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Properties

Label 168.2.ba.a
Level $168$
Weight $2$
Character orbit 168.ba
Analytic conductor $1.341$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,2,Mod(5,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 2) q^{3} + 2 \beta_{2} q^{4} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{3} - 2 \beta_1) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{7} + 2 \beta_{3} q^{8} + (3 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 2) q^{3} + 2 \beta_{2} q^{4} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{3} - 2 \beta_1) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{7} + 2 \beta_{3} q^{8} + (3 \beta_{2} + 3) q^{9} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{10} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{11} + ( - 2 \beta_{2} + 2) q^{12} + (\beta_{3} + 4 \beta_{2} + 2) q^{14} + (3 \beta_{3} - 3) q^{15} + ( - 4 \beta_{2} - 4) q^{16} + (3 \beta_{3} + 3 \beta_1) q^{18} + (2 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 2) q^{20} + ( - \beta_{2} - 3 \beta_1 + 1) q^{21} + (3 \beta_{3} + 4) q^{22} + ( - 2 \beta_{3} + 2 \beta_1) q^{24} + ( - 6 \beta_{3} - 4 \beta_{2} - 6 \beta_1) q^{25} + ( - 6 \beta_{2} - 3) q^{27} + (4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{28} + (\beta_{3} - 9) q^{29} + ( - 6 \beta_{2} - 3 \beta_1 - 6) q^{30} + ( - \beta_{3} - 5 \beta_{2} + \cdots - 10) q^{31}+ \cdots + ( - 6 \beta_{3} - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} - 4 q^{4} + 6 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} - 4 q^{4} + 6 q^{5} - 2 q^{7} + 6 q^{9} + 12 q^{10} - 6 q^{11} + 12 q^{12} - 12 q^{15} - 8 q^{16} + 6 q^{21} + 16 q^{22} + 8 q^{25} - 4 q^{28} - 36 q^{29} - 12 q^{30} - 30 q^{31} + 18 q^{33} + 24 q^{35} - 24 q^{36} - 24 q^{40} + 12 q^{42} - 12 q^{44} + 18 q^{45} + 10 q^{49} + 48 q^{50} - 6 q^{53} - 24 q^{56} - 4 q^{58} + 18 q^{59} + 12 q^{60} - 12 q^{63} + 32 q^{64} - 24 q^{66} - 24 q^{75} + 18 q^{77} + 10 q^{79} - 24 q^{80} - 18 q^{81} + 54 q^{87} - 16 q^{88} + 30 q^{93} - 24 q^{98} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(1 + \beta_{2}\) \(-1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−0.707107 + 1.22474i
0.707107 − 1.22474i
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 − 1.73205i −0.621320 − 0.358719i − 2.44949i −2.62132 + 0.358719i 2.82843 1.50000 − 2.59808i 0.878680 − 0.507306i
5.2 0.707107 − 1.22474i −1.50000 + 0.866025i −1.00000 − 1.73205i 3.62132 + 2.09077i 2.44949i 1.62132 − 2.09077i −2.82843 1.50000 − 2.59808i 5.12132 − 2.95680i
101.1 −0.707107 − 1.22474i −1.50000 − 0.866025i −1.00000 + 1.73205i −0.621320 + 0.358719i 2.44949i −2.62132 − 0.358719i 2.82843 1.50000 + 2.59808i 0.878680 + 0.507306i
101.2 0.707107 + 1.22474i −1.50000 − 0.866025i −1.00000 + 1.73205i 3.62132 − 2.09077i − 2.44949i 1.62132 + 2.09077i −2.82843 1.50000 + 2.59808i 5.12132 + 2.95680i
\(n\): e.g. 2-40 or 990-1000
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 168.2.ba.a ✓ 4
3.b odd 2 1 168.2.ba.b yes 4
4.b odd 2 1 672.2.bi.b 4
7.d odd 6 1 inner 168.2.ba.a ✓ 4
8.b even 2 1 168.2.ba.b yes 4
8.d odd 2 1 672.2.bi.a 4
12.b even 2 1 672.2.bi.a 4
21.g even 6 1 168.2.ba.b yes 4
24.f even 2 1 672.2.bi.b 4
24.h odd 2 1 CM 168.2.ba.a ✓ 4
28.f even 6 1 672.2.bi.b 4
56.j odd 6 1 168.2.ba.b yes 4
56.m even 6 1 672.2.bi.a 4
84.j odd 6 1 672.2.bi.a 4
168.ba even 6 1 inner 168.2.ba.a ✓ 4
168.be odd 6 1 672.2.bi.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.ba.a ✓ 4 1.a even 1 1 trivial
168.2.ba.a ✓ 4 7.d odd 6 1 inner
168.2.ba.a ✓ 4 24.h odd 2 1 CM
168.2.ba.a ✓ 4 168.ba even 6 1 inner
168.2.ba.b yes 4 3.b odd 2 1
168.2.ba.b yes 4 8.b even 2 1
168.2.ba.b yes 4 21.g even 6 1
168.2.ba.b yes 4 56.j odd 6 1
672.2.bi.a 4 8.d odd 2 1
672.2.bi.a 4 12.b even 2 1
672.2.bi.a 4 56.m even 6 1
672.2.bi.a 4 84.j odd 6 1
672.2.bi.b 4 4.b odd 2 1
672.2.bi.b 4 24.f even 2 1
672.2.bi.b 4 28.f even 6 1
672.2.bi.b 4 168.be odd 6 1

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 \(S_{2}^{\mathrm{new}}(168, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2T^{2} + 4 \) 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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