LMFDB - Embedded newform 393.2.a.e.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [393,2,Mod(1,393)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(393, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("393.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$393 = 3 \cdot 131$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	393.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$3.13812079944$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.12062776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{5} - 7x^{4} + 5x^{3} + 13x^{2} - 4x - 5$ x^6 - x^5 - 7x^4 + 5x^3 + 13x^2 - 4x - 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.18801$ of defining polynomial
Character	$\chi$	$=$	393.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.18801 q^{2} +1.00000 q^{3} +2.78738 q^{4} -0.982095 q^{5} +2.18801 q^{6} +1.75537 q^{7} +1.72279 q^{8} +1.00000 q^{9} -2.14883 q^{10} +0.0712998 q^{11} +2.78738 q^{12} -1.56066 q^{13} +3.84077 q^{14} -0.982095 q^{15} -1.80528 q^{16} +3.40814 q^{17} +2.18801 q^{18} -6.12422 q^{19} -2.73747 q^{20} +1.75537 q^{21} +0.156004 q^{22} +3.69020 q^{23} +1.72279 q^{24} -4.03549 q^{25} -3.41473 q^{26} +1.00000 q^{27} +4.89289 q^{28} -2.71064 q^{29} -2.14883 q^{30} +1.26333 q^{31} -7.39555 q^{32} +0.0712998 q^{33} +7.45703 q^{34} -1.72394 q^{35} +2.78738 q^{36} -2.95494 q^{37} -13.3998 q^{38} -1.56066 q^{39} -1.69194 q^{40} +10.3705 q^{41} +3.84077 q^{42} -4.92606 q^{43} +0.198739 q^{44} -0.982095 q^{45} +8.07419 q^{46} -0.369425 q^{47} -1.80528 q^{48} -3.91866 q^{49} -8.82968 q^{50} +3.40814 q^{51} -4.35014 q^{52} -6.75029 q^{53} +2.18801 q^{54} -0.0700231 q^{55} +3.02414 q^{56} -6.12422 q^{57} -5.93091 q^{58} -1.64029 q^{59} -2.73747 q^{60} +9.30265 q^{61} +2.76417 q^{62} +1.75537 q^{63} -12.5709 q^{64} +1.53271 q^{65} +0.156004 q^{66} -3.56331 q^{67} +9.49977 q^{68} +3.69020 q^{69} -3.77200 q^{70} +2.99790 q^{71} +1.72279 q^{72} +10.0038 q^{73} -6.46543 q^{74} -4.03549 q^{75} -17.0705 q^{76} +0.125158 q^{77} -3.41473 q^{78} +7.46316 q^{79} +1.77296 q^{80} +1.00000 q^{81} +22.6907 q^{82} +1.78854 q^{83} +4.89289 q^{84} -3.34712 q^{85} -10.7783 q^{86} -2.71064 q^{87} +0.122834 q^{88} +7.55631 q^{89} -2.14883 q^{90} -2.73954 q^{91} +10.2860 q^{92} +1.26333 q^{93} -0.808306 q^{94} +6.01456 q^{95} -7.39555 q^{96} -3.53954 q^{97} -8.57406 q^{98} +0.0712998 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q + q^{2} + 6 q^{3} + 3 q^{4} + 8 q^{5} + q^{6} + 4 q^{7} + 3 q^{8} + 6 q^{9} - 3 q^{10} + 6 q^{11} + 3 q^{12} - 3 q^{13} + q^{14} + 8 q^{15} - 11 q^{16} + 4 q^{17} + q^{18} - q^{19} + 4 q^{20} + 4 q^{21}+ \cdots + 6 q^{99}+O(q^{100})$ 6 * q + q^2 + 6 * q^3 + 3 * q^4 + 8 * q^5 + q^6 + 4 * q^7 + 3 * q^8 + 6 * q^9 - 3 * q^10 + 6 * q^11 + 3 * q^12 - 3 * q^13 + q^14 + 8 * q^15 - 11 * q^16 + 4 * q^17 + q^18 - q^19 + 4 * q^20 + 4 * q^21 - 4 * q^22 + 14 * q^23 + 3 * q^24 - 2 * q^25 - q^26 + 6 * q^27 - 4 * q^28 + 12 * q^29 - 3 * q^30 - 2 * q^31 - 8 * q^32 + 6 * q^33 - 10 * q^34 + 9 * q^35 + 3 * q^36 - 12 * q^37 + q^38 - 3 * q^39 + 2 * q^40 + 7 * q^41 + q^42 - 5 * q^43 - 2 * q^44 + 8 * q^45 - 9 * q^46 + 19 * q^47 - 11 * q^48 - 4 * q^49 - 4 * q^50 + 4 * q^51 - 13 * q^52 + 4 * q^53 + q^54 - 3 * q^55 + 11 * q^56 - q^57 - 12 * q^58 + 5 * q^59 + 4 * q^60 - 20 * q^61 - 11 * q^62 + 4 * q^63 - 13 * q^64 - 25 * q^65 - 4 * q^66 + 4 * q^67 + 4 * q^68 + 14 * q^69 - 27 * q^70 + 11 * q^71 + 3 * q^72 - 18 * q^73 + 4 * q^74 - 2 * q^75 - 25 * q^76 - 9 * q^77 - q^78 - 6 * q^79 - 32 * q^80 + 6 * q^81 + 8 * q^82 + 13 * q^83 - 4 * q^84 - 15 * q^85 - 22 * q^86 + 12 * q^87 - 13 * q^88 - 8 * q^89 - 3 * q^90 - 35 * q^91 + 14 * q^92 - 2 * q^93 + 3 * q^94 + 8 * q^95 - 8 * q^96 - 14 * q^97 - 29 * q^98 + 6 * 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

)

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

$2$	2.18801	1.54716	0.773578	−	0.633702i	$-0.218465\pi$
$2$	2.18801	1.54716	0.773578	+	0.633702i	$0.218465\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.78738	1.39369
$5$	−0.982095	−0.439206	−0.219603	−	0.975589i	$-0.570476\pi$
$5$	−0.982095	−0.439206	−0.219603	+	0.975589i	$0.570476\pi$
$6$	2.18801	0.893250
$7$	1.75537	0.663469	0.331735	−	0.943373i	$-0.392366\pi$
$7$	1.75537	0.663469	0.331735	+	0.943373i	$0.392366\pi$
$8$	1.72279	0.609097
$9$	1.00000	0.333333
$10$	−2.14883	−0.679520
$11$	0.0712998	0.0214977	0.0107488	−	0.999942i	$-0.496578\pi$
$11$	0.0712998	0.0214977	0.0107488	+	0.999942i	$0.496578\pi$
$12$	2.78738	0.804647
$13$	−1.56066	−0.432848	−0.216424	−	0.976299i	$-0.569439\pi$
$13$	−1.56066	−0.432848	−0.216424	+	0.976299i	$0.569439\pi$
$14$	3.84077	1.02649
$15$	−0.982095	−0.253576
$16$	−1.80528	−0.451321
$17$	3.40814	0.826595	0.413298	−	0.910596i	$-0.364377\pi$
$17$	3.40814	0.826595	0.413298	+	0.910596i	$0.364377\pi$
$18$	2.18801	0.515718
$19$	−6.12422	−1.40499	−0.702496	−	0.711688i	$-0.747931\pi$
$19$	−6.12422	−1.40499	−0.702496	+	0.711688i	$0.747931\pi$
$20$	−2.73747	−0.612117
$21$	1.75537	0.383054
$22$	0.156004	0.0332603
$23$	3.69020	0.769460	0.384730	−	0.923029i	$-0.374295\pi$
$23$	3.69020	0.769460	0.384730	+	0.923029i	$0.374295\pi$
$24$	1.72279	0.351663
$25$	−4.03549	−0.807098
$26$	−3.41473	−0.669683
$27$	1.00000	0.192450
$28$	4.89289	0.924669
$29$	−2.71064	−0.503354	−0.251677	−	0.967811i	$-0.580982\pi$
$29$	−2.71064	−0.503354	−0.251677	+	0.967811i	$0.580982\pi$
$30$	−2.14883	−0.392321
$31$	1.26333	0.226901	0.113450	−	0.993544i	$-0.463810\pi$
$31$	1.26333	0.226901	0.113450	+	0.993544i	$0.463810\pi$
$32$	−7.39555	−1.30736
$33$	0.0712998	0.0124117
$34$	7.45703	1.27887
$35$	−1.72394	−0.291400
$36$	2.78738	0.464563
$37$	−2.95494	−0.485789	−0.242895	−	0.970053i	$-0.578097\pi$
$37$	−2.95494	−0.485789	−0.242895	+	0.970053i	$0.578097\pi$
$38$	−13.3998	−2.17374
$39$	−1.56066	−0.249905
$40$	−1.69194	−0.267519
$41$	10.3705	1.61960	0.809798	−	0.586709i	$-0.199577\pi$
$41$	10.3705	1.61960	0.809798	+	0.586709i	$0.199577\pi$
$42$	3.84077	0.592644
$43$	−4.92606	−0.751217	−0.375609	−	0.926778i	$-0.622566\pi$
$43$	−4.92606	−0.751217	−0.375609	+	0.926778i	$0.622566\pi$
$44$	0.198739	0.0299611
$45$	−0.982095	−0.146402
$46$	8.07419	1.19047
$47$	−0.369425	−0.0538862	−0.0269431	−	0.999637i	$-0.508577\pi$
$47$	−0.369425	−0.0538862	−0.0269431	+	0.999637i	$0.508577\pi$
$48$	−1.80528	−0.260570
$49$	−3.91866	−0.559809
$50$	−8.82968	−1.24871
$51$	3.40814	0.477235
$52$	−4.35014	−0.603256
$53$	−6.75029	−0.927224	−0.463612	−	0.886038i	$-0.653447\pi$
$53$	−6.75029	−0.927224	−0.463612	+	0.886038i	$0.653447\pi$
$54$	2.18801	0.297750
$55$	−0.0700231	−0.00944192
$56$	3.02414	0.404117
$57$	−6.12422	−0.811172
$58$	−5.93091	−0.778767
$59$	−1.64029	−0.213548	−0.106774	−	0.994283i	$-0.534052\pi$
$59$	−1.64029	−0.213548	−0.106774	+	0.994283i	$0.534052\pi$
$60$	−2.73747	−0.353406
$61$	9.30265	1.19108	0.595541	−	0.803325i	$-0.296938\pi$
$61$	9.30265	1.19108	0.595541	+	0.803325i	$0.296938\pi$
$62$	2.76417	0.351050
$63$	1.75537	0.221156
$64$	−12.5709	−1.57137
$65$	1.53271	0.190110
$66$	0.156004	0.0192028
$67$	−3.56331	−0.435328	−0.217664	−	0.976024i	$-0.569844\pi$
$67$	−3.56331	−0.435328	−0.217664	+	0.976024i	$0.569844\pi$
$68$	9.49977	1.15202
$69$	3.69020	0.444248
$70$	−3.77200	−0.450841
$71$	2.99790	0.355785	0.177892	−	0.984050i	$-0.443072\pi$
$71$	2.99790	0.355785	0.177892	+	0.984050i	$0.443072\pi$
$72$	1.72279	0.203032
$73$	10.0038	1.17086	0.585428	−	0.810724i	$-0.300926\pi$
$73$	10.0038	1.17086	0.585428	+	0.810724i	$0.300926\pi$
$74$	−6.46543	−0.751592
$75$	−4.03549	−0.465978
$76$	−17.0705	−1.95812
$77$	0.125158	0.0142631
$78$	−3.41473	−0.386642
$79$	7.46316	0.839671	0.419836	−	0.907600i	$-0.362088\pi$
$79$	7.46316	0.839671	0.419836	+	0.907600i	$0.362088\pi$
$80$	1.77296	0.198223
$81$	1.00000	0.111111
$82$	22.6907	2.50576
$83$	1.78854	0.196318	0.0981591	−	0.995171i	$-0.468705\pi$
$83$	1.78854	0.196318	0.0981591	+	0.995171i	$0.468705\pi$
$84$	4.89289	0.533858
$85$	−3.34712	−0.363046
$86$	−10.7783	−1.16225
$87$	−2.71064	−0.290612
$88$	0.122834	0.0130942
$89$	7.55631	0.800968	0.400484	−	0.916304i	$-0.368842\pi$
$89$	7.55631	0.800968	0.400484	+	0.916304i	$0.368842\pi$
$90$	−2.14883	−0.226507
$91$	−2.73954	−0.287181
$92$	10.2860	1.07239
$93$	1.26333	0.131001
$94$	−0.808306	−0.0833703
$95$	6.01456	0.617081
$96$	−7.39555	−0.754805
$97$	−3.53954	−0.359386	−0.179693	−	0.983723i	$-0.557510\pi$
$97$	−3.53954	−0.359386	−0.179693	+	0.983723i	$0.557510\pi$
$98$	−8.57406	−0.866111
$99$	0.0712998	0.00716590
$100$	−11.2484	−1.12484
$101$	8.98488	0.894029	0.447014	−	0.894527i	$-0.352487\pi$
$101$	8.98488	0.894029	0.447014	+	0.894527i	$0.352487\pi$
$102$	7.45703	0.738356
$103$	−5.93497	−0.584790	−0.292395	−	0.956298i	$-0.594452\pi$
$103$	−5.93497	−0.584790	−0.292395	+	0.956298i	$0.594452\pi$
$104$	−2.68868	−0.263647
$105$	−1.72394	−0.168240
$106$	−14.7697	−1.43456
$107$	8.53190	0.824810	0.412405	−	0.911001i	$-0.364689\pi$
$107$	8.53190	0.824810	0.412405	+	0.911001i	$0.364689\pi$
$108$	2.78738	0.268216
$109$	−4.90026	−0.469359	−0.234680	−	0.972073i	$-0.575404\pi$
$109$	−4.90026	−0.469359	−0.234680	+	0.972073i	$0.575404\pi$
$110$	−0.153211	−0.0146081
$111$	−2.95494	−0.280471
$112$	−3.16895	−0.299437
$113$	8.58547	0.807653	0.403827	−	0.914836i	$-0.367680\pi$
$113$	8.58547	0.807653	0.403827	+	0.914836i	$0.367680\pi$
$114$	−13.3998	−1.25501
$115$	−3.62413	−0.337952
$116$	−7.55559	−0.701519
$117$	−1.56066	−0.144283
$118$	−3.58897	−0.330392
$119$	5.98256	0.548420
$120$	−1.69194	−0.154452
$121$	−10.9949	−0.999538
$122$	20.3543	1.84279
$123$	10.3705	0.935074
$124$	3.52138	0.316229
$125$	8.87371	0.793689
$126$	3.84077	0.342163
$127$	17.0084	1.50926	0.754628	−	0.656153i	$-0.227817\pi$
$127$	17.0084	1.50926	0.754628	+	0.656153i	$0.227817\pi$
$128$	−12.7142	−1.12379
$129$	−4.92606	−0.433715
$130$	3.35359	0.294129
$131$	−1.00000	−0.0873704
$131$	−1.00000	−0.0873704
$132$	0.198739	0.0172980
$133$	−10.7503	−0.932168
$134$	−7.79656	−0.673520
$135$	−0.982095	−0.0845253
$136$	5.87150	0.503477
$137$	−3.57762	−0.305657	−0.152828	−	0.988253i	$-0.548838\pi$
$137$	−3.57762	−0.305657	−0.152828	+	0.988253i	$0.548838\pi$
$138$	8.07419	0.687320
$139$	−11.1841	−0.948621	−0.474311	−	0.880358i	$-0.657303\pi$
$139$	−11.1841	−0.948621	−0.474311	+	0.880358i	$0.657303\pi$
$140$	−4.80528	−0.406120
$141$	−0.369425	−0.0311112
$142$	6.55942	0.550454
$143$	−0.111274	−0.00930524
$144$	−1.80528	−0.150440
$145$	2.66211	0.221076
$146$	21.8884	1.81150
$147$	−3.91866	−0.323206
$148$	−8.23654	−0.677039
$149$	10.1204	0.829093	0.414546	−	0.910028i	$-0.363940\pi$
$149$	10.1204	0.829093	0.414546	+	0.910028i	$0.363940\pi$
$150$	−8.82968	−0.720941
$151$	3.45909	0.281497	0.140749	−	0.990045i	$-0.455049\pi$
$151$	3.45909	0.281497	0.140749	+	0.990045i	$0.455049\pi$
$152$	−10.5507	−0.855777
$153$	3.40814	0.275532
$154$	0.273846	0.0220672
$155$	−1.24071	−0.0996562
$156$	−4.35014	−0.348290
$157$	0.374706	0.0299048	0.0149524	−	0.999888i	$-0.495240\pi$
$157$	0.374706	0.0299048	0.0149524	+	0.999888i	$0.495240\pi$
$158$	16.3294	1.29910
$159$	−6.75029	−0.535333
$160$	7.26313	0.574201
$161$	6.47768	0.510513
$162$	2.18801	0.171906
$163$	16.4849	1.29120	0.645599	−	0.763677i	$-0.276608\pi$
$163$	16.4849	1.29120	0.645599	+	0.763677i	$0.276608\pi$
$164$	28.9064	2.25721
$165$	−0.0700231	−0.00545129
$166$	3.91335	0.303735
$167$	2.46568	0.190800	0.0954002	−	0.995439i	$-0.469587\pi$
$167$	2.46568	0.190800	0.0954002	+	0.995439i	$0.469587\pi$
$168$	3.02414	0.233317
$169$	−10.5644	−0.812642
$170$	−7.32351	−0.561688
$171$	−6.12422	−0.468331
$172$	−13.7308	−1.04696
$173$	−18.5372	−1.40936	−0.704679	−	0.709526i	$-0.748909\pi$
$173$	−18.5372	−1.40936	−0.704679	+	0.709526i	$0.748909\pi$
$174$	−5.93091	−0.449621
$175$	−7.08379	−0.535484
$176$	−0.128716	−0.00970235
$177$	−1.64029	−0.123292
$178$	16.5333	1.23922
$179$	4.76645	0.356261	0.178131	−	0.984007i	$-0.442995\pi$
$179$	4.76645	0.356261	0.178131	+	0.984007i	$0.442995\pi$
$180$	−2.73747	−0.204039
$181$	8.10391	0.602359	0.301179	−	0.953567i	$-0.402620\pi$
$181$	8.10391	0.602359	0.301179	+	0.953567i	$0.402620\pi$
$182$	−5.99413	−0.444314
$183$	9.30265	0.687672
$184$	6.35743	0.468676
$185$	2.90203	0.213362
$186$	2.76417	0.202679
$187$	0.243000	0.0177699
$188$	−1.02973	−0.0751006
$189$	1.75537	0.127685
$190$	13.1599	0.954720
$191$	18.4208	1.33288	0.666440	−	0.745559i	$-0.267817\pi$
$191$	18.4208	1.33288	0.666440	+	0.745559i	$0.267817\pi$
$192$	−12.5709	−0.907230
$193$	−26.8519	−1.93284	−0.966419	−	0.256970i	$-0.917276\pi$
$193$	−26.8519	−1.93284	−0.966419	+	0.256970i	$0.917276\pi$
$194$	−7.74455	−0.556026
$195$	1.53271	0.109760
$196$	−10.9228	−0.780199
$197$	−17.2088	−1.22607	−0.613037	−	0.790054i	$-0.710052\pi$
$197$	−17.2088	−1.22607	−0.613037	+	0.790054i	$0.710052\pi$
$198$	0.156004	0.0110868
$199$	8.73022	0.618869	0.309435	−	0.950921i	$-0.399860\pi$
$199$	8.73022	0.618869	0.309435	+	0.950921i	$0.399860\pi$
$200$	−6.95229	−0.491601
$201$	−3.56331	−0.251337
$202$	19.6590	1.38320
$203$	−4.75819	−0.333960
$204$	9.49977	0.665117
$205$	−10.1848	−0.711336
$206$	−12.9858	−0.904761
$207$	3.69020	0.256487
$208$	2.81743	0.195353
$209$	−0.436655	−0.0302041
$210$	−3.77200	−0.260293
$211$	−8.77268	−0.603936	−0.301968	−	0.953318i	$-0.597644\pi$
$211$	−8.77268	−0.603936	−0.301968	+	0.953318i	$0.597644\pi$
$212$	−18.8156	−1.29226
$213$	2.99790	0.205412
$214$	18.6679	1.27611
$215$	4.83786	0.329939
$216$	1.72279	0.117221
$217$	2.21762	0.150542
$218$	−10.7218	−0.726172
$219$	10.0038	0.675995
$220$	−0.195181	−0.0131591
$221$	−5.31893	−0.357790
$222$	−6.46543	−0.433932
$223$	9.65707	0.646685	0.323343	−	0.946282i	$-0.395193\pi$
$223$	9.65707	0.646685	0.323343	+	0.946282i	$0.395193\pi$
$224$	−12.9820	−0.867393
$225$	−4.03549	−0.269033
$226$	18.7851	1.24956
$227$	7.25213	0.481341	0.240671	−	0.970607i	$-0.422633\pi$
$227$	7.25213	0.481341	0.240671	+	0.970607i	$0.422633\pi$
$228$	−17.0705	−1.13052
$229$	1.14442	0.0756253	0.0378126	−	0.999285i	$-0.487961\pi$
$229$	1.14442	0.0756253	0.0378126	+	0.999285i	$0.487961\pi$
$230$	−7.92962	−0.522864
$231$	0.125158	0.00823478
$232$	−4.66986	−0.306592
$233$	−9.96373	−0.652746	−0.326373	−	0.945241i	$-0.605827\pi$
$233$	−9.96373	−0.652746	−0.326373	+	0.945241i	$0.605827\pi$
$234$	−3.41473	−0.223228
$235$	0.362811	0.0236672
$236$	−4.57211	−0.297619
$237$	7.46316	0.484784
$238$	13.0899	0.848491
$239$	12.9401	0.837025	0.418513	−	0.908211i	$-0.362552\pi$
$239$	12.9401	0.837025	0.418513	+	0.908211i	$0.362552\pi$
$240$	1.77296	0.114444
$241$	−7.88922	−0.508189	−0.254095	−	0.967179i	$-0.581777\pi$
$241$	−7.88922	−0.508189	−0.254095	+	0.967179i	$0.581777\pi$
$242$	−24.0570	−1.54644
$243$	1.00000	0.0641500
$244$	25.9300	1.66000
$245$	3.84850	0.245871
$246$	22.6907	1.44670
$247$	9.55780	0.608148
$248$	2.17645	0.138205
$249$	1.78854	0.113344
$250$	19.4157	1.22796
$251$	13.0595	0.824307	0.412153	−	0.911114i	$-0.364777\pi$
$251$	13.0595	0.824307	0.412153	+	0.911114i	$0.364777\pi$
$252$	4.89289	0.308223
$253$	0.263111	0.0165416
$254$	37.2146	2.33505
$255$	−3.34712	−0.209605
$256$	−2.67695	−0.167309
$257$	6.53801	0.407830	0.203915	−	0.978989i	$-0.434633\pi$
$257$	6.53801	0.407830	0.203915	+	0.978989i	$0.434633\pi$
$258$	−10.7783	−0.671025
$259$	−5.18703	−0.322306
$260$	4.27225	0.264954
$261$	−2.71064	−0.167785
$262$	−2.18801	−0.135176
$263$	−22.6754	−1.39823	−0.699113	−	0.715011i	$-0.746422\pi$
$263$	−22.6754	−1.39823	−0.699113	+	0.715011i	$0.746422\pi$
$264$	0.122834	0.00755993
$265$	6.62943	0.407242
$266$	−23.5217	−1.44221
$267$	7.55631	0.462439
$268$	−9.93230	−0.606712
$269$	−10.3264	−0.629611	−0.314805	−	0.949156i	$-0.601939\pi$
$269$	−10.3264	−0.629611	−0.314805	+	0.949156i	$0.601939\pi$
$270$	−2.14883	−0.130774
$271$	24.0929	1.46354	0.731770	−	0.681552i	$-0.238694\pi$
$271$	24.0929	1.46354	0.731770	+	0.681552i	$0.238694\pi$
$272$	−6.15265	−0.373059
$273$	−2.73954	−0.165804
$274$	−7.82786	−0.472898
$275$	−0.287730	−0.0173507
$276$	10.2860	0.619143
$277$	8.02160	0.481971	0.240985	−	0.970529i	$-0.422529\pi$
$277$	8.02160	0.481971	0.240985	+	0.970529i	$0.422529\pi$
$278$	−24.4709	−1.46766
$279$	1.26333	0.0756335
$280$	−2.96999	−0.177491
$281$	−32.2015	−1.92098	−0.960491	−	0.278311i	$-0.910225\pi$
$281$	−32.2015	−1.92098	−0.960491	+	0.278311i	$0.910225\pi$
$282$	−0.808306	−0.0481339
$283$	2.20717	0.131203	0.0656014	−	0.997846i	$-0.479103\pi$
$283$	2.20717	0.131203	0.0656014	+	0.997846i	$0.479103\pi$
$284$	8.35627	0.495853
$285$	6.01456	0.356272
$286$	−0.243469	−0.0143966
$287$	18.2041	1.07455
$288$	−7.39555	−0.435787
$289$	−5.38459	−0.316741
$290$	5.82472	0.342039
$291$	−3.53954	−0.207492
$292$	27.8844	1.63181
$293$	29.3883	1.71688	0.858441	−	0.512913i	$-0.171434\pi$
$293$	29.3883	1.71688	0.858441	+	0.512913i	$0.171434\pi$
$294$	−8.57406	−0.500049
$295$	1.61092	0.0937916
$296$	−5.09074	−0.295893
$297$	0.0712998	0.00413723
$298$	22.1434	1.28273
$299$	−5.75914	−0.333059
$300$	−11.2484	−0.649429
$301$	−8.64708	−0.498409
$302$	7.56852	0.435520
$303$	8.98488	0.516168
$304$	11.0559	0.634102
$305$	−9.13609	−0.523131
$306$	7.45703	0.426290
$307$	29.8391	1.70301	0.851505	−	0.524347i	$-0.175690\pi$
$307$	29.8391	1.70301	0.851505	+	0.524347i	$0.175690\pi$
$308$	0.348862	0.0198783
$309$	−5.93497	−0.337629
$310$	−2.71468	−0.154184
$311$	−10.9460	−0.620690	−0.310345	−	0.950624i	$-0.600445\pi$
$311$	−10.9460	−0.620690	−0.310345	+	0.950624i	$0.600445\pi$
$312$	−2.68868	−0.152217
$313$	9.94061	0.561877	0.280938	−	0.959726i	$-0.409354\pi$
$313$	9.94061	0.561877	0.280938	+	0.959726i	$0.409354\pi$
$314$	0.819859	0.0462673
$315$	−1.72394	−0.0971332
$316$	20.8026	1.17024
$317$	15.4388	0.867132	0.433566	−	0.901122i	$-0.357255\pi$
$317$	15.4388	0.867132	0.433566	+	0.901122i	$0.357255\pi$
$318$	−14.7697	−0.828243
$319$	−0.193268	−0.0108209
$320$	12.3459	0.690155
$321$	8.53190	0.476204
$322$	14.1732	0.789843
$323$	−20.8722	−1.16136
$324$	2.78738	0.154854
$325$	6.29801	0.349351
$326$	36.0691	1.99768
$327$	−4.90026	−0.270985
$328$	17.8661	0.986491
$329$	−0.648480	−0.0357518
$330$	−0.153211	−0.00843400
$331$	−22.2466	−1.22279	−0.611393	−	0.791327i	$-0.709390\pi$
$331$	−22.2466	−1.22279	−0.611393	+	0.791327i	$0.709390\pi$
$332$	4.98535	0.273606
$333$	−2.95494	−0.161930
$334$	5.39493	0.295198
$335$	3.49951	0.191199
$336$	−3.16895	−0.172880
$337$	5.13082	0.279494	0.139747	−	0.990187i	$-0.455371\pi$
$337$	5.13082	0.279494	0.139747	+	0.990187i	$0.455371\pi$
$338$	−23.1149	−1.25728
$339$	8.58547	0.466299
$340$	−9.32967	−0.505973
$341$	0.0900751	0.00487784
$342$	−13.3998	−0.724580
$343$	−19.1663	−1.03488
$344$	−8.48655	−0.457564
$345$	−3.62413	−0.195116
$346$	−40.5596	−2.18049
$347$	−30.7748	−1.65208	−0.826039	−	0.563613i	$-0.809411\pi$
$347$	−30.7748	−1.65208	−0.826039	+	0.563613i	$0.809411\pi$
$348$	−7.55559	−0.405022
$349$	−29.0567	−1.55537	−0.777685	−	0.628654i	$-0.783606\pi$
$349$	−29.0567	−1.55537	−0.777685	+	0.628654i	$0.783606\pi$
$350$	−15.4994	−0.828478
$351$	−1.56066	−0.0833017
$352$	−0.527301	−0.0281052
$353$	−33.7641	−1.79708	−0.898540	−	0.438891i	$-0.855372\pi$
$353$	−33.7641	−1.79708	−0.898540	+	0.438891i	$0.855372\pi$
$354$	−3.58897	−0.190752
$355$	−2.94422	−0.156263
$356$	21.0623	1.11630
$357$	5.98256	0.316631
$358$	10.4290	0.551191
$359$	−18.8846	−0.996693	−0.498346	−	0.866978i	$-0.666059\pi$
$359$	−18.8846	−0.996693	−0.498346	+	0.866978i	$0.666059\pi$
$360$	−1.69194	−0.0891731
$361$	18.5060	0.974001
$362$	17.7314	0.931942
$363$	−10.9949	−0.577083
$364$	−7.63612	−0.400241
$365$	−9.82469	−0.514248
$366$	20.3543	1.06394
$367$	21.9289	1.14468	0.572340	−	0.820016i	$-0.306036\pi$
$367$	21.9289	1.14468	0.572340	+	0.820016i	$0.306036\pi$
$368$	−6.66185	−0.347273
$369$	10.3705	0.539865
$370$	6.34967	0.330104
$371$	−11.8493	−0.615184
$372$	3.52138	0.182575
$373$	−22.0932	−1.14394	−0.571972	−	0.820273i	$-0.693821\pi$
$373$	−22.0932	−1.14394	−0.571972	+	0.820273i	$0.693821\pi$
$374$	0.531685	0.0274928
$375$	8.87371	0.458236
$376$	−0.636441	−0.0328220
$377$	4.23038	0.217876
$378$	3.84077	0.197548
$379$	−35.8391	−1.84093	−0.920467	−	0.390821i	$-0.872191\pi$
$379$	−35.8391	−1.84093	−0.920467	+	0.390821i	$0.872191\pi$
$380$	16.7649	0.860019
$381$	17.0084	0.871369
$382$	40.3048	2.06217
$383$	−29.6573	−1.51542	−0.757710	−	0.652592i	$-0.773682\pi$
$383$	−29.6573	−1.51542	−0.757710	+	0.652592i	$0.773682\pi$
$384$	−12.7142	−0.648821
$385$	−0.122917	−0.00626442
$386$	−58.7521	−2.99040
$387$	−4.92606	−0.250406
$388$	−9.86604	−0.500872
$389$	34.5039	1.74942	0.874708	−	0.484651i	$-0.161053\pi$
$389$	34.5039	1.74942	0.874708	+	0.484651i	$0.161053\pi$
$390$	3.35359	0.169815
$391$	12.5767	0.636032
$392$	−6.75102	−0.340978
$393$	−1.00000	−0.0504433
$394$	−37.6529	−1.89693
$395$	−7.32953	−0.368789
$396$	0.198739	0.00998703
$397$	−36.1895	−1.81630	−0.908150	−	0.418646i	$-0.862505\pi$
$397$	−36.1895	−1.81630	−0.908150	+	0.418646i	$0.862505\pi$
$398$	19.1018	0.957487
$399$	−10.7503	−0.538188
$400$	7.28520	0.364260
$401$	−39.2359	−1.95935	−0.979673	−	0.200602i	$-0.935710\pi$
$401$	−39.2359	−1.95935	−0.979673	+	0.200602i	$0.935710\pi$
$402$	−7.79656	−0.388857
$403$	−1.97162	−0.0982135
$404$	25.0442	1.24600
$405$	−0.982095	−0.0488007
$406$	−10.4110	−0.516688
$407$	−0.210687	−0.0104434
$408$	5.87150	0.290682
$409$	−26.0833	−1.28974	−0.644869	−	0.764293i	$-0.723088\pi$
$409$	−26.0833	−1.28974	−0.644869	+	0.764293i	$0.723088\pi$
$410$	−22.2844	−1.10055
$411$	−3.57762	−0.176471
$412$	−16.5430	−0.815015
$413$	−2.87933	−0.141682
$414$	8.07419	0.396825
$415$	−1.75652	−0.0862242
$416$	11.5419	0.565889
$417$	−11.1841	−0.547687
$418$	−0.955405	−0.0467304
$419$	−5.32441	−0.260115	−0.130057	−	0.991506i	$-0.541516\pi$
$419$	−5.32441	−0.260115	−0.130057	+	0.991506i	$0.541516\pi$
$420$	−4.80528	−0.234474
$421$	−6.52344	−0.317933	−0.158967	−	0.987284i	$-0.550816\pi$
$421$	−6.52344	−0.317933	−0.158967	+	0.987284i	$0.550816\pi$
$422$	−19.1947	−0.934383
$423$	−0.369425	−0.0179621
$424$	−11.6293	−0.564770
$425$	−13.7535	−0.667143
$426$	6.55942	0.317805
$427$	16.3296	0.790247
$428$	23.7816	1.14953
$429$	−0.111274	−0.00537238
$430$	10.5853	0.510467
$431$	4.10434	0.197699	0.0988495	−	0.995102i	$-0.468484\pi$
$431$	4.10434	0.197699	0.0988495	+	0.995102i	$0.468484\pi$
$432$	−1.80528	−0.0868567
$433$	−38.0982	−1.83088	−0.915441	−	0.402453i	$-0.868158\pi$
$433$	−38.0982	−1.83088	−0.915441	+	0.402453i	$0.868158\pi$
$434$	4.85216	0.232911
$435$	2.66211	0.127638
$436$	−13.6589	−0.654141
$437$	−22.5996	−1.08108
$438$	21.8884	1.04587
$439$	16.7089	0.797472	0.398736	−	0.917066i	$-0.369449\pi$
$439$	16.7089	0.797472	0.398736	+	0.917066i	$0.369449\pi$
$440$	−0.120635	−0.00575105
$441$	−3.91866	−0.186603
$442$	−11.6379	−0.553557
$443$	27.0469	1.28504	0.642519	−	0.766270i	$-0.277889\pi$
$443$	27.0469	1.28504	0.642519	+	0.766270i	$0.277889\pi$
$444$	−8.23654	−0.390889
$445$	−7.42102	−0.351790
$446$	21.1297	1.00052
$447$	10.1204	0.478677
$448$	−22.0667	−1.04255
$449$	39.0600	1.84335	0.921677	−	0.387959i	$-0.126820\pi$
$449$	39.0600	1.84335	0.921677	+	0.387959i	$0.126820\pi$
$450$	−8.82968	−0.416235
$451$	0.739412	0.0348176
$452$	23.9309	1.12562
$453$	3.45909	0.162522
$454$	15.8677	0.744709
$455$	2.69048	0.126132
$456$	−10.5507	−0.494083
$457$	−27.6221	−1.29211	−0.646053	−	0.763292i	$-0.723582\pi$
$457$	−27.6221	−1.29211	−0.646053	+	0.763292i	$0.723582\pi$
$458$	2.50400	0.117004
$459$	3.40814	0.159078
$460$	−10.1018	−0.470999
$461$	−20.6202	−0.960378	−0.480189	−	0.877165i	$-0.659432\pi$
$461$	−20.6202	−0.960378	−0.480189	+	0.877165i	$0.659432\pi$
$462$	0.273846	0.0127405
$463$	−16.8533	−0.783241	−0.391621	−	0.920127i	$-0.628085\pi$
$463$	−16.8533	−0.783241	−0.391621	+	0.920127i	$0.628085\pi$
$464$	4.89348	0.227174
$465$	−1.24071	−0.0575365
$466$	−21.8007	−1.00990
$467$	4.84225	0.224073	0.112036	−	0.993704i	$-0.464263\pi$
$467$	4.84225	0.224073	0.112036	+	0.993704i	$0.464263\pi$
$468$	−4.35014	−0.201085
$469$	−6.25495	−0.288827
$470$	0.793833	0.0366168
$471$	0.374706	0.0172655
$472$	−2.82588	−0.130071
$473$	−0.351227	−0.0161494
$474$	16.3294	0.750036
$475$	24.7142	1.13397
$476$	16.6756	0.764327
$477$	−6.75029	−0.309075
$478$	28.3130	1.29501
$479$	4.89854	0.223820	0.111910	−	0.993718i	$-0.464303\pi$
$479$	4.89854	0.223820	0.111910	+	0.993718i	$0.464303\pi$
$480$	7.26313	0.331515
$481$	4.61165	0.210273
$482$	−17.2617	−0.786248
$483$	6.47768	0.294745
$484$	−30.6470	−1.39304
$485$	3.47617	0.157845
$486$	2.18801	0.0992500
$487$	40.7307	1.84568	0.922842	−	0.385179i	$-0.125860\pi$
$487$	40.7307	1.84568	0.922842	+	0.385179i	$0.125860\pi$
$488$	16.0265	0.725485
$489$	16.4849	0.745473
$490$	8.42054	0.380401
$491$	−5.80655	−0.262046	−0.131023	−	0.991379i	$-0.541826\pi$
$491$	−5.80655	−0.262046	−0.131023	+	0.991379i	$0.541826\pi$
$492$	28.9064	1.30320
$493$	−9.23825	−0.416070
$494$	20.9125	0.940899
$495$	−0.0700231	−0.00314731
$496$	−2.28067	−0.102405
$497$	5.26243	0.236052
$498$	3.91335	0.175361
$499$	−26.4723	−1.18506	−0.592532	−	0.805547i	$-0.701871\pi$
$499$	−26.4723	−1.18506	−0.592532	+	0.805547i	$0.701871\pi$
$500$	24.7344	1.10615
$501$	2.46568	0.110159
$502$	28.5742	1.27533
$503$	7.07069	0.315266	0.157633	−	0.987498i	$-0.449614\pi$
$503$	7.07069	0.315266	0.157633	+	0.987498i	$0.449614\pi$
$504$	3.02414	0.134706
$505$	−8.82400	−0.392663
$506$	0.575688	0.0255924
$507$	−10.5644	−0.469179
$508$	47.4090	2.10343
$509$	12.1085	0.536700	0.268350	−	0.963322i	$-0.413522\pi$
$509$	12.1085	0.536700	0.268350	+	0.963322i	$0.413522\pi$
$510$	−7.32351	−0.324291
$511$	17.5604	0.776827
$512$	19.5713	0.864937
$513$	−6.12422	−0.270391
$514$	14.3052	0.630976
$515$	5.82870	0.256843
$516$	−13.7308	−0.604464
$517$	−0.0263399	−0.00115843
$518$	−11.3493	−0.498658
$519$	−18.5372	−0.813693
$520$	2.64054	0.115795
$521$	15.2282	0.667161	0.333580	−	0.942722i	$-0.391743\pi$
$521$	15.2282	0.667161	0.333580	+	0.942722i	$0.391743\pi$
$522$	−5.93091	−0.259589
$523$	21.6263	0.945650	0.472825	−	0.881156i	$-0.343234\pi$
$523$	21.6263	0.945650	0.472825	+	0.881156i	$0.343234\pi$
$524$	−2.78738	−0.121767
$525$	−7.08379	−0.309162
$526$	−49.6140	−2.16327
$527$	4.30560	0.187555
$528$	−0.128716	−0.00560165
$529$	−9.38242	−0.407931
$530$	14.5052	0.630067
$531$	−1.64029	−0.0711826
$532$	−29.9651	−1.29915
$533$	−16.1847	−0.701039
$534$	16.5333	0.715465
$535$	−8.37914	−0.362262
$536$	−6.13883	−0.265157
$537$	4.76645	0.205687
$538$	−22.5942	−0.974106
$539$	−0.279400	−0.0120346
$540$	−2.73747	−0.117802
$541$	−5.65011	−0.242917	−0.121459	−	0.992596i	$-0.538757\pi$
$541$	−5.65011	−0.242917	−0.121459	+	0.992596i	$0.538757\pi$
$542$	52.7155	2.26432
$543$	8.10391	0.347772
$544$	−25.2050	−1.08066
$545$	4.81252	0.206146
$546$	−5.99413	−0.256525
$547$	18.2931	0.782155	0.391077	−	0.920358i	$-0.372103\pi$
$547$	18.2931	0.782155	0.391077	+	0.920358i	$0.372103\pi$
$548$	−9.97218	−0.425990
$549$	9.30265	0.397028
$550$	−0.629554	−0.0268443
$551$	16.6006	0.707208
$552$	6.35743	0.270590
$553$	13.1006	0.557096
$554$	17.5513	0.745684
$555$	2.90203	0.123184
$556$	−31.1743	−1.32208
$557$	4.06424	0.172208	0.0861038	−	0.996286i	$-0.472558\pi$
$557$	4.06424	0.172208	0.0861038	+	0.996286i	$0.472558\pi$
$558$	2.76417	0.117017
$559$	7.68789	0.325163
$560$	3.11221	0.131515
$561$	0.243000	0.0102594
$562$	−70.4572	−2.97206
$563$	−12.4894	−0.526366	−0.263183	−	0.964746i	$-0.584772\pi$
$563$	−12.4894	−0.526366	−0.263183	+	0.964746i	$0.584772\pi$
$564$	−1.02973	−0.0433594
$565$	−8.43175	−0.354726
$566$	4.82931	0.202991
$567$	1.75537	0.0737188
$568$	5.16474	0.216708
$569$	26.2283	1.09955	0.549773	−	0.835314i	$-0.314714\pi$
$569$	26.2283	1.09955	0.549773	+	0.835314i	$0.314714\pi$
$570$	13.1599	0.551208
$571$	3.01974	0.126372	0.0631861	−	0.998002i	$-0.479874\pi$
$571$	3.01974	0.126372	0.0631861	+	0.998002i	$0.479874\pi$
$572$	−0.310164	−0.0129686
$573$	18.4208	0.769538
$574$	39.8306	1.66250
$575$	−14.8918	−0.621030
$576$	−12.5709	−0.523789
$577$	8.82989	0.367593	0.183796	−	0.982964i	$-0.441161\pi$
$577$	8.82989	0.367593	0.183796	+	0.982964i	$0.441161\pi$
$578$	−11.7815	−0.490047
$579$	−26.8519	−1.11592
$580$	7.42030	0.308111
$581$	3.13956	0.130251
$582$	−7.74455	−0.321022
$583$	−0.481294	−0.0199332
$584$	17.2344	0.713166
$585$	1.53271	0.0633699
$586$	64.3018	2.65628
$587$	23.7243	0.979208	0.489604	−	0.871945i	$-0.337141\pi$
$587$	23.7243	0.979208	0.489604	+	0.871945i	$0.337141\pi$
$588$	−10.9228	−0.450448
$589$	−7.73690	−0.318793
$590$	3.52471	0.145110
$591$	−17.2088	−0.707875
$592$	5.33450	0.219247
$593$	−16.0544	−0.659276	−0.329638	−	0.944107i	$-0.606927\pi$
$593$	−16.0544	−0.659276	−0.329638	+	0.944107i	$0.606927\pi$
$594$	0.156004	0.00640094
$595$	−5.87544	−0.240870
$596$	28.2093	1.15550
$597$	8.73022	0.357304
$598$	−12.6010	−0.515295
$599$	−32.6118	−1.33248	−0.666241	−	0.745736i	$-0.732098\pi$
$599$	−32.6118	−1.33248	−0.666241	+	0.745736i	$0.732098\pi$
$600$	−6.95229	−0.283826
$601$	23.6436	0.964442	0.482221	−	0.876050i	$-0.339830\pi$
$601$	23.6436	0.964442	0.482221	+	0.876050i	$0.339830\pi$
$602$	−18.9199	−0.771116
$603$	−3.56331	−0.145109
$604$	9.64180	0.392319
$605$	10.7981	0.439003
$606$	19.6590	0.798592
$607$	−8.17045	−0.331628	−0.165814	−	0.986157i	$-0.553025\pi$
$607$	−8.17045	−0.331628	−0.165814	+	0.986157i	$0.553025\pi$
$608$	45.2919	1.83683
$609$	−4.75819	−0.192812
$610$	−19.9898	−0.809365
$611$	0.576546	0.0233246
$612$	9.49977	0.384005
$613$	−1.79929	−0.0726728	−0.0363364	−	0.999340i	$-0.511569\pi$
$613$	−1.79929	−0.0726728	−0.0363364	+	0.999340i	$0.511569\pi$
$614$	65.2883	2.63482
$615$	−10.1848	−0.410690
$616$	0.215620	0.00868759
$617$	−8.41418	−0.338742	−0.169371	−	0.985552i	$-0.554174\pi$
$617$	−8.41418	−0.338742	−0.169371	+	0.985552i	$0.554174\pi$
$618$	−12.9858	−0.522364
$619$	40.8408	1.64153	0.820765	−	0.571266i	$-0.193548\pi$
$619$	40.8408	1.64153	0.820765	+	0.571266i	$0.193548\pi$
$620$	−3.45832	−0.138890
$621$	3.69020	0.148083
$622$	−23.9499	−0.960304
$623$	13.2642	0.531417
$624$	2.81743	0.112787
$625$	11.4626	0.458505
$626$	21.7501	0.869310
$627$	−0.436655	−0.0174383
$628$	1.04445	0.0416779
$629$	−10.0709	−0.401551
$630$	−3.77200	−0.150280
$631$	−9.39439	−0.373985	−0.186992	−	0.982361i	$-0.559874\pi$
$631$	−9.39439	−0.373985	−0.186992	+	0.982361i	$0.559874\pi$
$632$	12.8574	0.511441
$633$	−8.77268	−0.348683
$634$	33.7803	1.34159
$635$	−16.7039	−0.662874
$636$	−18.8156	−0.746087
$637$	6.11568	0.242312
$638$	−0.422873	−0.0167417
$639$	2.99790	0.118595
$640$	12.4866	0.493576
$641$	−10.9887	−0.434029	−0.217014	−	0.976168i	$-0.569632\pi$
$641$	−10.9887	−0.434029	−0.217014	+	0.976168i	$0.569632\pi$
$642$	18.6679	0.736762
$643$	12.3351	0.486449	0.243224	−	0.969970i	$-0.421795\pi$
$643$	12.3351	0.486449	0.243224	+	0.969970i	$0.421795\pi$
$644$	18.0557	0.711496
$645$	4.83786	0.190490
$646$	−45.6685	−1.79680
$647$	2.56902	0.100999	0.0504994	−	0.998724i	$-0.483919\pi$
$647$	2.56902	0.100999	0.0504994	+	0.998724i	$0.483919\pi$
$648$	1.72279	0.0676775
$649$	−0.116952	−0.00459079
$650$	13.7801	0.540500
$651$	2.21762	0.0869152
$652$	45.9496	1.79953
$653$	−7.91072	−0.309571	−0.154785	−	0.987948i	$-0.549469\pi$
$653$	−7.91072	−0.309571	−0.154785	+	0.987948i	$0.549469\pi$
$654$	−10.7218	−0.419255
$655$	0.982095	0.0383736
$656$	−18.7216	−0.730957
$657$	10.0038	0.390286
$658$	−1.41888	−0.0553136
$659$	36.6058	1.42596	0.712980	−	0.701185i	$-0.247345\pi$
$659$	36.6058	1.42596	0.712980	+	0.701185i	$0.247345\pi$
$660$	−0.195181	−0.00759741
$661$	29.3593	1.14194	0.570972	−	0.820969i	$-0.306566\pi$
$661$	29.3593	1.14194	0.570972	+	0.820969i	$0.306566\pi$
$662$	−48.6758	−1.89184
$663$	−5.31893	−0.206570
$664$	3.08128	0.119577
$665$	10.5578	0.409414
$666$	−6.46543	−0.250531
$667$	−10.0028	−0.387311
$668$	6.87279	0.265916
$669$	9.65707	0.373364
$670$	7.65696	0.295814
$671$	0.663277	0.0256055
$672$	−12.9820	−0.500790
$673$	30.3127	1.16847	0.584234	−	0.811585i	$-0.301395\pi$
$673$	30.3127	1.16847	0.584234	+	0.811585i	$0.301395\pi$
$674$	11.2263	0.432420
$675$	−4.03549	−0.155326
$676$	−29.4468	−1.13257
$677$	2.05055	0.0788089	0.0394044	−	0.999223i	$-0.487454\pi$
$677$	2.05055	0.0788089	0.0394044	+	0.999223i	$0.487454\pi$
$678$	18.7851	0.721437
$679$	−6.21322	−0.238442
$680$	−5.76637	−0.221130
$681$	7.25213	0.277902
$682$	0.197085	0.00754677
$683$	−20.9734	−0.802526	−0.401263	−	0.915963i	$-0.631429\pi$
$683$	−20.9734	−0.802526	−0.401263	+	0.915963i	$0.631429\pi$
$684$	−17.0705	−0.652707
$685$	3.51356	0.134246
$686$	−41.9361	−1.60113
$687$	1.14442	0.0436623
$688$	8.89293	0.339040
$689$	10.5349	0.401347
$690$	−7.92962	−0.301875
$691$	16.2244	0.617206	0.308603	−	0.951191i	$-0.400139\pi$
$691$	16.2244	0.617206	0.308603	+	0.951191i	$0.400139\pi$
$692$	−51.6702	−1.96421
$693$	0.125158	0.00475435
$694$	−67.3355	−2.55602
$695$	10.9838	0.416640
$696$	−4.66986	−0.177011
$697$	35.3440	1.33875
$698$	−63.5763	−2.40640
$699$	−9.96373	−0.376863
$700$	−19.7452	−0.746299
$701$	24.2437	0.915672	0.457836	−	0.889037i	$-0.348625\pi$
$701$	24.2437	0.915672	0.457836	+	0.889037i	$0.348625\pi$
$702$	−3.41473	−0.128881
$703$	18.0967	0.682530
$704$	−0.896306	−0.0337808
$705$	0.362811	0.0136642
$706$	−73.8761	−2.78036
$707$	15.7718	0.593160
$708$	−4.57211	−0.171831
$709$	22.5966	0.848634	0.424317	−	0.905514i	$-0.360514\pi$
$709$	22.5966	0.848634	0.424317	+	0.905514i	$0.360514\pi$
$710$	−6.44197	−0.241763
$711$	7.46316	0.279890
$712$	13.0179	0.487867
$713$	4.66194	0.174591
$714$	13.0899	0.489877
$715$	0.109282	0.00408692
$716$	13.2859	0.496517
$717$	12.9401	0.483257
$718$	−41.3197	−1.54204
$719$	14.5287	0.541828	0.270914	−	0.962604i	$-0.412674\pi$
$719$	14.5287	0.541828	0.270914	+	0.962604i	$0.412674\pi$
$720$	1.77296	0.0660743
$721$	−10.4181	−0.387990
$722$	40.4913	1.50693
$723$	−7.88922	−0.293403
$724$	22.5887	0.839501
$725$	10.9388	0.406256
$726$	−24.0570	−0.892838
$727$	39.8199	1.47684	0.738419	−	0.674342i	$-0.235573\pi$
$727$	39.8199	1.47684	0.738419	+	0.674342i	$0.235573\pi$
$728$	−4.71964	−0.174921
$729$	1.00000	0.0370370
$730$	−21.4965	−0.795621
$731$	−16.7887	−0.620952
$732$	25.9300	0.958401
$733$	36.0769	1.33253	0.666266	−	0.745714i	$-0.267891\pi$
$733$	36.0769	1.33253	0.666266	+	0.745714i	$0.267891\pi$
$734$	47.9806	1.77100
$735$	3.84850	0.141954
$736$	−27.2911	−1.00596
$737$	−0.254064	−0.00935855
$738$	22.6907	0.835255
$739$	32.0781	1.18001	0.590005	−	0.807399i	$-0.299126\pi$
$739$	32.0781	1.18001	0.590005	+	0.807399i	$0.299126\pi$
$740$	8.08906	0.297360
$741$	9.55780	0.351114
$742$	−25.9263	−0.951785
$743$	25.2462	0.926194	0.463097	−	0.886308i	$-0.346738\pi$
$743$	25.2462	0.926194	0.463097	+	0.886308i	$0.346738\pi$
$744$	2.17645	0.0797924
$745$	−9.93916	−0.364143
$746$	−48.3401	−1.76986
$747$	1.78854	0.0654394
$748$	0.677331	0.0247657
$749$	14.9767	0.547236
$750$	19.4157	0.708963
$751$	−19.9241	−0.727043	−0.363521	−	0.931586i	$-0.618426\pi$
$751$	−19.9241	−0.727043	−0.363521	+	0.931586i	$0.618426\pi$
$752$	0.666917	0.0243200
$753$	13.0595	0.475914
$754$	9.25611	0.337088
$755$	−3.39716	−0.123635
$756$	4.89289	0.177953
$757$	11.4549	0.416334	0.208167	−	0.978093i	$-0.433250\pi$
$757$	11.4549	0.416334	0.208167	+	0.978093i	$0.433250\pi$
$758$	−78.4163	−2.84821
$759$	0.263111	0.00955031
$760$	10.3618	0.375862
$761$	−42.8720	−1.55411	−0.777055	−	0.629433i	$-0.783287\pi$
$761$	−42.8720	−1.55411	−0.777055	+	0.629433i	$0.783287\pi$
$762$	37.2146	1.34814
$763$	−8.60178	−0.311405
$764$	51.3456	1.85762
$765$	−3.34712	−0.121015
$766$	−64.8905	−2.34459
$767$	2.55993	0.0924338
$768$	−2.67695	−0.0965961
$769$	−40.3136	−1.45374	−0.726872	−	0.686773i	$-0.759027\pi$
$769$	−40.3136	−1.45374	−0.726872	+	0.686773i	$0.759027\pi$
$770$	−0.268943	−0.00969203
$771$	6.53801	0.235461
$772$	−74.8462	−2.69378
$773$	−29.6754	−1.06735	−0.533675	−	0.845689i	$-0.679190\pi$
$773$	−29.6754	−1.06735	−0.533675	+	0.845689i	$0.679190\pi$
$774$	−10.7783	−0.387416
$775$	−5.09815	−0.183131
$776$	−6.09788	−0.218901
$777$	−5.18703	−0.186084
$778$	75.4947	2.70662
$779$	−63.5110	−2.27552
$780$	4.27225	0.152971
$781$	0.213749	0.00764855
$782$	27.5179	0.984040
$783$	−2.71064	−0.0968705
$784$	7.07429	0.252653
$785$	−0.367996	−0.0131344
$786$	−2.18801	−0.0780436
$787$	−23.6101	−0.841607	−0.420804	−	0.907152i	$-0.638252\pi$
$787$	−23.6101	−0.841607	−0.420804	+	0.907152i	$0.638252\pi$
$788$	−47.9674	−1.70877
$789$	−22.6754	−0.807267
$790$	−16.0371	−0.570573
$791$	15.0707	0.535853
$792$	0.122834	0.00436473
$793$	−14.5182	−0.515558
$794$	−79.1829	−2.81010
$795$	6.62943	0.235122
$796$	24.3344	0.862511
$797$	50.0334	1.77227	0.886137	−	0.463423i	$-0.153379\pi$
$797$	50.0334	1.77227	0.886137	+	0.463423i	$0.153379\pi$
$798$	−23.5217	−0.832660
$799$	−1.25905	−0.0445421
$800$	29.8447	1.05517
$801$	7.55631	0.266989
$802$	−85.8484	−3.03141
$803$	0.713269	0.0251707
$804$	−9.93230	−0.350285
$805$	−6.36170	−0.224220
$806$	−4.31393	−0.151952
$807$	−10.3264	−0.363506
$808$	15.4790	0.544551
$809$	10.2664	0.360946	0.180473	−	0.983580i	$-0.442237\pi$
$809$	10.2664	0.360946	0.180473	+	0.983580i	$0.442237\pi$
$810$	−2.14883	−0.0755022
$811$	−30.2517	−1.06228	−0.531141	−	0.847284i	$-0.678236\pi$
$811$	−30.2517	−1.06228	−0.531141	+	0.847284i	$0.678236\pi$
$812$	−13.2629	−0.465436
$813$	24.0929	0.844975
$814$	−0.460984	−0.0161575
$815$	−16.1897	−0.567102
$816$	−6.15265	−0.215386
$817$	30.1683	1.05545
$818$	−57.0705	−1.99542
$819$	−2.73954	−0.0957271
$820$	−28.3888	−0.991381
$821$	−33.4354	−1.16690	−0.583452	−	0.812148i	$-0.698298\pi$
$821$	−33.4354	−1.16690	−0.583452	+	0.812148i	$0.698298\pi$
$822$	−7.82786	−0.273028
$823$	43.2687	1.50825	0.754126	−	0.656730i	$-0.228061\pi$
$823$	43.2687	1.50825	0.754126	+	0.656730i	$0.228061\pi$
$824$	−10.2247	−0.356194
$825$	−0.287730	−0.0100175
$826$	−6.29999	−0.219205
$827$	5.45949	0.189845	0.0949225	−	0.995485i	$-0.469740\pi$
$827$	5.45949	0.189845	0.0949225	+	0.995485i	$0.469740\pi$
$828$	10.2860	0.357463
$829$	−43.3123	−1.50430	−0.752149	−	0.658993i	$-0.770983\pi$
$829$	−43.3123	−1.50430	−0.752149	+	0.658993i	$0.770983\pi$
$830$	−3.84328	−0.133402
$831$	8.02160	0.278266
$832$	19.6189	0.680164
$833$	−13.3553	−0.462735
$834$	−24.4709	−0.847356
$835$	−2.42154	−0.0838007
$836$	−1.21712	−0.0420951
$837$	1.26333	0.0436670
$838$	−11.6499	−0.402437
$839$	−13.6741	−0.472082	−0.236041	−	0.971743i	$-0.575850\pi$
$839$	−13.6741	−0.472082	−0.236041	+	0.971743i	$0.575850\pi$
$840$	−2.96999	−0.102474
$841$	−21.6524	−0.746635
$842$	−14.2733	−0.491892
$843$	−32.2015	−1.10908
$844$	−24.4528	−0.841699
$845$	10.3752	0.356918
$846$	−0.808306	−0.0277901
$847$	−19.3002	−0.663162
$848$	12.1862	0.418475
$849$	2.20717	0.0757500
$850$	−30.0928	−1.03217
$851$	−10.9043	−0.373796
$852$	8.35627	0.286281
$853$	37.9455	1.29923	0.649614	−	0.760264i	$-0.274931\pi$
$853$	37.9455	1.29923	0.649614	+	0.760264i	$0.274931\pi$
$854$	35.7294	1.22263
$855$	6.01456	0.205694
$856$	14.6987	0.502390
$857$	−0.409912	−0.0140023	−0.00700117	−	0.999975i	$-0.502229\pi$
$857$	−0.409912	−0.0140023	−0.00700117	+	0.999975i	$0.502229\pi$
$858$	−0.243469	−0.00831191
$859$	−43.2893	−1.47701	−0.738505	−	0.674248i	$-0.764468\pi$
$859$	−43.2893	−1.47701	−0.738505	+	0.674248i	$0.764468\pi$
$860$	13.4849	0.459833
$861$	18.2041	0.620392
$862$	8.98032	0.305871
$863$	−21.8204	−0.742776	−0.371388	−	0.928478i	$-0.621118\pi$
$863$	−21.8204	−0.742776	−0.371388	+	0.928478i	$0.621118\pi$
$864$	−7.39555	−0.251602
$865$	18.2053	0.618999
$866$	−83.3591	−2.83266
$867$	−5.38459	−0.182870
$868$	6.18133	0.209808
$869$	0.532122	0.0180510
$870$	5.82472	0.197476
$871$	5.56111	0.188431
$872$	−8.44210	−0.285886
$873$	−3.53954	−0.119795
$874$	−49.4481	−1.67261
$875$	15.5767	0.526588
$876$	27.8844	0.942126
$877$	−7.88534	−0.266269	−0.133135	−	0.991098i	$-0.542504\pi$
$877$	−7.88534	−0.266269	−0.133135	+	0.991098i	$0.542504\pi$
$878$	36.5592	1.23381
$879$	29.3883	0.991242
$880$	0.126412	0.00426133
$881$	46.7497	1.57504	0.787519	−	0.616291i	$-0.211365\pi$
$881$	46.7497	1.57504	0.787519	+	0.616291i	$0.211365\pi$
$882$	−8.57406	−0.288704
$883$	8.40810	0.282955	0.141478	−	0.989941i	$-0.454815\pi$
$883$	8.40810	0.282955	0.141478	+	0.989941i	$0.454815\pi$
$884$	−14.8259	−0.498648
$885$	1.61092	0.0541506
$886$	59.1788	1.98815
$887$	5.24750	0.176194	0.0880968	−	0.996112i	$-0.471922\pi$
$887$	5.24750	0.176194	0.0880968	+	0.996112i	$0.471922\pi$
$888$	−5.09074	−0.170834
$889$	29.8562	1.00134
$890$	−16.2372	−0.544274
$891$	0.0712998	0.00238863
$892$	26.9179	0.901278
$893$	2.26244	0.0757097
$894$	22.1434	0.740587
$895$	−4.68111	−0.156472
$896$	−22.3182	−0.745600
$897$	−5.75914	−0.192292
$898$	85.4635	2.85195
$899$	−3.42444	−0.114211
$900$	−11.2484	−0.374948
$901$	−23.0059	−0.766438
$902$	1.61784	0.0538682
$903$	−8.64708	−0.287757
$904$	14.7909	0.491939
$905$	−7.95881	−0.264560
$906$	7.56852	0.251447
$907$	−34.5602	−1.14755	−0.573776	−	0.819012i	$-0.694522\pi$
$907$	−34.5602	−1.14755	−0.573776	+	0.819012i	$0.694522\pi$
$908$	20.2144	0.670840
$909$	8.98488	0.298010
$910$	5.88680	0.195146
$911$	−30.5958	−1.01368	−0.506842	−	0.862039i	$-0.669187\pi$
$911$	−30.5958	−1.01368	−0.506842	+	0.862039i	$0.669187\pi$
$912$	11.0559	0.366099
$913$	0.127523	0.00422039
$914$	−60.4373	−1.99909
$915$	−9.13609	−0.302030
$916$	3.18992	0.105398
$917$	−1.75537	−0.0579676
$918$	7.45703	0.246119
$919$	52.8058	1.74190	0.870952	−	0.491368i	$-0.163503\pi$
$919$	52.8058	1.74190	0.870952	+	0.491368i	$0.163503\pi$
$920$	−6.24360	−0.205845
$921$	29.8391	0.983233
$922$	−45.1171	−1.48585
$923$	−4.67868	−0.154001
$924$	0.348862	0.0114767
$925$	11.9246	0.392080
$926$	−36.8752	−1.21180
$927$	−5.93497	−0.194930
$928$	20.0467	0.658065
$929$	−27.0415	−0.887203	−0.443602	−	0.896224i	$-0.646300\pi$
$929$	−27.0415	−0.887203	−0.443602	+	0.896224i	$0.646300\pi$
$930$	−2.71468	−0.0890179
$931$	23.9987	0.786527
$932$	−27.7727	−0.909725
$933$	−10.9460	−0.358356
$934$	10.5949	0.346675
$935$	−0.238649	−0.00780464
$936$	−2.68868	−0.0878822
$937$	32.3989	1.05843	0.529213	−	0.848489i	$-0.322487\pi$
$937$	32.3989	1.05843	0.529213	+	0.848489i	$0.322487\pi$
$938$	−13.6859	−0.446860
$939$	9.94061	0.324400
$940$	1.01129	0.0329847
$941$	−21.2895	−0.694019	−0.347010	−	0.937862i	$-0.612803\pi$
$941$	−21.2895	−0.694019	−0.347010	+	0.937862i	$0.612803\pi$
$942$	0.819859	0.0267124
$943$	38.2691	1.24621
$944$	2.96119	0.0963786
$945$	−1.72394	−0.0560799
$946$	−0.768487	−0.0249857
$947$	12.4200	0.403596	0.201798	−	0.979427i	$-0.435322\pi$
$947$	12.4200	0.403596	0.201798	+	0.979427i	$0.435322\pi$
$948$	20.8026	0.675638
$949$	−15.6125	−0.506803
$950$	54.0749	1.75442
$951$	15.4388	0.500639
$952$	10.3067	0.334041
$953$	44.7344	1.44909	0.724545	−	0.689227i	$-0.242050\pi$
$953$	44.7344	1.44909	0.724545	+	0.689227i	$0.242050\pi$
$954$	−14.7697	−0.478186
$955$	−18.0909	−0.585409
$956$	36.0689	1.16655
$957$	−0.193268	−0.00624748
$958$	10.7180	0.346284
$959$	−6.28006	−0.202794
$960$	12.3459	0.398461
$961$	−29.4040	−0.948516
$962$	10.0903	0.325325
$963$	8.53190	0.274937
$964$	−21.9902	−0.708258
$965$	26.3711	0.848915
$966$	14.1732	0.456016
$967$	39.8071	1.28011	0.640055	−	0.768329i	$-0.278912\pi$
$967$	39.8071	1.28011	0.640055	+	0.768329i	$0.278912\pi$
$968$	−18.9419	−0.608816
$969$	−20.8722	−0.670511
$970$	7.60588	0.244210
$971$	−30.2762	−0.971610	−0.485805	−	0.874067i	$-0.661474\pi$
$971$	−30.2762	−0.971610	−0.485805	+	0.874067i	$0.661474\pi$
$972$	2.78738	0.0894052
$973$	−19.6322	−0.629381
$974$	89.1191	2.85556
$975$	6.29801	0.201698
$976$	−16.7939	−0.537560
$977$	5.16038	0.165095	0.0825476	−	0.996587i	$-0.473694\pi$
$977$	5.16038	0.165095	0.0825476	+	0.996587i	$0.473694\pi$
$978$	36.0691	1.15336
$979$	0.538764	0.0172190
$980$	10.7272	0.342668
$981$	−4.90026	−0.156453
$982$	−12.7048	−0.405426
$983$	56.1268	1.79017	0.895083	−	0.445899i	$-0.147116\pi$
$983$	56.1268	1.79017	0.895083	+	0.445899i	$0.147116\pi$
$984$	17.8661	0.569551
$985$	16.9007	0.538500
$986$	−20.2134	−0.643725
$987$	−0.648480	−0.0206413
$988$	26.6412	0.847569
$989$	−18.1782	−0.578032
$990$	−0.153211	−0.00486937
$991$	30.7459	0.976676	0.488338	−	0.872655i	$-0.337603\pi$
$991$	30.7459	0.976676	0.488338	+	0.872655i	$0.337603\pi$
$992$	−9.34301	−0.296641
$993$	−22.2466	−0.705975
$994$	11.5142	0.365209
$995$	−8.57391	−0.271811
$996$	4.98535	0.157967
$997$	−27.5857	−0.873649	−0.436824	−	0.899547i	$-0.643897\pi$
$997$	−27.5857	−0.873649	−0.436824	+	0.899547i	$0.643897\pi$
$998$	−57.9216	−1.83348
$999$	−2.95494	−0.0934902

Display

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a_n

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a_n

instead) (See

a_n

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a_n

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a_p

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	393.2.a.e.1.6	✓	6
3.2	odd	2		1179.2.a.f.1.1		6
4.3	odd	2		6288.2.a.bj.1.1		6
5.4	even	2		9825.2.a.r.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
393.2.a.e.1.6	✓	6	1.1	even	1	trivial
1179.2.a.f.1.1		6	3.2	odd	2
6288.2.a.bj.1.1		6	4.3	odd	2
9825.2.a.r.1.1		6	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 $\F_{q}$

Introduction

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Fields

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Groups

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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	393.2.a.e.1.6

Level	$393$
Weight	$2$
Character	393.1
Self dual	yes
Analytic conductor	$3.138$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$1$