Properties

Label 825.4.a.p
Level $825$
Weight $4$
Character orbit 825.a
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
Dimension $3$
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] = [825,4,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1957.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 q^{3} + (2 \beta_{2} + \beta_1 + 6) q^{4} + 3 \beta_1 q^{6} + ( - \beta_{2} + 3 \beta_1 - 1) q^{7} + ( - 2 \beta_{2} + 3 \beta_1 + 2) q^{8} + 9 q^{9} + 11 q^{11} + (6 \beta_{2} + 3 \beta_1 + 18) q^{12}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 9 q^{3} + 17 q^{4} - 3 q^{6} - 6 q^{7} + 3 q^{8} + 27 q^{9} + 33 q^{11} + 51 q^{12} + 20 q^{13} + 144 q^{14} + 25 q^{16} - 32 q^{17} - 9 q^{18} + 116 q^{19} - 18 q^{21} - 11 q^{22} - 240 q^{23}+ \cdots + 297 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 9x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + \nu + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{2} + \beta _1 + 13 ) / 2 \) 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
1.12946
−3.04096
2.91150
−4.59486 3.00000 13.1127 0 −13.7846 −20.6383 −23.4921 9.00000 0
1.2 −0.793499 3.00000 −7.37036 0 −2.38050 2.90793 12.1964 9.00000 0
1.3 4.38835 3.00000 11.2577 0 13.1651 11.7304 14.2958 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)
\(11\) \( -1 \)

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 825.4.a.p 3
3.b odd 2 1 2475.4.a.z 3
5.b even 2 1 165.4.a.g 3
5.c odd 4 2 825.4.c.m 6
15.d odd 2 1 495.4.a.i 3
55.d odd 2 1 1815.4.a.q 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.g 3 5.b even 2 1
495.4.a.i 3 15.d odd 2 1
825.4.a.p 3 1.a even 1 1 trivial
825.4.c.m 6 5.c odd 4 2
1815.4.a.q 3 55.d odd 2 1
2475.4.a.z 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(825))\):

\( T_{2}^{3} + T_{2}^{2} - 20T_{2} - 16 \) Copy content Toggle raw display
\( T_{7}^{3} + 6T_{7}^{2} - 268T_{7} + 704 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} + \cdots - 16 \) Copy content Toggle raw display
$3$ \( (T - 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 6 T^{2} + \cdots + 704 \) Copy content Toggle raw display
$11$ \( (T - 11)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 20 T^{2} + \cdots + 78104 \) Copy content Toggle raw display
$17$ \( T^{3} + 32 T^{2} + \cdots + 22424 \) Copy content Toggle raw display
$19$ \( T^{3} - 116 T^{2} + \cdots - 80 \) Copy content Toggle raw display
$23$ \( T^{3} + 240 T^{2} + \cdots - 180224 \) Copy content Toggle raw display
$29$ \( T^{3} - 238 T^{2} + \cdots + 428416 \) Copy content Toggle raw display
$31$ \( T^{3} - 92 T^{2} + \cdots + 6769664 \) Copy content Toggle raw display
$37$ \( T^{3} - 90 T^{2} + \cdots + 6364168 \) Copy content Toggle raw display
$41$ \( T^{3} + 46 T^{2} + \cdots - 245888 \) Copy content Toggle raw display
$43$ \( T^{3} - 134 T^{2} + \cdots + 2381360 \) Copy content Toggle raw display
$47$ \( T^{3} - 220 T^{2} + \cdots - 10980224 \) Copy content Toggle raw display
$53$ \( T^{3} - 798 T^{2} + \cdots + 17262968 \) Copy content Toggle raw display
$59$ \( T^{3} - 1236 T^{2} + \cdots - 32923904 \) Copy content Toggle raw display
$61$ \( T^{3} - 342 T^{2} + \cdots - 2655176 \) Copy content Toggle raw display
$67$ \( T^{3} + 764 T^{2} + \cdots - 153685184 \) Copy content Toggle raw display
$71$ \( T^{3} - 1816 T^{2} + \cdots - 198158720 \) Copy content Toggle raw display
$73$ \( T^{3} + 100 T^{2} + \cdots + 5132984 \) Copy content Toggle raw display
$79$ \( T^{3} + 96 T^{2} + \cdots - 167159872 \) Copy content Toggle raw display
$83$ \( T^{3} + 858 T^{2} + \cdots - 542136176 \) Copy content Toggle raw display
$89$ \( T^{3} - 838 T^{2} + \cdots + 693013592 \) Copy content Toggle raw display
$97$ \( T^{3} - 1322 T^{2} + \cdots + 354601256 \) Copy content Toggle raw display
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