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Properties

Label 525.1.be.a
Level $525$
Weight $1$
Character orbit 525.be
Analytic conductor $0.262$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -3
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,1,Mod(68,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.68");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 525.be (of order \(12\), degree \(4\), 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: \(0.262009131632\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.472696875.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{24}^{5} q^{3} + \zeta_{24}^{2} q^{4} - \zeta_{24}^{11} q^{7} + \zeta_{24}^{10} q^{9} - \zeta_{24}^{7} q^{12} + \zeta_{24}^{9} q^{13} + \zeta_{24}^{4} q^{16} + ( - \zeta_{24}^{6} - \zeta_{24}^{2}) q^{19} + \cdots - \zeta_{24}^{3} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{16} - 4 q^{21} - 8 q^{36} - 12 q^{61} + 4 q^{81} - 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-\zeta_{24}^{6}\) \(-1\) \(\zeta_{24}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
0 −0.965926 − 0.258819i −0.866025 + 0.500000i 0 0 0.258819 − 0.965926i 0 0.866025 + 0.500000i 0
68.2 0 0.965926 + 0.258819i −0.866025 + 0.500000i 0 0 −0.258819 + 0.965926i 0 0.866025 + 0.500000i 0
143.1 0 −0.258819 − 0.965926i 0.866025 + 0.500000i 0 0 0.965926 − 0.258819i 0 −0.866025 + 0.500000i 0
143.2 0 0.258819 + 0.965926i 0.866025 + 0.500000i 0 0 −0.965926 + 0.258819i 0 −0.866025 + 0.500000i 0
257.1 0 −0.258819 + 0.965926i 0.866025 − 0.500000i 0 0 0.965926 + 0.258819i 0 −0.866025 − 0.500000i 0
257.2 0 0.258819 − 0.965926i 0.866025 − 0.500000i 0 0 −0.965926 − 0.258819i 0 −0.866025 − 0.500000i 0
332.1 0 −0.965926 + 0.258819i −0.866025 − 0.500000i 0 0 0.258819 + 0.965926i 0 0.866025 − 0.500000i 0
332.2 0 0.965926 − 0.258819i −0.866025 − 0.500000i 0 0 −0.258819 − 0.965926i 0 0.866025 − 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
21.g even 6 1 inner
35.i odd 6 1 inner
35.k even 12 2 inner
105.p even 6 1 inner
105.w odd 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.1.be.a ✓ 8
3.b odd 2 1 CM 525.1.be.a ✓ 8
5.b even 2 1 inner 525.1.be.a ✓ 8
5.c odd 4 2 inner 525.1.be.a ✓ 8
7.b odd 2 1 3675.1.bf.b 8
7.c even 3 1 3675.1.k.b 8
7.c even 3 1 3675.1.bf.b 8
7.d odd 6 1 inner 525.1.be.a ✓ 8
7.d odd 6 1 3675.1.k.b 8
15.d odd 2 1 inner 525.1.be.a ✓ 8
15.e even 4 2 inner 525.1.be.a ✓ 8
21.c even 2 1 3675.1.bf.b 8
21.g even 6 1 inner 525.1.be.a ✓ 8
21.g even 6 1 3675.1.k.b 8
21.h odd 6 1 3675.1.k.b 8
21.h odd 6 1 3675.1.bf.b 8
35.c odd 2 1 3675.1.bf.b 8
35.f even 4 2 3675.1.bf.b 8
35.i odd 6 1 inner 525.1.be.a ✓ 8
35.i odd 6 1 3675.1.k.b 8
35.j even 6 1 3675.1.k.b 8
35.j even 6 1 3675.1.bf.b 8
35.k even 12 2 inner 525.1.be.a ✓ 8
35.k even 12 2 3675.1.k.b 8
35.l odd 12 2 3675.1.k.b 8
35.l odd 12 2 3675.1.bf.b 8
105.g even 2 1 3675.1.bf.b 8
105.k odd 4 2 3675.1.bf.b 8
105.o odd 6 1 3675.1.k.b 8
105.o odd 6 1 3675.1.bf.b 8
105.p even 6 1 inner 525.1.be.a ✓ 8
105.p even 6 1 3675.1.k.b 8
105.w odd 12 2 inner 525.1.be.a ✓ 8
105.w odd 12 2 3675.1.k.b 8
105.x even 12 2 3675.1.k.b 8
105.x even 12 2 3675.1.bf.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.1.be.a ✓ 8 1.a even 1 1 trivial
525.1.be.a ✓ 8 3.b odd 2 1 CM
525.1.be.a ✓ 8 5.b even 2 1 inner
525.1.be.a ✓ 8 5.c odd 4 2 inner
525.1.be.a ✓ 8 7.d odd 6 1 inner
525.1.be.a ✓ 8 15.d odd 2 1 inner
525.1.be.a ✓ 8 15.e even 4 2 inner
525.1.be.a ✓ 8 21.g even 6 1 inner
525.1.be.a ✓ 8 35.i odd 6 1 inner
525.1.be.a ✓ 8 35.k even 12 2 inner
525.1.be.a ✓ 8 105.p even 6 1 inner
525.1.be.a ✓ 8 105.w odd 12 2 inner
3675.1.k.b 8 7.c even 3 1
3675.1.k.b 8 7.d odd 6 1
3675.1.k.b 8 21.g even 6 1
3675.1.k.b 8 21.h odd 6 1
3675.1.k.b 8 35.i odd 6 1
3675.1.k.b 8 35.j even 6 1
3675.1.k.b 8 35.k even 12 2
3675.1.k.b 8 35.l odd 12 2
3675.1.k.b 8 105.o odd 6 1
3675.1.k.b 8 105.p even 6 1
3675.1.k.b 8 105.w odd 12 2
3675.1.k.b 8 105.x even 12 2
3675.1.bf.b 8 7.b odd 2 1
3675.1.bf.b 8 7.c even 3 1
3675.1.bf.b 8 21.c even 2 1
3675.1.bf.b 8 21.h odd 6 1
3675.1.bf.b 8 35.c odd 2 1
3675.1.bf.b 8 35.f even 4 2
3675.1.bf.b 8 35.j even 6 1
3675.1.bf.b 8 35.l odd 12 2
3675.1.bf.b 8 105.g even 2 1
3675.1.bf.b 8 105.k odd 4 2
3675.1.bf.b 8 105.o odd 6 1
3675.1.bf.b 8 105.x even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(525, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$79$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
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