LMFDB - Embedded newform 374.2.a.e.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [374,2,Mod(1,374)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("374.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(374, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$374 = 2 \cdot 11 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	374.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$2.98640503560$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.17417.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 2x^{3} - 5x^{2} + 3x + 4$ x^4 - 2x^3 - 5x^2 + 3*x + 4
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			1.3
Root			$3.15587$ of defining polynomial
Character	$\chi$	$=$	374.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.759376 q^{3} +1.00000 q^{4} -0.888387 q^{5} +0.759376 q^{6} +2.39649 q^{7} +1.00000 q^{8} -2.42335 q^{9} -0.888387 q^{10} +1.00000 q^{11} +0.759376 q^{12} +6.84788 q^{13} +2.39649 q^{14} -0.674620 q^{15} +1.00000 q^{16} +1.00000 q^{17} -2.42335 q^{18} -0.619718 q^{19} -0.888387 q^{20} +1.81984 q^{21} +1.00000 q^{22} -5.60726 q^{23} +0.759376 q^{24} -4.21077 q^{25} +6.84788 q^{26} -4.11836 q^{27} +2.39649 q^{28} -3.51875 q^{29} -0.674620 q^{30} -1.51875 q^{31} +1.00000 q^{32} +0.759376 q^{33} +1.00000 q^{34} -2.12901 q^{35} -2.42335 q^{36} +8.22698 q^{37} -0.619718 q^{38} +5.20012 q^{39} -0.888387 q^{40} -4.88839 q^{41} +1.81984 q^{42} -3.70267 q^{43} +1.00000 q^{44} +2.15287 q^{45} -5.60726 q^{46} +1.11161 q^{47} +0.759376 q^{48} -1.25683 q^{49} -4.21077 q^{50} +0.759376 q^{51} +6.84788 q^{52} -7.49746 q^{53} -4.11836 q^{54} -0.888387 q^{55} +2.39649 q^{56} -0.470599 q^{57} -3.51875 q^{58} -2.31173 q^{59} -0.674620 q^{60} -12.2307 q^{61} -1.51875 q^{62} -5.80753 q^{63} +1.00000 q^{64} -6.08357 q^{65} +0.759376 q^{66} +9.07230 q^{67} +1.00000 q^{68} -4.25802 q^{69} -2.12901 q^{70} +7.83049 q^{71} -2.42335 q^{72} +7.10590 q^{73} +8.22698 q^{74} -3.19756 q^{75} -0.619718 q^{76} +2.39649 q^{77} +5.20012 q^{78} -7.86409 q^{79} -0.888387 q^{80} +4.14266 q^{81} -4.88839 q^{82} +16.7967 q^{83} +1.81984 q^{84} -0.888387 q^{85} -3.70267 q^{86} -2.67206 q^{87} +1.00000 q^{88} -14.0038 q^{89} +2.15287 q^{90} +16.4109 q^{91} -5.60726 q^{92} -1.15330 q^{93} +1.11161 q^{94} +0.550549 q^{95} +0.759376 q^{96} +1.12601 q^{97} -1.25683 q^{98} -2.42335 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 4 q^{2} + q^{3} + 4 q^{4} - q^{5} + q^{6} + q^{7} + 4 q^{8} + 9 q^{9} - q^{10} + 4 q^{11} + q^{12} - q^{13} + q^{14} + 12 q^{15} + 4 q^{16} + 4 q^{17} + 9 q^{18} + q^{19} - q^{20} - 20 q^{21}+ \cdots + 9 q^{99}+O(q^{100})$ 4 * q + 4 * q^2 + q^3 + 4 * q^4 - q^5 + q^6 + q^7 + 4 * q^8 + 9 * q^9 - q^10 + 4 * q^11 + q^12 - q^13 + q^14 + 12 * q^15 + 4 * q^16 + 4 * q^17 + 9 * q^18 + q^19 - q^20 - 20 * q^21 + 4 * q^22 + 8 * q^23 + q^24 + 5 * q^25 - q^26 - 14 * q^27 + q^28 - 10 * q^29 + 12 * q^30 - 2 * q^31 + 4 * q^32 + q^33 + 4 * q^34 - 8 * q^35 + 9 * q^36 - q^37 + q^38 - 3 * q^39 - q^40 - 17 * q^41 - 20 * q^42 + q^43 + 4 * q^44 - 36 * q^45 + 8 * q^46 + 7 * q^47 + q^48 + 5 * q^49 + 5 * q^50 + q^51 - q^52 - 22 * q^53 - 14 * q^54 - q^55 + q^56 - 4 * q^57 - 10 * q^58 + 12 * q^59 + 12 * q^60 + 24 * q^61 - 2 * q^62 - 15 * q^63 + 4 * q^64 - 28 * q^65 + q^66 + 22 * q^67 + 4 * q^68 - 16 * q^69 - 8 * q^70 + 6 * q^71 + 9 * q^72 - q^73 - q^74 - 37 * q^75 + q^76 + q^77 - 3 * q^78 + 9 * q^79 - q^80 + 52 * q^81 - 17 * q^82 + 11 * q^83 - 20 * q^84 - q^85 + q^86 - 44 * q^87 + 4 * q^88 - 17 * q^89 - 36 * q^90 + 20 * q^91 + 8 * q^92 - 42 * q^93 + 7 * q^94 - 42 * q^95 + q^96 - 30 * q^97 + 5 * q^98 + 9 * 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
$3$	0.759376	0.438426	0.219213	−	0.975677i	$-0.429651\pi$
$3$	0.759376	0.438426	0.219213	+	0.975677i	$0.429651\pi$
$4$	1.00000	0.500000
$5$	−0.888387	−0.397299	−0.198649	−	0.980071i	$-0.563655\pi$
$5$	−0.888387	−0.397299	−0.198649	+	0.980071i	$0.563655\pi$
$6$	0.759376	0.310014
$7$	2.39649	0.905788	0.452894	−	0.891564i	$-0.350392\pi$
$7$	2.39649	0.905788	0.452894	+	0.891564i	$0.350392\pi$
$8$	1.00000	0.353553
$9$	−2.42335	−0.807783
$10$	−0.888387	−0.280933
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.759376	0.219213
$13$	6.84788	1.89926	0.949631	−	0.313372i	$-0.101459\pi$
$13$	6.84788	1.89926	0.949631	+	0.313372i	$0.101459\pi$
$14$	2.39649	0.640489
$15$	−0.674620	−0.174186
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−2.42335	−0.571189
$19$	−0.619718	−0.142173	−0.0710865	−	0.997470i	$-0.522647\pi$
$19$	−0.619718	−0.142173	−0.0710865	+	0.997470i	$0.522647\pi$
$20$	−0.888387	−0.198649
$21$	1.81984	0.397121
$22$	1.00000	0.213201
$23$	−5.60726	−1.16919	−0.584597	−	0.811324i	$-0.698747\pi$
$23$	−5.60726	−1.16919	−0.584597	+	0.811324i	$0.698747\pi$
$24$	0.759376	0.155007
$25$	−4.21077	−0.842154
$26$	6.84788	1.34298
$27$	−4.11836	−0.792579
$28$	2.39649	0.452894
$29$	−3.51875	−0.653416	−0.326708	−	0.945125i	$-0.605939\pi$
$29$	−3.51875	−0.653416	−0.326708	+	0.945125i	$0.605939\pi$
$30$	−0.674620	−0.123168
$31$	−1.51875	−0.272776	−0.136388	−	0.990655i	$-0.543549\pi$
$31$	−1.51875	−0.272776	−0.136388	+	0.990655i	$0.543549\pi$
$32$	1.00000	0.176777
$33$	0.759376	0.132190
$34$	1.00000	0.171499
$35$	−2.12901	−0.359868
$36$	−2.42335	−0.403891
$37$	8.22698	1.35251	0.676253	−	0.736669i	$-0.263602\pi$
$37$	8.22698	1.35251	0.676253	+	0.736669i	$0.263602\pi$
$38$	−0.619718	−0.100531
$39$	5.20012	0.832686
$40$	−0.888387	−0.140466
$41$	−4.88839	−0.763438	−0.381719	−	0.924278i	$-0.624668\pi$
$41$	−4.88839	−0.763438	−0.381719	+	0.924278i	$0.624668\pi$
$42$	1.81984	0.280807
$43$	−3.70267	−0.564651	−0.282326	−	0.959319i	$-0.591106\pi$
$43$	−3.70267	−0.564651	−0.282326	+	0.959319i	$0.591106\pi$
$44$	1.00000	0.150756
$45$	2.15287	0.320931
$46$	−5.60726	−0.826745
$47$	1.11161	0.162145	0.0810727	−	0.996708i	$-0.474165\pi$
$47$	1.11161	0.162145	0.0810727	+	0.996708i	$0.474165\pi$
$48$	0.759376	0.109607
$49$	−1.25683	−0.179547
$50$	−4.21077	−0.595493
$51$	0.759376	0.106334
$52$	6.84788	0.949631
$53$	−7.49746	−1.02985	−0.514927	−	0.857234i	$-0.672181\pi$
$53$	−7.49746	−1.02985	−0.514927	+	0.857234i	$0.672181\pi$
$54$	−4.11836	−0.560438
$55$	−0.888387	−0.119790
$56$	2.39649	0.320245
$57$	−0.470599	−0.0623323
$58$	−3.51875	−0.462035
$59$	−2.31173	−0.300962	−0.150481	−	0.988613i	$-0.548082\pi$
$59$	−2.31173	−0.300962	−0.150481	+	0.988613i	$0.548082\pi$
$60$	−0.674620	−0.0870930
$61$	−12.2307	−1.56598	−0.782992	−	0.622032i	$-0.786308\pi$
$61$	−12.2307	−1.56598	−0.782992	+	0.622032i	$0.786308\pi$
$62$	−1.51875	−0.192882
$63$	−5.80753	−0.731680
$64$	1.00000	0.125000
$65$	−6.08357	−0.754574
$66$	0.759376	0.0934728
$67$	9.07230	1.10836	0.554179	−	0.832398i	$-0.313032\pi$
$67$	9.07230	1.10836	0.554179	+	0.832398i	$0.313032\pi$
$68$	1.00000	0.121268
$69$	−4.25802	−0.512605
$70$	−2.12901	−0.254465
$71$	7.83049	0.929308	0.464654	−	0.885492i	$-0.346179\pi$
$71$	7.83049	0.929308	0.464654	+	0.885492i	$0.346179\pi$
$72$	−2.42335	−0.285594
$73$	7.10590	0.831683	0.415842	−	0.909437i	$-0.363487\pi$
$73$	7.10590	0.831683	0.415842	+	0.909437i	$0.363487\pi$
$74$	8.22698	0.956367
$75$	−3.19756	−0.369222
$76$	−0.619718	−0.0710865
$77$	2.39649	0.273105
$78$	5.20012	0.588798
$79$	−7.86409	−0.884779	−0.442390	−	0.896823i	$-0.645869\pi$
$79$	−7.86409	−0.884779	−0.442390	+	0.896823i	$0.645869\pi$
$80$	−0.888387	−0.0993246
$81$	4.14266	0.460295
$82$	−4.88839	−0.539832
$83$	16.7967	1.84368	0.921840	−	0.387570i	$-0.126685\pi$
$83$	16.7967	1.84368	0.921840	+	0.387570i	$0.126685\pi$
$84$	1.81984	0.198561
$85$	−0.888387	−0.0963591
$86$	−3.70267	−0.399269
$87$	−2.67206	−0.286475
$88$	1.00000	0.106600
$89$	−14.0038	−1.48439	−0.742197	−	0.670181i	$-0.766216\pi$
$89$	−14.0038	−1.48439	−0.742197	+	0.670181i	$0.766216\pi$
$90$	2.15287	0.226932
$91$	16.4109	1.72033
$92$	−5.60726	−0.584597
$93$	−1.15330	−0.119592
$94$	1.11161	0.114654
$95$	0.550549	0.0564851
$96$	0.759376	0.0775035
$97$	1.12601	0.114329	0.0571646	−	0.998365i	$-0.481794\pi$
$97$	1.12601	0.114329	0.0571646	+	0.998365i	$0.481794\pi$
$98$	−1.25683	−0.126959
$99$	−2.42335	−0.243556
$100$	−4.21077	−0.421077
$101$	−8.01440	−0.797463	−0.398731	−	0.917068i	$-0.630549\pi$
$101$	−8.01440	−0.797463	−0.398731	+	0.917068i	$0.630549\pi$
$102$	0.759376	0.0751895
$103$	6.98260	0.688016	0.344008	−	0.938967i	$-0.388215\pi$
$103$	6.98260	0.688016	0.344008	+	0.938967i	$0.388215\pi$
$104$	6.84788	0.671490
$105$	−1.61672	−0.157776
$106$	−7.49746	−0.728217
$107$	−12.7120	−1.22891	−0.614457	−	0.788951i	$-0.710625\pi$
$107$	−12.7120	−1.22891	−0.614457	+	0.788951i	$0.710625\pi$
$108$	−4.11836	−0.396290
$109$	−19.2469	−1.84352	−0.921761	−	0.387758i	$-0.873250\pi$
$109$	−19.2469	−1.84352	−0.921761	+	0.387758i	$0.873250\pi$
$110$	−0.888387	−0.0847043
$111$	6.24737	0.592974
$112$	2.39649	0.226447
$113$	1.72068	0.161868	0.0809341	−	0.996719i	$-0.474210\pi$
$113$	1.72068	0.161868	0.0809341	+	0.996719i	$0.474210\pi$
$114$	−0.470599	−0.0440756
$115$	4.98141	0.464519
$116$	−3.51875	−0.326708
$117$	−16.5948	−1.53419
$118$	−2.31173	−0.212812
$119$	2.39649	0.219686
$120$	−0.674620	−0.0615841
$121$	1.00000	0.0909091
$122$	−12.2307	−1.10732
$123$	−3.71213	−0.334711
$124$	−1.51875	−0.136388
$125$	8.18272	0.731885
$126$	−5.80753	−0.517376
$127$	20.7333	1.83978	0.919890	−	0.392176i	$-0.128278\pi$
$127$	20.7333	1.83978	0.919890	+	0.392176i	$0.128278\pi$
$128$	1.00000	0.0883883
$129$	−2.81172	−0.247558
$130$	−6.08357	−0.533564
$131$	−0.446454	−0.0390068	−0.0195034	−	0.999810i	$-0.506209\pi$
$131$	−0.446454	−0.0390068	−0.0195034	+	0.999810i	$0.506209\pi$
$132$	0.759376	0.0660952
$133$	−1.48515	−0.128779
$134$	9.07230	0.783727
$135$	3.65870	0.314891
$136$	1.00000	0.0857493
$137$	15.4714	1.32181	0.660903	−	0.750471i	$-0.270173\pi$
$137$	15.4714	1.32181	0.660903	+	0.750471i	$0.270173\pi$
$138$	−4.25802	−0.362467
$139$	−5.69577	−0.483109	−0.241554	−	0.970387i	$-0.577657\pi$
$139$	−5.69577	−0.483109	−0.241554	+	0.970387i	$0.577657\pi$
$140$	−2.12901	−0.179934
$141$	0.844133	0.0710888
$142$	7.83049	0.657120
$143$	6.84788	0.572649
$144$	−2.42335	−0.201946
$145$	3.12601	0.259601
$146$	7.10590	0.588089
$147$	−0.954408	−0.0787183
$148$	8.22698	0.676253
$149$	14.5089	1.18861	0.594306	−	0.804239i	$-0.297427\pi$
$149$	14.5089	1.18861	0.594306	+	0.804239i	$0.297427\pi$
$150$	−3.19756	−0.261080
$151$	19.7495	1.60719	0.803595	−	0.595177i	$-0.202918\pi$
$151$	19.7495	1.60719	0.803595	+	0.595177i	$0.202918\pi$
$152$	−0.619718	−0.0502657
$153$	−2.42335	−0.195916
$154$	2.39649	0.193115
$155$	1.34924	0.108374
$156$	5.20012	0.416343
$157$	−4.56975	−0.364706	−0.182353	−	0.983233i	$-0.558371\pi$
$157$	−4.56975	−0.364706	−0.182353	+	0.983233i	$0.558371\pi$
$158$	−7.86409	−0.625634
$159$	−5.69339	−0.451515
$160$	−0.888387	−0.0702331
$161$	−13.4377	−1.05904
$162$	4.14266	0.325478
$163$	−8.94448	−0.700586	−0.350293	−	0.936640i	$-0.613918\pi$
$163$	−8.94448	−0.700586	−0.350293	+	0.936640i	$0.613918\pi$
$164$	−4.88839	−0.381719
$165$	−0.674620	−0.0525191
$166$	16.7967	1.30368
$167$	−3.61416	−0.279672	−0.139836	−	0.990175i	$-0.544658\pi$
$167$	−3.61416	−0.279672	−0.139836	+	0.990175i	$0.544658\pi$
$168$	1.81984	0.140404
$169$	33.8935	2.60719
$170$	−0.888387	−0.0681361
$171$	1.50179	0.114845
$172$	−3.70267	−0.282326
$173$	−7.49746	−0.570021	−0.285011	−	0.958524i	$-0.591997\pi$
$173$	−7.49746	−0.570021	−0.285011	+	0.958524i	$0.591997\pi$
$174$	−2.67206	−0.202568
$175$	−10.0911	−0.762813
$176$	1.00000	0.0753778
$177$	−1.75548	−0.131950
$178$	−14.0038	−1.04963
$179$	23.3840	1.74780	0.873902	−	0.486102i	$-0.161581\pi$
$179$	23.3840	1.74780	0.873902	+	0.486102i	$0.161581\pi$
$180$	2.15287	0.160465
$181$	−3.55236	−0.264045	−0.132022	−	0.991247i	$-0.542147\pi$
$181$	−3.55236	−0.264045	−0.132022	+	0.991247i	$0.542147\pi$
$182$	16.4109	1.21646
$183$	−9.28773	−0.686568
$184$	−5.60726	−0.413373
$185$	−7.30874	−0.537349
$186$	−1.15330	−0.0845644
$187$	1.00000	0.0731272
$188$	1.11161	0.0810727
$189$	−9.86962	−0.717909
$190$	0.550549	0.0399410
$191$	13.6409	0.987018	0.493509	−	0.869741i	$-0.335714\pi$
$191$	13.6409	0.987018	0.493509	+	0.869741i	$0.335714\pi$
$192$	0.759376	0.0548033
$193$	19.5524	1.40741	0.703705	−	0.710493i	$-0.251528\pi$
$193$	19.5524	1.40741	0.703705	+	0.710493i	$0.251528\pi$
$194$	1.12601	0.0808430
$195$	−4.61972	−0.330825
$196$	−1.25683	−0.0897737
$197$	−9.29553	−0.662279	−0.331139	−	0.943582i	$-0.607433\pi$
$197$	−9.29553	−0.662279	−0.331139	+	0.943582i	$0.607433\pi$
$198$	−2.42335	−0.172220
$199$	−17.9751	−1.27422	−0.637110	−	0.770773i	$-0.719870\pi$
$199$	−17.9751	−1.27422	−0.637110	+	0.770773i	$0.719870\pi$
$200$	−4.21077	−0.297746
$201$	6.88929	0.485933
$202$	−8.01440	−0.563891
$203$	−8.43266	−0.591857
$204$	0.759376	0.0531670
$205$	4.34278	0.303313
$206$	6.98260	0.486501
$207$	13.5883	0.944455
$208$	6.84788	0.474815
$209$	−0.619718	−0.0428668
$210$	−1.61672	−0.111564
$211$	18.2520	1.25652	0.628261	−	0.778003i	$-0.283767\pi$
$211$	18.2520	1.25652	0.628261	+	0.778003i	$0.283767\pi$
$212$	−7.49746	−0.514927
$213$	5.94629	0.407433
$214$	−12.7120	−0.868973
$215$	3.28940	0.224335
$216$	−4.11836	−0.280219
$217$	−3.63968	−0.247077
$218$	−19.2469	−1.30357
$219$	5.39606	0.364632
$220$	−0.888387	−0.0598950
$221$	6.84788	0.460639
$222$	6.24737	0.419296
$223$	−14.2057	−0.951283	−0.475641	−	0.879639i	$-0.657784\pi$
$223$	−14.2057	−0.951283	−0.475641	+	0.879639i	$0.657784\pi$
$224$	2.39649	0.160122
$225$	10.2042	0.680277
$226$	1.72068	0.114458
$227$	12.3654	0.820724	0.410362	−	0.911923i	$-0.365402\pi$
$227$	12.3654	0.820724	0.410362	+	0.911923i	$0.365402\pi$
$228$	−0.470599	−0.0311662
$229$	6.34653	0.419390	0.209695	−	0.977767i	$-0.432753\pi$
$229$	6.34653	0.419390	0.209695	+	0.977767i	$0.432753\pi$
$230$	4.98141	0.328465
$231$	1.81984	0.119737
$232$	−3.51875	−0.231017
$233$	−7.48515	−0.490368	−0.245184	−	0.969477i	$-0.578848\pi$
$233$	−7.48515	−0.490368	−0.245184	+	0.969477i	$0.578848\pi$
$234$	−16.5948	−1.08484
$235$	−0.987542	−0.0644202
$236$	−2.31173	−0.150481
$237$	−5.97181	−0.387910
$238$	2.39649	0.155341
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−0.674620	−0.0435465
$241$	24.2094	1.55947	0.779733	−	0.626112i	$-0.215355\pi$
$241$	24.2094	1.55947	0.779733	+	0.626112i	$0.215355\pi$
$242$	1.00000	0.0642824
$243$	15.5009	0.994385
$244$	−12.2307	−0.782992
$245$	1.11655	0.0713339
$246$	−3.71213	−0.236676
$247$	−4.24375	−0.270024
$248$	−1.51875	−0.0964409
$249$	12.7550	0.808318
$250$	8.18272	0.517521
$251$	−3.53225	−0.222954	−0.111477	−	0.993767i	$-0.535558\pi$
$251$	−3.53225	−0.222954	−0.111477	+	0.993767i	$0.535558\pi$
$252$	−5.80753	−0.365840
$253$	−5.60726	−0.352525
$254$	20.7333	1.30092
$255$	−0.674620	−0.0422463
$256$	1.00000	0.0625000
$257$	−0.140847	−0.00878581	−0.00439291	−	0.999990i	$-0.501398\pi$
$257$	−0.140847	−0.00878581	−0.00439291	+	0.999990i	$0.501398\pi$
$258$	−2.81172	−0.175050
$259$	19.7159	1.22508
$260$	−6.08357	−0.377287
$261$	8.52716	0.527818
$262$	−0.446454	−0.0275820
$263$	−9.41645	−0.580643	−0.290322	−	0.956929i	$-0.593762\pi$
$263$	−9.41645	−0.580643	−0.290322	+	0.956929i	$0.593762\pi$
$264$	0.759376	0.0467364
$265$	6.66064	0.409160
$266$	−1.48515	−0.0910602
$267$	−10.6341	−0.650797
$268$	9.07230	0.554179
$269$	−20.2270	−1.23326	−0.616630	−	0.787253i	$-0.711503\pi$
$269$	−20.2270	−1.23326	−0.616630	+	0.787253i	$0.711503\pi$
$270$	3.65870	0.222661
$271$	13.4975	0.819912	0.409956	−	0.912105i	$-0.365544\pi$
$271$	13.4975	0.819912	0.409956	+	0.912105i	$0.365544\pi$
$272$	1.00000	0.0606339
$273$	12.4620	0.754237
$274$	15.4714	0.934659
$275$	−4.21077	−0.253919
$276$	−4.25802	−0.256303
$277$	−22.5448	−1.35459	−0.677294	−	0.735712i	$-0.736848\pi$
$277$	−22.5448	−1.35459	−0.677294	+	0.735712i	$0.736848\pi$
$278$	−5.69577	−0.341609
$279$	3.68047	0.220344
$280$	−2.12901	−0.127233
$281$	−13.3030	−0.793592	−0.396796	−	0.917907i	$-0.629878\pi$
$281$	−13.3030	−0.793592	−0.396796	+	0.917907i	$0.629878\pi$
$282$	0.844133	0.0502674
$283$	6.14460	0.365258	0.182629	−	0.983182i	$-0.441539\pi$
$283$	6.14460	0.365258	0.182629	+	0.983182i	$0.441539\pi$
$284$	7.83049	0.464654
$285$	0.418074	0.0247645
$286$	6.84788	0.404924
$287$	−11.7150	−0.691513
$288$	−2.42335	−0.142797
$289$	1.00000	0.0588235
$290$	3.12601	0.183566
$291$	0.855067	0.0501249
$292$	7.10590	0.415842
$293$	−28.4609	−1.66270	−0.831350	−	0.555749i	$-0.812432\pi$
$293$	−28.4609	−1.66270	−0.831350	+	0.555749i	$0.812432\pi$
$294$	−0.954408	−0.0556622
$295$	2.05371	0.119572
$296$	8.22698	0.478183
$297$	−4.11836	−0.238972
$298$	14.5089	0.840476
$299$	−38.3979	−2.22061
$300$	−3.19756	−0.184611
$301$	−8.87340	−0.511454
$302$	19.7495	1.13645
$303$	−6.08595	−0.349628
$304$	−0.619718	−0.0355432
$305$	10.8656	0.622163
$306$	−2.42335	−0.138534
$307$	22.3741	1.27696	0.638480	−	0.769639i	$-0.279564\pi$
$307$	22.3741	1.27696	0.638480	+	0.769639i	$0.279564\pi$
$308$	2.39649	0.136553
$309$	5.30242	0.301644
$310$	1.34924	0.0766317
$311$	22.7444	1.28972	0.644858	−	0.764302i	$-0.276916\pi$
$311$	22.7444	1.28972	0.644858	+	0.764302i	$0.276916\pi$
$312$	5.20012	0.294399
$313$	3.86528	0.218479	0.109239	−	0.994015i	$-0.465159\pi$
$313$	3.86528	0.218479	0.109239	+	0.994015i	$0.465159\pi$
$314$	−4.56975	−0.257886
$315$	5.15933	0.290695
$316$	−7.86409	−0.442390
$317$	−1.48396	−0.0833474	−0.0416737	−	0.999131i	$-0.513269\pi$
$317$	−1.48396	−0.0833474	−0.0416737	+	0.999131i	$0.513269\pi$
$318$	−5.69339	−0.319269
$319$	−3.51875	−0.197012
$320$	−0.888387	−0.0496623
$321$	−9.65317	−0.538788
$322$	−13.4377	−0.748856
$323$	−0.619718	−0.0344820
$324$	4.14266	0.230148
$325$	−28.8349	−1.59947
$326$	−8.94448	−0.495389
$327$	−14.6157	−0.808248
$328$	−4.88839	−0.269916
$329$	2.66397	0.146869
$330$	−0.674620	−0.0371366
$331$	−2.59105	−0.142417	−0.0712085	−	0.997461i	$-0.522686\pi$
$331$	−2.59105	−0.142417	−0.0712085	+	0.997461i	$0.522686\pi$
$332$	16.7967	0.921840
$333$	−19.9368	−1.09253
$334$	−3.61416	−0.197758
$335$	−8.05971	−0.440349
$336$	1.81984	0.0992803
$337$	27.0330	1.47258	0.736290	−	0.676666i	$-0.236576\pi$
$337$	27.0330	1.47258	0.736290	+	0.676666i	$0.236576\pi$
$338$	33.8935	1.84356
$339$	1.30665	0.0709672
$340$	−0.888387	−0.0481795
$341$	−1.51875	−0.0822451
$342$	1.50179	0.0812076
$343$	−19.7874	−1.06842
$344$	−3.70267	−0.199634
$345$	3.78277	0.203657
$346$	−7.49746	−0.403066
$347$	−12.7120	−0.682415	−0.341207	−	0.939988i	$-0.610836\pi$
$347$	−12.7120	−0.682415	−0.341207	+	0.939988i	$0.610836\pi$
$348$	−2.67206	−0.143237
$349$	−14.6235	−0.782776	−0.391388	−	0.920226i	$-0.628005\pi$
$349$	−14.6235	−0.782776	−0.391388	+	0.920226i	$0.628005\pi$
$350$	−10.0911	−0.539390
$351$	−28.2021	−1.50531
$352$	1.00000	0.0533002
$353$	−23.5679	−1.25440	−0.627198	−	0.778860i	$-0.715798\pi$
$353$	−23.5679	−1.25440	−0.627198	+	0.778860i	$0.715798\pi$
$354$	−1.75548	−0.0933025
$355$	−6.95650	−0.369213
$356$	−14.0038	−0.742197
$357$	1.81984	0.0963161
$358$	23.3840	1.23588
$359$	−28.1746	−1.48700	−0.743500	−	0.668736i	$-0.766836\pi$
$359$	−28.1746	−1.48700	−0.743500	+	0.668736i	$0.766836\pi$
$360$	2.15287	0.113466
$361$	−18.6160	−0.979787
$362$	−3.55236	−0.186708
$363$	0.759376	0.0398569
$364$	16.4109	0.860164
$365$	−6.31279	−0.330427
$366$	−9.28773	−0.485477
$367$	−2.39274	−0.124900	−0.0624500	−	0.998048i	$-0.519891\pi$
$367$	−2.39274	−0.124900	−0.0624500	+	0.998048i	$0.519891\pi$
$368$	−5.60726	−0.292299
$369$	11.8463	0.616692
$370$	−7.30874	−0.379963
$371$	−17.9676	−0.932830
$372$	−1.15330	−0.0597961
$373$	15.0737	0.780485	0.390243	−	0.920712i	$-0.372391\pi$
$373$	15.0737	0.780485	0.390243	+	0.920712i	$0.372391\pi$
$374$	1.00000	0.0517088
$375$	6.21377	0.320878
$376$	1.11161	0.0573271
$377$	−24.0960	−1.24101
$378$	−9.86962	−0.507638
$379$	36.3313	1.86621	0.933106	−	0.359602i	$-0.117088\pi$
$379$	36.3313	1.86621	0.933106	+	0.359602i	$0.117088\pi$
$380$	0.550549	0.0282426
$381$	15.7444	0.806608
$382$	13.6409	0.697927
$383$	35.3915	1.80842	0.904211	−	0.427085i	$-0.140459\pi$
$383$	35.3915	1.80842	0.904211	+	0.427085i	$0.140459\pi$
$384$	0.759376	0.0387518
$385$	−2.12901	−0.108504
$386$	19.5524	0.995189
$387$	8.97284	0.456115
$388$	1.12601	0.0571646
$389$	8.91628	0.452073	0.226037	−	0.974119i	$-0.427423\pi$
$389$	8.91628	0.452073	0.226037	+	0.974119i	$0.427423\pi$
$390$	−4.61972	−0.233928
$391$	−5.60726	−0.283571
$392$	−1.25683	−0.0634796
$393$	−0.339026	−0.0171016
$394$	−9.29553	−0.468302
$395$	6.98635	0.351522
$396$	−2.42335	−0.121778
$397$	−7.60351	−0.381609	−0.190805	−	0.981628i	$-0.561110\pi$
$397$	−7.60351	−0.381609	−0.190805	+	0.981628i	$0.561110\pi$
$398$	−17.9751	−0.901010
$399$	−1.12779	−0.0564599
$400$	−4.21077	−0.210538
$401$	−23.9940	−1.19820	−0.599102	−	0.800673i	$-0.704475\pi$
$401$	−23.9940	−1.19820	−0.599102	+	0.800673i	$0.704475\pi$
$402$	6.88929	0.343607
$403$	−10.4002	−0.518073
$404$	−8.01440	−0.398731
$405$	−3.68028	−0.182875
$406$	−8.43266	−0.418506
$407$	8.22698	0.407796
$408$	0.759376	0.0375947
$409$	−19.3780	−0.958182	−0.479091	−	0.877765i	$-0.659034\pi$
$409$	−19.3780	−0.958182	−0.479091	+	0.877765i	$0.659034\pi$
$410$	4.34278	0.214474
$411$	11.7486	0.579515
$412$	6.98260	0.344008
$413$	−5.54005	−0.272608
$414$	13.5883	0.667830
$415$	−14.9220	−0.732492
$416$	6.84788	0.335745
$417$	−4.32523	−0.211807
$418$	−0.619718	−0.0303114
$419$	−12.6410	−0.617554	−0.308777	−	0.951134i	$-0.599920\pi$
$419$	−12.6410	−0.617554	−0.308777	+	0.951134i	$0.599920\pi$
$420$	−1.61672	−0.0788879
$421$	19.7171	0.960951	0.480475	−	0.877008i	$-0.340464\pi$
$421$	19.7171	0.960951	0.480475	+	0.877008i	$0.340464\pi$
$422$	18.2520	0.888495
$423$	−2.69383	−0.130978
$424$	−7.49746	−0.364109
$425$	−4.21077	−0.204252
$426$	5.94629	0.288099
$427$	−29.3108	−1.41845
$428$	−12.7120	−0.614457
$429$	5.20012	0.251064
$430$	3.28940	0.158629
$431$	13.4340	0.647093	0.323546	−	0.946212i	$-0.395125\pi$
$431$	13.4340	0.647093	0.323546	+	0.946212i	$0.395125\pi$
$432$	−4.11836	−0.198145
$433$	−9.82721	−0.472265	−0.236133	−	0.971721i	$-0.575880\pi$
$433$	−9.82721	−0.472265	−0.236133	+	0.971721i	$0.575880\pi$
$434$	−3.63968	−0.174710
$435$	2.37382	0.113816
$436$	−19.2469	−0.921761
$437$	3.47492	0.166228
$438$	5.39606	0.257834
$439$	−23.8560	−1.13859	−0.569293	−	0.822135i	$-0.692783\pi$
$439$	−23.8560	−1.13859	−0.569293	+	0.822135i	$0.692783\pi$
$440$	−0.888387	−0.0423522
$441$	3.04574	0.145035
$442$	6.84788	0.325721
$443$	−26.7444	−1.27066	−0.635332	−	0.772239i	$-0.719137\pi$
$443$	−26.7444	−1.27066	−0.635332	+	0.772239i	$0.719137\pi$
$444$	6.24737	0.296487
$445$	12.4407	0.589748
$446$	−14.2057	−0.672658
$447$	11.0177	0.521119
$448$	2.39649	0.113224
$449$	15.9652	0.753445	0.376722	−	0.926326i	$-0.377051\pi$
$449$	15.9652	0.753445	0.376722	+	0.926326i	$0.377051\pi$
$450$	10.2042	0.481029
$451$	−4.88839	−0.230185
$452$	1.72068	0.0809341
$453$	14.9973	0.704634
$454$	12.3654	0.580339
$455$	−14.5792	−0.683484
$456$	−0.470599	−0.0220378
$457$	2.36545	0.110651	0.0553255	−	0.998468i	$-0.482380\pi$
$457$	2.36545	0.110651	0.0553255	+	0.998468i	$0.482380\pi$
$458$	6.34653	0.296554
$459$	−4.11836	−0.192229
$460$	4.98141	0.232260
$461$	−15.9874	−0.744607	−0.372304	−	0.928111i	$-0.621432\pi$
$461$	−15.9874	−0.744607	−0.372304	+	0.928111i	$0.621432\pi$
$462$	1.81984	0.0846665
$463$	−21.1920	−0.984877	−0.492439	−	0.870347i	$-0.663894\pi$
$463$	−21.1920	−0.984877	−0.492439	+	0.870347i	$0.663894\pi$
$464$	−3.51875	−0.163354
$465$	1.02458	0.0475138
$466$	−7.48515	−0.346743
$467$	15.4627	0.715527	0.357763	−	0.933812i	$-0.383539\pi$
$467$	15.4627	0.715527	0.357763	+	0.933812i	$0.383539\pi$
$468$	−16.5948	−0.767095
$469$	21.7417	1.00394
$470$	−0.987542	−0.0455519
$471$	−3.47016	−0.159897
$472$	−2.31173	−0.106406
$473$	−3.70267	−0.170249
$474$	−5.97181	−0.274294
$475$	2.60949	0.119732
$476$	2.39649	0.109843
$477$	18.1689	0.831899
$478$	0	0
$479$	−14.3453	−0.655455	−0.327728	−	0.944772i	$-0.606283\pi$
$479$	−14.3453	−0.655455	−0.327728	+	0.944772i	$0.606283\pi$
$480$	−0.674620	−0.0307920
$481$	56.3374	2.56876
$482$	24.2094	1.10271
$483$	−10.2043	−0.464312
$484$	1.00000	0.0454545
$485$	−1.00033	−0.0454229
$486$	15.5009	0.703136
$487$	24.1173	1.09286	0.546430	−	0.837505i	$-0.315987\pi$
$487$	24.1173	1.09286	0.546430	+	0.837505i	$0.315987\pi$
$488$	−12.2307	−0.553659
$489$	−6.79222	−0.307155
$490$	1.11655	0.0504407
$491$	8.92080	0.402590	0.201295	−	0.979531i	$-0.435485\pi$
$491$	8.92080	0.402590	0.201295	+	0.979531i	$0.435485\pi$
$492$	−3.71213	−0.167356
$493$	−3.51875	−0.158477
$494$	−4.24375	−0.190936
$495$	2.15287	0.0967643
$496$	−1.51875	−0.0681940
$497$	18.7657	0.841756
$498$	12.7550	0.571567
$499$	−13.5599	−0.607023	−0.303511	−	0.952828i	$-0.598159\pi$
$499$	−13.5599	−0.607023	−0.303511	+	0.952828i	$0.598159\pi$
$500$	8.18272	0.365943
$501$	−2.74451	−0.122615
$502$	−3.53225	−0.157652
$503$	−7.40158	−0.330020	−0.165010	−	0.986292i	$-0.552766\pi$
$503$	−7.40158	−0.330020	−0.165010	+	0.986292i	$0.552766\pi$
$504$	−5.80753	−0.258688
$505$	7.11988	0.316831
$506$	−5.60726	−0.249273
$507$	25.7379	1.14306
$508$	20.7333	0.919890
$509$	24.5290	1.08723	0.543614	−	0.839336i	$-0.317056\pi$
$509$	24.5290	1.08723	0.543614	+	0.839336i	$0.317056\pi$
$510$	−0.674620	−0.0298727
$511$	17.0292	0.753329
$512$	1.00000	0.0441942
$513$	2.55222	0.112683
$514$	−0.140847	−0.00621251
$515$	−6.20325	−0.273348
$516$	−2.81172	−0.123779
$517$	1.11161	0.0488887
$518$	19.7159	0.866266
$519$	−5.69339	−0.249912
$520$	−6.08357	−0.266782
$521$	31.0798	1.36163	0.680815	−	0.732455i	$-0.261626\pi$
$521$	31.0798	1.36163	0.680815	+	0.732455i	$0.261626\pi$
$522$	8.52716	0.373224
$523$	−25.9121	−1.13306	−0.566528	−	0.824042i	$-0.691714\pi$
$523$	−25.9121	−1.13306	−0.566528	+	0.824042i	$0.691714\pi$
$524$	−0.446454	−0.0195034
$525$	−7.66292	−0.334437
$526$	−9.41645	−0.410577
$527$	−1.51875	−0.0661579
$528$	0.759376	0.0330476
$529$	8.44137	0.367016
$530$	6.66064	0.289320
$531$	5.60214	0.243112
$532$	−1.48515	−0.0643893
$533$	−33.4751	−1.44997
$534$	−10.6341	−0.460183
$535$	11.2931	0.488245
$536$	9.07230	0.391864
$537$	17.7573	0.766283
$538$	−20.2270	−0.872047
$539$	−1.25683	−0.0541356
$540$	3.65870	0.157445
$541$	−3.39753	−0.146071	−0.0730356	−	0.997329i	$-0.523269\pi$
$541$	−3.39753	−0.146071	−0.0730356	+	0.997329i	$0.523269\pi$
$542$	13.4975	0.579765
$543$	−2.69758	−0.115764
$544$	1.00000	0.0428746
$545$	17.0987	0.732429
$546$	12.4620	0.533326
$547$	−26.7870	−1.14533	−0.572664	−	0.819790i	$-0.694090\pi$
$547$	−26.7870	−1.14533	−0.572664	+	0.819790i	$0.694090\pi$
$548$	15.4714	0.660903
$549$	29.6393	1.26497
$550$	−4.21077	−0.179548
$551$	2.18063	0.0928981
$552$	−4.25802	−0.181233
$553$	−18.8462	−0.801423
$554$	−22.5448	−0.957838
$555$	−5.55008	−0.235588
$556$	−5.69577	−0.241554
$557$	21.8560	0.926069	0.463034	−	0.886340i	$-0.346761\pi$
$557$	21.8560	0.926069	0.463034	+	0.886340i	$0.346761\pi$
$558$	3.68047	0.155807
$559$	−25.3554	−1.07242
$560$	−2.12901	−0.0899671
$561$	0.759376	0.0320609
$562$	−13.3030	−0.561154
$563$	−10.8050	−0.455377	−0.227688	−	0.973734i	$-0.573117\pi$
$563$	−10.8050	−0.455377	−0.227688	+	0.973734i	$0.573117\pi$
$564$	0.844133	0.0355444
$565$	−1.52863	−0.0643100
$566$	6.14460	0.258277
$567$	9.92784	0.416930
$568$	7.83049	0.328560
$569$	−9.63968	−0.404116	−0.202058	−	0.979374i	$-0.564763\pi$
$569$	−9.63968	−0.404116	−0.202058	+	0.979374i	$0.564763\pi$
$570$	0.418074	0.0175112
$571$	19.4915	0.815692	0.407846	−	0.913051i	$-0.366280\pi$
$571$	19.4915	0.815692	0.407846	+	0.913051i	$0.366280\pi$
$572$	6.84788	0.286324
$573$	10.3586	0.432735
$574$	−11.7150	−0.488974
$575$	23.6109	0.984642
$576$	−2.42335	−0.100973
$577$	−33.2258	−1.38321	−0.691604	−	0.722277i	$-0.743095\pi$
$577$	−33.2258	−1.38321	−0.691604	+	0.722277i	$0.743095\pi$
$578$	1.00000	0.0415945
$579$	14.8476	0.617045
$580$	3.12601	0.129801
$581$	40.2532	1.66998
$582$	0.855067	0.0354437
$583$	−7.49746	−0.310513
$584$	7.10590	0.294044
$585$	14.7426	0.609531
$586$	−28.4609	−1.17571
$587$	4.64477	0.191710	0.0958550	−	0.995395i	$-0.469441\pi$
$587$	4.64477	0.191710	0.0958550	+	0.995395i	$0.469441\pi$
$588$	−0.954408	−0.0393591
$589$	0.941198	0.0387814
$590$	2.05371	0.0845501
$591$	−7.05880	−0.290360
$592$	8.22698	0.338127
$593$	32.8542	1.34916	0.674580	−	0.738202i	$-0.264325\pi$
$593$	32.8542	1.34916	0.674580	+	0.738202i	$0.264325\pi$
$594$	−4.11836	−0.168978
$595$	−2.12901	−0.0872809
$596$	14.5089	0.594306
$597$	−13.6499	−0.558651
$598$	−38.3979	−1.57021
$599$	32.1770	1.31472	0.657359	−	0.753578i	$-0.271674\pi$
$599$	32.1770	1.31472	0.657359	+	0.753578i	$0.271674\pi$
$600$	−3.19756	−0.130540
$601$	11.7109	0.477699	0.238850	−	0.971057i	$-0.423230\pi$
$601$	11.7109	0.477699	0.238850	+	0.971057i	$0.423230\pi$
$602$	−8.87340	−0.361653
$603$	−21.9853	−0.895312
$604$	19.7495	0.803595
$605$	−0.888387	−0.0361181
$606$	−6.08595	−0.247225
$607$	5.11674	0.207682	0.103841	−	0.994594i	$-0.466887\pi$
$607$	5.11674	0.207682	0.103841	+	0.994594i	$0.466887\pi$
$608$	−0.619718	−0.0251329
$609$	−6.40356	−0.259485
$610$	10.8656	0.439936
$611$	7.61220	0.307957
$612$	−2.42335	−0.0979580
$613$	−29.0914	−1.17499	−0.587494	−	0.809228i	$-0.699886\pi$
$613$	−29.0914	−1.17499	−0.587494	+	0.809228i	$0.699886\pi$
$614$	22.3741	0.902947
$615$	3.29780	0.132980
$616$	2.39649	0.0965574
$617$	31.5476	1.27006	0.635028	−	0.772489i	$-0.280988\pi$
$617$	31.5476	1.27006	0.635028	+	0.772489i	$0.280988\pi$
$618$	5.30242	0.213295
$619$	−17.0462	−0.685144	−0.342572	−	0.939491i	$-0.611298\pi$
$619$	−17.0462	−0.685144	−0.342572	+	0.939491i	$0.611298\pi$
$620$	1.34924	0.0541868
$621$	23.0927	0.926679
$622$	22.7444	0.911967
$623$	−33.5599	−1.34455
$624$	5.20012	0.208171
$625$	13.7844	0.551377
$626$	3.86528	0.154488
$627$	−0.470599	−0.0187939
$628$	−4.56975	−0.182353
$629$	8.22698	0.328031
$630$	5.15933	0.205553
$631$	19.8929	0.791923	0.395962	−	0.918267i	$-0.370411\pi$
$631$	19.8929	0.791923	0.395962	+	0.918267i	$0.370411\pi$
$632$	−7.86409	−0.312817
$633$	13.8602	0.550892
$634$	−1.48396	−0.0589355
$635$	−18.4192	−0.730942
$636$	−5.69339	−0.225758
$637$	−8.60664	−0.341007
$638$	−3.51875	−0.139309
$639$	−18.9760	−0.750679
$640$	−0.888387	−0.0351166
$641$	27.1797	1.07353	0.536767	−	0.843730i	$-0.319645\pi$
$641$	27.1797	1.07353	0.536767	+	0.843730i	$0.319645\pi$
$642$	−9.65317	−0.380980
$643$	45.9400	1.81170	0.905848	−	0.423602i	$-0.139235\pi$
$643$	45.9400	1.81170	0.905848	+	0.423602i	$0.139235\pi$
$644$	−13.4377	−0.529521
$645$	2.49789	0.0983544
$646$	−0.619718	−0.0243825
$647$	31.4986	1.23834	0.619170	−	0.785257i	$-0.287469\pi$
$647$	31.4986	1.23834	0.619170	+	0.785257i	$0.287469\pi$
$648$	4.14266	0.162739
$649$	−2.31173	−0.0907435
$650$	−28.8349	−1.13100
$651$	−2.76388	−0.108325
$652$	−8.94448	−0.350293
$653$	0.573506	0.0224430	0.0112215	−	0.999937i	$-0.496428\pi$
$653$	0.573506	0.0224430	0.0112215	+	0.999937i	$0.496428\pi$
$654$	−14.6157	−0.571518
$655$	0.396623	0.0154974
$656$	−4.88839	−0.190859
$657$	−17.2201	−0.671819
$658$	2.66397	0.103852
$659$	10.5873	0.412423	0.206211	−	0.978507i	$-0.433887\pi$
$659$	10.5873	0.412423	0.206211	+	0.978507i	$0.433887\pi$
$660$	−0.674620	−0.0262595
$661$	16.9427	0.658995	0.329497	−	0.944156i	$-0.393121\pi$
$661$	16.9427	0.658995	0.329497	+	0.944156i	$0.393121\pi$
$662$	−2.59105	−0.100704
$663$	5.20012	0.201956
$664$	16.7967	0.651840
$665$	1.31939	0.0511636
$666$	−19.9368	−0.772536
$667$	19.7306	0.763970
$668$	−3.61416	−0.139836
$669$	−10.7875	−0.417067
$670$	−8.05971	−0.311374
$671$	−12.2307	−0.472162
$672$	1.81984	0.0702018
$673$	−9.98711	−0.384975	−0.192487	−	0.981299i	$-0.561655\pi$
$673$	−9.98711	−0.384975	−0.192487	+	0.981299i	$0.561655\pi$
$674$	27.0330	1.04127
$675$	17.3415	0.667474
$676$	33.8935	1.30360
$677$	−33.6610	−1.29370	−0.646848	−	0.762619i	$-0.723913\pi$
$677$	−33.6610	−1.29370	−0.646848	+	0.762619i	$0.723913\pi$
$678$	1.30665	0.0501814
$679$	2.69848	0.103558
$680$	−0.888387	−0.0340681
$681$	9.39003	0.359827
$682$	−1.51875	−0.0581560
$683$	48.1026	1.84060	0.920298	−	0.391219i	$-0.127947\pi$
$683$	48.1026	1.84060	0.920298	+	0.391219i	$0.127947\pi$
$684$	1.50179	0.0574224
$685$	−13.7445	−0.525152
$686$	−19.7874	−0.755487
$687$	4.81940	0.183872
$688$	−3.70267	−0.141163
$689$	−51.3417	−1.95596
$690$	3.78277	0.144008
$691$	−1.42649	−0.0542664	−0.0271332	−	0.999632i	$-0.508638\pi$
$691$	−1.42649	−0.0542664	−0.0271332	+	0.999632i	$0.508638\pi$
$692$	−7.49746	−0.285011
$693$	−5.80753	−0.220610
$694$	−12.7120	−0.482540
$695$	5.06004	0.191938
$696$	−2.67206	−0.101284
$697$	−4.88839	−0.185161
$698$	−14.6235	−0.553506
$699$	−5.68404	−0.214990
$700$	−10.0911	−0.381407
$701$	18.8393	0.711552	0.355776	−	0.934571i	$-0.384217\pi$
$701$	18.8393	0.711552	0.355776	+	0.934571i	$0.384217\pi$
$702$	−28.2021	−1.06442
$703$	−5.09840	−0.192290
$704$	1.00000	0.0376889
$705$	−0.749916	−0.0282435
$706$	−23.5679	−0.886991
$707$	−19.2064	−0.722332
$708$	−1.75548	−0.0659748
$709$	−28.3614	−1.06513	−0.532567	−	0.846388i	$-0.678773\pi$
$709$	−28.3614	−1.06513	−0.532567	+	0.846388i	$0.678773\pi$
$710$	−6.95650	−0.261073
$711$	19.0574	0.714709
$712$	−14.0038	−0.524813
$713$	8.51604	0.318928
$714$	1.81984	0.0681057
$715$	−6.08357	−0.227513
$716$	23.3840	0.873902
$717$	0	0
$718$	−28.1746	−1.05147
$719$	−43.5800	−1.62526	−0.812629	−	0.582781i	$-0.801965\pi$
$719$	−43.5800	−1.62526	−0.812629	+	0.582781i	$0.801965\pi$
$720$	2.15287	0.0802327
$721$	16.7337	0.623197
$722$	−18.6160	−0.692814
$723$	18.3841	0.683711
$724$	−3.55236	−0.132022
$725$	14.8167	0.550277
$726$	0.759376	0.0281831
$727$	−6.35676	−0.235759	−0.117880	−	0.993028i	$-0.537610\pi$
$727$	−6.35676	−0.235759	−0.117880	+	0.993028i	$0.537610\pi$
$728$	16.4109	0.608228
$729$	−0.656937	−0.0243310
$730$	−6.31279	−0.233647
$731$	−3.70267	−0.136948
$732$	−9.28773	−0.343284
$733$	26.7399	0.987660	0.493830	−	0.869559i	$-0.335597\pi$
$733$	26.7399	0.987660	0.493830	+	0.869559i	$0.335597\pi$
$734$	−2.39274	−0.0883176
$735$	0.847884	0.0312747
$736$	−5.60726	−0.206686
$737$	9.07230	0.334182
$738$	11.8463	0.436067
$739$	−7.60607	−0.279794	−0.139897	−	0.990166i	$-0.544677\pi$
$739$	−7.60607	−0.279794	−0.139897	+	0.990166i	$0.544677\pi$
$740$	−7.30874	−0.268675
$741$	−3.22261	−0.118385
$742$	−17.9676	−0.659611
$743$	−9.82359	−0.360393	−0.180196	−	0.983631i	$-0.557673\pi$
$743$	−9.82359	−0.360393	−0.180196	+	0.983631i	$0.557673\pi$
$744$	−1.15330	−0.0422822
$745$	−12.8895	−0.472234
$746$	15.0737	0.551886
$747$	−40.7043	−1.48929
$748$	1.00000	0.0365636
$749$	−30.4641	−1.11314
$750$	6.21377	0.226895
$751$	−21.2493	−0.775398	−0.387699	−	0.921786i	$-0.626730\pi$
$751$	−21.2493	−0.775398	−0.387699	+	0.921786i	$0.626730\pi$
$752$	1.11161	0.0405364
$753$	−2.68231	−0.0977487
$754$	−24.0960	−0.877525
$755$	−17.5452	−0.638534
$756$	−9.86962	−0.358954
$757$	−4.38524	−0.159384	−0.0796921	−	0.996820i	$-0.525394\pi$
$757$	−4.38524	−0.159384	−0.0796921	+	0.996820i	$0.525394\pi$
$758$	36.3313	1.31961
$759$	−4.25802	−0.154556
$760$	0.550549	0.0199705
$761$	5.48634	0.198880	0.0994398	−	0.995044i	$-0.468295\pi$
$761$	5.48634	0.198880	0.0994398	+	0.995044i	$0.468295\pi$
$762$	15.7444	0.570358
$763$	−46.1251	−1.66984
$764$	13.6409	0.493509
$765$	2.15287	0.0778372
$766$	35.3915	1.27875
$767$	−15.8305	−0.571606
$768$	0.759376	0.0274016
$769$	30.9700	1.11681	0.558403	−	0.829570i	$-0.311414\pi$
$769$	30.9700	1.11681	0.558403	+	0.829570i	$0.311414\pi$
$770$	−2.12901	−0.0767242
$771$	−0.106956	−0.00385193
$772$	19.5524	0.703705
$773$	−2.35795	−0.0848094	−0.0424047	−	0.999101i	$-0.513502\pi$
$773$	−2.35795	−0.0848094	−0.0424047	+	0.999101i	$0.513502\pi$
$774$	8.97284	0.322522
$775$	6.39512	0.229719
$776$	1.12601	0.0404215
$777$	14.9718	0.537109
$778$	8.91628	0.319664
$779$	3.02942	0.108540
$780$	−4.61972	−0.165412
$781$	7.83049	0.280197
$782$	−5.60726	−0.200515
$783$	14.4915	0.517884
$784$	−1.25683	−0.0448869
$785$	4.05971	0.144897
$786$	−0.339026	−0.0120927
$787$	−43.4564	−1.54905	−0.774526	−	0.632542i	$-0.782012\pi$
$787$	−43.4564	−1.54905	−0.774526	+	0.632542i	$0.782012\pi$
$788$	−9.29553	−0.331139
$789$	−7.15063	−0.254569
$790$	6.98635	0.248563
$791$	4.12360	0.146618
$792$	−2.42335	−0.0861099
$793$	−83.7546	−2.97421
$794$	−7.60351	−0.269838
$795$	5.05793	0.179386
$796$	−17.9751	−0.637110
$797$	−53.3915	−1.89123	−0.945613	−	0.325295i	$-0.894536\pi$
$797$	−53.3915	−1.89123	−0.945613	+	0.325295i	$0.894536\pi$
$798$	−1.12779	−0.0399232
$799$	1.11161	0.0393261
$800$	−4.21077	−0.148873
$801$	33.9360	1.19907
$802$	−23.9940	−0.847258
$803$	7.10590	0.250762
$804$	6.88929	0.242966
$805$	11.9379	0.420756
$806$	−10.4002	−0.366333
$807$	−15.3599	−0.540694
$808$	−8.01440	−0.281946
$809$	11.6286	0.408838	0.204419	−	0.978883i	$-0.434469\pi$
$809$	11.6286	0.408838	0.204419	+	0.978883i	$0.434469\pi$
$810$	−3.68028	−0.129312
$811$	−44.9316	−1.57776	−0.788881	−	0.614546i	$-0.789339\pi$
$811$	−44.9316	−1.57776	−0.788881	+	0.614546i	$0.789339\pi$
$812$	−8.43266	−0.295928
$813$	10.2496	0.359471
$814$	8.22698	0.288355
$815$	7.94615	0.278342
$816$	0.759376	0.0265835
$817$	2.29461	0.0802781
$818$	−19.3780	−0.677537
$819$	−39.7693	−1.38965
$820$	4.34278	0.151656
$821$	−13.8926	−0.484854	−0.242427	−	0.970170i	$-0.577944\pi$
$821$	−13.8926	−0.484854	−0.242427	+	0.970170i	$0.577944\pi$
$822$	11.7486	0.409779
$823$	−0.0672104	−0.00234281	−0.00117140	−	0.999999i	$-0.500373\pi$
$823$	−0.0672104	−0.00234281	−0.00117140	+	0.999999i	$0.500373\pi$
$824$	6.98260	0.243251
$825$	−3.19756	−0.111325
$826$	−5.54005	−0.192763
$827$	52.3654	1.82093	0.910463	−	0.413592i	$-0.135726\pi$
$827$	52.3654	1.82093	0.910463	+	0.413592i	$0.135726\pi$
$828$	13.5883	0.472227
$829$	37.7783	1.31209	0.656047	−	0.754720i	$-0.272227\pi$
$829$	37.7783	1.31209	0.656047	+	0.754720i	$0.272227\pi$
$830$	−14.9220	−0.517950
$831$	−17.1200	−0.593887
$832$	6.84788	0.237408
$833$	−1.25683	−0.0435466
$834$	−4.32523	−0.149770
$835$	3.21077	0.111113
$836$	−0.619718	−0.0214334
$837$	6.25477	0.216197
$838$	−12.6410	−0.436677
$839$	47.2622	1.63167	0.815837	−	0.578282i	$-0.196277\pi$
$839$	47.2622	1.63167	0.815837	+	0.578282i	$0.196277\pi$
$840$	−1.61672	−0.0557821
$841$	−16.6184	−0.573048
$842$	19.7171	0.679495
$843$	−10.1020	−0.347932
$844$	18.2520	0.628261
$845$	−30.1105	−1.03583
$846$	−2.69383	−0.0926156
$847$	2.39649	0.0823444
$848$	−7.49746	−0.257464
$849$	4.66606	0.160139
$850$	−4.21077	−0.144428
$851$	−46.1308	−1.58134
$852$	5.94629	0.203716
$853$	−28.1446	−0.963653	−0.481826	−	0.876267i	$-0.660026\pi$
$853$	−28.1446	−0.963653	−0.481826	+	0.876267i	$0.660026\pi$
$854$	−29.3108	−1.00300
$855$	−1.33417	−0.0456277
$856$	−12.7120	−0.434486
$857$	39.6185	1.35334	0.676671	−	0.736285i	$-0.263422\pi$
$857$	39.6185	1.35334	0.676671	+	0.736285i	$0.263422\pi$
$858$	5.20012	0.177529
$859$	−47.1311	−1.60809	−0.804047	−	0.594566i	$-0.797324\pi$
$859$	−47.1311	−1.60809	−0.804047	+	0.594566i	$0.797324\pi$
$860$	3.28940	0.112168
$861$	−8.89607	−0.303177
$862$	13.4340	0.457564
$863$	−8.34278	−0.283991	−0.141996	−	0.989867i	$-0.545352\pi$
$863$	−8.34278	−0.283991	−0.141996	+	0.989867i	$0.545352\pi$
$864$	−4.11836	−0.140110
$865$	6.66064	0.226469
$866$	−9.82721	−0.333942
$867$	0.759376	0.0257898
$868$	−3.63968	−0.123539
$869$	−7.86409	−0.266771
$870$	2.37382	0.0804800
$871$	62.1260	2.10506
$872$	−19.2469	−0.651784
$873$	−2.72872	−0.0923532
$874$	3.47492	0.117541
$875$	19.6098	0.662933
$876$	5.39606	0.182316
$877$	46.5017	1.57025	0.785125	−	0.619337i	$-0.212599\pi$
$877$	46.5017	1.57025	0.785125	+	0.619337i	$0.212599\pi$
$878$	−23.8560	−0.805101
$879$	−21.6125	−0.728971
$880$	−0.888387	−0.0299475
$881$	−19.2682	−0.649163	−0.324582	−	0.945858i	$-0.605224\pi$
$881$	−19.2682	−0.649163	−0.324582	+	0.945858i	$0.605224\pi$
$882$	3.04574	0.102555
$883$	−13.8842	−0.467241	−0.233620	−	0.972328i	$-0.575057\pi$
$883$	−13.8842	−0.467241	−0.233620	+	0.972328i	$0.575057\pi$
$884$	6.84788	0.230319
$885$	1.55954	0.0524234
$886$	−26.7444	−0.898495
$887$	45.9650	1.54336	0.771678	−	0.636014i	$-0.219418\pi$
$887$	45.9650	1.54336	0.771678	+	0.636014i	$0.219418\pi$
$888$	6.24737	0.209648
$889$	49.6871	1.66645
$890$	12.4407	0.417015
$891$	4.14266	0.138784
$892$	−14.2057	−0.475641
$893$	−0.688886	−0.0230527
$894$	11.0177	0.368486
$895$	−20.7741	−0.694400
$896$	2.39649	0.0800611
$897$	−29.1584	−0.973572
$898$	15.9652	0.532766
$899$	5.34411	0.178236
$900$	10.2042	0.340139
$901$	−7.49746	−0.249776
$902$	−4.88839	−0.162765
$903$	−6.73825	−0.224235
$904$	1.72068	0.0572290
$905$	3.15587	0.104905
$906$	14.9973	0.498252
$907$	20.1935	0.670514	0.335257	−	0.942127i	$-0.391177\pi$
$907$	20.1935	0.670514	0.335257	+	0.942127i	$0.391177\pi$
$908$	12.3654	0.410362
$909$	19.4217	0.644176
$910$	−14.5792	−0.483296
$911$	41.9589	1.39016	0.695080	−	0.718932i	$-0.255369\pi$
$911$	41.9589	1.39016	0.695080	+	0.718932i	$0.255369\pi$
$912$	−0.470599	−0.0155831
$913$	16.7967	0.555891
$914$	2.36545	0.0782421
$915$	8.25109	0.272773
$916$	6.34653	0.209695
$917$	−1.06992	−0.0353319
$918$	−4.11836	−0.135926
$919$	−8.33903	−0.275079	−0.137540	−	0.990496i	$-0.543919\pi$
$919$	−8.33903	−0.275079	−0.137540	+	0.990496i	$0.543919\pi$
$920$	4.98141	0.164232
$921$	16.9904	0.559852
$922$	−15.9874	−0.526517
$923$	53.6223	1.76500
$924$	1.81984	0.0598683
$925$	−34.6419	−1.13902
$926$	−21.1920	−0.696413
$927$	−16.9213	−0.555768
$928$	−3.51875	−0.115509
$929$	15.3469	0.503514	0.251757	−	0.967790i	$-0.418992\pi$
$929$	15.3469	0.503514	0.251757	+	0.967790i	$0.418992\pi$
$930$	1.02458	0.0335973
$931$	0.778881	0.0255268
$932$	−7.48515	−0.245184
$933$	17.2716	0.565445
$934$	15.4627	0.505954
$935$	−0.888387	−0.0290533
$936$	−16.5948	−0.542418
$937$	−1.86890	−0.0610543	−0.0305271	−	0.999534i	$-0.509719\pi$
$937$	−1.86890	−0.0610543	−0.0305271	+	0.999534i	$0.509719\pi$
$938$	21.7417	0.709891
$939$	2.93520	0.0957867
$940$	−0.987542	−0.0322101
$941$	−8.84670	−0.288394	−0.144197	−	0.989549i	$-0.546060\pi$
$941$	−8.84670	−0.288394	−0.144197	+	0.989549i	$0.546060\pi$
$942$	−3.47016	−0.113064
$943$	27.4105	0.892607
$944$	−2.31173	−0.0752405
$945$	8.76803	0.285224
$946$	−3.70267	−0.120384
$947$	−39.1492	−1.27218	−0.636088	−	0.771616i	$-0.719449\pi$
$947$	−39.1492	−1.27218	−0.636088	+	0.771616i	$0.719449\pi$
$948$	−5.97181	−0.193955
$949$	48.6604	1.57958
$950$	2.60949	0.0846630
$951$	−1.12688	−0.0365417
$952$	2.39649	0.0776707
$953$	−21.3144	−0.690443	−0.345221	−	0.938521i	$-0.612196\pi$
$953$	−21.3144	−0.690443	−0.345221	+	0.938521i	$0.612196\pi$
$954$	18.1689	0.588241
$955$	−12.1184	−0.392141
$956$	0	0
$957$	−2.67206	−0.0863754
$958$	−14.3453	−0.463477
$959$	37.0770	1.19728
$960$	−0.674620	−0.0217733
$961$	−28.6934	−0.925593
$962$	56.3374	1.81639
$963$	30.8055	0.992695
$964$	24.2094	0.779733
$965$	−17.3701	−0.559162
$966$	−10.2043	−0.328318
$967$	−50.7921	−1.63336	−0.816682	−	0.577088i	$-0.804189\pi$
$967$	−50.7921	−1.63336	−0.816682	+	0.577088i	$0.804189\pi$
$968$	1.00000	0.0321412
$969$	−0.470599	−0.0151178
$970$	−1.00033	−0.0321188
$971$	22.0973	0.709135	0.354567	−	0.935031i	$-0.384628\pi$
$971$	22.0973	0.709135	0.354567	+	0.935031i	$0.384628\pi$
$972$	15.5009	0.497192
$973$	−13.6499	−0.437594
$974$	24.1173	0.772768
$975$	−21.8965	−0.701250
$976$	−12.2307	−0.391496
$977$	48.0144	1.53612	0.768058	−	0.640380i	$-0.221223\pi$
$977$	48.0144	1.53612	0.768058	+	0.640380i	$0.221223\pi$
$978$	−6.79222	−0.217191
$979$	−14.0038	−0.447562
$980$	1.11655	0.0356670
$981$	46.6420	1.48917
$982$	8.92080	0.284674
$983$	21.8107	0.695653	0.347827	−	0.937559i	$-0.386920\pi$
$983$	21.8107	0.695653	0.347827	+	0.937559i	$0.386920\pi$
$984$	−3.71213	−0.118338
$985$	8.25802	0.263122
$986$	−3.51875	−0.112060
$987$	2.02296	0.0643914
$988$	−4.24375	−0.135012
$989$	20.7618	0.660187
$990$	2.15287	0.0684227
$991$	−49.6085	−1.57586	−0.787932	−	0.615762i	$-0.788848\pi$
$991$	−49.6085	−1.57586	−0.787932	+	0.615762i	$0.788848\pi$
$992$	−1.51875	−0.0482204
$993$	−1.96758	−0.0624394
$994$	18.7657	0.595212
$995$	15.9688	0.506246
$996$	12.7550	0.404159
$997$	−23.2682	−0.736912	−0.368456	−	0.929645i	$-0.620114\pi$
$997$	−23.2682	−0.736912	−0.368456	+	0.929645i	$0.620114\pi$
$998$	−13.5599	−0.429230
$999$	−33.8817	−1.07197

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	374.2.a.e.1.3	✓	4
3.2	odd	2		3366.2.a.bf.1.3		4
4.3	odd	2		2992.2.a.u.1.2		4
5.4	even	2		9350.2.a.cj.1.2		4
11.10	odd	2		4114.2.a.x.1.3		4
17.16	even	2		6358.2.a.x.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
374.2.a.e.1.3	✓	4	1.1	even	1	trivial
2992.2.a.u.1.2		4	4.3	odd	2
3366.2.a.bf.1.3		4	3.2	odd	2
4114.2.a.x.1.3		4	11.10	odd	2
6358.2.a.x.1.2		4	17.16	even	2
9350.2.a.cj.1.2		4	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	374.2.a.e.1.3

Level	$374$
Weight	$2$
Character	374.1
Self dual	yes
Analytic conductor	$2.986$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$1$