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Properties

Label 714.2.t.f
Level $714$
Weight $2$
Character orbit 714.t
Analytic conductor $5.701$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [714,2,Mod(67,714)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(714, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("714.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.t (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: \(5.70131870432\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

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

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} + (\beta_{3} - \beta_1) q^{3} - \beta_{2} q^{4} + (\beta_{6} + \beta_1) q^{5} + \beta_{3} q^{6} + (2 \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + ( - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} + (\beta_{3} - \beta_1) q^{3} - \beta_{2} q^{4} + (\beta_{6} + \beta_1) q^{5} + \beta_{3} q^{6} + (2 \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} + 4 q^{9} - 8 q^{15} - 4 q^{16} + 4 q^{17} - 4 q^{18} - 12 q^{19} - 4 q^{21} - 8 q^{25} - 4 q^{30} + 4 q^{32} + 8 q^{33} + 8 q^{34} + 8 q^{35} - 8 q^{36} + 12 q^{38} + 4 q^{42} - 8 q^{43} + 8 q^{47} - 20 q^{49} - 16 q^{50} + 16 q^{51} - 32 q^{53} + 16 q^{55} + 4 q^{59} + 4 q^{60} + 8 q^{64} - 8 q^{66} - 20 q^{67} + 4 q^{68} - 40 q^{69} + 28 q^{70} - 4 q^{72} + 24 q^{76} + 40 q^{77} - 4 q^{81} + 32 q^{83} + 8 q^{84} - 32 q^{85} - 4 q^{86} + 16 q^{87} + 28 q^{89} + 8 q^{93} - 8 q^{94} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/714\mathbb{Z}\right)^\times\).

\(n\) \(239\) \(409\) \(547\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.965926 + 0.258819i
0.965926 − 0.258819i
−0.258819 − 0.965926i
0.258819 + 0.965926i
−0.965926 − 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 − 0.965926i
0.500000 + 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i −0.358719 + 0.207107i − 1.00000i −0.358719 + 2.62132i −1.00000 0.500000 + 0.866025i −0.358719 − 0.207107i
67.2 0.500000 + 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i 2.09077 − 1.20711i − 1.00000i 2.09077 − 1.62132i −1.00000 0.500000 + 0.866025i 2.09077 + 1.20711i
67.3 0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i −2.09077 + 1.20711i 1.00000i −2.09077 + 1.62132i −1.00000 0.500000 + 0.866025i −2.09077 − 1.20711i
67.4 0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 0.358719 − 0.207107i 1.00000i 0.358719 − 2.62132i −1.00000 0.500000 + 0.866025i 0.358719 + 0.207107i
373.1 0.500000 − 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i −0.358719 − 0.207107i 1.00000i −0.358719 − 2.62132i −1.00000 0.500000 − 0.866025i −0.358719 + 0.207107i
373.2 0.500000 − 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i 2.09077 + 1.20711i 1.00000i 2.09077 + 1.62132i −1.00000 0.500000 − 0.866025i 2.09077 − 1.20711i
373.3 0.500000 − 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i −2.09077 − 1.20711i − 1.00000i −2.09077 − 1.62132i −1.00000 0.500000 − 0.866025i −2.09077 + 1.20711i
373.4 0.500000 − 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i 0.358719 + 0.207107i − 1.00000i 0.358719 + 2.62132i −1.00000 0.500000 − 0.866025i 0.358719 − 0.207107i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
17.b even 2 1 inner
119.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 714.2.t.f ✓ 8
7.c even 3 1 inner 714.2.t.f ✓ 8
17.b even 2 1 inner 714.2.t.f ✓ 8
119.j even 6 1 inner 714.2.t.f ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.t.f ✓ 8 1.a even 1 1 trivial
714.2.t.f ✓ 8 7.c even 3 1 inner
714.2.t.f ✓ 8 17.b even 2 1 inner
714.2.t.f ✓ 8 119.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\):

\( T_{5}^{8} - 6T_{5}^{6} + 35T_{5}^{4} - 6T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{8} - 24T_{11}^{6} + 560T_{11}^{4} - 384T_{11}^{2} + 256 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 6 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} + 10 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 24 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 2 T^{3} + \cdots + 289)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 6 T^{3} + 35 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 86 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$29$ \( (T^{4} + 48 T^{2} + 64)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 72 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
$37$ \( T^{8} - 38 T^{6} + \cdots + 83521 \) Copy content Toggle raw display
$41$ \( (T^{4} + 96 T^{2} + 256)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T - 31)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 4 T^{3} + \cdots + 15376)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 16 T^{3} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 2 T^{3} + 11 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 10 T^{3} + \cdots + 289)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 38 T^{2} + 289)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 1475789056 \) Copy content Toggle raw display
$79$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 8 T - 16)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 7 T + 49)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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