LMFDB - Embedded newform 1503.2.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,2,Mod(1,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, 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("1503.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1503.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:	$12.0015154238$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 28x^{5} + 9x^{4} - 64x^{3} + 17x^{2} + 23x + 1 $ x^8 - 3x^7 - 8x^6 + 28x^5 + 9x^4 - 64x^3 + 17x^2 + 23*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.0452510$ of defining polynomial
Character	$\chi$	$=$	1503.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0452510 q^{2} -1.99795 q^{4} -1.52778 q^{5} +0.428515 q^{7} -0.180911 q^{8} +O(q^{10})$	\(q+0.0452510 q^{2} -1.99795 q^{4} -1.52778 q^{5} +0.428515 q^{7} -0.180911 q^{8} -0.0691336 q^{10} +1.35070 q^{11} +5.07380 q^{13} +0.0193907 q^{14} +3.98772 q^{16} -3.18336 q^{17} -2.82446 q^{19} +3.05243 q^{20} +0.0611203 q^{22} -7.39390 q^{23} -2.66589 q^{25} +0.229595 q^{26} -0.856153 q^{28} +1.58226 q^{29} +9.02675 q^{31} +0.542271 q^{32} -0.144050 q^{34} -0.654678 q^{35} -4.97987 q^{37} -0.127810 q^{38} +0.276393 q^{40} +9.59877 q^{41} +4.94401 q^{43} -2.69862 q^{44} -0.334581 q^{46} -10.0293 q^{47} -6.81637 q^{49} -0.120634 q^{50} -10.1372 q^{52} -9.53888 q^{53} -2.06357 q^{55} -0.0775233 q^{56} +0.0715989 q^{58} -7.81712 q^{59} -11.5115 q^{61} +0.408469 q^{62} -7.95090 q^{64} -7.75166 q^{65} -6.88984 q^{67} +6.36019 q^{68} -0.0296248 q^{70} +10.6598 q^{71} -13.5809 q^{73} -0.225344 q^{74} +5.64314 q^{76} +0.578794 q^{77} -10.8615 q^{79} -6.09236 q^{80} +0.434354 q^{82} -10.2803 q^{83} +4.86347 q^{85} +0.223722 q^{86} -0.244356 q^{88} +8.75932 q^{89} +2.17420 q^{91} +14.7727 q^{92} -0.453834 q^{94} +4.31516 q^{95} +2.00584 q^{97} -0.308448 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{2} + 9 q^{4} - 7 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - 3 * q^2 + 9 * q^4 - 7 * q^5 - 4 * q^7 - 3 * q^8	$ 8 q - 3 q^{2} + 9 q^{4} - 7 q^{5} - 4 q^{7} - 3 q^{8} - q^{10} - 13 q^{11} - 5 q^{14} + 7 q^{16} - 11 q^{17} + 12 q^{19} - 9 q^{20} - 17 q^{22} - 7 q^{23} - 5 q^{25} - 3 q^{26} - 27 q^{28} - q^{29} - 2 q^{31} - 4 q^{32} - 14 q^{34} + 4 q^{35} - 9 q^{37} - 22 q^{40} - 4 q^{41} + 2 q^{43} - 3 q^{44} - 5 q^{46} - 17 q^{47} - 2 q^{49} + 4 q^{50} - 36 q^{52} - 9 q^{53} + 7 q^{55} - 9 q^{56} - 29 q^{58} - 29 q^{59} - 12 q^{61} + 34 q^{62} - 5 q^{64} - 8 q^{65} - 26 q^{68} + 5 q^{70} - 13 q^{71} - 20 q^{73} + 17 q^{74} + 30 q^{76} + 22 q^{77} + 8 q^{79} + 34 q^{80} + 15 q^{82} - 33 q^{83} - 31 q^{85} - 11 q^{86} - 44 q^{88} - 4 q^{89} + q^{91} + 33 q^{92} - 2 q^{94} - 3 q^{95} - 31 q^{97} + 57 q^{98}+O(q^{100}) $ 8 * q - 3 * q^2 + 9 * q^4 - 7 * q^5 - 4 * q^7 - 3 * q^8 - q^10 - 13 * q^11 - 5 * q^14 + 7 * q^16 - 11 * q^17 + 12 * q^19 - 9 * q^20 - 17 * q^22 - 7 * q^23 - 5 * q^25 - 3 * q^26 - 27 * q^28 - q^29 - 2 * q^31 - 4 * q^32 - 14 * q^34 + 4 * q^35 - 9 * q^37 - 22 * q^40 - 4 * q^41 + 2 * q^43 - 3 * q^44 - 5 * q^46 - 17 * q^47 - 2 * q^49 + 4 * q^50 - 36 * q^52 - 9 * q^53 + 7 * q^55 - 9 * q^56 - 29 * q^58 - 29 * q^59 - 12 * q^61 + 34 * q^62 - 5 * q^64 - 8 * q^65 - 26 * q^68 + 5 * q^70 - 13 * q^71 - 20 * q^73 + 17 * q^74 + 30 * q^76 + 22 * q^77 + 8 * q^79 + 34 * q^80 + 15 * q^82 - 33 * q^83 - 31 * q^85 - 11 * q^86 - 44 * q^88 - 4 * q^89 + q^91 + 33 * q^92 - 2 * q^94 - 3 * q^95 - 31 * q^97 + 57 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	0.0452510	0.0319973	0.0159986	−	0.999872i	$-0.494907\pi$
$2$	0.0452510	0.0319973	0.0159986	+	0.999872i	$0.494907\pi$
$3$	0	0
$3$	0	0
$4$	−1.99795	−0.998976
$5$	−1.52778	−0.683244	−0.341622	−	0.939837i	$-0.610976\pi$
$5$	−1.52778	−0.683244	−0.341622	+	0.939837i	$0.610976\pi$
$6$	0	0
$7$	0.428515	0.161964	0.0809818	−	0.996716i	$-0.474194\pi$
$7$	0.428515	0.161964	0.0809818	+	0.996716i	$0.474194\pi$
$8$	−0.180911	−0.0639618
$9$	0	0
$10$	−0.0691336	−0.0218620
$11$	1.35070	0.407250	0.203625	−	0.979049i	$-0.434728\pi$
$11$	1.35070	0.407250	0.203625	+	0.979049i	$0.434728\pi$
$12$	0	0
$13$	5.07380	1.40722	0.703610	−	0.710587i	$-0.251570\pi$
$13$	5.07380	1.40722	0.703610	+	0.710587i	$0.251570\pi$
$14$	0.0193907	0.00518239
$15$	0	0
$16$	3.98772	0.996930
$17$	−3.18336	−0.772077	−0.386039	−	0.922483i	$-0.626157\pi$
$17$	−3.18336	−0.772077	−0.386039	+	0.922483i	$0.626157\pi$
$18$	0	0
$19$	−2.82446	−0.647976	−0.323988	−	0.946061i	$-0.605024\pi$
$19$	−2.82446	−0.647976	−0.323988	+	0.946061i	$0.605024\pi$
$20$	3.05243	0.682545
$21$	0	0
$22$	0.0611203	0.0130309
$23$	−7.39390	−1.54173	−0.770867	−	0.636996i	$-0.780177\pi$
$23$	−7.39390	−1.54173	−0.770867	+	0.636996i	$0.780177\pi$
$24$	0	0
$25$	−2.66589	−0.533177
$26$	0.229595	0.0450272
$27$	0	0
$28$	−0.856153	−0.161798
$29$	1.58226	0.293819	0.146909	−	0.989150i	$-0.453067\pi$
$29$	1.58226	0.293819	0.146909	+	0.989150i	$0.453067\pi$
$30$	0	0
$31$	9.02675	1.62125	0.810626	−	0.585564i	$-0.199127\pi$
$31$	9.02675	1.62125	0.810626	+	0.585564i	$0.199127\pi$
$32$	0.542271	0.0958608
$33$	0	0
$34$	−0.144050	−0.0247044
$35$	−0.654678	−0.110661
$36$	0	0
$37$	−4.97987	−0.818685	−0.409343	−	0.912381i	$-0.634242\pi$
$37$	−4.97987	−0.818685	−0.409343	+	0.912381i	$0.634242\pi$
$38$	−0.127810	−0.0207335
$39$	0	0
$40$	0.276393	0.0437015
$41$	9.59877	1.49908	0.749538	−	0.661961i	$-0.230276\pi$
$41$	9.59877	1.49908	0.749538	+	0.661961i	$0.230276\pi$
$42$	0	0
$43$	4.94401	0.753955	0.376978	−	0.926222i	$-0.376963\pi$
$43$	4.94401	0.753955	0.376978	+	0.926222i	$0.376963\pi$
$44$	−2.69862	−0.406833
$45$	0	0
$46$	−0.334581	−0.0493313
$47$	−10.0293	−1.46292	−0.731459	−	0.681885i	$-0.761160\pi$
$47$	−10.0293	−1.46292	−0.731459	+	0.681885i	$0.761160\pi$
$48$	0	0
$49$	−6.81637	−0.973768
$50$	−0.120634	−0.0170602
$51$	0	0
$52$	−10.1372	−1.40578
$53$	−9.53888	−1.31027	−0.655133	−	0.755514i	$-0.727387\pi$
$53$	−9.53888	−1.31027	−0.655133	+	0.755514i	$0.727387\pi$
$54$	0	0
$55$	−2.06357	−0.278251
$56$	−0.0775233	−0.0103595
$57$	0	0
$58$	0.0715989	0.00940140
$59$	−7.81712	−1.01770	−0.508851	−	0.860855i	$-0.669930\pi$
$59$	−7.81712	−1.01770	−0.508851	+	0.860855i	$0.669930\pi$
$60$	0	0
$61$	−11.5115	−1.47389	−0.736947	−	0.675951i	$-0.763733\pi$
$61$	−11.5115	−1.47389	−0.736947	+	0.675951i	$0.763733\pi$
$62$	0.408469	0.0518756
$63$	0	0
$64$	−7.95090	−0.993862
$65$	−7.75166	−0.961475
$66$	0	0
$67$	−6.88984	−0.841728	−0.420864	−	0.907124i	$-0.638273\pi$
$67$	−6.88984	−0.841728	−0.420864	+	0.907124i	$0.638273\pi$
$68$	6.36019	0.771287
$69$	0	0
$70$	−0.0296248	−0.00354084
$71$	10.6598	1.26509	0.632544	−	0.774525i	$-0.282011\pi$
$71$	10.6598	1.26509	0.632544	+	0.774525i	$0.282011\pi$
$72$	0	0
$73$	−13.5809	−1.58952	−0.794760	−	0.606923i	$-0.792404\pi$
$73$	−13.5809	−1.58952	−0.794760	+	0.606923i	$0.792404\pi$
$74$	−0.225344	−0.0261957
$75$	0	0
$76$	5.64314	0.647312
$77$	0.578794	0.0659597
$78$	0	0
$79$	−10.8615	−1.22201	−0.611007	−	0.791625i	$-0.709235\pi$
$79$	−10.8615	−1.22201	−0.611007	+	0.791625i	$0.709235\pi$
$80$	−6.09236	−0.681147
$81$	0	0
$82$	0.434354	0.0479664
$83$	−10.2803	−1.12841	−0.564207	−	0.825634i	$-0.690818\pi$
$83$	−10.2803	−1.12841	−0.564207	+	0.825634i	$0.690818\pi$
$84$	0	0
$85$	4.86347	0.527517
$86$	0.223722	0.0241245
$87$	0	0
$88$	−0.244356	−0.0260484
$89$	8.75932	0.928486	0.464243	−	0.885708i	$-0.346326\pi$
$89$	8.75932	0.928486	0.464243	+	0.885708i	$0.346326\pi$
$90$	0	0
$91$	2.17420	0.227918
$92$	14.7727	1.54016
$93$	0	0
$94$	−0.453834	−0.0468094
$95$	4.31516	0.442726
$96$	0	0
$97$	2.00584	0.203662	0.101831	−	0.994802i	$-0.467530\pi$
$97$	2.00584	0.203662	0.101831	+	0.994802i	$0.467530\pi$
$98$	−0.308448	−0.0311579
$99$	0	0
$100$	5.32631	0.532631
$101$	−14.6567	−1.45840	−0.729198	−	0.684303i	$-0.760107\pi$
$101$	−14.6567	−1.45840	−0.729198	+	0.684303i	$0.760107\pi$
$102$	0	0
$103$	−10.2655	−1.01149	−0.505745	−	0.862683i	$-0.668782\pi$
$103$	−10.2655	−1.01149	−0.505745	+	0.862683i	$0.668782\pi$
$104$	−0.917908	−0.0900083
$105$	0	0
$106$	−0.431644	−0.0419249
$107$	−0.182196	−0.0176136	−0.00880679	−	0.999961i	$-0.502803\pi$
$107$	−0.182196	−0.0176136	−0.00880679	+	0.999961i	$0.502803\pi$
$108$	0	0
$109$	5.79915	0.555458	0.277729	−	0.960660i	$-0.410418\pi$
$109$	5.79915	0.555458	0.277729	+	0.960660i	$0.410418\pi$
$110$	−0.0933784	−0.00890328
$111$	0	0
$112$	1.70880	0.161466
$113$	0.650111	0.0611573	0.0305786	−	0.999532i	$-0.490265\pi$
$113$	0.650111	0.0611573	0.0305786	+	0.999532i	$0.490265\pi$
$114$	0	0
$115$	11.2963	1.05338
$116$	−3.16128	−0.293518
$117$	0	0
$118$	−0.353732	−0.0325637
$119$	−1.36412	−0.125048
$120$	0	0
$121$	−9.17562	−0.834148
$122$	−0.520906	−0.0471606
$123$	0	0
$124$	−18.0350	−1.61959
$125$	11.7118	1.04753
$126$	0	0
$127$	4.32909	0.384145	0.192072	−	0.981381i	$-0.438479\pi$
$127$	4.32909	0.384145	0.192072	+	0.981381i	$0.438479\pi$
$128$	−1.44433	−0.127662
$129$	0	0
$130$	−0.350770	−0.0307646
$131$	21.6759	1.89384	0.946918	−	0.321475i	$-0.104179\pi$
$131$	21.6759	1.89384	0.946918	+	0.321475i	$0.104179\pi$
$132$	0	0
$133$	−1.21033	−0.104948
$134$	−0.311772	−0.0269330
$135$	0	0
$136$	0.575905	0.0493834
$137$	−4.33341	−0.370229	−0.185114	−	0.982717i	$-0.559266\pi$
$137$	−4.33341	−0.370229	−0.185114	+	0.982717i	$0.559266\pi$
$138$	0	0
$139$	−16.5086	−1.40024	−0.700120	−	0.714025i	$-0.746870\pi$
$139$	−16.5086	−1.40024	−0.700120	+	0.714025i	$0.746870\pi$
$140$	1.30801	0.110547
$141$	0	0
$142$	0.482367	0.0404794
$143$	6.85316	0.573090
$144$	0	0
$145$	−2.41735	−0.200750
$146$	−0.614548	−0.0508603
$147$	0	0
$148$	9.94954	0.817847
$149$	12.5118	1.02501	0.512505	−	0.858684i	$-0.328718\pi$
$149$	12.5118	1.02501	0.512505	+	0.858684i	$0.328718\pi$
$150$	0	0
$151$	10.0102	0.814618	0.407309	−	0.913290i	$-0.366467\pi$
$151$	10.0102	0.814618	0.407309	+	0.913290i	$0.366467\pi$
$152$	0.510977	0.0414457
$153$	0	0
$154$	0.0261910	0.00211053
$155$	−13.7909	−1.10771
$156$	0	0
$157$	−11.3518	−0.905974	−0.452987	−	0.891517i	$-0.649642\pi$
$157$	−11.3518	−0.905974	−0.452987	+	0.891517i	$0.649642\pi$
$158$	−0.491493	−0.0391011
$159$	0	0
$160$	−0.828471	−0.0654964
$161$	−3.16840	−0.249705
$162$	0	0
$163$	14.5601	1.14043	0.570217	−	0.821494i	$-0.306859\pi$
$163$	14.5601	1.14043	0.570217	+	0.821494i	$0.306859\pi$
$164$	−19.1779	−1.49754
$165$	0	0
$166$	−0.465195	−0.0361062
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	12.7435	0.980267
$170$	0.220077	0.0168791
$171$	0	0
$172$	−9.87791	−0.753183
$173$	2.04279	0.155311	0.0776553	−	0.996980i	$-0.475257\pi$
$173$	2.04279	0.155311	0.0776553	+	0.996980i	$0.475257\pi$
$174$	0	0
$175$	−1.14237	−0.0863553
$176$	5.38619	0.405999
$177$	0	0
$178$	0.396368	0.0297090
$179$	3.86304	0.288737	0.144369	−	0.989524i	$-0.453885\pi$
$179$	3.86304	0.288737	0.144369	+	0.989524i	$0.453885\pi$
$180$	0	0
$181$	−16.3899	−1.21825	−0.609125	−	0.793074i	$-0.708479\pi$
$181$	−16.3899	−1.21825	−0.609125	+	0.793074i	$0.708479\pi$
$182$	0.0983848	0.00729276
$183$	0	0
$184$	1.33764	0.0986121
$185$	7.60815	0.559362
$186$	0	0
$187$	−4.29974	−0.314428
$188$	20.0380	1.46142
$189$	0	0
$190$	0.195265	0.0141660
$191$	−8.93681	−0.646645	−0.323323	−	0.946289i	$-0.604800\pi$
$191$	−8.93681	−0.646645	−0.323323	+	0.946289i	$0.604800\pi$
$192$	0	0
$193$	22.1282	1.59282	0.796410	−	0.604757i	$-0.206730\pi$
$193$	22.1282	1.59282	0.796410	+	0.604757i	$0.206730\pi$
$194$	0.0907662	0.00651663
$195$	0	0
$196$	13.6188	0.972771
$197$	19.9870	1.42401	0.712007	−	0.702172i	$-0.247786\pi$
$197$	19.9870	1.42401	0.712007	+	0.702172i	$0.247786\pi$
$198$	0	0
$199$	11.4337	0.810516	0.405258	−	0.914202i	$-0.367182\pi$
$199$	11.4337	0.810516	0.405258	+	0.914202i	$0.367182\pi$
$200$	0.482289	0.0341030
$201$	0	0
$202$	−0.663230	−0.0466647
$203$	0.678024	0.0475879
$204$	0	0
$205$	−14.6648	−1.02424
$206$	−0.464524	−0.0323649
$207$	0	0
$208$	20.2329	1.40290
$209$	−3.81499	−0.263888
$210$	0	0
$211$	−0.0960463	−0.00661210	−0.00330605	−	0.999995i	$-0.501052\pi$
$211$	−0.0960463	−0.00661210	−0.00330605	+	0.999995i	$0.501052\pi$
$212$	19.0582	1.30892
$213$	0	0
$214$	−0.00824456	−0.000563587 0
$215$	−7.55337	−0.515136
$216$	0	0
$217$	3.86810	0.262584
$218$	0.262417	0.0177731
$219$	0	0
$220$	4.12291	0.277966
$221$	−16.1517	−1.08648
$222$	0	0
$223$	7.46087	0.499617	0.249808	−	0.968295i	$-0.419632\pi$
$223$	7.46087	0.499617	0.249808	+	0.968295i	$0.419632\pi$
$224$	0.232371	0.0155260
$225$	0	0
$226$	0.0294181	0.00195687
$227$	−16.9614	−1.12576	−0.562882	−	0.826537i	$-0.690308\pi$
$227$	−16.9614	−1.12576	−0.562882	+	0.826537i	$0.690308\pi$
$228$	0	0
$229$	22.0619	1.45789	0.728946	−	0.684571i	$-0.240011\pi$
$229$	22.0619	1.45789	0.728946	+	0.684571i	$0.240011\pi$
$230$	0.511167	0.0337053
$231$	0	0
$232$	−0.286249	−0.0187932
$233$	−20.2551	−1.32695	−0.663476	−	0.748197i	$-0.730920\pi$
$233$	−20.2551	−1.32695	−0.663476	+	0.748197i	$0.730920\pi$
$234$	0	0
$235$	15.3225	0.999531
$236$	15.6182	1.01666
$237$	0	0
$238$	−0.0617276	−0.00400121
$239$	−13.3973	−0.866599	−0.433299	−	0.901250i	$-0.642651\pi$
$239$	−13.3973	−0.866599	−0.433299	+	0.901250i	$0.642651\pi$
$240$	0	0
$241$	−8.47987	−0.546236	−0.273118	−	0.961980i	$-0.588055\pi$
$241$	−8.47987	−0.546236	−0.273118	+	0.961980i	$0.588055\pi$
$242$	−0.415206	−0.0266904
$243$	0	0
$244$	22.9994	1.47238
$245$	10.4139	0.665321
$246$	0	0
$247$	−14.3308	−0.911844
$248$	−1.63304	−0.103698
$249$	0	0
$250$	0.529970	0.0335183
$251$	−21.0954	−1.33153	−0.665766	−	0.746160i	$-0.731895\pi$
$251$	−21.0954	−1.33153	−0.665766	+	0.746160i	$0.731895\pi$
$252$	0	0
$253$	−9.98690	−0.627871
$254$	0.195896	0.0122916
$255$	0	0
$256$	15.8364	0.989777
$257$	7.40473	0.461895	0.230947	−	0.972966i	$-0.425817\pi$
$257$	7.40473	0.461895	0.230947	+	0.972966i	$0.425817\pi$
$258$	0	0
$259$	−2.13395	−0.132597
$260$	15.4874	0.960490
$261$	0	0
$262$	0.980858	0.0605976
$263$	−6.45037	−0.397747	−0.198873	−	0.980025i	$-0.563728\pi$
$263$	−6.45037	−0.397747	−0.198873	+	0.980025i	$0.563728\pi$
$264$	0	0
$265$	14.5733	0.895232
$266$	−0.0547684	−0.00335807
$267$	0	0
$268$	13.7656	0.840866
$269$	8.02935	0.489558	0.244779	−	0.969579i	$-0.421285\pi$
$269$	8.02935	0.489558	0.244779	+	0.969579i	$0.421285\pi$
$270$	0	0
$271$	−13.3956	−0.813726	−0.406863	−	0.913489i	$-0.633377\pi$
$271$	−13.3956	−0.813726	−0.406863	+	0.913489i	$0.633377\pi$
$272$	−12.6943	−0.769707
$273$	0	0
$274$	−0.196091	−0.0118463
$275$	−3.60080	−0.217136
$276$	0	0
$277$	5.34307	0.321034	0.160517	−	0.987033i	$-0.448684\pi$
$277$	5.34307	0.321034	0.160517	+	0.987033i	$0.448684\pi$
$278$	−0.747030	−0.0448039
$279$	0	0
$280$	0.118439	0.00707806
$281$	−11.4830	−0.685017	−0.342509	−	0.939515i	$-0.611277\pi$
$281$	−11.4830	−0.685017	−0.342509	+	0.939515i	$0.611277\pi$
$282$	0	0
$283$	13.4572	0.799945	0.399973	−	0.916527i	$-0.369020\pi$
$283$	13.4572	0.799945	0.399973	+	0.916527i	$0.369020\pi$
$284$	−21.2978	−1.26379
$285$	0	0
$286$	0.310112	0.0183373
$287$	4.11322	0.242796
$288$	0	0
$289$	−6.86625	−0.403897
$290$	−0.109387	−0.00642345
$291$	0	0
$292$	27.1339	1.58789
$293$	−23.6428	−1.38123	−0.690614	−	0.723224i	$-0.742659\pi$
$293$	−23.6428	−1.38123	−0.690614	+	0.723224i	$0.742659\pi$
$294$	0	0
$295$	11.9428	0.695339
$296$	0.900914	0.0523646
$297$	0	0
$298$	0.566173	0.0327975
$299$	−37.5152	−2.16956
$300$	0	0
$301$	2.11859	0.122113
$302$	0.452971	0.0260655
$303$	0	0
$304$	−11.2632	−0.645986
$305$	17.5870	1.00703
$306$	0	0
$307$	9.57899	0.546702	0.273351	−	0.961914i	$-0.411868\pi$
$307$	9.57899	0.546702	0.273351	+	0.961914i	$0.411868\pi$
$308$	−1.15640	−0.0658921
$309$	0	0
$310$	−0.624051	−0.0354437
$311$	33.9104	1.92288	0.961441	−	0.275011i	$-0.0886816\pi$
$311$	33.9104	1.92288	0.961441	+	0.275011i	$0.0886816\pi$
$312$	0	0
$313$	2.35053	0.132860	0.0664299	−	0.997791i	$-0.478839\pi$
$313$	2.35053	0.132860	0.0664299	+	0.997791i	$0.478839\pi$
$314$	−0.513681	−0.0289887
$315$	0	0
$316$	21.7008	1.22076
$317$	1.81502	0.101942	0.0509708	−	0.998700i	$-0.483768\pi$
$317$	1.81502	0.101942	0.0509708	+	0.998700i	$0.483768\pi$
$318$	0	0
$319$	2.13715	0.119658
$320$	12.1472	0.679051
$321$	0	0
$322$	−0.143373	−0.00798987
$323$	8.99126	0.500287
$324$	0	0
$325$	−13.5262	−0.750297
$326$	0.658858	0.0364908
$327$	0	0
$328$	−1.73653	−0.0958836
$329$	−4.29769	−0.236939
$330$	0	0
$331$	2.06219	0.113348	0.0566741	−	0.998393i	$-0.481950\pi$
$331$	2.06219	0.113348	0.0566741	+	0.998393i	$0.481950\pi$
$332$	20.5396	1.12726
$333$	0	0
$334$	0.0452510	0.00247602
$335$	10.5262	0.575106
$336$	0	0
$337$	−11.9893	−0.653101	−0.326550	−	0.945180i	$-0.605886\pi$
$337$	−11.9893	−0.653101	−0.326550	+	0.945180i	$0.605886\pi$
$338$	0.576654	0.0313659
$339$	0	0
$340$	−9.71698	−0.526977
$341$	12.1924	0.660255
$342$	0	0
$343$	−5.92053	−0.319679
$344$	−0.894428	−0.0482243
$345$	0	0
$346$	0.0924384	0.00496952
$347$	6.76364	0.363091	0.181546	−	0.983383i	$-0.441890\pi$
$347$	6.76364	0.363091	0.181546	+	0.983383i	$0.441890\pi$
$348$	0	0
$349$	−32.0675	−1.71653	−0.858266	−	0.513205i	$-0.828458\pi$
$349$	−32.0675	−1.71653	−0.858266	+	0.513205i	$0.828458\pi$
$350$	−0.0516935	−0.00276313
$351$	0	0
$352$	0.732442	0.0390393
$353$	8.46900	0.450759	0.225380	−	0.974271i	$-0.427638\pi$
$353$	8.46900	0.450759	0.225380	+	0.974271i	$0.427638\pi$
$354$	0	0
$355$	−16.2859	−0.864364
$356$	−17.5007	−0.927536
$357$	0	0
$358$	0.174806	0.00923881
$359$	−20.7325	−1.09422	−0.547109	−	0.837061i	$-0.684272\pi$
$359$	−20.7325	−1.09422	−0.547109	+	0.837061i	$0.684272\pi$
$360$	0	0
$361$	−11.0224	−0.580127
$362$	−0.741658	−0.0389807
$363$	0	0
$364$	−4.34395	−0.227685
$365$	20.7486	1.08603
$366$	0	0
$367$	−25.1588	−1.31328	−0.656640	−	0.754204i	$-0.728023\pi$
$367$	−25.1588	−1.31328	−0.656640	+	0.754204i	$0.728023\pi$
$368$	−29.4848	−1.53700
$369$	0	0
$370$	0.344276	0.0178981
$371$	−4.08756	−0.212215
$372$	0	0
$373$	−3.84768	−0.199225	−0.0996126	−	0.995026i	$-0.531760\pi$
$373$	−3.84768	−0.199225	−0.0996126	+	0.995026i	$0.531760\pi$
$374$	−0.194568	−0.0100608
$375$	0	0
$376$	1.81441	0.0935709
$377$	8.02808	0.413467
$378$	0	0
$379$	−31.5552	−1.62088	−0.810441	−	0.585820i	$-0.800772\pi$
$379$	−31.5552	−1.62088	−0.810441	+	0.585820i	$0.800772\pi$
$380$	−8.62148	−0.442273
$381$	0	0
$382$	−0.404400	−0.0206909
$383$	−1.53734	−0.0785546	−0.0392773	−	0.999228i	$-0.512506\pi$
$383$	−1.53734	−0.0785546	−0.0392773	+	0.999228i	$0.512506\pi$
$384$	0	0
$385$	−0.884270	−0.0450666
$386$	1.00132	0.0509659
$387$	0	0
$388$	−4.00757	−0.203454
$389$	−28.0710	−1.42325	−0.711627	−	0.702557i	$-0.752042\pi$
$389$	−28.0710	−1.42325	−0.711627	+	0.702557i	$0.752042\pi$
$390$	0	0
$391$	23.5374	1.19034
$392$	1.23316	0.0622839
$393$	0	0
$394$	0.904431	0.0455646
$395$	16.5940	0.834934
$396$	0	0
$397$	−17.1362	−0.860039	−0.430019	−	0.902820i	$-0.641493\pi$
$397$	−17.1362	−0.860039	−0.430019	+	0.902820i	$0.641493\pi$
$398$	0.517387	0.0259343
$399$	0	0
$400$	−10.6308	−0.531540
$401$	18.5624	0.926963	0.463481	−	0.886107i	$-0.346600\pi$
$401$	18.5624	0.926963	0.463481	+	0.886107i	$0.346600\pi$
$402$	0	0
$403$	45.7999	2.28146
$404$	29.2834	1.45690
$405$	0	0
$406$	0.0306812	0.00152268
$407$	−6.72628	−0.333409
$408$	0	0
$409$	11.4358	0.565466	0.282733	−	0.959199i	$-0.408759\pi$
$409$	11.4358	0.565466	0.282733	+	0.959199i	$0.408759\pi$
$410$	−0.663598	−0.0327727
$411$	0	0
$412$	20.5100	1.01045
$413$	−3.34976	−0.164831
$414$	0	0
$415$	15.7061	0.770982
$416$	2.75137	0.134897
$417$	0	0
$418$	−0.172632	−0.00844370
$419$	34.2759	1.67449	0.837243	−	0.546832i	$-0.184166\pi$
$419$	34.2759	1.67449	0.837243	+	0.546832i	$0.184166\pi$
$420$	0	0
$421$	28.5085	1.38942	0.694709	−	0.719291i	$-0.255533\pi$
$421$	28.5085	1.38942	0.694709	+	0.719291i	$0.255533\pi$
$422$	−0.00434619	−0.000211569 0
$423$	0	0
$424$	1.72569	0.0838070
$425$	8.48646	0.411654
$426$	0	0
$427$	−4.93285	−0.238717
$428$	0.364019	0.0175955
$429$	0	0
$430$	−0.341797	−0.0164829
$431$	28.6887	1.38189	0.690943	−	0.722909i	$-0.257196\pi$
$431$	28.6887	1.38189	0.690943	+	0.722909i	$0.257196\pi$
$432$	0	0
$433$	14.6362	0.703370	0.351685	−	0.936118i	$-0.385609\pi$
$433$	14.6362	0.703370	0.351685	+	0.936118i	$0.385609\pi$
$434$	0.175035	0.00840196
$435$	0	0
$436$	−11.5864	−0.554889
$437$	20.8838	0.999007
$438$	0	0
$439$	13.6465	0.651314	0.325657	−	0.945488i	$-0.394415\pi$
$439$	13.6465	0.651314	0.325657	+	0.945488i	$0.394415\pi$
$440$	0.373322	0.0177974
$441$	0	0
$442$	−0.730881	−0.0347645
$443$	−9.50323	−0.451512	−0.225756	−	0.974184i	$-0.572485\pi$
$443$	−9.50323	−0.451512	−0.225756	+	0.974184i	$0.572485\pi$
$444$	0	0
$445$	−13.3823	−0.634383
$446$	0.337612	0.0159864
$447$	0	0
$448$	−3.40708	−0.160970
$449$	−11.6463	−0.549624	−0.274812	−	0.961498i	$-0.588616\pi$
$449$	−11.6463	−0.549624	−0.274812	+	0.961498i	$0.588616\pi$
$450$	0	0
$451$	12.9650	0.610499
$452$	−1.29889	−0.0610947
$453$	0	0
$454$	−0.767518	−0.0360214
$455$	−3.32170	−0.155724
$456$	0	0
$457$	−7.99627	−0.374050	−0.187025	−	0.982355i	$-0.559884\pi$
$457$	−7.99627	−0.374050	−0.187025	+	0.982355i	$0.559884\pi$
$458$	0.998323	0.0466486
$459$	0	0
$460$	−22.5694	−1.05230
$461$	13.8097	0.643184	0.321592	−	0.946878i	$-0.395782\pi$
$461$	13.8097	0.643184	0.321592	+	0.946878i	$0.395782\pi$
$462$	0	0
$463$	−16.9372	−0.787139	−0.393570	−	0.919295i	$-0.628760\pi$
$463$	−16.9372	−0.787139	−0.393570	+	0.919295i	$0.628760\pi$
$464$	6.30962	0.292917
$465$	0	0
$466$	−0.916561	−0.0424589
$467$	13.0936	0.605902	0.302951	−	0.953006i	$-0.402028\pi$
$467$	13.0936	0.605902	0.302951	+	0.953006i	$0.402028\pi$
$468$	0	0
$469$	−2.95240	−0.136329
$470$	0.693359	0.0319823
$471$	0	0
$472$	1.41420	0.0650940
$473$	6.67786	0.307048
$474$	0	0
$475$	7.52969	0.345486
$476$	2.72544	0.124920
$477$	0	0
$478$	−0.606241	−0.0277288
$479$	22.5926	1.03228	0.516141	−	0.856504i	$-0.327368\pi$
$479$	22.5926	1.03228	0.516141	+	0.856504i	$0.327368\pi$
$480$	0	0
$481$	−25.2669	−1.15207
$482$	−0.383722	−0.0174781
$483$	0	0
$484$	18.3325	0.833294
$485$	−3.06448	−0.139151
$486$	0	0
$487$	7.36059	0.333540	0.166770	−	0.985996i	$-0.446666\pi$
$487$	7.36059	0.333540	0.166770	+	0.985996i	$0.446666\pi$
$488$	2.08256	0.0942729
$489$	0	0
$490$	0.471240	0.0212885
$491$	−23.5438	−1.06252	−0.531258	−	0.847210i	$-0.678281\pi$
$491$	−23.5438	−1.06252	−0.531258	+	0.847210i	$0.678281\pi$
$492$	0	0
$493$	−5.03690	−0.226851
$494$	−0.648481	−0.0291765
$495$	0	0
$496$	35.9961	1.61627
$497$	4.56790	0.204898
$498$	0	0
$499$	0.474843	0.0212569	0.0106284	−	0.999944i	$-0.496617\pi$
$499$	0.474843	0.0212569	0.0106284	+	0.999944i	$0.496617\pi$
$500$	−23.3996	−1.04646
$501$	0	0
$502$	−0.954589	−0.0426054
$503$	15.1234	0.674320	0.337160	−	0.941447i	$-0.390534\pi$
$503$	15.1234	0.674320	0.337160	+	0.941447i	$0.390534\pi$
$504$	0	0
$505$	22.3922	0.996440
$506$	−0.451917	−0.0200902
$507$	0	0
$508$	−8.64932	−0.383752
$509$	−17.6144	−0.780745	−0.390373	−	0.920657i	$-0.627654\pi$
$509$	−17.6144	−0.780745	−0.390373	+	0.920657i	$0.627654\pi$
$510$	0	0
$511$	−5.81961	−0.257444
$512$	3.60527	0.159332
$513$	0	0
$514$	0.335072	0.0147794
$515$	15.6834	0.691094
$516$	0	0
$517$	−13.5465	−0.595773
$518$	−0.0965633	−0.00424275
$519$	0	0
$520$	1.40236	0.0614977
$521$	−37.8019	−1.65613	−0.828065	−	0.560632i	$-0.810558\pi$
$521$	−37.8019	−1.65613	−0.828065	+	0.560632i	$0.810558\pi$
$522$	0	0
$523$	−25.1194	−1.09840	−0.549198	−	0.835693i	$-0.685067\pi$
$523$	−25.1194	−1.09840	−0.549198	+	0.835693i	$0.685067\pi$
$524$	−43.3075	−1.89190
$525$	0	0
$526$	−0.291886	−0.0127268
$527$	−28.7353	−1.25173
$528$	0	0
$529$	31.6697	1.37694
$530$	0.659457	0.0286450
$531$	0	0
$532$	2.41817	0.104841
$533$	48.7023	2.10953
$534$	0	0
$535$	0.278356	0.0120344
$536$	1.24645	0.0538384
$537$	0	0
$538$	0.363336	0.0156645
$539$	−9.20684	−0.396567
$540$	0	0
$541$	−10.7401	−0.461753	−0.230877	−	0.972983i	$-0.574159\pi$
$541$	−10.7401	−0.461753	−0.230877	+	0.972983i	$0.574159\pi$
$542$	−0.606165	−0.0260370
$543$	0	0
$544$	−1.72624	−0.0740119
$545$	−8.85983	−0.379513
$546$	0	0
$547$	−20.9646	−0.896380	−0.448190	−	0.893938i	$-0.647931\pi$
$547$	−20.9646	−0.896380	−0.448190	+	0.893938i	$0.647931\pi$
$548$	8.65795	0.369849
$549$	0	0
$550$	−0.162940	−0.00694777
$551$	−4.46904	−0.190387
$552$	0	0
$553$	−4.65432	−0.197922
$554$	0.241779	0.0102722
$555$	0	0
$556$	32.9834	1.39881
$557$	−0.616207	−0.0261095	−0.0130548	−	0.999915i	$-0.504156\pi$
$557$	−0.616207	−0.0261095	−0.0130548	+	0.999915i	$0.504156\pi$
$558$	0	0
$559$	25.0850	1.06098
$560$	−2.61067	−0.110321
$561$	0	0
$562$	−0.519616	−0.0219187
$563$	24.5592	1.03504	0.517522	−	0.855670i	$-0.326854\pi$
$563$	24.5592	1.03504	0.517522	+	0.855670i	$0.326854\pi$
$564$	0	0
$565$	−0.993227	−0.0417854
$566$	0.608950	0.0255961
$567$	0	0
$568$	−1.92848	−0.0809173
$569$	24.9925	1.04774	0.523869	−	0.851799i	$-0.324488\pi$
$569$	24.9925	1.04774	0.523869	+	0.851799i	$0.324488\pi$
$570$	0	0
$571$	17.9085	0.749448	0.374724	−	0.927136i	$-0.377738\pi$
$571$	17.9085	0.749448	0.374724	+	0.927136i	$0.377738\pi$
$572$	−13.6923	−0.572503
$573$	0	0
$574$	0.186127	0.00776880
$575$	19.7113	0.822017
$576$	0	0
$577$	−5.55879	−0.231415	−0.115708	−	0.993283i	$-0.536914\pi$
$577$	−5.55879	−0.231415	−0.115708	+	0.993283i	$0.536914\pi$
$578$	−0.310704	−0.0129236
$579$	0	0
$580$	4.82975	0.200544
$581$	−4.40528	−0.182762
$582$	0	0
$583$	−12.8841	−0.533606
$584$	2.45693	0.101669
$585$	0	0
$586$	−1.06986	−0.0441955
$587$	8.59253	0.354652	0.177326	−	0.984152i	$-0.443255\pi$
$587$	8.59253	0.354652	0.177326	+	0.984152i	$0.443255\pi$
$588$	0	0
$589$	−25.4957	−1.05053
$590$	0.540425	0.0222490
$591$	0	0
$592$	−19.8583	−0.816171
$593$	24.3700	1.00076	0.500378	−	0.865807i	$-0.333194\pi$
$593$	24.3700	1.00076	0.500378	+	0.865807i	$0.333194\pi$
$594$	0	0
$595$	2.08407	0.0854386
$596$	−24.9981	−1.02396
$597$	0	0
$598$	−1.69760	−0.0694200
$599$	−21.0301	−0.859267	−0.429634	−	0.903003i	$-0.641357\pi$
$599$	−21.0301	−0.859267	−0.429634	+	0.903003i	$0.641357\pi$
$600$	0	0
$601$	−28.4759	−1.16156	−0.580778	−	0.814062i	$-0.697252\pi$
$601$	−28.4759	−1.16156	−0.580778	+	0.814062i	$0.697252\pi$
$602$	0.0958681	0.00390729
$603$	0	0
$604$	−19.9999	−0.813784
$605$	14.0183	0.569927
$606$	0	0
$607$	−47.7294	−1.93728	−0.968639	−	0.248474i	$-0.920071\pi$
$607$	−47.7294	−1.93728	−0.968639	+	0.248474i	$0.920071\pi$
$608$	−1.53162	−0.0621155
$609$	0	0
$610$	0.795830	0.0322222
$611$	−50.8865	−2.05865
$612$	0	0
$613$	−26.4193	−1.06707	−0.533533	−	0.845779i	$-0.679136\pi$
$613$	−26.4193	−1.06707	−0.533533	+	0.845779i	$0.679136\pi$
$614$	0.433459	0.0174930
$615$	0	0
$616$	−0.104710	−0.00421890
$617$	23.7402	0.955743	0.477871	−	0.878430i	$-0.341409\pi$
$617$	23.7402	0.955743	0.477871	+	0.878430i	$0.341409\pi$
$618$	0	0
$619$	35.8068	1.43920	0.719599	−	0.694390i	$-0.244326\pi$
$619$	35.8068	1.43920	0.719599	+	0.694390i	$0.244326\pi$
$620$	27.5535	1.10658
$621$	0	0
$622$	1.53448	0.0615270
$623$	3.75350	0.150381
$624$	0	0
$625$	−4.56363	−0.182545
$626$	0.106364	0.00425115
$627$	0	0
$628$	22.6804	0.905046
$629$	15.8527	0.632088
$630$	0	0
$631$	31.2994	1.24601	0.623005	−	0.782217i	$-0.285911\pi$
$631$	31.2994	1.24601	0.623005	+	0.782217i	$0.285911\pi$
$632$	1.96497	0.0781622
$633$	0	0
$634$	0.0821314	0.00326185
$635$	−6.61390	−0.262465
$636$	0	0
$637$	−34.5849	−1.37030
$638$	0.0967083	0.00382872
$639$	0	0
$640$	2.20662	0.0872241
$641$	−10.3111	−0.407263	−0.203632	−	0.979048i	$-0.565275\pi$
$641$	−10.3111	−0.407263	−0.203632	+	0.979048i	$0.565275\pi$
$642$	0	0
$643$	26.1882	1.03276	0.516381	−	0.856359i	$-0.327279\pi$
$643$	26.1882	1.03276	0.516381	+	0.856359i	$0.327279\pi$
$644$	6.33031	0.249449
$645$	0	0
$646$	0.406864	0.0160078
$647$	5.86683	0.230649	0.115324	−	0.993328i	$-0.463209\pi$
$647$	5.86683	0.230649	0.115324	+	0.993328i	$0.463209\pi$
$648$	0	0
$649$	−10.5585	−0.414459
$650$	−0.612073	−0.0240075
$651$	0	0
$652$	−29.0903	−1.13927
$653$	−47.0674	−1.84189	−0.920945	−	0.389694i	$-0.872581\pi$
$653$	−47.0674	−1.84189	−0.920945	+	0.389694i	$0.872581\pi$
$654$	0	0
$655$	−33.1161	−1.29395
$656$	38.2772	1.49447
$657$	0	0
$658$	−0.194475	−0.00758142
$659$	3.84232	0.149675	0.0748377	−	0.997196i	$-0.476156\pi$
$659$	3.84232	0.149675	0.0748377	+	0.997196i	$0.476156\pi$
$660$	0	0
$661$	36.0603	1.40258	0.701292	−	0.712874i	$-0.252607\pi$
$661$	36.0603	1.40258	0.701292	+	0.712874i	$0.252607\pi$
$662$	0.0933162	0.00362684
$663$	0	0
$664$	1.85983	0.0721754
$665$	1.84911	0.0717055
$666$	0	0
$667$	−11.6991	−0.452990
$668$	−1.99795	−0.0773031
$669$	0	0
$670$	0.476319	0.0184018
$671$	−15.5485	−0.600243
$672$	0	0
$673$	−26.3282	−1.01488	−0.507438	−	0.861688i	$-0.669408\pi$
$673$	−26.3282	−1.01488	−0.507438	+	0.861688i	$0.669408\pi$
$674$	−0.542529	−0.0208974
$675$	0	0
$676$	−25.4608	−0.979263
$677$	−32.8446	−1.26232	−0.631160	−	0.775653i	$-0.717421\pi$
$677$	−32.8446	−1.26232	−0.631160	+	0.775653i	$0.717421\pi$
$678$	0	0
$679$	0.859533	0.0329858
$680$	−0.879856	−0.0337410
$681$	0	0
$682$	0.551717	0.0211263
$683$	7.52568	0.287962	0.143981	−	0.989580i	$-0.454010\pi$
$683$	7.52568	0.287962	0.143981	+	0.989580i	$0.454010\pi$
$684$	0	0
$685$	6.62051	0.252957
$686$	−0.267910	−0.0102288
$687$	0	0
$688$	19.7153	0.751640
$689$	−48.3984	−1.84383
$690$	0	0
$691$	35.9858	1.36896	0.684481	−	0.729030i	$-0.260029\pi$
$691$	35.9858	1.36896	0.684481	+	0.729030i	$0.260029\pi$
$692$	−4.08140	−0.155152
$693$	0	0
$694$	0.306061	0.0116179
$695$	25.2215	0.956707
$696$	0	0
$697$	−30.5563	−1.15740
$698$	−1.45108	−0.0549244
$699$	0	0
$700$	2.28241	0.0862669
$701$	−0.0397788	−0.00150243	−0.000751213	−	1.00000i	$-0.500239\pi$
$701$	−0.0397788	−0.00150243	−0.000751213		1.00000i	$0.500239\pi$
$702$	0	0
$703$	14.0654	0.530488
$704$	−10.7392	−0.404750
$705$	0	0
$706$	0.383231	0.0144231
$707$	−6.28062	−0.236207
$708$	0	0
$709$	−17.0360	−0.639799	−0.319899	−	0.947452i	$-0.603649\pi$
$709$	−17.0360	−0.639799	−0.319899	+	0.947452i	$0.603649\pi$
$710$	−0.736952	−0.0276573
$711$	0	0
$712$	−1.58466	−0.0593876
$713$	−66.7429	−2.49954
$714$	0	0
$715$	−10.4701	−0.391560
$716$	−7.71817	−0.288442
$717$	0	0
$718$	−0.938165	−0.0350120
$719$	27.2936	1.01788	0.508940	−	0.860802i	$-0.330037\pi$
$719$	27.2936	1.01788	0.508940	+	0.860802i	$0.330037\pi$
$720$	0	0
$721$	−4.39892	−0.163824
$722$	−0.498775	−0.0185625
$723$	0	0
$724$	32.7462	1.21700
$725$	−4.21813	−0.156657
$726$	0	0
$727$	13.4446	0.498631	0.249315	−	0.968422i	$-0.419794\pi$
$727$	13.4446	0.498631	0.249315	+	0.968422i	$0.419794\pi$
$728$	−0.393338	−0.0145781
$729$	0	0
$730$	0.938894	0.0347500
$731$	−15.7386	−0.582112
$732$	0	0
$733$	16.9960	0.627761	0.313880	−	0.949463i	$-0.398371\pi$
$733$	16.9960	0.627761	0.313880	+	0.949463i	$0.398371\pi$
$734$	−1.13846	−0.0420214
$735$	0	0
$736$	−4.00949	−0.147792
$737$	−9.30607	−0.342793
$738$	0	0
$739$	5.95228	0.218958	0.109479	−	0.993989i	$-0.465082\pi$
$739$	5.95228	0.218958	0.109479	+	0.993989i	$0.465082\pi$
$740$	−15.2007	−0.558789
$741$	0	0
$742$	−0.184966	−0.00679031
$743$	17.4663	0.640776	0.320388	−	0.947286i	$-0.396187\pi$
$743$	17.4663	0.640776	0.320388	+	0.947286i	$0.396187\pi$
$744$	0	0
$745$	−19.1154	−0.700332
$746$	−0.174111	−0.00637466
$747$	0	0
$748$	8.59068	0.314106
$749$	−0.0780739	−0.00285276
$750$	0	0
$751$	20.2692	0.739632	0.369816	−	0.929105i	$-0.379421\pi$
$751$	20.2692	0.739632	0.369816	+	0.929105i	$0.379421\pi$
$752$	−39.9939	−1.45843
$753$	0	0
$754$	0.363279	0.0132298
$755$	−15.2934	−0.556583
$756$	0	0
$757$	23.5315	0.855267	0.427633	−	0.903952i	$-0.359347\pi$
$757$	23.5315	0.855267	0.427633	+	0.903952i	$0.359347\pi$
$758$	−1.42790	−0.0518638
$759$	0	0
$760$	−0.780661	−0.0283175
$761$	2.80302	0.101610	0.0508048	−	0.998709i	$-0.483821\pi$
$761$	2.80302	0.101610	0.0508048	+	0.998709i	$0.483821\pi$
$762$	0	0
$763$	2.48502	0.0899639
$764$	17.8553	0.645983
$765$	0	0
$766$	−0.0695663	−0.00251353
$767$	−39.6625	−1.43213
$768$	0	0
$769$	−16.0820	−0.579931	−0.289965	−	0.957037i	$-0.593644\pi$
$769$	−16.0820	−0.579931	−0.289965	+	0.957037i	$0.593644\pi$
$770$	−0.0400141	−0.00144201
$771$	0	0
$772$	−44.2110	−1.59119
$773$	49.9431	1.79633	0.898165	−	0.439658i	$-0.144900\pi$
$773$	49.9431	1.79633	0.898165	+	0.439658i	$0.144900\pi$
$774$	0	0
$775$	−24.0643	−0.864414
$776$	−0.362879	−0.0130266
$777$	0	0
$778$	−1.27024	−0.0455403
$779$	−27.1114	−0.971365
$780$	0	0
$781$	14.3982	0.515207
$782$	1.06509	0.0380876
$783$	0	0
$784$	−27.1818	−0.970778
$785$	17.3431	0.619002
$786$	0	0
$787$	33.0395	1.17773	0.588865	−	0.808231i	$-0.299575\pi$
$787$	33.0395	1.17773	0.588865	+	0.808231i	$0.299575\pi$
$788$	−39.9331	−1.42256
$789$	0	0
$790$	0.750894	0.0267156
$791$	0.278582	0.00990525
$792$	0	0
$793$	−58.4070	−2.07409
$794$	−0.775428	−0.0275189
$795$	0	0
$796$	−22.8440	−0.809686
$797$	−29.5260	−1.04586	−0.522931	−	0.852375i	$-0.675162\pi$
$797$	−29.5260	−1.04586	−0.522931	+	0.852375i	$0.675162\pi$
$798$	0	0
$799$	31.9267	1.12949
$800$	−1.44563	−0.0511108
$801$	0	0
$802$	0.839968	0.0296603
$803$	−18.3436	−0.647332
$804$	0	0
$805$	4.84062	0.170609
$806$	2.07249	0.0730004
$807$	0	0
$808$	2.65156	0.0932816
$809$	−15.3969	−0.541325	−0.270663	−	0.962674i	$-0.587243\pi$
$809$	−15.3969	−0.541325	−0.270663	+	0.962674i	$0.587243\pi$
$810$	0	0
$811$	49.3168	1.73175	0.865874	−	0.500263i	$-0.166763\pi$
$811$	49.3168	1.73175	0.865874	+	0.500263i	$0.166763\pi$
$812$	−1.35466	−0.0475392
$813$	0	0
$814$	−0.304371	−0.0106682
$815$	−22.2446	−0.779195
$816$	0	0
$817$	−13.9642	−0.488545
$818$	0.517483	0.0180934
$819$	0	0
$820$	29.2996	1.02319
$821$	23.7755	0.829771	0.414885	−	0.909874i	$-0.363822\pi$
$821$	23.7755	0.829771	0.414885	+	0.909874i	$0.363822\pi$
$822$	0	0
$823$	33.5377	1.16905	0.584525	−	0.811376i	$-0.301281\pi$
$823$	33.5377	1.16905	0.584525	+	0.811376i	$0.301281\pi$
$824$	1.85714	0.0646967
$825$	0	0
$826$	−0.151580	−0.00527413
$827$	−9.70721	−0.337553	−0.168776	−	0.985654i	$-0.553982\pi$
$827$	−9.70721	−0.337553	−0.168776	+	0.985654i	$0.553982\pi$
$828$	0	0
$829$	−32.5448	−1.13033	−0.565165	−	0.824978i	$-0.691187\pi$
$829$	−32.5448	−1.13033	−0.565165	+	0.824978i	$0.691187\pi$
$830$	0.710717	0.0246693
$831$	0	0
$832$	−40.3413	−1.39858
$833$	21.6989	0.751824
$834$	0	0
$835$	−1.52778	−0.0528710
$836$	7.62216	0.263618
$837$	0	0
$838$	1.55102	0.0535790
$839$	−52.3025	−1.80568	−0.902841	−	0.429975i	$-0.858522\pi$
$839$	−52.3025	−1.80568	−0.902841	+	0.429975i	$0.858522\pi$
$840$	0	0
$841$	−26.4964	−0.913671
$842$	1.29004	0.0444576
$843$	0	0
$844$	0.191896	0.00660533
$845$	−19.4692	−0.669762
$846$	0	0
$847$	−3.93190	−0.135102
$848$	−38.0384	−1.30624
$849$	0	0
$850$	0.384021	0.0131718
$851$	36.8206	1.26220
$852$	0	0
$853$	19.8753	0.680519	0.340259	−	0.940332i	$-0.389485\pi$
$853$	19.8753	0.680519	0.340259	+	0.940332i	$0.389485\pi$
$854$	−0.223216	−0.00763830
$855$	0	0
$856$	0.0329614	0.00112660
$857$	−36.8003	−1.25708	−0.628538	−	0.777779i	$-0.716346\pi$
$857$	−36.8003	−1.25708	−0.628538	+	0.777779i	$0.716346\pi$
$858$	0	0
$859$	32.0050	1.09200	0.545998	−	0.837786i	$-0.316150\pi$
$859$	32.0050	1.09200	0.545998	+	0.837786i	$0.316150\pi$
$860$	15.0913	0.514608
$861$	0	0
$862$	1.29819	0.0442166
$863$	56.9495	1.93858	0.969292	−	0.245914i	$-0.0790882\pi$
$863$	56.9495	1.93858	0.969292	+	0.245914i	$0.0790882\pi$
$864$	0	0
$865$	−3.12094	−0.106115
$866$	0.662301	0.0225059
$867$	0	0
$868$	−7.72828	−0.262315
$869$	−14.6706	−0.497665
$870$	0	0
$871$	−34.9577	−1.18450
$872$	−1.04913	−0.0355281
$873$	0	0
$874$	0.945012	0.0319655
$875$	5.01868	0.169662
$876$	0	0
$877$	−13.3400	−0.450459	−0.225229	−	0.974306i	$-0.572313\pi$
$877$	−13.3400	−0.450459	−0.225229	+	0.974306i	$0.572313\pi$
$878$	0.617519	0.0208403
$879$	0	0
$880$	−8.22892	−0.277397
$881$	−57.3314	−1.93154	−0.965772	−	0.259391i	$-0.916478\pi$
$881$	−57.3314	−1.93154	−0.965772	+	0.259391i	$0.916478\pi$
$882$	0	0
$883$	−15.5666	−0.523859	−0.261929	−	0.965087i	$-0.584359\pi$
$883$	−15.5666	−0.523859	−0.261929	+	0.965087i	$0.584359\pi$
$884$	32.2704	1.08537
$885$	0	0
$886$	−0.430031	−0.0144472
$887$	29.4516	0.988889	0.494444	−	0.869209i	$-0.335372\pi$
$887$	29.4516	0.988889	0.494444	+	0.869209i	$0.335372\pi$
$888$	0	0
$889$	1.85508	0.0622175
$890$	−0.605563	−0.0202985
$891$	0	0
$892$	−14.9065	−0.499105
$893$	28.3273	0.947936
$894$	0	0
$895$	−5.90188	−0.197278
$896$	−0.618916	−0.0206765
$897$	0	0
$898$	−0.527008	−0.0175865
$899$	14.2827	0.476354
$900$	0	0
$901$	30.3657	1.01163
$902$	0.586680	0.0195343
$903$	0	0
$904$	−0.117612	−0.00391173
$905$	25.0401	0.832363
$906$	0	0
$907$	−13.4630	−0.447033	−0.223516	−	0.974700i	$-0.571754\pi$
$907$	−13.4630	−0.447033	−0.223516	+	0.974700i	$0.571754\pi$
$908$	33.8880	1.12461
$909$	0	0
$910$	−0.150310	−0.00498274
$911$	29.4058	0.974258	0.487129	−	0.873330i	$-0.338044\pi$
$911$	29.4058	0.974258	0.487129	+	0.873330i	$0.338044\pi$
$912$	0	0
$913$	−13.8856	−0.459546
$914$	−0.361839	−0.0119686
$915$	0	0
$916$	−44.0786	−1.45640
$917$	9.28848	0.306732
$918$	0	0
$919$	1.27712	0.0421282	0.0210641	−	0.999778i	$-0.493295\pi$
$919$	1.27712	0.0421282	0.0210641	+	0.999778i	$0.493295\pi$
$920$	−2.04362	−0.0673762
$921$	0	0
$922$	0.624904	0.0205801
$923$	54.0858	1.78026
$924$	0	0
$925$	13.2758	0.436504
$926$	−0.766426	−0.0251863
$927$	0	0
$928$	0.858014	0.0281657
$929$	−16.3571	−0.536660	−0.268330	−	0.963327i	$-0.586472\pi$
$929$	−16.3571	−0.536660	−0.268330	+	0.963327i	$0.586472\pi$
$930$	0	0
$931$	19.2526	0.630978
$932$	40.4686	1.32559
$933$	0	0
$934$	0.592500	0.0193872
$935$	6.56906	0.214831
$936$	0	0
$937$	21.0383	0.687290	0.343645	−	0.939100i	$-0.388338\pi$
$937$	21.0383	0.687290	0.343645	+	0.939100i	$0.388338\pi$
$938$	−0.133599	−0.00436216
$939$	0	0
$940$	−30.6137	−0.998507
$941$	46.7227	1.52312	0.761558	−	0.648097i	$-0.224435\pi$
$941$	46.7227	1.52312	0.761558	+	0.648097i	$0.224435\pi$
$942$	0	0
$943$	−70.9723	−2.31118
$944$	−31.1725	−1.01458
$945$	0	0
$946$	0.302180	0.00982470
$947$	19.8669	0.645587	0.322794	−	0.946469i	$-0.395378\pi$
$947$	19.8669	0.645587	0.322794	+	0.946469i	$0.395378\pi$
$948$	0	0
$949$	−68.9067	−2.23680
$950$	0.340726	0.0110546
$951$	0	0
$952$	0.246784	0.00799832
$953$	12.9133	0.418303	0.209152	−	0.977883i	$-0.432930\pi$
$953$	12.9133	0.418303	0.209152	+	0.977883i	$0.432930\pi$
$954$	0	0
$955$	13.6535	0.441817
$956$	26.7671	0.865711
$957$	0	0
$958$	1.02234	0.0330302
$959$	−1.85693	−0.0599635
$960$	0	0
$961$	50.4822	1.62846
$962$	−1.14335	−0.0368631
$963$	0	0
$964$	16.9424	0.545677
$965$	−33.8070	−1.08829
$966$	0	0
$967$	51.9428	1.67037	0.835184	−	0.549971i	$-0.185361\pi$
$967$	51.9428	1.67037	0.835184	+	0.549971i	$0.185361\pi$
$968$	1.65997	0.0533536
$969$	0	0
$970$	−0.138671	−0.00445245
$971$	−15.2651	−0.489880	−0.244940	−	0.969538i	$-0.578768\pi$
$971$	−15.2651	−0.489880	−0.244940	+	0.969538i	$0.578768\pi$
$972$	0	0
$973$	−7.07419	−0.226788
$974$	0.333074	0.0106724
$975$	0	0
$976$	−45.9045	−1.46937
$977$	−56.9714	−1.82268	−0.911338	−	0.411658i	$-0.864950\pi$
$977$	−56.9714	−1.82268	−0.911338	+	0.411658i	$0.864950\pi$
$978$	0	0
$979$	11.8312	0.378126
$980$	−20.8065	−0.664640
$981$	0	0
$982$	−1.06538	−0.0339976
$983$	−59.1413	−1.88632	−0.943158	−	0.332346i	$-0.892160\pi$
$983$	−59.1413	−1.88632	−0.943158	+	0.332346i	$0.892160\pi$
$984$	0	0
$985$	−30.5357	−0.972950
$986$	−0.227925	−0.00725860
$987$	0	0
$988$	28.6322	0.910911
$989$	−36.5555	−1.16240
$990$	0	0
$991$	−37.8243	−1.20153	−0.600764	−	0.799426i	$-0.705137\pi$
$991$	−37.8243	−1.20153	−0.600764	+	0.799426i	$0.705137\pi$
$992$	4.89494	0.155415
$993$	0	0
$994$	0.206702	0.00655618
$995$	−17.4682	−0.553780
$996$	0	0
$997$	32.5173	1.02983	0.514917	−	0.857240i	$-0.327823\pi$
$997$	32.5173	1.02983	0.514917	+	0.857240i	$0.327823\pi$
$998$	0.0214871	0.000680162 0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.2.a.e.1.5		8
3.2	odd	2		501.2.a.e.1.4	✓	8
12.11	even	2		8016.2.a.x.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.e.1.4	✓	8	3.2	odd	2
1503.2.a.e.1.5		8	1.1	even	1	trivial
8016.2.a.x.1.5		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1503.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0452510 q^{2} -1.99795 q^{4} -1.52778 q^{5} +0.428515 q^{7} -0.180911 q^{8} +O(q^{10})\)	\(q+0.0452510 q^{2} -1.99795 q^{4} -1.52778 q^{5} +0.428515 q^{7} -0.180911 q^{8} -0.0691336 q^{10} +1.35070 q^{11} +5.07380 q^{13} +0.0193907 q^{14} +3.98772 q^{16} -3.18336 q^{17} -2.82446 q^{19} +3.05243 q^{20} +0.0611203 q^{22} -7.39390 q^{23} -2.66589 q^{25} +0.229595 q^{26} -0.856153 q^{28} +1.58226 q^{29} +9.02675 q^{31} +0.542271 q^{32} -0.144050 q^{34} -0.654678 q^{35} -4.97987 q^{37} -0.127810 q^{38} +0.276393 q^{40} +9.59877 q^{41} +4.94401 q^{43} -2.69862 q^{44} -0.334581 q^{46} -10.0293 q^{47} -6.81637 q^{49} -0.120634 q^{50} -10.1372 q^{52} -9.53888 q^{53} -2.06357 q^{55} -0.0775233 q^{56} +0.0715989 q^{58} -7.81712 q^{59} -11.5115 q^{61} +0.408469 q^{62} -7.95090 q^{64} -7.75166 q^{65} -6.88984 q^{67} +6.36019 q^{68} -0.0296248 q^{70} +10.6598 q^{71} -13.5809 q^{73} -0.225344 q^{74} +5.64314 q^{76} +0.578794 q^{77} -10.8615 q^{79} -6.09236 q^{80} +0.434354 q^{82} -10.2803 q^{83} +4.86347 q^{85} +0.223722 q^{86} -0.244356 q^{88} +8.75932 q^{89} +2.17420 q^{91} +14.7727 q^{92} -0.453834 q^{94} +4.31516 q^{95} +2.00584 q^{97} -0.308448 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} + 9 q^{4} - 7 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - 3 * q^2 + 9 * q^4 - 7 * q^5 - 4 * q^7 - 3 * q^8	\( 8 q - 3 q^{2} + 9 q^{4} - 7 q^{5} - 4 q^{7} - 3 q^{8} - q^{10} - 13 q^{11} - 5 q^{14} + 7 q^{16} - 11 q^{17} + 12 q^{19} - 9 q^{20} - 17 q^{22} - 7 q^{23} - 5 q^{25} - 3 q^{26} - 27 q^{28} - q^{29} - 2 q^{31} - 4 q^{32} - 14 q^{34} + 4 q^{35} - 9 q^{37} - 22 q^{40} - 4 q^{41} + 2 q^{43} - 3 q^{44} - 5 q^{46} - 17 q^{47} - 2 q^{49} + 4 q^{50} - 36 q^{52} - 9 q^{53} + 7 q^{55} - 9 q^{56} - 29 q^{58} - 29 q^{59} - 12 q^{61} + 34 q^{62} - 5 q^{64} - 8 q^{65} - 26 q^{68} + 5 q^{70} - 13 q^{71} - 20 q^{73} + 17 q^{74} + 30 q^{76} + 22 q^{77} + 8 q^{79} + 34 q^{80} + 15 q^{82} - 33 q^{83} - 31 q^{85} - 11 q^{86} - 44 q^{88} - 4 q^{89} + q^{91} + 33 q^{92} - 2 q^{94} - 3 q^{95} - 31 q^{97} + 57 q^{98}+O(q^{100}) \) 8 * q - 3 * q^2 + 9 * q^4 - 7 * q^5 - 4 * q^7 - 3 * q^8 - q^10 - 13 * q^11 - 5 * q^14 + 7 * q^16 - 11 * q^17 + 12 * q^19 - 9 * q^20 - 17 * q^22 - 7 * q^23 - 5 * q^25 - 3 * q^26 - 27 * q^28 - q^29 - 2 * q^31 - 4 * q^32 - 14 * q^34 + 4 * q^35 - 9 * q^37 - 22 * q^40 - 4 * q^41 + 2 * q^43 - 3 * q^44 - 5 * q^46 - 17 * q^47 - 2 * q^49 + 4 * q^50 - 36 * q^52 - 9 * q^53 + 7 * q^55 - 9 * q^56 - 29 * q^58 - 29 * q^59 - 12 * q^61 + 34 * q^62 - 5 * q^64 - 8 * q^65 - 26 * q^68 + 5 * q^70 - 13 * q^71 - 20 * q^73 + 17 * q^74 + 30 * q^76 + 22 * q^77 + 8 * q^79 + 34 * q^80 + 15 * q^82 - 33 * q^83 - 31 * q^85 - 11 * q^86 - 44 * q^88 - 4 * q^89 + q^91 + 33 * q^92 - 2 * q^94 - 3 * q^95 - 31 * q^97 + 57 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more