LMFDB - Embedded newform 7942.2.a.bn.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, 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("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7942 = 2 \cdot 11 \cdot 19^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7942.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:	$63.4171892853$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{7} - 19x^{6} + 14x^{5} + 116x^{4} - 65x^{3} - 235x^{2} + 120x + 80$ x^8 - x^7 - 19x^6 + 14x^5 + 116x^4 - 65x^3 - 235x^2 + 120x + 80
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$5$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.405236$ of defining polynomial
Character	$\chi$	$=$	7942.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.405236 q^{3} +1.00000 q^{4} -2.02327 q^{5} -0.405236 q^{6} -4.23745 q^{7} -1.00000 q^{8} -2.83578 q^{9} +2.02327 q^{10} +1.00000 q^{11} +0.405236 q^{12} +3.21775 q^{13} +4.23745 q^{14} -0.819901 q^{15} +1.00000 q^{16} -2.00085 q^{17} +2.83578 q^{18} -2.02327 q^{20} -1.71717 q^{21} -1.00000 q^{22} -2.37064 q^{23} -0.405236 q^{24} -0.906380 q^{25} -3.21775 q^{26} -2.36487 q^{27} -4.23745 q^{28} +2.15785 q^{29} +0.819901 q^{30} +6.20065 q^{31} -1.00000 q^{32} +0.405236 q^{33} +2.00085 q^{34} +8.57350 q^{35} -2.83578 q^{36} +11.1129 q^{37} +1.30395 q^{39} +2.02327 q^{40} +3.97211 q^{41} +1.71717 q^{42} +5.64641 q^{43} +1.00000 q^{44} +5.73756 q^{45} +2.37064 q^{46} -2.21586 q^{47} +0.405236 q^{48} +10.9560 q^{49} +0.906380 q^{50} -0.810818 q^{51} +3.21775 q^{52} -11.1217 q^{53} +2.36487 q^{54} -2.02327 q^{55} +4.23745 q^{56} -2.15785 q^{58} +1.13939 q^{59} -0.819901 q^{60} +2.99123 q^{61} -6.20065 q^{62} +12.0165 q^{63} +1.00000 q^{64} -6.51038 q^{65} -0.405236 q^{66} +1.86919 q^{67} -2.00085 q^{68} -0.960670 q^{69} -8.57350 q^{70} +4.23524 q^{71} +2.83578 q^{72} -5.29840 q^{73} -11.1129 q^{74} -0.367297 q^{75} -4.23745 q^{77} -1.30395 q^{78} +5.14495 q^{79} -2.02327 q^{80} +7.54902 q^{81} -3.97211 q^{82} -6.45404 q^{83} -1.71717 q^{84} +4.04827 q^{85} -5.64641 q^{86} +0.874437 q^{87} -1.00000 q^{88} +12.4620 q^{89} -5.73756 q^{90} -13.6351 q^{91} -2.37064 q^{92} +2.51273 q^{93} +2.21586 q^{94} -0.405236 q^{96} -18.9049 q^{97} -10.9560 q^{98} -2.83578 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 8 q^{2} - q^{3} + 8 q^{4} - 3 q^{5} + q^{6} - 8 q^{8} + 15 q^{9} + 3 q^{10} + 8 q^{11} - q^{12} - 3 q^{13} - 36 q^{15} + 8 q^{16} - 4 q^{17} - 15 q^{18} - 3 q^{20} - 3 q^{21} - 8 q^{22} - q^{23}+ \cdots + 15 q^{99}+O(q^{100})$ 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 3 * q^5 + q^6 - 8 * q^8 + 15 * q^9 + 3 * q^10 + 8 * q^11 - q^12 - 3 * q^13 - 36 * q^15 + 8 * q^16 - 4 * q^17 - 15 * q^18 - 3 * q^20 - 3 * q^21 - 8 * q^22 - q^23 + q^24 + 5 * q^25 + 3 * q^26 - 10 * q^27 + 4 * q^29 + 36 * q^30 + 3 * q^31 - 8 * q^32 - q^33 + 4 * q^34 + 3 * q^35 + 15 * q^36 + 26 * q^37 + 14 * q^39 + 3 * q^40 - 17 * q^41 + 3 * q^42 - 6 * q^43 + 8 * q^44 + 3 * q^45 + q^46 - q^48 + 36 * q^49 - 5 * q^50 + 12 * q^51 - 3 * q^52 - 49 * q^53 + 10 * q^54 - 3 * q^55 - 4 * q^58 - 15 * q^59 - 36 * q^60 - 14 * q^61 - 3 * q^62 + 17 * q^63 + 8 * q^64 + q^66 - 5 * q^67 - 4 * q^68 - 4 * q^69 - 3 * q^70 + q^71 - 15 * q^72 - 10 * q^73 - 26 * q^74 + 29 * q^75 - 14 * q^78 - 12 * q^79 - 3 * q^80 + 17 * q^82 - 7 * q^83 - 3 * q^84 - 20 * q^85 + 6 * q^86 + 45 * q^87 - 8 * q^88 - 8 * q^89 - 3 * q^90 - 17 * q^91 - q^92 - 33 * q^93 + q^96 - 54 * q^97 - 36 * q^98 + 15 * 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
$3$	0.405236	0.233963	0.116981	−	0.993134i	$-0.462678\pi$
$3$	0.405236	0.233963	0.116981	+	0.993134i	$0.462678\pi$
$4$	1.00000	0.500000
$5$	−2.02327	−0.904834	−0.452417	−	0.891807i	$-0.649438\pi$
$5$	−2.02327	−0.904834	−0.452417	+	0.891807i	$0.649438\pi$
$6$	−0.405236	−0.165437
$7$	−4.23745	−1.60161	−0.800803	−	0.598928i	$-0.795593\pi$
$7$	−4.23745	−1.60161	−0.800803	+	0.598928i	$0.795593\pi$
$8$	−1.00000	−0.353553
$9$	−2.83578	−0.945261
$10$	2.02327	0.639814
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.405236	0.116981
$13$	3.21775	0.892443	0.446222	−	0.894922i	$-0.352769\pi$
$13$	3.21775	0.892443	0.446222	+	0.894922i	$0.352769\pi$
$14$	4.23745	1.13251
$15$	−0.819901	−0.211698
$16$	1.00000	0.250000
$17$	−2.00085	−0.485278	−0.242639	−	0.970117i	$-0.578013\pi$
$17$	−2.00085	−0.485278	−0.242639	+	0.970117i	$0.578013\pi$
$18$	2.83578	0.668401
$19$	0	0
$19$	0	0
$20$	−2.02327	−0.452417
$21$	−1.71717	−0.374716
$22$	−1.00000	−0.213201
$23$	−2.37064	−0.494314	−0.247157	−	0.968975i	$-0.579496\pi$
$23$	−2.37064	−0.494314	−0.247157	+	0.968975i	$0.579496\pi$
$24$	−0.405236	−0.0827184
$25$	−0.906380	−0.181276
$26$	−3.21775	−0.631053
$27$	−2.36487	−0.455119
$28$	−4.23745	−0.800803
$29$	2.15785	0.400702	0.200351	−	0.979724i	$-0.435792\pi$
$29$	2.15785	0.400702	0.200351	+	0.979724i	$0.435792\pi$
$30$	0.819901	0.149693
$31$	6.20065	1.11367	0.556835	−	0.830623i	$-0.312016\pi$
$31$	6.20065	1.11367	0.556835	+	0.830623i	$0.312016\pi$
$32$	−1.00000	−0.176777
$33$	0.405236	0.0705425
$34$	2.00085	0.343144
$35$	8.57350	1.44919
$36$	−2.83578	−0.472631
$37$	11.1129	1.82695	0.913477	−	0.406890i	$-0.133387\pi$
$37$	11.1129	1.82695	0.913477	+	0.406890i	$0.133387\pi$
$38$	0	0
$39$	1.30395	0.208799
$40$	2.02327	0.319907
$41$	3.97211	0.620339	0.310170	−	0.950681i	$-0.399614\pi$
$41$	3.97211	0.620339	0.310170	+	0.950681i	$0.399614\pi$
$42$	1.71717	0.264964
$43$	5.64641	0.861069	0.430534	−	0.902574i	$-0.358325\pi$
$43$	5.64641	0.861069	0.430534	+	0.902574i	$0.358325\pi$
$44$	1.00000	0.150756
$45$	5.73756	0.855304
$46$	2.37064	0.349533
$47$	−2.21586	−0.323216	−0.161608	−	0.986855i	$-0.551668\pi$
$47$	−2.21586	−0.323216	−0.161608	+	0.986855i	$0.551668\pi$
$48$	0.405236	0.0584907
$49$	10.9560	1.56514
$50$	0.906380	0.128181
$51$	−0.810818	−0.113537
$52$	3.21775	0.446222
$53$	−11.1217	−1.52768	−0.763840	−	0.645406i	$-0.776688\pi$
$53$	−11.1217	−1.52768	−0.763840	+	0.645406i	$0.776688\pi$
$54$	2.36487	0.321818
$55$	−2.02327	−0.272818
$56$	4.23745	0.566253
$57$	0	0
$58$	−2.15785	−0.283339
$59$	1.13939	0.148335	0.0741677	−	0.997246i	$-0.476370\pi$
$59$	1.13939	0.148335	0.0741677	+	0.997246i	$0.476370\pi$
$60$	−0.819901	−0.105849
$61$	2.99123	0.382988	0.191494	−	0.981494i	$-0.438667\pi$
$61$	2.99123	0.382988	0.191494	+	0.981494i	$0.438667\pi$
$62$	−6.20065	−0.787484
$63$	12.0165	1.51394
$64$	1.00000	0.125000
$65$	−6.51038	−0.807513
$66$	−0.405236	−0.0498811
$67$	1.86919	0.228357	0.114179	−	0.993460i	$-0.463576\pi$
$67$	1.86919	0.228357	0.114179	+	0.993460i	$0.463576\pi$
$68$	−2.00085	−0.242639
$69$	−0.960670	−0.115651
$70$	−8.57350	−1.02473
$71$	4.23524	0.502631	0.251315	−	0.967905i	$-0.419137\pi$
$71$	4.23524	0.502631	0.251315	+	0.967905i	$0.419137\pi$
$72$	2.83578	0.334200
$73$	−5.29840	−0.620131	−0.310066	−	0.950715i	$-0.600351\pi$
$73$	−5.29840	−0.620131	−0.310066	+	0.950715i	$0.600351\pi$
$74$	−11.1129	−1.29185
$75$	−0.367297	−0.0424119
$76$	0	0
$77$	−4.23745	−0.482902
$78$	−1.30395	−0.147643
$79$	5.14495	0.578852	0.289426	−	0.957200i	$-0.406536\pi$
$79$	5.14495	0.578852	0.289426	+	0.957200i	$0.406536\pi$
$80$	−2.02327	−0.226208
$81$	7.54902	0.838780
$82$	−3.97211	−0.438646
$83$	−6.45404	−0.708423	−0.354212	−	0.935165i	$-0.615251\pi$
$83$	−6.45404	−0.708423	−0.354212	+	0.935165i	$0.615251\pi$
$84$	−1.71717	−0.187358
$85$	4.04827	0.439096
$86$	−5.64641	−0.608868
$87$	0.874437	0.0937494
$88$	−1.00000	−0.106600
$89$	12.4620	1.32097	0.660485	−	0.750839i	$-0.270351\pi$
$89$	12.4620	1.32097	0.660485	+	0.750839i	$0.270351\pi$
$90$	−5.73756	−0.604791
$91$	−13.6351	−1.42934
$92$	−2.37064	−0.247157
$93$	2.51273	0.260558
$94$	2.21586	0.228548
$95$	0	0
$96$	−0.405236	−0.0413592
$97$	−18.9049	−1.91950	−0.959752	−	0.280849i	$-0.909384\pi$
$97$	−18.9049	−1.91950	−0.959752	+	0.280849i	$0.909384\pi$
$98$	−10.9560	−1.10672
$99$	−2.83578	−0.285007
$100$	−0.906380	−0.0906380
$101$	16.0297	1.59501	0.797505	−	0.603312i	$-0.206153\pi$
$101$	16.0297	1.59501	0.797505	+	0.603312i	$0.206153\pi$
$102$	0.810818	0.0802829
$103$	2.67008	0.263091	0.131546	−	0.991310i	$-0.458006\pi$
$103$	2.67008	0.263091	0.131546	+	0.991310i	$0.458006\pi$
$104$	−3.21775	−0.315526
$105$	3.47429	0.339056
$106$	11.1217	1.08023
$107$	−2.42394	−0.234331	−0.117166	−	0.993112i	$-0.537381\pi$
$107$	−2.42394	−0.234331	−0.117166	+	0.993112i	$0.537381\pi$
$108$	−2.36487	−0.227560
$109$	−11.3556	−1.08767	−0.543833	−	0.839194i	$-0.683027\pi$
$109$	−11.3556	−1.08767	−0.543833	+	0.839194i	$0.683027\pi$
$110$	2.02327	0.192911
$111$	4.50336	0.427440
$112$	−4.23745	−0.400401
$113$	−3.22715	−0.303584	−0.151792	−	0.988412i	$-0.548504\pi$
$113$	−3.22715	−0.303584	−0.151792	+	0.988412i	$0.548504\pi$
$114$	0	0
$115$	4.79645	0.447272
$116$	2.15785	0.200351
$117$	−9.12484	−0.843592
$118$	−1.13939	−0.104889
$119$	8.47852	0.777224
$120$	0.819901	0.0748464
$121$	1.00000	0.0909091
$122$	−2.99123	−0.270813
$123$	1.60964	0.145136
$124$	6.20065	0.556835
$125$	11.9502	1.06886
$126$	−12.0165	−1.07051
$127$	1.27045	0.112734	0.0563669	−	0.998410i	$-0.482048\pi$
$127$	1.27045	0.112734	0.0563669	+	0.998410i	$0.482048\pi$
$128$	−1.00000	−0.0883883
$129$	2.28813	0.201458
$130$	6.51038	0.570998
$131$	−15.3191	−1.33844	−0.669219	−	0.743066i	$-0.733371\pi$
$131$	−15.3191	−1.33844	−0.669219	+	0.743066i	$0.733371\pi$
$132$	0.405236	0.0352712
$133$	0	0
$134$	−1.86919	−0.161473
$135$	4.78477	0.411807
$136$	2.00085	0.171572
$137$	−10.3502	−0.884281	−0.442140	−	0.896946i	$-0.645781\pi$
$137$	−10.3502	−0.884281	−0.442140	+	0.896946i	$0.645781\pi$
$138$	0.960670	0.0817777
$139$	9.86425	0.836675	0.418337	−	0.908292i	$-0.362613\pi$
$139$	9.86425	0.836675	0.418337	+	0.908292i	$0.362613\pi$
$140$	8.57350	0.724593
$141$	−0.897945	−0.0756206
$142$	−4.23524	−0.355414
$143$	3.21775	0.269082
$144$	−2.83578	−0.236315
$145$	−4.36591	−0.362569
$146$	5.29840	0.438499
$147$	4.43975	0.366185
$148$	11.1129	0.913477
$149$	−17.3586	−1.42207	−0.711034	−	0.703157i	$-0.751773\pi$
$149$	−17.3586	−1.42207	−0.711034	+	0.703157i	$0.751773\pi$
$150$	0.367297	0.0299897
$151$	19.9018	1.61958	0.809791	−	0.586718i	$-0.199580\pi$
$151$	19.9018	1.61958	0.809791	+	0.586718i	$0.199580\pi$
$152$	0	0
$153$	5.67399	0.458715
$154$	4.23745	0.341463
$155$	−12.5456	−1.00769
$156$	1.30395	0.104399
$157$	18.1197	1.44611	0.723055	−	0.690790i	$-0.242737\pi$
$157$	18.1197	1.44611	0.723055	+	0.690790i	$0.242737\pi$
$158$	−5.14495	−0.409310
$159$	−4.50690	−0.357421
$160$	2.02327	0.159954
$161$	10.0455	0.791695
$162$	−7.54902	−0.593107
$163$	−5.12549	−0.401459	−0.200730	−	0.979647i	$-0.564331\pi$
$163$	−5.12549	−0.401459	−0.200730	+	0.979647i	$0.564331\pi$
$164$	3.97211	0.310170
$165$	−0.819901	−0.0638292
$166$	6.45404	0.500931
$167$	−15.6892	−1.21406	−0.607032	−	0.794678i	$-0.707640\pi$
$167$	−15.6892	−1.21406	−0.607032	+	0.794678i	$0.707640\pi$
$168$	1.71717	0.132482
$169$	−2.64608	−0.203545
$170$	−4.04827	−0.310488
$171$	0	0
$172$	5.64641	0.430534
$173$	12.9086	0.981423	0.490712	−	0.871322i	$-0.336737\pi$
$173$	12.9086	0.981423	0.490712	+	0.871322i	$0.336737\pi$
$174$	−0.874437	−0.0662909
$175$	3.84074	0.290332
$176$	1.00000	0.0753778
$177$	0.461720	0.0347050
$178$	−12.4620	−0.934066
$179$	14.1628	1.05858	0.529288	−	0.848442i	$-0.322459\pi$
$179$	14.1628	1.05858	0.529288	+	0.848442i	$0.322459\pi$
$180$	5.73756	0.427652
$181$	−9.47135	−0.704000	−0.352000	−	0.936000i	$-0.614498\pi$
$181$	−9.47135	−0.704000	−0.352000	+	0.936000i	$0.614498\pi$
$182$	13.6351	1.01070
$183$	1.21215	0.0896050
$184$	2.37064	0.174766
$185$	−22.4845	−1.65309
$186$	−2.51273	−0.184242
$187$	−2.00085	−0.146317
$188$	−2.21586	−0.161608
$189$	10.0210	0.728921
$190$	0	0
$191$	−0.976211	−0.0706361	−0.0353181	−	0.999376i	$-0.511244\pi$
$191$	−0.976211	−0.0706361	−0.0353181	+	0.999376i	$0.511244\pi$
$192$	0.405236	0.0292454
$193$	−0.765511	−0.0551027	−0.0275514	−	0.999620i	$-0.508771\pi$
$193$	−0.765511	−0.0551027	−0.0275514	+	0.999620i	$0.508771\pi$
$194$	18.9049	1.35729
$195$	−2.63824	−0.188928
$196$	10.9560	0.782570
$197$	20.4736	1.45868	0.729341	−	0.684150i	$-0.239827\pi$
$197$	20.4736	1.45868	0.729341	+	0.684150i	$0.239827\pi$
$198$	2.83578	0.201530
$199$	7.97061	0.565021	0.282511	−	0.959264i	$-0.408833\pi$
$199$	7.97061	0.565021	0.282511	+	0.959264i	$0.408833\pi$
$200$	0.906380	0.0640907
$201$	0.757461	0.0534272
$202$	−16.0297	−1.12784
$203$	−9.14377	−0.641767
$204$	−0.810818	−0.0567686
$205$	−8.03665	−0.561304
$206$	−2.67008	−0.186034
$207$	6.72264	0.467256
$208$	3.21775	0.223111
$209$	0	0
$210$	−3.47429	−0.239749
$211$	−25.0656	−1.72559	−0.862794	−	0.505556i	$-0.831287\pi$
$211$	−25.0656	−1.72559	−0.862794	+	0.505556i	$0.831287\pi$
$212$	−11.1217	−0.763840
$213$	1.71627	0.117597
$214$	2.42394	0.165697
$215$	−11.4242	−0.779124
$216$	2.36487	0.160909
$217$	−26.2750	−1.78366
$218$	11.3556	0.769095
$219$	−2.14710	−0.145088
$220$	−2.02327	−0.136409
$221$	−6.43825	−0.433083
$222$	−4.50336	−0.302246
$223$	−24.0247	−1.60881	−0.804407	−	0.594078i	$-0.797517\pi$
$223$	−24.0247	−1.60881	−0.804407	+	0.594078i	$0.797517\pi$
$224$	4.23745	0.283127
$225$	2.57030	0.171353
$226$	3.22715	0.214667
$227$	6.55830	0.435290	0.217645	−	0.976028i	$-0.430163\pi$
$227$	6.55830	0.435290	0.217645	+	0.976028i	$0.430163\pi$
$228$	0	0
$229$	4.93753	0.326281	0.163141	−	0.986603i	$-0.447838\pi$
$229$	4.93753	0.326281	0.163141	+	0.986603i	$0.447838\pi$
$230$	−4.79645	−0.316269
$231$	−1.71717	−0.112981
$232$	−2.15785	−0.141670
$233$	−28.4901	−1.86645	−0.933223	−	0.359297i	$-0.883017\pi$
$233$	−28.4901	−1.86645	−0.933223	+	0.359297i	$0.883017\pi$
$234$	9.12484	0.596510
$235$	4.48328	0.292457
$236$	1.13939	0.0741677
$237$	2.08492	0.135430
$238$	−8.47852	−0.549581
$239$	−11.8669	−0.767603	−0.383801	−	0.923416i	$-0.625385\pi$
$239$	−11.8669	−0.767603	−0.383801	+	0.923416i	$0.625385\pi$
$240$	−0.819901	−0.0529244
$241$	−8.61238	−0.554772	−0.277386	−	0.960759i	$-0.589468\pi$
$241$	−8.61238	−0.554772	−0.277386	+	0.960759i	$0.589468\pi$
$242$	−1.00000	−0.0642824
$243$	10.1537	0.651363
$244$	2.99123	0.191494
$245$	−22.1669	−1.41619
$246$	−1.60964	−0.102627
$247$	0	0
$248$	−6.20065	−0.393742
$249$	−2.61541	−0.165745
$250$	−11.9502	−0.755797
$251$	25.1783	1.58924	0.794621	−	0.607106i	$-0.207670\pi$
$251$	25.1783	1.58924	0.794621	+	0.607106i	$0.207670\pi$
$252$	12.0165	0.756968
$253$	−2.37064	−0.149041
$254$	−1.27045	−0.0797149
$255$	1.64050	0.102732
$256$	1.00000	0.0625000
$257$	0.345739	0.0215666	0.0107833	−	0.999942i	$-0.496567\pi$
$257$	0.345739	0.0215666	0.0107833	+	0.999942i	$0.496567\pi$
$258$	−2.28813	−0.142452
$259$	−47.0905	−2.92606
$260$	−6.51038	−0.403756
$261$	−6.11919	−0.378768
$262$	15.3191	0.946418
$263$	−5.21616	−0.321642	−0.160821	−	0.986984i	$-0.551414\pi$
$263$	−5.21616	−0.321642	−0.160821	+	0.986984i	$0.551414\pi$
$264$	−0.405236	−0.0249405
$265$	22.5022	1.38230
$266$	0	0
$267$	5.05005	0.309058
$268$	1.86919	0.114179
$269$	−22.7443	−1.38675	−0.693373	−	0.720579i	$-0.743876\pi$
$269$	−22.7443	−1.38675	−0.693373	+	0.720579i	$0.743876\pi$
$270$	−4.78477	−0.291192
$271$	−3.77208	−0.229138	−0.114569	−	0.993415i	$-0.536549\pi$
$271$	−3.77208	−0.229138	−0.114569	+	0.993415i	$0.536549\pi$
$272$	−2.00085	−0.121320
$273$	−5.52541	−0.334413
$274$	10.3502	0.625281
$275$	−0.906380	−0.0546567
$276$	−0.960670	−0.0578255
$277$	−24.8242	−1.49154	−0.745770	−	0.666204i	$-0.767918\pi$
$277$	−24.8242	−1.49154	−0.745770	+	0.666204i	$0.767918\pi$
$278$	−9.86425	−0.591619
$279$	−17.5837	−1.05271
$280$	−8.57350	−0.512365
$281$	−5.69332	−0.339635	−0.169817	−	0.985476i	$-0.554318\pi$
$281$	−5.69332	−0.339635	−0.169817	+	0.985476i	$0.554318\pi$
$282$	0.897945	0.0534719
$283$	23.7525	1.41194	0.705970	−	0.708241i	$-0.250511\pi$
$283$	23.7525	1.41194	0.705970	+	0.708241i	$0.250511\pi$
$284$	4.23524	0.251315
$285$	0	0
$286$	−3.21775	−0.190270
$287$	−16.8316	−0.993539
$288$	2.83578	0.167100
$289$	−12.9966	−0.764505
$290$	4.36591	0.256375
$291$	−7.66095	−0.449093
$292$	−5.29840	−0.310066
$293$	−5.88842	−0.344005	−0.172003	−	0.985096i	$-0.555024\pi$
$293$	−5.88842	−0.344005	−0.172003	+	0.985096i	$0.555024\pi$
$294$	−4.43975	−0.258932
$295$	−2.30528	−0.134219
$296$	−11.1129	−0.645926
$297$	−2.36487	−0.137224
$298$	17.3586	1.00555
$299$	−7.62814	−0.441147
$300$	−0.367297	−0.0212059
$301$	−23.9264	−1.37909
$302$	−19.9018	−1.14522
$303$	6.49579	0.373173
$304$	0	0
$305$	−6.05207	−0.346540
$306$	−5.67399	−0.324360
$307$	−8.36781	−0.477576	−0.238788	−	0.971072i	$-0.576750\pi$
$307$	−8.36781	−0.477576	−0.238788	+	0.971072i	$0.576750\pi$
$308$	−4.23745	−0.241451
$309$	1.08201	0.0615536
$310$	12.5456	0.712542
$311$	−33.2576	−1.88586	−0.942931	−	0.332987i	$-0.891943\pi$
$311$	−33.2576	−1.88586	−0.942931	+	0.332987i	$0.891943\pi$
$312$	−1.30395	−0.0738215
$313$	13.6874	0.773657	0.386828	−	0.922152i	$-0.373571\pi$
$313$	13.6874	0.773657	0.386828	+	0.922152i	$0.373571\pi$
$314$	−18.1197	−1.02255
$315$	−24.3126	−1.36986
$316$	5.14495	0.289426
$317$	−3.03012	−0.170188	−0.0850942	−	0.996373i	$-0.527119\pi$
$317$	−3.03012	−0.170188	−0.0850942	+	0.996373i	$0.527119\pi$
$318$	4.50690	0.252734
$319$	2.15785	0.120816
$320$	−2.02327	−0.113104
$321$	−0.982267	−0.0548248
$322$	−10.0455	−0.559813
$323$	0	0
$324$	7.54902	0.419390
$325$	−2.91650	−0.161778
$326$	5.12549	0.283875
$327$	−4.60168	−0.254473
$328$	−3.97211	−0.219323
$329$	9.38959	0.517665
$330$	0.819901	0.0451341
$331$	−11.4410	−0.628855	−0.314428	−	0.949281i	$-0.601813\pi$
$331$	−11.4410	−0.628855	−0.314428	+	0.949281i	$0.601813\pi$
$332$	−6.45404	−0.354212
$333$	−31.5139	−1.72695
$334$	15.6892	0.858473
$335$	−3.78187	−0.206625
$336$	−1.71717	−0.0936791
$337$	−18.8740	−1.02813	−0.514067	−	0.857750i	$-0.671862\pi$
$337$	−18.8740	−1.02813	−0.514067	+	0.857750i	$0.671862\pi$
$338$	2.64608	0.143928
$339$	−1.30775	−0.0710275
$340$	4.04827	0.219548
$341$	6.20065	0.335784
$342$	0	0
$343$	−16.7633	−0.905131
$344$	−5.64641	−0.304434
$345$	1.94369	0.104645
$346$	−12.9086	−0.693971
$347$	36.1443	1.94033	0.970164	−	0.242451i	$-0.0779515\pi$
$347$	36.1443	1.94033	0.970164	+	0.242451i	$0.0779515\pi$
$348$	0.874437	0.0468747
$349$	−26.9542	−1.44283	−0.721413	−	0.692505i	$-0.756507\pi$
$349$	−26.9542	−1.44283	−0.721413	+	0.692505i	$0.756507\pi$
$350$	−3.84074	−0.205296
$351$	−7.60956	−0.406168
$352$	−1.00000	−0.0533002
$353$	29.3403	1.56162	0.780812	−	0.624766i	$-0.214806\pi$
$353$	29.3403	1.56162	0.780812	+	0.624766i	$0.214806\pi$
$354$	−0.461720	−0.0245401
$355$	−8.56904	−0.454797
$356$	12.4620	0.660485
$357$	3.43580	0.181842
$358$	−14.1628	−0.748526
$359$	−14.6477	−0.773076	−0.386538	−	0.922273i	$-0.626329\pi$
$359$	−14.6477	−0.773076	−0.386538	+	0.922273i	$0.626329\pi$
$360$	−5.73756	−0.302396
$361$	0	0
$362$	9.47135	0.497803
$363$	0.405236	0.0212694
$364$	−13.6351	−0.714671
$365$	10.7201	0.561116
$366$	−1.21215	−0.0633603
$367$	−32.1003	−1.67562	−0.837811	−	0.545961i	$-0.816165\pi$
$367$	−32.1003	−1.67562	−0.837811	+	0.545961i	$0.816165\pi$
$368$	−2.37064	−0.123578
$369$	−11.2640	−0.586383
$370$	22.4845	1.16891
$371$	47.1276	2.44674
$372$	2.51273	0.130279
$373$	35.7170	1.84936	0.924678	−	0.380751i	$-0.124334\pi$
$373$	35.7170	1.84936	0.924678	+	0.380751i	$0.124334\pi$
$374$	2.00085	0.103462
$375$	4.84265	0.250073
$376$	2.21586	0.114274
$377$	6.94341	0.357604
$378$	−10.0210	−0.515425
$379$	35.1473	1.80540	0.902699	−	0.430273i	$-0.141583\pi$
$379$	35.1473	1.80540	0.902699	+	0.430273i	$0.141583\pi$
$380$	0	0
$381$	0.514830	0.0263756
$382$	0.976211	0.0499473
$383$	−22.9790	−1.17417	−0.587085	−	0.809525i	$-0.699725\pi$
$383$	−22.9790	−1.17417	−0.587085	+	0.809525i	$0.699725\pi$
$384$	−0.405236	−0.0206796
$385$	8.57350	0.436946
$386$	0.765511	0.0389635
$387$	−16.0120	−0.813935
$388$	−18.9049	−0.959752
$389$	−16.9298	−0.858376	−0.429188	−	0.903215i	$-0.641200\pi$
$389$	−16.9298	−0.858376	−0.429188	+	0.903215i	$0.641200\pi$
$390$	2.63824	0.133592
$391$	4.74331	0.239880
$392$	−10.9560	−0.553361
$393$	−6.20785	−0.313145
$394$	−20.4736	−1.03144
$395$	−10.4096	−0.523765
$396$	−2.83578	−0.142504
$397$	14.8799	0.746803	0.373402	−	0.927670i	$-0.378191\pi$
$397$	14.8799	0.746803	0.373402	+	0.927670i	$0.378191\pi$
$398$	−7.97061	−0.399530
$399$	0	0
$400$	−0.906380	−0.0453190
$401$	−27.9982	−1.39816	−0.699081	−	0.715042i	$-0.746408\pi$
$401$	−27.9982	−1.39816	−0.699081	+	0.715042i	$0.746408\pi$
$402$	−0.757461	−0.0377787
$403$	19.9521	0.993887
$404$	16.0297	0.797505
$405$	−15.2737	−0.758957
$406$	9.14377	0.453798
$407$	11.1129	0.550847
$408$	0.810818	0.0401414
$409$	17.1697	0.848986	0.424493	−	0.905431i	$-0.360452\pi$
$409$	17.1697	0.848986	0.424493	+	0.905431i	$0.360452\pi$
$410$	8.03665	0.396902
$411$	−4.19429	−0.206889
$412$	2.67008	0.131546
$413$	−4.82809	−0.237575
$414$	−6.72264	−0.330400
$415$	13.0583	0.641005
$416$	−3.21775	−0.157763
$417$	3.99735	0.195751
$418$	0	0
$419$	−9.55038	−0.466566	−0.233283	−	0.972409i	$-0.574947\pi$
$419$	−9.55038	−0.466566	−0.233283	+	0.972409i	$0.574947\pi$
$420$	3.47429	0.169528
$421$	26.0989	1.27198	0.635992	−	0.771696i	$-0.280591\pi$
$421$	26.0989	1.27198	0.635992	+	0.771696i	$0.280591\pi$
$422$	25.0656	1.22017
$423$	6.28370	0.305524
$424$	11.1217	0.540116
$425$	1.81353	0.0879693
$426$	−1.71627	−0.0831536
$427$	−12.6752	−0.613396
$428$	−2.42394	−0.117166
$429$	1.30395	0.0629552
$430$	11.4242	0.550924
$431$	−24.2576	−1.16845	−0.584224	−	0.811592i	$-0.698601\pi$
$431$	−24.2576	−1.16845	−0.584224	+	0.811592i	$0.698601\pi$
$432$	−2.36487	−0.113780
$433$	35.3246	1.69759	0.848797	−	0.528719i	$-0.177327\pi$
$433$	35.3246	1.69759	0.848797	+	0.528719i	$0.177327\pi$
$434$	26.2750	1.26124
$435$	−1.76922	−0.0848277
$436$	−11.3556	−0.543833
$437$	0	0
$438$	2.14710	0.102593
$439$	−16.6205	−0.793254	−0.396627	−	0.917980i	$-0.629819\pi$
$439$	−16.6205	−0.793254	−0.396627	+	0.917980i	$0.629819\pi$
$440$	2.02327	0.0964556
$441$	−31.0688	−1.47947
$442$	6.43825	0.306236
$443$	20.5710	0.977357	0.488678	−	0.872464i	$-0.337479\pi$
$443$	20.5710	0.977357	0.488678	+	0.872464i	$0.337479\pi$
$444$	4.50336	0.213720
$445$	−25.2140	−1.19526
$446$	24.0247	1.13760
$447$	−7.03431	−0.332711
$448$	−4.23745	−0.200201
$449$	33.5618	1.58388	0.791939	−	0.610600i	$-0.209072\pi$
$449$	33.5618	1.58388	0.791939	+	0.610600i	$0.209072\pi$
$450$	−2.57030	−0.121165
$451$	3.97211	0.187039
$452$	−3.22715	−0.151792
$453$	8.06491	0.378922
$454$	−6.55830	−0.307796
$455$	27.5874	1.29332
$456$	0	0
$457$	−25.4237	−1.18927	−0.594636	−	0.803995i	$-0.702704\pi$
$457$	−25.4237	−1.18927	−0.594636	+	0.803995i	$0.702704\pi$
$458$	−4.93753	−0.230716
$459$	4.73176	0.220859
$460$	4.79645	0.223636
$461$	−24.9356	−1.16137	−0.580684	−	0.814129i	$-0.697215\pi$
$461$	−24.9356	−1.16137	−0.580684	+	0.814129i	$0.697215\pi$
$462$	1.71717	0.0798898
$463$	−35.0253	−1.62776	−0.813881	−	0.581032i	$-0.802649\pi$
$463$	−35.0253	−1.62776	−0.813881	+	0.581032i	$0.802649\pi$
$464$	2.15785	0.100176
$465$	−5.08392	−0.235761
$466$	28.4901	1.31978
$467$	15.0345	0.695715	0.347857	−	0.937547i	$-0.386909\pi$
$467$	15.0345	0.695715	0.347857	+	0.937547i	$0.386909\pi$
$468$	−9.12484	−0.421796
$469$	−7.92058	−0.365738
$470$	−4.48328	−0.206798
$471$	7.34275	0.338336
$472$	−1.13939	−0.0524445
$473$	5.64641	0.259622
$474$	−2.08492	−0.0957634
$475$	0	0
$476$	8.47852	0.388612
$477$	31.5387	1.44406
$478$	11.8669	0.542777
$479$	−28.0302	−1.28073	−0.640367	−	0.768069i	$-0.721218\pi$
$479$	−28.0302	−1.28073	−0.640367	+	0.768069i	$0.721218\pi$
$480$	0.819901	0.0374232
$481$	35.7586	1.63045
$482$	8.61238	0.392283
$483$	4.07079	0.185227
$484$	1.00000	0.0454545
$485$	38.2498	1.73683
$486$	−10.1537	−0.460583
$487$	31.1365	1.41093	0.705464	−	0.708746i	$-0.250739\pi$
$487$	31.1365	1.41093	0.705464	+	0.708746i	$0.250739\pi$
$488$	−2.99123	−0.135407
$489$	−2.07703	−0.0939266
$490$	22.1669	1.00140
$491$	35.4580	1.60020	0.800100	−	0.599867i	$-0.204780\pi$
$491$	35.4580	1.60020	0.800100	+	0.599867i	$0.204780\pi$
$492$	1.60964	0.0725682
$493$	−4.31754	−0.194452
$494$	0	0
$495$	5.73756	0.257884
$496$	6.20065	0.278418
$497$	−17.9466	−0.805016
$498$	2.61541	0.117199
$499$	33.3097	1.49115	0.745574	−	0.666423i	$-0.232176\pi$
$499$	33.3097	1.49115	0.745574	+	0.666423i	$0.232176\pi$
$500$	11.9502	0.534429
$501$	−6.35781	−0.284046
$502$	−25.1783	−1.12376
$503$	14.1546	0.631124	0.315562	−	0.948905i	$-0.397807\pi$
$503$	14.1546	0.631124	0.315562	+	0.948905i	$0.397807\pi$
$504$	−12.0165	−0.535257
$505$	−32.4323	−1.44322
$506$	2.37064	0.105388
$507$	−1.07229	−0.0476220
$508$	1.27045	0.0563669
$509$	3.10819	0.137768	0.0688841	−	0.997625i	$-0.478056\pi$
$509$	3.10819	0.137768	0.0688841	+	0.997625i	$0.478056\pi$
$510$	−1.64050	−0.0726427
$511$	22.4517	0.993205
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−0.345739	−0.0152499
$515$	−5.40230	−0.238054
$516$	2.28813	0.100729
$517$	−2.21586	−0.0974533
$518$	47.0905	2.06904
$519$	5.23103	0.229617
$520$	6.51038	0.285499
$521$	−20.5080	−0.898473	−0.449236	−	0.893413i	$-0.648304\pi$
$521$	−20.5080	−0.898473	−0.449236	+	0.893413i	$0.648304\pi$
$522$	6.11919	0.267830
$523$	−11.7677	−0.514567	−0.257284	−	0.966336i	$-0.582827\pi$
$523$	−11.7677	−0.514567	−0.257284	+	0.966336i	$0.582827\pi$
$524$	−15.3191	−0.669219
$525$	1.55640	0.0679270
$526$	5.21616	0.227435
$527$	−12.4066	−0.540440
$528$	0.405236	0.0176356
$529$	−17.3800	−0.755654
$530$	−22.5022	−0.977431
$531$	−3.23105	−0.140216
$532$	0	0
$533$	12.7813	0.553618
$534$	−5.05005	−0.218537
$535$	4.90428	0.212031
$536$	−1.86919	−0.0807365
$537$	5.73927	0.247668
$538$	22.7443	0.980577
$539$	10.9560	0.471907
$540$	4.78477	0.205904
$541$	−12.6284	−0.542939	−0.271470	−	0.962447i	$-0.587510\pi$
$541$	−12.6284	−0.542939	−0.271470	+	0.962447i	$0.587510\pi$
$542$	3.77208	0.162025
$543$	−3.83813	−0.164710
$544$	2.00085	0.0857859
$545$	22.9754	0.984156
$546$	5.52541	0.236466
$547$	−19.7404	−0.844039	−0.422020	−	0.906587i	$-0.638679\pi$
$547$	−19.7404	−0.844039	−0.422020	+	0.906587i	$0.638679\pi$
$548$	−10.3502	−0.442140
$549$	−8.48249	−0.362024
$550$	0.906380	0.0386482
$551$	0	0
$552$	0.960670	0.0408888
$553$	−21.8015	−0.927092
$554$	24.8242	1.05468
$555$	−9.11151	−0.386762
$556$	9.86425	0.418337
$557$	−34.6068	−1.46634	−0.733168	−	0.680047i	$-0.761959\pi$
$557$	−34.6068	−1.46634	−0.733168	+	0.680047i	$0.761959\pi$
$558$	17.5837	0.744378
$559$	18.1687	0.768455
$560$	8.57350	0.362297
$561$	−0.810818	−0.0342327
$562$	5.69332	0.240158
$563$	23.7811	1.00225	0.501127	−	0.865374i	$-0.332919\pi$
$563$	23.7811	1.00225	0.501127	+	0.865374i	$0.332919\pi$
$564$	−0.897945	−0.0378103
$565$	6.52939	0.274693
$566$	−23.7525	−0.998393
$567$	−31.9886	−1.34340
$568$	−4.23524	−0.177707
$569$	10.9111	0.457415	0.228708	−	0.973495i	$-0.426550\pi$
$569$	10.9111	0.457415	0.228708	+	0.973495i	$0.426550\pi$
$570$	0	0
$571$	30.1936	1.26356	0.631781	−	0.775147i	$-0.282324\pi$
$571$	30.1936	1.26356	0.631781	+	0.775147i	$0.282324\pi$
$572$	3.21775	0.134541
$573$	−0.395595	−0.0165262
$574$	16.8316	0.702538
$575$	2.14870	0.0896072
$576$	−2.83578	−0.118158
$577$	−9.07937	−0.377979	−0.188990	−	0.981979i	$-0.560521\pi$
$577$	−9.07937	−0.377979	−0.188990	+	0.981979i	$0.560521\pi$
$578$	12.9966	0.540587
$579$	−0.310213	−0.0128920
$580$	−4.36591	−0.181284
$581$	27.3487	1.13461
$582$	7.66095	0.317557
$583$	−11.1217	−0.460613
$584$	5.29840	0.219249
$585$	18.4620	0.763311
$586$	5.88842	0.243248
$587$	−44.5960	−1.84067	−0.920337	−	0.391127i	$-0.872085\pi$
$587$	−44.5960	−1.84067	−0.920337	+	0.391127i	$0.872085\pi$
$588$	4.43975	0.183092
$589$	0	0
$590$	2.30528	0.0949070
$591$	8.29663	0.341278
$592$	11.1129	0.456739
$593$	−21.8531	−0.897398	−0.448699	−	0.893683i	$-0.648112\pi$
$593$	−21.8531	−0.897398	−0.448699	+	0.893683i	$0.648112\pi$
$594$	2.36487	0.0970317
$595$	−17.1543	−0.703259
$596$	−17.3586	−0.711034
$597$	3.22997	0.132194
$598$	7.62814	0.311938
$599$	−19.5950	−0.800631	−0.400316	−	0.916377i	$-0.631100\pi$
$599$	−19.5950	−0.800631	−0.400316	+	0.916377i	$0.631100\pi$
$600$	0.367297	0.0149949
$601$	−9.78831	−0.399274	−0.199637	−	0.979870i	$-0.563976\pi$
$601$	−9.78831	−0.399274	−0.199637	+	0.979870i	$0.563976\pi$
$602$	23.9264	0.975166
$603$	−5.30061	−0.215857
$604$	19.9018	0.809791
$605$	−2.02327	−0.0822576
$606$	−6.49579	−0.263873
$607$	−20.7831	−0.843560	−0.421780	−	0.906698i	$-0.638595\pi$
$607$	−20.7831	−0.843560	−0.421780	+	0.906698i	$0.638595\pi$
$608$	0	0
$609$	−3.70538	−0.150150
$610$	6.05207	0.245041
$611$	−7.13008	−0.288452
$612$	5.67399	0.229357
$613$	0.783957	0.0316637	0.0158319	−	0.999875i	$-0.494960\pi$
$613$	0.783957	0.0316637	0.0158319	+	0.999875i	$0.494960\pi$
$614$	8.36781	0.337697
$615$	−3.25674	−0.131324
$616$	4.23745	0.170732
$617$	21.8237	0.878591	0.439295	−	0.898343i	$-0.355228\pi$
$617$	21.8237	0.878591	0.439295	+	0.898343i	$0.355228\pi$
$618$	−1.08201	−0.0435250
$619$	34.1019	1.37067	0.685335	−	0.728228i	$-0.259656\pi$
$619$	34.1019	1.37067	0.685335	+	0.728228i	$0.259656\pi$
$620$	−12.5456	−0.503843
$621$	5.60626	0.224972
$622$	33.2576	1.33351
$623$	−52.8071	−2.11567
$624$	1.30395	0.0521997
$625$	−19.6466	−0.785863
$626$	−13.6874	−0.547058
$627$	0	0
$628$	18.1197	0.723055
$629$	−22.2353	−0.886581
$630$	24.3126	0.968637
$631$	−33.1428	−1.31940	−0.659698	−	0.751531i	$-0.729316\pi$
$631$	−33.1428	−1.31940	−0.659698	+	0.751531i	$0.729316\pi$
$632$	−5.14495	−0.204655
$633$	−10.1575	−0.403724
$634$	3.03012	0.120341
$635$	−2.57046	−0.102005
$636$	−4.50690	−0.178710
$637$	35.2536	1.39680
$638$	−2.15785	−0.0854300
$639$	−12.0102	−0.475118
$640$	2.02327	0.0799768
$641$	1.65951	0.0655467	0.0327733	−	0.999463i	$-0.489566\pi$
$641$	1.65951	0.0655467	0.0327733	+	0.999463i	$0.489566\pi$
$642$	0.982267	0.0387670
$643$	41.1209	1.62165	0.810825	−	0.585289i	$-0.199019\pi$
$643$	41.1209	1.62165	0.810825	+	0.585289i	$0.199019\pi$
$644$	10.0455	0.395848
$645$	−4.62950	−0.182286
$646$	0	0
$647$	−36.3298	−1.42827	−0.714135	−	0.700008i	$-0.753180\pi$
$647$	−36.3298	−1.42827	−0.714135	+	0.700008i	$0.753180\pi$
$648$	−7.54902	−0.296554
$649$	1.13939	0.0447248
$650$	2.91650	0.114395
$651$	−10.6475	−0.417310
$652$	−5.12549	−0.200730
$653$	5.76318	0.225531	0.112765	−	0.993622i	$-0.464029\pi$
$653$	5.76318	0.225531	0.112765	+	0.993622i	$0.464029\pi$
$654$	4.60168	0.179940
$655$	30.9947	1.21106
$656$	3.97211	0.155085
$657$	15.0251	0.586186
$658$	−9.38959	−0.366044
$659$	−4.73210	−0.184337	−0.0921683	−	0.995743i	$-0.529380\pi$
$659$	−4.73210	−0.184337	−0.0921683	+	0.995743i	$0.529380\pi$
$660$	−0.819901	−0.0319146
$661$	26.8507	1.04437	0.522186	−	0.852832i	$-0.325117\pi$
$661$	26.8507	1.04437	0.522186	+	0.852832i	$0.325117\pi$
$662$	11.4410	0.444668
$663$	−2.60901	−0.101325
$664$	6.45404	0.250465
$665$	0	0
$666$	31.5139	1.22114
$667$	−5.11549	−0.198072
$668$	−15.6892	−0.607032
$669$	−9.73568	−0.376403
$670$	3.78187	0.146106
$671$	2.99123	0.115475
$672$	1.71717	0.0662411
$673$	−47.7268	−1.83973	−0.919866	−	0.392233i	$-0.871703\pi$
$673$	−47.7268	−1.83973	−0.919866	+	0.392233i	$0.871703\pi$
$674$	18.8740	0.727000
$675$	2.14347	0.0825021
$676$	−2.64608	−0.101772
$677$	−12.2652	−0.471391	−0.235695	−	0.971827i	$-0.575737\pi$
$677$	−12.2652	−0.471391	−0.235695	+	0.971827i	$0.575737\pi$
$678$	1.30775	0.0502240
$679$	80.1086	3.07429
$680$	−4.04827	−0.155244
$681$	2.65766	0.101842
$682$	−6.20065	−0.237435
$683$	5.34681	0.204590	0.102295	−	0.994754i	$-0.467381\pi$
$683$	5.34681	0.204590	0.102295	+	0.994754i	$0.467381\pi$
$684$	0	0
$685$	20.9413	0.800127
$686$	16.7633	0.640024
$687$	2.00086	0.0763377
$688$	5.64641	0.215267
$689$	−35.7868	−1.36337
$690$	−1.94369	−0.0739952
$691$	−12.0130	−0.456996	−0.228498	−	0.973544i	$-0.573381\pi$
$691$	−12.0130	−0.456996	−0.228498	+	0.973544i	$0.573381\pi$
$692$	12.9086	0.490712
$693$	12.0165	0.456469
$694$	−36.1443	−1.37202
$695$	−19.9580	−0.757052
$696$	−0.874437	−0.0331454
$697$	−7.94761	−0.301037
$698$	26.9542	1.02023
$699$	−11.5452	−0.436679
$700$	3.84074	0.145166
$701$	−15.0463	−0.568291	−0.284146	−	0.958781i	$-0.591710\pi$
$701$	−15.0463	−0.568291	−0.284146	+	0.958781i	$0.591710\pi$
$702$	7.60956	0.287204
$703$	0	0
$704$	1.00000	0.0376889
$705$	1.81679	0.0684241
$706$	−29.3403	−1.10424
$707$	−67.9249	−2.55458
$708$	0.461720	0.0173525
$709$	−8.24255	−0.309556	−0.154778	−	0.987949i	$-0.549466\pi$
$709$	−8.24255	−0.309556	−0.154778	+	0.987949i	$0.549466\pi$
$710$	8.56904	0.321590
$711$	−14.5900	−0.547166
$712$	−12.4620	−0.467033
$713$	−14.6995	−0.550502
$714$	−3.43580	−0.128582
$715$	−6.51038	−0.243474
$716$	14.1628	0.529288
$717$	−4.80887	−0.179591
$718$	14.6477	0.546648
$719$	−46.4425	−1.73201	−0.866007	−	0.500031i	$-0.833322\pi$
$719$	−46.4425	−1.73201	−0.866007	+	0.500031i	$0.833322\pi$
$720$	5.73756	0.213826
$721$	−11.3143	−0.421368
$722$	0	0
$723$	−3.49004	−0.129796
$724$	−9.47135	−0.352000
$725$	−1.95583	−0.0726376
$726$	−0.405236	−0.0150397
$727$	−14.3843	−0.533483	−0.266742	−	0.963768i	$-0.585947\pi$
$727$	−14.3843	−0.533483	−0.266742	+	0.963768i	$0.585947\pi$
$728$	13.6351	0.505349
$729$	−18.5324	−0.686386
$730$	−10.7201	−0.396769
$731$	−11.2976	−0.417858
$732$	1.21215	0.0448025
$733$	2.26250	0.0835672	0.0417836	−	0.999127i	$-0.486696\pi$
$733$	2.26250	0.0835672	0.0417836	+	0.999127i	$0.486696\pi$
$734$	32.1003	1.18484
$735$	−8.98282	−0.331336
$736$	2.37064	0.0873831
$737$	1.86919	0.0688524
$738$	11.2640	0.414635
$739$	12.9882	0.477777	0.238888	−	0.971047i	$-0.423217\pi$
$739$	12.9882	0.477777	0.238888	+	0.971047i	$0.423217\pi$
$740$	−22.4845	−0.826545
$741$	0	0
$742$	−47.1276	−1.73011
$743$	12.4356	0.456216	0.228108	−	0.973636i	$-0.426746\pi$
$743$	12.4356	0.456216	0.228108	+	0.973636i	$0.426746\pi$
$744$	−2.51273	−0.0921210
$745$	35.1210	1.28674
$746$	−35.7170	−1.30769
$747$	18.3023	0.669645
$748$	−2.00085	−0.0731585
$749$	10.2713	0.375306
$750$	−4.84265	−0.176829
$751$	22.9170	0.836253	0.418126	−	0.908389i	$-0.362687\pi$
$751$	22.9170	0.836253	0.418126	+	0.908389i	$0.362687\pi$
$752$	−2.21586	−0.0808040
$753$	10.2032	0.371824
$754$	−6.94341	−0.252864
$755$	−40.2666	−1.46545
$756$	10.0210	0.364461
$757$	−14.0288	−0.509886	−0.254943	−	0.966956i	$-0.582057\pi$
$757$	−14.0288	−0.509886	−0.254943	+	0.966956i	$0.582057\pi$
$758$	−35.1473	−1.27661
$759$	−0.960670	−0.0348701
$760$	0	0
$761$	−20.9226	−0.758443	−0.379222	−	0.925306i	$-0.623808\pi$
$761$	−20.9226	−0.758443	−0.379222	+	0.925306i	$0.623808\pi$
$762$	−0.514830	−0.0186503
$763$	48.1186	1.74201
$764$	−0.976211	−0.0353181
$765$	−11.4800	−0.415061
$766$	22.9790	0.830264
$767$	3.66626	0.132381
$768$	0.405236	0.0146227
$769$	23.4730	0.846459	0.423229	−	0.906023i	$-0.360896\pi$
$769$	23.4730	0.846459	0.423229	+	0.906023i	$0.360896\pi$
$770$	−8.57350	−0.308968
$771$	0.140106	0.00504579
$772$	−0.765511	−0.0275514
$773$	29.0521	1.04493	0.522465	−	0.852661i	$-0.325012\pi$
$773$	29.0521	1.04493	0.522465	+	0.852661i	$0.325012\pi$
$774$	16.0120	0.575539
$775$	−5.62014	−0.201882
$776$	18.9049	0.678647
$777$	−19.0827	−0.684590
$778$	16.9298	0.606964
$779$	0	0
$780$	−2.63824	−0.0944640
$781$	4.23524	0.151549
$782$	−4.74331	−0.169621
$783$	−5.10302	−0.182367
$784$	10.9560	0.391285
$785$	−36.6611	−1.30849
$786$	6.20785	0.221427
$787$	25.6558	0.914530	0.457265	−	0.889330i	$-0.348829\pi$
$787$	25.6558	0.914530	0.457265	+	0.889330i	$0.348829\pi$
$788$	20.4736	0.729341
$789$	−2.11378	−0.0752524
$790$	10.4096	0.370358
$791$	13.6749	0.486222
$792$	2.83578	0.100765
$793$	9.62503	0.341795
$794$	−14.8799	−0.528070
$795$	9.11868	0.323406
$796$	7.97061	0.282511
$797$	−30.7576	−1.08949	−0.544746	−	0.838601i	$-0.683374\pi$
$797$	−30.7576	−1.08949	−0.544746	+	0.838601i	$0.683374\pi$
$798$	0	0
$799$	4.43361	0.156850
$800$	0.906380	0.0320454
$801$	−35.3395	−1.24866
$802$	27.9982	0.988650
$803$	−5.29840	−0.186977
$804$	0.757461	0.0267136
$805$	−20.3247	−0.716353
$806$	−19.9521	−0.702784
$807$	−9.21681	−0.324447
$808$	−16.0297	−0.563921
$809$	10.6369	0.373972	0.186986	−	0.982363i	$-0.440128\pi$
$809$	10.6369	0.373972	0.186986	+	0.982363i	$0.440128\pi$
$810$	15.2737	0.536663
$811$	−19.9177	−0.699405	−0.349703	−	0.936861i	$-0.613717\pi$
$811$	−19.9177	−0.699405	−0.349703	+	0.936861i	$0.613717\pi$
$812$	−9.14377	−0.320883
$813$	−1.52858	−0.0536098
$814$	−11.1129	−0.389508
$815$	10.3703	0.363254
$816$	−0.810818	−0.0283843
$817$	0	0
$818$	−17.1697	−0.600324
$819$	38.6661	1.35110
$820$	−8.03665	−0.280652
$821$	−2.44878	−0.0854631	−0.0427315	−	0.999087i	$-0.513606\pi$
$821$	−2.44878	−0.0854631	−0.0427315	+	0.999087i	$0.513606\pi$
$822$	4.19429	0.146293
$823$	30.7341	1.07132	0.535662	−	0.844433i	$-0.320062\pi$
$823$	30.7341	1.07132	0.535662	+	0.844433i	$0.320062\pi$
$824$	−2.67008	−0.0930168
$825$	−0.367297	−0.0127877
$826$	4.82809	0.167991
$827$	−18.8819	−0.656588	−0.328294	−	0.944576i	$-0.606474\pi$
$827$	−18.8819	−0.656588	−0.328294	+	0.944576i	$0.606474\pi$
$828$	6.72264	0.233628
$829$	35.1837	1.22198	0.610989	−	0.791639i	$-0.290772\pi$
$829$	35.1837	1.22198	0.610989	+	0.791639i	$0.290772\pi$
$830$	−13.0583	−0.453259
$831$	−10.0596	−0.348965
$832$	3.21775	0.111555
$833$	−21.9213	−0.759529
$834$	−3.99735	−0.138417
$835$	31.7434	1.09853
$836$	0	0
$837$	−14.6637	−0.506853
$838$	9.55038	0.329912
$839$	20.8157	0.718637	0.359319	−	0.933215i	$-0.383009\pi$
$839$	20.8157	0.718637	0.359319	+	0.933215i	$0.383009\pi$
$840$	−3.47429	−0.119874
$841$	−24.3437	−0.839438
$842$	−26.0989	−0.899428
$843$	−2.30714	−0.0794620
$844$	−25.0656	−0.862794
$845$	5.35374	0.184174
$846$	−6.28370	−0.216038
$847$	−4.23745	−0.145600
$848$	−11.1217	−0.381920
$849$	9.62537	0.330342
$850$	−1.81353	−0.0622037
$851$	−26.3448	−0.903088
$852$	1.71627	0.0587985
$853$	42.1522	1.44326	0.721632	−	0.692277i	$-0.243392\pi$
$853$	42.1522	1.44326	0.721632	+	0.692277i	$0.243392\pi$
$854$	12.6752	0.433736
$855$	0	0
$856$	2.42394	0.0828486
$857$	3.06453	0.104682	0.0523412	−	0.998629i	$-0.483332\pi$
$857$	3.06453	0.104682	0.0523412	+	0.998629i	$0.483332\pi$
$858$	−1.30395	−0.0445160
$859$	36.3872	1.24151	0.620757	−	0.784003i	$-0.286825\pi$
$859$	36.3872	1.24151	0.620757	+	0.784003i	$0.286825\pi$
$860$	−11.4242	−0.389562
$861$	−6.82077	−0.232451
$862$	24.2576	0.826218
$863$	−19.3750	−0.659531	−0.329766	−	0.944063i	$-0.606970\pi$
$863$	−19.3750	−0.659531	−0.329766	+	0.944063i	$0.606970\pi$
$864$	2.36487	0.0804545
$865$	−26.1176	−0.888025
$866$	−35.3246	−1.20038
$867$	−5.26668	−0.178866
$868$	−26.2750	−0.891830
$869$	5.14495	0.174530
$870$	1.76922	0.0599822
$871$	6.01457	0.203796
$872$	11.3556	0.384548
$873$	53.6103	1.81443
$874$	0	0
$875$	−50.6384	−1.71189
$876$	−2.14710	−0.0725439
$877$	−7.06281	−0.238494	−0.119247	−	0.992865i	$-0.538048\pi$
$877$	−7.06281	−0.238494	−0.119247	+	0.992865i	$0.538048\pi$
$878$	16.6205	0.560915
$879$	−2.38620	−0.0804845
$880$	−2.02327	−0.0682044
$881$	21.6102	0.728067	0.364033	−	0.931386i	$-0.381399\pi$
$881$	21.6102	0.728067	0.364033	+	0.931386i	$0.381399\pi$
$882$	31.0688	1.04614
$883$	−12.1597	−0.409206	−0.204603	−	0.978845i	$-0.565590\pi$
$883$	−12.1597	−0.409206	−0.204603	+	0.978845i	$0.565590\pi$
$884$	−6.43825	−0.216542
$885$	−0.934183	−0.0314022
$886$	−20.5710	−0.691096
$887$	−47.3042	−1.58832	−0.794160	−	0.607709i	$-0.792089\pi$
$887$	−47.3042	−1.58832	−0.794160	+	0.607709i	$0.792089\pi$
$888$	−4.50336	−0.151123
$889$	−5.38345	−0.180555
$890$	25.2140	0.845175
$891$	7.54902	0.252902
$892$	−24.0247	−0.804407
$893$	0	0
$894$	7.03431	0.235262
$895$	−28.6551	−0.957835
$896$	4.23745	0.141563
$897$	−3.09120	−0.103212
$898$	−33.5618	−1.11997
$899$	13.3801	0.446250
$900$	2.57030	0.0856766
$901$	22.2529	0.741350
$902$	−3.97211	−0.132257
$903$	−9.69582	−0.322657
$904$	3.22715	0.107333
$905$	19.1631	0.637003
$906$	−8.06491	−0.267939
$907$	15.5394	0.515977	0.257988	−	0.966148i	$-0.416940\pi$
$907$	15.5394	0.515977	0.257988	+	0.966148i	$0.416940\pi$
$908$	6.55830	0.217645
$909$	−45.4566	−1.50770
$910$	−27.5874	−0.914513
$911$	−52.4184	−1.73670	−0.868349	−	0.495954i	$-0.834818\pi$
$911$	−52.4184	−1.73670	−0.868349	+	0.495954i	$0.834818\pi$
$912$	0	0
$913$	−6.45404	−0.213598
$914$	25.4237	0.840942
$915$	−2.45251	−0.0810776
$916$	4.93753	0.163141
$917$	64.9140	2.14365
$918$	−4.73176	−0.156171
$919$	35.8907	1.18392	0.591962	−	0.805966i	$-0.298354\pi$
$919$	35.8907	1.18392	0.591962	+	0.805966i	$0.298354\pi$
$920$	−4.79645	−0.158134
$921$	−3.39093	−0.111735
$922$	24.9356	0.821211
$923$	13.6280	0.448570
$924$	−1.71717	−0.0564906
$925$	−10.0725	−0.331183
$926$	35.0253	1.15100
$927$	−7.57178	−0.248690
$928$	−2.15785	−0.0708348
$929$	−10.4081	−0.341477	−0.170739	−	0.985316i	$-0.554615\pi$
$929$	−10.4081	−0.341477	−0.170739	+	0.985316i	$0.554615\pi$
$930$	5.08392	0.166708
$931$	0	0
$932$	−28.4901	−0.933223
$933$	−13.4771	−0.441222
$934$	−15.0345	−0.491945
$935$	4.04827	0.132392
$936$	9.12484	0.298255
$937$	−53.1072	−1.73494	−0.867469	−	0.497492i	$-0.834254\pi$
$937$	−53.1072	−1.73494	−0.867469	+	0.497492i	$0.834254\pi$
$938$	7.92058	0.258616
$939$	5.54662	0.181007
$940$	4.48328	0.146228
$941$	−6.13305	−0.199932	−0.0999658	−	0.994991i	$-0.531873\pi$
$941$	−6.13305	−0.199932	−0.0999658	+	0.994991i	$0.531873\pi$
$942$	−7.34275	−0.239240
$943$	−9.41646	−0.306642
$944$	1.13939	0.0370838
$945$	−20.2752	−0.659553
$946$	−5.64641	−0.183581
$947$	22.1095	0.718463	0.359231	−	0.933249i	$-0.383039\pi$
$947$	22.1095	0.718463	0.359231	+	0.933249i	$0.383039\pi$
$948$	2.08492	0.0677150
$949$	−17.0489	−0.553432
$950$	0	0
$951$	−1.22791	−0.0398178
$952$	−8.47852	−0.274790
$953$	−47.0512	−1.52414	−0.762069	−	0.647496i	$-0.775816\pi$
$953$	−47.0512	−1.52414	−0.762069	+	0.647496i	$0.775816\pi$
$954$	−31.5387	−1.02110
$955$	1.97514	0.0639139
$956$	−11.8669	−0.383801
$957$	0.874437	0.0282665
$958$	28.0302	0.905616
$959$	43.8586	1.41627
$960$	−0.819901	−0.0264622
$961$	7.44809	0.240261
$962$	−35.7586	−1.15290
$963$	6.87377	0.221504
$964$	−8.61238	−0.277386
$965$	1.54884	0.0498588
$966$	−4.07079	−0.130976
$967$	−23.4958	−0.755574	−0.377787	−	0.925893i	$-0.623315\pi$
$967$	−23.4958	−0.755574	−0.377787	+	0.925893i	$0.623315\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−38.2498	−1.22813
$971$	−20.8584	−0.669379	−0.334689	−	0.942329i	$-0.608631\pi$
$971$	−20.8584	−0.669379	−0.334689	+	0.942329i	$0.608631\pi$
$972$	10.1537	0.325681
$973$	−41.7993	−1.34002
$974$	−31.1365	−0.997677
$975$	−1.18187	−0.0378502
$976$	2.99123	0.0957470
$977$	27.5388	0.881043	0.440522	−	0.897742i	$-0.354794\pi$
$977$	27.5388	0.881043	0.440522	+	0.897742i	$0.354794\pi$
$978$	2.07703	0.0664162
$979$	12.4620	0.398287
$980$	−22.1669	−0.708096
$981$	32.2019	1.02813
$982$	−35.4580	−1.13151
$983$	49.0052	1.56302	0.781511	−	0.623892i	$-0.214449\pi$
$983$	49.0052	1.56302	0.781511	+	0.623892i	$0.214449\pi$
$984$	−1.60964	−0.0513135
$985$	−41.4236	−1.31987
$986$	4.31754	0.137498
$987$	3.80500	0.121114
$988$	0	0
$989$	−13.3856	−0.425638
$990$	−5.73756	−0.182351
$991$	41.5666	1.32041	0.660203	−	0.751088i	$-0.270470\pi$
$991$	41.5666	1.32041	0.660203	+	0.751088i	$0.270470\pi$
$992$	−6.20065	−0.196871
$993$	−4.63631	−0.147129
$994$	17.9466	0.569233
$995$	−16.1267	−0.511250
$996$	−2.61541	−0.0828724
$997$	−35.3686	−1.12013	−0.560067	−	0.828447i	$-0.689225\pi$
$997$	−35.3686	−1.12013	−0.560067	+	0.828447i	$0.689225\pi$
$998$	−33.3097	−1.05440
$999$	−26.2806	−0.831482

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	7942.2.a.bn.1.5	✓	8
19.18	odd	2		7942.2.a.bq.1.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7942.2.a.bn.1.5	✓	8	1.1	even	1	trivial
7942.2.a.bq.1.4	yes	8	19.18	odd	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	7942.2.a.bn.1.5

Level	$7942$
Weight	$2$
Character	7942.1
Self dual	yes
Analytic conductor	$63.417$
Analytic rank	$1$
Dimension	$8$
CM	no
Inner twists	$1$