LMFDB - Embedded newform 6003.2.a.o.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:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 11 x^{11} + 58 x^{10} + 24 x^{9} - 298 x^{8} + 97 x^{7} + 641 x^{6} - 402 x^{5} + \cdots - 40 $ x^13 - 4x^12 - 11x^11 + 58x^10 + 24x^9 - 298x^8 + 97x^7 + 641x^6 - 402x^5 - 547x^4 + 352x^3 + 219x^2 - 88x - 40
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
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			$-0.806558$ 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)

$f(q)$	$=$	$q+0.806558 q^{2} -1.34946 q^{4} -2.04515 q^{5} +4.05430 q^{7} -2.70154 q^{8} +O(q^{10})$	\(q+0.806558 q^{2} -1.34946 q^{4} -2.04515 q^{5} +4.05430 q^{7} -2.70154 q^{8} -1.64953 q^{10} +3.00186 q^{11} +1.84298 q^{13} +3.27003 q^{14} +0.519985 q^{16} -7.58612 q^{17} -2.21989 q^{19} +2.75986 q^{20} +2.42117 q^{22} +1.00000 q^{23} -0.817347 q^{25} +1.48647 q^{26} -5.47114 q^{28} -1.00000 q^{29} +2.91860 q^{31} +5.82247 q^{32} -6.11864 q^{34} -8.29167 q^{35} -4.19589 q^{37} -1.79047 q^{38} +5.52506 q^{40} -7.56150 q^{41} +1.21770 q^{43} -4.05090 q^{44} +0.806558 q^{46} -2.01776 q^{47} +9.43736 q^{49} -0.659237 q^{50} -2.48703 q^{52} -4.03062 q^{53} -6.13926 q^{55} -10.9528 q^{56} -0.806558 q^{58} -4.61650 q^{59} +6.09622 q^{61} +2.35402 q^{62} +3.65619 q^{64} -3.76917 q^{65} +3.87921 q^{67} +10.2372 q^{68} -6.68771 q^{70} +14.7800 q^{71} +6.28677 q^{73} -3.38423 q^{74} +2.99566 q^{76} +12.1704 q^{77} +8.58201 q^{79} -1.06345 q^{80} -6.09879 q^{82} +2.36242 q^{83} +15.5148 q^{85} +0.982147 q^{86} -8.10962 q^{88} +9.84922 q^{89} +7.47198 q^{91} -1.34946 q^{92} -1.62744 q^{94} +4.54001 q^{95} -16.9001 q^{97} +7.61177 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8} + 10 q^{10} - 10 q^{11} + 7 q^{13} + 12 q^{14} + 2 q^{16} - 26 q^{17} - 25 q^{20} - 15 q^{22} + 13 q^{23} + 19 q^{25} + 15 q^{26} + 5 q^{28} - 13 q^{29} - 6 q^{31} - 16 q^{32} + 11 q^{34} - q^{35} + 15 q^{37} - 8 q^{38} + 14 q^{40} - 9 q^{41} + q^{43} - 29 q^{44} - 4 q^{46} - 15 q^{47} + 4 q^{49} - 31 q^{50} - 8 q^{52} - 43 q^{53} - 3 q^{55} + 5 q^{56} + 4 q^{58} + 9 q^{59} + 20 q^{61} - 11 q^{62} - 16 q^{64} + 25 q^{65} + q^{67} - 21 q^{68} - 2 q^{70} - 17 q^{71} + 26 q^{73} - 11 q^{74} + 8 q^{76} - 17 q^{77} + 5 q^{79} - 10 q^{80} - 25 q^{82} - 4 q^{83} + 20 q^{85} + 13 q^{86} - 32 q^{88} - 48 q^{89} - 9 q^{91} + 12 q^{92} - 65 q^{94} - 8 q^{95} + 30 q^{97} - 2 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8 + 10 * q^10 - 10 * q^11 + 7 * q^13 + 12 * q^14 + 2 * q^16 - 26 * q^17 - 25 * q^20 - 15 * q^22 + 13 * q^23 + 19 * q^25 + 15 * q^26 + 5 * q^28 - 13 * q^29 - 6 * q^31 - 16 * q^32 + 11 * q^34 - q^35 + 15 * q^37 - 8 * q^38 + 14 * q^40 - 9 * q^41 + q^43 - 29 * q^44 - 4 * q^46 - 15 * q^47 + 4 * q^49 - 31 * q^50 - 8 * q^52 - 43 * q^53 - 3 * q^55 + 5 * q^56 + 4 * q^58 + 9 * q^59 + 20 * q^61 - 11 * q^62 - 16 * q^64 + 25 * q^65 + q^67 - 21 * q^68 - 2 * q^70 - 17 * q^71 + 26 * q^73 - 11 * q^74 + 8 * q^76 - 17 * q^77 + 5 * q^79 - 10 * q^80 - 25 * q^82 - 4 * q^83 + 20 * q^85 + 13 * q^86 - 32 * q^88 - 48 * q^89 - 9 * q^91 + 12 * q^92 - 65 * q^94 - 8 * q^95 + 30 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.806558	0.570322	0.285161	−	0.958480i	$-0.407953\pi$
$2$	0.806558	0.570322	0.285161	+	0.958480i	$0.407953\pi$
$3$	0	0
$3$	0	0
$4$	−1.34946	−0.674732
$5$	−2.04515	−0.914620	−0.457310	−	0.889307i	$-0.651187\pi$
$5$	−2.04515	−0.914620	−0.457310	+	0.889307i	$0.651187\pi$
$6$	0	0
$7$	4.05430	1.53238	0.766191	−	0.642613i	$-0.222150\pi$
$7$	4.05430	1.53238	0.766191	+	0.642613i	$0.222150\pi$
$8$	−2.70154	−0.955137
$9$	0	0
$10$	−1.64953	−0.521628
$11$	3.00186	0.905094	0.452547	−	0.891741i	$-0.350516\pi$
$11$	3.00186	0.905094	0.452547	+	0.891741i	$0.350516\pi$
$12$	0	0
$13$	1.84298	0.511150	0.255575	−	0.966789i	$-0.417735\pi$
$13$	1.84298	0.511150	0.255575	+	0.966789i	$0.417735\pi$
$14$	3.27003	0.873952
$15$	0	0
$16$	0.519985	0.129996
$17$	−7.58612	−1.83990	−0.919952	−	0.392032i	$-0.871772\pi$
$17$	−7.58612	−1.83990	−0.919952	+	0.392032i	$0.871772\pi$
$18$	0	0
$19$	−2.21989	−0.509278	−0.254639	−	0.967036i	$-0.581957\pi$
$19$	−2.21989	−0.509278	−0.254639	+	0.967036i	$0.581957\pi$
$20$	2.75986	0.617124
$21$	0	0
$22$	2.42117	0.516195
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−0.817347	−0.163469
$26$	1.48647	0.291520
$27$	0	0
$28$	−5.47114	−1.03395
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	2.91860	0.524195	0.262098	−	0.965041i	$-0.415586\pi$
$31$	2.91860	0.524195	0.262098	+	0.965041i	$0.415586\pi$
$32$	5.82247	1.02928
$33$	0	0
$34$	−6.11864	−1.04934
$35$	−8.29167	−1.40155
$36$	0	0
$37$	−4.19589	−0.689800	−0.344900	−	0.938639i	$-0.612087\pi$
$37$	−4.19589	−0.689800	−0.344900	+	0.938639i	$0.612087\pi$
$38$	−1.79047	−0.290452
$39$	0	0
$40$	5.52506	0.873588
$41$	−7.56150	−1.18091	−0.590454	−	0.807071i	$-0.701051\pi$
$41$	−7.56150	−1.18091	−0.590454	+	0.807071i	$0.701051\pi$
$42$	0	0
$43$	1.21770	0.185698	0.0928489	−	0.995680i	$-0.470403\pi$
$43$	1.21770	0.185698	0.0928489	+	0.995680i	$0.470403\pi$
$44$	−4.05090	−0.610696
$45$	0	0
$46$	0.806558	0.118920
$47$	−2.01776	−0.294321	−0.147160	−	0.989113i	$-0.547013\pi$
$47$	−2.01776	−0.294321	−0.147160	+	0.989113i	$0.547013\pi$
$48$	0	0
$49$	9.43736	1.34819
$50$	−0.659237	−0.0932303
$51$	0	0
$52$	−2.48703	−0.344889
$53$	−4.03062	−0.553649	−0.276824	−	0.960921i	$-0.589282\pi$
$53$	−4.03062	−0.553649	−0.276824	+	0.960921i	$0.589282\pi$
$54$	0	0
$55$	−6.13926	−0.827817
$56$	−10.9528	−1.46364
$57$	0	0
$58$	−0.806558	−0.105906
$59$	−4.61650	−0.601017	−0.300509	−	0.953779i	$-0.597156\pi$
$59$	−4.61650	−0.601017	−0.300509	+	0.953779i	$0.597156\pi$
$60$	0	0
$61$	6.09622	0.780541	0.390270	−	0.920700i	$-0.372382\pi$
$61$	6.09622	0.780541	0.390270	+	0.920700i	$0.372382\pi$
$62$	2.35402	0.298960
$63$	0	0
$64$	3.65619	0.457023
$65$	−3.76917	−0.467508
$66$	0	0
$67$	3.87921	0.473920	0.236960	−	0.971519i	$-0.423849\pi$
$67$	3.87921	0.473920	0.236960	+	0.971519i	$0.423849\pi$
$68$	10.2372	1.24144
$69$	0	0
$70$	−6.68771	−0.799334
$71$	14.7800	1.75406	0.877031	−	0.480434i	$-0.159521\pi$
$71$	14.7800	1.75406	0.877031	+	0.480434i	$0.159521\pi$
$72$	0	0
$73$	6.28677	0.735811	0.367906	−	0.929863i	$-0.380075\pi$
$73$	6.28677	0.735811	0.367906	+	0.929863i	$0.380075\pi$
$74$	−3.38423	−0.393408
$75$	0	0
$76$	2.99566	0.343626
$77$	12.1704	1.38695
$78$	0	0
$79$	8.58201	0.965552	0.482776	−	0.875744i	$-0.339629\pi$
$79$	8.58201	0.965552	0.482776	+	0.875744i	$0.339629\pi$
$80$	−1.06345	−0.118897
$81$	0	0
$82$	−6.09879	−0.673499
$83$	2.36242	0.259310	0.129655	−	0.991559i	$-0.458613\pi$
$83$	2.36242	0.259310	0.129655	+	0.991559i	$0.458613\pi$
$84$	0	0
$85$	15.5148	1.68281
$86$	0.982147	0.105908
$87$	0	0
$88$	−8.10962	−0.864489
$89$	9.84922	1.04402	0.522008	−	0.852941i	$-0.325183\pi$
$89$	9.84922	1.04402	0.522008	+	0.852941i	$0.325183\pi$
$90$	0	0
$91$	7.47198	0.783277
$92$	−1.34946	−0.140691
$93$	0	0
$94$	−1.62744	−0.167858
$95$	4.54001	0.465796
$96$	0	0
$97$	−16.9001	−1.71595	−0.857973	−	0.513695i	$-0.828276\pi$
$97$	−16.9001	−1.71595	−0.857973	+	0.513695i	$0.828276\pi$
$98$	7.61177	0.768905
$99$	0	0
$100$	1.10298	0.110298
$101$	−14.9861	−1.49118	−0.745589	−	0.666407i	$-0.767832\pi$
$101$	−14.9861	−1.49118	−0.745589	+	0.666407i	$0.767832\pi$
$102$	0	0
$103$	−18.3682	−1.80987	−0.904937	−	0.425546i	$-0.860082\pi$
$103$	−18.3682	−1.80987	−0.904937	+	0.425546i	$0.860082\pi$
$104$	−4.97887	−0.488218
$105$	0	0
$106$	−3.25093	−0.315758
$107$	−12.2465	−1.18392	−0.591959	−	0.805968i	$-0.701645\pi$
$107$	−12.2465	−1.18392	−0.591959	+	0.805968i	$0.701645\pi$
$108$	0	0
$109$	−12.4689	−1.19430	−0.597151	−	0.802129i	$-0.703701\pi$
$109$	−12.4689	−1.19430	−0.597151	+	0.802129i	$0.703701\pi$
$110$	−4.95166	−0.472123
$111$	0	0
$112$	2.10818	0.199204
$113$	20.4069	1.91972	0.959862	−	0.280473i	$-0.0904912\pi$
$113$	20.4069	1.91972	0.959862	+	0.280473i	$0.0904912\pi$
$114$	0	0
$115$	−2.04515	−0.190712
$116$	1.34946	0.125295
$117$	0	0
$118$	−3.72347	−0.342774
$119$	−30.7564	−2.81943
$120$	0	0
$121$	−1.98886	−0.180806
$122$	4.91695	0.445160
$123$	0	0
$124$	−3.93854	−0.353692
$125$	11.8974	1.06413
$126$	0	0
$127$	−9.90186	−0.878648	−0.439324	−	0.898329i	$-0.644782\pi$
$127$	−9.90186	−0.878648	−0.439324	+	0.898329i	$0.644782\pi$
$128$	−8.69602	−0.768626
$129$	0	0
$130$	−3.04005	−0.266630
$131$	−16.9397	−1.48003	−0.740014	−	0.672591i	$-0.765181\pi$
$131$	−16.9397	−1.48003	−0.740014	+	0.672591i	$0.765181\pi$
$132$	0	0
$133$	−9.00010	−0.780408
$134$	3.12880	0.270287
$135$	0	0
$136$	20.4942	1.75736
$137$	−17.8614	−1.52600	−0.763002	−	0.646396i	$-0.776276\pi$
$137$	−17.8614	−1.52600	−0.763002	+	0.646396i	$0.776276\pi$
$138$	0	0
$139$	−9.50597	−0.806286	−0.403143	−	0.915137i	$-0.632082\pi$
$139$	−9.50597	−0.806286	−0.403143	+	0.915137i	$0.632082\pi$
$140$	11.1893	0.945670
$141$	0	0
$142$	11.9209	1.00038
$143$	5.53235	0.462638
$144$	0	0
$145$	2.04515	0.169841
$146$	5.07064	0.419650
$147$	0	0
$148$	5.66221	0.465431
$149$	15.2689	1.25088	0.625438	−	0.780274i	$-0.284920\pi$
$149$	15.2689	1.25088	0.625438	+	0.780274i	$0.284920\pi$
$150$	0	0
$151$	−10.5774	−0.860777	−0.430388	−	0.902644i	$-0.641623\pi$
$151$	−10.5774	−0.860777	−0.430388	+	0.902644i	$0.641623\pi$
$152$	5.99711	0.486430
$153$	0	0
$154$	9.81615	0.791008
$155$	−5.96898	−0.479440
$156$	0	0
$157$	−5.77058	−0.460543	−0.230271	−	0.973126i	$-0.573961\pi$
$157$	−5.77058	−0.460543	−0.230271	+	0.973126i	$0.573961\pi$
$158$	6.92189	0.550676
$159$	0	0
$160$	−11.9078	−0.941398
$161$	4.05430	0.319524
$162$	0	0
$163$	−16.3991	−1.28447	−0.642237	−	0.766506i	$-0.721994\pi$
$163$	−16.3991	−1.28447	−0.642237	+	0.766506i	$0.721994\pi$
$164$	10.2040	0.796797
$165$	0	0
$166$	1.90543	0.147890
$167$	5.15361	0.398799	0.199399	−	0.979918i	$-0.436101\pi$
$167$	5.15361	0.398799	0.199399	+	0.979918i	$0.436101\pi$
$168$	0	0
$169$	−9.60344	−0.738726
$170$	12.5136	0.959746
$171$	0	0
$172$	−1.64325	−0.125296
$173$	7.04728	0.535795	0.267897	−	0.963447i	$-0.413671\pi$
$173$	7.04728	0.535795	0.267897	+	0.963447i	$0.413671\pi$
$174$	0	0
$175$	−3.31377	−0.250498
$176$	1.56092	0.117659
$177$	0	0
$178$	7.94397	0.595425
$179$	−0.527490	−0.0394265	−0.0197132	−	0.999806i	$-0.506275\pi$
$179$	−0.527490	−0.0394265	−0.0197132	+	0.999806i	$0.506275\pi$
$180$	0	0
$181$	−1.18927	−0.0883975	−0.0441988	−	0.999023i	$-0.514073\pi$
$181$	−1.18927	−0.0883975	−0.0441988	+	0.999023i	$0.514073\pi$
$182$	6.02658	0.446720
$183$	0	0
$184$	−2.70154	−0.199160
$185$	8.58124	0.630905
$186$	0	0
$187$	−22.7724	−1.66528
$188$	2.72290	0.198588
$189$	0	0
$190$	3.66178	0.265654
$191$	3.48800	0.252383	0.126192	−	0.992006i	$-0.459725\pi$
$191$	3.48800	0.252383	0.126192	+	0.992006i	$0.459725\pi$
$192$	0	0
$193$	0.0117890	0.000848590 0	0.000424295	−	1.00000i	$-0.499865\pi$
$193$	0.0117890	0.000848590 0	0.000424295		1.00000i	$0.499865\pi$
$194$	−13.6309	−0.978642
$195$	0	0
$196$	−12.7354	−0.909670
$197$	−22.8749	−1.62977	−0.814886	−	0.579621i	$-0.803201\pi$
$197$	−22.8749	−1.62977	−0.814886	+	0.579621i	$0.803201\pi$
$198$	0	0
$199$	−20.7363	−1.46995	−0.734977	−	0.678092i	$-0.762807\pi$
$199$	−20.7363	−1.46995	−0.734977	+	0.678092i	$0.762807\pi$
$200$	2.20809	0.156136
$201$	0	0
$202$	−12.0872	−0.850452
$203$	−4.05430	−0.284556
$204$	0	0
$205$	15.4644	1.08008
$206$	−14.8150	−1.03221
$207$	0	0
$208$	0.958321	0.0664476
$209$	−6.66379	−0.460944
$210$	0	0
$211$	−10.8507	−0.746992	−0.373496	−	0.927632i	$-0.621841\pi$
$211$	−10.8507	−0.746992	−0.373496	+	0.927632i	$0.621841\pi$
$212$	5.43918	0.373565
$213$	0	0
$214$	−9.87755	−0.675215
$215$	−2.49039	−0.169843
$216$	0	0
$217$	11.8329	0.803267
$218$	−10.0569	−0.681137
$219$	0	0
$220$	8.28471	0.558555
$221$	−13.9810	−0.940466
$222$	0	0
$223$	−7.03248	−0.470930	−0.235465	−	0.971883i	$-0.575661\pi$
$223$	−7.03248	−0.470930	−0.235465	+	0.971883i	$0.575661\pi$
$224$	23.6060	1.57725
$225$	0	0
$226$	16.4594	1.09486
$227$	−14.3071	−0.949597	−0.474798	−	0.880095i	$-0.657479\pi$
$227$	−14.3071	−0.949597	−0.474798	+	0.880095i	$0.657479\pi$
$228$	0	0
$229$	6.10416	0.403374	0.201687	−	0.979450i	$-0.435358\pi$
$229$	6.10416	0.403374	0.201687	+	0.979450i	$0.435358\pi$
$230$	−1.64953	−0.108767
$231$	0	0
$232$	2.70154	0.177365
$233$	−22.1337	−1.45002	−0.725012	−	0.688736i	$-0.758166\pi$
$233$	−22.1337	−1.45002	−0.725012	+	0.688736i	$0.758166\pi$
$234$	0	0
$235$	4.12663	0.269192
$236$	6.22981	0.405526
$237$	0	0
$238$	−24.8068	−1.60799
$239$	14.8240	0.958882	0.479441	−	0.877574i	$-0.340839\pi$
$239$	14.8240	0.958882	0.479441	+	0.877574i	$0.340839\pi$
$240$	0	0
$241$	−1.77555	−0.114373	−0.0571865	−	0.998364i	$-0.518213\pi$
$241$	−1.77555	−0.114373	−0.0571865	+	0.998364i	$0.518213\pi$
$242$	−1.60413	−0.103117
$243$	0	0
$244$	−8.22663	−0.526656
$245$	−19.3008	−1.23309
$246$	0	0
$247$	−4.09120	−0.260317
$248$	−7.88469	−0.500678
$249$	0	0
$250$	9.59591	0.606899
$251$	−14.9548	−0.943936	−0.471968	−	0.881616i	$-0.656456\pi$
$251$	−14.9548	−0.943936	−0.471968	+	0.881616i	$0.656456\pi$
$252$	0	0
$253$	3.00186	0.188725
$254$	−7.98642	−0.501113
$255$	0	0
$256$	−14.3262	−0.895388
$257$	−2.91061	−0.181559	−0.0907794	−	0.995871i	$-0.528936\pi$
$257$	−2.91061	−0.181559	−0.0907794	+	0.995871i	$0.528936\pi$
$258$	0	0
$259$	−17.0114	−1.05704
$260$	5.08636	0.315443
$261$	0	0
$262$	−13.6628	−0.844093
$263$	30.6896	1.89240	0.946201	−	0.323580i	$-0.104886\pi$
$263$	30.6896	1.89240	0.946201	+	0.323580i	$0.104886\pi$
$264$	0	0
$265$	8.24324	0.506378
$266$	−7.25910	−0.445084
$267$	0	0
$268$	−5.23485	−0.319769
$269$	18.0587	1.10106	0.550530	−	0.834815i	$-0.314426\pi$
$269$	18.0587	1.10106	0.550530	+	0.834815i	$0.314426\pi$
$270$	0	0
$271$	7.82283	0.475203	0.237601	−	0.971363i	$-0.423639\pi$
$271$	7.82283	0.475203	0.237601	+	0.971363i	$0.423639\pi$
$272$	−3.94467	−0.239181
$273$	0	0
$274$	−14.4063	−0.870315
$275$	−2.45356	−0.147955
$276$	0	0
$277$	19.2194	1.15478	0.577390	−	0.816469i	$-0.304071\pi$
$277$	19.2194	1.15478	0.577390	+	0.816469i	$0.304071\pi$
$278$	−7.66711	−0.459843
$279$	0	0
$280$	22.4002	1.33867
$281$	−22.5950	−1.34791	−0.673953	−	0.738775i	$-0.735405\pi$
$281$	−22.5950	−1.34791	−0.673953	+	0.738775i	$0.735405\pi$
$282$	0	0
$283$	−4.59462	−0.273122	−0.136561	−	0.990632i	$-0.543605\pi$
$283$	−4.59462	−0.273122	−0.136561	+	0.990632i	$0.543605\pi$
$284$	−19.9451	−1.18352
$285$	0	0
$286$	4.46216	0.263853
$287$	−30.6566	−1.80960
$288$	0	0
$289$	40.5492	2.38525
$290$	1.64953	0.0968640
$291$	0	0
$292$	−8.48378	−0.496476
$293$	−6.38423	−0.372971	−0.186485	−	0.982458i	$-0.559710\pi$
$293$	−6.38423	−0.372971	−0.186485	+	0.982458i	$0.559710\pi$
$294$	0	0
$295$	9.44145	0.549703
$296$	11.3354	0.658854
$297$	0	0
$298$	12.3152	0.713403
$299$	1.84298	0.106582
$300$	0	0
$301$	4.93693	0.284560
$302$	−8.53128	−0.490920
$303$	0	0
$304$	−1.15431	−0.0662042
$305$	−12.4677	−0.713899
$306$	0	0
$307$	−11.3337	−0.646849	−0.323425	−	0.946254i	$-0.604834\pi$
$307$	−11.3337	−0.646849	−0.323425	+	0.946254i	$0.604834\pi$
$308$	−16.4236	−0.935819
$309$	0	0
$310$	−4.81432	−0.273435
$311$	3.14192	0.178162	0.0890808	−	0.996024i	$-0.471607\pi$
$311$	3.14192	0.178162	0.0890808	+	0.996024i	$0.471607\pi$
$312$	0	0
$313$	−14.5798	−0.824100	−0.412050	−	0.911161i	$-0.635187\pi$
$313$	−14.5798	−0.824100	−0.412050	+	0.911161i	$0.635187\pi$
$314$	−4.65431	−0.262658
$315$	0	0
$316$	−11.5811	−0.651489
$317$	0.503716	0.0282915	0.0141458	−	0.999900i	$-0.495497\pi$
$317$	0.503716	0.0282915	0.0141458	+	0.999900i	$0.495497\pi$
$318$	0	0
$319$	−3.00186	−0.168072
$320$	−7.47746	−0.418003
$321$	0	0
$322$	3.27003	0.182231
$323$	16.8403	0.937022
$324$	0	0
$325$	−1.50635	−0.0835573
$326$	−13.2268	−0.732564
$327$	0	0
$328$	20.4277	1.12793
$329$	−8.18062	−0.451012
$330$	0	0
$331$	21.9714	1.20766	0.603829	−	0.797114i	$-0.293641\pi$
$331$	21.9714	1.20766	0.603829	+	0.797114i	$0.293641\pi$
$332$	−3.18801	−0.174965
$333$	0	0
$334$	4.15669	0.227444
$335$	−7.93357	−0.433457
$336$	0	0
$337$	−9.59646	−0.522752	−0.261376	−	0.965237i	$-0.584176\pi$
$337$	−9.59646	−0.522752	−0.261376	+	0.965237i	$0.584176\pi$
$338$	−7.74573	−0.421312
$339$	0	0
$340$	−20.9366	−1.13545
$341$	8.76120	0.474446
$342$	0	0
$343$	9.88178	0.533566
$344$	−3.28967	−0.177367
$345$	0	0
$346$	5.68404	0.305576
$347$	10.0974	0.542059	0.271029	−	0.962571i	$-0.412636\pi$
$347$	10.0974	0.542059	0.271029	+	0.962571i	$0.412636\pi$
$348$	0	0
$349$	−9.00185	−0.481858	−0.240929	−	0.970543i	$-0.577452\pi$
$349$	−9.00185	−0.481858	−0.240929	+	0.970543i	$0.577452\pi$
$350$	−2.67275	−0.142864
$351$	0	0
$352$	17.4782	0.931592
$353$	15.2438	0.811344	0.405672	−	0.914019i	$-0.367038\pi$
$353$	15.2438	0.811344	0.405672	+	0.914019i	$0.367038\pi$
$354$	0	0
$355$	−30.2274	−1.60430
$356$	−13.2912	−0.704431
$357$	0	0
$358$	−0.425451	−0.0224858
$359$	−8.30291	−0.438211	−0.219105	−	0.975701i	$-0.570314\pi$
$359$	−8.30291	−0.438211	−0.219105	+	0.975701i	$0.570314\pi$
$360$	0	0
$361$	−14.0721	−0.740636
$362$	−0.959212	−0.0504151
$363$	0	0
$364$	−10.0832	−0.528502
$365$	−12.8574	−0.672988
$366$	0	0
$367$	−26.7249	−1.39503	−0.697514	−	0.716571i	$-0.745711\pi$
$367$	−26.7249	−1.39503	−0.697514	+	0.716571i	$0.745711\pi$
$368$	0.519985	0.0271061
$369$	0	0
$370$	6.92126	0.359819
$371$	−16.3414	−0.848401
$372$	0	0
$373$	27.1260	1.40453	0.702265	−	0.711915i	$-0.252172\pi$
$373$	27.1260	1.40453	0.702265	+	0.711915i	$0.252172\pi$
$374$	−18.3673	−0.949749
$375$	0	0
$376$	5.45106	0.281117
$377$	−1.84298	−0.0949181
$378$	0	0
$379$	26.1448	1.34297	0.671483	−	0.741020i	$-0.265658\pi$
$379$	26.1448	1.34297	0.671483	+	0.741020i	$0.265658\pi$
$380$	−6.12659	−0.314287
$381$	0	0
$382$	2.81328	0.143940
$383$	−9.77660	−0.499561	−0.249781	−	0.968302i	$-0.580358\pi$
$383$	−9.77660	−0.499561	−0.249781	+	0.968302i	$0.580358\pi$
$384$	0	0
$385$	−24.8904	−1.26853
$386$	0.00950850	0.000483970 0
$387$	0	0
$388$	22.8061	1.15780
$389$	−33.3393	−1.69037	−0.845185	−	0.534474i	$-0.820510\pi$
$389$	−33.3393	−1.69037	−0.845185	+	0.534474i	$0.820510\pi$
$390$	0	0
$391$	−7.58612	−0.383646
$392$	−25.4954	−1.28771
$393$	0	0
$394$	−18.4500	−0.929496
$395$	−17.5515	−0.883114
$396$	0	0
$397$	−3.29783	−0.165514	−0.0827568	−	0.996570i	$-0.526372\pi$
$397$	−3.29783	−0.165514	−0.0827568	+	0.996570i	$0.526372\pi$
$398$	−16.7250	−0.838348
$399$	0	0
$400$	−0.425009	−0.0212504
$401$	6.69026	0.334095	0.167048	−	0.985949i	$-0.446577\pi$
$401$	6.69026	0.334095	0.167048	+	0.985949i	$0.446577\pi$
$402$	0	0
$403$	5.37890	0.267942
$404$	20.2233	1.00615
$405$	0	0
$406$	−3.27003	−0.162289
$407$	−12.5955	−0.624334
$408$	0	0
$409$	−1.88547	−0.0932303	−0.0466151	−	0.998913i	$-0.514843\pi$
$409$	−1.88547	−0.0932303	−0.0466151	+	0.998913i	$0.514843\pi$
$410$	12.4730	0.615996
$411$	0	0
$412$	24.7873	1.22118
$413$	−18.7167	−0.920988
$414$	0	0
$415$	−4.83152	−0.237170
$416$	10.7307	0.526115
$417$	0	0
$418$	−5.37473	−0.262887
$419$	18.4700	0.902316	0.451158	−	0.892444i	$-0.351011\pi$
$419$	18.4700	0.902316	0.451158	+	0.892444i	$0.351011\pi$
$420$	0	0
$421$	37.7969	1.84211	0.921054	−	0.389434i	$-0.127329\pi$
$421$	37.7969	1.84211	0.921054	+	0.389434i	$0.127329\pi$
$422$	−8.75170	−0.426026
$423$	0	0
$424$	10.8889	0.528810
$425$	6.20049	0.300768
$426$	0	0
$427$	24.7159	1.19609
$428$	16.5263	0.798828
$429$	0	0
$430$	−2.00864	−0.0968653
$431$	−17.9306	−0.863685	−0.431843	−	0.901949i	$-0.642136\pi$
$431$	−17.9306	−0.863685	−0.431843	+	0.901949i	$0.642136\pi$
$432$	0	0
$433$	−6.66248	−0.320178	−0.160089	−	0.987103i	$-0.551178\pi$
$433$	−6.66248	−0.320178	−0.160089	+	0.987103i	$0.551178\pi$
$434$	9.54389	0.458121
$435$	0	0
$436$	16.8263	0.805834
$437$	−2.21989	−0.106192
$438$	0	0
$439$	26.7654	1.27744	0.638721	−	0.769439i	$-0.279464\pi$
$439$	26.7654	1.27744	0.638721	+	0.769439i	$0.279464\pi$
$440$	16.5854	0.790679
$441$	0	0
$442$	−11.2765	−0.536369
$443$	0.322044	0.0153008	0.00765040	−	0.999971i	$-0.497565\pi$
$443$	0.322044	0.0153008	0.00765040	+	0.999971i	$0.497565\pi$
$444$	0	0
$445$	−20.1432	−0.954878
$446$	−5.67210	−0.268582
$447$	0	0
$448$	14.8233	0.700334
$449$	37.4180	1.76587	0.882933	−	0.469499i	$-0.155565\pi$
$449$	37.4180	1.76587	0.882933	+	0.469499i	$0.155565\pi$
$450$	0	0
$451$	−22.6985	−1.06883
$452$	−27.5385	−1.29530
$453$	0	0
$454$	−11.5395	−0.541576
$455$	−15.2813	−0.716401
$456$	0	0
$457$	−7.20356	−0.336968	−0.168484	−	0.985704i	$-0.553887\pi$
$457$	−7.20356	−0.336968	−0.168484	+	0.985704i	$0.553887\pi$
$458$	4.92335	0.230053
$459$	0	0
$460$	2.75986	0.128679
$461$	−2.27025	−0.105736	−0.0528681	−	0.998602i	$-0.516836\pi$
$461$	−2.27025	−0.105736	−0.0528681	+	0.998602i	$0.516836\pi$
$462$	0	0
$463$	−14.6651	−0.681544	−0.340772	−	0.940146i	$-0.610688\pi$
$463$	−14.6651	−0.681544	−0.340772	+	0.940146i	$0.610688\pi$
$464$	−0.519985	−0.0241397
$465$	0	0
$466$	−17.8521	−0.826981
$467$	−18.9504	−0.876921	−0.438461	−	0.898750i	$-0.644476\pi$
$467$	−18.9504	−0.876921	−0.438461	+	0.898750i	$0.644476\pi$
$468$	0	0
$469$	15.7275	0.726227
$470$	3.32837	0.153526
$471$	0	0
$472$	12.4716	0.574054
$473$	3.65537	0.168074
$474$	0	0
$475$	1.81442	0.0832513
$476$	41.5047	1.90236
$477$	0	0
$478$	11.9564	0.546872
$479$	−20.2734	−0.926317	−0.463158	−	0.886275i	$-0.653284\pi$
$479$	−20.2734	−0.926317	−0.463158	+	0.886275i	$0.653284\pi$
$480$	0	0
$481$	−7.73293	−0.352591
$482$	−1.43208	−0.0652295
$483$	0	0
$484$	2.68390	0.121995
$485$	34.5633	1.56944
$486$	0	0
$487$	6.62022	0.299991	0.149995	−	0.988687i	$-0.452074\pi$
$487$	6.62022	0.299991	0.149995	+	0.988687i	$0.452074\pi$
$488$	−16.4692	−0.745524
$489$	0	0
$490$	−15.5672	−0.703256
$491$	4.57913	0.206653	0.103327	−	0.994647i	$-0.467051\pi$
$491$	4.57913	0.206653	0.103327	+	0.994647i	$0.467051\pi$
$492$	0	0
$493$	7.58612	0.341662
$494$	−3.29979	−0.148465
$495$	0	0
$496$	1.51763	0.0681435
$497$	59.9225	2.68789
$498$	0	0
$499$	−33.4375	−1.49687	−0.748435	−	0.663208i	$-0.769194\pi$
$499$	−33.4375	−1.49687	−0.748435	+	0.663208i	$0.769194\pi$
$500$	−16.0551	−0.718005
$501$	0	0
$502$	−12.0619	−0.538348
$503$	26.2354	1.16978	0.584890	−	0.811112i	$-0.301138\pi$
$503$	26.2354	1.16978	0.584890	+	0.811112i	$0.301138\pi$
$504$	0	0
$505$	30.6490	1.36386
$506$	2.42117	0.107634
$507$	0	0
$508$	13.3622	0.592853
$509$	−11.3542	−0.503266	−0.251633	−	0.967823i	$-0.580968\pi$
$509$	−11.3542	−0.503266	−0.251633	+	0.967823i	$0.580968\pi$
$510$	0	0
$511$	25.4885	1.12754
$512$	5.83712	0.257967
$513$	0	0
$514$	−2.34757	−0.103547
$515$	37.5658	1.65535
$516$	0	0
$517$	−6.05703	−0.266388
$518$	−13.7207	−0.602852
$519$	0	0
$520$	10.1825	0.446534
$521$	−1.35393	−0.0593168	−0.0296584	−	0.999560i	$-0.509442\pi$
$521$	−1.35393	−0.0593168	−0.0296584	+	0.999560i	$0.509442\pi$
$522$	0	0
$523$	−34.4624	−1.50694	−0.753468	−	0.657484i	$-0.771621\pi$
$523$	−34.4624	−1.50694	−0.753468	+	0.657484i	$0.771621\pi$
$524$	22.8595	0.998623
$525$	0	0
$526$	24.7529	1.07928
$527$	−22.1408	−0.964469
$528$	0	0
$529$	1.00000	0.0434783
$530$	6.64865	0.288799
$531$	0	0
$532$	12.1453	0.526566
$533$	−13.9357	−0.603621
$534$	0	0
$535$	25.0461	1.08284
$536$	−10.4798	−0.452659
$537$	0	0
$538$	14.5654	0.627959
$539$	28.3296	1.22024
$540$	0	0
$541$	31.8239	1.36822	0.684109	−	0.729380i	$-0.260191\pi$
$541$	31.8239	1.36822	0.684109	+	0.729380i	$0.260191\pi$
$542$	6.30956	0.271019
$543$	0	0
$544$	−44.1699	−1.89377
$545$	25.5008	1.09233
$546$	0	0
$547$	−32.5223	−1.39056	−0.695278	−	0.718741i	$-0.744719\pi$
$547$	−32.5223	−1.39056	−0.695278	+	0.718741i	$0.744719\pi$
$548$	24.1034	1.02965
$549$	0	0
$550$	−1.97894	−0.0843821
$551$	2.21989	0.0945705
$552$	0	0
$553$	34.7941	1.47959
$554$	15.5015	0.658597
$555$	0	0
$556$	12.8280	0.544027
$557$	−13.1453	−0.556985	−0.278492	−	0.960438i	$-0.589835\pi$
$557$	−13.1453	−0.556985	−0.278492	+	0.960438i	$0.589835\pi$
$558$	0	0
$559$	2.24420	0.0949194
$560$	−4.31155	−0.182196
$561$	0	0
$562$	−18.2242	−0.768740
$563$	−0.527928	−0.0222495	−0.0111247	−	0.999938i	$-0.503541\pi$
$563$	−0.527928	−0.0222495	−0.0111247	+	0.999938i	$0.503541\pi$
$564$	0	0
$565$	−41.7353	−1.75582
$566$	−3.70582	−0.155767
$567$	0	0
$568$	−39.9287	−1.67537
$569$	−33.0756	−1.38660	−0.693301	−	0.720648i	$-0.743844\pi$
$569$	−33.0756	−1.38660	−0.693301	+	0.720648i	$0.743844\pi$
$570$	0	0
$571$	30.1395	1.26130	0.630648	−	0.776069i	$-0.282789\pi$
$571$	30.1395	1.26130	0.630648	+	0.776069i	$0.282789\pi$
$572$	−7.46571	−0.312157
$573$	0	0
$574$	−24.7263	−1.03206
$575$	−0.817347	−0.0340857
$576$	0	0
$577$	33.4063	1.39072	0.695362	−	0.718660i	$-0.255244\pi$
$577$	33.4063	1.39072	0.695362	+	0.718660i	$0.255244\pi$
$578$	32.7052	1.36036
$579$	0	0
$580$	−2.75986	−0.114597
$581$	9.57798	0.397362
$582$	0	0
$583$	−12.0993	−0.501104
$584$	−16.9839	−0.702801
$585$	0	0
$586$	−5.14925	−0.212714
$587$	22.3434	0.922210	0.461105	−	0.887346i	$-0.347453\pi$
$587$	22.3434	0.922210	0.461105	+	0.887346i	$0.347453\pi$
$588$	0	0
$589$	−6.47896	−0.266961
$590$	7.61508	0.313508
$591$	0	0
$592$	−2.18180	−0.0896715
$593$	−25.0660	−1.02934	−0.514668	−	0.857389i	$-0.672085\pi$
$593$	−25.0660	−1.02934	−0.514668	+	0.857389i	$0.672085\pi$
$594$	0	0
$595$	62.9016	2.57871
$596$	−20.6048	−0.844007
$597$	0	0
$598$	1.48647	0.0607861
$599$	−3.08054	−0.125868	−0.0629338	−	0.998018i	$-0.520046\pi$
$599$	−3.08054	−0.125868	−0.0629338	+	0.998018i	$0.520046\pi$
$600$	0	0
$601$	1.72624	0.0704146	0.0352073	−	0.999380i	$-0.488791\pi$
$601$	1.72624	0.0704146	0.0352073	+	0.999380i	$0.488791\pi$
$602$	3.98192	0.162291
$603$	0	0
$604$	14.2738	0.580794
$605$	4.06753	0.165368
$606$	0	0
$607$	−15.5236	−0.630085	−0.315043	−	0.949078i	$-0.602019\pi$
$607$	−15.5236	−0.630085	−0.315043	+	0.949078i	$0.602019\pi$
$608$	−12.9252	−0.524188
$609$	0	0
$610$	−10.0559	−0.407152
$611$	−3.71869	−0.150442
$612$	0	0
$613$	19.2422	0.777185	0.388592	−	0.921410i	$-0.372961\pi$
$613$	19.2422	0.777185	0.388592	+	0.921410i	$0.372961\pi$
$614$	−9.14129	−0.368912
$615$	0	0
$616$	−32.8789	−1.32473
$617$	−35.7408	−1.43887	−0.719435	−	0.694560i	$-0.755599\pi$
$617$	−35.7408	−1.43887	−0.719435	+	0.694560i	$0.755599\pi$
$618$	0	0
$619$	−1.95266	−0.0784838	−0.0392419	−	0.999230i	$-0.512494\pi$
$619$	−1.95266	−0.0784838	−0.0392419	+	0.999230i	$0.512494\pi$
$620$	8.05492	0.323494
$621$	0	0
$622$	2.53414	0.101610
$623$	39.9317	1.59983
$624$	0	0
$625$	−20.2452	−0.809808
$626$	−11.7595	−0.470003
$627$	0	0
$628$	7.78720	0.310743
$629$	31.8305	1.26917
$630$	0	0
$631$	−19.3286	−0.769459	−0.384729	−	0.923029i	$-0.625705\pi$
$631$	−19.3286	−0.769459	−0.384729	+	0.923029i	$0.625705\pi$
$632$	−23.1846	−0.922235
$633$	0	0
$634$	0.406276	0.0161353
$635$	20.2508	0.803630
$636$	0	0
$637$	17.3928	0.689129
$638$	−2.42117	−0.0958550
$639$	0	0
$640$	17.7847	0.703002
$641$	−10.7786	−0.425727	−0.212864	−	0.977082i	$-0.568279\pi$
$641$	−10.7786	−0.425727	−0.212864	+	0.977082i	$0.568279\pi$
$642$	0	0
$643$	−27.2439	−1.07439	−0.537197	−	0.843457i	$-0.680517\pi$
$643$	−27.2439	−1.07439	−0.537197	+	0.843457i	$0.680517\pi$
$644$	−5.47114	−0.215593
$645$	0	0
$646$	13.5827	0.534404
$647$	−21.8677	−0.859706	−0.429853	−	0.902899i	$-0.641435\pi$
$647$	−21.8677	−0.859706	−0.429853	+	0.902899i	$0.641435\pi$
$648$	0	0
$649$	−13.8581	−0.543977
$650$	−1.21496	−0.0476546
$651$	0	0
$652$	22.1300	0.866676
$653$	37.1875	1.45526	0.727631	−	0.685969i	$-0.240622\pi$
$653$	37.1875	1.45526	0.727631	+	0.685969i	$0.240622\pi$
$654$	0	0
$655$	34.6443	1.35366
$656$	−3.93187	−0.153514
$657$	0	0
$658$	−6.59814	−0.257222
$659$	−0.0830271	−0.00323428	−0.00161714	−	0.999999i	$-0.500515\pi$
$659$	−0.0830271	−0.00323428	−0.00161714	+	0.999999i	$0.500515\pi$
$660$	0	0
$661$	11.9274	0.463921	0.231960	−	0.972725i	$-0.425486\pi$
$661$	11.9274	0.463921	0.231960	+	0.972725i	$0.425486\pi$
$662$	17.7212	0.688754
$663$	0	0
$664$	−6.38218	−0.247676
$665$	18.4066	0.713777
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−6.95462	−0.269082
$669$	0	0
$670$	−6.39888	−0.247210
$671$	18.3000	0.706463
$672$	0	0
$673$	7.06226	0.272230	0.136115	−	0.990693i	$-0.456538\pi$
$673$	7.06226	0.272230	0.136115	+	0.990693i	$0.456538\pi$
$674$	−7.74010	−0.298137
$675$	0	0
$676$	12.9595	0.498442
$677$	17.6531	0.678464	0.339232	−	0.940703i	$-0.389833\pi$
$677$	17.6531	0.678464	0.339232	+	0.940703i	$0.389833\pi$
$678$	0	0
$679$	−68.5181	−2.62948
$680$	−41.9137	−1.60732
$681$	0	0
$682$	7.06641	0.270587
$683$	−30.2672	−1.15814	−0.579072	−	0.815277i	$-0.696585\pi$
$683$	−30.2672	−1.15814	−0.579072	+	0.815277i	$0.696585\pi$
$684$	0	0
$685$	36.5294	1.39572
$686$	7.97023	0.304305
$687$	0	0
$688$	0.633187	0.0241400
$689$	−7.42834	−0.282997
$690$	0	0
$691$	7.10897	0.270438	0.135219	−	0.990816i	$-0.456826\pi$
$691$	7.10897	0.270438	0.135219	+	0.990816i	$0.456826\pi$
$692$	−9.51006	−0.361518
$693$	0	0
$694$	8.14417	0.309148
$695$	19.4412	0.737446
$696$	0	0
$697$	57.3625	2.17276
$698$	−7.26051	−0.274814
$699$	0	0
$700$	4.47182	0.169019
$701$	−38.2389	−1.44426	−0.722131	−	0.691756i	$-0.756837\pi$
$701$	−38.2389	−1.44426	−0.722131	+	0.691756i	$0.756837\pi$
$702$	0	0
$703$	9.31441	0.351300
$704$	10.9753	0.413649
$705$	0	0
$706$	12.2950	0.462727
$707$	−60.7583	−2.28505
$708$	0	0
$709$	16.8926	0.634413	0.317207	−	0.948356i	$-0.397255\pi$
$709$	16.8926	0.634413	0.317207	+	0.948356i	$0.397255\pi$
$710$	−24.3801	−0.914969
$711$	0	0
$712$	−26.6080	−0.997178
$713$	2.91860	0.109302
$714$	0	0
$715$	−11.3145	−0.423138
$716$	0.711829	0.0266023
$717$	0	0
$718$	−6.69677	−0.249921
$719$	44.6469	1.66505	0.832524	−	0.553989i	$-0.186895\pi$
$719$	44.6469	1.66505	0.832524	+	0.553989i	$0.186895\pi$
$720$	0	0
$721$	−74.4702	−2.77342
$722$	−11.3500	−0.422401
$723$	0	0
$724$	1.60487	0.0596447
$725$	0.817347	0.0303555
$726$	0	0
$727$	53.7250	1.99255	0.996274	−	0.0862392i	$-0.0274849\pi$
$727$	53.7250	1.99255	0.996274	+	0.0862392i	$0.0274849\pi$
$728$	−20.1858	−0.748137
$729$	0	0
$730$	−10.3702	−0.383820
$731$	−9.23763	−0.341666
$732$	0	0
$733$	11.9235	0.440403	0.220202	−	0.975454i	$-0.429328\pi$
$733$	11.9235	0.440403	0.220202	+	0.975454i	$0.429328\pi$
$734$	−21.5552	−0.795616
$735$	0	0
$736$	5.82247	0.214619
$737$	11.6448	0.428942
$738$	0	0
$739$	−31.0283	−1.14139	−0.570697	−	0.821161i	$-0.693327\pi$
$739$	−31.0283	−1.14139	−0.570697	+	0.821161i	$0.693327\pi$
$740$	−11.5801	−0.425692
$741$	0	0
$742$	−13.1802	−0.483862
$743$	37.1174	1.36171	0.680853	−	0.732420i	$-0.261609\pi$
$743$	37.1174	1.36171	0.680853	+	0.732420i	$0.261609\pi$
$744$	0	0
$745$	−31.2272	−1.14408
$746$	21.8787	0.801035
$747$	0	0
$748$	30.7306	1.12362
$749$	−49.6512	−1.81422
$750$	0	0
$751$	−7.93504	−0.289554	−0.144777	−	0.989464i	$-0.546246\pi$
$751$	−7.93504	−0.289554	−0.144777	+	0.989464i	$0.546246\pi$
$752$	−1.04921	−0.0382606
$753$	0	0
$754$	−1.48647	−0.0541339
$755$	21.6324	0.787284
$756$	0	0
$757$	26.9655	0.980077	0.490038	−	0.871701i	$-0.336983\pi$
$757$	26.9655	0.980077	0.490038	+	0.871701i	$0.336983\pi$
$758$	21.0873	0.765924
$759$	0	0
$760$	−12.2650	−0.444899
$761$	16.1655	0.586000	0.293000	−	0.956112i	$-0.405346\pi$
$761$	16.1655	0.586000	0.293000	+	0.956112i	$0.405346\pi$
$762$	0	0
$763$	−50.5526	−1.83013
$764$	−4.70694	−0.170291
$765$	0	0
$766$	−7.88539	−0.284911
$767$	−8.50810	−0.307210
$768$	0	0
$769$	47.1411	1.69995	0.849976	−	0.526821i	$-0.176616\pi$
$769$	47.1411	1.69995	0.849976	+	0.526821i	$0.176616\pi$
$770$	−20.0755	−0.723472
$771$	0	0
$772$	−0.0159088	−0.000572571 0
$773$	29.2704	1.05278	0.526391	−	0.850243i	$-0.323545\pi$
$773$	29.2704	1.05278	0.526391	+	0.850243i	$0.323545\pi$
$774$	0	0
$775$	−2.38551	−0.0856899
$776$	45.6562	1.63896
$777$	0	0
$778$	−26.8901	−0.964056
$779$	16.7857	0.601410
$780$	0	0
$781$	44.3674	1.58759
$782$	−6.11864	−0.218802
$783$	0	0
$784$	4.90729	0.175260
$785$	11.8017	0.421222
$786$	0	0
$787$	−41.1128	−1.46551	−0.732756	−	0.680491i	$-0.761767\pi$
$787$	−41.1128	−1.46551	−0.732756	+	0.680491i	$0.761767\pi$
$788$	30.8689	1.09966
$789$	0	0
$790$	−14.1563	−0.503659
$791$	82.7359	2.94175
$792$	0	0
$793$	11.2352	0.398973
$794$	−2.65989	−0.0943961
$795$	0	0
$796$	27.9829	0.991826
$797$	5.29125	0.187426	0.0937128	−	0.995599i	$-0.470126\pi$
$797$	5.29125	0.187426	0.0937128	+	0.995599i	$0.470126\pi$
$798$	0	0
$799$	15.3070	0.541522
$800$	−4.75898	−0.168255
$801$	0	0
$802$	5.39608	0.190542
$803$	18.8720	0.665978
$804$	0	0
$805$	−8.29167	−0.292243
$806$	4.33839	0.152813
$807$	0	0
$808$	40.4856	1.42428
$809$	51.6941	1.81747	0.908734	−	0.417375i	$-0.137050\pi$
$809$	51.6941	1.81747	0.908734	+	0.417375i	$0.137050\pi$
$810$	0	0
$811$	−52.2658	−1.83530	−0.917650	−	0.397390i	$-0.869916\pi$
$811$	−52.2658	−1.83530	−0.917650	+	0.397390i	$0.869916\pi$
$812$	5.47114	0.191999
$813$	0	0
$814$	−10.1590	−0.356071
$815$	33.5386	1.17481
$816$	0	0
$817$	−2.70316	−0.0945718
$818$	−1.52074	−0.0531713
$819$	0	0
$820$	−20.8687	−0.728767
$821$	47.3575	1.65279	0.826394	−	0.563093i	$-0.190389\pi$
$821$	47.3575	1.65279	0.826394	+	0.563093i	$0.190389\pi$
$822$	0	0
$823$	40.3942	1.40805	0.704027	−	0.710174i	$-0.251384\pi$
$823$	40.3942	1.40805	0.704027	+	0.710174i	$0.251384\pi$
$824$	49.6224	1.72868
$825$	0	0
$826$	−15.0961	−0.525260
$827$	18.6652	0.649053	0.324526	−	0.945877i	$-0.394795\pi$
$827$	18.6652	0.649053	0.324526	+	0.945877i	$0.394795\pi$
$828$	0	0
$829$	5.05460	0.175553	0.0877767	−	0.996140i	$-0.472024\pi$
$829$	5.05460	0.175553	0.0877767	+	0.996140i	$0.472024\pi$
$830$	−3.89690	−0.135263
$831$	0	0
$832$	6.73827	0.233607
$833$	−71.5929	−2.48055
$834$	0	0
$835$	−10.5399	−0.364749
$836$	8.99255	0.311014
$837$	0	0
$838$	14.8971	0.514611
$839$	−11.3499	−0.391841	−0.195920	−	0.980620i	$-0.562769\pi$
$839$	−11.3499	−0.391841	−0.195920	+	0.980620i	$0.562769\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	30.4854	1.05060
$843$	0	0
$844$	14.6426	0.504019
$845$	19.6405	0.675654
$846$	0	0
$847$	−8.06344	−0.277063
$848$	−2.09586	−0.0719723
$849$	0	0
$850$	5.00105	0.171535
$851$	−4.19589	−0.143833
$852$	0	0
$853$	6.25961	0.214325	0.107162	−	0.994242i	$-0.465824\pi$
$853$	6.25961	0.214325	0.107162	+	0.994242i	$0.465824\pi$
$854$	19.9348	0.682155
$855$	0	0
$856$	33.0845	1.13080
$857$	−47.0510	−1.60723	−0.803615	−	0.595149i	$-0.797093\pi$
$857$	−47.0510	−1.60723	−0.803615	+	0.595149i	$0.797093\pi$
$858$	0	0
$859$	−25.1552	−0.858284	−0.429142	−	0.903237i	$-0.641184\pi$
$859$	−25.1552	−0.858284	−0.429142	+	0.903237i	$0.641184\pi$
$860$	3.36069	0.114599
$861$	0	0
$862$	−14.4620	−0.492579
$863$	−40.7578	−1.38741	−0.693705	−	0.720259i	$-0.744023\pi$
$863$	−40.7578	−1.38741	−0.693705	+	0.720259i	$0.744023\pi$
$864$	0	0
$865$	−14.4128	−0.490049
$866$	−5.37367	−0.182605
$867$	0	0
$868$	−15.9680	−0.541990
$869$	25.7620	0.873915
$870$	0	0
$871$	7.14928	0.242244
$872$	33.6851	1.14072
$873$	0	0
$874$	−1.79047	−0.0605635
$875$	48.2355	1.63066
$876$	0	0
$877$	−40.6135	−1.37142	−0.685710	−	0.727875i	$-0.740508\pi$
$877$	−40.6135	−1.37142	−0.685710	+	0.727875i	$0.740508\pi$
$878$	21.5878	0.728553
$879$	0	0
$880$	−3.19232	−0.107613
$881$	31.5751	1.06379	0.531896	−	0.846809i	$-0.321480\pi$
$881$	31.5751	1.06379	0.531896	+	0.846809i	$0.321480\pi$
$882$	0	0
$883$	17.2242	0.579641	0.289820	−	0.957081i	$-0.406404\pi$
$883$	17.2242	0.579641	0.289820	+	0.957081i	$0.406404\pi$
$884$	18.8669	0.634563
$885$	0	0
$886$	0.259747	0.00872638
$887$	−16.1007	−0.540608	−0.270304	−	0.962775i	$-0.587124\pi$
$887$	−16.1007	−0.540608	−0.270304	+	0.962775i	$0.587124\pi$
$888$	0	0
$889$	−40.1451	−1.34642
$890$	−16.2466	−0.544588
$891$	0	0
$892$	9.49009	0.317752
$893$	4.47921	0.149891
$894$	0	0
$895$	1.07880	0.0360603
$896$	−35.2563	−1.17783
$897$	0	0
$898$	30.1798	1.00711
$899$	−2.91860	−0.0973406
$900$	0	0
$901$	30.5768	1.01866
$902$	−18.3077	−0.609579
$903$	0	0
$904$	−55.1301	−1.83360
$905$	2.43223	0.0808502
$906$	0	0
$907$	1.20503	0.0400124	0.0200062	−	0.999800i	$-0.493631\pi$
$907$	1.20503	0.0400124	0.0200062	+	0.999800i	$0.493631\pi$
$908$	19.3070	0.640724
$909$	0	0
$910$	−12.3253	−0.408579
$911$	−28.2695	−0.936610	−0.468305	−	0.883567i	$-0.655135\pi$
$911$	−28.2695	−0.936610	−0.468305	+	0.883567i	$0.655135\pi$
$912$	0	0
$913$	7.09166	0.234700
$914$	−5.81008	−0.192180
$915$	0	0
$916$	−8.23734	−0.272169
$917$	−68.6787	−2.26797
$918$	0	0
$919$	25.7108	0.848122	0.424061	−	0.905634i	$-0.360604\pi$
$919$	25.7108	0.848122	0.424061	+	0.905634i	$0.360604\pi$
$920$	5.52506	0.182156
$921$	0	0
$922$	−1.83109	−0.0603038
$923$	27.2392	0.896588
$924$	0	0
$925$	3.42950	0.112761
$926$	−11.8282	−0.388700
$927$	0	0
$928$	−5.82247	−0.191132
$929$	30.0952	0.987390	0.493695	−	0.869635i	$-0.335646\pi$
$929$	30.0952	0.987390	0.493695	+	0.869635i	$0.335646\pi$
$930$	0	0
$931$	−20.9499	−0.686605
$932$	29.8686	0.978379
$933$	0	0
$934$	−15.2846	−0.500128
$935$	46.5731	1.52310
$936$	0	0
$937$	−58.4410	−1.90918	−0.954592	−	0.297915i	$-0.903709\pi$
$937$	−58.4410	−1.90918	−0.954592	+	0.297915i	$0.903709\pi$
$938$	12.6851	0.414183
$939$	0	0
$940$	−5.56875	−0.181633
$941$	−14.8023	−0.482540	−0.241270	−	0.970458i	$-0.577564\pi$
$941$	−14.8023	−0.482540	−0.241270	+	0.970458i	$0.577564\pi$
$942$	0	0
$943$	−7.56150	−0.246236
$944$	−2.40051	−0.0781300
$945$	0	0
$946$	2.94826	0.0958563
$947$	−51.8728	−1.68564	−0.842819	−	0.538196i	$-0.819106\pi$
$947$	−51.8728	−1.68564	−0.842819	+	0.538196i	$0.819106\pi$
$948$	0	0
$949$	11.5864	0.376110
$950$	1.46343	0.0474801
$951$	0	0
$952$	83.0895	2.69295
$953$	−27.8978	−0.903697	−0.451849	−	0.892095i	$-0.649235\pi$
$953$	−27.8978	−0.903697	−0.451849	+	0.892095i	$0.649235\pi$
$954$	0	0
$955$	−7.13350	−0.230835
$956$	−20.0044	−0.646989
$957$	0	0
$958$	−16.3517	−0.528299
$959$	−72.4156	−2.33842
$960$	0	0
$961$	−22.4818	−0.725219
$962$	−6.23705	−0.201091
$963$	0	0
$964$	2.39604	0.0771712
$965$	−0.0241103	−0.000776138 0
$966$	0	0
$967$	36.0174	1.15824	0.579120	−	0.815242i	$-0.303396\pi$
$967$	36.0174	1.15824	0.579120	+	0.815242i	$0.303396\pi$
$968$	5.37298	0.172694
$969$	0	0
$970$	27.8773	0.895086
$971$	−10.4186	−0.334347	−0.167174	−	0.985927i	$-0.553464\pi$
$971$	−10.4186	−0.334347	−0.167174	+	0.985927i	$0.553464\pi$
$972$	0	0
$973$	−38.5401	−1.23554
$974$	5.33959	0.171091
$975$	0	0
$976$	3.16994	0.101467
$977$	−1.30072	−0.0416137	−0.0208068	−	0.999784i	$-0.506624\pi$
$977$	−1.30072	−0.0416137	−0.0208068	+	0.999784i	$0.506624\pi$
$978$	0	0
$979$	29.5659	0.944932
$980$	26.0458	0.832003
$981$	0	0
$982$	3.69333	0.117859
$983$	−17.6168	−0.561887	−0.280944	−	0.959724i	$-0.590647\pi$
$983$	−17.6168	−0.561887	−0.280944	+	0.959724i	$0.590647\pi$
$984$	0	0
$985$	46.7828	1.49062
$986$	6.11864	0.194857
$987$	0	0
$988$	5.52094	0.175644
$989$	1.21770	0.0387207
$990$	0	0
$991$	19.6932	0.625576	0.312788	−	0.949823i	$-0.398737\pi$
$991$	19.6932	0.625576	0.312788	+	0.949823i	$0.398737\pi$
$992$	16.9934	0.539542
$993$	0	0
$994$	48.3310	1.53297
$995$	42.4088	1.34445
$996$	0	0
$997$	50.3107	1.59336	0.796678	−	0.604404i	$-0.206589\pi$
$997$	50.3107	1.59336	0.796678	+	0.604404i	$0.206589\pi$
$998$	−26.9693	−0.853698
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.o.1.10		13
3.2	odd	2		667.2.a.c.1.4	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.c.1.4	✓	13	3.2	odd	2
6003.2.a.o.1.10		13	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)

\(f(q)\)	\(=\)	\(q+0.806558 q^{2} -1.34946 q^{4} -2.04515 q^{5} +4.05430 q^{7} -2.70154 q^{8} +O(q^{10})\)	\(q+0.806558 q^{2} -1.34946 q^{4} -2.04515 q^{5} +4.05430 q^{7} -2.70154 q^{8} -1.64953 q^{10} +3.00186 q^{11} +1.84298 q^{13} +3.27003 q^{14} +0.519985 q^{16} -7.58612 q^{17} -2.21989 q^{19} +2.75986 q^{20} +2.42117 q^{22} +1.00000 q^{23} -0.817347 q^{25} +1.48647 q^{26} -5.47114 q^{28} -1.00000 q^{29} +2.91860 q^{31} +5.82247 q^{32} -6.11864 q^{34} -8.29167 q^{35} -4.19589 q^{37} -1.79047 q^{38} +5.52506 q^{40} -7.56150 q^{41} +1.21770 q^{43} -4.05090 q^{44} +0.806558 q^{46} -2.01776 q^{47} +9.43736 q^{49} -0.659237 q^{50} -2.48703 q^{52} -4.03062 q^{53} -6.13926 q^{55} -10.9528 q^{56} -0.806558 q^{58} -4.61650 q^{59} +6.09622 q^{61} +2.35402 q^{62} +3.65619 q^{64} -3.76917 q^{65} +3.87921 q^{67} +10.2372 q^{68} -6.68771 q^{70} +14.7800 q^{71} +6.28677 q^{73} -3.38423 q^{74} +2.99566 q^{76} +12.1704 q^{77} +8.58201 q^{79} -1.06345 q^{80} -6.09879 q^{82} +2.36242 q^{83} +15.5148 q^{85} +0.982147 q^{86} -8.10962 q^{88} +9.84922 q^{89} +7.47198 q^{91} -1.34946 q^{92} -1.62744 q^{94} +4.54001 q^{95} -16.9001 q^{97} +7.61177 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8} + 10 q^{10} - 10 q^{11} + 7 q^{13} + 12 q^{14} + 2 q^{16} - 26 q^{17} - 25 q^{20} - 15 q^{22} + 13 q^{23} + 19 q^{25} + 15 q^{26} + 5 q^{28} - 13 q^{29} - 6 q^{31} - 16 q^{32} + 11 q^{34} - q^{35} + 15 q^{37} - 8 q^{38} + 14 q^{40} - 9 q^{41} + q^{43} - 29 q^{44} - 4 q^{46} - 15 q^{47} + 4 q^{49} - 31 q^{50} - 8 q^{52} - 43 q^{53} - 3 q^{55} + 5 q^{56} + 4 q^{58} + 9 q^{59} + 20 q^{61} - 11 q^{62} - 16 q^{64} + 25 q^{65} + q^{67} - 21 q^{68} - 2 q^{70} - 17 q^{71} + 26 q^{73} - 11 q^{74} + 8 q^{76} - 17 q^{77} + 5 q^{79} - 10 q^{80} - 25 q^{82} - 4 q^{83} + 20 q^{85} + 13 q^{86} - 32 q^{88} - 48 q^{89} - 9 q^{91} + 12 q^{92} - 65 q^{94} - 8 q^{95} + 30 q^{97} - 2 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8 + 10 * q^10 - 10 * q^11 + 7 * q^13 + 12 * q^14 + 2 * q^16 - 26 * q^17 - 25 * q^20 - 15 * q^22 + 13 * q^23 + 19 * q^25 + 15 * q^26 + 5 * q^28 - 13 * q^29 - 6 * q^31 - 16 * q^32 + 11 * q^34 - q^35 + 15 * q^37 - 8 * q^38 + 14 * q^40 - 9 * q^41 + q^43 - 29 * q^44 - 4 * q^46 - 15 * q^47 + 4 * q^49 - 31 * q^50 - 8 * q^52 - 43 * q^53 - 3 * q^55 + 5 * q^56 + 4 * q^58 + 9 * q^59 + 20 * q^61 - 11 * q^62 - 16 * q^64 + 25 * q^65 + q^67 - 21 * q^68 - 2 * q^70 - 17 * q^71 + 26 * q^73 - 11 * q^74 + 8 * q^76 - 17 * q^77 + 5 * q^79 - 10 * q^80 - 25 * q^82 - 4 * q^83 + 20 * q^85 + 13 * q^86 - 32 * q^88 - 48 * q^89 - 9 * q^91 + 12 * q^92 - 65 * q^94 - 8 * q^95 + 30 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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