LMFDB - Embedded newform 2898.2.a.bh.1.4

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.271296.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 18x^{2} - 8x + 24 $ x^4 - 18x^2 - 8x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.30859$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +4.30859 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +4.30859 q^{5} +1.00000 q^{7} -1.00000 q^{8} -4.30859 q^{10} +2.30859 q^{11} -1.36594 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.36594 q^{17} +2.94264 q^{19} +4.30859 q^{20} -2.30859 q^{22} -1.00000 q^{23} +13.5639 q^{25} +1.36594 q^{26} +1.00000 q^{28} -2.00000 q^{29} -7.19798 q^{31} -1.00000 q^{32} +1.36594 q^{34} +4.30859 q^{35} -1.52345 q^{37} -2.94264 q^{38} -4.30859 q^{40} +2.00000 q^{41} +9.04047 q^{43} +2.30859 q^{44} +1.00000 q^{46} +11.1980 q^{47} +1.00000 q^{49} -13.5639 q^{50} -1.36594 q^{52} -5.52345 q^{53} +9.94675 q^{55} -1.00000 q^{56} +2.00000 q^{58} -4.61718 q^{59} +12.9258 q^{61} +7.19798 q^{62} +1.00000 q^{64} -5.88529 q^{65} +16.1939 q^{67} -1.36594 q^{68} -4.30859 q^{70} -0.785135 q^{71} +9.83204 q^{73} +1.52345 q^{74} +2.94264 q^{76} +2.30859 q^{77} +13.8320 q^{79} +4.30859 q^{80} -2.00000 q^{82} -15.4533 q^{83} -5.88529 q^{85} -9.04047 q^{86} -2.30859 q^{88} -8.41285 q^{89} -1.36594 q^{91} -1.00000 q^{92} -11.1980 q^{94} +12.6786 q^{95} -17.0300 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^7 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8} - 8 q^{11} - 4 q^{14} + 4 q^{16} + 8 q^{22} - 4 q^{23} + 16 q^{25} + 4 q^{28} - 8 q^{29} + 4 q^{31} - 4 q^{32} + 4 q^{37} + 8 q^{41} + 8 q^{43} - 8 q^{44} + 4 q^{46} + 12 q^{47} + 4 q^{49} - 16 q^{50} - 12 q^{53} + 36 q^{55} - 4 q^{56} + 8 q^{58} + 16 q^{59} - 4 q^{62} + 4 q^{64} + 24 q^{67} + 4 q^{71} + 12 q^{73} - 4 q^{74} - 8 q^{77} + 28 q^{79} - 8 q^{82} - 8 q^{83} - 8 q^{86} + 8 q^{88} - 8 q^{89} - 4 q^{92} - 12 q^{94} + 36 q^{95} - 8 q^{97} - 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^7 - 4 * q^8 - 8 * q^11 - 4 * q^14 + 4 * q^16 + 8 * q^22 - 4 * q^23 + 16 * q^25 + 4 * q^28 - 8 * q^29 + 4 * q^31 - 4 * q^32 + 4 * q^37 + 8 * q^41 + 8 * q^43 - 8 * q^44 + 4 * q^46 + 12 * q^47 + 4 * q^49 - 16 * q^50 - 12 * q^53 + 36 * q^55 - 4 * q^56 + 8 * q^58 + 16 * q^59 - 4 * q^62 + 4 * q^64 + 24 * q^67 + 4 * q^71 + 12 * q^73 - 4 * q^74 - 8 * q^77 + 28 * q^79 - 8 * q^82 - 8 * q^83 - 8 * q^86 + 8 * q^88 - 8 * q^89 - 4 * q^92 - 12 * q^94 + 36 * q^95 - 8 * q^97 - 4 * 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
$5$	4.30859	1.92686	0.963429	−	0.267962i	$-0.0863502\pi$
$5$	4.30859	1.92686	0.963429	+	0.267962i	$0.0863502\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−4.30859	−1.36250
$11$	2.30859	0.696065	0.348033	−	0.937482i	$-0.386850\pi$
$11$	2.30859	0.696065	0.348033	+	0.937482i	$0.386850\pi$
$12$	0	0
$13$	−1.36594	−0.378845	−0.189422	−	0.981896i	$-0.560662\pi$
$13$	−1.36594	−0.378845	−0.189422	+	0.981896i	$0.560662\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.36594	−0.331290	−0.165645	−	0.986185i	$-0.552971\pi$
$17$	−1.36594	−0.331290	−0.165645	+	0.986185i	$0.552971\pi$
$18$	0	0
$19$	2.94264	0.675089	0.337544	−	0.941310i	$-0.390404\pi$
$19$	2.94264	0.675089	0.337544	+	0.941310i	$0.390404\pi$
$20$	4.30859	0.963429
$21$	0	0
$22$	−2.30859	−0.492193
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	13.5639	2.71279
$26$	1.36594	0.267884
$27$	0	0
$28$	1.00000	0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−7.19798	−1.29280	−0.646398	−	0.763000i	$-0.723725\pi$
$31$	−7.19798	−1.29280	−0.646398	+	0.763000i	$0.723725\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.36594	0.234257
$35$	4.30859	0.728284
$36$	0	0
$37$	−1.52345	−0.250454	−0.125227	−	0.992128i	$-0.539966\pi$
$37$	−1.52345	−0.250454	−0.125227	+	0.992128i	$0.539966\pi$
$38$	−2.94264	−0.477360
$39$	0	0
$40$	−4.30859	−0.681248
$41$	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
$43$	9.04047	1.37866	0.689330	−	0.724448i	$-0.257905\pi$
$43$	9.04047	1.37866	0.689330	+	0.724448i	$0.257905\pi$
$44$	2.30859	0.348033
$45$	0	0
$46$	1.00000	0.147442
$47$	11.1980	1.63339	0.816697	−	0.577067i	$-0.195803\pi$
$47$	11.1980	1.63339	0.816697	+	0.577067i	$0.195803\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−13.5639	−1.91823
$51$	0	0
$52$	−1.36594	−0.189422
$53$	−5.52345	−0.758705	−0.379352	−	0.925252i	$-0.623853\pi$
$53$	−5.52345	−0.758705	−0.379352	+	0.925252i	$0.623853\pi$
$54$	0	0
$55$	9.94675	1.34122
$56$	−1.00000	−0.133631
$57$	0	0
$58$	2.00000	0.262613
$59$	−4.61718	−0.601105	−0.300553	−	0.953765i	$-0.597171\pi$
$59$	−4.61718	−0.601105	−0.300553	+	0.953765i	$0.597171\pi$
$60$	0	0
$61$	12.9258	1.65497	0.827487	−	0.561485i	$-0.189770\pi$
$61$	12.9258	1.65497	0.827487	+	0.561485i	$0.189770\pi$
$62$	7.19798	0.914145
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.88529	−0.729980
$66$	0	0
$67$	16.1939	1.97840	0.989198	−	0.146583i	$-0.0468275\pi$
$67$	16.1939	1.97840	0.989198	+	0.146583i	$0.0468275\pi$
$68$	−1.36594	−0.165645
$69$	0	0
$70$	−4.30859	−0.514975
$71$	−0.785135	−0.0931784	−0.0465892	−	0.998914i	$-0.514835\pi$
$71$	−0.785135	−0.0931784	−0.0465892	+	0.998914i	$0.514835\pi$
$72$	0	0
$73$	9.83204	1.15075	0.575377	−	0.817889i	$-0.304855\pi$
$73$	9.83204	1.15075	0.575377	+	0.817889i	$0.304855\pi$
$74$	1.52345	0.177098
$75$	0	0
$76$	2.94264	0.337544
$77$	2.30859	0.263088
$78$	0	0
$79$	13.8320	1.55623	0.778113	−	0.628124i	$-0.216177\pi$
$79$	13.8320	1.55623	0.778113	+	0.628124i	$0.216177\pi$
$80$	4.30859	0.481715
$81$	0	0
$82$	−2.00000	−0.220863
$83$	−15.4533	−1.69622	−0.848111	−	0.529818i	$-0.822260\pi$
$83$	−15.4533	−1.69622	−0.848111	+	0.529818i	$0.822260\pi$
$84$	0	0
$85$	−5.88529	−0.638349
$86$	−9.04047	−0.974859
$87$	0	0
$88$	−2.30859	−0.246096
$89$	−8.41285	−0.891760	−0.445880	−	0.895093i	$-0.647109\pi$
$89$	−8.41285	−0.891760	−0.445880	+	0.895093i	$0.647109\pi$
$90$	0	0
$91$	−1.36594	−0.143190
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−11.1980	−1.15498
$95$	12.6786	1.30080
$96$	0	0
$97$	−17.0300	−1.72914	−0.864568	−	0.502515i	$-0.832408\pi$
$97$	−17.0300	−1.72914	−0.864568	+	0.502515i	$0.832408\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	13.5639	1.35639
$101$	−16.7150	−1.66321	−0.831603	−	0.555371i	$-0.812576\pi$
$101$	−16.7150	−1.66321	−0.831603	+	0.555371i	$0.812576\pi$
$102$	0	0
$103$	1.88529	0.185763	0.0928815	−	0.995677i	$-0.470392\pi$
$103$	1.88529	0.185763	0.0928815	+	0.995677i	$0.470392\pi$
$104$	1.36594	0.133942
$105$	0	0
$106$	5.52345	0.536485
$107$	2.30859	0.223180	0.111590	−	0.993754i	$-0.464406\pi$
$107$	2.30859	0.223180	0.111590	+	0.993754i	$0.464406\pi$
$108$	0	0
$109$	−10.0874	−0.966196	−0.483098	−	0.875566i	$-0.660488\pi$
$109$	−10.0874	−0.966196	−0.483098	+	0.875566i	$0.660488\pi$
$110$	−9.94675	−0.948386
$111$	0	0
$112$	1.00000	0.0944911
$113$	3.16162	0.297420	0.148710	−	0.988881i	$-0.452488\pi$
$113$	3.16162	0.297420	0.148710	+	0.988881i	$0.452488\pi$
$114$	0	0
$115$	−4.30859	−0.401778
$116$	−2.00000	−0.185695
$117$	0	0
$118$	4.61718	0.425045
$119$	−1.36594	−0.125216
$120$	0	0
$121$	−5.67042	−0.515493
$122$	−12.9258	−1.17024
$123$	0	0
$124$	−7.19798	−0.646398
$125$	36.8984	3.30030
$126$	0	0
$127$	1.26811	0.112527	0.0562634	−	0.998416i	$-0.482081\pi$
$127$	1.26811	0.112527	0.0562634	+	0.998416i	$0.482081\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	5.88529	0.516174
$131$	−8.61718	−0.752886	−0.376443	−	0.926440i	$-0.622853\pi$
$131$	−8.61718	−0.752886	−0.376443	+	0.926440i	$0.622853\pi$
$132$	0	0
$133$	2.94264	0.255160
$134$	−16.1939	−1.39894
$135$	0	0
$136$	1.36594	0.117129
$137$	−10.6172	−0.907086	−0.453543	−	0.891234i	$-0.649840\pi$
$137$	−10.6172	−0.907086	−0.453543	+	0.891234i	$0.649840\pi$
$138$	0	0
$139$	−1.88529	−0.159908	−0.0799540	−	0.996799i	$-0.525477\pi$
$139$	−1.88529	−0.159908	−0.0799540	+	0.996799i	$0.525477\pi$
$140$	4.30859	0.364142
$141$	0	0
$142$	0.785135	0.0658871
$143$	−3.15340	−0.263701
$144$	0	0
$145$	−8.61718	−0.715617
$146$	−9.83204	−0.813705
$147$	0	0
$148$	−1.52345	−0.125227
$149$	7.04047	0.576778	0.288389	−	0.957513i	$-0.406880\pi$
$149$	7.04047	0.576778	0.288389	+	0.957513i	$0.406880\pi$
$150$	0	0
$151$	−8.44921	−0.687587	−0.343794	−	0.939045i	$-0.611712\pi$
$151$	−8.44921	−0.687587	−0.343794	+	0.939045i	$0.611712\pi$
$152$	−2.94264	−0.238680
$153$	0	0
$154$	−2.30859	−0.186031
$155$	−31.0131	−2.49104
$156$	0	0
$157$	9.47020	0.755805	0.377902	−	0.925845i	$-0.376646\pi$
$157$	9.47020	0.755805	0.377902	+	0.925845i	$0.376646\pi$
$158$	−13.8320	−1.10042
$159$	0	0
$160$	−4.30859	−0.340624
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−1.21486	−0.0951555	−0.0475778	−	0.998868i	$-0.515150\pi$
$163$	−1.21486	−0.0951555	−0.0475778	+	0.998868i	$0.515150\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	15.4533	1.19941
$167$	15.3045	1.18430	0.592148	−	0.805829i	$-0.298280\pi$
$167$	15.3045	1.18430	0.592148	+	0.805829i	$0.298280\pi$
$168$	0	0
$169$	−11.1342	−0.856477
$170$	5.88529	0.451381
$171$	0	0
$172$	9.04047	0.689330
$173$	19.7619	1.50247	0.751235	−	0.660035i	$-0.229458\pi$
$173$	19.7619	1.50247	0.751235	+	0.660035i	$0.229458\pi$
$174$	0	0
$175$	13.5639	1.02534
$176$	2.30859	0.174016
$177$	0	0
$178$	8.41285	0.630570
$179$	3.15340	0.235696	0.117848	−	0.993032i	$-0.462400\pi$
$179$	3.15340	0.235696	0.117848	+	0.993032i	$0.462400\pi$
$180$	0	0
$181$	6.73832	0.500855	0.250428	−	0.968135i	$-0.419429\pi$
$181$	6.73832	0.500855	0.250428	+	0.968135i	$0.419429\pi$
$182$	1.36594	0.101250
$183$	0	0
$184$	1.00000	0.0737210
$185$	−6.56393	−0.482590
$186$	0	0
$187$	−3.15340	−0.230599
$188$	11.1980	0.816697
$189$	0	0
$190$	−12.6786	−0.919805
$191$	−6.78514	−0.490955	−0.245478	−	0.969402i	$-0.578945\pi$
$191$	−6.78514	−0.490955	−0.245478	+	0.969402i	$0.578945\pi$
$192$	0	0
$193$	1.43607	0.103371	0.0516854	−	0.998663i	$-0.483541\pi$
$193$	1.43607	0.103371	0.0516854	+	0.998663i	$0.483541\pi$
$194$	17.0300	1.22268
$195$	0	0
$196$	1.00000	0.0714286
$197$	−10.9322	−0.778886	−0.389443	−	0.921050i	$-0.627333\pi$
$197$	−10.9322	−0.778886	−0.389443	+	0.921050i	$0.627333\pi$
$198$	0	0
$199$	23.9662	1.69892	0.849461	−	0.527652i	$-0.176927\pi$
$199$	23.9662	1.69892	0.849461	+	0.527652i	$0.176927\pi$
$200$	−13.5639	−0.959114
$201$	0	0
$202$	16.7150	1.17606
$203$	−2.00000	−0.140372
$204$	0	0
$205$	8.61718	0.601850
$206$	−1.88529	−0.131354
$207$	0	0
$208$	−1.36594	−0.0947112
$209$	6.79335	0.469906
$210$	0	0
$211$	7.40231	0.509596	0.254798	−	0.966994i	$-0.417991\pi$
$211$	7.40231	0.509596	0.254798	+	0.966994i	$0.417991\pi$
$212$	−5.52345	−0.379352
$213$	0	0
$214$	−2.30859	−0.157812
$215$	38.9517	2.65648
$216$	0	0
$217$	−7.19798	−0.488631
$218$	10.0874	0.683204
$219$	0	0
$220$	9.94675	0.670610
$221$	1.86580	0.125507
$222$	0	0
$223$	−20.6618	−1.38361	−0.691807	−	0.722083i	$-0.743185\pi$
$223$	−20.6618	−1.38361	−0.691807	+	0.722083i	$0.743185\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−3.16162	−0.210308
$227$	12.1043	0.803388	0.401694	−	0.915774i	$-0.368422\pi$
$227$	12.1043	0.803388	0.401694	+	0.915774i	$0.368422\pi$
$228$	0	0
$229$	−20.9258	−1.38281	−0.691407	−	0.722466i	$-0.743009\pi$
$229$	−20.9258	−1.38281	−0.691407	+	0.722466i	$0.743009\pi$
$230$	4.30859	0.284100
$231$	0	0
$232$	2.00000	0.131306
$233$	12.6704	0.830067	0.415034	−	0.909806i	$-0.363770\pi$
$233$	12.6704	0.830067	0.415034	+	0.909806i	$0.363770\pi$
$234$	0	0
$235$	48.2475	3.14732
$236$	−4.61718	−0.300553
$237$	0	0
$238$	1.36594	0.0885410
$239$	−10.3960	−0.672459	−0.336230	−	0.941780i	$-0.609152\pi$
$239$	−10.3960	−0.672459	−0.336230	+	0.941780i	$0.609152\pi$
$240$	0	0
$241$	−3.79567	−0.244501	−0.122250	−	0.992499i	$-0.539011\pi$
$241$	−3.79567	−0.244501	−0.122250	+	0.992499i	$0.539011\pi$
$242$	5.67042	0.364509
$243$	0	0
$244$	12.9258	0.827487
$245$	4.30859	0.275266
$246$	0	0
$247$	−4.01949	−0.255754
$248$	7.19798	0.457072
$249$	0	0
$250$	−36.8984	−2.33366
$251$	−6.18521	−0.390407	−0.195204	−	0.980763i	$-0.562537\pi$
$251$	−6.18521	−0.390407	−0.195204	+	0.980763i	$0.562537\pi$
$252$	0	0
$253$	−2.30859	−0.145140
$254$	−1.26811	−0.0795685
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.16162	−0.197216	−0.0986081	−	0.995126i	$-0.531439\pi$
$257$	−3.16162	−0.197216	−0.0986081	+	0.995126i	$0.531439\pi$
$258$	0	0
$259$	−1.52345	−0.0946627
$260$	−5.88529	−0.364990
$261$	0	0
$262$	8.61718	0.532371
$263$	−7.66408	−0.472587	−0.236294	−	0.971682i	$-0.575933\pi$
$263$	−7.66408	−0.472587	−0.236294	+	0.971682i	$0.575933\pi$
$264$	0	0
$265$	−23.7983	−1.46192
$266$	−2.94264	−0.180425
$267$	0	0
$268$	16.1939	0.989198
$269$	−9.36594	−0.571052	−0.285526	−	0.958371i	$-0.592168\pi$
$269$	−9.36594	−0.571052	−0.285526	+	0.958371i	$0.592168\pi$
$270$	0	0
$271$	−10.7488	−0.652941	−0.326471	−	0.945207i	$-0.605859\pi$
$271$	−10.7488	−0.652941	−0.326471	+	0.945207i	$0.605859\pi$
$272$	−1.36594	−0.0828225
$273$	0	0
$274$	10.6172	0.641407
$275$	31.3135	1.88828
$276$	0	0
$277$	22.5107	1.35254	0.676268	−	0.736656i	$-0.263596\pi$
$277$	22.5107	1.35254	0.676268	+	0.736656i	$0.263596\pi$
$278$	1.88529	0.113072
$279$	0	0
$280$	−4.30859	−0.257487
$281$	−24.6981	−1.47337	−0.736683	−	0.676238i	$-0.763609\pi$
$281$	−24.6981	−1.47337	−0.736683	+	0.676238i	$0.763609\pi$
$282$	0	0
$283$	22.2917	1.32510	0.662552	−	0.749016i	$-0.269473\pi$
$283$	22.2917	1.32510	0.662552	+	0.749016i	$0.269473\pi$
$284$	−0.785135	−0.0465892
$285$	0	0
$286$	3.15340	0.186464
$287$	2.00000	0.118056
$288$	0	0
$289$	−15.1342	−0.890247
$290$	8.61718	0.506018
$291$	0	0
$292$	9.83204	0.575377
$293$	28.8193	1.68364	0.841820	−	0.539759i	$-0.181485\pi$
$293$	28.8193	1.68364	0.841820	+	0.539759i	$0.181485\pi$
$294$	0	0
$295$	−19.8935	−1.15824
$296$	1.52345	0.0885489
$297$	0	0
$298$	−7.04047	−0.407844
$299$	1.36594	0.0789946
$300$	0	0
$301$	9.04047	0.521084
$302$	8.44921	0.486198
$303$	0	0
$304$	2.94264	0.168772
$305$	55.6918	3.18890
$306$	0	0
$307$	−10.5025	−0.599407	−0.299704	−	0.954032i	$-0.596888\pi$
$307$	−10.5025	−0.599407	−0.299704	+	0.954032i	$0.596888\pi$
$308$	2.30859	0.131544
$309$	0	0
$310$	31.0131	1.76143
$311$	7.05093	0.399821	0.199911	−	0.979814i	$-0.435935\pi$
$311$	7.05093	0.399821	0.199911	+	0.979814i	$0.435935\pi$
$312$	0	0
$313$	14.6003	0.825257	0.412629	−	0.910899i	$-0.364611\pi$
$313$	14.6003	0.825257	0.412629	+	0.910899i	$0.364611\pi$
$314$	−9.47020	−0.534435
$315$	0	0
$316$	13.8320	0.778113
$317$	−20.7319	−1.16442	−0.582209	−	0.813039i	$-0.697812\pi$
$317$	−20.7319	−1.16442	−0.582209	+	0.813039i	$0.697812\pi$
$318$	0	0
$319$	−4.61718	−0.258512
$320$	4.30859	0.240857
$321$	0	0
$322$	1.00000	0.0557278
$323$	−4.01949	−0.223650
$324$	0	0
$325$	−18.5276	−1.02772
$326$	1.21486	0.0672851
$327$	0	0
$328$	−2.00000	−0.110432
$329$	11.1980	0.617365
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−15.4533	−0.848111
$333$	0	0
$334$	−15.3045	−0.837424
$335$	69.7727	3.81209
$336$	0	0
$337$	12.7319	0.693550	0.346775	−	0.937948i	$-0.387277\pi$
$337$	12.7319	0.693550	0.346775	+	0.937948i	$0.387277\pi$
$338$	11.1342	0.605621
$339$	0	0
$340$	−5.88529	−0.319175
$341$	−16.6172	−0.899870
$342$	0	0
$343$	1.00000	0.0539949
$344$	−9.04047	−0.487430
$345$	0	0
$346$	−19.7619	−1.06241
$347$	−6.44921	−0.346212	−0.173106	−	0.984903i	$-0.555380\pi$
$347$	−6.44921	−0.346212	−0.173106	+	0.984903i	$0.555380\pi$
$348$	0	0
$349$	28.3791	1.51910	0.759549	−	0.650450i	$-0.225420\pi$
$349$	28.3791	1.51910	0.759549	+	0.650450i	$0.225420\pi$
$350$	−13.5639	−0.725022
$351$	0	0
$352$	−2.30859	−0.123048
$353$	−3.32958	−0.177215	−0.0886077	−	0.996067i	$-0.528242\pi$
$353$	−3.32958	−0.177215	−0.0886077	+	0.996067i	$0.528242\pi$
$354$	0	0
$355$	−3.38282	−0.179542
$356$	−8.41285	−0.445880
$357$	0	0
$358$	−3.15340	−0.166662
$359$	26.3427	1.39032	0.695158	−	0.718857i	$-0.255335\pi$
$359$	26.3427	1.39032	0.695158	+	0.718857i	$0.255335\pi$
$360$	0	0
$361$	−10.3408	−0.544255
$362$	−6.73832	−0.354158
$363$	0	0
$364$	−1.36594	−0.0715949
$365$	42.3622	2.21734
$366$	0	0
$367$	5.16162	0.269434	0.134717	−	0.990884i	$-0.456987\pi$
$367$	5.16162	0.269434	0.134717	+	0.990884i	$0.456987\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	6.56393	0.341242
$371$	−5.52345	−0.286763
$372$	0	0
$373$	−15.6044	−0.807965	−0.403983	−	0.914767i	$-0.632374\pi$
$373$	−15.6044	−0.807965	−0.403983	+	0.914767i	$0.632374\pi$
$374$	3.15340	0.163058
$375$	0	0
$376$	−11.1980	−0.577492
$377$	2.73189	0.140699
$378$	0	0
$379$	18.5898	0.954896	0.477448	−	0.878660i	$-0.341562\pi$
$379$	18.5898	0.954896	0.477448	+	0.878660i	$0.341562\pi$
$380$	12.6786	0.650400
$381$	0	0
$382$	6.78514	0.347158
$383$	0.335921	0.0171647	0.00858237	−	0.999963i	$-0.497268\pi$
$383$	0.335921	0.0171647	0.00858237	+	0.999963i	$0.497268\pi$
$384$	0	0
$385$	9.94675	0.506933
$386$	−1.43607	−0.0730942
$387$	0	0
$388$	−17.0300	−0.864568
$389$	−32.1151	−1.62830	−0.814150	−	0.580655i	$-0.802796\pi$
$389$	−32.1151	−1.62830	−0.814150	+	0.580655i	$0.802796\pi$
$390$	0	0
$391$	1.36594	0.0690787
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	10.9322	0.550756
$395$	59.5966	2.99863
$396$	0	0
$397$	−22.6341	−1.13597	−0.567985	−	0.823039i	$-0.692277\pi$
$397$	−22.6341	−1.13597	−0.567985	+	0.823039i	$0.692277\pi$
$398$	−23.9662	−1.20132
$399$	0	0
$400$	13.5639	0.678196
$401$	0.187446	0.00936061	0.00468030	−	0.999989i	$-0.498510\pi$
$401$	0.187446	0.00936061	0.00468030	+	0.999989i	$0.498510\pi$
$402$	0	0
$403$	9.83204	0.489769
$404$	−16.7150	−0.831603
$405$	0	0
$406$	2.00000	0.0992583
$407$	−3.51702	−0.174332
$408$	0	0
$409$	−19.1279	−0.945812	−0.472906	−	0.881113i	$-0.656795\pi$
$409$	−19.1279	−0.945812	−0.472906	+	0.881113i	$0.656795\pi$
$410$	−8.61718	−0.425572
$411$	0	0
$412$	1.88529	0.0928815
$413$	−4.61718	−0.227196
$414$	0	0
$415$	−66.5820	−3.26838
$416$	1.36594	0.0669709
$417$	0	0
$418$	−6.79335	−0.332274
$419$	38.6067	1.88606	0.943031	−	0.332705i	$-0.107961\pi$
$419$	38.6067	1.88606	0.943031	+	0.332705i	$0.107961\pi$
$420$	0	0
$421$	−16.5042	−0.804368	−0.402184	−	0.915559i	$-0.631749\pi$
$421$	−16.5042	−0.804368	−0.402184	+	0.915559i	$0.631749\pi$
$422$	−7.40231	−0.360339
$423$	0	0
$424$	5.52345	0.268243
$425$	−18.5276	−0.898719
$426$	0	0
$427$	12.9258	0.625522
$428$	2.30859	0.111590
$429$	0	0
$430$	−38.9517	−1.87842
$431$	33.7450	1.62544	0.812720	−	0.582654i	$-0.197986\pi$
$431$	33.7450	1.62544	0.812720	+	0.582654i	$0.197986\pi$
$432$	0	0
$433$	−15.8684	−0.762587	−0.381293	−	0.924454i	$-0.624521\pi$
$433$	−15.8684	−0.762587	−0.381293	+	0.924454i	$0.624521\pi$
$434$	7.19798	0.345514
$435$	0	0
$436$	−10.0874	−0.483098
$437$	−2.94264	−0.140766
$438$	0	0
$439$	−29.2512	−1.39609	−0.698043	−	0.716056i	$-0.745945\pi$
$439$	−29.2512	−1.39609	−0.698043	+	0.716056i	$0.745945\pi$
$440$	−9.94675	−0.474193
$441$	0	0
$442$	−1.86580	−0.0887472
$443$	−31.6836	−1.50533	−0.752666	−	0.658403i	$-0.771232\pi$
$443$	−31.6836	−1.50533	−0.752666	+	0.658403i	$0.771232\pi$
$444$	0	0
$445$	−36.2475	−1.71830
$446$	20.6618	0.978363
$447$	0	0
$448$	1.00000	0.0472456
$449$	−1.89350	−0.0893600	−0.0446800	−	0.999001i	$-0.514227\pi$
$449$	−1.89350	−0.0893600	−0.0446800	+	0.999001i	$0.514227\pi$
$450$	0	0
$451$	4.61718	0.217414
$452$	3.16162	0.148710
$453$	0	0
$454$	−12.1043	−0.568081
$455$	−5.88529	−0.275907
$456$	0	0
$457$	−2.51068	−0.117445	−0.0587223	−	0.998274i	$-0.518703\pi$
$457$	−2.51068	−0.117445	−0.0587223	+	0.998274i	$0.518703\pi$
$458$	20.9258	0.977797
$459$	0	0
$460$	−4.30859	−0.200889
$461$	−17.2257	−0.802280	−0.401140	−	0.916017i	$-0.631386\pi$
$461$	−17.2257	−0.802280	−0.401140	+	0.916017i	$0.631386\pi$
$462$	0	0
$463$	9.12785	0.424207	0.212104	−	0.977247i	$-0.431969\pi$
$463$	9.12785	0.424207	0.212104	+	0.977247i	$0.431969\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−12.6704	−0.586946
$467$	9.95579	0.460699	0.230349	−	0.973108i	$-0.426013\pi$
$467$	9.95579	0.460699	0.230349	+	0.973108i	$0.426013\pi$
$468$	0	0
$469$	16.1939	0.747764
$470$	−48.2475	−2.22549
$471$	0	0
$472$	4.61718	0.212523
$473$	20.8707	0.959637
$474$	0	0
$475$	39.9138	1.83137
$476$	−1.36594	−0.0626079
$477$	0	0
$478$	10.3960	0.475500
$479$	−9.26811	−0.423471	−0.211735	−	0.977327i	$-0.567912\pi$
$479$	−9.26811	−0.423471	−0.211735	+	0.977327i	$0.567912\pi$
$480$	0	0
$481$	2.08095	0.0948831
$482$	3.79567	0.172888
$483$	0	0
$484$	−5.67042	−0.257747
$485$	−73.3753	−3.33180
$486$	0	0
$487$	25.3408	1.14830	0.574152	−	0.818749i	$-0.305332\pi$
$487$	25.3408	1.14830	0.574152	+	0.818749i	$0.305332\pi$
$488$	−12.9258	−0.585122
$489$	0	0
$490$	−4.30859	−0.194642
$491$	−8.98544	−0.405507	−0.202754	−	0.979230i	$-0.564989\pi$
$491$	−8.98544	−0.405507	−0.202754	+	0.979230i	$0.564989\pi$
$492$	0	0
$493$	2.73189	0.123038
$494$	4.01949	0.180845
$495$	0	0
$496$	−7.19798	−0.323199
$497$	−0.785135	−0.0352181
$498$	0	0
$499$	−10.6786	−0.478042	−0.239021	−	0.971014i	$-0.576826\pi$
$499$	−10.6786	−0.478042	−0.239021	+	0.971014i	$0.576826\pi$
$500$	36.8984	1.65015
$501$	0	0
$502$	6.18521	0.276059
$503$	−5.74503	−0.256158	−0.128079	−	0.991764i	$-0.540881\pi$
$503$	−5.74503	−0.256158	−0.128079	+	0.991764i	$0.540881\pi$
$504$	0	0
$505$	−72.0181	−3.20476
$506$	2.30859	0.102629
$507$	0	0
$508$	1.26811	0.0562634
$509$	−25.4387	−1.12755	−0.563775	−	0.825929i	$-0.690651\pi$
$509$	−25.4387	−1.12755	−0.563775	+	0.825929i	$0.690651\pi$
$510$	0	0
$511$	9.83204	0.434944
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	3.16162	0.139453
$515$	8.12293	0.357939
$516$	0	0
$517$	25.8515	1.13695
$518$	1.52345	0.0669366
$519$	0	0
$520$	5.88529	0.258087
$521$	−24.1706	−1.05893	−0.529466	−	0.848331i	$-0.677608\pi$
$521$	−24.1706	−1.05893	−0.529466	+	0.848331i	$0.677608\pi$
$522$	0	0
$523$	−15.5598	−0.680383	−0.340192	−	0.940356i	$-0.610492\pi$
$523$	−15.5598	−0.680383	−0.340192	+	0.940356i	$0.610492\pi$
$524$	−8.61718	−0.376443
$525$	0	0
$526$	7.66408	0.334170
$527$	9.83204	0.428290
$528$	0	0
$529$	1.00000	0.0434783
$530$	23.7983	1.03373
$531$	0	0
$532$	2.94264	0.127580
$533$	−2.73189	−0.118331
$534$	0	0
$535$	9.94675	0.430036
$536$	−16.1939	−0.699469
$537$	0	0
$538$	9.36594	0.403795
$539$	2.30859	0.0994379
$540$	0	0
$541$	−1.48932	−0.0640309	−0.0320155	−	0.999487i	$-0.510193\pi$
$541$	−1.48932	−0.0640309	−0.0320155	+	0.999487i	$0.510193\pi$
$542$	10.7488	0.461699
$543$	0	0
$544$	1.36594	0.0585644
$545$	−43.4624	−1.86172
$546$	0	0
$547$	22.5916	0.965948	0.482974	−	0.875635i	$-0.339557\pi$
$547$	22.5916	0.965948	0.482974	+	0.875635i	$0.339557\pi$
$548$	−10.6172	−0.453543
$549$	0	0
$550$	−31.3135	−1.33521
$551$	−5.88529	−0.250722
$552$	0	0
$553$	13.8320	0.588198
$554$	−22.5107	−0.956387
$555$	0	0
$556$	−1.88529	−0.0799540
$557$	−3.77701	−0.160037	−0.0800184	−	0.996793i	$-0.525498\pi$
$557$	−3.77701	−0.160037	−0.0800184	+	0.996793i	$0.525498\pi$
$558$	0	0
$559$	−12.3488	−0.522298
$560$	4.30859	0.182071
$561$	0	0
$562$	24.6981	1.04183
$563$	−5.97669	−0.251887	−0.125944	−	0.992037i	$-0.540196\pi$
$563$	−5.97669	−0.251887	−0.125944	+	0.992037i	$0.540196\pi$
$564$	0	0
$565$	13.6221	0.573086
$566$	−22.2917	−0.936990
$567$	0	0
$568$	0.785135	0.0329436
$569$	−42.3878	−1.77699	−0.888493	−	0.458889i	$-0.848247\pi$
$569$	−42.3878	−1.77699	−0.888493	+	0.458889i	$0.848247\pi$
$570$	0	0
$571$	−5.65765	−0.236765	−0.118383	−	0.992968i	$-0.537771\pi$
$571$	−5.65765	−0.236765	−0.118383	+	0.992968i	$0.537771\pi$
$572$	−3.15340	−0.131850
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	−13.5639	−0.565655
$576$	0	0
$577$	16.0323	0.667435	0.333718	−	0.942673i	$-0.391697\pi$
$577$	16.0323	0.667435	0.333718	+	0.942673i	$0.391697\pi$
$578$	15.1342	0.629500
$579$	0	0
$580$	−8.61718	−0.357809
$581$	−15.4533	−0.641112
$582$	0	0
$583$	−12.7514	−0.528108
$584$	−9.83204	−0.406853
$585$	0	0
$586$	−28.8193	−1.19051
$587$	−16.9193	−0.698336	−0.349168	−	0.937060i	$-0.613536\pi$
$587$	−16.9193	−0.698336	−0.349168	+	0.937060i	$0.613536\pi$
$588$	0	0
$589$	−21.1811	−0.872752
$590$	19.8935	0.819003
$591$	0	0
$592$	−1.52345	−0.0626135
$593$	−47.3071	−1.94267	−0.971335	−	0.237717i	$-0.923601\pi$
$593$	−47.3071	−1.94267	−0.971335	+	0.237717i	$0.923601\pi$
$594$	0	0
$595$	−5.88529	−0.241273
$596$	7.04047	0.288389
$597$	0	0
$598$	−1.36594	−0.0558576
$599$	−20.7919	−0.849535	−0.424768	−	0.905302i	$-0.639644\pi$
$599$	−20.7919	−0.849535	−0.424768	+	0.905302i	$0.639644\pi$
$600$	0	0
$601$	−14.1680	−0.577923	−0.288962	−	0.957341i	$-0.593310\pi$
$601$	−14.1680	−0.577923	−0.288962	+	0.957341i	$0.593310\pi$
$602$	−9.04047	−0.368462
$603$	0	0
$604$	−8.44921	−0.343794
$605$	−24.4315	−0.993282
$606$	0	0
$607$	−16.7683	−0.680602	−0.340301	−	0.940316i	$-0.610529\pi$
$607$	−16.7683	−0.680602	−0.340301	+	0.940316i	$0.610529\pi$
$608$	−2.94264	−0.119340
$609$	0	0
$610$	−55.6918	−2.25489
$611$	−15.2958	−0.618802
$612$	0	0
$613$	21.1214	0.853086	0.426543	−	0.904467i	$-0.359731\pi$
$613$	21.1214	0.853086	0.426543	+	0.904467i	$0.359731\pi$
$614$	10.5025	0.423845
$615$	0	0
$616$	−2.30859	−0.0930156
$617$	18.8984	0.760822	0.380411	−	0.924818i	$-0.375783\pi$
$617$	18.8984	0.760822	0.380411	+	0.924818i	$0.375783\pi$
$618$	0	0
$619$	21.4451	0.861952	0.430976	−	0.902363i	$-0.358169\pi$
$619$	21.4451	0.861952	0.430976	+	0.902363i	$0.358169\pi$
$620$	−31.0131	−1.24552
$621$	0	0
$622$	−7.05093	−0.282716
$623$	−8.41285	−0.337054
$624$	0	0
$625$	91.1605	3.64642
$626$	−14.6003	−0.583545
$627$	0	0
$628$	9.47020	0.377902
$629$	2.08095	0.0829729
$630$	0	0
$631$	21.1473	0.841862	0.420931	−	0.907093i	$-0.361703\pi$
$631$	21.1473	0.841862	0.420931	+	0.907093i	$0.361703\pi$
$632$	−13.8320	−0.550209
$633$	0	0
$634$	20.7319	0.823368
$635$	5.46377	0.216823
$636$	0	0
$637$	−1.36594	−0.0541207
$638$	4.61718	0.182796
$639$	0	0
$640$	−4.30859	−0.170312
$641$	48.9794	1.93457	0.967285	−	0.253694i	$-0.0816455\pi$
$641$	48.9794	1.93457	0.967285	+	0.253694i	$0.0816455\pi$
$642$	0	0
$643$	−37.0236	−1.46007	−0.730034	−	0.683411i	$-0.760496\pi$
$643$	−37.0236	−1.46007	−0.730034	+	0.683411i	$0.760496\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	4.01949	0.158145
$647$	−11.8766	−0.466918	−0.233459	−	0.972367i	$-0.575004\pi$
$647$	−11.8766	−0.466918	−0.233459	+	0.972367i	$0.575004\pi$
$648$	0	0
$649$	−10.6592	−0.418408
$650$	18.5276	0.726711
$651$	0	0
$652$	−1.21486	−0.0475778
$653$	33.9453	1.32838	0.664192	−	0.747562i	$-0.268776\pi$
$653$	33.9453	1.32838	0.664192	+	0.747562i	$0.268776\pi$
$654$	0	0
$655$	−37.1279	−1.45071
$656$	2.00000	0.0780869
$657$	0	0
$658$	−11.1980	−0.436543
$659$	46.6790	1.81836	0.909178	−	0.416408i	$-0.136711\pi$
$659$	46.6790	1.81836	0.909178	+	0.416408i	$0.136711\pi$
$660$	0	0
$661$	23.3426	0.907923	0.453962	−	0.891021i	$-0.350010\pi$
$661$	23.3426	0.907923	0.453962	+	0.891021i	$0.350010\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	15.4533	0.599705
$665$	12.6786	0.491657
$666$	0	0
$667$	2.00000	0.0774403
$668$	15.3045	0.592148
$669$	0	0
$670$	−69.7727	−2.69556
$671$	29.8403	1.15197
$672$	0	0
$673$	−37.0049	−1.42644	−0.713218	−	0.700943i	$-0.752763\pi$
$673$	−37.0049	−1.42644	−0.713218	+	0.700943i	$0.752763\pi$
$674$	−12.7319	−0.490414
$675$	0	0
$676$	−11.1342	−0.428238
$677$	−8.64451	−0.332235	−0.166118	−	0.986106i	$-0.553123\pi$
$677$	−8.64451	−0.332235	−0.166118	+	0.986106i	$0.553123\pi$
$678$	0	0
$679$	−17.0300	−0.653552
$680$	5.88529	0.225690
$681$	0	0
$682$	16.6172	0.636305
$683$	−11.4023	−0.436297	−0.218149	−	0.975916i	$-0.570002\pi$
$683$	−11.4023	−0.436297	−0.218149	+	0.975916i	$0.570002\pi$
$684$	0	0
$685$	−45.7450	−1.74783
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	9.04047	0.344665
$689$	7.54472	0.287431
$690$	0	0
$691$	−28.8128	−1.09609	−0.548046	−	0.836448i	$-0.684628\pi$
$691$	−28.8128	−1.09609	−0.548046	+	0.836448i	$0.684628\pi$
$692$	19.7619	0.751235
$693$	0	0
$694$	6.44921	0.244809
$695$	−8.12293	−0.308120
$696$	0	0
$697$	−2.73189	−0.103478
$698$	−28.3791	−1.07416
$699$	0	0
$700$	13.5639	0.512668
$701$	7.93890	0.299848	0.149924	−	0.988698i	$-0.452097\pi$
$701$	7.93890	0.299848	0.149924	+	0.988698i	$0.452097\pi$
$702$	0	0
$703$	−4.48298	−0.169079
$704$	2.30859	0.0870082
$705$	0	0
$706$	3.32958	0.125310
$707$	−16.7150	−0.628633
$708$	0	0
$709$	37.6854	1.41530	0.707652	−	0.706562i	$-0.249755\pi$
$709$	37.6854	1.41530	0.707652	+	0.706562i	$0.249755\pi$
$710$	3.38282	0.126955
$711$	0	0
$712$	8.41285	0.315285
$713$	7.19798	0.269567
$714$	0	0
$715$	−13.5867	−0.508114
$716$	3.15340	0.117848
$717$	0	0
$718$	−26.3427	−0.983102
$719$	−6.37909	−0.237900	−0.118950	−	0.992900i	$-0.537953\pi$
$719$	−6.37909	−0.237900	−0.118950	+	0.992900i	$0.537953\pi$
$720$	0	0
$721$	1.88529	0.0702118
$722$	10.3408	0.384846
$723$	0	0
$724$	6.73832	0.250428
$725$	−27.1279	−1.00750
$726$	0	0
$727$	−41.2006	−1.52805	−0.764023	−	0.645189i	$-0.776778\pi$
$727$	−41.2006	−1.52805	−0.764023	+	0.645189i	$0.776778\pi$
$728$	1.36594	0.0506252
$729$	0	0
$730$	−42.3622	−1.56790
$731$	−12.3488	−0.456736
$732$	0	0
$733$	0.853029	0.0315073	0.0157537	−	0.999876i	$-0.494985\pi$
$733$	0.853029	0.0315073	0.0157537	+	0.999876i	$0.494985\pi$
$734$	−5.16162	−0.190519
$735$	0	0
$736$	1.00000	0.0368605
$737$	37.3850	1.37709
$738$	0	0
$739$	4.01949	0.147859	0.0739296	−	0.997263i	$-0.476446\pi$
$739$	4.01949	0.147859	0.0739296	+	0.997263i	$0.476446\pi$
$740$	−6.56393	−0.241295
$741$	0	0
$742$	5.52345	0.202772
$743$	12.9726	0.475918	0.237959	−	0.971275i	$-0.423522\pi$
$743$	12.9726	0.475918	0.237959	+	0.971275i	$0.423522\pi$
$744$	0	0
$745$	30.3345	1.11137
$746$	15.6044	0.571318
$747$	0	0
$748$	−3.15340	−0.115300
$749$	2.30859	0.0843540
$750$	0	0
$751$	3.38282	0.123441	0.0617205	−	0.998093i	$-0.480341\pi$
$751$	3.38282	0.123441	0.0617205	+	0.998093i	$0.480341\pi$
$752$	11.1980	0.408348
$753$	0	0
$754$	−2.73189	−0.0994895
$755$	−36.4042	−1.32488
$756$	0	0
$757$	−37.1214	−1.34920	−0.674601	−	0.738183i	$-0.735684\pi$
$757$	−37.1214	−1.34920	−0.674601	+	0.738183i	$0.735684\pi$
$758$	−18.5898	−0.675213
$759$	0	0
$760$	−12.6786	−0.459903
$761$	−34.3299	−1.24446	−0.622228	−	0.782836i	$-0.713772\pi$
$761$	−34.3299	−1.24446	−0.622228	+	0.782836i	$0.713772\pi$
$762$	0	0
$763$	−10.0874	−0.365188
$764$	−6.78514	−0.245478
$765$	0	0
$766$	−0.335921	−0.0121373
$767$	6.30680	0.227725
$768$	0	0
$769$	44.9625	1.62139	0.810695	−	0.585469i	$-0.199090\pi$
$769$	44.9625	1.62139	0.810695	+	0.585469i	$0.199090\pi$
$770$	−9.94675	−0.358456
$771$	0	0
$772$	1.43607	0.0516854
$773$	21.2279	0.763515	0.381758	−	0.924262i	$-0.375319\pi$
$773$	21.2279	0.763515	0.381758	+	0.924262i	$0.375319\pi$
$774$	0	0
$775$	−97.6329	−3.50708
$776$	17.0300	0.611342
$777$	0	0
$778$	32.1151	1.15138
$779$	5.88529	0.210862
$780$	0	0
$781$	−1.81255	−0.0648583
$782$	−1.36594	−0.0488460
$783$	0	0
$784$	1.00000	0.0357143
$785$	40.8032	1.45633
$786$	0	0
$787$	54.2579	1.93409	0.967043	−	0.254612i	$-0.0819476\pi$
$787$	54.2579	1.93409	0.967043	+	0.254612i	$0.0819476\pi$
$788$	−10.9322	−0.389443
$789$	0	0
$790$	−59.5966	−2.12035
$791$	3.16162	0.112414
$792$	0	0
$793$	−17.6559	−0.626978
$794$	22.6341	0.803253
$795$	0	0
$796$	23.9662	0.849461
$797$	−29.7515	−1.05385	−0.526925	−	0.849912i	$-0.676655\pi$
$797$	−29.7515	−1.05385	−0.526925	+	0.849912i	$0.676655\pi$
$798$	0	0
$799$	−15.2958	−0.541127
$800$	−13.5639	−0.479557
$801$	0	0
$802$	−0.187446	−0.00661895
$803$	22.6981	0.800999
$804$	0	0
$805$	−4.30859	−0.151858
$806$	−9.83204	−0.346319
$807$	0	0
$808$	16.7150	0.588032
$809$	49.4624	1.73900	0.869502	−	0.493930i	$-0.164440\pi$
$809$	49.4624	1.73900	0.869502	+	0.493930i	$0.164440\pi$
$810$	0	0
$811$	−36.6172	−1.28580	−0.642901	−	0.765949i	$-0.722270\pi$
$811$	−36.6172	−1.28580	−0.642901	+	0.765949i	$0.722270\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	3.51702	0.123272
$815$	−5.23435	−0.183351
$816$	0	0
$817$	26.6029	0.930718
$818$	19.1279	0.668790
$819$	0	0
$820$	8.61718	0.300925
$821$	−27.9918	−0.976920	−0.488460	−	0.872586i	$-0.662441\pi$
$821$	−27.9918	−0.976920	−0.488460	+	0.872586i	$0.662441\pi$
$822$	0	0
$823$	30.6495	1.06838	0.534188	−	0.845366i	$-0.320618\pi$
$823$	30.6495	1.06838	0.534188	+	0.845366i	$0.320618\pi$
$824$	−1.88529	−0.0656771
$825$	0	0
$826$	4.61718	0.160652
$827$	27.5429	0.957762	0.478881	−	0.877880i	$-0.341043\pi$
$827$	27.5429	0.957762	0.478881	+	0.877880i	$0.341043\pi$
$828$	0	0
$829$	−2.35280	−0.0817162	−0.0408581	−	0.999165i	$-0.513009\pi$
$829$	−2.35280	−0.0817162	−0.0408581	+	0.999165i	$0.513009\pi$
$830$	66.5820	2.31109
$831$	0	0
$832$	−1.36594	−0.0473556
$833$	−1.36594	−0.0473271
$834$	0	0
$835$	65.9407	2.28197
$836$	6.79335	0.234953
$837$	0	0
$838$	−38.6067	−1.33365
$839$	−46.4687	−1.60428	−0.802139	−	0.597138i	$-0.796305\pi$
$839$	−46.4687	−1.60428	−0.802139	+	0.597138i	$0.796305\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	16.5042	0.568774
$843$	0	0
$844$	7.40231	0.254798
$845$	−47.9727	−1.65031
$846$	0	0
$847$	−5.67042	−0.194838
$848$	−5.52345	−0.189676
$849$	0	0
$850$	18.5276	0.635490
$851$	1.52345	0.0522233
$852$	0	0
$853$	12.1706	0.416712	0.208356	−	0.978053i	$-0.433189\pi$
$853$	12.1706	0.416712	0.208356	+	0.978053i	$0.433189\pi$
$854$	−12.9258	−0.442310
$855$	0	0
$856$	−2.30859	−0.0789059
$857$	24.3345	0.831251	0.415625	−	0.909536i	$-0.363563\pi$
$857$	24.3345	0.831251	0.415625	+	0.909536i	$0.363563\pi$
$858$	0	0
$859$	−14.6090	−0.498451	−0.249226	−	0.968445i	$-0.580176\pi$
$859$	−14.6090	−0.498451	−0.249226	+	0.968445i	$0.580176\pi$
$860$	38.9517	1.32824
$861$	0	0
$862$	−33.7450	−1.14936
$863$	−25.2006	−0.857838	−0.428919	−	0.903343i	$-0.641105\pi$
$863$	−25.2006	−0.857838	−0.428919	+	0.903343i	$0.641105\pi$
$864$	0	0
$865$	85.1459	2.89505
$866$	15.8684	0.539230
$867$	0	0
$868$	−7.19798	−0.244315
$869$	31.9325	1.08324
$870$	0	0
$871$	−22.1199	−0.749505
$872$	10.0874	0.341602
$873$	0	0
$874$	2.94264	0.0995364
$875$	36.8984	1.24739
$876$	0	0
$877$	−43.4818	−1.46828	−0.734139	−	0.678999i	$-0.762414\pi$
$877$	−43.4818	−1.46828	−0.734139	+	0.678999i	$0.762414\pi$
$878$	29.2512	0.987181
$879$	0	0
$880$	9.94675	0.335305
$881$	−23.4807	−0.791083	−0.395542	−	0.918448i	$-0.629443\pi$
$881$	−23.4807	−0.791083	−0.395542	+	0.918448i	$0.629443\pi$
$882$	0	0
$883$	22.4364	0.755043	0.377522	−	0.926001i	$-0.376776\pi$
$883$	22.4364	0.755043	0.377522	+	0.926001i	$0.376776\pi$
$884$	1.86580	0.0627537
$885$	0	0
$886$	31.6836	1.06443
$887$	−36.5215	−1.22627	−0.613136	−	0.789977i	$-0.710092\pi$
$887$	−36.5215	−1.22627	−0.613136	+	0.789977i	$0.710092\pi$
$888$	0	0
$889$	1.26811	0.0425311
$890$	36.2475	1.21502
$891$	0	0
$892$	−20.6618	−0.691807
$893$	32.9517	1.10269
$894$	0	0
$895$	13.5867	0.454153
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	1.89350	0.0631870
$899$	14.3960	0.480132
$900$	0	0
$901$	7.54472	0.251351
$902$	−4.61718	−0.153735
$903$	0	0
$904$	−3.16162	−0.105154
$905$	29.0326	0.965077
$906$	0	0
$907$	−41.6367	−1.38253	−0.691263	−	0.722604i	$-0.742945\pi$
$907$	−41.6367	−1.38253	−0.691263	+	0.722604i	$0.742945\pi$
$908$	12.1043	0.401694
$909$	0	0
$910$	5.88529	0.195095
$911$	53.2538	1.76438	0.882189	−	0.470895i	$-0.156069\pi$
$911$	53.2538	1.76438	0.882189	+	0.470895i	$0.156069\pi$
$912$	0	0
$913$	−35.6753	−1.18068
$914$	2.51068	0.0830459
$915$	0	0
$916$	−20.9258	−0.691407
$917$	−8.61718	−0.284564
$918$	0	0
$919$	13.6190	0.449251	0.224625	−	0.974445i	$-0.427884\pi$
$919$	13.6190	0.449251	0.224625	+	0.974445i	$0.427884\pi$
$920$	4.30859	0.142050
$921$	0	0
$922$	17.2257	0.567298
$923$	1.07245	0.0353001
$924$	0	0
$925$	−20.6640	−0.679428
$926$	−9.12785	−0.299960
$927$	0	0
$928$	2.00000	0.0656532
$929$	−33.3282	−1.09346	−0.546731	−	0.837309i	$-0.684128\pi$
$929$	−33.3282	−1.09346	−0.546731	+	0.837309i	$0.684128\pi$
$930$	0	0
$931$	2.94264	0.0964413
$932$	12.6704	0.415034
$933$	0	0
$934$	−9.95579	−0.325763
$935$	−13.5867	−0.444333
$936$	0	0
$937$	16.4128	0.536184	0.268092	−	0.963393i	$-0.413607\pi$
$937$	16.4128	0.536184	0.268092	+	0.963393i	$0.413607\pi$
$938$	−16.1939	−0.528749
$939$	0	0
$940$	48.2475	1.57366
$941$	−32.1857	−1.04922	−0.524611	−	0.851342i	$-0.675789\pi$
$941$	−32.1857	−1.04922	−0.524611	+	0.851342i	$0.675789\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	−4.61718	−0.150276
$945$	0	0
$946$	−20.8707	−0.678566
$947$	−51.2411	−1.66511	−0.832557	−	0.553940i	$-0.813124\pi$
$947$	−51.2411	−1.66511	−0.832557	+	0.553940i	$0.813124\pi$
$948$	0	0
$949$	−13.4300	−0.435957
$950$	−39.9138	−1.29497
$951$	0	0
$952$	1.36594	0.0442705
$953$	−54.1410	−1.75380	−0.876899	−	0.480674i	$-0.840392\pi$
$953$	−54.1410	−1.75380	−0.876899	+	0.480674i	$0.840392\pi$
$954$	0	0
$955$	−29.2344	−0.946001
$956$	−10.3960	−0.336230
$957$	0	0
$958$	9.26811	0.299439
$959$	−10.6172	−0.342846
$960$	0	0
$961$	20.8110	0.671321
$962$	−2.08095	−0.0670925
$963$	0	0
$964$	−3.79567	−0.122250
$965$	6.18745	0.199181
$966$	0	0
$967$	−40.1860	−1.29230	−0.646148	−	0.763212i	$-0.723621\pi$
$967$	−40.1860	−1.29230	−0.646148	+	0.763212i	$0.723621\pi$
$968$	5.67042	0.182254
$969$	0	0
$970$	73.3753	2.35594
$971$	−12.5258	−0.401971	−0.200986	−	0.979594i	$-0.564414\pi$
$971$	−12.5258	−0.401971	−0.200986	+	0.979594i	$0.564414\pi$
$972$	0	0
$973$	−1.88529	−0.0604396
$974$	−25.3408	−0.811973
$975$	0	0
$976$	12.9258	0.413744
$977$	−42.3878	−1.35610	−0.678052	−	0.735014i	$-0.737176\pi$
$977$	−42.3878	−1.35610	−0.678052	+	0.735014i	$0.737176\pi$
$978$	0	0
$979$	−19.4218	−0.620723
$980$	4.30859	0.137633
$981$	0	0
$982$	8.98544	0.286737
$983$	−53.8515	−1.71760	−0.858798	−	0.512314i	$-0.828789\pi$
$983$	−53.8515	−1.71760	−0.858798	+	0.512314i	$0.828789\pi$
$984$	0	0
$985$	−47.1023	−1.50080
$986$	−2.73189	−0.0870010
$987$	0	0
$988$	−4.01949	−0.127877
$989$	−9.04047	−0.287470
$990$	0	0
$991$	−33.6836	−1.06999	−0.534997	−	0.844854i	$-0.679687\pi$
$991$	−33.6836	−1.06999	−0.534997	+	0.844854i	$0.679687\pi$
$992$	7.19798	0.228536
$993$	0	0
$994$	0.785135	0.0249030
$995$	103.261	3.27358
$996$	0	0
$997$	27.5872	0.873694	0.436847	−	0.899536i	$-0.356095\pi$
$997$	27.5872	0.873694	0.436847	+	0.899536i	$0.356095\pi$
$998$	10.6786	0.338026
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.bh.1.4	✓	4
3.2	odd	2		2898.2.a.bi.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2898.2.a.bh.1.4	✓	4	1.1	even	1	trivial
2898.2.a.bi.1.1	yes	4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2898.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +4.30859 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +4.30859 q^{5} +1.00000 q^{7} -1.00000 q^{8} -4.30859 q^{10} +2.30859 q^{11} -1.36594 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.36594 q^{17} +2.94264 q^{19} +4.30859 q^{20} -2.30859 q^{22} -1.00000 q^{23} +13.5639 q^{25} +1.36594 q^{26} +1.00000 q^{28} -2.00000 q^{29} -7.19798 q^{31} -1.00000 q^{32} +1.36594 q^{34} +4.30859 q^{35} -1.52345 q^{37} -2.94264 q^{38} -4.30859 q^{40} +2.00000 q^{41} +9.04047 q^{43} +2.30859 q^{44} +1.00000 q^{46} +11.1980 q^{47} +1.00000 q^{49} -13.5639 q^{50} -1.36594 q^{52} -5.52345 q^{53} +9.94675 q^{55} -1.00000 q^{56} +2.00000 q^{58} -4.61718 q^{59} +12.9258 q^{61} +7.19798 q^{62} +1.00000 q^{64} -5.88529 q^{65} +16.1939 q^{67} -1.36594 q^{68} -4.30859 q^{70} -0.785135 q^{71} +9.83204 q^{73} +1.52345 q^{74} +2.94264 q^{76} +2.30859 q^{77} +13.8320 q^{79} +4.30859 q^{80} -2.00000 q^{82} -15.4533 q^{83} -5.88529 q^{85} -9.04047 q^{86} -2.30859 q^{88} -8.41285 q^{89} -1.36594 q^{91} -1.00000 q^{92} -11.1980 q^{94} +12.6786 q^{95} -17.0300 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^7 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8} - 8 q^{11} - 4 q^{14} + 4 q^{16} + 8 q^{22} - 4 q^{23} + 16 q^{25} + 4 q^{28} - 8 q^{29} + 4 q^{31} - 4 q^{32} + 4 q^{37} + 8 q^{41} + 8 q^{43} - 8 q^{44} + 4 q^{46} + 12 q^{47} + 4 q^{49} - 16 q^{50} - 12 q^{53} + 36 q^{55} - 4 q^{56} + 8 q^{58} + 16 q^{59} - 4 q^{62} + 4 q^{64} + 24 q^{67} + 4 q^{71} + 12 q^{73} - 4 q^{74} - 8 q^{77} + 28 q^{79} - 8 q^{82} - 8 q^{83} - 8 q^{86} + 8 q^{88} - 8 q^{89} - 4 q^{92} - 12 q^{94} + 36 q^{95} - 8 q^{97} - 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^7 - 4 * q^8 - 8 * q^11 - 4 * q^14 + 4 * q^16 + 8 * q^22 - 4 * q^23 + 16 * q^25 + 4 * q^28 - 8 * q^29 + 4 * q^31 - 4 * q^32 + 4 * q^37 + 8 * q^41 + 8 * q^43 - 8 * q^44 + 4 * q^46 + 12 * q^47 + 4 * q^49 - 16 * q^50 - 12 * q^53 + 36 * q^55 - 4 * q^56 + 8 * q^58 + 16 * q^59 - 4 * q^62 + 4 * q^64 + 24 * q^67 + 4 * q^71 + 12 * q^73 - 4 * q^74 - 8 * q^77 + 28 * q^79 - 8 * q^82 - 8 * q^83 - 8 * q^86 + 8 * q^88 - 8 * q^89 - 4 * q^92 - 12 * q^94 + 36 * q^95 - 8 * q^97 - 4 * q^98

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