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Properties

Label 825.2.cw.c
Level $825$
Weight $2$
Character orbit 825.cw
Analytic conductor $6.588$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(7,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 5, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.cw (of order \(20\), 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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 64 q^{11} + 32 q^{16} + 16 q^{26} - 8 q^{31} + 48 q^{36} + 240 q^{41} - 80 q^{51} + 64 q^{56} - 160 q^{61} - 160 q^{66} + 48 q^{71} + 32 q^{81} + 176 q^{86} + 48 q^{91} + 200 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.72955 + 0.432319i −0.453990 − 0.891007i 5.36144 − 1.74204i 0 1.62439 + 2.23578i −0.0897771 − 0.0457437i −8.95649 + 4.56356i −0.587785 + 0.809017i 0
7.2 −2.15894 + 0.341943i 0.453990 + 0.891007i 2.64199 − 0.858434i 0 −1.28481 − 1.76839i −3.95050 − 2.01288i −1.51515 + 0.772006i −0.587785 + 0.809017i 0
7.3 −1.94949 + 0.308769i 0.453990 + 0.891007i 1.80306 − 0.585850i 0 −1.16016 − 1.59683i 4.53800 + 2.31222i 0.183166 − 0.0933280i −0.587785 + 0.809017i 0
7.4 −1.39692 + 0.221251i −0.453990 − 0.891007i 0.000333510 0 0.000108364i 0 0.831327 + 1.14422i 2.41494 + 1.23047i 2.51993 − 1.28397i −0.587785 + 0.809017i 0
7.5 −1.23386 + 0.195424i −0.453990 − 0.891007i −0.417900 + 0.135784i 0 0.734283 + 1.01065i 0.557140 + 0.283877i 2.71525 − 1.38349i −0.587785 + 0.809017i 0
7.6 −0.851835 + 0.134917i 0.453990 + 0.891007i −1.19469 + 0.388179i 0 −0.506937 − 0.697740i −2.09083 − 1.06533i 2.50221 − 1.27494i −0.587785 + 0.809017i 0
7.7 −0.379421 + 0.0600944i 0.453990 + 0.891007i −1.76176 + 0.572432i 0 −0.225798 − 0.310784i −0.0303552 − 0.0154668i 1.31861 − 0.671866i −0.587785 + 0.809017i 0
7.8 −0.0206462 + 0.00327004i 0.453990 + 0.891007i −1.90170 + 0.617899i 0 −0.0122868 − 0.0169114i 4.41599 + 2.25006i 0.0744928 − 0.0379560i −0.587785 + 0.809017i 0
7.9 0.0206462 − 0.00327004i −0.453990 − 0.891007i −1.90170 + 0.617899i 0 −0.0122868 − 0.0169114i −4.41599 − 2.25006i −0.0744928 + 0.0379560i −0.587785 + 0.809017i 0
7.10 0.379421 − 0.0600944i −0.453990 − 0.891007i −1.76176 + 0.572432i 0 −0.225798 − 0.310784i 0.0303552 + 0.0154668i −1.31861 + 0.671866i −0.587785 + 0.809017i 0
7.11 0.851835 − 0.134917i −0.453990 − 0.891007i −1.19469 + 0.388179i 0 −0.506937 − 0.697740i 2.09083 + 1.06533i −2.50221 + 1.27494i −0.587785 + 0.809017i 0
7.12 1.23386 − 0.195424i 0.453990 + 0.891007i −0.417900 + 0.135784i 0 0.734283 + 1.01065i −0.557140 − 0.283877i −2.71525 + 1.38349i −0.587785 + 0.809017i 0
7.13 1.39692 − 0.221251i 0.453990 + 0.891007i 0.000333510 0 0.000108364i 0 0.831327 + 1.14422i −2.41494 − 1.23047i −2.51993 + 1.28397i −0.587785 + 0.809017i 0
7.14 1.94949 − 0.308769i −0.453990 − 0.891007i 1.80306 − 0.585850i 0 −1.16016 − 1.59683i −4.53800 − 2.31222i −0.183166 + 0.0933280i −0.587785 + 0.809017i 0
7.15 2.15894 − 0.341943i −0.453990 − 0.891007i 2.64199 − 0.858434i 0 −1.28481 − 1.76839i 3.95050 + 2.01288i 1.51515 − 0.772006i −0.587785 + 0.809017i 0
7.16 2.72955 − 0.432319i 0.453990 + 0.891007i 5.36144 − 1.74204i 0 1.62439 + 2.23578i 0.0897771 + 0.0457437i 8.95649 − 4.56356i −0.587785 + 0.809017i 0
118.1 −2.72955 − 0.432319i −0.453990 + 0.891007i 5.36144 + 1.74204i 0 1.62439 − 2.23578i −0.0897771 + 0.0457437i −8.95649 − 4.56356i −0.587785 − 0.809017i 0
118.2 −2.15894 − 0.341943i 0.453990 − 0.891007i 2.64199 + 0.858434i 0 −1.28481 + 1.76839i −3.95050 + 2.01288i −1.51515 − 0.772006i −0.587785 − 0.809017i 0
118.3 −1.94949 − 0.308769i 0.453990 − 0.891007i 1.80306 + 0.585850i 0 −1.16016 + 1.59683i 4.53800 − 2.31222i 0.183166 + 0.0933280i −0.587785 − 0.809017i 0
118.4 −1.39692 − 0.221251i −0.453990 + 0.891007i 0.000333510 0 0.000108364i 0 0.831327 − 1.14422i 2.41494 − 1.23047i 2.51993 + 1.28397i −0.587785 − 0.809017i 0
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.16
Significant digits:
Format:

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
11.d odd 10 1 inner
55.h odd 10 1 inner
55.l even 20 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.cw.c ✓ 128
5.b even 2 1 inner 825.2.cw.c ✓ 128
5.c odd 4 2 inner 825.2.cw.c ✓ 128
11.d odd 10 1 inner 825.2.cw.c ✓ 128
55.h odd 10 1 inner 825.2.cw.c ✓ 128
55.l even 20 2 inner 825.2.cw.c ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.cw.c ✓ 128 1.a even 1 1 trivial
825.2.cw.c ✓ 128 5.b even 2 1 inner
825.2.cw.c ✓ 128 5.c odd 4 2 inner
825.2.cw.c ✓ 128 11.d odd 10 1 inner
825.2.cw.c ✓ 128 55.h odd 10 1 inner
825.2.cw.c ✓ 128 55.l even 20 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{128} - 84 T_{2}^{124} + 6734 T_{2}^{120} - 470784 T_{2}^{116} + 29747929 T_{2}^{112} - 1104843700 T_{2}^{108} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\). Copy content Toggle raw display