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Properties

Label 1020.2.bp.a
Level $1020$
Weight $2$
Character orbit 1020.bp
Analytic conductor $8.145$
Analytic rank $0$
Dimension $24$
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] = [1020,2,Mod(121,1020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1020, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1020.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1020.bp (of order \(8\), degree \(4\), 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: \(8.14474100617\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{7} - 24 q^{17} - 8 q^{19} + 16 q^{23} - 16 q^{29} + 16 q^{31} + 16 q^{33} + 32 q^{37} + 8 q^{39} - 16 q^{41} - 24 q^{49} - 8 q^{51} - 40 q^{53} + 32 q^{57} + 24 q^{59} + 8 q^{61} + 8 q^{63} + 16 q^{69} + 32 q^{71} - 16 q^{73} + 56 q^{77} + 40 q^{79} + 24 q^{83} - 8 q^{85} - 24 q^{87} + 80 q^{91} + 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1 0 −0.923880 + 0.382683i 0 −0.382683 − 0.923880i 0 1.01333 − 2.44641i 0 0.707107 − 0.707107i 0
121.2 0 −0.923880 + 0.382683i 0 −0.382683 − 0.923880i 0 −1.53535 + 3.70666i 0 0.707107 − 0.707107i 0
121.3 0 −0.923880 + 0.382683i 0 −0.382683 − 0.923880i 0 −0.312076 + 0.753418i 0 0.707107 − 0.707107i 0
121.4 0 0.923880 − 0.382683i 0 0.382683 + 0.923880i 0 0.552673 − 1.33427i 0 0.707107 − 0.707107i 0
121.5 0 0.923880 − 0.382683i 0 0.382683 + 0.923880i 0 −1.31008 + 3.16280i 0 0.707107 − 0.707107i 0
121.6 0 0.923880 − 0.382683i 0 0.382683 + 0.923880i 0 1.00571 − 2.42799i 0 0.707107 − 0.707107i 0
661.1 0 −0.382683 − 0.923880i 0 0.923880 − 0.382683i 0 −0.174159 − 0.0721390i 0 −0.707107 + 0.707107i 0
661.2 0 −0.382683 − 0.923880i 0 0.923880 − 0.382683i 0 0.0846675 + 0.0350704i 0 −0.707107 + 0.707107i 0
661.3 0 −0.382683 − 0.923880i 0 0.923880 − 0.382683i 0 −2.92418 − 1.21123i 0 −0.707107 + 0.707107i 0
661.4 0 0.382683 + 0.923880i 0 −0.923880 + 0.382683i 0 −1.34426 − 0.556809i 0 −0.707107 + 0.707107i 0
661.5 0 0.382683 + 0.923880i 0 −0.923880 + 0.382683i 0 −2.04257 − 0.846059i 0 −0.707107 + 0.707107i 0
661.6 0 0.382683 + 0.923880i 0 −0.923880 + 0.382683i 0 2.98628 + 1.23696i 0 −0.707107 + 0.707107i 0
841.1 0 −0.382683 + 0.923880i 0 0.923880 + 0.382683i 0 −0.174159 + 0.0721390i 0 −0.707107 − 0.707107i 0
841.2 0 −0.382683 + 0.923880i 0 0.923880 + 0.382683i 0 0.0846675 − 0.0350704i 0 −0.707107 − 0.707107i 0
841.3 0 −0.382683 + 0.923880i 0 0.923880 + 0.382683i 0 −2.92418 + 1.21123i 0 −0.707107 − 0.707107i 0
841.4 0 0.382683 − 0.923880i 0 −0.923880 − 0.382683i 0 −1.34426 + 0.556809i 0 −0.707107 − 0.707107i 0
841.5 0 0.382683 − 0.923880i 0 −0.923880 − 0.382683i 0 −2.04257 + 0.846059i 0 −0.707107 − 0.707107i 0
841.6 0 0.382683 − 0.923880i 0 −0.923880 − 0.382683i 0 2.98628 − 1.23696i 0 −0.707107 − 0.707107i 0
961.1 0 −0.923880 − 0.382683i 0 −0.382683 + 0.923880i 0 1.01333 + 2.44641i 0 0.707107 + 0.707107i 0
961.2 0 −0.923880 − 0.382683i 0 −0.382683 + 0.923880i 0 −1.53535 − 3.70666i 0 0.707107 + 0.707107i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1020.2.bp.a ✓ 24
17.d even 8 1 inner 1020.2.bp.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1020.2.bp.a ✓ 24 1.a even 1 1 trivial
1020.2.bp.a ✓ 24 17.d even 8 1 inner