Properties

Label 252.6.a.g
Level $252$
Weight $6$
Character orbit 252.a
Self dual yes
Analytic conductor $40.417$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{91}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 91 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{91}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} + 49 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} + 49 q^{7} - 9 \beta q^{11} - 670 q^{13} - 17 \beta q^{17} + 284 q^{19} - 31 \beta q^{23} + 151 q^{25} + 120 \beta q^{29} + 1532 q^{31} + 49 \beta q^{35} - 15118 q^{37} + 93 \beta q^{41} - 10996 q^{43} - 338 \beta q^{47} + 2401 q^{49} + 410 \beta q^{53} - 29484 q^{55} - 414 \beta q^{59} - 14602 q^{61} - 670 \beta q^{65} - 36628 q^{67} + 1187 \beta q^{71} - 54802 q^{73} - 441 \beta q^{77} - 31768 q^{79} - 1296 \beta q^{83} - 55692 q^{85} - 15 \beta q^{89} - 32830 q^{91} + 284 \beta q^{95} + 14126 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 98 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 98 q^{7} - 1340 q^{13} + 568 q^{19} + 302 q^{25} + 3064 q^{31} - 30236 q^{37} - 21992 q^{43} + 4802 q^{49} - 58968 q^{55} - 29204 q^{61} - 73256 q^{67} - 109604 q^{73} - 63536 q^{79} - 111384 q^{85} - 65660 q^{91} + 28252 q^{97}+O(q^{100}) \) 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
−9.53939
9.53939
0 0 0 −57.2364 0 49.0000 0 0 0
1.2 0 0 0 57.2364 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

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.6.a.g 2
3.b odd 2 1 inner 252.6.a.g 2
4.b odd 2 1 1008.6.a.bk 2
12.b even 2 1 1008.6.a.bk 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.6.a.g 2 1.a even 1 1 trivial
252.6.a.g 2 3.b odd 2 1 inner
1008.6.a.bk 2 4.b odd 2 1
1008.6.a.bk 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 3276 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(252))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3276 \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 265356 \) Copy content Toggle raw display
$13$ \( (T + 670)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 946764 \) Copy content Toggle raw display
$19$ \( (T - 284)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3148236 \) Copy content Toggle raw display
$29$ \( T^{2} - 47174400 \) Copy content Toggle raw display
$31$ \( (T - 1532)^{2} \) Copy content Toggle raw display
$37$ \( (T + 15118)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 28334124 \) Copy content Toggle raw display
$43$ \( (T + 10996)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 374263344 \) Copy content Toggle raw display
$53$ \( T^{2} - 550695600 \) Copy content Toggle raw display
$59$ \( T^{2} - 561493296 \) Copy content Toggle raw display
$61$ \( (T + 14602)^{2} \) Copy content Toggle raw display
$67$ \( (T + 36628)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 4615782444 \) Copy content Toggle raw display
$73$ \( (T + 54802)^{2} \) Copy content Toggle raw display
$79$ \( (T + 31768)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 5502422016 \) Copy content Toggle raw display
$89$ \( T^{2} - 737100 \) Copy content Toggle raw display
$97$ \( (T - 14126)^{2} \) Copy content Toggle raw display
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