Properties

Label 252.2.f.a
Level $252$
Weight $2$
Character orbit 252.f
Analytic conductor $2.012$
Analytic rank $0$
Dimension $4$
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] = [252,2,Mod(125,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("252.125");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.f (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: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + ( - \beta_1 - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + ( - \beta_1 - 1) q^{7} - \beta_{2} q^{11} - 2 \beta_1 q^{13} + \beta_{3} q^{17} + 2 \beta_1 q^{19} + \beta_{2} q^{23} + 7 q^{25} - \beta_{2} q^{29} + ( - \beta_{3} + 2 \beta_{2}) q^{35} - 8 q^{37} - \beta_{3} q^{41} - 2 q^{43} - 2 \beta_{3} q^{47} + (2 \beta_1 - 5) q^{49} - 3 \beta_{2} q^{53} + 6 \beta_1 q^{55} - 4 \beta_{3} q^{59} - 4 \beta_1 q^{61} + 4 \beta_{2} q^{65} + 8 q^{67} + \beta_{2} q^{71} + 2 \beta_1 q^{73} + (3 \beta_{3} + \beta_{2}) q^{77} - 4 q^{79} + 2 \beta_{3} q^{83} + 12 q^{85} - 3 \beta_{3} q^{89} + (2 \beta_1 - 12) q^{91} - 4 \beta_{2} q^{95} - 2 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{7} + 28 q^{25} - 32 q^{37} - 8 q^{43} - 20 q^{49} + 32 q^{67} - 16 q^{79} + 48 q^{85} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -3\nu^{3} - 9\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{2} - 9\beta_1 ) / 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
1.93185i
1.93185i
0.517638i
0.517638i
0 0 0 −3.46410 0 −1.00000 2.44949i 0 0 0
125.2 0 0 0 −3.46410 0 −1.00000 + 2.44949i 0 0 0
125.3 0 0 0 3.46410 0 −1.00000 2.44949i 0 0 0
125.4 0 0 0 3.46410 0 −1.00000 + 2.44949i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.f.a 4
3.b odd 2 1 inner 252.2.f.a 4
4.b odd 2 1 1008.2.k.b 4
5.b even 2 1 6300.2.d.c 4
5.c odd 4 2 6300.2.f.b 8
7.b odd 2 1 inner 252.2.f.a 4
7.c even 3 2 1764.2.t.b 8
7.d odd 6 2 1764.2.t.b 8
8.b even 2 1 4032.2.k.a 4
8.d odd 2 1 4032.2.k.d 4
9.c even 3 2 2268.2.x.i 8
9.d odd 6 2 2268.2.x.i 8
12.b even 2 1 1008.2.k.b 4
15.d odd 2 1 6300.2.d.c 4
15.e even 4 2 6300.2.f.b 8
21.c even 2 1 inner 252.2.f.a 4
21.g even 6 2 1764.2.t.b 8
21.h odd 6 2 1764.2.t.b 8
24.f even 2 1 4032.2.k.d 4
24.h odd 2 1 4032.2.k.a 4
28.d even 2 1 1008.2.k.b 4
35.c odd 2 1 6300.2.d.c 4
35.f even 4 2 6300.2.f.b 8
56.e even 2 1 4032.2.k.d 4
56.h odd 2 1 4032.2.k.a 4
63.l odd 6 2 2268.2.x.i 8
63.o even 6 2 2268.2.x.i 8
84.h odd 2 1 1008.2.k.b 4
105.g even 2 1 6300.2.d.c 4
105.k odd 4 2 6300.2.f.b 8
168.e odd 2 1 4032.2.k.d 4
168.i even 2 1 4032.2.k.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.f.a 4 1.a even 1 1 trivial
252.2.f.a 4 3.b odd 2 1 inner
252.2.f.a 4 7.b odd 2 1 inner
252.2.f.a 4 21.c even 2 1 inner
1008.2.k.b 4 4.b odd 2 1
1008.2.k.b 4 12.b even 2 1
1008.2.k.b 4 28.d even 2 1
1008.2.k.b 4 84.h odd 2 1
1764.2.t.b 8 7.c even 3 2
1764.2.t.b 8 7.d odd 6 2
1764.2.t.b 8 21.g even 6 2
1764.2.t.b 8 21.h odd 6 2
2268.2.x.i 8 9.c even 3 2
2268.2.x.i 8 9.d odd 6 2
2268.2.x.i 8 63.l odd 6 2
2268.2.x.i 8 63.o even 6 2
4032.2.k.a 4 8.b even 2 1
4032.2.k.a 4 24.h odd 2 1
4032.2.k.a 4 56.h odd 2 1
4032.2.k.a 4 168.i even 2 1
4032.2.k.d 4 8.d odd 2 1
4032.2.k.d 4 24.f even 2 1
4032.2.k.d 4 56.e even 2 1
4032.2.k.d 4 168.e odd 2 1
6300.2.d.c 4 5.b even 2 1
6300.2.d.c 4 15.d odd 2 1
6300.2.d.c 4 35.c odd 2 1
6300.2.d.c 4 105.g even 2 1
6300.2.f.b 8 5.c odd 4 2
6300.2.f.b 8 15.e even 4 2
6300.2.f.b 8 35.f even 4 2
6300.2.f.b 8 105.k odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T + 8)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$43$ \( (T + 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$67$ \( (T - 8)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
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