LMFDB - Embedded newform 6003.2.a.v.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-1.96558 q^{2} +1.86349 q^{4} -2.31356 q^{5} +5.13967 q^{7} +0.268317 q^{8} +O(q^{10})$	\(q-1.96558 q^{2} +1.86349 q^{4} -2.31356 q^{5} +5.13967 q^{7} +0.268317 q^{8} +4.54748 q^{10} +2.78317 q^{11} +1.47321 q^{13} -10.1024 q^{14} -4.25438 q^{16} +1.99602 q^{17} +2.47180 q^{19} -4.31130 q^{20} -5.47053 q^{22} +1.00000 q^{23} +0.352569 q^{25} -2.89570 q^{26} +9.57774 q^{28} -1.00000 q^{29} +5.51730 q^{31} +7.82568 q^{32} -3.92334 q^{34} -11.8910 q^{35} +7.95358 q^{37} -4.85851 q^{38} -0.620768 q^{40} -2.86730 q^{41} +6.65301 q^{43} +5.18641 q^{44} -1.96558 q^{46} +1.23566 q^{47} +19.4162 q^{49} -0.693001 q^{50} +2.74531 q^{52} +1.25738 q^{53} -6.43903 q^{55} +1.37906 q^{56} +1.96558 q^{58} -3.65301 q^{59} +7.80980 q^{61} -10.8447 q^{62} -6.87321 q^{64} -3.40835 q^{65} -4.27263 q^{67} +3.71957 q^{68} +23.3726 q^{70} +3.71934 q^{71} -2.30243 q^{73} -15.6334 q^{74} +4.60618 q^{76} +14.3046 q^{77} +9.24788 q^{79} +9.84278 q^{80} +5.63590 q^{82} +1.97525 q^{83} -4.61792 q^{85} -13.0770 q^{86} +0.746771 q^{88} +4.99866 q^{89} +7.57180 q^{91} +1.86349 q^{92} -2.42878 q^{94} -5.71866 q^{95} -2.35337 q^{97} -38.1641 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) $ 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8	$ 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8} + 8 q^{10} + 36 q^{13} - 7 q^{14} + 47 q^{16} - 18 q^{17} + 16 q^{19} + 25 q^{22} + 30 q^{23} + 56 q^{25} - 11 q^{26} + 27 q^{28} - 30 q^{29} + 14 q^{31} + 7 q^{32} + 3 q^{34} + 22 q^{35} + 40 q^{37} - 6 q^{38} + 30 q^{40} - 14 q^{41} + 34 q^{43} - 5 q^{44} - q^{46} + 2 q^{47} + 74 q^{49} + 21 q^{50} + 71 q^{52} - 16 q^{53} + 22 q^{55} - 14 q^{56} + q^{58} + 32 q^{59} + 46 q^{61} - 20 q^{62} + 68 q^{64} - 12 q^{65} + 14 q^{67} - 27 q^{68} + 32 q^{71} + 50 q^{73} + 26 q^{74} + 56 q^{76} - 34 q^{77} + 16 q^{79} - 2 q^{80} + 38 q^{82} + 14 q^{83} + 38 q^{85} - 10 q^{86} + 40 q^{88} + 2 q^{89} + 32 q^{91} + 37 q^{92} + 29 q^{94} + 28 q^{95} + 56 q^{97} - 8 q^{98}+O(q^{100}) $ 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8 + 8 * q^10 + 36 * q^13 - 7 * q^14 + 47 * q^16 - 18 * q^17 + 16 * q^19 + 25 * q^22 + 30 * q^23 + 56 * q^25 - 11 * q^26 + 27 * q^28 - 30 * q^29 + 14 * q^31 + 7 * q^32 + 3 * q^34 + 22 * q^35 + 40 * q^37 - 6 * q^38 + 30 * q^40 - 14 * q^41 + 34 * q^43 - 5 * q^44 - q^46 + 2 * q^47 + 74 * q^49 + 21 * q^50 + 71 * q^52 - 16 * q^53 + 22 * q^55 - 14 * q^56 + q^58 + 32 * q^59 + 46 * q^61 - 20 * q^62 + 68 * q^64 - 12 * q^65 + 14 * q^67 - 27 * q^68 + 32 * q^71 + 50 * q^73 + 26 * q^74 + 56 * q^76 - 34 * q^77 + 16 * q^79 - 2 * q^80 + 38 * q^82 + 14 * q^83 + 38 * q^85 - 10 * q^86 + 40 * q^88 + 2 * q^89 + 32 * q^91 + 37 * q^92 + 29 * q^94 + 28 * q^95 + 56 * q^97 - 8 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.96558	−1.38987	−0.694936	−	0.719071i	$-0.744567\pi$
$2$	−1.96558	−1.38987	−0.694936	+	0.719071i	$0.744567\pi$
$3$	0	0
$3$	0	0
$4$	1.86349	0.931746
$5$	−2.31356	−1.03466	−0.517328	−	0.855787i	$-0.673073\pi$
$5$	−2.31356	−1.03466	−0.517328	+	0.855787i	$0.673073\pi$
$6$	0	0
$7$	5.13967	1.94261	0.971307	−	0.237829i	$-0.0764358\pi$
$7$	5.13967	1.94261	0.971307	+	0.237829i	$0.0764358\pi$
$8$	0.268317	0.0948644
$9$	0	0
$10$	4.54748	1.43804
$11$	2.78317	0.839156	0.419578	−	0.907719i	$-0.362178\pi$
$11$	2.78317	0.839156	0.419578	+	0.907719i	$0.362178\pi$
$12$	0	0
$13$	1.47321	0.408594	0.204297	−	0.978909i	$-0.434509\pi$
$13$	1.47321	0.408594	0.204297	+	0.978909i	$0.434509\pi$
$14$	−10.1024	−2.69999
$15$	0	0
$16$	−4.25438	−1.06360
$17$	1.99602	0.484107	0.242053	−	0.970263i	$-0.422179\pi$
$17$	1.99602	0.484107	0.242053	+	0.970263i	$0.422179\pi$
$18$	0	0
$19$	2.47180	0.567069	0.283535	−	0.958962i	$-0.408493\pi$
$19$	2.47180	0.567069	0.283535	+	0.958962i	$0.408493\pi$
$20$	−4.31130	−0.964037
$21$	0	0
$22$	−5.47053	−1.16632
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0.352569	0.0705138
$26$	−2.89570	−0.567894
$27$	0	0
$28$	9.57774	1.81002
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.51730	0.990937	0.495468	−	0.868626i	$-0.334996\pi$
$31$	5.51730	0.990937	0.495468	+	0.868626i	$0.334996\pi$
$32$	7.82568	1.38340
$33$	0	0
$34$	−3.92334	−0.672847
$35$	−11.8910	−2.00994
$36$	0	0
$37$	7.95358	1.30756	0.653780	−	0.756685i	$-0.273182\pi$
$37$	7.95358	1.30756	0.653780	+	0.756685i	$0.273182\pi$
$38$	−4.85851	−0.788154
$39$	0	0
$40$	−0.620768	−0.0981520
$41$	−2.86730	−0.447797	−0.223899	−	0.974612i	$-0.571878\pi$
$41$	−2.86730	−0.447797	−0.223899	+	0.974612i	$0.571878\pi$
$42$	0	0
$43$	6.65301	1.01457	0.507287	−	0.861777i	$-0.330648\pi$
$43$	6.65301	1.01457	0.507287	+	0.861777i	$0.330648\pi$
$44$	5.18641	0.781881
$45$	0	0
$46$	−1.96558	−0.289808
$47$	1.23566	0.180239	0.0901194	−	0.995931i	$-0.471275\pi$
$47$	1.23566	0.180239	0.0901194	+	0.995931i	$0.471275\pi$
$48$	0	0
$49$	19.4162	2.77375
$50$	−0.693001	−0.0980052
$51$	0	0
$52$	2.74531	0.380706
$53$	1.25738	0.172714	0.0863569	−	0.996264i	$-0.472477\pi$
$53$	1.25738	0.172714	0.0863569	+	0.996264i	$0.472477\pi$
$54$	0	0
$55$	−6.43903	−0.868239
$56$	1.37906	0.184285
$57$	0	0
$58$	1.96558	0.258093
$59$	−3.65301	−0.475582	−0.237791	−	0.971316i	$-0.576423\pi$
$59$	−3.65301	−0.475582	−0.237791	+	0.971316i	$0.576423\pi$
$60$	0	0
$61$	7.80980	0.999943	0.499971	−	0.866042i	$-0.333344\pi$
$61$	7.80980	0.999943	0.499971	+	0.866042i	$0.333344\pi$
$62$	−10.8447	−1.37728
$63$	0	0
$64$	−6.87321	−0.859151
$65$	−3.40835	−0.422754
$66$	0	0
$67$	−4.27263	−0.521985	−0.260993	−	0.965341i	$-0.584050\pi$
$67$	−4.27263	−0.521985	−0.260993	+	0.965341i	$0.584050\pi$
$68$	3.71957	0.451065
$69$	0	0
$70$	23.3726	2.79356
$71$	3.71934	0.441405	0.220702	−	0.975341i	$-0.429165\pi$
$71$	3.71934	0.441405	0.220702	+	0.975341i	$0.429165\pi$
$72$	0	0
$73$	−2.30243	−0.269479	−0.134739	−	0.990881i	$-0.543020\pi$
$73$	−2.30243	−0.269479	−0.134739	+	0.990881i	$0.543020\pi$
$74$	−15.6334	−1.81734
$75$	0	0
$76$	4.60618	0.528365
$77$	14.3046	1.63016
$78$	0	0
$79$	9.24788	1.04047	0.520234	−	0.854024i	$-0.325845\pi$
$79$	9.24788	1.04047	0.520234	+	0.854024i	$0.325845\pi$
$80$	9.84278	1.10046
$81$	0	0
$82$	5.63590	0.622381
$83$	1.97525	0.216812	0.108406	−	0.994107i	$-0.465425\pi$
$83$	1.97525	0.216812	0.108406	+	0.994107i	$0.465425\pi$
$84$	0	0
$85$	−4.61792	−0.500884
$86$	−13.0770	−1.41013
$87$	0	0
$88$	0.746771	0.0796060
$89$	4.99866	0.529856	0.264928	−	0.964268i	$-0.414652\pi$
$89$	4.99866	0.529856	0.264928	+	0.964268i	$0.414652\pi$
$90$	0	0
$91$	7.57180	0.793740
$92$	1.86349	0.194282
$93$	0	0
$94$	−2.42878	−0.250509
$95$	−5.71866	−0.586722
$96$	0	0
$97$	−2.35337	−0.238949	−0.119474	−	0.992837i	$-0.538121\pi$
$97$	−2.35337	−0.238949	−0.119474	+	0.992837i	$0.538121\pi$
$98$	−38.1641	−3.85516
$99$	0	0
$100$	0.657009	0.0657009
$101$	−2.77789	−0.276411	−0.138205	−	0.990404i	$-0.544133\pi$
$101$	−2.77789	−0.276411	−0.138205	+	0.990404i	$0.544133\pi$
$102$	0	0
$103$	−2.13884	−0.210746	−0.105373	−	0.994433i	$-0.533604\pi$
$103$	−2.13884	−0.210746	−0.105373	+	0.994433i	$0.533604\pi$
$104$	0.395286	0.0387610
$105$	0	0
$106$	−2.47147	−0.240050
$107$	−0.489276	−0.0473001	−0.0236500	−	0.999720i	$-0.507529\pi$
$107$	−0.489276	−0.0473001	−0.0236500	+	0.999720i	$0.507529\pi$
$108$	0	0
$109$	5.05915	0.484579	0.242289	−	0.970204i	$-0.422102\pi$
$109$	5.05915	0.484579	0.242289	+	0.970204i	$0.422102\pi$
$110$	12.6564	1.20674
$111$	0	0
$112$	−21.8661	−2.06616
$113$	−3.90746	−0.367583	−0.183791	−	0.982965i	$-0.558837\pi$
$113$	−3.90746	−0.367583	−0.183791	+	0.982965i	$0.558837\pi$
$114$	0	0
$115$	−2.31356	−0.215741
$116$	−1.86349	−0.173021
$117$	0	0
$118$	7.18027	0.660998
$119$	10.2589	0.940433
$120$	0	0
$121$	−3.25398	−0.295817
$122$	−15.3508	−1.38979
$123$	0	0
$124$	10.2815	0.923302
$125$	10.7521	0.961699
$126$	0	0
$127$	−8.36002	−0.741832	−0.370916	−	0.928667i	$-0.620956\pi$
$127$	−8.36002	−0.741832	−0.370916	+	0.928667i	$0.620956\pi$
$128$	−2.14154	−0.189287
$129$	0	0
$130$	6.69938	0.587575
$131$	2.07624	0.181402	0.0907011	−	0.995878i	$-0.471089\pi$
$131$	2.07624	0.181402	0.0907011	+	0.995878i	$0.471089\pi$
$132$	0	0
$133$	12.7042	1.10160
$134$	8.39819	0.725493
$135$	0	0
$136$	0.535567	0.0459245
$137$	4.68581	0.400336	0.200168	−	0.979762i	$-0.435851\pi$
$137$	4.68581	0.400336	0.200168	+	0.979762i	$0.435851\pi$
$138$	0	0
$139$	−6.85953	−0.581817	−0.290909	−	0.956751i	$-0.593958\pi$
$139$	−6.85953	−0.581817	−0.290909	+	0.956751i	$0.593958\pi$
$140$	−22.1587	−1.87275
$141$	0	0
$142$	−7.31065	−0.613496
$143$	4.10018	0.342874
$144$	0	0
$145$	2.31356	0.192131
$146$	4.52560	0.374541
$147$	0	0
$148$	14.8214	1.21831
$149$	3.83853	0.314464	0.157232	−	0.987562i	$-0.449743\pi$
$149$	3.83853	0.314464	0.157232	+	0.987562i	$0.449743\pi$
$150$	0	0
$151$	−1.94182	−0.158023	−0.0790114	−	0.996874i	$-0.525176\pi$
$151$	−1.94182	−0.158023	−0.0790114	+	0.996874i	$0.525176\pi$
$152$	0.663225	0.0537947
$153$	0	0
$154$	−28.1167	−2.26571
$155$	−12.7646	−1.02528
$156$	0	0
$157$	−19.8593	−1.58494	−0.792472	−	0.609909i	$-0.791206\pi$
$157$	−19.8593	−1.58494	−0.792472	+	0.609909i	$0.791206\pi$
$158$	−18.1774	−1.44612
$159$	0	0
$160$	−18.1052	−1.43134
$161$	5.13967	0.405063
$162$	0	0
$163$	5.43646	0.425817	0.212908	−	0.977072i	$-0.431706\pi$
$163$	5.43646	0.425817	0.212908	+	0.977072i	$0.431706\pi$
$164$	−5.34320	−0.417233
$165$	0	0
$166$	−3.88251	−0.301341
$167$	−12.9734	−1.00391	−0.501956	−	0.864893i	$-0.667386\pi$
$167$	−12.9734	−1.00391	−0.501956	+	0.864893i	$0.667386\pi$
$168$	0	0
$169$	−10.8297	−0.833051
$170$	9.07688	0.696165
$171$	0	0
$172$	12.3978	0.945326
$173$	−12.8934	−0.980271	−0.490135	−	0.871646i	$-0.663053\pi$
$173$	−12.8934	−0.980271	−0.490135	+	0.871646i	$0.663053\pi$
$174$	0	0
$175$	1.81209	0.136981
$176$	−11.8407	−0.892523
$177$	0	0
$178$	−9.82524	−0.736433
$179$	23.5816	1.76257	0.881285	−	0.472585i	$-0.156679\pi$
$179$	23.5816	1.76257	0.881285	+	0.472585i	$0.156679\pi$
$180$	0	0
$181$	−24.9065	−1.85128	−0.925641	−	0.378403i	$-0.876473\pi$
$181$	−24.9065	−1.85128	−0.925641	+	0.378403i	$0.876473\pi$
$182$	−14.8830	−1.10320
$183$	0	0
$184$	0.268317	0.0197806
$185$	−18.4011	−1.35288
$186$	0	0
$187$	5.55527	0.406241
$188$	2.30263	0.167937
$189$	0	0
$190$	11.2405	0.815469
$191$	5.53021	0.400152	0.200076	−	0.979780i	$-0.435881\pi$
$191$	5.53021	0.400152	0.200076	+	0.979780i	$0.435881\pi$
$192$	0	0
$193$	−24.3558	−1.75317	−0.876583	−	0.481251i	$-0.840183\pi$
$193$	−24.3558	−1.75317	−0.876583	+	0.481251i	$0.840183\pi$
$194$	4.62574	0.332109
$195$	0	0
$196$	36.1820	2.58443
$197$	−16.1018	−1.14721	−0.573604	−	0.819132i	$-0.694455\pi$
$197$	−16.1018	−1.14721	−0.573604	+	0.819132i	$0.694455\pi$
$198$	0	0
$199$	−14.4878	−1.02701	−0.513507	−	0.858085i	$-0.671654\pi$
$199$	−14.4878	−1.02701	−0.513507	+	0.858085i	$0.671654\pi$
$200$	0.0946002	0.00668925
$201$	0	0
$202$	5.46016	0.384176
$203$	−5.13967	−0.360734
$204$	0	0
$205$	6.63368	0.463316
$206$	4.20406	0.292911
$207$	0	0
$208$	−6.26758	−0.434579
$209$	6.87943	0.475860
$210$	0	0
$211$	−14.9323	−1.02798	−0.513991	−	0.857796i	$-0.671833\pi$
$211$	−14.9323	−1.02798	−0.513991	+	0.857796i	$0.671833\pi$
$212$	2.34311	0.160925
$213$	0	0
$214$	0.961709	0.0657411
$215$	−15.3922	−1.04974
$216$	0	0
$217$	28.3571	1.92501
$218$	−9.94415	−0.673503
$219$	0	0
$220$	−11.9991	−0.808978
$221$	2.94055	0.197803
$222$	0	0
$223$	26.1693	1.75243	0.876214	−	0.481923i	$-0.160061\pi$
$223$	26.1693	1.75243	0.876214	+	0.481923i	$0.160061\pi$
$224$	40.2214	2.68741
$225$	0	0
$226$	7.68041	0.510893
$227$	23.0660	1.53094	0.765471	−	0.643471i	$-0.222506\pi$
$227$	23.0660	1.53094	0.765471	+	0.643471i	$0.222506\pi$
$228$	0	0
$229$	2.49890	0.165132	0.0825659	−	0.996586i	$-0.473689\pi$
$229$	2.49890	0.165132	0.0825659	+	0.996586i	$0.473689\pi$
$230$	4.54748	0.299852
$231$	0	0
$232$	−0.268317	−0.0176159
$233$	−14.2159	−0.931318	−0.465659	−	0.884964i	$-0.654183\pi$
$233$	−14.2159	−0.931318	−0.465659	+	0.884964i	$0.654183\pi$
$234$	0	0
$235$	−2.85877	−0.186485
$236$	−6.80736	−0.443121
$237$	0	0
$238$	−20.1647	−1.30708
$239$	−8.77873	−0.567849	−0.283924	−	0.958847i	$-0.591636\pi$
$239$	−8.77873	−0.567849	−0.283924	+	0.958847i	$0.591636\pi$
$240$	0	0
$241$	11.7782	0.758699	0.379350	−	0.925253i	$-0.376148\pi$
$241$	11.7782	0.758699	0.379350	+	0.925253i	$0.376148\pi$
$242$	6.39595	0.411147
$243$	0	0
$244$	14.5535	0.931693
$245$	−44.9207	−2.86988
$246$	0	0
$247$	3.64147	0.231701
$248$	1.48039	0.0940046
$249$	0	0
$250$	−21.1341	−1.33664
$251$	29.9612	1.89113	0.945566	−	0.325431i	$-0.105509\pi$
$251$	29.9612	1.89113	0.945566	+	0.325431i	$0.105509\pi$
$252$	0	0
$253$	2.78317	0.174976
$254$	16.4323	1.03105
$255$	0	0
$256$	17.9558	1.12224
$257$	7.04706	0.439584	0.219792	−	0.975547i	$-0.429462\pi$
$257$	7.04706	0.439584	0.219792	+	0.975547i	$0.429462\pi$
$258$	0	0
$259$	40.8788	2.54008
$260$	−6.35144	−0.393900
$261$	0	0
$262$	−4.08102	−0.252126
$263$	−2.11869	−0.130644	−0.0653221	−	0.997864i	$-0.520807\pi$
$263$	−2.11869	−0.130644	−0.0653221	+	0.997864i	$0.520807\pi$
$264$	0	0
$265$	−2.90902	−0.178700
$266$	−24.9712	−1.53108
$267$	0	0
$268$	−7.96201	−0.486357
$269$	−0.482188	−0.0293995	−0.0146998	−	0.999892i	$-0.504679\pi$
$269$	−0.482188	−0.0293995	−0.0146998	+	0.999892i	$0.504679\pi$
$270$	0	0
$271$	1.06502	0.0646952	0.0323476	−	0.999477i	$-0.489702\pi$
$271$	1.06502	0.0646952	0.0323476	+	0.999477i	$0.489702\pi$
$272$	−8.49184	−0.514894
$273$	0	0
$274$	−9.21033	−0.556416
$275$	0.981258	0.0591721
$276$	0	0
$277$	−4.56914	−0.274533	−0.137267	−	0.990534i	$-0.543832\pi$
$277$	−4.56914	−0.274533	−0.137267	+	0.990534i	$0.543832\pi$
$278$	13.4829	0.808652
$279$	0	0
$280$	−3.19054	−0.190671
$281$	13.1315	0.783361	0.391681	−	0.920101i	$-0.371894\pi$
$281$	13.1315	0.783361	0.391681	+	0.920101i	$0.371894\pi$
$282$	0	0
$283$	−11.8409	−0.703871	−0.351935	−	0.936024i	$-0.614476\pi$
$283$	−11.8409	−0.703871	−0.351935	+	0.936024i	$0.614476\pi$
$284$	6.93096	0.411277
$285$	0	0
$286$	−8.05922	−0.476551
$287$	−14.7370	−0.869898
$288$	0	0
$289$	−13.0159	−0.765641
$290$	−4.54748	−0.267037
$291$	0	0
$292$	−4.29056	−0.251086
$293$	−13.6383	−0.796756	−0.398378	−	0.917221i	$-0.630427\pi$
$293$	−13.6383	−0.796756	−0.398378	+	0.917221i	$0.630427\pi$
$294$	0	0
$295$	8.45147	0.492064
$296$	2.13408	0.124041
$297$	0	0
$298$	−7.54492	−0.437065
$299$	1.47321	0.0851977
$300$	0	0
$301$	34.1943	1.97093
$302$	3.81679	0.219632
$303$	0	0
$304$	−10.5160	−0.603132
$305$	−18.0685	−1.03460
$306$	0	0
$307$	22.7551	1.29870	0.649352	−	0.760488i	$-0.275040\pi$
$307$	22.7551	1.29870	0.649352	+	0.760488i	$0.275040\pi$
$308$	26.6565	1.51889
$309$	0	0
$310$	25.0898	1.42501
$311$	−29.8822	−1.69446	−0.847231	−	0.531225i	$-0.821732\pi$
$311$	−29.8822	−1.69446	−0.847231	+	0.531225i	$0.821732\pi$
$312$	0	0
$313$	−2.08206	−0.117685	−0.0588426	−	0.998267i	$-0.518741\pi$
$313$	−2.08206	−0.117685	−0.0588426	+	0.998267i	$0.518741\pi$
$314$	39.0349	2.20287
$315$	0	0
$316$	17.2334	0.969452
$317$	−22.0016	−1.23573	−0.617866	−	0.786284i	$-0.712003\pi$
$317$	−22.0016	−1.23573	−0.617866	+	0.786284i	$0.712003\pi$
$318$	0	0
$319$	−2.78317	−0.155827
$320$	15.9016	0.888926
$321$	0	0
$322$	−10.1024	−0.562986
$323$	4.93377	0.274522
$324$	0	0
$325$	0.519407	0.0288115
$326$	−10.6858	−0.591831
$327$	0	0
$328$	−0.769346	−0.0424800
$329$	6.35086	0.350135
$330$	0	0
$331$	9.91328	0.544883	0.272441	−	0.962172i	$-0.412169\pi$
$331$	9.91328	0.544883	0.272441	+	0.962172i	$0.412169\pi$
$332$	3.68087	0.202014
$333$	0	0
$334$	25.5002	1.39531
$335$	9.88500	0.540075
$336$	0	0
$337$	14.6453	0.797778	0.398889	−	0.916999i	$-0.369396\pi$
$337$	14.6453	0.797778	0.398889	+	0.916999i	$0.369396\pi$
$338$	21.2865	1.15783
$339$	0	0
$340$	−8.60546	−0.466697
$341$	15.3556	0.831551
$342$	0	0
$343$	63.8154	3.44571
$344$	1.78512	0.0962470
$345$	0	0
$346$	25.3431	1.36245
$347$	10.2667	0.551146	0.275573	−	0.961280i	$-0.411132\pi$
$347$	10.2667	0.551146	0.275573	+	0.961280i	$0.411132\pi$
$348$	0	0
$349$	−14.5744	−0.780148	−0.390074	−	0.920783i	$-0.627551\pi$
$349$	−14.5744	−0.780148	−0.390074	+	0.920783i	$0.627551\pi$
$350$	−3.56180	−0.190386
$351$	0	0
$352$	21.7802	1.16089
$353$	34.5905	1.84107	0.920533	−	0.390665i	$-0.127755\pi$
$353$	34.5905	1.84107	0.920533	+	0.390665i	$0.127755\pi$
$354$	0	0
$355$	−8.60493	−0.456702
$356$	9.31495	0.493692
$357$	0	0
$358$	−46.3514	−2.44975
$359$	−12.0264	−0.634730	−0.317365	−	0.948303i	$-0.602798\pi$
$359$	−12.0264	−0.634730	−0.317365	+	0.948303i	$0.602798\pi$
$360$	0	0
$361$	−12.8902	−0.678432
$362$	48.9555	2.57305
$363$	0	0
$364$	14.1100	0.739564
$365$	5.32681	0.278818
$366$	0	0
$367$	−17.7960	−0.928946	−0.464473	−	0.885587i	$-0.653756\pi$
$367$	−17.7960	−0.928946	−0.464473	+	0.885587i	$0.653756\pi$
$368$	−4.25438	−0.221775
$369$	0	0
$370$	36.1688	1.88032
$371$	6.46250	0.335516
$372$	0	0
$373$	25.2980	1.30988	0.654940	−	0.755681i	$-0.272694\pi$
$373$	25.2980	1.30988	0.654940	+	0.755681i	$0.272694\pi$
$374$	−10.9193	−0.564624
$375$	0	0
$376$	0.331547	0.0170982
$377$	−1.47321	−0.0758740
$378$	0	0
$379$	15.1434	0.777863	0.388932	−	0.921267i	$-0.372844\pi$
$379$	15.1434	0.777863	0.388932	+	0.921267i	$0.372844\pi$
$380$	−10.6567	−0.546676
$381$	0	0
$382$	−10.8700	−0.556160
$383$	15.6805	0.801236	0.400618	−	0.916245i	$-0.368796\pi$
$383$	15.6805	0.801236	0.400618	+	0.916245i	$0.368796\pi$
$384$	0	0
$385$	−33.0945	−1.68665
$386$	47.8731	2.43668
$387$	0	0
$388$	−4.38549	−0.222640
$389$	19.8617	1.00703	0.503514	−	0.863987i	$-0.332040\pi$
$389$	19.8617	1.00703	0.503514	+	0.863987i	$0.332040\pi$
$390$	0	0
$391$	1.99602	0.100943
$392$	5.20971	0.263130
$393$	0	0
$394$	31.6494	1.59447
$395$	−21.3955	−1.07653
$396$	0	0
$397$	−11.5053	−0.577435	−0.288718	−	0.957414i	$-0.593229\pi$
$397$	−11.5053	−0.577435	−0.288718	+	0.957414i	$0.593229\pi$
$398$	28.4769	1.42742
$399$	0	0
$400$	−1.49996	−0.0749981
$401$	−18.2766	−0.912692	−0.456346	−	0.889803i	$-0.650842\pi$
$401$	−18.2766	−0.912692	−0.456346	+	0.889803i	$0.650842\pi$
$402$	0	0
$403$	8.12813	0.404891
$404$	−5.17658	−0.257545
$405$	0	0
$406$	10.1024	0.501375
$407$	22.1361	1.09725
$408$	0	0
$409$	7.04911	0.348556	0.174278	−	0.984697i	$-0.444241\pi$
$409$	7.04911	0.348556	0.174278	+	0.984697i	$0.444241\pi$
$410$	−13.0390	−0.643951
$411$	0	0
$412$	−3.98572	−0.196362
$413$	−18.7753	−0.923871
$414$	0	0
$415$	−4.56987	−0.224326
$416$	11.5288	0.565248
$417$	0	0
$418$	−13.5220	−0.661385
$419$	34.9985	1.70979	0.854894	−	0.518803i	$-0.173622\pi$
$419$	34.9985	1.70979	0.854894	+	0.518803i	$0.173622\pi$
$420$	0	0
$421$	−7.76739	−0.378559	−0.189280	−	0.981923i	$-0.560615\pi$
$421$	−7.76739	−0.378559	−0.189280	+	0.981923i	$0.560615\pi$
$422$	29.3506	1.42876
$423$	0	0
$424$	0.337375	0.0163844
$425$	0.703736	0.0341362
$426$	0	0
$427$	40.1398	1.94250
$428$	−0.911762	−0.0440717
$429$	0	0
$430$	30.2545	1.45900
$431$	−8.51899	−0.410345	−0.205173	−	0.978726i	$-0.565776\pi$
$431$	−8.51899	−0.410345	−0.205173	+	0.978726i	$0.565776\pi$
$432$	0	0
$433$	−19.9350	−0.958013	−0.479006	−	0.877811i	$-0.659003\pi$
$433$	−19.9350	−0.958013	−0.479006	+	0.877811i	$0.659003\pi$
$434$	−55.7381	−2.67552
$435$	0	0
$436$	9.42769	0.451505
$437$	2.47180	0.118242
$438$	0	0
$439$	4.79992	0.229088	0.114544	−	0.993418i	$-0.463459\pi$
$439$	4.79992	0.229088	0.114544	+	0.993418i	$0.463459\pi$
$440$	−1.72770	−0.0823649
$441$	0	0
$442$	−5.77988	−0.274921
$443$	−4.24297	−0.201590	−0.100795	−	0.994907i	$-0.532139\pi$
$443$	−4.24297	−0.201590	−0.100795	+	0.994907i	$0.532139\pi$
$444$	0	0
$445$	−11.5647	−0.548219
$446$	−51.4378	−2.43565
$447$	0	0
$448$	−35.3261	−1.66900
$449$	−23.8275	−1.12449	−0.562244	−	0.826971i	$-0.690062\pi$
$449$	−23.8275	−1.12449	−0.562244	+	0.826971i	$0.690062\pi$
$450$	0	0
$451$	−7.98018	−0.375772
$452$	−7.28152	−0.342494
$453$	0	0
$454$	−45.3379	−2.12781
$455$	−17.5178	−0.821248
$456$	0	0
$457$	8.50278	0.397743	0.198872	−	0.980026i	$-0.436272\pi$
$457$	8.50278	0.397743	0.198872	+	0.980026i	$0.436272\pi$
$458$	−4.91177	−0.229512
$459$	0	0
$460$	−4.31130	−0.201016
$461$	−5.94430	−0.276853	−0.138427	−	0.990373i	$-0.544205\pi$
$461$	−5.94430	−0.276853	−0.138427	+	0.990373i	$0.544205\pi$
$462$	0	0
$463$	10.3856	0.482662	0.241331	−	0.970443i	$-0.422416\pi$
$463$	10.3856	0.482662	0.241331	+	0.970443i	$0.422416\pi$
$464$	4.25438	0.197505
$465$	0	0
$466$	27.9425	1.29441
$467$	15.0284	0.695429	0.347715	−	0.937600i	$-0.386958\pi$
$467$	15.0284	0.695429	0.347715	+	0.937600i	$0.386958\pi$
$468$	0	0
$469$	−21.9599	−1.01402
$470$	5.61912	0.259191
$471$	0	0
$472$	−0.980165	−0.0451157
$473$	18.5164	0.851387
$474$	0	0
$475$	0.871479	0.0399862
$476$	19.1174	0.876244
$477$	0	0
$478$	17.2553	0.789237
$479$	19.0577	0.870768	0.435384	−	0.900245i	$-0.356613\pi$
$479$	19.0577	0.870768	0.435384	+	0.900245i	$0.356613\pi$
$480$	0	0
$481$	11.7173	0.534261
$482$	−23.1509	−1.05450
$483$	0	0
$484$	−6.06377	−0.275626
$485$	5.44468	0.247230
$486$	0	0
$487$	−39.1608	−1.77455	−0.887273	−	0.461244i	$-0.847403\pi$
$487$	−39.1608	−1.77455	−0.887273	+	0.461244i	$0.847403\pi$
$488$	2.09550	0.0948590
$489$	0	0
$490$	88.2951	3.98876
$491$	−25.0588	−1.13089	−0.565443	−	0.824787i	$-0.691295\pi$
$491$	−25.0588	−1.13089	−0.565443	+	0.824787i	$0.691295\pi$
$492$	0	0
$493$	−1.99602	−0.0898964
$494$	−7.15759	−0.322035
$495$	0	0
$496$	−23.4727	−1.05396
$497$	19.1162	0.857479
$498$	0	0
$499$	24.0721	1.07762	0.538808	−	0.842428i	$-0.318875\pi$
$499$	24.0721	1.07762	0.538808	+	0.842428i	$0.318875\pi$
$500$	20.0365	0.896059
$501$	0	0
$502$	−58.8910	−2.62843
$503$	0.643213	0.0286795	0.0143397	−	0.999897i	$-0.495435\pi$
$503$	0.643213	0.0286795	0.0143397	+	0.999897i	$0.495435\pi$
$504$	0	0
$505$	6.42683	0.285990
$506$	−5.47053	−0.243195
$507$	0	0
$508$	−15.5788	−0.691199
$509$	31.8840	1.41323	0.706617	−	0.707596i	$-0.250220\pi$
$509$	31.8840	1.41323	0.706617	+	0.707596i	$0.250220\pi$
$510$	0	0
$511$	−11.8337	−0.523493
$512$	−31.0104	−1.37048
$513$	0	0
$514$	−13.8515	−0.610965
$515$	4.94834	0.218050
$516$	0	0
$517$	3.43904	0.151249
$518$	−80.3504	−3.53039
$519$	0	0
$520$	−0.914519	−0.0401043
$521$	−34.8631	−1.52738	−0.763689	−	0.645584i	$-0.776614\pi$
$521$	−34.8631	−1.52738	−0.763689	+	0.645584i	$0.776614\pi$
$522$	0	0
$523$	−32.4818	−1.42033	−0.710164	−	0.704036i	$-0.751379\pi$
$523$	−32.4818	−1.42033	−0.710164	+	0.704036i	$0.751379\pi$
$524$	3.86906	0.169021
$525$	0	0
$526$	4.16445	0.181579
$527$	11.0127	0.479719
$528$	0	0
$529$	1.00000	0.0434783
$530$	5.71790	0.248370
$531$	0	0
$532$	23.6742	1.02641
$533$	−4.22413	−0.182967
$534$	0	0
$535$	1.13197	0.0489393
$536$	−1.14642	−0.0495178
$537$	0	0
$538$	0.947778	0.0408616
$539$	54.0386	2.32761
$540$	0	0
$541$	−28.3375	−1.21832	−0.609162	−	0.793045i	$-0.708494\pi$
$541$	−28.3375	−1.21832	−0.609162	+	0.793045i	$0.708494\pi$
$542$	−2.09337	−0.0899181
$543$	0	0
$544$	15.6202	0.669712
$545$	−11.7047	−0.501373
$546$	0	0
$547$	−15.1711	−0.648669	−0.324335	−	0.945942i	$-0.605140\pi$
$547$	−15.1711	−0.648669	−0.324335	+	0.945942i	$0.605140\pi$
$548$	8.73198	0.373011
$549$	0	0
$550$	−1.92874	−0.0822417
$551$	−2.47180	−0.105302
$552$	0	0
$553$	47.5311	2.02123
$554$	8.98100	0.381566
$555$	0	0
$556$	−12.7827	−0.542106
$557$	−5.04104	−0.213596	−0.106798	−	0.994281i	$-0.534060\pi$
$557$	−5.04104	−0.213596	−0.106798	+	0.994281i	$0.534060\pi$
$558$	0	0
$559$	9.80126	0.414549
$560$	50.5887	2.13776
$561$	0	0
$562$	−25.8110	−1.08877
$563$	−4.51075	−0.190105	−0.0950527	−	0.995472i	$-0.530302\pi$
$563$	−4.51075	−0.190105	−0.0950527	+	0.995472i	$0.530302\pi$
$564$	0	0
$565$	9.04015	0.380322
$566$	23.2743	0.978291
$567$	0	0
$568$	0.997963	0.0418736
$569$	15.2375	0.638788	0.319394	−	0.947622i	$-0.396521\pi$
$569$	15.2375	0.638788	0.319394	+	0.947622i	$0.396521\pi$
$570$	0	0
$571$	21.7895	0.911862	0.455931	−	0.890015i	$-0.349306\pi$
$571$	21.7895	0.911862	0.455931	+	0.890015i	$0.349306\pi$
$572$	7.64065	0.319472
$573$	0	0
$574$	28.9667	1.20905
$575$	0.352569	0.0147031
$576$	0	0
$577$	−27.8186	−1.15810	−0.579052	−	0.815291i	$-0.696577\pi$
$577$	−27.8186	−1.15810	−0.579052	+	0.815291i	$0.696577\pi$
$578$	25.5837	1.06414
$579$	0	0
$580$	4.31130	0.179017
$581$	10.1522	0.421182
$582$	0	0
$583$	3.49949	0.144934
$584$	−0.617781	−0.0255639
$585$	0	0
$586$	26.8071	1.10739
$587$	−0.822686	−0.0339559	−0.0169779	−	0.999856i	$-0.505405\pi$
$587$	−0.822686	−0.0339559	−0.0169779	+	0.999856i	$0.505405\pi$
$588$	0	0
$589$	13.6377	0.561930
$590$	−16.6120	−0.683906
$591$	0	0
$592$	−33.8375	−1.39071
$593$	0.610153	0.0250560	0.0125280	−	0.999922i	$-0.496012\pi$
$593$	0.610153	0.0250560	0.0125280	+	0.999922i	$0.496012\pi$
$594$	0	0
$595$	−23.7346	−0.973025
$596$	7.15306	0.293001
$597$	0	0
$598$	−2.89570	−0.118414
$599$	−2.71145	−0.110787	−0.0553934	−	0.998465i	$-0.517641\pi$
$599$	−2.71145	−0.110787	−0.0553934	+	0.998465i	$0.517641\pi$
$600$	0	0
$601$	18.6069	0.758993	0.379496	−	0.925193i	$-0.376097\pi$
$601$	18.6069	0.758993	0.379496	+	0.925193i	$0.376097\pi$
$602$	−67.2115	−2.73934
$603$	0	0
$604$	−3.61856	−0.147237
$605$	7.52829	0.306068
$606$	0	0
$607$	−3.79000	−0.153832	−0.0769158	−	0.997038i	$-0.524507\pi$
$607$	−3.79000	−0.153832	−0.0769158	+	0.997038i	$0.524507\pi$
$608$	19.3435	0.784483
$609$	0	0
$610$	35.5150	1.43796
$611$	1.82038	0.0736445
$612$	0	0
$613$	−4.85555	−0.196114	−0.0980569	−	0.995181i	$-0.531263\pi$
$613$	−4.85555	−0.196114	−0.0980569	+	0.995181i	$0.531263\pi$
$614$	−44.7270	−1.80503
$615$	0	0
$616$	3.83816	0.154644
$617$	12.5673	0.505939	0.252969	−	0.967474i	$-0.418593\pi$
$617$	12.5673	0.505939	0.252969	+	0.967474i	$0.418593\pi$
$618$	0	0
$619$	−19.9816	−0.803127	−0.401563	−	0.915831i	$-0.631533\pi$
$619$	−19.9816	−0.803127	−0.401563	+	0.915831i	$0.631533\pi$
$620$	−23.7868	−0.955300
$621$	0	0
$622$	58.7357	2.35509
$623$	25.6915	1.02931
$624$	0	0
$625$	−26.6385	−1.06554
$626$	4.09245	0.163567
$627$	0	0
$628$	−37.0076	−1.47676
$629$	15.8755	0.632999
$630$	0	0
$631$	40.6792	1.61941	0.809706	−	0.586835i	$-0.199626\pi$
$631$	40.6792	1.61941	0.809706	+	0.586835i	$0.199626\pi$
$632$	2.48136	0.0987033
$633$	0	0
$634$	43.2458	1.71751
$635$	19.3414	0.767541
$636$	0	0
$637$	28.6041	1.13334
$638$	5.47053	0.216580
$639$	0	0
$640$	4.95458	0.195847
$641$	−49.2572	−1.94554	−0.972772	−	0.231766i	$-0.925550\pi$
$641$	−49.2572	−1.94554	−0.972772	+	0.231766i	$0.925550\pi$
$642$	0	0
$643$	47.3971	1.86916	0.934579	−	0.355756i	$-0.115777\pi$
$643$	47.3971	1.86916	0.934579	+	0.355756i	$0.115777\pi$
$644$	9.57774	0.377416
$645$	0	0
$646$	−9.69770	−0.381551
$647$	39.2208	1.54193	0.770964	−	0.636878i	$-0.219775\pi$
$647$	39.2208	1.54193	0.770964	+	0.636878i	$0.219775\pi$
$648$	0	0
$649$	−10.1669	−0.399087
$650$	−1.02093	−0.0400443
$651$	0	0
$652$	10.1308	0.396753
$653$	−35.5185	−1.38995	−0.694974	−	0.719035i	$-0.744584\pi$
$653$	−35.5185	−1.38995	−0.694974	+	0.719035i	$0.744584\pi$
$654$	0	0
$655$	−4.80352	−0.187689
$656$	12.1986	0.476275
$657$	0	0
$658$	−12.4831	−0.486642
$659$	41.3766	1.61180	0.805902	−	0.592049i	$-0.201681\pi$
$659$	41.3766	1.61180	0.805902	+	0.592049i	$0.201681\pi$
$660$	0	0
$661$	−17.5323	−0.681928	−0.340964	−	0.940076i	$-0.610753\pi$
$661$	−17.5323	−0.681928	−0.340964	+	0.940076i	$0.610753\pi$
$662$	−19.4853	−0.757318
$663$	0	0
$664$	0.529994	0.0205677
$665$	−29.3920	−1.13977
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−24.1758	−0.935390
$669$	0	0
$670$	−19.4297	−0.750636
$671$	21.7360	0.839109
$672$	0	0
$673$	15.5716	0.600241	0.300121	−	0.953901i	$-0.402973\pi$
$673$	15.5716	0.600241	0.300121	+	0.953901i	$0.402973\pi$
$674$	−28.7864	−1.10881
$675$	0	0
$676$	−20.1810	−0.776192
$677$	37.2526	1.43173	0.715867	−	0.698237i	$-0.246032\pi$
$677$	37.2526	1.43173	0.715867	+	0.698237i	$0.246032\pi$
$678$	0	0
$679$	−12.0956	−0.464185
$680$	−1.23907	−0.0475161
$681$	0	0
$682$	−30.1826	−1.15575
$683$	−21.5109	−0.823091	−0.411545	−	0.911389i	$-0.635011\pi$
$683$	−21.5109	−0.823091	−0.411545	+	0.911389i	$0.635011\pi$
$684$	0	0
$685$	−10.8409	−0.414210
$686$	−125.434	−4.78910
$687$	0	0
$688$	−28.3045	−1.07910
$689$	1.85237	0.0705698
$690$	0	0
$691$	37.4216	1.42358	0.711792	−	0.702390i	$-0.247884\pi$
$691$	37.4216	1.42358	0.711792	+	0.702390i	$0.247884\pi$
$692$	−24.0268	−0.913363
$693$	0	0
$694$	−20.1800	−0.766023
$695$	15.8699	0.601981
$696$	0	0
$697$	−5.72320	−0.216782
$698$	28.6471	1.08431
$699$	0	0
$700$	3.37681	0.127632
$701$	27.1943	1.02712	0.513558	−	0.858055i	$-0.328327\pi$
$701$	27.1943	1.02712	0.513558	+	0.858055i	$0.328327\pi$
$702$	0	0
$703$	19.6596	0.741477
$704$	−19.1293	−0.720962
$705$	0	0
$706$	−67.9903	−2.55885
$707$	−14.2775	−0.536959
$708$	0	0
$709$	−11.8906	−0.446561	−0.223280	−	0.974754i	$-0.571677\pi$
$709$	−11.8906	−0.446561	−0.223280	+	0.974754i	$0.571677\pi$
$710$	16.9136	0.634758
$711$	0	0
$712$	1.34122	0.0502645
$713$	5.51730	0.206625
$714$	0	0
$715$	−9.48602	−0.354757
$716$	43.9441	1.64227
$717$	0	0
$718$	23.6389	0.882194
$719$	18.7429	0.698992	0.349496	−	0.936938i	$-0.386353\pi$
$719$	18.7429	0.698992	0.349496	+	0.936938i	$0.386353\pi$
$720$	0	0
$721$	−10.9929	−0.409399
$722$	25.3367	0.942934
$723$	0	0
$724$	−46.4130	−1.72492
$725$	−0.352569	−0.0130941
$726$	0	0
$727$	11.4760	0.425621	0.212810	−	0.977094i	$-0.431738\pi$
$727$	11.4760	0.425621	0.212810	+	0.977094i	$0.431738\pi$
$728$	2.03164	0.0752977
$729$	0	0
$730$	−10.4703	−0.387522
$731$	13.2796	0.491163
$732$	0	0
$733$	15.6446	0.577847	0.288923	−	0.957352i	$-0.406703\pi$
$733$	15.6446	0.577847	0.288923	+	0.957352i	$0.406703\pi$
$734$	34.9795	1.29112
$735$	0	0
$736$	7.82568	0.288458
$737$	−11.8914	−0.438027
$738$	0	0
$739$	16.1482	0.594023	0.297011	−	0.954874i	$-0.404010\pi$
$739$	16.1482	0.594023	0.297011	+	0.954874i	$0.404010\pi$
$740$	−34.2903	−1.26054
$741$	0	0
$742$	−12.7025	−0.466325
$743$	46.3451	1.70024	0.850119	−	0.526590i	$-0.176530\pi$
$743$	46.3451	1.70024	0.850119	+	0.526590i	$0.176530\pi$
$744$	0	0
$745$	−8.88067	−0.325362
$746$	−49.7252	−1.82057
$747$	0	0
$748$	10.3522	0.378514
$749$	−2.51472	−0.0918858
$750$	0	0
$751$	−13.0617	−0.476627	−0.238314	−	0.971188i	$-0.576595\pi$
$751$	−13.0617	−0.476627	−0.238314	+	0.971188i	$0.576595\pi$
$752$	−5.25695	−0.191701
$753$	0	0
$754$	2.89570	0.105455
$755$	4.49251	0.163499
$756$	0	0
$757$	26.7849	0.973513	0.486757	−	0.873538i	$-0.338180\pi$
$757$	26.7849	0.973513	0.486757	+	0.873538i	$0.338180\pi$
$758$	−29.7655	−1.08113
$759$	0	0
$760$	−1.53441	−0.0556590
$761$	10.5087	0.380939	0.190470	−	0.981693i	$-0.438999\pi$
$761$	10.5087	0.380939	0.190470	+	0.981693i	$0.438999\pi$
$762$	0	0
$763$	26.0024	0.941350
$764$	10.3055	0.372840
$765$	0	0
$766$	−30.8212	−1.11362
$767$	−5.38164	−0.194320
$768$	0	0
$769$	9.74104	0.351271	0.175636	−	0.984455i	$-0.443802\pi$
$769$	9.74104	0.351271	0.175636	+	0.984455i	$0.443802\pi$
$770$	65.0498	2.34423
$771$	0	0
$772$	−45.3868	−1.63351
$773$	4.42680	0.159221	0.0796104	−	0.996826i	$-0.474632\pi$
$773$	4.42680	0.159221	0.0796104	+	0.996826i	$0.474632\pi$
$774$	0	0
$775$	1.94523	0.0698747
$776$	−0.631450	−0.0226677
$777$	0	0
$778$	−39.0397	−1.39964
$779$	−7.08739	−0.253932
$780$	0	0
$781$	10.3516	0.370408
$782$	−3.92334	−0.140298
$783$	0	0
$784$	−82.6041	−2.95015
$785$	45.9457	1.63987
$786$	0	0
$787$	37.4858	1.33623	0.668113	−	0.744060i	$-0.267102\pi$
$787$	37.4858	1.33623	0.668113	+	0.744060i	$0.267102\pi$
$788$	−30.0057	−1.06891
$789$	0	0
$790$	42.0546	1.49624
$791$	−20.0831	−0.714072
$792$	0	0
$793$	11.5055	0.408571
$794$	22.6146	0.802561
$795$	0	0
$796$	−26.9979	−0.956916
$797$	26.6996	0.945750	0.472875	−	0.881130i	$-0.343216\pi$
$797$	26.6996	0.945750	0.472875	+	0.881130i	$0.343216\pi$
$798$	0	0
$799$	2.46640	0.0872548
$800$	2.75909	0.0975486
$801$	0	0
$802$	35.9241	1.26853
$803$	−6.40804	−0.226135
$804$	0	0
$805$	−11.8910	−0.419101
$806$	−15.9765	−0.562747
$807$	0	0
$808$	−0.745356	−0.0262215
$809$	−55.4565	−1.94975	−0.974874	−	0.222758i	$-0.928494\pi$
$809$	−55.4565	−1.94975	−0.974874	+	0.222758i	$0.928494\pi$
$810$	0	0
$811$	−18.1663	−0.637904	−0.318952	−	0.947771i	$-0.603331\pi$
$811$	−18.1663	−0.637904	−0.318952	+	0.947771i	$0.603331\pi$
$812$	−9.57774	−0.336113
$813$	0	0
$814$	−43.5103	−1.52503
$815$	−12.5776	−0.440574
$816$	0	0
$817$	16.4449	0.575334
$818$	−13.8556	−0.484448
$819$	0	0
$820$	12.3618	0.431693
$821$	13.0282	0.454689	0.227344	−	0.973814i	$-0.426996\pi$
$821$	13.0282	0.454689	0.227344	+	0.973814i	$0.426996\pi$
$822$	0	0
$823$	−43.0752	−1.50151	−0.750753	−	0.660582i	$-0.770309\pi$
$823$	−43.0752	−1.50151	−0.750753	+	0.660582i	$0.770309\pi$
$824$	−0.573888	−0.0199923
$825$	0	0
$826$	36.9043	1.28406
$827$	−29.6621	−1.03145	−0.515727	−	0.856753i	$-0.672478\pi$
$827$	−29.6621	−1.03145	−0.515727	+	0.856753i	$0.672478\pi$
$828$	0	0
$829$	22.3408	0.775927	0.387963	−	0.921675i	$-0.373179\pi$
$829$	22.3408	0.775927	0.387963	+	0.921675i	$0.373179\pi$
$830$	8.98243	0.311785
$831$	0	0
$832$	−10.1257	−0.351044
$833$	38.7553	1.34279
$834$	0	0
$835$	30.0147	1.03870
$836$	12.8198	0.443381
$837$	0	0
$838$	−68.7922	−2.37639
$839$	41.0618	1.41761	0.708806	−	0.705403i	$-0.249234\pi$
$839$	41.0618	1.41761	0.708806	+	0.705403i	$0.249234\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	15.2674	0.526149
$843$	0	0
$844$	−27.8262	−0.957818
$845$	25.0551	0.861922
$846$	0	0
$847$	−16.7244	−0.574657
$848$	−5.34936	−0.183698
$849$	0	0
$850$	−1.38325	−0.0474450
$851$	7.95358	0.272645
$852$	0	0
$853$	19.9953	0.684626	0.342313	−	0.939586i	$-0.388790\pi$
$853$	19.9953	0.684626	0.342313	+	0.939586i	$0.388790\pi$
$854$	−78.8979	−2.69983
$855$	0	0
$856$	−0.131281	−0.00448709
$857$	−28.5925	−0.976701	−0.488350	−	0.872648i	$-0.662401\pi$
$857$	−28.5925	−0.976701	−0.488350	+	0.872648i	$0.662401\pi$
$858$	0	0
$859$	−8.81431	−0.300740	−0.150370	−	0.988630i	$-0.548047\pi$
$859$	−8.81431	−0.300740	−0.150370	+	0.988630i	$0.548047\pi$
$860$	−28.6832	−0.978088
$861$	0	0
$862$	16.7447	0.570328
$863$	45.5335	1.54998	0.774990	−	0.631974i	$-0.217755\pi$
$863$	45.5335	1.54998	0.774990	+	0.631974i	$0.217755\pi$
$864$	0	0
$865$	29.8298	1.01424
$866$	39.1837	1.33152
$867$	0	0
$868$	52.8433	1.79362
$869$	25.7384	0.873115
$870$	0	0
$871$	−6.29447	−0.213280
$872$	1.35746	0.0459693
$873$	0	0
$874$	−4.85851	−0.164342
$875$	55.2624	1.86821
$876$	0	0
$877$	−4.57380	−0.154446	−0.0772232	−	0.997014i	$-0.524605\pi$
$877$	−4.57380	−0.154446	−0.0772232	+	0.997014i	$0.524605\pi$
$878$	−9.43462	−0.318403
$879$	0	0
$880$	27.3941	0.923454
$881$	−50.1900	−1.69094	−0.845472	−	0.534019i	$-0.820681\pi$
$881$	−50.1900	−1.69094	−0.845472	+	0.534019i	$0.820681\pi$
$882$	0	0
$883$	−50.4005	−1.69611	−0.848056	−	0.529906i	$-0.822227\pi$
$883$	−50.4005	−1.69611	−0.848056	+	0.529906i	$0.822227\pi$
$884$	5.47970	0.184302
$885$	0	0
$886$	8.33988	0.280184
$887$	39.0538	1.31130	0.655649	−	0.755066i	$-0.272395\pi$
$887$	39.0538	1.31130	0.655649	+	0.755066i	$0.272395\pi$
$888$	0	0
$889$	−42.9678	−1.44109
$890$	22.7313	0.761955
$891$	0	0
$892$	48.7663	1.63282
$893$	3.05429	0.102208
$894$	0	0
$895$	−54.5575	−1.82365
$896$	−11.0068	−0.367711
$897$	0	0
$898$	46.8347	1.56289
$899$	−5.51730	−0.184012
$900$	0	0
$901$	2.50975	0.0836120
$902$	15.6857	0.522275
$903$	0	0
$904$	−1.04844	−0.0348705
$905$	57.6226	1.91544
$906$	0	0
$907$	−48.0631	−1.59591	−0.797955	−	0.602717i	$-0.794085\pi$
$907$	−48.0631	−1.59591	−0.797955	+	0.602717i	$0.794085\pi$
$908$	42.9832	1.42645
$909$	0	0
$910$	34.4326	1.14143
$911$	−25.4367	−0.842756	−0.421378	−	0.906885i	$-0.638453\pi$
$911$	−25.4367	−0.842756	−0.421378	+	0.906885i	$0.638453\pi$
$912$	0	0
$913$	5.49746	0.181939
$914$	−16.7129	−0.552813
$915$	0	0
$916$	4.65668	0.153861
$917$	10.6712	0.352395
$918$	0	0
$919$	29.6164	0.976955	0.488478	−	0.872576i	$-0.337552\pi$
$919$	29.6164	0.976955	0.488478	+	0.872576i	$0.337552\pi$
$920$	−0.620768	−0.0204661
$921$	0	0
$922$	11.6840	0.384791
$923$	5.47936	0.180355
$924$	0	0
$925$	2.80418	0.0922010
$926$	−20.4138	−0.670839
$927$	0	0
$928$	−7.82568	−0.256891
$929$	16.1346	0.529359	0.264679	−	0.964336i	$-0.414734\pi$
$929$	16.1346	0.529359	0.264679	+	0.964336i	$0.414734\pi$
$930$	0	0
$931$	47.9930	1.57291
$932$	−26.4913	−0.867752
$933$	0	0
$934$	−29.5394	−0.966558
$935$	−12.8525	−0.420320
$936$	0	0
$937$	38.2847	1.25071	0.625354	−	0.780341i	$-0.284955\pi$
$937$	38.2847	1.25071	0.625354	+	0.780341i	$0.284955\pi$
$938$	43.1639	1.40935
$939$	0	0
$940$	−5.32729	−0.173757
$941$	41.1416	1.34118	0.670589	−	0.741829i	$-0.266041\pi$
$941$	41.1416	1.34118	0.670589	+	0.741829i	$0.266041\pi$
$942$	0	0
$943$	−2.86730	−0.0933722
$944$	15.5413	0.505826
$945$	0	0
$946$	−36.3955	−1.18332
$947$	14.8093	0.481238	0.240619	−	0.970620i	$-0.422650\pi$
$947$	14.8093	0.481238	0.240619	+	0.970620i	$0.422650\pi$
$948$	0	0
$949$	−3.39195	−0.110107
$950$	−1.71296	−0.0555757
$951$	0	0
$952$	2.75264	0.0892135
$953$	−1.74853	−0.0566405	−0.0283202	−	0.999599i	$-0.509016\pi$
$953$	−1.74853	−0.0566405	−0.0283202	+	0.999599i	$0.509016\pi$
$954$	0	0
$955$	−12.7945	−0.414020
$956$	−16.3591	−0.529091
$957$	0	0
$958$	−37.4593	−1.21026
$959$	24.0835	0.777698
$960$	0	0
$961$	−0.559365	−0.0180440
$962$	−23.0312	−0.742555
$963$	0	0
$964$	21.9485	0.706915
$965$	56.3486	1.81392
$966$	0	0
$967$	7.60556	0.244578	0.122289	−	0.992495i	$-0.460976\pi$
$967$	7.60556	0.244578	0.122289	+	0.992495i	$0.460976\pi$
$968$	−0.873098	−0.0280624
$969$	0	0
$970$	−10.7019	−0.343618
$971$	33.3049	1.06881	0.534403	−	0.845230i	$-0.320537\pi$
$971$	33.3049	1.06881	0.534403	+	0.845230i	$0.320537\pi$
$972$	0	0
$973$	−35.2557	−1.13025
$974$	76.9736	2.46639
$975$	0	0
$976$	−33.2259	−1.06353
$977$	36.0257	1.15257	0.576283	−	0.817250i	$-0.304503\pi$
$977$	36.0257	1.15257	0.576283	+	0.817250i	$0.304503\pi$
$978$	0	0
$979$	13.9121	0.444632
$980$	−83.7093	−2.67400
$981$	0	0
$982$	49.2550	1.57179
$983$	−5.29775	−0.168972	−0.0844859	−	0.996425i	$-0.526925\pi$
$983$	−5.29775	−0.168972	−0.0844859	+	0.996425i	$0.526925\pi$
$984$	0	0
$985$	37.2526	1.18697
$986$	3.92334	0.124944
$987$	0	0
$988$	6.78585	0.215887
$989$	6.65301	0.211553
$990$	0	0
$991$	−52.1180	−1.65558	−0.827791	−	0.561037i	$-0.810403\pi$
$991$	−52.1180	−1.65558	−0.827791	+	0.561037i	$0.810403\pi$
$992$	43.1766	1.37086
$993$	0	0
$994$	−37.5744	−1.19179
$995$	33.5185	1.06261
$996$	0	0
$997$	−4.31530	−0.136667	−0.0683335	−	0.997663i	$-0.521768\pi$
$997$	−4.31530	−0.136667	−0.0683335	+	0.997663i	$0.521768\pi$
$998$	−47.3156	−1.49775
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.v.1.7	✓	30
3.2	odd	2		6003.2.a.w.1.24	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.7	✓	30	1.1	even	1	trivial
6003.2.a.w.1.24	yes	30	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-1.96558 q^{2} +1.86349 q^{4} -2.31356 q^{5} +5.13967 q^{7} +0.268317 q^{8} +O(q^{10})\)	\(q-1.96558 q^{2} +1.86349 q^{4} -2.31356 q^{5} +5.13967 q^{7} +0.268317 q^{8} +4.54748 q^{10} +2.78317 q^{11} +1.47321 q^{13} -10.1024 q^{14} -4.25438 q^{16} +1.99602 q^{17} +2.47180 q^{19} -4.31130 q^{20} -5.47053 q^{22} +1.00000 q^{23} +0.352569 q^{25} -2.89570 q^{26} +9.57774 q^{28} -1.00000 q^{29} +5.51730 q^{31} +7.82568 q^{32} -3.92334 q^{34} -11.8910 q^{35} +7.95358 q^{37} -4.85851 q^{38} -0.620768 q^{40} -2.86730 q^{41} +6.65301 q^{43} +5.18641 q^{44} -1.96558 q^{46} +1.23566 q^{47} +19.4162 q^{49} -0.693001 q^{50} +2.74531 q^{52} +1.25738 q^{53} -6.43903 q^{55} +1.37906 q^{56} +1.96558 q^{58} -3.65301 q^{59} +7.80980 q^{61} -10.8447 q^{62} -6.87321 q^{64} -3.40835 q^{65} -4.27263 q^{67} +3.71957 q^{68} +23.3726 q^{70} +3.71934 q^{71} -2.30243 q^{73} -15.6334 q^{74} +4.60618 q^{76} +14.3046 q^{77} +9.24788 q^{79} +9.84278 q^{80} +5.63590 q^{82} +1.97525 q^{83} -4.61792 q^{85} -13.0770 q^{86} +0.746771 q^{88} +4.99866 q^{89} +7.57180 q^{91} +1.86349 q^{92} -2.42878 q^{94} -5.71866 q^{95} -2.35337 q^{97} -38.1641 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) \) 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8} + 8 q^{10} + 36 q^{13} - 7 q^{14} + 47 q^{16} - 18 q^{17} + 16 q^{19} + 25 q^{22} + 30 q^{23} + 56 q^{25} - 11 q^{26} + 27 q^{28} - 30 q^{29} + 14 q^{31} + 7 q^{32} + 3 q^{34} + 22 q^{35} + 40 q^{37} - 6 q^{38} + 30 q^{40} - 14 q^{41} + 34 q^{43} - 5 q^{44} - q^{46} + 2 q^{47} + 74 q^{49} + 21 q^{50} + 71 q^{52} - 16 q^{53} + 22 q^{55} - 14 q^{56} + q^{58} + 32 q^{59} + 46 q^{61} - 20 q^{62} + 68 q^{64} - 12 q^{65} + 14 q^{67} - 27 q^{68} + 32 q^{71} + 50 q^{73} + 26 q^{74} + 56 q^{76} - 34 q^{77} + 16 q^{79} - 2 q^{80} + 38 q^{82} + 14 q^{83} + 38 q^{85} - 10 q^{86} + 40 q^{88} + 2 q^{89} + 32 q^{91} + 37 q^{92} + 29 q^{94} + 28 q^{95} + 56 q^{97} - 8 q^{98}+O(q^{100}) \) 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8 + 8 * q^10 + 36 * q^13 - 7 * q^14 + 47 * q^16 - 18 * q^17 + 16 * q^19 + 25 * q^22 + 30 * q^23 + 56 * q^25 - 11 * q^26 + 27 * q^28 - 30 * q^29 + 14 * q^31 + 7 * q^32 + 3 * q^34 + 22 * q^35 + 40 * q^37 - 6 * q^38 + 30 * q^40 - 14 * q^41 + 34 * q^43 - 5 * q^44 - q^46 + 2 * q^47 + 74 * q^49 + 21 * q^50 + 71 * q^52 - 16 * q^53 + 22 * q^55 - 14 * q^56 + q^58 + 32 * q^59 + 46 * q^61 - 20 * q^62 + 68 * q^64 - 12 * q^65 + 14 * q^67 - 27 * q^68 + 32 * q^71 + 50 * q^73 + 26 * q^74 + 56 * q^76 - 34 * q^77 + 16 * q^79 - 2 * q^80 + 38 * q^82 + 14 * q^83 + 38 * q^85 - 10 * q^86 + 40 * q^88 + 2 * q^89 + 32 * q^91 + 37 * q^92 + 29 * q^94 + 28 * q^95 + 56 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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