LMFDB - Embedded newform 6003.2.a.u.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-1.93234 q^{2} +1.73393 q^{4} -2.82056 q^{5} -3.97268 q^{7} +0.514145 q^{8} +O(q^{10})$	\(q-1.93234 q^{2} +1.73393 q^{4} -2.82056 q^{5} -3.97268 q^{7} +0.514145 q^{8} +5.45026 q^{10} -4.70087 q^{11} -5.71953 q^{13} +7.67656 q^{14} -4.46135 q^{16} +5.11092 q^{17} +0.224637 q^{19} -4.89063 q^{20} +9.08366 q^{22} -1.00000 q^{23} +2.95554 q^{25} +11.0521 q^{26} -6.88834 q^{28} -1.00000 q^{29} +5.16107 q^{31} +7.59255 q^{32} -9.87602 q^{34} +11.2052 q^{35} -5.77440 q^{37} -0.434074 q^{38} -1.45017 q^{40} -0.469446 q^{41} +2.68640 q^{43} -8.15096 q^{44} +1.93234 q^{46} +12.2820 q^{47} +8.78220 q^{49} -5.71109 q^{50} -9.91724 q^{52} -3.63780 q^{53} +13.2591 q^{55} -2.04253 q^{56} +1.93234 q^{58} +8.78187 q^{59} +4.68255 q^{61} -9.97293 q^{62} -5.74865 q^{64} +16.1323 q^{65} -15.9642 q^{67} +8.86195 q^{68} -21.6522 q^{70} +10.0253 q^{71} +8.41059 q^{73} +11.1581 q^{74} +0.389504 q^{76} +18.6751 q^{77} -12.9991 q^{79} +12.5835 q^{80} +0.907129 q^{82} +11.6167 q^{83} -14.4156 q^{85} -5.19102 q^{86} -2.41693 q^{88} +10.7508 q^{89} +22.7219 q^{91} -1.73393 q^{92} -23.7330 q^{94} -0.633601 q^{95} -5.53357 q^{97} -16.9702 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) $ 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8	$ 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8} - 12 q^{10} - 28 q^{13} + q^{14} + 3 q^{16} + 10 q^{17} - 8 q^{19} - 11 q^{22} - 22 q^{23} - 11 q^{26} - 21 q^{28} - 22 q^{29} - 18 q^{31} - 5 q^{32} - 33 q^{34} - 2 q^{35} - 28 q^{37} - 14 q^{38} - 30 q^{40} + 10 q^{41} - 14 q^{43} - 37 q^{44} - 3 q^{46} + 18 q^{47} + 2 q^{49} - 7 q^{50} - 57 q^{52} - 20 q^{53} - 42 q^{55} + 2 q^{56} - 3 q^{58} + 20 q^{59} - 38 q^{61} - 4 q^{62} - 24 q^{64} - 12 q^{65} - 50 q^{67} - 11 q^{68} - 48 q^{70} - 12 q^{71} - 46 q^{73} + 6 q^{74} - 16 q^{76} + 14 q^{77} - 20 q^{79} + 58 q^{80} - 42 q^{82} - 22 q^{83} - 66 q^{85} - 22 q^{86} - 68 q^{88} + 14 q^{89} - 16 q^{91} - 17 q^{92} - 27 q^{94} + 20 q^{95} - 48 q^{97} + 28 q^{98}+O(q^{100}) $ 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8 - 12 * q^10 - 28 * q^13 + q^14 + 3 * q^16 + 10 * q^17 - 8 * q^19 - 11 * q^22 - 22 * q^23 - 11 * q^26 - 21 * q^28 - 22 * q^29 - 18 * q^31 - 5 * q^32 - 33 * q^34 - 2 * q^35 - 28 * q^37 - 14 * q^38 - 30 * q^40 + 10 * q^41 - 14 * q^43 - 37 * q^44 - 3 * q^46 + 18 * q^47 + 2 * q^49 - 7 * q^50 - 57 * q^52 - 20 * q^53 - 42 * q^55 + 2 * q^56 - 3 * q^58 + 20 * q^59 - 38 * q^61 - 4 * q^62 - 24 * q^64 - 12 * q^65 - 50 * q^67 - 11 * q^68 - 48 * q^70 - 12 * q^71 - 46 * q^73 + 6 * q^74 - 16 * q^76 + 14 * q^77 - 20 * q^79 + 58 * q^80 - 42 * q^82 - 22 * q^83 - 66 * q^85 - 22 * q^86 - 68 * q^88 + 14 * q^89 - 16 * q^91 - 17 * q^92 - 27 * q^94 + 20 * q^95 - 48 * q^97 + 28 * 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 $

$2$	−1.93234	−1.36637	−0.683184	−	0.730246i	$-0.739405\pi$
$2$	−1.93234	−1.36637	−0.683184	+	0.730246i	$0.739405\pi$
$3$	0	0
$3$	0	0
$4$	1.73393	0.866963
$5$	−2.82056	−1.26139	−0.630696	−	0.776030i	$-0.717230\pi$
$5$	−2.82056	−1.26139	−0.630696	+	0.776030i	$0.717230\pi$
$6$	0	0
$7$	−3.97268	−1.50153	−0.750766	−	0.660568i	$-0.770316\pi$
$7$	−3.97268	−1.50153	−0.750766	+	0.660568i	$0.770316\pi$
$8$	0.514145	0.181778
$9$	0	0
$10$	5.45026	1.72352
$11$	−4.70087	−1.41737	−0.708683	−	0.705527i	$-0.750710\pi$
$11$	−4.70087	−1.41737	−0.708683	+	0.705527i	$0.750710\pi$
$12$	0	0
$13$	−5.71953	−1.58631	−0.793156	−	0.609018i	$-0.791564\pi$
$13$	−5.71953	−1.58631	−0.793156	+	0.609018i	$0.791564\pi$
$14$	7.67656	2.05165
$15$	0	0
$16$	−4.46135	−1.11534
$17$	5.11092	1.23958	0.619790	−	0.784768i	$-0.287218\pi$
$17$	5.11092	1.23958	0.619790	+	0.784768i	$0.287218\pi$
$18$	0	0
$19$	0.224637	0.0515353	0.0257676	−	0.999668i	$-0.491797\pi$
$19$	0.224637	0.0515353	0.0257676	+	0.999668i	$0.491797\pi$
$20$	−4.89063	−1.09358
$21$	0	0
$22$	9.08366	1.93664
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	2.95554	0.591107
$26$	11.0521	2.16749
$27$	0	0
$28$	−6.88834	−1.30177
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.16107	0.926956	0.463478	−	0.886108i	$-0.346601\pi$
$31$	5.16107	0.926956	0.463478	+	0.886108i	$0.346601\pi$
$32$	7.59255	1.34219
$33$	0	0
$34$	−9.87602	−1.69372
$35$	11.2052	1.89402
$36$	0	0
$37$	−5.77440	−0.949305	−0.474652	−	0.880173i	$-0.657426\pi$
$37$	−5.77440	−0.949305	−0.474652	+	0.880173i	$0.657426\pi$
$38$	−0.434074	−0.0704162
$39$	0	0
$40$	−1.45017	−0.229293
$41$	−0.469446	−0.0733152	−0.0366576	−	0.999328i	$-0.511671\pi$
$41$	−0.469446	−0.0733152	−0.0366576	+	0.999328i	$0.511671\pi$
$42$	0	0
$43$	2.68640	0.409672	0.204836	−	0.978796i	$-0.434334\pi$
$43$	2.68640	0.409672	0.204836	+	0.978796i	$0.434334\pi$
$44$	−8.15096	−1.22880
$45$	0	0
$46$	1.93234	0.284908
$47$	12.2820	1.79152	0.895759	−	0.444540i	$-0.146633\pi$
$47$	12.2820	1.79152	0.895759	+	0.444540i	$0.146633\pi$
$48$	0	0
$49$	8.78220	1.25460
$50$	−5.71109	−0.807670
$51$	0	0
$52$	−9.91724	−1.37527
$53$	−3.63780	−0.499690	−0.249845	−	0.968286i	$-0.580380\pi$
$53$	−3.63780	−0.499690	−0.249845	+	0.968286i	$0.580380\pi$
$54$	0	0
$55$	13.2591	1.78785
$56$	−2.04253	−0.272945
$57$	0	0
$58$	1.93234	0.253728
$59$	8.78187	1.14330	0.571651	−	0.820497i	$-0.306303\pi$
$59$	8.78187	1.14330	0.571651	+	0.820497i	$0.306303\pi$
$60$	0	0
$61$	4.68255	0.599539	0.299770	−	0.954012i	$-0.403090\pi$
$61$	4.68255	0.599539	0.299770	+	0.954012i	$0.403090\pi$
$62$	−9.97293	−1.26656
$63$	0	0
$64$	−5.74865	−0.718582
$65$	16.1323	2.00096
$66$	0	0
$67$	−15.9642	−1.95034	−0.975170	−	0.221457i	$-0.928919\pi$
$67$	−15.9642	−1.95034	−0.975170	+	0.221457i	$0.928919\pi$
$68$	8.86195	1.07467
$69$	0	0
$70$	−21.6522	−2.58793
$71$	10.0253	1.18979	0.594894	−	0.803804i	$-0.297194\pi$
$71$	10.0253	1.18979	0.594894	+	0.803804i	$0.297194\pi$
$72$	0	0
$73$	8.41059	0.984385	0.492193	−	0.870486i	$-0.336196\pi$
$73$	8.41059	0.984385	0.492193	+	0.870486i	$0.336196\pi$
$74$	11.1581	1.29710
$75$	0	0
$76$	0.389504	0.0446792
$77$	18.6751	2.12822
$78$	0	0
$79$	−12.9991	−1.46251	−0.731254	−	0.682105i	$-0.761065\pi$
$79$	−12.9991	−1.46251	−0.731254	+	0.682105i	$0.761065\pi$
$80$	12.5835	1.40688
$81$	0	0
$82$	0.907129	0.100176
$83$	11.6167	1.27509	0.637547	−	0.770412i	$-0.279949\pi$
$83$	11.6167	1.27509	0.637547	+	0.770412i	$0.279949\pi$
$84$	0	0
$85$	−14.4156	−1.56360
$86$	−5.19102	−0.559762
$87$	0	0
$88$	−2.41693	−0.257645
$89$	10.7508	1.13958	0.569790	−	0.821790i	$-0.307025\pi$
$89$	10.7508	1.13958	0.569790	+	0.821790i	$0.307025\pi$
$90$	0	0
$91$	22.7219	2.38190
$92$	−1.73393	−0.180774
$93$	0	0
$94$	−23.7330	−2.44787
$95$	−0.633601	−0.0650061
$96$	0	0
$97$	−5.53357	−0.561849	−0.280924	−	0.959730i	$-0.590641\pi$
$97$	−5.53357	−0.561849	−0.280924	+	0.959730i	$0.590641\pi$
$98$	−16.9702	−1.71425
$99$	0	0
$100$	5.12468	0.512468
$101$	−0.789577	−0.0785659	−0.0392829	−	0.999228i	$-0.512507\pi$
$101$	−0.789577	−0.0785659	−0.0392829	+	0.999228i	$0.512507\pi$
$102$	0	0
$103$	−1.12240	−0.110593	−0.0552967	−	0.998470i	$-0.517610\pi$
$103$	−1.12240	−0.110593	−0.0552967	+	0.998470i	$0.517610\pi$
$104$	−2.94067	−0.288356
$105$	0	0
$106$	7.02945	0.682761
$107$	−9.39437	−0.908188	−0.454094	−	0.890954i	$-0.650037\pi$
$107$	−9.39437	−0.908188	−0.454094	+	0.890954i	$0.650037\pi$
$108$	0	0
$109$	8.34828	0.799620	0.399810	−	0.916598i	$-0.369076\pi$
$109$	8.34828	0.799620	0.399810	+	0.916598i	$0.369076\pi$
$110$	−25.6210	−2.44286
$111$	0	0
$112$	17.7235	1.67472
$113$	0.894306	0.0841292	0.0420646	−	0.999115i	$-0.486606\pi$
$113$	0.894306	0.0841292	0.0420646	+	0.999115i	$0.486606\pi$
$114$	0	0
$115$	2.82056	0.263018
$116$	−1.73393	−0.160991
$117$	0	0
$118$	−16.9695	−1.56217
$119$	−20.3041	−1.86127
$120$	0	0
$121$	11.0982	1.00893
$122$	−9.04827	−0.819192
$123$	0	0
$124$	8.94892	0.803637
$125$	5.76652	0.515774
$126$	0	0
$127$	−10.1089	−0.897018	−0.448509	−	0.893778i	$-0.648045\pi$
$127$	−10.1089	−0.897018	−0.448509	+	0.893778i	$0.648045\pi$
$128$	−4.07676	−0.360338
$129$	0	0
$130$	−31.1730	−2.73405
$131$	−14.7583	−1.28944	−0.644718	−	0.764420i	$-0.723025\pi$
$131$	−14.7583	−1.28944	−0.644718	+	0.764420i	$0.723025\pi$
$132$	0	0
$133$	−0.892411	−0.0773819
$134$	30.8483	2.66488
$135$	0	0
$136$	2.62775	0.225328
$137$	−9.23341	−0.788864	−0.394432	−	0.918925i	$-0.629059\pi$
$137$	−9.23341	−0.788864	−0.394432	+	0.918925i	$0.629059\pi$
$138$	0	0
$139$	10.4024	0.882319	0.441160	−	0.897429i	$-0.354567\pi$
$139$	10.4024	0.882319	0.441160	+	0.897429i	$0.354567\pi$
$140$	19.4289	1.64204
$141$	0	0
$142$	−19.3723	−1.62569
$143$	26.8868	2.24838
$144$	0	0
$145$	2.82056	0.234234
$146$	−16.2521	−1.34503
$147$	0	0
$148$	−10.0124	−0.823012
$149$	13.6004	1.11419	0.557093	−	0.830450i	$-0.311917\pi$
$149$	13.6004	1.11419	0.557093	+	0.830450i	$0.311917\pi$
$150$	0	0
$151$	−2.89036	−0.235214	−0.117607	−	0.993060i	$-0.537522\pi$
$151$	−2.89036	−0.235214	−0.117607	+	0.993060i	$0.537522\pi$
$152$	0.115496	0.00936796
$153$	0	0
$154$	−36.0865	−2.90793
$155$	−14.5571	−1.16925
$156$	0	0
$157$	11.9053	0.950150	0.475075	−	0.879945i	$-0.342421\pi$
$157$	11.9053	0.950150	0.475075	+	0.879945i	$0.342421\pi$
$158$	25.1186	1.99833
$159$	0	0
$160$	−21.4152	−1.69302
$161$	3.97268	0.313091
$162$	0	0
$163$	7.05697	0.552745	0.276372	−	0.961051i	$-0.410868\pi$
$163$	7.05697	0.552745	0.276372	+	0.961051i	$0.410868\pi$
$164$	−0.813985	−0.0635616
$165$	0	0
$166$	−22.4473	−1.74225
$167$	12.2598	0.948689	0.474344	−	0.880339i	$-0.342685\pi$
$167$	12.2598	0.948689	0.474344	+	0.880339i	$0.342685\pi$
$168$	0	0
$169$	19.7130	1.51639
$170$	27.8559	2.13645
$171$	0	0
$172$	4.65801	0.355170
$173$	−2.41040	−0.183259	−0.0916296	−	0.995793i	$-0.529208\pi$
$173$	−2.41040	−0.183259	−0.0916296	+	0.995793i	$0.529208\pi$
$174$	0	0
$175$	−11.7414	−0.887567
$176$	20.9722	1.58084
$177$	0	0
$178$	−20.7741	−1.55709
$179$	−5.21330	−0.389661	−0.194830	−	0.980837i	$-0.562416\pi$
$179$	−5.21330	−0.389661	−0.194830	+	0.980837i	$0.562416\pi$
$180$	0	0
$181$	−15.8370	−1.17715	−0.588576	−	0.808442i	$-0.700311\pi$
$181$	−15.8370	−1.17715	−0.588576	+	0.808442i	$0.700311\pi$
$182$	−43.9063	−3.25455
$183$	0	0
$184$	−0.514145	−0.0379033
$185$	16.2870	1.19744
$186$	0	0
$187$	−24.0258	−1.75694
$188$	21.2961	1.55318
$189$	0	0
$190$	1.22433	0.0888223
$191$	−25.5745	−1.85051	−0.925254	−	0.379349i	$-0.876148\pi$
$191$	−25.5745	−1.85051	−0.925254	+	0.379349i	$0.876148\pi$
$192$	0	0
$193$	9.38745	0.675723	0.337862	−	0.941196i	$-0.390296\pi$
$193$	9.38745	0.675723	0.337862	+	0.941196i	$0.390296\pi$
$194$	10.6927	0.767692
$195$	0	0
$196$	15.2277	1.08769
$197$	10.9347	0.779066	0.389533	−	0.921012i	$-0.372636\pi$
$197$	10.9347	0.779066	0.389533	+	0.921012i	$0.372636\pi$
$198$	0	0
$199$	19.0802	1.35256	0.676281	−	0.736644i	$-0.263590\pi$
$199$	19.0802	1.35256	0.676281	+	0.736644i	$0.263590\pi$
$200$	1.51957	0.107450
$201$	0	0
$202$	1.52573	0.107350
$203$	3.97268	0.278828
$204$	0	0
$205$	1.32410	0.0924792
$206$	2.16886	0.151111
$207$	0	0
$208$	25.5168	1.76927
$209$	−1.05599	−0.0730443
$210$	0	0
$211$	−9.96719	−0.686170	−0.343085	−	0.939304i	$-0.611472\pi$
$211$	−9.96719	−0.686170	−0.343085	+	0.939304i	$0.611472\pi$
$212$	−6.30767	−0.433213
$213$	0	0
$214$	18.1531	1.24092
$215$	−7.57713	−0.516756
$216$	0	0
$217$	−20.5033	−1.39185
$218$	−16.1317	−1.09258
$219$	0	0
$220$	22.9902	1.55000
$221$	−29.2321	−1.96636
$222$	0	0
$223$	12.5991	0.843695	0.421847	−	0.906667i	$-0.361382\pi$
$223$	12.5991	0.843695	0.421847	+	0.906667i	$0.361382\pi$
$224$	−30.1628	−2.01534
$225$	0	0
$226$	−1.72810	−0.114952
$227$	−8.62488	−0.572453	−0.286227	−	0.958162i	$-0.592401\pi$
$227$	−8.62488	−0.572453	−0.286227	+	0.958162i	$0.592401\pi$
$228$	0	0
$229$	−26.6654	−1.76210	−0.881050	−	0.473022i	$-0.843163\pi$
$229$	−26.6654	−1.76210	−0.881050	+	0.473022i	$0.843163\pi$
$230$	−5.45026	−0.359380
$231$	0	0
$232$	−0.514145	−0.0337553
$233$	−3.28507	−0.215212	−0.107606	−	0.994194i	$-0.534319\pi$
$233$	−3.28507	−0.215212	−0.107606	+	0.994194i	$0.534319\pi$
$234$	0	0
$235$	−34.6421	−2.25980
$236$	15.2271	0.991200
$237$	0	0
$238$	39.2343	2.54318
$239$	−13.3297	−0.862224	−0.431112	−	0.902298i	$-0.641879\pi$
$239$	−13.3297	−0.862224	−0.431112	+	0.902298i	$0.641879\pi$
$240$	0	0
$241$	6.39449	0.411906	0.205953	−	0.978562i	$-0.433971\pi$
$241$	6.39449	0.411906	0.205953	+	0.978562i	$0.433971\pi$
$242$	−21.4454	−1.37856
$243$	0	0
$244$	8.11920	0.519778
$245$	−24.7707	−1.58254
$246$	0	0
$247$	−1.28482	−0.0817510
$248$	2.65354	0.168500
$249$	0	0
$250$	−11.1429	−0.704737
$251$	15.6412	0.987262	0.493631	−	0.869671i	$-0.335669\pi$
$251$	15.6412	0.987262	0.493631	+	0.869671i	$0.335669\pi$
$252$	0	0
$253$	4.70087	0.295541
$254$	19.5338	1.22566
$255$	0	0
$256$	19.3750	1.21094
$257$	−9.59553	−0.598552	−0.299276	−	0.954167i	$-0.596745\pi$
$257$	−9.59553	−0.598552	−0.299276	+	0.954167i	$0.596745\pi$
$258$	0	0
$259$	22.9398	1.42541
$260$	27.9721	1.73476
$261$	0	0
$262$	28.5180	1.76185
$263$	8.20978	0.506237	0.253118	−	0.967435i	$-0.418544\pi$
$263$	8.20978	0.506237	0.253118	+	0.967435i	$0.418544\pi$
$264$	0	0
$265$	10.2606	0.630305
$266$	1.72444	0.105732
$267$	0	0
$268$	−27.6808	−1.69087
$269$	2.60278	0.158694	0.0793471	−	0.996847i	$-0.474716\pi$
$269$	2.60278	0.158694	0.0793471	+	0.996847i	$0.474716\pi$
$270$	0	0
$271$	14.9550	0.908452	0.454226	−	0.890887i	$-0.349916\pi$
$271$	14.9550	0.908452	0.454226	+	0.890887i	$0.349916\pi$
$272$	−22.8016	−1.38255
$273$	0	0
$274$	17.8421	1.07788
$275$	−13.8936	−0.837815
$276$	0	0
$277$	−18.0554	−1.08484	−0.542422	−	0.840106i	$-0.682493\pi$
$277$	−18.0554	−1.08484	−0.542422	+	0.840106i	$0.682493\pi$
$278$	−20.1009	−1.20557
$279$	0	0
$280$	5.76108	0.344291
$281$	6.43104	0.383644	0.191822	−	0.981430i	$-0.438560\pi$
$281$	6.43104	0.383644	0.191822	+	0.981430i	$0.438560\pi$
$282$	0	0
$283$	−6.16593	−0.366526	−0.183263	−	0.983064i	$-0.558666\pi$
$283$	−6.16593	−0.366526	−0.183263	+	0.983064i	$0.558666\pi$
$284$	17.3832	1.03150
$285$	0	0
$286$	−51.9543	−3.07212
$287$	1.86496	0.110085
$288$	0	0
$289$	9.12150	0.536559
$290$	−5.45026	−0.320051
$291$	0	0
$292$	14.5833	0.853425
$293$	6.44220	0.376357	0.188179	−	0.982135i	$-0.439742\pi$
$293$	6.44220	0.376357	0.188179	+	0.982135i	$0.439742\pi$
$294$	0	0
$295$	−24.7697	−1.44215
$296$	−2.96888	−0.172562
$297$	0	0
$298$	−26.2805	−1.52239
$299$	5.71953	0.330769
$300$	0	0
$301$	−10.6722	−0.615135
$302$	5.58514	0.321389
$303$	0	0
$304$	−1.00218	−0.0574792
$305$	−13.2074	−0.756253
$306$	0	0
$307$	15.0115	0.856749	0.428374	−	0.903601i	$-0.359086\pi$
$307$	15.0115	0.856749	0.428374	+	0.903601i	$0.359086\pi$
$308$	32.3812	1.84509
$309$	0	0
$310$	28.1292	1.59763
$311$	22.3886	1.26954	0.634771	−	0.772700i	$-0.281094\pi$
$311$	22.3886	1.26954	0.634771	+	0.772700i	$0.281094\pi$
$312$	0	0
$313$	−13.3858	−0.756613	−0.378306	−	0.925680i	$-0.623493\pi$
$313$	−13.3858	−0.756613	−0.378306	+	0.925680i	$0.623493\pi$
$314$	−23.0051	−1.29825
$315$	0	0
$316$	−22.5394	−1.26794
$317$	−20.2458	−1.13712	−0.568559	−	0.822642i	$-0.692499\pi$
$317$	−20.2458	−1.13712	−0.568559	+	0.822642i	$0.692499\pi$
$318$	0	0
$319$	4.70087	0.263198
$320$	16.2144	0.906412
$321$	0	0
$322$	−7.67656	−0.427798
$323$	1.14810	0.0638821
$324$	0	0
$325$	−16.9043	−0.937681
$326$	−13.6364	−0.755253
$327$	0	0
$328$	−0.241364	−0.0133271
$329$	−48.7926	−2.69002
$330$	0	0
$331$	28.2462	1.55255	0.776275	−	0.630394i	$-0.217107\pi$
$331$	28.2462	1.55255	0.776275	+	0.630394i	$0.217107\pi$
$332$	20.1424	1.10546
$333$	0	0
$334$	−23.6900	−1.29626
$335$	45.0280	2.46014
$336$	0	0
$337$	13.9484	0.759818	0.379909	−	0.925024i	$-0.375955\pi$
$337$	13.9484	0.759818	0.379909	+	0.925024i	$0.375955\pi$
$338$	−38.0922	−2.07194
$339$	0	0
$340$	−24.9956	−1.35558
$341$	−24.2615	−1.31384
$342$	0	0
$343$	−7.08012	−0.382291
$344$	1.38120	0.0744692
$345$	0	0
$346$	4.65770	0.250400
$347$	13.0430	0.700186	0.350093	−	0.936715i	$-0.386150\pi$
$347$	13.0430	0.700186	0.350093	+	0.936715i	$0.386150\pi$
$348$	0	0
$349$	−25.9323	−1.38813	−0.694063	−	0.719914i	$-0.744181\pi$
$349$	−25.9323	−1.38813	−0.694063	+	0.719914i	$0.744181\pi$
$350$	22.6884	1.21274
$351$	0	0
$352$	−35.6916	−1.90237
$353$	19.7060	1.04884	0.524422	−	0.851459i	$-0.324282\pi$
$353$	19.7060	1.04884	0.524422	+	0.851459i	$0.324282\pi$
$354$	0	0
$355$	−28.2770	−1.50079
$356$	18.6410	0.987973
$357$	0	0
$358$	10.0739	0.532420
$359$	−35.4959	−1.87340	−0.936701	−	0.350129i	$-0.886138\pi$
$359$	−35.4959	−1.87340	−0.936701	+	0.350129i	$0.886138\pi$
$360$	0	0
$361$	−18.9495	−0.997344
$362$	30.6023	1.60842
$363$	0	0
$364$	39.3980	2.06502
$365$	−23.7225	−1.24169
$366$	0	0
$367$	−13.8149	−0.721134	−0.360567	−	0.932733i	$-0.617417\pi$
$367$	−13.8149	−0.721134	−0.360567	+	0.932733i	$0.617417\pi$
$368$	4.46135	0.232564
$369$	0	0
$370$	−31.4720	−1.63615
$371$	14.4518	0.750301
$372$	0	0
$373$	26.9343	1.39460	0.697301	−	0.716778i	$-0.254384\pi$
$373$	26.9343	1.39460	0.697301	+	0.716778i	$0.254384\pi$
$374$	46.4259	2.40062
$375$	0	0
$376$	6.31474	0.325658
$377$	5.71953	0.294571
$378$	0	0
$379$	32.3899	1.66376	0.831880	−	0.554956i	$-0.187265\pi$
$379$	32.3899	1.66376	0.831880	+	0.554956i	$0.187265\pi$
$380$	−1.09862	−0.0563579
$381$	0	0
$382$	49.4186	2.52848
$383$	−36.9727	−1.88922	−0.944608	−	0.328200i	$-0.893558\pi$
$383$	−36.9727	−1.88922	−0.944608	+	0.328200i	$0.893558\pi$
$384$	0	0
$385$	−52.6741	−2.68452
$386$	−18.1397	−0.923287
$387$	0	0
$388$	−9.59479	−0.487102
$389$	4.21678	0.213799	0.106900	−	0.994270i	$-0.465908\pi$
$389$	4.21678	0.213799	0.106900	+	0.994270i	$0.465908\pi$
$390$	0	0
$391$	−5.11092	−0.258470
$392$	4.51533	0.228058
$393$	0	0
$394$	−21.1296	−1.06449
$395$	36.6646	1.84480
$396$	0	0
$397$	−1.67840	−0.0842364	−0.0421182	−	0.999113i	$-0.513411\pi$
$397$	−1.67840	−0.0842364	−0.0421182	+	0.999113i	$0.513411\pi$
$398$	−36.8694	−1.84810
$399$	0	0
$400$	−13.1857	−0.659285
$401$	35.0056	1.74810	0.874048	−	0.485839i	$-0.161486\pi$
$401$	35.0056	1.74810	0.874048	+	0.485839i	$0.161486\pi$
$402$	0	0
$403$	−29.5189	−1.47044
$404$	−1.36907	−0.0681137
$405$	0	0
$406$	−7.67656	−0.380981
$407$	27.1447	1.34551
$408$	0	0
$409$	7.76285	0.383848	0.191924	−	0.981410i	$-0.438527\pi$
$409$	7.76285	0.383848	0.191924	+	0.981410i	$0.438527\pi$
$410$	−2.55861	−0.126361
$411$	0	0
$412$	−1.94616	−0.0958804
$413$	−34.8876	−1.71670
$414$	0	0
$415$	−32.7654	−1.60839
$416$	−43.4258	−2.12913
$417$	0	0
$418$	2.04053	0.0998054
$419$	4.61054	0.225240	0.112620	−	0.993638i	$-0.464076\pi$
$419$	4.61054	0.225240	0.112620	+	0.993638i	$0.464076\pi$
$420$	0	0
$421$	−32.4865	−1.58329	−0.791647	−	0.610979i	$-0.790776\pi$
$421$	−32.4865	−1.58329	−0.791647	+	0.610979i	$0.790776\pi$
$422$	19.2600	0.937560
$423$	0	0
$424$	−1.87036	−0.0908325
$425$	15.1055	0.732725
$426$	0	0
$427$	−18.6023	−0.900228
$428$	−16.2891	−0.787365
$429$	0	0
$430$	14.6416	0.706079
$431$	−0.775560	−0.0373574	−0.0186787	−	0.999826i	$-0.505946\pi$
$431$	−0.775560	−0.0373574	−0.0186787	+	0.999826i	$0.505946\pi$
$432$	0	0
$433$	−6.57931	−0.316181	−0.158091	−	0.987425i	$-0.550534\pi$
$433$	−6.57931	−0.316181	−0.158091	+	0.987425i	$0.550534\pi$
$434$	39.6193	1.90179
$435$	0	0
$436$	14.4753	0.693241
$437$	−0.224637	−0.0107458
$438$	0	0
$439$	35.1830	1.67919	0.839595	−	0.543212i	$-0.182792\pi$
$439$	35.1830	1.67919	0.839595	+	0.543212i	$0.182792\pi$
$440$	6.81708	0.324992
$441$	0	0
$442$	56.4862	2.68677
$443$	−24.4074	−1.15963	−0.579815	−	0.814748i	$-0.696875\pi$
$443$	−24.4074	−1.15963	−0.579815	+	0.814748i	$0.696875\pi$
$444$	0	0
$445$	−30.3232	−1.43746
$446$	−24.3456	−1.15280
$447$	0	0
$448$	22.8376	1.07897
$449$	−5.62450	−0.265437	−0.132718	−	0.991154i	$-0.542371\pi$
$449$	−5.62450	−0.265437	−0.132718	+	0.991154i	$0.542371\pi$
$450$	0	0
$451$	2.20681	0.103914
$452$	1.55066	0.0729369
$453$	0	0
$454$	16.6662	0.782182
$455$	−64.0883	−3.00451
$456$	0	0
$457$	6.51862	0.304928	0.152464	−	0.988309i	$-0.451279\pi$
$457$	6.51862	0.304928	0.152464	+	0.988309i	$0.451279\pi$
$458$	51.5266	2.40768
$459$	0	0
$460$	4.89063	0.228027
$461$	4.33185	0.201754	0.100877	−	0.994899i	$-0.467835\pi$
$461$	4.33185	0.201754	0.100877	+	0.994899i	$0.467835\pi$
$462$	0	0
$463$	−22.2599	−1.03451	−0.517254	−	0.855832i	$-0.673046\pi$
$463$	−22.2599	−1.03451	−0.517254	+	0.855832i	$0.673046\pi$
$464$	4.46135	0.207113
$465$	0	0
$466$	6.34786	0.294059
$467$	4.41272	0.204196	0.102098	−	0.994774i	$-0.467444\pi$
$467$	4.41272	0.204196	0.102098	+	0.994774i	$0.467444\pi$
$468$	0	0
$469$	63.4208	2.92850
$470$	66.9403	3.08773
$471$	0	0
$472$	4.51515	0.207827
$473$	−12.6284	−0.580654
$474$	0	0
$475$	0.663923	0.0304629
$476$	−35.2057	−1.61365
$477$	0	0
$478$	25.7574	1.17812
$479$	9.33238	0.426407	0.213204	−	0.977008i	$-0.431610\pi$
$479$	9.33238	0.426407	0.213204	+	0.977008i	$0.431610\pi$
$480$	0	0
$481$	33.0268	1.50589
$482$	−12.3563	−0.562815
$483$	0	0
$484$	19.2434	0.874701
$485$	15.6077	0.708711
$486$	0	0
$487$	6.00747	0.272225	0.136112	−	0.990693i	$-0.456539\pi$
$487$	6.00747	0.272225	0.136112	+	0.990693i	$0.456539\pi$
$488$	2.40751	0.108983
$489$	0	0
$490$	47.8653	2.16233
$491$	−16.0494	−0.724299	−0.362149	−	0.932120i	$-0.617957\pi$
$491$	−16.0494	−0.724299	−0.362149	+	0.932120i	$0.617957\pi$
$492$	0	0
$493$	−5.11092	−0.230184
$494$	2.48270	0.111702
$495$	0	0
$496$	−23.0254	−1.03387
$497$	−39.8275	−1.78651
$498$	0	0
$499$	−24.7809	−1.10935	−0.554674	−	0.832068i	$-0.687157\pi$
$499$	−24.7809	−1.10935	−0.554674	+	0.832068i	$0.687157\pi$
$500$	9.99872	0.447157
$501$	0	0
$502$	−30.2240	−1.34896
$503$	38.0583	1.69694	0.848469	−	0.529245i	$-0.177525\pi$
$503$	38.0583	1.69694	0.848469	+	0.529245i	$0.177525\pi$
$504$	0	0
$505$	2.22705	0.0991023
$506$	−9.08366	−0.403818
$507$	0	0
$508$	−17.5281	−0.777682
$509$	19.2146	0.851671	0.425836	−	0.904800i	$-0.359980\pi$
$509$	19.2146	0.851671	0.425836	+	0.904800i	$0.359980\pi$
$510$	0	0
$511$	−33.4126	−1.47809
$512$	−29.2855	−1.29425
$513$	0	0
$514$	18.5418	0.817843
$515$	3.16579	0.139502
$516$	0	0
$517$	−57.7362	−2.53924
$518$	−44.3275	−1.94764
$519$	0	0
$520$	8.29432	0.363730
$521$	19.4926	0.853988	0.426994	−	0.904254i	$-0.359573\pi$
$521$	19.4926	0.853988	0.426994	+	0.904254i	$0.359573\pi$
$522$	0	0
$523$	28.4432	1.24373	0.621866	−	0.783124i	$-0.286375\pi$
$523$	28.4432	1.24373	0.621866	+	0.783124i	$0.286375\pi$
$524$	−25.5898	−1.11789
$525$	0	0
$526$	−15.8641	−0.691706
$527$	26.3778	1.14904
$528$	0	0
$529$	1.00000	0.0434783
$530$	−19.8270	−0.861228
$531$	0	0
$532$	−1.54737	−0.0670872
$533$	2.68501	0.116301
$534$	0	0
$535$	26.4973	1.14558
$536$	−8.20793	−0.354528
$537$	0	0
$538$	−5.02945	−0.216835
$539$	−41.2840	−1.77823
$540$	0	0
$541$	−6.26466	−0.269339	−0.134669	−	0.990891i	$-0.542997\pi$
$541$	−6.26466	−0.269339	−0.134669	+	0.990891i	$0.542997\pi$
$542$	−28.8981	−1.24128
$543$	0	0
$544$	38.8049	1.66375
$545$	−23.5468	−1.00863
$546$	0	0
$547$	−32.9269	−1.40785	−0.703927	−	0.710272i	$-0.748572\pi$
$547$	−32.9269	−1.40785	−0.703927	+	0.710272i	$0.748572\pi$
$548$	−16.0101	−0.683916
$549$	0	0
$550$	26.8471	1.14476
$551$	−0.224637	−0.00956986
$552$	0	0
$553$	51.6411	2.19600
$554$	34.8891	1.48230
$555$	0	0
$556$	18.0370	0.764938
$557$	−36.0931	−1.52931	−0.764657	−	0.644438i	$-0.777092\pi$
$557$	−36.0931	−1.52931	−0.764657	+	0.644438i	$0.777092\pi$
$558$	0	0
$559$	−15.3649	−0.649867
$560$	−49.9902	−2.11247
$561$	0	0
$562$	−12.4269	−0.524199
$563$	−24.8551	−1.04752	−0.523758	−	0.851867i	$-0.675471\pi$
$563$	−24.8551	−1.04752	−0.523758	+	0.851867i	$0.675471\pi$
$564$	0	0
$565$	−2.52244	−0.106120
$566$	11.9147	0.500810
$567$	0	0
$568$	5.15448	0.216277
$569$	24.2965	1.01856	0.509281	−	0.860600i	$-0.329911\pi$
$569$	24.2965	1.01856	0.509281	+	0.860600i	$0.329911\pi$
$570$	0	0
$571$	36.3348	1.52056	0.760282	−	0.649593i	$-0.225061\pi$
$571$	36.3348	1.52056	0.760282	+	0.649593i	$0.225061\pi$
$572$	46.6197	1.94927
$573$	0	0
$574$	−3.60373	−0.150417
$575$	−2.95554	−0.123254
$576$	0	0
$577$	6.50544	0.270825	0.135412	−	0.990789i	$-0.456764\pi$
$577$	6.50544	0.270825	0.135412	+	0.990789i	$0.456764\pi$
$578$	−17.6258	−0.733137
$579$	0	0
$580$	4.89063	0.203073
$581$	−46.1493	−1.91459
$582$	0	0
$583$	17.1008	0.708243
$584$	4.32426	0.178939
$585$	0	0
$586$	−12.4485	−0.514242
$587$	44.9379	1.85479	0.927394	−	0.374087i	$-0.122044\pi$
$587$	44.9379	1.85479	0.927394	+	0.374087i	$0.122044\pi$
$588$	0	0
$589$	1.15937	0.0477709
$590$	47.8635	1.97051
$591$	0	0
$592$	25.7616	1.05880
$593$	−48.0280	−1.97227	−0.986136	−	0.165939i	$-0.946934\pi$
$593$	−48.0280	−1.97227	−0.986136	+	0.165939i	$0.946934\pi$
$594$	0	0
$595$	57.2687	2.34779
$596$	23.5821	0.965958
$597$	0	0
$598$	−11.0521	−0.451952
$599$	8.51888	0.348072	0.174036	−	0.984739i	$-0.444319\pi$
$599$	8.51888	0.348072	0.174036	+	0.984739i	$0.444319\pi$
$600$	0	0
$601$	−21.6288	−0.882257	−0.441129	−	0.897444i	$-0.645422\pi$
$601$	−21.6288	−0.882257	−0.441129	+	0.897444i	$0.645422\pi$
$602$	20.6223	0.840501
$603$	0	0
$604$	−5.01166	−0.203922
$605$	−31.3030	−1.27265
$606$	0	0
$607$	−2.96277	−0.120255	−0.0601275	−	0.998191i	$-0.519151\pi$
$607$	−2.96277	−0.120255	−0.0601275	+	0.998191i	$0.519151\pi$
$608$	1.70557	0.0691699
$609$	0	0
$610$	25.5211	1.03332
$611$	−70.2474	−2.84191
$612$	0	0
$613$	−33.7840	−1.36452	−0.682262	−	0.731108i	$-0.739004\pi$
$613$	−33.7840	−1.36452	−0.682262	+	0.731108i	$0.739004\pi$
$614$	−29.0072	−1.17063
$615$	0	0
$616$	9.60169	0.386863
$617$	−16.0811	−0.647400	−0.323700	−	0.946160i	$-0.604927\pi$
$617$	−16.0811	−0.647400	−0.323700	+	0.946160i	$0.604927\pi$
$618$	0	0
$619$	28.1296	1.13062	0.565312	−	0.824877i	$-0.308756\pi$
$619$	28.1296	1.13062	0.565312	+	0.824877i	$0.308756\pi$
$620$	−25.2409	−1.01370
$621$	0	0
$622$	−43.2624	−1.73466
$623$	−42.7094	−1.71112
$624$	0	0
$625$	−31.0425	−1.24170
$626$	25.8660	1.03381
$627$	0	0
$628$	20.6430	0.823745
$629$	−29.5125	−1.17674
$630$	0	0
$631$	26.6943	1.06268	0.531341	−	0.847158i	$-0.321688\pi$
$631$	26.6943	1.06268	0.531341	+	0.847158i	$0.321688\pi$
$632$	−6.68340	−0.265851
$633$	0	0
$634$	39.1218	1.55372
$635$	28.5127	1.13149
$636$	0	0
$637$	−50.2301	−1.99019
$638$	−9.08366	−0.359626
$639$	0	0
$640$	11.4987	0.454527
$641$	−15.2742	−0.603296	−0.301648	−	0.953419i	$-0.597537\pi$
$641$	−15.2742	−0.603296	−0.301648	+	0.953419i	$0.597537\pi$
$642$	0	0
$643$	−37.1583	−1.46538	−0.732690	−	0.680562i	$-0.761736\pi$
$643$	−37.1583	−1.46538	−0.732690	+	0.680562i	$0.761736\pi$
$644$	6.88834	0.271438
$645$	0	0
$646$	−2.21852	−0.0872865
$647$	4.05766	0.159523	0.0797615	−	0.996814i	$-0.474584\pi$
$647$	4.05766	0.159523	0.0797615	+	0.996814i	$0.474584\pi$
$648$	0	0
$649$	−41.2824	−1.62048
$650$	32.6648	1.28122
$651$	0	0
$652$	12.2363	0.479209
$653$	−39.9199	−1.56219	−0.781093	−	0.624415i	$-0.785337\pi$
$653$	−39.9199	−1.56219	−0.781093	+	0.624415i	$0.785337\pi$
$654$	0	0
$655$	41.6265	1.62648
$656$	2.09437	0.0817713
$657$	0	0
$658$	94.2837	3.67556
$659$	−8.93986	−0.348248	−0.174124	−	0.984724i	$-0.555709\pi$
$659$	−8.93986	−0.348248	−0.174124	+	0.984724i	$0.555709\pi$
$660$	0	0
$661$	−19.2868	−0.750171	−0.375086	−	0.926990i	$-0.622387\pi$
$661$	−19.2868	−0.750171	−0.375086	+	0.926990i	$0.622387\pi$
$662$	−54.5811	−2.12136
$663$	0	0
$664$	5.97265	0.231784
$665$	2.51710	0.0976088
$666$	0	0
$667$	1.00000	0.0387202
$668$	21.2575	0.822478
$669$	0	0
$670$	−87.0092	−3.36146
$671$	−22.0121	−0.849766
$672$	0	0
$673$	6.74414	0.259967	0.129984	−	0.991516i	$-0.458507\pi$
$673$	6.74414	0.259967	0.129984	+	0.991516i	$0.458507\pi$
$674$	−26.9530	−1.03819
$675$	0	0
$676$	34.1809	1.31465
$677$	30.5313	1.17341	0.586706	−	0.809800i	$-0.300425\pi$
$677$	30.5313	1.17341	0.586706	+	0.809800i	$0.300425\pi$
$678$	0	0
$679$	21.9831	0.843634
$680$	−7.41173	−0.284227
$681$	0	0
$682$	46.8815	1.79518
$683$	50.6678	1.93875	0.969376	−	0.245583i	$-0.0789793\pi$
$683$	50.6678	1.93875	0.969376	+	0.245583i	$0.0789793\pi$
$684$	0	0
$685$	26.0434	0.995066
$686$	13.6812	0.522350
$687$	0	0
$688$	−11.9850	−0.456922
$689$	20.8065	0.792664
$690$	0	0
$691$	5.90345	0.224578	0.112289	−	0.993676i	$-0.464182\pi$
$691$	5.90345	0.224578	0.112289	+	0.993676i	$0.464182\pi$
$692$	−4.17945	−0.158879
$693$	0	0
$694$	−25.2035	−0.956713
$695$	−29.3405	−1.11295
$696$	0	0
$697$	−2.39930	−0.0908801
$698$	50.1100	1.89669
$699$	0	0
$700$	−20.3587	−0.769488
$701$	−41.5271	−1.56846	−0.784228	−	0.620473i	$-0.786941\pi$
$701$	−41.5271	−1.56846	−0.784228	+	0.620473i	$0.786941\pi$
$702$	0	0
$703$	−1.29714	−0.0489227
$704$	27.0237	1.01849
$705$	0	0
$706$	−38.0786	−1.43311
$707$	3.13674	0.117969
$708$	0	0
$709$	35.6554	1.33907	0.669534	−	0.742781i	$-0.266494\pi$
$709$	35.6554	1.33907	0.669534	+	0.742781i	$0.266494\pi$
$710$	54.6407	2.05063
$711$	0	0
$712$	5.52746	0.207150
$713$	−5.16107	−0.193284
$714$	0	0
$715$	−75.8356	−2.83609
$716$	−9.03948	−0.337821
$717$	0	0
$718$	68.5901	2.55976
$719$	−8.21784	−0.306474	−0.153237	−	0.988189i	$-0.548970\pi$
$719$	−8.21784	−0.306474	−0.153237	+	0.988189i	$0.548970\pi$
$720$	0	0
$721$	4.45894	0.166060
$722$	36.6169	1.36274
$723$	0	0
$724$	−27.4601	−1.02055
$725$	−2.95554	−0.109766
$726$	0	0
$727$	−41.0160	−1.52120	−0.760599	−	0.649222i	$-0.775095\pi$
$727$	−41.0160	−1.52120	−0.760599	+	0.649222i	$0.775095\pi$
$728$	11.6823	0.432976
$729$	0	0
$730$	45.8399	1.69661
$731$	13.7300	0.507821
$732$	0	0
$733$	−14.0254	−0.518040	−0.259020	−	0.965872i	$-0.583400\pi$
$733$	−14.0254	−0.518040	−0.259020	+	0.965872i	$0.583400\pi$
$734$	26.6951	0.985334
$735$	0	0
$736$	−7.59255	−0.279865
$737$	75.0457	2.76434
$738$	0	0
$739$	−28.3370	−1.04239	−0.521197	−	0.853437i	$-0.674514\pi$
$739$	−28.3370	−1.04239	−0.521197	+	0.853437i	$0.674514\pi$
$740$	28.2405	1.03814
$741$	0	0
$742$	−27.9258	−1.02519
$743$	−13.0956	−0.480433	−0.240216	−	0.970719i	$-0.577218\pi$
$743$	−13.0956	−0.480433	−0.240216	+	0.970719i	$0.577218\pi$
$744$	0	0
$745$	−38.3606	−1.40543
$746$	−52.0460	−1.90554
$747$	0	0
$748$	−41.6589	−1.52320
$749$	37.3208	1.36367
$750$	0	0
$751$	−40.4975	−1.47778	−0.738888	−	0.673828i	$-0.764649\pi$
$751$	−40.4975	−1.47778	−0.738888	+	0.673828i	$0.764649\pi$
$752$	−54.7945	−1.99815
$753$	0	0
$754$	−11.0521	−0.402492
$755$	8.15241	0.296697
$756$	0	0
$757$	−43.3831	−1.57679	−0.788394	−	0.615171i	$-0.789087\pi$
$757$	−43.3831	−1.57679	−0.788394	+	0.615171i	$0.789087\pi$
$758$	−62.5883	−2.27331
$759$	0	0
$760$	−0.325763	−0.0118167
$761$	33.3792	1.21000	0.604998	−	0.796227i	$-0.293174\pi$
$761$	33.3792	1.21000	0.604998	+	0.796227i	$0.293174\pi$
$762$	0	0
$763$	−33.1650	−1.20066
$764$	−44.3443	−1.60432
$765$	0	0
$766$	71.4437	2.58137
$767$	−50.2281	−1.81363
$768$	0	0
$769$	15.9543	0.575327	0.287663	−	0.957732i	$-0.407122\pi$
$769$	15.9543	0.575327	0.287663	+	0.957732i	$0.407122\pi$
$770$	101.784	3.66804
$771$	0	0
$772$	16.2771	0.585827
$773$	−36.1871	−1.30156	−0.650780	−	0.759266i	$-0.725558\pi$
$773$	−36.1871	−1.30156	−0.650780	+	0.759266i	$0.725558\pi$
$774$	0	0
$775$	15.2537	0.547931
$776$	−2.84506	−0.102132
$777$	0	0
$778$	−8.14824	−0.292129
$779$	−0.105455	−0.00377832
$780$	0	0
$781$	−47.1278	−1.68637
$782$	9.87602	0.353166
$783$	0	0
$784$	−39.1805	−1.39930
$785$	−33.5797	−1.19851
$786$	0	0
$787$	32.4245	1.15581	0.577905	−	0.816104i	$-0.303870\pi$
$787$	32.4245	1.15581	0.577905	+	0.816104i	$0.303870\pi$
$788$	18.9600	0.675421
$789$	0	0
$790$	−70.8483	−2.52067
$791$	−3.55279	−0.126323
$792$	0	0
$793$	−26.7820	−0.951056
$794$	3.24323	0.115098
$795$	0	0
$796$	33.0837	1.17262
$797$	−43.9980	−1.55849	−0.779245	−	0.626720i	$-0.784397\pi$
$797$	−43.9980	−1.55849	−0.779245	+	0.626720i	$0.784397\pi$
$798$	0	0
$799$	62.7724	2.22073
$800$	22.4401	0.793376
$801$	0	0
$802$	−67.6426	−2.38854
$803$	−39.5371	−1.39523
$804$	0	0
$805$	−11.2052	−0.394930
$806$	57.0405	2.00917
$807$	0	0
$808$	−0.405957	−0.0142815
$809$	−0.740559	−0.0260367	−0.0130183	−	0.999915i	$-0.504144\pi$
$809$	−0.740559	−0.0260367	−0.0130183	+	0.999915i	$0.504144\pi$
$810$	0	0
$811$	−48.6291	−1.70760	−0.853800	−	0.520602i	$-0.825708\pi$
$811$	−48.6291	−1.70760	−0.853800	+	0.520602i	$0.825708\pi$
$812$	6.88834	0.241733
$813$	0	0
$814$	−52.4527	−1.83847
$815$	−19.9046	−0.697227
$816$	0	0
$817$	0.603464	0.0211125
$818$	−15.0004	−0.524478
$819$	0	0
$820$	2.29589	0.0801760
$821$	3.00633	0.104922	0.0524608	−	0.998623i	$-0.483294\pi$
$821$	3.00633	0.104922	0.0524608	+	0.998623i	$0.483294\pi$
$822$	0	0
$823$	11.4507	0.399146	0.199573	−	0.979883i	$-0.436045\pi$
$823$	11.4507	0.399146	0.199573	+	0.979883i	$0.436045\pi$
$824$	−0.577077	−0.0201034
$825$	0	0
$826$	67.4145	2.34565
$827$	−24.1302	−0.839089	−0.419544	−	0.907735i	$-0.637810\pi$
$827$	−24.1302	−0.839089	−0.419544	+	0.907735i	$0.637810\pi$
$828$	0	0
$829$	−8.02376	−0.278677	−0.139338	−	0.990245i	$-0.544498\pi$
$829$	−8.02376	−0.278677	−0.139338	+	0.990245i	$0.544498\pi$
$830$	63.3138	2.19766
$831$	0	0
$832$	32.8796	1.13989
$833$	44.8851	1.55518
$834$	0	0
$835$	−34.5793	−1.19667
$836$	−1.83101	−0.0633267
$837$	0	0
$838$	−8.90912	−0.307760
$839$	29.8644	1.03103	0.515516	−	0.856880i	$-0.327600\pi$
$839$	29.8644	1.03103	0.515516	+	0.856880i	$0.327600\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	62.7748	2.16336
$843$	0	0
$844$	−17.2824	−0.594884
$845$	−55.6017	−1.91276
$846$	0	0
$847$	−44.0895	−1.51493
$848$	16.2295	0.557323
$849$	0	0
$850$	−29.1889	−1.00117
$851$	5.77440	0.197944
$852$	0	0
$853$	−51.1122	−1.75005	−0.875023	−	0.484081i	$-0.839154\pi$
$853$	−51.1122	−1.75005	−0.875023	+	0.484081i	$0.839154\pi$
$854$	35.9459	1.23004
$855$	0	0
$856$	−4.83007	−0.165088
$857$	13.5277	0.462096	0.231048	−	0.972942i	$-0.425785\pi$
$857$	13.5277	0.462096	0.231048	+	0.972942i	$0.425785\pi$
$858$	0	0
$859$	42.5821	1.45288	0.726440	−	0.687229i	$-0.241173\pi$
$859$	42.5821	1.45288	0.726440	+	0.687229i	$0.241173\pi$
$860$	−13.1382	−0.448008
$861$	0	0
$862$	1.49864	0.0510440
$863$	−40.9531	−1.39406	−0.697031	−	0.717041i	$-0.745496\pi$
$863$	−40.9531	−1.39406	−0.697031	+	0.717041i	$0.745496\pi$
$864$	0	0
$865$	6.79867	0.231162
$866$	12.7134	0.432020
$867$	0	0
$868$	−35.5512	−1.20669
$869$	61.1069	2.07291
$870$	0	0
$871$	91.3078	3.09385
$872$	4.29222	0.145353
$873$	0	0
$874$	0.434074	0.0146828
$875$	−22.9086	−0.774451
$876$	0	0
$877$	24.6726	0.833136	0.416568	−	0.909105i	$-0.363233\pi$
$877$	24.6726	0.833136	0.416568	+	0.909105i	$0.363233\pi$
$878$	−67.9853	−2.29439
$879$	0	0
$880$	−59.1534	−1.99406
$881$	44.4515	1.49761	0.748804	−	0.662791i	$-0.230628\pi$
$881$	44.4515	1.49761	0.748804	+	0.662791i	$0.230628\pi$
$882$	0	0
$883$	−0.759374	−0.0255550	−0.0127775	−	0.999918i	$-0.504067\pi$
$883$	−0.759374	−0.0255550	−0.0127775	+	0.999918i	$0.504067\pi$
$884$	−50.6862	−1.70476
$885$	0	0
$886$	47.1633	1.58448
$887$	−39.0346	−1.31065	−0.655326	−	0.755346i	$-0.727469\pi$
$887$	−39.0346	−1.31065	−0.655326	+	0.755346i	$0.727469\pi$
$888$	0	0
$889$	40.1594	1.34690
$890$	58.5946	1.96409
$891$	0	0
$892$	21.8458	0.731452
$893$	2.75900	0.0923263
$894$	0	0
$895$	14.7044	0.491514
$896$	16.1957	0.541060
$897$	0	0
$898$	10.8684	0.362684
$899$	−5.16107	−0.172131
$900$	0	0
$901$	−18.5925	−0.619406
$902$	−4.26429	−0.141985
$903$	0	0
$904$	0.459803	0.0152928
$905$	44.6690	1.48485
$906$	0	0
$907$	0.394908	0.0131127	0.00655636	−	0.999979i	$-0.497913\pi$
$907$	0.394908	0.0131127	0.00655636	+	0.999979i	$0.497913\pi$
$908$	−14.9549	−0.496296
$909$	0	0
$910$	123.840	4.10526
$911$	24.8847	0.824468	0.412234	−	0.911078i	$-0.364749\pi$
$911$	24.8847	0.824468	0.412234	+	0.911078i	$0.364749\pi$
$912$	0	0
$913$	−54.6084	−1.80727
$914$	−12.5962	−0.416644
$915$	0	0
$916$	−46.2359	−1.52768
$917$	58.6299	1.93613
$918$	0	0
$919$	−48.2769	−1.59251	−0.796254	−	0.604963i	$-0.793188\pi$
$919$	−48.2769	−1.59251	−0.796254	+	0.604963i	$0.793188\pi$
$920$	1.45017	0.0478108
$921$	0	0
$922$	−8.37060	−0.275671
$923$	−57.3402	−1.88738
$924$	0	0
$925$	−17.0664	−0.561141
$926$	43.0137	1.41352
$927$	0	0
$928$	−7.59255	−0.249238
$929$	15.2969	0.501874	0.250937	−	0.968003i	$-0.419261\pi$
$929$	15.2969	0.501874	0.250937	+	0.968003i	$0.419261\pi$
$930$	0	0
$931$	1.97281	0.0646562
$932$	−5.69607	−0.186581
$933$	0	0
$934$	−8.52686	−0.279007
$935$	67.7660	2.21619
$936$	0	0
$937$	−39.1270	−1.27822	−0.639111	−	0.769114i	$-0.720698\pi$
$937$	−39.1270	−1.27822	−0.639111	+	0.769114i	$0.720698\pi$
$938$	−122.550	−4.00141
$939$	0	0
$940$	−60.0669	−1.95917
$941$	0.719765	0.0234637	0.0117318	−	0.999931i	$-0.496266\pi$
$941$	0.719765	0.0234637	0.0117318	+	0.999931i	$0.496266\pi$
$942$	0	0
$943$	0.469446	0.0152873
$944$	−39.1790	−1.27517
$945$	0	0
$946$	24.4023	0.793388
$947$	29.9337	0.972714	0.486357	−	0.873760i	$-0.338325\pi$
$947$	29.9337	0.972714	0.486357	+	0.873760i	$0.338325\pi$
$948$	0	0
$949$	−48.1046	−1.56154
$950$	−1.28292	−0.0416235
$951$	0	0
$952$	−10.4392	−0.338337
$953$	46.7453	1.51423	0.757114	−	0.653282i	$-0.226609\pi$
$953$	46.7453	1.51423	0.757114	+	0.653282i	$0.226609\pi$
$954$	0	0
$955$	72.1344	2.33421
$956$	−23.1126	−0.747516
$957$	0	0
$958$	−18.0333	−0.582629
$959$	36.6814	1.18450
$960$	0	0
$961$	−4.36332	−0.140752
$962$	−63.8190	−2.05761
$963$	0	0
$964$	11.0876	0.357107
$965$	−26.4778	−0.852351
$966$	0	0
$967$	36.9325	1.18767	0.593834	−	0.804587i	$-0.297613\pi$
$967$	36.9325	1.18767	0.593834	+	0.804587i	$0.297613\pi$
$968$	5.70607	0.183400
$969$	0	0
$970$	−30.1594	−0.968360
$971$	10.7651	0.345469	0.172735	−	0.984968i	$-0.444740\pi$
$971$	10.7651	0.345469	0.172735	+	0.984968i	$0.444740\pi$
$972$	0	0
$973$	−41.3254	−1.32483
$974$	−11.6085	−0.371959
$975$	0	0
$976$	−20.8905	−0.668689
$977$	−8.98778	−0.287545	−0.143772	−	0.989611i	$-0.545923\pi$
$977$	−8.98778	−0.287545	−0.143772	+	0.989611i	$0.545923\pi$
$978$	0	0
$979$	−50.5380	−1.61520
$980$	−42.9505	−1.37200
$981$	0	0
$982$	31.0128	0.989659
$983$	−41.4053	−1.32062	−0.660312	−	0.750991i	$-0.729576\pi$
$983$	−41.4053	−1.32062	−0.660312	+	0.750991i	$0.729576\pi$
$984$	0	0
$985$	−30.8420	−0.982707
$986$	9.87602	0.314516
$987$	0	0
$988$	−2.22778	−0.0708751
$989$	−2.68640	−0.0854224
$990$	0	0
$991$	40.0411	1.27195	0.635974	−	0.771710i	$-0.280598\pi$
$991$	40.0411	1.27195	0.635974	+	0.771710i	$0.280598\pi$
$992$	39.1857	1.24415
$993$	0	0
$994$	76.9601	2.44103
$995$	−53.8169	−1.70611
$996$	0	0
$997$	−11.3402	−0.359149	−0.179575	−	0.983744i	$-0.557472\pi$
$997$	−11.3402	−0.359149	−0.179575	+	0.983744i	$0.557472\pi$
$998$	47.8851	1.51578
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.u.1.4	yes	22
3.2	odd	2		6003.2.a.t.1.19	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.19	✓	22	3.2	odd	2
6003.2.a.u.1.4	yes	22	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)

\(f(q)\)	\(=\)	\(q-1.93234 q^{2} +1.73393 q^{4} -2.82056 q^{5} -3.97268 q^{7} +0.514145 q^{8} +O(q^{10})\)	\(q-1.93234 q^{2} +1.73393 q^{4} -2.82056 q^{5} -3.97268 q^{7} +0.514145 q^{8} +5.45026 q^{10} -4.70087 q^{11} -5.71953 q^{13} +7.67656 q^{14} -4.46135 q^{16} +5.11092 q^{17} +0.224637 q^{19} -4.89063 q^{20} +9.08366 q^{22} -1.00000 q^{23} +2.95554 q^{25} +11.0521 q^{26} -6.88834 q^{28} -1.00000 q^{29} +5.16107 q^{31} +7.59255 q^{32} -9.87602 q^{34} +11.2052 q^{35} -5.77440 q^{37} -0.434074 q^{38} -1.45017 q^{40} -0.469446 q^{41} +2.68640 q^{43} -8.15096 q^{44} +1.93234 q^{46} +12.2820 q^{47} +8.78220 q^{49} -5.71109 q^{50} -9.91724 q^{52} -3.63780 q^{53} +13.2591 q^{55} -2.04253 q^{56} +1.93234 q^{58} +8.78187 q^{59} +4.68255 q^{61} -9.97293 q^{62} -5.74865 q^{64} +16.1323 q^{65} -15.9642 q^{67} +8.86195 q^{68} -21.6522 q^{70} +10.0253 q^{71} +8.41059 q^{73} +11.1581 q^{74} +0.389504 q^{76} +18.6751 q^{77} -12.9991 q^{79} +12.5835 q^{80} +0.907129 q^{82} +11.6167 q^{83} -14.4156 q^{85} -5.19102 q^{86} -2.41693 q^{88} +10.7508 q^{89} +22.7219 q^{91} -1.73393 q^{92} -23.7330 q^{94} -0.633601 q^{95} -5.53357 q^{97} -16.9702 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) \) 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8	\( 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8} - 12 q^{10} - 28 q^{13} + q^{14} + 3 q^{16} + 10 q^{17} - 8 q^{19} - 11 q^{22} - 22 q^{23} - 11 q^{26} - 21 q^{28} - 22 q^{29} - 18 q^{31} - 5 q^{32} - 33 q^{34} - 2 q^{35} - 28 q^{37} - 14 q^{38} - 30 q^{40} + 10 q^{41} - 14 q^{43} - 37 q^{44} - 3 q^{46} + 18 q^{47} + 2 q^{49} - 7 q^{50} - 57 q^{52} - 20 q^{53} - 42 q^{55} + 2 q^{56} - 3 q^{58} + 20 q^{59} - 38 q^{61} - 4 q^{62} - 24 q^{64} - 12 q^{65} - 50 q^{67} - 11 q^{68} - 48 q^{70} - 12 q^{71} - 46 q^{73} + 6 q^{74} - 16 q^{76} + 14 q^{77} - 20 q^{79} + 58 q^{80} - 42 q^{82} - 22 q^{83} - 66 q^{85} - 22 q^{86} - 68 q^{88} + 14 q^{89} - 16 q^{91} - 17 q^{92} - 27 q^{94} + 20 q^{95} - 48 q^{97} + 28 q^{98}+O(q^{100}) \) 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8 - 12 * q^10 - 28 * q^13 + q^14 + 3 * q^16 + 10 * q^17 - 8 * q^19 - 11 * q^22 - 22 * q^23 - 11 * q^26 - 21 * q^28 - 22 * q^29 - 18 * q^31 - 5 * q^32 - 33 * q^34 - 2 * q^35 - 28 * q^37 - 14 * q^38 - 30 * q^40 + 10 * q^41 - 14 * q^43 - 37 * q^44 - 3 * q^46 + 18 * q^47 + 2 * q^49 - 7 * q^50 - 57 * q^52 - 20 * q^53 - 42 * q^55 + 2 * q^56 - 3 * q^58 + 20 * q^59 - 38 * q^61 - 4 * q^62 - 24 * q^64 - 12 * q^65 - 50 * q^67 - 11 * q^68 - 48 * q^70 - 12 * q^71 - 46 * q^73 + 6 * q^74 - 16 * q^76 + 14 * q^77 - 20 * q^79 + 58 * q^80 - 42 * q^82 - 22 * q^83 - 66 * q^85 - 22 * q^86 - 68 * q^88 + 14 * q^89 - 16 * q^91 - 17 * q^92 - 27 * q^94 + 20 * q^95 - 48 * q^97 + 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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