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Properties

Label 1287.2.b.b
Level $1287$
Weight $2$
Character orbit 1287.b
Analytic conductor $10.277$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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] = [1287,2,Mod(298,1287)] 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("1287.298"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1287, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1287.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.2767467401\)
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: \( x^{12} + 19x^{10} + 133x^{8} + 423x^{6} + 601x^{4} + 312x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{5} + \beta_{4} - 1) q^{4} + ( - \beta_{8} - \beta_{6}) q^{5} + ( - \beta_{6} - \beta_{2}) q^{7} + ( - \beta_{7} + 2 \beta_{6} + \cdots - \beta_1) q^{8} + (\beta_{11} + \beta_{9}) q^{10}+ \cdots + ( - 2 \beta_{11} + 2 \beta_{9} + \cdots + 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 14 q^{4} + 8 q^{10} - 8 q^{13} + 10 q^{16} + 12 q^{17} - 2 q^{22} + 4 q^{23} - 16 q^{25} - 10 q^{26} - 8 q^{29} - 16 q^{35} - 18 q^{38} + 16 q^{40} + 12 q^{43} + 32 q^{49} + 36 q^{52} + 20 q^{53}+ \cdots + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 19x^{10} + 133x^{8} + 423x^{6} + 601x^{4} + 312x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 205\nu^{9} + 2767\nu^{7} + 11877\nu^{5} + 15235\nu^{3} - 1482\nu ) / 2088 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} + 31\nu^{8} + 331\nu^{6} + 1437\nu^{4} + 2185\nu^{2} + 432 ) / 174 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{10} + 167\nu^{8} + 857\nu^{6} + 1887\nu^{4} + 2285\nu^{2} + 1446 ) / 348 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{10} + 167\nu^{8} + 857\nu^{6} + 1887\nu^{4} + 1937\nu^{2} + 402 ) / 348 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -67\nu^{11} - 1207\nu^{9} - 7909\nu^{7} - 23199\nu^{5} - 28945\nu^{3} - 9282\nu ) / 2088 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -15\nu^{11} - 291\nu^{9} - 2065\nu^{7} - 6475\nu^{5} - 8357\nu^{3} - 3058\nu ) / 232 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -55\nu^{11} - 835\nu^{9} - 3937\nu^{7} - 4911\nu^{5} + 5627\nu^{3} + 8430\nu ) / 1044 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 50 \nu^{11} - 81 \nu^{10} + 854 \nu^{9} - 1293 \nu^{8} + 5066 \nu^{7} - 6975 \nu^{6} + 12342 \nu^{5} + \cdots - 1062 ) / 696 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25\nu^{10} + 427\nu^{8} + 2533\nu^{6} + 6171\nu^{4} + 5557\nu^{2} + 1230 ) / 174 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 50 \nu^{11} - 81 \nu^{10} - 854 \nu^{9} - 1293 \nu^{8} - 5066 \nu^{7} - 6975 \nu^{6} - 12342 \nu^{5} + \cdots - 1062 ) / 696 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + 2\beta_{6} - \beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{9} + 9\beta_{5} - 6\beta_{4} - \beta_{3} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{11} + 2\beta_{9} + 3\beta_{8} + 10\beta_{7} - 16\beta_{6} + 8\beta_{2} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{11} - 12\beta_{10} - 11\beta_{9} - 67\beta_{5} + 38\beta_{4} + 14\beta_{3} - 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 26\beta_{11} - 26\beta_{9} - 36\beta_{8} - 81\beta_{7} + 105\beta_{6} - 60\beta_{2} - 195\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 96\beta_{11} + 112\beta_{10} + 96\beta_{9} + 477\beta_{5} - 255\beta_{4} - 133\beta_{3} + 340 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -245\beta_{11} + 245\beta_{9} + 325\beta_{8} + 610\beta_{7} - 659\beta_{6} + 447\beta_{2} + 1317\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -772\beta_{11} - 937\beta_{10} - 772\beta_{9} - 3358\beta_{5} + 1764\beta_{4} + 1100\beta_{3} - 1914 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2037\beta_{11} - 2037\beta_{9} - 2644\beta_{8} - 4458\beta_{7} + 4122\beta_{6} - 3308\beta_{2} - 9073\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1287\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(937\) \(1145\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
298.1
− 2.66546i
− 2.23753i
− 1.92437i
− 1.50594i
− 0.871160i
− 0.398488i
0.398488i
0.871160i
1.50594i
1.92437i
2.23753i
2.66546i
− 2.66546i 0 −5.10468 0.458686i 0 1.17072i 8.27540i 0 1.22261
298.2 − 2.23753i 0 −3.00653 − 1.83227i 0 − 2.75439i 2.25214i 0 −4.09976
298.3 − 1.92437i 0 −1.70321 3.67785i 0 3.13208i − 0.571131i 0 7.07756
298.4 − 1.50594i 0 −0.267864 − 2.54333i 0 − 0.340634i − 2.60850i 0 −3.83012
298.5 − 0.871160i 0 1.24108 3.32707i 0 − 2.06458i − 2.82350i 0 2.89841
298.6 − 0.398488i 0 1.84121 1.83517i 0 − 1.68947i − 1.53067i 0 0.731293
298.7 0.398488i 0 1.84121 − 1.83517i 0 1.68947i 1.53067i 0 0.731293
298.8 0.871160i 0 1.24108 − 3.32707i 0 2.06458i 2.82350i 0 2.89841
298.9 1.50594i 0 −0.267864 2.54333i 0 0.340634i 2.60850i 0 −3.83012
298.10 1.92437i 0 −1.70321 − 3.67785i 0 − 3.13208i 0.571131i 0 7.07756
298.11 2.23753i 0 −3.00653 1.83227i 0 2.75439i − 2.25214i 0 −4.09976
298.12 2.66546i 0 −5.10468 − 0.458686i 0 − 1.17072i − 8.27540i 0 1.22261
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 298.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1287.2.b.b 12
3.b odd 2 1 143.2.b.a ✓ 12
12.b even 2 1 2288.2.j.k 12
13.b even 2 1 inner 1287.2.b.b 12
39.d odd 2 1 143.2.b.a ✓ 12
39.f even 4 1 1859.2.a.j 6
39.f even 4 1 1859.2.a.n 6
156.h even 2 1 2288.2.j.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.b.a ✓ 12 3.b odd 2 1
143.2.b.a ✓ 12 39.d odd 2 1
1287.2.b.b 12 1.a even 1 1 trivial
1287.2.b.b 12 13.b even 2 1 inner
1859.2.a.j 6 39.f even 4 1
1859.2.a.n 6 39.f even 4 1
2288.2.j.k 12 12.b even 2 1
2288.2.j.k 12 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 19T_{2}^{10} + 133T_{2}^{8} + 423T_{2}^{6} + 601T_{2}^{4} + 312T_{2}^{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(1287, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 19 T^{10} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 38 T^{10} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( T^{12} + 26 T^{10} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{12} + 8 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} - 6 T^{5} - 34 T^{4} + \cdots - 96)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 118 T^{10} + \cdots + 16112196 \) Copy content Toggle raw display
$23$ \( (T^{6} - 2 T^{5} + \cdots + 17625)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 4 T^{5} - 42 T^{4} + \cdots - 96)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 162 T^{10} + \cdots + 1937664 \) Copy content Toggle raw display
$37$ \( T^{12} + 226 T^{10} + \cdots + 53934336 \) Copy content Toggle raw display
$41$ \( T^{12} + 142 T^{10} + \cdots + 5062500 \) Copy content Toggle raw display
$43$ \( (T^{6} - 6 T^{5} - 86 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 220 T^{10} + \cdots + 36864 \) Copy content Toggle raw display
$53$ \( (T^{6} - 10 T^{5} + \cdots + 6024)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 2477650176 \) Copy content Toggle raw display
$61$ \( (T^{6} + 6 T^{5} + \cdots + 27296)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 296941824 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 93401139456 \) Copy content Toggle raw display
$73$ \( T^{12} + 234 T^{10} + \cdots + 9216 \) Copy content Toggle raw display
$79$ \( (T^{6} + 24 T^{5} + \cdots - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 478 T^{10} + \cdots + 2762244 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 38598103296 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 152176896 \) Copy content Toggle raw display
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