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Properties

Label 7803.2.a.bs
Level $7803$
Weight $2$
Character orbit 7803.a
Self dual yes
Analytic conductor $62.307$
Analytic rank $1$
Dimension $10$
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] = [7803,2,Mod(1,7803)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7803, 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("7803.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 7803 = 3^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7803.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,12,2,0,0,0,0,0,-22,0,0,0,0,16,0,0,-10,-8,0,0,-28,0,8, 0,0,0,-6,0,0,0,0,0,0,0,0,0,0,0,-28,0,-8,-76,0,0,0,0,2,0,0,-4,0,0,42,64, 0,0,0,0,0,-52,0,-36,-32,0,18,0,0,-72,50,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(73)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(62.3072686972\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 16x^{8} + 86x^{6} - 170x^{4} + 73x^{2} - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 459)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{4} q^{5} + \beta_{6} q^{7} + ( - \beta_{7} + \beta_{6} + \beta_1) q^{8} + (\beta_{7} - \beta_{6} - \beta_{3}) q^{10} + ( - \beta_{5} - \beta_{4} - \beta_{2} - 2) q^{11}+ \cdots + (\beta_{8} - 5 \beta_{7} + \cdots - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 12 q^{4} + 2 q^{5} - 22 q^{11} + 16 q^{16} - 10 q^{19} - 8 q^{20} - 28 q^{23} + 8 q^{25} - 6 q^{29} - 28 q^{41} - 8 q^{43} - 76 q^{44} + 2 q^{49} - 4 q^{52} + 42 q^{55} + 64 q^{56} - 52 q^{62} - 36 q^{64}+ \cdots - 38 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 16x^{8} + 86x^{6} - 170x^{4} + 73x^{2} - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{7} - 11\nu^{5} + 29\nu^{3} - 2\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} - 11\nu^{4} + 30\nu^{2} - 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{8} - 12\nu^{6} + 39\nu^{4} - 26\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} - 14\nu^{7} + 62\nu^{5} - 90\nu^{3} + 11\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} - 14\nu^{7} + 62\nu^{5} - 92\nu^{3} + 21\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -2\nu^{7} + 23\nu^{5} - 67\nu^{3} + 21\nu \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( \nu^{8} - 13\nu^{6} + 51\nu^{4} - 62\nu^{2} + 14 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{5} + \beta_{4} + 6\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} - 9\beta_{7} + 9\beta_{6} + 2\beta_{3} + 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{9} - 11\beta_{5} + 12\beta_{4} + 36\beta_{2} + 93 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{8} - 70\beta_{7} + 70\beta_{6} + 23\beta_{3} + 165\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 93\beta_{9} - 92\beta_{5} + 105\beta_{4} + 224\beta_{2} + 565 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 92\beta_{8} - 512\beta_{7} + 514\beta_{6} + 198\beta_{3} + 1013\beta_1 \) 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
−2.60924
−2.09890
−2.06212
−0.599899
−0.417492
0.417492
0.599899
2.06212
2.09890
2.60924
−2.60924 0 4.80812 −2.94788 0 −3.30360 −7.32705 0 7.69173
1.2 −2.09890 0 2.40538 2.82282 0 2.66227 −0.850843 0 −5.92481
1.3 −2.06212 0 2.25232 1.44351 0 0.127581 −0.520322 0 −2.97669
1.4 −0.599899 0 −1.64012 −2.41834 0 4.19782 2.18371 0 1.45076
1.5 −0.417492 0 −1.82570 2.09990 0 0.600478 1.59720 0 −0.876691
1.6 0.417492 0 −1.82570 2.09990 0 −0.600478 −1.59720 0 0.876691
1.7 0.599899 0 −1.64012 −2.41834 0 −4.19782 −2.18371 0 −1.45076
1.8 2.06212 0 2.25232 1.44351 0 −0.127581 0.520322 0 2.97669
1.9 2.09890 0 2.40538 2.82282 0 −2.66227 0.850843 0 5.92481
1.10 2.60924 0 4.80812 −2.94788 0 3.30360 7.32705 0 −7.69173
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(17\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
51.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7803.2.a.bs 10
3.b odd 2 1 7803.2.a.br 10
17.b even 2 1 7803.2.a.br 10
17.d even 8 2 459.2.f.b ✓ 20
51.c odd 2 1 inner 7803.2.a.bs 10
51.g odd 8 2 459.2.f.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
459.2.f.b ✓ 20 17.d even 8 2
459.2.f.b ✓ 20 51.g odd 8 2
7803.2.a.br 10 3.b odd 2 1
7803.2.a.br 10 17.b even 2 1
7803.2.a.bs 10 1.a even 1 1 trivial
7803.2.a.bs 10 51.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7803))\):

\( T_{2}^{10} - 16T_{2}^{8} + 86T_{2}^{6} - 170T_{2}^{4} + 73T_{2}^{2} - 8 \) Copy content Toggle raw display
\( T_{5}^{5} - T_{5}^{4} - 14T_{5}^{3} + 16T_{5}^{2} + 47T_{5} - 61 \) Copy content Toggle raw display
\( T_{7}^{10} - 36T_{7}^{8} + 408T_{7}^{6} - 1512T_{7}^{4} + 516T_{7}^{2} - 8 \) Copy content Toggle raw display
\( T_{11}^{5} + 11T_{11}^{4} + 22T_{11}^{3} - 64T_{11}^{2} - 76T_{11} + 124 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 16 T^{8} + \cdots - 8 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T^{5} - T^{4} - 14 T^{3} + \cdots - 61)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} - 36 T^{8} + \cdots - 8 \) Copy content Toggle raw display
$11$ \( (T^{5} + 11 T^{4} + \cdots + 124)^{2} \) Copy content Toggle raw display
$13$ \( (T^{5} - 30 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} \) Copy content Toggle raw display
$19$ \( (T^{5} + 5 T^{4} - 8 T^{3} + \cdots + 15)^{2} \) Copy content Toggle raw display
$23$ \( (T^{5} + 14 T^{4} + \cdots - 842)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 3 T^{4} + \cdots + 2839)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 208 T^{8} + \cdots - 492032 \) Copy content Toggle raw display
$37$ \( T^{10} - 86 T^{8} + \cdots - 30752 \) Copy content Toggle raw display
$41$ \( (T^{5} + 14 T^{4} + \cdots + 3074)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 4 T^{4} + \cdots + 11262)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 168 T^{8} + \cdots - 109512 \) Copy content Toggle raw display
$53$ \( T^{10} - 346 T^{8} + \cdots - 51200 \) Copy content Toggle raw display
$59$ \( T^{10} - 242 T^{8} + \cdots - 9248 \) Copy content Toggle raw display
$61$ \( T^{10} - 264 T^{8} + \cdots - 42632 \) Copy content Toggle raw display
$67$ \( (T^{5} - 9 T^{4} + \cdots + 1769)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 25 T^{4} + \cdots + 55331)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 242088008 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 123496328 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 144636032 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 928977408 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 232330568 \) Copy content Toggle raw display
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