LMFDB - Embedded newform 2898.2.a.ba.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(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.56155 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.56155 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.56155 q^{10} -5.12311 q^{11} -3.56155 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.12311 q^{17} -5.12311 q^{19} +1.56155 q^{20} -5.12311 q^{22} -1.00000 q^{23} -2.56155 q^{25} -3.56155 q^{26} -1.00000 q^{28} -7.56155 q^{29} +3.12311 q^{31} +1.00000 q^{32} +1.12311 q^{34} -1.56155 q^{35} -1.56155 q^{37} -5.12311 q^{38} +1.56155 q^{40} -3.56155 q^{41} +6.68466 q^{43} -5.12311 q^{44} -1.00000 q^{46} -2.43845 q^{47} +1.00000 q^{49} -2.56155 q^{50} -3.56155 q^{52} -14.2462 q^{53} -8.00000 q^{55} -1.00000 q^{56} -7.56155 q^{58} -4.87689 q^{59} +0.876894 q^{61} +3.12311 q^{62} +1.00000 q^{64} -5.56155 q^{65} -1.12311 q^{67} +1.12311 q^{68} -1.56155 q^{70} +9.36932 q^{71} +9.12311 q^{73} -1.56155 q^{74} -5.12311 q^{76} +5.12311 q^{77} +14.2462 q^{79} +1.56155 q^{80} -3.56155 q^{82} +9.12311 q^{83} +1.75379 q^{85} +6.68466 q^{86} -5.12311 q^{88} -14.0000 q^{89} +3.56155 q^{91} -1.00000 q^{92} -2.43845 q^{94} -8.00000 q^{95} -12.4384 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} + 2 q^{8} - q^{10} - 2 q^{11} - 3 q^{13} - 2 q^{14} + 2 q^{16} - 6 q^{17} - 2 q^{19} - q^{20} - 2 q^{22} - 2 q^{23} - q^{25} - 3 q^{26} - 2 q^{28} - 11 q^{29} - 2 q^{31} + 2 q^{32} - 6 q^{34} + q^{35} + q^{37} - 2 q^{38} - q^{40} - 3 q^{41} + q^{43} - 2 q^{44} - 2 q^{46} - 9 q^{47} + 2 q^{49} - q^{50} - 3 q^{52} - 12 q^{53} - 16 q^{55} - 2 q^{56} - 11 q^{58} - 18 q^{59} + 10 q^{61} - 2 q^{62} + 2 q^{64} - 7 q^{65} + 6 q^{67} - 6 q^{68} + q^{70} - 6 q^{71} + 10 q^{73} + q^{74} - 2 q^{76} + 2 q^{77} + 12 q^{79} - q^{80} - 3 q^{82} + 10 q^{83} + 20 q^{85} + q^{86} - 2 q^{88} - 28 q^{89} + 3 q^{91} - 2 q^{92} - 9 q^{94} - 16 q^{95} - 29 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 + 2 * q^8 - q^10 - 2 * q^11 - 3 * q^13 - 2 * q^14 + 2 * q^16 - 6 * q^17 - 2 * q^19 - q^20 - 2 * q^22 - 2 * q^23 - q^25 - 3 * q^26 - 2 * q^28 - 11 * q^29 - 2 * q^31 + 2 * q^32 - 6 * q^34 + q^35 + q^37 - 2 * q^38 - q^40 - 3 * q^41 + q^43 - 2 * q^44 - 2 * q^46 - 9 * q^47 + 2 * q^49 - q^50 - 3 * q^52 - 12 * q^53 - 16 * q^55 - 2 * q^56 - 11 * q^58 - 18 * q^59 + 10 * q^61 - 2 * q^62 + 2 * q^64 - 7 * q^65 + 6 * q^67 - 6 * q^68 + q^70 - 6 * q^71 + 10 * q^73 + q^74 - 2 * q^76 + 2 * q^77 + 12 * q^79 - q^80 - 3 * q^82 + 10 * q^83 + 20 * q^85 + q^86 - 2 * q^88 - 28 * q^89 + 3 * q^91 - 2 * q^92 - 9 * q^94 - 16 * q^95 - 29 * 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
$5$	1.56155	0.698348	0.349174	−	0.937058i	$-0.386462\pi$
$5$	1.56155	0.698348	0.349174	+	0.937058i	$0.386462\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	1.56155	0.493806
$11$	−5.12311	−1.54467	−0.772337	−	0.635213i	$-0.780912\pi$
$11$	−5.12311	−1.54467	−0.772337	+	0.635213i	$0.780912\pi$
$12$	0	0
$13$	−3.56155	−0.987797	−0.493899	−	0.869520i	$-0.664429\pi$
$13$	−3.56155	−0.987797	−0.493899	+	0.869520i	$0.664429\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.12311	0.272393	0.136197	−	0.990682i	$-0.456512\pi$
$17$	1.12311	0.272393	0.136197	+	0.990682i	$0.456512\pi$
$18$	0	0
$19$	−5.12311	−1.17532	−0.587661	−	0.809108i	$-0.699951\pi$
$19$	−5.12311	−1.17532	−0.587661	+	0.809108i	$0.699951\pi$
$20$	1.56155	0.349174
$21$	0	0
$22$	−5.12311	−1.09225
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−2.56155	−0.512311
$26$	−3.56155	−0.698478
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−7.56155	−1.40415	−0.702073	−	0.712105i	$-0.747742\pi$
$29$	−7.56155	−1.40415	−0.702073	+	0.712105i	$0.747742\pi$
$30$	0	0
$31$	3.12311	0.560926	0.280463	−	0.959865i	$-0.409512\pi$
$31$	3.12311	0.560926	0.280463	+	0.959865i	$0.409512\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.12311	0.192611
$35$	−1.56155	−0.263951
$36$	0	0
$37$	−1.56155	−0.256718	−0.128359	−	0.991728i	$-0.540971\pi$
$37$	−1.56155	−0.256718	−0.128359	+	0.991728i	$0.540971\pi$
$38$	−5.12311	−0.831077
$39$	0	0
$40$	1.56155	0.246903
$41$	−3.56155	−0.556221	−0.278111	−	0.960549i	$-0.589708\pi$
$41$	−3.56155	−0.556221	−0.278111	+	0.960549i	$0.589708\pi$
$42$	0	0
$43$	6.68466	1.01940	0.509700	−	0.860352i	$-0.329756\pi$
$43$	6.68466	1.01940	0.509700	+	0.860352i	$0.329756\pi$
$44$	−5.12311	−0.772337
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−2.43845	−0.355684	−0.177842	−	0.984059i	$-0.556912\pi$
$47$	−2.43845	−0.355684	−0.177842	+	0.984059i	$0.556912\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.56155	−0.362258
$51$	0	0
$52$	−3.56155	−0.493899
$53$	−14.2462	−1.95687	−0.978434	−	0.206561i	$-0.933773\pi$
$53$	−14.2462	−1.95687	−0.978434	+	0.206561i	$0.933773\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−7.56155	−0.992881
$59$	−4.87689	−0.634918	−0.317459	−	0.948272i	$-0.602830\pi$
$59$	−4.87689	−0.634918	−0.317459	+	0.948272i	$0.602830\pi$
$60$	0	0
$61$	0.876894	0.112275	0.0561374	−	0.998423i	$-0.482122\pi$
$61$	0.876894	0.112275	0.0561374	+	0.998423i	$0.482122\pi$
$62$	3.12311	0.396635
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.56155	−0.689826
$66$	0	0
$67$	−1.12311	−0.137209	−0.0686046	−	0.997644i	$-0.521855\pi$
$67$	−1.12311	−0.137209	−0.0686046	+	0.997644i	$0.521855\pi$
$68$	1.12311	0.136197
$69$	0	0
$70$	−1.56155	−0.186641
$71$	9.36932	1.11193	0.555967	−	0.831205i	$-0.312348\pi$
$71$	9.36932	1.11193	0.555967	+	0.831205i	$0.312348\pi$
$72$	0	0
$73$	9.12311	1.06778	0.533889	−	0.845554i	$-0.320730\pi$
$73$	9.12311	1.06778	0.533889	+	0.845554i	$0.320730\pi$
$74$	−1.56155	−0.181527
$75$	0	0
$76$	−5.12311	−0.587661
$77$	5.12311	0.583832
$78$	0	0
$79$	14.2462	1.60282	0.801412	−	0.598113i	$-0.204082\pi$
$79$	14.2462	1.60282	0.801412	+	0.598113i	$0.204082\pi$
$80$	1.56155	0.174587
$81$	0	0
$82$	−3.56155	−0.393308
$83$	9.12311	1.00139	0.500695	−	0.865624i	$-0.333078\pi$
$83$	9.12311	1.00139	0.500695	+	0.865624i	$0.333078\pi$
$84$	0	0
$85$	1.75379	0.190225
$86$	6.68466	0.720825
$87$	0	0
$88$	−5.12311	−0.546125
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	3.56155	0.373352
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−2.43845	−0.251507
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−12.4384	−1.26293	−0.631466	−	0.775403i	$-0.717547\pi$
$97$	−12.4384	−1.26293	−0.631466	+	0.775403i	$0.717547\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−2.56155	−0.256155
$101$	−16.2462	−1.61656	−0.808279	−	0.588799i	$-0.799601\pi$
$101$	−16.2462	−1.61656	−0.808279	+	0.588799i	$0.799601\pi$
$102$	0	0
$103$	18.9309	1.86531	0.932657	−	0.360764i	$-0.117484\pi$
$103$	18.9309	1.86531	0.932657	+	0.360764i	$0.117484\pi$
$104$	−3.56155	−0.349239
$105$	0	0
$106$	−14.2462	−1.38371
$107$	5.12311	0.495269	0.247635	−	0.968853i	$-0.420347\pi$
$107$	5.12311	0.495269	0.247635	+	0.968853i	$0.420347\pi$
$108$	0	0
$109$	10.4384	0.999822	0.499911	−	0.866077i	$-0.333366\pi$
$109$	10.4384	0.999822	0.499911	+	0.866077i	$0.333366\pi$
$110$	−8.00000	−0.762770
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	2.68466	0.252551	0.126276	−	0.991995i	$-0.459698\pi$
$113$	2.68466	0.252551	0.126276	+	0.991995i	$0.459698\pi$
$114$	0	0
$115$	−1.56155	−0.145616
$116$	−7.56155	−0.702073
$117$	0	0
$118$	−4.87689	−0.448955
$119$	−1.12311	−0.102955
$120$	0	0
$121$	15.2462	1.38602
$122$	0.876894	0.0793903
$123$	0	0
$124$	3.12311	0.280463
$125$	−11.8078	−1.05612
$126$	0	0
$127$	−16.6847	−1.48052	−0.740262	−	0.672318i	$-0.765299\pi$
$127$	−16.6847	−1.48052	−0.740262	+	0.672318i	$0.765299\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−5.56155	−0.487780
$131$	−4.87689	−0.426096	−0.213048	−	0.977042i	$-0.568339\pi$
$131$	−4.87689	−0.426096	−0.213048	+	0.977042i	$0.568339\pi$
$132$	0	0
$133$	5.12311	0.444230
$134$	−1.12311	−0.0970215
$135$	0	0
$136$	1.12311	0.0963055
$137$	0.438447	0.0374591	0.0187295	−	0.999825i	$-0.494038\pi$
$137$	0.438447	0.0374591	0.0187295	+	0.999825i	$0.494038\pi$
$138$	0	0
$139$	3.80776	0.322970	0.161485	−	0.986875i	$-0.448372\pi$
$139$	3.80776	0.322970	0.161485	+	0.986875i	$0.448372\pi$
$140$	−1.56155	−0.131975
$141$	0	0
$142$	9.36932	0.786256
$143$	18.2462	1.52582
$144$	0	0
$145$	−11.8078	−0.980581
$146$	9.12311	0.755034
$147$	0	0
$148$	−1.56155	−0.128359
$149$	6.24621	0.511710	0.255855	−	0.966715i	$-0.417643\pi$
$149$	6.24621	0.511710	0.255855	+	0.966715i	$0.417643\pi$
$150$	0	0
$151$	7.31534	0.595314	0.297657	−	0.954673i	$-0.403795\pi$
$151$	7.31534	0.595314	0.297657	+	0.954673i	$0.403795\pi$
$152$	−5.12311	−0.415539
$153$	0	0
$154$	5.12311	0.412832
$155$	4.87689	0.391722
$156$	0	0
$157$	14.2462	1.13697	0.568486	−	0.822693i	$-0.307529\pi$
$157$	14.2462	1.13697	0.568486	+	0.822693i	$0.307529\pi$
$158$	14.2462	1.13337
$159$	0	0
$160$	1.56155	0.123452
$161$	1.00000	0.0788110
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−3.56155	−0.278111
$165$	0	0
$166$	9.12311	0.708090
$167$	14.2462	1.10240	0.551202	−	0.834372i	$-0.314169\pi$
$167$	14.2462	1.10240	0.551202	+	0.834372i	$0.314169\pi$
$168$	0	0
$169$	−0.315342	−0.0242570
$170$	1.75379	0.134509
$171$	0	0
$172$	6.68466	0.509700
$173$	−18.4924	−1.40595	−0.702976	−	0.711213i	$-0.748146\pi$
$173$	−18.4924	−1.40595	−0.702976	+	0.711213i	$0.748146\pi$
$174$	0	0
$175$	2.56155	0.193635
$176$	−5.12311	−0.386169
$177$	0	0
$178$	−14.0000	−1.04934
$179$	3.80776	0.284606	0.142303	−	0.989823i	$-0.454549\pi$
$179$	3.80776	0.284606	0.142303	+	0.989823i	$0.454549\pi$
$180$	0	0
$181$	−15.1231	−1.12409	−0.562046	−	0.827106i	$-0.689986\pi$
$181$	−15.1231	−1.12409	−0.562046	+	0.827106i	$0.689986\pi$
$182$	3.56155	0.264000
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−2.43845	−0.179278
$186$	0	0
$187$	−5.75379	−0.420759
$188$	−2.43845	−0.177842
$189$	0	0
$190$	−8.00000	−0.580381
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	2.68466	0.193246	0.0966230	−	0.995321i	$-0.469196\pi$
$193$	2.68466	0.193246	0.0966230	+	0.995321i	$0.469196\pi$
$194$	−12.4384	−0.893028
$195$	0	0
$196$	1.00000	0.0714286
$197$	−24.9309	−1.77625	−0.888125	−	0.459601i	$-0.847992\pi$
$197$	−24.9309	−1.77625	−0.888125	+	0.459601i	$0.847992\pi$
$198$	0	0
$199$	−21.1771	−1.50120	−0.750602	−	0.660755i	$-0.770236\pi$
$199$	−21.1771	−1.50120	−0.750602	+	0.660755i	$0.770236\pi$
$200$	−2.56155	−0.181129
$201$	0	0
$202$	−16.2462	−1.14308
$203$	7.56155	0.530717
$204$	0	0
$205$	−5.56155	−0.388436
$206$	18.9309	1.31898
$207$	0	0
$208$	−3.56155	−0.246949
$209$	26.2462	1.81549
$210$	0	0
$211$	−16.4924	−1.13539	−0.567693	−	0.823241i	$-0.692164\pi$
$211$	−16.4924	−1.13539	−0.567693	+	0.823241i	$0.692164\pi$
$212$	−14.2462	−0.978434
$213$	0	0
$214$	5.12311	0.350208
$215$	10.4384	0.711896
$216$	0	0
$217$	−3.12311	−0.212010
$218$	10.4384	0.706981
$219$	0	0
$220$	−8.00000	−0.539360
$221$	−4.00000	−0.269069
$222$	0	0
$223$	3.12311	0.209139	0.104569	−	0.994518i	$-0.466654\pi$
$223$	3.12311	0.209139	0.104569	+	0.994518i	$0.466654\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	2.68466	0.178581
$227$	6.19224	0.410993	0.205497	−	0.978658i	$-0.434119\pi$
$227$	6.19224	0.410993	0.205497	+	0.978658i	$0.434119\pi$
$228$	0	0
$229$	2.24621	0.148434	0.0742169	−	0.997242i	$-0.476354\pi$
$229$	2.24621	0.148434	0.0742169	+	0.997242i	$0.476354\pi$
$230$	−1.56155	−0.102966
$231$	0	0
$232$	−7.56155	−0.496440
$233$	18.4924	1.21148	0.605739	−	0.795663i	$-0.292877\pi$
$233$	18.4924	1.21148	0.605739	+	0.795663i	$0.292877\pi$
$234$	0	0
$235$	−3.80776	−0.248391
$236$	−4.87689	−0.317459
$237$	0	0
$238$	−1.12311	−0.0728001
$239$	−4.49242	−0.290591	−0.145295	−	0.989388i	$-0.546413\pi$
$239$	−4.49242	−0.290591	−0.145295	+	0.989388i	$0.546413\pi$
$240$	0	0
$241$	−20.0540	−1.29179	−0.645895	−	0.763426i	$-0.723516\pi$
$241$	−20.0540	−1.29179	−0.645895	+	0.763426i	$0.723516\pi$
$242$	15.2462	0.980064
$243$	0	0
$244$	0.876894	0.0561374
$245$	1.56155	0.0997639
$246$	0	0
$247$	18.2462	1.16098
$248$	3.12311	0.198317
$249$	0	0
$250$	−11.8078	−0.746789
$251$	0.438447	0.0276745	0.0138373	−	0.999904i	$-0.495595\pi$
$251$	0.438447	0.0276745	0.0138373	+	0.999904i	$0.495595\pi$
$252$	0	0
$253$	5.12311	0.322087
$254$	−16.6847	−1.04689
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	1.56155	0.0970302
$260$	−5.56155	−0.344913
$261$	0	0
$262$	−4.87689	−0.301296
$263$	−22.9309	−1.41398	−0.706989	−	0.707225i	$-0.749947\pi$
$263$	−22.9309	−1.41398	−0.706989	+	0.707225i	$0.749947\pi$
$264$	0	0
$265$	−22.2462	−1.36657
$266$	5.12311	0.314118
$267$	0	0
$268$	−1.12311	−0.0686046
$269$	−12.2462	−0.746665	−0.373332	−	0.927698i	$-0.621785\pi$
$269$	−12.2462	−0.746665	−0.373332	+	0.927698i	$0.621785\pi$
$270$	0	0
$271$	14.2462	0.865396	0.432698	−	0.901539i	$-0.357562\pi$
$271$	14.2462	0.865396	0.432698	+	0.901539i	$0.357562\pi$
$272$	1.12311	0.0680983
$273$	0	0
$274$	0.438447	0.0264876
$275$	13.1231	0.791353
$276$	0	0
$277$	13.1231	0.788491	0.394245	−	0.919005i	$-0.371006\pi$
$277$	13.1231	0.788491	0.394245	+	0.919005i	$0.371006\pi$
$278$	3.80776	0.228375
$279$	0	0
$280$	−1.56155	−0.0933206
$281$	−4.93087	−0.294151	−0.147076	−	0.989125i	$-0.546986\pi$
$281$	−4.93087	−0.294151	−0.147076	+	0.989125i	$0.546986\pi$
$282$	0	0
$283$	−10.4924	−0.623710	−0.311855	−	0.950130i	$-0.600950\pi$
$283$	−10.4924	−0.623710	−0.311855	+	0.950130i	$0.600950\pi$
$284$	9.36932	0.555967
$285$	0	0
$286$	18.2462	1.07892
$287$	3.56155	0.210232
$288$	0	0
$289$	−15.7386	−0.925802
$290$	−11.8078	−0.693376
$291$	0	0
$292$	9.12311	0.533889
$293$	−27.6155	−1.61332	−0.806658	−	0.591018i	$-0.798726\pi$
$293$	−27.6155	−1.61332	−0.806658	+	0.591018i	$0.798726\pi$
$294$	0	0
$295$	−7.61553	−0.443393
$296$	−1.56155	−0.0907634
$297$	0	0
$298$	6.24621	0.361833
$299$	3.56155	0.205970
$300$	0	0
$301$	−6.68466	−0.385297
$302$	7.31534	0.420951
$303$	0	0
$304$	−5.12311	−0.293830
$305$	1.36932	0.0784069
$306$	0	0
$307$	2.93087	0.167274	0.0836368	−	0.996496i	$-0.473346\pi$
$307$	2.93087	0.167274	0.0836368	+	0.996496i	$0.473346\pi$
$308$	5.12311	0.291916
$309$	0	0
$310$	4.87689	0.276989
$311$	−18.7386	−1.06257	−0.531285	−	0.847193i	$-0.678291\pi$
$311$	−18.7386	−1.06257	−0.531285	+	0.847193i	$0.678291\pi$
$312$	0	0
$313$	4.24621	0.240010	0.120005	−	0.992773i	$-0.461709\pi$
$313$	4.24621	0.240010	0.120005	+	0.992773i	$0.461709\pi$
$314$	14.2462	0.803960
$315$	0	0
$316$	14.2462	0.801412
$317$	17.8078	1.00018	0.500092	−	0.865972i	$-0.333300\pi$
$317$	17.8078	1.00018	0.500092	+	0.865972i	$0.333300\pi$
$318$	0	0
$319$	38.7386	2.16895
$320$	1.56155	0.0872935
$321$	0	0
$322$	1.00000	0.0557278
$323$	−5.75379	−0.320149
$324$	0	0
$325$	9.12311	0.506059
$326$	−8.00000	−0.443079
$327$	0	0
$328$	−3.56155	−0.196654
$329$	2.43845	0.134436
$330$	0	0
$331$	−24.4924	−1.34623	−0.673113	−	0.739540i	$-0.735043\pi$
$331$	−24.4924	−1.34623	−0.673113	+	0.739540i	$0.735043\pi$
$332$	9.12311	0.500695
$333$	0	0
$334$	14.2462	0.779518
$335$	−1.75379	−0.0958197
$336$	0	0
$337$	27.3693	1.49090	0.745451	−	0.666561i	$-0.232234\pi$
$337$	27.3693	1.49090	0.745451	+	0.666561i	$0.232234\pi$
$338$	−0.315342	−0.0171523
$339$	0	0
$340$	1.75379	0.0951125
$341$	−16.0000	−0.866449
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	6.68466	0.360413
$345$	0	0
$346$	−18.4924	−0.994159
$347$	−14.9309	−0.801531	−0.400766	−	0.916181i	$-0.631256\pi$
$347$	−14.9309	−0.801531	−0.400766	+	0.916181i	$0.631256\pi$
$348$	0	0
$349$	−16.2462	−0.869640	−0.434820	−	0.900517i	$-0.643188\pi$
$349$	−16.2462	−0.869640	−0.434820	+	0.900517i	$0.643188\pi$
$350$	2.56155	0.136921
$351$	0	0
$352$	−5.12311	−0.273062
$353$	33.8078	1.79941	0.899703	−	0.436503i	$-0.143783\pi$
$353$	33.8078	1.79941	0.899703	+	0.436503i	$0.143783\pi$
$354$	0	0
$355$	14.6307	0.776516
$356$	−14.0000	−0.741999
$357$	0	0
$358$	3.80776	0.201247
$359$	6.43845	0.339808	0.169904	−	0.985461i	$-0.445654\pi$
$359$	6.43845	0.339808	0.169904	+	0.985461i	$0.445654\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	−15.1231	−0.794853
$363$	0	0
$364$	3.56155	0.186676
$365$	14.2462	0.745681
$366$	0	0
$367$	−18.4384	−0.962479	−0.481240	−	0.876589i	$-0.659813\pi$
$367$	−18.4384	−0.962479	−0.481240	+	0.876589i	$0.659813\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−2.43845	−0.126769
$371$	14.2462	0.739626
$372$	0	0
$373$	15.1231	0.783045	0.391522	−	0.920169i	$-0.371949\pi$
$373$	15.1231	0.783045	0.391522	+	0.920169i	$0.371949\pi$
$374$	−5.75379	−0.297521
$375$	0	0
$376$	−2.43845	−0.125753
$377$	26.9309	1.38701
$378$	0	0
$379$	20.0540	1.03010	0.515052	−	0.857159i	$-0.327773\pi$
$379$	20.0540	1.03010	0.515052	+	0.857159i	$0.327773\pi$
$380$	−8.00000	−0.410391
$381$	0	0
$382$	20.0000	1.02329
$383$	6.24621	0.319166	0.159583	−	0.987184i	$-0.448985\pi$
$383$	6.24621	0.319166	0.159583	+	0.987184i	$0.448985\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	2.68466	0.136646
$387$	0	0
$388$	−12.4384	−0.631466
$389$	29.3693	1.48908	0.744542	−	0.667576i	$-0.232668\pi$
$389$	29.3693	1.48908	0.744542	+	0.667576i	$0.232668\pi$
$390$	0	0
$391$	−1.12311	−0.0567979
$392$	1.00000	0.0505076
$393$	0	0
$394$	−24.9309	−1.25600
$395$	22.2462	1.11933
$396$	0	0
$397$	−7.75379	−0.389152	−0.194576	−	0.980887i	$-0.562333\pi$
$397$	−7.75379	−0.389152	−0.194576	+	0.980887i	$0.562333\pi$
$398$	−21.1771	−1.06151
$399$	0	0
$400$	−2.56155	−0.128078
$401$	−16.2462	−0.811297	−0.405649	−	0.914029i	$-0.632954\pi$
$401$	−16.2462	−0.811297	−0.405649	+	0.914029i	$0.632954\pi$
$402$	0	0
$403$	−11.1231	−0.554081
$404$	−16.2462	−0.808279
$405$	0	0
$406$	7.56155	0.375274
$407$	8.00000	0.396545
$408$	0	0
$409$	33.6155	1.66218	0.831090	−	0.556137i	$-0.187717\pi$
$409$	33.6155	1.66218	0.831090	+	0.556137i	$0.187717\pi$
$410$	−5.56155	−0.274666
$411$	0	0
$412$	18.9309	0.932657
$413$	4.87689	0.239976
$414$	0	0
$415$	14.2462	0.699319
$416$	−3.56155	−0.174619
$417$	0	0
$418$	26.2462	1.28374
$419$	−23.8617	−1.16572	−0.582861	−	0.812572i	$-0.698067\pi$
$419$	−23.8617	−1.16572	−0.582861	+	0.812572i	$0.698067\pi$
$420$	0	0
$421$	33.1771	1.61695	0.808476	−	0.588529i	$-0.200293\pi$
$421$	33.1771	1.61695	0.808476	+	0.588529i	$0.200293\pi$
$422$	−16.4924	−0.802839
$423$	0	0
$424$	−14.2462	−0.691857
$425$	−2.87689	−0.139550
$426$	0	0
$427$	−0.876894	−0.0424359
$428$	5.12311	0.247635
$429$	0	0
$430$	10.4384	0.503387
$431$	3.31534	0.159694	0.0798472	−	0.996807i	$-0.474557\pi$
$431$	3.31534	0.159694	0.0798472	+	0.996807i	$0.474557\pi$
$432$	0	0
$433$	14.6847	0.705700	0.352850	−	0.935680i	$-0.385213\pi$
$433$	14.6847	0.705700	0.352850	+	0.935680i	$0.385213\pi$
$434$	−3.12311	−0.149914
$435$	0	0
$436$	10.4384	0.499911
$437$	5.12311	0.245071
$438$	0	0
$439$	−23.6155	−1.12711	−0.563554	−	0.826079i	$-0.690566\pi$
$439$	−23.6155	−1.12711	−0.563554	+	0.826079i	$0.690566\pi$
$440$	−8.00000	−0.381385
$441$	0	0
$442$	−4.00000	−0.190261
$443$	−9.06913	−0.430887	−0.215444	−	0.976516i	$-0.569120\pi$
$443$	−9.06913	−0.430887	−0.215444	+	0.976516i	$0.569120\pi$
$444$	0	0
$445$	−21.8617	−1.03635
$446$	3.12311	0.147883
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	0.246211	0.0116194	0.00580971	−	0.999983i	$-0.498151\pi$
$449$	0.246211	0.0116194	0.00580971	+	0.999983i	$0.498151\pi$
$450$	0	0
$451$	18.2462	0.859181
$452$	2.68466	0.126276
$453$	0	0
$454$	6.19224	0.290616
$455$	5.56155	0.260730
$456$	0	0
$457$	7.36932	0.344722	0.172361	−	0.985034i	$-0.444860\pi$
$457$	7.36932	0.344722	0.172361	+	0.985034i	$0.444860\pi$
$458$	2.24621	0.104959
$459$	0	0
$460$	−1.56155	−0.0728078
$461$	−40.7386	−1.89739	−0.948694	−	0.316197i	$-0.897594\pi$
$461$	−40.7386	−1.89739	−0.948694	+	0.316197i	$0.897594\pi$
$462$	0	0
$463$	−13.5616	−0.630259	−0.315129	−	0.949049i	$-0.602048\pi$
$463$	−13.5616	−0.630259	−0.315129	+	0.949049i	$0.602048\pi$
$464$	−7.56155	−0.351036
$465$	0	0
$466$	18.4924	0.856645
$467$	−2.68466	−0.124231	−0.0621156	−	0.998069i	$-0.519785\pi$
$467$	−2.68466	−0.124231	−0.0621156	+	0.998069i	$0.519785\pi$
$468$	0	0
$469$	1.12311	0.0518602
$470$	−3.80776	−0.175639
$471$	0	0
$472$	−4.87689	−0.224477
$473$	−34.2462	−1.57464
$474$	0	0
$475$	13.1231	0.602129
$476$	−1.12311	−0.0514775
$477$	0	0
$478$	−4.49242	−0.205479
$479$	−31.6155	−1.44455	−0.722275	−	0.691606i	$-0.756904\pi$
$479$	−31.6155	−1.44455	−0.722275	+	0.691606i	$0.756904\pi$
$480$	0	0
$481$	5.56155	0.253585
$482$	−20.0540	−0.913434
$483$	0	0
$484$	15.2462	0.693010
$485$	−19.4233	−0.881966
$486$	0	0
$487$	−5.56155	−0.252018	−0.126009	−	0.992029i	$-0.540217\pi$
$487$	−5.56155	−0.252018	−0.126009	+	0.992029i	$0.540217\pi$
$488$	0.876894	0.0396951
$489$	0	0
$490$	1.56155	0.0705438
$491$	26.7386	1.20670	0.603349	−	0.797477i	$-0.293833\pi$
$491$	26.7386	1.20670	0.603349	+	0.797477i	$0.293833\pi$
$492$	0	0
$493$	−8.49242	−0.382479
$494$	18.2462	0.820936
$495$	0	0
$496$	3.12311	0.140232
$497$	−9.36932	−0.420271
$498$	0	0
$499$	41.3693	1.85194	0.925972	−	0.377591i	$-0.123247\pi$
$499$	41.3693	1.85194	0.925972	+	0.377591i	$0.123247\pi$
$500$	−11.8078	−0.528059
$501$	0	0
$502$	0.438447	0.0195689
$503$	32.1080	1.43162	0.715811	−	0.698294i	$-0.246057\pi$
$503$	32.1080	1.43162	0.715811	+	0.698294i	$0.246057\pi$
$504$	0	0
$505$	−25.3693	−1.12892
$506$	5.12311	0.227750
$507$	0	0
$508$	−16.6847	−0.740262
$509$	3.36932	0.149342	0.0746712	−	0.997208i	$-0.476209\pi$
$509$	3.36932	0.149342	0.0746712	+	0.997208i	$0.476209\pi$
$510$	0	0
$511$	−9.12311	−0.403582
$512$	1.00000	0.0441942
$513$	0	0
$514$	2.00000	0.0882162
$515$	29.5616	1.30264
$516$	0	0
$517$	12.4924	0.549416
$518$	1.56155	0.0686107
$519$	0	0
$520$	−5.56155	−0.243890
$521$	−0.246211	−0.0107867	−0.00539336	−	0.999985i	$-0.501717\pi$
$521$	−0.246211	−0.0107867	−0.00539336	+	0.999985i	$0.501717\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	−4.87689	−0.213048
$525$	0	0
$526$	−22.9309	−0.999833
$527$	3.50758	0.152792
$528$	0	0
$529$	1.00000	0.0434783
$530$	−22.2462	−0.966314
$531$	0	0
$532$	5.12311	0.222115
$533$	12.6847	0.549434
$534$	0	0
$535$	8.00000	0.345870
$536$	−1.12311	−0.0485108
$537$	0	0
$538$	−12.2462	−0.527972
$539$	−5.12311	−0.220668
$540$	0	0
$541$	−15.3693	−0.660779	−0.330389	−	0.943845i	$-0.607180\pi$
$541$	−15.3693	−0.660779	−0.330389	+	0.943845i	$0.607180\pi$
$542$	14.2462	0.611927
$543$	0	0
$544$	1.12311	0.0481528
$545$	16.3002	0.698223
$546$	0	0
$547$	−3.61553	−0.154589	−0.0772944	−	0.997008i	$-0.524628\pi$
$547$	−3.61553	−0.154589	−0.0772944	+	0.997008i	$0.524628\pi$
$548$	0.438447	0.0187295
$549$	0	0
$550$	13.1231	0.559571
$551$	38.7386	1.65032
$552$	0	0
$553$	−14.2462	−0.605811
$554$	13.1231	0.557547
$555$	0	0
$556$	3.80776	0.161485
$557$	28.9848	1.22813	0.614064	−	0.789257i	$-0.289534\pi$
$557$	28.9848	1.22813	0.614064	+	0.789257i	$0.289534\pi$
$558$	0	0
$559$	−23.8078	−1.00696
$560$	−1.56155	−0.0659877
$561$	0	0
$562$	−4.93087	−0.207996
$563$	−19.5616	−0.824421	−0.412211	−	0.911089i	$-0.635243\pi$
$563$	−19.5616	−0.824421	−0.412211	+	0.911089i	$0.635243\pi$
$564$	0	0
$565$	4.19224	0.176369
$566$	−10.4924	−0.441029
$567$	0	0
$568$	9.36932	0.393128
$569$	44.5464	1.86748	0.933741	−	0.357949i	$-0.116524\pi$
$569$	44.5464	1.86748	0.933741	+	0.357949i	$0.116524\pi$
$570$	0	0
$571$	43.3693	1.81495	0.907475	−	0.420107i	$-0.138007\pi$
$571$	43.3693	1.81495	0.907475	+	0.420107i	$0.138007\pi$
$572$	18.2462	0.762912
$573$	0	0
$574$	3.56155	0.148656
$575$	2.56155	0.106824
$576$	0	0
$577$	−19.7538	−0.822361	−0.411180	−	0.911554i	$-0.634883\pi$
$577$	−19.7538	−0.822361	−0.411180	+	0.911554i	$0.634883\pi$
$578$	−15.7386	−0.654641
$579$	0	0
$580$	−11.8078	−0.490291
$581$	−9.12311	−0.378490
$582$	0	0
$583$	72.9848	3.02272
$584$	9.12311	0.377517
$585$	0	0
$586$	−27.6155	−1.14079
$587$	24.9848	1.03123	0.515617	−	0.856819i	$-0.327563\pi$
$587$	24.9848	1.03123	0.515617	+	0.856819i	$0.327563\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	−7.61553	−0.313526
$591$	0	0
$592$	−1.56155	−0.0641794
$593$	−30.3002	−1.24428	−0.622140	−	0.782906i	$-0.713736\pi$
$593$	−30.3002	−1.24428	−0.622140	+	0.782906i	$0.713736\pi$
$594$	0	0
$595$	−1.75379	−0.0718983
$596$	6.24621	0.255855
$597$	0	0
$598$	3.56155	0.145643
$599$	10.7386	0.438769	0.219384	−	0.975639i	$-0.429595\pi$
$599$	10.7386	0.438769	0.219384	+	0.975639i	$0.429595\pi$
$600$	0	0
$601$	−3.36932	−0.137437	−0.0687187	−	0.997636i	$-0.521891\pi$
$601$	−3.36932	−0.137437	−0.0687187	+	0.997636i	$0.521891\pi$
$602$	−6.68466	−0.272446
$603$	0	0
$604$	7.31534	0.297657
$605$	23.8078	0.967923
$606$	0	0
$607$	−47.6155	−1.93265	−0.966327	−	0.257316i	$-0.917162\pi$
$607$	−47.6155	−1.93265	−0.966327	+	0.257316i	$0.917162\pi$
$608$	−5.12311	−0.207769
$609$	0	0
$610$	1.36932	0.0554420
$611$	8.68466	0.351344
$612$	0	0
$613$	3.80776	0.153794	0.0768971	−	0.997039i	$-0.475499\pi$
$613$	3.80776	0.153794	0.0768971	+	0.997039i	$0.475499\pi$
$614$	2.93087	0.118280
$615$	0	0
$616$	5.12311	0.206416
$617$	−31.7538	−1.27836	−0.639180	−	0.769057i	$-0.720726\pi$
$617$	−31.7538	−1.27836	−0.639180	+	0.769057i	$0.720726\pi$
$618$	0	0
$619$	8.24621	0.331443	0.165722	−	0.986173i	$-0.447005\pi$
$619$	8.24621	0.331443	0.165722	+	0.986173i	$0.447005\pi$
$620$	4.87689	0.195861
$621$	0	0
$622$	−18.7386	−0.751351
$623$	14.0000	0.560898
$624$	0	0
$625$	−5.63068	−0.225227
$626$	4.24621	0.169713
$627$	0	0
$628$	14.2462	0.568486
$629$	−1.75379	−0.0699281
$630$	0	0
$631$	8.49242	0.338078	0.169039	−	0.985609i	$-0.445934\pi$
$631$	8.49242	0.338078	0.169039	+	0.985609i	$0.445934\pi$
$632$	14.2462	0.566684
$633$	0	0
$634$	17.8078	0.707237
$635$	−26.0540	−1.03392
$636$	0	0
$637$	−3.56155	−0.141114
$638$	38.7386	1.53368
$639$	0	0
$640$	1.56155	0.0617258
$641$	13.3153	0.525924	0.262962	−	0.964806i	$-0.415301\pi$
$641$	13.3153	0.525924	0.262962	+	0.964806i	$0.415301\pi$
$642$	0	0
$643$	−38.0000	−1.49857	−0.749287	−	0.662246i	$-0.769604\pi$
$643$	−38.0000	−1.49857	−0.749287	+	0.662246i	$0.769604\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	−5.75379	−0.226380
$647$	3.50758	0.137897	0.0689486	−	0.997620i	$-0.478036\pi$
$647$	3.50758	0.137897	0.0689486	+	0.997620i	$0.478036\pi$
$648$	0	0
$649$	24.9848	0.980741
$650$	9.12311	0.357838
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−36.5464	−1.43017	−0.715086	−	0.699037i	$-0.753612\pi$
$653$	−36.5464	−1.43017	−0.715086	+	0.699037i	$0.753612\pi$
$654$	0	0
$655$	−7.61553	−0.297563
$656$	−3.56155	−0.139055
$657$	0	0
$658$	2.43845	0.0950606
$659$	−5.61553	−0.218750	−0.109375	−	0.994001i	$-0.534885\pi$
$659$	−5.61553	−0.218750	−0.109375	+	0.994001i	$0.534885\pi$
$660$	0	0
$661$	32.8769	1.27876	0.639381	−	0.768890i	$-0.279190\pi$
$661$	32.8769	1.27876	0.639381	+	0.768890i	$0.279190\pi$
$662$	−24.4924	−0.951925
$663$	0	0
$664$	9.12311	0.354045
$665$	8.00000	0.310227
$666$	0	0
$667$	7.56155	0.292784
$668$	14.2462	0.551202
$669$	0	0
$670$	−1.75379	−0.0677548
$671$	−4.49242	−0.173428
$672$	0	0
$673$	22.6847	0.874429	0.437215	−	0.899357i	$-0.355965\pi$
$673$	22.6847	0.874429	0.437215	+	0.899357i	$0.355965\pi$
$674$	27.3693	1.05423
$675$	0	0
$676$	−0.315342	−0.0121285
$677$	−2.63068	−0.101105	−0.0505527	−	0.998721i	$-0.516098\pi$
$677$	−2.63068	−0.101105	−0.0505527	+	0.998721i	$0.516098\pi$
$678$	0	0
$679$	12.4384	0.477344
$680$	1.75379	0.0672547
$681$	0	0
$682$	−16.0000	−0.612672
$683$	7.50758	0.287269	0.143635	−	0.989631i	$-0.454121\pi$
$683$	7.50758	0.287269	0.143635	+	0.989631i	$0.454121\pi$
$684$	0	0
$685$	0.684658	0.0261595
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	6.68466	0.254850
$689$	50.7386	1.93299
$690$	0	0
$691$	−50.0540	−1.90414	−0.952071	−	0.305876i	$-0.901051\pi$
$691$	−50.0540	−1.90414	−0.952071	+	0.305876i	$0.901051\pi$
$692$	−18.4924	−0.702976
$693$	0	0
$694$	−14.9309	−0.566768
$695$	5.94602	0.225546
$696$	0	0
$697$	−4.00000	−0.151511
$698$	−16.2462	−0.614928
$699$	0	0
$700$	2.56155	0.0968176
$701$	−34.2462	−1.29346	−0.646731	−	0.762718i	$-0.723864\pi$
$701$	−34.2462	−1.29346	−0.646731	+	0.762718i	$0.723864\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	−5.12311	−0.193084
$705$	0	0
$706$	33.8078	1.27237
$707$	16.2462	0.611002
$708$	0	0
$709$	10.6307	0.399244	0.199622	−	0.979873i	$-0.436029\pi$
$709$	10.6307	0.399244	0.199622	+	0.979873i	$0.436029\pi$
$710$	14.6307	0.549080
$711$	0	0
$712$	−14.0000	−0.524672
$713$	−3.12311	−0.116961
$714$	0	0
$715$	28.4924	1.06556
$716$	3.80776	0.142303
$717$	0	0
$718$	6.43845	0.240281
$719$	−10.4384	−0.389288	−0.194644	−	0.980874i	$-0.562355\pi$
$719$	−10.4384	−0.389288	−0.194644	+	0.980874i	$0.562355\pi$
$720$	0	0
$721$	−18.9309	−0.705022
$722$	7.24621	0.269676
$723$	0	0
$724$	−15.1231	−0.562046
$725$	19.3693	0.719358
$726$	0	0
$727$	−1.75379	−0.0650444	−0.0325222	−	0.999471i	$-0.510354\pi$
$727$	−1.75379	−0.0650444	−0.0325222	+	0.999471i	$0.510354\pi$
$728$	3.56155	0.132000
$729$	0	0
$730$	14.2462	0.527276
$731$	7.50758	0.277678
$732$	0	0
$733$	25.7538	0.951238	0.475619	−	0.879651i	$-0.342224\pi$
$733$	25.7538	0.951238	0.475619	+	0.879651i	$0.342224\pi$
$734$	−18.4384	−0.680576
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	5.75379	0.211944
$738$	0	0
$739$	23.6155	0.868711	0.434356	−	0.900741i	$-0.356976\pi$
$739$	23.6155	0.868711	0.434356	+	0.900741i	$0.356976\pi$
$740$	−2.43845	−0.0896391
$741$	0	0
$742$	14.2462	0.522995
$743$	24.9848	0.916605	0.458303	−	0.888796i	$-0.348458\pi$
$743$	24.9848	0.916605	0.458303	+	0.888796i	$0.348458\pi$
$744$	0	0
$745$	9.75379	0.357351
$746$	15.1231	0.553696
$747$	0	0
$748$	−5.75379	−0.210379
$749$	−5.12311	−0.187194
$750$	0	0
$751$	24.4924	0.893741	0.446871	−	0.894599i	$-0.352538\pi$
$751$	24.4924	0.893741	0.446871	+	0.894599i	$0.352538\pi$
$752$	−2.43845	−0.0889210
$753$	0	0
$754$	26.9309	0.980764
$755$	11.4233	0.415736
$756$	0	0
$757$	−25.3693	−0.922064	−0.461032	−	0.887384i	$-0.652521\pi$
$757$	−25.3693	−0.922064	−0.461032	+	0.887384i	$0.652521\pi$
$758$	20.0540	0.728393
$759$	0	0
$760$	−8.00000	−0.290191
$761$	3.26137	0.118224	0.0591122	−	0.998251i	$-0.481173\pi$
$761$	3.26137	0.118224	0.0591122	+	0.998251i	$0.481173\pi$
$762$	0	0
$763$	−10.4384	−0.377897
$764$	20.0000	0.723575
$765$	0	0
$766$	6.24621	0.225685
$767$	17.3693	0.627170
$768$	0	0
$769$	−47.1771	−1.70125	−0.850625	−	0.525774i	$-0.823776\pi$
$769$	−47.1771	−1.70125	−0.850625	+	0.525774i	$0.823776\pi$
$770$	8.00000	0.288300
$771$	0	0
$772$	2.68466	0.0966230
$773$	32.3002	1.16176	0.580878	−	0.813990i	$-0.302709\pi$
$773$	32.3002	1.16176	0.580878	+	0.813990i	$0.302709\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	−12.4384	−0.446514
$777$	0	0
$778$	29.3693	1.05294
$779$	18.2462	0.653738
$780$	0	0
$781$	−48.0000	−1.71758
$782$	−1.12311	−0.0401622
$783$	0	0
$784$	1.00000	0.0357143
$785$	22.2462	0.794001
$786$	0	0
$787$	−10.8769	−0.387719	−0.193860	−	0.981029i	$-0.562101\pi$
$787$	−10.8769	−0.387719	−0.193860	+	0.981029i	$0.562101\pi$
$788$	−24.9309	−0.888125
$789$	0	0
$790$	22.2462	0.791485
$791$	−2.68466	−0.0954555
$792$	0	0
$793$	−3.12311	−0.110905
$794$	−7.75379	−0.275172
$795$	0	0
$796$	−21.1771	−0.750602
$797$	−27.4233	−0.971383	−0.485691	−	0.874130i	$-0.661432\pi$
$797$	−27.4233	−0.971383	−0.485691	+	0.874130i	$0.661432\pi$
$798$	0	0
$799$	−2.73863	−0.0968859
$800$	−2.56155	−0.0905646
$801$	0	0
$802$	−16.2462	−0.573674
$803$	−46.7386	−1.64937
$804$	0	0
$805$	1.56155	0.0550375
$806$	−11.1231	−0.391795
$807$	0	0
$808$	−16.2462	−0.571540
$809$	−23.8617	−0.838934	−0.419467	−	0.907771i	$-0.637783\pi$
$809$	−23.8617	−0.838934	−0.419467	+	0.907771i	$0.637783\pi$
$810$	0	0
$811$	−24.6847	−0.866796	−0.433398	−	0.901203i	$-0.642685\pi$
$811$	−24.6847	−0.866796	−0.433398	+	0.901203i	$0.642685\pi$
$812$	7.56155	0.265358
$813$	0	0
$814$	8.00000	0.280400
$815$	−12.4924	−0.437590
$816$	0	0
$817$	−34.2462	−1.19812
$818$	33.6155	1.17534
$819$	0	0
$820$	−5.56155	−0.194218
$821$	55.4773	1.93617	0.968085	−	0.250622i	$-0.0806352\pi$
$821$	55.4773	1.93617	0.968085	+	0.250622i	$0.0806352\pi$
$822$	0	0
$823$	23.3153	0.812722	0.406361	−	0.913713i	$-0.366798\pi$
$823$	23.3153	0.812722	0.406361	+	0.913713i	$0.366798\pi$
$824$	18.9309	0.659488
$825$	0	0
$826$	4.87689	0.169689
$827$	−48.7386	−1.69481	−0.847404	−	0.530948i	$-0.821836\pi$
$827$	−48.7386	−1.69481	−0.847404	+	0.530948i	$0.821836\pi$
$828$	0	0
$829$	24.7386	0.859208	0.429604	−	0.903017i	$-0.358653\pi$
$829$	24.7386	0.859208	0.429604	+	0.903017i	$0.358653\pi$
$830$	14.2462	0.494493
$831$	0	0
$832$	−3.56155	−0.123475
$833$	1.12311	0.0389133
$834$	0	0
$835$	22.2462	0.769862
$836$	26.2462	0.907744
$837$	0	0
$838$	−23.8617	−0.824290
$839$	−7.61553	−0.262917	−0.131459	−	0.991322i	$-0.541966\pi$
$839$	−7.61553	−0.262917	−0.131459	+	0.991322i	$0.541966\pi$
$840$	0	0
$841$	28.1771	0.971623
$842$	33.1771	1.14336
$843$	0	0
$844$	−16.4924	−0.567693
$845$	−0.492423	−0.0169398
$846$	0	0
$847$	−15.2462	−0.523866
$848$	−14.2462	−0.489217
$849$	0	0
$850$	−2.87689	−0.0986767
$851$	1.56155	0.0535293
$852$	0	0
$853$	−0.438447	−0.0150121	−0.00750607	−	0.999972i	$-0.502389\pi$
$853$	−0.438447	−0.0150121	−0.00750607	+	0.999972i	$0.502389\pi$
$854$	−0.876894	−0.0300067
$855$	0	0
$856$	5.12311	0.175104
$857$	−1.31534	−0.0449312	−0.0224656	−	0.999748i	$-0.507152\pi$
$857$	−1.31534	−0.0449312	−0.0224656	+	0.999748i	$0.507152\pi$
$858$	0	0
$859$	−33.1771	−1.13199	−0.565994	−	0.824410i	$-0.691507\pi$
$859$	−33.1771	−1.13199	−0.565994	+	0.824410i	$0.691507\pi$
$860$	10.4384	0.355948
$861$	0	0
$862$	3.31534	0.112921
$863$	−10.7386	−0.365547	−0.182774	−	0.983155i	$-0.558508\pi$
$863$	−10.7386	−0.365547	−0.182774	+	0.983155i	$0.558508\pi$
$864$	0	0
$865$	−28.8769	−0.981844
$866$	14.6847	0.499005
$867$	0	0
$868$	−3.12311	−0.106005
$869$	−72.9848	−2.47584
$870$	0	0
$871$	4.00000	0.135535
$872$	10.4384	0.353490
$873$	0	0
$874$	5.12311	0.173292
$875$	11.8078	0.399175
$876$	0	0
$877$	−35.8617	−1.21096	−0.605482	−	0.795859i	$-0.707020\pi$
$877$	−35.8617	−1.21096	−0.605482	+	0.795859i	$0.707020\pi$
$878$	−23.6155	−0.796985
$879$	0	0
$880$	−8.00000	−0.269680
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	−8.87689	−0.298731	−0.149366	−	0.988782i	$-0.547723\pi$
$883$	−8.87689	−0.298731	−0.149366	+	0.988782i	$0.547723\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−9.06913	−0.304683
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	16.6847	0.559585
$890$	−21.8617	−0.732807
$891$	0	0
$892$	3.12311	0.104569
$893$	12.4924	0.418043
$894$	0	0
$895$	5.94602	0.198754
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	0.246211	0.00821618
$899$	−23.6155	−0.787622
$900$	0	0
$901$	−16.0000	−0.533037
$902$	18.2462	0.607532
$903$	0	0
$904$	2.68466	0.0892904
$905$	−23.6155	−0.785007
$906$	0	0
$907$	52.5464	1.74477	0.872387	−	0.488815i	$-0.162571\pi$
$907$	52.5464	1.74477	0.872387	+	0.488815i	$0.162571\pi$
$908$	6.19224	0.205497
$909$	0	0
$910$	5.56155	0.184364
$911$	1.94602	0.0644747	0.0322373	−	0.999480i	$-0.489737\pi$
$911$	1.94602	0.0644747	0.0322373	+	0.999480i	$0.489737\pi$
$912$	0	0
$913$	−46.7386	−1.54682
$914$	7.36932	0.243755
$915$	0	0
$916$	2.24621	0.0742169
$917$	4.87689	0.161049
$918$	0	0
$919$	−50.7386	−1.67371	−0.836857	−	0.547422i	$-0.815609\pi$
$919$	−50.7386	−1.67371	−0.836857	+	0.547422i	$0.815609\pi$
$920$	−1.56155	−0.0514829
$921$	0	0
$922$	−40.7386	−1.34166
$923$	−33.3693	−1.09836
$924$	0	0
$925$	4.00000	0.131519
$926$	−13.5616	−0.445660
$927$	0	0
$928$	−7.56155	−0.248220
$929$	36.4384	1.19551	0.597753	−	0.801680i	$-0.296060\pi$
$929$	36.4384	1.19551	0.597753	+	0.801680i	$0.296060\pi$
$930$	0	0
$931$	−5.12311	−0.167903
$932$	18.4924	0.605739
$933$	0	0
$934$	−2.68466	−0.0878447
$935$	−8.98485	−0.293836
$936$	0	0
$937$	−27.1771	−0.887837	−0.443918	−	0.896067i	$-0.646412\pi$
$937$	−27.1771	−0.887837	−0.443918	+	0.896067i	$0.646412\pi$
$938$	1.12311	0.0366707
$939$	0	0
$940$	−3.80776	−0.124196
$941$	−1.94602	−0.0634386	−0.0317193	−	0.999497i	$-0.510098\pi$
$941$	−1.94602	−0.0634386	−0.0317193	+	0.999497i	$0.510098\pi$
$942$	0	0
$943$	3.56155	0.115980
$944$	−4.87689	−0.158729
$945$	0	0
$946$	−34.2462	−1.11344
$947$	54.9309	1.78501	0.892507	−	0.451034i	$-0.148945\pi$
$947$	54.9309	1.78501	0.892507	+	0.451034i	$0.148945\pi$
$948$	0	0
$949$	−32.4924	−1.05475
$950$	13.1231	0.425770
$951$	0	0
$952$	−1.12311	−0.0364001
$953$	−50.0000	−1.61966	−0.809829	−	0.586665i	$-0.800440\pi$
$953$	−50.0000	−1.61966	−0.809829	+	0.586665i	$0.800440\pi$
$954$	0	0
$955$	31.2311	1.01061
$956$	−4.49242	−0.145295
$957$	0	0
$958$	−31.6155	−1.02145
$959$	−0.438447	−0.0141582
$960$	0	0
$961$	−21.2462	−0.685362
$962$	5.56155	0.179312
$963$	0	0
$964$	−20.0540	−0.645895
$965$	4.19224	0.134953
$966$	0	0
$967$	−9.75379	−0.313661	−0.156830	−	0.987626i	$-0.550128\pi$
$967$	−9.75379	−0.313661	−0.156830	+	0.987626i	$0.550128\pi$
$968$	15.2462	0.490032
$969$	0	0
$970$	−19.4233	−0.623644
$971$	1.61553	0.0518448	0.0259224	−	0.999664i	$-0.491748\pi$
$971$	1.61553	0.0518448	0.0259224	+	0.999664i	$0.491748\pi$
$972$	0	0
$973$	−3.80776	−0.122071
$974$	−5.56155	−0.178204
$975$	0	0
$976$	0.876894	0.0280687
$977$	26.1922	0.837964	0.418982	−	0.907995i	$-0.362387\pi$
$977$	26.1922	0.837964	0.418982	+	0.907995i	$0.362387\pi$
$978$	0	0
$979$	71.7235	2.29229
$980$	1.56155	0.0498820
$981$	0	0
$982$	26.7386	0.853264
$983$	−33.8617	−1.08002	−0.540011	−	0.841658i	$-0.681580\pi$
$983$	−33.8617	−1.08002	−0.540011	+	0.841658i	$0.681580\pi$
$984$	0	0
$985$	−38.9309	−1.24044
$986$	−8.49242	−0.270454
$987$	0	0
$988$	18.2462	0.580489
$989$	−6.68466	−0.212560
$990$	0	0
$991$	−6.24621	−0.198417	−0.0992087	−	0.995067i	$-0.531631\pi$
$991$	−6.24621	−0.198417	−0.0992087	+	0.995067i	$0.531631\pi$
$992$	3.12311	0.0991587
$993$	0	0
$994$	−9.36932	−0.297177
$995$	−33.0691	−1.04836
$996$	0	0
$997$	51.4773	1.63030	0.815151	−	0.579249i	$-0.196654\pi$
$997$	51.4773	1.63030	0.815151	+	0.579249i	$0.196654\pi$
$998$	41.3693	1.30952
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.ba.1.2		2
3.2	odd	2		966.2.a.l.1.1	✓	2
12.11	even	2		7728.2.a.bm.1.1		2
21.20	even	2		6762.2.a.bw.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.l.1.1	✓	2	3.2	odd	2
2898.2.a.ba.1.2		2	1.1	even	1	trivial
6762.2.a.bw.1.2		2	21.20	even	2
7728.2.a.bm.1.1		2	12.11	even	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.56155 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.56155 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.56155 q^{10} -5.12311 q^{11} -3.56155 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.12311 q^{17} -5.12311 q^{19} +1.56155 q^{20} -5.12311 q^{22} -1.00000 q^{23} -2.56155 q^{25} -3.56155 q^{26} -1.00000 q^{28} -7.56155 q^{29} +3.12311 q^{31} +1.00000 q^{32} +1.12311 q^{34} -1.56155 q^{35} -1.56155 q^{37} -5.12311 q^{38} +1.56155 q^{40} -3.56155 q^{41} +6.68466 q^{43} -5.12311 q^{44} -1.00000 q^{46} -2.43845 q^{47} +1.00000 q^{49} -2.56155 q^{50} -3.56155 q^{52} -14.2462 q^{53} -8.00000 q^{55} -1.00000 q^{56} -7.56155 q^{58} -4.87689 q^{59} +0.876894 q^{61} +3.12311 q^{62} +1.00000 q^{64} -5.56155 q^{65} -1.12311 q^{67} +1.12311 q^{68} -1.56155 q^{70} +9.36932 q^{71} +9.12311 q^{73} -1.56155 q^{74} -5.12311 q^{76} +5.12311 q^{77} +14.2462 q^{79} +1.56155 q^{80} -3.56155 q^{82} +9.12311 q^{83} +1.75379 q^{85} +6.68466 q^{86} -5.12311 q^{88} -14.0000 q^{89} +3.56155 q^{91} -1.00000 q^{92} -2.43845 q^{94} -8.00000 q^{95} -12.4384 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} + 2 q^{8} - q^{10} - 2 q^{11} - 3 q^{13} - 2 q^{14} + 2 q^{16} - 6 q^{17} - 2 q^{19} - q^{20} - 2 q^{22} - 2 q^{23} - q^{25} - 3 q^{26} - 2 q^{28} - 11 q^{29} - 2 q^{31} + 2 q^{32} - 6 q^{34} + q^{35} + q^{37} - 2 q^{38} - q^{40} - 3 q^{41} + q^{43} - 2 q^{44} - 2 q^{46} - 9 q^{47} + 2 q^{49} - q^{50} - 3 q^{52} - 12 q^{53} - 16 q^{55} - 2 q^{56} - 11 q^{58} - 18 q^{59} + 10 q^{61} - 2 q^{62} + 2 q^{64} - 7 q^{65} + 6 q^{67} - 6 q^{68} + q^{70} - 6 q^{71} + 10 q^{73} + q^{74} - 2 q^{76} + 2 q^{77} + 12 q^{79} - q^{80} - 3 q^{82} + 10 q^{83} + 20 q^{85} + q^{86} - 2 q^{88} - 28 q^{89} + 3 q^{91} - 2 q^{92} - 9 q^{94} - 16 q^{95} - 29 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 + 2 * q^8 - q^10 - 2 * q^11 - 3 * q^13 - 2 * q^14 + 2 * q^16 - 6 * q^17 - 2 * q^19 - q^20 - 2 * q^22 - 2 * q^23 - q^25 - 3 * q^26 - 2 * q^28 - 11 * q^29 - 2 * q^31 + 2 * q^32 - 6 * q^34 + q^35 + q^37 - 2 * q^38 - q^40 - 3 * q^41 + q^43 - 2 * q^44 - 2 * q^46 - 9 * q^47 + 2 * q^49 - q^50 - 3 * q^52 - 12 * q^53 - 16 * q^55 - 2 * q^56 - 11 * q^58 - 18 * q^59 + 10 * q^61 - 2 * q^62 + 2 * q^64 - 7 * q^65 + 6 * q^67 - 6 * q^68 + q^70 - 6 * q^71 + 10 * q^73 + q^74 - 2 * q^76 + 2 * q^77 + 12 * q^79 - q^80 - 3 * q^82 + 10 * q^83 + 20 * q^85 + q^86 - 2 * q^88 - 28 * q^89 + 3 * q^91 - 2 * q^92 - 9 * q^94 - 16 * q^95 - 29 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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