LMFDB - Embedded newform 6003.2.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53344 q^{2} +4.41832 q^{4} -4.44002 q^{5} -2.94217 q^{7} -6.12666 q^{8} +O(q^{10})$	\(q-2.53344 q^{2} +4.41832 q^{4} -4.44002 q^{5} -2.94217 q^{7} -6.12666 q^{8} +11.2485 q^{10} +2.11774 q^{11} +4.95904 q^{13} +7.45382 q^{14} +6.68489 q^{16} +1.10516 q^{17} +3.34374 q^{19} -19.6174 q^{20} -5.36516 q^{22} +1.00000 q^{23} +14.7137 q^{25} -12.5634 q^{26} -12.9995 q^{28} -1.00000 q^{29} +0.450034 q^{31} -4.68245 q^{32} -2.79986 q^{34} +13.0633 q^{35} -3.22088 q^{37} -8.47117 q^{38} +27.2025 q^{40} -12.6572 q^{41} +3.22624 q^{43} +9.35684 q^{44} -2.53344 q^{46} +12.1886 q^{47} +1.65639 q^{49} -37.2764 q^{50} +21.9106 q^{52} +8.26682 q^{53} -9.40279 q^{55} +18.0257 q^{56} +2.53344 q^{58} +11.3155 q^{59} +3.82843 q^{61} -1.14013 q^{62} -1.50708 q^{64} -22.0182 q^{65} +12.7528 q^{67} +4.88296 q^{68} -33.0951 q^{70} +6.51894 q^{71} -11.8183 q^{73} +8.15990 q^{74} +14.7737 q^{76} -6.23075 q^{77} +7.44994 q^{79} -29.6810 q^{80} +32.0663 q^{82} -7.31271 q^{83} -4.90694 q^{85} -8.17348 q^{86} -12.9747 q^{88} -12.2352 q^{89} -14.5904 q^{91} +4.41832 q^{92} -30.8791 q^{94} -14.8463 q^{95} -8.07116 q^{97} -4.19635 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) $ 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8	$ 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8} + 8 q^{10} + 36 q^{13} - 7 q^{14} + 47 q^{16} - 18 q^{17} + 16 q^{19} + 25 q^{22} + 30 q^{23} + 56 q^{25} - 11 q^{26} + 27 q^{28} - 30 q^{29} + 14 q^{31} + 7 q^{32} + 3 q^{34} + 22 q^{35} + 40 q^{37} - 6 q^{38} + 30 q^{40} - 14 q^{41} + 34 q^{43} - 5 q^{44} - q^{46} + 2 q^{47} + 74 q^{49} + 21 q^{50} + 71 q^{52} - 16 q^{53} + 22 q^{55} - 14 q^{56} + q^{58} + 32 q^{59} + 46 q^{61} - 20 q^{62} + 68 q^{64} - 12 q^{65} + 14 q^{67} - 27 q^{68} + 32 q^{71} + 50 q^{73} + 26 q^{74} + 56 q^{76} - 34 q^{77} + 16 q^{79} - 2 q^{80} + 38 q^{82} + 14 q^{83} + 38 q^{85} - 10 q^{86} + 40 q^{88} + 2 q^{89} + 32 q^{91} + 37 q^{92} + 29 q^{94} + 28 q^{95} + 56 q^{97} - 8 q^{98}+O(q^{100}) $ 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8 + 8 * q^10 + 36 * q^13 - 7 * q^14 + 47 * q^16 - 18 * q^17 + 16 * q^19 + 25 * q^22 + 30 * q^23 + 56 * q^25 - 11 * q^26 + 27 * q^28 - 30 * q^29 + 14 * q^31 + 7 * q^32 + 3 * q^34 + 22 * q^35 + 40 * q^37 - 6 * q^38 + 30 * q^40 - 14 * q^41 + 34 * q^43 - 5 * q^44 - q^46 + 2 * q^47 + 74 * q^49 + 21 * q^50 + 71 * q^52 - 16 * q^53 + 22 * q^55 - 14 * q^56 + q^58 + 32 * q^59 + 46 * q^61 - 20 * q^62 + 68 * q^64 - 12 * q^65 + 14 * q^67 - 27 * q^68 + 32 * q^71 + 50 * q^73 + 26 * q^74 + 56 * q^76 - 34 * q^77 + 16 * q^79 - 2 * q^80 + 38 * q^82 + 14 * q^83 + 38 * q^85 - 10 * q^86 + 40 * q^88 + 2 * q^89 + 32 * q^91 + 37 * q^92 + 29 * q^94 + 28 * q^95 + 56 * q^97 - 8 * 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 $

$2$	−2.53344	−1.79141	−0.895706	−	0.444646i	$-0.853329\pi$
$2$	−2.53344	−1.79141	−0.895706	+	0.444646i	$0.853329\pi$
$3$	0	0
$3$	0	0
$4$	4.41832	2.20916
$5$	−4.44002	−1.98564	−0.992818	−	0.119636i	$-0.961827\pi$
$5$	−4.44002	−1.98564	−0.992818	+	0.119636i	$0.961827\pi$
$6$	0	0
$7$	−2.94217	−1.11204	−0.556019	−	0.831170i	$-0.687672\pi$
$7$	−2.94217	−1.11204	−0.556019	+	0.831170i	$0.687672\pi$
$8$	−6.12666	−2.16610
$9$	0	0
$10$	11.2485	3.55709
$11$	2.11774	0.638522	0.319261	−	0.947667i	$-0.396565\pi$
$11$	2.11774	0.638522	0.319261	+	0.947667i	$0.396565\pi$
$12$	0	0
$13$	4.95904	1.37539	0.687695	−	0.726000i	$-0.258623\pi$
$13$	4.95904	1.37539	0.687695	+	0.726000i	$0.258623\pi$
$14$	7.45382	1.99212
$15$	0	0
$16$	6.68489	1.67122
$17$	1.10516	0.268041	0.134021	−	0.990979i	$-0.457211\pi$
$17$	1.10516	0.268041	0.134021	+	0.990979i	$0.457211\pi$
$18$	0	0
$19$	3.34374	0.767107	0.383554	−	0.923519i	$-0.374700\pi$
$19$	3.34374	0.767107	0.383554	+	0.923519i	$0.374700\pi$
$20$	−19.6174	−4.38658
$21$	0	0
$22$	−5.36516	−1.14386
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	14.7137	2.94275
$26$	−12.5634	−2.46389
$27$	0	0
$28$	−12.9995	−2.45667
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	0.450034	0.0808284	0.0404142	−	0.999183i	$-0.487132\pi$
$31$	0.450034	0.0808284	0.0404142	+	0.999183i	$0.487132\pi$
$32$	−4.68245	−0.827748
$33$	0	0
$34$	−2.79986	−0.480173
$35$	13.0633	2.20810
$36$	0	0
$37$	−3.22088	−0.529509	−0.264755	−	0.964316i	$-0.585291\pi$
$37$	−3.22088	−0.529509	−0.264755	+	0.964316i	$0.585291\pi$
$38$	−8.47117	−1.37421
$39$	0	0
$40$	27.2025	4.30109
$41$	−12.6572	−1.97673	−0.988363	−	0.152112i	$-0.951392\pi$
$41$	−12.6572	−1.97673	−0.988363	+	0.152112i	$0.951392\pi$
$42$	0	0
$43$	3.22624	0.491996	0.245998	−	0.969270i	$-0.420884\pi$
$43$	3.22624	0.491996	0.245998	+	0.969270i	$0.420884\pi$
$44$	9.35684	1.41060
$45$	0	0
$46$	−2.53344	−0.373535
$47$	12.1886	1.77789	0.888944	−	0.458015i	$-0.151440\pi$
$47$	12.1886	1.77789	0.888944	+	0.458015i	$0.151440\pi$
$48$	0	0
$49$	1.65639	0.236626
$50$	−37.2764	−5.27168
$51$	0	0
$52$	21.9106	3.03845
$53$	8.26682	1.13554	0.567768	−	0.823189i	$-0.307807\pi$
$53$	8.26682	1.13554	0.567768	+	0.823189i	$0.307807\pi$
$54$	0	0
$55$	−9.40279	−1.26787
$56$	18.0257	2.40879
$57$	0	0
$58$	2.53344	0.332657
$59$	11.3155	1.47315	0.736573	−	0.676358i	$-0.236443\pi$
$59$	11.3155	1.47315	0.736573	+	0.676358i	$0.236443\pi$
$60$	0	0
$61$	3.82843	0.490180	0.245090	−	0.969500i	$-0.421183\pi$
$61$	3.82843	0.490180	0.245090	+	0.969500i	$0.421183\pi$
$62$	−1.14013	−0.144797
$63$	0	0
$64$	−1.50708	−0.188385
$65$	−22.0182	−2.73102
$66$	0	0
$67$	12.7528	1.55800	0.778999	−	0.627025i	$-0.215728\pi$
$67$	12.7528	1.55800	0.778999	+	0.627025i	$0.215728\pi$
$68$	4.88296	0.592146
$69$	0	0
$70$	−33.0951	−3.95562
$71$	6.51894	0.773656	0.386828	−	0.922152i	$-0.373571\pi$
$71$	6.51894	0.773656	0.386828	+	0.922152i	$0.373571\pi$
$72$	0	0
$73$	−11.8183	−1.38323	−0.691615	−	0.722266i	$-0.743101\pi$
$73$	−11.8183	−1.38323	−0.691615	+	0.722266i	$0.743101\pi$
$74$	8.15990	0.948569
$75$	0	0
$76$	14.7737	1.69466
$77$	−6.23075	−0.710060
$78$	0	0
$79$	7.44994	0.838184	0.419092	−	0.907944i	$-0.362349\pi$
$79$	7.44994	0.838184	0.419092	+	0.907944i	$0.362349\pi$
$80$	−29.6810	−3.31844
$81$	0	0
$82$	32.0663	3.54113
$83$	−7.31271	−0.802674	−0.401337	−	0.915930i	$-0.631454\pi$
$83$	−7.31271	−0.802674	−0.401337	+	0.915930i	$0.631454\pi$
$84$	0	0
$85$	−4.90694	−0.532233
$86$	−8.17348	−0.881369
$87$	0	0
$88$	−12.9747	−1.38310
$89$	−12.2352	−1.29693	−0.648463	−	0.761246i	$-0.724588\pi$
$89$	−12.2352	−1.29693	−0.648463	+	0.761246i	$0.724588\pi$
$90$	0	0
$91$	−14.5904	−1.52948
$92$	4.41832	0.460641
$93$	0	0
$94$	−30.8791	−3.18493
$95$	−14.8463	−1.52320
$96$	0	0
$97$	−8.07116	−0.819502	−0.409751	−	0.912197i	$-0.634384\pi$
$97$	−8.07116	−0.819502	−0.409751	+	0.912197i	$0.634384\pi$
$98$	−4.19635	−0.423896
$99$	0	0
$100$	65.0100	6.50100
$101$	1.22599	0.121991	0.0609955	−	0.998138i	$-0.480572\pi$
$101$	1.22599	0.121991	0.0609955	+	0.998138i	$0.480572\pi$
$102$	0	0
$103$	12.5667	1.23823	0.619116	−	0.785300i	$-0.287491\pi$
$103$	12.5667	1.23823	0.619116	+	0.785300i	$0.287491\pi$
$104$	−30.3823	−2.97923
$105$	0	0
$106$	−20.9435	−2.03421
$107$	−11.2796	−1.09044	−0.545218	−	0.838295i	$-0.683553\pi$
$107$	−11.2796	−1.09044	−0.545218	+	0.838295i	$0.683553\pi$
$108$	0	0
$109$	−18.8465	−1.80517	−0.902584	−	0.430514i	$-0.858332\pi$
$109$	−18.8465	−1.80517	−0.902584	+	0.430514i	$0.858332\pi$
$110$	23.8214	2.27128
$111$	0	0
$112$	−19.6681	−1.85846
$113$	0.830218	0.0781003	0.0390502	−	0.999237i	$-0.487567\pi$
$113$	0.830218	0.0781003	0.0390502	+	0.999237i	$0.487567\pi$
$114$	0	0
$115$	−4.44002	−0.414034
$116$	−4.41832	−0.410230
$117$	0	0
$118$	−28.6670	−2.63901
$119$	−3.25158	−0.298072
$120$	0	0
$121$	−6.51519	−0.592290
$122$	−9.69909	−0.878114
$123$	0	0
$124$	1.98839	0.178563
$125$	−43.1292	−3.85759
$126$	0	0
$127$	8.48959	0.753329	0.376664	−	0.926350i	$-0.377071\pi$
$127$	8.48959	0.753329	0.376664	+	0.926350i	$0.377071\pi$
$128$	13.1830	1.16522
$129$	0	0
$130$	55.7818	4.89239
$131$	8.06851	0.704949	0.352475	−	0.935821i	$-0.385340\pi$
$131$	8.06851	0.704949	0.352475	+	0.935821i	$0.385340\pi$
$132$	0	0
$133$	−9.83787	−0.853052
$134$	−32.3084	−2.79102
$135$	0	0
$136$	−6.77096	−0.580605
$137$	0.0218169	0.00186394	0.000931971	−	1.00000i	$-0.499703\pi$
$137$	0.0218169	0.00186394	0.000931971		1.00000i	$0.499703\pi$
$138$	0	0
$139$	−14.9810	−1.27068	−0.635338	−	0.772234i	$-0.719139\pi$
$139$	−14.9810	−1.27068	−0.635338	+	0.772234i	$0.719139\pi$
$140$	57.7178	4.87804
$141$	0	0
$142$	−16.5154	−1.38594
$143$	10.5019	0.878217
$144$	0	0
$145$	4.44002	0.368723
$146$	29.9410	2.47794
$147$	0	0
$148$	−14.2309	−1.16977
$149$	−9.68343	−0.793298	−0.396649	−	0.917970i	$-0.629827\pi$
$149$	−9.68343	−0.793298	−0.396649	+	0.917970i	$0.629827\pi$
$150$	0	0
$151$	−4.56663	−0.371627	−0.185814	−	0.982585i	$-0.559492\pi$
$151$	−4.56663	−0.371627	−0.185814	+	0.982585i	$0.559492\pi$
$152$	−20.4860	−1.66163
$153$	0	0
$154$	15.7852	1.27201
$155$	−1.99816	−0.160496
$156$	0	0
$157$	11.0244	0.879845	0.439922	−	0.898036i	$-0.355006\pi$
$157$	11.0244	0.879845	0.439922	+	0.898036i	$0.355006\pi$
$158$	−18.8740	−1.50153
$159$	0	0
$160$	20.7902	1.64361
$161$	−2.94217	−0.231876
$162$	0	0
$163$	−22.0639	−1.72818	−0.864090	−	0.503337i	$-0.832106\pi$
$163$	−22.0639	−1.72818	−0.864090	+	0.503337i	$0.832106\pi$
$164$	−55.9236	−4.36690
$165$	0	0
$166$	18.5263	1.43792
$167$	23.6472	1.82987	0.914937	−	0.403598i	$-0.132240\pi$
$167$	23.6472	1.82987	0.914937	+	0.403598i	$0.132240\pi$
$168$	0	0
$169$	11.5921	0.891697
$170$	12.4314	0.953448
$171$	0	0
$172$	14.2545	1.08690
$173$	−0.440563	−0.0334954	−0.0167477	−	0.999860i	$-0.505331\pi$
$173$	−0.440563	−0.0334954	−0.0167477	+	0.999860i	$0.505331\pi$
$174$	0	0
$175$	−43.2904	−3.27245
$176$	14.1569	1.06711
$177$	0	0
$178$	30.9971	2.32333
$179$	−22.0425	−1.64753	−0.823766	−	0.566929i	$-0.808131\pi$
$179$	−22.0425	−1.64753	−0.823766	+	0.566929i	$0.808131\pi$
$180$	0	0
$181$	5.02523	0.373523	0.186761	−	0.982405i	$-0.440201\pi$
$181$	5.02523	0.373523	0.186761	+	0.982405i	$0.440201\pi$
$182$	36.9638	2.73994
$183$	0	0
$184$	−6.12666	−0.451663
$185$	14.3008	1.05141
$186$	0	0
$187$	2.34045	0.171150
$188$	53.8531	3.92764
$189$	0	0
$190$	37.6121	2.72867
$191$	4.39845	0.318261	0.159130	−	0.987258i	$-0.449131\pi$
$191$	4.39845	0.318261	0.159130	+	0.987258i	$0.449131\pi$
$192$	0	0
$193$	25.1737	1.81204	0.906022	−	0.423230i	$-0.139104\pi$
$193$	25.1737	1.81204	0.906022	+	0.423230i	$0.139104\pi$
$194$	20.4478	1.46807
$195$	0	0
$196$	7.31844	0.522745
$197$	−10.0048	−0.712814	−0.356407	−	0.934331i	$-0.615998\pi$
$197$	−10.0048	−0.712814	−0.356407	+	0.934331i	$0.615998\pi$
$198$	0	0
$199$	−17.8870	−1.26797	−0.633987	−	0.773344i	$-0.718583\pi$
$199$	−17.8870	−1.26797	−0.633987	+	0.773344i	$0.718583\pi$
$200$	−90.1461	−6.37429
$201$	0	0
$202$	−3.10598	−0.218536
$203$	2.94217	0.206500
$204$	0	0
$205$	56.1983	3.92506
$206$	−31.8369	−2.21818
$207$	0	0
$208$	33.1506	2.29858
$209$	7.08117	0.489815
$210$	0	0
$211$	−2.71963	−0.187227	−0.0936136	−	0.995609i	$-0.529842\pi$
$211$	−2.71963	−0.187227	−0.0936136	+	0.995609i	$0.529842\pi$
$212$	36.5254	2.50858
$213$	0	0
$214$	28.5761	1.95342
$215$	−14.3245	−0.976926
$216$	0	0
$217$	−1.32408	−0.0898842
$218$	47.7465	3.23380
$219$	0	0
$220$	−41.5445	−2.80093
$221$	5.48055	0.368661
$222$	0	0
$223$	−17.6250	−1.18026	−0.590129	−	0.807309i	$-0.700923\pi$
$223$	−17.6250	−1.18026	−0.590129	+	0.807309i	$0.700923\pi$
$224$	13.7766	0.920486
$225$	0	0
$226$	−2.10331	−0.139910
$227$	0.819513	0.0543930	0.0271965	−	0.999630i	$-0.491342\pi$
$227$	0.819513	0.0543930	0.0271965	+	0.999630i	$0.491342\pi$
$228$	0	0
$229$	10.2088	0.674618	0.337309	−	0.941394i	$-0.390483\pi$
$229$	10.2088	0.674618	0.337309	+	0.941394i	$0.390483\pi$
$230$	11.2485	0.741705
$231$	0	0
$232$	6.12666	0.402235
$233$	24.7668	1.62253	0.811263	−	0.584681i	$-0.198780\pi$
$233$	24.7668	1.62253	0.811263	+	0.584681i	$0.198780\pi$
$234$	0	0
$235$	−54.1175	−3.53024
$236$	49.9953	3.25441
$237$	0	0
$238$	8.23769	0.533970
$239$	4.62792	0.299355	0.149678	−	0.988735i	$-0.452176\pi$
$239$	4.62792	0.299355	0.149678	+	0.988735i	$0.452176\pi$
$240$	0	0
$241$	1.30386	0.0839891	0.0419946	−	0.999118i	$-0.486629\pi$
$241$	1.30386	0.0839891	0.0419946	+	0.999118i	$0.486629\pi$
$242$	16.5058	1.06103
$243$	0	0
$244$	16.9152	1.08288
$245$	−7.35438	−0.469854
$246$	0	0
$247$	16.5817	1.05507
$248$	−2.75720	−0.175083
$249$	0	0
$250$	109.265	6.91054
$251$	21.8275	1.37774	0.688869	−	0.724886i	$-0.258108\pi$
$251$	21.8275	1.37774	0.688869	+	0.724886i	$0.258108\pi$
$252$	0	0
$253$	2.11774	0.133141
$254$	−21.5079	−1.34952
$255$	0	0
$256$	−30.3842	−1.89901
$257$	15.9666	0.995966	0.497983	−	0.867187i	$-0.334074\pi$
$257$	15.9666	0.995966	0.497983	+	0.867187i	$0.334074\pi$
$258$	0	0
$259$	9.47638	0.588834
$260$	−97.2834	−6.03326
$261$	0	0
$262$	−20.4411	−1.26285
$263$	−7.14851	−0.440796	−0.220398	−	0.975410i	$-0.570736\pi$
$263$	−7.14851	−0.440796	−0.220398	+	0.975410i	$0.570736\pi$
$264$	0	0
$265$	−36.7048	−2.25476
$266$	24.9237	1.52817
$267$	0	0
$268$	56.3458	3.44187
$269$	25.5977	1.56072	0.780358	−	0.625333i	$-0.215037\pi$
$269$	25.5977	1.56072	0.780358	+	0.625333i	$0.215037\pi$
$270$	0	0
$271$	6.99729	0.425055	0.212528	−	0.977155i	$-0.431830\pi$
$271$	6.99729	0.425055	0.212528	+	0.977155i	$0.431830\pi$
$272$	7.38790	0.447957
$273$	0	0
$274$	−0.0552717	−0.00333909
$275$	31.1599	1.87901
$276$	0	0
$277$	1.90160	0.114256	0.0571281	−	0.998367i	$-0.481806\pi$
$277$	1.90160	0.114256	0.0571281	+	0.998367i	$0.481806\pi$
$278$	37.9536	2.27630
$279$	0	0
$280$	−80.0344	−4.78297
$281$	14.3399	0.855448	0.427724	−	0.903909i	$-0.359315\pi$
$281$	14.3399	0.855448	0.427724	+	0.903909i	$0.359315\pi$
$282$	0	0
$283$	−19.8651	−1.18086	−0.590430	−	0.807089i	$-0.701042\pi$
$283$	−19.8651	−1.18086	−0.590430	+	0.807089i	$0.701042\pi$
$284$	28.8028	1.70913
$285$	0	0
$286$	−26.6060	−1.57325
$287$	37.2398	2.19819
$288$	0	0
$289$	−15.7786	−0.928154
$290$	−11.2485	−0.660535
$291$	0	0
$292$	−52.2171	−3.05578
$293$	−5.64553	−0.329815	−0.164908	−	0.986309i	$-0.552733\pi$
$293$	−5.64553	−0.329815	−0.164908	+	0.986309i	$0.552733\pi$
$294$	0	0
$295$	−50.2408	−2.92513
$296$	19.7332	1.14697
$297$	0	0
$298$	24.5324	1.42112
$299$	4.95904	0.286789
$300$	0	0
$301$	−9.49215	−0.547118
$302$	11.5693	0.665738
$303$	0	0
$304$	22.3526	1.28201
$305$	−16.9983	−0.973318
$306$	0	0
$307$	8.42372	0.480767	0.240384	−	0.970678i	$-0.422727\pi$
$307$	8.42372	0.480767	0.240384	+	0.970678i	$0.422727\pi$
$308$	−27.5294	−1.56864
$309$	0	0
$310$	5.06221	0.287514
$311$	19.9578	1.13170	0.565851	−	0.824508i	$-0.308548\pi$
$311$	19.9578	1.13170	0.565851	+	0.824508i	$0.308548\pi$
$312$	0	0
$313$	25.5893	1.44639	0.723196	−	0.690643i	$-0.242672\pi$
$313$	25.5893	1.44639	0.723196	+	0.690643i	$0.242672\pi$
$314$	−27.9297	−1.57616
$315$	0	0
$316$	32.9162	1.85168
$317$	−22.6572	−1.27255	−0.636277	−	0.771461i	$-0.719526\pi$
$317$	−22.6572	−1.27255	−0.636277	+	0.771461i	$0.719526\pi$
$318$	0	0
$319$	−2.11774	−0.118571
$320$	6.69146	0.374064
$321$	0	0
$322$	7.45382	0.415385
$323$	3.69538	0.205617
$324$	0	0
$325$	72.9660	4.04743
$326$	55.8976	3.09588
$327$	0	0
$328$	77.5465	4.28179
$329$	−35.8609	−1.97708
$330$	0	0
$331$	9.61045	0.528238	0.264119	−	0.964490i	$-0.414919\pi$
$331$	9.61045	0.528238	0.264119	+	0.964490i	$0.414919\pi$
$332$	−32.3099	−1.77323
$333$	0	0
$334$	−59.9087	−3.27806
$335$	−56.6225	−3.09362
$336$	0	0
$337$	−18.1093	−0.986477	−0.493238	−	0.869894i	$-0.664187\pi$
$337$	−18.1093	−0.986477	−0.493238	+	0.869894i	$0.664187\pi$
$338$	−29.3678	−1.59740
$339$	0	0
$340$	−21.6804	−1.17579
$341$	0.953054	0.0516107
$342$	0	0
$343$	15.7218	0.848900
$344$	−19.7661	−1.06571
$345$	0	0
$346$	1.11614	0.0600041
$347$	−0.717163	−0.0384993	−0.0192497	−	0.999815i	$-0.506128\pi$
$347$	−0.717163	−0.0384993	−0.0192497	+	0.999815i	$0.506128\pi$
$348$	0	0
$349$	3.00291	0.160742	0.0803710	−	0.996765i	$-0.474389\pi$
$349$	3.00291	0.160742	0.0803710	+	0.996765i	$0.474389\pi$
$350$	109.674	5.86230
$351$	0	0
$352$	−9.91620	−0.528535
$353$	20.6107	1.09700	0.548499	−	0.836151i	$-0.315200\pi$
$353$	20.6107	1.09700	0.548499	+	0.836151i	$0.315200\pi$
$354$	0	0
$355$	−28.9442	−1.53620
$356$	−54.0589	−2.86511
$357$	0	0
$358$	55.8433	2.95141
$359$	−19.5454	−1.03157	−0.515783	−	0.856719i	$-0.672499\pi$
$359$	−19.5454	−1.03157	−0.515783	+	0.856719i	$0.672499\pi$
$360$	0	0
$361$	−7.81938	−0.411547
$362$	−12.7311	−0.669133
$363$	0	0
$364$	−64.4648	−3.37887
$365$	52.4736	2.74659
$366$	0	0
$367$	3.70867	0.193591	0.0967955	−	0.995304i	$-0.469141\pi$
$367$	3.70867	0.193591	0.0967955	+	0.995304i	$0.469141\pi$
$368$	6.68489	0.348474
$369$	0	0
$370$	−36.2301	−1.88351
$371$	−24.3224	−1.26276
$372$	0	0
$373$	−25.7046	−1.33093	−0.665466	−	0.746428i	$-0.731767\pi$
$373$	−25.7046	−1.33093	−0.665466	+	0.746428i	$0.731767\pi$
$374$	−5.92938	−0.306601
$375$	0	0
$376$	−74.6753	−3.85109
$377$	−4.95904	−0.255403
$378$	0	0
$379$	0.294547	0.0151299	0.00756494	−	0.999971i	$-0.497592\pi$
$379$	0.294547	0.0151299	0.00756494	+	0.999971i	$0.497592\pi$
$380$	−65.5955	−3.36498
$381$	0	0
$382$	−11.1432	−0.570136
$383$	−1.81469	−0.0927262	−0.0463631	−	0.998925i	$-0.514763\pi$
$383$	−1.81469	−0.0927262	−0.0463631	+	0.998925i	$0.514763\pi$
$384$	0	0
$385$	27.6646	1.40992
$386$	−63.7761	−3.24612
$387$	0	0
$388$	−35.6610	−1.81041
$389$	−20.1583	−1.02207	−0.511033	−	0.859561i	$-0.670737\pi$
$389$	−20.1583	−1.02207	−0.511033	+	0.859561i	$0.670737\pi$
$390$	0	0
$391$	1.10516	0.0558905
$392$	−10.1481	−0.512557
$393$	0	0
$394$	25.3466	1.27694
$395$	−33.0778	−1.66433
$396$	0	0
$397$	18.3145	0.919178	0.459589	−	0.888132i	$-0.347997\pi$
$397$	18.3145	0.919178	0.459589	+	0.888132i	$0.347997\pi$
$398$	45.3155	2.27146
$399$	0	0
$400$	98.3598	4.91799
$401$	21.1607	1.05672	0.528359	−	0.849021i	$-0.322808\pi$
$401$	21.1607	1.05672	0.528359	+	0.849021i	$0.322808\pi$
$402$	0	0
$403$	2.23173	0.111171
$404$	5.41683	0.269497
$405$	0	0
$406$	−7.45382	−0.369927
$407$	−6.82098	−0.338103
$408$	0	0
$409$	−5.51231	−0.272566	−0.136283	−	0.990670i	$-0.543516\pi$
$409$	−5.51231	−0.272566	−0.136283	+	0.990670i	$0.543516\pi$
$410$	−142.375	−7.03140
$411$	0	0
$412$	55.5236	2.73545
$413$	−33.2920	−1.63819
$414$	0	0
$415$	32.4685	1.59382
$416$	−23.2204	−1.13848
$417$	0	0
$418$	−17.9397	−0.877460
$419$	18.1880	0.888543	0.444272	−	0.895892i	$-0.353462\pi$
$419$	18.1880	0.888543	0.444272	+	0.895892i	$0.353462\pi$
$420$	0	0
$421$	−10.1946	−0.496856	−0.248428	−	0.968650i	$-0.579914\pi$
$421$	−10.1946	−0.496856	−0.248428	+	0.968650i	$0.579914\pi$
$422$	6.89002	0.335401
$423$	0	0
$424$	−50.6480	−2.45969
$425$	16.2611	0.788779
$426$	0	0
$427$	−11.2639	−0.545098
$428$	−49.8366	−2.40894
$429$	0	0
$430$	36.2904	1.75008
$431$	0.641700	0.0309096	0.0154548	−	0.999881i	$-0.495080\pi$
$431$	0.641700	0.0309096	0.0154548	+	0.999881i	$0.495080\pi$
$432$	0	0
$433$	2.59124	0.124527	0.0622636	−	0.998060i	$-0.480168\pi$
$433$	2.59124	0.124527	0.0622636	+	0.998060i	$0.480168\pi$
$434$	3.35447	0.161020
$435$	0	0
$436$	−83.2699	−3.98790
$437$	3.34374	0.159953
$438$	0	0
$439$	0.552183	0.0263543	0.0131771	−	0.999913i	$-0.495805\pi$
$439$	0.552183	0.0263543	0.0131771	+	0.999913i	$0.495805\pi$
$440$	57.6077	2.74634
$441$	0	0
$442$	−13.8846	−0.660425
$443$	39.7633	1.88921	0.944605	−	0.328210i	$-0.106445\pi$
$443$	39.7633	1.88921	0.944605	+	0.328210i	$0.106445\pi$
$444$	0	0
$445$	54.3244	2.57522
$446$	44.6519	2.11433
$447$	0	0
$448$	4.43409	0.209491
$449$	−17.3139	−0.817092	−0.408546	−	0.912738i	$-0.633964\pi$
$449$	−17.3139	−0.817092	−0.408546	+	0.912738i	$0.633964\pi$
$450$	0	0
$451$	−26.8047	−1.26218
$452$	3.66817	0.172536
$453$	0	0
$454$	−2.07619	−0.0974403
$455$	64.7814	3.03700
$456$	0	0
$457$	0.841205	0.0393499	0.0196750	−	0.999806i	$-0.493737\pi$
$457$	0.841205	0.0393499	0.0196750	+	0.999806i	$0.493737\pi$
$458$	−25.8634	−1.20852
$459$	0	0
$460$	−19.6174	−0.914666
$461$	−15.8075	−0.736229	−0.368114	−	0.929780i	$-0.619997\pi$
$461$	−15.8075	−0.736229	−0.368114	+	0.929780i	$0.619997\pi$
$462$	0	0
$463$	38.4993	1.78922	0.894608	−	0.446852i	$-0.147455\pi$
$463$	38.4993	1.78922	0.894608	+	0.446852i	$0.147455\pi$
$464$	−6.68489	−0.310338
$465$	0	0
$466$	−62.7452	−2.90661
$467$	−20.1242	−0.931238	−0.465619	−	0.884985i	$-0.654168\pi$
$467$	−20.1242	−0.931238	−0.465619	+	0.884985i	$0.654168\pi$
$468$	0	0
$469$	−37.5209	−1.73255
$470$	137.104	6.32411
$471$	0	0
$472$	−69.3260	−3.19099
$473$	6.83232	0.314151
$474$	0	0
$475$	49.1990	2.25740
$476$	−14.3665	−0.658488
$477$	0	0
$478$	−11.7246	−0.536269
$479$	7.71639	0.352571	0.176286	−	0.984339i	$-0.443592\pi$
$479$	7.71639	0.352571	0.176286	+	0.984339i	$0.443592\pi$
$480$	0	0
$481$	−15.9725	−0.728281
$482$	−3.30326	−0.150459
$483$	0	0
$484$	−28.7862	−1.30846
$485$	35.8361	1.62723
$486$	0	0
$487$	7.68391	0.348191	0.174096	−	0.984729i	$-0.444300\pi$
$487$	7.68391	0.348191	0.174096	+	0.984729i	$0.444300\pi$
$488$	−23.4555	−1.06178
$489$	0	0
$490$	18.6319	0.841702
$491$	5.92167	0.267241	0.133621	−	0.991033i	$-0.457340\pi$
$491$	5.92167	0.267241	0.133621	+	0.991033i	$0.457340\pi$
$492$	0	0
$493$	−1.10516	−0.0497740
$494$	−42.0089	−1.89007
$495$	0	0
$496$	3.00843	0.135082
$497$	−19.1799	−0.860335
$498$	0	0
$499$	12.9844	0.581262	0.290631	−	0.956835i	$-0.406135\pi$
$499$	12.9844	0.581262	0.290631	+	0.956835i	$0.406135\pi$
$500$	−190.558	−8.52203
$501$	0	0
$502$	−55.2986	−2.46810
$503$	10.6449	0.474632	0.237316	−	0.971433i	$-0.423732\pi$
$503$	10.6449	0.474632	0.237316	+	0.971433i	$0.423732\pi$
$504$	0	0
$505$	−5.44343	−0.242230
$506$	−5.36516	−0.238511
$507$	0	0
$508$	37.5097	1.66422
$509$	4.30842	0.190967	0.0954836	−	0.995431i	$-0.469560\pi$
$509$	4.30842	0.190967	0.0954836	+	0.995431i	$0.469560\pi$
$510$	0	0
$511$	34.7716	1.53820
$512$	50.6105	2.23669
$513$	0	0
$514$	−40.4503	−1.78419
$515$	−55.7962	−2.45868
$516$	0	0
$517$	25.8122	1.13522
$518$	−24.0078	−1.05484
$519$	0	0
$520$	134.898	5.91567
$521$	19.1517	0.839052	0.419526	−	0.907743i	$-0.362196\pi$
$521$	19.1517	0.839052	0.419526	+	0.907743i	$0.362196\pi$
$522$	0	0
$523$	−10.7988	−0.472196	−0.236098	−	0.971729i	$-0.575869\pi$
$523$	−10.7988	−0.472196	−0.236098	+	0.971729i	$0.575869\pi$
$524$	35.6492	1.55734
$525$	0	0
$526$	18.1103	0.789648
$527$	0.497361	0.0216654
$528$	0	0
$529$	1.00000	0.0434783
$530$	92.9895	4.03920
$531$	0	0
$532$	−43.4668	−1.88453
$533$	−62.7677	−2.71877
$534$	0	0
$535$	50.0814	2.16521
$536$	−78.1319	−3.37478
$537$	0	0
$538$	−64.8501	−2.79589
$539$	3.50779	0.151091
$540$	0	0
$541$	−12.3527	−0.531083	−0.265542	−	0.964099i	$-0.585551\pi$
$541$	−12.3527	−0.531083	−0.265542	+	0.964099i	$0.585551\pi$
$542$	−17.7272	−0.761449
$543$	0	0
$544$	−5.17487	−0.221871
$545$	83.6788	3.58441
$546$	0	0
$547$	42.9206	1.83515	0.917576	−	0.397559i	$-0.130143\pi$
$547$	42.9206	1.83515	0.917576	+	0.397559i	$0.130143\pi$
$548$	0.0963939	0.00411774
$549$	0	0
$550$	−78.9416	−3.36608
$551$	−3.34374	−0.142448
$552$	0	0
$553$	−21.9190	−0.932091
$554$	−4.81759	−0.204680
$555$	0	0
$556$	−66.1910	−2.80712
$557$	−29.4259	−1.24681	−0.623407	−	0.781898i	$-0.714252\pi$
$557$	−29.4259	−1.24681	−0.623407	+	0.781898i	$0.714252\pi$
$558$	0	0
$559$	15.9990	0.676687
$560$	87.3267	3.69023
$561$	0	0
$562$	−36.3293	−1.53246
$563$	−27.2699	−1.14929	−0.574644	−	0.818404i	$-0.694859\pi$
$563$	−27.2699	−1.14929	−0.574644	+	0.818404i	$0.694859\pi$
$564$	0	0
$565$	−3.68618	−0.155079
$566$	50.3271	2.11541
$567$	0	0
$568$	−39.9394	−1.67582
$569$	6.63812	0.278285	0.139142	−	0.990272i	$-0.455565\pi$
$569$	6.63812	0.278285	0.139142	+	0.990272i	$0.455565\pi$
$570$	0	0
$571$	33.8991	1.41863	0.709317	−	0.704890i	$-0.249004\pi$
$571$	33.8991	1.41863	0.709317	+	0.704890i	$0.249004\pi$
$572$	46.4009	1.94012
$573$	0	0
$574$	−94.3447	−3.93787
$575$	14.7137	0.613606
$576$	0	0
$577$	26.4782	1.10230	0.551151	−	0.834405i	$-0.314189\pi$
$577$	26.4782	1.10230	0.551151	+	0.834405i	$0.314189\pi$
$578$	39.9742	1.66271
$579$	0	0
$580$	19.6174	0.814568
$581$	21.5153	0.892603
$582$	0	0
$583$	17.5070	0.725065
$584$	72.4069	2.99622
$585$	0	0
$586$	14.3026	0.590835
$587$	5.63371	0.232528	0.116264	−	0.993218i	$-0.462908\pi$
$587$	5.63371	0.232528	0.116264	+	0.993218i	$0.462908\pi$
$588$	0	0
$589$	1.50480	0.0620041
$590$	127.282	5.24012
$591$	0	0
$592$	−21.5312	−0.884928
$593$	16.2741	0.668297	0.334148	−	0.942521i	$-0.391551\pi$
$593$	16.2741	0.668297	0.334148	+	0.942521i	$0.391551\pi$
$594$	0	0
$595$	14.4371	0.591862
$596$	−42.7845	−1.75252
$597$	0	0
$598$	−12.5634	−0.513757
$599$	24.6590	1.00754	0.503769	−	0.863839i	$-0.331946\pi$
$599$	24.6590	1.00754	0.503769	+	0.863839i	$0.331946\pi$
$600$	0	0
$601$	−4.87303	−0.198775	−0.0993875	−	0.995049i	$-0.531688\pi$
$601$	−4.87303	−0.198775	−0.0993875	+	0.995049i	$0.531688\pi$
$602$	24.0478	0.980115
$603$	0	0
$604$	−20.1768	−0.820983
$605$	28.9275	1.17607
$606$	0	0
$607$	27.7980	1.12829	0.564143	−	0.825677i	$-0.309207\pi$
$607$	27.7980	1.12829	0.564143	+	0.825677i	$0.309207\pi$
$608$	−15.6569	−0.634971
$609$	0	0
$610$	43.0641	1.74361
$611$	60.4437	2.44529
$612$	0	0
$613$	3.64314	0.147145	0.0735726	−	0.997290i	$-0.476560\pi$
$613$	3.64314	0.147145	0.0735726	+	0.997290i	$0.476560\pi$
$614$	−21.3410	−0.861252
$615$	0	0
$616$	38.1737	1.53806
$617$	8.12398	0.327059	0.163529	−	0.986538i	$-0.447712\pi$
$617$	8.12398	0.327059	0.163529	+	0.986538i	$0.447712\pi$
$618$	0	0
$619$	−29.1336	−1.17098	−0.585490	−	0.810680i	$-0.699098\pi$
$619$	−29.1336	−1.17098	−0.585490	+	0.810680i	$0.699098\pi$
$620$	−8.82849	−0.354561
$621$	0	0
$622$	−50.5618	−2.02734
$623$	35.9980	1.44223
$624$	0	0
$625$	117.926	4.71702
$626$	−64.8289	−2.59108
$627$	0	0
$628$	48.7094	1.94372
$629$	−3.55960	−0.141930
$630$	0	0
$631$	−26.5387	−1.05649	−0.528244	−	0.849092i	$-0.677150\pi$
$631$	−26.5387	−1.05649	−0.528244	+	0.849092i	$0.677150\pi$
$632$	−45.6432	−1.81559
$633$	0	0
$634$	57.4006	2.27967
$635$	−37.6939	−1.49584
$636$	0	0
$637$	8.21408	0.325454
$638$	5.36516	0.212409
$639$	0	0
$640$	−58.5327	−2.31371
$641$	−23.1736	−0.915302	−0.457651	−	0.889132i	$-0.651309\pi$
$641$	−23.1736	−0.915302	−0.457651	+	0.889132i	$0.651309\pi$
$642$	0	0
$643$	−19.5316	−0.770249	−0.385125	−	0.922865i	$-0.625842\pi$
$643$	−19.5316	−0.770249	−0.385125	+	0.922865i	$0.625842\pi$
$644$	−12.9995	−0.512250
$645$	0	0
$646$	−9.36203	−0.368344
$647$	40.3682	1.58704	0.793519	−	0.608545i	$-0.208246\pi$
$647$	40.3682	1.58704	0.793519	+	0.608545i	$0.208246\pi$
$648$	0	0
$649$	23.9632	0.940637
$650$	−184.855	−7.25061
$651$	0	0
$652$	−97.4855	−3.81783
$653$	4.39476	0.171980	0.0859901	−	0.996296i	$-0.472595\pi$
$653$	4.39476	0.171980	0.0859901	+	0.996296i	$0.472595\pi$
$654$	0	0
$655$	−35.8243	−1.39977
$656$	−84.6122	−3.30355
$657$	0	0
$658$	90.8515	3.54176
$659$	7.66543	0.298603	0.149301	−	0.988792i	$-0.452298\pi$
$659$	7.66543	0.298603	0.149301	+	0.988792i	$0.452298\pi$
$660$	0	0
$661$	−47.1898	−1.83547	−0.917735	−	0.397194i	$-0.869984\pi$
$661$	−47.1898	−1.83547	−0.917735	+	0.397194i	$0.869984\pi$
$662$	−24.3475	−0.946293
$663$	0	0
$664$	44.8025	1.73867
$665$	43.6803	1.69385
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	104.481	4.04248
$669$	0	0
$670$	143.450	5.54194
$671$	8.10760	0.312991
$672$	0	0
$673$	23.4936	0.905613	0.452806	−	0.891609i	$-0.350423\pi$
$673$	23.4936	0.905613	0.452806	+	0.891609i	$0.350423\pi$
$674$	45.8788	1.76719
$675$	0	0
$676$	51.2174	1.96990
$677$	19.2955	0.741586	0.370793	−	0.928716i	$-0.379086\pi$
$677$	19.2955	0.741586	0.370793	+	0.928716i	$0.379086\pi$
$678$	0	0
$679$	23.7468	0.911317
$680$	30.0632	1.15287
$681$	0	0
$682$	−2.41450	−0.0924561
$683$	3.82541	0.146375	0.0731876	−	0.997318i	$-0.476683\pi$
$683$	3.82541	0.146375	0.0731876	+	0.997318i	$0.476683\pi$
$684$	0	0
$685$	−0.0968673	−0.00370111
$686$	−39.8303	−1.52073
$687$	0	0
$688$	21.5670	0.822236
$689$	40.9955	1.56180
$690$	0	0
$691$	16.7012	0.635342	0.317671	−	0.948201i	$-0.397099\pi$
$691$	16.7012	0.635342	0.317671	+	0.948201i	$0.397099\pi$
$692$	−1.94655	−0.0739967
$693$	0	0
$694$	1.81689	0.0689682
$695$	66.5161	2.52310
$696$	0	0
$697$	−13.9883	−0.529845
$698$	−7.60769	−0.287955
$699$	0	0
$700$	−191.271	−7.22935
$701$	14.9920	0.566241	0.283120	−	0.959084i	$-0.408630\pi$
$701$	14.9920	0.566241	0.283120	+	0.959084i	$0.408630\pi$
$702$	0	0
$703$	−10.7698	−0.406190
$704$	−3.19160	−0.120288
$705$	0	0
$706$	−52.2160	−1.96517
$707$	−3.60709	−0.135658
$708$	0	0
$709$	39.9822	1.50156	0.750781	−	0.660551i	$-0.229677\pi$
$709$	39.9822	1.50156	0.750781	+	0.660551i	$0.229677\pi$
$710$	73.3284	2.75197
$711$	0	0
$712$	74.9607	2.80927
$713$	0.450034	0.0168539
$714$	0	0
$715$	−46.6288	−1.74382
$716$	−97.3907	−3.63966
$717$	0	0
$718$	49.5170	1.84796
$719$	−25.3557	−0.945607	−0.472803	−	0.881168i	$-0.656758\pi$
$719$	−25.3557	−0.945607	−0.472803	+	0.881168i	$0.656758\pi$
$720$	0	0
$721$	−36.9733	−1.37696
$722$	19.8099	0.737250
$723$	0	0
$724$	22.2031	0.825171
$725$	−14.7137	−0.546455
$726$	0	0
$727$	−20.5063	−0.760535	−0.380267	−	0.924877i	$-0.624168\pi$
$727$	−20.5063	−0.760535	−0.380267	+	0.924877i	$0.624168\pi$
$728$	89.3901	3.31302
$729$	0	0
$730$	−132.939	−4.92028
$731$	3.56552	0.131875
$732$	0	0
$733$	41.5274	1.53385	0.766925	−	0.641737i	$-0.221786\pi$
$733$	41.5274	1.53385	0.766925	+	0.641737i	$0.221786\pi$
$734$	−9.39569	−0.346801
$735$	0	0
$736$	−4.68245	−0.172597
$737$	27.0070	0.994817
$738$	0	0
$739$	−37.2473	−1.37016	−0.685082	−	0.728465i	$-0.740234\pi$
$739$	−37.2473	−1.37016	−0.685082	+	0.728465i	$0.740234\pi$
$740$	63.1853	2.32274
$741$	0	0
$742$	61.6194	2.26212
$743$	−22.5568	−0.827530	−0.413765	−	0.910384i	$-0.635786\pi$
$743$	−22.5568	−0.827530	−0.413765	+	0.910384i	$0.635786\pi$
$744$	0	0
$745$	42.9946	1.57520
$746$	65.1210	2.38425
$747$	0	0
$748$	10.3408	0.378098
$749$	33.1864	1.21260
$750$	0	0
$751$	4.01903	0.146656	0.0733282	−	0.997308i	$-0.476638\pi$
$751$	4.01903	0.146656	0.0733282	+	0.997308i	$0.476638\pi$
$752$	81.4794	2.97125
$753$	0	0
$754$	12.5634	0.457533
$755$	20.2759	0.737916
$756$	0	0
$757$	35.7160	1.29812	0.649060	−	0.760737i	$-0.275162\pi$
$757$	35.7160	1.29812	0.649060	+	0.760737i	$0.275162\pi$
$758$	−0.746218	−0.0271039
$759$	0	0
$760$	90.9581	3.29940
$761$	−14.2618	−0.516990	−0.258495	−	0.966013i	$-0.583227\pi$
$761$	−14.2618	−0.516990	−0.258495	+	0.966013i	$0.583227\pi$
$762$	0	0
$763$	55.4497	2.00741
$764$	19.4337	0.703088
$765$	0	0
$766$	4.59740	0.166111
$767$	56.1138	2.02615
$768$	0	0
$769$	8.12092	0.292848	0.146424	−	0.989222i	$-0.453224\pi$
$769$	8.12092	0.292848	0.146424	+	0.989222i	$0.453224\pi$
$770$	−70.0867	−2.52575
$771$	0	0
$772$	111.226	4.00309
$773$	39.1617	1.40855	0.704275	−	0.709927i	$-0.251272\pi$
$773$	39.1617	1.40855	0.704275	+	0.709927i	$0.251272\pi$
$774$	0	0
$775$	6.62168	0.237858
$776$	49.4493	1.77513
$777$	0	0
$778$	51.0698	1.83094
$779$	−42.3225	−1.51636
$780$	0	0
$781$	13.8054	0.493997
$782$	−2.79986	−0.100123
$783$	0	0
$784$	11.0728	0.395456
$785$	−48.9486	−1.74705
$786$	0	0
$787$	−8.45412	−0.301357	−0.150678	−	0.988583i	$-0.548146\pi$
$787$	−8.45412	−0.301357	−0.150678	+	0.988583i	$0.548146\pi$
$788$	−44.2045	−1.57472
$789$	0	0
$790$	83.8007	2.98150
$791$	−2.44265	−0.0868505
$792$	0	0
$793$	18.9853	0.674188
$794$	−46.3987	−1.64663
$795$	0	0
$796$	−79.0303	−2.80115
$797$	−32.7938	−1.16162	−0.580809	−	0.814040i	$-0.697264\pi$
$797$	−32.7938	−1.16162	−0.580809	+	0.814040i	$0.697264\pi$
$798$	0	0
$799$	13.4704	0.476548
$800$	−68.8964	−2.43585
$801$	0	0
$802$	−53.6095	−1.89302
$803$	−25.0281	−0.883223
$804$	0	0
$805$	13.0633	0.460421
$806$	−5.65396	−0.199152
$807$	0	0
$808$	−7.51125	−0.264245
$809$	−27.8798	−0.980203	−0.490101	−	0.871665i	$-0.663040\pi$
$809$	−27.8798	−0.980203	−0.490101	+	0.871665i	$0.663040\pi$
$810$	0	0
$811$	20.5996	0.723351	0.361676	−	0.932304i	$-0.382205\pi$
$811$	20.5996	0.723351	0.361676	+	0.932304i	$0.382205\pi$
$812$	12.9995	0.456191
$813$	0	0
$814$	17.2805	0.605682
$815$	97.9642	3.43154
$816$	0	0
$817$	10.7877	0.377414
$818$	13.9651	0.488279
$819$	0	0
$820$	248.302	8.67108
$821$	52.9011	1.84626	0.923130	−	0.384487i	$-0.125622\pi$
$821$	52.9011	1.84626	0.923130	+	0.384487i	$0.125622\pi$
$822$	0	0
$823$	−23.8386	−0.830962	−0.415481	−	0.909602i	$-0.636387\pi$
$823$	−23.8386	−0.830962	−0.415481	+	0.909602i	$0.636387\pi$
$824$	−76.9918	−2.68213
$825$	0	0
$826$	84.3434	2.93468
$827$	17.8912	0.622136	0.311068	−	0.950388i	$-0.399313\pi$
$827$	17.8912	0.622136	0.311068	+	0.950388i	$0.399313\pi$
$828$	0	0
$829$	41.2215	1.43168	0.715840	−	0.698264i	$-0.246044\pi$
$829$	41.2215	1.43168	0.715840	+	0.698264i	$0.246044\pi$
$830$	−82.2571	−2.85519
$831$	0	0
$832$	−7.47367	−0.259103
$833$	1.83058	0.0634257
$834$	0	0
$835$	−104.994	−3.63346
$836$	31.2869	1.08208
$837$	0	0
$838$	−46.0783	−1.59175
$839$	−39.4436	−1.36174	−0.680872	−	0.732403i	$-0.738399\pi$
$839$	−39.4436	−1.36174	−0.680872	+	0.732403i	$0.738399\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	25.8275	0.890075
$843$	0	0
$844$	−12.0162	−0.413615
$845$	−51.4689	−1.77059
$846$	0	0
$847$	19.1688	0.658648
$848$	55.2628	1.89773
$849$	0	0
$850$	−41.1965	−1.41303
$851$	−3.22088	−0.110410
$852$	0	0
$853$	−6.71865	−0.230042	−0.115021	−	0.993363i	$-0.536694\pi$
$853$	−6.71865	−0.230042	−0.115021	+	0.993363i	$0.536694\pi$
$854$	28.5364	0.976495
$855$	0	0
$856$	69.1060	2.36199
$857$	−28.4733	−0.972628	−0.486314	−	0.873784i	$-0.661659\pi$
$857$	−28.4733	−0.972628	−0.486314	+	0.873784i	$0.661659\pi$
$858$	0	0
$859$	−31.9843	−1.09129	−0.545645	−	0.838017i	$-0.683715\pi$
$859$	−31.9843	−1.09129	−0.545645	+	0.838017i	$0.683715\pi$
$860$	−63.2904	−2.15818
$861$	0	0
$862$	−1.62571	−0.0553718
$863$	−51.8924	−1.76644	−0.883218	−	0.468962i	$-0.844628\pi$
$863$	−51.8924	−1.76644	−0.883218	+	0.468962i	$0.844628\pi$
$864$	0	0
$865$	1.95611	0.0665097
$866$	−6.56476	−0.223080
$867$	0	0
$868$	−5.85019	−0.198568
$869$	15.7770	0.535199
$870$	0	0
$871$	63.2415	2.14286
$872$	115.466	3.91018
$873$	0	0
$874$	−8.47117	−0.286542
$875$	126.894	4.28978
$876$	0	0
$877$	13.6092	0.459551	0.229775	−	0.973244i	$-0.426201\pi$
$877$	13.6092	0.459551	0.229775	+	0.973244i	$0.426201\pi$
$878$	−1.39892	−0.0472114
$879$	0	0
$880$	−62.8566	−2.11890
$881$	−52.6757	−1.77469	−0.887345	−	0.461106i	$-0.847453\pi$
$881$	−52.6757	−1.77469	−0.887345	+	0.461106i	$0.847453\pi$
$882$	0	0
$883$	−26.9435	−0.906720	−0.453360	−	0.891328i	$-0.649775\pi$
$883$	−26.9435	−0.906720	−0.453360	+	0.891328i	$0.649775\pi$
$884$	24.2148	0.814432
$885$	0	0
$886$	−100.738	−3.38435
$887$	−0.584619	−0.0196296	−0.00981480	−	0.999952i	$-0.503124\pi$
$887$	−0.584619	−0.0196296	−0.00981480	+	0.999952i	$0.503124\pi$
$888$	0	0
$889$	−24.9778	−0.837730
$890$	−137.627	−4.61328
$891$	0	0
$892$	−77.8729	−2.60738
$893$	40.7555	1.36383
$894$	0	0
$895$	97.8690	3.27140
$896$	−38.7867	−1.29577
$897$	0	0
$898$	43.8637	1.46375
$899$	−0.450034	−0.0150095
$900$	0	0
$901$	9.13619	0.304371
$902$	67.9081	2.26109
$903$	0	0
$904$	−5.08646	−0.169173
$905$	−22.3121	−0.741680
$906$	0	0
$907$	−36.4128	−1.20907	−0.604534	−	0.796580i	$-0.706641\pi$
$907$	−36.4128	−1.20907	−0.604534	+	0.796580i	$0.706641\pi$
$908$	3.62087	0.120163
$909$	0	0
$910$	−164.120	−5.44052
$911$	−22.2410	−0.736877	−0.368439	−	0.929652i	$-0.620108\pi$
$911$	−22.2410	−0.736877	−0.368439	+	0.929652i	$0.620108\pi$
$912$	0	0
$913$	−15.4864	−0.512525
$914$	−2.13114	−0.0704919
$915$	0	0
$916$	45.1058	1.49034
$917$	−23.7390	−0.783930
$918$	0	0
$919$	−36.8297	−1.21490	−0.607450	−	0.794358i	$-0.707808\pi$
$919$	−36.8297	−1.21490	−0.607450	+	0.794358i	$0.707808\pi$
$920$	27.2025	0.896839
$921$	0	0
$922$	40.0474	1.31889
$923$	32.3277	1.06408
$924$	0	0
$925$	−47.3912	−1.55821
$926$	−97.5358	−3.20522
$927$	0	0
$928$	4.68245	0.153709
$929$	−22.9737	−0.753741	−0.376871	−	0.926266i	$-0.623000\pi$
$929$	−22.9737	−0.753741	−0.376871	+	0.926266i	$0.623000\pi$
$930$	0	0
$931$	5.53853	0.181518
$932$	109.428	3.58442
$933$	0	0
$934$	50.9835	1.66823
$935$	−10.3916	−0.339842
$936$	0	0
$937$	−35.7106	−1.16661	−0.583307	−	0.812252i	$-0.698242\pi$
$937$	−35.7106	−1.16661	−0.583307	+	0.812252i	$0.698242\pi$
$938$	95.0568	3.10372
$939$	0	0
$940$	−239.108	−7.79886
$941$	22.7389	0.741266	0.370633	−	0.928779i	$-0.379141\pi$
$941$	22.7389	0.741266	0.370633	+	0.928779i	$0.379141\pi$
$942$	0	0
$943$	−12.6572	−0.412176
$944$	75.6426	2.46196
$945$	0	0
$946$	−17.3093	−0.562773
$947$	−31.7380	−1.03135	−0.515673	−	0.856786i	$-0.672458\pi$
$947$	−31.7380	−1.03135	−0.515673	+	0.856786i	$0.672458\pi$
$948$	0	0
$949$	−58.6075	−1.90248
$950$	−124.643	−4.04394
$951$	0	0
$952$	19.9213	0.645654
$953$	−51.4807	−1.66762	−0.833811	−	0.552049i	$-0.813846\pi$
$953$	−51.4807	−1.66762	−0.833811	+	0.552049i	$0.813846\pi$
$954$	0	0
$955$	−19.5292	−0.631950
$956$	20.4476	0.661323
$957$	0	0
$958$	−19.5490	−0.631600
$959$	−0.0641890	−0.00207277
$960$	0	0
$961$	−30.7975	−0.993467
$962$	40.4653	1.30465
$963$	0	0
$964$	5.76088	0.185545
$965$	−111.772	−3.59806
$966$	0	0
$967$	−0.0572233	−0.00184018	−0.000920088	−	1.00000i	$-0.500293\pi$
$967$	−0.0572233	−0.00184018	−0.000920088		1.00000i	$0.500293\pi$
$968$	39.9163	1.28296
$969$	0	0
$970$	−90.7886	−2.91505
$971$	−24.4473	−0.784552	−0.392276	−	0.919847i	$-0.628312\pi$
$971$	−24.4473	−0.784552	−0.392276	+	0.919847i	$0.628312\pi$
$972$	0	0
$973$	44.0768	1.41304
$974$	−19.4667	−0.623754
$975$	0	0
$976$	25.5926	0.819200
$977$	−35.7213	−1.14283	−0.571413	−	0.820662i	$-0.693605\pi$
$977$	−35.7213	−1.14283	−0.571413	+	0.820662i	$0.693605\pi$
$978$	0	0
$979$	−25.9109	−0.828116
$980$	−32.4940	−1.03798
$981$	0	0
$982$	−15.0022	−0.478739
$983$	4.24384	0.135357	0.0676787	−	0.997707i	$-0.478441\pi$
$983$	4.24384	0.135357	0.0676787	+	0.997707i	$0.478441\pi$
$984$	0	0
$985$	44.4216	1.41539
$986$	2.79986	0.0891659
$987$	0	0
$988$	73.2634	2.33082
$989$	3.22624	0.102588
$990$	0	0
$991$	15.5288	0.493290	0.246645	−	0.969106i	$-0.420672\pi$
$991$	15.5288	0.493290	0.246645	+	0.969106i	$0.420672\pi$
$992$	−2.10726	−0.0669056
$993$	0	0
$994$	48.5910	1.54121
$995$	79.4184	2.51773
$996$	0	0
$997$	26.3595	0.834813	0.417407	−	0.908720i	$-0.362939\pi$
$997$	26.3595	0.834813	0.417407	+	0.908720i	$0.362939\pi$
$998$	−32.8952	−1.04128
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.v.1.3	✓	30
3.2	odd	2		6003.2.a.w.1.28	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.3	✓	30	1.1	even	1	trivial
6003.2.a.w.1.28	yes	30	3.2	odd	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.53344 q^{2} +4.41832 q^{4} -4.44002 q^{5} -2.94217 q^{7} -6.12666 q^{8} +O(q^{10})\)	\(q-2.53344 q^{2} +4.41832 q^{4} -4.44002 q^{5} -2.94217 q^{7} -6.12666 q^{8} +11.2485 q^{10} +2.11774 q^{11} +4.95904 q^{13} +7.45382 q^{14} +6.68489 q^{16} +1.10516 q^{17} +3.34374 q^{19} -19.6174 q^{20} -5.36516 q^{22} +1.00000 q^{23} +14.7137 q^{25} -12.5634 q^{26} -12.9995 q^{28} -1.00000 q^{29} +0.450034 q^{31} -4.68245 q^{32} -2.79986 q^{34} +13.0633 q^{35} -3.22088 q^{37} -8.47117 q^{38} +27.2025 q^{40} -12.6572 q^{41} +3.22624 q^{43} +9.35684 q^{44} -2.53344 q^{46} +12.1886 q^{47} +1.65639 q^{49} -37.2764 q^{50} +21.9106 q^{52} +8.26682 q^{53} -9.40279 q^{55} +18.0257 q^{56} +2.53344 q^{58} +11.3155 q^{59} +3.82843 q^{61} -1.14013 q^{62} -1.50708 q^{64} -22.0182 q^{65} +12.7528 q^{67} +4.88296 q^{68} -33.0951 q^{70} +6.51894 q^{71} -11.8183 q^{73} +8.15990 q^{74} +14.7737 q^{76} -6.23075 q^{77} +7.44994 q^{79} -29.6810 q^{80} +32.0663 q^{82} -7.31271 q^{83} -4.90694 q^{85} -8.17348 q^{86} -12.9747 q^{88} -12.2352 q^{89} -14.5904 q^{91} +4.41832 q^{92} -30.8791 q^{94} -14.8463 q^{95} -8.07116 q^{97} -4.19635 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) \) 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8} + 8 q^{10} + 36 q^{13} - 7 q^{14} + 47 q^{16} - 18 q^{17} + 16 q^{19} + 25 q^{22} + 30 q^{23} + 56 q^{25} - 11 q^{26} + 27 q^{28} - 30 q^{29} + 14 q^{31} + 7 q^{32} + 3 q^{34} + 22 q^{35} + 40 q^{37} - 6 q^{38} + 30 q^{40} - 14 q^{41} + 34 q^{43} - 5 q^{44} - q^{46} + 2 q^{47} + 74 q^{49} + 21 q^{50} + 71 q^{52} - 16 q^{53} + 22 q^{55} - 14 q^{56} + q^{58} + 32 q^{59} + 46 q^{61} - 20 q^{62} + 68 q^{64} - 12 q^{65} + 14 q^{67} - 27 q^{68} + 32 q^{71} + 50 q^{73} + 26 q^{74} + 56 q^{76} - 34 q^{77} + 16 q^{79} - 2 q^{80} + 38 q^{82} + 14 q^{83} + 38 q^{85} - 10 q^{86} + 40 q^{88} + 2 q^{89} + 32 q^{91} + 37 q^{92} + 29 q^{94} + 28 q^{95} + 56 q^{97} - 8 q^{98}+O(q^{100}) \) 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8 + 8 * q^10 + 36 * q^13 - 7 * q^14 + 47 * q^16 - 18 * q^17 + 16 * q^19 + 25 * q^22 + 30 * q^23 + 56 * q^25 - 11 * q^26 + 27 * q^28 - 30 * q^29 + 14 * q^31 + 7 * q^32 + 3 * q^34 + 22 * q^35 + 40 * q^37 - 6 * q^38 + 30 * q^40 - 14 * q^41 + 34 * q^43 - 5 * q^44 - q^46 + 2 * q^47 + 74 * q^49 + 21 * q^50 + 71 * q^52 - 16 * q^53 + 22 * q^55 - 14 * q^56 + q^58 + 32 * q^59 + 46 * q^61 - 20 * q^62 + 68 * q^64 - 12 * q^65 + 14 * q^67 - 27 * q^68 + 32 * q^71 + 50 * q^73 + 26 * q^74 + 56 * q^76 - 34 * q^77 + 16 * q^79 - 2 * q^80 + 38 * q^82 + 14 * q^83 + 38 * q^85 - 10 * q^86 + 40 * q^88 + 2 * q^89 + 32 * q^91 + 37 * q^92 + 29 * q^94 + 28 * q^95 + 56 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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