26.4209	−1.19724	−0.598622	−	0.801032i	$-0.704285\pi$
$487$	−26.4209	−1.19724	−0.598622	+	0.801032i	$0.704285\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 3.70049i	− 0.167001i	−0.996508	−	0.0835003i	$-0.973390\pi$
$491$	− 3.70049i	− 0.167001i	0.996508	−	0.0835003i	$-0.0266100\pi$
$492$	0	0
$493$	8.18554i	0.368658i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.54642	0.0693664
$498$	0	0
$499$	2.12488i	0.0951229i	0.998868	+	0.0475614i	$0.0151450\pi$
$499$	2.12488i	0.0951229i	−0.998868	+	0.0475614i	$0.984855\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−43.3846	−1.93443	−0.967213	−	0.253968i	$-0.918264\pi$
$503$	−43.3846	−1.93443	−0.967213	+	0.253968i	$0.918264\pi$
$504$	0	0
$505$	−3.33785	−0.148532
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 2.34482i	− 0.103932i	−0.998649	−	0.0519662i	$-0.983451\pi$
$509$	− 2.34482i	− 0.103932i	0.998649	−	0.0519662i	$-0.0165488\pi$
$510$	0	0
$511$	2.96871	0.131328
$512$	0	0
$513$	0	0
$514$	0	0
$515$	14.4174i	0.635306i
$516$	0	0
$517$	− 55.8307i	− 2.45543i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−8.53436	−0.373897	−0.186949	−	0.982370i	$-0.559860\pi$
$521$	−8.53436	−0.373897	−0.186949	+	0.982370i	$0.559860\pi$
$522$	0	0
$523$	− 30.8607i	− 1.34944i	−0.738072	−	0.674722i	$-0.764264\pi$
$523$	− 30.8607i	− 1.34944i	0.738072	−	0.674722i	$-0.235736\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.31160	−0.405620
$528$	0	0
$529$	−14.6796	−0.638241
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.7611i	0.466116i
$534$	0	0
$535$	2.27244	0.0982461
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.98632i	0.257849i
$540$	0	0
$541$	− 15.2568i	− 0.655940i	−0.944688	−	0.327970i	$-0.893636\pi$
$541$	− 15.2568i	− 0.655940i	0.944688	−	0.327970i	$-0.106364\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.45817	0.276638
$546$	0	0
$547$	35.6416i	1.52393i	0.647621	+	0.761963i	$0.275764\pi$
$547$	35.6416i	1.52393i	−0.647621	+	0.761963i	$0.724236\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	18.0743	0.769992
$552$	0	0
$553$	15.9515	0.678326
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 28.3871i	− 1.20280i	−0.798949	−	0.601399i	$-0.794610\pi$
$557$	− 28.3871i	− 1.20280i	0.798949	−	0.601399i	$-0.205390\pi$
$558$	0	0
$559$	−47.2328	−1.99773
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 13.0963i	− 0.551942i	−0.961166	−	0.275971i	$-0.911001\pi$
$563$	− 13.0963i	− 0.551942i	0.961166	−	0.275971i	$-0.0889993\pi$
$564$	0	0
$565$	− 14.5336i	− 0.611435i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.0434	−0.840263	−0.420132	−	0.907463i	$-0.638016\pi$
$569$	−20.0434	−0.840263	−0.420132	+	0.907463i	$0.638016\pi$
$570$	0	0
$571$	6.91621i	0.289434i	0.989473	+	0.144717i	$0.0462272\pi$
$571$	6.91621i	0.289434i	−0.989473	+	0.144717i	$0.953773\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	11.8714	0.495071
$576$	0	0
$577$	23.2279	0.966992	0.483496	−	0.875347i	$-0.339367\pi$
$577$	23.2279	0.966992	0.483496	+	0.875347i	$0.339367\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 8.74782i	− 0.362921i
$582$	0	0
$583$	44.9599	1.86205
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.76774i	− 0.0729624i	−0.999334	−	0.0364812i	$-0.988385\pi$
$587$	− 1.76774i	− 0.0729624i	0.999334	−	0.0364812i	$-0.0116149\pi$
$588$	0	0
$589$	20.5607i	0.847191i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14.8807	−0.611079	−0.305540	−	0.952179i	$-0.598837\pi$
$593$	−14.8807	−0.611079	−0.305540	+	0.952179i	$0.598837\pi$
$594$	0	0
$595$	2.48490i	0.101871i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.47561	0.0602916	0.0301458	−	0.999546i	$-0.490403\pi$
$599$	1.47561	0.0602916	0.0301458	+	0.999546i	$0.490403\pi$
$600$	0	0
$601$	−11.1957	−0.456684	−0.228342	−	0.973581i	$-0.573330\pi$
$601$	−11.1957	−0.456684	−0.228342	+	0.973581i	$0.573330\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	23.3570i	0.949598i
$606$	0	0
$607$	31.4554	1.27674	0.638368	−	0.769731i	$-0.279610\pi$
$607$	31.4554	1.27674	0.638368	+	0.769731i	$0.279610\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 61.4625i	− 2.48651i
$612$	0	0
$613$	18.9222i	0.764262i	0.924108	+	0.382131i	$0.124810\pi$
$613$	18.9222i	0.764262i	−0.924108	+	0.382131i	$0.875190\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.1549	0.489339	0.244669	−	0.969607i	$-0.421321\pi$
$617$	12.1549	0.489339	0.244669	+	0.969607i	$0.421321\pi$
$618$	0	0
$619$	14.9619i	0.601371i	0.953723	+	0.300686i	$0.0972155\pi$
$619$	14.9619i	0.601371i	−0.953723	+	0.300686i	$0.902784\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.50339	0.300617
$624$	0	0
$625$	12.5156	0.500623
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.564278i	0.0224992i
$630$	0	0
$631$	8.71232	0.346832	0.173416	−	0.984849i	$-0.444519\pi$
$631$	8.71232	0.346832	0.173416	+	0.984849i	$0.444519\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.49313i	0.297356i
$636$	0	0
$637$	6.59017i	0.261112i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.31021	−0.288736	−0.144368	−	0.989524i	$-0.546115\pi$
$641$	−7.31021	−0.288736	−0.144368	+	0.989524i	$0.546115\pi$
$642$	0	0
$643$	− 11.1766i	− 0.440763i	−0.975414	−	0.220381i	$-0.929270\pi$
$643$	− 11.1766i	− 0.440763i	0.975414	−	0.220381i	$-0.0707302\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7.13625	−0.280555	−0.140277	−	0.990112i	$-0.544799\pi$
$647$	−7.13625	−0.280555	−0.140277	+	0.990112i	$0.544799\pi$
$648$	0	0
$649$	71.4046	2.80288
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 15.6385i	− 0.611982i	−0.952034	−	0.305991i	$-0.901012\pi$
$653$	− 15.6385i	− 0.611982i	0.952034	−	0.305991i	$-0.0989878\pi$
$654$	0	0
$655$	−1.44622	−0.0565086
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.73739i	0.0676791i	0.999427	+	0.0338395i	$0.0107735\pi$
$659$	1.73739i	0.0676791i	−0.999427	+	0.0338395i	$0.989226\pi$
$660$	0	0
$661$	22.7968i	0.886691i	0.896351	+	0.443345i	$0.146208\pi$
$661$	22.7968i	0.886691i	−0.896351	+	0.443345i	$0.853792\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.48686	0.212771
$666$	0	0
$667$	8.93608i	0.346006i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.87795	0.342729
$672$	0	0
$673$	−13.1667	−0.507538	−0.253769	−	0.967265i	$-0.581670\pi$
$673$	−13.1667	−0.507538	−0.253769	+	0.967265i	$0.581670\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.3183i	1.43426i	0.696941	+	0.717129i	$0.254544\pi$
$677$	37.3183i	1.43426i	−0.696941	+	0.717129i	$0.745456\pi$
$678$	0	0
$679$	−10.7529	−0.412657
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.72259i	0.257233i	0.991694	+	0.128616i	$0.0410536\pi$
$683$	6.72259i	0.257233i	−0.991694	+	0.128616i	$0.958946\pi$
$684$	0	0
$685$	5.96285i	0.227829i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	49.4951	1.88561
$690$	0	0
$691$	8.18845i	0.311503i	0.987796	+	0.155752i	$0.0497800\pi$
$691$	8.18845i	0.311503i	−0.987796	+	0.155752i	$0.950220\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.65205	0.290259
$696$	0	0
$697$	4.31455	0.163425
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5.85226i	0.221037i	0.993874	+	0.110518i	$0.0352511\pi$
$701$	5.85226i	0.221037i	−0.993874	+	0.110518i	$0.964749\pi$
$702$	0	0
$703$	1.24597	0.0469927
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.54920i	0.133482i
$708$	0	0
$709$	45.3444i	1.70294i	0.524399	+	0.851472i	$0.324290\pi$
$709$	45.3444i	1.70294i	−0.524399	+	0.851472i	$0.675710\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−10.1654	−0.380697
$714$	0	0
$715$	37.1015i	1.38752i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	29.1182	1.08592	0.542962	−	0.839757i	$-0.317303\pi$
$719$	29.1182	1.08592	0.542962	+	0.839757i	$0.317303\pi$
$720$	0	0
$721$	15.3303	0.570931
$722$	0	0
$723$	0	0
$724$	0	0
$725$	12.7498i	0.473514i
$726$	0	0
$727$	42.5099	1.57661	0.788303	−	0.615287i	$-0.210960\pi$
$727$	42.5099	1.57661	0.788303	+	0.615287i	$0.210960\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	18.9374i	0.700426i
$732$	0	0
$733$	2.79234i	0.103137i	0.998669	+	0.0515687i	$0.0164221\pi$
$733$	2.79234i	0.103137i	−0.998669	+	0.0515687i	$0.983578\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−78.1544	−2.87886
$738$	0	0
$739$	− 15.4715i	− 0.569129i	−0.958657	−	0.284565i	$-0.908151\pi$
$739$	− 15.4715i	− 0.569129i	0.958657	−	0.284565i	$-0.0918490\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−18.2897	−0.670984	−0.335492	−	0.942043i	$-0.608903\pi$
$743$	−18.2897	−0.670984	−0.335492	+	0.942043i	$0.608903\pi$
$744$	0	0
$745$	14.3507	0.525768
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 2.41633i	− 0.0882908i
$750$	0	0
$751$	−11.1167	−0.405655	−0.202828	−	0.979214i	$-0.565013\pi$
$751$	−11.1167	−0.405655	−0.202828	+	0.979214i	$0.565013\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 0.685062i	− 0.0249320i
$756$	0	0
$757$	11.8173i	0.429507i	0.976668	+	0.214753i	$0.0688948\pi$
$757$	11.8173i	0.429507i	−0.976668	+	0.214753i	$0.931105\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.2039	0.442390	0.221195	−	0.975230i	$-0.429004\pi$
$761$	12.2039	0.442390	0.221195	+	0.975230i	$0.429004\pi$
$762$	0	0
$763$	− 6.86711i	− 0.248606i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	78.6073	2.83835
$768$	0	0
$769$	−7.32578	−0.264175	−0.132087	−	0.991238i	$-0.542168\pi$
$769$	−7.32578	−0.264175	−0.132087	+	0.991238i	$0.542168\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.26470i	0.261293i	0.991429	+	0.130647i	$0.0417053\pi$
$773$	7.26470i	0.261293i	−0.991429	+	0.130647i	$0.958295\pi$
$774$	0	0
$775$	−14.5037	−0.520988
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 9.52686i	− 0.341335i
$780$	0	0
$781$	− 9.25737i	− 0.331255i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11.3989	0.406844
$786$	0	0
$787$	− 3.02431i	− 0.107805i	−0.998546	−	0.0539025i	$-0.982834\pi$
$787$	− 3.02431i	− 0.107805i	0.998546	−	0.0539025i	$-0.0171660\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−15.4539	−0.549478
$792$	0	0
$793$	9.77348	0.347067
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 20.6317i	− 0.730813i	−0.930848	−	0.365406i	$-0.880930\pi$
$797$	− 20.6317i	− 0.730813i	0.930848	−	0.365406i	$-0.119070\pi$
$798$	0	0
$799$	−24.6426	−0.871794
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 17.7716i	− 0.627148i
$804$	0	0
$805$	2.71274i	0.0956117i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−29.5889	−1.04029	−0.520145	−	0.854078i	$-0.674122\pi$
$809$	−29.5889	−1.04029	−0.520145	+	0.854078i	$0.674122\pi$
$810$	0	0
$811$	− 42.6230i	− 1.49670i	−0.663307	−	0.748348i	$-0.730847\pi$
$811$	− 42.6230i	− 1.49670i	0.663307	−	0.748348i	$-0.269153\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.12608	−0.179559
$816$	0	0
$817$	41.8153	1.46293
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.6214i	1.69690i	0.529276	+	0.848450i	$0.322464\pi$
$821$	48.6214i	1.69690i	−0.529276	+	0.848450i	$0.677536\pi$
$822$	0	0
$823$	−23.7228	−0.826924	−0.413462	−	0.910521i	$-0.635680\pi$
$823$	−23.7228	−0.826924	−0.413462	+	0.910521i	$0.635680\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.3889i	1.02195i	0.859595	+	0.510976i	$0.170716\pi$
$827$	29.3889i	1.02195i	−0.859595	+	0.510976i	$0.829284\pi$
$828$	0	0
$829$	− 26.4785i	− 0.919635i	−0.888013	−	0.459818i	$-0.847915\pi$
$829$	− 26.4785i	− 0.919635i	0.888013	−	0.459818i	$-0.152085\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.64225	0.0915485
$834$	0	0
$835$	1.85045i	0.0640375i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.51211	−0.121252	−0.0606258	−	0.998161i	$-0.519310\pi$
$839$	−3.51211	−0.121252	−0.0606258	+	0.998161i	$0.519310\pi$
$840$	0	0
$841$	19.4027	0.669060
$842$	0	0
$843$	0	0
$844$	0	0
$845$	28.6182i	0.984496i
$846$	0	0
$847$	24.8360	0.853376
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.616017i	0.0211168i
$852$	0	0
$853$	− 28.2378i	− 0.966843i	−0.875388	−	0.483422i	$-0.839394\pi$
$853$	− 28.2378i	− 0.966843i	0.875388	−	0.483422i	$-0.160606\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.8126	0.437669	0.218834	−	0.975762i	$-0.429775\pi$
$857$	12.8126	0.437669	0.218834	+	0.975762i	$0.429775\pi$
$858$	0	0
$859$	− 35.4496i	− 1.20952i	−0.796406	−	0.604762i	$-0.793268\pi$
$859$	− 35.4496i	− 1.20952i	0.796406	−	0.604762i	$-0.206732\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.3418	−1.10093	−0.550465	−	0.834858i	$-0.685549\pi$
$863$	−32.3418	−1.10093	−0.550465	+	0.834858i	$0.685549\pi$
$864$	0	0
$865$	−4.21887	−0.143446
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 95.4907i	− 3.23930i
$870$	0	0
$871$	−86.0380	−2.91529
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.57272i	0.289811i
$876$	0	0
$877$	− 6.53592i	− 0.220702i	−0.993893	−	0.110351i	$-0.964802\pi$
$877$	− 6.53592i	− 0.220702i	0.993893	−	0.110351i	$-0.0351976\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	53.0057	1.78581	0.892904	−	0.450248i	$-0.148664\pi$
$881$	53.0057	1.78581	0.892904	+	0.450248i	$0.148664\pi$
$882$	0	0
$883$	28.0481i	0.943894i	0.881627	+	0.471947i	$0.156449\pi$
$883$	28.0481i	0.943894i	−0.881627	+	0.471947i	$0.843551\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.58627	0.153992	0.0769959	−	0.997031i	$-0.475467\pi$
$887$	4.58627	0.153992	0.0769959	+	0.997031i	$0.475467\pi$
$888$	0	0
$889$	7.96760	0.267225
$890$	0	0
$891$	0	0
$892$	0	0
$893$	54.4129i	1.82086i
$894$	0	0
$895$	−23.3312	−0.779876
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 10.9175i	− 0.364120i
$900$	0	0
$901$	− 19.8445i	− 0.661115i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	21.2649	0.706870
$906$	0	0
$907$	− 31.8338i	− 1.05703i	−0.848925	−	0.528513i	$-0.822750\pi$
$907$	− 31.8338i	− 1.05703i	0.848925	−	0.528513i	$-0.177250\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	55.9123	1.85246	0.926228	−	0.376964i	$-0.123032\pi$
$911$	55.9123	1.85246	0.926228	+	0.376964i	$0.123032\pi$
$912$	0	0
$913$	−52.3673	−1.73310
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.53780i	0.0507826i
$918$	0	0
$919$	51.6749	1.70460	0.852299	−	0.523055i	$-0.175208\pi$
$919$	51.6749	1.70460	0.852299	+	0.523055i	$0.175208\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 10.1912i	− 0.335447i
$924$	0	0
$925$	0.878917i	0.0288986i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10.6988	−0.351016	−0.175508	−	0.984478i	$-0.556157\pi$
$929$	−10.6988	−0.351016	−0.175508	+	0.984478i	$0.556157\pi$
$930$	0	0
$931$	− 5.83430i	− 0.191211i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	14.8754	0.486478
$936$	0	0
$937$	29.3422	0.958568	0.479284	−	0.877660i	$-0.340896\pi$
$937$	29.3422	0.958568	0.479284	+	0.877660i	$0.340896\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	37.2035i	1.21280i	0.795160	+	0.606399i	$0.207387\pi$
$941$	37.2035i	1.21280i	−0.795160	+	0.606399i	$0.792613\pi$
$942$	0	0
$943$	4.71015	0.153384
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 25.7353i	− 0.836283i	−0.908382	−	0.418142i	$-0.862682\pi$
$947$	− 25.7353i	− 0.836283i	0.908382	−	0.418142i	$-0.137318\pi$
$948$	0	0
$949$	− 19.5643i	− 0.635084i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	17.6526	0.571824	0.285912	−	0.958256i	$-0.407703\pi$
$953$	17.6526	0.571824	0.285912	+	0.958256i	$0.407703\pi$
$954$	0	0
$955$	− 15.3326i	− 0.496150i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.34042	0.204743
$960$	0	0
$961$	−18.5806	−0.599374
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 6.56799i	− 0.211431i
$966$	0	0
$967$	−9.22421	−0.296631	−0.148315	−	0.988940i	$-0.547385\pi$
$967$	−9.22421	−0.296631	−0.148315	+	0.988940i	$0.547385\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 0.945630i	− 0.0303467i	−0.999885	−	0.0151734i	$-0.995170\pi$
$971$	− 0.945630i	− 0.0303467i	0.999885	−	0.0151734i	$-0.00483002\pi$
$972$	0	0
$973$	− 8.13659i	− 0.260847i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.7944	−1.11317	−0.556585	−	0.830791i	$-0.687889\pi$
$977$	−34.7944	−1.11317	−0.556585	+	0.830791i	$0.687889\pi$
$978$	0	0
$979$	− 44.9177i	− 1.43558i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	23.7133	0.756336	0.378168	−	0.925737i	$-0.376554\pi$
$983$	23.7133	0.756336	0.378168	+	0.925737i	$0.376554\pi$
$984$	0	0
$985$	12.7799	0.407200
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20.6738i	0.657389i
$990$	0	0
$991$	24.1109	0.765908	0.382954	−	0.923767i	$-0.374907\pi$
$991$	24.1109	0.765908	0.382954	+	0.923767i	$0.374907\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	19.0258i	0.603158i
$996$	0	0
$997$	− 30.0398i	− 0.951371i	−0.879615	−	0.475685i	$-0.842200\pi$
$997$	− 30.0398i	− 0.951371i	0.879615	−	0.475685i	$-0.157800\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.f.3025.12		24
3.2	odd	2	inner	6048.2.c.f.3025.14		24
4.3	odd	2		1512.2.c.g.757.6	yes	24
8.3	odd	2		1512.2.c.g.757.5	✓	24
8.5	even	2	inner	6048.2.c.f.3025.13		24
12.11	even	2		1512.2.c.g.757.19	yes	24
24.5	odd	2	inner	6048.2.c.f.3025.11		24
24.11	even	2		1512.2.c.g.757.20	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.g.757.5	✓	24	8.3	odd	2
1512.2.c.g.757.6	yes	24	4.3	odd	2
1512.2.c.g.757.19	yes	24	12.11	even	2
1512.2.c.g.757.20	yes	24	24.11	even	2
6048.2.c.f.3025.11		24	24.5	odd	2	inner
6048.2.c.f.3025.12		24	1.1	even	1	trivial
6048.2.c.f.3025.13		24	8.5	even	2	inner
6048.2.c.f.3025.14		24	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.940450i q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-0.940450i q^{5} -1.00000 q^{7} +5.98632i q^{11} +6.59017i q^{13} +2.64225 q^{17} -5.83430i q^{19} +2.88452 q^{23} +4.11555 q^{25} +3.09794i q^{29} -3.52412 q^{31} +0.940450i q^{35} +0.213560i q^{37} +1.63291 q^{41} +7.16716i q^{43} -9.32639 q^{47} +1.00000 q^{49} -7.51044i q^{53} +5.62983 q^{55} -11.9280i q^{59} -1.48304i q^{61} +6.19772 q^{65} +13.0555i q^{67} -1.54642 q^{71} -2.96871 q^{73} -5.98632i q^{77} -15.9515 q^{79} +8.74782i q^{83} -2.48490i q^{85} -7.50339 q^{89} -6.59017i q^{91} -5.48686 q^{95} +10.7529 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{7}+O(q^{10}) \) 24 * q - 24 * q^7	\( 24 q - 24 q^{7} - 24 q^{25} - 8 q^{31} + 24 q^{49} - 16 q^{55} - 8 q^{79}+O(q^{100}) \) 24 * q - 24 * q^7 - 24 * q^25 - 8 * q^31 + 24 * q^49 - 16 * q^55 - 8 * q^79

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