LMFDB - Embedded newform 6048.2.j.a.5615.2

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			5615.2
Root			$0.500000 + 1.19293i$ of defining polynomial
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.a.5615.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.38587 q^{5} +1.00000i q^{7} +O(q^{10})$	$q-4.38587 q^{5} +1.00000i q^{7} -5.11792i q^{11} -5.21068i q^{13} -2.00000i q^{17} +6.50379 q^{19} -4.63929 q^{23} +14.2358 q^{25} -1.26795 q^{29} -4.21068i q^{31} -4.38587i q^{35} -1.98547i q^{37} +7.37134i q^{41} -3.71753 q^{43} +2.57558 q^{47} -1.00000 q^{49} +10.2358 q^{53} +22.4465i q^{55} +3.50379i q^{59} -4.53590i q^{61} +22.8533i q^{65} -5.21068 q^{67} -1.36071 q^{71} +15.4465 q^{73} +5.11792 q^{77} -3.98241i q^{79} -11.6893i q^{83} +8.77174i q^{85} -14.8645i q^{89} +5.21068 q^{91} -28.5247 q^{95} +4.42136 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−4.38587	−1.96142	−0.980710	−	0.195469i	$-0.937377\pi$
$5$	−4.38587	−1.96142	−0.980710	+	0.195469i	$0.937377\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.11792i	− 1.54311i	−0.636162	−	0.771555i	$-0.719479\pi$
$11$	− 5.11792i	− 1.54311i	0.636162	−	0.771555i	$-0.280521\pi$
$12$	0	0
$13$	− 5.21068i	− 1.44518i	−0.691276	−	0.722591i	$-0.742951\pi$
$13$	− 5.21068i	− 1.44518i	0.691276	−	0.722591i	$-0.257049\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	6.50379	1.49207	0.746035	−	0.665906i	$-0.231955\pi$
$19$	6.50379	1.49207	0.746035	+	0.665906i	$0.231955\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.63929	−0.967359	−0.483680	−	0.875245i	$-0.660700\pi$
$23$	−4.63929	−0.967359	−0.483680	+	0.875245i	$0.660700\pi$
$24$	0	0
$25$	14.2358	2.84717
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.26795	−0.235452	−0.117726	−	0.993046i	$-0.537560\pi$
$29$	−1.26795	−0.235452	−0.117726	+	0.993046i	$0.537560\pi$
$30$	0	0
$31$	− 4.21068i	− 0.756260i	−0.925752	−	0.378130i	$-0.876567\pi$
$31$	− 4.21068i	− 0.756260i	0.925752	−	0.378130i	$-0.123433\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 4.38587i	− 0.741347i
$36$	0	0
$37$	− 1.98547i	− 0.326410i	−0.986592	−	0.163205i	$-0.947817\pi$
$37$	− 1.98547i	− 0.326410i	0.986592	−	0.163205i	$-0.0521832\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.37134i	1.15121i	0.817728	+	0.575605i	$0.195233\pi$
$41$	7.37134i	1.15121i	−0.817728	+	0.575605i	$0.804767\pi$
$42$	0	0
$43$	−3.71753	−0.566917	−0.283459	−	0.958984i	$-0.591482\pi$
$43$	−3.71753	−0.566917	−0.283459	+	0.958984i	$0.591482\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.57558	0.375687	0.187844	−	0.982199i	$-0.439850\pi$
$47$	2.57558	0.375687	0.187844	+	0.982199i	$0.439850\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.2358	1.40600	0.703000	−	0.711190i	$-0.251843\pi$
$53$	10.2358	1.40600	0.703000	+	0.711190i	$0.251843\pi$
$54$	0	0
$55$	22.4465i	3.02669i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.50379i	0.456154i	0.973643	+	0.228077i	$0.0732438\pi$
$59$	3.50379i	0.456154i	−0.973643	+	0.228077i	$0.926756\pi$
$60$	0	0
$61$	− 4.53590i	− 0.580762i	−0.956911	−	0.290381i	$-0.906218\pi$
$61$	− 4.53590i	− 0.580762i	0.956911	−	0.290381i	$-0.0937821\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	22.8533i	2.83461i
$66$	0	0
$67$	−5.21068	−0.636586	−0.318293	−	0.947992i	$-0.603110\pi$
$67$	−5.21068	−0.636586	−0.318293	+	0.947992i	$0.603110\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.36071	−0.161486	−0.0807432	−	0.996735i	$-0.525729\pi$
$71$	−1.36071	−0.161486	−0.0807432	+	0.996735i	$0.525729\pi$
$72$	0	0
$73$	15.4465	1.80788	0.903939	−	0.427662i	$-0.140662\pi$
$73$	15.4465	1.80788	0.903939	+	0.427662i	$0.140662\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.11792	0.583241
$78$	0	0
$79$	− 3.98241i	− 0.448057i	−0.974583	−	0.224028i	$-0.928079\pi$
$79$	− 3.98241i	− 0.448057i	0.974583	−	0.224028i	$-0.0719208\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 11.6893i	− 1.28307i	−0.767095	−	0.641534i	$-0.778298\pi$
$83$	− 11.6893i	− 1.28307i	0.767095	−	0.641534i	$-0.221702\pi$
$84$	0	0
$85$	8.77174i	0.951428i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 14.8645i	− 1.57563i	−0.615910	−	0.787817i	$-0.711211\pi$
$89$	− 14.8645i	− 1.57563i	0.615910	−	0.787817i	$-0.288789\pi$
$90$	0	0
$91$	5.21068	0.546227
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−28.5247	−2.92658
$96$	0	0
$97$	4.42136	0.448921	0.224460	−	0.974483i	$-0.427938\pi$
$97$	4.42136	0.448921	0.224460	+	0.974483i	$0.427938\pi$
$98$	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
$103$	− 1.67478i	− 0.165021i	−0.996590	−	0.0825105i	$-0.973706\pi$
$103$	− 1.67478i	− 0.165021i	0.996590	−	0.0825105i	$-0.0262938\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.2649i	0.992344i	0.868224	+	0.496172i	$0.165262\pi$
$107$	10.2649i	0.992344i	−0.868224	+	0.496172i	$0.834738\pi$
$108$	0	0
$109$	1.29311i	0.123857i	0.998081	+	0.0619287i	$0.0197251\pi$
$109$	1.29311i	0.123857i	−0.998081	+	0.0619287i	$0.980275\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 8.07937i	− 0.760043i	−0.924978	−	0.380022i	$-0.875917\pi$
$113$	− 8.07937i	− 0.760043i	0.924978	−	0.380022i	$-0.124083\pi$
$114$	0	0
$115$	20.3473	1.89740
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	−15.1931	−1.38119
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−40.5072	−3.62307
$126$	0	0
$127$	13.4465i	1.19319i	0.802544	+	0.596593i	$0.203479\pi$
$127$	13.4465i	1.19319i	−0.802544	+	0.596593i	$0.796521\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.77480i	− 0.155065i	−0.996990	−	0.0775323i	$-0.975296\pi$
$131$	− 1.77480i	− 0.155065i	0.996990	−	0.0775323i	$-0.0247041\pi$
$132$	0	0
$133$	6.50379i	0.563950i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14.5756i	− 1.24528i	−0.782510	−	0.622638i	$-0.786061\pi$
$137$	− 14.5756i	− 1.24528i	0.782510	−	0.622638i	$-0.213939\pi$
$138$	0	0
$139$	−15.0076	−1.27293	−0.636463	−	0.771307i	$-0.719603\pi$
$139$	−15.0076	−1.27293	−0.636463	+	0.771307i	$0.719603\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−26.6678	−2.23008
$144$	0	0
$145$	5.56106	0.461821
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.33669	0.437198	0.218599	−	0.975815i	$-0.429851\pi$
$149$	5.33669	0.437198	0.218599	+	0.975815i	$0.429851\pi$
$150$	0	0
$151$	17.5725i	1.43003i	0.699108	+	0.715016i	$0.253581\pi$
$151$	17.5725i	1.43003i	−0.699108	+	0.715016i	$0.746419\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	18.4675i	1.48334i
$156$	0	0
$157$	− 13.8259i	− 1.10343i	−0.834032	−	0.551715i	$-0.813973\pi$
$157$	− 13.8259i	− 1.10343i	0.834032	−	0.551715i	$-0.186027\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 4.63929i	− 0.365627i
$162$	0	0
$163$	−8.67478	−0.679461	−0.339731	−	0.940523i	$-0.610336\pi$
$163$	−8.67478	−0.679461	−0.339731	+	0.940523i	$0.610336\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.8855	−1.22925	−0.614627	−	0.788818i	$-0.710693\pi$
$167$	−15.8855	−1.22925	−0.614627	+	0.788818i	$0.710693\pi$
$168$	0	0
$169$	−14.1512	−1.08855
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.22940	−0.321555	−0.160778	−	0.986991i	$-0.551400\pi$
$173$	−4.22940	−0.321555	−0.160778	+	0.986991i	$0.551400\pi$
$174$	0	0
$175$	14.2358i	1.07613i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 6.45041i	− 0.482126i	−0.970509	−	0.241063i	$-0.922504\pi$
$179$	− 6.45041i	− 0.482126i	0.970509	−	0.241063i	$-0.0774961\pi$
$180$	0	0
$181$	− 10.5068i	− 0.780968i	−0.920610	−	0.390484i	$-0.872308\pi$
$181$	− 10.5068i	− 0.780968i	0.920610	−	0.390484i	$-0.127692\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.70803i	0.640227i
$186$	0	0
$187$	−10.2358	−0.748519
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.25986	0.597663	0.298831	−	0.954306i	$-0.403403\pi$
$191$	8.25986	0.597663	0.298831	+	0.954306i	$0.403403\pi$
$192$	0	0
$193$	−8.07937	−0.581566	−0.290783	−	0.956789i	$-0.593916\pi$
$193$	−8.07937	−0.581566	−0.290783	+	0.956789i	$0.593916\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.04275	−0.501775	−0.250887	−	0.968016i	$-0.580722\pi$
$197$	−7.04275	−0.501775	−0.250887	+	0.968016i	$0.580722\pi$
$198$	0	0
$199$	− 20.1336i	− 1.42723i	−0.700537	−	0.713616i	$-0.747056\pi$
$199$	− 20.1336i	− 1.42723i	0.700537	−	0.713616i	$-0.252944\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.26795i	− 0.0889926i
$204$	0	0
$205$	− 32.3297i	− 2.25801i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 33.2859i	− 2.30243i
$210$	0	0
$211$	−7.54347	−0.519314	−0.259657	−	0.965701i	$-0.583610\pi$
$211$	−7.54347	−0.519314	−0.259657	+	0.965701i	$0.583610\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.3046	1.11196
$216$	0	0
$217$	4.21068	0.285839
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−10.4214	−0.701016
$222$	0	0
$223$	− 5.48926i	− 0.367588i	−0.982965	−	0.183794i	$-0.941162\pi$
$223$	− 5.48926i	− 0.367588i	0.982965	−	0.183794i	$-0.0588380\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.127416i	0.00845689i	0.999991	+	0.00422845i	$0.00134596\pi$
$227$	0.127416i	0.00845689i	−0.999991	+	0.00422845i	$0.998654\pi$
$228$	0	0
$229$	− 6.53590i	− 0.431904i	−0.976404	−	0.215952i	$-0.930714\pi$
$229$	− 6.53590i	− 0.431904i	0.976404	−	0.215952i	$-0.0692855\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 10.5359i	− 0.690230i	−0.938560	−	0.345115i	$-0.887840\pi$
$233$	− 10.5359i	− 0.690230i	0.938560	−	0.345115i	$-0.112160\pi$
$234$	0	0
$235$	−11.2962	−0.736881
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.3862	1.57741	0.788706	−	0.614771i	$-0.210752\pi$
$239$	24.3862	1.57741	0.788706	+	0.614771i	$0.210752\pi$
$240$	0	0
$241$	−10.8603	−0.699573	−0.349787	−	0.936829i	$-0.613746\pi$
$241$	−10.8603	−0.699573	−0.349787	+	0.936829i	$0.613746\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.38587	0.280203
$246$	0	0
$247$	− 33.8891i	− 2.15631i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 3.73511i	− 0.235758i	−0.993028	−	0.117879i	$-0.962390\pi$
$251$	− 3.73511i	− 0.235758i	0.993028	−	0.117879i	$-0.0376095\pi$
$252$	0	0
$253$	23.7435i	1.49274i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 7.52781i	− 0.469572i	−0.972047	−	0.234786i	$-0.924561\pi$
$257$	− 7.52781i	− 0.469572i	0.972047	−	0.234786i	$-0.0754389\pi$
$258$	0	0
$259$	1.98547	0.123371
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.7187	−1.03092	−0.515458	−	0.856915i	$-0.672378\pi$
$263$	−16.7187	−1.03092	−0.515458	+	0.856915i	$0.672378\pi$
$264$	0	0
$265$	−44.8930	−2.75776
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.11486	0.250887	0.125444	−	0.992101i	$-0.459965\pi$
$269$	4.11486	0.250887	0.125444	+	0.992101i	$0.459965\pi$
$270$	0	0
$271$	4.31293i	0.261992i	0.991383	+	0.130996i	$0.0418175\pi$
$271$	4.31293i	0.261992i	−0.991383	+	0.130996i	$0.958182\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 72.8579i	− 4.39349i
$276$	0	0
$277$	27.7564i	1.66772i	0.551976	+	0.833860i	$0.313874\pi$
$277$	27.7564i	1.66772i	−0.551976	+	0.833860i	$0.686126\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.8961i	1.30621i	0.757267	+	0.653106i	$0.226534\pi$
$281$	21.8961i	1.30621i	−0.757267	+	0.653106i	$0.773466\pi$
$282$	0	0
$283$	−15.3786	−0.914164	−0.457082	−	0.889425i	$-0.651105\pi$
$283$	−15.3786	−0.914164	−0.457082	+	0.889425i	$0.651105\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.37134	−0.435117
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0.408482	0.0238638	0.0119319	−	0.999929i	$-0.496202\pi$
$293$	0.408482	0.0238638	0.0119319	+	0.999929i	$0.496202\pi$
$294$	0	0
$295$	− 15.3671i	− 0.894710i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	24.1739i	1.39801i
$300$	0	0
$301$	− 3.71753i	− 0.214275i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	19.8939i	1.13912i
$306$	0	0
$307$	−9.99694	−0.570555	−0.285278	−	0.958445i	$-0.592086\pi$
$307$	−9.99694	−0.570555	−0.285278	+	0.958445i	$0.592086\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.9603	−1.13185	−0.565923	−	0.824458i	$-0.691480\pi$
$311$	−19.9603	−1.13185	−0.565923	+	0.824458i	$0.691480\pi$
$312$	0	0
$313$	23.4465	1.32528	0.662638	−	0.748940i	$-0.269437\pi$
$313$	23.4465	1.32528	0.662638	+	0.748940i	$0.269437\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.7740	−1.16678	−0.583391	−	0.812191i	$-0.698275\pi$
$317$	−20.7740	−1.16678	−0.583391	+	0.812191i	$0.698275\pi$
$318$	0	0
$319$	6.48926i	0.363329i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 13.0076i	− 0.723761i
$324$	0	0
$325$	− 74.1784i	− 4.11468i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.57558i	0.141997i
$330$	0	0
$331$	−19.0366	−1.04635	−0.523174	−	0.852226i	$-0.675252\pi$
$331$	−19.0366	−1.04635	−0.523174	+	0.852226i	$0.675252\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	22.8533	1.24861
$336$	0	0
$337$	2.88546	0.157181	0.0785905	−	0.996907i	$-0.474958\pi$
$337$	2.88546	0.157181	0.0785905	+	0.996907i	$0.474958\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−21.5499	−1.16699
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.7033i	1.64824i	0.566415	+	0.824120i	$0.308330\pi$
$347$	30.7033i	1.64824i	−0.566415	+	0.824120i	$0.691670\pi$
$348$	0	0
$349$	26.5007i	1.41855i	0.704931	+	0.709276i	$0.250978\pi$
$349$	26.5007i	1.41855i	−0.704931	+	0.709276i	$0.749022\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.29955i	0.335291i	0.985847	+	0.167645i	$0.0536164\pi$
$353$	6.29955i	0.335291i	−0.985847	+	0.167645i	$0.946384\pi$
$354$	0	0
$355$	5.96789	0.316743
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.60770	−0.0848509	−0.0424255	−	0.999100i	$-0.513508\pi$
$359$	−1.60770	−0.0848509	−0.0424255	+	0.999100i	$0.513508\pi$
$360$	0	0
$361$	23.2992	1.22628
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−67.7464	−3.54601
$366$	0	0
$367$	− 17.1763i	− 0.896597i	−0.893884	−	0.448298i	$-0.852030\pi$
$367$	− 17.1763i	− 0.896597i	0.893884	−	0.448298i	$-0.147970\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.2358i	0.531418i
$372$	0	0
$373$	− 2.27941i	− 0.118024i	−0.998257	−	0.0590118i	$-0.981205\pi$
$373$	− 2.27941i	− 0.118024i	0.998257	−	0.0590118i	$-0.0187950\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.60688i	0.340271i
$378$	0	0
$379$	−20.1389	−1.03446	−0.517232	−	0.855845i	$-0.673038\pi$
$379$	−20.1389	−1.03446	−0.517232	+	0.855845i	$0.673038\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−22.3068	−1.13982	−0.569912	−	0.821705i	$-0.693023\pi$
$383$	−22.3068	−1.13982	−0.569912	+	0.821705i	$0.693023\pi$
$384$	0	0
$385$	−22.4465	−1.14398
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.57864	−0.0800404	−0.0400202	−	0.999199i	$-0.512742\pi$
$389$	−1.57864	−0.0800404	−0.0400202	+	0.999199i	$0.512742\pi$
$390$	0	0
$391$	9.27858i	0.469238i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	17.4663i	0.878827i
$396$	0	0
$397$	− 31.4289i	− 1.57737i	−0.614796	−	0.788686i	$-0.710762\pi$
$397$	− 31.4289i	− 1.57737i	0.614796	−	0.788686i	$-0.289238\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 26.5404i	− 1.32536i	−0.748901	−	0.662682i	$-0.769418\pi$
$401$	− 26.5404i	− 1.32536i	0.748901	−	0.662682i	$-0.230582\pi$
$402$	0	0
$403$	−21.9405	−1.09293
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.1615	−0.503687
$408$	0	0
$409$	32.0037	1.58248	0.791240	−	0.611506i	$-0.209436\pi$
$409$	32.0037	1.58248	0.791240	+	0.611506i	$0.209436\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.50379	−0.172410
$414$	0	0
$415$	51.2678i	2.51663i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 20.6572i	− 1.00917i	−0.863362	−	0.504585i	$-0.831646\pi$
$419$	− 20.6572i	− 1.00917i	0.863362	−	0.504585i	$-0.168354\pi$
$420$	0	0
$421$	32.9930i	1.60798i	0.594641	+	0.803991i	$0.297294\pi$
$421$	32.9930i	1.60798i	−0.594641	+	0.803991i	$0.702706\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 28.4717i	− 1.38108i
$426$	0	0
$427$	4.53590	0.219508
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.48282	0.312267	0.156133	−	0.987736i	$-0.450097\pi$
$431$	6.48282	0.312267	0.156133	+	0.987736i	$0.450097\pi$
$432$	0	0
$433$	−18.4038	−0.884429	−0.442214	−	0.896909i	$-0.645807\pi$
$433$	−18.4038	−0.884429	−0.442214	+	0.896909i	$0.645807\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−30.1730	−1.44337
$438$	0	0
$439$	− 4.31293i	− 0.205845i	−0.994689	−	0.102923i	$-0.967181\pi$
$439$	− 4.31293i	− 0.205845i	0.994689	−	0.102923i	$-0.0328194\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 5.54540i	− 0.263470i	−0.991285	−	0.131735i	$-0.957945\pi$
$443$	− 5.54540i	− 0.263470i	0.991285	−	0.131735i	$-0.0420547\pi$
$444$	0	0
$445$	65.1937i	3.09048i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	30.2007i	1.42526i	0.701541	+	0.712629i	$0.252496\pi$
$449$	30.2007i	1.42526i	−0.701541	+	0.712629i	$0.747504\pi$
$450$	0	0
$451$	37.7259	1.77644
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−22.8533	−1.07138
$456$	0	0
$457$	−24.3205	−1.13767	−0.568833	−	0.822453i	$-0.692605\pi$
$457$	−24.3205	−1.13767	−0.568833	+	0.822453i	$0.692605\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	37.3141	1.73789	0.868945	−	0.494909i	$-0.164799\pi$
$461$	37.3141	1.73789	0.868945	+	0.494909i	$0.164799\pi$
$462$	0	0
$463$	35.2853i	1.63985i	0.572472	+	0.819924i	$0.305985\pi$
$463$	35.2853i	1.63985i	−0.572472	+	0.819924i	$0.694015\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.7824i	0.961693i	0.876805	+	0.480847i	$0.159671\pi$
$467$	20.7824i	0.961693i	−0.876805	+	0.480847i	$0.840329\pi$
$468$	0	0
$469$	− 5.21068i	− 0.240607i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	19.0260i	0.874816i
$474$	0	0
$475$	92.5868	4.24818
$476$	0	0
$477$	0	0
$478$	0	0
$479$	17.1640	0.784245	0.392123	−	0.919913i	$-0.371741\pi$
$479$	17.1640	0.784245	0.392123	+	0.919913i	$0.371741\pi$
$480$	0	0
$481$	−10.3457	−0.471722
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−19.3915	−0.880522
$486$	0	0
$487$	− 38.7961i	− 1.75802i	−0.476805	−	0.879009i	$-0.658205\pi$
$487$	− 38.7961i	− 1.75802i	0.476805	−	0.879009i	$-0.341795\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	31.1973i	1.40791i	0.710243	+	0.703957i	$0.248585\pi$
$491$	31.1973i	1.40791i	−0.710243	+	0.703957i	$0.751415\pi$
$492$	0	0
$493$	2.53590i	0.114211i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.36071i	− 0.0610361i
$498$	0	0
$499$	8.78320	0.393190	0.196595	−	0.980485i	$-0.437012\pi$
$499$	8.78320	0.393190	0.196595	+	0.980485i	$0.437012\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	11.2679	0.502413	0.251207	−	0.967934i	$-0.419173\pi$
$503$	11.2679	0.502413	0.251207	+	0.967934i	$0.419173\pi$
$504$	0	0
$505$	8.77174	0.390337
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.99388	0.354322	0.177161	−	0.984182i	$-0.443309\pi$
$509$	7.99388	0.354322	0.177161	+	0.984182i	$0.443309\pi$
$510$	0	0
$511$	15.4465i	0.683314i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7.34536i	0.323675i
$516$	0	0
$517$	− 13.1816i	− 0.579727i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 15.1797i	− 0.665035i	−0.943097	−	0.332517i	$-0.892102\pi$
$521$	− 15.1797i	− 0.665035i	0.943097	−	0.332517i	$-0.107898\pi$
$522$	0	0
$523$	5.58846	0.244366	0.122183	−	0.992508i	$-0.461010\pi$
$523$	5.58846	0.244366	0.122183	+	0.992508i	$0.461010\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.42136	−0.366840
$528$	0	0
$529$	−1.47698	−0.0642163
$530$	0	0
$531$	0	0
$532$	0	0
$533$	38.4097	1.66371
$534$	0	0
$535$	− 45.0204i	− 1.94640i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.11792i	0.220444i
$540$	0	0
$541$	10.1229i	0.435219i	0.976036	+	0.217610i	$0.0698260\pi$
$541$	10.1229i	0.435219i	−0.976036	+	0.217610i	$0.930174\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 5.67140i	− 0.242936i
$546$	0	0
$547$	−7.03662	−0.300864	−0.150432	−	0.988620i	$-0.548067\pi$
$547$	−7.03662	−0.300864	−0.150432	+	0.988620i	$0.548067\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.24647	−0.351311
$552$	0	0
$553$	3.98241	0.169349
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11.0427	−0.467896	−0.233948	−	0.972249i	$-0.575165\pi$
$557$	−11.0427	−0.467896	−0.233948	+	0.972249i	$0.575165\pi$
$558$	0	0
$559$	19.3708i	0.819299i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.1115i	0.805453i	0.915320	+	0.402726i	$0.131937\pi$
$563$	19.1115i	0.805453i	−0.915320	+	0.402726i	$0.868063\pi$
$564$	0	0
$565$	35.4351i	1.49076i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 12.6443i	− 0.530077i	−0.964238	−	0.265039i	$-0.914615\pi$
$569$	− 12.6443i	− 0.530077i	0.964238	−	0.265039i	$-0.0853847\pi$
$570$	0	0
$571$	−19.4351	−0.813332	−0.406666	−	0.913577i	$-0.633309\pi$
$571$	−19.4351	−0.813332	−0.406666	+	0.913577i	$0.633309\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−66.0442	−2.75423
$576$	0	0
$577$	−42.8137	−1.78236	−0.891178	−	0.453654i	$-0.850120\pi$
$577$	−42.8137	−1.78236	−0.891178	+	0.453654i	$0.850120\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11.6893	0.484954
$582$	0	0
$583$	− 52.3862i	− 2.16961i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.9221i	1.19374i	0.802337	+	0.596871i	$0.203590\pi$
$587$	28.9221i	1.19374i	−0.802337	+	0.596871i	$0.796410\pi$
$588$	0	0
$589$	− 27.3854i	− 1.12839i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 41.9140i	− 1.72120i	−0.509280	−	0.860601i	$-0.670088\pi$
$593$	− 41.9140i	− 1.72120i	0.509280	−	0.860601i	$-0.329912\pi$
$594$	0	0
$595$	−8.77174	−0.359606
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20.5829	0.840993	0.420496	−	0.907294i	$-0.361856\pi$
$599$	20.5829	0.840993	0.420496	+	0.907294i	$0.361856\pi$
$600$	0	0
$601$	5.44652	0.222168	0.111084	−	0.993811i	$-0.464568\pi$
$601$	5.44652	0.222168	0.111084	+	0.993811i	$0.464568\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	66.6349	2.70909
$606$	0	0
$607$	− 3.40600i	− 0.138245i	−0.997608	−	0.0691226i	$-0.977980\pi$
$607$	− 3.40600i	− 0.138245i	0.997608	−	0.0691226i	$-0.0220200\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 13.4205i	− 0.542937i
$612$	0	0
$613$	23.0992i	0.932968i	0.884530	+	0.466484i	$0.154479\pi$
$613$	23.0992i	0.932968i	−0.884530	+	0.466484i	$0.845521\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25.4457i	1.02440i	0.858865	+	0.512202i	$0.171170\pi$
$617$	25.4457i	1.02440i	−0.858865	+	0.512202i	$0.828830\pi$
$618$	0	0
$619$	18.7167	0.752287	0.376144	−	0.926561i	$-0.377250\pi$
$619$	18.7167	0.752287	0.376144	+	0.926561i	$0.377250\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.8645	0.595533
$624$	0	0
$625$	106.480	4.25920
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.97095	−0.158332
$630$	0	0
$631$	31.5549i	1.25618i	0.778140	+	0.628091i	$0.216163\pi$
$631$	31.5549i	1.25618i	−0.778140	+	0.628091i	$0.783837\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 58.9746i	− 2.34034i
$636$	0	0
$637$	5.21068i	0.206455i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9.53508i	0.376613i	0.982110	+	0.188306i	$0.0602998\pi$
$641$	9.53508i	0.376613i	−0.982110	+	0.188306i	$0.939700\pi$
$642$	0	0
$643$	37.8243	1.49164	0.745822	−	0.666145i	$-0.232057\pi$
$643$	37.8243	1.49164	0.745822	+	0.666145i	$0.232057\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19.4663	−0.765301	−0.382650	−	0.923893i	$-0.624989\pi$
$647$	−19.4663	−0.765301	−0.382650	+	0.923893i	$0.624989\pi$
$648$	0	0
$649$	17.9321	0.703896
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.58928	−0.375257	−0.187629	−	0.982240i	$-0.560080\pi$
$653$	−9.58928	−0.375257	−0.187629	+	0.982240i	$0.560080\pi$
$654$	0	0
$655$	7.78402i	0.304147i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 13.0615i	− 0.508803i	−0.967099	−	0.254402i	$-0.918122\pi$
$659$	− 13.0615i	− 0.508803i	0.967099	−	0.254402i	$-0.0818785\pi$
$660$	0	0
$661$	39.8807i	1.55118i	0.631236	+	0.775591i	$0.282548\pi$
$661$	39.8807i	1.55118i	−0.631236	+	0.775591i	$0.717452\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 28.5247i	− 1.10614i
$666$	0	0
$667$	5.88239	0.227767
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−23.2144	−0.896180
$672$	0	0
$673$	−12.8992	−0.497226	−0.248613	−	0.968603i	$-0.579975\pi$
$673$	−12.8992	−0.497226	−0.248613	+	0.968603i	$0.579975\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.1713	1.15958	0.579789	−	0.814767i	$-0.303135\pi$
$677$	30.1713	1.15958	0.579789	+	0.814767i	$0.303135\pi$
$678$	0	0
$679$	4.42136i	0.169676i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 6.61719i	− 0.253200i	−0.991954	−	0.126600i	$-0.959594\pi$
$683$	− 6.61719i	− 0.253200i	0.991954	−	0.126600i	$-0.0404064\pi$
$684$	0	0
$685$	63.9266i	2.44251i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 53.3357i	− 2.03193i
$690$	0	0
$691$	−40.6701	−1.54716	−0.773581	−	0.633697i	$-0.781537\pi$
$691$	−40.6701	−1.54716	−0.773581	+	0.633697i	$0.781537\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	65.8212	2.49674
$696$	0	0
$697$	14.7427	0.558419
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.4320	0.696167	0.348083	−	0.937464i	$-0.386833\pi$
$701$	18.4320	0.696167	0.348083	+	0.937464i	$0.386833\pi$
$702$	0	0
$703$	− 12.9131i	− 0.487027i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2.00000i	− 0.0752177i
$708$	0	0
$709$	− 21.1221i	− 0.793258i	−0.917979	−	0.396629i	$-0.870180\pi$
$709$	− 21.1221i	− 0.793258i	0.917979	−	0.396629i	$-0.129820\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	19.5346i	0.731575i
$714$	0	0
$715$	116.962	4.37411
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24.1319	−0.899969	−0.449985	−	0.893036i	$-0.648571\pi$
$719$	−24.1319	−0.899969	−0.449985	+	0.893036i	$0.648571\pi$
$720$	0	0
$721$	1.67478	0.0623721
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−18.0503	−0.670372
$726$	0	0
$727$	− 18.3923i	− 0.682133i	−0.940039	−	0.341066i	$-0.889212\pi$
$727$	− 18.3923i	− 0.682133i	0.940039	−	0.341066i	$-0.110788\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7.43505i	0.274995i
$732$	0	0
$733$	34.5602i	1.27651i	0.769824	+	0.638256i	$0.220344\pi$
$733$	34.5602i	1.27651i	−0.769824	+	0.638256i	$0.779656\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26.6678i	0.982322i
$738$	0	0
$739$	−45.0366	−1.65670	−0.828350	−	0.560212i	$-0.810720\pi$
$739$	−45.0366	−1.65670	−0.828350	+	0.560212i	$0.810720\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7.83850	−0.287567	−0.143783	−	0.989609i	$-0.545927\pi$
$743$	−7.83850	−0.287567	−0.143783	+	0.989609i	$0.545927\pi$
$744$	0	0
$745$	−23.4060	−0.857529
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−10.2649	−0.375071
$750$	0	0
$751$	− 38.4328i	− 1.40243i	−0.712948	−	0.701217i	$-0.752641\pi$
$751$	− 38.4328i	− 1.40243i	0.712948	−	0.701217i	$-0.247359\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 77.0708i	− 2.80489i
$756$	0	0
$757$	2.24423i	0.0815680i	0.999168	+	0.0407840i	$0.0129855\pi$
$757$	2.24423i	0.0815680i	−0.999168	+	0.0407840i	$0.987014\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	28.3946i	1.02930i	0.857400	+	0.514651i	$0.172079\pi$
$761$	28.3946i	1.02930i	−0.857400	+	0.514651i	$0.827921\pi$
$762$	0	0
$763$	−1.29311	−0.0468137
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.2571	0.659226
$768$	0	0
$769$	27.2968	0.984348	0.492174	−	0.870497i	$-0.336202\pi$
$769$	27.2968	0.984348	0.492174	+	0.870497i	$0.336202\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−31.6583	−1.13867	−0.569335	−	0.822105i	$-0.692799\pi$
$773$	−31.6583	−1.13867	−0.569335	+	0.822105i	$0.692799\pi$
$774$	0	0
$775$	− 59.9425i	− 2.15320i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	47.9416i	1.71769i
$780$	0	0
$781$	6.96400i	0.249191i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	60.6388i	2.16429i
$786$	0	0
$787$	24.1375	0.860408	0.430204	−	0.902732i	$-0.358442\pi$
$787$	24.1375	0.860408	0.430204	+	0.902732i	$0.358442\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.07937	0.287269
$792$	0	0
$793$	−23.6351	−0.839307
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.1867	0.821313	0.410657	−	0.911790i	$-0.365299\pi$
$797$	23.1867	0.821313	0.410657	+	0.911790i	$0.365299\pi$
$798$	0	0
$799$	− 5.15117i	− 0.182235i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 79.0540i	− 2.78976i
$804$	0	0
$805$	20.3473i	0.717149i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 24.2823i	− 0.853719i	−0.904318	−	0.426860i	$-0.859620\pi$
$809$	− 24.2823i	− 0.853719i	0.904318	−	0.426860i	$-0.140380\pi$
$810$	0	0
$811$	−29.1610	−1.02398	−0.511990	−	0.858991i	$-0.671092\pi$
$811$	−29.1610	−1.02398	−0.511990	+	0.858991i	$0.671092\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	38.0464	1.33271
$816$	0	0
$817$	−24.1780	−0.845881
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.6701	−1.48920	−0.744598	−	0.667513i	$-0.767359\pi$
$821$	−42.6701	−1.48920	−0.744598	+	0.667513i	$0.767359\pi$
$822$	0	0
$823$	− 18.4487i	− 0.643083i	−0.946896	−	0.321541i	$-0.895799\pi$
$823$	− 18.4487i	− 0.643083i	0.946896	−	0.321541i	$-0.104201\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.8393i	1.31580i	0.753104	+	0.657901i	$0.228556\pi$
$827$	37.8393i	1.31580i	−0.753104	+	0.657901i	$0.771444\pi$
$828$	0	0
$829$	− 46.5463i	− 1.61662i	−0.588756	−	0.808310i	$-0.700382\pi$
$829$	− 46.5463i	− 1.61662i	0.588756	−	0.808310i	$-0.299618\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.00000i	0.0692959i
$834$	0	0
$835$	69.6715	2.41108
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12.9388	0.446698	0.223349	−	0.974739i	$-0.428301\pi$
$839$	12.9388	0.446698	0.223349	+	0.974739i	$0.428301\pi$
$840$	0	0
$841$	−27.3923	−0.944562
$842$	0	0
$843$	0	0
$844$	0	0
$845$	62.0651	2.13511
$846$	0	0
$847$	− 15.1931i	− 0.522041i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.21119i	0.315756i
$852$	0	0
$853$	− 49.9509i	− 1.71029i	−0.518391	−	0.855144i	$-0.673469\pi$
$853$	− 49.9509i	− 1.71029i	0.518391	−	0.855144i	$-0.326531\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32.4208i	1.10747i	0.832691	+	0.553737i	$0.186799\pi$
$857$	32.4208i	1.10747i	−0.832691	+	0.553737i	$0.813201\pi$
$858$	0	0
$859$	18.3311	0.625450	0.312725	−	0.949844i	$-0.398758\pi$
$859$	18.3311	0.625450	0.312725	+	0.949844i	$0.398758\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.77704	−0.128572	−0.0642859	−	0.997932i	$-0.520477\pi$
$863$	−3.77704	−0.128572	−0.0642859	+	0.997932i	$0.520477\pi$
$864$	0	0
$865$	18.5496	0.630705
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−20.3817	−0.691401
$870$	0	0
$871$	27.1512i	0.919982i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 40.5072i	− 1.36939i
$876$	0	0
$877$	9.61218i	0.324580i	0.986743	+	0.162290i	$0.0518880\pi$
$877$	9.61218i	0.324580i	−0.986743	+	0.162290i	$0.948112\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.4722i	0.757107i	0.925579	+	0.378554i	$0.123578\pi$
$881$	22.4722i	0.757107i	−0.925579	+	0.378554i	$0.876422\pi$
$882$	0	0
$883$	9.77704	0.329023	0.164512	−	0.986375i	$-0.447395\pi$
$883$	9.77704	0.329023	0.164512	+	0.986375i	$0.447395\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−59.1043	−1.98453	−0.992265	−	0.124141i	$-0.960383\pi$
$887$	−59.1043	−1.98453	−0.992265	+	0.124141i	$0.960383\pi$
$888$	0	0
$889$	−13.4465	−0.450982
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16.7510	0.560552
$894$	0	0
$895$	28.2906i	0.945652i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.33893i	0.178063i
$900$	0	0
$901$	− 20.4717i	− 0.682010i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	46.0816i	1.53181i
$906$	0	0
$907$	31.9861	1.06208	0.531040	−	0.847346i	$-0.321801\pi$
$907$	31.9861	1.06208	0.531040	+	0.847346i	$0.321801\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	26.4052	0.874843	0.437421	−	0.899257i	$-0.355892\pi$
$911$	26.4052	0.874843	0.437421	+	0.899257i	$0.355892\pi$
$912$	0	0
$913$	−59.8249	−1.97992
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.77480	0.0586089
$918$	0	0
$919$	46.4656i	1.53276i	0.642389	+	0.766379i	$0.277943\pi$
$919$	46.4656i	1.53276i	−0.642389	+	0.766379i	$0.722057\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.09021i	0.233377i
$924$	0	0
$925$	− 28.2649i	− 0.929344i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	16.7656i	0.550062i	0.961435	+	0.275031i	$0.0886881\pi$
$929$	16.7656i	0.550062i	−0.961435	+	0.275031i	$0.911312\pi$
$930$	0	0
$931$	−6.50379	−0.213153
$932$	0	0
$933$	0	0
$934$	0	0
$935$	44.8930	1.46816
$936$	0	0
$937$	−32.3923	−1.05821	−0.529105	−	0.848556i	$-0.677472\pi$
$937$	−32.3923	−1.05821	−0.529105	+	0.848556i	$0.677472\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	40.1216	1.30793	0.653964	−	0.756526i	$-0.273105\pi$
$941$	40.1216	1.30793	0.653964	+	0.756526i	$0.273105\pi$
$942$	0	0
$943$	− 34.1978i	− 1.11363i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 38.9905i	− 1.26702i	−0.773734	−	0.633511i	$-0.781613\pi$
$947$	− 38.9905i	− 1.26702i	0.773734	−	0.633511i	$-0.218387\pi$
$948$	0	0
$949$	− 80.4868i	− 2.61271i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1.39764i	− 0.0452739i	−0.999744	−	0.0226370i	$-0.992794\pi$
$953$	− 1.39764i	− 0.0452739i	0.999744	−	0.0226370i	$-0.00720619\pi$
$954$	0	0
$955$	−36.2267	−1.17227
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.5756	0.470670
$960$	0	0
$961$	13.2702	0.428071
$962$	0	0
$963$	0	0
$964$	0	0
$965$	35.4351	1.14069
$966$	0	0
$967$	21.9709i	0.706538i	0.935522	+	0.353269i	$0.114930\pi$
$967$	21.9709i	0.706538i	−0.935522	+	0.353269i	$0.885070\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 42.7098i	− 1.37062i	−0.728251	−	0.685311i	$-0.759666\pi$
$971$	− 42.7098i	− 1.37062i	0.728251	−	0.685311i	$-0.240334\pi$
$972$	0	0
$973$	− 15.0076i	− 0.481121i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 60.3885i	− 1.93200i	−0.258546	−	0.965999i	$-0.583243\pi$
$977$	− 60.3885i	− 1.93200i	0.258546	−	0.965999i	$-0.416757\pi$
$978$	0	0
$979$	−76.0753	−2.43138
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−19.6251	−0.625943	−0.312971	−	0.949763i	$-0.601324\pi$
$983$	−19.6251	−0.625943	−0.312971	+	0.949763i	$0.601324\pi$
$984$	0	0
$985$	30.8886	0.984191
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17.2467	0.548413
$990$	0	0
$991$	7.84660i	0.249256i	0.992204	+	0.124628i	$0.0397737\pi$
$991$	7.84660i	0.249256i	−0.992204	+	0.124628i	$0.960226\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	88.3032i	2.79940i
$996$	0	0
$997$	− 19.9128i	− 0.630646i	−0.948984	−	0.315323i	$-0.897887\pi$
$997$	− 19.9128i	− 0.630646i	0.948984	−	0.315323i	$-0.102113\pi$
$998$	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.5615.2		8
3.2	odd	2		6048.2.j.b.5615.8		8
4.3	odd	2		1512.2.j.a.323.7	yes	8
8.3	odd	2		6048.2.j.b.5615.7		8
8.5	even	2		1512.2.j.b.323.4	yes	8
12.11	even	2		1512.2.j.b.323.2	yes	8
24.5	odd	2		1512.2.j.a.323.5	✓	8
24.11	even	2	inner	6048.2.j.a.5615.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.a.323.5	✓	8	24.5	odd	2
1512.2.j.a.323.7	yes	8	4.3	odd	2
1512.2.j.b.323.2	yes	8	12.11	even	2
1512.2.j.b.323.4	yes	8	8.5	even	2
6048.2.j.a.5615.1		8	24.11	even	2	inner
6048.2.j.a.5615.2		8	1.1	even	1	trivial
6048.2.j.b.5615.7		8	8.3	odd	2
6048.2.j.b.5615.8		8	3.2	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)

\(f(q)\)	\(=\)	\(q-4.38587 q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q-4.38587 q^{5} +1.00000i q^{7} -5.11792i q^{11} -5.21068i q^{13} -2.00000i q^{17} +6.50379 q^{19} -4.63929 q^{23} +14.2358 q^{25} -1.26795 q^{29} -4.21068i q^{31} -4.38587i q^{35} -1.98547i q^{37} +7.37134i q^{41} -3.71753 q^{43} +2.57558 q^{47} -1.00000 q^{49} +10.2358 q^{53} +22.4465i q^{55} +3.50379i q^{59} -4.53590i q^{61} +22.8533i q^{65} -5.21068 q^{67} -1.36071 q^{71} +15.4465 q^{73} +5.11792 q^{77} -3.98241i q^{79} -11.6893i q^{83} +8.77174i q^{85} -14.8645i q^{89} +5.21068 q^{91} -28.5247 q^{95} +4.42136 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 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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