LMFDB - Embedded newform 6048.2.j.a.5615.7

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			5615.7
Root			$0.500000 + 2.19293i$ of defining polynomial
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.a.5615.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.38587 q^{5} -1.00000i q^{7} +O(q^{10})$	$q+2.38587 q^{5} -1.00000i q^{7} -1.65382i q^{11} +0.253423i q^{13} +2.00000i q^{17} -7.03969 q^{19} -2.82481 q^{23} +0.692366 q^{25} -1.26795 q^{29} -0.746577i q^{31} -2.38587i q^{35} +6.94273i q^{37} -5.55686i q^{41} -8.67478 q^{43} -10.9679 q^{47} -1.00000 q^{49} -3.30763 q^{53} -3.94579i q^{55} +10.0397i q^{59} +4.53590i q^{61} +0.604635i q^{65} -0.253423 q^{67} -3.17519 q^{71} -3.05421 q^{73} -1.65382 q^{77} -14.5183i q^{79} +1.77480i q^{83} +4.77174i q^{85} +3.13550i q^{89} +0.253423 q^{91} -16.7958 q^{95} -5.49315 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.38587	1.06699	0.533496	−	0.845802i	$-0.320878\pi$
$5$	2.38587	1.06699	0.533496	+	0.845802i	$0.320878\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.65382i	− 0.498645i	−0.968421	−	0.249322i	$-0.919792\pi$
$11$	− 1.65382i	− 0.498645i	0.968421	−	0.249322i	$-0.0802079\pi$
$12$	0	0
$13$	0.253423i	0.0702870i	0.999382	+	0.0351435i	$0.0111888\pi$
$13$	0.253423i	0.0702870i	−0.999382	+	0.0351435i	$0.988811\pi$
$14$	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
$19$	−7.03969	−1.61501	−0.807507	−	0.589858i	$-0.799184\pi$
$19$	−7.03969	−1.61501	−0.807507	+	0.589858i	$0.799184\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.82481	−0.589014	−0.294507	−	0.955649i	$-0.595155\pi$
$23$	−2.82481	−0.589014	−0.294507	+	0.955649i	$0.595155\pi$
$24$	0	0
$25$	0.692366	0.138473
$26$	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
$31$	− 0.746577i	− 0.134089i	−0.997750	−	0.0670446i	$-0.978643\pi$
$31$	− 0.746577i	− 0.134089i	0.997750	−	0.0670446i	$-0.0213570\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.38587i	− 0.403285i
$36$	0	0
$37$	6.94273i	1.14138i	0.821166	+	0.570689i	$0.193324\pi$
$37$	6.94273i	1.14138i	−0.821166	+	0.570689i	$0.806676\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.55686i	− 0.867836i	−0.900952	−	0.433918i	$-0.857131\pi$
$41$	− 5.55686i	− 0.867836i	0.900952	−	0.433918i	$-0.142869\pi$
$42$	0	0
$43$	−8.67478	−1.32289	−0.661446	−	0.749993i	$-0.730057\pi$
$43$	−8.67478	−1.32289	−0.661446	+	0.749993i	$0.730057\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.9679	−1.59983	−0.799915	−	0.600113i	$-0.795122\pi$
$47$	−10.9679	−1.59983	−0.799915	+	0.600113i	$0.795122\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.30763	−0.454339	−0.227169	−	0.973855i	$-0.572947\pi$
$53$	−3.30763	−0.454339	−0.227169	+	0.973855i	$0.572947\pi$
$54$	0	0
$55$	− 3.94579i	− 0.532050i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.0397i	1.30706i	0.756902	+	0.653528i	$0.226712\pi$
$59$	10.0397i	1.30706i	−0.756902	+	0.653528i	$0.773288\pi$
$60$	0	0
$61$	4.53590i	0.580762i	0.956911	+	0.290381i	$0.0937821\pi$
$61$	4.53590i	0.580762i	−0.956911	+	0.290381i	$0.906218\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.604635i	0.0749957i
$66$	0	0
$67$	−0.253423	−0.0309606	−0.0154803	−	0.999880i	$-0.504928\pi$
$67$	−0.253423	−0.0309606	−0.0154803	+	0.999880i	$0.504928\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.17519	−0.376826	−0.188413	−	0.982090i	$-0.560334\pi$
$71$	−3.17519	−0.376826	−0.188413	+	0.982090i	$0.560334\pi$
$72$	0	0
$73$	−3.05421	−0.357468	−0.178734	−	0.983897i	$-0.557200\pi$
$73$	−3.05421	−0.357468	−0.178734	+	0.983897i	$0.557200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.65382	−0.188470
$78$	0	0
$79$	− 14.5183i	− 1.63344i	−0.577036	−	0.816719i	$-0.695791\pi$
$79$	− 14.5183i	− 1.63344i	0.577036	−	0.816719i	$-0.304209\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.77480i	0.194809i	0.995245	+	0.0974046i	$0.0310541\pi$
$83$	1.77480i	0.194809i	−0.995245	+	0.0974046i	$0.968946\pi$
$84$	0	0
$85$	4.77174i	0.517567i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.13550i	0.332363i	0.986095	+	0.166181i	$0.0531437\pi$
$89$	3.13550i	0.332363i	−0.986095	+	0.166181i	$0.946856\pi$
$90$	0	0
$91$	0.253423	0.0265660
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.7958	−1.72321
$96$	0	0
$97$	−5.49315	−0.557745	−0.278873	−	0.960328i	$-0.589961\pi$
$97$	−5.49315	−0.557745	−0.278873	+	0.960328i	$0.589961\pi$
$98$	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
$103$	− 3.28247i	− 0.323432i	−0.986837	−	0.161716i	$-0.948297\pi$
$103$	− 3.28247i	− 0.323432i	0.986837	−	0.161716i	$-0.0517028\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.1931i	1.27542i	0.770275	+	0.637712i	$0.220119\pi$
$107$	13.1931i	1.27542i	−0.770275	+	0.637712i	$0.779881\pi$
$108$	0	0
$109$	7.29311i	0.698553i	0.937020	+	0.349277i	$0.113573\pi$
$109$	7.29311i	0.698553i	−0.937020	+	0.349277i	$0.886427\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 19.0076i	− 1.78808i	−0.447985	−	0.894041i	$-0.647858\pi$
$113$	− 19.0076i	− 1.78808i	0.447985	−	0.894041i	$-0.352142\pi$
$114$	0	0
$115$	−6.73962	−0.628473
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	8.26489	0.751354
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.2774	−0.919243
$126$	0	0
$127$	5.05421i	0.448489i	0.974533	+	0.224244i	$0.0719914\pi$
$127$	5.05421i	0.448489i	−0.974533	+	0.224244i	$0.928009\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.6893i	1.02130i	0.859789	+	0.510650i	$0.170595\pi$
$131$	11.6893i	1.02130i	−0.859789	+	0.510650i	$0.829405\pi$
$132$	0	0
$133$	7.03969i	0.610418i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.03211i	0.0881793i	0.999028	+	0.0440896i	$0.0140387\pi$
$137$	1.03211i	0.0881793i	−0.999028	+	0.0440896i	$0.985961\pi$
$138$	0	0
$139$	12.0794	1.02456	0.512279	−	0.858819i	$-0.328801\pi$
$139$	12.0794	1.02456	0.512279	+	0.858819i	$0.328801\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.419116	0.0350482
$144$	0	0
$145$	−3.02516	−0.251226
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.1213	−1.48455	−0.742277	−	0.670093i	$-0.766254\pi$
$149$	−18.1213	−1.48455	−0.742277	+	0.670093i	$0.766254\pi$
$150$	0	0
$151$	19.4289i	1.58110i	0.612395	+	0.790552i	$0.290206\pi$
$151$	19.4289i	1.58110i	−0.612395	+	0.790552i	$0.709794\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.78123i	− 0.143072i
$156$	0	0
$157$	− 18.2183i	− 1.45397i	−0.686651	−	0.726987i	$-0.740920\pi$
$157$	− 18.2183i	− 1.45397i	0.686651	−	0.726987i	$-0.259080\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.82481i	0.222626i
$162$	0	0
$163$	−3.71753	−0.291179	−0.145590	−	0.989345i	$-0.546508\pi$
$163$	−3.71753	−0.291179	−0.145590	+	0.989345i	$0.546508\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.97095	−0.462046	−0.231023	−	0.972948i	$-0.574207\pi$
$167$	−5.97095	−0.462046	−0.231023	+	0.972948i	$0.574207\pi$
$168$	0	0
$169$	12.9358	0.995060
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.0858	1.22298	0.611491	−	0.791252i	$-0.290570\pi$
$173$	16.0858	1.22298	0.611491	+	0.791252i	$0.290570\pi$
$174$	0	0
$175$	− 0.692366i	− 0.0523379i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 13.3786i	− 0.999964i	−0.866036	−	0.499982i	$-0.833340\pi$
$179$	− 13.3786i	− 0.999964i	0.866036	−	0.499982i	$-0.166660\pi$
$180$	0	0
$181$	20.4214i	1.51791i	0.651145	+	0.758954i	$0.274289\pi$
$181$	20.4214i	1.51791i	−0.651145	+	0.758954i	$0.725711\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.5644i	1.21784i
$186$	0	0
$187$	3.30763	0.241878
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.9889	1.44634	0.723171	−	0.690669i	$-0.242684\pi$
$191$	19.9889	1.44634	0.723171	+	0.690669i	$0.242684\pi$
$192$	0	0
$193$	19.0076	1.36820	0.684098	−	0.729391i	$-0.260196\pi$
$193$	19.0076	1.36820	0.684098	+	0.729391i	$0.260196\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.9573	−1.20815	−0.604077	−	0.796926i	$-0.706458\pi$
$197$	−16.9573	−1.20815	−0.604077	+	0.796926i	$0.706458\pi$
$198$	0	0
$199$	− 25.4541i	− 1.80439i	−0.431326	−	0.902196i	$-0.641954\pi$
$199$	− 25.4541i	− 1.80439i	0.431326	−	0.902196i	$-0.358046\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.26795i	0.0889926i
$204$	0	0
$205$	− 13.2579i	− 0.925974i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	11.6424i	0.805318i
$210$	0	0
$211$	19.5435	1.34543	0.672714	−	0.739903i	$-0.265128\pi$
$211$	19.5435	1.34543	0.672714	+	0.739903i	$0.265128\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−20.6969	−1.41152
$216$	0	0
$217$	−0.746577	−0.0506809
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.506847	−0.0340942
$222$	0	0
$223$	− 3.09696i	− 0.207388i	−0.994609	−	0.103694i	$-0.966934\pi$
$223$	− 3.09696i	− 0.207388i	0.994609	−	0.103694i	$-0.0330662\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 23.5854i	− 1.56542i	−0.622388	−	0.782709i	$-0.713837\pi$
$227$	− 23.5854i	− 1.56542i	0.622388	−	0.782709i	$-0.286163\pi$
$228$	0	0
$229$	6.53590i	0.431904i	0.976404	+	0.215952i	$0.0692855\pi$
$229$	6.53590i	0.431904i	−0.976404	+	0.215952i	$0.930714\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.5359i	0.690230i	0.938560	+	0.345115i	$0.112160\pi$
$233$	10.5359i	0.690230i	−0.938560	+	0.345115i	$0.887840\pi$
$234$	0	0
$235$	−26.1679	−1.70701
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−22.5298	−1.45733	−0.728665	−	0.684870i	$-0.759859\pi$
$239$	−22.5298	−1.45733	−0.728665	+	0.684870i	$0.759859\pi$
$240$	0	0
$241$	−9.53201	−0.614010	−0.307005	−	0.951708i	$-0.599327\pi$
$241$	−9.53201	−0.614010	−0.307005	+	0.951708i	$0.599327\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.38587	−0.152428
$246$	0	0
$247$	− 1.78402i	− 0.113515i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	27.1931i	1.71641i	0.513305	+	0.858206i	$0.328421\pi$
$251$	27.1931i	1.71641i	−0.513305	+	0.858206i	$0.671579\pi$
$252$	0	0
$253$	4.67172i	0.293708i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.2568i	1.20121i	0.799547	+	0.600603i	$0.205073\pi$
$257$	19.2568i	1.20121i	−0.799547	+	0.600603i	$0.794927\pi$
$258$	0	0
$259$	6.94273	0.431400
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.1828	0.751221	0.375611	−	0.926778i	$-0.377433\pi$
$263$	12.1828	0.751221	0.375611	+	0.926778i	$0.377433\pi$
$264$	0	0
$265$	−7.89158	−0.484776
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−26.1149	−1.59225	−0.796126	−	0.605132i	$-0.793121\pi$
$269$	−26.1149	−1.59225	−0.796126	+	0.605132i	$0.793121\pi$
$270$	0	0
$271$	− 31.3999i	− 1.90741i	−0.300749	−	0.953703i	$-0.597237\pi$
$271$	− 31.3999i	− 1.90741i	0.300749	−	0.953703i	$-0.402763\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.14505i	− 0.0690489i
$276$	0	0
$277$	− 12.8846i	− 0.774162i	−0.922046	−	0.387081i	$-0.873483\pi$
$277$	− 12.8846i	− 0.774162i	0.922046	−	0.387081i	$-0.126517\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 8.35262i	− 0.498276i	−0.968468	−	0.249138i	$-0.919853\pi$
$281$	− 8.35262i	− 0.498276i	0.968468	−	0.249138i	$-0.0801472\pi$
$282$	0	0
$283$	4.45041	0.264549	0.132275	−	0.991213i	$-0.457772\pi$
$283$	4.45041	0.264549	0.132275	+	0.991213i	$0.457772\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.55686	−0.328011
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−23.0495	−1.34657	−0.673283	−	0.739385i	$-0.735116\pi$
$293$	−23.0495	−1.34657	−0.673283	+	0.739385i	$0.735116\pi$
$294$	0	0
$295$	23.9534i	1.39462i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 0.715873i	− 0.0414000i
$300$	0	0
$301$	8.67478i	0.500006i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.8221i	0.619669i
$306$	0	0
$307$	13.4610	0.768262	0.384131	−	0.923279i	$-0.374501\pi$
$307$	13.4610	0.768262	0.384131	+	0.923279i	$0.374501\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−33.5038	−1.89983	−0.949913	−	0.312515i	$-0.898828\pi$
$311$	−33.5038	−1.89983	−0.949913	+	0.312515i	$0.898828\pi$
$312$	0	0
$313$	4.94579	0.279553	0.139776	−	0.990183i	$-0.455362\pi$
$313$	4.94579	0.279553	0.139776	+	0.990183i	$0.455362\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.4029	−1.37061	−0.685303	−	0.728258i	$-0.740330\pi$
$317$	−24.4029	−1.37061	−0.685303	+	0.728258i	$0.740330\pi$
$318$	0	0
$319$	2.09696i	0.117407i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 14.0794i	− 0.783397i
$324$	0	0
$325$	0.175462i	0.00973287i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10.9679i	0.604679i
$330$	0	0
$331$	17.9648	0.987436	0.493718	−	0.869622i	$-0.335637\pi$
$331$	17.9648	0.987436	0.493718	+	0.869622i	$0.335637\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.604635	−0.0330347
$336$	0	0
$337$	−7.02905	−0.382897	−0.191448	−	0.981503i	$-0.561318\pi$
$337$	−7.02905	−0.382897	−0.191448	+	0.981503i	$0.561318\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.23470	−0.0668628
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 0.473599i	− 0.0254241i	−0.999919	−	0.0127121i	$-0.995954\pi$
$347$	− 0.473599i	− 0.0254241i	0.999919	−	0.0127121i	$-0.00404648\pi$
$348$	0	0
$349$	10.5007i	0.562091i	0.959694	+	0.281045i	$0.0906812\pi$
$349$	10.5007i	0.562091i	−0.959694	+	0.281045i	$0.909319\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 4.48506i	− 0.238716i	−0.992851	−	0.119358i	$-0.961916\pi$
$353$	− 4.48506i	− 0.238716i	0.992851	−	0.119358i	$-0.0380836\pi$
$354$	0	0
$355$	−7.57558	−0.402070
$356$	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
$361$	30.5572	1.60827
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.28694	−0.381416
$366$	0	0
$367$	− 18.4968i	− 0.965527i	−0.875751	−	0.482763i	$-0.839633\pi$
$367$	− 18.4968i	− 0.965527i	0.875751	−	0.482763i	$-0.160367\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.30763i	0.171724i
$372$	0	0
$373$	− 26.1358i	− 1.35326i	−0.736322	−	0.676631i	$-0.763439\pi$
$373$	− 26.1358i	− 1.35326i	0.736322	−	0.676631i	$-0.236561\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 0.321328i	− 0.0165492i
$378$	0	0
$379$	−15.1816	−0.779828	−0.389914	−	0.920851i	$-0.627495\pi$
$379$	−15.1816	−0.779828	−0.389914	+	0.920851i	$0.627495\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.47780	−0.126609	−0.0633047	−	0.997994i	$-0.520164\pi$
$383$	−2.47780	−0.126609	−0.0633047	+	0.997994i	$0.520164\pi$
$384$	0	0
$385$	−3.94579	−0.201096
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−11.4932	−0.582726	−0.291363	−	0.956613i	$-0.594109\pi$
$389$	−11.4932	−0.582726	−0.291363	+	0.956613i	$0.594109\pi$
$390$	0	0
$391$	− 5.64962i	− 0.285714i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 34.6388i	− 1.74287i
$396$	0	0
$397$	− 5.57252i	− 0.279677i	−0.990174	−	0.139838i	$-0.955342\pi$
$397$	− 5.57252i	− 0.279677i	0.990174	−	0.139838i	$-0.0446583\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 24.0045i	− 1.19873i	−0.800477	−	0.599364i	$-0.795420\pi$
$401$	− 24.0045i	− 1.19873i	0.800477	−	0.599364i	$-0.204580\pi$
$402$	0	0
$403$	0.189200	0.00942472
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.4820	0.569142
$408$	0	0
$409$	6.24503	0.308797	0.154398	−	0.988009i	$-0.450656\pi$
$409$	6.24503	0.308797	0.154398	+	0.988009i	$0.450656\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.0397	0.494021
$414$	0	0
$415$	4.23443i	0.207860i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 2.80079i	− 0.136827i	−0.997657	−	0.0684137i	$-0.978206\pi$
$419$	− 2.80079i	− 0.136827i	0.997657	−	0.0684137i	$-0.0217938\pi$
$420$	0	0
$421$	− 10.8634i	− 0.529448i	−0.964324	−	0.264724i	$-0.914719\pi$
$421$	− 10.8634i	− 0.529448i	0.964324	−	0.264724i	$-0.0852808\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.38473i	0.0671693i
$426$	0	0
$427$	4.53590	0.219508
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.87513	−0.427500	−0.213750	−	0.976888i	$-0.568568\pi$
$431$	−8.87513	−0.427500	−0.213750	+	0.976888i	$0.568568\pi$
$432$	0	0
$433$	10.0115	0.481120	0.240560	−	0.970634i	$-0.422669\pi$
$433$	10.0115	0.481120	0.240560	+	0.970634i	$0.422669\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.8858	0.951266
$438$	0	0
$439$	31.3999i	1.49863i	0.662211	+	0.749317i	$0.269618\pi$
$439$	31.3999i	1.49863i	−0.662211	+	0.749317i	$0.730382\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	35.7751i	1.69973i	0.527003	+	0.849863i	$0.323316\pi$
$443$	35.7751i	1.69973i	−0.527003	+	0.849863i	$0.676684\pi$
$444$	0	0
$445$	7.48090i	0.354629i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.3443i	0.960105i	0.877240	+	0.480052i	$0.159382\pi$
$449$	20.3443i	0.960105i	−0.877240	+	0.480052i	$0.840618\pi$
$450$	0	0
$451$	−9.19003	−0.432742
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.604635	0.0283457
$456$	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
$461$	30.5423	1.42250	0.711249	−	0.702940i	$-0.248130\pi$
$461$	30.5423	1.42250	0.711249	+	0.702940i	$0.248130\pi$
$462$	0	0
$463$	1.71612i	0.0797547i	0.999205	+	0.0398774i	$0.0126967\pi$
$463$	1.71612i	0.0797547i	−0.999205	+	0.0398774i	$0.987303\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.60994i	− 0.167048i	−0.996506	−	0.0835239i	$-0.973382\pi$
$467$	− 3.60994i	− 0.167048i	0.996506	−	0.0835239i	$-0.0266175\pi$
$468$	0	0
$469$	0.253423i	0.0117020i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	14.3465i	0.659653i
$474$	0	0
$475$	−4.87404	−0.223636
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.62057	0.165428	0.0827140	−	0.996573i	$-0.473641\pi$
$479$	3.62057	0.165428	0.0827140	+	0.996573i	$0.473641\pi$
$480$	0	0
$481$	−1.75945	−0.0802240
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13.1059	−0.595110
$486$	0	0
$487$	10.3808i	0.470401i	0.971947	+	0.235200i	$0.0755746\pi$
$487$	10.3808i	0.470401i	−0.971947	+	0.235200i	$0.924425\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.66139i	0.120107i	0.998195	+	0.0600534i	$0.0191271\pi$
$491$	2.66139i	0.120107i	−0.998195	+	0.0600534i	$0.980873\pi$
$492$	0	0
$493$	− 2.53590i	− 0.114211i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.17519i	0.142427i
$498$	0	0
$499$	−33.1755	−1.48514	−0.742570	−	0.669769i	$-0.766393\pi$
$499$	−33.1755	−1.48514	−0.742570	+	0.669769i	$0.766393\pi$
$500$	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
$505$	−4.77174	−0.212339
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−38.9221	−1.72519	−0.862595	−	0.505894i	$-0.831163\pi$
$509$	−38.9221	−1.72519	−0.862595	+	0.505894i	$0.831163\pi$
$510$	0	0
$511$	3.05421i	0.135110i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 7.83155i	− 0.345099i
$516$	0	0
$517$	18.1389i	0.797747i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 37.1797i	− 1.62887i	−0.580253	−	0.814436i	$-0.697046\pi$
$521$	− 37.1797i	− 1.62887i	0.580253	−	0.814436i	$-0.302954\pi$
$522$	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
$527$	1.49315	0.0650428
$528$	0	0
$529$	−15.0204	−0.653063
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.40824	0.0609976
$534$	0	0
$535$	31.4770i	1.36087i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.65382i	0.0712349i
$540$	0	0
$541$	31.8358i	1.36873i	0.729141	+	0.684363i	$0.239920\pi$
$541$	31.8358i	1.36873i	−0.729141	+	0.684363i	$0.760080\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	17.4004i	0.745351i
$546$	0	0
$547$	29.9648	1.28120	0.640602	−	0.767873i	$-0.278685\pi$
$547$	29.9648	1.28120	0.640602	+	0.767873i	$0.278685\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.92596	0.380259
$552$	0	0
$553$	−14.5183	−0.617381
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−20.9573	−0.887987	−0.443994	−	0.896030i	$-0.646439\pi$
$557$	−20.9573	−0.887987	−0.443994	+	0.896030i	$0.646439\pi$
$558$	0	0
$559$	− 2.19839i	− 0.0929821i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 5.56801i	− 0.234664i	−0.993093	−	0.117332i	$-0.962566\pi$
$563$	− 5.56801i	− 0.234664i	0.993093	−	0.117332i	$-0.0374341\pi$
$564$	0	0
$565$	− 45.3496i	− 1.90787i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 24.3571i	− 1.02110i	−0.859847	−	0.510552i	$-0.829441\pi$
$569$	− 24.3571i	− 1.02110i	0.859847	−	0.510552i	$-0.170559\pi$
$570$	0	0
$571$	−29.3496	−1.22824	−0.614120	−	0.789212i	$-0.710489\pi$
$571$	−29.3496	−1.22824	−0.614120	+	0.789212i	$0.710489\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.95580	−0.0815626
$576$	0	0
$577$	−32.8992	−1.36961	−0.684805	−	0.728727i	$-0.740113\pi$
$577$	−32.8992	−1.36961	−0.684805	+	0.728727i	$0.740113\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.77480	0.0736309
$582$	0	0
$583$	5.47022i	0.226553i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.9939i	0.742687i	0.928496	+	0.371343i	$0.121103\pi$
$587$	17.9939i	0.742687i	−0.928496	+	0.371343i	$0.878897\pi$
$588$	0	0
$589$	5.25566i	0.216556i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.72702i	0.276246i	0.990415	+	0.138123i	$0.0441069\pi$
$593$	6.72702i	0.276246i	−0.990415	+	0.138123i	$0.955893\pi$
$594$	0	0
$595$	4.77174	0.195622
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10.9752	−0.448433	−0.224216	−	0.974539i	$-0.571982\pi$
$599$	−10.9752	−0.448433	−0.224216	+	0.974539i	$0.571982\pi$
$600$	0	0
$601$	−13.0542	−0.532492	−0.266246	−	0.963905i	$-0.585783\pi$
$601$	−13.0542	−0.532492	−0.266246	+	0.963905i	$0.585783\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	19.7189	0.801689
$606$	0	0
$607$	23.2350i	0.943080i	0.881845	+	0.471540i	$0.156302\pi$
$607$	23.2350i	0.943080i	−0.881845	+	0.471540i	$0.843698\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 2.77952i	− 0.112447i
$612$	0	0
$613$	− 31.6854i	− 1.27976i	−0.768474	−	0.639881i	$-0.778984\pi$
$613$	− 31.6854i	− 1.27976i	0.768474	−	0.639881i	$-0.221016\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 31.7312i	− 1.27745i	−0.769435	−	0.638726i	$-0.779462\pi$
$617$	− 31.7312i	− 1.27745i	0.769435	−	0.638726i	$-0.220538\pi$
$618$	0	0
$619$	48.4602	1.94778	0.973890	−	0.227019i	$-0.0728979\pi$
$619$	48.4602	1.94778	0.973890	+	0.227019i	$0.0728979\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.13550	0.125621
$624$	0	0
$625$	−27.9825	−1.11930
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.8855	−0.553649
$630$	0	0
$631$	23.9472i	0.953325i	0.879086	+	0.476662i	$0.158154\pi$
$631$	23.9472i	0.953325i	−0.879086	+	0.476662i	$0.841846\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.0587i	0.478534i
$636$	0	0
$637$	− 0.253423i	− 0.0100410i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 3.24953i	− 0.128349i	−0.997939	−	0.0641744i	$-0.979559\pi$
$641$	− 3.24953i	− 0.128349i	0.997939	−	0.0641744i	$-0.0204414\pi$
$642$	0	0
$643$	24.2808	0.957542	0.478771	−	0.877940i	$-0.341082\pi$
$643$	24.2808	0.957542	0.478771	+	0.877940i	$0.341082\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−36.6388	−1.44042	−0.720209	−	0.693757i	$-0.755954\pi$
$647$	−36.6388	−1.44042	−0.720209	+	0.693757i	$0.755954\pi$
$648$	0	0
$649$	16.6038	0.651756
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15.8748	−0.621230	−0.310615	−	0.950536i	$-0.600535\pi$
$653$	−15.8748	−0.621230	−0.310615	+	0.950536i	$0.600535\pi$
$654$	0	0
$655$	27.8891i	1.08972i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 23.4538i	− 0.913630i	−0.889562	−	0.456815i	$-0.848990\pi$
$659$	− 23.4538i	− 0.913630i	0.889562	−	0.456815i	$-0.151010\pi$
$660$	0	0
$661$	− 25.0090i	− 0.972737i	−0.873754	−	0.486368i	$-0.838321\pi$
$661$	− 25.0090i	− 0.972737i	0.873754	−	0.486368i	$-0.161679\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.7958i	0.651312i
$666$	0	0
$667$	3.58172	0.138685
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.50155	0.289594
$672$	0	0
$673$	−22.8137	−0.879402	−0.439701	−	0.898144i	$-0.644916\pi$
$673$	−22.8137	−0.879402	−0.439701	+	0.898144i	$0.644916\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.6851	1.14089	0.570446	−	0.821335i	$-0.306770\pi$
$677$	29.6851	1.14089	0.570446	+	0.821335i	$0.306770\pi$
$678$	0	0
$679$	5.49315i	0.210808i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36.8469i	1.40991i	0.709253	+	0.704954i	$0.249032\pi$
$683$	36.8469i	1.40991i	−0.709253	+	0.704954i	$0.750968\pi$
$684$	0	0
$685$	2.46248i	0.0940866i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 0.838232i	− 0.0319341i
$690$	0	0
$691$	−30.7556	−1.17000	−0.584998	−	0.811035i	$-0.698905\pi$
$691$	−30.7556	−1.17000	−0.584998	+	0.811035i	$0.698905\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	28.8198	1.09320
$696$	0	0
$697$	11.1137	0.420962
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.88852	0.184637	0.0923184	−	0.995730i	$-0.470572\pi$
$701$	4.88852	0.184637	0.0923184	+	0.995730i	$0.470572\pi$
$702$	0	0
$703$	− 48.8746i	− 1.84334i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.00000i	0.0752177i
$708$	0	0
$709$	− 27.1221i	− 1.01859i	−0.860591	−	0.509296i	$-0.829906\pi$
$709$	− 27.1221i	− 1.01859i	0.860591	−	0.509296i	$-0.170094\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.10894i	0.0789803i
$714$	0	0
$715$	0.999956	0.0373962
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.95501	0.110203	0.0551017	−	0.998481i	$-0.482452\pi$
$719$	2.95501	0.110203	0.0551017	+	0.998481i	$0.482452\pi$
$720$	0	0
$721$	−3.28247	−0.122246
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.877885	−0.0326038
$726$	0	0
$727$	18.3923i	0.682133i	0.940039	+	0.341066i	$0.110788\pi$
$727$	18.3923i	0.682133i	−0.940039	+	0.341066i	$0.889212\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 17.3496i	− 0.641697i
$732$	0	0
$733$	− 19.6885i	− 0.727210i	−0.931553	−	0.363605i	$-0.881546\pi$
$733$	− 19.6885i	− 0.727210i	0.931553	−	0.363605i	$-0.118454\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.419116i	0.0154383i
$738$	0	0
$739$	−8.03517	−0.295579	−0.147789	−	0.989019i	$-0.547216\pi$
$739$	−8.03517	−0.295579	−0.147789	+	0.989019i	$0.547216\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29.4820	−1.08159	−0.540795	−	0.841154i	$-0.681876\pi$
$743$	−29.4820	−1.08159	−0.540795	+	0.841154i	$0.681876\pi$
$744$	0	0
$745$	−43.2350	−1.58401
$746$	0	0
$747$	0	0
$748$	0	0
$749$	13.1931	0.482065
$750$	0	0
$751$	0.103077i	0.00376132i	0.999998	+	0.00188066i	$0.000598633\pi$
$751$	0.103077i	0.00376132i	−0.999998	+	0.00188066i	$0.999401\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	46.3549i	1.68703i
$756$	0	0
$757$	32.1006i	1.16672i	0.812215	+	0.583359i	$0.198262\pi$
$757$	32.1006i	1.16672i	−0.812215	+	0.583359i	$0.801738\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	39.3228i	1.42545i	0.701444	+	0.712725i	$0.252539\pi$
$761$	39.3228i	1.42545i	−0.701444	+	0.712725i	$0.747461\pi$
$762$	0	0
$763$	7.29311	0.264028
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.54429	−0.0918690
$768$	0	0
$769$	−38.1199	−1.37464	−0.687319	−	0.726356i	$-0.741213\pi$
$769$	−38.1199	−1.37464	−0.687319	+	0.726356i	$0.741213\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	25.6583	0.922866	0.461433	−	0.887175i	$-0.347335\pi$
$773$	25.6583	0.922866	0.461433	+	0.887175i	$0.347335\pi$
$774$	0	0
$775$	− 0.516904i	− 0.0185677i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	39.1186i	1.40157i
$780$	0	0
$781$	5.25118i	0.187902i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 43.4663i	− 1.55138i
$786$	0	0
$787$	−22.7785	−0.811965	−0.405983	−	0.913881i	$-0.633071\pi$
$787$	−22.7785	−0.811965	−0.405983	+	0.913881i	$0.633071\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−19.0076	−0.675832
$792$	0	0
$793$	−1.14950	−0.0408201
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−7.04306	−0.249478	−0.124739	−	0.992190i	$-0.539809\pi$
$797$	−7.04306	−0.249478	−0.124739	+	0.992190i	$0.539809\pi$
$798$	0	0
$799$	− 21.9358i	− 0.776032i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.05111i	0.178250i
$804$	0	0
$805$	6.73962i	0.237541i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 36.1772i	− 1.27192i	−0.771722	−	0.635961i	$-0.780604\pi$
$809$	− 36.1772i	− 1.27192i	0.771722	−	0.635961i	$-0.219396\pi$
$810$	0	0
$811$	7.84047	0.275316	0.137658	−	0.990480i	$-0.456042\pi$
$811$	7.84047	0.275316	0.137658	+	0.990480i	$0.456042\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.86952	−0.310686
$816$	0	0
$817$	61.0677	2.13649
$818$	0	0
$819$	0	0
$820$	0	0
$821$	52.1342	1.81950	0.909748	−	0.415161i	$-0.136275\pi$
$821$	52.1342	1.81950	0.909748	+	0.415161i	$0.136275\pi$
$822$	0	0
$823$	48.1923i	1.67988i	0.542682	+	0.839939i	$0.317409\pi$
$823$	48.1923i	1.67988i	−0.542682	+	0.839939i	$0.682591\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 34.6966i	− 1.20652i	−0.797545	−	0.603259i	$-0.793869\pi$
$827$	− 34.6966i	− 1.20652i	0.797545	−	0.603259i	$-0.206131\pi$
$828$	0	0
$829$	− 12.5848i	− 0.437088i	−0.975827	−	0.218544i	$-0.929869\pi$
$829$	− 12.5848i	− 0.437088i	0.975827	−	0.218544i	$-0.0701308\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2.00000i	− 0.0692959i
$834$	0	0
$835$	−14.2459	−0.493000
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9.30987	0.321413	0.160706	−	0.987002i	$-0.448623\pi$
$839$	9.30987	0.321413	0.160706	+	0.987002i	$0.448623\pi$
$840$	0	0
$841$	−27.3923	−0.944562
$842$	0	0
$843$	0	0
$844$	0	0
$845$	30.8631	1.06172
$846$	0	0
$847$	− 8.26489i	− 0.283985i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 19.6119i	− 0.672287i
$852$	0	0
$853$	− 31.3099i	− 1.07203i	−0.844208	−	0.536015i	$-0.819929\pi$
$853$	− 31.3099i	− 1.07203i	0.844208	−	0.536015i	$-0.180071\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 7.14838i	− 0.244184i	−0.992519	−	0.122092i	$-0.961040\pi$
$857$	− 7.14838i	− 0.244184i	0.992519	−	0.122092i	$-0.0389603\pi$
$858$	0	0
$859$	14.7022	0.501632	0.250816	−	0.968035i	$-0.419301\pi$
$859$	14.7022	0.501632	0.250816	+	0.968035i	$0.419301\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.8640	−1.05062	−0.525311	−	0.850910i	$-0.676051\pi$
$863$	−30.8640	−1.05062	−0.525311	+	0.850910i	$0.676051\pi$
$864$	0	0
$865$	38.3786	1.30491
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−24.0106	−0.814505
$870$	0	0
$871$	− 0.0642234i	− 0.00217613i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.2774i	0.347441i
$876$	0	0
$877$	− 43.9570i	− 1.48432i	−0.670221	−	0.742162i	$-0.733801\pi$
$877$	− 43.9570i	− 1.48432i	0.670221	−	0.742162i	$-0.266199\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 10.7432i	− 0.361948i	−0.983488	−	0.180974i	$-0.942075\pi$
$881$	− 10.7432i	− 0.361948i	0.983488	−	0.180974i	$-0.0579249\pi$
$882$	0	0
$883$	36.8640	1.24057	0.620286	−	0.784376i	$-0.287017\pi$
$883$	36.8640	1.24057	0.620286	+	0.784376i	$0.287017\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.0710	1.07684	0.538420	−	0.842677i	$-0.319022\pi$
$887$	32.0710	1.07684	0.538420	+	0.842677i	$0.319022\pi$
$888$	0	0
$889$	5.05421	0.169513
$890$	0	0
$891$	0	0
$892$	0	0
$893$	77.2105	2.58375
$894$	0	0
$895$	− 31.9196i	− 1.06695i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.946621i	0.0315716i
$900$	0	0
$901$	− 6.61527i	− 0.220387i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	48.7227i	1.61960i
$906$	0	0
$907$	−12.2733	−0.407528	−0.203764	−	0.979020i	$-0.565317\pi$
$907$	−12.2733	−0.407528	−0.203764	+	0.979020i	$0.565317\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39.9487	1.32356	0.661779	−	0.749699i	$-0.269802\pi$
$911$	39.9487	1.32356	0.661779	+	0.749699i	$0.269802\pi$
$912$	0	0
$913$	2.93519	0.0971406
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.6893	0.386015
$918$	0	0
$919$	27.5374i	0.908373i	0.890907	+	0.454187i	$0.150070\pi$
$919$	27.5374i	0.908373i	−0.890907	+	0.454187i	$0.849930\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 0.804668i	− 0.0264860i
$924$	0	0
$925$	4.80691i	0.158050i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	43.6938i	1.43355i	0.697306	+	0.716774i	$0.254382\pi$
$929$	43.6938i	1.43355i	−0.697306	+	0.716774i	$0.745618\pi$
$930$	0	0
$931$	7.03969	0.230716
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.89158	0.258082
$936$	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
$941$	−23.4806	−0.765446	−0.382723	−	0.923863i	$-0.625014\pi$
$941$	−23.4806	−0.765446	−0.382723	+	0.923863i	$0.625014\pi$
$942$	0	0
$943$	15.6971i	0.511167i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.76079i	0.284687i	0.989817	+	0.142344i	$0.0454638\pi$
$947$	8.76079i	0.284687i	−0.989817	+	0.142344i	$0.954536\pi$
$948$	0	0
$949$	− 0.774009i	− 0.0251254i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 42.8617i	− 1.38843i	−0.719769	−	0.694214i	$-0.755752\pi$
$953$	− 42.8617i	− 1.38843i	0.719769	−	0.694214i	$-0.244248\pi$
$954$	0	0
$955$	47.6908	1.54324
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.03211	0.0333286
$960$	0	0
$961$	30.4426	0.982020
$962$	0	0
$963$	0	0
$964$	0	0
$965$	45.3496	1.45985
$966$	0	0
$967$	− 31.8855i	− 1.02537i	−0.858578	−	0.512684i	$-0.828651\pi$
$967$	− 31.8855i	− 1.02537i	0.858578	−	0.512684i	$-0.171349\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19.2518i	0.617819i	0.951091	+	0.308909i	$0.0999640\pi$
$971$	19.2518i	0.617819i	−0.951091	+	0.308909i	$0.900036\pi$
$972$	0	0
$973$	− 12.0794i	− 0.387247i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 54.2449i	− 1.73545i	−0.497047	−	0.867723i	$-0.665582\pi$
$977$	− 54.2449i	− 1.73545i	0.497047	−	0.867723i	$-0.334418\pi$
$978$	0	0
$979$	5.18555	0.165731
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17.3764	0.554220	0.277110	−	0.960838i	$-0.410623\pi$
$983$	17.3764	0.554220	0.277110	+	0.960838i	$0.410623\pi$
$984$	0	0
$985$	−40.4578	−1.28909
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24.5046	0.779201
$990$	0	0
$991$	13.3107i	0.422829i	0.977397	+	0.211414i	$0.0678069\pi$
$991$	13.3107i	0.422829i	−0.977397	+	0.211414i	$0.932193\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 60.7301i	− 1.92527i
$996$	0	0
$997$	49.6564i	1.57263i	0.617824	+	0.786317i	$0.288014\pi$
$997$	49.6564i	1.57263i	−0.617824	+	0.786317i	$0.711986\pi$
$998$	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.5615.7		8
3.2	odd	2		6048.2.j.b.5615.1		8
4.3	odd	2		1512.2.j.a.323.6	✓	8
8.3	odd	2		6048.2.j.b.5615.2		8
8.5	even	2		1512.2.j.b.323.1	yes	8
12.11	even	2		1512.2.j.b.323.3	yes	8
24.5	odd	2		1512.2.j.a.323.8	yes	8
24.11	even	2	inner	6048.2.j.a.5615.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.a.323.6	✓	8	4.3	odd	2
1512.2.j.a.323.8	yes	8	24.5	odd	2
1512.2.j.b.323.1	yes	8	8.5	even	2
1512.2.j.b.323.3	yes	8	12.11	even	2
6048.2.j.a.5615.7		8	1.1	even	1	trivial
6048.2.j.a.5615.8		8	24.11	even	2	inner
6048.2.j.b.5615.1		8	3.2	odd	2
6048.2.j.b.5615.2		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)

\(f(q)\)	\(=\)	\(q+2.38587 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+2.38587 q^{5} -1.00000i q^{7} -1.65382i q^{11} +0.253423i q^{13} +2.00000i q^{17} -7.03969 q^{19} -2.82481 q^{23} +0.692366 q^{25} -1.26795 q^{29} -0.746577i q^{31} -2.38587i q^{35} +6.94273i q^{37} -5.55686i q^{41} -8.67478 q^{43} -10.9679 q^{47} -1.00000 q^{49} -3.30763 q^{53} -3.94579i q^{55} +10.0397i q^{59} +4.53590i q^{61} +0.604635i q^{65} -0.253423 q^{67} -3.17519 q^{71} -3.05421 q^{73} -1.65382 q^{77} -14.5183i q^{79} +1.77480i q^{83} +4.77174i q^{85} +3.13550i q^{89} +0.253423 q^{91} -16.7958 q^{95} -5.49315 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 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

Introduction

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