LMFDB - Embedded newform 6003.2.a.w.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-1.20001 q^{2} -0.559985 q^{4} +3.40098 q^{5} +4.71373 q^{7} +3.07200 q^{8} +O(q^{10})$	\(q-1.20001 q^{2} -0.559985 q^{4} +3.40098 q^{5} +4.71373 q^{7} +3.07200 q^{8} -4.08120 q^{10} +1.68704 q^{11} -2.33398 q^{13} -5.65651 q^{14} -2.56645 q^{16} +4.01822 q^{17} +0.666367 q^{19} -1.90450 q^{20} -2.02446 q^{22} -1.00000 q^{23} +6.56669 q^{25} +2.80079 q^{26} -2.63962 q^{28} +1.00000 q^{29} -4.38298 q^{31} -3.06425 q^{32} -4.82188 q^{34} +16.0313 q^{35} +1.15254 q^{37} -0.799644 q^{38} +10.4478 q^{40} -10.3676 q^{41} -4.96331 q^{43} -0.944717 q^{44} +1.20001 q^{46} +12.7990 q^{47} +15.2193 q^{49} -7.88007 q^{50} +1.30699 q^{52} -7.60435 q^{53} +5.73759 q^{55} +14.4806 q^{56} -1.20001 q^{58} -2.67174 q^{59} +13.7587 q^{61} +5.25960 q^{62} +8.81000 q^{64} -7.93781 q^{65} +7.64919 q^{67} -2.25014 q^{68} -19.2377 q^{70} +2.23124 q^{71} +3.18073 q^{73} -1.38306 q^{74} -0.373156 q^{76} +7.95225 q^{77} -10.6718 q^{79} -8.72844 q^{80} +12.4411 q^{82} -0.691279 q^{83} +13.6659 q^{85} +5.95600 q^{86} +5.18258 q^{88} +18.4224 q^{89} -11.0017 q^{91} +0.559985 q^{92} -15.3589 q^{94} +2.26630 q^{95} +15.4956 q^{97} -18.2632 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.20001	−0.848532	−0.424266	−	0.905538i	$-0.639468\pi$
$2$	−1.20001	−0.848532	−0.424266	+	0.905538i	$0.639468\pi$
$3$	0	0
$3$	0	0
$4$	−0.559985	−0.279993
$5$	3.40098	1.52097	0.760483	−	0.649358i	$-0.224962\pi$
$5$	3.40098	1.52097	0.760483	+	0.649358i	$0.224962\pi$
$6$	0	0
$7$	4.71373	1.78162	0.890812	−	0.454372i	$-0.150136\pi$
$7$	4.71373	1.78162	0.890812	+	0.454372i	$0.150136\pi$
$8$	3.07200	1.08612
$9$	0	0
$10$	−4.08120	−1.29059
$11$	1.68704	0.508661	0.254331	−	0.967117i	$-0.418145\pi$
$11$	1.68704	0.508661	0.254331	+	0.967117i	$0.418145\pi$
$12$	0	0
$13$	−2.33398	−0.647328	−0.323664	−	0.946172i	$-0.604915\pi$
$13$	−2.33398	−0.647328	−0.323664	+	0.946172i	$0.604915\pi$
$14$	−5.65651	−1.51177
$15$	0	0
$16$	−2.56645	−0.641611
$17$	4.01822	0.974560	0.487280	−	0.873246i	$-0.337989\pi$
$17$	4.01822	0.974560	0.487280	+	0.873246i	$0.337989\pi$
$18$	0	0
$19$	0.666367	0.152875	0.0764375	−	0.997074i	$-0.475645\pi$
$19$	0.666367	0.152875	0.0764375	+	0.997074i	$0.475645\pi$
$20$	−1.90450	−0.425859
$21$	0	0
$22$	−2.02446	−0.431616
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	6.56669	1.31334
$26$	2.80079	0.549279
$27$	0	0
$28$	−2.63962	−0.498842
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.38298	−0.787206	−0.393603	−	0.919280i	$-0.628772\pi$
$31$	−4.38298	−0.787206	−0.393603	+	0.919280i	$0.628772\pi$
$32$	−3.06425	−0.541687
$33$	0	0
$34$	−4.82188	−0.826946
$35$	16.0313	2.70979
$36$	0	0
$37$	1.15254	0.189477	0.0947384	−	0.995502i	$-0.469799\pi$
$37$	1.15254	0.189477	0.0947384	+	0.995502i	$0.469799\pi$
$38$	−0.799644	−0.129719
$39$	0	0
$40$	10.4478	1.65194
$41$	−10.3676	−1.61914	−0.809571	−	0.587022i	$-0.800300\pi$
$41$	−10.3676	−1.61914	−0.809571	+	0.587022i	$0.800300\pi$
$42$	0	0
$43$	−4.96331	−0.756897	−0.378449	−	0.925622i	$-0.623542\pi$
$43$	−4.96331	−0.756897	−0.378449	+	0.925622i	$0.623542\pi$
$44$	−0.944717	−0.142421
$45$	0	0
$46$	1.20001	0.176931
$47$	12.7990	1.86693	0.933463	−	0.358673i	$-0.116771\pi$
$47$	12.7990	1.86693	0.933463	+	0.358673i	$0.116771\pi$
$48$	0	0
$49$	15.2193	2.17418
$50$	−7.88007	−1.11441
$51$	0	0
$52$	1.30699	0.181247
$53$	−7.60435	−1.04454	−0.522269	−	0.852781i	$-0.674914\pi$
$53$	−7.60435	−1.04454	−0.522269	+	0.852781i	$0.674914\pi$
$54$	0	0
$55$	5.73759	0.773657
$56$	14.4806	1.93505
$57$	0	0
$58$	−1.20001	−0.157569
$59$	−2.67174	−0.347831	−0.173915	−	0.984761i	$-0.555642\pi$
$59$	−2.67174	−0.347831	−0.173915	+	0.984761i	$0.555642\pi$
$60$	0	0
$61$	13.7587	1.76162	0.880812	−	0.473466i	$-0.156997\pi$
$61$	13.7587	1.76162	0.880812	+	0.473466i	$0.156997\pi$
$62$	5.25960	0.667970
$63$	0	0
$64$	8.81000	1.10125
$65$	−7.93781	−0.984565
$66$	0	0
$67$	7.64919	0.934497	0.467248	−	0.884126i	$-0.345245\pi$
$67$	7.64919	0.934497	0.467248	+	0.884126i	$0.345245\pi$
$68$	−2.25014	−0.272870
$69$	0	0
$70$	−19.2377	−2.29934
$71$	2.23124	0.264800	0.132400	−	0.991196i	$-0.457732\pi$
$71$	2.23124	0.264800	0.132400	+	0.991196i	$0.457732\pi$
$72$	0	0
$73$	3.18073	0.372276	0.186138	−	0.982524i	$-0.440403\pi$
$73$	3.18073	0.372276	0.186138	+	0.982524i	$0.440403\pi$
$74$	−1.38306	−0.160777
$75$	0	0
$76$	−0.373156	−0.0428039
$77$	7.95225	0.906243
$78$	0	0
$79$	−10.6718	−1.20067	−0.600334	−	0.799750i	$-0.704966\pi$
$79$	−10.6718	−1.20067	−0.600334	+	0.799750i	$0.704966\pi$
$80$	−8.72844	−0.975869
$81$	0	0
$82$	12.4411	1.37389
$83$	−0.691279	−0.0758778	−0.0379389	−	0.999280i	$-0.512079\pi$
$83$	−0.691279	−0.0758778	−0.0379389	+	0.999280i	$0.512079\pi$
$84$	0	0
$85$	13.6659	1.48227
$86$	5.95600	0.642252
$87$	0	0
$88$	5.18258	0.552465
$89$	18.4224	1.95277	0.976387	−	0.216027i	$-0.0693101\pi$
$89$	18.4224	1.95277	0.976387	+	0.216027i	$0.0693101\pi$
$90$	0	0
$91$	−11.0017	−1.15330
$92$	0.559985	0.0583825
$93$	0	0
$94$	−15.3589	−1.58415
$95$	2.26630	0.232518
$96$	0	0
$97$	15.4956	1.57334	0.786670	−	0.617374i	$-0.211803\pi$
$97$	15.4956	1.57334	0.786670	+	0.617374i	$0.211803\pi$
$98$	−18.2632	−1.84486
$99$	0	0
$100$	−3.67725	−0.367725
$101$	−4.19835	−0.417751	−0.208875	−	0.977942i	$-0.566980\pi$
$101$	−4.19835	−0.417751	−0.208875	+	0.977942i	$0.566980\pi$
$102$	0	0
$103$	2.46250	0.242638	0.121319	−	0.992614i	$-0.461288\pi$
$103$	2.46250	0.242638	0.121319	+	0.992614i	$0.461288\pi$
$104$	−7.16997	−0.703073
$105$	0	0
$106$	9.12526	0.886324
$107$	15.5894	1.50708	0.753542	−	0.657399i	$-0.228344\pi$
$107$	15.5894	1.50708	0.753542	+	0.657399i	$0.228344\pi$
$108$	0	0
$109$	10.5233	1.00795	0.503977	−	0.863717i	$-0.331870\pi$
$109$	10.5233	1.00795	0.503977	+	0.863717i	$0.331870\pi$
$110$	−6.88514	−0.656473
$111$	0	0
$112$	−12.0975	−1.14311
$113$	−6.76893	−0.636768	−0.318384	−	0.947962i	$-0.603140\pi$
$113$	−6.76893	−0.636768	−0.318384	+	0.947962i	$0.603140\pi$
$114$	0	0
$115$	−3.40098	−0.317143
$116$	−0.559985	−0.0519933
$117$	0	0
$118$	3.20610	0.295146
$119$	18.9408	1.73630
$120$	0	0
$121$	−8.15390	−0.741264
$122$	−16.5106	−1.49479
$123$	0	0
$124$	2.45441	0.220412
$125$	5.32829	0.476577
$126$	0	0
$127$	−10.7746	−0.956089	−0.478045	−	0.878336i	$-0.658654\pi$
$127$	−10.7746	−0.956089	−0.478045	+	0.878336i	$0.658654\pi$
$128$	−4.44357	−0.392759
$129$	0	0
$130$	9.52543	0.835435
$131$	−7.38872	−0.645556	−0.322778	−	0.946475i	$-0.604617\pi$
$131$	−7.38872	−0.645556	−0.322778	+	0.946475i	$0.604617\pi$
$132$	0	0
$133$	3.14108	0.272366
$134$	−9.17907	−0.792951
$135$	0	0
$136$	12.3439	1.05848
$137$	1.85093	0.158136	0.0790678	−	0.996869i	$-0.474806\pi$
$137$	1.85093	0.158136	0.0790678	+	0.996869i	$0.474806\pi$
$138$	0	0
$139$	−1.02911	−0.0872879	−0.0436440	−	0.999047i	$-0.513897\pi$
$139$	−1.02911	−0.0872879	−0.0436440	+	0.999047i	$0.513897\pi$
$140$	−8.97731	−0.758721
$141$	0	0
$142$	−2.67750	−0.224691
$143$	−3.93751	−0.329271
$144$	0	0
$145$	3.40098	0.282436
$146$	−3.81689	−0.315888
$147$	0	0
$148$	−0.645407	−0.0530522
$149$	−9.73121	−0.797212	−0.398606	−	0.917122i	$-0.630506\pi$
$149$	−9.73121	−0.797212	−0.398606	+	0.917122i	$0.630506\pi$
$150$	0	0
$151$	19.7523	1.60742	0.803712	−	0.595019i	$-0.202855\pi$
$151$	19.7523	1.60742	0.803712	+	0.595019i	$0.202855\pi$
$152$	2.04708	0.166040
$153$	0	0
$154$	−9.54275	−0.768977
$155$	−14.9064	−1.19731
$156$	0	0
$157$	4.89912	0.390993	0.195496	−	0.980704i	$-0.437368\pi$
$157$	4.89912	0.390993	0.195496	+	0.980704i	$0.437368\pi$
$158$	12.8062	1.01881
$159$	0	0
$160$	−10.4215	−0.823888
$161$	−4.71373	−0.371494
$162$	0	0
$163$	10.0360	0.786081	0.393041	−	0.919521i	$-0.371423\pi$
$163$	10.0360	0.786081	0.393041	+	0.919521i	$0.371423\pi$
$164$	5.80569	0.453348
$165$	0	0
$166$	0.829539	0.0643847
$167$	−18.5181	−1.43297	−0.716485	−	0.697602i	$-0.754250\pi$
$167$	−18.5181	−1.43297	−0.716485	+	0.697602i	$0.754250\pi$
$168$	0	0
$169$	−7.55256	−0.580966
$170$	−16.3991	−1.25776
$171$	0	0
$172$	2.77938	0.211926
$173$	−15.1863	−1.15460	−0.577298	−	0.816534i	$-0.695893\pi$
$173$	−15.1863	−1.15460	−0.577298	+	0.816534i	$0.695893\pi$
$174$	0	0
$175$	30.9536	2.33987
$176$	−4.32969	−0.326363
$177$	0	0
$178$	−22.1070	−1.65699
$179$	−2.47303	−0.184843	−0.0924216	−	0.995720i	$-0.529461\pi$
$179$	−2.47303	−0.184843	−0.0924216	+	0.995720i	$0.529461\pi$
$180$	0	0
$181$	25.7565	1.91447	0.957233	−	0.289318i	$-0.0934285\pi$
$181$	25.7565	1.91447	0.957233	+	0.289318i	$0.0934285\pi$
$182$	13.2022	0.978609
$183$	0	0
$184$	−3.07200	−0.226471
$185$	3.91978	0.288188
$186$	0	0
$187$	6.77888	0.495721
$188$	−7.16726	−0.522726
$189$	0	0
$190$	−2.71958	−0.197299
$191$	−7.49796	−0.542533	−0.271266	−	0.962504i	$-0.587442\pi$
$191$	−7.49796	−0.542533	−0.271266	+	0.962504i	$0.587442\pi$
$192$	0	0
$193$	−4.13126	−0.297374	−0.148687	−	0.988884i	$-0.547505\pi$
$193$	−4.13126	−0.297374	−0.148687	+	0.988884i	$0.547505\pi$
$194$	−18.5948	−1.33503
$195$	0	0
$196$	−8.52258	−0.608756
$197$	−10.4334	−0.743352	−0.371676	−	0.928363i	$-0.621217\pi$
$197$	−10.4334	−0.743352	−0.371676	+	0.928363i	$0.621217\pi$
$198$	0	0
$199$	12.7858	0.906363	0.453182	−	0.891418i	$-0.350289\pi$
$199$	12.7858	0.906363	0.453182	+	0.891418i	$0.350289\pi$
$200$	20.1729	1.42644
$201$	0	0
$202$	5.03804	0.354475
$203$	4.71373	0.330839
$204$	0	0
$205$	−35.2599	−2.46266
$206$	−2.95502	−0.205886
$207$	0	0
$208$	5.99002	0.415333
$209$	1.12419	0.0777616
$210$	0	0
$211$	27.3289	1.88140	0.940698	−	0.339244i	$-0.110171\pi$
$211$	27.3289	1.88140	0.940698	+	0.339244i	$0.110171\pi$
$212$	4.25832	0.292463
$213$	0	0
$214$	−18.7074	−1.27881
$215$	−16.8801	−1.15122
$216$	0	0
$217$	−20.6602	−1.40251
$218$	−12.6281	−0.855281
$219$	0	0
$220$	−3.21297	−0.216618
$221$	−9.37842	−0.630861
$222$	0	0
$223$	−21.6168	−1.44757	−0.723785	−	0.690026i	$-0.757599\pi$
$223$	−21.6168	−1.44757	−0.723785	+	0.690026i	$0.757599\pi$
$224$	−14.4440	−0.965083
$225$	0	0
$226$	8.12276	0.540318
$227$	−22.9761	−1.52498	−0.762488	−	0.647002i	$-0.776023\pi$
$227$	−22.9761	−1.52498	−0.762488	+	0.647002i	$0.776023\pi$
$228$	0	0
$229$	−5.24050	−0.346302	−0.173151	−	0.984895i	$-0.555395\pi$
$229$	−5.24050	−0.346302	−0.173151	+	0.984895i	$0.555395\pi$
$230$	4.08120	0.269106
$231$	0	0
$232$	3.07200	0.201687
$233$	−24.1445	−1.58176	−0.790880	−	0.611971i	$-0.790377\pi$
$233$	−24.1445	−1.58176	−0.790880	+	0.611971i	$0.790377\pi$
$234$	0	0
$235$	43.5292	2.83953
$236$	1.49614	0.0973901
$237$	0	0
$238$	−22.7291	−1.47331
$239$	9.08853	0.587888	0.293944	−	0.955823i	$-0.405032\pi$
$239$	9.08853	0.587888	0.293944	+	0.955823i	$0.405032\pi$
$240$	0	0
$241$	−10.1452	−0.653507	−0.326753	−	0.945110i	$-0.605955\pi$
$241$	−10.1452	−0.653507	−0.326753	+	0.945110i	$0.605955\pi$
$242$	9.78473	0.628986
$243$	0	0
$244$	−7.70468	−0.493242
$245$	51.7605	3.30686
$246$	0	0
$247$	−1.55528	−0.0989604
$248$	−13.4645	−0.854997
$249$	0	0
$250$	−6.39398	−0.404391
$251$	−6.13836	−0.387450	−0.193725	−	0.981056i	$-0.562057\pi$
$251$	−6.13836	−0.387450	−0.193725	+	0.981056i	$0.562057\pi$
$252$	0	0
$253$	−1.68704	−0.106063
$254$	12.9296	0.811273
$255$	0	0
$256$	−12.2877	−0.767981
$257$	−6.75018	−0.421065	−0.210532	−	0.977587i	$-0.567520\pi$
$257$	−6.75018	−0.421065	−0.210532	+	0.977587i	$0.567520\pi$
$258$	0	0
$259$	5.43278	0.337577
$260$	4.44506	0.275671
$261$	0	0
$262$	8.86651	0.547775
$263$	21.6381	1.33426	0.667130	−	0.744941i	$-0.267522\pi$
$263$	21.6381	1.33426	0.667130	+	0.744941i	$0.267522\pi$
$264$	0	0
$265$	−25.8623	−1.58871
$266$	−3.76931	−0.231111
$267$	0	0
$268$	−4.28343	−0.261652
$269$	−18.0943	−1.10323	−0.551614	−	0.834100i	$-0.685988\pi$
$269$	−18.0943	−1.10323	−0.551614	+	0.834100i	$0.685988\pi$
$270$	0	0
$271$	−3.59791	−0.218557	−0.109279	−	0.994011i	$-0.534854\pi$
$271$	−3.59791	−0.218557	−0.109279	+	0.994011i	$0.534854\pi$
$272$	−10.3125	−0.625289
$273$	0	0
$274$	−2.22113	−0.134183
$275$	11.0783	0.668044
$276$	0	0
$277$	−7.60814	−0.457129	−0.228564	−	0.973529i	$-0.573403\pi$
$277$	−7.60814	−0.457129	−0.228564	+	0.973529i	$0.573403\pi$
$278$	1.23494	0.0740666
$279$	0	0
$280$	49.2482	2.94314
$281$	−14.3110	−0.853720	−0.426860	−	0.904318i	$-0.640380\pi$
$281$	−14.3110	−0.853720	−0.426860	+	0.904318i	$0.640380\pi$
$282$	0	0
$283$	−8.43236	−0.501252	−0.250626	−	0.968084i	$-0.580636\pi$
$283$	−8.43236	−0.501252	−0.250626	+	0.968084i	$0.580636\pi$
$284$	−1.24946	−0.0741420
$285$	0	0
$286$	4.72503	0.279397
$287$	−48.8700	−2.88470
$288$	0	0
$289$	−0.853947	−0.0502322
$290$	−4.08120	−0.239656
$291$	0	0
$292$	−1.78116	−0.104235
$293$	−16.1128	−0.941323	−0.470661	−	0.882314i	$-0.655985\pi$
$293$	−16.1128	−0.941323	−0.470661	+	0.882314i	$0.655985\pi$
$294$	0	0
$295$	−9.08654	−0.529039
$296$	3.54061	0.205794
$297$	0	0
$298$	11.6775	0.676460
$299$	2.33398	0.134977
$300$	0	0
$301$	−23.3957	−1.34851
$302$	−23.7029	−1.36395
$303$	0	0
$304$	−1.71019	−0.0980864
$305$	46.7932	2.67937
$306$	0	0
$307$	15.7393	0.898292	0.449146	−	0.893458i	$-0.351728\pi$
$307$	15.7393	0.898292	0.449146	+	0.893458i	$0.351728\pi$
$308$	−4.45314	−0.253741
$309$	0	0
$310$	17.8878	1.01596
$311$	2.20489	0.125028	0.0625139	−	0.998044i	$-0.480088\pi$
$311$	2.20489	0.125028	0.0625139	+	0.998044i	$0.480088\pi$
$312$	0	0
$313$	28.8786	1.63232	0.816159	−	0.577828i	$-0.196099\pi$
$313$	28.8786	1.63232	0.816159	+	0.577828i	$0.196099\pi$
$314$	−5.87898	−0.331770
$315$	0	0
$316$	5.97603	0.336178
$317$	21.0195	1.18057	0.590287	−	0.807194i	$-0.299015\pi$
$317$	21.0195	1.18057	0.590287	+	0.807194i	$0.299015\pi$
$318$	0	0
$319$	1.68704	0.0944560
$320$	29.9627	1.67496
$321$	0	0
$322$	5.65651	0.315225
$323$	2.67761	0.148986
$324$	0	0
$325$	−15.3265	−0.850161
$326$	−12.0433	−0.667015
$327$	0	0
$328$	−31.8492	−1.75858
$329$	60.3311	3.32616
$330$	0	0
$331$	8.10244	0.445350	0.222675	−	0.974893i	$-0.428521\pi$
$331$	8.10244	0.445350	0.222675	+	0.974893i	$0.428521\pi$
$332$	0.387106	0.0212452
$333$	0	0
$334$	22.2218	1.21592
$335$	26.0148	1.42134
$336$	0	0
$337$	34.0761	1.85625	0.928123	−	0.372274i	$-0.121422\pi$
$337$	34.0761	1.85625	0.928123	+	0.372274i	$0.121422\pi$
$338$	9.06311	0.492968
$339$	0	0
$340$	−7.65270	−0.415026
$341$	−7.39426	−0.400421
$342$	0	0
$343$	38.7435	2.09195
$344$	−15.2473	−0.822078
$345$	0	0
$346$	18.2237	0.979711
$347$	−1.19568	−0.0641876	−0.0320938	−	0.999485i	$-0.510218\pi$
$347$	−1.19568	−0.0641876	−0.0320938	+	0.999485i	$0.510218\pi$
$348$	0	0
$349$	−11.2766	−0.603621	−0.301810	−	0.953368i	$-0.597591\pi$
$349$	−11.2766	−0.603621	−0.301810	+	0.953368i	$0.597591\pi$
$350$	−37.1445	−1.98546
$351$	0	0
$352$	−5.16950	−0.275535
$353$	12.3799	0.658917	0.329459	−	0.944170i	$-0.393134\pi$
$353$	12.3799	0.658917	0.329459	+	0.944170i	$0.393134\pi$
$354$	0	0
$355$	7.58842	0.402751
$356$	−10.3163	−0.546763
$357$	0	0
$358$	2.96765	0.156845
$359$	32.1559	1.69713	0.848563	−	0.529094i	$-0.177468\pi$
$359$	32.1559	1.69713	0.848563	+	0.529094i	$0.177468\pi$
$360$	0	0
$361$	−18.5560	−0.976629
$362$	−30.9080	−1.62449
$363$	0	0
$364$	6.16082	0.322914
$365$	10.8176	0.566219
$366$	0	0
$367$	5.88840	0.307372	0.153686	−	0.988120i	$-0.450886\pi$
$367$	5.88840	0.307372	0.153686	+	0.988120i	$0.450886\pi$
$368$	2.56645	0.133785
$369$	0	0
$370$	−4.70376	−0.244537
$371$	−35.8449	−1.86097
$372$	0	0
$373$	23.5972	1.22182	0.610908	−	0.791701i	$-0.290804\pi$
$373$	23.5972	1.22182	0.610908	+	0.791701i	$0.290804\pi$
$374$	−8.13470	−0.420635
$375$	0	0
$376$	39.3185	2.02770
$377$	−2.33398	−0.120206
$378$	0	0
$379$	−30.2909	−1.55594	−0.777971	−	0.628300i	$-0.783751\pi$
$379$	−30.2909	−1.55594	−0.777971	+	0.628300i	$0.783751\pi$
$380$	−1.26910	−0.0651033
$381$	0	0
$382$	8.99759	0.460357
$383$	−3.83922	−0.196175	−0.0980875	−	0.995178i	$-0.531273\pi$
$383$	−3.83922	−0.196175	−0.0980875	+	0.995178i	$0.531273\pi$
$384$	0	0
$385$	27.0455	1.37836
$386$	4.95753	0.252332
$387$	0	0
$388$	−8.67731	−0.440524
$389$	14.5578	0.738109	0.369054	−	0.929408i	$-0.379682\pi$
$389$	14.5578	0.738109	0.369054	+	0.929408i	$0.379682\pi$
$390$	0	0
$391$	−4.01822	−0.203210
$392$	46.7536	2.36141
$393$	0	0
$394$	12.5202	0.630758
$395$	−36.2945	−1.82617
$396$	0	0
$397$	17.9270	0.899731	0.449865	−	0.893096i	$-0.351472\pi$
$397$	17.9270	0.899731	0.449865	+	0.893096i	$0.351472\pi$
$398$	−15.3431	−0.769079
$399$	0	0
$400$	−16.8531	−0.842653
$401$	24.1365	1.20532	0.602659	−	0.797999i	$-0.294108\pi$
$401$	24.1365	1.20532	0.602659	+	0.797999i	$0.294108\pi$
$402$	0	0
$403$	10.2298	0.509581
$404$	2.35101	0.116967
$405$	0	0
$406$	−5.65651	−0.280728
$407$	1.94438	0.0963795
$408$	0	0
$409$	−7.38670	−0.365249	−0.182625	−	0.983183i	$-0.558459\pi$
$409$	−7.38670	−0.365249	−0.182625	+	0.983183i	$0.558459\pi$
$410$	42.3121	2.08965
$411$	0	0
$412$	−1.37897	−0.0679368
$413$	−12.5939	−0.619704
$414$	0	0
$415$	−2.35103	−0.115408
$416$	7.15188	0.350650
$417$	0	0
$418$	−1.34903	−0.0659832
$419$	26.3086	1.28526	0.642629	−	0.766178i	$-0.277844\pi$
$419$	26.3086	1.28526	0.642629	+	0.766178i	$0.277844\pi$
$420$	0	0
$421$	−16.7376	−0.815739	−0.407869	−	0.913040i	$-0.633728\pi$
$421$	−16.7376	−0.815739	−0.407869	+	0.913040i	$0.633728\pi$
$422$	−32.7948	−1.59643
$423$	0	0
$424$	−23.3605	−1.13449
$425$	26.3864	1.27993
$426$	0	0
$427$	64.8550	3.13855
$428$	−8.72984	−0.421973
$429$	0	0
$430$	20.2563	0.976843
$431$	−32.6268	−1.57158	−0.785788	−	0.618495i	$-0.787743\pi$
$431$	−32.6268	−1.57158	−0.785788	+	0.618495i	$0.787743\pi$
$432$	0	0
$433$	−11.7337	−0.563885	−0.281943	−	0.959431i	$-0.590979\pi$
$433$	−11.7337	−0.563885	−0.281943	+	0.959431i	$0.590979\pi$
$434$	24.7924	1.19007
$435$	0	0
$436$	−5.89292	−0.282220
$437$	−0.666367	−0.0318767
$438$	0	0
$439$	3.11853	0.148839	0.0744196	−	0.997227i	$-0.476290\pi$
$439$	3.11853	0.148839	0.0744196	+	0.997227i	$0.476290\pi$
$440$	17.6259	0.840280
$441$	0	0
$442$	11.2542	0.535306
$443$	22.2472	1.05700	0.528499	−	0.848934i	$-0.322755\pi$
$443$	22.2472	1.05700	0.528499	+	0.848934i	$0.322755\pi$
$444$	0	0
$445$	62.6544	2.97010
$446$	25.9403	1.22831
$447$	0	0
$448$	41.5280	1.96201
$449$	−15.8159	−0.746401	−0.373200	−	0.927751i	$-0.621740\pi$
$449$	−15.8159	−0.746401	−0.373200	+	0.927751i	$0.621740\pi$
$450$	0	0
$451$	−17.4905	−0.823595
$452$	3.79050	0.178290
$453$	0	0
$454$	27.5714	1.29399
$455$	−37.4167	−1.75412
$456$	0	0
$457$	−19.0581	−0.891502	−0.445751	−	0.895157i	$-0.647063\pi$
$457$	−19.0581	−0.891502	−0.445751	+	0.895157i	$0.647063\pi$
$458$	6.28863	0.293848
$459$	0	0
$460$	1.90450	0.0887978
$461$	−3.63378	−0.169242	−0.0846209	−	0.996413i	$-0.526968\pi$
$461$	−3.63378	−0.169242	−0.0846209	+	0.996413i	$0.526968\pi$
$462$	0	0
$463$	−35.3741	−1.64397	−0.821986	−	0.569507i	$-0.807134\pi$
$463$	−35.3741	−1.64397	−0.821986	+	0.569507i	$0.807134\pi$
$464$	−2.56645	−0.119144
$465$	0	0
$466$	28.9736	1.34217
$467$	−29.0372	−1.34368	−0.671841	−	0.740695i	$-0.734496\pi$
$467$	−29.0372	−1.34368	−0.671841	+	0.740695i	$0.734496\pi$
$468$	0	0
$469$	36.0562	1.66492
$470$	−52.2353	−2.40944
$471$	0	0
$472$	−8.20758	−0.377784
$473$	−8.37329	−0.385004
$474$	0	0
$475$	4.37583	0.200777
$476$	−10.6066	−0.486151
$477$	0	0
$478$	−10.9063	−0.498842
$479$	29.9087	1.36656	0.683281	−	0.730156i	$-0.260553\pi$
$479$	29.9087	1.36656	0.683281	+	0.730156i	$0.260553\pi$
$480$	0	0
$481$	−2.69001	−0.122654
$482$	12.1742	0.554522
$483$	0	0
$484$	4.56607	0.207548
$485$	52.7003	2.39300
$486$	0	0
$487$	3.82616	0.173380	0.0866898	−	0.996235i	$-0.472371\pi$
$487$	3.82616	0.173380	0.0866898	+	0.996235i	$0.472371\pi$
$488$	42.2668	1.91333
$489$	0	0
$490$	−62.1130	−2.80598
$491$	9.80744	0.442604	0.221302	−	0.975205i	$-0.428969\pi$
$491$	9.80744	0.442604	0.221302	+	0.975205i	$0.428969\pi$
$492$	0	0
$493$	4.01822	0.180971
$494$	1.86635	0.0839711
$495$	0	0
$496$	11.2487	0.505081
$497$	10.5175	0.471773
$498$	0	0
$499$	26.3964	1.18166	0.590832	−	0.806794i	$-0.298799\pi$
$499$	26.3964	1.18166	0.590832	+	0.806794i	$0.298799\pi$
$500$	−2.98376	−0.133438
$501$	0	0
$502$	7.36607	0.328764
$503$	−4.26570	−0.190198	−0.0950991	−	0.995468i	$-0.530317\pi$
$503$	−4.26570	−0.190198	−0.0950991	+	0.995468i	$0.530317\pi$
$504$	0	0
$505$	−14.2785	−0.635385
$506$	2.02446	0.0899981
$507$	0	0
$508$	6.03361	0.267698
$509$	12.1846	0.540072	0.270036	−	0.962850i	$-0.412964\pi$
$509$	12.1846	0.540072	0.270036	+	0.962850i	$0.412964\pi$
$510$	0	0
$511$	14.9931	0.663256
$512$	23.6325	1.04442
$513$	0	0
$514$	8.10026	0.357287
$515$	8.37493	0.369044
$516$	0	0
$517$	21.5924	0.949633
$518$	−6.51937	−0.286445
$519$	0	0
$520$	−24.3850	−1.06935
$521$	38.2286	1.67482	0.837412	−	0.546572i	$-0.184067\pi$
$521$	38.2286	1.67482	0.837412	+	0.546572i	$0.184067\pi$
$522$	0	0
$523$	−10.7018	−0.467959	−0.233979	−	0.972242i	$-0.575175\pi$
$523$	−10.7018	−0.467959	−0.233979	+	0.972242i	$0.575175\pi$
$524$	4.13758	0.180751
$525$	0	0
$526$	−25.9658	−1.13216
$527$	−17.6118	−0.767180
$528$	0	0
$529$	1.00000	0.0434783
$530$	31.0349	1.34807
$531$	0	0
$532$	−1.75896	−0.0762605
$533$	24.1977	1.04812
$534$	0	0
$535$	53.0193	2.29222
$536$	23.4983	1.01497
$537$	0	0
$538$	21.7132	0.936125
$539$	25.6755	1.10592
$540$	0	0
$541$	19.2234	0.826479	0.413240	−	0.910622i	$-0.364397\pi$
$541$	19.2234	0.826479	0.413240	+	0.910622i	$0.364397\pi$
$542$	4.31751	0.185453
$543$	0	0
$544$	−12.3128	−0.527907
$545$	35.7897	1.53306
$546$	0	0
$547$	−31.5601	−1.34941	−0.674707	−	0.738086i	$-0.735730\pi$
$547$	−31.5601	−1.34941	−0.674707	+	0.738086i	$0.735730\pi$
$548$	−1.03649	−0.0442768
$549$	0	0
$550$	−13.2940	−0.566857
$551$	0.666367	0.0283882
$552$	0	0
$553$	−50.3039	−2.13914
$554$	9.12981	0.387889
$555$	0	0
$556$	0.576286	0.0244400
$557$	11.3207	0.479675	0.239837	−	0.970813i	$-0.422906\pi$
$557$	11.3207	0.479675	0.239837	+	0.970813i	$0.422906\pi$
$558$	0	0
$559$	11.5842	0.489961
$560$	−41.1435	−1.73863
$561$	0	0
$562$	17.1732	0.724409
$563$	−15.4891	−0.652788	−0.326394	−	0.945234i	$-0.605834\pi$
$563$	−15.4891	−0.652788	−0.326394	+	0.945234i	$0.605834\pi$
$564$	0	0
$565$	−23.0210	−0.968502
$566$	10.1189	0.425329
$567$	0	0
$568$	6.85437	0.287603
$569$	−36.6441	−1.53620	−0.768100	−	0.640330i	$-0.778797\pi$
$569$	−36.6441	−1.53620	−0.768100	+	0.640330i	$0.778797\pi$
$570$	0	0
$571$	−27.7270	−1.16034	−0.580170	−	0.814495i	$-0.697014\pi$
$571$	−27.7270	−1.16034	−0.580170	+	0.814495i	$0.697014\pi$
$572$	2.20495	0.0921935
$573$	0	0
$574$	58.6442	2.44776
$575$	−6.56669	−0.273850
$576$	0	0
$577$	−41.2565	−1.71753	−0.858765	−	0.512370i	$-0.828768\pi$
$577$	−41.2565	−1.71753	−0.858765	+	0.512370i	$0.828768\pi$
$578$	1.02474	0.0426236
$579$	0	0
$580$	−1.90450	−0.0790801
$581$	−3.25851	−0.135186
$582$	0	0
$583$	−12.8288	−0.531316
$584$	9.77119	0.404335
$585$	0	0
$586$	19.3355	0.798743
$587$	−19.5882	−0.808493	−0.404246	−	0.914650i	$-0.632466\pi$
$587$	−19.5882	−0.808493	−0.404246	+	0.914650i	$0.632466\pi$
$588$	0	0
$589$	−2.92067	−0.120344
$590$	10.9039	0.448907
$591$	0	0
$592$	−2.95794	−0.121571
$593$	−11.8088	−0.484928	−0.242464	−	0.970160i	$-0.577956\pi$
$593$	−11.8088	−0.484928	−0.242464	+	0.970160i	$0.577956\pi$
$594$	0	0
$595$	64.4173	2.64085
$596$	5.44934	0.223214
$597$	0	0
$598$	−2.80079	−0.114533
$599$	−0.892995	−0.0364868	−0.0182434	−	0.999834i	$-0.505807\pi$
$599$	−0.892995	−0.0364868	−0.0182434	+	0.999834i	$0.505807\pi$
$600$	0	0
$601$	12.8302	0.523357	0.261678	−	0.965155i	$-0.415724\pi$
$601$	12.8302	0.523357	0.261678	+	0.965155i	$0.415724\pi$
$602$	28.0750	1.14425
$603$	0	0
$604$	−11.0610	−0.450067
$605$	−27.7313	−1.12744
$606$	0	0
$607$	2.83648	0.115129	0.0575646	−	0.998342i	$-0.481666\pi$
$607$	2.83648	0.115129	0.0575646	+	0.998342i	$0.481666\pi$
$608$	−2.04191	−0.0828105
$609$	0	0
$610$	−56.1521	−2.27353
$611$	−29.8726	−1.20851
$612$	0	0
$613$	23.5328	0.950481	0.475241	−	0.879856i	$-0.342361\pi$
$613$	23.5328	0.950481	0.475241	+	0.879856i	$0.342361\pi$
$614$	−18.8873	−0.762229
$615$	0	0
$616$	24.4293	0.984284
$617$	−4.25178	−0.171170	−0.0855850	−	0.996331i	$-0.527276\pi$
$617$	−4.25178	−0.171170	−0.0855850	+	0.996331i	$0.527276\pi$
$618$	0	0
$619$	0.559468	0.0224869	0.0112435	−	0.999937i	$-0.496421\pi$
$619$	0.559468	0.0224869	0.0112435	+	0.999937i	$0.496421\pi$
$620$	8.34739	0.335239
$621$	0	0
$622$	−2.64588	−0.106090
$623$	86.8385	3.47911
$624$	0	0
$625$	−14.7120	−0.588481
$626$	−34.6545	−1.38507
$627$	0	0
$628$	−2.74344	−0.109475
$629$	4.63117	0.184657
$630$	0	0
$631$	−13.4967	−0.537297	−0.268648	−	0.963238i	$-0.586577\pi$
$631$	−13.4967	−0.537297	−0.268648	+	0.963238i	$0.586577\pi$
$632$	−32.7836	−1.30406
$633$	0	0
$634$	−25.2235	−1.00175
$635$	−36.6442	−1.45418
$636$	0	0
$637$	−35.5214	−1.40741
$638$	−2.02446	−0.0801490
$639$	0	0
$640$	−15.1125	−0.597374
$641$	−1.13418	−0.0447974	−0.0223987	−	0.999749i	$-0.507130\pi$
$641$	−1.13418	−0.0447974	−0.0223987	+	0.999749i	$0.507130\pi$
$642$	0	0
$643$	−24.1459	−0.952221	−0.476111	−	0.879385i	$-0.657954\pi$
$643$	−24.1459	−0.952221	−0.476111	+	0.879385i	$0.657954\pi$
$644$	2.63962	0.104016
$645$	0	0
$646$	−3.21314	−0.126419
$647$	−35.7333	−1.40482	−0.702410	−	0.711772i	$-0.747893\pi$
$647$	−35.7333	−1.40482	−0.702410	+	0.711772i	$0.747893\pi$
$648$	0	0
$649$	−4.50733	−0.176928
$650$	18.3919	0.721389
$651$	0	0
$652$	−5.62002	−0.220097
$653$	47.6682	1.86540	0.932701	−	0.360651i	$-0.117446\pi$
$653$	47.6682	1.86540	0.932701	+	0.360651i	$0.117446\pi$
$654$	0	0
$655$	−25.1289	−0.981868
$656$	26.6078	1.03886
$657$	0	0
$658$	−72.3977	−2.82236
$659$	−21.5575	−0.839761	−0.419881	−	0.907579i	$-0.637928\pi$
$659$	−21.5575	−0.839761	−0.419881	+	0.907579i	$0.637928\pi$
$660$	0	0
$661$	32.2523	1.25447	0.627235	−	0.778830i	$-0.284187\pi$
$661$	32.2523	1.25447	0.627235	+	0.778830i	$0.284187\pi$
$662$	−9.72298	−0.377894
$663$	0	0
$664$	−2.12361	−0.0824120
$665$	10.6827	0.414259
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	10.3698	0.401221
$669$	0	0
$670$	−31.2179	−1.20605
$671$	23.2115	0.896070
$672$	0	0
$673$	45.7173	1.76227	0.881137	−	0.472862i	$-0.156779\pi$
$673$	45.7173	1.76227	0.881137	+	0.472862i	$0.156779\pi$
$674$	−40.8916	−1.57508
$675$	0	0
$676$	4.22932	0.162666
$677$	−12.5398	−0.481944	−0.240972	−	0.970532i	$-0.577466\pi$
$677$	−12.5398	−0.481944	−0.240972	+	0.970532i	$0.577466\pi$
$678$	0	0
$679$	73.0421	2.80310
$680$	41.9816	1.60992
$681$	0	0
$682$	8.87315	0.339771
$683$	−22.1288	−0.846735	−0.423367	−	0.905958i	$-0.639152\pi$
$683$	−22.1288	−0.846735	−0.423367	+	0.905958i	$0.639152\pi$
$684$	0	0
$685$	6.29498	0.240519
$686$	−46.4924	−1.77509
$687$	0	0
$688$	12.7381	0.485634
$689$	17.7484	0.676159
$690$	0	0
$691$	−6.35727	−0.241842	−0.120921	−	0.992662i	$-0.538585\pi$
$691$	−6.35727	−0.241842	−0.120921	+	0.992662i	$0.538585\pi$
$692$	8.50412	0.323278
$693$	0	0
$694$	1.43483	0.0544652
$695$	−3.49998	−0.132762
$696$	0	0
$697$	−41.6591	−1.57795
$698$	13.5320	0.512192
$699$	0	0
$700$	−17.3336	−0.655148
$701$	34.3393	1.29698	0.648488	−	0.761225i	$-0.275402\pi$
$701$	34.3393	1.29698	0.648488	+	0.761225i	$0.275402\pi$
$702$	0	0
$703$	0.768016	0.0289663
$704$	14.8628	0.560163
$705$	0	0
$706$	−14.8560	−0.559113
$707$	−19.7899	−0.744275
$708$	0	0
$709$	17.3034	0.649841	0.324921	−	0.945741i	$-0.394662\pi$
$709$	17.3034	0.649841	0.324921	+	0.945741i	$0.394662\pi$
$710$	−9.10615	−0.341748
$711$	0	0
$712$	56.5937	2.12094
$713$	4.38298	0.164144
$714$	0	0
$715$	−13.3914	−0.500810
$716$	1.38486	0.0517547
$717$	0	0
$718$	−38.5873	−1.44007
$719$	9.73697	0.363128	0.181564	−	0.983379i	$-0.441884\pi$
$719$	9.73697	0.363128	0.181564	+	0.983379i	$0.441884\pi$
$720$	0	0
$721$	11.6076	0.432289
$722$	22.2673	0.828702
$723$	0	0
$724$	−14.4233	−0.536037
$725$	6.56669	0.243881
$726$	0	0
$727$	−35.0537	−1.30007	−0.650035	−	0.759904i	$-0.725246\pi$
$727$	−35.0537	−1.30007	−0.650035	+	0.759904i	$0.725246\pi$
$728$	−33.7973	−1.25261
$729$	0	0
$730$	−12.9812	−0.480456
$731$	−19.9436	−0.737642
$732$	0	0
$733$	15.8403	0.585076	0.292538	−	0.956254i	$-0.405500\pi$
$733$	15.8403	0.585076	0.292538	+	0.956254i	$0.405500\pi$
$734$	−7.06611	−0.260815
$735$	0	0
$736$	3.06425	0.112950
$737$	12.9045	0.475342
$738$	0	0
$739$	39.2829	1.44504	0.722522	−	0.691348i	$-0.242983\pi$
$739$	39.2829	1.44504	0.722522	+	0.691348i	$0.242983\pi$
$740$	−2.19502	−0.0806905
$741$	0	0
$742$	43.0140	1.57910
$743$	−29.5318	−1.08342	−0.541708	−	0.840567i	$-0.682222\pi$
$743$	−29.5318	−1.08342	−0.541708	+	0.840567i	$0.682222\pi$
$744$	0	0
$745$	−33.0957	−1.21253
$746$	−28.3168	−1.03675
$747$	0	0
$748$	−3.79608	−0.138798
$749$	73.4843	2.68506
$750$	0	0
$751$	−6.99607	−0.255290	−0.127645	−	0.991820i	$-0.540742\pi$
$751$	−6.99607	−0.255290	−0.127645	+	0.991820i	$0.540742\pi$
$752$	−32.8479	−1.19784
$753$	0	0
$754$	2.80079	0.101999
$755$	67.1774	2.44484
$756$	0	0
$757$	−5.79327	−0.210560	−0.105280	−	0.994443i	$-0.533574\pi$
$757$	−5.79327	−0.210560	−0.105280	+	0.994443i	$0.533574\pi$
$758$	36.3493	1.32027
$759$	0	0
$760$	6.96208	0.252541
$761$	−33.1283	−1.20090	−0.600450	−	0.799662i	$-0.705012\pi$
$761$	−33.1283	−1.20090	−0.600450	+	0.799662i	$0.705012\pi$
$762$	0	0
$763$	49.6042	1.79579
$764$	4.19875	0.151905
$765$	0	0
$766$	4.60709	0.166461
$767$	6.23578	0.225161
$768$	0	0
$769$	−52.3541	−1.88794	−0.943968	−	0.330037i	$-0.892939\pi$
$769$	−52.3541	−1.88794	−0.943968	+	0.330037i	$0.892939\pi$
$770$	−32.4547	−1.16959
$771$	0	0
$772$	2.31344	0.0832627
$773$	−53.7465	−1.93313	−0.966563	−	0.256429i	$-0.917454\pi$
$773$	−53.7465	−1.93313	−0.966563	+	0.256429i	$0.917454\pi$
$774$	0	0
$775$	−28.7817	−1.03387
$776$	47.6025	1.70883
$777$	0	0
$778$	−17.4694	−0.626309
$779$	−6.90861	−0.247526
$780$	0	0
$781$	3.76419	0.134693
$782$	4.82188	0.172430
$783$	0	0
$784$	−39.0595	−1.39498
$785$	16.6618	0.594686
$786$	0	0
$787$	0.641404	0.0228636	0.0114318	−	0.999935i	$-0.496361\pi$
$787$	0.641404	0.0228636	0.0114318	+	0.999935i	$0.496361\pi$
$788$	5.84257	0.208133
$789$	0	0
$790$	43.5536	1.54957
$791$	−31.9069	−1.13448
$792$	0	0
$793$	−32.1125	−1.14035
$794$	−21.5125	−0.763451
$795$	0	0
$796$	−7.15988	−0.253775
$797$	−12.9649	−0.459240	−0.229620	−	0.973280i	$-0.573748\pi$
$797$	−12.9649	−0.459240	−0.229620	+	0.973280i	$0.573748\pi$
$798$	0	0
$799$	51.4292	1.81943
$800$	−20.1220	−0.711419
$801$	0	0
$802$	−28.9639	−1.02275
$803$	5.36601	0.189362
$804$	0	0
$805$	−16.0313	−0.565030
$806$	−12.2758	−0.432396
$807$	0	0
$808$	−12.8973	−0.453726
$809$	18.7074	0.657719	0.328859	−	0.944379i	$-0.393336\pi$
$809$	18.7074	0.657719	0.328859	+	0.944379i	$0.393336\pi$
$810$	0	0
$811$	7.50908	0.263679	0.131840	−	0.991271i	$-0.457912\pi$
$811$	7.50908	0.263679	0.131840	+	0.991271i	$0.457912\pi$
$812$	−2.63962	−0.0926326
$813$	0	0
$814$	−2.33327	−0.0817812
$815$	34.1323	1.19560
$816$	0	0
$817$	−3.30738	−0.115711
$818$	8.86409	0.309926
$819$	0	0
$820$	19.7451	0.689527
$821$	−21.9545	−0.766217	−0.383108	−	0.923703i	$-0.625146\pi$
$821$	−21.9545	−0.766217	−0.383108	+	0.923703i	$0.625146\pi$
$822$	0	0
$823$	−11.2046	−0.390567	−0.195284	−	0.980747i	$-0.562563\pi$
$823$	−11.2046	−0.390567	−0.195284	+	0.980747i	$0.562563\pi$
$824$	7.56481	0.263533
$825$	0	0
$826$	15.1127	0.525839
$827$	21.9758	0.764172	0.382086	−	0.924127i	$-0.375206\pi$
$827$	21.9758	0.764172	0.382086	+	0.924127i	$0.375206\pi$
$828$	0	0
$829$	8.71280	0.302608	0.151304	−	0.988487i	$-0.451653\pi$
$829$	8.71280	0.302608	0.151304	+	0.988487i	$0.451653\pi$
$830$	2.82125	0.0979270
$831$	0	0
$832$	−20.5623	−0.712871
$833$	61.1544	2.11887
$834$	0	0
$835$	−62.9796	−2.17950
$836$	−0.629528	−0.0217727
$837$	0	0
$838$	−31.5704	−1.09058
$839$	32.2052	1.11185	0.555924	−	0.831233i	$-0.312365\pi$
$839$	32.2052	1.11185	0.555924	+	0.831233i	$0.312365\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	20.0852	0.692181
$843$	0	0
$844$	−15.3038	−0.526777
$845$	−25.6861	−0.883629
$846$	0	0
$847$	−38.4353	−1.32065
$848$	19.5161	0.670187
$849$	0	0
$850$	−31.6638	−1.08606
$851$	−1.15254	−0.0395087
$852$	0	0
$853$	−41.5747	−1.42349	−0.711746	−	0.702437i	$-0.752095\pi$
$853$	−41.5747	−1.42349	−0.711746	+	0.702437i	$0.752095\pi$
$854$	−77.8263	−2.66316
$855$	0	0
$856$	47.8906	1.63687
$857$	−9.16308	−0.313005	−0.156502	−	0.987678i	$-0.550022\pi$
$857$	−9.16308	−0.313005	−0.156502	+	0.987678i	$0.550022\pi$
$858$	0	0
$859$	−5.91441	−0.201797	−0.100899	−	0.994897i	$-0.532172\pi$
$859$	−5.91441	−0.201797	−0.100899	+	0.994897i	$0.532172\pi$
$860$	9.45263	0.322332
$861$	0	0
$862$	39.1523	1.33353
$863$	−27.1483	−0.924138	−0.462069	−	0.886844i	$-0.652893\pi$
$863$	−27.1483	−0.924138	−0.462069	+	0.886844i	$0.652893\pi$
$864$	0	0
$865$	−51.6485	−1.75610
$866$	14.0805	0.478475
$867$	0	0
$868$	11.5694	0.392691
$869$	−18.0037	−0.610733
$870$	0	0
$871$	−17.8530	−0.604926
$872$	32.3277	1.09475
$873$	0	0
$874$	0.799644	0.0270484
$875$	25.1161	0.849080
$876$	0	0
$877$	−40.1031	−1.35418	−0.677092	−	0.735898i	$-0.736760\pi$
$877$	−40.1031	−1.35418	−0.677092	+	0.735898i	$0.736760\pi$
$878$	−3.74225	−0.126295
$879$	0	0
$880$	−14.7252	−0.496387
$881$	−5.11124	−0.172202	−0.0861010	−	0.996286i	$-0.527441\pi$
$881$	−5.11124	−0.172202	−0.0861010	+	0.996286i	$0.527441\pi$
$882$	0	0
$883$	−52.3717	−1.76245	−0.881224	−	0.472699i	$-0.843280\pi$
$883$	−52.3717	−1.76245	−0.881224	+	0.472699i	$0.843280\pi$
$884$	5.25178	0.176636
$885$	0	0
$886$	−26.6968	−0.896897
$887$	1.99354	0.0669366	0.0334683	−	0.999440i	$-0.489345\pi$
$887$	1.99354	0.0669366	0.0334683	+	0.999440i	$0.489345\pi$
$888$	0	0
$889$	−50.7885	−1.70339
$890$	−75.1857	−2.52023
$891$	0	0
$892$	12.1051	0.405309
$893$	8.52883	0.285407
$894$	0	0
$895$	−8.41074	−0.281140
$896$	−20.9458	−0.699750
$897$	0	0
$898$	18.9792	0.633345
$899$	−4.38298	−0.146181
$900$	0	0
$901$	−30.5559	−1.01796
$902$	20.9887	0.698847
$903$	0	0
$904$	−20.7941	−0.691603
$905$	87.5975	2.91184
$906$	0	0
$907$	−53.3044	−1.76994	−0.884972	−	0.465645i	$-0.845822\pi$
$907$	−53.3044	−1.76994	−0.884972	+	0.465645i	$0.845822\pi$
$908$	12.8663	0.426982
$909$	0	0
$910$	44.9003	1.48843
$911$	13.4627	0.446038	0.223019	−	0.974814i	$-0.428409\pi$
$911$	13.4627	0.446038	0.223019	+	0.974814i	$0.428409\pi$
$912$	0	0
$913$	−1.16621	−0.0385961
$914$	22.8699	0.756468
$915$	0	0
$916$	2.93460	0.0969620
$917$	−34.8285	−1.15014
$918$	0	0
$919$	−51.1630	−1.68771	−0.843856	−	0.536570i	$-0.819720\pi$
$919$	−51.1630	−1.68771	−0.843856	+	0.536570i	$0.819720\pi$
$920$	−10.4478	−0.344454
$921$	0	0
$922$	4.36055	0.143607
$923$	−5.20766	−0.171412
$924$	0	0
$925$	7.56839	0.248847
$926$	42.4491	1.39496
$927$	0	0
$928$	−3.06425	−0.100589
$929$	24.7320	0.811429	0.405715	−	0.914000i	$-0.367023\pi$
$929$	24.7320	0.811429	0.405715	+	0.914000i	$0.367023\pi$
$930$	0	0
$931$	10.1416	0.332378
$932$	13.5206	0.442881
$933$	0	0
$934$	34.8448	1.14016
$935$	23.0549	0.753975
$936$	0	0
$937$	25.2196	0.823887	0.411944	−	0.911209i	$-0.364850\pi$
$937$	25.2196	0.823887	0.411944	+	0.911209i	$0.364850\pi$
$938$	−43.2677	−1.41274
$939$	0	0
$940$	−24.3757	−0.795048
$941$	−10.9290	−0.356275	−0.178137	−	0.984006i	$-0.557007\pi$
$941$	−10.9290	−0.356275	−0.178137	+	0.984006i	$0.557007\pi$
$942$	0	0
$943$	10.3676	0.337615
$944$	6.85687	0.223172
$945$	0	0
$946$	10.0480	0.326689
$947$	43.0723	1.39966	0.699831	−	0.714309i	$-0.253259\pi$
$947$	43.0723	1.39966	0.699831	+	0.714309i	$0.253259\pi$
$948$	0	0
$949$	−7.42375	−0.240985
$950$	−5.25102	−0.170365
$951$	0	0
$952$	58.1861	1.88582
$953$	56.3699	1.82600	0.913000	−	0.407959i	$-0.133759\pi$
$953$	56.3699	1.82600	0.913000	+	0.407959i	$0.133759\pi$
$954$	0	0
$955$	−25.5004	−0.825174
$956$	−5.08944	−0.164604
$957$	0	0
$958$	−35.8906	−1.15957
$959$	8.72479	0.281738
$960$	0	0
$961$	−11.7895	−0.380306
$962$	3.22803	0.104076
$963$	0	0
$964$	5.68114	0.182977
$965$	−14.0503	−0.452296
$966$	0	0
$967$	11.8068	0.379681	0.189841	−	0.981815i	$-0.439203\pi$
$967$	11.8068	0.379681	0.189841	+	0.981815i	$0.439203\pi$
$968$	−25.0488	−0.805098
$969$	0	0
$970$	−63.2407	−2.03054
$971$	−30.1555	−0.967736	−0.483868	−	0.875141i	$-0.660769\pi$
$971$	−30.1555	−0.967736	−0.483868	+	0.875141i	$0.660769\pi$
$972$	0	0
$973$	−4.85095	−0.155514
$974$	−4.59141	−0.147118
$975$	0	0
$976$	−35.3110	−1.13028
$977$	−14.9369	−0.477874	−0.238937	−	0.971035i	$-0.576799\pi$
$977$	−14.9369	−0.477874	−0.238937	+	0.971035i	$0.576799\pi$
$978$	0	0
$979$	31.0794	0.993301
$980$	−28.9851	−0.925897
$981$	0	0
$982$	−11.7690	−0.375564
$983$	34.6885	1.10639	0.553196	−	0.833051i	$-0.313408\pi$
$983$	34.6885	1.10639	0.553196	+	0.833051i	$0.313408\pi$
$984$	0	0
$985$	−35.4840	−1.13061
$986$	−4.82188	−0.153560
$987$	0	0
$988$	0.870937	0.0277082
$989$	4.96331	0.157824
$990$	0	0
$991$	51.0374	1.62126	0.810628	−	0.585561i	$-0.199126\pi$
$991$	51.0374	1.62126	0.810628	+	0.585561i	$0.199126\pi$
$992$	13.4305	0.426420
$993$	0	0
$994$	−12.6210	−0.400315
$995$	43.4844	1.37855
$996$	0	0
$997$	49.5504	1.56928	0.784638	−	0.619954i	$-0.212849\pi$
$997$	49.5504	1.56928	0.784638	+	0.619954i	$0.212849\pi$
$998$	−31.6758	−1.00268
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.w.1.10	yes	30
3.2	odd	2		6003.2.a.v.1.21	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.21	✓	30	3.2	odd	2
6003.2.a.w.1.10	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)

\(f(q)\)	\(=\)	\(q-1.20001 q^{2} -0.559985 q^{4} +3.40098 q^{5} +4.71373 q^{7} +3.07200 q^{8} +O(q^{10})\)	\(q-1.20001 q^{2} -0.559985 q^{4} +3.40098 q^{5} +4.71373 q^{7} +3.07200 q^{8} -4.08120 q^{10} +1.68704 q^{11} -2.33398 q^{13} -5.65651 q^{14} -2.56645 q^{16} +4.01822 q^{17} +0.666367 q^{19} -1.90450 q^{20} -2.02446 q^{22} -1.00000 q^{23} +6.56669 q^{25} +2.80079 q^{26} -2.63962 q^{28} +1.00000 q^{29} -4.38298 q^{31} -3.06425 q^{32} -4.82188 q^{34} +16.0313 q^{35} +1.15254 q^{37} -0.799644 q^{38} +10.4478 q^{40} -10.3676 q^{41} -4.96331 q^{43} -0.944717 q^{44} +1.20001 q^{46} +12.7990 q^{47} +15.2193 q^{49} -7.88007 q^{50} +1.30699 q^{52} -7.60435 q^{53} +5.73759 q^{55} +14.4806 q^{56} -1.20001 q^{58} -2.67174 q^{59} +13.7587 q^{61} +5.25960 q^{62} +8.81000 q^{64} -7.93781 q^{65} +7.64919 q^{67} -2.25014 q^{68} -19.2377 q^{70} +2.23124 q^{71} +3.18073 q^{73} -1.38306 q^{74} -0.373156 q^{76} +7.95225 q^{77} -10.6718 q^{79} -8.72844 q^{80} +12.4411 q^{82} -0.691279 q^{83} +13.6659 q^{85} +5.95600 q^{86} +5.18258 q^{88} +18.4224 q^{89} -11.0017 q^{91} +0.559985 q^{92} -15.3589 q^{94} +2.26630 q^{95} +15.4956 q^{97} -18.2632 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) 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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