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Properties

Label 9065.2.a.m
Level $9065$
Weight $2$
Character orbit 9065.a
Self dual yes
Analytic conductor $72.384$
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] = [9065,2,Mod(1,9065)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
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")
 
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);
 
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 = [12,5,-4,15,-12,-6,0,21,18,-5,-4] 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: \(0\)
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: \( x^{12} - 5 x^{11} - 7 x^{10} + 63 x^{9} - 11 x^{8} - 279 x^{7} + 171 x^{6} + 503 x^{5} - 367 x^{4} + \cdots - 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 1295)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} - q^{5} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \cdots - 1) q^{6} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{8} + ( - \beta_{9} + \beta_{7} - \beta_{3} + \cdots + 2) q^{9}+ \cdots + ( - 2 \beta_{11} - \beta_{9} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 5 q^{2} - 4 q^{3} + 15 q^{4} - 12 q^{5} - 6 q^{6} + 21 q^{8} + 18 q^{9} - 5 q^{10} - 4 q^{11} - 8 q^{12} - 6 q^{13} + 4 q^{15} + 21 q^{16} - 20 q^{17} + 14 q^{18} + 4 q^{19} - 15 q^{20} + 2 q^{22}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} - 7 x^{10} + 63 x^{9} - 11 x^{8} - 279 x^{7} + 171 x^{6} + 503 x^{5} - 367 x^{4} + \cdots - 26 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3 \nu^{11} - 72 \nu^{10} + 62 \nu^{9} + 1067 \nu^{8} - 1031 \nu^{7} - 5663 \nu^{6} + 4282 \nu^{5} + \cdots + 1335 ) / 257 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27 \nu^{11} + 134 \nu^{10} + 213 \nu^{9} - 1636 \nu^{8} - 230 \nu^{7} + 7020 \nu^{6} - 1273 \nu^{5} + \cdots - 707 ) / 257 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 36 \nu^{11} + 93 \nu^{10} + 541 \nu^{9} - 1239 \nu^{8} - 3048 \nu^{7} + 5762 \nu^{6} + 7469 \nu^{5} + \cdots - 1628 ) / 257 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 38 \nu^{11} - 141 \nu^{10} - 414 \nu^{9} + 1779 \nu^{8} + 1504 \nu^{7} - 8081 \nu^{6} - 1873 \nu^{5} + \cdots + 1490 ) / 257 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 40 \nu^{11} + 189 \nu^{10} + 287 \nu^{9} - 2319 \nu^{8} + 297 \nu^{7} + 9886 \nu^{6} - 5779 \nu^{5} + \cdots - 581 ) / 257 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 49 \nu^{11} - 148 \nu^{10} - 615 \nu^{9} + 1922 \nu^{8} + 2778 \nu^{7} - 8885 \nu^{6} - 5533 \nu^{5} + \cdots + 988 ) / 257 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 73 \nu^{11} - 210 \nu^{10} - 890 \nu^{9} + 2491 \nu^{8} + 3782 \nu^{7} - 9728 \nu^{6} - 6229 \nu^{5} + \cdots - 411 ) / 257 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 89 \nu^{11} - 337 \nu^{10} - 902 \nu^{9} + 4241 \nu^{8} + 2481 \nu^{7} - 18771 \nu^{6} + 246 \nu^{5} + \cdots + 2597 ) / 257 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{9} + \beta_{8} + \beta_{3} + 7\beta_{2} + 8\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{11} + 2 \beta_{10} - 11 \beta_{9} + 12 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15 \beta_{11} + 12 \beta_{10} - 25 \beta_{9} + 28 \beta_{8} + 22 \beta_{7} + 13 \beta_{6} + 10 \beta_{5} + \cdots + 103 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 82 \beta_{11} + 27 \beta_{10} - 95 \beta_{9} + 110 \beta_{8} + 31 \beta_{7} + 29 \beta_{6} + 16 \beta_{5} + \cdots + 502 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 157 \beta_{11} + 108 \beta_{10} - 234 \beta_{9} + 277 \beta_{8} + 187 \beta_{7} + 125 \beta_{6} + \cdots + 801 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 640 \beta_{11} + 260 \beta_{10} - 759 \beta_{9} + 915 \beta_{8} + 328 \beta_{7} + 298 \beta_{6} + \cdots + 3155 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1417 \beta_{11} + 877 \beta_{10} - 1955 \beta_{9} + 2394 \beta_{8} + 1463 \beta_{7} + 1070 \beta_{6} + \cdots + 5993 \) 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
−2.17563
−2.01828
−1.60300
−0.876141
−0.412986
0.366100
0.657665
1.13623
2.21408
2.31196
2.68572
2.71430
−2.17563 1.68814 2.73336 −1.00000 −3.67276 0 −1.59553 −0.150199 2.17563
1.2 −2.01828 −3.10507 2.07347 −1.00000 6.26691 0 −0.148289 6.64146 2.01828
1.3 −1.60300 1.65152 0.569623 −1.00000 −2.64740 0 2.29290 −0.272472 1.60300
1.4 −0.876141 −0.718988 −1.23238 −1.00000 0.629935 0 2.83202 −2.48306 0.876141
1.5 −0.412986 −1.12400 −1.82944 −1.00000 0.464196 0 1.58151 −1.73663 0.412986
1.6 0.366100 2.82387 −1.86597 −1.00000 1.03382 0 −1.41533 4.97422 −0.366100
1.7 0.657665 −2.65005 −1.56748 −1.00000 −1.74285 0 −2.34621 4.02279 −0.657665
1.8 1.13623 0.962552 −0.708987 −1.00000 1.09368 0 −3.07803 −2.07349 −1.13623
1.9 2.21408 −2.15744 2.90215 −1.00000 −4.77674 0 1.99744 1.65453 −2.21408
1.10 2.31196 −2.84984 3.34514 −1.00000 −6.58871 0 3.10990 5.12159 −2.31196
1.11 2.68572 2.63795 5.21310 −1.00000 7.08481 0 8.62948 3.95881 −2.68572
1.12 2.71430 −1.15864 5.36741 −1.00000 −3.14491 0 9.14014 −1.65754 −2.71430
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

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.m 12
7.b odd 2 1 1295.2.a.h ✓ 12
35.c odd 2 1 6475.2.a.r 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1295.2.a.h ✓ 12 7.b odd 2 1
6475.2.a.r 12 35.c odd 2 1
9065.2.a.m 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9065))\):

\( T_{2}^{12} - 5 T_{2}^{11} - 7 T_{2}^{10} + 63 T_{2}^{9} - 11 T_{2}^{8} - 279 T_{2}^{7} + 171 T_{2}^{6} + \cdots - 26 \) Copy content Toggle raw display
\( T_{3}^{12} + 4 T_{3}^{11} - 19 T_{3}^{10} - 86 T_{3}^{9} + 114 T_{3}^{8} + 666 T_{3}^{7} - 169 T_{3}^{6} + \cdots - 947 \) Copy content Toggle raw display
\( T_{11}^{12} + 4 T_{11}^{11} - 69 T_{11}^{10} - 220 T_{11}^{9} + 1550 T_{11}^{8} + 3144 T_{11}^{7} + \cdots - 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 5 T^{11} + \cdots - 26 \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{11} + \cdots - 947 \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + \cdots - 144 \) Copy content Toggle raw display
$13$ \( T^{12} + 6 T^{11} + \cdots - 3509504 \) Copy content Toggle raw display
$17$ \( T^{12} + 20 T^{11} + \cdots + 2193152 \) Copy content Toggle raw display
$19$ \( T^{12} - 4 T^{11} + \cdots - 162518 \) Copy content Toggle raw display
$23$ \( T^{12} - 20 T^{11} + \cdots + 89776282 \) Copy content Toggle raw display
$29$ \( T^{12} + 12 T^{11} + \cdots - 85865728 \) Copy content Toggle raw display
$31$ \( T^{12} - 229 T^{10} + \cdots + 58378144 \) Copy content Toggle raw display
$37$ \( (T + 1)^{12} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 837402624 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 537600096 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 222893537 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 229897728 \) Copy content Toggle raw display
$59$ \( T^{12} + 4 T^{11} + \cdots - 3923478 \) Copy content Toggle raw display
$61$ \( T^{12} - 349 T^{10} + \cdots - 36390232 \) Copy content Toggle raw display
$67$ \( T^{12} - 6 T^{11} + \cdots + 114176 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 42266925396 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 684598989376 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 14356327424 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 103223851168 \) Copy content Toggle raw display
$89$ \( T^{12} - 259 T^{10} + \cdots + 93740288 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 1075459072 \) Copy content Toggle raw display
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