Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,6,0,0,-2,0,0,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.4754596827\)
Analytic rank: \(0\)
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: \( x^{10} - 13x^{8} + 53x^{6} - 84x^{4} + 45x^{2} - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{7} - \beta_{2} - 1) q^{7} + (\beta_{9} + \beta_{8} + \beta_1) q^{8} + ( - \beta_{4} - \beta_1) q^{11} + ( - \beta_{3} + 1) q^{13} + ( - \beta_{9} - 2 \beta_{8} + \cdots - 2 \beta_1) q^{14}+ \cdots + (2 \beta_{9} - 3 \beta_{4} + 12 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 6 q^{4} - 2 q^{7} + 14 q^{13} + 10 q^{16} + 28 q^{19} - 28 q^{22} - 38 q^{28} + 28 q^{31} - 42 q^{34} + 10 q^{37} + 38 q^{43} + 4 q^{46} + 64 q^{49} - 4 q^{52} + 36 q^{58} + 30 q^{61} + 48 q^{64}+ \cdots + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 13x^{8} + 53x^{6} - 84x^{4} + 45x^{2} - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{6} - 10\nu^{4} + 22\nu^{2} - 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{7} - 10\nu^{5} + 22\nu^{3} - 8\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{7} + 11\nu^{5} - 31\nu^{3} + 22\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{8} + 11\nu^{6} - 31\nu^{4} + 22\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{8} - 11\nu^{6} + 32\nu^{4} - 30\nu^{2} + 7 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{9} - 12\nu^{7} + 42\nu^{5} - 52\nu^{3} + 16\nu \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -\nu^{9} + 12\nu^{7} - 42\nu^{5} + 53\nu^{3} - 21\nu \) 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
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} + 8\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{9} + 9\beta_{8} + \beta_{5} + \beta_{4} + 31\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{7} + 10\beta_{6} + \beta_{3} + 58\beta_{2} + 102 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 68\beta_{9} + 68\beta_{8} + 10\beta_{5} + 11\beta_{4} + 208\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 79\beta_{7} + 78\beta_{6} + 11\beta_{3} + 412\beta_{2} + 693 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 490\beta_{9} + 491\beta_{8} + 78\beta_{5} + 90\beta_{4} + 1438\beta_1 \) 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
−2.65283
−1.67209
−1.49190
−0.911555
−0.331543
0.331543
0.911555
1.49190
1.67209
2.65283
−2.65283 0 5.03751 0 0 −4.66099 −8.05799 0 0
1.2 −1.67209 0 0.795879 0 0 5.24127 2.01340 0 0
1.3 −1.49190 0 0.225760 0 0 −2.23264 2.64699 0 0
1.4 −0.911555 0 −1.16907 0 0 2.83684 2.88878 0 0
1.5 −0.331543 0 −1.89008 0 0 −2.18448 1.28973 0 0
1.6 0.331543 0 −1.89008 0 0 −2.18448 −1.28973 0 0
1.7 0.911555 0 −1.16907 0 0 2.83684 −2.88878 0 0
1.8 1.49190 0 0.225760 0 0 −2.23264 −2.64699 0 0
1.9 1.67209 0 0.795879 0 0 5.24127 −2.01340 0 0
1.10 2.65283 0 5.03751 0 0 −4.66099 8.05799 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)
\(37\) \( -1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8325.2.a.cs ✓ 10
3.b odd 2 1 inner 8325.2.a.cs ✓ 10
5.b even 2 1 8325.2.a.ct yes 10
15.d odd 2 1 8325.2.a.ct yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8325.2.a.cs ✓ 10 1.a even 1 1 trivial
8325.2.a.cs ✓ 10 3.b odd 2 1 inner
8325.2.a.ct yes 10 5.b even 2 1
8325.2.a.ct yes 10 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8325))\):

\( T_{2}^{10} - 13T_{2}^{8} + 53T_{2}^{6} - 84T_{2}^{4} + 45T_{2}^{2} - 4 \) Copy content Toggle raw display
\( T_{7}^{5} + T_{7}^{4} - 33T_{7}^{3} - 48T_{7}^{2} + 195T_{7} + 338 \) Copy content Toggle raw display
\( T_{11}^{10} - 67T_{11}^{8} + 1604T_{11}^{6} - 16723T_{11}^{4} + 74236T_{11}^{2} - 102400 \) Copy content Toggle raw display
\( T_{13}^{5} - 7T_{13}^{4} - 11T_{13}^{3} + 82T_{13}^{2} + 49T_{13} - 98 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 13 T^{8} + \cdots - 4 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( (T^{5} + T^{4} - 33 T^{3} + \cdots + 338)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} - 67 T^{8} + \cdots - 102400 \) Copy content Toggle raw display
$13$ \( (T^{5} - 7 T^{4} - 11 T^{3} + \cdots - 98)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} - 133 T^{8} + \cdots - 23104 \) Copy content Toggle raw display
$19$ \( (T^{5} - 14 T^{4} + \cdots + 1315)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} - 143 T^{8} + \cdots - 743044 \) Copy content Toggle raw display
$29$ \( T^{10} - 161 T^{8} + \cdots - 6400 \) Copy content Toggle raw display
$31$ \( (T^{5} - 14 T^{4} + \cdots + 4064)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1)^{10} \) Copy content Toggle raw display
$41$ \( T^{10} - 293 T^{8} + \cdots - 7695076 \) Copy content Toggle raw display
$43$ \( (T^{5} - 19 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 283 T^{8} + \cdots - 1784896 \) Copy content Toggle raw display
$53$ \( T^{10} - 178 T^{8} + \cdots - 33547264 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 2872316836 \) Copy content Toggle raw display
$61$ \( (T^{5} - 15 T^{4} + \cdots + 856)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} - 8 T^{4} + \cdots - 79138)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} - 241 T^{8} + \cdots - 16384 \) Copy content Toggle raw display
$73$ \( (T^{5} - 14 T^{4} + \cdots - 14138)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 15 T^{4} + \cdots - 18016)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} - 229 T^{8} + \cdots - 10240000 \) Copy content Toggle raw display
$89$ \( T^{10} - 169 T^{8} + \cdots - 5550736 \) Copy content Toggle raw display
$97$ \( (T^{5} - 3 T^{4} + \cdots + 9088)^{2} \) Copy content Toggle raw display
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