Properties

Label 252.8.a.e
Level $252$
Weight $8$
Character orbit 252.a
Self dual yes
Analytic conductor $78.721$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3529}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 882 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{3529}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 21) q^{5} + 343 q^{7} + (14 \beta - 3714) q^{11} + (3 \beta + 5915) q^{13} + ( - 106 \beta - 7896) q^{17} + (9 \beta + 13307) q^{19} + (504 \beta - 16320) q^{23} + (42 \beta - 45923) q^{25}+ \cdots + ( - 47790 \beta - 8688736) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 42 q^{5} + 686 q^{7} - 7428 q^{11} + 11830 q^{13} - 15792 q^{17} + 26614 q^{19} - 32640 q^{23} - 91846 q^{25} + 158016 q^{29} - 180740 q^{31} - 14406 q^{35} - 45824 q^{37} + 321720 q^{41} + 1023868 q^{43}+ \cdots - 17377472 q^{97}+O(q^{100}) \) 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
30.2027
−29.2027
0 0 0 −199.216 0 343.000 0 0 0
1.2 0 0 0 157.216 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.e 2
3.b odd 2 1 28.8.a.a 2
12.b even 2 1 112.8.a.i 2
21.c even 2 1 196.8.a.b 2
21.g even 6 2 196.8.e.a 4
21.h odd 6 2 196.8.e.d 4
24.f even 2 1 448.8.a.n 2
24.h odd 2 1 448.8.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.8.a.a 2 3.b odd 2 1
112.8.a.i 2 12.b even 2 1
196.8.a.b 2 21.c even 2 1
196.8.e.a 4 21.g even 6 2
196.8.e.d 4 21.h odd 6 2
252.8.a.e 2 1.a even 1 1 trivial
448.8.a.n 2 24.f even 2 1
448.8.a.p 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 42T_{5} - 31320 \) 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
$5$ \( T^{2} + 42T - 31320 \) Copy content Toggle raw display
$7$ \( (T - 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 7428 T + 7568640 \) Copy content Toggle raw display
$13$ \( T^{2} - 11830 T + 34701376 \) Copy content Toggle raw display
$17$ \( T^{2} + 15792 T - 294519780 \) Copy content Toggle raw display
$19$ \( T^{2} - 26614 T + 174503608 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 7801459776 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 5489020188 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 16540907264 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 207949289540 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 208069107156 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 127557024352 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 662231593152 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 15217199100 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 951114840120 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 676162372040 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 7466582740400 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 4028431841280 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 15928428632500 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 52062570791552 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 14563855983960 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 54854655982020 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 2955690377596 \) Copy content Toggle raw display
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