LMFDB - Embedded newform 2898.2.a.be.1.1

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:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.470683$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -4.24914 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -4.24914 q^{5} -1.00000 q^{7} -1.00000 q^{8} +4.24914 q^{10} +1.05863 q^{11} +0.941367 q^{13} +1.00000 q^{14} +1.00000 q^{16} -5.30777 q^{17} +5.55691 q^{19} -4.24914 q^{20} -1.05863 q^{22} +1.00000 q^{23} +13.0552 q^{25} -0.941367 q^{26} -1.00000 q^{28} +9.43965 q^{29} -3.30777 q^{31} -1.00000 q^{32} +5.30777 q^{34} +4.24914 q^{35} -8.61555 q^{37} -5.55691 q^{38} +4.24914 q^{40} +9.11383 q^{41} -1.88273 q^{43} +1.05863 q^{44} -1.00000 q^{46} -3.30777 q^{47} +1.00000 q^{49} -13.0552 q^{50} +0.941367 q^{52} -4.61555 q^{53} -4.49828 q^{55} +1.00000 q^{56} -9.43965 q^{58} +6.74742 q^{59} -6.13187 q^{61} +3.30777 q^{62} +1.00000 q^{64} -4.00000 q^{65} +1.05863 q^{67} -5.30777 q^{68} -4.24914 q^{70} +14.6155 q^{71} -6.73281 q^{73} +8.61555 q^{74} +5.55691 q^{76} -1.05863 q^{77} +10.5535 q^{79} -4.24914 q^{80} -9.11383 q^{82} -9.05863 q^{83} +22.5535 q^{85} +1.88273 q^{86} -1.05863 q^{88} -3.19051 q^{89} -0.941367 q^{91} +1.00000 q^{92} +3.30777 q^{94} -23.6121 q^{95} -3.68879 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} + 4 q^{10} + 4 q^{11} + 2 q^{13} + 3 q^{14} + 3 q^{16} - 8 q^{17} - 4 q^{20} - 4 q^{22} + 3 q^{23} + 5 q^{25} - 2 q^{26} - 3 q^{28} + 10 q^{29} - 2 q^{31} - 3 q^{32} + 8 q^{34} + 4 q^{35} - 10 q^{37} + 4 q^{40} - 6 q^{41} - 4 q^{43} + 4 q^{44} - 3 q^{46} - 2 q^{47} + 3 q^{49} - 5 q^{50} + 2 q^{52} + 2 q^{53} + 4 q^{55} + 3 q^{56} - 10 q^{58} - 6 q^{59} - 8 q^{61} + 2 q^{62} + 3 q^{64} - 12 q^{65} + 4 q^{67} - 8 q^{68} - 4 q^{70} + 28 q^{71} - 6 q^{73} + 10 q^{74} - 4 q^{77} - 20 q^{79} - 4 q^{80} + 6 q^{82} - 28 q^{83} + 16 q^{85} + 4 q^{86} - 4 q^{88} - 2 q^{91} + 3 q^{92} + 2 q^{94} - 20 q^{95} + 16 q^{97} - 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8 + 4 * q^10 + 4 * q^11 + 2 * q^13 + 3 * q^14 + 3 * q^16 - 8 * q^17 - 4 * q^20 - 4 * q^22 + 3 * q^23 + 5 * q^25 - 2 * q^26 - 3 * q^28 + 10 * q^29 - 2 * q^31 - 3 * q^32 + 8 * q^34 + 4 * q^35 - 10 * q^37 + 4 * q^40 - 6 * q^41 - 4 * q^43 + 4 * q^44 - 3 * q^46 - 2 * q^47 + 3 * q^49 - 5 * q^50 + 2 * q^52 + 2 * q^53 + 4 * q^55 + 3 * q^56 - 10 * q^58 - 6 * q^59 - 8 * q^61 + 2 * q^62 + 3 * q^64 - 12 * q^65 + 4 * q^67 - 8 * q^68 - 4 * q^70 + 28 * q^71 - 6 * q^73 + 10 * q^74 - 4 * q^77 - 20 * q^79 - 4 * q^80 + 6 * q^82 - 28 * q^83 + 16 * q^85 + 4 * q^86 - 4 * q^88 - 2 * q^91 + 3 * q^92 + 2 * q^94 - 20 * q^95 + 16 * q^97 - 3 * 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
$5$	−4.24914	−1.90027	−0.950137	−	0.311834i	$-0.899057\pi$
$5$	−4.24914	−1.90027	−0.950137	+	0.311834i	$0.899057\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	4.24914	1.34370
$11$	1.05863	0.319190	0.159595	−	0.987183i	$-0.448981\pi$
$11$	1.05863	0.319190	0.159595	+	0.987183i	$0.448981\pi$
$12$	0	0
$13$	0.941367	0.261088	0.130544	−	0.991443i	$-0.458328\pi$
$13$	0.941367	0.261088	0.130544	+	0.991443i	$0.458328\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.30777	−1.28732	−0.643662	−	0.765310i	$-0.722586\pi$
$17$	−5.30777	−1.28732	−0.643662	+	0.765310i	$0.722586\pi$
$18$	0	0
$19$	5.55691	1.27484	0.637422	−	0.770515i	$-0.280001\pi$
$19$	5.55691	1.27484	0.637422	+	0.770515i	$0.280001\pi$
$20$	−4.24914	−0.950137
$21$	0	0
$22$	−1.05863	−0.225701
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	13.0552	2.61104
$26$	−0.941367	−0.184617
$27$	0	0
$28$	−1.00000	−0.188982
$29$	9.43965	1.75290	0.876449	−	0.481494i	$-0.159906\pi$
$29$	9.43965	1.75290	0.876449	+	0.481494i	$0.159906\pi$
$30$	0	0
$31$	−3.30777	−0.594094	−0.297047	−	0.954863i	$-0.596002\pi$
$31$	−3.30777	−0.594094	−0.297047	+	0.954863i	$0.596002\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.30777	0.910276
$35$	4.24914	0.718236
$36$	0	0
$37$	−8.61555	−1.41639	−0.708194	−	0.706018i	$-0.750490\pi$
$37$	−8.61555	−1.41639	−0.708194	+	0.706018i	$0.750490\pi$
$38$	−5.55691	−0.901451
$39$	0	0
$40$	4.24914	0.671848
$41$	9.11383	1.42334	0.711670	−	0.702513i	$-0.247939\pi$
$41$	9.11383	1.42334	0.711670	+	0.702513i	$0.247939\pi$
$42$	0	0
$43$	−1.88273	−0.287114	−0.143557	−	0.989642i	$-0.545854\pi$
$43$	−1.88273	−0.287114	−0.143557	+	0.989642i	$0.545854\pi$
$44$	1.05863	0.159595
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−3.30777	−0.482488	−0.241244	−	0.970464i	$-0.577555\pi$
$47$	−3.30777	−0.482488	−0.241244	+	0.970464i	$0.577555\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−13.0552	−1.84628
$51$	0	0
$52$	0.941367	0.130544
$53$	−4.61555	−0.633994	−0.316997	−	0.948427i	$-0.602675\pi$
$53$	−4.61555	−0.633994	−0.316997	+	0.948427i	$0.602675\pi$
$54$	0	0
$55$	−4.49828	−0.606548
$56$	1.00000	0.133631
$57$	0	0
$58$	−9.43965	−1.23949
$59$	6.74742	0.878439	0.439220	−	0.898380i	$-0.355255\pi$
$59$	6.74742	0.878439	0.439220	+	0.898380i	$0.355255\pi$
$60$	0	0
$61$	−6.13187	−0.785106	−0.392553	−	0.919729i	$-0.628408\pi$
$61$	−6.13187	−0.785106	−0.392553	+	0.919729i	$0.628408\pi$
$62$	3.30777	0.420088
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	1.05863	0.129333	0.0646663	−	0.997907i	$-0.479402\pi$
$67$	1.05863	0.129333	0.0646663	+	0.997907i	$0.479402\pi$
$68$	−5.30777	−0.643662
$69$	0	0
$70$	−4.24914	−0.507869
$71$	14.6155	1.73455	0.867273	−	0.497833i	$-0.165871\pi$
$71$	14.6155	1.73455	0.867273	+	0.497833i	$0.165871\pi$
$72$	0	0
$73$	−6.73281	−0.788016	−0.394008	−	0.919107i	$-0.628912\pi$
$73$	−6.73281	−0.788016	−0.394008	+	0.919107i	$0.628912\pi$
$74$	8.61555	1.00154
$75$	0	0
$76$	5.55691	0.637422
$77$	−1.05863	−0.120642
$78$	0	0
$79$	10.5535	1.18736	0.593679	−	0.804702i	$-0.297675\pi$
$79$	10.5535	1.18736	0.593679	+	0.804702i	$0.297675\pi$
$80$	−4.24914	−0.475068
$81$	0	0
$82$	−9.11383	−1.00645
$83$	−9.05863	−0.994314	−0.497157	−	0.867661i	$-0.665623\pi$
$83$	−9.05863	−0.994314	−0.497157	+	0.867661i	$0.665623\pi$
$84$	0	0
$85$	22.5535	2.44627
$86$	1.88273	0.203020
$87$	0	0
$88$	−1.05863	−0.112851
$89$	−3.19051	−0.338193	−0.169097	−	0.985599i	$-0.554085\pi$
$89$	−3.19051	−0.338193	−0.169097	+	0.985599i	$0.554085\pi$
$90$	0	0
$91$	−0.941367	−0.0986821
$92$	1.00000	0.104257
$93$	0	0
$94$	3.30777	0.341171
$95$	−23.6121	−2.42255
$96$	0	0
$97$	−3.68879	−0.374540	−0.187270	−	0.982309i	$-0.559964\pi$
$97$	−3.68879	−0.374540	−0.187270	+	0.982309i	$0.559964\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	13.0552	1.30552
$101$	−8.94137	−0.889699	−0.444850	−	0.895605i	$-0.646743\pi$
$101$	−8.94137	−0.889699	−0.444850	+	0.895605i	$0.646743\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−0.941367	−0.0923086
$105$	0	0
$106$	4.61555	0.448302
$107$	0.234533	0.0226731	0.0113366	−	0.999936i	$-0.496391\pi$
$107$	0.234533	0.0226731	0.0113366	+	0.999936i	$0.496391\pi$
$108$	0	0
$109$	−13.1138	−1.25608	−0.628038	−	0.778182i	$-0.716142\pi$
$109$	−13.1138	−1.25608	−0.628038	+	0.778182i	$0.716142\pi$
$110$	4.49828	0.428894
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−14.4983	−1.36388	−0.681942	−	0.731406i	$-0.738864\pi$
$113$	−14.4983	−1.36388	−0.681942	+	0.731406i	$0.738864\pi$
$114$	0	0
$115$	−4.24914	−0.396234
$116$	9.43965	0.876449
$117$	0	0
$118$	−6.74742	−0.621151
$119$	5.30777	0.486563
$120$	0	0
$121$	−9.87930	−0.898118
$122$	6.13187	0.555154
$123$	0	0
$124$	−3.30777	−0.297047
$125$	−34.2277	−3.06141
$126$	0	0
$127$	−3.50172	−0.310727	−0.155364	−	0.987857i	$-0.549655\pi$
$127$	−3.50172	−0.310727	−0.155364	+	0.987857i	$0.549655\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	4.00000	0.350823
$131$	−7.86813	−0.687441	−0.343721	−	0.939072i	$-0.611687\pi$
$131$	−7.86813	−0.687441	−0.343721	+	0.939072i	$0.611687\pi$
$132$	0	0
$133$	−5.55691	−0.481846
$134$	−1.05863	−0.0914520
$135$	0	0
$136$	5.30777	0.455138
$137$	15.7294	1.34385	0.671926	−	0.740619i	$-0.265467\pi$
$137$	15.7294	1.34385	0.671926	+	0.740619i	$0.265467\pi$
$138$	0	0
$139$	8.13187	0.689737	0.344868	−	0.938651i	$-0.387924\pi$
$139$	8.13187	0.689737	0.344868	+	0.938651i	$0.387924\pi$
$140$	4.24914	0.359118
$141$	0	0
$142$	−14.6155	−1.22651
$143$	0.996562	0.0833367
$144$	0	0
$145$	−40.1104	−3.33099
$146$	6.73281	0.557212
$147$	0	0
$148$	−8.61555	−0.708194
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−13.4948	−1.09819	−0.549097	−	0.835758i	$-0.685028\pi$
$151$	−13.4948	−1.09819	−0.549097	+	0.835758i	$0.685028\pi$
$152$	−5.55691	−0.450725
$153$	0	0
$154$	1.05863	0.0853071
$155$	14.0552	1.12894
$156$	0	0
$157$	12.2491	0.977588	0.488794	−	0.872399i	$-0.337437\pi$
$157$	12.2491	0.977588	0.488794	+	0.872399i	$0.337437\pi$
$158$	−10.5535	−0.839590
$159$	0	0
$160$	4.24914	0.335924
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	3.61211	0.282922	0.141461	−	0.989944i	$-0.454820\pi$
$163$	3.61211	0.282922	0.141461	+	0.989944i	$0.454820\pi$
$164$	9.11383	0.711670
$165$	0	0
$166$	9.05863	0.703086
$167$	11.3078	0.875022	0.437511	−	0.899213i	$-0.355860\pi$
$167$	11.3078	0.875022	0.437511	+	0.899213i	$0.355860\pi$
$168$	0	0
$169$	−12.1138	−0.931833
$170$	−22.5535	−1.72977
$171$	0	0
$172$	−1.88273	−0.143557
$173$	−23.5569	−1.79100	−0.895500	−	0.445062i	$-0.853181\pi$
$173$	−23.5569	−1.79100	−0.895500	+	0.445062i	$0.853181\pi$
$174$	0	0
$175$	−13.0552	−0.986880
$176$	1.05863	0.0797975
$177$	0	0
$178$	3.19051	0.239139
$179$	−18.8793	−1.41110	−0.705552	−	0.708658i	$-0.749301\pi$
$179$	−18.8793	−1.41110	−0.705552	+	0.708658i	$0.749301\pi$
$180$	0	0
$181$	−11.3630	−0.844603	−0.422301	−	0.906455i	$-0.638778\pi$
$181$	−11.3630	−0.844603	−0.422301	+	0.906455i	$0.638778\pi$
$182$	0.941367	0.0697788
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	36.6087	2.69152
$186$	0	0
$187$	−5.61899	−0.410901
$188$	−3.30777	−0.241244
$189$	0	0
$190$	23.6121	1.71300
$191$	−22.2277	−1.60834	−0.804168	−	0.594402i	$-0.797389\pi$
$191$	−22.2277	−1.60834	−0.804168	+	0.594402i	$0.797389\pi$
$192$	0	0
$193$	26.9966	1.94326	0.971628	−	0.236516i	$-0.0760057\pi$
$193$	26.9966	1.94326	0.971628	+	0.236516i	$0.0760057\pi$
$194$	3.68879	0.264840
$195$	0	0
$196$	1.00000	0.0714286
$197$	2.99656	0.213496	0.106748	−	0.994286i	$-0.465956\pi$
$197$	2.99656	0.213496	0.106748	+	0.994286i	$0.465956\pi$
$198$	0	0
$199$	−16.9966	−1.20485	−0.602427	−	0.798174i	$-0.705800\pi$
$199$	−16.9966	−1.20485	−0.602427	+	0.798174i	$0.705800\pi$
$200$	−13.0552	−0.923142
$201$	0	0
$202$	8.94137	0.629112
$203$	−9.43965	−0.662533
$204$	0	0
$205$	−38.7259	−2.70474
$206$	8.00000	0.557386
$207$	0	0
$208$	0.941367	0.0652720
$209$	5.88273	0.406917
$210$	0	0
$211$	23.1138	1.59122	0.795611	−	0.605808i	$-0.207150\pi$
$211$	23.1138	1.59122	0.795611	+	0.605808i	$0.207150\pi$
$212$	−4.61555	−0.316997
$213$	0	0
$214$	−0.234533	−0.0160323
$215$	8.00000	0.545595
$216$	0	0
$217$	3.30777	0.224546
$218$	13.1138	0.888181
$219$	0	0
$220$	−4.49828	−0.303274
$221$	−4.99656	−0.336105
$222$	0	0
$223$	−0.926759	−0.0620604	−0.0310302	−	0.999518i	$-0.509879\pi$
$223$	−0.926759	−0.0620604	−0.0310302	+	0.999518i	$0.509879\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	14.4983	0.964411
$227$	22.5535	1.49693	0.748463	−	0.663176i	$-0.230792\pi$
$227$	22.5535	1.49693	0.748463	+	0.663176i	$0.230792\pi$
$228$	0	0
$229$	10.8647	0.717959	0.358979	−	0.933345i	$-0.383125\pi$
$229$	10.8647	0.717959	0.358979	+	0.933345i	$0.383125\pi$
$230$	4.24914	0.280180
$231$	0	0
$232$	−9.43965	−0.619743
$233$	−10.8241	−0.709110	−0.354555	−	0.935035i	$-0.615368\pi$
$233$	−10.8241	−0.709110	−0.354555	+	0.935035i	$0.615368\pi$
$234$	0	0
$235$	14.0552	0.916860
$236$	6.74742	0.439220
$237$	0	0
$238$	−5.30777	−0.344052
$239$	8.26375	0.534537	0.267269	−	0.963622i	$-0.413879\pi$
$239$	8.26375	0.534537	0.267269	+	0.963622i	$0.413879\pi$
$240$	0	0
$241$	−6.45769	−0.415977	−0.207988	−	0.978131i	$-0.566692\pi$
$241$	−6.45769	−0.415977	−0.207988	+	0.978131i	$0.566692\pi$
$242$	9.87930	0.635065
$243$	0	0
$244$	−6.13187	−0.392553
$245$	−4.24914	−0.271468
$246$	0	0
$247$	5.23109	0.332847
$248$	3.30777	0.210044
$249$	0	0
$250$	34.2277	2.16475
$251$	20.4362	1.28992	0.644961	−	0.764215i	$-0.276874\pi$
$251$	20.4362	1.28992	0.644961	+	0.764215i	$0.276874\pi$
$252$	0	0
$253$	1.05863	0.0665557
$254$	3.50172	0.219717
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.8793	−1.05290	−0.526451	−	0.850206i	$-0.676478\pi$
$257$	−16.8793	−1.05290	−0.526451	+	0.850206i	$0.676478\pi$
$258$	0	0
$259$	8.61555	0.535344
$260$	−4.00000	−0.248069
$261$	0	0
$262$	7.86813	0.486094
$263$	2.55348	0.157454	0.0787270	−	0.996896i	$-0.474914\pi$
$263$	2.55348	0.157454	0.0787270	+	0.996896i	$0.474914\pi$
$264$	0	0
$265$	19.6121	1.20476
$266$	5.55691	0.340716
$267$	0	0
$268$	1.05863	0.0646663
$269$	−28.3189	−1.72664	−0.863318	−	0.504660i	$-0.831618\pi$
$269$	−28.3189	−1.72664	−0.863318	+	0.504660i	$0.831618\pi$
$270$	0	0
$271$	−19.3078	−1.17286	−0.586432	−	0.809999i	$-0.699468\pi$
$271$	−19.3078	−1.17286	−0.586432	+	0.809999i	$0.699468\pi$
$272$	−5.30777	−0.321831
$273$	0	0
$274$	−15.7294	−0.950246
$275$	13.8207	0.833417
$276$	0	0
$277$	16.5535	0.994602	0.497301	−	0.867578i	$-0.334324\pi$
$277$	16.5535	0.994602	0.497301	+	0.867578i	$0.334324\pi$
$278$	−8.13187	−0.487717
$279$	0	0
$280$	−4.24914	−0.253935
$281$	−12.3810	−0.738589	−0.369295	−	0.929312i	$-0.620401\pi$
$281$	−12.3810	−0.738589	−0.369295	+	0.929312i	$0.620401\pi$
$282$	0	0
$283$	10.7880	0.641281	0.320640	−	0.947201i	$-0.396102\pi$
$283$	10.7880	0.641281	0.320640	+	0.947201i	$0.396102\pi$
$284$	14.6155	0.867273
$285$	0	0
$286$	−0.996562	−0.0589280
$287$	−9.11383	−0.537972
$288$	0	0
$289$	11.1725	0.657204
$290$	40.1104	2.35536
$291$	0	0
$292$	−6.73281	−0.394008
$293$	−4.24914	−0.248237	−0.124119	−	0.992267i	$-0.539610\pi$
$293$	−4.24914	−0.248237	−0.124119	+	0.992267i	$0.539610\pi$
$294$	0	0
$295$	−28.6707	−1.66928
$296$	8.61555	0.500769
$297$	0	0
$298$	−2.00000	−0.115857
$299$	0.941367	0.0544406
$300$	0	0
$301$	1.88273	0.108519
$302$	13.4948	0.776541
$303$	0	0
$304$	5.55691	0.318711
$305$	26.0552	1.49192
$306$	0	0
$307$	−5.75086	−0.328219	−0.164109	−	0.986442i	$-0.552475\pi$
$307$	−5.75086	−0.328219	−0.164109	+	0.986442i	$0.552475\pi$
$308$	−1.05863	−0.0603212
$309$	0	0
$310$	−14.0552	−0.798281
$311$	−29.3009	−1.66150	−0.830751	−	0.556645i	$-0.812089\pi$
$311$	−29.3009	−1.66150	−0.830751	+	0.556645i	$0.812089\pi$
$312$	0	0
$313$	−18.6922	−1.05655	−0.528274	−	0.849074i	$-0.677160\pi$
$313$	−18.6922	−1.05655	−0.528274	+	0.849074i	$0.677160\pi$
$314$	−12.2491	−0.691259
$315$	0	0
$316$	10.5535	0.593679
$317$	5.20512	0.292348	0.146174	−	0.989259i	$-0.453304\pi$
$317$	5.20512	0.292348	0.146174	+	0.989259i	$0.453304\pi$
$318$	0	0
$319$	9.99312	0.559508
$320$	−4.24914	−0.237534
$321$	0	0
$322$	1.00000	0.0557278
$323$	−29.4948	−1.64114
$324$	0	0
$325$	12.2897	0.681711
$326$	−3.61211	−0.200056
$327$	0	0
$328$	−9.11383	−0.503227
$329$	3.30777	0.182363
$330$	0	0
$331$	−8.96735	−0.492890	−0.246445	−	0.969157i	$-0.579262\pi$
$331$	−8.96735	−0.492890	−0.246445	+	0.969157i	$0.579262\pi$
$332$	−9.05863	−0.497157
$333$	0	0
$334$	−11.3078	−0.618734
$335$	−4.49828	−0.245767
$336$	0	0
$337$	−17.1138	−0.932250	−0.466125	−	0.884719i	$-0.654350\pi$
$337$	−17.1138	−0.932250	−0.466125	+	0.884719i	$0.654350\pi$
$338$	12.1138	0.658905
$339$	0	0
$340$	22.5535	1.22313
$341$	−3.50172	−0.189629
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	1.88273	0.101510
$345$	0	0
$346$	23.5569	1.26643
$347$	−20.6087	−1.10633	−0.553166	−	0.833071i	$-0.686580\pi$
$347$	−20.6087	−1.10633	−0.553166	+	0.833071i	$0.686580\pi$
$348$	0	0
$349$	−22.2017	−1.18843	−0.594214	−	0.804307i	$-0.702537\pi$
$349$	−22.2017	−1.18843	−0.594214	+	0.804307i	$0.702537\pi$
$350$	13.0552	0.697830
$351$	0	0
$352$	−1.05863	−0.0564253
$353$	3.14992	0.167653	0.0838267	−	0.996480i	$-0.473286\pi$
$353$	3.14992	0.167653	0.0838267	+	0.996480i	$0.473286\pi$
$354$	0	0
$355$	−62.1035	−3.29611
$356$	−3.19051	−0.169097
$357$	0	0
$358$	18.8793	0.997802
$359$	−3.67418	−0.193916	−0.0969579	−	0.995288i	$-0.530911\pi$
$359$	−3.67418	−0.193916	−0.0969579	+	0.995288i	$0.530911\pi$
$360$	0	0
$361$	11.8793	0.625226
$362$	11.3630	0.597224
$363$	0	0
$364$	−0.941367	−0.0493410
$365$	28.6087	1.49745
$366$	0	0
$367$	−16.7328	−0.873446	−0.436723	−	0.899596i	$-0.643861\pi$
$367$	−16.7328	−0.873446	−0.436723	+	0.899596i	$0.643861\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−36.6087	−1.90319
$371$	4.61555	0.239627
$372$	0	0
$373$	9.76547	0.505637	0.252818	−	0.967514i	$-0.418642\pi$
$373$	9.76547	0.505637	0.252818	+	0.967514i	$0.418642\pi$
$374$	5.61899	0.290551
$375$	0	0
$376$	3.30777	0.170585
$377$	8.88617	0.457661
$378$	0	0
$379$	−26.0552	−1.33837	−0.669183	−	0.743098i	$-0.733356\pi$
$379$	−26.0552	−1.33837	−0.669183	+	0.743098i	$0.733356\pi$
$380$	−23.6121	−1.21128
$381$	0	0
$382$	22.2277	1.13727
$383$	−14.1465	−0.722851	−0.361426	−	0.932401i	$-0.617710\pi$
$383$	−14.1465	−0.722851	−0.361426	+	0.932401i	$0.617710\pi$
$384$	0	0
$385$	4.49828	0.229254
$386$	−26.9966	−1.37409
$387$	0	0
$388$	−3.68879	−0.187270
$389$	32.8793	1.66705	0.833523	−	0.552484i	$-0.186320\pi$
$389$	32.8793	1.66705	0.833523	+	0.552484i	$0.186320\pi$
$390$	0	0
$391$	−5.30777	−0.268426
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−2.99656	−0.150965
$395$	−44.8432	−2.25631
$396$	0	0
$397$	7.16902	0.359803	0.179901	−	0.983685i	$-0.442422\pi$
$397$	7.16902	0.359803	0.179901	+	0.983685i	$0.442422\pi$
$398$	16.9966	0.851961
$399$	0	0
$400$	13.0552	0.652760
$401$	21.3484	1.06609	0.533043	−	0.846088i	$-0.321048\pi$
$401$	21.3484	1.06609	0.533043	+	0.846088i	$0.321048\pi$
$402$	0	0
$403$	−3.11383	−0.155111
$404$	−8.94137	−0.444850
$405$	0	0
$406$	9.43965	0.468482
$407$	−9.12070	−0.452097
$408$	0	0
$409$	3.38445	0.167350	0.0836752	−	0.996493i	$-0.473334\pi$
$409$	3.38445	0.167350	0.0836752	+	0.996493i	$0.473334\pi$
$410$	38.7259	1.91254
$411$	0	0
$412$	−8.00000	−0.394132
$413$	−6.74742	−0.332019
$414$	0	0
$415$	38.4914	1.88947
$416$	−0.941367	−0.0461543
$417$	0	0
$418$	−5.88273	−0.287734
$419$	−7.20512	−0.351993	−0.175996	−	0.984391i	$-0.556315\pi$
$419$	−7.20512	−0.351993	−0.175996	+	0.984391i	$0.556315\pi$
$420$	0	0
$421$	13.6121	0.663414	0.331707	−	0.943383i	$-0.392376\pi$
$421$	13.6121	0.663414	0.331707	+	0.943383i	$0.392376\pi$
$422$	−23.1138	−1.12516
$423$	0	0
$424$	4.61555	0.224151
$425$	−69.2940	−3.36125
$426$	0	0
$427$	6.13187	0.296742
$428$	0.234533	0.0113366
$429$	0	0
$430$	−8.00000	−0.385794
$431$	33.9931	1.63739	0.818696	−	0.574228i	$-0.194698\pi$
$431$	33.9931	1.63739	0.818696	+	0.574228i	$0.194698\pi$
$432$	0	0
$433$	18.5389	0.890921	0.445461	−	0.895302i	$-0.353040\pi$
$433$	18.5389	0.890921	0.445461	+	0.895302i	$0.353040\pi$
$434$	−3.30777	−0.158778
$435$	0	0
$436$	−13.1138	−0.628038
$437$	5.55691	0.265823
$438$	0	0
$439$	13.3009	0.634817	0.317409	−	0.948289i	$-0.397187\pi$
$439$	13.3009	0.634817	0.317409	+	0.948289i	$0.397187\pi$
$440$	4.49828	0.214447
$441$	0	0
$442$	4.99656	0.237662
$443$	16.1104	0.765428	0.382714	−	0.923867i	$-0.374990\pi$
$443$	16.1104	0.765428	0.382714	+	0.923867i	$0.374990\pi$
$444$	0	0
$445$	13.5569	0.642659
$446$	0.926759	0.0438833
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	23.8207	1.12417	0.562083	−	0.827081i	$-0.310000\pi$
$449$	23.8207	1.12417	0.562083	+	0.827081i	$0.310000\pi$
$450$	0	0
$451$	9.64820	0.454316
$452$	−14.4983	−0.681942
$453$	0	0
$454$	−22.5535	−1.05849
$455$	4.00000	0.187523
$456$	0	0
$457$	7.61899	0.356401	0.178201	−	0.983994i	$-0.442972\pi$
$457$	7.61899	0.356401	0.178201	+	0.983994i	$0.442972\pi$
$458$	−10.8647	−0.507674
$459$	0	0
$460$	−4.24914	−0.198117
$461$	16.3189	0.760049	0.380024	−	0.924976i	$-0.375916\pi$
$461$	16.3189	0.760049	0.380024	+	0.924976i	$0.375916\pi$
$462$	0	0
$463$	−31.2242	−1.45111	−0.725556	−	0.688163i	$-0.758417\pi$
$463$	−31.2242	−1.45111	−0.725556	+	0.688163i	$0.758417\pi$
$464$	9.43965	0.438225
$465$	0	0
$466$	10.8241	0.501417
$467$	−2.44309	−0.113053	−0.0565263	−	0.998401i	$-0.518002\pi$
$467$	−2.44309	−0.113053	−0.0565263	+	0.998401i	$0.518002\pi$
$468$	0	0
$469$	−1.05863	−0.0488831
$470$	−14.0552	−0.648318
$471$	0	0
$472$	−6.74742	−0.310575
$473$	−1.99312	−0.0916440
$474$	0	0
$475$	72.5466	3.32867
$476$	5.30777	0.243281
$477$	0	0
$478$	−8.26375	−0.377975
$479$	−23.8759	−1.09092	−0.545458	−	0.838138i	$-0.683644\pi$
$479$	−23.8759	−1.09092	−0.545458	+	0.838138i	$0.683644\pi$
$480$	0	0
$481$	−8.11039	−0.369802
$482$	6.45769	0.294140
$483$	0	0
$484$	−9.87930	−0.449059
$485$	15.6742	0.711728
$486$	0	0
$487$	−23.2242	−1.05239	−0.526195	−	0.850364i	$-0.676382\pi$
$487$	−23.2242	−1.05239	−0.526195	+	0.850364i	$0.676382\pi$
$488$	6.13187	0.277577
$489$	0	0
$490$	4.24914	0.191957
$491$	−27.2242	−1.22861	−0.614306	−	0.789068i	$-0.710564\pi$
$491$	−27.2242	−1.22861	−0.614306	+	0.789068i	$0.710564\pi$
$492$	0	0
$493$	−50.1035	−2.25655
$494$	−5.23109	−0.235358
$495$	0	0
$496$	−3.30777	−0.148523
$497$	−14.6155	−0.655597
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−34.2277	−1.53071
$501$	0	0
$502$	−20.4362	−0.912113
$503$	20.1104	0.896678	0.448339	−	0.893864i	$-0.352016\pi$
$503$	20.1104	0.896678	0.448339	+	0.893864i	$0.352016\pi$
$504$	0	0
$505$	37.9931	1.69067
$506$	−1.05863	−0.0470620
$507$	0	0
$508$	−3.50172	−0.155364
$509$	−35.6673	−1.58093	−0.790463	−	0.612510i	$-0.790160\pi$
$509$	−35.6673	−1.58093	−0.790463	+	0.612510i	$0.790160\pi$
$510$	0	0
$511$	6.73281	0.297842
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	16.8793	0.744514
$515$	33.9931	1.49792
$516$	0	0
$517$	−3.50172	−0.154005
$518$	−8.61555	−0.378545
$519$	0	0
$520$	4.00000	0.175412
$521$	−15.3009	−0.670345	−0.335172	−	0.942157i	$-0.608795\pi$
$521$	−15.3009	−0.670345	−0.335172	+	0.942157i	$0.608795\pi$
$522$	0	0
$523$	22.5535	0.986195	0.493097	−	0.869974i	$-0.335865\pi$
$523$	22.5535	0.986195	0.493097	+	0.869974i	$0.335865\pi$
$524$	−7.86813	−0.343721
$525$	0	0
$526$	−2.55348	−0.111337
$527$	17.5569	0.764791
$528$	0	0
$529$	1.00000	0.0434783
$530$	−19.6121	−0.851896
$531$	0	0
$532$	−5.55691	−0.240923
$533$	8.57946	0.371617
$534$	0	0
$535$	−0.996562	−0.0430851
$536$	−1.05863	−0.0457260
$537$	0	0
$538$	28.3189	1.22092
$539$	1.05863	0.0455986
$540$	0	0
$541$	−5.20512	−0.223785	−0.111893	−	0.993720i	$-0.535691\pi$
$541$	−5.20512	−0.223785	−0.111893	+	0.993720i	$0.535691\pi$
$542$	19.3078	0.829340
$543$	0	0
$544$	5.30777	0.227569
$545$	55.7225	2.38689
$546$	0	0
$547$	−24.8432	−1.06222	−0.531109	−	0.847303i	$-0.678225\pi$
$547$	−24.8432	−1.06222	−0.531109	+	0.847303i	$0.678225\pi$
$548$	15.7294	0.671926
$549$	0	0
$550$	−13.8207	−0.589315
$551$	52.4553	2.23467
$552$	0	0
$553$	−10.5535	−0.448779
$554$	−16.5535	−0.703290
$555$	0	0
$556$	8.13187	0.344868
$557$	−25.4588	−1.07872	−0.539361	−	0.842075i	$-0.681334\pi$
$557$	−25.4588	−1.07872	−0.539361	+	0.842075i	$0.681334\pi$
$558$	0	0
$559$	−1.77234	−0.0749621
$560$	4.24914	0.179559
$561$	0	0
$562$	12.3810	0.522262
$563$	20.9053	0.881052	0.440526	−	0.897740i	$-0.354792\pi$
$563$	20.9053	0.881052	0.440526	+	0.897740i	$0.354792\pi$
$564$	0	0
$565$	61.6052	2.59175
$566$	−10.7880	−0.453454
$567$	0	0
$568$	−14.6155	−0.613255
$569$	−1.53093	−0.0641801	−0.0320901	−	0.999485i	$-0.510216\pi$
$569$	−1.53093	−0.0641801	−0.0320901	+	0.999485i	$0.510216\pi$
$570$	0	0
$571$	5.12070	0.214295	0.107147	−	0.994243i	$-0.465828\pi$
$571$	5.12070	0.214295	0.107147	+	0.994243i	$0.465828\pi$
$572$	0.996562	0.0416684
$573$	0	0
$574$	9.11383	0.380404
$575$	13.0552	0.544439
$576$	0	0
$577$	−19.4948	−0.811581	−0.405790	−	0.913966i	$-0.633004\pi$
$577$	−19.4948	−0.811581	−0.405790	+	0.913966i	$0.633004\pi$
$578$	−11.1725	−0.464713
$579$	0	0
$580$	−40.1104	−1.66549
$581$	9.05863	0.375815
$582$	0	0
$583$	−4.88617	−0.202365
$584$	6.73281	0.278606
$585$	0	0
$586$	4.24914	0.175530
$587$	−39.3561	−1.62440	−0.812200	−	0.583379i	$-0.801730\pi$
$587$	−39.3561	−1.62440	−0.812200	+	0.583379i	$0.801730\pi$
$588$	0	0
$589$	−18.3810	−0.757377
$590$	28.6707	1.18036
$591$	0	0
$592$	−8.61555	−0.354097
$593$	−7.88273	−0.323705	−0.161853	−	0.986815i	$-0.551747\pi$
$593$	−7.88273	−0.323705	−0.161853	+	0.986815i	$0.551747\pi$
$594$	0	0
$595$	−22.5535	−0.924602
$596$	2.00000	0.0819232
$597$	0	0
$598$	−0.941367	−0.0384954
$599$	−38.1035	−1.55687	−0.778434	−	0.627727i	$-0.783985\pi$
$599$	−38.1035	−1.55687	−0.778434	+	0.627727i	$0.783985\pi$
$600$	0	0
$601$	−14.2637	−0.581830	−0.290915	−	0.956749i	$-0.593960\pi$
$601$	−14.2637	−0.581830	−0.290915	+	0.956749i	$0.593960\pi$
$602$	−1.88273	−0.0767345
$603$	0	0
$604$	−13.4948	−0.549097
$605$	41.9785	1.70667
$606$	0	0
$607$	48.8838	1.98413	0.992066	−	0.125719i	$-0.0401236\pi$
$607$	48.8838	1.98413	0.992066	+	0.125719i	$0.0401236\pi$
$608$	−5.55691	−0.225363
$609$	0	0
$610$	−26.0552	−1.05494
$611$	−3.11383	−0.125972
$612$	0	0
$613$	−9.34836	−0.377577	−0.188788	−	0.982018i	$-0.560456\pi$
$613$	−9.34836	−0.377577	−0.188788	+	0.982018i	$0.560456\pi$
$614$	5.75086	0.232086
$615$	0	0
$616$	1.05863	0.0426535
$617$	19.7586	0.795451	0.397725	−	0.917504i	$-0.369800\pi$
$617$	19.7586	0.795451	0.397725	+	0.917504i	$0.369800\pi$
$618$	0	0
$619$	26.4001	1.06111	0.530555	−	0.847650i	$-0.321983\pi$
$619$	26.4001	1.06111	0.530555	+	0.847650i	$0.321983\pi$
$620$	14.0552	0.564470
$621$	0	0
$622$	29.3009	1.17486
$623$	3.19051	0.127825
$624$	0	0
$625$	80.1621	3.20649
$626$	18.6922	0.747092
$627$	0	0
$628$	12.2491	0.488794
$629$	45.7294	1.82335
$630$	0	0
$631$	−27.2051	−1.08302	−0.541509	−	0.840695i	$-0.682147\pi$
$631$	−27.2051	−1.08302	−0.541509	+	0.840695i	$0.682147\pi$
$632$	−10.5535	−0.419795
$633$	0	0
$634$	−5.20512	−0.206722
$635$	14.8793	0.590467
$636$	0	0
$637$	0.941367	0.0372983
$638$	−9.99312	−0.395632
$639$	0	0
$640$	4.24914	0.167962
$641$	7.77234	0.306989	0.153495	−	0.988149i	$-0.450947\pi$
$641$	7.77234	0.306989	0.153495	+	0.988149i	$0.450947\pi$
$642$	0	0
$643$	43.4396	1.71309	0.856546	−	0.516070i	$-0.172606\pi$
$643$	43.4396	1.71309	0.856546	+	0.516070i	$0.172606\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	29.4948	1.16046
$647$	13.3009	0.522912	0.261456	−	0.965215i	$-0.415797\pi$
$647$	13.3009	0.522912	0.261456	+	0.965215i	$0.415797\pi$
$648$	0	0
$649$	7.14304	0.280389
$650$	−12.2897	−0.482043
$651$	0	0
$652$	3.61211	0.141461
$653$	10.6516	0.416831	0.208415	−	0.978040i	$-0.433169\pi$
$653$	10.6516	0.416831	0.208415	+	0.978040i	$0.433169\pi$
$654$	0	0
$655$	33.4328	1.30633
$656$	9.11383	0.355835
$657$	0	0
$658$	−3.30777	−0.128950
$659$	−46.3380	−1.80507	−0.902537	−	0.430612i	$-0.858298\pi$
$659$	−46.3380	−1.80507	−0.902537	+	0.430612i	$0.858298\pi$
$660$	0	0
$661$	10.8647	0.422587	0.211294	−	0.977423i	$-0.432232\pi$
$661$	10.8647	0.422587	0.211294	+	0.977423i	$0.432232\pi$
$662$	8.96735	0.348526
$663$	0	0
$664$	9.05863	0.351543
$665$	23.6121	0.915638
$666$	0	0
$667$	9.43965	0.365505
$668$	11.3078	0.437511
$669$	0	0
$670$	4.49828	0.173784
$671$	−6.49141	−0.250598
$672$	0	0
$673$	−1.93793	−0.0747017	−0.0373508	−	0.999302i	$-0.511892\pi$
$673$	−1.93793	−0.0747017	−0.0373508	+	0.999302i	$0.511892\pi$
$674$	17.1138	0.659200
$675$	0	0
$676$	−12.1138	−0.465916
$677$	−36.9820	−1.42133	−0.710666	−	0.703530i	$-0.751606\pi$
$677$	−36.9820	−1.42133	−0.710666	+	0.703530i	$0.751606\pi$
$678$	0	0
$679$	3.68879	0.141563
$680$	−22.5535	−0.864886
$681$	0	0
$682$	3.50172	0.134088
$683$	9.88273	0.378152	0.189076	−	0.981962i	$-0.439451\pi$
$683$	9.88273	0.378152	0.189076	+	0.981962i	$0.439451\pi$
$684$	0	0
$685$	−66.8363	−2.55368
$686$	1.00000	0.0381802
$687$	0	0
$688$	−1.88273	−0.0717785
$689$	−4.34492	−0.165528
$690$	0	0
$691$	8.78351	0.334141	0.167070	−	0.985945i	$-0.446569\pi$
$691$	8.78351	0.334141	0.167070	+	0.985945i	$0.446569\pi$
$692$	−23.5569	−0.895500
$693$	0	0
$694$	20.6087	0.782294
$695$	−34.5535	−1.31069
$696$	0	0
$697$	−48.3741	−1.83230
$698$	22.2017	0.840346
$699$	0	0
$700$	−13.0552	−0.493440
$701$	12.3810	0.467624	0.233812	−	0.972282i	$-0.424880\pi$
$701$	12.3810	0.467624	0.233812	+	0.972282i	$0.424880\pi$
$702$	0	0
$703$	−47.8759	−1.80567
$704$	1.05863	0.0398987
$705$	0	0
$706$	−3.14992	−0.118549
$707$	8.94137	0.336275
$708$	0	0
$709$	13.2672	0.498260	0.249130	−	0.968470i	$-0.419855\pi$
$709$	13.2672	0.498260	0.249130	+	0.968470i	$0.419855\pi$
$710$	62.1035	2.33070
$711$	0	0
$712$	3.19051	0.119569
$713$	−3.30777	−0.123877
$714$	0	0
$715$	−4.23453	−0.158363
$716$	−18.8793	−0.705552
$717$	0	0
$718$	3.67418	0.137119
$719$	−7.15442	−0.266815	−0.133407	−	0.991061i	$-0.542592\pi$
$719$	−7.15442	−0.266815	−0.133407	+	0.991061i	$0.542592\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	−11.8793	−0.442102
$723$	0	0
$724$	−11.3630	−0.422301
$725$	123.236	4.57689
$726$	0	0
$727$	−15.8759	−0.588803	−0.294401	−	0.955682i	$-0.595120\pi$
$727$	−15.8759	−0.588803	−0.294401	+	0.955682i	$0.595120\pi$
$728$	0.941367	0.0348894
$729$	0	0
$730$	−28.6087	−1.05885
$731$	9.99312	0.369609
$732$	0	0
$733$	33.0112	1.21930	0.609648	−	0.792672i	$-0.291311\pi$
$733$	33.0112	1.21930	0.609648	+	0.792672i	$0.291311\pi$
$734$	16.7328	0.617619
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	1.12070	0.0412817
$738$	0	0
$739$	30.2569	1.11302	0.556509	−	0.830842i	$-0.312141\pi$
$739$	30.2569	1.11302	0.556509	+	0.830842i	$0.312141\pi$
$740$	36.6087	1.34576
$741$	0	0
$742$	−4.61555	−0.169442
$743$	−8.09129	−0.296841	−0.148420	−	0.988924i	$-0.547419\pi$
$743$	−8.09129	−0.296841	−0.148420	+	0.988924i	$0.547419\pi$
$744$	0	0
$745$	−8.49828	−0.311353
$746$	−9.76547	−0.357539
$747$	0	0
$748$	−5.61899	−0.205450
$749$	−0.234533	−0.00856964
$750$	0	0
$751$	7.43965	0.271477	0.135738	−	0.990745i	$-0.456659\pi$
$751$	7.43965	0.271477	0.135738	+	0.990745i	$0.456659\pi$
$752$	−3.30777	−0.120622
$753$	0	0
$754$	−8.88617	−0.323615
$755$	57.3415	2.08687
$756$	0	0
$757$	18.2345	0.662745	0.331373	−	0.943500i	$-0.392488\pi$
$757$	18.2345	0.662745	0.331373	+	0.943500i	$0.392488\pi$
$758$	26.0552	0.946367
$759$	0	0
$760$	23.6121	0.856501
$761$	−43.0777	−1.56157	−0.780783	−	0.624802i	$-0.785180\pi$
$761$	−43.0777	−1.56157	−0.780783	+	0.624802i	$0.785180\pi$
$762$	0	0
$763$	13.1138	0.474752
$764$	−22.2277	−0.804168
$765$	0	0
$766$	14.1465	0.511133
$767$	6.35180	0.229350
$768$	0	0
$769$	12.8387	0.462976	0.231488	−	0.972838i	$-0.425641\pi$
$769$	12.8387	0.462976	0.231488	+	0.972838i	$0.425641\pi$
$770$	−4.49828	−0.162107
$771$	0	0
$772$	26.9966	0.971628
$773$	14.8647	0.534646	0.267323	−	0.963607i	$-0.413861\pi$
$773$	14.8647	0.534646	0.267323	+	0.963607i	$0.413861\pi$
$774$	0	0
$775$	−43.1836	−1.55120
$776$	3.68879	0.132420
$777$	0	0
$778$	−32.8793	−1.17878
$779$	50.6448	1.81454
$780$	0	0
$781$	15.4725	0.553650
$782$	5.30777	0.189806
$783$	0	0
$784$	1.00000	0.0357143
$785$	−52.0483	−1.85768
$786$	0	0
$787$	11.8275	0.421606	0.210803	−	0.977529i	$-0.432392\pi$
$787$	11.8275	0.421606	0.210803	+	0.977529i	$0.432392\pi$
$788$	2.99656	0.106748
$789$	0	0
$790$	44.8432	1.59545
$791$	14.4983	0.515500
$792$	0	0
$793$	−5.77234	−0.204982
$794$	−7.16902	−0.254419
$795$	0	0
$796$	−16.9966	−0.602427
$797$	35.3630	1.25262	0.626310	−	0.779574i	$-0.284564\pi$
$797$	35.3630	1.25262	0.626310	+	0.779574i	$0.284564\pi$
$798$	0	0
$799$	17.5569	0.621119
$800$	−13.0552	−0.461571
$801$	0	0
$802$	−21.3484	−0.753837
$803$	−7.12758	−0.251527
$804$	0	0
$805$	4.24914	0.149763
$806$	3.11383	0.109680
$807$	0	0
$808$	8.94137	0.314556
$809$	−2.99656	−0.105354	−0.0526768	−	0.998612i	$-0.516775\pi$
$809$	−2.99656	−0.105354	−0.0526768	+	0.998612i	$0.516775\pi$
$810$	0	0
$811$	2.63703	0.0925987	0.0462993	−	0.998928i	$-0.485257\pi$
$811$	2.63703	0.0925987	0.0462993	+	0.998928i	$0.485257\pi$
$812$	−9.43965	−0.331267
$813$	0	0
$814$	9.12070	0.319681
$815$	−15.3484	−0.537630
$816$	0	0
$817$	−10.4622	−0.366026
$818$	−3.38445	−0.118335
$819$	0	0
$820$	−38.7259	−1.35237
$821$	38.5466	1.34529	0.672643	−	0.739967i	$-0.265159\pi$
$821$	38.5466	1.34529	0.672643	+	0.739967i	$0.265159\pi$
$822$	0	0
$823$	10.1984	0.355495	0.177748	−	0.984076i	$-0.443119\pi$
$823$	10.1984	0.355495	0.177748	+	0.984076i	$0.443119\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	6.74742	0.234773
$827$	−20.5275	−0.713811	−0.356906	−	0.934140i	$-0.616168\pi$
$827$	−20.5275	−0.713811	−0.356906	+	0.934140i	$0.616168\pi$
$828$	0	0
$829$	−32.6639	−1.13446	−0.567231	−	0.823558i	$-0.691986\pi$
$829$	−32.6639	−1.13446	−0.567231	+	0.823558i	$0.691986\pi$
$830$	−38.4914	−1.33606
$831$	0	0
$832$	0.941367	0.0326360
$833$	−5.30777	−0.183903
$834$	0	0
$835$	−48.0483	−1.66278
$836$	5.88273	0.203459
$837$	0	0
$838$	7.20512	0.248897
$839$	−45.4948	−1.57066	−0.785328	−	0.619080i	$-0.787506\pi$
$839$	−45.4948	−1.57066	−0.785328	+	0.619080i	$0.787506\pi$
$840$	0	0
$841$	60.1070	2.07265
$842$	−13.6121	−0.469104
$843$	0	0
$844$	23.1138	0.795611
$845$	51.4734	1.77074
$846$	0	0
$847$	9.87930	0.339457
$848$	−4.61555	−0.158499
$849$	0	0
$850$	69.2940	2.37677
$851$	−8.61555	−0.295337
$852$	0	0
$853$	−2.64153	−0.0904442	−0.0452221	−	0.998977i	$-0.514400\pi$
$853$	−2.64153	−0.0904442	−0.0452221	+	0.998977i	$0.514400\pi$
$854$	−6.13187	−0.209828
$855$	0	0
$856$	−0.234533	−0.00801616
$857$	32.9897	1.12691	0.563453	−	0.826148i	$-0.309473\pi$
$857$	32.9897	1.12691	0.563453	+	0.826148i	$0.309473\pi$
$858$	0	0
$859$	7.35609	0.250987	0.125493	−	0.992094i	$-0.459949\pi$
$859$	7.35609	0.250987	0.125493	+	0.992094i	$0.459949\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−33.9931	−1.15781
$863$	−6.61555	−0.225196	−0.112598	−	0.993641i	$-0.535917\pi$
$863$	−6.61555	−0.225196	−0.112598	+	0.993641i	$0.535917\pi$
$864$	0	0
$865$	100.097	3.40339
$866$	−18.5389	−0.629976
$867$	0	0
$868$	3.30777	0.112273
$869$	11.1723	0.378993
$870$	0	0
$871$	0.996562	0.0337672
$872$	13.1138	0.444090
$873$	0	0
$874$	−5.55691	−0.187965
$875$	34.2277	1.15711
$876$	0	0
$877$	−6.67074	−0.225255	−0.112627	−	0.993637i	$-0.535927\pi$
$877$	−6.67074	−0.225255	−0.112627	+	0.993637i	$0.535927\pi$
$878$	−13.3009	−0.448884
$879$	0	0
$880$	−4.49828	−0.151637
$881$	−35.2717	−1.18833	−0.594167	−	0.804342i	$-0.702518\pi$
$881$	−35.2717	−1.18833	−0.594167	+	0.804342i	$0.702518\pi$
$882$	0	0
$883$	11.3484	0.381903	0.190951	−	0.981599i	$-0.438843\pi$
$883$	11.3484	0.381903	0.190951	+	0.981599i	$0.438843\pi$
$884$	−4.99656	−0.168053
$885$	0	0
$886$	−16.1104	−0.541239
$887$	−51.1836	−1.71858	−0.859289	−	0.511490i	$-0.829094\pi$
$887$	−51.1836	−1.71858	−0.859289	+	0.511490i	$0.829094\pi$
$888$	0	0
$889$	3.50172	0.117444
$890$	−13.5569	−0.454429
$891$	0	0
$892$	−0.926759	−0.0310302
$893$	−18.3810	−0.615097
$894$	0	0
$895$	80.2208	2.68148
$896$	1.00000	0.0334077
$897$	0	0
$898$	−23.8207	−0.794906
$899$	−31.2242	−1.04139
$900$	0	0
$901$	24.4983	0.816156
$902$	−9.64820	−0.321250
$903$	0	0
$904$	14.4983	0.482206
$905$	48.2829	1.60498
$906$	0	0
$907$	−45.2173	−1.50142	−0.750709	−	0.660633i	$-0.770288\pi$
$907$	−45.2173	−1.50142	−0.750709	+	0.660633i	$0.770288\pi$
$908$	22.5535	0.748463
$909$	0	0
$910$	−4.00000	−0.132599
$911$	−44.5466	−1.47589	−0.737947	−	0.674858i	$-0.764205\pi$
$911$	−44.5466	−1.47589	−0.737947	+	0.674858i	$0.764205\pi$
$912$	0	0
$913$	−9.58977	−0.317375
$914$	−7.61899	−0.252014
$915$	0	0
$916$	10.8647	0.358979
$917$	7.86813	0.259828
$918$	0	0
$919$	−47.7846	−1.57627	−0.788134	−	0.615504i	$-0.788953\pi$
$919$	−47.7846	−1.57627	−0.788134	+	0.615504i	$0.788953\pi$
$920$	4.24914	0.140090
$921$	0	0
$922$	−16.3189	−0.537436
$923$	13.7586	0.452870
$924$	0	0
$925$	−112.478	−3.69824
$926$	31.2242	1.02609
$927$	0	0
$928$	−9.43965	−0.309872
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	5.55691	0.182121
$932$	−10.8241	−0.354555
$933$	0	0
$934$	2.44309	0.0799402
$935$	23.8759	0.780824
$936$	0	0
$937$	7.81293	0.255237	0.127619	−	0.991823i	$-0.459267\pi$
$937$	7.81293	0.255237	0.127619	+	0.991823i	$0.459267\pi$
$938$	1.05863	0.0345656
$939$	0	0
$940$	14.0552	0.458430
$941$	4.48367	0.146164	0.0730818	−	0.997326i	$-0.476717\pi$
$941$	4.48367	0.146164	0.0730818	+	0.997326i	$0.476717\pi$
$942$	0	0
$943$	9.11383	0.296787
$944$	6.74742	0.219610
$945$	0	0
$946$	1.99312	0.0648021
$947$	21.7294	0.706110	0.353055	−	0.935603i	$-0.385143\pi$
$947$	21.7294	0.706110	0.353055	+	0.935603i	$0.385143\pi$
$948$	0	0
$949$	−6.33805	−0.205742
$950$	−72.5466	−2.35372
$951$	0	0
$952$	−5.30777	−0.172026
$953$	−31.8827	−1.03278	−0.516392	−	0.856353i	$-0.672725\pi$
$953$	−31.8827	−1.03278	−0.516392	+	0.856353i	$0.672725\pi$
$954$	0	0
$955$	94.4484	3.05628
$956$	8.26375	0.267269
$957$	0	0
$958$	23.8759	0.771394
$959$	−15.7294	−0.507928
$960$	0	0
$961$	−20.0586	−0.647053
$962$	8.11039	0.261489
$963$	0	0
$964$	−6.45769	−0.207988
$965$	−114.712	−3.69272
$966$	0	0
$967$	9.12070	0.293302	0.146651	−	0.989188i	$-0.453151\pi$
$967$	9.12070	0.293302	0.146651	+	0.989188i	$0.453151\pi$
$968$	9.87930	0.317533
$969$	0	0
$970$	−15.6742	−0.503268
$971$	39.1621	1.25677	0.628387	−	0.777901i	$-0.283716\pi$
$971$	39.1621	1.25677	0.628387	+	0.777901i	$0.283716\pi$
$972$	0	0
$973$	−8.13187	−0.260696
$974$	23.2242	0.744152
$975$	0	0
$976$	−6.13187	−0.196277
$977$	3.14992	0.100775	0.0503874	−	0.998730i	$-0.483954\pi$
$977$	3.14992	0.100775	0.0503874	+	0.998730i	$0.483954\pi$
$978$	0	0
$979$	−3.37758	−0.107948
$980$	−4.24914	−0.135734
$981$	0	0
$982$	27.2242	0.868760
$983$	−6.49141	−0.207044	−0.103522	−	0.994627i	$-0.533011\pi$
$983$	−6.49141	−0.207044	−0.103522	+	0.994627i	$0.533011\pi$
$984$	0	0
$985$	−12.7328	−0.405701
$986$	50.1035	1.59562
$987$	0	0
$988$	5.23109	0.166423
$989$	−1.88273	−0.0598674
$990$	0	0
$991$	−34.9085	−1.10891	−0.554453	−	0.832215i	$-0.687072\pi$
$991$	−34.9085	−1.10891	−0.554453	+	0.832215i	$0.687072\pi$
$992$	3.30777	0.105022
$993$	0	0
$994$	14.6155	0.463577
$995$	72.2208	2.28955
$996$	0	0
$997$	37.7846	1.19665	0.598325	−	0.801254i	$-0.295833\pi$
$997$	37.7846	1.19665	0.598325	+	0.801254i	$0.295833\pi$
$998$	−4.00000	−0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.be.1.1		3
3.2	odd	2		322.2.a.g.1.1	✓	3
12.11	even	2		2576.2.a.w.1.3		3
15.14	odd	2		8050.2.a.bh.1.3		3
21.20	even	2		2254.2.a.p.1.3		3
69.68	even	2		7406.2.a.x.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.a.g.1.1	✓	3	3.2	odd	2
2254.2.a.p.1.3		3	21.20	even	2
2576.2.a.w.1.3		3	12.11	even	2
2898.2.a.be.1.1		3	1.1	even	1	trivial
7406.2.a.x.1.1		3	69.68	even	2
8050.2.a.bh.1.3		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.24914 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.24914 q^{5} -1.00000 q^{7} -1.00000 q^{8} +4.24914 q^{10} +1.05863 q^{11} +0.941367 q^{13} +1.00000 q^{14} +1.00000 q^{16} -5.30777 q^{17} +5.55691 q^{19} -4.24914 q^{20} -1.05863 q^{22} +1.00000 q^{23} +13.0552 q^{25} -0.941367 q^{26} -1.00000 q^{28} +9.43965 q^{29} -3.30777 q^{31} -1.00000 q^{32} +5.30777 q^{34} +4.24914 q^{35} -8.61555 q^{37} -5.55691 q^{38} +4.24914 q^{40} +9.11383 q^{41} -1.88273 q^{43} +1.05863 q^{44} -1.00000 q^{46} -3.30777 q^{47} +1.00000 q^{49} -13.0552 q^{50} +0.941367 q^{52} -4.61555 q^{53} -4.49828 q^{55} +1.00000 q^{56} -9.43965 q^{58} +6.74742 q^{59} -6.13187 q^{61} +3.30777 q^{62} +1.00000 q^{64} -4.00000 q^{65} +1.05863 q^{67} -5.30777 q^{68} -4.24914 q^{70} +14.6155 q^{71} -6.73281 q^{73} +8.61555 q^{74} +5.55691 q^{76} -1.05863 q^{77} +10.5535 q^{79} -4.24914 q^{80} -9.11383 q^{82} -9.05863 q^{83} +22.5535 q^{85} +1.88273 q^{86} -1.05863 q^{88} -3.19051 q^{89} -0.941367 q^{91} +1.00000 q^{92} +3.30777 q^{94} -23.6121 q^{95} -3.68879 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} + 4 q^{10} + 4 q^{11} + 2 q^{13} + 3 q^{14} + 3 q^{16} - 8 q^{17} - 4 q^{20} - 4 q^{22} + 3 q^{23} + 5 q^{25} - 2 q^{26} - 3 q^{28} + 10 q^{29} - 2 q^{31} - 3 q^{32} + 8 q^{34} + 4 q^{35} - 10 q^{37} + 4 q^{40} - 6 q^{41} - 4 q^{43} + 4 q^{44} - 3 q^{46} - 2 q^{47} + 3 q^{49} - 5 q^{50} + 2 q^{52} + 2 q^{53} + 4 q^{55} + 3 q^{56} - 10 q^{58} - 6 q^{59} - 8 q^{61} + 2 q^{62} + 3 q^{64} - 12 q^{65} + 4 q^{67} - 8 q^{68} - 4 q^{70} + 28 q^{71} - 6 q^{73} + 10 q^{74} - 4 q^{77} - 20 q^{79} - 4 q^{80} + 6 q^{82} - 28 q^{83} + 16 q^{85} + 4 q^{86} - 4 q^{88} - 2 q^{91} + 3 q^{92} + 2 q^{94} - 20 q^{95} + 16 q^{97} - 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8 + 4 * q^10 + 4 * q^11 + 2 * q^13 + 3 * q^14 + 3 * q^16 - 8 * q^17 - 4 * q^20 - 4 * q^22 + 3 * q^23 + 5 * q^25 - 2 * q^26 - 3 * q^28 + 10 * q^29 - 2 * q^31 - 3 * q^32 + 8 * q^34 + 4 * q^35 - 10 * q^37 + 4 * q^40 - 6 * q^41 - 4 * q^43 + 4 * q^44 - 3 * q^46 - 2 * q^47 + 3 * q^49 - 5 * q^50 + 2 * q^52 + 2 * q^53 + 4 * q^55 + 3 * q^56 - 10 * q^58 - 6 * q^59 - 8 * q^61 + 2 * q^62 + 3 * q^64 - 12 * q^65 + 4 * q^67 - 8 * q^68 - 4 * q^70 + 28 * q^71 - 6 * q^73 + 10 * q^74 - 4 * q^77 - 20 * q^79 - 4 * q^80 + 6 * q^82 - 28 * q^83 + 16 * q^85 + 4 * q^86 - 4 * q^88 - 2 * q^91 + 3 * q^92 + 2 * q^94 - 20 * q^95 + 16 * q^97 - 3 * q^98

Introduction

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