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Properties

Label 1020.3.bh.a
Level $1020$
Weight $3$
Character orbit 1020.bh
Analytic conductor $27.793$
Analytic rank $0$
Dimension $64$
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] = [1020,3,Mod(613,1020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1020, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1020.613");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1020.bh (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: \(27.7929869648\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 16 q^{7} + 32 q^{11} + 4 q^{13} + 48 q^{15} + 48 q^{21} + 16 q^{23} - 112 q^{31} - 84 q^{33} - 24 q^{35} - 72 q^{37} + 32 q^{41} - 196 q^{43} + 104 q^{47} + 200 q^{53} + 264 q^{55} + 256 q^{61} - 48 q^{63} - 456 q^{65} + 64 q^{67} - 112 q^{71} - 104 q^{73} + 72 q^{77} - 576 q^{81} + 488 q^{83} + 48 q^{87} - 784 q^{91} - 600 q^{95} - 664 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
613.1 0 −1.22474 − 1.22474i 0 −2.47449 − 4.34476i 0 −6.49825 + 6.49825i 0 3.00000i 0
613.2 0 −1.22474 − 1.22474i 0 2.91009 − 4.06588i 0 −4.76072 + 4.76072i 0 3.00000i 0
613.3 0 −1.22474 − 1.22474i 0 −4.99999 − 0.0103074i 0 4.90706 − 4.90706i 0 3.00000i 0
613.4 0 −1.22474 − 1.22474i 0 −1.04916 + 4.88869i 0 −4.13588 + 4.13588i 0 3.00000i 0
613.5 0 −1.22474 − 1.22474i 0 4.60277 + 1.95309i 0 −0.291980 + 0.291980i 0 3.00000i 0
613.6 0 −1.22474 − 1.22474i 0 4.23184 + 2.66300i 0 0.550605 − 0.550605i 0 3.00000i 0
613.7 0 −1.22474 − 1.22474i 0 −4.85985 − 1.17552i 0 −1.31020 + 1.31020i 0 3.00000i 0
613.8 0 −1.22474 − 1.22474i 0 −3.48327 + 3.58704i 0 1.25279 − 1.25279i 0 3.00000i 0
613.9 0 −1.22474 − 1.22474i 0 1.12804 + 4.87109i 0 2.80919 − 2.80919i 0 3.00000i 0
613.10 0 −1.22474 − 1.22474i 0 −1.00778 − 4.89738i 0 2.39249 − 2.39249i 0 3.00000i 0
613.11 0 −1.22474 − 1.22474i 0 −1.01846 − 4.89518i 0 3.95716 − 3.95716i 0 3.00000i 0
613.12 0 −1.22474 − 1.22474i 0 4.92540 − 0.860502i 0 6.57155 − 6.57155i 0 3.00000i 0
613.13 0 −1.22474 − 1.22474i 0 4.71173 − 1.67319i 0 −6.50826 + 6.50826i 0 3.00000i 0
613.14 0 −1.22474 − 1.22474i 0 −4.79774 + 1.40772i 0 −7.72432 + 7.72432i 0 3.00000i 0
613.15 0 −1.22474 − 1.22474i 0 −3.31433 + 3.74369i 0 9.26585 − 9.26585i 0 3.00000i 0
613.16 0 −1.22474 − 1.22474i 0 0.820975 + 4.93214i 0 −9.37606 + 9.37606i 0 3.00000i 0
613.17 0 1.22474 + 1.22474i 0 −4.35837 + 2.45044i 0 −8.12800 + 8.12800i 0 3.00000i 0
613.18 0 1.22474 + 1.22474i 0 −3.17543 − 3.86221i 0 3.55223 − 3.55223i 0 3.00000i 0
613.19 0 1.22474 + 1.22474i 0 −1.96329 − 4.59842i 0 −2.65606 + 2.65606i 0 3.00000i 0
613.20 0 1.22474 + 1.22474i 0 −3.30459 − 3.75229i 0 2.09791 − 2.09791i 0 3.00000i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 613.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1020.3.bh.a ✓ 64
5.c odd 4 1 inner 1020.3.bh.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1020.3.bh.a ✓ 64 1.a even 1 1 trivial
1020.3.bh.a ✓ 64 5.c odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1020, [\chi])\).