Properties

Label 1002.2.d.a
Level $1002$
Weight $2$
Character orbit 1002.d
Analytic conductor $8.001$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1002,2,Mod(1001,1002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1002, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1002.1001");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1002 = 2 \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1002.d (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: \(8.00101028253\)
Analytic rank: \(0\)
Dimension: \(56\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 56 q^{4} + 4 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 56 q^{4} + 4 q^{6} + 4 q^{9} + 56 q^{16} + 8 q^{18} + 8 q^{19} + 16 q^{21} - 4 q^{24} + 64 q^{25} + 12 q^{27} + 16 q^{31} + 40 q^{33} - 4 q^{36} - 20 q^{42} + 32 q^{49} - 16 q^{54} + 36 q^{57} + 16 q^{61} + 16 q^{63} - 56 q^{64} + 8 q^{66} - 8 q^{72} - 24 q^{75} - 8 q^{76} + 36 q^{81} - 16 q^{84} - 64 q^{85} - 4 q^{87} + 64 q^{93} - 40 q^{94} + 4 q^{96} - 72 q^{97} - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1001.1 1.00000i −1.63328 + 0.576543i −1.00000 3.86291 0.576543 + 1.63328i 2.63317 1.00000i 2.33520 1.88331i 3.86291i
1001.2 1.00000i −1.63328 0.576543i −1.00000 3.86291 0.576543 1.63328i 2.63317 1.00000i 2.33520 + 1.88331i 3.86291i
1001.3 1.00000i −1.44295 0.958070i −1.00000 −3.09196 −0.958070 + 1.44295i −4.02418 1.00000i 1.16420 + 2.76489i 3.09196i
1001.4 1.00000i −1.44295 + 0.958070i −1.00000 −3.09196 −0.958070 1.44295i −4.02418 1.00000i 1.16420 2.76489i 3.09196i
1001.5 1.00000i 0.387921 + 1.68805i −1.00000 2.66678 1.68805 0.387921i 2.13474 1.00000i −2.69904 + 1.30966i 2.66678i
1001.6 1.00000i 0.387921 1.68805i −1.00000 2.66678 1.68805 + 0.387921i 2.13474 1.00000i −2.69904 1.30966i 2.66678i
1001.7 1.00000i 1.16629 1.28053i −1.00000 1.79742 −1.28053 1.16629i 2.83390 1.00000i −0.279525 2.98695i 1.79742i
1001.8 1.00000i 1.16629 + 1.28053i −1.00000 1.79742 −1.28053 + 1.16629i 2.83390 1.00000i −0.279525 + 2.98695i 1.79742i
1001.9 1.00000i 1.70354 + 0.312980i −1.00000 3.03132 0.312980 1.70354i −1.54520 1.00000i 2.80409 + 1.06635i 3.03132i
1001.10 1.00000i 1.70354 0.312980i −1.00000 3.03132 0.312980 + 1.70354i −1.54520 1.00000i 2.80409 1.06635i 3.03132i
1001.11 1.00000i −1.63328 + 0.576543i −1.00000 −3.86291 0.576543 + 1.63328i 2.63317 1.00000i 2.33520 1.88331i 3.86291i
1001.12 1.00000i −1.63328 0.576543i −1.00000 −3.86291 0.576543 1.63328i 2.63317 1.00000i 2.33520 + 1.88331i 3.86291i
1001.13 1.00000i 0.387921 + 1.68805i −1.00000 −2.66678 1.68805 0.387921i 2.13474 1.00000i −2.69904 + 1.30966i 2.66678i
1001.14 1.00000i 0.387921 1.68805i −1.00000 −2.66678 1.68805 + 0.387921i 2.13474 1.00000i −2.69904 1.30966i 2.66678i
1001.15 1.00000i −1.32193 1.11916i −1.00000 0.490913 −1.11916 + 1.32193i 2.44549 1.00000i 0.494978 + 2.95888i 0.490913i
1001.16 1.00000i −1.32193 + 1.11916i −1.00000 0.490913 −1.11916 1.32193i 2.44549 1.00000i 0.494978 2.95888i 0.490913i
1001.17 1.00000i −0.183033 + 1.72235i −1.00000 3.52388 1.72235 + 0.183033i −5.07645 1.00000i −2.93300 0.630495i 3.52388i
1001.18 1.00000i −0.183033 1.72235i −1.00000 3.52388 1.72235 0.183033i −5.07645 1.00000i −2.93300 + 0.630495i 3.52388i
1001.19 1.00000i 1.67234 0.450849i −1.00000 2.07212 −0.450849 1.67234i −2.63137 1.00000i 2.59347 1.50795i 2.07212i
1001.20 1.00000i 1.67234 + 0.450849i −1.00000 2.07212 −0.450849 + 1.67234i −2.63137 1.00000i 2.59347 + 1.50795i 2.07212i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1001.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
167.b odd 2 1 inner
501.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1002.2.d.a 56
3.b odd 2 1 inner 1002.2.d.a 56
167.b odd 2 1 inner 1002.2.d.a 56
501.c even 2 1 inner 1002.2.d.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.d.a 56 1.a even 1 1 trivial
1002.2.d.a 56 3.b odd 2 1 inner
1002.2.d.a 56 167.b odd 2 1 inner
1002.2.d.a 56 501.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1002, [\chi])\).