↑

Properties

Label 532.2.j.b
Level $532$
Weight $2$
Character orbit 532.j
Analytic conductor $4.248$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [532,2,Mod(197,532)] 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("532.197"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(532, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 532 = 2^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 532.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.24804138753\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.102293147889.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 9x^{6} + 31x^{4} - 9x^{3} + 45x^{2} + 20x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1) q^{3} + ( - \beta_{6} - \beta_1) q^{5} + q^{7} + ( - \beta_{6} + \beta_{4} + \beta_{3} + \cdots - 1) q^{9} + (\beta_{7} + \beta_{2}) q^{11} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + 1) q^{13}+ \cdots + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 2 q^{5} + 8 q^{7} - 2 q^{9} - 2 q^{11} + 3 q^{13} - 9 q^{15} - 2 q^{17} - 3 q^{19} + 2 q^{21} + 3 q^{23} - 10 q^{25} - 10 q^{27} + 17 q^{29} - 50 q^{31} + 2 q^{33} - 2 q^{35} - 22 q^{37}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 9x^{6} + 31x^{4} - 9x^{3} + 45x^{2} + 20x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -110\nu^{7} + 389\nu^{6} - 682\nu^{5} - 1166\nu^{4} + 2809\nu^{3} - 2420\nu^{2} - 1232\nu - 24659 ) / 11069 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -155\nu^{7} + 45\nu^{6} - 961\nu^{5} - 1643\nu^{4} - 7614\nu^{3} - 3410\nu^{2} - 1736\nu - 6068 ) / 11069 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1517\nu^{7} + 3654\nu^{6} - 13833\nu^{5} + 3844\nu^{4} - 40455\nu^{3} + 44109\nu^{2} - 54625\nu + 20880 ) / 44276 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2815 \nu^{7} - 10458 \nu^{6} + 39591 \nu^{5} - 47644 \nu^{4} + 115785 \nu^{3} - 126243 \nu^{2} + \cdots - 59760 ) / 44276 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3491 \nu^{7} + 9226 \nu^{6} - 34927 \nu^{5} + 22768 \nu^{4} - 102145 \nu^{3} + 111371 \nu^{2} + \cdots + 52720 ) / 44276 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1150\nu^{7} - 1048\nu^{6} + 7130\nu^{5} + 12190\nu^{4} + 28997\nu^{3} + 25300\nu^{2} + 12880\nu + 52519 ) / 11069 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + 3\beta_{4} - \beta_{3} - \beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - 6\beta_{3} - 2\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{6} + 2\beta_{5} - 17\beta_{4} - 12\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{7} + 23\beta_{6} + 9\beta_{5} - 34\beta_{4} + 45\beta_{3} + 23\beta_{2} - 45\beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 23\beta_{7} + 116\beta_{3} + 77\beta_{2} + 126 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -216\beta_{6} - 77\beta_{5} + 325\beta_{4} + 373\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/532\mathbb{Z}\right)^\times\).

\(n\) \(267\) \(381\) \(477\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
1.48208 + 2.56704i
0.676992 + 1.17258i
−0.285090 − 0.493791i
−0.873982 − 1.51378i
1.48208 − 2.56704i
0.676992 − 1.17258i
−0.285090 + 0.493791i
−0.873982 + 1.51378i
0 −0.982080 − 1.70101i 0 −0.0710378 − 0.123041i 0 1.00000 0 −0.428962 + 0.742984i 0
197.2 0 −0.176992 − 0.306559i 0 −1.93735 − 3.35558i 0 1.00000 0 1.43735 − 2.48956i 0
197.3 0 0.785090 + 1.35982i 0 −0.767267 − 1.32894i 0 1.00000 0 0.267267 − 0.462919i 0
197.4 0 1.37398 + 2.37981i 0 1.77565 + 3.07552i 0 1.00000 0 −2.27565 + 3.94154i 0
505.1 0 −0.982080 + 1.70101i 0 −0.0710378 + 0.123041i 0 1.00000 0 −0.428962 − 0.742984i 0
505.2 0 −0.176992 + 0.306559i 0 −1.93735 + 3.35558i 0 1.00000 0 1.43735 + 2.48956i 0
505.3 0 0.785090 − 1.35982i 0 −0.767267 + 1.32894i 0 1.00000 0 0.267267 + 0.462919i 0
505.4 0 1.37398 − 2.37981i 0 1.77565 − 3.07552i 0 1.00000 0 −2.27565 − 3.94154i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 532.2.j.b ✓ 8
3.b odd 2 1 4788.2.w.f 8
19.c even 3 1 inner 532.2.j.b ✓ 8
57.h odd 6 1 4788.2.w.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.2.j.b ✓ 8 1.a even 1 1 trivial
532.2.j.b ✓ 8 19.c even 3 1 inner
4788.2.w.f 8 3.b odd 2 1
4788.2.w.f 8 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 2T_{3}^{7} + 9T_{3}^{6} - 4T_{3}^{5} + 36T_{3}^{4} - 23T_{3}^{3} + 64T_{3}^{2} + 21T_{3} + 9 \) acting on \(S_{2}^{\mathrm{new}}(532, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( (T - 1)^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + T^{3} - 24 T^{2} + \cdots - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 3 T^{7} + \cdots + 6241 \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} + \cdots + 110889 \) Copy content Toggle raw display
$19$ \( T^{8} + 3 T^{7} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} - 3 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$29$ \( T^{8} - 17 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$31$ \( (T^{4} + 25 T^{3} + \cdots - 332)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 11 T^{3} + \cdots - 18)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$43$ \( T^{8} - 7 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{8} + 11 T^{7} + \cdots + 46656 \) Copy content Toggle raw display
$53$ \( T^{8} - 18 T^{7} + \cdots + 145161 \) Copy content Toggle raw display
$59$ \( T^{8} + 9 T^{7} + \cdots + 62001 \) Copy content Toggle raw display
$61$ \( T^{8} + 82 T^{6} + \cdots + 39601 \) Copy content Toggle raw display
$67$ \( T^{8} - 6 T^{7} + \cdots + 8982009 \) Copy content Toggle raw display
$71$ \( T^{8} - 15 T^{7} + \cdots + 859329 \) Copy content Toggle raw display
$73$ \( T^{8} + 8 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$79$ \( T^{8} - 10 T^{7} + \cdots + 17447329 \) Copy content Toggle raw display
$83$ \( (T^{4} - 14 T^{3} + \cdots - 24)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 17 T^{7} + \cdots + 46656 \) Copy content Toggle raw display
$97$ \( T^{8} + 9 T^{7} + \cdots + 145161 \) Copy content Toggle raw display
show more
show less