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Properties

Label 3484.2.f.a
Level $3484$
Weight $2$
Character orbit 3484.f
Analytic conductor $27.820$
Analytic rank $0$
Dimension $78$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3484,2,Mod(805,3484)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3484, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3484.805");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3484 = 2^{2} \cdot 13 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3484.f (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: \(27.8198800642\)
Analytic rank: \(0\)
Dimension: \(78\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 78 q + 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 78 q + 74 q^{9} - 4 q^{13} - 8 q^{17} + 4 q^{23} - 78 q^{25} + 4 q^{29} + 12 q^{35} - 28 q^{39} - 4 q^{43} - 78 q^{49} + 52 q^{51} - 4 q^{53} - 12 q^{55} - 10 q^{65} + 28 q^{69} + 32 q^{75} - 44 q^{77} + 28 q^{79} + 46 q^{81} - 8 q^{87} + 14 q^{91} - 16 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
805.1 0 −3.26212 0 2.35111i 0 − 0.827456i 0 7.64145 0
805.2 0 −3.26212 0 − 2.35111i 0 0.827456i 0 7.64145 0
805.3 0 −3.21013 0 0.969967i 0 − 2.99060i 0 7.30491 0
805.4 0 −3.21013 0 − 0.969967i 0 2.99060i 0 7.30491 0
805.5 0 −2.91071 0 0.623416i 0 − 4.23953i 0 5.47224 0
805.6 0 −2.91071 0 − 0.623416i 0 4.23953i 0 5.47224 0
805.7 0 −2.82295 0 − 3.42654i 0 1.36463i 0 4.96903 0
805.8 0 −2.82295 0 3.42654i 0 − 1.36463i 0 4.96903 0
805.9 0 −2.52269 0 3.35477i 0 3.39146i 0 3.36396 0
805.10 0 −2.52269 0 − 3.35477i 0 − 3.39146i 0 3.36396 0
805.11 0 −2.47033 0 − 2.46639i 0 − 0.729903i 0 3.10252 0
805.12 0 −2.47033 0 2.46639i 0 0.729903i 0 3.10252 0
805.13 0 −2.26364 0 0.696325i 0 − 1.56653i 0 2.12406 0
805.14 0 −2.26364 0 − 0.696325i 0 1.56653i 0 2.12406 0
805.15 0 −2.10123 0 − 3.71130i 0 − 4.40414i 0 1.41517 0
805.16 0 −2.10123 0 3.71130i 0 4.40414i 0 1.41517 0
805.17 0 −1.98500 0 0.0799547i 0 3.50116i 0 0.940236 0
805.18 0 −1.98500 0 − 0.0799547i 0 − 3.50116i 0 0.940236 0
805.19 0 −1.86140 0 − 4.25967i 0 2.32754i 0 0.464826 0
805.20 0 −1.86140 0 4.25967i 0 − 2.32754i 0 0.464826 0
See all 78 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 805.78
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3484.2.f.a ✓ 78
13.b even 2 1 inner 3484.2.f.a ✓ 78
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3484.2.f.a ✓ 78 1.a even 1 1 trivial
3484.2.f.a ✓ 78 13.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(3484, [\chi])\).