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Properties

Label 9065.2.a.bd
Level $9065$
Weight $2$
Character orbit 9065.a
Self dual yes
Analytic conductor $72.384$
Analytic rank $1$
Dimension $38$
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] = [9065,2,Mod(1,9065)] 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("9065.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9065, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9065 = 5 \cdot 7^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9065.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [38,-2,-6,42,-38,-8,0,-6,48,2,34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.3843894323\)
Analytic rank: \(1\)
Dimension: \(38\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 2 q^{2} - 6 q^{3} + 42 q^{4} - 38 q^{5} - 8 q^{6} - 6 q^{8} + 48 q^{9} + 2 q^{10} + 34 q^{11} - 20 q^{12} - 22 q^{13} + 6 q^{15} + 46 q^{16} - 22 q^{17} - 36 q^{18} - 40 q^{19} - 42 q^{20} - 4 q^{22}+ \cdots + 92 q^{99}+O(q^{100}) \) 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 −2.76493 −3.05664 5.64486 −1.00000 8.45140 0 −10.0778 6.34304 2.76493
1.2 −2.67194 −0.492037 5.13928 −1.00000 1.31469 0 −8.38797 −2.75790 2.67194
1.3 −2.64003 2.83159 4.96978 −1.00000 −7.47550 0 −7.84034 5.01791 2.64003
1.4 −2.54479 2.35441 4.47595 −1.00000 −5.99147 0 −6.30078 2.54323 2.54479
1.5 −2.35581 1.15914 3.54986 −1.00000 −2.73072 0 −3.65118 −1.65639 2.35581
1.6 −2.27347 −3.21093 3.16869 −1.00000 7.29997 0 −2.65698 7.31008 2.27347
1.7 −2.10125 0.620531 2.41526 −1.00000 −1.30389 0 −0.872557 −2.61494 2.10125
1.8 −1.80708 −0.658880 1.26553 −1.00000 1.19065 0 1.32724 −2.56588 1.80708
1.9 −1.80074 −1.52571 1.24267 −1.00000 2.74742 0 1.36376 −0.672196 1.80074
1.10 −1.70429 −2.61912 0.904597 −1.00000 4.46374 0 1.86688 3.85980 1.70429
1.11 −1.52102 3.00220 0.313495 −1.00000 −4.56640 0 2.56520 6.01320 1.52102
1.12 −1.48432 3.10364 0.203196 −1.00000 −4.60679 0 2.66703 6.63259 1.48432
1.13 −1.19919 −2.34157 −0.561938 −1.00000 2.80799 0 3.07226 2.48295 1.19919
1.14 −1.08209 −2.34188 −0.829090 −1.00000 2.53412 0 3.06132 2.48441 1.08209
1.15 −0.996254 −0.129416 −1.00748 −1.00000 0.128931 0 2.99621 −2.98325 0.996254
1.16 −0.727934 1.69863 −1.47011 −1.00000 −1.23649 0 2.52601 −0.114646 0.727934
1.17 −0.333833 −0.136318 −1.88856 −1.00000 0.0455075 0 1.29813 −2.98142 0.333833
1.18 −0.101692 −0.616177 −1.98966 −1.00000 0.0626600 0 0.405715 −2.62033 0.101692
1.19 −0.0451191 2.73072 −1.99796 −1.00000 −0.123207 0 0.180384 4.45683 0.0451191
1.20 −0.0412902 −1.58988 −1.99830 −1.00000 0.0656462 0 0.165090 −0.472294 0.0412902
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.38
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(7\) \( +1 \)
\(37\) \( +1 \)

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 9065.2.a.bd ✓ 38
7.b odd 2 1 9065.2.a.be yes 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9065.2.a.bd ✓ 38 1.a even 1 1 trivial
9065.2.a.be yes 38 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9065))\):

\( T_{2}^{38} + 2 T_{2}^{37} - 57 T_{2}^{36} - 112 T_{2}^{35} + 1482 T_{2}^{34} + 2860 T_{2}^{33} + \cdots + 14 \) Copy content Toggle raw display
\( T_{3}^{38} + 6 T_{3}^{37} - 63 T_{3}^{36} - 428 T_{3}^{35} + 1715 T_{3}^{34} + 13842 T_{3}^{33} + \cdots - 487424 \) Copy content Toggle raw display
\( T_{11}^{38} - 34 T_{11}^{37} + 371 T_{11}^{36} + 144 T_{11}^{35} - 33179 T_{11}^{34} + \cdots - 2651707928576 \) Copy content Toggle raw display