LMFDB - Newform orbit 289.3.e.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [289,3,Mod(40,289)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [8,0,8,-8,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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} + ( - \zeta_{16}^{7} + 2 \zeta_{16}^{6} + \cdots + 1) q^{3} + (\zeta_{16}^{6} + \zeta_{16}^{4} + \cdots - 1) q^{4} + (\zeta_{16}^{7} - \zeta_{16}^{6} + \cdots + 1) q^{5}+ \cdots + (15 \zeta_{16}^{7} + 18 \zeta_{16}^{6} + \cdots - 32) q^{99}+O(q^{100}) $ q + (z^7 - z^4 + z^2) * q^2 + (-z^7 + 2z^6 - 2z^3 + z^2 + z + 1) * q^3 + (z^6 + z^4 + 2z^3 - 2z - 1) * q^4 + (z^7 - z^6 + z^5 - 3z^4 + 2z^3 + 2z^2 - 3z + 1) * q^5 + (3z^7 - 5z^5 + 5z^2 - 3) q^6 + (2z^7 + z^6 + z^5 + 2z^4 + z^3 + 2z^2 - 2z - 1) * 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 + (-6z^6 + 6z^5 + z^3 - 2z^2 - 2z + 1) * q^10 + (-3z^6 - 5z^4 - 5z^3 - 3z) * q^11 + (-4z^7 - 4z^6 + 4z^5 + 5z^4 + 3z^3 - 3z^2 - 5z - 4) q^12 + (7z^7 - 7z^5 - 2z^2) q^13 + (-z^7 - 2z^6 + 2z^5 + 2z^4 - 2z^3 - z^2 - 3z + 3) 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 + 5z^5 - 5z^4 - 6z^3 + 6z^2) * q^20 + (-3z^7 + 2z^6 - 3z^5 + 8z^4 + 3z^3 - 3z - 8) * q^21 + (5z^7 - 5z^6 + z^5 + 2z^3 + 2z^2 + 1) * q^22 + (4z^7 + 13z^6 + 2z^5 + z^4 - z^3 - 2z^2 - 13z - 4) q^23 + (-3z^7 - 3z^4 + 6z^3 - 11z^2 + 11z - 6) q^24 + (-17z^6 + 5z^5 - 17z^4 + 2z^2 - 2) * q^25 + (-7z^7 - 5z^6 + 5z^4 + 7z^3 - 12z) q^26 + (12z^7 + 4z^6 - 4z^5 - 12z^4 - 14z^3 + 8z^2 + 8z - 14) q^27 + (-z^7 - 5z^5 - 10z^4 - 10z^3 - 5z^2 - 1) * q^28 + (18z^7 + 18z^6 + 3z^4 - 3z) * q^29 + (-5z^7 + 5z^5 + 21z^4 - 17z^3 - 2z^2 - 17z + 21) * q^30 + (18z^7 - 11z^6 + 4z^5 + 4z^4 - 11z^3 + 18z^2 + 3z - 3) 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 + (-6z^7 + 3z^6 + 21z^5 - 21z^4 - 3z^3 + 6z^2 + 7z + 7) q^37 + (-9z^7 + 26z^6 - 9z^5 - z^4 - 3z^3 + 3z + 1) q^38 + (-17z^5 - 9z^4 + 12z^3 + 12z^2 - 9z - 17) q^39 + (3z^7 - z^6 - 12z^5 + 25z^4 - 25z^3 + 12z^2 + z - 3) q^40 + (26z^6 + 26z^5 - 7z^3 - 18z^2 + 18z + 7) 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 + (-12z^7 - 7z^6 + 7z^5 + 12z^4 + 5z^3 + 14z^2 + 14z + 5) q^44 + (-32z^7 - 6z^6 + 11z^4 + 11z^3 - 6z - 32) q^45 + (-z^7 - z^6 - z^5 - 10z^3 + 10z^2 + 1) * q^46 + (3z^7 - 3z^5 - 10z^4 - 11z^3 - 4z^2 - 11z - 10) * q^47 + (-32z^7 - 5z^6 + 11z^5 + 11z^4 - 5z^3 - 32z^2 + 2z - 2) 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 + (-4z^7 - 32z^6 - 26z^5 + 26z^4 + 32z^3 + 4z^2 - 24z - 24) q^54 + (17z^7 - 26z^6 + 17z^5 + z^4 - 3z^3 + 3z - 1) q^55 + (-8z^7 + 8z^6 - 6z^5 + 9z^4 + z^3 + z^2 + 9z - 6) q^56 + (-21z^7 - 5z^6 + 25z^5 - 31z^4 + 31z^3 - 25z^2 + 5z + 21) q^57 + (-15z^6 - 15z^5 + 12z^3 + 18z^2 - 18z - 12) 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 + (33z^7 - 2z^6 + 2z^5 - 33z^4 + 7z^3 - 11z^2 - 11z + 7) q^61 + (12z^7 - 32z^6 - 3z^5 + 17z^4 + 17z^3 - 3z^2 - 32z + 12) q^62 + (-27z^7 - 27z^6 - 2z^5 + z^4 + 12z^3 - 12z^2 - z + 2) q^63 + (2z^7 - 2z^5 - 15z^4 + 36z^3 + 15z^2 + 36z - 15) * q^64 + (-9z^7 + 20z^6 - 4z^5 - 4z^4 + 20z^3 - 9z^2 - 33z + 33) 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 + (41z^7 + 6z^6 - 4z^5 + 4z^4 - 6z^3 - 41z^2 + 17z + 17) q^71 + (22z^7 - 21z^6 + 22z^5 - 3z^4 - 3z^3 + 3z + 3) * q^72 + (28z^7 - 28z^6 + 30z^5 + 3z^4 - 20z^3 - 20z^2 + 3z + 30) q^73 + (31z^7 - 21z^6 - 13z^5 - 22z^4 + 22z^3 + 13z^2 + 21z - 31) q^74 + (24z^7 - 33z^6 - 33z^5 + 24z^4 - 21z^3 + 34z^2 - 34z + 21) 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 + (-46z^7 - 21z^6 + 21z^5 + 46z^4 - 29z^2 - 29z) * q^78 + (41z^7 + 44z^6 + 17z^5 + 12z^4 + 12z^3 + 17z^2 + 44z + 41) q^79 + (10z^7 + 10z^6 - z^5 + 3z^4 - 41z^3 + 41z^2 - 3z + 1) * q^80 + (3z^7 - 3z^5 - 28z^4 + 35z^3 + 9z^2 + 35z - 28) * q^81 + (40z^7 + 18z^6 - 51z^5 - 51z^4 + 18z^3 + 40z^2 + 44z - 44) 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 + (7z^7 + 14z^6 - 22z^5 + 22z^4 - 14z^3 - 7z^2 - 3z - 3) q^88 + (-47z^7 - 2z^6 - 47z^5 - 36z^4 - 8z^3 + 8z + 36) * q^89 + (-43z^7 + 43z^6 + 23z^5 + 32z^4 - 49z^3 - 49z^2 + 32z + 23) q^90 + (-23z^7 - 4z^6 - 2z^5 + 3z^4 - 3z^3 + 2z^2 + 4z + 23) q^91 + (-42z^7 - 25z^6 - 25z^5 - 42z^4 - 9z^3 + 6z^2 - 6z + 9) 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 + (-19z^7 + 100z^6 - 100z^5 + 19z^4 + 27z^3 - 18z^2 - 18z + 27) q^95 + (58z^7 + 51z^6 + 10z^5 - 41z^4 - 41z^3 + 10z^2 + 51z + 58) q^96 + (-26z^7 - 26z^6 - 35z^5 - 43z^4 + 63z^3 - 63z^2 + 43z + 35) q^97 + (-40z^7 + 40z^5 + 2z^4 - z^3 - 31z^2 - z + 2) * q^98 + (15z^7 + 18z^6 + 17z^5 + 17z^4 + 18z^3 + 15z^2 + 32z - 32) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 8 q^{4} + 8 q^{5} - 24 q^{6} - 8 q^{7} - 16 q^{8} - 8 q^{9} + 8 q^{10} - 32 q^{12} + 24 q^{14} - 32 q^{15} - 136 q^{18} + 32 q^{19} - 64 q^{21} + 8 q^{22} - 32 q^{23} - 48 q^{24} - 16 q^{25}+ \cdots - 256 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 - 8 * q^4 + 8 * q^5 - 24 * q^6 - 8 * q^7 - 16 * q^8 - 8 * q^9 + 8 * q^10 - 32 * q^12 + 24 * q^14 - 32 * q^15 - 136 * q^18 + 32 * q^19 - 64 * q^21 + 8 * q^22 - 32 * q^23 - 48 * q^24 - 16 * q^25 - 112 * q^27 - 8 * q^28 + 168 * q^30 - 24 * q^31 + 24 * q^32 + 80 * q^35 + 104 * q^36 + 56 * q^37 + 8 * q^38 - 136 * q^39 - 24 * q^40 + 56 * q^41 + 72 * q^42 - 96 * q^43 + 40 * q^44 - 256 * q^45 + 8 * q^46 - 80 * q^47 - 16 * q^48 - 8 * q^49 + 240 * q^52 - 96 * q^53 - 192 * q^54 - 8 * q^55 - 48 * q^56 + 168 * q^57 - 96 * q^58 - 8 * q^59 - 16 * q^60 + 56 * q^61 + 96 * q^62 + 16 * q^63 - 120 * q^64 + 264 * q^65 - 8 * q^66 - 208 * q^69 + 80 * q^70 + 136 * q^71 + 24 * q^72 + 240 * q^73 - 248 * q^74 + 168 * q^75 + 80 * q^76 + 216 * q^77 + 328 * q^79 + 8 * q^80 - 224 * q^81 - 352 * q^82 + 88 * q^83 + 288 * q^86 - 312 * q^87 - 24 * q^88 + 288 * q^89 + 184 * q^90 + 184 * q^91 + 72 * q^92 - 280 * q^93 + 8 * q^94 + 216 * q^95 + 464 * q^96 + 280 * q^97 + 16 * q^98 - 256 * q^99

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

−0.216773

−

0.0897902i

1.37529

−

0.273561i

−2.78950

−

2.78950i

−0.191514

0.286621i

−0.322688

−

0.0641865i

−3.98841

−

5.96908i

0.713379

1.72225i

−6.49834

2.69170i

0.0672509

−

0.0449356i

65.1

1.63099

−

0.675577i

−0.789499

3.96908i

−0.624715

0.624715i

6.43416

−

4.29916i

1.39376

7.00688i

3.40262

2.27356i

−3.29916

7.96489i

−6.81537

−

2.82302i

7.58960

−

11.3586i

75.1

−0.324423

0.783227i

−0.906019

1.35595i

2.32023

2.32023i

1.33329

6.70292i

−0.768086

−

1.14952i

−0.176373

0.886687i

−5.70292

2.36223i

2.42641

5.85788i

−5.68246

−

1.13031i

131.1

−1.08979

−

2.63099i

4.32023

−

2.88669i

−2.90602

2.90602i

−3.57593

−

0.711297i

−12.3030

−

8.22059i

−3.23784

0.644047i

0.288703

0.119585i

6.88730

−

16.6274i

2.02560

10.1834i

158.1