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Properties

Label 3780.2.q.d
Level $3780$
Weight $2$
Character orbit 3780.q
Analytic conductor $30.183$
Analytic rank $0$
Dimension $26$
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] = [3780,2,Mod(2881,3780)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3780, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3780.2881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3780 = 2^{2} \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3780.q (of order \(3\), degree \(2\), not 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: \(30.1834519640\)
Analytic rank: \(0\)
Dimension: \(26\)
Relative dimension: \(13\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 1260)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 13 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q + 13 q^{5} - 6 q^{7} - q^{11} + 7 q^{13} - 8 q^{17} - 9 q^{19} + 3 q^{23} - 13 q^{25} + 7 q^{29} + 2 q^{31} - 3 q^{35} + 10 q^{37} - 13 q^{41} - 8 q^{43} + 12 q^{47} - 28 q^{49} - 17 q^{53} - 2 q^{55} - 16 q^{59} - 20 q^{61} + 14 q^{65} - 48 q^{67} - 4 q^{71} - 17 q^{73} - 19 q^{77} - 38 q^{79} - 9 q^{83} + 8 q^{85} - q^{89} - 2 q^{91} - 18 q^{95} + 9 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2881.1 0 0 0 0.500000 − 0.866025i 0 −2.51245 + 0.829223i 0 0 0
2881.2 0 0 0 0.500000 − 0.866025i 0 −2.45595 + 0.984024i 0 0 0
2881.3 0 0 0 0.500000 − 0.866025i 0 −2.23092 − 1.42233i 0 0 0
2881.4 0 0 0 0.500000 − 0.866025i 0 −1.66760 − 2.05405i 0 0 0
2881.5 0 0 0 0.500000 − 0.866025i 0 −1.16872 − 2.37363i 0 0 0
2881.6 0 0 0 0.500000 − 0.866025i 0 −1.01878 + 2.44174i 0 0 0
2881.7 0 0 0 0.500000 − 0.866025i 0 −0.205366 + 2.63777i 0 0 0
2881.8 0 0 0 0.500000 − 0.866025i 0 0.00201961 + 2.64575i 0 0 0
2881.9 0 0 0 0.500000 − 0.866025i 0 0.776620 − 2.52920i 0 0 0
2881.10 0 0 0 0.500000 − 0.866025i 0 1.04803 − 2.42933i 0 0 0
2881.11 0 0 0 0.500000 − 0.866025i 0 1.63057 + 2.08357i 0 0 0
2881.12 0 0 0 0.500000 − 0.866025i 0 2.22863 − 1.42590i 0 0 0
2881.13 0 0 0 0.500000 − 0.866025i 0 2.57391 + 0.612365i 0 0 0
3061.1 0 0 0 0.500000 + 0.866025i 0 −2.51245 − 0.829223i 0 0 0
3061.2 0 0 0 0.500000 + 0.866025i 0 −2.45595 − 0.984024i 0 0 0
3061.3 0 0 0 0.500000 + 0.866025i 0 −2.23092 + 1.42233i 0 0 0
3061.4 0 0 0 0.500000 + 0.866025i 0 −1.66760 + 2.05405i 0 0 0
3061.5 0 0 0 0.500000 + 0.866025i 0 −1.16872 + 2.37363i 0 0 0
3061.6 0 0 0 0.500000 + 0.866025i 0 −1.01878 − 2.44174i 0 0 0
3061.7 0 0 0 0.500000 + 0.866025i 0 −0.205366 − 2.63777i 0 0 0
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2881.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3780.2.q.d 26
3.b odd 2 1 1260.2.q.d ✓ 26
7.c even 3 1 3780.2.t.d 26
9.c even 3 1 3780.2.t.d 26
9.d odd 6 1 1260.2.t.d yes 26
21.h odd 6 1 1260.2.t.d yes 26
63.h even 3 1 inner 3780.2.q.d 26
63.j odd 6 1 1260.2.q.d ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.q.d ✓ 26 3.b odd 2 1
1260.2.q.d ✓ 26 63.j odd 6 1
1260.2.t.d yes 26 9.d odd 6 1
1260.2.t.d yes 26 21.h odd 6 1
3780.2.q.d 26 1.a even 1 1 trivial
3780.2.q.d 26 63.h even 3 1 inner
3780.2.t.d 26 7.c even 3 1
3780.2.t.d 26 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{26} + T_{11}^{25} + 67 T_{11}^{24} + 6 T_{11}^{23} + 2997 T_{11}^{22} - 453 T_{11}^{21} + \cdots + 241864704 \) acting on \(S_{2}^{\mathrm{new}}(3780, [\chi])\). Copy content Toggle raw display