LMFDB - Embedded newform 8550.2.a.cf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 570)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.07838 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +1.07838 q^{7} -1.00000 q^{8} -3.41855 q^{11} +0.921622 q^{13} -1.07838 q^{14} +1.00000 q^{16} +0.340173 q^{17} -1.00000 q^{19} +3.41855 q^{22} -0.921622 q^{23} -0.921622 q^{26} +1.07838 q^{28} +1.41855 q^{29} +7.26180 q^{31} -1.00000 q^{32} -0.340173 q^{34} -5.60197 q^{37} +1.00000 q^{38} +1.07838 q^{41} -0.738205 q^{43} -3.41855 q^{44} +0.921622 q^{46} -7.75872 q^{47} -5.83710 q^{49} +0.921622 q^{52} +2.68035 q^{53} -1.07838 q^{56} -1.41855 q^{58} -8.34017 q^{59} +2.00000 q^{61} -7.26180 q^{62} +1.00000 q^{64} +4.68035 q^{67} +0.340173 q^{68} +10.8371 q^{71} +6.83710 q^{73} +5.60197 q^{74} -1.00000 q^{76} -3.68649 q^{77} +4.73820 q^{79} -1.07838 q^{82} +11.0205 q^{83} +0.738205 q^{86} +3.41855 q^{88} -9.75872 q^{89} +0.993857 q^{91} -0.921622 q^{92} +7.75872 q^{94} -16.2557 q^{97} +5.83710 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} + 4 q^{11} + 6 q^{13} + 3 q^{16} - 10 q^{17} - 3 q^{19} - 4 q^{22} - 6 q^{23} - 6 q^{26} - 10 q^{29} + 14 q^{31} - 3 q^{32} + 10 q^{34} + 2 q^{37} + 3 q^{38} - 10 q^{43} + 4 q^{44} + 6 q^{46} + 2 q^{47} + 11 q^{49} + 6 q^{52} - 14 q^{53} + 10 q^{58} - 14 q^{59} + 6 q^{61} - 14 q^{62} + 3 q^{64} - 8 q^{67} - 10 q^{68} + 4 q^{71} - 8 q^{73} - 2 q^{74} - 3 q^{76} - 24 q^{77} + 22 q^{79} + 10 q^{86} - 4 q^{88} - 4 q^{89} - 32 q^{91} - 6 q^{92} - 2 q^{94} - 6 q^{97} - 11 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 + 4 * q^11 + 6 * q^13 + 3 * q^16 - 10 * q^17 - 3 * q^19 - 4 * q^22 - 6 * q^23 - 6 * q^26 - 10 * q^29 + 14 * q^31 - 3 * q^32 + 10 * q^34 + 2 * q^37 + 3 * q^38 - 10 * q^43 + 4 * q^44 + 6 * q^46 + 2 * q^47 + 11 * q^49 + 6 * q^52 - 14 * q^53 + 10 * q^58 - 14 * q^59 + 6 * q^61 - 14 * q^62 + 3 * q^64 - 8 * q^67 - 10 * q^68 + 4 * q^71 - 8 * q^73 - 2 * q^74 - 3 * q^76 - 24 * q^77 + 22 * q^79 + 10 * q^86 - 4 * q^88 - 4 * q^89 - 32 * q^91 - 6 * q^92 - 2 * q^94 - 6 * q^97 - 11 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.07838	0.407588	0.203794	−	0.979014i	$-0.434673\pi$
$7$	1.07838	0.407588	0.203794	+	0.979014i	$0.434673\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−3.41855	−1.03073	−0.515366	−	0.856970i	$-0.672344\pi$
$11$	−3.41855	−1.03073	−0.515366	+	0.856970i	$0.672344\pi$
$12$	0	0
$13$	0.921622	0.255612	0.127806	−	0.991799i	$-0.459207\pi$
$13$	0.921622	0.255612	0.127806	+	0.991799i	$0.459207\pi$
$14$	−1.07838	−0.288209
$15$	0	0
$16$	1.00000	0.250000
$17$	0.340173	0.0825041	0.0412520	−	0.999149i	$-0.486865\pi$
$17$	0.340173	0.0825041	0.0412520	+	0.999149i	$0.486865\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	3.41855	0.728837
$23$	−0.921622	−0.192172	−0.0960858	−	0.995373i	$-0.530632\pi$
$23$	−0.921622	−0.192172	−0.0960858	+	0.995373i	$0.530632\pi$
$24$	0	0
$25$	0	0
$26$	−0.921622	−0.180745
$27$	0	0
$28$	1.07838	0.203794
$29$	1.41855	0.263418	0.131709	−	0.991288i	$-0.457954\pi$
$29$	1.41855	0.263418	0.131709	+	0.991288i	$0.457954\pi$
$30$	0	0
$31$	7.26180	1.30426	0.652128	−	0.758108i	$-0.273876\pi$
$31$	7.26180	1.30426	0.652128	+	0.758108i	$0.273876\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.340173	−0.0583392
$35$	0	0
$36$	0	0
$37$	−5.60197	−0.920958	−0.460479	−	0.887671i	$-0.652322\pi$
$37$	−5.60197	−0.920958	−0.460479	+	0.887671i	$0.652322\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	1.07838	0.168414	0.0842072	−	0.996448i	$-0.473164\pi$
$41$	1.07838	0.168414	0.0842072	+	0.996448i	$0.473164\pi$
$42$	0	0
$43$	−0.738205	−0.112575	−0.0562876	−	0.998415i	$-0.517926\pi$
$43$	−0.738205	−0.112575	−0.0562876	+	0.998415i	$0.517926\pi$
$44$	−3.41855	−0.515366
$45$	0	0
$46$	0.921622	0.135886
$47$	−7.75872	−1.13173	−0.565863	−	0.824499i	$-0.691457\pi$
$47$	−7.75872	−1.13173	−0.565863	+	0.824499i	$0.691457\pi$
$48$	0	0
$49$	−5.83710	−0.833872
$50$	0	0
$51$	0	0
$52$	0.921622	0.127806
$53$	2.68035	0.368174	0.184087	−	0.982910i	$-0.441067\pi$
$53$	2.68035	0.368174	0.184087	+	0.982910i	$0.441067\pi$
$54$	0	0
$55$	0	0
$56$	−1.07838	−0.144104
$57$	0	0
$58$	−1.41855	−0.186265
$59$	−8.34017	−1.08580	−0.542899	−	0.839798i	$-0.682673\pi$
$59$	−8.34017	−1.08580	−0.542899	+	0.839798i	$0.682673\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−7.26180	−0.922249
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	4.68035	0.571795	0.285898	−	0.958260i	$-0.407708\pi$
$67$	4.68035	0.571795	0.285898	+	0.958260i	$0.407708\pi$
$68$	0.340173	0.0412520
$69$	0	0
$70$	0	0
$71$	10.8371	1.28613	0.643064	−	0.765813i	$-0.277663\pi$
$71$	10.8371	1.28613	0.643064	+	0.765813i	$0.277663\pi$
$72$	0	0
$73$	6.83710	0.800222	0.400111	−	0.916467i	$-0.368972\pi$
$73$	6.83710	0.800222	0.400111	+	0.916467i	$0.368972\pi$
$74$	5.60197	0.651216
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−3.68649	−0.420114
$78$	0	0
$79$	4.73820	0.533090	0.266545	−	0.963823i	$-0.414118\pi$
$79$	4.73820	0.533090	0.266545	+	0.963823i	$0.414118\pi$
$80$	0	0
$81$	0	0
$82$	−1.07838	−0.119087
$83$	11.0205	1.20966	0.604830	−	0.796355i	$-0.293241\pi$
$83$	11.0205	1.20966	0.604830	+	0.796355i	$0.293241\pi$
$84$	0	0
$85$	0	0
$86$	0.738205	0.0796027
$87$	0	0
$88$	3.41855	0.364419
$89$	−9.75872	−1.03442	−0.517211	−	0.855858i	$-0.673030\pi$
$89$	−9.75872	−1.03442	−0.517211	+	0.855858i	$0.673030\pi$
$90$	0	0
$91$	0.993857	0.104185
$92$	−0.921622	−0.0960858
$93$	0	0
$94$	7.75872	0.800251
$95$	0	0
$96$	0	0
$97$	−16.2557	−1.65051	−0.825256	−	0.564759i	$-0.808969\pi$
$97$	−16.2557	−1.65051	−0.825256	+	0.564759i	$0.808969\pi$
$98$	5.83710	0.589636
$99$	0	0
$100$	0	0
$101$	13.0205	1.29559	0.647795	−	0.761815i	$-0.275691\pi$
$101$	13.0205	1.29559	0.647795	+	0.761815i	$0.275691\pi$
$102$	0	0
$103$	−14.3402	−1.41298	−0.706490	−	0.707723i	$-0.749722\pi$
$103$	−14.3402	−1.41298	−0.706490	+	0.707723i	$0.749722\pi$
$104$	−0.921622	−0.0903725
$105$	0	0
$106$	−2.68035	−0.260338
$107$	−10.8371	−1.04766	−0.523831	−	0.851822i	$-0.675498\pi$
$107$	−10.8371	−1.04766	−0.523831	+	0.851822i	$0.675498\pi$
$108$	0	0
$109$	18.6225	1.78371	0.891855	−	0.452321i	$-0.149404\pi$
$109$	18.6225	1.78371	0.891855	+	0.452321i	$0.149404\pi$
$110$	0	0
$111$	0	0
$112$	1.07838	0.101897
$113$	−13.5174	−1.27161	−0.635807	−	0.771848i	$-0.719333\pi$
$113$	−13.5174	−1.27161	−0.635807	+	0.771848i	$0.719333\pi$
$114$	0	0
$115$	0	0
$116$	1.41855	0.131709
$117$	0	0
$118$	8.34017	0.767775
$119$	0.366835	0.0336277
$120$	0	0
$121$	0.686489	0.0624081
$122$	−2.00000	−0.181071
$123$	0	0
$124$	7.26180	0.652128
$125$	0	0
$126$	0	0
$127$	−16.4969	−1.46387	−0.731933	−	0.681377i	$-0.761382\pi$
$127$	−16.4969	−1.46387	−0.731933	+	0.681377i	$0.761382\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−0.581449	−0.0508015	−0.0254007	−	0.999677i	$-0.508086\pi$
$131$	−0.581449	−0.0508015	−0.0254007	+	0.999677i	$0.508086\pi$
$132$	0	0
$133$	−1.07838	−0.0935072
$134$	−4.68035	−0.404320
$135$	0	0
$136$	−0.340173	−0.0291696
$137$	−5.81658	−0.496944	−0.248472	−	0.968639i	$-0.579928\pi$
$137$	−5.81658	−0.496944	−0.248472	+	0.968639i	$0.579928\pi$
$138$	0	0
$139$	6.15676	0.522209	0.261105	−	0.965311i	$-0.415913\pi$
$139$	6.15676	0.522209	0.261105	+	0.965311i	$0.415913\pi$
$140$	0	0
$141$	0	0
$142$	−10.8371	−0.909429
$143$	−3.15061	−0.263467
$144$	0	0
$145$	0	0
$146$	−6.83710	−0.565843
$147$	0	0
$148$	−5.60197	−0.460479
$149$	−8.34017	−0.683254	−0.341627	−	0.939836i	$-0.610978\pi$
$149$	−8.34017	−0.683254	−0.341627	+	0.939836i	$0.610978\pi$
$150$	0	0
$151$	16.2557	1.32287	0.661433	−	0.750004i	$-0.269949\pi$
$151$	16.2557	1.32287	0.661433	+	0.750004i	$0.269949\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	3.68649	0.297066
$155$	0	0
$156$	0	0
$157$	−3.65983	−0.292086	−0.146043	−	0.989278i	$-0.546654\pi$
$157$	−3.65983	−0.292086	−0.146043	+	0.989278i	$0.546654\pi$
$158$	−4.73820	−0.376951
$159$	0	0
$160$	0	0
$161$	−0.993857	−0.0783269
$162$	0	0
$163$	−6.89496	−0.540055	−0.270027	−	0.962853i	$-0.587033\pi$
$163$	−6.89496	−0.540055	−0.270027	+	0.962853i	$0.587033\pi$
$164$	1.07838	0.0842072
$165$	0	0
$166$	−11.0205	−0.855358
$167$	−4.68035	−0.362176	−0.181088	−	0.983467i	$-0.557962\pi$
$167$	−4.68035	−0.362176	−0.181088	+	0.983467i	$0.557962\pi$
$168$	0	0
$169$	−12.1506	−0.934662
$170$	0	0
$171$	0	0
$172$	−0.738205	−0.0562876
$173$	−4.52359	−0.343922	−0.171961	−	0.985104i	$-0.555010\pi$
$173$	−4.52359	−0.343922	−0.171961	+	0.985104i	$0.555010\pi$
$174$	0	0
$175$	0	0
$176$	−3.41855	−0.257683
$177$	0	0
$178$	9.75872	0.731447
$179$	−14.4969	−1.08355	−0.541776	−	0.840523i	$-0.682248\pi$
$179$	−14.4969	−1.08355	−0.541776	+	0.840523i	$0.682248\pi$
$180$	0	0
$181$	−12.7792	−0.949874	−0.474937	−	0.880020i	$-0.657529\pi$
$181$	−12.7792	−0.949874	−0.474937	+	0.880020i	$0.657529\pi$
$182$	−0.993857	−0.0736696
$183$	0	0
$184$	0.921622	0.0679429
$185$	0	0
$186$	0	0
$187$	−1.16290	−0.0850396
$188$	−7.75872	−0.565863
$189$	0	0
$190$	0	0
$191$	4.39803	0.318230	0.159115	−	0.987260i	$-0.449136\pi$
$191$	4.39803	0.318230	0.159115	+	0.987260i	$0.449136\pi$
$192$	0	0
$193$	−24.6225	−1.77237	−0.886183	−	0.463336i	$-0.846652\pi$
$193$	−24.6225	−1.77237	−0.886183	+	0.463336i	$0.846652\pi$
$194$	16.2557	1.16709
$195$	0	0
$196$	−5.83710	−0.416936
$197$	−3.91548	−0.278966	−0.139483	−	0.990224i	$-0.544544\pi$
$197$	−3.91548	−0.278966	−0.139483	+	0.990224i	$0.544544\pi$
$198$	0	0
$199$	8.99386	0.637558	0.318779	−	0.947829i	$-0.396727\pi$
$199$	8.99386	0.637558	0.318779	+	0.947829i	$0.396727\pi$
$200$	0	0
$201$	0	0
$202$	−13.0205	−0.916121
$203$	1.52973	0.107366
$204$	0	0
$205$	0	0
$206$	14.3402	0.999127
$207$	0	0
$208$	0.921622	0.0639030
$209$	3.41855	0.236466
$210$	0	0
$211$	−4.68035	−0.322208	−0.161104	−	0.986937i	$-0.551506\pi$
$211$	−4.68035	−0.322208	−0.161104	+	0.986937i	$0.551506\pi$
$212$	2.68035	0.184087
$213$	0	0
$214$	10.8371	0.740809
$215$	0	0
$216$	0	0
$217$	7.83096	0.531600
$218$	−18.6225	−1.26127
$219$	0	0
$220$	0	0
$221$	0.313511	0.0210890
$222$	0	0
$223$	−14.3402	−0.960289	−0.480145	−	0.877189i	$-0.659416\pi$
$223$	−14.3402	−0.960289	−0.480145	+	0.877189i	$0.659416\pi$
$224$	−1.07838	−0.0720521
$225$	0	0
$226$	13.5174	0.899167
$227$	19.2039	1.27461	0.637305	−	0.770612i	$-0.280049\pi$
$227$	19.2039	1.27461	0.637305	+	0.770612i	$0.280049\pi$
$228$	0	0
$229$	2.31351	0.152881	0.0764406	−	0.997074i	$-0.475644\pi$
$229$	2.31351	0.152881	0.0764406	+	0.997074i	$0.475644\pi$
$230$	0	0
$231$	0	0
$232$	−1.41855	−0.0931324
$233$	25.6475	1.68023	0.840113	−	0.542411i	$-0.182489\pi$
$233$	25.6475	1.68023	0.840113	+	0.542411i	$0.182489\pi$
$234$	0	0
$235$	0	0
$236$	−8.34017	−0.542899
$237$	0	0
$238$	−0.366835	−0.0237784
$239$	−3.60197	−0.232992	−0.116496	−	0.993191i	$-0.537166\pi$
$239$	−3.60197	−0.232992	−0.116496	+	0.993191i	$0.537166\pi$
$240$	0	0
$241$	−2.36683	−0.152461	−0.0762306	−	0.997090i	$-0.524289\pi$
$241$	−2.36683	−0.152461	−0.0762306	+	0.997090i	$0.524289\pi$
$242$	−0.686489	−0.0441292
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	−0.921622	−0.0586414
$248$	−7.26180	−0.461124
$249$	0	0
$250$	0	0
$251$	−6.73820	−0.425312	−0.212656	−	0.977127i	$-0.568211\pi$
$251$	−6.73820	−0.425312	−0.212656	+	0.977127i	$0.568211\pi$
$252$	0	0
$253$	3.15061	0.198077
$254$	16.4969	1.03511
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.20394	0.574126	0.287063	−	0.957912i	$-0.407321\pi$
$257$	9.20394	0.574126	0.287063	+	0.957912i	$0.407321\pi$
$258$	0	0
$259$	−6.04104	−0.375372
$260$	0	0
$261$	0	0
$262$	0.581449	0.0359221
$263$	25.7998	1.59088	0.795441	−	0.606031i	$-0.207239\pi$
$263$	25.7998	1.59088	0.795441	+	0.606031i	$0.207239\pi$
$264$	0	0
$265$	0	0
$266$	1.07838	0.0661196
$267$	0	0
$268$	4.68035	0.285898
$269$	−4.73820	−0.288893	−0.144447	−	0.989513i	$-0.546140\pi$
$269$	−4.73820	−0.288893	−0.144447	+	0.989513i	$0.546140\pi$
$270$	0	0
$271$	16.3668	0.994214	0.497107	−	0.867689i	$-0.334396\pi$
$271$	16.3668	0.994214	0.497107	+	0.867689i	$0.334396\pi$
$272$	0.340173	0.0206260
$273$	0	0
$274$	5.81658	0.351393
$275$	0	0
$276$	0	0
$277$	−12.6537	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$277$	−12.6537	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$278$	−6.15676	−0.369258
$279$	0	0
$280$	0	0
$281$	−9.44521	−0.563454	−0.281727	−	0.959495i	$-0.590907\pi$
$281$	−9.44521	−0.563454	−0.281727	+	0.959495i	$0.590907\pi$
$282$	0	0
$283$	−15.8888	−0.944492	−0.472246	−	0.881467i	$-0.656557\pi$
$283$	−15.8888	−0.944492	−0.472246	+	0.881467i	$0.656557\pi$
$284$	10.8371	0.643064
$285$	0	0
$286$	3.15061	0.186300
$287$	1.16290	0.0686437
$288$	0	0
$289$	−16.8843	−0.993193
$290$	0	0
$291$	0	0
$292$	6.83710	0.400111
$293$	4.47027	0.261156	0.130578	−	0.991438i	$-0.458317\pi$
$293$	4.47027	0.261156	0.130578	+	0.991438i	$0.458317\pi$
$294$	0	0
$295$	0	0
$296$	5.60197	0.325608
$297$	0	0
$298$	8.34017	0.483133
$299$	−0.849388	−0.0491214
$300$	0	0
$301$	−0.796064	−0.0458843
$302$	−16.2557	−0.935408
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	−15.3197	−0.874339	−0.437169	−	0.899379i	$-0.644019\pi$
$307$	−15.3197	−0.874339	−0.437169	+	0.899379i	$0.644019\pi$
$308$	−3.68649	−0.210057
$309$	0	0
$310$	0	0
$311$	4.39803	0.249390	0.124695	−	0.992195i	$-0.460205\pi$
$311$	4.39803	0.249390	0.124695	+	0.992195i	$0.460205\pi$
$312$	0	0
$313$	−8.31351	−0.469907	−0.234954	−	0.972007i	$-0.575494\pi$
$313$	−8.31351	−0.469907	−0.234954	+	0.972007i	$0.575494\pi$
$314$	3.65983	0.206536
$315$	0	0
$316$	4.73820	0.266545
$317$	0.470266	0.0264128	0.0132064	−	0.999913i	$-0.495796\pi$
$317$	0.470266	0.0264128	0.0132064	+	0.999913i	$0.495796\pi$
$318$	0	0
$319$	−4.84939	−0.271514
$320$	0	0
$321$	0	0
$322$	0.993857	0.0553855
$323$	−0.340173	−0.0189277
$324$	0	0
$325$	0	0
$326$	6.89496	0.381877
$327$	0	0
$328$	−1.07838	−0.0595435
$329$	−8.36683	−0.461279
$330$	0	0
$331$	31.5174	1.73236	0.866178	−	0.499736i	$-0.166570\pi$
$331$	31.5174	1.73236	0.866178	+	0.499736i	$0.166570\pi$
$332$	11.0205	0.604830
$333$	0	0
$334$	4.68035	0.256097
$335$	0	0
$336$	0	0
$337$	−0.0578588	−0.00315177	−0.00157589	−	0.999999i	$-0.500502\pi$
$337$	−0.0578588	−0.00315177	−0.00157589	+	0.999999i	$0.500502\pi$
$338$	12.1506	0.660906
$339$	0	0
$340$	0	0
$341$	−24.8248	−1.34434
$342$	0	0
$343$	−13.8432	−0.747465
$344$	0.738205	0.0398013
$345$	0	0
$346$	4.52359	0.243190
$347$	−6.34017	−0.340358	−0.170179	−	0.985413i	$-0.554435\pi$
$347$	−6.34017	−0.340358	−0.170179	+	0.985413i	$0.554435\pi$
$348$	0	0
$349$	−20.5236	−1.09860	−0.549301	−	0.835624i	$-0.685106\pi$
$349$	−20.5236	−1.09860	−0.549301	+	0.835624i	$0.685106\pi$
$350$	0	0
$351$	0	0
$352$	3.41855	0.182209
$353$	−22.3812	−1.19123	−0.595616	−	0.803269i	$-0.703092\pi$
$353$	−22.3812	−1.19123	−0.595616	+	0.803269i	$0.703092\pi$
$354$	0	0
$355$	0	0
$356$	−9.75872	−0.517211
$357$	0	0
$358$	14.4969	0.766186
$359$	0.282314	0.0149000	0.00744999	−	0.999972i	$-0.497629\pi$
$359$	0.282314	0.0149000	0.00744999	+	0.999972i	$0.497629\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	12.7792	0.671662
$363$	0	0
$364$	0.993857	0.0520923
$365$	0	0
$366$	0	0
$367$	−27.9155	−1.45718	−0.728588	−	0.684952i	$-0.759823\pi$
$367$	−27.9155	−1.45718	−0.728588	+	0.684952i	$0.759823\pi$
$368$	−0.921622	−0.0480429
$369$	0	0
$370$	0	0
$371$	2.89043	0.150063
$372$	0	0
$373$	−3.44521	−0.178386	−0.0891932	−	0.996014i	$-0.528429\pi$
$373$	−3.44521	−0.178386	−0.0891932	+	0.996014i	$0.528429\pi$
$374$	1.16290	0.0601321
$375$	0	0
$376$	7.75872	0.400126
$377$	1.30737	0.0673329
$378$	0	0
$379$	−28.5113	−1.46453	−0.732264	−	0.681021i	$-0.761536\pi$
$379$	−28.5113	−1.46453	−0.732264	+	0.681021i	$0.761536\pi$
$380$	0	0
$381$	0	0
$382$	−4.39803	−0.225023
$383$	2.63931	0.134862	0.0674312	−	0.997724i	$-0.478520\pi$
$383$	2.63931	0.134862	0.0674312	+	0.997724i	$0.478520\pi$
$384$	0	0
$385$	0	0
$386$	24.6225	1.25325
$387$	0	0
$388$	−16.2557	−0.825256
$389$	22.1834	1.12474	0.562372	−	0.826884i	$-0.309889\pi$
$389$	22.1834	1.12474	0.562372	+	0.826884i	$0.309889\pi$
$390$	0	0
$391$	−0.313511	−0.0158549
$392$	5.83710	0.294818
$393$	0	0
$394$	3.91548	0.197259
$395$	0	0
$396$	0	0
$397$	−24.2245	−1.21579	−0.607895	−	0.794017i	$-0.707986\pi$
$397$	−24.2245	−1.21579	−0.607895	+	0.794017i	$0.707986\pi$
$398$	−8.99386	−0.450821
$399$	0	0
$400$	0	0
$401$	−12.7649	−0.637447	−0.318724	−	0.947848i	$-0.603254\pi$
$401$	−12.7649	−0.637447	−0.318724	+	0.947848i	$0.603254\pi$
$402$	0	0
$403$	6.69263	0.333384
$404$	13.0205	0.647795
$405$	0	0
$406$	−1.52973	−0.0759194
$407$	19.1506	0.949261
$408$	0	0
$409$	22.8781	1.13125	0.565626	−	0.824662i	$-0.308635\pi$
$409$	22.8781	1.13125	0.565626	+	0.824662i	$0.308635\pi$
$410$	0	0
$411$	0	0
$412$	−14.3402	−0.706490
$413$	−8.99386	−0.442559
$414$	0	0
$415$	0	0
$416$	−0.921622	−0.0451862
$417$	0	0
$418$	−3.41855	−0.167207
$419$	−3.47187	−0.169612	−0.0848061	−	0.996397i	$-0.527027\pi$
$419$	−3.47187	−0.169612	−0.0848061	+	0.996397i	$0.527027\pi$
$420$	0	0
$421$	−20.4657	−0.997439	−0.498719	−	0.866764i	$-0.666196\pi$
$421$	−20.4657	−0.997439	−0.498719	+	0.866764i	$0.666196\pi$
$422$	4.68035	0.227836
$423$	0	0
$424$	−2.68035	−0.130169
$425$	0	0
$426$	0	0
$427$	2.15676	0.104373
$428$	−10.8371	−0.523831
$429$	0	0
$430$	0	0
$431$	33.1917	1.59879	0.799393	−	0.600809i	$-0.205155\pi$
$431$	33.1917	1.59879	0.799393	+	0.600809i	$0.205155\pi$
$432$	0	0
$433$	41.1338	1.97676	0.988382	−	0.151991i	$-0.0485684\pi$
$433$	41.1338	1.97676	0.988382	+	0.151991i	$0.0485684\pi$
$434$	−7.83096	−0.375898
$435$	0	0
$436$	18.6225	0.891855
$437$	0.921622	0.0440872
$438$	0	0
$439$	19.1461	0.913792	0.456896	−	0.889520i	$-0.348961\pi$
$439$	19.1461	0.913792	0.456896	+	0.889520i	$0.348961\pi$
$440$	0	0
$441$	0	0
$442$	−0.313511	−0.0149122
$443$	−37.4908	−1.78124	−0.890620	−	0.454747i	$-0.849730\pi$
$443$	−37.4908	−1.78124	−0.890620	+	0.454747i	$0.849730\pi$
$444$	0	0
$445$	0	0
$446$	14.3402	0.679027
$447$	0	0
$448$	1.07838	0.0509486
$449$	−6.43907	−0.303878	−0.151939	−	0.988390i	$-0.548552\pi$
$449$	−6.43907	−0.303878	−0.151939	+	0.988390i	$0.548552\pi$
$450$	0	0
$451$	−3.68649	−0.173590
$452$	−13.5174	−0.635807
$453$	0	0
$454$	−19.2039	−0.901285
$455$	0	0
$456$	0	0
$457$	13.6742	0.639652	0.319826	−	0.947476i	$-0.396376\pi$
$457$	13.6742	0.639652	0.319826	+	0.947476i	$0.396376\pi$
$458$	−2.31351	−0.108103
$459$	0	0
$460$	0	0
$461$	−27.5441	−1.28286	−0.641429	−	0.767183i	$-0.721658\pi$
$461$	−27.5441	−1.28286	−0.641429	+	0.767183i	$0.721658\pi$
$462$	0	0
$463$	−28.5958	−1.32896	−0.664480	−	0.747306i	$-0.731347\pi$
$463$	−28.5958	−1.32896	−0.664480	+	0.747306i	$0.731347\pi$
$464$	1.41855	0.0658546
$465$	0	0
$466$	−25.6475	−1.18810
$467$	35.5318	1.64422	0.822108	−	0.569331i	$-0.192798\pi$
$467$	35.5318	1.64422	0.822108	+	0.569331i	$0.192798\pi$
$468$	0	0
$469$	5.04718	0.233057
$470$	0	0
$471$	0	0
$472$	8.34017	0.383888
$473$	2.52359	0.116035
$474$	0	0
$475$	0	0
$476$	0.366835	0.0168139
$477$	0	0
$478$	3.60197	0.164750
$479$	−24.6491	−1.12625	−0.563124	−	0.826372i	$-0.690401\pi$
$479$	−24.6491	−1.12625	−0.563124	+	0.826372i	$0.690401\pi$
$480$	0	0
$481$	−5.16290	−0.235408
$482$	2.36683	0.107806
$483$	0	0
$484$	0.686489	0.0312040
$485$	0	0
$486$	0	0
$487$	15.3340	0.694851	0.347426	−	0.937708i	$-0.387056\pi$
$487$	15.3340	0.694851	0.347426	+	0.937708i	$0.387056\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	0	0
$491$	32.7792	1.47931	0.739653	−	0.672988i	$-0.234990\pi$
$491$	32.7792	1.47931	0.739653	+	0.672988i	$0.234990\pi$
$492$	0	0
$493$	0.482553	0.0217331
$494$	0.921622	0.0414657
$495$	0	0
$496$	7.26180	0.326064
$497$	11.6865	0.524211
$498$	0	0
$499$	−25.6742	−1.14934	−0.574668	−	0.818387i	$-0.694869\pi$
$499$	−25.6742	−1.14934	−0.574668	+	0.818387i	$0.694869\pi$
$500$	0	0
$501$	0	0
$502$	6.73820	0.300741
$503$	24.7526	1.10366	0.551832	−	0.833956i	$-0.313929\pi$
$503$	24.7526	1.10366	0.551832	+	0.833956i	$0.313929\pi$
$504$	0	0
$505$	0	0
$506$	−3.15061	−0.140062
$507$	0	0
$508$	−16.4969	−0.731933
$509$	20.9360	0.927972	0.463986	−	0.885843i	$-0.346419\pi$
$509$	20.9360	0.927972	0.463986	+	0.885843i	$0.346419\pi$
$510$	0	0
$511$	7.37298	0.326161
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−9.20394	−0.405968
$515$	0	0
$516$	0	0
$517$	26.5236	1.16651
$518$	6.04104	0.265428
$519$	0	0
$520$	0	0
$521$	−23.5486	−1.03168	−0.515842	−	0.856683i	$-0.672521\pi$
$521$	−23.5486	−1.03168	−0.515842	+	0.856683i	$0.672521\pi$
$522$	0	0
$523$	14.7838	0.646449	0.323225	−	0.946322i	$-0.395233\pi$
$523$	14.7838	0.646449	0.323225	+	0.946322i	$0.395233\pi$
$524$	−0.581449	−0.0254007
$525$	0	0
$526$	−25.7998	−1.12492
$527$	2.47027	0.107606
$528$	0	0
$529$	−22.1506	−0.963070
$530$	0	0
$531$	0	0
$532$	−1.07838	−0.0467536
$533$	0.993857	0.0430487
$534$	0	0
$535$	0	0
$536$	−4.68035	−0.202160
$537$	0	0
$538$	4.73820	0.204279
$539$	19.9544	0.859498
$540$	0	0
$541$	−40.8371	−1.75572	−0.877862	−	0.478914i	$-0.841031\pi$
$541$	−40.8371	−1.75572	−0.877862	+	0.478914i	$0.841031\pi$
$542$	−16.3668	−0.703016
$543$	0	0
$544$	−0.340173	−0.0145848
$545$	0	0
$546$	0	0
$547$	−42.0410	−1.79754	−0.898772	−	0.438415i	$-0.855540\pi$
$547$	−42.0410	−1.79754	−0.898772	+	0.438415i	$0.855540\pi$
$548$	−5.81658	−0.248472
$549$	0	0
$550$	0	0
$551$	−1.41855	−0.0604323
$552$	0	0
$553$	5.10957	0.217281
$554$	12.6537	0.537604
$555$	0	0
$556$	6.15676	0.261105
$557$	−26.7526	−1.13354	−0.566772	−	0.823875i	$-0.691808\pi$
$557$	−26.7526	−1.13354	−0.566772	+	0.823875i	$0.691808\pi$
$558$	0	0
$559$	−0.680346	−0.0287756
$560$	0	0
$561$	0	0
$562$	9.44521	0.398422
$563$	−21.9877	−0.926672	−0.463336	−	0.886183i	$-0.653348\pi$
$563$	−21.9877	−0.926672	−0.463336	+	0.886183i	$0.653348\pi$
$564$	0	0
$565$	0	0
$566$	15.8888	0.667857
$567$	0	0
$568$	−10.8371	−0.454715
$569$	32.5958	1.36649	0.683244	−	0.730190i	$-0.260569\pi$
$569$	32.5958	1.36649	0.683244	+	0.730190i	$0.260569\pi$
$570$	0	0
$571$	−32.1445	−1.34520	−0.672602	−	0.740004i	$-0.734823\pi$
$571$	−32.1445	−1.34520	−0.672602	+	0.740004i	$0.734823\pi$
$572$	−3.15061	−0.131734
$573$	0	0
$574$	−1.16290	−0.0485384
$575$	0	0
$576$	0	0
$577$	13.6742	0.569265	0.284632	−	0.958637i	$-0.408129\pi$
$577$	13.6742	0.569265	0.284632	+	0.958637i	$0.408129\pi$
$578$	16.8843	0.702294
$579$	0	0
$580$	0	0
$581$	11.8843	0.493043
$582$	0	0
$583$	−9.16290	−0.379488
$584$	−6.83710	−0.282921
$585$	0	0
$586$	−4.47027	−0.184665
$587$	−30.9672	−1.27815	−0.639076	−	0.769143i	$-0.720683\pi$
$587$	−30.9672	−1.27815	−0.639076	+	0.769143i	$0.720683\pi$
$588$	0	0
$589$	−7.26180	−0.299217
$590$	0	0
$591$	0	0
$592$	−5.60197	−0.230239
$593$	−42.3812	−1.74039	−0.870194	−	0.492709i	$-0.836007\pi$
$593$	−42.3812	−1.74039	−0.870194	+	0.492709i	$0.836007\pi$
$594$	0	0
$595$	0	0
$596$	−8.34017	−0.341627
$597$	0	0
$598$	0.849388	0.0347340
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	11.4764	0.468133	0.234066	−	0.972221i	$-0.424797\pi$
$601$	11.4764	0.468133	0.234066	+	0.972221i	$0.424797\pi$
$602$	0.796064	0.0324451
$603$	0	0
$604$	16.2557	0.661433
$605$	0	0
$606$	0	0
$607$	15.3340	0.622389	0.311195	−	0.950346i	$-0.399271\pi$
$607$	15.3340	0.622389	0.311195	+	0.950346i	$0.399271\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	−7.15061	−0.289283
$612$	0	0
$613$	−4.34017	−0.175298	−0.0876490	−	0.996151i	$-0.527935\pi$
$613$	−4.34017	−0.175298	−0.0876490	+	0.996151i	$0.527935\pi$
$614$	15.3197	0.618251
$615$	0	0
$616$	3.68649	0.148533
$617$	−30.1834	−1.21514	−0.607569	−	0.794267i	$-0.707855\pi$
$617$	−30.1834	−1.21514	−0.607569	+	0.794267i	$0.707855\pi$
$618$	0	0
$619$	−7.94668	−0.319404	−0.159702	−	0.987165i	$-0.551053\pi$
$619$	−7.94668	−0.319404	−0.159702	+	0.987165i	$0.551053\pi$
$620$	0	0
$621$	0	0
$622$	−4.39803	−0.176345
$623$	−10.5236	−0.421619
$624$	0	0
$625$	0	0
$626$	8.31351	0.332275
$627$	0	0
$628$	−3.65983	−0.146043
$629$	−1.90564	−0.0759828
$630$	0	0
$631$	8.68035	0.345559	0.172780	−	0.984961i	$-0.444725\pi$
$631$	8.68035	0.345559	0.172780	+	0.984961i	$0.444725\pi$
$632$	−4.73820	−0.188476
$633$	0	0
$634$	−0.470266	−0.0186767
$635$	0	0
$636$	0	0
$637$	−5.37960	−0.213148
$638$	4.84939	0.191989
$639$	0	0
$640$	0	0
$641$	41.6430	1.64480	0.822400	−	0.568910i	$-0.192635\pi$
$641$	41.6430	1.64480	0.822400	+	0.568910i	$0.192635\pi$
$642$	0	0
$643$	29.7854	1.17462	0.587310	−	0.809362i	$-0.300187\pi$
$643$	29.7854	1.17462	0.587310	+	0.809362i	$0.300187\pi$
$644$	−0.993857	−0.0391634
$645$	0	0
$646$	0.340173	0.0133839
$647$	28.9216	1.13703	0.568513	−	0.822674i	$-0.307519\pi$
$647$	28.9216	1.13703	0.568513	+	0.822674i	$0.307519\pi$
$648$	0	0
$649$	28.5113	1.11917
$650$	0	0
$651$	0	0
$652$	−6.89496	−0.270027
$653$	41.9565	1.64189	0.820943	−	0.571011i	$-0.193448\pi$
$653$	41.9565	1.64189	0.820943	+	0.571011i	$0.193448\pi$
$654$	0	0
$655$	0	0
$656$	1.07838	0.0421036
$657$	0	0
$658$	8.36683	0.326173
$659$	−43.0082	−1.67536	−0.837681	−	0.546159i	$-0.816089\pi$
$659$	−43.0082	−1.67536	−0.837681	+	0.546159i	$0.816089\pi$
$660$	0	0
$661$	−34.3090	−1.33446	−0.667232	−	0.744850i	$-0.732521\pi$
$661$	−34.3090	−1.33446	−0.667232	+	0.744850i	$0.732521\pi$
$662$	−31.5174	−1.22496
$663$	0	0
$664$	−11.0205	−0.427679
$665$	0	0
$666$	0	0
$667$	−1.30737	−0.0506215
$668$	−4.68035	−0.181088
$669$	0	0
$670$	0	0
$671$	−6.83710	−0.263943
$672$	0	0
$673$	31.0928	1.19854	0.599269	−	0.800548i	$-0.295458\pi$
$673$	31.0928	1.19854	0.599269	+	0.800548i	$0.295458\pi$
$674$	0.0578588	0.00222864
$675$	0	0
$676$	−12.1506	−0.467331
$677$	−35.9877	−1.38312	−0.691560	−	0.722319i	$-0.743076\pi$
$677$	−35.9877	−1.38312	−0.691560	+	0.722319i	$0.743076\pi$
$678$	0	0
$679$	−17.5297	−0.672729
$680$	0	0
$681$	0	0
$682$	24.8248	0.950591
$683$	−27.4017	−1.04850	−0.524249	−	0.851565i	$-0.675654\pi$
$683$	−27.4017	−1.04850	−0.524249	+	0.851565i	$0.675654\pi$
$684$	0	0
$685$	0	0
$686$	13.8432	0.528538
$687$	0	0
$688$	−0.738205	−0.0281438
$689$	2.47027	0.0941097
$690$	0	0
$691$	3.83096	0.145737	0.0728683	−	0.997342i	$-0.476785\pi$
$691$	3.83096	0.145737	0.0728683	+	0.997342i	$0.476785\pi$
$692$	−4.52359	−0.171961
$693$	0	0
$694$	6.34017	0.240670
$695$	0	0
$696$	0	0
$697$	0.366835	0.0138949
$698$	20.5236	0.776829
$699$	0	0
$700$	0	0
$701$	−46.5790	−1.75926	−0.879632	−	0.475654i	$-0.842211\pi$
$701$	−46.5790	−1.75926	−0.879632	+	0.475654i	$0.842211\pi$
$702$	0	0
$703$	5.60197	0.211282
$704$	−3.41855	−0.128841
$705$	0	0
$706$	22.3812	0.842328
$707$	14.0410	0.528068
$708$	0	0
$709$	15.1917	0.570534	0.285267	−	0.958448i	$-0.407918\pi$
$709$	15.1917	0.570534	0.285267	+	0.958448i	$0.407918\pi$
$710$	0	0
$711$	0	0
$712$	9.75872	0.365724
$713$	−6.69263	−0.250641
$714$	0	0
$715$	0	0
$716$	−14.4969	−0.541776
$717$	0	0
$718$	−0.282314	−0.0105359
$719$	8.45136	0.315182	0.157591	−	0.987504i	$-0.449627\pi$
$719$	8.45136	0.315182	0.157591	+	0.987504i	$0.449627\pi$
$720$	0	0
$721$	−15.4641	−0.575914
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−12.7792	−0.474937
$725$	0	0
$726$	0	0
$727$	−39.9688	−1.48236	−0.741180	−	0.671306i	$-0.765734\pi$
$727$	−39.9688	−1.48236	−0.741180	+	0.671306i	$0.765734\pi$
$728$	−0.993857	−0.0368348
$729$	0	0
$730$	0	0
$731$	−0.251117	−0.00928791
$732$	0	0
$733$	−38.7480	−1.43119	−0.715596	−	0.698515i	$-0.753845\pi$
$733$	−38.7480	−1.43119	−0.715596	+	0.698515i	$0.753845\pi$
$734$	27.9155	1.03038
$735$	0	0
$736$	0.921622	0.0339714
$737$	−16.0000	−0.589368
$738$	0	0
$739$	−15.1506	−0.557324	−0.278662	−	0.960389i	$-0.589891\pi$
$739$	−15.1506	−0.557324	−0.278662	+	0.960389i	$0.589891\pi$
$740$	0	0
$741$	0	0
$742$	−2.89043	−0.106111
$743$	28.8781	1.05944	0.529718	−	0.848174i	$-0.322298\pi$
$743$	28.8781	1.05944	0.529718	+	0.848174i	$0.322298\pi$
$744$	0	0
$745$	0	0
$746$	3.44521	0.126138
$747$	0	0
$748$	−1.16290	−0.0425198
$749$	−11.6865	−0.427015
$750$	0	0
$751$	−14.2679	−0.520644	−0.260322	−	0.965522i	$-0.583829\pi$
$751$	−14.2679	−0.520644	−0.260322	+	0.965522i	$0.583829\pi$
$752$	−7.75872	−0.282932
$753$	0	0
$754$	−1.30737	−0.0476115
$755$	0	0
$756$	0	0
$757$	−22.2122	−0.807315	−0.403658	−	0.914910i	$-0.632261\pi$
$757$	−22.2122	−0.807315	−0.403658	+	0.914910i	$0.632261\pi$
$758$	28.5113	1.03558
$759$	0	0
$760$	0	0
$761$	−2.48255	−0.0899925	−0.0449962	−	0.998987i	$-0.514328\pi$
$761$	−2.48255	−0.0899925	−0.0449962	+	0.998987i	$0.514328\pi$
$762$	0	0
$763$	20.0821	0.727020
$764$	4.39803	0.159115
$765$	0	0
$766$	−2.63931	−0.0953621
$767$	−7.68649	−0.277543
$768$	0	0
$769$	−22.3135	−0.804646	−0.402323	−	0.915498i	$-0.631797\pi$
$769$	−22.3135	−0.804646	−0.402323	+	0.915498i	$0.631797\pi$
$770$	0	0
$771$	0	0
$772$	−24.6225	−0.886183
$773$	24.2101	0.870776	0.435388	−	0.900243i	$-0.356611\pi$
$773$	24.2101	0.870776	0.435388	+	0.900243i	$0.356611\pi$
$774$	0	0
$775$	0	0
$776$	16.2557	0.583544
$777$	0	0
$778$	−22.1834	−0.795314
$779$	−1.07838	−0.0386369
$780$	0	0
$781$	−37.0472	−1.32565
$782$	0.313511	0.0112111
$783$	0	0
$784$	−5.83710	−0.208468
$785$	0	0
$786$	0	0
$787$	−31.0349	−1.10627	−0.553137	−	0.833090i	$-0.686569\pi$
$787$	−31.0349	−1.10627	−0.553137	+	0.833090i	$0.686569\pi$
$788$	−3.91548	−0.139483
$789$	0	0
$790$	0	0
$791$	−14.5769	−0.518295
$792$	0	0
$793$	1.84324	0.0654555
$794$	24.2245	0.859694
$795$	0	0
$796$	8.99386	0.318779
$797$	−25.2039	−0.892769	−0.446385	−	0.894841i	$-0.647289\pi$
$797$	−25.2039	−0.892769	−0.446385	+	0.894841i	$0.647289\pi$
$798$	0	0
$799$	−2.63931	−0.0933720
$800$	0	0
$801$	0	0
$802$	12.7649	0.450743
$803$	−23.3730	−0.824814
$804$	0	0
$805$	0	0
$806$	−6.69263	−0.235738
$807$	0	0
$808$	−13.0205	−0.458060
$809$	−17.9467	−0.630972	−0.315486	−	0.948930i	$-0.602167\pi$
$809$	−17.9467	−0.630972	−0.315486	+	0.948930i	$0.602167\pi$
$810$	0	0
$811$	40.5113	1.42254	0.711272	−	0.702917i	$-0.248119\pi$
$811$	40.5113	1.42254	0.711272	+	0.702917i	$0.248119\pi$
$812$	1.52973	0.0536831
$813$	0	0
$814$	−19.1506	−0.671229
$815$	0	0
$816$	0	0
$817$	0.738205	0.0258265
$818$	−22.8781	−0.799915
$819$	0	0
$820$	0	0
$821$	−49.3340	−1.72177	−0.860885	−	0.508800i	$-0.830089\pi$
$821$	−49.3340	−1.72177	−0.860885	+	0.508800i	$0.830089\pi$
$822$	0	0
$823$	5.24742	0.182914	0.0914568	−	0.995809i	$-0.470848\pi$
$823$	5.24742	0.182914	0.0914568	+	0.995809i	$0.470848\pi$
$824$	14.3402	0.499564
$825$	0	0
$826$	8.99386	0.312936
$827$	22.5236	0.783222	0.391611	−	0.920131i	$-0.371918\pi$
$827$	22.5236	0.783222	0.391611	+	0.920131i	$0.371918\pi$
$828$	0	0
$829$	43.2495	1.50212	0.751059	−	0.660235i	$-0.229543\pi$
$829$	43.2495	1.50212	0.751059	+	0.660235i	$0.229543\pi$
$830$	0	0
$831$	0	0
$832$	0.921622	0.0319515
$833$	−1.98562	−0.0687978
$834$	0	0
$835$	0	0
$836$	3.41855	0.118233
$837$	0	0
$838$	3.47187	0.119934
$839$	34.6681	1.19687	0.598437	−	0.801170i	$-0.295789\pi$
$839$	34.6681	1.19687	0.598437	+	0.801170i	$0.295789\pi$
$840$	0	0
$841$	−26.9877	−0.930611
$842$	20.4657	0.705296
$843$	0	0
$844$	−4.68035	−0.161104
$845$	0	0
$846$	0	0
$847$	0.740294	0.0254368
$848$	2.68035	0.0920435
$849$	0	0
$850$	0	0
$851$	5.16290	0.176982
$852$	0	0
$853$	28.0554	0.960599	0.480300	−	0.877105i	$-0.340528\pi$
$853$	28.0554	0.960599	0.480300	+	0.877105i	$0.340528\pi$
$854$	−2.15676	−0.0738027
$855$	0	0
$856$	10.8371	0.370405
$857$	−49.0882	−1.67682	−0.838411	−	0.545039i	$-0.816515\pi$
$857$	−49.0882	−1.67682	−0.838411	+	0.545039i	$0.816515\pi$
$858$	0	0
$859$	−37.7275	−1.28725	−0.643623	−	0.765342i	$-0.722570\pi$
$859$	−37.7275	−1.28725	−0.643623	+	0.765342i	$0.722570\pi$
$860$	0	0
$861$	0	0
$862$	−33.1917	−1.13051
$863$	44.9939	1.53161	0.765804	−	0.643074i	$-0.222341\pi$
$863$	44.9939	1.53161	0.765804	+	0.643074i	$0.222341\pi$
$864$	0	0
$865$	0	0
$866$	−41.1338	−1.39778
$867$	0	0
$868$	7.83096	0.265800
$869$	−16.1978	−0.549473
$870$	0	0
$871$	4.31351	0.146158
$872$	−18.6225	−0.630637
$873$	0	0
$874$	−0.921622	−0.0311743
$875$	0	0
$876$	0	0
$877$	2.64915	0.0894554	0.0447277	−	0.998999i	$-0.485758\pi$
$877$	2.64915	0.0894554	0.0447277	+	0.998999i	$0.485758\pi$
$878$	−19.1461	−0.646149
$879$	0	0
$880$	0	0
$881$	−30.2511	−1.01919	−0.509593	−	0.860416i	$-0.670204\pi$
$881$	−30.2511	−1.01919	−0.509593	+	0.860416i	$0.670204\pi$
$882$	0	0
$883$	−19.9955	−0.672901	−0.336450	−	0.941701i	$-0.609226\pi$
$883$	−19.9955	−0.672901	−0.336450	+	0.941701i	$0.609226\pi$
$884$	0.313511	0.0105445
$885$	0	0
$886$	37.4908	1.25953
$887$	−57.0759	−1.91642	−0.958211	−	0.286062i	$-0.907654\pi$
$887$	−57.0759	−1.91642	−0.958211	+	0.286062i	$0.907654\pi$
$888$	0	0
$889$	−17.7899	−0.596655
$890$	0	0
$891$	0	0
$892$	−14.3402	−0.480145
$893$	7.75872	0.259636
$894$	0	0
$895$	0	0
$896$	−1.07838	−0.0360261
$897$	0	0
$898$	6.43907	0.214875
$899$	10.3012	0.343565
$900$	0	0
$901$	0.911781	0.0303758
$902$	3.68649	0.122747
$903$	0	0
$904$	13.5174	0.449584
$905$	0	0
$906$	0	0
$907$	−15.4017	−0.511406	−0.255703	−	0.966755i	$-0.582307\pi$
$907$	−15.4017	−0.511406	−0.255703	+	0.966755i	$0.582307\pi$
$908$	19.2039	0.637305
$909$	0	0
$910$	0	0
$911$	37.5585	1.24437	0.622184	−	0.782871i	$-0.286246\pi$
$911$	37.5585	1.24437	0.622184	+	0.782871i	$0.286246\pi$
$912$	0	0
$913$	−37.6742	−1.24683
$914$	−13.6742	−0.452302
$915$	0	0
$916$	2.31351	0.0764406
$917$	−0.627022	−0.0207061
$918$	0	0
$919$	−10.0410	−0.331223	−0.165612	−	0.986191i	$-0.552960\pi$
$919$	−10.0410	−0.331223	−0.165612	+	0.986191i	$0.552960\pi$
$920$	0	0
$921$	0	0
$922$	27.5441	0.907117
$923$	9.98771	0.328750
$924$	0	0
$925$	0	0
$926$	28.5958	0.939717
$927$	0	0
$928$	−1.41855	−0.0465662
$929$	−1.26633	−0.0415469	−0.0207735	−	0.999784i	$-0.506613\pi$
$929$	−1.26633	−0.0415469	−0.0207735	+	0.999784i	$0.506613\pi$
$930$	0	0
$931$	5.83710	0.191303
$932$	25.6475	0.840113
$933$	0	0
$934$	−35.5318	−1.16264
$935$	0	0
$936$	0	0
$937$	14.1568	0.462481	0.231241	−	0.972897i	$-0.425722\pi$
$937$	14.1568	0.462481	0.231241	+	0.972897i	$0.425722\pi$
$938$	−5.04718	−0.164796
$939$	0	0
$940$	0	0
$941$	−15.2085	−0.495782	−0.247891	−	0.968788i	$-0.579737\pi$
$941$	−15.2085	−0.495782	−0.247891	+	0.968788i	$0.579737\pi$
$942$	0	0
$943$	−0.993857	−0.0323644
$944$	−8.34017	−0.271450
$945$	0	0
$946$	−2.52359	−0.0820490
$947$	−2.53797	−0.0824728	−0.0412364	−	0.999149i	$-0.513130\pi$
$947$	−2.53797	−0.0824728	−0.0412364	+	0.999149i	$0.513130\pi$
$948$	0	0
$949$	6.30122	0.204546
$950$	0	0
$951$	0	0
$952$	−0.366835	−0.0118892
$953$	55.0759	1.78408	0.892042	−	0.451952i	$-0.149272\pi$
$953$	55.0759	1.78408	0.892042	+	0.451952i	$0.149272\pi$
$954$	0	0
$955$	0	0
$956$	−3.60197	−0.116496
$957$	0	0
$958$	24.6491	0.796378
$959$	−6.27247	−0.202549
$960$	0	0
$961$	21.7337	0.701086
$962$	5.16290	0.166459
$963$	0	0
$964$	−2.36683	−0.0762306
$965$	0	0
$966$	0	0
$967$	12.6491	0.406769	0.203385	−	0.979099i	$-0.434806\pi$
$967$	12.6491	0.406769	0.203385	+	0.979099i	$0.434806\pi$
$968$	−0.686489	−0.0220646
$969$	0	0
$970$	0	0
$971$	−38.0677	−1.22165	−0.610825	−	0.791765i	$-0.709162\pi$
$971$	−38.0677	−1.22165	−0.610825	+	0.791765i	$0.709162\pi$
$972$	0	0
$973$	6.63931	0.212846
$974$	−15.3340	−0.491334
$975$	0	0
$976$	2.00000	0.0640184
$977$	−6.16904	−0.197365	−0.0986826	−	0.995119i	$-0.531463\pi$
$977$	−6.16904	−0.197365	−0.0986826	+	0.995119i	$0.531463\pi$
$978$	0	0
$979$	33.3607	1.06621
$980$	0	0
$981$	0	0
$982$	−32.7792	−1.04603
$983$	−1.98771	−0.0633982	−0.0316991	−	0.999497i	$-0.510092\pi$
$983$	−1.98771	−0.0633982	−0.0316991	+	0.999497i	$0.510092\pi$
$984$	0	0
$985$	0	0
$986$	−0.482553	−0.0153676
$987$	0	0
$988$	−0.921622	−0.0293207
$989$	0.680346	0.0216337
$990$	0	0
$991$	−0.907246	−0.0288196	−0.0144098	−	0.999896i	$-0.504587\pi$
$991$	−0.907246	−0.0288196	−0.0144098	+	0.999896i	$0.504587\pi$
$992$	−7.26180	−0.230562
$993$	0	0
$994$	−11.6865	−0.370673
$995$	0	0
$996$	0	0
$997$	17.3874	0.550663	0.275332	−	0.961349i	$-0.411212\pi$
$997$	17.3874	0.550663	0.275332	+	0.961349i	$0.411212\pi$
$998$	25.6742	0.812703
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.cf.1.2		3
3.2	odd	2		2850.2.a.bn.1.2		3
5.2	odd	4		1710.2.d.e.1369.1		6
5.3	odd	4		1710.2.d.e.1369.4		6
5.4	even	2		8550.2.a.cr.1.2		3
15.2	even	4		570.2.d.d.229.6	yes	6
15.8	even	4		570.2.d.d.229.3	✓	6
15.14	odd	2		2850.2.a.bk.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.d.d.229.3	✓	6	15.8	even	4
570.2.d.d.229.6	yes	6	15.2	even	4
1710.2.d.e.1369.1		6	5.2	odd	4
1710.2.d.e.1369.4		6	5.3	odd	4
2850.2.a.bk.1.2		3	15.14	odd	2
2850.2.a.bn.1.2		3	3.2	odd	2
8550.2.a.cf.1.2		3	1.1	even	1	trivial
8550.2.a.cr.1.2		3	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.07838 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.07838 q^{7} -1.00000 q^{8} -3.41855 q^{11} +0.921622 q^{13} -1.07838 q^{14} +1.00000 q^{16} +0.340173 q^{17} -1.00000 q^{19} +3.41855 q^{22} -0.921622 q^{23} -0.921622 q^{26} +1.07838 q^{28} +1.41855 q^{29} +7.26180 q^{31} -1.00000 q^{32} -0.340173 q^{34} -5.60197 q^{37} +1.00000 q^{38} +1.07838 q^{41} -0.738205 q^{43} -3.41855 q^{44} +0.921622 q^{46} -7.75872 q^{47} -5.83710 q^{49} +0.921622 q^{52} +2.68035 q^{53} -1.07838 q^{56} -1.41855 q^{58} -8.34017 q^{59} +2.00000 q^{61} -7.26180 q^{62} +1.00000 q^{64} +4.68035 q^{67} +0.340173 q^{68} +10.8371 q^{71} +6.83710 q^{73} +5.60197 q^{74} -1.00000 q^{76} -3.68649 q^{77} +4.73820 q^{79} -1.07838 q^{82} +11.0205 q^{83} +0.738205 q^{86} +3.41855 q^{88} -9.75872 q^{89} +0.993857 q^{91} -0.921622 q^{92} +7.75872 q^{94} -16.2557 q^{97} +5.83710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} + 4 q^{11} + 6 q^{13} + 3 q^{16} - 10 q^{17} - 3 q^{19} - 4 q^{22} - 6 q^{23} - 6 q^{26} - 10 q^{29} + 14 q^{31} - 3 q^{32} + 10 q^{34} + 2 q^{37} + 3 q^{38} - 10 q^{43} + 4 q^{44} + 6 q^{46} + 2 q^{47} + 11 q^{49} + 6 q^{52} - 14 q^{53} + 10 q^{58} - 14 q^{59} + 6 q^{61} - 14 q^{62} + 3 q^{64} - 8 q^{67} - 10 q^{68} + 4 q^{71} - 8 q^{73} - 2 q^{74} - 3 q^{76} - 24 q^{77} + 22 q^{79} + 10 q^{86} - 4 q^{88} - 4 q^{89} - 32 q^{91} - 6 q^{92} - 2 q^{94} - 6 q^{97} - 11 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 + 4 * q^11 + 6 * q^13 + 3 * q^16 - 10 * q^17 - 3 * q^19 - 4 * q^22 - 6 * q^23 - 6 * q^26 - 10 * q^29 + 14 * q^31 - 3 * q^32 + 10 * q^34 + 2 * q^37 + 3 * q^38 - 10 * q^43 + 4 * q^44 + 6 * q^46 + 2 * q^47 + 11 * q^49 + 6 * q^52 - 14 * q^53 + 10 * q^58 - 14 * q^59 + 6 * q^61 - 14 * q^62 + 3 * q^64 - 8 * q^67 - 10 * q^68 + 4 * q^71 - 8 * q^73 - 2 * q^74 - 3 * q^76 - 24 * q^77 + 22 * q^79 + 10 * q^86 - 4 * q^88 - 4 * q^89 - 32 * q^91 - 6 * q^92 - 2 * q^94 - 6 * q^97 - 11 * q^98

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