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Properties

Label 3762.2.b.a
Level $3762$
Weight $2$
Character orbit 3762.b
Analytic conductor $30.040$
Analytic rank $0$
Dimension $36$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3762,2,Mod(989,3762)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3762, 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("3762.989");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3762.b (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: \(30.0397212404\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 36 q^{2} + 36 q^{4} - 36 q^{8} + 4 q^{11} + 36 q^{16} - 16 q^{17} - 4 q^{22} - 28 q^{25} - 8 q^{31} - 36 q^{32} + 16 q^{34} - 16 q^{35} + 16 q^{37} - 24 q^{41} + 4 q^{44} - 68 q^{49} + 28 q^{50}+ \cdots + 68 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
989.1 −1.00000 0 1.00000 − 4.07403i 0 − 3.38003i −1.00000 0 4.07403i
989.2 −1.00000 0 1.00000 − 3.97332i 0 − 3.75123i −1.00000 0 3.97332i
989.3 −1.00000 0 1.00000 − 3.66682i 0 2.53190i −1.00000 0 3.66682i
989.4 −1.00000 0 1.00000 − 3.17738i 0 − 1.82716i −1.00000 0 3.17738i
989.5 −1.00000 0 1.00000 − 3.00309i 0 3.31087i −1.00000 0 3.00309i
989.6 −1.00000 0 1.00000 − 2.60173i 0 1.40935i −1.00000 0 2.60173i
989.7 −1.00000 0 1.00000 − 2.59021i 0 4.42739i −1.00000 0 2.59021i
989.8 −1.00000 0 1.00000 − 2.52146i 0 − 2.04168i −1.00000 0 2.52146i
989.9 −1.00000 0 1.00000 − 2.27036i 0 − 1.52634i −1.00000 0 2.27036i
989.10 −1.00000 0 1.00000 − 1.98219i 0 0.669307i −1.00000 0 1.98219i
989.11 −1.00000 0 1.00000 − 1.63530i 0 − 4.47731i −1.00000 0 1.63530i
989.12 −1.00000 0 1.00000 − 1.60040i 0 − 1.65399i −1.00000 0 1.60040i
989.13 −1.00000 0 1.00000 − 1.52959i 0 3.87698i −1.00000 0 1.52959i
989.14 −1.00000 0 1.00000 − 1.25865i 0 4.43558i −1.00000 0 1.25865i
989.15 −1.00000 0 1.00000 − 0.580945i 0 − 5.13734i −1.00000 0 0.580945i
989.16 −1.00000 0 1.00000 − 0.557246i 0 0.737186i −1.00000 0 0.557246i
989.17 −1.00000 0 1.00000 − 0.504839i 0 0.909795i −1.00000 0 0.504839i
989.18 −1.00000 0 1.00000 − 0.270972i 0 − 0.0881778i −1.00000 0 0.270972i
989.19 −1.00000 0 1.00000 0.270972i 0 0.0881778i −1.00000 0 − 0.270972i
989.20 −1.00000 0 1.00000 0.504839i 0 − 0.909795i −1.00000 0 − 0.504839i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 989.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3762.2.b.a ✓ 36
3.b odd 2 1 3762.2.b.b yes 36
11.b odd 2 1 3762.2.b.b yes 36
33.d even 2 1 inner 3762.2.b.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3762.2.b.a ✓ 36 1.a even 1 1 trivial
3762.2.b.a ✓ 36 33.d even 2 1 inner
3762.2.b.b yes 36 3.b odd 2 1
3762.2.b.b yes 36 11.b odd 2 1