LMFDB - Embedded newform 6003.2.a.r.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + \cdots - 192 $ x^16 - x^15 - 28x^14 + 27x^13 + 316x^12 - 295x^11 - 1835x^10 + 1665x^9 + 5778x^8 - 5124x^7 - 9405x^6 + 8288x^5 + 6405x^4 - 6032x^3 - 400x^2 + 1088x - 192
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.563182$ 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)

$f(q)$	$=$	$q+0.563182 q^{2} -1.68283 q^{4} +1.59433 q^{5} +4.95339 q^{7} -2.07410 q^{8} +O(q^{10})$	\(q+0.563182 q^{2} -1.68283 q^{4} +1.59433 q^{5} +4.95339 q^{7} -2.07410 q^{8} +0.897897 q^{10} +0.616624 q^{11} +6.48438 q^{13} +2.78966 q^{14} +2.19756 q^{16} +1.67611 q^{17} +2.69708 q^{19} -2.68298 q^{20} +0.347272 q^{22} +1.00000 q^{23} -2.45812 q^{25} +3.65188 q^{26} -8.33570 q^{28} +1.00000 q^{29} -3.94303 q^{31} +5.38583 q^{32} +0.943955 q^{34} +7.89733 q^{35} +5.81974 q^{37} +1.51895 q^{38} -3.30680 q^{40} +6.01020 q^{41} +0.472349 q^{43} -1.03767 q^{44} +0.563182 q^{46} -4.27730 q^{47} +17.5361 q^{49} -1.38437 q^{50} -10.9121 q^{52} +7.82304 q^{53} +0.983101 q^{55} -10.2738 q^{56} +0.563182 q^{58} -11.6621 q^{59} +10.3582 q^{61} -2.22064 q^{62} -1.36191 q^{64} +10.3382 q^{65} -6.36345 q^{67} -2.82060 q^{68} +4.44764 q^{70} -8.81789 q^{71} -2.13920 q^{73} +3.27757 q^{74} -4.53872 q^{76} +3.05438 q^{77} -11.9951 q^{79} +3.50362 q^{80} +3.38484 q^{82} -10.2126 q^{83} +2.67227 q^{85} +0.266019 q^{86} -1.27894 q^{88} +6.30439 q^{89} +32.1197 q^{91} -1.68283 q^{92} -2.40890 q^{94} +4.30003 q^{95} +11.9110 q^{97} +9.87603 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.563182	0.398230	0.199115	−	0.979976i	$-0.436193\pi$
$2$	0.563182	0.398230	0.199115	+	0.979976i	$0.436193\pi$
$3$	0	0
$3$	0	0
$4$	−1.68283	−0.841413
$5$	1.59433	0.713005	0.356503	−	0.934294i	$-0.383969\pi$
$5$	1.59433	0.713005	0.356503	+	0.934294i	$0.383969\pi$
$6$	0	0
$7$	4.95339	1.87221	0.936104	−	0.351724i	$-0.114405\pi$
$7$	4.95339	1.87221	0.936104	+	0.351724i	$0.114405\pi$
$8$	−2.07410	−0.733306
$9$	0	0
$10$	0.897897	0.283940
$11$	0.616624	0.185919	0.0929596	−	0.995670i	$-0.470367\pi$
$11$	0.616624	0.185919	0.0929596	+	0.995670i	$0.470367\pi$
$12$	0	0
$13$	6.48438	1.79844	0.899221	−	0.437494i	$-0.144134\pi$
$13$	6.48438	1.79844	0.899221	+	0.437494i	$0.144134\pi$
$14$	2.78966	0.745569
$15$	0	0
$16$	2.19756	0.549389
$17$	1.67611	0.406516	0.203258	−	0.979125i	$-0.434847\pi$
$17$	1.67611	0.406516	0.203258	+	0.979125i	$0.434847\pi$
$18$	0	0
$19$	2.69708	0.618753	0.309376	−	0.950940i	$-0.399880\pi$
$19$	2.69708	0.618753	0.309376	+	0.950940i	$0.399880\pi$
$20$	−2.68298	−0.599932
$21$	0	0
$22$	0.347272	0.0740386
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.45812	−0.491624
$26$	3.65188	0.716193
$27$	0	0
$28$	−8.33570	−1.57530
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−3.94303	−0.708188	−0.354094	−	0.935210i	$-0.615211\pi$
$31$	−3.94303	−0.708188	−0.354094	+	0.935210i	$0.615211\pi$
$32$	5.38583	0.952089
$33$	0	0
$34$	0.943955	0.161887
$35$	7.89733	1.33489
$36$	0	0
$37$	5.81974	0.956759	0.478379	−	0.878153i	$-0.341224\pi$
$37$	5.81974	0.956759	0.478379	+	0.878153i	$0.341224\pi$
$38$	1.51895	0.246406
$39$	0	0
$40$	−3.30680	−0.522851
$41$	6.01020	0.938636	0.469318	−	0.883029i	$-0.344500\pi$
$41$	6.01020	0.938636	0.469318	+	0.883029i	$0.344500\pi$
$42$	0	0
$43$	0.472349	0.0720326	0.0360163	−	0.999351i	$-0.488533\pi$
$43$	0.472349	0.0720326	0.0360163	+	0.999351i	$0.488533\pi$
$44$	−1.03767	−0.156435
$45$	0	0
$46$	0.563182	0.0830367
$47$	−4.27730	−0.623909	−0.311954	−	0.950097i	$-0.600984\pi$
$47$	−4.27730	−0.623909	−0.311954	+	0.950097i	$0.600984\pi$
$48$	0	0
$49$	17.5361	2.50516
$50$	−1.38437	−0.195779
$51$	0	0
$52$	−10.9121	−1.51323
$53$	7.82304	1.07458	0.537288	−	0.843399i	$-0.319449\pi$
$53$	7.82304	1.07458	0.537288	+	0.843399i	$0.319449\pi$
$54$	0	0
$55$	0.983101	0.132561
$56$	−10.2738	−1.37290
$57$	0	0
$58$	0.563182	0.0739494
$59$	−11.6621	−1.51827	−0.759137	−	0.650930i	$-0.774379\pi$
$59$	−11.6621	−1.51827	−0.759137	+	0.650930i	$0.774379\pi$
$60$	0	0
$61$	10.3582	1.32623	0.663115	−	0.748517i	$-0.269234\pi$
$61$	10.3582	1.32623	0.663115	+	0.748517i	$0.269234\pi$
$62$	−2.22064	−0.282022
$63$	0	0
$64$	−1.36191	−0.170239
$65$	10.3382	1.28230
$66$	0	0
$67$	−6.36345	−0.777420	−0.388710	−	0.921360i	$-0.627079\pi$
$67$	−6.36345	−0.777420	−0.388710	+	0.921360i	$0.627079\pi$
$68$	−2.82060	−0.342048
$69$	0	0
$70$	4.44764	0.531594
$71$	−8.81789	−1.04649	−0.523246	−	0.852182i	$-0.675279\pi$
$71$	−8.81789	−1.04649	−0.523246	+	0.852182i	$0.675279\pi$
$72$	0	0
$73$	−2.13920	−0.250375	−0.125187	−	0.992133i	$-0.539953\pi$
$73$	−2.13920	−0.250375	−0.125187	+	0.992133i	$0.539953\pi$
$74$	3.27757	0.381010
$75$	0	0
$76$	−4.53872	−0.520626
$77$	3.05438	0.348079
$78$	0	0
$79$	−11.9951	−1.34956	−0.674780	−	0.738019i	$-0.735761\pi$
$79$	−11.9951	−1.34956	−0.674780	+	0.738019i	$0.735761\pi$
$80$	3.50362	0.391717
$81$	0	0
$82$	3.38484	0.373793
$83$	−10.2126	−1.12098	−0.560490	−	0.828161i	$-0.689387\pi$
$83$	−10.2126	−1.12098	−0.560490	+	0.828161i	$0.689387\pi$
$84$	0	0
$85$	2.67227	0.289848
$86$	0.266019	0.0286855
$87$	0	0
$88$	−1.27894	−0.136336
$89$	6.30439	0.668264	0.334132	−	0.942526i	$-0.391557\pi$
$89$	6.30439	0.668264	0.334132	+	0.942526i	$0.391557\pi$
$90$	0	0
$91$	32.1197	3.36706
$92$	−1.68283	−0.175447
$93$	0	0
$94$	−2.40890	−0.248459
$95$	4.30003	0.441174
$96$	0	0
$97$	11.9110	1.20938	0.604691	−	0.796461i	$-0.293297\pi$
$97$	11.9110	1.20938	0.604691	+	0.796461i	$0.293297\pi$
$98$	9.87603	0.997630
$99$	0	0
$100$	4.13659	0.413659
$101$	−4.59685	−0.457404	−0.228702	−	0.973496i	$-0.573448\pi$
$101$	−4.59685	−0.457404	−0.228702	+	0.973496i	$0.573448\pi$
$102$	0	0
$103$	4.58092	0.451372	0.225686	−	0.974200i	$-0.427538\pi$
$103$	4.58092	0.451372	0.225686	+	0.974200i	$0.427538\pi$
$104$	−13.4493	−1.31881
$105$	0	0
$106$	4.40579	0.427929
$107$	−9.04500	−0.874413	−0.437207	−	0.899361i	$-0.644032\pi$
$107$	−9.04500	−0.874413	−0.437207	+	0.899361i	$0.644032\pi$
$108$	0	0
$109$	−16.0639	−1.53864	−0.769320	−	0.638864i	$-0.779405\pi$
$109$	−16.0639	−1.53864	−0.769320	+	0.638864i	$0.779405\pi$
$110$	0.553665	0.0527899
$111$	0	0
$112$	10.8854	1.02857
$113$	−18.3897	−1.72996	−0.864978	−	0.501809i	$-0.832668\pi$
$113$	−18.3897	−1.72996	−0.864978	+	0.501809i	$0.832668\pi$
$114$	0	0
$115$	1.59433	0.148672
$116$	−1.68283	−0.156246
$117$	0	0
$118$	−6.56788	−0.604622
$119$	8.30243	0.761083
$120$	0	0
$121$	−10.6198	−0.965434
$122$	5.83355	0.528144
$123$	0	0
$124$	6.63543	0.595879
$125$	−11.8907	−1.06354
$126$	0	0
$127$	4.34065	0.385171	0.192585	−	0.981280i	$-0.438313\pi$
$127$	4.34065	0.385171	0.192585	+	0.981280i	$0.438313\pi$
$128$	−11.5387	−1.01988
$129$	0	0
$130$	5.82230	0.510650
$131$	−14.2331	−1.24355	−0.621776	−	0.783195i	$-0.713589\pi$
$131$	−14.2331	−1.24355	−0.621776	+	0.783195i	$0.713589\pi$
$132$	0	0
$133$	13.3597	1.15843
$134$	−3.58378	−0.309592
$135$	0	0
$136$	−3.47642	−0.298101
$137$	−10.4871	−0.895977	−0.447988	−	0.894039i	$-0.647859\pi$
$137$	−10.4871	−0.895977	−0.447988	+	0.894039i	$0.647859\pi$
$138$	0	0
$139$	−19.5982	−1.66230	−0.831151	−	0.556047i	$-0.812317\pi$
$139$	−19.5982	−1.66230	−0.831151	+	0.556047i	$0.812317\pi$
$140$	−13.2898	−1.12320
$141$	0	0
$142$	−4.96608	−0.416744
$143$	3.99842	0.334365
$144$	0	0
$145$	1.59433	0.132402
$146$	−1.20476	−0.0997067
$147$	0	0
$148$	−9.79360	−0.805029
$149$	6.73460	0.551720	0.275860	−	0.961198i	$-0.411037\pi$
$149$	6.73460	0.551720	0.275860	+	0.961198i	$0.411037\pi$
$150$	0	0
$151$	4.73376	0.385228	0.192614	−	0.981275i	$-0.438304\pi$
$151$	4.73376	0.385228	0.192614	+	0.981275i	$0.438304\pi$
$152$	−5.59402	−0.453735
$153$	0	0
$154$	1.72017	0.138616
$155$	−6.28648	−0.504942
$156$	0	0
$157$	21.3212	1.70162	0.850810	−	0.525473i	$-0.176112\pi$
$157$	21.3212	1.70162	0.850810	+	0.525473i	$0.176112\pi$
$158$	−6.75545	−0.537435
$159$	0	0
$160$	8.58677	0.678844
$161$	4.95339	0.390382
$162$	0	0
$163$	4.09507	0.320750	0.160375	−	0.987056i	$-0.448730\pi$
$163$	4.09507	0.320750	0.160375	+	0.987056i	$0.448730\pi$
$164$	−10.1141	−0.789780
$165$	0	0
$166$	−5.75156	−0.446408
$167$	10.3928	0.804217	0.402108	−	0.915592i	$-0.368277\pi$
$167$	10.3928	0.804217	0.402108	+	0.915592i	$0.368277\pi$
$168$	0	0
$169$	29.0471	2.23440
$170$	1.50497	0.115426
$171$	0	0
$172$	−0.794882	−0.0606091
$173$	4.60624	0.350206	0.175103	−	0.984550i	$-0.443974\pi$
$173$	4.60624	0.350206	0.175103	+	0.984550i	$0.443974\pi$
$174$	0	0
$175$	−12.1760	−0.920422
$176$	1.35507	0.102142
$177$	0	0
$178$	3.55052	0.266123
$179$	25.3247	1.89286	0.946430	−	0.322910i	$-0.104661\pi$
$179$	25.3247	1.89286	0.946430	+	0.322910i	$0.104661\pi$
$180$	0	0
$181$	−22.1243	−1.64449	−0.822245	−	0.569134i	$-0.807279\pi$
$181$	−22.1243	−1.64449	−0.822245	+	0.569134i	$0.807279\pi$
$182$	18.0892	1.34086
$183$	0	0
$184$	−2.07410	−0.152905
$185$	9.27857	0.682174
$186$	0	0
$187$	1.03353	0.0755792
$188$	7.19796	0.524965
$189$	0	0
$190$	2.42170	0.175689
$191$	20.9666	1.51709	0.758545	−	0.651620i	$-0.225910\pi$
$191$	20.9666	1.51709	0.758545	+	0.651620i	$0.225910\pi$
$192$	0	0
$193$	20.7423	1.49306	0.746530	−	0.665351i	$-0.231718\pi$
$193$	20.7423	1.49306	0.746530	+	0.665351i	$0.231718\pi$
$194$	6.70807	0.481612
$195$	0	0
$196$	−29.5102	−2.10787
$197$	22.2579	1.58581	0.792906	−	0.609344i	$-0.208567\pi$
$197$	22.2579	1.58581	0.792906	+	0.609344i	$0.208567\pi$
$198$	0	0
$199$	0.316336	0.0224245	0.0112122	−	0.999937i	$-0.496431\pi$
$199$	0.316336	0.0224245	0.0112122	+	0.999937i	$0.496431\pi$
$200$	5.09839	0.360511
$201$	0	0
$202$	−2.58887	−0.182152
$203$	4.95339	0.347660
$204$	0	0
$205$	9.58223	0.669252
$206$	2.57989	0.179750
$207$	0	0
$208$	14.2498	0.988044
$209$	1.66308	0.115038
$210$	0	0
$211$	−0.730479	−0.0502883	−0.0251441	−	0.999684i	$-0.508004\pi$
$211$	−0.730479	−0.0502883	−0.0251441	+	0.999684i	$0.508004\pi$
$212$	−13.1648	−0.904163
$213$	0	0
$214$	−5.09398	−0.348217
$215$	0.753079	0.0513596
$216$	0	0
$217$	−19.5314	−1.32588
$218$	−9.04688	−0.612732
$219$	0	0
$220$	−1.65439	−0.111539
$221$	10.8685	0.731096
$222$	0	0
$223$	−0.992801	−0.0664829	−0.0332414	−	0.999447i	$-0.510583\pi$
$223$	−0.992801	−0.0664829	−0.0332414	+	0.999447i	$0.510583\pi$
$224$	26.6781	1.78251
$225$	0	0
$226$	−10.3567	−0.688920
$227$	6.80592	0.451725	0.225862	−	0.974159i	$-0.427480\pi$
$227$	6.80592	0.451725	0.225862	+	0.974159i	$0.427480\pi$
$228$	0	0
$229$	16.5937	1.09654	0.548272	−	0.836300i	$-0.315286\pi$
$229$	16.5937	1.09654	0.548272	+	0.836300i	$0.315286\pi$
$230$	0.897897	0.0592056
$231$	0	0
$232$	−2.07410	−0.136171
$233$	−25.8874	−1.69594	−0.847971	−	0.530042i	$-0.822176\pi$
$233$	−25.8874	−1.69594	−0.847971	+	0.530042i	$0.822176\pi$
$234$	0	0
$235$	−6.81942	−0.444850
$236$	19.6253	1.27750
$237$	0	0
$238$	4.67578	0.303086
$239$	−6.18941	−0.400360	−0.200180	−	0.979759i	$-0.564153\pi$
$239$	−6.18941	−0.400360	−0.200180	+	0.979759i	$0.564153\pi$
$240$	0	0
$241$	20.6273	1.32872	0.664360	−	0.747413i	$-0.268704\pi$
$241$	20.6273	1.32872	0.664360	+	0.747413i	$0.268704\pi$
$242$	−5.98087	−0.384465
$243$	0	0
$244$	−17.4310	−1.11591
$245$	27.9583	1.78619
$246$	0	0
$247$	17.4889	1.11279
$248$	8.17824	0.519319
$249$	0	0
$250$	−6.69662	−0.423531
$251$	16.1608	1.02006	0.510029	−	0.860157i	$-0.329635\pi$
$251$	16.1608	1.02006	0.510029	+	0.860157i	$0.329635\pi$
$252$	0	0
$253$	0.616624	0.0387668
$254$	2.44458	0.153386
$255$	0	0
$256$	−3.77454	−0.235909
$257$	7.71144	0.481026	0.240513	−	0.970646i	$-0.422684\pi$
$257$	7.71144	0.481026	0.240513	+	0.970646i	$0.422684\pi$
$258$	0	0
$259$	28.8275	1.79125
$260$	−17.3974	−1.07894
$261$	0	0
$262$	−8.01583	−0.495220
$263$	3.66457	0.225967	0.112984	−	0.993597i	$-0.463959\pi$
$263$	3.66457	0.225967	0.112984	+	0.993597i	$0.463959\pi$
$264$	0	0
$265$	12.4725	0.766179
$266$	7.52394	0.461323
$267$	0	0
$268$	10.7086	0.654131
$269$	−18.5829	−1.13302	−0.566510	−	0.824055i	$-0.691707\pi$
$269$	−18.5829	−1.13302	−0.566510	+	0.824055i	$0.691707\pi$
$270$	0	0
$271$	−11.7264	−0.712326	−0.356163	−	0.934424i	$-0.615915\pi$
$271$	−11.7264	−0.712326	−0.356163	+	0.934424i	$0.615915\pi$
$272$	3.68334	0.223336
$273$	0	0
$274$	−5.90617	−0.356805
$275$	−1.51574	−0.0914023
$276$	0	0
$277$	−25.5590	−1.53569	−0.767846	−	0.640634i	$-0.778672\pi$
$277$	−25.5590	−1.53569	−0.767846	+	0.640634i	$0.778672\pi$
$278$	−11.0374	−0.661978
$279$	0	0
$280$	−16.3799	−0.978885
$281$	7.56252	0.451142	0.225571	−	0.974227i	$-0.427575\pi$
$281$	7.56252	0.451142	0.225571	+	0.974227i	$0.427575\pi$
$282$	0	0
$283$	15.6951	0.932980	0.466490	−	0.884527i	$-0.345518\pi$
$283$	15.6951	0.932980	0.466490	+	0.884527i	$0.345518\pi$
$284$	14.8390	0.880531
$285$	0	0
$286$	2.25184	0.133154
$287$	29.7709	1.75732
$288$	0	0
$289$	−14.1907	−0.834744
$290$	0.897897	0.0527263
$291$	0	0
$292$	3.59991	0.210669
$293$	−27.4297	−1.60246	−0.801229	−	0.598358i	$-0.795820\pi$
$293$	−27.4297	−1.60246	−0.801229	+	0.598358i	$0.795820\pi$
$294$	0	0
$295$	−18.5932	−1.08254
$296$	−12.0707	−0.701596
$297$	0	0
$298$	3.79281	0.219711
$299$	6.48438	0.375001
$300$	0	0
$301$	2.33973	0.134860
$302$	2.66597	0.153409
$303$	0	0
$304$	5.92698	0.339936
$305$	16.5143	0.945609
$306$	0	0
$307$	12.9197	0.737365	0.368683	−	0.929555i	$-0.379809\pi$
$307$	12.9197	0.737365	0.368683	+	0.929555i	$0.379809\pi$
$308$	−5.14000	−0.292878
$309$	0	0
$310$	−3.54043	−0.201083
$311$	3.77938	0.214309	0.107155	−	0.994242i	$-0.465826\pi$
$311$	3.77938	0.214309	0.107155	+	0.994242i	$0.465826\pi$
$312$	0	0
$313$	4.35313	0.246054	0.123027	−	0.992403i	$-0.460740\pi$
$313$	4.35313	0.246054	0.123027	+	0.992403i	$0.460740\pi$
$314$	12.0077	0.677636
$315$	0	0
$316$	20.1857	1.13554
$317$	−0.496952	−0.0279116	−0.0139558	−	0.999903i	$-0.504442\pi$
$317$	−0.496952	−0.0279116	−0.0139558	+	0.999903i	$0.504442\pi$
$318$	0	0
$319$	0.616624	0.0345243
$320$	−2.17133	−0.121381
$321$	0	0
$322$	2.78966	0.155462
$323$	4.52060	0.251533
$324$	0	0
$325$	−15.9394	−0.884157
$326$	2.30627	0.127732
$327$	0	0
$328$	−12.4658	−0.688307
$329$	−21.1872	−1.16809
$330$	0	0
$331$	−5.64763	−0.310422	−0.155211	−	0.987881i	$-0.549606\pi$
$331$	−5.64763	−0.310422	−0.155211	+	0.987881i	$0.549606\pi$
$332$	17.1861	0.943207
$333$	0	0
$334$	5.85302	0.320263
$335$	−10.1454	−0.554304
$336$	0	0
$337$	−7.46517	−0.406654	−0.203327	−	0.979111i	$-0.565175\pi$
$337$	−7.46517	−0.406654	−0.203327	+	0.979111i	$0.565175\pi$
$338$	16.3588	0.889803
$339$	0	0
$340$	−4.49696	−0.243882
$341$	−2.43137	−0.131666
$342$	0	0
$343$	52.1896	2.81797
$344$	−0.979700	−0.0528219
$345$	0	0
$346$	2.59415	0.139462
$347$	−14.1931	−0.761923	−0.380961	−	0.924591i	$-0.624407\pi$
$347$	−14.1931	−0.761923	−0.380961	+	0.924591i	$0.624407\pi$
$348$	0	0
$349$	−18.3724	−0.983455	−0.491727	−	0.870749i	$-0.663634\pi$
$349$	−18.3724	−0.983455	−0.491727	+	0.870749i	$0.663634\pi$
$350$	−6.85732	−0.366539
$351$	0	0
$352$	3.32103	0.177012
$353$	29.4962	1.56992	0.784962	−	0.619544i	$-0.212682\pi$
$353$	29.4962	1.56992	0.784962	+	0.619544i	$0.212682\pi$
$354$	0	0
$355$	−14.0586	−0.746154
$356$	−10.6092	−0.562286
$357$	0	0
$358$	14.2624	0.753793
$359$	−1.45373	−0.0767248	−0.0383624	−	0.999264i	$-0.512214\pi$
$359$	−1.45373	−0.0767248	−0.0383624	+	0.999264i	$0.512214\pi$
$360$	0	0
$361$	−11.7258	−0.617145
$362$	−12.4600	−0.654885
$363$	0	0
$364$	−54.0518	−2.83309
$365$	−3.41059	−0.178518
$366$	0	0
$367$	−10.1531	−0.529990	−0.264995	−	0.964250i	$-0.585370\pi$
$367$	−10.1531	−0.529990	−0.264995	+	0.964250i	$0.585370\pi$
$368$	2.19756	0.114555
$369$	0	0
$370$	5.22552	0.271662
$371$	38.7506	2.01183
$372$	0	0
$373$	10.0893	0.522406	0.261203	−	0.965284i	$-0.415881\pi$
$373$	10.0893	0.522406	0.261203	+	0.965284i	$0.415881\pi$
$374$	0.582065	0.0300979
$375$	0	0
$376$	8.87156	0.457516
$377$	6.48438	0.333962
$378$	0	0
$379$	−14.5482	−0.747293	−0.373646	−	0.927571i	$-0.621893\pi$
$379$	−14.5482	−0.747293	−0.373646	+	0.927571i	$0.621893\pi$
$380$	−7.23620	−0.371209
$381$	0	0
$382$	11.8080	0.604151
$383$	−14.7357	−0.752960	−0.376480	−	0.926425i	$-0.622866\pi$
$383$	−14.7357	−0.752960	−0.376480	+	0.926425i	$0.622866\pi$
$384$	0	0
$385$	4.86969	0.248182
$386$	11.6817	0.594581
$387$	0	0
$388$	−20.0442	−1.01759
$389$	5.48698	0.278201	0.139101	−	0.990278i	$-0.455579\pi$
$389$	5.48698	0.278201	0.139101	+	0.990278i	$0.455579\pi$
$390$	0	0
$391$	1.67611	0.0847645
$392$	−36.3717	−1.83705
$393$	0	0
$394$	12.5353	0.631518
$395$	−19.1242	−0.962243
$396$	0	0
$397$	−8.85454	−0.444397	−0.222198	−	0.975002i	$-0.571323\pi$
$397$	−8.85454	−0.444397	−0.222198	+	0.975002i	$0.571323\pi$
$398$	0.178155	0.00893009
$399$	0	0
$400$	−5.40185	−0.270093
$401$	1.93901	0.0968298	0.0484149	−	0.998827i	$-0.484583\pi$
$401$	1.93901	0.0968298	0.0484149	+	0.998827i	$0.484583\pi$
$402$	0	0
$403$	−25.5681	−1.27364
$404$	7.73571	0.384866
$405$	0	0
$406$	2.78966	0.138449
$407$	3.58859	0.177880
$408$	0	0
$409$	6.64781	0.328713	0.164357	−	0.986401i	$-0.447445\pi$
$409$	6.64781	0.328713	0.164357	+	0.986401i	$0.447445\pi$
$410$	5.39654	0.266516
$411$	0	0
$412$	−7.70890	−0.379790
$413$	−57.7669	−2.84253
$414$	0	0
$415$	−16.2823	−0.799264
$416$	34.9237	1.71228
$417$	0	0
$418$	0.936620	0.0458116
$419$	−0.802297	−0.0391948	−0.0195974	−	0.999808i	$-0.506238\pi$
$419$	−0.802297	−0.0391948	−0.0195974	+	0.999808i	$0.506238\pi$
$420$	0	0
$421$	21.2976	1.03798	0.518990	−	0.854781i	$-0.326308\pi$
$421$	21.2976	1.03798	0.518990	+	0.854781i	$0.326308\pi$
$422$	−0.411393	−0.0200263
$423$	0	0
$424$	−16.2258	−0.787993
$425$	−4.12008	−0.199853
$426$	0	0
$427$	51.3082	2.48298
$428$	15.2212	0.735743
$429$	0	0
$430$	0.424121	0.0204529
$431$	21.0438	1.01364	0.506822	−	0.862051i	$-0.330820\pi$
$431$	21.0438	1.01364	0.506822	+	0.862051i	$0.330820\pi$
$432$	0	0
$433$	3.07000	0.147535	0.0737675	−	0.997275i	$-0.476498\pi$
$433$	3.07000	0.147535	0.0737675	+	0.997275i	$0.476498\pi$
$434$	−10.9997	−0.528003
$435$	0	0
$436$	27.0327	1.29463
$437$	2.69708	0.129019
$438$	0	0
$439$	22.8075	1.08854	0.544271	−	0.838909i	$-0.316806\pi$
$439$	22.8075	1.08854	0.544271	+	0.838909i	$0.316806\pi$
$440$	−2.03905	−0.0972080
$441$	0	0
$442$	6.12096	0.291144
$443$	−1.82887	−0.0868922	−0.0434461	−	0.999056i	$-0.513834\pi$
$443$	−1.82887	−0.0868922	−0.0434461	+	0.999056i	$0.513834\pi$
$444$	0	0
$445$	10.0513	0.476476
$446$	−0.559128	−0.0264755
$447$	0	0
$448$	−6.74608	−0.318722
$449$	−32.1557	−1.51752	−0.758760	−	0.651371i	$-0.774194\pi$
$449$	−32.1557	−1.51752	−0.758760	+	0.651371i	$0.774194\pi$
$450$	0	0
$451$	3.70604	0.174510
$452$	30.9466	1.45561
$453$	0	0
$454$	3.83297	0.179890
$455$	51.2093	2.40073
$456$	0	0
$457$	−17.6731	−0.826714	−0.413357	−	0.910569i	$-0.635644\pi$
$457$	−17.6731	−0.826714	−0.413357	+	0.910569i	$0.635644\pi$
$458$	9.34528	0.436676
$459$	0	0
$460$	−2.68298	−0.125094
$461$	−35.1978	−1.63932	−0.819662	−	0.572848i	$-0.805839\pi$
$461$	−35.1978	−1.63932	−0.819662	+	0.572848i	$0.805839\pi$
$462$	0	0
$463$	−9.53381	−0.443074	−0.221537	−	0.975152i	$-0.571107\pi$
$463$	−9.53381	−0.443074	−0.221537	+	0.975152i	$0.571107\pi$
$464$	2.19756	0.102019
$465$	0	0
$466$	−14.5793	−0.675375
$467$	20.9801	0.970841	0.485421	−	0.874281i	$-0.338667\pi$
$467$	20.9801	0.970841	0.485421	+	0.874281i	$0.338667\pi$
$468$	0	0
$469$	−31.5207	−1.45549
$470$	−3.84058	−0.177153
$471$	0	0
$472$	24.1884	1.11336
$473$	0.291262	0.0133922
$474$	0	0
$475$	−6.62974	−0.304194
$476$	−13.9716	−0.640385
$477$	0	0
$478$	−3.48577	−0.159435
$479$	29.5362	1.34954	0.674772	−	0.738027i	$-0.264242\pi$
$479$	29.5362	1.34954	0.674772	+	0.738027i	$0.264242\pi$
$480$	0	0
$481$	37.7374	1.72068
$482$	11.6169	0.529136
$483$	0	0
$484$	17.8712	0.812329
$485$	18.9901	0.862295
$486$	0	0
$487$	−4.69157	−0.212595	−0.106298	−	0.994334i	$-0.533900\pi$
$487$	−4.69157	−0.212595	−0.106298	+	0.994334i	$0.533900\pi$
$488$	−21.4839	−0.972532
$489$	0	0
$490$	15.7456	0.711315
$491$	17.4315	0.786674	0.393337	−	0.919394i	$-0.371321\pi$
$491$	17.4315	0.786674	0.393337	+	0.919394i	$0.371321\pi$
$492$	0	0
$493$	1.67611	0.0754882
$494$	9.84942	0.443147
$495$	0	0
$496$	−8.66502	−0.389071
$497$	−43.6785	−1.95925
$498$	0	0
$499$	9.04617	0.404962	0.202481	−	0.979286i	$-0.435100\pi$
$499$	9.04617	0.404962	0.202481	+	0.979286i	$0.435100\pi$
$500$	20.0100	0.894872
$501$	0	0
$502$	9.10145	0.406217
$503$	−25.7136	−1.14651	−0.573255	−	0.819377i	$-0.694320\pi$
$503$	−25.7136	−1.14651	−0.573255	+	0.819377i	$0.694320\pi$
$504$	0	0
$505$	−7.32889	−0.326131
$506$	0.347272	0.0154381
$507$	0	0
$508$	−7.30456	−0.324088
$509$	10.4056	0.461220	0.230610	−	0.973046i	$-0.425928\pi$
$509$	10.4056	0.461220	0.230610	+	0.973046i	$0.425928\pi$
$510$	0	0
$511$	−10.5963	−0.468753
$512$	20.9516	0.925937
$513$	0	0
$514$	4.34294	0.191559
$515$	7.30349	0.321830
$516$	0	0
$517$	−2.63749	−0.115997
$518$	16.2351	0.713329
$519$	0	0
$520$	−21.4425	−0.940317
$521$	44.3130	1.94139	0.970693	−	0.240323i	$-0.0772534\pi$
$521$	44.3130	1.94139	0.970693	+	0.240323i	$0.0772534\pi$
$522$	0	0
$523$	2.54821	0.111425	0.0557127	−	0.998447i	$-0.482257\pi$
$523$	2.54821	0.111425	0.0557127	+	0.998447i	$0.482257\pi$
$524$	23.9518	1.04634
$525$	0	0
$526$	2.06382	0.0899869
$527$	−6.60894	−0.287890
$528$	0	0
$529$	1.00000	0.0434783
$530$	7.02428	0.305115
$531$	0	0
$532$	−22.4821	−0.974721
$533$	38.9724	1.68808
$534$	0	0
$535$	−14.4207	−0.623461
$536$	13.1984	0.570086
$537$	0	0
$538$	−10.4656	−0.451203
$539$	10.8132	0.465757
$540$	0	0
$541$	−9.57940	−0.411851	−0.205925	−	0.978568i	$-0.566020\pi$
$541$	−9.57940	−0.411851	−0.205925	+	0.978568i	$0.566020\pi$
$542$	−6.60408	−0.283670
$543$	0	0
$544$	9.02724	0.387040
$545$	−25.6111	−1.09706
$546$	0	0
$547$	7.62918	0.326200	0.163100	−	0.986610i	$-0.447851\pi$
$547$	7.62918	0.326200	0.163100	+	0.986610i	$0.447851\pi$
$548$	17.6480	0.753887
$549$	0	0
$550$	−0.853635	−0.0363991
$551$	2.69708	0.114899
$552$	0	0
$553$	−59.4167	−2.52665
$554$	−14.3944	−0.611559
$555$	0	0
$556$	32.9804	1.39868
$557$	29.7588	1.26092	0.630460	−	0.776222i	$-0.282866\pi$
$557$	29.7588	1.26092	0.630460	+	0.776222i	$0.282866\pi$
$558$	0	0
$559$	3.06289	0.129546
$560$	17.3548	0.733375
$561$	0	0
$562$	4.25907	0.179658
$563$	−35.5508	−1.49829	−0.749145	−	0.662407i	$-0.769535\pi$
$563$	−35.5508	−1.49829	−0.749145	+	0.662407i	$0.769535\pi$
$564$	0	0
$565$	−29.3192	−1.23347
$566$	8.83923	0.371540
$567$	0	0
$568$	18.2892	0.767398
$569$	16.4596	0.690023	0.345011	−	0.938599i	$-0.387875\pi$
$569$	16.4596	0.690023	0.345011	+	0.938599i	$0.387875\pi$
$570$	0	0
$571$	−32.1451	−1.34523	−0.672616	−	0.739992i	$-0.734829\pi$
$571$	−32.1451	−1.34523	−0.672616	+	0.739992i	$0.734829\pi$
$572$	−6.72865	−0.281339
$573$	0	0
$574$	16.7664	0.699818
$575$	−2.45812	−0.102511
$576$	0	0
$577$	−22.2212	−0.925080	−0.462540	−	0.886598i	$-0.653062\pi$
$577$	−22.2212	−0.925080	−0.462540	+	0.886598i	$0.653062\pi$
$578$	−7.99192	−0.332420
$579$	0	0
$580$	−2.68298	−0.111405
$581$	−50.5871	−2.09871
$582$	0	0
$583$	4.82387	0.199784
$584$	4.43692	0.183601
$585$	0	0
$586$	−15.4479	−0.638146
$587$	−8.38053	−0.345902	−0.172951	−	0.984930i	$-0.555330\pi$
$587$	−8.38053	−0.345902	−0.172951	+	0.984930i	$0.555330\pi$
$588$	0	0
$589$	−10.6347	−0.438193
$590$	−10.4714	−0.431099
$591$	0	0
$592$	12.7892	0.525633
$593$	−41.5343	−1.70561	−0.852805	−	0.522230i	$-0.825100\pi$
$593$	−41.5343	−1.70561	−0.852805	+	0.522230i	$0.825100\pi$
$594$	0	0
$595$	13.2368	0.542656
$596$	−11.3332	−0.464224
$597$	0	0
$598$	3.65188	0.149337
$599$	41.6783	1.70293	0.851465	−	0.524412i	$-0.175715\pi$
$599$	41.6783	1.70293	0.851465	+	0.524412i	$0.175715\pi$
$600$	0	0
$601$	−15.7600	−0.642862	−0.321431	−	0.946933i	$-0.604164\pi$
$601$	−15.7600	−0.642862	−0.321431	+	0.946933i	$0.604164\pi$
$602$	1.31770	0.0537052
$603$	0	0
$604$	−7.96609	−0.324136
$605$	−16.9314	−0.688359
$606$	0	0
$607$	−28.2457	−1.14646	−0.573229	−	0.819395i	$-0.694309\pi$
$607$	−28.2457	−1.14646	−0.573229	+	0.819395i	$0.694309\pi$
$608$	14.5260	0.589107
$609$	0	0
$610$	9.30058	0.376570
$611$	−27.7356	−1.12206
$612$	0	0
$613$	37.9241	1.53174	0.765870	−	0.642996i	$-0.222309\pi$
$613$	37.9241	1.53174	0.765870	+	0.642996i	$0.222309\pi$
$614$	7.27613	0.293641
$615$	0	0
$616$	−6.33510	−0.255249
$617$	−12.9445	−0.521128	−0.260564	−	0.965457i	$-0.583908\pi$
$617$	−12.9445	−0.521128	−0.260564	+	0.965457i	$0.583908\pi$
$618$	0	0
$619$	−44.2979	−1.78048	−0.890242	−	0.455487i	$-0.849465\pi$
$619$	−44.2979	−1.78048	−0.890242	+	0.455487i	$0.849465\pi$
$620$	10.5790	0.424865
$621$	0	0
$622$	2.12848	0.0853443
$623$	31.2281	1.25113
$624$	0	0
$625$	−6.66705	−0.266682
$626$	2.45161	0.0979859
$627$	0	0
$628$	−35.8800	−1.43177
$629$	9.75452	0.388938
$630$	0	0
$631$	1.21447	0.0483474	0.0241737	−	0.999708i	$-0.492305\pi$
$631$	1.21447	0.0483474	0.0241737	+	0.999708i	$0.492305\pi$
$632$	24.8791	0.989639
$633$	0	0
$634$	−0.279875	−0.0111152
$635$	6.92042	0.274629
$636$	0	0
$637$	113.711	4.50539
$638$	0.347272	0.0137486
$639$	0	0
$640$	−18.3964	−0.727182
$641$	41.6001	1.64310	0.821552	−	0.570134i	$-0.193109\pi$
$641$	41.6001	1.64310	0.821552	+	0.570134i	$0.193109\pi$
$642$	0	0
$643$	−33.3387	−1.31475	−0.657374	−	0.753564i	$-0.728333\pi$
$643$	−33.3387	−1.31475	−0.657374	+	0.753564i	$0.728333\pi$
$644$	−8.33570	−0.328473
$645$	0	0
$646$	2.54592	0.100168
$647$	16.2174	0.637570	0.318785	−	0.947827i	$-0.396725\pi$
$647$	16.2174	0.637570	0.318785	+	0.947827i	$0.396725\pi$
$648$	0	0
$649$	−7.19113	−0.282276
$650$	−8.97677	−0.352098
$651$	0	0
$652$	−6.89128	−0.269883
$653$	19.3624	0.757708	0.378854	−	0.925456i	$-0.376318\pi$
$653$	19.3624	0.757708	0.378854	+	0.925456i	$0.376318\pi$
$654$	0	0
$655$	−22.6922	−0.886659
$656$	13.2078	0.515676
$657$	0	0
$658$	−11.9322	−0.465167
$659$	29.7211	1.15777	0.578885	−	0.815409i	$-0.303488\pi$
$659$	29.7211	1.15777	0.578885	+	0.815409i	$0.303488\pi$
$660$	0	0
$661$	16.5459	0.643562	0.321781	−	0.946814i	$-0.395719\pi$
$661$	16.5459	0.643562	0.321781	+	0.946814i	$0.395719\pi$
$662$	−3.18065	−0.123619
$663$	0	0
$664$	21.1820	0.822021
$665$	21.2997	0.825969
$666$	0	0
$667$	1.00000	0.0387202
$668$	−17.4892	−0.676679
$669$	0	0
$670$	−5.71372	−0.220740
$671$	6.38711	0.246572
$672$	0	0
$673$	−34.0750	−1.31349	−0.656746	−	0.754112i	$-0.728068\pi$
$673$	−34.0750	−1.31349	−0.656746	+	0.754112i	$0.728068\pi$
$674$	−4.20425	−0.161942
$675$	0	0
$676$	−48.8813	−1.88005
$677$	20.9027	0.803357	0.401678	−	0.915781i	$-0.368427\pi$
$677$	20.9027	0.803357	0.401678	+	0.915781i	$0.368427\pi$
$678$	0	0
$679$	59.0000	2.26421
$680$	−5.54255	−0.212547
$681$	0	0
$682$	−1.36930	−0.0524333
$683$	−47.7573	−1.82738	−0.913692	−	0.406408i	$-0.866781\pi$
$683$	−47.7573	−1.82738	−0.913692	+	0.406408i	$0.866781\pi$
$684$	0	0
$685$	−16.7199	−0.638836
$686$	29.3922	1.12220
$687$	0	0
$688$	1.03801	0.0395739
$689$	50.7275	1.93256
$690$	0	0
$691$	−0.154264	−0.00586849	−0.00293424	−	0.999996i	$-0.500934\pi$
$691$	−0.154264	−0.00586849	−0.00293424	+	0.999996i	$0.500934\pi$
$692$	−7.75150	−0.294668
$693$	0	0
$694$	−7.99327	−0.303420
$695$	−31.2460	−1.18523
$696$	0	0
$697$	10.0738	0.381571
$698$	−10.3470	−0.391641
$699$	0	0
$700$	20.4901	0.774455
$701$	30.8317	1.16450	0.582248	−	0.813011i	$-0.302173\pi$
$701$	30.8317	1.16450	0.582248	+	0.813011i	$0.302173\pi$
$702$	0	0
$703$	15.6963	0.591997
$704$	−0.839787	−0.0316507
$705$	0	0
$706$	16.6117	0.625191
$707$	−22.7700	−0.856355
$708$	0	0
$709$	−3.56458	−0.133871	−0.0669353	−	0.997757i	$-0.521322\pi$
$709$	−3.56458	−0.133871	−0.0669353	+	0.997757i	$0.521322\pi$
$710$	−7.91756	−0.297141
$711$	0	0
$712$	−13.0759	−0.490042
$713$	−3.94303	−0.147667
$714$	0	0
$715$	6.37480	0.238404
$716$	−42.6171	−1.59268
$717$	0	0
$718$	−0.818713	−0.0305541
$719$	−20.9959	−0.783016	−0.391508	−	0.920175i	$-0.628046\pi$
$719$	−20.9959	−0.783016	−0.391508	+	0.920175i	$0.628046\pi$
$720$	0	0
$721$	22.6911	0.845062
$722$	−6.60374	−0.245766
$723$	0	0
$724$	37.2314	1.38370
$725$	−2.45812	−0.0912923
$726$	0	0
$727$	19.3196	0.716524	0.358262	−	0.933621i	$-0.383369\pi$
$727$	19.3196	0.716524	0.358262	+	0.933621i	$0.383369\pi$
$728$	−66.6195	−2.46908
$729$	0	0
$730$	−1.92078	−0.0710914
$731$	0.791709	0.0292824
$732$	0	0
$733$	0.320498	0.0118379	0.00591893	−	0.999982i	$-0.498116\pi$
$733$	0.320498	0.0118379	0.00591893	+	0.999982i	$0.498116\pi$
$734$	−5.71807	−0.211058
$735$	0	0
$736$	5.38583	0.198524
$737$	−3.92386	−0.144537
$738$	0	0
$739$	−29.9692	−1.10243	−0.551217	−	0.834362i	$-0.685837\pi$
$739$	−29.9692	−1.10243	−0.551217	+	0.834362i	$0.685837\pi$
$740$	−15.6142	−0.573990
$741$	0	0
$742$	21.8236	0.801171
$743$	−14.3052	−0.524808	−0.262404	−	0.964958i	$-0.584515\pi$
$743$	−14.3052	−0.524808	−0.262404	+	0.964958i	$0.584515\pi$
$744$	0	0
$745$	10.7372	0.393379
$746$	5.68214	0.208038
$747$	0	0
$748$	−1.73925	−0.0635933
$749$	−44.8035	−1.63708
$750$	0	0
$751$	40.9102	1.49283	0.746417	−	0.665479i	$-0.231773\pi$
$751$	40.9102	1.49283	0.746417	+	0.665479i	$0.231773\pi$
$752$	−9.39961	−0.342769
$753$	0	0
$754$	3.65188	0.132994
$755$	7.54716	0.274669
$756$	0	0
$757$	−6.22635	−0.226300	−0.113150	−	0.993578i	$-0.536094\pi$
$757$	−6.22635	−0.226300	−0.113150	+	0.993578i	$0.536094\pi$
$758$	−8.19331	−0.297594
$759$	0	0
$760$	−8.91870	−0.323515
$761$	13.1589	0.477011	0.238505	−	0.971141i	$-0.423343\pi$
$761$	13.1589	0.477011	0.238505	+	0.971141i	$0.423343\pi$
$762$	0	0
$763$	−79.5707	−2.88065
$764$	−35.2832	−1.27650
$765$	0	0
$766$	−8.29889	−0.299851
$767$	−75.6214	−2.73053
$768$	0	0
$769$	15.8580	0.571854	0.285927	−	0.958251i	$-0.407698\pi$
$769$	15.8580	0.571854	0.285927	+	0.958251i	$0.407698\pi$
$770$	2.74252	0.0988336
$771$	0	0
$772$	−34.9056	−1.25628
$773$	−25.0338	−0.900402	−0.450201	−	0.892927i	$-0.648648\pi$
$773$	−25.0338	−0.900402	−0.450201	+	0.892927i	$0.648648\pi$
$774$	0	0
$775$	9.69243	0.348162
$776$	−24.7047	−0.886846
$777$	0	0
$778$	3.09017	0.110788
$779$	16.2100	0.580783
$780$	0	0
$781$	−5.43733	−0.194563
$782$	0.943955	0.0337558
$783$	0	0
$784$	38.5366	1.37631
$785$	33.9931	1.21326
$786$	0	0
$787$	−22.1812	−0.790674	−0.395337	−	0.918536i	$-0.629372\pi$
$787$	−22.1812	−0.790674	−0.395337	+	0.918536i	$0.629372\pi$
$788$	−37.4562	−1.33432
$789$	0	0
$790$	−10.7704	−0.383194
$791$	−91.0914	−3.23884
$792$	0	0
$793$	67.1664	2.38515
$794$	−4.98672	−0.176972
$795$	0	0
$796$	−0.532338	−0.0188682
$797$	30.2552	1.07169	0.535846	−	0.844315i	$-0.319993\pi$
$797$	30.2552	1.07169	0.535846	+	0.844315i	$0.319993\pi$
$798$	0	0
$799$	−7.16923	−0.253629
$800$	−13.2390	−0.468069
$801$	0	0
$802$	1.09202	0.0385605
$803$	−1.31908	−0.0465495
$804$	0	0
$805$	7.89733	0.278344
$806$	−14.3995	−0.507200
$807$	0	0
$808$	9.53434	0.335417
$809$	32.5292	1.14367	0.571833	−	0.820370i	$-0.306233\pi$
$809$	32.5292	1.14367	0.571833	+	0.820370i	$0.306233\pi$
$810$	0	0
$811$	−17.2748	−0.606599	−0.303300	−	0.952895i	$-0.598088\pi$
$811$	−17.2748	−0.606599	−0.303300	+	0.952895i	$0.598088\pi$
$812$	−8.33570	−0.292526
$813$	0	0
$814$	2.02103	0.0708371
$815$	6.52888	0.228697
$816$	0	0
$817$	1.27396	0.0445703
$818$	3.74393	0.130903
$819$	0	0
$820$	−16.1252	−0.563117
$821$	−11.3961	−0.397728	−0.198864	−	0.980027i	$-0.563725\pi$
$821$	−11.3961	−0.397728	−0.198864	+	0.980027i	$0.563725\pi$
$822$	0	0
$823$	16.8758	0.588255	0.294128	−	0.955766i	$-0.404971\pi$
$823$	16.8758	0.588255	0.294128	+	0.955766i	$0.404971\pi$
$824$	−9.50130	−0.330993
$825$	0	0
$826$	−32.5333	−1.13198
$827$	45.9867	1.59911	0.799557	−	0.600590i	$-0.205068\pi$
$827$	45.9867	1.59911	0.799557	+	0.600590i	$0.205068\pi$
$828$	0	0
$829$	−53.2178	−1.84833	−0.924165	−	0.381995i	$-0.875237\pi$
$829$	−53.2178	−1.84833	−0.924165	+	0.381995i	$0.875237\pi$
$830$	−9.16987	−0.318291
$831$	0	0
$832$	−8.83114	−0.306165
$833$	29.3925	1.01839
$834$	0	0
$835$	16.5695	0.573411
$836$	−2.79868	−0.0967945
$837$	0	0
$838$	−0.451839	−0.0156085
$839$	−13.5867	−0.469064	−0.234532	−	0.972108i	$-0.575356\pi$
$839$	−13.5867	−0.469064	−0.234532	+	0.972108i	$0.575356\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	11.9944	0.413354
$843$	0	0
$844$	1.22927	0.0423132
$845$	46.3107	1.59314
$846$	0	0
$847$	−52.6039	−1.80749
$848$	17.1916	0.590361
$849$	0	0
$850$	−2.32035	−0.0795875
$851$	5.81974	0.199498
$852$	0	0
$853$	−6.73050	−0.230448	−0.115224	−	0.993340i	$-0.536759\pi$
$853$	−6.73050	−0.230448	−0.115224	+	0.993340i	$0.536759\pi$
$854$	28.8959	0.988796
$855$	0	0
$856$	18.7603	0.641212
$857$	−35.0261	−1.19647	−0.598234	−	0.801321i	$-0.704131\pi$
$857$	−35.0261	−1.19647	−0.598234	+	0.801321i	$0.704131\pi$
$858$	0	0
$859$	1.65393	0.0564314	0.0282157	−	0.999602i	$-0.491017\pi$
$859$	1.65393	0.0564314	0.0282157	+	0.999602i	$0.491017\pi$
$860$	−1.26730	−0.0432146
$861$	0	0
$862$	11.8515	0.403663
$863$	−31.4926	−1.07202	−0.536010	−	0.844212i	$-0.680069\pi$
$863$	−31.4926	−1.07202	−0.536010	+	0.844212i	$0.680069\pi$
$864$	0	0
$865$	7.34385	0.249698
$866$	1.72897	0.0587528
$867$	0	0
$868$	32.8679	1.11561
$869$	−7.39650	−0.250909
$870$	0	0
$871$	−41.2630	−1.39814
$872$	33.3181	1.12829
$873$	0	0
$874$	1.51895	0.0513791
$875$	−58.8993	−1.99116
$876$	0	0
$877$	42.2692	1.42733	0.713665	−	0.700487i	$-0.247034\pi$
$877$	42.2692	1.42733	0.713665	+	0.700487i	$0.247034\pi$
$878$	12.8448	0.433490
$879$	0	0
$880$	2.16042	0.0728277
$881$	32.5253	1.09581	0.547903	−	0.836542i	$-0.315426\pi$
$881$	32.5253	1.09581	0.547903	+	0.836542i	$0.315426\pi$
$882$	0	0
$883$	−20.9256	−0.704202	−0.352101	−	0.935962i	$-0.614533\pi$
$883$	−20.9256	−0.704202	−0.352101	+	0.935962i	$0.614533\pi$
$884$	−18.2898	−0.615154
$885$	0	0
$886$	−1.02999	−0.0346031
$887$	−3.39320	−0.113933	−0.0569663	−	0.998376i	$-0.518143\pi$
$887$	−3.39320	−0.113933	−0.0569663	+	0.998376i	$0.518143\pi$
$888$	0	0
$889$	21.5010	0.721119
$890$	5.66069	0.189747
$891$	0	0
$892$	1.67071	0.0559395
$893$	−11.5362	−0.386045
$894$	0	0
$895$	40.3759	1.34962
$896$	−57.1555	−1.90943
$897$	0	0
$898$	−18.1095	−0.604321
$899$	−3.94303	−0.131507
$900$	0	0
$901$	13.1123	0.436833
$902$	2.08717	0.0694953
$903$	0	0
$904$	38.1421	1.26859
$905$	−35.2735	−1.17253
$906$	0	0
$907$	46.3463	1.53890	0.769452	−	0.638704i	$-0.220529\pi$
$907$	46.3463	1.53890	0.769452	+	0.638704i	$0.220529\pi$
$908$	−11.4532	−0.380087
$909$	0	0
$910$	28.8402	0.956042
$911$	−34.0270	−1.12737	−0.563683	−	0.825991i	$-0.690616\pi$
$911$	−34.0270	−1.12737	−0.563683	+	0.825991i	$0.690616\pi$
$912$	0	0
$913$	−6.29735	−0.208412
$914$	−9.95319	−0.329222
$915$	0	0
$916$	−27.9243	−0.922646
$917$	−70.5022	−2.32819
$918$	0	0
$919$	8.92150	0.294293	0.147147	−	0.989115i	$-0.452991\pi$
$919$	8.92150	0.294293	0.147147	+	0.989115i	$0.452991\pi$
$920$	−3.30680	−0.109022
$921$	0	0
$922$	−19.8227	−0.652827
$923$	−57.1785	−1.88205
$924$	0	0
$925$	−14.3056	−0.470365
$926$	−5.36927	−0.176445
$927$	0	0
$928$	5.38583	0.176798
$929$	3.51706	0.115391	0.0576955	−	0.998334i	$-0.481625\pi$
$929$	3.51706	0.115391	0.0576955	+	0.998334i	$0.481625\pi$
$930$	0	0
$931$	47.2963	1.55007
$932$	43.5641	1.42699
$933$	0	0
$934$	11.8156	0.386618
$935$	1.64779	0.0538883
$936$	0	0
$937$	16.7996	0.548819	0.274409	−	0.961613i	$-0.411518\pi$
$937$	16.7996	0.548819	0.274409	+	0.961613i	$0.411518\pi$
$938$	−17.7519	−0.579620
$939$	0	0
$940$	11.4759	0.374303
$941$	−2.96990	−0.0968159	−0.0484080	−	0.998828i	$-0.515415\pi$
$941$	−2.96990	−0.0968159	−0.0484080	+	0.998828i	$0.515415\pi$
$942$	0	0
$943$	6.01020	0.195719
$944$	−25.6281	−0.834123
$945$	0	0
$946$	0.164034	0.00533319
$947$	25.1490	0.817233	0.408617	−	0.912706i	$-0.366011\pi$
$947$	25.1490	0.817233	0.408617	+	0.912706i	$0.366011\pi$
$948$	0	0
$949$	−13.8714	−0.450285
$950$	−3.73375	−0.121139
$951$	0	0
$952$	−17.2201	−0.558106
$953$	0.976881	0.0316443	0.0158221	−	0.999875i	$-0.494963\pi$
$953$	0.976881	0.0316443	0.0158221	+	0.999875i	$0.494963\pi$
$954$	0	0
$955$	33.4276	1.08169
$956$	10.4157	0.336868
$957$	0	0
$958$	16.6343	0.537428
$959$	−51.9469	−1.67745
$960$	0	0
$961$	−15.4525	−0.498469
$962$	21.2530	0.685224
$963$	0	0
$964$	−34.7121	−1.11800
$965$	33.0700	1.06456
$966$	0	0
$967$	−50.3884	−1.62038	−0.810191	−	0.586166i	$-0.800637\pi$
$967$	−50.3884	−1.62038	−0.810191	+	0.586166i	$0.800637\pi$
$968$	22.0265	0.707958
$969$	0	0
$970$	10.6949	0.343391
$971$	−47.9179	−1.53776	−0.768879	−	0.639394i	$-0.779185\pi$
$971$	−47.9179	−1.53776	−0.768879	+	0.639394i	$0.779185\pi$
$972$	0	0
$973$	−97.0779	−3.11217
$974$	−2.64221	−0.0846617
$975$	0	0
$976$	22.7627	0.728616
$977$	−25.9762	−0.831053	−0.415527	−	0.909581i	$-0.636403\pi$
$977$	−25.9762	−0.831053	−0.415527	+	0.909581i	$0.636403\pi$
$978$	0	0
$979$	3.88744	0.124243
$980$	−47.0490	−1.50293
$981$	0	0
$982$	9.81712	0.313277
$983$	−50.0104	−1.59509	−0.797543	−	0.603263i	$-0.793867\pi$
$983$	−50.0104	−1.59509	−0.797543	+	0.603263i	$0.793867\pi$
$984$	0	0
$985$	35.4865	1.13069
$986$	0.943955	0.0300616
$987$	0	0
$988$	−29.4307	−0.936317
$989$	0.472349	0.0150198
$990$	0	0
$991$	−47.1683	−1.49835	−0.749176	−	0.662371i	$-0.769550\pi$
$991$	−47.1683	−1.49835	−0.749176	+	0.662371i	$0.769550\pi$
$992$	−21.2365	−0.674258
$993$	0	0
$994$	−24.5989	−0.780231
$995$	0.504343	0.0159888
$996$	0	0
$997$	−5.86063	−0.185608	−0.0928040	−	0.995684i	$-0.529583\pi$
$997$	−5.86063	−0.185608	−0.0928040	+	0.995684i	$0.529583\pi$
$998$	5.09464	0.161268
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.r.1.10		16
3.2	odd	2		2001.2.a.n.1.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.n.1.7	✓	16	3.2	odd	2
6003.2.a.r.1.10		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.563182 q^{2} -1.68283 q^{4} +1.59433 q^{5} +4.95339 q^{7} -2.07410 q^{8} +O(q^{10})\)	\(q+0.563182 q^{2} -1.68283 q^{4} +1.59433 q^{5} +4.95339 q^{7} -2.07410 q^{8} +0.897897 q^{10} +0.616624 q^{11} +6.48438 q^{13} +2.78966 q^{14} +2.19756 q^{16} +1.67611 q^{17} +2.69708 q^{19} -2.68298 q^{20} +0.347272 q^{22} +1.00000 q^{23} -2.45812 q^{25} +3.65188 q^{26} -8.33570 q^{28} +1.00000 q^{29} -3.94303 q^{31} +5.38583 q^{32} +0.943955 q^{34} +7.89733 q^{35} +5.81974 q^{37} +1.51895 q^{38} -3.30680 q^{40} +6.01020 q^{41} +0.472349 q^{43} -1.03767 q^{44} +0.563182 q^{46} -4.27730 q^{47} +17.5361 q^{49} -1.38437 q^{50} -10.9121 q^{52} +7.82304 q^{53} +0.983101 q^{55} -10.2738 q^{56} +0.563182 q^{58} -11.6621 q^{59} +10.3582 q^{61} -2.22064 q^{62} -1.36191 q^{64} +10.3382 q^{65} -6.36345 q^{67} -2.82060 q^{68} +4.44764 q^{70} -8.81789 q^{71} -2.13920 q^{73} +3.27757 q^{74} -4.53872 q^{76} +3.05438 q^{77} -11.9951 q^{79} +3.50362 q^{80} +3.38484 q^{82} -10.2126 q^{83} +2.67227 q^{85} +0.266019 q^{86} -1.27894 q^{88} +6.30439 q^{89} +32.1197 q^{91} -1.68283 q^{92} -2.40890 q^{94} +4.30003 q^{95} +11.9110 q^{97} +9.87603 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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