LMFDB - Embedded newform 2736.2.d.b.2015.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(2015,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("2736.2015");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.d (of order $2$, degree $1$, 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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2015.11
Character	$\chi$	$=$	2736.2015
Dual form			2736.2.d.b.2015.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.11119i q^{5} -1.19380i q^{7} +O(q^{10})$	$q-1.11119i q^{5} -1.19380i q^{7} +3.72045 q^{11} +0.113448 q^{13} +2.89258i q^{17} +1.00000i q^{19} +1.35549 q^{23} +3.76526 q^{25} +0.860408i q^{29} +9.01283i q^{31} -1.32654 q^{35} +5.64396 q^{37} -10.5323i q^{41} +2.47195i q^{43} -5.05623 q^{47} +5.57485 q^{49} -9.51146i q^{53} -4.13413i q^{55} +8.73964 q^{59} -8.52172 q^{61} -0.126062i q^{65} +0.857080i q^{67} -1.91011 q^{71} +3.06213 q^{73} -4.44146i q^{77} +7.62732i q^{79} +10.0020 q^{83} +3.21420 q^{85} -13.1659i q^{89} -0.135434i q^{91} +1.11119 q^{95} +11.6659 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 8 q^{25} - 32 q^{37} - 32 q^{49} + 8 q^{73} + 40 q^{85} + 16 q^{97}+O(q^{100}) $ 24 * q + 8 * q^25 - 32 * q^37 - 32 * q^49 + 8 * q^73 + 40 * q^85 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$5$	− 1.11119i	− 0.496939i	−0.968640	−	0.248470i	$-0.920072\pi$
$5$	− 1.11119i	− 0.496939i	0.968640	−	0.248470i	$-0.0799276\pi$
$6$	0	0
$7$	− 1.19380i	− 0.451213i	−0.974219	−	0.225606i	$-0.927564\pi$
$7$	− 1.19380i	− 0.451213i	0.974219	−	0.225606i	$-0.0724363\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.72045	1.12176	0.560879	−	0.827898i	$-0.310463\pi$
$11$	3.72045	1.12176	0.560879	+	0.827898i	$0.310463\pi$
$12$	0	0
$13$	0.113448	0.0314648	0.0157324	−	0.999876i	$-0.494992\pi$
$13$	0.113448	0.0314648	0.0157324	+	0.999876i	$0.494992\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.89258i	0.701553i	0.936459	+	0.350776i	$0.114082\pi$
$17$	2.89258i	0.701553i	−0.936459	+	0.350776i	$0.885918\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.35549	0.282639	0.141319	−	0.989964i	$-0.454866\pi$
$23$	1.35549	0.282639	0.141319	+	0.989964i	$0.454866\pi$
$24$	0	0
$25$	3.76526	0.753051
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.860408i	0.159774i	0.996804	+	0.0798869i	$0.0254559\pi$
$29$	0.860408i	0.159774i	−0.996804	+	0.0798869i	$0.974544\pi$
$30$	0	0
$31$	9.01283i	1.61875i	0.587290	+	0.809376i	$0.300195\pi$
$31$	9.01283i	1.61875i	−0.587290	+	0.809376i	$0.699805\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.32654	−0.224225
$36$	0	0
$37$	5.64396	0.927862	0.463931	−	0.885871i	$-0.346439\pi$
$37$	5.64396	0.927862	0.463931	+	0.885871i	$0.346439\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 10.5323i	− 1.64487i	−0.568859	−	0.822435i	$-0.692615\pi$
$41$	− 10.5323i	− 1.64487i	0.568859	−	0.822435i	$-0.307385\pi$
$42$	0	0
$43$	2.47195i	0.376969i	0.982076	+	0.188484i	$0.0603575\pi$
$43$	2.47195i	0.376969i	−0.982076	+	0.188484i	$0.939643\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.05623	−0.737527	−0.368764	−	0.929523i	$-0.620219\pi$
$47$	−5.05623	−0.737527	−0.368764	+	0.929523i	$0.620219\pi$
$48$	0	0
$49$	5.57485	0.796407
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 9.51146i	− 1.30650i	−0.757143	−	0.653250i	$-0.773405\pi$
$53$	− 9.51146i	− 1.30650i	0.757143	−	0.653250i	$-0.226595\pi$
$54$	0	0
$55$	− 4.13413i	− 0.557446i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.73964	1.13780	0.568902	−	0.822405i	$-0.307368\pi$
$59$	8.73964	1.13780	0.568902	+	0.822405i	$0.307368\pi$
$60$	0	0
$61$	−8.52172	−1.09109	−0.545547	−	0.838080i	$-0.683678\pi$
$61$	−8.52172	−1.09109	−0.545547	+	0.838080i	$0.683678\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 0.126062i	− 0.0156361i
$66$	0	0
$67$	0.857080i	0.104709i	0.998629	+	0.0523545i	$0.0166726\pi$
$67$	0.857080i	0.104709i	−0.998629	+	0.0523545i	$0.983327\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.91011	−0.226689	−0.113344	−	0.993556i	$-0.536156\pi$
$71$	−1.91011	−0.226689	−0.113344	+	0.993556i	$0.536156\pi$
$72$	0	0
$73$	3.06213	0.358395	0.179198	−	0.983813i	$-0.442650\pi$
$73$	3.06213	0.358395	0.179198	+	0.983813i	$0.442650\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.44146i	− 0.506152i
$78$	0	0
$79$	7.62732i	0.858141i	0.903271	+	0.429071i	$0.141159\pi$
$79$	7.62732i	0.858141i	−0.903271	+	0.429071i	$0.858841\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.0020	1.09787	0.548933	−	0.835867i	$-0.315034\pi$
$83$	10.0020	1.09787	0.548933	+	0.835867i	$0.315034\pi$
$84$	0	0
$85$	3.21420	0.348629
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 13.1659i	− 1.39558i	−0.716300	−	0.697792i	$-0.754166\pi$
$89$	− 13.1659i	− 1.39558i	0.716300	−	0.697792i	$-0.245834\pi$
$90$	0	0
$91$	− 0.135434i	− 0.0141973i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.11119	0.114006
$96$	0	0
$97$	11.6659	1.18450	0.592249	−	0.805755i	$-0.298240\pi$
$97$	11.6659	1.18450	0.592249	+	0.805755i	$0.298240\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.64199i	− 0.163384i	−0.996658	−	0.0816921i	$-0.973968\pi$
$101$	− 1.64199i	− 0.163384i	0.996658	−	0.0816921i	$-0.0260324\pi$
$102$	0	0
$103$	− 2.09681i	− 0.206605i	−0.994650	−	0.103302i	$-0.967059\pi$
$103$	− 2.09681i	− 0.206605i	0.994650	−	0.103302i	$-0.0329410\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.03841	−0.197061	−0.0985303	−	0.995134i	$-0.531414\pi$
$107$	−2.03841	−0.197061	−0.0985303	+	0.995134i	$0.531414\pi$
$108$	0	0
$109$	−2.32276	−0.222480	−0.111240	−	0.993794i	$-0.535482\pi$
$109$	−2.32276	−0.222480	−0.111240	+	0.993794i	$0.535482\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 5.91983i	− 0.556891i	−0.960452	−	0.278445i	$-0.910181\pi$
$113$	− 5.91983i	− 0.556891i	0.960452	−	0.278445i	$-0.0898191\pi$
$114$	0	0
$115$	− 1.50621i	− 0.140454i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.45315	0.316550
$120$	0	0
$121$	2.84176	0.258342
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 9.73987i	− 0.871160i
$126$	0	0
$127$	− 12.4051i	− 1.10078i	−0.834909	−	0.550388i	$-0.814480\pi$
$127$	− 12.4051i	− 1.10078i	0.834909	−	0.550388i	$-0.185520\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.9820	1.04687	0.523434	−	0.852066i	$-0.324651\pi$
$131$	11.9820	1.04687	0.523434	+	0.852066i	$0.324651\pi$
$132$	0	0
$133$	1.19380	0.103515
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.69469i	− 0.315658i	−0.987466	−	0.157829i	$-0.949550\pi$
$137$	− 3.69469i	− 0.315658i	0.987466	−	0.157829i	$-0.0504495\pi$
$138$	0	0
$139$	0.965743i	0.0819133i	0.999161	+	0.0409566i	$0.0130406\pi$
$139$	0.965743i	0.0819133i	−0.999161	+	0.0409566i	$0.986959\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.422078	0.0352960
$144$	0	0
$145$	0.956077	0.0793978
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.59705i	0.294682i	0.989086	+	0.147341i	$0.0470715\pi$
$149$	3.59705i	0.294682i	−0.989086	+	0.147341i	$0.952929\pi$
$150$	0	0
$151$	− 2.17609i	− 0.177088i	−0.996072	−	0.0885440i	$-0.971779\pi$
$151$	− 2.17609i	− 0.177088i	0.996072	−	0.0885440i	$-0.0282214\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.0150	0.804422
$156$	0	0
$157$	24.4292	1.94966	0.974832	−	0.222943i	$-0.0715663\pi$
$157$	24.4292	1.94966	0.974832	+	0.222943i	$0.0715663\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.61818i	− 0.127530i
$162$	0	0
$163$	18.6476i	1.46059i	0.683129	+	0.730297i	$0.260619\pi$
$163$	18.6476i	1.46059i	−0.683129	+	0.730297i	$0.739381\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.46847	0.732692	0.366346	−	0.930479i	$-0.380609\pi$
$167$	9.46847	0.732692	0.366346	+	0.930479i	$0.380609\pi$
$168$	0	0
$169$	−12.9871	−0.999010
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 8.52499i	− 0.648143i	−0.946033	−	0.324071i	$-0.894948\pi$
$173$	− 8.52499i	− 0.648143i	0.946033	−	0.324071i	$-0.105052\pi$
$174$	0	0
$175$	− 4.49495i	− 0.339786i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−5.16062	−0.385723	−0.192861	−	0.981226i	$-0.561777\pi$
$179$	−5.16062	−0.385723	−0.192861	+	0.981226i	$0.561777\pi$
$180$	0	0
$181$	17.3864	1.29232	0.646160	−	0.763202i	$-0.276374\pi$
$181$	17.3864	1.29232	0.646160	+	0.763202i	$0.276374\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 6.27151i	− 0.461091i
$186$	0	0
$187$	10.7617i	0.786973i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.768626	0.0556159	0.0278079	−	0.999613i	$-0.491147\pi$
$191$	0.768626	0.0556159	0.0278079	+	0.999613i	$0.491147\pi$
$192$	0	0
$193$	−12.1274	−0.872951	−0.436475	−	0.899716i	$-0.643773\pi$
$193$	−12.1274	−0.872951	−0.436475	+	0.899716i	$0.643773\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.95333i	− 0.210416i	−0.994450	−	0.105208i	$-0.966449\pi$
$197$	− 2.95333i	− 0.210416i	0.994450	−	0.105208i	$-0.0335509\pi$
$198$	0	0
$199$	17.3465i	1.22966i	0.788660	+	0.614829i	$0.210775\pi$
$199$	17.3465i	1.22966i	−0.788660	+	0.614829i	$0.789225\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.02715	0.0720920
$204$	0	0
$205$	−11.7034	−0.817400
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.72045i	0.257349i
$210$	0	0
$211$	2.32941i	0.160363i	0.996780	+	0.0801815i	$0.0255500\pi$
$211$	2.32941i	0.160363i	−0.996780	+	0.0801815i	$0.974450\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.74680	0.187331
$216$	0	0
$217$	10.7595	0.730402
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.328157i	0.0220742i
$222$	0	0
$223$	− 10.1883i	− 0.682257i	−0.940017	−	0.341129i	$-0.889191\pi$
$223$	− 10.1883i	− 0.682257i	0.940017	−	0.341129i	$-0.110809\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.2877	0.948305	0.474153	−	0.880443i	$-0.342754\pi$
$227$	14.2877	0.948305	0.474153	+	0.880443i	$0.342754\pi$
$228$	0	0
$229$	4.84008	0.319842	0.159921	−	0.987130i	$-0.448876\pi$
$229$	4.84008	0.319842	0.159921	+	0.987130i	$0.448876\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.5288i	0.951813i	0.879496	+	0.475907i	$0.157880\pi$
$233$	14.5288i	0.951813i	−0.879496	+	0.475907i	$0.842120\pi$
$234$	0	0
$235$	5.61843i	0.366506i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−29.9420	−1.93679	−0.968395	−	0.249422i	$-0.919759\pi$
$239$	−29.9420	−1.93679	−0.968395	+	0.249422i	$0.919759\pi$
$240$	0	0
$241$	−10.0657	−0.648390	−0.324195	−	0.945990i	$-0.605093\pi$
$241$	−10.0657	−0.648390	−0.324195	+	0.945990i	$0.605093\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 6.19472i	− 0.395766i
$246$	0	0
$247$	0.113448i	0.00721853i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	16.6664	1.05197	0.525986	−	0.850493i	$-0.323696\pi$
$251$	16.6664	1.05197	0.525986	+	0.850493i	$0.323696\pi$
$252$	0	0
$253$	5.04303	0.317053
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 21.7438i	− 1.35634i	−0.734906	−	0.678169i	$-0.762774\pi$
$257$	− 21.7438i	− 1.35634i	0.734906	−	0.678169i	$-0.237226\pi$
$258$	0	0
$259$	− 6.73774i	− 0.418663i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−30.8883	−1.90466	−0.952328	−	0.305075i	$-0.901319\pi$
$263$	−30.8883	−1.90466	−0.952328	+	0.305075i	$0.901319\pi$
$264$	0	0
$265$	−10.5690	−0.649251
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.71830i	0.287680i	0.989601	+	0.143840i	$0.0459450\pi$
$269$	4.71830i	0.287680i	−0.989601	+	0.143840i	$0.954055\pi$
$270$	0	0
$271$	4.77033i	0.289777i	0.989448	+	0.144888i	$0.0462823\pi$
$271$	4.77033i	0.289777i	−0.989448	+	0.144888i	$0.953718\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	14.0085	0.844742
$276$	0	0
$277$	−5.73084	−0.344333	−0.172166	−	0.985068i	$-0.555077\pi$
$277$	−5.73084	−0.344333	−0.172166	+	0.985068i	$0.555077\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 12.7428i	− 0.760171i	−0.924951	−	0.380085i	$-0.875895\pi$
$281$	− 12.7428i	− 0.760171i	0.924951	−	0.380085i	$-0.124105\pi$
$282$	0	0
$283$	24.1940i	1.43818i	0.694914	+	0.719092i	$0.255442\pi$
$283$	24.1940i	1.43818i	−0.694914	+	0.719092i	$0.744558\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.5734	−0.742186
$288$	0	0
$289$	8.63300	0.507824
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8.35727i	0.488237i	0.969745	+	0.244118i	$0.0784986\pi$
$293$	8.35727i	0.488237i	−0.969745	+	0.244118i	$0.921501\pi$
$294$	0	0
$295$	− 9.71140i	− 0.565420i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.153778	0.00889319
$300$	0	0
$301$	2.95100	0.170093
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.46925i	0.542208i
$306$	0	0
$307$	− 13.0461i	− 0.744578i	−0.928117	−	0.372289i	$-0.878573\pi$
$307$	− 13.0461i	− 0.744578i	0.928117	−	0.372289i	$-0.121427\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.71584	0.210706	0.105353	−	0.994435i	$-0.466403\pi$
$311$	3.71584	0.210706	0.105353	+	0.994435i	$0.466403\pi$
$312$	0	0
$313$	7.32643	0.414114	0.207057	−	0.978329i	$-0.433611\pi$
$313$	7.32643	0.414114	0.207057	+	0.978329i	$0.433611\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.42123i	0.248321i	0.992262	+	0.124161i	$0.0396238\pi$
$317$	4.42123i	0.248321i	−0.992262	+	0.124161i	$0.960376\pi$
$318$	0	0
$319$	3.20111i	0.179228i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.89258	−0.160947
$324$	0	0
$325$	0.427161	0.0236946
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.03611i	0.332782i
$330$	0	0
$331$	1.97474i	0.108541i	0.998526	+	0.0542707i	$0.0172834\pi$
$331$	1.97474i	0.108541i	−0.998526	+	0.0542707i	$0.982717\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.952379	0.0520340
$336$	0	0
$337$	24.4795	1.33348	0.666741	−	0.745290i	$-0.267689\pi$
$337$	24.4795	1.33348	0.666741	+	0.745290i	$0.267689\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	33.5318i	1.81585i
$342$	0	0
$343$	− 15.0118i	− 0.810562i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.8988	1.01454	0.507271	−	0.861787i	$-0.330654\pi$
$347$	18.8988	1.01454	0.507271	+	0.861787i	$0.330654\pi$
$348$	0	0
$349$	−10.1508	−0.543361	−0.271680	−	0.962388i	$-0.587579\pi$
$349$	−10.1508	−0.543361	−0.271680	+	0.962388i	$0.587579\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.6591i	1.09957i	0.835306	+	0.549785i	$0.185291\pi$
$353$	20.6591i	1.09957i	−0.835306	+	0.549785i	$0.814709\pi$
$354$	0	0
$355$	2.12250i	0.112650i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.15543	0.0609810	0.0304905	−	0.999535i	$-0.490293\pi$
$359$	1.15543	0.0609810	0.0304905	+	0.999535i	$0.490293\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3.40261i	− 0.178101i
$366$	0	0
$367$	− 25.9993i	− 1.35715i	−0.734530	−	0.678576i	$-0.762597\pi$
$367$	− 25.9993i	− 1.35715i	0.734530	−	0.678576i	$-0.237403\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.3547	−0.589509
$372$	0	0
$373$	18.6006	0.963102	0.481551	−	0.876418i	$-0.340074\pi$
$373$	18.6006	0.963102	0.481551	+	0.876418i	$0.340074\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.0976116i	0.00502726i
$378$	0	0
$379$	6.01294i	0.308864i	0.988003	+	0.154432i	$0.0493547\pi$
$379$	6.01294i	0.308864i	−0.988003	+	0.154432i	$0.950645\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.6395	−0.543653	−0.271827	−	0.962346i	$-0.587628\pi$
$383$	−10.6395	−0.543653	−0.271827	+	0.962346i	$0.587628\pi$
$384$	0	0
$385$	−4.93531	−0.251527
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 15.2848i	− 0.774971i	−0.921876	−	0.387485i	$-0.873344\pi$
$389$	− 15.2848i	− 0.774971i	0.921876	−	0.387485i	$-0.126656\pi$
$390$	0	0
$391$	3.92085i	0.198286i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.47541	0.426444
$396$	0	0
$397$	−11.7056	−0.587487	−0.293743	−	0.955884i	$-0.594901\pi$
$397$	−11.7056	−0.587487	−0.293743	+	0.955884i	$0.594901\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.79270i	0.189399i	0.995506	+	0.0946993i	$0.0301890\pi$
$401$	3.79270i	0.189399i	−0.995506	+	0.0946993i	$0.969811\pi$
$402$	0	0
$403$	1.02249i	0.0509338i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.9981	1.04084
$408$	0	0
$409$	0.556303	0.0275074	0.0137537	−	0.999905i	$-0.495622\pi$
$409$	0.556303	0.0275074	0.0137537	+	0.999905i	$0.495622\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 10.4334i	− 0.513392i
$414$	0	0
$415$	− 11.1142i	− 0.545572i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−26.4184	−1.29063	−0.645313	−	0.763918i	$-0.723273\pi$
$419$	−26.4184	−1.29063	−0.645313	+	0.763918i	$0.723273\pi$
$420$	0	0
$421$	−24.0288	−1.17109	−0.585546	−	0.810639i	$-0.699120\pi$
$421$	−24.0288	−1.17109	−0.585546	+	0.810639i	$0.699120\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	10.8913i	0.528305i
$426$	0	0
$427$	10.1732i	0.492316i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−30.3393	−1.46139	−0.730697	−	0.682702i	$-0.760805\pi$
$431$	−30.3393	−1.46139	−0.730697	+	0.682702i	$0.760805\pi$
$432$	0	0
$433$	−15.1246	−0.726843	−0.363421	−	0.931625i	$-0.618391\pi$
$433$	−15.1246	−0.726843	−0.363421	+	0.931625i	$0.618391\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.35549i	0.0648418i
$438$	0	0
$439$	− 28.6582i	− 1.36778i	−0.729585	−	0.683890i	$-0.760287\pi$
$439$	− 28.6582i	− 1.36778i	0.729585	−	0.683890i	$-0.239713\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−36.9951	−1.75769	−0.878845	−	0.477108i	$-0.841685\pi$
$443$	−36.9951	−1.75769	−0.878845	+	0.477108i	$0.841685\pi$
$444$	0	0
$445$	−14.6298	−0.693521
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17.4824i	0.825044i	0.910948	+	0.412522i	$0.135352\pi$
$449$	17.4824i	0.825044i	−0.910948	+	0.412522i	$0.864648\pi$
$450$	0	0
$451$	− 39.1849i	− 1.84515i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.150493	−0.00705522
$456$	0	0
$457$	−33.2001	−1.55303	−0.776517	−	0.630096i	$-0.783015\pi$
$457$	−33.2001	−1.55303	−0.776517	+	0.630096i	$0.783015\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13.2270i	0.616043i	0.951379	+	0.308022i	$0.0996669\pi$
$461$	13.2270i	0.616043i	−0.951379	+	0.308022i	$0.900333\pi$
$462$	0	0
$463$	− 19.7916i	− 0.919794i	−0.887972	−	0.459897i	$-0.847886\pi$
$463$	− 19.7916i	− 0.919794i	0.887972	−	0.459897i	$-0.152114\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.7076	0.588038	0.294019	−	0.955800i	$-0.405007\pi$
$467$	12.7076	0.588038	0.294019	+	0.955800i	$0.405007\pi$
$468$	0	0
$469$	1.02318	0.0472460
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.19677i	0.422868i
$474$	0	0
$475$	3.76526i	0.172762i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6.28722	−0.287271	−0.143635	−	0.989631i	$-0.545879\pi$
$479$	−6.28722	−0.287271	−0.143635	+	0.989631i	$0.545879\pi$
$480$	0	0
$481$	0.640297	0.0291950
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 12.9631i	− 0.588623i
$486$	0	0
$487$	18.6477i	0.845008i	0.906361	+	0.422504i	$0.138849\pi$
$487$	18.6477i	0.845008i	−0.906361	+	0.422504i	$0.861151\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.8097	0.713480	0.356740	−	0.934204i	$-0.383888\pi$
$491$	15.8097	0.713480	0.356740	+	0.934204i	$0.383888\pi$
$492$	0	0
$493$	−2.48880	−0.112090
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.28029i	0.102285i
$498$	0	0
$499$	13.1361i	0.588052i	0.955797	+	0.294026i	$0.0949952\pi$
$499$	13.1361i	0.588052i	−0.955797	+	0.294026i	$0.905005\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.07107	0.136932	0.0684661	−	0.997653i	$-0.478189\pi$
$503$	3.07107	0.136932	0.0684661	+	0.997653i	$0.478189\pi$
$504$	0	0
$505$	−1.82456	−0.0811920
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21.1047i	0.935449i	0.883874	+	0.467725i	$0.154926\pi$
$509$	21.1047i	0.935449i	−0.883874	+	0.467725i	$0.845074\pi$
$510$	0	0
$511$	− 3.65556i	− 0.161713i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.32995	−0.102670
$516$	0	0
$517$	−18.8115	−0.827327
$518$	0	0
$519$	0	0
$520$	0	0
$521$	19.1133i	0.837367i	0.908132	+	0.418684i	$0.137508\pi$
$521$	19.1133i	0.837367i	−0.908132	+	0.418684i	$0.862492\pi$
$522$	0	0
$523$	29.0588i	1.27065i	0.772244	+	0.635326i	$0.219134\pi$
$523$	29.0588i	1.27065i	−0.772244	+	0.635326i	$0.780866\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−26.0703	−1.13564
$528$	0	0
$529$	−21.1627	−0.920115
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1.19487i	− 0.0517556i
$534$	0	0
$535$	2.26506i	0.0979272i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	20.7410	0.893376
$540$	0	0
$541$	−5.90913	−0.254053	−0.127027	−	0.991899i	$-0.540543\pi$
$541$	−5.90913	−0.254053	−0.127027	+	0.991899i	$0.540543\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.58103i	0.110559i
$546$	0	0
$547$	13.7241i	0.586802i	0.955990	+	0.293401i	$0.0947870\pi$
$547$	13.7241i	0.586802i	−0.955990	+	0.293401i	$0.905213\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.860408	−0.0366546
$552$	0	0
$553$	9.10548	0.387204
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.35820i	− 0.311777i	−0.987775	−	0.155889i	$-0.950176\pi$
$557$	− 7.35820i	− 0.311777i	0.987775	−	0.155889i	$-0.0498241\pi$
$558$	0	0
$559$	0.280438i	0.0118613i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.41425	0.396763	0.198382	−	0.980125i	$-0.436431\pi$
$563$	9.41425	0.396763	0.198382	+	0.980125i	$0.436431\pi$
$564$	0	0
$565$	−6.57805	−0.276741
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.6116i	0.612551i	0.951943	+	0.306276i	$0.0990829\pi$
$569$	14.6116i	0.612551i	−0.951943	+	0.306276i	$0.900917\pi$
$570$	0	0
$571$	39.1353i	1.63776i	0.573963	+	0.818881i	$0.305405\pi$
$571$	39.1353i	1.63776i	−0.573963	+	0.818881i	$0.694595\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.10376	0.212842
$576$	0	0
$577$	−1.36547	−0.0568451	−0.0284226	−	0.999596i	$-0.509048\pi$
$577$	−1.36547	−0.0568451	−0.0284226	+	0.999596i	$0.509048\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 11.9404i	− 0.495371i
$582$	0	0
$583$	− 35.3869i	− 1.46558i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.0683	0.910855	0.455427	−	0.890273i	$-0.349486\pi$
$587$	22.0683	0.910855	0.455427	+	0.890273i	$0.349486\pi$
$588$	0	0
$589$	−9.01283	−0.371367
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.0401i	0.494429i	0.968961	+	0.247215i	$0.0795153\pi$
$593$	12.0401i	0.494429i	−0.968961	+	0.247215i	$0.920485\pi$
$594$	0	0
$595$	− 3.83710i	− 0.157306i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−40.0405	−1.63601	−0.818006	−	0.575210i	$-0.804920\pi$
$599$	−40.0405	−1.63601	−0.818006	+	0.575210i	$0.804920\pi$
$600$	0	0
$601$	−32.8278	−1.33907	−0.669537	−	0.742778i	$-0.733508\pi$
$601$	−32.8278	−1.33907	−0.669537	+	0.742778i	$0.733508\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 3.15774i	− 0.128380i
$606$	0	0
$607$	− 4.55151i	− 0.184740i	−0.995725	−	0.0923700i	$-0.970556\pi$
$607$	− 4.55151i	− 0.184740i	0.995725	−	0.0923700i	$-0.0294443\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.573620	−0.0232062
$612$	0	0
$613$	−42.4038	−1.71267	−0.856336	−	0.516419i	$-0.827265\pi$
$613$	−42.4038	−1.71267	−0.856336	+	0.516419i	$0.827265\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.9498i	0.682373i	0.939996	+	0.341187i	$0.110829\pi$
$617$	16.9498i	0.682373i	−0.939996	+	0.341187i	$0.889171\pi$
$618$	0	0
$619$	− 22.7120i	− 0.912872i	−0.889756	−	0.456436i	$-0.849126\pi$
$619$	− 22.7120i	− 0.912872i	0.889756	−	0.456436i	$-0.150874\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−15.7174	−0.629705
$624$	0	0
$625$	8.00344	0.320138
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.3256i	0.650944i
$630$	0	0
$631$	0.732322i	0.0291533i	0.999894	+	0.0145766i	$0.00464005\pi$
$631$	0.732322i	0.0291533i	−0.999894	+	0.0145766i	$0.995360\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.7844	−0.547019
$636$	0	0
$637$	0.632456	0.0250588
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.90566i	0.312255i	0.987737	+	0.156127i	$0.0499010\pi$
$641$	7.90566i	0.312255i	−0.987737	+	0.156127i	$0.950099\pi$
$642$	0	0
$643$	− 36.9435i	− 1.45691i	−0.685095	−	0.728454i	$-0.740239\pi$
$643$	− 36.9435i	− 1.45691i	0.685095	−	0.728454i	$-0.259761\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.47909	−0.136777	−0.0683886	−	0.997659i	$-0.521786\pi$
$647$	−3.47909	−0.136777	−0.0683886	+	0.997659i	$0.521786\pi$
$648$	0	0
$649$	32.5154	1.27634
$650$	0	0
$651$	0	0
$652$	0	0
$653$	37.0587i	1.45022i	0.688634	+	0.725109i	$0.258210\pi$
$653$	37.0587i	1.45022i	−0.688634	+	0.725109i	$0.741790\pi$
$654$	0	0
$655$	− 13.3142i	− 0.520230i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−38.8706	−1.51418	−0.757091	−	0.653309i	$-0.773380\pi$
$659$	−38.8706	−1.51418	−0.757091	+	0.653309i	$0.773380\pi$
$660$	0	0
$661$	17.6229	0.685450	0.342725	−	0.939436i	$-0.388650\pi$
$661$	17.6229	0.685450	0.342725	+	0.939436i	$0.388650\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.32654i	− 0.0514408i
$666$	0	0
$667$	1.16627i	0.0451583i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−31.7047	−1.22394
$672$	0	0
$673$	−20.3151	−0.783090	−0.391545	−	0.920159i	$-0.628059\pi$
$673$	−20.3151	−0.783090	−0.391545	+	0.920159i	$0.628059\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11.8396i	0.455032i	0.973774	+	0.227516i	$0.0730604\pi$
$677$	11.8396i	0.455032i	−0.973774	+	0.227516i	$0.926940\pi$
$678$	0	0
$679$	− 13.9268i	− 0.534460i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	50.6817	1.93928	0.969640	−	0.244535i	$-0.0786355\pi$
$683$	50.6817	1.93928	0.969640	+	0.244535i	$0.0786355\pi$
$684$	0	0
$685$	−4.10550	−0.156863
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1.07906i	− 0.0411088i
$690$	0	0
$691$	32.6962i	1.24382i	0.783088	+	0.621910i	$0.213643\pi$
$691$	32.6962i	1.24382i	−0.783088	+	0.621910i	$0.786357\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.07312	0.0407059
$696$	0	0
$697$	30.4655	1.15396
$698$	0	0
$699$	0	0
$700$	0	0
$701$	50.1158i	1.89285i	0.322924	+	0.946425i	$0.395334\pi$
$701$	50.1158i	1.89285i	−0.322924	+	0.946425i	$0.604666\pi$
$702$	0	0
$703$	5.64396i	0.212866i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.96020	−0.0737210
$708$	0	0
$709$	−12.4838	−0.468840	−0.234420	−	0.972135i	$-0.575319\pi$
$709$	−12.4838	−0.468840	−0.234420	+	0.972135i	$0.575319\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	12.2168i	0.457522i
$714$	0	0
$715$	− 0.469009i	− 0.0175399i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.4623	−0.763116	−0.381558	−	0.924345i	$-0.624612\pi$
$719$	−20.4623	−0.763116	−0.381558	+	0.924345i	$0.624612\pi$
$720$	0	0
$721$	−2.50317	−0.0932228
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.23966i	0.120318i
$726$	0	0
$727$	− 20.7963i	− 0.771292i	−0.922647	−	0.385646i	$-0.873979\pi$
$727$	− 20.7963i	− 0.771292i	0.922647	−	0.385646i	$-0.126021\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7.15030	−0.264463
$732$	0	0
$733$	13.3029	0.491352	0.245676	−	0.969352i	$-0.420990\pi$
$733$	13.3029	0.491352	0.245676	+	0.969352i	$0.420990\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.18873i	0.117458i
$738$	0	0
$739$	6.39272i	0.235160i	0.993063	+	0.117580i	$0.0375136\pi$
$739$	6.39272i	0.235160i	−0.993063	+	0.117580i	$0.962486\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	34.2605	1.25689	0.628447	−	0.777852i	$-0.283691\pi$
$743$	34.2605	1.25689	0.628447	+	0.777852i	$0.283691\pi$
$744$	0	0
$745$	3.99701	0.146439
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.43345i	0.0889163i
$750$	0	0
$751$	− 45.5988i	− 1.66392i	−0.554833	−	0.831962i	$-0.687218\pi$
$751$	− 45.5988i	− 1.66392i	0.554833	−	0.831962i	$-0.312782\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2.41805	−0.0880020
$756$	0	0
$757$	25.1493	0.914066	0.457033	−	0.889450i	$-0.348912\pi$
$757$	25.1493	0.914066	0.457033	+	0.889450i	$0.348912\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 12.3805i	− 0.448793i	−0.974498	−	0.224396i	$-0.927959\pi$
$761$	− 12.3805i	− 0.448793i	0.974498	−	0.224396i	$-0.0720411\pi$
$762$	0	0
$763$	2.77291i	0.100386i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.991496	0.0358008
$768$	0	0
$769$	−37.9127	−1.36717	−0.683584	−	0.729872i	$-0.739580\pi$
$769$	−37.9127	−1.36717	−0.683584	+	0.729872i	$0.739580\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 41.1007i	− 1.47829i	−0.673546	−	0.739145i	$-0.735230\pi$
$773$	− 41.1007i	− 1.47829i	0.673546	−	0.739145i	$-0.264770\pi$
$774$	0	0
$775$	33.9356i	1.21900i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.5323	0.377359
$780$	0	0
$781$	−7.10648	−0.254290
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 27.1455i	− 0.968864i
$786$	0	0
$787$	− 0.408338i	− 0.0145557i	−0.999974	−	0.00727783i	$-0.997683\pi$
$787$	− 0.408338i	− 0.0145557i	0.999974	−	0.00727783i	$-0.00231663\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.06707	−0.251276
$792$	0	0
$793$	−0.966773	−0.0343311
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.1914i	0.502686i	0.967898	+	0.251343i	$0.0808723\pi$
$797$	14.1914i	0.502686i	−0.967898	+	0.251343i	$0.919128\pi$
$798$	0	0
$799$	− 14.6255i	− 0.517414i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.3925	0.402033
$804$	0	0
$805$	−1.79810	−0.0633748
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 15.2844i	− 0.537370i	−0.963228	−	0.268685i	$-0.913411\pi$
$809$	− 15.2844i	− 0.537370i	0.963228	−	0.268685i	$-0.0865890\pi$
$810$	0	0
$811$	− 1.35096i	− 0.0474386i	−0.999719	−	0.0237193i	$-0.992449\pi$
$811$	− 1.35096i	− 0.0474386i	0.999719	−	0.0237193i	$-0.00755080\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20.7211	0.725827
$816$	0	0
$817$	−2.47195	−0.0864825
$818$	0	0
$819$	0	0
$820$	0	0
$821$	45.3093i	1.58131i	0.612264	+	0.790653i	$0.290259\pi$
$821$	45.3093i	1.58131i	−0.612264	+	0.790653i	$0.709741\pi$
$822$	0	0
$823$	− 2.98709i	− 0.104123i	−0.998644	−	0.0520617i	$-0.983421\pi$
$823$	− 2.98709i	− 0.104123i	0.998644	−	0.0520617i	$-0.0165793\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.0319	−0.696579	−0.348289	−	0.937387i	$-0.613237\pi$
$827$	−20.0319	−0.696579	−0.348289	+	0.937387i	$0.613237\pi$
$828$	0	0
$829$	−17.6585	−0.613304	−0.306652	−	0.951822i	$-0.599209\pi$
$829$	−17.6585	−0.613304	−0.306652	+	0.951822i	$0.599209\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	16.1257i	0.558722i
$834$	0	0
$835$	− 10.5213i	− 0.364103i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9.22764	0.318574	0.159287	−	0.987232i	$-0.449081\pi$
$839$	9.22764	0.318574	0.159287	+	0.987232i	$0.449081\pi$
$840$	0	0
$841$	28.2597	0.974472
$842$	0	0
$843$	0	0
$844$	0	0
$845$	14.4312i	0.496447i
$846$	0	0
$847$	− 3.39249i	− 0.116567i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.65033	0.262250
$852$	0	0
$853$	−34.7289	−1.18909	−0.594547	−	0.804061i	$-0.702669\pi$
$853$	−34.7289	−1.18909	−0.594547	+	0.804061i	$0.702669\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 28.3620i	− 0.968828i	−0.874839	−	0.484414i	$-0.839033\pi$
$857$	− 28.3620i	− 0.968828i	0.874839	−	0.484414i	$-0.160967\pi$
$858$	0	0
$859$	13.7817i	0.470226i	0.971968	+	0.235113i	$0.0755460\pi$
$859$	13.7817i	0.470226i	−0.971968	+	0.235113i	$0.924454\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.1403	−1.23023	−0.615114	−	0.788438i	$-0.710890\pi$
$863$	−36.1403	−1.23023	−0.615114	+	0.788438i	$0.710890\pi$
$864$	0	0
$865$	−9.47288	−0.322088
$866$	0	0
$867$	0	0
$868$	0	0
$869$	28.3771i	0.962627i
$870$	0	0
$871$	0.0972341i	0.00329465i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−11.6274	−0.393079
$876$	0	0
$877$	34.2580	1.15681	0.578405	−	0.815750i	$-0.303675\pi$
$877$	34.2580	1.15681	0.578405	+	0.815750i	$0.303675\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 21.4623i	− 0.723085i	−0.932356	−	0.361542i	$-0.882250\pi$
$881$	− 21.4623i	− 0.723085i	0.932356	−	0.361542i	$-0.117750\pi$
$882$	0	0
$883$	5.60029i	0.188465i	0.995550	+	0.0942323i	$0.0300396\pi$
$883$	5.60029i	0.188465i	−0.995550	+	0.0942323i	$0.969960\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38.9684	−1.30843	−0.654216	−	0.756308i	$-0.727001\pi$
$887$	−38.9684	−1.30843	−0.654216	+	0.756308i	$0.727001\pi$
$888$	0	0
$889$	−14.8092	−0.496685
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 5.05623i	− 0.169200i
$894$	0	0
$895$	5.73443i	0.191681i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.75471	−0.258634
$900$	0	0
$901$	27.5126	0.916578
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 19.3196i	− 0.642205i
$906$	0	0
$907$	− 23.8500i	− 0.791927i	−0.918266	−	0.395964i	$-0.870411\pi$
$907$	− 23.8500i	− 0.791927i	0.918266	−	0.395964i	$-0.129589\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.5268	0.415032	0.207516	−	0.978232i	$-0.433462\pi$
$911$	12.5268	0.415032	0.207516	+	0.978232i	$0.433462\pi$
$912$	0	0
$913$	37.2121	1.23154
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 14.3040i	− 0.472360i
$918$	0	0
$919$	22.4838i	0.741671i	0.928699	+	0.370836i	$0.120929\pi$
$919$	22.4838i	0.741671i	−0.928699	+	0.370836i	$0.879071\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−0.216699	−0.00713272
$924$	0	0
$925$	21.2510	0.698727
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34.9715i	1.14738i	0.819073	+	0.573689i	$0.194488\pi$
$929$	34.9715i	1.14738i	−0.819073	+	0.573689i	$0.805512\pi$
$930$	0	0
$931$	5.57485i	0.182708i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11.9583	0.391078
$936$	0	0
$937$	19.0472	0.622245	0.311122	−	0.950370i	$-0.399295\pi$
$937$	19.0472	0.622245	0.311122	+	0.950370i	$0.399295\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 21.1518i	− 0.689527i	−0.938690	−	0.344764i	$-0.887959\pi$
$941$	− 21.1518i	− 0.689527i	0.938690	−	0.344764i	$-0.112041\pi$
$942$	0	0
$943$	− 14.2764i	− 0.464904i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.6426	−0.670793	−0.335396	−	0.942077i	$-0.608870\pi$
$947$	−20.6426	−0.670793	−0.335396	+	0.942077i	$0.608870\pi$
$948$	0	0
$949$	0.347393	0.0112768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	48.3617i	1.56659i	0.621650	+	0.783295i	$0.286463\pi$
$953$	48.3617i	1.56659i	−0.621650	+	0.783295i	$0.713537\pi$
$954$	0	0
$955$	− 0.854090i	− 0.0276377i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.41071	−0.142429
$960$	0	0
$961$	−50.2312	−1.62036
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.4759i	0.433804i
$966$	0	0
$967$	62.1680i	1.99919i	0.0285042	+	0.999594i	$0.490926\pi$
$967$	62.1680i	1.99919i	−0.0285042	+	0.999594i	$0.509074\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−39.5108	−1.26796	−0.633981	−	0.773348i	$-0.718580\pi$
$971$	−39.5108	−1.26796	−0.633981	+	0.773348i	$0.718580\pi$
$972$	0	0
$973$	1.15290	0.0369603
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 36.3792i	− 1.16387i	−0.813234	−	0.581937i	$-0.802295\pi$
$977$	− 36.3792i	− 1.16387i	0.813234	−	0.581937i	$-0.197705\pi$
$978$	0	0
$979$	− 48.9832i	− 1.56551i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−51.8584	−1.65403	−0.827013	−	0.562182i	$-0.809962\pi$
$983$	−51.8584	−1.65403	−0.827013	+	0.562182i	$0.809962\pi$
$984$	0	0
$985$	−3.28171	−0.104564
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.35070i	0.106546i
$990$	0	0
$991$	51.5075i	1.63619i	0.575083	+	0.818095i	$0.304970\pi$
$991$	51.5075i	1.63619i	−0.575083	+	0.818095i	$0.695030\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	19.2752	0.611066
$996$	0	0
$997$	−19.9320	−0.631252	−0.315626	−	0.948884i	$-0.602214\pi$
$997$	−19.9320	−0.631252	−0.315626	+	0.948884i	$0.602214\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.d.b.2015.11	✓	24
3.2	odd	2	inner	2736.2.d.b.2015.13	yes	24
4.3	odd	2	inner	2736.2.d.b.2015.12	yes	24
12.11	even	2	inner	2736.2.d.b.2015.14	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.d.b.2015.11	✓	24	1.1	even	1	trivial
2736.2.d.b.2015.12	yes	24	4.3	odd	2	inner
2736.2.d.b.2015.13	yes	24	3.2	odd	2	inner
2736.2.d.b.2015.14	yes	24	12.11	even	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.11119i q^{5} -1.19380i q^{7} +O(q^{10})\)	\(q-1.11119i q^{5} -1.19380i q^{7} +3.72045 q^{11} +0.113448 q^{13} +2.89258i q^{17} +1.00000i q^{19} +1.35549 q^{23} +3.76526 q^{25} +0.860408i q^{29} +9.01283i q^{31} -1.32654 q^{35} +5.64396 q^{37} -10.5323i q^{41} +2.47195i q^{43} -5.05623 q^{47} +5.57485 q^{49} -9.51146i q^{53} -4.13413i q^{55} +8.73964 q^{59} -8.52172 q^{61} -0.126062i q^{65} +0.857080i q^{67} -1.91011 q^{71} +3.06213 q^{73} -4.44146i q^{77} +7.62732i q^{79} +10.0020 q^{83} +3.21420 q^{85} -13.1659i q^{89} -0.135434i q^{91} +1.11119 q^{95} +11.6659 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 8 q^{25} - 32 q^{37} - 32 q^{49} + 8 q^{73} + 40 q^{85} + 16 q^{97}+O(q^{100}) \) 24 * q + 8 * q^25 - 32 * q^37 - 32 * q^49 + 8 * q^73 + 40 * q^85 + 16 * q^97

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