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Properties

Label 2883.2.a.n
Level $2883$
Weight $2$
Character orbit 2883.a
Self dual yes
Analytic conductor $23.021$
Analytic rank $1$
Dimension $8$
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] = [2883,2,Mod(1,2883)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2883, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("2883.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2883 = 3 \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2883.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-5,8,5,-6] 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: \(23.0208709027\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.1697203125.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 5x^{6} + 12x^{5} + 9x^{4} - 12x^{3} - 5x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 93)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + q^{3} + (\beta_{3} + \beta_{2} + 1) q^{4} + ( - \beta_{7} + \beta_{4} - 1) q^{5} + (\beta_1 - 1) q^{6} + (\beta_{6} - \beta_{3} - \beta_1) q^{7} + (\beta_{6} + \beta_{4} - \beta_{2}) q^{8}+ \cdots + ( - \beta_{6} - \beta_{4} - \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{2} + 8 q^{3} + 5 q^{4} - 6 q^{5} - 5 q^{6} - 6 q^{7} + 8 q^{9} + q^{10} + 5 q^{12} - 12 q^{13} - 3 q^{14} - 6 q^{15} + 3 q^{16} - 2 q^{17} - 5 q^{18} - 8 q^{19} - 5 q^{20} - 6 q^{21} - 4 q^{22}+ \cdots + 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 5x^{6} + 12x^{5} + 9x^{4} - 12x^{3} - 5x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{6} + 3\nu^{5} + 4\nu^{4} - 9\nu^{3} - 5\nu^{2} + 3\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{6} - 3\nu^{5} - 4\nu^{4} + 9\nu^{3} + 6\nu^{2} - 5\nu - 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{7} + 2\nu^{6} + 8\nu^{5} - 8\nu^{4} - 17\nu^{3} + 4\nu^{2} + 6\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{7} - 2\nu^{6} - 7\nu^{5} + 4\nu^{4} + 17\nu^{3} + 5\nu^{2} - 9\nu - 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - 3\nu^{6} - 5\nu^{5} + 12\nu^{4} + 9\nu^{3} - 12\nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\nu^{7} - 6\nu^{6} - 9\nu^{5} + 21\nu^{4} + 14\nu^{3} - 15\nu^{2} - 4\nu + 1 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} + 3\beta_{3} + 2\beta_{2} + 7\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 2\beta_{6} - \beta_{5} + 3\beta_{4} + 12\beta_{3} + 8\beta_{2} + 21\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} + 8\beta_{6} - 3\beta_{5} + 13\beta_{4} + 39\beta_{3} + 23\beta_{2} + 69\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16\beta_{7} + 23\beta_{6} - 13\beta_{5} + 42\beta_{4} + 133\beta_{3} + 77\beta_{2} + 221\beta _1 + 123 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 56\beta_{7} + 77\beta_{6} - 42\beta_{5} + 146\beta_{4} + 435\beta_{3} + 244\beta_{2} + 721\beta _1 + 388 \) 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
−1.49568
−1.18333
−0.563998
−0.306487
0.668591
0.845071
1.77306
3.26278
−2.49568 1.00000 4.22843 −2.30796 −2.49568 −2.75147 −5.56145 1.00000 5.75994
1.2 −2.18333 1.00000 2.76694 0.544022 −2.18333 0.576411 −1.67449 1.00000 −1.18778
1.3 −1.56400 1.00000 0.446091 1.37281 −1.56400 −0.139513 2.43031 1.00000 −2.14707
1.4 −1.30649 1.00000 −0.293092 −1.68211 −1.30649 2.45220 2.99589 1.00000 2.19765
1.5 −0.331409 1.00000 −1.89017 1.92600 −0.331409 −1.41388 1.28924 1.00000 −0.638292
1.6 −0.154929 1.00000 −1.97600 −3.16206 −0.154929 −2.70562 0.615997 1.00000 0.489894
1.7 0.773055 1.00000 −1.40239 −1.75478 0.773055 1.30487 −2.63023 1.00000 −1.35654
1.8 2.26278 1.00000 3.12018 −0.935927 2.26278 −3.32300 2.53473 1.00000 −2.11780
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(31\) \( -1 \)

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 2883.2.a.n 8
3.b odd 2 1 8649.2.a.bi 8
31.b odd 2 1 2883.2.a.m 8
31.h odd 30 2 93.2.m.a ✓ 16
93.c even 2 1 8649.2.a.bj 8
93.p even 30 2 279.2.y.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.2.m.a ✓ 16 31.h odd 30 2
279.2.y.b 16 93.p even 30 2
2883.2.a.m 8 31.b odd 2 1
2883.2.a.n 8 1.a even 1 1 trivial
8649.2.a.bi 8 3.b odd 2 1
8649.2.a.bj 8 93.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2883))\):

\( T_{2}^{8} + 5T_{2}^{7} + 2T_{2}^{6} - 25T_{2}^{5} - 41T_{2}^{4} - 5T_{2}^{3} + 23T_{2}^{2} + 10T_{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{8} - 38T_{11}^{6} - 15T_{11}^{5} + 354T_{11}^{4} + 75T_{11}^{3} - 752T_{11}^{2} + 75T_{11} + 211 \) Copy content Toggle raw display
\( T_{13}^{8} + 12T_{13}^{7} + 36T_{13}^{6} - 39T_{13}^{5} - 246T_{13}^{4} - 27T_{13}^{3} + 324T_{13}^{2} - 81T_{13} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{7} + \cdots - 29 \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + \cdots - 9 \) Copy content Toggle raw display
$11$ \( T^{8} - 38 T^{6} + \cdots + 211 \) Copy content Toggle raw display
$13$ \( T^{8} + 12 T^{7} + \cdots - 9 \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} + \cdots - 659 \) Copy content Toggle raw display
$19$ \( T^{8} + 8 T^{7} + \cdots + 24721 \) Copy content Toggle raw display
$23$ \( T^{8} + 4 T^{7} + \cdots - 16619 \) Copy content Toggle raw display
$29$ \( T^{8} - 6 T^{7} + \cdots + 51991 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 8 T^{7} + \cdots + 388051 \) Copy content Toggle raw display
$41$ \( T^{8} + 20 T^{7} + \cdots + 1655401 \) Copy content Toggle raw display
$43$ \( T^{8} + 14 T^{7} + \cdots + 9811 \) Copy content Toggle raw display
$47$ \( T^{8} - 6 T^{7} + \cdots + 523801 \) Copy content Toggle raw display
$53$ \( T^{8} - 30 T^{7} + \cdots + 665811 \) Copy content Toggle raw display
$59$ \( T^{8} + 20 T^{7} + \cdots - 12597749 \) Copy content Toggle raw display
$61$ \( T^{8} + 13 T^{7} + \cdots - 452849 \) Copy content Toggle raw display
$67$ \( T^{8} - 32 T^{7} + \cdots - 1287089 \) Copy content Toggle raw display
$71$ \( T^{8} + 17 T^{7} + \cdots - 19949 \) Copy content Toggle raw display
$73$ \( T^{8} - 12 T^{7} + \cdots + 48344841 \) Copy content Toggle raw display
$79$ \( T^{8} + 30 T^{7} + \cdots + 8491221 \) Copy content Toggle raw display
$83$ \( T^{8} + 12 T^{7} + \cdots + 6271 \) Copy content Toggle raw display
$89$ \( T^{8} + 6 T^{7} + \cdots - 117659 \) Copy content Toggle raw display
$97$ \( T^{8} + 39 T^{7} + \cdots - 2834199 \) Copy content Toggle raw display
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