LMFDB - Embedded newform 5148.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5148,2,Mod(1,5148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5148, 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("5148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5148.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:	$41.1069869606$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1620.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 12x - 14 $ x^3 - 12*x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1716)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.94338$ of defining polynomial
Character	$\chi$	$=$	5148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.94338 q^{5} -0.336509 q^{7} +O(q^{10})$	$q-3.94338 q^{5} -0.336509 q^{7} +1.00000 q^{11} -1.00000 q^{13} -5.60687 q^{17} +0.336509 q^{19} +0.336509 q^{23} +10.5503 q^{25} -4.27989 q^{29} -1.94338 q^{31} +1.32698 q^{35} +4.00000 q^{37} -2.33651 q^{41} -4.27989 q^{43} -9.21374 q^{47} -6.88676 q^{49} +5.21374 q^{53} -3.94338 q^{55} -11.8868 q^{59} -3.21374 q^{61} +3.94338 q^{65} -2.61640 q^{67} +0.653965 q^{71} -0.223271 q^{73} -0.336509 q^{77} +11.6069 q^{79} +12.5598 q^{83} +22.1100 q^{85} +9.83014 q^{89} +0.336509 q^{91} -1.32698 q^{95} +8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q+O(q^{10}) $ 3 * q	$ 3 q + 3 q^{11} - 3 q^{13} - 6 q^{17} + 9 q^{25} + 6 q^{31} + 6 q^{35} + 12 q^{37} - 6 q^{41} - 6 q^{47} + 3 q^{49} - 6 q^{53} - 12 q^{59} + 12 q^{61} + 6 q^{67} + 6 q^{71} + 24 q^{73} + 24 q^{79} + 12 q^{83} + 18 q^{85} - 6 q^{89} - 6 q^{95} + 24 q^{97}+O(q^{100}) $ 3 * q + 3 * q^11 - 3 * q^13 - 6 * q^17 + 9 * q^25 + 6 * q^31 + 6 * q^35 + 12 * q^37 - 6 * q^41 - 6 * q^47 + 3 * q^49 - 6 * q^53 - 12 * q^59 + 12 * q^61 + 6 * q^67 + 6 * q^71 + 24 * q^73 + 24 * q^79 + 12 * q^83 + 18 * q^85 - 6 * q^89 - 6 * q^95 + 24 * 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
$5$	−3.94338	−1.76353	−0.881767	−	0.471686i	$-0.843646\pi$
$5$	−3.94338	−1.76353	−0.881767	+	0.471686i	$0.843646\pi$
$6$	0	0
$7$	−0.336509	−0.127188	−0.0635942	−	0.997976i	$-0.520256\pi$
$7$	−0.336509	−0.127188	−0.0635942	+	0.997976i	$0.520256\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.60687	−1.35987	−0.679933	−	0.733274i	$-0.737991\pi$
$17$	−5.60687	−1.35987	−0.679933	+	0.733274i	$0.737991\pi$
$18$	0	0
$19$	0.336509	0.0772004	0.0386002	−	0.999255i	$-0.487710\pi$
$19$	0.336509	0.0772004	0.0386002	+	0.999255i	$0.487710\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.336509	0.0701669	0.0350835	−	0.999384i	$-0.488830\pi$
$23$	0.336509	0.0701669	0.0350835	+	0.999384i	$0.488830\pi$
$24$	0	0
$25$	10.5503	2.11005
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.27989	−0.794756	−0.397378	−	0.917655i	$-0.630080\pi$
$29$	−4.27989	−0.794756	−0.397378	+	0.917655i	$0.630080\pi$
$30$	0	0
$31$	−1.94338	−0.349042	−0.174521	−	0.984653i	$-0.555838\pi$
$31$	−1.94338	−0.349042	−0.174521	+	0.984653i	$0.555838\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.32698	0.224301
$36$	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
$41$	−2.33651	−0.364901	−0.182451	−	0.983215i	$-0.558403\pi$
$41$	−2.33651	−0.364901	−0.182451	+	0.983215i	$0.558403\pi$
$42$	0	0
$43$	−4.27989	−0.652677	−0.326339	−	0.945253i	$-0.605815\pi$
$43$	−4.27989	−0.652677	−0.326339	+	0.945253i	$0.605815\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.21374	−1.34396	−0.671981	−	0.740568i	$-0.734557\pi$
$47$	−9.21374	−1.34396	−0.671981	+	0.740568i	$0.734557\pi$
$48$	0	0
$49$	−6.88676	−0.983823
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.21374	0.716163	0.358081	−	0.933690i	$-0.383431\pi$
$53$	5.21374	0.716163	0.358081	+	0.933690i	$0.383431\pi$
$54$	0	0
$55$	−3.94338	−0.531725
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.8868	−1.54752	−0.773762	−	0.633476i	$-0.781628\pi$
$59$	−11.8868	−1.54752	−0.773762	+	0.633476i	$0.781628\pi$
$60$	0	0
$61$	−3.21374	−0.411478	−0.205739	−	0.978607i	$-0.565960\pi$
$61$	−3.21374	−0.411478	−0.205739	+	0.978607i	$0.565960\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.94338	0.489116
$66$	0	0
$67$	−2.61640	−0.319644	−0.159822	−	0.987146i	$-0.551092\pi$
$67$	−2.61640	−0.319644	−0.159822	+	0.987146i	$0.551092\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.653965	0.0776113	0.0388057	−	0.999247i	$-0.487645\pi$
$71$	0.653965	0.0776113	0.0388057	+	0.999247i	$0.487645\pi$
$72$	0	0
$73$	−0.223271	−0.0261319	−0.0130659	−	0.999915i	$-0.504159\pi$
$73$	−0.223271	−0.0261319	−0.0130659	+	0.999915i	$0.504159\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.336509	−0.0383487
$78$	0	0
$79$	11.6069	1.30588	0.652938	−	0.757412i	$-0.273536\pi$
$79$	11.6069	1.30588	0.652938	+	0.757412i	$0.273536\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.5598	1.37861	0.689307	−	0.724469i	$-0.257915\pi$
$83$	12.5598	1.37861	0.689307	+	0.724469i	$0.257915\pi$
$84$	0	0
$85$	22.1100	2.39817
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.83014	1.04199	0.520997	−	0.853559i	$-0.325560\pi$
$89$	9.83014	1.04199	0.520997	+	0.853559i	$0.325560\pi$
$90$	0	0
$91$	0.336509	0.0352757
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.32698	−0.136146
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.27989	0.624872	0.312436	−	0.949939i	$-0.398855\pi$
$101$	6.27989	0.624872	0.312436	+	0.949939i	$0.398855\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.113238	0.0109471	0.00547357	−	0.999985i	$-0.498258\pi$
$107$	0.113238	0.0109471	0.00547357	+	0.999985i	$0.498258\pi$
$108$	0	0
$109$	18.8772	1.80811	0.904056	−	0.427415i	$-0.140576\pi$
$109$	18.8772	1.80811	0.904056	+	0.427415i	$0.140576\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.8868	−1.30636	−0.653178	−	0.757204i	$-0.726565\pi$
$113$	−13.8868	−1.30636	−0.653178	+	0.757204i	$0.726565\pi$
$114$	0	0
$115$	−1.32698	−0.123742
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.88676	0.172959
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−21.8868	−1.95761
$126$	0	0
$127$	−9.49363	−0.842424	−0.421212	−	0.906962i	$-0.638395\pi$
$127$	−9.49363	−0.842424	−0.421212	+	0.906962i	$0.638395\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−17.7545	−1.55122	−0.775608	−	0.631215i	$-0.782556\pi$
$131$	−17.7545	−1.55122	−0.775608	+	0.631215i	$0.782556\pi$
$132$	0	0
$133$	−0.113238	−0.00981900
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.94338	−0.507777	−0.253889	−	0.967233i	$-0.581710\pi$
$137$	−5.94338	−0.507777	−0.253889	+	0.967233i	$0.581710\pi$
$138$	0	0
$139$	−4.82062	−0.408879	−0.204440	−	0.978879i	$-0.565537\pi$
$139$	−4.82062	−0.408879	−0.204440	+	0.978879i	$0.565537\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	16.8772	1.40158
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.7831	1.53877	0.769384	−	0.638787i	$-0.220563\pi$
$149$	18.7831	1.53877	0.769384	+	0.638787i	$0.220563\pi$
$150$	0	0
$151$	1.68254	0.136923	0.0684617	−	0.997654i	$-0.478191\pi$
$151$	1.68254	0.136923	0.0684617	+	0.997654i	$0.478191\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.66349	0.615546
$156$	0	0
$157$	15.5693	1.24257	0.621283	−	0.783586i	$-0.286612\pi$
$157$	15.5693	1.24257	0.621283	+	0.783586i	$0.286612\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.113238	−0.00892442
$162$	0	0
$163$	20.3899	1.59706	0.798531	−	0.601954i	$-0.205611\pi$
$163$	20.3899	1.59706	0.798531	+	0.601954i	$0.205611\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.4465	1.89173	0.945865	−	0.324560i	$-0.105216\pi$
$167$	24.4465	1.89173	0.945865	+	0.324560i	$0.105216\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.9529	0.832734	0.416367	−	0.909197i	$-0.363303\pi$
$173$	10.9529	0.832734	0.416367	+	0.909197i	$0.363303\pi$
$174$	0	0
$175$	−3.55025	−0.268374
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.87724	−0.215055	−0.107527	−	0.994202i	$-0.534293\pi$
$179$	−2.87724	−0.215055	−0.107527	+	0.994202i	$0.534293\pi$
$180$	0	0
$181$	4.20422	0.312497	0.156249	−	0.987718i	$-0.450060\pi$
$181$	4.20422	0.312497	0.156249	+	0.987718i	$0.450060\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−15.7735	−1.15969
$186$	0	0
$187$	−5.60687	−0.410015
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.5598	−0.908794	−0.454397	−	0.890799i	$-0.650145\pi$
$191$	−12.5598	−0.908794	−0.454397	+	0.890799i	$0.650145\pi$
$192$	0	0
$193$	−11.5503	−0.831405	−0.415703	−	0.909501i	$-0.636464\pi$
$193$	−11.5503	−0.831405	−0.415703	+	0.909501i	$0.636464\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.877235	0.0625004	0.0312502	−	0.999512i	$-0.490051\pi$
$197$	0.877235	0.0625004	0.0312502	+	0.999512i	$0.490051\pi$
$198$	0	0
$199$	15.7735	1.11816	0.559078	−	0.829115i	$-0.311155\pi$
$199$	15.7735	1.11816	0.559078	+	0.829115i	$0.311155\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.44022	0.101084
$204$	0	0
$205$	9.21374	0.643516
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.336509	0.0232768
$210$	0	0
$211$	−5.06615	−0.348768	−0.174384	−	0.984678i	$-0.555793\pi$
$211$	−5.06615	−0.348768	−0.174384	+	0.984678i	$0.555793\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.8772	1.15102
$216$	0	0
$217$	0.653965	0.0443940
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.60687	0.377159
$222$	0	0
$223$	−4.48411	−0.300278	−0.150139	−	0.988665i	$-0.547972\pi$
$223$	−4.48411	−0.300278	−0.150139	+	0.988665i	$0.547972\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.11003	−0.405537	−0.202769	−	0.979227i	$-0.564994\pi$
$227$	−6.11003	−0.405537	−0.202769	+	0.979227i	$0.564994\pi$
$228$	0	0
$229$	−14.9873	−0.990387	−0.495193	−	0.868783i	$-0.664903\pi$
$229$	−14.9873	−0.990387	−0.495193	+	0.868783i	$0.664903\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.2608	0.934258	0.467129	−	0.884189i	$-0.345288\pi$
$233$	14.2608	0.934258	0.467129	+	0.884189i	$0.345288\pi$
$234$	0	0
$235$	36.3333	2.37012
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.7640	1.60185	0.800925	−	0.598765i	$-0.204342\pi$
$239$	24.7640	1.60185	0.800925	+	0.598765i	$0.204342\pi$
$240$	0	0
$241$	11.2328	0.723568	0.361784	−	0.932262i	$-0.382168\pi$
$241$	11.2328	0.723568	0.361784	+	0.932262i	$0.382168\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	27.1571	1.73501
$246$	0	0
$247$	−0.336509	−0.0214115
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.6412	1.36598	0.682991	−	0.730427i	$-0.260679\pi$
$251$	21.6412	1.36598	0.682991	+	0.730427i	$0.260679\pi$
$252$	0	0
$253$	0.336509	0.0211561
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.4465	0.651637	0.325819	−	0.945432i	$-0.394360\pi$
$257$	10.4465	0.651637	0.325819	+	0.945432i	$0.394360\pi$
$258$	0	0
$259$	−1.34604	−0.0836386
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.11324	0.500284	0.250142	−	0.968209i	$-0.419523\pi$
$263$	8.11324	0.500284	0.250142	+	0.968209i	$0.419523\pi$
$264$	0	0
$265$	−20.5598	−1.26298
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−13.2137	−0.805656	−0.402828	−	0.915276i	$-0.631973\pi$
$269$	−13.2137	−0.805656	−0.402828	+	0.915276i	$0.631973\pi$
$270$	0	0
$271$	−25.3047	−1.53715	−0.768576	−	0.639758i	$-0.779034\pi$
$271$	−25.3047	−1.53715	−0.768576	+	0.639758i	$0.779034\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.5503	0.636204
$276$	0	0
$277$	13.1005	0.787133	0.393567	−	0.919296i	$-0.371241\pi$
$277$	13.1005	0.787133	0.393567	+	0.919296i	$0.371241\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8.87724	−0.529571	−0.264786	−	0.964307i	$-0.585301\pi$
$281$	−8.87724	−0.529571	−0.264786	+	0.964307i	$0.585301\pi$
$282$	0	0
$283$	−18.2799	−1.08663	−0.543313	−	0.839530i	$-0.682830\pi$
$283$	−18.2799	−1.08663	−0.543313	+	0.839530i	$0.682830\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.786256	0.0464112
$288$	0	0
$289$	14.4370	0.849236
$290$	0	0
$291$	0	0
$292$	0	0
$293$	13.9968	0.817701	0.408851	−	0.912601i	$-0.365930\pi$
$293$	13.9968	0.817701	0.408851	+	0.912601i	$0.365930\pi$
$294$	0	0
$295$	46.8740	2.72911
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.336509	−0.0194608
$300$	0	0
$301$	1.44022	0.0830129
$302$	0	0
$303$	0	0
$304$	0	0
$305$	12.6730	0.725655
$306$	0	0
$307$	2.31746	0.132264	0.0661321	−	0.997811i	$-0.478934\pi$
$307$	2.31746	0.132264	0.0661321	+	0.997811i	$0.478934\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.00953	0.284064	0.142032	−	0.989862i	$-0.454636\pi$
$311$	5.00953	0.284064	0.142032	+	0.989862i	$0.454636\pi$
$312$	0	0
$313$	−21.4370	−1.21169	−0.605846	−	0.795582i	$-0.707165\pi$
$313$	−21.4370	−1.21169	−0.605846	+	0.795582i	$0.707165\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	34.9497	1.96297	0.981485	−	0.191537i	$-0.0613472\pi$
$317$	34.9497	1.96297	0.981485	+	0.191537i	$0.0613472\pi$
$318$	0	0
$319$	−4.27989	−0.239628
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.88676	−0.104982
$324$	0	0
$325$	−10.5503	−0.585223
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.10051	0.170936
$330$	0	0
$331$	7.71690	0.424159	0.212080	−	0.977252i	$-0.431976\pi$
$331$	7.71690	0.424159	0.212080	+	0.977252i	$0.431976\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.3175	0.563703
$336$	0	0
$337$	−19.0063	−1.03534	−0.517670	−	0.855580i	$-0.673201\pi$
$337$	−19.0063	−1.03534	−0.517670	+	0.855580i	$0.673201\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.94338	−0.105240
$342$	0	0
$343$	4.67302	0.252319
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.7735	1.38360	0.691798	−	0.722091i	$-0.256819\pi$
$347$	25.7735	1.38360	0.691798	+	0.722091i	$0.256819\pi$
$348$	0	0
$349$	16.3143	0.873282	0.436641	−	0.899636i	$-0.356168\pi$
$349$	16.3143	0.873282	0.436641	+	0.899636i	$0.356168\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.94338	0.422784	0.211392	−	0.977401i	$-0.432200\pi$
$353$	7.94338	0.422784	0.211392	+	0.977401i	$0.432200\pi$
$354$	0	0
$355$	−2.57883	−0.136870
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.00953	0.158837	0.0794184	−	0.996841i	$-0.474694\pi$
$359$	3.00953	0.158837	0.0794184	+	0.996841i	$0.474694\pi$
$360$	0	0
$361$	−18.8868	−0.994040
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.880441	0.0460844
$366$	0	0
$367$	−12.5598	−0.655615	−0.327808	−	0.944745i	$-0.606310\pi$
$367$	−12.5598	−0.655615	−0.327808	+	0.944745i	$0.606310\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.75447	−0.0910876
$372$	0	0
$373$	−21.6603	−1.12153	−0.560764	−	0.827976i	$-0.689492\pi$
$373$	−21.6603	−1.12153	−0.560764	+	0.827976i	$0.689492\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.27989	0.220426
$378$	0	0
$379$	25.8111	1.32583	0.662913	−	0.748696i	$-0.269320\pi$
$379$	25.8111	1.32583	0.662913	+	0.748696i	$0.269320\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9.90581	0.506163	0.253082	−	0.967445i	$-0.418556\pi$
$383$	9.90581	0.506163	0.253082	+	0.967445i	$0.418556\pi$
$384$	0	0
$385$	1.32698	0.0676293
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.3143	1.23278	0.616391	−	0.787440i	$-0.288594\pi$
$389$	24.3143	1.23278	0.616391	+	0.787440i	$0.288594\pi$
$390$	0	0
$391$	−1.88676	−0.0954176
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−45.7703	−2.30295
$396$	0	0
$397$	−33.0815	−1.66031	−0.830155	−	0.557532i	$-0.811748\pi$
$397$	−33.0815	−1.66031	−0.830155	+	0.557532i	$0.811748\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.50316	−0.424628	−0.212314	−	0.977202i	$-0.568100\pi$
$401$	−8.50316	−0.424628	−0.212314	+	0.977202i	$0.568100\pi$
$402$	0	0
$403$	1.94338	0.0968067
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	−14.8772	−0.735632	−0.367816	−	0.929899i	$-0.619894\pi$
$409$	−14.8772	−0.735632	−0.367816	+	0.929899i	$0.619894\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	−49.5280	−2.43123
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.12276	0.0548506	0.0274253	−	0.999624i	$-0.491269\pi$
$419$	1.12276	0.0548506	0.0274253	+	0.999624i	$0.491269\pi$
$420$	0	0
$421$	4.01905	0.195877	0.0979383	−	0.995192i	$-0.468775\pi$
$421$	4.01905	0.195877	0.0979383	+	0.995192i	$0.468775\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−59.1539	−2.86939
$426$	0	0
$427$	1.08145	0.0523352
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.1005	0.727366	0.363683	−	0.931523i	$-0.381519\pi$
$431$	15.1005	0.727366	0.363683	+	0.931523i	$0.381519\pi$
$432$	0	0
$433$	38.8931	1.86908	0.934541	−	0.355855i	$-0.115810\pi$
$433$	38.8931	1.86908	0.934541	+	0.355855i	$0.115810\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.113238	0.00541692
$438$	0	0
$439$	−6.93385	−0.330935	−0.165467	−	0.986215i	$-0.552913\pi$
$439$	−6.93385	−0.330935	−0.165467	+	0.986215i	$0.552913\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	34.4275	1.63570	0.817850	−	0.575432i	$-0.195166\pi$
$443$	34.4275	1.63570	0.817850	+	0.575432i	$0.195166\pi$
$444$	0	0
$445$	−38.7640	−1.83759
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−35.2704	−1.66451	−0.832256	−	0.554392i	$-0.812951\pi$
$449$	−35.2704	−1.66451	−0.832256	+	0.554392i	$0.812951\pi$
$450$	0	0
$451$	−2.33651	−0.110022
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.32698	−0.0622099
$456$	0	0
$457$	25.1005	1.17415	0.587076	−	0.809532i	$-0.300279\pi$
$457$	25.1005	1.17415	0.587076	+	0.809532i	$0.300279\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	31.8645	1.48408	0.742039	−	0.670357i	$-0.233859\pi$
$461$	31.8645	1.48408	0.742039	+	0.670357i	$0.233859\pi$
$462$	0	0
$463$	26.4090	1.22733	0.613665	−	0.789567i	$-0.289695\pi$
$463$	26.4090	1.22733	0.613665	+	0.789567i	$0.289695\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.7831	−0.591529	−0.295765	−	0.955261i	$-0.595574\pi$
$467$	−12.7831	−0.591529	−0.295765	+	0.955261i	$0.595574\pi$
$468$	0	0
$469$	0.880441	0.0406550
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.27989	−0.196790
$474$	0	0
$475$	3.55025	0.162897
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−25.8836	−1.18265	−0.591325	−	0.806433i	$-0.701395\pi$
$479$	−25.8836	−1.18265	−0.591325	+	0.806433i	$0.701395\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−31.5470	−1.43248
$486$	0	0
$487$	25.8111	1.16961	0.584806	−	0.811173i	$-0.301171\pi$
$487$	25.8111	1.16961	0.584806	+	0.811173i	$0.301171\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−15.7926	−0.712709	−0.356354	−	0.934351i	$-0.615980\pi$
$491$	−15.7926	−0.712709	−0.356354	+	0.934351i	$0.615980\pi$
$492$	0	0
$493$	23.9968	1.08076
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.220065	−0.00987126
$498$	0	0
$499$	14.0566	0.629261	0.314630	−	0.949214i	$-0.398119\pi$
$499$	14.0566	0.629261	0.314630	+	0.949214i	$0.398119\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−42.3143	−1.88670	−0.943350	−	0.331800i	$-0.892344\pi$
$503$	−42.3143	−1.88670	−0.943350	+	0.331800i	$0.892344\pi$
$504$	0	0
$505$	−24.7640	−1.10198
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−38.9116	−1.72473	−0.862363	−	0.506290i	$-0.831016\pi$
$509$	−38.9116	−1.72473	−0.862363	+	0.506290i	$0.831016\pi$
$510$	0	0
$511$	0.0751325	0.00332367
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−31.5470	−1.39013
$516$	0	0
$517$	−9.21374	−0.405220
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0.427488	0.0187286	0.00936430	−	0.999956i	$-0.497019\pi$
$521$	0.427488	0.0187286	0.00936430	+	0.999956i	$0.497019\pi$
$522$	0	0
$523$	−15.4936	−0.677489	−0.338745	−	0.940878i	$-0.610002\pi$
$523$	−15.4936	−0.677489	−0.338745	+	0.940878i	$0.610002\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.8963	0.474650
$528$	0	0
$529$	−22.8868	−0.995077
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.33651	0.101205
$534$	0	0
$535$	−0.446541	−0.0193057
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.88676	−0.296634
$540$	0	0
$541$	−6.00000	−0.257960	−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000	−0.257960	−0.128980	+	0.991647i	$0.541170\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−74.4401	−3.18866
$546$	0	0
$547$	−1.60687	−0.0687049	−0.0343524	−	0.999410i	$-0.510937\pi$
$547$	−1.60687	−0.0687049	−0.0343524	+	0.999410i	$0.510937\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.44022	−0.0613555
$552$	0	0
$553$	−3.90581	−0.166092
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.99047	−0.211453	−0.105727	−	0.994395i	$-0.533717\pi$
$557$	−4.99047	−0.211453	−0.105727	+	0.994395i	$0.533717\pi$
$558$	0	0
$559$	4.27989	0.181020
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.5280	1.58162	0.790808	−	0.612064i	$-0.209661\pi$
$563$	37.5280	1.58162	0.790808	+	0.612064i	$0.209661\pi$
$564$	0	0
$565$	54.7608	2.30380
$566$	0	0
$567$	0	0
$568$	0	0
$569$	36.0725	1.51224	0.756118	−	0.654435i	$-0.227093\pi$
$569$	36.0725	1.51224	0.756118	+	0.654435i	$0.227093\pi$
$570$	0	0
$571$	−16.3931	−0.686031	−0.343016	−	0.939330i	$-0.611448\pi$
$571$	−16.3931	−0.686031	−0.343016	+	0.939330i	$0.611448\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.55025	0.148056
$576$	0	0
$577$	−40.8359	−1.70002	−0.850011	−	0.526765i	$-0.823405\pi$
$577$	−40.8359	−1.70002	−0.850011	+	0.526765i	$0.823405\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.22648	−0.175344
$582$	0	0
$583$	5.21374	0.215931
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.7417	1.35140	0.675698	−	0.737178i	$-0.263842\pi$
$587$	32.7417	1.35140	0.675698	+	0.737178i	$0.263842\pi$
$588$	0	0
$589$	−0.653965	−0.0269461
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−34.8963	−1.43302	−0.716509	−	0.697577i	$-0.754261\pi$
$593$	−34.8963	−1.43302	−0.716509	+	0.697577i	$0.754261\pi$
$594$	0	0
$595$	−7.44022	−0.305019
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−19.2137	−0.785052	−0.392526	−	0.919741i	$-0.628399\pi$
$599$	−19.2137	−0.785052	−0.392526	+	0.919741i	$0.628399\pi$
$600$	0	0
$601$	−4.65396	−0.189839	−0.0949196	−	0.995485i	$-0.530259\pi$
$601$	−4.65396	−0.189839	−0.0949196	+	0.995485i	$0.530259\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.94338	−0.160321
$606$	0	0
$607$	−0.147599	−0.00599085	−0.00299542	−	0.999996i	$-0.500953\pi$
$607$	−0.147599	−0.00599085	−0.00299542	+	0.999996i	$0.500953\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.21374	0.372748
$612$	0	0
$613$	3.55025	0.143393	0.0716967	−	0.997426i	$-0.477159\pi$
$613$	3.55025	0.143393	0.0716967	+	0.997426i	$0.477159\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.5032	1.30853	0.654264	−	0.756266i	$-0.272979\pi$
$617$	32.5032	1.30853	0.654264	+	0.756266i	$0.272979\pi$
$618$	0	0
$619$	−15.4904	−0.622613	−0.311306	−	0.950310i	$-0.600767\pi$
$619$	−15.4904	−0.622613	−0.311306	+	0.950310i	$0.600767\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.30793	−0.132529
$624$	0	0
$625$	33.5566	1.34226
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−22.4275	−0.894243
$630$	0	0
$631$	14.5973	0.581111	0.290556	−	0.956858i	$-0.406160\pi$
$631$	14.5973	0.581111	0.290556	+	0.956858i	$0.406160\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	37.4370	1.48564
$636$	0	0
$637$	6.88676	0.272863
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−15.7926	−0.623769	−0.311885	−	0.950120i	$-0.600960\pi$
$641$	−15.7926	−0.623769	−0.311885	+	0.950120i	$0.600960\pi$
$642$	0	0
$643$	−35.3581	−1.39439	−0.697194	−	0.716882i	$-0.745569\pi$
$643$	−35.3581	−1.39439	−0.697194	+	0.716882i	$0.745569\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.90581	−0.232182	−0.116091	−	0.993239i	$-0.537036\pi$
$647$	−5.90581	−0.232182	−0.116091	+	0.993239i	$0.537036\pi$
$648$	0	0
$649$	−11.8868	−0.466596
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.67302	0.261135	0.130568	−	0.991439i	$-0.458320\pi$
$653$	6.67302	0.261135	0.130568	+	0.991439i	$0.458320\pi$
$654$	0	0
$655$	70.0126	2.73562
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−8.13229	−0.316789	−0.158395	−	0.987376i	$-0.550632\pi$
$659$	−8.13229	−0.316789	−0.158395	+	0.987376i	$0.550632\pi$
$660$	0	0
$661$	−25.2518	−0.982183	−0.491092	−	0.871108i	$-0.663402\pi$
$661$	−25.2518	−0.982183	−0.491092	+	0.871108i	$0.663402\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.446541	0.0173161
$666$	0	0
$667$	−1.44022	−0.0557656
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.21374	−0.124065
$672$	0	0
$673$	39.3206	1.51570	0.757848	−	0.652431i	$-0.226251\pi$
$673$	39.3206	1.51570	0.757848	+	0.652431i	$0.226251\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	44.0534	1.69311	0.846555	−	0.532301i	$-0.178673\pi$
$677$	44.0534	1.69311	0.846555	+	0.532301i	$0.178673\pi$
$678$	0	0
$679$	−2.69207	−0.103312
$680$	0	0
$681$	0	0
$682$	0	0
$683$	45.3206	1.73414	0.867072	−	0.498183i	$-0.165999\pi$
$683$	45.3206	1.73414	0.867072	+	0.498183i	$0.165999\pi$
$684$	0	0
$685$	23.4370	0.895482
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.21374	−0.198628
$690$	0	0
$691$	34.3709	1.30753	0.653765	−	0.756698i	$-0.273189\pi$
$691$	34.3709	1.30753	0.653765	+	0.756698i	$0.273189\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	19.0095	0.721072
$696$	0	0
$697$	13.1005	0.496217
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−38.3486	−1.44841	−0.724204	−	0.689586i	$-0.757792\pi$
$701$	−38.3486	−1.44841	−0.724204	+	0.689586i	$0.757792\pi$
$702$	0	0
$703$	1.34604	0.0507667
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.11324	−0.0794765
$708$	0	0
$709$	−14.3143	−0.537583	−0.268791	−	0.963198i	$-0.586624\pi$
$709$	−14.3143	−0.537583	−0.268791	+	0.963198i	$0.586624\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.653965	−0.0244912
$714$	0	0
$715$	3.94338	0.147474
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.20101	0.231259	0.115629	−	0.993292i	$-0.463112\pi$
$719$	6.20101	0.231259	0.115629	+	0.993292i	$0.463112\pi$
$720$	0	0
$721$	−2.69207	−0.100258
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−45.1539	−1.67697
$726$	0	0
$727$	−33.6793	−1.24910	−0.624549	−	0.780986i	$-0.714717\pi$
$727$	−33.6793	−1.24910	−0.624549	+	0.780986i	$0.714717\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	23.9968	0.887553
$732$	0	0
$733$	−9.30472	−0.343678	−0.171839	−	0.985125i	$-0.554971\pi$
$733$	−9.30472	−0.343678	−0.171839	+	0.985125i	$0.554971\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.61640	−0.0963763
$738$	0	0
$739$	−37.8645	−1.39287	−0.696434	−	0.717621i	$-0.745231\pi$
$739$	−37.8645	−1.39287	−0.696434	+	0.717621i	$0.745231\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9.42117	−0.345629	−0.172815	−	0.984954i	$-0.555286\pi$
$743$	−9.42117	−0.345629	−0.172815	+	0.984954i	$0.555286\pi$
$744$	0	0
$745$	−74.0687	−2.71367
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−0.0381056	−0.00139235
$750$	0	0
$751$	−26.9873	−0.984779	−0.492390	−	0.870375i	$-0.663877\pi$
$751$	−26.9873	−0.984779	−0.492390	+	0.870375i	$0.663877\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.63491	−0.241469
$756$	0	0
$757$	−22.8963	−0.832180	−0.416090	−	0.909323i	$-0.636600\pi$
$757$	−22.8963	−0.832180	−0.416090	+	0.909323i	$0.636600\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.6698	0.821780	0.410890	−	0.911685i	$-0.365218\pi$
$761$	22.6698	0.821780	0.410890	+	0.911685i	$0.365218\pi$
$762$	0	0
$763$	−6.35236	−0.229971
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.8868	0.429206
$768$	0	0
$769$	23.3460	0.841880	0.420940	−	0.907089i	$-0.361700\pi$
$769$	23.3460	0.841880	0.420940	+	0.907089i	$0.361700\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32.2576	1.16023	0.580113	−	0.814536i	$-0.303008\pi$
$773$	32.2576	1.16023	0.580113	+	0.814536i	$0.303008\pi$
$774$	0	0
$775$	−20.5032	−0.736495
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.786256	−0.0281705
$780$	0	0
$781$	0.653965	0.0234007
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−61.3957	−2.19131
$786$	0	0
$787$	21.1915	0.755395	0.377697	−	0.925929i	$-0.376716\pi$
$787$	21.1915	0.755395	0.377697	+	0.925929i	$0.376716\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.67302	0.166153
$792$	0	0
$793$	3.21374	0.114123
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.9745	1.41597	0.707985	−	0.706227i	$-0.249604\pi$
$797$	39.9745	1.41597	0.707985	+	0.706227i	$0.249604\pi$
$798$	0	0
$799$	51.6603	1.82761
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−0.223271	−0.00787905
$804$	0	0
$805$	0.446541	0.0157385
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.26084	0.149803	0.0749015	−	0.997191i	$-0.476136\pi$
$809$	4.26084	0.149803	0.0749015	+	0.997191i	$0.476136\pi$
$810$	0	0
$811$	3.43701	0.120690	0.0603450	−	0.998178i	$-0.480780\pi$
$811$	3.43701	0.120690	0.0603450	+	0.998178i	$0.480780\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−80.4052	−2.81647
$816$	0	0
$817$	−1.44022	−0.0503869
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−33.9968	−1.18650	−0.593248	−	0.805020i	$-0.702155\pi$
$821$	−33.9968	−1.18650	−0.593248	+	0.805020i	$0.702155\pi$
$822$	0	0
$823$	2.13229	0.0743270	0.0371635	−	0.999309i	$-0.488168\pi$
$823$	2.13229	0.0743270	0.0371635	+	0.999309i	$0.488168\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.0095	0.800120	0.400060	−	0.916489i	$-0.368989\pi$
$827$	23.0095	0.800120	0.400060	+	0.916489i	$0.368989\pi$
$828$	0	0
$829$	33.7735	1.17300	0.586501	−	0.809948i	$-0.300505\pi$
$829$	33.7735	1.17300	0.586501	+	0.809948i	$0.300505\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	38.6132	1.33787
$834$	0	0
$835$	−96.4020	−3.33613
$836$	0	0
$837$	0	0
$838$	0	0
$839$	42.6476	1.47236	0.736179	−	0.676787i	$-0.236628\pi$
$839$	42.6476	1.47236	0.736179	+	0.676787i	$0.236628\pi$
$840$	0	0
$841$	−10.6825	−0.368364
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.94338	−0.135656
$846$	0	0
$847$	−0.336509	−0.0115626
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.34604	0.0461415
$852$	0	0
$853$	12.0191	0.411525	0.205762	−	0.978602i	$-0.434033\pi$
$853$	12.0191	0.411525	0.205762	+	0.978602i	$0.434033\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.70738	−0.229120	−0.114560	−	0.993416i	$-0.536546\pi$
$857$	−6.70738	−0.229120	−0.114560	+	0.993416i	$0.536546\pi$
$858$	0	0
$859$	−11.1756	−0.381308	−0.190654	−	0.981657i	$-0.561061\pi$
$859$	−11.1756	−0.381308	−0.190654	+	0.981657i	$0.561061\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.5598	−1.24451	−0.622255	−	0.782815i	$-0.713783\pi$
$863$	−36.5598	−1.24451	−0.622255	+	0.782815i	$0.713783\pi$
$864$	0	0
$865$	−43.1915	−1.46855
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.6069	0.393736
$870$	0	0
$871$	2.61640	0.0886533
$872$	0	0
$873$	0	0
$874$	0	0
$875$	7.36509	0.248985
$876$	0	0
$877$	−25.8868	−0.874134	−0.437067	−	0.899429i	$-0.643983\pi$
$877$	−25.8868	−0.874134	−0.437067	+	0.899429i	$0.643983\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−8.91855	−0.300473	−0.150237	−	0.988650i	$-0.548004\pi$
$881$	−8.91855	−0.300473	−0.150237	+	0.988650i	$0.548004\pi$
$882$	0	0
$883$	6.91855	0.232828	0.116414	−	0.993201i	$-0.462860\pi$
$883$	6.91855	0.232828	0.116414	+	0.993201i	$0.462860\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25.0254	−0.840270	−0.420135	−	0.907462i	$-0.638017\pi$
$887$	−25.0254	−0.840270	−0.420135	+	0.907462i	$0.638017\pi$
$888$	0	0
$889$	3.19469	0.107147
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.10051	−0.103754
$894$	0	0
$895$	11.3460	0.379256
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.31746	0.277403
$900$	0	0
$901$	−29.2328	−0.973886
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16.5788	−0.551099
$906$	0	0
$907$	−24.7863	−0.823014	−0.411507	−	0.911407i	$-0.634997\pi$
$907$	−24.7863	−0.823014	−0.411507	+	0.911407i	$0.634997\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−40.5598	−1.34381	−0.671903	−	0.740639i	$-0.734523\pi$
$911$	−40.5598	−1.34381	−0.671903	+	0.740639i	$0.734523\pi$
$912$	0	0
$913$	12.5598	0.415668
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.97454	0.197297
$918$	0	0
$919$	39.6259	1.30714	0.653569	−	0.756867i	$-0.273271\pi$
$919$	39.6259	1.30714	0.653569	+	0.756867i	$0.273271\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−0.653965	−0.0215255
$924$	0	0
$925$	42.2010	1.38756
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−34.6164	−1.13573	−0.567864	−	0.823123i	$-0.692230\pi$
$929$	−34.6164	−1.13573	−0.567864	+	0.823123i	$0.692230\pi$
$930$	0	0
$931$	−2.31746	−0.0759516
$932$	0	0
$933$	0	0
$934$	0	0
$935$	22.1100	0.723075
$936$	0	0
$937$	−11.1005	−0.362638	−0.181319	−	0.983424i	$-0.558037\pi$
$937$	−11.1005	−0.362638	−0.181319	+	0.983424i	$0.558037\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−28.8021	−0.938922	−0.469461	−	0.882953i	$-0.655552\pi$
$941$	−28.8021	−0.938922	−0.469461	+	0.882953i	$0.655552\pi$
$942$	0	0
$943$	−0.786256	−0.0256040
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.1947	0.753726	0.376863	−	0.926269i	$-0.377003\pi$
$947$	23.1947	0.753726	0.376863	+	0.926269i	$0.377003\pi$
$948$	0	0
$949$	0.223271	0.00724767
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−22.2608	−0.721099	−0.360550	−	0.932740i	$-0.617411\pi$
$953$	−22.2608	−0.721099	−0.360550	+	0.932740i	$0.617411\pi$
$954$	0	0
$955$	49.5280	1.60269
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	−27.2233	−0.878170
$962$	0	0
$963$	0	0
$964$	0	0
$965$	45.5470	1.46621
$966$	0	0
$967$	46.4243	1.49290	0.746452	−	0.665439i	$-0.231756\pi$
$967$	46.4243	1.49290	0.746452	+	0.665439i	$0.231756\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.87091	−0.188407	−0.0942033	−	0.995553i	$-0.530030\pi$
$971$	−5.87091	−0.188407	−0.0942033	+	0.995553i	$0.530030\pi$
$972$	0	0
$973$	1.62218	0.0520047
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.7232	−1.62278	−0.811390	−	0.584505i	$-0.801289\pi$
$977$	−50.7232	−1.62278	−0.811390	+	0.584505i	$0.801289\pi$
$978$	0	0
$979$	9.83014	0.314173
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−20.2201	−0.644920	−0.322460	−	0.946583i	$-0.604510\pi$
$983$	−20.2201	−0.644920	−0.322460	+	0.946583i	$0.604510\pi$
$984$	0	0
$985$	−3.45927	−0.110222
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.44022	−0.0457963
$990$	0	0
$991$	41.8613	1.32977	0.664884	−	0.746947i	$-0.268481\pi$
$991$	41.8613	1.32977	0.664884	+	0.746947i	$0.268481\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−62.2010	−1.97190
$996$	0	0
$997$	5.84866	0.185229	0.0926144	−	0.995702i	$-0.470478\pi$
$997$	5.84866	0.185229	0.0926144	+	0.995702i	$0.470478\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.n.1.1		3
3.2	odd	2		1716.2.a.f.1.3	✓	3
12.11	even	2		6864.2.a.bv.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.f.1.3	✓	3	3.2	odd	2
5148.2.a.n.1.1		3	1.1	even	1	trivial
6864.2.a.bv.1.3		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5148.a (trivial)

\(f(q)\)	\(=\)	\(q-3.94338 q^{5} -0.336509 q^{7} +O(q^{10})\)	\(q-3.94338 q^{5} -0.336509 q^{7} +1.00000 q^{11} -1.00000 q^{13} -5.60687 q^{17} +0.336509 q^{19} +0.336509 q^{23} +10.5503 q^{25} -4.27989 q^{29} -1.94338 q^{31} +1.32698 q^{35} +4.00000 q^{37} -2.33651 q^{41} -4.27989 q^{43} -9.21374 q^{47} -6.88676 q^{49} +5.21374 q^{53} -3.94338 q^{55} -11.8868 q^{59} -3.21374 q^{61} +3.94338 q^{65} -2.61640 q^{67} +0.653965 q^{71} -0.223271 q^{73} -0.336509 q^{77} +11.6069 q^{79} +12.5598 q^{83} +22.1100 q^{85} +9.83014 q^{89} +0.336509 q^{91} -1.32698 q^{95} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q+O(q^{10}) \) 3 * q	\( 3 q + 3 q^{11} - 3 q^{13} - 6 q^{17} + 9 q^{25} + 6 q^{31} + 6 q^{35} + 12 q^{37} - 6 q^{41} - 6 q^{47} + 3 q^{49} - 6 q^{53} - 12 q^{59} + 12 q^{61} + 6 q^{67} + 6 q^{71} + 24 q^{73} + 24 q^{79} + 12 q^{83} + 18 q^{85} - 6 q^{89} - 6 q^{95} + 24 q^{97}+O(q^{100}) \) 3 * q + 3 * q^11 - 3 * q^13 - 6 * q^17 + 9 * q^25 + 6 * q^31 + 6 * q^35 + 12 * q^37 - 6 * q^41 - 6 * q^47 + 3 * q^49 - 6 * q^53 - 12 * q^59 + 12 * q^61 + 6 * q^67 + 6 * q^71 + 24 * q^73 + 24 * q^79 + 12 * q^83 + 18 * q^85 - 6 * q^89 - 6 * q^95 + 24 * q^97

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