LMFDB - Embedded newform 6003.2.a.v.1.9

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			1.9
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)

$f(q)$	$=$	$q-1.58386 q^{2} +0.508607 q^{4} +2.90489 q^{5} +0.0968084 q^{7} +2.36216 q^{8} +O(q^{10})$	\(q-1.58386 q^{2} +0.508607 q^{4} +2.90489 q^{5} +0.0968084 q^{7} +2.36216 q^{8} -4.60094 q^{10} +4.36295 q^{11} -3.92383 q^{13} -0.153331 q^{14} -4.75853 q^{16} +1.52637 q^{17} +5.57221 q^{19} +1.47745 q^{20} -6.91030 q^{22} +1.00000 q^{23} +3.43841 q^{25} +6.21480 q^{26} +0.0492374 q^{28} -1.00000 q^{29} +4.44461 q^{31} +2.81253 q^{32} -2.41755 q^{34} +0.281218 q^{35} -8.67239 q^{37} -8.82559 q^{38} +6.86181 q^{40} +3.40493 q^{41} +7.68843 q^{43} +2.21903 q^{44} -1.58386 q^{46} +8.61378 q^{47} -6.99063 q^{49} -5.44595 q^{50} -1.99569 q^{52} -2.21815 q^{53} +12.6739 q^{55} +0.228677 q^{56} +1.58386 q^{58} -8.80739 q^{59} -1.48378 q^{61} -7.03964 q^{62} +5.06242 q^{64} -11.3983 q^{65} -2.72698 q^{67} +0.776321 q^{68} -0.445410 q^{70} -2.50491 q^{71} +15.6794 q^{73} +13.7358 q^{74} +2.83406 q^{76} +0.422370 q^{77} +16.2533 q^{79} -13.8230 q^{80} -5.39292 q^{82} +14.8238 q^{83} +4.43394 q^{85} -12.1774 q^{86} +10.3060 q^{88} -2.04744 q^{89} -0.379860 q^{91} +0.508607 q^{92} -13.6430 q^{94} +16.1867 q^{95} -14.3399 q^{97} +11.0722 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.58386	−1.11996	−0.559978	−	0.828507i	$-0.689191\pi$
$2$	−1.58386	−1.11996	−0.559978	+	0.828507i	$0.689191\pi$
$3$	0	0
$3$	0	0
$4$	0.508607	0.254303
$5$	2.90489	1.29911	0.649554	−	0.760315i	$-0.274956\pi$
$5$	2.90489	1.29911	0.649554	+	0.760315i	$0.274956\pi$
$6$	0	0
$7$	0.0968084	0.0365902	0.0182951	−	0.999833i	$-0.494176\pi$
$7$	0.0968084	0.0365902	0.0182951	+	0.999833i	$0.494176\pi$
$8$	2.36216	0.835148
$9$	0	0
$10$	−4.60094	−1.45495
$11$	4.36295	1.31548	0.657740	−	0.753245i	$-0.271513\pi$
$11$	4.36295	1.31548	0.657740	+	0.753245i	$0.271513\pi$
$12$	0	0
$13$	−3.92383	−1.08828	−0.544138	−	0.838996i	$-0.683143\pi$
$13$	−3.92383	−1.08828	−0.544138	+	0.838996i	$0.683143\pi$
$14$	−0.153331	−0.0409794
$15$	0	0
$16$	−4.75853	−1.18963
$17$	1.52637	0.370199	0.185099	−	0.982720i	$-0.440739\pi$
$17$	1.52637	0.370199	0.185099	+	0.982720i	$0.440739\pi$
$18$	0	0
$19$	5.57221	1.27835	0.639177	−	0.769060i	$-0.279275\pi$
$19$	5.57221	1.27835	0.639177	+	0.769060i	$0.279275\pi$
$20$	1.47745	0.330367
$21$	0	0
$22$	−6.91030	−1.47328
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	3.43841	0.687682
$26$	6.21480	1.21882
$27$	0	0
$28$	0.0492374	0.00930500
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.44461	0.798276	0.399138	−	0.916891i	$-0.369309\pi$
$31$	4.44461	0.798276	0.399138	+	0.916891i	$0.369309\pi$
$32$	2.81253	0.497190
$33$	0	0
$34$	−2.41755	−0.414607
$35$	0.281218	0.0475346
$36$	0	0
$37$	−8.67239	−1.42573	−0.712866	−	0.701300i	$-0.752603\pi$
$37$	−8.67239	−1.42573	−0.712866	+	0.701300i	$0.752603\pi$
$38$	−8.82559	−1.43170
$39$	0	0
$40$	6.86181	1.08495
$41$	3.40493	0.531760	0.265880	−	0.964006i	$-0.414337\pi$
$41$	3.40493	0.531760	0.265880	+	0.964006i	$0.414337\pi$
$42$	0	0
$43$	7.68843	1.17247	0.586237	−	0.810139i	$-0.300609\pi$
$43$	7.68843	1.17247	0.586237	+	0.810139i	$0.300609\pi$
$44$	2.21903	0.334531
$45$	0	0
$46$	−1.58386	−0.233527
$47$	8.61378	1.25645	0.628225	−	0.778032i	$-0.283782\pi$
$47$	8.61378	1.25645	0.628225	+	0.778032i	$0.283782\pi$
$48$	0	0
$49$	−6.99063	−0.998661
$50$	−5.44595	−0.770174
$51$	0	0
$52$	−1.99569	−0.276752
$53$	−2.21815	−0.304686	−0.152343	−	0.988328i	$-0.548682\pi$
$53$	−2.21815	−0.304686	−0.152343	+	0.988328i	$0.548682\pi$
$54$	0	0
$55$	12.6739	1.70895
$56$	0.228677	0.0305582
$57$	0	0
$58$	1.58386	0.207971
$59$	−8.80739	−1.14662	−0.573312	−	0.819337i	$-0.694342\pi$
$59$	−8.80739	−1.14662	−0.573312	+	0.819337i	$0.694342\pi$
$60$	0	0
$61$	−1.48378	−0.189979	−0.0949893	−	0.995478i	$-0.530282\pi$
$61$	−1.48378	−0.189979	−0.0949893	+	0.995478i	$0.530282\pi$
$62$	−7.03964	−0.894035
$63$	0	0
$64$	5.06242	0.632802
$65$	−11.3983	−1.41379
$66$	0	0
$67$	−2.72698	−0.333153	−0.166577	−	0.986029i	$-0.553271\pi$
$67$	−2.72698	−0.333153	−0.166577	+	0.986029i	$0.553271\pi$
$68$	0.776321	0.0941428
$69$	0	0
$70$	−0.445410	−0.0532367
$71$	−2.50491	−0.297278	−0.148639	−	0.988892i	$-0.547489\pi$
$71$	−2.50491	−0.297278	−0.148639	+	0.988892i	$0.547489\pi$
$72$	0	0
$73$	15.6794	1.83514	0.917569	−	0.397575i	$-0.130148\pi$
$73$	15.6794	1.83514	0.917569	+	0.397575i	$0.130148\pi$
$74$	13.7358	1.59676
$75$	0	0
$76$	2.83406	0.325089
$77$	0.422370	0.0481336
$78$	0	0
$79$	16.2533	1.82864	0.914321	−	0.404990i	$-0.132725\pi$
$79$	16.2533	1.82864	0.914321	+	0.404990i	$0.132725\pi$
$80$	−13.8230	−1.54546
$81$	0	0
$82$	−5.39292	−0.595548
$83$	14.8238	1.62713	0.813563	−	0.581476i	$-0.197525\pi$
$83$	14.8238	1.62713	0.813563	+	0.581476i	$0.197525\pi$
$84$	0	0
$85$	4.43394	0.480928
$86$	−12.1774	−1.31312
$87$	0	0
$88$	10.3060	1.09862
$89$	−2.04744	−0.217028	−0.108514	−	0.994095i	$-0.534609\pi$
$89$	−2.04744	−0.217028	−0.108514	+	0.994095i	$0.534609\pi$
$90$	0	0
$91$	−0.379860	−0.0398202
$92$	0.508607	0.0530259
$93$	0	0
$94$	−13.6430	−1.40717
$95$	16.1867	1.66072
$96$	0	0
$97$	−14.3399	−1.45599	−0.727996	−	0.685581i	$-0.759548\pi$
$97$	−14.3399	−1.45599	−0.727996	+	0.685581i	$0.759548\pi$
$98$	11.0722	1.11846
$99$	0	0
$100$	1.74880	0.174880
$101$	1.99454	0.198464	0.0992319	−	0.995064i	$-0.468361\pi$
$101$	1.99454	0.198464	0.0992319	+	0.995064i	$0.468361\pi$
$102$	0	0
$103$	17.1173	1.68662	0.843311	−	0.537426i	$-0.180603\pi$
$103$	17.1173	1.68662	0.843311	+	0.537426i	$0.180603\pi$
$104$	−9.26871	−0.908871
$105$	0	0
$106$	3.51323	0.341235
$107$	−18.2227	−1.76165	−0.880826	−	0.473439i	$-0.843012\pi$
$107$	−18.2227	−1.76165	−0.880826	+	0.473439i	$0.843012\pi$
$108$	0	0
$109$	0.356775	0.0341729	0.0170864	−	0.999854i	$-0.494561\pi$
$109$	0.356775	0.0341729	0.0170864	+	0.999854i	$0.494561\pi$
$110$	−20.0737	−1.91395
$111$	0	0
$112$	−0.460666	−0.0435289
$113$	3.28856	0.309362	0.154681	−	0.987965i	$-0.450565\pi$
$113$	3.28856	0.309362	0.154681	+	0.987965i	$0.450565\pi$
$114$	0	0
$115$	2.90489	0.270883
$116$	−0.508607	−0.0472229
$117$	0	0
$118$	13.9497	1.28417
$119$	0.147765	0.0135456
$120$	0	0
$121$	8.03534	0.730485
$122$	2.35010	0.212768
$123$	0	0
$124$	2.26056	0.203004
$125$	−4.53625	−0.405735
$126$	0	0
$127$	16.5645	1.46986	0.734929	−	0.678144i	$-0.237215\pi$
$127$	16.5645	1.46986	0.734929	+	0.678144i	$0.237215\pi$
$128$	−13.6432	−1.20590
$129$	0	0
$130$	18.0533	1.58338
$131$	1.42520	0.124520	0.0622602	−	0.998060i	$-0.480169\pi$
$131$	1.42520	0.124520	0.0622602	+	0.998060i	$0.480169\pi$
$132$	0	0
$133$	0.539437	0.0467751
$134$	4.31914	0.373117
$135$	0	0
$136$	3.60552	0.309171
$137$	11.7775	1.00622	0.503110	−	0.864222i	$-0.332189\pi$
$137$	11.7775	1.00622	0.503110	+	0.864222i	$0.332189\pi$
$138$	0	0
$139$	11.7774	0.998949	0.499475	−	0.866329i	$-0.333526\pi$
$139$	11.7774	0.998949	0.499475	+	0.866329i	$0.333526\pi$
$140$	0.143029	0.0120882
$141$	0	0
$142$	3.96742	0.332938
$143$	−17.1195	−1.43160
$144$	0	0
$145$	−2.90489	−0.241238
$146$	−24.8340	−2.05528
$147$	0	0
$148$	−4.41084	−0.362568
$149$	−8.64979	−0.708618	−0.354309	−	0.935128i	$-0.615284\pi$
$149$	−8.64979	−0.708618	−0.354309	+	0.935128i	$0.615284\pi$
$150$	0	0
$151$	−18.5565	−1.51011	−0.755055	−	0.655662i	$-0.772390\pi$
$151$	−18.5565	−1.51011	−0.755055	+	0.655662i	$0.772390\pi$
$152$	13.1624	1.06761
$153$	0	0
$154$	−0.668975	−0.0539075
$155$	12.9111	1.03705
$156$	0	0
$157$	18.9439	1.51188	0.755942	−	0.654638i	$-0.227179\pi$
$157$	18.9439	1.51188	0.755942	+	0.654638i	$0.227179\pi$
$158$	−25.7430	−2.04800
$159$	0	0
$160$	8.17010	0.645903
$161$	0.0968084	0.00762957
$162$	0	0
$163$	4.23576	0.331771	0.165885	−	0.986145i	$-0.446952\pi$
$163$	4.23576	0.331771	0.165885	+	0.986145i	$0.446952\pi$
$164$	1.73177	0.135228
$165$	0	0
$166$	−23.4788	−1.82231
$167$	−16.6112	−1.28541	−0.642705	−	0.766113i	$-0.722188\pi$
$167$	−16.6112	−1.28541	−0.642705	+	0.766113i	$0.722188\pi$
$168$	0	0
$169$	2.39647	0.184344
$170$	−7.02273	−0.538619
$171$	0	0
$172$	3.91039	0.298164
$173$	11.7545	0.893680	0.446840	−	0.894614i	$-0.352549\pi$
$173$	11.7545	0.893680	0.446840	+	0.894614i	$0.352549\pi$
$174$	0	0
$175$	0.332867	0.0251624
$176$	−20.7612	−1.56494
$177$	0	0
$178$	3.24285	0.243062
$179$	3.73448	0.279128	0.139564	−	0.990213i	$-0.455430\pi$
$179$	3.73448	0.279128	0.139564	+	0.990213i	$0.455430\pi$
$180$	0	0
$181$	−3.23482	−0.240442	−0.120221	−	0.992747i	$-0.538360\pi$
$181$	−3.23482	−0.240442	−0.120221	+	0.992747i	$0.538360\pi$
$182$	0.601645	0.0445969
$183$	0	0
$184$	2.36216	0.174140
$185$	−25.1924	−1.85218
$186$	0	0
$187$	6.65947	0.486989
$188$	4.38103	0.319519
$189$	0	0
$190$	−25.6374	−1.85993
$191$	−7.85342	−0.568254	−0.284127	−	0.958787i	$-0.591704\pi$
$191$	−7.85342	−0.568254	−0.284127	+	0.958787i	$0.591704\pi$
$192$	0	0
$193$	−20.3460	−1.46454	−0.732270	−	0.681014i	$-0.761539\pi$
$193$	−20.3460	−1.46454	−0.732270	+	0.681014i	$0.761539\pi$
$194$	22.7123	1.63065
$195$	0	0
$196$	−3.55548	−0.253963
$197$	0.764728	0.0544846	0.0272423	−	0.999629i	$-0.491327\pi$
$197$	0.764728	0.0544846	0.0272423	+	0.999629i	$0.491327\pi$
$198$	0	0
$199$	−23.5625	−1.67030	−0.835152	−	0.550020i	$-0.814620\pi$
$199$	−23.5625	−1.67030	−0.835152	+	0.550020i	$0.814620\pi$
$200$	8.12206	0.574316
$201$	0	0
$202$	−3.15906	−0.222271
$203$	−0.0968084	−0.00679462
$204$	0	0
$205$	9.89095	0.690814
$206$	−27.1114	−1.88894
$207$	0	0
$208$	18.6717	1.29465
$209$	24.3113	1.68165
$210$	0	0
$211$	16.8510	1.16007	0.580036	−	0.814591i	$-0.303038\pi$
$211$	16.8510	1.16007	0.580036	+	0.814591i	$0.303038\pi$
$212$	−1.12816	−0.0774827
$213$	0	0
$214$	28.8621	1.97297
$215$	22.3341	1.52317
$216$	0	0
$217$	0.430276	0.0292090
$218$	−0.565082	−0.0382722
$219$	0	0
$220$	6.44603	0.434592
$221$	−5.98922	−0.402878
$222$	0	0
$223$	0.609811	0.0408360	0.0204180	−	0.999792i	$-0.493500\pi$
$223$	0.609811	0.0408360	0.0204180	+	0.999792i	$0.493500\pi$
$224$	0.272277	0.0181922
$225$	0	0
$226$	−5.20861	−0.346472
$227$	14.7656	0.980027	0.490014	−	0.871715i	$-0.336992\pi$
$227$	14.7656	0.980027	0.490014	+	0.871715i	$0.336992\pi$
$228$	0	0
$229$	−7.78830	−0.514666	−0.257333	−	0.966323i	$-0.582844\pi$
$229$	−7.78830	−0.514666	−0.257333	+	0.966323i	$0.582844\pi$
$230$	−4.60094	−0.303377
$231$	0	0
$232$	−2.36216	−0.155083
$233$	−13.6189	−0.892203	−0.446101	−	0.894982i	$-0.647188\pi$
$233$	−13.6189	−0.892203	−0.446101	+	0.894982i	$0.647188\pi$
$234$	0	0
$235$	25.0221	1.63226
$236$	−4.47950	−0.291590
$237$	0	0
$238$	−0.234039	−0.0151705
$239$	−1.28716	−0.0832594	−0.0416297	−	0.999133i	$-0.513255\pi$
$239$	−1.28716	−0.0832594	−0.0416297	+	0.999133i	$0.513255\pi$
$240$	0	0
$241$	−23.6117	−1.52096	−0.760480	−	0.649361i	$-0.775036\pi$
$241$	−23.6117	−1.52096	−0.760480	+	0.649361i	$0.775036\pi$
$242$	−12.7268	−0.818112
$243$	0	0
$244$	−0.754660	−0.0483122
$245$	−20.3070	−1.29737
$246$	0	0
$247$	−21.8644	−1.39120
$248$	10.4989	0.666679
$249$	0	0
$250$	7.18478	0.454406
$251$	−21.6398	−1.36589	−0.682946	−	0.730469i	$-0.739302\pi$
$251$	−21.6398	−1.36589	−0.682946	+	0.730469i	$0.739302\pi$
$252$	0	0
$253$	4.36295	0.274296
$254$	−26.2358	−1.64618
$255$	0	0
$256$	11.4841	0.717755
$257$	19.7096	1.22945	0.614726	−	0.788740i	$-0.289266\pi$
$257$	19.7096	1.22945	0.614726	+	0.788740i	$0.289266\pi$
$258$	0	0
$259$	−0.839561	−0.0521678
$260$	−5.79726	−0.359531
$261$	0	0
$262$	−2.25732	−0.139457
$263$	−15.5944	−0.961593	−0.480797	−	0.876832i	$-0.659653\pi$
$263$	−15.5944	−0.961593	−0.480797	+	0.876832i	$0.659653\pi$
$264$	0	0
$265$	−6.44348	−0.395820
$266$	−0.854392	−0.0523861
$267$	0	0
$268$	−1.38696	−0.0847219
$269$	2.42431	0.147813	0.0739063	−	0.997265i	$-0.476453\pi$
$269$	2.42431	0.147813	0.0739063	+	0.997265i	$0.476453\pi$
$270$	0	0
$271$	−17.4887	−1.06236	−0.531182	−	0.847257i	$-0.678252\pi$
$271$	−17.4887	−1.06236	−0.531182	+	0.847257i	$0.678252\pi$
$272$	−7.26327	−0.440401
$273$	0	0
$274$	−18.6539	−1.12692
$275$	15.0016	0.904631
$276$	0	0
$277$	20.1549	1.21099	0.605494	−	0.795850i	$-0.292975\pi$
$277$	20.1549	1.21099	0.605494	+	0.795850i	$0.292975\pi$
$278$	−18.6538	−1.11878
$279$	0	0
$280$	0.664281	0.0396984
$281$	6.42574	0.383327	0.191664	−	0.981461i	$-0.438612\pi$
$281$	6.42574	0.383327	0.191664	+	0.981461i	$0.438612\pi$
$282$	0	0
$283$	31.2285	1.85634	0.928170	−	0.372156i	$-0.121381\pi$
$283$	31.2285	1.85634	0.928170	+	0.372156i	$0.121381\pi$
$284$	−1.27401	−0.0755987
$285$	0	0
$286$	27.1149	1.60333
$287$	0.329626	0.0194572
$288$	0	0
$289$	−14.6702	−0.862953
$290$	4.60094	0.270177
$291$	0	0
$292$	7.97466	0.466682
$293$	7.41080	0.432943	0.216472	−	0.976289i	$-0.430545\pi$
$293$	7.41080	0.432943	0.216472	+	0.976289i	$0.430545\pi$
$294$	0	0
$295$	−25.5845	−1.48959
$296$	−20.4855	−1.19070
$297$	0	0
$298$	13.7000	0.793622
$299$	−3.92383	−0.226921
$300$	0	0
$301$	0.744305	0.0429010
$302$	29.3909	1.69126
$303$	0	0
$304$	−26.5156	−1.52077
$305$	−4.31022	−0.246803
$306$	0	0
$307$	−28.1194	−1.60486	−0.802429	−	0.596747i	$-0.796460\pi$
$307$	−28.1194	−1.60486	−0.802429	+	0.596747i	$0.796460\pi$
$308$	0.214820	0.0122405
$309$	0	0
$310$	−20.4494	−1.16145
$311$	13.9070	0.788595	0.394297	−	0.918983i	$-0.370988\pi$
$311$	13.9070	0.788595	0.394297	+	0.918983i	$0.370988\pi$
$312$	0	0
$313$	5.36582	0.303294	0.151647	−	0.988435i	$-0.451542\pi$
$313$	5.36582	0.303294	0.151647	+	0.988435i	$0.451542\pi$
$314$	−30.0044	−1.69325
$315$	0	0
$316$	8.26655	0.465030
$317$	21.1720	1.18914	0.594568	−	0.804045i	$-0.297323\pi$
$317$	21.1720	1.18914	0.594568	+	0.804045i	$0.297323\pi$
$318$	0	0
$319$	−4.36295	−0.244278
$320$	14.7058	0.822079
$321$	0	0
$322$	−0.153331	−0.00854479
$323$	8.50525	0.473245
$324$	0	0
$325$	−13.4918	−0.748388
$326$	−6.70885	−0.371569
$327$	0	0
$328$	8.04297	0.444098
$329$	0.833887	0.0459737
$330$	0	0
$331$	−2.36292	−0.129878	−0.0649390	−	0.997889i	$-0.520685\pi$
$331$	−2.36292	−0.129878	−0.0649390	+	0.997889i	$0.520685\pi$
$332$	7.53950	0.413784
$333$	0	0
$334$	26.3097	1.43960
$335$	−7.92157	−0.432802
$336$	0	0
$337$	9.11704	0.496637	0.248318	−	0.968678i	$-0.420122\pi$
$337$	9.11704	0.496637	0.248318	+	0.968678i	$0.420122\pi$
$338$	−3.79567	−0.206457
$339$	0	0
$340$	2.25513	0.122302
$341$	19.3916	1.05012
$342$	0	0
$343$	−1.35441	−0.0731313
$344$	18.1613	0.979190
$345$	0	0
$346$	−18.6175	−1.00088
$347$	−1.41418	−0.0759171	−0.0379585	−	0.999279i	$-0.512085\pi$
$347$	−1.41418	−0.0759171	−0.0379585	+	0.999279i	$0.512085\pi$
$348$	0	0
$349$	−2.27278	−0.121659	−0.0608295	−	0.998148i	$-0.519375\pi$
$349$	−2.27278	−0.121659	−0.0608295	+	0.998148i	$0.519375\pi$
$350$	−0.527214	−0.0281808
$351$	0	0
$352$	12.2709	0.654043
$353$	−13.3967	−0.713032	−0.356516	−	0.934289i	$-0.616035\pi$
$353$	−13.3967	−0.713032	−0.356516	+	0.934289i	$0.616035\pi$
$354$	0	0
$355$	−7.27649	−0.386196
$356$	−1.04134	−0.0551909
$357$	0	0
$358$	−5.91489	−0.312612
$359$	14.9573	0.789413	0.394707	−	0.918807i	$-0.370846\pi$
$359$	14.9573	0.789413	0.394707	+	0.918807i	$0.370846\pi$
$360$	0	0
$361$	12.0495	0.634186
$362$	5.12349	0.269285
$363$	0	0
$364$	−0.193199	−0.0101264
$365$	45.5471	2.38404
$366$	0	0
$367$	28.4967	1.48752	0.743758	−	0.668449i	$-0.233041\pi$
$367$	28.4967	1.48752	0.743758	+	0.668449i	$0.233041\pi$
$368$	−4.75853	−0.248056
$369$	0	0
$370$	39.9012	2.07436
$371$	−0.214735	−0.0111485
$372$	0	0
$373$	18.8175	0.974332	0.487166	−	0.873309i	$-0.338031\pi$
$373$	18.8175	0.974332	0.487166	+	0.873309i	$0.338031\pi$
$374$	−10.5477	−0.545406
$375$	0	0
$376$	20.3471	1.04932
$377$	3.92383	0.202088
$378$	0	0
$379$	−4.26908	−0.219288	−0.109644	−	0.993971i	$-0.534971\pi$
$379$	−4.26908	−0.219288	−0.109644	+	0.993971i	$0.534971\pi$
$380$	8.23265	0.422326
$381$	0	0
$382$	12.4387	0.636420
$383$	−30.7470	−1.57110	−0.785549	−	0.618799i	$-0.787619\pi$
$383$	−30.7470	−1.57110	−0.785549	+	0.618799i	$0.787619\pi$
$384$	0	0
$385$	1.22694	0.0625307
$386$	32.2252	1.64022
$387$	0	0
$388$	−7.29335	−0.370264
$389$	−6.84915	−0.347266	−0.173633	−	0.984810i	$-0.555551\pi$
$389$	−6.84915	−0.347266	−0.173633	+	0.984810i	$0.555551\pi$
$390$	0	0
$391$	1.52637	0.0771918
$392$	−16.5130	−0.834030
$393$	0	0
$394$	−1.21122	−0.0610204
$395$	47.2142	2.37560
$396$	0	0
$397$	29.9704	1.50417	0.752086	−	0.659065i	$-0.229048\pi$
$397$	29.9704	1.50417	0.752086	+	0.659065i	$0.229048\pi$
$398$	37.3197	1.87067
$399$	0	0
$400$	−16.3618	−0.818089
$401$	39.3655	1.96582	0.982911	−	0.184084i	$-0.0589319\pi$
$401$	39.3655	1.96582	0.982911	+	0.184084i	$0.0589319\pi$
$402$	0	0
$403$	−17.4399	−0.868745
$404$	1.01443	0.0504700
$405$	0	0
$406$	0.153331	0.00760968
$407$	−37.8372	−1.87552
$408$	0	0
$409$	20.3285	1.00518	0.502589	−	0.864526i	$-0.332381\pi$
$409$	20.3285	1.00518	0.502589	+	0.864526i	$0.332381\pi$
$410$	−15.6659	−0.773682
$411$	0	0
$412$	8.70599	0.428913
$413$	−0.852630	−0.0419552
$414$	0	0
$415$	43.0616	2.11381
$416$	−11.0359	−0.541079
$417$	0	0
$418$	−38.5056	−1.88337
$419$	−39.1322	−1.91173	−0.955867	−	0.293799i	$-0.905080\pi$
$419$	−39.1322	−1.91173	−0.955867	+	0.293799i	$0.905080\pi$
$420$	0	0
$421$	18.0319	0.878819	0.439410	−	0.898287i	$-0.355188\pi$
$421$	18.0319	0.878819	0.439410	+	0.898287i	$0.355188\pi$
$422$	−26.6896	−1.29923
$423$	0	0
$424$	−5.23961	−0.254458
$425$	5.24828	0.254579
$426$	0	0
$427$	−0.143642	−0.00695134
$428$	−9.26817	−0.447994
$429$	0	0
$430$	−35.3740	−1.70589
$431$	32.6230	1.57139	0.785697	−	0.618611i	$-0.212304\pi$
$431$	32.6230	1.57139	0.785697	+	0.618611i	$0.212304\pi$
$432$	0	0
$433$	−39.7919	−1.91228	−0.956139	−	0.292915i	$-0.905375\pi$
$433$	−39.7919	−1.91228	−0.956139	+	0.292915i	$0.905375\pi$
$434$	−0.681496	−0.0327129
$435$	0	0
$436$	0.181458	0.00869028
$437$	5.57221	0.266555
$438$	0	0
$439$	−20.5386	−0.980253	−0.490127	−	0.871651i	$-0.663049\pi$
$439$	−20.5386	−0.980253	−0.490127	+	0.871651i	$0.663049\pi$
$440$	29.9377	1.42723
$441$	0	0
$442$	9.48607	0.451206
$443$	30.8866	1.46747	0.733734	−	0.679437i	$-0.237776\pi$
$443$	30.8866	1.46747	0.733734	+	0.679437i	$0.237776\pi$
$444$	0	0
$445$	−5.94758	−0.281943
$446$	−0.965854	−0.0457345
$447$	0	0
$448$	0.490085	0.0231543
$449$	−3.92637	−0.185297	−0.0926485	−	0.995699i	$-0.529533\pi$
$449$	−3.92637	−0.185297	−0.0926485	+	0.995699i	$0.529533\pi$
$450$	0	0
$451$	14.8555	0.699519
$452$	1.67258	0.0786716
$453$	0	0
$454$	−23.3866	−1.09759
$455$	−1.10345	−0.0517307
$456$	0	0
$457$	7.12358	0.333227	0.166613	−	0.986022i	$-0.446717\pi$
$457$	7.12358	0.333227	0.166613	+	0.986022i	$0.446717\pi$
$458$	12.3356	0.576403
$459$	0	0
$460$	1.47745	0.0688864
$461$	−19.9339	−0.928413	−0.464207	−	0.885727i	$-0.653661\pi$
$461$	−19.9339	−0.928413	−0.464207	+	0.885727i	$0.653661\pi$
$462$	0	0
$463$	26.0682	1.21149	0.605746	−	0.795658i	$-0.292875\pi$
$463$	26.0682	1.21149	0.605746	+	0.795658i	$0.292875\pi$
$464$	4.75853	0.220909
$465$	0	0
$466$	21.5704	0.999228
$467$	−18.4633	−0.854381	−0.427190	−	0.904162i	$-0.640497\pi$
$467$	−18.4633	−0.854381	−0.427190	+	0.904162i	$0.640497\pi$
$468$	0	0
$469$	−0.263994	−0.0121901
$470$	−39.6315	−1.82807
$471$	0	0
$472$	−20.8044	−0.957601
$473$	33.5442	1.54237
$474$	0	0
$475$	19.1595	0.879100
$476$	0.0751544	0.00344470
$477$	0	0
$478$	2.03868	0.0932469
$479$	−1.98652	−0.0907665	−0.0453833	−	0.998970i	$-0.514451\pi$
$479$	−1.98652	−0.0907665	−0.0453833	+	0.998970i	$0.514451\pi$
$480$	0	0
$481$	34.0290	1.55159
$482$	37.3975	1.70341
$483$	0	0
$484$	4.08683	0.185765
$485$	−41.6558	−1.89149
$486$	0	0
$487$	−9.06545	−0.410795	−0.205397	−	0.978679i	$-0.565849\pi$
$487$	−9.06545	−0.410795	−0.205397	+	0.978679i	$0.565849\pi$
$488$	−3.50492	−0.158660
$489$	0	0
$490$	32.1635	1.45300
$491$	−41.6936	−1.88161	−0.940804	−	0.338951i	$-0.889928\pi$
$491$	−41.6936	−1.88161	−0.940804	+	0.338951i	$0.889928\pi$
$492$	0	0
$493$	−1.52637	−0.0687442
$494$	34.6302	1.55808
$495$	0	0
$496$	−21.1498	−0.949656
$497$	−0.242496	−0.0108774
$498$	0	0
$499$	−6.87004	−0.307545	−0.153773	−	0.988106i	$-0.549142\pi$
$499$	−6.87004	−0.307545	−0.153773	+	0.988106i	$0.549142\pi$
$500$	−2.30717	−0.103180
$501$	0	0
$502$	34.2744	1.52974
$503$	12.9064	0.575466	0.287733	−	0.957711i	$-0.407098\pi$
$503$	12.9064	0.575466	0.287733	+	0.957711i	$0.407098\pi$
$504$	0	0
$505$	5.79392	0.257826
$506$	−6.91030	−0.307200
$507$	0	0
$508$	8.42479	0.373790
$509$	17.6686	0.783147	0.391573	−	0.920147i	$-0.371931\pi$
$509$	17.6686	0.783147	0.391573	+	0.920147i	$0.371931\pi$
$510$	0	0
$511$	1.51790	0.0671480
$512$	9.09728	0.402047
$513$	0	0
$514$	−31.2173	−1.37693
$515$	49.7240	2.19110
$516$	0	0
$517$	37.5815	1.65283
$518$	1.32975	0.0584256
$519$	0	0
$520$	−26.9246	−1.18072
$521$	−37.5554	−1.64533	−0.822666	−	0.568525i	$-0.807514\pi$
$521$	−37.5554	−1.64533	−0.822666	+	0.568525i	$0.807514\pi$
$522$	0	0
$523$	23.2036	1.01462	0.507311	−	0.861763i	$-0.330639\pi$
$523$	23.2036	1.01462	0.507311	+	0.861763i	$0.330639\pi$
$524$	0.724867	0.0316660
$525$	0	0
$526$	24.6994	1.07694
$527$	6.78412	0.295521
$528$	0	0
$529$	1.00000	0.0434783
$530$	10.2056	0.443301
$531$	0	0
$532$	0.274361	0.0118951
$533$	−13.3604	−0.578702
$534$	0	0
$535$	−52.9349	−2.28858
$536$	−6.44154	−0.278232
$537$	0	0
$538$	−3.83976	−0.165544
$539$	−30.4998	−1.31372
$540$	0	0
$541$	17.5245	0.753436	0.376718	−	0.926328i	$-0.377053\pi$
$541$	17.5245	0.753436	0.376718	+	0.926328i	$0.377053\pi$
$542$	27.6997	1.18980
$543$	0	0
$544$	4.29296	0.184059
$545$	1.03639	0.0443943
$546$	0	0
$547$	4.23553	0.181098	0.0905492	−	0.995892i	$-0.471138\pi$
$547$	4.23553	0.181098	0.0905492	+	0.995892i	$0.471138\pi$
$548$	5.99012	0.255885
$549$	0	0
$550$	−23.7604	−1.01315
$551$	−5.57221	−0.237384
$552$	0	0
$553$	1.57346	0.0669103
$554$	−31.9225	−1.35625
$555$	0	0
$556$	5.99008	0.254036
$557$	−16.8012	−0.711891	−0.355946	−	0.934507i	$-0.615841\pi$
$557$	−16.8012	−0.711891	−0.355946	+	0.934507i	$0.615841\pi$
$558$	0	0
$559$	−30.1681	−1.27598
$560$	−1.33819	−0.0565487
$561$	0	0
$562$	−10.1775	−0.429310
$563$	−6.12837	−0.258280	−0.129140	−	0.991626i	$-0.541222\pi$
$563$	−6.12837	−0.258280	−0.129140	+	0.991626i	$0.541222\pi$
$564$	0	0
$565$	9.55291	0.401894
$566$	−49.4615	−2.07902
$567$	0	0
$568$	−5.91698	−0.248271
$569$	26.9326	1.12908	0.564538	−	0.825407i	$-0.309054\pi$
$569$	26.9326	1.12908	0.564538	+	0.825407i	$0.309054\pi$
$570$	0	0
$571$	27.1474	1.13608	0.568042	−	0.822999i	$-0.307701\pi$
$571$	27.1474	1.13608	0.568042	+	0.822999i	$0.307701\pi$
$572$	−8.70709	−0.364062
$573$	0	0
$574$	−0.522080	−0.0217912
$575$	3.43841	0.143392
$576$	0	0
$577$	5.49869	0.228913	0.114457	−	0.993428i	$-0.463487\pi$
$577$	5.49869	0.228913	0.114457	+	0.993428i	$0.463487\pi$
$578$	23.2355	0.966470
$579$	0	0
$580$	−1.47745	−0.0613477
$581$	1.43507	0.0595368
$582$	0	0
$583$	−9.67767	−0.400808
$584$	37.0373	1.53261
$585$	0	0
$586$	−11.7377	−0.484878
$587$	27.7447	1.14515	0.572573	−	0.819854i	$-0.305945\pi$
$587$	27.7447	1.14515	0.572573	+	0.819854i	$0.305945\pi$
$588$	0	0
$589$	24.7663	1.02048
$590$	40.5223	1.66828
$591$	0	0
$592$	41.2679	1.69610
$593$	−24.1118	−0.990154	−0.495077	−	0.868849i	$-0.664860\pi$
$593$	−24.1118	−0.990154	−0.495077	+	0.868849i	$0.664860\pi$
$594$	0	0
$595$	0.429243	0.0175972
$596$	−4.39934	−0.180204
$597$	0	0
$598$	6.21480	0.254142
$599$	41.5579	1.69801	0.849004	−	0.528387i	$-0.177203\pi$
$599$	41.5579	1.69801	0.849004	+	0.528387i	$0.177203\pi$
$600$	0	0
$601$	−42.4855	−1.73302	−0.866511	−	0.499158i	$-0.833643\pi$
$601$	−42.4855	−1.73302	−0.866511	+	0.499158i	$0.833643\pi$
$602$	−1.17887	−0.0480473
$603$	0	0
$604$	−9.43797	−0.384026
$605$	23.3418	0.948980
$606$	0	0
$607$	42.6081	1.72941	0.864706	−	0.502279i	$-0.167505\pi$
$607$	42.6081	1.72941	0.864706	+	0.502279i	$0.167505\pi$
$608$	15.6720	0.635584
$609$	0	0
$610$	6.82678	0.276408
$611$	−33.7991	−1.36736
$612$	0	0
$613$	44.7421	1.80712	0.903559	−	0.428464i	$-0.140945\pi$
$613$	44.7421	1.80712	0.903559	+	0.428464i	$0.140945\pi$
$614$	44.5371	1.79737
$615$	0	0
$616$	0.997705	0.0401987
$617$	−6.92432	−0.278763	−0.139381	−	0.990239i	$-0.544511\pi$
$617$	−6.92432	−0.278763	−0.139381	+	0.990239i	$0.544511\pi$
$618$	0	0
$619$	14.1440	0.568494	0.284247	−	0.958751i	$-0.408256\pi$
$619$	14.1440	0.568494	0.284247	+	0.958751i	$0.408256\pi$
$620$	6.56669	0.263724
$621$	0	0
$622$	−22.0267	−0.883192
$623$	−0.198209	−0.00794108
$624$	0	0
$625$	−30.3694	−1.21478
$626$	−8.49870	−0.339676
$627$	0	0
$628$	9.63497	0.384477
$629$	−13.2373	−0.527804
$630$	0	0
$631$	24.3216	0.968228	0.484114	−	0.875005i	$-0.339142\pi$
$631$	24.3216	0.968228	0.484114	+	0.875005i	$0.339142\pi$
$632$	38.3929	1.52719
$633$	0	0
$634$	−33.5334	−1.33178
$635$	48.1180	1.90950
$636$	0	0
$637$	27.4301	1.08682
$638$	6.91030	0.273581
$639$	0	0
$640$	−39.6321	−1.56660
$641$	40.2588	1.59013	0.795064	−	0.606525i	$-0.207437\pi$
$641$	40.2588	1.59013	0.795064	+	0.606525i	$0.207437\pi$
$642$	0	0
$643$	8.90630	0.351230	0.175615	−	0.984459i	$-0.443809\pi$
$643$	8.90630	0.351230	0.175615	+	0.984459i	$0.443809\pi$
$644$	0.0492374	0.00194023
$645$	0	0
$646$	−13.4711	−0.530014
$647$	−25.6208	−1.00726	−0.503628	−	0.863921i	$-0.668002\pi$
$647$	−25.6208	−1.00726	−0.503628	+	0.863921i	$0.668002\pi$
$648$	0	0
$649$	−38.4262	−1.50836
$650$	21.3690	0.838162
$651$	0	0
$652$	2.15434	0.0843704
$653$	27.7689	1.08668	0.543340	−	0.839513i	$-0.317159\pi$
$653$	27.7689	1.08668	0.543340	+	0.839513i	$0.317159\pi$
$654$	0	0
$655$	4.14006	0.161765
$656$	−16.2025	−0.632599
$657$	0	0
$658$	−1.32076	−0.0514885
$659$	21.2075	0.826126	0.413063	−	0.910702i	$-0.364459\pi$
$659$	21.2075	0.826126	0.413063	+	0.910702i	$0.364459\pi$
$660$	0	0
$661$	−20.4915	−0.797028	−0.398514	−	0.917162i	$-0.630474\pi$
$661$	−20.4915	−0.797028	−0.398514	+	0.917162i	$0.630474\pi$
$662$	3.74253	0.145458
$663$	0	0
$664$	35.0162	1.35889
$665$	1.56701	0.0607660
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−8.44855	−0.326884
$669$	0	0
$670$	12.5466	0.484719
$671$	−6.47366	−0.249913
$672$	0	0
$673$	−12.6572	−0.487899	−0.243950	−	0.969788i	$-0.578443\pi$
$673$	−12.6572	−0.487899	−0.243950	+	0.969788i	$0.578443\pi$
$674$	−14.4401	−0.556212
$675$	0	0
$676$	1.21886	0.0468793
$677$	−19.4221	−0.746450	−0.373225	−	0.927741i	$-0.621748\pi$
$677$	−19.4221	−0.746450	−0.373225	+	0.927741i	$0.621748\pi$
$678$	0	0
$679$	−1.38822	−0.0532750
$680$	10.4737	0.401646
$681$	0	0
$682$	−30.7136	−1.17608
$683$	3.06562	0.117303	0.0586514	−	0.998279i	$-0.481320\pi$
$683$	3.06562	0.117303	0.0586514	+	0.998279i	$0.481320\pi$
$684$	0	0
$685$	34.2124	1.30719
$686$	2.14519	0.0819039
$687$	0	0
$688$	−36.5856	−1.39481
$689$	8.70364	0.331582
$690$	0	0
$691$	−6.86721	−0.261241	−0.130621	−	0.991432i	$-0.541697\pi$
$691$	−6.86721	−0.261241	−0.130621	+	0.991432i	$0.541697\pi$
$692$	5.97843	0.227266
$693$	0	0
$694$	2.23986	0.0850238
$695$	34.2122	1.29774
$696$	0	0
$697$	5.19717	0.196857
$698$	3.59976	0.136253
$699$	0	0
$700$	0.169298	0.00639888
$701$	8.80645	0.332615	0.166308	−	0.986074i	$-0.446816\pi$
$701$	8.80645	0.332615	0.166308	+	0.986074i	$0.446816\pi$
$702$	0	0
$703$	−48.3244	−1.82259
$704$	22.0871	0.832438
$705$	0	0
$706$	21.2184	0.798565
$707$	0.193088	0.00726182
$708$	0	0
$709$	21.2991	0.799905	0.399952	−	0.916536i	$-0.369027\pi$
$709$	21.2991	0.799905	0.399952	+	0.916536i	$0.369027\pi$
$710$	11.5249	0.432523
$711$	0	0
$712$	−4.83636	−0.181250
$713$	4.44461	0.166452
$714$	0	0
$715$	−49.7303	−1.85981
$716$	1.89938	0.0709832
$717$	0	0
$718$	−23.6902	−0.884109
$719$	6.58400	0.245542	0.122771	−	0.992435i	$-0.460822\pi$
$719$	6.58400	0.245542	0.122771	+	0.992435i	$0.460822\pi$
$720$	0	0
$721$	1.65710	0.0617137
$722$	−19.0848	−0.710261
$723$	0	0
$724$	−1.64525	−0.0611452
$725$	−3.43841	−0.127699
$726$	0	0
$727$	20.3458	0.754584	0.377292	−	0.926094i	$-0.376855\pi$
$727$	20.3458	0.754584	0.377292	+	0.926094i	$0.376855\pi$
$728$	−0.897289	−0.0332557
$729$	0	0
$730$	−72.1401	−2.67003
$731$	11.7354	0.434049
$732$	0	0
$733$	32.3845	1.19615	0.598074	−	0.801441i	$-0.295933\pi$
$733$	32.3845	1.19615	0.598074	+	0.801441i	$0.295933\pi$
$734$	−45.1348	−1.66595
$735$	0	0
$736$	2.81253	0.103671
$737$	−11.8977	−0.438256
$738$	0	0
$739$	7.18250	0.264213	0.132106	−	0.991236i	$-0.457826\pi$
$739$	7.18250	0.264213	0.132106	+	0.991236i	$0.457826\pi$
$740$	−12.8130	−0.471016
$741$	0	0
$742$	0.340110	0.0124858
$743$	−18.5111	−0.679106	−0.339553	−	0.940587i	$-0.610276\pi$
$743$	−18.5111	−0.679106	−0.339553	+	0.940587i	$0.610276\pi$
$744$	0	0
$745$	−25.1267	−0.920571
$746$	−29.8042	−1.09121
$747$	0	0
$748$	3.38705	0.123843
$749$	−1.76411	−0.0644591
$750$	0	0
$751$	43.0785	1.57196	0.785978	−	0.618255i	$-0.212160\pi$
$751$	43.0785	1.57196	0.785978	+	0.618255i	$0.212160\pi$
$752$	−40.9890	−1.49471
$753$	0	0
$754$	−6.21480	−0.226330
$755$	−53.9048	−1.96180
$756$	0	0
$757$	−10.4225	−0.378813	−0.189406	−	0.981899i	$-0.560656\pi$
$757$	−10.4225	−0.378813	−0.189406	+	0.981899i	$0.560656\pi$
$758$	6.76162	0.245593
$759$	0	0
$760$	38.2355	1.38695
$761$	25.2406	0.914970	0.457485	−	0.889217i	$-0.348750\pi$
$761$	25.2406	0.914970	0.457485	+	0.889217i	$0.348750\pi$
$762$	0	0
$763$	0.0345389	0.00125039
$764$	−3.99430	−0.144509
$765$	0	0
$766$	48.6989	1.75956
$767$	34.5587	1.24784
$768$	0	0
$769$	23.1983	0.836553	0.418277	−	0.908320i	$-0.362634\pi$
$769$	23.1983	0.836553	0.418277	+	0.908320i	$0.362634\pi$
$770$	−1.94330	−0.0700317
$771$	0	0
$772$	−10.3481	−0.372437
$773$	−9.56064	−0.343872	−0.171936	−	0.985108i	$-0.555002\pi$
$773$	−9.56064	−0.343872	−0.171936	+	0.985108i	$0.555002\pi$
$774$	0	0
$775$	15.2824	0.548960
$776$	−33.8730	−1.21597
$777$	0	0
$778$	10.8481	0.388923
$779$	18.9730	0.679777
$780$	0	0
$781$	−10.9288	−0.391063
$782$	−2.41755	−0.0864515
$783$	0	0
$784$	33.2651	1.18804
$785$	55.0299	1.96410
$786$	0	0
$787$	−44.0106	−1.56881	−0.784404	−	0.620250i	$-0.787031\pi$
$787$	−44.0106	−1.56881	−0.784404	+	0.620250i	$0.787031\pi$
$788$	0.388945	0.0138556
$789$	0	0
$790$	−74.7806	−2.66057
$791$	0.318360	0.0113196
$792$	0	0
$793$	5.82211	0.206749
$794$	−47.4689	−1.68461
$795$	0	0
$796$	−11.9841	−0.424764
$797$	4.52871	0.160415	0.0802076	−	0.996778i	$-0.474442\pi$
$797$	4.52871	0.160415	0.0802076	+	0.996778i	$0.474442\pi$
$798$	0	0
$799$	13.1478	0.465136
$800$	9.67063	0.341908
$801$	0	0
$802$	−62.3494	−2.20163
$803$	68.4086	2.41409
$804$	0	0
$805$	0.281218	0.00991164
$806$	27.6224	0.972956
$807$	0	0
$808$	4.71140	0.165747
$809$	17.8990	0.629296	0.314648	−	0.949208i	$-0.398114\pi$
$809$	17.8990	0.629296	0.314648	+	0.949208i	$0.398114\pi$
$810$	0	0
$811$	−46.9914	−1.65009	−0.825045	−	0.565067i	$-0.808850\pi$
$811$	−46.9914	−1.65009	−0.825045	+	0.565067i	$0.808850\pi$
$812$	−0.0492374	−0.00172789
$813$	0	0
$814$	59.9288	2.10050
$815$	12.3044	0.431006
$816$	0	0
$817$	42.8416	1.49884
$818$	−32.1974	−1.12576
$819$	0	0
$820$	5.03060	0.175676
$821$	23.8602	0.832725	0.416363	−	0.909199i	$-0.363305\pi$
$821$	23.8602	0.832725	0.416363	+	0.909199i	$0.363305\pi$
$822$	0	0
$823$	19.1320	0.666901	0.333450	−	0.942768i	$-0.391787\pi$
$823$	19.1320	0.666901	0.333450	+	0.942768i	$0.391787\pi$
$824$	40.4338	1.40858
$825$	0	0
$826$	1.35044	0.0469880
$827$	−41.3582	−1.43817	−0.719083	−	0.694924i	$-0.755438\pi$
$827$	−41.3582	−1.43817	−0.719083	+	0.694924i	$0.755438\pi$
$828$	0	0
$829$	−12.2569	−0.425699	−0.212849	−	0.977085i	$-0.568274\pi$
$829$	−12.2569	−0.425699	−0.212849	+	0.977085i	$0.568274\pi$
$830$	−68.2035	−2.36738
$831$	0	0
$832$	−19.8641	−0.688663
$833$	−10.6703	−0.369703
$834$	0	0
$835$	−48.2537	−1.66989
$836$	12.3649	0.427648
$837$	0	0
$838$	61.9799	2.14106
$839$	43.9192	1.51626	0.758130	−	0.652104i	$-0.226113\pi$
$839$	43.9192	1.51626	0.758130	+	0.652104i	$0.226113\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−28.5599	−0.984240
$843$	0	0
$844$	8.57054	0.295010
$845$	6.96150	0.239483
$846$	0	0
$847$	0.777889	0.0267286
$848$	10.5551	0.362465
$849$	0	0
$850$	−8.31253	−0.285118
$851$	−8.67239	−0.297286
$852$	0	0
$853$	−11.5581	−0.395740	−0.197870	−	0.980228i	$-0.563402\pi$
$853$	−11.5581	−0.395740	−0.197870	+	0.980228i	$0.563402\pi$
$854$	0.227509	0.00778520
$855$	0	0
$856$	−43.0448	−1.47124
$857$	−31.6176	−1.08004	−0.540018	−	0.841653i	$-0.681583\pi$
$857$	−31.6176	−1.08004	−0.540018	+	0.841653i	$0.681583\pi$
$858$	0	0
$859$	−45.8035	−1.56280	−0.781398	−	0.624033i	$-0.785493\pi$
$859$	−45.8035	−1.56280	−0.781398	+	0.624033i	$0.785493\pi$
$860$	11.3593	0.387347
$861$	0	0
$862$	−51.6702	−1.75989
$863$	2.72163	0.0926454	0.0463227	−	0.998927i	$-0.485250\pi$
$863$	2.72163	0.0926454	0.0463227	+	0.998927i	$0.485250\pi$
$864$	0	0
$865$	34.1456	1.16099
$866$	63.0247	2.14167
$867$	0	0
$868$	0.218841	0.00742796
$869$	70.9125	2.40554
$870$	0	0
$871$	10.7002	0.362562
$872$	0.842759	0.0285394
$873$	0	0
$874$	−8.82559	−0.298530
$875$	−0.439148	−0.0148459
$876$	0	0
$877$	−38.1813	−1.28929	−0.644645	−	0.764482i	$-0.722995\pi$
$877$	−38.1813	−1.28929	−0.644645	+	0.764482i	$0.722995\pi$
$878$	32.5302	1.09784
$879$	0	0
$880$	−60.3092	−2.03302
$881$	24.2288	0.816289	0.408144	−	0.912917i	$-0.366176\pi$
$881$	24.2288	0.816289	0.408144	+	0.912917i	$0.366176\pi$
$882$	0	0
$883$	−26.8175	−0.902481	−0.451241	−	0.892402i	$-0.649018\pi$
$883$	−26.8175	−0.902481	−0.451241	+	0.892402i	$0.649018\pi$
$884$	−3.04616	−0.102453
$885$	0	0
$886$	−48.9200	−1.64350
$887$	−21.8632	−0.734096	−0.367048	−	0.930202i	$-0.619631\pi$
$887$	−21.8632	−0.734096	−0.367048	+	0.930202i	$0.619631\pi$
$888$	0	0
$889$	1.60358	0.0537823
$890$	9.42013	0.315763
$891$	0	0
$892$	0.310154	0.0103847
$893$	47.9978	1.60619
$894$	0	0
$895$	10.8483	0.362618
$896$	−1.32078	−0.0441241
$897$	0	0
$898$	6.21882	0.207525
$899$	−4.44461	−0.148236
$900$	0	0
$901$	−3.38571	−0.112794
$902$	−23.5290	−0.783431
$903$	0	0
$904$	7.76809	0.258363
$905$	−9.39680	−0.312360
$906$	0	0
$907$	−27.7820	−0.922485	−0.461242	−	0.887274i	$-0.652596\pi$
$907$	−27.7820	−0.922485	−0.461242	+	0.887274i	$0.652596\pi$
$908$	7.50988	0.249224
$909$	0	0
$910$	1.74771	0.0579362
$911$	−9.80864	−0.324975	−0.162487	−	0.986711i	$-0.551952\pi$
$911$	−9.80864	−0.324975	−0.162487	+	0.986711i	$0.551952\pi$
$912$	0	0
$913$	64.6756	2.14045
$914$	−11.2827	−0.373200
$915$	0	0
$916$	−3.96118	−0.130881
$917$	0.137972	0.00455622
$918$	0	0
$919$	11.0060	0.363056	0.181528	−	0.983386i	$-0.441896\pi$
$919$	11.0060	0.363056	0.181528	+	0.983386i	$0.441896\pi$
$920$	6.86181	0.226227
$921$	0	0
$922$	31.5724	1.03978
$923$	9.82884	0.323520
$924$	0	0
$925$	−29.8192	−0.980450
$926$	−41.2884	−1.35682
$927$	0	0
$928$	−2.81253	−0.0923258
$929$	−27.3752	−0.898150	−0.449075	−	0.893494i	$-0.648246\pi$
$929$	−27.3752	−0.898150	−0.449075	+	0.893494i	$0.648246\pi$
$930$	0	0
$931$	−38.9533	−1.27664
$932$	−6.92665	−0.226890
$933$	0	0
$934$	29.2433	0.956869
$935$	19.3451	0.632651
$936$	0	0
$937$	29.6395	0.968279	0.484140	−	0.874991i	$-0.339133\pi$
$937$	29.6395	0.968279	0.484140	+	0.874991i	$0.339133\pi$
$938$	0.418129	0.0136524
$939$	0	0
$940$	12.7264	0.415090
$941$	−14.7181	−0.479796	−0.239898	−	0.970798i	$-0.577114\pi$
$941$	−14.7181	−0.479796	−0.239898	+	0.970798i	$0.577114\pi$
$942$	0	0
$943$	3.40493	0.110880
$944$	41.9103	1.36406
$945$	0	0
$946$	−53.1293	−1.72738
$947$	−17.6693	−0.574177	−0.287088	−	0.957904i	$-0.592687\pi$
$947$	−17.6693	−0.574177	−0.287088	+	0.957904i	$0.592687\pi$
$948$	0	0
$949$	−61.5235	−1.99714
$950$	−30.3460	−0.984554
$951$	0	0
$952$	0.349045	0.0113126
$953$	17.4849	0.566392	0.283196	−	0.959062i	$-0.408605\pi$
$953$	17.4849	0.566392	0.283196	+	0.959062i	$0.408605\pi$
$954$	0	0
$955$	−22.8134	−0.738223
$956$	−0.654658	−0.0211731
$957$	0	0
$958$	3.14637	0.101655
$959$	1.14016	0.0368178
$960$	0	0
$961$	−11.2454	−0.362755
$962$	−53.8971	−1.73771
$963$	0	0
$964$	−12.0090	−0.386785
$965$	−59.1031	−1.90260
$966$	0	0
$967$	44.5854	1.43377	0.716884	−	0.697192i	$-0.245568\pi$
$967$	44.5854	1.43377	0.716884	+	0.697192i	$0.245568\pi$
$968$	18.9807	0.610064
$969$	0	0
$970$	65.9768	2.11839
$971$	−53.0079	−1.70110	−0.850552	−	0.525890i	$-0.823732\pi$
$971$	−53.0079	−1.70110	−0.850552	+	0.525890i	$0.823732\pi$
$972$	0	0
$973$	1.14016	0.0365517
$974$	14.3584	0.460072
$975$	0	0
$976$	7.06061	0.226005
$977$	0.586001	0.0187478	0.00937392	−	0.999956i	$-0.497016\pi$
$977$	0.586001	0.0187478	0.00937392	+	0.999956i	$0.497016\pi$
$978$	0	0
$979$	−8.93286	−0.285495
$980$	−10.3283	−0.329925
$981$	0	0
$982$	66.0368	2.10732
$983$	53.2258	1.69764	0.848819	−	0.528684i	$-0.177314\pi$
$983$	53.2258	1.69764	0.848819	+	0.528684i	$0.177314\pi$
$984$	0	0
$985$	2.22145	0.0707814
$986$	2.41755	0.0769905
$987$	0	0
$988$	−11.1204	−0.353787
$989$	7.68843	0.244478
$990$	0	0
$991$	30.5555	0.970626	0.485313	−	0.874340i	$-0.338706\pi$
$991$	30.5555	0.970626	0.485313	+	0.874340i	$0.338706\pi$
$992$	12.5006	0.396895
$993$	0	0
$994$	0.384080	0.0121823
$995$	−68.4467	−2.16991
$996$	0	0
$997$	38.6503	1.22407	0.612034	−	0.790831i	$-0.290352\pi$
$997$	38.6503	1.22407	0.612034	+	0.790831i	$0.290352\pi$
$998$	10.8812	0.344438
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.v.1.9	✓	30
3.2	odd	2		6003.2.a.w.1.22	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.9	✓	30	1.1	even	1	trivial
6003.2.a.w.1.22	yes	30	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.58386 q^{2} +0.508607 q^{4} +2.90489 q^{5} +0.0968084 q^{7} +2.36216 q^{8} +O(q^{10})\)	\(q-1.58386 q^{2} +0.508607 q^{4} +2.90489 q^{5} +0.0968084 q^{7} +2.36216 q^{8} -4.60094 q^{10} +4.36295 q^{11} -3.92383 q^{13} -0.153331 q^{14} -4.75853 q^{16} +1.52637 q^{17} +5.57221 q^{19} +1.47745 q^{20} -6.91030 q^{22} +1.00000 q^{23} +3.43841 q^{25} +6.21480 q^{26} +0.0492374 q^{28} -1.00000 q^{29} +4.44461 q^{31} +2.81253 q^{32} -2.41755 q^{34} +0.281218 q^{35} -8.67239 q^{37} -8.82559 q^{38} +6.86181 q^{40} +3.40493 q^{41} +7.68843 q^{43} +2.21903 q^{44} -1.58386 q^{46} +8.61378 q^{47} -6.99063 q^{49} -5.44595 q^{50} -1.99569 q^{52} -2.21815 q^{53} +12.6739 q^{55} +0.228677 q^{56} +1.58386 q^{58} -8.80739 q^{59} -1.48378 q^{61} -7.03964 q^{62} +5.06242 q^{64} -11.3983 q^{65} -2.72698 q^{67} +0.776321 q^{68} -0.445410 q^{70} -2.50491 q^{71} +15.6794 q^{73} +13.7358 q^{74} +2.83406 q^{76} +0.422370 q^{77} +16.2533 q^{79} -13.8230 q^{80} -5.39292 q^{82} +14.8238 q^{83} +4.43394 q^{85} -12.1774 q^{86} +10.3060 q^{88} -2.04744 q^{89} -0.379860 q^{91} +0.508607 q^{92} -13.6430 q^{94} +16.1867 q^{95} -14.3399 q^{97} +11.0722 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) \) 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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