LMFDB - Embedded newform 9054.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9054,2,Mod(1,9054)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9054, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9054.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9054 = 2 \cdot 3^{2} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9054.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:	$72.2965539901$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.36497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 $ x^5 - 2x^4 - 3x^3 + 5*x^2 + x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.55629$ of defining polynomial
Character	$\chi$	$=$	9054.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.813221 q^{5} -3.14341 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -0.813221 q^{5} -3.14341 q^{7} +1.00000 q^{8} -0.813221 q^{10} +0.935416 q^{11} +6.27767 q^{13} -3.14341 q^{14} +1.00000 q^{16} +4.52254 q^{17} +3.63511 q^{19} -0.813221 q^{20} +0.935416 q^{22} +2.08440 q^{23} -4.33867 q^{25} +6.27767 q^{26} -3.14341 q^{28} +7.17238 q^{29} -3.77203 q^{31} +1.00000 q^{32} +4.52254 q^{34} +2.55629 q^{35} -8.05157 q^{37} +3.63511 q^{38} -0.813221 q^{40} -0.104212 q^{41} +0.412877 q^{43} +0.935416 q^{44} +2.08440 q^{46} -9.24902 q^{47} +2.88102 q^{49} -4.33867 q^{50} +6.27767 q^{52} +4.95783 q^{53} -0.760700 q^{55} -3.14341 q^{56} +7.17238 q^{58} -1.98123 q^{59} -11.8824 q^{61} -3.77203 q^{62} +1.00000 q^{64} -5.10513 q^{65} +4.21544 q^{67} +4.52254 q^{68} +2.55629 q^{70} +13.4960 q^{71} -11.3749 q^{73} -8.05157 q^{74} +3.63511 q^{76} -2.94040 q^{77} +14.7420 q^{79} -0.813221 q^{80} -0.104212 q^{82} +7.06816 q^{83} -3.67782 q^{85} +0.412877 q^{86} +0.935416 q^{88} +7.76652 q^{89} -19.7333 q^{91} +2.08440 q^{92} -9.24902 q^{94} -2.95615 q^{95} +11.1338 q^{97} +2.88102 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} + q^{5} + 3 q^{7} + 5 q^{8}+O(q^{10}) $ 5 * q + 5 * q^2 + 5 * q^4 + q^5 + 3 * q^7 + 5 * q^8	$ 5 q + 5 q^{2} + 5 q^{4} + q^{5} + 3 q^{7} + 5 q^{8} + q^{10} + 11 q^{11} + 5 q^{13} + 3 q^{14} + 5 q^{16} + 20 q^{17} - 4 q^{19} + q^{20} + 11 q^{22} + 4 q^{23} - 6 q^{25} + 5 q^{26} + 3 q^{28} + 6 q^{29} - 3 q^{31} + 5 q^{32} + 20 q^{34} + 3 q^{35} - 10 q^{37} - 4 q^{38} + q^{40} + 6 q^{41} + 11 q^{43} + 11 q^{44} + 4 q^{46} + 9 q^{47} - 4 q^{49} - 6 q^{50} + 5 q^{52} + 22 q^{53} + 14 q^{55} + 3 q^{56} + 6 q^{58} + 10 q^{59} - 11 q^{61} - 3 q^{62} + 5 q^{64} + 12 q^{65} + 14 q^{67} + 20 q^{68} + 3 q^{70} + 26 q^{71} - 7 q^{73} - 10 q^{74} - 4 q^{76} + 26 q^{77} - 15 q^{79} + q^{80} + 6 q^{82} + 12 q^{83} + 12 q^{85} + 11 q^{86} + 11 q^{88} + 5 q^{89} - 22 q^{91} + 4 q^{92} + 9 q^{94} + 10 q^{95} + 6 q^{97} - 4 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 + q^5 + 3 * q^7 + 5 * q^8 + q^10 + 11 * q^11 + 5 * q^13 + 3 * q^14 + 5 * q^16 + 20 * q^17 - 4 * q^19 + q^20 + 11 * q^22 + 4 * q^23 - 6 * q^25 + 5 * q^26 + 3 * q^28 + 6 * q^29 - 3 * q^31 + 5 * q^32 + 20 * q^34 + 3 * q^35 - 10 * q^37 - 4 * q^38 + q^40 + 6 * q^41 + 11 * q^43 + 11 * q^44 + 4 * q^46 + 9 * q^47 - 4 * q^49 - 6 * q^50 + 5 * q^52 + 22 * q^53 + 14 * q^55 + 3 * q^56 + 6 * q^58 + 10 * q^59 - 11 * q^61 - 3 * q^62 + 5 * q^64 + 12 * q^65 + 14 * q^67 + 20 * q^68 + 3 * q^70 + 26 * q^71 - 7 * q^73 - 10 * q^74 - 4 * q^76 + 26 * q^77 - 15 * q^79 + q^80 + 6 * q^82 + 12 * q^83 + 12 * q^85 + 11 * q^86 + 11 * q^88 + 5 * q^89 - 22 * q^91 + 4 * q^92 + 9 * q^94 + 10 * q^95 + 6 * 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
$5$	−0.813221	−0.363683	−0.181842	−	0.983328i	$-0.558206\pi$
$5$	−0.813221	−0.363683	−0.181842	+	0.983328i	$0.558206\pi$
$6$	0	0
$7$	−3.14341	−1.18810	−0.594049	−	0.804429i	$-0.702471\pi$
$7$	−3.14341	−1.18810	−0.594049	+	0.804429i	$0.702471\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.813221	−0.257163
$11$	0.935416	0.282039	0.141019	−	0.990007i	$-0.454962\pi$
$11$	0.935416	0.282039	0.141019	+	0.990007i	$0.454962\pi$
$12$	0	0
$13$	6.27767	1.74111	0.870556	−	0.492070i	$-0.163759\pi$
$13$	6.27767	1.74111	0.870556	+	0.492070i	$0.163759\pi$
$14$	−3.14341	−0.840112
$15$	0	0
$16$	1.00000	0.250000
$17$	4.52254	1.09688	0.548438	−	0.836191i	$-0.315223\pi$
$17$	4.52254	1.09688	0.548438	+	0.836191i	$0.315223\pi$
$18$	0	0
$19$	3.63511	0.833952	0.416976	−	0.908917i	$-0.363090\pi$
$19$	3.63511	0.833952	0.416976	+	0.908917i	$0.363090\pi$
$20$	−0.813221	−0.181842
$21$	0	0
$22$	0.935416	0.199431
$23$	2.08440	0.434627	0.217313	−	0.976102i	$-0.430271\pi$
$23$	2.08440	0.434627	0.217313	+	0.976102i	$0.430271\pi$
$24$	0	0
$25$	−4.33867	−0.867734
$26$	6.27767	1.23115
$27$	0	0
$28$	−3.14341	−0.594049
$29$	7.17238	1.33188	0.665938	−	0.746007i	$-0.268031\pi$
$29$	7.17238	1.33188	0.665938	+	0.746007i	$0.268031\pi$
$30$	0	0
$31$	−3.77203	−0.677477	−0.338738	−	0.940881i	$-0.610000\pi$
$31$	−3.77203	−0.677477	−0.338738	+	0.940881i	$0.610000\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.52254	0.775609
$35$	2.55629	0.432091
$36$	0	0
$37$	−8.05157	−1.32367	−0.661835	−	0.749650i	$-0.730222\pi$
$37$	−8.05157	−1.32367	−0.661835	+	0.749650i	$0.730222\pi$
$38$	3.63511	0.589693
$39$	0	0
$40$	−0.813221	−0.128582
$41$	−0.104212	−0.0162751	−0.00813755	−	0.999967i	$-0.502590\pi$
$41$	−0.104212	−0.0162751	−0.00813755	+	0.999967i	$0.502590\pi$
$42$	0	0
$43$	0.412877	0.0629632	0.0314816	−	0.999504i	$-0.489977\pi$
$43$	0.412877	0.0629632	0.0314816	+	0.999504i	$0.489977\pi$
$44$	0.935416	0.141019
$45$	0	0
$46$	2.08440	0.307328
$47$	−9.24902	−1.34911	−0.674554	−	0.738225i	$-0.735664\pi$
$47$	−9.24902	−1.34911	−0.674554	+	0.738225i	$0.735664\pi$
$48$	0	0
$49$	2.88102	0.411575
$50$	−4.33867	−0.613581
$51$	0	0
$52$	6.27767	0.870556
$53$	4.95783	0.681011	0.340505	−	0.940243i	$-0.389402\pi$
$53$	4.95783	0.681011	0.340505	+	0.940243i	$0.389402\pi$
$54$	0	0
$55$	−0.760700	−0.102573
$56$	−3.14341	−0.420056
$57$	0	0
$58$	7.17238	0.941779
$59$	−1.98123	−0.257934	−0.128967	−	0.991649i	$-0.541166\pi$
$59$	−1.98123	−0.257934	−0.128967	+	0.991649i	$0.541166\pi$
$60$	0	0
$61$	−11.8824	−1.52139	−0.760694	−	0.649110i	$-0.775141\pi$
$61$	−11.8824	−1.52139	−0.760694	+	0.649110i	$0.775141\pi$
$62$	−3.77203	−0.479048
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.10513	−0.633214
$66$	0	0
$67$	4.21544	0.514997	0.257499	−	0.966279i	$-0.417102\pi$
$67$	4.21544	0.514997	0.257499	+	0.966279i	$0.417102\pi$
$68$	4.52254	0.548438
$69$	0	0
$70$	2.55629	0.305535
$71$	13.4960	1.60168	0.800841	−	0.598877i	$-0.204386\pi$
$71$	13.4960	1.60168	0.800841	+	0.598877i	$0.204386\pi$
$72$	0	0
$73$	−11.3749	−1.33133	−0.665667	−	0.746249i	$-0.731853\pi$
$73$	−11.3749	−1.33133	−0.665667	+	0.746249i	$0.731853\pi$
$74$	−8.05157	−0.935976
$75$	0	0
$76$	3.63511	0.416976
$77$	−2.94040	−0.335089
$78$	0	0
$79$	14.7420	1.65860	0.829300	−	0.558804i	$-0.188740\pi$
$79$	14.7420	1.65860	0.829300	+	0.558804i	$0.188740\pi$
$80$	−0.813221	−0.0909209
$81$	0	0
$82$	−0.104212	−0.0115082
$83$	7.06816	0.775832	0.387916	−	0.921695i	$-0.373195\pi$
$83$	7.06816	0.775832	0.387916	+	0.921695i	$0.373195\pi$
$84$	0	0
$85$	−3.67782	−0.398916
$86$	0.412877	0.0445217
$87$	0	0
$88$	0.935416	0.0997157
$89$	7.76652	0.823250	0.411625	−	0.911353i	$-0.364961\pi$
$89$	7.76652	0.823250	0.411625	+	0.911353i	$0.364961\pi$
$90$	0	0
$91$	−19.7333	−2.06861
$92$	2.08440	0.217313
$93$	0	0
$94$	−9.24902	−0.953964
$95$	−2.95615	−0.303295
$96$	0	0
$97$	11.1338	1.13046	0.565232	−	0.824932i	$-0.308786\pi$
$97$	11.1338	1.13046	0.565232	+	0.824932i	$0.308786\pi$
$98$	2.88102	0.291027
$99$	0	0
$100$	−4.33867	−0.433867
$101$	9.86429	0.981534	0.490767	−	0.871291i	$-0.336717\pi$
$101$	9.86429	0.981534	0.490767	+	0.871291i	$0.336717\pi$
$102$	0	0
$103$	−7.33892	−0.723126	−0.361563	−	0.932348i	$-0.617757\pi$
$103$	−7.33892	−0.723126	−0.361563	+	0.932348i	$0.617757\pi$
$104$	6.27767	0.615576
$105$	0	0
$106$	4.95783	0.481547
$107$	−1.78748	−0.172802	−0.0864009	−	0.996260i	$-0.527537\pi$
$107$	−1.78748	−0.172802	−0.0864009	+	0.996260i	$0.527537\pi$
$108$	0	0
$109$	1.54188	0.147686	0.0738428	−	0.997270i	$-0.476474\pi$
$109$	1.54188	0.147686	0.0738428	+	0.997270i	$0.476474\pi$
$110$	−0.760700	−0.0725299
$111$	0	0
$112$	−3.14341	−0.297024
$113$	−15.5871	−1.46631	−0.733155	−	0.680061i	$-0.761953\pi$
$113$	−15.5871	−1.46631	−0.733155	+	0.680061i	$0.761953\pi$
$114$	0	0
$115$	−1.69508	−0.158067
$116$	7.17238	0.665938
$117$	0	0
$118$	−1.98123	−0.182387
$119$	−14.2162	−1.30320
$120$	0	0
$121$	−10.1250	−0.920454
$122$	−11.8824	−1.07578
$123$	0	0
$124$	−3.77203	−0.338738
$125$	7.59440	0.679264
$126$	0	0
$127$	9.83548	0.872757	0.436379	−	0.899763i	$-0.356261\pi$
$127$	9.83548	0.872757	0.436379	+	0.899763i	$0.356261\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−5.10513	−0.447750
$131$	10.0639	0.879291	0.439645	−	0.898171i	$-0.355104\pi$
$131$	10.0639	0.879291	0.439645	+	0.898171i	$0.355104\pi$
$132$	0	0
$133$	−11.4266	−0.990816
$134$	4.21544	0.364158
$135$	0	0
$136$	4.52254	0.387805
$137$	2.95002	0.252037	0.126019	−	0.992028i	$-0.459780\pi$
$137$	2.95002	0.252037	0.126019	+	0.992028i	$0.459780\pi$
$138$	0	0
$139$	5.88956	0.499546	0.249773	−	0.968304i	$-0.419644\pi$
$139$	5.88956	0.499546	0.249773	+	0.968304i	$0.419644\pi$
$140$	2.55629	0.216046
$141$	0	0
$142$	13.4960	1.13256
$143$	5.87223	0.491061
$144$	0	0
$145$	−5.83273	−0.484382
$146$	−11.3749	−0.941395
$147$	0	0
$148$	−8.05157	−0.661835
$149$	1.57392	0.128941	0.0644703	−	0.997920i	$-0.479464\pi$
$149$	1.57392	0.128941	0.0644703	+	0.997920i	$0.479464\pi$
$150$	0	0
$151$	18.2020	1.48125	0.740627	−	0.671916i	$-0.234529\pi$
$151$	18.2020	1.48125	0.740627	+	0.671916i	$0.234529\pi$
$152$	3.63511	0.294847
$153$	0	0
$154$	−2.94040	−0.236944
$155$	3.06750	0.246387
$156$	0	0
$157$	−6.36868	−0.508276	−0.254138	−	0.967168i	$-0.581792\pi$
$157$	−6.36868	−0.508276	−0.254138	+	0.967168i	$0.581792\pi$
$158$	14.7420	1.17281
$159$	0	0
$160$	−0.813221	−0.0642908
$161$	−6.55212	−0.516379
$162$	0	0
$163$	19.6461	1.53880	0.769400	−	0.638767i	$-0.220555\pi$
$163$	19.6461	1.53880	0.769400	+	0.638767i	$0.220555\pi$
$164$	−0.104212	−0.00813755
$165$	0	0
$166$	7.06816	0.548596
$167$	14.1270	1.09318	0.546591	−	0.837400i	$-0.315925\pi$
$167$	14.1270	1.09318	0.546591	+	0.837400i	$0.315925\pi$
$168$	0	0
$169$	26.4091	2.03147
$170$	−3.67782	−0.282076
$171$	0	0
$172$	0.412877	0.0314816
$173$	−8.56018	−0.650818	−0.325409	−	0.945573i	$-0.605502\pi$
$173$	−8.56018	−0.650818	−0.325409	+	0.945573i	$0.605502\pi$
$174$	0	0
$175$	13.6382	1.03095
$176$	0.935416	0.0705097
$177$	0	0
$178$	7.76652	0.582126
$179$	6.04290	0.451667	0.225834	−	0.974166i	$-0.427489\pi$
$179$	6.04290	0.451667	0.225834	+	0.974166i	$0.427489\pi$
$180$	0	0
$181$	16.5698	1.23162	0.615810	−	0.787895i	$-0.288829\pi$
$181$	16.5698	1.23162	0.615810	+	0.787895i	$0.288829\pi$
$182$	−19.7333	−1.46273
$183$	0	0
$184$	2.08440	0.153664
$185$	6.54771	0.481397
$186$	0	0
$187$	4.23046	0.309362
$188$	−9.24902	−0.674554
$189$	0	0
$190$	−2.95615	−0.214462
$191$	−0.168957	−0.0122253	−0.00611263	−	0.999981i	$-0.501946\pi$
$191$	−0.168957	−0.0122253	−0.00611263	+	0.999981i	$0.501946\pi$
$192$	0	0
$193$	5.62723	0.405057	0.202528	−	0.979276i	$-0.435084\pi$
$193$	5.62723	0.405057	0.202528	+	0.979276i	$0.435084\pi$
$194$	11.1338	0.799359
$195$	0	0
$196$	2.88102	0.205787
$197$	−15.9688	−1.13773	−0.568864	−	0.822431i	$-0.692617\pi$
$197$	−15.9688	−1.13773	−0.568864	+	0.822431i	$0.692617\pi$
$198$	0	0
$199$	17.5515	1.24420	0.622098	−	0.782940i	$-0.286281\pi$
$199$	17.5515	1.24420	0.622098	+	0.782940i	$0.286281\pi$
$200$	−4.33867	−0.306790
$201$	0	0
$202$	9.86429	0.694049
$203$	−22.5457	−1.58240
$204$	0	0
$205$	0.0847470	0.00591899
$206$	−7.33892	−0.511327
$207$	0	0
$208$	6.27767	0.435278
$209$	3.40034	0.235207
$210$	0	0
$211$	−20.5889	−1.41740	−0.708700	−	0.705510i	$-0.750718\pi$
$211$	−20.5889	−1.41740	−0.708700	+	0.705510i	$0.750718\pi$
$212$	4.95783	0.340505
$213$	0	0
$214$	−1.78748	−0.122189
$215$	−0.335760	−0.0228987
$216$	0	0
$217$	11.8570	0.804908
$218$	1.54188	0.104429
$219$	0	0
$220$	−0.760700	−0.0512864
$221$	28.3910	1.90979
$222$	0	0
$223$	−4.38001	−0.293307	−0.146654	−	0.989188i	$-0.546850\pi$
$223$	−4.38001	−0.293307	−0.146654	+	0.989188i	$0.546850\pi$
$224$	−3.14341	−0.210028
$225$	0	0
$226$	−15.5871	−1.03684
$227$	25.8731	1.71726	0.858628	−	0.512599i	$-0.171317\pi$
$227$	25.8731	1.71726	0.858628	+	0.512599i	$0.171317\pi$
$228$	0	0
$229$	−14.6201	−0.966123	−0.483061	−	0.875586i	$-0.660475\pi$
$229$	−14.6201	−0.966123	−0.483061	+	0.875586i	$0.660475\pi$
$230$	−1.69508	−0.111770
$231$	0	0
$232$	7.17238	0.470890
$233$	17.8827	1.17154	0.585768	−	0.810478i	$-0.300793\pi$
$233$	17.8827	1.17154	0.585768	+	0.810478i	$0.300793\pi$
$234$	0	0
$235$	7.52150	0.490648
$236$	−1.98123	−0.128967
$237$	0	0
$238$	−14.2162	−0.921499
$239$	2.21652	0.143375	0.0716873	−	0.997427i	$-0.477162\pi$
$239$	2.21652	0.143375	0.0716873	+	0.997427i	$0.477162\pi$
$240$	0	0
$241$	−2.40173	−0.154709	−0.0773546	−	0.997004i	$-0.524647\pi$
$241$	−2.40173	−0.154709	−0.0773546	+	0.997004i	$0.524647\pi$
$242$	−10.1250	−0.650859
$243$	0	0
$244$	−11.8824	−0.760694
$245$	−2.34291	−0.149683
$246$	0	0
$247$	22.8200	1.45200
$248$	−3.77203	−0.239524
$249$	0	0
$250$	7.59440	0.480312
$251$	2.82570	0.178357	0.0891783	−	0.996016i	$-0.471576\pi$
$251$	2.82570	0.178357	0.0891783	+	0.996016i	$0.471576\pi$
$252$	0	0
$253$	1.94978	0.122582
$254$	9.83548	0.617133
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.8910	−0.804117	−0.402058	−	0.915614i	$-0.631705\pi$
$257$	−12.8910	−0.804117	−0.402058	+	0.915614i	$0.631705\pi$
$258$	0	0
$259$	25.3094	1.57265
$260$	−5.10513	−0.316607
$261$	0	0
$262$	10.0639	0.621752
$263$	19.4774	1.20103	0.600515	−	0.799613i	$-0.294962\pi$
$263$	19.4774	1.20103	0.600515	+	0.799613i	$0.294962\pi$
$264$	0	0
$265$	−4.03181	−0.247672
$266$	−11.4266	−0.700613
$267$	0	0
$268$	4.21544	0.257499
$269$	30.6652	1.86969	0.934845	−	0.355055i	$-0.115538\pi$
$269$	30.6652	1.86969	0.934845	+	0.355055i	$0.115538\pi$
$270$	0	0
$271$	7.27064	0.441660	0.220830	−	0.975312i	$-0.429123\pi$
$271$	7.27064	0.441660	0.220830	+	0.975312i	$0.429123\pi$
$272$	4.52254	0.274219
$273$	0	0
$274$	2.95002	0.178217
$275$	−4.05846	−0.244735
$276$	0	0
$277$	13.8174	0.830209	0.415104	−	0.909774i	$-0.363745\pi$
$277$	13.8174	0.830209	0.415104	+	0.909774i	$0.363745\pi$
$278$	5.88956	0.353232
$279$	0	0
$280$	2.55629	0.152767
$281$	−10.7396	−0.640672	−0.320336	−	0.947304i	$-0.603796\pi$
$281$	−10.7396	−0.640672	−0.320336	+	0.947304i	$0.603796\pi$
$282$	0	0
$283$	18.4761	1.09829	0.549145	−	0.835727i	$-0.314954\pi$
$283$	18.4761	1.09829	0.549145	+	0.835727i	$0.314954\pi$
$284$	13.4960	0.800841
$285$	0	0
$286$	5.87223	0.347232
$287$	0.327580	0.0193364
$288$	0	0
$289$	3.45336	0.203139
$290$	−5.83273	−0.342509
$291$	0	0
$292$	−11.3749	−0.665667
$293$	−22.6917	−1.32567	−0.662833	−	0.748767i	$-0.730646\pi$
$293$	−22.6917	−1.32567	−0.662833	+	0.748767i	$0.730646\pi$
$294$	0	0
$295$	1.61118	0.0938062
$296$	−8.05157	−0.467988
$297$	0	0
$298$	1.57392	0.0911748
$299$	13.0852	0.756734
$300$	0	0
$301$	−1.29784	−0.0748064
$302$	18.2020	1.04740
$303$	0	0
$304$	3.63511	0.208488
$305$	9.66304	0.553304
$306$	0	0
$307$	16.6848	0.952251	0.476126	−	0.879377i	$-0.342041\pi$
$307$	16.6848	0.952251	0.476126	+	0.879377i	$0.342041\pi$
$308$	−2.94040	−0.167545
$309$	0	0
$310$	3.06750	0.174222
$311$	−18.9279	−1.07330	−0.536652	−	0.843804i	$-0.680311\pi$
$311$	−18.9279	−1.07330	−0.536652	+	0.843804i	$0.680311\pi$
$312$	0	0
$313$	18.6899	1.05642	0.528208	−	0.849115i	$-0.322864\pi$
$313$	18.6899	1.05642	0.528208	+	0.849115i	$0.322864\pi$
$314$	−6.36868	−0.359405
$315$	0	0
$316$	14.7420	0.829300
$317$	−31.8168	−1.78701	−0.893504	−	0.449054i	$-0.851761\pi$
$317$	−31.8168	−1.78701	−0.893504	+	0.449054i	$0.851761\pi$
$318$	0	0
$319$	6.70916	0.375641
$320$	−0.813221	−0.0454604
$321$	0	0
$322$	−6.55212	−0.365135
$323$	16.4399	0.914743
$324$	0	0
$325$	−27.2367	−1.51082
$326$	19.6461	1.08810
$327$	0	0
$328$	−0.104212	−0.00575412
$329$	29.0735	1.60287
$330$	0	0
$331$	−13.4665	−0.740184	−0.370092	−	0.928995i	$-0.620674\pi$
$331$	−13.4665	−0.740184	−0.370092	+	0.928995i	$0.620674\pi$
$332$	7.06816	0.387916
$333$	0	0
$334$	14.1270	0.772996
$335$	−3.42808	−0.187296
$336$	0	0
$337$	−27.9427	−1.52214	−0.761068	−	0.648672i	$-0.775325\pi$
$337$	−27.9427	−1.52214	−0.761068	+	0.648672i	$0.775325\pi$
$338$	26.4091	1.43647
$339$	0	0
$340$	−3.67782	−0.199458
$341$	−3.52842	−0.191075
$342$	0	0
$343$	12.9476	0.699106
$344$	0.412877	0.0222608
$345$	0	0
$346$	−8.56018	−0.460198
$347$	−3.49880	−0.187825	−0.0939127	−	0.995580i	$-0.529937\pi$
$347$	−3.49880	−0.187825	−0.0939127	+	0.995580i	$0.529937\pi$
$348$	0	0
$349$	−2.41464	−0.129253	−0.0646263	−	0.997910i	$-0.520586\pi$
$349$	−2.41464	−0.129253	−0.0646263	+	0.997910i	$0.520586\pi$
$350$	13.6382	0.728994
$351$	0	0
$352$	0.935416	0.0498579
$353$	6.00091	0.319396	0.159698	−	0.987166i	$-0.448948\pi$
$353$	6.00091	0.319396	0.159698	+	0.987166i	$0.448948\pi$
$354$	0	0
$355$	−10.9752	−0.582505
$356$	7.76652	0.411625
$357$	0	0
$358$	6.04290	0.319377
$359$	−10.6230	−0.560663	−0.280332	−	0.959903i	$-0.590444\pi$
$359$	−10.6230	−0.560663	−0.280332	+	0.959903i	$0.590444\pi$
$360$	0	0
$361$	−5.78595	−0.304524
$362$	16.5698	0.870887
$363$	0	0
$364$	−19.7333	−1.03431
$365$	9.25032	0.484184
$366$	0	0
$367$	23.3767	1.22025	0.610127	−	0.792304i	$-0.291119\pi$
$367$	23.3767	1.22025	0.610127	+	0.792304i	$0.291119\pi$
$368$	2.08440	0.108657
$369$	0	0
$370$	6.54771	0.340399
$371$	−15.5845	−0.809107
$372$	0	0
$373$	4.27846	0.221530	0.110765	−	0.993847i	$-0.464670\pi$
$373$	4.27846	0.221530	0.110765	+	0.993847i	$0.464670\pi$
$374$	4.23046	0.218752
$375$	0	0
$376$	−9.24902	−0.476982
$377$	45.0258	2.31895
$378$	0	0
$379$	13.4590	0.691343	0.345671	−	0.938356i	$-0.387651\pi$
$379$	13.4590	0.691343	0.345671	+	0.938356i	$0.387651\pi$
$380$	−2.95615	−0.151647
$381$	0	0
$382$	−0.168957	−0.00864457
$383$	−12.2710	−0.627020	−0.313510	−	0.949585i	$-0.601505\pi$
$383$	−12.2710	−0.627020	−0.313510	+	0.949585i	$0.601505\pi$
$384$	0	0
$385$	2.39119	0.121866
$386$	5.62723	0.286419
$387$	0	0
$388$	11.1338	0.565232
$389$	−24.3325	−1.23371	−0.616854	−	0.787078i	$-0.711593\pi$
$389$	−24.3325	−1.23371	−0.616854	+	0.787078i	$0.711593\pi$
$390$	0	0
$391$	9.42677	0.476732
$392$	2.88102	0.145514
$393$	0	0
$394$	−15.9688	−0.804496
$395$	−11.9885	−0.603205
$396$	0	0
$397$	13.5772	0.681420	0.340710	−	0.940168i	$-0.389333\pi$
$397$	13.5772	0.681420	0.340710	+	0.940168i	$0.389333\pi$
$398$	17.5515	0.879779
$399$	0	0
$400$	−4.33867	−0.216934
$401$	11.3626	0.567421	0.283711	−	0.958910i	$-0.408434\pi$
$401$	11.3626	0.567421	0.283711	+	0.958910i	$0.408434\pi$
$402$	0	0
$403$	−23.6796	−1.17956
$404$	9.86429	0.490767
$405$	0	0
$406$	−22.5457	−1.11893
$407$	−7.53157	−0.373326
$408$	0	0
$409$	−24.8484	−1.22867	−0.614337	−	0.789044i	$-0.710576\pi$
$409$	−24.8484	−1.22867	−0.614337	+	0.789044i	$0.710576\pi$
$410$	0.0847470	0.00418536
$411$	0	0
$412$	−7.33892	−0.361563
$413$	6.22781	0.306450
$414$	0	0
$415$	−5.74798	−0.282157
$416$	6.27767	0.307788
$417$	0	0
$418$	3.40034	0.166316
$419$	10.4331	0.509689	0.254844	−	0.966982i	$-0.417976\pi$
$419$	10.4331	0.509689	0.254844	+	0.966982i	$0.417976\pi$
$420$	0	0
$421$	3.33127	0.162356	0.0811782	−	0.996700i	$-0.474132\pi$
$421$	3.33127	0.162356	0.0811782	+	0.996700i	$0.474132\pi$
$422$	−20.5889	−1.00225
$423$	0	0
$424$	4.95783	0.240774
$425$	−19.6218	−0.951798
$426$	0	0
$427$	37.3513	1.80756
$428$	−1.78748	−0.0864009
$429$	0	0
$430$	−0.335760	−0.0161918
$431$	31.1223	1.49911	0.749554	−	0.661943i	$-0.230268\pi$
$431$	31.1223	1.49911	0.749554	+	0.661943i	$0.230268\pi$
$432$	0	0
$433$	−33.6809	−1.61860	−0.809301	−	0.587395i	$-0.800154\pi$
$433$	−33.6809	−1.61860	−0.809301	+	0.587395i	$0.800154\pi$
$434$	11.8570	0.569156
$435$	0	0
$436$	1.54188	0.0738428
$437$	7.57702	0.362458
$438$	0	0
$439$	9.72891	0.464336	0.232168	−	0.972676i	$-0.425418\pi$
$439$	9.72891	0.464336	0.232168	+	0.972676i	$0.425418\pi$
$440$	−0.760700	−0.0362650
$441$	0	0
$442$	28.3910	1.35042
$443$	−36.9189	−1.75407	−0.877035	−	0.480427i	$-0.840482\pi$
$443$	−36.9189	−1.75407	−0.877035	+	0.480427i	$0.840482\pi$
$444$	0	0
$445$	−6.31590	−0.299402
$446$	−4.38001	−0.207399
$447$	0	0
$448$	−3.14341	−0.148512
$449$	32.4703	1.53237	0.766185	−	0.642620i	$-0.222153\pi$
$449$	32.4703	1.53237	0.766185	+	0.642620i	$0.222153\pi$
$450$	0	0
$451$	−0.0974812	−0.00459021
$452$	−15.5871	−0.733155
$453$	0	0
$454$	25.8731	1.21428
$455$	16.0475	0.752319
$456$	0	0
$457$	9.95036	0.465458	0.232729	−	0.972542i	$-0.425234\pi$
$457$	9.95036	0.465458	0.232729	+	0.972542i	$0.425234\pi$
$458$	−14.6201	−0.683152
$459$	0	0
$460$	−1.69508	−0.0790333
$461$	1.65594	0.0771246	0.0385623	−	0.999256i	$-0.487722\pi$
$461$	1.65594	0.0771246	0.0385623	+	0.999256i	$0.487722\pi$
$462$	0	0
$463$	36.4531	1.69412	0.847059	−	0.531498i	$-0.178371\pi$
$463$	36.4531	1.69412	0.847059	+	0.531498i	$0.178371\pi$
$464$	7.17238	0.332969
$465$	0	0
$466$	17.8827	0.828402
$467$	−3.67241	−0.169939	−0.0849694	−	0.996384i	$-0.527079\pi$
$467$	−3.67241	−0.169939	−0.0849694	+	0.996384i	$0.527079\pi$
$468$	0	0
$469$	−13.2508	−0.611867
$470$	7.52150	0.346941
$471$	0	0
$472$	−1.98123	−0.0911934
$473$	0.386212	0.0177580
$474$	0	0
$475$	−15.7716	−0.723649
$476$	−14.2162	−0.651598
$477$	0	0
$478$	2.21652	0.101381
$479$	19.1667	0.875748	0.437874	−	0.899036i	$-0.355732\pi$
$479$	19.1667	0.875748	0.437874	+	0.899036i	$0.355732\pi$
$480$	0	0
$481$	−50.5451	−2.30466
$482$	−2.40173	−0.109396
$483$	0	0
$484$	−10.1250	−0.460227
$485$	−9.05423	−0.411131
$486$	0	0
$487$	−36.7939	−1.66729	−0.833646	−	0.552298i	$-0.813751\pi$
$487$	−36.7939	−1.66729	−0.833646	+	0.552298i	$0.813751\pi$
$488$	−11.8824	−0.537892
$489$	0	0
$490$	−2.34291	−0.105842
$491$	−13.9423	−0.629205	−0.314603	−	0.949223i	$-0.601871\pi$
$491$	−13.9423	−0.629205	−0.314603	+	0.949223i	$0.601871\pi$
$492$	0	0
$493$	32.4373	1.46090
$494$	22.8200	1.02672
$495$	0	0
$496$	−3.77203	−0.169369
$497$	−42.4235	−1.90295
$498$	0	0
$499$	−12.0514	−0.539496	−0.269748	−	0.962931i	$-0.586940\pi$
$499$	−12.0514	−0.539496	−0.269748	+	0.962931i	$0.586940\pi$
$500$	7.59440	0.339632
$501$	0	0
$502$	2.82570	0.126117
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−8.02185	−0.356968
$506$	1.94978	0.0866783
$507$	0	0
$508$	9.83548	0.436379
$509$	13.3438	0.591452	0.295726	−	0.955273i	$-0.404438\pi$
$509$	13.3438	0.591452	0.295726	+	0.955273i	$0.404438\pi$
$510$	0	0
$511$	35.7560	1.58175
$512$	1.00000	0.0441942
$513$	0	0
$514$	−12.8910	−0.568597
$515$	5.96817	0.262989
$516$	0	0
$517$	−8.65169	−0.380501
$518$	25.3094	1.11203
$519$	0	0
$520$	−5.10513	−0.223875
$521$	17.4349	0.763835	0.381918	−	0.924196i	$-0.375264\pi$
$521$	17.4349	0.763835	0.381918	+	0.924196i	$0.375264\pi$
$522$	0	0
$523$	−40.2946	−1.76196	−0.880981	−	0.473152i	$-0.843116\pi$
$523$	−40.2946	−1.76196	−0.880981	+	0.473152i	$0.843116\pi$
$524$	10.0639	0.439645
$525$	0	0
$526$	19.4774	0.849257
$527$	−17.0592	−0.743109
$528$	0	0
$529$	−18.6553	−0.811099
$530$	−4.03181	−0.175131
$531$	0	0
$532$	−11.4266	−0.495408
$533$	−0.654205	−0.0283368
$534$	0	0
$535$	1.45361	0.0628452
$536$	4.21544	0.182079
$537$	0	0
$538$	30.6652	1.32207
$539$	2.69496	0.116080
$540$	0	0
$541$	−24.7221	−1.06288	−0.531442	−	0.847094i	$-0.678350\pi$
$541$	−24.7221	−1.06288	−0.531442	+	0.847094i	$0.678350\pi$
$542$	7.27064	0.312301
$543$	0	0
$544$	4.52254	0.193902
$545$	−1.25389	−0.0537108
$546$	0	0
$547$	−32.4623	−1.38799	−0.693994	−	0.719980i	$-0.744151\pi$
$547$	−32.4623	−1.38799	−0.693994	+	0.719980i	$0.744151\pi$
$548$	2.95002	0.126019
$549$	0	0
$550$	−4.05846	−0.173053
$551$	26.0724	1.11072
$552$	0	0
$553$	−46.3400	−1.97058
$554$	13.8174	0.587046
$555$	0	0
$556$	5.88956	0.249773
$557$	38.2242	1.61961	0.809806	−	0.586698i	$-0.199573\pi$
$557$	38.2242	1.61961	0.809806	+	0.586698i	$0.199573\pi$
$558$	0	0
$559$	2.59190	0.109626
$560$	2.55629	0.108023
$561$	0	0
$562$	−10.7396	−0.453024
$563$	8.20606	0.345844	0.172922	−	0.984936i	$-0.444679\pi$
$563$	8.20606	0.345844	0.172922	+	0.984936i	$0.444679\pi$
$564$	0	0
$565$	12.6758	0.533273
$566$	18.4761	0.776608
$567$	0	0
$568$	13.4960	0.566280
$569$	−9.53803	−0.399855	−0.199928	−	0.979811i	$-0.564071\pi$
$569$	−9.53803	−0.399855	−0.199928	+	0.979811i	$0.564071\pi$
$570$	0	0
$571$	−32.6514	−1.36642	−0.683210	−	0.730222i	$-0.739417\pi$
$571$	−32.6514	−1.36642	−0.683210	+	0.730222i	$0.739417\pi$
$572$	5.87223	0.245530
$573$	0	0
$574$	0.327580	0.0136729
$575$	−9.04352	−0.377141
$576$	0	0
$577$	18.3529	0.764042	0.382021	−	0.924154i	$-0.375228\pi$
$577$	18.3529	0.764042	0.382021	+	0.924154i	$0.375228\pi$
$578$	3.45336	0.143641
$579$	0	0
$580$	−5.83273	−0.242191
$581$	−22.2181	−0.921764
$582$	0	0
$583$	4.63764	0.192071
$584$	−11.3749	−0.470697
$585$	0	0
$586$	−22.6917	−0.937387
$587$	14.9281	0.616150	0.308075	−	0.951362i	$-0.400315\pi$
$587$	14.9281	0.616150	0.308075	+	0.951362i	$0.400315\pi$
$588$	0	0
$589$	−13.7118	−0.564983
$590$	1.61118	0.0663310
$591$	0	0
$592$	−8.05157	−0.330917
$593$	−24.3426	−0.999630	−0.499815	−	0.866132i	$-0.666599\pi$
$593$	−24.3426	−0.999630	−0.499815	+	0.866132i	$0.666599\pi$
$594$	0	0
$595$	11.5609	0.473951
$596$	1.57392	0.0644703
$597$	0	0
$598$	13.0852	0.535092
$599$	18.2919	0.747386	0.373693	−	0.927552i	$-0.378091\pi$
$599$	18.2919	0.747386	0.373693	+	0.927552i	$0.378091\pi$
$600$	0	0
$601$	−26.1164	−1.06531	−0.532655	−	0.846333i	$-0.678806\pi$
$601$	−26.1164	−1.06531	−0.532655	+	0.846333i	$0.678806\pi$
$602$	−1.29784	−0.0528961
$603$	0	0
$604$	18.2020	0.740627
$605$	8.23386	0.334754
$606$	0	0
$607$	−12.8673	−0.522269	−0.261135	−	0.965302i	$-0.584097\pi$
$607$	−12.8673	−0.522269	−0.261135	+	0.965302i	$0.584097\pi$
$608$	3.63511	0.147423
$609$	0	0
$610$	9.66304	0.391245
$611$	−58.0623	−2.34895
$612$	0	0
$613$	−10.5919	−0.427804	−0.213902	−	0.976855i	$-0.568617\pi$
$613$	−10.5919	−0.427804	−0.213902	+	0.976855i	$0.568617\pi$
$614$	16.6848	0.673343
$615$	0	0
$616$	−2.94040	−0.118472
$617$	2.94095	0.118398	0.0591990	−	0.998246i	$-0.481145\pi$
$617$	2.94095	0.118398	0.0591990	+	0.998246i	$0.481145\pi$
$618$	0	0
$619$	10.1326	0.407263	0.203632	−	0.979048i	$-0.434725\pi$
$619$	10.1326	0.407263	0.203632	+	0.979048i	$0.434725\pi$
$620$	3.06750	0.123194
$621$	0	0
$622$	−18.9279	−0.758940
$623$	−24.4134	−0.978101
$624$	0	0
$625$	15.5174	0.620697
$626$	18.6899	0.746999
$627$	0	0
$628$	−6.36868	−0.254138
$629$	−36.4135	−1.45190
$630$	0	0
$631$	20.3719	0.810993	0.405497	−	0.914097i	$-0.367099\pi$
$631$	20.3719	0.810993	0.405497	+	0.914097i	$0.367099\pi$
$632$	14.7420	0.586403
$633$	0	0
$634$	−31.8168	−1.26361
$635$	−7.99842	−0.317407
$636$	0	0
$637$	18.0861	0.716598
$638$	6.70916	0.265618
$639$	0	0
$640$	−0.813221	−0.0321454
$641$	15.2866	0.603783	0.301892	−	0.953342i	$-0.402382\pi$
$641$	15.2866	0.603783	0.301892	+	0.953342i	$0.402382\pi$
$642$	0	0
$643$	−16.8979	−0.666390	−0.333195	−	0.942858i	$-0.608127\pi$
$643$	−16.8979	−0.666390	−0.333195	+	0.942858i	$0.608127\pi$
$644$	−6.55212	−0.258190
$645$	0	0
$646$	16.4399	0.646821
$647$	−6.90398	−0.271423	−0.135712	−	0.990748i	$-0.543332\pi$
$647$	−6.90398	−0.271423	−0.135712	+	0.990748i	$0.543332\pi$
$648$	0	0
$649$	−1.85327	−0.0727473
$650$	−27.2367	−1.06831
$651$	0	0
$652$	19.6461	0.769400
$653$	−27.7005	−1.08400	−0.542002	−	0.840377i	$-0.682333\pi$
$653$	−27.7005	−1.08400	−0.542002	+	0.840377i	$0.682333\pi$
$654$	0	0
$655$	−8.18421	−0.319783
$656$	−0.104212	−0.00406878
$657$	0	0
$658$	29.0735	1.13340
$659$	−19.9962	−0.778942	−0.389471	−	0.921039i	$-0.627342\pi$
$659$	−19.9962	−0.778942	−0.389471	+	0.921039i	$0.627342\pi$
$660$	0	0
$661$	−6.81293	−0.264992	−0.132496	−	0.991184i	$-0.542299\pi$
$661$	−6.81293	−0.264992	−0.132496	+	0.991184i	$0.542299\pi$
$662$	−13.4665	−0.523389
$663$	0	0
$664$	7.06816	0.274298
$665$	9.29239	0.360343
$666$	0	0
$667$	14.9501	0.578869
$668$	14.1270	0.546591
$669$	0	0
$670$	−3.42808	−0.132438
$671$	−11.1150	−0.429090
$672$	0	0
$673$	1.57260	0.0606194	0.0303097	−	0.999541i	$-0.490351\pi$
$673$	1.57260	0.0606194	0.0303097	+	0.999541i	$0.490351\pi$
$674$	−27.9427	−1.07631
$675$	0	0
$676$	26.4091	1.01574
$677$	38.0167	1.46110	0.730551	−	0.682858i	$-0.239263\pi$
$677$	38.0167	1.46110	0.730551	+	0.682858i	$0.239263\pi$
$678$	0	0
$679$	−34.9981	−1.34310
$680$	−3.67782	−0.141038
$681$	0	0
$682$	−3.52842	−0.135110
$683$	38.6081	1.47730	0.738649	−	0.674090i	$-0.235464\pi$
$683$	38.6081	1.47730	0.738649	+	0.674090i	$0.235464\pi$
$684$	0	0
$685$	−2.39902	−0.0916617
$686$	12.9476	0.494343
$687$	0	0
$688$	0.412877	0.0157408
$689$	31.1236	1.18572
$690$	0	0
$691$	−37.9253	−1.44275	−0.721374	−	0.692546i	$-0.756489\pi$
$691$	−37.9253	−1.44275	−0.721374	+	0.692546i	$0.756489\pi$
$692$	−8.56018	−0.325409
$693$	0	0
$694$	−3.49880	−0.132813
$695$	−4.78951	−0.181677
$696$	0	0
$697$	−0.471301	−0.0178518
$698$	−2.41464	−0.0913954
$699$	0	0
$700$	13.6382	0.515476
$701$	−35.7661	−1.35087	−0.675433	−	0.737421i	$-0.736043\pi$
$701$	−35.7661	−1.35087	−0.675433	+	0.737421i	$0.736043\pi$
$702$	0	0
$703$	−29.2684	−1.10388
$704$	0.935416	0.0352548
$705$	0	0
$706$	6.00091	0.225847
$707$	−31.0075	−1.16616
$708$	0	0
$709$	21.2634	0.798564	0.399282	−	0.916828i	$-0.369260\pi$
$709$	21.2634	0.798564	0.399282	+	0.916828i	$0.369260\pi$
$710$	−10.9752	−0.411894
$711$	0	0
$712$	7.76652	0.291063
$713$	−7.86241	−0.294450
$714$	0	0
$715$	−4.77542	−0.178591
$716$	6.04290	0.225834
$717$	0	0
$718$	−10.6230	−0.396449
$719$	−45.4811	−1.69616	−0.848079	−	0.529870i	$-0.822241\pi$
$719$	−45.4811	−1.69616	−0.848079	+	0.529870i	$0.822241\pi$
$720$	0	0
$721$	23.0692	0.859143
$722$	−5.78595	−0.215331
$723$	0	0
$724$	16.5698	0.615810
$725$	−31.1186	−1.15572
$726$	0	0
$727$	43.4589	1.61180	0.805901	−	0.592051i	$-0.201681\pi$
$727$	43.4589	1.61180	0.805901	+	0.592051i	$0.201681\pi$
$728$	−19.7333	−0.731364
$729$	0	0
$730$	9.25032	0.342370
$731$	1.86725	0.0690628
$732$	0	0
$733$	−7.44886	−0.275130	−0.137565	−	0.990493i	$-0.543928\pi$
$733$	−7.44886	−0.275130	−0.137565	+	0.990493i	$0.543928\pi$
$734$	23.3767	0.862849
$735$	0	0
$736$	2.08440	0.0768319
$737$	3.94319	0.145249
$738$	0	0
$739$	4.31285	0.158651	0.0793253	−	0.996849i	$-0.474723\pi$
$739$	4.31285	0.158651	0.0793253	+	0.996849i	$0.474723\pi$
$740$	6.54771	0.240698
$741$	0	0
$742$	−15.5845	−0.572125
$743$	−37.3328	−1.36961	−0.684805	−	0.728727i	$-0.740112\pi$
$743$	−37.3328	−1.36961	−0.684805	+	0.728727i	$0.740112\pi$
$744$	0	0
$745$	−1.27995	−0.0468936
$746$	4.27846	0.156645
$747$	0	0
$748$	4.23046	0.154681
$749$	5.61877	0.205305
$750$	0	0
$751$	5.53928	0.202131	0.101066	−	0.994880i	$-0.467775\pi$
$751$	5.53928	0.202131	0.101066	+	0.994880i	$0.467775\pi$
$752$	−9.24902	−0.337277
$753$	0	0
$754$	45.0258	1.63974
$755$	−14.8022	−0.538708
$756$	0	0
$757$	−25.5143	−0.927333	−0.463666	−	0.886010i	$-0.653466\pi$
$757$	−25.5143	−0.927333	−0.463666	+	0.886010i	$0.653466\pi$
$758$	13.4590	0.488853
$759$	0	0
$760$	−2.95615	−0.107231
$761$	31.5088	1.14219	0.571097	−	0.820883i	$-0.306518\pi$
$761$	31.5088	1.14219	0.571097	+	0.820883i	$0.306518\pi$
$762$	0	0
$763$	−4.84677	−0.175465
$764$	−0.168957	−0.00611263
$765$	0	0
$766$	−12.2710	−0.443370
$767$	−12.4375	−0.449091
$768$	0	0
$769$	−3.95793	−0.142727	−0.0713633	−	0.997450i	$-0.522735\pi$
$769$	−3.95793	−0.142727	−0.0713633	+	0.997450i	$0.522735\pi$
$770$	2.39119	0.0861726
$771$	0	0
$772$	5.62723	0.202528
$773$	9.80022	0.352489	0.176245	−	0.984346i	$-0.443605\pi$
$773$	9.80022	0.352489	0.176245	+	0.984346i	$0.443605\pi$
$774$	0	0
$775$	16.3656	0.587870
$776$	11.1338	0.399680
$777$	0	0
$778$	−24.3325	−0.872363
$779$	−0.378821	−0.0135727
$780$	0	0
$781$	12.6244	0.451736
$782$	9.42677	0.337101
$783$	0	0
$784$	2.88102	0.102894
$785$	5.17914	0.184852
$786$	0	0
$787$	−6.53777	−0.233046	−0.116523	−	0.993188i	$-0.537175\pi$
$787$	−6.53777	−0.233046	−0.116523	+	0.993188i	$0.537175\pi$
$788$	−15.9688	−0.568864
$789$	0	0
$790$	−11.9885	−0.426530
$791$	48.9966	1.74212
$792$	0	0
$793$	−74.5939	−2.64891
$794$	13.5772	0.481837
$795$	0	0
$796$	17.5515	0.622098
$797$	−30.9321	−1.09567	−0.547836	−	0.836586i	$-0.684548\pi$
$797$	−30.9321	−1.09567	−0.547836	+	0.836586i	$0.684548\pi$
$798$	0	0
$799$	−41.8291	−1.47981
$800$	−4.33867	−0.153395
$801$	0	0
$802$	11.3626	0.401227
$803$	−10.6403	−0.375487
$804$	0	0
$805$	5.32832	0.187799
$806$	−23.6796	−0.834077
$807$	0	0
$808$	9.86429	0.347025
$809$	32.4321	1.14025	0.570125	−	0.821558i	$-0.306895\pi$
$809$	32.4321	1.14025	0.570125	+	0.821558i	$0.306895\pi$
$810$	0	0
$811$	−12.3578	−0.433942	−0.216971	−	0.976178i	$-0.569618\pi$
$811$	−12.3578	−0.433942	−0.216971	+	0.976178i	$0.569618\pi$
$812$	−22.5457	−0.791199
$813$	0	0
$814$	−7.53157	−0.263981
$815$	−15.9766	−0.559636
$816$	0	0
$817$	1.50085	0.0525083
$818$	−24.8484	−0.868803
$819$	0	0
$820$	0.0847470	0.00295949
$821$	40.0699	1.39845	0.699225	−	0.714902i	$-0.253529\pi$
$821$	40.0699	1.39845	0.699225	+	0.714902i	$0.253529\pi$
$822$	0	0
$823$	27.5103	0.958948	0.479474	−	0.877556i	$-0.340827\pi$
$823$	27.5103	0.958948	0.479474	+	0.877556i	$0.340827\pi$
$824$	−7.33892	−0.255663
$825$	0	0
$826$	6.22781	0.216693
$827$	32.8789	1.14331	0.571656	−	0.820493i	$-0.306301\pi$
$827$	32.8789	1.14331	0.571656	+	0.820493i	$0.306301\pi$
$828$	0	0
$829$	−21.7701	−0.756108	−0.378054	−	0.925784i	$-0.623407\pi$
$829$	−21.7701	−0.756108	−0.378054	+	0.925784i	$0.623407\pi$
$830$	−5.74798	−0.199515
$831$	0	0
$832$	6.27767	0.217639
$833$	13.0295	0.451447
$834$	0	0
$835$	−11.4884	−0.397572
$836$	3.40034	0.117603
$837$	0	0
$838$	10.4331	0.360405
$839$	−15.5554	−0.537032	−0.268516	−	0.963275i	$-0.586533\pi$
$839$	−15.5554	−0.537032	−0.268516	+	0.963275i	$0.586533\pi$
$840$	0	0
$841$	22.4430	0.773895
$842$	3.33127	0.114803
$843$	0	0
$844$	−20.5889	−0.708700
$845$	−21.4764	−0.738812
$846$	0	0
$847$	31.8270	1.09359
$848$	4.95783	0.170253
$849$	0	0
$850$	−19.6218	−0.673023
$851$	−16.7827	−0.575303
$852$	0	0
$853$	−51.2275	−1.75400	−0.876998	−	0.480493i	$-0.840458\pi$
$853$	−51.2275	−1.75400	−0.876998	+	0.480493i	$0.840458\pi$
$854$	37.3513	1.27814
$855$	0	0
$856$	−1.78748	−0.0610947
$857$	−57.7319	−1.97208	−0.986042	−	0.166497i	$-0.946754\pi$
$857$	−57.7319	−1.97208	−0.986042	+	0.166497i	$0.946754\pi$
$858$	0	0
$859$	6.86388	0.234192	0.117096	−	0.993121i	$-0.462641\pi$
$859$	6.86388	0.234192	0.117096	+	0.993121i	$0.462641\pi$
$860$	−0.335760	−0.0114493
$861$	0	0
$862$	31.1223	1.06003
$863$	−44.8679	−1.52732	−0.763661	−	0.645618i	$-0.776600\pi$
$863$	−44.8679	−1.52732	−0.763661	+	0.645618i	$0.776600\pi$
$864$	0	0
$865$	6.96131	0.236692
$866$	−33.6809	−1.14452
$867$	0	0
$868$	11.8570	0.402454
$869$	13.7899	0.467789
$870$	0	0
$871$	26.4631	0.896668
$872$	1.54188	0.0522147
$873$	0	0
$874$	7.57702	0.256297
$875$	−23.8723	−0.807032
$876$	0	0
$877$	44.6205	1.50673	0.753363	−	0.657605i	$-0.228430\pi$
$877$	44.6205	1.50673	0.753363	+	0.657605i	$0.228430\pi$
$878$	9.72891	0.328335
$879$	0	0
$880$	−0.760700	−0.0256432
$881$	11.9111	0.401295	0.200647	−	0.979664i	$-0.435695\pi$
$881$	11.9111	0.401295	0.200647	+	0.979664i	$0.435695\pi$
$882$	0	0
$883$	11.3860	0.383170	0.191585	−	0.981476i	$-0.438637\pi$
$883$	11.3860	0.383170	0.191585	+	0.981476i	$0.438637\pi$
$884$	28.3910	0.954893
$885$	0	0
$886$	−36.9189	−1.24031
$887$	32.5755	1.09378	0.546889	−	0.837205i	$-0.315812\pi$
$887$	32.5755	1.09378	0.546889	+	0.837205i	$0.315812\pi$
$888$	0	0
$889$	−30.9169	−1.03692
$890$	−6.31590	−0.211709
$891$	0	0
$892$	−4.38001	−0.146654
$893$	−33.6212	−1.12509
$894$	0	0
$895$	−4.91421	−0.164264
$896$	−3.14341	−0.105014
$897$	0	0
$898$	32.4703	1.08355
$899$	−27.0544	−0.902316
$900$	0	0
$901$	22.4220	0.746985
$902$	−0.0974812	−0.00324577
$903$	0	0
$904$	−15.5871	−0.518419
$905$	−13.4749	−0.447920
$906$	0	0
$907$	32.6995	1.08577	0.542885	−	0.839807i	$-0.317332\pi$
$907$	32.6995	1.08577	0.542885	+	0.839807i	$0.317332\pi$
$908$	25.8731	0.858628
$909$	0	0
$910$	16.0475	0.531970
$911$	50.7353	1.68093	0.840467	−	0.541863i	$-0.182281\pi$
$911$	50.7353	1.68093	0.840467	+	0.541863i	$0.182281\pi$
$912$	0	0
$913$	6.61168	0.218815
$914$	9.95036	0.329129
$915$	0	0
$916$	−14.6201	−0.483061
$917$	−31.6351	−1.04468
$918$	0	0
$919$	30.2301	0.997199	0.498599	−	0.866833i	$-0.333848\pi$
$919$	30.2301	0.997199	0.498599	+	0.866833i	$0.333848\pi$
$920$	−1.69508	−0.0558850
$921$	0	0
$922$	1.65594	0.0545353
$923$	84.7235	2.78871
$924$	0	0
$925$	34.9331	1.14859
$926$	36.4531	1.19792
$927$	0	0
$928$	7.17238	0.235445
$929$	5.70119	0.187050	0.0935250	−	0.995617i	$-0.470186\pi$
$929$	5.70119	0.187050	0.0935250	+	0.995617i	$0.470186\pi$
$930$	0	0
$931$	10.4729	0.343234
$932$	17.8827	0.585768
$933$	0	0
$934$	−3.67241	−0.120165
$935$	−3.44030	−0.112510
$936$	0	0
$937$	30.1864	0.986147	0.493073	−	0.869988i	$-0.335873\pi$
$937$	30.1864	0.986147	0.493073	+	0.869988i	$0.335873\pi$
$938$	−13.2508	−0.432655
$939$	0	0
$940$	7.52150	0.245324
$941$	30.2132	0.984922	0.492461	−	0.870334i	$-0.336097\pi$
$941$	30.2132	0.984922	0.492461	+	0.870334i	$0.336097\pi$
$942$	0	0
$943$	−0.217218	−0.00707360
$944$	−1.98123	−0.0644834
$945$	0	0
$946$	0.386212	0.0125568
$947$	60.8276	1.97663	0.988316	−	0.152422i	$-0.0487073\pi$
$947$	60.8276	1.97663	0.988316	+	0.152422i	$0.0487073\pi$
$948$	0	0
$949$	−71.4080	−2.31800
$950$	−15.7716	−0.511697
$951$	0	0
$952$	−14.2162	−0.460749
$953$	44.0330	1.42637	0.713185	−	0.700976i	$-0.247252\pi$
$953$	44.0330	1.42637	0.713185	+	0.700976i	$0.247252\pi$
$954$	0	0
$955$	0.137399	0.00444613
$956$	2.21652	0.0716873
$957$	0	0
$958$	19.1667	0.619247
$959$	−9.27312	−0.299445
$960$	0	0
$961$	−16.7718	−0.541025
$962$	−50.5451	−1.62964
$963$	0	0
$964$	−2.40173	−0.0773546
$965$	−4.57618	−0.147313
$966$	0	0
$967$	−50.4567	−1.62258	−0.811290	−	0.584644i	$-0.801234\pi$
$967$	−50.4567	−1.62258	−0.811290	+	0.584644i	$0.801234\pi$
$968$	−10.1250	−0.325430
$969$	0	0
$970$	−9.05423	−0.290714
$971$	−42.8065	−1.37373	−0.686863	−	0.726787i	$-0.741013\pi$
$971$	−42.8065	−1.37373	−0.686863	+	0.726787i	$0.741013\pi$
$972$	0	0
$973$	−18.5133	−0.593509
$974$	−36.7939	−1.17895
$975$	0	0
$976$	−11.8824	−0.380347
$977$	−21.1059	−0.675238	−0.337619	−	0.941283i	$-0.609621\pi$
$977$	−21.1059	−0.675238	−0.337619	+	0.941283i	$0.609621\pi$
$978$	0	0
$979$	7.26493	0.232188
$980$	−2.34291	−0.0748415
$981$	0	0
$982$	−13.9423	−0.444915
$983$	17.4280	0.555867	0.277933	−	0.960600i	$-0.410351\pi$
$983$	17.4280	0.555867	0.277933	+	0.960600i	$0.410351\pi$
$984$	0	0
$985$	12.9861	0.413773
$986$	32.4373	1.03302
$987$	0	0
$988$	22.8200	0.726002
$989$	0.860600	0.0273655
$990$	0	0
$991$	−27.7294	−0.880853	−0.440426	−	0.897789i	$-0.645173\pi$
$991$	−27.7294	−0.880853	−0.440426	+	0.897789i	$0.645173\pi$
$992$	−3.77203	−0.119762
$993$	0	0
$994$	−42.4235	−1.34559
$995$	−14.2733	−0.452493
$996$	0	0
$997$	−26.3758	−0.835331	−0.417665	−	0.908601i	$-0.637152\pi$
$997$	−26.3758	−0.835331	−0.417665	+	0.908601i	$0.637152\pi$
$998$	−12.0514	−0.381481
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9054.2.a.bb.1.2		5
3.2	odd	2		1006.2.a.g.1.4	✓	5
12.11	even	2		8048.2.a.n.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.g.1.4	✓	5	3.2	odd	2
8048.2.a.n.1.2		5	12.11	even	2
9054.2.a.bb.1.2		5	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:	\( N \)	\(=\)	\( 9054 = 2 \cdot 3^{2} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9054.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.813221 q^{5} -3.14341 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.813221 q^{5} -3.14341 q^{7} +1.00000 q^{8} -0.813221 q^{10} +0.935416 q^{11} +6.27767 q^{13} -3.14341 q^{14} +1.00000 q^{16} +4.52254 q^{17} +3.63511 q^{19} -0.813221 q^{20} +0.935416 q^{22} +2.08440 q^{23} -4.33867 q^{25} +6.27767 q^{26} -3.14341 q^{28} +7.17238 q^{29} -3.77203 q^{31} +1.00000 q^{32} +4.52254 q^{34} +2.55629 q^{35} -8.05157 q^{37} +3.63511 q^{38} -0.813221 q^{40} -0.104212 q^{41} +0.412877 q^{43} +0.935416 q^{44} +2.08440 q^{46} -9.24902 q^{47} +2.88102 q^{49} -4.33867 q^{50} +6.27767 q^{52} +4.95783 q^{53} -0.760700 q^{55} -3.14341 q^{56} +7.17238 q^{58} -1.98123 q^{59} -11.8824 q^{61} -3.77203 q^{62} +1.00000 q^{64} -5.10513 q^{65} +4.21544 q^{67} +4.52254 q^{68} +2.55629 q^{70} +13.4960 q^{71} -11.3749 q^{73} -8.05157 q^{74} +3.63511 q^{76} -2.94040 q^{77} +14.7420 q^{79} -0.813221 q^{80} -0.104212 q^{82} +7.06816 q^{83} -3.67782 q^{85} +0.412877 q^{86} +0.935416 q^{88} +7.76652 q^{89} -19.7333 q^{91} +2.08440 q^{92} -9.24902 q^{94} -2.95615 q^{95} +11.1338 q^{97} +2.88102 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} + q^{5} + 3 q^{7} + 5 q^{8}+O(q^{10}) \) 5 * q + 5 * q^2 + 5 * q^4 + q^5 + 3 * q^7 + 5 * q^8	\( 5 q + 5 q^{2} + 5 q^{4} + q^{5} + 3 q^{7} + 5 q^{8} + q^{10} + 11 q^{11} + 5 q^{13} + 3 q^{14} + 5 q^{16} + 20 q^{17} - 4 q^{19} + q^{20} + 11 q^{22} + 4 q^{23} - 6 q^{25} + 5 q^{26} + 3 q^{28} + 6 q^{29} - 3 q^{31} + 5 q^{32} + 20 q^{34} + 3 q^{35} - 10 q^{37} - 4 q^{38} + q^{40} + 6 q^{41} + 11 q^{43} + 11 q^{44} + 4 q^{46} + 9 q^{47} - 4 q^{49} - 6 q^{50} + 5 q^{52} + 22 q^{53} + 14 q^{55} + 3 q^{56} + 6 q^{58} + 10 q^{59} - 11 q^{61} - 3 q^{62} + 5 q^{64} + 12 q^{65} + 14 q^{67} + 20 q^{68} + 3 q^{70} + 26 q^{71} - 7 q^{73} - 10 q^{74} - 4 q^{76} + 26 q^{77} - 15 q^{79} + q^{80} + 6 q^{82} + 12 q^{83} + 12 q^{85} + 11 q^{86} + 11 q^{88} + 5 q^{89} - 22 q^{91} + 4 q^{92} + 9 q^{94} + 10 q^{95} + 6 q^{97} - 4 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 + q^5 + 3 * q^7 + 5 * q^8 + q^10 + 11 * q^11 + 5 * q^13 + 3 * q^14 + 5 * q^16 + 20 * q^17 - 4 * q^19 + q^20 + 11 * q^22 + 4 * q^23 - 6 * q^25 + 5 * q^26 + 3 * q^28 + 6 * q^29 - 3 * q^31 + 5 * q^32 + 20 * q^34 + 3 * q^35 - 10 * q^37 - 4 * q^38 + q^40 + 6 * q^41 + 11 * q^43 + 11 * q^44 + 4 * q^46 + 9 * q^47 - 4 * q^49 - 6 * q^50 + 5 * q^52 + 22 * q^53 + 14 * q^55 + 3 * q^56 + 6 * q^58 + 10 * q^59 - 11 * q^61 - 3 * q^62 + 5 * q^64 + 12 * q^65 + 14 * q^67 + 20 * q^68 + 3 * q^70 + 26 * q^71 - 7 * q^73 - 10 * q^74 - 4 * q^76 + 26 * q^77 - 15 * q^79 + q^80 + 6 * q^82 + 12 * q^83 + 12 * q^85 + 11 * q^86 + 11 * q^88 + 5 * q^89 - 22 * q^91 + 4 * q^92 + 9 * q^94 + 10 * q^95 + 6 * q^97 - 4 * q^98

Introduction

L-functions

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Database

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