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Properties

Label 9386.2.a.bw
Level $9386$
Weight $2$
Character orbit 9386.a
Self dual yes
Analytic conductor $74.948$
Analytic rank $1$
Dimension $15$
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 = [15,-15,-3,15,3] 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: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 3 x^{14} - 27 x^{13} + 70 x^{12} + 306 x^{11} - 609 x^{10} - 1854 x^{9} + 2346 x^{8} + \cdots - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} - \beta_{6} q^{5} + \beta_1 q^{6} + ( - \beta_{12} + \beta_1) q^{7} - q^{8} + (\beta_{10} + \beta_{8} - \beta_{4} + \cdots + 1) q^{9} + \beta_{6} q^{10} + (\beta_{11} + \beta_{8} + \beta_{6}) q^{11}+ \cdots + ( - 2 \beta_{14} + \beta_{13} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 15 q^{2} - 3 q^{3} + 15 q^{4} + 3 q^{5} + 3 q^{6} - 15 q^{8} + 18 q^{9} - 3 q^{10} - 3 q^{11} - 3 q^{12} + 15 q^{13} + 15 q^{16} + 3 q^{17} - 18 q^{18} + 3 q^{20} - 33 q^{21} + 3 q^{22} + 9 q^{23}+ \cdots + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 3 x^{14} - 27 x^{13} + 70 x^{12} + 306 x^{11} - 609 x^{10} - 1854 x^{9} + 2346 x^{8} + \cdots - 24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 30512799 \nu^{14} + 1369031611 \nu^{13} + 486452335 \nu^{12} - 35781723640 \nu^{11} + \cdots + 226777799140 ) / 48867386416 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 300979031 \nu^{14} + 2206996283 \nu^{13} + 4689259951 \nu^{12} - 50214501344 \nu^{11} + \cdots - 171680941004 ) / 48867386416 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 597185531 \nu^{14} - 1850936507 \nu^{13} - 15428772207 \nu^{12} + 42954158116 \nu^{11} + \cdots - 120871397468 ) / 48867386416 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 167919537 \nu^{14} - 386308715 \nu^{13} - 4342645423 \nu^{12} + 8425477542 \nu^{11} + \cdots - 6201110136 ) / 12216846604 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 354105151 \nu^{14} - 1014412411 \nu^{13} - 8741935039 \nu^{12} + 22596023928 \nu^{11} + \cdots + 28087907228 ) / 24433693208 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 365529031 \nu^{14} + 1120235995 \nu^{13} + 9260876319 \nu^{12} - 25765160648 \nu^{11} + \cdots + 19568446644 ) / 24433693208 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 258379589 \nu^{14} + 943058304 \nu^{13} + 6589940188 \nu^{12} - 22429216653 \nu^{11} + \cdots + 26173469474 ) / 12216846604 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 899718135 \nu^{14} + 2823786483 \nu^{13} + 22957817847 \nu^{12} - 64473598840 \nu^{11} + \cdots + 178576528916 ) / 24433693208 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 965841459 \nu^{14} - 3915083003 \nu^{13} - 23238896455 \nu^{12} + 91442763036 \nu^{11} + \cdots - 124676940012 ) / 24433693208 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 160984963 \nu^{14} + 461302382 \nu^{13} + 4337658249 \nu^{12} - 10688112081 \nu^{11} + \cdots + 11580415096 ) / 3054211651 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 323573495 \nu^{14} - 998710914 \nu^{13} - 8797374628 \nu^{12} + 23903704083 \nu^{11} + \cdots - 30874277060 ) / 6108423302 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2774911355 \nu^{14} - 9167156495 \nu^{13} - 74890898611 \nu^{12} + 222141906320 \nu^{11} + \cdots - 504986569252 ) / 48867386416 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 398054606 \nu^{14} + 1174754547 \nu^{13} + 10255137015 \nu^{12} - 26609692203 \nu^{11} + \cdots + 32690767658 ) / 6108423302 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{8} - \beta_{4} + \beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{5} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + 9\beta_{10} + 11\beta_{8} + \beta_{6} + \beta_{5} - 13\beta_{4} + 8\beta_{3} + 12\beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{14} + 10 \beta_{13} - 9 \beta_{12} + 13 \beta_{10} + 9 \beta_{9} + 26 \beta_{8} - 13 \beta_{7} + \cdots + 39 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{14} + 17 \beta_{13} - 3 \beta_{12} + \beta_{11} + 79 \beta_{10} + 111 \beta_{8} - 7 \beta_{7} + \cdots + 223 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 70 \beta_{14} + 92 \beta_{13} - 74 \beta_{12} - \beta_{11} + 140 \beta_{10} + 66 \beta_{9} + \cdots + 416 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 44 \beta_{14} + 209 \beta_{13} - 55 \beta_{12} + 10 \beta_{11} + 706 \beta_{10} - 7 \beta_{9} + \cdots + 1893 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 535 \beta_{14} + 859 \beta_{13} - 611 \beta_{12} - 15 \beta_{11} + 1433 \beta_{10} + 438 \beta_{9} + \cdots + 4165 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 481 \beta_{14} + 2293 \beta_{13} - 700 \beta_{12} + 79 \beta_{11} + 6406 \beta_{10} - 193 \beta_{9} + \cdots + 16676 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4131 \beta_{14} + 8202 \beta_{13} - 5115 \beta_{12} - 117 \beta_{11} + 14358 \beta_{10} + \cdots + 40659 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4786 \beta_{14} + 23902 \beta_{13} - 7673 \beta_{12} + 725 \beta_{11} + 58829 \beta_{10} + \cdots + 150497 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 32559 \beta_{14} + 79586 \beta_{13} - 43313 \beta_{12} - 146 \beta_{11} + 142245 \beta_{10} + \cdots + 392885 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 46083 \beta_{14} + 242831 \beta_{13} - 77678 \beta_{12} + 8759 \beta_{11} + 545782 \beta_{10} + \cdots + 1381952 \) 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
3.17787
2.98215
2.90071
2.83525
1.73559
0.992722
0.118241
−0.177355
−0.276311
−0.966809
−1.24330
−1.91682
−2.07938
−2.39853
−2.68404
−1.00000 −3.17787 1.00000 1.77302 3.17787 1.17923 −1.00000 7.09889 −1.77302
1.2 −1.00000 −2.98215 1.00000 3.77078 2.98215 2.92386 −1.00000 5.89324 −3.77078
1.3 −1.00000 −2.90071 1.00000 −3.22232 2.90071 1.61433 −1.00000 5.41411 3.22232
1.4 −1.00000 −2.83525 1.00000 −2.30728 2.83525 −1.33915 −1.00000 5.03866 2.30728
1.5 −1.00000 −1.73559 1.00000 1.81197 1.73559 4.67756 −1.00000 0.0122745 −1.81197
1.6 −1.00000 −0.992722 1.00000 2.61889 0.992722 −2.44340 −1.00000 −2.01450 −2.61889
1.7 −1.00000 −0.118241 1.00000 −0.845181 0.118241 −3.20510 −1.00000 −2.98602 0.845181
1.8 −1.00000 0.177355 1.00000 −2.51963 −0.177355 3.59860 −1.00000 −2.96855 2.51963
1.9 −1.00000 0.276311 1.00000 −3.22926 −0.276311 −1.92690 −1.00000 −2.92365 3.22926
1.10 −1.00000 0.966809 1.00000 3.48601 −0.966809 0.573182 −1.00000 −2.06528 −3.48601
1.11 −1.00000 1.24330 1.00000 1.00990 −1.24330 1.78833 −1.00000 −1.45422 −1.00990
1.12 −1.00000 1.91682 1.00000 −1.89821 −1.91682 −3.52843 −1.00000 0.674200 1.89821
1.13 −1.00000 2.07938 1.00000 −2.14853 −2.07938 1.43364 −1.00000 1.32381 2.14853
1.14 −1.00000 2.39853 1.00000 4.06197 −2.39853 −3.19984 −1.00000 2.75296 −4.06197
1.15 −1.00000 2.68404 1.00000 0.637875 −2.68404 −2.14592 −1.00000 4.20407 −0.637875
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.bw 15
19.b odd 2 1 9386.2.a.bz 15
19.f odd 18 2 494.2.x.d ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.x.d ✓ 30 19.f odd 18 2
9386.2.a.bw 15 1.a even 1 1 trivial
9386.2.a.bz 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9386))\):

\( T_{3}^{15} + 3 T_{3}^{14} - 27 T_{3}^{13} - 70 T_{3}^{12} + 306 T_{3}^{11} + 609 T_{3}^{10} - 1854 T_{3}^{9} + \cdots + 24 \) Copy content Toggle raw display
\( T_{5}^{15} - 3 T_{5}^{14} - 45 T_{5}^{13} + 117 T_{5}^{12} + 834 T_{5}^{11} - 1803 T_{5}^{10} + \cdots + 60344 \) Copy content Toggle raw display
\( T_{7}^{15} - 51 T_{7}^{13} - 15 T_{7}^{12} + 1005 T_{7}^{11} + 423 T_{7}^{10} - 9890 T_{7}^{9} + \cdots + 67411 \) Copy content Toggle raw display
\( T_{29}^{15} + 12 T_{29}^{14} - 132 T_{29}^{13} - 2019 T_{29}^{12} + 2733 T_{29}^{11} + 108639 T_{29}^{10} + \cdots + 5298301 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + 3 T^{14} + \cdots + 24 \) Copy content Toggle raw display
$5$ \( T^{15} - 3 T^{14} + \cdots + 60344 \) Copy content Toggle raw display
$7$ \( T^{15} - 51 T^{13} + \cdots + 67411 \) Copy content Toggle raw display
$11$ \( T^{15} + 3 T^{14} + \cdots - 1865971 \) Copy content Toggle raw display
$13$ \( (T - 1)^{15} \) Copy content Toggle raw display
$17$ \( T^{15} - 3 T^{14} + \cdots - 7017201 \) Copy content Toggle raw display
$19$ \( T^{15} \) Copy content Toggle raw display
$23$ \( T^{15} - 9 T^{14} + \cdots + 71189056 \) Copy content Toggle raw display
$29$ \( T^{15} + 12 T^{14} + \cdots + 5298301 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 30012301441 \) Copy content Toggle raw display
$37$ \( T^{15} + 30 T^{14} + \cdots - 71663616 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 6081142464 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 5866971832 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 136821032 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 103322943 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 4482688589 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 554317759 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 421704034471 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 26364117677059 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 47715597485632 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 217497079232 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 57183042277329 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 125796441336 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 440274493592 \) Copy content Toggle raw display
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