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Properties

Label 5239.2.a.t
Level $5239$
Weight $2$
Character orbit 5239.a
Self dual yes
Analytic conductor $41.834$
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] = [5239,2,Mod(1,5239)] 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("5239.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5239, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5239 = 13^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5239.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,2,-5,28,5] 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: \(41.8336256189\)
Analytic rank: \(1\)
Dimension: \(36\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{2} - 5 q^{3} + 28 q^{4} + 5 q^{5} - 3 q^{6} + 5 q^{7} + 3 q^{8} + 5 q^{9} - 15 q^{10} - q^{11} - 13 q^{12} - 19 q^{14} - 10 q^{15} + 4 q^{16} - 46 q^{17} - 9 q^{18} + 8 q^{19} - 5 q^{20} - 16 q^{21}+ \cdots - 47 q^{99}+O(q^{100}) \) 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 −2.56895 −1.20237 4.59948 3.54314 3.08883 −2.49913 −6.67792 −1.55430 −9.10212
1.2 −2.54656 −2.14666 4.48499 −1.97365 5.46662 3.56450 −6.32819 1.60816 5.02604
1.3 −2.51712 −2.55317 4.33588 0.350569 6.42663 1.25323 −5.87968 3.51869 −0.882423
1.4 −2.18595 1.24527 2.77837 −1.95275 −2.72209 4.61231 −1.70147 −1.44930 4.26861
1.5 −2.10547 1.13587 2.43300 0.844720 −2.39153 −0.705223 −0.911672 −1.70981 −1.77853
1.6 −1.92927 1.67028 1.72208 2.56491 −3.22241 0.681322 0.536185 −0.210175 −4.94841
1.7 −1.88690 0.0658892 1.56038 −0.400631 −0.124326 −2.24315 0.829521 −2.99566 0.755948
1.8 −1.80595 −0.302872 1.26147 −1.76874 0.546973 −2.82549 1.33376 −2.90827 3.19426
1.9 −1.40921 2.45063 −0.0141145 3.50259 −3.45347 −4.31661 2.83832 3.00561 −4.93590
1.10 −1.33705 2.56782 −0.212302 −1.42532 −3.43329 2.10054 2.95795 3.59368 1.90573
1.11 −1.29188 −1.72668 −0.331049 2.63778 2.23067 2.01903 3.01143 −0.0185657 −3.40769
1.12 −0.911653 2.69491 −1.16889 −0.577106 −2.45682 3.79528 2.88893 4.26252 0.526121
1.13 −0.836363 −1.14473 −1.30050 2.65923 0.957409 −3.97709 2.76041 −1.68960 −2.22408
1.14 −0.709075 −2.87369 −1.49721 −2.11332 2.03766 −0.783955 2.47979 5.25811 1.49850
1.15 −0.643360 0.244979 −1.58609 1.48910 −0.157609 1.17289 2.30715 −2.93999 −0.958027
1.16 −0.552625 −1.71891 −1.69461 −0.0841389 0.949914 3.82547 2.04173 −0.0453470 0.0464973
1.17 −0.314789 1.77594 −1.90091 −0.987732 −0.559047 −0.624388 1.22796 0.153977 0.310927
1.18 −0.159901 −1.41029 −1.97443 2.72970 0.225506 −2.29044 0.635516 −1.01110 −0.436482
1.19 0.0200502 0.797351 −1.99960 −4.00544 0.0159871 2.30092 −0.0801929 −2.36423 −0.0803100
1.20 0.466152 2.51297 −1.78270 0.786932 1.17142 1.93750 −1.76331 3.31500 0.366830
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.36
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(13\) \( +1 \)
\(31\) \( +1 \)

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 5239.2.a.t yes 36
13.b even 2 1 5239.2.a.s ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5239.2.a.s ✓ 36 13.b even 2 1
5239.2.a.t yes 36 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5239))\):

\( T_{2}^{36} - 2 T_{2}^{35} - 48 T_{2}^{34} + 95 T_{2}^{33} + 1043 T_{2}^{32} - 2041 T_{2}^{31} + \cdots + 169 \) Copy content Toggle raw display
\( T_{5}^{36} - 5 T_{5}^{35} - 80 T_{5}^{34} + 415 T_{5}^{33} + 2829 T_{5}^{32} - 15272 T_{5}^{31} + \cdots + 79939 \) Copy content Toggle raw display