LMFDB - Embedded newform 6048.2.j.d.5615.22

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:	$48$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5615.22
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.d.5615.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.530075 q^{5} +1.00000i q^{7} +O(q^{10})$	$q-0.530075 q^{5} +1.00000i q^{7} -0.905144i q^{11} -4.38398i q^{13} -1.25430i q^{17} -0.00364927 q^{19} -0.778929 q^{23} -4.71902 q^{25} +5.43228 q^{29} +3.49953i q^{31} -0.530075i q^{35} -5.34322i q^{37} +1.74538i q^{41} +0.259493 q^{43} +3.75927 q^{47} -1.00000 q^{49} +3.08087 q^{53} +0.479794i q^{55} +5.73302i q^{59} +0.555629i q^{61} +2.32384i q^{65} -11.5465 q^{67} -16.0043 q^{71} -13.1552 q^{73} +0.905144 q^{77} -1.96889i q^{79} -5.82561i q^{83} +0.664872i q^{85} -1.94491i q^{89} +4.38398 q^{91} +0.00193439 q^{95} -13.8042 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 16 q^{19} + 48 q^{25} - 64 q^{43} - 48 q^{49} - 32 q^{67}+O(q^{100}) $ 48 * q - 16 * q^19 + 48 * q^25 - 64 * q^43 - 48 * q^49 - 32 * q^67

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−0.530075	−0.237057	−0.118528	−	0.992951i	$-0.537818\pi$
$5$	−0.530075	−0.237057	−0.118528	+	0.992951i	$0.537818\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 0.905144i	− 0.272911i	−0.990646	−	0.136456i	$-0.956429\pi$
$11$	− 0.905144i	− 0.272911i	0.990646	−	0.136456i	$-0.0435711\pi$
$12$	0	0
$13$	− 4.38398i	− 1.21590i	−0.793976	−	0.607949i	$-0.791992\pi$
$13$	− 4.38398i	− 1.21590i	0.793976	−	0.607949i	$-0.208008\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.25430i	− 0.304212i	−0.988364	−	0.152106i	$-0.951395\pi$
$17$	− 1.25430i	− 0.304212i	0.988364	−	0.152106i	$-0.0486055\pi$
$18$	0	0
$19$	−0.00364927	−0.000837200 0	−0.000418600	−	1.00000i	$-0.500133\pi$
$19$	−0.00364927	−0.000837200 0	−0.000418600		1.00000i	$0.500133\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.778929	−0.162418	−0.0812090	−	0.996697i	$-0.525878\pi$
$23$	−0.778929	−0.162418	−0.0812090	+	0.996697i	$0.525878\pi$
$24$	0	0
$25$	−4.71902	−0.943804
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.43228	1.00875	0.504375	−	0.863485i	$-0.331723\pi$
$29$	5.43228	1.00875	0.504375	+	0.863485i	$0.331723\pi$
$30$	0	0
$31$	3.49953i	0.628533i	0.949335	+	0.314267i	$0.101759\pi$
$31$	3.49953i	0.628533i	−0.949335	+	0.314267i	$0.898241\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 0.530075i	− 0.0895990i
$36$	0	0
$37$	− 5.34322i	− 0.878421i	−0.898384	−	0.439210i	$-0.855258\pi$
$37$	− 5.34322i	− 0.878421i	0.898384	−	0.439210i	$-0.144742\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.74538i	0.272582i	0.990669	+	0.136291i	$0.0435183\pi$
$41$	1.74538i	0.272582i	−0.990669	+	0.136291i	$0.956482\pi$
$42$	0	0
$43$	0.259493	0.0395723	0.0197862	−	0.999804i	$-0.493701\pi$
$43$	0.259493	0.0395723	0.0197862	+	0.999804i	$0.493701\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.75927	0.548346	0.274173	−	0.961680i	$-0.411596\pi$
$47$	3.75927	0.548346	0.274173	+	0.961680i	$0.411596\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.08087	0.423190	0.211595	−	0.977357i	$-0.432134\pi$
$53$	3.08087	0.423190	0.211595	+	0.977357i	$0.432134\pi$
$54$	0	0
$55$	0.479794i	0.0646955i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.73302i	0.746376i	0.927756	+	0.373188i	$0.121735\pi$
$59$	5.73302i	0.746376i	−0.927756	+	0.373188i	$0.878265\pi$
$60$	0	0
$61$	0.555629i	0.0711410i	0.999367	+	0.0355705i	$0.0113248\pi$
$61$	0.555629i	0.0711410i	−0.999367	+	0.0355705i	$0.988675\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.32384i	0.288237i
$66$	0	0
$67$	−11.5465	−1.41063	−0.705313	−	0.708896i	$-0.749194\pi$
$67$	−11.5465	−1.41063	−0.705313	+	0.708896i	$0.749194\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−16.0043	−1.89936	−0.949681	−	0.313218i	$-0.898593\pi$
$71$	−16.0043	−1.89936	−0.949681	+	0.313218i	$0.898593\pi$
$72$	0	0
$73$	−13.1552	−1.53970	−0.769851	−	0.638224i	$-0.779669\pi$
$73$	−13.1552	−1.53970	−0.769851	+	0.638224i	$0.779669\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.905144	0.103151
$78$	0	0
$79$	− 1.96889i	− 0.221517i	−0.993847	−	0.110759i	$-0.964672\pi$
$79$	− 1.96889i	− 0.221517i	0.993847	−	0.110759i	$-0.0353280\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 5.82561i	− 0.639444i	−0.947511	−	0.319722i	$-0.896411\pi$
$83$	− 5.82561i	− 0.639444i	0.947511	−	0.319722i	$-0.103589\pi$
$84$	0	0
$85$	0.664872i	0.0721155i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1.94491i	− 0.206160i	−0.994673	−	0.103080i	$-0.967130\pi$
$89$	− 1.94491i	− 0.206160i	0.994673	−	0.103080i	$-0.0328697\pi$
$90$	0	0
$91$	4.38398	0.459566
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.00193439	0.000198464 0
$96$	0	0
$97$	−13.8042	−1.40161	−0.700803	−	0.713355i	$-0.747175\pi$
$97$	−13.8042	−1.40161	−0.700803	+	0.713355i	$0.747175\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.41639	0.538951	0.269475	−	0.963007i	$-0.413150\pi$
$101$	5.41639	0.538951	0.269475	+	0.963007i	$0.413150\pi$
$102$	0	0
$103$	2.14390i	0.211245i	0.994406	+	0.105623i	$0.0336835\pi$
$103$	2.14390i	0.211245i	−0.994406	+	0.105623i	$0.966316\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 9.10325i	− 0.880045i	−0.897987	−	0.440022i	$-0.854971\pi$
$107$	− 9.10325i	− 0.880045i	0.897987	−	0.440022i	$-0.145029\pi$
$108$	0	0
$109$	− 1.29567i	− 0.124103i	−0.998073	−	0.0620515i	$-0.980236\pi$
$109$	− 1.29567i	− 0.124103i	0.998073	−	0.0620515i	$-0.0197643\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.0728i	1.32386i	0.749567	+	0.661928i	$0.230262\pi$
$113$	14.0728i	1.32386i	−0.749567	+	0.661928i	$0.769738\pi$
$114$	0	0
$115$	0.412891	0.0385023
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.25430	0.114981
$120$	0	0
$121$	10.1807	0.925519
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.15181	0.460792
$126$	0	0
$127$	11.5015i	1.02060i	0.859997	+	0.510298i	$0.170465\pi$
$127$	11.5015i	1.02060i	−0.859997	+	0.510298i	$0.829535\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 10.2828i	− 0.898411i	−0.893429	−	0.449205i	$-0.851707\pi$
$131$	− 10.2828i	− 0.898411i	0.893429	−	0.449205i	$-0.148293\pi$
$132$	0	0
$133$	− 0.00364927i	0 0.000316432i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.3027i	1.64914i	0.565761	+	0.824570i	$0.308583\pi$
$137$	19.3027i	1.64914i	−0.565761	+	0.824570i	$0.691417\pi$
$138$	0	0
$139$	−15.3098	−1.29856	−0.649280	−	0.760550i	$-0.724930\pi$
$139$	−15.3098	−1.29856	−0.649280	+	0.760550i	$0.724930\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.96814	−0.331832
$144$	0	0
$145$	−2.87952	−0.239131
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.53793	0.125992	0.0629962	−	0.998014i	$-0.479934\pi$
$149$	1.53793	0.125992	0.0629962	+	0.998014i	$0.479934\pi$
$150$	0	0
$151$	− 10.1304i	− 0.824398i	−0.911094	−	0.412199i	$-0.864761\pi$
$151$	− 10.1304i	− 0.824398i	0.911094	−	0.412199i	$-0.135239\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.85501i	− 0.148998i
$156$	0	0
$157$	− 2.61643i	− 0.208814i	−0.994535	−	0.104407i	$-0.966706\pi$
$157$	− 2.61643i	− 0.208814i	0.994535	−	0.104407i	$-0.0332945\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 0.778929i	− 0.0613882i
$162$	0	0
$163$	−14.4604	−1.13262	−0.566312	−	0.824191i	$-0.691630\pi$
$163$	−14.4604	−1.13262	−0.566312	+	0.824191i	$0.691630\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.65107	−0.592058	−0.296029	−	0.955179i	$-0.595662\pi$
$167$	−7.65107	−0.592058	−0.296029	+	0.955179i	$0.595662\pi$
$168$	0	0
$169$	−6.21931	−0.478409
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.98644	−0.759255	−0.379628	−	0.925139i	$-0.623948\pi$
$173$	−9.98644	−0.759255	−0.379628	+	0.925139i	$0.623948\pi$
$174$	0	0
$175$	− 4.71902i	− 0.356724i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 16.1143i	− 1.20444i	−0.798331	−	0.602219i	$-0.794283\pi$
$179$	− 16.1143i	− 1.20444i	0.798331	−	0.602219i	$-0.205717\pi$
$180$	0	0
$181$	− 20.4077i	− 1.51689i	−0.651736	−	0.758446i	$-0.725959\pi$
$181$	− 20.4077i	− 1.51689i	0.651736	−	0.758446i	$-0.274041\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.83231i	0.208236i
$186$	0	0
$187$	−1.13532	−0.0830228
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.05526	0.293428	0.146714	−	0.989179i	$-0.453130\pi$
$191$	4.05526	0.293428	0.146714	+	0.989179i	$0.453130\pi$
$192$	0	0
$193$	−5.82186	−0.419066	−0.209533	−	0.977802i	$-0.567194\pi$
$193$	−5.82186	−0.419066	−0.209533	+	0.977802i	$0.567194\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.4557	1.10118	0.550588	−	0.834777i	$-0.314404\pi$
$197$	15.4557	1.10118	0.550588	+	0.834777i	$0.314404\pi$
$198$	0	0
$199$	− 13.6803i	− 0.969771i	−0.874578	−	0.484885i	$-0.838861\pi$
$199$	− 13.6803i	− 0.969771i	0.874578	−	0.484885i	$-0.161139\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.43228i	0.381271i
$204$	0	0
$205$	− 0.925182i	− 0.0646175i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.00330311i	0 0.000228481i
$210$	0	0
$211$	−10.9961	−0.757000	−0.378500	−	0.925601i	$-0.623560\pi$
$211$	−10.9961	−0.757000	−0.378500	+	0.925601i	$0.623560\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.137551	−0.00938089
$216$	0	0
$217$	−3.49953	−0.237563
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.49882	−0.369891
$222$	0	0
$223$	− 18.7956i	− 1.25864i	−0.777144	−	0.629322i	$-0.783332\pi$
$223$	− 18.7956i	− 1.25864i	0.777144	−	0.629322i	$-0.216668\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 18.8278i	− 1.24965i	−0.780767	−	0.624823i	$-0.785171\pi$
$227$	− 18.8278i	− 1.24965i	0.780767	−	0.624823i	$-0.214829\pi$
$228$	0	0
$229$	− 15.7721i	− 1.04225i	−0.853480	−	0.521126i	$-0.825512\pi$
$229$	− 15.7721i	− 1.04225i	0.853480	−	0.521126i	$-0.174488\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 4.23089i	− 0.277175i	−0.990350	−	0.138587i	$-0.955744\pi$
$233$	− 4.23089i	− 0.277175i	0.990350	−	0.138587i	$-0.0442562\pi$
$234$	0	0
$235$	−1.99269	−0.129989
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−7.80160	−0.504644	−0.252322	−	0.967643i	$-0.581194\pi$
$239$	−7.80160	−0.504644	−0.252322	+	0.967643i	$0.581194\pi$
$240$	0	0
$241$	−20.2165	−1.30226	−0.651129	−	0.758967i	$-0.725704\pi$
$241$	−20.2165	−1.30226	−0.651129	+	0.758967i	$0.725704\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.530075	0.0338653
$246$	0	0
$247$	0.0159983i	0.00101795i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.7749i	0.806345i	0.915124	+	0.403172i	$0.132093\pi$
$251$	12.7749i	0.806345i	−0.915124	+	0.403172i	$0.867907\pi$
$252$	0	0
$253$	0.705043i	0.0443257i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.3263i	1.26792i	0.773365	+	0.633961i	$0.218572\pi$
$257$	20.3263i	1.26792i	−0.773365	+	0.633961i	$0.781428\pi$
$258$	0	0
$259$	5.34322	0.332012
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.85204	0.299190	0.149595	−	0.988747i	$-0.452203\pi$
$263$	4.85204	0.299190	0.149595	+	0.988747i	$0.452203\pi$
$264$	0	0
$265$	−1.63309	−0.100320
$266$	0	0
$267$	0	0
$268$	0	0
$269$	31.3976	1.91435	0.957174	−	0.289514i	$-0.0934937\pi$
$269$	31.3976	1.91435	0.957174	+	0.289514i	$0.0934937\pi$
$270$	0	0
$271$	− 24.4088i	− 1.48273i	−0.671103	−	0.741364i	$-0.734179\pi$
$271$	− 24.4088i	− 1.48273i	0.671103	−	0.741364i	$-0.265821\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.27139i	0.257575i
$276$	0	0
$277$	12.8387i	0.771402i	0.922624	+	0.385701i	$0.126040\pi$
$277$	12.8387i	0.771402i	−0.922624	+	0.385701i	$0.873960\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 27.8988i	− 1.66431i	−0.554547	−	0.832153i	$-0.687108\pi$
$281$	− 27.8988i	− 1.66431i	0.554547	−	0.832153i	$-0.312892\pi$
$282$	0	0
$283$	−6.65773	−0.395761	−0.197880	−	0.980226i	$-0.563406\pi$
$283$	−6.65773	−0.395761	−0.197880	+	0.980226i	$0.563406\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.74538	−0.103026
$288$	0	0
$289$	15.4267	0.907455
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.0566	−1.40540	−0.702699	−	0.711487i	$-0.748022\pi$
$293$	−24.0566	−1.40540	−0.702699	+	0.711487i	$0.748022\pi$
$294$	0	0
$295$	− 3.03893i	− 0.176933i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.41481i	0.197484i
$300$	0	0
$301$	0.259493i	0.0149569i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 0.294525i	− 0.0168645i
$306$	0	0
$307$	−23.6311	−1.34870	−0.674350	−	0.738411i	$-0.735576\pi$
$307$	−23.6311	−1.34870	−0.674350	+	0.738411i	$0.735576\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−25.7037	−1.45752	−0.728761	−	0.684768i	$-0.759904\pi$
$311$	−25.7037	−1.45752	−0.728761	+	0.684768i	$0.759904\pi$
$312$	0	0
$313$	−9.99902	−0.565178	−0.282589	−	0.959241i	$-0.591193\pi$
$313$	−9.99902	−0.565178	−0.282589	+	0.959241i	$0.591193\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.7177	−1.33212	−0.666059	−	0.745899i	$-0.732020\pi$
$317$	−23.7177	−1.33212	−0.666059	+	0.745899i	$0.732020\pi$
$318$	0	0
$319$	− 4.91700i	− 0.275299i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.00457727i	0 0.000254686i
$324$	0	0
$325$	20.6881i	1.14757i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.75927i	0.207255i
$330$	0	0
$331$	−12.8674	−0.707254	−0.353627	−	0.935386i	$-0.615052\pi$
$331$	−12.8674	−0.707254	−0.353627	+	0.935386i	$0.615052\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.12050	0.334398
$336$	0	0
$337$	−10.2878	−0.560412	−0.280206	−	0.959940i	$-0.590403\pi$
$337$	−10.2878	−0.560412	−0.280206	+	0.959940i	$0.590403\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.16758	0.171534
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.21263i	0.118780i	0.998235	+	0.0593901i	$0.0189156\pi$
$347$	2.21263i	0.118780i	−0.998235	+	0.0593901i	$0.981084\pi$
$348$	0	0
$349$	8.10593i	0.433900i	0.976183	+	0.216950i	$0.0696109\pi$
$349$	8.10593i	0.433900i	−0.976183	+	0.216950i	$0.930389\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 16.8821i	− 0.898546i	−0.893395	−	0.449273i	$-0.851683\pi$
$353$	− 16.8821i	− 0.898546i	0.893395	−	0.449273i	$-0.148317\pi$
$354$	0	0
$355$	8.48349	0.450257
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.06059	0.319866	0.159933	−	0.987128i	$-0.448872\pi$
$359$	6.06059	0.319866	0.159933	+	0.987128i	$0.448872\pi$
$360$	0	0
$361$	−19.0000	−0.999999
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.97325	0.364997
$366$	0	0
$367$	4.36701i	0.227956i	0.993483	+	0.113978i	$0.0363594\pi$
$367$	4.36701i	0.227956i	−0.993483	+	0.113978i	$0.963641\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.08087i	0.159951i
$372$	0	0
$373$	− 10.3873i	− 0.537835i	−0.963163	−	0.268918i	$-0.913334\pi$
$373$	− 10.3873i	− 0.537835i	0.963163	−	0.268918i	$-0.0866660\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 23.8150i	− 1.22654i
$378$	0	0
$379$	−28.3525	−1.45637	−0.728186	−	0.685380i	$-0.759636\pi$
$379$	−28.3525	−1.45637	−0.728186	+	0.685380i	$0.759636\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	26.3164	1.34470	0.672352	−	0.740232i	$-0.265284\pi$
$383$	26.3164	1.34470	0.672352	+	0.740232i	$0.265284\pi$
$384$	0	0
$385$	−0.479794	−0.0244526
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−17.5966	−0.892182	−0.446091	−	0.894988i	$-0.647184\pi$
$389$	−17.5966	−0.892182	−0.446091	+	0.894988i	$0.647184\pi$
$390$	0	0
$391$	0.977009i	0.0494095i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.04366i	0.0525121i
$396$	0	0
$397$	5.06951i	0.254431i	0.991875	+	0.127216i	$0.0406040\pi$
$397$	5.06951i	0.254431i	−0.991875	+	0.127216i	$0.959396\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.6708i	0.832498i	0.909251	+	0.416249i	$0.136655\pi$
$401$	16.6708i	0.832498i	−0.909251	+	0.416249i	$0.863345\pi$
$402$	0	0
$403$	15.3419	0.764233
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.83639	−0.239731
$408$	0	0
$409$	−5.09197	−0.251782	−0.125891	−	0.992044i	$-0.540179\pi$
$409$	−5.09197	−0.251782	−0.125891	+	0.992044i	$0.540179\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.73302	−0.282104
$414$	0	0
$415$	3.08801i	0.151584i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	37.4571i	1.82990i	0.403570	+	0.914949i	$0.367769\pi$
$419$	37.4571i	1.82990i	−0.403570	+	0.914949i	$0.632231\pi$
$420$	0	0
$421$	35.9612i	1.75264i	0.481728	+	0.876321i	$0.340009\pi$
$421$	35.9612i	1.75264i	−0.481728	+	0.876321i	$0.659991\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.91906i	0.287116i
$426$	0	0
$427$	−0.555629	−0.0268888
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.6573	0.995025	0.497513	−	0.867457i	$-0.334247\pi$
$431$	20.6573	0.995025	0.497513	+	0.867457i	$0.334247\pi$
$432$	0	0
$433$	23.8116	1.14431	0.572157	−	0.820144i	$-0.306107\pi$
$433$	23.8116	1.14431	0.572157	+	0.820144i	$0.306107\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.00284252	0.000135976 0
$438$	0	0
$439$	− 26.5607i	− 1.26767i	−0.773467	−	0.633836i	$-0.781479\pi$
$439$	− 26.5607i	− 1.26767i	0.773467	−	0.633836i	$-0.218521\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.346564i	0.0164658i	0.999966	+	0.00823288i	$0.00262064\pi$
$443$	0.346564i	0.0164658i	−0.999966	+	0.00823288i	$0.997379\pi$
$444$	0	0
$445$	1.03095i	0.0488716i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 10.4601i	− 0.493641i	−0.969061	−	0.246821i	$-0.920614\pi$
$449$	− 10.4601i	− 0.493641i	0.969061	−	0.246821i	$-0.0793859\pi$
$450$	0	0
$451$	1.57982	0.0743908
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.32384	−0.108943
$456$	0	0
$457$	−11.0889	−0.518717	−0.259359	−	0.965781i	$-0.583511\pi$
$457$	−11.0889	−0.518717	−0.259359	+	0.965781i	$0.583511\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	37.8447	1.76261	0.881303	−	0.472553i	$-0.156667\pi$
$461$	37.8447	1.76261	0.881303	+	0.472553i	$0.156667\pi$
$462$	0	0
$463$	18.1959i	0.845634i	0.906215	+	0.422817i	$0.138959\pi$
$463$	18.1959i	0.845634i	−0.906215	+	0.422817i	$0.861041\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.19184i	− 0.286524i	−0.989685	−	0.143262i	$-0.954241\pi$
$467$	− 6.19184i	− 0.286524i	0.989685	−	0.143262i	$-0.0457592\pi$
$468$	0	0
$469$	− 11.5465i	− 0.533167i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 0.234879i	− 0.0107997i
$474$	0	0
$475$	0.0172210	0.000790152 0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.17517	−0.190769	−0.0953843	−	0.995441i	$-0.530408\pi$
$479$	−4.17517	−0.190769	−0.0953843	+	0.995441i	$0.530408\pi$
$480$	0	0
$481$	−23.4246	−1.06807
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.31727	0.332260
$486$	0	0
$487$	16.4966i	0.747534i	0.927523	+	0.373767i	$0.121934\pi$
$487$	16.4966i	0.747534i	−0.927523	+	0.373767i	$0.878066\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.1039i	1.13292i	0.824089	+	0.566461i	$0.191688\pi$
$491$	25.1039i	1.13292i	−0.824089	+	0.566461i	$0.808312\pi$
$492$	0	0
$493$	− 6.81370i	− 0.306873i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 16.0043i	− 0.717892i
$498$	0	0
$499$	−37.1869	−1.66472	−0.832358	−	0.554238i	$-0.813010\pi$
$499$	−37.1869	−1.66472	−0.832358	+	0.554238i	$0.813010\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−8.24722	−0.367725	−0.183863	−	0.982952i	$-0.558860\pi$
$503$	−8.24722	−0.367725	−0.183863	+	0.982952i	$0.558860\pi$
$504$	0	0
$505$	−2.87109	−0.127762
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−29.7128	−1.31700	−0.658498	−	0.752582i	$-0.728808\pi$
$509$	−29.7128	−1.31700	−0.658498	+	0.752582i	$0.728808\pi$
$510$	0	0
$511$	− 13.1552i	− 0.581953i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1.13643i	− 0.0500771i
$516$	0	0
$517$	− 3.40268i	− 0.149650i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 9.22757i	− 0.404267i	−0.979358	−	0.202134i	$-0.935212\pi$
$521$	− 9.22757i	− 0.404267i	0.979358	−	0.202134i	$-0.0647875\pi$
$522$	0	0
$523$	−1.55380	−0.0679431	−0.0339716	−	0.999423i	$-0.510816\pi$
$523$	−1.55380	−0.0679431	−0.0339716	+	0.999423i	$0.510816\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.38945	0.191207
$528$	0	0
$529$	−22.3933	−0.973620
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.65171	0.331433
$534$	0	0
$535$	4.82541i	0.208621i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.905144i	0.0389873i
$540$	0	0
$541$	− 7.33522i	− 0.315366i	−0.987490	−	0.157683i	$-0.949598\pi$
$541$	− 7.33522i	− 0.315366i	0.987490	−	0.157683i	$-0.0504024\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.686805i	0.0294195i
$546$	0	0
$547$	−21.4488	−0.917086	−0.458543	−	0.888672i	$-0.651628\pi$
$547$	−21.4488	−0.917086	−0.458543	+	0.888672i	$0.651628\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.0198238	−0.000844524 0
$552$	0	0
$553$	1.96889	0.0837256
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−29.9101	−1.26733	−0.633666	−	0.773607i	$-0.718451\pi$
$557$	−29.9101	−1.26733	−0.633666	+	0.773607i	$0.718451\pi$
$558$	0	0
$559$	− 1.13761i	− 0.0481159i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	42.7436i	1.80143i	0.434414	+	0.900713i	$0.356956\pi$
$563$	42.7436i	1.80143i	−0.434414	+	0.900713i	$0.643044\pi$
$564$	0	0
$565$	− 7.45963i	− 0.313829i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 20.9180i	− 0.876927i	−0.898749	−	0.438464i	$-0.855523\pi$
$569$	− 20.9180i	− 0.876927i	0.898749	−	0.438464i	$-0.144477\pi$
$570$	0	0
$571$	5.23949	0.219266	0.109633	−	0.993972i	$-0.465032\pi$
$571$	5.23949	0.219266	0.109633	+	0.993972i	$0.465032\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.67578	0.153291
$576$	0	0
$577$	−5.71213	−0.237799	−0.118900	−	0.992906i	$-0.537937\pi$
$577$	−5.71213	−0.237799	−0.118900	+	0.992906i	$0.537937\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.82561	0.241687
$582$	0	0
$583$	− 2.78863i	− 0.115493i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 28.8731i	− 1.19172i	−0.803088	−	0.595860i	$-0.796811\pi$
$587$	− 28.8731i	− 1.19172i	0.803088	−	0.595860i	$-0.203189\pi$
$588$	0	0
$589$	− 0.0127707i	0 0.000526208i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 32.7752i	− 1.34592i	−0.739680	−	0.672958i	$-0.765023\pi$
$593$	− 32.7752i	− 1.34592i	0.739680	−	0.672958i	$-0.234977\pi$
$594$	0	0
$595$	−0.664872	−0.0272571
$596$	0	0
$597$	0	0
$598$	0	0
$599$	18.9096	0.772625	0.386313	−	0.922368i	$-0.373749\pi$
$599$	18.9096	0.772625	0.386313	+	0.922368i	$0.373749\pi$
$600$	0	0
$601$	36.4186	1.48555	0.742773	−	0.669543i	$-0.233510\pi$
$601$	36.4186	1.48555	0.742773	+	0.669543i	$0.233510\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5.39654	−0.219401
$606$	0	0
$607$	8.21476i	0.333427i	0.986005	+	0.166713i	$0.0533155\pi$
$607$	8.21476i	0.333427i	−0.986005	+	0.166713i	$0.946685\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 16.4806i	− 0.666732i
$612$	0	0
$613$	− 27.6348i	− 1.11616i	−0.829787	−	0.558080i	$-0.811538\pi$
$613$	− 27.6348i	− 1.11616i	0.829787	−	0.558080i	$-0.188462\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23.3314i	0.939287i	0.882856	+	0.469643i	$0.155617\pi$
$617$	23.3314i	0.939287i	−0.882856	+	0.469643i	$0.844383\pi$
$618$	0	0
$619$	22.9819	0.923719	0.461859	−	0.886953i	$-0.347182\pi$
$619$	22.9819	0.923719	0.461859	+	0.886953i	$0.347182\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.94491	0.0779211
$624$	0	0
$625$	20.8643	0.834570
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.70199	−0.267226
$630$	0	0
$631$	44.4382i	1.76906i	0.466486	+	0.884528i	$0.345520\pi$
$631$	44.4382i	1.76906i	−0.466486	+	0.884528i	$0.654480\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 6.09668i	− 0.241939i
$636$	0	0
$637$	4.38398i	0.173700i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	37.6567i	1.48735i	0.668541	+	0.743675i	$0.266919\pi$
$641$	37.6567i	1.48735i	−0.668541	+	0.743675i	$0.733081\pi$
$642$	0	0
$643$	22.1342	0.872886	0.436443	−	0.899732i	$-0.356238\pi$
$643$	22.1342	0.872886	0.436443	+	0.899732i	$0.356238\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.8560	0.819936	0.409968	−	0.912100i	$-0.365540\pi$
$647$	20.8560	0.819936	0.409968	+	0.912100i	$0.365540\pi$
$648$	0	0
$649$	5.18921	0.203694
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.5052	−0.489366	−0.244683	−	0.969603i	$-0.578684\pi$
$653$	−12.5052	−0.489366	−0.244683	+	0.969603i	$0.578684\pi$
$654$	0	0
$655$	5.45065i	0.212974i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27.7704i	1.08178i	0.841092	+	0.540891i	$0.181913\pi$
$659$	27.7704i	1.08178i	−0.841092	+	0.540891i	$0.818087\pi$
$660$	0	0
$661$	17.8134i	0.692862i	0.938075	+	0.346431i	$0.112607\pi$
$661$	17.8134i	0.692862i	−0.938075	+	0.346431i	$0.887393\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.00193439i	0 7.50123e-5i
$666$	0	0
$667$	−4.23136	−0.163839
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.502924	0.0194152
$672$	0	0
$673$	−16.2911	−0.627975	−0.313987	−	0.949427i	$-0.601665\pi$
$673$	−16.2911	−0.627975	−0.313987	+	0.949427i	$0.601665\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.61803	−0.177485	−0.0887426	−	0.996055i	$-0.528285\pi$
$677$	−4.61803	−0.177485	−0.0887426	+	0.996055i	$0.528285\pi$
$678$	0	0
$679$	− 13.8042i	− 0.529757i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 30.9791i	− 1.18538i	−0.805429	−	0.592692i	$-0.798065\pi$
$683$	− 30.9791i	− 1.18538i	0.805429	−	0.592692i	$-0.201935\pi$
$684$	0	0
$685$	− 10.2319i	− 0.390940i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 13.5065i	− 0.514556i
$690$	0	0
$691$	29.0382	1.10467	0.552333	−	0.833624i	$-0.313738\pi$
$691$	29.0382	1.10467	0.552333	+	0.833624i	$0.313738\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.11534	0.307832
$696$	0	0
$697$	2.18923	0.0829228
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−27.6010	−1.04248	−0.521238	−	0.853411i	$-0.674530\pi$
$701$	−27.6010	−1.04248	−0.521238	+	0.853411i	$0.674530\pi$
$702$	0	0
$703$	0.0194989i	0 0.000735413i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.41639i	0.203704i
$708$	0	0
$709$	− 37.8212i	− 1.42040i	−0.703998	−	0.710202i	$-0.748604\pi$
$709$	− 37.8212i	− 1.42040i	0.703998	−	0.710202i	$-0.251396\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.72588i	− 0.102085i
$714$	0	0
$715$	2.10341	0.0786631
$716$	0	0
$717$	0	0
$718$	0	0
$719$	41.2974	1.54013	0.770067	−	0.637963i	$-0.220223\pi$
$719$	41.2974	1.54013	0.770067	+	0.637963i	$0.220223\pi$
$720$	0	0
$721$	−2.14390	−0.0798432
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−25.6350	−0.952061
$726$	0	0
$727$	− 11.9168i	− 0.441971i	−0.975277	−	0.220985i	$-0.929073\pi$
$727$	− 11.9168i	− 0.441971i	0.975277	−	0.220985i	$-0.0709273\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 0.325482i	− 0.0120384i
$732$	0	0
$733$	− 47.2142i	− 1.74390i	−0.489597	−	0.871949i	$-0.662856\pi$
$733$	− 47.2142i	− 1.74390i	0.489597	−	0.871949i	$-0.337144\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.4512i	0.384976i
$738$	0	0
$739$	13.4529	0.494874	0.247437	−	0.968904i	$-0.420412\pi$
$739$	13.4529	0.494874	0.247437	+	0.968904i	$0.420412\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.7698	−0.395106	−0.197553	−	0.980292i	$-0.563299\pi$
$743$	−10.7698	−0.395106	−0.197553	+	0.980292i	$0.563299\pi$
$744$	0	0
$745$	−0.815220	−0.0298673
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.10325	0.332626
$750$	0	0
$751$	− 14.1004i	− 0.514530i	−0.966341	−	0.257265i	$-0.917179\pi$
$751$	− 14.1004i	− 0.514530i	0.966341	−	0.257265i	$-0.0828213\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.36986i	0.195429i
$756$	0	0
$757$	13.8374i	0.502929i	0.967866	+	0.251465i	$0.0809122\pi$
$757$	13.8374i	0.502929i	−0.967866	+	0.251465i	$0.919088\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.1367i	1.20121i	0.799548	+	0.600603i	$0.205073\pi$
$761$	33.1367i	1.20121i	−0.799548	+	0.600603i	$0.794927\pi$
$762$	0	0
$763$	1.29567	0.0469066
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25.1335	0.907517
$768$	0	0
$769$	30.1190	1.08612	0.543060	−	0.839694i	$-0.317266\pi$
$769$	30.1190	1.08612	0.543060	+	0.839694i	$0.317266\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−16.0579	−0.577561	−0.288780	−	0.957395i	$-0.593250\pi$
$773$	−16.0579	−0.577561	−0.288780	+	0.957395i	$0.593250\pi$
$774$	0	0
$775$	− 16.5143i	− 0.593212i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 0.00636936i	0 0.000228206i
$780$	0	0
$781$	14.4862i	0.518357i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.38691i	0.0495008i
$786$	0	0
$787$	−11.2848	−0.402258	−0.201129	−	0.979565i	$-0.564461\pi$
$787$	−11.2848	−0.402258	−0.201129	+	0.979565i	$0.564461\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−14.0728	−0.500371
$792$	0	0
$793$	2.43587	0.0865002
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.3545	1.39401	0.697004	−	0.717068i	$-0.254516\pi$
$797$	39.3545	1.39401	0.697004	+	0.717068i	$0.254516\pi$
$798$	0	0
$799$	− 4.71524i	− 0.166813i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.9074i	0.420202i
$804$	0	0
$805$	0.412891i	0.0145525i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0.183323i	0.00644529i	0.999995	+	0.00322265i	$0.00102580\pi$
$809$	0.183323i	0.00644529i	−0.999995	+	0.00322265i	$0.998974\pi$
$810$	0	0
$811$	46.7136	1.64033	0.820167	−	0.572124i	$-0.193880\pi$
$811$	46.7136	1.64033	0.820167	+	0.572124i	$0.193880\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.66508	0.268496
$816$	0	0
$817$	−0.000946960 0	−3.31299e−5 0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	19.7408	0.688959	0.344480	−	0.938794i	$-0.388055\pi$
$821$	19.7408	0.688959	0.344480	+	0.938794i	$0.388055\pi$
$822$	0	0
$823$	11.6881i	0.407421i	0.979031	+	0.203711i	$0.0653002\pi$
$823$	11.6881i	0.407421i	−0.979031	+	0.203711i	$0.934700\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 30.8036i	− 1.07115i	−0.844489	−	0.535573i	$-0.820096\pi$
$827$	− 30.8036i	− 1.07115i	0.844489	−	0.535573i	$-0.179904\pi$
$828$	0	0
$829$	33.2395i	1.15446i	0.816583	+	0.577228i	$0.195866\pi$
$829$	33.2395i	1.15446i	−0.816583	+	0.577228i	$0.804134\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.25430i	0.0434588i
$834$	0	0
$835$	4.05564	0.140351
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−42.9568	−1.48303	−0.741517	−	0.670934i	$-0.765893\pi$
$839$	−42.9568	−1.48303	−0.741517	+	0.670934i	$0.765893\pi$
$840$	0	0
$841$	0.509663	0.0175746
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.29670	0.113410
$846$	0	0
$847$	10.1807i	0.349813i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.16199i	0.142671i
$852$	0	0
$853$	− 31.5917i	− 1.08168i	−0.841126	−	0.540840i	$-0.818107\pi$
$853$	− 31.5917i	− 1.08168i	0.841126	−	0.540840i	$-0.181893\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 37.4806i	− 1.28031i	−0.768245	−	0.640156i	$-0.778870\pi$
$857$	− 37.4806i	− 1.28031i	0.768245	−	0.640156i	$-0.221130\pi$
$858$	0	0
$859$	41.4525	1.41434	0.707171	−	0.707042i	$-0.249971\pi$
$859$	41.4525	1.41434	0.707171	+	0.707042i	$0.249971\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	43.8203	1.49166	0.745830	−	0.666136i	$-0.232053\pi$
$863$	43.8203	1.49166	0.745830	+	0.666136i	$0.232053\pi$
$864$	0	0
$865$	5.29356	0.179987
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.78213	−0.0604545
$870$	0	0
$871$	50.6195i	1.71518i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.15181i	0.174163i
$876$	0	0
$877$	− 33.3035i	− 1.12458i	−0.826940	−	0.562289i	$-0.809921\pi$
$877$	− 33.3035i	− 1.12458i	0.826940	−	0.562289i	$-0.190079\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 35.6627i	− 1.20151i	−0.799434	−	0.600754i	$-0.794867\pi$
$881$	− 35.6627i	− 1.20151i	0.799434	−	0.600754i	$-0.205133\pi$
$882$	0	0
$883$	−8.32201	−0.280058	−0.140029	−	0.990147i	$-0.544720\pi$
$883$	−8.32201	−0.280058	−0.140029	+	0.990147i	$0.544720\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−43.3551	−1.45572	−0.727861	−	0.685725i	$-0.759485\pi$
$887$	−43.3551	−1.45572	−0.727861	+	0.685725i	$0.759485\pi$
$888$	0	0
$889$	−11.5015	−0.385749
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.0137186	−0.000459075 0
$894$	0	0
$895$	8.54178i	0.285520i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19.0104i	0.634032i
$900$	0	0
$901$	− 3.86433i	− 0.128739i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.8176i	0.359590i
$906$	0	0
$907$	7.89354	0.262101	0.131050	−	0.991376i	$-0.458165\pi$
$907$	7.89354	0.262101	0.131050	+	0.991376i	$0.458165\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	47.9212	1.58770	0.793850	−	0.608113i	$-0.208073\pi$
$911$	47.9212	1.58770	0.793850	+	0.608113i	$0.208073\pi$
$912$	0	0
$913$	−5.27302	−0.174511
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.2828	0.339567
$918$	0	0
$919$	− 22.2192i	− 0.732943i	−0.930429	−	0.366471i	$-0.880566\pi$
$919$	− 22.2192i	− 0.732943i	0.930429	−	0.366471i	$-0.119434\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	70.1627i	2.30943i
$924$	0	0
$925$	25.2148i	0.829057i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 34.2658i	− 1.12422i	−0.827061	−	0.562112i	$-0.809989\pi$
$929$	− 34.2658i	− 1.12422i	0.827061	−	0.562112i	$-0.190011\pi$
$930$	0	0
$931$	0.00364927	0.000119600 0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.601805	0.0196811
$936$	0	0
$937$	5.43172	0.177447	0.0887233	−	0.996056i	$-0.471721\pi$
$937$	5.43172	0.177447	0.0887233	+	0.996056i	$0.471721\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18.1273	−0.590932	−0.295466	−	0.955353i	$-0.595475\pi$
$941$	−18.1273	−0.590932	−0.295466	+	0.955353i	$0.595475\pi$
$942$	0	0
$943$	− 1.35953i	− 0.0442723i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11.9799i	0.389295i	0.980873	+	0.194647i	$0.0623562\pi$
$947$	11.9799i	0.389295i	−0.980873	+	0.194647i	$0.937644\pi$
$948$	0	0
$949$	57.6723i	1.87212i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 22.7578i	− 0.737197i	−0.929589	−	0.368599i	$-0.879838\pi$
$953$	− 22.7578i	− 0.737197i	0.929589	−	0.368599i	$-0.120162\pi$
$954$	0	0
$955$	−2.14959	−0.0695592
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−19.3027	−0.623316
$960$	0	0
$961$	18.7533	0.604946
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.08602	0.0993425
$966$	0	0
$967$	41.9989i	1.35060i	0.737545	+	0.675298i	$0.235985\pi$
$967$	41.9989i	1.35060i	−0.737545	+	0.675298i	$0.764015\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 36.5508i	− 1.17297i	−0.809959	−	0.586486i	$-0.800511\pi$
$971$	− 36.5508i	− 1.17297i	0.809959	−	0.586486i	$-0.199489\pi$
$972$	0	0
$973$	− 15.3098i	− 0.490809i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.6094i	0.979281i	0.871925	+	0.489640i	$0.162872\pi$
$977$	30.6094i	0.979281i	−0.871925	+	0.489640i	$0.837128\pi$
$978$	0	0
$979$	−1.76042	−0.0562633
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−36.1930	−1.15438	−0.577189	−	0.816610i	$-0.695850\pi$
$983$	−36.1930	−1.15438	−0.577189	+	0.816610i	$0.695850\pi$
$984$	0	0
$985$	−8.19270	−0.261041
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.202127	−0.00642726
$990$	0	0
$991$	52.1154i	1.65550i	0.561097	+	0.827750i	$0.310379\pi$
$991$	52.1154i	1.65550i	−0.561097	+	0.827750i	$0.689621\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.25159i	0.229891i
$996$	0	0
$997$	20.2906i	0.642609i	0.946976	+	0.321305i	$0.104121\pi$
$997$	20.2906i	0.642609i	−0.946976	+	0.321305i	$0.895879\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.j.d.5615.22		48
3.2	odd	2	inner	6048.2.j.d.5615.27		48
4.3	odd	2		1512.2.j.d.323.43	yes	48
8.3	odd	2	inner	6048.2.j.d.5615.28		48
8.5	even	2		1512.2.j.d.323.5	✓	48
12.11	even	2		1512.2.j.d.323.6	yes	48
24.5	odd	2		1512.2.j.d.323.44	yes	48
24.11	even	2	inner	6048.2.j.d.5615.21		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.d.323.5	✓	48	8.5	even	2
1512.2.j.d.323.6	yes	48	12.11	even	2
1512.2.j.d.323.43	yes	48	4.3	odd	2
1512.2.j.d.323.44	yes	48	24.5	odd	2
6048.2.j.d.5615.21		48	24.11	even	2	inner
6048.2.j.d.5615.22		48	1.1	even	1	trivial
6048.2.j.d.5615.27		48	3.2	odd	2	inner
6048.2.j.d.5615.28		48	8.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.530075 q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q-0.530075 q^{5} +1.00000i q^{7} -0.905144i q^{11} -4.38398i q^{13} -1.25430i q^{17} -0.00364927 q^{19} -0.778929 q^{23} -4.71902 q^{25} +5.43228 q^{29} +3.49953i q^{31} -0.530075i q^{35} -5.34322i q^{37} +1.74538i q^{41} +0.259493 q^{43} +3.75927 q^{47} -1.00000 q^{49} +3.08087 q^{53} +0.479794i q^{55} +5.73302i q^{59} +0.555629i q^{61} +2.32384i q^{65} -11.5465 q^{67} -16.0043 q^{71} -13.1552 q^{73} +0.905144 q^{77} -1.96889i q^{79} -5.82561i q^{83} +0.664872i q^{85} -1.94491i q^{89} +4.38398 q^{91} +0.00193439 q^{95} -13.8042 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 16 q^{19} + 48 q^{25} - 64 q^{43} - 48 q^{49} - 32 q^{67}+O(q^{100}) \) 48 * q - 16 * q^19 + 48 * q^25 - 64 * q^43 - 48 * q^49 - 32 * q^67

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more