LMFDB - Embedded newform 374.2.a.d.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)

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")

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);

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:	4.4.55585.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 10x^{2} + 9x + 16$ x^4 - x^3 - 10x^2 + 9x + 16
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			$2.22696$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +2.22696 q^{3} +1.00000 q^{4} +0.959363 q^{5} -2.22696 q^{6} +4.62570 q^{7} -1.00000 q^{8} +1.95936 q^{9} -0.959363 q^{10} +1.00000 q^{11} +2.22696 q^{12} -6.94316 q^{13} -4.62570 q^{14} +2.13646 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.95936 q^{18} +2.17177 q^{19} +0.959363 q^{20} +10.3013 q^{21} -1.00000 q^{22} -2.45393 q^{23} -2.22696 q^{24} -4.07962 q^{25} +6.94316 q^{26} -2.31746 q^{27} +4.62570 q^{28} -7.17012 q^{29} -2.13646 q^{30} +9.17012 q^{31} -1.00000 q^{32} +2.22696 q^{33} +1.00000 q^{34} +4.43772 q^{35} +1.95936 q^{36} +10.6257 q^{37} -2.17177 q^{38} -15.4622 q^{39} -0.959363 q^{40} -10.5834 q^{41} -10.3013 q^{42} -1.75683 q^{43} +1.00000 q^{44} +1.87974 q^{45} +2.45393 q^{46} -0.324441 q^{47} +2.22696 q^{48} +14.3971 q^{49} +4.07962 q^{50} -2.22696 q^{51} -6.94316 q^{52} +4.71620 q^{53} +2.31746 q^{54} +0.959363 q^{55} -4.62570 q^{56} +4.83646 q^{57} +7.17012 q^{58} +13.7053 q^{59} +2.13646 q^{60} -0.453925 q^{61} -9.17012 q^{62} +9.06342 q^{63} +1.00000 q^{64} -6.66101 q^{65} -2.22696 q^{66} -0.453925 q^{67} -1.00000 q^{68} -5.46480 q^{69} -4.43772 q^{70} -13.8050 q^{71} -1.95936 q^{72} +0.861884 q^{73} -10.6257 q^{74} -9.08517 q^{75} +2.17177 q^{76} +4.62570 q^{77} +15.4622 q^{78} -3.10571 q^{79} +0.959363 q^{80} -11.0390 q^{81} +10.5834 q^{82} +3.26062 q^{83} +10.3013 q^{84} -0.959363 q^{85} +1.75683 q^{86} -15.9676 q^{87} -1.00000 q^{88} -9.26062 q^{89} -1.87974 q^{90} -32.1170 q^{91} -2.45393 q^{92} +20.4215 q^{93} +0.324441 q^{94} +2.08352 q^{95} -2.22696 q^{96} -13.6240 q^{97} -14.3971 q^{98} +1.95936 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{2} + q^{3} + 4 q^{4} + 5 q^{5} - q^{6} + q^{7} - 4 q^{8} + 9 q^{9} - 5 q^{10} + 4 q^{11} + q^{12} - 3 q^{13} - q^{14} + 4 q^{16} - 4 q^{17} - 9 q^{18} + 7 q^{19} + 5 q^{20} + 8 q^{21} - 4 q^{22}+ \cdots + 9 q^{99}+O(q^{100})$ 4 * q - 4 * q^2 + q^3 + 4 * q^4 + 5 * q^5 - q^6 + q^7 - 4 * q^8 + 9 * q^9 - 5 * q^10 + 4 * q^11 + q^12 - 3 * q^13 - q^14 + 4 * q^16 - 4 * q^17 - 9 * q^18 + 7 * q^19 + 5 * q^20 + 8 * q^21 - 4 * q^22 + 6 * q^23 - q^24 + 17 * q^25 + 3 * q^26 - 2 * q^27 + q^28 + 4 * q^29 + 4 * q^31 - 4 * q^32 + q^33 + 4 * q^34 - 24 * q^35 + 9 * q^36 + 25 * q^37 - 7 * q^38 + 7 * q^39 - 5 * q^40 + 5 * q^41 - 8 * q^42 + 11 * q^43 + 4 * q^44 + 42 * q^45 - 6 * q^46 - 17 * q^47 + q^48 + 17 * q^49 - 17 * q^50 - q^51 - 3 * q^52 + 2 * q^53 + 2 * q^54 + 5 * q^55 - q^56 - 32 * q^57 - 4 * q^58 + 4 * q^59 + 14 * q^61 - 4 * q^62 - 23 * q^63 + 4 * q^64 - 16 * q^65 - q^66 + 14 * q^67 - 4 * q^68 - 40 * q^69 + 24 * q^70 - 8 * q^71 - 9 * q^72 - 19 * q^73 - 25 * q^74 - 45 * q^75 + 7 * q^76 + q^77 - 7 * q^78 + 17 * q^79 + 5 * q^80 - 12 * q^81 - 5 * q^82 - 19 * q^83 + 8 * q^84 - 5 * q^85 - 11 * q^86 - 12 * q^87 - 4 * q^88 - 5 * q^89 - 42 * q^90 - 32 * q^91 + 6 * q^92 + 14 * q^93 + 17 * q^94 - 14 * q^95 - q^96 - 6 * q^97 - 17 * 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
$3$	2.22696	1.28574	0.642869	−	0.765976i	$-0.277744\pi$
$3$	2.22696	1.28574	0.642869	+	0.765976i	$0.277744\pi$
$4$	1.00000	0.500000
$5$	0.959363	0.429040	0.214520	−	0.976720i	$-0.431181\pi$
$5$	0.959363	0.429040	0.214520	+	0.976720i	$0.431181\pi$
$6$	−2.22696	−0.909154
$7$	4.62570	1.74835	0.874175	−	0.485611i	$-0.161403\pi$
$7$	4.62570	1.74835	0.874175	+	0.485611i	$0.161403\pi$
$8$	−1.00000	−0.353553
$9$	1.95936	0.653121
$10$	−0.959363	−0.303377
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.22696	0.642869
$13$	−6.94316	−1.92569	−0.962843	−	0.270062i	$-0.912956\pi$
$13$	−6.94316	−1.92569	−0.962843	+	0.270062i	$0.912956\pi$
$14$	−4.62570	−1.23627
$15$	2.13646	0.551633
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.95936	−0.461826
$19$	2.17177	0.498239	0.249119	−	0.968473i	$-0.419859\pi$
$19$	2.17177	0.498239	0.249119	+	0.968473i	$0.419859\pi$
$20$	0.959363	0.214520
$21$	10.3013	2.24792
$22$	−1.00000	−0.213201
$23$	−2.45393	−0.511679	−0.255839	−	0.966719i	$-0.582352\pi$
$23$	−2.45393	−0.511679	−0.255839	+	0.966719i	$0.582352\pi$
$24$	−2.22696	−0.454577
$25$	−4.07962	−0.815925
$26$	6.94316	1.36167
$27$	−2.31746	−0.445996
$28$	4.62570	0.874175
$29$	−7.17012	−1.33146	−0.665729	−	0.746194i	$-0.731879\pi$
$29$	−7.17012	−1.33146	−0.665729	+	0.746194i	$0.731879\pi$
$30$	−2.13646	−0.390063
$31$	9.17012	1.64700	0.823501	−	0.567314i	$-0.192018\pi$
$31$	9.17012	1.64700	0.823501	+	0.567314i	$0.192018\pi$
$32$	−1.00000	−0.176777
$33$	2.22696	0.387664
$34$	1.00000	0.171499
$35$	4.43772	0.750112
$36$	1.95936	0.326560
$37$	10.6257	1.74685	0.873427	−	0.486955i	$-0.161892\pi$
$37$	10.6257	1.74685	0.873427	+	0.486955i	$0.161892\pi$
$38$	−2.17177	−0.352308
$39$	−15.4622	−2.47593
$40$	−0.959363	−0.151689
$41$	−10.5834	−1.65285	−0.826425	−	0.563046i	$-0.809629\pi$
$41$	−10.5834	−1.65285	−0.826425	+	0.563046i	$0.809629\pi$
$42$	−10.3013	−1.58952
$43$	−1.75683	−0.267915	−0.133957	−	0.990987i	$-0.542769\pi$
$43$	−1.75683	−0.267915	−0.133957	+	0.990987i	$0.542769\pi$
$44$	1.00000	0.150756
$45$	1.87974	0.280215
$46$	2.45393	0.361812
$47$	−0.324441	−0.0473246	−0.0236623	−	0.999720i	$-0.507533\pi$
$47$	−0.324441	−0.0473246	−0.0236623	+	0.999720i	$0.507533\pi$
$48$	2.22696	0.321434
$49$	14.3971	2.05673
$50$	4.07962	0.576946
$51$	−2.22696	−0.311837
$52$	−6.94316	−0.962843
$53$	4.71620	0.647819	0.323910	−	0.946088i	$-0.395003\pi$
$53$	4.71620	0.647819	0.323910	+	0.946088i	$0.395003\pi$
$54$	2.31746	0.315366
$55$	0.959363	0.129360
$56$	−4.62570	−0.618135
$57$	4.83646	0.640604
$58$	7.17012	0.941483
$59$	13.7053	1.78428	0.892140	−	0.451758i	$-0.149203\pi$
$59$	13.7053	1.78428	0.892140	+	0.451758i	$0.149203\pi$
$60$	2.13646	0.275816
$61$	−0.453925	−0.0581192	−0.0290596	−	0.999578i	$-0.509251\pi$
$61$	−0.453925	−0.0581192	−0.0290596	+	0.999578i	$0.509251\pi$
$62$	−9.17012	−1.16461
$63$	9.06342	1.14188
$64$	1.00000	0.125000
$65$	−6.66101	−0.826196
$66$	−2.22696	−0.274120
$67$	−0.453925	−0.0554558	−0.0277279	−	0.999616i	$-0.508827\pi$
$67$	−0.453925	−0.0554558	−0.0277279	+	0.999616i	$0.508827\pi$
$68$	−1.00000	−0.121268
$69$	−5.46480	−0.657885
$70$	−4.43772	−0.530409
$71$	−13.8050	−1.63836	−0.819179	−	0.573539i	$-0.805570\pi$
$71$	−13.8050	−1.63836	−0.819179	+	0.573539i	$0.805570\pi$
$72$	−1.95936	−0.230913
$73$	0.861884	0.100876	0.0504379	−	0.998727i	$-0.483938\pi$
$73$	0.861884	0.100876	0.0504379	+	0.998727i	$0.483938\pi$
$74$	−10.6257	−1.23521
$75$	−9.08517	−1.04906
$76$	2.17177	0.249119
$77$	4.62570	0.527147
$78$	15.4622	1.75074
$79$	−3.10571	−0.349420	−0.174710	−	0.984620i	$-0.555899\pi$
$79$	−3.10571	−0.349420	−0.174710	+	0.984620i	$0.555899\pi$
$80$	0.959363	0.107260
$81$	−11.0390	−1.22655
$82$	10.5834	1.16874
$83$	3.26062	0.357900	0.178950	−	0.983858i	$-0.442730\pi$
$83$	3.26062	0.357900	0.178950	+	0.983858i	$0.442730\pi$
$84$	10.3013	1.12396
$85$	−0.959363	−0.104057
$86$	1.75683	0.189444
$87$	−15.9676	−1.71191
$88$	−1.00000	−0.106600
$89$	−9.26062	−0.981624	−0.490812	−	0.871266i	$-0.663300\pi$
$89$	−9.26062	−0.981624	−0.490812	+	0.871266i	$0.663300\pi$
$90$	−1.87974	−0.198142
$91$	−32.1170	−3.36677
$92$	−2.45393	−0.255839
$93$	20.4215	2.11761
$94$	0.324441	0.0334635
$95$	2.08352	0.213764
$96$	−2.22696	−0.227288
$97$	−13.6240	−1.38331	−0.691656	−	0.722227i	$-0.743119\pi$
$97$	−13.6240	−1.38331	−0.691656	+	0.722227i	$0.743119\pi$
$98$	−14.3971	−1.45433
$99$	1.95936	0.196923
$100$	−4.07962	−0.407962
$101$	−6.95936	−0.692482	−0.346241	−	0.938146i	$-0.612542\pi$
$101$	−6.95936	−0.692482	−0.346241	+	0.938146i	$0.612542\pi$
$102$	2.22696	0.220502
$103$	−13.2867	−1.30918	−0.654589	−	0.755985i	$-0.727158\pi$
$103$	−13.2867	−1.30918	−0.654589	+	0.755985i	$0.727158\pi$
$104$	6.94316	0.680833
$105$	9.88264	0.964447
$106$	−4.71620	−0.458077
$107$	−1.28380	−0.124110	−0.0620550	−	0.998073i	$-0.519765\pi$
$107$	−1.28380	−0.124110	−0.0620550	+	0.998073i	$0.519765\pi$
$108$	−2.31746	−0.222998
$109$	11.6240	1.11338	0.556691	−	0.830720i	$-0.312071\pi$
$109$	11.6240	1.11338	0.556691	+	0.830720i	$0.312071\pi$
$110$	−0.959363	−0.0914716
$111$	23.6630	2.24600
$112$	4.62570	0.437087
$113$	2.79747	0.263164	0.131582	−	0.991305i	$-0.457994\pi$
$113$	2.79747	0.263164	0.131582	+	0.991305i	$0.457994\pi$
$114$	−4.83646	−0.452976
$115$	−2.35420	−0.219531
$116$	−7.17012	−0.665729
$117$	−13.6042	−1.25771
$118$	−13.7053	−1.26168
$119$	−4.62570	−0.424037
$120$	−2.13646	−0.195032
$121$	1.00000	0.0909091
$122$	0.453925	0.0410965
$123$	−23.5689	−2.12513
$124$	9.17012	0.823501
$125$	−8.71065	−0.779104
$126$	−9.06342	−0.807434
$127$	16.4539	1.46005	0.730025	−	0.683421i	$-0.239508\pi$
$127$	16.4539	1.46005	0.730025	+	0.683421i	$0.239508\pi$
$128$	−1.00000	−0.0883883
$129$	−3.91240	−0.344468
$130$	6.66101	0.584209
$131$	9.43239	0.824112	0.412056	−	0.911159i	$-0.364811\pi$
$131$	9.43239	0.824112	0.412056	+	0.911159i	$0.364811\pi$
$132$	2.22696	0.193832
$133$	10.0460	0.871096
$134$	0.453925	0.0392132
$135$	−2.22329	−0.191350
$136$	1.00000	0.0857493
$137$	−6.48923	−0.554413	−0.277206	−	0.960810i	$-0.589409\pi$
$137$	−6.48923	−0.554413	−0.277206	+	0.960810i	$0.589409\pi$
$138$	5.46480	0.465195
$139$	−4.29138	−0.363990	−0.181995	−	0.983299i	$-0.558255\pi$
$139$	−4.29138	−0.363990	−0.181995	+	0.983299i	$0.558255\pi$
$140$	4.43772	0.375056
$141$	−0.722518	−0.0608470
$142$	13.8050	1.15849
$143$	−6.94316	−0.580616
$144$	1.95936	0.163280
$145$	−6.87875	−0.571249
$146$	−0.861884	−0.0713300
$147$	32.0618	2.64441
$148$	10.6257	0.873427
$149$	10.1272	0.829656	0.414828	−	0.909900i	$-0.363842\pi$
$149$	10.1272	0.829656	0.414828	+	0.909900i	$0.363842\pi$
$150$	9.08517	0.741801
$151$	6.71620	0.546556	0.273278	−	0.961935i	$-0.411892\pi$
$151$	6.71620	0.546556	0.273278	+	0.961935i	$0.411892\pi$
$152$	−2.17177	−0.176154
$153$	−1.95936	−0.158405
$154$	−4.62570	−0.372749
$155$	8.79747	0.706630
$156$	−15.4622	−1.23796
$157$	0.424821	0.0339044	0.0169522	−	0.999856i	$-0.494604\pi$
$157$	0.424821	0.0339044	0.0169522	+	0.999856i	$0.494604\pi$
$158$	3.10571	0.247077
$159$	10.5028	0.832926
$160$	−0.959363	−0.0758443
$161$	−11.3511	−0.894593
$162$	11.0390	0.867305
$163$	−17.3809	−1.36138	−0.680688	−	0.732574i	$-0.738319\pi$
$163$	−17.3809	−1.36138	−0.680688	+	0.732574i	$0.738319\pi$
$164$	−10.5834	−0.826425
$165$	2.13646	0.166324
$166$	−3.26062	−0.253073
$167$	1.83811	0.142237	0.0711186	−	0.997468i	$-0.477343\pi$
$167$	1.83811	0.142237	0.0711186	+	0.997468i	$0.477343\pi$
$168$	−10.3013	−0.794759
$169$	35.2075	2.70827
$170$	0.959363	0.0735797
$171$	4.25529	0.325410
$172$	−1.75683	−0.133957
$173$	16.9079	1.28548	0.642740	−	0.766084i	$-0.277798\pi$
$173$	16.9079	1.28548	0.642740	+	0.766084i	$0.277798\pi$
$174$	15.9676	1.21050
$175$	−18.8711	−1.42652
$176$	1.00000	0.0753778
$177$	30.5212	2.29412
$178$	9.26062	0.694113
$179$	0.343546	0.0256778	0.0128389	−	0.999918i	$-0.495913\pi$
$179$	0.343546	0.0256778	0.0128389	+	0.999918i	$0.495913\pi$
$180$	1.87974	0.140107
$181$	16.3082	1.21218	0.606091	−	0.795395i	$-0.292737\pi$
$181$	16.3082	1.21218	0.606091	+	0.795395i	$0.292737\pi$
$182$	32.1170	2.38067
$183$	−1.01087	−0.0747260
$184$	2.45393	0.180906
$185$	10.1939	0.749470
$186$	−20.4215	−1.49738
$187$	−1.00000	−0.0731272
$188$	−0.324441	−0.0236623
$189$	−10.7199	−0.779756
$190$	−2.08352	−0.151154
$191$	−5.74063	−0.415377	−0.207689	−	0.978195i	$-0.566594\pi$
$191$	−5.74063	−0.415377	−0.207689	+	0.978195i	$0.566594\pi$
$192$	2.22696	0.160717
$193$	−12.4080	−0.893144	−0.446572	−	0.894748i	$-0.647355\pi$
$193$	−12.4080	−0.893144	−0.446572	+	0.894748i	$0.647355\pi$
$194$	13.6240	0.978150
$195$	−14.8338	−1.06227
$196$	14.3971	1.02836
$197$	6.74860	0.480818	0.240409	−	0.970672i	$-0.422718\pi$
$197$	6.74860	0.480818	0.240409	+	0.970672i	$0.422718\pi$
$198$	−1.95936	−0.139246
$199$	8.53520	0.605044	0.302522	−	0.953142i	$-0.402171\pi$
$199$	8.53520	0.605044	0.302522	+	0.953142i	$0.402171\pi$
$200$	4.07962	0.288473
$201$	−1.01087	−0.0713016
$202$	6.95936	0.489659
$203$	−33.1668	−2.32785
$204$	−2.22696	−0.155919
$205$	−10.1533	−0.709139
$206$	13.2867	0.925729
$207$	−4.80813	−0.334188
$208$	−6.94316	−0.481421
$209$	2.17177	0.150225
$210$	−9.88264	−0.681967
$211$	−0.162550	−0.0111904	−0.00559519	−	0.999984i	$-0.501781\pi$
$211$	−0.162550	−0.0111904	−0.00559519	+	0.999984i	$0.501781\pi$
$212$	4.71620	0.323910
$213$	−30.7433	−2.10650
$214$	1.28380	0.0877590
$215$	−1.68544	−0.114946
$216$	2.31746	0.157683
$217$	42.4182	2.87954
$218$	−11.6240	−0.787280
$219$	1.91938	0.129700
$220$	0.959363	0.0646802
$221$	6.94316	0.467047
$222$	−23.6630	−1.58816
$223$	−10.1718	−0.681152	−0.340576	−	0.940217i	$-0.610622\pi$
$223$	−10.1718	−0.681152	−0.340576	+	0.940217i	$0.610622\pi$
$224$	−4.62570	−0.309067
$225$	−7.99346	−0.532897
$226$	−2.79747	−0.186085
$227$	−29.0380	−1.92732	−0.963660	−	0.267133i	$-0.913924\pi$
$227$	−29.0380	−1.92732	−0.963660	+	0.267133i	$0.913924\pi$
$228$	4.83646	0.320302
$229$	15.9967	1.05709	0.528546	−	0.848905i	$-0.322738\pi$
$229$	15.9967	1.05709	0.528546	+	0.848905i	$0.322738\pi$
$230$	2.35420	0.155232
$231$	10.3013	0.677773
$232$	7.17012	0.470742
$233$	5.93228	0.388637	0.194318	−	0.980939i	$-0.437750\pi$
$233$	5.93228	0.388637	0.194318	+	0.980939i	$0.437750\pi$
$234$	13.6042	0.889332
$235$	−0.311257	−0.0203041
$236$	13.7053	0.892140
$237$	−6.91630	−0.449262
$238$	4.62570	0.299839
$239$	21.8157	1.41114	0.705570	−	0.708640i	$-0.250691\pi$
$239$	21.8157	1.41114	0.705570	+	0.708640i	$0.250691\pi$
$240$	2.13646	0.137908
$241$	6.90785	0.444974	0.222487	−	0.974936i	$-0.428583\pi$
$241$	6.90785	0.444974	0.222487	+	0.974936i	$0.428583\pi$
$242$	−1.00000	−0.0642824
$243$	−17.6310	−1.13103
$244$	−0.453925	−0.0290596
$245$	13.8120	0.882418
$246$	23.5689	1.50270
$247$	−15.0790	−0.959452
$248$	−9.17012	−0.582303
$249$	7.26128	0.460165
$250$	8.71065	0.550910
$251$	1.57518	0.0994244	0.0497122	−	0.998764i	$-0.484170\pi$
$251$	1.57518	0.0994244	0.0497122	+	0.998764i	$0.484170\pi$
$252$	9.06342	0.570942
$253$	−2.45393	−0.154277
$254$	−16.4539	−1.03241
$255$	−2.13646	−0.133791
$256$	1.00000	0.0625000
$257$	4.73608	0.295428	0.147714	−	0.989030i	$-0.452808\pi$
$257$	4.73608	0.295428	0.147714	+	0.989030i	$0.452808\pi$
$258$	3.91240	0.243576
$259$	49.1513	3.05411
$260$	−6.66101	−0.413098
$261$	−14.0489	−0.869603
$262$	−9.43239	−0.582735
$263$	14.6838	0.905441	0.452721	−	0.891652i	$-0.350453\pi$
$263$	14.6838	0.905441	0.452721	+	0.891652i	$0.350453\pi$
$264$	−2.22696	−0.137060
$265$	4.52454	0.277940
$266$	−10.0460	−0.615958
$267$	−20.6231	−1.26211
$268$	−0.453925	−0.0277279
$269$	9.37430	0.571561	0.285781	−	0.958295i	$-0.407747\pi$
$269$	9.37430	0.571561	0.285781	+	0.958295i	$0.407747\pi$
$270$	2.22329	0.135305
$271$	2.87875	0.174871	0.0874357	−	0.996170i	$-0.472133\pi$
$271$	2.87875	0.174871	0.0874357	+	0.996170i	$0.472133\pi$
$272$	−1.00000	−0.0606339
$273$	−71.5233	−4.32878
$274$	6.48923	0.392029
$275$	−4.07962	−0.246011
$276$	−5.46480	−0.328942
$277$	−14.4506	−0.868254	−0.434127	−	0.900852i	$-0.642943\pi$
$277$	−14.4506	−0.868254	−0.434127	+	0.900852i	$0.642943\pi$
$278$	4.29138	0.257380
$279$	17.9676	1.07569
$280$	−4.43772	−0.265205
$281$	−14.5319	−0.866900	−0.433450	−	0.901178i	$-0.642704\pi$
$281$	−14.5319	−0.866900	−0.433450	+	0.901178i	$0.642704\pi$
$282$	0.722518	0.0430253
$283$	−5.26984	−0.313260	−0.156630	−	0.987657i	$-0.550063\pi$
$283$	−5.26984	−0.313260	−0.156630	+	0.987657i	$0.550063\pi$
$284$	−13.8050	−0.819179
$285$	4.63992	0.274845
$286$	6.94316	0.410558
$287$	−48.9557	−2.88976
$288$	−1.95936	−0.115457
$289$	1.00000	0.0588235
$290$	6.87875	0.403934
$291$	−30.3402	−1.77858
$292$	0.861884	0.0504379
$293$	16.1262	0.942102	0.471051	−	0.882106i	$-0.343875\pi$
$293$	16.1262	0.942102	0.471051	+	0.882106i	$0.343875\pi$
$294$	−32.0618	−1.86988
$295$	13.1484	0.765528
$296$	−10.6257	−0.617606
$297$	−2.31746	−0.134473
$298$	−10.1272	−0.586655
$299$	17.0380	0.985333
$300$	−9.08517	−0.524532
$301$	−8.12658	−0.468408
$302$	−6.71620	−0.386474
$303$	−15.4982	−0.890351
$304$	2.17177	0.124560
$305$	−0.435479	−0.0249354
$306$	1.95936	0.112009
$307$	2.03201	0.115973	0.0579863	−	0.998317i	$-0.481532\pi$
$307$	2.03201	0.115973	0.0579863	+	0.998317i	$0.481532\pi$
$308$	4.62570	0.263574
$309$	−29.5890	−1.68326
$310$	−8.79747	−0.499663
$311$	14.7453	0.836129	0.418065	−	0.908417i	$-0.362709\pi$
$311$	14.7453	0.836129	0.418065	+	0.908417i	$0.362709\pi$
$312$	15.4622	0.875372
$313$	−25.4628	−1.43924	−0.719622	−	0.694366i	$-0.755685\pi$
$313$	−25.4628	−1.43924	−0.719622	+	0.694366i	$0.755685\pi$
$314$	−0.424821	−0.0239740
$315$	8.69511	0.489914
$316$	−3.10571	−0.174710
$317$	−3.62405	−0.203547	−0.101773	−	0.994808i	$-0.532452\pi$
$317$	−3.62405	−0.203547	−0.101773	+	0.994808i	$0.532452\pi$
$318$	−10.5028	−0.588967
$319$	−7.17012	−0.401450
$320$	0.959363	0.0536300
$321$	−2.85898	−0.159573
$322$	11.3511	0.632573
$323$	−2.17177	−0.120841
$324$	−11.0390	−0.613277
$325$	28.3255	1.57121
$326$	17.3809	0.962638
$327$	25.8863	1.43152
$328$	10.5834	0.584371
$329$	−1.50077	−0.0827399
$330$	−2.13646	−0.117609
$331$	2.40506	0.132194	0.0660970	−	0.997813i	$-0.478945\pi$
$331$	2.40506	0.132194	0.0660970	+	0.997813i	$0.478945\pi$
$332$	3.26062	0.178950
$333$	20.8196	1.14091
$334$	−1.83811	−0.100577
$335$	−0.435479	−0.0237928
$336$	10.3013	0.561980
$337$	4.24317	0.231140	0.115570	−	0.993299i	$-0.463131\pi$
$337$	4.24317	0.231140	0.115570	+	0.993299i	$0.463131\pi$
$338$	−35.2075	−1.91503
$339$	6.22986	0.338360
$340$	−0.959363	−0.0520287
$341$	9.17012	0.496590
$342$	−4.25529	−0.230100
$343$	34.2167	1.84753
$344$	1.75683	0.0947221
$345$	−5.24272	−0.282259
$346$	−16.9079	−0.908972
$347$	−10.7162	−0.575276	−0.287638	−	0.957739i	$-0.592870\pi$
$347$	−10.7162	−0.575276	−0.287638	+	0.957739i	$0.592870\pi$
$348$	−15.9676	−0.855953
$349$	−13.8375	−0.740702	−0.370351	−	0.928892i	$-0.620763\pi$
$349$	−13.8375	−0.740702	−0.370351	+	0.928892i	$0.620763\pi$
$350$	18.8711	1.00870
$351$	16.0905	0.858847
$352$	−1.00000	−0.0533002
$353$	9.29961	0.494968	0.247484	−	0.968892i	$-0.420396\pi$
$353$	9.29961	0.494968	0.247484	+	0.968892i	$0.420396\pi$
$354$	−30.5212	−1.62219
$355$	−13.2440	−0.702921
$356$	−9.26062	−0.490812
$357$	−10.3013	−0.545200
$358$	−0.343546	−0.0181570
$359$	3.54607	0.187155	0.0935773	−	0.995612i	$-0.470170\pi$
$359$	3.54607	0.187155	0.0935773	+	0.995612i	$0.470170\pi$
$360$	−1.87974	−0.0990709
$361$	−14.2834	−0.751758
$362$	−16.3082	−0.857142
$363$	2.22696	0.116885
$364$	−32.1170	−1.68339
$365$	0.826859	0.0432798
$366$	1.01087	0.0528393
$367$	18.6165	0.971772	0.485886	−	0.874022i	$-0.338497\pi$
$367$	18.6165	0.971772	0.485886	+	0.874022i	$0.338497\pi$
$368$	−2.45393	−0.127920
$369$	−20.7367	−1.07951
$370$	−10.1939	−0.529955
$371$	21.8157	1.13261
$372$	20.4215	1.05881
$373$	−16.8986	−0.874978	−0.437489	−	0.899224i	$-0.644132\pi$
$373$	−16.8986	−0.874978	−0.437489	+	0.899224i	$0.644132\pi$
$374$	1.00000	0.0517088
$375$	−19.3983	−1.00172
$376$	0.324441	0.0167318
$377$	49.7833	2.56397
$378$	10.7199	0.551371
$379$	17.0558	0.876097	0.438048	−	0.898951i	$-0.355670\pi$
$379$	17.0558	0.876097	0.438048	+	0.898951i	$0.355670\pi$
$380$	2.08352	0.106882
$381$	36.6423	1.87724
$382$	5.74063	0.293716
$383$	−7.09215	−0.362392	−0.181196	−	0.983447i	$-0.557997\pi$
$383$	−7.09215	−0.362392	−0.181196	+	0.983447i	$0.557997\pi$
$384$	−2.22696	−0.113644
$385$	4.43772	0.226167
$386$	12.4080	0.631548
$387$	−3.44227	−0.174981
$388$	−13.6240	−0.691656
$389$	0.826576	0.0419090	0.0209545	−	0.999780i	$-0.493329\pi$
$389$	0.826576	0.0419090	0.0209545	+	0.999780i	$0.493329\pi$
$390$	14.8338	0.751139
$391$	2.45393	0.124100
$392$	−14.3971	−0.727163
$393$	21.0056	1.05959
$394$	−6.74860	−0.339990
$395$	−2.97950	−0.149915
$396$	1.95936	0.0984617
$397$	30.2173	1.51656	0.758282	−	0.651926i	$-0.226039\pi$
$397$	30.2173	1.51656	0.758282	+	0.651926i	$0.226039\pi$
$398$	−8.53520	−0.427831
$399$	22.3720	1.12000
$400$	−4.07962	−0.203981
$401$	18.1301	0.905376	0.452688	−	0.891669i	$-0.350465\pi$
$401$	18.1301	0.905376	0.452688	+	0.891669i	$0.350465\pi$
$402$	1.01087	0.0504178
$403$	−63.6696	−3.17161
$404$	−6.95936	−0.346241
$405$	−10.5904	−0.526241
$406$	33.1668	1.64604
$407$	10.6257	0.526696
$408$	2.22696	0.110251
$409$	−9.78660	−0.483916	−0.241958	−	0.970287i	$-0.577790\pi$
$409$	−9.78660	−0.483916	−0.241958	+	0.970287i	$0.577790\pi$
$410$	10.1533	0.501437
$411$	−14.4513	−0.712829
$412$	−13.2867	−0.654589
$413$	63.3967	3.11955
$414$	4.80813	0.236307
$415$	3.12812	0.153553
$416$	6.94316	0.340416
$417$	−9.55673	−0.467995
$418$	−2.17177	−0.106225
$419$	25.5811	1.24972	0.624859	−	0.780737i	$-0.285156\pi$
$419$	25.5811	1.24972	0.624859	+	0.780737i	$0.285156\pi$
$420$	9.88264	0.482223
$421$	−33.5592	−1.63558	−0.817788	−	0.575519i	$-0.804800\pi$
$421$	−33.5592	−1.63558	−0.817788	+	0.575519i	$0.804800\pi$
$422$	0.162550	0.00791280
$423$	−0.635698	−0.0309087
$424$	−4.71620	−0.229039
$425$	4.07962	0.197891
$426$	30.7433	1.48952
$427$	−2.09972	−0.101613
$428$	−1.28380	−0.0620550
$429$	−15.4622	−0.746520
$430$	1.68544	0.0812792
$431$	4.10116	0.197546	0.0987729	−	0.995110i	$-0.468508\pi$
$431$	4.10116	0.197546	0.0987729	+	0.995110i	$0.468508\pi$
$432$	−2.31746	−0.111499
$433$	−28.9758	−1.39249	−0.696245	−	0.717805i	$-0.745147\pi$
$433$	−28.9758	−1.39249	−0.696245	+	0.717805i	$0.745147\pi$
$434$	−42.4182	−2.03614
$435$	−15.3187	−0.734476
$436$	11.6240	0.556691
$437$	−5.32937	−0.254938
$438$	−1.91938	−0.0917117
$439$	−23.1675	−1.10572	−0.552862	−	0.833273i	$-0.686464\pi$
$439$	−23.1675	−1.10572	−0.552862	+	0.833273i	$0.686464\pi$
$440$	−0.959363	−0.0457358
$441$	28.2091	1.34329
$442$	−6.94316	−0.330252
$443$	−12.3693	−0.587685	−0.293843	−	0.955854i	$-0.594934\pi$
$443$	−12.3693	−0.587685	−0.293843	+	0.955854i	$0.594934\pi$
$444$	23.6630	1.12300
$445$	−8.88429	−0.421156
$446$	10.1718	0.481647
$447$	22.5530	1.06672
$448$	4.62570	0.218544
$449$	16.2114	0.765064	0.382532	−	0.923942i	$-0.375052\pi$
$449$	16.2114	0.765064	0.382532	+	0.923942i	$0.375052\pi$
$450$	7.99346	0.376815
$451$	−10.5834	−0.498353
$452$	2.79747	0.131582
$453$	14.9567	0.702728
$454$	29.0380	1.36282
$455$	−30.8118	−1.44448
$456$	−4.83646	−0.226488
$457$	−18.8754	−0.882956	−0.441478	−	0.897272i	$-0.645546\pi$
$457$	−18.8754	−0.882956	−0.441478	+	0.897272i	$0.645546\pi$
$458$	−15.9967	−0.747477
$459$	2.31746	0.108170
$460$	−2.35420	−0.109765
$461$	24.9583	1.16242	0.581212	−	0.813752i	$-0.302579\pi$
$461$	24.9583	1.16242	0.581212	+	0.813752i	$0.302579\pi$
$462$	−10.3013	−0.479258
$463$	14.5197	0.674786	0.337393	−	0.941364i	$-0.390455\pi$
$463$	14.5197	0.674786	0.337393	+	0.941364i	$0.390455\pi$
$464$	−7.17012	−0.332865
$465$	19.5916	0.908541
$466$	−5.93228	−0.274808
$467$	25.8355	1.19552	0.597761	−	0.801674i	$-0.296057\pi$
$467$	25.8355	1.19552	0.597761	+	0.801674i	$0.296057\pi$
$468$	−13.6042	−0.628853
$469$	−2.09972	−0.0969561
$470$	0.311257	0.0143572
$471$	0.946060	0.0435922
$472$	−13.7053	−0.630839
$473$	−1.75683	−0.0807793
$474$	6.91630	0.317676
$475$	−8.86002	−0.406525
$476$	−4.62570	−0.212019
$477$	9.24074	0.423104
$478$	−21.8157	−0.997827
$479$	4.01356	0.183384	0.0916921	−	0.995787i	$-0.470772\pi$
$479$	4.01356	0.183384	0.0916921	+	0.995787i	$0.470772\pi$
$480$	−2.13646	−0.0975158
$481$	−73.7759	−3.36389
$482$	−6.90785	−0.314644
$483$	−25.2785	−1.15021
$484$	1.00000	0.0454545
$485$	−13.0704	−0.593496
$486$	17.6310	0.799760
$487$	−11.7193	−0.531051	−0.265526	−	0.964104i	$-0.585546\pi$
$487$	−11.7193	−0.531051	−0.265526	+	0.964104i	$0.585546\pi$
$488$	0.453925	0.0205482
$489$	−38.7066	−1.75037
$490$	−13.8120	−0.623964
$491$	17.6538	0.796705	0.398353	−	0.917232i	$-0.369582\pi$
$491$	17.6538	0.796705	0.398353	+	0.917232i	$0.369582\pi$
$492$	−23.5689	−1.06257
$493$	7.17012	0.322926
$494$	15.0790	0.678435
$495$	1.87974	0.0844880
$496$	9.17012	0.411751
$497$	−63.8580	−2.86442
$498$	−7.26128	−0.325386
$499$	3.36798	0.150771	0.0753857	−	0.997154i	$-0.475981\pi$
$499$	3.36798	0.150771	0.0753857	+	0.997154i	$0.475981\pi$
$500$	−8.71065	−0.389552
$501$	4.09340	0.182880
$502$	−1.57518	−0.0703037
$503$	26.9475	1.20153	0.600765	−	0.799426i	$-0.294863\pi$
$503$	26.9475	1.20153	0.600765	+	0.799426i	$0.294863\pi$
$504$	−9.06342	−0.403717
$505$	−6.67655	−0.297103
$506$	2.45393	0.109090
$507$	78.4057	3.48212
$508$	16.4539	0.730025
$509$	−12.9045	−0.571984	−0.285992	−	0.958232i	$-0.592323\pi$
$509$	−12.9045	−0.571984	−0.285992	+	0.958232i	$0.592323\pi$
$510$	2.13646	0.0946042
$511$	3.98682	0.176366
$512$	−1.00000	−0.0441942
$513$	−5.03300	−0.222212
$514$	−4.73608	−0.208899
$515$	−12.7468	−0.561690
$516$	−3.91240	−0.172234
$517$	−0.324441	−0.0142689
$518$	−49.1513	−2.15958
$519$	37.6532	1.65279
$520$	6.66101	0.292104
$521$	−42.4076	−1.85791	−0.928954	−	0.370194i	$-0.879291\pi$
$521$	−42.4076	−1.85791	−0.928954	+	0.370194i	$0.879291\pi$
$522$	14.0489	0.614902
$523$	−8.84898	−0.386939	−0.193470	−	0.981106i	$-0.561974\pi$
$523$	−8.84898	−0.386939	−0.193470	+	0.981106i	$0.561974\pi$
$524$	9.43239	0.412056
$525$	−42.0253	−1.83413
$526$	−14.6838	−0.640244
$527$	−9.17012	−0.399457
$528$	2.22696	0.0969161
$529$	−16.9783	−0.738185
$530$	−4.52454	−0.196534
$531$	26.8537	1.16535
$532$	10.0460	0.435548
$533$	73.4823	3.18287
$534$	20.6231	0.892447
$535$	−1.23163	−0.0532481
$536$	0.453925	0.0196066
$537$	0.765064	0.0330149
$538$	−9.37430	−0.404155
$539$	14.3971	0.620126
$540$	−2.22329	−0.0956750
$541$	43.4291	1.86716	0.933581	−	0.358366i	$-0.116666\pi$
$541$	43.4291	1.86716	0.933581	+	0.358366i	$0.116666\pi$
$542$	−2.87875	−0.123653
$543$	36.3178	1.55855
$544$	1.00000	0.0428746
$545$	11.1517	0.477685
$546$	71.5233	3.06091
$547$	23.6240	1.01009	0.505046	−	0.863093i	$-0.331476\pi$
$547$	23.6240	1.01009	0.505046	+	0.863093i	$0.331476\pi$
$548$	−6.48923	−0.277206
$549$	−0.889404	−0.0379588
$550$	4.07962	0.173956
$551$	−15.5719	−0.663384
$552$	5.46480	0.232597
$553$	−14.3661	−0.610908
$554$	14.4506	0.613948
$555$	22.7014	0.963622
$556$	−4.29138	−0.181995
$557$	0.819879	0.0347394	0.0173697	−	0.999849i	$-0.494471\pi$
$557$	0.819879	0.0347394	0.0173697	+	0.999849i	$0.494471\pi$
$558$	−17.9676	−0.760629
$559$	12.1980	0.515919
$560$	4.43772	0.187528
$561$	−2.22696	−0.0940224
$562$	14.5319	0.612991
$563$	7.15102	0.301379	0.150690	−	0.988581i	$-0.451851\pi$
$563$	7.15102	0.301379	0.150690	+	0.988581i	$0.451851\pi$
$564$	−0.722518	−0.0304235
$565$	2.68379	0.112908
$566$	5.26984	0.221508
$567$	−51.0630	−2.14445
$568$	13.8050	0.579247
$569$	−1.39749	−0.0585857	−0.0292928	−	0.999571i	$-0.509326\pi$
$569$	−1.39749	−0.0585857	−0.0292928	+	0.999571i	$0.509326\pi$
$570$	−4.63992	−0.194345
$571$	26.9045	1.12592	0.562960	−	0.826484i	$-0.309662\pi$
$571$	26.9045	1.12592	0.562960	+	0.826484i	$0.309662\pi$
$572$	−6.94316	−0.290308
$573$	−12.7842	−0.534066
$574$	48.9557	2.04337
$575$	10.0111	0.417491
$576$	1.95936	0.0816401
$577$	20.6713	0.860556	0.430278	−	0.902696i	$-0.358416\pi$
$577$	20.6713	0.860556	0.430278	+	0.902696i	$0.358416\pi$
$578$	−1.00000	−0.0415945
$579$	−27.6321	−1.14835
$580$	−6.87875	−0.285624
$581$	15.0826	0.625733
$582$	30.3402	1.25764
$583$	4.71620	0.195325
$584$	−0.861884	−0.0356650
$585$	−13.0513	−0.539606
$586$	−16.1262	−0.666166
$587$	−17.9003	−0.738824	−0.369412	−	0.929266i	$-0.620441\pi$
$587$	−17.9003	−0.738824	−0.369412	+	0.929266i	$0.620441\pi$
$588$	32.0618	1.32221
$589$	19.9154	0.820601
$590$	−13.1484	−0.541310
$591$	15.0289	0.618206
$592$	10.6257	0.436714
$593$	−13.1701	−0.540832	−0.270416	−	0.962744i	$-0.587161\pi$
$593$	−13.1701	−0.540832	−0.270416	+	0.962744i	$0.587161\pi$
$594$	2.31746	0.0950866
$595$	−4.43772	−0.181929
$596$	10.1272	0.414828
$597$	19.0076	0.777928
$598$	−17.0380	−0.696735
$599$	21.8157	0.891365	0.445683	−	0.895191i	$-0.352961\pi$
$599$	21.8157	0.891365	0.445683	+	0.895191i	$0.352961\pi$
$600$	9.08517	0.370900
$601$	−10.0721	−0.410848	−0.205424	−	0.978673i	$-0.565857\pi$
$601$	−10.0721	−0.410848	−0.205424	+	0.978673i	$0.565857\pi$
$602$	8.12658	0.331215
$603$	−0.889404	−0.0362193
$604$	6.71620	0.273278
$605$	0.959363	0.0390036
$606$	15.4982	0.629573
$607$	−22.0971	−0.896893	−0.448446	−	0.893810i	$-0.648022\pi$
$607$	−22.0971	−0.896893	−0.448446	+	0.893810i	$0.648022\pi$
$608$	−2.17177	−0.0880770
$609$	−73.8613	−2.99301
$610$	0.435479	0.0176320
$611$	2.25265	0.0911323
$612$	−1.95936	−0.0792025
$613$	36.8523	1.48845	0.744224	−	0.667930i	$-0.232819\pi$
$613$	36.8523	1.48845	0.744224	+	0.667930i	$0.232819\pi$
$614$	−2.03201	−0.0820051
$615$	−22.6111	−0.911767
$616$	−4.62570	−0.186375
$617$	−28.5286	−1.14852	−0.574259	−	0.818674i	$-0.694710\pi$
$617$	−28.5286	−1.14852	−0.574259	+	0.818674i	$0.694710\pi$
$618$	29.5890	1.19024
$619$	11.8219	0.475162	0.237581	−	0.971368i	$-0.423645\pi$
$619$	11.8219	0.475162	0.237581	+	0.971368i	$0.423645\pi$
$620$	8.79747	0.353315
$621$	5.68688	0.228206
$622$	−14.7453	−0.591233
$623$	−42.8368	−1.71622
$624$	−15.4622	−0.618982
$625$	12.0414	0.481658
$626$	25.4628	1.01770
$627$	4.83646	0.193149
$628$	0.424821	0.0169522
$629$	−10.6257	−0.423674
$630$	−8.69511	−0.346421
$631$	−2.06573	−0.0822354	−0.0411177	−	0.999154i	$-0.513092\pi$
$631$	−2.06573	−0.0822354	−0.0411177	+	0.999154i	$0.513092\pi$
$632$	3.10571	0.123538
$633$	−0.361992	−0.0143879
$634$	3.62405	0.143929
$635$	15.7853	0.626420
$636$	10.5028	0.416463
$637$	−99.9612	−3.96061
$638$	7.17012	0.283868
$639$	−27.0491	−1.07005
$640$	−0.959363	−0.0379221
$641$	−8.63162	−0.340928	−0.170464	−	0.985364i	$-0.554527\pi$
$641$	−8.63162	−0.340928	−0.170464	+	0.985364i	$0.554527\pi$
$642$	2.85898	0.112835
$643$	−29.0704	−1.14642	−0.573212	−	0.819407i	$-0.694303\pi$
$643$	−29.0704	−1.14642	−0.573212	+	0.819407i	$0.694303\pi$
$644$	−11.3511	−0.447297
$645$	−3.75341	−0.147790
$646$	2.17177	0.0854473
$647$	36.4060	1.43127	0.715633	−	0.698476i	$-0.246138\pi$
$647$	36.4060	1.43127	0.715633	+	0.698476i	$0.246138\pi$
$648$	11.0390	0.433652
$649$	13.7053	0.537981
$650$	−28.3255	−1.11102
$651$	94.4638	3.70233
$652$	−17.3809	−0.680688
$653$	−47.1436	−1.84487	−0.922436	−	0.386149i	$-0.873805\pi$
$653$	−47.1436	−1.84487	−0.922436	+	0.386149i	$0.873805\pi$
$654$	−25.8863	−1.01224
$655$	9.04908	0.353577
$656$	−10.5834	−0.413213
$657$	1.68874	0.0658842
$658$	1.50077	0.0585060
$659$	12.4936	0.486680	0.243340	−	0.969941i	$-0.421757\pi$
$659$	12.4936	0.486680	0.243340	+	0.969941i	$0.421757\pi$
$660$	2.13646	0.0831618
$661$	2.97847	0.115849	0.0579245	−	0.998321i	$-0.481552\pi$
$661$	2.97847	0.115849	0.0579245	+	0.998321i	$0.481552\pi$
$662$	−2.40506	−0.0934752
$663$	15.4622	0.600500
$664$	−3.26062	−0.126537
$665$	9.63772	0.373735
$666$	−20.8196	−0.806743
$667$	17.5949	0.681279
$668$	1.83811	0.0711186
$669$	−22.6522	−0.875783
$670$	0.435479	0.0168240
$671$	−0.453925	−0.0175236
$672$	−10.3013	−0.397380
$673$	26.7460	1.03098	0.515490	−	0.856895i	$-0.327610\pi$
$673$	26.7460	1.03098	0.515490	+	0.856895i	$0.327610\pi$
$674$	−4.24317	−0.163441
$675$	9.45437	0.363899
$676$	35.2075	1.35413
$677$	26.7096	1.02653	0.513266	−	0.858229i	$-0.328435\pi$
$677$	26.7096	1.02653	0.513266	+	0.858229i	$0.328435\pi$
$678$	−6.22986	−0.239256
$679$	−63.0207	−2.41851
$680$	0.959363	0.0367899
$681$	−64.6665	−2.47803
$682$	−9.17012	−0.351142
$683$	13.8024	0.528134	0.264067	−	0.964504i	$-0.414936\pi$
$683$	13.8024	0.528134	0.264067	+	0.964504i	$0.414936\pi$
$684$	4.25529	0.162705
$685$	−6.22553	−0.237865
$686$	−34.2167	−1.30640
$687$	35.6240	1.35914
$688$	−1.75683	−0.0669787
$689$	−32.7453	−1.24750
$690$	5.24272	0.199587
$691$	30.8140	1.17222	0.586111	−	0.810231i	$-0.300658\pi$
$691$	30.8140	1.17222	0.586111	+	0.810231i	$0.300658\pi$
$692$	16.9079	0.642740
$693$	9.06342	0.344291
$694$	10.7162	0.406781
$695$	−4.11698	−0.156166
$696$	15.9676	0.605250
$697$	10.5834	0.400875
$698$	13.8375	0.523755
$699$	13.2110	0.499685
$700$	−18.8711	−0.713261
$701$	35.0181	1.32262	0.661308	−	0.750115i	$-0.270002\pi$
$701$	35.0181	1.32262	0.661308	+	0.750115i	$0.270002\pi$
$702$	−16.0905	−0.607297
$703$	23.0766	0.870351
$704$	1.00000	0.0376889
$705$	−0.693157	−0.0261058
$706$	−9.29961	−0.349995
$707$	−32.1919	−1.21070
$708$	30.5212	1.14706
$709$	20.0979	0.754791	0.377395	−	0.926052i	$-0.376820\pi$
$709$	20.0979	0.754791	0.377395	+	0.926052i	$0.376820\pi$
$710$	13.2440	0.497040
$711$	−6.08521	−0.228213
$712$	9.26062	0.347056
$713$	−22.5028	−0.842736
$714$	10.3013	0.385515
$715$	−6.66101	−0.249108
$716$	0.343546	0.0128389
$717$	48.5828	1.81436
$718$	−3.54607	−0.132338
$719$	22.3369	0.833027	0.416514	−	0.909129i	$-0.363252\pi$
$719$	22.3369	0.833027	0.416514	+	0.909129i	$0.363252\pi$
$720$	1.87974	0.0700537
$721$	−61.4603	−2.28890
$722$	14.2834	0.531573
$723$	15.3835	0.572119
$724$	16.3082	0.606091
$725$	29.2514	1.08637
$726$	−2.22696	−0.0826503
$727$	−0.0168621	−0.000625380 0	−0.000312690	−	1.00000i	$-0.500100\pi$
$727$	−0.0168621	−0.000625380 0	−0.000312690		1.00000i	$0.500100\pi$
$728$	32.1170	1.19033
$729$	−6.14668	−0.227655
$730$	−0.826859	−0.0306034
$731$	1.75683	0.0649788
$732$	−1.01087	−0.0373630
$733$	−17.2032	−0.635414	−0.317707	−	0.948189i	$-0.602913\pi$
$733$	−17.2032	−0.635414	−0.317707	+	0.948189i	$0.602913\pi$
$734$	−18.6165	−0.687147
$735$	30.7589	1.13456
$736$	2.45393	0.0904529
$737$	−0.453925	−0.0167206
$738$	20.7367	0.763330
$739$	32.1457	1.18250	0.591249	−	0.806489i	$-0.298635\pi$
$739$	32.1457	1.18250	0.591249	+	0.806489i	$0.298635\pi$
$740$	10.1939	0.374735
$741$	−33.5803	−1.23360
$742$	−21.8157	−0.800880
$743$	−50.2596	−1.84385	−0.921923	−	0.387372i	$-0.873383\pi$
$743$	−50.2596	−1.84385	−0.921923	+	0.387372i	$0.873383\pi$
$744$	−20.4215	−0.748689
$745$	9.71570	0.355956
$746$	16.8986	0.618703
$747$	6.38874	0.233752
$748$	−1.00000	−0.0365636
$749$	−5.93849	−0.216988
$750$	19.3983	0.708326
$751$	−5.55343	−0.202648	−0.101324	−	0.994854i	$-0.532308\pi$
$751$	−5.55343	−0.202648	−0.101324	+	0.994854i	$0.532308\pi$
$752$	−0.324441	−0.0118312
$753$	3.50787	0.127834
$754$	−49.7833	−1.81300
$755$	6.44327	0.234495
$756$	−10.7199	−0.389878
$757$	−42.7400	−1.55341	−0.776706	−	0.629863i	$-0.783111\pi$
$757$	−42.7400	−1.55341	−0.776706	+	0.629863i	$0.783111\pi$
$758$	−17.0558	−0.619494
$759$	−5.46480	−0.198360
$760$	−2.08352	−0.0755771
$761$	−1.18408	−0.0429230	−0.0214615	−	0.999770i	$-0.506832\pi$
$761$	−1.18408	−0.0429230	−0.0214615	+	0.999770i	$0.506832\pi$
$762$	−36.6423	−1.32741
$763$	53.7693	1.94658
$764$	−5.74063	−0.207689
$765$	−1.87974	−0.0679621
$766$	7.09215	0.256250
$767$	−95.1582	−3.43596
$768$	2.22696	0.0803586
$769$	18.3369	0.661247	0.330623	−	0.943763i	$-0.392741\pi$
$769$	18.3369	0.661247	0.330623	+	0.943763i	$0.392741\pi$
$770$	−4.43772	−0.159924
$771$	10.5471	0.379843
$772$	−12.4080	−0.446572
$773$	7.08436	0.254807	0.127403	−	0.991851i	$-0.459336\pi$
$773$	7.08436	0.254807	0.127403	+	0.991851i	$0.459336\pi$
$774$	3.44227	0.123730
$775$	−37.4106	−1.34383
$776$	13.6240	0.489075
$777$	109.458	3.92679
$778$	−0.826576	−0.0296342
$779$	−22.9848	−0.823515
$780$	−14.8338	−0.531136
$781$	−13.8050	−0.493983
$782$	−2.45393	−0.0877522
$783$	16.6165	0.593824
$784$	14.3971	0.514182
$785$	0.407557	0.0145463
$786$	−21.0056	−0.749244
$787$	−35.9352	−1.28095	−0.640476	−	0.767979i	$-0.721263\pi$
$787$	−35.9352	−1.28095	−0.640476	+	0.767979i	$0.721263\pi$
$788$	6.74860	0.240409
$789$	32.7002	1.16416
$790$	2.97950	0.106006
$791$	12.9403	0.460103
$792$	−1.95936	−0.0696229
$793$	3.15168	0.111919
$794$	−30.2173	−1.07237
$795$	10.0760	0.357358
$796$	8.53520	0.302522
$797$	−24.1777	−0.856418	−0.428209	−	0.903680i	$-0.640855\pi$
$797$	−24.1777	−0.856418	−0.428209	+	0.903680i	$0.640855\pi$
$798$	−22.3720	−0.791960
$799$	0.324441	0.0114779
$800$	4.07962	0.144236
$801$	−18.1449	−0.641119
$802$	−18.1301	−0.640198
$803$	0.861884	0.0304152
$804$	−1.01087	−0.0356508
$805$	−10.8898	−0.383816
$806$	63.6696	2.24267
$807$	20.8762	0.734878
$808$	6.95936	0.244830
$809$	−35.1681	−1.23645	−0.618223	−	0.786003i	$-0.712147\pi$
$809$	−35.1681	−1.23645	−0.618223	+	0.786003i	$0.712147\pi$
$810$	10.5904	0.372108
$811$	23.3967	0.821569	0.410784	−	0.911733i	$-0.365255\pi$
$811$	23.3967	0.821569	0.410784	+	0.911733i	$0.365255\pi$
$812$	−33.1668	−1.16393
$813$	6.41086	0.224839
$814$	−10.6257	−0.372431
$815$	−16.6746	−0.584085
$816$	−2.22696	−0.0779593
$817$	−3.81544	−0.133485
$818$	9.78660	0.342180
$819$	−62.9288	−2.19891
$820$	−10.1533	−0.354570
$821$	18.7744	0.655231	0.327616	−	0.944811i	$-0.393755\pi$
$821$	18.7744	0.655231	0.327616	+	0.944811i	$0.393755\pi$
$822$	14.4513	0.504046
$823$	29.3047	1.02150	0.510750	−	0.859729i	$-0.329368\pi$
$823$	29.3047	1.02150	0.510750	+	0.859729i	$0.329368\pi$
$824$	13.2867	0.462864
$825$	−9.08517	−0.316305
$826$	−63.3967	−2.20585
$827$	−23.4431	−0.815195	−0.407597	−	0.913162i	$-0.633633\pi$
$827$	−23.4431	−0.815195	−0.407597	+	0.913162i	$0.633633\pi$
$828$	−4.80813	−0.167094
$829$	−54.4658	−1.89167	−0.945837	−	0.324642i	$-0.894756\pi$
$829$	−54.4658	−1.89167	−0.945837	+	0.324642i	$0.894756\pi$
$830$	−3.12812	−0.108579
$831$	−32.1810	−1.11635
$832$	−6.94316	−0.240711
$833$	−14.3971	−0.498829
$834$	9.55673	0.330923
$835$	1.76341	0.0610254
$836$	2.17177	0.0751123
$837$	−21.2514	−0.734556
$838$	−25.5811	−0.883684
$839$	−35.5288	−1.22659	−0.613295	−	0.789854i	$-0.710156\pi$
$839$	−35.5288	−1.22659	−0.613295	+	0.789854i	$0.710156\pi$
$840$	−9.88264	−0.340983
$841$	22.4106	0.772781
$842$	33.5592	1.15653
$843$	−32.3620	−1.11461
$844$	−0.162550	−0.00559519
$845$	33.7767	1.16195
$846$	0.635698	0.0218557
$847$	4.62570	0.158941
$848$	4.71620	0.161955
$849$	−11.7357	−0.402770
$850$	−4.07962	−0.139930
$851$	−26.0747	−0.893828
$852$	−30.7433	−1.05325
$853$	−25.4184	−0.870311	−0.435155	−	0.900355i	$-0.643307\pi$
$853$	−25.4184	−0.870311	−0.435155	+	0.900355i	$0.643307\pi$
$854$	2.09972	0.0718510
$855$	4.08237	0.139614
$856$	1.28380	0.0438795
$857$	47.4794	1.62187	0.810933	−	0.585140i	$-0.198960\pi$
$857$	47.4794	1.62187	0.810933	+	0.585140i	$0.198960\pi$
$858$	15.4622	0.527869
$859$	−19.6762	−0.671344	−0.335672	−	0.941979i	$-0.608963\pi$
$859$	−19.6762	−0.671344	−0.335672	+	0.941979i	$0.608963\pi$
$860$	−1.68544	−0.0574730
$861$	−109.022	−3.71547
$862$	−4.10116	−0.139686
$863$	−33.2389	−1.13146	−0.565732	−	0.824589i	$-0.691406\pi$
$863$	−33.2389	−1.13146	−0.565732	+	0.824589i	$0.691406\pi$
$864$	2.31746	0.0788416
$865$	16.2208	0.551522
$866$	28.9758	0.984639
$867$	2.22696	0.0756316
$868$	42.4182	1.43977
$869$	−3.10571	−0.105354
$870$	15.3187	0.519353
$871$	3.15168	0.106790
$872$	−11.6240	−0.393640
$873$	−26.6944	−0.903470
$874$	5.32937	0.180269
$875$	−40.2928	−1.36215
$876$	1.91938	0.0648500
$877$	11.3327	0.382677	0.191339	−	0.981524i	$-0.438717\pi$
$877$	11.3327	0.382677	0.191339	+	0.981524i	$0.438717\pi$
$878$	23.1675	0.781864
$879$	35.9124	1.21130
$880$	0.959363	0.0323401
$881$	36.9707	1.24557	0.622787	−	0.782392i	$-0.286000\pi$
$881$	36.9707	1.24557	0.622787	+	0.782392i	$0.286000\pi$
$882$	−28.2091	−0.949850
$883$	−7.18857	−0.241915	−0.120957	−	0.992658i	$-0.538596\pi$
$883$	−7.18857	−0.241915	−0.120957	+	0.992658i	$0.538596\pi$
$884$	6.94316	0.233524
$885$	29.2809	0.984268
$886$	12.3693	0.415556
$887$	−17.7145	−0.594796	−0.297398	−	0.954754i	$-0.596119\pi$
$887$	−17.7145	−0.594796	−0.297398	+	0.954754i	$0.596119\pi$
$888$	−23.6630	−0.794079
$889$	76.1109	2.55268
$890$	8.88429	0.297802
$891$	−11.0390	−0.369820
$892$	−10.1718	−0.340576
$893$	−0.704612	−0.0235790
$894$	−22.5530	−0.754285
$895$	0.329585	0.0110168
$896$	−4.62570	−0.154534
$897$	37.9430	1.26688
$898$	−16.2114	−0.540982
$899$	−65.7509	−2.19291
$900$	−7.99346	−0.266449
$901$	−4.71620	−0.157119
$902$	10.5834	0.352389
$903$	−18.0976	−0.602250
$904$	−2.79747	−0.0930425
$905$	15.6455	0.520074
$906$	−14.9567	−0.496904
$907$	−12.6378	−0.419632	−0.209816	−	0.977741i	$-0.567287\pi$
$907$	−12.6378	−0.419632	−0.209816	+	0.977741i	$0.567287\pi$
$908$	−29.0380	−0.963660
$909$	−13.6359	−0.452275
$910$	30.8118	1.02140
$911$	−25.0921	−0.831340	−0.415670	−	0.909516i	$-0.636453\pi$
$911$	−25.0921	−0.831340	−0.415670	+	0.909516i	$0.636453\pi$
$912$	4.83646	0.160151
$913$	3.26062	0.107911
$914$	18.8754	0.624344
$915$	−0.969795	−0.0320604
$916$	15.9967	0.528546
$917$	43.6314	1.44084
$918$	−2.31746	−0.0764876
$919$	11.6380	0.383902	0.191951	−	0.981404i	$-0.438518\pi$
$919$	11.6380	0.383902	0.191951	+	0.981404i	$0.438518\pi$
$920$	2.35420	0.0776158
$921$	4.52520	0.149110
$922$	−24.9583	−0.821958
$923$	95.8506	3.15496
$924$	10.3013	0.338886
$925$	−43.3488	−1.42530
$926$	−14.5197	−0.477145
$927$	−26.0335	−0.855051
$928$	7.17012	0.235371
$929$	53.0023	1.73895	0.869474	−	0.493978i	$-0.164458\pi$
$929$	53.0023	1.73895	0.869474	+	0.493978i	$0.164458\pi$
$930$	−19.5916	−0.642435
$931$	31.2672	1.02474
$932$	5.93228	0.194318
$933$	32.8372	1.07504
$934$	−25.8355	−0.845362
$935$	−0.959363	−0.0313745
$936$	13.6042	0.444666
$937$	4.65799	0.152170	0.0760849	−	0.997101i	$-0.475758\pi$
$937$	4.65799	0.152170	0.0760849	+	0.997101i	$0.475758\pi$
$938$	2.09972	0.0685583
$939$	−56.7047	−1.85049
$940$	−0.311257	−0.0101521
$941$	19.4997	0.635672	0.317836	−	0.948146i	$-0.397044\pi$
$941$	19.4997	0.635672	0.317836	+	0.948146i	$0.397044\pi$
$942$	−0.946060	−0.0308243
$943$	25.9709	0.845729
$944$	13.7053	0.446070
$945$	−10.2842	−0.334547
$946$	1.75683	0.0571196
$947$	56.3634	1.83156	0.915782	−	0.401675i	$-0.131572\pi$
$947$	56.3634	1.83156	0.915782	+	0.401675i	$0.131572\pi$
$948$	−6.91630	−0.224631
$949$	−5.98420	−0.194255
$950$	8.86002	0.287457
$951$	−8.07062	−0.261708
$952$	4.62570	0.149920
$953$	36.0944	1.16921	0.584607	−	0.811317i	$-0.301249\pi$
$953$	36.0944	1.16921	0.584607	+	0.811317i	$0.301249\pi$
$954$	−9.24074	−0.299180
$955$	−5.50735	−0.178214
$956$	21.8157	0.705570
$957$	−15.9676	−0.516159
$958$	−4.01356	−0.129672
$959$	−30.0172	−0.969307
$960$	2.13646	0.0689541
$961$	53.0911	1.71262
$962$	73.7759	2.37863
$963$	−2.51544	−0.0810588
$964$	6.90785	0.222487
$965$	−11.9037	−0.383195
$966$	25.2785	0.813323
$967$	13.8605	0.445724	0.222862	−	0.974850i	$-0.428460\pi$
$967$	13.8605	0.445724	0.222862	+	0.974850i	$0.428460\pi$
$968$	−1.00000	−0.0321412
$969$	−4.83646	−0.155369
$970$	13.0704	0.419665
$971$	36.2603	1.16365	0.581824	−	0.813315i	$-0.302339\pi$
$971$	36.2603	1.16365	0.581824	+	0.813315i	$0.302339\pi$
$972$	−17.6310	−0.565515
$973$	−19.8506	−0.636381
$974$	11.7193	0.375510
$975$	63.0798	2.02017
$976$	−0.453925	−0.0145298
$977$	−24.3549	−0.779181	−0.389591	−	0.920988i	$-0.627383\pi$
$977$	−24.3549	−0.779181	−0.389591	+	0.920988i	$0.627383\pi$
$978$	38.7066	1.23770
$979$	−9.26062	−0.295971
$980$	13.8120	0.441209
$981$	22.7757	0.727173
$982$	−17.6538	−0.563356
$983$	−53.3076	−1.70025	−0.850125	−	0.526582i	$-0.823473\pi$
$983$	−53.3076	−1.70025	−0.850125	+	0.526582i	$0.823473\pi$
$984$	23.5689	0.751348
$985$	6.47436	0.206290
$986$	−7.17012	−0.228343
$987$	−3.34215	−0.106382
$988$	−15.0790	−0.479726
$989$	4.31114	0.137086
$990$	−1.87974	−0.0597420
$991$	−59.7833	−1.89908	−0.949539	−	0.313648i	$-0.898449\pi$
$991$	−59.7833	−1.89908	−0.949539	+	0.313648i	$0.898449\pi$
$992$	−9.17012	−0.291152
$993$	5.35597	0.169967
$994$	63.8580	2.02545
$995$	8.18835	0.259588
$996$	7.26128	0.230082
$997$	−28.4473	−0.900936	−0.450468	−	0.892793i	$-0.648743\pi$
$997$	−28.4473	−0.900936	−0.450468	+	0.892793i	$0.648743\pi$
$998$	−3.36798	−0.106612
$999$	−24.6246	−0.779089

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.d.1.3	✓	4
3.2	odd	2		3366.2.a.bg.1.3		4
4.3	odd	2		2992.2.a.w.1.2		4
5.4	even	2		9350.2.a.cl.1.2		4
11.10	odd	2		4114.2.a.bc.1.3		4
17.16	even	2		6358.2.a.t.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
374.2.a.d.1.3	✓	4	1.1	even	1	trivial
2992.2.a.w.1.2		4	4.3	odd	2
3366.2.a.bg.1.3		4	3.2	odd	2
4114.2.a.bc.1.3		4	11.10	odd	2
6358.2.a.t.1.2		4	17.16	even	2
9350.2.a.cl.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.d.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	$1$