LMFDB - Embedded newform 6003.2.a.l.1.10

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 10x^{8} + 32x^{7} + 32x^{6} - 118x^{5} - 29x^{4} + 182x^{3} - 28x^{2} - 101x + 43 $ x^10 - 3x^9 - 10x^8 + 32x^7 + 32x^6 - 118x^5 - 29x^4 + 182x^3 - 28x^2 - 101*x + 43
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.67549$ of defining polynomial
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)

$f(q)$	$=$	$q+2.67549 q^{2} +5.15827 q^{4} -1.93239 q^{5} +0.721979 q^{7} +8.44992 q^{8} +O(q^{10})$	\(q+2.67549 q^{2} +5.15827 q^{4} -1.93239 q^{5} +0.721979 q^{7} +8.44992 q^{8} -5.17011 q^{10} +1.21766 q^{11} +0.759191 q^{13} +1.93165 q^{14} +12.2912 q^{16} +7.43542 q^{17} +0.975858 q^{19} -9.96781 q^{20} +3.25785 q^{22} -1.00000 q^{23} -1.26585 q^{25} +2.03121 q^{26} +3.72416 q^{28} -1.00000 q^{29} +4.89136 q^{31} +15.9851 q^{32} +19.8934 q^{34} -1.39515 q^{35} -7.11770 q^{37} +2.61090 q^{38} -16.3286 q^{40} -6.93952 q^{41} +4.20028 q^{43} +6.28103 q^{44} -2.67549 q^{46} +6.78394 q^{47} -6.47875 q^{49} -3.38678 q^{50} +3.91611 q^{52} +7.29810 q^{53} -2.35301 q^{55} +6.10066 q^{56} -2.67549 q^{58} +6.54924 q^{59} -7.08057 q^{61} +13.0868 q^{62} +18.1857 q^{64} -1.46706 q^{65} +1.32547 q^{67} +38.3539 q^{68} -3.73271 q^{70} +12.3498 q^{71} -4.80670 q^{73} -19.0434 q^{74} +5.03373 q^{76} +0.879128 q^{77} +14.4789 q^{79} -23.7514 q^{80} -18.5667 q^{82} +8.13628 q^{83} -14.3682 q^{85} +11.2378 q^{86} +10.2892 q^{88} +0.456964 q^{89} +0.548120 q^{91} -5.15827 q^{92} +18.1504 q^{94} -1.88574 q^{95} -6.57680 q^{97} -17.3338 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) $ 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8	$ 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8} - 6 q^{10} - 13 q^{13} + 12 q^{14} - 5 q^{16} + 22 q^{17} - 2 q^{19} - 3 q^{20} + 3 q^{22} - 10 q^{23} + 10 q^{25} + 25 q^{26} + 19 q^{28} - 10 q^{29} - 22 q^{31} + 31 q^{32} + 13 q^{34} + 15 q^{35} - 9 q^{37} + 10 q^{38} - 6 q^{40} + 25 q^{41} + 3 q^{43} + 27 q^{44} - 3 q^{46} + 17 q^{47} + 17 q^{49} - 2 q^{50} - 18 q^{52} + 43 q^{53} - 11 q^{55} + 7 q^{56} - 3 q^{58} + 7 q^{59} - 6 q^{61} - 3 q^{62} + 33 q^{64} - 11 q^{65} + 11 q^{67} + 51 q^{68} + 34 q^{70} + 17 q^{71} - 44 q^{73} - 9 q^{74} + 24 q^{76} + 71 q^{77} + 5 q^{79} - 38 q^{80} + 33 q^{82} + 32 q^{83} + 16 q^{85} + 9 q^{86} + 18 q^{88} + 10 q^{89} - 3 q^{91} - 9 q^{92} + 47 q^{94} + 8 q^{95} + 6 q^{97} + 73 q^{98}+O(q^{100}) $ 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8 - 6 * q^10 - 13 * q^13 + 12 * q^14 - 5 * q^16 + 22 * q^17 - 2 * q^19 - 3 * q^20 + 3 * q^22 - 10 * q^23 + 10 * q^25 + 25 * q^26 + 19 * q^28 - 10 * q^29 - 22 * q^31 + 31 * q^32 + 13 * q^34 + 15 * q^35 - 9 * q^37 + 10 * q^38 - 6 * q^40 + 25 * q^41 + 3 * q^43 + 27 * q^44 - 3 * q^46 + 17 * q^47 + 17 * q^49 - 2 * q^50 - 18 * q^52 + 43 * q^53 - 11 * q^55 + 7 * q^56 - 3 * q^58 + 7 * q^59 - 6 * q^61 - 3 * q^62 + 33 * q^64 - 11 * q^65 + 11 * q^67 + 51 * q^68 + 34 * q^70 + 17 * q^71 - 44 * q^73 - 9 * q^74 + 24 * q^76 + 71 * q^77 + 5 * q^79 - 38 * q^80 + 33 * q^82 + 32 * q^83 + 16 * q^85 + 9 * q^86 + 18 * q^88 + 10 * q^89 - 3 * q^91 - 9 * q^92 + 47 * q^94 + 8 * q^95 + 6 * q^97 + 73 * 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 $

$2$	2.67549	1.89186	0.945930	−	0.324371i	$-0.105153\pi$
$2$	2.67549	1.89186	0.945930	+	0.324371i	$0.105153\pi$
$3$	0	0
$3$	0	0
$4$	5.15827	2.57913
$5$	−1.93239	−0.864193	−0.432097	−	0.901827i	$-0.642226\pi$
$5$	−1.93239	−0.864193	−0.432097	+	0.901827i	$0.642226\pi$
$6$	0	0
$7$	0.721979	0.272882	0.136441	−	0.990648i	$-0.456434\pi$
$7$	0.721979	0.272882	0.136441	+	0.990648i	$0.456434\pi$
$8$	8.44992	2.98750
$9$	0	0
$10$	−5.17011	−1.63493
$11$	1.21766	0.367139	0.183570	−	0.983007i	$-0.441235\pi$
$11$	1.21766	0.367139	0.183570	+	0.983007i	$0.441235\pi$
$12$	0	0
$13$	0.759191	0.210562	0.105281	−	0.994443i	$-0.466426\pi$
$13$	0.759191	0.210562	0.105281	+	0.994443i	$0.466426\pi$
$14$	1.93165	0.516255
$15$	0	0
$16$	12.2912	3.07279
$17$	7.43542	1.80335	0.901677	−	0.432409i	$-0.142336\pi$
$17$	7.43542	1.80335	0.901677	+	0.432409i	$0.142336\pi$
$18$	0	0
$19$	0.975858	0.223877	0.111939	−	0.993715i	$-0.464294\pi$
$19$	0.975858	0.223877	0.111939	+	0.993715i	$0.464294\pi$
$20$	−9.96781	−2.22887
$21$	0	0
$22$	3.25785	0.694576
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.26585	−0.253170
$26$	2.03121	0.398353
$27$	0	0
$28$	3.72416	0.703800
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.89136	0.878515	0.439257	−	0.898361i	$-0.355242\pi$
$31$	4.89136	0.878515	0.439257	+	0.898361i	$0.355242\pi$
$32$	15.9851	2.82580
$33$	0	0
$34$	19.8934	3.41169
$35$	−1.39515	−0.235823
$36$	0	0
$37$	−7.11770	−1.17014	−0.585071	−	0.810982i	$-0.698933\pi$
$37$	−7.11770	−1.17014	−0.585071	+	0.810982i	$0.698933\pi$
$38$	2.61090	0.423544
$39$	0	0
$40$	−16.3286	−2.58178
$41$	−6.93952	−1.08377	−0.541886	−	0.840452i	$-0.682289\pi$
$41$	−6.93952	−1.08377	−0.541886	+	0.840452i	$0.682289\pi$
$42$	0	0
$43$	4.20028	0.640537	0.320268	−	0.947327i	$-0.396227\pi$
$43$	4.20028	0.640537	0.320268	+	0.947327i	$0.396227\pi$
$44$	6.28103	0.946901
$45$	0	0
$46$	−2.67549	−0.394480
$47$	6.78394	0.989540	0.494770	−	0.869024i	$-0.335252\pi$
$47$	6.78394	0.989540	0.494770	+	0.869024i	$0.335252\pi$
$48$	0	0
$49$	−6.47875	−0.925535
$50$	−3.38678	−0.478963
$51$	0	0
$52$	3.91611	0.543067
$53$	7.29810	1.00247	0.501236	−	0.865311i	$-0.332879\pi$
$53$	7.29810	1.00247	0.501236	+	0.865311i	$0.332879\pi$
$54$	0	0
$55$	−2.35301	−0.317279
$56$	6.10066	0.815236
$57$	0	0
$58$	−2.67549	−0.351310
$59$	6.54924	0.852638	0.426319	−	0.904573i	$-0.359810\pi$
$59$	6.54924	0.852638	0.426319	+	0.904573i	$0.359810\pi$
$60$	0	0
$61$	−7.08057	−0.906574	−0.453287	−	0.891365i	$-0.649749\pi$
$61$	−7.08057	−0.906574	−0.453287	+	0.891365i	$0.649749\pi$
$62$	13.0868	1.66203
$63$	0	0
$64$	18.1857	2.27322
$65$	−1.46706	−0.181966
$66$	0	0
$67$	1.32547	0.161932	0.0809660	−	0.996717i	$-0.474199\pi$
$67$	1.32547	0.161932	0.0809660	+	0.996717i	$0.474199\pi$
$68$	38.3539	4.65109
$69$	0	0
$70$	−3.73271	−0.446144
$71$	12.3498	1.46565	0.732825	−	0.680417i	$-0.238201\pi$
$71$	12.3498	1.46565	0.732825	+	0.680417i	$0.238201\pi$
$72$	0	0
$73$	−4.80670	−0.562581	−0.281291	−	0.959623i	$-0.590762\pi$
$73$	−4.80670	−0.562581	−0.281291	+	0.959623i	$0.590762\pi$
$74$	−19.0434	−2.21374
$75$	0	0
$76$	5.03373	0.577409
$77$	0.879128	0.100186
$78$	0	0
$79$	14.4789	1.62901	0.814504	−	0.580159i	$-0.197009\pi$
$79$	14.4789	1.62901	0.814504	+	0.580159i	$0.197009\pi$
$80$	−23.7514	−2.65549
$81$	0	0
$82$	−18.5667	−2.05034
$83$	8.13628	0.893073	0.446536	−	0.894765i	$-0.352657\pi$
$83$	8.13628	0.893073	0.446536	+	0.894765i	$0.352657\pi$
$84$	0	0
$85$	−14.3682	−1.55845
$86$	11.2378	1.21181
$87$	0	0
$88$	10.2892	1.09683
$89$	0.456964	0.0484381	0.0242191	−	0.999707i	$-0.492290\pi$
$89$	0.456964	0.0484381	0.0242191	+	0.999707i	$0.492290\pi$
$90$	0	0
$91$	0.548120	0.0574586
$92$	−5.15827	−0.537786
$93$	0	0
$94$	18.1504	1.87207
$95$	−1.88574	−0.193473
$96$	0	0
$97$	−6.57680	−0.667773	−0.333886	−	0.942613i	$-0.608360\pi$
$97$	−6.57680	−0.667773	−0.333886	+	0.942613i	$0.608360\pi$
$98$	−17.3338	−1.75098
$99$	0	0
$100$	−6.52960	−0.652960
$101$	−8.50531	−0.846310	−0.423155	−	0.906057i	$-0.639077\pi$
$101$	−8.50531	−0.846310	−0.423155	+	0.906057i	$0.639077\pi$
$102$	0	0
$103$	6.73914	0.664027	0.332013	−	0.943275i	$-0.392272\pi$
$103$	6.73914	0.664027	0.332013	+	0.943275i	$0.392272\pi$
$104$	6.41511	0.629053
$105$	0	0
$106$	19.5260	1.89654
$107$	−10.7615	−1.04036	−0.520178	−	0.854058i	$-0.674134\pi$
$107$	−10.7615	−1.04036	−0.520178	+	0.854058i	$0.674134\pi$
$108$	0	0
$109$	1.68415	0.161312	0.0806562	−	0.996742i	$-0.474298\pi$
$109$	1.68415	0.161312	0.0806562	+	0.996742i	$0.474298\pi$
$110$	−6.29546	−0.600248
$111$	0	0
$112$	8.87397	0.838511
$113$	9.47905	0.891714	0.445857	−	0.895104i	$-0.352899\pi$
$113$	9.47905	0.891714	0.445857	+	0.895104i	$0.352899\pi$
$114$	0	0
$115$	1.93239	0.180197
$116$	−5.15827	−0.478933
$117$	0	0
$118$	17.5224	1.61307
$119$	5.36822	0.492104
$120$	0	0
$121$	−9.51729	−0.865209
$122$	−18.9440	−1.71511
$123$	0	0
$124$	25.2309	2.26581
$125$	12.1081	1.08298
$126$	0	0
$127$	−7.40313	−0.656922	−0.328461	−	0.944518i	$-0.606530\pi$
$127$	−7.40313	−0.656922	−0.328461	+	0.944518i	$0.606530\pi$
$128$	16.6856	1.47481
$129$	0	0
$130$	−3.92510	−0.344254
$131$	13.1875	1.15220	0.576099	−	0.817380i	$-0.304574\pi$
$131$	13.1875	1.15220	0.576099	+	0.817380i	$0.304574\pi$
$132$	0	0
$133$	0.704549	0.0610921
$134$	3.54629	0.306353
$135$	0	0
$136$	62.8287	5.38752
$137$	14.7027	1.25614	0.628069	−	0.778158i	$-0.283845\pi$
$137$	14.7027	1.25614	0.628069	+	0.778158i	$0.283845\pi$
$138$	0	0
$139$	−2.25950	−0.191649	−0.0958243	−	0.995398i	$-0.530549\pi$
$139$	−2.25950	−0.191649	−0.0958243	+	0.995398i	$0.530549\pi$
$140$	−7.19655	−0.608219
$141$	0	0
$142$	33.0418	2.77280
$143$	0.924440	0.0773055
$144$	0	0
$145$	1.93239	0.160477
$146$	−12.8603	−1.06432
$147$	0	0
$148$	−36.7150	−3.01795
$149$	8.31996	0.681598	0.340799	−	0.940136i	$-0.389303\pi$
$149$	8.31996	0.681598	0.340799	+	0.940136i	$0.389303\pi$
$150$	0	0
$151$	−2.15704	−0.175537	−0.0877685	−	0.996141i	$-0.527974\pi$
$151$	−2.15704	−0.175537	−0.0877685	+	0.996141i	$0.527974\pi$
$152$	8.24592	0.668833
$153$	0	0
$154$	2.35210	0.189538
$155$	−9.45204	−0.759206
$156$	0	0
$157$	13.5196	1.07898	0.539489	−	0.841993i	$-0.318617\pi$
$157$	13.5196	1.07898	0.539489	+	0.841993i	$0.318617\pi$
$158$	38.7383	3.08185
$159$	0	0
$160$	−30.8896	−2.44203
$161$	−0.721979	−0.0568999
$162$	0	0
$163$	−16.2609	−1.27365	−0.636827	−	0.771007i	$-0.719754\pi$
$163$	−16.2609	−1.27365	−0.636827	+	0.771007i	$0.719754\pi$
$164$	−35.7959	−2.79519
$165$	0	0
$166$	21.7686	1.68957
$167$	0.0282352	0.00218490	0.00109245	−	0.999999i	$-0.499652\pi$
$167$	0.0282352	0.00218490	0.00109245	+	0.999999i	$0.499652\pi$
$168$	0	0
$169$	−12.4236	−0.955664
$170$	−38.4419	−2.94836
$171$	0	0
$172$	21.6662	1.65203
$173$	4.70262	0.357534	0.178767	−	0.983891i	$-0.442789\pi$
$173$	4.70262	0.357534	0.178767	+	0.983891i	$0.442789\pi$
$174$	0	0
$175$	−0.913918	−0.0690857
$176$	14.9665	1.12814
$177$	0	0
$178$	1.22260	0.0916381
$179$	14.7323	1.10114	0.550570	−	0.834789i	$-0.314410\pi$
$179$	14.7323	1.10114	0.550570	+	0.834789i	$0.314410\pi$
$180$	0	0
$181$	−9.13899	−0.679296	−0.339648	−	0.940553i	$-0.610308\pi$
$181$	−9.13899	−0.679296	−0.339648	+	0.940553i	$0.610308\pi$
$182$	1.46649	0.108704
$183$	0	0
$184$	−8.44992	−0.622936
$185$	13.7542	1.01123
$186$	0	0
$187$	9.05385	0.662083
$188$	34.9934	2.55216
$189$	0	0
$190$	−5.04529	−0.366024
$191$	−8.88101	−0.642608	−0.321304	−	0.946976i	$-0.604121\pi$
$191$	−8.88101	−0.642608	−0.321304	+	0.946976i	$0.604121\pi$
$192$	0	0
$193$	−22.3157	−1.60632	−0.803160	−	0.595763i	$-0.796850\pi$
$193$	−22.3157	−1.60632	−0.803160	+	0.595763i	$0.796850\pi$
$194$	−17.5962	−1.26333
$195$	0	0
$196$	−33.4191	−2.38708
$197$	−14.0126	−0.998354	−0.499177	−	0.866500i	$-0.666364\pi$
$197$	−14.0126	−0.998354	−0.499177	+	0.866500i	$0.666364\pi$
$198$	0	0
$199$	−26.3908	−1.87079	−0.935397	−	0.353600i	$-0.884957\pi$
$199$	−26.3908	−1.87079	−0.935397	+	0.353600i	$0.884957\pi$
$200$	−10.6963	−0.756345
$201$	0	0
$202$	−22.7559	−1.60110
$203$	−0.721979	−0.0506730
$204$	0	0
$205$	13.4099	0.936588
$206$	18.0305	1.25625
$207$	0	0
$208$	9.33135	0.647013
$209$	1.18827	0.0821941
$210$	0	0
$211$	9.86160	0.678900	0.339450	−	0.940624i	$-0.389759\pi$
$211$	9.86160	0.678900	0.339450	+	0.940624i	$0.389759\pi$
$212$	37.6456	2.58551
$213$	0	0
$214$	−28.7924	−1.96821
$215$	−8.11660	−0.553547
$216$	0	0
$217$	3.53146	0.239731
$218$	4.50593	0.305180
$219$	0	0
$220$	−12.1374	−0.818306
$221$	5.64491	0.379718
$222$	0	0
$223$	24.6137	1.64825	0.824126	−	0.566406i	$-0.191667\pi$
$223$	24.6137	1.64825	0.824126	+	0.566406i	$0.191667\pi$
$224$	11.5409	0.771110
$225$	0	0
$226$	25.3611	1.68700
$227$	12.2803	0.815069	0.407535	−	0.913190i	$-0.366389\pi$
$227$	12.2803	0.815069	0.407535	+	0.913190i	$0.366389\pi$
$228$	0	0
$229$	4.04487	0.267293	0.133646	−	0.991029i	$-0.457331\pi$
$229$	4.04487	0.267293	0.133646	+	0.991029i	$0.457331\pi$
$230$	5.17011	0.340907
$231$	0	0
$232$	−8.44992	−0.554764
$233$	−17.1248	−1.12188	−0.560942	−	0.827855i	$-0.689561\pi$
$233$	−17.1248	−1.12188	−0.560942	+	0.827855i	$0.689561\pi$
$234$	0	0
$235$	−13.1093	−0.855154
$236$	33.7827	2.19907
$237$	0	0
$238$	14.3626	0.930991
$239$	−24.7785	−1.60279	−0.801393	−	0.598138i	$-0.795907\pi$
$239$	−24.7785	−1.60279	−0.801393	+	0.598138i	$0.795907\pi$
$240$	0	0
$241$	19.2639	1.24090	0.620450	−	0.784246i	$-0.286950\pi$
$241$	19.2639	1.24090	0.620450	+	0.784246i	$0.286950\pi$
$242$	−25.4635	−1.63685
$243$	0	0
$244$	−36.5235	−2.33817
$245$	12.5195	0.799841
$246$	0	0
$247$	0.740863	0.0471400
$248$	41.3316	2.62456
$249$	0	0
$250$	32.3951	2.04885
$251$	6.40532	0.404300	0.202150	−	0.979355i	$-0.435207\pi$
$251$	6.40532	0.404300	0.202150	+	0.979355i	$0.435207\pi$
$252$	0	0
$253$	−1.21766	−0.0765539
$254$	−19.8070	−1.24280
$255$	0	0
$256$	8.27069	0.516918
$257$	−11.5345	−0.719501	−0.359751	−	0.933049i	$-0.617138\pi$
$257$	−11.5345	−0.719501	−0.359751	+	0.933049i	$0.617138\pi$
$258$	0	0
$259$	−5.13883	−0.319311
$260$	−7.56747	−0.469315
$261$	0	0
$262$	35.2831	2.17980
$263$	−29.4709	−1.81725	−0.908626	−	0.417610i	$-0.862868\pi$
$263$	−29.4709	−1.81725	−0.908626	+	0.417610i	$0.862868\pi$
$264$	0	0
$265$	−14.1028	−0.866329
$266$	1.88502	0.115578
$267$	0	0
$268$	6.83713	0.417644
$269$	−7.08827	−0.432180	−0.216090	−	0.976373i	$-0.569330\pi$
$269$	−7.08827	−0.432180	−0.216090	+	0.976373i	$0.569330\pi$
$270$	0	0
$271$	−15.1303	−0.919101	−0.459550	−	0.888152i	$-0.651989\pi$
$271$	−15.1303	−0.919101	−0.459550	+	0.888152i	$0.651989\pi$
$272$	91.3901	5.54134
$273$	0	0
$274$	39.3370	2.37644
$275$	−1.54138	−0.0929488
$276$	0	0
$277$	−23.8949	−1.43571	−0.717854	−	0.696194i	$-0.754875\pi$
$277$	−23.8949	−1.43571	−0.717854	+	0.696194i	$0.754875\pi$
$278$	−6.04529	−0.362572
$279$	0	0
$280$	−11.7889	−0.704521
$281$	6.56691	0.391749	0.195874	−	0.980629i	$-0.437246\pi$
$281$	6.56691	0.391749	0.195874	+	0.980629i	$0.437246\pi$
$282$	0	0
$283$	−26.1297	−1.55325	−0.776624	−	0.629965i	$-0.783069\pi$
$283$	−26.1297	−1.55325	−0.776624	+	0.629965i	$0.783069\pi$
$284$	63.7035	3.78011
$285$	0	0
$286$	2.47333	0.146251
$287$	−5.01019	−0.295742
$288$	0	0
$289$	38.2855	2.25209
$290$	5.17011	0.303599
$291$	0	0
$292$	−24.7942	−1.45097
$293$	−23.9338	−1.39823	−0.699114	−	0.715010i	$-0.746422\pi$
$293$	−23.9338	−1.39823	−0.699114	+	0.715010i	$0.746422\pi$
$294$	0	0
$295$	−12.6557	−0.736844
$296$	−60.1440	−3.49580
$297$	0	0
$298$	22.2600	1.28949
$299$	−0.759191	−0.0439052
$300$	0	0
$301$	3.03251	0.174791
$302$	−5.77113	−0.332091
$303$	0	0
$304$	11.9944	0.687928
$305$	13.6825	0.783455
$306$	0	0
$307$	15.7887	0.901109	0.450554	−	0.892749i	$-0.351226\pi$
$307$	15.7887	0.901109	0.450554	+	0.892749i	$0.351226\pi$
$308$	4.53477	0.258393
$309$	0	0
$310$	−25.2889	−1.43631
$311$	10.0741	0.571249	0.285624	−	0.958342i	$-0.407799\pi$
$311$	10.0741	0.571249	0.285624	+	0.958342i	$0.407799\pi$
$312$	0	0
$313$	15.6828	0.886447	0.443223	−	0.896411i	$-0.353835\pi$
$313$	15.6828	0.886447	0.443223	+	0.896411i	$0.353835\pi$
$314$	36.1715	2.04128
$315$	0	0
$316$	74.6862	4.20143
$317$	1.15227	0.0647178	0.0323589	−	0.999476i	$-0.489698\pi$
$317$	1.15227	0.0647178	0.0323589	+	0.999476i	$0.489698\pi$
$318$	0	0
$319$	−1.21766	−0.0681761
$320$	−35.1420	−1.96450
$321$	0	0
$322$	−1.93165	−0.107647
$323$	7.25592	0.403730
$324$	0	0
$325$	−0.961023	−0.0533080
$326$	−43.5060	−2.40958
$327$	0	0
$328$	−58.6384	−3.23777
$329$	4.89786	0.270028
$330$	0	0
$331$	−32.9192	−1.80941	−0.904703	−	0.426044i	$-0.859907\pi$
$331$	−32.9192	−1.80941	−0.904703	+	0.426044i	$0.859907\pi$
$332$	41.9691	2.30335
$333$	0	0
$334$	0.0755431	0.00413353
$335$	−2.56133	−0.139941
$336$	0	0
$337$	−28.4389	−1.54916	−0.774582	−	0.632474i	$-0.782040\pi$
$337$	−28.4389	−1.54916	−0.774582	+	0.632474i	$0.782040\pi$
$338$	−33.2393	−1.80798
$339$	0	0
$340$	−74.1148	−4.01944
$341$	5.95603	0.322537
$342$	0	0
$343$	−9.73137	−0.525445
$344$	35.4920	1.91360
$345$	0	0
$346$	12.5818	0.676404
$347$	−21.2888	−1.14284	−0.571421	−	0.820657i	$-0.693608\pi$
$347$	−21.2888	−1.14284	−0.571421	+	0.820657i	$0.693608\pi$
$348$	0	0
$349$	27.4019	1.46679	0.733395	−	0.679803i	$-0.237935\pi$
$349$	27.4019	1.46679	0.733395	+	0.679803i	$0.237935\pi$
$350$	−2.44518	−0.130700
$351$	0	0
$352$	19.4645	1.03746
$353$	−19.2067	−1.02227	−0.511135	−	0.859501i	$-0.670775\pi$
$353$	−19.2067	−1.02227	−0.511135	+	0.859501i	$0.670775\pi$
$354$	0	0
$355$	−23.8647	−1.26660
$356$	2.35714	0.124928
$357$	0	0
$358$	39.4161	2.08320
$359$	−26.0413	−1.37440	−0.687202	−	0.726466i	$-0.741161\pi$
$359$	−26.0413	−1.37440	−0.687202	+	0.726466i	$0.741161\pi$
$360$	0	0
$361$	−18.0477	−0.949879
$362$	−24.4513	−1.28513
$363$	0	0
$364$	2.82735	0.148193
$365$	9.28843	0.486179
$366$	0	0
$367$	−17.8240	−0.930403	−0.465201	−	0.885205i	$-0.654018\pi$
$367$	−17.8240	−0.930403	−0.465201	+	0.885205i	$0.654018\pi$
$368$	−12.2912	−0.640722
$369$	0	0
$370$	36.7993	1.91310
$371$	5.26908	0.273557
$372$	0	0
$373$	25.1002	1.29964	0.649819	−	0.760089i	$-0.274845\pi$
$373$	25.1002	1.29964	0.649819	+	0.760089i	$0.274845\pi$
$374$	24.2235	1.25257
$375$	0	0
$376$	57.3238	2.95625
$377$	−0.759191	−0.0391003
$378$	0	0
$379$	29.1521	1.49744	0.748720	−	0.662886i	$-0.230669\pi$
$379$	29.1521	1.49744	0.748720	+	0.662886i	$0.230669\pi$
$380$	−9.72716	−0.498993
$381$	0	0
$382$	−23.7611	−1.21572
$383$	−20.5470	−1.04990	−0.524952	−	0.851132i	$-0.675917\pi$
$383$	−20.5470	−1.04990	−0.524952	+	0.851132i	$0.675917\pi$
$384$	0	0
$385$	−1.69882	−0.0865800
$386$	−59.7056	−3.03893
$387$	0	0
$388$	−33.9249	−1.72227
$389$	−0.146769	−0.00744146	−0.00372073	−	0.999993i	$-0.501184\pi$
$389$	−0.146769	−0.00744146	−0.00372073	+	0.999993i	$0.501184\pi$
$390$	0	0
$391$	−7.43542	−0.376025
$392$	−54.7449	−2.76503
$393$	0	0
$394$	−37.4905	−1.88875
$395$	−27.9790	−1.40778
$396$	0	0
$397$	−33.1878	−1.66565	−0.832823	−	0.553539i	$-0.813277\pi$
$397$	−33.1878	−1.66565	−0.832823	+	0.553539i	$0.813277\pi$
$398$	−70.6084	−3.53928
$399$	0	0
$400$	−15.5588	−0.777940
$401$	−21.7592	−1.08660	−0.543302	−	0.839537i	$-0.682826\pi$
$401$	−21.7592	−1.08660	−0.543302	+	0.839537i	$0.682826\pi$
$402$	0	0
$403$	3.71348	0.184982
$404$	−43.8727	−2.18275
$405$	0	0
$406$	−1.93165	−0.0958662
$407$	−8.66696	−0.429605
$408$	0	0
$409$	−29.6837	−1.46777	−0.733883	−	0.679276i	$-0.762294\pi$
$409$	−29.6837	−1.46777	−0.733883	+	0.679276i	$0.762294\pi$
$410$	35.8781	1.77189
$411$	0	0
$412$	34.7623	1.71261
$413$	4.72841	0.232670
$414$	0	0
$415$	−15.7225	−0.771787
$416$	12.1358	0.595005
$417$	0	0
$418$	3.17920	0.155500
$419$	18.6118	0.909244	0.454622	−	0.890684i	$-0.349774\pi$
$419$	18.6118	0.909244	0.454622	+	0.890684i	$0.349774\pi$
$420$	0	0
$421$	−9.48608	−0.462323	−0.231162	−	0.972915i	$-0.574253\pi$
$421$	−9.48608	−0.462323	−0.231162	+	0.972915i	$0.574253\pi$
$422$	26.3846	1.28438
$423$	0	0
$424$	61.6684	2.99488
$425$	−9.41214	−0.456556
$426$	0	0
$427$	−5.11202	−0.247388
$428$	−55.5108	−2.68322
$429$	0	0
$430$	−21.7159	−1.04723
$431$	−26.2928	−1.26648	−0.633240	−	0.773956i	$-0.718275\pi$
$431$	−26.2928	−1.26648	−0.633240	+	0.773956i	$0.718275\pi$
$432$	0	0
$433$	33.6618	1.61768	0.808842	−	0.588026i	$-0.200095\pi$
$433$	33.6618	1.61768	0.808842	+	0.588026i	$0.200095\pi$
$434$	9.44840	0.453538
$435$	0	0
$436$	8.68729	0.416046
$437$	−0.975858	−0.0466816
$438$	0	0
$439$	−20.1034	−0.959481	−0.479741	−	0.877410i	$-0.659269\pi$
$439$	−20.1034	−0.959481	−0.479741	+	0.877410i	$0.659269\pi$
$440$	−19.8827	−0.947872
$441$	0	0
$442$	15.1029	0.718372
$443$	−12.9472	−0.615142	−0.307571	−	0.951525i	$-0.599516\pi$
$443$	−12.9472	−0.615142	−0.307571	+	0.951525i	$0.599516\pi$
$444$	0	0
$445$	−0.883035	−0.0418599
$446$	65.8537	3.11826
$447$	0	0
$448$	13.1297	0.620321
$449$	−15.5981	−0.736119	−0.368059	−	0.929802i	$-0.619978\pi$
$449$	−15.5981	−0.736119	−0.368059	+	0.929802i	$0.619978\pi$
$450$	0	0
$451$	−8.45001	−0.397895
$452$	48.8954	2.29985
$453$	0	0
$454$	32.8558	1.54200
$455$	−1.05918	−0.0496553
$456$	0	0
$457$	−16.3556	−0.765084	−0.382542	−	0.923938i	$-0.624951\pi$
$457$	−16.3556	−0.765084	−0.382542	+	0.923938i	$0.624951\pi$
$458$	10.8220	0.505680
$459$	0	0
$460$	9.96781	0.464751
$461$	40.7099	1.89605	0.948025	−	0.318196i	$-0.103077\pi$
$461$	40.7099	1.89605	0.948025	+	0.318196i	$0.103077\pi$
$462$	0	0
$463$	34.9631	1.62487	0.812437	−	0.583049i	$-0.198140\pi$
$463$	34.9631	1.62487	0.812437	+	0.583049i	$0.198140\pi$
$464$	−12.2912	−0.570603
$465$	0	0
$466$	−45.8174	−2.12245
$467$	−7.18417	−0.332444	−0.166222	−	0.986088i	$-0.553157\pi$
$467$	−7.18417	−0.332444	−0.166222	+	0.986088i	$0.553157\pi$
$468$	0	0
$469$	0.956962	0.0441884
$470$	−35.0737	−1.61783
$471$	0	0
$472$	55.3405	2.54725
$473$	5.11453	0.235166
$474$	0	0
$475$	−1.23529	−0.0566790
$476$	27.6907	1.26920
$477$	0	0
$478$	−66.2946	−3.03225
$479$	14.9608	0.683577	0.341788	−	0.939777i	$-0.388967\pi$
$479$	14.9608	0.683577	0.341788	+	0.939777i	$0.388967\pi$
$480$	0	0
$481$	−5.40369	−0.246387
$482$	51.5406	2.34761
$483$	0	0
$484$	−49.0927	−2.23149
$485$	12.7090	0.577084
$486$	0	0
$487$	33.3897	1.51303	0.756515	−	0.653976i	$-0.226900\pi$
$487$	33.3897	1.51303	0.756515	+	0.653976i	$0.226900\pi$
$488$	−59.8302	−2.70839
$489$	0	0
$490$	33.4958	1.51319
$491$	14.9913	0.676548	0.338274	−	0.941048i	$-0.390157\pi$
$491$	14.9913	0.676548	0.338274	+	0.941048i	$0.390157\pi$
$492$	0	0
$493$	−7.43542	−0.334875
$494$	1.98217	0.0891822
$495$	0	0
$496$	60.1206	2.69949
$497$	8.91629	0.399950
$498$	0	0
$499$	11.3162	0.506584	0.253292	−	0.967390i	$-0.418487\pi$
$499$	11.3162	0.506584	0.253292	+	0.967390i	$0.418487\pi$
$500$	62.4568	2.79315
$501$	0	0
$502$	17.1374	0.764879
$503$	−28.5521	−1.27307	−0.636537	−	0.771246i	$-0.719634\pi$
$503$	−28.5521	−1.27307	−0.636537	+	0.771246i	$0.719634\pi$
$504$	0	0
$505$	16.4356	0.731376
$506$	−3.25785	−0.144829
$507$	0	0
$508$	−38.1873	−1.69429
$509$	−6.23210	−0.276233	−0.138116	−	0.990416i	$-0.544105\pi$
$509$	−6.23210	−0.276233	−0.138116	+	0.990416i	$0.544105\pi$
$510$	0	0
$511$	−3.47033	−0.153518
$512$	−11.2430	−0.496874
$513$	0	0
$514$	−30.8604	−1.36120
$515$	−13.0227	−0.573848
$516$	0	0
$517$	8.26056	0.363299
$518$	−13.7489	−0.604092
$519$	0	0
$520$	−12.3965	−0.543623
$521$	13.0252	0.570645	0.285322	−	0.958432i	$-0.407899\pi$
$521$	13.0252	0.570645	0.285322	+	0.958432i	$0.407899\pi$
$522$	0	0
$523$	−32.9039	−1.43879	−0.719394	−	0.694602i	$-0.755580\pi$
$523$	−32.9039	−1.43879	−0.719394	+	0.694602i	$0.755580\pi$
$524$	68.0247	2.97167
$525$	0	0
$526$	−78.8492	−3.43799
$527$	36.3693	1.58427
$528$	0	0
$529$	1.00000	0.0434783
$530$	−37.7320	−1.63897
$531$	0	0
$532$	3.63425	0.157565
$533$	−5.26843	−0.228201
$534$	0	0
$535$	20.7955	0.899069
$536$	11.2001	0.483772
$537$	0	0
$538$	−18.9646	−0.817623
$539$	−7.88894	−0.339801
$540$	0	0
$541$	−17.4943	−0.752140	−0.376070	−	0.926591i	$-0.622725\pi$
$541$	−17.4943	−0.752140	−0.376070	+	0.926591i	$0.622725\pi$
$542$	−40.4810	−1.73881
$543$	0	0
$544$	118.856	5.09591
$545$	−3.25444	−0.139405
$546$	0	0
$547$	2.73202	0.116813	0.0584063	−	0.998293i	$-0.481398\pi$
$547$	2.73202	0.116813	0.0584063	+	0.998293i	$0.481398\pi$
$548$	75.8406	3.23975
$549$	0	0
$550$	−4.12396	−0.175846
$551$	−0.975858	−0.0415729
$552$	0	0
$553$	10.4535	0.444527
$554$	−63.9307	−2.71616
$555$	0	0
$556$	−11.6551	−0.494287
$557$	26.0332	1.10306	0.551531	−	0.834155i	$-0.314044\pi$
$557$	26.0332	1.10306	0.551531	+	0.834155i	$0.314044\pi$
$558$	0	0
$559$	3.18882	0.134873
$560$	−17.1480	−0.724636
$561$	0	0
$562$	17.5697	0.741134
$563$	33.1978	1.39912	0.699561	−	0.714573i	$-0.253379\pi$
$563$	33.1978	1.39912	0.699561	+	0.714573i	$0.253379\pi$
$564$	0	0
$565$	−18.3173	−0.770613
$566$	−69.9098	−2.93853
$567$	0	0
$568$	104.355	4.37863
$569$	−33.6879	−1.41227	−0.706135	−	0.708077i	$-0.749563\pi$
$569$	−33.6879	−1.41227	−0.706135	+	0.708077i	$0.749563\pi$
$570$	0	0
$571$	−31.9158	−1.33564	−0.667818	−	0.744324i	$-0.732772\pi$
$571$	−31.9158	−1.33564	−0.667818	+	0.744324i	$0.732772\pi$
$572$	4.76851	0.199381
$573$	0	0
$574$	−13.4047	−0.559503
$575$	1.26585	0.0527896
$576$	0	0
$577$	12.9808	0.540397	0.270198	−	0.962805i	$-0.412911\pi$
$577$	12.9808	0.540397	0.270198	+	0.962805i	$0.412911\pi$
$578$	102.433	4.26064
$579$	0	0
$580$	9.96781	0.413891
$581$	5.87422	0.243704
$582$	0	0
$583$	8.88664	0.368047
$584$	−40.6162	−1.68071
$585$	0	0
$586$	−64.0347	−2.64525
$587$	34.4468	1.42177	0.710885	−	0.703308i	$-0.248295\pi$
$587$	34.4468	1.42177	0.710885	+	0.703308i	$0.248295\pi$
$588$	0	0
$589$	4.77327	0.196679
$590$	−33.8603	−1.39401
$591$	0	0
$592$	−87.4849	−3.59561
$593$	−31.9913	−1.31373	−0.656863	−	0.754010i	$-0.728117\pi$
$593$	−31.9913	−1.31373	−0.656863	+	0.754010i	$0.728117\pi$
$594$	0	0
$595$	−10.3735	−0.425273
$596$	42.9166	1.75793
$597$	0	0
$598$	−2.03121	−0.0830624
$599$	−5.18325	−0.211782	−0.105891	−	0.994378i	$-0.533769\pi$
$599$	−5.18325	−0.211782	−0.105891	+	0.994378i	$0.533769\pi$
$600$	0	0
$601$	28.5632	1.16512	0.582560	−	0.812788i	$-0.302051\pi$
$601$	28.5632	1.16512	0.582560	+	0.812788i	$0.302051\pi$
$602$	8.11347	0.330680
$603$	0	0
$604$	−11.1266	−0.452733
$605$	18.3912	0.747707
$606$	0	0
$607$	37.5625	1.52461	0.762307	−	0.647215i	$-0.224066\pi$
$607$	37.5625	1.52461	0.762307	+	0.647215i	$0.224066\pi$
$608$	15.5992	0.632631
$609$	0	0
$610$	36.6073	1.48219
$611$	5.15031	0.208359
$612$	0	0
$613$	21.8684	0.883257	0.441628	−	0.897198i	$-0.354401\pi$
$613$	21.8684	0.883257	0.441628	+	0.897198i	$0.354401\pi$
$614$	42.2426	1.70477
$615$	0	0
$616$	7.42856	0.299305
$617$	6.11377	0.246131	0.123066	−	0.992399i	$-0.460727\pi$
$617$	6.11377	0.246131	0.123066	+	0.992399i	$0.460727\pi$
$618$	0	0
$619$	16.0179	0.643812	0.321906	−	0.946772i	$-0.395677\pi$
$619$	16.0179	0.643812	0.321906	+	0.946772i	$0.395677\pi$
$620$	−48.7561	−1.95809
$621$	0	0
$622$	26.9531	1.08072
$623$	0.329918	0.0132179
$624$	0	0
$625$	−17.0684	−0.682735
$626$	41.9594	1.67703
$627$	0	0
$628$	69.7375	2.78283
$629$	−52.9231	−2.11018
$630$	0	0
$631$	22.0014	0.875860	0.437930	−	0.899009i	$-0.355712\pi$
$631$	22.0014	0.875860	0.437930	+	0.899009i	$0.355712\pi$
$632$	122.346	4.86666
$633$	0	0
$634$	3.08289	0.122437
$635$	14.3058	0.567707
$636$	0	0
$637$	−4.91861	−0.194882
$638$	−3.25785	−0.128980
$639$	0	0
$640$	−32.2431	−1.27452
$641$	34.4264	1.35976	0.679881	−	0.733322i	$-0.262031\pi$
$641$	34.4264	1.35976	0.679881	+	0.733322i	$0.262031\pi$
$642$	0	0
$643$	21.9206	0.864464	0.432232	−	0.901762i	$-0.357726\pi$
$643$	21.9206	0.864464	0.432232	+	0.901762i	$0.357726\pi$
$644$	−3.72416	−0.146752
$645$	0	0
$646$	19.4132	0.763800
$647$	−23.9747	−0.942542	−0.471271	−	0.881988i	$-0.656205\pi$
$647$	−23.9747	−0.942542	−0.471271	+	0.881988i	$0.656205\pi$
$648$	0	0
$649$	7.97477	0.313037
$650$	−2.57121	−0.100851
$651$	0	0
$652$	−83.8782	−3.28492
$653$	−30.8057	−1.20552	−0.602760	−	0.797922i	$-0.705932\pi$
$653$	−30.8057	−1.20552	−0.602760	+	0.797922i	$0.705932\pi$
$654$	0	0
$655$	−25.4835	−0.995722
$656$	−85.2949	−3.33021
$657$	0	0
$658$	13.1042	0.510855
$659$	18.1149	0.705656	0.352828	−	0.935688i	$-0.385220\pi$
$659$	18.1149	0.705656	0.352828	+	0.935688i	$0.385220\pi$
$660$	0	0
$661$	30.7408	1.19568	0.597840	−	0.801616i	$-0.296026\pi$
$661$	30.7408	1.19568	0.597840	+	0.801616i	$0.296026\pi$
$662$	−88.0752	−3.42314
$663$	0	0
$664$	68.7509	2.66805
$665$	−1.36147	−0.0527954
$666$	0	0
$667$	1.00000	0.0387202
$668$	0.145645	0.00563516
$669$	0	0
$670$	−6.85283	−0.264748
$671$	−8.62175	−0.332839
$672$	0	0
$673$	−13.7528	−0.530132	−0.265066	−	0.964230i	$-0.585394\pi$
$673$	−13.7528	−0.530132	−0.265066	+	0.964230i	$0.585394\pi$
$674$	−76.0880	−2.93080
$675$	0	0
$676$	−64.0844	−2.46478
$677$	−22.5089	−0.865086	−0.432543	−	0.901613i	$-0.642384\pi$
$677$	−22.5089	−0.865086	−0.432543	+	0.901613i	$0.642384\pi$
$678$	0	0
$679$	−4.74831	−0.182223
$680$	−121.410	−4.65586
$681$	0	0
$682$	15.9353	0.610195
$683$	−7.72190	−0.295470	−0.147735	−	0.989027i	$-0.547198\pi$
$683$	−7.72190	−0.295470	−0.147735	+	0.989027i	$0.547198\pi$
$684$	0	0
$685$	−28.4115	−1.08555
$686$	−26.0362	−0.994068
$687$	0	0
$688$	51.6264	1.96824
$689$	5.54066	0.211082
$690$	0	0
$691$	9.51693	0.362041	0.181020	−	0.983479i	$-0.442060\pi$
$691$	9.51693	0.362041	0.181020	+	0.983479i	$0.442060\pi$
$692$	24.2574	0.922127
$693$	0	0
$694$	−56.9580	−2.16210
$695$	4.36625	0.165621
$696$	0	0
$697$	−51.5983	−1.95442
$698$	73.3136	2.77496
$699$	0	0
$700$	−4.71423	−0.178181
$701$	−4.42170	−0.167005	−0.0835026	−	0.996508i	$-0.526611\pi$
$701$	−4.42170	−0.167005	−0.0835026	+	0.996508i	$0.526611\pi$
$702$	0	0
$703$	−6.94586	−0.261968
$704$	22.1441	0.834588
$705$	0	0
$706$	−51.3874	−1.93399
$707$	−6.14066	−0.230943
$708$	0	0
$709$	−0.791672	−0.0297319	−0.0148659	−	0.999889i	$-0.504732\pi$
$709$	−0.791672	−0.0297319	−0.0148659	+	0.999889i	$0.504732\pi$
$710$	−63.8497	−2.39624
$711$	0	0
$712$	3.86131	0.144709
$713$	−4.89136	−0.183183
$714$	0	0
$715$	−1.78638	−0.0668069
$716$	75.9929	2.83999
$717$	0	0
$718$	−69.6732	−2.60018
$719$	19.2556	0.718112	0.359056	−	0.933316i	$-0.383099\pi$
$719$	19.2556	0.718112	0.359056	+	0.933316i	$0.383099\pi$
$720$	0	0
$721$	4.86551	0.181201
$722$	−48.2865	−1.79704
$723$	0	0
$724$	−47.1413	−1.75199
$725$	1.26585	0.0470125
$726$	0	0
$727$	−13.2064	−0.489800	−0.244900	−	0.969548i	$-0.578755\pi$
$727$	−13.2064	−0.489800	−0.244900	+	0.969548i	$0.578755\pi$
$728$	4.63157	0.171657
$729$	0	0
$730$	24.8511	0.919782
$731$	31.2309	1.15511
$732$	0	0
$733$	−10.3235	−0.381309	−0.190654	−	0.981657i	$-0.561061\pi$
$733$	−10.3235	−0.381309	−0.190654	+	0.981657i	$0.561061\pi$
$734$	−47.6879	−1.76019
$735$	0	0
$736$	−15.9851	−0.589219
$737$	1.61398	0.0594517
$738$	0	0
$739$	−18.6857	−0.687363	−0.343682	−	0.939086i	$-0.611674\pi$
$739$	−18.6857	−0.687363	−0.343682	+	0.939086i	$0.611674\pi$
$740$	70.9478	2.60809
$741$	0	0
$742$	14.0974	0.517531
$743$	13.6337	0.500171	0.250085	−	0.968224i	$-0.419541\pi$
$743$	13.6337	0.500171	0.250085	+	0.968224i	$0.419541\pi$
$744$	0	0
$745$	−16.0774	−0.589032
$746$	67.1554	2.45873
$747$	0	0
$748$	46.7021	1.70760
$749$	−7.76960	−0.283895
$750$	0	0
$751$	1.12990	0.0412307	0.0206153	−	0.999787i	$-0.493437\pi$
$751$	1.12990	0.0412307	0.0206153	+	0.999787i	$0.493437\pi$
$752$	83.3826	3.04065
$753$	0	0
$754$	−2.03121	−0.0739724
$755$	4.16824	0.151698
$756$	0	0
$757$	−18.0367	−0.655554	−0.327777	−	0.944755i	$-0.606300\pi$
$757$	−18.0367	−0.655554	−0.327777	+	0.944755i	$0.606300\pi$
$758$	77.9962	2.83295
$759$	0	0
$760$	−15.9344	−0.578000
$761$	−41.9255	−1.51980	−0.759899	−	0.650041i	$-0.774752\pi$
$761$	−41.9255	−1.51980	−0.759899	+	0.650041i	$0.774752\pi$
$762$	0	0
$763$	1.21592	0.0440193
$764$	−45.8106	−1.65737
$765$	0	0
$766$	−54.9735	−1.98627
$767$	4.97212	0.179533
$768$	0	0
$769$	−19.9423	−0.719138	−0.359569	−	0.933118i	$-0.617076\pi$
$769$	−19.9423	−0.719138	−0.359569	+	0.933118i	$0.617076\pi$
$770$	−4.54519	−0.163797
$771$	0	0
$772$	−115.110	−4.14291
$773$	53.2127	1.91393	0.956964	−	0.290206i	$-0.0937237\pi$
$773$	53.2127	1.91393	0.956964	+	0.290206i	$0.0937237\pi$
$774$	0	0
$775$	−6.19174	−0.222414
$776$	−55.5734	−1.99497
$777$	0	0
$778$	−0.392678	−0.0140782
$779$	−6.77199	−0.242632
$780$	0	0
$781$	15.0379	0.538098
$782$	−19.8934	−0.711387
$783$	0	0
$784$	−79.6314	−2.84398
$785$	−26.1251	−0.932446
$786$	0	0
$787$	−34.6938	−1.23670	−0.618351	−	0.785902i	$-0.712199\pi$
$787$	−34.6938	−1.23670	−0.618351	+	0.785902i	$0.712199\pi$
$788$	−72.2805	−2.57489
$789$	0	0
$790$	−74.8577	−2.66332
$791$	6.84367	0.243333
$792$	0	0
$793$	−5.37551	−0.190890
$794$	−88.7936	−3.15117
$795$	0	0
$796$	−136.131	−4.82502
$797$	25.4424	0.901218	0.450609	−	0.892721i	$-0.351207\pi$
$797$	25.4424	0.901218	0.450609	+	0.892721i	$0.351207\pi$
$798$	0	0
$799$	50.4415	1.78449
$800$	−20.2348	−0.715408
$801$	0	0
$802$	−58.2167	−2.05570
$803$	−5.85294	−0.206546
$804$	0	0
$805$	1.39515	0.0491725
$806$	9.93539	0.349959
$807$	0	0
$808$	−71.8692	−2.52835
$809$	−5.25710	−0.184830	−0.0924149	−	0.995721i	$-0.529459\pi$
$809$	−5.25710	−0.184830	−0.0924149	+	0.995721i	$0.529459\pi$
$810$	0	0
$811$	23.1032	0.811261	0.405631	−	0.914037i	$-0.367052\pi$
$811$	23.1032	0.811261	0.405631	+	0.914037i	$0.367052\pi$
$812$	−3.72416	−0.130692
$813$	0	0
$814$	−23.1884	−0.812753
$815$	31.4225	1.10068
$816$	0	0
$817$	4.09888	0.143402
$818$	−79.4186	−2.77681
$819$	0	0
$820$	69.1718	2.41558
$821$	23.2924	0.812912	0.406456	−	0.913670i	$-0.366765\pi$
$821$	23.2924	0.812912	0.406456	+	0.913670i	$0.366765\pi$
$822$	0	0
$823$	50.0863	1.74590	0.872950	−	0.487810i	$-0.162204\pi$
$823$	50.0863	1.74590	0.872950	+	0.487810i	$0.162204\pi$
$824$	56.9452	1.98378
$825$	0	0
$826$	12.6508	0.440179
$827$	−19.9497	−0.693719	−0.346859	−	0.937917i	$-0.612752\pi$
$827$	−19.9497	−0.693719	−0.346859	+	0.937917i	$0.612752\pi$
$828$	0	0
$829$	16.1619	0.561325	0.280663	−	0.959806i	$-0.409446\pi$
$829$	16.1619	0.561325	0.280663	+	0.959806i	$0.409446\pi$
$830$	−42.0655	−1.46011
$831$	0	0
$832$	13.8065	0.478653
$833$	−48.1722	−1.66907
$834$	0	0
$835$	−0.0545615	−0.00188818
$836$	6.12940	0.211990
$837$	0	0
$838$	49.7957	1.72016
$839$	−8.99321	−0.310480	−0.155240	−	0.987877i	$-0.549615\pi$
$839$	−8.99321	−0.310480	−0.155240	+	0.987877i	$0.549615\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−25.3799	−0.874650
$843$	0	0
$844$	50.8687	1.75097
$845$	24.0074	0.825878
$846$	0	0
$847$	−6.87129	−0.236100
$848$	89.7023	3.08039
$849$	0	0
$850$	−25.1821	−0.863739
$851$	7.11770	0.243992
$852$	0	0
$853$	−14.6568	−0.501841	−0.250920	−	0.968008i	$-0.580733\pi$
$853$	−14.6568	−0.501841	−0.250920	+	0.968008i	$0.580733\pi$
$854$	−13.6772	−0.468024
$855$	0	0
$856$	−90.9340	−3.10806
$857$	49.1741	1.67976	0.839878	−	0.542775i	$-0.182627\pi$
$857$	49.1741	1.67976	0.839878	+	0.542775i	$0.182627\pi$
$858$	0	0
$859$	−49.7842	−1.69862	−0.849308	−	0.527898i	$-0.822980\pi$
$859$	−49.7842	−1.69862	−0.849308	+	0.527898i	$0.822980\pi$
$860$	−41.8676	−1.42767
$861$	0	0
$862$	−70.3462	−2.39600
$863$	−19.2285	−0.654546	−0.327273	−	0.944930i	$-0.606130\pi$
$863$	−19.2285	−0.654546	−0.327273	+	0.944930i	$0.606130\pi$
$864$	0	0
$865$	−9.08732	−0.308978
$866$	90.0620	3.06043
$867$	0	0
$868$	18.2162	0.618299
$869$	17.6305	0.598073
$870$	0	0
$871$	1.00629	0.0340967
$872$	14.2309	0.481920
$873$	0	0
$874$	−2.61090	−0.0883151
$875$	8.74179	0.295526
$876$	0	0
$877$	−11.6483	−0.393333	−0.196667	−	0.980470i	$-0.563012\pi$
$877$	−11.6483	−0.393333	−0.196667	+	0.980470i	$0.563012\pi$
$878$	−53.7864	−1.81520
$879$	0	0
$880$	−28.9212	−0.974934
$881$	57.7298	1.94497	0.972483	−	0.232975i	$-0.0748462\pi$
$881$	57.7298	1.94497	0.972483	+	0.232975i	$0.0748462\pi$
$882$	0	0
$883$	−57.9128	−1.94892	−0.974460	−	0.224561i	$-0.927905\pi$
$883$	−57.9128	−1.94892	−0.974460	+	0.224561i	$0.927905\pi$
$884$	29.1179	0.979342
$885$	0	0
$886$	−34.6403	−1.16376
$887$	−2.99750	−0.100646	−0.0503231	−	0.998733i	$-0.516025\pi$
$887$	−2.99750	−0.100646	−0.0503231	+	0.998733i	$0.516025\pi$
$888$	0	0
$889$	−5.34491	−0.179262
$890$	−2.36255	−0.0791930
$891$	0	0
$892$	126.964	4.25106
$893$	6.62016	0.221535
$894$	0	0
$895$	−28.4685	−0.951598
$896$	12.0466	0.402450
$897$	0	0
$898$	−41.7325	−1.39263
$899$	−4.89136	−0.163136
$900$	0	0
$901$	54.2645	1.80781
$902$	−22.6079	−0.752762
$903$	0	0
$904$	80.0972	2.66399
$905$	17.6601	0.587043
$906$	0	0
$907$	3.99187	0.132548	0.0662740	−	0.997801i	$-0.478889\pi$
$907$	3.99187	0.132548	0.0662740	+	0.997801i	$0.478889\pi$
$908$	63.3448	2.10217
$909$	0	0
$910$	−2.83384	−0.0939409
$911$	40.9059	1.35527	0.677637	−	0.735397i	$-0.263004\pi$
$911$	40.9059	1.35527	0.677637	+	0.735397i	$0.263004\pi$
$912$	0	0
$913$	9.90726	0.327882
$914$	−43.7594	−1.44743
$915$	0	0
$916$	20.8645	0.689383
$917$	9.52111	0.314415
$918$	0	0
$919$	49.8939	1.64585	0.822925	−	0.568150i	$-0.192341\pi$
$919$	49.8939	1.64585	0.822925	+	0.568150i	$0.192341\pi$
$920$	16.3286	0.538337
$921$	0	0
$922$	108.919	3.58706
$923$	9.37585	0.308610
$924$	0	0
$925$	9.00994	0.296245
$926$	93.5436	3.07403
$927$	0	0
$928$	−15.9851	−0.524737
$929$	−7.25862	−0.238148	−0.119074	−	0.992885i	$-0.537993\pi$
$929$	−7.25862	−0.238148	−0.119074	+	0.992885i	$0.537993\pi$
$930$	0	0
$931$	−6.32234	−0.207206
$932$	−88.3344	−2.89349
$933$	0	0
$934$	−19.2212	−0.628937
$935$	−17.4956	−0.572167
$936$	0	0
$937$	−18.9134	−0.617872	−0.308936	−	0.951083i	$-0.599973\pi$
$937$	−18.9134	−0.617872	−0.308936	+	0.951083i	$0.599973\pi$
$938$	2.56035	0.0835983
$939$	0	0
$940$	−67.6210	−2.20555
$941$	−21.8323	−0.711712	−0.355856	−	0.934541i	$-0.615811\pi$
$941$	−21.8323	−0.711712	−0.355856	+	0.934541i	$0.615811\pi$
$942$	0	0
$943$	6.93952	0.225982
$944$	80.4978	2.61998
$945$	0	0
$946$	13.6839	0.444902
$947$	−7.36345	−0.239280	−0.119640	−	0.992817i	$-0.538174\pi$
$947$	−7.36345	−0.239280	−0.119640	+	0.992817i	$0.538174\pi$
$948$	0	0
$949$	−3.64920	−0.118458
$950$	−3.30501	−0.107229
$951$	0	0
$952$	45.3610	1.47016
$953$	−11.8527	−0.383945	−0.191973	−	0.981400i	$-0.561488\pi$
$953$	−11.8527	−0.383945	−0.191973	+	0.981400i	$0.561488\pi$
$954$	0	0
$955$	17.1616	0.555337
$956$	−127.814	−4.13380
$957$	0	0
$958$	40.0275	1.29323
$959$	10.6151	0.342778
$960$	0	0
$961$	−7.07458	−0.228212
$962$	−14.4575	−0.466130
$963$	0	0
$964$	99.3686	3.20045
$965$	43.1228	1.38817
$966$	0	0
$967$	14.5412	0.467614	0.233807	−	0.972283i	$-0.424882\pi$
$967$	14.5412	0.467614	0.233807	+	0.972283i	$0.424882\pi$
$968$	−80.4204	−2.58481
$969$	0	0
$970$	34.0028	1.09176
$971$	0.384819	0.0123494	0.00617471	−	0.999981i	$-0.498035\pi$
$971$	0.384819	0.0123494	0.00617471	+	0.999981i	$0.498035\pi$
$972$	0	0
$973$	−1.63131	−0.0522975
$974$	89.3338	2.86244
$975$	0	0
$976$	−87.0285	−2.78572
$977$	−38.9574	−1.24636	−0.623179	−	0.782079i	$-0.714159\pi$
$977$	−38.9574	−1.24636	−0.623179	+	0.782079i	$0.714159\pi$
$978$	0	0
$979$	0.556429	0.0177835
$980$	64.5789	2.06290
$981$	0	0
$982$	40.1091	1.27993
$983$	52.8096	1.68436	0.842182	−	0.539193i	$-0.181271\pi$
$983$	52.8096	1.68436	0.842182	+	0.539193i	$0.181271\pi$
$984$	0	0
$985$	27.0778	0.862771
$986$	−19.8934	−0.633536
$987$	0	0
$988$	3.82157	0.121580
$989$	−4.20028	−0.133561
$990$	0	0
$991$	−12.9556	−0.411547	−0.205773	−	0.978600i	$-0.565971\pi$
$991$	−12.9556	−0.411547	−0.205773	+	0.978600i	$0.565971\pi$
$992$	78.1890	2.48250
$993$	0	0
$994$	23.8555	0.756649
$995$	50.9974	1.61673
$996$	0	0
$997$	31.5432	0.998982	0.499491	−	0.866319i	$-0.333520\pi$
$997$	31.5432	0.998982	0.499491	+	0.866319i	$0.333520\pi$
$998$	30.2765	0.958386
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.l.1.10		10
3.2	odd	2		667.2.a.a.1.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.a.1.1	✓	10	3.2	odd	2
6003.2.a.l.1.10		10	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+2.67549 q^{2} +5.15827 q^{4} -1.93239 q^{5} +0.721979 q^{7} +8.44992 q^{8} +O(q^{10})\)	\(q+2.67549 q^{2} +5.15827 q^{4} -1.93239 q^{5} +0.721979 q^{7} +8.44992 q^{8} -5.17011 q^{10} +1.21766 q^{11} +0.759191 q^{13} +1.93165 q^{14} +12.2912 q^{16} +7.43542 q^{17} +0.975858 q^{19} -9.96781 q^{20} +3.25785 q^{22} -1.00000 q^{23} -1.26585 q^{25} +2.03121 q^{26} +3.72416 q^{28} -1.00000 q^{29} +4.89136 q^{31} +15.9851 q^{32} +19.8934 q^{34} -1.39515 q^{35} -7.11770 q^{37} +2.61090 q^{38} -16.3286 q^{40} -6.93952 q^{41} +4.20028 q^{43} +6.28103 q^{44} -2.67549 q^{46} +6.78394 q^{47} -6.47875 q^{49} -3.38678 q^{50} +3.91611 q^{52} +7.29810 q^{53} -2.35301 q^{55} +6.10066 q^{56} -2.67549 q^{58} +6.54924 q^{59} -7.08057 q^{61} +13.0868 q^{62} +18.1857 q^{64} -1.46706 q^{65} +1.32547 q^{67} +38.3539 q^{68} -3.73271 q^{70} +12.3498 q^{71} -4.80670 q^{73} -19.0434 q^{74} +5.03373 q^{76} +0.879128 q^{77} +14.4789 q^{79} -23.7514 q^{80} -18.5667 q^{82} +8.13628 q^{83} -14.3682 q^{85} +11.2378 q^{86} +10.2892 q^{88} +0.456964 q^{89} +0.548120 q^{91} -5.15827 q^{92} +18.1504 q^{94} -1.88574 q^{95} -6.57680 q^{97} -17.3338 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) \) 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8	\( 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8} - 6 q^{10} - 13 q^{13} + 12 q^{14} - 5 q^{16} + 22 q^{17} - 2 q^{19} - 3 q^{20} + 3 q^{22} - 10 q^{23} + 10 q^{25} + 25 q^{26} + 19 q^{28} - 10 q^{29} - 22 q^{31} + 31 q^{32} + 13 q^{34} + 15 q^{35} - 9 q^{37} + 10 q^{38} - 6 q^{40} + 25 q^{41} + 3 q^{43} + 27 q^{44} - 3 q^{46} + 17 q^{47} + 17 q^{49} - 2 q^{50} - 18 q^{52} + 43 q^{53} - 11 q^{55} + 7 q^{56} - 3 q^{58} + 7 q^{59} - 6 q^{61} - 3 q^{62} + 33 q^{64} - 11 q^{65} + 11 q^{67} + 51 q^{68} + 34 q^{70} + 17 q^{71} - 44 q^{73} - 9 q^{74} + 24 q^{76} + 71 q^{77} + 5 q^{79} - 38 q^{80} + 33 q^{82} + 32 q^{83} + 16 q^{85} + 9 q^{86} + 18 q^{88} + 10 q^{89} - 3 q^{91} - 9 q^{92} + 47 q^{94} + 8 q^{95} + 6 q^{97} + 73 q^{98}+O(q^{100}) \) 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8 - 6 * q^10 - 13 * q^13 + 12 * q^14 - 5 * q^16 + 22 * q^17 - 2 * q^19 - 3 * q^20 + 3 * q^22 - 10 * q^23 + 10 * q^25 + 25 * q^26 + 19 * q^28 - 10 * q^29 - 22 * q^31 + 31 * q^32 + 13 * q^34 + 15 * q^35 - 9 * q^37 + 10 * q^38 - 6 * q^40 + 25 * q^41 + 3 * q^43 + 27 * q^44 - 3 * q^46 + 17 * q^47 + 17 * q^49 - 2 * q^50 - 18 * q^52 + 43 * q^53 - 11 * q^55 + 7 * q^56 - 3 * q^58 + 7 * q^59 - 6 * q^61 - 3 * q^62 + 33 * q^64 - 11 * q^65 + 11 * q^67 + 51 * q^68 + 34 * q^70 + 17 * q^71 - 44 * q^73 - 9 * q^74 + 24 * q^76 + 71 * q^77 + 5 * q^79 - 38 * q^80 + 33 * q^82 + 32 * q^83 + 16 * q^85 + 9 * q^86 + 18 * q^88 + 10 * q^89 - 3 * q^91 - 9 * q^92 + 47 * q^94 + 8 * q^95 + 6 * q^97 + 73 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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