LMFDB - Embedded newform 8100.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8100,2,Mod(1,8100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8100.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.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:	$64.6788256372$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.3981.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 2x + 1 $ x^4 - x^3 - 4x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 900)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.785261$ of defining polynomial
Character	$\chi$	$=$	8100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.680426 q^{7} +O(q^{10})$	$q-0.680426 q^{7} +1.68043 q^{11} -5.14503 q^{13} -1.31957 q^{17} -0.324642 q^{19} +3.78924 q^{23} +8.64584 q^{29} -4.14503 q^{31} -1.35578 q^{37} +7.15009 q^{41} +7.29005 q^{43} -12.9654 q^{47} -6.53702 q^{49} -8.83052 q^{53} -8.81532 q^{59} +9.97048 q^{61} +4.17617 q^{67} +0.891185 q^{71} +7.82038 q^{73} -1.14341 q^{77} +9.65597 q^{79} +4.85659 q^{83} -17.4764 q^{89} +3.50081 q^{91} -8.93427 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{7}+O(q^{10}) $ 4 * q + q^7	$ 4 q + q^{7} + 3 q^{11} - 2 q^{13} - 9 q^{17} - 4 q^{19} + 3 q^{23} - 9 q^{29} + 2 q^{31} + q^{37} + 9 q^{41} - 8 q^{43} - 12 q^{47} + 9 q^{49} - 12 q^{53} - 15 q^{59} - q^{61} - 11 q^{67} + 12 q^{71} + 10 q^{73} - 36 q^{77} - 7 q^{79} - 12 q^{83} - 3 q^{89} - 11 q^{91} - 5 q^{97}+O(q^{100}) $ 4 * q + q^7 + 3 * q^11 - 2 * q^13 - 9 * q^17 - 4 * q^19 + 3 * q^23 - 9 * q^29 + 2 * q^31 + q^37 + 9 * q^41 - 8 * q^43 - 12 * q^47 + 9 * q^49 - 12 * q^53 - 15 * q^59 - q^61 - 11 * q^67 + 12 * q^71 + 10 * q^73 - 36 * q^77 - 7 * q^79 - 12 * q^83 - 3 * q^89 - 11 * q^91 - 5 * 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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.680426	−0.257177	−0.128588	−	0.991698i	$-0.541045\pi$
$7$	−0.680426	−0.257177	−0.128588	+	0.991698i	$0.541045\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.68043	0.506668	0.253334	−	0.967379i	$-0.418473\pi$
$11$	1.68043	0.506668	0.253334	+	0.967379i	$0.418473\pi$
$12$	0	0
$13$	−5.14503	−1.42697	−0.713487	−	0.700669i	$-0.752885\pi$
$13$	−5.14503	−1.42697	−0.713487	+	0.700669i	$0.752885\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.31957	−0.320044	−0.160022	−	0.987113i	$-0.551156\pi$
$17$	−1.31957	−0.320044	−0.160022	+	0.987113i	$0.551156\pi$
$18$	0	0
$19$	−0.324642	−0.0744779	−0.0372390	−	0.999306i	$-0.511856\pi$
$19$	−0.324642	−0.0744779	−0.0372390	+	0.999306i	$0.511856\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.78924	0.790111	0.395056	−	0.918657i	$-0.370725\pi$
$23$	3.78924	0.790111	0.395056	+	0.918657i	$0.370725\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.64584	1.60549	0.802746	−	0.596322i	$-0.203372\pi$
$29$	8.64584	1.60549	0.802746	+	0.596322i	$0.203372\pi$
$30$	0	0
$31$	−4.14503	−0.744469	−0.372234	−	0.928139i	$-0.621408\pi$
$31$	−4.14503	−0.744469	−0.372234	+	0.928139i	$0.621408\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.35578	−0.222890	−0.111445	−	0.993771i	$-0.535548\pi$
$37$	−1.35578	−0.222890	−0.111445	+	0.993771i	$0.535548\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.15009	1.11666	0.558328	−	0.829620i	$-0.311443\pi$
$41$	7.15009	1.11666	0.558328	+	0.829620i	$0.311443\pi$
$42$	0	0
$43$	7.29005	1.11172	0.555861	−	0.831275i	$-0.312389\pi$
$43$	7.29005	1.11172	0.555861	+	0.831275i	$0.312389\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.9654	−1.89120	−0.945600	−	0.325333i	$-0.894524\pi$
$47$	−12.9654	−1.89120	−0.945600	+	0.325333i	$0.894524\pi$
$48$	0	0
$49$	−6.53702	−0.933860
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.83052	−1.21297	−0.606483	−	0.795097i	$-0.707420\pi$
$53$	−8.83052	−1.21297	−0.606483	+	0.795097i	$0.707420\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.81532	−1.14766	−0.573828	−	0.818976i	$-0.694542\pi$
$59$	−8.81532	−1.14766	−0.573828	+	0.818976i	$0.694542\pi$
$60$	0	0
$61$	9.97048	1.27659	0.638294	−	0.769792i	$-0.279640\pi$
$61$	9.97048	1.27659	0.638294	+	0.769792i	$0.279640\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.17617	0.510200	0.255100	−	0.966915i	$-0.417892\pi$
$67$	4.17617	0.510200	0.255100	+	0.966915i	$0.417892\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.891185	0.105764	0.0528821	−	0.998601i	$-0.483159\pi$
$71$	0.891185	0.105764	0.0528821	+	0.998601i	$0.483159\pi$
$72$	0	0
$73$	7.82038	0.915307	0.457653	−	0.889131i	$-0.348690\pi$
$73$	7.82038	0.915307	0.457653	+	0.889131i	$0.348690\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.14341	−0.130303
$78$	0	0
$79$	9.65597	1.08638	0.543191	−	0.839609i	$-0.317216\pi$
$79$	9.65597	1.08638	0.543191	+	0.839609i	$0.317216\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.85659	0.533080	0.266540	−	0.963824i	$-0.414119\pi$
$83$	4.85659	0.533080	0.266540	+	0.963824i	$0.414119\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−17.4764	−1.85249	−0.926245	−	0.376922i	$-0.876982\pi$
$89$	−17.4764	−1.85249	−0.926245	+	0.376922i	$0.876982\pi$
$90$	0	0
$91$	3.50081	0.366985
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.93427	−0.907137	−0.453569	−	0.891221i	$-0.649849\pi$
$97$	−8.93427	−0.907137	−0.453569	+	0.891221i	$0.649849\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.8566	1.08027	0.540136	−	0.841578i	$-0.318373\pi$
$101$	10.8566	1.08027	0.540136	+	0.841578i	$0.318373\pi$
$102$	0	0
$103$	−11.6820	−1.15107	−0.575533	−	0.817778i	$-0.695205\pi$
$103$	−11.6820	−1.15107	−0.575533	+	0.817778i	$0.695205\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.11550	0.494534	0.247267	−	0.968947i	$-0.420467\pi$
$107$	5.11550	0.494534	0.247267	+	0.968947i	$0.420467\pi$
$108$	0	0
$109$	7.17110	0.686867	0.343433	−	0.939177i	$-0.388410\pi$
$109$	7.17110	0.686867	0.343433	+	0.939177i	$0.388410\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.6458	−1.65998	−0.829990	−	0.557778i	$-0.811654\pi$
$113$	−17.6458	−1.65998	−0.829990	+	0.557778i	$0.811654\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.897873	0.0823078
$120$	0	0
$121$	−8.17617	−0.743288
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.13489	−0.278176	−0.139088	−	0.990280i	$-0.544417\pi$
$127$	−3.13489	−0.278176	−0.139088	+	0.990280i	$0.544417\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.11550	−0.709055	−0.354527	−	0.935046i	$-0.615358\pi$
$131$	−8.11550	−0.709055	−0.354527	+	0.935046i	$0.615358\pi$
$132$	0	0
$133$	0.220895	0.0191540
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.89119	−0.588754	−0.294377	−	0.955689i	$-0.595112\pi$
$137$	−6.89119	−0.588754	−0.294377	+	0.955689i	$0.595112\pi$
$138$	0	0
$139$	−20.2555	−1.71805	−0.859023	−	0.511937i	$-0.828928\pi$
$139$	−20.2555	−1.71805	−0.859023	+	0.511937i	$0.828928\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.64584	−0.723001
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.0674	−0.824750	−0.412375	−	0.911014i	$-0.635301\pi$
$149$	−10.0674	−0.824750	−0.412375	+	0.911014i	$0.635301\pi$
$150$	0	0
$151$	−18.7909	−1.52918	−0.764589	−	0.644518i	$-0.777058\pi$
$151$	−18.7909	−1.52918	−0.764589	+	0.644518i	$0.777058\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	19.4814	1.55479	0.777393	−	0.629015i	$-0.216541\pi$
$157$	19.4814	1.55479	0.777393	+	0.629015i	$0.216541\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.57830	−0.203198
$162$	0	0
$163$	−12.2640	−0.960589	−0.480294	−	0.877107i	$-0.659470\pi$
$163$	−12.2640	−0.960589	−0.480294	+	0.877107i	$0.659470\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.5109	−1.04551	−0.522754	−	0.852483i	$-0.675095\pi$
$167$	−13.5109	−1.04551	−0.522754	+	0.852483i	$0.675095\pi$
$168$	0	0
$169$	13.4713	1.03625
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.75465	0.133404	0.0667018	−	0.997773i	$-0.478752\pi$
$173$	1.75465	0.133404	0.0667018	+	0.997773i	$0.478752\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−22.9393	−1.71457	−0.857283	−	0.514845i	$-0.827849\pi$
$179$	−22.9393	−1.71457	−0.857283	+	0.514845i	$0.827849\pi$
$180$	0	0
$181$	−12.4452	−0.925045	−0.462523	−	0.886607i	$-0.653056\pi$
$181$	−12.4452	−0.925045	−0.462523	+	0.886607i	$0.653056\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.21745	−0.162156
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.5785	1.19958	0.599788	−	0.800159i	$-0.295251\pi$
$191$	16.5785	1.19958	0.599788	+	0.800159i	$0.295251\pi$
$192$	0	0
$193$	15.6147	1.12397	0.561985	−	0.827147i	$-0.310038\pi$
$193$	15.6147	1.12397	0.561985	+	0.827147i	$0.310038\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.8153	−1.26929	−0.634644	−	0.772804i	$-0.718853\pi$
$197$	−17.8153	−1.26929	−0.634644	+	0.772804i	$0.718853\pi$
$198$	0	0
$199$	8.07749	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$199$	8.07749	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.88285	−0.412895
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.545537	−0.0377356
$210$	0	0
$211$	14.3887	0.990561	0.495281	−	0.868733i	$-0.335065\pi$
$211$	14.3887	0.990561	0.495281	+	0.868733i	$0.335065\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.82038	0.191460
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.78924	0.456694
$222$	0	0
$223$	2.03621	0.136355	0.0681774	−	0.997673i	$-0.478282\pi$
$223$	2.03621	0.136355	0.0681774	+	0.997673i	$0.478282\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−27.3675	−1.81645	−0.908224	−	0.418485i	$-0.862561\pi$
$227$	−27.3675	−1.81645	−0.908224	+	0.418485i	$0.862561\pi$
$228$	0	0
$229$	−9.43001	−0.623152	−0.311576	−	0.950221i	$-0.600857\pi$
$229$	−9.43001	−0.623152	−0.311576	+	0.950221i	$0.600857\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.191372	0.0125372	0.00626859	−	0.999980i	$-0.498005\pi$
$233$	0.191372	0.0125372	0.00626859	+	0.999980i	$0.498005\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.21076	−0.143002	−0.0715011	−	0.997441i	$-0.522779\pi$
$239$	−2.21076	−0.143002	−0.0715011	+	0.997441i	$0.522779\pi$
$240$	0	0
$241$	22.5075	1.44984	0.724918	−	0.688836i	$-0.241878\pi$
$241$	22.5075	1.44984	0.724918	+	0.688836i	$0.241878\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.67029	0.106278
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.9654	1.57580	0.787901	−	0.615802i	$-0.211168\pi$
$251$	24.9654	1.57580	0.787901	+	0.615802i	$0.211168\pi$
$252$	0	0
$253$	6.36754	0.400324
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.3196	−1.01799	−0.508994	−	0.860770i	$-0.669982\pi$
$257$	−16.3196	−1.01799	−0.508994	+	0.860770i	$0.669982\pi$
$258$	0	0
$259$	0.922511	0.0573221
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.6871	1.02897	0.514486	−	0.857499i	$-0.327983\pi$
$263$	16.6871	1.02897	0.514486	+	0.857499i	$0.327983\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.36085	−0.204915	−0.102457	−	0.994737i	$-0.532671\pi$
$269$	−3.36085	−0.204915	−0.102457	+	0.994737i	$0.532671\pi$
$270$	0	0
$271$	20.0498	1.21794	0.608968	−	0.793195i	$-0.291584\pi$
$271$	20.0498	1.21794	0.608968	+	0.793195i	$0.291584\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.21238	0.493434	0.246717	−	0.969088i	$-0.420648\pi$
$277$	8.21238	0.493434	0.246717	+	0.969088i	$0.420648\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−0.376056	−0.0224336	−0.0112168	−	0.999937i	$-0.503570\pi$
$281$	−0.376056	−0.0224336	−0.0112168	+	0.999937i	$0.503570\pi$
$282$	0	0
$283$	−0.704881	−0.0419008	−0.0209504	−	0.999781i	$-0.506669\pi$
$283$	−0.704881	−0.0419008	−0.0209504	+	0.999781i	$0.506669\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.86511	−0.287178
$288$	0	0
$289$	−15.2587	−0.897572
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−18.1914	−1.06275	−0.531376	−	0.847136i	$-0.678325\pi$
$293$	−18.1914	−1.06275	−0.531376	+	0.847136i	$0.678325\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−19.4957	−1.12747
$300$	0	0
$301$	−4.96034	−0.285909
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−4.04128	−0.230648	−0.115324	−	0.993328i	$-0.536791\pi$
$307$	−4.04128	−0.230648	−0.115324	+	0.993328i	$0.536791\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.16948	0.519954	0.259977	−	0.965615i	$-0.416285\pi$
$311$	9.16948	0.519954	0.259977	+	0.965615i	$0.416285\pi$
$312$	0	0
$313$	−17.2225	−0.973474	−0.486737	−	0.873549i	$-0.661813\pi$
$313$	−17.2225	−0.973474	−0.486737	+	0.873549i	$0.661813\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.46298	−0.475328	−0.237664	−	0.971347i	$-0.576382\pi$
$317$	−8.46298	−0.475328	−0.237664	+	0.971347i	$0.576382\pi$
$318$	0	0
$319$	14.5287	0.813450
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.428389	0.0238362
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.82200	0.486373
$330$	0	0
$331$	−7.03783	−0.386834	−0.193417	−	0.981117i	$-0.561957\pi$
$331$	−7.03783	−0.386834	−0.193417	+	0.981117i	$0.561957\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−24.5488	−1.33726	−0.668629	−	0.743597i	$-0.733118\pi$
$337$	−24.5488	−1.33726	−0.668629	+	0.743597i	$0.733118\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.96541	−0.377198
$342$	0	0
$343$	9.21094	0.497344
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−25.3263	−1.35958	−0.679792	−	0.733405i	$-0.737930\pi$
$347$	−25.3263	−1.35958	−0.679792	+	0.733405i	$0.737930\pi$
$348$	0	0
$349$	−10.8879	−0.582817	−0.291409	−	0.956599i	$-0.594124\pi$
$349$	−10.8879	−0.582817	−0.291409	+	0.956599i	$0.594124\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−26.4612	−1.40838	−0.704192	−	0.710009i	$-0.748691\pi$
$353$	−26.4612	−1.40838	−0.704192	+	0.710009i	$0.748691\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.7007	1.30365	0.651826	−	0.758369i	$-0.274003\pi$
$359$	24.7007	1.30365	0.651826	+	0.758369i	$0.274003\pi$
$360$	0	0
$361$	−18.8946	−0.994453
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−15.1349	−0.790035	−0.395017	−	0.918674i	$-0.629261\pi$
$367$	−15.1349	−0.790035	−0.395017	+	0.918674i	$0.629261\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.00852	0.311947
$372$	0	0
$373$	−8.94947	−0.463386	−0.231693	−	0.972789i	$-0.574426\pi$
$373$	−8.94947	−0.463386	−0.231693	+	0.972789i	$0.574426\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−44.4830	−2.29099
$378$	0	0
$379$	7.22270	0.371005	0.185503	−	0.982644i	$-0.440609\pi$
$379$	7.22270	0.371005	0.185503	+	0.982644i	$0.440609\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	35.7267	1.82555	0.912776	−	0.408461i	$-0.133934\pi$
$383$	35.7267	1.82555	0.912776	+	0.408461i	$0.133934\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−0.706501	−0.0358210	−0.0179105	−	0.999840i	$-0.505701\pi$
$389$	−0.706501	−0.0358210	−0.0179105	+	0.999840i	$0.505701\pi$
$390$	0	0
$391$	−5.00018	−0.252870
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.44014	−0.323222	−0.161611	−	0.986855i	$-0.551669\pi$
$397$	−6.44014	−0.323222	−0.161611	+	0.986855i	$0.551669\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.7201	1.33434	0.667168	−	0.744907i	$-0.267506\pi$
$401$	26.7201	1.33434	0.667168	+	0.744907i	$0.267506\pi$
$402$	0	0
$403$	21.3263	1.06234
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.27830	−0.112931
$408$	0	0
$409$	24.0927	1.19131	0.595653	−	0.803242i	$-0.296893\pi$
$409$	24.0927	1.19131	0.595653	+	0.803242i	$0.296893\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.99817	0.295151
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−23.9133	−1.16824	−0.584120	−	0.811668i	$-0.698560\pi$
$419$	−23.9133	−1.16824	−0.584120	+	0.811668i	$0.698560\pi$
$420$	0	0
$421$	−9.50933	−0.463456	−0.231728	−	0.972781i	$-0.574438\pi$
$421$	−9.50933	−0.463456	−0.231728	+	0.972781i	$0.574438\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.78417	−0.328309
$428$	0	0
$429$	0	0
$430$	0	0
$431$	29.8287	1.43680	0.718399	−	0.695632i	$-0.244875\pi$
$431$	29.8287	1.43680	0.718399	+	0.695632i	$0.244875\pi$
$432$	0	0
$433$	−14.2385	−0.684256	−0.342128	−	0.939653i	$-0.611148\pi$
$433$	−14.2385	−0.684256	−0.342128	+	0.939653i	$0.611148\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.23015	−0.0588459
$438$	0	0
$439$	−32.9527	−1.57275	−0.786374	−	0.617751i	$-0.788044\pi$
$439$	−32.9527	−1.57275	−0.786374	+	0.617751i	$0.788044\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−34.4224	−1.63546	−0.817728	−	0.575604i	$-0.804767\pi$
$443$	−34.4224	−1.63546	−0.817728	+	0.575604i	$0.804767\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.91326	0.137485	0.0687426	−	0.997634i	$-0.478101\pi$
$449$	2.91326	0.137485	0.0687426	+	0.997634i	$0.478101\pi$
$450$	0	0
$451$	12.0152	0.565774
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.8289	1.11467	0.557334	−	0.830288i	$-0.311824\pi$
$457$	23.8289	1.11467	0.557334	+	0.830288i	$0.311824\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.32644	−0.294652	−0.147326	−	0.989088i	$-0.547067\pi$
$461$	−6.32644	−0.294652	−0.147326	+	0.989088i	$0.547067\pi$
$462$	0	0
$463$	4.31777	0.200664	0.100332	−	0.994954i	$-0.468010\pi$
$463$	4.31777	0.200664	0.100332	+	0.994954i	$0.468010\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.352336	0.0163042	0.00815208	−	0.999967i	$-0.497405\pi$
$467$	0.352336	0.0163042	0.00815208	+	0.999967i	$0.497405\pi$
$468$	0	0
$469$	−2.84157	−0.131212
$470$	0	0
$471$	0	0
$472$	0	0
$473$	12.2504	0.563274
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−28.0740	−1.28274	−0.641368	−	0.767234i	$-0.721633\pi$
$479$	−28.0740	−1.28274	−0.641368	+	0.767234i	$0.721633\pi$
$480$	0	0
$481$	6.97554	0.318057
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	27.9426	1.26620	0.633099	−	0.774070i	$-0.281782\pi$
$487$	27.9426	1.26620	0.633099	+	0.774070i	$0.281782\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.5303	0.565486	0.282743	−	0.959196i	$-0.408756\pi$
$491$	12.5303	0.565486	0.282743	+	0.959196i	$0.408756\pi$
$492$	0	0
$493$	−11.4088	−0.513827
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.606386	−0.0272001
$498$	0	0
$499$	42.8700	1.91912	0.959562	−	0.281498i	$-0.0908314\pi$
$499$	42.8700	1.91912	0.959562	+	0.281498i	$0.0908314\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.8635	−0.573554	−0.286777	−	0.957997i	$-0.592584\pi$
$503$	−12.8635	−0.573554	−0.286777	+	0.957997i	$0.592584\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13.6527	−0.605146	−0.302573	−	0.953126i	$-0.597846\pi$
$509$	−13.6527	−0.605146	−0.302573	+	0.953126i	$0.597846\pi$
$510$	0	0
$511$	−5.32119	−0.235396
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−21.7874	−0.958209
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−32.8962	−1.44121	−0.720605	−	0.693346i	$-0.756136\pi$
$521$	−32.8962	−1.44121	−0.720605	+	0.693346i	$0.756136\pi$
$522$	0	0
$523$	−18.0641	−0.789887	−0.394944	−	0.918705i	$-0.629236\pi$
$523$	−18.0641	−0.789887	−0.394944	+	0.918705i	$0.629236\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.46967	0.238262
$528$	0	0
$529$	−8.64165	−0.375724
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−36.7874	−1.59344
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−10.9850	−0.473157
$540$	0	0
$541$	−2.25204	−0.0968226	−0.0484113	−	0.998827i	$-0.515416\pi$
$541$	−2.25204	−0.0968226	−0.0484113	+	0.998827i	$0.515416\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.79431	0.376018	0.188009	−	0.982167i	$-0.439797\pi$
$547$	8.79431	0.376018	0.188009	+	0.982167i	$0.439797\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.80680	−0.119574
$552$	0	0
$553$	−6.57018	−0.279392
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.4199	−1.03470	−0.517352	−	0.855773i	$-0.673082\pi$
$557$	−24.4199	−1.03470	−0.517352	+	0.855773i	$0.673082\pi$
$558$	0	0
$559$	−37.5075	−1.58640
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.1568	−0.428057	−0.214029	−	0.976827i	$-0.568659\pi$
$563$	−10.1568	−0.428057	−0.214029	+	0.976827i	$0.568659\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−45.7074	−1.91615	−0.958076	−	0.286514i	$-0.907504\pi$
$569$	−45.7074	−1.91615	−0.958076	+	0.286514i	$0.907504\pi$
$570$	0	0
$571$	10.7326	0.449144	0.224572	−	0.974457i	$-0.427902\pi$
$571$	10.7326	0.449144	0.224572	+	0.974457i	$0.427902\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−15.9857	−0.665493	−0.332746	−	0.943016i	$-0.607975\pi$
$577$	−15.9857	−0.665493	−0.332746	+	0.943016i	$0.607975\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.30455	−0.137096
$582$	0	0
$583$	−14.8390	−0.614570
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.3465	1.50018	0.750090	−	0.661335i	$-0.230010\pi$
$587$	36.3465	1.50018	0.750090	+	0.661335i	$0.230010\pi$
$588$	0	0
$589$	1.34565	0.0554465
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.27142	−0.380732	−0.190366	−	0.981713i	$-0.560967\pi$
$593$	−9.27142	−0.380732	−0.190366	+	0.981713i	$0.560967\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16.2328	−0.663256	−0.331628	−	0.943410i	$-0.607598\pi$
$599$	−16.2328	−0.663256	−0.331628	+	0.943410i	$0.607598\pi$
$600$	0	0
$601$	15.3885	0.627712	0.313856	−	0.949471i	$-0.398379\pi$
$601$	15.3885	0.627712	0.313856	+	0.949471i	$0.398379\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−9.64077	−0.391307	−0.195653	−	0.980673i	$-0.562683\pi$
$607$	−9.64077	−0.391307	−0.195653	+	0.980673i	$0.562683\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	66.7074	2.69869
$612$	0	0
$613$	40.2967	1.62757	0.813785	−	0.581166i	$-0.197403\pi$
$613$	40.2967	1.62757	0.813785	+	0.581166i	$0.197403\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15.2696	−0.614731	−0.307365	−	0.951592i	$-0.599447\pi$
$617$	−15.2696	−0.614731	−0.307365	+	0.951592i	$0.599447\pi$
$618$	0	0
$619$	−26.0085	−1.04537	−0.522685	−	0.852526i	$-0.675070\pi$
$619$	−26.0085	−1.04537	−0.522685	+	0.852526i	$0.675070\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11.8914	0.476418
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.78906	0.0713344
$630$	0	0
$631$	−4.84484	−0.192870	−0.0964350	−	0.995339i	$-0.530744\pi$
$631$	−4.84484	−0.192870	−0.0964350	+	0.995339i	$0.530744\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	33.6331	1.33259
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13.3263	−0.526356	−0.263178	−	0.964747i	$-0.584771\pi$
$641$	−13.3263	−0.526356	−0.263178	+	0.964747i	$0.584771\pi$
$642$	0	0
$643$	36.0202	1.42050	0.710250	−	0.703950i	$-0.248582\pi$
$643$	36.0202	1.42050	0.710250	+	0.703950i	$0.248582\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.82219	−0.189580	−0.0947899	−	0.995497i	$-0.530218\pi$
$647$	−4.82219	−0.189580	−0.0947899	+	0.995497i	$0.530218\pi$
$648$	0	0
$649$	−14.8135	−0.581480
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−42.2723	−1.65424	−0.827121	−	0.562024i	$-0.810023\pi$
$653$	−42.2723	−1.65424	−0.827121	+	0.562024i	$0.810023\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−37.5091	−1.46115	−0.730574	−	0.682834i	$-0.760747\pi$
$659$	−37.5091	−1.46115	−0.730574	+	0.682834i	$0.760747\pi$
$660$	0	0
$661$	−6.35905	−0.247338	−0.123669	−	0.992324i	$-0.539466\pi$
$661$	−6.35905	−0.247338	−0.123669	+	0.992324i	$0.539466\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	32.7612	1.26852
$668$	0	0
$669$	0	0
$670$	0	0
$671$	16.7547	0.646806
$672$	0	0
$673$	−0.899676	−0.0346799	−0.0173400	−	0.999850i	$-0.505520\pi$
$673$	−0.899676	−0.0346799	−0.0173400	+	0.999850i	$0.505520\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−32.0288	−1.23097	−0.615483	−	0.788150i	$-0.711039\pi$
$677$	−32.0288	−1.23097	−0.615483	+	0.788150i	$0.711039\pi$
$678$	0	0
$679$	6.07911	0.233295
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14.7371	0.563899	0.281950	−	0.959429i	$-0.409019\pi$
$683$	14.7371	0.563899	0.281950	+	0.959429i	$0.409019\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	45.4332	1.73087
$690$	0	0
$691$	−22.5437	−0.857603	−0.428802	−	0.903399i	$-0.641064\pi$
$691$	−22.5437	−0.857603	−0.428802	+	0.903399i	$0.641064\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−9.43508	−0.357379
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−19.9984	−0.755327	−0.377664	−	0.925943i	$-0.623272\pi$
$701$	−19.9984	−0.755327	−0.377664	+	0.925943i	$0.623272\pi$
$702$	0	0
$703$	0.440144	0.0166004
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7.38711	−0.277821
$708$	0	0
$709$	15.0170	0.563976	0.281988	−	0.959418i	$-0.409006\pi$
$709$	15.0170	0.563976	0.281988	+	0.959418i	$0.409006\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−15.7065	−0.588213
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	25.8289	0.963254	0.481627	−	0.876376i	$-0.340046\pi$
$719$	25.8289	0.963254	0.481627	+	0.876376i	$0.340046\pi$
$720$	0	0
$721$	7.94877	0.296028
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−10.6266	−0.394120	−0.197060	−	0.980391i	$-0.563139\pi$
$727$	−10.6266	−0.394120	−0.197060	+	0.980391i	$0.563139\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9.61976	−0.355800
$732$	0	0
$733$	41.7613	1.54249	0.771245	−	0.636538i	$-0.219634\pi$
$733$	41.7613	1.54249	0.771245	+	0.636538i	$0.219634\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.01774	0.258502
$738$	0	0
$739$	49.1030	1.80628	0.903141	−	0.429343i	$-0.141255\pi$
$739$	49.1030	1.80628	0.903141	+	0.429343i	$0.141255\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	19.3111	0.708454	0.354227	−	0.935159i	$-0.384744\pi$
$743$	19.3111	0.708454	0.354227	+	0.935159i	$0.384744\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.48072	−0.127183
$750$	0	0
$751$	8.31451	0.303401	0.151700	−	0.988427i	$-0.451525\pi$
$751$	8.31451	0.303401	0.151700	+	0.988427i	$0.451525\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−33.8667	−1.23091	−0.615453	−	0.788173i	$-0.711027\pi$
$757$	−33.8667	−1.23091	−0.615453	+	0.788173i	$0.711027\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19.0481	−0.690495	−0.345247	−	0.938512i	$-0.612205\pi$
$761$	−19.0481	−0.690495	−0.345247	+	0.938512i	$0.612205\pi$
$762$	0	0
$763$	−4.87940	−0.176646
$764$	0	0
$765$	0	0
$766$	0	0
$767$	45.3550	1.63767
$768$	0	0
$769$	33.6627	1.21391	0.606953	−	0.794738i	$-0.292392\pi$
$769$	33.6627	1.21391	0.606953	+	0.794738i	$0.292392\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	23.5354	0.846509	0.423254	−	0.906011i	$-0.360888\pi$
$773$	23.5354	0.846509	0.423254	+	0.906011i	$0.360888\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.32122	−0.0831663
$780$	0	0
$781$	1.49757	0.0535873
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	23.5321	0.838830	0.419415	−	0.907795i	$-0.362235\pi$
$787$	23.5321	0.838830	0.419415	+	0.907795i	$0.362235\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.0067	0.426909
$792$	0	0
$793$	−51.2984	−1.82166
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.53052	−0.0896355	−0.0448177	−	0.998995i	$-0.514271\pi$
$797$	−2.53052	−0.0896355	−0.0448177	+	0.998995i	$0.514271\pi$
$798$	0	0
$799$	17.1088	0.605266
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13.1416	0.463756
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.5178	0.756526	0.378263	−	0.925698i	$-0.376521\pi$
$809$	21.5178	0.756526	0.378263	+	0.925698i	$0.376521\pi$
$810$	0	0
$811$	−45.0566	−1.58215	−0.791076	−	0.611717i	$-0.790479\pi$
$811$	−45.0566	−1.58215	−0.791076	+	0.611717i	$0.790479\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.36666	−0.0827988
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29.8095	1.04036	0.520179	−	0.854057i	$-0.325865\pi$
$821$	29.8095	1.04036	0.520179	+	0.854057i	$0.325865\pi$
$822$	0	0
$823$	−12.9122	−0.450091	−0.225045	−	0.974348i	$-0.572253\pi$
$823$	−12.9122	−0.450091	−0.225045	+	0.974348i	$0.572253\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.92395	−0.310316	−0.155158	−	0.987890i	$-0.549589\pi$
$827$	−8.92395	−0.310316	−0.155158	+	0.987890i	$0.549589\pi$
$828$	0	0
$829$	23.2259	0.806670	0.403335	−	0.915052i	$-0.367851\pi$
$829$	23.2259	0.806670	0.403335	+	0.915052i	$0.367851\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8.62608	0.298876
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	26.3677	0.910315	0.455157	−	0.890411i	$-0.349583\pi$
$839$	26.3677	0.910315	0.455157	+	0.890411i	$0.349583\pi$
$840$	0	0
$841$	45.7505	1.57760
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.56328	0.191157
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.13739	−0.176108
$852$	0	0
$853$	−54.9223	−1.88050	−0.940252	−	0.340480i	$-0.889411\pi$
$853$	−54.9223	−1.88050	−0.940252	+	0.340480i	$0.889411\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.93933	−0.271202	−0.135601	−	0.990763i	$-0.543297\pi$
$857$	−7.93933	−0.271202	−0.135601	+	0.990763i	$0.543297\pi$
$858$	0	0
$859$	−18.8769	−0.644070	−0.322035	−	0.946728i	$-0.604367\pi$
$859$	−18.8769	−0.644070	−0.322035	+	0.946728i	$0.604367\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19.4215	0.661116	0.330558	−	0.943786i	$-0.392763\pi$
$863$	19.4215	0.661116	0.330558	+	0.943786i	$0.392763\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	16.2261	0.550434
$870$	0	0
$871$	−21.4865	−0.728042
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	12.3811	0.418080	0.209040	−	0.977907i	$-0.432966\pi$
$877$	12.3811	0.418080	0.209040	+	0.977907i	$0.432966\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−16.8910	−0.569072	−0.284536	−	0.958665i	$-0.591840\pi$
$881$	−16.8910	−0.569072	−0.284536	+	0.958665i	$0.591840\pi$
$882$	0	0
$883$	−17.8457	−0.600556	−0.300278	−	0.953852i	$-0.597079\pi$
$883$	−17.8457	−0.600556	−0.300278	+	0.953852i	$0.597079\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.98730	−0.234611	−0.117305	−	0.993096i	$-0.537426\pi$
$887$	−6.98730	−0.234611	−0.117305	+	0.993096i	$0.537426\pi$
$888$	0	0
$889$	2.13306	0.0715406
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.20911	0.140853
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−35.8372	−1.19524
$900$	0	0
$901$	11.6525	0.388202
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−36.0085	−1.19564	−0.597821	−	0.801630i	$-0.703967\pi$
$907$	−36.0085	−1.19564	−0.597821	+	0.801630i	$0.703967\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.41987	0.312094	0.156047	−	0.987750i	$-0.450125\pi$
$911$	9.41987	0.312094	0.156047	+	0.987750i	$0.450125\pi$
$912$	0	0
$913$	8.16115	0.270095
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.52200	0.182353
$918$	0	0
$919$	−9.54389	−0.314824	−0.157412	−	0.987533i	$-0.550315\pi$
$919$	−9.54389	−0.314824	−0.157412	+	0.987533i	$0.550315\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4.58517	−0.150923
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.01939	0.263108	0.131554	−	0.991309i	$-0.458003\pi$
$929$	8.01939	0.263108	0.131554	+	0.991309i	$0.458003\pi$
$930$	0	0
$931$	2.12219	0.0695520
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	36.1070	1.17956	0.589782	−	0.807563i	$-0.299214\pi$
$937$	36.1070	1.17956	0.589782	+	0.807563i	$0.299214\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−20.8153	−0.678560	−0.339280	−	0.940685i	$-0.610183\pi$
$941$	−20.8153	−0.678560	−0.339280	+	0.940685i	$0.610183\pi$
$942$	0	0
$943$	27.0934	0.882283
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.92413	−0.160013	−0.0800064	−	0.996794i	$-0.525494\pi$
$947$	−4.92413	−0.160013	−0.0800064	+	0.996794i	$0.525494\pi$
$948$	0	0
$949$	−40.2361	−1.30612
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−6.23702	−0.202037	−0.101018	−	0.994885i	$-0.532210\pi$
$953$	−6.23702	−0.202037	−0.101018	+	0.994885i	$0.532210\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.68894	0.151414
$960$	0	0
$961$	−13.8188	−0.445767
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−17.7648	−0.571277	−0.285639	−	0.958337i	$-0.592206\pi$
$967$	−17.7648	−0.571277	−0.285639	+	0.958337i	$0.592206\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−9.98061	−0.320293	−0.160147	−	0.987093i	$-0.551197\pi$
$971$	−9.98061	−0.320293	−0.160147	+	0.987093i	$0.551197\pi$
$972$	0	0
$973$	13.7823	0.441842
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.5328	−0.528932	−0.264466	−	0.964395i	$-0.585196\pi$
$977$	−16.5328	−0.528932	−0.264466	+	0.964395i	$0.585196\pi$
$978$	0	0
$979$	−29.3677	−0.938597
$980$	0	0
$981$	0	0
$982$	0	0
$983$	57.4418	1.83211	0.916054	−	0.401055i	$-0.131356\pi$
$983$	57.4418	1.83211	0.916054	+	0.401055i	$0.131356\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	27.6238	0.878384
$990$	0	0
$991$	55.5294	1.76395	0.881975	−	0.471297i	$-0.156214\pi$
$991$	55.5294	1.76395	0.881975	+	0.471297i	$0.156214\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.11969	0.257153	0.128577	−	0.991700i	$-0.458959\pi$
$997$	8.11969	0.257153	0.128577	+	0.991700i	$0.458959\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.a.ba.1.2		4
3.2	odd	2		8100.2.a.z.1.2		4
5.2	odd	4		8100.2.d.s.649.3		8
5.3	odd	4		8100.2.d.s.649.6		8
5.4	even	2		8100.2.a.y.1.3		4
9.2	odd	6		900.2.i.e.301.2	yes	8
9.4	even	3		2700.2.i.d.1801.3		8
9.5	odd	6		900.2.i.e.601.2	yes	8
9.7	even	3		2700.2.i.d.901.3		8
15.2	even	4		8100.2.d.q.649.3		8
15.8	even	4		8100.2.d.q.649.6		8
15.14	odd	2		8100.2.a.x.1.3		4
45.2	even	12		900.2.s.d.49.1		16
45.4	even	6		2700.2.i.e.1801.2		8
45.7	odd	12		2700.2.s.d.1549.3		16
45.13	odd	12		2700.2.s.d.2449.3		16
45.14	odd	6		900.2.i.d.601.3	yes	8
45.22	odd	12		2700.2.s.d.2449.6		16
45.23	even	12		900.2.s.d.349.1		16
45.29	odd	6		900.2.i.d.301.3	✓	8
45.32	even	12		900.2.s.d.349.8		16
45.34	even	6		2700.2.i.e.901.2		8
45.38	even	12		900.2.s.d.49.8		16
45.43	odd	12		2700.2.s.d.1549.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.2.i.d.301.3	✓	8	45.29	odd	6
900.2.i.d.601.3	yes	8	45.14	odd	6
900.2.i.e.301.2	yes	8	9.2	odd	6
900.2.i.e.601.2	yes	8	9.5	odd	6
900.2.s.d.49.1		16	45.2	even	12
900.2.s.d.49.8		16	45.38	even	12
900.2.s.d.349.1		16	45.23	even	12
900.2.s.d.349.8		16	45.32	even	12
2700.2.i.d.901.3		8	9.7	even	3
2700.2.i.d.1801.3		8	9.4	even	3
2700.2.i.e.901.2		8	45.34	even	6
2700.2.i.e.1801.2		8	45.4	even	6
2700.2.s.d.1549.3		16	45.7	odd	12
2700.2.s.d.1549.6		16	45.43	odd	12
2700.2.s.d.2449.3		16	45.13	odd	12
2700.2.s.d.2449.6		16	45.22	odd	12
8100.2.a.x.1.3		4	15.14	odd	2
8100.2.a.y.1.3		4	5.4	even	2
8100.2.a.z.1.2		4	3.2	odd	2
8100.2.a.ba.1.2		4	1.1	even	1	trivial
8100.2.d.q.649.3		8	15.2	even	4
8100.2.d.q.649.6		8	15.8	even	4
8100.2.d.s.649.3		8	5.2	odd	4
8100.2.d.s.649.6		8	5.3	odd	4

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:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.a (trivial)

\(f(q)\)	\(=\)	\(q-0.680426 q^{7} +O(q^{10})\)	\(q-0.680426 q^{7} +1.68043 q^{11} -5.14503 q^{13} -1.31957 q^{17} -0.324642 q^{19} +3.78924 q^{23} +8.64584 q^{29} -4.14503 q^{31} -1.35578 q^{37} +7.15009 q^{41} +7.29005 q^{43} -12.9654 q^{47} -6.53702 q^{49} -8.83052 q^{53} -8.81532 q^{59} +9.97048 q^{61} +4.17617 q^{67} +0.891185 q^{71} +7.82038 q^{73} -1.14341 q^{77} +9.65597 q^{79} +4.85659 q^{83} -17.4764 q^{89} +3.50081 q^{91} -8.93427 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{7}+O(q^{10}) \) 4 * q + q^7	\( 4 q + q^{7} + 3 q^{11} - 2 q^{13} - 9 q^{17} - 4 q^{19} + 3 q^{23} - 9 q^{29} + 2 q^{31} + q^{37} + 9 q^{41} - 8 q^{43} - 12 q^{47} + 9 q^{49} - 12 q^{53} - 15 q^{59} - q^{61} - 11 q^{67} + 12 q^{71} + 10 q^{73} - 36 q^{77} - 7 q^{79} - 12 q^{83} - 3 q^{89} - 11 q^{91} - 5 q^{97}+O(q^{100}) \) 4 * q + q^7 + 3 * q^11 - 2 * q^13 - 9 * q^17 - 4 * q^19 + 3 * q^23 - 9 * q^29 + 2 * q^31 + q^37 + 9 * q^41 - 8 * q^43 - 12 * q^47 + 9 * q^49 - 12 * q^53 - 15 * q^59 - q^61 - 11 * q^67 + 12 * q^71 + 10 * q^73 - 36 * q^77 - 7 * q^79 - 12 * q^83 - 3 * q^89 - 11 * q^91 - 5 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more