LMFDB - Newform orbit 289.3.e.m

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} + \cdots + 1) q^{2}+ \cdots + ( - 4 \zeta_{16}^{7} - \zeta_{16}^{6} + \cdots + 2) q^{9}+O(q^{10}) $ q + (-z^7 - z^6 - z^5 + z^4 - z^2 + z + 1) * q^2 + (-z^7 - z^5 + z^4 + z^2 + z + 1) * q^3 + (-2z^7 - 3z^6 - 2z^5 - 2z^3 + 2z) q^4 + (-2z^7 + 2z^6 + 2z^5 - 2z^4 - 2z + 2) q^5 + (-3z^7 - 4z^6 - z^5 - z^4 + z^3 + z^2 + 4z + 3) q^6 + (z^7 + z^6 + z^5 + z^4 + z^3 + 5z^2 - 5z - 1) * q^7 + (-3z^7 - 5z^6 + z^5 - 5z^4 - 3z^3 - 3z^2 + 3) q^8 + (-4z^7 - z^6 + z^4 + 4z^3 + 2z^2 - 3z + 2) * q^9	$ q + ( - \zeta_{16}^{7} - \zeta_{16}^{6} + \cdots + 1) q^{2}+ \cdots + ( - 27 \zeta_{16}^{7} + 16 \zeta_{16}^{6} + \cdots + 34) q^{99}+O(q^{100}) $ q + (-z^7 - z^6 - z^5 + z^4 - z^2 + z + 1) * q^2 + (-z^7 - z^5 + z^4 + z^2 + z + 1) * q^3 + (-2z^7 - 3z^6 - 2z^5 - 2z^3 + 2z) q^4 + (-2z^7 + 2z^6 + 2z^5 - 2z^4 - 2z + 2) q^5 + (-3z^7 - 4z^6 - z^5 - z^4 + z^3 + z^2 + 4z + 3) q^6 + (z^7 + z^6 + z^5 + z^4 + z^3 + 5z^2 - 5z - 1) * q^7 + (-3z^7 - 5z^6 + z^5 - 5z^4 - 3z^3 - 3z^2 + 3) q^8 + (-4z^7 - z^6 + z^4 + 4z^3 + 2z^2 - 3z + 2) * q^9 + (-2z^7 + 4z^6 - 4z^5 + 2z^4 + 6z^3 - 6z^2 - 6z + 6) q^10 + (6z^7 + 3z^6 + z^5 - z^4 - z^3 + z^2 + 3z + 6) q^11 + (-9z^7 - 9z^6 - 5z^5 - 6z^4 - z^3 + z^2 + 6z + 5) q^12 + (2z^7 - 2z^5 + 2z^4 - 10z^3 - 10z + 2) q^13 + (3z^7 + 13z^6 - z^5 - z^4 + 13z^3 + 3z^2 + z - 1) * q^14 + (2z^6 - 2z^5 - 2z + 2) q^15 + (-4z^7 - 8z^6 - 4z^5 + 3z^4 - 4z^3 - 8z^2 - 4z) q^16 + (-2z^7 - 3z^6 - 13z^5 + 13z^3 + 3z^2 + 2z + 7) * q^18 + (-4z^7 + 7z^5 + 6z^4 - 6z^2 - 7z) q^19 + (10z^7 - 2z^6 - 10z^5 + 10z^4 + 2z^3 - 10z^2 + 4z + 4) q^20 + (14z^6 + 8z^4 + 2z^3 - 2z - 8) * q^21 + (-3z^7 + 3z^6 + 7z^5 + 13z^4 - 6z^3 - 6z^2 + 13z + 7) q^22 + (-5z^7 - 3z^6 + 7z^5 - 7z^4 + 7z^3 - 7z^2 + 3z + 5) q^23 + (-10z^7 - 15z^6 - 15z^5 - 10z^4 - 13z^3 + 3z^2 - 3z + 13) q^24 + (-16z^7 - 4z^6 - z^5 - 4z^4 - 16z^3 + 8z^2 - 8) q^25 + (2z^7 + 6z^6 - 6z^4 - 2z^3 - 22z^2 - 12z - 22) * q^26 + (-5z^7 + 10z^6 - 10z^5 + 5z^4 - 2z^3 - 3z^2 - 3z - 2) q^27 + (7z^7 + 15z^6 - z^5 + 19z^4 + 19z^3 - z^2 + 15z + 7) q^28 + (16z^7 + 16z^6 + 8z^5 - 6z^4 + 2z^3 - 2z^2 + 6z - 8) q^29 + (4z^7 - 4z^5 + 2z^4 + 2z^3 - 8z^2 + 2z + 2) * q^30 + (-5z^7 - 5z^6 - 3z^5 - 3z^4 - 5z^3 - 5z^2 + 5z - 5) q^31 + (-11z^6 - z^5 + 11z^4 - 17z^3 + 11z^2 - z - 11) * q^32 + (2z^7 + 7z^6 - z^5 + 12z^4 - z^3 + 7z^2 + 2z) q^33 + (-20z^6 + 8z^5 - 8z^3 + 20z^2 - 20) * q^35 + (-3z^7 - 16z^6 - 20z^5 + 3z^4 - 3z^2 + 20z + 16) * q^36 + (-2z^7 - 12z^6 + 4z^5 - 4z^4 + 12z^3 + 2z^2 - 16z - 16) q^37 + (2z^7 - 2z^6 + 2z^5 + 15z^4 + 18z^3 - 18z - 15) * q^38 + (-12z^7 + 12z^6 - 22z^5 - 6z^4 - 20z^3 - 20z^2 - 6z - 22) q^39 + (-2z^7 - 6z^6 + 14z^5 + 8z^4 - 8z^3 - 14z^2 + 6z + 2) q^40 + (6z^7 - 5z^6 - 5z^5 + 6z^4 + 14z^3 - 6z^2 + 6z - 14) q^41 + (26z^7 + 16z^6 + 26z^5 + 16z^4 + 26z^3 + 2z^2 - 2) * q^42 + (2z^7 + 14z^6 - 14z^4 - 2z^3 + 29z^2 + 22z + 29) * q^43 + (-15z^7 - 8z^6 + 8z^5 + 15z^4 - 3z^3 + 15z^2 + 15z - 3) q^44 + (-20z^7 - 16z^6 + 14z^5 + 14z^2 - 16z - 20) q^45 + (-9z^7 - 9z^6 - 17z^5 + 11z^4 - z^3 + z^2 - 11z + 17) q^46 + (22z^7 - 22z^5 + 24z^4 + 8z^3 + 2z^2 + 8z + 24) * q^47 + (-12z^7 - 17z^6 - 17z^5 - 17z^4 - 17z^3 - 12z^2 - z + 1) * q^48 + (-2z^6 - 4z^5 + 10z^4 - 7z^3 + 10z^2 - 4z - 2) * q^49 + (-35z^7 - z^6 - z^5 - 57z^4 - z^3 - z^2 - 35z) q^50 + (22z^7 + 34z^6 + 6z^5 - 6z^3 - 34z^2 - 22z - 48) * q^52 + (2z^7 - 4z^6 - 18z^5 + 16z^4 - 16z^2 + 18z + 4) * q^53 + (21z^7 - 8z^6 + z^5 - z^4 + 8z^3 - 21z^2) * q^54 + (4z^7 + 32z^6 + 4z^5 - 28z^4 - 2z^3 + 2z + 28) * q^55 + (25z^7 - 25z^6 + 17z^5 + 15z^4 + 33z^3 + 33z^2 + 15z + 17) q^56 + (9z^7 + 10z^6 + 6z^5 + 3z^4 - 3z^3 - 6z^2 - 10z - 9) q^57 + (30z^7 + 46z^6 + 46z^5 + 30z^4 - 4z^3 + 4) q^58 + (-10z^7 - 15z^6 - 42z^5 - 15z^4 - 10z^3 - 6z^2 + 6) * q^59 + (6z^7 - 6z^6 + 6z^4 - 6z^3 - 8z^2 + 8z - 8) * q^60 + (20z^7 - 12z^6 + 12z^5 - 20z^4 + 8z^3 + 36z^2 + 36z + 8) q^61 + (-7z^7 - 15z^6 - z^5 - 21z^4 - 21z^3 - z^2 - 15z - 7) q^62 + (15z^7 + 15z^6 + 21z^5 - 7z^4 - 23z^3 + 23z^2 + 7z - 21) q^63 + (-42z^7 + 42z^5 - 8z^4 + 10z^3 + 13z^2 + 10z - 8) * q^64 + (-8z^6 + 12z^5 + 12z^4 - 8z^3 - 12z + 12) q^65 + (z^6 + 23z^5 + 12z^4 + 20z^3 + 12z^2 + 23z + 1) q^66 + (33z^7 + 4z^6 - 26z^5 + 42z^4 - 26z^3 + 4z^2 + 33z) q^67 + (8z^7 - 18z^6 + 2z^5 - 2z^3 + 18z^2 - 8z + 30) * q^69 + (-36z^7 + 28z^6 + 16z^5 - 60z^4 + 60z^2 - 16z - 28) * q^70 + (-35z^7 + 43z^6 + 5z^5 - 5z^4 - 43z^3 + 35z^2 - 11z - 11) q^71 + (-24z^7 - 53z^6 - 24z^5 - 5z^4 - z^3 + z + 5) * q^72 + (25z^7 - 25z^6 - 6z^5 - 6z^4 - 26z^3 - 26z^2 - 6z - 6) q^73 + (24z^7 + 28z^6 - 26z^5 - 20z^4 + 20z^3 + 26z^2 - 28z - 24) q^74 + (-37z^7 - 17z^6 - 17z^5 - 37z^4 - 13z^2 + 13z) * q^75 + (21z^7 + 20z^6 + 12z^5 + 20z^4 + 21z^3 + 10z^2 - 10) * q^76 + (-8z^7 - 10z^6 + 10z^4 + 8z^3 + 6z^2 - 66z + 6) * q^77 + (50z^7 - 8z^6 + 8z^5 - 50z^4 - 38z^3 - 64z^2 - 64z - 38) q^78 + (21z^7 - 3z^6 - 41z^5 - 17z^4 - 17z^3 - 41z^2 - 3z + 21) q^79 + (-8z^7 - 8z^6 - 14z^5 + 30z^4 + 30z^3 - 30z^2 - 30z + 14) q^80 + (7z^7 - 7z^5 - 53z^4 - 6z^3 - 3z^2 - 6z - 53) * q^81 + (34z^7 - 8z^6 + 8z^5 + 8z^4 - 8z^3 + 34z^2 + 17z - 17) q^82 + (33z^6 + 20z^5 - 42z^4 + 54z^3 - 42z^2 + 20z + 33) * q^83 + (34z^7 + 24z^6 + 60z^5 + 50z^4 + 60z^3 + 24z^2 + 34z) q^84 + (-66z^7 + 8z^6 - 3z^5 + 3z^3 - 8z^2 + 66z + 104) * q^86 + (52z^7 + 26z^6 + 34z^5 + 10z^4 - 10z^2 - 34z - 26) * q^87 + (-9z^7 + z^6 + 33z^5 - 33z^4 - z^3 + 9z^2 + 40z + 40) q^88 + (29z^7 - 8z^6 + 29z^5 - 20z^4 - 46z^3 + 46z + 20) * q^89 + (-28z^7 + 28z^6 - 18z^5 - 56z^4 + 48z^3 + 48z^2 - 56z - 18) q^90 + (-28z^7 - 68z^5 + 40z^4 - 40z^3 + 68z^2 + 28) q^91 + (3z^7 - 31z^6 - 31z^5 + 3z^4 - 23z^3 + 3z^2 - 3z + 23) q^92 + (-8z^7 - 26z^6 - 6z^5 - 26z^4 - 8z^3 - 8z^2 + 8) * q^93 + (18z^7 - 54z^6 + 54z^4 - 18z^3 - 4z^2 + 110z - 4) * q^94 + (-34z^7 - 24z^6 + 24z^5 + 34z^4 + 2z^3 - 24z^2 - 24z + 2) q^95 + (-29z^7 + 11z^6 - 8z^5 - 7z^4 - 7z^3 - 8z^2 + 11z - 29) q^96 + (-30z^7 - 30z^6 + 13z^5 - 6z^4 + 22z^3 - 22z^2 + 6z - 13) q^97 + (-9z^7 + 9z^5 - 15z^4 + 11z^3 + 9z^2 + 11z - 15) * q^98 + (-27z^7 + 16z^6 + 28z^5 + 28z^4 + 16z^3 - 27z^2 - 34z + 34) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{3} + 16 q^{5} + 24 q^{6} - 8 q^{7} + 24 q^{8} + 16 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^3 + 16 * q^5 + 24 * q^6 - 8 * q^7 + 24 * q^8 + 16 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 16 q^{5} + 24 q^{6} - 8 q^{7} + 24 q^{8} + 16 q^{9} + 48 q^{10} + 48 q^{11} + 40 q^{12} + 16 q^{13} - 8 q^{14} + 16 q^{15} + 56 q^{18} + 32 q^{20} - 64 q^{21} + 56 q^{22} + 40 q^{23} + 104 q^{24} - 64 q^{25} - 176 q^{26} - 16 q^{27} + 56 q^{28} - 64 q^{29} + 16 q^{30} - 40 q^{31} - 88 q^{32} - 160 q^{35} + 128 q^{36} - 128 q^{37} - 120 q^{38} - 176 q^{39} + 16 q^{40} - 112 q^{41} - 16 q^{42} + 232 q^{43} - 24 q^{44} - 160 q^{45} + 136 q^{46} + 192 q^{47} + 8 q^{48} - 16 q^{49} - 384 q^{52} + 32 q^{53} + 224 q^{55} + 136 q^{56} - 72 q^{57} + 32 q^{58} + 48 q^{59} - 64 q^{60} + 64 q^{61} - 56 q^{62} - 168 q^{63} - 64 q^{64} + 96 q^{65} + 8 q^{66} + 240 q^{69} - 224 q^{70} - 88 q^{71} + 40 q^{72} - 48 q^{73} - 192 q^{74} - 80 q^{76} + 48 q^{77} - 304 q^{78} + 168 q^{79} + 112 q^{80} - 424 q^{81} - 136 q^{82} + 264 q^{83} + 832 q^{86} - 208 q^{87} + 320 q^{88} + 160 q^{89} - 144 q^{90} + 224 q^{91} + 184 q^{92} + 64 q^{93} - 32 q^{94} + 16 q^{95} - 232 q^{96} - 104 q^{97} - 120 q^{98} + 272 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 16 * q^5 + 24 * q^6 - 8 * q^7 + 24 * q^8 + 16 * q^9 + 48 * q^10 + 48 * q^11 + 40 * q^12 + 16 * q^13 - 8 * q^14 + 16 * q^15 + 56 * q^18 + 32 * q^20 - 64 * q^21 + 56 * q^22 + 40 * q^23 + 104 * q^24 - 64 * q^25 - 176 * q^26 - 16 * q^27 + 56 * q^28 - 64 * q^29 + 16 * q^30 - 40 * q^31 - 88 * q^32 - 160 * q^35 + 128 * q^36 - 128 * q^37 - 120 * q^38 - 176 * q^39 + 16 * q^40 - 112 * q^41 - 16 * q^42 + 232 * q^43 - 24 * q^44 - 160 * q^45 + 136 * q^46 + 192 * q^47 + 8 * q^48 - 16 * q^49 - 384 * q^52 + 32 * q^53 + 224 * q^55 + 136 * q^56 - 72 * q^57 + 32 * q^58 + 48 * q^59 - 64 * q^60 + 64 * q^61 - 56 * q^62 - 168 * q^63 - 64 * q^64 + 96 * q^65 + 8 * q^66 + 240 * q^69 - 224 * q^70 - 88 * q^71 + 40 * q^72 - 48 * q^73 - 192 * q^74 - 80 * q^76 + 48 * q^77 - 304 * q^78 + 168 * q^79 + 112 * q^80 - 424 * q^81 - 136 * q^82 + 264 * q^83 + 832 * q^86 - 208 * q^87 + 320 * q^88 + 160 * q^89 - 144 * q^90 + 224 * q^91 + 184 * q^92 + 64 * q^93 - 32 * q^94 + 16 * q^95 - 232 * q^96 - 104 * q^97 - 120 * q^98 + 272 * 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

3.23044

1.33809i

3.93755

−

0.783227i

5.81684

5.81684i

−0.179580

0.268761i

13.7681

2.73864i

−3.71485

−

5.55967i

5.65512

13.6527i

6.57593

−

2.72384i

−0.939752

0.627922i

65.1

−1.23044

0.509666i

−0.523336

2.63099i

−1.57420

1.57420i

1.35115

−

0.902812i

−0.696990

−

3.50400i

7.37170

4.92562i

3.17331

−

7.66104i

1.66671

0.690373i

−1.20238

1.79949i

75.1

1.15851

−

2.79690i

0.451406

−

0.675577i

−3.65205

−

3.65205i

1.56645

7.87510i

−1.36656

−

2.04520i

−1.53233

7.70353i

−3.25778

1.34942i

3.19151

7.70500i

23.8406

4.74219i

131.1

0.841487

2.03153i

0.134381

−

0.0897902i

−0.590587

0.590587i

5.26197

1.04667i

0.295491

0.197441i

−6.12453

1.21824i

6.42935

2.66313i

−3.43416

8.29078i

2.30154

11.5706i

158.1