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Properties

Label 9386.2.a.bo
Level $9386$
Weight $2$
Character orbit 9386.a
Self dual yes
Analytic conductor $74.948$
Analytic rank $0$
Dimension $9$
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] = [9386,2,Mod(1,9386)] 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("9386.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9386, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9386 = 2 \cdot 13 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9386.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,-9,3,9,-3] 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: \(74.9475873372\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 12x^{7} + 34x^{6} + 51x^{5} - 123x^{4} - 81x^{3} + 138x^{2} + 15x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 494)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + \beta_{8} q^{5} - \beta_1 q^{6} + ( - \beta_{8} - \beta_{4} + \beta_1 - 1) q^{7} - q^{8} + (\beta_{3} + \beta_{2} + \beta_1) q^{9} - \beta_{8} q^{10} + (\beta_{6} + \beta_{4} - 1) q^{11}+ \cdots + (\beta_{7} - 2 \beta_{6} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{2} + 3 q^{3} + 9 q^{4} - 3 q^{5} - 3 q^{6} - 6 q^{7} - 9 q^{8} + 6 q^{9} + 3 q^{10} - 6 q^{11} + 3 q^{12} - 9 q^{13} + 6 q^{14} + 3 q^{15} + 9 q^{16} - 6 q^{18} - 3 q^{20} + 9 q^{21} + 6 q^{22}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 12x^{7} + 34x^{6} + 51x^{5} - 123x^{4} - 81x^{3} + 138x^{2} + 15x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{8} - \nu^{7} + 8\nu^{6} + 35\nu^{5} - 22\nu^{4} - 187\nu^{3} + 36\nu^{2} + 191\nu - 65 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} + \nu^{7} - 8\nu^{6} - 35\nu^{5} + 22\nu^{4} + 187\nu^{3} + \nu^{2} - 228\nu - 46 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{8} + 4\nu^{7} - 69\nu^{6} - 29\nu^{5} + 347\nu^{4} + 8\nu^{3} - 551\nu^{2} + 198\nu + 75 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6\nu^{8} - 31\nu^{7} - 11\nu^{6} + 234\nu^{5} - 164\nu^{4} - 395\nu^{3} + 413\nu^{2} - 147\nu + 57 ) / 37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{8} + 29\nu^{7} + 64\nu^{6} - 275\nu^{5} - 139\nu^{4} + 798\nu^{3} + 66\nu^{2} - 655\nu - 39 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{6} - 2\nu^{5} - 9\nu^{4} + 15\nu^{3} + 21\nu^{2} - 27\nu - 3 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -4\nu^{8} - 4\nu^{7} + 106\nu^{6} - 45\nu^{5} - 643\nu^{4} + 473\nu^{3} + 1143\nu^{2} - 901\nu - 112 ) / 37 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{2} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{7} + 2\beta_{6} + 2\beta_{5} + 3\beta_{4} + 5\beta_{3} + 9\beta_{2} + 9\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{8} - \beta_{7} + 9\beta_{6} + 9\beta_{5} + 12\beta_{4} + 22\beta_{2} + 39\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{8} - 10\beta_{7} + 21\beta_{6} + 21\beta_{5} + 36\beta_{4} + 24\beta_{3} + 74\beta_{2} + 75\beta _1 + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 29\beta_{8} - 14\beta_{7} + 72\beta_{6} + 71\beta_{5} + 115\beta_{4} + \beta_{3} + 195\beta_{2} + 269\beta _1 + 123 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 123 \beta_{8} - 79 \beta_{7} + 180 \beta_{6} + 181 \beta_{5} + 340 \beta_{4} + 117 \beta_{3} + \cdots + 546 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.09391
−1.96062
−1.76508
−0.147896
0.0469692
1.18602
2.14447
2.76122
2.82883
−1.00000 −2.09391 1.00000 1.51831 2.09391 −3.68205 −1.00000 1.38445 −1.51831
1.2 −1.00000 −1.96062 1.00000 −3.27542 1.96062 0.728538 −1.00000 0.844016 3.27542
1.3 −1.00000 −1.76508 1.00000 0.274708 1.76508 −0.515534 −1.00000 0.115499 −0.274708
1.4 −1.00000 −0.147896 1.00000 1.20058 0.147896 −3.26215 −1.00000 −2.97813 −1.20058
1.5 −1.00000 0.0469692 1.00000 −4.10140 −0.0469692 0.902778 −1.00000 −2.99779 4.10140
1.6 −1.00000 1.18602 1.00000 2.82669 −1.18602 −2.73454 −1.00000 −1.59336 −2.82669
1.7 −1.00000 2.14447 1.00000 −2.06899 −2.14447 3.74089 −1.00000 1.59874 2.06899
1.8 −1.00000 2.76122 1.00000 1.07484 −2.76122 2.41300 −1.00000 4.62431 −1.07484
1.9 −1.00000 2.82883 1.00000 −0.449319 −2.82883 −3.59093 −1.00000 5.00226 0.449319
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(13\) \( +1 \)
\(19\) \( -1 \)

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 9386.2.a.bo 9
19.b odd 2 1 9386.2.a.bp 9
19.f odd 18 2 494.2.x.b ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.x.b ✓ 18 19.f odd 18 2
9386.2.a.bo 9 1.a even 1 1 trivial
9386.2.a.bp 9 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9386))\):

\( T_{3}^{9} - 3T_{3}^{8} - 12T_{3}^{7} + 34T_{3}^{6} + 51T_{3}^{5} - 123T_{3}^{4} - 81T_{3}^{3} + 138T_{3}^{2} + 15T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{9} + 3T_{5}^{8} - 18T_{5}^{7} - 35T_{5}^{6} + 117T_{5}^{5} + 66T_{5}^{4} - 276T_{5}^{3} + 102T_{5}^{2} + 60T_{5} - 19 \) Copy content Toggle raw display
\( T_{7}^{9} + 6T_{7}^{8} - 15T_{7}^{7} - 134T_{7}^{6} - 15T_{7}^{5} + 795T_{7}^{4} + 407T_{7}^{3} - 1353T_{7}^{2} + 361 \) Copy content Toggle raw display
\( T_{29}^{9} - 21 T_{29}^{8} + 66 T_{29}^{7} + 1387 T_{29}^{6} - 12984 T_{29}^{5} + 35904 T_{29}^{4} + \cdots + 89281 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 3 T^{8} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{9} + 3 T^{8} + \cdots - 19 \) Copy content Toggle raw display
$7$ \( T^{9} + 6 T^{8} + \cdots + 361 \) Copy content Toggle raw display
$11$ \( T^{9} + 6 T^{8} + \cdots - 19 \) Copy content Toggle raw display
$13$ \( (T + 1)^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - 69 T^{7} + \cdots - 19 \) Copy content Toggle raw display
$19$ \( T^{9} \) Copy content Toggle raw display
$23$ \( T^{9} + 3 T^{8} + \cdots - 6841 \) Copy content Toggle raw display
$29$ \( T^{9} - 21 T^{8} + \cdots + 89281 \) Copy content Toggle raw display
$31$ \( T^{9} - 3 T^{8} + \cdots - 379 \) Copy content Toggle raw display
$37$ \( T^{9} - 12 T^{8} + \cdots + 97831 \) Copy content Toggle raw display
$41$ \( T^{9} - 81 T^{7} + \cdots - 1061 \) Copy content Toggle raw display
$43$ \( T^{9} + 3 T^{8} + \cdots + 16021 \) Copy content Toggle raw display
$47$ \( T^{9} - 258 T^{7} + \cdots - 253009 \) Copy content Toggle raw display
$53$ \( T^{9} + 6 T^{8} + \cdots + 1220977 \) Copy content Toggle raw display
$59$ \( T^{9} - 12 T^{8} + \cdots + 16939081 \) Copy content Toggle raw display
$61$ \( T^{9} + 15 T^{8} + \cdots - 12307807 \) Copy content Toggle raw display
$67$ \( T^{9} - 21 T^{8} + \cdots - 5248459 \) Copy content Toggle raw display
$71$ \( T^{9} - 21 T^{8} + \cdots - 36963 \) Copy content Toggle raw display
$73$ \( T^{9} + 45 T^{8} + \cdots - 26867699 \) Copy content Toggle raw display
$79$ \( T^{9} + 9 T^{8} + \cdots + 19855457 \) Copy content Toggle raw display
$83$ \( T^{9} - 15 T^{8} + \cdots + 2391643 \) Copy content Toggle raw display
$89$ \( T^{9} + 3 T^{8} + \cdots + 2345939 \) Copy content Toggle raw display
$97$ \( T^{9} + 3 T^{8} + \cdots + 71289793 \) Copy content Toggle raw display
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