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Properties

Label 5819.2.a.j
Level $5819$
Weight $2$
Character orbit 5819.a
Self dual yes
Analytic conductor $46.465$
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] = [5819,2,Mod(1,5819)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5819, 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("5819.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 5819 = 11 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5819.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,-1,-1,13,-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: \(46.4649489362\)
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: \( x^{9} - x^{8} - 15x^{7} + 11x^{6} + 77x^{5} - 32x^{4} - 159x^{3} + 24x^{2} + 102x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{4} + \beta_{3} + 1) q^{4} + (\beta_{6} - 1) q^{5} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{6} + ( - \beta_{2} - 1) q^{7} + ( - \beta_{7} - \beta_{6} - \beta_1 - 1) q^{8}+ \cdots + ( - \beta_{6} + \beta_{3} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{2} - q^{3} + 13 q^{4} - 6 q^{5} - 3 q^{6} - 5 q^{7} - 9 q^{8} + 8 q^{9} - 12 q^{10} + 9 q^{11} - 23 q^{12} + 8 q^{13} - 4 q^{14} + 2 q^{15} + 9 q^{16} - 3 q^{17} + 30 q^{18} - 9 q^{19} - 10 q^{20}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 15x^{7} + 11x^{6} + 77x^{5} - 32x^{4} - 159x^{3} + 24x^{2} + 102x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{8} + 4\nu^{7} + 12\nu^{6} - 47\nu^{5} - 41\nu^{4} + 158\nu^{3} + 36\nu^{2} - 153\nu + 21 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{8} + 5\nu^{7} + 24\nu^{6} - 58\nu^{5} - 85\nu^{4} + 190\nu^{3} + 96\nu^{2} - 177\nu + 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{8} - 5\nu^{7} - 24\nu^{6} + 58\nu^{5} + 85\nu^{4} - 190\nu^{3} - 93\nu^{2} + 177\nu - 12 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{8} - 2\nu^{7} - 12\nu^{6} + 23\nu^{5} + 42\nu^{4} - 74\nu^{3} - 44\nu^{2} + 68\nu - 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 3\nu^{8} - 7\nu^{7} - 35\nu^{6} + 80\nu^{5} + 117\nu^{4} - 255\nu^{3} - 115\nu^{2} + 232\nu - 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -3\nu^{8} + 7\nu^{7} + 35\nu^{6} - 80\nu^{5} - 117\nu^{4} + 256\nu^{3} + 115\nu^{2} - 237\nu + 19 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -10\nu^{8} + 25\nu^{7} + 117\nu^{6} - 284\nu^{5} - 395\nu^{4} + 896\nu^{3} + 405\nu^{2} - 798\nu + 54 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{5} + 8\beta_{4} + 6\beta_{3} + \beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} + 9\beta_{7} + 10\beta_{6} - \beta_{5} - 2\beta_{3} + 29\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{8} + 2\beta_{6} - 12\beta_{5} + 55\beta_{4} + 36\beta_{3} + 10\beta_{2} + 7\beta _1 + 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{8} + 66\beta_{7} + 78\beta_{6} - 11\beta_{5} - 23\beta_{3} + \beta_{2} + 180\beta _1 + 62 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13\beta_{8} - \beta_{7} + 24\beta_{6} - 100\beta_{5} + 368\beta_{4} + 224\beta_{3} + 80\beta_{2} + 37\beta _1 + 553 \) 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
2.58142
2.50432
1.90019
0.990726
0.0874926
−1.36992
−1.40341
−1.69105
−2.59978
−2.58142 −0.803669 4.66374 2.63276 2.07461 −4.37209 −6.87623 −2.35412 −6.79627
1.2 −2.50432 −1.75949 4.27162 −1.08798 4.40632 3.30887 −5.68887 0.0957939 2.72465
1.3 −1.90019 2.37383 1.61073 −1.55069 −4.51073 −0.380765 0.739684 2.63505 2.94660
1.4 −0.990726 0.319959 −1.01846 0.208397 −0.316992 2.93163 2.99047 −2.89763 −0.206464
1.5 −0.0874926 −0.884099 −1.99235 −1.74557 0.0773522 −3.66413 0.349301 −2.21837 0.152724
1.6 1.36992 3.18553 −0.123327 −3.71549 4.36391 −2.67104 −2.90878 7.14761 −5.08992
1.7 1.40341 1.35940 −0.0304410 2.88440 1.90779 −1.15620 −2.84954 −1.15204 4.04800
1.8 1.69105 −1.60054 0.859651 0.388420 −2.70660 2.71500 −1.92839 −0.438262 0.656837
1.9 2.59978 −3.19092 4.75884 −4.01426 −8.29567 −1.71126 7.17235 7.18195 −10.4362
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( -1 \)
\(23\) \( -1 \)

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 5819.2.a.j ✓ 9
23.b odd 2 1 5819.2.a.k yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5819.2.a.j ✓ 9 1.a even 1 1 trivial
5819.2.a.k yes 9 23.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5819))\):

\( T_{2}^{9} + T_{2}^{8} - 15T_{2}^{7} - 11T_{2}^{6} + 77T_{2}^{5} + 32T_{2}^{4} - 159T_{2}^{3} - 24T_{2}^{2} + 102T_{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{9} + 6T_{5}^{8} - 8T_{5}^{7} - 92T_{5}^{6} - 54T_{5}^{5} + 347T_{5}^{4} + 462T_{5}^{3} - 27T_{5}^{2} - 147T_{5} + 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + T^{8} - 15 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{9} + T^{8} + \cdots + 21 \) Copy content Toggle raw display
$5$ \( T^{9} + 6 T^{8} + \cdots + 27 \) Copy content Toggle raw display
$7$ \( T^{9} + 5 T^{8} + \cdots - 849 \) Copy content Toggle raw display
$11$ \( (T - 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{9} - 8 T^{8} + \cdots - 1123 \) Copy content Toggle raw display
$17$ \( T^{9} + 3 T^{8} + \cdots + 3411 \) Copy content Toggle raw display
$19$ \( T^{9} + 9 T^{8} + \cdots - 158807 \) Copy content Toggle raw display
$23$ \( T^{9} \) Copy content Toggle raw display
$29$ \( T^{9} - T^{8} + \cdots + 10449 \) Copy content Toggle raw display
$31$ \( T^{9} + 26 T^{8} + \cdots - 234613 \) Copy content Toggle raw display
$37$ \( T^{9} + T^{8} + \cdots + 56349 \) Copy content Toggle raw display
$41$ \( T^{9} - 9 T^{8} + \cdots - 2354649 \) Copy content Toggle raw display
$43$ \( T^{9} + 9 T^{8} + \cdots + 9481 \) Copy content Toggle raw display
$47$ \( T^{9} - 8 T^{8} + \cdots + 20865105 \) Copy content Toggle raw display
$53$ \( T^{9} - 4 T^{8} + \cdots + 93441 \) Copy content Toggle raw display
$59$ \( T^{9} + 16 T^{8} + \cdots + 2106837 \) Copy content Toggle raw display
$61$ \( T^{9} + 50 T^{8} + \cdots - 2168067 \) Copy content Toggle raw display
$67$ \( T^{9} - 8 T^{8} + \cdots - 3687 \) Copy content Toggle raw display
$71$ \( T^{9} + 32 T^{8} + \cdots - 21037119 \) Copy content Toggle raw display
$73$ \( T^{9} - T^{8} + \cdots - 225427 \) Copy content Toggle raw display
$79$ \( T^{9} - 7 T^{8} + \cdots - 35148617 \) Copy content Toggle raw display
$83$ \( T^{9} - 22 T^{8} + \cdots - 11907663 \) Copy content Toggle raw display
$89$ \( T^{9} + 9 T^{8} + \cdots - 8413599 \) Copy content Toggle raw display
$97$ \( T^{9} + 10 T^{8} + \cdots + 156211 \) Copy content Toggle raw display
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