LMFDB - Embedded newform 6003.2.a.j.1.6

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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 5x^{5} + 18x^{4} + 4x^{3} - 26x^{2} + x + 8 $ x^7 - 3x^6 - 5x^5 + 18x^4 + 4x^3 - 26*x^2 + x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.22973$ 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)

$f(q)$	$=$	$q+2.22973 q^{2} +2.97167 q^{4} -0.537118 q^{5} -1.88241 q^{7} +2.16657 q^{8} +O(q^{10})$	\(q+2.22973 q^{2} +2.97167 q^{4} -0.537118 q^{5} -1.88241 q^{7} +2.16657 q^{8} -1.19762 q^{10} -1.11723 q^{11} +1.56781 q^{13} -4.19726 q^{14} -1.11250 q^{16} +3.41741 q^{17} -2.62638 q^{19} -1.59614 q^{20} -2.49110 q^{22} -1.00000 q^{23} -4.71150 q^{25} +3.49579 q^{26} -5.59392 q^{28} +1.00000 q^{29} -2.26219 q^{31} -6.81370 q^{32} +7.61989 q^{34} +1.01108 q^{35} +1.64906 q^{37} -5.85610 q^{38} -1.16370 q^{40} -5.06667 q^{41} -10.0332 q^{43} -3.32003 q^{44} -2.22973 q^{46} +6.06906 q^{47} -3.45652 q^{49} -10.5054 q^{50} +4.65903 q^{52} -10.2961 q^{53} +0.600081 q^{55} -4.07837 q^{56} +2.22973 q^{58} +8.80232 q^{59} -3.73087 q^{61} -5.04406 q^{62} -12.9677 q^{64} -0.842100 q^{65} -7.67562 q^{67} +10.1554 q^{68} +2.25442 q^{70} +2.43146 q^{71} +1.02242 q^{73} +3.67695 q^{74} -7.80475 q^{76} +2.10308 q^{77} +1.24162 q^{79} +0.597544 q^{80} -11.2973 q^{82} -13.0672 q^{83} -1.83555 q^{85} -22.3712 q^{86} -2.42054 q^{88} +13.1482 q^{89} -2.95127 q^{91} -2.97167 q^{92} +13.5323 q^{94} +1.41068 q^{95} -7.41766 q^{97} -7.70710 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8}+O(q^{10}) $ 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8	$ 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8} + 3 q^{10} + 4 q^{11} - 18 q^{13} + 2 q^{14} - 7 q^{16} + 3 q^{17} - 4 q^{19} + 2 q^{20} - 26 q^{22} - 7 q^{23} - 8 q^{25} + 7 q^{26} - 6 q^{28} + 7 q^{29} - 22 q^{31} - 5 q^{32} + 9 q^{34} - 3 q^{35} - 25 q^{37} - 14 q^{38} - 10 q^{40} + 13 q^{41} - 2 q^{43} - 4 q^{44} - 3 q^{46} + 25 q^{47} - 8 q^{49} - 19 q^{50} - 12 q^{52} + 5 q^{53} - 15 q^{55} - 18 q^{56} + 3 q^{58} - 11 q^{59} - 33 q^{61} - 28 q^{62} - 14 q^{64} + 2 q^{65} + 8 q^{67} - 12 q^{68} - 22 q^{70} + 6 q^{71} + 15 q^{73} - 34 q^{74} - 28 q^{76} + q^{77} - 15 q^{79} + 12 q^{80} - 14 q^{82} - 21 q^{83} - 28 q^{85} + 12 q^{86} - 13 q^{88} - 8 q^{89} + 6 q^{91} - 5 q^{92} - 35 q^{94} + 25 q^{95} + 13 q^{97} - q^{98}+O(q^{100}) $ 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8 + 3 * q^10 + 4 * q^11 - 18 * q^13 + 2 * q^14 - 7 * q^16 + 3 * q^17 - 4 * q^19 + 2 * q^20 - 26 * q^22 - 7 * q^23 - 8 * q^25 + 7 * q^26 - 6 * q^28 + 7 * q^29 - 22 * q^31 - 5 * q^32 + 9 * q^34 - 3 * q^35 - 25 * q^37 - 14 * q^38 - 10 * q^40 + 13 * q^41 - 2 * q^43 - 4 * q^44 - 3 * q^46 + 25 * q^47 - 8 * q^49 - 19 * q^50 - 12 * q^52 + 5 * q^53 - 15 * q^55 - 18 * q^56 + 3 * q^58 - 11 * q^59 - 33 * q^61 - 28 * q^62 - 14 * q^64 + 2 * q^65 + 8 * q^67 - 12 * q^68 - 22 * q^70 + 6 * q^71 + 15 * q^73 - 34 * q^74 - 28 * q^76 + q^77 - 15 * q^79 + 12 * q^80 - 14 * q^82 - 21 * q^83 - 28 * q^85 + 12 * q^86 - 13 * q^88 - 8 * q^89 + 6 * q^91 - 5 * q^92 - 35 * q^94 + 25 * q^95 + 13 * q^97 - 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 $

$2$	2.22973	1.57665	0.788327	−	0.615257i	$-0.210948\pi$
$2$	2.22973	1.57665	0.788327	+	0.615257i	$0.210948\pi$
$3$	0	0
$3$	0	0
$4$	2.97167	1.48584
$5$	−0.537118	−0.240206	−0.120103	−	0.992761i	$-0.538323\pi$
$5$	−0.537118	−0.240206	−0.120103	+	0.992761i	$0.538323\pi$
$6$	0	0
$7$	−1.88241	−0.711485	−0.355743	−	0.934584i	$-0.615772\pi$
$7$	−1.88241	−0.711485	−0.355743	+	0.934584i	$0.615772\pi$
$8$	2.16657	0.765997
$9$	0	0
$10$	−1.19762	−0.378722
$11$	−1.11723	−0.336856	−0.168428	−	0.985714i	$-0.553869\pi$
$11$	−1.11723	−0.336856	−0.168428	+	0.985714i	$0.553869\pi$
$12$	0	0
$13$	1.56781	0.434833	0.217417	−	0.976079i	$-0.430237\pi$
$13$	1.56781	0.434833	0.217417	+	0.976079i	$0.430237\pi$
$14$	−4.19726	−1.12177
$15$	0	0
$16$	−1.11250	−0.278125
$17$	3.41741	0.828844	0.414422	−	0.910085i	$-0.363984\pi$
$17$	3.41741	0.828844	0.414422	+	0.910085i	$0.363984\pi$
$18$	0	0
$19$	−2.62638	−0.602533	−0.301266	−	0.953540i	$-0.597409\pi$
$19$	−2.62638	−0.602533	−0.301266	+	0.953540i	$0.597409\pi$
$20$	−1.59614	−0.356908
$21$	0	0
$22$	−2.49110	−0.531105
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.71150	−0.942301
$26$	3.49579	0.685581
$27$	0	0
$28$	−5.59392	−1.05715
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−2.26219	−0.406301	−0.203150	−	0.979148i	$-0.565118\pi$
$31$	−2.26219	−0.406301	−0.203150	+	0.979148i	$0.565118\pi$
$32$	−6.81370	−1.20450
$33$	0	0
$34$	7.61989	1.30680
$35$	1.01108	0.170903
$36$	0	0
$37$	1.64906	0.271104	0.135552	−	0.990770i	$-0.456719\pi$
$37$	1.64906	0.271104	0.135552	+	0.990770i	$0.456719\pi$
$38$	−5.85610	−0.949986
$39$	0	0
$40$	−1.16370	−0.183997
$41$	−5.06667	−0.791280	−0.395640	−	0.918406i	$-0.629477\pi$
$41$	−5.06667	−0.791280	−0.395640	+	0.918406i	$0.629477\pi$
$42$	0	0
$43$	−10.0332	−1.53004	−0.765021	−	0.644005i	$-0.777272\pi$
$43$	−10.0332	−1.53004	−0.765021	+	0.644005i	$0.777272\pi$
$44$	−3.32003	−0.500513
$45$	0	0
$46$	−2.22973	−0.328755
$47$	6.06906	0.885263	0.442631	−	0.896704i	$-0.354045\pi$
$47$	6.06906	0.885263	0.442631	+	0.896704i	$0.354045\pi$
$48$	0	0
$49$	−3.45652	−0.493789
$50$	−10.5054	−1.48568
$51$	0	0
$52$	4.65903	0.646091
$53$	−10.2961	−1.41427	−0.707137	−	0.707077i	$-0.750013\pi$
$53$	−10.2961	−1.41427	−0.707137	+	0.707077i	$0.750013\pi$
$54$	0	0
$55$	0.600081	0.0809150
$56$	−4.07837	−0.544995
$57$	0	0
$58$	2.22973	0.292777
$59$	8.80232	1.14596	0.572982	−	0.819568i	$-0.305786\pi$
$59$	8.80232	1.14596	0.572982	+	0.819568i	$0.305786\pi$
$60$	0	0
$61$	−3.73087	−0.477689	−0.238845	−	0.971058i	$-0.576769\pi$
$61$	−3.73087	−0.477689	−0.238845	+	0.971058i	$0.576769\pi$
$62$	−5.04406	−0.640596
$63$	0	0
$64$	−12.9677	−1.62096
$65$	−0.842100	−0.104450
$66$	0	0
$67$	−7.67562	−0.937726	−0.468863	−	0.883271i	$-0.655336\pi$
$67$	−7.67562	−0.937726	−0.468863	+	0.883271i	$0.655336\pi$
$68$	10.1554	1.23153
$69$	0	0
$70$	2.25442	0.269455
$71$	2.43146	0.288561	0.144281	−	0.989537i	$-0.453913\pi$
$71$	2.43146	0.288561	0.144281	+	0.989537i	$0.453913\pi$
$72$	0	0
$73$	1.02242	0.119665	0.0598326	−	0.998208i	$-0.480943\pi$
$73$	1.02242	0.119665	0.0598326	+	0.998208i	$0.480943\pi$
$74$	3.67695	0.427437
$75$	0	0
$76$	−7.80475	−0.895266
$77$	2.10308	0.239668
$78$	0	0
$79$	1.24162	0.139693	0.0698466	−	0.997558i	$-0.477749\pi$
$79$	1.24162	0.139693	0.0698466	+	0.997558i	$0.477749\pi$
$80$	0.597544	0.0668074
$81$	0	0
$82$	−11.2973	−1.24758
$83$	−13.0672	−1.43432	−0.717158	−	0.696910i	$-0.754558\pi$
$83$	−13.0672	−1.43432	−0.717158	+	0.696910i	$0.754558\pi$
$84$	0	0
$85$	−1.83555	−0.199094
$86$	−22.3712	−2.41235
$87$	0	0
$88$	−2.42054	−0.258031
$89$	13.1482	1.39370	0.696851	−	0.717216i	$-0.254584\pi$
$89$	13.1482	1.39370	0.696851	+	0.717216i	$0.254584\pi$
$90$	0	0
$91$	−2.95127	−0.309377
$92$	−2.97167	−0.309818
$93$	0	0
$94$	13.5323	1.39575
$95$	1.41068	0.144732
$96$	0	0
$97$	−7.41766	−0.753149	−0.376575	−	0.926386i	$-0.622898\pi$
$97$	−7.41766	−0.753149	−0.376575	+	0.926386i	$0.622898\pi$
$98$	−7.70710	−0.778534
$99$	0	0
$100$	−14.0011	−1.40011
$101$	2.80104	0.278714	0.139357	−	0.990242i	$-0.455496\pi$
$101$	2.80104	0.278714	0.139357	+	0.990242i	$0.455496\pi$
$102$	0	0
$103$	10.0128	0.986589	0.493295	−	0.869862i	$-0.335792\pi$
$103$	10.0128	0.986589	0.493295	+	0.869862i	$0.335792\pi$
$104$	3.39677	0.333081
$105$	0	0
$106$	−22.9574	−2.22982
$107$	5.25354	0.507878	0.253939	−	0.967220i	$-0.418274\pi$
$107$	5.25354	0.507878	0.253939	+	0.967220i	$0.418274\pi$
$108$	0	0
$109$	−5.08675	−0.487223	−0.243611	−	0.969873i	$-0.578332\pi$
$109$	−5.08675	−0.487223	−0.243611	+	0.969873i	$0.578332\pi$
$110$	1.33802	0.127575
$111$	0	0
$112$	2.09418	0.197882
$113$	−18.3795	−1.72900	−0.864500	−	0.502633i	$-0.832365\pi$
$113$	−18.3795	−1.72900	−0.864500	+	0.502633i	$0.832365\pi$
$114$	0	0
$115$	0.537118	0.0500865
$116$	2.97167	0.275913
$117$	0	0
$118$	19.6268	1.80679
$119$	−6.43298	−0.589710
$120$	0	0
$121$	−9.75181	−0.886528
$122$	−8.31882	−0.753151
$123$	0	0
$124$	−6.72249	−0.603697
$125$	5.21622	0.466553
$126$	0	0
$127$	4.33301	0.384492	0.192246	−	0.981347i	$-0.438423\pi$
$127$	4.33301	0.384492	0.192246	+	0.981347i	$0.438423\pi$
$128$	−15.2870	−1.35119
$129$	0	0
$130$	−1.87765	−0.164681
$131$	−19.3607	−1.69155	−0.845777	−	0.533537i	$-0.820863\pi$
$131$	−19.3607	−1.69155	−0.845777	+	0.533537i	$0.820863\pi$
$132$	0	0
$133$	4.94393	0.428693
$134$	−17.1145	−1.47847
$135$	0	0
$136$	7.40405	0.634892
$137$	10.2352	0.874456	0.437228	−	0.899351i	$-0.355960\pi$
$137$	10.2352	0.874456	0.437228	+	0.899351i	$0.355960\pi$
$138$	0	0
$139$	−0.289212	−0.0245306	−0.0122653	−	0.999925i	$-0.503904\pi$
$139$	−0.289212	−0.0245306	−0.0122653	+	0.999925i	$0.503904\pi$
$140$	3.00459	0.253934
$141$	0	0
$142$	5.42149	0.454961
$143$	−1.75160	−0.146476
$144$	0	0
$145$	−0.537118	−0.0446052
$146$	2.27972	0.188671
$147$	0	0
$148$	4.90047	0.402816
$149$	−20.1266	−1.64884	−0.824419	−	0.565979i	$-0.808498\pi$
$149$	−20.1266	−1.64884	−0.824419	+	0.565979i	$0.808498\pi$
$150$	0	0
$151$	0.370644	0.0301626	0.0150813	−	0.999886i	$-0.495199\pi$
$151$	0.370644	0.0301626	0.0150813	+	0.999886i	$0.495199\pi$
$152$	−5.69023	−0.461538
$153$	0	0
$154$	4.68929	0.377874
$155$	1.21506	0.0975961
$156$	0	0
$157$	−7.03038	−0.561086	−0.280543	−	0.959842i	$-0.590514\pi$
$157$	−7.03038	−0.561086	−0.280543	+	0.959842i	$0.590514\pi$
$158$	2.76847	0.220248
$159$	0	0
$160$	3.65976	0.289330
$161$	1.88241	0.148355
$162$	0	0
$163$	−19.9434	−1.56209	−0.781044	−	0.624476i	$-0.785312\pi$
$163$	−19.9434	−1.56209	−0.781044	+	0.624476i	$0.785312\pi$
$164$	−15.0565	−1.17571
$165$	0	0
$166$	−29.1364	−2.26142
$167$	12.4215	0.961201	0.480601	−	0.876940i	$-0.340419\pi$
$167$	12.4215	0.961201	0.480601	+	0.876940i	$0.340419\pi$
$168$	0	0
$169$	−10.5420	−0.810920
$170$	−4.09278	−0.313902
$171$	0	0
$172$	−29.8153	−2.27339
$173$	12.5473	0.953954	0.476977	−	0.878916i	$-0.341733\pi$
$173$	12.5473	0.953954	0.476977	+	0.878916i	$0.341733\pi$
$174$	0	0
$175$	8.86900	0.670433
$176$	1.24291	0.0936881
$177$	0	0
$178$	29.3168	2.19739
$179$	23.4139	1.75003	0.875017	−	0.484092i	$-0.160850\pi$
$179$	23.4139	1.75003	0.875017	+	0.484092i	$0.160850\pi$
$180$	0	0
$181$	−10.0534	−0.747263	−0.373632	−	0.927577i	$-0.621888\pi$
$181$	−10.0534	−0.747263	−0.373632	+	0.927577i	$0.621888\pi$
$182$	−6.58052	−0.487781
$183$	0	0
$184$	−2.16657	−0.159721
$185$	−0.885740	−0.0651209
$186$	0	0
$187$	−3.81802	−0.279201
$188$	18.0353	1.31536
$189$	0	0
$190$	3.14542	0.228193
$191$	20.7174	1.49905	0.749527	−	0.661973i	$-0.230281\pi$
$191$	20.7174	1.49905	0.749527	+	0.661973i	$0.230281\pi$
$192$	0	0
$193$	8.55796	0.616015	0.308008	−	0.951384i	$-0.400338\pi$
$193$	8.55796	0.616015	0.308008	+	0.951384i	$0.400338\pi$
$194$	−16.5393	−1.18746
$195$	0	0
$196$	−10.2717	−0.733690
$197$	−5.84888	−0.416715	−0.208358	−	0.978053i	$-0.566812\pi$
$197$	−5.84888	−0.416715	−0.208358	+	0.978053i	$0.566812\pi$
$198$	0	0
$199$	9.16433	0.649642	0.324821	−	0.945775i	$-0.394696\pi$
$199$	9.16433	0.649642	0.324821	+	0.945775i	$0.394696\pi$
$200$	−10.2078	−0.721800
$201$	0	0
$202$	6.24555	0.439436
$203$	−1.88241	−0.132119
$204$	0	0
$205$	2.72140	0.190071
$206$	22.3258	1.55551
$207$	0	0
$208$	−1.74419	−0.120938
$209$	2.93426	0.202967
$210$	0	0
$211$	−0.0926588	−0.00637889	−0.00318945	−	0.999995i	$-0.501015\pi$
$211$	−0.0926588	−0.00637889	−0.00318945	+	0.999995i	$0.501015\pi$
$212$	−30.5966	−2.10138
$213$	0	0
$214$	11.7139	0.800749
$215$	5.38899	0.367526
$216$	0	0
$217$	4.25837	0.289077
$218$	−11.3421	−0.768182
$219$	0	0
$220$	1.78325	0.120226
$221$	5.35786	0.360409
$222$	0	0
$223$	−12.8223	−0.858643	−0.429322	−	0.903152i	$-0.641247\pi$
$223$	−12.8223	−0.858643	−0.429322	+	0.903152i	$0.641247\pi$
$224$	12.8262	0.856987
$225$	0	0
$226$	−40.9813	−2.72603
$227$	−14.6393	−0.971643	−0.485821	−	0.874058i	$-0.661479\pi$
$227$	−14.6393	−0.971643	−0.485821	+	0.874058i	$0.661479\pi$
$228$	0	0
$229$	−24.1483	−1.59577	−0.797883	−	0.602812i	$-0.794047\pi$
$229$	−24.1483	−1.59577	−0.797883	+	0.602812i	$0.794047\pi$
$230$	1.19762	0.0789691
$231$	0	0
$232$	2.16657	0.142242
$233$	−16.5564	−1.08465	−0.542324	−	0.840170i	$-0.682455\pi$
$233$	−16.5564	−1.08465	−0.542324	+	0.840170i	$0.682455\pi$
$234$	0	0
$235$	−3.25980	−0.212646
$236$	26.1576	1.70272
$237$	0	0
$238$	−14.3438	−0.929769
$239$	18.3128	1.18455	0.592277	−	0.805735i	$-0.298229\pi$
$239$	18.3128	1.18455	0.592277	+	0.805735i	$0.298229\pi$
$240$	0	0
$241$	16.0413	1.03331	0.516655	−	0.856194i	$-0.327177\pi$
$241$	16.0413	1.03331	0.516655	+	0.856194i	$0.327177\pi$
$242$	−21.7439	−1.39775
$243$	0	0
$244$	−11.0869	−0.709768
$245$	1.85656	0.118611
$246$	0	0
$247$	−4.11767	−0.262001
$248$	−4.90118	−0.311225
$249$	0	0
$250$	11.6307	0.735593
$251$	16.2544	1.02597	0.512983	−	0.858399i	$-0.328540\pi$
$251$	16.2544	1.02597	0.512983	+	0.858399i	$0.328540\pi$
$252$	0	0
$253$	1.11723	0.0702393
$254$	9.66141	0.606211
$255$	0	0
$256$	−8.15037	−0.509398
$257$	−2.52528	−0.157523	−0.0787613	−	0.996894i	$-0.525096\pi$
$257$	−2.52528	−0.157523	−0.0787613	+	0.996894i	$0.525096\pi$
$258$	0	0
$259$	−3.10421	−0.192886
$260$	−2.50245	−0.155195
$261$	0	0
$262$	−43.1691	−2.66699
$263$	28.6908	1.76915	0.884575	−	0.466399i	$-0.154449\pi$
$263$	28.6908	1.76915	0.884575	+	0.466399i	$0.154449\pi$
$264$	0	0
$265$	5.53020	0.339717
$266$	11.0236	0.675901
$267$	0	0
$268$	−22.8094	−1.39331
$269$	18.4710	1.12620	0.563099	−	0.826389i	$-0.309609\pi$
$269$	18.4710	1.12620	0.563099	+	0.826389i	$0.309609\pi$
$270$	0	0
$271$	−7.71437	−0.468614	−0.234307	−	0.972163i	$-0.575282\pi$
$271$	−7.71437	−0.468614	−0.234307	+	0.972163i	$0.575282\pi$
$272$	−3.80187	−0.230522
$273$	0	0
$274$	22.8218	1.37871
$275$	5.26381	0.317420
$276$	0	0
$277$	−0.702588	−0.0422144	−0.0211072	−	0.999777i	$-0.506719\pi$
$277$	−0.702588	−0.0422144	−0.0211072	+	0.999777i	$0.506719\pi$
$278$	−0.644863	−0.0386763
$279$	0	0
$280$	2.19057	0.130911
$281$	−2.46040	−0.146775	−0.0733876	−	0.997303i	$-0.523381\pi$
$281$	−2.46040	−0.146775	−0.0733876	+	0.997303i	$0.523381\pi$
$282$	0	0
$283$	−15.1893	−0.902908	−0.451454	−	0.892294i	$-0.649095\pi$
$283$	−15.1893	−0.902908	−0.451454	+	0.892294i	$0.649095\pi$
$284$	7.22551	0.428755
$285$	0	0
$286$	−3.90559	−0.230942
$287$	9.53756	0.562984
$288$	0	0
$289$	−5.32130	−0.313018
$290$	−1.19762	−0.0703270
$291$	0	0
$292$	3.03830	0.177803
$293$	−8.84429	−0.516689	−0.258344	−	0.966053i	$-0.583177\pi$
$293$	−8.84429	−0.516689	−0.258344	+	0.966053i	$0.583177\pi$
$294$	0	0
$295$	−4.72788	−0.275268
$296$	3.57280	0.207665
$297$	0	0
$298$	−44.8769	−2.59965
$299$	−1.56781	−0.0906690
$300$	0	0
$301$	18.8865	1.08860
$302$	0.826433	0.0475559
$303$	0	0
$304$	2.92185	0.167579
$305$	2.00392	0.114744
$306$	0	0
$307$	5.52326	0.315229	0.157614	−	0.987501i	$-0.449620\pi$
$307$	5.52326	0.315229	0.157614	+	0.987501i	$0.449620\pi$
$308$	6.24966	0.356108
$309$	0	0
$310$	2.70925	0.153875
$311$	−0.0662747	−0.00375809	−0.00187905	−	0.999998i	$-0.500598\pi$
$311$	−0.0662747	−0.00375809	−0.00187905	+	0.999998i	$0.500598\pi$
$312$	0	0
$313$	17.7785	1.00490	0.502450	−	0.864606i	$-0.332432\pi$
$313$	17.7785	1.00490	0.502450	+	0.864606i	$0.332432\pi$
$314$	−15.6758	−0.884638
$315$	0	0
$316$	3.68969	0.207561
$317$	8.12451	0.456318	0.228159	−	0.973624i	$-0.426729\pi$
$317$	8.12451	0.456318	0.228159	+	0.973624i	$0.426729\pi$
$318$	0	0
$319$	−1.11723	−0.0625526
$320$	6.96517	0.389365
$321$	0	0
$322$	4.19726	0.233904
$323$	−8.97542	−0.499406
$324$	0	0
$325$	−7.38676	−0.409744
$326$	−44.4683	−2.46287
$327$	0	0
$328$	−10.9773	−0.606118
$329$	−11.4245	−0.629851
$330$	0	0
$331$	20.2364	1.11229	0.556146	−	0.831085i	$-0.312280\pi$
$331$	20.2364	1.11229	0.556146	+	0.831085i	$0.312280\pi$
$332$	−38.8316	−2.13116
$333$	0	0
$334$	27.6964	1.51548
$335$	4.12271	0.225248
$336$	0	0
$337$	19.0760	1.03914	0.519568	−	0.854429i	$-0.326093\pi$
$337$	19.0760	1.03914	0.519568	+	0.854429i	$0.326093\pi$
$338$	−23.5057	−1.27854
$339$	0	0
$340$	−5.45466	−0.295821
$341$	2.52737	0.136865
$342$	0	0
$343$	19.6835	1.06281
$344$	−21.7375	−1.17201
$345$	0	0
$346$	27.9770	1.50405
$347$	29.5028	1.58379	0.791897	−	0.610655i	$-0.209094\pi$
$347$	29.5028	1.58379	0.791897	+	0.610655i	$0.209094\pi$
$348$	0	0
$349$	−11.6279	−0.622426	−0.311213	−	0.950340i	$-0.600735\pi$
$349$	−11.6279	−0.622426	−0.311213	+	0.950340i	$0.600735\pi$
$350$	19.7754	1.05704
$351$	0	0
$352$	7.61244	0.405744
$353$	0.349829	0.0186195	0.00930975	−	0.999957i	$-0.497037\pi$
$353$	0.349829	0.0186195	0.00930975	+	0.999957i	$0.497037\pi$
$354$	0	0
$355$	−1.30598	−0.0693143
$356$	39.0721	2.07082
$357$	0	0
$358$	52.2065	2.75920
$359$	17.4984	0.923528	0.461764	−	0.887003i	$-0.347217\pi$
$359$	17.4984	0.923528	0.461764	+	0.887003i	$0.347217\pi$
$360$	0	0
$361$	−12.1021	−0.636954
$362$	−22.4163	−1.17818
$363$	0	0
$364$	−8.77022	−0.459684
$365$	−0.549160	−0.0287444
$366$	0	0
$367$	−33.9848	−1.77399	−0.886997	−	0.461775i	$-0.847213\pi$
$367$	−33.9848	−1.77399	−0.886997	+	0.461775i	$0.847213\pi$
$368$	1.11250	0.0579931
$369$	0	0
$370$	−1.97496	−0.102673
$371$	19.3814	1.00623
$372$	0	0
$373$	19.9955	1.03533	0.517664	−	0.855584i	$-0.326802\pi$
$373$	19.9955	1.03533	0.517664	+	0.855584i	$0.326802\pi$
$374$	−8.51313	−0.440203
$375$	0	0
$376$	13.1490	0.678109
$377$	1.56781	0.0807465
$378$	0	0
$379$	23.0774	1.18541	0.592704	−	0.805420i	$-0.298060\pi$
$379$	23.0774	1.18541	0.592704	+	0.805420i	$0.298060\pi$
$380$	4.19207	0.215049
$381$	0	0
$382$	46.1940	2.36349
$383$	−32.8565	−1.67889	−0.839445	−	0.543444i	$-0.817120\pi$
$383$	−32.8565	−1.67889	−0.839445	+	0.543444i	$0.817120\pi$
$384$	0	0
$385$	−1.12960	−0.0575698
$386$	19.0819	0.971243
$387$	0	0
$388$	−22.0429	−1.11906
$389$	−13.9353	−0.706547	−0.353274	−	0.935520i	$-0.614932\pi$
$389$	−13.9353	−0.706547	−0.353274	+	0.935520i	$0.614932\pi$
$390$	0	0
$391$	−3.41741	−0.172826
$392$	−7.48879	−0.378241
$393$	0	0
$394$	−13.0414	−0.657016
$395$	−0.666896	−0.0335552
$396$	0	0
$397$	13.9272	0.698988	0.349494	−	0.936939i	$-0.386353\pi$
$397$	13.9272	0.698988	0.349494	+	0.936939i	$0.386353\pi$
$398$	20.4339	1.02426
$399$	0	0
$400$	5.24155	0.262077
$401$	−5.07639	−0.253503	−0.126752	−	0.991935i	$-0.540455\pi$
$401$	−5.07639	−0.253503	−0.126752	+	0.991935i	$0.540455\pi$
$402$	0	0
$403$	−3.54669	−0.176673
$404$	8.32378	0.414124
$405$	0	0
$406$	−4.19726	−0.208307
$407$	−1.84237	−0.0913230
$408$	0	0
$409$	−20.8607	−1.03150	−0.515748	−	0.856740i	$-0.672486\pi$
$409$	−20.8607	−1.03150	−0.515748	+	0.856740i	$0.672486\pi$
$410$	6.06797	0.299676
$411$	0	0
$412$	29.7547	1.46591
$413$	−16.5696	−0.815336
$414$	0	0
$415$	7.01865	0.344532
$416$	−10.6826	−0.523758
$417$	0	0
$418$	6.54259	0.320008
$419$	−3.26760	−0.159633	−0.0798163	−	0.996810i	$-0.525433\pi$
$419$	−3.26760	−0.159633	−0.0798163	+	0.996810i	$0.525433\pi$
$420$	0	0
$421$	−1.97217	−0.0961175	−0.0480587	−	0.998845i	$-0.515303\pi$
$421$	−1.97217	−0.0961175	−0.0480587	+	0.998845i	$0.515303\pi$
$422$	−0.206604	−0.0100573
$423$	0	0
$424$	−22.3071	−1.08333
$425$	−16.1011	−0.781020
$426$	0	0
$427$	7.02304	0.339869
$428$	15.6118	0.754625
$429$	0	0
$430$	12.0160	0.579461
$431$	3.63225	0.174959	0.0874797	−	0.996166i	$-0.472119\pi$
$431$	3.63225	0.174959	0.0874797	+	0.996166i	$0.472119\pi$
$432$	0	0
$433$	−1.31543	−0.0632156	−0.0316078	−	0.999500i	$-0.510063\pi$
$433$	−1.31543	−0.0632156	−0.0316078	+	0.999500i	$0.510063\pi$
$434$	9.49500	0.455774
$435$	0	0
$436$	−15.1162	−0.723934
$437$	2.62638	0.125637
$438$	0	0
$439$	17.1910	0.820480	0.410240	−	0.911978i	$-0.365445\pi$
$439$	17.1910	0.820480	0.410240	+	0.911978i	$0.365445\pi$
$440$	1.30012	0.0619806
$441$	0	0
$442$	11.9466	0.568240
$443$	9.59576	0.455908	0.227954	−	0.973672i	$-0.426796\pi$
$443$	9.59576	0.455908	0.227954	+	0.973672i	$0.426796\pi$
$444$	0	0
$445$	−7.06211	−0.334776
$446$	−28.5902	−1.35378
$447$	0	0
$448$	24.4105	1.15329
$449$	−19.5494	−0.922594	−0.461297	−	0.887246i	$-0.652616\pi$
$449$	−19.5494	−0.922594	−0.461297	+	0.887246i	$0.652616\pi$
$450$	0	0
$451$	5.66061	0.266548
$452$	−54.6179	−2.56901
$453$	0	0
$454$	−32.6416	−1.53194
$455$	1.58518	0.0743144
$456$	0	0
$457$	−12.3018	−0.575453	−0.287726	−	0.957713i	$-0.592899\pi$
$457$	−12.3018	−0.575453	−0.287726	+	0.957713i	$0.592899\pi$
$458$	−53.8441	−2.51597
$459$	0	0
$460$	1.59614	0.0744204
$461$	33.7959	1.57403	0.787016	−	0.616933i	$-0.211625\pi$
$461$	33.7959	1.57403	0.787016	+	0.616933i	$0.211625\pi$
$462$	0	0
$463$	32.9615	1.53185	0.765925	−	0.642930i	$-0.222282\pi$
$463$	32.9615	1.53185	0.765925	+	0.642930i	$0.222282\pi$
$464$	−1.11250	−0.0516465
$465$	0	0
$466$	−36.9163	−1.71011
$467$	11.5793	0.535826	0.267913	−	0.963443i	$-0.413666\pi$
$467$	11.5793	0.535826	0.267913	+	0.963443i	$0.413666\pi$
$468$	0	0
$469$	14.4487	0.667178
$470$	−7.26845	−0.335269
$471$	0	0
$472$	19.0708	0.877805
$473$	11.2093	0.515404
$474$	0	0
$475$	12.3742	0.567767
$476$	−19.1167	−0.876213
$477$	0	0
$478$	40.8324	1.86763
$479$	−14.2331	−0.650326	−0.325163	−	0.945658i	$-0.605419\pi$
$479$	−14.2331	−0.650326	−0.325163	+	0.945658i	$0.605419\pi$
$480$	0	0
$481$	2.58542	0.117885
$482$	35.7676	1.62917
$483$	0	0
$484$	−28.9792	−1.31724
$485$	3.98416	0.180911
$486$	0	0
$487$	25.8118	1.16964	0.584821	−	0.811162i	$-0.301165\pi$
$487$	25.8118	1.16964	0.584821	+	0.811162i	$0.301165\pi$
$488$	−8.08318	−0.365909
$489$	0	0
$490$	4.13962	0.187009
$491$	35.5608	1.60484	0.802419	−	0.596761i	$-0.203546\pi$
$491$	35.5608	1.60484	0.802419	+	0.596761i	$0.203546\pi$
$492$	0	0
$493$	3.41741	0.153912
$494$	−9.18128	−0.413085
$495$	0	0
$496$	2.51668	0.113002
$497$	−4.57701	−0.205307
$498$	0	0
$499$	15.5456	0.695918	0.347959	−	0.937510i	$-0.386875\pi$
$499$	15.5456	0.695918	0.347959	+	0.937510i	$0.386875\pi$
$500$	15.5009	0.693222
$501$	0	0
$502$	36.2428	1.61759
$503$	−19.4816	−0.868641	−0.434321	−	0.900758i	$-0.643011\pi$
$503$	−19.4816	−0.868641	−0.434321	+	0.900758i	$0.643011\pi$
$504$	0	0
$505$	−1.50449	−0.0669489
$506$	2.49110	0.110743
$507$	0	0
$508$	12.8763	0.571293
$509$	−21.8713	−0.969428	−0.484714	−	0.874673i	$-0.661076\pi$
$509$	−21.8713	−0.969428	−0.484714	+	0.874673i	$0.661076\pi$
$510$	0	0
$511$	−1.92462	−0.0851401
$512$	12.4009	0.548046
$513$	0	0
$514$	−5.63068	−0.248359
$515$	−5.37805	−0.236985
$516$	0	0
$517$	−6.78050	−0.298206
$518$	−6.92154	−0.304115
$519$	0	0
$520$	−1.82447	−0.0800082
$521$	−27.2480	−1.19375	−0.596877	−	0.802333i	$-0.703592\pi$
$521$	−27.2480	−1.19375	−0.596877	+	0.802333i	$0.703592\pi$
$522$	0	0
$523$	16.5965	0.725715	0.362858	−	0.931845i	$-0.381801\pi$
$523$	16.5965	0.725715	0.362858	+	0.931845i	$0.381801\pi$
$524$	−57.5337	−2.51337
$525$	0	0
$526$	63.9726	2.78934
$527$	−7.73083	−0.336760
$528$	0	0
$529$	1.00000	0.0434783
$530$	12.3308	0.535617
$531$	0	0
$532$	14.6917	0.636968
$533$	−7.94359	−0.344075
$534$	0	0
$535$	−2.82177	−0.121996
$536$	−16.6297	−0.718295
$537$	0	0
$538$	41.1853	1.77563
$539$	3.86171	0.166336
$540$	0	0
$541$	−32.7816	−1.40939	−0.704696	−	0.709509i	$-0.748917\pi$
$541$	−32.7816	−1.40939	−0.704696	+	0.709509i	$0.748917\pi$
$542$	−17.2009	−0.738843
$543$	0	0
$544$	−23.2852	−0.998346
$545$	2.73219	0.117034
$546$	0	0
$547$	32.1520	1.37472	0.687359	−	0.726318i	$-0.258770\pi$
$547$	32.1520	1.37472	0.687359	+	0.726318i	$0.258770\pi$
$548$	30.4158	1.29930
$549$	0	0
$550$	11.7369	0.500461
$551$	−2.62638	−0.111888
$552$	0	0
$553$	−2.33724	−0.0993896
$554$	−1.56658	−0.0665575
$555$	0	0
$556$	−0.859444	−0.0364485
$557$	4.00666	0.169768	0.0848839	−	0.996391i	$-0.472948\pi$
$557$	4.00666	0.169768	0.0848839	+	0.996391i	$0.472948\pi$
$558$	0	0
$559$	−15.7301	−0.665313
$560$	−1.12482	−0.0475325
$561$	0	0
$562$	−5.48602	−0.231414
$563$	−2.38095	−0.100345	−0.0501726	−	0.998741i	$-0.515977\pi$
$563$	−2.38095	−0.100345	−0.0501726	+	0.998741i	$0.515977\pi$
$564$	0	0
$565$	9.87196	0.415317
$566$	−33.8679	−1.42357
$567$	0	0
$568$	5.26792	0.221037
$569$	−42.7445	−1.79194	−0.895971	−	0.444113i	$-0.853519\pi$
$569$	−42.7445	−1.79194	−0.895971	+	0.444113i	$0.853519\pi$
$570$	0	0
$571$	−29.5201	−1.23538	−0.617688	−	0.786423i	$-0.711930\pi$
$571$	−29.5201	−1.23538	−0.617688	+	0.786423i	$0.711930\pi$
$572$	−5.20519	−0.217640
$573$	0	0
$574$	21.2661	0.887631
$575$	4.71150	0.196483
$576$	0	0
$577$	31.3969	1.30707	0.653535	−	0.756896i	$-0.273285\pi$
$577$	31.3969	1.30707	0.653535	+	0.756896i	$0.273285\pi$
$578$	−11.8650	−0.493521
$579$	0	0
$580$	−1.59614	−0.0662761
$581$	24.5980	1.02049
$582$	0	0
$583$	11.5030	0.476406
$584$	2.21514	0.0916633
$585$	0	0
$586$	−19.7203	−0.814640
$587$	−44.3332	−1.82983	−0.914913	−	0.403652i	$-0.867741\pi$
$587$	−44.3332	−1.82983	−0.914913	+	0.403652i	$0.867741\pi$
$588$	0	0
$589$	5.94136	0.244810
$590$	−10.5419	−0.434002
$591$	0	0
$592$	−1.83458	−0.0754008
$593$	7.80918	0.320685	0.160342	−	0.987061i	$-0.448740\pi$
$593$	7.80918	0.320685	0.160342	+	0.987061i	$0.448740\pi$
$594$	0	0
$595$	3.45527	0.141652
$596$	−59.8098	−2.44991
$597$	0	0
$598$	−3.49579	−0.142954
$599$	26.4951	1.08256	0.541280	−	0.840843i	$-0.317940\pi$
$599$	26.4951	1.08256	0.541280	+	0.840843i	$0.317940\pi$
$600$	0	0
$601$	44.4503	1.81317	0.906583	−	0.422027i	$-0.138681\pi$
$601$	44.4503	1.81317	0.906583	+	0.422027i	$0.138681\pi$
$602$	42.1118	1.71635
$603$	0	0
$604$	1.10143	0.0448166
$605$	5.23787	0.212950
$606$	0	0
$607$	−31.2745	−1.26939	−0.634697	−	0.772761i	$-0.718875\pi$
$607$	−31.2745	−1.26939	−0.634697	+	0.772761i	$0.718875\pi$
$608$	17.8954	0.725753
$609$	0	0
$610$	4.46819	0.180912
$611$	9.51515	0.384942
$612$	0	0
$613$	8.39427	0.339041	0.169521	−	0.985527i	$-0.445778\pi$
$613$	8.39427	0.339041	0.169521	+	0.985527i	$0.445778\pi$
$614$	12.3153	0.497007
$615$	0	0
$616$	4.55646	0.183585
$617$	−9.87343	−0.397489	−0.198745	−	0.980051i	$-0.563686\pi$
$617$	−9.87343	−0.397489	−0.198745	+	0.980051i	$0.563686\pi$
$618$	0	0
$619$	36.9182	1.48387	0.741935	−	0.670472i	$-0.233908\pi$
$619$	36.9182	1.48387	0.741935	+	0.670472i	$0.233908\pi$
$620$	3.61077	0.145012
$621$	0	0
$622$	−0.147774	−0.00592521
$623$	−24.7503	−0.991599
$624$	0	0
$625$	20.7558	0.830232
$626$	39.6411	1.58438
$627$	0	0
$628$	−20.8920	−0.833682
$629$	5.63552	0.224703
$630$	0	0
$631$	15.9457	0.634788	0.317394	−	0.948294i	$-0.397192\pi$
$631$	15.9457	0.634788	0.317394	+	0.948294i	$0.397192\pi$
$632$	2.69005	0.107005
$633$	0	0
$634$	18.1154	0.719455
$635$	−2.32734	−0.0923575
$636$	0	0
$637$	−5.41918	−0.214716
$638$	−2.49110	−0.0986238
$639$	0	0
$640$	8.21090	0.324564
$641$	2.07492	0.0819542	0.0409771	−	0.999160i	$-0.486953\pi$
$641$	2.07492	0.0819542	0.0409771	+	0.999160i	$0.486953\pi$
$642$	0	0
$643$	7.94610	0.313364	0.156682	−	0.987649i	$-0.449920\pi$
$643$	7.94610	0.313364	0.156682	+	0.987649i	$0.449920\pi$
$644$	5.59392	0.220431
$645$	0	0
$646$	−20.0127	−0.787390
$647$	−16.9460	−0.666217	−0.333109	−	0.942888i	$-0.608098\pi$
$647$	−16.9460	−0.666217	−0.333109	+	0.942888i	$0.608098\pi$
$648$	0	0
$649$	−9.83417	−0.386025
$650$	−16.4704	−0.646024
$651$	0	0
$652$	−59.2653	−2.32101
$653$	−21.8874	−0.856520	−0.428260	−	0.903656i	$-0.640873\pi$
$653$	−21.8874	−0.856520	−0.428260	+	0.903656i	$0.640873\pi$
$654$	0	0
$655$	10.3990	0.406322
$656$	5.63667	0.220075
$657$	0	0
$658$	−25.4734	−0.993057
$659$	−28.5542	−1.11231	−0.556157	−	0.831077i	$-0.687725\pi$
$659$	−28.5542	−1.11231	−0.556157	+	0.831077i	$0.687725\pi$
$660$	0	0
$661$	−17.1280	−0.666202	−0.333101	−	0.942891i	$-0.608095\pi$
$661$	−17.1280	−0.666202	−0.333101	+	0.942891i	$0.608095\pi$
$662$	45.1215	1.75370
$663$	0	0
$664$	−28.3111	−1.09868
$665$	−2.65547	−0.102975
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	36.9125	1.42819
$669$	0	0
$670$	9.19251	0.355138
$671$	4.16822	0.160912
$672$	0	0
$673$	5.47704	0.211124	0.105562	−	0.994413i	$-0.466336\pi$
$673$	5.47704	0.211124	0.105562	+	0.994413i	$0.466336\pi$
$674$	42.5342	1.63836
$675$	0	0
$676$	−31.3273	−1.20490
$677$	42.4886	1.63297	0.816485	−	0.577367i	$-0.195920\pi$
$677$	42.4886	1.63297	0.816485	+	0.577367i	$0.195920\pi$
$678$	0	0
$679$	13.9631	0.535855
$680$	−3.97685	−0.152505
$681$	0	0
$682$	5.63535	0.215789
$683$	25.1897	0.963857	0.481929	−	0.876210i	$-0.339936\pi$
$683$	25.1897	0.963857	0.481929	+	0.876210i	$0.339936\pi$
$684$	0	0
$685$	−5.49753	−0.210050
$686$	43.8888	1.67568
$687$	0	0
$688$	11.1619	0.425543
$689$	−16.1423	−0.614973
$690$	0	0
$691$	5.79765	0.220553	0.110277	−	0.993901i	$-0.464826\pi$
$691$	5.79765	0.220553	0.110277	+	0.993901i	$0.464826\pi$
$692$	37.2865	1.41742
$693$	0	0
$694$	65.7832	2.49710
$695$	0.155341	0.00589241
$696$	0	0
$697$	−17.3149	−0.655848
$698$	−25.9270	−0.981351
$699$	0	0
$700$	26.3558	0.996154
$701$	−13.4101	−0.506494	−0.253247	−	0.967402i	$-0.581499\pi$
$701$	−13.4101	−0.506494	−0.253247	+	0.967402i	$0.581499\pi$
$702$	0	0
$703$	−4.33106	−0.163349
$704$	14.4878	0.546030
$705$	0	0
$706$	0.780022	0.0293565
$707$	−5.27271	−0.198301
$708$	0	0
$709$	−18.0256	−0.676965	−0.338482	−	0.940973i	$-0.609914\pi$
$709$	−18.0256	−0.676965	−0.338482	+	0.940973i	$0.609914\pi$
$710$	−2.91198	−0.109285
$711$	0	0
$712$	28.4864	1.06757
$713$	2.26219	0.0847196
$714$	0	0
$715$	0.940816	0.0351845
$716$	69.5784	2.60027
$717$	0	0
$718$	39.0165	1.45608
$719$	20.6072	0.768518	0.384259	−	0.923225i	$-0.374457\pi$
$719$	20.6072	0.768518	0.384259	+	0.923225i	$0.374457\pi$
$720$	0	0
$721$	−18.8482	−0.701944
$722$	−26.9844	−1.00426
$723$	0	0
$724$	−29.8754	−1.11031
$725$	−4.71150	−0.174981
$726$	0	0
$727$	−8.38118	−0.310841	−0.155420	−	0.987848i	$-0.549673\pi$
$727$	−8.38118	−0.310841	−0.155420	+	0.987848i	$0.549673\pi$
$728$	−6.39413	−0.236982
$729$	0	0
$730$	−1.22448	−0.0453199
$731$	−34.2874	−1.26817
$732$	0	0
$733$	−9.51389	−0.351404	−0.175702	−	0.984443i	$-0.556219\pi$
$733$	−9.51389	−0.351404	−0.175702	+	0.984443i	$0.556219\pi$
$734$	−75.7768	−2.79697
$735$	0	0
$736$	6.81370	0.251156
$737$	8.57539	0.315879
$738$	0	0
$739$	−31.0800	−1.14330	−0.571648	−	0.820499i	$-0.693696\pi$
$739$	−31.0800	−1.14330	−0.571648	+	0.820499i	$0.693696\pi$
$740$	−2.63213	−0.0967590
$741$	0	0
$742$	43.2153	1.58648
$743$	−45.6581	−1.67503	−0.837517	−	0.546412i	$-0.815993\pi$
$743$	−45.6581	−1.67503	−0.837517	+	0.546412i	$0.815993\pi$
$744$	0	0
$745$	10.8104	0.396062
$746$	44.5844	1.63235
$747$	0	0
$748$	−11.3459	−0.414847
$749$	−9.88932	−0.361348
$750$	0	0
$751$	−30.5101	−1.11333	−0.556664	−	0.830737i	$-0.687919\pi$
$751$	−30.5101	−1.11333	−0.556664	+	0.830737i	$0.687919\pi$
$752$	−6.75183	−0.246214
$753$	0	0
$754$	3.49579	0.127309
$755$	−0.199079	−0.00724524
$756$	0	0
$757$	−30.5168	−1.10915	−0.554576	−	0.832133i	$-0.687120\pi$
$757$	−30.5168	−1.10915	−0.554576	+	0.832133i	$0.687120\pi$
$758$	51.4563	1.86898
$759$	0	0
$760$	3.05632	0.110864
$761$	20.6725	0.749377	0.374689	−	0.927151i	$-0.377750\pi$
$761$	20.6725	0.749377	0.374689	+	0.927151i	$0.377750\pi$
$762$	0	0
$763$	9.57537	0.346652
$764$	61.5652	2.22735
$765$	0	0
$766$	−73.2611	−2.64703
$767$	13.8004	0.498303
$768$	0	0
$769$	32.3606	1.16695	0.583476	−	0.812130i	$-0.301692\pi$
$769$	32.3606	1.16695	0.583476	+	0.812130i	$0.301692\pi$
$770$	−2.51870	−0.0907676
$771$	0	0
$772$	25.4315	0.915299
$773$	−8.63105	−0.310437	−0.155219	−	0.987880i	$-0.549608\pi$
$773$	−8.63105	−0.310437	−0.155219	+	0.987880i	$0.549608\pi$
$774$	0	0
$775$	10.6583	0.382858
$776$	−16.0709	−0.576910
$777$	0	0
$778$	−31.0719	−1.11398
$779$	13.3070	0.476772
$780$	0	0
$781$	−2.71649	−0.0972037
$782$	−7.61989	−0.272487
$783$	0	0
$784$	3.84538	0.137335
$785$	3.77614	0.134776
$786$	0	0
$787$	17.5702	0.626310	0.313155	−	0.949702i	$-0.398614\pi$
$787$	17.5702	0.626310	0.313155	+	0.949702i	$0.398614\pi$
$788$	−17.3810	−0.619171
$789$	0	0
$790$	−1.48700	−0.0529049
$791$	34.5978	1.23016
$792$	0	0
$793$	−5.84931	−0.207715
$794$	31.0539	1.10206
$795$	0	0
$796$	27.2334	0.965262
$797$	−15.4807	−0.548355	−0.274178	−	0.961679i	$-0.588406\pi$
$797$	−15.4807	−0.548355	−0.274178	+	0.961679i	$0.588406\pi$
$798$	0	0
$799$	20.7405	0.733745
$800$	32.1028	1.13501
$801$	0	0
$802$	−11.3190	−0.399686
$803$	−1.14227	−0.0403100
$804$	0	0
$805$	−1.01108	−0.0356358
$806$	−7.90814	−0.278552
$807$	0	0
$808$	6.06864	0.213494
$809$	19.9422	0.701131	0.350565	−	0.936538i	$-0.385989\pi$
$809$	19.9422	0.701131	0.350565	+	0.936538i	$0.385989\pi$
$810$	0	0
$811$	−24.0964	−0.846139	−0.423069	−	0.906097i	$-0.639047\pi$
$811$	−24.0964	−0.846139	−0.423069	+	0.906097i	$0.639047\pi$
$812$	−5.59392	−0.196308
$813$	0	0
$814$	−4.10798	−0.143985
$815$	10.7120	0.375223
$816$	0	0
$817$	26.3509	0.921901
$818$	−46.5137	−1.62631
$819$	0	0
$820$	8.08710	0.282414
$821$	−29.8257	−1.04092	−0.520461	−	0.853885i	$-0.674240\pi$
$821$	−29.8257	−1.04092	−0.520461	+	0.853885i	$0.674240\pi$
$822$	0	0
$823$	−4.38398	−0.152816	−0.0764080	−	0.997077i	$-0.524345\pi$
$823$	−4.38398	−0.152816	−0.0764080	+	0.997077i	$0.524345\pi$
$824$	21.6934	0.755725
$825$	0	0
$826$	−36.9456	−1.28550
$827$	−33.8262	−1.17625	−0.588126	−	0.808769i	$-0.700134\pi$
$827$	−33.8262	−1.17625	−0.588126	+	0.808769i	$0.700134\pi$
$828$	0	0
$829$	19.9346	0.692358	0.346179	−	0.938169i	$-0.387479\pi$
$829$	19.9346	0.692358	0.346179	+	0.938169i	$0.387479\pi$
$830$	15.6497	0.543208
$831$	0	0
$832$	−20.3309	−0.704848
$833$	−11.8124	−0.409274
$834$	0	0
$835$	−6.67179	−0.230887
$836$	8.71966	0.301576
$837$	0	0
$838$	−7.28584	−0.251685
$839$	9.52467	0.328828	0.164414	−	0.986391i	$-0.447427\pi$
$839$	9.52467	0.328828	0.164414	+	0.986391i	$0.447427\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−4.39739	−0.151544
$843$	0	0
$844$	−0.275352	−0.00947800
$845$	5.66227	0.194788
$846$	0	0
$847$	18.3569	0.630751
$848$	11.4544	0.393345
$849$	0	0
$850$	−35.9011	−1.23140
$851$	−1.64906	−0.0565291
$852$	0	0
$853$	45.9120	1.57200	0.785998	−	0.618229i	$-0.212150\pi$
$853$	45.9120	1.57200	0.785998	+	0.618229i	$0.212150\pi$
$854$	15.6594	0.535855
$855$	0	0
$856$	11.3821	0.389033
$857$	−54.7299	−1.86954	−0.934768	−	0.355258i	$-0.884393\pi$
$857$	−54.7299	−1.86954	−0.934768	+	0.355258i	$0.884393\pi$
$858$	0	0
$859$	−39.0054	−1.33085	−0.665423	−	0.746466i	$-0.731749\pi$
$859$	−39.0054	−1.33085	−0.665423	+	0.746466i	$0.731749\pi$
$860$	16.0143	0.546084
$861$	0	0
$862$	8.09892	0.275850
$863$	−14.2092	−0.483685	−0.241843	−	0.970316i	$-0.577752\pi$
$863$	−14.2092	−0.483685	−0.241843	+	0.970316i	$0.577752\pi$
$864$	0	0
$865$	−6.73938	−0.229146
$866$	−2.93305	−0.0996692
$867$	0	0
$868$	12.6545	0.429521
$869$	−1.38717	−0.0470565
$870$	0	0
$871$	−12.0339	−0.407754
$872$	−11.0208	−0.373211
$873$	0	0
$874$	5.85610	0.198086
$875$	−9.81908	−0.331946
$876$	0	0
$877$	17.7516	0.599431	0.299715	−	0.954029i	$-0.403108\pi$
$877$	17.7516	0.599431	0.299715	+	0.954029i	$0.403108\pi$
$878$	38.3311	1.29361
$879$	0	0
$880$	−0.667591	−0.0225045
$881$	−10.8019	−0.363926	−0.181963	−	0.983305i	$-0.558245\pi$
$881$	−10.8019	−0.363926	−0.181963	+	0.983305i	$0.558245\pi$
$882$	0	0
$883$	−23.4523	−0.789232	−0.394616	−	0.918846i	$-0.629122\pi$
$883$	−23.4523	−0.789232	−0.394616	+	0.918846i	$0.629122\pi$
$884$	15.9218	0.535509
$885$	0	0
$886$	21.3959	0.718809
$887$	39.1075	1.31310	0.656550	−	0.754283i	$-0.272015\pi$
$887$	39.1075	1.31310	0.656550	+	0.754283i	$0.272015\pi$
$888$	0	0
$889$	−8.15651	−0.273560
$890$	−15.7466	−0.527826
$891$	0	0
$892$	−38.1036	−1.27580
$893$	−15.9396	−0.533400
$894$	0	0
$895$	−12.5760	−0.420369
$896$	28.7764	0.961351
$897$	0	0
$898$	−43.5898	−1.45461
$899$	−2.26219	−0.0754482
$900$	0	0
$901$	−35.1859	−1.17221
$902$	12.6216	0.420253
$903$	0	0
$904$	−39.8204	−1.32441
$905$	5.39986	0.179497
$906$	0	0
$907$	0.854437	0.0283711	0.0141856	−	0.999899i	$-0.495484\pi$
$907$	0.854437	0.0283711	0.0141856	+	0.999899i	$0.495484\pi$
$908$	−43.5032	−1.44370
$909$	0	0
$910$	3.53452	0.117168
$911$	−27.2030	−0.901276	−0.450638	−	0.892707i	$-0.648803\pi$
$911$	−27.2030	−0.901276	−0.450638	+	0.892707i	$0.648803\pi$
$912$	0	0
$913$	14.5991	0.483158
$914$	−27.4296	−0.907290
$915$	0	0
$916$	−71.7609	−2.37105
$917$	36.4449	1.20352
$918$	0	0
$919$	−46.5672	−1.53611	−0.768055	−	0.640384i	$-0.778775\pi$
$919$	−46.5672	−1.53611	−0.768055	+	0.640384i	$0.778775\pi$
$920$	1.16370	0.0383661
$921$	0	0
$922$	75.3556	2.48170
$923$	3.81208	0.125476
$924$	0	0
$925$	−7.76956	−0.255461
$926$	73.4950	2.41520
$927$	0	0
$928$	−6.81370	−0.223671
$929$	−31.6499	−1.03840	−0.519200	−	0.854653i	$-0.673770\pi$
$929$	−31.6499	−1.03840	−0.519200	+	0.854653i	$0.673770\pi$
$930$	0	0
$931$	9.07814	0.297524
$932$	−49.2003	−1.61161
$933$	0	0
$934$	25.8186	0.844812
$935$	2.05072	0.0670659
$936$	0	0
$937$	17.7619	0.580255	0.290128	−	0.956988i	$-0.406302\pi$
$937$	17.7619	0.580255	0.290128	+	0.956988i	$0.406302\pi$
$938$	32.2166	1.05191
$939$	0	0
$940$	−9.68706	−0.315957
$941$	−52.0711	−1.69747	−0.848734	−	0.528820i	$-0.822635\pi$
$941$	−52.0711	−1.69747	−0.848734	+	0.528820i	$0.822635\pi$
$942$	0	0
$943$	5.06667	0.164993
$944$	−9.79258	−0.318721
$945$	0	0
$946$	24.9937	0.812614
$947$	7.10267	0.230806	0.115403	−	0.993319i	$-0.463184\pi$
$947$	7.10267	0.230806	0.115403	+	0.993319i	$0.463184\pi$
$948$	0	0
$949$	1.60296	0.0520344
$950$	27.5911	0.895172
$951$	0	0
$952$	−13.9375	−0.451716
$953$	−11.9585	−0.387373	−0.193687	−	0.981063i	$-0.562044\pi$
$953$	−11.9585	−0.387373	−0.193687	+	0.981063i	$0.562044\pi$
$954$	0	0
$955$	−11.1277	−0.360083
$956$	54.4195	1.76005
$957$	0	0
$958$	−31.7359	−1.02534
$959$	−19.2670	−0.622163
$960$	0	0
$961$	−25.8825	−0.834920
$962$	5.76478	0.185864
$963$	0	0
$964$	47.6694	1.53533
$965$	−4.59663	−0.147971
$966$	0	0
$967$	10.3118	0.331604	0.165802	−	0.986159i	$-0.446979\pi$
$967$	10.3118	0.331604	0.165802	+	0.986159i	$0.446979\pi$
$968$	−21.1279	−0.679078
$969$	0	0
$970$	8.88358	0.285234
$971$	11.4796	0.368398	0.184199	−	0.982889i	$-0.441031\pi$
$971$	11.4796	0.368398	0.184199	+	0.982889i	$0.441031\pi$
$972$	0	0
$973$	0.544416	0.0174532
$974$	57.5531	1.84412
$975$	0	0
$976$	4.15060	0.132857
$977$	12.8478	0.411036	0.205518	−	0.978653i	$-0.434112\pi$
$977$	12.8478	0.411036	0.205518	+	0.978653i	$0.434112\pi$
$978$	0	0
$979$	−14.6895	−0.469477
$980$	5.51709	0.176237
$981$	0	0
$982$	79.2909	2.53027
$983$	−1.92074	−0.0612622	−0.0306311	−	0.999531i	$-0.509752\pi$
$983$	−1.92074	−0.0612622	−0.0306311	+	0.999531i	$0.509752\pi$
$984$	0	0
$985$	3.14154	0.100098
$986$	7.61989	0.242667
$987$	0	0
$988$	−12.2364	−0.389291
$989$	10.0332	0.319036
$990$	0	0
$991$	−2.06517	−0.0656022	−0.0328011	−	0.999462i	$-0.510443\pi$
$991$	−2.06517	−0.0656022	−0.0328011	+	0.999462i	$0.510443\pi$
$992$	15.4139	0.489391
$993$	0	0
$994$	−10.2055	−0.323698
$995$	−4.92232	−0.156048
$996$	0	0
$997$	−40.2717	−1.27542	−0.637709	−	0.770277i	$-0.720118\pi$
$997$	−40.2717	−1.27542	−0.637709	+	0.770277i	$0.720118\pi$
$998$	34.6625	1.09722
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.j.1.6		7
3.2	odd	2		2001.2.a.i.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.i.1.2	✓	7	3.2	odd	2
6003.2.a.j.1.6		7	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)

\(f(q)\)	\(=\)	\(q+2.22973 q^{2} +2.97167 q^{4} -0.537118 q^{5} -1.88241 q^{7} +2.16657 q^{8} +O(q^{10})\)	\(q+2.22973 q^{2} +2.97167 q^{4} -0.537118 q^{5} -1.88241 q^{7} +2.16657 q^{8} -1.19762 q^{10} -1.11723 q^{11} +1.56781 q^{13} -4.19726 q^{14} -1.11250 q^{16} +3.41741 q^{17} -2.62638 q^{19} -1.59614 q^{20} -2.49110 q^{22} -1.00000 q^{23} -4.71150 q^{25} +3.49579 q^{26} -5.59392 q^{28} +1.00000 q^{29} -2.26219 q^{31} -6.81370 q^{32} +7.61989 q^{34} +1.01108 q^{35} +1.64906 q^{37} -5.85610 q^{38} -1.16370 q^{40} -5.06667 q^{41} -10.0332 q^{43} -3.32003 q^{44} -2.22973 q^{46} +6.06906 q^{47} -3.45652 q^{49} -10.5054 q^{50} +4.65903 q^{52} -10.2961 q^{53} +0.600081 q^{55} -4.07837 q^{56} +2.22973 q^{58} +8.80232 q^{59} -3.73087 q^{61} -5.04406 q^{62} -12.9677 q^{64} -0.842100 q^{65} -7.67562 q^{67} +10.1554 q^{68} +2.25442 q^{70} +2.43146 q^{71} +1.02242 q^{73} +3.67695 q^{74} -7.80475 q^{76} +2.10308 q^{77} +1.24162 q^{79} +0.597544 q^{80} -11.2973 q^{82} -13.0672 q^{83} -1.83555 q^{85} -22.3712 q^{86} -2.42054 q^{88} +13.1482 q^{89} -2.95127 q^{91} -2.97167 q^{92} +13.5323 q^{94} +1.41068 q^{95} -7.41766 q^{97} -7.70710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8}+O(q^{10}) \) 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8	\( 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8} + 3 q^{10} + 4 q^{11} - 18 q^{13} + 2 q^{14} - 7 q^{16} + 3 q^{17} - 4 q^{19} + 2 q^{20} - 26 q^{22} - 7 q^{23} - 8 q^{25} + 7 q^{26} - 6 q^{28} + 7 q^{29} - 22 q^{31} - 5 q^{32} + 9 q^{34} - 3 q^{35} - 25 q^{37} - 14 q^{38} - 10 q^{40} + 13 q^{41} - 2 q^{43} - 4 q^{44} - 3 q^{46} + 25 q^{47} - 8 q^{49} - 19 q^{50} - 12 q^{52} + 5 q^{53} - 15 q^{55} - 18 q^{56} + 3 q^{58} - 11 q^{59} - 33 q^{61} - 28 q^{62} - 14 q^{64} + 2 q^{65} + 8 q^{67} - 12 q^{68} - 22 q^{70} + 6 q^{71} + 15 q^{73} - 34 q^{74} - 28 q^{76} + q^{77} - 15 q^{79} + 12 q^{80} - 14 q^{82} - 21 q^{83} - 28 q^{85} + 12 q^{86} - 13 q^{88} - 8 q^{89} + 6 q^{91} - 5 q^{92} - 35 q^{94} + 25 q^{95} + 13 q^{97} - q^{98}+O(q^{100}) \) 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8 + 3 * q^10 + 4 * q^11 - 18 * q^13 + 2 * q^14 - 7 * q^16 + 3 * q^17 - 4 * q^19 + 2 * q^20 - 26 * q^22 - 7 * q^23 - 8 * q^25 + 7 * q^26 - 6 * q^28 + 7 * q^29 - 22 * q^31 - 5 * q^32 + 9 * q^34 - 3 * q^35 - 25 * q^37 - 14 * q^38 - 10 * q^40 + 13 * q^41 - 2 * q^43 - 4 * q^44 - 3 * q^46 + 25 * q^47 - 8 * q^49 - 19 * q^50 - 12 * q^52 + 5 * q^53 - 15 * q^55 - 18 * q^56 + 3 * q^58 - 11 * q^59 - 33 * q^61 - 28 * q^62 - 14 * q^64 + 2 * q^65 + 8 * q^67 - 12 * q^68 - 22 * q^70 + 6 * q^71 + 15 * q^73 - 34 * q^74 - 28 * q^76 + q^77 - 15 * q^79 + 12 * q^80 - 14 * q^82 - 21 * q^83 - 28 * q^85 + 12 * q^86 - 13 * q^88 - 8 * q^89 + 6 * q^91 - 5 * q^92 - 35 * q^94 + 25 * q^95 + 13 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more