LMFDB - Newform orbit 289.3.e.n

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} - \zeta_{16}^{6} + 1) q^{2} + (\zeta_{16}^{6} - 2 \zeta_{16}^{5} + \cdots + 1) q^{3}+ \cdots + ( - 2 \zeta_{16}^{7} - \zeta_{16}^{6} + \cdots - 7) q^{9}+O(q^{10}) $ q + (z^7 - z^6 + 1) * q^2 + (z^6 - 2z^5 + 2z^4 - z^3 + z + 1) * q^3 + (2z^7 + z^6 + 2z^5 - z^4 + 1) * q^4 + (z^7 - z^6 - 2z^5 + z^4 - 3z^3 - 3z^2 + z - 2) q^5 + (-3z^5 + 5z^4 - 5z^3 + 3z^2) * q^6 + (-2z^7 + 2z^6 + 2z^5 - 2z^4 - z^3 - z^2 + z + 1) * q^7 + (3z^7 + 2z^6 - z^5 + 2z^4 + 3z^3 - z^2 + 1) * q^8 + (-2z^7 - z^6 + z^4 + 2z^3 - 7z^2 + 3z - 7) * q^9	$ q + (\zeta_{16}^{7} - \zeta_{16}^{6} + 1) q^{2} + (\zeta_{16}^{6} - 2 \zeta_{16}^{5} + \cdots + 1) q^{3}+ \cdots + ( - 17 \zeta_{16}^{7} - 32 \zeta_{16}^{6} + \cdots + 15) q^{99}+O(q^{100}) $ q + (z^7 - z^6 + 1) * q^2 + (z^6 - 2z^5 + 2z^4 - z^3 + z + 1) * q^3 + (2z^7 + z^6 + 2z^5 - z^4 + 1) * q^4 + (z^7 - z^6 - 2z^5 + z^4 - 3z^3 - 3z^2 + z - 2) q^5 + (-3z^5 + 5z^4 - 5z^3 + 3z^2) * q^6 + (-2z^7 + 2z^6 + 2z^5 - 2z^4 - z^3 - z^2 + z + 1) * q^7 + (3z^7 + 2z^6 - z^5 + 2z^4 + 3z^3 - z^2 + 1) * q^8 + (-2z^7 - z^6 + z^4 + 2z^3 - 7z^2 + 3z - 7) * q^9 + (-2z^7 + 2z^4 - 6z^3 + z^2 + z - 6) q^10 + (3z^7 + 5z^6 + 5z + 3) q^11 + (4z^7 + 4z^6 - 3z^5 + 4z^4 - 5z^3 + 5z^2 - 4z + 3) q^12 + (-7z^3 - 2z^2 - 7z) q^13 + (-2z^7 + 3z^6 - 2z^5 - 2z^4 + 3z^3 - 2z^2 + z - 1) * q^14 + (-10z^6 + 7z^5 - 4z^4 - 6z^3 - 4z^2 + 7z - 10) * q^15 + (4z^7 + 2z^6 - 8z^5 + z^4 - 8z^3 + 2z^2 + 4z) * q^16 + (-12z^7 + 8z^6 - z^5 + z^3 - 8z^2 + 12z - 17) * q^18 + (-6z^7 + 10z^6 + 13z^5 + 4z^4 - 4z^2 - 13z - 10) * q^19 + (-5z^7 - 6z^5 + 6z^4 + 5z^2 + 6z + 6) q^20 + (3z^7 + 2z^6 + 3z^5 - 8z^4 + 3z^3 - 3z + 8) * q^21 + (z^7 - z^6 - 2z^5 + 5z^4 + 5z - 2) q^22 + (13z^7 - z^6 - 4z^5 - 2z^4 + 2z^3 + 4z^2 + z - 13) q^23 + (11z^7 - 3z^6 - 3z^5 + 11z^4 - 6z^2 + 6z) * q^24 + (2z^6 + 5z^5 + 2z^4 + 17z^2 - 17) * q^25 + (7z^7 - 7z^3 + 5z^2 - 12z + 5) * q^26 + (8z^7 - 12z^6 + 12z^5 - 8z^4 + 4z^3 - 14z^2 - 14z + 4) q^27 + (10z^6 + z^5 - 5z^4 - 5z^3 + z^2 + 10z) * q^28 + (-18z^4 - 3z^3 + 3z^2 + 18z) * q^29 + (-17z^7 + 17z^5 - 21z^4 + 5z^3 - 2z^2 + 5z - 21) * q^30 + (-4z^7 - 3z^6 - 11z^5 - 11z^4 - 3z^3 - 4z^2 - 18z + 18) q^31 + (2z^6 - 18z^5 + 3z^4 - 13z^3 + 3z^2 - 18z + 2) * q^32 + (13z^7 - 2z^6 + z^5 + 2z^4 + z^3 - 2z^2 + 13z) q^33 + (7z^7 + 5z^6 - 5z^5 + 5z^3 - 5z^2 - 7z + 10) * q^35 + (-29z^7 + 9z^6 - 13z^5 + 13z^4 - 13z^2 + 13z - 9) * q^36 + (-21z^7 + 7z^6 - 3z^5 + 3z^4 - 7z^3 + 21z^2 + 6z + 6) q^37 + (-3z^7 + 26z^6 - 3z^5 + z^4 + 9z^3 - 9z - 1) q^38 + (-17z^7 + 17z^6 - 12z^5 - 9z^3 - 9z^2 - 12) q^39 + (-z^7 - 25z^6 - 3z^5 + 12z^4 - 12z^3 + 3z^2 + 25z + 1) * q^40 + (18z^7 + 18z^4 - 26z^3 + 7z^2 - 7z + 26) q^41 + (14z^7 - 9z^6 + 4z^5 - 9z^4 + 14z^3 - 11z^2 + 11) * q^42 + (7z^7 - 12z^6 + 12z^4 - 7z^3 + 6z^2 - 24z + 6) * q^43 + (14z^7 + 12z^6 - 12z^5 - 14z^4 - 7z^3 + 5z^2 + 5z - 7) q^44 + (6z^7 - 11z^6 + 32z^5 + 32z^2 - 11z + 6) q^45 + (-z^7 - z^6 + 10z^5 + z^4 - z - 10) q^46 + (-11z^7 + 11z^5 + 10z^4 - 3z^3 - 4z^2 - 3z + 10) * q^47 + (-11z^7 - 2z^6 - 5z^5 - 5z^4 - 2z^3 - 11z^2 + 32z - 32) q^48 + (9z^6 + 10z^5 - z^4 + 29z^3 - z^2 + 10z + 9) * q^49 + (-17z^7 + 19z^6 + 3z^5 - z^4 + 3z^3 + 19z^2 - 17z) * q^50 + (-4z^7 - 12z^6 + 7z^5 - 7z^3 + 12z^2 + 4z + 30) * q^52 + (36z^7 - 11z^6 - 6z^5 - 12z^4 + 12z^2 + 6z + 11) * q^53 + (26z^7 - 24z^6 + 32z^5 - 32z^4 + 24z^3 - 26z^2 + 4z + 4) q^54 + (-3z^7 - 26z^6 - 3z^5 - z^4 - 17z^3 + 17z + 1) q^55 + (-6z^7 + 6z^6 - z^5 - 8z^4 + 9z^3 + 9z^2 - 8z - 1) * q^56 + (-5z^7 + 31z^6 + 21z^5 - 25z^4 + 25z^3 - 21z^2 - 31z + 5) q^57 + (-18z^7 - 18z^4 + 15z^3 - 12z^2 + 12z - 15) q^58 + (9z^7 + z^6 + 16z^5 + z^4 + 9z^3 + 5z^2 - 5) * q^59 + (-15z^7 - 2z^6 + 2z^4 + 15z^3 - 12z^2 + 36z - 12) * q^60 + (-11z^7 - 33z^6 + 33z^5 + 11z^4 - 2z^3 + 7z^2 + 7z - 2) q^61 + (32z^7 - 17z^6 - 12z^5 - 3z^4 - 3z^3 - 12z^2 - 17z + 32) q^62 + (-2z^7 - 2z^6 - 12z^5 + 27z^4 - z^3 + z^2 - 27z + 12) q^63 + (36z^7 - 36z^5 + 15z^4 - 2z^3 + 15z^2 - 2z + 15) * q^64 + (4z^7 + 33z^6 + 20z^5 + 20z^4 + 33z^3 + 4z^2 + 9z - 9) q^65 + (-15z^6 + 16z^5 - z^4 - z^2 + 16z - 15) q^66 + (-25z^7 - 8z^6 - 3z^5 + 4z^4 - 3z^3 - 8z^2 - 25z) q^67 + (11z^7 - 14z^6 + 5z^5 - 5z^3 + 14z^2 - 11z - 26) * q^69 + (24z^7 - 12z^6 - 3z^5 + 10z^4 - 10z^2 + 3z + 12) * q^70 + (4z^7 + 17z^6 - 6z^5 + 6z^4 - 17z^3 - 4z^2 - 41z - 41) q^71 + (-3z^7 - 21z^6 - 3z^5 + 3z^4 - 22z^3 + 22z - 3) * q^72 + (30z^7 - 30z^6 + 20z^5 + 28z^4 + 3z^3 + 3z^2 + 28z + 20) q^73 + (-21z^7 + 22z^6 - 31z^5 + 13z^4 - 13z^3 + 31z^2 - 22z + 21) q^74 + (-34z^7 + 24z^6 + 24z^5 - 34z^4 + 33z^3 + 21z^2 - 21z - 33) q^75 + (-47z^7 - 10z^6 - 8z^5 - 10z^4 - 47z^3 - 24z^2 + 24) * q^76 + (12z^7 + 27z^6 - 27z^4 - 12z^3 + 15z^2 + 16z + 15) * q^77 + (-29z^7 + 46z^6 - 46z^5 + 29z^4 - 21z^3 - 21) q^78 + (-44z^7 - 12z^6 - 41z^5 + 17z^4 + 17z^3 - 41z^2 - 12z - 44) q^79 + (-z^7 - z^6 + 41z^5 - 10z^4 - 3z^3 + 3z^2 + 10z - 41) q^80 + (35z^7 - 35z^5 + 28z^4 - 3z^3 + 9z^2 - 3z + 28) * q^81 + (51z^7 - 44z^6 + 18z^5 + 18z^4 - 44z^3 + 51z^2 - 40z + 40) q^82 + (63z^6 - 11z^5 + 11z^4 - 10z^3 + 11z^2 - 11z + 63) * q^83 + (13z^7 - 2z^6 + 15z^5 - 30z^4 + 15z^3 - 2z^2 + 13z) q^84 + (37z^7 - 25z^6 + 19z^5 - 19z^3 + 25z^2 - 37z + 36) * q^86 + (24z^7 - 27z^6 + 15z^5 - 39z^4 + 39z^2 - 15z + 27) * q^87 + (22z^7 - 3z^6 - 14z^5 + 14z^4 + 3z^3 - 22z^2 - 7z - 7) q^88 + (-8z^7 - 2z^6 - 8z^5 + 36z^4 + 47z^3 - 47z - 36) * q^89 + (23z^7 - 23z^6 + 49z^5 - 43z^4 + 32z^3 + 32z^2 - 43z + 49) q^90 + (-4z^7 - 3z^6 + 23z^5 + 2z^4 - 2z^3 - 23z^2 + 3z + 4) q^91 + (-6z^7 - 42z^6 - 42z^5 - 6z^4 + 25z^3 + 9z^2 - 9z - 25) q^92 + (-12z^7 + 35z^6 - 86z^5 + 35z^4 - 12z^3 - 31z^2 + 31) * q^93 + (2z^7 + z^6 - z^4 - 2z^3 + 9z^2 - 2z + 9) * q^94 + (-18z^7 + 19z^6 - 19z^5 + 18z^4 + 100z^3 + 27z^2 + 27z + 100) q^95 + (-51z^7 + 41z^6 - 58z^5 + 10z^4 + 10z^3 - 58z^2 + 41z - 51) q^96 + (-35z^7 - 35z^6 - 63z^5 + 26z^4 + 43z^3 - 43z^2 - 26z + 63) q^97 + (-z^7 + z^5 - 2z^4 + 40z^3 - 31z^2 + 40z - 2) * q^98 + (-17z^7 - 32z^6 + 18z^5 + 18z^4 - 32z^3 - 17z^2 - 15z + 15) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 16 q^{5} + 8 q^{7} + 8 q^{8} - 56 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 - 16 * q^5 + 8 * q^7 + 8 * q^8 - 56 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 16 q^{5} + 8 q^{7} + 8 q^{8} - 56 q^{9} - 48 q^{10} + 24 q^{11} + 24 q^{12} - 8 q^{14} - 80 q^{15} - 136 q^{18} - 80 q^{19} + 48 q^{20} + 64 q^{21} - 16 q^{22} - 104 q^{23} - 136 q^{25} + 40 q^{26} + 32 q^{27} - 168 q^{30} + 144 q^{31} + 16 q^{32} + 80 q^{35} - 72 q^{36} + 48 q^{37} - 8 q^{38} - 96 q^{39} + 8 q^{40} + 208 q^{41} + 88 q^{42} + 48 q^{43} - 56 q^{44} + 48 q^{45} - 80 q^{46} + 80 q^{47} - 256 q^{48} + 72 q^{49} + 240 q^{52} + 88 q^{53} + 32 q^{54} + 8 q^{55} - 8 q^{56} + 40 q^{57} - 120 q^{58} - 40 q^{59} - 96 q^{60} - 16 q^{61} + 256 q^{62} + 96 q^{63} + 120 q^{64} - 72 q^{65} - 120 q^{66} - 208 q^{69} + 96 q^{70} - 328 q^{71} - 24 q^{72} + 160 q^{73} + 168 q^{74} - 264 q^{75} + 192 q^{76} + 120 q^{77} - 168 q^{78} - 352 q^{79} - 328 q^{80} + 224 q^{81} + 320 q^{82} + 504 q^{83} + 288 q^{86} + 216 q^{87} - 56 q^{88} - 288 q^{89} + 392 q^{90} + 32 q^{91} - 200 q^{92} + 248 q^{93} + 72 q^{94} + 800 q^{95} - 408 q^{96} + 504 q^{97} - 16 q^{98} + 120 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 - 16 * q^5 + 8 * q^7 + 8 * q^8 - 56 * q^9 - 48 * q^10 + 24 * q^11 + 24 * q^12 - 8 * q^14 - 80 * q^15 - 136 * q^18 - 80 * q^19 + 48 * q^20 + 64 * q^21 - 16 * q^22 - 104 * q^23 - 136 * q^25 + 40 * q^26 + 32 * q^27 - 168 * q^30 + 144 * q^31 + 16 * q^32 + 80 * q^35 - 72 * q^36 + 48 * q^37 - 8 * q^38 - 96 * q^39 + 8 * q^40 + 208 * q^41 + 88 * q^42 + 48 * q^43 - 56 * q^44 + 48 * q^45 - 80 * q^46 + 80 * q^47 - 256 * q^48 + 72 * q^49 + 240 * q^52 + 88 * q^53 + 32 * q^54 + 8 * q^55 - 8 * q^56 + 40 * q^57 - 120 * q^58 - 40 * q^59 - 96 * q^60 - 16 * q^61 + 256 * q^62 + 96 * q^63 + 120 * q^64 - 72 * q^65 - 120 * q^66 - 208 * q^69 + 96 * q^70 - 328 * q^71 - 24 * q^72 + 160 * q^73 + 168 * q^74 - 264 * q^75 + 192 * q^76 + 120 * q^77 - 168 * q^78 - 352 * q^79 - 328 * q^80 + 224 * q^81 + 320 * q^82 + 504 * q^83 + 288 * q^86 + 216 * q^87 - 56 * q^88 - 288 * q^89 + 392 * q^90 + 32 * q^91 - 200 * q^92 + 248 * q^93 + 72 * q^94 + 800 * q^95 - 408 * q^96 + 504 * q^97 - 16 * q^98 + 120 * 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

0.783227

0.324423i

1.59946

−

0.318152i

−2.32023

−

2.32023i

−3.79690

5.68246i

1.35595

0.269716i

0.502268

0.751697i

−2.36223

−

5.70292i

−5.85788

2.42641i

−4.81736

3.21885i

65.1

2.63099

−

1.08979i

−1.01367

5.09606i

2.90602

−

2.90602i

−3.03153

2.02560i

2.88669

14.5124i

−2.74491

−

1.83409i

0.119585

−

0.288703i

−16.6274

−

6.88730i

−5.76843

8.63307i

75.1

0.675577

−

1.63099i

2.24830

−

3.36482i

0.624715

0.624715i

−1.50967

−

7.58960i

−3.96908

−

5.94015i

−0.798369

4.01367i

7.96489

−

3.29916i

−2.82302

−

6.81537i

−13.3984

−

2.66511i

131.1

−0.0897902

−

0.216773i

1.16591

−

0.779037i

2.78950

−

2.78950i

0.338093

0.0672509i

−0.273561

−

0.182788i

7.04101

−

1.40054i

−1.72225

−

0.713379i

−2.69170

6.49834i

−0.0157793

−

0.0793278i

158.1