LMFDB - Embedded newform 9610.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9610,2,Mod(1,9610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9610, 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("9610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9610 = 2 \cdot 5 \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9610.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:	$76.7362363425$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.874032$ of defining polynomial
Character	$\chi$	$=$	9610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -0.874032 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.874032 q^{6} +3.23607 q^{7} -1.00000 q^{8} -2.23607 q^{9} +1.00000 q^{10} +5.11667 q^{11} -0.874032 q^{12} +2.28825 q^{13} -3.23607 q^{14} +0.874032 q^{15} +1.00000 q^{16} -6.32456 q^{17} +2.23607 q^{18} -7.23607 q^{19} -1.00000 q^{20} -2.82843 q^{21} -5.11667 q^{22} +3.16228 q^{23} +0.874032 q^{24} +1.00000 q^{25} -2.28825 q^{26} +4.57649 q^{27} +3.23607 q^{28} -2.62210 q^{29} -0.874032 q^{30} -1.00000 q^{32} -4.47214 q^{33} +6.32456 q^{34} -3.23607 q^{35} -2.23607 q^{36} +2.28825 q^{37} +7.23607 q^{38} -2.00000 q^{39} +1.00000 q^{40} +3.23607 q^{41} +2.82843 q^{42} +0.206331 q^{43} +5.11667 q^{44} +2.23607 q^{45} -3.16228 q^{46} -11.7082 q^{47} -0.874032 q^{48} +3.47214 q^{49} -1.00000 q^{50} +5.52786 q^{51} +2.28825 q^{52} -1.62054 q^{53} -4.57649 q^{54} -5.11667 q^{55} -3.23607 q^{56} +6.32456 q^{57} +2.62210 q^{58} +9.70820 q^{59} +0.874032 q^{60} -8.94665 q^{61} -7.23607 q^{63} +1.00000 q^{64} -2.28825 q^{65} +4.47214 q^{66} +5.23607 q^{67} -6.32456 q^{68} -2.76393 q^{69} +3.23607 q^{70} +8.94427 q^{71} +2.23607 q^{72} +3.90879 q^{73} -2.28825 q^{74} -0.874032 q^{75} -7.23607 q^{76} +16.5579 q^{77} +2.00000 q^{78} +3.90879 q^{79} -1.00000 q^{80} +2.70820 q^{81} -3.23607 q^{82} -12.8554 q^{83} -2.82843 q^{84} +6.32456 q^{85} -0.206331 q^{86} +2.29180 q^{87} -5.11667 q^{88} +16.2241 q^{89} -2.23607 q^{90} +7.40492 q^{91} +3.16228 q^{92} +11.7082 q^{94} +7.23607 q^{95} +0.874032 q^{96} -11.4164 q^{97} -3.47214 q^{98} -11.4412 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8} + 4 q^{10} - 4 q^{14} + 4 q^{16} - 20 q^{19} - 4 q^{20} + 4 q^{25} + 4 q^{28} - 4 q^{32} - 4 q^{35} + 20 q^{38} - 8 q^{39} + 4 q^{40} + 4 q^{41} - 20 q^{47}+ \cdots + 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 - 4 * q^8 + 4 * q^10 - 4 * q^14 + 4 * q^16 - 20 * q^19 - 4 * q^20 + 4 * q^25 + 4 * q^28 - 4 * q^32 - 4 * q^35 + 20 * q^38 - 8 * q^39 + 4 * q^40 + 4 * q^41 - 20 * q^47 - 4 * q^49 - 4 * q^50 + 40 * q^51 - 4 * q^56 + 12 * q^59 - 20 * q^63 + 4 * q^64 + 12 * q^67 - 20 * q^69 + 4 * q^70 - 20 * q^76 + 8 * q^78 - 4 * q^80 - 16 * q^81 - 4 * q^82 + 36 * q^87 + 20 * q^94 + 20 * q^95 + 8 * q^97 + 4 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.874032	−0.504623	−0.252311	−	0.967646i	$-0.581191\pi$
$3$	−0.874032	−0.504623	−0.252311	+	0.967646i	$0.581191\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.874032	0.356822
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	−1.00000	−0.353553
$9$	−2.23607	−0.745356
$10$	1.00000	0.316228
$11$	5.11667	1.54273	0.771367	−	0.636390i	$-0.219573\pi$
$11$	5.11667	1.54273	0.771367	+	0.636390i	$0.219573\pi$
$12$	−0.874032	−0.252311
$13$	2.28825	0.634645	0.317323	−	0.948318i	$-0.397216\pi$
$13$	2.28825	0.634645	0.317323	+	0.948318i	$0.397216\pi$
$14$	−3.23607	−0.864876
$15$	0.874032	0.225674
$16$	1.00000	0.250000
$17$	−6.32456	−1.53393	−0.766965	−	0.641689i	$-0.778234\pi$
$17$	−6.32456	−1.53393	−0.766965	+	0.641689i	$0.778234\pi$
$18$	2.23607	0.527046
$19$	−7.23607	−1.66007	−0.830034	−	0.557713i	$-0.811679\pi$
$19$	−7.23607	−1.66007	−0.830034	+	0.557713i	$0.811679\pi$
$20$	−1.00000	−0.223607
$21$	−2.82843	−0.617213
$22$	−5.11667	−1.09088
$23$	3.16228	0.659380	0.329690	−	0.944089i	$-0.393056\pi$
$23$	3.16228	0.659380	0.329690	+	0.944089i	$0.393056\pi$
$24$	0.874032	0.178411
$25$	1.00000	0.200000
$26$	−2.28825	−0.448762
$27$	4.57649	0.880746
$28$	3.23607	0.611559
$29$	−2.62210	−0.486911	−0.243456	−	0.969912i	$-0.578281\pi$
$29$	−2.62210	−0.486911	−0.243456	+	0.969912i	$0.578281\pi$
$30$	−0.874032	−0.159576
$31$	0	0
$31$	0	0
$32$	−1.00000	−0.176777
$33$	−4.47214	−0.778499
$34$	6.32456	1.08465
$35$	−3.23607	−0.546995
$36$	−2.23607	−0.372678
$37$	2.28825	0.376185	0.188093	−	0.982151i	$-0.439769\pi$
$37$	2.28825	0.376185	0.188093	+	0.982151i	$0.439769\pi$
$38$	7.23607	1.17385
$39$	−2.00000	−0.320256
$40$	1.00000	0.158114
$41$	3.23607	0.505389	0.252694	−	0.967546i	$-0.418683\pi$
$41$	3.23607	0.505389	0.252694	+	0.967546i	$0.418683\pi$
$42$	2.82843	0.436436
$43$	0.206331	0.0314652	0.0157326	−	0.999876i	$-0.494992\pi$
$43$	0.206331	0.0314652	0.0157326	+	0.999876i	$0.494992\pi$
$44$	5.11667	0.771367
$45$	2.23607	0.333333
$46$	−3.16228	−0.466252
$47$	−11.7082	−1.70782	−0.853909	−	0.520423i	$-0.825774\pi$
$47$	−11.7082	−1.70782	−0.853909	+	0.520423i	$0.825774\pi$
$48$	−0.874032	−0.126156
$49$	3.47214	0.496019
$50$	−1.00000	−0.141421
$51$	5.52786	0.774056
$52$	2.28825	0.317323
$53$	−1.62054	−0.222599	−0.111299	−	0.993787i	$-0.535501\pi$
$53$	−1.62054	−0.222599	−0.111299	+	0.993787i	$0.535501\pi$
$54$	−4.57649	−0.622782
$55$	−5.11667	−0.689932
$56$	−3.23607	−0.432438
$57$	6.32456	0.837708
$58$	2.62210	0.344298
$59$	9.70820	1.26390	0.631950	−	0.775009i	$-0.282255\pi$
$59$	9.70820	1.26390	0.631950	+	0.775009i	$0.282255\pi$
$60$	0.874032	0.112837
$61$	−8.94665	−1.14550	−0.572751	−	0.819730i	$-0.694124\pi$
$61$	−8.94665	−1.14550	−0.572751	+	0.819730i	$0.694124\pi$
$62$	0	0
$63$	−7.23607	−0.911659
$64$	1.00000	0.125000
$65$	−2.28825	−0.283822
$66$	4.47214	0.550482
$67$	5.23607	0.639688	0.319844	−	0.947470i	$-0.396370\pi$
$67$	5.23607	0.639688	0.319844	+	0.947470i	$0.396370\pi$
$68$	−6.32456	−0.766965
$69$	−2.76393	−0.332738
$70$	3.23607	0.386784
$71$	8.94427	1.06149	0.530745	−	0.847532i	$-0.321912\pi$
$71$	8.94427	1.06149	0.530745	+	0.847532i	$0.321912\pi$
$72$	2.23607	0.263523
$73$	3.90879	0.457489	0.228745	−	0.973486i	$-0.426538\pi$
$73$	3.90879	0.457489	0.228745	+	0.973486i	$0.426538\pi$
$74$	−2.28825	−0.266003
$75$	−0.874032	−0.100925
$76$	−7.23607	−0.830034
$77$	16.5579	1.88695
$78$	2.00000	0.226455
$79$	3.90879	0.439773	0.219887	−	0.975525i	$-0.429431\pi$
$79$	3.90879	0.439773	0.219887	+	0.975525i	$0.429431\pi$
$80$	−1.00000	−0.111803
$81$	2.70820	0.300912
$82$	−3.23607	−0.357364
$83$	−12.8554	−1.41107	−0.705534	−	0.708676i	$-0.749293\pi$
$83$	−12.8554	−1.41107	−0.705534	+	0.708676i	$0.749293\pi$
$84$	−2.82843	−0.308607
$85$	6.32456	0.685994
$86$	−0.206331	−0.0222492
$87$	2.29180	0.245706
$88$	−5.11667	−0.545439
$89$	16.2241	1.71975	0.859873	−	0.510508i	$-0.170543\pi$
$89$	16.2241	1.71975	0.859873	+	0.510508i	$0.170543\pi$
$90$	−2.23607	−0.235702
$91$	7.40492	0.776246
$92$	3.16228	0.329690
$93$	0	0
$94$	11.7082	1.20761
$95$	7.23607	0.742405
$96$	0.874032	0.0892055
$97$	−11.4164	−1.15916	−0.579580	−	0.814915i	$-0.696783\pi$
$97$	−11.4164	−1.15916	−0.579580	+	0.814915i	$0.696783\pi$
$98$	−3.47214	−0.350739
$99$	−11.4412	−1.14989
$100$	1.00000	0.100000
$101$	18.9443	1.88503	0.942513	−	0.334170i	$-0.108456\pi$
$101$	18.9443	1.88503	0.942513	+	0.334170i	$0.108456\pi$
$102$	−5.52786	−0.547340
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−2.28825	−0.224381
$105$	2.82843	0.276026
$106$	1.62054	0.157401
$107$	16.9443	1.63806	0.819032	−	0.573747i	$-0.194511\pi$
$107$	16.9443	1.63806	0.819032	+	0.573747i	$0.194511\pi$
$108$	4.57649	0.440373
$109$	−17.7082	−1.69614	−0.848069	−	0.529886i	$-0.822235\pi$
$109$	−17.7082	−1.69614	−0.848069	+	0.529886i	$0.822235\pi$
$110$	5.11667	0.487856
$111$	−2.00000	−0.189832
$112$	3.23607	0.305780
$113$	9.52786	0.896306	0.448153	−	0.893957i	$-0.352082\pi$
$113$	9.52786	0.896306	0.448153	+	0.893957i	$0.352082\pi$
$114$	−6.32456	−0.592349
$115$	−3.16228	−0.294884
$116$	−2.62210	−0.243456
$117$	−5.11667	−0.473037
$118$	−9.70820	−0.893713
$119$	−20.4667	−1.87618
$120$	−0.874032	−0.0797878
$121$	15.1803	1.38003
$122$	8.94665	0.809992
$123$	−2.82843	−0.255031
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	7.23607	0.644640
$127$	3.16228	0.280607	0.140303	−	0.990109i	$-0.455192\pi$
$127$	3.16228	0.280607	0.140303	+	0.990109i	$0.455192\pi$
$128$	−1.00000	−0.0883883
$129$	−0.180340	−0.0158780
$130$	2.28825	0.200692
$131$	−1.70820	−0.149246	−0.0746232	−	0.997212i	$-0.523775\pi$
$131$	−1.70820	−0.149246	−0.0746232	+	0.997212i	$0.523775\pi$
$132$	−4.47214	−0.389249
$133$	−23.4164	−2.03046
$134$	−5.23607	−0.452327
$135$	−4.57649	−0.393882
$136$	6.32456	0.542326
$137$	14.1421	1.20824	0.604122	−	0.796892i	$-0.293524\pi$
$137$	14.1421	1.20824	0.604122	+	0.796892i	$0.293524\pi$
$138$	2.76393	0.235282
$139$	−12.1089	−1.02707	−0.513533	−	0.858070i	$-0.671664\pi$
$139$	−12.1089	−1.02707	−0.513533	+	0.858070i	$0.671664\pi$
$140$	−3.23607	−0.273498
$141$	10.2333	0.861803
$142$	−8.94427	−0.750587
$143$	11.7082	0.979089
$144$	−2.23607	−0.186339
$145$	2.62210	0.217753
$146$	−3.90879	−0.323494
$147$	−3.03476	−0.250303
$148$	2.28825	0.188093
$149$	11.2361	0.920495	0.460247	−	0.887791i	$-0.347761\pi$
$149$	11.2361	0.920495	0.460247	+	0.887791i	$0.347761\pi$
$150$	0.874032	0.0713644
$151$	−13.0618	−1.06295	−0.531476	−	0.847073i	$-0.678362\pi$
$151$	−13.0618	−1.06295	−0.531476	+	0.847073i	$0.678362\pi$
$152$	7.23607	0.586923
$153$	14.1421	1.14332
$154$	−16.5579	−1.33427
$155$	0	0
$156$	−2.00000	−0.160128
$157$	6.94427	0.554213	0.277107	−	0.960839i	$-0.410624\pi$
$157$	6.94427	0.554213	0.277107	+	0.960839i	$0.410624\pi$
$158$	−3.90879	−0.310967
$159$	1.41641	0.112328
$160$	1.00000	0.0790569
$161$	10.2333	0.806501
$162$	−2.70820	−0.212777
$163$	14.1803	1.11069	0.555345	−	0.831620i	$-0.312586\pi$
$163$	14.1803	1.11069	0.555345	+	0.831620i	$0.312586\pi$
$164$	3.23607	0.252694
$165$	4.47214	0.348155
$166$	12.8554	0.997776
$167$	22.9613	1.77680	0.888398	−	0.459074i	$-0.151819\pi$
$167$	22.9613	1.77680	0.888398	+	0.459074i	$0.151819\pi$
$168$	2.82843	0.218218
$169$	−7.76393	−0.597226
$170$	−6.32456	−0.485071
$171$	16.1803	1.23734
$172$	0.206331	0.0157326
$173$	9.41641	0.715916	0.357958	−	0.933738i	$-0.383473\pi$
$173$	9.41641	0.715916	0.357958	+	0.933738i	$0.383473\pi$
$174$	−2.29180	−0.173741
$175$	3.23607	0.244624
$176$	5.11667	0.385684
$177$	−8.48528	−0.637793
$178$	−16.2241	−1.21604
$179$	−11.4412	−0.855158	−0.427579	−	0.903978i	$-0.640633\pi$
$179$	−11.4412	−0.855158	−0.427579	+	0.903978i	$0.640633\pi$
$180$	2.23607	0.166667
$181$	16.7642	1.24608	0.623038	−	0.782192i	$-0.285898\pi$
$181$	16.7642	1.24608	0.623038	+	0.782192i	$0.285898\pi$
$182$	−7.40492	−0.548889
$183$	7.81966	0.578046
$184$	−3.16228	−0.233126
$185$	−2.28825	−0.168235
$186$	0	0
$187$	−32.3607	−2.36645
$188$	−11.7082	−0.853909
$189$	14.8098	1.07726
$190$	−7.23607	−0.524960
$191$	17.4164	1.26021	0.630104	−	0.776511i	$-0.283012\pi$
$191$	17.4164	1.26021	0.630104	+	0.776511i	$0.283012\pi$
$192$	−0.874032	−0.0630778
$193$	2.94427	0.211933	0.105967	−	0.994370i	$-0.466206\pi$
$193$	2.94427	0.211933	0.105967	+	0.994370i	$0.466206\pi$
$194$	11.4164	0.819650
$195$	2.00000	0.143223
$196$	3.47214	0.248010
$197$	−11.8539	−0.844555	−0.422277	−	0.906467i	$-0.638769\pi$
$197$	−11.8539	−0.844555	−0.422277	+	0.906467i	$0.638769\pi$
$198$	11.4412	0.813093
$199$	−13.3168	−0.944004	−0.472002	−	0.881598i	$-0.656468\pi$
$199$	−13.3168	−0.944004	−0.472002	+	0.881598i	$0.656468\pi$
$200$	−1.00000	−0.0707107
$201$	−4.57649	−0.322801
$202$	−18.9443	−1.33291
$203$	−8.48528	−0.595550
$204$	5.52786	0.387028
$205$	−3.23607	−0.226017
$206$	4.00000	0.278693
$207$	−7.07107	−0.491473
$208$	2.28825	0.158661
$209$	−37.0246	−2.56104
$210$	−2.82843	−0.195180
$211$	10.4721	0.720932	0.360466	−	0.932772i	$-0.382618\pi$
$211$	10.4721	0.720932	0.360466	+	0.932772i	$0.382618\pi$
$212$	−1.62054	−0.111299
$213$	−7.81758	−0.535652
$214$	−16.9443	−1.15829
$215$	−0.206331	−0.0140717
$216$	−4.57649	−0.311391
$217$	0	0
$218$	17.7082	1.19935
$219$	−3.41641	−0.230859
$220$	−5.11667	−0.344966
$221$	−14.4721	−0.973501
$222$	2.00000	0.134231
$223$	−3.82998	−0.256474	−0.128237	−	0.991744i	$-0.540932\pi$
$223$	−3.82998	−0.256474	−0.128237	+	0.991744i	$0.540932\pi$
$224$	−3.23607	−0.216219
$225$	−2.23607	−0.149071
$226$	−9.52786	−0.633784
$227$	−21.5967	−1.43343	−0.716713	−	0.697368i	$-0.754354\pi$
$227$	−21.5967	−1.43343	−0.716713	+	0.697368i	$0.754354\pi$
$228$	6.32456	0.418854
$229$	−7.86629	−0.519819	−0.259909	−	0.965633i	$-0.583693\pi$
$229$	−7.86629	−0.519819	−0.259909	+	0.965633i	$0.583693\pi$
$230$	3.16228	0.208514
$231$	−14.4721	−0.952197
$232$	2.62210	0.172149
$233$	29.8885	1.95806	0.979032	−	0.203707i	$-0.0652991\pi$
$233$	29.8885	1.95806	0.979032	+	0.203707i	$0.0652991\pi$
$234$	5.11667	0.334487
$235$	11.7082	0.763759
$236$	9.70820	0.631950
$237$	−3.41641	−0.221920
$238$	20.4667	1.32666
$239$	−13.0618	−0.844896	−0.422448	−	0.906387i	$-0.638829\pi$
$239$	−13.0618	−0.844896	−0.422448	+	0.906387i	$0.638829\pi$
$240$	0.874032	0.0564185
$241$	−21.8809	−1.40947	−0.704736	−	0.709469i	$-0.748935\pi$
$241$	−21.8809	−1.40947	−0.704736	+	0.709469i	$0.748935\pi$
$242$	−15.1803	−0.975829
$243$	−16.0965	−1.03259
$244$	−8.94665	−0.572751
$245$	−3.47214	−0.221827
$246$	2.82843	0.180334
$247$	−16.5579	−1.05355
$248$	0	0
$249$	11.2361	0.712057
$250$	1.00000	0.0632456
$251$	−17.3531	−1.09532	−0.547660	−	0.836701i	$-0.684481\pi$
$251$	−17.3531	−1.09532	−0.547660	+	0.836701i	$0.684481\pi$
$252$	−7.23607	−0.455829
$253$	16.1803	1.01725
$254$	−3.16228	−0.198419
$255$	−5.52786	−0.346168
$256$	1.00000	0.0625000
$257$	5.41641	0.337866	0.168933	−	0.985628i	$-0.445968\pi$
$257$	5.41641	0.337866	0.168933	+	0.985628i	$0.445968\pi$
$258$	0.180340	0.0112275
$259$	7.40492	0.460119
$260$	−2.28825	−0.141911
$261$	5.86319	0.362922
$262$	1.70820	0.105533
$263$	−24.2967	−1.49820	−0.749098	−	0.662459i	$-0.769513\pi$
$263$	−24.2967	−1.49820	−0.749098	+	0.662459i	$0.769513\pi$
$264$	4.47214	0.275241
$265$	1.62054	0.0995493
$266$	23.4164	1.43575
$267$	−14.1803	−0.867823
$268$	5.23607	0.319844
$269$	14.1908	0.865231	0.432616	−	0.901579i	$-0.357591\pi$
$269$	14.1908	0.865231	0.432616	+	0.901579i	$0.357591\pi$
$270$	4.57649	0.278516
$271$	−18.0509	−1.09652	−0.548258	−	0.836309i	$-0.684709\pi$
$271$	−18.0509	−1.09652	−0.548258	+	0.836309i	$0.684709\pi$
$272$	−6.32456	−0.383482
$273$	−6.47214	−0.391711
$274$	−14.1421	−0.854358
$275$	5.11667	0.308547
$276$	−2.76393	−0.166369
$277$	2.70091	0.162282	0.0811409	−	0.996703i	$-0.474144\pi$
$277$	2.70091	0.162282	0.0811409	+	0.996703i	$0.474144\pi$
$278$	12.1089	0.726245
$279$	0	0
$280$	3.23607	0.193392
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	−10.2333	−0.609387
$283$	−1.23607	−0.0734766	−0.0367383	−	0.999325i	$-0.511697\pi$
$283$	−1.23607	−0.0734766	−0.0367383	+	0.999325i	$0.511697\pi$
$284$	8.94427	0.530745
$285$	−6.32456	−0.374634
$286$	−11.7082	−0.692321
$287$	10.4721	0.618151
$288$	2.23607	0.131762
$289$	23.0000	1.35294
$290$	−2.62210	−0.153975
$291$	9.97831	0.584939
$292$	3.90879	0.228745
$293$	−8.65248	−0.505483	−0.252742	−	0.967534i	$-0.581332\pi$
$293$	−8.65248	−0.505483	−0.252742	+	0.967534i	$0.581332\pi$
$294$	3.03476	0.176991
$295$	−9.70820	−0.565233
$296$	−2.28825	−0.133002
$297$	23.4164	1.35876
$298$	−11.2361	−0.650888
$299$	7.23607	0.418473
$300$	−0.874032	−0.0504623
$301$	0.667701	0.0384856
$302$	13.0618	0.751621
$303$	−16.5579	−0.951227
$304$	−7.23607	−0.415017
$305$	8.94665	0.512284
$306$	−14.1421	−0.808452
$307$	24.0000	1.36975	0.684876	−	0.728659i	$-0.259856\pi$
$307$	24.0000	1.36975	0.684876	+	0.728659i	$0.259856\pi$
$308$	16.5579	0.943474
$309$	3.49613	0.198888
$310$	0	0
$311$	−7.41641	−0.420546	−0.210273	−	0.977643i	$-0.567435\pi$
$311$	−7.41641	−0.420546	−0.210273	+	0.977643i	$0.567435\pi$
$312$	2.00000	0.113228
$313$	25.4558	1.43885	0.719425	−	0.694570i	$-0.244406\pi$
$313$	25.4558	1.43885	0.719425	+	0.694570i	$0.244406\pi$
$314$	−6.94427	−0.391888
$315$	7.23607	0.407706
$316$	3.90879	0.219887
$317$	15.7082	0.882261	0.441130	−	0.897443i	$-0.354578\pi$
$317$	15.7082	0.882261	0.441130	+	0.897443i	$0.354578\pi$
$318$	−1.41641	−0.0794282
$319$	−13.4164	−0.751175
$320$	−1.00000	−0.0559017
$321$	−14.8098	−0.826604
$322$	−10.2333	−0.570282
$323$	45.7649	2.54643
$324$	2.70820	0.150456
$325$	2.28825	0.126929
$326$	−14.1803	−0.785376
$327$	15.4775	0.855910
$328$	−3.23607	−0.178682
$329$	−37.8885	−2.08886
$330$	−4.47214	−0.246183
$331$	22.3423	1.22804	0.614021	−	0.789290i	$-0.289551\pi$
$331$	22.3423	1.22804	0.614021	+	0.789290i	$0.289551\pi$
$332$	−12.8554	−0.705534
$333$	−5.11667	−0.280392
$334$	−22.9613	−1.25638
$335$	−5.23607	−0.286077
$336$	−2.82843	−0.154303
$337$	0.412662	0.0224791	0.0112396	−	0.999937i	$-0.496422\pi$
$337$	0.412662	0.0224791	0.0112396	+	0.999937i	$0.496422\pi$
$338$	7.76393	0.422302
$339$	−8.32766	−0.452296
$340$	6.32456	0.342997
$341$	0	0
$342$	−16.1803	−0.874933
$343$	−11.4164	−0.616428
$344$	−0.206331	−0.0111246
$345$	2.76393	0.148805
$346$	−9.41641	−0.506229
$347$	16.7642	0.899951	0.449976	−	0.893041i	$-0.351433\pi$
$347$	16.7642	0.899951	0.449976	+	0.893041i	$0.351433\pi$
$348$	2.29180	0.122853
$349$	−7.23607	−0.387338	−0.193669	−	0.981067i	$-0.562039\pi$
$349$	−7.23607	−0.387338	−0.193669	+	0.981067i	$0.562039\pi$
$350$	−3.23607	−0.172975
$351$	10.4721	0.558961
$352$	−5.11667	−0.272720
$353$	−0.255039	−0.0135744	−0.00678718	−	0.999977i	$-0.502160\pi$
$353$	−0.255039	−0.0135744	−0.00678718	+	0.999977i	$0.502160\pi$
$354$	8.48528	0.450988
$355$	−8.94427	−0.474713
$356$	16.2241	0.859873
$357$	17.8885	0.946762
$358$	11.4412	0.604688
$359$	−3.88854	−0.205229	−0.102615	−	0.994721i	$-0.532721\pi$
$359$	−3.88854	−0.205229	−0.102615	+	0.994721i	$0.532721\pi$
$360$	−2.23607	−0.117851
$361$	33.3607	1.75583
$362$	−16.7642	−0.881108
$363$	−13.2681	−0.696395
$364$	7.40492	0.388123
$365$	−3.90879	−0.204595
$366$	−7.81966	−0.408740
$367$	27.9504	1.45900	0.729500	−	0.683981i	$-0.239753\pi$
$367$	27.9504	1.45900	0.729500	+	0.683981i	$0.239753\pi$
$368$	3.16228	0.164845
$369$	−7.23607	−0.376695
$370$	2.28825	0.118960
$371$	−5.24419	−0.272265
$372$	0	0
$373$	4.65248	0.240896	0.120448	−	0.992720i	$-0.461567\pi$
$373$	4.65248	0.240896	0.120448	+	0.992720i	$0.461567\pi$
$374$	32.3607	1.67333
$375$	0.874032	0.0451348
$376$	11.7082	0.603805
$377$	−6.00000	−0.309016
$378$	−14.8098	−0.761736
$379$	−25.7082	−1.32054	−0.660271	−	0.751028i	$-0.729559\pi$
$379$	−25.7082	−1.32054	−0.660271	+	0.751028i	$0.729559\pi$
$380$	7.23607	0.371202
$381$	−2.76393	−0.141601
$382$	−17.4164	−0.891101
$383$	8.81913	0.450637	0.225318	−	0.974285i	$-0.427658\pi$
$383$	8.81913	0.450637	0.225318	+	0.974285i	$0.427658\pi$
$384$	0.874032	0.0446028
$385$	−16.5579	−0.843869
$386$	−2.94427	−0.149859
$387$	−0.461370	−0.0234528
$388$	−11.4164	−0.579580
$389$	−21.3407	−1.08202	−0.541009	−	0.841017i	$-0.681957\pi$
$389$	−21.3407	−1.08202	−0.541009	+	0.841017i	$0.681957\pi$
$390$	−2.00000	−0.101274
$391$	−20.0000	−1.01144
$392$	−3.47214	−0.175369
$393$	1.49302	0.0753131
$394$	11.8539	0.597190
$395$	−3.90879	−0.196673
$396$	−11.4412	−0.574943
$397$	23.1246	1.16059	0.580295	−	0.814406i	$-0.302937\pi$
$397$	23.1246	1.16059	0.580295	+	0.814406i	$0.302937\pi$
$398$	13.3168	0.667511
$399$	20.4667	1.02462
$400$	1.00000	0.0500000
$401$	6.40337	0.319769	0.159884	−	0.987136i	$-0.448888\pi$
$401$	6.40337	0.319769	0.159884	+	0.987136i	$0.448888\pi$
$402$	4.57649	0.228255
$403$	0	0
$404$	18.9443	0.942513
$405$	−2.70820	−0.134572
$406$	8.48528	0.421117
$407$	11.7082	0.580354
$408$	−5.52786	−0.273670
$409$	37.1034	1.83465	0.917323	−	0.398145i	$-0.130346\pi$
$409$	37.1034	1.83465	0.917323	+	0.398145i	$0.130346\pi$
$410$	3.23607	0.159818
$411$	−12.3607	−0.609707
$412$	−4.00000	−0.197066
$413$	31.4164	1.54590
$414$	7.07107	0.347524
$415$	12.8554	0.631049
$416$	−2.28825	−0.112190
$417$	10.5836	0.518281
$418$	37.0246	1.81093
$419$	−19.2361	−0.939743	−0.469872	−	0.882735i	$-0.655700\pi$
$419$	−19.2361	−0.939743	−0.469872	+	0.882735i	$0.655700\pi$
$420$	2.82843	0.138013
$421$	−6.65248	−0.324222	−0.162111	−	0.986773i	$-0.551830\pi$
$421$	−6.65248	−0.324222	−0.162111	+	0.986773i	$0.551830\pi$
$422$	−10.4721	−0.509776
$423$	26.1803	1.27293
$424$	1.62054	0.0787006
$425$	−6.32456	−0.306786
$426$	7.81758	0.378763
$427$	−28.9520	−1.40108
$428$	16.9443	0.819032
$429$	−10.2333	−0.494071
$430$	0.206331	0.00995016
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	4.57649	0.220187
$433$	25.2982	1.21575	0.607877	−	0.794031i	$-0.292021\pi$
$433$	25.2982	1.21575	0.607877	+	0.794031i	$0.292021\pi$
$434$	0	0
$435$	−2.29180	−0.109883
$436$	−17.7082	−0.848069
$437$	−22.8825	−1.09462
$438$	3.41641	0.163242
$439$	8.47214	0.404353	0.202176	−	0.979349i	$-0.435199\pi$
$439$	8.47214	0.404353	0.202176	+	0.979349i	$0.435199\pi$
$440$	5.11667	0.243928
$441$	−7.76393	−0.369711
$442$	14.4721	0.688369
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−2.00000	−0.0949158
$445$	−16.2241	−0.769094
$446$	3.82998	0.181355
$447$	−9.82068	−0.464502
$448$	3.23607	0.152890
$449$	−1.15917	−0.0547048	−0.0273524	−	0.999626i	$-0.508708\pi$
$449$	−1.15917	−0.0547048	−0.0273524	+	0.999626i	$0.508708\pi$
$450$	2.23607	0.105409
$451$	16.5579	0.779681
$452$	9.52786	0.448153
$453$	11.4164	0.536390
$454$	21.5967	1.01359
$455$	−7.40492	−0.347148
$456$	−6.32456	−0.296174
$457$	−18.9737	−0.887551	−0.443775	−	0.896138i	$-0.646361\pi$
$457$	−18.9737	−0.887551	−0.443775	+	0.896138i	$0.646361\pi$
$458$	7.86629	0.367568
$459$	−28.9443	−1.35100
$460$	−3.16228	−0.147442
$461$	13.2681	0.617957	0.308979	−	0.951069i	$-0.400013\pi$
$461$	13.2681	0.617957	0.308979	+	0.951069i	$0.400013\pi$
$462$	14.4721	0.673305
$463$	24.5517	1.14101	0.570507	−	0.821293i	$-0.306747\pi$
$463$	24.5517	1.14101	0.570507	+	0.821293i	$0.306747\pi$
$464$	−2.62210	−0.121728
$465$	0	0
$466$	−29.8885	−1.38456
$467$	7.05573	0.326500	0.163250	−	0.986585i	$-0.447802\pi$
$467$	7.05573	0.326500	0.163250	+	0.986585i	$0.447802\pi$
$468$	−5.11667	−0.236518
$469$	16.9443	0.782414
$470$	−11.7082	−0.540059
$471$	−6.06952	−0.279669
$472$	−9.70820	−0.446856
$473$	1.05573	0.0485424
$474$	3.41641	0.156921
$475$	−7.23607	−0.332014
$476$	−20.4667	−0.938089
$477$	3.62365	0.165915
$478$	13.0618	0.597432
$479$	25.8885	1.18288	0.591439	−	0.806350i	$-0.298560\pi$
$479$	25.8885	1.18288	0.591439	+	0.806350i	$0.298560\pi$
$480$	−0.874032	−0.0398939
$481$	5.23607	0.238744
$482$	21.8809	0.996648
$483$	−8.94427	−0.406978
$484$	15.1803	0.690015
$485$	11.4164	0.518392
$486$	16.0965	0.730153
$487$	8.81913	0.399633	0.199817	−	0.979833i	$-0.435965\pi$
$487$	8.81913	0.399633	0.199817	+	0.979833i	$0.435965\pi$
$488$	8.94665	0.404996
$489$	−12.3941	−0.560479
$490$	3.47214	0.156855
$491$	−28.2542	−1.27509	−0.637546	−	0.770412i	$-0.720051\pi$
$491$	−28.2542	−1.27509	−0.637546	+	0.770412i	$0.720051\pi$
$492$	−2.82843	−0.127515
$493$	16.5836	0.746887
$494$	16.5579	0.744975
$495$	11.4412	0.514245
$496$	0	0
$497$	28.9443	1.29833
$498$	−11.2361	−0.503500
$499$	7.53244	0.337198	0.168599	−	0.985685i	$-0.446076\pi$
$499$	7.53244	0.337198	0.168599	+	0.985685i	$0.446076\pi$
$500$	−1.00000	−0.0447214
$501$	−20.0689	−0.896612
$502$	17.3531	0.774508
$503$	2.11146	0.0941452	0.0470726	−	0.998891i	$-0.485011\pi$
$503$	2.11146	0.0941452	0.0470726	+	0.998891i	$0.485011\pi$
$504$	7.23607	0.322320
$505$	−18.9443	−0.843009
$506$	−16.1803	−0.719304
$507$	6.78593	0.301374
$508$	3.16228	0.140303
$509$	32.2418	1.42909	0.714546	−	0.699589i	$-0.246633\pi$
$509$	32.2418	1.42909	0.714546	+	0.699589i	$0.246633\pi$
$510$	5.52786	0.244778
$511$	12.6491	0.559564
$512$	−1.00000	−0.0441942
$513$	−33.1158	−1.46210
$514$	−5.41641	−0.238908
$515$	4.00000	0.176261
$516$	−0.180340	−0.00793902
$517$	−59.9070	−2.63471
$518$	−7.40492	−0.325353
$519$	−8.23024	−0.361268
$520$	2.28825	0.100346
$521$	−16.3607	−0.716774	−0.358387	−	0.933573i	$-0.616673\pi$
$521$	−16.3607	−0.716774	−0.358387	+	0.933573i	$0.616673\pi$
$522$	−5.86319	−0.256625
$523$	−21.0857	−0.922013	−0.461006	−	0.887397i	$-0.652511\pi$
$523$	−21.0857	−0.922013	−0.461006	+	0.887397i	$0.652511\pi$
$524$	−1.70820	−0.0746232
$525$	−2.82843	−0.123443
$526$	24.2967	1.05939
$527$	0	0
$528$	−4.47214	−0.194625
$529$	−13.0000	−0.565217
$530$	−1.62054	−0.0703920
$531$	−21.7082	−0.942056
$532$	−23.4164	−1.01523
$533$	7.40492	0.320743
$534$	14.1803	0.613643
$535$	−16.9443	−0.732565
$536$	−5.23607	−0.226164
$537$	10.0000	0.431532
$538$	−14.1908	−0.611811
$539$	17.7658	0.765226
$540$	−4.57649	−0.196941
$541$	42.0689	1.80868	0.904341	−	0.426810i	$-0.140363\pi$
$541$	42.0689	1.80868	0.904341	+	0.426810i	$0.140363\pi$
$542$	18.0509	0.775354
$543$	−14.6525	−0.628798
$544$	6.32456	0.271163
$545$	17.7082	0.758536
$546$	6.47214	0.276982
$547$	16.2918	0.696587	0.348293	−	0.937386i	$-0.386761\pi$
$547$	16.2918	0.696587	0.348293	+	0.937386i	$0.386761\pi$
$548$	14.1421	0.604122
$549$	20.0053	0.853806
$550$	−5.11667	−0.218176
$551$	18.9737	0.808305
$552$	2.76393	0.117641
$553$	12.6491	0.537895
$554$	−2.70091	−0.114751
$555$	2.00000	0.0848953
$556$	−12.1089	−0.513533
$557$	17.7658	0.752760	0.376380	−	0.926465i	$-0.377169\pi$
$557$	17.7658	0.752760	0.376380	+	0.926465i	$0.377169\pi$
$558$	0	0
$559$	0.472136	0.0199692
$560$	−3.23607	−0.136749
$561$	28.2843	1.19416
$562$	26.0000	1.09674
$563$	24.6525	1.03898	0.519489	−	0.854477i	$-0.326122\pi$
$563$	24.6525	1.03898	0.519489	+	0.854477i	$0.326122\pi$
$564$	10.2333	0.430902
$565$	−9.52786	−0.400840
$566$	1.23607	0.0519558
$567$	8.76393	0.368051
$568$	−8.94427	−0.375293
$569$	37.1034	1.55546	0.777728	−	0.628601i	$-0.216372\pi$
$569$	37.1034	1.55546	0.777728	+	0.628601i	$0.216372\pi$
$570$	6.32456	0.264906
$571$	18.4335	0.771417	0.385709	−	0.922621i	$-0.373957\pi$
$571$	18.4335	0.771417	0.385709	+	0.922621i	$0.373957\pi$
$572$	11.7082	0.489545
$573$	−15.2225	−0.635929
$574$	−10.4721	−0.437099
$575$	3.16228	0.131876
$576$	−2.23607	−0.0931695
$577$	−16.0000	−0.666089	−0.333044	−	0.942911i	$-0.608076\pi$
$577$	−16.0000	−0.666089	−0.333044	+	0.942911i	$0.608076\pi$
$578$	−23.0000	−0.956674
$579$	−2.57339	−0.106946
$580$	2.62210	0.108877
$581$	−41.6011	−1.72590
$582$	−9.97831	−0.413614
$583$	−8.29180	−0.343411
$584$	−3.90879	−0.161747
$585$	5.11667	0.211548
$586$	8.65248	0.357430
$587$	−27.9203	−1.15239	−0.576197	−	0.817311i	$-0.695464\pi$
$587$	−27.9203	−1.15239	−0.576197	+	0.817311i	$0.695464\pi$
$588$	−3.03476	−0.125151
$589$	0	0
$590$	9.70820	0.399680
$591$	10.3607	0.426181
$592$	2.28825	0.0940463
$593$	21.5279	0.884043	0.442022	−	0.897004i	$-0.354261\pi$
$593$	21.5279	0.884043	0.442022	+	0.897004i	$0.354261\pi$
$594$	−23.4164	−0.960787
$595$	20.4667	0.839053
$596$	11.2361	0.460247
$597$	11.6393	0.476366
$598$	−7.23607	−0.295905
$599$	33.4164	1.36536	0.682679	−	0.730719i	$-0.260815\pi$
$599$	33.4164	1.36536	0.682679	+	0.730719i	$0.260815\pi$
$600$	0.874032	0.0356822
$601$	−8.81913	−0.359740	−0.179870	−	0.983690i	$-0.557568\pi$
$601$	−8.81913	−0.359740	−0.179870	+	0.983690i	$0.557568\pi$
$602$	−0.667701	−0.0272135
$603$	−11.7082	−0.476795
$604$	−13.0618	−0.531476
$605$	−15.1803	−0.617169
$606$	16.5579	0.672619
$607$	6.94427	0.281859	0.140930	−	0.990020i	$-0.454991\pi$
$607$	6.94427	0.281859	0.140930	+	0.990020i	$0.454991\pi$
$608$	7.23607	0.293461
$609$	7.41641	0.300528
$610$	−8.94665	−0.362239
$611$	−26.7912	−1.08386
$612$	14.1421	0.571662
$613$	−26.6637	−1.07694	−0.538469	−	0.842645i	$-0.680997\pi$
$613$	−26.6637	−1.07694	−0.538469	+	0.842645i	$0.680997\pi$
$614$	−24.0000	−0.968561
$615$	2.82843	0.114053
$616$	−16.5579	−0.667137
$617$	38.9443	1.56784	0.783919	−	0.620864i	$-0.213218\pi$
$617$	38.9443	1.56784	0.783919	+	0.620864i	$0.213218\pi$
$618$	−3.49613	−0.140635
$619$	−44.9697	−1.80748	−0.903742	−	0.428077i	$-0.859191\pi$
$619$	−44.9697	−1.80748	−0.903742	+	0.428077i	$0.859191\pi$
$620$	0	0
$621$	14.4721	0.580747
$622$	7.41641	0.297371
$623$	52.5021	2.10345
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	−25.4558	−1.01742
$627$	32.3607	1.29236
$628$	6.94427	0.277107
$629$	−14.4721	−0.577042
$630$	−7.23607	−0.288292
$631$	9.40802	0.374527	0.187264	−	0.982310i	$-0.440038\pi$
$631$	9.40802	0.374527	0.187264	+	0.982310i	$0.440038\pi$
$632$	−3.90879	−0.155483
$633$	−9.15298	−0.363798
$634$	−15.7082	−0.623852
$635$	−3.16228	−0.125491
$636$	1.41641	0.0561642
$637$	7.94510	0.314796
$638$	13.4164	0.531161
$639$	−20.0000	−0.791188
$640$	1.00000	0.0395285
$641$	25.3770	1.00233	0.501166	−	0.865351i	$-0.332905\pi$
$641$	25.3770	1.00233	0.501166	+	0.865351i	$0.332905\pi$
$642$	14.8098	0.584498
$643$	29.6684	1.17001	0.585003	−	0.811031i	$-0.301093\pi$
$643$	29.6684	1.17001	0.585003	+	0.811031i	$0.301093\pi$
$644$	10.2333	0.403250
$645$	0.180340	0.00710088
$646$	−45.7649	−1.80060
$647$	−36.5331	−1.43627	−0.718133	−	0.695906i	$-0.755003\pi$
$647$	−36.5331	−1.43627	−0.718133	+	0.695906i	$0.755003\pi$
$648$	−2.70820	−0.106388
$649$	49.6737	1.94986
$650$	−2.28825	−0.0897524
$651$	0	0
$652$	14.1803	0.555345
$653$	40.8328	1.59791	0.798956	−	0.601390i	$-0.205386\pi$
$653$	40.8328	1.59791	0.798956	+	0.601390i	$0.205386\pi$
$654$	−15.4775	−0.605220
$655$	1.70820	0.0667451
$656$	3.23607	0.126347
$657$	−8.74032	−0.340992
$658$	37.8885	1.47705
$659$	−27.2361	−1.06097	−0.530483	−	0.847695i	$-0.677990\pi$
$659$	−27.2361	−1.06097	−0.530483	+	0.847695i	$0.677990\pi$
$660$	4.47214	0.174078
$661$	−14.6525	−0.569915	−0.284958	−	0.958540i	$-0.591980\pi$
$661$	−14.6525	−0.569915	−0.284958	+	0.958540i	$0.591980\pi$
$662$	−22.3423	−0.868357
$663$	12.6491	0.491251
$664$	12.8554	0.498888
$665$	23.4164	0.908049
$666$	5.11667	0.198267
$667$	−8.29180	−0.321060
$668$	22.9613	0.888398
$669$	3.34752	0.129423
$670$	5.23607	0.202287
$671$	−45.7771	−1.76720
$672$	2.82843	0.109109
$673$	16.8129	0.648091	0.324046	−	0.946041i	$-0.394957\pi$
$673$	16.8129	0.648091	0.324046	+	0.946041i	$0.394957\pi$
$674$	−0.412662	−0.0158951
$675$	4.57649	0.176149
$676$	−7.76393	−0.298613
$677$	−41.3159	−1.58790	−0.793950	−	0.607983i	$-0.791979\pi$
$677$	−41.3159	−1.58790	−0.793950	+	0.607983i	$0.791979\pi$
$678$	8.32766	0.319822
$679$	−36.9443	−1.41779
$680$	−6.32456	−0.242536
$681$	18.8762	0.723339
$682$	0	0
$683$	−2.11146	−0.0807926	−0.0403963	−	0.999184i	$-0.512862\pi$
$683$	−2.11146	−0.0807926	−0.0403963	+	0.999184i	$0.512862\pi$
$684$	16.1803	0.618671
$685$	−14.1421	−0.540343
$686$	11.4164	0.435880
$687$	6.87539	0.262312
$688$	0.206331	0.00786629
$689$	−3.70820	−0.141271
$690$	−2.76393	−0.105221
$691$	47.7771	1.81753	0.908763	−	0.417313i	$-0.137028\pi$
$691$	47.7771	1.81753	0.908763	+	0.417313i	$0.137028\pi$
$692$	9.41641	0.357958
$693$	−37.0246	−1.40645
$694$	−16.7642	−0.636362
$695$	12.1089	0.459318
$696$	−2.29180	−0.0868703
$697$	−20.4667	−0.775231
$698$	7.23607	0.273889
$699$	−26.1235	−0.988083
$700$	3.23607	0.122312
$701$	5.05573	0.190952	0.0954761	−	0.995432i	$-0.469563\pi$
$701$	5.05573	0.190952	0.0954761	+	0.995432i	$0.469563\pi$
$702$	−10.4721	−0.395245
$703$	−16.5579	−0.624493
$704$	5.11667	0.192842
$705$	−10.2333	−0.385410
$706$	0.255039	0.00959852
$707$	61.3050	2.30561
$708$	−8.48528	−0.318896
$709$	−2.20943	−0.0829770	−0.0414885	−	0.999139i	$-0.513210\pi$
$709$	−2.20943	−0.0829770	−0.0414885	+	0.999139i	$0.513210\pi$
$710$	8.94427	0.335673
$711$	−8.74032	−0.327788
$712$	−16.2241	−0.608022
$713$	0	0
$714$	−17.8885	−0.669462
$715$	−11.7082	−0.437862
$716$	−11.4412	−0.427579
$717$	11.4164	0.426354
$718$	3.88854	0.145119
$719$	30.2874	1.12953	0.564764	−	0.825252i	$-0.308967\pi$
$719$	30.2874	1.12953	0.564764	+	0.825252i	$0.308967\pi$
$720$	2.23607	0.0833333
$721$	−12.9443	−0.482070
$722$	−33.3607	−1.24156
$723$	19.1246	0.711252
$724$	16.7642	0.623038
$725$	−2.62210	−0.0973822
$726$	13.2681	0.492426
$727$	−1.41641	−0.0525317	−0.0262658	−	0.999655i	$-0.508362\pi$
$727$	−1.41641	−0.0525317	−0.0262658	+	0.999655i	$0.508362\pi$
$728$	−7.40492	−0.274445
$729$	5.94427	0.220158
$730$	3.90879	0.144671
$731$	−1.30495	−0.0482654
$732$	7.81966	0.289023
$733$	−30.7639	−1.13629	−0.568146	−	0.822928i	$-0.692339\pi$
$733$	−30.7639	−1.13629	−0.568146	+	0.822928i	$0.692339\pi$
$734$	−27.9504	−1.03167
$735$	3.03476	0.111939
$736$	−3.16228	−0.116563
$737$	26.7912	0.986868
$738$	7.23607	0.266363
$739$	−11.0286	−0.405692	−0.202846	−	0.979211i	$-0.565019\pi$
$739$	−11.0286	−0.405692	−0.202846	+	0.979211i	$0.565019\pi$
$740$	−2.28825	−0.0841176
$741$	14.4721	0.531647
$742$	5.24419	0.192520
$743$	9.74187	0.357395	0.178697	−	0.983904i	$-0.442812\pi$
$743$	9.74187	0.357395	0.178697	+	0.983904i	$0.442812\pi$
$744$	0	0
$745$	−11.2361	−0.411658
$746$	−4.65248	−0.170339
$747$	28.7456	1.05175
$748$	−32.3607	−1.18322
$749$	54.8328	2.00355
$750$	−0.874032	−0.0319151
$751$	−23.3050	−0.850410	−0.425205	−	0.905097i	$-0.639798\pi$
$751$	−23.3050	−0.850410	−0.425205	+	0.905097i	$0.639798\pi$
$752$	−11.7082	−0.426954
$753$	15.1672	0.552723
$754$	6.00000	0.218507
$755$	13.0618	0.475367
$756$	14.8098	0.538629
$757$	31.4953	1.14471	0.572357	−	0.820004i	$-0.306029\pi$
$757$	31.4953	1.14471	0.572357	+	0.820004i	$0.306029\pi$
$758$	25.7082	0.933764
$759$	−14.1421	−0.513327
$760$	−7.23607	−0.262480
$761$	−51.5006	−1.86689	−0.933447	−	0.358715i	$-0.883215\pi$
$761$	−51.5006	−1.86689	−0.933447	+	0.358715i	$0.883215\pi$
$762$	2.76393	0.100127
$763$	−57.3050	−2.07458
$764$	17.4164	0.630104
$765$	−14.1421	−0.511310
$766$	−8.81913	−0.318648
$767$	22.2148	0.802128
$768$	−0.874032	−0.0315389
$769$	−15.3475	−0.553446	−0.276723	−	0.960950i	$-0.589248\pi$
$769$	−15.3475	−0.553446	−0.276723	+	0.960950i	$0.589248\pi$
$770$	16.5579	0.596705
$771$	−4.73411	−0.170495
$772$	2.94427	0.105967
$773$	27.0764	0.973870	0.486935	−	0.873438i	$-0.338115\pi$
$773$	27.0764	0.973870	0.486935	+	0.873438i	$0.338115\pi$
$774$	0.461370	0.0165836
$775$	0	0
$776$	11.4164	0.409825
$777$	−6.47214	−0.232187
$778$	21.3407	0.765102
$779$	−23.4164	−0.838980
$780$	2.00000	0.0716115
$781$	45.7649	1.63760
$782$	20.0000	0.715199
$783$	−12.0000	−0.428845
$784$	3.47214	0.124005
$785$	−6.94427	−0.247852
$786$	−1.49302	−0.0532544
$787$	−13.6808	−0.487667	−0.243833	−	0.969817i	$-0.578405\pi$
$787$	−13.6808	−0.487667	−0.243833	+	0.969817i	$0.578405\pi$
$788$	−11.8539	−0.422277
$789$	21.2361	0.756024
$790$	3.90879	0.139069
$791$	30.8328	1.09629
$792$	11.4412	0.406546
$793$	−20.4721	−0.726987
$794$	−23.1246	−0.820662
$795$	−1.41641	−0.0502348
$796$	−13.3168	−0.472002
$797$	−12.5216	−0.443538	−0.221769	−	0.975099i	$-0.571183\pi$
$797$	−12.5216	−0.443538	−0.221769	+	0.975099i	$0.571183\pi$
$798$	−20.4667	−0.724513
$799$	74.0492	2.61967
$800$	−1.00000	−0.0353553
$801$	−36.2781	−1.28182
$802$	−6.40337	−0.226111
$803$	20.0000	0.705785
$804$	−4.57649	−0.161400
$805$	−10.2333	−0.360678
$806$	0	0
$807$	−12.4033	−0.436615
$808$	−18.9443	−0.666457
$809$	−36.9458	−1.29894	−0.649472	−	0.760385i	$-0.725010\pi$
$809$	−36.9458	−1.29894	−0.649472	+	0.760385i	$0.725010\pi$
$810$	2.70820	0.0951566
$811$	56.5410	1.98542	0.992712	−	0.120512i	$-0.0384536\pi$
$811$	56.5410	1.98542	0.992712	+	0.120512i	$0.0384536\pi$
$812$	−8.48528	−0.297775
$813$	15.7771	0.553327
$814$	−11.7082	−0.410372
$815$	−14.1803	−0.496716
$816$	5.52786	0.193514
$817$	−1.49302	−0.0522343
$818$	−37.1034	−1.29729
$819$	−16.5579	−0.578580
$820$	−3.23607	−0.113008
$821$	1.79677	0.0627078	0.0313539	−	0.999508i	$-0.490018\pi$
$821$	1.79677	0.0627078	0.0313539	+	0.999508i	$0.490018\pi$
$822$	12.3607	0.431128
$823$	1.25659	0.0438020	0.0219010	−	0.999760i	$-0.493028\pi$
$823$	1.25659	0.0438020	0.0219010	+	0.999760i	$0.493028\pi$
$824$	4.00000	0.139347
$825$	−4.47214	−0.155700
$826$	−31.4164	−1.09312
$827$	−0.874032	−0.0303931	−0.0151965	−	0.999885i	$-0.504837\pi$
$827$	−0.874032	−0.0303931	−0.0151965	+	0.999885i	$0.504837\pi$
$828$	−7.07107	−0.245737
$829$	−37.6436	−1.30742	−0.653708	−	0.756747i	$-0.726787\pi$
$829$	−37.6436	−1.30742	−0.653708	+	0.756747i	$0.726787\pi$
$830$	−12.8554	−0.446219
$831$	−2.36068	−0.0818911
$832$	2.28825	0.0793306
$833$	−21.9597	−0.760859
$834$	−10.5836	−0.366480
$835$	−22.9613	−0.794607
$836$	−37.0246	−1.28052
$837$	0	0
$838$	19.2361	0.664499
$839$	30.9443	1.06831	0.534157	−	0.845385i	$-0.320629\pi$
$839$	30.9443	1.06831	0.534157	+	0.845385i	$0.320629\pi$
$840$	−2.82843	−0.0975900
$841$	−22.1246	−0.762918
$842$	6.65248	0.229259
$843$	22.7248	0.782685
$844$	10.4721	0.360466
$845$	7.76393	0.267087
$846$	−26.1803	−0.900099
$847$	49.1246	1.68794
$848$	−1.62054	−0.0556497
$849$	1.08036	0.0370780
$850$	6.32456	0.216930
$851$	7.23607	0.248049
$852$	−7.81758	−0.267826
$853$	14.9443	0.511682	0.255841	−	0.966719i	$-0.417648\pi$
$853$	14.9443	0.511682	0.255841	+	0.966719i	$0.417648\pi$
$854$	28.9520	0.990716
$855$	−16.1803	−0.553356
$856$	−16.9443	−0.579143
$857$	−7.52786	−0.257147	−0.128573	−	0.991700i	$-0.541040\pi$
$857$	−7.52786	−0.257147	−0.128573	+	0.991700i	$0.541040\pi$
$858$	10.2333	0.349361
$859$	21.0069	0.716745	0.358373	−	0.933579i	$-0.383332\pi$
$859$	21.0069	0.716745	0.358373	+	0.933579i	$0.383332\pi$
$860$	−0.206331	−0.00703583
$861$	−9.15298	−0.311933
$862$	−18.0000	−0.613082
$863$	−0.491473	−0.0167299	−0.00836497	−	0.999965i	$-0.502663\pi$
$863$	−0.491473	−0.0167299	−0.00836497	+	0.999965i	$0.502663\pi$
$864$	−4.57649	−0.155695
$865$	−9.41641	−0.320167
$866$	−25.2982	−0.859669
$867$	−20.1027	−0.682725
$868$	0	0
$869$	20.0000	0.678454
$870$	2.29180	0.0776992
$871$	11.9814	0.405975
$872$	17.7082	0.599675
$873$	25.5279	0.863987
$874$	22.8825	0.774011
$875$	−3.23607	−0.109399
$876$	−3.41641	−0.115430
$877$	5.23607	0.176809	0.0884047	−	0.996085i	$-0.471823\pi$
$877$	5.23607	0.176809	0.0884047	+	0.996085i	$0.471823\pi$
$878$	−8.47214	−0.285921
$879$	7.56254	0.255078
$880$	−5.11667	−0.172483
$881$	−36.9458	−1.24473	−0.622367	−	0.782725i	$-0.713829\pi$
$881$	−36.9458	−1.24473	−0.622367	+	0.782725i	$0.713829\pi$
$882$	7.76393	0.261425
$883$	54.0439	1.81872	0.909360	−	0.416009i	$-0.136572\pi$
$883$	54.0439	1.81872	0.909360	+	0.416009i	$0.136572\pi$
$884$	−14.4721	−0.486751
$885$	8.48528	0.285230
$886$	−24.0000	−0.806296
$887$	37.2361	1.25026	0.625132	−	0.780519i	$-0.285045\pi$
$887$	37.2361	1.25026	0.625132	+	0.780519i	$0.285045\pi$
$888$	2.00000	0.0671156
$889$	10.2333	0.343215
$890$	16.2241	0.543831
$891$	13.8570	0.464227
$892$	−3.82998	−0.128237
$893$	84.7214	2.83509
$894$	9.82068	0.328453
$895$	11.4412	0.382438
$896$	−3.23607	−0.108109
$897$	−6.32456	−0.211171
$898$	1.15917	0.0386822
$899$	0	0
$900$	−2.23607	−0.0745356
$901$	10.2492	0.341451
$902$	−16.5579	−0.551318
$903$	−0.583592	−0.0194207
$904$	−9.52786	−0.316892
$905$	−16.7642	−0.557262
$906$	−11.4164	−0.379285
$907$	34.4721	1.14463	0.572314	−	0.820034i	$-0.306046\pi$
$907$	34.4721	1.14463	0.572314	+	0.820034i	$0.306046\pi$
$908$	−21.5967	−0.716713
$909$	−42.3607	−1.40502
$910$	7.40492	0.245471
$911$	−21.9597	−0.727558	−0.363779	−	0.931485i	$-0.618514\pi$
$911$	−21.9597	−0.727558	−0.363779	+	0.931485i	$0.618514\pi$
$912$	6.32456	0.209427
$913$	−65.7771	−2.17690
$914$	18.9737	0.627593
$915$	−7.81966	−0.258510
$916$	−7.86629	−0.259909
$917$	−5.52786	−0.182546
$918$	28.9443	0.955303
$919$	11.6393	0.383946	0.191973	−	0.981400i	$-0.438511\pi$
$919$	11.6393	0.383946	0.191973	+	0.981400i	$0.438511\pi$
$920$	3.16228	0.104257
$921$	−20.9768	−0.691208
$922$	−13.2681	−0.436962
$923$	20.4667	0.673669
$924$	−14.4721	−0.476098
$925$	2.28825	0.0752371
$926$	−24.5517	−0.806819
$927$	8.94427	0.293768
$928$	2.62210	0.0860745
$929$	14.7310	0.483309	0.241655	−	0.970362i	$-0.422310\pi$
$929$	14.7310	0.483309	0.241655	+	0.970362i	$0.422310\pi$
$930$	0	0
$931$	−25.1246	−0.823426
$932$	29.8885	0.979032
$933$	6.48218	0.212217
$934$	−7.05573	−0.230870
$935$	32.3607	1.05831
$936$	5.11667	0.167244
$937$	−55.3050	−1.80673	−0.903367	−	0.428868i	$-0.858912\pi$
$937$	−55.3050	−1.80673	−0.903367	+	0.428868i	$0.858912\pi$
$938$	−16.9443	−0.553250
$939$	−22.2492	−0.726076
$940$	11.7082	0.381880
$941$	−8.27895	−0.269886	−0.134943	−	0.990853i	$-0.543085\pi$
$941$	−8.27895	−0.269886	−0.134943	+	0.990853i	$0.543085\pi$
$942$	6.06952	0.197756
$943$	10.2333	0.333244
$944$	9.70820	0.315975
$945$	−14.8098	−0.481764
$946$	−1.05573	−0.0343247
$947$	19.5927	0.636676	0.318338	−	0.947977i	$-0.396875\pi$
$947$	19.5927	0.636676	0.318338	+	0.947977i	$0.396875\pi$
$948$	−3.41641	−0.110960
$949$	8.94427	0.290343
$950$	7.23607	0.234769
$951$	−13.7295	−0.445209
$952$	20.4667	0.663329
$953$	54.2502	1.75734	0.878668	−	0.477433i	$-0.158433\pi$
$953$	54.2502	1.75734	0.878668	+	0.477433i	$0.158433\pi$
$954$	−3.62365	−0.117320
$955$	−17.4164	−0.563582
$956$	−13.0618	−0.422448
$957$	11.7264	0.379060
$958$	−25.8885	−0.836421
$959$	45.7649	1.47783
$960$	0.874032	0.0282093
$961$	0	0
$962$	−5.23607	−0.168818
$963$	−37.8885	−1.22094
$964$	−21.8809	−0.704736
$965$	−2.94427	−0.0947795
$966$	8.94427	0.287777
$967$	−45.4311	−1.46096	−0.730482	−	0.682932i	$-0.760705\pi$
$967$	−45.4311	−1.46096	−0.730482	+	0.682932i	$0.760705\pi$
$968$	−15.1803	−0.487915
$969$	−40.0000	−1.28499
$970$	−11.4164	−0.366559
$971$	−6.83282	−0.219275	−0.109638	−	0.993972i	$-0.534969\pi$
$971$	−6.83282	−0.219275	−0.109638	+	0.993972i	$0.534969\pi$
$972$	−16.0965	−0.516296
$973$	−39.1853	−1.25622
$974$	−8.81913	−0.282583
$975$	−2.00000	−0.0640513
$976$	−8.94665	−0.286375
$977$	−17.4164	−0.557200	−0.278600	−	0.960407i	$-0.589870\pi$
$977$	−17.4164	−0.557200	−0.278600	+	0.960407i	$0.589870\pi$
$978$	12.3941	0.396319
$979$	83.0132	2.65311
$980$	−3.47214	−0.110913
$981$	39.5967	1.26423
$982$	28.2542	0.901627
$983$	−41.0122	−1.30809	−0.654043	−	0.756457i	$-0.726928\pi$
$983$	−41.0122	−1.30809	−0.654043	+	0.756457i	$0.726928\pi$
$984$	2.82843	0.0901670
$985$	11.8539	0.377696
$986$	−16.5836	−0.528129
$987$	33.1158	1.05409
$988$	−16.5579	−0.526777
$989$	0.652476	0.0207475
$990$	−11.4412	−0.363626
$991$	55.7432	1.77074	0.885371	−	0.464885i	$-0.153904\pi$
$991$	55.7432	1.77074	0.885371	+	0.464885i	$0.153904\pi$
$992$	0	0
$993$	−19.5279	−0.619698
$994$	−28.9443	−0.918057
$995$	13.3168	0.422171
$996$	11.2361	0.356028
$997$	12.2918	0.389285	0.194643	−	0.980874i	$-0.437645\pi$
$997$	12.2918	0.389285	0.194643	+	0.980874i	$0.437645\pi$
$998$	−7.53244	−0.238435
$999$	10.4721	0.331324

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9610.2.a.be.1.2	✓	4
31.30	odd	2	inner	9610.2.a.be.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9610.2.a.be.1.2	✓	4	1.1	even	1	trivial
9610.2.a.be.1.3	yes	4	31.30	odd	2	inner

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:	\( N \)	\(=\)	\( 9610 = 2 \cdot 5 \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9610.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.874032 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.874032 q^{6} +3.23607 q^{7} -1.00000 q^{8} -2.23607 q^{9} +1.00000 q^{10} +5.11667 q^{11} -0.874032 q^{12} +2.28825 q^{13} -3.23607 q^{14} +0.874032 q^{15} +1.00000 q^{16} -6.32456 q^{17} +2.23607 q^{18} -7.23607 q^{19} -1.00000 q^{20} -2.82843 q^{21} -5.11667 q^{22} +3.16228 q^{23} +0.874032 q^{24} +1.00000 q^{25} -2.28825 q^{26} +4.57649 q^{27} +3.23607 q^{28} -2.62210 q^{29} -0.874032 q^{30} -1.00000 q^{32} -4.47214 q^{33} +6.32456 q^{34} -3.23607 q^{35} -2.23607 q^{36} +2.28825 q^{37} +7.23607 q^{38} -2.00000 q^{39} +1.00000 q^{40} +3.23607 q^{41} +2.82843 q^{42} +0.206331 q^{43} +5.11667 q^{44} +2.23607 q^{45} -3.16228 q^{46} -11.7082 q^{47} -0.874032 q^{48} +3.47214 q^{49} -1.00000 q^{50} +5.52786 q^{51} +2.28825 q^{52} -1.62054 q^{53} -4.57649 q^{54} -5.11667 q^{55} -3.23607 q^{56} +6.32456 q^{57} +2.62210 q^{58} +9.70820 q^{59} +0.874032 q^{60} -8.94665 q^{61} -7.23607 q^{63} +1.00000 q^{64} -2.28825 q^{65} +4.47214 q^{66} +5.23607 q^{67} -6.32456 q^{68} -2.76393 q^{69} +3.23607 q^{70} +8.94427 q^{71} +2.23607 q^{72} +3.90879 q^{73} -2.28825 q^{74} -0.874032 q^{75} -7.23607 q^{76} +16.5579 q^{77} +2.00000 q^{78} +3.90879 q^{79} -1.00000 q^{80} +2.70820 q^{81} -3.23607 q^{82} -12.8554 q^{83} -2.82843 q^{84} +6.32456 q^{85} -0.206331 q^{86} +2.29180 q^{87} -5.11667 q^{88} +16.2241 q^{89} -2.23607 q^{90} +7.40492 q^{91} +3.16228 q^{92} +11.7082 q^{94} +7.23607 q^{95} +0.874032 q^{96} -11.4164 q^{97} -3.47214 q^{98} -11.4412 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8} + 4 q^{10} - 4 q^{14} + 4 q^{16} - 20 q^{19} - 4 q^{20} + 4 q^{25} + 4 q^{28} - 4 q^{32} - 4 q^{35} + 20 q^{38} - 8 q^{39} + 4 q^{40} + 4 q^{41} - 20 q^{47}+ \cdots + 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 - 4 * q^8 + 4 * q^10 - 4 * q^14 + 4 * q^16 - 20 * q^19 - 4 * q^20 + 4 * q^25 + 4 * q^28 - 4 * q^32 - 4 * q^35 + 20 * q^38 - 8 * q^39 + 4 * q^40 + 4 * q^41 - 20 * q^47 - 4 * q^49 - 4 * q^50 + 40 * q^51 - 4 * q^56 + 12 * q^59 - 20 * q^63 + 4 * q^64 + 12 * q^67 - 20 * q^69 + 4 * q^70 - 20 * q^76 + 8 * q^78 - 4 * q^80 - 16 * q^81 - 4 * q^82 + 36 * q^87 + 20 * q^94 + 20 * q^95 + 8 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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