LMFDB - Embedded newform 9610.2.a.k.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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 1$ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 310)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +0.236068 q^{7} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} -4.23607 q^{13} -0.236068 q^{14} +1.00000 q^{15} +1.00000 q^{16} -7.85410 q^{17} +2.00000 q^{18} +2.47214 q^{19} +1.00000 q^{20} +0.236068 q^{21} -3.00000 q^{22} +7.85410 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.23607 q^{26} -5.00000 q^{27} +0.236068 q^{28} -3.00000 q^{29} -1.00000 q^{30} -1.00000 q^{32} +3.00000 q^{33} +7.85410 q^{34} +0.236068 q^{35} -2.00000 q^{36} +9.94427 q^{37} -2.47214 q^{38} -4.23607 q^{39} -1.00000 q^{40} +4.14590 q^{41} -0.236068 q^{42} +3.23607 q^{43} +3.00000 q^{44} -2.00000 q^{45} -7.85410 q^{46} -9.00000 q^{47} +1.00000 q^{48} -6.94427 q^{49} -1.00000 q^{50} -7.85410 q^{51} -4.23607 q^{52} +13.4164 q^{53} +5.00000 q^{54} +3.00000 q^{55} -0.236068 q^{56} +2.47214 q^{57} +3.00000 q^{58} +6.00000 q^{59} +1.00000 q^{60} +0.618034 q^{61} -0.472136 q^{63} +1.00000 q^{64} -4.23607 q^{65} -3.00000 q^{66} -9.79837 q^{67} -7.85410 q^{68} +7.85410 q^{69} -0.236068 q^{70} +4.85410 q^{71} +2.00000 q^{72} -10.1803 q^{73} -9.94427 q^{74} +1.00000 q^{75} +2.47214 q^{76} +0.708204 q^{77} +4.23607 q^{78} +6.61803 q^{79} +1.00000 q^{80} +1.00000 q^{81} -4.14590 q^{82} -12.7082 q^{83} +0.236068 q^{84} -7.85410 q^{85} -3.23607 q^{86} -3.00000 q^{87} -3.00000 q^{88} +12.0000 q^{89} +2.00000 q^{90} -1.00000 q^{91} +7.85410 q^{92} +9.00000 q^{94} +2.47214 q^{95} -1.00000 q^{96} -1.23607 q^{97} +6.94427 q^{98} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 4 q^{7} - 2 q^{8} - 4 q^{9} - 2 q^{10} + 6 q^{11} + 2 q^{12} - 4 q^{13} + 4 q^{14} + 2 q^{15} + 2 q^{16} - 9 q^{17} + 4 q^{18} - 4 q^{19} + 2 q^{20}+ \cdots - 12 q^{99}+O(q^{100})$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 4 * q^7 - 2 * q^8 - 4 * q^9 - 2 * q^10 + 6 * q^11 + 2 * q^12 - 4 * q^13 + 4 * q^14 + 2 * q^15 + 2 * q^16 - 9 * q^17 + 4 * q^18 - 4 * q^19 + 2 * q^20 - 4 * q^21 - 6 * q^22 + 9 * q^23 - 2 * q^24 + 2 * q^25 + 4 * q^26 - 10 * q^27 - 4 * q^28 - 6 * q^29 - 2 * q^30 - 2 * q^32 + 6 * q^33 + 9 * q^34 - 4 * q^35 - 4 * q^36 + 2 * q^37 + 4 * q^38 - 4 * q^39 - 2 * q^40 + 15 * q^41 + 4 * q^42 + 2 * q^43 + 6 * q^44 - 4 * q^45 - 9 * q^46 - 18 * q^47 + 2 * q^48 + 4 * q^49 - 2 * q^50 - 9 * q^51 - 4 * q^52 + 10 * q^54 + 6 * q^55 + 4 * q^56 - 4 * q^57 + 6 * q^58 + 12 * q^59 + 2 * q^60 - q^61 + 8 * q^63 + 2 * q^64 - 4 * q^65 - 6 * q^66 + 5 * q^67 - 9 * q^68 + 9 * q^69 + 4 * q^70 + 3 * q^71 + 4 * q^72 + 2 * q^73 - 2 * q^74 + 2 * q^75 - 4 * q^76 - 12 * q^77 + 4 * q^78 + 11 * q^79 + 2 * q^80 + 2 * q^81 - 15 * q^82 - 12 * q^83 - 4 * q^84 - 9 * q^85 - 2 * q^86 - 6 * q^87 - 6 * q^88 + 24 * q^89 + 4 * q^90 - 2 * q^91 + 9 * q^92 + 18 * q^94 - 4 * q^95 - 2 * q^96 + 2 * q^97 - 4 * q^98 - 12 * q^99

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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	−1.00000	−0.353553
$9$	−2.00000	−0.666667
$10$	−1.00000	−0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	1.00000	0.288675
$13$	−4.23607	−1.17487	−0.587437	−	0.809270i	$-0.699863\pi$
$13$	−4.23607	−1.17487	−0.587437	+	0.809270i	$0.699863\pi$
$14$	−0.236068	−0.0630918
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−7.85410	−1.90490	−0.952450	−	0.304696i	$-0.901445\pi$
$17$	−7.85410	−1.90490	−0.952450	+	0.304696i	$0.901445\pi$
$18$	2.00000	0.471405
$19$	2.47214	0.567147	0.283573	−	0.958951i	$-0.408480\pi$
$19$	2.47214	0.567147	0.283573	+	0.958951i	$0.408480\pi$
$20$	1.00000	0.223607
$21$	0.236068	0.0515143
$22$	−3.00000	−0.639602
$23$	7.85410	1.63769	0.818847	−	0.574012i	$-0.194614\pi$
$23$	7.85410	1.63769	0.818847	+	0.574012i	$0.194614\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	4.23607	0.830761
$27$	−5.00000	−0.962250
$28$	0.236068	0.0446127
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	−1.00000	−0.182574
$31$	0	0
$31$	0	0
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	7.85410	1.34697
$35$	0.236068	0.0399028
$36$	−2.00000	−0.333333
$37$	9.94427	1.63483	0.817414	−	0.576050i	$-0.195407\pi$
$37$	9.94427	1.63483	0.817414	+	0.576050i	$0.195407\pi$
$38$	−2.47214	−0.401033
$39$	−4.23607	−0.678314
$40$	−1.00000	−0.158114
$41$	4.14590	0.647480	0.323740	−	0.946146i	$-0.395060\pi$
$41$	4.14590	0.647480	0.323740	+	0.946146i	$0.395060\pi$
$42$	−0.236068	−0.0364261
$43$	3.23607	0.493496	0.246748	−	0.969080i	$-0.420638\pi$
$43$	3.23607	0.493496	0.246748	+	0.969080i	$0.420638\pi$
$44$	3.00000	0.452267
$45$	−2.00000	−0.298142
$46$	−7.85410	−1.15802
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	1.00000	0.144338
$49$	−6.94427	−0.992039
$50$	−1.00000	−0.141421
$51$	−7.85410	−1.09979
$52$	−4.23607	−0.587437
$53$	13.4164	1.84289	0.921443	−	0.388514i	$-0.127012\pi$
$53$	13.4164	1.84289	0.921443	+	0.388514i	$0.127012\pi$
$54$	5.00000	0.680414
$55$	3.00000	0.404520
$56$	−0.236068	−0.0315459
$57$	2.47214	0.327442
$58$	3.00000	0.393919
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	1.00000	0.129099
$61$	0.618034	0.0791311	0.0395656	−	0.999217i	$-0.487403\pi$
$61$	0.618034	0.0791311	0.0395656	+	0.999217i	$0.487403\pi$
$62$	0	0
$63$	−0.472136	−0.0594835
$64$	1.00000	0.125000
$65$	−4.23607	−0.525420
$66$	−3.00000	−0.369274
$67$	−9.79837	−1.19706	−0.598531	−	0.801100i	$-0.704249\pi$
$67$	−9.79837	−1.19706	−0.598531	+	0.801100i	$0.704249\pi$
$68$	−7.85410	−0.952450
$69$	7.85410	0.945523
$70$	−0.236068	−0.0282155
$71$	4.85410	0.576076	0.288038	−	0.957619i	$-0.406997\pi$
$71$	4.85410	0.576076	0.288038	+	0.957619i	$0.406997\pi$
$72$	2.00000	0.235702
$73$	−10.1803	−1.19152	−0.595759	−	0.803163i	$-0.703149\pi$
$73$	−10.1803	−1.19152	−0.595759	+	0.803163i	$0.703149\pi$
$74$	−9.94427	−1.15600
$75$	1.00000	0.115470
$76$	2.47214	0.283573
$77$	0.708204	0.0807073
$78$	4.23607	0.479640
$79$	6.61803	0.744587	0.372293	−	0.928115i	$-0.378571\pi$
$79$	6.61803	0.744587	0.372293	+	0.928115i	$0.378571\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−4.14590	−0.457838
$83$	−12.7082	−1.39491	−0.697453	−	0.716630i	$-0.745684\pi$
$83$	−12.7082	−1.39491	−0.697453	+	0.716630i	$0.745684\pi$
$84$	0.236068	0.0257571
$85$	−7.85410	−0.851897
$86$	−3.23607	−0.348954
$87$	−3.00000	−0.321634
$88$	−3.00000	−0.319801
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	2.00000	0.210819
$91$	−1.00000	−0.104828
$92$	7.85410	0.818847
$93$	0	0
$94$	9.00000	0.928279
$95$	2.47214	0.253636
$96$	−1.00000	−0.102062
$97$	−1.23607	−0.125504	−0.0627518	−	0.998029i	$-0.519988\pi$
$97$	−1.23607	−0.125504	−0.0627518	+	0.998029i	$0.519988\pi$
$98$	6.94427	0.701477
$99$	−6.00000	−0.603023
$100$	1.00000	0.100000
$101$	−6.70820	−0.667491	−0.333746	−	0.942663i	$-0.608313\pi$
$101$	−6.70820	−0.667491	−0.333746	+	0.942663i	$0.608313\pi$
$102$	7.85410	0.777672
$103$	15.2361	1.50125	0.750627	−	0.660726i	$-0.229751\pi$
$103$	15.2361	1.50125	0.750627	+	0.660726i	$0.229751\pi$
$104$	4.23607	0.415381
$105$	0.236068	0.0230379
$106$	−13.4164	−1.30312
$107$	−9.70820	−0.938527	−0.469264	−	0.883058i	$-0.655481\pi$
$107$	−9.70820	−0.938527	−0.469264	+	0.883058i	$0.655481\pi$
$108$	−5.00000	−0.481125
$109$	17.0902	1.63694	0.818471	−	0.574548i	$-0.194822\pi$
$109$	17.0902	1.63694	0.818471	+	0.574548i	$0.194822\pi$
$110$	−3.00000	−0.286039
$111$	9.94427	0.943869
$112$	0.236068	0.0223063
$113$	12.7082	1.19549	0.597744	−	0.801687i	$-0.296064\pi$
$113$	12.7082	1.19549	0.597744	+	0.801687i	$0.296064\pi$
$114$	−2.47214	−0.231537
$115$	7.85410	0.732399
$116$	−3.00000	−0.278543
$117$	8.47214	0.783249
$118$	−6.00000	−0.552345
$119$	−1.85410	−0.169965
$120$	−1.00000	−0.0912871
$121$	−2.00000	−0.181818
$122$	−0.618034	−0.0559542
$123$	4.14590	0.373823
$124$	0	0
$125$	1.00000	0.0894427
$126$	0.472136	0.0420612
$127$	4.05573	0.359888	0.179944	−	0.983677i	$-0.442408\pi$
$127$	4.05573	0.359888	0.179944	+	0.983677i	$0.442408\pi$
$128$	−1.00000	−0.0883883
$129$	3.23607	0.284920
$130$	4.23607	0.371528
$131$	−4.85410	−0.424105	−0.212052	−	0.977258i	$-0.568015\pi$
$131$	−4.85410	−0.424105	−0.212052	+	0.977258i	$0.568015\pi$
$132$	3.00000	0.261116
$133$	0.583592	0.0506039
$134$	9.79837	0.846451
$135$	−5.00000	−0.430331
$136$	7.85410	0.673484
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	−7.85410	−0.668586
$139$	−2.76393	−0.234434	−0.117217	−	0.993106i	$-0.537397\pi$
$139$	−2.76393	−0.234434	−0.117217	+	0.993106i	$0.537397\pi$
$140$	0.236068	0.0199514
$141$	−9.00000	−0.757937
$142$	−4.85410	−0.407347
$143$	−12.7082	−1.06271
$144$	−2.00000	−0.166667
$145$	−3.00000	−0.249136
$146$	10.1803	0.842531
$147$	−6.94427	−0.572754
$148$	9.94427	0.817414
$149$	1.85410	0.151894	0.0759470	−	0.997112i	$-0.475802\pi$
$149$	1.85410	0.151894	0.0759470	+	0.997112i	$0.475802\pi$
$150$	−1.00000	−0.0816497
$151$	15.9443	1.29753	0.648763	−	0.760990i	$-0.275287\pi$
$151$	15.9443	1.29753	0.648763	+	0.760990i	$0.275287\pi$
$152$	−2.47214	−0.200517
$153$	15.7082	1.26993
$154$	−0.708204	−0.0570687
$155$	0	0
$156$	−4.23607	−0.339157
$157$	20.0344	1.59892	0.799461	−	0.600718i	$-0.205118\pi$
$157$	20.0344	1.59892	0.799461	+	0.600718i	$0.205118\pi$
$158$	−6.61803	−0.526503
$159$	13.4164	1.06399
$160$	−1.00000	−0.0790569
$161$	1.85410	0.146124
$162$	−1.00000	−0.0785674
$163$	0.236068	0.0184903	0.00924514	−	0.999957i	$-0.497057\pi$
$163$	0.236068	0.0184903	0.00924514	+	0.999957i	$0.497057\pi$
$164$	4.14590	0.323740
$165$	3.00000	0.233550
$166$	12.7082	0.986348
$167$	13.4164	1.03819	0.519096	−	0.854716i	$-0.326269\pi$
$167$	13.4164	1.03819	0.519096	+	0.854716i	$0.326269\pi$
$168$	−0.236068	−0.0182130
$169$	4.94427	0.380329
$170$	7.85410	0.602382
$171$	−4.94427	−0.378098
$172$	3.23607	0.246748
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	3.00000	0.227429
$175$	0.236068	0.0178451
$176$	3.00000	0.226134
$177$	6.00000	0.450988
$178$	−12.0000	−0.899438
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	−2.00000	−0.149071
$181$	2.09017	0.155361	0.0776806	−	0.996978i	$-0.475249\pi$
$181$	2.09017	0.155361	0.0776806	+	0.996978i	$0.475249\pi$
$182$	1.00000	0.0741249
$183$	0.618034	0.0456864
$184$	−7.85410	−0.579012
$185$	9.94427	0.731117
$186$	0	0
$187$	−23.5623	−1.72305
$188$	−9.00000	−0.656392
$189$	−1.18034	−0.0858571
$190$	−2.47214	−0.179348
$191$	−6.43769	−0.465815	−0.232908	−	0.972499i	$-0.574824\pi$
$191$	−6.43769	−0.465815	−0.232908	+	0.972499i	$0.574824\pi$
$192$	1.00000	0.0721688
$193$	8.03444	0.578332	0.289166	−	0.957279i	$-0.406622\pi$
$193$	8.03444	0.578332	0.289166	+	0.957279i	$0.406622\pi$
$194$	1.23607	0.0887445
$195$	−4.23607	−0.303351
$196$	−6.94427	−0.496019
$197$	12.7082	0.905422	0.452711	−	0.891657i	$-0.350457\pi$
$197$	12.7082	0.905422	0.452711	+	0.891657i	$0.350457\pi$
$198$	6.00000	0.426401
$199$	−10.8885	−0.771868	−0.385934	−	0.922526i	$-0.626121\pi$
$199$	−10.8885	−0.771868	−0.385934	+	0.922526i	$0.626121\pi$
$200$	−1.00000	−0.0707107
$201$	−9.79837	−0.691124
$202$	6.70820	0.471988
$203$	−0.708204	−0.0497062
$204$	−7.85410	−0.549897
$205$	4.14590	0.289562
$206$	−15.2361	−1.06155
$207$	−15.7082	−1.09180
$208$	−4.23607	−0.293718
$209$	7.41641	0.513004
$210$	−0.236068	−0.0162902
$211$	5.09017	0.350422	0.175211	−	0.984531i	$-0.443939\pi$
$211$	5.09017	0.350422	0.175211	+	0.984531i	$0.443939\pi$
$212$	13.4164	0.921443
$213$	4.85410	0.332598
$214$	9.70820	0.663639
$215$	3.23607	0.220698
$216$	5.00000	0.340207
$217$	0	0
$218$	−17.0902	−1.15749
$219$	−10.1803	−0.687924
$220$	3.00000	0.202260
$221$	33.2705	2.23802
$222$	−9.94427	−0.667416
$223$	−9.79837	−0.656148	−0.328074	−	0.944652i	$-0.606399\pi$
$223$	−9.79837	−0.656148	−0.328074	+	0.944652i	$0.606399\pi$
$224$	−0.236068	−0.0157730
$225$	−2.00000	−0.133333
$226$	−12.7082	−0.845337
$227$	4.14590	0.275173	0.137586	−	0.990490i	$-0.456066\pi$
$227$	4.14590	0.275173	0.137586	+	0.990490i	$0.456066\pi$
$228$	2.47214	0.163721
$229$	15.6180	1.03207	0.516034	−	0.856568i	$-0.327408\pi$
$229$	15.6180	1.03207	0.516034	+	0.856568i	$0.327408\pi$
$230$	−7.85410	−0.517884
$231$	0.708204	0.0465964
$232$	3.00000	0.196960
$233$	−17.5623	−1.15054	−0.575272	−	0.817962i	$-0.695104\pi$
$233$	−17.5623	−1.15054	−0.575272	+	0.817962i	$0.695104\pi$
$234$	−8.47214	−0.553841
$235$	−9.00000	−0.587095
$236$	6.00000	0.390567
$237$	6.61803	0.429888
$238$	1.85410	0.120184
$239$	18.9787	1.22763	0.613815	−	0.789450i	$-0.289634\pi$
$239$	18.9787	1.22763	0.613815	+	0.789450i	$0.289634\pi$
$240$	1.00000	0.0645497
$241$	−12.1246	−0.781015	−0.390507	−	0.920600i	$-0.627700\pi$
$241$	−12.1246	−0.781015	−0.390507	+	0.920600i	$0.627700\pi$
$242$	2.00000	0.128565
$243$	16.0000	1.02640
$244$	0.618034	0.0395656
$245$	−6.94427	−0.443653
$246$	−4.14590	−0.264333
$247$	−10.4721	−0.666326
$248$	0	0
$249$	−12.7082	−0.805350
$250$	−1.00000	−0.0632456
$251$	25.8541	1.63190	0.815948	−	0.578125i	$-0.196215\pi$
$251$	25.8541	1.63190	0.815948	+	0.578125i	$0.196215\pi$
$252$	−0.472136	−0.0297418
$253$	23.5623	1.48135
$254$	−4.05573	−0.254479
$255$	−7.85410	−0.491843
$256$	1.00000	0.0625000
$257$	5.29180	0.330093	0.165047	−	0.986286i	$-0.447223\pi$
$257$	5.29180	0.330093	0.165047	+	0.986286i	$0.447223\pi$
$258$	−3.23607	−0.201469
$259$	2.34752	0.145868
$260$	−4.23607	−0.262710
$261$	6.00000	0.371391
$262$	4.85410	0.299887
$263$	−8.56231	−0.527974	−0.263987	−	0.964526i	$-0.585038\pi$
$263$	−8.56231	−0.527974	−0.263987	+	0.964526i	$0.585038\pi$
$264$	−3.00000	−0.184637
$265$	13.4164	0.824163
$266$	−0.583592	−0.0357823
$267$	12.0000	0.734388
$268$	−9.79837	−0.598531
$269$	1.14590	0.0698666	0.0349333	−	0.999390i	$-0.488878\pi$
$269$	1.14590	0.0698666	0.0349333	+	0.999390i	$0.488878\pi$
$270$	5.00000	0.304290
$271$	15.9443	0.968546	0.484273	−	0.874917i	$-0.339084\pi$
$271$	15.9443	0.968546	0.484273	+	0.874917i	$0.339084\pi$
$272$	−7.85410	−0.476225
$273$	−1.00000	−0.0605228
$274$	15.0000	0.906183
$275$	3.00000	0.180907
$276$	7.85410	0.472761
$277$	−15.5623	−0.935048	−0.467524	−	0.883980i	$-0.654854\pi$
$277$	−15.5623	−0.935048	−0.467524	+	0.883980i	$0.654854\pi$
$278$	2.76393	0.165770
$279$	0	0
$280$	−0.236068	−0.0141078
$281$	7.85410	0.468536	0.234268	−	0.972172i	$-0.424731\pi$
$281$	7.85410	0.468536	0.234268	+	0.972172i	$0.424731\pi$
$282$	9.00000	0.535942
$283$	−7.18034	−0.426827	−0.213413	−	0.976962i	$-0.568458\pi$
$283$	−7.18034	−0.426827	−0.213413	+	0.976962i	$0.568458\pi$
$284$	4.85410	0.288038
$285$	2.47214	0.146437
$286$	12.7082	0.751452
$287$	0.978714	0.0577716
$288$	2.00000	0.117851
$289$	44.6869	2.62864
$290$	3.00000	0.176166
$291$	−1.23607	−0.0724596
$292$	−10.1803	−0.595759
$293$	0.708204	0.0413737	0.0206869	−	0.999786i	$-0.493415\pi$
$293$	0.708204	0.0413737	0.0206869	+	0.999786i	$0.493415\pi$
$294$	6.94427	0.404998
$295$	6.00000	0.349334
$296$	−9.94427	−0.577999
$297$	−15.0000	−0.870388
$298$	−1.85410	−0.107405
$299$	−33.2705	−1.92408
$300$	1.00000	0.0577350
$301$	0.763932	0.0440323
$302$	−15.9443	−0.917490
$303$	−6.70820	−0.385376
$304$	2.47214	0.141787
$305$	0.618034	0.0353885
$306$	−15.7082	−0.897978
$307$	−2.65248	−0.151385	−0.0756924	−	0.997131i	$-0.524117\pi$
$307$	−2.65248	−0.151385	−0.0756924	+	0.997131i	$0.524117\pi$
$308$	0.708204	0.0403537
$309$	15.2361	0.866750
$310$	0	0
$311$	15.7082	0.890731	0.445365	−	0.895349i	$-0.353074\pi$
$311$	15.7082	0.890731	0.445365	+	0.895349i	$0.353074\pi$
$312$	4.23607	0.239820
$313$	29.7984	1.68430	0.842152	−	0.539240i	$-0.181289\pi$
$313$	29.7984	1.68430	0.842152	+	0.539240i	$0.181289\pi$
$314$	−20.0344	−1.13061
$315$	−0.472136	−0.0266018
$316$	6.61803	0.372293
$317$	7.41641	0.416547	0.208273	−	0.978071i	$-0.433216\pi$
$317$	7.41641	0.416547	0.208273	+	0.978071i	$0.433216\pi$
$318$	−13.4164	−0.752355
$319$	−9.00000	−0.503903
$320$	1.00000	0.0559017
$321$	−9.70820	−0.541859
$322$	−1.85410	−0.103325
$323$	−19.4164	−1.08036
$324$	1.00000	0.0555556
$325$	−4.23607	−0.234975
$326$	−0.236068	−0.0130746
$327$	17.0902	0.945089
$328$	−4.14590	−0.228919
$329$	−2.12461	−0.117134
$330$	−3.00000	−0.165145
$331$	−15.5623	−0.855382	−0.427691	−	0.903925i	$-0.640673\pi$
$331$	−15.5623	−0.855382	−0.427691	+	0.903925i	$0.640673\pi$
$332$	−12.7082	−0.697453
$333$	−19.8885	−1.08989
$334$	−13.4164	−0.734113
$335$	−9.79837	−0.535342
$336$	0.236068	0.0128786
$337$	12.8885	0.702084	0.351042	−	0.936360i	$-0.385827\pi$
$337$	12.8885	0.702084	0.351042	+	0.936360i	$0.385827\pi$
$338$	−4.94427	−0.268933
$339$	12.7082	0.690215
$340$	−7.85410	−0.425948
$341$	0	0
$342$	4.94427	0.267356
$343$	−3.29180	−0.177740
$344$	−3.23607	−0.174477
$345$	7.85410	0.422851
$346$	−9.00000	−0.483843
$347$	12.2705	0.658715	0.329358	−	0.944205i	$-0.393168\pi$
$347$	12.2705	0.658715	0.329358	+	0.944205i	$0.393168\pi$
$348$	−3.00000	−0.160817
$349$	5.52786	0.295900	0.147950	−	0.988995i	$-0.452733\pi$
$349$	5.52786	0.295900	0.147950	+	0.988995i	$0.452733\pi$
$350$	−0.236068	−0.0126184
$351$	21.1803	1.13052
$352$	−3.00000	−0.159901
$353$	10.4164	0.554409	0.277205	−	0.960811i	$-0.410592\pi$
$353$	10.4164	0.554409	0.277205	+	0.960811i	$0.410592\pi$
$354$	−6.00000	−0.318896
$355$	4.85410	0.257629
$356$	12.0000	0.635999
$357$	−1.85410	−0.0981295
$358$	−24.0000	−1.26844
$359$	−33.2705	−1.75595	−0.877975	−	0.478706i	$-0.841106\pi$
$359$	−33.2705	−1.75595	−0.877975	+	0.478706i	$0.841106\pi$
$360$	2.00000	0.105409
$361$	−12.8885	−0.678344
$362$	−2.09017	−0.109857
$363$	−2.00000	−0.104973
$364$	−1.00000	−0.0524142
$365$	−10.1803	−0.532863
$366$	−0.618034	−0.0323052
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	7.85410	0.409423
$369$	−8.29180	−0.431654
$370$	−9.94427	−0.516978
$371$	3.16718	0.164432
$372$	0	0
$373$	−17.6525	−0.914011	−0.457005	−	0.889464i	$-0.651078\pi$
$373$	−17.6525	−0.914011	−0.457005	+	0.889464i	$0.651078\pi$
$374$	23.5623	1.21838
$375$	1.00000	0.0516398
$376$	9.00000	0.464140
$377$	12.7082	0.654506
$378$	1.18034	0.0607101
$379$	−7.50658	−0.385587	−0.192794	−	0.981239i	$-0.561755\pi$
$379$	−7.50658	−0.385587	−0.192794	+	0.981239i	$0.561755\pi$
$380$	2.47214	0.126818
$381$	4.05573	0.207781
$382$	6.43769	0.329381
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	−1.00000	−0.0510310
$385$	0.708204	0.0360934
$386$	−8.03444	−0.408942
$387$	−6.47214	−0.328997
$388$	−1.23607	−0.0627518
$389$	11.2918	0.572517	0.286258	−	0.958152i	$-0.407588\pi$
$389$	11.2918	0.572517	0.286258	+	0.958152i	$0.407588\pi$
$390$	4.23607	0.214502
$391$	−61.6869	−3.11964
$392$	6.94427	0.350739
$393$	−4.85410	−0.244857
$394$	−12.7082	−0.640230
$395$	6.61803	0.332989
$396$	−6.00000	−0.301511
$397$	11.9098	0.597737	0.298869	−	0.954294i	$-0.403391\pi$
$397$	11.9098	0.597737	0.298869	+	0.954294i	$0.403391\pi$
$398$	10.8885	0.545793
$399$	0.583592	0.0292161
$400$	1.00000	0.0500000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	9.79837	0.488698
$403$	0	0
$404$	−6.70820	−0.333746
$405$	1.00000	0.0496904
$406$	0.708204	0.0351476
$407$	29.8328	1.47876
$408$	7.85410	0.388836
$409$	19.5623	0.967294	0.483647	−	0.875263i	$-0.339312\pi$
$409$	19.5623	0.967294	0.483647	+	0.875263i	$0.339312\pi$
$410$	−4.14590	−0.204751
$411$	−15.0000	−0.739895
$412$	15.2361	0.750627
$413$	1.41641	0.0696969
$414$	15.7082	0.772016
$415$	−12.7082	−0.623821
$416$	4.23607	0.207690
$417$	−2.76393	−0.135350
$418$	−7.41641	−0.362748
$419$	−15.7082	−0.767396	−0.383698	−	0.923459i	$-0.625350\pi$
$419$	−15.7082	−0.767396	−0.383698	+	0.923459i	$0.625350\pi$
$420$	0.236068	0.0115189
$421$	14.5279	0.708045	0.354022	−	0.935237i	$-0.384814\pi$
$421$	14.5279	0.708045	0.354022	+	0.935237i	$0.384814\pi$
$422$	−5.09017	−0.247786
$423$	18.0000	0.875190
$424$	−13.4164	−0.651558
$425$	−7.85410	−0.380980
$426$	−4.85410	−0.235182
$427$	0.145898	0.00706050
$428$	−9.70820	−0.469264
$429$	−12.7082	−0.613558
$430$	−3.23607	−0.156057
$431$	31.1459	1.50025	0.750123	−	0.661299i	$-0.229994\pi$
$431$	31.1459	1.50025	0.750123	+	0.661299i	$0.229994\pi$
$432$	−5.00000	−0.240563
$433$	1.11146	0.0534132	0.0267066	−	0.999643i	$-0.491498\pi$
$433$	1.11146	0.0534132	0.0267066	+	0.999643i	$0.491498\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	17.0902	0.818471
$437$	19.4164	0.928813
$438$	10.1803	0.486435
$439$	−19.6180	−0.936318	−0.468159	−	0.883644i	$-0.655082\pi$
$439$	−19.6180	−0.936318	−0.468159	+	0.883644i	$0.655082\pi$
$440$	−3.00000	−0.143019
$441$	13.8885	0.661359
$442$	−33.2705	−1.58252
$443$	−15.2705	−0.725524	−0.362762	−	0.931882i	$-0.618166\pi$
$443$	−15.2705	−0.725524	−0.362762	+	0.931882i	$0.618166\pi$
$444$	9.94427	0.471934
$445$	12.0000	0.568855
$446$	9.79837	0.463966
$447$	1.85410	0.0876960
$448$	0.236068	0.0111532
$449$	34.8541	1.64487	0.822433	−	0.568861i	$-0.192616\pi$
$449$	34.8541	1.64487	0.822433	+	0.568861i	$0.192616\pi$
$450$	2.00000	0.0942809
$451$	12.4377	0.585668
$452$	12.7082	0.597744
$453$	15.9443	0.749127
$454$	−4.14590	−0.194577
$455$	−1.00000	−0.0468807
$456$	−2.47214	−0.115768
$457$	4.56231	0.213416	0.106708	−	0.994290i	$-0.465969\pi$
$457$	4.56231	0.213416	0.106708	+	0.994290i	$0.465969\pi$
$458$	−15.6180	−0.729783
$459$	39.2705	1.83299
$460$	7.85410	0.366199
$461$	11.8328	0.551109	0.275555	−	0.961285i	$-0.411139\pi$
$461$	11.8328	0.551109	0.275555	+	0.961285i	$0.411139\pi$
$462$	−0.708204	−0.0329486
$463$	−0.527864	−0.0245319	−0.0122660	−	0.999925i	$-0.503904\pi$
$463$	−0.527864	−0.0245319	−0.0122660	+	0.999925i	$0.503904\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	17.5623	0.813558
$467$	−25.8541	−1.19639	−0.598193	−	0.801352i	$-0.704114\pi$
$467$	−25.8541	−1.19639	−0.598193	+	0.801352i	$0.704114\pi$
$468$	8.47214	0.391625
$469$	−2.31308	−0.106808
$470$	9.00000	0.415139
$471$	20.0344	0.923138
$472$	−6.00000	−0.276172
$473$	9.70820	0.446384
$474$	−6.61803	−0.303976
$475$	2.47214	0.113429
$476$	−1.85410	−0.0849826
$477$	−26.8328	−1.22859
$478$	−18.9787	−0.868066
$479$	−27.9787	−1.27838	−0.639190	−	0.769049i	$-0.720730\pi$
$479$	−27.9787	−1.27838	−0.639190	+	0.769049i	$0.720730\pi$
$480$	−1.00000	−0.0456435
$481$	−42.1246	−1.92072
$482$	12.1246	0.552261
$483$	1.85410	0.0843646
$484$	−2.00000	−0.0909091
$485$	−1.23607	−0.0561270
$486$	−16.0000	−0.725775
$487$	14.7984	0.670578	0.335289	−	0.942115i	$-0.391166\pi$
$487$	14.7984	0.670578	0.335289	+	0.942115i	$0.391166\pi$
$488$	−0.618034	−0.0279771
$489$	0.236068	0.0106754
$490$	6.94427	0.313710
$491$	−22.4164	−1.01164	−0.505819	−	0.862640i	$-0.668810\pi$
$491$	−22.4164	−1.01164	−0.505819	+	0.862640i	$0.668810\pi$
$492$	4.14590	0.186912
$493$	23.5623	1.06119
$494$	10.4721	0.471164
$495$	−6.00000	−0.269680
$496$	0	0
$497$	1.14590	0.0514006
$498$	12.7082	0.569468
$499$	2.03444	0.0910741	0.0455371	−	0.998963i	$-0.485500\pi$
$499$	2.03444	0.0910741	0.0455371	+	0.998963i	$0.485500\pi$
$500$	1.00000	0.0447214
$501$	13.4164	0.599401
$502$	−25.8541	−1.15393
$503$	44.5623	1.98694	0.993468	−	0.114115i	$-0.0364033\pi$
$503$	44.5623	1.98694	0.993468	+	0.114115i	$0.0364033\pi$
$504$	0.472136	0.0210306
$505$	−6.70820	−0.298511
$506$	−23.5623	−1.04747
$507$	4.94427	0.219583
$508$	4.05573	0.179944
$509$	0.875388	0.0388009	0.0194004	−	0.999812i	$-0.493824\pi$
$509$	0.875388	0.0388009	0.0194004	+	0.999812i	$0.493824\pi$
$510$	7.85410	0.347785
$511$	−2.40325	−0.106314
$512$	−1.00000	−0.0441942
$513$	−12.3607	−0.545737
$514$	−5.29180	−0.233411
$515$	15.2361	0.671381
$516$	3.23607	0.142460
$517$	−27.0000	−1.18746
$518$	−2.34752	−0.103144
$519$	9.00000	0.395056
$520$	4.23607	0.185764
$521$	−16.4164	−0.719216	−0.359608	−	0.933103i	$-0.617090\pi$
$521$	−16.4164	−0.719216	−0.359608	+	0.933103i	$0.617090\pi$
$522$	−6.00000	−0.262613
$523$	−0.472136	−0.0206451	−0.0103225	−	0.999947i	$-0.503286\pi$
$523$	−0.472136	−0.0206451	−0.0103225	+	0.999947i	$0.503286\pi$
$524$	−4.85410	−0.212052
$525$	0.236068	0.0103029
$526$	8.56231	0.373334
$527$	0	0
$528$	3.00000	0.130558
$529$	38.6869	1.68204
$530$	−13.4164	−0.582772
$531$	−12.0000	−0.520756
$532$	0.583592	0.0253019
$533$	−17.5623	−0.760708
$534$	−12.0000	−0.519291
$535$	−9.70820	−0.419722
$536$	9.79837	0.423225
$537$	24.0000	1.03568
$538$	−1.14590	−0.0494032
$539$	−20.8328	−0.897333
$540$	−5.00000	−0.215166
$541$	−28.2361	−1.21396	−0.606982	−	0.794716i	$-0.707620\pi$
$541$	−28.2361	−1.21396	−0.606982	+	0.794716i	$0.707620\pi$
$542$	−15.9443	−0.684865
$543$	2.09017	0.0896978
$544$	7.85410	0.336742
$545$	17.0902	0.732062
$546$	1.00000	0.0427960
$547$	16.3262	0.698060	0.349030	−	0.937112i	$-0.386511\pi$
$547$	16.3262	0.698060	0.349030	+	0.937112i	$0.386511\pi$
$548$	−15.0000	−0.640768
$549$	−1.23607	−0.0527541
$550$	−3.00000	−0.127920
$551$	−7.41641	−0.315950
$552$	−7.85410	−0.334293
$553$	1.56231	0.0664360
$554$	15.5623	0.661179
$555$	9.94427	0.422111
$556$	−2.76393	−0.117217
$557$	−26.2918	−1.11402	−0.557010	−	0.830506i	$-0.688051\pi$
$557$	−26.2918	−1.11402	−0.557010	+	0.830506i	$0.688051\pi$
$558$	0	0
$559$	−13.7082	−0.579795
$560$	0.236068	0.00997569
$561$	−23.5623	−0.994801
$562$	−7.85410	−0.331305
$563$	7.41641	0.312564	0.156282	−	0.987712i	$-0.450049\pi$
$563$	7.41641	0.312564	0.156282	+	0.987712i	$0.450049\pi$
$564$	−9.00000	−0.378968
$565$	12.7082	0.534638
$566$	7.18034	0.301812
$567$	0.236068	0.00991392
$568$	−4.85410	−0.203674
$569$	−9.70820	−0.406989	−0.203495	−	0.979076i	$-0.565230\pi$
$569$	−9.70820	−0.406989	−0.203495	+	0.979076i	$0.565230\pi$
$570$	−2.47214	−0.103546
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	−12.7082	−0.531357
$573$	−6.43769	−0.268939
$574$	−0.978714	−0.0408507
$575$	7.85410	0.327539
$576$	−2.00000	−0.0833333
$577$	11.7984	0.491173	0.245586	−	0.969375i	$-0.421020\pi$
$577$	11.7984	0.491173	0.245586	+	0.969375i	$0.421020\pi$
$578$	−44.6869	−1.85873
$579$	8.03444	0.333900
$580$	−3.00000	−0.124568
$581$	−3.00000	−0.124461
$582$	1.23607	0.0512367
$583$	40.2492	1.66695
$584$	10.1803	0.421265
$585$	8.47214	0.350280
$586$	−0.708204	−0.0292556
$587$	47.8328	1.97427	0.987136	−	0.159884i	$-0.0511120\pi$
$587$	47.8328	1.97427	0.987136	+	0.159884i	$0.0511120\pi$
$588$	−6.94427	−0.286377
$589$	0	0
$590$	−6.00000	−0.247016
$591$	12.7082	0.522746
$592$	9.94427	0.408707
$593$	−24.2705	−0.996670	−0.498335	−	0.866984i	$-0.666055\pi$
$593$	−24.2705	−0.996670	−0.498335	+	0.866984i	$0.666055\pi$
$594$	15.0000	0.615457
$595$	−1.85410	−0.0760108
$596$	1.85410	0.0759470
$597$	−10.8885	−0.445638
$598$	33.2705	1.36053
$599$	−35.1246	−1.43515	−0.717576	−	0.696480i	$-0.754749\pi$
$599$	−35.1246	−1.43515	−0.717576	+	0.696480i	$0.754749\pi$
$600$	−1.00000	−0.0408248
$601$	3.94427	0.160890	0.0804451	−	0.996759i	$-0.474366\pi$
$601$	3.94427	0.160890	0.0804451	+	0.996759i	$0.474366\pi$
$602$	−0.763932	−0.0311355
$603$	19.5967	0.798041
$604$	15.9443	0.648763
$605$	−2.00000	−0.0813116
$606$	6.70820	0.272502
$607$	30.6180	1.24275	0.621374	−	0.783514i	$-0.286575\pi$
$607$	30.6180	1.24275	0.621374	+	0.783514i	$0.286575\pi$
$608$	−2.47214	−0.100258
$609$	−0.708204	−0.0286979
$610$	−0.618034	−0.0250235
$611$	38.1246	1.54236
$612$	15.7082	0.634967
$613$	21.4508	0.866392	0.433196	−	0.901300i	$-0.357386\pi$
$613$	21.4508	0.866392	0.433196	+	0.901300i	$0.357386\pi$
$614$	2.65248	0.107045
$615$	4.14590	0.167179
$616$	−0.708204	−0.0285343
$617$	−7.14590	−0.287683	−0.143842	−	0.989601i	$-0.545946\pi$
$617$	−7.14590	−0.287683	−0.143842	+	0.989601i	$0.545946\pi$
$618$	−15.2361	−0.612885
$619$	9.88854	0.397454	0.198727	−	0.980055i	$-0.436319\pi$
$619$	9.88854	0.397454	0.198727	+	0.980055i	$0.436319\pi$
$620$	0	0
$621$	−39.2705	−1.57587
$622$	−15.7082	−0.629842
$623$	2.83282	0.113494
$624$	−4.23607	−0.169578
$625$	1.00000	0.0400000
$626$	−29.7984	−1.19098
$627$	7.41641	0.296183
$628$	20.0344	0.799461
$629$	−78.1033	−3.11418
$630$	0.472136	0.0188103
$631$	−19.0689	−0.759120	−0.379560	−	0.925167i	$-0.623925\pi$
$631$	−19.0689	−0.759120	−0.379560	+	0.925167i	$0.623925\pi$
$632$	−6.61803	−0.263251
$633$	5.09017	0.202316
$634$	−7.41641	−0.294543
$635$	4.05573	0.160947
$636$	13.4164	0.531995
$637$	29.4164	1.16552
$638$	9.00000	0.356313
$639$	−9.70820	−0.384051
$640$	−1.00000	−0.0395285
$641$	−11.8328	−0.467368	−0.233684	−	0.972313i	$-0.575078\pi$
$641$	−11.8328	−0.467368	−0.233684	+	0.972313i	$0.575078\pi$
$642$	9.70820	0.383152
$643$	−35.9230	−1.41666	−0.708332	−	0.705879i	$-0.750552\pi$
$643$	−35.9230	−1.41666	−0.708332	+	0.705879i	$0.750552\pi$
$644$	1.85410	0.0730619
$645$	3.23607	0.127420
$646$	19.4164	0.763928
$647$	−38.5623	−1.51604	−0.758020	−	0.652231i	$-0.773833\pi$
$647$	−38.5623	−1.51604	−0.758020	+	0.652231i	$0.773833\pi$
$648$	−1.00000	−0.0392837
$649$	18.0000	0.706562
$650$	4.23607	0.166152
$651$	0	0
$652$	0.236068	0.00924514
$653$	33.2705	1.30198	0.650988	−	0.759088i	$-0.274355\pi$
$653$	33.2705	1.30198	0.650988	+	0.759088i	$0.274355\pi$
$654$	−17.0902	−0.668279
$655$	−4.85410	−0.189665
$656$	4.14590	0.161870
$657$	20.3607	0.794346
$658$	2.12461	0.0828260
$659$	−27.0000	−1.05177	−0.525885	−	0.850555i	$-0.676266\pi$
$659$	−27.0000	−1.05177	−0.525885	+	0.850555i	$0.676266\pi$
$660$	3.00000	0.116775
$661$	−8.76393	−0.340877	−0.170439	−	0.985368i	$-0.554518\pi$
$661$	−8.76393	−0.340877	−0.170439	+	0.985368i	$0.554518\pi$
$662$	15.5623	0.604846
$663$	33.2705	1.29212
$664$	12.7082	0.493174
$665$	0.583592	0.0226307
$666$	19.8885	0.770665
$667$	−23.5623	−0.912336
$668$	13.4164	0.519096
$669$	−9.79837	−0.378827
$670$	9.79837	0.378544
$671$	1.85410	0.0715768
$672$	−0.236068	−0.00910652
$673$	22.2148	0.856317	0.428158	−	0.903704i	$-0.359163\pi$
$673$	22.2148	0.856317	0.428158	+	0.903704i	$0.359163\pi$
$674$	−12.8885	−0.496448
$675$	−5.00000	−0.192450
$676$	4.94427	0.190164
$677$	−20.5623	−0.790274	−0.395137	−	0.918622i	$-0.629303\pi$
$677$	−20.5623	−0.790274	−0.395137	+	0.918622i	$0.629303\pi$
$678$	−12.7082	−0.488056
$679$	−0.291796	−0.0111981
$680$	7.85410	0.301191
$681$	4.14590	0.158871
$682$	0	0
$683$	−33.5410	−1.28341	−0.641706	−	0.766951i	$-0.721773\pi$
$683$	−33.5410	−1.28341	−0.641706	+	0.766951i	$0.721773\pi$
$684$	−4.94427	−0.189049
$685$	−15.0000	−0.573121
$686$	3.29180	0.125681
$687$	15.6180	0.595865
$688$	3.23607	0.123374
$689$	−56.8328	−2.16516
$690$	−7.85410	−0.299001
$691$	−23.2148	−0.883132	−0.441566	−	0.897229i	$-0.645577\pi$
$691$	−23.2148	−0.883132	−0.441566	+	0.897229i	$0.645577\pi$
$692$	9.00000	0.342129
$693$	−1.41641	−0.0538049
$694$	−12.2705	−0.465782
$695$	−2.76393	−0.104842
$696$	3.00000	0.113715
$697$	−32.5623	−1.23339
$698$	−5.52786	−0.209233
$699$	−17.5623	−0.664267
$700$	0.236068	0.00892253
$701$	29.8328	1.12677	0.563385	−	0.826195i	$-0.309499\pi$
$701$	29.8328	1.12677	0.563385	+	0.826195i	$0.309499\pi$
$702$	−21.1803	−0.799400
$703$	24.5836	0.927188
$704$	3.00000	0.113067
$705$	−9.00000	−0.338960
$706$	−10.4164	−0.392027
$707$	−1.58359	−0.0595571
$708$	6.00000	0.225494
$709$	2.43769	0.0915495	0.0457748	−	0.998952i	$-0.485424\pi$
$709$	2.43769	0.0915495	0.0457748	+	0.998952i	$0.485424\pi$
$710$	−4.85410	−0.182171
$711$	−13.2361	−0.496391
$712$	−12.0000	−0.449719
$713$	0	0
$714$	1.85410	0.0693880
$715$	−12.7082	−0.475260
$716$	24.0000	0.896922
$717$	18.9787	0.708773
$718$	33.2705	1.24164
$719$	32.5623	1.21437	0.607185	−	0.794561i	$-0.292299\pi$
$719$	32.5623	1.21437	0.607185	+	0.794561i	$0.292299\pi$
$720$	−2.00000	−0.0745356
$721$	3.59675	0.133950
$722$	12.8885	0.479662
$723$	−12.1246	−0.450919
$724$	2.09017	0.0776806
$725$	−3.00000	−0.111417
$726$	2.00000	0.0742270
$727$	1.12461	0.0417095	0.0208548	−	0.999783i	$-0.493361\pi$
$727$	1.12461	0.0417095	0.0208548	+	0.999783i	$0.493361\pi$
$728$	1.00000	0.0370625
$729$	13.0000	0.481481
$730$	10.1803	0.376791
$731$	−25.4164	−0.940060
$732$	0.618034	0.0228432
$733$	−53.2492	−1.96680	−0.983402	−	0.181438i	$-0.941925\pi$
$733$	−53.2492	−1.96680	−0.983402	+	0.181438i	$0.941925\pi$
$734$	28.0000	1.03350
$735$	−6.94427	−0.256143
$736$	−7.85410	−0.289506
$737$	−29.3951	−1.08278
$738$	8.29180	0.305225
$739$	51.1803	1.88270	0.941350	−	0.337433i	$-0.109559\pi$
$739$	51.1803	1.88270	0.941350	+	0.337433i	$0.109559\pi$
$740$	9.94427	0.365559
$741$	−10.4721	−0.384704
$742$	−3.16718	−0.116271
$743$	36.9787	1.35662	0.678309	−	0.734777i	$-0.262713\pi$
$743$	36.9787	1.35662	0.678309	+	0.734777i	$0.262713\pi$
$744$	0	0
$745$	1.85410	0.0679290
$746$	17.6525	0.646303
$747$	25.4164	0.929938
$748$	−23.5623	−0.861523
$749$	−2.29180	−0.0837404
$750$	−1.00000	−0.0365148
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	−9.00000	−0.328196
$753$	25.8541	0.942176
$754$	−12.7082	−0.462805
$755$	15.9443	0.580271
$756$	−1.18034	−0.0429285
$757$	−17.6525	−0.641590	−0.320795	−	0.947149i	$-0.603950\pi$
$757$	−17.6525	−0.641590	−0.320795	+	0.947149i	$0.603950\pi$
$758$	7.50658	0.272651
$759$	23.5623	0.855258
$760$	−2.47214	−0.0896738
$761$	4.85410	0.175961	0.0879805	−	0.996122i	$-0.471959\pi$
$761$	4.85410	0.175961	0.0879805	+	0.996122i	$0.471959\pi$
$762$	−4.05573	−0.146924
$763$	4.03444	0.146057
$764$	−6.43769	−0.232908
$765$	15.7082	0.567931
$766$	−9.00000	−0.325183
$767$	−25.4164	−0.917733
$768$	1.00000	0.0360844
$769$	22.8197	0.822898	0.411449	−	0.911433i	$-0.365023\pi$
$769$	22.8197	0.822898	0.411449	+	0.911433i	$0.365023\pi$
$770$	−0.708204	−0.0255219
$771$	5.29180	0.190579
$772$	8.03444	0.289166
$773$	4.85410	0.174590	0.0872950	−	0.996183i	$-0.472178\pi$
$773$	4.85410	0.174590	0.0872950	+	0.996183i	$0.472178\pi$
$774$	6.47214	0.232636
$775$	0	0
$776$	1.23607	0.0443723
$777$	2.34752	0.0842170
$778$	−11.2918	−0.404831
$779$	10.2492	0.367217
$780$	−4.23607	−0.151676
$781$	14.5623	0.521080
$782$	61.6869	2.20592
$783$	15.0000	0.536056
$784$	−6.94427	−0.248010
$785$	20.0344	0.715060
$786$	4.85410	0.173140
$787$	30.6869	1.09387	0.546935	−	0.837175i	$-0.315794\pi$
$787$	30.6869	1.09387	0.546935	+	0.837175i	$0.315794\pi$
$788$	12.7082	0.452711
$789$	−8.56231	−0.304826
$790$	−6.61803	−0.235459
$791$	3.00000	0.106668
$792$	6.00000	0.213201
$793$	−2.61803	−0.0929691
$794$	−11.9098	−0.422664
$795$	13.4164	0.475831
$796$	−10.8885	−0.385934
$797$	−8.83282	−0.312874	−0.156437	−	0.987688i	$-0.550001\pi$
$797$	−8.83282	−0.312874	−0.156437	+	0.987688i	$0.550001\pi$
$798$	−0.583592	−0.0206589
$799$	70.6869	2.50072
$800$	−1.00000	−0.0353553
$801$	−24.0000	−0.847998
$802$	−30.0000	−1.05934
$803$	−30.5410	−1.07777
$804$	−9.79837	−0.345562
$805$	1.85410	0.0653485
$806$	0	0
$807$	1.14590	0.0403375
$808$	6.70820	0.235994
$809$	22.1459	0.778608	0.389304	−	0.921109i	$-0.372716\pi$
$809$	22.1459	0.778608	0.389304	+	0.921109i	$0.372716\pi$
$810$	−1.00000	−0.0351364
$811$	47.0000	1.65039	0.825197	−	0.564846i	$-0.191064\pi$
$811$	47.0000	1.65039	0.825197	+	0.564846i	$0.191064\pi$
$812$	−0.708204	−0.0248531
$813$	15.9443	0.559190
$814$	−29.8328	−1.04564
$815$	0.236068	0.00826910
$816$	−7.85410	−0.274949
$817$	8.00000	0.279885
$818$	−19.5623	−0.683980
$819$	2.00000	0.0698857
$820$	4.14590	0.144781
$821$	−18.2705	−0.637645	−0.318823	−	0.947814i	$-0.603287\pi$
$821$	−18.2705	−0.637645	−0.318823	+	0.947814i	$0.603287\pi$
$822$	15.0000	0.523185
$823$	10.3820	0.361893	0.180946	−	0.983493i	$-0.442084\pi$
$823$	10.3820	0.361893	0.180946	+	0.983493i	$0.442084\pi$
$824$	−15.2361	−0.530774
$825$	3.00000	0.104447
$826$	−1.41641	−0.0492831
$827$	18.4377	0.641141	0.320571	−	0.947225i	$-0.396125\pi$
$827$	18.4377	0.641141	0.320571	+	0.947225i	$0.396125\pi$
$828$	−15.7082	−0.545898
$829$	31.7639	1.10321	0.551603	−	0.834106i	$-0.314016\pi$
$829$	31.7639	1.10321	0.551603	+	0.834106i	$0.314016\pi$
$830$	12.7082	0.441108
$831$	−15.5623	−0.539850
$832$	−4.23607	−0.146859
$833$	54.5410	1.88973
$834$	2.76393	0.0957071
$835$	13.4164	0.464294
$836$	7.41641	0.256502
$837$	0	0
$838$	15.7082	0.542631
$839$	8.12461	0.280493	0.140246	−	0.990117i	$-0.455211\pi$
$839$	8.12461	0.280493	0.140246	+	0.990117i	$0.455211\pi$
$840$	−0.236068	−0.00814512
$841$	−20.0000	−0.689655
$842$	−14.5279	−0.500663
$843$	7.85410	0.270510
$844$	5.09017	0.175211
$845$	4.94427	0.170088
$846$	−18.0000	−0.618853
$847$	−0.472136	−0.0162228
$848$	13.4164	0.460721
$849$	−7.18034	−0.246429
$850$	7.85410	0.269393
$851$	78.1033	2.67735
$852$	4.85410	0.166299
$853$	−55.0000	−1.88316	−0.941582	−	0.336784i	$-0.890661\pi$
$853$	−55.0000	−1.88316	−0.941582	+	0.336784i	$0.890661\pi$
$854$	−0.145898	−0.00499253
$855$	−4.94427	−0.169091
$856$	9.70820	0.331820
$857$	−2.12461	−0.0725754	−0.0362877	−	0.999341i	$-0.511553\pi$
$857$	−2.12461	−0.0725754	−0.0362877	+	0.999341i	$0.511553\pi$
$858$	12.7082	0.433851
$859$	−52.0132	−1.77467	−0.887333	−	0.461129i	$-0.847444\pi$
$859$	−52.0132	−1.77467	−0.887333	+	0.461129i	$0.847444\pi$
$860$	3.23607	0.110349
$861$	0.978714	0.0333545
$862$	−31.1459	−1.06083
$863$	−42.1033	−1.43321	−0.716607	−	0.697477i	$-0.754306\pi$
$863$	−42.1033	−1.43321	−0.716607	+	0.697477i	$0.754306\pi$
$864$	5.00000	0.170103
$865$	9.00000	0.306009
$866$	−1.11146	−0.0377688
$867$	44.6869	1.51765
$868$	0	0
$869$	19.8541	0.673504
$870$	3.00000	0.101710
$871$	41.5066	1.40640
$872$	−17.0902	−0.578746
$873$	2.47214	0.0836691
$874$	−19.4164	−0.656770
$875$	0.236068	0.00798055
$876$	−10.1803	−0.343962
$877$	−13.0000	−0.438979	−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000	−0.438979	−0.219489	+	0.975615i	$0.570439\pi$
$878$	19.6180	0.662077
$879$	0.708204	0.0238871
$880$	3.00000	0.101130
$881$	51.5410	1.73646	0.868231	−	0.496161i	$-0.165257\pi$
$881$	51.5410	1.73646	0.868231	+	0.496161i	$0.165257\pi$
$882$	−13.8885	−0.467652
$883$	−27.4721	−0.924511	−0.462255	−	0.886747i	$-0.652960\pi$
$883$	−27.4721	−0.924511	−0.462255	+	0.886747i	$0.652960\pi$
$884$	33.2705	1.11901
$885$	6.00000	0.201688
$886$	15.2705	0.513023
$887$	12.9787	0.435783	0.217891	−	0.975973i	$-0.430082\pi$
$887$	12.9787	0.435783	0.217891	+	0.975973i	$0.430082\pi$
$888$	−9.94427	−0.333708
$889$	0.957428	0.0321111
$890$	−12.0000	−0.402241
$891$	3.00000	0.100504
$892$	−9.79837	−0.328074
$893$	−22.2492	−0.744542
$894$	−1.85410	−0.0620104
$895$	24.0000	0.802232
$896$	−0.236068	−0.00788648
$897$	−33.2705	−1.11087
$898$	−34.8541	−1.16310
$899$	0	0
$900$	−2.00000	−0.0666667
$901$	−105.374	−3.51051
$902$	−12.4377	−0.414130
$903$	0.763932	0.0254221
$904$	−12.7082	−0.422669
$905$	2.09017	0.0694796
$906$	−15.9443	−0.529713
$907$	−43.9443	−1.45915	−0.729573	−	0.683903i	$-0.760281\pi$
$907$	−43.9443	−1.45915	−0.729573	+	0.683903i	$0.760281\pi$
$908$	4.14590	0.137586
$909$	13.4164	0.444994
$910$	1.00000	0.0331497
$911$	43.4164	1.43845	0.719225	−	0.694777i	$-0.244497\pi$
$911$	43.4164	1.43845	0.719225	+	0.694777i	$0.244497\pi$
$912$	2.47214	0.0818606
$913$	−38.1246	−1.26174
$914$	−4.56231	−0.150908
$915$	0.618034	0.0204316
$916$	15.6180	0.516034
$917$	−1.14590	−0.0378409
$918$	−39.2705	−1.29612
$919$	19.5967	0.646437	0.323219	−	0.946324i	$-0.395235\pi$
$919$	19.5967	0.646437	0.323219	+	0.946324i	$0.395235\pi$
$920$	−7.85410	−0.258942
$921$	−2.65248	−0.0874021
$922$	−11.8328	−0.389693
$923$	−20.5623	−0.676817
$924$	0.708204	0.0232982
$925$	9.94427	0.326966
$926$	0.527864	0.0173467
$927$	−30.4721	−1.00084
$928$	3.00000	0.0984798
$929$	5.56231	0.182493	0.0912467	−	0.995828i	$-0.470915\pi$
$929$	5.56231	0.182493	0.0912467	+	0.995828i	$0.470915\pi$
$930$	0	0
$931$	−17.1672	−0.562632
$932$	−17.5623	−0.575272
$933$	15.7082	0.514264
$934$	25.8541	0.845972
$935$	−23.5623	−0.770570
$936$	−8.47214	−0.276920
$937$	−38.7639	−1.26636	−0.633181	−	0.774004i	$-0.718251\pi$
$937$	−38.7639	−1.26636	−0.633181	+	0.774004i	$0.718251\pi$
$938$	2.31308	0.0755248
$939$	29.7984	0.972433
$940$	−9.00000	−0.293548
$941$	−8.72949	−0.284573	−0.142287	−	0.989825i	$-0.545445\pi$
$941$	−8.72949	−0.284573	−0.142287	+	0.989825i	$0.545445\pi$
$942$	−20.0344	−0.652757
$943$	32.5623	1.06037
$944$	6.00000	0.195283
$945$	−1.18034	−0.0383965
$946$	−9.70820	−0.315641
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	6.61803	0.214944
$949$	43.1246	1.39988
$950$	−2.47214	−0.0802067
$951$	7.41641	0.240494
$952$	1.85410	0.0600918
$953$	18.7082	0.606018	0.303009	−	0.952988i	$-0.402009\pi$
$953$	18.7082	0.606018	0.303009	+	0.952988i	$0.402009\pi$
$954$	26.8328	0.868744
$955$	−6.43769	−0.208319
$956$	18.9787	0.613815
$957$	−9.00000	−0.290929
$958$	27.9787	0.903951
$959$	−3.54102	−0.114345
$960$	1.00000	0.0322749
$961$	0	0
$962$	42.1246	1.35815
$963$	19.4164	0.625685
$964$	−12.1246	−0.390507
$965$	8.03444	0.258638
$966$	−1.85410	−0.0596548
$967$	−4.88854	−0.157205	−0.0786025	−	0.996906i	$-0.525046\pi$
$967$	−4.88854	−0.157205	−0.0786025	+	0.996906i	$0.525046\pi$
$968$	2.00000	0.0642824
$969$	−19.4164	−0.623745
$970$	1.23607	0.0396878
$971$	−20.8328	−0.668557	−0.334278	−	0.942474i	$-0.608493\pi$
$971$	−20.8328	−0.668557	−0.334278	+	0.942474i	$0.608493\pi$
$972$	16.0000	0.513200
$973$	−0.652476	−0.0209174
$974$	−14.7984	−0.474170
$975$	−4.23607	−0.135663
$976$	0.618034	0.0197828
$977$	−50.8328	−1.62629	−0.813143	−	0.582064i	$-0.802245\pi$
$977$	−50.8328	−1.62629	−0.813143	+	0.582064i	$0.802245\pi$
$978$	−0.236068	−0.00754862
$979$	36.0000	1.15056
$980$	−6.94427	−0.221827
$981$	−34.1803	−1.09129
$982$	22.4164	0.715336
$983$	12.8754	0.410661	0.205331	−	0.978693i	$-0.434173\pi$
$983$	12.8754	0.410661	0.205331	+	0.978693i	$0.434173\pi$
$984$	−4.14590	−0.132166
$985$	12.7082	0.404917
$986$	−23.5623	−0.750377
$987$	−2.12461	−0.0676271
$988$	−10.4721	−0.333163
$989$	25.4164	0.808195
$990$	6.00000	0.190693
$991$	9.34752	0.296934	0.148467	−	0.988917i	$-0.452566\pi$
$991$	9.34752	0.296934	0.148467	+	0.988917i	$0.452566\pi$
$992$	0	0
$993$	−15.5623	−0.493855
$994$	−1.14590	−0.0363457
$995$	−10.8885	−0.345190
$996$	−12.7082	−0.402675
$997$	−35.5967	−1.12736	−0.563680	−	0.825993i	$-0.690615\pi$
$997$	−35.5967	−1.12736	−0.563680	+	0.825993i	$0.690615\pi$
$998$	−2.03444	−0.0643991
$999$	−49.7214	−1.57311

Display

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a_n

instead) (See

a_n

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a_n

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9610.2.a.k.1.2		2
31.4	even	5		310.2.h.a.171.1	✓	4
31.8	even	5		310.2.h.a.281.1	yes	4
31.30	odd	2		9610.2.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.h.a.171.1	✓	4	31.4	even	5
310.2.h.a.281.1	yes	4	31.8	even	5
9610.2.a.c.1.2		2	31.30	odd	2
9610.2.a.k.1.2		2	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

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Varieties

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Groups

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Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	9610.2.a.k.1.2

Level	$9610$
Weight	$2$
Character	9610.1
Self dual	yes
Analytic conductor	$76.736$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$