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Properties

Label 8649.2.a.bu
Level $8649$
Weight $2$
Character orbit 8649.a
Self dual yes
Analytic conductor $69.063$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8649,2,Mod(1,8649)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8649, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8649.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8649 = 3^{2} \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8649.a (trivial)

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: yes
Analytic conductor: \(69.0626127082\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 2883)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8} - 8 q^{10} - 16 q^{11} + 32 q^{13} + 24 q^{14} + 48 q^{16} - 32 q^{17} + 32 q^{19} + 24 q^{20} + 32 q^{22} - 32 q^{23} + 40 q^{25} - 16 q^{26} + 8 q^{28} - 48 q^{29}+ \cdots + 24 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.64601 0 5.00135 0.929477 0 −4.25556 −7.94161 0 −2.45940
1.2 −2.46988 0 4.10029 0.240205 0 −1.25177 −5.18746 0 −0.593276
1.3 −2.39335 0 3.72812 3.81642 0 3.08197 −4.13599 0 −9.13404
1.4 −2.22192 0 2.93691 −1.02426 0 0.246895 −2.08174 0 2.27581
1.5 −1.86657 0 1.48408 3.06179 0 3.20321 0.963008 0 −5.71504
1.6 −1.73783 0 1.02004 −3.85240 0 −2.44804 1.70300 0 6.69480
1.7 −1.62315 0 0.634610 −1.02432 0 1.99793 2.21623 0 1.66262
1.8 −1.21162 0 −0.531985 −1.58010 0 4.11574 3.06780 0 1.91448
1.9 −1.14647 0 −0.685607 −2.53076 0 −1.30842 3.07897 0 2.90144
1.10 −0.959557 0 −1.07925 4.21749 0 −1.12111 2.95472 0 −4.04692
1.11 −0.583104 0 −1.65999 −2.31932 0 −3.54758 2.13416 0 1.35240
1.12 −0.294138 0 −1.91348 1.97505 0 −3.94391 1.15111 0 −0.580938
1.13 −0.286378 0 −1.91799 3.36659 0 5.00540 1.12203 0 −0.964118
1.14 0.256648 0 −1.93413 −2.52738 0 4.03776 −1.00969 0 −0.648646
1.15 0.544587 0 −1.70342 −0.493609 0 −0.226241 −2.01684 0 −0.268813
1.16 0.849623 0 −1.27814 −0.881731 0 3.72139 −2.78518 0 −0.749139
1.17 1.14050 0 −0.699260 −2.66755 0 0.198186 −3.07851 0 −3.04235
1.18 1.75069 0 1.06491 1.92100 0 4.93164 −1.63706 0 3.36307
1.19 2.00987 0 2.03956 −3.19190 0 0.292061 0.0795053 0 −6.41528
1.20 2.33041 0 3.43080 4.15417 0 1.65165 3.33434 0 9.68090
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(31\) \( +1 \)

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 8649.2.a.bu 24
3.b odd 2 1 2883.2.a.v yes 24
31.b odd 2 1 8649.2.a.bv 24
93.c even 2 1 2883.2.a.u ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2883.2.a.u ✓ 24 93.c even 2 1
2883.2.a.v yes 24 3.b odd 2 1
8649.2.a.bu 24 1.a even 1 1 trivial
8649.2.a.bv 24 31.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8649))\):

\( T_{2}^{24} - 40 T_{2}^{22} - 8 T_{2}^{21} + 692 T_{2}^{20} + 272 T_{2}^{19} - 6756 T_{2}^{18} + \cdots - 641 \) Copy content Toggle raw display
\( T_{5}^{24} - 80 T_{5}^{22} - 32 T_{5}^{21} + 2764 T_{5}^{20} + 2112 T_{5}^{19} - 53768 T_{5}^{18} + \cdots - 7106564 \) Copy content Toggle raw display
\( T_{7}^{24} - 16 T_{7}^{23} + 16 T_{7}^{22} + 1008 T_{7}^{21} - 4640 T_{7}^{20} - 20672 T_{7}^{19} + \cdots + 1052672 \) Copy content Toggle raw display
\( T_{11}^{24} + 16 T_{11}^{23} - 32 T_{11}^{22} - 1792 T_{11}^{21} - 5212 T_{11}^{20} + \cdots + 206235170816 \) Copy content Toggle raw display
\( T_{13}^{24} - 32 T_{13}^{23} + 336 T_{13}^{22} - 192 T_{13}^{21} - 21728 T_{13}^{20} + \cdots - 45247233016 \) Copy content Toggle raw display