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Properties

Label 1320.2.bu.b
Level $1320$
Weight $2$
Character orbit 1320.bu
Analytic conductor $10.540$
Analytic rank $0$
Dimension $36$
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] = [1320,2,Mod(1033,1320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1320.1033");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1320.bu (of order \(4\), degree \(2\), 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: \(10.5402530668\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{7} - 4 q^{11} - 4 q^{15} + 16 q^{17} - 16 q^{19} + 24 q^{23} - 8 q^{25} - 8 q^{31} + 24 q^{35} + 8 q^{37} + 8 q^{39} - 24 q^{43} - 16 q^{47} + 4 q^{55} - 4 q^{63} + 32 q^{65} + 16 q^{67} + 12 q^{73} + 8 q^{75} - 24 q^{77} - 32 q^{79} - 36 q^{81} - 32 q^{83} + 16 q^{85} - 4 q^{87} + 16 q^{91} + 24 q^{93} - 64 q^{95} + 12 q^{97} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1033.1 0 −0.707107 − 0.707107i 0 0.142732 − 2.23151i 0 3.30841 + 3.30841i 0 1.00000i 0
1033.2 0 −0.707107 − 0.707107i 0 1.09838 + 1.94771i 0 2.01481 + 2.01481i 0 1.00000i 0
1033.3 0 −0.707107 − 0.707107i 0 2.19933 − 0.403680i 0 0.974249 + 0.974249i 0 1.00000i 0
1033.4 0 −0.707107 − 0.707107i 0 −0.528637 + 2.17268i 0 −1.11929 − 1.11929i 0 1.00000i 0
1033.5 0 −0.707107 − 0.707107i 0 −1.73236 − 1.41383i 0 −0.153594 − 0.153594i 0 1.00000i 0
1033.6 0 −0.707107 − 0.707107i 0 −1.79153 + 1.33806i 0 −0.297397 − 0.297397i 0 1.00000i 0
1033.7 0 −0.707107 − 0.707107i 0 1.17482 − 1.90258i 0 0.360804 + 0.360804i 0 1.00000i 0
1033.8 0 −0.707107 − 0.707107i 0 −1.77826 − 1.35566i 0 −1.79509 − 1.79509i 0 1.00000i 0
1033.9 0 −0.707107 − 0.707107i 0 1.92263 + 1.14170i 0 −3.70711 − 3.70711i 0 1.00000i 0
1033.10 0 0.707107 + 0.707107i 0 −0.734221 + 2.11209i 0 1.89725 + 1.89725i 0 1.00000i 0
1033.11 0 0.707107 + 0.707107i 0 2.08432 + 0.809694i 0 1.71136 + 1.71136i 0 1.00000i 0
1033.12 0 0.707107 + 0.707107i 0 1.08448 − 1.95548i 0 2.33654 + 2.33654i 0 1.00000i 0
1033.13 0 0.707107 + 0.707107i 0 2.20569 + 0.367330i 0 −0.856156 − 0.856156i 0 1.00000i 0
1033.14 0 0.707107 + 0.707107i 0 −0.177741 − 2.22899i 0 −0.396204 − 0.396204i 0 1.00000i 0
1033.15 0 0.707107 + 0.707107i 0 −1.51868 − 1.64122i 0 −0.965779 − 0.965779i 0 1.00000i 0
1033.16 0 0.707107 + 0.707107i 0 −2.23474 − 0.0769910i 0 −1.70779 − 1.70779i 0 1.00000i 0
1033.17 0 0.707107 + 0.707107i 0 0.505894 + 2.17809i 0 −2.52103 − 2.52103i 0 1.00000i 0
1033.18 0 0.707107 + 0.707107i 0 −1.92210 + 1.14259i 0 2.91602 + 2.91602i 0 1.00000i 0
1297.1 0 −0.707107 + 0.707107i 0 0.142732 + 2.23151i 0 3.30841 − 3.30841i 0 − 1.00000i 0
1297.2 0 −0.707107 + 0.707107i 0 1.09838 − 1.94771i 0 2.01481 − 2.01481i 0 − 1.00000i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1033.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1320.2.bu.b yes 36
5.c odd 4 1 1320.2.bu.a ✓ 36
11.b odd 2 1 1320.2.bu.a ✓ 36
55.e even 4 1 inner 1320.2.bu.b yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.bu.a ✓ 36 5.c odd 4 1
1320.2.bu.a ✓ 36 11.b odd 2 1
1320.2.bu.b yes 36 1.a even 1 1 trivial
1320.2.bu.b yes 36 55.e even 4 1 inner