Properties

Label 6845.2.a.t
Level $6845$
Weight $2$
Character orbit 6845.a
Self dual yes
Analytic conductor $54.658$
Analytic rank $1$
Dimension $36$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6845,2,Mod(1,6845)] 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("6845.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6845, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6845 = 5 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6845.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,-6,-6,30,36,-12,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.6576001836\)
Analytic rank: \(1\)
Dimension: \(36\)
Twist minimal: no (minimal twist has level 185)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 6 q^{2} - 6 q^{3} + 30 q^{4} + 36 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8} + 30 q^{9} - 6 q^{10} - 12 q^{12} - 24 q^{13} + 24 q^{14} - 6 q^{15} + 18 q^{16} - 12 q^{17} - 30 q^{18} - 36 q^{19} + 30 q^{20}+ \cdots + 24 q^{99}+O(q^{100}) \) 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 −2.66862 −2.71326 5.12153 1.00000 7.24065 −3.92337 −8.33019 4.36176 −2.66862
1.2 −2.64329 1.68613 4.98696 1.00000 −4.45691 −1.58892 −7.89537 −0.156982 −2.64329
1.3 −2.62725 −0.448343 4.90244 1.00000 1.17791 −2.11283 −7.62544 −2.79899 −2.62725
1.4 −2.44229 3.11584 3.96477 1.00000 −7.60978 −3.97157 −4.79854 6.70846 −2.44229
1.5 −2.41271 1.50825 3.82115 1.00000 −3.63895 3.09300 −4.39389 −0.725194 −2.41271
1.6 −2.22094 −2.47243 2.93257 1.00000 5.49111 0.592708 −2.07119 3.11289 −2.22094
1.7 −1.99991 −0.736256 1.99962 1.00000 1.47244 4.71407 0.000750647 0 −2.45793 −1.99991
1.8 −1.99223 2.77088 1.96898 1.00000 −5.52022 2.61149 0.0617914 4.67775 −1.99223
1.9 −1.83579 −0.969521 1.37012 1.00000 1.77983 −1.56082 1.15633 −2.06003 −1.83579
1.10 −1.59103 −3.31890 0.531373 1.00000 5.28047 −3.56780 2.33663 8.01512 −1.59103
1.11 −1.45397 0.250557 0.114040 1.00000 −0.364304 −4.77166 2.74214 −2.93722 −1.45397
1.12 −1.26319 1.34965 −0.404360 1.00000 −1.70486 3.69870 3.03715 −1.17844 −1.26319
1.13 −0.917930 0.790346 −1.15741 1.00000 −0.725482 −3.40028 2.89828 −2.37535 −0.917930
1.14 −0.885860 2.45318 −1.21525 1.00000 −2.17317 −3.59261 2.84826 3.01807 −0.885860
1.15 −0.857829 −2.11542 −1.26413 1.00000 1.81467 0.123375 2.80007 1.47501 −0.857829
1.16 −0.832321 1.09284 −1.30724 1.00000 −0.909592 0.752660 2.75269 −1.80571 −0.832321
1.17 −0.549278 −0.739383 −1.69829 1.00000 0.406126 2.37087 2.03139 −2.45331 −0.549278
1.18 −0.310157 −1.87338 −1.90380 1.00000 0.581042 −1.69942 1.21079 0.509555 −0.310157
1.19 −0.107323 −2.83553 −1.98848 1.00000 0.304317 −3.93158 0.428056 5.04022 −0.107323
1.20 0.147547 −2.96412 −1.97823 1.00000 −0.437347 1.08371 −0.586975 5.78601 0.147547
See all 36 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.36
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(37\) \( -1 \)

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 6845.2.a.t 36
37.b even 2 1 6845.2.a.u 36
37.i odd 36 2 185.2.w.a ✓ 72
185.z even 36 2 925.2.ba.c 72
185.ba odd 36 2 925.2.bb.b 72
185.bc even 36 2 925.2.ba.b 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.w.a ✓ 72 37.i odd 36 2
925.2.ba.b 72 185.bc even 36 2
925.2.ba.c 72 185.z even 36 2
925.2.bb.b 72 185.ba odd 36 2
6845.2.a.t 36 1.a even 1 1 trivial
6845.2.a.u 36 37.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6845))\):

\( T_{2}^{36} + 6 T_{2}^{35} - 33 T_{2}^{34} - 256 T_{2}^{33} + 399 T_{2}^{32} + 4902 T_{2}^{31} + \cdots - 107 \) Copy content Toggle raw display
\( T_{7}^{36} + 12 T_{7}^{35} - 75 T_{7}^{34} - 1470 T_{7}^{33} + 120 T_{7}^{32} + 76242 T_{7}^{31} + \cdots - 99836492183 \) Copy content Toggle raw display
\( T_{17}^{36} + 12 T_{17}^{35} - 279 T_{17}^{34} - 3798 T_{17}^{33} + 32070 T_{17}^{32} + \cdots - 18\!\cdots\!23 \) Copy content Toggle raw display