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Properties

Label 9386.2.a.bu
Level $9386$
Weight $2$
Character orbit 9386.a
Self dual yes
Analytic conductor $74.948$
Analytic rank $0$
Dimension $12$
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 = [12,12,0,12,4] 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 26x^{10} + 247x^{8} - x^{7} - 1029x^{6} + 23x^{5} + 1691x^{4} - 84x^{3} - 491x^{2} + 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_{8} q^{5} + \beta_1 q^{6} + (\beta_{8} - \beta_{6}) q^{7} + q^{8} + (\beta_{10} + \beta_{8} - \beta_{5} + 1) q^{9} + \beta_{8} q^{10} + (\beta_{11} + \beta_{5} + \beta_1) q^{11}+ \cdots + ( - 2 \beta_{11} - \beta_{10} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 12 q^{4} + 4 q^{5} + 4 q^{7} + 12 q^{8} + 16 q^{9} + 4 q^{10} + 8 q^{11} + 12 q^{13} + 4 q^{14} + 2 q^{15} + 12 q^{16} + 4 q^{17} + 16 q^{18} + 4 q^{20} - 9 q^{21} + 8 q^{22} - 8 q^{23}+ \cdots - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 26x^{10} + 247x^{8} - x^{7} - 1029x^{6} + 23x^{5} + 1691x^{4} - 84x^{3} - 491x^{2} + 5x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6761 \nu^{11} - 26584 \nu^{10} - 167752 \nu^{9} + 663460 \nu^{8} + 1553382 \nu^{7} - 5800774 \nu^{6} + \cdots + 3749725 ) / 796765 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 376 \nu^{11} + 164 \nu^{10} - 7920 \nu^{9} - 1670 \nu^{8} + 55562 \nu^{7} - 4693 \nu^{6} + \cdots + 24475 ) / 41935 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13694 \nu^{11} + 37558 \nu^{10} - 382668 \nu^{9} - 800573 \nu^{8} + 3845611 \nu^{7} + 5683429 \nu^{6} + \cdots + 53800 ) / 796765 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14634 \nu^{11} - 3967 \nu^{10} - 377307 \nu^{9} + 109435 \nu^{8} + 3573553 \nu^{7} - 1067258 \nu^{6} + \cdots + 1016590 ) / 796765 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15928 \nu^{11} + 25530 \nu^{10} + 376369 \nu^{9} - 578267 \nu^{8} - 3108174 \nu^{7} + \cdots - 5406965 ) / 796765 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 979 \nu^{11} - 376 \nu^{10} - 25618 \nu^{9} + 7920 \nu^{8} + 243483 \nu^{7} - 56541 \nu^{6} + \cdots + 30825 ) / 41935 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20063 \nu^{11} + 32738 \nu^{10} + 409577 \nu^{9} - 631414 \nu^{8} - 2699697 \nu^{7} + \cdots - 242085 ) / 796765 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 29407 \nu^{11} - 21703 \nu^{10} - 700082 \nu^{9} + 484986 \nu^{8} + 5990388 \nu^{7} - 3725922 \nu^{6} + \cdots + 824440 ) / 796765 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 34697 \nu^{11} - 36705 \nu^{10} - 786884 \nu^{9} + 740849 \nu^{8} + 6273250 \nu^{7} + \cdots - 1928385 ) / 796765 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 42431 \nu^{11} - 319 \nu^{10} - 1050971 \nu^{9} - 79292 \nu^{8} + 9450139 \nu^{7} + 1552899 \nu^{6} + \cdots - 1859250 ) / 796765 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{8} - \beta_{5} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} - \beta_{7} + \beta_{4} - \beta_{3} + \beta_{2} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + 8\beta_{10} - 2\beta_{9} + 9\beta_{8} - \beta_{6} - 10\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{11} + 12 \beta_{10} - 10 \beta_{9} - \beta_{8} - 13 \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26 \beta_{11} + 63 \beta_{10} - 27 \beta_{9} + 77 \beta_{8} + 7 \beta_{7} - 11 \beta_{6} - 92 \beta_{5} + \cdots + 228 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12 \beta_{11} + 120 \beta_{10} - 90 \beta_{9} - 16 \beta_{8} - 149 \beta_{7} + 16 \beta_{6} + \cdots + 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 266 \beta_{11} + 503 \beta_{10} - 281 \beta_{9} + 663 \beta_{8} + 134 \beta_{7} - 99 \beta_{6} + \cdots + 1879 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 119 \beta_{11} + 1121 \beta_{10} - 779 \beta_{9} - 209 \beta_{8} - 1597 \beta_{7} + 192 \beta_{6} + \cdots + 90 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2538 \beta_{11} + 4073 \beta_{10} - 2678 \beta_{9} + 5791 \beta_{8} + 1789 \beta_{7} - 849 \beta_{6} + \cdots + 15998 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1164 \beta_{11} + 10136 \beta_{10} - 6619 \beta_{9} - 2547 \beta_{8} - 16393 \beta_{7} + 2076 \beta_{6} + \cdots + 35 \) 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
−3.09601
−2.49196
−2.42410
−2.01376
−0.499612
−0.255279
0.254044
0.569991
1.91116
2.20765
2.88390
2.95398
1.00000 −3.09601 1.00000 3.12355 −3.09601 1.32981 1.00000 6.58529 3.12355
1.2 1.00000 −2.49196 1.00000 −3.57989 −2.49196 −3.10112 1.00000 3.20985 −3.57989
1.3 1.00000 −2.42410 1.00000 4.30744 −2.42410 2.46983 1.00000 2.87628 4.30744
1.4 1.00000 −2.01376 1.00000 −1.55662 −2.01376 0.146295 1.00000 1.05523 −1.55662
1.5 1.00000 −0.499612 1.00000 −0.651928 −0.499612 −0.915355 1.00000 −2.75039 −0.651928
1.6 1.00000 −0.255279 1.00000 −0.115724 −0.255279 5.07592 1.00000 −2.93483 −0.115724
1.7 1.00000 0.254044 1.00000 −0.894447 0.254044 3.51840 1.00000 −2.93546 −0.894447
1.8 1.00000 0.569991 1.00000 −2.58040 0.569991 −3.31293 1.00000 −2.67511 −2.58040
1.9 1.00000 1.91116 1.00000 4.37902 1.91116 −1.37614 1.00000 0.652514 4.37902
1.10 1.00000 2.20765 1.00000 2.05710 2.20765 2.93175 1.00000 1.87370 2.05710
1.11 1.00000 2.88390 1.00000 0.986243 2.88390 2.37312 1.00000 5.31691 0.986243
1.12 1.00000 2.95398 1.00000 −1.47433 2.95398 −5.13956 1.00000 5.72602 −1.47433
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.bu yes 12
19.b odd 2 1 9386.2.a.br ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9386.2.a.br ✓ 12 19.b odd 2 1
9386.2.a.bu yes 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9386))\):

\( T_{3}^{12} - 26 T_{3}^{10} + 247 T_{3}^{8} - T_{3}^{7} - 1029 T_{3}^{6} + 23 T_{3}^{5} + 1691 T_{3}^{4} + \cdots + 25 \) Copy content Toggle raw display
\( T_{5}^{12} - 4 T_{5}^{11} - 31 T_{5}^{10} + 101 T_{5}^{9} + 375 T_{5}^{8} - 756 T_{5}^{7} - 2211 T_{5}^{6} + \cdots - 171 \) Copy content Toggle raw display
\( T_{7}^{12} - 4 T_{7}^{11} - 47 T_{7}^{10} + 201 T_{7}^{9} + 647 T_{7}^{8} - 3206 T_{7}^{7} - 2621 T_{7}^{6} + \cdots - 3971 \) Copy content Toggle raw display
\( T_{29}^{12} - 13 T_{29}^{11} - 167 T_{29}^{10} + 2812 T_{29}^{9} + 4134 T_{29}^{8} - 196107 T_{29}^{7} + \cdots + 3800000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 26 T^{10} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} + \cdots - 171 \) Copy content Toggle raw display
$7$ \( T^{12} - 4 T^{11} + \cdots - 3971 \) Copy content Toggle raw display
$11$ \( T^{12} - 8 T^{11} + \cdots + 19456 \) Copy content Toggle raw display
$13$ \( (T - 1)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 4 T^{11} + \cdots + 44276 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 178975744 \) Copy content Toggle raw display
$29$ \( T^{12} - 13 T^{11} + \cdots + 3800000 \) Copy content Toggle raw display
$31$ \( T^{12} + 9 T^{11} + \cdots + 5105 \) Copy content Toggle raw display
$37$ \( T^{12} + 12 T^{11} + \cdots + 33074155 \) Copy content Toggle raw display
$41$ \( T^{12} + 6 T^{11} + \cdots - 318400 \) Copy content Toggle raw display
$43$ \( T^{12} - 25 T^{11} + \cdots - 13205 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 685676864 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 14988597184 \) Copy content Toggle raw display
$59$ \( T^{12} + 20 T^{11} + \cdots + 61281280 \) Copy content Toggle raw display
$61$ \( T^{12} + 2 T^{11} + \cdots + 89418304 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 3475853120 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 2552518025 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 48919347520 \) Copy content Toggle raw display
$79$ \( T^{12} - 12 T^{11} + \cdots - 41053120 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 102842624 \) Copy content Toggle raw display
$89$ \( T^{12} - 32 T^{11} + \cdots - 94974400 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 12575951424 \) Copy content Toggle raw display
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