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Properties

Label 117.16.b.f
Level $117$
Weight $16$
Character orbit 117.b
Analytic conductor $166.951$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,16,Mod(64,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.64");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 117.b (of order \(2\), degree \(1\), 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: \(166.951400967\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 458752 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 458752 q^{4} - 228861840 q^{10} + 803257676 q^{13} - 7451384960 q^{16} - 30544905456 q^{22} - 238197886180 q^{25} + 3510668297280 q^{40} + 10830121503776 q^{43} - 19674845564108 q^{49} + 30846257429728 q^{52} + 53012410974240 q^{55} - 23758117088504 q^{61} + 327062953083392 q^{64} - 829003210390736 q^{79} - 13\!\cdots\!56 q^{82}+ \cdots + 81131101347024 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 − 315.663i 0 −66875.2 − 103626.i 0 − 730363.i 1.07664e7i 0 −3.27109e7
64.2 − 315.663i 0 −66875.2 − 103626.i 0 730363.i 1.07664e7i 0 −3.27109e7
64.3 − 281.203i 0 −46307.3 − 271057.i 0 − 3.98113e6i 3.80730e6i 0 −7.62223e7
64.4 − 281.203i 0 −46307.3 − 271057.i 0 3.98113e6i 3.80730e6i 0 −7.62223e7
64.5 − 256.142i 0 −32840.9 255459.i 0 − 2.70274e6i 18675.7i 0 6.54339e7
64.6 − 256.142i 0 −32840.9 255459.i 0 2.70274e6i 18675.7i 0 6.54339e7
64.7 − 234.538i 0 −22239.9 99432.1i 0 − 2.97332e6i − 2.46923e6i 0 2.33206e7
64.8 − 234.538i 0 −22239.9 99432.1i 0 2.97332e6i − 2.46923e6i 0 2.33206e7
64.9 − 151.203i 0 9905.69 − 46372.2i 0 − 143330.i − 6.45238e6i 0 −7.01161e6
64.10 − 151.203i 0 9905.69 − 46372.2i 0 143330.i − 6.45238e6i 0 −7.01161e6
64.11 − 141.961i 0 12615.2 − 269224.i 0 − 1.58849e6i − 6.44262e6i 0 −3.82193e7
64.12 − 141.961i 0 12615.2 − 269224.i 0 1.58849e6i − 6.44262e6i 0 −3.82193e7
64.13 − 41.3948i 0 31054.5 197949.i 0 − 1.75486e6i − 2.64192e6i 0 8.19407e6
64.14 − 41.3948i 0 31054.5 197949.i 0 1.75486e6i − 2.64192e6i 0 8.19407e6
64.15 41.3948i 0 31054.5 − 197949.i 0 − 1.75486e6i 2.64192e6i 0 8.19407e6
64.16 41.3948i 0 31054.5 − 197949.i 0 1.75486e6i 2.64192e6i 0 8.19407e6
64.17 141.961i 0 12615.2 269224.i 0 − 1.58849e6i 6.44262e6i 0 −3.82193e7
64.18 141.961i 0 12615.2 269224.i 0 1.58849e6i 6.44262e6i 0 −3.82193e7
64.19 151.203i 0 9905.69 46372.2i 0 − 143330.i 6.45238e6i 0 −7.01161e6
64.20 151.203i 0 9905.69 46372.2i 0 143330.i 6.45238e6i 0 −7.01161e6
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.16.b.f ✓ 28
3.b odd 2 1 inner 117.16.b.f ✓ 28
13.b even 2 1 inner 117.16.b.f ✓ 28
39.d odd 2 1 inner 117.16.b.f ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.16.b.f ✓ 28 1.a even 1 1 trivial
117.16.b.f ✓ 28 3.b odd 2 1 inner
117.16.b.f ✓ 28 13.b even 2 1 inner
117.16.b.f ✓ 28 39.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 344064 T_{2}^{12} + 46968103824 T_{2}^{10} + \cdots + 22\!\cdots\!00 \) acting on \(S_{16}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display