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Properties

Label 2394.2.f.a
Level $2394$
Weight $2$
Character orbit 2394.f
Analytic conductor $19.116$
Analytic rank $0$
Dimension $24$
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] = [2394,2,Mod(2015,2394)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2394, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("2394.2015");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.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: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} + 4 q^{7} - 4 q^{14} + 24 q^{16} + 32 q^{17} - 8 q^{22} + 16 q^{25} - 4 q^{28} - 24 q^{35} + 24 q^{38} - 8 q^{41} + 16 q^{43} - 8 q^{46} - 16 q^{49} + 4 q^{56} + 16 q^{58} + 16 q^{59} - 16 q^{62} - 24 q^{64} - 24 q^{67} - 32 q^{68} - 16 q^{70} + 8 q^{77} - 40 q^{79} - 64 q^{83} + 40 q^{85} + 8 q^{88} - 64 q^{89} + 8 q^{91} - 16 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2015.1 − 1.00000i 0 −1.00000 −1.90763 0 −0.638569 + 2.56753i 1.00000i 0 1.90763i
2015.2 1.00000i 0 −1.00000 −1.90763 0 −0.638569 − 2.56753i − 1.00000i 0 − 1.90763i
2015.3 − 1.00000i 0 −1.00000 −4.23211 0 2.64176 + 0.145265i 1.00000i 0 4.23211i
2015.4 1.00000i 0 −1.00000 −4.23211 0 2.64176 − 0.145265i − 1.00000i 0 − 4.23211i
2015.5 − 1.00000i 0 −1.00000 3.41122 0 0.143354 − 2.64186i 1.00000i 0 − 3.41122i
2015.6 1.00000i 0 −1.00000 3.41122 0 0.143354 + 2.64186i − 1.00000i 0 3.41122i
2015.7 − 1.00000i 0 −1.00000 1.26222 0 1.09287 + 2.40949i 1.00000i 0 − 1.26222i
2015.8 1.00000i 0 −1.00000 1.26222 0 1.09287 − 2.40949i − 1.00000i 0 1.26222i
2015.9 − 1.00000i 0 −1.00000 1.42576 0 2.62657 − 0.317999i 1.00000i 0 − 1.42576i
2015.10 1.00000i 0 −1.00000 1.42576 0 2.62657 + 0.317999i − 1.00000i 0 1.42576i
2015.11 − 1.00000i 0 −1.00000 −1.62775 0 −2.31472 − 1.28144i 1.00000i 0 1.62775i
2015.12 1.00000i 0 −1.00000 −1.62775 0 −2.31472 + 1.28144i − 1.00000i 0 − 1.62775i
2015.13 − 1.00000i 0 −1.00000 −0.335231 0 −2.35234 − 1.21100i 1.00000i 0 0.335231i
2015.14 1.00000i 0 −1.00000 −0.335231 0 −2.35234 + 1.21100i − 1.00000i 0 − 0.335231i
2015.15 − 1.00000i 0 −1.00000 3.11350 0 −0.672508 − 2.55885i 1.00000i 0 − 3.11350i
2015.16 1.00000i 0 −1.00000 3.11350 0 −0.672508 + 2.55885i − 1.00000i 0 3.11350i
2015.17 − 1.00000i 0 −1.00000 2.79254 0 −1.85020 + 1.89123i 1.00000i 0 − 2.79254i
2015.18 1.00000i 0 −1.00000 2.79254 0 −1.85020 − 1.89123i − 1.00000i 0 2.79254i
2015.19 − 1.00000i 0 −1.00000 −3.20606 0 1.74307 − 1.99040i 1.00000i 0 3.20606i
2015.20 1.00000i 0 −1.00000 −3.20606 0 1.74307 + 1.99040i − 1.00000i 0 − 3.20606i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2015.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2394.2.f.a ✓ 24
3.b odd 2 1 2394.2.f.b yes 24
7.b odd 2 1 2394.2.f.b yes 24
21.c even 2 1 inner 2394.2.f.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.f.a ✓ 24 1.a even 1 1 trivial
2394.2.f.a ✓ 24 21.c even 2 1 inner
2394.2.f.b yes 24 3.b odd 2 1
2394.2.f.b yes 24 7.b odd 2 1