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Properties

Label 3751.2.a.p
Level $3751$
Weight $2$
Character orbit 3751.a
Self dual yes
Analytic conductor $29.952$
Analytic rank $1$
Dimension $30$
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] = [3751,2,Mod(1,3751)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3751, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3751.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3751 = 11^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3751.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30,-9,-1,29,-1] 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: \(29.9518857982\)
Analytic rank: \(1\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 341)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 9 q^{2} - q^{3} + 29 q^{4} - q^{5} - 11 q^{6} - 11 q^{7} - 27 q^{8} + 25 q^{9} - 8 q^{10} - 16 q^{12} - 2 q^{13} - 3 q^{14} - 11 q^{15} + 35 q^{16} - 32 q^{17} - 4 q^{18} - 17 q^{19} - 4 q^{20}+ \cdots - 40 q^{98}+O(q^{100}) \) 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 −2.76765 1.60773 5.65988 0.677993 −4.44964 −1.90866 −10.1293 −0.415189 −1.87645
1.2 −2.73185 −2.74061 5.46303 −2.19792 7.48694 0.378355 −9.46050 4.51093 6.00441
1.3 −2.59478 −1.09142 4.73286 3.97879 2.83200 0.943415 −7.09117 −1.80879 −10.3241
1.4 −2.54880 −1.44819 4.49636 −3.26096 3.69114 2.64997 −6.36271 −0.902745 8.31151
1.5 −2.37611 2.08434 3.64591 1.68294 −4.95263 −4.00915 −3.91087 1.34449 −3.99886
1.6 −2.27751 0.815461 3.18707 2.96819 −1.85722 3.93457 −2.70357 −2.33502 −6.76008
1.7 −2.10191 −1.86503 2.41802 0.760179 3.92012 −4.66326 −0.878647 0.478328 −1.59783
1.8 −2.06445 3.38749 2.26196 −3.45782 −6.99330 −0.112095 −0.540812 8.47506 7.13851
1.9 −1.32197 −0.888436 −0.252403 −1.72129 1.17448 3.29883 2.97760 −2.21068 2.27549
1.10 −1.27391 0.0599919 −0.377165 −3.80804 −0.0764240 −4.34950 3.02828 −2.99640 4.85109
1.11 −1.20486 2.07810 −0.548310 2.89499 −2.50382 −1.93395 3.07036 1.31850 −3.48806
1.12 −1.18669 −1.13252 −0.591757 1.29761 1.34395 3.67025 3.07562 −1.71740 −1.53987
1.13 −1.11867 2.77147 −0.748568 1.51120 −3.10038 −1.74043 3.07475 4.68107 −1.69054
1.14 −0.659531 −1.83958 −1.56502 3.05360 1.21326 −2.48935 2.35124 0.384062 −2.01394
1.15 −0.490769 2.36606 −1.75915 −2.52890 −1.16119 −3.36169 1.84487 2.59826 1.24111
1.16 −0.425035 1.56959 −1.81935 −1.96906 −0.667132 2.78724 1.62335 −0.536378 0.836917
1.17 0.0184858 −2.94603 −1.99966 1.83598 −0.0544597 3.52926 −0.0739368 5.67910 0.0339395
1.18 0.150344 −0.145795 −1.97740 1.52515 −0.0219194 1.13667 −0.597977 −2.97874 0.229296
1.19 0.151111 −2.39368 −1.97717 −0.259539 −0.361712 −3.23244 −0.600995 2.72968 −0.0392192
1.20 0.740862 1.64049 −1.45112 −4.14413 1.21537 1.71028 −2.55681 −0.308806 −3.07023
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(11\) \( -1 \)
\(31\) \( -1 \)

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 3751.2.a.p 30
11.b odd 2 1 3751.2.a.s 30
11.d odd 10 2 341.2.h.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
341.2.h.a ✓ 60 11.d odd 10 2
3751.2.a.p 30 1.a even 1 1 trivial
3751.2.a.s 30 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3751))\):

\( T_{2}^{30} + 9 T_{2}^{29} - 4 T_{2}^{28} - 258 T_{2}^{27} - 462 T_{2}^{26} + 2995 T_{2}^{25} + 9193 T_{2}^{24} + \cdots + 16 \) Copy content Toggle raw display
\( T_{3}^{30} + T_{3}^{29} - 57 T_{3}^{28} - 57 T_{3}^{27} + 1444 T_{3}^{26} + 1436 T_{3}^{25} + \cdots - 10709 \) Copy content Toggle raw display