↑

Properties

Label 6223.2.a.q
Level $6223$
Weight $2$
Character orbit 6223.a
Self dual yes
Analytic conductor $49.691$
Analytic rank $1$
Dimension $40$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6223,2,Mod(1,6223)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6223.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6223, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6223 = 7^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6223.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,-3,-6,39,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(49.6909051778\)
Analytic rank: \(1\)
Dimension: \(40\)
Twist minimal: no (minimal twist has level 889)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{2} - 6 q^{3} + 39 q^{4} - 24 q^{5} - 9 q^{8} + 42 q^{9} - 15 q^{10} + 4 q^{11} - 29 q^{12} - 7 q^{13} + 8 q^{15} + 29 q^{16} - 56 q^{17} - 12 q^{18} - 6 q^{19} - 45 q^{20} + 3 q^{22} - 7 q^{23}+ \cdots + 65 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.73783 −3.09982 5.49572 2.24078 8.48678 0 −9.57068 6.60889 −6.13487
1.2 −2.68664 −0.656933 5.21805 −1.45033 1.76494 0 −8.64575 −2.56844 3.89652
1.3 −2.67543 2.21467 5.15792 −3.77035 −5.92518 0 −8.44878 1.90474 10.0873
1.4 −2.60936 −2.27618 4.80875 −0.761982 5.93936 0 −7.32905 2.18098 1.98829
1.5 −2.24401 2.80584 3.03558 1.28157 −6.29634 0 −2.32386 4.87274 −2.87586
1.6 −2.15797 1.79179 2.65682 −0.285167 −3.86663 0 −1.41741 0.210520 0.615381
1.7 −1.89455 −1.68668 1.58933 −0.447365 3.19551 0 0.778030 −0.155104 0.847556
1.8 −1.82190 1.06573 1.31931 1.64925 −1.94165 0 1.24014 −1.86422 −3.00476
1.9 −1.79801 −0.144112 1.23286 −4.04161 0.259116 0 1.37934 −2.97923 7.26687
1.10 −1.77970 −3.07662 1.16735 −3.89409 5.47548 0 1.48188 6.46560 6.93033
1.11 −1.73616 −2.58223 1.01424 1.04688 4.48316 0 1.71143 3.66791 −1.81755
1.12 −1.60118 2.10158 0.563789 3.69203 −3.36501 0 2.29964 1.41662 −5.91162
1.13 −1.28300 2.26059 −0.353920 −1.66860 −2.90033 0 3.02007 2.11025 2.14080
1.14 −1.14887 0.105959 −0.680100 0.464834 −0.121733 0 3.07908 −2.98877 −0.534033
1.15 −1.09299 −0.661876 −0.805363 −0.633566 0.723427 0 3.06625 −2.56192 0.692484
1.16 −1.01110 0.720309 −0.977683 2.17925 −0.728302 0 3.01073 −2.48115 −2.20344
1.17 −0.783388 −2.52435 −1.38630 −3.31043 1.97755 0 2.65279 3.37237 2.59335
1.18 −0.628033 −1.21971 −1.60557 2.87801 0.766017 0 2.26442 −1.51232 −1.80749
1.19 −0.156733 −3.33897 −1.97543 0.705840 0.523328 0 0.623083 8.14873 −0.110629
1.20 −0.138409 −0.525573 −1.98084 −2.27925 0.0727438 0 0.550983 −2.72377 0.315468
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( +1 \)
\(127\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6223.2.a.q 40
7.b odd 2 1 6223.2.a.r 40
7.c even 3 2 889.2.f.d ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
889.2.f.d ✓ 80 7.c even 3 2
6223.2.a.q 40 1.a even 1 1 trivial
6223.2.a.r 40 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6223))\):

\( T_{2}^{40} + 3 T_{2}^{39} - 55 T_{2}^{38} - 167 T_{2}^{37} + 1381 T_{2}^{36} + 4255 T_{2}^{35} + \cdots + 789 \) Copy content Toggle raw display
\( T_{3}^{40} + 6 T_{3}^{39} - 63 T_{3}^{38} - 420 T_{3}^{37} + 1764 T_{3}^{36} + 13397 T_{3}^{35} + \cdots - 16527 \) Copy content Toggle raw display
\( T_{5}^{40} + 24 T_{5}^{39} + 164 T_{5}^{38} - 556 T_{5}^{37} - 12110 T_{5}^{36} - 31411 T_{5}^{35} + \cdots + 25258724 \) Copy content Toggle raw display