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Properties

Label 3680.2.i.b
Level $3680$
Weight $2$
Character orbit 3680.i
Analytic conductor $29.385$
Analytic rank $0$
Dimension $48$
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] = [3680,2,Mod(1471,3680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3680.1471");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3680.i (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: \(29.3849479438\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{7} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{7} - 48 q^{9} + 16 q^{11} + 20 q^{23} - 48 q^{25} - 8 q^{29} + 8 q^{41} + 56 q^{49} - 24 q^{51} - 120 q^{63} - 32 q^{67} - 20 q^{69} + 32 q^{77} + 72 q^{81} + 64 q^{83} + 40 q^{91} - 32 q^{93} - 144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1 0 − 3.43521i 0 1.00000i 0 4.41643 0 −8.80064 0
1471.2 0 − 3.29100i 0 − 1.00000i 0 1.56144 0 −7.83068 0
1471.3 0 − 3.12862i 0 − 1.00000i 0 2.84993 0 −6.78825 0
1471.4 0 − 2.85069i 0 1.00000i 0 0.905359 0 −5.12646 0
1471.5 0 − 2.77635i 0 − 1.00000i 0 −2.12267 0 −4.70810 0
1471.6 0 − 2.77298i 0 1.00000i 0 −2.77688 0 −4.68941 0
1471.7 0 − 2.21949i 0 − 1.00000i 0 −4.12023 0 −1.92614 0
1471.8 0 − 2.15220i 0 1.00000i 0 0.557589 0 −1.63197 0
1471.9 0 − 2.13983i 0 1.00000i 0 2.13998 0 −1.57886 0
1471.10 0 − 2.08047i 0 1.00000i 0 −1.84866 0 −1.32834 0
1471.11 0 − 2.00496i 0 − 1.00000i 0 −1.21687 0 −1.01987 0
1471.12 0 − 1.89612i 0 1.00000i 0 4.39085 0 −0.595278 0
1471.13 0 − 1.87710i 0 − 1.00000i 0 2.98502 0 −0.523497 0
1471.14 0 − 1.81994i 0 − 1.00000i 0 −2.60997 0 −0.312172 0
1471.15 0 − 1.49951i 0 − 1.00000i 0 4.97041 0 0.751457 0
1471.16 0 − 1.19368i 0 1.00000i 0 0.108875 0 1.57512 0
1471.17 0 − 0.868297i 0 1.00000i 0 −3.63028 0 2.24606 0
1471.18 0 − 0.847139i 0 − 1.00000i 0 −0.543298 0 2.28236 0
1471.19 0 − 0.816645i 0 1.00000i 0 3.72066 0 2.33309 0
1471.20 0 − 0.707214i 0 − 1.00000i 0 −1.57768 0 2.49985 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1471.48
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3680.2.i.b yes 48
4.b odd 2 1 3680.2.i.a ✓ 48
23.b odd 2 1 3680.2.i.a ✓ 48
92.b even 2 1 inner 3680.2.i.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3680.2.i.a ✓ 48 4.b odd 2 1
3680.2.i.a ✓ 48 23.b odd 2 1
3680.2.i.b yes 48 1.a even 1 1 trivial
3680.2.i.b yes 48 92.b even 2 1 inner