LMFDB - Embedded newform 8649.2.a.bo.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 19x^{10} + 129x^{8} - 379x^{6} + 473x^{4} - 210x^{2} + 9 $ x^12 - 19x^10 + 129x^8 - 379x^6 + 473x^4 - 210*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 279)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$2.69750$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.69750 q^{2} +5.27650 q^{4} -3.42101 q^{5} -2.44227 q^{7} +8.83837 q^{8} +O(q^{10})$	$q+2.69750 q^{2} +5.27650 q^{4} -3.42101 q^{5} -2.44227 q^{7} +8.83837 q^{8} -9.22818 q^{10} -4.40082 q^{11} +4.26106 q^{13} -6.58802 q^{14} +13.2885 q^{16} +3.71404 q^{17} +1.44227 q^{19} -18.0510 q^{20} -11.8712 q^{22} +2.28652 q^{23} +6.70333 q^{25} +11.4942 q^{26} -12.8867 q^{28} -2.25462 q^{29} +18.1689 q^{32} +10.0186 q^{34} +8.35504 q^{35} -2.58589 q^{37} +3.89052 q^{38} -30.2362 q^{40} -4.77507 q^{41} +5.45845 q^{43} -23.2209 q^{44} +6.16788 q^{46} -0.225810 q^{47} -1.03531 q^{49} +18.0822 q^{50} +22.4835 q^{52} +11.0775 q^{53} +15.0553 q^{55} -21.5857 q^{56} -6.08183 q^{58} -3.56133 q^{59} +9.41939 q^{61} +22.4338 q^{64} -14.5771 q^{65} +8.78757 q^{67} +19.5971 q^{68} +22.5377 q^{70} +2.28774 q^{71} +6.09087 q^{73} -6.97543 q^{74} +7.61015 q^{76} +10.7480 q^{77} +16.8462 q^{79} -45.4601 q^{80} -12.8808 q^{82} +7.31383 q^{83} -12.7058 q^{85} +14.7242 q^{86} -38.8961 q^{88} +14.2515 q^{89} -10.4067 q^{91} +12.0648 q^{92} -0.609122 q^{94} -4.93403 q^{95} +15.9124 q^{97} -2.79276 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 14 q^{4} - 4 q^{7}+O(q^{10}) $ 12 * q + 14 * q^4 - 4 * q^7	$ 12 q + 14 q^{4} - 4 q^{7} - 16 q^{10} + 10 q^{13} + 26 q^{16} - 8 q^{19} - 20 q^{22} + 14 q^{25} - 24 q^{28} + 56 q^{34} + 14 q^{37} - 82 q^{40} + 54 q^{43} + 18 q^{46} + 12 q^{49} - 2 q^{52} + 16 q^{55} + 56 q^{58} + 22 q^{61} + 42 q^{64} - 38 q^{67} + 42 q^{70} + 16 q^{73} + 10 q^{76} + 94 q^{79} - 26 q^{82} - 8 q^{85} - 60 q^{88} + 8 q^{91} + 86 q^{94} + 68 q^{97}+O(q^{100}) $ 12 * q + 14 * q^4 - 4 * q^7 - 16 * q^10 + 10 * q^13 + 26 * q^16 - 8 * q^19 - 20 * q^22 + 14 * q^25 - 24 * q^28 + 56 * q^34 + 14 * q^37 - 82 * q^40 + 54 * q^43 + 18 * q^46 + 12 * q^49 - 2 * q^52 + 16 * q^55 + 56 * q^58 + 22 * q^61 + 42 * q^64 - 38 * q^67 + 42 * q^70 + 16 * q^73 + 10 * q^76 + 94 * q^79 - 26 * q^82 - 8 * q^85 - 60 * q^88 + 8 * q^91 + 86 * q^94 + 68 * 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 $

$2$	2.69750	1.90742	0.953710	−	0.300727i	$-0.0972295\pi$
$2$	2.69750	1.90742	0.953710	+	0.300727i	$0.0972295\pi$
$3$	0	0
$3$	0	0
$4$	5.27650	2.63825
$5$	−3.42101	−1.52992	−0.764962	−	0.644076i	$-0.777242\pi$
$5$	−3.42101	−1.52992	−0.764962	+	0.644076i	$0.777242\pi$
$6$	0	0
$7$	−2.44227	−0.923092	−0.461546	−	0.887116i	$-0.652705\pi$
$7$	−2.44227	−0.923092	−0.461546	+	0.887116i	$0.652705\pi$
$8$	8.83837	3.12483
$9$	0	0
$10$	−9.22818	−2.91821
$11$	−4.40082	−1.32690	−0.663448	−	0.748222i	$-0.730908\pi$
$11$	−4.40082	−1.32690	−0.663448	+	0.748222i	$0.730908\pi$
$12$	0	0
$13$	4.26106	1.18181	0.590903	−	0.806743i	$-0.298772\pi$
$13$	4.26106	1.18181	0.590903	+	0.806743i	$0.298772\pi$
$14$	−6.58802	−1.76072
$15$	0	0
$16$	13.2885	3.32212
$17$	3.71404	0.900786	0.450393	−	0.892830i	$-0.351284\pi$
$17$	3.71404	0.900786	0.450393	+	0.892830i	$0.351284\pi$
$18$	0	0
$19$	1.44227	0.330880	0.165440	−	0.986220i	$-0.447096\pi$
$19$	1.44227	0.330880	0.165440	+	0.986220i	$0.447096\pi$
$20$	−18.0510	−4.03632
$21$	0	0
$22$	−11.8712	−2.53095
$23$	2.28652	0.476772	0.238386	−	0.971171i	$-0.423382\pi$
$23$	2.28652	0.476772	0.238386	+	0.971171i	$0.423382\pi$
$24$	0	0
$25$	6.70333	1.34067
$26$	11.4942	2.25420
$27$	0	0
$28$	−12.8867	−2.43535
$29$	−2.25462	−0.418672	−0.209336	−	0.977844i	$-0.567130\pi$
$29$	−2.25462	−0.418672	−0.209336	+	0.977844i	$0.567130\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	18.1689	3.21185
$33$	0	0
$34$	10.0186	1.71818
$35$	8.35504	1.41226
$36$	0	0
$37$	−2.58589	−0.425117	−0.212558	−	0.977148i	$-0.568180\pi$
$37$	−2.58589	−0.425117	−0.212558	+	0.977148i	$0.568180\pi$
$38$	3.89052	0.631126
$39$	0	0
$40$	−30.2362	−4.78076
$41$	−4.77507	−0.745741	−0.372870	−	0.927883i	$-0.621626\pi$
$41$	−4.77507	−0.745741	−0.372870	+	0.927883i	$0.621626\pi$
$42$	0	0
$43$	5.45845	0.832405	0.416203	−	0.909272i	$-0.363361\pi$
$43$	5.45845	0.832405	0.416203	+	0.909272i	$0.363361\pi$
$44$	−23.2209	−3.50069
$45$	0	0
$46$	6.16788	0.909404
$47$	−0.225810	−0.0329377	−0.0164689	−	0.999864i	$-0.505242\pi$
$47$	−0.225810	−0.0329377	−0.0164689	+	0.999864i	$0.505242\pi$
$48$	0	0
$49$	−1.03531	−0.147902
$50$	18.0822	2.55721
$51$	0	0
$52$	22.4835	3.11790
$53$	11.0775	1.52162	0.760808	−	0.648977i	$-0.224803\pi$
$53$	11.0775	1.52162	0.760808	+	0.648977i	$0.224803\pi$
$54$	0	0
$55$	15.0553	2.03005
$56$	−21.5857	−2.88451
$57$	0	0
$58$	−6.08183	−0.798584
$59$	−3.56133	−0.463646	−0.231823	−	0.972758i	$-0.574469\pi$
$59$	−3.56133	−0.463646	−0.231823	+	0.972758i	$0.574469\pi$
$60$	0	0
$61$	9.41939	1.20603	0.603015	−	0.797730i	$-0.293966\pi$
$61$	9.41939	1.20603	0.603015	+	0.797730i	$0.293966\pi$
$62$	0	0
$63$	0	0
$64$	22.4338	2.80422
$65$	−14.5771	−1.80807
$66$	0	0
$67$	8.78757	1.07357	0.536786	−	0.843718i	$-0.319638\pi$
$67$	8.78757	1.07357	0.536786	+	0.843718i	$0.319638\pi$
$68$	19.5971	2.37650
$69$	0	0
$70$	22.5377	2.69377
$71$	2.28774	0.271505	0.135753	−	0.990743i	$-0.456655\pi$
$71$	2.28774	0.271505	0.135753	+	0.990743i	$0.456655\pi$
$72$	0	0
$73$	6.09087	0.712882	0.356441	−	0.934318i	$-0.383990\pi$
$73$	6.09087	0.712882	0.356441	+	0.934318i	$0.383990\pi$
$74$	−6.97543	−0.810877
$75$	0	0
$76$	7.61015	0.872944
$77$	10.7480	1.22485
$78$	0	0
$79$	16.8462	1.89535	0.947674	−	0.319241i	$-0.103428\pi$
$79$	16.8462	1.89535	0.947674	+	0.319241i	$0.103428\pi$
$80$	−45.4601	−5.08259
$81$	0	0
$82$	−12.8808	−1.42244
$83$	7.31383	0.802797	0.401398	−	0.915904i	$-0.368524\pi$
$83$	7.31383	0.802797	0.401398	+	0.915904i	$0.368524\pi$
$84$	0	0
$85$	−12.7058	−1.37813
$86$	14.7242	1.58775
$87$	0	0
$88$	−38.8961	−4.14633
$89$	14.2515	1.51065	0.755326	−	0.655350i	$-0.227479\pi$
$89$	14.2515	1.51065	0.755326	+	0.655350i	$0.227479\pi$
$90$	0	0
$91$	−10.4067	−1.09091
$92$	12.0648	1.25784
$93$	0	0
$94$	−0.609122	−0.0628261
$95$	−4.93403	−0.506220
$96$	0	0
$97$	15.9124	1.61566	0.807828	−	0.589418i	$-0.200643\pi$
$97$	15.9124	1.61566	0.807828	+	0.589418i	$0.200643\pi$
$98$	−2.79276	−0.282111
$99$	0	0
$100$	35.3701	3.53701
$101$	1.41546	0.140844	0.0704220	−	0.997517i	$-0.477565\pi$
$101$	1.41546	0.140844	0.0704220	+	0.997517i	$0.477565\pi$
$102$	0	0
$103$	2.72003	0.268013	0.134006	−	0.990980i	$-0.457216\pi$
$103$	2.72003	0.268013	0.134006	+	0.990980i	$0.457216\pi$
$104$	37.6608	3.69295
$105$	0	0
$106$	29.8816	2.90236
$107$	4.09255	0.395642	0.197821	−	0.980238i	$-0.436614\pi$
$107$	4.09255	0.395642	0.197821	+	0.980238i	$0.436614\pi$
$108$	0	0
$109$	−19.7674	−1.89337	−0.946687	−	0.322155i	$-0.895593\pi$
$109$	−19.7674	−1.89337	−0.946687	+	0.322155i	$0.895593\pi$
$110$	40.6115	3.87216
$111$	0	0
$112$	−32.4541	−3.06662
$113$	−0.203600	−0.0191531	−0.00957654	−	0.999954i	$-0.503048\pi$
$113$	−0.203600	−0.0191531	−0.00957654	+	0.999954i	$0.503048\pi$
$114$	0	0
$115$	−7.82220	−0.729424
$116$	−11.8965	−1.10456
$117$	0	0
$118$	−9.60668	−0.884367
$119$	−9.07068	−0.831508
$120$	0	0
$121$	8.36720	0.760655
$122$	25.4088	2.30040
$123$	0	0
$124$	0	0
$125$	−5.82711	−0.521193
$126$	0	0
$127$	−10.3738	−0.920524	−0.460262	−	0.887783i	$-0.652245\pi$
$127$	−10.3738	−0.920524	−0.460262	+	0.887783i	$0.652245\pi$
$128$	24.1772	2.13698
$129$	0	0
$130$	−39.3218	−3.44875
$131$	−6.97346	−0.609274	−0.304637	−	0.952469i	$-0.598535\pi$
$131$	−6.97346	−0.609274	−0.304637	+	0.952469i	$0.598535\pi$
$132$	0	0
$133$	−3.52242	−0.305432
$134$	23.7045	2.04775
$135$	0	0
$136$	32.8260	2.81481
$137$	−2.21294	−0.189065	−0.0945323	−	0.995522i	$-0.530136\pi$
$137$	−2.21294	−0.189065	−0.0945323	+	0.995522i	$0.530136\pi$
$138$	0	0
$139$	−1.62047	−0.137446	−0.0687232	−	0.997636i	$-0.521893\pi$
$139$	−1.62047	−0.137446	−0.0687232	+	0.997636i	$0.521893\pi$
$140$	44.0854	3.72590
$141$	0	0
$142$	6.17118	0.517874
$143$	−18.7521	−1.56813
$144$	0	0
$145$	7.71308	0.640536
$146$	16.4301	1.35977
$147$	0	0
$148$	−13.6444	−1.12157
$149$	−18.3673	−1.50471	−0.752353	−	0.658760i	$-0.771081\pi$
$149$	−18.3673	−1.50471	−0.752353	+	0.658760i	$0.771081\pi$
$150$	0	0
$151$	16.2476	1.32221	0.661105	−	0.750293i	$-0.270088\pi$
$151$	16.2476	1.32221	0.661105	+	0.750293i	$0.270088\pi$
$152$	12.7473	1.03394
$153$	0	0
$154$	28.9927	2.33630
$155$	0	0
$156$	0	0
$157$	−5.18702	−0.413969	−0.206984	−	0.978344i	$-0.566365\pi$
$157$	−5.18702	−0.413969	−0.206984	+	0.978344i	$0.566365\pi$
$158$	45.4427	3.61522
$159$	0	0
$160$	−62.1562	−4.91388
$161$	−5.58429	−0.440104
$162$	0	0
$163$	−0.663141	−0.0519412	−0.0259706	−	0.999663i	$-0.508268\pi$
$163$	−0.663141	−0.0519412	−0.0259706	+	0.999663i	$0.508268\pi$
$164$	−25.1957	−1.96745
$165$	0	0
$166$	19.7290	1.53127
$167$	−15.9195	−1.23189	−0.615944	−	0.787790i	$-0.711225\pi$
$167$	−15.9195	−1.23189	−0.615944	+	0.787790i	$0.711225\pi$
$168$	0	0
$169$	5.15662	0.396663
$170$	−34.2738	−2.62868
$171$	0	0
$172$	28.8015	2.19609
$173$	4.10014	0.311728	0.155864	−	0.987779i	$-0.450184\pi$
$173$	4.10014	0.311728	0.155864	+	0.987779i	$0.450184\pi$
$174$	0	0
$175$	−16.3713	−1.23756
$176$	−58.4802	−4.40811
$177$	0	0
$178$	38.4433	2.88145
$179$	3.04817	0.227831	0.113915	−	0.993490i	$-0.463661\pi$
$179$	3.04817	0.227831	0.113915	+	0.993490i	$0.463661\pi$
$180$	0	0
$181$	21.1703	1.57357	0.786787	−	0.617225i	$-0.211743\pi$
$181$	21.1703	1.57357	0.786787	+	0.617225i	$0.211743\pi$
$182$	−28.0720	−2.08083
$183$	0	0
$184$	20.2091	1.48983
$185$	8.84635	0.650396
$186$	0	0
$187$	−16.3448	−1.19525
$188$	−1.19149	−0.0868981
$189$	0	0
$190$	−13.3095	−0.965575
$191$	26.4740	1.91559	0.957796	−	0.287450i	$-0.0928075\pi$
$191$	26.4740	1.91559	0.957796	+	0.287450i	$0.0928075\pi$
$192$	0	0
$193$	13.6394	0.981783	0.490891	−	0.871221i	$-0.336671\pi$
$193$	13.6394	0.981783	0.490891	+	0.871221i	$0.336671\pi$
$194$	42.9236	3.08174
$195$	0	0
$196$	−5.46284	−0.390203
$197$	−3.50773	−0.249916	−0.124958	−	0.992162i	$-0.539880\pi$
$197$	−3.50773	−0.249916	−0.124958	+	0.992162i	$0.539880\pi$
$198$	0	0
$199$	5.30549	0.376096	0.188048	−	0.982160i	$-0.439784\pi$
$199$	5.30549	0.376096	0.188048	+	0.982160i	$0.439784\pi$
$200$	59.2465	4.18936
$201$	0	0
$202$	3.81821	0.268649
$203$	5.50639	0.386473
$204$	0	0
$205$	16.3356	1.14093
$206$	7.33729	0.511213
$207$	0	0
$208$	56.6230	3.92610
$209$	−6.34717	−0.439043
$210$	0	0
$211$	17.3644	1.19542	0.597709	−	0.801713i	$-0.296078\pi$
$211$	17.3644	1.19542	0.597709	+	0.801713i	$0.296078\pi$
$212$	58.4506	4.01441
$213$	0	0
$214$	11.0397	0.754656
$215$	−18.6734	−1.27352
$216$	0	0
$217$	0	0
$218$	−53.3226	−3.61146
$219$	0	0
$220$	79.4391	5.35578
$221$	15.8257	1.06455
$222$	0	0
$223$	16.1458	1.08120	0.540601	−	0.841279i	$-0.318197\pi$
$223$	16.1458	1.08120	0.540601	+	0.841279i	$0.318197\pi$
$224$	−44.3735	−2.96483
$225$	0	0
$226$	−0.549211	−0.0365330
$227$	21.8243	1.44853	0.724265	−	0.689522i	$-0.242179\pi$
$227$	21.8243	1.44853	0.724265	+	0.689522i	$0.242179\pi$
$228$	0	0
$229$	6.95640	0.459692	0.229846	−	0.973227i	$-0.426178\pi$
$229$	6.95640	0.459692	0.229846	+	0.973227i	$0.426178\pi$
$230$	−21.1004	−1.39132
$231$	0	0
$232$	−19.9271	−1.30828
$233$	−27.5074	−1.80207	−0.901035	−	0.433747i	$-0.857191\pi$
$233$	−27.5074	−1.80207	−0.901035	+	0.433747i	$0.857191\pi$
$234$	0	0
$235$	0.772498	0.0503922
$236$	−18.7914	−1.22321
$237$	0	0
$238$	−24.4682	−1.58604
$239$	−19.1995	−1.24191	−0.620956	−	0.783845i	$-0.713256\pi$
$239$	−19.1995	−1.24191	−0.620956	+	0.783845i	$0.713256\pi$
$240$	0	0
$241$	2.24103	0.144358	0.0721789	−	0.997392i	$-0.477005\pi$
$241$	2.24103	0.144358	0.0721789	+	0.997392i	$0.477005\pi$
$242$	22.5705	1.45089
$243$	0	0
$244$	49.7014	3.18181
$245$	3.54182	0.226279
$246$	0	0
$247$	6.14560	0.391035
$248$	0	0
$249$	0	0
$250$	−15.7186	−0.994134
$251$	−21.0165	−1.32655	−0.663275	−	0.748375i	$-0.730834\pi$
$251$	−21.0165	−1.32655	−0.663275	+	0.748375i	$0.730834\pi$
$252$	0	0
$253$	−10.0625	−0.632627
$254$	−27.9833	−1.75583
$255$	0	0
$256$	20.3504	1.27190
$257$	9.51291	0.593399	0.296699	−	0.954971i	$-0.404114\pi$
$257$	9.51291	0.593399	0.296699	+	0.954971i	$0.404114\pi$
$258$	0	0
$259$	6.31543	0.392422
$260$	−76.9163	−4.77015
$261$	0	0
$262$	−18.8109	−1.16214
$263$	5.53730	0.341445	0.170722	−	0.985319i	$-0.445390\pi$
$263$	5.53730	0.341445	0.170722	+	0.985319i	$0.445390\pi$
$264$	0	0
$265$	−37.8964	−2.32796
$266$	−9.50171	−0.582587
$267$	0	0
$268$	46.3676	2.83235
$269$	−20.6446	−1.25872	−0.629361	−	0.777113i	$-0.716683\pi$
$269$	−20.6446	−1.25872	−0.629361	+	0.777113i	$0.716683\pi$
$270$	0	0
$271$	−3.49168	−0.212104	−0.106052	−	0.994361i	$-0.533821\pi$
$271$	−3.49168	−0.212104	−0.106052	+	0.994361i	$0.533821\pi$
$272$	49.3539	2.99252
$273$	0	0
$274$	−5.96942	−0.360626
$275$	−29.5001	−1.77893
$276$	0	0
$277$	−18.6415	−1.12006	−0.560029	−	0.828473i	$-0.689210\pi$
$277$	−18.6415	−1.12006	−0.560029	+	0.828473i	$0.689210\pi$
$278$	−4.37122	−0.262168
$279$	0	0
$280$	73.8449	4.41308
$281$	−11.8714	−0.708189	−0.354094	−	0.935210i	$-0.615211\pi$
$281$	−11.8714	−0.708189	−0.354094	+	0.935210i	$0.615211\pi$
$282$	0	0
$283$	−9.50240	−0.564859	−0.282430	−	0.959288i	$-0.591140\pi$
$283$	−9.50240	−0.564859	−0.282430	+	0.959288i	$0.591140\pi$
$284$	12.0713	0.716299
$285$	0	0
$286$	−50.5839	−2.99109
$287$	11.6620	0.688387
$288$	0	0
$289$	−3.20594	−0.188584
$290$	20.8060	1.22177
$291$	0	0
$292$	32.1385	1.88076
$293$	8.48047	0.495434	0.247717	−	0.968832i	$-0.420320\pi$
$293$	8.48047	0.495434	0.247717	+	0.968832i	$0.420320\pi$
$294$	0	0
$295$	12.1834	0.709342
$296$	−22.8550	−1.32842
$297$	0	0
$298$	−49.5457	−2.87011
$299$	9.74298	0.563451
$300$	0	0
$301$	−13.3310	−0.768386
$302$	43.8279	2.52201
$303$	0	0
$304$	19.1656	1.09922
$305$	−32.2239	−1.84513
$306$	0	0
$307$	7.77976	0.444015	0.222007	−	0.975045i	$-0.428739\pi$
$307$	7.77976	0.444015	0.222007	+	0.975045i	$0.428739\pi$
$308$	56.7118	3.23146
$309$	0	0
$310$	0	0
$311$	−0.451108	−0.0255800	−0.0127900	−	0.999918i	$-0.504071\pi$
$311$	−0.451108	−0.0255800	−0.0127900	+	0.999918i	$0.504071\pi$
$312$	0	0
$313$	−21.2990	−1.20389	−0.601944	−	0.798538i	$-0.705607\pi$
$313$	−21.2990	−1.20389	−0.601944	+	0.798538i	$0.705607\pi$
$314$	−13.9920	−0.789613
$315$	0	0
$316$	88.8891	5.00040
$317$	15.8104	0.888001	0.444001	−	0.896027i	$-0.353559\pi$
$317$	15.8104	0.888001	0.444001	+	0.896027i	$0.353559\pi$
$318$	0	0
$319$	9.92217	0.555535
$320$	−76.7462	−4.29024
$321$	0	0
$322$	−15.0636	−0.839463
$323$	5.35665	0.298052
$324$	0	0
$325$	28.5633	1.58441
$326$	−1.78882	−0.0990737
$327$	0	0
$328$	−42.2038	−2.33032
$329$	0.551489	0.0304046
$330$	0	0
$331$	−6.00917	−0.330294	−0.165147	−	0.986269i	$-0.552810\pi$
$331$	−6.00917	−0.330294	−0.165147	+	0.986269i	$0.552810\pi$
$332$	38.5914	2.11798
$333$	0	0
$334$	−42.9428	−2.34973
$335$	−30.0624	−1.64248
$336$	0	0
$337$	0.569512	0.0310233	0.0155117	−	0.999880i	$-0.495062\pi$
$337$	0.569512	0.0310233	0.0155117	+	0.999880i	$0.495062\pi$
$338$	13.9100	0.756603
$339$	0	0
$340$	−67.0420	−3.63586
$341$	0	0
$342$	0	0
$343$	19.6244	1.05962
$344$	48.2437	2.60113
$345$	0	0
$346$	11.0601	0.594596
$347$	−23.5963	−1.26671	−0.633357	−	0.773859i	$-0.718324\pi$
$347$	−23.5963	−1.26671	−0.633357	+	0.773859i	$0.718324\pi$
$348$	0	0
$349$	−11.9652	−0.640483	−0.320242	−	0.947336i	$-0.603764\pi$
$349$	−11.9652	−0.640483	−0.320242	+	0.947336i	$0.603764\pi$
$350$	−44.1617	−2.36054
$351$	0	0
$352$	−79.9582	−4.26179
$353$	9.68075	0.515254	0.257627	−	0.966244i	$-0.417059\pi$
$353$	9.68075	0.515254	0.257627	+	0.966244i	$0.417059\pi$
$354$	0	0
$355$	−7.82640	−0.415382
$356$	75.1979	3.98548
$357$	0	0
$358$	8.22244	0.434569
$359$	−26.0119	−1.37286	−0.686428	−	0.727198i	$-0.740822\pi$
$359$	−26.0119	−1.37286	−0.686428	+	0.727198i	$0.740822\pi$
$360$	0	0
$361$	−16.9199	−0.890519
$362$	57.1068	3.00147
$363$	0	0
$364$	−54.9108	−2.87811
$365$	−20.8369	−1.09066
$366$	0	0
$367$	27.2442	1.42213	0.711067	−	0.703124i	$-0.248212\pi$
$367$	27.2442	1.42213	0.711067	+	0.703124i	$0.248212\pi$
$368$	30.3843	1.58389
$369$	0	0
$370$	23.8630	1.24058
$371$	−27.0543	−1.40459
$372$	0	0
$373$	8.99458	0.465722	0.232861	−	0.972510i	$-0.425191\pi$
$373$	8.99458	0.465722	0.232861	+	0.972510i	$0.425191\pi$
$374$	−44.0901	−2.27984
$375$	0	0
$376$	−1.99579	−0.102925
$377$	−9.60706	−0.494789
$378$	0	0
$379$	−14.7704	−0.758705	−0.379353	−	0.925252i	$-0.623853\pi$
$379$	−14.7704	−0.758705	−0.379353	+	0.925252i	$0.623853\pi$
$380$	−26.0344	−1.33554
$381$	0	0
$382$	71.4136	3.65384
$383$	−17.3827	−0.888216	−0.444108	−	0.895973i	$-0.646479\pi$
$383$	−17.3827	−0.888216	−0.444108	+	0.895973i	$0.646479\pi$
$384$	0	0
$385$	−36.7690	−1.87392
$386$	36.7922	1.87267
$387$	0	0
$388$	83.9617	4.26251
$389$	23.7125	1.20227	0.601135	−	0.799148i	$-0.294715\pi$
$389$	23.7125	1.20227	0.601135	+	0.799148i	$0.294715\pi$
$390$	0	0
$391$	8.49220	0.429469
$392$	−9.15049	−0.462169
$393$	0	0
$394$	−9.46211	−0.476694
$395$	−57.6311	−2.89974
$396$	0	0
$397$	−12.9547	−0.650176	−0.325088	−	0.945684i	$-0.605394\pi$
$397$	−12.9547	−0.650176	−0.325088	+	0.945684i	$0.605394\pi$
$398$	14.3115	0.717373
$399$	0	0
$400$	89.0771	4.45385
$401$	−11.1389	−0.556252	−0.278126	−	0.960545i	$-0.589713\pi$
$401$	−11.1389	−0.556252	−0.278126	+	0.960545i	$0.589713\pi$
$402$	0	0
$403$	0	0
$404$	7.46870	0.371582
$405$	0	0
$406$	14.8535	0.737166
$407$	11.3800	0.564086
$408$	0	0
$409$	1.29793	0.0641784	0.0320892	−	0.999485i	$-0.489784\pi$
$409$	1.29793	0.0641784	0.0320892	+	0.999485i	$0.489784\pi$
$410$	44.0652	2.17623
$411$	0	0
$412$	14.3523	0.707085
$413$	8.69773	0.427987
$414$	0	0
$415$	−25.0207	−1.22822
$416$	77.4190	3.79578
$417$	0	0
$418$	−17.1215	−0.837440
$419$	1.75685	0.0858279	0.0429140	−	0.999079i	$-0.486336\pi$
$419$	1.75685	0.0858279	0.0429140	+	0.999079i	$0.486336\pi$
$420$	0	0
$421$	−29.5521	−1.44028	−0.720140	−	0.693829i	$-0.755923\pi$
$421$	−29.5521	−1.44028	−0.720140	+	0.693829i	$0.755923\pi$
$422$	46.8406	2.28016
$423$	0	0
$424$	97.9073	4.75480
$425$	24.8964	1.20765
$426$	0	0
$427$	−23.0047	−1.11328
$428$	21.5944	1.04380
$429$	0	0
$430$	−50.3715	−2.42913
$431$	37.3444	1.79881	0.899407	−	0.437112i	$-0.143999\pi$
$431$	37.3444	1.79881	0.899407	+	0.437112i	$0.143999\pi$
$432$	0	0
$433$	−31.6910	−1.52297	−0.761486	−	0.648181i	$-0.775530\pi$
$433$	−31.6910	−1.52297	−0.761486	+	0.648181i	$0.775530\pi$
$434$	0	0
$435$	0	0
$436$	−104.303	−4.99520
$437$	3.29778	0.157754
$438$	0	0
$439$	5.00816	0.239026	0.119513	−	0.992833i	$-0.461867\pi$
$439$	5.00816	0.239026	0.119513	+	0.992833i	$0.461867\pi$
$440$	133.064	6.34357
$441$	0	0
$442$	42.6899	2.03055
$443$	4.95968	0.235641	0.117821	−	0.993035i	$-0.462409\pi$
$443$	4.95968	0.235641	0.117821	+	0.993035i	$0.462409\pi$
$444$	0	0
$445$	−48.7544	−2.31118
$446$	43.5533	2.06231
$447$	0	0
$448$	−54.7893	−2.58855
$449$	−17.6297	−0.831997	−0.415999	−	0.909365i	$-0.636568\pi$
$449$	−17.6297	−0.831997	−0.415999	+	0.909365i	$0.636568\pi$
$450$	0	0
$451$	21.0142	0.989521
$452$	−1.07430	−0.0505307
$453$	0	0
$454$	58.8710	2.76295
$455$	35.6013	1.66902
$456$	0	0
$457$	17.2351	0.806225	0.403112	−	0.915150i	$-0.367928\pi$
$457$	17.2351	0.806225	0.403112	+	0.915150i	$0.367928\pi$
$458$	18.7649	0.876825
$459$	0	0
$460$	−41.2739	−1.92440
$461$	33.9010	1.57892	0.789462	−	0.613799i	$-0.210359\pi$
$461$	33.9010	1.57892	0.789462	+	0.613799i	$0.210359\pi$
$462$	0	0
$463$	29.8935	1.38927	0.694634	−	0.719364i	$-0.255567\pi$
$463$	29.8935	1.38927	0.694634	+	0.719364i	$0.255567\pi$
$464$	−29.9605	−1.39088
$465$	0	0
$466$	−74.2012	−3.43730
$467$	29.8045	1.37919	0.689595	−	0.724195i	$-0.257789\pi$
$467$	29.8045	1.37919	0.689595	+	0.724195i	$0.257789\pi$
$468$	0	0
$469$	−21.4616	−0.991006
$470$	2.08381	0.0961192
$471$	0	0
$472$	−31.4763	−1.44882
$473$	−24.0216	−1.10452
$474$	0	0
$475$	9.66802	0.443599
$476$	−47.8615	−2.19373
$477$	0	0
$478$	−51.7906	−2.36885
$479$	−32.7216	−1.49509	−0.747545	−	0.664211i	$-0.768768\pi$
$479$	−32.7216	−1.49509	−0.747545	+	0.664211i	$0.768768\pi$
$480$	0	0
$481$	−11.0186	−0.502405
$482$	6.04519	0.275351
$483$	0	0
$484$	44.1496	2.00680
$485$	−54.4364	−2.47183
$486$	0	0
$487$	37.0973	1.68104	0.840519	−	0.541782i	$-0.182250\pi$
$487$	37.0973	1.68104	0.840519	+	0.541782i	$0.182250\pi$
$488$	83.2520	3.76864
$489$	0	0
$490$	9.55406	0.431609
$491$	−28.8846	−1.30354	−0.651772	−	0.758415i	$-0.725974\pi$
$491$	−28.8846	−1.30354	−0.651772	+	0.758415i	$0.725974\pi$
$492$	0	0
$493$	−8.37373	−0.377134
$494$	16.5778	0.745868
$495$	0	0
$496$	0	0
$497$	−5.58729	−0.250624
$498$	0	0
$499$	−26.7462	−1.19732	−0.598662	−	0.801002i	$-0.704301\pi$
$499$	−26.7462	−1.19732	−0.598662	+	0.801002i	$0.704301\pi$
$500$	−30.7468	−1.37504
$501$	0	0
$502$	−56.6921	−2.53029
$503$	20.8283	0.928687	0.464344	−	0.885655i	$-0.346290\pi$
$503$	20.8283	0.928687	0.464344	+	0.885655i	$0.346290\pi$
$504$	0	0
$505$	−4.84232	−0.215480
$506$	−27.1437	−1.20668
$507$	0	0
$508$	−54.7373	−2.42857
$509$	−28.4770	−1.26222	−0.631111	−	0.775693i	$-0.717401\pi$
$509$	−28.4770	−1.26222	−0.631111	+	0.775693i	$0.717401\pi$
$510$	0	0
$511$	−14.8755	−0.658056
$512$	6.54076	0.289064
$513$	0	0
$514$	25.6611	1.13186
$515$	−9.30527	−0.410039
$516$	0	0
$517$	0.993748	0.0437050
$518$	17.0359	0.748513
$519$	0	0
$520$	−128.838	−5.64992
$521$	−28.2660	−1.23836	−0.619178	−	0.785250i	$-0.712534\pi$
$521$	−28.2660	−1.23836	−0.619178	+	0.785250i	$0.712534\pi$
$522$	0	0
$523$	−16.3966	−0.716972	−0.358486	−	0.933535i	$-0.616707\pi$
$523$	−16.3966	−0.716972	−0.358486	+	0.933535i	$0.616707\pi$
$524$	−36.7955	−1.60742
$525$	0	0
$526$	14.9369	0.651278
$527$	0	0
$528$	0	0
$529$	−17.7718	−0.772689
$530$	−102.225	−4.44039
$531$	0	0
$532$	−18.5860	−0.805807
$533$	−20.3469	−0.881320
$534$	0	0
$535$	−14.0007	−0.605302
$536$	77.6678	3.35474
$537$	0	0
$538$	−55.6887	−2.40091
$539$	4.55623	0.196251
$540$	0	0
$541$	−31.6709	−1.36164	−0.680818	−	0.732452i	$-0.738376\pi$
$541$	−31.6709	−1.36164	−0.680818	+	0.732452i	$0.738376\pi$
$542$	−9.41880	−0.404572
$543$	0	0
$544$	67.4801	2.89319
$545$	67.6245	2.89672
$546$	0	0
$547$	−11.9668	−0.511665	−0.255833	−	0.966721i	$-0.582350\pi$
$547$	−11.9668	−0.511665	−0.255833	+	0.966721i	$0.582350\pi$
$548$	−11.6766	−0.498800
$549$	0	0
$550$	−79.5766	−3.39316
$551$	−3.25177	−0.138530
$552$	0	0
$553$	−41.1430	−1.74958
$554$	−50.2854	−2.13642
$555$	0	0
$556$	−8.55042	−0.362618
$557$	4.56973	0.193626	0.0968129	−	0.995303i	$-0.469135\pi$
$557$	4.56973	0.193626	0.0968129	+	0.995303i	$0.469135\pi$
$558$	0	0
$559$	23.2588	0.983741
$560$	111.026	4.69170
$561$	0	0
$562$	−32.0231	−1.35081
$563$	15.4106	0.649479	0.324740	−	0.945803i	$-0.394723\pi$
$563$	15.4106	0.649479	0.324740	+	0.945803i	$0.394723\pi$
$564$	0	0
$565$	0.696519	0.0293028
$566$	−25.6327	−1.07742
$567$	0	0
$568$	20.2199	0.848409
$569$	−25.7391	−1.07904	−0.539520	−	0.841972i	$-0.681394\pi$
$569$	−25.7391	−1.07904	−0.539520	+	0.841972i	$0.681394\pi$
$570$	0	0
$571$	−7.44664	−0.311632	−0.155816	−	0.987786i	$-0.549801\pi$
$571$	−7.44664	−0.311632	−0.155816	+	0.987786i	$0.549801\pi$
$572$	−98.9458	−4.13713
$573$	0	0
$574$	31.4583	1.31304
$575$	15.3273	0.639191
$576$	0	0
$577$	−18.2692	−0.760556	−0.380278	−	0.924872i	$-0.624172\pi$
$577$	−18.2692	−0.760556	−0.380278	+	0.924872i	$0.624172\pi$
$578$	−8.64801	−0.359710
$579$	0	0
$580$	40.6981	1.68990
$581$	−17.8623	−0.741055
$582$	0	0
$583$	−48.7502	−2.01903
$584$	53.8333	2.22764
$585$	0	0
$586$	22.8761	0.945001
$587$	28.0767	1.15885	0.579425	−	0.815026i	$-0.303277\pi$
$587$	28.0767	1.15885	0.579425	+	0.815026i	$0.303277\pi$
$588$	0	0
$589$	0	0
$590$	32.8646	1.35301
$591$	0	0
$592$	−34.3625	−1.41229
$593$	12.2441	0.502805	0.251402	−	0.967883i	$-0.419108\pi$
$593$	12.2441	0.502805	0.251402	+	0.967883i	$0.419108\pi$
$594$	0	0
$595$	31.0309	1.27214
$596$	−96.9150	−3.96979
$597$	0	0
$598$	26.2817	1.07474
$599$	32.2385	1.31723	0.658615	−	0.752480i	$-0.271142\pi$
$599$	32.2385	1.31723	0.658615	+	0.752480i	$0.271142\pi$
$600$	0	0
$601$	8.94091	0.364707	0.182354	−	0.983233i	$-0.441628\pi$
$601$	8.94091	0.364707	0.182354	+	0.983233i	$0.441628\pi$
$602$	−35.9604	−1.46564
$603$	0	0
$604$	85.7305	3.48833
$605$	−28.6243	−1.16374
$606$	0	0
$607$	34.4739	1.39925	0.699626	−	0.714510i	$-0.253350\pi$
$607$	34.4739	1.39925	0.699626	+	0.714510i	$0.253350\pi$
$608$	26.2045	1.06273
$609$	0	0
$610$	−86.9238	−3.51944
$611$	−0.962189	−0.0389260
$612$	0	0
$613$	15.8778	0.641299	0.320649	−	0.947198i	$-0.396099\pi$
$613$	15.8778	0.641299	0.320649	+	0.947198i	$0.396099\pi$
$614$	20.9859	0.846922
$615$	0	0
$616$	94.9947	3.82744
$617$	7.41898	0.298677	0.149338	−	0.988786i	$-0.452286\pi$
$617$	7.41898	0.298677	0.149338	+	0.988786i	$0.452286\pi$
$618$	0	0
$619$	−38.1293	−1.53255	−0.766273	−	0.642516i	$-0.777891\pi$
$619$	−38.1293	−1.53255	−0.766273	+	0.642516i	$0.777891\pi$
$620$	0	0
$621$	0	0
$622$	−1.21686	−0.0487918
$623$	−34.8059	−1.39447
$624$	0	0
$625$	−13.5820	−0.543281
$626$	−57.4539	−2.29632
$627$	0	0
$628$	−27.3693	−1.09215
$629$	−9.60407	−0.382939
$630$	0	0
$631$	−10.3347	−0.411418	−0.205709	−	0.978613i	$-0.565950\pi$
$631$	−10.3347	−0.411418	−0.205709	+	0.978613i	$0.565950\pi$
$632$	148.893	5.92265
$633$	0	0
$634$	42.6486	1.69379
$635$	35.4888	1.40833
$636$	0	0
$637$	−4.41153	−0.174791
$638$	26.7650	1.05964
$639$	0	0
$640$	−82.7104	−3.26941
$641$	−28.4417	−1.12338	−0.561690	−	0.827348i	$-0.689849\pi$
$641$	−28.4417	−1.12338	−0.561690	+	0.827348i	$0.689849\pi$
$642$	0	0
$643$	24.3141	0.958853	0.479426	−	0.877582i	$-0.340845\pi$
$643$	24.3141	0.958853	0.479426	+	0.877582i	$0.340845\pi$
$644$	−29.4655	−1.16110
$645$	0	0
$646$	14.4495	0.568510
$647$	−6.87160	−0.270151	−0.135075	−	0.990835i	$-0.543128\pi$
$647$	−6.87160	−0.270151	−0.135075	+	0.990835i	$0.543128\pi$
$648$	0	0
$649$	15.6728	0.615210
$650$	77.0494	3.02213
$651$	0	0
$652$	−3.49907	−0.137034
$653$	−25.2385	−0.987658	−0.493829	−	0.869559i	$-0.664403\pi$
$653$	−25.2385	−0.987658	−0.493829	+	0.869559i	$0.664403\pi$
$654$	0	0
$655$	23.8563	0.932142
$656$	−63.4535	−2.47744
$657$	0	0
$658$	1.48764	0.0579943
$659$	3.07095	0.119627	0.0598137	−	0.998210i	$-0.480949\pi$
$659$	3.07095	0.119627	0.0598137	+	0.998210i	$0.480949\pi$
$660$	0	0
$661$	−35.0072	−1.36162	−0.680811	−	0.732459i	$-0.738372\pi$
$661$	−35.0072	−1.36162	−0.680811	+	0.732459i	$0.738372\pi$
$662$	−16.2097	−0.630009
$663$	0	0
$664$	64.6423	2.50861
$665$	12.0502	0.467288
$666$	0	0
$667$	−5.15522	−0.199611
$668$	−83.9993	−3.25003
$669$	0	0
$670$	−81.0933	−3.13291
$671$	−41.4530	−1.60028
$672$	0	0
$673$	36.7497	1.41660	0.708298	−	0.705914i	$-0.249464\pi$
$673$	36.7497	1.41660	0.708298	+	0.705914i	$0.249464\pi$
$674$	1.53626	0.0591745
$675$	0	0
$676$	27.2089	1.04650
$677$	21.4827	0.825648	0.412824	−	0.910811i	$-0.364542\pi$
$677$	21.4827	0.825648	0.412824	+	0.910811i	$0.364542\pi$
$678$	0	0
$679$	−38.8623	−1.49140
$680$	−112.298	−4.30644
$681$	0	0
$682$	0	0
$683$	−2.76928	−0.105963	−0.0529817	−	0.998595i	$-0.516872\pi$
$683$	−2.76928	−0.105963	−0.0529817	+	0.998595i	$0.516872\pi$
$684$	0	0
$685$	7.57051	0.289254
$686$	52.9368	2.02114
$687$	0	0
$688$	72.5345	2.76535
$689$	47.2020	1.79825
$690$	0	0
$691$	19.0641	0.725232	0.362616	−	0.931939i	$-0.381884\pi$
$691$	19.0641	0.725232	0.362616	+	0.931939i	$0.381884\pi$
$692$	21.6344	0.822416
$693$	0	0
$694$	−63.6510	−2.41616
$695$	5.54365	0.210283
$696$	0	0
$697$	−17.7348	−0.671753
$698$	−32.2762	−1.22167
$699$	0	0
$700$	−86.3835	−3.26499
$701$	−44.2887	−1.67276	−0.836381	−	0.548149i	$-0.815333\pi$
$701$	−44.2887	−1.67276	−0.836381	+	0.548149i	$0.815333\pi$
$702$	0	0
$703$	−3.72955	−0.140663
$704$	−98.7269	−3.72091
$705$	0	0
$706$	26.1138	0.982806
$707$	−3.45695	−0.130012
$708$	0	0
$709$	−18.4730	−0.693769	−0.346885	−	0.937908i	$-0.612760\pi$
$709$	−18.4730	−0.693769	−0.346885	+	0.937908i	$0.612760\pi$
$710$	−21.1117	−0.792308
$711$	0	0
$712$	125.960	4.72054
$713$	0	0
$714$	0	0
$715$	64.1513	2.39912
$716$	16.0837	0.601075
$717$	0	0
$718$	−70.1671	−2.61861
$719$	−18.7959	−0.700970	−0.350485	−	0.936568i	$-0.613983\pi$
$719$	−18.7959	−0.700970	−0.350485	+	0.936568i	$0.613983\pi$
$720$	0	0
$721$	−6.64306	−0.247400
$722$	−45.6413	−1.69859
$723$	0	0
$724$	111.705	4.15148
$725$	−15.1134	−0.561299
$726$	0	0
$727$	14.4617	0.536355	0.268178	−	0.963369i	$-0.413579\pi$
$727$	14.4617	0.536355	0.268178	+	0.963369i	$0.413579\pi$
$728$	−91.9779	−3.40893
$729$	0	0
$730$	−56.2076	−2.08034
$731$	20.2729	0.749819
$732$	0	0
$733$	41.0580	1.51651	0.758256	−	0.651957i	$-0.226052\pi$
$733$	41.0580	1.51651	0.758256	+	0.651957i	$0.226052\pi$
$734$	73.4911	2.71261
$735$	0	0
$736$	41.5436	1.53132
$737$	−38.6725	−1.42452
$738$	0	0
$739$	−6.32146	−0.232539	−0.116269	−	0.993218i	$-0.537094\pi$
$739$	−6.32146	−0.232539	−0.116269	+	0.993218i	$0.537094\pi$
$740$	46.6778	1.71591
$741$	0	0
$742$	−72.9790	−2.67914
$743$	−18.1307	−0.665151	−0.332575	−	0.943077i	$-0.607918\pi$
$743$	−18.1307	−0.665151	−0.332575	+	0.943077i	$0.607918\pi$
$744$	0	0
$745$	62.8347	2.30208
$746$	24.2629	0.888327
$747$	0	0
$748$	−86.2434	−3.15337
$749$	−9.99512	−0.365214
$750$	0	0
$751$	2.56898	0.0937435	0.0468717	−	0.998901i	$-0.485075\pi$
$751$	2.56898	0.0937435	0.0468717	+	0.998901i	$0.485075\pi$
$752$	−3.00067	−0.109423
$753$	0	0
$754$	−25.9150	−0.943770
$755$	−55.5832	−2.02288
$756$	0	0
$757$	−11.8154	−0.429439	−0.214719	−	0.976676i	$-0.568884\pi$
$757$	−11.8154	−0.429439	−0.214719	+	0.976676i	$0.568884\pi$
$758$	−39.8432	−1.44717
$759$	0	0
$760$	−43.6087	−1.58186
$761$	18.9669	0.687548	0.343774	−	0.939052i	$-0.388295\pi$
$761$	18.9669	0.687548	0.343774	+	0.939052i	$0.388295\pi$
$762$	0	0
$763$	48.2773	1.74776
$764$	139.690	5.05381
$765$	0	0
$766$	−46.8899	−1.69420
$767$	−15.1750	−0.547939
$768$	0	0
$769$	18.9894	0.684776	0.342388	−	0.939559i	$-0.388764\pi$
$769$	18.9894	0.684776	0.342388	+	0.939559i	$0.388764\pi$
$770$	−99.1844	−3.57436
$771$	0	0
$772$	71.9681	2.59019
$773$	−16.8519	−0.606122	−0.303061	−	0.952971i	$-0.598009\pi$
$773$	−16.8519	−0.606122	−0.303061	+	0.952971i	$0.598009\pi$
$774$	0	0
$775$	0	0
$776$	140.639	5.04866
$777$	0	0
$778$	63.9644	2.29323
$779$	−6.88694	−0.246750
$780$	0	0
$781$	−10.0679	−0.360259
$782$	22.9077	0.819178
$783$	0	0
$784$	−13.7578	−0.491348
$785$	17.7448	0.633341
$786$	0	0
$787$	54.3176	1.93621	0.968107	−	0.250536i	$-0.0806070\pi$
$787$	54.3176	1.93621	0.968107	+	0.250536i	$0.0806070\pi$
$788$	−18.5086	−0.659340
$789$	0	0
$790$	−155.460	−5.53102
$791$	0.497247	0.0176801
$792$	0	0
$793$	40.1366	1.42529
$794$	−34.9452	−1.24016
$795$	0	0
$796$	27.9944	0.992236
$797$	46.3454	1.64164	0.820819	−	0.571189i	$-0.193518\pi$
$797$	46.3454	1.64164	0.820819	+	0.571189i	$0.193518\pi$
$798$	0	0
$799$	−0.838666	−0.0296699
$800$	121.792	4.30601
$801$	0	0
$802$	−30.0473	−1.06101
$803$	−26.8048	−0.945921
$804$	0	0
$805$	19.1039	0.673325
$806$	0	0
$807$	0	0
$808$	12.5104	0.440114
$809$	19.9112	0.700039	0.350019	−	0.936742i	$-0.386175\pi$
$809$	19.9112	0.700039	0.350019	+	0.936742i	$0.386175\pi$
$810$	0	0
$811$	0.0153026	0.000537349 0	0.000268674	−	1.00000i	$-0.499914\pi$
$811$	0.0153026	0.000537349 0	0.000268674		1.00000i	$0.499914\pi$
$812$	29.0545	1.01961
$813$	0	0
$814$	30.6976	1.07595
$815$	2.26861	0.0794661
$816$	0	0
$817$	7.87256	0.275426
$818$	3.50116	0.122415
$819$	0	0
$820$	86.1947	3.01005
$821$	−14.9304	−0.521073	−0.260537	−	0.965464i	$-0.583899\pi$
$821$	−14.9304	−0.521073	−0.260537	+	0.965464i	$0.583899\pi$
$822$	0	0
$823$	−12.4051	−0.432416	−0.216208	−	0.976347i	$-0.569369\pi$
$823$	−12.4051	−0.432416	−0.216208	+	0.976347i	$0.569369\pi$
$824$	24.0406	0.837496
$825$	0	0
$826$	23.4621	0.816352
$827$	37.7983	1.31437	0.657187	−	0.753727i	$-0.271746\pi$
$827$	37.7983	1.31437	0.657187	+	0.753727i	$0.271746\pi$
$828$	0	0
$829$	39.1904	1.36114	0.680570	−	0.732683i	$-0.261732\pi$
$829$	39.1904	1.36114	0.680570	+	0.732683i	$0.261732\pi$
$830$	−67.4933	−2.34273
$831$	0	0
$832$	95.5916	3.31404
$833$	−3.84519	−0.133228
$834$	0	0
$835$	54.4608	1.88469
$836$	−33.4909	−1.15831
$837$	0	0
$838$	4.73911	0.163710
$839$	−18.3970	−0.635135	−0.317567	−	0.948236i	$-0.602866\pi$
$839$	−18.3970	−0.635135	−0.317567	+	0.948236i	$0.602866\pi$
$840$	0	0
$841$	−23.9167	−0.824714
$842$	−79.7167	−2.74722
$843$	0	0
$844$	91.6235	3.15381
$845$	−17.6409	−0.606864
$846$	0	0
$847$	−20.4350	−0.702154
$848$	147.204	5.05499
$849$	0	0
$850$	67.1580	2.30350
$851$	−5.91267	−0.202684
$852$	0	0
$853$	17.3889	0.595386	0.297693	−	0.954662i	$-0.403783\pi$
$853$	17.3889	0.595386	0.297693	+	0.954662i	$0.403783\pi$
$854$	−62.0552	−2.12348
$855$	0	0
$856$	36.1715	1.23632
$857$	47.7190	1.63005	0.815025	−	0.579426i	$-0.196723\pi$
$857$	47.7190	1.63005	0.815025	+	0.579426i	$0.196723\pi$
$858$	0	0
$859$	21.3817	0.729533	0.364767	−	0.931099i	$-0.381149\pi$
$859$	21.3817	0.729533	0.364767	+	0.931099i	$0.381149\pi$
$860$	−98.5303	−3.35986
$861$	0	0
$862$	100.736	3.43109
$863$	−41.3756	−1.40844	−0.704221	−	0.709981i	$-0.748703\pi$
$863$	−41.3756	−1.40844	−0.704221	+	0.709981i	$0.748703\pi$
$864$	0	0
$865$	−14.0266	−0.476919
$866$	−85.4865	−2.90495
$867$	0	0
$868$	0	0
$869$	−74.1371	−2.51493
$870$	0	0
$871$	37.4443	1.26875
$872$	−174.712	−5.91648
$873$	0	0
$874$	8.89575	0.300903
$875$	14.2314	0.481109
$876$	0	0
$877$	36.2161	1.22293	0.611465	−	0.791271i	$-0.290580\pi$
$877$	36.2161	1.22293	0.611465	+	0.791271i	$0.290580\pi$
$878$	13.5095	0.455924
$879$	0	0
$880$	200.062	6.74407
$881$	40.1608	1.35305	0.676525	−	0.736419i	$-0.263485\pi$
$881$	40.1608	1.35305	0.676525	+	0.736419i	$0.263485\pi$
$882$	0	0
$883$	−18.1181	−0.609723	−0.304862	−	0.952397i	$-0.598610\pi$
$883$	−18.1181	−0.609723	−0.304862	+	0.952397i	$0.598610\pi$
$884$	83.5045	2.80856
$885$	0	0
$886$	13.3787	0.449467
$887$	46.1033	1.54800	0.773999	−	0.633187i	$-0.218253\pi$
$887$	46.1033	1.54800	0.773999	+	0.633187i	$0.218253\pi$
$888$	0	0
$889$	25.3356	0.849728
$890$	−131.515	−4.40839
$891$	0	0
$892$	85.1934	2.85249
$893$	−0.325679	−0.0108984
$894$	0	0
$895$	−10.4278	−0.348564
$896$	−59.0472	−1.97263
$897$	0	0
$898$	−47.5561	−1.58697
$899$	0	0
$900$	0	0
$901$	41.1423	1.37065
$902$	56.6858	1.88743
$903$	0	0
$904$	−1.79949	−0.0598502
$905$	−72.4237	−2.40745
$906$	0	0
$907$	43.6641	1.44984	0.724921	−	0.688832i	$-0.241876\pi$
$907$	43.6641	1.44984	0.724921	+	0.688832i	$0.241876\pi$
$908$	115.156	3.82159
$909$	0	0
$910$	96.0345	3.18351
$911$	−23.5994	−0.781885	−0.390942	−	0.920415i	$-0.627851\pi$
$911$	−23.5994	−0.781885	−0.390942	+	0.920415i	$0.627851\pi$
$912$	0	0
$913$	−32.1868	−1.06523
$914$	46.4917	1.53781
$915$	0	0
$916$	36.7055	1.21278
$917$	17.0311	0.562416
$918$	0	0
$919$	−10.7600	−0.354940	−0.177470	−	0.984126i	$-0.556791\pi$
$919$	−10.7600	−0.354940	−0.177470	+	0.984126i	$0.556791\pi$
$920$	−69.1355	−2.27933
$921$	0	0
$922$	91.4478	3.01167
$923$	9.74820	0.320866
$924$	0	0
$925$	−17.3340	−0.569940
$926$	80.6376	2.64992
$927$	0	0
$928$	−40.9640	−1.34471
$929$	19.4549	0.638294	0.319147	−	0.947705i	$-0.396604\pi$
$929$	19.4549	0.638294	0.319147	+	0.947705i	$0.396604\pi$
$930$	0	0
$931$	−1.49320	−0.0489377
$932$	−145.143	−4.75431
$933$	0	0
$934$	80.3977	2.63069
$935$	55.9158	1.82864
$936$	0	0
$937$	12.0718	0.394370	0.197185	−	0.980366i	$-0.436820\pi$
$937$	12.0718	0.394370	0.197185	+	0.980366i	$0.436820\pi$
$938$	−57.8927	−1.89026
$939$	0	0
$940$	4.07609	0.132947
$941$	38.1458	1.24352	0.621758	−	0.783209i	$-0.286419\pi$
$941$	38.1458	1.24352	0.621758	+	0.783209i	$0.286419\pi$
$942$	0	0
$943$	−10.9183	−0.355548
$944$	−47.3247	−1.54029
$945$	0	0
$946$	−64.7983	−2.10678
$947$	33.2556	1.08066	0.540331	−	0.841453i	$-0.318299\pi$
$947$	33.2556	1.08066	0.540331	+	0.841453i	$0.318299\pi$
$948$	0	0
$949$	25.9535	0.842488
$950$	26.0795	0.846130
$951$	0	0
$952$	−80.1700	−2.59833
$953$	3.48755	0.112973	0.0564864	−	0.998403i	$-0.482010\pi$
$953$	3.48755	0.112973	0.0564864	+	0.998403i	$0.482010\pi$
$954$	0	0
$955$	−90.5679	−2.93071
$956$	−101.306	−3.27648
$957$	0	0
$958$	−88.2666	−2.85176
$959$	5.40461	0.174524
$960$	0	0
$961$	0	0
$962$	−29.7227	−0.958298
$963$	0	0
$964$	11.8248	0.380852
$965$	−46.6604	−1.50205
$966$	0	0
$967$	8.79415	0.282801	0.141400	−	0.989952i	$-0.454839\pi$
$967$	8.79415	0.282801	0.141400	+	0.989952i	$0.454839\pi$
$968$	73.9524	2.37692
$969$	0	0
$970$	−146.842	−4.71482
$971$	16.4981	0.529450	0.264725	−	0.964324i	$-0.414719\pi$
$971$	16.4981	0.529450	0.264725	+	0.964324i	$0.414719\pi$
$972$	0	0
$973$	3.95763	0.126876
$974$	100.070	3.20645
$975$	0	0
$976$	125.169	4.00658
$977$	−54.9537	−1.75812	−0.879062	−	0.476708i	$-0.841830\pi$
$977$	−54.9537	−1.75812	−0.879062	+	0.476708i	$0.841830\pi$
$978$	0	0
$979$	−62.7181	−2.00448
$980$	18.6884	0.596980
$981$	0	0
$982$	−77.9163	−2.48641
$983$	−54.4216	−1.73578	−0.867889	−	0.496758i	$-0.834524\pi$
$983$	−54.4216	−1.73578	−0.867889	+	0.496758i	$0.834524\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	−22.5881	−0.719353
$987$	0	0
$988$	32.4273	1.03165
$989$	12.4808	0.396867
$990$	0	0
$991$	34.9449	1.11006	0.555031	−	0.831830i	$-0.312706\pi$
$991$	34.9449	1.11006	0.555031	+	0.831830i	$0.312706\pi$
$992$	0	0
$993$	0	0
$994$	−15.0717	−0.478045
$995$	−18.1501	−0.575398
$996$	0	0
$997$	−36.9542	−1.17035	−0.585176	−	0.810906i	$-0.698975\pi$
$997$	−36.9542	−1.17035	−0.585176	+	0.810906i	$0.698975\pi$
$998$	−72.1479	−2.28380
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bo.1.12		12
3.2	odd	2	inner	8649.2.a.bo.1.1		12
31.15	odd	10		279.2.i.d.163.6	yes	24
31.29	odd	10		279.2.i.d.190.6	yes	24
31.30	odd	2		8649.2.a.bn.1.12		12
93.29	even	10		279.2.i.d.190.1	yes	24
93.77	even	10		279.2.i.d.163.1	✓	24
93.92	even	2		8649.2.a.bn.1.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
279.2.i.d.163.1	✓	24	93.77	even	10
279.2.i.d.163.6	yes	24	31.15	odd	10
279.2.i.d.190.1	yes	24	93.29	even	10
279.2.i.d.190.6	yes	24	31.29	odd	10
8649.2.a.bn.1.1		12	93.92	even	2
8649.2.a.bn.1.12		12	31.30	odd	2
8649.2.a.bo.1.1		12	3.2	odd	2	inner
8649.2.a.bo.1.12		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q+2.69750 q^{2} +5.27650 q^{4} -3.42101 q^{5} -2.44227 q^{7} +8.83837 q^{8} +O(q^{10})\)	\(q+2.69750 q^{2} +5.27650 q^{4} -3.42101 q^{5} -2.44227 q^{7} +8.83837 q^{8} -9.22818 q^{10} -4.40082 q^{11} +4.26106 q^{13} -6.58802 q^{14} +13.2885 q^{16} +3.71404 q^{17} +1.44227 q^{19} -18.0510 q^{20} -11.8712 q^{22} +2.28652 q^{23} +6.70333 q^{25} +11.4942 q^{26} -12.8867 q^{28} -2.25462 q^{29} +18.1689 q^{32} +10.0186 q^{34} +8.35504 q^{35} -2.58589 q^{37} +3.89052 q^{38} -30.2362 q^{40} -4.77507 q^{41} +5.45845 q^{43} -23.2209 q^{44} +6.16788 q^{46} -0.225810 q^{47} -1.03531 q^{49} +18.0822 q^{50} +22.4835 q^{52} +11.0775 q^{53} +15.0553 q^{55} -21.5857 q^{56} -6.08183 q^{58} -3.56133 q^{59} +9.41939 q^{61} +22.4338 q^{64} -14.5771 q^{65} +8.78757 q^{67} +19.5971 q^{68} +22.5377 q^{70} +2.28774 q^{71} +6.09087 q^{73} -6.97543 q^{74} +7.61015 q^{76} +10.7480 q^{77} +16.8462 q^{79} -45.4601 q^{80} -12.8808 q^{82} +7.31383 q^{83} -12.7058 q^{85} +14.7242 q^{86} -38.8961 q^{88} +14.2515 q^{89} -10.4067 q^{91} +12.0648 q^{92} -0.609122 q^{94} -4.93403 q^{95} +15.9124 q^{97} -2.79276 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 14 q^{4} - 4 q^{7}+O(q^{10}) \) 12 * q + 14 * q^4 - 4 * q^7	\( 12 q + 14 q^{4} - 4 q^{7} - 16 q^{10} + 10 q^{13} + 26 q^{16} - 8 q^{19} - 20 q^{22} + 14 q^{25} - 24 q^{28} + 56 q^{34} + 14 q^{37} - 82 q^{40} + 54 q^{43} + 18 q^{46} + 12 q^{49} - 2 q^{52} + 16 q^{55} + 56 q^{58} + 22 q^{61} + 42 q^{64} - 38 q^{67} + 42 q^{70} + 16 q^{73} + 10 q^{76} + 94 q^{79} - 26 q^{82} - 8 q^{85} - 60 q^{88} + 8 q^{91} + 86 q^{94} + 68 q^{97}+O(q^{100}) \) 12 * q + 14 * q^4 - 4 * q^7 - 16 * q^10 + 10 * q^13 + 26 * q^16 - 8 * q^19 - 20 * q^22 + 14 * q^25 - 24 * q^28 + 56 * q^34 + 14 * q^37 - 82 * q^40 + 54 * q^43 + 18 * q^46 + 12 * q^49 - 2 * q^52 + 16 * q^55 + 56 * q^58 + 22 * q^61 + 42 * q^64 - 38 * q^67 + 42 * q^70 + 16 * q^73 + 10 * q^76 + 94 * q^79 - 26 * q^82 - 8 * q^85 - 60 * q^88 + 8 * q^91 + 86 * q^94 + 68 * q^97

Introduction

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