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Properties

Label 252.2.bf.d
Level $252$
Weight $2$
Character orbit 252.bf
Analytic conductor $2.012$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,2,Mod(19,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.bf (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: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{2} + \beta_1 + 1) q^{4} + (\beta_{2} - 2) q^{5} + (\beta_{2} + 2 \beta_1 - 1) q^{7} + (\beta_{3} - \beta_{2} - \beta_1 + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_{2} + \beta_1 + 1) q^{4} + (\beta_{2} - 2) q^{5} + (\beta_{2} + 2 \beta_1 - 1) q^{7} + (\beta_{3} - \beta_{2} - \beta_1 + 3) q^{8} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{10} + (\beta_{2} + 2 \beta_1 - 1) q^{11} + ( - 4 \beta_{2} + 2) q^{13} + ( - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{14} + (3 \beta_{3} - 2 \beta_{2}) q^{16} + ( - 4 \beta_{2} - 4) q^{17} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 - 1) q^{20} + ( - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{22} + ( - 4 \beta_{3} + 2 \beta_{2}) q^{23} + (2 \beta_{2} - 2) q^{25} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 4) q^{26} + (3 \beta_{3} + 2 \beta_{2}) q^{28} - 5 q^{29} + ( - 4 \beta_{3} + 3 \beta_{2} + \cdots - 1) q^{31}+ \cdots + (7 \beta_{2} + 7 \beta_1 - 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 3 q^{4} - 6 q^{5} + 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 3 q^{4} - 6 q^{5} + 10 q^{8} - 3 q^{10} + 14 q^{14} - q^{16} - 24 q^{17} + 14 q^{22} - 4 q^{25} + 6 q^{26} + 7 q^{28} - 20 q^{29} + 11 q^{32} - 15 q^{40} + 7 q^{44} - 14 q^{46} + 14 q^{49} - 4 q^{50} - 18 q^{52} + 14 q^{53} + 7 q^{56} - 5 q^{58} - 36 q^{61} + 18 q^{64} + 12 q^{65} - 36 q^{68} - 21 q^{70} - 24 q^{73} + 14 q^{77} + 3 q^{80} - 6 q^{82} + 48 q^{85} - 28 q^{86} + 7 q^{88} + 36 q^{89} - 28 q^{92} + 42 q^{94} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - x^{2} - 2x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} - \nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + \nu^{2} + \nu + 2 ) / 2 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.895644 + 1.09445i
1.39564 − 0.228425i
−0.895644 − 1.09445i
1.39564 + 0.228425i
−0.895644 − 1.09445i 0 −0.395644 + 1.96048i −1.50000 − 0.866025i 0 −2.29129 + 1.32288i 2.50000 − 1.32288i 0 0.395644 + 2.41733i
19.2 1.39564 + 0.228425i 0 1.89564 + 0.637600i −1.50000 − 0.866025i 0 2.29129 − 1.32288i 2.50000 + 1.32288i 0 −1.89564 − 1.55130i
199.1 −0.895644 + 1.09445i 0 −0.395644 − 1.96048i −1.50000 + 0.866025i 0 −2.29129 − 1.32288i 2.50000 + 1.32288i 0 0.395644 − 2.41733i
199.2 1.39564 − 0.228425i 0 1.89564 − 0.637600i −1.50000 + 0.866025i 0 2.29129 + 1.32288i 2.50000 − 1.32288i 0 −1.89564 + 1.55130i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.bf.d yes 4
3.b odd 2 1 252.2.bf.a ✓ 4
4.b odd 2 1 inner 252.2.bf.d yes 4
7.c even 3 1 1764.2.b.b 4
7.d odd 6 1 inner 252.2.bf.d yes 4
7.d odd 6 1 1764.2.b.b 4
12.b even 2 1 252.2.bf.a ✓ 4
21.g even 6 1 252.2.bf.a ✓ 4
21.g even 6 1 1764.2.b.h 4
21.h odd 6 1 1764.2.b.h 4
28.f even 6 1 inner 252.2.bf.d yes 4
28.f even 6 1 1764.2.b.b 4
28.g odd 6 1 1764.2.b.b 4
84.j odd 6 1 252.2.bf.a ✓ 4
84.j odd 6 1 1764.2.b.h 4
84.n even 6 1 1764.2.b.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.bf.a ✓ 4 3.b odd 2 1
252.2.bf.a ✓ 4 12.b even 2 1
252.2.bf.a ✓ 4 21.g even 6 1
252.2.bf.a ✓ 4 84.j odd 6 1
252.2.bf.d yes 4 1.a even 1 1 trivial
252.2.bf.d yes 4 4.b odd 2 1 inner
252.2.bf.d yes 4 7.d odd 6 1 inner
252.2.bf.d yes 4 28.f even 6 1 inner
1764.2.b.b 4 7.c even 3 1
1764.2.b.b 4 7.d odd 6 1
1764.2.b.b 4 28.f even 6 1
1764.2.b.b 4 28.g odd 6 1
1764.2.b.h 4 21.g even 6 1
1764.2.b.h 4 21.h odd 6 1
1764.2.b.h 4 84.j odd 6 1
1764.2.b.h 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\):

\( T_{5}^{2} + 3T_{5} + 3 \) Copy content Toggle raw display
\( T_{11}^{4} - 7T_{11}^{2} + 49 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - T^{2} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 7T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 7T^{2} + 49 \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 28T^{2} + 784 \) Copy content Toggle raw display
$29$ \( (T + 5)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 21T^{2} + 441 \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 112)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 84T^{2} + 7056 \) Copy content Toggle raw display
$53$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 189 T^{2} + 35721 \) Copy content Toggle raw display
$61$ \( (T^{2} + 18 T + 108)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 63T^{2} + 3969 \) Copy content Toggle raw display
$83$ \( (T^{2} - 21)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 18 T + 108)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
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