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Properties

Label 825.2.f.b
Level $825$
Weight $2$
Character orbit 825.f
Analytic conductor $6.588$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(626,825)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.626");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.f (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - 2 q^{4} + (\beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - 2 q^{4} + (\beta_{2} + 3) q^{9} + (2 \beta_{2} + 1) q^{11} + 2 \beta_1 q^{12} + 4 q^{16} + ( - 3 \beta_{3} - 3 \beta_1) q^{23} + ( - \beta_{3} - 3 \beta_1) q^{27} - 5 q^{31} + ( - 2 \beta_{3} - \beta_1) q^{33} + ( - 2 \beta_{2} - 6) q^{36} + (\beta_{3} - 5 \beta_1) q^{37} + ( - 4 \beta_{2} - 2) q^{44} + ( - 4 \beta_{3} - 4 \beta_1) q^{47} - 4 \beta_1 q^{48} + 7 q^{49} + ( - 2 \beta_{3} - 2 \beta_1) q^{53} + ( - 2 \beta_{2} - 1) q^{59} - 8 q^{64} + ( - \beta_{3} + 5 \beta_1) q^{67} + 9 \beta_{2} q^{69} + ( - 10 \beta_{2} - 5) q^{71} + (5 \beta_{2} + 6) q^{81} + ( - 10 \beta_{2} - 5) q^{89} + (6 \beta_{3} + 6 \beta_1) q^{92} + 5 \beta_1 q^{93} + ( - \beta_{3} + 5 \beta_1) q^{97} + (5 \beta_{2} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 10 q^{9} + 16 q^{16} - 20 q^{31} - 20 q^{36} + 28 q^{49} - 32 q^{64} - 18 q^{69} + 14 q^{81} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
626.1
1.65831 + 0.500000i
1.65831 − 0.500000i
−1.65831 + 0.500000i
−1.65831 − 0.500000i
0 −1.65831 − 0.500000i −2.00000 0 0 0 0 2.50000 + 1.65831i 0
626.2 0 −1.65831 + 0.500000i −2.00000 0 0 0 0 2.50000 − 1.65831i 0
626.3 0 1.65831 − 0.500000i −2.00000 0 0 0 0 2.50000 − 1.65831i 0
626.4 0 1.65831 + 0.500000i −2.00000 0 0 0 0 2.50000 + 1.65831i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
33.d even 2 1 inner
55.d odd 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.f.b 4
3.b odd 2 1 inner 825.2.f.b 4
5.b even 2 1 inner 825.2.f.b 4
5.c odd 4 1 165.2.d.a ✓ 2
5.c odd 4 1 165.2.d.b yes 2
11.b odd 2 1 CM 825.2.f.b 4
15.d odd 2 1 inner 825.2.f.b 4
15.e even 4 1 165.2.d.a ✓ 2
15.e even 4 1 165.2.d.b yes 2
33.d even 2 1 inner 825.2.f.b 4
55.d odd 2 1 inner 825.2.f.b 4
55.e even 4 1 165.2.d.a ✓ 2
55.e even 4 1 165.2.d.b yes 2
165.d even 2 1 inner 825.2.f.b 4
165.l odd 4 1 165.2.d.a ✓ 2
165.l odd 4 1 165.2.d.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.d.a ✓ 2 5.c odd 4 1
165.2.d.a ✓ 2 15.e even 4 1
165.2.d.a ✓ 2 55.e even 4 1
165.2.d.a ✓ 2 165.l odd 4 1
165.2.d.b yes 2 5.c odd 4 1
165.2.d.b yes 2 15.e even 4 1
165.2.d.b yes 2 55.e even 4 1
165.2.d.b yes 2 165.l odd 4 1
825.2.f.b 4 1.a even 1 1 trivial
825.2.f.b 4 3.b odd 2 1 inner
825.2.f.b 4 5.b even 2 1 inner
825.2.f.b 4 11.b odd 2 1 CM
825.2.f.b 4 15.d odd 2 1 inner
825.2.f.b 4 33.d even 2 1 inner
825.2.f.b 4 55.d odd 2 1 inner
825.2.f.b 4 165.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{23}^{2} + 81 \) Copy content Toggle raw display
\( T_{37}^{2} - 99 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 5T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T + 5)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 99)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 11)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 99)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 275)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 275)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 99)^{2} \) Copy content Toggle raw display
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