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Properties

Label 168.2.i.d
Level $168$
Weight $2$
Character orbit 168.i
Analytic conductor $1.341$
Analytic rank $0$
Dimension $8$
CM discriminant -56
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,2,Mod(125,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.125");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.i (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.10070523904.11
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 10x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - \beta_1 q^{3} - 2 q^{4} + ( - \beta_{7} + \beta_{6}) q^{5} - \beta_{7} q^{6} - \beta_{3} q^{7} + 2 \beta_{2} q^{8} + (\beta_{3} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - \beta_1 q^{3} - 2 q^{4} + ( - \beta_{7} + \beta_{6}) q^{5} - \beta_{7} q^{6} - \beta_{3} q^{7} + 2 \beta_{2} q^{8} + (\beta_{3} + \beta_{2}) q^{9} + (\beta_{5} + \beta_1) q^{10} + 2 \beta_1 q^{12} + ( - \beta_{7} - \beta_{6} - 2 \beta_1) q^{13} + \beta_{4} q^{14} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} - 1) q^{15} + 4 q^{16} + ( - \beta_{4} + 2) q^{18} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_1) q^{19} + (2 \beta_{7} - 2 \beta_{6}) q^{20} + (\beta_{6} + \beta_{5}) q^{21} + 2 \beta_{4} q^{23} + 2 \beta_{7} q^{24} + ( - 2 \beta_{3} - 5) q^{25} + ( - 2 \beta_{7} - \beta_{5} + 3 \beta_1) q^{26} + (\beta_{7} - \beta_{6} - \beta_{5}) q^{27} + 2 \beta_{3} q^{28} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} - 4) q^{30} - 4 \beta_{2} q^{32} + (3 \beta_{7} - \beta_{6} + \cdots - 3 \beta_1) q^{35}+ \cdots - 7 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} - 8 q^{15} + 32 q^{16} + 16 q^{18} - 40 q^{25} - 32 q^{30} + 40 q^{39} + 56 q^{49} - 8 q^{57} + 16 q^{60} - 56 q^{63} - 64 q^{64} - 32 q^{72} + 64 q^{78} + 40 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 10x^{4} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - \nu^{2} ) / 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 19\nu^{2} ) / 18 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 9\nu^{5} + 37\nu^{3} + 63\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{7} + 9\nu^{5} + 20\nu^{3} - 63\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + \nu^{3} ) / 18 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{4} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} + 4\beta_{6} - 2\beta_{5} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{3} + 19\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -19\beta_{7} + \beta_{6} + \beta_{5} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1
1.68014 + 0.420861i
0.420861 + 1.68014i
−0.420861 − 1.68014i
−1.68014 − 0.420861i
1.68014 − 0.420861i
0.420861 − 1.68014i
−0.420861 + 1.68014i
−1.68014 + 0.420861i
− 1.41421i −1.68014 − 0.420861i −2.00000 3.91044i −0.595188 + 2.37608i −2.64575 2.82843i 2.64575 + 1.41421i 5.53019
125.2 − 1.41421i −0.420861 − 1.68014i −2.00000 − 2.16991i −2.37608 + 0.595188i 2.64575 2.82843i −2.64575 + 1.41421i −3.06871
125.3 − 1.41421i 0.420861 + 1.68014i −2.00000 2.16991i 2.37608 − 0.595188i 2.64575 2.82843i −2.64575 + 1.41421i 3.06871
125.4 − 1.41421i 1.68014 + 0.420861i −2.00000 − 3.91044i 0.595188 − 2.37608i −2.64575 2.82843i 2.64575 + 1.41421i −5.53019
125.5 1.41421i −1.68014 + 0.420861i −2.00000 − 3.91044i −0.595188 − 2.37608i −2.64575 − 2.82843i 2.64575 − 1.41421i 5.53019
125.6 1.41421i −0.420861 + 1.68014i −2.00000 2.16991i −2.37608 − 0.595188i 2.64575 − 2.82843i −2.64575 − 1.41421i −3.06871
125.7 1.41421i 0.420861 − 1.68014i −2.00000 − 2.16991i 2.37608 + 0.595188i 2.64575 − 2.82843i −2.64575 − 1.41421i 3.06871
125.8 1.41421i 1.68014 − 0.420861i −2.00000 3.91044i 0.595188 + 2.37608i −2.64575 − 2.82843i 2.64575 − 1.41421i −5.53019
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
3.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
24.h odd 2 1 inner
168.i even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.2.i.d ✓ 8
3.b odd 2 1 inner 168.2.i.d ✓ 8
4.b odd 2 1 672.2.i.e 8
7.b odd 2 1 inner 168.2.i.d ✓ 8
8.b even 2 1 inner 168.2.i.d ✓ 8
8.d odd 2 1 672.2.i.e 8
12.b even 2 1 672.2.i.e 8
21.c even 2 1 inner 168.2.i.d ✓ 8
24.f even 2 1 672.2.i.e 8
24.h odd 2 1 inner 168.2.i.d ✓ 8
28.d even 2 1 672.2.i.e 8
56.e even 2 1 672.2.i.e 8
56.h odd 2 1 CM 168.2.i.d ✓ 8
84.h odd 2 1 672.2.i.e 8
168.e odd 2 1 672.2.i.e 8
168.i even 2 1 inner 168.2.i.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.i.d ✓ 8 1.a even 1 1 trivial
168.2.i.d ✓ 8 3.b odd 2 1 inner
168.2.i.d ✓ 8 7.b odd 2 1 inner
168.2.i.d ✓ 8 8.b even 2 1 inner
168.2.i.d ✓ 8 21.c even 2 1 inner
168.2.i.d ✓ 8 24.h odd 2 1 inner
168.2.i.d ✓ 8 56.h odd 2 1 CM
168.2.i.d ✓ 8 168.i even 2 1 inner
672.2.i.e 8 4.b odd 2 1
672.2.i.e 8 8.d odd 2 1
672.2.i.e 8 12.b even 2 1
672.2.i.e 8 24.f even 2 1
672.2.i.e 8 28.d even 2 1
672.2.i.e 8 56.e even 2 1
672.2.i.e 8 84.h odd 2 1
672.2.i.e 8 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(168, [\chi])\):

\( T_{5}^{4} + 20T_{5}^{2} + 72 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{4} - 52T_{13}^{2} + 648 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - 10T^{4} + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} + 20 T^{2} + 72)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - 52 T^{2} + 648)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} - 76 T^{2} + 72)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 56)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 236 T^{2} + 5832)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 244 T^{2} + 72)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 332 T^{2} + 648)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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