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Properties

Label 9409.2.a.n
Level $9409$
Weight $2$
Character orbit 9409.a
Self dual yes
Analytic conductor $75.131$
Analytic rank $0$
Dimension $56$
Inner twists $2$

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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] = [9409,2,Mod(1,9409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9409, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("9409.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 9409 = 97^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9409.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56,8,16,40,0] 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: \(75.1312432618\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: no (minimal twist has level 97)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 8 q^{2} + 16 q^{3} + 40 q^{4} + 16 q^{6} + 24 q^{8} + 40 q^{9} + 32 q^{11} + 48 q^{12} + 8 q^{16} + 24 q^{18} + 16 q^{22} + 8 q^{25} + 64 q^{27} + 48 q^{31} + 56 q^{32} + 80 q^{33} + 48 q^{35} + 8 q^{36}+ \cdots - 16 q^{99}+O(q^{100}) \) 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 −2.50643 2.42359 4.28220 −3.37181 −6.07457 −3.94032 −5.72019 2.87380 8.45122
1.2 −2.50643 2.42359 4.28220 3.37181 −6.07457 3.94032 −5.72019 2.87380 −8.45122
1.3 −2.36394 0.429115 3.58823 −1.05721 −1.01440 1.83323 −3.75448 −2.81586 2.49919
1.4 −2.36394 0.429115 3.58823 1.05721 −1.01440 −1.83323 −3.75448 −2.81586 −2.49919
1.5 −2.15107 −2.03471 2.62711 −3.21461 4.37680 4.00609 −1.34897 1.14003 6.91487
1.6 −2.15107 −2.03471 2.62711 3.21461 4.37680 −4.00609 −1.34897 1.14003 −6.91487
1.7 −2.05110 2.67677 2.20701 −0.288263 −5.49031 2.74678 −0.424591 4.16508 0.591255
1.8 −2.05110 2.67677 2.20701 0.288263 −5.49031 −2.74678 −0.424591 4.16508 −0.591255
1.9 −1.78055 −1.12743 1.17037 −4.01496 2.00745 0.537014 1.47720 −1.72890 7.14886
1.10 −1.78055 −1.12743 1.17037 4.01496 2.00745 −0.537014 1.47720 −1.72890 −7.14886
1.11 −1.49760 −2.30048 0.242806 −0.558865 3.44521 3.77554 2.63157 2.29223 0.836956
1.12 −1.49760 −2.30048 0.242806 0.558865 3.44521 −3.77554 2.63157 2.29223 −0.836956
1.13 −1.44014 1.00093 0.0740135 −1.32109 −1.44148 −2.31617 2.77370 −1.99814 1.90256
1.14 −1.44014 1.00093 0.0740135 1.32109 −1.44148 2.31617 2.77370 −1.99814 −1.90256
1.15 −1.24646 −1.29743 −0.446331 −1.36689 1.61720 0.317442 3.04926 −1.31667 1.70377
1.16 −1.24646 −1.29743 −0.446331 1.36689 1.61720 −0.317442 3.04926 −1.31667 −1.70377
1.17 −1.23640 1.85903 −0.471304 −1.37988 −2.29851 −1.12799 3.05553 0.455981 1.70609
1.18 −1.23640 1.85903 −0.471304 1.37988 −2.29851 1.12799 3.05553 0.455981 −1.70609
1.19 −1.14420 −0.245064 −0.690818 −0.515668 0.280401 3.83456 3.07882 −2.93994 0.590025
1.20 −1.14420 −0.245064 −0.690818 0.515668 0.280401 −3.83456 3.07882 −2.93994 −0.590025
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.56
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(97\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
97.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9409.2.a.n 56
97.b even 2 1 inner 9409.2.a.n 56
97.j odd 32 2 97.2.h.a ✓ 56
291.s even 32 2 873.2.bh.b 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
97.2.h.a ✓ 56 97.j odd 32 2
873.2.bh.b 56 291.s even 32 2
9409.2.a.n 56 1.a even 1 1 trivial
9409.2.a.n 56 97.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9409))\):

\( T_{2}^{28} - 4 T_{2}^{27} - 30 T_{2}^{26} + 132 T_{2}^{25} + 379 T_{2}^{24} - 1904 T_{2}^{23} + \cdots + 31 \) Copy content Toggle raw display
\( T_{5}^{56} - 144 T_{5}^{54} + 9664 T_{5}^{52} - 401616 T_{5}^{50} + 11582992 T_{5}^{48} - 246262520 T_{5}^{46} + \cdots + 32258 \) Copy content Toggle raw display