↑

Properties

Label 396.2.w.a
Level $396$
Weight $2$
Character orbit 396.w
Analytic conductor $3.162$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [396,2,Mod(71,396)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(396, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("396.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 396.w (of order \(10\), degree \(4\), 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: \(3.16207592004\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 16 q^{10} + 32 q^{16} + 60 q^{22} + 8 q^{25} - 12 q^{28} + 24 q^{34} + 16 q^{40} - 40 q^{46} + 40 q^{49} - 32 q^{52} - 80 q^{58} - 72 q^{64} - 48 q^{70} - 24 q^{73} - 136 q^{76} - 132 q^{82} - 48 q^{85} - 64 q^{88} - 80 q^{94} - 80 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −1.41073 − 0.0991802i 0 1.98033 + 0.279833i −1.34195 − 0.436027i 0 0.246271 + 0.338963i −2.76596 − 0.591179i 0 1.84989 + 0.748212i
71.2 −1.40725 + 0.140204i 0 1.96069 − 0.394603i 1.96829 + 0.639537i 0 2.79916 + 3.85272i −2.70384 + 0.830199i 0 −2.85954 − 0.624025i
71.3 −1.38746 − 0.273803i 0 1.85006 + 0.759779i −3.05406 − 0.992323i 0 −1.53508 − 2.11286i −2.35885 − 1.56071i 0 3.96566 + 2.21301i
71.4 −1.22090 + 0.713732i 0 0.981175 − 1.74278i 1.96829 + 0.639537i 0 −2.79916 − 3.85272i 0.0459679 + 2.82805i 0 −2.85954 + 0.624025i
71.5 −1.12090 − 0.862311i 0 0.512838 + 1.93313i 1.73701 + 0.564387i 0 1.29028 + 1.77592i 1.09212 − 2.60908i 0 −1.46033 − 2.13046i
71.6 −1.08301 + 0.909446i 0 0.345817 − 1.96988i −1.34195 − 0.436027i 0 −0.246271 − 0.338963i 1.41697 + 2.44790i 0 1.84989 − 0.748212i
71.7 −0.961537 + 1.03704i 0 −0.150891 − 1.99430i −3.05406 − 0.992323i 0 1.53508 + 2.11286i 2.21325 + 1.76111i 0 3.96566 − 2.21301i
71.8 −0.768885 − 1.18694i 0 −0.817630 + 1.82523i −1.74237 − 0.566131i 0 1.15540 + 1.59027i 2.79510 − 0.432922i 0 0.667724 + 2.50337i
71.9 −0.688992 − 1.23503i 0 −1.05058 + 1.70185i 3.22319 + 1.04728i 0 −1.48774 − 2.04769i 2.82567 + 0.124937i 0 −0.927335 − 4.70229i
71.10 −0.399974 + 1.35647i 0 −1.68004 − 1.08511i 1.73701 + 0.564387i 0 −1.29028 − 1.77592i 2.14389 − 1.84492i 0 −1.46033 + 2.13046i
71.11 −0.168524 − 1.40414i 0 −1.94320 + 0.473262i −3.22319 − 1.04728i 0 1.48774 + 2.04769i 0.992001 + 2.64876i 0 −0.927335 + 4.70229i
71.12 −0.0756216 − 1.41219i 0 −1.98856 + 0.213584i 1.74237 + 0.566131i 0 −1.15540 − 1.59027i 0.452000 + 2.79208i 0 0.667724 − 2.50337i
71.13 0.0756216 + 1.41219i 0 −1.98856 + 0.213584i −1.74237 − 0.566131i 0 −1.15540 − 1.59027i −0.452000 − 2.79208i 0 0.667724 − 2.50337i
71.14 0.168524 + 1.40414i 0 −1.94320 + 0.473262i 3.22319 + 1.04728i 0 1.48774 + 2.04769i −0.992001 − 2.64876i 0 −0.927335 + 4.70229i
71.15 0.399974 − 1.35647i 0 −1.68004 − 1.08511i −1.73701 − 0.564387i 0 −1.29028 − 1.77592i −2.14389 + 1.84492i 0 −1.46033 + 2.13046i
71.16 0.688992 + 1.23503i 0 −1.05058 + 1.70185i −3.22319 − 1.04728i 0 −1.48774 − 2.04769i −2.82567 − 0.124937i 0 −0.927335 − 4.70229i
71.17 0.768885 + 1.18694i 0 −0.817630 + 1.82523i 1.74237 + 0.566131i 0 1.15540 + 1.59027i −2.79510 + 0.432922i 0 0.667724 + 2.50337i
71.18 0.961537 − 1.03704i 0 −0.150891 − 1.99430i 3.05406 + 0.992323i 0 1.53508 + 2.11286i −2.21325 − 1.76111i 0 3.96566 − 2.21301i
71.19 1.08301 − 0.909446i 0 0.345817 − 1.96988i 1.34195 + 0.436027i 0 −0.246271 − 0.338963i −1.41697 − 2.44790i 0 1.84989 − 0.748212i
71.20 1.12090 + 0.862311i 0 0.512838 + 1.93313i −1.73701 − 0.564387i 0 1.29028 + 1.77592i −1.09212 + 2.60908i 0 −1.46033 − 2.13046i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
11.c even 5 1 inner
12.b even 2 1 inner
33.h odd 10 1 inner
44.h odd 10 1 inner
132.o even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 396.2.w.a ✓ 96
3.b odd 2 1 inner 396.2.w.a ✓ 96
4.b odd 2 1 inner 396.2.w.a ✓ 96
11.c even 5 1 inner 396.2.w.a ✓ 96
12.b even 2 1 inner 396.2.w.a ✓ 96
33.h odd 10 1 inner 396.2.w.a ✓ 96
44.h odd 10 1 inner 396.2.w.a ✓ 96
132.o even 10 1 inner 396.2.w.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
396.2.w.a ✓ 96 1.a even 1 1 trivial
396.2.w.a ✓ 96 3.b odd 2 1 inner
396.2.w.a ✓ 96 4.b odd 2 1 inner
396.2.w.a ✓ 96 11.c even 5 1 inner
396.2.w.a ✓ 96 12.b even 2 1 inner
396.2.w.a ✓ 96 33.h odd 10 1 inner
396.2.w.a ✓ 96 44.h odd 10 1 inner
396.2.w.a ✓ 96 132.o even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(396, [\chi])\).