LMFDB - Embedded newform 2898.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2898,2,Mod(1,2898)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2898, 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("2898.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2898.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:	$23.1406465058$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	2898.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +0.732051 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +0.732051 q^{5} +1.00000 q^{7} -1.00000 q^{8} -0.732051 q^{10} -3.46410 q^{11} +5.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.73205 q^{17} -2.00000 q^{19} +0.732051 q^{20} +3.46410 q^{22} -1.00000 q^{23} -4.46410 q^{25} -5.46410 q^{26} +1.00000 q^{28} -8.00000 q^{29} -10.1962 q^{31} -1.00000 q^{32} +2.73205 q^{34} +0.732051 q^{35} +7.46410 q^{37} +2.00000 q^{38} -0.732051 q^{40} +2.00000 q^{41} -3.46410 q^{44} +1.00000 q^{46} -2.19615 q^{47} +1.00000 q^{49} +4.46410 q^{50} +5.46410 q^{52} +3.46410 q^{53} -2.53590 q^{55} -1.00000 q^{56} +8.00000 q^{58} -5.66025 q^{59} -2.19615 q^{61} +10.1962 q^{62} +1.00000 q^{64} +4.00000 q^{65} -11.4641 q^{67} -2.73205 q^{68} -0.732051 q^{70} +12.3923 q^{71} -0.535898 q^{73} -7.46410 q^{74} -2.00000 q^{76} -3.46410 q^{77} -12.9282 q^{79} +0.732051 q^{80} -2.00000 q^{82} -14.3923 q^{83} -2.00000 q^{85} +3.46410 q^{88} -3.80385 q^{89} +5.46410 q^{91} -1.00000 q^{92} +2.19615 q^{94} -1.46410 q^{95} -5.26795 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} + 4 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} - 4 q^{19} - 2 q^{20} - 2 q^{23} - 2 q^{25} - 4 q^{26} + 2 q^{28} - 16 q^{29} - 10 q^{31} - 2 q^{32} + 2 q^{34} - 2 q^{35} + 8 q^{37} + 4 q^{38} + 2 q^{40} + 4 q^{41} + 2 q^{46} + 6 q^{47} + 2 q^{49} + 2 q^{50} + 4 q^{52} - 12 q^{55} - 2 q^{56} + 16 q^{58} + 6 q^{59} + 6 q^{61} + 10 q^{62} + 2 q^{64} + 8 q^{65} - 16 q^{67} - 2 q^{68} + 2 q^{70} + 4 q^{71} - 8 q^{73} - 8 q^{74} - 4 q^{76} - 12 q^{79} - 2 q^{80} - 4 q^{82} - 8 q^{83} - 4 q^{85} - 18 q^{89} + 4 q^{91} - 2 q^{92} - 6 q^{94} + 4 q^{95} - 14 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 2 * q^10 + 4 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 - 4 * q^19 - 2 * q^20 - 2 * q^23 - 2 * q^25 - 4 * q^26 + 2 * q^28 - 16 * q^29 - 10 * q^31 - 2 * q^32 + 2 * q^34 - 2 * q^35 + 8 * q^37 + 4 * q^38 + 2 * q^40 + 4 * q^41 + 2 * q^46 + 6 * q^47 + 2 * q^49 + 2 * q^50 + 4 * q^52 - 12 * q^55 - 2 * q^56 + 16 * q^58 + 6 * q^59 + 6 * q^61 + 10 * q^62 + 2 * q^64 + 8 * q^65 - 16 * q^67 - 2 * q^68 + 2 * q^70 + 4 * q^71 - 8 * q^73 - 8 * q^74 - 4 * q^76 - 12 * q^79 - 2 * q^80 - 4 * q^82 - 8 * q^83 - 4 * q^85 - 18 * q^89 + 4 * q^91 - 2 * q^92 - 6 * q^94 + 4 * q^95 - 14 * q^97 - 2 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.732051	0.327383	0.163692	−	0.986512i	$-0.447660\pi$
$5$	0.732051	0.327383	0.163692	+	0.986512i	$0.447660\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.732051	−0.231495
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	5.46410	1.51547	0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410	1.51547	0.757735	+	0.652563i	$0.226306\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.73205	−0.662620	−0.331310	−	0.943522i	$-0.607491\pi$
$17$	−2.73205	−0.662620	−0.331310	+	0.943522i	$0.607491\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0.732051	0.163692
$21$	0	0
$22$	3.46410	0.738549
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.46410	−0.892820
$26$	−5.46410	−1.07160
$27$	0	0
$28$	1.00000	0.188982
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	−10.1962	−1.83128	−0.915642	−	0.401996i	$-0.868317\pi$
$31$	−10.1962	−1.83128	−0.915642	+	0.401996i	$0.868317\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.73205	0.468543
$35$	0.732051	0.123739
$36$	0	0
$37$	7.46410	1.22709	0.613545	−	0.789659i	$-0.289743\pi$
$37$	7.46410	1.22709	0.613545	+	0.789659i	$0.289743\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	−0.732051	−0.115747
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−3.46410	−0.522233
$45$	0	0
$46$	1.00000	0.147442
$47$	−2.19615	−0.320342	−0.160171	−	0.987089i	$-0.551205\pi$
$47$	−2.19615	−0.320342	−0.160171	+	0.987089i	$0.551205\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.46410	0.631319
$51$	0	0
$52$	5.46410	0.757735
$53$	3.46410	0.475831	0.237915	−	0.971286i	$-0.423536\pi$
$53$	3.46410	0.475831	0.237915	+	0.971286i	$0.423536\pi$
$54$	0	0
$55$	−2.53590	−0.341940
$56$	−1.00000	−0.133631
$57$	0	0
$58$	8.00000	1.05045
$59$	−5.66025	−0.736902	−0.368451	−	0.929647i	$-0.620112\pi$
$59$	−5.66025	−0.736902	−0.368451	+	0.929647i	$0.620112\pi$
$60$	0	0
$61$	−2.19615	−0.281189	−0.140594	−	0.990067i	$-0.544901\pi$
$61$	−2.19615	−0.281189	−0.140594	+	0.990067i	$0.544901\pi$
$62$	10.1962	1.29491
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	0	0
$67$	−11.4641	−1.40056	−0.700281	−	0.713867i	$-0.746942\pi$
$67$	−11.4641	−1.40056	−0.700281	+	0.713867i	$0.746942\pi$
$68$	−2.73205	−0.331310
$69$	0	0
$70$	−0.732051	−0.0874968
$71$	12.3923	1.47070	0.735348	−	0.677690i	$-0.237019\pi$
$71$	12.3923	1.47070	0.735348	+	0.677690i	$0.237019\pi$
$72$	0	0
$73$	−0.535898	−0.0627222	−0.0313611	−	0.999508i	$-0.509984\pi$
$73$	−0.535898	−0.0627222	−0.0313611	+	0.999508i	$0.509984\pi$
$74$	−7.46410	−0.867684
$75$	0	0
$76$	−2.00000	−0.229416
$77$	−3.46410	−0.394771
$78$	0	0
$79$	−12.9282	−1.45454	−0.727268	−	0.686353i	$-0.759210\pi$
$79$	−12.9282	−1.45454	−0.727268	+	0.686353i	$0.759210\pi$
$80$	0.732051	0.0818458
$81$	0	0
$82$	−2.00000	−0.220863
$83$	−14.3923	−1.57976	−0.789880	−	0.613261i	$-0.789857\pi$
$83$	−14.3923	−1.57976	−0.789880	+	0.613261i	$0.789857\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	0	0
$88$	3.46410	0.369274
$89$	−3.80385	−0.403207	−0.201604	−	0.979467i	$-0.564615\pi$
$89$	−3.80385	−0.403207	−0.201604	+	0.979467i	$0.564615\pi$
$90$	0	0
$91$	5.46410	0.572793
$92$	−1.00000	−0.104257
$93$	0	0
$94$	2.19615	0.226516
$95$	−1.46410	−0.150214
$96$	0	0
$97$	−5.26795	−0.534879	−0.267440	−	0.963575i	$-0.586178\pi$
$97$	−5.26795	−0.534879	−0.267440	+	0.963575i	$0.586178\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−4.46410	−0.446410
$101$	−5.46410	−0.543698	−0.271849	−	0.962340i	$-0.587635\pi$
$101$	−5.46410	−0.543698	−0.271849	+	0.962340i	$0.587635\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−5.46410	−0.535799
$105$	0	0
$106$	−3.46410	−0.336463
$107$	6.92820	0.669775	0.334887	−	0.942258i	$-0.391302\pi$
$107$	6.92820	0.669775	0.334887	+	0.942258i	$0.391302\pi$
$108$	0	0
$109$	4.92820	0.472036	0.236018	−	0.971749i	$-0.424158\pi$
$109$	4.92820	0.472036	0.236018	+	0.971749i	$0.424158\pi$
$110$	2.53590	0.241788
$111$	0	0
$112$	1.00000	0.0944911
$113$	−12.5359	−1.17928	−0.589639	−	0.807667i	$-0.700730\pi$
$113$	−12.5359	−1.17928	−0.589639	+	0.807667i	$0.700730\pi$
$114$	0	0
$115$	−0.732051	−0.0682641
$116$	−8.00000	−0.742781
$117$	0	0
$118$	5.66025	0.521069
$119$	−2.73205	−0.250447
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.19615	0.198830
$123$	0	0
$124$	−10.1962	−0.915642
$125$	−6.92820	−0.619677
$126$	0	0
$127$	19.3205	1.71442	0.857209	−	0.514969i	$-0.172196\pi$
$127$	19.3205	1.71442	0.857209	+	0.514969i	$0.172196\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−4.00000	−0.350823
$131$	6.73205	0.588182	0.294091	−	0.955777i	$-0.404983\pi$
$131$	6.73205	0.588182	0.294091	+	0.955777i	$0.404983\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	11.4641	0.990348
$135$	0	0
$136$	2.73205	0.234271
$137$	17.3205	1.47979	0.739895	−	0.672722i	$-0.234875\pi$
$137$	17.3205	1.47979	0.739895	+	0.672722i	$0.234875\pi$
$138$	0	0
$139$	−12.1962	−1.03446	−0.517232	−	0.855845i	$-0.673038\pi$
$139$	−12.1962	−1.03446	−0.517232	+	0.855845i	$0.673038\pi$
$140$	0.732051	0.0618696
$141$	0	0
$142$	−12.3923	−1.03994
$143$	−18.9282	−1.58286
$144$	0	0
$145$	−5.85641	−0.486348
$146$	0.535898	0.0443513
$147$	0	0
$148$	7.46410	0.613545
$149$	18.7846	1.53890	0.769448	−	0.638710i	$-0.220532\pi$
$149$	18.7846	1.53890	0.769448	+	0.638710i	$0.220532\pi$
$150$	0	0
$151$	−10.5359	−0.857399	−0.428700	−	0.903447i	$-0.641028\pi$
$151$	−10.5359	−0.857399	−0.428700	+	0.903447i	$0.641028\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	3.46410	0.279145
$155$	−7.46410	−0.599531
$156$	0	0
$157$	2.19615	0.175272	0.0876360	−	0.996153i	$-0.472069\pi$
$157$	2.19615	0.175272	0.0876360	+	0.996153i	$0.472069\pi$
$158$	12.9282	1.02851
$159$	0	0
$160$	−0.732051	−0.0578737
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−8.39230	−0.657336	−0.328668	−	0.944446i	$-0.606600\pi$
$163$	−8.39230	−0.657336	−0.328668	+	0.944446i	$0.606600\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	14.3923	1.11706
$167$	16.0526	1.24218	0.621092	−	0.783738i	$-0.286689\pi$
$167$	16.0526	1.24218	0.621092	+	0.783738i	$0.286689\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	2.00000	0.153393
$171$	0	0
$172$	0	0
$173$	6.92820	0.526742	0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820	0.526742	0.263371	+	0.964695i	$0.415166\pi$
$174$	0	0
$175$	−4.46410	−0.337454
$176$	−3.46410	−0.261116
$177$	0	0
$178$	3.80385	0.285110
$179$	−13.8564	−1.03568	−0.517838	−	0.855479i	$-0.673263\pi$
$179$	−13.8564	−1.03568	−0.517838	+	0.855479i	$0.673263\pi$
$180$	0	0
$181$	−24.0526	−1.78781	−0.893906	−	0.448254i	$-0.852046\pi$
$181$	−24.0526	−1.78781	−0.893906	+	0.448254i	$0.852046\pi$
$182$	−5.46410	−0.405026
$183$	0	0
$184$	1.00000	0.0737210
$185$	5.46410	0.401729
$186$	0	0
$187$	9.46410	0.692084
$188$	−2.19615	−0.160171
$189$	0	0
$190$	1.46410	0.106217
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−8.92820	−0.642666	−0.321333	−	0.946966i	$-0.604131\pi$
$193$	−8.92820	−0.642666	−0.321333	+	0.946966i	$0.604131\pi$
$194$	5.26795	0.378217
$195$	0	0
$196$	1.00000	0.0714286
$197$	4.92820	0.351120	0.175560	−	0.984469i	$-0.443826\pi$
$197$	4.92820	0.351120	0.175560	+	0.984469i	$0.443826\pi$
$198$	0	0
$199$	16.7846	1.18983	0.594915	−	0.803789i	$-0.297186\pi$
$199$	16.7846	1.18983	0.594915	+	0.803789i	$0.297186\pi$
$200$	4.46410	0.315660
$201$	0	0
$202$	5.46410	0.384453
$203$	−8.00000	−0.561490
$204$	0	0
$205$	1.46410	0.102257
$206$	−16.0000	−1.11477
$207$	0	0
$208$	5.46410	0.378867
$209$	6.92820	0.479234
$210$	0	0
$211$	−27.7128	−1.90783	−0.953914	−	0.300079i	$-0.902987\pi$
$211$	−27.7128	−1.90783	−0.953914	+	0.300079i	$0.902987\pi$
$212$	3.46410	0.237915
$213$	0	0
$214$	−6.92820	−0.473602
$215$	0	0
$216$	0	0
$217$	−10.1962	−0.692160
$218$	−4.92820	−0.333780
$219$	0	0
$220$	−2.53590	−0.170970
$221$	−14.9282	−1.00418
$222$	0	0
$223$	−24.7321	−1.65618	−0.828090	−	0.560595i	$-0.810573\pi$
$223$	−24.7321	−1.65618	−0.828090	+	0.560595i	$0.810573\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	12.5359	0.833876
$227$	−12.9282	−0.858075	−0.429037	−	0.903287i	$-0.641147\pi$
$227$	−12.9282	−0.858075	−0.429037	+	0.903287i	$0.641147\pi$
$228$	0	0
$229$	−10.1962	−0.673781	−0.336890	−	0.941544i	$-0.609375\pi$
$229$	−10.1962	−0.673781	−0.336890	+	0.941544i	$0.609375\pi$
$230$	0.732051	0.0482700
$231$	0	0
$232$	8.00000	0.525226
$233$	4.39230	0.287749	0.143875	−	0.989596i	$-0.454044\pi$
$233$	4.39230	0.287749	0.143875	+	0.989596i	$0.454044\pi$
$234$	0	0
$235$	−1.60770	−0.104874
$236$	−5.66025	−0.368451
$237$	0	0
$238$	2.73205	0.177093
$239$	−16.3923	−1.06033	−0.530165	−	0.847894i	$-0.677870\pi$
$239$	−16.3923	−1.06033	−0.530165	+	0.847894i	$0.677870\pi$
$240$	0	0
$241$	−3.12436	−0.201257	−0.100629	−	0.994924i	$-0.532085\pi$
$241$	−3.12436	−0.201257	−0.100629	+	0.994924i	$0.532085\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−2.19615	−0.140594
$245$	0.732051	0.0467690
$246$	0	0
$247$	−10.9282	−0.695345
$248$	10.1962	0.647456
$249$	0	0
$250$	6.92820	0.438178
$251$	−19.8564	−1.25333	−0.626663	−	0.779291i	$-0.715580\pi$
$251$	−19.8564	−1.25333	−0.626663	+	0.779291i	$0.715580\pi$
$252$	0	0
$253$	3.46410	0.217786
$254$	−19.3205	−1.21228
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.85641	−0.240556	−0.120278	−	0.992740i	$-0.538379\pi$
$257$	−3.85641	−0.240556	−0.120278	+	0.992740i	$0.538379\pi$
$258$	0	0
$259$	7.46410	0.463797
$260$	4.00000	0.248069
$261$	0	0
$262$	−6.73205	−0.415907
$263$	−4.92820	−0.303886	−0.151943	−	0.988389i	$-0.548553\pi$
$263$	−4.92820	−0.303886	−0.151943	+	0.988389i	$0.548553\pi$
$264$	0	0
$265$	2.53590	0.155779
$266$	2.00000	0.122628
$267$	0	0
$268$	−11.4641	−0.700281
$269$	25.8564	1.57649	0.788246	−	0.615360i	$-0.210989\pi$
$269$	25.8564	1.57649	0.788246	+	0.615360i	$0.210989\pi$
$270$	0	0
$271$	−7.26795	−0.441496	−0.220748	−	0.975331i	$-0.570850\pi$
$271$	−7.26795	−0.441496	−0.220748	+	0.975331i	$0.570850\pi$
$272$	−2.73205	−0.165655
$273$	0	0
$274$	−17.3205	−1.04637
$275$	15.4641	0.932520
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	12.1962	0.731477
$279$	0	0
$280$	−0.732051	−0.0437484
$281$	12.5359	0.747829	0.373915	−	0.927463i	$-0.378015\pi$
$281$	12.5359	0.747829	0.373915	+	0.927463i	$0.378015\pi$
$282$	0	0
$283$	24.9282	1.48183	0.740914	−	0.671600i	$-0.234393\pi$
$283$	24.9282	1.48183	0.740914	+	0.671600i	$0.234393\pi$
$284$	12.3923	0.735348
$285$	0	0
$286$	18.9282	1.11925
$287$	2.00000	0.118056
$288$	0	0
$289$	−9.53590	−0.560935
$290$	5.85641	0.343900
$291$	0	0
$292$	−0.535898	−0.0313611
$293$	−23.2679	−1.35933	−0.679664	−	0.733524i	$-0.737874\pi$
$293$	−23.2679	−1.35933	−0.679664	+	0.733524i	$0.737874\pi$
$294$	0	0
$295$	−4.14359	−0.241249
$296$	−7.46410	−0.433842
$297$	0	0
$298$	−18.7846	−1.08816
$299$	−5.46410	−0.315997
$300$	0	0
$301$	0	0
$302$	10.5359	0.606273
$303$	0	0
$304$	−2.00000	−0.114708
$305$	−1.60770	−0.0920564
$306$	0	0
$307$	−30.0526	−1.71519	−0.857595	−	0.514325i	$-0.828042\pi$
$307$	−30.0526	−1.71519	−0.857595	+	0.514325i	$0.828042\pi$
$308$	−3.46410	−0.197386
$309$	0	0
$310$	7.46410	0.423932
$311$	24.7321	1.40243	0.701213	−	0.712952i	$-0.252642\pi$
$311$	24.7321	1.40243	0.701213	+	0.712952i	$0.252642\pi$
$312$	0	0
$313$	−19.1244	−1.08097	−0.540486	−	0.841353i	$-0.681760\pi$
$313$	−19.1244	−1.08097	−0.540486	+	0.841353i	$0.681760\pi$
$314$	−2.19615	−0.123936
$315$	0	0
$316$	−12.9282	−0.727268
$317$	2.14359	0.120396	0.0601981	−	0.998186i	$-0.480827\pi$
$317$	2.14359	0.120396	0.0601981	+	0.998186i	$0.480827\pi$
$318$	0	0
$319$	27.7128	1.55162
$320$	0.732051	0.0409229
$321$	0	0
$322$	1.00000	0.0557278
$323$	5.46410	0.304031
$324$	0	0
$325$	−24.3923	−1.35304
$326$	8.39230	0.464807
$327$	0	0
$328$	−2.00000	−0.110432
$329$	−2.19615	−0.121078
$330$	0	0
$331$	20.3923	1.12086	0.560431	−	0.828201i	$-0.310635\pi$
$331$	20.3923	1.12086	0.560431	+	0.828201i	$0.310635\pi$
$332$	−14.3923	−0.789880
$333$	0	0
$334$	−16.0526	−0.878357
$335$	−8.39230	−0.458521
$336$	0	0
$337$	−4.14359	−0.225716	−0.112858	−	0.993611i	$-0.536001\pi$
$337$	−4.14359	−0.225716	−0.112858	+	0.993611i	$0.536001\pi$
$338$	−16.8564	−0.916868
$339$	0	0
$340$	−2.00000	−0.108465
$341$	35.3205	1.91271
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	−6.92820	−0.372463
$347$	−26.5359	−1.42452	−0.712261	−	0.701915i	$-0.752329\pi$
$347$	−26.5359	−1.42452	−0.712261	+	0.701915i	$0.752329\pi$
$348$	0	0
$349$	12.7846	0.684344	0.342172	−	0.939637i	$-0.388837\pi$
$349$	12.7846	0.684344	0.342172	+	0.939637i	$0.388837\pi$
$350$	4.46410	0.238616
$351$	0	0
$352$	3.46410	0.184637
$353$	−5.60770	−0.298467	−0.149234	−	0.988802i	$-0.547681\pi$
$353$	−5.60770	−0.298467	−0.149234	+	0.988802i	$0.547681\pi$
$354$	0	0
$355$	9.07180	0.481481
$356$	−3.80385	−0.201604
$357$	0	0
$358$	13.8564	0.732334
$359$	−12.9282	−0.682324	−0.341162	−	0.940004i	$-0.610821\pi$
$359$	−12.9282	−0.682324	−0.341162	+	0.940004i	$0.610821\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	24.0526	1.26417
$363$	0	0
$364$	5.46410	0.286397
$365$	−0.392305	−0.0205342
$366$	0	0
$367$	22.2487	1.16137	0.580687	−	0.814127i	$-0.302784\pi$
$367$	22.2487	1.16137	0.580687	+	0.814127i	$0.302784\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−5.46410	−0.284065
$371$	3.46410	0.179847
$372$	0	0
$373$	−4.92820	−0.255173	−0.127586	−	0.991827i	$-0.540723\pi$
$373$	−4.92820	−0.255173	−0.127586	+	0.991827i	$0.540723\pi$
$374$	−9.46410	−0.489377
$375$	0	0
$376$	2.19615	0.113258
$377$	−43.7128	−2.25132
$378$	0	0
$379$	−7.46410	−0.383405	−0.191703	−	0.981453i	$-0.561401\pi$
$379$	−7.46410	−0.383405	−0.191703	+	0.981453i	$0.561401\pi$
$380$	−1.46410	−0.0751068
$381$	0	0
$382$	16.0000	0.818631
$383$	25.4641	1.30115	0.650577	−	0.759440i	$-0.274527\pi$
$383$	25.4641	1.30115	0.650577	+	0.759440i	$0.274527\pi$
$384$	0	0
$385$	−2.53590	−0.129241
$386$	8.92820	0.454434
$387$	0	0
$388$	−5.26795	−0.267440
$389$	−20.9282	−1.06110	−0.530551	−	0.847653i	$-0.678015\pi$
$389$	−20.9282	−1.06110	−0.530551	+	0.847653i	$0.678015\pi$
$390$	0	0
$391$	2.73205	0.138166
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−4.92820	−0.248279
$395$	−9.46410	−0.476191
$396$	0	0
$397$	3.32051	0.166652	0.0833258	−	0.996522i	$-0.473446\pi$
$397$	3.32051	0.166652	0.0833258	+	0.996522i	$0.473446\pi$
$398$	−16.7846	−0.841336
$399$	0	0
$400$	−4.46410	−0.223205
$401$	−25.7128	−1.28404	−0.642018	−	0.766689i	$-0.721903\pi$
$401$	−25.7128	−1.28404	−0.642018	+	0.766689i	$0.721903\pi$
$402$	0	0
$403$	−55.7128	−2.77525
$404$	−5.46410	−0.271849
$405$	0	0
$406$	8.00000	0.397033
$407$	−25.8564	−1.28165
$408$	0	0
$409$	0.535898	0.0264985	0.0132492	−	0.999912i	$-0.495783\pi$
$409$	0.535898	0.0264985	0.0132492	+	0.999912i	$0.495783\pi$
$410$	−1.46410	−0.0723068
$411$	0	0
$412$	16.0000	0.788263
$413$	−5.66025	−0.278523
$414$	0	0
$415$	−10.5359	−0.517187
$416$	−5.46410	−0.267900
$417$	0	0
$418$	−6.92820	−0.338869
$419$	24.9282	1.21782	0.608911	−	0.793238i	$-0.291607\pi$
$419$	24.9282	1.21782	0.608911	+	0.793238i	$0.291607\pi$
$420$	0	0
$421$	39.4641	1.92336	0.961681	−	0.274170i	$-0.0884030\pi$
$421$	39.4641	1.92336	0.961681	+	0.274170i	$0.0884030\pi$
$422$	27.7128	1.34904
$423$	0	0
$424$	−3.46410	−0.168232
$425$	12.1962	0.591600
$426$	0	0
$427$	−2.19615	−0.106279
$428$	6.92820	0.334887
$429$	0	0
$430$	0	0
$431$	27.7128	1.33488	0.667440	−	0.744664i	$-0.267390\pi$
$431$	27.7128	1.33488	0.667440	+	0.744664i	$0.267390\pi$
$432$	0	0
$433$	32.5885	1.56610	0.783051	−	0.621958i	$-0.213663\pi$
$433$	32.5885	1.56610	0.783051	+	0.621958i	$0.213663\pi$
$434$	10.1962	0.489431
$435$	0	0
$436$	4.92820	0.236018
$437$	2.00000	0.0956730
$438$	0	0
$439$	15.2679	0.728699	0.364350	−	0.931262i	$-0.381291\pi$
$439$	15.2679	0.728699	0.364350	+	0.931262i	$0.381291\pi$
$440$	2.53590	0.120894
$441$	0	0
$442$	14.9282	0.710062
$443$	−21.0718	−1.00115	−0.500576	−	0.865693i	$-0.666878\pi$
$443$	−21.0718	−1.00115	−0.500576	+	0.865693i	$0.666878\pi$
$444$	0	0
$445$	−2.78461	−0.132003
$446$	24.7321	1.17110
$447$	0	0
$448$	1.00000	0.0472456
$449$	−11.3205	−0.534248	−0.267124	−	0.963662i	$-0.586073\pi$
$449$	−11.3205	−0.534248	−0.267124	+	0.963662i	$0.586073\pi$
$450$	0	0
$451$	−6.92820	−0.326236
$452$	−12.5359	−0.589639
$453$	0	0
$454$	12.9282	0.606751
$455$	4.00000	0.187523
$456$	0	0
$457$	14.3923	0.673244	0.336622	−	0.941640i	$-0.390716\pi$
$457$	14.3923	0.673244	0.336622	+	0.941640i	$0.390716\pi$
$458$	10.1962	0.476435
$459$	0	0
$460$	−0.732051	−0.0341320
$461$	19.7128	0.918117	0.459059	−	0.888406i	$-0.348187\pi$
$461$	19.7128	0.918117	0.459059	+	0.888406i	$0.348187\pi$
$462$	0	0
$463$	−2.92820	−0.136085	−0.0680426	−	0.997682i	$-0.521675\pi$
$463$	−2.92820	−0.136085	−0.0680426	+	0.997682i	$0.521675\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	−4.39230	−0.203470
$467$	3.85641	0.178453	0.0892266	−	0.996011i	$-0.471560\pi$
$467$	3.85641	0.178453	0.0892266	+	0.996011i	$0.471560\pi$
$468$	0	0
$469$	−11.4641	−0.529363
$470$	1.60770	0.0741574
$471$	0	0
$472$	5.66025	0.260534
$473$	0	0
$474$	0	0
$475$	8.92820	0.409654
$476$	−2.73205	−0.125223
$477$	0	0
$478$	16.3923	0.749767
$479$	−22.9282	−1.04762	−0.523808	−	0.851836i	$-0.675489\pi$
$479$	−22.9282	−1.04762	−0.523808	+	0.851836i	$0.675489\pi$
$480$	0	0
$481$	40.7846	1.85962
$482$	3.12436	0.142311
$483$	0	0
$484$	1.00000	0.0454545
$485$	−3.85641	−0.175110
$486$	0	0
$487$	−32.7846	−1.48561	−0.742806	−	0.669506i	$-0.766506\pi$
$487$	−32.7846	−1.48561	−0.742806	+	0.669506i	$0.766506\pi$
$488$	2.19615	0.0994151
$489$	0	0
$490$	−0.732051	−0.0330707
$491$	28.7846	1.29903	0.649516	−	0.760348i	$-0.274972\pi$
$491$	28.7846	1.29903	0.649516	+	0.760348i	$0.274972\pi$
$492$	0	0
$493$	21.8564	0.984363
$494$	10.9282	0.491683
$495$	0	0
$496$	−10.1962	−0.457821
$497$	12.3923	0.555871
$498$	0	0
$499$	25.8564	1.15749	0.578746	−	0.815508i	$-0.303542\pi$
$499$	25.8564	1.15749	0.578746	+	0.815508i	$0.303542\pi$
$500$	−6.92820	−0.309839
$501$	0	0
$502$	19.8564	0.886235
$503$	26.6410	1.18786	0.593932	−	0.804515i	$-0.297575\pi$
$503$	26.6410	1.18786	0.593932	+	0.804515i	$0.297575\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	−3.46410	−0.153998
$507$	0	0
$508$	19.3205	0.857209
$509$	40.0000	1.77297	0.886484	−	0.462758i	$-0.153140\pi$
$509$	40.0000	1.77297	0.886484	+	0.462758i	$0.153140\pi$
$510$	0	0
$511$	−0.535898	−0.0237067
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	3.85641	0.170099
$515$	11.7128	0.516128
$516$	0	0
$517$	7.60770	0.334586
$518$	−7.46410	−0.327954
$519$	0	0
$520$	−4.00000	−0.175412
$521$	−5.66025	−0.247980	−0.123990	−	0.992283i	$-0.539569\pi$
$521$	−5.66025	−0.247980	−0.123990	+	0.992283i	$0.539569\pi$
$522$	0	0
$523$	0.928203	0.0405875	0.0202937	−	0.999794i	$-0.493540\pi$
$523$	0.928203	0.0405875	0.0202937	+	0.999794i	$0.493540\pi$
$524$	6.73205	0.294091
$525$	0	0
$526$	4.92820	0.214880
$527$	27.8564	1.21344
$528$	0	0
$529$	1.00000	0.0434783
$530$	−2.53590	−0.110152
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	10.9282	0.473353
$534$	0	0
$535$	5.07180	0.219273
$536$	11.4641	0.495174
$537$	0	0
$538$	−25.8564	−1.11475
$539$	−3.46410	−0.149209
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	7.26795	0.312185
$543$	0	0
$544$	2.73205	0.117136
$545$	3.60770	0.154537
$546$	0	0
$547$	23.6077	1.00939	0.504696	−	0.863297i	$-0.331605\pi$
$547$	23.6077	1.00939	0.504696	+	0.863297i	$0.331605\pi$
$548$	17.3205	0.739895
$549$	0	0
$550$	−15.4641	−0.659392
$551$	16.0000	0.681623
$552$	0	0
$553$	−12.9282	−0.549763
$554$	8.00000	0.339887
$555$	0	0
$556$	−12.1962	−0.517232
$557$	4.14359	0.175570	0.0877848	−	0.996139i	$-0.472021\pi$
$557$	4.14359	0.175570	0.0877848	+	0.996139i	$0.472021\pi$
$558$	0	0
$559$	0	0
$560$	0.732051	0.0309348
$561$	0	0
$562$	−12.5359	−0.528795
$563$	−33.7128	−1.42083	−0.710413	−	0.703785i	$-0.751492\pi$
$563$	−33.7128	−1.42083	−0.710413	+	0.703785i	$0.751492\pi$
$564$	0	0
$565$	−9.17691	−0.386076
$566$	−24.9282	−1.04781
$567$	0	0
$568$	−12.3923	−0.519970
$569$	−7.85641	−0.329358	−0.164679	−	0.986347i	$-0.552659\pi$
$569$	−7.85641	−0.329358	−0.164679	+	0.986347i	$0.552659\pi$
$570$	0	0
$571$	26.9282	1.12691	0.563455	−	0.826147i	$-0.309472\pi$
$571$	26.9282	1.12691	0.563455	+	0.826147i	$0.309472\pi$
$572$	−18.9282	−0.791428
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	4.46410	0.186166
$576$	0	0
$577$	−17.3205	−0.721062	−0.360531	−	0.932747i	$-0.617405\pi$
$577$	−17.3205	−0.721062	−0.360531	+	0.932747i	$0.617405\pi$
$578$	9.53590	0.396641
$579$	0	0
$580$	−5.85641	−0.243174
$581$	−14.3923	−0.597093
$582$	0	0
$583$	−12.0000	−0.496989
$584$	0.535898	0.0221756
$585$	0	0
$586$	23.2679	0.961190
$587$	−22.7321	−0.938252	−0.469126	−	0.883131i	$-0.655431\pi$
$587$	−22.7321	−0.938252	−0.469126	+	0.883131i	$0.655431\pi$
$588$	0	0
$589$	20.3923	0.840250
$590$	4.14359	0.170589
$591$	0	0
$592$	7.46410	0.306773
$593$	−14.7846	−0.607131	−0.303566	−	0.952811i	$-0.598177\pi$
$593$	−14.7846	−0.607131	−0.303566	+	0.952811i	$0.598177\pi$
$594$	0	0
$595$	−2.00000	−0.0819920
$596$	18.7846	0.769448
$597$	0	0
$598$	5.46410	0.223444
$599$	−20.7846	−0.849236	−0.424618	−	0.905373i	$-0.639592\pi$
$599$	−20.7846	−0.849236	−0.424618	+	0.905373i	$0.639592\pi$
$600$	0	0
$601$	−0.535898	−0.0218598	−0.0109299	−	0.999940i	$-0.503479\pi$
$601$	−0.535898	−0.0218598	−0.0109299	+	0.999940i	$0.503479\pi$
$602$	0	0
$603$	0	0
$604$	−10.5359	−0.428700
$605$	0.732051	0.0297621
$606$	0	0
$607$	−9.80385	−0.397926	−0.198963	−	0.980007i	$-0.563757\pi$
$607$	−9.80385	−0.397926	−0.198963	+	0.980007i	$0.563757\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	1.60770	0.0650937
$611$	−12.0000	−0.485468
$612$	0	0
$613$	−38.7846	−1.56650	−0.783248	−	0.621710i	$-0.786438\pi$
$613$	−38.7846	−1.56650	−0.783248	+	0.621710i	$0.786438\pi$
$614$	30.0526	1.21282
$615$	0	0
$616$	3.46410	0.139573
$617$	−21.7128	−0.874125	−0.437062	−	0.899431i	$-0.643981\pi$
$617$	−21.7128	−0.874125	−0.437062	+	0.899431i	$0.643981\pi$
$618$	0	0
$619$	32.2487	1.29619	0.648093	−	0.761562i	$-0.275567\pi$
$619$	32.2487	1.29619	0.648093	+	0.761562i	$0.275567\pi$
$620$	−7.46410	−0.299766
$621$	0	0
$622$	−24.7321	−0.991665
$623$	−3.80385	−0.152398
$624$	0	0
$625$	17.2487	0.689948
$626$	19.1244	0.764363
$627$	0	0
$628$	2.19615	0.0876360
$629$	−20.3923	−0.813094
$630$	0	0
$631$	30.7846	1.22552	0.612758	−	0.790271i	$-0.290060\pi$
$631$	30.7846	1.22552	0.612758	+	0.790271i	$0.290060\pi$
$632$	12.9282	0.514256
$633$	0	0
$634$	−2.14359	−0.0851330
$635$	14.1436	0.561271
$636$	0	0
$637$	5.46410	0.216496
$638$	−27.7128	−1.09716
$639$	0	0
$640$	−0.732051	−0.0289368
$641$	−45.7128	−1.80555	−0.902774	−	0.430116i	$-0.858473\pi$
$641$	−45.7128	−1.80555	−0.902774	+	0.430116i	$0.858473\pi$
$642$	0	0
$643$	42.7846	1.68726	0.843630	−	0.536925i	$-0.180414\pi$
$643$	42.7846	1.68726	0.843630	+	0.536925i	$0.180414\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	−5.46410	−0.214982
$647$	−9.51666	−0.374139	−0.187069	−	0.982347i	$-0.559899\pi$
$647$	−9.51666	−0.374139	−0.187069	+	0.982347i	$0.559899\pi$
$648$	0	0
$649$	19.6077	0.769669
$650$	24.3923	0.956745
$651$	0	0
$652$	−8.39230	−0.328668
$653$	25.7128	1.00622	0.503110	−	0.864222i	$-0.332189\pi$
$653$	25.7128	1.00622	0.503110	+	0.864222i	$0.332189\pi$
$654$	0	0
$655$	4.92820	0.192561
$656$	2.00000	0.0780869
$657$	0	0
$658$	2.19615	0.0856149
$659$	−19.7128	−0.767902	−0.383951	−	0.923353i	$-0.625437\pi$
$659$	−19.7128	−0.767902	−0.383951	+	0.923353i	$0.625437\pi$
$660$	0	0
$661$	28.4449	1.10638	0.553188	−	0.833056i	$-0.313411\pi$
$661$	28.4449	1.10638	0.553188	+	0.833056i	$0.313411\pi$
$662$	−20.3923	−0.792569
$663$	0	0
$664$	14.3923	0.558530
$665$	−1.46410	−0.0567754
$666$	0	0
$667$	8.00000	0.309761
$668$	16.0526	0.621092
$669$	0	0
$670$	8.39230	0.324223
$671$	7.60770	0.293692
$672$	0	0
$673$	−31.3205	−1.20732	−0.603658	−	0.797243i	$-0.706291\pi$
$673$	−31.3205	−1.20732	−0.603658	+	0.797243i	$0.706291\pi$
$674$	4.14359	0.159605
$675$	0	0
$676$	16.8564	0.648323
$677$	46.1962	1.77546	0.887731	−	0.460362i	$-0.152280\pi$
$677$	46.1962	1.77546	0.887731	+	0.460362i	$0.152280\pi$
$678$	0	0
$679$	−5.26795	−0.202165
$680$	2.00000	0.0766965
$681$	0	0
$682$	−35.3205	−1.35249
$683$	2.92820	0.112045	0.0560223	−	0.998430i	$-0.482158\pi$
$683$	2.92820	0.112045	0.0560223	+	0.998430i	$0.482158\pi$
$684$	0	0
$685$	12.6795	0.484458
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	0	0
$689$	18.9282	0.721107
$690$	0	0
$691$	−5.26795	−0.200402	−0.100201	−	0.994967i	$-0.531949\pi$
$691$	−5.26795	−0.200402	−0.100201	+	0.994967i	$0.531949\pi$
$692$	6.92820	0.263371
$693$	0	0
$694$	26.5359	1.00729
$695$	−8.92820	−0.338666
$696$	0	0
$697$	−5.46410	−0.206968
$698$	−12.7846	−0.483905
$699$	0	0
$700$	−4.46410	−0.168727
$701$	52.2487	1.97341	0.986703	−	0.162532i	$-0.0519660\pi$
$701$	52.2487	1.97341	0.986703	+	0.162532i	$0.0519660\pi$
$702$	0	0
$703$	−14.9282	−0.563028
$704$	−3.46410	−0.130558
$705$	0	0
$706$	5.60770	0.211048
$707$	−5.46410	−0.205499
$708$	0	0
$709$	−7.46410	−0.280320	−0.140160	−	0.990129i	$-0.544762\pi$
$709$	−7.46410	−0.280320	−0.140160	+	0.990129i	$0.544762\pi$
$710$	−9.07180	−0.340458
$711$	0	0
$712$	3.80385	0.142555
$713$	10.1962	0.381849
$714$	0	0
$715$	−13.8564	−0.518200
$716$	−13.8564	−0.517838
$717$	0	0
$718$	12.9282	0.482476
$719$	26.1962	0.976952	0.488476	−	0.872577i	$-0.337553\pi$
$719$	26.1962	0.976952	0.488476	+	0.872577i	$0.337553\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	15.0000	0.558242
$723$	0	0
$724$	−24.0526	−0.893906
$725$	35.7128	1.32634
$726$	0	0
$727$	−46.9282	−1.74047	−0.870235	−	0.492636i	$-0.836033\pi$
$727$	−46.9282	−1.74047	−0.870235	+	0.492636i	$0.836033\pi$
$728$	−5.46410	−0.202513
$729$	0	0
$730$	0.392305	0.0145199
$731$	0	0
$732$	0	0
$733$	−3.66025	−0.135195	−0.0675973	−	0.997713i	$-0.521533\pi$
$733$	−3.66025	−0.135195	−0.0675973	+	0.997713i	$0.521533\pi$
$734$	−22.2487	−0.821215
$735$	0	0
$736$	1.00000	0.0368605
$737$	39.7128	1.46284
$738$	0	0
$739$	15.3205	0.563574	0.281787	−	0.959477i	$-0.409073\pi$
$739$	15.3205	0.563574	0.281787	+	0.959477i	$0.409073\pi$
$740$	5.46410	0.200864
$741$	0	0
$742$	−3.46410	−0.127171
$743$	−0.928203	−0.0340525	−0.0170262	−	0.999855i	$-0.505420\pi$
$743$	−0.928203	−0.0340525	−0.0170262	+	0.999855i	$0.505420\pi$
$744$	0	0
$745$	13.7513	0.503808
$746$	4.92820	0.180434
$747$	0	0
$748$	9.46410	0.346042
$749$	6.92820	0.253151
$750$	0	0
$751$	−20.6410	−0.753201	−0.376601	−	0.926376i	$-0.622907\pi$
$751$	−20.6410	−0.753201	−0.376601	+	0.926376i	$0.622907\pi$
$752$	−2.19615	−0.0800854
$753$	0	0
$754$	43.7128	1.59193
$755$	−7.71281	−0.280698
$756$	0	0
$757$	46.7846	1.70042	0.850208	−	0.526447i	$-0.176476\pi$
$757$	46.7846	1.70042	0.850208	+	0.526447i	$0.176476\pi$
$758$	7.46410	0.271108
$759$	0	0
$760$	1.46410	0.0531085
$761$	6.67949	0.242131	0.121066	−	0.992644i	$-0.461369\pi$
$761$	6.67949	0.242131	0.121066	+	0.992644i	$0.461369\pi$
$762$	0	0
$763$	4.92820	0.178413
$764$	−16.0000	−0.578860
$765$	0	0
$766$	−25.4641	−0.920055
$767$	−30.9282	−1.11675
$768$	0	0
$769$	−3.12436	−0.112667	−0.0563336	−	0.998412i	$-0.517941\pi$
$769$	−3.12436	−0.112667	−0.0563336	+	0.998412i	$0.517941\pi$
$770$	2.53590	0.0913874
$771$	0	0
$772$	−8.92820	−0.321333
$773$	7.66025	0.275520	0.137760	−	0.990466i	$-0.456010\pi$
$773$	7.66025	0.275520	0.137760	+	0.990466i	$0.456010\pi$
$774$	0	0
$775$	45.5167	1.63501
$776$	5.26795	0.189108
$777$	0	0
$778$	20.9282	0.750312
$779$	−4.00000	−0.143315
$780$	0	0
$781$	−42.9282	−1.53609
$782$	−2.73205	−0.0976979
$783$	0	0
$784$	1.00000	0.0357143
$785$	1.60770	0.0573811
$786$	0	0
$787$	−32.2487	−1.14954	−0.574771	−	0.818314i	$-0.694909\pi$
$787$	−32.2487	−1.14954	−0.574771	+	0.818314i	$0.694909\pi$
$788$	4.92820	0.175560
$789$	0	0
$790$	9.46410	0.336718
$791$	−12.5359	−0.445725
$792$	0	0
$793$	−12.0000	−0.426132
$794$	−3.32051	−0.117840
$795$	0	0
$796$	16.7846	0.594915
$797$	43.7654	1.55025	0.775125	−	0.631809i	$-0.217687\pi$
$797$	43.7654	1.55025	0.775125	+	0.631809i	$0.217687\pi$
$798$	0	0
$799$	6.00000	0.212265
$800$	4.46410	0.157830
$801$	0	0
$802$	25.7128	0.907951
$803$	1.85641	0.0655112
$804$	0	0
$805$	−0.732051	−0.0258014
$806$	55.7128	1.96240
$807$	0	0
$808$	5.46410	0.192226
$809$	−10.7846	−0.379167	−0.189583	−	0.981865i	$-0.560714\pi$
$809$	−10.7846	−0.379167	−0.189583	+	0.981865i	$0.560714\pi$
$810$	0	0
$811$	15.9090	0.558639	0.279320	−	0.960198i	$-0.409891\pi$
$811$	15.9090	0.558639	0.279320	+	0.960198i	$0.409891\pi$
$812$	−8.00000	−0.280745
$813$	0	0
$814$	25.8564	0.906267
$815$	−6.14359	−0.215201
$816$	0	0
$817$	0	0
$818$	−0.535898	−0.0187372
$819$	0	0
$820$	1.46410	0.0511286
$821$	55.7128	1.94439	0.972195	−	0.234172i	$-0.0752377\pi$
$821$	55.7128	1.94439	0.972195	+	0.234172i	$0.0752377\pi$
$822$	0	0
$823$	18.5359	0.646121	0.323060	−	0.946378i	$-0.395288\pi$
$823$	18.5359	0.646121	0.323060	+	0.946378i	$0.395288\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	5.66025	0.196945
$827$	17.8564	0.620928	0.310464	−	0.950585i	$-0.399516\pi$
$827$	17.8564	0.620928	0.310464	+	0.950585i	$0.399516\pi$
$828$	0	0
$829$	−1.07180	−0.0372250	−0.0186125	−	0.999827i	$-0.505925\pi$
$829$	−1.07180	−0.0372250	−0.0186125	+	0.999827i	$0.505925\pi$
$830$	10.5359	0.365706
$831$	0	0
$832$	5.46410	0.189434
$833$	−2.73205	−0.0946600
$834$	0	0
$835$	11.7513	0.406670
$836$	6.92820	0.239617
$837$	0	0
$838$	−24.9282	−0.861130
$839$	19.3205	0.667018	0.333509	−	0.942747i	$-0.391767\pi$
$839$	19.3205	0.667018	0.333509	+	0.942747i	$0.391767\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−39.4641	−1.36002
$843$	0	0
$844$	−27.7128	−0.953914
$845$	12.3397	0.424500
$846$	0	0
$847$	1.00000	0.0343604
$848$	3.46410	0.118958
$849$	0	0
$850$	−12.1962	−0.418325
$851$	−7.46410	−0.255866
$852$	0	0
$853$	9.46410	0.324045	0.162022	−	0.986787i	$-0.448198\pi$
$853$	9.46410	0.324045	0.162022	+	0.986787i	$0.448198\pi$
$854$	2.19615	0.0751508
$855$	0	0
$856$	−6.92820	−0.236801
$857$	−39.0718	−1.33467	−0.667334	−	0.744759i	$-0.732564\pi$
$857$	−39.0718	−1.33467	−0.667334	+	0.744759i	$0.732564\pi$
$858$	0	0
$859$	−23.1244	−0.788993	−0.394496	−	0.918897i	$-0.629081\pi$
$859$	−23.1244	−0.788993	−0.394496	+	0.918897i	$0.629081\pi$
$860$	0	0
$861$	0	0
$862$	−27.7128	−0.943902
$863$	−14.5359	−0.494808	−0.247404	−	0.968912i	$-0.579577\pi$
$863$	−14.5359	−0.494808	−0.247404	+	0.968912i	$0.579577\pi$
$864$	0	0
$865$	5.07180	0.172446
$866$	−32.5885	−1.10740
$867$	0	0
$868$	−10.1962	−0.346080
$869$	44.7846	1.51921
$870$	0	0
$871$	−62.6410	−2.12251
$872$	−4.92820	−0.166890
$873$	0	0
$874$	−2.00000	−0.0676510
$875$	−6.92820	−0.234216
$876$	0	0
$877$	37.0718	1.25183	0.625913	−	0.779893i	$-0.284727\pi$
$877$	37.0718	1.25183	0.625913	+	0.779893i	$0.284727\pi$
$878$	−15.2679	−0.515268
$879$	0	0
$880$	−2.53590	−0.0854851
$881$	53.6603	1.80786	0.903930	−	0.427681i	$-0.140669\pi$
$881$	53.6603	1.80786	0.903930	+	0.427681i	$0.140669\pi$
$882$	0	0
$883$	−40.0000	−1.34611	−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000	−1.34611	−0.673054	+	0.739594i	$0.735018\pi$
$884$	−14.9282	−0.502090
$885$	0	0
$886$	21.0718	0.707921
$887$	13.5167	0.453845	0.226923	−	0.973913i	$-0.427134\pi$
$887$	13.5167	0.453845	0.226923	+	0.973913i	$0.427134\pi$
$888$	0	0
$889$	19.3205	0.647989
$890$	2.78461	0.0933403
$891$	0	0
$892$	−24.7321	−0.828090
$893$	4.39230	0.146983
$894$	0	0
$895$	−10.1436	−0.339063
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	11.3205	0.377770
$899$	81.5692	2.72049
$900$	0	0
$901$	−9.46410	−0.315295
$902$	6.92820	0.230684
$903$	0	0
$904$	12.5359	0.416938
$905$	−17.6077	−0.585300
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−12.9282	−0.429037
$909$	0	0
$910$	−4.00000	−0.132599
$911$	7.07180	0.234299	0.117150	−	0.993114i	$-0.462624\pi$
$911$	7.07180	0.234299	0.117150	+	0.993114i	$0.462624\pi$
$912$	0	0
$913$	49.8564	1.65001
$914$	−14.3923	−0.476055
$915$	0	0
$916$	−10.1962	−0.336890
$917$	6.73205	0.222312
$918$	0	0
$919$	−35.8564	−1.18279	−0.591397	−	0.806381i	$-0.701423\pi$
$919$	−35.8564	−1.18279	−0.591397	+	0.806381i	$0.701423\pi$
$920$	0.732051	0.0241350
$921$	0	0
$922$	−19.7128	−0.649207
$923$	67.7128	2.22879
$924$	0	0
$925$	−33.3205	−1.09557
$926$	2.92820	0.0962267
$927$	0	0
$928$	8.00000	0.262613
$929$	−15.8564	−0.520232	−0.260116	−	0.965577i	$-0.583761\pi$
$929$	−15.8564	−0.520232	−0.260116	+	0.965577i	$0.583761\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	4.39230	0.143875
$933$	0	0
$934$	−3.85641	−0.126185
$935$	6.92820	0.226576
$936$	0	0
$937$	48.9808	1.60013	0.800066	−	0.599912i	$-0.204798\pi$
$937$	48.9808	1.60013	0.800066	+	0.599912i	$0.204798\pi$
$938$	11.4641	0.374316
$939$	0	0
$940$	−1.60770	−0.0524372
$941$	−27.3731	−0.892336	−0.446168	−	0.894949i	$-0.647212\pi$
$941$	−27.3731	−0.892336	−0.446168	+	0.894949i	$0.647212\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	−5.66025	−0.184226
$945$	0	0
$946$	0	0
$947$	−47.3205	−1.53771	−0.768855	−	0.639423i	$-0.779173\pi$
$947$	−47.3205	−1.53771	−0.768855	+	0.639423i	$0.779173\pi$
$948$	0	0
$949$	−2.92820	−0.0950535
$950$	−8.92820	−0.289669
$951$	0	0
$952$	2.73205	0.0885463
$953$	31.0718	1.00651	0.503257	−	0.864137i	$-0.332135\pi$
$953$	31.0718	1.00651	0.503257	+	0.864137i	$0.332135\pi$
$954$	0	0
$955$	−11.7128	−0.379018
$956$	−16.3923	−0.530165
$957$	0	0
$958$	22.9282	0.740777
$959$	17.3205	0.559308
$960$	0	0
$961$	72.9615	2.35360
$962$	−40.7846	−1.31495
$963$	0	0
$964$	−3.12436	−0.100629
$965$	−6.53590	−0.210398
$966$	0	0
$967$	9.85641	0.316961	0.158480	−	0.987362i	$-0.449341\pi$
$967$	9.85641	0.316961	0.158480	+	0.987362i	$0.449341\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	3.85641	0.123822
$971$	−42.1051	−1.35122	−0.675609	−	0.737260i	$-0.736119\pi$
$971$	−42.1051	−1.35122	−0.675609	+	0.737260i	$0.736119\pi$
$972$	0	0
$973$	−12.1962	−0.390991
$974$	32.7846	1.05049
$975$	0	0
$976$	−2.19615	−0.0702971
$977$	20.5359	0.657002	0.328501	−	0.944504i	$-0.393457\pi$
$977$	20.5359	0.657002	0.328501	+	0.944504i	$0.393457\pi$
$978$	0	0
$979$	13.1769	0.421136
$980$	0.732051	0.0233845
$981$	0	0
$982$	−28.7846	−0.918554
$983$	24.3923	0.777994	0.388997	−	0.921239i	$-0.372822\pi$
$983$	24.3923	0.777994	0.388997	+	0.921239i	$0.372822\pi$
$984$	0	0
$985$	3.60770	0.114951
$986$	−21.8564	−0.696050
$987$	0	0
$988$	−10.9282	−0.347672
$989$	0	0
$990$	0	0
$991$	−33.4641	−1.06302	−0.531511	−	0.847051i	$-0.678376\pi$
$991$	−33.4641	−1.06302	−0.531511	+	0.847051i	$0.678376\pi$
$992$	10.1962	0.323728
$993$	0	0
$994$	−12.3923	−0.393060
$995$	12.2872	0.389530
$996$	0	0
$997$	44.7846	1.41834	0.709171	−	0.705036i	$-0.249069\pi$
$997$	44.7846	1.41834	0.709171	+	0.705036i	$0.249069\pi$
$998$	−25.8564	−0.818470
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.w.1.2		2
3.2	odd	2		322.2.a.f.1.1	✓	2
12.11	even	2		2576.2.a.u.1.2		2
15.14	odd	2		8050.2.a.x.1.2		2
21.20	even	2		2254.2.a.o.1.2		2
69.68	even	2		7406.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.a.f.1.1	✓	2	3.2	odd	2
2254.2.a.o.1.2		2	21.20	even	2
2576.2.a.u.1.2		2	12.11	even	2
2898.2.a.w.1.2		2	1.1	even	1	trivial
7406.2.a.s.1.1		2	69.68	even	2
8050.2.a.x.1.2		2	15.14	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:	\( N \)	\(=\)	\( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2898.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.732051 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.732051 q^{5} +1.00000 q^{7} -1.00000 q^{8} -0.732051 q^{10} -3.46410 q^{11} +5.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.73205 q^{17} -2.00000 q^{19} +0.732051 q^{20} +3.46410 q^{22} -1.00000 q^{23} -4.46410 q^{25} -5.46410 q^{26} +1.00000 q^{28} -8.00000 q^{29} -10.1962 q^{31} -1.00000 q^{32} +2.73205 q^{34} +0.732051 q^{35} +7.46410 q^{37} +2.00000 q^{38} -0.732051 q^{40} +2.00000 q^{41} -3.46410 q^{44} +1.00000 q^{46} -2.19615 q^{47} +1.00000 q^{49} +4.46410 q^{50} +5.46410 q^{52} +3.46410 q^{53} -2.53590 q^{55} -1.00000 q^{56} +8.00000 q^{58} -5.66025 q^{59} -2.19615 q^{61} +10.1962 q^{62} +1.00000 q^{64} +4.00000 q^{65} -11.4641 q^{67} -2.73205 q^{68} -0.732051 q^{70} +12.3923 q^{71} -0.535898 q^{73} -7.46410 q^{74} -2.00000 q^{76} -3.46410 q^{77} -12.9282 q^{79} +0.732051 q^{80} -2.00000 q^{82} -14.3923 q^{83} -2.00000 q^{85} +3.46410 q^{88} -3.80385 q^{89} +5.46410 q^{91} -1.00000 q^{92} +2.19615 q^{94} -1.46410 q^{95} -5.26795 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} + 4 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} - 4 q^{19} - 2 q^{20} - 2 q^{23} - 2 q^{25} - 4 q^{26} + 2 q^{28} - 16 q^{29} - 10 q^{31} - 2 q^{32} + 2 q^{34} - 2 q^{35} + 8 q^{37} + 4 q^{38} + 2 q^{40} + 4 q^{41} + 2 q^{46} + 6 q^{47} + 2 q^{49} + 2 q^{50} + 4 q^{52} - 12 q^{55} - 2 q^{56} + 16 q^{58} + 6 q^{59} + 6 q^{61} + 10 q^{62} + 2 q^{64} + 8 q^{65} - 16 q^{67} - 2 q^{68} + 2 q^{70} + 4 q^{71} - 8 q^{73} - 8 q^{74} - 4 q^{76} - 12 q^{79} - 2 q^{80} - 4 q^{82} - 8 q^{83} - 4 q^{85} - 18 q^{89} + 4 q^{91} - 2 q^{92} - 6 q^{94} + 4 q^{95} - 14 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 2 * q^10 + 4 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 - 4 * q^19 - 2 * q^20 - 2 * q^23 - 2 * q^25 - 4 * q^26 + 2 * q^28 - 16 * q^29 - 10 * q^31 - 2 * q^32 + 2 * q^34 - 2 * q^35 + 8 * q^37 + 4 * q^38 + 2 * q^40 + 4 * q^41 + 2 * q^46 + 6 * q^47 + 2 * q^49 + 2 * q^50 + 4 * q^52 - 12 * q^55 - 2 * q^56 + 16 * q^58 + 6 * q^59 + 6 * q^61 + 10 * q^62 + 2 * q^64 + 8 * q^65 - 16 * q^67 - 2 * q^68 + 2 * q^70 + 4 * q^71 - 8 * q^73 - 8 * q^74 - 4 * q^76 - 12 * q^79 - 2 * q^80 - 4 * q^82 - 8 * q^83 - 4 * q^85 - 18 * q^89 + 4 * q^91 - 2 * q^92 - 6 * q^94 + 4 * q^95 - 14 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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