LMFDB - Embedded newform 6048.2.j.a.5615.3

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:	$8$
Coefficient field:	8.0.56070144.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 63x^4 - 74x^3 + 70x^2 - 38*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5615.3
Root			$0.500000 - 0.564882i$ of defining polynomial
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.a.5615.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.12976 q^{5} -1.00000i q^{7} +O(q^{10})$	$q-3.12976 q^{5} -1.00000i q^{7} +0.397714i q^{11} -6.55068i q^{13} +2.00000i q^{17} +0.527479 q^{19} -8.21634 q^{23} +4.79543 q^{25} -4.73205 q^{29} -7.55068i q^{31} +3.12976i q^{35} +3.35452i q^{37} -7.48429i q^{41} -1.62247 q^{43} +10.4557 q^{47} -1.00000 q^{49} +0.795428 q^{53} -1.24475i q^{55} +2.47252i q^{59} +11.4641i q^{61} +20.5021i q^{65} +6.55068 q^{67} +2.21634 q^{71} -5.75525 q^{73} +0.397714 q^{77} -10.2911i q^{79} -8.36930i q^{83} -6.25953i q^{85} +5.31114i q^{89} -6.55068 q^{91} -1.65089 q^{95} -19.1014 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} - 16 q^{19} - 16 q^{23} + 32 q^{25} - 24 q^{29} - 8 q^{43} + 8 q^{47} - 8 q^{49} - 8 q^{67} - 32 q^{71} + 8 q^{73} + 8 q^{91} - 112 q^{95} - 32 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 - 16 * q^19 - 16 * q^23 + 32 * q^25 - 24 * q^29 - 8 * q^43 + 8 * q^47 - 8 * q^49 - 8 * q^67 - 32 * q^71 + 8 * q^73 + 8 * q^91 - 112 * q^95 - 32 * 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
$5$	−3.12976	−1.39967	−0.699837	−	0.714303i	$-0.746744\pi$
$5$	−3.12976	−1.39967	−0.699837	+	0.714303i	$0.746744\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.397714i	0.119915i	0.998201	+	0.0599577i	$0.0190966\pi$
$11$	0.397714i	0.119915i	−0.998201	+	0.0599577i	$0.980903\pi$
$12$	0	0
$13$	− 6.55068i	− 1.81683i	−0.418069	−	0.908415i	$-0.637293\pi$
$13$	− 6.55068i	− 1.81683i	0.418069	−	0.908415i	$-0.362707\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	0.527479	0.121012	0.0605060	−	0.998168i	$-0.480729\pi$
$19$	0.527479	0.121012	0.0605060	+	0.998168i	$0.480729\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.21634	−1.71323	−0.856613	−	0.515960i	$-0.827435\pi$
$23$	−8.21634	−1.71323	−0.856613	+	0.515960i	$0.827435\pi$
$24$	0	0
$25$	4.79543	0.959086
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.73205	−0.878720	−0.439360	−	0.898311i	$-0.644795\pi$
$29$	−4.73205	−0.878720	−0.439360	+	0.898311i	$0.644795\pi$
$30$	0	0
$31$	− 7.55068i	− 1.35614i	−0.734997	−	0.678071i	$-0.762816\pi$
$31$	− 7.55068i	− 1.35614i	0.734997	−	0.678071i	$-0.237184\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.12976i	0.529027i
$36$	0	0
$37$	3.35452i	0.551480i	0.961232	+	0.275740i	$0.0889229\pi$
$37$	3.35452i	0.551480i	−0.961232	+	0.275740i	$0.911077\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 7.48429i	− 1.16885i	−0.811448	−	0.584425i	$-0.801320\pi$
$41$	− 7.48429i	− 1.16885i	0.811448	−	0.584425i	$-0.198680\pi$
$42$	0	0
$43$	−1.62247	−0.247425	−0.123712	−	0.992318i	$-0.539480\pi$
$43$	−1.62247	−0.247425	−0.123712	+	0.992318i	$0.539480\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.4557	1.52512	0.762559	−	0.646919i	$-0.223943\pi$
$47$	10.4557	1.52512	0.762559	+	0.646919i	$0.223943\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.795428	0.109260	0.0546302	−	0.998507i	$-0.482602\pi$
$53$	0.795428	0.109260	0.0546302	+	0.998507i	$0.482602\pi$
$54$	0	0
$55$	− 1.24475i	− 0.167842i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.47252i	0.321895i	0.986963	+	0.160947i	$0.0514550\pi$
$59$	2.47252i	0.321895i	−0.986963	+	0.160947i	$0.948545\pi$
$60$	0	0
$61$	11.4641i	1.46783i	0.679243	+	0.733914i	$0.262308\pi$
$61$	11.4641i	1.46783i	−0.679243	+	0.733914i	$0.737692\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	20.5021i	2.54297i
$66$	0	0
$67$	6.55068	0.800293	0.400146	−	0.916451i	$-0.368959\pi$
$67$	6.55068	0.800293	0.400146	+	0.916451i	$0.368959\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.21634	0.263031	0.131516	−	0.991314i	$-0.458016\pi$
$71$	2.21634	0.263031	0.131516	+	0.991314i	$0.458016\pi$
$72$	0	0
$73$	−5.75525	−0.673601	−0.336800	−	0.941576i	$-0.609345\pi$
$73$	−5.75525	−0.673601	−0.336800	+	0.941576i	$0.609345\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.397714	0.0453237
$78$	0	0
$79$	− 10.2911i	− 1.15784i	−0.815383	−	0.578922i	$-0.803473\pi$
$79$	− 10.2911i	− 1.15784i	0.815383	−	0.578922i	$-0.196527\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 8.36930i	− 0.918650i	−0.888268	−	0.459325i	$-0.848091\pi$
$83$	− 8.36930i	− 0.918650i	0.888268	−	0.459325i	$-0.151909\pi$
$84$	0	0
$85$	− 6.25953i	− 0.678941i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.31114i	0.562980i	0.959564	+	0.281490i	$0.0908286\pi$
$89$	5.31114i	0.562980i	−0.959564	+	0.281490i	$0.909171\pi$
$90$	0	0
$91$	−6.55068	−0.686697
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.65089	−0.169377
$96$	0	0
$97$	−19.1014	−1.93945	−0.969724	−	0.244202i	$-0.921474\pi$
$97$	−19.1014	−1.93945	−0.969724	+	0.244202i	$0.921474\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	− 17.0148i	− 1.67652i	−0.545274	−	0.838258i	$-0.683574\pi$
$103$	− 17.0148i	− 1.67652i	0.545274	−	0.838258i	$-0.316426\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.91362i	0.184997i	0.995713	+	0.0924983i	$0.0294853\pi$
$107$	1.91362i	0.184997i	−0.995713	+	0.0924983i	$0.970515\pi$
$108$	0	0
$109$	− 7.07816i	− 0.677964i	−0.940793	−	0.338982i	$-0.889917\pi$
$109$	− 7.07816i	− 0.677964i	0.940793	−	0.338982i	$-0.110083\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.98316i	0.939137i	0.882896	+	0.469568i	$0.155590\pi$
$113$	9.98316i	0.939137i	−0.882896	+	0.469568i	$0.844410\pi$
$114$	0	0
$115$	25.7152	2.39796
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	10.8418	0.985620
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0.640262	0.0572668
$126$	0	0
$127$	7.75525i	0.688167i	0.938939	+	0.344084i	$0.111810\pi$
$127$	7.75525i	0.688167i	−0.938939	+	0.344084i	$0.888190\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.9052i	1.30227i	0.758960	+	0.651137i	$0.225708\pi$
$131$	14.9052i	1.30227i	−0.758960	+	0.651137i	$0.774292\pi$
$132$	0	0
$133$	− 0.527479i	− 0.0457382i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.4557i	1.91852i	0.282527	+	0.959259i	$0.408827\pi$
$137$	22.4557i	1.91852i	−0.282527	+	0.959259i	$0.591173\pi$
$138$	0	0
$139$	−3.05496	−0.259118	−0.129559	−	0.991572i	$-0.541356\pi$
$139$	−3.05496	−0.259118	−0.129559	+	0.991572i	$0.541356\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.60530	0.217866
$144$	0	0
$145$	14.8102	1.22992
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.01458	0.574657	0.287329	−	0.957832i	$-0.407233\pi$
$149$	7.01458	0.574657	0.287329	+	0.957832i	$0.407233\pi$
$150$	0	0
$151$	− 9.81001i	− 0.798327i	−0.916880	−	0.399164i	$-0.869301\pi$
$151$	− 9.81001i	− 0.798327i	0.916880	−	0.399164i	$-0.130699\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	23.6318i	1.89816i
$156$	0	0
$157$	10.8966i	0.869642i	0.900517	+	0.434821i	$0.143188\pi$
$157$	10.8966i	0.869642i	−0.900517	+	0.434821i	$0.856812\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.21634i	0.647538i
$162$	0	0
$163$	10.0148	0.784418	0.392209	−	0.919876i	$-0.371711\pi$
$163$	10.0148	0.784418	0.392209	+	0.919876i	$0.371711\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.5655	1.12711	0.563554	−	0.826079i	$-0.309434\pi$
$167$	14.5655	1.12711	0.563554	+	0.826079i	$0.309434\pi$
$168$	0	0
$169$	−29.9114	−2.30087
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.3175	−1.08854	−0.544270	−	0.838910i	$-0.683193\pi$
$173$	−14.3175	−1.08854	−0.544270	+	0.838910i	$0.683193\pi$
$174$	0	0
$175$	− 4.79543i	− 0.362500i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 19.8104i	− 1.48070i	−0.672222	−	0.740349i	$-0.734660\pi$
$179$	− 19.8104i	− 1.48070i	0.672222	−	0.740349i	$-0.265340\pi$
$180$	0	0
$181$	20.1732i	1.49946i	0.661745	+	0.749729i	$0.269816\pi$
$181$	20.1732i	1.49946i	−0.661745	+	0.749729i	$0.730184\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 10.4989i	− 0.771892i
$186$	0	0
$187$	−0.795428	−0.0581675
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.43549	−0.465656	−0.232828	−	0.972518i	$-0.574798\pi$
$191$	−6.43549	−0.465656	−0.232828	+	0.972518i	$0.574798\pi$
$192$	0	0
$193$	−9.98316	−0.718604	−0.359302	−	0.933221i	$-0.616985\pi$
$193$	−9.98316	−0.718604	−0.359302	+	0.933221i	$0.616985\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.6373	−1.68408	−0.842042	−	0.539412i	$-0.818647\pi$
$197$	−23.6373	−1.68408	−0.842042	+	0.539412i	$0.818647\pi$
$198$	0	0
$199$	21.6202i	1.53262i	0.642473	+	0.766308i	$0.277908\pi$
$199$	21.6202i	1.53262i	−0.642473	+	0.766308i	$0.722092\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.73205i	0.332125i
$204$	0	0
$205$	23.4241i	1.63601i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.209786i	0.0145112i
$210$	0	0
$211$	−2.51906	−0.173419	−0.0867096	−	0.996234i	$-0.527635\pi$
$211$	−2.51906	−0.173419	−0.0867096	+	0.996234i	$0.527635\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.07796	0.346314
$216$	0	0
$217$	−7.55068	−0.512573
$218$	0	0
$219$	0	0
$220$	0	0
$221$	13.1014	0.881292
$222$	0	0
$223$	0.882003i	0.0590633i	0.999564	+	0.0295316i	$0.00940158\pi$
$223$	0.882003i	0.0590633i	−0.999564	+	0.0295316i	$0.990598\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.47868i	0.562750i	0.959598	+	0.281375i	$0.0907905\pi$
$227$	8.47868i	0.562750i	−0.959598	+	0.281375i	$0.909210\pi$
$228$	0	0
$229$	13.4641i	0.889733i	0.895597	+	0.444866i	$0.146749\pi$
$229$	13.4641i	0.889733i	−0.895597	+	0.444866i	$0.853251\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.4641i	1.14411i	0.820215	+	0.572056i	$0.193854\pi$
$233$	17.4641i	1.14411i	−0.820215	+	0.572056i	$0.806146\pi$
$234$	0	0
$235$	−32.7238	−2.13467
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−27.6836	−1.79071	−0.895353	−	0.445357i	$-0.853077\pi$
$239$	−27.6836	−1.79071	−0.895353	+	0.445357i	$0.853077\pi$
$240$	0	0
$241$	21.9116	1.41145	0.705724	−	0.708487i	$-0.250622\pi$
$241$	21.9116	1.41145	0.705724	+	0.708487i	$0.250622\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.12976	0.199953
$246$	0	0
$247$	− 3.45534i	− 0.219858i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.9136i	1.00446i	0.864734	+	0.502229i	$0.167487\pi$
$251$	15.9136i	1.00446i	−0.864734	+	0.502229i	$0.832513\pi$
$252$	0	0
$253$	− 3.26775i	− 0.205442i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.70344i	− 0.231014i	−0.993307	−	0.115507i	$-0.963151\pi$
$257$	− 3.70344i	− 0.231014i	0.993307	−	0.115507i	$-0.0368493\pi$
$258$	0	0
$259$	3.35452	0.208440
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−22.1995	−1.36888	−0.684440	−	0.729069i	$-0.739953\pi$
$263$	−22.1995	−1.36888	−0.684440	+	0.729069i	$0.739953\pi$
$264$	0	0
$265$	−2.48950	−0.152929
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−16.2480	−0.990655	−0.495328	−	0.868706i	$-0.664952\pi$
$269$	−16.2480	−0.990655	−0.495328	+	0.868706i	$0.664952\pi$
$270$	0	0
$271$	18.3755i	1.11623i	0.829764	+	0.558115i	$0.188475\pi$
$271$	18.3755i	1.11623i	−0.829764	+	0.558115i	$0.811525\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.90721i	0.115009i
$276$	0	0
$277$	31.7764i	1.90926i	0.297798	+	0.954629i	$0.403748\pi$
$277$	31.7764i	1.90926i	−0.297798	+	0.954629i	$0.596252\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.86483i	0.290211i	0.989416	+	0.145106i	$0.0463522\pi$
$281$	4.86483i	0.290211i	−0.989416	+	0.145106i	$0.953648\pi$
$282$	0	0
$283$	24.7386	1.47056	0.735279	−	0.677765i	$-0.237051\pi$
$283$	24.7386	1.47056	0.735279	+	0.677765i	$0.237051\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.48429	−0.441784
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.9428	0.931388	0.465694	−	0.884946i	$-0.345805\pi$
$293$	15.9428	0.931388	0.465694	+	0.884946i	$0.345805\pi$
$294$	0	0
$295$	− 7.73841i	− 0.450548i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	53.8226i	3.11264i
$300$	0	0
$301$	1.62247i	0.0935178i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 35.8799i	− 2.05448i
$306$	0	0
$307$	5.64567	0.322215	0.161108	−	0.986937i	$-0.448493\pi$
$307$	5.64567	0.322215	0.161108	+	0.986937i	$0.448493\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.0084	−1.07787	−0.538934	−	0.842348i	$-0.681173\pi$
$311$	−19.0084	−1.07787	−0.538934	+	0.842348i	$0.681173\pi$
$312$	0	0
$313$	2.24475	0.126881	0.0634404	−	0.997986i	$-0.479793\pi$
$313$	2.24475	0.126881	0.0634404	+	0.997986i	$0.479793\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.4852	1.37523	0.687614	−	0.726076i	$-0.258658\pi$
$317$	24.4852	1.37523	0.687614	+	0.726076i	$0.258658\pi$
$318$	0	0
$319$	− 1.88200i	− 0.105372i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.05496i	0.0586994i
$324$	0	0
$325$	− 31.4133i	− 1.74250i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 10.4557i	− 0.576440i
$330$	0	0
$331$	−4.34591	−0.238873	−0.119436	−	0.992842i	$-0.538109\pi$
$331$	−4.34591	−0.238873	−0.119436	+	0.992842i	$0.538109\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−20.5021	−1.12015
$336$	0	0
$337$	−27.5655	−1.50159	−0.750793	−	0.660538i	$-0.770328\pi$
$337$	−27.5655	−1.50159	−0.750793	+	0.660538i	$0.770328\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.00301	0.162622
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.8364i	1.11856i	0.828980	+	0.559279i	$0.188922\pi$
$347$	20.8364i	1.11856i	−0.828980	+	0.559279i	$0.811078\pi$
$348$	0	0
$349$	− 4.88181i	− 0.261317i	−0.991427	−	0.130659i	$-0.958291\pi$
$349$	− 4.88181i	− 0.261317i	0.991427	−	0.130659i	$-0.0417092\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.44391i	0.396200i	0.980182	+	0.198100i	$0.0634770\pi$
$353$	7.44391i	0.396200i	−0.980182	+	0.198100i	$0.936523\pi$
$354$	0	0
$355$	−6.93662	−0.368158
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−22.3923	−1.18182	−0.590910	−	0.806737i	$-0.701231\pi$
$359$	−22.3923	−1.18182	−0.590910	+	0.806737i	$0.701231\pi$
$360$	0	0
$361$	−18.7218	−0.985356
$362$	0	0
$363$	0	0
$364$	0	0
$365$	18.0126	0.942821
$366$	0	0
$367$	35.2575i	1.84042i	0.391419	+	0.920212i	$0.371984\pi$
$367$	35.2575i	1.84042i	−0.391419	+	0.920212i	$0.628016\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 0.795428i	− 0.0412966i
$372$	0	0
$373$	− 11.2681i	− 0.583442i	−0.956503	−	0.291721i	$-0.905772\pi$
$373$	− 11.2681i	− 0.583442i	0.956503	−	0.291721i	$-0.0942279\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	30.9981i	1.59649i
$378$	0	0
$379$	5.47888	0.281431	0.140716	−	0.990050i	$-0.455060\pi$
$379$	5.47888	0.281431	0.140716	+	0.990050i	$0.455060\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	31.6668	1.61810	0.809049	−	0.587741i	$-0.199983\pi$
$383$	31.6668	1.61810	0.809049	+	0.587741i	$0.199983\pi$
$384$	0	0
$385$	−1.24475	−0.0634384
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−25.1014	−1.27269	−0.636345	−	0.771405i	$-0.719554\pi$
$389$	−25.1014	−1.27269	−0.636345	+	0.771405i	$0.719554\pi$
$390$	0	0
$391$	− 16.4327i	− 0.831036i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	32.2089i	1.62060i
$396$	0	0
$397$	− 4.04640i	− 0.203083i	−0.994831	−	0.101541i	$-0.967623\pi$
$397$	− 4.04640i	− 0.203083i	0.994831	−	0.101541i	$-0.0323774\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.87339i	0.293303i	0.989188	+	0.146652i	$0.0468496\pi$
$401$	5.87339i	0.293303i	−0.989188	+	0.146652i	$0.953150\pi$
$402$	0	0
$403$	−49.4620	−2.46388
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.33414	−0.0661309
$408$	0	0
$409$	25.1101	1.24162	0.620808	−	0.783963i	$-0.286805\pi$
$409$	25.1101	1.24162	0.620808	+	0.783963i	$0.286805\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.47252	0.121665
$414$	0	0
$415$	26.1939i	1.28581i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 12.3059i	− 0.601184i	−0.953753	−	0.300592i	$-0.902816\pi$
$419$	− 12.3059i	− 0.601184i	0.953753	−	0.300592i	$-0.0971842\pi$
$420$	0	0
$421$	− 22.4095i	− 1.09217i	−0.837729	−	0.546086i	$-0.816117\pi$
$421$	− 22.4095i	− 1.09217i	0.837729	−	0.546086i	$-0.183883\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9.59086i	0.465225i
$426$	0	0
$427$	11.4641	0.554787
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.4041	1.03100	0.515499	−	0.856890i	$-0.327607\pi$
$431$	21.4041	1.03100	0.515499	+	0.856890i	$0.327607\pi$
$432$	0	0
$433$	19.3925	0.931944	0.465972	−	0.884799i	$-0.345705\pi$
$433$	19.3925	0.931944	0.465972	+	0.884799i	$0.345705\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.33395	−0.207321
$438$	0	0
$439$	− 18.3755i	− 0.877013i	−0.898728	−	0.438507i	$-0.855508\pi$
$439$	− 18.3755i	− 0.877013i	0.898728	−	0.438507i	$-0.144492\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.58770i	0.408014i	0.978969	+	0.204007i	$0.0653965\pi$
$443$	8.58770i	0.408014i	−0.978969	+	0.204007i	$0.934603\pi$
$444$	0	0
$445$	− 16.6226i	− 0.787988i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.78687i	0.367485i	0.982974	+	0.183742i	$0.0588212\pi$
$449$	7.78687i	0.367485i	−0.982974	+	0.183742i	$0.941179\pi$
$450$	0	0
$451$	2.97661	0.140163
$452$	0	0
$453$	0	0
$454$	0	0
$455$	20.5021	0.961152
$456$	0	0
$457$	10.3205	0.482773	0.241387	−	0.970429i	$-0.422398\pi$
$457$	10.3205	0.482773	0.241387	+	0.970429i	$0.422398\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.2016	1.03403	0.517015	−	0.855976i	$-0.327043\pi$
$461$	22.2016	1.03403	0.517015	+	0.855976i	$0.327043\pi$
$462$	0	0
$463$	27.9028i	1.29675i	0.761320	+	0.648377i	$0.224552\pi$
$463$	27.9028i	1.29675i	−0.761320	+	0.648377i	$0.775448\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 21.9602i	− 1.01619i	−0.861300	−	0.508097i	$-0.830349\pi$
$467$	− 21.9602i	− 1.01619i	0.861300	−	0.508097i	$-0.169651\pi$
$468$	0	0
$469$	− 6.55068i	− 0.302482i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 0.645280i	− 0.0296700i
$474$	0	0
$475$	2.52949	0.116061
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6.13278	−0.280214	−0.140107	−	0.990136i	$-0.544745\pi$
$479$	−6.13278	−0.280214	−0.140107	+	0.990136i	$0.544745\pi$
$480$	0	0
$481$	21.9744	1.00195
$482$	0	0
$483$	0	0
$484$	0	0
$485$	59.7827	2.71459
$486$	0	0
$487$	− 19.7848i	− 0.896535i	−0.893899	−	0.448268i	$-0.852041\pi$
$487$	− 19.7848i	− 0.896535i	0.893899	−	0.448268i	$-0.147959\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 28.3809i	− 1.28081i	−0.768037	−	0.640405i	$-0.778766\pi$
$491$	− 28.3809i	− 1.28081i	0.768037	−	0.640405i	$-0.221234\pi$
$492$	0	0
$493$	− 9.46410i	− 0.426242i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.21634i	− 0.0994164i
$498$	0	0
$499$	−10.7407	−0.480818	−0.240409	−	0.970672i	$-0.577282\pi$
$499$	−10.7407	−0.480818	−0.240409	+	0.970672i	$0.577282\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14.7321	0.656870	0.328435	−	0.944527i	$-0.393479\pi$
$503$	14.7321	0.656870	0.328435	+	0.944527i	$0.393479\pi$
$504$	0	0
$505$	6.25953	0.278545
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−23.2913	−1.03237	−0.516185	−	0.856477i	$-0.672648\pi$
$509$	−23.2913	−1.03237	−0.516185	+	0.856477i	$0.672648\pi$
$510$	0	0
$511$	5.75525i	0.254597i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	53.2523i	2.34657i
$516$	0	0
$517$	4.15837i	0.182885i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 1.91027i	− 0.0836905i	−0.999124	−	0.0418453i	$-0.986676\pi$
$521$	− 1.91027i	− 0.0836905i	0.999124	−	0.0418453i	$-0.0133237\pi$
$522$	0	0
$523$	−25.5885	−1.11891	−0.559453	−	0.828862i	$-0.688989\pi$
$523$	−25.5885	−1.11891	−0.559453	+	0.828862i	$0.688989\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15.1014	0.657825
$528$	0	0
$529$	44.5082	1.93514
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−49.0272	−2.12360
$534$	0	0
$535$	− 5.98918i	− 0.258935i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 0.397714i	− 0.0171308i
$540$	0	0
$541$	− 7.91959i	− 0.340490i	−0.985402	−	0.170245i	$-0.945544\pi$
$541$	− 7.91959i	− 0.340490i	0.985402	−	0.170245i	$-0.0544559\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.1530i	0.948929i
$546$	0	0
$547$	7.65409	0.327265	0.163633	−	0.986521i	$-0.447679\pi$
$547$	7.65409	0.327265	0.163633	+	0.986521i	$0.447679\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.49606	−0.106336
$552$	0	0
$553$	−10.2911	−0.437624
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−27.6373	−1.17103	−0.585514	−	0.810662i	$-0.699107\pi$
$557$	−27.6373	−1.17103	−0.585514	+	0.810662i	$0.699107\pi$
$558$	0	0
$559$	10.6283i	0.449529i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 33.9198i	− 1.42955i	−0.699355	−	0.714774i	$-0.746529\pi$
$563$	− 33.9198i	− 1.42955i	0.699355	−	0.714774i	$-0.253471\pi$
$564$	0	0
$565$	− 31.2449i	− 1.31448i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.7382i	0.785547i	0.919635	+	0.392773i	$0.128484\pi$
$569$	18.7382i	0.785547i	−0.919635	+	0.392773i	$0.871516\pi$
$570$	0	0
$571$	−15.2449	−0.637981	−0.318991	−	0.947758i	$-0.603344\pi$
$571$	−15.2449	−0.637981	−0.318991	+	0.947758i	$0.603344\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−39.4009	−1.64313
$576$	0	0
$577$	1.49366	0.0621818	0.0310909	−	0.999517i	$-0.490102\pi$
$577$	1.49366	0.0621818	0.0310909	+	0.999517i	$0.490102\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.36930	−0.347217
$582$	0	0
$583$	0.316353i	0.0131020i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.2195i	0.669452i	0.942315	+	0.334726i	$0.108644\pi$
$587$	16.2195i	0.669452i	−0.942315	+	0.334726i	$0.891356\pi$
$588$	0	0
$589$	− 3.98282i	− 0.164109i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 21.3871i	− 0.878263i	−0.898423	−	0.439131i	$-0.855286\pi$
$593$	− 21.3871i	− 0.878263i	0.898423	−	0.439131i	$-0.144714\pi$
$594$	0	0
$595$	−6.25953	−0.256616
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−11.7672	−0.480795	−0.240398	−	0.970674i	$-0.577278\pi$
$599$	−11.7672	−0.480795	−0.240398	+	0.970674i	$0.577278\pi$
$600$	0	0
$601$	−15.7552	−0.642670	−0.321335	−	0.946966i	$-0.604132\pi$
$601$	−15.7552	−0.642670	−0.321335	+	0.946966i	$0.604132\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−33.9324	−1.37955
$606$	0	0
$607$	1.95400i	0.0793102i	0.999213	+	0.0396551i	$0.0126259\pi$
$607$	1.95400i	0.0793102i	−0.999213	+	0.0396551i	$0.987374\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 68.4918i	− 2.77088i
$612$	0	0
$613$	3.47046i	0.140171i	0.997541	+	0.0700853i	$0.0223271\pi$
$613$	3.47046i	0.140171i	−0.997541	+	0.0700853i	$0.977673\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 24.9456i	− 1.00427i	−0.864789	−	0.502136i	$-0.832548\pi$
$617$	− 24.9456i	− 1.00427i	0.864789	−	0.502136i	$-0.167452\pi$
$618$	0	0
$619$	37.3233	1.50015	0.750075	−	0.661353i	$-0.230017\pi$
$619$	37.3233	1.50015	0.750075	+	0.661353i	$0.230017\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.31114	0.212786
$624$	0	0
$625$	−25.9810	−1.03924
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.70905	−0.267507
$630$	0	0
$631$	− 9.51886i	− 0.378940i	−0.981887	−	0.189470i	$-0.939323\pi$
$631$	− 9.51886i	− 0.378940i	0.981887	−	0.189470i	$-0.0606770\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 24.2721i	− 0.963209i
$636$	0	0
$637$	6.55068i	0.259547i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	41.9263i	1.65599i	0.560735	+	0.827995i	$0.310519\pi$
$641$	41.9263i	1.65599i	−0.560735	+	0.827995i	$0.689481\pi$
$642$	0	0
$643$	−2.79303	−0.110146	−0.0550732	−	0.998482i	$-0.517539\pi$
$643$	−2.79303	−0.110146	−0.0550732	+	0.998482i	$0.517539\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30.2089	1.18763	0.593817	−	0.804600i	$-0.297620\pi$
$647$	30.2089	1.18763	0.593817	+	0.804600i	$0.297620\pi$
$648$	0	0
$649$	−0.983356	−0.0386001
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−36.8020	−1.44017	−0.720086	−	0.693884i	$-0.755898\pi$
$653$	−36.8020	−1.44017	−0.720086	+	0.693884i	$0.755898\pi$
$654$	0	0
$655$	− 46.6498i	− 1.82276i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 27.5858i	− 1.07459i	−0.843394	−	0.537296i	$-0.819446\pi$
$659$	− 27.5858i	− 1.07459i	0.843394	−	0.537296i	$-0.180554\pi$
$660$	0	0
$661$	43.9007i	1.70754i	0.520650	+	0.853770i	$0.325690\pi$
$661$	43.9007i	1.70754i	−0.520650	+	0.853770i	$0.674310\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.65089i	0.0640186i
$666$	0	0
$667$	38.8801	1.50544
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.55943	−0.176015
$672$	0	0
$673$	−1.78085	−0.0686465	−0.0343233	−	0.999411i	$-0.510928\pi$
$673$	−1.78085	−0.0686465	−0.0343233	+	0.999411i	$0.510928\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	45.7356	1.75776	0.878881	−	0.477041i	$-0.158291\pi$
$677$	45.7356	1.75776	0.878881	+	0.477041i	$0.158291\pi$
$678$	0	0
$679$	19.1014i	0.733043i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	23.5159i	0.899811i	0.893076	+	0.449906i	$0.148542\pi$
$683$	23.5159i	0.899811i	−0.893076	+	0.449906i	$0.851458\pi$
$684$	0	0
$685$	− 70.2810i	− 2.68530i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 5.21059i	− 0.198508i
$690$	0	0
$691$	31.3501	1.19261	0.596306	−	0.802757i	$-0.296634\pi$
$691$	31.3501	1.19261	0.596306	+	0.802757i	$0.296634\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.56130	0.362681
$696$	0	0
$697$	14.9686	0.566975
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.40072	−0.0529046	−0.0264523	−	0.999650i	$-0.508421\pi$
$701$	−1.40072	−0.0529046	−0.0264523	+	0.999650i	$0.508421\pi$
$702$	0	0
$703$	1.76944i	0.0667357i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.00000i	0.0752177i
$708$	0	0
$709$	− 39.4709i	− 1.48236i	−0.671307	−	0.741179i	$-0.734267\pi$
$709$	− 39.4709i	− 1.48236i	0.671307	−	0.741179i	$-0.265733\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	62.0389i	2.32338i
$714$	0	0
$715$	−8.15396	−0.304941
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12.0694	0.450113	0.225056	−	0.974346i	$-0.427743\pi$
$719$	12.0694	0.450113	0.225056	+	0.974346i	$0.427743\pi$
$720$	0	0
$721$	−17.0148	−0.633663
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−22.6922	−0.842768
$726$	0	0
$727$	− 2.39230i	− 0.0887257i	−0.999015	−	0.0443628i	$-0.985874\pi$
$727$	− 2.39230i	− 0.0887257i	0.999015	−	0.0443628i	$-0.0141258\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 3.24495i	− 0.120019i
$732$	0	0
$733$	14.5802i	0.538533i	0.963066	+	0.269267i	$0.0867813\pi$
$733$	14.5802i	0.538533i	−0.963066	+	0.269267i	$0.913219\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.60530i	0.0959673i
$738$	0	0
$739$	−30.3459	−1.11629	−0.558146	−	0.829743i	$-0.688487\pi$
$739$	−30.3459	−1.11629	−0.558146	+	0.829743i	$0.688487\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16.6659	−0.611411	−0.305706	−	0.952126i	$-0.598892\pi$
$743$	−16.6659	−0.611411	−0.305706	+	0.952126i	$0.598892\pi$
$744$	0	0
$745$	−21.9540	−0.804332
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.91362	0.0699222
$750$	0	0
$751$	− 2.10155i	− 0.0766866i	−0.999265	−	0.0383433i	$-0.987792\pi$
$751$	− 2.10155i	− 0.0766866i	0.999265	−	0.0383433i	$-0.0122080\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	30.7030i	1.11740i
$756$	0	0
$757$	− 39.2408i	− 1.42623i	−0.701046	−	0.713116i	$-0.747283\pi$
$757$	− 39.2408i	− 1.42623i	0.701046	−	0.713116i	$-0.252717\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 22.7618i	− 0.825113i	−0.910932	−	0.412556i	$-0.864636\pi$
$761$	− 22.7618i	− 0.825113i	0.910932	−	0.412556i	$-0.135364\pi$
$762$	0	0
$763$	−7.07816	−0.256246
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.1967	0.584828
$768$	0	0
$769$	−52.9030	−1.90773	−0.953865	−	0.300234i	$-0.902935\pi$
$769$	−52.9030	−1.90773	−0.953865	+	0.300234i	$0.902935\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.27110	−0.225556	−0.112778	−	0.993620i	$-0.535975\pi$
$773$	−6.27110	−0.225556	−0.112778	+	0.993620i	$0.535975\pi$
$774$	0	0
$775$	− 36.2087i	− 1.30066i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 3.94780i	− 0.141445i
$780$	0	0
$781$	0.881470i	0.0315415i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 34.1038i	− 1.21722i
$786$	0	0
$787$	20.5651	0.733065	0.366533	−	0.930405i	$-0.380545\pi$
$787$	20.5651	0.733065	0.366533	+	0.930405i	$0.380545\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.98316	0.354960
$792$	0	0
$793$	75.0976	2.66679
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.6802	0.590845	0.295422	−	0.955367i	$-0.404540\pi$
$797$	16.6802	0.590845	0.295422	+	0.955367i	$0.404540\pi$
$798$	0	0
$799$	20.9114i	0.739791i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 2.28894i	− 0.0807751i
$804$	0	0
$805$	− 25.7152i	− 0.906342i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 54.5485i	− 1.91782i	−0.283707	−	0.958911i	$-0.591564\pi$
$809$	− 54.5485i	− 1.91782i	0.283707	−	0.958911i	$-0.408436\pi$
$810$	0	0
$811$	9.77845	0.343368	0.171684	−	0.985152i	$-0.445079\pi$
$811$	9.77845	0.343368	0.171684	+	0.985152i	$0.445079\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−31.3439	−1.09793
$816$	0	0
$817$	−0.855821	−0.0299414
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.5488	−0.996359	−0.498179	−	0.867074i	$-0.665998\pi$
$821$	−28.5488	−0.996359	−0.498179	+	0.867074i	$0.665998\pi$
$822$	0	0
$823$	33.5912i	1.17092i	0.810702	+	0.585459i	$0.199086\pi$
$823$	33.5912i	1.17092i	−0.810702	+	0.585459i	$0.800914\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 25.9650i	− 0.902893i	−0.892298	−	0.451446i	$-0.850908\pi$
$827$	− 25.9650i	− 0.902893i	0.892298	−	0.451446i	$-0.149092\pi$
$828$	0	0
$829$	− 23.7613i	− 0.825263i	−0.910898	−	0.412631i	$-0.864610\pi$
$829$	− 23.7613i	− 0.825263i	0.910898	−	0.412631i	$-0.135390\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2.00000i	− 0.0692959i
$834$	0	0
$835$	−45.5864	−1.57758
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.77243	0.0957148	0.0478574	−	0.998854i	$-0.484761\pi$
$839$	2.77243	0.0957148	0.0478574	+	0.998854i	$0.484761\pi$
$840$	0	0
$841$	−6.60770	−0.227852
$842$	0	0
$843$	0	0
$844$	0	0
$845$	93.6155	3.22047
$846$	0	0
$847$	− 10.8418i	− 0.372529i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 27.5619i	− 0.944810i
$852$	0	0
$853$	0.236670i	0.00810344i	0.999992	+	0.00405172i	$0.00128971\pi$
$853$	0.236670i	0.00810344i	−0.999992	+	0.00405172i	$0.998710\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.2139i	0.724654i	0.932051	+	0.362327i	$0.118018\pi$
$857$	21.2139i	0.724654i	−0.932051	+	0.362327i	$0.881982\pi$
$858$	0	0
$859$	−12.6199	−0.430585	−0.215292	−	0.976550i	$-0.569070\pi$
$859$	−12.6199	−0.430585	−0.215292	+	0.976550i	$0.569070\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	25.8396	0.879589	0.439795	−	0.898098i	$-0.355051\pi$
$863$	25.8396	0.879589	0.439795	+	0.898098i	$0.355051\pi$
$864$	0	0
$865$	44.8104	1.52360
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.09293	0.138843
$870$	0	0
$871$	− 42.9114i	− 1.45400i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 0.640262i	− 0.0216448i
$876$	0	0
$877$	55.0972i	1.86050i	0.366925	+	0.930251i	$0.380411\pi$
$877$	55.0972i	1.86050i	−0.366925	+	0.930251i	$0.619589\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 33.7034i	− 1.13550i	−0.823202	−	0.567749i	$-0.807814\pi$
$881$	− 33.7034i	− 1.13550i	0.823202	−	0.567749i	$-0.192186\pi$
$882$	0	0
$883$	−19.8396	−0.667655	−0.333827	−	0.942634i	$-0.608340\pi$
$883$	−19.8396	−0.667655	−0.333827	+	0.942634i	$0.608340\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.5967	0.590840	0.295420	−	0.955367i	$-0.404540\pi$
$887$	17.5967	0.590840	0.295420	+	0.955367i	$0.404540\pi$
$888$	0	0
$889$	7.75525	0.260103
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.51515	0.184558
$894$	0	0
$895$	62.0019i	2.07249i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	35.7302i	1.19167i
$900$	0	0
$901$	1.59086i	0.0529991i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 63.1372i	− 2.09875i
$906$	0	0
$907$	10.8190	0.359238	0.179619	−	0.983736i	$-0.442514\pi$
$907$	10.8190	0.359238	0.179619	+	0.983736i	$0.442514\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−33.4364	−1.10780	−0.553899	−	0.832584i	$-0.686861\pi$
$911$	−33.4364	−1.10780	−0.553899	+	0.832584i	$0.686861\pi$
$912$	0	0
$913$	3.32859	0.110160
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.9052	0.492213
$918$	0	0
$919$	3.70049i	0.122068i	0.998136	+	0.0610339i	$0.0194398\pi$
$919$	3.70049i	0.122068i	−0.998136	+	0.0610339i	$0.980560\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 14.5185i	− 0.477883i
$924$	0	0
$925$	16.0864i	0.528917i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17.0318i	0.558796i	0.960175	+	0.279398i	$0.0901348\pi$
$929$	17.0318i	0.558796i	−0.960175	+	0.279398i	$0.909865\pi$
$930$	0	0
$931$	−0.527479	−0.0172874
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.48950	0.0814155
$936$	0	0
$937$	−11.6077	−0.379207	−0.189603	−	0.981861i	$-0.560720\pi$
$937$	−11.6077	−0.379207	−0.189603	+	0.981861i	$0.560720\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−50.5834	−1.64897	−0.824486	−	0.565882i	$-0.808536\pi$
$941$	−50.5834	−1.64897	−0.824486	+	0.565882i	$0.808536\pi$
$942$	0	0
$943$	61.4935i	2.00250i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.8764i	1.39330i	0.717413	+	0.696648i	$0.245326\pi$
$947$	42.8764i	1.39330i	−0.717413	+	0.696648i	$0.754674\pi$
$948$	0	0
$949$	37.7008i	1.22382i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	11.4074i	0.369523i	0.982783	+	0.184761i	$0.0591512\pi$
$953$	11.4074i	0.369523i	−0.982783	+	0.184761i	$0.940849\pi$
$954$	0	0
$955$	20.1416	0.651766
$956$	0	0
$957$	0	0
$958$	0	0
$959$	22.4557	0.725132
$960$	0	0
$961$	−26.0127	−0.839120
$962$	0	0
$963$	0	0
$964$	0	0
$965$	31.2449	1.00581
$966$	0	0
$967$	− 24.7090i	− 0.794589i	−0.917691	−	0.397295i	$-0.869949\pi$
$967$	− 24.7090i	− 0.794589i	0.917691	−	0.397295i	$-0.130051\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 28.3585i	− 0.910067i	−0.890474	−	0.455034i	$-0.849627\pi$
$971$	− 28.3585i	− 0.910067i	0.890474	−	0.455034i	$-0.150373\pi$
$972$	0	0
$973$	3.05496i	0.0979375i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.4704i	0.750885i	0.926846	+	0.375442i	$0.122509\pi$
$977$	23.4704i	0.750885i	−0.926846	+	0.375442i	$0.877491\pi$
$978$	0	0
$979$	−2.11231	−0.0675099
$980$	0	0
$981$	0	0
$982$	0	0
$983$	26.2425	0.837007	0.418504	−	0.908215i	$-0.362555\pi$
$983$	26.2425	0.837007	0.418504	+	0.908215i	$0.362555\pi$
$984$	0	0
$985$	73.9790	2.35717
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13.3308	0.423895
$990$	0	0
$991$	44.2579i	1.40590i	0.711241	+	0.702949i	$0.248134\pi$
$991$	44.2579i	1.40590i	−0.711241	+	0.702949i	$0.751866\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 67.6662i	− 2.14516i
$996$	0	0
$997$	28.1271i	0.890796i	0.895333	+	0.445398i	$0.146938\pi$
$997$	28.1271i	0.890796i	−0.895333	+	0.445398i	$0.853062\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.j.a.5615.3		8
3.2	odd	2		6048.2.j.b.5615.5		8
4.3	odd	2		1512.2.j.a.323.3	yes	8
8.3	odd	2		6048.2.j.b.5615.6		8
8.5	even	2		1512.2.j.b.323.8	yes	8
12.11	even	2		1512.2.j.b.323.6	yes	8
24.5	odd	2		1512.2.j.a.323.1	✓	8
24.11	even	2	inner	6048.2.j.a.5615.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.a.323.1	✓	8	24.5	odd	2
1512.2.j.a.323.3	yes	8	4.3	odd	2
1512.2.j.b.323.6	yes	8	12.11	even	2
1512.2.j.b.323.8	yes	8	8.5	even	2
6048.2.j.a.5615.3		8	1.1	even	1	trivial
6048.2.j.a.5615.4		8	24.11	even	2	inner
6048.2.j.b.5615.5		8	3.2	odd	2
6048.2.j.b.5615.6		8	8.3	odd	2

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)

\(f(q)\)	\(=\)	\(q-3.12976 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-3.12976 q^{5} -1.00000i q^{7} +0.397714i q^{11} -6.55068i q^{13} +2.00000i q^{17} +0.527479 q^{19} -8.21634 q^{23} +4.79543 q^{25} -4.73205 q^{29} -7.55068i q^{31} +3.12976i q^{35} +3.35452i q^{37} -7.48429i q^{41} -1.62247 q^{43} +10.4557 q^{47} -1.00000 q^{49} +0.795428 q^{53} -1.24475i q^{55} +2.47252i q^{59} +11.4641i q^{61} +20.5021i q^{65} +6.55068 q^{67} +2.21634 q^{71} -5.75525 q^{73} +0.397714 q^{77} -10.2911i q^{79} -8.36930i q^{83} -6.25953i q^{85} +5.31114i q^{89} -6.55068 q^{91} -1.65089 q^{95} -19.1014 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} - 16 q^{19} - 16 q^{23} + 32 q^{25} - 24 q^{29} - 8 q^{43} + 8 q^{47} - 8 q^{49} - 8 q^{67} - 32 q^{71} + 8 q^{73} + 8 q^{91} - 112 q^{95} - 32 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 - 16 * q^19 - 16 * q^23 + 32 * q^25 - 24 * q^29 - 8 * q^43 + 8 * q^47 - 8 * q^49 - 8 * q^67 - 32 * q^71 + 8 * q^73 + 8 * q^91 - 112 * q^95 - 32 * q^97

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