LMFDB - Embedded newform 1503.4.a.b.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,4,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, 4, names="a")

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

chi := DirichletCharacter("1503.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 4 $
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:	$88.6798707386$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - x^{18} - 105 x^{17} + 101 x^{16} + 4534 x^{15} - 4163 x^{14} - 103845 x^{13} + 89794 x^{12} + \cdots - 362016 $ x^19 - x^18 - 105x^17 + 101x^16 + 4534x^15 - 4163x^14 - 103845x^13 + 89794x^12 + 1356761x^11 - 1079599x^10 - 10105015x^9 + 7121073x^8 + 40212888x^7 - 23631389x^6 - 71822499x^5 + 31594800x^4 + 33681568x^3 + 488616x^2 - 2731104*x - 362016
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.13862$ 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)

$f(q)$	$=$	$q-1.13862 q^{2} -6.70355 q^{4} +7.78898 q^{5} +30.6595 q^{7} +16.7417 q^{8} +O(q^{10})$	\(q-1.13862 q^{2} -6.70355 q^{4} +7.78898 q^{5} +30.6595 q^{7} +16.7417 q^{8} -8.86867 q^{10} +31.3983 q^{11} +53.7510 q^{13} -34.9094 q^{14} +34.5660 q^{16} -34.0219 q^{17} -136.577 q^{19} -52.2139 q^{20} -35.7506 q^{22} -192.660 q^{23} -64.3317 q^{25} -61.2018 q^{26} -205.528 q^{28} -198.833 q^{29} -297.918 q^{31} -173.291 q^{32} +38.7380 q^{34} +238.806 q^{35} -207.325 q^{37} +155.509 q^{38} +130.401 q^{40} +162.111 q^{41} -37.7473 q^{43} -210.480 q^{44} +219.365 q^{46} +474.460 q^{47} +597.006 q^{49} +73.2492 q^{50} -360.323 q^{52} +58.6290 q^{53} +244.561 q^{55} +513.293 q^{56} +226.395 q^{58} +466.502 q^{59} -842.621 q^{61} +339.214 q^{62} -79.2161 q^{64} +418.666 q^{65} -858.756 q^{67} +228.068 q^{68} -271.909 q^{70} -753.392 q^{71} -624.100 q^{73} +236.063 q^{74} +915.550 q^{76} +962.656 q^{77} -445.525 q^{79} +269.234 q^{80} -184.582 q^{82} +789.023 q^{83} -264.996 q^{85} +42.9797 q^{86} +525.661 q^{88} -1164.54 q^{89} +1647.98 q^{91} +1291.50 q^{92} -540.228 q^{94} -1063.80 q^{95} -24.5196 q^{97} -679.761 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - q^{2} + 59 q^{4} - 12 q^{5} - 28 q^{7} + 3 q^{8}+O(q^{10}) $ 19 * q - q^2 + 59 * q^4 - 12 * q^5 - 28 * q^7 + 3 * q^8	$ 19 q - q^{2} + 59 q^{4} - 12 q^{5} - 28 q^{7} + 3 q^{8} - 69 q^{10} + 120 q^{11} - 92 q^{13} + 69 q^{14} + 83 q^{16} - 54 q^{17} - 400 q^{19} + 93 q^{20} - 273 q^{22} - 114 q^{23} + 171 q^{25} - q^{26} - 521 q^{28} - 72 q^{29} - 712 q^{31} - 42 q^{32} - 468 q^{34} + 306 q^{35} - 448 q^{37} - 50 q^{38} - 1030 q^{40} + 310 q^{41} - 852 q^{43} + 757 q^{44} - 2187 q^{46} + 1244 q^{47} + 47 q^{49} + 1670 q^{50} - 2548 q^{52} - 58 q^{53} - 2190 q^{55} + 2541 q^{56} - 3481 q^{58} + 2736 q^{59} - 2922 q^{61} - 486 q^{62} - 3677 q^{64} + 380 q^{65} - 2658 q^{67} + 1558 q^{68} - 4887 q^{70} + 636 q^{71} - 2304 q^{73} + 3137 q^{74} - 6536 q^{76} - 230 q^{77} - 2666 q^{79} + 1644 q^{80} - 1949 q^{82} + 2552 q^{83} - 2816 q^{85} + 4825 q^{86} - 5144 q^{88} + 1136 q^{89} - 6128 q^{91} + 755 q^{92} - 1776 q^{94} + 468 q^{95} - 3560 q^{97} - 1635 q^{98}+O(q^{100}) $ 19 * q - q^2 + 59 * q^4 - 12 * q^5 - 28 * q^7 + 3 * q^8 - 69 * q^10 + 120 * q^11 - 92 * q^13 + 69 * q^14 + 83 * q^16 - 54 * q^17 - 400 * q^19 + 93 * q^20 - 273 * q^22 - 114 * q^23 + 171 * q^25 - q^26 - 521 * q^28 - 72 * q^29 - 712 * q^31 - 42 * q^32 - 468 * q^34 + 306 * q^35 - 448 * q^37 - 50 * q^38 - 1030 * q^40 + 310 * q^41 - 852 * q^43 + 757 * q^44 - 2187 * q^46 + 1244 * q^47 + 47 * q^49 + 1670 * q^50 - 2548 * q^52 - 58 * q^53 - 2190 * q^55 + 2541 * q^56 - 3481 * q^58 + 2736 * q^59 - 2922 * q^61 - 486 * q^62 - 3677 * q^64 + 380 * q^65 - 2658 * q^67 + 1558 * q^68 - 4887 * q^70 + 636 * q^71 - 2304 * q^73 + 3137 * q^74 - 6536 * q^76 - 230 * q^77 - 2666 * q^79 + 1644 * q^80 - 1949 * q^82 + 2552 * q^83 - 2816 * q^85 + 4825 * q^86 - 5144 * q^88 + 1136 * q^89 - 6128 * q^91 + 755 * q^92 - 1776 * q^94 + 468 * q^95 - 3560 * q^97 - 1635 * 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 $

$2$	−1.13862	−0.402562	−0.201281	−	0.979534i	$-0.564510\pi$
$2$	−1.13862	−0.402562	−0.201281	+	0.979534i	$0.564510\pi$
$3$	0	0
$3$	0	0
$4$	−6.70355	−0.837944
$5$	7.78898	0.696668	0.348334	−	0.937370i	$-0.386748\pi$
$5$	7.78898	0.696668	0.348334	+	0.937370i	$0.386748\pi$
$6$	0	0
$7$	30.6595	1.65546	0.827729	−	0.561128i	$-0.189632\pi$
$7$	30.6595	1.65546	0.827729	+	0.561128i	$0.189632\pi$
$8$	16.7417	0.739886
$9$	0	0
$10$	−8.86867	−0.280452
$11$	31.3983	0.860631	0.430315	−	0.902679i	$-0.358402\pi$
$11$	31.3983	0.860631	0.430315	+	0.902679i	$0.358402\pi$
$12$	0	0
$13$	53.7510	1.14676	0.573378	−	0.819291i	$-0.305632\pi$
$13$	53.7510	1.14676	0.573378	+	0.819291i	$0.305632\pi$
$14$	−34.9094	−0.666424
$15$	0	0
$16$	34.5660	0.540094
$17$	−34.0219	−0.485384	−0.242692	−	0.970103i	$-0.578031\pi$
$17$	−34.0219	−0.485384	−0.242692	+	0.970103i	$0.578031\pi$
$18$	0	0
$19$	−136.577	−1.64910	−0.824549	−	0.565790i	$-0.808571\pi$
$19$	−136.577	−1.64910	−0.824549	+	0.565790i	$0.808571\pi$
$20$	−52.2139	−0.583769
$21$	0	0
$22$	−35.7506	−0.346457
$23$	−192.660	−1.74662	−0.873311	−	0.487163i	$-0.838032\pi$
$23$	−192.660	−1.74662	−0.873311	+	0.487163i	$0.838032\pi$
$24$	0	0
$25$	−64.3317	−0.514654
$26$	−61.2018	−0.461641
$27$	0	0
$28$	−205.528	−1.38718
$29$	−198.833	−1.27319	−0.636594	−	0.771199i	$-0.719657\pi$
$29$	−198.833	−1.27319	−0.636594	+	0.771199i	$0.719657\pi$
$30$	0	0
$31$	−297.918	−1.72605	−0.863026	−	0.505160i	$-0.831434\pi$
$31$	−297.918	−1.72605	−0.863026	+	0.505160i	$0.831434\pi$
$32$	−173.291	−0.957307
$33$	0	0
$34$	38.7380	0.195397
$35$	238.806	1.15330
$36$	0	0
$37$	−207.325	−0.921188	−0.460594	−	0.887611i	$-0.652364\pi$
$37$	−207.325	−0.921188	−0.460594	+	0.887611i	$0.652364\pi$
$38$	155.509	0.663864
$39$	0	0
$40$	130.401	0.515455
$41$	162.111	0.617500	0.308750	−	0.951143i	$-0.400089\pi$
$41$	162.111	0.617500	0.308750	+	0.951143i	$0.400089\pi$
$42$	0	0
$43$	−37.7473	−0.133870	−0.0669351	−	0.997757i	$-0.521322\pi$
$43$	−37.7473	−0.133870	−0.0669351	+	0.997757i	$0.521322\pi$
$44$	−210.480	−0.721160
$45$	0	0
$46$	219.365	0.703123
$47$	474.460	1.47249	0.736246	−	0.676714i	$-0.236597\pi$
$47$	474.460	1.47249	0.736246	+	0.676714i	$0.236597\pi$
$48$	0	0
$49$	597.006	1.74054
$50$	73.2492	0.207180
$51$	0	0
$52$	−360.323	−0.960918
$53$	58.6290	0.151949	0.0759747	−	0.997110i	$-0.475793\pi$
$53$	58.6290	0.151949	0.0759747	+	0.997110i	$0.475793\pi$
$54$	0	0
$55$	244.561	0.599574
$56$	513.293	1.22485
$57$	0	0
$58$	226.395	0.512537
$59$	466.502	1.02938	0.514690	−	0.857376i	$-0.327907\pi$
$59$	466.502	1.02938	0.514690	+	0.857376i	$0.327907\pi$
$60$	0	0
$61$	−842.621	−1.76863	−0.884317	−	0.466888i	$-0.845375\pi$
$61$	−842.621	−1.76863	−0.884317	+	0.466888i	$0.845375\pi$
$62$	339.214	0.694842
$63$	0	0
$64$	−79.2161	−0.154719
$65$	418.666	0.798909
$66$	0	0
$67$	−858.756	−1.56588	−0.782938	−	0.622100i	$-0.786280\pi$
$67$	−858.756	−1.56588	−0.782938	+	0.622100i	$0.786280\pi$
$68$	228.068	0.406725
$69$	0	0
$70$	−271.909	−0.464276
$71$	−753.392	−1.25931	−0.629656	−	0.776874i	$-0.716804\pi$
$71$	−753.392	−1.25931	−0.629656	+	0.776874i	$0.716804\pi$
$72$	0	0
$73$	−624.100	−1.00062	−0.500311	−	0.865846i	$-0.666781\pi$
$73$	−624.100	−1.00062	−0.500311	+	0.865846i	$0.666781\pi$
$74$	236.063	0.370835
$75$	0	0
$76$	915.550	1.38185
$77$	962.656	1.42474
$78$	0	0
$79$	−445.525	−0.634500	−0.317250	−	0.948342i	$-0.602759\pi$
$79$	−445.525	−0.634500	−0.317250	+	0.948342i	$0.602759\pi$
$80$	269.234	0.376266
$81$	0	0
$82$	−184.582	−0.248582
$83$	789.023	1.04345	0.521726	−	0.853113i	$-0.325288\pi$
$83$	789.023	1.04345	0.521726	+	0.853113i	$0.325288\pi$
$84$	0	0
$85$	−264.996	−0.338152
$86$	42.9797	0.0538910
$87$	0	0
$88$	525.661	0.636769
$89$	−1164.54	−1.38698	−0.693490	−	0.720466i	$-0.743928\pi$
$89$	−1164.54	−1.38698	−0.693490	+	0.720466i	$0.743928\pi$
$90$	0	0
$91$	1647.98	1.89841
$92$	1291.50	1.46357
$93$	0	0
$94$	−540.228	−0.592769
$95$	−1063.80	−1.14887
$96$	0	0
$97$	−24.5196	−0.0256658	−0.0128329	−	0.999918i	$-0.504085\pi$
$97$	−24.5196	−0.0256658	−0.0128329	+	0.999918i	$0.504085\pi$
$98$	−679.761	−0.700675
$99$	0	0
$100$	431.251	0.431251
$101$	−841.771	−0.829300	−0.414650	−	0.909981i	$-0.636096\pi$
$101$	−841.771	−0.829300	−0.414650	+	0.909981i	$0.636096\pi$
$102$	0	0
$103$	541.124	0.517656	0.258828	−	0.965923i	$-0.416664\pi$
$103$	541.124	0.517656	0.258828	+	0.965923i	$0.416664\pi$
$104$	899.883	0.848469
$105$	0	0
$106$	−66.7560	−0.0611690
$107$	−2012.21	−1.81802	−0.909008	−	0.416779i	$-0.863159\pi$
$107$	−2012.21	−1.81802	−0.909008	+	0.416779i	$0.863159\pi$
$108$	0	0
$109$	243.954	0.214372	0.107186	−	0.994239i	$-0.465816\pi$
$109$	243.954	0.214372	0.107186	+	0.994239i	$0.465816\pi$
$110$	−278.461	−0.241366
$111$	0	0
$112$	1059.78	0.894103
$113$	642.610	0.534971	0.267485	−	0.963562i	$-0.413807\pi$
$113$	642.610	0.534971	0.267485	+	0.963562i	$0.413807\pi$
$114$	0	0
$115$	−1500.62	−1.21682
$116$	1332.89	1.06686
$117$	0	0
$118$	−531.167	−0.414389
$119$	−1043.10	−0.803533
$120$	0	0
$121$	−345.148	−0.259315
$122$	959.423	0.711984
$123$	0	0
$124$	1997.11	1.44633
$125$	−1474.70	−1.05521
$126$	0	0
$127$	−70.7744	−0.0494505	−0.0247252	−	0.999694i	$-0.507871\pi$
$127$	−70.7744	−0.0494505	−0.0247252	+	0.999694i	$0.507871\pi$
$128$	1476.53	1.01959
$129$	0	0
$130$	−476.700	−0.321610
$131$	563.198	0.375625	0.187812	−	0.982205i	$-0.439860\pi$
$131$	563.198	0.375625	0.187812	+	0.982205i	$0.439860\pi$
$132$	0	0
$133$	−4187.38	−2.73001
$134$	977.794	0.630362
$135$	0	0
$136$	−569.586	−0.359129
$137$	971.081	0.605584	0.302792	−	0.953057i	$-0.402081\pi$
$137$	971.081	0.605584	0.302792	+	0.953057i	$0.402081\pi$
$138$	0	0
$139$	−385.260	−0.235089	−0.117544	−	0.993068i	$-0.537502\pi$
$139$	−385.260	−0.235089	−0.117544	+	0.993068i	$0.537502\pi$
$140$	−1600.85	−0.966405
$141$	0	0
$142$	857.824	0.506951
$143$	1687.69	0.986934
$144$	0	0
$145$	−1548.71	−0.886989
$146$	710.610	0.402812
$147$	0	0
$148$	1389.81	0.771904
$149$	673.772	0.370453	0.185227	−	0.982696i	$-0.440698\pi$
$149$	673.772	0.370453	0.185227	+	0.982696i	$0.440698\pi$
$150$	0	0
$151$	2383.10	1.28433	0.642164	−	0.766567i	$-0.278037\pi$
$151$	2383.10	1.28433	0.642164	+	0.766567i	$0.278037\pi$
$152$	−2286.53	−1.22015
$153$	0	0
$154$	−1096.10	−0.573545
$155$	−2320.48	−1.20248
$156$	0	0
$157$	−413.494	−0.210194	−0.105097	−	0.994462i	$-0.533515\pi$
$157$	−413.494	−0.210194	−0.105097	+	0.994462i	$0.533515\pi$
$158$	507.282	0.255425
$159$	0	0
$160$	−1349.76	−0.666925
$161$	−5906.85	−2.89146
$162$	0	0
$163$	−3251.05	−1.56222	−0.781110	−	0.624394i	$-0.785346\pi$
$163$	−3251.05	−1.56222	−0.781110	+	0.624394i	$0.785346\pi$
$164$	−1086.72	−0.517431
$165$	0	0
$166$	−898.394	−0.420054
$167$	−167.000	−0.0773823
$167$	−167.000	−0.0773823
$168$	0	0
$169$	692.168	0.315051
$170$	301.729	0.136127
$171$	0	0
$172$	253.041	0.112176
$173$	2743.22	1.20557	0.602783	−	0.797905i	$-0.294059\pi$
$173$	2743.22	1.20557	0.602783	+	0.797905i	$0.294059\pi$
$174$	0	0
$175$	−1972.38	−0.851988
$176$	1085.31	0.464822
$177$	0	0
$178$	1325.97	0.558345
$179$	1949.31	0.813959	0.406979	−	0.913437i	$-0.366582\pi$
$179$	1949.31	0.813959	0.406979	+	0.913437i	$0.366582\pi$
$180$	0	0
$181$	3620.01	1.48659	0.743296	−	0.668962i	$-0.233261\pi$
$181$	3620.01	1.48659	0.743296	+	0.668962i	$0.233261\pi$
$182$	−1876.42	−0.764227
$183$	0	0
$184$	−3225.45	−1.29230
$185$	−1614.85	−0.641762
$186$	0	0
$187$	−1068.23	−0.417737
$188$	−3180.57	−1.23387
$189$	0	0
$190$	1211.25	0.462493
$191$	3657.79	1.38570	0.692849	−	0.721083i	$-0.256355\pi$
$191$	3657.79	1.38570	0.692849	+	0.721083i	$0.256355\pi$
$192$	0	0
$193$	557.230	0.207825	0.103913	−	0.994586i	$-0.466864\pi$
$193$	557.230	0.207825	0.103913	+	0.994586i	$0.466864\pi$
$194$	27.9184	0.0103321
$195$	0	0
$196$	−4002.06	−1.45848
$197$	5440.87	1.96775	0.983873	−	0.178870i	$-0.0572440\pi$
$197$	5440.87	1.96775	0.983873	+	0.178870i	$0.0572440\pi$
$198$	0	0
$199$	−5031.11	−1.79219	−0.896095	−	0.443862i	$-0.853608\pi$
$199$	−5031.11	−1.79219	−0.896095	+	0.443862i	$0.853608\pi$
$200$	−1077.02	−0.380785
$201$	0	0
$202$	958.454	0.333844
$203$	−6096.13	−2.10771
$204$	0	0
$205$	1262.68	0.430193
$206$	−616.133	−0.208388
$207$	0	0
$208$	1857.96	0.619357
$209$	−4288.28	−1.41927
$210$	0	0
$211$	4201.99	1.37098	0.685490	−	0.728082i	$-0.259588\pi$
$211$	4201.99	1.37098	0.685490	+	0.728082i	$0.259588\pi$
$212$	−393.023	−0.127325
$213$	0	0
$214$	2291.14	0.731864
$215$	−294.013	−0.0932630
$216$	0	0
$217$	−9134.01	−2.85741
$218$	−277.770	−0.0862979
$219$	0	0
$220$	−1639.43	−0.502409
$221$	−1828.71	−0.556618
$222$	0	0
$223$	3496.49	1.04996	0.524982	−	0.851113i	$-0.324072\pi$
$223$	3496.49	1.04996	0.524982	+	0.851113i	$0.324072\pi$
$224$	−5313.02	−1.58478
$225$	0	0
$226$	−731.687	−0.215359
$227$	−1469.58	−0.429690	−0.214845	−	0.976648i	$-0.568925\pi$
$227$	−1469.58	−0.429690	−0.214845	+	0.976648i	$0.568925\pi$
$228$	0	0
$229$	−3481.75	−1.00472	−0.502358	−	0.864659i	$-0.667534\pi$
$229$	−3481.75	−1.00472	−0.502358	+	0.864659i	$0.667534\pi$
$230$	1708.63	0.489844
$231$	0	0
$232$	−3328.81	−0.942013
$233$	1028.97	0.289314	0.144657	−	0.989482i	$-0.453792\pi$
$233$	1028.97	0.289314	0.144657	+	0.989482i	$0.453792\pi$
$234$	0	0
$235$	3695.56	1.02584
$236$	−3127.22	−0.862563
$237$	0	0
$238$	1187.69	0.323472
$239$	2404.80	0.650851	0.325426	−	0.945568i	$-0.394492\pi$
$239$	2404.80	0.650851	0.325426	+	0.945568i	$0.394492\pi$
$240$	0	0
$241$	−5013.41	−1.34001	−0.670004	−	0.742357i	$-0.733708\pi$
$241$	−5013.41	−1.34001	−0.670004	+	0.742357i	$0.733708\pi$
$242$	392.991	0.104390
$243$	0	0
$244$	5648.56	1.48202
$245$	4650.07	1.21258
$246$	0	0
$247$	−7341.14	−1.89112
$248$	−4987.65	−1.27708
$249$	0	0
$250$	1679.12	0.424788
$251$	−1314.42	−0.330539	−0.165270	−	0.986248i	$-0.552849\pi$
$251$	−1314.42	−0.330539	−0.165270	+	0.986248i	$0.552849\pi$
$252$	0	0
$253$	−6049.18	−1.50320
$254$	80.5849	0.0199069
$255$	0	0
$256$	−1047.47	−0.255730
$257$	−4067.37	−0.987220	−0.493610	−	0.869683i	$-0.664323\pi$
$257$	−4067.37	−0.987220	−0.493610	+	0.869683i	$0.664323\pi$
$258$	0	0
$259$	−6356.47	−1.52499
$260$	−2806.55	−0.669441
$261$	0	0
$262$	−641.266	−0.151212
$263$	86.9630	0.0203892	0.0101946	−	0.999948i	$-0.496755\pi$
$263$	86.9630	0.0203892	0.0101946	+	0.999948i	$0.496755\pi$
$264$	0	0
$265$	456.661	0.105858
$266$	4767.82	1.09900
$267$	0	0
$268$	5756.71	1.31212
$269$	2444.27	0.554013	0.277007	−	0.960868i	$-0.410658\pi$
$269$	2444.27	0.554013	0.277007	+	0.960868i	$0.410658\pi$
$270$	0	0
$271$	5227.97	1.17187	0.585935	−	0.810358i	$-0.300727\pi$
$271$	5227.97	1.17187	0.585935	+	0.810358i	$0.300727\pi$
$272$	−1176.00	−0.262153
$273$	0	0
$274$	−1105.69	−0.243785
$275$	−2019.91	−0.442927
$276$	0	0
$277$	4118.90	0.893432	0.446716	−	0.894676i	$-0.352593\pi$
$277$	4118.90	0.893432	0.446716	+	0.894676i	$0.352593\pi$
$278$	438.664	0.0946378
$279$	0	0
$280$	3998.03	0.853314
$281$	−5961.29	−1.26555	−0.632777	−	0.774334i	$-0.718085\pi$
$281$	−5961.29	−1.26555	−0.632777	+	0.774334i	$0.718085\pi$
$282$	0	0
$283$	4875.91	1.02418	0.512089	−	0.858932i	$-0.328872\pi$
$283$	4875.91	1.02418	0.512089	+	0.858932i	$0.328872\pi$
$284$	5050.40	1.05523
$285$	0	0
$286$	−1921.63	−0.397302
$287$	4970.25	1.02225
$288$	0	0
$289$	−3755.51	−0.764402
$290$	1763.39	0.357068
$291$	0	0
$292$	4183.69	0.838465
$293$	6813.92	1.35861	0.679306	−	0.733855i	$-0.262281\pi$
$293$	6813.92	1.35861	0.679306	+	0.733855i	$0.262281\pi$
$294$	0	0
$295$	3633.58	0.717136
$296$	−3470.97	−0.681574
$297$	0	0
$298$	−767.168	−0.149130
$299$	−10355.6	−2.00295
$300$	0	0
$301$	−1157.31	−0.221616
$302$	−2713.43	−0.517022
$303$	0	0
$304$	−4720.92	−0.890669
$305$	−6563.17	−1.23215
$306$	0	0
$307$	−655.877	−0.121931	−0.0609656	−	0.998140i	$-0.519418\pi$
$307$	−655.877	−0.121931	−0.0609656	+	0.998140i	$0.519418\pi$
$308$	−6453.21	−1.19385
$309$	0	0
$310$	2642.13	0.484074
$311$	6460.22	1.17789	0.588947	−	0.808171i	$-0.299543\pi$
$311$	6460.22	1.17789	0.588947	+	0.808171i	$0.299543\pi$
$312$	0	0
$313$	6817.56	1.23115	0.615577	−	0.788077i	$-0.288923\pi$
$313$	6817.56	1.23115	0.615577	+	0.788077i	$0.288923\pi$
$314$	470.812	0.0846160
$315$	0	0
$316$	2986.60	0.531675
$317$	6519.93	1.15519	0.577596	−	0.816323i	$-0.303991\pi$
$317$	6519.93	1.15519	0.577596	+	0.816323i	$0.303991\pi$
$318$	0	0
$319$	−6243.03	−1.09574
$320$	−617.013	−0.107788
$321$	0	0
$322$	6725.64	1.16399
$323$	4646.61	0.800447
$324$	0	0
$325$	−3457.89	−0.590183
$326$	3701.70	0.628890
$327$	0	0
$328$	2714.02	0.456880
$329$	14546.7	2.43765
$330$	0	0
$331$	3085.09	0.512302	0.256151	−	0.966637i	$-0.417546\pi$
$331$	3085.09	0.512302	0.256151	+	0.966637i	$0.417546\pi$
$332$	−5289.25	−0.874354
$333$	0	0
$334$	190.149	0.0311512
$335$	−6688.84	−1.09090
$336$	0	0
$337$	8163.23	1.31952	0.659762	−	0.751475i	$-0.270657\pi$
$337$	8163.23	1.31952	0.659762	+	0.751475i	$0.270657\pi$
$338$	−788.114	−0.126828
$339$	0	0
$340$	1776.42	0.283352
$341$	−9354.10	−1.48549
$342$	0	0
$343$	7787.69	1.22594
$344$	−631.955	−0.0990486
$345$	0	0
$346$	−3123.47	−0.485315
$347$	4113.08	0.636316	0.318158	−	0.948038i	$-0.396936\pi$
$347$	4113.08	0.636316	0.318158	+	0.948038i	$0.396936\pi$
$348$	0	0
$349$	−10714.7	−1.64339	−0.821696	−	0.569926i	$-0.806972\pi$
$349$	−10714.7	−1.64339	−0.821696	+	0.569926i	$0.806972\pi$
$350$	2245.78	0.342978
$351$	0	0
$352$	−5441.04	−0.823888
$353$	965.216	0.145533	0.0727666	−	0.997349i	$-0.476817\pi$
$353$	965.216	0.145533	0.0727666	+	0.997349i	$0.476817\pi$
$354$	0	0
$355$	−5868.16	−0.877322
$356$	7806.57	1.16221
$357$	0	0
$358$	−2219.52	−0.327669
$359$	−3841.80	−0.564798	−0.282399	−	0.959297i	$-0.591130\pi$
$359$	−3841.80	−0.564798	−0.282399	+	0.959297i	$0.591130\pi$
$360$	0	0
$361$	11794.2	1.71953
$362$	−4121.80	−0.598445
$363$	0	0
$364$	−11047.3	−1.59076
$365$	−4861.10	−0.697101
$366$	0	0
$367$	−6986.06	−0.993649	−0.496825	−	0.867851i	$-0.665501\pi$
$367$	−6986.06	−0.993649	−0.496825	+	0.867851i	$0.665501\pi$
$368$	−6659.48	−0.943341
$369$	0	0
$370$	1838.69	0.258349
$371$	1797.54	0.251546
$372$	0	0
$373$	869.512	0.120701	0.0603507	−	0.998177i	$-0.480778\pi$
$373$	869.512	0.120701	0.0603507	+	0.998177i	$0.480778\pi$
$374$	1216.31	0.168165
$375$	0	0
$376$	7943.27	1.08948
$377$	−10687.5	−1.46004
$378$	0	0
$379$	−4838.60	−0.655784	−0.327892	−	0.944715i	$-0.606338\pi$
$379$	−4838.60	−0.655784	−0.327892	+	0.944715i	$0.606338\pi$
$380$	7131.21	0.962692
$381$	0	0
$382$	−4164.82	−0.557829
$383$	−5316.56	−0.709304	−0.354652	−	0.934998i	$-0.615401\pi$
$383$	−5316.56	−0.709304	−0.354652	+	0.934998i	$0.615401\pi$
$384$	0	0
$385$	7498.11	0.992569
$386$	−634.471	−0.0836625
$387$	0	0
$388$	164.368	0.0215065
$389$	−10242.1	−1.33495	−0.667475	−	0.744632i	$-0.732625\pi$
$389$	−10242.1	−1.33495	−0.667475	+	0.744632i	$0.732625\pi$
$390$	0	0
$391$	6554.66	0.847783
$392$	9994.90	1.28780
$393$	0	0
$394$	−6195.06	−0.792139
$395$	−3470.19	−0.442036
$396$	0	0
$397$	−764.215	−0.0966117	−0.0483059	−	0.998833i	$-0.515382\pi$
$397$	−764.215	−0.0966117	−0.0483059	+	0.998833i	$0.515382\pi$
$398$	5728.50	0.721467
$399$	0	0
$400$	−2223.69	−0.277961
$401$	453.962	0.0565331	0.0282666	−	0.999600i	$-0.491001\pi$
$401$	453.962	0.0565331	0.0282666	+	0.999600i	$0.491001\pi$
$402$	0	0
$403$	−16013.4	−1.97936
$404$	5642.85	0.694907
$405$	0	0
$406$	6941.16	0.848483
$407$	−6509.64	−0.792803
$408$	0	0
$409$	−1026.41	−0.124089	−0.0620447	−	0.998073i	$-0.519762\pi$
$409$	−1026.41	−0.124089	−0.0620447	+	0.998073i	$0.519762\pi$
$410$	−1437.71	−0.173179
$411$	0	0
$412$	−3627.45	−0.433766
$413$	14302.7	1.70410
$414$	0	0
$415$	6145.68	0.726939
$416$	−9314.57	−1.09780
$417$	0	0
$418$	4882.71	0.571342
$419$	−6338.52	−0.739038	−0.369519	−	0.929223i	$-0.620478\pi$
$419$	−6338.52	−0.739038	−0.369519	+	0.929223i	$0.620478\pi$
$420$	0	0
$421$	−5834.74	−0.675458	−0.337729	−	0.941243i	$-0.609659\pi$
$421$	−5834.74	−0.675458	−0.337729	+	0.941243i	$0.609659\pi$
$422$	−4784.45	−0.551904
$423$	0	0
$424$	981.550	0.112425
$425$	2188.69	0.249805
$426$	0	0
$427$	−25834.4	−2.92790
$428$	13489.0	1.52340
$429$	0	0
$430$	334.769	0.0375441
$431$	1340.73	0.149840	0.0749198	−	0.997190i	$-0.476130\pi$
$431$	1340.73	0.149840	0.0749198	+	0.997190i	$0.476130\pi$
$432$	0	0
$433$	12263.0	1.36102	0.680509	−	0.732739i	$-0.261759\pi$
$433$	12263.0	1.36102	0.680509	+	0.732739i	$0.261759\pi$
$434$	10400.1	1.15028
$435$	0	0
$436$	−1635.36	−0.179632
$437$	26312.8	2.88035
$438$	0	0
$439$	−14261.0	−1.55043	−0.775217	−	0.631694i	$-0.782360\pi$
$439$	−14261.0	−1.55043	−0.775217	+	0.631694i	$0.782360\pi$
$440$	4094.36	0.443616
$441$	0	0
$442$	2082.20	0.224073
$443$	12976.1	1.39168	0.695839	−	0.718198i	$-0.255033\pi$
$443$	12976.1	1.39168	0.695839	+	0.718198i	$0.255033\pi$
$444$	0	0
$445$	−9070.60	−0.966265
$446$	−3981.16	−0.422675
$447$	0	0
$448$	−2428.73	−0.256131
$449$	−6707.70	−0.705025	−0.352512	−	0.935807i	$-0.614673\pi$
$449$	−6707.70	−0.705025	−0.352512	+	0.935807i	$0.614673\pi$
$450$	0	0
$451$	5090.01	0.531440
$452$	−4307.77	−0.448275
$453$	0	0
$454$	1673.29	0.172977
$455$	12836.1	1.32256
$456$	0	0
$457$	8156.62	0.834903	0.417451	−	0.908699i	$-0.362923\pi$
$457$	8156.62	0.834903	0.417451	+	0.908699i	$0.362923\pi$
$458$	3964.37	0.404461
$459$	0	0
$460$	10059.5	1.01962
$461$	−12494.1	−1.26228	−0.631139	−	0.775670i	$-0.717412\pi$
$461$	−12494.1	−1.26228	−0.631139	+	0.775670i	$0.717412\pi$
$462$	0	0
$463$	−5454.60	−0.547509	−0.273755	−	0.961800i	$-0.588266\pi$
$463$	−5454.60	−0.547509	−0.273755	+	0.961800i	$0.588266\pi$
$464$	−6872.88	−0.687641
$465$	0	0
$466$	−1171.61	−0.116467
$467$	18883.8	1.87117	0.935585	−	0.353101i	$-0.114873\pi$
$467$	18883.8	1.87117	0.935585	+	0.353101i	$0.114873\pi$
$468$	0	0
$469$	−26329.0	−2.59224
$470$	−4207.83	−0.412963
$471$	0	0
$472$	7810.05	0.761624
$473$	−1185.20	−0.115213
$474$	0	0
$475$	8786.22	0.848715
$476$	6992.45	0.673316
$477$	0	0
$478$	−2738.14	−0.262008
$479$	−11913.0	−1.13636	−0.568181	−	0.822904i	$-0.692353\pi$
$479$	−11913.0	−1.13636	−0.568181	+	0.822904i	$0.692353\pi$
$480$	0	0
$481$	−11143.9	−1.05638
$482$	5708.35	0.539436
$483$	0	0
$484$	2313.72	0.217291
$485$	−190.982	−0.0178805
$486$	0	0
$487$	−8343.64	−0.776358	−0.388179	−	0.921584i	$-0.626896\pi$
$487$	−8343.64	−0.776358	−0.388179	+	0.921584i	$0.626896\pi$
$488$	−14106.9	−1.30859
$489$	0	0
$490$	−5294.64	−0.488138
$491$	−7779.69	−0.715056	−0.357528	−	0.933902i	$-0.616380\pi$
$491$	−7779.69	−0.715056	−0.357528	+	0.933902i	$0.616380\pi$
$492$	0	0
$493$	6764.70	0.617985
$494$	8358.74	0.761291
$495$	0	0
$496$	−10297.8	−0.932230
$497$	−23098.6	−2.08474
$498$	0	0
$499$	−15617.7	−1.40109	−0.700544	−	0.713610i	$-0.747059\pi$
$499$	−15617.7	−1.40109	−0.700544	+	0.713610i	$0.747059\pi$
$500$	9885.74	0.884208
$501$	0	0
$502$	1496.62	0.133062
$503$	1055.81	0.0935907	0.0467953	−	0.998904i	$-0.485099\pi$
$503$	1055.81	0.0935907	0.0467953	+	0.998904i	$0.485099\pi$
$504$	0	0
$505$	−6556.54	−0.577747
$506$	6887.70	0.605130
$507$	0	0
$508$	474.440	0.0414367
$509$	−88.7993	−0.00773273	−0.00386637	−	0.999993i	$-0.501231\pi$
$509$	−88.7993	−0.00773273	−0.00386637	+	0.999993i	$0.501231\pi$
$510$	0	0
$511$	−19134.6	−1.65649
$512$	−10619.5	−0.916644
$513$	0	0
$514$	4631.18	0.397417
$515$	4214.81	0.360634
$516$	0	0
$517$	14897.2	1.26727
$518$	7237.59	0.613902
$519$	0	0
$520$	7009.18	0.591101
$521$	−13563.0	−1.14051	−0.570256	−	0.821467i	$-0.693156\pi$
$521$	−13563.0	−1.14051	−0.570256	+	0.821467i	$0.693156\pi$
$522$	0	0
$523$	−2822.42	−0.235977	−0.117988	−	0.993015i	$-0.537645\pi$
$523$	−2822.42	−0.235977	−0.117988	+	0.993015i	$0.537645\pi$
$524$	−3775.43	−0.314752
$525$	0	0
$526$	−99.0175	−0.00820793
$527$	10135.7	0.837798
$528$	0	0
$529$	24950.7	2.05069
$530$	−519.961	−0.0426145
$531$	0	0
$532$	28070.3	2.28760
$533$	8713.63	0.708123
$534$	0	0
$535$	−15673.1	−1.26655
$536$	−14377.0	−1.15857
$537$	0	0
$538$	−2783.08	−0.223025
$539$	18745.0	1.49796
$540$	0	0
$541$	17414.8	1.38395	0.691977	−	0.721919i	$-0.256740\pi$
$541$	17414.8	1.38395	0.691977	+	0.721919i	$0.256740\pi$
$542$	−5952.65	−0.471750
$543$	0	0
$544$	5895.70	0.464662
$545$	1900.15	0.149346
$546$	0	0
$547$	1912.20	0.149470	0.0747348	−	0.997203i	$-0.476189\pi$
$547$	1912.20	0.149470	0.0747348	+	0.997203i	$0.476189\pi$
$548$	−6509.69	−0.507446
$549$	0	0
$550$	2299.90	0.178305
$551$	27156.0	2.09961
$552$	0	0
$553$	−13659.6	−1.05039
$554$	−4689.85	−0.359662
$555$	0	0
$556$	2582.61	0.196991
$557$	−10659.5	−0.810877	−0.405438	−	0.914122i	$-0.632881\pi$
$557$	−10659.5	−0.810877	−0.405438	+	0.914122i	$0.632881\pi$
$558$	0	0
$559$	−2028.96	−0.153516
$560$	8254.59	0.622893
$561$	0	0
$562$	6787.62	0.509464
$563$	−21721.7	−1.62604	−0.813022	−	0.582234i	$-0.802179\pi$
$563$	−21721.7	−1.62604	−0.813022	+	0.582234i	$0.802179\pi$
$564$	0	0
$565$	5005.28	0.372697
$566$	−5551.79	−0.412295
$567$	0	0
$568$	−12613.1	−0.931747
$569$	−2870.58	−0.211495	−0.105748	−	0.994393i	$-0.533724\pi$
$569$	−2870.58	−0.211495	−0.105748	+	0.994393i	$0.533724\pi$
$570$	0	0
$571$	6697.66	0.490873	0.245437	−	0.969413i	$-0.421069\pi$
$571$	6697.66	0.490873	0.245437	+	0.969413i	$0.421069\pi$
$572$	−11313.5	−0.826996
$573$	0	0
$574$	−5659.21	−0.411517
$575$	12394.1	0.898906
$576$	0	0
$577$	15882.6	1.14593	0.572966	−	0.819579i	$-0.305793\pi$
$577$	15882.6	1.14593	0.572966	+	0.819579i	$0.305793\pi$
$578$	4276.08	0.307719
$579$	0	0
$580$	10381.9	0.743247
$581$	24191.0	1.72739
$582$	0	0
$583$	1840.85	0.130772
$584$	−10448.5	−0.740346
$585$	0	0
$586$	−7758.44	−0.546925
$587$	3449.49	0.242548	0.121274	−	0.992619i	$-0.461302\pi$
$587$	3449.49	0.242548	0.121274	+	0.992619i	$0.461302\pi$
$588$	0	0
$589$	40688.7	2.84643
$590$	−4137.25	−0.288692
$591$	0	0
$592$	−7166.39	−0.497528
$593$	18682.7	1.29377	0.646884	−	0.762588i	$-0.276072\pi$
$593$	18682.7	1.29377	0.646884	+	0.762588i	$0.276072\pi$
$594$	0	0
$595$	−8124.66	−0.559796
$596$	−4516.67	−0.310419
$597$	0	0
$598$	11791.1	0.806312
$599$	2826.68	0.192813	0.0964067	−	0.995342i	$-0.469265\pi$
$599$	2826.68	0.192813	0.0964067	+	0.995342i	$0.469265\pi$
$600$	0	0
$601$	16942.3	1.14990	0.574952	−	0.818187i	$-0.305021\pi$
$601$	16942.3	1.14990	0.574952	+	0.818187i	$0.305021\pi$
$602$	1317.74	0.0892143
$603$	0	0
$604$	−15975.2	−1.07620
$605$	−2688.35	−0.180656
$606$	0	0
$607$	−13046.0	−0.872360	−0.436180	−	0.899859i	$-0.643669\pi$
$607$	−13046.0	−0.872360	−0.436180	+	0.899859i	$0.643669\pi$
$608$	23667.6	1.57869
$609$	0	0
$610$	7472.93	0.496016
$611$	25502.7	1.68859
$612$	0	0
$613$	8024.22	0.528703	0.264352	−	0.964426i	$-0.414842\pi$
$613$	8024.22	0.528703	0.264352	+	0.964426i	$0.414842\pi$
$614$	746.792	0.0490848
$615$	0	0
$616$	16116.5	1.05414
$617$	−7942.38	−0.518231	−0.259115	−	0.965846i	$-0.583431\pi$
$617$	−7942.38	−0.518231	−0.259115	+	0.965846i	$0.583431\pi$
$618$	0	0
$619$	−25525.1	−1.65742	−0.828708	−	0.559681i	$-0.810924\pi$
$619$	−25525.1	−1.65742	−0.828708	+	0.559681i	$0.810924\pi$
$620$	15555.4	1.00761
$621$	0	0
$622$	−7355.71	−0.474175
$623$	−35704.3	−2.29609
$624$	0	0
$625$	−3444.96	−0.220478
$626$	−7762.58	−0.495615
$627$	0	0
$628$	2771.88	0.176131
$629$	7053.59	0.447130
$630$	0	0
$631$	−10860.6	−0.685189	−0.342595	−	0.939483i	$-0.611306\pi$
$631$	−10860.6	−0.685189	−0.342595	+	0.939483i	$0.611306\pi$
$632$	−7458.85	−0.469457
$633$	0	0
$634$	−7423.70	−0.465036
$635$	−551.261	−0.0344506
$636$	0	0
$637$	32089.6	1.99598
$638$	7108.41	0.441105
$639$	0	0
$640$	11500.6	0.710317
$641$	10065.5	0.620223	0.310111	−	0.950700i	$-0.399634\pi$
$641$	10065.5	0.620223	0.310111	+	0.950700i	$0.399634\pi$
$642$	0	0
$643$	−17762.1	−1.08938	−0.544689	−	0.838638i	$-0.683352\pi$
$643$	−17762.1	−1.08938	−0.544689	+	0.838638i	$0.683352\pi$
$644$	39596.9	2.42288
$645$	0	0
$646$	−5290.71	−0.322229
$647$	23008.7	1.39809	0.699047	−	0.715076i	$-0.253608\pi$
$647$	23008.7	1.39809	0.699047	+	0.715076i	$0.253608\pi$
$648$	0	0
$649$	14647.4	0.885916
$650$	3937.21	0.237585
$651$	0	0
$652$	21793.6	1.30905
$653$	−21831.4	−1.30832	−0.654158	−	0.756358i	$-0.726977\pi$
$653$	−21831.4	−1.30832	−0.654158	+	0.756358i	$0.726977\pi$
$654$	0	0
$655$	4386.74	0.261686
$656$	5603.54	0.333508
$657$	0	0
$658$	−16563.1	−0.981304
$659$	8326.08	0.492167	0.246083	−	0.969249i	$-0.420856\pi$
$659$	8326.08	0.492167	0.246083	+	0.969249i	$0.420856\pi$
$660$	0	0
$661$	−10871.1	−0.639691	−0.319846	−	0.947470i	$-0.603631\pi$
$661$	−10871.1	−0.639691	−0.319846	+	0.947470i	$0.603631\pi$
$662$	−3512.74	−0.206233
$663$	0	0
$664$	13209.6	0.772035
$665$	−32615.4	−1.90191
$666$	0	0
$667$	38307.2	2.22378
$668$	1119.49	0.0648421
$669$	0	0
$670$	7616.02	0.439153
$671$	−26456.9	−1.52214
$672$	0	0
$673$	14437.3	0.826917	0.413458	−	0.910523i	$-0.364321\pi$
$673$	14437.3	0.826917	0.413458	+	0.910523i	$0.364321\pi$
$674$	−9294.79	−0.531190
$675$	0	0
$676$	−4639.99	−0.263996
$677$	−12990.9	−0.737490	−0.368745	−	0.929531i	$-0.620212\pi$
$677$	−12990.9	−0.737490	−0.368745	+	0.929531i	$0.620212\pi$
$678$	0	0
$679$	−751.758	−0.0424887
$680$	−4436.49	−0.250194
$681$	0	0
$682$	10650.7	0.598003
$683$	604.874	0.0338871	0.0169435	−	0.999856i	$-0.494606\pi$
$683$	604.874	0.0338871	0.0169435	+	0.999856i	$0.494606\pi$
$684$	0	0
$685$	7563.73	0.421891
$686$	−8867.19	−0.493515
$687$	0	0
$688$	−1304.78	−0.0723025
$689$	3151.37	0.174249
$690$	0	0
$691$	10878.6	0.598904	0.299452	−	0.954111i	$-0.403196\pi$
$691$	10878.6	0.598904	0.299452	+	0.954111i	$0.403196\pi$
$692$	−18389.3	−1.01020
$693$	0	0
$694$	−4683.22	−0.256157
$695$	−3000.79	−0.163779
$696$	0	0
$697$	−5515.34	−0.299725
$698$	12199.9	0.661567
$699$	0	0
$700$	13221.9	0.713918
$701$	−10181.3	−0.548561	−0.274280	−	0.961650i	$-0.588440\pi$
$701$	−10181.3	−0.548561	−0.274280	+	0.961650i	$0.588440\pi$
$702$	0	0
$703$	28315.8	1.51913
$704$	−2487.25	−0.133156
$705$	0	0
$706$	−1099.01	−0.0585861
$707$	−25808.3	−1.37287
$708$	0	0
$709$	−5968.59	−0.316157	−0.158078	−	0.987427i	$-0.550530\pi$
$709$	−5968.59	−0.316157	−0.158078	+	0.987427i	$0.550530\pi$
$710$	6681.58	0.353176
$711$	0	0
$712$	−19496.4	−1.02621
$713$	57396.7	3.01476
$714$	0	0
$715$	13145.4	0.687566
$716$	−13067.3	−0.682052
$717$	0	0
$718$	4374.34	0.227366
$719$	−27408.5	−1.42165	−0.710824	−	0.703370i	$-0.751678\pi$
$719$	−27408.5	−1.42165	−0.710824	+	0.703370i	$0.751678\pi$
$720$	0	0
$721$	16590.6	0.856957
$722$	−13429.1	−0.692216
$723$	0	0
$724$	−24266.9	−1.24568
$725$	12791.3	0.655251
$726$	0	0
$727$	31067.6	1.58492	0.792458	−	0.609927i	$-0.208801\pi$
$727$	31067.6	1.58492	0.792458	+	0.609927i	$0.208801\pi$
$728$	27590.0	1.40461
$729$	0	0
$730$	5534.93	0.280626
$731$	1284.24	0.0649785
$732$	0	0
$733$	−8221.82	−0.414297	−0.207149	−	0.978309i	$-0.566418\pi$
$733$	−8221.82	−0.414297	−0.207149	+	0.978309i	$0.566418\pi$
$734$	7954.44	0.400005
$735$	0	0
$736$	33386.2	1.67205
$737$	−26963.5	−1.34764
$738$	0	0
$739$	2949.53	0.146821	0.0734103	−	0.997302i	$-0.476612\pi$
$739$	2949.53	0.146821	0.0734103	+	0.997302i	$0.476612\pi$
$740$	10825.2	0.537761
$741$	0	0
$742$	−2046.71	−0.101263
$743$	8386.16	0.414076	0.207038	−	0.978333i	$-0.433618\pi$
$743$	8386.16	0.414076	0.207038	+	0.978333i	$0.433618\pi$
$744$	0	0
$745$	5248.00	0.258083
$746$	−990.040	−0.0485898
$747$	0	0
$748$	7160.94	0.350040
$749$	−61693.4	−3.00965
$750$	0	0
$751$	−3683.19	−0.178963	−0.0894816	−	0.995988i	$-0.528521\pi$
$751$	−3683.19	−0.178963	−0.0894816	+	0.995988i	$0.528521\pi$
$752$	16400.2	0.795284
$753$	0	0
$754$	12169.0	0.587755
$755$	18561.9	0.894751
$756$	0	0
$757$	15706.1	0.754091	0.377045	−	0.926195i	$-0.376940\pi$
$757$	15706.1	0.754091	0.377045	+	0.926195i	$0.376940\pi$
$758$	5509.31	0.263993
$759$	0	0
$760$	−17809.7	−0.850036
$761$	−16038.1	−0.763969	−0.381985	−	0.924169i	$-0.624759\pi$
$761$	−16038.1	−0.763969	−0.381985	+	0.924169i	$0.624759\pi$
$762$	0	0
$763$	7479.50	0.354884
$764$	−24520.2	−1.16114
$765$	0	0
$766$	6053.52	0.285539
$767$	25075.0	1.18045
$768$	0	0
$769$	14997.5	0.703282	0.351641	−	0.936135i	$-0.385624\pi$
$769$	14997.5	0.703282	0.351641	+	0.936135i	$0.385624\pi$
$770$	−8537.48	−0.399570
$771$	0	0
$772$	−3735.42	−0.174146
$773$	8378.86	0.389866	0.194933	−	0.980817i	$-0.437551\pi$
$773$	8378.86	0.389866	0.194933	+	0.980817i	$0.437551\pi$
$774$	0	0
$775$	19165.6	0.888319
$776$	−410.499	−0.0189898
$777$	0	0
$778$	11661.8	0.537400
$779$	−22140.6	−1.01832
$780$	0	0
$781$	−23655.2	−1.08380
$782$	−7463.24	−0.341285
$783$	0	0
$784$	20636.1	0.940056
$785$	−3220.70	−0.146435
$786$	0	0
$787$	−32322.1	−1.46399	−0.731995	−	0.681310i	$-0.761411\pi$
$787$	−32322.1	−1.46399	−0.731995	+	0.681310i	$0.761411\pi$
$788$	−36473.1	−1.64886
$789$	0	0
$790$	3951.21	0.177947
$791$	19702.1	0.885621
$792$	0	0
$793$	−45291.7	−2.02819
$794$	870.148	0.0388922
$795$	0	0
$796$	33726.3	1.50175
$797$	24625.7	1.09446	0.547232	−	0.836981i	$-0.315681\pi$
$797$	24625.7	1.09446	0.547232	+	0.836981i	$0.315681\pi$
$798$	0	0
$799$	−16142.1	−0.714725
$800$	11148.1	0.492682
$801$	0	0
$802$	−516.889	−0.0227581
$803$	−19595.7	−0.861165
$804$	0	0
$805$	−46008.4	−2.01439
$806$	18233.1	0.796815
$807$	0	0
$808$	−14092.7	−0.613587
$809$	−14324.5	−0.622523	−0.311261	−	0.950324i	$-0.600751\pi$
$809$	−14324.5	−0.622523	−0.311261	+	0.950324i	$0.600751\pi$
$810$	0	0
$811$	19442.0	0.841803	0.420901	−	0.907106i	$-0.361714\pi$
$811$	19442.0	0.841803	0.420901	+	0.907106i	$0.361714\pi$
$812$	40865.8	1.76614
$813$	0	0
$814$	7411.98	0.319152
$815$	−25322.4	−1.08835
$816$	0	0
$817$	5155.41	0.220765
$818$	1168.68	0.0499537
$819$	0	0
$820$	−8464.45	−0.360477
$821$	−36615.5	−1.55650	−0.778252	−	0.627952i	$-0.783894\pi$
$821$	−36615.5	−1.55650	−0.778252	+	0.627952i	$0.783894\pi$
$822$	0	0
$823$	45586.4	1.93079	0.965396	−	0.260789i	$-0.0839828\pi$
$823$	45586.4	1.93079	0.965396	+	0.260789i	$0.0839828\pi$
$824$	9059.34	0.383006
$825$	0	0
$826$	−16285.3	−0.686004
$827$	−103.414	−0.00434830	−0.00217415	−	0.999998i	$-0.500692\pi$
$827$	−103.414	−0.00434830	−0.00217415	+	0.999998i	$0.500692\pi$
$828$	0	0
$829$	16905.1	0.708251	0.354125	−	0.935198i	$-0.384779\pi$
$829$	16905.1	0.708251	0.354125	+	0.935198i	$0.384779\pi$
$830$	−6997.58	−0.292638
$831$	0	0
$832$	−4257.94	−0.177425
$833$	−20311.3	−0.844832
$834$	0	0
$835$	−1300.76	−0.0539098
$836$	28746.7	1.18926
$837$	0	0
$838$	7217.15	0.297509
$839$	12689.1	0.522141	0.261070	−	0.965320i	$-0.415924\pi$
$839$	12689.1	0.522141	0.261070	+	0.965320i	$0.415924\pi$
$840$	0	0
$841$	15145.7	0.621006
$842$	6643.53	0.271914
$843$	0	0
$844$	−28168.2	−1.14880
$845$	5391.29	0.219486
$846$	0	0
$847$	−10582.1	−0.429285
$848$	2026.57	0.0820670
$849$	0	0
$850$	−2492.08	−0.100562
$851$	39943.1	1.60897
$852$	0	0
$853$	−22808.3	−0.915524	−0.457762	−	0.889075i	$-0.651349\pi$
$853$	−22808.3	−0.915524	−0.457762	+	0.889075i	$0.651349\pi$
$854$	29415.4	1.17866
$855$	0	0
$856$	−33687.8	−1.34512
$857$	11022.1	0.439331	0.219666	−	0.975575i	$-0.429503\pi$
$857$	11022.1	0.439331	0.219666	+	0.975575i	$0.429503\pi$
$858$	0	0
$859$	6703.21	0.266252	0.133126	−	0.991099i	$-0.457498\pi$
$859$	6703.21	0.266252	0.133126	+	0.991099i	$0.457498\pi$
$860$	1970.93	0.0781492
$861$	0	0
$862$	−1526.58	−0.0603197
$863$	21097.1	0.832161	0.416080	−	0.909328i	$-0.363404\pi$
$863$	21097.1	0.832161	0.416080	+	0.909328i	$0.363404\pi$
$864$	0	0
$865$	21366.9	0.839879
$866$	−13962.8	−0.547894
$867$	0	0
$868$	61230.3	2.39435
$869$	−13988.7	−0.546070
$870$	0	0
$871$	−46159.0	−1.79568
$872$	4084.20	0.158611
$873$	0	0
$874$	−29960.2	−1.15952
$875$	−45213.6	−1.74686
$876$	0	0
$877$	−6161.84	−0.237252	−0.118626	−	0.992939i	$-0.537849\pi$
$877$	−6161.84	−0.237252	−0.118626	+	0.992939i	$0.537849\pi$
$878$	16237.8	0.624146
$879$	0	0
$880$	8453.49	0.323826
$881$	24168.0	0.924223	0.462111	−	0.886822i	$-0.347092\pi$
$881$	24168.0	0.924223	0.462111	+	0.886822i	$0.347092\pi$
$882$	0	0
$883$	31183.9	1.18847	0.594237	−	0.804290i	$-0.297454\pi$
$883$	31183.9	1.18847	0.594237	+	0.804290i	$0.297454\pi$
$884$	12258.9	0.466415
$885$	0	0
$886$	−14774.8	−0.560236
$887$	39183.5	1.48326	0.741631	−	0.670808i	$-0.234053\pi$
$887$	39183.5	1.48326	0.741631	+	0.670808i	$0.234053\pi$
$888$	0	0
$889$	−2169.91	−0.0818632
$890$	10327.9	0.388981
$891$	0	0
$892$	−23438.9	−0.879811
$893$	−64800.3	−2.42829
$894$	0	0
$895$	15183.2	0.567059
$896$	45269.6	1.68789
$897$	0	0
$898$	7637.50	0.283816
$899$	59236.0	2.19759
$900$	0	0
$901$	−1994.67	−0.0737539
$902$	−5795.57	−0.213937
$903$	0	0
$904$	10758.4	0.395817
$905$	28196.2	1.03566
$906$	0	0
$907$	11847.0	0.433709	0.216855	−	0.976204i	$-0.430420\pi$
$907$	11847.0	0.433709	0.216855	+	0.976204i	$0.430420\pi$
$908$	9851.43	0.360056
$909$	0	0
$910$	−14615.4	−0.532412
$911$	−15538.5	−0.565108	−0.282554	−	0.959251i	$-0.591182\pi$
$911$	−15538.5	−0.565108	−0.282554	+	0.959251i	$0.591182\pi$
$912$	0	0
$913$	24774.0	0.898027
$914$	−9287.26	−0.336100
$915$	0	0
$916$	23340.1	0.841897
$917$	17267.4	0.621831
$918$	0	0
$919$	−22944.3	−0.823571	−0.411786	−	0.911281i	$-0.635095\pi$
$919$	−22944.3	−0.823571	−0.411786	+	0.911281i	$0.635095\pi$
$920$	−25123.0	−0.900305
$921$	0	0
$922$	14226.0	0.508145
$923$	−40495.5	−1.44412
$924$	0	0
$925$	13337.6	0.474093
$926$	6210.70	0.220406
$927$	0	0
$928$	34456.1	1.21883
$929$	−17702.8	−0.625199	−0.312599	−	0.949885i	$-0.601200\pi$
$929$	−17702.8	−0.625199	−0.312599	+	0.949885i	$0.601200\pi$
$930$	0	0
$931$	−81537.2	−2.87032
$932$	−6897.78	−0.242429
$933$	0	0
$934$	−21501.4	−0.753262
$935$	−8320.43	−0.291024
$936$	0	0
$937$	−56334.3	−1.96410	−0.982049	−	0.188626i	$-0.939597\pi$
$937$	−56334.3	−1.96410	−0.982049	+	0.188626i	$0.939597\pi$
$938$	29978.7	1.04354
$939$	0	0
$940$	−24773.4	−0.859595
$941$	−31715.4	−1.09872	−0.549358	−	0.835587i	$-0.685128\pi$
$941$	−31715.4	−1.09872	−0.549358	+	0.835587i	$0.685128\pi$
$942$	0	0
$943$	−31232.3	−1.07854
$944$	16125.1	0.555962
$945$	0	0
$946$	1349.49	0.0463802
$947$	−25434.8	−0.872776	−0.436388	−	0.899758i	$-0.643743\pi$
$947$	−25434.8	−0.872776	−0.436388	+	0.899758i	$0.643743\pi$
$948$	0	0
$949$	−33546.0	−1.14747
$950$	−10004.1	−0.341660
$951$	0	0
$952$	−17463.2	−0.594523
$953$	−12373.4	−0.420580	−0.210290	−	0.977639i	$-0.567441\pi$
$953$	−12373.4	−0.420580	−0.210290	+	0.977639i	$0.567441\pi$
$954$	0	0
$955$	28490.5	0.965371
$956$	−16120.7	−0.545377
$957$	0	0
$958$	13564.3	0.457456
$959$	29772.9	1.00252
$960$	0	0
$961$	58964.0	1.97925
$962$	12688.6	0.425258
$963$	0	0
$964$	33607.6	1.12285
$965$	4340.25	0.144785
$966$	0	0
$967$	44702.7	1.48660	0.743299	−	0.668959i	$-0.233260\pi$
$967$	44702.7	1.48660	0.743299	+	0.668959i	$0.233260\pi$
$968$	−5778.37	−0.191863
$969$	0	0
$970$	217.456	0.00719803
$971$	49592.6	1.63903	0.819517	−	0.573054i	$-0.194242\pi$
$971$	49592.6	1.63903	0.819517	+	0.573054i	$0.194242\pi$
$972$	0	0
$973$	−11811.9	−0.389180
$974$	9500.20	0.312532
$975$	0	0
$976$	−29126.1	−0.955228
$977$	−28695.9	−0.939676	−0.469838	−	0.882753i	$-0.655688\pi$
$977$	−28695.9	−0.939676	−0.469838	+	0.882753i	$0.655688\pi$
$978$	0	0
$979$	−36564.6	−1.19368
$980$	−31172.0	−1.01607
$981$	0	0
$982$	8858.09	0.287854
$983$	13934.6	0.452130	0.226065	−	0.974112i	$-0.427414\pi$
$983$	13934.6	0.452130	0.226065	+	0.974112i	$0.427414\pi$
$984$	0	0
$985$	42378.8	1.37087
$986$	−7702.40	−0.248777
$987$	0	0
$988$	49211.7	1.58465
$989$	7272.39	0.233821
$990$	0	0
$991$	43393.1	1.39095	0.695473	−	0.718552i	$-0.255195\pi$
$991$	43393.1	1.39095	0.695473	+	0.718552i	$0.255195\pi$
$992$	51626.5	1.65236
$993$	0	0
$994$	26300.5	0.839236
$995$	−39187.2	−1.24856
$996$	0	0
$997$	12897.4	0.409694	0.204847	−	0.978794i	$-0.434330\pi$
$997$	12897.4	0.409694	0.204847	+	0.978794i	$0.434330\pi$
$998$	17782.5	0.564024
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.4.a.b.1.8		19
3.2	odd	2		501.4.a.b.1.12	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.4.a.b.1.12	✓	19	3.2	odd	2
1503.4.a.b.1.8		19	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1503.a (trivial)

\(f(q)\)	\(=\)	\(q-1.13862 q^{2} -6.70355 q^{4} +7.78898 q^{5} +30.6595 q^{7} +16.7417 q^{8} +O(q^{10})\)	\(q-1.13862 q^{2} -6.70355 q^{4} +7.78898 q^{5} +30.6595 q^{7} +16.7417 q^{8} -8.86867 q^{10} +31.3983 q^{11} +53.7510 q^{13} -34.9094 q^{14} +34.5660 q^{16} -34.0219 q^{17} -136.577 q^{19} -52.2139 q^{20} -35.7506 q^{22} -192.660 q^{23} -64.3317 q^{25} -61.2018 q^{26} -205.528 q^{28} -198.833 q^{29} -297.918 q^{31} -173.291 q^{32} +38.7380 q^{34} +238.806 q^{35} -207.325 q^{37} +155.509 q^{38} +130.401 q^{40} +162.111 q^{41} -37.7473 q^{43} -210.480 q^{44} +219.365 q^{46} +474.460 q^{47} +597.006 q^{49} +73.2492 q^{50} -360.323 q^{52} +58.6290 q^{53} +244.561 q^{55} +513.293 q^{56} +226.395 q^{58} +466.502 q^{59} -842.621 q^{61} +339.214 q^{62} -79.2161 q^{64} +418.666 q^{65} -858.756 q^{67} +228.068 q^{68} -271.909 q^{70} -753.392 q^{71} -624.100 q^{73} +236.063 q^{74} +915.550 q^{76} +962.656 q^{77} -445.525 q^{79} +269.234 q^{80} -184.582 q^{82} +789.023 q^{83} -264.996 q^{85} +42.9797 q^{86} +525.661 q^{88} -1164.54 q^{89} +1647.98 q^{91} +1291.50 q^{92} -540.228 q^{94} -1063.80 q^{95} -24.5196 q^{97} -679.761 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - q^{2} + 59 q^{4} - 12 q^{5} - 28 q^{7} + 3 q^{8}+O(q^{10}) \) 19 * q - q^2 + 59 * q^4 - 12 * q^5 - 28 * q^7 + 3 * q^8	\( 19 q - q^{2} + 59 q^{4} - 12 q^{5} - 28 q^{7} + 3 q^{8} - 69 q^{10} + 120 q^{11} - 92 q^{13} + 69 q^{14} + 83 q^{16} - 54 q^{17} - 400 q^{19} + 93 q^{20} - 273 q^{22} - 114 q^{23} + 171 q^{25} - q^{26} - 521 q^{28} - 72 q^{29} - 712 q^{31} - 42 q^{32} - 468 q^{34} + 306 q^{35} - 448 q^{37} - 50 q^{38} - 1030 q^{40} + 310 q^{41} - 852 q^{43} + 757 q^{44} - 2187 q^{46} + 1244 q^{47} + 47 q^{49} + 1670 q^{50} - 2548 q^{52} - 58 q^{53} - 2190 q^{55} + 2541 q^{56} - 3481 q^{58} + 2736 q^{59} - 2922 q^{61} - 486 q^{62} - 3677 q^{64} + 380 q^{65} - 2658 q^{67} + 1558 q^{68} - 4887 q^{70} + 636 q^{71} - 2304 q^{73} + 3137 q^{74} - 6536 q^{76} - 230 q^{77} - 2666 q^{79} + 1644 q^{80} - 1949 q^{82} + 2552 q^{83} - 2816 q^{85} + 4825 q^{86} - 5144 q^{88} + 1136 q^{89} - 6128 q^{91} + 755 q^{92} - 1776 q^{94} + 468 q^{95} - 3560 q^{97} - 1635 q^{98}+O(q^{100}) \) 19 * q - q^2 + 59 * q^4 - 12 * q^5 - 28 * q^7 + 3 * q^8 - 69 * q^10 + 120 * q^11 - 92 * q^13 + 69 * q^14 + 83 * q^16 - 54 * q^17 - 400 * q^19 + 93 * q^20 - 273 * q^22 - 114 * q^23 + 171 * q^25 - q^26 - 521 * q^28 - 72 * q^29 - 712 * q^31 - 42 * q^32 - 468 * q^34 + 306 * q^35 - 448 * q^37 - 50 * q^38 - 1030 * q^40 + 310 * q^41 - 852 * q^43 + 757 * q^44 - 2187 * q^46 + 1244 * q^47 + 47 * q^49 + 1670 * q^50 - 2548 * q^52 - 58 * q^53 - 2190 * q^55 + 2541 * q^56 - 3481 * q^58 + 2736 * q^59 - 2922 * q^61 - 486 * q^62 - 3677 * q^64 + 380 * q^65 - 2658 * q^67 + 1558 * q^68 - 4887 * q^70 + 636 * q^71 - 2304 * q^73 + 3137 * q^74 - 6536 * q^76 - 230 * q^77 - 2666 * q^79 + 1644 * q^80 - 1949 * q^82 + 2552 * q^83 - 2816 * q^85 + 4825 * q^86 - 5144 * q^88 + 1136 * q^89 - 6128 * q^91 + 755 * q^92 - 1776 * q^94 + 468 * q^95 - 3560 * q^97 - 1635 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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