LMFDB - Embedded newform 6003.2.a.w.1.15

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			1.15
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)

$f(q)$	$=$	$q-0.395940 q^{2} -1.84323 q^{4} +0.579308 q^{5} -2.59556 q^{7} +1.52169 q^{8} +O(q^{10})$	\(q-0.395940 q^{2} -1.84323 q^{4} +0.579308 q^{5} -2.59556 q^{7} +1.52169 q^{8} -0.229371 q^{10} -1.87640 q^{11} -6.20966 q^{13} +1.02769 q^{14} +3.08396 q^{16} -4.01505 q^{17} -2.79036 q^{19} -1.06780 q^{20} +0.742943 q^{22} -1.00000 q^{23} -4.66440 q^{25} +2.45865 q^{26} +4.78422 q^{28} +1.00000 q^{29} -3.08724 q^{31} -4.26445 q^{32} +1.58972 q^{34} -1.50363 q^{35} -3.22901 q^{37} +1.10481 q^{38} +0.881527 q^{40} +8.40860 q^{41} -6.56696 q^{43} +3.45864 q^{44} +0.395940 q^{46} +0.0783194 q^{47} -0.263051 q^{49} +1.84683 q^{50} +11.4458 q^{52} -9.39482 q^{53} -1.08701 q^{55} -3.94964 q^{56} -0.395940 q^{58} +7.94475 q^{59} -9.00728 q^{61} +1.22236 q^{62} -4.47946 q^{64} -3.59730 q^{65} +0.596834 q^{67} +7.40067 q^{68} +0.595348 q^{70} -4.73876 q^{71} +0.703822 q^{73} +1.27850 q^{74} +5.14327 q^{76} +4.87032 q^{77} +2.70957 q^{79} +1.78656 q^{80} -3.32931 q^{82} +9.68330 q^{83} -2.32595 q^{85} +2.60012 q^{86} -2.85530 q^{88} -9.96833 q^{89} +16.1176 q^{91} +1.84323 q^{92} -0.0310098 q^{94} -1.61648 q^{95} +12.0713 q^{97} +0.104153 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.395940	−0.279972	−0.139986	−	0.990153i	$-0.544706\pi$
$2$	−0.395940	−0.279972	−0.139986	+	0.990153i	$0.544706\pi$
$3$	0	0
$3$	0	0
$4$	−1.84323	−0.921616
$5$	0.579308	0.259074	0.129537	−	0.991575i	$-0.458651\pi$
$5$	0.579308	0.259074	0.129537	+	0.991575i	$0.458651\pi$
$6$	0	0
$7$	−2.59556	−0.981031	−0.490515	−	0.871433i	$-0.663191\pi$
$7$	−2.59556	−0.981031	−0.490515	+	0.871433i	$0.663191\pi$
$8$	1.52169	0.537999
$9$	0	0
$10$	−0.229371	−0.0725336
$11$	−1.87640	−0.565756	−0.282878	−	0.959156i	$-0.591289\pi$
$11$	−1.87640	−0.565756	−0.282878	+	0.959156i	$0.591289\pi$
$12$	0	0
$13$	−6.20966	−1.72225	−0.861124	−	0.508394i	$-0.830239\pi$
$13$	−6.20966	−1.72225	−0.861124	+	0.508394i	$0.830239\pi$
$14$	1.02769	0.274661
$15$	0	0
$16$	3.08396	0.770991
$17$	−4.01505	−0.973794	−0.486897	−	0.873459i	$-0.661871\pi$
$17$	−4.01505	−0.973794	−0.486897	+	0.873459i	$0.661871\pi$
$18$	0	0
$19$	−2.79036	−0.640152	−0.320076	−	0.947392i	$-0.603708\pi$
$19$	−2.79036	−0.640152	−0.320076	+	0.947392i	$0.603708\pi$
$20$	−1.06780	−0.238767
$21$	0	0
$22$	0.742943	0.158396
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.66440	−0.932880
$26$	2.45865	0.482182
$27$	0	0
$28$	4.78422	0.904133
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−3.08724	−0.554485	−0.277242	−	0.960800i	$-0.589420\pi$
$31$	−3.08724	−0.554485	−0.277242	+	0.960800i	$0.589420\pi$
$32$	−4.26445	−0.753855
$33$	0	0
$34$	1.58972	0.272635
$35$	−1.50363	−0.254160
$36$	0	0
$37$	−3.22901	−0.530846	−0.265423	−	0.964132i	$-0.585512\pi$
$37$	−3.22901	−0.530846	−0.265423	+	0.964132i	$0.585512\pi$
$38$	1.10481	0.179225
$39$	0	0
$40$	0.881527	0.139382
$41$	8.40860	1.31320	0.656602	−	0.754238i	$-0.271993\pi$
$41$	8.40860	1.31320	0.656602	+	0.754238i	$0.271993\pi$
$42$	0	0
$43$	−6.56696	−1.00145	−0.500726	−	0.865606i	$-0.666934\pi$
$43$	−6.56696	−1.00145	−0.500726	+	0.865606i	$0.666934\pi$
$44$	3.45864	0.521410
$45$	0	0
$46$	0.395940	0.0583782
$47$	0.0783194	0.0114241	0.00571203	−	0.999984i	$-0.498182\pi$
$47$	0.0783194	0.0114241	0.00571203	+	0.999984i	$0.498182\pi$
$48$	0	0
$49$	−0.263051	−0.0375788
$50$	1.84683	0.261181
$51$	0	0
$52$	11.4458	1.58725
$53$	−9.39482	−1.29048	−0.645239	−	0.763981i	$-0.723242\pi$
$53$	−9.39482	−1.29048	−0.645239	+	0.763981i	$0.723242\pi$
$54$	0	0
$55$	−1.08701	−0.146573
$56$	−3.94964	−0.527793
$57$	0	0
$58$	−0.395940	−0.0519895
$59$	7.94475	1.03432	0.517159	−	0.855889i	$-0.326989\pi$
$59$	7.94475	1.03432	0.517159	+	0.855889i	$0.326989\pi$
$60$	0	0
$61$	−9.00728	−1.15326	−0.576632	−	0.817004i	$-0.695633\pi$
$61$	−9.00728	−1.15326	−0.576632	+	0.817004i	$0.695633\pi$
$62$	1.22236	0.155240
$63$	0	0
$64$	−4.47946	−0.559933
$65$	−3.59730	−0.446190
$66$	0	0
$67$	0.596834	0.0729149	0.0364574	−	0.999335i	$-0.488393\pi$
$67$	0.596834	0.0729149	0.0364574	+	0.999335i	$0.488393\pi$
$68$	7.40067	0.897463
$69$	0	0
$70$	0.595348	0.0711577
$71$	−4.73876	−0.562387	−0.281194	−	0.959651i	$-0.590730\pi$
$71$	−4.73876	−0.562387	−0.281194	+	0.959651i	$0.590730\pi$
$72$	0	0
$73$	0.703822	0.0823762	0.0411881	−	0.999151i	$-0.486886\pi$
$73$	0.703822	0.0823762	0.0411881	+	0.999151i	$0.486886\pi$
$74$	1.27850	0.148622
$75$	0	0
$76$	5.14327	0.589974
$77$	4.87032	0.555024
$78$	0	0
$79$	2.70957	0.304851	0.152425	−	0.988315i	$-0.451292\pi$
$79$	2.70957	0.304851	0.152425	+	0.988315i	$0.451292\pi$
$80$	1.78656	0.199744
$81$	0	0
$82$	−3.32931	−0.367660
$83$	9.68330	1.06288	0.531440	−	0.847096i	$-0.321651\pi$
$83$	9.68330	1.06288	0.531440	+	0.847096i	$0.321651\pi$
$84$	0	0
$85$	−2.32595	−0.252285
$86$	2.60012	0.280379
$87$	0	0
$88$	−2.85530	−0.304376
$89$	−9.96833	−1.05664	−0.528321	−	0.849045i	$-0.677178\pi$
$89$	−9.96833	−1.05664	−0.528321	+	0.849045i	$0.677178\pi$
$90$	0	0
$91$	16.1176	1.68958
$92$	1.84323	0.192170
$93$	0	0
$94$	−0.0310098	−0.00319842
$95$	−1.61648	−0.165847
$96$	0	0
$97$	12.0713	1.22566	0.612829	−	0.790215i	$-0.290031\pi$
$97$	12.0713	1.22566	0.612829	+	0.790215i	$0.290031\pi$
$98$	0.104153	0.0105210
$99$	0	0
$100$	8.59757	0.859757
$101$	1.15152	0.114581	0.0572903	−	0.998358i	$-0.481754\pi$
$101$	1.15152	0.114581	0.0572903	+	0.998358i	$0.481754\pi$
$102$	0	0
$103$	8.06395	0.794565	0.397282	−	0.917696i	$-0.369953\pi$
$103$	8.06395	0.794565	0.397282	+	0.917696i	$0.369953\pi$
$104$	−9.44917	−0.926568
$105$	0	0
$106$	3.71979	0.361298
$107$	−6.95139	−0.672016	−0.336008	−	0.941859i	$-0.609077\pi$
$107$	−6.95139	−0.672016	−0.336008	+	0.941859i	$0.609077\pi$
$108$	0	0
$109$	8.35459	0.800225	0.400112	−	0.916466i	$-0.368971\pi$
$109$	8.35459	0.800225	0.400112	+	0.916466i	$0.368971\pi$
$110$	0.430393	0.0410363
$111$	0	0
$112$	−8.00462	−0.756366
$113$	−3.30753	−0.311146	−0.155573	−	0.987824i	$-0.549722\pi$
$113$	−3.30753	−0.311146	−0.155573	+	0.987824i	$0.549722\pi$
$114$	0	0
$115$	−0.579308	−0.0540207
$116$	−1.84323	−0.171140
$117$	0	0
$118$	−3.14565	−0.289580
$119$	10.4213	0.955321
$120$	0	0
$121$	−7.47912	−0.679920
$122$	3.56635	0.322882
$123$	0	0
$124$	5.69050	0.511022
$125$	−5.59866	−0.500760
$126$	0	0
$127$	1.82193	0.161670	0.0808350	−	0.996727i	$-0.474241\pi$
$127$	1.82193	0.161670	0.0808350	+	0.996727i	$0.474241\pi$
$128$	10.3025	0.910620
$129$	0	0
$130$	1.42432	0.124921
$131$	−19.4572	−1.69999	−0.849993	−	0.526794i	$-0.823394\pi$
$131$	−19.4572	−1.69999	−0.849993	+	0.526794i	$0.823394\pi$
$132$	0	0
$133$	7.24255	0.628008
$134$	−0.236311	−0.0204141
$135$	0	0
$136$	−6.10967	−0.523900
$137$	15.9089	1.35919	0.679593	−	0.733589i	$-0.262156\pi$
$137$	15.9089	1.35919	0.679593	+	0.733589i	$0.262156\pi$
$138$	0	0
$139$	−7.26742	−0.616414	−0.308207	−	0.951319i	$-0.599729\pi$
$139$	−7.26742	−0.616414	−0.308207	+	0.951319i	$0.599729\pi$
$140$	2.77154	0.234238
$141$	0	0
$142$	1.87627	0.157453
$143$	11.6518	0.974373
$144$	0	0
$145$	0.579308	0.0481089
$146$	−0.278672	−0.0230630
$147$	0	0
$148$	5.95181	0.489236
$149$	−19.9493	−1.63431	−0.817155	−	0.576418i	$-0.804450\pi$
$149$	−19.9493	−1.63431	−0.817155	+	0.576418i	$0.804450\pi$
$150$	0	0
$151$	−11.9350	−0.971257	−0.485629	−	0.874165i	$-0.661409\pi$
$151$	−11.9350	−0.971257	−0.485629	+	0.874165i	$0.661409\pi$
$152$	−4.24606	−0.344401
$153$	0	0
$154$	−1.92836	−0.155391
$155$	−1.78846	−0.143653
$156$	0	0
$157$	8.42143	0.672103	0.336051	−	0.941844i	$-0.390908\pi$
$157$	8.42143	0.672103	0.336051	+	0.941844i	$0.390908\pi$
$158$	−1.07283	−0.0853497
$159$	0	0
$160$	−2.47043	−0.195304
$161$	2.59556	0.204559
$162$	0	0
$163$	23.6060	1.84896	0.924482	−	0.381225i	$-0.124498\pi$
$163$	23.6060	1.84896	0.924482	+	0.381225i	$0.124498\pi$
$164$	−15.4990	−1.21027
$165$	0	0
$166$	−3.83401	−0.297577
$167$	−18.5080	−1.43219	−0.716095	−	0.698002i	$-0.754072\pi$
$167$	−18.5080	−1.43219	−0.716095	+	0.698002i	$0.754072\pi$
$168$	0	0
$169$	25.5598	1.96614
$170$	0.920938	0.0706327
$171$	0	0
$172$	12.1044	0.922954
$173$	−9.70767	−0.738060	−0.369030	−	0.929417i	$-0.620310\pi$
$173$	−9.70767	−0.738060	−0.369030	+	0.929417i	$0.620310\pi$
$174$	0	0
$175$	12.1068	0.915184
$176$	−5.78676	−0.436193
$177$	0	0
$178$	3.94687	0.295830
$179$	−1.12703	−0.0842380	−0.0421190	−	0.999113i	$-0.513411\pi$
$179$	−1.12703	−0.0842380	−0.0421190	+	0.999113i	$0.513411\pi$
$180$	0	0
$181$	21.6177	1.60683	0.803414	−	0.595420i	$-0.203014\pi$
$181$	21.6177	1.60683	0.803414	+	0.595420i	$0.203014\pi$
$182$	−6.38159	−0.473035
$183$	0	0
$184$	−1.52169	−0.112180
$185$	−1.87059	−0.137529
$186$	0	0
$187$	7.53385	0.550930
$188$	−0.144361	−0.0105286
$189$	0	0
$190$	0.640028	0.0464325
$191$	2.62732	0.190106	0.0950529	−	0.995472i	$-0.469698\pi$
$191$	2.62732	0.190106	0.0950529	+	0.995472i	$0.469698\pi$
$192$	0	0
$193$	−21.5354	−1.55015	−0.775076	−	0.631868i	$-0.782288\pi$
$193$	−21.5354	−1.55015	−0.775076	+	0.631868i	$0.782288\pi$
$194$	−4.77953	−0.343150
$195$	0	0
$196$	0.484865	0.0346332
$197$	19.5960	1.39616	0.698080	−	0.716019i	$-0.254038\pi$
$197$	19.5960	1.39616	0.698080	+	0.716019i	$0.254038\pi$
$198$	0	0
$199$	18.6926	1.32508	0.662541	−	0.749026i	$-0.269478\pi$
$199$	18.6926	1.32508	0.662541	+	0.749026i	$0.269478\pi$
$200$	−7.09778	−0.501889
$201$	0	0
$202$	−0.455933	−0.0320794
$203$	−2.59556	−0.182173
$204$	0	0
$205$	4.87117	0.340217
$206$	−3.19284	−0.222456
$207$	0	0
$208$	−19.1504	−1.32784
$209$	5.23583	0.362170
$210$	0	0
$211$	−11.2932	−0.777456	−0.388728	−	0.921353i	$-0.627085\pi$
$211$	−11.2932	−0.777456	−0.388728	+	0.921353i	$0.627085\pi$
$212$	17.3168	1.18932
$213$	0	0
$214$	2.75233	0.188146
$215$	−3.80429	−0.259451
$216$	0	0
$217$	8.01312	0.543966
$218$	−3.30792	−0.224041
$219$	0	0
$220$	2.00362	0.135084
$221$	24.9321	1.67711
$222$	0	0
$223$	−2.83799	−0.190046	−0.0950228	−	0.995475i	$-0.530292\pi$
$223$	−2.83799	−0.190046	−0.0950228	+	0.995475i	$0.530292\pi$
$224$	11.0686	0.739555
$225$	0	0
$226$	1.30958	0.0871121
$227$	−4.57810	−0.303859	−0.151930	−	0.988391i	$-0.548549\pi$
$227$	−4.57810	−0.303859	−0.151930	+	0.988391i	$0.548549\pi$
$228$	0	0
$229$	−3.97141	−0.262438	−0.131219	−	0.991353i	$-0.541889\pi$
$229$	−3.97141	−0.262438	−0.131219	+	0.991353i	$0.541889\pi$
$230$	0.229371	0.0151243
$231$	0	0
$232$	1.52169	0.0999039
$233$	15.7838	1.03403	0.517017	−	0.855975i	$-0.327042\pi$
$233$	15.7838	1.03403	0.517017	+	0.855975i	$0.327042\pi$
$234$	0	0
$235$	0.0453711	0.00295968
$236$	−14.6440	−0.953244
$237$	0	0
$238$	−4.12622	−0.267463
$239$	−9.82510	−0.635533	−0.317766	−	0.948169i	$-0.602933\pi$
$239$	−9.82510	−0.635533	−0.317766	+	0.948169i	$0.602933\pi$
$240$	0	0
$241$	−11.2125	−0.722262	−0.361131	−	0.932515i	$-0.617609\pi$
$241$	−11.2125	−0.722262	−0.361131	+	0.932515i	$0.617609\pi$
$242$	2.96128	0.190359
$243$	0	0
$244$	16.6025	1.06287
$245$	−0.152388	−0.00973570
$246$	0	0
$247$	17.3272	1.10250
$248$	−4.69782	−0.298312
$249$	0	0
$250$	2.21674	0.140199
$251$	−12.8806	−0.813016	−0.406508	−	0.913647i	$-0.633254\pi$
$251$	−12.8806	−0.813016	−0.406508	+	0.913647i	$0.633254\pi$
$252$	0	0
$253$	1.87640	0.117968
$254$	−0.721375	−0.0452631
$255$	0	0
$256$	4.87975	0.304984
$257$	−29.2876	−1.82691	−0.913456	−	0.406937i	$-0.866597\pi$
$257$	−29.2876	−1.82691	−0.913456	+	0.406937i	$0.866597\pi$
$258$	0	0
$259$	8.38110	0.520776
$260$	6.63066	0.411216
$261$	0	0
$262$	7.70390	0.475949
$263$	22.3171	1.37613	0.688066	−	0.725648i	$-0.258460\pi$
$263$	22.3171	1.37613	0.688066	+	0.725648i	$0.258460\pi$
$264$	0	0
$265$	−5.44249	−0.334330
$266$	−2.86762	−0.175825
$267$	0	0
$268$	−1.10010	−0.0671995
$269$	0.821814	0.0501069	0.0250534	−	0.999686i	$-0.492024\pi$
$269$	0.821814	0.0501069	0.0250534	+	0.999686i	$0.492024\pi$
$270$	0	0
$271$	−0.477020	−0.0289769	−0.0144884	−	0.999895i	$-0.504612\pi$
$271$	−0.477020	−0.0289769	−0.0144884	+	0.999895i	$0.504612\pi$
$272$	−12.3823	−0.750786
$273$	0	0
$274$	−6.29896	−0.380534
$275$	8.75229	0.527783
$276$	0	0
$277$	26.0380	1.56447	0.782235	−	0.622983i	$-0.214080\pi$
$277$	26.0380	1.56447	0.782235	+	0.622983i	$0.214080\pi$
$278$	2.87746	0.172579
$279$	0	0
$280$	−2.28806	−0.136738
$281$	−23.0168	−1.37307	−0.686534	−	0.727098i	$-0.740869\pi$
$281$	−23.0168	−1.37307	−0.686534	+	0.727098i	$0.740869\pi$
$282$	0	0
$283$	−1.75884	−0.104552	−0.0522761	−	0.998633i	$-0.516648\pi$
$283$	−1.75884	−0.104552	−0.0522761	+	0.998633i	$0.516648\pi$
$284$	8.73463	0.518305
$285$	0	0
$286$	−4.61342	−0.272797
$287$	−21.8251	−1.28829
$288$	0	0
$289$	−0.879340	−0.0517259
$290$	−0.229371	−0.0134691
$291$	0	0
$292$	−1.29731	−0.0759192
$293$	23.1523	1.35257	0.676285	−	0.736640i	$-0.263589\pi$
$293$	23.1523	1.35257	0.676285	+	0.736640i	$0.263589\pi$
$294$	0	0
$295$	4.60246	0.267965
$296$	−4.91355	−0.285594
$297$	0	0
$298$	7.89873	0.457561
$299$	6.20966	0.359114
$300$	0	0
$301$	17.0450	0.982455
$302$	4.72555	0.271925
$303$	0	0
$304$	−8.60536	−0.493551
$305$	−5.21799	−0.298781
$306$	0	0
$307$	−21.9150	−1.25076	−0.625379	−	0.780321i	$-0.715056\pi$
$307$	−21.9150	−1.25076	−0.625379	+	0.780321i	$0.715056\pi$
$308$	−8.97713	−0.511519
$309$	0	0
$310$	0.708124	0.0402187
$311$	−16.2132	−0.919364	−0.459682	−	0.888084i	$-0.652037\pi$
$311$	−16.2132	−0.919364	−0.459682	+	0.888084i	$0.652037\pi$
$312$	0	0
$313$	12.2318	0.691380	0.345690	−	0.938349i	$-0.387645\pi$
$313$	12.2318	0.691380	0.345690	+	0.938349i	$0.387645\pi$
$314$	−3.33438	−0.188170
$315$	0	0
$316$	−4.99437	−0.280955
$317$	33.9140	1.90480	0.952401	−	0.304848i	$-0.0986057\pi$
$317$	33.9140	1.90480	0.952401	+	0.304848i	$0.0986057\pi$
$318$	0	0
$319$	−1.87640	−0.105058
$320$	−2.59499	−0.145064
$321$	0	0
$322$	−1.02769	−0.0572708
$323$	11.2034	0.623376
$324$	0	0
$325$	28.9643	1.60665
$326$	−9.34657	−0.517659
$327$	0	0
$328$	12.7953	0.706502
$329$	−0.203283	−0.0112074
$330$	0	0
$331$	8.36860	0.459980	0.229990	−	0.973193i	$-0.426131\pi$
$331$	8.36860	0.459980	0.229990	+	0.973193i	$0.426131\pi$
$332$	−17.8486	−0.979567
$333$	0	0
$334$	7.32806	0.400973
$335$	0.345751	0.0188904
$336$	0	0
$337$	−1.51180	−0.0823532	−0.0411766	−	0.999152i	$-0.513111\pi$
$337$	−1.51180	−0.0823532	−0.0411766	+	0.999152i	$0.513111\pi$
$338$	−10.1202	−0.550465
$339$	0	0
$340$	4.28727	0.232510
$341$	5.79290	0.313703
$342$	0	0
$343$	18.8517	1.01790
$344$	−9.99288	−0.538780
$345$	0	0
$346$	3.84366	0.206636
$347$	32.8768	1.76492	0.882459	−	0.470390i	$-0.155887\pi$
$347$	32.8768	1.76492	0.882459	+	0.470390i	$0.155887\pi$
$348$	0	0
$349$	11.4424	0.612500	0.306250	−	0.951951i	$-0.400926\pi$
$349$	11.4424	0.612500	0.306250	+	0.951951i	$0.400926\pi$
$350$	−4.79355	−0.256226
$351$	0	0
$352$	8.00181	0.426498
$353$	19.9449	1.06156	0.530779	−	0.847510i	$-0.321899\pi$
$353$	19.9449	1.06156	0.530779	+	0.847510i	$0.321899\pi$
$354$	0	0
$355$	−2.74520	−0.145700
$356$	18.3739	0.973817
$357$	0	0
$358$	0.446236	0.0235843
$359$	−32.3232	−1.70595	−0.852977	−	0.521949i	$-0.825205\pi$
$359$	−32.3232	−1.70595	−0.852977	+	0.521949i	$0.825205\pi$
$360$	0	0
$361$	−11.2139	−0.590206
$362$	−8.55931	−0.449867
$363$	0	0
$364$	−29.7084	−1.55714
$365$	0.407730	0.0213416
$366$	0	0
$367$	2.91262	0.152038	0.0760188	−	0.997106i	$-0.475779\pi$
$367$	2.91262	0.152038	0.0760188	+	0.997106i	$0.475779\pi$
$368$	−3.08396	−0.160763
$369$	0	0
$370$	0.740642	0.0385042
$371$	24.3848	1.26600
$372$	0	0
$373$	13.6742	0.708023	0.354011	−	0.935241i	$-0.384817\pi$
$373$	13.6742	0.708023	0.354011	+	0.935241i	$0.384817\pi$
$374$	−2.98296	−0.154245
$375$	0	0
$376$	0.119178	0.00614613
$377$	−6.20966	−0.319814
$378$	0	0
$379$	20.5159	1.05383	0.526916	−	0.849917i	$-0.323348\pi$
$379$	20.5159	1.05383	0.526916	+	0.849917i	$0.323348\pi$
$380$	2.97954	0.152847
$381$	0	0
$382$	−1.04026	−0.0532243
$383$	34.4819	1.76194	0.880971	−	0.473170i	$-0.156891\pi$
$383$	34.4819	1.76194	0.880971	+	0.473170i	$0.156891\pi$
$384$	0	0
$385$	2.82141	0.143793
$386$	8.52673	0.433999
$387$	0	0
$388$	−22.2503	−1.12959
$389$	21.1458	1.07214	0.536068	−	0.844175i	$-0.319909\pi$
$389$	21.1458	1.07214	0.536068	+	0.844175i	$0.319909\pi$
$390$	0	0
$391$	4.01505	0.203050
$392$	−0.400283	−0.0202173
$393$	0	0
$394$	−7.75887	−0.390886
$395$	1.56968	0.0789790
$396$	0	0
$397$	36.7136	1.84260	0.921301	−	0.388851i	$-0.127128\pi$
$397$	36.7136	1.84260	0.921301	+	0.388851i	$0.127128\pi$
$398$	−7.40114	−0.370986
$399$	0	0
$400$	−14.3848	−0.719242
$401$	−24.2930	−1.21314	−0.606568	−	0.795031i	$-0.707454\pi$
$401$	−24.2930	−1.21314	−0.606568	+	0.795031i	$0.707454\pi$
$402$	0	0
$403$	19.1707	0.954960
$404$	−2.12252	−0.105599
$405$	0	0
$406$	1.02769	0.0510033
$407$	6.05892	0.300330
$408$	0	0
$409$	−19.5923	−0.968776	−0.484388	−	0.874853i	$-0.660958\pi$
$409$	−19.5923	−0.968776	−0.484388	+	0.874853i	$0.660958\pi$
$410$	−1.92869	−0.0952513
$411$	0	0
$412$	−14.8637	−0.732283
$413$	−20.6211	−1.01470
$414$	0	0
$415$	5.60961	0.275365
$416$	26.4807	1.29833
$417$	0	0
$418$	−2.07308	−0.101397
$419$	−26.2797	−1.28385	−0.641923	−	0.766769i	$-0.721863\pi$
$419$	−26.2797	−1.28385	−0.641923	+	0.766769i	$0.721863\pi$
$420$	0	0
$421$	9.32333	0.454391	0.227196	−	0.973849i	$-0.427044\pi$
$421$	9.32333	0.454391	0.227196	+	0.973849i	$0.427044\pi$
$422$	4.47143	0.217666
$423$	0	0
$424$	−14.2960	−0.694275
$425$	18.7278	0.908433
$426$	0	0
$427$	23.3790	1.13139
$428$	12.8130	0.619340
$429$	0	0
$430$	1.50627	0.0726389
$431$	16.4763	0.793636	0.396818	−	0.917897i	$-0.370114\pi$
$431$	16.4763	0.793636	0.396818	+	0.917897i	$0.370114\pi$
$432$	0	0
$433$	3.86255	0.185622	0.0928111	−	0.995684i	$-0.470415\pi$
$433$	3.86255	0.185622	0.0928111	+	0.995684i	$0.470415\pi$
$434$	−3.17272	−0.152295
$435$	0	0
$436$	−15.3994	−0.737499
$437$	2.79036	0.133481
$438$	0	0
$439$	−8.55822	−0.408462	−0.204231	−	0.978923i	$-0.565469\pi$
$439$	−8.55822	−0.408462	−0.204231	+	0.978923i	$0.565469\pi$
$440$	−1.65410	−0.0788561
$441$	0	0
$442$	−9.87163	−0.469545
$443$	6.08169	0.288950	0.144475	−	0.989508i	$-0.453851\pi$
$443$	6.08169	0.288950	0.144475	+	0.989508i	$0.453851\pi$
$444$	0	0
$445$	−5.77473	−0.273749
$446$	1.12367	0.0532075
$447$	0	0
$448$	11.6267	0.549311
$449$	14.6440	0.691092	0.345546	−	0.938402i	$-0.387694\pi$
$449$	14.6440	0.691092	0.345546	+	0.938402i	$0.387694\pi$
$450$	0	0
$451$	−15.7779	−0.742953
$452$	6.09653	0.286757
$453$	0	0
$454$	1.81266	0.0850721
$455$	9.33703	0.437727
$456$	0	0
$457$	24.8053	1.16034	0.580171	−	0.814495i	$-0.302986\pi$
$457$	24.8053	1.16034	0.580171	+	0.814495i	$0.302986\pi$
$458$	1.57244	0.0734753
$459$	0	0
$460$	1.06780	0.0497864
$461$	−32.1947	−1.49946	−0.749728	−	0.661747i	$-0.769815\pi$
$461$	−32.1947	−1.49946	−0.749728	+	0.661747i	$0.769815\pi$
$462$	0	0
$463$	−16.6435	−0.773488	−0.386744	−	0.922187i	$-0.626400\pi$
$463$	−16.6435	−0.773488	−0.386744	+	0.922187i	$0.626400\pi$
$464$	3.08396	0.143169
$465$	0	0
$466$	−6.24946	−0.289501
$467$	−28.4988	−1.31877	−0.659383	−	0.751807i	$-0.729182\pi$
$467$	−28.4988	−1.31877	−0.659383	+	0.751807i	$0.729182\pi$
$468$	0	0
$469$	−1.54912	−0.0715318
$470$	−0.0179642	−0.000828628 0
$471$	0	0
$472$	12.0895	0.556462
$473$	12.3223	0.566578
$474$	0	0
$475$	13.0153	0.597185
$476$	−19.2089	−0.880439
$477$	0	0
$478$	3.89015	0.177931
$479$	32.6240	1.49063	0.745314	−	0.666714i	$-0.232300\pi$
$479$	32.6240	1.49063	0.745314	+	0.666714i	$0.232300\pi$
$480$	0	0
$481$	20.0510	0.914249
$482$	4.43949	0.202213
$483$	0	0
$484$	13.7857	0.626625
$485$	6.99302	0.317537
$486$	0	0
$487$	8.93284	0.404786	0.202393	−	0.979304i	$-0.435128\pi$
$487$	8.93284	0.404786	0.202393	+	0.979304i	$0.435128\pi$
$488$	−13.7063	−0.620455
$489$	0	0
$490$	0.0603365	0.00272572
$491$	−5.36028	−0.241906	−0.120953	−	0.992658i	$-0.538595\pi$
$491$	−5.36028	−0.241906	−0.120953	+	0.992658i	$0.538595\pi$
$492$	0	0
$493$	−4.01505	−0.180829
$494$	−6.86052	−0.308669
$495$	0	0
$496$	−9.52093	−0.427503
$497$	12.2997	0.551719
$498$	0	0
$499$	−5.44297	−0.243661	−0.121830	−	0.992551i	$-0.538876\pi$
$499$	−5.44297	−0.243661	−0.121830	+	0.992551i	$0.538876\pi$
$500$	10.3196	0.461508
$501$	0	0
$502$	5.09995	0.227622
$503$	−13.8756	−0.618683	−0.309341	−	0.950951i	$-0.600109\pi$
$503$	−13.8756	−0.618683	−0.309341	+	0.950951i	$0.600109\pi$
$504$	0	0
$505$	0.667085	0.0296849
$506$	−0.742943	−0.0330279
$507$	0	0
$508$	−3.35823	−0.148998
$509$	−6.70153	−0.297040	−0.148520	−	0.988909i	$-0.547451\pi$
$509$	−6.70153	−0.297040	−0.148520	+	0.988909i	$0.547451\pi$
$510$	0	0
$511$	−1.82682	−0.0808136
$512$	−22.5371	−0.996007
$513$	0	0
$514$	11.5962	0.511485
$515$	4.67151	0.205851
$516$	0	0
$517$	−0.146959	−0.00646324
$518$	−3.31842	−0.145803
$519$	0	0
$520$	−5.47398	−0.240050
$521$	−2.66765	−0.116872	−0.0584359	−	0.998291i	$-0.518611\pi$
$521$	−2.66765	−0.116872	−0.0584359	+	0.998291i	$0.518611\pi$
$522$	0	0
$523$	−17.2171	−0.752852	−0.376426	−	0.926447i	$-0.622847\pi$
$523$	−17.2171	−0.752852	−0.376426	+	0.926447i	$0.622847\pi$
$524$	35.8642	1.56673
$525$	0	0
$526$	−8.83624	−0.385279
$527$	12.3954	0.539953
$528$	0	0
$529$	1.00000	0.0434783
$530$	2.15490	0.0936030
$531$	0	0
$532$	−13.3497	−0.578782
$533$	−52.2145	−2.26166
$534$	0	0
$535$	−4.02699	−0.174102
$536$	0.908197	0.0392281
$537$	0	0
$538$	−0.325389	−0.0140285
$539$	0.493590	0.0212604
$540$	0	0
$541$	−17.3783	−0.747152	−0.373576	−	0.927600i	$-0.621868\pi$
$541$	−17.3783	−0.747152	−0.373576	+	0.927600i	$0.621868\pi$
$542$	0.188871	0.00811272
$543$	0	0
$544$	17.1220	0.734099
$545$	4.83988	0.207318
$546$	0	0
$547$	−38.8067	−1.65925	−0.829627	−	0.558318i	$-0.811447\pi$
$547$	−38.8067	−1.65925	−0.829627	+	0.558318i	$0.811447\pi$
$548$	−29.3237	−1.25265
$549$	0	0
$550$	−3.46539	−0.147765
$551$	−2.79036	−0.118873
$552$	0	0
$553$	−7.03287	−0.299068
$554$	−10.3095	−0.438008
$555$	0	0
$556$	13.3955	0.568097
$557$	45.7836	1.93991	0.969956	−	0.243279i	$-0.0782229\pi$
$557$	45.7836	1.93991	0.969956	+	0.243279i	$0.0782229\pi$
$558$	0	0
$559$	40.7786	1.72475
$560$	−4.63714	−0.195955
$561$	0	0
$562$	9.11329	0.384421
$563$	−5.07837	−0.214028	−0.107014	−	0.994258i	$-0.534129\pi$
$563$	−5.07837	−0.214028	−0.107014	+	0.994258i	$0.534129\pi$
$564$	0	0
$565$	−1.91608	−0.0806099
$566$	0.696395	0.0292717
$567$	0	0
$568$	−7.21092	−0.302564
$569$	8.39350	0.351874	0.175937	−	0.984401i	$-0.443705\pi$
$569$	8.39350	0.351874	0.175937	+	0.984401i	$0.443705\pi$
$570$	0	0
$571$	0.836227	0.0349950	0.0174975	−	0.999847i	$-0.494430\pi$
$571$	0.836227	0.0349950	0.0174975	+	0.999847i	$0.494430\pi$
$572$	−21.4770	−0.897998
$573$	0	0
$574$	8.64142	0.360686
$575$	4.66440	0.194519
$576$	0	0
$577$	4.30822	0.179353	0.0896767	−	0.995971i	$-0.471417\pi$
$577$	4.30822	0.179353	0.0896767	+	0.995971i	$0.471417\pi$
$578$	0.348166	0.0144818
$579$	0	0
$580$	−1.06780	−0.0443379
$581$	−25.1336	−1.04272
$582$	0	0
$583$	17.6285	0.730096
$584$	1.07100	0.0443183
$585$	0	0
$586$	−9.16692	−0.378682
$587$	−45.2320	−1.86692	−0.933462	−	0.358676i	$-0.883228\pi$
$587$	−45.2320	−1.86692	−0.933462	+	0.358676i	$0.883228\pi$
$588$	0	0
$589$	8.61450	0.354954
$590$	−1.82230	−0.0750228
$591$	0	0
$592$	−9.95815	−0.409277
$593$	28.7912	1.18231	0.591156	−	0.806557i	$-0.298672\pi$
$593$	28.7912	1.18231	0.591156	+	0.806557i	$0.298672\pi$
$594$	0	0
$595$	6.03716	0.247499
$596$	36.7712	1.50621
$597$	0	0
$598$	−2.45865	−0.100542
$599$	9.30640	0.380249	0.190125	−	0.981760i	$-0.439111\pi$
$599$	9.30640	0.380249	0.190125	+	0.981760i	$0.439111\pi$
$600$	0	0
$601$	−21.2220	−0.865662	−0.432831	−	0.901475i	$-0.642485\pi$
$601$	−21.2220	−0.865662	−0.432831	+	0.901475i	$0.642485\pi$
$602$	−6.74879	−0.275060
$603$	0	0
$604$	21.9990	0.895126
$605$	−4.33271	−0.176150
$606$	0	0
$607$	2.64480	0.107349	0.0536745	−	0.998558i	$-0.482907\pi$
$607$	2.64480	0.107349	0.0536745	+	0.998558i	$0.482907\pi$
$608$	11.8993	0.482581
$609$	0	0
$610$	2.06601	0.0836504
$611$	−0.486337	−0.0196751
$612$	0	0
$613$	8.97028	0.362306	0.181153	−	0.983455i	$-0.442017\pi$
$613$	8.97028	0.362306	0.181153	+	0.983455i	$0.442017\pi$
$614$	8.67705	0.350177
$615$	0	0
$616$	7.41112	0.298602
$617$	−11.9943	−0.482874	−0.241437	−	0.970416i	$-0.577619\pi$
$617$	−11.9943	−0.482874	−0.241437	+	0.970416i	$0.577619\pi$
$618$	0	0
$619$	45.9586	1.84723	0.923616	−	0.383318i	$-0.125219\pi$
$619$	45.9586	1.84723	0.923616	+	0.383318i	$0.125219\pi$
$620$	3.29655	0.132393
$621$	0	0
$622$	6.41944	0.257396
$623$	25.8734	1.03660
$624$	0	0
$625$	20.0787	0.803146
$626$	−4.84305	−0.193567
$627$	0	0
$628$	−15.5226	−0.619421
$629$	12.9646	0.516934
$630$	0	0
$631$	−30.6300	−1.21936	−0.609680	−	0.792648i	$-0.708702\pi$
$631$	−30.6300	−1.21936	−0.609680	+	0.792648i	$0.708702\pi$
$632$	4.12313	0.164009
$633$	0	0
$634$	−13.4279	−0.533291
$635$	1.05546	0.0418845
$636$	0	0
$637$	1.63346	0.0647200
$638$	0.742943	0.0294134
$639$	0	0
$640$	5.96831	0.235918
$641$	−3.35392	−0.132472	−0.0662360	−	0.997804i	$-0.521099\pi$
$641$	−3.35392	−0.132472	−0.0662360	+	0.997804i	$0.521099\pi$
$642$	0	0
$643$	35.8942	1.41553	0.707764	−	0.706449i	$-0.249704\pi$
$643$	35.8942	1.41553	0.707764	+	0.706449i	$0.249704\pi$
$644$	−4.78422	−0.188525
$645$	0	0
$646$	−4.43589	−0.174528
$647$	−16.8868	−0.663887	−0.331943	−	0.943299i	$-0.607704\pi$
$647$	−16.8868	−0.663887	−0.331943	+	0.943299i	$0.607704\pi$
$648$	0	0
$649$	−14.9075	−0.585172
$650$	−11.4681	−0.449818
$651$	0	0
$652$	−43.5113	−1.70403
$653$	−31.5936	−1.23635	−0.618177	−	0.786039i	$-0.712128\pi$
$653$	−31.5936	−1.23635	−0.618177	+	0.786039i	$0.712128\pi$
$654$	0	0
$655$	−11.2717	−0.440423
$656$	25.9318	1.01247
$657$	0	0
$658$	0.0804879	0.00313775
$659$	−4.95273	−0.192931	−0.0964656	−	0.995336i	$-0.530754\pi$
$659$	−4.95273	−0.192931	−0.0964656	+	0.995336i	$0.530754\pi$
$660$	0	0
$661$	−2.25253	−0.0876131	−0.0438066	−	0.999040i	$-0.513949\pi$
$661$	−2.25253	−0.0876131	−0.0438066	+	0.999040i	$0.513949\pi$
$662$	−3.31347	−0.128782
$663$	0	0
$664$	14.7350	0.571828
$665$	4.19566	0.162701
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	34.1145	1.31993
$669$	0	0
$670$	−0.136897	−0.00528878
$671$	16.9013	0.652467
$672$	0	0
$673$	40.7757	1.57179	0.785893	−	0.618363i	$-0.212204\pi$
$673$	40.7757	1.57179	0.785893	+	0.618363i	$0.212204\pi$
$674$	0.598584	0.0230566
$675$	0	0
$676$	−47.1127	−1.81203
$677$	−40.2860	−1.54832	−0.774158	−	0.632992i	$-0.781827\pi$
$677$	−40.2860	−1.54832	−0.774158	+	0.632992i	$0.781827\pi$
$678$	0	0
$679$	−31.3319	−1.20241
$680$	−3.53938	−0.135729
$681$	0	0
$682$	−2.29364	−0.0878281
$683$	26.0249	0.995816	0.497908	−	0.867230i	$-0.334102\pi$
$683$	26.0249	0.995816	0.497908	+	0.867230i	$0.334102\pi$
$684$	0	0
$685$	9.21613	0.352130
$686$	−7.46415	−0.284983
$687$	0	0
$688$	−20.2523	−0.772111
$689$	58.3386	2.22252
$690$	0	0
$691$	−26.9608	−1.02564	−0.512819	−	0.858497i	$-0.671399\pi$
$691$	−26.9608	−1.02564	−0.512819	+	0.858497i	$0.671399\pi$
$692$	17.8935	0.680208
$693$	0	0
$694$	−13.0172	−0.494128
$695$	−4.21007	−0.159697
$696$	0	0
$697$	−33.7610	−1.27879
$698$	−4.53053	−0.171483
$699$	0	0
$700$	−22.3155	−0.843448
$701$	−15.6620	−0.591546	−0.295773	−	0.955258i	$-0.595577\pi$
$701$	−15.6620	−0.591546	−0.295773	+	0.955258i	$0.595577\pi$
$702$	0	0
$703$	9.01009	0.339822
$704$	8.40527	0.316786
$705$	0	0
$706$	−7.89698	−0.297207
$707$	−2.98884	−0.112407
$708$	0	0
$709$	−26.5765	−0.998101	−0.499050	−	0.866573i	$-0.666318\pi$
$709$	−26.5765	−0.998101	−0.499050	+	0.866573i	$0.666318\pi$
$710$	1.08694	0.0407919
$711$	0	0
$712$	−15.1687	−0.568472
$713$	3.08724	0.115618
$714$	0	0
$715$	6.74998	0.252435
$716$	2.07737	0.0776351
$717$	0	0
$718$	12.7981	0.477619
$719$	31.8619	1.18825	0.594124	−	0.804373i	$-0.297499\pi$
$719$	31.8619	1.18825	0.594124	+	0.804373i	$0.297499\pi$
$720$	0	0
$721$	−20.9305	−0.779492
$722$	4.44004	0.165241
$723$	0	0
$724$	−39.8464	−1.48088
$725$	−4.66440	−0.173232
$726$	0	0
$727$	3.67239	0.136201	0.0681007	−	0.997678i	$-0.478306\pi$
$727$	3.67239	0.136201	0.0681007	+	0.997678i	$0.478306\pi$
$728$	24.5259	0.908991
$729$	0	0
$730$	−0.161437	−0.00597504
$731$	26.3667	0.975208
$732$	0	0
$733$	−34.8917	−1.28875	−0.644377	−	0.764708i	$-0.722883\pi$
$733$	−34.8917	−1.28875	−0.644377	+	0.764708i	$0.722883\pi$
$734$	−1.15322	−0.0425663
$735$	0	0
$736$	4.26445	0.157190
$737$	−1.11990	−0.0412521
$738$	0	0
$739$	36.6332	1.34757	0.673787	−	0.738925i	$-0.264666\pi$
$739$	36.6332	1.34757	0.673787	+	0.738925i	$0.264666\pi$
$740$	3.44793	0.126748
$741$	0	0
$742$	−9.65494	−0.354444
$743$	30.0312	1.10174	0.550869	−	0.834592i	$-0.314297\pi$
$743$	30.0312	1.10174	0.550869	+	0.834592i	$0.314297\pi$
$744$	0	0
$745$	−11.5568	−0.423408
$746$	−5.41416	−0.198227
$747$	0	0
$748$	−13.8866	−0.507746
$749$	18.0428	0.659268
$750$	0	0
$751$	−49.0657	−1.79043	−0.895217	−	0.445630i	$-0.852980\pi$
$751$	−49.0657	−1.79043	−0.895217	+	0.445630i	$0.852980\pi$
$752$	0.241534	0.00880785
$753$	0	0
$754$	2.45865	0.0895389
$755$	−6.91405	−0.251628
$756$	0	0
$757$	−31.6108	−1.14891	−0.574457	−	0.818535i	$-0.694787\pi$
$757$	−31.6108	−1.14891	−0.574457	+	0.818535i	$0.694787\pi$
$758$	−8.12309	−0.295044
$759$	0	0
$760$	−2.45978	−0.0892254
$761$	39.4073	1.42852	0.714258	−	0.699883i	$-0.246765\pi$
$761$	39.4073	1.42852	0.714258	+	0.699883i	$0.246765\pi$
$762$	0	0
$763$	−21.6849	−0.785045
$764$	−4.84275	−0.175205
$765$	0	0
$766$	−13.6528	−0.493295
$767$	−49.3342	−1.78135
$768$	0	0
$769$	2.31376	0.0834363	0.0417182	−	0.999129i	$-0.486717\pi$
$769$	2.31376	0.0834363	0.0417182	+	0.999129i	$0.486717\pi$
$770$	−1.11711	−0.0402579
$771$	0	0
$772$	39.6947	1.42864
$773$	−9.67989	−0.348161	−0.174081	−	0.984731i	$-0.555695\pi$
$773$	−9.67989	−0.348161	−0.174081	+	0.984731i	$0.555695\pi$
$774$	0	0
$775$	14.4001	0.517268
$776$	18.3688	0.659403
$777$	0	0
$778$	−8.37248	−0.300168
$779$	−23.4630	−0.840649
$780$	0	0
$781$	8.89181	0.318174
$782$	−1.58972	−0.0568483
$783$	0	0
$784$	−0.811241	−0.0289729
$785$	4.87860	0.174125
$786$	0	0
$787$	22.4981	0.801970	0.400985	−	0.916085i	$-0.368668\pi$
$787$	22.4981	0.801970	0.400985	+	0.916085i	$0.368668\pi$
$788$	−36.1200	−1.28672
$789$	0	0
$790$	−0.621498	−0.0221119
$791$	8.58489	0.305244
$792$	0	0
$793$	55.9321	1.98621
$794$	−14.5364	−0.515877
$795$	0	0
$796$	−34.4547	−1.22122
$797$	−31.7414	−1.12434	−0.562169	−	0.827023i	$-0.690033\pi$
$797$	−31.7414	−1.12434	−0.562169	+	0.827023i	$0.690033\pi$
$798$	0	0
$799$	−0.314457	−0.0111247
$800$	19.8911	0.703256
$801$	0	0
$802$	9.61860	0.339644
$803$	−1.32065	−0.0466049
$804$	0	0
$805$	1.50363	0.0529960
$806$	−7.59045	−0.267362
$807$	0	0
$808$	1.75226	0.0616442
$809$	−47.1833	−1.65888	−0.829439	−	0.558597i	$-0.811340\pi$
$809$	−47.1833	−1.65888	−0.829439	+	0.558597i	$0.811340\pi$
$810$	0	0
$811$	−37.6766	−1.32300	−0.661502	−	0.749943i	$-0.730081\pi$
$811$	−37.6766	−1.32300	−0.661502	+	0.749943i	$0.730081\pi$
$812$	4.78422	0.167893
$813$	0	0
$814$	−2.39897	−0.0840839
$815$	13.6751	0.479019
$816$	0	0
$817$	18.3242	0.641081
$818$	7.75737	0.271230
$819$	0	0
$820$	−8.97869	−0.313550
$821$	28.6192	0.998817	0.499409	−	0.866367i	$-0.333551\pi$
$821$	28.6192	0.998817	0.499409	+	0.866367i	$0.333551\pi$
$822$	0	0
$823$	10.8950	0.379775	0.189887	−	0.981806i	$-0.439188\pi$
$823$	10.8950	0.379775	0.189887	+	0.981806i	$0.439188\pi$
$824$	12.2708	0.427475
$825$	0	0
$826$	8.16473	0.284087
$827$	−40.0272	−1.39188	−0.695941	−	0.718099i	$-0.745013\pi$
$827$	−40.0272	−1.39188	−0.695941	+	0.718099i	$0.745013\pi$
$828$	0	0
$829$	31.3663	1.08939	0.544697	−	0.838633i	$-0.316644\pi$
$829$	31.3663	1.08939	0.544697	+	0.838633i	$0.316644\pi$
$830$	−2.22107	−0.0770945
$831$	0	0
$832$	27.8159	0.964343
$833$	1.05617	0.0365940
$834$	0	0
$835$	−10.7218	−0.371044
$836$	−9.65085	−0.333781
$837$	0	0
$838$	10.4052	0.359441
$839$	−7.36911	−0.254410	−0.127205	−	0.991876i	$-0.540601\pi$
$839$	−7.36911	−0.254410	−0.127205	+	0.991876i	$0.540601\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−3.69148	−0.127217
$843$	0	0
$844$	20.8160	0.716516
$845$	14.8070	0.509377
$846$	0	0
$847$	19.4125	0.667022
$848$	−28.9733	−0.994946
$849$	0	0
$850$	−7.41510	−0.254336
$851$	3.22901	0.110689
$852$	0	0
$853$	2.23153	0.0764061	0.0382031	−	0.999270i	$-0.487837\pi$
$853$	2.23153	0.0764061	0.0382031	+	0.999270i	$0.487837\pi$
$854$	−9.25668	−0.316757
$855$	0	0
$856$	−10.5779	−0.361544
$857$	−3.20003	−0.109311	−0.0546555	−	0.998505i	$-0.517406\pi$
$857$	−3.20003	−0.109311	−0.0546555	+	0.998505i	$0.517406\pi$
$858$	0	0
$859$	−55.8487	−1.90553	−0.952766	−	0.303706i	$-0.901776\pi$
$859$	−55.8487	−1.90553	−0.952766	+	0.303706i	$0.901776\pi$
$860$	7.01219	0.239114
$861$	0	0
$862$	−6.52363	−0.222196
$863$	−45.5794	−1.55154	−0.775770	−	0.631015i	$-0.782638\pi$
$863$	−45.5794	−1.55154	−0.775770	+	0.631015i	$0.782638\pi$
$864$	0	0
$865$	−5.62373	−0.191212
$866$	−1.52934	−0.0519690
$867$	0	0
$868$	−14.7700	−0.501328
$869$	−5.08425	−0.172471
$870$	0	0
$871$	−3.70613	−0.125578
$872$	12.7131	0.430520
$873$	0	0
$874$	−1.10481	−0.0373709
$875$	14.5317	0.491261
$876$	0	0
$877$	52.0667	1.75817	0.879084	−	0.476667i	$-0.158155\pi$
$877$	52.0667	1.75817	0.879084	+	0.476667i	$0.158155\pi$
$878$	3.38855	0.114358
$879$	0	0
$880$	−3.35231	−0.113006
$881$	−20.6461	−0.695586	−0.347793	−	0.937571i	$-0.613069\pi$
$881$	−20.6461	−0.695586	−0.347793	+	0.937571i	$0.613069\pi$
$882$	0	0
$883$	−57.1293	−1.92255	−0.961277	−	0.275585i	$-0.911129\pi$
$883$	−57.1293	−1.92255	−0.961277	+	0.275585i	$0.911129\pi$
$884$	−45.9556	−1.54566
$885$	0	0
$886$	−2.40799	−0.0808979
$887$	−30.4357	−1.02193	−0.510966	−	0.859601i	$-0.670712\pi$
$887$	−30.4357	−1.02193	−0.510966	+	0.859601i	$0.670712\pi$
$888$	0	0
$889$	−4.72893	−0.158603
$890$	2.28645	0.0766420
$891$	0	0
$892$	5.23107	0.175149
$893$	−0.218539	−0.00731313
$894$	0	0
$895$	−0.652896	−0.0218239
$896$	−26.7408	−0.893346
$897$	0	0
$898$	−5.79814	−0.193486
$899$	−3.08724	−0.102965
$900$	0	0
$901$	37.7207	1.25666
$902$	6.24711	0.208006
$903$	0	0
$904$	−5.03303	−0.167396
$905$	12.5233	0.416288
$906$	0	0
$907$	39.6703	1.31723	0.658615	−	0.752480i	$-0.271143\pi$
$907$	39.6703	1.31723	0.658615	+	0.752480i	$0.271143\pi$
$908$	8.43850	0.280041
$909$	0	0
$910$	−3.69691	−0.122551
$911$	17.5930	0.582882	0.291441	−	0.956589i	$-0.405865\pi$
$911$	17.5930	0.582882	0.291441	+	0.956589i	$0.405865\pi$
$912$	0	0
$913$	−18.1698	−0.601331
$914$	−9.82141	−0.324863
$915$	0	0
$916$	7.32022	0.241867
$917$	50.5025	1.66774
$918$	0	0
$919$	−22.9102	−0.755737	−0.377868	−	0.925859i	$-0.623343\pi$
$919$	−22.9102	−0.755737	−0.377868	+	0.925859i	$0.623343\pi$
$920$	−0.881527	−0.0290631
$921$	0	0
$922$	12.7472	0.419806
$923$	29.4261	0.968570
$924$	0	0
$925$	15.0614	0.495216
$926$	6.58982	0.216555
$927$	0	0
$928$	−4.26445	−0.139987
$929$	−11.8132	−0.387578	−0.193789	−	0.981043i	$-0.562078\pi$
$929$	−11.8132	−0.387578	−0.193789	+	0.981043i	$0.562078\pi$
$930$	0	0
$931$	0.734008	0.0240561
$932$	−29.0933	−0.952982
$933$	0	0
$934$	11.2838	0.369218
$935$	4.36442	0.142732
$936$	0	0
$937$	−19.9816	−0.652771	−0.326386	−	0.945237i	$-0.605831\pi$
$937$	−19.9816	−0.652771	−0.326386	+	0.945237i	$0.605831\pi$
$938$	0.613359	0.0200269
$939$	0	0
$940$	−0.0836293	−0.00272769
$941$	−40.1760	−1.30970	−0.654850	−	0.755759i	$-0.727268\pi$
$941$	−40.1760	−1.30970	−0.654850	+	0.755759i	$0.727268\pi$
$942$	0	0
$943$	−8.40860	−0.273822
$944$	24.5013	0.797450
$945$	0	0
$946$	−4.87888	−0.158626
$947$	−22.2543	−0.723168	−0.361584	−	0.932339i	$-0.617764\pi$
$947$	−22.2543	−0.723168	−0.361584	+	0.932339i	$0.617764\pi$
$948$	0	0
$949$	−4.37049	−0.141872
$950$	−5.15330	−0.167195
$951$	0	0
$952$	15.8580	0.513962
$953$	6.40819	0.207582	0.103791	−	0.994599i	$-0.466903\pi$
$953$	6.40819	0.207582	0.103791	+	0.994599i	$0.466903\pi$
$954$	0	0
$955$	1.52202	0.0492516
$956$	18.1099	0.585717
$957$	0	0
$958$	−12.9171	−0.417334
$959$	−41.2925	−1.33340
$960$	0	0
$961$	−21.4690	−0.692547
$962$	−7.93902	−0.255964
$963$	0	0
$964$	20.6673	0.665648
$965$	−12.4756	−0.401605
$966$	0	0
$967$	−44.7836	−1.44014	−0.720072	−	0.693899i	$-0.755891\pi$
$967$	−44.7836	−1.44014	−0.720072	+	0.693899i	$0.755891\pi$
$968$	−11.3809	−0.365796
$969$	0	0
$970$	−2.76882	−0.0889014
$971$	38.1900	1.22558	0.612788	−	0.790248i	$-0.290048\pi$
$971$	38.1900	1.22558	0.612788	+	0.790248i	$0.290048\pi$
$972$	0	0
$973$	18.8630	0.604721
$974$	−3.53687	−0.113329
$975$	0	0
$976$	−27.7781	−0.889157
$977$	−1.05944	−0.0338944	−0.0169472	−	0.999856i	$-0.505395\pi$
$977$	−1.05944	−0.0338944	−0.0169472	+	0.999856i	$0.505395\pi$
$978$	0	0
$979$	18.7046	0.597802
$980$	0.280886	0.00897257
$981$	0	0
$982$	2.12235	0.0677269
$983$	−26.8563	−0.856584	−0.428292	−	0.903640i	$-0.640884\pi$
$983$	−26.8563	−0.856584	−0.428292	+	0.903640i	$0.640884\pi$
$984$	0	0
$985$	11.3521	0.361709
$986$	1.58972	0.0506271
$987$	0	0
$988$	−31.9380	−1.01608
$989$	6.56696	0.208817
$990$	0	0
$991$	−29.6798	−0.942810	−0.471405	−	0.881917i	$-0.656253\pi$
$991$	−29.6798	−0.942810	−0.471405	+	0.881917i	$0.656253\pi$
$992$	13.1654	0.418001
$993$	0	0
$994$	−4.86996	−0.154466
$995$	10.8288	0.343295
$996$	0	0
$997$	48.9821	1.55128	0.775640	−	0.631175i	$-0.217427\pi$
$997$	48.9821	1.55128	0.775640	+	0.631175i	$0.217427\pi$
$998$	2.15509	0.0682182
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.w.1.15	yes	30
3.2	odd	2		6003.2.a.v.1.16	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.16	✓	30	3.2	odd	2
6003.2.a.w.1.15	yes	30	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)

\(f(q)\)	\(=\)	\(q-0.395940 q^{2} -1.84323 q^{4} +0.579308 q^{5} -2.59556 q^{7} +1.52169 q^{8} +O(q^{10})\)	\(q-0.395940 q^{2} -1.84323 q^{4} +0.579308 q^{5} -2.59556 q^{7} +1.52169 q^{8} -0.229371 q^{10} -1.87640 q^{11} -6.20966 q^{13} +1.02769 q^{14} +3.08396 q^{16} -4.01505 q^{17} -2.79036 q^{19} -1.06780 q^{20} +0.742943 q^{22} -1.00000 q^{23} -4.66440 q^{25} +2.45865 q^{26} +4.78422 q^{28} +1.00000 q^{29} -3.08724 q^{31} -4.26445 q^{32} +1.58972 q^{34} -1.50363 q^{35} -3.22901 q^{37} +1.10481 q^{38} +0.881527 q^{40} +8.40860 q^{41} -6.56696 q^{43} +3.45864 q^{44} +0.395940 q^{46} +0.0783194 q^{47} -0.263051 q^{49} +1.84683 q^{50} +11.4458 q^{52} -9.39482 q^{53} -1.08701 q^{55} -3.94964 q^{56} -0.395940 q^{58} +7.94475 q^{59} -9.00728 q^{61} +1.22236 q^{62} -4.47946 q^{64} -3.59730 q^{65} +0.596834 q^{67} +7.40067 q^{68} +0.595348 q^{70} -4.73876 q^{71} +0.703822 q^{73} +1.27850 q^{74} +5.14327 q^{76} +4.87032 q^{77} +2.70957 q^{79} +1.78656 q^{80} -3.32931 q^{82} +9.68330 q^{83} -2.32595 q^{85} +2.60012 q^{86} -2.85530 q^{88} -9.96833 q^{89} +16.1176 q^{91} +1.84323 q^{92} -0.0310098 q^{94} -1.61648 q^{95} +12.0713 q^{97} +0.104153 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) 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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