Properties

Label 825.6.a.f
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
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] = [825,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(132.316651346\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.307532.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 76x + 168 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{2} + 9 q^{3} + (\beta_{2} + 2 \beta_1 + 24) q^{4} + ( - 9 \beta_1 - 18) q^{6} + ( - 4 \beta_{2} + 6 \beta_1 - 34) q^{7} + ( - 7 \beta_{2} - 10 \beta_1 - 76) q^{8} + 81 q^{9} + 121 q^{11}+ \cdots + 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 7 q^{2} + 27 q^{3} + 73 q^{4} - 63 q^{6} - 92 q^{7} - 231 q^{8} + 243 q^{9} + 363 q^{11} + 657 q^{12} + 90 q^{13} - 784 q^{14} - 415 q^{16} - 1934 q^{17} - 567 q^{18} + 2084 q^{19} - 828 q^{21} - 847 q^{22}+ \cdots + 29403 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 76x + 168 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2\nu - 52 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2\beta _1 + 52 \) 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
7.91848
2.30119
−9.21967
−9.91848 9.00000 66.3762 0 −89.2663 −92.6461 −340.959 81.0000 0
1.2 −4.30119 9.00000 −13.4998 0 −38.7107 148.216 195.703 81.0000 0
1.3 7.21967 9.00000 20.1236 0 64.9770 −147.570 −85.7438 81.0000 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 825.6.a.f 3
5.b even 2 1 165.6.a.e 3
15.d odd 2 1 495.6.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.e 3 5.b even 2 1
495.6.a.a 3 15.d odd 2 1
825.6.a.f 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 7T_{2}^{2} - 60T_{2} - 308 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 7 T^{2} + \cdots - 308 \) Copy content Toggle raw display
$3$ \( (T - 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 92 T^{2} + \cdots - 2026368 \) Copy content Toggle raw display
$11$ \( (T - 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 90 T^{2} + \cdots - 210673232 \) Copy content Toggle raw display
$17$ \( T^{3} + 1934 T^{2} + \cdots - 123909408 \) Copy content Toggle raw display
$19$ \( T^{3} - 2084 T^{2} + \cdots + 14437440 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 2404328128 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 80705567064 \) Copy content Toggle raw display
$31$ \( T^{3} + 10688 T^{2} + \cdots + 593236224 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 165140256344 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 127610311752 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 5672527691040 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 5576540180928 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 850686588776 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 7185357358784 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 2993338614376 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 6432661987328 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 31171869026560 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 152306824713328 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 283704612543488 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 200710881230832 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 340522035911288 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 93893760682568 \) Copy content Toggle raw display
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