LMFDB - Newform orbit 289.3.e.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [289,3,Mod(40,289)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(289, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([15]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("289.40");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 289 = 17^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	289.e (of order $16$, degree $8$, minimal)

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:	no
Analytic conductor:	$7.87467964001$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{16}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\zeta_{16}^{7} + \cdots - \zeta_{16}^{2}) q^{2}+ \cdots + (2 \zeta_{16}^{7} - 7 \zeta_{16}^{6} + \cdots + 1) q^{9}+O(q^{10}) $ q + (z^7 + z^4 - z^2) * q^2 + (2z^7 + z^6 - z^5 + z^4 - z^3 - 2z^2) * q^3 + (z^6 + z^4 - 2z^3 + 2z - 1) * q^4 + (-2z^7 + 2z^6 + 3z^5 + z^4 + z^3 + z^2 + z + 3) q^5 + (-5z^6 + 3z^4 - 3z^3 + 5z) * q^6 + (z^7 - 2z^6 - 2z^5 + z^4 - 2z^3 + z^2 - z + 2) q^7 + (-3z^7 - z^6 - z^5 - z^4 - 3z^3 - 2z^2 + 2) q^8 + (2z^7 - 7z^6 + 7z^4 - 2z^3 + z^2 + 3z + 1) q^9	$ q + (\zeta_{16}^{7} + \cdots - \zeta_{16}^{2}) q^{2}+ \cdots + ( - 18 \zeta_{16}^{7} + 15 \zeta_{16}^{6} + \cdots - 17) q^{99}+O(q^{100}) $ q + (z^7 + z^4 - z^2) * q^2 + (2z^7 + z^6 - z^5 + z^4 - z^3 - 2z^2) * q^3 + (z^6 + z^4 - 2z^3 + 2z - 1) * q^4 + (-2z^7 + 2z^6 + 3z^5 + z^4 + z^3 + z^2 + z + 3) q^5 + (-5z^6 + 3z^4 - 3z^3 + 5z) * q^6 + (z^7 - 2z^6 - 2z^5 + z^4 - 2z^3 + z^2 - z + 2) q^7 + (-3z^7 - z^6 - z^5 - z^4 - 3z^3 - 2z^2 + 2) q^8 + (2z^7 - 7z^6 + 7z^4 - 2z^3 + z^2 + 3z + 1) q^9 + (z^7 + 2z^6 - 2z^5 - z^4 - 6z^2 - 6z) * q^10 + (-5z^7 - 3z^5 - 3z^2 - 5) q^11 + (-3z^7 - 3z^6 + 5z^5 - 4z^4 - 4z^3 + 4z^2 + 4z - 5) q^12 + (-7z^7 + 7z^5 - 2z^2) q^13 + (2z^7 - z^6 + 3z^5 + 3z^4 - z^3 + 2z^2 + 2z - 2) q^14 + (4z^6 - 7z^5 - 10z^4 - 6z^3 - 10z^2 - 7z + 4) * q^15 + (8z^7 - 2z^6 + 4z^5 + z^4 + 4z^3 - 2z^2 + 8z) * q^16 + (-z^7 - 8z^6 + 12z^5 - 12z^3 + 8z^2 + z - 17) * q^18 + (-6z^7 + 4z^6 - 13z^5 - 10z^4 + 10z^2 + 13z - 4) * q^19 + (6z^7 + 6z^6 - 6z^3 - 6z^2 + 5z + 5) q^20 + (3z^7 + 2z^6 + 3z^5 + 8z^4 - 3z^3 + 3z - 8) * q^21 + (-2z^7 + 2z^6 + z^4 + 5z^3 + 5z^2 + z) * q^22 + (-z^7 + 2z^6 - 13z^5 + 4z^4 - 4z^3 + 13z^2 - 2z + 1) * q^23 + (6z^7 + 11z^6 + 11z^5 + 6z^4 + 3z^3 - 3) q^24 + (17z^6 + 5z^5 + 17z^4 - 2z^2 + 2) * q^25 + (-7z^7 + 5z^6 - 5z^4 + 7z^3 - 12z) q^26 + (-14z^7 - 8z^6 + 8z^5 + 14z^4 - 12z^3 + 4z^2 + 4z - 12) q^27 + (-10z^7 + 5z^6 + z^4 + z^3 + 5z - 10) q^28 + (3z^5 + 18z^3 - 18z^2 - 3) q^29 + (5z^7 - 5z^5 + 21z^4 + 17z^3 - 2z^2 + 17z + 21) * q^30 + (11z^7 + 18z^6 - 3z^5 - 3z^4 + 18z^3 + 11z^2 + 4z - 4) q^31 + (-3z^6 + 18z^5 + 2z^4 - 13z^3 + 2z^2 + 18z - 3) * q^32 + (-z^7 + 2z^6 + 13z^5 + 2z^4 + 13z^3 + 2z^2 - z) q^33 + (-5z^7 - 5z^6 - 7z^5 + 7z^3 + 5z^2 + 5z + 10) * q^35 + (-29z^7 + 13z^6 + 13z^5 - 9z^4 + 9z^2 - 13z - 13) * q^36 + (3z^7 + 6z^6 - 7z^5 + 7z^4 - 6z^3 - 3z^2 + 21z + 21) q^37 + (9z^7 + 26z^6 + 9z^5 - z^4 + 3z^3 - 3z + 1) q^38 + (-12z^7 + 12z^6 + 9z^5 - 17z^4 - 17z + 9) q^39 + (-25z^7 - 12z^6 + z^5 + 3z^4 - 3z^3 - z^2 + 12z + 25) q^40 + (-7z^7 + 18z^6 + 18z^5 - 7z^4 + 26z^2 - 26z) * q^41 + (-14z^7 - 11z^6 + 4z^5 - 11z^4 - 14z^3 + 9z^2 - 9) * q^42 + (-7z^7 + 6z^6 - 6z^4 + 7z^3 + 12z^2 - 24z + 12) * q^43 + (5z^7 - 14z^6 + 14z^5 - 5z^4 + 12z^3 - 7z^2 - 7z + 12) q^44 + (11z^7 - 6z^5 + 32z^4 + 32z^3 - 6z^2 + 11) q^45 + (10z^7 + 10z^6 + z^4 - z^3 + z^2 - z) * q^46 + (-3z^7 + 3z^5 - 10z^4 + 11z^3 - 4z^2 + 11z - 10) * q^47 + (5z^7 - 32z^6 - 2z^5 - 2z^4 - 32z^3 + 5z^2 + 11z - 11) q^48 + (z^6 - 10z^5 + 9z^4 + 29z^3 + 9z^2 - 10z + 1) q^49 + (-3z^7 - 19z^6 - 17z^5 - z^4 - 17z^3 - 19z^2 - 3z) * q^50 + (7z^7 + 12z^6 + 4z^5 - 4z^3 - 12z^2 - 7z + 30) * q^52 + (36z^7 - 12z^6 + 6z^5 + 11z^4 - 11z^2 - 6z + 12) * q^53 + (-32z^7 + 4z^6 + 24z^5 - 24z^4 - 4z^3 + 32z^2 - 26z - 26) q^54 + (-17z^7 - 26z^6 - 17z^5 + z^4 + 3z^3 - 3z - 1) q^55 + (-z^7 + z^6 - 9z^5 - 6z^4 - 8z^3 - 8z^2 - 6z - 9) q^56 + (31z^7 + 25z^6 + 5z^5 - 21z^4 + 21z^3 - 5z^2 - 25z - 31) q^57 + (12z^7 - 18z^6 - 18z^5 + 12z^4 - 15z^2 + 15z) * q^58 + (-9z^7 + 5z^6 + 16z^5 + 5z^4 - 9z^3 - z^2 + 1) q^59 + (15z^7 - 12z^6 + 12z^4 - 15z^3 + 2z^2 + 36z + 2) * q^60 + (7z^7 + 11z^6 - 11z^5 - 7z^4 - 33z^3 - 2z^2 - 2z - 33) q^61 + (17z^7 + 3z^6 - 32z^5 - 12z^4 - 12z^3 - 32z^2 + 3z + 17) q^62 + (-12z^7 - 12z^6 + z^5 + 2z^4 - 27z^3 + 27z^2 - 2z - 1) * q^63 + (-2z^7 + 2z^5 - 15z^4 - 36z^3 + 15z^2 - 36z - 15) * q^64 + (-20z^7 - 9z^6 + 33z^5 + 33z^4 - 9z^3 - 20z^2 - 4z + 4) q^65 + (z^6 - 16z^5 - 15z^4 - 15z^2 - 16z + 1) * q^66 + (3z^7 + 8z^6 - 25z^5 + 4z^4 - 25z^3 + 8z^2 + 3z) q^67 + (5z^7 + 14z^6 - 11z^5 + 11z^3 - 14z^2 - 5z - 26) * q^69 + (24z^7 + 10z^6 + 3z^5 + 12z^4 - 12z^2 - 3z - 10) * q^70 + (6z^7 - 41z^6 - 17z^5 + 17z^4 + 41z^3 - 6z^2 - 4z - 4) q^71 + (-22z^7 - 21z^6 - 22z^5 - 3z^4 + 3z^3 - 3z + 3) * q^72 + (20z^7 - 20z^6 - 3z^5 + 30z^4 + 28z^3 + 28z^2 + 30z - 3) q^73 + (22z^7 - 13z^6 + 21z^5 + 31z^4 - 31z^3 - 21z^2 + 13z - 22) q^74 + (-21z^7 - 34z^6 - 34z^5 - 21z^4 - 24z^3 - 33z^2 + 33z + 24) q^75 + (47z^7 - 24z^6 - 8z^5 - 24z^4 + 47z^3 + 10z^2 - 10) * q^76 + (-12z^7 + 15z^6 - 15z^4 + 12z^3 - 27z^2 + 16z - 27) * q^77 + (29z^6 - 29z^5 + 46z^3 - 21z^2 - 21z + 46) q^78 + (12z^7 - 17z^6 + 44z^5 - 41z^4 - 41z^3 + 44z^2 - 17z + 12) q^79 + (41z^7 + 41z^6 + 3z^5 + z^4 + 10z^3 - 10z^2 - z - 3) q^80 + (-3z^7 + 3z^5 - 28z^4 - 35z^3 + 9z^2 - 35z - 28) * q^81 + (-18z^7 + 40z^6 - 44z^5 - 44z^4 + 40z^3 - 18z^2 - 51z + 51) q^82 + (-11z^6 + 11z^5 + 63z^4 - 10z^3 + 63z^2 + 11z - 11) * q^83 + (-15z^7 + 2z^6 + 13z^5 - 30z^4 + 13z^3 + 2z^2 - 15z) q^84 + (19z^7 + 25z^6 - 37z^5 + 37z^3 - 25z^2 - 19z + 36) * q^86 + (24z^7 - 39z^6 - 15z^5 + 27z^4 - 27z^2 + 15z + 39) * q^87 + (14z^7 - 7z^6 + 3z^5 - 3z^4 + 7z^3 - 14z^2 - 22z - 22) q^88 + (47z^7 - 2z^6 + 47z^5 - 36z^4 + 8z^3 - 8z + 36) * q^89 + (49z^7 - 49z^6 - 32z^5 + 23z^4 - 43z^3 - 43z^2 + 23z - 32) q^90 + (-3z^7 - 2z^6 + 4z^5 - 23z^4 + 23z^3 - 4z^2 + 2z + 3) q^91 + (-9z^7 - 6z^6 - 6z^5 - 9z^4 + 42z^3 - 25z^2 + 25z - 42) q^92 + (12z^7 - 31z^6 - 86z^5 - 31z^4 + 12z^3 - 35z^2 + 35) * q^93 + (-2z^7 + 9z^6 - 9z^4 + 2z^3 - z^2 - 2z - 1) q^94 + (27z^7 + 18z^6 - 18z^5 - 27z^4 + 19z^3 + 100z^2 + 100z + 19) q^95 + (-41z^7 - 10z^6 + 51z^5 - 58z^4 - 58z^3 + 51z^2 - 10z - 41) q^96 + (-63z^7 - 63z^6 - 43z^5 + 35z^4 - 26z^3 + 26z^2 - 35z + 43) q^97 + (40z^7 - 40z^5 + 2z^4 + z^3 - 31z^2 + z + 2) * q^98 + (-18z^7 + 15z^6 - 32z^5 - 32z^4 + 15z^3 - 18z^2 + 17z - 17) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} + 24 q^{5} + 16 q^{7} + 16 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^4 + 24 * q^5 + 16 * q^7 + 16 * q^8 + 8 * q^9	$ 8 q - 8 q^{4} + 24 q^{5} + 16 q^{7} + 16 q^{8} + 8 q^{9} - 40 q^{11} - 40 q^{12} - 16 q^{14} + 32 q^{15} - 136 q^{18} - 32 q^{19} + 40 q^{20} - 64 q^{21} + 8 q^{23} - 24 q^{24} + 16 q^{25} - 96 q^{27} - 80 q^{28} - 24 q^{29} + 168 q^{30} - 32 q^{31} - 24 q^{32} + 80 q^{35} - 104 q^{36} + 168 q^{37} + 8 q^{38} + 72 q^{39} + 200 q^{40} - 72 q^{42} + 96 q^{43} + 96 q^{44} + 88 q^{45} - 80 q^{47} - 88 q^{48} + 8 q^{49} + 240 q^{52} + 96 q^{53} - 208 q^{54} - 8 q^{55} - 72 q^{56} - 248 q^{57} + 8 q^{59} + 16 q^{60} - 264 q^{61} + 136 q^{62} - 8 q^{63} - 120 q^{64} + 32 q^{65} + 8 q^{66} - 208 q^{69} - 80 q^{70} - 32 q^{71} + 24 q^{72} - 24 q^{73} - 176 q^{74} + 192 q^{75} - 80 q^{76} - 216 q^{77} + 368 q^{78} + 96 q^{79} - 24 q^{80} - 224 q^{81} + 408 q^{82} - 88 q^{83} + 288 q^{86} + 312 q^{87} - 176 q^{88} + 288 q^{89} - 256 q^{90} + 24 q^{91} - 336 q^{92} + 280 q^{93} - 8 q^{94} + 152 q^{95} - 328 q^{96} + 344 q^{97} + 16 q^{98} - 136 q^{99}+O(q^{100}) $ 8 * q - 8 * q^4 + 24 * q^5 + 16 * q^7 + 16 * q^8 + 8 * q^9 - 40 * q^11 - 40 * q^12 - 16 * q^14 + 32 * q^15 - 136 * q^18 - 32 * q^19 + 40 * q^20 - 64 * q^21 + 8 * q^23 - 24 * q^24 + 16 * q^25 - 96 * q^27 - 80 * q^28 - 24 * q^29 + 168 * q^30 - 32 * q^31 - 24 * q^32 + 80 * q^35 - 104 * q^36 + 168 * q^37 + 8 * q^38 + 72 * q^39 + 200 * q^40 - 72 * q^42 + 96 * q^43 + 96 * q^44 + 88 * q^45 - 80 * q^47 - 88 * q^48 + 8 * q^49 + 240 * q^52 + 96 * q^53 - 208 * q^54 - 8 * q^55 - 72 * q^56 - 248 * q^57 + 8 * q^59 + 16 * q^60 - 264 * q^61 + 136 * q^62 - 8 * q^63 - 120 * q^64 + 32 * q^65 + 8 * q^66 - 208 * q^69 - 80 * q^70 - 32 * q^71 + 24 * q^72 - 24 * q^73 - 176 * q^74 + 192 * q^75 - 80 * q^76 - 216 * q^77 + 368 * q^78 + 96 * q^79 - 24 * q^80 - 224 * q^81 + 408 * q^82 - 88 * q^83 + 288 * q^86 + 312 * q^87 - 176 * q^88 + 288 * q^89 - 256 * q^90 + 24 * q^91 - 336 * q^92 + 280 * q^93 - 8 * q^94 + 152 * q^95 - 328 * q^96 + 344 * q^97 + 16 * q^98 - 136 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/289\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$\zeta_{16}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

40.1

0.923880	−	0.382683i
−0.923880	−	0.382683i
−0.382683	−	0.923880i
0.382683	−	0.923880i
−0.382683	+	0.923880i
0.382683	+	0.923880i
0.923880	+	0.382683i
−0.923880	+	0.382683i

−1.63099

−

0.675577i

−3.96908

0.789499i

−0.624715

−

0.624715i

4.29916

−

6.43416i

7.00688

1.39376i

2.27356

3.40262i

3.29916

7.96489i

6.81537

−

2.82302i

−11.3586

7.58960i

65.1

0.216773

−

0.0897902i

−0.273561

1.37529i

−2.78950

2.78950i

0.286621

−

0.191514i

0.0641865

0.322688i

5.96908

3.98841i

−0.713379

1.72225i

6.49834

2.69170i

0.0449356

−

0.0672509i

75.1

1.08979

−

2.63099i

2.88669

−

4.32023i

−2.90602

−

2.90602i

0.711297

3.57593i

−8.22059

−

12.3030i

0.644047

−

3.23784i

−0.288703

0.119585i

−6.88730

−

16.6274i

10.1834

2.02560i

131.1

0.324423

0.783227i

1.35595

−

0.906019i

2.32023

−

2.32023i

6.70292

1.33329i

1.14952

0.768086i

−0.886687

0.176373i

5.70292

2.36223i

−2.42641

5.85788i

1.13031

5.68246i

158.1