LMFDB - Embedded newform 161.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [161,2,Mod(1,161)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(161, 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("161.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$161 = 7 \cdot 23$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	161.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:	$1.28559147254$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	161.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70928 q^{2} -1.17009 q^{3} +5.34017 q^{4} -1.17009 q^{5} +3.17009 q^{6} -1.00000 q^{7} -9.04945 q^{8} -1.63090 q^{9} +3.17009 q^{10} +3.70928 q^{11} -6.24846 q^{12} +4.34017 q^{13} +2.70928 q^{14} +1.36910 q^{15} +13.8371 q^{16} +3.17009 q^{17} +4.41855 q^{18} +5.26180 q^{19} -6.24846 q^{20} +1.17009 q^{21} -10.0494 q^{22} +1.00000 q^{23} +10.5886 q^{24} -3.63090 q^{25} -11.7587 q^{26} +5.41855 q^{27} -5.34017 q^{28} +0.630898 q^{29} -3.70928 q^{30} +9.32684 q^{31} -19.3896 q^{32} -4.34017 q^{33} -8.58864 q^{34} +1.17009 q^{35} -8.70928 q^{36} -3.07838 q^{37} -14.2557 q^{38} -5.07838 q^{39} +10.5886 q^{40} +2.68035 q^{41} -3.17009 q^{42} -8.49693 q^{43} +19.8082 q^{44} +1.90829 q^{45} -2.70928 q^{46} +6.09171 q^{47} -16.1906 q^{48} +1.00000 q^{49} +9.83710 q^{50} -3.70928 q^{51} +23.1773 q^{52} +4.15676 q^{53} -14.6803 q^{54} -4.34017 q^{55} +9.04945 q^{56} -6.15676 q^{57} -1.70928 q^{58} -8.40522 q^{59} +7.31124 q^{60} -6.92881 q^{61} -25.2690 q^{62} +1.63090 q^{63} +24.8576 q^{64} -5.07838 q^{65} +11.7587 q^{66} +9.86603 q^{67} +16.9288 q^{68} -1.17009 q^{69} -3.17009 q^{70} +10.8371 q^{71} +14.7587 q^{72} -13.0205 q^{73} +8.34017 q^{74} +4.24846 q^{75} +28.0989 q^{76} -3.70928 q^{77} +13.7587 q^{78} -4.38962 q^{79} -16.1906 q^{80} -1.44748 q^{81} -7.26180 q^{82} +14.6537 q^{83} +6.24846 q^{84} -3.70928 q^{85} +23.0205 q^{86} -0.738205 q^{87} -33.5669 q^{88} -6.77205 q^{89} -5.17009 q^{90} -4.34017 q^{91} +5.34017 q^{92} -10.9132 q^{93} -16.5041 q^{94} -6.15676 q^{95} +22.6875 q^{96} +8.58864 q^{97} -2.70928 q^{98} -6.04945 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q - q^{2} + 2 q^{3} + 5 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} - 9 q^{8} - q^{9} + 4 q^{10} + 4 q^{11} - 10 q^{12} + 2 q^{13} + q^{14} + 8 q^{15} + 13 q^{16} + 4 q^{17} - q^{18} + 8 q^{19} - 10 q^{20}+ \cdots - q^{98}+O(q^{100})$ 3 * q - q^2 + 2 * q^3 + 5 * q^4 + 2 * q^5 + 4 * q^6 - 3 * q^7 - 9 * q^8 - q^9 + 4 * q^10 + 4 * q^11 - 10 * q^12 + 2 * q^13 + q^14 + 8 * q^15 + 13 * q^16 + 4 * q^17 - q^18 + 8 * q^19 - 10 * q^20 - 2 * q^21 - 12 * q^22 + 3 * q^23 + 12 * q^24 - 7 * q^25 - 10 * q^26 + 2 * q^27 - 5 * q^28 - 2 * q^29 - 4 * q^30 + 16 * q^31 - 29 * q^32 - 2 * q^33 - 6 * q^34 - 2 * q^35 - 19 * q^36 - 6 * q^37 - 12 * q^39 + 12 * q^40 - 14 * q^41 - 4 * q^42 - 8 * q^43 + 16 * q^44 + 8 * q^45 - q^46 + 16 * q^47 - 10 * q^48 + 3 * q^49 + q^50 - 4 * q^51 + 30 * q^52 + 6 * q^53 - 22 * q^54 - 2 * q^55 + 9 * q^56 - 12 * q^57 + 2 * q^58 - 10 * q^59 - 4 * q^60 + 10 * q^61 - 34 * q^62 + q^63 + 13 * q^64 - 12 * q^65 + 10 * q^66 + 16 * q^67 + 20 * q^68 + 2 * q^69 - 4 * q^70 + 4 * q^71 + 19 * q^72 - 6 * q^73 + 14 * q^74 + 4 * q^75 + 48 * q^76 - 4 * q^77 + 16 * q^78 + 16 * q^79 - 10 * q^80 - 5 * q^81 - 14 * q^82 + 20 * q^83 + 10 * q^84 - 4 * q^85 + 36 * q^86 - 10 * q^87 - 32 * q^88 + 4 * q^89 - 10 * q^90 - 2 * q^91 + 5 * q^92 + 12 * q^93 + 2 * q^94 - 12 * q^95 + 12 * q^96 + 6 * q^97 - 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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−2.70928	−1.91575	−0.957873	−	0.287190i	$-0.907279\pi$
$2$	−2.70928	−1.91575	−0.957873	+	0.287190i	$0.907279\pi$
$3$	−1.17009	−0.675550	−0.337775	−	0.941227i	$-0.609674\pi$
$3$	−1.17009	−0.675550	−0.337775	+	0.941227i	$0.609674\pi$
$4$	5.34017	2.67009
$5$	−1.17009	−0.523279	−0.261639	−	0.965166i	$-0.584263\pi$
$5$	−1.17009	−0.523279	−0.261639	+	0.965166i	$0.584263\pi$
$6$	3.17009	1.29418
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−9.04945	−3.19946
$9$	−1.63090	−0.543633
$10$	3.17009	1.00247
$11$	3.70928	1.11839	0.559194	−	0.829037i	$-0.311111\pi$
$11$	3.70928	1.11839	0.559194	+	0.829037i	$0.311111\pi$
$12$	−6.24846	−1.80378
$13$	4.34017	1.20375	0.601874	−	0.798591i	$-0.294421\pi$
$13$	4.34017	1.20375	0.601874	+	0.798591i	$0.294421\pi$
$14$	2.70928	0.724084
$15$	1.36910	0.353501
$16$	13.8371	3.45928
$17$	3.17009	0.768859	0.384429	−	0.923154i	$-0.374398\pi$
$17$	3.17009	0.768859	0.384429	+	0.923154i	$0.374398\pi$
$18$	4.41855	1.04146
$19$	5.26180	1.20714	0.603569	−	0.797311i	$-0.293745\pi$
$19$	5.26180	1.20714	0.603569	+	0.797311i	$0.293745\pi$
$20$	−6.24846	−1.39720
$21$	1.17009	0.255334
$22$	−10.0494	−2.14255
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	10.5886	2.16140
$25$	−3.63090	−0.726180
$26$	−11.7587	−2.30608
$27$	5.41855	1.04280
$28$	−5.34017	−1.00920
$29$	0.630898	0.117155	0.0585774	−	0.998283i	$-0.481344\pi$
$29$	0.630898	0.117155	0.0585774	+	0.998283i	$0.481344\pi$
$30$	−3.70928	−0.677218
$31$	9.32684	1.67515	0.837575	−	0.546322i	$-0.183973\pi$
$31$	9.32684	1.67515	0.837575	+	0.546322i	$0.183973\pi$
$32$	−19.3896	−3.42763
$33$	−4.34017	−0.755527
$34$	−8.58864	−1.47294
$35$	1.17009	0.197781
$36$	−8.70928	−1.45155
$37$	−3.07838	−0.506082	−0.253041	−	0.967456i	$-0.581431\pi$
$37$	−3.07838	−0.506082	−0.253041	+	0.967456i	$0.581431\pi$
$38$	−14.2557	−2.31257
$39$	−5.07838	−0.813191
$40$	10.5886	1.67421
$41$	2.68035	0.418600	0.209300	−	0.977852i	$-0.432882\pi$
$41$	2.68035	0.418600	0.209300	+	0.977852i	$0.432882\pi$
$42$	−3.17009	−0.489155
$43$	−8.49693	−1.29577	−0.647885	−	0.761738i	$-0.724346\pi$
$43$	−8.49693	−1.29577	−0.647885	+	0.761738i	$0.724346\pi$
$44$	19.8082	2.98619
$45$	1.90829	0.284471
$46$	−2.70928	−0.399461
$47$	6.09171	0.888567	0.444284	−	0.895886i	$-0.353458\pi$
$47$	6.09171	0.888567	0.444284	+	0.895886i	$0.353458\pi$
$48$	−16.1906	−2.33691
$49$	1.00000	0.142857
$50$	9.83710	1.39118
$51$	−3.70928	−0.519402
$52$	23.1773	3.21411
$53$	4.15676	0.570974	0.285487	−	0.958383i	$-0.407845\pi$
$53$	4.15676	0.570974	0.285487	+	0.958383i	$0.407845\pi$
$54$	−14.6803	−1.99774
$55$	−4.34017	−0.585229
$56$	9.04945	1.20928
$57$	−6.15676	−0.815482
$58$	−1.70928	−0.224439
$59$	−8.40522	−1.09427	−0.547133	−	0.837046i	$-0.684281\pi$
$59$	−8.40522	−1.09427	−0.547133	+	0.837046i	$0.684281\pi$
$60$	7.31124	0.943877
$61$	−6.92881	−0.887143	−0.443572	−	0.896239i	$-0.646289\pi$
$61$	−6.92881	−0.887143	−0.443572	+	0.896239i	$0.646289\pi$
$62$	−25.2690	−3.20916
$63$	1.63090	0.205474
$64$	24.8576	3.10720
$65$	−5.07838	−0.629895
$66$	11.7587	1.44740
$67$	9.86603	1.20533	0.602664	−	0.797995i	$-0.294106\pi$
$67$	9.86603	1.20533	0.602664	+	0.797995i	$0.294106\pi$
$68$	16.9288	2.05292
$69$	−1.17009	−0.140862
$70$	−3.17009	−0.378898
$71$	10.8371	1.28613	0.643064	−	0.765813i	$-0.277663\pi$
$71$	10.8371	1.28613	0.643064	+	0.765813i	$0.277663\pi$
$72$	14.7587	1.73933
$73$	−13.0205	−1.52394	−0.761968	−	0.647614i	$-0.775767\pi$
$73$	−13.0205	−1.52394	−0.761968	+	0.647614i	$0.775767\pi$
$74$	8.34017	0.969525
$75$	4.24846	0.490570
$76$	28.0989	3.22316
$77$	−3.70928	−0.422711
$78$	13.7587	1.55787
$79$	−4.38962	−0.493871	−0.246935	−	0.969032i	$-0.579424\pi$
$79$	−4.38962	−0.493871	−0.246935	+	0.969032i	$0.579424\pi$
$80$	−16.1906	−1.81016
$81$	−1.44748	−0.160831
$82$	−7.26180	−0.801931
$83$	14.6537	1.60845	0.804225	−	0.594324i	$-0.202580\pi$
$83$	14.6537	1.60845	0.804225	+	0.594324i	$0.202580\pi$
$84$	6.24846	0.681763
$85$	−3.70928	−0.402327
$86$	23.0205	2.48237
$87$	−0.738205	−0.0791439
$88$	−33.5669	−3.57824
$89$	−6.77205	−0.717836	−0.358918	−	0.933369i	$-0.616854\pi$
$89$	−6.77205	−0.717836	−0.358918	+	0.933369i	$0.616854\pi$
$90$	−5.17009	−0.544975
$91$	−4.34017	−0.454974
$92$	5.34017	0.556752
$93$	−10.9132	−1.13165
$94$	−16.5041	−1.70227
$95$	−6.15676	−0.631670
$96$	22.6875	2.31554
$97$	8.58864	0.872044	0.436022	−	0.899936i	$-0.356387\pi$
$97$	8.58864	0.872044	0.436022	+	0.899936i	$0.356387\pi$
$98$	−2.70928	−0.273678
$99$	−6.04945	−0.607992
$100$	−19.3896	−1.93896
$101$	−4.92162	−0.489720	−0.244860	−	0.969558i	$-0.578742\pi$
$101$	−4.92162	−0.489720	−0.244860	+	0.969558i	$0.578742\pi$
$102$	10.0494	0.995044
$103$	16.6803	1.64356	0.821782	−	0.569803i	$-0.192980\pi$
$103$	16.6803	1.64356	0.821782	+	0.569803i	$0.192980\pi$
$104$	−39.2762	−3.85135
$105$	−1.36910	−0.133611
$106$	−11.2618	−1.09384
$107$	−15.3340	−1.48240	−0.741198	−	0.671286i	$-0.765742\pi$
$107$	−15.3340	−1.48240	−0.741198	+	0.671286i	$0.765742\pi$
$108$	28.9360	2.78437
$109$	−11.1773	−1.07059	−0.535294	−	0.844666i	$-0.679799\pi$
$109$	−11.1773	−1.07059	−0.535294	+	0.844666i	$0.679799\pi$
$110$	11.7587	1.12115
$111$	3.60197	0.341884
$112$	−13.8371	−1.30748
$113$	13.5174	1.27161	0.635807	−	0.771848i	$-0.280667\pi$
$113$	13.5174	1.27161	0.635807	+	0.771848i	$0.280667\pi$
$114$	16.6803	1.56226
$115$	−1.17009	−0.109111
$116$	3.36910	0.312813
$117$	−7.07838	−0.654396
$118$	22.7721	2.09634
$119$	−3.17009	−0.290601
$120$	−12.3896	−1.13101
$121$	2.75872	0.250793
$122$	18.7721	1.69954
$123$	−3.13624	−0.282785
$124$	49.8069	4.47280
$125$	10.0989	0.903273
$126$	−4.41855	−0.393636
$127$	2.92162	0.259252	0.129626	−	0.991563i	$-0.458622\pi$
$127$	2.92162	0.259252	0.129626	+	0.991563i	$0.458622\pi$
$128$	−28.5669	−2.52498
$129$	9.94214	0.875357
$130$	13.7587	1.20672
$131$	−6.27513	−0.548260	−0.274130	−	0.961693i	$-0.588390\pi$
$131$	−6.27513	−0.548260	−0.274130	+	0.961693i	$0.588390\pi$
$132$	−23.1773	−2.01732
$133$	−5.26180	−0.456256
$134$	−26.7298	−2.30910
$135$	−6.34017	−0.545675
$136$	−28.6875	−2.45994
$137$	7.75872	0.662873	0.331436	−	0.943478i	$-0.392467\pi$
$137$	7.75872	0.662873	0.331436	+	0.943478i	$0.392467\pi$
$138$	3.17009	0.269856
$139$	−2.92881	−0.248418	−0.124209	−	0.992256i	$-0.539639\pi$
$139$	−2.92881	−0.248418	−0.124209	+	0.992256i	$0.539639\pi$
$140$	6.24846	0.528092
$141$	−7.12783	−0.600271
$142$	−29.3607	−2.46389
$143$	16.0989	1.34626
$144$	−22.5669	−1.88057
$145$	−0.738205	−0.0613046
$146$	35.2762	2.91948
$147$	−1.17009	−0.0965071
$148$	−16.4391	−1.35128
$149$	0.837101	0.0685780	0.0342890	−	0.999412i	$-0.489083\pi$
$149$	0.837101	0.0685780	0.0342890	+	0.999412i	$0.489083\pi$
$150$	−11.5103	−0.939809
$151$	15.9155	1.29518	0.647592	−	0.761988i	$-0.275776\pi$
$151$	15.9155	1.29518	0.647592	+	0.761988i	$0.275776\pi$
$152$	−47.6163	−3.86220
$153$	−5.17009	−0.417977
$154$	10.0494	0.809808
$155$	−10.9132	−0.876570
$156$	−27.1194	−2.17129
$157$	−11.1701	−0.891470	−0.445735	−	0.895165i	$-0.647058\pi$
$157$	−11.1701	−0.891470	−0.445735	+	0.895165i	$0.647058\pi$
$158$	11.8927	0.946132
$159$	−4.86376	−0.385722
$160$	22.6875	1.79361
$161$	−1.00000	−0.0788110
$162$	3.92162	0.308112
$163$	14.6225	1.14532	0.572661	−	0.819792i	$-0.305911\pi$
$163$	14.6225	1.14532	0.572661	+	0.819792i	$0.305911\pi$
$164$	14.3135	1.11770
$165$	5.07838	0.395351
$166$	−39.7009	−3.08138
$167$	6.00719	0.464850	0.232425	−	0.972614i	$-0.425334\pi$
$167$	6.00719	0.464850	0.232425	+	0.972614i	$0.425334\pi$
$168$	−10.5886	−0.816931
$169$	5.83710	0.449008
$170$	10.0494	0.770758
$171$	−8.58145	−0.656240
$172$	−45.3751	−3.45982
$173$	14.0989	1.07192	0.535960	−	0.844244i	$-0.319950\pi$
$173$	14.0989	1.07192	0.535960	+	0.844244i	$0.319950\pi$
$174$	2.00000	0.151620
$175$	3.63090	0.274470
$176$	51.3256	3.86881
$177$	9.83483	0.739231
$178$	18.3474	1.37519
$179$	−18.1256	−1.35477	−0.677384	−	0.735630i	$-0.736886\pi$
$179$	−18.1256	−1.35477	−0.677384	+	0.735630i	$0.736886\pi$
$180$	10.1906	0.759563
$181$	12.4319	0.924054	0.462027	−	0.886866i	$-0.347122\pi$
$181$	12.4319	0.924054	0.462027	+	0.886866i	$0.347122\pi$
$182$	11.7587	0.871615
$183$	8.10731	0.599309
$184$	−9.04945	−0.667134
$185$	3.60197	0.264822
$186$	29.5669	2.16795
$187$	11.7587	0.859883
$188$	32.5308	2.37255
$189$	−5.41855	−0.394142
$190$	16.6803	1.21012
$191$	−4.86376	−0.351930	−0.175965	−	0.984396i	$-0.556304\pi$
$191$	−4.86376	−0.351930	−0.175965	+	0.984396i	$0.556304\pi$
$192$	−29.0856	−2.09907
$193$	−17.2351	−1.24061	−0.620306	−	0.784360i	$-0.712992\pi$
$193$	−17.2351	−1.24061	−0.620306	+	0.784360i	$0.712992\pi$
$194$	−23.2690	−1.67062
$195$	5.94214	0.425526
$196$	5.34017	0.381441
$197$	−12.5236	−0.892269	−0.446134	−	0.894966i	$-0.647200\pi$
$197$	−12.5236	−0.892269	−0.446134	+	0.894966i	$0.647200\pi$
$198$	16.3896	1.16476
$199$	1.16290	0.0824357	0.0412178	−	0.999150i	$-0.486876\pi$
$199$	1.16290	0.0824357	0.0412178	+	0.999150i	$0.486876\pi$
$200$	32.8576	2.32338
$201$	−11.5441	−0.814259
$202$	13.3340	0.938179
$203$	−0.630898	−0.0442803
$204$	−19.8082	−1.38685
$205$	−3.13624	−0.219044
$206$	−45.1917	−3.14865
$207$	−1.63090	−0.113355
$208$	60.0554	4.16409
$209$	19.5174	1.35005
$210$	3.70928	0.255964
$211$	4.76487	0.328027	0.164013	−	0.986458i	$-0.447556\pi$
$211$	4.76487	0.328027	0.164013	+	0.986458i	$0.447556\pi$
$212$	22.1978	1.52455
$213$	−12.6803	−0.868843
$214$	41.5441	2.83990
$215$	9.94214	0.678048
$216$	−49.0349	−3.33640
$217$	−9.32684	−0.633147
$218$	30.2823	2.05098
$219$	15.2351	1.02949
$220$	−23.1773	−1.56261
$221$	13.7587	0.925512
$222$	−9.75872	−0.654963
$223$	−21.3679	−1.43090	−0.715450	−	0.698664i	$-0.753778\pi$
$223$	−21.3679	−1.43090	−0.715450	+	0.698664i	$0.753778\pi$
$224$	19.3896	1.29552
$225$	5.92162	0.394775
$226$	−36.6225	−2.43609
$227$	−5.84324	−0.387830	−0.193915	−	0.981018i	$-0.562119\pi$
$227$	−5.84324	−0.387830	−0.193915	+	0.981018i	$0.562119\pi$
$228$	−32.8781	−2.17741
$229$	2.85658	0.188768	0.0943839	−	0.995536i	$-0.469912\pi$
$229$	2.85658	0.188768	0.0943839	+	0.995536i	$0.469912\pi$
$230$	3.17009	0.209029
$231$	4.34017	0.285562
$232$	−5.70928	−0.374832
$233$	−13.5259	−0.886108	−0.443054	−	0.896495i	$-0.646105\pi$
$233$	−13.5259	−0.886108	−0.443054	+	0.896495i	$0.646105\pi$
$234$	19.1773	1.25366
$235$	−7.12783	−0.464968
$236$	−44.8853	−2.92179
$237$	5.13624	0.333634
$238$	8.58864	0.556719
$239$	4.76487	0.308214	0.154107	−	0.988054i	$-0.450750\pi$
$239$	4.76487	0.308214	0.154107	+	0.988054i	$0.450750\pi$
$240$	18.9444	1.22286
$241$	24.4775	1.57673	0.788366	−	0.615207i	$-0.210928\pi$
$241$	24.4775	1.57673	0.788366	+	0.615207i	$0.210928\pi$
$242$	−7.47414	−0.480456
$243$	−14.5620	−0.934151
$244$	−37.0010	−2.36875
$245$	−1.17009	−0.0747541
$246$	8.49693	0.541744
$247$	22.8371	1.45309
$248$	−84.4028	−5.35958
$249$	−17.1461	−1.08659
$250$	−27.3607	−1.73044
$251$	−26.2557	−1.65724	−0.828621	−	0.559810i	$-0.810874\pi$
$251$	−26.2557	−1.65724	−0.828621	+	0.559810i	$0.810874\pi$
$252$	8.70928	0.548633
$253$	3.70928	0.233200
$254$	−7.91548	−0.496661
$255$	4.34017	0.271792
$256$	27.6803	1.73002
$257$	7.84324	0.489248	0.244624	−	0.969618i	$-0.421335\pi$
$257$	7.84324	0.489248	0.244624	+	0.969618i	$0.421335\pi$
$258$	−26.9360	−1.67696
$259$	3.07838	0.191281
$260$	−27.1194	−1.68187
$261$	−1.02893	−0.0636891
$262$	17.0010	1.05033
$263$	10.9132	0.672937	0.336469	−	0.941695i	$-0.390767\pi$
$263$	10.9132	0.672937	0.336469	+	0.941695i	$0.390767\pi$
$264$	39.2762	2.41728
$265$	−4.86376	−0.298779
$266$	14.2557	0.874070
$267$	7.92389	0.484934
$268$	52.6863	3.21833
$269$	−3.57531	−0.217990	−0.108995	−	0.994042i	$-0.534763\pi$
$269$	−3.57531	−0.217990	−0.108995	+	0.994042i	$0.534763\pi$
$270$	17.1773	1.04538
$271$	−5.98053	−0.363291	−0.181646	−	0.983364i	$-0.558142\pi$
$271$	−5.98053	−0.363291	−0.181646	+	0.983364i	$0.558142\pi$
$272$	43.8648	2.65969
$273$	5.07838	0.307357
$274$	−21.0205	−1.26990
$275$	−13.4680	−0.812151
$276$	−6.24846	−0.376113
$277$	−13.3112	−0.799795	−0.399898	−	0.916560i	$-0.630954\pi$
$277$	−13.3112	−0.799795	−0.399898	+	0.916560i	$0.630954\pi$
$278$	7.93495	0.475907
$279$	−15.2111	−0.910666
$280$	−10.5886	−0.632792
$281$	−2.13009	−0.127071	−0.0635354	−	0.997980i	$-0.520238\pi$
$281$	−2.13009	−0.127071	−0.0635354	+	0.997980i	$0.520238\pi$
$282$	19.3112	1.14997
$283$	13.6742	0.812847	0.406423	−	0.913685i	$-0.366776\pi$
$283$	13.6742	0.812847	0.406423	+	0.913685i	$0.366776\pi$
$284$	57.8720	3.43407
$285$	7.20394	0.426724
$286$	−43.6163	−2.57909
$287$	−2.68035	−0.158216
$288$	31.6225	1.86337
$289$	−6.95055	−0.408856
$290$	2.00000	0.117444
$291$	−10.0494	−0.589109
$292$	−69.5318	−4.06904
$293$	−13.2690	−0.775182	−0.387591	−	0.921831i	$-0.626693\pi$
$293$	−13.2690	−0.775182	−0.387591	+	0.921831i	$0.626693\pi$
$294$	3.17009	0.184883
$295$	9.83483	0.572606
$296$	27.8576	1.61919
$297$	20.0989	1.16626
$298$	−2.26794	−0.131378
$299$	4.34017	0.250999
$300$	22.6875	1.30987
$301$	8.49693	0.489755
$302$	−43.1194	−2.48124
$303$	5.75872	0.330830
$304$	72.8080	4.17582
$305$	8.10731	0.464223
$306$	14.0072	0.800738
$307$	−17.5103	−0.999363	−0.499682	−	0.866209i	$-0.666550\pi$
$307$	−17.5103	−0.999363	−0.499682	+	0.866209i	$0.666550\pi$
$308$	−19.8082	−1.12868
$309$	−19.5174	−1.11031
$310$	29.5669	1.67929
$311$	2.03385	0.115329	0.0576645	−	0.998336i	$-0.481635\pi$
$311$	2.03385	0.115329	0.0576645	+	0.998336i	$0.481635\pi$
$312$	45.9565	2.60178
$313$	12.3786	0.699677	0.349839	−	0.936810i	$-0.386236\pi$
$313$	12.3786	0.699677	0.349839	+	0.936810i	$0.386236\pi$
$314$	30.2628	1.70783
$315$	−1.90829	−0.107520
$316$	−23.4413	−1.31868
$317$	−20.1483	−1.13164	−0.565822	−	0.824527i	$-0.691441\pi$
$317$	−20.1483	−1.13164	−0.565822	+	0.824527i	$0.691441\pi$
$318$	13.1773	0.738945
$319$	2.34017	0.131025
$320$	−29.0856	−1.62593
$321$	17.9421	1.00143
$322$	2.70928	0.150982
$323$	16.6803	0.928119
$324$	−7.72979	−0.429433
$325$	−15.7587	−0.874137
$326$	−39.6163	−2.19415
$327$	13.0784	0.723236
$328$	−24.2557	−1.33929
$329$	−6.09171	−0.335847
$330$	−13.7587	−0.757393
$331$	−6.70701	−0.368650	−0.184325	−	0.982865i	$-0.559010\pi$
$331$	−6.70701	−0.368650	−0.184325	+	0.982865i	$0.559010\pi$
$332$	78.2532	4.29470
$333$	5.02052	0.275123
$334$	−16.2751	−0.890535
$335$	−11.5441	−0.630722
$336$	16.1906	0.883270
$337$	−29.0205	−1.58085	−0.790424	−	0.612560i	$-0.790140\pi$
$337$	−29.0205	−1.58085	−0.790424	+	0.612560i	$0.790140\pi$
$338$	−15.8143	−0.860185
$339$	−15.8166	−0.859039
$340$	−19.8082	−1.07425
$341$	34.5958	1.87347
$342$	23.2495	1.25719
$343$	−1.00000	−0.0539949
$344$	76.8925	4.14577
$345$	1.36910	0.0737100
$346$	−38.1978	−2.05353
$347$	−18.2413	−0.979243	−0.489622	−	0.871935i	$-0.662865\pi$
$347$	−18.2413	−0.979243	−0.489622	+	0.871935i	$0.662865\pi$
$348$	−3.94214	−0.211321
$349$	−8.35455	−0.447209	−0.223604	−	0.974680i	$-0.571782\pi$
$349$	−8.35455	−0.447209	−0.223604	+	0.974680i	$0.571782\pi$
$350$	−9.83710	−0.525815
$351$	23.5174	1.25527
$352$	−71.9214	−3.83343
$353$	−31.5897	−1.68135	−0.840675	−	0.541541i	$-0.817841\pi$
$353$	−31.5897	−1.68135	−0.840675	+	0.541541i	$0.817841\pi$
$354$	−26.6453	−1.41618
$355$	−12.6803	−0.673003
$356$	−36.1639	−1.91669
$357$	3.70928	0.196316
$358$	49.1071	2.59539
$359$	−4.48852	−0.236895	−0.118447	−	0.992960i	$-0.537792\pi$
$359$	−4.48852	−0.236895	−0.118447	+	0.992960i	$0.537792\pi$
$360$	−17.2690	−0.910155
$361$	8.68649	0.457184
$362$	−33.6814	−1.77025
$363$	−3.22795	−0.169423
$364$	−23.1773	−1.21482
$365$	15.2351	0.797443
$366$	−21.9649	−1.14813
$367$	−18.7526	−0.978877	−0.489438	−	0.872038i	$-0.662798\pi$
$367$	−18.7526	−0.978877	−0.489438	+	0.872038i	$0.662798\pi$
$368$	13.8371	0.721309
$369$	−4.37137	−0.227564
$370$	−9.75872	−0.507332
$371$	−4.15676	−0.215808
$372$	−58.2784	−3.02160
$373$	22.5113	1.16559	0.582796	−	0.812619i	$-0.301959\pi$
$373$	22.5113	1.16559	0.582796	+	0.812619i	$0.301959\pi$
$374$	−31.8576	−1.64732
$375$	−11.8166	−0.610206
$376$	−55.1266	−2.84294
$377$	2.73820	0.141025
$378$	14.6803	0.755076
$379$	−22.3318	−1.14711	−0.573553	−	0.819169i	$-0.694435\pi$
$379$	−22.3318	−1.14711	−0.573553	+	0.819169i	$0.694435\pi$
$380$	−32.8781	−1.68661
$381$	−3.41855	−0.175138
$382$	13.1773	0.674208
$383$	15.2351	0.778479	0.389239	−	0.921137i	$-0.372738\pi$
$383$	15.2351	0.778479	0.389239	+	0.921137i	$0.372738\pi$
$384$	33.4257	1.70575
$385$	4.34017	0.221196
$386$	46.6947	2.37670
$387$	13.8576	0.704422
$388$	45.8648	2.32843
$389$	21.3874	1.08438	0.542191	−	0.840255i	$-0.317595\pi$
$389$	21.3874	1.08438	0.542191	+	0.840255i	$0.317595\pi$
$390$	−16.0989	−0.815199
$391$	3.17009	0.160318
$392$	−9.04945	−0.457066
$393$	7.34244	0.370377
$394$	33.9299	1.70936
$395$	5.13624	0.258432
$396$	−32.3051	−1.62339
$397$	20.6537	1.03658	0.518289	−	0.855205i	$-0.326569\pi$
$397$	20.6537	1.03658	0.518289	+	0.855205i	$0.326569\pi$
$398$	−3.15061	−0.157926
$399$	6.15676	0.308223
$400$	−50.2411	−2.51205
$401$	24.3402	1.21549	0.607745	−	0.794132i	$-0.292074\pi$
$401$	24.3402	1.21549	0.607745	+	0.794132i	$0.292074\pi$
$402$	31.2762	1.55991
$403$	40.4801	2.01646
$404$	−26.2823	−1.30759
$405$	1.69368	0.0841595
$406$	1.70928	0.0848299
$407$	−11.4186	−0.565997
$408$	33.5669	1.66181
$409$	17.2885	0.854859	0.427430	−	0.904049i	$-0.359419\pi$
$409$	17.2885	0.854859	0.427430	+	0.904049i	$0.359419\pi$
$410$	8.49693	0.419633
$411$	−9.07838	−0.447803
$412$	89.0759	4.38846
$413$	8.40522	0.413594
$414$	4.41855	0.217160
$415$	−17.1461	−0.841668
$416$	−84.1543	−4.12600
$417$	3.42696	0.167819
$418$	−52.8781	−2.58635
$419$	−3.05172	−0.149086	−0.0745430	−	0.997218i	$-0.523750\pi$
$419$	−3.05172	−0.149086	−0.0745430	+	0.997218i	$0.523750\pi$
$420$	−7.31124	−0.356752
$421$	13.8843	0.676679	0.338339	−	0.941024i	$-0.390135\pi$
$421$	13.8843	0.676679	0.338339	+	0.941024i	$0.390135\pi$
$422$	−12.9093	−0.628417
$423$	−9.93495	−0.483054
$424$	−37.6163	−1.82681
$425$	−11.5103	−0.558330
$426$	34.3545	1.66448
$427$	6.92881	0.335309
$428$	−81.8864	−3.95813
$429$	−18.8371	−0.909464
$430$	−26.9360	−1.29897
$431$	15.2039	0.732348	0.366174	−	0.930546i	$-0.380668\pi$
$431$	15.2039	0.732348	0.366174	+	0.930546i	$0.380668\pi$
$432$	74.9770	3.60733
$433$	12.8566	0.617848	0.308924	−	0.951087i	$-0.400031\pi$
$433$	12.8566	0.617848	0.308924	+	0.951087i	$0.400031\pi$
$434$	25.2690	1.21295
$435$	0.863763	0.0414143
$436$	−59.6886	−2.85856
$437$	5.26180	0.251706
$438$	−41.2762	−1.97225
$439$	14.5620	0.695005	0.347503	−	0.937679i	$-0.387030\pi$
$439$	14.5620	0.695005	0.347503	+	0.937679i	$0.387030\pi$
$440$	39.2762	1.87242
$441$	−1.63090	−0.0776618
$442$	−37.2762	−1.77305
$443$	−30.2557	−1.43749	−0.718745	−	0.695274i	$-0.755283\pi$
$443$	−30.2557	−1.43749	−0.718745	+	0.695274i	$0.755283\pi$
$444$	19.2351	0.912859
$445$	7.92389	0.375628
$446$	57.8915	2.74124
$447$	−0.979481	−0.0463279
$448$	−24.8576	−1.17441
$449$	−4.68422	−0.221062	−0.110531	−	0.993873i	$-0.535255\pi$
$449$	−4.68422	−0.221062	−0.110531	+	0.993873i	$0.535255\pi$
$450$	−16.0433	−0.756289
$451$	9.94214	0.468157
$452$	72.1855	3.39532
$453$	−18.6225	−0.874961
$454$	15.8310	0.742984
$455$	5.07838	0.238078
$456$	55.7152	2.60911
$457$	23.4596	1.09739	0.548697	−	0.836022i	$-0.315124\pi$
$457$	23.4596	1.09739	0.548697	+	0.836022i	$0.315124\pi$
$458$	−7.73925	−0.361631
$459$	17.1773	0.801767
$460$	−6.24846	−0.291336
$461$	6.58145	0.306529	0.153264	−	0.988185i	$-0.451021\pi$
$461$	6.58145	0.306529	0.153264	+	0.988185i	$0.451021\pi$
$462$	−11.7587	−0.547065
$463$	20.0144	0.930147	0.465073	−	0.885272i	$-0.346028\pi$
$463$	20.0144	0.930147	0.465073	+	0.885272i	$0.346028\pi$
$464$	8.72979	0.405271
$465$	12.7694	0.592167
$466$	36.6453	1.69756
$467$	21.4596	0.993031	0.496516	−	0.868028i	$-0.334613\pi$
$467$	21.4596	0.993031	0.496516	+	0.868028i	$0.334613\pi$
$468$	−37.7998	−1.74729
$469$	−9.86603	−0.455571
$470$	19.3112	0.890761
$471$	13.0700	0.602232
$472$	76.0626	3.50107
$473$	−31.5174	−1.44917
$474$	−13.9155	−0.639159
$475$	−19.1050	−0.876599
$476$	−16.9288	−0.775931
$477$	−6.77924	−0.310400
$478$	−12.9093	−0.590459
$479$	−14.3090	−0.653794	−0.326897	−	0.945060i	$-0.606003\pi$
$479$	−14.3090	−0.653794	−0.326897	+	0.945060i	$0.606003\pi$
$480$	−26.5464	−1.21167
$481$	−13.3607	−0.609195
$482$	−66.3162	−3.02062
$483$	1.17009	0.0532408
$484$	14.7321	0.669639
$485$	−10.0494	−0.456322
$486$	39.4524	1.78960
$487$	−5.36069	−0.242916	−0.121458	−	0.992597i	$-0.538757\pi$
$487$	−5.36069	−0.242916	−0.121458	+	0.992597i	$0.538757\pi$
$488$	62.7019	2.83838
$489$	−17.1096	−0.773722
$490$	3.17009	0.143210
$491$	8.99386	0.405887	0.202944	−	0.979190i	$-0.434949\pi$
$491$	8.99386	0.405887	0.202944	+	0.979190i	$0.434949\pi$
$492$	−16.7480	−0.755060
$493$	2.00000	0.0900755
$494$	−61.8720	−2.78375
$495$	7.07838	0.318149
$496$	129.056	5.79481
$497$	−10.8371	−0.486110
$498$	46.4534	2.08163
$499$	2.02666	0.0907259	0.0453629	−	0.998971i	$-0.485556\pi$
$499$	2.02666	0.0907259	0.0453629	+	0.998971i	$0.485556\pi$
$500$	53.9299	2.41182
$501$	−7.02893	−0.314029
$502$	71.1338	3.17486
$503$	−32.4124	−1.44520	−0.722599	−	0.691268i	$-0.757053\pi$
$503$	−32.4124	−1.44520	−0.722599	+	0.691268i	$0.757053\pi$
$504$	−14.7587	−0.657406
$505$	5.75872	0.256260
$506$	−10.0494	−0.446752
$507$	−6.82991	−0.303327
$508$	15.6020	0.692225
$509$	−10.4826	−0.464631	−0.232315	−	0.972640i	$-0.574630\pi$
$509$	−10.4826	−0.464631	−0.232315	+	0.972640i	$0.574630\pi$
$510$	−11.7587	−0.520685
$511$	13.0205	0.575994
$512$	−17.8599	−0.789303
$513$	28.5113	1.25880
$514$	−21.2495	−0.937276
$515$	−19.5174	−0.860041
$516$	53.0928	2.33728
$517$	22.5958	0.993763
$518$	−8.34017	−0.366446
$519$	−16.4969	−0.724135
$520$	45.9565	2.01533
$521$	−24.6030	−1.07788	−0.538939	−	0.842345i	$-0.681175\pi$
$521$	−24.6030	−1.07788	−0.538939	+	0.842345i	$0.681175\pi$
$522$	2.78765	0.122012
$523$	12.1978	0.533372	0.266686	−	0.963783i	$-0.414071\pi$
$523$	12.1978	0.533372	0.266686	+	0.963783i	$0.414071\pi$
$524$	−33.5103	−1.46390
$525$	−4.24846	−0.185418
$526$	−29.5669	−1.28918
$527$	29.5669	1.28795
$528$	−60.0554	−2.61358
$529$	1.00000	0.0434783
$530$	13.1773	0.572384
$531$	13.7081	0.594879
$532$	−28.0989	−1.21824
$533$	11.6332	0.503888
$534$	−21.4680	−0.929011
$535$	17.9421	0.775706
$536$	−89.2821	−3.85640
$537$	21.2085	0.915213
$538$	9.68649	0.417614
$539$	3.70928	0.159770
$540$	−33.8576	−1.45700
$541$	17.5259	0.753495	0.376748	−	0.926316i	$-0.377042\pi$
$541$	17.5259	0.753495	0.376748	+	0.926316i	$0.377042\pi$
$542$	16.2029	0.695974
$543$	−14.5464	−0.624245
$544$	−61.4668	−2.63537
$545$	13.0784	0.560216
$546$	−13.7587	−0.588819
$547$	12.5958	0.538559	0.269279	−	0.963062i	$-0.413215\pi$
$547$	12.5958	0.538559	0.269279	+	0.963062i	$0.413215\pi$
$548$	41.4329	1.76993
$549$	11.3002	0.482280
$550$	36.4885	1.55588
$551$	3.31965	0.141422
$552$	10.5886	0.450682
$553$	4.38962	0.186666
$554$	36.0638	1.53221
$555$	−4.21461	−0.178900
$556$	−15.6404	−0.663299
$557$	−26.5113	−1.12332	−0.561660	−	0.827368i	$-0.689837\pi$
$557$	−26.5113	−1.12332	−0.561660	+	0.827368i	$0.689837\pi$
$558$	41.2111	1.74461
$559$	−36.8781	−1.55978
$560$	16.1906	0.684178
$561$	−13.7587	−0.580894
$562$	5.77101	0.243435
$563$	−17.0472	−0.718453	−0.359227	−	0.933250i	$-0.616959\pi$
$563$	−17.0472	−0.718453	−0.359227	+	0.933250i	$0.616959\pi$
$564$	−38.0638	−1.60278
$565$	−15.8166	−0.665409
$566$	−37.0472	−1.55721
$567$	1.44748	0.0607885
$568$	−98.0698	−4.11492
$569$	−25.1194	−1.05306	−0.526530	−	0.850156i	$-0.676507\pi$
$569$	−25.1194	−1.05306	−0.526530	+	0.850156i	$0.676507\pi$
$570$	−19.5174	−0.817496
$571$	−12.0144	−0.502786	−0.251393	−	0.967885i	$-0.580889\pi$
$571$	−12.0144	−0.502786	−0.251393	+	0.967885i	$0.580889\pi$
$572$	85.9709	3.59462
$573$	5.69102	0.237746
$574$	7.26180	0.303101
$575$	−3.63090	−0.151419
$576$	−40.5402	−1.68918
$577$	−10.1133	−0.421021	−0.210511	−	0.977592i	$-0.567513\pi$
$577$	−10.1133	−0.421021	−0.210511	+	0.977592i	$0.567513\pi$
$578$	18.8310	0.783265
$579$	20.1666	0.838095
$580$	−3.94214	−0.163689
$581$	−14.6537	−0.607937
$582$	27.2267	1.12858
$583$	15.4186	0.638571
$584$	117.829	4.87578
$585$	8.28231	0.342432
$586$	35.9493	1.48505
$587$	−14.2485	−0.588097	−0.294049	−	0.955790i	$-0.595003\pi$
$587$	−14.2485	−0.588097	−0.294049	+	0.955790i	$0.595003\pi$
$588$	−6.24846	−0.257682
$589$	49.0759	2.02214
$590$	−26.6453	−1.09697
$591$	14.6537	0.602772
$592$	−42.5958	−1.75068
$593$	−20.2557	−0.831800	−0.415900	−	0.909410i	$-0.636533\pi$
$593$	−20.2557	−0.831800	−0.415900	+	0.909410i	$0.636533\pi$
$594$	−54.4534	−2.23425
$595$	3.70928	0.152065
$596$	4.47027	0.183109
$597$	−1.36069	−0.0556894
$598$	−11.7587	−0.480850
$599$	−19.2351	−0.785926	−0.392963	−	0.919554i	$-0.628550\pi$
$599$	−19.2351	−0.785926	−0.392963	+	0.919554i	$0.628550\pi$
$600$	−38.4463	−1.56956
$601$	−21.4329	−0.874267	−0.437134	−	0.899397i	$-0.644006\pi$
$601$	−21.4329	−0.874267	−0.437134	+	0.899397i	$0.644006\pi$
$602$	−23.0205	−0.938246
$603$	−16.0905	−0.655255
$604$	84.9914	3.45825
$605$	−3.22795	−0.131235
$606$	−15.6020	−0.633787
$607$	−37.7926	−1.53395	−0.766977	−	0.641675i	$-0.778240\pi$
$607$	−37.7926	−1.53395	−0.766977	+	0.641675i	$0.778240\pi$
$608$	−102.024	−4.13763
$609$	0.738205	0.0299136
$610$	−21.9649	−0.889334
$611$	26.4391	1.06961
$612$	−27.6092	−1.11603
$613$	14.6947	0.593514	0.296757	−	0.954953i	$-0.404095\pi$
$613$	14.6947	0.593514	0.296757	+	0.954953i	$0.404095\pi$
$614$	47.4401	1.91453
$615$	3.66967	0.147975
$616$	33.5669	1.35245
$617$	12.1256	0.488157	0.244078	−	0.969756i	$-0.421515\pi$
$617$	12.1256	0.488157	0.244078	+	0.969756i	$0.421515\pi$
$618$	52.8781	2.12707
$619$	−12.0144	−0.482899	−0.241449	−	0.970413i	$-0.577623\pi$
$619$	−12.0144	−0.482899	−0.241449	+	0.970413i	$0.577623\pi$
$620$	−58.2784	−2.34052
$621$	5.41855	0.217439
$622$	−5.51026	−0.220941
$623$	6.77205	0.271317
$624$	−70.2700	−2.81305
$625$	6.33791	0.253516
$626$	−33.5369	−1.34040
$627$	−22.8371	−0.912026
$628$	−59.6502	−2.38030
$629$	−9.75872	−0.389106
$630$	5.17009	0.205981
$631$	10.9300	0.435118	0.217559	−	0.976047i	$-0.430191\pi$
$631$	10.9300	0.435118	0.217559	+	0.976047i	$0.430191\pi$
$632$	39.7237	1.58012
$633$	−5.57531	−0.221599
$634$	54.5874	2.16794
$635$	−3.41855	−0.135661
$636$	−25.9733	−1.02991
$637$	4.34017	0.171964
$638$	−6.34017	−0.251010
$639$	−17.6742	−0.699181
$640$	33.4257	1.32127
$641$	−33.1194	−1.30814	−0.654069	−	0.756435i	$-0.726939\pi$
$641$	−33.1194	−1.30814	−0.654069	+	0.756435i	$0.726939\pi$
$642$	−48.6102	−1.91849
$643$	−17.5297	−0.691305	−0.345653	−	0.938363i	$-0.612342\pi$
$643$	−17.5297	−0.691305	−0.345653	+	0.938363i	$0.612342\pi$
$644$	−5.34017	−0.210432
$645$	−11.6332	−0.458055
$646$	−45.1917	−1.77804
$647$	−6.88777	−0.270786	−0.135393	−	0.990792i	$-0.543230\pi$
$647$	−6.88777	−0.270786	−0.135393	+	0.990792i	$0.543230\pi$
$648$	13.0989	0.514573
$649$	−31.1773	−1.22382
$650$	42.6947	1.67462
$651$	10.9132	0.427722
$652$	78.0866	3.05811
$653$	−36.3545	−1.42266	−0.711332	−	0.702856i	$-0.751908\pi$
$653$	−36.3545	−1.42266	−0.711332	+	0.702856i	$0.751908\pi$
$654$	−35.4329	−1.38554
$655$	7.34244	0.286893
$656$	37.0882	1.44805
$657$	21.2351	0.828461
$658$	16.5041	0.643397
$659$	−2.63931	−0.102813	−0.0514064	−	0.998678i	$-0.516370\pi$
$659$	−2.63931	−0.102813	−0.0514064	+	0.998678i	$0.516370\pi$
$660$	27.1194	1.05562
$661$	21.0010	0.816846	0.408423	−	0.912793i	$-0.366079\pi$
$661$	21.0010	0.816846	0.408423	+	0.912793i	$0.366079\pi$
$662$	18.1711	0.706241
$663$	−16.0989	−0.625229
$664$	−132.608	−5.14618
$665$	6.15676	0.238749
$666$	−13.6020	−0.527066
$667$	0.630898	0.0244285
$668$	32.0794	1.24119
$669$	25.0023	0.966644
$670$	31.2762	1.20830
$671$	−25.7009	−0.992171
$672$	−22.6875	−0.875191
$673$	−51.2534	−1.97567	−0.987836	−	0.155497i	$-0.950302\pi$
$673$	−51.2534	−1.97567	−0.987836	+	0.155497i	$0.950302\pi$
$674$	78.6246	3.02851
$675$	−19.6742	−0.757260
$676$	31.1711	1.19889
$677$	41.4668	1.59370	0.796849	−	0.604179i	$-0.206499\pi$
$677$	41.4668	1.59370	0.796849	+	0.604179i	$0.206499\pi$
$678$	42.8515	1.64570
$679$	−8.58864	−0.329602
$680$	33.5669	1.28723
$681$	6.83710	0.261998
$682$	−93.7296	−3.58909
$683$	−26.2557	−1.00464	−0.502322	−	0.864680i	$-0.667521\pi$
$683$	−26.2557	−1.00464	−0.502322	+	0.864680i	$0.667521\pi$
$684$	−45.8264	−1.75222
$685$	−9.07838	−0.346867
$686$	2.70928	0.103441
$687$	−3.34244	−0.127522
$688$	−117.573	−4.48242
$689$	18.0410	0.687309
$690$	−3.70928	−0.141210
$691$	−12.0338	−0.457789	−0.228895	−	0.973451i	$-0.573511\pi$
$691$	−12.0338	−0.457789	−0.228895	+	0.973451i	$0.573511\pi$
$692$	75.2905	2.86212
$693$	6.04945	0.229800
$694$	49.4206	1.87598
$695$	3.42696	0.129992
$696$	6.68035	0.253218
$697$	8.49693	0.321844
$698$	22.6348	0.856739
$699$	15.8264	0.598610
$700$	19.3896	0.732859
$701$	34.6491	1.30868	0.654340	−	0.756200i	$-0.272946\pi$
$701$	34.6491	1.30868	0.654340	+	0.756200i	$0.272946\pi$
$702$	−63.7152	−2.40478
$703$	−16.1978	−0.610911
$704$	92.2038	3.47506
$705$	8.34017	0.314109
$706$	85.5851	3.22104
$707$	4.92162	0.185097
$708$	52.5197	1.97381
$709$	7.65983	0.287671	0.143835	−	0.989602i	$-0.454056\pi$
$709$	7.65983	0.287671	0.143835	+	0.989602i	$0.454056\pi$
$710$	34.3545	1.28930
$711$	7.15902	0.268484
$712$	61.2834	2.29669
$713$	9.32684	0.349293
$714$	−10.0494	−0.376091
$715$	−18.8371	−0.704468
$716$	−96.7936	−3.61735
$717$	−5.57531	−0.208214
$718$	12.1606	0.453831
$719$	22.4196	0.836110	0.418055	−	0.908422i	$-0.362712\pi$
$719$	22.4196	0.836110	0.418055	+	0.908422i	$0.362712\pi$
$720$	26.4052	0.984064
$721$	−16.6803	−0.621209
$722$	−23.5341	−0.875848
$723$	−28.6407	−1.06516
$724$	66.3884	2.46731
$725$	−2.29072	−0.0850754
$726$	8.74539	0.324572
$727$	34.8371	1.29204	0.646018	−	0.763322i	$-0.276433\pi$
$727$	34.8371	1.29204	0.646018	+	0.763322i	$0.276433\pi$
$728$	39.2762	1.45567
$729$	21.3812	0.791897
$730$	−41.2762	−1.52770
$731$	−26.9360	−0.996264
$732$	43.2944	1.60021
$733$	−38.6752	−1.42850	−0.714251	−	0.699889i	$-0.753233\pi$
$733$	−38.6752	−1.42850	−0.714251	+	0.699889i	$0.753233\pi$
$734$	50.8059	1.87528
$735$	1.36910	0.0505001
$736$	−19.3896	−0.714711
$737$	36.5958	1.34802
$738$	11.8432	0.435956
$739$	−51.3484	−1.88888	−0.944441	−	0.328682i	$-0.893396\pi$
$739$	−51.3484	−1.88888	−0.944441	+	0.328682i	$0.893396\pi$
$740$	19.2351	0.707098
$741$	−26.7214	−0.981635
$742$	11.2618	0.413434
$743$	8.00597	0.293710	0.146855	−	0.989158i	$-0.453085\pi$
$743$	8.00597	0.293710	0.146855	+	0.989158i	$0.453085\pi$
$744$	98.7585	3.62066
$745$	−0.979481	−0.0358854
$746$	−60.9893	−2.23298
$747$	−23.8987	−0.874406
$748$	62.7936	2.29596
$749$	15.3340	0.560293
$750$	32.0144	1.16900
$751$	23.3958	0.853724	0.426862	−	0.904317i	$-0.359619\pi$
$751$	23.3958	0.853724	0.426862	+	0.904317i	$0.359619\pi$
$752$	84.2916	3.07380
$753$	30.7214	1.11955
$754$	−7.41855	−0.270168
$755$	−18.6225	−0.677742
$756$	−28.9360	−1.05239
$757$	−29.1194	−1.05836	−0.529182	−	0.848509i	$-0.677501\pi$
$757$	−29.1194	−1.05836	−0.529182	+	0.848509i	$0.677501\pi$
$758$	60.5029	2.19756
$759$	−4.34017	−0.157538
$760$	55.7152	2.02100
$761$	−10.1834	−0.369149	−0.184574	−	0.982819i	$-0.559091\pi$
$761$	−10.1834	−0.369149	−0.184574	+	0.982819i	$0.559091\pi$
$762$	9.26180	0.335519
$763$	11.1773	0.404645
$764$	−25.9733	−0.939682
$765$	6.04945	0.218718
$766$	−41.2762	−1.49137
$767$	−36.4801	−1.31722
$768$	−32.3884	−1.16872
$769$	0.431882	0.0155741	0.00778703	−	0.999970i	$-0.497521\pi$
$769$	0.431882	0.0155741	0.00778703	+	0.999970i	$0.497521\pi$
$770$	−11.7587	−0.423755
$771$	−9.17727	−0.330511
$772$	−92.0386	−3.31254
$773$	−0.0650468	−0.00233957	−0.00116978	−	0.999999i	$-0.500372\pi$
$773$	−0.0650468	−0.00233957	−0.00116978	+	0.999999i	$0.500372\pi$
$774$	−37.5441	−1.34950
$775$	−33.8648	−1.21646
$776$	−77.7224	−2.79007
$777$	−3.60197	−0.129220
$778$	−57.9442	−2.07740
$779$	14.1034	0.505308
$780$	31.7321	1.13619
$781$	40.1978	1.43839
$782$	−8.58864	−0.307129
$783$	3.41855	0.122169
$784$	13.8371	0.494182
$785$	13.0700	0.466487
$786$	−19.8927	−0.709549
$787$	31.5486	1.12459	0.562294	−	0.826937i	$-0.309919\pi$
$787$	31.5486	1.12459	0.562294	+	0.826937i	$0.309919\pi$
$788$	−66.8781	−2.38244
$789$	−12.7694	−0.454603
$790$	−13.9155	−0.495091
$791$	−13.5174	−0.480625
$792$	54.7442	1.94525
$793$	−30.0722	−1.06790
$794$	−55.9565	−1.98582
$795$	5.69102	0.201840
$796$	6.21008	0.220110
$797$	51.8336	1.83604	0.918020	−	0.396533i	$-0.129787\pi$
$797$	51.8336	1.83604	0.918020	+	0.396533i	$0.129787\pi$
$798$	−16.6803	−0.590478
$799$	19.3112	0.683183
$800$	70.4017	2.48908
$801$	11.0445	0.390239
$802$	−65.9442	−2.32857
$803$	−48.2967	−1.70435
$804$	−61.6475	−2.17414
$805$	1.17009	0.0412401
$806$	−109.672	−3.86302
$807$	4.18342	0.147263
$808$	44.5380	1.56684
$809$	44.4391	1.56239	0.781197	−	0.624284i	$-0.214609\pi$
$809$	44.4391	1.56239	0.781197	+	0.624284i	$0.214609\pi$
$810$	−4.58864	−0.161228
$811$	−23.6358	−0.829966	−0.414983	−	0.909829i	$-0.636212\pi$
$811$	−23.6358	−0.829966	−0.414983	+	0.909829i	$0.636212\pi$
$812$	−3.36910	−0.118232
$813$	6.99773	0.245421
$814$	30.9360	1.08431
$815$	−17.1096	−0.599322
$816$	−51.3256	−1.79676
$817$	−44.7091	−1.56417
$818$	−46.8392	−1.63769
$819$	7.07838	0.247339
$820$	−16.7480	−0.584867
$821$	−14.9399	−0.521405	−0.260703	−	0.965419i	$-0.583954\pi$
$821$	−14.9399	−0.521405	−0.260703	+	0.965419i	$0.583954\pi$
$822$	24.5958	0.857878
$823$	30.7670	1.07247	0.536234	−	0.844069i	$-0.319846\pi$
$823$	30.7670	1.07247	0.536234	+	0.844069i	$0.319846\pi$
$824$	−150.948	−5.25852
$825$	15.7587	0.548648
$826$	−22.7721	−0.792341
$827$	−24.5113	−0.852342	−0.426171	−	0.904643i	$-0.640138\pi$
$827$	−24.5113	−0.852342	−0.426171	+	0.904643i	$0.640138\pi$
$828$	−8.70928	−0.302668
$829$	43.8720	1.52374	0.761869	−	0.647731i	$-0.224282\pi$
$829$	43.8720	1.52374	0.761869	+	0.647731i	$0.224282\pi$
$830$	46.4534	1.61242
$831$	15.5753	0.540301
$832$	107.886	3.74029
$833$	3.17009	0.109837
$834$	−9.28458	−0.321499
$835$	−7.02893	−0.243246
$836$	104.227	3.60475
$837$	50.5380	1.74685
$838$	8.26794	0.285611
$839$	−5.39189	−0.186149	−0.0930743	−	0.995659i	$-0.529669\pi$
$839$	−5.39189	−0.186149	−0.0930743	+	0.995659i	$0.529669\pi$
$840$	12.3896	0.427483
$841$	−28.6020	−0.986275
$842$	−37.6163	−1.29634
$843$	2.49239	0.0858426
$844$	25.4452	0.875860
$845$	−6.82991	−0.234956
$846$	26.9165	0.925409
$847$	−2.75872	−0.0947909
$848$	57.5174	1.97516
$849$	−16.0000	−0.549119
$850$	31.1845	1.06962
$851$	−3.07838	−0.105525
$852$	−67.7152	−2.31989
$853$	−1.33403	−0.0456763	−0.0228382	−	0.999739i	$-0.507270\pi$
$853$	−1.33403	−0.0456763	−0.0228382	+	0.999739i	$0.507270\pi$
$854$	−18.7721	−0.642366
$855$	10.0410	0.343396
$856$	138.765	4.74287
$857$	12.1034	0.413445	0.206723	−	0.978400i	$-0.433720\pi$
$857$	12.1034	0.413445	0.206723	+	0.978400i	$0.433720\pi$
$858$	51.0349	1.74230
$859$	40.5718	1.38429	0.692146	−	0.721757i	$-0.256665\pi$
$859$	40.5718	1.38429	0.692146	+	0.721757i	$0.256665\pi$
$860$	53.0928	1.81045
$861$	3.13624	0.106883
$862$	−41.1917	−1.40299
$863$	34.0944	1.16059	0.580293	−	0.814408i	$-0.302938\pi$
$863$	34.0944	1.16059	0.580293	+	0.814408i	$0.302938\pi$
$864$	−105.064	−3.57434
$865$	−16.4969	−0.560912
$866$	−34.8320	−1.18364
$867$	8.13275	0.276203
$868$	−49.8069	−1.69056
$869$	−16.2823	−0.552340
$870$	−2.34017	−0.0793393
$871$	42.8203	1.45091
$872$	101.148	3.42531
$873$	−14.0072	−0.474071
$874$	−14.2557	−0.482205
$875$	−10.0989	−0.341405
$876$	81.3582	2.74884
$877$	−20.4163	−0.689409	−0.344704	−	0.938711i	$-0.612021\pi$
$877$	−20.4163	−0.689409	−0.344704	+	0.938711i	$0.612021\pi$
$878$	−39.4524	−1.33145
$879$	15.5259	0.523674
$880$	−60.0554	−2.02447
$881$	−35.4668	−1.19491	−0.597453	−	0.801904i	$-0.703821\pi$
$881$	−35.4668	−1.19491	−0.597453	+	0.801904i	$0.703821\pi$
$882$	4.41855	0.148780
$883$	7.91548	0.266377	0.133189	−	0.991091i	$-0.457478\pi$
$883$	7.91548	0.266377	0.133189	+	0.991091i	$0.457478\pi$
$884$	73.4740	2.47120
$885$	−11.5076	−0.386824
$886$	81.9709	2.75387
$887$	−8.10608	−0.272176	−0.136088	−	0.990697i	$-0.543453\pi$
$887$	−8.10608	−0.272176	−0.136088	+	0.990697i	$0.543453\pi$
$888$	−32.5958	−1.09384
$889$	−2.92162	−0.0979881
$890$	−21.4680	−0.719609
$891$	−5.36910	−0.179872
$892$	−114.108	−3.82062
$893$	32.0533	1.07262
$894$	2.65368	0.0887525
$895$	21.2085	0.708921
$896$	28.5669	0.954353
$897$	−5.07838	−0.169562
$898$	12.6908	0.423499
$899$	5.88428	0.196252
$900$	31.6225	1.05408
$901$	13.1773	0.438999
$902$	−26.9360	−0.896871
$903$	−9.94214	−0.330854
$904$	−122.325	−4.06848
$905$	−14.5464	−0.483538
$906$	50.4534	1.67620
$907$	−18.8371	−0.625476	−0.312738	−	0.949839i	$-0.601246\pi$
$907$	−18.8371	−0.625476	−0.312738	+	0.949839i	$0.601246\pi$
$908$	−31.2039	−1.03554
$909$	8.02666	0.266228
$910$	−13.7587	−0.456097
$911$	33.6514	1.11492	0.557461	−	0.830203i	$-0.311776\pi$
$911$	33.6514	1.11492	0.557461	+	0.830203i	$0.311776\pi$
$912$	−85.1917	−2.82098
$913$	54.3545	1.79887
$914$	−63.5585	−2.10233
$915$	−9.48625	−0.313606
$916$	15.2546	0.504026
$917$	6.27513	0.207223
$918$	−46.5380	−1.53598
$919$	42.5464	1.40348	0.701738	−	0.712435i	$-0.252408\pi$
$919$	42.5464	1.40348	0.701738	+	0.712435i	$0.252408\pi$
$920$	10.5886	0.349097
$921$	20.4885	0.675120
$922$	−17.8310	−0.587231
$923$	47.0349	1.54817
$924$	23.1773	0.762476
$925$	11.1773	0.367507
$926$	−54.2245	−1.78193
$927$	−27.2039	−0.893495
$928$	−12.2329	−0.401563
$929$	36.3012	1.19100	0.595502	−	0.803354i	$-0.296953\pi$
$929$	36.3012	1.19100	0.595502	+	0.803354i	$0.296953\pi$
$930$	−34.5958	−1.13444
$931$	5.26180	0.172448
$932$	−72.2304	−2.36599
$933$	−2.37978	−0.0779105
$934$	−58.1399	−1.90240
$935$	−13.7587	−0.449958
$936$	64.0554	2.09372
$937$	6.50412	0.212480	0.106240	−	0.994341i	$-0.466119\pi$
$937$	6.50412	0.212480	0.106240	+	0.994341i	$0.466119\pi$
$938$	26.7298	0.872759
$939$	−14.4840	−0.472667
$940$	−38.0638	−1.24151
$941$	−2.35965	−0.0769223	−0.0384611	−	0.999260i	$-0.512246\pi$
$941$	−2.35965	−0.0769223	−0.0384611	+	0.999260i	$0.512246\pi$
$942$	−35.4101	−1.15372
$943$	2.68035	0.0872841
$944$	−116.304	−3.78537
$945$	6.34017	0.206246
$946$	85.3894	2.77625
$947$	1.17727	0.0382563	0.0191281	−	0.999817i	$-0.493911\pi$
$947$	1.17727	0.0382563	0.0191281	+	0.999817i	$0.493911\pi$
$948$	27.4284	0.890833
$949$	−56.5113	−1.83443
$950$	51.7608	1.67934
$951$	23.5753	0.764482
$952$	28.6875	0.929768
$953$	−31.8264	−1.03096	−0.515479	−	0.856902i	$-0.672386\pi$
$953$	−31.8264	−1.03096	−0.515479	+	0.856902i	$0.672386\pi$
$954$	18.3668	0.594648
$955$	5.69102	0.184157
$956$	25.4452	0.822957
$957$	−2.73820	−0.0885136
$958$	38.7670	1.25250
$959$	−7.75872	−0.250542
$960$	34.0326	1.09840
$961$	55.9900	1.80613
$962$	36.1978	1.16706
$963$	25.0082	0.805879
$964$	130.714	4.21001
$965$	20.1666	0.649186
$966$	−3.17009	−0.101996
$967$	42.6225	1.37065	0.685323	−	0.728239i	$-0.259661\pi$
$967$	42.6225	1.37065	0.685323	+	0.728239i	$0.259661\pi$
$968$	−24.9649	−0.802403
$969$	−19.5174	−0.626991
$970$	27.2267	0.874197
$971$	55.7998	1.79070	0.895350	−	0.445364i	$-0.146926\pi$
$971$	55.7998	1.79070	0.895350	+	0.445364i	$0.146926\pi$
$972$	−77.7635	−2.49426
$973$	2.92881	0.0938933
$974$	14.5236	0.465366
$975$	18.4391	0.590523
$976$	−95.8746	−3.06887
$977$	52.2245	1.67081	0.835404	−	0.549636i	$-0.185234\pi$
$977$	52.2245	1.67081	0.835404	+	0.549636i	$0.185234\pi$
$978$	46.3545	1.48226
$979$	−25.1194	−0.802820
$980$	−6.24846	−0.199600
$981$	18.2290	0.582007
$982$	−24.3668	−0.777577
$983$	−16.7649	−0.534716	−0.267358	−	0.963597i	$-0.586151\pi$
$983$	−16.7649	−0.534716	−0.267358	+	0.963597i	$0.586151\pi$
$984$	28.3812	0.904760
$985$	14.6537	0.466905
$986$	−5.41855	−0.172562
$987$	7.12783	0.226881
$988$	121.954	3.87988
$989$	−8.49693	−0.270187
$990$	−19.1773	−0.609494
$991$	−39.0349	−1.23998	−0.619992	−	0.784608i	$-0.712864\pi$
$991$	−39.0349	−1.23998	−0.619992	+	0.784608i	$0.712864\pi$
$992$	−180.844	−5.74180
$993$	7.84778	0.249042
$994$	29.3607	0.931265
$995$	−1.36069	−0.0431368
$996$	−91.5630	−2.90129
$997$	−11.2618	−0.356665	−0.178332	−	0.983970i	$-0.557070\pi$
$997$	−11.2618	−0.356665	−0.178332	+	0.983970i	$0.557070\pi$
$998$	−5.49079	−0.173808
$999$	−16.6803	−0.527743

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.2.a.c.1.1	✓	3
3.2	odd	2		1449.2.a.m.1.3		3
4.3	odd	2		2576.2.a.v.1.3		3
5.4	even	2		4025.2.a.j.1.3		3
7.6	odd	2		1127.2.a.f.1.1		3
23.22	odd	2		3703.2.a.c.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.c.1.1	✓	3	1.1	even	1	trivial
1127.2.a.f.1.1		3	7.6	odd	2
1449.2.a.m.1.3		3	3.2	odd	2
2576.2.a.v.1.3		3	4.3	odd	2
3703.2.a.c.1.1		3	23.22	odd	2
4025.2.a.j.1.3		3	5.4	even	2

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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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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	161.2.a.c.1.1

Level	$161$
Weight	$2$
Character	161.1
Self dual	yes
Analytic conductor	$1.286$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$