Properties

Label 252.8.a.d
Level $252$
Weight $8$
Character orbit 252.a
Self dual yes
Analytic conductor $78.721$
Analytic rank $0$
Dimension $2$
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] = [252,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(78.7210264220\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21961}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5490 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 84)
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 \(\beta = 2\sqrt{21961}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 48) q^{5} - 343 q^{7} + ( - 17 \beta - 2070) q^{11} + ( - 24 \beta + 4614) q^{13} + (25 \beta - 15264) q^{17} + ( - 78 \beta - 352) q^{19} + (91 \beta - 19314) q^{23} + (96 \beta + 12023) q^{25}+ \cdots + ( - 42 \beta + 3695086) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 96 q^{5} - 686 q^{7} - 4140 q^{11} + 9228 q^{13} - 30528 q^{17} - 704 q^{19} - 38628 q^{23} + 24046 q^{25} - 131988 q^{29} + 165384 q^{31} + 32928 q^{35} + 381004 q^{37} - 379128 q^{41} + 916216 q^{43}+ \cdots + 7390172 q^{97}+O(q^{100}) \) 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
74.5962
−73.5962
0 0 0 −344.385 0 −343.000 0 0 0
1.2 0 0 0 248.385 0 −343.000 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 252.8.a.d 2
3.b odd 2 1 84.8.a.d 2
12.b even 2 1 336.8.a.k 2
21.c even 2 1 588.8.a.e 2
21.g even 6 2 588.8.i.l 4
21.h odd 6 2 588.8.i.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.8.a.d 2 3.b odd 2 1
252.8.a.d 2 1.a even 1 1 trivial
336.8.a.k 2 12.b even 2 1
588.8.a.e 2 21.c even 2 1
588.8.i.i 4 21.h odd 6 2
588.8.i.l 4 21.g even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 96T_{5} - 85540 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(252))\). 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
$5$ \( T^{2} + 96T - 85540 \) Copy content Toggle raw display
$7$ \( (T + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4140 T - 21102016 \) Copy content Toggle raw display
$13$ \( T^{2} - 9228 T - 29309148 \) Copy content Toggle raw display
$17$ \( T^{2} + 30528 T + 178087196 \) Copy content Toggle raw display
$19$ \( T^{2} + 704 T - 534318992 \) Copy content Toggle raw display
$23$ \( T^{2} + 38628 T - 354405568 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 28331544364 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 3394130688 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 1973834396 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 35838847980 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 209546701264 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 151386418944 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 970659712188 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 168887355056 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 9309192081228 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 26574813392 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 9479579570240 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 28271426136140 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 16619921043840 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 12879756857936 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 25558993834668 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 13653505590580 \) Copy content Toggle raw display
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