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Properties

Label 3344.2.o.a
Level $3344$
Weight $2$
Character orbit 3344.o
Analytic conductor $26.702$
Analytic rank $0$
Dimension $36$
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] = [3344,2,Mod(1519,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3344.1519");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.o (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: \(26.7019744359\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 44 q^{9} - 16 q^{17} + 36 q^{25} - 32 q^{45} - 28 q^{49} + 24 q^{57} - 48 q^{61} - 24 q^{73} + 52 q^{81} + 24 q^{85} - 64 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1519.1 0 −3.25299 0 −2.27354 0 4.28093i 0 7.58195 0
1519.2 0 −3.25299 0 −2.27354 0 − 4.28093i 0 7.58195 0
1519.3 0 −3.02651 0 2.92273 0 2.58786i 0 6.15975 0
1519.4 0 −3.02651 0 2.92273 0 − 2.58786i 0 6.15975 0
1519.5 0 −2.50928 0 −1.52863 0 0.415002i 0 3.29648 0
1519.6 0 −2.50928 0 −1.52863 0 − 0.415002i 0 3.29648 0
1519.7 0 −2.23461 0 1.13396 0 3.35736i 0 1.99347 0
1519.8 0 −2.23461 0 1.13396 0 − 3.35736i 0 1.99347 0
1519.9 0 −1.71785 0 −3.66778 0 2.65082i 0 −0.0489799 0
1519.10 0 −1.71785 0 −3.66778 0 − 2.65082i 0 −0.0489799 0
1519.11 0 −1.31411 0 1.36581 0 4.52438i 0 −1.27312 0
1519.12 0 −1.31411 0 1.36581 0 − 4.52438i 0 −1.27312 0
1519.13 0 −0.963498 0 3.73336 0 0.145369i 0 −2.07167 0
1519.14 0 −0.963498 0 3.73336 0 − 0.145369i 0 −2.07167 0
1519.15 0 −0.924782 0 −2.57340 0 0.765529i 0 −2.14478 0
1519.16 0 −0.924782 0 −2.57340 0 − 0.765529i 0 −2.14478 0
1519.17 0 −0.711963 0 0.887508 0 − 2.32992i 0 −2.49311 0
1519.18 0 −0.711963 0 0.887508 0 2.32992i 0 −2.49311 0
1519.19 0 0.711963 0 0.887508 0 2.32992i 0 −2.49311 0
1519.20 0 0.711963 0 0.887508 0 − 2.32992i 0 −2.49311 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1519.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3344.2.o.a ✓ 36
4.b odd 2 1 inner 3344.2.o.a ✓ 36
19.b odd 2 1 inner 3344.2.o.a ✓ 36
76.d even 2 1 inner 3344.2.o.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3344.2.o.a ✓ 36 1.a even 1 1 trivial
3344.2.o.a ✓ 36 4.b odd 2 1 inner
3344.2.o.a ✓ 36 19.b odd 2 1 inner
3344.2.o.a ✓ 36 76.d even 2 1 inner