LMFDB - Embedded newform 374.2.a.b.1.1

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 = [3,-3] 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:	$3$
Coefficient field:	3.3.785.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 6x + 5$ x^3 - x^2 - 6*x + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.57577$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.57577 q^{3} +1.00000 q^{4} -2.63461 q^{5} +2.57577 q^{6} -4.21038 q^{7} -1.00000 q^{8} +3.63461 q^{9} +2.63461 q^{10} -1.00000 q^{11} -2.57577 q^{12} +1.42423 q^{13} +4.21038 q^{14} +6.78616 q^{15} +1.00000 q^{16} +1.00000 q^{17} -3.63461 q^{18} +1.05884 q^{19} -2.63461 q^{20} +10.8450 q^{21} +1.00000 q^{22} -7.15155 q^{23} +2.57577 q^{24} +1.94116 q^{25} -1.42423 q^{26} -1.63461 q^{27} -4.21038 q^{28} +7.15155 q^{29} -6.78616 q^{30} +9.15155 q^{31} -1.00000 q^{32} +2.57577 q^{33} -1.00000 q^{34} +11.0927 q^{35} +3.63461 q^{36} +4.09271 q^{37} -1.05884 q^{38} -3.66848 q^{39} +2.63461 q^{40} -7.90383 q^{41} -10.8450 q^{42} +4.63461 q^{43} -1.00000 q^{44} -9.57577 q^{45} +7.15155 q^{46} -5.66848 q^{47} -2.57577 q^{48} +10.7273 q^{49} -1.94116 q^{50} -2.57577 q^{51} +1.42423 q^{52} +10.5384 q^{53} +1.63461 q^{54} +2.63461 q^{55} +4.21038 q^{56} -2.72732 q^{57} -7.15155 q^{58} -11.2692 q^{59} +6.78616 q^{60} -5.15155 q^{61} -9.15155 q^{62} -15.3031 q^{63} +1.00000 q^{64} -3.75228 q^{65} -2.57577 q^{66} -2.84845 q^{67} +1.00000 q^{68} +18.4208 q^{69} -11.0927 q^{70} +0.420765 q^{71} -3.63461 q^{72} -3.84499 q^{73} -4.09271 q^{74} -5.00000 q^{75} +1.05884 q^{76} +4.21038 q^{77} +3.66848 q^{78} +2.57577 q^{79} -2.63461 q^{80} -6.69344 q^{81} +7.90383 q^{82} -5.05884 q^{83} +10.8450 q^{84} -2.63461 q^{85} -4.63461 q^{86} -18.4208 q^{87} +1.00000 q^{88} -15.2443 q^{89} +9.57577 q^{90} -5.99654 q^{91} -7.15155 q^{92} -23.5723 q^{93} +5.66848 q^{94} -2.78962 q^{95} +2.57577 q^{96} +10.3031 q^{97} -10.7273 q^{98} -3.63461 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 3 q^{2} - q^{3} + 3 q^{4} - q^{5} + q^{6} + q^{7} - 3 q^{8} + 4 q^{9} + q^{10} - 3 q^{11} - q^{12} + 11 q^{13} - q^{14} + 3 q^{16} + 3 q^{17} - 4 q^{18} + 3 q^{19} - q^{20} + 12 q^{21} + 3 q^{22}+ \cdots - 4 q^{99}+O(q^{100})$ 3 * q - 3 * q^2 - q^3 + 3 * q^4 - q^5 + q^6 + q^7 - 3 * q^8 + 4 * q^9 + q^10 - 3 * q^11 - q^12 + 11 * q^13 - q^14 + 3 * q^16 + 3 * q^17 - 4 * q^18 + 3 * q^19 - q^20 + 12 * q^21 + 3 * q^22 - 8 * q^23 + q^24 + 6 * q^25 - 11 * q^26 + 2 * q^27 + q^28 + 8 * q^29 + 14 * q^31 - 3 * q^32 + q^33 - 3 * q^34 + 20 * q^35 + 4 * q^36 - q^37 - 3 * q^38 + 9 * q^39 + q^40 - 3 * q^41 - 12 * q^42 + 7 * q^43 - 3 * q^44 - 22 * q^45 + 8 * q^46 + 3 * q^47 - q^48 + 12 * q^49 - 6 * q^50 - q^51 + 11 * q^52 + 4 * q^53 - 2 * q^54 + q^55 - q^56 + 12 * q^57 - 8 * q^58 - 20 * q^59 - 2 * q^61 - 14 * q^62 - 19 * q^63 + 3 * q^64 - 4 * q^65 - q^66 - 22 * q^67 + 3 * q^68 + 28 * q^69 - 20 * q^70 - 26 * q^71 - 4 * q^72 + 9 * q^73 + q^74 - 15 * q^75 + 3 * q^76 - q^77 - 9 * q^78 + q^79 - q^80 - 13 * q^81 + 3 * q^82 - 15 * q^83 + 12 * q^84 - q^85 - 7 * q^86 - 28 * q^87 + 3 * q^88 - 19 * q^89 + 22 * q^90 + 16 * q^91 - 8 * q^92 - 30 * q^93 - 3 * q^94 - 22 * q^95 + q^96 + 4 * q^97 - 12 * q^98 - 4 * 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
$3$	−2.57577	−1.48712	−0.743562	−	0.668667i	$-0.766865\pi$
$3$	−2.57577	−1.48712	−0.743562	+	0.668667i	$0.766865\pi$
$4$	1.00000	0.500000
$5$	−2.63461	−1.17823	−0.589116	−	0.808048i	$-0.700524\pi$
$5$	−2.63461	−1.17823	−0.589116	+	0.808048i	$0.700524\pi$
$6$	2.57577	1.05156
$7$	−4.21038	−1.59137	−0.795687	−	0.605707i	$-0.792890\pi$
$7$	−4.21038	−1.59137	−0.795687	+	0.605707i	$0.792890\pi$
$8$	−1.00000	−0.353553
$9$	3.63461	1.21154
$10$	2.63461	0.833137
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.57577	−0.743562
$13$	1.42423	0.395009	0.197505	−	0.980302i	$-0.436716\pi$
$13$	1.42423	0.395009	0.197505	+	0.980302i	$0.436716\pi$
$14$	4.21038	1.12527
$15$	6.78616	1.75218
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−3.63461	−0.856686
$19$	1.05884	0.242914	0.121457	−	0.992597i	$-0.461243\pi$
$19$	1.05884	0.242914	0.121457	+	0.992597i	$0.461243\pi$
$20$	−2.63461	−0.589116
$21$	10.8450	2.36657
$22$	1.00000	0.213201
$23$	−7.15155	−1.49120	−0.745600	−	0.666393i	$-0.767837\pi$
$23$	−7.15155	−1.49120	−0.745600	+	0.666393i	$0.767837\pi$
$24$	2.57577	0.525778
$25$	1.94116	0.388233
$26$	−1.42423	−0.279314
$27$	−1.63461	−0.314581
$28$	−4.21038	−0.795687
$29$	7.15155	1.32801	0.664004	−	0.747729i	$-0.268856\pi$
$29$	7.15155	1.32801	0.664004	+	0.747729i	$0.268856\pi$
$30$	−6.78616	−1.23898
$31$	9.15155	1.64367	0.821833	−	0.569728i	$-0.192952\pi$
$31$	9.15155	1.64367	0.821833	+	0.569728i	$0.192952\pi$
$32$	−1.00000	−0.176777
$33$	2.57577	0.448385
$34$	−1.00000	−0.171499
$35$	11.0927	1.87501
$36$	3.63461	0.605768
$37$	4.09271	0.672838	0.336419	−	0.941712i	$-0.390784\pi$
$37$	4.09271	0.672838	0.336419	+	0.941712i	$0.390784\pi$
$38$	−1.05884	−0.171766
$39$	−3.66848	−0.587428
$40$	2.63461	0.416568
$41$	−7.90383	−1.23437	−0.617185	−	0.786818i	$-0.711727\pi$
$41$	−7.90383	−1.23437	−0.617185	+	0.786818i	$0.711727\pi$
$42$	−10.8450	−1.67342
$43$	4.63461	0.706771	0.353386	−	0.935478i	$-0.385030\pi$
$43$	4.63461	0.706771	0.353386	+	0.935478i	$0.385030\pi$
$44$	−1.00000	−0.150756
$45$	−9.57577	−1.42747
$46$	7.15155	1.05444
$47$	−5.66848	−0.826834	−0.413417	−	0.910542i	$-0.635665\pi$
$47$	−5.66848	−0.826834	−0.413417	+	0.910542i	$0.635665\pi$
$48$	−2.57577	−0.371781
$49$	10.7273	1.53247
$50$	−1.94116	−0.274522
$51$	−2.57577	−0.360680
$52$	1.42423	0.197505
$53$	10.5384	1.44757	0.723783	−	0.690028i	$-0.242402\pi$
$53$	10.5384	1.44757	0.723783	+	0.690028i	$0.242402\pi$
$54$	1.63461	0.222442
$55$	2.63461	0.355251
$56$	4.21038	0.562636
$57$	−2.72732	−0.361242
$58$	−7.15155	−0.939044
$59$	−11.2692	−1.46713	−0.733564	−	0.679621i	$-0.762144\pi$
$59$	−11.2692	−1.46713	−0.733564	+	0.679621i	$0.762144\pi$
$60$	6.78616	0.876089
$61$	−5.15155	−0.659588	−0.329794	−	0.944053i	$-0.606979\pi$
$61$	−5.15155	−0.659588	−0.329794	+	0.944053i	$0.606979\pi$
$62$	−9.15155	−1.16225
$63$	−15.3031	−1.92801
$64$	1.00000	0.125000
$65$	−3.75228	−0.465413
$66$	−2.57577	−0.317056
$67$	−2.84845	−0.347994	−0.173997	−	0.984746i	$-0.555668\pi$
$67$	−2.84845	−0.347994	−0.173997	+	0.984746i	$0.555668\pi$
$68$	1.00000	0.121268
$69$	18.4208	2.21760
$70$	−11.0927	−1.32583
$71$	0.420765	0.0499356	0.0249678	−	0.999688i	$-0.492052\pi$
$71$	0.420765	0.0499356	0.0249678	+	0.999688i	$0.492052\pi$
$72$	−3.63461	−0.428343
$73$	−3.84499	−0.450022	−0.225011	−	0.974356i	$-0.572242\pi$
$73$	−3.84499	−0.450022	−0.225011	+	0.974356i	$0.572242\pi$
$74$	−4.09271	−0.475768
$75$	−5.00000	−0.577350
$76$	1.05884	0.121457
$77$	4.21038	0.479818
$78$	3.66848	0.415374
$79$	2.57577	0.289797	0.144899	−	0.989447i	$-0.453714\pi$
$79$	2.57577	0.289797	0.144899	+	0.989447i	$0.453714\pi$
$80$	−2.63461	−0.294558
$81$	−6.69344	−0.743716
$82$	7.90383	0.872832
$83$	−5.05884	−0.555279	−0.277640	−	0.960685i	$-0.589552\pi$
$83$	−5.05884	−0.555279	−0.277640	+	0.960685i	$0.589552\pi$
$84$	10.8450	1.18329
$85$	−2.63461	−0.285763
$86$	−4.63461	−0.499763
$87$	−18.4208	−1.97491
$88$	1.00000	0.106600
$89$	−15.2443	−1.61589	−0.807944	−	0.589259i	$-0.799420\pi$
$89$	−15.2443	−1.61589	−0.807944	+	0.589259i	$0.799420\pi$
$90$	9.57577	1.00938
$91$	−5.99654	−0.628608
$92$	−7.15155	−0.745600
$93$	−23.5723	−2.44433
$94$	5.66848	0.584660
$95$	−2.78962	−0.286209
$96$	2.57577	0.262889
$97$	10.3031	1.04612	0.523060	−	0.852296i	$-0.324790\pi$
$97$	10.3031	1.04612	0.523060	+	0.852296i	$0.324790\pi$
$98$	−10.7273	−1.08362
$99$	−3.63461	−0.365292
$100$	1.94116	0.194116
$101$	17.6685	1.75808	0.879040	−	0.476748i	$-0.158185\pi$
$101$	17.6685	1.75808	0.879040	+	0.476748i	$0.158185\pi$
$102$	2.57577	0.255040
$103$	7.84499	0.772990	0.386495	−	0.922292i	$-0.373686\pi$
$103$	7.84499	0.772990	0.386495	+	0.922292i	$0.373686\pi$
$104$	−1.42423	−0.139657
$105$	−28.5723	−2.78837
$106$	−10.5384	−1.02358
$107$	18.8415	1.82148	0.910740	−	0.412981i	$-0.135512\pi$
$107$	18.8415	1.82148	0.910740	+	0.412981i	$0.135512\pi$
$108$	−1.63461	−0.157290
$109$	16.3031	1.56155	0.780777	−	0.624810i	$-0.214824\pi$
$109$	16.3031	1.56155	0.780777	+	0.624810i	$0.214824\pi$
$110$	−2.63461	−0.251200
$111$	−10.5419	−1.00059
$112$	−4.21038	−0.397844
$113$	−6.73078	−0.633179	−0.316589	−	0.948563i	$-0.602538\pi$
$113$	−6.73078	−0.633179	−0.316589	+	0.948563i	$0.602538\pi$
$114$	2.72732	0.255437
$115$	18.8415	1.75698
$116$	7.15155	0.664004
$117$	5.17651	0.478568
$118$	11.2692	1.03742
$119$	−4.21038	−0.385965
$120$	−6.78616	−0.619488
$121$	1.00000	0.0909091
$122$	5.15155	0.466399
$123$	20.3585	1.83566
$124$	9.15155	0.821833
$125$	8.05884	0.720804
$126$	15.3031	1.36331
$127$	7.68998	0.682376	0.341188	−	0.939995i	$-0.389171\pi$
$127$	7.68998	0.682376	0.341188	+	0.939995i	$0.389171\pi$
$128$	−1.00000	−0.0883883
$129$	−11.9377	−1.05106
$130$	3.75228	0.329097
$131$	−10.5384	−0.920747	−0.460374	−	0.887725i	$-0.652285\pi$
$131$	−10.5384	−0.920747	−0.460374	+	0.887725i	$0.652285\pi$
$132$	2.57577	0.224192
$133$	−4.45810	−0.386567
$134$	2.84845	0.246069
$135$	4.30656	0.370649
$136$	−1.00000	−0.0857493
$137$	6.57577	0.561806	0.280903	−	0.959736i	$-0.409366\pi$
$137$	6.57577	0.561806	0.280903	+	0.959736i	$0.409366\pi$
$138$	−18.4208	−1.56808
$139$	−7.45464	−0.632294	−0.316147	−	0.948710i	$-0.602389\pi$
$139$	−7.45464	−0.632294	−0.316147	+	0.948710i	$0.602389\pi$
$140$	11.0927	0.937505
$141$	14.6007	1.22960
$142$	−0.420765	−0.0353098
$143$	−1.42423	−0.119100
$144$	3.63461	0.302884
$145$	−18.8415	−1.56470
$146$	3.84499	0.318214
$147$	−27.6311	−2.27898
$148$	4.09271	0.336419
$149$	18.2658	1.49639	0.748195	−	0.663479i	$-0.230921\pi$
$149$	18.2658	1.49639	0.748195	+	0.663479i	$0.230921\pi$
$150$	5.00000	0.408248
$151$	18.6062	1.51415	0.757075	−	0.653328i	$-0.226628\pi$
$151$	18.6062	1.51415	0.757075	+	0.653328i	$0.226628\pi$
$152$	−1.05884	−0.0858829
$153$	3.63461	0.293841
$154$	−4.21038	−0.339282
$155$	−24.1107	−1.93662
$156$	−3.66848	−0.293714
$157$	−17.9931	−1.43600	−0.718002	−	0.696041i	$-0.754943\pi$
$157$	−17.9931	−1.43600	−0.718002	+	0.696041i	$0.754943\pi$
$158$	−2.57577	−0.204918
$159$	−27.1446	−2.15271
$160$	2.63461	0.208284
$161$	30.1107	2.37306
$162$	6.69344	0.525887
$163$	−17.1730	−1.34510	−0.672548	−	0.740053i	$-0.734800\pi$
$163$	−17.1730	−1.34510	−0.672548	+	0.740053i	$0.734800\pi$
$164$	−7.90383	−0.617185
$165$	−6.78616	−0.528302
$166$	5.05884	0.392642
$167$	7.90383	0.611616	0.305808	−	0.952093i	$-0.401073\pi$
$167$	7.90383	0.611616	0.305808	+	0.952093i	$0.401073\pi$
$168$	−10.8450	−0.836709
$169$	−10.9716	−0.843968
$170$	2.63461	0.202065
$171$	3.84845	0.294299
$172$	4.63461	0.353386
$173$	−1.46156	−0.111121	−0.0555603	−	0.998455i	$-0.517695\pi$
$173$	−1.46156	−0.111121	−0.0555603	+	0.998455i	$0.517695\pi$
$174$	18.4208	1.39647
$175$	−8.17304	−0.617824
$176$	−1.00000	−0.0753778
$177$	29.0270	2.18180
$178$	15.2443	1.14261
$179$	12.4208	0.928372	0.464186	−	0.885738i	$-0.346347\pi$
$179$	12.4208	0.928372	0.464186	+	0.885738i	$0.346347\pi$
$180$	−9.57577	−0.713736
$181$	17.2319	1.28084	0.640418	−	0.768027i	$-0.278761\pi$
$181$	17.2319	1.28084	0.640418	+	0.768027i	$0.278761\pi$
$182$	5.99654	0.444493
$183$	13.2692	0.980889
$184$	7.15155	0.527219
$185$	−10.7827	−0.792759
$186$	23.5723	1.72841
$187$	−1.00000	−0.0731272
$188$	−5.66848	−0.413417
$189$	6.88233	0.500616
$190$	2.78962	0.202380
$191$	−1.30656	−0.0945390	−0.0472695	−	0.998882i	$-0.515052\pi$
$191$	−1.30656	−0.0945390	−0.0472695	+	0.998882i	$0.515052\pi$
$192$	−2.57577	−0.185890
$193$	11.2996	0.813365	0.406683	−	0.913569i	$-0.366685\pi$
$193$	11.2996	0.813365	0.406683	+	0.913569i	$0.366685\pi$
$194$	−10.3031	−0.739719
$195$	9.66502	0.692127
$196$	10.7273	0.766237
$197$	20.6561	1.47169	0.735843	−	0.677152i	$-0.236786\pi$
$197$	20.6561	1.47169	0.735843	+	0.677152i	$0.236786\pi$
$198$	3.63461	0.258300
$199$	6.11767	0.433670	0.216835	−	0.976208i	$-0.430427\pi$
$199$	6.11767	0.433670	0.216835	+	0.976208i	$0.430427\pi$
$200$	−1.94116	−0.137261
$201$	7.33697	0.517510
$202$	−17.6685	−1.24315
$203$	−30.1107	−2.11336
$204$	−2.57577	−0.180340
$205$	20.8235	1.45438
$206$	−7.84499	−0.546586
$207$	−25.9931	−1.80664
$208$	1.42423	0.0987523
$209$	−1.05884	−0.0732412
$210$	28.5723	1.97168
$211$	14.7738	1.01707	0.508534	−	0.861042i	$-0.330187\pi$
$211$	14.7738	1.01707	0.508534	+	0.861042i	$0.330187\pi$
$212$	10.5384	0.723783
$213$	−1.08380	−0.0742604
$214$	−18.8415	−1.28798
$215$	−12.2104	−0.832741
$216$	1.63461	0.111221
$217$	−38.5315	−2.61569
$218$	−16.3031	−1.10418
$219$	9.90383	0.669239
$220$	2.63461	0.177625
$221$	1.42423	0.0958038
$222$	10.5419	0.707526
$223$	−7.59727	−0.508751	−0.254375	−	0.967106i	$-0.581870\pi$
$223$	−7.59727	−0.508751	−0.254375	+	0.967106i	$0.581870\pi$
$224$	4.21038	0.281318
$225$	7.05537	0.470358
$226$	6.73078	0.447725
$227$	7.88233	0.523169	0.261584	−	0.965181i	$-0.415755\pi$
$227$	7.88233	0.523169	0.261584	+	0.965181i	$0.415755\pi$
$228$	−2.72732	−0.180621
$229$	−22.4208	−1.48161	−0.740803	−	0.671722i	$-0.765555\pi$
$229$	−22.4208	−1.48161	−0.740803	+	0.671722i	$0.765555\pi$
$230$	−18.8415	−1.24237
$231$	−10.8450	−0.713548
$232$	−7.15155	−0.469522
$233$	−3.60965	−0.236476	−0.118238	−	0.992985i	$-0.537725\pi$
$233$	−3.60965	−0.236476	−0.118238	+	0.992985i	$0.537725\pi$
$234$	−5.17651	−0.338399
$235$	14.9342	0.974202
$236$	−11.2692	−0.733564
$237$	−6.63461	−0.430964
$238$	4.21038	0.272919
$239$	1.46156	0.0945407	0.0472704	−	0.998882i	$-0.484948\pi$
$239$	1.46156	0.0945407	0.0472704	+	0.998882i	$0.484948\pi$
$240$	6.78616	0.438044
$241$	4.30309	0.277187	0.138593	−	0.990349i	$-0.455742\pi$
$241$	4.30309	0.277187	0.138593	+	0.990349i	$0.455742\pi$
$242$	−1.00000	−0.0642824
$243$	22.1446	1.42058
$244$	−5.15155	−0.329794
$245$	−28.2623	−1.80561
$246$	−20.3585	−1.29801
$247$	1.50802	0.0959531
$248$	−9.15155	−0.581124
$249$	13.0304	0.825769
$250$	−8.05884	−0.509686
$251$	25.2193	1.59183	0.795914	−	0.605410i	$-0.206991\pi$
$251$	25.2193	1.59183	0.795914	+	0.605410i	$0.206991\pi$
$252$	−15.3031	−0.964004
$253$	7.15155	0.449614
$254$	−7.68998	−0.482512
$255$	6.78616	0.424966
$256$	1.00000	0.0625000
$257$	22.0180	1.37345	0.686724	−	0.726919i	$-0.259048\pi$
$257$	22.0180	1.37345	0.686724	+	0.726919i	$0.259048\pi$
$258$	11.9377	0.743209
$259$	−17.2319	−1.07074
$260$	−3.75228	−0.232707
$261$	25.9931	1.60893
$262$	10.5384	0.651067
$263$	−11.5792	−0.714006	−0.357003	−	0.934103i	$-0.616201\pi$
$263$	−11.5792	−0.714006	−0.357003	+	0.934103i	$0.616201\pi$
$264$	−2.57577	−0.158528
$265$	−27.7647	−1.70557
$266$	4.45810	0.273344
$267$	39.2658	2.40303
$268$	−2.84845	−0.173997
$269$	−29.1696	−1.77850	−0.889250	−	0.457421i	$-0.848773\pi$
$269$	−29.1696	−1.77850	−0.889250	+	0.457421i	$0.848773\pi$
$270$	−4.30656	−0.262089
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	1.00000	0.0606339
$273$	15.4457	0.934818
$274$	−6.57577	−0.397257
$275$	−1.94116	−0.117057
$276$	18.4208	1.10880
$277$	−19.8754	−1.19420	−0.597099	−	0.802168i	$-0.703680\pi$
$277$	−19.8754	−1.19420	−0.597099	+	0.802168i	$0.703680\pi$
$278$	7.45464	0.447100
$279$	33.2623	1.99136
$280$	−11.0927	−0.662916
$281$	9.69691	0.578469	0.289234	−	0.957258i	$-0.406599\pi$
$281$	9.69691	0.578469	0.289234	+	0.957258i	$0.406599\pi$
$282$	−14.6007	−0.869461
$283$	8.84153	0.525574	0.262787	−	0.964854i	$-0.415358\pi$
$283$	8.84153	0.525574	0.262787	+	0.964854i	$0.415358\pi$
$284$	0.420765	0.0249678
$285$	7.18542	0.425628
$286$	1.42423	0.0842163
$287$	33.2781	1.96435
$288$	−3.63461	−0.214171
$289$	1.00000	0.0588235
$290$	18.8415	1.10641
$291$	−26.5384	−1.55571
$292$	−3.84499	−0.225011
$293$	−21.4761	−1.25465	−0.627325	−	0.778758i	$-0.715850\pi$
$293$	−21.4761	−1.25465	−0.627325	+	0.778758i	$0.715850\pi$
$294$	27.6311	1.61148
$295$	29.6900	1.72862
$296$	−4.09271	−0.237884
$297$	1.63461	0.0948496
$298$	−18.2658	−1.05811
$299$	−10.1854	−0.589038
$300$	−5.00000	−0.288675
$301$	−19.5135	−1.12474
$302$	−18.6062	−1.07067
$303$	−45.5100	−2.61448
$304$	1.05884	0.0607284
$305$	13.5723	0.777148
$306$	−3.63461	−0.207777
$307$	32.9965	1.88321	0.941606	−	0.336717i	$-0.109317\pi$
$307$	32.9965	1.88321	0.941606	+	0.336717i	$0.109317\pi$
$308$	4.21038	0.239909
$309$	−20.2069	−1.14953
$310$	24.1107	1.36940
$311$	12.7738	0.724334	0.362167	−	0.932113i	$-0.382037\pi$
$311$	12.7738	0.724334	0.362167	+	0.932113i	$0.382037\pi$
$312$	3.66848	0.207687
$313$	−13.0339	−0.736718	−0.368359	−	0.929684i	$-0.620080\pi$
$313$	−13.0339	−0.736718	−0.368359	+	0.929684i	$0.620080\pi$
$314$	17.9931	1.01541
$315$	40.3177	2.27164
$316$	2.57577	0.144899
$317$	9.76466	0.548438	0.274219	−	0.961667i	$-0.411581\pi$
$317$	9.76466	0.548438	0.274219	+	0.961667i	$0.411581\pi$
$318$	27.1446	1.52219
$319$	−7.15155	−0.400410
$320$	−2.63461	−0.147279
$321$	−48.5315	−2.70876
$322$	−30.1107	−1.67801
$323$	1.05884	0.0589152
$324$	−6.69344	−0.371858
$325$	2.76466	0.153356
$326$	17.1730	0.951127
$327$	−41.9931	−2.32222
$328$	7.90383	0.436416
$329$	23.8665	1.31580
$330$	6.78616	0.373566
$331$	−30.3031	−1.66561	−0.832804	−	0.553567i	$-0.813266\pi$
$331$	−30.3031	−1.66561	−0.832804	+	0.553567i	$0.813266\pi$
$332$	−5.05884	−0.277640
$333$	14.8754	0.815167
$334$	−7.90383	−0.432478
$335$	7.50456	0.410018
$336$	10.8450	0.591643
$337$	9.13005	0.497345	0.248673	−	0.968588i	$-0.420006\pi$
$337$	9.13005	0.497345	0.248673	+	0.968588i	$0.420006\pi$
$338$	10.9716	0.596775
$339$	17.3370	0.941615
$340$	−2.63461	−0.142882
$341$	−9.15155	−0.495584
$342$	−3.84845	−0.208101
$343$	−15.6934	−0.847366
$344$	−4.63461	−0.249881
$345$	−48.5315	−2.61285
$346$	1.46156	0.0785742
$347$	0.0677519	0.00363711	0.00181855	−	0.999998i	$-0.499421\pi$
$347$	0.0677519	0.00363711	0.00181855	+	0.999998i	$0.499421\pi$
$348$	−18.4208	−0.987457
$349$	−25.1446	−1.34596	−0.672980	−	0.739660i	$-0.734986\pi$
$349$	−25.1446	−1.34596	−0.672980	+	0.739660i	$0.734986\pi$
$350$	8.17304	0.436868
$351$	−2.32805	−0.124262
$352$	1.00000	0.0533002
$353$	−17.6685	−0.940399	−0.470199	−	0.882560i	$-0.655818\pi$
$353$	−17.6685	−0.940399	−0.470199	+	0.882560i	$0.655818\pi$
$354$	−29.0270	−1.54277
$355$	−1.10855	−0.0588358
$356$	−15.2443	−0.807944
$357$	10.8450	0.573978
$358$	−12.4208	−0.656458
$359$	−3.45464	−0.182329	−0.0911645	−	0.995836i	$-0.529059\pi$
$359$	−3.45464	−0.182329	−0.0911645	+	0.995836i	$0.529059\pi$
$360$	9.57577	0.504688
$361$	−17.8789	−0.940993
$362$	−17.2319	−0.905688
$363$	−2.57577	−0.135193
$364$	−5.99654	−0.314304
$365$	10.1300	0.530231
$366$	−13.2692	−0.693593
$367$	30.2962	1.58145	0.790724	−	0.612173i	$-0.209705\pi$
$367$	30.2962	1.58145	0.790724	+	0.612173i	$0.209705\pi$
$368$	−7.15155	−0.372800
$369$	−28.7273	−1.49548
$370$	10.7827	0.560566
$371$	−44.3708	−2.30362
$372$	−23.5723	−1.22217
$373$	16.1357	0.835475	0.417738	−	0.908568i	$-0.362823\pi$
$373$	16.1357	0.835475	0.417738	+	0.908568i	$0.362823\pi$
$374$	1.00000	0.0517088
$375$	−20.7577	−1.07192
$376$	5.66848	0.292330
$377$	10.1854	0.524576
$378$	−6.88233	−0.353989
$379$	12.0962	0.621339	0.310669	−	0.950518i	$-0.399447\pi$
$379$	12.0962	0.621339	0.310669	+	0.950518i	$0.399447\pi$
$380$	−2.78962	−0.143104
$381$	−19.8077	−1.01478
$382$	1.30656	0.0668492
$383$	−23.3800	−1.19466	−0.597330	−	0.801995i	$-0.703772\pi$
$383$	−23.3800	−1.19466	−0.597330	+	0.801995i	$0.703772\pi$
$384$	2.57577	0.131444
$385$	−11.0927	−0.565337
$386$	−11.2996	−0.575136
$387$	16.8450	0.856279
$388$	10.3031	0.523060
$389$	−3.87541	−0.196491	−0.0982454	−	0.995162i	$-0.531323\pi$
$389$	−3.87541	−0.196491	−0.0982454	+	0.995162i	$0.531323\pi$
$390$	−9.66502	−0.489407
$391$	−7.15155	−0.361669
$392$	−10.7273	−0.541812
$393$	27.1446	1.36927
$394$	−20.6561	−1.04064
$395$	−6.78616	−0.341449
$396$	−3.63461	−0.182646
$397$	1.36885	0.0687007	0.0343504	−	0.999410i	$-0.489064\pi$
$397$	1.36885	0.0687007	0.0343504	+	0.999410i	$0.489064\pi$
$398$	−6.11767	−0.306651
$399$	11.4831	0.574872
$400$	1.94116	0.0970582
$401$	−8.65611	−0.432265	−0.216133	−	0.976364i	$-0.569344\pi$
$401$	−8.65611	−0.432265	−0.216133	+	0.976364i	$0.569344\pi$
$402$	−7.33697	−0.365935
$403$	13.0339	0.649264
$404$	17.6685	0.879040
$405$	17.6346	0.876271
$406$	30.1107	1.49437
$407$	−4.09271	−0.202868
$408$	2.57577	0.127520
$409$	6.53844	0.323305	0.161652	−	0.986848i	$-0.448318\pi$
$409$	6.53844	0.323305	0.161652	+	0.986848i	$0.448318\pi$
$410$	−20.8235	−1.02840
$411$	−16.9377	−0.835475
$412$	7.84499	0.386495
$413$	47.4477	2.33475
$414$	25.9931	1.27749
$415$	13.3281	0.654248
$416$	−1.42423	−0.0698285
$417$	19.2015	0.940300
$418$	1.05884	0.0517893
$419$	−8.86649	−0.433156	−0.216578	−	0.976265i	$-0.569490\pi$
$419$	−8.86649	−0.433156	−0.216578	+	0.976265i	$0.569490\pi$
$420$	−28.5723	−1.39419
$421$	−6.30309	−0.307194	−0.153597	−	0.988134i	$-0.549086\pi$
$421$	−6.30309	−0.307194	−0.153597	+	0.988134i	$0.549086\pi$
$422$	−14.7738	−0.719176
$423$	−20.6027	−1.00174
$424$	−10.5384	−0.511792
$425$	1.94116	0.0941603
$426$	1.08380	0.0525100
$427$	21.6900	1.04965
$428$	18.8415	0.910740
$429$	3.66848	0.177116
$430$	12.2104	0.588837
$431$	−35.5903	−1.71433	−0.857163	−	0.515045i	$-0.827775\pi$
$431$	−35.5903	−1.71433	−0.857163	+	0.515045i	$0.827775\pi$
$432$	−1.63461	−0.0786452
$433$	−36.8131	−1.76913	−0.884563	−	0.466422i	$-0.845543\pi$
$433$	−36.8131	−1.76913	−0.884563	+	0.466422i	$0.845543\pi$
$434$	38.5315	1.84957
$435$	48.5315	2.32691
$436$	16.3031	0.780777
$437$	−7.57231	−0.362233
$438$	−9.90383	−0.473223
$439$	−24.7024	−1.17898	−0.589490	−	0.807776i	$-0.700671\pi$
$439$	−24.7024	−1.17898	−0.589490	+	0.807776i	$0.700671\pi$
$440$	−2.63461	−0.125600
$441$	38.9896	1.85665
$442$	−1.42423	−0.0677435
$443$	24.9093	1.18348	0.591738	−	0.806130i	$-0.298442\pi$
$443$	24.9093	1.18348	0.591738	+	0.806130i	$0.298442\pi$
$444$	−10.5419	−0.500296
$445$	40.1627	1.90389
$446$	7.59727	0.359741
$447$	−47.0485	−2.22532
$448$	−4.21038	−0.198922
$449$	−0.377769	−0.0178280	−0.00891401	−	0.999960i	$-0.502837\pi$
$449$	−0.377769	−0.0178280	−0.00891401	+	0.999960i	$0.502837\pi$
$450$	−7.05537	−0.332594
$451$	7.90383	0.372177
$452$	−6.73078	−0.316589
$453$	−47.9253	−2.25173
$454$	−7.88233	−0.369936
$455$	15.7985	0.740647
$456$	2.72732	0.127718
$457$	0.185423	0.00867372	0.00433686	−	0.999991i	$-0.498620\pi$
$457$	0.185423	0.00867372	0.00433686	+	0.999991i	$0.498620\pi$
$458$	22.4208	1.04765
$459$	−1.63461	−0.0762970
$460$	18.8415	0.878491
$461$	37.4851	1.74585	0.872927	−	0.487852i	$-0.162219\pi$
$461$	37.4851	1.74585	0.872927	+	0.487852i	$0.162219\pi$
$462$	10.8450	0.504555
$463$	15.0643	0.700097	0.350048	−	0.936732i	$-0.386165\pi$
$463$	15.0643	0.700097	0.350048	+	0.936732i	$0.386165\pi$
$464$	7.15155	0.332002
$465$	62.1038	2.88000
$466$	3.60965	0.167214
$467$	16.8485	0.779653	0.389827	−	0.920888i	$-0.372535\pi$
$467$	16.8485	0.779653	0.389827	+	0.920888i	$0.372535\pi$
$468$	5.17651	0.239284
$469$	11.9931	0.553789
$470$	−14.9342	−0.688865
$471$	46.3461	2.13552
$472$	11.2692	0.518708
$473$	−4.63461	−0.213100
$474$	6.63461	0.304738
$475$	2.05537	0.0943070
$476$	−4.21038	−0.192983
$477$	38.3031	1.75378
$478$	−1.46156	−0.0668504
$479$	36.3335	1.66012	0.830060	−	0.557674i	$-0.188306\pi$
$479$	36.3335	1.66012	0.830060	+	0.557674i	$0.188306\pi$
$480$	−6.78616	−0.309744
$481$	5.82895	0.265777
$482$	−4.30309	−0.196001
$483$	−77.5585	−3.52903
$484$	1.00000	0.0454545
$485$	−27.1446	−1.23257
$486$	−22.1446	−1.00450
$487$	15.6401	0.708719	0.354359	−	0.935109i	$-0.384699\pi$
$487$	15.6401	0.708719	0.354359	+	0.935109i	$0.384699\pi$
$488$	5.15155	0.233200
$489$	44.2339	2.00032
$490$	28.2623	1.27676
$491$	−6.51001	−0.293793	−0.146896	−	0.989152i	$-0.546928\pi$
$491$	−6.51001	−0.293793	−0.146896	+	0.989152i	$0.546928\pi$
$492$	20.3585	0.917831
$493$	7.15155	0.322089
$494$	−1.50802	−0.0678491
$495$	9.57577	0.430399
$496$	9.15155	0.410917
$497$	−1.77158	−0.0794663
$498$	−13.0304	−0.583907
$499$	20.6435	0.924131	0.462066	−	0.886846i	$-0.347108\pi$
$499$	20.6435	0.924131	0.462066	+	0.886846i	$0.347108\pi$
$500$	8.05884	0.360402
$501$	−20.3585	−0.909549
$502$	−25.2193	−1.12559
$503$	28.8166	1.28487	0.642434	−	0.766341i	$-0.277925\pi$
$503$	28.8166	1.28487	0.642434	+	0.766341i	$0.277925\pi$
$504$	15.3031	0.681654
$505$	−46.5495	−2.07143
$506$	−7.15155	−0.317925
$507$	28.2603	1.25508
$508$	7.68998	0.341188
$509$	−11.6470	−0.516244	−0.258122	−	0.966112i	$-0.583104\pi$
$509$	−11.6470	−0.516244	−0.258122	+	0.966112i	$0.583104\pi$
$510$	−6.78616	−0.300496
$511$	16.1889	0.716154
$512$	−1.00000	−0.0441942
$513$	−1.73078	−0.0764159
$514$	−22.0180	−0.971174
$515$	−20.6685	−0.910762
$516$	−11.9377	−0.525528
$517$	5.66848	0.249300
$518$	17.2319	0.757125
$519$	3.76466	0.165250
$520$	3.75228	0.164548
$521$	−9.87541	−0.432649	−0.216325	−	0.976321i	$-0.569407\pi$
$521$	−9.87541	−0.432649	−0.216325	+	0.976321i	$0.569407\pi$
$522$	−25.9931	−1.13769
$523$	2.33152	0.101950	0.0509750	−	0.998700i	$-0.483767\pi$
$523$	2.33152	0.101950	0.0509750	+	0.998700i	$0.483767\pi$
$524$	−10.5384	−0.460374
$525$	21.0519	0.918781
$526$	11.5792	0.504879
$527$	9.15155	0.398648
$528$	2.57577	0.112096
$529$	28.1446	1.22368
$530$	27.7647	1.20602
$531$	−40.9592	−1.77748
$532$	−4.45810	−0.193283
$533$	−11.2568	−0.487588
$534$	−39.2658	−1.69920
$535$	−49.6401	−2.14613
$536$	2.84845	0.123034
$537$	−31.9931	−1.38060
$538$	29.1696	1.25759
$539$	−10.7273	−0.462058
$540$	4.30656	0.185325
$541$	6.95920	0.299199	0.149600	−	0.988747i	$-0.452202\pi$
$541$	6.95920	0.299199	0.149600	+	0.988747i	$0.452202\pi$
$542$	−2.00000	−0.0859074
$543$	−44.3854	−1.90476
$544$	−1.00000	−0.0428746
$545$	−42.9523	−1.83987
$546$	−15.4457	−0.661016
$547$	−4.77378	−0.204112	−0.102056	−	0.994779i	$-0.532542\pi$
$547$	−4.77378	−0.204112	−0.102056	+	0.994779i	$0.532542\pi$
$548$	6.57577	0.280903
$549$	−18.7239	−0.799115
$550$	1.94116	0.0827715
$551$	7.57231	0.322591
$552$	−18.4208	−0.784040
$553$	−10.8450	−0.461176
$554$	19.8754	0.844425
$555$	27.7738	1.17893
$556$	−7.45464	−0.316147
$557$	−9.47614	−0.401517	−0.200758	−	0.979641i	$-0.564341\pi$
$557$	−9.47614	−0.401517	−0.200758	+	0.979641i	$0.564341\pi$
$558$	−33.2623	−1.40811
$559$	6.60073	0.279181
$560$	11.0927	0.468753
$561$	2.57577	0.108749
$562$	−9.69691	−0.409039
$563$	44.0146	1.85499	0.927497	−	0.373832i	$-0.121956\pi$
$563$	44.0146	1.85499	0.927497	+	0.373832i	$0.121956\pi$
$564$	14.6007	0.614802
$565$	17.7330	0.746032
$566$	−8.84153	−0.371637
$567$	28.1820	1.18353
$568$	−0.420765	−0.0176549
$569$	21.3869	0.896585	0.448293	−	0.893887i	$-0.352032\pi$
$569$	21.3869	0.896585	0.448293	+	0.893887i	$0.352032\pi$
$570$	−7.18542	−0.300964
$571$	−14.9592	−0.626023	−0.313012	−	0.949749i	$-0.601338\pi$
$571$	−14.9592	−0.626023	−0.313012	+	0.949749i	$0.601338\pi$
$572$	−1.42423	−0.0595499
$573$	3.36539	0.140591
$574$	−33.2781	−1.38900
$575$	−13.8823	−0.578933
$576$	3.63461	0.151442
$577$	26.3889	1.09858	0.549292	−	0.835631i	$-0.314898\pi$
$577$	26.3889	1.09858	0.549292	+	0.835631i	$0.314898\pi$
$578$	−1.00000	−0.0415945
$579$	−29.1053	−1.20957
$580$	−18.8415	−0.782352
$581$	21.2996	0.883658
$582$	26.5384	1.10005
$583$	−10.5384	−0.436457
$584$	3.84499	0.159107
$585$	−13.6381	−0.563865
$586$	21.4761	0.887171
$587$	19.1515	0.790469	0.395234	−	0.918580i	$-0.370663\pi$
$587$	19.1515	0.790469	0.395234	+	0.918580i	$0.370663\pi$
$588$	−27.6311	−1.13949
$589$	9.68998	0.399269
$590$	−29.6900	−1.22232
$591$	−53.2055	−2.18858
$592$	4.09271	0.168209
$593$	−13.9931	−0.574627	−0.287313	−	0.957837i	$-0.592762\pi$
$593$	−13.9931	−0.574627	−0.287313	+	0.957837i	$0.592762\pi$
$594$	−1.63461	−0.0670688
$595$	11.0927	0.454757
$596$	18.2658	0.748195
$597$	−15.7577	−0.644921
$598$	10.1854	0.416513
$599$	24.1355	0.986150	0.493075	−	0.869987i	$-0.335873\pi$
$599$	24.1355	0.986150	0.493075	+	0.869987i	$0.335873\pi$
$600$	5.00000	0.204124
$601$	−22.6311	−0.923144	−0.461572	−	0.887103i	$-0.652714\pi$
$601$	−22.6311	−0.923144	−0.461572	+	0.887103i	$0.652714\pi$
$602$	19.5135	0.795310
$603$	−10.3530	−0.421607
$604$	18.6062	0.757075
$605$	−2.63461	−0.107112
$606$	45.5100	1.84872
$607$	−42.0146	−1.70532	−0.852659	−	0.522467i	$-0.825012\pi$
$607$	−42.0146	−1.70532	−0.852659	+	0.522467i	$0.825012\pi$
$608$	−1.05884	−0.0429415
$609$	77.5585	3.14283
$610$	−13.5723	−0.549527
$611$	−8.07321	−0.326607
$612$	3.63461	0.146920
$613$	−13.3120	−0.537667	−0.268834	−	0.963187i	$-0.586638\pi$
$613$	−13.3120	−0.537667	−0.268834	+	0.963187i	$0.586638\pi$
$614$	−32.9965	−1.33163
$615$	−53.6366	−2.16284
$616$	−4.21038	−0.169641
$617$	23.8007	0.958181	0.479091	−	0.877765i	$-0.340967\pi$
$617$	23.8007	0.958181	0.479091	+	0.877765i	$0.340967\pi$
$618$	20.2069	0.812842
$619$	7.35647	0.295682	0.147841	−	0.989011i	$-0.452768\pi$
$619$	7.35647	0.295682	0.147841	+	0.989011i	$0.452768\pi$
$620$	−24.1107	−0.968311
$621$	11.6900	0.469103
$622$	−12.7738	−0.512182
$623$	64.1842	2.57148
$624$	−3.66848	−0.146857
$625$	−30.9377	−1.23751
$626$	13.0339	0.520938
$627$	2.72732	0.108919
$628$	−17.9931	−0.718002
$629$	4.09271	0.163187
$630$	−40.3177	−1.60629
$631$	−9.91274	−0.394620	−0.197310	−	0.980341i	$-0.563221\pi$
$631$	−9.91274	−0.394620	−0.197310	+	0.980341i	$0.563221\pi$
$632$	−2.57577	−0.102459
$633$	−38.0539	−1.51251
$634$	−9.76466	−0.387804
$635$	−20.2601	−0.803997
$636$	−27.1446	−1.07635
$637$	15.2781	0.605342
$638$	7.15155	0.283132
$639$	1.52932	0.0604988
$640$	2.63461	0.104142
$641$	−2.54536	−0.100536	−0.0502678	−	0.998736i	$-0.516007\pi$
$641$	−2.54536	−0.100536	−0.0502678	+	0.998736i	$0.516007\pi$
$642$	48.5315	1.91539
$643$	−28.8415	−1.13740	−0.568699	−	0.822545i	$-0.692553\pi$
$643$	−28.8415	−1.13740	−0.568699	+	0.822545i	$0.692553\pi$
$644$	30.1107	1.18653
$645$	31.4512	1.23839
$646$	−1.05884	−0.0416593
$647$	41.2927	1.62338	0.811692	−	0.584086i	$-0.198547\pi$
$647$	41.2927	1.62338	0.811692	+	0.584086i	$0.198547\pi$
$648$	6.69344	0.262943
$649$	11.2692	0.442355
$650$	−2.76466	−0.108439
$651$	99.2484	3.88985
$652$	−17.1730	−0.672548
$653$	−23.2373	−0.909347	−0.454674	−	0.890658i	$-0.650244\pi$
$653$	−23.2373	−0.909347	−0.454674	+	0.890658i	$0.650244\pi$
$654$	41.9931	1.64206
$655$	27.7647	1.08485
$656$	−7.90383	−0.308593
$657$	−13.9750	−0.545218
$658$	−23.8665	−0.930413
$659$	−14.8912	−0.580081	−0.290040	−	0.957014i	$-0.593669\pi$
$659$	−14.8912	−0.580081	−0.290040	+	0.957014i	$0.593669\pi$
$660$	−6.78616	−0.264151
$661$	30.2962	1.17838	0.589192	−	0.807993i	$-0.299446\pi$
$661$	30.2962	1.17838	0.589192	+	0.807993i	$0.299446\pi$
$662$	30.3031	1.17776
$663$	−3.66848	−0.142472
$664$	5.05884	0.196321
$665$	11.7454	0.455465
$666$	−14.8754	−0.576410
$667$	−51.1446	−1.98033
$668$	7.90383	0.305808
$669$	19.5689	0.756575
$670$	−7.50456	−0.289926
$671$	5.15155	0.198873
$672$	−10.8450	−0.418355
$673$	−2.02842	−0.0781899	−0.0390949	−	0.999236i	$-0.512447\pi$
$673$	−2.02842	−0.0781899	−0.0390949	+	0.999236i	$0.512447\pi$
$674$	−9.13005	−0.351676
$675$	−3.17304	−0.122131
$676$	−10.9716	−0.421984
$677$	35.4477	1.36237	0.681183	−	0.732113i	$-0.261466\pi$
$677$	35.4477	1.36237	0.681183	+	0.732113i	$0.261466\pi$
$678$	−17.3370	−0.665822
$679$	−43.3800	−1.66477
$680$	2.63461	0.101033
$681$	−20.3031	−0.778016
$682$	9.15155	0.350431
$683$	23.6685	0.905649	0.452825	−	0.891600i	$-0.350416\pi$
$683$	23.6685	0.905649	0.452825	+	0.891600i	$0.350416\pi$
$684$	3.84845	0.147149
$685$	−17.3246	−0.661939
$686$	15.6934	0.599178
$687$	57.7508	2.20333
$688$	4.63461	0.176693
$689$	15.0091	0.571802
$690$	48.5315	1.84756
$691$	−10.5634	−0.401851	−0.200925	−	0.979607i	$-0.564395\pi$
$691$	−10.5634	−0.401851	−0.200925	+	0.979607i	$0.564395\pi$
$692$	−1.46156	−0.0555603
$693$	15.3031	0.581316
$694$	−0.0677519	−0.00257183
$695$	19.6401	0.744990
$696$	18.4208	0.698237
$697$	−7.90383	−0.299379
$698$	25.1446	0.951738
$699$	9.29764	0.351669
$700$	−8.17304	−0.308912
$701$	33.1765	1.25306	0.626530	−	0.779398i	$-0.284475\pi$
$701$	33.1765	1.25306	0.626530	+	0.779398i	$0.284475\pi$
$702$	2.32805	0.0878667
$703$	4.33351	0.163441
$704$	−1.00000	−0.0376889
$705$	−38.4672	−1.44876
$706$	17.6685	0.664962
$707$	−74.3911	−2.79776
$708$	29.0270	1.09090
$709$	−18.8665	−0.708546	−0.354273	−	0.935142i	$-0.615272\pi$
$709$	−18.8665	−0.708546	−0.354273	+	0.935142i	$0.615272\pi$
$710$	1.10855	0.0416032
$711$	9.36193	0.351100
$712$	15.2443	0.571303
$713$	−65.4477	−2.45104
$714$	−10.8450	−0.405864
$715$	3.75228	0.140327
$716$	12.4208	0.464186
$717$	−3.76466	−0.140594
$718$	3.45464	0.128926
$719$	−3.41164	−0.127233	−0.0636164	−	0.997974i	$-0.520263\pi$
$719$	−3.41164	−0.127233	−0.0636164	+	0.997974i	$0.520263\pi$
$720$	−9.57577	−0.356868
$721$	−33.0304	−1.23012
$722$	17.8789	0.665383
$723$	−11.0838	−0.412211
$724$	17.2319	0.640418
$725$	13.8823	0.515577
$726$	2.57577	0.0955959
$727$	−10.2228	−0.379141	−0.189571	−	0.981867i	$-0.560710\pi$
$727$	−10.2228	−0.379141	−0.189571	+	0.981867i	$0.560710\pi$
$728$	5.99654	0.222246
$729$	−36.9592	−1.36886
$730$	−10.1300	−0.374930
$731$	4.63461	0.171417
$732$	13.2692	0.490444
$733$	−8.20692	−0.303130	−0.151565	−	0.988447i	$-0.548431\pi$
$733$	−8.20692	−0.303130	−0.151565	+	0.988447i	$0.548431\pi$
$734$	−30.2962	−1.11825
$735$	72.7973	2.68517
$736$	7.15155	0.263610
$737$	2.84845	0.104924
$738$	28.7273	1.05747
$739$	−17.8381	−0.656184	−0.328092	−	0.944646i	$-0.606406\pi$
$739$	−17.8381	−0.656184	−0.328092	+	0.944646i	$0.606406\pi$
$740$	−10.7827	−0.396380
$741$	−3.88432	−0.142694
$742$	44.3708	1.62890
$743$	10.2069	0.374456	0.187228	−	0.982317i	$-0.440050\pi$
$743$	10.2069	0.374456	0.187228	+	0.982317i	$0.440050\pi$
$744$	23.5723	0.864203
$745$	−48.1231	−1.76309
$746$	−16.1357	−0.590770
$747$	−18.3869	−0.672741
$748$	−1.00000	−0.0365636
$749$	−79.3300	−2.89866
$750$	20.7577	0.757965
$751$	10.6131	0.387278	0.193639	−	0.981073i	$-0.437971\pi$
$751$	10.6131	0.387278	0.193639	+	0.981073i	$0.437971\pi$
$752$	−5.66848	−0.206708
$753$	−64.9592	−2.36724
$754$	−10.1854	−0.370931
$755$	−49.0200	−1.78402
$756$	6.88233	0.250308
$757$	−34.8845	−1.26790	−0.633950	−	0.773374i	$-0.718567\pi$
$757$	−34.8845	−1.26790	−0.633950	+	0.773374i	$0.718567\pi$
$758$	−12.0962	−0.439353
$759$	−18.4208	−0.668631
$760$	2.78962	0.101190
$761$	43.9931	1.59475	0.797374	−	0.603486i	$-0.206222\pi$
$761$	43.9931	1.59475	0.797374	+	0.603486i	$0.206222\pi$
$762$	19.8077	0.717556
$763$	−68.6423	−2.48502
$764$	−1.30656	−0.0472695
$765$	−9.57577	−0.346213
$766$	23.3800	0.844753
$767$	−16.0499	−0.579529
$768$	−2.57577	−0.0929452
$769$	20.4885	0.738835	0.369417	−	0.929264i	$-0.379557\pi$
$769$	20.4885	0.738835	0.369417	+	0.929264i	$0.379557\pi$
$770$	11.0927	0.399754
$771$	−56.7135	−2.04249
$772$	11.2996	0.406683
$773$	−13.6831	−0.492145	−0.246073	−	0.969251i	$-0.579140\pi$
$773$	−13.6831	−0.492145	−0.246073	+	0.969251i	$0.579140\pi$
$774$	−16.8450	−0.605481
$775$	17.7647	0.638125
$776$	−10.3031	−0.369860
$777$	44.3854	1.59232
$778$	3.87541	0.138940
$779$	−8.36885	−0.299845
$780$	9.66502	0.346063
$781$	−0.420765	−0.0150562
$782$	7.15155	0.255739
$783$	−11.6900	−0.417766
$784$	10.7273	0.383119
$785$	47.4047	1.69195
$786$	−27.1446	−0.968217
$787$	18.5384	0.660824	0.330412	−	0.943837i	$-0.392812\pi$
$787$	18.5384	0.660824	0.330412	+	0.943837i	$0.392812\pi$
$788$	20.6561	0.735843
$789$	29.8255	1.06182
$790$	6.78616	0.241441
$791$	28.3392	1.00762
$792$	3.63461	0.129150
$793$	−7.33697	−0.260543
$794$	−1.36885	−0.0485788
$795$	71.5155	2.53639
$796$	6.11767	0.216835
$797$	47.4477	1.68068	0.840342	−	0.542056i	$-0.182354\pi$
$797$	47.4477	1.68068	0.840342	+	0.542056i	$0.182354\pi$
$798$	−11.4831	−0.406496
$799$	−5.66848	−0.200537
$800$	−1.94116	−0.0686305
$801$	−55.4069	−1.95771
$802$	8.65611	0.305658
$803$	3.84499	0.135687
$804$	7.33697	0.258755
$805$	−79.3300	−2.79602
$806$	−13.0339	−0.459099
$807$	75.1342	2.64485
$808$	−17.6685	−0.621575
$809$	−16.5384	−0.581460	−0.290730	−	0.956805i	$-0.593898\pi$
$809$	−16.5384	−0.581460	−0.290730	+	0.956805i	$0.593898\pi$
$810$	−17.6346	−0.619617
$811$	23.0769	0.810339	0.405169	−	0.914242i	$-0.367213\pi$
$811$	23.0769	0.810339	0.405169	+	0.914242i	$0.367213\pi$
$812$	−30.1107	−1.05668
$813$	−5.15155	−0.180673
$814$	4.09271	0.143449
$815$	45.2443	1.58484
$816$	−2.57577	−0.0901701
$817$	4.90729	0.171684
$818$	−6.53844	−0.228611
$819$	−21.7951	−0.761581
$820$	20.8235	0.727188
$821$	−28.5384	−0.995998	−0.497999	−	0.867178i	$-0.665932\pi$
$821$	−28.5384	−0.995998	−0.497999	+	0.867178i	$0.665932\pi$
$822$	16.9377	0.590770
$823$	−31.4546	−1.09644	−0.548220	−	0.836334i	$-0.684694\pi$
$823$	−31.4546	−1.09644	−0.548220	+	0.836334i	$0.684694\pi$
$824$	−7.84499	−0.273293
$825$	5.00000	0.174078
$826$	−47.4477	−1.65092
$827$	−50.7916	−1.76620	−0.883099	−	0.469187i	$-0.844547\pi$
$827$	−50.7916	−1.76620	−0.883099	+	0.469187i	$0.844547\pi$
$828$	−25.9931	−0.903322
$829$	9.64006	0.334813	0.167407	−	0.985888i	$-0.446461\pi$
$829$	9.64006	0.334813	0.167407	+	0.985888i	$0.446461\pi$
$830$	−13.3281	−0.462623
$831$	51.1945	1.77592
$832$	1.42423	0.0493762
$833$	10.7273	0.371680
$834$	−19.2015	−0.664892
$835$	−20.8235	−0.720627
$836$	−1.05884	−0.0366206
$837$	−14.9592	−0.517066
$838$	8.86649	0.306288
$839$	−32.9661	−1.13812	−0.569058	−	0.822297i	$-0.692692\pi$
$839$	−32.9661	−1.13812	−0.569058	+	0.822297i	$0.692692\pi$
$840$	28.5723	0.985838
$841$	22.1446	0.763608
$842$	6.30309	0.217219
$843$	−24.9770	−0.860255
$844$	14.7738	0.508534
$845$	28.9058	0.994390
$846$	20.6027	0.708336
$847$	−4.21038	−0.144670
$848$	10.5384	0.361891
$849$	−22.7738	−0.781594
$850$	−1.94116	−0.0665814
$851$	−29.2692	−1.00334
$852$	−1.08380	−0.0371302
$853$	29.9862	1.02671	0.513353	−	0.858177i	$-0.328403\pi$
$853$	29.9862	1.02671	0.513353	+	0.858177i	$0.328403\pi$
$854$	−21.6900	−0.742216
$855$	−10.1392	−0.346752
$856$	−18.8415	−0.643990
$857$	−15.4865	−0.529010	−0.264505	−	0.964384i	$-0.585208\pi$
$857$	−15.4865	−0.529010	−0.264505	+	0.964384i	$0.585208\pi$
$858$	−3.66848	−0.125240
$859$	7.33697	0.250334	0.125167	−	0.992136i	$-0.460053\pi$
$859$	7.33697	0.250334	0.125167	+	0.992136i	$0.460053\pi$
$860$	−12.2104	−0.416371
$861$	−85.7169	−2.92123
$862$	35.5903	1.21221
$863$	42.7419	1.45495	0.727476	−	0.686134i	$-0.240693\pi$
$863$	42.7419	1.45495	0.727476	+	0.686134i	$0.240693\pi$
$864$	1.63461	0.0556105
$865$	3.85065	0.130926
$866$	36.8131	1.25096
$867$	−2.57577	−0.0874779
$868$	−38.5315	−1.30784
$869$	−2.57577	−0.0873771
$870$	−48.5315	−1.64537
$871$	−4.05684	−0.137461
$872$	−16.3031	−0.552092
$873$	37.4477	1.26741
$874$	7.57231	0.256137
$875$	−33.9308	−1.14707
$876$	9.90383	0.334619
$877$	−31.5224	−1.06444	−0.532218	−	0.846607i	$-0.678641\pi$
$877$	−31.5224	−1.06444	−0.532218	+	0.846607i	$0.678641\pi$
$878$	24.7024	0.833664
$879$	55.3177	1.86582
$880$	2.63461	0.0888127
$881$	8.54536	0.287900	0.143950	−	0.989585i	$-0.454019\pi$
$881$	8.54536	0.287900	0.143950	+	0.989585i	$0.454019\pi$
$882$	−38.9896	−1.31285
$883$	21.5046	0.723686	0.361843	−	0.932239i	$-0.382148\pi$
$883$	21.5046	0.723686	0.361843	+	0.932239i	$0.382148\pi$
$884$	1.42423	0.0479019
$885$	−76.4747	−2.57067
$886$	−24.9093	−0.836844
$887$	44.2465	1.48565	0.742825	−	0.669485i	$-0.233485\pi$
$887$	44.2465	1.48565	0.742825	+	0.669485i	$0.233485\pi$
$888$	10.5419	0.353763
$889$	−32.3778	−1.08592
$890$	−40.1627	−1.34626
$891$	6.69344	0.224239
$892$	−7.59727	−0.254375
$893$	−6.00199	−0.200849
$894$	47.0485	1.57354
$895$	−32.7239	−1.09384
$896$	4.21038	0.140659
$897$	26.2353	0.875973
$898$	0.377769	0.0126063
$899$	65.4477	2.18280
$900$	7.05537	0.235179
$901$	10.5384	0.351086
$902$	−7.90383	−0.263169
$903$	50.2623	1.67262
$904$	6.73078	0.223862
$905$	−45.3993	−1.50912
$906$	47.9253	1.59221
$907$	25.4242	0.844198	0.422099	−	0.906550i	$-0.361294\pi$
$907$	25.4242	0.844198	0.422099	+	0.906550i	$0.361294\pi$
$908$	7.88233	0.261584
$909$	64.2180	2.12998
$910$	−15.7985	−0.523716
$911$	51.2124	1.69674	0.848371	−	0.529402i	$-0.177584\pi$
$911$	51.2124	1.69674	0.848371	+	0.529402i	$0.177584\pi$
$912$	−2.72732	−0.0903106
$913$	5.05884	0.167423
$914$	−0.185423	−0.00613324
$915$	−34.9592	−1.15572
$916$	−22.4208	−0.740803
$917$	44.3708	1.46525
$918$	1.63461	0.0539501
$919$	35.1446	1.15931	0.579657	−	0.814861i	$-0.303187\pi$
$919$	35.1446	1.15931	0.579657	+	0.814861i	$0.303187\pi$
$920$	−18.8415	−0.621187
$921$	−84.9916	−2.80057
$922$	−37.4851	−1.23450
$923$	0.599265	0.0197250
$924$	−10.8450	−0.356774
$925$	7.94463	0.261218
$926$	−15.0643	−0.495043
$927$	28.5135	0.936505
$928$	−7.15155	−0.234761
$929$	−54.6423	−1.79275	−0.896377	−	0.443292i	$-0.853810\pi$
$929$	−54.6423	−1.79275	−0.896377	+	0.443292i	$0.853810\pi$
$930$	−62.1038	−2.03646
$931$	11.3585	0.372259
$932$	−3.60965	−0.118238
$933$	−32.9024	−1.07717
$934$	−16.8485	−0.551298
$935$	2.63461	0.0861609
$936$	−5.17651	−0.169199
$937$	4.23534	0.138363	0.0691813	−	0.997604i	$-0.477961\pi$
$937$	4.23534	0.138363	0.0691813	+	0.997604i	$0.477961\pi$
$938$	−11.9931	−0.391588
$939$	33.5723	1.09559
$940$	14.9342	0.487101
$941$	−45.1876	−1.47307	−0.736537	−	0.676397i	$-0.763540\pi$
$941$	−45.1876	−1.47307	−0.736537	+	0.676397i	$0.763540\pi$
$942$	−46.3461	−1.51004
$943$	56.5246	1.84069
$944$	−11.2692	−0.366782
$945$	−18.1322	−0.589842
$946$	4.63461	0.150684
$947$	−4.09617	−0.133108	−0.0665539	−	0.997783i	$-0.521200\pi$
$947$	−4.09617	−0.133108	−0.0665539	+	0.997783i	$0.521200\pi$
$948$	−6.63461	−0.215482
$949$	−5.47614	−0.177763
$950$	−2.05537	−0.0666851
$951$	−25.1515	−0.815595
$952$	4.21038	0.136459
$953$	−33.4299	−1.08290	−0.541450	−	0.840733i	$-0.682124\pi$
$953$	−33.4299	−1.08290	−0.541450	+	0.840733i	$0.682124\pi$
$954$	−38.3031	−1.24011
$955$	3.44226	0.111389
$956$	1.46156	0.0472704
$957$	18.4208	0.595459
$958$	−36.3335	−1.17388
$959$	−27.6865	−0.894044
$960$	6.78616	0.219022
$961$	52.7508	1.70164
$962$	−5.82895	−0.187933
$963$	68.4816	2.20679
$964$	4.30309	0.138593
$965$	−29.7701	−0.958334
$966$	77.5585	2.49540
$967$	−20.6492	−0.664033	−0.332016	−	0.943274i	$-0.607729\pi$
$967$	−20.6492	−0.664033	−0.332016	+	0.943274i	$0.607729\pi$
$968$	−1.00000	−0.0321412
$969$	−2.72732	−0.0876142
$970$	27.1446	0.871561
$971$	−37.6831	−1.20931	−0.604653	−	0.796489i	$-0.706688\pi$
$971$	−37.6831	−1.20931	−0.604653	+	0.796489i	$0.706688\pi$
$972$	22.1446	0.710289
$973$	31.3869	1.00622
$974$	−15.6401	−0.501140
$975$	−7.12113	−0.228059
$976$	−5.15155	−0.164897
$977$	−17.0055	−0.544053	−0.272026	−	0.962290i	$-0.587694\pi$
$977$	−17.0055	−0.544053	−0.272026	+	0.962290i	$0.587694\pi$
$978$	−44.2339	−1.41444
$979$	15.2443	0.487209
$980$	−28.2623	−0.902806
$981$	59.2554	1.89188
$982$	6.51001	0.207743
$983$	3.03388	0.0967656	0.0483828	−	0.998829i	$-0.484593\pi$
$983$	3.03388	0.0967656	0.0483828	+	0.998829i	$0.484593\pi$
$984$	−20.3585	−0.649004
$985$	−54.4208	−1.73399
$986$	−7.15155	−0.227752
$987$	−61.4747	−1.95676
$988$	1.50802	0.0479766
$989$	−33.1446	−1.05394
$990$	−9.57577	−0.304338
$991$	−16.4208	−0.521623	−0.260811	−	0.965390i	$-0.583990\pi$
$991$	−16.4208	−0.521623	−0.260811	+	0.965390i	$0.583990\pi$
$992$	−9.15155	−0.290562
$993$	78.0539	2.47697
$994$	1.77158	0.0561911
$995$	−16.1177	−0.510965
$996$	13.0304	0.412884
$997$	−27.8255	−0.881242	−0.440621	−	0.897693i	$-0.645242\pi$
$997$	−27.8255	−0.881242	−0.440621	+	0.897693i	$0.645242\pi$
$998$	−20.6435	−0.653459
$999$	−6.68998	−0.211662

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p

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

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n

up to: 50 250 1000 (See only

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a_p

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	374.2.a.b.1.1	✓	3
3.2	odd	2		3366.2.a.bd.1.3		3
4.3	odd	2		2992.2.a.q.1.3		3
5.4	even	2		9350.2.a.cf.1.3		3
11.10	odd	2		4114.2.a.s.1.1		3
17.16	even	2		6358.2.a.o.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
374.2.a.b.1.1	✓	3	1.1	even	1	trivial
2992.2.a.q.1.3		3	4.3	odd	2
3366.2.a.bd.1.3		3	3.2	odd	2
4114.2.a.s.1.1		3	11.10	odd	2
6358.2.a.o.1.3		3	17.16	even	2
9350.2.a.cf.1.3		3	5.4	even	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

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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.b.1.1

Level	$374$
Weight	$2$
Character	374.1
Self dual	yes
Analytic conductor	$2.986$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$