LMFDB - Embedded newform 2736.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1,2736)]

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([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2736.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.a (trivial)

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:	yes
Analytic conductor:	$21.8470699930$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 684)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	2736.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{5} -3.00000 q^{7} +O(q^{10})$	$q+2.64575 q^{5} -3.00000 q^{7} -2.64575 q^{11} +2.00000 q^{13} -7.93725 q^{17} +1.00000 q^{19} +5.29150 q^{23} +2.00000 q^{25} +5.29150 q^{29} -10.0000 q^{31} -7.93725 q^{35} +4.00000 q^{37} -10.5830 q^{41} -1.00000 q^{43} -2.64575 q^{47} +2.00000 q^{49} -5.29150 q^{53} -7.00000 q^{55} -7.00000 q^{61} +5.29150 q^{65} -12.0000 q^{67} -10.5830 q^{71} -3.00000 q^{73} +7.93725 q^{77} +4.00000 q^{79} +15.8745 q^{83} -21.0000 q^{85} +15.8745 q^{89} -6.00000 q^{91} +2.64575 q^{95} -14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{7}+O(q^{10}) $ 2 * q - 6 * q^7	$ 2 q - 6 q^{7} + 4 q^{13} + 2 q^{19} + 4 q^{25} - 20 q^{31} + 8 q^{37} - 2 q^{43} + 4 q^{49} - 14 q^{55} - 14 q^{61} - 24 q^{67} - 6 q^{73} + 8 q^{79} - 42 q^{85} - 12 q^{91} - 28 q^{97}+O(q^{100}) $ 2 * q - 6 * q^7 + 4 * q^13 + 2 * q^19 + 4 * q^25 - 20 * q^31 + 8 * q^37 - 2 * q^43 + 4 * q^49 - 14 * q^55 - 14 * q^61 - 24 * q^67 - 6 * q^73 + 8 * q^79 - 42 * q^85 - 12 * q^91 - 28 * q^97

Display 100 coefficients 10 coefficients

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
$5$	2.64575	1.18322	0.591608	−	0.806226i	$-0.298493\pi$
$5$	2.64575	1.18322	0.591608	+	0.806226i	$0.298493\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.64575	−0.797724	−0.398862	−	0.917011i	$-0.630595\pi$
$11$	−2.64575	−0.797724	−0.398862	+	0.917011i	$0.630595\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.93725	−1.92507	−0.962533	−	0.271163i	$-0.912592\pi$
$17$	−7.93725	−1.92507	−0.962533	+	0.271163i	$0.912592\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.29150	1.10335	0.551677	−	0.834058i	$-0.313988\pi$
$23$	5.29150	1.10335	0.551677	+	0.834058i	$0.313988\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.29150	0.982607	0.491304	−	0.870988i	$-0.336521\pi$
$29$	5.29150	0.982607	0.491304	+	0.870988i	$0.336521\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−7.93725	−1.34164
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.5830	−1.65279	−0.826394	−	0.563093i	$-0.809611\pi$
$41$	−10.5830	−1.65279	−0.826394	+	0.563093i	$0.809611\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.64575	−0.385922	−0.192961	−	0.981206i	$-0.561809\pi$
$47$	−2.64575	−0.385922	−0.192961	+	0.981206i	$0.561809\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.29150	−0.726844	−0.363422	−	0.931625i	$-0.618392\pi$
$53$	−5.29150	−0.726844	−0.363422	+	0.931625i	$0.618392\pi$
$54$	0	0
$55$	−7.00000	−0.943880
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.29150	0.656330
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.5830	−1.25597	−0.627986	−	0.778225i	$-0.716120\pi$
$71$	−10.5830	−1.25597	−0.627986	+	0.778225i	$0.716120\pi$
$72$	0	0
$73$	−3.00000	−0.351123	−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000	−0.351123	−0.175562	+	0.984468i	$0.556174\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.93725	0.904534
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.8745	1.74245	0.871227	−	0.490881i	$-0.163325\pi$
$83$	15.8745	1.74245	0.871227	+	0.490881i	$0.163325\pi$
$84$	0	0
$85$	−21.0000	−2.27777
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.8745	1.68269	0.841347	−	0.540495i	$-0.181763\pi$
$89$	15.8745	1.68269	0.841347	+	0.540495i	$0.181763\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.64575	0.271448
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.5830	1.05305	0.526524	−	0.850160i	$-0.323495\pi$
$101$	10.5830	1.05305	0.526524	+	0.850160i	$0.323495\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.29150	0.511549	0.255774	−	0.966736i	$-0.417670\pi$
$107$	5.29150	0.511549	0.255774	+	0.966736i	$0.417670\pi$
$108$	0	0
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.29150	−0.497783	−0.248891	−	0.968531i	$-0.580066\pi$
$113$	−5.29150	−0.497783	−0.248891	+	0.968531i	$0.580066\pi$
$114$	0	0
$115$	14.0000	1.30551
$116$	0	0
$117$	0	0
$118$	0	0
$119$	23.8118	2.18282
$120$	0	0
$121$	−4.00000	−0.363636
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.93725	−0.709930
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.93725	0.693481	0.346741	−	0.937961i	$-0.387288\pi$
$131$	7.93725	0.693481	0.346741	+	0.937961i	$0.387288\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.64575	−0.226042	−0.113021	−	0.993593i	$-0.536053\pi$
$137$	−2.64575	−0.226042	−0.113021	+	0.993593i	$0.536053\pi$
$138$	0	0
$139$	9.00000	0.763370	0.381685	−	0.924292i	$-0.375344\pi$
$139$	9.00000	0.763370	0.381685	+	0.924292i	$0.375344\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.29150	−0.442498
$144$	0	0
$145$	14.0000	1.16264
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.93725	−0.650245	−0.325123	−	0.945672i	$-0.605406\pi$
$149$	−7.93725	−0.650245	−0.325123	+	0.945672i	$0.605406\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−26.4575	−2.12512
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−15.8745	−1.25109
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.8745	−1.22841	−0.614203	−	0.789148i	$-0.710522\pi$
$167$	−15.8745	−1.22841	−0.614203	+	0.789148i	$0.710522\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.8745	1.20692	0.603458	−	0.797395i	$-0.293789\pi$
$173$	15.8745	1.20692	0.603458	+	0.797395i	$0.293789\pi$
$174$	0	0
$175$	−6.00000	−0.453557
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.8745	1.18652	0.593258	−	0.805012i	$-0.297841\pi$
$179$	15.8745	1.18652	0.593258	+	0.805012i	$0.297841\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.5830	0.778078
$186$	0	0
$187$	21.0000	1.53567
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.64575	0.191440	0.0957199	−	0.995408i	$-0.469485\pi$
$191$	2.64575	0.191440	0.0957199	+	0.995408i	$0.469485\pi$
$192$	0	0
$193$	−12.0000	−0.863779	−0.431889	−	0.901927i	$-0.642153\pi$
$193$	−12.0000	−0.863779	−0.431889	+	0.901927i	$0.642153\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.5830	0.754008	0.377004	−	0.926212i	$-0.376954\pi$
$197$	10.5830	0.754008	0.377004	+	0.926212i	$0.376954\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−15.8745	−1.11417
$204$	0	0
$205$	−28.0000	−1.95560
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.64575	−0.183010
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.64575	−0.180439
$216$	0	0
$217$	30.0000	2.03653
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−15.8745	−1.06783
$222$	0	0
$223$	28.0000	1.87502	0.937509	−	0.347960i	$-0.113126\pi$
$223$	28.0000	1.87502	0.937509	+	0.347960i	$0.113126\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.4575	1.75605	0.878023	−	0.478618i	$-0.158862\pi$
$227$	26.4575	1.75605	0.878023	+	0.478618i	$0.158862\pi$
$228$	0	0
$229$	−1.00000	−0.0660819	−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000	−0.0660819	−0.0330409	+	0.999454i	$0.510519\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.93725	0.519987	0.259993	−	0.965610i	$-0.416280\pi$
$233$	7.93725	0.519987	0.259993	+	0.965610i	$0.416280\pi$
$234$	0	0
$235$	−7.00000	−0.456630
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−7.93725	−0.513418	−0.256709	−	0.966489i	$-0.582638\pi$
$239$	−7.93725	−0.513418	−0.256709	+	0.966489i	$0.582638\pi$
$240$	0	0
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.29150	0.338062
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−18.5203	−1.16899	−0.584494	−	0.811398i	$-0.698707\pi$
$251$	−18.5203	−1.16899	−0.584494	+	0.811398i	$0.698707\pi$
$252$	0	0
$253$	−14.0000	−0.880172
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.5830	0.660150	0.330075	−	0.943955i	$-0.392926\pi$
$257$	10.5830	0.660150	0.330075	+	0.943955i	$0.392926\pi$
$258$	0	0
$259$	−12.0000	−0.745644
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.93725	−0.489432	−0.244716	−	0.969595i	$-0.578695\pi$
$263$	−7.93725	−0.489432	−0.244716	+	0.969595i	$0.578695\pi$
$264$	0	0
$265$	−14.0000	−0.860013
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−15.8745	−0.967886	−0.483943	−	0.875100i	$-0.660796\pi$
$269$	−15.8745	−0.967886	−0.483943	+	0.875100i	$0.660796\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.29150	−0.319090
$276$	0	0
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−26.4575	−1.57832	−0.789161	−	0.614186i	$-0.789485\pi$
$281$	−26.4575	−1.57832	−0.789161	+	0.614186i	$0.789485\pi$
$282$	0	0
$283$	3.00000	0.178331	0.0891657	−	0.996017i	$-0.471580\pi$
$283$	3.00000	0.178331	0.0891657	+	0.996017i	$0.471580\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	31.7490	1.87409
$288$	0	0
$289$	46.0000	2.70588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10.5830	0.612031
$300$	0	0
$301$	3.00000	0.172917
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−18.5203	−1.06047
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.2288	−0.750134	−0.375067	−	0.926998i	$-0.622380\pi$
$311$	−13.2288	−0.750134	−0.375067	+	0.926998i	$0.622380\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−31.7490	−1.78320	−0.891601	−	0.452822i	$-0.850417\pi$
$317$	−31.7490	−1.78320	−0.891601	+	0.452822i	$0.850417\pi$
$318$	0	0
$319$	−14.0000	−0.783850
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7.93725	−0.441641
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.93725	0.437595
$330$	0	0
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−31.7490	−1.73463
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	26.4575	1.43275
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.2288	−0.710157	−0.355078	−	0.934837i	$-0.615546\pi$
$347$	−13.2288	−0.710157	−0.355078	+	0.934837i	$0.615546\pi$
$348$	0	0
$349$	−35.0000	−1.87351	−0.936754	−	0.349990i	$-0.886185\pi$
$349$	−35.0000	−1.87351	−0.936754	+	0.349990i	$0.886185\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	−28.0000	−1.48609
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.93725	−0.418912	−0.209456	−	0.977818i	$-0.567169\pi$
$359$	−7.93725	−0.418912	−0.209456	+	0.977818i	$0.567169\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.93725	−0.415455
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	15.8745	0.824163
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10.5830	0.545053
$378$	0	0
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	26.4575	1.35192	0.675958	−	0.736940i	$-0.263730\pi$
$383$	26.4575	1.35192	0.675958	+	0.736940i	$0.263730\pi$
$384$	0	0
$385$	21.0000	1.07026
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.93725	0.402435	0.201217	−	0.979547i	$-0.435510\pi$
$389$	7.93725	0.402435	0.201217	+	0.979547i	$0.435510\pi$
$390$	0	0
$391$	−42.0000	−2.12403
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10.5830	0.532489
$396$	0	0
$397$	29.0000	1.45547	0.727734	−	0.685859i	$-0.240573\pi$
$397$	29.0000	1.45547	0.727734	+	0.685859i	$0.240573\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.1660	1.05698	0.528490	−	0.848939i	$-0.322758\pi$
$401$	21.1660	1.05698	0.528490	+	0.848939i	$0.322758\pi$
$402$	0	0
$403$	−20.0000	−0.996271
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.5830	−0.524580
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	42.0000	2.06170
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−15.8745	−0.775520	−0.387760	−	0.921760i	$-0.626751\pi$
$419$	−15.8745	−0.775520	−0.387760	+	0.921760i	$0.626751\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−15.8745	−0.770027
$426$	0	0
$427$	21.0000	1.01626
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.8745	0.764648	0.382324	−	0.924028i	$-0.375124\pi$
$431$	15.8745	0.764648	0.382324	+	0.924028i	$0.375124\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.29150	0.253127
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.5203	0.879924	0.439962	−	0.898016i	$-0.354992\pi$
$443$	18.5203	0.879924	0.439962	+	0.898016i	$0.354992\pi$
$444$	0	0
$445$	42.0000	1.99099
$446$	0	0
$447$	0	0
$448$	0	0
$449$	21.1660	0.998886	0.499443	−	0.866347i	$-0.333538\pi$
$449$	21.1660	0.998886	0.499443	+	0.866347i	$0.333538\pi$
$450$	0	0
$451$	28.0000	1.31847
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−15.8745	−0.744208
$456$	0	0
$457$	11.0000	0.514558	0.257279	−	0.966337i	$-0.417174\pi$
$457$	11.0000	0.514558	0.257279	+	0.966337i	$0.417174\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	34.3948	1.60192	0.800962	−	0.598715i	$-0.204322\pi$
$461$	34.3948	1.60192	0.800962	+	0.598715i	$0.204322\pi$
$462$	0	0
$463$	29.0000	1.34774	0.673872	−	0.738848i	$-0.264630\pi$
$463$	29.0000	1.34774	0.673872	+	0.738848i	$0.264630\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−29.1033	−1.34674	−0.673369	−	0.739306i	$-0.735154\pi$
$467$	−29.1033	−1.34674	−0.673369	+	0.739306i	$0.735154\pi$
$468$	0	0
$469$	36.0000	1.66233
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.64575	0.121652
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	26.4575	1.20887	0.604437	−	0.796653i	$-0.293398\pi$
$479$	26.4575	1.20887	0.604437	+	0.796653i	$0.293398\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−37.0405	−1.68192
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	37.0405	1.67162	0.835808	−	0.549022i	$-0.185000\pi$
$491$	37.0405	1.67162	0.835808	+	0.549022i	$0.185000\pi$
$492$	0	0
$493$	−42.0000	−1.89158
$494$	0	0
$495$	0	0
$496$	0	0
$497$	31.7490	1.42414
$498$	0	0
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−15.8745	−0.707809	−0.353905	−	0.935282i	$-0.615146\pi$
$503$	−15.8745	−0.707809	−0.353905	+	0.935282i	$0.615146\pi$
$504$	0	0
$505$	28.0000	1.24598
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.8745	0.703625	0.351813	−	0.936070i	$-0.385565\pi$
$509$	15.8745	0.703625	0.351813	+	0.936070i	$0.385565\pi$
$510$	0	0
$511$	9.00000	0.398137
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−15.8745	−0.699514
$516$	0	0
$517$	7.00000	0.307860
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.5830	0.463650	0.231825	−	0.972758i	$-0.425530\pi$
$521$	10.5830	0.463650	0.231825	+	0.972758i	$0.425530\pi$
$522$	0	0
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	79.3725	3.45752
$528$	0	0
$529$	5.00000	0.217391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−21.1660	−0.916802
$534$	0	0
$535$	14.0000	0.605273
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.29150	−0.227921
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−31.7490	−1.35998
$546$	0	0
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.29150	0.225426
$552$	0	0
$553$	−12.0000	−0.510292
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−34.3948	−1.45735	−0.728677	−	0.684858i	$-0.759864\pi$
$557$	−34.3948	−1.45735	−0.728677	+	0.684858i	$0.759864\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	0	0
$562$	0	0
$563$	31.7490	1.33806	0.669031	−	0.743235i	$-0.266709\pi$
$563$	31.7490	1.33806	0.669031	+	0.743235i	$0.266709\pi$
$564$	0	0
$565$	−14.0000	−0.588984
$566$	0	0
$567$	0	0
$568$	0	0
$569$	37.0405	1.55282	0.776410	−	0.630229i	$-0.217039\pi$
$569$	37.0405	1.55282	0.776410	+	0.630229i	$0.217039\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	10.5830	0.441342
$576$	0	0
$577$	17.0000	0.707719	0.353860	−	0.935299i	$-0.384869\pi$
$577$	17.0000	0.707719	0.353860	+	0.935299i	$0.384869\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−47.6235	−1.97576
$582$	0	0
$583$	14.0000	0.579821
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−39.6863	−1.63803	−0.819014	−	0.573774i	$-0.805479\pi$
$587$	−39.6863	−1.63803	−0.819014	+	0.573774i	$0.805479\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.1660	0.869184	0.434592	−	0.900627i	$-0.356893\pi$
$593$	21.1660	0.869184	0.434592	+	0.900627i	$0.356893\pi$
$594$	0	0
$595$	63.0000	2.58275
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.7490	−1.29723	−0.648615	−	0.761117i	$-0.724651\pi$
$599$	−31.7490	−1.29723	−0.648615	+	0.761117i	$0.724651\pi$
$600$	0	0
$601$	40.0000	1.63163	0.815817	−	0.578310i	$-0.196288\pi$
$601$	40.0000	1.63163	0.815817	+	0.578310i	$0.196288\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.5830	−0.430260
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.29150	−0.214071
$612$	0	0
$613$	−17.0000	−0.686624	−0.343312	−	0.939222i	$-0.611549\pi$
$613$	−17.0000	−0.686624	−0.343312	+	0.939222i	$0.611549\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.93725	0.319542	0.159771	−	0.987154i	$-0.448924\pi$
$617$	7.93725	0.319542	0.159771	+	0.987154i	$0.448924\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−47.6235	−1.90800
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−31.7490	−1.26592
$630$	0	0
$631$	−23.0000	−0.915616	−0.457808	−	0.889051i	$-0.651365\pi$
$631$	−23.0000	−0.915616	−0.457808	+	0.889051i	$0.651365\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−15.8745	−0.629961
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	47.6235	1.88102	0.940508	−	0.339771i	$-0.110350\pi$
$641$	47.6235	1.88102	0.940508	+	0.339771i	$0.110350\pi$
$642$	0	0
$643$	47.0000	1.85350	0.926750	−	0.375680i	$-0.122591\pi$
$643$	47.0000	1.85350	0.926750	+	0.375680i	$0.122591\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−34.3948	−1.35220	−0.676099	−	0.736811i	$-0.736331\pi$
$647$	−34.3948	−1.35220	−0.676099	+	0.736811i	$0.736331\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	39.6863	1.55304	0.776522	−	0.630090i	$-0.216982\pi$
$653$	39.6863	1.55304	0.776522	+	0.630090i	$0.216982\pi$
$654$	0	0
$655$	21.0000	0.820538
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.29150	0.206128	0.103064	−	0.994675i	$-0.467135\pi$
$659$	5.29150	0.206128	0.103064	+	0.994675i	$0.467135\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.93725	−0.307794
$666$	0	0
$667$	28.0000	1.08416
$668$	0	0
$669$	0	0
$670$	0	0
$671$	18.5203	0.714967
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.29150	0.203369	0.101684	−	0.994817i	$-0.467577\pi$
$677$	5.29150	0.203369	0.101684	+	0.994817i	$0.467577\pi$
$678$	0	0
$679$	42.0000	1.61181
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.4575	1.01237	0.506184	−	0.862425i	$-0.331056\pi$
$683$	26.4575	1.01237	0.506184	+	0.862425i	$0.331056\pi$
$684$	0	0
$685$	−7.00000	−0.267456
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−10.5830	−0.403180
$690$	0	0
$691$	−15.0000	−0.570627	−0.285313	−	0.958434i	$-0.592098\pi$
$691$	−15.0000	−0.570627	−0.285313	+	0.958434i	$0.592098\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	23.8118	0.903232
$696$	0	0
$697$	84.0000	3.18173
$698$	0	0
$699$	0	0
$700$	0	0
$701$	10.5830	0.399715	0.199857	−	0.979825i	$-0.435952\pi$
$701$	10.5830	0.399715	0.199857	+	0.979825i	$0.435952\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−31.7490	−1.19404
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−52.9150	−1.98168
$714$	0	0
$715$	−14.0000	−0.523570
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−13.2288	−0.493349	−0.246675	−	0.969098i	$-0.579338\pi$
$719$	−13.2288	−0.493349	−0.246675	+	0.969098i	$0.579338\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	0	0
$723$	0	0
$724$	0	0
$725$	10.5830	0.393043
$726$	0	0
$727$	5.00000	0.185440	0.0927199	−	0.995692i	$-0.470444\pi$
$727$	5.00000	0.185440	0.0927199	+	0.995692i	$0.470444\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7.93725	0.293570
$732$	0	0
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	31.7490	1.16949
$738$	0	0
$739$	−47.0000	−1.72892	−0.864461	−	0.502699i	$-0.832340\pi$
$739$	−47.0000	−1.72892	−0.864461	+	0.502699i	$0.832340\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.1660	0.776506	0.388253	−	0.921553i	$-0.373079\pi$
$743$	21.1660	0.776506	0.388253	+	0.921553i	$0.373079\pi$
$744$	0	0
$745$	−21.0000	−0.769380
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−15.8745	−0.580042
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	10.5830	0.385155
$756$	0	0
$757$	−23.0000	−0.835949	−0.417975	−	0.908459i	$-0.637260\pi$
$757$	−23.0000	−0.835949	−0.417975	+	0.908459i	$0.637260\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.93725	−0.287725	−0.143863	−	0.989598i	$-0.545952\pi$
$761$	−7.93725	−0.287725	−0.143863	+	0.989598i	$0.545952\pi$
$762$	0	0
$763$	36.0000	1.30329
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−47.0000	−1.69486	−0.847432	−	0.530904i	$-0.821852\pi$
$769$	−47.0000	−1.69486	−0.847432	+	0.530904i	$0.821852\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.7490	1.14193	0.570966	−	0.820973i	$-0.306569\pi$
$773$	31.7490	1.14193	0.570966	+	0.820973i	$0.306569\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10.5830	−0.379176
$780$	0	0
$781$	28.0000	1.00192
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−37.0405	−1.32203
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15.8745	0.564433
$792$	0	0
$793$	−14.0000	−0.497155
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	21.0000	0.742927
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.93725	0.280100
$804$	0	0
$805$	−42.0000	−1.48031
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.93725	0.279059	0.139529	−	0.990218i	$-0.455441\pi$
$809$	7.93725	0.279059	0.139529	+	0.990218i	$0.455441\pi$
$810$	0	0
$811$	42.0000	1.47482	0.737410	−	0.675446i	$-0.236049\pi$
$811$	42.0000	1.47482	0.737410	+	0.675446i	$0.236049\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−21.1660	−0.741413
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.6863	1.38506	0.692530	−	0.721389i	$-0.256496\pi$
$821$	39.6863	1.38506	0.692530	+	0.721389i	$0.256496\pi$
$822$	0	0
$823$	−11.0000	−0.383436	−0.191718	−	0.981450i	$-0.561406\pi$
$823$	−11.0000	−0.383436	−0.191718	+	0.981450i	$0.561406\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	21.1660	0.736014	0.368007	−	0.929823i	$-0.380040\pi$
$827$	21.1660	0.736014	0.368007	+	0.929823i	$0.380040\pi$
$828$	0	0
$829$	24.0000	0.833554	0.416777	−	0.909009i	$-0.363160\pi$
$829$	24.0000	0.833554	0.416777	+	0.909009i	$0.363160\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−15.8745	−0.550019
$834$	0	0
$835$	−42.0000	−1.45347
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.29150	0.182683	0.0913415	−	0.995820i	$-0.470885\pi$
$839$	5.29150	0.182683	0.0913415	+	0.995820i	$0.470885\pi$
$840$	0	0
$841$	−1.00000	−0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−23.8118	−0.819150
$846$	0	0
$847$	12.0000	0.412325
$848$	0	0
$849$	0	0
$850$	0	0
$851$	21.1660	0.725561
$852$	0	0
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31.7490	−1.08453	−0.542263	−	0.840209i	$-0.682432\pi$
$857$	−31.7490	−1.08453	−0.542263	+	0.840209i	$0.682432\pi$
$858$	0	0
$859$	−7.00000	−0.238837	−0.119418	−	0.992844i	$-0.538103\pi$
$859$	−7.00000	−0.238837	−0.119418	+	0.992844i	$0.538103\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	42.0000	1.42804
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−10.5830	−0.359004
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	0	0
$874$	0	0
$875$	23.8118	0.804984
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.93725	0.267413	0.133706	−	0.991021i	$-0.457312\pi$
$881$	7.93725	0.267413	0.133706	+	0.991021i	$0.457312\pi$
$882$	0	0
$883$	41.0000	1.37976	0.689880	−	0.723924i	$-0.257663\pi$
$883$	41.0000	1.37976	0.689880	+	0.723924i	$0.257663\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−21.1660	−0.710685	−0.355343	−	0.934736i	$-0.615636\pi$
$887$	−21.1660	−0.710685	−0.355343	+	0.934736i	$0.615636\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.64575	−0.0885367
$894$	0	0
$895$	42.0000	1.40391
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−52.9150	−1.76481
$900$	0	0
$901$	42.0000	1.39922
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.29150	−0.175895
$906$	0	0
$907$	−2.00000	−0.0664089	−0.0332045	−	0.999449i	$-0.510571\pi$
$907$	−2.00000	−0.0664089	−0.0332045	+	0.999449i	$0.510571\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−47.6235	−1.57784	−0.788919	−	0.614497i	$-0.789359\pi$
$911$	−47.6235	−1.57784	−0.788919	+	0.614497i	$0.789359\pi$
$912$	0	0
$913$	−42.0000	−1.39000
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−23.8118	−0.786334
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−21.1660	−0.696688
$924$	0	0
$925$	8.00000	0.263038
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−52.9150	−1.73609	−0.868043	−	0.496489i	$-0.834622\pi$
$929$	−52.9150	−1.73609	−0.868043	+	0.496489i	$0.834622\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	0	0
$934$	0	0
$935$	55.5608	1.81703
$936$	0	0
$937$	5.00000	0.163343	0.0816714	−	0.996659i	$-0.473974\pi$
$937$	5.00000	0.163343	0.0816714	+	0.996659i	$0.473974\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	37.0405	1.20749	0.603743	−	0.797179i	$-0.293675\pi$
$941$	37.0405	1.20749	0.603743	+	0.797179i	$0.293675\pi$
$942$	0	0
$943$	−56.0000	−1.82361
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−15.8745	−0.515852	−0.257926	−	0.966165i	$-0.583039\pi$
$947$	−15.8745	−0.515852	−0.257926	+	0.966165i	$0.583039\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.5830	0.342817	0.171409	−	0.985200i	$-0.445168\pi$
$953$	10.5830	0.342817	0.171409	+	0.985200i	$0.445168\pi$
$954$	0	0
$955$	7.00000	0.226515
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.93725	0.256307
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−31.7490	−1.02204
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42.3320	−1.35850	−0.679250	−	0.733907i	$-0.737695\pi$
$971$	−42.3320	−1.35850	−0.679250	+	0.733907i	$0.737695\pi$
$972$	0	0
$973$	−27.0000	−0.865580
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−37.0405	−1.18503	−0.592516	−	0.805559i	$-0.701865\pi$
$977$	−37.0405	−1.18503	−0.592516	+	0.805559i	$0.701865\pi$
$978$	0	0
$979$	−42.0000	−1.34233
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−31.7490	−1.01264	−0.506318	−	0.862347i	$-0.668994\pi$
$983$	−31.7490	−1.01264	−0.506318	+	0.862347i	$0.668994\pi$
$984$	0	0
$985$	28.0000	0.892154
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.29150	−0.168260
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	18.5203	0.587132
$996$	0	0
$997$	−17.0000	−0.538395	−0.269198	−	0.963085i	$-0.586759\pi$
$997$	−17.0000	−0.538395	−0.269198	+	0.963085i	$0.586759\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.a.ba.1.2		2
3.2	odd	2	inner	2736.2.a.ba.1.1		2
4.3	odd	2		684.2.a.e.1.2	yes	2
12.11	even	2		684.2.a.e.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.2.a.e.1.1	✓	2	12.11	even	2
684.2.a.e.1.2	yes	2	4.3	odd	2
2736.2.a.ba.1.1		2	3.2	odd	2	inner
2736.2.a.ba.1.2		2	1.1	even	1	trivial

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.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{5} -3.00000 q^{7} +O(q^{10})\)	\(q+2.64575 q^{5} -3.00000 q^{7} -2.64575 q^{11} +2.00000 q^{13} -7.93725 q^{17} +1.00000 q^{19} +5.29150 q^{23} +2.00000 q^{25} +5.29150 q^{29} -10.0000 q^{31} -7.93725 q^{35} +4.00000 q^{37} -10.5830 q^{41} -1.00000 q^{43} -2.64575 q^{47} +2.00000 q^{49} -5.29150 q^{53} -7.00000 q^{55} -7.00000 q^{61} +5.29150 q^{65} -12.0000 q^{67} -10.5830 q^{71} -3.00000 q^{73} +7.93725 q^{77} +4.00000 q^{79} +15.8745 q^{83} -21.0000 q^{85} +15.8745 q^{89} -6.00000 q^{91} +2.64575 q^{95} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{7}+O(q^{10}) \) 2 * q - 6 * q^7	\( 2 q - 6 q^{7} + 4 q^{13} + 2 q^{19} + 4 q^{25} - 20 q^{31} + 8 q^{37} - 2 q^{43} + 4 q^{49} - 14 q^{55} - 14 q^{61} - 24 q^{67} - 6 q^{73} + 8 q^{79} - 42 q^{85} - 12 q^{91} - 28 q^{97}+O(q^{100}) \) 2 * q - 6 * q^7 + 4 * q^13 + 2 * q^19 + 4 * q^25 - 20 * q^31 + 8 * q^37 - 2 * q^43 + 4 * q^49 - 14 * q^55 - 14 * q^61 - 24 * q^67 - 6 * q^73 + 8 * q^79 - 42 * q^85 - 12 * q^91 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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