LMFDB - Embedded newform 6003.2.a.i.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 9x^{5} + 10x^{4} + 19x^{3} - 20x^{2} - 5x + 1 $ x^7 - x^6 - 9x^5 + 10x^4 + 19x^3 - 20x^2 - 5*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.81932$ of defining polynomial
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)

$f(q)$	$=$	$q+2.37229 q^{2} +3.62775 q^{4} -2.09663 q^{5} -1.30993 q^{7} +3.86149 q^{8} +O(q^{10})$	\(q+2.37229 q^{2} +3.62775 q^{4} -2.09663 q^{5} -1.30993 q^{7} +3.86149 q^{8} -4.97382 q^{10} +4.73037 q^{11} -2.36326 q^{13} -3.10753 q^{14} +1.90506 q^{16} -4.59351 q^{17} -1.09166 q^{19} -7.60606 q^{20} +11.2218 q^{22} +1.00000 q^{23} -0.604124 q^{25} -5.60633 q^{26} -4.75209 q^{28} -1.00000 q^{29} -2.06348 q^{31} -3.20362 q^{32} -10.8971 q^{34} +2.74644 q^{35} -1.42713 q^{37} -2.58974 q^{38} -8.09613 q^{40} -5.43495 q^{41} -5.29682 q^{43} +17.1606 q^{44} +2.37229 q^{46} +3.95047 q^{47} -5.28409 q^{49} -1.43316 q^{50} -8.57332 q^{52} +12.6640 q^{53} -9.91786 q^{55} -5.05827 q^{56} -2.37229 q^{58} +0.349383 q^{59} -3.89988 q^{61} -4.89516 q^{62} -11.4100 q^{64} +4.95489 q^{65} -4.91966 q^{67} -16.6641 q^{68} +6.51535 q^{70} +0.192690 q^{71} -0.0751700 q^{73} -3.38557 q^{74} -3.96028 q^{76} -6.19645 q^{77} -7.88002 q^{79} -3.99422 q^{80} -12.8933 q^{82} -3.95860 q^{83} +9.63091 q^{85} -12.5656 q^{86} +18.2663 q^{88} -8.32098 q^{89} +3.09570 q^{91} +3.62775 q^{92} +9.37166 q^{94} +2.28882 q^{95} -7.82125 q^{97} -12.5354 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8}+O(q^{10}) $ 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8	$ 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8} - 11 q^{10} + 12 q^{11} - 13 q^{13} + 3 q^{14} - 13 q^{16} + 12 q^{17} - 5 q^{19} + 8 q^{20} - q^{22} + 7 q^{23} - 4 q^{25} - 2 q^{26} - 21 q^{28} - 7 q^{29} - 8 q^{31} + 5 q^{32} - 28 q^{34} - 5 q^{35} - 24 q^{37} + 6 q^{38} - 20 q^{40} - 9 q^{41} - q^{43} + 23 q^{44} + q^{46} - 27 q^{47} - 14 q^{49} - 7 q^{50} - 9 q^{52} + q^{53} - 11 q^{55} + 20 q^{56} - q^{58} - 8 q^{59} + q^{61} + 3 q^{64} - 12 q^{65} - 16 q^{67} - 15 q^{68} + 40 q^{70} + 13 q^{71} - 23 q^{73} + 8 q^{74} - 2 q^{76} - 13 q^{77} - 44 q^{79} - 30 q^{80} - 10 q^{82} - 21 q^{83} - 6 q^{86} + 21 q^{88} + 5 q^{89} - 18 q^{91} + 7 q^{92} + 28 q^{94} - 9 q^{95} - 55 q^{97} - 36 q^{98}+O(q^{100}) $ 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8 - 11 * q^10 + 12 * q^11 - 13 * q^13 + 3 * q^14 - 13 * q^16 + 12 * q^17 - 5 * q^19 + 8 * q^20 - q^22 + 7 * q^23 - 4 * q^25 - 2 * q^26 - 21 * q^28 - 7 * q^29 - 8 * q^31 + 5 * q^32 - 28 * q^34 - 5 * q^35 - 24 * q^37 + 6 * q^38 - 20 * q^40 - 9 * q^41 - q^43 + 23 * q^44 + q^46 - 27 * q^47 - 14 * q^49 - 7 * q^50 - 9 * q^52 + q^53 - 11 * q^55 + 20 * q^56 - q^58 - 8 * q^59 + q^61 + 3 * q^64 - 12 * q^65 - 16 * q^67 - 15 * q^68 + 40 * q^70 + 13 * q^71 - 23 * q^73 + 8 * q^74 - 2 * q^76 - 13 * q^77 - 44 * q^79 - 30 * q^80 - 10 * q^82 - 21 * q^83 - 6 * q^86 + 21 * q^88 + 5 * q^89 - 18 * q^91 + 7 * q^92 + 28 * q^94 - 9 * q^95 - 55 * q^97 - 36 * 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 $

$2$	2.37229	1.67746	0.838730	−	0.544547i	$-0.183298\pi$
$2$	2.37229	1.67746	0.838730	+	0.544547i	$0.183298\pi$
$3$	0	0
$3$	0	0
$4$	3.62775	1.81387
$5$	−2.09663	−0.937643	−0.468822	−	0.883293i	$-0.655321\pi$
$5$	−2.09663	−0.937643	−0.468822	+	0.883293i	$0.655321\pi$
$6$	0	0
$7$	−1.30993	−0.495107	−0.247553	−	0.968874i	$-0.579627\pi$
$7$	−1.30993	−0.495107	−0.247553	+	0.968874i	$0.579627\pi$
$8$	3.86149	1.36524
$9$	0	0
$10$	−4.97382	−1.57286
$11$	4.73037	1.42626	0.713130	−	0.701032i	$-0.247277\pi$
$11$	4.73037	1.42626	0.713130	+	0.701032i	$0.247277\pi$
$12$	0	0
$13$	−2.36326	−0.655451	−0.327725	−	0.944773i	$-0.606282\pi$
$13$	−2.36326	−0.655451	−0.327725	+	0.944773i	$0.606282\pi$
$14$	−3.10753	−0.830522
$15$	0	0
$16$	1.90506	0.476266
$17$	−4.59351	−1.11409	−0.557045	−	0.830482i	$-0.688065\pi$
$17$	−4.59351	−1.11409	−0.557045	+	0.830482i	$0.688065\pi$
$18$	0	0
$19$	−1.09166	−0.250445	−0.125222	−	0.992129i	$-0.539964\pi$
$19$	−1.09166	−0.250445	−0.125222	+	0.992129i	$0.539964\pi$
$20$	−7.60606	−1.70077
$21$	0	0
$22$	11.2218	2.39250
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−0.604124	−0.120825
$26$	−5.60633	−1.09949
$27$	0	0
$28$	−4.75209	−0.898061
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−2.06348	−0.370611	−0.185306	−	0.982681i	$-0.559328\pi$
$31$	−2.06348	−0.370611	−0.185306	+	0.982681i	$0.559328\pi$
$32$	−3.20362	−0.566325
$33$	0	0
$34$	−10.8971	−1.86884
$35$	2.74644	0.464233
$36$	0	0
$37$	−1.42713	−0.234619	−0.117309	−	0.993095i	$-0.537427\pi$
$37$	−1.42713	−0.234619	−0.117309	+	0.993095i	$0.537427\pi$
$38$	−2.58974	−0.420111
$39$	0	0
$40$	−8.09613	−1.28011
$41$	−5.43495	−0.848797	−0.424398	−	0.905476i	$-0.639514\pi$
$41$	−5.43495	−0.848797	−0.424398	+	0.905476i	$0.639514\pi$
$42$	0	0
$43$	−5.29682	−0.807757	−0.403879	−	0.914813i	$-0.632338\pi$
$43$	−5.29682	−0.807757	−0.403879	+	0.914813i	$0.632338\pi$
$44$	17.1606	2.58706
$45$	0	0
$46$	2.37229	0.349775
$47$	3.95047	0.576236	0.288118	−	0.957595i	$-0.406970\pi$
$47$	3.95047	0.576236	0.288118	+	0.957595i	$0.406970\pi$
$48$	0	0
$49$	−5.28409	−0.754869
$50$	−1.43316	−0.202679
$51$	0	0
$52$	−8.57332	−1.18891
$53$	12.6640	1.73953	0.869765	−	0.493467i	$-0.164271\pi$
$53$	12.6640	1.73953	0.869765	+	0.493467i	$0.164271\pi$
$54$	0	0
$55$	−9.91786	−1.33732
$56$	−5.05827	−0.675940
$57$	0	0
$58$	−2.37229	−0.311497
$59$	0.349383	0.0454857	0.0227429	−	0.999741i	$-0.492760\pi$
$59$	0.349383	0.0454857	0.0227429	+	0.999741i	$0.492760\pi$
$60$	0	0
$61$	−3.89988	−0.499329	−0.249664	−	0.968332i	$-0.580320\pi$
$61$	−3.89988	−0.499329	−0.249664	+	0.968332i	$0.580320\pi$
$62$	−4.89516	−0.621686
$63$	0	0
$64$	−11.4100	−1.42625
$65$	4.95489	0.614579
$66$	0	0
$67$	−4.91966	−0.601033	−0.300516	−	0.953777i	$-0.597159\pi$
$67$	−4.91966	−0.601033	−0.300516	+	0.953777i	$0.597159\pi$
$68$	−16.6641	−2.02082
$69$	0	0
$70$	6.51535	0.778733
$71$	0.192690	0.0228681	0.0114341	−	0.999935i	$-0.496360\pi$
$71$	0.192690	0.0228681	0.0114341	+	0.999935i	$0.496360\pi$
$72$	0	0
$73$	−0.0751700	−0.00879798	−0.00439899	−	0.999990i	$-0.501400\pi$
$73$	−0.0751700	−0.00879798	−0.00439899	+	0.999990i	$0.501400\pi$
$74$	−3.38557	−0.393564
$75$	0	0
$76$	−3.96028	−0.454275
$77$	−6.19645	−0.706151
$78$	0	0
$79$	−7.88002	−0.886571	−0.443286	−	0.896380i	$-0.646187\pi$
$79$	−7.88002	−0.886571	−0.443286	+	0.896380i	$0.646187\pi$
$80$	−3.99422	−0.446567
$81$	0	0
$82$	−12.8933	−1.42382
$83$	−3.95860	−0.434513	−0.217256	−	0.976115i	$-0.569711\pi$
$83$	−3.95860	−0.434513	−0.217256	+	0.976115i	$0.569711\pi$
$84$	0	0
$85$	9.63091	1.04462
$86$	−12.5656	−1.35498
$87$	0	0
$88$	18.2663	1.94719
$89$	−8.32098	−0.882022	−0.441011	−	0.897502i	$-0.645380\pi$
$89$	−8.32098	−0.882022	−0.441011	+	0.897502i	$0.645380\pi$
$90$	0	0
$91$	3.09570	0.324518
$92$	3.62775	0.378219
$93$	0	0
$94$	9.37166	0.966613
$95$	2.28882	0.234828
$96$	0	0
$97$	−7.82125	−0.794128	−0.397064	−	0.917791i	$-0.629971\pi$
$97$	−7.82125	−0.794128	−0.397064	+	0.917791i	$0.629971\pi$
$98$	−12.5354	−1.26626
$99$	0	0
$100$	−2.19161	−0.219161
$101$	−6.21680	−0.618595	−0.309297	−	0.950965i	$-0.600094\pi$
$101$	−6.21680	−0.618595	−0.309297	+	0.950965i	$0.600094\pi$
$102$	0	0
$103$	−6.74168	−0.664277	−0.332139	−	0.943231i	$-0.607770\pi$
$103$	−6.74168	−0.664277	−0.332139	+	0.943231i	$0.607770\pi$
$104$	−9.12570	−0.894849
$105$	0	0
$106$	30.0426	2.91799
$107$	−2.51900	−0.243521	−0.121760	−	0.992560i	$-0.538854\pi$
$107$	−2.51900	−0.243521	−0.121760	+	0.992560i	$0.538854\pi$
$108$	0	0
$109$	7.87610	0.754393	0.377197	−	0.926133i	$-0.376888\pi$
$109$	7.87610	0.754393	0.377197	+	0.926133i	$0.376888\pi$
$110$	−23.5280	−2.24331
$111$	0	0
$112$	−2.49550	−0.235802
$113$	−8.00452	−0.753002	−0.376501	−	0.926416i	$-0.622873\pi$
$113$	−8.00452	−0.753002	−0.376501	+	0.926416i	$0.622873\pi$
$114$	0	0
$115$	−2.09663	−0.195512
$116$	−3.62775	−0.336828
$117$	0	0
$118$	0.828836	0.0763005
$119$	6.01717	0.551593
$120$	0	0
$121$	11.3764	1.03422
$122$	−9.25165	−0.837605
$123$	0	0
$124$	−7.48577	−0.672242
$125$	11.7498	1.05093
$126$	0	0
$127$	−11.0488	−0.980424	−0.490212	−	0.871603i	$-0.663081\pi$
$127$	−11.0488	−0.980424	−0.490212	+	0.871603i	$0.663081\pi$
$128$	−20.6606	−1.82616
$129$	0	0
$130$	11.7544	1.03093
$131$	9.61978	0.840484	0.420242	−	0.907412i	$-0.361945\pi$
$131$	9.61978	0.840484	0.420242	+	0.907412i	$0.361945\pi$
$132$	0	0
$133$	1.43000	0.123997
$134$	−11.6709	−1.00821
$135$	0	0
$136$	−17.7378	−1.52100
$137$	6.60764	0.564529	0.282264	−	0.959337i	$-0.408914\pi$
$137$	6.60764	0.564529	0.282264	+	0.959337i	$0.408914\pi$
$138$	0	0
$139$	−5.52325	−0.468476	−0.234238	−	0.972179i	$-0.575259\pi$
$139$	−5.52325	−0.468476	−0.234238	+	0.972179i	$0.575259\pi$
$140$	9.96340	0.842061
$141$	0	0
$142$	0.457117	0.0383604
$143$	−11.1791	−0.934843
$144$	0	0
$145$	2.09663	0.174116
$146$	−0.178325	−0.0147583
$147$	0	0
$148$	−5.17727	−0.425569
$149$	20.2927	1.66244	0.831220	−	0.555943i	$-0.187643\pi$
$149$	20.2927	1.66244	0.831220	+	0.555943i	$0.187643\pi$
$150$	0	0
$151$	−0.340828	−0.0277362	−0.0138681	−	0.999904i	$-0.504414\pi$
$151$	−0.340828	−0.0277362	−0.0138681	+	0.999904i	$0.504414\pi$
$152$	−4.21544	−0.341918
$153$	0	0
$154$	−14.6998	−1.18454
$155$	4.32636	0.347501
$156$	0	0
$157$	9.25110	0.738318	0.369159	−	0.929366i	$-0.379646\pi$
$157$	9.25110	0.738318	0.369159	+	0.929366i	$0.379646\pi$
$158$	−18.6937	−1.48719
$159$	0	0
$160$	6.71682	0.531011
$161$	−1.30993	−0.103237
$162$	0	0
$163$	7.32914	0.574063	0.287031	−	0.957921i	$-0.407332\pi$
$163$	7.32914	0.574063	0.287031	+	0.957921i	$0.407332\pi$
$164$	−19.7166	−1.53961
$165$	0	0
$166$	−9.39094	−0.728878
$167$	−15.4637	−1.19662	−0.598308	−	0.801266i	$-0.704160\pi$
$167$	−15.4637	−1.19662	−0.598308	+	0.801266i	$0.704160\pi$
$168$	0	0
$169$	−7.41500	−0.570384
$170$	22.8473	1.75231
$171$	0	0
$172$	−19.2155	−1.46517
$173$	−5.22400	−0.397173	−0.198586	−	0.980083i	$-0.563635\pi$
$173$	−5.22400	−0.397173	−0.198586	+	0.980083i	$0.563635\pi$
$174$	0	0
$175$	0.791359	0.0598211
$176$	9.01165	0.679279
$177$	0	0
$178$	−19.7398	−1.47956
$179$	5.76333	0.430772	0.215386	−	0.976529i	$-0.430899\pi$
$179$	5.76333	0.430772	0.215386	+	0.976529i	$0.430899\pi$
$180$	0	0
$181$	−12.4134	−0.922681	−0.461340	−	0.887223i	$-0.652631\pi$
$181$	−12.4134	−0.922681	−0.461340	+	0.887223i	$0.652631\pi$
$182$	7.34390	0.544366
$183$	0	0
$184$	3.86149	0.284673
$185$	2.99217	0.219989
$186$	0	0
$187$	−21.7290	−1.58898
$188$	14.3313	1.04522
$189$	0	0
$190$	5.42974	0.393914
$191$	−4.01165	−0.290272	−0.145136	−	0.989412i	$-0.546362\pi$
$191$	−4.01165	−0.290272	−0.145136	+	0.989412i	$0.546362\pi$
$192$	0	0
$193$	22.8492	1.64473	0.822363	−	0.568964i	$-0.192656\pi$
$193$	22.8492	1.64473	0.822363	+	0.568964i	$0.192656\pi$
$194$	−18.5543	−1.33212
$195$	0	0
$196$	−19.1693	−1.36924
$197$	−22.8093	−1.62510	−0.812548	−	0.582895i	$-0.801920\pi$
$197$	−22.8093	−1.62510	−0.812548	+	0.582895i	$0.801920\pi$
$198$	0	0
$199$	−20.2299	−1.43406	−0.717029	−	0.697044i	$-0.754498\pi$
$199$	−20.2299	−1.43406	−0.717029	+	0.697044i	$0.754498\pi$
$200$	−2.33282	−0.164955
$201$	0	0
$202$	−14.7480	−1.03767
$203$	1.30993	0.0919390
$204$	0	0
$205$	11.3951	0.795869
$206$	−15.9932	−1.11430
$207$	0	0
$208$	−4.50216	−0.312169
$209$	−5.16397	−0.357199
$210$	0	0
$211$	0.285230	0.0196361	0.00981803	−	0.999952i	$-0.496875\pi$
$211$	0.285230	0.0196361	0.00981803	+	0.999952i	$0.496875\pi$
$212$	45.9417	3.15529
$213$	0	0
$214$	−5.97579	−0.408496
$215$	11.1055	0.757388
$216$	0	0
$217$	2.70301	0.183492
$218$	18.6844	1.26547
$219$	0	0
$220$	−35.9795	−2.42574
$221$	10.8557	0.730231
$222$	0	0
$223$	−17.1981	−1.15167	−0.575835	−	0.817566i	$-0.695323\pi$
$223$	−17.1981	−1.15167	−0.575835	+	0.817566i	$0.695323\pi$
$224$	4.19651	0.280391
$225$	0	0
$226$	−18.9890	−1.26313
$227$	−0.927871	−0.0615849	−0.0307925	−	0.999526i	$-0.509803\pi$
$227$	−0.927871	−0.0615849	−0.0307925	+	0.999526i	$0.509803\pi$
$228$	0	0
$229$	26.7379	1.76689	0.883447	−	0.468532i	$-0.155217\pi$
$229$	26.7379	1.76689	0.883447	+	0.468532i	$0.155217\pi$
$230$	−4.97382	−0.327964
$231$	0	0
$232$	−3.86149	−0.253519
$233$	13.1076	0.858709	0.429354	−	0.903136i	$-0.358741\pi$
$233$	13.1076	0.858709	0.429354	+	0.903136i	$0.358741\pi$
$234$	0	0
$235$	−8.28270	−0.540304
$236$	1.26747	0.0825054
$237$	0	0
$238$	14.2745	0.925276
$239$	18.3178	1.18488	0.592442	−	0.805613i	$-0.298164\pi$
$239$	18.3178	1.18488	0.592442	+	0.805613i	$0.298164\pi$
$240$	0	0
$241$	−9.46544	−0.609723	−0.304861	−	0.952397i	$-0.598610\pi$
$241$	−9.46544	−0.609723	−0.304861	+	0.952397i	$0.598610\pi$
$242$	26.9881	1.73486
$243$	0	0
$244$	−14.1478	−0.905720
$245$	11.0788	0.707798
$246$	0	0
$247$	2.57988	0.164154
$248$	−7.96809	−0.505974
$249$	0	0
$250$	27.8739	1.76290
$251$	18.9396	1.19545	0.597727	−	0.801700i	$-0.296071\pi$
$251$	18.9396	1.19545	0.597727	+	0.801700i	$0.296071\pi$
$252$	0	0
$253$	4.73037	0.297396
$254$	−26.2110	−1.64462
$255$	0	0
$256$	−26.1929	−1.63706
$257$	14.4326	0.900282	0.450141	−	0.892957i	$-0.351374\pi$
$257$	14.4326	0.900282	0.450141	+	0.892957i	$0.351374\pi$
$258$	0	0
$259$	1.86944	0.116161
$260$	17.9751	1.11477
$261$	0	0
$262$	22.8209	1.40988
$263$	3.02472	0.186512	0.0932561	−	0.995642i	$-0.470272\pi$
$263$	3.02472	0.186512	0.0932561	+	0.995642i	$0.470272\pi$
$264$	0	0
$265$	−26.5517	−1.63106
$266$	3.39237	0.208000
$267$	0	0
$268$	−17.8473	−1.09020
$269$	13.6336	0.831253	0.415627	−	0.909535i	$-0.363562\pi$
$269$	13.6336	0.831253	0.415627	+	0.909535i	$0.363562\pi$
$270$	0	0
$271$	−25.8561	−1.57065	−0.785324	−	0.619085i	$-0.787504\pi$
$271$	−25.8561	−1.57065	−0.785324	+	0.619085i	$0.787504\pi$
$272$	−8.75092	−0.530602
$273$	0	0
$274$	15.6752	0.946975
$275$	−2.85773	−0.172328
$276$	0	0
$277$	−16.5742	−0.995848	−0.497924	−	0.867221i	$-0.665904\pi$
$277$	−16.5742	−0.995848	−0.497924	+	0.867221i	$0.665904\pi$
$278$	−13.1027	−0.785849
$279$	0	0
$280$	10.6054	0.633791
$281$	4.83197	0.288251	0.144126	−	0.989559i	$-0.453963\pi$
$281$	4.83197	0.288251	0.144126	+	0.989559i	$0.453963\pi$
$282$	0	0
$283$	28.8993	1.71788	0.858942	−	0.512073i	$-0.171122\pi$
$283$	28.8993	1.71788	0.858942	+	0.512073i	$0.171122\pi$
$284$	0.699032	0.0414799
$285$	0	0
$286$	−26.5200	−1.56816
$287$	7.11940	0.420245
$288$	0	0
$289$	4.10032	0.241195
$290$	4.97382	0.292073
$291$	0	0
$292$	−0.272698	−0.0159584
$293$	−1.67650	−0.0979423	−0.0489712	−	0.998800i	$-0.515594\pi$
$293$	−1.67650	−0.0979423	−0.0489712	+	0.998800i	$0.515594\pi$
$294$	0	0
$295$	−0.732528	−0.0426494
$296$	−5.51085	−0.320312
$297$	0	0
$298$	48.1401	2.78868
$299$	−2.36326	−0.136671
$300$	0	0
$301$	6.93845	0.399926
$302$	−0.808541	−0.0465263
$303$	0	0
$304$	−2.07969	−0.119278
$305$	8.17663	0.468193
$306$	0	0
$307$	4.47374	0.255330	0.127665	−	0.991817i	$-0.459252\pi$
$307$	4.47374	0.255330	0.127665	+	0.991817i	$0.459252\pi$
$308$	−22.4792	−1.28087
$309$	0	0
$310$	10.2634	0.582920
$311$	7.78691	0.441555	0.220778	−	0.975324i	$-0.429140\pi$
$311$	7.78691	0.441555	0.220778	+	0.975324i	$0.429140\pi$
$312$	0	0
$313$	−8.69093	−0.491240	−0.245620	−	0.969366i	$-0.578992\pi$
$313$	−8.69093	−0.491240	−0.245620	+	0.969366i	$0.578992\pi$
$314$	21.9463	1.23850
$315$	0	0
$316$	−28.5867	−1.60813
$317$	−27.1083	−1.52255	−0.761277	−	0.648427i	$-0.775427\pi$
$317$	−27.1083	−1.52255	−0.761277	+	0.648427i	$0.775427\pi$
$318$	0	0
$319$	−4.73037	−0.264850
$320$	23.9227	1.33732
$321$	0	0
$322$	−3.10753	−0.173176
$323$	5.01456	0.279018
$324$	0	0
$325$	1.42770	0.0791946
$326$	17.3868	0.962968
$327$	0	0
$328$	−20.9870	−1.15881
$329$	−5.17484	−0.285298
$330$	0	0
$331$	−19.4019	−1.06643	−0.533213	−	0.845981i	$-0.679016\pi$
$331$	−19.4019	−1.06643	−0.533213	+	0.845981i	$0.679016\pi$
$332$	−14.3608	−0.788152
$333$	0	0
$334$	−36.6843	−2.00728
$335$	10.3147	0.563554
$336$	0	0
$337$	9.78867	0.533223	0.266612	−	0.963804i	$-0.414096\pi$
$337$	9.78867	0.533223	0.266612	+	0.963804i	$0.414096\pi$
$338$	−17.5905	−0.956797
$339$	0	0
$340$	34.9385	1.89481
$341$	−9.76101	−0.528588
$342$	0	0
$343$	16.0913	0.868847
$344$	−20.4536	−1.10278
$345$	0	0
$346$	−12.3928	−0.666242
$347$	−0.788572	−0.0423328	−0.0211664	−	0.999776i	$-0.506738\pi$
$347$	−0.788572	−0.0423328	−0.0211664	+	0.999776i	$0.506738\pi$
$348$	0	0
$349$	1.22972	0.0658255	0.0329127	−	0.999458i	$-0.489522\pi$
$349$	1.22972	0.0658255	0.0329127	+	0.999458i	$0.489522\pi$
$350$	1.87733	0.100348
$351$	0	0
$352$	−15.1543	−0.807727
$353$	5.60029	0.298073	0.149037	−	0.988832i	$-0.452383\pi$
$353$	5.60029	0.298073	0.149037	+	0.988832i	$0.452383\pi$
$354$	0	0
$355$	−0.404001	−0.0214422
$356$	−30.1864	−1.59988
$357$	0	0
$358$	13.6723	0.722603
$359$	−0.365086	−0.0192685	−0.00963424	−	0.999954i	$-0.503067\pi$
$359$	−0.365086	−0.0192685	−0.00963424	+	0.999954i	$0.503067\pi$
$360$	0	0
$361$	−17.8083	−0.937277
$362$	−29.4482	−1.54776
$363$	0	0
$364$	11.2304	0.588635
$365$	0.157604	0.00824937
$366$	0	0
$367$	34.3964	1.79548	0.897738	−	0.440530i	$-0.145209\pi$
$367$	34.3964	1.79548	0.897738	+	0.440530i	$0.145209\pi$
$368$	1.90506	0.0993082
$369$	0	0
$370$	7.09829	0.369023
$371$	−16.5889	−0.861252
$372$	0	0
$373$	18.8593	0.976496	0.488248	−	0.872705i	$-0.337636\pi$
$373$	18.8593	0.976496	0.488248	+	0.872705i	$0.337636\pi$
$374$	−51.5474	−2.66545
$375$	0	0
$376$	15.2547	0.786701
$377$	2.36326	0.121714
$378$	0	0
$379$	5.89233	0.302668	0.151334	−	0.988483i	$-0.451643\pi$
$379$	5.89233	0.302668	0.151334	+	0.988483i	$0.451643\pi$
$380$	8.30326	0.425948
$381$	0	0
$382$	−9.51678	−0.486921
$383$	−14.8967	−0.761188	−0.380594	−	0.924742i	$-0.624280\pi$
$383$	−14.8967	−0.761188	−0.380594	+	0.924742i	$0.624280\pi$
$384$	0	0
$385$	12.9917	0.662118
$386$	54.2050	2.75896
$387$	0	0
$388$	−28.3735	−1.44045
$389$	19.4013	0.983683	0.491841	−	0.870685i	$-0.336324\pi$
$389$	19.4013	0.983683	0.491841	+	0.870685i	$0.336324\pi$
$390$	0	0
$391$	−4.59351	−0.232304
$392$	−20.4044	−1.03058
$393$	0	0
$394$	−54.1102	−2.72603
$395$	16.5215	0.831288
$396$	0	0
$397$	25.8765	1.29870	0.649352	−	0.760488i	$-0.275040\pi$
$397$	25.8765	1.29870	0.649352	+	0.760488i	$0.275040\pi$
$398$	−47.9911	−2.40558
$399$	0	0
$400$	−1.15089	−0.0575446
$401$	14.2999	0.714102	0.357051	−	0.934085i	$-0.383782\pi$
$401$	14.2999	0.714102	0.357051	+	0.934085i	$0.383782\pi$
$402$	0	0
$403$	4.87653	0.242917
$404$	−22.5530	−1.12205
$405$	0	0
$406$	3.10753	0.154224
$407$	−6.75086	−0.334628
$408$	0	0
$409$	29.7667	1.47187	0.735935	−	0.677053i	$-0.236743\pi$
$409$	29.7667	1.47187	0.735935	+	0.677053i	$0.236743\pi$
$410$	27.0325	1.33504
$411$	0	0
$412$	−24.4571	−1.20492
$413$	−0.457666	−0.0225203
$414$	0	0
$415$	8.29974	0.407418
$416$	7.57099	0.371198
$417$	0	0
$418$	−12.2504	−0.599188
$419$	−21.4046	−1.04568	−0.522841	−	0.852430i	$-0.675128\pi$
$419$	−21.4046	−1.04568	−0.522841	+	0.852430i	$0.675128\pi$
$420$	0	0
$421$	14.7543	0.719080	0.359540	−	0.933130i	$-0.382934\pi$
$421$	14.7543	0.719080	0.359540	+	0.933130i	$0.382934\pi$
$422$	0.676648	0.0329387
$423$	0	0
$424$	48.9017	2.37488
$425$	2.77505	0.134610
$426$	0	0
$427$	5.10857	0.247221
$428$	−9.13829	−0.441716
$429$	0	0
$430$	26.3454	1.27049
$431$	16.5410	0.796750	0.398375	−	0.917223i	$-0.369574\pi$
$431$	16.5410	0.796750	0.398375	+	0.917223i	$0.369574\pi$
$432$	0	0
$433$	−6.94392	−0.333704	−0.166852	−	0.985982i	$-0.553360\pi$
$433$	−6.94392	−0.333704	−0.166852	+	0.985982i	$0.553360\pi$
$434$	6.41231	0.307801
$435$	0	0
$436$	28.5725	1.36837
$437$	−1.09166	−0.0522213
$438$	0	0
$439$	−35.4709	−1.69293	−0.846467	−	0.532440i	$-0.821275\pi$
$439$	−35.4709	−1.69293	−0.846467	+	0.532440i	$0.821275\pi$
$440$	−38.2977	−1.82577
$441$	0	0
$442$	25.7527	1.22493
$443$	−29.6590	−1.40914	−0.704571	−	0.709634i	$-0.748860\pi$
$443$	−29.6590	−1.40914	−0.704571	+	0.709634i	$0.748860\pi$
$444$	0	0
$445$	17.4461	0.827022
$446$	−40.7988	−1.93188
$447$	0	0
$448$	14.9463	0.706148
$449$	−17.9546	−0.847328	−0.423664	−	0.905819i	$-0.639256\pi$
$449$	−17.9546	−0.847328	−0.423664	+	0.905819i	$0.639256\pi$
$450$	0	0
$451$	−25.7093	−1.21060
$452$	−29.0384	−1.36585
$453$	0	0
$454$	−2.20118	−0.103306
$455$	−6.49056	−0.304282
$456$	0	0
$457$	19.5239	0.913291	0.456645	−	0.889649i	$-0.349051\pi$
$457$	19.5239	0.913291	0.456645	+	0.889649i	$0.349051\pi$
$458$	63.4301	2.96389
$459$	0	0
$460$	−7.60606	−0.354635
$461$	−27.7062	−1.29041	−0.645203	−	0.764011i	$-0.723227\pi$
$461$	−27.7062	−1.29041	−0.645203	+	0.764011i	$0.723227\pi$
$462$	0	0
$463$	10.0214	0.465734	0.232867	−	0.972509i	$-0.425189\pi$
$463$	10.0214	0.465734	0.232867	+	0.972509i	$0.425189\pi$
$464$	−1.90506	−0.0884403
$465$	0	0
$466$	31.0950	1.44045
$467$	14.1233	0.653548	0.326774	−	0.945103i	$-0.394038\pi$
$467$	14.1233	0.653548	0.326774	+	0.945103i	$0.394038\pi$
$468$	0	0
$469$	6.44441	0.297575
$470$	−19.6489	−0.906338
$471$	0	0
$472$	1.34914	0.0620990
$473$	−25.0559	−1.15207
$474$	0	0
$475$	0.659499	0.0302599
$476$	21.8288	1.00052
$477$	0	0
$478$	43.4552	1.98759
$479$	13.5034	0.616985	0.308492	−	0.951227i	$-0.400176\pi$
$479$	13.5034	0.616985	0.308492	+	0.951227i	$0.400176\pi$
$480$	0	0
$481$	3.37268	0.153781
$482$	−22.4548	−1.02279
$483$	0	0
$484$	41.2707	1.87594
$485$	16.3983	0.744609
$486$	0	0
$487$	−36.3392	−1.64669	−0.823343	−	0.567544i	$-0.807894\pi$
$487$	−36.3392	−1.64669	−0.823343	+	0.567544i	$0.807894\pi$
$488$	−15.0594	−0.681705
$489$	0	0
$490$	26.2821	1.18730
$491$	−21.8423	−0.985729	−0.492864	−	0.870106i	$-0.664050\pi$
$491$	−21.8423	−0.985729	−0.492864	+	0.870106i	$0.664050\pi$
$492$	0	0
$493$	4.59351	0.206881
$494$	6.12023	0.275362
$495$	0	0
$496$	−3.93105	−0.176509
$497$	−0.252411	−0.0113222
$498$	0	0
$499$	13.4141	0.600496	0.300248	−	0.953861i	$-0.402931\pi$
$499$	13.4141	0.600496	0.300248	+	0.953861i	$0.402931\pi$
$500$	42.6253	1.90626
$501$	0	0
$502$	44.9301	2.00533
$503$	9.42420	0.420205	0.210102	−	0.977679i	$-0.432620\pi$
$503$	9.42420	0.420205	0.210102	+	0.977679i	$0.432620\pi$
$504$	0	0
$505$	13.0344	0.580021
$506$	11.2218	0.498870
$507$	0	0
$508$	−40.0823	−1.77837
$509$	−8.72802	−0.386863	−0.193431	−	0.981114i	$-0.561962\pi$
$509$	−8.72802	−0.386863	−0.193431	+	0.981114i	$0.561962\pi$
$510$	0	0
$511$	0.0984673	0.00435594
$512$	−20.8158	−0.919939
$513$	0	0
$514$	34.2383	1.51019
$515$	14.1348	0.622855
$516$	0	0
$517$	18.6872	0.821862
$518$	4.43485	0.194856
$519$	0	0
$520$	19.1333	0.839049
$521$	23.5678	1.03253	0.516263	−	0.856430i	$-0.327323\pi$
$521$	23.5678	1.03253	0.516263	+	0.856430i	$0.327323\pi$
$522$	0	0
$523$	3.08811	0.135033	0.0675167	−	0.997718i	$-0.478492\pi$
$523$	3.08811	0.135033	0.0675167	+	0.997718i	$0.478492\pi$
$524$	34.8981	1.52453
$525$	0	0
$526$	7.17551	0.312867
$527$	9.47860	0.412894
$528$	0	0
$529$	1.00000	0.0434783
$530$	−62.9883	−2.73604
$531$	0	0
$532$	5.18768	0.224915
$533$	12.8442	0.556344
$534$	0	0
$535$	5.28142	0.228336
$536$	−18.9972	−0.820555
$537$	0	0
$538$	32.3427	1.39439
$539$	−24.9957	−1.07664
$540$	0	0
$541$	28.0181	1.20459	0.602296	−	0.798273i	$-0.294253\pi$
$541$	28.0181	1.20459	0.602296	+	0.798273i	$0.294253\pi$
$542$	−61.3382	−2.63470
$543$	0	0
$544$	14.7159	0.630937
$545$	−16.5133	−0.707352
$546$	0	0
$547$	12.7714	0.546066	0.273033	−	0.962005i	$-0.411973\pi$
$547$	12.7714	0.546066	0.273033	+	0.962005i	$0.411973\pi$
$548$	23.9709	1.02398
$549$	0	0
$550$	−6.77935	−0.289073
$551$	1.09166	0.0465064
$552$	0	0
$553$	10.3223	0.438947
$554$	−39.3188	−1.67050
$555$	0	0
$556$	−20.0369	−0.849756
$557$	−20.1928	−0.855595	−0.427797	−	0.903875i	$-0.640710\pi$
$557$	−20.1928	−0.855595	−0.427797	+	0.903875i	$0.640710\pi$
$558$	0	0
$559$	12.5178	0.529445
$560$	5.23214	0.221098
$561$	0	0
$562$	11.4628	0.483530
$563$	9.91284	0.417776	0.208888	−	0.977940i	$-0.433016\pi$
$563$	9.91284	0.417776	0.208888	+	0.977940i	$0.433016\pi$
$564$	0	0
$565$	16.7825	0.706047
$566$	68.5574	2.88168
$567$	0	0
$568$	0.744072	0.0312206
$569$	21.2341	0.890181	0.445090	−	0.895486i	$-0.353172\pi$
$569$	21.2341	0.890181	0.445090	+	0.895486i	$0.353172\pi$
$570$	0	0
$571$	26.7191	1.11816	0.559080	−	0.829114i	$-0.311155\pi$
$571$	26.7191	1.11816	0.559080	+	0.829114i	$0.311155\pi$
$572$	−40.5550	−1.69569
$573$	0	0
$574$	16.8893	0.704944
$575$	−0.604124	−0.0251937
$576$	0	0
$577$	−22.8576	−0.951573	−0.475787	−	0.879561i	$-0.657837\pi$
$577$	−22.8576	−0.951573	−0.475787	+	0.879561i	$0.657837\pi$
$578$	9.72714	0.404596
$579$	0	0
$580$	7.60606	0.315825
$581$	5.18549	0.215130
$582$	0	0
$583$	59.9052	2.48102
$584$	−0.290268	−0.0120114
$585$	0	0
$586$	−3.97715	−0.164294
$587$	−19.1573	−0.790706	−0.395353	−	0.918529i	$-0.629378\pi$
$587$	−19.1573	−0.790706	−0.395353	+	0.918529i	$0.629378\pi$
$588$	0	0
$589$	2.25262	0.0928176
$590$	−1.73777	−0.0715427
$591$	0	0
$592$	−2.71877	−0.111741
$593$	−12.0914	−0.496536	−0.248268	−	0.968691i	$-0.579861\pi$
$593$	−12.0914	−0.496536	−0.248268	+	0.968691i	$0.579861\pi$
$594$	0	0
$595$	−12.6158	−0.517198
$596$	73.6167	3.01546
$597$	0	0
$598$	−5.60633	−0.229260
$599$	0.779835	0.0318632	0.0159316	−	0.999873i	$-0.494929\pi$
$599$	0.779835	0.0318632	0.0159316	+	0.999873i	$0.494929\pi$
$600$	0	0
$601$	0.763990	0.0311638	0.0155819	−	0.999879i	$-0.495040\pi$
$601$	0.763990	0.0311638	0.0155819	+	0.999879i	$0.495040\pi$
$602$	16.4600	0.670860
$603$	0	0
$604$	−1.23644	−0.0503099
$605$	−23.8522	−0.969729
$606$	0	0
$607$	−24.9119	−1.01114	−0.505572	−	0.862784i	$-0.668719\pi$
$607$	−24.9119	−1.01114	−0.505572	+	0.862784i	$0.668719\pi$
$608$	3.49727	0.141833
$609$	0	0
$610$	19.3973	0.785375
$611$	−9.33600	−0.377694
$612$	0	0
$613$	−24.3565	−0.983749	−0.491875	−	0.870666i	$-0.663688\pi$
$613$	−24.3565	−0.983749	−0.491875	+	0.870666i	$0.663688\pi$
$614$	10.6130	0.428306
$615$	0	0
$616$	−23.9275	−0.964067
$617$	27.8312	1.12044	0.560220	−	0.828344i	$-0.310716\pi$
$617$	27.8312	1.12044	0.560220	+	0.828344i	$0.310716\pi$
$618$	0	0
$619$	2.71267	0.109032	0.0545158	−	0.998513i	$-0.482638\pi$
$619$	2.71267	0.109032	0.0545158	+	0.998513i	$0.482638\pi$
$620$	15.6949	0.630324
$621$	0	0
$622$	18.4728	0.740692
$623$	10.8999	0.436695
$624$	0	0
$625$	−21.6144	−0.864577
$626$	−20.6174	−0.824036
$627$	0	0
$628$	33.5606	1.33922
$629$	6.55554	0.261386
$630$	0	0
$631$	36.4187	1.44981	0.724903	−	0.688851i	$-0.241884\pi$
$631$	36.4187	1.44981	0.724903	+	0.688851i	$0.241884\pi$
$632$	−30.4286	−1.21038
$633$	0	0
$634$	−64.3087	−2.55402
$635$	23.1653	0.919288
$636$	0	0
$637$	12.4877	0.494780
$638$	−11.2218	−0.444275
$639$	0	0
$640$	43.3178	1.71229
$641$	31.4183	1.24095	0.620475	−	0.784226i	$-0.286940\pi$
$641$	31.4183	1.24095	0.620475	+	0.784226i	$0.286940\pi$
$642$	0	0
$643$	−19.5450	−0.770778	−0.385389	−	0.922754i	$-0.625933\pi$
$643$	−19.5450	−0.770778	−0.385389	+	0.922754i	$0.625933\pi$
$644$	−4.75209	−0.187259
$645$	0	0
$646$	11.8960	0.468041
$647$	−26.8041	−1.05378	−0.526889	−	0.849934i	$-0.676642\pi$
$647$	−26.8041	−1.05378	−0.526889	+	0.849934i	$0.676642\pi$
$648$	0	0
$649$	1.65271	0.0648745
$650$	3.38692	0.132846
$651$	0	0
$652$	26.5883	1.04128
$653$	−34.9209	−1.36656	−0.683280	−	0.730156i	$-0.739447\pi$
$653$	−34.9209	−1.36656	−0.683280	+	0.730156i	$0.739447\pi$
$654$	0	0
$655$	−20.1692	−0.788075
$656$	−10.3539	−0.404253
$657$	0	0
$658$	−12.2762	−0.478576
$659$	19.6238	0.764435	0.382217	−	0.924072i	$-0.375161\pi$
$659$	19.6238	0.764435	0.382217	+	0.924072i	$0.375161\pi$
$660$	0	0
$661$	−26.6062	−1.03486	−0.517430	−	0.855725i	$-0.673111\pi$
$661$	−26.6062	−1.03486	−0.517430	+	0.855725i	$0.673111\pi$
$662$	−46.0270	−1.78889
$663$	0	0
$664$	−15.2861	−0.593215
$665$	−2.99819	−0.116265
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−56.0984	−2.17051
$669$	0	0
$670$	24.4695	0.945340
$671$	−18.4479	−0.712173
$672$	0	0
$673$	43.6519	1.68266	0.841329	−	0.540523i	$-0.181774\pi$
$673$	43.6519	1.68266	0.841329	+	0.540523i	$0.181774\pi$
$674$	23.2216	0.894461
$675$	0	0
$676$	−26.8997	−1.03461
$677$	22.5454	0.866489	0.433244	−	0.901276i	$-0.357369\pi$
$677$	22.5454	0.866489	0.433244	+	0.901276i	$0.357369\pi$
$678$	0	0
$679$	10.2453	0.393178
$680$	37.1896	1.42616
$681$	0	0
$682$	−23.1559	−0.886686
$683$	51.5139	1.97112	0.985562	−	0.169315i	$-0.0541555\pi$
$683$	51.5139	1.97112	0.985562	+	0.169315i	$0.0541555\pi$
$684$	0	0
$685$	−13.8538	−0.529327
$686$	38.1731	1.45746
$687$	0	0
$688$	−10.0908	−0.384707
$689$	−29.9283	−1.14018
$690$	0	0
$691$	1.75250	0.0666683	0.0333342	−	0.999444i	$-0.489387\pi$
$691$	1.75250	0.0666683	0.0333342	+	0.999444i	$0.489387\pi$
$692$	−18.9513	−0.720422
$693$	0	0
$694$	−1.87072	−0.0710116
$695$	11.5802	0.439263
$696$	0	0
$697$	24.9655	0.945635
$698$	2.91725	0.110420
$699$	0	0
$700$	2.87085	0.108508
$701$	34.9864	1.32142	0.660708	−	0.750643i	$-0.270256\pi$
$701$	34.9864	1.32142	0.660708	+	0.750643i	$0.270256\pi$
$702$	0	0
$703$	1.55795	0.0587591
$704$	−53.9737	−2.03421
$705$	0	0
$706$	13.2855	0.500006
$707$	8.14357	0.306270
$708$	0	0
$709$	−34.7908	−1.30659	−0.653297	−	0.757102i	$-0.726615\pi$
$709$	−34.7908	−1.30659	−0.653297	+	0.757102i	$0.726615\pi$
$710$	−0.958408	−0.0359684
$711$	0	0
$712$	−32.1314	−1.20417
$713$	−2.06348	−0.0772778
$714$	0	0
$715$	23.4385	0.876550
$716$	20.9079	0.781366
$717$	0	0
$718$	−0.866088	−0.0323221
$719$	35.6595	1.32987	0.664937	−	0.746899i	$-0.268458\pi$
$719$	35.6595	1.32987	0.664937	+	0.746899i	$0.268458\pi$
$720$	0	0
$721$	8.83112	0.328888
$722$	−42.2463	−1.57225
$723$	0	0
$724$	−45.0327	−1.67363
$725$	0.604124	0.0224366
$726$	0	0
$727$	−11.4221	−0.423624	−0.211812	−	0.977310i	$-0.567936\pi$
$727$	−11.4221	−0.423624	−0.211812	+	0.977310i	$0.567936\pi$
$728$	11.9540	0.443046
$729$	0	0
$730$	0.373882	0.0138380
$731$	24.3310	0.899914
$732$	0	0
$733$	−42.8745	−1.58361	−0.791803	−	0.610777i	$-0.790857\pi$
$733$	−42.8745	−1.58361	−0.791803	+	0.610777i	$0.790857\pi$
$734$	81.5981	3.01184
$735$	0	0
$736$	−3.20362	−0.118087
$737$	−23.2718	−0.857229
$738$	0	0
$739$	4.33158	0.159340	0.0796699	−	0.996821i	$-0.474613\pi$
$739$	4.33158	0.159340	0.0796699	+	0.996821i	$0.474613\pi$
$740$	10.8548	0.399032
$741$	0	0
$742$	−39.3536	−1.44472
$743$	−9.50481	−0.348698	−0.174349	−	0.984684i	$-0.555782\pi$
$743$	−9.50481	−0.348698	−0.174349	+	0.984684i	$0.555782\pi$
$744$	0	0
$745$	−42.5463	−1.55878
$746$	44.7396	1.63803
$747$	0	0
$748$	−78.8273	−2.88221
$749$	3.29971	0.120569
$750$	0	0
$751$	−4.90475	−0.178977	−0.0894884	−	0.995988i	$-0.528523\pi$
$751$	−4.90475	−0.178977	−0.0894884	+	0.995988i	$0.528523\pi$
$752$	7.52590	0.274441
$753$	0	0
$754$	5.60633	0.204171
$755$	0.714591	0.0260066
$756$	0	0
$757$	32.8319	1.19329	0.596647	−	0.802504i	$-0.296499\pi$
$757$	32.8319	1.19329	0.596647	+	0.802504i	$0.296499\pi$
$758$	13.9783	0.507714
$759$	0	0
$760$	8.83824	0.320597
$761$	22.5870	0.818779	0.409390	−	0.912360i	$-0.365742\pi$
$761$	22.5870	0.818779	0.409390	+	0.912360i	$0.365742\pi$
$762$	0	0
$763$	−10.3171	−0.373505
$764$	−14.5532	−0.526518
$765$	0	0
$766$	−35.3393	−1.27686
$767$	−0.825682	−0.0298137
$768$	0	0
$769$	31.9554	1.15234	0.576170	−	0.817330i	$-0.304547\pi$
$769$	31.9554	1.15234	0.576170	+	0.817330i	$0.304547\pi$
$770$	30.8200	1.11068
$771$	0	0
$772$	82.8913	2.98332
$773$	−29.3514	−1.05570	−0.527848	−	0.849339i	$-0.677001\pi$
$773$	−29.3514	−1.05570	−0.527848	+	0.849339i	$0.677001\pi$
$774$	0	0
$775$	1.24660	0.0447790
$776$	−30.2017	−1.08418
$777$	0	0
$778$	46.0254	1.65009
$779$	5.93313	0.212577
$780$	0	0
$781$	0.911497	0.0326159
$782$	−10.8971	−0.389680
$783$	0	0
$784$	−10.0665	−0.359518
$785$	−19.3962	−0.692279
$786$	0	0
$787$	2.73285	0.0974157	0.0487078	−	0.998813i	$-0.484490\pi$
$787$	2.73285	0.0974157	0.0487078	+	0.998813i	$0.484490\pi$
$788$	−82.7464	−2.94772
$789$	0	0
$790$	39.1938	1.39445
$791$	10.4853	0.372816
$792$	0	0
$793$	9.21644	0.327286
$794$	61.3865	2.17853
$795$	0	0
$796$	−73.3889	−2.60120
$797$	50.5084	1.78910	0.894549	−	0.446969i	$-0.147497\pi$
$797$	50.5084	1.78910	0.894549	+	0.446969i	$0.147497\pi$
$798$	0	0
$799$	−18.1465	−0.641978
$800$	1.93538	0.0684261
$801$	0	0
$802$	33.9234	1.19788
$803$	−0.355582	−0.0125482
$804$	0	0
$805$	2.74644	0.0967994
$806$	11.5685	0.407484
$807$	0	0
$808$	−24.0061	−0.844532
$809$	−28.9427	−1.01757	−0.508785	−	0.860893i	$-0.669905\pi$
$809$	−28.9427	−1.01757	−0.508785	+	0.860893i	$0.669905\pi$
$810$	0	0
$811$	7.66225	0.269058	0.134529	−	0.990910i	$-0.457048\pi$
$811$	7.66225	0.269058	0.134529	+	0.990910i	$0.457048\pi$
$812$	4.75209	0.166766
$813$	0	0
$814$	−16.0150	−0.561325
$815$	−15.3665	−0.538266
$816$	0	0
$817$	5.78234	0.202298
$818$	70.6152	2.46900
$819$	0	0
$820$	41.3386	1.44361
$821$	−21.3310	−0.744456	−0.372228	−	0.928141i	$-0.621406\pi$
$821$	−21.3310	−0.744456	−0.372228	+	0.928141i	$0.621406\pi$
$822$	0	0
$823$	−1.39369	−0.0485810	−0.0242905	−	0.999705i	$-0.507733\pi$
$823$	−1.39369	−0.0485810	−0.0242905	+	0.999705i	$0.507733\pi$
$824$	−26.0329	−0.906899
$825$	0	0
$826$	−1.08572	−0.0377769
$827$	−26.9029	−0.935505	−0.467753	−	0.883859i	$-0.654936\pi$
$827$	−26.9029	−0.935505	−0.467753	+	0.883859i	$0.654936\pi$
$828$	0	0
$829$	−13.5073	−0.469129	−0.234565	−	0.972101i	$-0.575366\pi$
$829$	−13.5073	−0.469129	−0.234565	+	0.972101i	$0.575366\pi$
$830$	19.6894	0.683428
$831$	0	0
$832$	26.9649	0.934839
$833$	24.2725	0.840992
$834$	0	0
$835$	32.4217	1.12200
$836$	−18.7336	−0.647915
$837$	0	0
$838$	−50.7778	−1.75409
$839$	−46.8116	−1.61612	−0.808058	−	0.589102i	$-0.799481\pi$
$839$	−46.8116	−1.61612	−0.808058	+	0.589102i	$0.799481\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	35.0014	1.20623
$843$	0	0
$844$	1.03474	0.0356174
$845$	15.5465	0.534817
$846$	0	0
$847$	−14.9023	−0.512049
$848$	24.1256	0.828478
$849$	0	0
$850$	6.58321	0.225802
$851$	−1.42713	−0.0489214
$852$	0	0
$853$	−14.9266	−0.511078	−0.255539	−	0.966799i	$-0.582253\pi$
$853$	−14.9266	−0.511078	−0.255539	+	0.966799i	$0.582253\pi$
$854$	12.1190	0.414704
$855$	0	0
$856$	−9.72708	−0.332465
$857$	4.67545	0.159710	0.0798552	−	0.996806i	$-0.474554\pi$
$857$	4.67545	0.159710	0.0798552	+	0.996806i	$0.474554\pi$
$858$	0	0
$859$	34.3787	1.17299	0.586494	−	0.809954i	$-0.300508\pi$
$859$	34.3787	1.17299	0.586494	+	0.809954i	$0.300508\pi$
$860$	40.2879	1.37381
$861$	0	0
$862$	39.2399	1.33652
$863$	−15.9653	−0.543467	−0.271733	−	0.962373i	$-0.587597\pi$
$863$	−15.9653	−0.543467	−0.271733	+	0.962373i	$0.587597\pi$
$864$	0	0
$865$	10.9528	0.372407
$866$	−16.4730	−0.559775
$867$	0	0
$868$	9.80583	0.332832
$869$	−37.2754	−1.26448
$870$	0	0
$871$	11.6265	0.393947
$872$	30.4135	1.02993
$873$	0	0
$874$	−2.58974	−0.0875992
$875$	−15.3914	−0.520324
$876$	0	0
$877$	7.73776	0.261286	0.130643	−	0.991429i	$-0.458296\pi$
$877$	7.73776	0.261286	0.130643	+	0.991429i	$0.458296\pi$
$878$	−84.1472	−2.83983
$879$	0	0
$880$	−18.8941	−0.636921
$881$	−26.6734	−0.898651	−0.449326	−	0.893368i	$-0.648336\pi$
$881$	−26.6734	−0.898651	−0.449326	+	0.893368i	$0.648336\pi$
$882$	0	0
$883$	36.4408	1.22633	0.613165	−	0.789955i	$-0.289896\pi$
$883$	36.4408	1.22633	0.613165	+	0.789955i	$0.289896\pi$
$884$	39.3816	1.32455
$885$	0	0
$886$	−70.3597	−2.36378
$887$	−18.3314	−0.615507	−0.307754	−	0.951466i	$-0.599577\pi$
$887$	−18.3314	−0.615507	−0.307754	+	0.951466i	$0.599577\pi$
$888$	0	0
$889$	14.4732	0.485414
$890$	41.3871	1.38730
$891$	0	0
$892$	−62.3904	−2.08898
$893$	−4.31259	−0.144315
$894$	0	0
$895$	−12.0836	−0.403910
$896$	27.0640	0.904144
$897$	0	0
$898$	−42.5934	−1.42136
$899$	2.06348	0.0688208
$900$	0	0
$901$	−58.1720	−1.93799
$902$	−60.9899	−2.03074
$903$	0	0
$904$	−30.9093	−1.02803
$905$	26.0264	0.865146
$906$	0	0
$907$	−36.0197	−1.19602	−0.598008	−	0.801490i	$-0.704041\pi$
$907$	−36.0197	−1.19602	−0.598008	+	0.801490i	$0.704041\pi$
$908$	−3.36608	−0.111707
$909$	0	0
$910$	−15.3975	−0.510421
$911$	−5.71034	−0.189192	−0.0945960	−	0.995516i	$-0.530156\pi$
$911$	−5.71034	−0.189192	−0.0945960	+	0.995516i	$0.530156\pi$
$912$	0	0
$913$	−18.7256	−0.619729
$914$	46.3164	1.53201
$915$	0	0
$916$	96.9985	3.20492
$917$	−12.6012	−0.416129
$918$	0	0
$919$	19.0910	0.629753	0.314877	−	0.949133i	$-0.398037\pi$
$919$	19.0910	0.629753	0.314877	+	0.949133i	$0.398037\pi$
$920$	−8.09613	−0.266921
$921$	0	0
$922$	−65.7270	−2.16460
$923$	−0.455378	−0.0149889
$924$	0	0
$925$	0.862164	0.0283478
$926$	23.7737	0.781251
$927$	0	0
$928$	3.20362	0.105164
$929$	18.7134	0.613968	0.306984	−	0.951715i	$-0.400680\pi$
$929$	18.7134	0.613968	0.306984	+	0.951715i	$0.400680\pi$
$930$	0	0
$931$	5.76844	0.189053
$932$	47.5511	1.55759
$933$	0	0
$934$	33.5045	1.09630
$935$	45.5578	1.48990
$936$	0	0
$937$	−50.9813	−1.66549	−0.832743	−	0.553660i	$-0.813231\pi$
$937$	−50.9813	−1.66549	−0.832743	+	0.553660i	$0.813231\pi$
$938$	15.2880	0.499171
$939$	0	0
$940$	−30.0476	−0.980043
$941$	−24.9407	−0.813045	−0.406522	−	0.913641i	$-0.633259\pi$
$941$	−24.9407	−0.813045	−0.406522	+	0.913641i	$0.633259\pi$
$942$	0	0
$943$	−5.43495	−0.176986
$944$	0.665596	0.0216633
$945$	0	0
$946$	−59.4398	−1.93256
$947$	−29.6639	−0.963949	−0.481974	−	0.876185i	$-0.660080\pi$
$947$	−29.6639	−0.963949	−0.481974	+	0.876185i	$0.660080\pi$
$948$	0	0
$949$	0.177646	0.00576664
$950$	1.56452	0.0507598
$951$	0	0
$952$	23.2352	0.753058
$953$	−32.3280	−1.04721	−0.523604	−	0.851962i	$-0.675413\pi$
$953$	−32.3280	−1.04721	−0.523604	+	0.851962i	$0.675413\pi$
$954$	0	0
$955$	8.41096	0.272172
$956$	66.4525	2.14923
$957$	0	0
$958$	32.0339	1.03497
$959$	−8.65554	−0.279502
$960$	0	0
$961$	−26.7421	−0.862647
$962$	8.00098	0.257962
$963$	0	0
$964$	−34.3382	−1.10596
$965$	−47.9065	−1.54217
$966$	0	0
$967$	−7.49040	−0.240875	−0.120437	−	0.992721i	$-0.538430\pi$
$967$	−7.49040	−0.240875	−0.120437	+	0.992721i	$0.538430\pi$
$968$	43.9299	1.41196
$969$	0	0
$970$	38.9015	1.24905
$971$	1.69031	0.0542446	0.0271223	−	0.999632i	$-0.491366\pi$
$971$	1.69031	0.0542446	0.0271223	+	0.999632i	$0.491366\pi$
$972$	0	0
$973$	7.23506	0.231945
$974$	−86.2070	−2.76225
$975$	0	0
$976$	−7.42952	−0.237813
$977$	19.2221	0.614969	0.307484	−	0.951553i	$-0.400513\pi$
$977$	19.2221	0.614969	0.307484	+	0.951553i	$0.400513\pi$
$978$	0	0
$979$	−39.3613	−1.25799
$980$	40.1911	1.28386
$981$	0	0
$982$	−51.8162	−1.65352
$983$	−44.4773	−1.41861	−0.709303	−	0.704903i	$-0.750990\pi$
$983$	−44.4773	−1.41861	−0.709303	+	0.704903i	$0.750990\pi$
$984$	0	0
$985$	47.8228	1.52376
$986$	10.8971	0.347035
$987$	0	0
$988$	9.35917	0.297755
$989$	−5.29682	−0.168429
$990$	0	0
$991$	−1.47217	−0.0467650	−0.0233825	−	0.999727i	$-0.507444\pi$
$991$	−1.47217	−0.0467650	−0.0233825	+	0.999727i	$0.507444\pi$
$992$	6.61060	0.209887
$993$	0	0
$994$	−0.598791	−0.0189925
$995$	42.4146	1.34463
$996$	0	0
$997$	55.3671	1.75349	0.876747	−	0.480951i	$-0.159709\pi$
$997$	55.3671	1.75349	0.876747	+	0.480951i	$0.159709\pi$
$998$	31.8220	1.00731
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.i.1.7		7
3.2	odd	2		2001.2.a.j.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.j.1.1	✓	7	3.2	odd	2
6003.2.a.i.1.7		7	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)

\(f(q)\)	\(=\)	\(q+2.37229 q^{2} +3.62775 q^{4} -2.09663 q^{5} -1.30993 q^{7} +3.86149 q^{8} +O(q^{10})\)	\(q+2.37229 q^{2} +3.62775 q^{4} -2.09663 q^{5} -1.30993 q^{7} +3.86149 q^{8} -4.97382 q^{10} +4.73037 q^{11} -2.36326 q^{13} -3.10753 q^{14} +1.90506 q^{16} -4.59351 q^{17} -1.09166 q^{19} -7.60606 q^{20} +11.2218 q^{22} +1.00000 q^{23} -0.604124 q^{25} -5.60633 q^{26} -4.75209 q^{28} -1.00000 q^{29} -2.06348 q^{31} -3.20362 q^{32} -10.8971 q^{34} +2.74644 q^{35} -1.42713 q^{37} -2.58974 q^{38} -8.09613 q^{40} -5.43495 q^{41} -5.29682 q^{43} +17.1606 q^{44} +2.37229 q^{46} +3.95047 q^{47} -5.28409 q^{49} -1.43316 q^{50} -8.57332 q^{52} +12.6640 q^{53} -9.91786 q^{55} -5.05827 q^{56} -2.37229 q^{58} +0.349383 q^{59} -3.89988 q^{61} -4.89516 q^{62} -11.4100 q^{64} +4.95489 q^{65} -4.91966 q^{67} -16.6641 q^{68} +6.51535 q^{70} +0.192690 q^{71} -0.0751700 q^{73} -3.38557 q^{74} -3.96028 q^{76} -6.19645 q^{77} -7.88002 q^{79} -3.99422 q^{80} -12.8933 q^{82} -3.95860 q^{83} +9.63091 q^{85} -12.5656 q^{86} +18.2663 q^{88} -8.32098 q^{89} +3.09570 q^{91} +3.62775 q^{92} +9.37166 q^{94} +2.28882 q^{95} -7.82125 q^{97} -12.5354 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8}+O(q^{10}) \) 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8	\( 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8} - 11 q^{10} + 12 q^{11} - 13 q^{13} + 3 q^{14} - 13 q^{16} + 12 q^{17} - 5 q^{19} + 8 q^{20} - q^{22} + 7 q^{23} - 4 q^{25} - 2 q^{26} - 21 q^{28} - 7 q^{29} - 8 q^{31} + 5 q^{32} - 28 q^{34} - 5 q^{35} - 24 q^{37} + 6 q^{38} - 20 q^{40} - 9 q^{41} - q^{43} + 23 q^{44} + q^{46} - 27 q^{47} - 14 q^{49} - 7 q^{50} - 9 q^{52} + q^{53} - 11 q^{55} + 20 q^{56} - q^{58} - 8 q^{59} + q^{61} + 3 q^{64} - 12 q^{65} - 16 q^{67} - 15 q^{68} + 40 q^{70} + 13 q^{71} - 23 q^{73} + 8 q^{74} - 2 q^{76} - 13 q^{77} - 44 q^{79} - 30 q^{80} - 10 q^{82} - 21 q^{83} - 6 q^{86} + 21 q^{88} + 5 q^{89} - 18 q^{91} + 7 q^{92} + 28 q^{94} - 9 q^{95} - 55 q^{97} - 36 q^{98}+O(q^{100}) \) 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8 - 11 * q^10 + 12 * q^11 - 13 * q^13 + 3 * q^14 - 13 * q^16 + 12 * q^17 - 5 * q^19 + 8 * q^20 - q^22 + 7 * q^23 - 4 * q^25 - 2 * q^26 - 21 * q^28 - 7 * q^29 - 8 * q^31 + 5 * q^32 - 28 * q^34 - 5 * q^35 - 24 * q^37 + 6 * q^38 - 20 * q^40 - 9 * q^41 - q^43 + 23 * q^44 + q^46 - 27 * q^47 - 14 * q^49 - 7 * q^50 - 9 * q^52 + q^53 - 11 * q^55 + 20 * q^56 - q^58 - 8 * q^59 + q^61 + 3 * q^64 - 12 * q^65 - 16 * q^67 - 15 * q^68 + 40 * q^70 + 13 * q^71 - 23 * q^73 + 8 * q^74 - 2 * q^76 - 13 * q^77 - 44 * q^79 - 30 * q^80 - 10 * q^82 - 21 * q^83 - 6 * q^86 + 21 * q^88 + 5 * q^89 - 18 * q^91 + 7 * q^92 + 28 * q^94 - 9 * q^95 - 55 * q^97 - 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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