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Properties

Label 1320.2.ba.a
Level $1320$
Weight $2$
Character orbit 1320.ba
Analytic conductor $10.540$
Analytic rank $0$
Dimension $72$
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] = [1320,2,Mod(329,1320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1320.329");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1320.ba (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: \(10.5402530668\)
Analytic rank: \(0\)
Dimension: \(72\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 6 q^{15} - 4 q^{25} + 34 q^{45} + 96 q^{49} - 28 q^{55} + 28 q^{69} + 14 q^{75} + 16 q^{81} - 48 q^{91} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
329.1 0 −1.72947 − 0.0944935i 0 1.89340 + 1.18956i 0 −2.85425 0 2.98214 + 0.326848i 0
329.2 0 −1.72947 − 0.0944935i 0 1.89340 + 1.18956i 0 2.85425 0 2.98214 + 0.326848i 0
329.3 0 −1.72947 + 0.0944935i 0 1.89340 − 1.18956i 0 −2.85425 0 2.98214 − 0.326848i 0
329.4 0 −1.72947 + 0.0944935i 0 1.89340 − 1.18956i 0 2.85425 0 2.98214 − 0.326848i 0
329.5 0 −1.72010 − 0.203093i 0 −0.910442 − 2.04233i 0 −4.06802 0 2.91751 + 0.698681i 0
329.6 0 −1.72010 − 0.203093i 0 −0.910442 − 2.04233i 0 4.06802 0 2.91751 + 0.698681i 0
329.7 0 −1.72010 + 0.203093i 0 −0.910442 + 2.04233i 0 −4.06802 0 2.91751 − 0.698681i 0
329.8 0 −1.72010 + 0.203093i 0 −0.910442 + 2.04233i 0 4.06802 0 2.91751 − 0.698681i 0
329.9 0 −1.54444 − 0.784039i 0 −2.21025 + 0.338784i 0 −1.57281 0 1.77056 + 2.42180i 0
329.10 0 −1.54444 − 0.784039i 0 −2.21025 + 0.338784i 0 1.57281 0 1.77056 + 2.42180i 0
329.11 0 −1.54444 + 0.784039i 0 −2.21025 − 0.338784i 0 −1.57281 0 1.77056 − 2.42180i 0
329.12 0 −1.54444 + 0.784039i 0 −2.21025 − 0.338784i 0 1.57281 0 1.77056 − 2.42180i 0
329.13 0 −1.28679 − 1.15938i 0 2.05055 − 0.891762i 0 −2.43485 0 0.311664 + 2.98377i 0
329.14 0 −1.28679 − 1.15938i 0 2.05055 − 0.891762i 0 2.43485 0 0.311664 + 2.98377i 0
329.15 0 −1.28679 + 1.15938i 0 2.05055 + 0.891762i 0 −2.43485 0 0.311664 − 2.98377i 0
329.16 0 −1.28679 + 1.15938i 0 2.05055 + 0.891762i 0 2.43485 0 0.311664 − 2.98377i 0
329.17 0 −1.28574 − 1.16055i 0 0.277937 + 2.21873i 0 −1.10479 0 0.306234 + 2.98433i 0
329.18 0 −1.28574 − 1.16055i 0 0.277937 + 2.21873i 0 1.10479 0 0.306234 + 2.98433i 0
329.19 0 −1.28574 + 1.16055i 0 0.277937 − 2.21873i 0 −1.10479 0 0.306234 − 2.98433i 0
329.20 0 −1.28574 + 1.16055i 0 0.277937 − 2.21873i 0 1.10479 0 0.306234 − 2.98433i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 329.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
11.b odd 2 1 inner
15.d odd 2 1 inner
33.d even 2 1 inner
55.d odd 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1320.2.ba.a ✓ 72
3.b odd 2 1 inner 1320.2.ba.a ✓ 72
5.b even 2 1 inner 1320.2.ba.a ✓ 72
11.b odd 2 1 inner 1320.2.ba.a ✓ 72
15.d odd 2 1 inner 1320.2.ba.a ✓ 72
33.d even 2 1 inner 1320.2.ba.a ✓ 72
55.d odd 2 1 inner 1320.2.ba.a ✓ 72
165.d even 2 1 inner 1320.2.ba.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.ba.a ✓ 72 1.a even 1 1 trivial
1320.2.ba.a ✓ 72 3.b odd 2 1 inner
1320.2.ba.a ✓ 72 5.b even 2 1 inner
1320.2.ba.a ✓ 72 11.b odd 2 1 inner
1320.2.ba.a ✓ 72 15.d odd 2 1 inner
1320.2.ba.a ✓ 72 33.d even 2 1 inner
1320.2.ba.a ✓ 72 55.d odd 2 1 inner
1320.2.ba.a ✓ 72 165.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1320, [\chi])\).