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Properties

Label 125.5.c.a
Level $125$
Weight $5$
Character orbit 125.c
Analytic conductor $12.921$
Analytic rank $0$
Dimension $32$
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] = [125,5,Mod(57,125)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(125, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("125.57");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 125.c (of order \(4\), degree \(2\), 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: \(12.9212453855\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 236 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 236 q^{6} - 36 q^{11} - 1028 q^{16} - 2236 q^{21} + 4944 q^{26} - 4836 q^{31} + 752 q^{36} + 17844 q^{41} + 30284 q^{46} - 27116 q^{51} - 70500 q^{56} + 444 q^{61} + 91828 q^{66} + 26424 q^{71} - 87500 q^{76} - 127728 q^{81} - 62796 q^{86} + 171484 q^{91} + 280064 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1 −5.38338 − 5.38338i 8.40023 − 8.40023i 41.9616i 0 −90.4433 −28.8863 − 28.8863i 139.761 − 139.761i − 60.1277i 0
57.2 −4.37808 − 4.37808i −0.00860182 + 0.00860182i 22.3351i 0 0.0753188 −36.7243 − 36.7243i 27.7355 − 27.7355i 80.9999i 0
57.3 −3.68186 − 3.68186i −4.51746 + 4.51746i 11.1122i 0 33.2654 −10.6844 − 10.6844i −17.9961 + 17.9961i 40.1850i 0
57.4 −3.39378 − 3.39378i −11.0381 + 11.0381i 7.03547i 0 74.9215 46.5875 + 46.5875i −30.4236 + 30.4236i − 162.677i 0
57.5 −2.85507 − 2.85507i 10.2630 − 10.2630i 0.302903i 0 −58.6034 17.6184 + 17.6184i −44.8164 + 44.8164i − 129.660i 0
57.6 −1.93233 − 1.93233i 0.841844 − 0.841844i − 8.53219i 0 −3.25344 32.4472 + 32.4472i −47.4043 + 47.4043i 79.5826i 0
57.7 −0.626346 − 0.626346i 11.9008 − 11.9008i − 15.2154i 0 −14.9080 9.48254 + 9.48254i −19.5516 + 19.5516i − 202.258i 0
57.8 −0.0120374 − 0.0120374i 2.24108 − 2.24108i − 15.9997i 0 −0.0539535 48.1115 + 48.1115i −0.385194 + 0.385194i 70.9551i 0
57.9 0.0120374 + 0.0120374i −2.24108 + 2.24108i − 15.9997i 0 −0.0539535 −48.1115 − 48.1115i 0.385194 − 0.385194i 70.9551i 0
57.10 0.626346 + 0.626346i −11.9008 + 11.9008i − 15.2154i 0 −14.9080 −9.48254 − 9.48254i 19.5516 − 19.5516i − 202.258i 0
57.11 1.93233 + 1.93233i −0.841844 + 0.841844i − 8.53219i 0 −3.25344 −32.4472 − 32.4472i 47.4043 − 47.4043i 79.5826i 0
57.12 2.85507 + 2.85507i −10.2630 + 10.2630i 0.302903i 0 −58.6034 −17.6184 − 17.6184i 44.8164 − 44.8164i − 129.660i 0
57.13 3.39378 + 3.39378i 11.0381 − 11.0381i 7.03547i 0 74.9215 −46.5875 − 46.5875i 30.4236 − 30.4236i − 162.677i 0
57.14 3.68186 + 3.68186i 4.51746 − 4.51746i 11.1122i 0 33.2654 10.6844 + 10.6844i 17.9961 − 17.9961i 40.1850i 0
57.15 4.37808 + 4.37808i 0.00860182 − 0.00860182i 22.3351i 0 0.0753188 36.7243 + 36.7243i −27.7355 + 27.7355i 80.9999i 0
57.16 5.38338 + 5.38338i −8.40023 + 8.40023i 41.9616i 0 −90.4433 28.8863 + 28.8863i −139.761 + 139.761i − 60.1277i 0
68.1 −5.38338 + 5.38338i 8.40023 + 8.40023i − 41.9616i 0 −90.4433 −28.8863 + 28.8863i 139.761 + 139.761i 60.1277i 0
68.2 −4.37808 + 4.37808i −0.00860182 − 0.00860182i − 22.3351i 0 0.0753188 −36.7243 + 36.7243i 27.7355 + 27.7355i − 80.9999i 0
68.3 −3.68186 + 3.68186i −4.51746 − 4.51746i − 11.1122i 0 33.2654 −10.6844 + 10.6844i −17.9961 − 17.9961i − 40.1850i 0
68.4 −3.39378 + 3.39378i −11.0381 − 11.0381i − 7.03547i 0 74.9215 46.5875 − 46.5875i −30.4236 − 30.4236i 162.677i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 125.5.c.a ✓ 32
5.b even 2 1 inner 125.5.c.a ✓ 32
5.c odd 4 2 inner 125.5.c.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.5.c.a ✓ 32 1.a even 1 1 trivial
125.5.c.a ✓ 32 5.b even 2 1 inner
125.5.c.a ✓ 32 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 6417 T_{2}^{28} + 13418008 T_{2}^{24} + 11909534889 T_{2}^{20} + 4717673986545 T_{2}^{16} + \cdots + 1475789056 \) acting on \(S_{5}^{\mathrm{new}}(125, [\chi])\). Copy content Toggle raw display