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Properties

Label 1767.2.a.c
Level $1767$
Weight $2$
Character orbit 1767.a
Self dual yes
Analytic conductor $14.110$
Analytic rank $1$
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] = [1767,2,Mod(1,1767)] 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("1767.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1767, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1767 = 3 \cdot 19 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1767.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9] 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: yes
Analytic conductor: \(14.1095660372\)
Analytic rank: \(1\)
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: \( x^{9} - 2x^{8} - 11x^{7} + 19x^{6} + 40x^{5} - 59x^{4} - 49x^{3} + 63x^{2} + 7x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{4} q^{5} + \beta_1 q^{6} + \beta_{6} q^{7} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{8} + q^{9} + ( - \beta_{8} - \beta_{5} - \beta_{3} + \cdots - 1) q^{10}+ \cdots + ( - \beta_{7} - \beta_{6} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 2 q^{2} - 9 q^{3} + 8 q^{4} + 2 q^{6} + 4 q^{7} - 9 q^{8} + 9 q^{9} - 5 q^{10} - 8 q^{12} - 5 q^{13} - 12 q^{14} + 2 q^{16} - 9 q^{17} - 2 q^{18} - 9 q^{19} + 3 q^{20} - 4 q^{21} - 10 q^{22} - 6 q^{23}+ \cdots - 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 11x^{7} + 19x^{6} + 40x^{5} - 59x^{4} - 49x^{3} + 63x^{2} + 7x - 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} + 14\nu^{6} + \nu^{5} - 58\nu^{4} - 6\nu^{3} + 69\nu^{2} + \nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - \nu^{7} - 12\nu^{6} + 7\nu^{5} + 47\nu^{4} - 14\nu^{3} - 59\nu^{2} + 12\nu + 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{8} + 14\nu^{6} + \nu^{5} - 58\nu^{4} - 8\nu^{3} + 71\nu^{2} + 9\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{8} - \nu^{7} - 11\nu^{6} + 5\nu^{5} + 39\nu^{4} - \nu^{3} - 42\nu^{2} - 7\nu + 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{8} - 3\nu^{7} - 20\nu^{6} + 20\nu^{5} + 63\nu^{4} - 34\nu^{3} - 58\nu^{2} + 15\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} + \nu^{7} - 16\nu^{6} - 9\nu^{5} + 71\nu^{4} + 24\nu^{3} - 91\nu^{2} - 10\nu + 7 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{5} + \beta_{4} + 3\beta_{3} + 7\beta_{2} + 2\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{8} + \beta_{7} - \beta_{6} - 8\beta_{5} + \beta_{4} + 11\beta_{3} + 12\beta_{2} + 20\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{8} + 2\beta_{7} - \beta_{6} - 11\beta_{5} + 8\beta_{4} + 33\beta_{3} + 50\beta_{2} + 23\beta _1 + 80 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 29\beta_{8} + 12\beta_{7} - 10\beta_{6} - 55\beta_{5} + 11\beta_{4} + 99\beta_{3} + 109\beta_{2} + 117\beta _1 + 133 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 112 \beta_{8} + 29 \beta_{7} - 15 \beta_{6} - 98 \beta_{5} + 55 \beta_{4} + 291 \beta_{3} + 369 \beta_{2} + \cdots + 521 \) 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
2.77650
2.15311
1.34102
1.26217
0.259984
−0.317738
−1.57702
−1.87501
−2.02301
−2.77650 −1.00000 5.70894 1.93824 2.77650 1.37307 −10.2979 1.00000 −5.38152
1.2 −2.15311 −1.00000 2.63589 −2.68542 2.15311 4.18572 −1.36914 1.00000 5.78201
1.3 −1.34102 −1.00000 −0.201679 0.773023 1.34102 2.16706 2.95248 1.00000 −1.03664
1.4 −1.26217 −1.00000 −0.406927 2.73544 1.26217 −2.89689 3.03795 1.00000 −3.45259
1.5 −0.259984 −1.00000 −1.93241 −3.05258 0.259984 −1.49564 1.02236 1.00000 0.793621
1.6 0.317738 −1.00000 −1.89904 0.937904 −0.317738 1.38645 −1.23887 1.00000 0.298008
1.7 1.57702 −1.00000 0.486994 1.69998 −1.57702 −0.734725 −2.38604 1.00000 2.68090
1.8 1.87501 −1.00000 1.51566 −0.428152 −1.87501 −2.52305 −0.908147 1.00000 −0.802789
1.9 2.02301 −1.00000 2.09257 −1.91843 −2.02301 2.53800 0.187276 1.00000 −3.88101
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(19\) \( +1 \)
\(31\) \( +1 \)

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 1767.2.a.c ✓ 9
3.b odd 2 1 5301.2.a.f 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1767.2.a.c ✓ 9 1.a even 1 1 trivial
5301.2.a.f 9 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} + 2T_{2}^{8} - 11T_{2}^{7} - 19T_{2}^{6} + 40T_{2}^{5} + 59T_{2}^{4} - 49T_{2}^{3} - 63T_{2}^{2} + 7T_{2} + 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1767))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 2 T^{8} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( (T + 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - 18 T^{7} + \cdots - 44 \) Copy content Toggle raw display
$7$ \( T^{9} - 4 T^{8} + \cdots - 352 \) Copy content Toggle raw display
$11$ \( T^{9} - 51 T^{7} + \cdots + 1322 \) Copy content Toggle raw display
$13$ \( T^{9} + 5 T^{8} + \cdots - 482 \) Copy content Toggle raw display
$17$ \( T^{9} + 9 T^{8} + \cdots - 112 \) Copy content Toggle raw display
$19$ \( (T + 1)^{9} \) Copy content Toggle raw display
$23$ \( T^{9} + 6 T^{8} + \cdots + 5354 \) Copy content Toggle raw display
$29$ \( T^{9} + 9 T^{8} + \cdots + 730 \) Copy content Toggle raw display
$31$ \( (T + 1)^{9} \) Copy content Toggle raw display
$37$ \( T^{9} + 11 T^{8} + \cdots - 324454 \) Copy content Toggle raw display
$41$ \( T^{9} + 21 T^{8} + \cdots - 603616 \) Copy content Toggle raw display
$43$ \( T^{9} - 12 T^{8} + \cdots - 50 \) Copy content Toggle raw display
$47$ \( T^{9} + 13 T^{8} + \cdots + 3563408 \) Copy content Toggle raw display
$53$ \( T^{9} + 31 T^{8} + \cdots - 7250326 \) Copy content Toggle raw display
$59$ \( T^{9} + 20 T^{8} + \cdots - 13107508 \) Copy content Toggle raw display
$61$ \( T^{9} - 7 T^{8} + \cdots + 47504 \) Copy content Toggle raw display
$67$ \( T^{9} + 4 T^{8} + \cdots - 853972 \) Copy content Toggle raw display
$71$ \( T^{9} + 13 T^{8} + \cdots - 13796 \) Copy content Toggle raw display
$73$ \( T^{9} + 5 T^{8} + \cdots - 154672 \) Copy content Toggle raw display
$79$ \( T^{9} + 6 T^{8} + \cdots - 1983616 \) Copy content Toggle raw display
$83$ \( T^{9} - 12 T^{8} + \cdots - 456106 \) Copy content Toggle raw display
$89$ \( T^{9} + 18 T^{8} + \cdots - 4887190 \) Copy content Toggle raw display
$97$ \( T^{9} + 44 T^{8} + \cdots + 118384 \) Copy content Toggle raw display
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