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Properties

Label 2340.2.g.a
Level $2340$
Weight $2$
Character orbit 2340.g
Analytic conductor $18.685$
Analytic rank $0$
Dimension $48$
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] = [2340,2,Mod(1691,2340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2340, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2340.1691");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2340.g (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: \(18.6849940730\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{4} - 4 q^{10} - 48 q^{13} + 28 q^{16} - 8 q^{22} - 48 q^{25} + 8 q^{28} + 8 q^{34} - 16 q^{37} + 4 q^{40} - 16 q^{46} - 32 q^{49} + 4 q^{52} - 56 q^{58} - 32 q^{61} - 28 q^{64} - 24 q^{70} + 48 q^{73} + 120 q^{76} + 48 q^{82} + 32 q^{85} + 136 q^{88} + 8 q^{94} + 32 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1691.1 −1.41299 − 0.0588515i 0 1.99307 + 0.166313i − 1.00000i 0 1.54479i −2.80640 − 0.352294i 0 −0.0588515 + 1.41299i
1691.2 −1.41299 + 0.0588515i 0 1.99307 − 0.166313i 1.00000i 0 − 1.54479i −2.80640 + 0.352294i 0 −0.0588515 − 1.41299i
1691.3 −1.37172 − 0.344063i 0 1.76324 + 0.943918i − 1.00000i 0 4.61967i −2.09391 − 1.90146i 0 −0.344063 + 1.37172i
1691.4 −1.37172 + 0.344063i 0 1.76324 − 0.943918i 1.00000i 0 − 4.61967i −2.09391 + 1.90146i 0 −0.344063 − 1.37172i
1691.5 −1.35285 − 0.412066i 0 1.66040 + 1.11493i 1.00000i 0 − 2.68642i −1.78685 − 2.19252i 0 0.412066 − 1.35285i
1691.6 −1.35285 + 0.412066i 0 1.66040 − 1.11493i − 1.00000i 0 2.68642i −1.78685 + 2.19252i 0 0.412066 + 1.35285i
1691.7 −1.32797 − 0.486315i 0 1.52700 + 1.29162i − 1.00000i 0 − 1.01640i −1.39967 − 2.45783i 0 −0.486315 + 1.32797i
1691.8 −1.32797 + 0.486315i 0 1.52700 − 1.29162i 1.00000i 0 1.01640i −1.39967 + 2.45783i 0 −0.486315 − 1.32797i
1691.9 −1.17406 − 0.788402i 0 0.756845 + 1.85127i 1.00000i 0 0.625037i 0.570959 − 2.77020i 0 0.788402 − 1.17406i
1691.10 −1.17406 + 0.788402i 0 0.756845 − 1.85127i − 1.00000i 0 − 0.625037i 0.570959 + 2.77020i 0 0.788402 + 1.17406i
1691.11 −0.996129 − 1.00386i 0 −0.0154550 + 1.99994i 1.00000i 0 2.55463i 2.02305 − 1.97668i 0 1.00386 − 0.996129i
1691.12 −0.996129 + 1.00386i 0 −0.0154550 − 1.99994i − 1.00000i 0 − 2.55463i 2.02305 + 1.97668i 0 1.00386 + 0.996129i
1691.13 −0.879289 − 1.10763i 0 −0.453703 + 1.94786i − 1.00000i 0 − 1.56236i 2.55645 − 1.21019i 0 −1.10763 + 0.879289i
1691.14 −0.879289 + 1.10763i 0 −0.453703 − 1.94786i 1.00000i 0 1.56236i 2.55645 + 1.21019i 0 −1.10763 − 0.879289i
1691.15 −0.785430 − 1.17605i 0 −0.766200 + 1.84741i − 1.00000i 0 − 3.49394i 2.77445 − 0.549922i 0 −1.17605 + 0.785430i
1691.16 −0.785430 + 1.17605i 0 −0.766200 − 1.84741i 1.00000i 0 3.49394i 2.77445 + 0.549922i 0 −1.17605 − 0.785430i
1691.17 −0.370924 − 1.36470i 0 −1.72483 + 1.01240i 1.00000i 0 − 3.32217i 2.02141 + 1.97836i 0 1.36470 − 0.370924i
1691.18 −0.370924 + 1.36470i 0 −1.72483 − 1.01240i − 1.00000i 0 3.32217i 2.02141 − 1.97836i 0 1.36470 + 0.370924i
1691.19 −0.264599 − 1.38924i 0 −1.85997 + 0.735183i − 1.00000i 0 3.29169i 1.51349 + 2.38942i 0 −1.38924 + 0.264599i
1691.20 −0.264599 + 1.38924i 0 −1.85997 − 0.735183i 1.00000i 0 − 3.29169i 1.51349 − 2.38942i 0 −1.38924 − 0.264599i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1691.48
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
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2340.2.g.a ✓ 48
3.b odd 2 1 inner 2340.2.g.a ✓ 48
4.b odd 2 1 inner 2340.2.g.a ✓ 48
12.b even 2 1 inner 2340.2.g.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2340.2.g.a ✓ 48 1.a even 1 1 trivial
2340.2.g.a ✓ 48 3.b odd 2 1 inner
2340.2.g.a ✓ 48 4.b odd 2 1 inner
2340.2.g.a ✓ 48 12.b even 2 1 inner