Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(175,522)] 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("522.175"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-4,-1,-4,2] 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: no
Analytic conductor: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.2091141441.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{6} + 3x^{5} - 15x^{4} + 9x^{3} + 9x^{2} - 27x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} - \beta_1 q^{3} + \beta_{2} q^{4} + (\beta_{6} + \beta_{3} - \beta_1) q^{5} + ( - \beta_{6} + \beta_1) q^{6} + (2 \beta_{2} + 2) q^{7} + q^{8} + (\beta_{5} - \beta_{4}) q^{9}+ \cdots + (\beta_{7} + 3 \beta_{6} - \beta_{5} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - q^{3} - 4 q^{4} + 2 q^{5} - q^{6} + 8 q^{7} + 8 q^{8} - q^{9} - 4 q^{10} + 5 q^{11} + 2 q^{12} + 8 q^{14} - 26 q^{15} - 4 q^{16} - 14 q^{17} + 2 q^{18} + 12 q^{19} + 2 q^{20} + 2 q^{21}+ \cdots + 41 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + x^{6} + 3x^{5} - 15x^{4} + 9x^{3} + 9x^{2} - 27x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 2\nu^{6} - 2\nu^{5} - 3\nu^{4} + 3\nu^{3} - 18\nu^{2} + 9\nu + 54 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + \nu^{6} - \nu^{5} - 3\nu^{4} + 15\nu^{3} - 9\nu^{2} - 9\nu + 27 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 5\nu^{6} - 4\nu^{5} + 12\nu^{4} - 21\nu^{3} + 9\nu^{2} + 72\nu - 81 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} - 5\nu^{6} - 4\nu^{5} + 12\nu^{4} - 21\nu^{3} + 36\nu^{2} + 72\nu - 81 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 2\nu^{5} - 6\nu^{4} + 9\nu^{3} - 27\nu + 27 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 8\nu^{6} + 8\nu^{5} + 15\nu^{4} - 12\nu^{3} + 72\nu^{2} - 9\nu - 216 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 3\beta_{6} - \beta_{5} - 2\beta_{4} + 2\beta_{3} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{7} - 4\beta_{5} + \beta_{4} - 3\beta_{3} + 6\beta_{2} + 6\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} - 6\beta_{6} + 3\beta_{5} - 12\beta_{4} - 2\beta_{3} - 3\beta_{2} + 6\beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} + 3\beta_{6} + \beta_{5} + 2\beta_{4} - 2\beta_{3} + 36\beta_{2} - 9\beta _1 - 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/522\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(407\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
175.1
1.65525 + 0.510048i
0.335492 + 1.69925i
0.199732 − 1.72050i
−1.69047 + 0.377226i
1.65525 − 0.510048i
0.335492 − 1.69925i
0.199732 + 1.72050i
−1.69047 − 0.377226i
−0.500000 − 0.866025i −1.65525 − 0.510048i −0.500000 + 0.866025i 1.26934 − 2.19856i 0.385911 + 1.68851i 1.00000 + 1.73205i 1.00000 2.47970 + 1.68851i −2.53868
175.2 −0.500000 − 0.866025i −0.335492 − 1.69925i −0.500000 + 0.866025i 1.63934 − 2.83942i −1.30385 + 1.14017i 1.00000 + 1.73205i 1.00000 −2.77489 + 1.14017i −3.27868
175.3 −0.500000 − 0.866025i −0.199732 + 1.72050i −0.500000 + 0.866025i −1.39013 + 2.40777i 1.58986 − 0.687275i 1.00000 + 1.73205i 1.00000 −2.92021 − 0.687275i 2.78026
175.4 −0.500000 − 0.866025i 1.69047 − 0.377226i −0.500000 + 0.866025i −0.518550 + 0.898154i −1.17192 − 1.27538i 1.00000 + 1.73205i 1.00000 2.71540 − 1.27538i 1.03710
349.1 −0.500000 + 0.866025i −1.65525 + 0.510048i −0.500000 − 0.866025i 1.26934 + 2.19856i 0.385911 − 1.68851i 1.00000 − 1.73205i 1.00000 2.47970 − 1.68851i −2.53868
349.2 −0.500000 + 0.866025i −0.335492 + 1.69925i −0.500000 − 0.866025i 1.63934 + 2.83942i −1.30385 − 1.14017i 1.00000 − 1.73205i 1.00000 −2.77489 − 1.14017i −3.27868
349.3 −0.500000 + 0.866025i −0.199732 − 1.72050i −0.500000 − 0.866025i −1.39013 − 2.40777i 1.58986 + 0.687275i 1.00000 − 1.73205i 1.00000 −2.92021 + 0.687275i 2.78026
349.4 −0.500000 + 0.866025i 1.69047 + 0.377226i −0.500000 − 0.866025i −0.518550 − 0.898154i −1.17192 + 1.27538i 1.00000 − 1.73205i 1.00000 2.71540 + 1.27538i 1.03710
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 175.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 522.2.e.g ✓ 8
3.b odd 2 1 1566.2.e.g 8
9.c even 3 1 inner 522.2.e.g ✓ 8
9.c even 3 1 4698.2.a.bc 4
9.d odd 6 1 1566.2.e.g 8
9.d odd 6 1 4698.2.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
522.2.e.g ✓ 8 1.a even 1 1 trivial
522.2.e.g ✓ 8 9.c even 3 1 inner
1566.2.e.g 8 3.b odd 2 1
1566.2.e.g 8 9.d odd 6 1
4698.2.a.bb 4 9.d odd 6 1
4698.2.a.bc 4 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(522, [\chi])\):

\( T_{5}^{8} - 2T_{5}^{7} + 15T_{5}^{6} - 8T_{5}^{5} + 127T_{5}^{4} - 69T_{5}^{3} + 489T_{5}^{2} + 360T_{5} + 576 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + \cdots + 576 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} - 5 T^{7} + \cdots + 38809 \) Copy content Toggle raw display
$13$ \( T^{8} + 21 T^{6} + \cdots + 2916 \) Copy content Toggle raw display
$17$ \( (T^{4} + 7 T^{3} + \cdots - 648)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + \cdots + 103)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} + \cdots + 57600 \) Copy content Toggle raw display
$29$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + 2 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$37$ \( (T^{4} + 10 T^{3} + \cdots + 192)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 13 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$43$ \( T^{8} + 17 T^{7} + \cdots + 8649 \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} + \cdots + 10355524 \) Copy content Toggle raw display
$53$ \( (T^{4} + 12 T^{3} + \cdots - 1112)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 13 T^{7} + \cdots + 36864 \) Copy content Toggle raw display
$61$ \( T^{8} - 28 T^{7} + \cdots + 331776 \) Copy content Toggle raw display
$67$ \( T^{8} - 17 T^{7} + \cdots + 4562496 \) Copy content Toggle raw display
$71$ \( (T^{4} - 6 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - T^{3} - 146 T^{2} + \cdots + 4776)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 18 T^{7} + \cdots + 1600 \) Copy content Toggle raw display
$83$ \( (T^{2} - 12 T + 144)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 24 T^{3} + \cdots - 432)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 3 T^{7} + \cdots + 46656 \) Copy content Toggle raw display
show more
show less