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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [672,2,Mod(17,672)] 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("672.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bi (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{5} + (2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} - 3 \beta_{2} q^{9} + (3 \beta_{2} - 2 \beta_1 + 3) q^{11} + ( - 3 \beta_{3} + 3) q^{15}+ \cdots + (6 \beta_{3} + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 6 q^{5} + 2 q^{7} + 6 q^{9} + 6 q^{11} + 12 q^{15} + 6 q^{21} + 8 q^{25} - 36 q^{29} + 30 q^{31} + 18 q^{33} - 24 q^{35} + 18 q^{45} + 10 q^{49} - 6 q^{53} - 18 q^{59} + 12 q^{63} + 24 q^{75}+ \cdots + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.707107 + 1.22474i
0.707107 − 1.22474i
−0.707107 − 1.22474i
0.707107 + 1.22474i
0 1.50000 + 0.866025i 0 −0.621320 + 0.358719i 0 2.62132 + 0.358719i 0 1.50000 + 2.59808i 0
17.2 0 1.50000 + 0.866025i 0 3.62132 − 2.09077i 0 −1.62132 − 2.09077i 0 1.50000 + 2.59808i 0
593.1 0 1.50000 − 0.866025i 0 −0.621320 − 0.358719i 0 2.62132 − 0.358719i 0 1.50000 − 2.59808i 0
593.2 0 1.50000 − 0.866025i 0 3.62132 + 2.09077i 0 −1.62132 + 2.09077i 0 1.50000 − 2.59808i 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
7.d odd 6 1 inner
168.ba even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.bi.b 4
3.b odd 2 1 672.2.bi.a 4
4.b odd 2 1 168.2.ba.a ✓ 4
7.d odd 6 1 inner 672.2.bi.b 4
8.b even 2 1 672.2.bi.a 4
8.d odd 2 1 168.2.ba.b yes 4
12.b even 2 1 168.2.ba.b yes 4
21.g even 6 1 672.2.bi.a 4
24.f even 2 1 168.2.ba.a ✓ 4
24.h odd 2 1 CM 672.2.bi.b 4
28.f even 6 1 168.2.ba.a ✓ 4
56.j odd 6 1 672.2.bi.a 4
56.m even 6 1 168.2.ba.b yes 4
84.j odd 6 1 168.2.ba.b yes 4
168.ba even 6 1 inner 672.2.bi.b 4
168.be odd 6 1 168.2.ba.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.ba.a ✓ 4 4.b odd 2 1
168.2.ba.a ✓ 4 24.f even 2 1
168.2.ba.a ✓ 4 28.f even 6 1
168.2.ba.a ✓ 4 168.be odd 6 1
168.2.ba.b yes 4 8.d odd 2 1
168.2.ba.b yes 4 12.b even 2 1
168.2.ba.b yes 4 56.m even 6 1
168.2.ba.b yes 4 84.j odd 6 1
672.2.bi.a 4 3.b odd 2 1
672.2.bi.a 4 8.b even 2 1
672.2.bi.a 4 21.g even 6 1
672.2.bi.a 4 56.j odd 6 1
672.2.bi.b 4 1.a even 1 1 trivial
672.2.bi.b 4 7.d odd 6 1 inner
672.2.bi.b 4 24.h odd 2 1 CM
672.2.bi.b 4 168.ba even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 6T_{5}^{3} + 9T_{5}^{2} + 18T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 18 T + 79)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 30 T^{3} + \cdots + 4761 \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$59$ \( T^{4} + 18 T^{3} + \cdots + 4761 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 96T^{2} + 9216 \) Copy content Toggle raw display
$79$ \( T^{4} + 10 T^{3} + \cdots + 18769 \) Copy content Toggle raw display
$83$ \( T^{4} + 198T^{2} + 2601 \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 198T^{2} + 8649 \) Copy content Toggle raw display
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