↑

Properties

Label 6897.2.a.bq
Level $6897$
Weight $2$
Character orbit 6897.a
Self dual yes
Analytic conductor $55.073$
Analytic rank $0$
Dimension $24$
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] = [6897,2,Mod(1,6897)] 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("6897.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6897, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6897 = 3 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6897.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,1,24,37,9,1,3,0,24,-1,0,37,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(55.0728222741\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 627)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{2} + 24 q^{3} + 37 q^{4} + 9 q^{5} + q^{6} + 3 q^{7} + 24 q^{9} - q^{10} + 37 q^{12} + 3 q^{13} - 2 q^{14} + 9 q^{15} + 55 q^{16} - 3 q^{17} + q^{18} + 24 q^{19} + 12 q^{20} + 3 q^{21} + 30 q^{23}+ \cdots + 59 q^{98}+O(q^{100}) \) 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.

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 −2.76536 1.00000 5.64724 2.69630 −2.76536 −1.13028 −10.0860 1.00000 −7.45626
1.2 −2.61041 1.00000 4.81426 −4.15044 −2.61041 2.99689 −7.34638 1.00000 10.8344
1.3 −2.54909 1.00000 4.49787 0.760543 −2.54909 −2.13586 −6.36729 1.00000 −1.93869
1.4 −2.25860 1.00000 3.10126 −3.25212 −2.25860 −1.96713 −2.48730 1.00000 7.34524
1.5 −2.16306 1.00000 2.67881 2.61770 −2.16306 −3.61470 −1.46830 1.00000 −5.66223
1.6 −2.13213 1.00000 2.54599 4.23712 −2.13213 1.55219 −1.16412 1.00000 −9.03410
1.7 −1.82592 1.00000 1.33398 −1.84387 −1.82592 4.05444 1.21611 1.00000 3.36675
1.8 −0.966731 1.00000 −1.06543 0.660294 −0.966731 5.06994 2.96345 1.00000 −0.638327
1.9 −0.924354 1.00000 −1.14557 0.893038 −0.924354 −1.30127 2.90762 1.00000 −0.825483
1.10 −0.788043 1.00000 −1.37899 4.07665 −0.788043 3.14519 2.66279 1.00000 −3.21257
1.11 −0.727577 1.00000 −1.47063 1.26102 −0.727577 −4.18031 2.52515 1.00000 −0.917489
1.12 0.0812681 1.00000 −1.99340 −2.68459 0.0812681 1.83809 −0.324536 1.00000 −0.218171
1.13 0.259683 1.00000 −1.93256 −1.30335 0.259683 1.57502 −1.02122 1.00000 −0.338459
1.14 0.328666 1.00000 −1.89198 2.76006 0.328666 −2.77967 −1.27916 1.00000 0.907136
1.15 1.27260 1.00000 −0.380477 −0.389881 1.27260 −3.02973 −3.02941 1.00000 −0.496164
1.16 1.47839 1.00000 0.185643 2.70475 1.47839 1.84363 −2.68233 1.00000 3.99868
1.17 1.52666 1.00000 0.330702 3.33513 1.52666 1.07821 −2.54846 1.00000 5.09162
1.18 1.53897 1.00000 0.368440 −3.54243 1.53897 −4.90874 −2.51093 1.00000 −5.45170
1.19 1.61883 1.00000 0.620622 −3.40265 1.61883 4.29833 −2.23298 1.00000 −5.50832
1.20 2.03955 1.00000 2.15975 −0.677824 2.03955 2.93506 0.325824 1.00000 −1.38245
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
\(3\) \( -1 \)
\(11\) \( -1 \)
\(19\) \( -1 \)

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 6897.2.a.bq 24
11.b odd 2 1 6897.2.a.bp 24
11.d odd 10 2 627.2.j.d ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
627.2.j.d ✓ 48 11.d odd 10 2
6897.2.a.bp 24 11.b odd 2 1
6897.2.a.bq 24 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6897))\):

\( T_{2}^{24} - T_{2}^{23} - 42 T_{2}^{22} + 41 T_{2}^{21} + 766 T_{2}^{20} - 730 T_{2}^{19} - 7958 T_{2}^{18} + \cdots - 880 \) Copy content Toggle raw display
\( T_{5}^{24} - 9 T_{5}^{23} - 45 T_{5}^{22} + 607 T_{5}^{21} + 324 T_{5}^{20} - 16940 T_{5}^{19} + \cdots - 5403584 \) Copy content Toggle raw display
\( T_{7}^{24} - 3 T_{7}^{23} - 109 T_{7}^{22} + 324 T_{7}^{21} + 5070 T_{7}^{20} - 14912 T_{7}^{19} + \cdots - 312840000 \) Copy content Toggle raw display
\( T_{13}^{24} - 3 T_{13}^{23} - 190 T_{13}^{22} + 394 T_{13}^{21} + 15581 T_{13}^{20} - 17309 T_{13}^{19} + \cdots + 747556864 \) Copy content Toggle raw display