LMFDB - Embedded newform 357.2.a.h.1.1

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)

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")

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);

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			1.1
Root			$-0.652223$ 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)

$f(q)$	$=$	$q-2.06644 q^{2} -1.00000 q^{3} +2.27016 q^{4} -0.238009 q^{5} +2.06644 q^{6} +1.00000 q^{7} -0.558268 q^{8} +1.00000 q^{9} +0.491831 q^{10} -5.40303 q^{11} -2.27016 q^{12} +0.238009 q^{13} -2.06644 q^{14} +0.238009 q^{15} -3.38669 q^{16} -1.00000 q^{17} -2.06644 q^{18} +7.89486 q^{19} -0.540319 q^{20} -1.00000 q^{21} +11.1650 q^{22} +6.38669 q^{23} +0.558268 q^{24} -4.94335 q^{25} -0.491831 q^{26} -1.00000 q^{27} +2.27016 q^{28} +4.13287 q^{29} -0.491831 q^{30} +6.18136 q^{31} +8.11492 q^{32} +5.40303 q^{33} +2.06644 q^{34} -0.238009 q^{35} +2.27016 q^{36} +5.64104 q^{37} -16.3142 q^{38} -0.238009 q^{39} +0.132873 q^{40} -6.30231 q^{41} +2.06644 q^{42} +10.8627 q^{43} -12.2657 q^{44} -0.238009 q^{45} -13.1977 q^{46} +6.18136 q^{47} +3.38669 q^{48} +1.00000 q^{49} +10.2151 q^{50} +1.00000 q^{51} +0.540319 q^{52} +12.1814 q^{53} +2.06644 q^{54} +1.28597 q^{55} -0.558268 q^{56} -7.89486 q^{57} -8.54032 q^{58} -4.14869 q^{59} +0.540319 q^{60} -2.55613 q^{61} -12.7734 q^{62} +1.00000 q^{63} -9.99559 q^{64} -0.0566484 q^{65} -11.1650 q^{66} +5.45968 q^{67} -2.27016 q^{68} -6.38669 q^{69} +0.491831 q^{70} +9.72543 q^{71} -0.558268 q^{72} -14.9389 q^{73} -11.6569 q^{74} +4.94335 q^{75} +17.9226 q^{76} -5.40303 q^{77} +0.491831 q^{78} -16.8904 q^{79} +0.806065 q^{80} +1.00000 q^{81} +13.0233 q^{82} +5.45968 q^{83} -2.27016 q^{84} +0.238009 q^{85} -22.4471 q^{86} -4.13287 q^{87} +3.01634 q^{88} -6.47602 q^{89} +0.491831 q^{90} +0.238009 q^{91} +14.4988 q^{92} -6.18136 q^{93} -12.7734 q^{94} -1.87905 q^{95} -8.11492 q^{96} -2.06857 q^{97} -2.06644 q^{98} -5.40303 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9} + 4 q^{10} + 2 q^{11} - 6 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{15} + 6 q^{16} - 4 q^{17} + 2 q^{18} + 10 q^{19} + 4 q^{20}+ \cdots + 2 q^{99}+O(q^{100})$ 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

)

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

$2$	−2.06644	−1.46119	−0.730596	−	0.682810i	$-0.760757\pi$
$2$	−2.06644	−1.46119	−0.730596	+	0.682810i	$0.760757\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.27016	1.13508
$5$	−0.238009	−0.106441	−0.0532205	−	0.998583i	$-0.516949\pi$
$5$	−0.238009	−0.106441	−0.0532205	+	0.998583i	$0.516949\pi$
$6$	2.06644	0.843619
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.558268	−0.197377
$9$	1.00000	0.333333
$10$	0.491831	0.155531
$11$	−5.40303	−1.62908	−0.814538	−	0.580110i	$-0.803009\pi$
$11$	−5.40303	−1.62908	−0.814538	+	0.580110i	$0.803009\pi$
$12$	−2.27016	−0.655339
$13$	0.238009	0.0660119	0.0330060	−	0.999455i	$-0.489492\pi$
$13$	0.238009	0.0660119	0.0330060	+	0.999455i	$0.489492\pi$
$14$	−2.06644	−0.552278
$15$	0.238009	0.0614537
$16$	−3.38669	−0.846674
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.06644	−0.487064
$19$	7.89486	1.81121	0.905603	−	0.424126i	$-0.139419\pi$
$19$	7.89486	1.81121	0.905603	+	0.424126i	$0.139419\pi$
$20$	−0.540319	−0.120819
$21$	−1.00000	−0.218218
$22$	11.1650	2.38039
$23$	6.38669	1.33172	0.665859	−	0.746078i	$-0.268065\pi$
$23$	6.38669	1.33172	0.665859	+	0.746078i	$0.268065\pi$
$24$	0.558268	0.113956
$25$	−4.94335	−0.988670
$26$	−0.491831	−0.0964560
$27$	−1.00000	−0.192450
$28$	2.27016	0.429020
$29$	4.13287	0.767455	0.383728	−	0.923446i	$-0.374640\pi$
$29$	4.13287	0.767455	0.383728	+	0.923446i	$0.374640\pi$
$30$	−0.491831	−0.0897957
$31$	6.18136	1.11021	0.555103	−	0.831782i	$-0.312679\pi$
$31$	6.18136	1.11021	0.555103	+	0.831782i	$0.312679\pi$
$32$	8.11492	1.43453
$33$	5.40303	0.940547
$34$	2.06644	0.354391
$35$	−0.238009	−0.0402309
$36$	2.27016	0.378360
$37$	5.64104	0.927382	0.463691	−	0.885997i	$-0.346525\pi$
$37$	5.64104	0.927382	0.463691	+	0.885997i	$0.346525\pi$
$38$	−16.3142	−2.64652
$39$	−0.238009	−0.0381120
$40$	0.132873	0.0210090
$41$	−6.30231	−0.984255	−0.492128	−	0.870523i	$-0.663781\pi$
$41$	−6.30231	−0.984255	−0.492128	+	0.870523i	$0.663781\pi$
$42$	2.06644	0.318858
$43$	10.8627	1.65655	0.828274	−	0.560323i	$-0.189323\pi$
$43$	10.8627	1.65655	0.828274	+	0.560323i	$0.189323\pi$
$44$	−12.2657	−1.84913
$45$	−0.238009	−0.0354803
$46$	−13.1977	−1.94589
$47$	6.18136	0.901644	0.450822	−	0.892614i	$-0.351131\pi$
$47$	6.18136	0.901644	0.450822	+	0.892614i	$0.351131\pi$
$48$	3.38669	0.488827
$49$	1.00000	0.142857
$50$	10.2151	1.44464
$51$	1.00000	0.140028
$52$	0.540319	0.0749288
$53$	12.1814	1.67324	0.836619	−	0.547785i	$-0.184529\pi$
$53$	12.1814	1.67324	0.836619	+	0.547785i	$0.184529\pi$
$54$	2.06644	0.281206
$55$	1.28597	0.173400
$56$	−0.558268	−0.0746016
$57$	−7.89486	−1.04570
$58$	−8.54032	−1.12140
$59$	−4.14869	−0.540113	−0.270056	−	0.962845i	$-0.587042\pi$
$59$	−4.14869	−0.540113	−0.270056	+	0.962845i	$0.587042\pi$
$60$	0.540319	0.0697549
$61$	−2.55613	−0.327279	−0.163640	−	0.986520i	$-0.552323\pi$
$61$	−2.55613	−0.327279	−0.163640	+	0.986520i	$0.552323\pi$
$62$	−12.7734	−1.62222
$63$	1.00000	0.125988
$64$	−9.99559	−1.24945
$65$	−0.0566484	−0.00702637
$66$	−11.1650	−1.37432
$67$	5.45968	0.667006	0.333503	−	0.942749i	$-0.391769\pi$
$67$	5.45968	0.667006	0.333503	+	0.942749i	$0.391769\pi$
$68$	−2.27016	−0.275297
$69$	−6.38669	−0.768868
$70$	0.491831	0.0587851
$71$	9.72543	1.15420	0.577098	−	0.816675i	$-0.304185\pi$
$71$	9.72543	1.15420	0.577098	+	0.816675i	$0.304185\pi$
$72$	−0.558268	−0.0657925
$73$	−14.9389	−1.74847	−0.874235	−	0.485503i	$-0.838637\pi$
$73$	−14.9389	−1.74847	−0.874235	+	0.485503i	$0.838637\pi$
$74$	−11.6569	−1.35508
$75$	4.94335	0.570809
$76$	17.9226	2.05586
$77$	−5.40303	−0.615733
$78$	0.491831	0.0556889
$79$	−16.8904	−1.90032	−0.950162	−	0.311756i	$-0.899083\pi$
$79$	−16.8904	−1.90032	−0.950162	+	0.311756i	$0.899083\pi$
$80$	0.806065	0.0901208
$81$	1.00000	0.111111
$82$	13.0233	1.43819
$83$	5.45968	0.599278	0.299639	−	0.954053i	$-0.403134\pi$
$83$	5.45968	0.599278	0.299639	+	0.954053i	$0.403134\pi$
$84$	−2.27016	−0.247695
$85$	0.238009	0.0258157
$86$	−22.4471	−2.42053
$87$	−4.13287	−0.443090
$88$	3.01634	0.321543
$89$	−6.47602	−0.686457	−0.343228	−	0.939252i	$-0.611520\pi$
$89$	−6.47602	−0.686457	−0.343228	+	0.939252i	$0.611520\pi$
$90$	0.491831	0.0518436
$91$	0.238009	0.0249502
$92$	14.4988	1.51161
$93$	−6.18136	−0.640977
$94$	−12.7734	−1.31747
$95$	−1.87905	−0.192787
$96$	−8.11492	−0.828226
$97$	−2.06857	−0.210032	−0.105016	−	0.994471i	$-0.533489\pi$
$97$	−2.06857	−0.210032	−0.105016	+	0.994471i	$0.533489\pi$
$98$	−2.06644	−0.208742
$99$	−5.40303	−0.543025
$100$	−11.2222	−1.12222
$101$	9.43077	0.938397	0.469198	−	0.883093i	$-0.344543\pi$
$101$	9.43077	0.938397	0.469198	+	0.883093i	$0.344543\pi$
$102$	−2.06644	−0.204608
$103$	−8.46786	−0.834363	−0.417181	−	0.908823i	$-0.636982\pi$
$103$	−8.46786	−0.834363	−0.417181	+	0.908823i	$0.636982\pi$
$104$	−0.132873	−0.0130293
$105$	0.238009	0.0232273
$106$	−25.1720	−2.44492
$107$	15.4673	1.49528	0.747642	−	0.664102i	$-0.231186\pi$
$107$	15.4673	1.49528	0.747642	+	0.664102i	$0.231186\pi$
$108$	−2.27016	−0.218446
$109$	1.37530	0.131729	0.0658647	−	0.997829i	$-0.479019\pi$
$109$	1.37530	0.131729	0.0658647	+	0.997829i	$0.479019\pi$
$110$	−2.65738	−0.253371
$111$	−5.64104	−0.535424
$112$	−3.38669	−0.320013
$113$	10.6165	0.998720	0.499360	−	0.866394i	$-0.333568\pi$
$113$	10.6165	0.998720	0.499360	+	0.866394i	$0.333568\pi$
$114$	16.3142	1.52797
$115$	−1.52009	−0.141749
$116$	9.38228	0.871123
$117$	0.238009	0.0220040
$118$	8.57299	0.789208
$119$	−1.00000	−0.0916698
$120$	−0.132873	−0.0121296
$121$	18.1928	1.65389
$122$	5.28208	0.478217
$123$	6.30231	0.568260
$124$	14.0327	1.26017
$125$	2.36661	0.211676
$126$	−2.06644	−0.184093
$127$	−2.38669	−0.211785	−0.105892	−	0.994378i	$-0.533770\pi$
$127$	−2.38669	−0.211785	−0.105892	+	0.994378i	$0.533770\pi$
$128$	4.42539	0.391153
$129$	−10.8627	−0.956409
$130$	0.117060	0.0102669
$131$	−13.0757	−1.14243	−0.571215	−	0.820801i	$-0.693528\pi$
$131$	−13.0757	−1.14243	−0.571215	+	0.820801i	$0.693528\pi$
$132$	12.2657	1.06760
$133$	7.89486	0.684571
$134$	−11.2821	−0.974624
$135$	0.238009	0.0204846
$136$	0.558268	0.0478710
$137$	−6.24566	−0.533603	−0.266801	−	0.963752i	$-0.585967\pi$
$137$	−6.24566	−0.533603	−0.266801	+	0.963752i	$0.585967\pi$
$138$	13.1977	1.12346
$139$	12.6046	1.06911	0.534555	−	0.845134i	$-0.320479\pi$
$139$	12.6046	1.06911	0.534555	+	0.845134i	$0.320479\pi$
$140$	−0.540319	−0.0456653
$141$	−6.18136	−0.520564
$142$	−20.0970	−1.68650
$143$	−1.28597	−0.107538
$144$	−3.38669	−0.282225
$145$	−0.983662	−0.0816887
$146$	30.8704	2.55485
$147$	−1.00000	−0.0824786
$148$	12.8061	1.05265
$149$	−2.09698	−0.171791	−0.0858955	−	0.996304i	$-0.527375\pi$
$149$	−2.09698	−0.171791	−0.0858955	+	0.996304i	$0.527375\pi$
$150$	−10.2151	−0.834061
$151$	8.60462	0.700234	0.350117	−	0.936706i	$-0.386142\pi$
$151$	8.60462	0.700234	0.350117	+	0.936706i	$0.386142\pi$
$152$	−4.40745	−0.357491
$153$	−1.00000	−0.0808452
$154$	11.1650	0.899703
$155$	−1.47122	−0.118171
$156$	−0.540319	−0.0432602
$157$	13.1084	1.04616	0.523081	−	0.852283i	$-0.324782\pi$
$157$	13.1084	1.04616	0.523081	+	0.852283i	$0.324782\pi$
$158$	34.9030	2.77674
$159$	−12.1814	−0.966045
$160$	−1.93143	−0.152693
$161$	6.38669	0.503342
$162$	−2.06644	−0.162355
$163$	−2.54032	−0.198973	−0.0994866	−	0.995039i	$-0.531720\pi$
$163$	−2.54032	−0.198973	−0.0994866	+	0.995039i	$0.531720\pi$
$164$	−14.3072	−1.11721
$165$	−1.28597	−0.100113
$166$	−11.2821	−0.875660
$167$	14.5038	1.12233	0.561167	−	0.827703i	$-0.310352\pi$
$167$	14.5038	1.12233	0.561167	+	0.827703i	$0.310352\pi$
$168$	0.558268	0.0430713
$169$	−12.9434	−0.995642
$170$	−0.491831	−0.0377217
$171$	7.89486	0.603735
$172$	24.6601	1.88031
$173$	−9.58439	−0.728688	−0.364344	−	0.931264i	$-0.618707\pi$
$173$	−9.58439	−0.728688	−0.364344	+	0.931264i	$0.618707\pi$
$174$	8.54032	0.647440
$175$	−4.94335	−0.373682
$176$	18.2984	1.37930
$177$	4.14869	0.311834
$178$	13.3823	1.00304
$179$	−10.3986	−0.777229	−0.388615	−	0.921400i	$-0.627046\pi$
$179$	−10.3986	−0.777229	−0.388615	+	0.921400i	$0.627046\pi$
$180$	−0.540319	−0.0402730
$181$	−18.9389	−1.40772	−0.703860	−	0.710339i	$-0.748542\pi$
$181$	−18.9389	−1.40772	−0.703860	+	0.710339i	$0.748542\pi$
$182$	−0.491831	−0.0364569
$183$	2.55613	0.188955
$184$	−3.56548	−0.262851
$185$	−1.34262	−0.0987114
$186$	12.7734	0.936590
$187$	5.40303	0.395109
$188$	14.0327	1.02344
$189$	−1.00000	−0.0727393
$190$	3.88294	0.281698
$191$	−11.2451	−0.813669	−0.406835	−	0.913502i	$-0.633368\pi$
$191$	−11.2451	−0.813669	−0.406835	+	0.913502i	$0.633368\pi$
$192$	9.99559	0.721369
$193$	3.10072	0.223195	0.111597	−	0.993753i	$-0.464403\pi$
$193$	3.10072	0.223195	0.111597	+	0.993753i	$0.464403\pi$
$194$	4.27457	0.306896
$195$	0.0566484	0.00405668
$196$	2.27016	0.162154
$197$	6.51957	0.464500	0.232250	−	0.972656i	$-0.425391\pi$
$197$	6.51957	0.464500	0.232250	+	0.972656i	$0.425391\pi$
$198$	11.1650	0.793464
$199$	−7.64104	−0.541659	−0.270830	−	0.962627i	$-0.587298\pi$
$199$	−7.64104	−0.541659	−0.270830	+	0.962627i	$0.587298\pi$
$200$	2.75971	0.195141
$201$	−5.45968	−0.385096
$202$	−19.4881	−1.37118
$203$	4.13287	0.290071
$204$	2.27016	0.158943
$205$	1.50001	0.104765
$206$	17.4983	1.21916
$207$	6.38669	0.443906
$208$	−0.806065	−0.0558905
$209$	−42.6562	−2.95059
$210$	−0.491831	−0.0339396
$211$	−2.29466	−0.157971	−0.0789854	−	0.996876i	$-0.525168\pi$
$211$	−2.29466	−0.157971	−0.0789854	+	0.996876i	$0.525168\pi$
$212$	27.6536	1.89926
$213$	−9.72543	−0.666375
$214$	−31.9623	−2.18490
$215$	−2.58543	−0.176325
$216$	0.558268	0.0379853
$217$	6.18136	0.419618
$218$	−2.84196	−0.192482
$219$	14.9389	1.00948
$220$	2.91936	0.196823
$221$	−0.238009	−0.0160102
$222$	11.6569	0.782357
$223$	−17.2729	−1.15668	−0.578339	−	0.815797i	$-0.696299\pi$
$223$	−17.2729	−1.15668	−0.578339	+	0.815797i	$0.696299\pi$
$224$	8.11492	0.542201
$225$	−4.94335	−0.329557
$226$	−21.9384	−1.45932
$227$	12.7783	0.848127	0.424064	−	0.905632i	$-0.360603\pi$
$227$	12.7783	0.848127	0.424064	+	0.905632i	$0.360603\pi$
$228$	−17.9226	−1.18695
$229$	18.8704	1.24699	0.623494	−	0.781828i	$-0.285712\pi$
$229$	18.8704	1.24699	0.623494	+	0.781828i	$0.285712\pi$
$230$	3.14118	0.207123
$231$	5.40303	0.355493
$232$	−2.30725	−0.151478
$233$	−9.09151	−0.595605	−0.297802	−	0.954628i	$-0.596254\pi$
$233$	−9.09151	−0.595605	−0.297802	+	0.954628i	$0.596254\pi$
$234$	−0.491831	−0.0321520
$235$	−1.47122	−0.0959719
$236$	−9.41818	−0.613071
$237$	16.8904	1.09715
$238$	2.06644	0.133947
$239$	16.8219	1.08812	0.544058	−	0.839047i	$-0.316887\pi$
$239$	16.8219	1.08812	0.544058	+	0.839047i	$0.316887\pi$
$240$	−0.806065	−0.0520313
$241$	−8.10125	−0.521847	−0.260924	−	0.965359i	$-0.584027\pi$
$241$	−8.10125	−0.521847	−0.260924	+	0.965359i	$0.584027\pi$
$242$	−37.5942	−2.41665
$243$	−1.00000	−0.0641500
$244$	−5.80283	−0.371488
$245$	−0.238009	−0.0152059
$246$	−13.0233	−0.830337
$247$	1.87905	0.119561
$248$	−3.45085	−0.219129
$249$	−5.45968	−0.345993
$250$	−4.89045	−0.309299
$251$	−2.68900	−0.169728	−0.0848642	−	0.996393i	$-0.527046\pi$
$251$	−2.68900	−0.169728	−0.0848642	+	0.996393i	$0.527046\pi$
$252$	2.27016	0.143007
$253$	−34.5075	−2.16947
$254$	4.93195	0.309458
$255$	−0.238009	−0.0149047
$256$	10.8464	0.677898
$257$	−20.5441	−1.28150	−0.640752	−	0.767748i	$-0.721377\pi$
$257$	−20.5441	−1.28150	−0.640752	+	0.767748i	$0.721377\pi$
$258$	22.4471	1.39750
$259$	5.64104	0.350517
$260$	−0.128601	−0.00797549
$261$	4.13287	0.255818
$262$	27.0201	1.66931
$263$	11.4554	0.706371	0.353185	−	0.935553i	$-0.385098\pi$
$263$	11.4554	0.706371	0.353185	+	0.935553i	$0.385098\pi$
$264$	−3.01634	−0.185643
$265$	−2.89928	−0.178101
$266$	−16.3142	−1.00029
$267$	6.47602	0.396326
$268$	12.3943	0.757105
$269$	−23.5201	−1.43405	−0.717023	−	0.697050i	$-0.754496\pi$
$269$	−23.5201	−1.43405	−0.717023	+	0.697050i	$0.754496\pi$
$270$	−0.491831	−0.0299319
$271$	27.4743	1.66895	0.834473	−	0.551049i	$-0.185772\pi$
$271$	27.4743	1.66895	0.834473	+	0.551049i	$0.185772\pi$
$272$	3.38669	0.205349
$273$	−0.238009	−0.0144050
$274$	12.9063	0.779696
$275$	26.7091	1.61062
$276$	−14.4988	−0.872726
$277$	9.01634	0.541739	0.270870	−	0.962616i	$-0.412689\pi$
$277$	9.01634	0.541739	0.270870	+	0.962616i	$0.412689\pi$
$278$	−26.0466	−1.56217
$279$	6.18136	0.370068
$280$	0.132873	0.00794067
$281$	9.78973	0.584006	0.292003	−	0.956417i	$-0.405678\pi$
$281$	9.78973	0.584006	0.292003	+	0.956417i	$0.405678\pi$
$282$	12.7734	0.760644
$283$	−12.1487	−0.722164	−0.361082	−	0.932534i	$-0.617593\pi$
$283$	−12.1487	−0.722164	−0.361082	+	0.932534i	$0.617593\pi$
$284$	22.0783	1.31010
$285$	1.87905	0.111305
$286$	2.65738	0.157134
$287$	−6.30231	−0.372014
$288$	8.11492	0.478177
$289$	1.00000	0.0588235
$290$	2.03268	0.119363
$291$	2.06857	0.121262
$292$	−33.9138	−1.98465
$293$	−27.7254	−1.61974	−0.809868	−	0.586612i	$-0.800462\pi$
$293$	−27.7254	−1.61974	−0.809868	+	0.586612i	$0.800462\pi$
$294$	2.06644	0.120517
$295$	0.987426	0.0574902
$296$	−3.14921	−0.183044
$297$	5.40303	0.313516
$298$	4.33327	0.251019
$299$	1.52009	0.0879092
$300$	11.2222	0.647914
$301$	10.8627	0.626116
$302$	−17.7809	−1.02318
$303$	−9.43077	−0.541784
$304$	−26.7375	−1.53350
$305$	0.608383	0.0348359
$306$	2.06644	0.118130
$307$	3.29518	0.188066	0.0940330	−	0.995569i	$-0.470024\pi$
$307$	3.29518	0.188066	0.0940330	+	0.995569i	$0.470024\pi$
$308$	−12.2657	−0.698906
$309$	8.46786	0.481720
$310$	3.04019	0.172671
$311$	7.18511	0.407430	0.203715	−	0.979030i	$-0.434698\pi$
$311$	7.18511	0.407430	0.203715	+	0.979030i	$0.434698\pi$
$312$	0.132873	0.00752245
$313$	−30.3697	−1.71660	−0.858299	−	0.513150i	$-0.828478\pi$
$313$	−30.3697	−1.71660	−0.858299	+	0.513150i	$0.828478\pi$
$314$	−27.0876	−1.52864
$315$	−0.238009	−0.0134103
$316$	−38.3440	−2.15702
$317$	−16.8746	−0.947774	−0.473887	−	0.880586i	$-0.657149\pi$
$317$	−16.8746	−0.947774	−0.473887	+	0.880586i	$0.657149\pi$
$318$	25.1720	1.41158
$319$	−22.3300	−1.25024
$320$	2.37904	0.132993
$321$	−15.4673	−0.863302
$322$	−13.1977	−0.735479
$323$	−7.89486	−0.439282
$324$	2.27016	0.126120
$325$	−1.17656	−0.0652640
$326$	5.24941	0.290738
$327$	−1.37530	−0.0760540
$328$	3.51838	0.194270
$329$	6.18136	0.340789
$330$	2.65738	0.146284
$331$	30.1764	1.65865	0.829323	−	0.558769i	$-0.188726\pi$
$331$	30.1764	1.65865	0.829323	+	0.558769i	$0.188726\pi$
$332$	12.3943	0.680228
$333$	5.64104	0.309127
$334$	−29.9711	−1.63994
$335$	−1.29945	−0.0709968
$336$	3.38669	0.184759
$337$	0.478732	0.0260782	0.0130391	−	0.999915i	$-0.495849\pi$
$337$	0.478732	0.0260782	0.0130391	+	0.999915i	$0.495849\pi$
$338$	26.7466	1.45482
$339$	−10.6165	−0.576611
$340$	0.540319	0.0293029
$341$	−33.3981	−1.80861
$342$	−16.3142	−0.882173
$343$	1.00000	0.0539949
$344$	−6.06430	−0.326965
$345$	1.52009	0.0818390
$346$	19.8055	1.06475
$347$	−5.82240	−0.312563	−0.156281	−	0.987713i	$-0.549951\pi$
$347$	−5.82240	−0.312563	−0.156281	+	0.987713i	$0.549951\pi$
$348$	−9.38228	−0.502943
$349$	−0.416657	−0.0223031	−0.0111516	−	0.999938i	$-0.503550\pi$
$349$	−0.416657	−0.0223031	−0.0111516	+	0.999938i	$0.503550\pi$
$350$	10.2151	0.546021
$351$	−0.238009	−0.0127040
$352$	−43.8452	−2.33696
$353$	1.37530	0.0731996	0.0365998	−	0.999330i	$-0.488347\pi$
$353$	1.37530	0.0731996	0.0365998	+	0.999330i	$0.488347\pi$
$354$	−8.57299	−0.455650
$355$	−2.31474	−0.122854
$356$	−14.7016	−0.779183
$357$	1.00000	0.0529256
$358$	21.4881	1.13568
$359$	−5.41119	−0.285592	−0.142796	−	0.989752i	$-0.545609\pi$
$359$	−5.41119	−0.285592	−0.142796	+	0.989752i	$0.545609\pi$
$360$	0.132873	0.00700302
$361$	43.3289	2.28047
$362$	39.1361	2.05695
$363$	−18.1928	−0.954872
$364$	0.540319	0.0283204
$365$	3.55561	0.186109
$366$	−5.28208	−0.276099
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−21.6298	−1.12753
$369$	−6.30231	−0.328085
$370$	2.77444	0.144236
$371$	12.1814	0.632425
$372$	−14.0327	−0.727560
$373$	16.8061	0.870185	0.435093	−	0.900386i	$-0.356716\pi$
$373$	16.8061	0.870185	0.435093	+	0.900386i	$0.356716\pi$
$374$	−11.1650	−0.577330
$375$	−2.36661	−0.122211
$376$	−3.45085	−0.177964
$377$	0.983662	0.0506612
$378$	2.06644	0.106286
$379$	−13.9673	−0.717453	−0.358727	−	0.933443i	$-0.616789\pi$
$379$	−13.9673	−0.717453	−0.358727	+	0.933443i	$0.616789\pi$
$380$	−4.26575	−0.218828
$381$	2.38669	0.122274
$382$	23.2374	1.18893
$383$	−19.7696	−1.01018	−0.505091	−	0.863066i	$-0.668541\pi$
$383$	−19.7696	−1.01018	−0.505091	+	0.863066i	$0.668541\pi$
$384$	−4.42539	−0.225832
$385$	1.28597	0.0655392
$386$	−6.40745	−0.326130
$387$	10.8627	0.552183
$388$	−4.69599	−0.238403
$389$	19.8224	1.00504	0.502518	−	0.864567i	$-0.332407\pi$
$389$	19.8224	1.00504	0.502518	+	0.864567i	$0.332407\pi$
$390$	−0.117060	−0.00592758
$391$	−6.38669	−0.322989
$392$	−0.558268	−0.0281968
$393$	13.0757	0.659582
$394$	−13.4723	−0.678723
$395$	4.02008	0.202272
$396$	−12.2657	−0.616377
$397$	34.2411	1.71851	0.859256	−	0.511546i	$-0.170927\pi$
$397$	34.2411	1.71851	0.859256	+	0.511546i	$0.170927\pi$
$398$	15.7897	0.791468
$399$	−7.89486	−0.395238
$400$	16.7416	0.837081
$401$	2.96396	0.148013	0.0740066	−	0.997258i	$-0.476421\pi$
$401$	2.96396	0.148013	0.0740066	+	0.997258i	$0.476421\pi$
$402$	11.2821	0.562699
$403$	1.47122	0.0732868
$404$	21.4094	1.06515
$405$	−0.238009	−0.0118268
$406$	−8.54032	−0.423849
$407$	−30.4787	−1.51077
$408$	−0.558268	−0.0276384
$409$	35.2128	1.74116	0.870582	−	0.492024i	$-0.163743\pi$
$409$	35.2128	1.74116	0.870582	+	0.492024i	$0.163743\pi$
$410$	−3.09967	−0.153082
$411$	6.24566	0.308076
$412$	−19.2234	−0.947068
$413$	−4.14869	−0.204143
$414$	−13.1977	−0.648632
$415$	−1.29945	−0.0637877
$416$	1.93143	0.0946960
$417$	−12.6046	−0.617251
$418$	88.1463	4.31138
$419$	−13.0391	−0.637003	−0.318502	−	0.947922i	$-0.603180\pi$
$419$	−13.0391	−0.637003	−0.318502	+	0.947922i	$0.603180\pi$
$420$	0.540319	0.0263649
$421$	−24.6763	−1.20265	−0.601324	−	0.799005i	$-0.705360\pi$
$421$	−24.6763	−1.20265	−0.601324	+	0.799005i	$0.705360\pi$
$422$	4.74176	0.230825
$423$	6.18136	0.300548
$424$	−6.80046	−0.330259
$425$	4.94335	0.239788
$426$	20.0970	0.973702
$427$	−2.55613	−0.123700
$428$	35.1133	1.69727
$429$	1.28597	0.0620873
$430$	5.34262	0.257644
$431$	0.265746	0.0128005	0.00640026	−	0.999980i	$-0.497963\pi$
$431$	0.265746	0.0128005	0.00640026	+	0.999980i	$0.497963\pi$
$432$	3.38669	0.162942
$433$	15.9961	0.768724	0.384362	−	0.923182i	$-0.374421\pi$
$433$	15.9961	0.768724	0.384362	+	0.923182i	$0.374421\pi$
$434$	−12.7734	−0.613142
$435$	0.983662	0.0471630
$436$	3.12214	0.149523
$437$	50.4221	2.41202
$438$	−30.8704	−1.47504
$439$	6.82615	0.325794	0.162897	−	0.986643i	$-0.447916\pi$
$439$	6.82615	0.325794	0.162897	+	0.986643i	$0.447916\pi$
$440$	−0.717916	−0.0342253
$441$	1.00000	0.0476190
$442$	0.491831	0.0233940
$443$	21.3295	1.01340	0.506698	−	0.862124i	$-0.330866\pi$
$443$	21.3295	1.01340	0.506698	+	0.862124i	$0.330866\pi$
$444$	−12.8061	−0.607749
$445$	1.54135	0.0730671
$446$	35.6933	1.69013
$447$	2.09698	0.0991836
$448$	−9.99559	−0.472247
$449$	4.49559	0.212160	0.106080	−	0.994358i	$-0.466170\pi$
$449$	4.49559	0.212160	0.106080	+	0.994358i	$0.466170\pi$
$450$	10.2151	0.481545
$451$	34.0516	1.60343
$452$	24.1012	1.13363
$453$	−8.60462	−0.404280
$454$	−26.4056	−1.23928
$455$	−0.0566484	−0.00265572
$456$	4.40745	0.206398
$457$	4.69394	0.219573	0.109787	−	0.993955i	$-0.464983\pi$
$457$	4.69394	0.219573	0.109787	+	0.993955i	$0.464983\pi$
$458$	−38.9944	−1.82209
$459$	1.00000	0.0466760
$460$	−3.45085	−0.160897
$461$	12.9548	0.603363	0.301681	−	0.953409i	$-0.402452\pi$
$461$	12.9548	0.603363	0.301681	+	0.953409i	$0.402452\pi$
$462$	−11.1650	−0.519444
$463$	−22.7744	−1.05842	−0.529209	−	0.848492i	$-0.677511\pi$
$463$	−22.7744	−1.05842	−0.529209	+	0.848492i	$0.677511\pi$
$464$	−13.9968	−0.649784
$465$	1.47122	0.0682263
$466$	18.7870	0.870292
$467$	−40.4498	−1.87179	−0.935897	−	0.352273i	$-0.885409\pi$
$467$	−40.4498	−1.87179	−0.935897	+	0.352273i	$0.885409\pi$
$468$	0.540319	0.0249763
$469$	5.45968	0.252105
$470$	3.04019	0.140233
$471$	−13.1084	−0.604002
$472$	2.31608	0.106606
$473$	−58.6916	−2.69864
$474$	−34.9030	−1.60315
$475$	−39.0271	−1.79069
$476$	−2.27016	−0.104053
$477$	12.1814	0.557746
$478$	−34.7613	−1.58995
$479$	7.31865	0.334398	0.167199	−	0.985923i	$-0.446528\pi$
$479$	7.31865	0.334398	0.167199	+	0.985923i	$0.446528\pi$
$480$	1.93143	0.0881572
$481$	1.34262	0.0612182
$482$	16.7407	0.762519
$483$	−6.38669	−0.290605
$484$	41.3005	1.87729
$485$	0.492339	0.0223560
$486$	2.06644	0.0937355
$487$	25.0391	1.13463	0.567316	−	0.823500i	$-0.307982\pi$
$487$	25.0391	1.13463	0.567316	+	0.823500i	$0.307982\pi$
$488$	1.42701	0.0645975
$489$	2.54032	0.114877
$490$	0.491831	0.0222187
$491$	35.5109	1.60258	0.801292	−	0.598274i	$-0.204146\pi$
$491$	35.5109	1.60258	0.801292	+	0.598274i	$0.204146\pi$
$492$	14.3072	0.645021
$493$	−4.13287	−0.186135
$494$	−3.88294	−0.174702
$495$	1.28597	0.0578001
$496$	−20.9344	−0.939982
$497$	9.72543	0.436245
$498$	11.2821	0.505562
$499$	21.1246	0.945666	0.472833	−	0.881152i	$-0.343231\pi$
$499$	21.1246	0.945666	0.472833	+	0.881152i	$0.343231\pi$
$500$	5.37258	0.240269
$501$	−14.5038	−0.647980
$502$	5.55666	0.248006
$503$	31.4296	1.40138	0.700688	−	0.713468i	$-0.252876\pi$
$503$	31.4296	1.40138	0.700688	+	0.713468i	$0.252876\pi$
$504$	−0.558268	−0.0248672
$505$	−2.24461	−0.0998839
$506$	71.3076	3.17001
$507$	12.9434	0.574834
$508$	−5.41818	−0.240393
$509$	−3.19394	−0.141569	−0.0707843	−	0.997492i	$-0.522550\pi$
$509$	−3.19394	−0.141569	−0.0707843	+	0.997492i	$0.522550\pi$
$510$	0.491831	0.0217786
$511$	−14.9389	−0.660860
$512$	−31.2641	−1.38169
$513$	−7.89486	−0.348567
$514$	42.4530	1.87252
$515$	2.01543	0.0888104
$516$	−24.6601	−1.08560
$517$	−33.3981	−1.46885
$518$	−11.6569	−0.512173
$519$	9.58439	0.420708
$520$	0.0316250	0.00138685
$521$	19.2782	0.844593	0.422297	−	0.906458i	$-0.361224\pi$
$521$	19.2782	0.844593	0.422297	+	0.906458i	$0.361224\pi$
$522$	−8.54032	−0.373800
$523$	6.51192	0.284746	0.142373	−	0.989813i	$-0.454527\pi$
$523$	6.51192	0.284746	0.142373	+	0.989813i	$0.454527\pi$
$524$	−29.6839	−1.29675
$525$	4.94335	0.215746
$526$	−23.6719	−1.03214
$527$	−6.18136	−0.269264
$528$	−18.2984	−0.796337
$529$	17.7899	0.773473
$530$	5.99117	0.260240
$531$	−4.14869	−0.180038
$532$	17.9226	0.777043
$533$	−1.50001	−0.0649726
$534$	−13.3823	−0.579108
$535$	−3.68137	−0.159159
$536$	−3.04796	−0.131652
$537$	10.3986	0.448734
$538$	48.6028	2.09541
$539$	−5.40303	−0.232725
$540$	0.540319	0.0232516
$541$	−17.7265	−0.762121	−0.381060	−	0.924550i	$-0.624441\pi$
$541$	−17.7265	−0.762121	−0.381060	+	0.924550i	$0.624441\pi$
$542$	−56.7739	−2.43865
$543$	18.9389	0.812748
$544$	−8.11492	−0.347925
$545$	−0.327333	−0.0140214
$546$	0.491831	0.0210484
$547$	−15.2620	−0.652556	−0.326278	−	0.945274i	$-0.605795\pi$
$547$	−15.2620	−0.652556	−0.326278	+	0.945274i	$0.605795\pi$
$548$	−14.1786	−0.605682
$549$	−2.55613	−0.109093
$550$	−55.1926	−2.35342
$551$	32.6285	1.39002
$552$	3.56548	0.151757
$553$	−16.8904	−0.718255
$554$	−18.6317	−0.791585
$555$	1.34262	0.0569911
$556$	28.6145	1.21353
$557$	−33.0102	−1.39869	−0.699344	−	0.714785i	$-0.746524\pi$
$557$	−33.0102	−1.39869	−0.699344	+	0.714785i	$0.746524\pi$
$558$	−12.7734	−0.540741
$559$	2.58543	0.109352
$560$	0.806065	0.0340625
$561$	−5.40303	−0.228116
$562$	−20.2298	−0.853345
$563$	1.49131	0.0628510	0.0314255	−	0.999506i	$-0.489995\pi$
$563$	1.49131	0.0628510	0.0314255	+	0.999506i	$0.489995\pi$
$564$	−14.0327	−0.590882
$565$	−2.52684	−0.106305
$566$	25.1045	1.05522
$567$	1.00000	0.0419961
$568$	−5.42939	−0.227812
$569$	0.242948	0.0101849	0.00509246	−	0.999987i	$-0.498379\pi$
$569$	0.242948	0.0101849	0.00509246	+	0.999987i	$0.498379\pi$
$570$	−3.88294	−0.162638
$571$	−23.2505	−0.973001	−0.486501	−	0.873680i	$-0.661727\pi$
$571$	−23.2505	−0.973001	−0.486501	+	0.873680i	$0.661727\pi$
$572$	−2.91936	−0.122065
$573$	11.2451	0.469772
$574$	13.0233	0.543583
$575$	−31.5717	−1.31663
$576$	−9.99559	−0.416483
$577$	−10.5364	−0.438637	−0.219319	−	0.975653i	$-0.570383\pi$
$577$	−10.5364	−0.438637	−0.219319	+	0.975653i	$0.570383\pi$
$578$	−2.06644	−0.0859524
$579$	−3.10072	−0.128862
$580$	−2.23307	−0.0927232
$581$	5.45968	0.226506
$582$	−4.27457	−0.177187
$583$	−65.8163	−2.72583
$584$	8.33992	0.345109
$585$	−0.0566484	−0.00234212
$586$	57.2928	2.36675
$587$	43.8752	1.81092	0.905461	−	0.424430i	$-0.139525\pi$
$587$	43.8752	1.81092	0.905461	+	0.424430i	$0.139525\pi$
$588$	−2.27016	−0.0936198
$589$	48.8010	2.01081
$590$	−2.04045	−0.0840041
$591$	−6.51957	−0.268179
$592$	−19.1045	−0.785190
$593$	45.3175	1.86097	0.930483	−	0.366336i	$-0.119388\pi$
$593$	45.3175	1.86097	0.930483	+	0.366336i	$0.119388\pi$
$594$	−11.1650	−0.458106
$595$	0.238009	0.00975743
$596$	−4.76047	−0.194996
$597$	7.64104	0.312727
$598$	−3.14118	−0.128452
$599$	−7.25368	−0.296377	−0.148189	−	0.988959i	$-0.547344\pi$
$599$	−7.25368	−0.296377	−0.148189	+	0.988959i	$0.547344\pi$
$600$	−2.75971	−0.112665
$601$	−4.18458	−0.170693	−0.0853463	−	0.996351i	$-0.527200\pi$
$601$	−4.18458	−0.170693	−0.0853463	+	0.996351i	$0.527200\pi$
$602$	−22.4471	−0.914876
$603$	5.45968	0.222335
$604$	19.5339	0.794821
$605$	−4.33005	−0.176041
$606$	19.4881	0.791649
$607$	−34.6812	−1.40767	−0.703834	−	0.710365i	$-0.748530\pi$
$607$	−34.6812	−1.40767	−0.703834	+	0.710365i	$0.748530\pi$
$608$	64.0662	2.59823
$609$	−4.13287	−0.167472
$610$	−1.25719	−0.0509019
$611$	1.47122	0.0595192
$612$	−2.27016	−0.0917658
$613$	1.98352	0.0801136	0.0400568	−	0.999197i	$-0.487246\pi$
$613$	1.98352	0.0801136	0.0400568	+	0.999197i	$0.487246\pi$
$614$	−6.80929	−0.274800
$615$	−1.50001	−0.0604862
$616$	3.01634	0.121532
$617$	−11.1808	−0.450123	−0.225062	−	0.974345i	$-0.572258\pi$
$617$	−11.1808	−0.450123	−0.225062	+	0.974345i	$0.572258\pi$
$618$	−17.4983	−0.703884
$619$	−40.8425	−1.64160	−0.820799	−	0.571217i	$-0.806472\pi$
$619$	−40.8425	−1.64160	−0.820799	+	0.571217i	$0.806472\pi$
$620$	−3.33991	−0.134134
$621$	−6.38669	−0.256289
$622$	−14.8476	−0.595333
$623$	−6.47602	−0.259456
$624$	0.806065	0.0322684
$625$	24.1535	0.966139
$626$	62.7571	2.50828
$627$	42.6562	1.70352
$628$	29.7581	1.18748
$629$	−5.64104	−0.224923
$630$	0.491831	0.0195950
$631$	11.8702	0.472546	0.236273	−	0.971687i	$-0.424074\pi$
$631$	11.8702	0.472546	0.236273	+	0.971687i	$0.424074\pi$
$632$	9.42939	0.375081
$633$	2.29466	0.0912045
$634$	34.8704	1.38488
$635$	0.568056	0.0225426
$636$	−27.6536	−1.09654
$637$	0.238009	0.00943027
$638$	46.1436	1.82684
$639$	9.72543	0.384732
$640$	−1.05329	−0.0416348
$641$	−37.2440	−1.47105	−0.735524	−	0.677499i	$-0.763064\pi$
$641$	−37.2440	−1.47105	−0.735524	+	0.677499i	$0.763064\pi$
$642$	31.9623	1.26145
$643$	44.7254	1.76380	0.881900	−	0.471437i	$-0.156265\pi$
$643$	44.7254	1.76380	0.881900	+	0.471437i	$0.156265\pi$
$644$	14.4988	0.571333
$645$	2.58543	0.101801
$646$	16.3142	0.641875
$647$	−4.67641	−0.183849	−0.0919244	−	0.995766i	$-0.529302\pi$
$647$	−4.67641	−0.183849	−0.0919244	+	0.995766i	$0.529302\pi$
$648$	−0.558268	−0.0219308
$649$	22.4155	0.879885
$650$	2.43129	0.0953632
$651$	−6.18136	−0.242267
$652$	−5.76693	−0.225850
$653$	−35.0511	−1.37165	−0.685827	−	0.727765i	$-0.740559\pi$
$653$	−35.0511	−1.37165	−0.685827	+	0.727765i	$0.740559\pi$
$654$	2.84196	0.111129
$655$	3.11214	0.121601
$656$	21.3440	0.833343
$657$	−14.9389	−0.582823
$658$	−12.7734	−0.497959
$659$	−50.0635	−1.95020	−0.975099	−	0.221771i	$-0.928816\pi$
$659$	−50.0635	−1.95020	−0.975099	+	0.221771i	$0.928816\pi$
$660$	−2.91936	−0.113636
$661$	−12.4221	−0.483163	−0.241582	−	0.970381i	$-0.577666\pi$
$661$	−12.4221	−0.483163	−0.241582	+	0.970381i	$0.577666\pi$
$662$	−62.3577	−2.42360
$663$	0.238009	0.00924352
$664$	−3.04796	−0.118284
$665$	−1.87905	−0.0728665
$666$	−11.6569	−0.451694
$667$	26.3954	1.02203
$668$	32.9258	1.27394
$669$	17.2729	0.667808
$670$	2.68524	0.103740
$671$	13.8109	0.533162
$672$	−8.11492	−0.313040
$673$	24.4270	0.941593	0.470796	−	0.882242i	$-0.343967\pi$
$673$	24.4270	0.941593	0.470796	+	0.882242i	$0.343967\pi$
$674$	−0.989268	−0.0381052
$675$	4.94335	0.190270
$676$	−29.3835	−1.13013
$677$	−0.375154	−0.0144183	−0.00720917	−	0.999974i	$-0.502295\pi$
$677$	−0.375154	−0.0144183	−0.00720917	+	0.999974i	$0.502295\pi$
$678$	21.9384	0.842540
$679$	−2.06857	−0.0793845
$680$	−0.132873	−0.00509544
$681$	−12.7783	−0.489667
$682$	69.0150	2.64272
$683$	5.26589	0.201494	0.100747	−	0.994912i	$-0.467877\pi$
$683$	5.26589	0.201494	0.100747	+	0.994912i	$0.467877\pi$
$684$	17.9226	0.685288
$685$	1.48653	0.0567972
$686$	−2.06644	−0.0788969
$687$	−18.8704	−0.719949
$688$	−36.7887	−1.40256
$689$	2.89928	0.110454
$690$	−3.14118	−0.119582
$691$	−39.3491	−1.49691	−0.748455	−	0.663185i	$-0.769204\pi$
$691$	−39.3491	−1.49691	−0.748455	+	0.663185i	$0.769204\pi$
$692$	−21.7581	−0.827119
$693$	−5.40303	−0.205244
$694$	12.0316	0.456714
$695$	−3.00002	−0.113797
$696$	2.30725	0.0874560
$697$	6.30231	0.238717
$698$	0.860996	0.0325892
$699$	9.09151	0.343873
$700$	−11.2222	−0.424159
$701$	3.68526	0.139190	0.0695951	−	0.997575i	$-0.477829\pi$
$701$	3.68526	0.139190	0.0695951	+	0.997575i	$0.477829\pi$
$702$	0.491831	0.0185630
$703$	44.5353	1.67968
$704$	54.0065	2.03545
$705$	1.47122	0.0554094
$706$	−2.84196	−0.106959
$707$	9.43077	0.354681
$708$	9.41818	0.353957
$709$	15.9673	0.599665	0.299833	−	0.953992i	$-0.403069\pi$
$709$	15.9673	0.599665	0.299833	+	0.953992i	$0.403069\pi$
$710$	4.78327	0.179513
$711$	−16.8904	−0.633441
$712$	3.61535	0.135491
$713$	39.4785	1.47848
$714$	−2.06644	−0.0773344
$715$	0.306073	0.0114465
$716$	−23.6065	−0.882217
$717$	−16.8219	−0.628225
$718$	11.1819	0.417304
$719$	−24.2206	−0.903277	−0.451639	−	0.892201i	$-0.649160\pi$
$719$	−24.2206	−0.903277	−0.451639	+	0.892201i	$0.649160\pi$
$720$	0.806065	0.0300403
$721$	−8.46786	−0.315360
$722$	−89.5364	−3.33220
$723$	8.10125	0.301289
$724$	−42.9944	−1.59787
$725$	−20.4302	−0.758760
$726$	37.5942	1.39525
$727$	31.9869	1.18633	0.593164	−	0.805081i	$-0.297878\pi$
$727$	31.9869	1.18633	0.593164	+	0.805081i	$0.297878\pi$
$728$	−0.132873	−0.00492460
$729$	1.00000	0.0370370
$730$	−7.34743	−0.271941
$731$	−10.8627	−0.401772
$732$	5.80283	0.214479
$733$	−13.6850	−0.505466	−0.252733	−	0.967536i	$-0.581329\pi$
$733$	−13.6850	−0.505466	−0.252733	+	0.967536i	$0.581329\pi$
$734$	0	0
$735$	0.238009	0.00877911
$736$	51.8275	1.91039
$737$	−29.4988	−1.08660
$738$	13.0233	0.479395
$739$	5.27443	0.194023	0.0970115	−	0.995283i	$-0.469072\pi$
$739$	5.27443	0.194023	0.0970115	+	0.995283i	$0.469072\pi$
$740$	−3.04796	−0.112045
$741$	−1.87905	−0.0690287
$742$	−25.1720	−0.924093
$743$	−8.67641	−0.318307	−0.159153	−	0.987254i	$-0.550876\pi$
$743$	−8.67641	−0.318307	−0.159153	+	0.987254i	$0.550876\pi$
$744$	3.45085	0.126514
$745$	0.499100	0.0182856
$746$	−34.7287	−1.27151
$747$	5.45968	0.199759
$748$	12.2657	0.448480
$749$	15.4673	0.565164
$750$	4.89045	0.178574
$751$	3.27832	0.119628	0.0598138	−	0.998210i	$-0.480949\pi$
$751$	3.27832	0.119628	0.0598138	+	0.998210i	$0.480949\pi$
$752$	−20.9344	−0.763398
$753$	2.68900	0.0979928
$754$	−2.03268	−0.0740257
$755$	−2.04798	−0.0745336
$756$	−2.27016	−0.0825649
$757$	0.273398	0.00993681	0.00496841	−	0.999988i	$-0.498419\pi$
$757$	0.273398	0.00993681	0.00496841	+	0.999988i	$0.498419\pi$
$758$	28.8626	1.04834
$759$	34.5075	1.25254
$760$	1.04901	0.0380517
$761$	13.5795	0.492255	0.246127	−	0.969237i	$-0.420842\pi$
$761$	13.5795	0.492255	0.246127	+	0.969237i	$0.420842\pi$
$762$	−4.93195	−0.178666
$763$	1.37530	0.0497891
$764$	−25.5283	−0.923580
$765$	0.238009	0.00860524
$766$	40.8527	1.47607
$767$	−0.987426	−0.0356539
$768$	−10.8464	−0.391385
$769$	19.6072	0.707053	0.353527	−	0.935424i	$-0.384982\pi$
$769$	19.6072	0.707053	0.353527	+	0.935424i	$0.384982\pi$
$770$	−2.65738	−0.0957653
$771$	20.5441	0.739877
$772$	7.03914	0.253344
$773$	−27.4634	−0.987791	−0.493896	−	0.869521i	$-0.664428\pi$
$773$	−27.4634	−0.987791	−0.493896	+	0.869521i	$0.664428\pi$
$774$	−22.4471	−0.806845
$775$	−30.5566	−1.09763
$776$	1.15482	0.0414555
$777$	−5.64104	−0.202371
$778$	−40.9617	−1.46855
$779$	−49.7559	−1.78269
$780$	0.128601	0.00460465
$781$	−52.5468	−1.88027
$782$	13.1977	0.471949
$783$	−4.13287	−0.147697
$784$	−3.38669	−0.120953
$785$	−3.11992	−0.111355
$786$	−27.0201	−0.963775
$787$	−32.5754	−1.16119	−0.580594	−	0.814193i	$-0.697180\pi$
$787$	−32.5754	−1.16119	−0.580594	+	0.814193i	$0.697180\pi$
$788$	14.8005	0.527245
$789$	−11.4554	−0.407823
$790$	−8.30725	−0.295559
$791$	10.6165	0.377481
$792$	3.01634	0.107181
$793$	−0.608383	−0.0216043
$794$	−70.7571	−2.51107
$795$	2.89928	0.102827
$796$	−17.3464	−0.614826
$797$	−12.4032	−0.439343	−0.219671	−	0.975574i	$-0.570499\pi$
$797$	−12.4032	−0.439343	−0.219671	+	0.975574i	$0.570499\pi$
$798$	16.3142	0.577518
$799$	−6.18136	−0.218681
$800$	−40.1149	−1.41828
$801$	−6.47602	−0.228819
$802$	−6.12484	−0.216276
$803$	80.7156	2.84839
$804$	−12.3943	−0.437115
$805$	−1.52009	−0.0535762
$806$	−3.04019	−0.107086
$807$	23.5201	0.827946
$808$	−5.26489	−0.185218
$809$	−45.1807	−1.58847	−0.794235	−	0.607611i	$-0.792128\pi$
$809$	−45.1807	−1.58847	−0.794235	+	0.607611i	$0.792128\pi$
$810$	0.491831	0.0172812
$811$	11.6928	0.410588	0.205294	−	0.978700i	$-0.434185\pi$
$811$	11.6928	0.410588	0.205294	+	0.978700i	$0.434185\pi$
$812$	9.38228	0.329254
$813$	−27.4743	−0.963566
$814$	62.9824	2.20753
$815$	0.604620	0.0211789
$816$	−3.38669	−0.118558
$817$	85.7596	3.00035
$818$	−72.7651	−2.54417
$819$	0.238009	0.00831672
$820$	3.40526	0.118917
$821$	39.1317	1.36571	0.682853	−	0.730556i	$-0.260739\pi$
$821$	39.1317	1.36571	0.682853	+	0.730556i	$0.260739\pi$
$822$	−12.9063	−0.450158
$823$	10.8415	0.377909	0.188955	−	0.981986i	$-0.439490\pi$
$823$	10.8415	0.377909	0.188955	+	0.981986i	$0.439490\pi$
$824$	4.72733	0.164684
$825$	−26.7091	−0.929891
$826$	8.57299	0.298293
$827$	29.8375	1.03755	0.518777	−	0.854910i	$-0.326388\pi$
$827$	29.8375	1.03755	0.518777	+	0.854910i	$0.326388\pi$
$828$	14.4988	0.503869
$829$	−37.8452	−1.31442	−0.657209	−	0.753708i	$-0.728263\pi$
$829$	−37.8452	−1.31442	−0.657209	+	0.753708i	$0.728263\pi$
$830$	2.68524	0.0932061
$831$	−9.01634	−0.312773
$832$	−2.37904	−0.0824785
$833$	−1.00000	−0.0346479
$834$	26.0466	0.901922
$835$	−3.45203	−0.119462
$836$	−96.8364	−3.34916
$837$	−6.18136	−0.213659
$838$	26.9445	0.930784
$839$	−39.4296	−1.36126	−0.680630	−	0.732627i	$-0.738294\pi$
$839$	−39.4296	−1.36126	−0.680630	+	0.732627i	$0.738294\pi$
$840$	−0.132873	−0.00458455
$841$	−11.9194	−0.411012
$842$	50.9920	1.75730
$843$	−9.78973	−0.337176
$844$	−5.20924	−0.179309
$845$	3.08064	0.105977
$846$	−12.7734	−0.439158
$847$	18.1928	0.625111
$848$	−41.2546	−1.41669
$849$	12.1487	0.416942
$850$	−10.2151	−0.350376
$851$	36.0276	1.23501
$852$	−22.0783	−0.756389
$853$	−2.44283	−0.0836411	−0.0418205	−	0.999125i	$-0.513316\pi$
$853$	−2.44283	−0.0836411	−0.0418205	+	0.999125i	$0.513316\pi$
$854$	5.28208	0.180749
$855$	−1.87905	−0.0642622
$856$	−8.63491	−0.295135
$857$	32.4018	1.10683	0.553413	−	0.832907i	$-0.313325\pi$
$857$	32.4018	1.10683	0.553413	+	0.832907i	$0.313325\pi$
$858$	−2.65738	−0.0907214
$859$	−49.1414	−1.67668	−0.838342	−	0.545145i	$-0.816475\pi$
$859$	−49.1414	−1.67668	−0.838342	+	0.545145i	$0.816475\pi$
$860$	−5.86933	−0.200143
$861$	6.30231	0.214782
$862$	−0.549147	−0.0187040
$863$	12.9842	0.441987	0.220994	−	0.975275i	$-0.429070\pi$
$863$	12.9842	0.441987	0.220994	+	0.975275i	$0.429070\pi$
$864$	−8.11492	−0.276075
$865$	2.28117	0.0775623
$866$	−33.0549	−1.12325
$867$	−1.00000	−0.0339618
$868$	14.0327	0.476300
$869$	91.2596	3.09577
$870$	−2.03268	−0.0689142
$871$	1.29945	0.0440303
$872$	−0.767783	−0.0260004
$873$	−2.06857	−0.0700106
$874$	−104.194	−3.52442
$875$	2.36661	0.0800060
$876$	33.9138	1.14584
$877$	−20.2130	−0.682544	−0.341272	−	0.939965i	$-0.610858\pi$
$877$	−20.2130	−0.682544	−0.341272	+	0.939965i	$0.610858\pi$
$878$	−14.1058	−0.476048
$879$	27.7254	0.935155
$880$	−4.35519	−0.146814
$881$	24.7329	0.833274	0.416637	−	0.909073i	$-0.363209\pi$
$881$	24.7329	0.833274	0.416637	+	0.909073i	$0.363209\pi$
$882$	−2.06644	−0.0695805
$883$	1.87051	0.0629476	0.0314738	−	0.999505i	$-0.489980\pi$
$883$	1.87051	0.0629476	0.0314738	+	0.999505i	$0.489980\pi$
$884$	−0.540319	−0.0181729
$885$	−0.987426	−0.0331920
$886$	−44.0761	−1.48077
$887$	−58.4459	−1.96242	−0.981211	−	0.192937i	$-0.938199\pi$
$887$	−58.4459	−1.96242	−0.981211	+	0.192937i	$0.938199\pi$
$888$	3.14921	0.105681
$889$	−2.38669	−0.0800472
$890$	−3.18511	−0.106765
$891$	−5.40303	−0.181008
$892$	−39.2122	−1.31292
$893$	48.8010	1.63306
$894$	−4.33327	−0.144926
$895$	2.47497	0.0827291
$896$	4.42539	0.147842
$897$	−1.52009	−0.0507544
$898$	−9.28986	−0.310007
$899$	25.5468	0.852033
$900$	−11.2222	−0.374073
$901$	−12.1814	−0.405820
$902$	−70.3654	−2.34291
$903$	−10.8627	−0.361488
$904$	−5.92687	−0.197125
$905$	4.50764	0.149839
$906$	17.7809	0.590731
$907$	−7.38684	−0.245276	−0.122638	−	0.992451i	$-0.539135\pi$
$907$	−7.38684	−0.245276	−0.122638	+	0.992451i	$0.539135\pi$
$908$	29.0088	0.962692
$909$	9.43077	0.312799
$910$	0.117060	0.00388051
$911$	−57.4106	−1.90210	−0.951048	−	0.309042i	$-0.899992\pi$
$911$	−57.4106	−1.90210	−0.951048	+	0.309042i	$0.899992\pi$
$912$	26.7375	0.885367
$913$	−29.4988	−0.976269
$914$	−9.69974	−0.320839
$915$	−0.608383	−0.0201125
$916$	42.8387	1.41543
$917$	−13.0757	−0.431798
$918$	−2.06644	−0.0682026
$919$	−39.6776	−1.30884	−0.654422	−	0.756130i	$-0.727088\pi$
$919$	−39.6776	−1.30884	−0.654422	+	0.756130i	$0.727088\pi$
$920$	0.848619	0.0279781
$921$	−3.29518	−0.108580
$922$	−26.7702	−0.881629
$923$	2.31474	0.0761907
$924$	12.2657	0.403513
$925$	−27.8857	−0.916875
$926$	47.0619	1.54655
$927$	−8.46786	−0.278121
$928$	33.5380	1.10094
$929$	5.89942	0.193554	0.0967768	−	0.995306i	$-0.469147\pi$
$929$	5.89942	0.193554	0.0967768	+	0.995306i	$0.469147\pi$
$930$	−3.04019	−0.0996916
$931$	7.89486	0.258744
$932$	−20.6392	−0.676059
$933$	−7.18511	−0.235230
$934$	83.5870	2.73505
$935$	−1.28597	−0.0420558
$936$	−0.132873	−0.00434309
$937$	−32.4825	−1.06116	−0.530578	−	0.847636i	$-0.678025\pi$
$937$	−32.4825	−1.06116	−0.530578	+	0.847636i	$0.678025\pi$
$938$	−11.2821	−0.368373
$939$	30.3697	0.991078
$940$	−3.33991	−0.108936
$941$	−11.2746	−0.367541	−0.183771	−	0.982969i	$-0.558830\pi$
$941$	−11.2746	−0.367541	−0.183771	+	0.982969i	$0.558830\pi$
$942$	27.0876	0.882562
$943$	−40.2509	−1.31075
$944$	14.0503	0.457299
$945$	0.238009	0.00774244
$946$	121.282	3.94323
$947$	17.6199	0.572570	0.286285	−	0.958145i	$-0.407580\pi$
$947$	17.6199	0.572570	0.286285	+	0.958145i	$0.407580\pi$
$948$	38.3440	1.24536
$949$	−3.55561	−0.115420
$950$	80.6470	2.61653
$951$	16.8746	0.547198
$952$	0.558268	0.0180936
$953$	−20.5026	−0.664144	−0.332072	−	0.943254i	$-0.607748\pi$
$953$	−20.5026	−0.664144	−0.332072	+	0.943254i	$0.607748\pi$
$954$	−25.1720	−0.814974
$955$	2.67645	0.0866078
$956$	38.1883	1.23510
$957$	22.3300	0.721828
$958$	−15.1235	−0.488619
$959$	−6.24566	−0.201683
$960$	−2.37904	−0.0767833
$961$	7.20922	0.232556
$962$	−2.77444	−0.0894515
$963$	15.4673	0.498428
$964$	−18.3911	−0.592338
$965$	−0.738001	−0.0237571
$966$	13.1977	0.424629
$967$	−36.0543	−1.15943	−0.579714	−	0.814820i	$-0.696836\pi$
$967$	−36.0543	−1.15943	−0.579714	+	0.814820i	$0.696836\pi$
$968$	−10.1564	−0.326440
$969$	7.89486	0.253620
$970$	−1.01739	−0.0326664
$971$	−1.13069	−0.0362854	−0.0181427	−	0.999835i	$-0.505775\pi$
$971$	−1.13069	−0.0362854	−0.0181427	+	0.999835i	$0.505775\pi$
$972$	−2.27016	−0.0728154
$973$	12.6046	0.404086
$974$	−51.7418	−1.65791
$975$	1.17656	0.0376802
$976$	8.65684	0.277099
$977$	11.3062	0.361718	0.180859	−	0.983509i	$-0.442112\pi$
$977$	11.3062	0.361718	0.180859	+	0.983509i	$0.442112\pi$
$978$	−5.24941	−0.167858
$979$	34.9901	1.11829
$980$	−0.540319	−0.0172599
$981$	1.37530	0.0439098
$982$	−73.3810	−2.34168
$983$	−25.7140	−0.820151	−0.410075	−	0.912052i	$-0.634498\pi$
$983$	−25.7140	−0.820151	−0.410075	+	0.912052i	$0.634498\pi$
$984$	−3.51838	−0.112162
$985$	−1.55172	−0.0494418
$986$	8.54032	0.271979
$987$	−6.18136	−0.196755
$988$	4.26575	0.135711
$989$	69.3768	2.20605
$990$	−2.65738	−0.0844571
$991$	10.4356	0.331497	0.165748	−	0.986168i	$-0.446996\pi$
$991$	10.4356	0.331497	0.165748	+	0.986168i	$0.446996\pi$
$992$	50.1613	1.59262
$993$	−30.1764	−0.957620
$994$	−20.0970	−0.637437
$995$	1.81864	0.0576547
$996$	−12.3943	−0.392730
$997$	28.6480	0.907293	0.453646	−	0.891182i	$-0.350123\pi$
$997$	28.6480	0.907293	0.453646	+	0.891182i	$0.350123\pi$
$998$	−43.6526	−1.38180
$999$	−5.64104	−0.178475

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.1.1	✓	4
3.2	odd	2		1071.2.a.j.1.4		4
4.3	odd	2		5712.2.a.bx.1.3		4
5.4	even	2		8925.2.a.bs.1.4		4
7.6	odd	2		2499.2.a.z.1.1		4
17.16	even	2		6069.2.a.s.1.1		4
21.20	even	2		7497.2.a.be.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.a.h.1.1	✓	4	1.1	even	1	trivial
1071.2.a.j.1.4		4	3.2	odd	2
2499.2.a.z.1.1		4	7.6	odd	2
5712.2.a.bx.1.3		4	4.3	odd	2
6069.2.a.s.1.1		4	17.16	even	2
7497.2.a.be.1.4		4	21.20	even	2
8925.2.a.bs.1.4		4	5.4	even	2

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Classical	Maass
Hilbert	Bianchi

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

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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.1.1

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