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Properties

Label 252.2.j.a
Level $252$
Weight $2$
Character orbit 252.j
Analytic conductor $2.012$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,2,Mod(85,252)]
 
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([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.85");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.j (of order \(3\), 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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{2}) q^{3} + (\beta_{5} - \beta_1) q^{5} + (\beta_{4} + 1) q^{7} + ( - \beta_{5} + 2 \beta_{3} + \beta_{2} + \cdots + 2) q^{9} + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots - 1) q^{11}+ \cdots + (8 \beta_{5} + 5 \beta_{4} - 3 \beta_{2} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - q^{5} + 3 q^{7} + 8 q^{9} - 2 q^{11} - 3 q^{13} + q^{15} + 4 q^{17} + 6 q^{19} - 4 q^{21} - 14 q^{23} + 6 q^{25} + 7 q^{27} - q^{29} + 3 q^{31} + 8 q^{33} - 2 q^{35} - 6 q^{37} - 24 q^{39}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} + 5\nu^{3} + \nu^{2} + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 5\nu^{3} + 2\nu^{2} - 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 6\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{5} - 2\beta_{4} - 11\beta_{3} - 6\beta_{2} + 8\beta _1 + 7 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1
0.500000 + 1.41036i
0.500000 − 2.05195i
0.500000 − 0.224437i
0.500000 − 1.41036i
0.500000 + 2.05195i
0.500000 + 0.224437i
0 −1.71053 − 0.272169i 0 −0.119562 + 0.207087i 0 0.500000 + 0.866025i 0 2.85185 + 0.931107i 0
85.2 0 −0.933463 + 1.45899i 0 −1.23025 + 2.13086i 0 0.500000 + 0.866025i 0 −1.25729 − 2.72382i 0
85.3 0 1.64400 + 0.545231i 0 0.849814 − 1.47192i 0 0.500000 + 0.866025i 0 2.40545 + 1.79272i 0
169.1 0 −1.71053 + 0.272169i 0 −0.119562 − 0.207087i 0 0.500000 − 0.866025i 0 2.85185 − 0.931107i 0
169.2 0 −0.933463 − 1.45899i 0 −1.23025 − 2.13086i 0 0.500000 − 0.866025i 0 −1.25729 + 2.72382i 0
169.3 0 1.64400 − 0.545231i 0 0.849814 + 1.47192i 0 0.500000 − 0.866025i 0 2.40545 − 1.79272i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.j.a ✓ 6
3.b odd 2 1 756.2.j.b 6
4.b odd 2 1 1008.2.r.j 6
7.b odd 2 1 1764.2.j.e 6
7.c even 3 1 1764.2.i.g 6
7.c even 3 1 1764.2.l.e 6
7.d odd 6 1 1764.2.i.d 6
7.d odd 6 1 1764.2.l.f 6
9.c even 3 1 inner 252.2.j.a ✓ 6
9.c even 3 1 2268.2.a.i 3
9.d odd 6 1 756.2.j.b 6
9.d odd 6 1 2268.2.a.h 3
12.b even 2 1 3024.2.r.j 6
21.c even 2 1 5292.2.j.d 6
21.g even 6 1 5292.2.i.e 6
21.g even 6 1 5292.2.l.f 6
21.h odd 6 1 5292.2.i.f 6
21.h odd 6 1 5292.2.l.e 6
36.f odd 6 1 1008.2.r.j 6
36.f odd 6 1 9072.2.a.by 3
36.h even 6 1 3024.2.r.j 6
36.h even 6 1 9072.2.a.bv 3
63.g even 3 1 1764.2.i.g 6
63.h even 3 1 1764.2.l.e 6
63.i even 6 1 5292.2.l.f 6
63.j odd 6 1 5292.2.l.e 6
63.k odd 6 1 1764.2.i.d 6
63.l odd 6 1 1764.2.j.e 6
63.n odd 6 1 5292.2.i.f 6
63.o even 6 1 5292.2.j.d 6
63.s even 6 1 5292.2.i.e 6
63.t odd 6 1 1764.2.l.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.j.a ✓ 6 1.a even 1 1 trivial
252.2.j.a ✓ 6 9.c even 3 1 inner
756.2.j.b 6 3.b odd 2 1
756.2.j.b 6 9.d odd 6 1
1008.2.r.j 6 4.b odd 2 1
1008.2.r.j 6 36.f odd 6 1
1764.2.i.d 6 7.d odd 6 1
1764.2.i.d 6 63.k odd 6 1
1764.2.i.g 6 7.c even 3 1
1764.2.i.g 6 63.g even 3 1
1764.2.j.e 6 7.b odd 2 1
1764.2.j.e 6 63.l odd 6 1
1764.2.l.e 6 7.c even 3 1
1764.2.l.e 6 63.h even 3 1
1764.2.l.f 6 7.d odd 6 1
1764.2.l.f 6 63.t odd 6 1
2268.2.a.h 3 9.d odd 6 1
2268.2.a.i 3 9.c even 3 1
3024.2.r.j 6 12.b even 2 1
3024.2.r.j 6 36.h even 6 1
5292.2.i.e 6 21.g even 6 1
5292.2.i.e 6 63.s even 6 1
5292.2.i.f 6 21.h odd 6 1
5292.2.i.f 6 63.n odd 6 1
5292.2.j.d 6 21.c even 2 1
5292.2.j.d 6 63.o even 6 1
5292.2.l.e 6 21.h odd 6 1
5292.2.l.e 6 63.j odd 6 1
5292.2.l.f 6 21.g even 6 1
5292.2.l.f 6 63.i even 6 1
9072.2.a.bv 3 36.h even 6 1
9072.2.a.by 3 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + T_{5}^{5} + 5T_{5}^{4} - 2T_{5}^{3} + 17T_{5}^{2} + 4T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 2 T^{5} + \cdots + 27 \) Copy content Toggle raw display
$5$ \( T^{6} + T^{5} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 2 T^{5} + \cdots + 3481 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$17$ \( (T^{3} - 2 T^{2} - 19 T + 47)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 3 T^{2} - 24 T + 79)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 14 T^{5} + \cdots + 961 \) Copy content Toggle raw display
$29$ \( T^{6} + T^{5} + \cdots + 11881 \) Copy content Toggle raw display
$31$ \( T^{6} - 3 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$37$ \( (T^{3} + 3 T^{2} - 30 T - 23)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 33 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} + \cdots + 78961 \) Copy content Toggle raw display
$47$ \( T^{6} + 21 T^{5} + \cdots + 42849 \) Copy content Toggle raw display
$53$ \( (T^{3} - 6 T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 31 T^{5} + \cdots + 978121 \) Copy content Toggle raw display
$61$ \( T^{6} + 6 T^{5} + \cdots + 1185921 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$71$ \( (T^{3} - 17 T^{2} + \cdots + 1907)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 3 T^{2} - 24 T - 79)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 9 T^{5} + \cdots + 136161 \) Copy content Toggle raw display
$83$ \( T^{6} + 20 T^{5} + \cdots + 16129 \) Copy content Toggle raw display
$89$ \( (T^{3} - 12 T^{2} + \cdots + 711)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 9 T^{5} + \cdots + 5329 \) Copy content Toggle raw display
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