Properties

Label 252.6.a.a
Level $252$
Weight $6$
Character orbit 252.a
Self dual yes
Analytic conductor $40.417$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,6,Mod(1,252)]
 
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, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 252.a (trivial)

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: yes
Analytic conductor: \(40.4167225929\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 16 q^{5} - 49 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 16 q^{5} - 49 q^{7} - 8 q^{11} + 684 q^{13} + 2218 q^{17} - 2698 q^{19} - 3344 q^{23} - 2869 q^{25} + 3254 q^{29} + 4788 q^{31} + 784 q^{35} - 11470 q^{37} - 13350 q^{41} - 928 q^{43} - 1212 q^{47} + 2401 q^{49} - 13110 q^{53} + 128 q^{55} - 34702 q^{59} - 1032 q^{61} - 10944 q^{65} + 10108 q^{67} - 62720 q^{71} - 18926 q^{73} + 392 q^{77} + 11400 q^{79} - 88958 q^{83} - 35488 q^{85} - 19722 q^{89} - 33516 q^{91} + 43168 q^{95} + 17062 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 −16.0000 0 −49.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.6.a.a 1
3.b odd 2 1 28.6.a.b 1
4.b odd 2 1 1008.6.a.l 1
12.b even 2 1 112.6.a.b 1
15.d odd 2 1 700.6.a.b 1
15.e even 4 2 700.6.e.b 2
21.c even 2 1 196.6.a.a 1
21.g even 6 2 196.6.e.i 2
21.h odd 6 2 196.6.e.a 2
24.f even 2 1 448.6.a.o 1
24.h odd 2 1 448.6.a.b 1
84.h odd 2 1 784.6.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.a.b 1 3.b odd 2 1
112.6.a.b 1 12.b even 2 1
196.6.a.a 1 21.c even 2 1
196.6.e.a 2 21.h odd 6 2
196.6.e.i 2 21.g even 6 2
252.6.a.a 1 1.a even 1 1 trivial
448.6.a.b 1 24.h odd 2 1
448.6.a.o 1 24.f even 2 1
700.6.a.b 1 15.d odd 2 1
700.6.e.b 2 15.e even 4 2
784.6.a.m 1 84.h odd 2 1
1008.6.a.l 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 16 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(252))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 16 \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T + 8 \) Copy content Toggle raw display
$13$ \( T - 684 \) Copy content Toggle raw display
$17$ \( T - 2218 \) Copy content Toggle raw display
$19$ \( T + 2698 \) Copy content Toggle raw display
$23$ \( T + 3344 \) Copy content Toggle raw display
$29$ \( T - 3254 \) Copy content Toggle raw display
$31$ \( T - 4788 \) Copy content Toggle raw display
$37$ \( T + 11470 \) Copy content Toggle raw display
$41$ \( T + 13350 \) Copy content Toggle raw display
$43$ \( T + 928 \) Copy content Toggle raw display
$47$ \( T + 1212 \) Copy content Toggle raw display
$53$ \( T + 13110 \) Copy content Toggle raw display
$59$ \( T + 34702 \) Copy content Toggle raw display
$61$ \( T + 1032 \) Copy content Toggle raw display
$67$ \( T - 10108 \) Copy content Toggle raw display
$71$ \( T + 62720 \) Copy content Toggle raw display
$73$ \( T + 18926 \) Copy content Toggle raw display
$79$ \( T - 11400 \) Copy content Toggle raw display
$83$ \( T + 88958 \) Copy content Toggle raw display
$89$ \( T + 19722 \) Copy content Toggle raw display
$97$ \( T - 17062 \) Copy content Toggle raw display
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