LMFDB - Embedded newform 2268.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(1,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, 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("2268.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.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:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 252)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.69963$ of defining polynomial
Character	$\chi$	$=$	2268.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.69963 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-1.69963 q^{5} -1.00000 q^{7} +2.47710 q^{11} +0.777472 q^{13} +2.81089 q^{17} -4.98762 q^{19} +0.712008 q^{23} -2.11126 q^{25} +4.51052 q^{29} +5.09888 q^{31} +1.69963 q^{35} -6.87636 q^{37} -5.87636 q^{41} -4.65383 q^{43} +12.9876 q^{47} +1.00000 q^{49} +1.88874 q^{53} -4.21015 q^{55} +14.2880 q^{59} +14.3090 q^{61} -1.32141 q^{65} +7.98762 q^{67} -10.2632 q^{71} +4.98762 q^{73} -2.47710 q^{77} -9.21015 q^{79} +8.81089 q^{83} -4.77747 q^{85} +9.65383 q^{89} -0.777472 q^{91} +8.47710 q^{95} +8.64145 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{5} - 3 q^{7}+O(q^{10}) $ 3 * q + q^5 - 3 * q^7	$ 3 q + q^{5} - 3 q^{7} + 2 q^{11} + 3 q^{13} + 2 q^{17} + 3 q^{19} + 14 q^{23} - 6 q^{25} + q^{29} - 3 q^{31} - q^{35} - 3 q^{37} + 3 q^{43} + 21 q^{47} + 3 q^{49} + 6 q^{53} + 6 q^{55} + 31 q^{59} + 6 q^{61} + 15 q^{65} + 6 q^{67} + 17 q^{71} - 3 q^{73} - 2 q^{77} - 9 q^{79} + 20 q^{83} - 15 q^{85} + 12 q^{89} - 3 q^{91} + 20 q^{95} - 9 q^{97}+O(q^{100}) $ 3 * q + q^5 - 3 * q^7 + 2 * q^11 + 3 * q^13 + 2 * q^17 + 3 * q^19 + 14 * q^23 - 6 * q^25 + q^29 - 3 * q^31 - q^35 - 3 * q^37 + 3 * q^43 + 21 * q^47 + 3 * q^49 + 6 * q^53 + 6 * q^55 + 31 * q^59 + 6 * q^61 + 15 * q^65 + 6 * q^67 + 17 * q^71 - 3 * q^73 - 2 * q^77 - 9 * q^79 + 20 * q^83 - 15 * q^85 + 12 * q^89 - 3 * q^91 + 20 * q^95 - 9 * 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
$5$	−1.69963	−0.760097	−0.380048	−	0.924967i	$-0.624093\pi$
$5$	−1.69963	−0.760097	−0.380048	+	0.924967i	$0.624093\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.47710	0.746874	0.373437	−	0.927656i	$-0.378179\pi$
$11$	2.47710	0.746874	0.373437	+	0.927656i	$0.378179\pi$
$12$	0	0
$13$	0.777472	0.215632	0.107816	−	0.994171i	$-0.465614\pi$
$13$	0.777472	0.215632	0.107816	+	0.994171i	$0.465614\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.81089	0.681742	0.340871	−	0.940110i	$-0.389278\pi$
$17$	2.81089	0.681742	0.340871	+	0.940110i	$0.389278\pi$
$18$	0	0
$19$	−4.98762	−1.14424	−0.572119	−	0.820170i	$-0.693879\pi$
$19$	−4.98762	−1.14424	−0.572119	+	0.820170i	$0.693879\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.712008	0.148464	0.0742320	−	0.997241i	$-0.476349\pi$
$23$	0.712008	0.148464	0.0742320	+	0.997241i	$0.476349\pi$
$24$	0	0
$25$	−2.11126	−0.422253
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.51052	0.837583	0.418791	−	0.908083i	$-0.362454\pi$
$29$	4.51052	0.837583	0.418791	+	0.908083i	$0.362454\pi$
$30$	0	0
$31$	5.09888	0.915787	0.457893	−	0.889007i	$-0.348604\pi$
$31$	5.09888	0.915787	0.457893	+	0.889007i	$0.348604\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.69963	0.287290
$36$	0	0
$37$	−6.87636	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$37$	−6.87636	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.87636	−0.917733	−0.458866	−	0.888505i	$-0.651744\pi$
$41$	−5.87636	−0.917733	−0.458866	+	0.888505i	$0.651744\pi$
$42$	0	0
$43$	−4.65383	−0.709702	−0.354851	−	0.934923i	$-0.615468\pi$
$43$	−4.65383	−0.709702	−0.354851	+	0.934923i	$0.615468\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.9876	1.89444	0.947220	−	0.320586i	$-0.103880\pi$
$47$	12.9876	1.89444	0.947220	+	0.320586i	$0.103880\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.88874	0.259438	0.129719	−	0.991551i	$-0.458593\pi$
$53$	1.88874	0.259438	0.129719	+	0.991551i	$0.458593\pi$
$54$	0	0
$55$	−4.21015	−0.567696
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.2880	1.86014	0.930069	−	0.367385i	$-0.119747\pi$
$59$	14.2880	1.86014	0.930069	+	0.367385i	$0.119747\pi$
$60$	0	0
$61$	14.3090	1.83208	0.916042	−	0.401082i	$-0.131366\pi$
$61$	14.3090	1.83208	0.916042	+	0.401082i	$0.131366\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.32141	−0.163901
$66$	0	0
$67$	7.98762	0.975843	0.487922	−	0.872887i	$-0.337755\pi$
$67$	7.98762	0.975843	0.487922	+	0.872887i	$0.337755\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.2632	−1.21802	−0.609011	−	0.793162i	$-0.708433\pi$
$71$	−10.2632	−1.21802	−0.609011	+	0.793162i	$0.708433\pi$
$72$	0	0
$73$	4.98762	0.583757	0.291878	−	0.956455i	$-0.405720\pi$
$73$	4.98762	0.583757	0.291878	+	0.956455i	$0.405720\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.47710	−0.282292
$78$	0	0
$79$	−9.21015	−1.03622	−0.518111	−	0.855313i	$-0.673365\pi$
$79$	−9.21015	−1.03622	−0.518111	+	0.855313i	$0.673365\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.81089	0.967121	0.483561	−	0.875311i	$-0.339343\pi$
$83$	8.81089	0.967121	0.483561	+	0.875311i	$0.339343\pi$
$84$	0	0
$85$	−4.77747	−0.518190
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.65383	1.02330	0.511652	−	0.859193i	$-0.329034\pi$
$89$	9.65383	1.02330	0.511652	+	0.859193i	$0.329034\pi$
$90$	0	0
$91$	−0.777472	−0.0815012
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.47710	0.869732
$96$	0	0
$97$	8.64145	0.877406	0.438703	−	0.898632i	$-0.355438\pi$
$97$	8.64145	0.877406	0.438703	+	0.898632i	$0.355438\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.41164	0.239967	0.119983	−	0.992776i	$-0.461716\pi$
$101$	2.41164	0.239967	0.119983	+	0.992776i	$0.461716\pi$
$102$	0	0
$103$	−4.33379	−0.427021	−0.213511	−	0.976941i	$-0.568490\pi$
$103$	−4.33379	−0.427021	−0.213511	+	0.976941i	$0.568490\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.1978	1.85592	0.927959	−	0.372682i	$-0.121562\pi$
$107$	19.1978	1.85592	0.927959	+	0.372682i	$0.121562\pi$
$108$	0	0
$109$	18.9629	1.81631	0.908156	−	0.418631i	$-0.137490\pi$
$109$	18.9629	1.81631	0.908156	+	0.418631i	$0.137490\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.9294	1.21630	0.608150	−	0.793822i	$-0.291912\pi$
$113$	12.9294	1.21630	0.608150	+	0.793822i	$0.291912\pi$
$114$	0	0
$115$	−1.21015	−0.112847
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.81089	−0.257674
$120$	0	0
$121$	−4.86398	−0.442180
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0865	1.08105
$126$	0	0
$127$	17.6291	1.56433	0.782163	−	0.623073i	$-0.214116\pi$
$127$	17.6291	1.56433	0.782163	+	0.623073i	$0.214116\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.68725	−0.496897	−0.248449	−	0.968645i	$-0.579921\pi$
$131$	−5.68725	−0.496897	−0.248449	+	0.968645i	$0.579921\pi$
$132$	0	0
$133$	4.98762	0.432482
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.4523	−1.66193	−0.830963	−	0.556328i	$-0.812210\pi$
$137$	−19.4523	−1.66193	−0.830963	+	0.556328i	$0.812210\pi$
$138$	0	0
$139$	−2.98762	−0.253407	−0.126703	−	0.991941i	$-0.540440\pi$
$139$	−2.98762	−0.253407	−0.126703	+	0.991941i	$0.540440\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.92587	0.161050
$144$	0	0
$145$	−7.66621	−0.636644
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.09888	0.663486	0.331743	−	0.943370i	$-0.392363\pi$
$149$	8.09888	0.663486	0.331743	+	0.943370i	$0.392363\pi$
$150$	0	0
$151$	−8.86398	−0.721340	−0.360670	−	0.932693i	$-0.617452\pi$
$151$	−8.86398	−0.721340	−0.360670	+	0.932693i	$0.617452\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.66621	−0.696087
$156$	0	0
$157$	−8.76509	−0.699530	−0.349765	−	0.936837i	$-0.613739\pi$
$157$	−8.76509	−0.699530	−0.349765	+	0.936837i	$0.613739\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.712008	−0.0561141
$162$	0	0
$163$	−1.98762	−0.155682	−0.0778412	−	0.996966i	$-0.524803\pi$
$163$	−1.98762	−0.155682	−0.0778412	+	0.996966i	$0.524803\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.62178	0.202880	0.101440	−	0.994842i	$-0.467655\pi$
$167$	2.62178	0.202880	0.101440	+	0.994842i	$0.467655\pi$
$168$	0	0
$169$	−12.3955	−0.953503
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.22981	−0.397615	−0.198808	−	0.980039i	$-0.563707\pi$
$173$	−5.22981	−0.397615	−0.198808	+	0.980039i	$0.563707\pi$
$174$	0	0
$175$	2.11126	0.159597
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.76509	−0.356160	−0.178080	−	0.984016i	$-0.556989\pi$
$179$	−4.76509	−0.356160	−0.178080	+	0.984016i	$0.556989\pi$
$180$	0	0
$181$	−10.4313	−0.775352	−0.387676	−	0.921796i	$-0.626722\pi$
$181$	−10.4313	−0.775352	−0.387676	+	0.921796i	$0.626722\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	11.6872	0.859264
$186$	0	0
$187$	6.96286	0.509175
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.3214	0.963904	0.481952	−	0.876198i	$-0.339928\pi$
$191$	13.3214	0.963904	0.481952	+	0.876198i	$0.339928\pi$
$192$	0	0
$193$	−14.6414	−1.05391	−0.526957	−	0.849892i	$-0.676667\pi$
$193$	−14.6414	−1.05391	−0.526957	+	0.849892i	$0.676667\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.4858	−1.31706	−0.658528	−	0.752556i	$-0.728821\pi$
$197$	−18.4858	−1.31706	−0.658528	+	0.752556i	$0.728821\pi$
$198$	0	0
$199$	23.6167	1.67414	0.837071	−	0.547094i	$-0.184266\pi$
$199$	23.6167	1.67414	0.837071	+	0.547094i	$0.184266\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.51052	−0.316576
$204$	0	0
$205$	9.98762	0.697566
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−12.3548	−0.854602
$210$	0	0
$211$	−14.5549	−1.00200	−0.501002	−	0.865446i	$-0.667035\pi$
$211$	−14.5549	−1.00200	−0.501002	+	0.865446i	$0.667035\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.90978	0.539442
$216$	0	0
$217$	−5.09888	−0.346135
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.18539	0.147005
$222$	0	0
$223$	9.44506	0.632488	0.316244	−	0.948678i	$-0.397578\pi$
$223$	9.44506	0.632488	0.316244	+	0.948678i	$0.397578\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.1113	1.26846	0.634230	−	0.773145i	$-0.281317\pi$
$227$	19.1113	1.26846	0.634230	+	0.773145i	$0.281317\pi$
$228$	0	0
$229$	11.4451	0.756311	0.378155	−	0.925742i	$-0.376559\pi$
$229$	11.4451	0.756311	0.378155	+	0.925742i	$0.376559\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.19049	−0.0779913	−0.0389956	−	0.999239i	$-0.512416\pi$
$233$	−1.19049	−0.0779913	−0.0389956	+	0.999239i	$0.512416\pi$
$234$	0	0
$235$	−22.0741	−1.43996
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.2829	1.57073	0.785365	−	0.619033i	$-0.212475\pi$
$239$	24.2829	1.57073	0.785365	+	0.619033i	$0.212475\pi$
$240$	0	0
$241$	−21.4189	−1.37971	−0.689857	−	0.723946i	$-0.742327\pi$
$241$	−21.4189	−1.37971	−0.689857	+	0.723946i	$0.742327\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.69963	−0.108585
$246$	0	0
$247$	−3.87773	−0.246734
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.67996	−0.169158	−0.0845789	−	0.996417i	$-0.526955\pi$
$251$	−2.67996	−0.169158	−0.0845789	+	0.996417i	$0.526955\pi$
$252$	0	0
$253$	1.76371	0.110884
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.0851	−0.691471	−0.345736	−	0.938332i	$-0.612371\pi$
$257$	−11.0851	−0.691471	−0.345736	+	0.938332i	$0.612371\pi$
$258$	0	0
$259$	6.87636	0.427276
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.4079	0.826768	0.413384	−	0.910557i	$-0.364347\pi$
$263$	13.4079	0.826768	0.413384	+	0.910557i	$0.364347\pi$
$264$	0	0
$265$	−3.21015	−0.197198
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.09022	0.249385	0.124693	−	0.992195i	$-0.460206\pi$
$269$	4.09022	0.249385	0.124693	+	0.992195i	$0.460206\pi$
$270$	0	0
$271$	6.12364	0.371985	0.185992	−	0.982551i	$-0.440450\pi$
$271$	6.12364	0.371985	0.185992	+	0.982551i	$0.440450\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.22981	−0.315370
$276$	0	0
$277$	15.7651	0.947233	0.473616	−	0.880731i	$-0.342948\pi$
$277$	15.7651	0.947233	0.473616	+	0.880731i	$0.342948\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−21.1891	−1.26404	−0.632018	−	0.774954i	$-0.717773\pi$
$281$	−21.1891	−1.26404	−0.632018	+	0.774954i	$0.717773\pi$
$282$	0	0
$283$	6.87636	0.408757	0.204378	−	0.978892i	$-0.434483\pi$
$283$	6.87636	0.408757	0.204378	+	0.978892i	$0.434483\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.87636	0.346870
$288$	0	0
$289$	−9.09888	−0.535228
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−27.5068	−1.60696	−0.803482	−	0.595329i	$-0.797022\pi$
$293$	−27.5068	−1.60696	−0.803482	+	0.595329i	$0.797022\pi$
$294$	0	0
$295$	−24.2843	−1.41389
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.553566	0.0320135
$300$	0	0
$301$	4.65383	0.268242
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−24.3200	−1.39256
$306$	0	0
$307$	−21.5178	−1.22809	−0.614043	−	0.789273i	$-0.710458\pi$
$307$	−21.5178	−1.22809	−0.614043	+	0.789273i	$0.710458\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.3855	1.04255	0.521273	−	0.853390i	$-0.325457\pi$
$311$	18.3855	1.04255	0.521273	+	0.853390i	$0.325457\pi$
$312$	0	0
$313$	−0.00137742	−7.78563e−5 0	−3.89281e−5	−	1.00000i	$-0.500012\pi$
$313$	−0.00137742	−7.78563e−5 0	−3.89281e−5		1.00000i	$0.500012\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.0989	0.791872	0.395936	−	0.918278i	$-0.370420\pi$
$317$	14.0989	0.791872	0.395936	+	0.918278i	$0.370420\pi$
$318$	0	0
$319$	11.1730	0.625568
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−14.0197	−0.780075
$324$	0	0
$325$	−1.64145	−0.0910512
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.9876	−0.716031
$330$	0	0
$331$	13.9629	0.767468	0.383734	−	0.923444i	$-0.374638\pi$
$331$	13.9629	0.767468	0.383734	+	0.923444i	$0.374638\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−13.5760	−0.741735
$336$	0	0
$337$	24.1964	1.31806	0.659031	−	0.752116i	$-0.270967\pi$
$337$	24.1964	1.31806	0.659031	+	0.752116i	$0.270967\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	12.6304	0.683977
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−32.7156	−1.75626	−0.878132	−	0.478418i	$-0.841210\pi$
$347$	−32.7156	−1.75626	−0.878132	+	0.478418i	$0.841210\pi$
$348$	0	0
$349$	−23.7775	−1.27278	−0.636389	−	0.771368i	$-0.719573\pi$
$349$	−23.7775	−1.27278	−0.636389	+	0.771368i	$0.719573\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.0617	1.06778	0.533889	−	0.845554i	$-0.320730\pi$
$353$	20.0617	1.06778	0.533889	+	0.845554i	$0.320730\pi$
$354$	0	0
$355$	17.4437	0.925814
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.30175	0.438150	0.219075	−	0.975708i	$-0.429696\pi$
$359$	8.30175	0.438150	0.219075	+	0.975708i	$0.429696\pi$
$360$	0	0
$361$	5.87636	0.309282
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.47710	−0.443712
$366$	0	0
$367$	11.5439	0.602589	0.301294	−	0.953531i	$-0.402581\pi$
$367$	11.5439	0.602589	0.301294	+	0.953531i	$0.402581\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.88874	−0.0980583
$372$	0	0
$373$	2.85160	0.147650	0.0738250	−	0.997271i	$-0.476479\pi$
$373$	2.85160	0.147650	0.0738250	+	0.997271i	$0.476479\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.50680	0.180609
$378$	0	0
$379$	35.9519	1.84672	0.923361	−	0.383932i	$-0.125430\pi$
$379$	35.9519	1.84672	0.923361	+	0.383932i	$0.125430\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.83056	−0.0935370	−0.0467685	−	0.998906i	$-0.514892\pi$
$383$	−1.83056	−0.0935370	−0.0467685	+	0.998906i	$0.514892\pi$
$384$	0	0
$385$	4.21015	0.214569
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−11.3906	−0.577526	−0.288763	−	0.957401i	$-0.593244\pi$
$389$	−11.3906	−0.577526	−0.288763	+	0.957401i	$0.593244\pi$
$390$	0	0
$391$	2.00138	0.101214
$392$	0	0
$393$	0	0
$394$	0	0
$395$	15.6538	0.787630
$396$	0	0
$397$	−10.4313	−0.523532	−0.261766	−	0.965131i	$-0.584305\pi$
$397$	−10.4313	−0.523532	−0.261766	+	0.965131i	$0.584305\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−34.0741	−1.70158	−0.850790	−	0.525505i	$-0.823876\pi$
$401$	−34.0741	−1.70158	−0.850790	+	0.525505i	$0.823876\pi$
$402$	0	0
$403$	3.96424	0.197473
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−17.0334	−0.844315
$408$	0	0
$409$	3.97524	0.196563	0.0982815	−	0.995159i	$-0.468665\pi$
$409$	3.97524	0.196563	0.0982815	+	0.995159i	$0.468665\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14.2880	−0.703066
$414$	0	0
$415$	−14.9752	−0.735106
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9.44368	−0.461354	−0.230677	−	0.973030i	$-0.574094\pi$
$419$	−9.44368	−0.461354	−0.230677	+	0.973030i	$0.574094\pi$
$420$	0	0
$421$	−6.32004	−0.308020	−0.154010	−	0.988069i	$-0.549219\pi$
$421$	−6.32004	−0.308020	−0.154010	+	0.988069i	$0.549219\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.93454	−0.287867
$426$	0	0
$427$	−14.3090	−0.692463
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−27.7541	−1.33687	−0.668434	−	0.743772i	$-0.733035\pi$
$431$	−27.7541	−1.33687	−0.668434	+	0.743772i	$0.733035\pi$
$432$	0	0
$433$	11.2473	0.540510	0.270255	−	0.962789i	$-0.412892\pi$
$433$	11.2473	0.540510	0.270255	+	0.962789i	$0.412892\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.55122	−0.169878
$438$	0	0
$439$	−15.0865	−0.720040	−0.360020	−	0.932945i	$-0.617230\pi$
$439$	−15.0865	−0.720040	−0.360020	+	0.932945i	$0.617230\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.93316	0.376916	0.188458	−	0.982081i	$-0.439651\pi$
$443$	7.93316	0.376916	0.188458	+	0.982081i	$0.439651\pi$
$444$	0	0
$445$	−16.4079	−0.777810
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−32.5636	−1.53677	−0.768386	−	0.639987i	$-0.778940\pi$
$449$	−32.5636	−1.53677	−0.768386	+	0.639987i	$0.778940\pi$
$450$	0	0
$451$	−14.5563	−0.685430
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.32141	0.0619488
$456$	0	0
$457$	11.4079	0.533640	0.266820	−	0.963746i	$-0.414027\pi$
$457$	11.4079	0.533640	0.266820	+	0.963746i	$0.414027\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.91706	0.229010	0.114505	−	0.993423i	$-0.463472\pi$
$461$	4.91706	0.229010	0.114505	+	0.993423i	$0.463472\pi$
$462$	0	0
$463$	15.1991	0.706364	0.353182	−	0.935555i	$-0.385100\pi$
$463$	15.1991	0.706364	0.353182	+	0.935555i	$0.385100\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.7810	1.10046	0.550228	−	0.835015i	$-0.314541\pi$
$467$	23.7810	1.10046	0.550228	+	0.835015i	$0.314541\pi$
$468$	0	0
$469$	−7.98762	−0.368834
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11.5280	−0.530058
$474$	0	0
$475$	10.5302	0.483158
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.05818	0.276805	0.138403	−	0.990376i	$-0.455803\pi$
$479$	6.05818	0.276805	0.138403	+	0.990376i	$0.455803\pi$
$480$	0	0
$481$	−5.34617	−0.243764
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.6872	−0.666914
$486$	0	0
$487$	1.13602	0.0514781	0.0257391	−	0.999669i	$-0.491806\pi$
$487$	1.13602	0.0514781	0.0257391	+	0.999669i	$0.491806\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−32.8764	−1.48369	−0.741845	−	0.670572i	$-0.766049\pi$
$491$	−32.8764	−1.48369	−0.741845	+	0.670572i	$0.766049\pi$
$492$	0	0
$493$	12.6786	0.571015
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.2632	0.460369
$498$	0	0
$499$	26.1978	1.17277	0.586387	−	0.810031i	$-0.300550\pi$
$499$	26.1978	1.17277	0.586387	+	0.810031i	$0.300550\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−25.8516	−1.15267	−0.576333	−	0.817215i	$-0.695517\pi$
$503$	−25.8516	−1.15267	−0.576333	+	0.817215i	$0.695517\pi$
$504$	0	0
$505$	−4.09888	−0.182398
$506$	0	0
$507$	0	0
$508$	0	0
$509$	35.1716	1.55896	0.779478	−	0.626430i	$-0.215485\pi$
$509$	35.1716	1.55896	0.779478	+	0.626430i	$0.215485\pi$
$510$	0	0
$511$	−4.98762	−0.220639
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7.36584	0.324578
$516$	0	0
$517$	32.1716	1.41491
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−17.8626	−0.782575	−0.391287	−	0.920269i	$-0.627970\pi$
$521$	−17.8626	−0.782575	−0.391287	+	0.920269i	$0.627970\pi$
$522$	0	0
$523$	−22.8640	−0.999772	−0.499886	−	0.866091i	$-0.666625\pi$
$523$	−22.8640	−0.999772	−0.499886	+	0.866091i	$0.666625\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.3324	0.624330
$528$	0	0
$529$	−22.4930	−0.977958
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.56870	−0.197892
$534$	0	0
$535$	−32.6291	−1.41068
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.47710	0.106696
$540$	0	0
$541$	22.3077	0.959081	0.479541	−	0.877520i	$-0.340803\pi$
$541$	22.3077	0.959081	0.479541	+	0.877520i	$0.340803\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−32.2298	−1.38057
$546$	0	0
$547$	21.6167	0.924263	0.462131	−	0.886811i	$-0.347085\pi$
$547$	21.6167	0.924263	0.462131	+	0.886811i	$0.347085\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−22.4968	−0.958394
$552$	0	0
$553$	9.21015	0.391655
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.17535	0.134544	0.0672720	−	0.997735i	$-0.478570\pi$
$557$	3.17535	0.134544	0.0672720	+	0.997735i	$0.478570\pi$
$558$	0	0
$559$	−3.61822	−0.153034
$560$	0	0
$561$	0	0
$562$	0	0
$563$	43.7628	1.84438	0.922190	−	0.386737i	$-0.126398\pi$
$563$	43.7628	1.84438	0.922190	+	0.386737i	$0.126398\pi$
$564$	0	0
$565$	−21.9752	−0.924505
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.8626	−1.00037	−0.500186	−	0.865918i	$-0.666735\pi$
$569$	−23.8626	−1.00037	−0.500186	+	0.865918i	$0.666735\pi$
$570$	0	0
$571$	10.2212	0.427742	0.213871	−	0.976862i	$-0.431393\pi$
$571$	10.2212	0.427742	0.213871	+	0.976862i	$0.431393\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.50324	−0.0626893
$576$	0	0
$577$	−36.0370	−1.50024	−0.750120	−	0.661302i	$-0.770004\pi$
$577$	−36.0370	−1.50024	−0.750120	+	0.661302i	$0.770004\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.81089	−0.365537
$582$	0	0
$583$	4.67859	0.193767
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21.0283	−0.867932	−0.433966	−	0.900929i	$-0.642886\pi$
$587$	−21.0283	−0.867932	−0.433966	+	0.900929i	$0.642886\pi$
$588$	0	0
$589$	−25.4313	−1.04788
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−25.1606	−1.03322	−0.516612	−	0.856220i	$-0.672807\pi$
$593$	−25.1606	−1.03322	−0.516612	+	0.856220i	$0.672807\pi$
$594$	0	0
$595$	4.77747	0.195857
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.22253	−0.0908100	−0.0454050	−	0.998969i	$-0.514458\pi$
$599$	−2.22253	−0.0908100	−0.0454050	+	0.998969i	$0.514458\pi$
$600$	0	0
$601$	28.0989	1.14618	0.573089	−	0.819493i	$-0.305745\pi$
$601$	28.0989	1.14618	0.573089	+	0.819493i	$0.305745\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.26695	0.336099
$606$	0	0
$607$	−6.53018	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$607$	−6.53018	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.0975	0.408501
$612$	0	0
$613$	10.7280	0.433298	0.216649	−	0.976250i	$-0.430487\pi$
$613$	10.7280	0.433298	0.216649	+	0.976250i	$0.430487\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.0531	−1.25015	−0.625075	−	0.780565i	$-0.714932\pi$
$617$	−31.0531	−1.25015	−0.625075	+	0.780565i	$0.714932\pi$
$618$	0	0
$619$	−1.44643	−0.0581371	−0.0290685	−	0.999577i	$-0.509254\pi$
$619$	−1.44643	−0.0581371	−0.0290685	+	0.999577i	$0.509254\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−9.65383	−0.386772
$624$	0	0
$625$	−9.98624	−0.399450
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−19.3287	−0.770686
$630$	0	0
$631$	0.0741250	0.00295087	0.00147544	−	0.999999i	$-0.499530\pi$
$631$	0.0741250	0.00295087	0.00147544	+	0.999999i	$0.499530\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−29.9629	−1.18904
$636$	0	0
$637$	0.777472	0.0308045
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−47.0407	−1.85800	−0.928998	−	0.370085i	$-0.879329\pi$
$641$	−47.0407	−1.85800	−0.928998	+	0.370085i	$0.879329\pi$
$642$	0	0
$643$	−33.7293	−1.33015	−0.665077	−	0.746774i	$-0.731601\pi$
$643$	−33.7293	−1.33015	−0.665077	+	0.746774i	$0.731601\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.9629	1.76767	0.883836	−	0.467796i	$-0.154952\pi$
$647$	44.9629	1.76767	0.883836	+	0.467796i	$0.154952\pi$
$648$	0	0
$649$	35.3928	1.38929
$650$	0	0
$651$	0	0
$652$	0	0
$653$	41.7156	1.63246	0.816228	−	0.577730i	$-0.196061\pi$
$653$	41.7156	1.63246	0.816228	+	0.577730i	$0.196061\pi$
$654$	0	0
$655$	9.66621	0.377690
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−21.0517	−0.820058	−0.410029	−	0.912072i	$-0.634481\pi$
$659$	−21.0517	−0.820058	−0.410029	+	0.912072i	$0.634481\pi$
$660$	0	0
$661$	22.4437	0.872958	0.436479	−	0.899714i	$-0.356225\pi$
$661$	22.4437	0.872958	0.436479	+	0.899714i	$0.356225\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.47710	−0.328728
$666$	0	0
$667$	3.21153	0.124351
$668$	0	0
$669$	0	0
$670$	0	0
$671$	35.4449	1.36834
$672$	0	0
$673$	−11.6786	−0.450176	−0.225088	−	0.974338i	$-0.572267\pi$
$673$	−11.6786	−0.450176	−0.225088	+	0.974338i	$0.572267\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.4684	0.402335	0.201167	−	0.979557i	$-0.435526\pi$
$677$	10.4684	0.402335	0.201167	+	0.979557i	$0.435526\pi$
$678$	0	0
$679$	−8.64145	−0.331628
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.8158	1.25566	0.627832	−	0.778349i	$-0.283943\pi$
$683$	32.8158	1.25566	0.627832	+	0.778349i	$0.283943\pi$
$684$	0	0
$685$	33.0617	1.26322
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.46844	0.0559431
$690$	0	0
$691$	5.90112	0.224489	0.112245	−	0.993681i	$-0.464196\pi$
$691$	5.90112	0.224489	0.112245	+	0.993681i	$0.464196\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.07784	0.192614
$696$	0	0
$697$	−16.5178	−0.625656
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12.3782	−0.467519	−0.233759	−	0.972294i	$-0.575103\pi$
$701$	−12.3782	−0.467519	−0.233759	+	0.972294i	$0.575103\pi$
$702$	0	0
$703$	34.2967	1.29352
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.41164	−0.0906989
$708$	0	0
$709$	−13.2829	−0.498850	−0.249425	−	0.968394i	$-0.580242\pi$
$709$	−13.2829	−0.498850	−0.249425	+	0.968394i	$0.580242\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.63045	0.135961
$714$	0	0
$715$	−3.27327	−0.122413
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24.3694	−0.908825	−0.454413	−	0.890791i	$-0.650151\pi$
$719$	−24.3694	−0.908825	−0.454413	+	0.890791i	$0.650151\pi$
$720$	0	0
$721$	4.33379	0.161399
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.52290	−0.353672
$726$	0	0
$727$	−15.9890	−0.592999	−0.296500	−	0.955033i	$-0.595819\pi$
$727$	−15.9890	−0.592999	−0.296500	+	0.955033i	$0.595819\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13.0814	−0.483833
$732$	0	0
$733$	−42.2829	−1.56175	−0.780877	−	0.624685i	$-0.785228\pi$
$733$	−42.2829	−1.56175	−0.780877	+	0.624685i	$0.785228\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19.7861	0.728832
$738$	0	0
$739$	−3.08650	−0.113539	−0.0567695	−	0.998387i	$-0.518080\pi$
$739$	−3.08650	−0.113539	−0.0567695	+	0.998387i	$0.518080\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.63045	0.243247	0.121624	−	0.992576i	$-0.461190\pi$
$743$	6.63045	0.243247	0.121624	+	0.992576i	$0.461190\pi$
$744$	0	0
$745$	−13.7651	−0.504314
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−19.1978	−0.701471
$750$	0	0
$751$	−42.7403	−1.55962	−0.779808	−	0.626018i	$-0.784684\pi$
$751$	−42.7403	−1.55962	−0.779808	+	0.626018i	$0.784684\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15.0655	0.548288
$756$	0	0
$757$	−31.0232	−1.12756	−0.563779	−	0.825926i	$-0.690653\pi$
$757$	−31.0232	−1.12756	−0.563779	+	0.825926i	$0.690653\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23.6364	0.856817	0.428409	−	0.903585i	$-0.359074\pi$
$761$	23.6364	0.856817	0.428409	+	0.903585i	$0.359074\pi$
$762$	0	0
$763$	−18.9629	−0.686502
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.1085	0.401105
$768$	0	0
$769$	−3.46844	−0.125075	−0.0625375	−	0.998043i	$-0.519919\pi$
$769$	−3.46844	−0.125075	−0.0625375	+	0.998043i	$0.519919\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	34.5970	1.24437	0.622184	−	0.782871i	$-0.286245\pi$
$773$	34.5970	1.24437	0.622184	+	0.782871i	$0.286245\pi$
$774$	0	0
$775$	−10.7651	−0.386694
$776$	0	0
$777$	0	0
$778$	0	0
$779$	29.3090	1.05011
$780$	0	0
$781$	−25.4231	−0.909708
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.8974	0.531711
$786$	0	0
$787$	−12.1593	−0.433431	−0.216715	−	0.976235i	$-0.569534\pi$
$787$	−12.1593	−0.433431	−0.216715	+	0.976235i	$0.569534\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.9294	−0.459718
$792$	0	0
$793$	11.1249	0.395056
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.78985	0.205087	0.102544	−	0.994729i	$-0.467302\pi$
$797$	5.78985	0.205087	0.102544	+	0.994729i	$0.467302\pi$
$798$	0	0
$799$	36.5068	1.29152
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.3548	0.435993
$804$	0	0
$805$	1.21015	0.0426521
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−49.1817	−1.72914	−0.864568	−	0.502516i	$-0.832408\pi$
$809$	−49.1817	−1.72914	−0.864568	+	0.502516i	$0.832408\pi$
$810$	0	0
$811$	40.7266	1.43010	0.715052	−	0.699072i	$-0.246403\pi$
$811$	40.7266	1.43010	0.715052	+	0.699072i	$0.246403\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.37822	0.118334
$816$	0	0
$817$	23.2115	0.812069
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.0938	0.526777	0.263388	−	0.964690i	$-0.415160\pi$
$821$	15.0938	0.526777	0.263388	+	0.964690i	$0.415160\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.2348	1.22523	0.612616	−	0.790381i	$-0.290117\pi$
$827$	35.2348	1.22523	0.612616	+	0.790381i	$0.290117\pi$
$828$	0	0
$829$	−3.23491	−0.112353	−0.0561765	−	0.998421i	$-0.517891\pi$
$829$	−3.23491	−0.112353	−0.0561765	+	0.998421i	$0.517891\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.81089	0.0973916
$834$	0	0
$835$	−4.45606	−0.154208
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31.0393	−1.07160	−0.535798	−	0.844346i	$-0.679989\pi$
$839$	−31.0393	−1.07160	−0.535798	+	0.844346i	$0.679989\pi$
$840$	0	0
$841$	−8.65521	−0.298455
$842$	0	0
$843$	0	0
$844$	0	0
$845$	21.0678	0.724755
$846$	0	0
$847$	4.86398	0.167128
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.89602	−0.167833
$852$	0	0
$853$	−16.0727	−0.550320	−0.275160	−	0.961398i	$-0.588731\pi$
$853$	−16.0727	−0.550320	−0.275160	+	0.961398i	$0.588731\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.2212	0.656582	0.328291	−	0.944577i	$-0.393527\pi$
$857$	19.2212	0.656582	0.328291	+	0.944577i	$0.393527\pi$
$858$	0	0
$859$	−14.8022	−0.505046	−0.252523	−	0.967591i	$-0.581260\pi$
$859$	−14.8022	−0.505046	−0.252523	+	0.967591i	$0.581260\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	14.7688	0.502736	0.251368	−	0.967892i	$-0.419120\pi$
$863$	14.7688	0.502736	0.251368	+	0.967892i	$0.419120\pi$
$864$	0	0
$865$	8.88874	0.302226
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−22.8145	−0.773927
$870$	0	0
$871$	6.21015	0.210423
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12.0865	−0.408598
$876$	0	0
$877$	52.3832	1.76885	0.884427	−	0.466679i	$-0.154550\pi$
$877$	52.3832	1.76885	0.884427	+	0.466679i	$0.154550\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−31.3214	−1.05525	−0.527623	−	0.849479i	$-0.676916\pi$
$881$	−31.3214	−1.05525	−0.527623	+	0.849479i	$0.676916\pi$
$882$	0	0
$883$	−43.0494	−1.44873	−0.724363	−	0.689419i	$-0.757866\pi$
$883$	−43.0494	−1.44873	−0.724363	+	0.689419i	$0.757866\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.9766	−0.502866	−0.251433	−	0.967875i	$-0.580902\pi$
$887$	−14.9766	−0.502866	−0.251433	+	0.967875i	$0.580902\pi$
$888$	0	0
$889$	−17.6291	−0.591260
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−64.7773	−2.16769
$894$	0	0
$895$	8.09888	0.270716
$896$	0	0
$897$	0	0
$898$	0	0
$899$	22.9986	0.767047
$900$	0	0
$901$	5.30903	0.176870
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.7293	0.589343
$906$	0	0
$907$	30.4559	1.01127	0.505636	−	0.862747i	$-0.331258\pi$
$907$	30.4559	1.01127	0.505636	+	0.862747i	$0.331258\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−19.9519	−0.661035	−0.330517	−	0.943800i	$-0.607223\pi$
$911$	−19.9519	−0.661035	−0.330517	+	0.943800i	$0.607223\pi$
$912$	0	0
$913$	21.8255	0.722317
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.68725	0.187809
$918$	0	0
$919$	45.6291	1.50516	0.752582	−	0.658498i	$-0.228808\pi$
$919$	45.6291	1.50516	0.752582	+	0.658498i	$0.228808\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7.97937	−0.262644
$924$	0	0
$925$	14.5178	0.477342
$926$	0	0
$927$	0	0
$928$	0	0
$929$	56.3722	1.84951	0.924755	−	0.380562i	$-0.124270\pi$
$929$	56.3722	1.84951	0.924755	+	0.380562i	$0.124270\pi$
$930$	0	0
$931$	−4.98762	−0.163463
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−11.8343	−0.387022
$936$	0	0
$937$	−36.8530	−1.20393	−0.601967	−	0.798521i	$-0.705616\pi$
$937$	−36.8530	−1.20393	−0.601967	+	0.798521i	$0.705616\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8.76000	−0.285568	−0.142784	−	0.989754i	$-0.545605\pi$
$941$	−8.76000	−0.285568	−0.142784	+	0.989754i	$0.545605\pi$
$942$	0	0
$943$	−4.18401	−0.136250
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.6452	0.865852	0.432926	−	0.901430i	$-0.357481\pi$
$947$	26.6452	0.865852	0.432926	+	0.901430i	$0.357481\pi$
$948$	0	0
$949$	3.87773	0.125877
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24.3039	0.787282	0.393641	−	0.919264i	$-0.371215\pi$
$953$	24.3039	0.787282	0.393641	+	0.919264i	$0.371215\pi$
$954$	0	0
$955$	−22.6414	−0.732660
$956$	0	0
$957$	0	0
$958$	0	0
$959$	19.4523	0.628149
$960$	0	0
$961$	−5.00138	−0.161335
$962$	0	0
$963$	0	0
$964$	0	0
$965$	24.8850	0.801077
$966$	0	0
$967$	−10.4574	−0.336288	−0.168144	−	0.985762i	$-0.553777\pi$
$967$	−10.4574	−0.336288	−0.168144	+	0.985762i	$0.553777\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−41.7156	−1.33872	−0.669358	−	0.742940i	$-0.733431\pi$
$971$	−41.7156	−1.33872	−0.669358	+	0.742940i	$0.733431\pi$
$972$	0	0
$973$	2.98762	0.0957787
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.89011	0.188441	0.0942207	−	0.995551i	$-0.469964\pi$
$977$	5.89011	0.188441	0.0942207	+	0.995551i	$0.469964\pi$
$978$	0	0
$979$	23.9135	0.764279
$980$	0	0
$981$	0	0
$982$	0	0
$983$	41.8392	1.33446	0.667232	−	0.744850i	$-0.267479\pi$
$983$	41.8392	1.33446	0.667232	+	0.744850i	$0.267479\pi$
$984$	0	0
$985$	31.4189	1.00109
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.31356	−0.105365
$990$	0	0
$991$	−54.7156	−1.73810	−0.869049	−	0.494726i	$-0.835268\pi$
$991$	−54.7156	−1.73810	−0.869049	+	0.494726i	$0.835268\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−40.1396	−1.27251
$996$	0	0
$997$	−18.0495	−0.571634	−0.285817	−	0.958284i	$-0.592265\pi$
$997$	−18.0495	−0.571634	−0.285817	+	0.958284i	$0.592265\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.a.i.1.1		3
3.2	odd	2		2268.2.a.h.1.3		3
4.3	odd	2		9072.2.a.by.1.1		3
9.2	odd	6		756.2.j.b.253.1		6
9.4	even	3		252.2.j.a.169.3	yes	6
9.5	odd	6		756.2.j.b.505.1		6
9.7	even	3		252.2.j.a.85.3	✓	6
12.11	even	2		9072.2.a.bv.1.3		3
36.7	odd	6		1008.2.r.j.337.1		6
36.11	even	6		3024.2.r.j.1009.1		6
36.23	even	6		3024.2.r.j.2017.1		6
36.31	odd	6		1008.2.r.j.673.1		6
63.2	odd	6		5292.2.l.e.361.3		6
63.4	even	3		1764.2.l.e.961.1		6
63.5	even	6		5292.2.i.e.2125.3		6
63.11	odd	6		5292.2.i.f.1549.1		6
63.13	odd	6		1764.2.j.e.1177.1		6
63.16	even	3		1764.2.l.e.949.1		6
63.20	even	6		5292.2.j.d.1765.3		6
63.23	odd	6		5292.2.i.f.2125.1		6
63.25	even	3		1764.2.i.g.373.1		6
63.31	odd	6		1764.2.l.f.961.3		6
63.32	odd	6		5292.2.l.e.3313.3		6
63.34	odd	6		1764.2.j.e.589.1		6
63.38	even	6		5292.2.i.e.1549.3		6
63.40	odd	6		1764.2.i.d.1537.3		6
63.41	even	6		5292.2.j.d.3529.3		6
63.47	even	6		5292.2.l.f.361.1		6
63.52	odd	6		1764.2.i.d.373.3		6
63.58	even	3		1764.2.i.g.1537.1		6
63.59	even	6		5292.2.l.f.3313.1		6
63.61	odd	6		1764.2.l.f.949.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.j.a.85.3	✓	6	9.7	even	3
252.2.j.a.169.3	yes	6	9.4	even	3
756.2.j.b.253.1		6	9.2	odd	6
756.2.j.b.505.1		6	9.5	odd	6
1008.2.r.j.337.1		6	36.7	odd	6
1008.2.r.j.673.1		6	36.31	odd	6
1764.2.i.d.373.3		6	63.52	odd	6
1764.2.i.d.1537.3		6	63.40	odd	6
1764.2.i.g.373.1		6	63.25	even	3
1764.2.i.g.1537.1		6	63.58	even	3
1764.2.j.e.589.1		6	63.34	odd	6
1764.2.j.e.1177.1		6	63.13	odd	6
1764.2.l.e.949.1		6	63.16	even	3
1764.2.l.e.961.1		6	63.4	even	3
1764.2.l.f.949.3		6	63.61	odd	6
1764.2.l.f.961.3		6	63.31	odd	6
2268.2.a.h.1.3		3	3.2	odd	2
2268.2.a.i.1.1		3	1.1	even	1	trivial
3024.2.r.j.1009.1		6	36.11	even	6
3024.2.r.j.2017.1		6	36.23	even	6
5292.2.i.e.1549.3		6	63.38	even	6
5292.2.i.e.2125.3		6	63.5	even	6
5292.2.i.f.1549.1		6	63.11	odd	6
5292.2.i.f.2125.1		6	63.23	odd	6
5292.2.j.d.1765.3		6	63.20	even	6
5292.2.j.d.3529.3		6	63.41	even	6
5292.2.l.e.361.3		6	63.2	odd	6
5292.2.l.e.3313.3		6	63.32	odd	6
5292.2.l.f.361.1		6	63.47	even	6
5292.2.l.f.3313.1		6	63.59	even	6
9072.2.a.bv.1.3		3	12.11	even	2
9072.2.a.by.1.1		3	4.3	odd	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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.a (trivial)

\(f(q)\)	\(=\)	\(q-1.69963 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-1.69963 q^{5} -1.00000 q^{7} +2.47710 q^{11} +0.777472 q^{13} +2.81089 q^{17} -4.98762 q^{19} +0.712008 q^{23} -2.11126 q^{25} +4.51052 q^{29} +5.09888 q^{31} +1.69963 q^{35} -6.87636 q^{37} -5.87636 q^{41} -4.65383 q^{43} +12.9876 q^{47} +1.00000 q^{49} +1.88874 q^{53} -4.21015 q^{55} +14.2880 q^{59} +14.3090 q^{61} -1.32141 q^{65} +7.98762 q^{67} -10.2632 q^{71} +4.98762 q^{73} -2.47710 q^{77} -9.21015 q^{79} +8.81089 q^{83} -4.77747 q^{85} +9.65383 q^{89} -0.777472 q^{91} +8.47710 q^{95} +8.64145 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{5} - 3 q^{7}+O(q^{10}) \) 3 * q + q^5 - 3 * q^7	\( 3 q + q^{5} - 3 q^{7} + 2 q^{11} + 3 q^{13} + 2 q^{17} + 3 q^{19} + 14 q^{23} - 6 q^{25} + q^{29} - 3 q^{31} - q^{35} - 3 q^{37} + 3 q^{43} + 21 q^{47} + 3 q^{49} + 6 q^{53} + 6 q^{55} + 31 q^{59} + 6 q^{61} + 15 q^{65} + 6 q^{67} + 17 q^{71} - 3 q^{73} - 2 q^{77} - 9 q^{79} + 20 q^{83} - 15 q^{85} + 12 q^{89} - 3 q^{91} + 20 q^{95} - 9 q^{97}+O(q^{100}) \) 3 * q + q^5 - 3 * q^7 + 2 * q^11 + 3 * q^13 + 2 * q^17 + 3 * q^19 + 14 * q^23 - 6 * q^25 + q^29 - 3 * q^31 - q^35 - 3 * q^37 + 3 * q^43 + 21 * q^47 + 3 * q^49 + 6 * q^53 + 6 * q^55 + 31 * q^59 + 6 * q^61 + 15 * q^65 + 6 * q^67 + 17 * q^71 - 3 * q^73 - 2 * q^77 - 9 * q^79 + 20 * q^83 - 15 * q^85 + 12 * q^89 - 3 * q^91 + 20 * q^95 - 9 * q^97

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