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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 = [8,0,0,12,0,0,-6,0,0,0,0,0,2] 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: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.77658083584.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} + 41x^{4} - 39x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_1 + 1) q^{4} + ( - \beta_{4} - 1) q^{7} + \beta_{6} q^{8} + ( - \beta_{7} - \beta_{2}) q^{11} + (\beta_{4} - \beta_{3} + 1) q^{13} + (2 \beta_{7} - \beta_{5} + 2 \beta_{2}) q^{14}+ \cdots + ( - 6 \beta_{7} + 3 \beta_{6} + \cdots - 3 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{4} - 6 q^{7} + 2 q^{13} - 12 q^{16} - 18 q^{19} + 16 q^{22} - 10 q^{28} - 24 q^{31} - 22 q^{34} - 8 q^{37} + 32 q^{43} - 18 q^{46} - 10 q^{49} - 4 q^{52} - 16 q^{58} - 26 q^{61} - 34 q^{64}+ \cdots + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 12x^{6} + 41x^{4} - 39x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} - 6\nu^{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 9\nu^{3} + 16\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 9\nu^{2} - 10 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 10\nu^{4} + 22\nu^{2} - 4 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 11\nu^{5} + 34\nu^{3} - 32\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 11\nu^{5} - 31\nu^{3} + 20\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 12\nu^{5} + 40\nu^{3} - 30\nu ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 3\beta_{6} + \beta_{5} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{3} + 9\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{7} + 19\beta_{6} + 9\beta_{5} + 13\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{4} + 38\beta_{3} + 68\beta _1 + 108 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 58\beta_{7} + 123\beta_{6} + 68\beta_{5} + 91\beta_{2} \) 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
−1.92922
1.16541
0.341036
2.60838
−2.60838
−0.341036
−1.16541
1.92922
−2.34371 0 3.49296 0 0 2.02856 −3.49906 0 0
1.2 −2.18361 0 2.76814 0 0 −4.31293 −1.67731 0 0
1.3 −1.70140 0 0.894768 0 0 −0.475007 1.88044 0 0
1.4 −0.918767 0 −1.15587 0 0 −0.240624 2.89951 0 0
1.5 0.918767 0 −1.15587 0 0 −0.240624 −2.89951 0 0
1.6 1.70140 0 0.894768 0 0 −0.475007 −1.88044 0 0
1.7 2.18361 0 2.76814 0 0 −4.31293 1.67731 0 0
1.8 2.34371 0 3.49296 0 0 2.02856 3.49906 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.8
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.co ✓ 8
3.b odd 2 1 inner 8325.2.a.co ✓ 8
5.b even 2 1 8325.2.a.cp yes 8
15.d odd 2 1 8325.2.a.cp yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8325.2.a.co ✓ 8 1.a even 1 1 trivial
8325.2.a.co ✓ 8 3.b odd 2 1 inner
8325.2.a.cp yes 8 5.b even 2 1
8325.2.a.cp yes 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8325))\):

\( T_{2}^{8} - 14T_{2}^{6} + 67T_{2}^{4} - 123T_{2}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{4} + 3T_{7}^{3} - 7T_{7}^{2} - 6T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{8} - 23T_{11}^{6} + 120T_{11}^{4} - 139T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} - T_{13}^{3} - 29T_{13}^{2} + 40T_{13} + 107 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 14 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 3 T^{3} - 7 T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 23 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{3} - 29 T^{2} + \cdots + 107)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 65 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$19$ \( (T^{4} + 9 T^{3} + \cdots + 325)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 76 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{8} - 29 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$31$ \( (T + 3)^{8} \) Copy content Toggle raw display
$37$ \( (T + 1)^{8} \) Copy content Toggle raw display
$41$ \( T^{8} - 122 T^{6} + \cdots + 25600 \) Copy content Toggle raw display
$43$ \( (T^{4} - 16 T^{3} + \cdots - 6725)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 239 T^{6} + \cdots + 1373584 \) Copy content Toggle raw display
$53$ \( T^{8} - 307 T^{6} + \cdots + 2560000 \) Copy content Toggle raw display
$59$ \( T^{8} - 257 T^{6} + \cdots + 1607824 \) Copy content Toggle raw display
$61$ \( (T^{4} + 13 T^{3} + \cdots + 1775)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 2 T^{3} + \cdots + 137)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 493 T^{6} + \cdots + 2755600 \) Copy content Toggle raw display
$73$ \( (T^{4} + 3 T^{3} + \cdots + 2390)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 16 T^{3} + \cdots - 6620)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 577 T^{6} + \cdots + 6760000 \) Copy content Toggle raw display
$89$ \( T^{8} - 433 T^{6} + \cdots + 18318400 \) Copy content Toggle raw display
$97$ \( (T^{4} - T^{3} - 98 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
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