LMFDB - Embedded newform 357.2.a.h.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [357,2,Mod(1,357)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("357.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(357, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$357 = 3 \cdot 7 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	357.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,2,-4,6,-2,-2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.85065935216$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7232.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 2x^{3} - 5x^{2} + 4x + 4$ x^4 - 2x^3 - 5x^2 + 4*x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$2$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.22219$ of defining polynomial
Character	$\chi$	$=$	357.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.63640 q^{2} -1.00000 q^{3} +4.95063 q^{4} -1.19202 q^{5} -2.63640 q^{6} +1.00000 q^{7} +7.77906 q^{8} +1.00000 q^{9} -3.14265 q^{10} +1.32218 q^{11} -4.95063 q^{12} +1.19202 q^{13} +2.63640 q^{14} +1.19202 q^{15} +10.6075 q^{16} -1.00000 q^{17} +2.63640 q^{18} -2.46483 q^{19} -5.90126 q^{20} -1.00000 q^{21} +3.48580 q^{22} -7.60749 q^{23} -7.77906 q^{24} -3.57908 q^{25} +3.14265 q^{26} -1.00000 q^{27} +4.95063 q^{28} -5.27281 q^{29} +3.14265 q^{30} +5.77111 q^{31} +12.4075 q^{32} -1.32218 q^{33} -2.63640 q^{34} -1.19202 q^{35} +4.95063 q^{36} -0.130157 q^{37} -6.49830 q^{38} -1.19202 q^{39} -9.27281 q^{40} -10.7092 q^{41} -2.63640 q^{42} -1.22344 q^{43} +6.54562 q^{44} -1.19202 q^{45} -20.0564 q^{46} +5.77111 q^{47} -10.6075 q^{48} +1.00000 q^{49} -9.43591 q^{50} +1.00000 q^{51} +5.90126 q^{52} +11.7711 q^{53} -2.63640 q^{54} -1.57607 q^{55} +7.77906 q^{56} +2.46483 q^{57} -13.9013 q^{58} +10.7995 q^{59} +5.90126 q^{60} -2.37456 q^{61} +15.2150 q^{62} +1.00000 q^{63} +11.4963 q^{64} -1.42092 q^{65} -3.48580 q^{66} +0.0987380 q^{67} -4.95063 q^{68} +7.60749 q^{69} -3.14265 q^{70} -14.4469 q^{71} +7.77906 q^{72} +7.91717 q^{73} -0.343146 q^{74} +3.57908 q^{75} -12.2025 q^{76} +1.32218 q^{77} -3.14265 q^{78} +14.9611 q^{79} -12.6444 q^{80} +1.00000 q^{81} -28.2339 q^{82} +0.0987380 q^{83} -4.95063 q^{84} +1.19202 q^{85} -3.22549 q^{86} +5.27281 q^{87} +10.2853 q^{88} -8.38404 q^{89} -3.14265 q^{90} +1.19202 q^{91} -37.6619 q^{92} -5.77111 q^{93} +15.2150 q^{94} +2.93813 q^{95} -12.4075 q^{96} +10.7900 q^{97} +2.63640 q^{98} +1.32218 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9} + 4 q^{10} + 2 q^{11} - 6 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{15} + 6 q^{16} - 4 q^{17} + 2 q^{18} + 10 q^{19} + 4 q^{20}+ \cdots + 2 q^{99}+O(q^{100})$ 4 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 + 4 * q^7 + 6 * q^8 + 4 * q^9 + 4 * q^10 + 2 * q^11 - 6 * q^12 + 2 * q^13 + 2 * q^14 + 2 * q^15 + 6 * q^16 - 4 * q^17 + 2 * q^18 + 10 * q^19 + 4 * q^20 - 4 * q^21 + 20 * q^22 + 6 * q^23 - 6 * q^24 + 10 * q^25 - 4 * q^26 - 4 * q^27 + 6 * q^28 - 4 * q^29 - 4 * q^30 - 4 * q^31 + 14 * q^32 - 2 * q^33 - 2 * q^34 - 2 * q^35 + 6 * q^36 - 16 * q^38 - 2 * q^39 - 20 * q^40 - 18 * q^41 - 2 * q^42 + 26 * q^43 - 8 * q^44 - 2 * q^45 - 20 * q^46 - 4 * q^47 - 6 * q^48 + 4 * q^49 + 10 * q^50 + 4 * q^51 - 4 * q^52 + 20 * q^53 - 2 * q^54 + 2 * q^55 + 6 * q^56 - 10 * q^57 - 28 * q^58 + 4 * q^59 - 4 * q^60 - 4 * q^61 - 12 * q^62 + 4 * q^63 - 2 * q^64 - 30 * q^65 - 20 * q^66 + 28 * q^67 - 6 * q^68 - 6 * q^69 + 4 * q^70 + 4 * q^71 + 6 * q^72 + 8 * q^73 - 24 * q^74 - 10 * q^75 + 8 * q^76 + 2 * q^77 + 4 * q^78 - 8 * q^79 - 44 * q^80 + 4 * q^81 - 28 * q^82 + 28 * q^83 - 6 * q^84 + 2 * q^85 - 20 * q^86 + 4 * q^87 + 8 * q^88 - 28 * q^89 + 4 * q^90 + 2 * q^91 - 16 * q^92 + 4 * q^93 - 12 * q^94 + 14 * q^95 - 14 * q^96 + 4 * q^97 + 2 * q^98 + 2 * 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

)

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

$2$	2.63640	1.86422	0.932110	−	0.362176i	$-0.117966\pi$
$2$	2.63640	1.86422	0.932110	+	0.362176i	$0.117966\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.95063	2.47532
$5$	−1.19202	−0.533089	−0.266544	−	0.963823i	$-0.585882\pi$
$5$	−1.19202	−0.533089	−0.266544	+	0.963823i	$0.585882\pi$
$6$	−2.63640	−1.07631
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	7.77906	2.75031
$9$	1.00000	0.333333
$10$	−3.14265	−0.993794
$11$	1.32218	0.398652	0.199326	−	0.979933i	$-0.436125\pi$
$11$	1.32218	0.398652	0.199326	+	0.979933i	$0.436125\pi$
$12$	−4.95063	−1.42912
$13$	1.19202	0.330607	0.165304	−	0.986243i	$-0.447140\pi$
$13$	1.19202	0.330607	0.165304	+	0.986243i	$0.447140\pi$
$14$	2.63640	0.704609
$15$	1.19202	0.307779
$16$	10.6075	2.65187
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	2.63640	0.621407
$19$	−2.46483	−0.565471	−0.282736	−	0.959198i	$-0.591242\pi$
$19$	−2.46483	−0.565471	−0.282736	+	0.959198i	$0.591242\pi$
$20$	−5.90126	−1.31956
$21$	−1.00000	−0.218218
$22$	3.48580	0.743175
$23$	−7.60749	−1.58627	−0.793135	−	0.609046i	$-0.791553\pi$
$23$	−7.60749	−1.58627	−0.793135	+	0.609046i	$0.791553\pi$
$24$	−7.77906	−1.58789
$25$	−3.57908	−0.715817
$26$	3.14265	0.616325
$27$	−1.00000	−0.192450
$28$	4.95063	0.935581
$29$	−5.27281	−0.979136	−0.489568	−	0.871965i	$-0.662846\pi$
$29$	−5.27281	−0.979136	−0.489568	+	0.871965i	$0.662846\pi$
$30$	3.14265	0.573767
$31$	5.77111	1.03652	0.518261	−	0.855223i	$-0.326580\pi$
$31$	5.77111	1.03652	0.518261	+	0.855223i	$0.326580\pi$
$32$	12.4075	2.19336
$33$	−1.32218	−0.230162
$34$	−2.63640	−0.452140
$35$	−1.19202	−0.201489
$36$	4.95063	0.825105
$37$	−0.130157	−0.0213976	−0.0106988	−	0.999943i	$-0.503406\pi$
$37$	−0.130157	−0.0213976	−0.0106988	+	0.999943i	$0.503406\pi$
$38$	−6.49830	−1.05416
$39$	−1.19202	−0.190876
$40$	−9.27281	−1.46616
$41$	−10.7092	−1.67250	−0.836251	−	0.548347i	$-0.815257\pi$
$41$	−10.7092	−1.67250	−0.836251	+	0.548347i	$0.815257\pi$
$42$	−2.63640	−0.406806
$43$	−1.22344	−0.186573	−0.0932865	−	0.995639i	$-0.529737\pi$
$43$	−1.22344	−0.186573	−0.0932865	+	0.995639i	$0.529737\pi$
$44$	6.54562	0.986789
$45$	−1.19202	−0.177696
$46$	−20.0564	−2.95716
$47$	5.77111	0.841802	0.420901	−	0.907107i	$-0.361714\pi$
$47$	5.77111	0.841802	0.420901	+	0.907107i	$0.361714\pi$
$48$	−10.6075	−1.53106
$49$	1.00000	0.142857
$50$	−9.43591	−1.33444
$51$	1.00000	0.140028
$52$	5.90126	0.818358
$53$	11.7711	1.61689	0.808443	−	0.588575i	$-0.200311\pi$
$53$	11.7711	1.61689	0.808443	+	0.588575i	$0.200311\pi$
$54$	−2.63640	−0.358769
$55$	−1.57607	−0.212517
$56$	7.77906	1.03952
$57$	2.46483	0.326475
$58$	−13.9013	−1.82533
$59$	10.7995	1.40598	0.702988	−	0.711202i	$-0.251849\pi$
$59$	10.7995	1.40598	0.702988	+	0.711202i	$0.251849\pi$
$60$	5.90126	0.761850
$61$	−2.37456	−0.304032	−0.152016	−	0.988378i	$-0.548576\pi$
$61$	−2.37456	−0.304032	−0.152016	+	0.988378i	$0.548576\pi$
$62$	15.2150	1.93230
$63$	1.00000	0.125988
$64$	11.4963	1.43703
$65$	−1.42092	−0.176243
$66$	−3.48580	−0.429072
$67$	0.0987380	0.0120628	0.00603138	−	0.999982i	$-0.498080\pi$
$67$	0.0987380	0.0120628	0.00603138	+	0.999982i	$0.498080\pi$
$68$	−4.95063	−0.600352
$69$	7.60749	0.915834
$70$	−3.14265	−0.375619
$71$	−14.4469	−1.71453	−0.857265	−	0.514876i	$-0.827838\pi$
$71$	−14.4469	−1.71453	−0.857265	+	0.514876i	$0.827838\pi$
$72$	7.77906	0.916771
$73$	7.91717	0.926634	0.463317	−	0.886193i	$-0.346659\pi$
$73$	7.91717	0.926634	0.463317	+	0.886193i	$0.346659\pi$
$74$	−0.343146	−0.0398899
$75$	3.57908	0.413277
$76$	−12.2025	−1.39972
$77$	1.32218	0.150676
$78$	−3.14265	−0.355835
$79$	14.9611	1.68325	0.841627	−	0.540060i	$-0.181598\pi$
$79$	14.9611	1.68325	0.841627	+	0.540060i	$0.181598\pi$
$80$	−12.6444	−1.41368
$81$	1.00000	0.111111
$82$	−28.2339	−3.11791
$83$	0.0987380	0.0108379	0.00541895	−	0.999985i	$-0.498275\pi$
$83$	0.0987380	0.0108379	0.00541895	+	0.999985i	$0.498275\pi$
$84$	−4.95063	−0.540158
$85$	1.19202	0.129293
$86$	−3.22549	−0.347813
$87$	5.27281	0.565305
$88$	10.2853	1.09642
$89$	−8.38404	−0.888707	−0.444353	−	0.895852i	$-0.646567\pi$
$89$	−8.38404	−0.888707	−0.444353	+	0.895852i	$0.646567\pi$
$90$	−3.14265	−0.331265
$91$	1.19202	0.124958
$92$	−37.6619	−3.92652
$93$	−5.77111	−0.598436
$94$	15.2150	1.56930
$95$	2.93813	0.301446
$96$	−12.4075	−1.26634
$97$	10.7900	1.09556	0.547781	−	0.836622i	$-0.315473\pi$
$97$	10.7900	1.09556	0.547781	+	0.836622i	$0.315473\pi$
$98$	2.63640	0.266317
$99$	1.32218	0.132884
$100$	−17.7187	−1.77187
$101$	−17.0598	−1.69752	−0.848758	−	0.528782i	$-0.822649\pi$
$101$	−17.0598	−1.69752	−0.848758	+	0.528782i	$0.822649\pi$
$102$	2.63640	0.261043
$103$	−18.0070	−1.77429	−0.887143	−	0.461494i	$-0.847314\pi$
$103$	−18.0070	−1.77429	−0.887143	+	0.461494i	$0.847314\pi$
$104$	9.27281	0.909274
$105$	1.19202	0.116329
$106$	31.0334	3.01423
$107$	12.1950	1.17894	0.589469	−	0.807791i	$-0.299337\pi$
$107$	12.1950	1.17894	0.589469	+	0.807791i	$0.299337\pi$
$108$	−4.95063	−0.476375
$109$	14.4155	1.38075	0.690375	−	0.723451i	$-0.257445\pi$
$109$	14.4155	1.38075	0.690375	+	0.723451i	$0.257445\pi$
$110$	−4.15515	−0.396178
$111$	0.130157	0.0123539
$112$	10.6075	1.00231
$113$	5.20754	0.489884	0.244942	−	0.969538i	$-0.421231\pi$
$113$	5.20754	0.489884	0.244942	+	0.969538i	$0.421231\pi$
$114$	6.49830	0.608621
$115$	9.06829	0.845623
$116$	−26.1037	−2.42367
$117$	1.19202	0.110202
$118$	28.4719	2.62105
$119$	−1.00000	−0.0916698
$120$	9.27281	0.846488
$121$	−9.25184	−0.841077
$122$	−6.26031	−0.566782
$123$	10.7092	0.965620
$124$	28.5706	2.56572
$125$	10.2265	0.914682
$126$	2.63640	0.234870
$127$	11.6075	1.03000	0.514999	−	0.857191i	$-0.327792\pi$
$127$	11.6075	1.03000	0.514999	+	0.857191i	$0.327792\pi$
$128$	5.49375	0.485584
$129$	1.22344	0.107718
$130$	−3.74611	−0.328556
$131$	10.5057	0.917890	0.458945	−	0.888465i	$-0.348228\pi$
$131$	10.5057	0.917890	0.458945	+	0.888465i	$0.348228\pi$
$132$	−6.54562	−0.569723
$133$	−2.46483	−0.213728
$134$	0.260313	0.0224877
$135$	1.19202	0.102593
$136$	−7.77906	−0.667049
$137$	−9.28832	−0.793555	−0.396777	−	0.917915i	$-0.629872\pi$
$137$	−9.28832	−0.793555	−0.396777	+	0.917915i	$0.629872\pi$
$138$	20.0564	1.70732
$139$	21.4185	1.81669	0.908346	−	0.418220i	$-0.137346\pi$
$139$	21.4185	1.81669	0.908346	+	0.418220i	$0.137346\pi$
$140$	−5.90126	−0.498748
$141$	−5.77111	−0.486015
$142$	−38.0878	−3.19626
$143$	1.57607	0.131797
$144$	10.6075	0.883957
$145$	6.28531	0.521966
$146$	20.8729	1.72745
$147$	−1.00000	−0.0824786
$148$	−0.644358	−0.0529659
$149$	−20.0878	−1.64566	−0.822830	−	0.568288i	$-0.807606\pi$
$149$	−20.0878	−1.64566	−0.822830	+	0.568288i	$0.807606\pi$
$150$	9.43591	0.770439
$151$	17.4185	1.41750	0.708748	−	0.705462i	$-0.249260\pi$
$151$	17.4185	1.41750	0.708748	+	0.705462i	$0.249260\pi$
$152$	−19.1741	−1.55522
$153$	−1.00000	−0.0808452
$154$	3.48580	0.280894
$155$	−6.87929	−0.552558
$156$	−5.90126	−0.472479
$157$	4.06488	0.324413	0.162206	−	0.986757i	$-0.448139\pi$
$157$	4.06488	0.324413	0.162206	+	0.986757i	$0.448139\pi$
$158$	39.4435	3.13795
$159$	−11.7711	−0.933509
$160$	−14.7900	−1.16925
$161$	−7.60749	−0.599554
$162$	2.63640	0.207136
$163$	−7.90126	−0.618875	−0.309437	−	0.950920i	$-0.600141\pi$
$163$	−7.90126	−0.618875	−0.309437	+	0.950920i	$0.600141\pi$
$164$	−53.0175	−4.13997
$165$	1.57607	0.122697
$166$	0.260313	0.0202042
$167$	−3.35360	−0.259509	−0.129755	−	0.991546i	$-0.541419\pi$
$167$	−3.35360	−0.259509	−0.129755	+	0.991546i	$0.541419\pi$
$168$	−7.77906	−0.600167
$169$	−11.5791	−0.890699
$170$	3.14265	0.241030
$171$	−2.46483	−0.188490
$172$	−6.05680	−0.461827
$173$	−2.44893	−0.186188	−0.0930942	−	0.995657i	$-0.529676\pi$
$173$	−2.44893	−0.186188	−0.0930942	+	0.995657i	$0.529676\pi$
$174$	13.9013	1.05385
$175$	−3.57908	−0.270553
$176$	14.0250	1.05717
$177$	−10.7995	−0.811741
$178$	−22.1037	−1.65675
$179$	17.8184	1.33181	0.665906	−	0.746036i	$-0.268045\pi$
$179$	17.8184	1.33181	0.665906	+	0.746036i	$0.268045\pi$
$180$	−5.90126	−0.439854
$181$	3.91717	0.291161	0.145580	−	0.989346i	$-0.453495\pi$
$181$	3.91717	0.291161	0.145580	+	0.989346i	$0.453495\pi$
$182$	3.14265	0.232949
$183$	2.37456	0.175533
$184$	−59.1791	−4.36274
$185$	0.155150	0.0114068
$186$	−15.2150	−1.11562
$187$	−1.32218	−0.0966873
$188$	28.5706	2.08373
$189$	−1.00000	−0.0727393
$190$	7.74611	0.561962
$191$	−1.47632	−0.106823	−0.0534113	−	0.998573i	$-0.517009\pi$
$191$	−1.47632	−0.106823	−0.0534113	+	0.998573i	$0.517009\pi$
$192$	−11.4963	−0.829670
$193$	−8.03142	−0.578114	−0.289057	−	0.957312i	$-0.593342\pi$
$193$	−8.03142	−0.578114	−0.289057	+	0.957312i	$0.593342\pi$
$194$	28.4469	2.04237
$195$	1.42092	0.101754
$196$	4.95063	0.353616
$197$	−16.8803	−1.20267	−0.601336	−	0.798997i	$-0.705365\pi$
$197$	−16.8803	−1.20267	−0.601336	+	0.798997i	$0.705365\pi$
$198$	3.48580	0.247725
$199$	−1.86984	−0.132550	−0.0662748	−	0.997801i	$-0.521111\pi$
$199$	−1.86984	−0.132550	−0.0662748	+	0.997801i	$0.521111\pi$
$200$	−27.8419	−1.96872
$201$	−0.0987380	−0.00696444
$202$	−44.9766	−3.16454
$203$	−5.27281	−0.370079
$204$	4.95063	0.346613
$205$	12.7657	0.891592
$206$	−47.4739	−3.30766
$207$	−7.60749	−0.528757
$208$	12.6444	0.876728
$209$	−3.25895	−0.225426
$210$	3.14265	0.216864
$211$	−4.61294	−0.317568	−0.158784	−	0.987313i	$-0.550757\pi$
$211$	−4.61294	−0.317568	−0.158784	+	0.987313i	$0.550757\pi$
$212$	58.2744	4.00230
$213$	14.4469	0.989884
$214$	32.1511	2.19780
$215$	1.45837	0.0994599
$216$	−7.77906	−0.529298
$217$	5.77111	0.391768
$218$	38.0050	2.57402
$219$	−7.91717	−0.534993
$220$	−7.80252	−0.526046
$221$	−1.19202	−0.0801841
$222$	0.343146	0.0230304
$223$	12.2613	0.821079	0.410539	−	0.911843i	$-0.365340\pi$
$223$	12.2613	0.821079	0.410539	+	0.911843i	$0.365340\pi$
$224$	12.4075	0.829012
$225$	−3.57908	−0.238606
$226$	13.7292	0.913251
$227$	19.0933	1.26727	0.633633	−	0.773634i	$-0.281563\pi$
$227$	19.0933	1.26727	0.633633	+	0.773634i	$0.281563\pi$
$228$	12.2025	0.808129
$229$	8.87286	0.586335	0.293168	−	0.956061i	$-0.405291\pi$
$229$	8.87286	0.586335	0.293168	+	0.956061i	$0.405291\pi$
$230$	23.9077	1.57643
$231$	−1.32218	−0.0869930
$232$	−41.0175	−2.69293
$233$	20.0324	1.31237	0.656184	−	0.754601i	$-0.272170\pi$
$233$	20.0324	1.31237	0.656184	+	0.754601i	$0.272170\pi$
$234$	3.14265	0.205442
$235$	−6.87929	−0.448755
$236$	53.4644	3.48023
$237$	−14.9611	−0.971827
$238$	−2.63640	−0.170893
$239$	−2.17106	−0.140434	−0.0702169	−	0.997532i	$-0.522369\pi$
$239$	−2.17106	−0.140434	−0.0702169	+	0.997532i	$0.522369\pi$
$240$	12.6444	0.816190
$241$	−9.78059	−0.630023	−0.315011	−	0.949088i	$-0.602008\pi$
$241$	−9.78059	−0.630023	−0.315011	+	0.949088i	$0.602008\pi$
$242$	−24.3916	−1.56795
$243$	−1.00000	−0.0641500
$244$	−11.7556	−0.752575
$245$	−1.19202	−0.0761555
$246$	28.2339	1.80013
$247$	−2.93813	−0.186949
$248$	44.8938	2.85076
$249$	−0.0987380	−0.00625727
$250$	26.9611	1.70517
$251$	6.89825	0.435413	0.217707	−	0.976014i	$-0.430142\pi$
$251$	6.89825	0.435413	0.217707	+	0.976014i	$0.430142\pi$
$252$	4.95063	0.311860
$253$	−10.0585	−0.632370
$254$	30.6020	1.92014
$255$	−1.19202	−0.0746473
$256$	−8.50875	−0.531797
$257$	−19.3133	−1.20473	−0.602366	−	0.798220i	$-0.705775\pi$
$257$	−19.3133	−1.20473	−0.602366	+	0.798220i	$0.705775\pi$
$258$	3.22549	0.200810
$259$	−0.130157	−0.00808754
$260$	−7.03444	−0.436257
$261$	−5.27281	−0.326379
$262$	27.6974	1.71115
$263$	22.4060	1.38161	0.690806	−	0.723040i	$-0.257256\pi$
$263$	22.4060	1.38161	0.690806	+	0.723040i	$0.257256\pi$
$264$	−10.2853	−0.633017
$265$	−14.0314	−0.861943
$266$	−6.49830	−0.398436
$267$	8.38404	0.513095
$268$	0.488815	0.0298592
$269$	−12.9317	−0.788460	−0.394230	−	0.919012i	$-0.628989\pi$
$269$	−12.9317	−0.788460	−0.394230	+	0.919012i	$0.628989\pi$
$270$	3.14265	0.191256
$271$	−24.3242	−1.47759	−0.738794	−	0.673932i	$-0.764604\pi$
$271$	−24.3242	−1.47759	−0.738794	+	0.673932i	$0.764604\pi$
$272$	−10.6075	−0.643173
$273$	−1.19202	−0.0721445
$274$	−24.4878	−1.47936
$275$	−4.73219	−0.285362
$276$	37.6619	2.26698
$277$	16.2853	0.978489	0.489245	−	0.872147i	$-0.337273\pi$
$277$	16.2853	0.978489	0.489245	+	0.872147i	$0.337273\pi$
$278$	56.4678	3.38671
$279$	5.77111	0.345507
$280$	−9.27281	−0.554156
$281$	−10.9297	−0.652009	−0.326005	−	0.945368i	$-0.605702\pi$
$281$	−10.9297	−0.652009	−0.326005	+	0.945368i	$0.605702\pi$
$282$	−15.2150	−0.906038
$283$	2.79951	0.166413	0.0832067	−	0.996532i	$-0.473484\pi$
$283$	2.79951	0.166413	0.0832067	+	0.996532i	$0.473484\pi$
$284$	−71.5212	−4.24400
$285$	−2.93813	−0.174040
$286$	4.15515	0.245699
$287$	−10.7092	−0.632146
$288$	12.4075	0.731120
$289$	1.00000	0.0588235
$290$	16.5706	0.973060
$291$	−10.7900	−0.632523
$292$	39.1950	2.29371
$293$	−3.55312	−0.207575	−0.103788	−	0.994599i	$-0.533096\pi$
$293$	−3.55312	−0.207575	−0.103788	+	0.994599i	$0.533096\pi$
$294$	−2.63640	−0.153758
$295$	−12.8733	−0.749510
$296$	−1.01250	−0.0588502
$297$	−1.32218	−0.0767206
$298$	−52.9597	−3.06787
$299$	−9.06829	−0.524433
$300$	17.7187	1.02299
$301$	−1.22344	−0.0705180
$302$	45.9222	2.64252
$303$	17.0598	0.980061
$304$	−26.1457	−1.49956
$305$	2.83053	0.162076
$306$	−2.63640	−0.150713
$307$	18.4249	1.05157	0.525784	−	0.850618i	$-0.323772\pi$
$307$	18.4249	1.05157	0.525784	+	0.850618i	$0.323772\pi$
$308$	6.54562	0.372971
$309$	18.0070	1.02438
$310$	−18.1366	−1.03009
$311$	−22.3481	−1.26725	−0.633623	−	0.773642i	$-0.718433\pi$
$311$	−22.3481	−1.26725	−0.633623	+	0.773642i	$0.718433\pi$
$312$	−9.27281	−0.524969
$313$	18.9770	1.07264	0.536321	−	0.844014i	$-0.319813\pi$
$313$	18.9770	1.07264	0.536321	+	0.844014i	$0.319813\pi$
$314$	10.7167	0.604777
$315$	−1.19202	−0.0671628
$316$	74.0668	4.16658
$317$	9.43439	0.529888	0.264944	−	0.964264i	$-0.414647\pi$
$317$	9.43439	0.529888	0.264944	+	0.964264i	$0.414647\pi$
$318$	−31.0334	−1.74027
$319$	−6.97160	−0.390335
$320$	−13.7038	−0.766065
$321$	−12.1950	−0.680661
$322$	−20.0564	−1.11770
$323$	2.46483	0.137147
$324$	4.95063	0.275035
$325$	−4.26635	−0.236654
$326$	−20.8309	−1.15372
$327$	−14.4155	−0.797177
$328$	−83.3078	−4.59990
$329$	5.77111	0.318171
$330$	4.15515	0.228733
$331$	−4.53715	−0.249384	−0.124692	−	0.992195i	$-0.539794\pi$
$331$	−4.53715	−0.249384	−0.124692	+	0.992195i	$0.539794\pi$
$332$	0.488815	0.0268272
$333$	−0.130157	−0.00713254
$334$	−8.84144	−0.483782
$335$	−0.117698	−0.00643052
$336$	−10.6075	−0.578686
$337$	−29.8279	−1.62483	−0.812415	−	0.583080i	$-0.801847\pi$
$337$	−29.8279	−1.62483	−0.812415	+	0.583080i	$0.801847\pi$
$338$	−30.5272	−1.66046
$339$	−5.20754	−0.282834
$340$	5.90126	0.320041
$341$	7.63043	0.413211
$342$	−6.49830	−0.351388
$343$	1.00000	0.0539949
$344$	−9.51722	−0.513134
$345$	−9.06829	−0.488220
$346$	−6.45636	−0.347096
$347$	0.359051	0.0192749	0.00963744	−	0.999954i	$-0.496932\pi$
$347$	0.359051	0.0192749	0.00963744	+	0.999954i	$0.496932\pi$
$348$	26.1037	1.39931
$349$	−33.1751	−1.77582	−0.887911	−	0.460016i	$-0.847844\pi$
$349$	−33.1751	−1.77582	−0.887911	+	0.460016i	$0.847844\pi$
$350$	−9.43591	−0.504371
$351$	−1.19202	−0.0636254
$352$	16.4049	0.874387
$353$	14.4155	0.767258	0.383629	−	0.923487i	$-0.374674\pi$
$353$	14.4155	0.767258	0.383629	+	0.923487i	$0.374674\pi$
$354$	−28.4719	−1.51326
$355$	17.2210	0.913996
$356$	−41.5063	−2.19983
$357$	1.00000	0.0529256
$358$	46.9766	2.48279
$359$	8.94518	0.472108	0.236054	−	0.971740i	$-0.424146\pi$
$359$	8.94518	0.472108	0.236054	+	0.971740i	$0.424146\pi$
$360$	−9.27281	−0.488720
$361$	−12.9246	−0.680242
$362$	10.3272	0.542788
$363$	9.25184	0.485596
$364$	5.90126	0.309310
$365$	−9.43744	−0.493978
$366$	6.26031	0.327232
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−80.6963	−4.20658
$369$	−10.7092	−0.557501
$370$	0.409037	0.0212648
$371$	11.7711	0.611125
$372$	−28.5706	−1.48132
$373$	3.35564	0.173749	0.0868743	−	0.996219i	$-0.472312\pi$
$373$	3.35564	0.173749	0.0868743	+	0.996219i	$0.472312\pi$
$374$	−3.48580	−0.180246
$375$	−10.2265	−0.528092
$376$	44.8938	2.31522
$377$	−6.28531	−0.323710
$378$	−2.63640	−0.135602
$379$	0.570613	0.0293104	0.0146552	−	0.999893i	$-0.495335\pi$
$379$	0.570613	0.0293104	0.0146552	+	0.999893i	$0.495335\pi$
$380$	14.5456	0.746175
$381$	−11.6075	−0.594669
$382$	−3.89217	−0.199141
$383$	−20.9043	−1.06816	−0.534079	−	0.845434i	$-0.679342\pi$
$383$	−20.9043	−1.06816	−0.534079	+	0.845434i	$0.679342\pi$
$384$	−5.49375	−0.280352
$385$	−1.57607	−0.0803238
$386$	−21.1741	−1.07773
$387$	−1.22344	−0.0621910
$388$	53.4174	2.71186
$389$	13.6409	0.691624	0.345812	−	0.938304i	$-0.387604\pi$
$389$	13.6409	0.691624	0.345812	+	0.938304i	$0.387604\pi$
$390$	3.74611	0.189692
$391$	7.60749	0.384727
$392$	7.77906	0.392902
$393$	−10.5057	−0.529944
$394$	−44.5033	−2.24204
$395$	−17.8339	−0.897323
$396$	6.54562	0.328930
$397$	−22.0114	−1.10472	−0.552361	−	0.833605i	$-0.686273\pi$
$397$	−22.0114	−1.10472	−0.552361	+	0.833605i	$0.686273\pi$
$398$	−4.92966	−0.247102
$399$	2.46483	0.123396
$400$	−37.9651	−1.89825
$401$	−7.44286	−0.371678	−0.185839	−	0.982580i	$-0.559500\pi$
$401$	−7.44286	−0.371678	−0.185839	+	0.982580i	$0.559500\pi$
$402$	−0.260313	−0.0129833
$403$	6.87929	0.342682
$404$	−84.4569	−4.20189
$405$	−1.19202	−0.0592321
$406$	−13.9013	−0.689908
$407$	−0.172090	−0.00853021
$408$	7.77906	0.385121
$409$	−14.0858	−0.696497	−0.348249	−	0.937402i	$-0.613224\pi$
$409$	−14.0858	−0.696497	−0.348249	+	0.937402i	$0.613224\pi$
$410$	33.6554	1.66212
$411$	9.28832	0.458159
$412$	−89.1462	−4.39192
$413$	10.7995	0.531409
$414$	−20.0564	−0.985719
$415$	−0.117698	−0.00577756
$416$	14.7900	0.725141
$417$	−21.4185	−1.04887
$418$	−8.59191	−0.420244
$419$	33.7606	1.64931	0.824656	−	0.565634i	$-0.191369\pi$
$419$	33.7606	1.64931	0.824656	+	0.565634i	$0.191369\pi$
$420$	5.90126	0.287952
$421$	36.5750	1.78256	0.891278	−	0.453457i	$-0.149810\pi$
$421$	36.5750	1.78256	0.891278	+	0.453457i	$0.149810\pi$
$422$	−12.1616	−0.592016
$423$	5.77111	0.280601
$424$	91.5681	4.44694
$425$	3.57908	0.173611
$426$	38.0878	1.84536
$427$	−2.37456	−0.114913
$428$	60.3731	2.91825
$429$	−1.57607	−0.0760932
$430$	3.84485	0.185415
$431$	−18.5456	−0.893311	−0.446656	−	0.894706i	$-0.647385\pi$
$431$	−18.5456	−0.893311	−0.446656	+	0.894706i	$0.647385\pi$
$432$	−10.6075	−0.510353
$433$	7.31575	0.351573	0.175786	−	0.984428i	$-0.443753\pi$
$433$	7.31575	0.351573	0.175786	+	0.984428i	$0.443753\pi$
$434$	15.2150	0.730342
$435$	−6.28531	−0.301357
$436$	71.3656	3.41779
$437$	18.7512	0.896990
$438$	−20.8729	−0.997344
$439$	−28.4783	−1.35920	−0.679598	−	0.733585i	$-0.737846\pi$
$439$	−28.4783	−1.35920	−0.679598	+	0.733585i	$0.737846\pi$
$440$	−12.2603	−0.584488
$441$	1.00000	0.0476190
$442$	−3.14265	−0.149481
$443$	−6.84041	−0.324997	−0.162499	−	0.986709i	$-0.551955\pi$
$443$	−6.84041	−0.324997	−0.162499	+	0.986709i	$0.551955\pi$
$444$	0.644358	0.0305799
$445$	9.99397	0.473759
$446$	32.3258	1.53067
$447$	20.0878	0.950122
$448$	11.4963	0.543147
$449$	−5.73060	−0.270444	−0.135222	−	0.990815i	$-0.543175\pi$
$449$	−5.73060	−0.270444	−0.135222	+	0.990815i	$0.543175\pi$
$450$	−9.43591	−0.444813
$451$	−14.1595	−0.666746
$452$	25.7806	1.21262
$453$	−17.4185	−0.818392
$454$	50.3376	2.36246
$455$	−1.42092	−0.0666136
$456$	19.1741	0.897908
$457$	29.4100	1.37574	0.687871	−	0.725833i	$-0.258545\pi$
$457$	29.4100	1.37574	0.687871	+	0.725833i	$0.258545\pi$
$458$	23.3925	1.09306
$459$	1.00000	0.0466760
$460$	44.8938	2.09318
$461$	−15.4439	−0.719293	−0.359646	−	0.933089i	$-0.617103\pi$
$461$	−15.4439	−0.719293	−0.359646	+	0.933089i	$0.617103\pi$
$462$	−3.48580	−0.162174
$463$	−20.4090	−0.948488	−0.474244	−	0.880393i	$-0.657279\pi$
$463$	−20.4090	−0.948488	−0.474244	+	0.880393i	$0.657279\pi$
$464$	−55.9313	−2.59654
$465$	6.87929	0.319019
$466$	52.8136	2.44654
$467$	10.9865	0.508393	0.254197	−	0.967153i	$-0.418189\pi$
$467$	10.9865	0.508393	0.254197	+	0.967153i	$0.418189\pi$
$468$	5.90126	0.272786
$469$	0.0987380	0.00455930
$470$	−18.1366	−0.836578
$471$	−4.06488	−0.187300
$472$	84.0100	3.86687
$473$	−1.61761	−0.0743777
$474$	−39.4435	−1.81170
$475$	8.82184	0.404774
$476$	−4.95063	−0.226912
$477$	11.7711	0.538962
$478$	−5.72378	−0.261800
$479$	18.9945	0.867883	0.433941	−	0.900941i	$-0.357122\pi$
$479$	18.9945	0.867883	0.433941	+	0.900941i	$0.357122\pi$
$480$	14.7900	0.675069
$481$	−0.155150	−0.00707422
$482$	−25.7856	−1.17450
$483$	7.60749	0.346153
$484$	−45.8025	−2.08193
$485$	−12.8620	−0.584031
$486$	−2.63640	−0.119590
$487$	−21.7606	−0.986066	−0.493033	−	0.870010i	$-0.664112\pi$
$487$	−21.7606	−0.986066	−0.493033	+	0.870010i	$0.664112\pi$
$488$	−18.4719	−0.836182
$489$	7.90126	0.357307
$490$	−3.14265	−0.141971
$491$	6.93070	0.312778	0.156389	−	0.987696i	$-0.450015\pi$
$491$	6.93070	0.312778	0.156389	+	0.987696i	$0.450015\pi$
$492$	53.0175	2.39021
$493$	5.27281	0.237475
$494$	−7.74611	−0.348514
$495$	−1.57607	−0.0708389
$496$	61.2169	2.74872
$497$	−14.4469	−0.648031
$498$	−0.260313	−0.0116649
$499$	−18.4533	−0.826083	−0.413042	−	0.910712i	$-0.635534\pi$
$499$	−18.4533	−0.826083	−0.413042	+	0.910712i	$0.635534\pi$
$500$	50.6274	2.26413
$501$	3.35360	0.149828
$502$	18.1866	0.811706
$503$	−35.9560	−1.60320	−0.801600	−	0.597861i	$-0.796018\pi$
$503$	−35.9560	−1.60320	−0.801600	+	0.597861i	$0.796018\pi$
$504$	7.77906	0.346507
$505$	20.3357	0.904926
$506$	−26.5182	−1.17888
$507$	11.5791	0.514245
$508$	57.4644	2.54957
$509$	−16.6444	−0.737748	−0.368874	−	0.929479i	$-0.620257\pi$
$509$	−16.6444	−0.737748	−0.368874	+	0.929479i	$0.620257\pi$
$510$	−3.14265	−0.139159
$511$	7.91717	0.350235
$512$	−33.4200	−1.47697
$513$	2.46483	0.108825
$514$	−50.9177	−2.24588
$515$	21.4648	0.945852
$516$	6.05680	0.266636
$517$	7.63043	0.335586
$518$	−0.343146	−0.0150770
$519$	2.44893	0.107496
$520$	−11.0534	−0.484723
$521$	−0.944560	−0.0413819	−0.0206910	−	0.999786i	$-0.506587\pi$
$521$	−0.944560	−0.0413819	−0.0206910	+	0.999786i	$0.506587\pi$
$522$	−13.9013	−0.608442
$523$	−18.9766	−0.829789	−0.414894	−	0.909870i	$-0.636181\pi$
$523$	−18.9766	−0.829789	−0.414894	+	0.909870i	$0.636181\pi$
$524$	52.0100	2.27207
$525$	3.57908	0.156204
$526$	59.0712	2.57563
$527$	−5.77111	−0.251393
$528$	−14.0250	−0.610359
$529$	34.8738	1.51625
$530$	−36.9925	−1.60685
$531$	10.7995	0.468659
$532$	−12.2025	−0.529044
$533$	−12.7657	−0.552942
$534$	22.1037	0.956522
$535$	−14.5368	−0.628479
$536$	0.768089	0.0331764
$537$	−17.8184	−0.768922
$538$	−34.0932	−1.46986
$539$	1.32218	0.0569503
$540$	5.90126	0.253950
$541$	−19.1771	−0.824489	−0.412245	−	0.911073i	$-0.635255\pi$
$541$	−19.1771	−0.824489	−0.412245	+	0.911073i	$0.635255\pi$
$542$	−64.1283	−2.75455
$543$	−3.91717	−0.168102
$544$	−12.4075	−0.531968
$545$	−17.1836	−0.736062
$546$	−3.14265	−0.134493
$547$	−25.5736	−1.09345	−0.546725	−	0.837312i	$-0.684126\pi$
$547$	−25.5736	−1.09345	−0.546725	+	0.837312i	$0.684126\pi$
$548$	−45.9831	−1.96430
$549$	−2.37456	−0.101344
$550$	−12.4760	−0.531977
$551$	12.9966	0.553673
$552$	59.1791	2.51883
$553$	14.9611	0.636210
$554$	42.9347	1.82412
$555$	−0.155150	−0.00658574
$556$	106.035	4.49689
$557$	34.9191	1.47957	0.739786	−	0.672842i	$-0.234927\pi$
$557$	34.9191	1.47957	0.739786	+	0.672842i	$0.234927\pi$
$558$	15.2150	0.644101
$559$	−1.45837	−0.0616824
$560$	−12.6444	−0.534322
$561$	1.32218	0.0558224
$562$	−28.8150	−1.21549
$563$	−14.9547	−0.630264	−0.315132	−	0.949048i	$-0.602049\pi$
$563$	−14.9547	−0.630264	−0.315132	+	0.949048i	$0.602049\pi$
$564$	−28.5706	−1.20304
$565$	−6.20750	−0.261151
$566$	7.38064	0.310231
$567$	1.00000	0.0419961
$568$	−112.383	−4.71549
$569$	35.5003	1.48825	0.744125	−	0.668041i	$-0.232867\pi$
$569$	35.5003	1.48825	0.744125	+	0.668041i	$0.232867\pi$
$570$	−7.74611	−0.324449
$571$	−22.7931	−0.953861	−0.476930	−	0.878941i	$-0.658251\pi$
$571$	−22.7931	−0.953861	−0.476930	+	0.878941i	$0.658251\pi$
$572$	7.80252	0.326240
$573$	1.47632	0.0616741
$574$	−28.2339	−1.17846
$575$	27.2278	1.13548
$576$	11.4963	0.479010
$577$	−7.21702	−0.300448	−0.150224	−	0.988652i	$-0.548000\pi$
$577$	−7.21702	−0.300448	−0.150224	+	0.988652i	$0.548000\pi$
$578$	2.63640	0.109660
$579$	8.03142	0.333774
$580$	31.1162	1.29203
$581$	0.0987380	0.00409634
$582$	−28.4469	−1.17916
$583$	15.5635	0.644575
$584$	61.5881	2.54853
$585$	−1.42092	−0.0587477
$586$	−9.36746	−0.386966
$587$	30.3776	1.25382	0.626909	−	0.779092i	$-0.284320\pi$
$587$	30.3776	1.25382	0.626909	+	0.779092i	$0.284320\pi$
$588$	−4.95063	−0.204161
$589$	−14.2248	−0.586123
$590$	−33.9391	−1.39725
$591$	16.8803	0.694363
$592$	−1.38064	−0.0567438
$593$	16.0983	0.661080	0.330540	−	0.943792i	$-0.392769\pi$
$593$	16.0983	0.661080	0.330540	+	0.943792i	$0.392769\pi$
$594$	−3.48580	−0.143024
$595$	1.19202	0.0488681
$596$	−99.4474	−4.07353
$597$	1.86984	0.0765276
$598$	−23.9077	−0.977658
$599$	35.1382	1.43571	0.717853	−	0.696194i	$-0.245125\pi$
$599$	35.1382	1.43571	0.717853	+	0.696194i	$0.245125\pi$
$600$	27.8419	1.13664
$601$	38.1601	1.55658	0.778292	−	0.627902i	$-0.216086\pi$
$601$	38.1601	1.55658	0.778292	+	0.627902i	$0.216086\pi$
$602$	−3.22549	−0.131461
$603$	0.0987380	0.00402092
$604$	86.2325	3.50875
$605$	11.0284	0.448368
$606$	44.9766	1.82705
$607$	−7.73326	−0.313883	−0.156942	−	0.987608i	$-0.550163\pi$
$607$	−7.73326	−0.313883	−0.156942	+	0.987608i	$0.550163\pi$
$608$	−30.5824	−1.24028
$609$	5.27281	0.213665
$610$	7.46243	0.302145
$611$	6.87929	0.278306
$612$	−4.95063	−0.200117
$613$	−43.0888	−1.74034	−0.870170	−	0.492751i	$-0.835991\pi$
$613$	−43.0888	−1.74034	−0.870170	+	0.492751i	$0.835991\pi$
$614$	48.5756	1.96035
$615$	−12.7657	−0.514761
$616$	10.2853	0.414407
$617$	2.04090	0.0821635	0.0410817	−	0.999156i	$-0.486920\pi$
$617$	2.04090	0.0821635	0.0410817	+	0.999156i	$0.486920\pi$
$618$	47.4739	1.90968
$619$	−35.3383	−1.42037	−0.710183	−	0.704017i	$-0.751388\pi$
$619$	−35.3383	−1.42037	−0.710183	+	0.704017i	$0.751388\pi$
$620$	−34.0568	−1.36775
$621$	7.60749	0.305278
$622$	−58.9188	−2.36243
$623$	−8.38404	−0.335900
$624$	−12.6444	−0.506179
$625$	5.70525	0.228210
$626$	50.0310	1.99964
$627$	3.25895	0.130150
$628$	20.1237	0.803024
$629$	0.130157	0.00518969
$630$	−3.14265	−0.125206
$631$	−35.9306	−1.43038	−0.715188	−	0.698932i	$-0.753659\pi$
$631$	−35.9306	−1.43038	−0.715188	+	0.698932i	$0.753659\pi$
$632$	116.383	4.62947
$633$	4.61294	0.183348
$634$	24.8729	0.987827
$635$	−13.8364	−0.549080
$636$	−58.2744	−2.31073
$637$	1.19202	0.0472296
$638$	−18.3800	−0.727669
$639$	−14.4469	−0.571510
$640$	−6.54867	−0.258859
$641$	13.4199	0.530053	0.265027	−	0.964241i	$-0.414619\pi$
$641$	13.4199	0.530053	0.265027	+	0.964241i	$0.414619\pi$
$642$	−32.1511	−1.26890
$643$	43.0844	1.69908	0.849542	−	0.527521i	$-0.176878\pi$
$643$	43.0844	1.69908	0.849542	+	0.527521i	$0.176878\pi$
$644$	−37.6619	−1.48408
$645$	−1.45837	−0.0574232
$646$	6.49830	0.255672
$647$	41.3028	1.62378	0.811890	−	0.583810i	$-0.198439\pi$
$647$	41.3028	1.62378	0.811890	+	0.583810i	$0.198439\pi$
$648$	7.77906	0.305590
$649$	14.2789	0.560495
$650$	−11.2478	−0.441176
$651$	−5.77111	−0.226187
$652$	−39.1162	−1.53191
$653$	25.9715	1.01634	0.508172	−	0.861255i	$-0.330321\pi$
$653$	25.9715	1.01634	0.508172	+	0.861255i	$0.330321\pi$
$654$	−38.0050	−1.48611
$655$	−12.5231	−0.489317
$656$	−113.598	−4.43526
$657$	7.91717	0.308878
$658$	15.2150	0.593141
$659$	12.3705	0.481885	0.240943	−	0.970539i	$-0.422543\pi$
$659$	12.3705	0.481885	0.240943	+	0.970539i	$0.422543\pi$
$660$	7.80252	0.303713
$661$	19.2488	0.748693	0.374346	−	0.927289i	$-0.377867\pi$
$661$	19.2488	0.748693	0.374346	+	0.927289i	$0.377867\pi$
$662$	−11.9618	−0.464907
$663$	1.19202	0.0462943
$664$	0.768089	0.0298076
$665$	2.93813	0.113936
$666$	−0.343146	−0.0132966
$667$	40.1128	1.55317
$668$	−16.6024	−0.642367
$669$	−12.2613	−0.474050
$670$	−0.310299	−0.0119879
$671$	−3.13960	−0.121203
$672$	−12.4075	−0.478630
$673$	27.0594	1.04306	0.521532	−	0.853232i	$-0.325361\pi$
$673$	27.0594	1.04306	0.521532	+	0.853232i	$0.325361\pi$
$674$	−78.6384	−3.02904
$675$	3.57908	0.137759
$676$	−57.3238	−2.20476
$677$	24.3880	0.937308	0.468654	−	0.883382i	$-0.344739\pi$
$677$	24.3880	0.937308	0.468654	+	0.883382i	$0.344739\pi$
$678$	−13.7292	−0.527266
$679$	10.7900	0.414083
$680$	9.27281	0.355596
$681$	−19.0933	−0.731656
$682$	20.1169	0.770316
$683$	24.2579	0.928202	0.464101	−	0.885782i	$-0.346377\pi$
$683$	24.2579	0.928202	0.464101	+	0.885782i	$0.346377\pi$
$684$	−12.2025	−0.466573
$685$	11.0719	0.423035
$686$	2.63640	0.100658
$687$	−8.87286	−0.338521
$688$	−12.9776	−0.494768
$689$	14.0314	0.534554
$690$	−23.9077	−0.910150
$691$	0.955050	0.0363318	0.0181659	−	0.999835i	$-0.494217\pi$
$691$	0.955050	0.0363318	0.0181659	+	0.999835i	$0.494217\pi$
$692$	−12.1237	−0.460875
$693$	1.32218	0.0502254
$694$	0.946605	0.0359326
$695$	−25.5313	−0.968458
$696$	41.0175	1.55476
$697$	10.7092	0.405641
$698$	−87.4629	−3.31052
$699$	−20.0324	−0.757696
$700$	−17.7187	−0.669705
$701$	23.2210	0.877045	0.438523	−	0.898720i	$-0.355502\pi$
$701$	23.2210	0.877045	0.438523	+	0.898720i	$0.355502\pi$
$702$	−3.14265	−0.118612
$703$	0.320814	0.0120997
$704$	15.2001	0.572875
$705$	6.87929	0.259089
$706$	38.0050	1.43034
$707$	−17.0598	−0.641601
$708$	−53.4644	−2.00931
$709$	1.42939	0.0536818	0.0268409	−	0.999640i	$-0.491455\pi$
$709$	1.42939	0.0536818	0.0268409	+	0.999640i	$0.491455\pi$
$710$	45.4015	1.70389
$711$	14.9611	0.561084
$712$	−65.2200	−2.44422
$713$	−43.9036	−1.64420
$714$	2.63640	0.0986650
$715$	−1.87871	−0.0702596
$716$	88.2125	3.29665
$717$	2.17106	0.0810795
$718$	23.5831	0.880114
$719$	−14.8140	−0.552469	−0.276235	−	0.961090i	$-0.589087\pi$
$719$	−14.8140	−0.552469	−0.276235	+	0.961090i	$0.589087\pi$
$720$	−12.6444	−0.471227
$721$	−18.0070	−0.670617
$722$	−34.0745	−1.26812
$723$	9.78059	0.363744
$724$	19.3925	0.720715
$725$	18.8718	0.700882
$726$	24.3916	0.905257
$727$	5.31474	0.197113	0.0985565	−	0.995131i	$-0.468577\pi$
$727$	5.31474	0.197113	0.0985565	+	0.995131i	$0.468577\pi$
$728$	9.27281	0.343673
$729$	1.00000	0.0370370
$730$	−24.8809	−0.920884
$731$	1.22344	0.0452506
$732$	11.7556	0.434499
$733$	42.3860	1.56556	0.782781	−	0.622297i	$-0.213800\pi$
$733$	42.3860	1.56556	0.782781	+	0.622297i	$0.213800\pi$
$734$	0	0
$735$	1.19202	0.0439684
$736$	−94.3900	−3.47926
$737$	0.130549	0.00480885
$738$	−28.2339	−1.03930
$739$	−8.35661	−0.307403	−0.153702	−	0.988117i	$-0.549119\pi$
$739$	−8.35661	−0.307403	−0.153702	+	0.988117i	$0.549119\pi$
$740$	0.768089	0.0282355
$741$	2.93813	0.107935
$742$	31.0334	1.13927
$743$	37.3028	1.36851	0.684254	−	0.729244i	$-0.260128\pi$
$743$	37.3028	1.36851	0.684254	+	0.729244i	$0.260128\pi$
$744$	−44.8938	−1.64589
$745$	23.9451	0.877282
$746$	8.84683	0.323906
$747$	0.0987380	0.00361264
$748$	−6.54562	−0.239332
$749$	12.1950	0.445597
$750$	−26.9611	−0.984480
$751$	−1.67237	−0.0610255	−0.0305128	−	0.999534i	$-0.509714\pi$
$751$	−1.67237	−0.0610255	−0.0305128	+	0.999534i	$0.509714\pi$
$752$	61.2169	2.23235
$753$	−6.89825	−0.251386
$754$	−16.5706	−0.603466
$755$	−20.7632	−0.755651
$756$	−4.95063	−0.180053
$757$	−16.4493	−0.597861	−0.298930	−	0.954275i	$-0.596630\pi$
$757$	−16.4493	−0.597861	−0.298930	+	0.954275i	$0.596630\pi$
$758$	1.50437	0.0546410
$759$	10.0585	0.365099
$760$	22.8559	0.829071
$761$	−27.8593	−1.00990	−0.504950	−	0.863149i	$-0.668489\pi$
$761$	−27.8593	−1.00990	−0.504950	+	0.863149i	$0.668489\pi$
$762$	−30.6020	−1.10859
$763$	14.4155	0.521875
$764$	−7.30871	−0.264420
$765$	1.19202	0.0430977
$766$	−55.1121	−1.99128
$767$	12.8733	0.464826
$768$	8.50875	0.307033
$769$	−41.5970	−1.50003	−0.750013	−	0.661424i	$-0.769953\pi$
$769$	−41.5970	−1.50003	−0.750013	+	0.661424i	$0.769953\pi$
$770$	−4.15515	−0.149741
$771$	19.3133	0.695552
$772$	−39.7606	−1.43101
$773$	−15.5108	−0.557884	−0.278942	−	0.960308i	$-0.589984\pi$
$773$	−15.5108	−0.557884	−0.278942	+	0.960308i	$0.589984\pi$
$774$	−3.22549	−0.115938
$775$	−20.6553	−0.741959
$776$	83.9363	3.01314
$777$	0.130157	0.00466935
$778$	35.9631	1.28934
$779$	26.3965	0.945752
$780$	7.03444	0.251873
$781$	−19.1014	−0.683500
$782$	20.0564	0.717216
$783$	5.27281	0.188435
$784$	10.6075	0.378839
$785$	−4.84543	−0.172941
$786$	−27.6974	−0.987932
$787$	55.3471	1.97291	0.986455	−	0.164032i	$-0.0524501\pi$
$787$	55.3471	1.97291	0.986455	+	0.164032i	$0.0524501\pi$
$788$	−83.5681	−2.97699
$789$	−22.4060	−0.797674
$790$	−47.0175	−1.67281
$791$	5.20754	0.185159
$792$	10.2853	0.365472
$793$	−2.83053	−0.100515
$794$	−58.0310	−2.05944
$795$	14.0314	0.497643
$796$	−9.25690	−0.328102
$797$	−43.4813	−1.54019	−0.770094	−	0.637931i	$-0.779791\pi$
$797$	−43.4813	−1.54019	−0.770094	+	0.637931i	$0.779791\pi$
$798$	6.49830	0.230037
$799$	−5.77111	−0.204167
$800$	−44.4075	−1.57004
$801$	−8.38404	−0.296236
$802$	−19.6224	−0.692890
$803$	10.4679	0.369405
$804$	−0.488815	−0.0172392
$805$	9.06829	0.319615
$806$	18.1366	0.638834
$807$	12.9317	0.455218
$808$	−132.709	−4.66870
$809$	−16.6869	−0.586680	−0.293340	−	0.956008i	$-0.594767\pi$
$809$	−16.6869	−0.586680	−0.293340	+	0.956008i	$0.594767\pi$
$810$	−3.14265	−0.110422
$811$	−27.0175	−0.948713	−0.474356	−	0.880333i	$-0.657319\pi$
$811$	−27.0175	−0.948713	−0.474356	+	0.880333i	$0.657319\pi$
$812$	−26.1037	−0.916062
$813$	24.3242	0.853086
$814$	−0.453700	−0.0159022
$815$	9.41848	0.329915
$816$	10.6075	0.371336
$817$	3.01558	0.105502
$818$	−37.1358	−1.29842
$819$	1.19202	0.0416526
$820$	63.1980	2.20697
$821$	11.3623	0.396547	0.198273	−	0.980147i	$-0.436467\pi$
$821$	11.3623	0.396547	0.198273	+	0.980147i	$0.436467\pi$
$822$	24.4878	0.854109
$823$	−20.2857	−0.707115	−0.353558	−	0.935413i	$-0.615028\pi$
$823$	−20.2857	−0.707115	−0.353558	+	0.935413i	$0.615028\pi$
$824$	−140.078	−4.87984
$825$	4.73219	0.164754
$826$	28.4719	0.990663
$827$	−32.5012	−1.13018	−0.565090	−	0.825030i	$-0.691158\pi$
$827$	−32.5012	−1.13018	−0.565090	+	0.825030i	$0.691158\pi$
$828$	−37.6619	−1.30884
$829$	22.4049	0.778156	0.389078	−	0.921205i	$-0.372794\pi$
$829$	22.4049	0.778156	0.389078	+	0.921205i	$0.372794\pi$
$830$	−0.310299	−0.0107706
$831$	−16.2853	−0.564931
$832$	13.7038	0.475093
$833$	−1.00000	−0.0346479
$834$	−56.4678	−1.95532
$835$	3.99756	0.138341
$836$	−16.1339	−0.558001
$837$	−5.77111	−0.199479
$838$	89.0066	3.07468
$839$	27.9560	0.965149	0.482575	−	0.875855i	$-0.339702\pi$
$839$	27.9560	0.965149	0.482575	+	0.875855i	$0.339702\pi$
$840$	9.27281	0.319942
$841$	−1.19748	−0.0412923
$842$	96.4265	3.32308
$843$	10.9297	0.376438
$844$	−22.8370	−0.786081
$845$	13.8025	0.474821
$846$	15.2150	0.523101
$847$	−9.25184	−0.317897
$848$	124.862	4.28777
$849$	−2.79951	−0.0960788
$850$	9.43591	0.323649
$851$	0.990165	0.0339424
$852$	71.5212	2.45028
$853$	0.467269	0.0159990	0.00799950	−	0.999968i	$-0.497454\pi$
$853$	0.467269	0.0159990	0.00799950	+	0.999968i	$0.497454\pi$
$854$	−6.26031	−0.214224
$855$	2.93813	0.100482
$856$	94.8659	3.24245
$857$	−37.7497	−1.28950	−0.644752	−	0.764392i	$-0.723039\pi$
$857$	−37.7497	−1.28950	−0.644752	+	0.764392i	$0.723039\pi$
$858$	−4.15515	−0.141854
$859$	−29.6440	−1.01144	−0.505720	−	0.862698i	$-0.668773\pi$
$859$	−29.6440	−1.01144	−0.505720	+	0.862698i	$0.668773\pi$
$860$	7.21985	0.246195
$861$	10.7092	0.364970
$862$	−48.8938	−1.66533
$863$	41.0580	1.39763	0.698815	−	0.715302i	$-0.253711\pi$
$863$	41.0580	1.39763	0.698815	+	0.715302i	$0.253711\pi$
$864$	−12.4075	−0.422112
$865$	2.91917	0.0992549
$866$	19.2873	0.655409
$867$	−1.00000	−0.0339618
$868$	28.5706	0.969750
$869$	19.7812	0.671032
$870$	−16.5706	−0.561796
$871$	0.117698	0.00398804
$872$	112.139	3.79750
$873$	10.7900	0.365187
$874$	49.4357	1.67219
$875$	10.2265	0.345717
$876$	−39.1950	−1.32428
$877$	−8.71771	−0.294376	−0.147188	−	0.989109i	$-0.547022\pi$
$877$	−8.71771	−0.294376	−0.147188	+	0.989109i	$0.547022\pi$
$878$	−75.0803	−2.53384
$879$	3.55312	0.119844
$880$	−16.7181	−0.563567
$881$	−35.1541	−1.18437	−0.592185	−	0.805802i	$-0.701735\pi$
$881$	−35.1541	−1.18437	−0.592185	+	0.805802i	$0.701735\pi$
$882$	2.63640	0.0887724
$883$	29.6764	0.998689	0.499344	−	0.866404i	$-0.333574\pi$
$883$	29.6764	0.998689	0.499344	+	0.866404i	$0.333574\pi$
$884$	−5.90126	−0.198481
$885$	12.8733	0.432730
$886$	−18.0341	−0.605867
$887$	1.67072	0.0560971	0.0280486	−	0.999607i	$-0.491071\pi$
$887$	1.67072	0.0560971	0.0280486	+	0.999607i	$0.491071\pi$
$888$	1.01250	0.0339772
$889$	11.6075	0.389303
$890$	26.3481	0.883192
$891$	1.32218	0.0442947
$892$	60.7013	2.03243
$893$	−14.2248	−0.476015
$894$	52.9597	1.77124
$895$	−21.2400	−0.709974
$896$	5.49375	0.183533
$897$	9.06829	0.302781
$898$	−15.1082	−0.504167
$899$	−30.4299	−1.01490
$900$	−17.7187	−0.590624
$901$	−11.7711	−0.392152
$902$	−37.3303	−1.24296
$903$	1.22344	0.0407136
$904$	40.5097	1.34733
$905$	−4.66935	−0.155214
$906$	−45.9222	−1.52566
$907$	−31.1960	−1.03585	−0.517923	−	0.855427i	$-0.673295\pi$
$907$	−31.1960	−1.03585	−0.517923	+	0.855427i	$0.673295\pi$
$908$	94.5238	3.13688
$909$	−17.0598	−0.565839
$910$	−3.74611	−0.124182
$911$	−37.5019	−1.24249	−0.621247	−	0.783615i	$-0.713374\pi$
$911$	−37.5019	−1.24249	−0.621247	+	0.783615i	$0.713374\pi$
$912$	26.1457	0.865770
$913$	0.130549	0.00432055
$914$	77.5367	2.56469
$915$	−2.83053	−0.0935746
$916$	43.9263	1.45136
$917$	10.5057	0.346930
$918$	2.63640	0.0870143
$919$	−57.1247	−1.88437	−0.942185	−	0.335093i	$-0.891232\pi$
$919$	−57.1247	−1.88437	−0.942185	+	0.335093i	$0.891232\pi$
$920$	70.5428	2.32573
$921$	−18.4249	−0.607123
$922$	−40.7163	−1.34092
$923$	−17.2210	−0.566836
$924$	−6.54562	−0.215335
$925$	0.465842	0.0153168
$926$	−53.8065	−1.76819
$927$	−18.0070	−0.591429
$928$	−65.4224	−2.14760
$929$	54.8349	1.79908	0.899538	−	0.436843i	$-0.143904\pi$
$929$	54.8349	1.79908	0.899538	+	0.436843i	$0.143904\pi$
$930$	18.1366	0.594722
$931$	−2.46483	−0.0807816
$932$	99.1732	3.24852
$933$	22.3481	0.731645
$934$	28.9648	0.947757
$935$	1.57607	0.0515429
$936$	9.27281	0.303091
$937$	4.41586	0.144260	0.0721299	−	0.997395i	$-0.477020\pi$
$937$	4.41586	0.144260	0.0721299	+	0.997395i	$0.477020\pi$
$938$	0.260313	0.00849953
$939$	−18.9770	−0.619291
$940$	−34.0568	−1.11081
$941$	−57.9782	−1.89003	−0.945017	−	0.327020i	$-0.893956\pi$
$941$	−57.9782	−1.89003	−0.945017	+	0.327020i	$0.893956\pi$
$942$	−10.7167	−0.349168
$943$	81.4704	2.65304
$944$	114.556	3.72847
$945$	1.19202	0.0387765
$946$	−4.26467	−0.138656
$947$	8.07978	0.262558	0.131279	−	0.991345i	$-0.458092\pi$
$947$	8.07978	0.262558	0.131279	+	0.991345i	$0.458092\pi$
$948$	−74.0668	−2.40558
$949$	9.43744	0.306352
$950$	23.2579	0.754587
$951$	−9.43439	−0.305931
$952$	−7.77906	−0.252121
$953$	38.2498	1.23903	0.619516	−	0.784984i	$-0.287329\pi$
$953$	38.2498	1.23903	0.619516	+	0.784984i	$0.287329\pi$
$954$	31.0334	1.00474
$955$	1.75980	0.0569460
$956$	−10.7481	−0.347618
$957$	6.97160	0.225360
$958$	50.0773	1.61792
$959$	−9.28832	−0.299936
$960$	13.7038	0.442288
$961$	2.30566	0.0743760
$962$	−0.409037	−0.0131879
$963$	12.1950	0.392980
$964$	−48.4201	−1.55951
$965$	9.57363	0.308186
$966$	20.0564	0.645304
$967$	44.3715	1.42689	0.713445	−	0.700711i	$-0.247134\pi$
$967$	44.3715	1.42689	0.713445	+	0.700711i	$0.247134\pi$
$968$	−71.9706	−2.31322
$969$	−2.46483	−0.0791818
$970$	−33.9093	−1.08876
$971$	−36.7511	−1.17940	−0.589700	−	0.807622i	$-0.700754\pi$
$971$	−36.7511	−1.17940	−0.589700	+	0.807622i	$0.700754\pi$
$972$	−4.95063	−0.158792
$973$	21.4185	0.686645
$974$	−57.3697	−1.83824
$975$	4.26635	0.136632
$976$	−25.1882	−0.806253
$977$	46.9248	1.50126	0.750629	−	0.660724i	$-0.229751\pi$
$977$	46.9248	1.50126	0.750629	+	0.660724i	$0.229751\pi$
$978$	20.8309	0.666100
$979$	−11.0852	−0.354285
$980$	−5.90126	−0.188509
$981$	14.4155	0.460250
$982$	18.2721	0.583087
$983$	−51.1074	−1.63007	−0.815036	−	0.579410i	$-0.803283\pi$
$983$	−51.1074	−1.63007	−0.815036	+	0.579410i	$0.803283\pi$
$984$	83.3078	2.65576
$985$	20.1217	0.641130
$986$	13.9013	0.442706
$987$	−5.77111	−0.183696
$988$	−14.5456	−0.462758
$989$	9.30731	0.295955
$990$	−4.15515	−0.132059
$991$	−19.5551	−0.621187	−0.310594	−	0.950543i	$-0.600528\pi$
$991$	−19.5551	−0.621187	−0.310594	+	0.950543i	$0.600528\pi$
$992$	71.6050	2.27346
$993$	4.53715	0.143982
$994$	−38.0878	−1.20807
$995$	2.22889	0.0706607
$996$	−0.488815	−0.0154887
$997$	−3.11805	−0.0987497	−0.0493749	−	0.998780i	$-0.515723\pi$
$997$	−3.11805	−0.0987497	−0.0493749	+	0.998780i	$0.515723\pi$
$998$	−48.6504	−1.54000
$999$	0.130157	0.00411798

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	357.2.a.h.1.4	✓	4
3.2	odd	2		1071.2.a.j.1.1		4
4.3	odd	2		5712.2.a.bx.1.2		4
5.4	even	2		8925.2.a.bs.1.1		4
7.6	odd	2		2499.2.a.z.1.4		4
17.16	even	2		6069.2.a.s.1.4		4
21.20	even	2		7497.2.a.be.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.a.h.1.4	✓	4	1.1	even	1	trivial
1071.2.a.j.1.1		4	3.2	odd	2
2499.2.a.z.1.4		4	7.6	odd	2
5712.2.a.bx.1.2		4	4.3	odd	2
6069.2.a.s.1.4		4	17.16	even	2
7497.2.a.be.1.1		4	21.20	even	2
8925.2.a.bs.1.1		4	5.4	even	2

Overview	Random
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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

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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	357.2.a.h.1.4

Level	$357$
Weight	$2$
Character	357.1
Self dual	yes
Analytic conductor	$2.851$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$1$