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Properties

Label 128.7.f.b
Level $128$
Weight $7$
Character orbit 128.f
Analytic conductor $29.447$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [128,7,Mod(31,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.31");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 128.f (of order \(4\), degree \(2\), not 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: \(29.4469227033\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{3} + 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{3} + 2 q^{5} - 4 q^{7} - 1358 q^{11} + 2 q^{13} - 4 q^{17} - 3934 q^{19} + 1460 q^{21} - 13124 q^{23} - 35776 q^{27} + 33202 q^{29} - 4 q^{33} + 112420 q^{35} - 3598 q^{37} + 254396 q^{39} + 267986 q^{43} - 32706 q^{45} + 168066 q^{49} - 301788 q^{51} + 221842 q^{53} - 232708 q^{55} - 39150 q^{59} - 326494 q^{61} + 186412 q^{65} + 122786 q^{67} + 543188 q^{69} - 267012 q^{71} + 275278 q^{75} - 231180 q^{77} - 354298 q^{81} + 288322 q^{83} - 340748 q^{85} + 2029884 q^{87} - 302396 q^{91} + 1173344 q^{93} - 4 q^{97} + 271522 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −31.6770 − 31.6770i 0 −69.9376 − 69.9376i 0 −445.243 0 1277.86i 0
31.2 0 −25.6946 − 25.6946i 0 141.826 + 141.826i 0 411.977 0 591.425i 0
31.3 0 −23.4827 − 23.4827i 0 −55.3108 − 55.3108i 0 386.163 0 373.878i 0
31.4 0 −14.8973 − 14.8973i 0 −41.0202 − 41.0202i 0 −67.2599 0 − 285.141i 0
31.5 0 −8.95029 − 8.95029i 0 127.202 + 127.202i 0 −458.680 0 − 568.785i 0
31.6 0 6.79913 + 6.79913i 0 −158.437 − 158.437i 0 −213.507 0 − 636.544i 0
31.7 0 8.66104 + 8.66104i 0 38.1039 + 38.1039i 0 −108.944 0 − 578.973i 0
31.8 0 9.43202 + 9.43202i 0 −46.6419 − 46.6419i 0 647.751 0 − 551.074i 0
31.9 0 15.4507 + 15.4507i 0 66.8872 + 66.8872i 0 −121.702 0 − 251.553i 0
31.10 0 31.9540 + 31.9540i 0 −95.0213 − 95.0213i 0 −293.582 0 1313.12i 0
31.11 0 33.4050 + 33.4050i 0 93.3489 + 93.3489i 0 261.028 0 1502.79i 0
95.1 0 −31.6770 + 31.6770i 0 −69.9376 + 69.9376i 0 −445.243 0 − 1277.86i 0
95.2 0 −25.6946 + 25.6946i 0 141.826 − 141.826i 0 411.977 0 − 591.425i 0
95.3 0 −23.4827 + 23.4827i 0 −55.3108 + 55.3108i 0 386.163 0 − 373.878i 0
95.4 0 −14.8973 + 14.8973i 0 −41.0202 + 41.0202i 0 −67.2599 0 285.141i 0
95.5 0 −8.95029 + 8.95029i 0 127.202 − 127.202i 0 −458.680 0 568.785i 0
95.6 0 6.79913 − 6.79913i 0 −158.437 + 158.437i 0 −213.507 0 636.544i 0
95.7 0 8.66104 − 8.66104i 0 38.1039 − 38.1039i 0 −108.944 0 578.973i 0
95.8 0 9.43202 − 9.43202i 0 −46.6419 + 46.6419i 0 647.751 0 551.074i 0
95.9 0 15.4507 − 15.4507i 0 66.8872 − 66.8872i 0 −121.702 0 251.553i 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.7.f.b 22
4.b odd 2 1 128.7.f.a 22
8.b even 2 1 16.7.f.a ✓ 22
8.d odd 2 1 64.7.f.a 22
16.e even 4 1 64.7.f.a 22
16.e even 4 1 128.7.f.a 22
16.f odd 4 1 16.7.f.a ✓ 22
16.f odd 4 1 inner 128.7.f.b 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.7.f.a ✓ 22 8.b even 2 1
16.7.f.a ✓ 22 16.f odd 4 1
64.7.f.a 22 8.d odd 2 1
64.7.f.a 22 16.e even 4 1
128.7.f.a 22 4.b odd 2 1
128.7.f.a 22 16.e even 4 1
128.7.f.b 22 1.a even 1 1 trivial
128.7.f.b 22 16.f odd 4 1 inner