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Properties

Label 525.1.k.a
Level $525$
Weight $1$
Character orbit 525.k
Analytic conductor $0.262$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -3, -35, 105
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(293,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.293");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 525.k (of order \(4\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{-35})\)
Artin image: $\OD_{16}:C_2$
Artin field: Galois closure of 16.0.96148443603515625.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{8} q^{3} + \zeta_{8}^{2} q^{4} + \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{3} + \zeta_{8}^{2} q^{4} + \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} - \zeta_{8}^{3} q^{12} + 2 \zeta_{8} q^{13} - q^{16} + q^{21} - \zeta_{8}^{3} q^{27} - \zeta_{8} q^{28} - q^{36} - 2 \zeta_{8}^{2} q^{39} + \zeta_{8} q^{48} - \zeta_{8}^{2} q^{49} + 2 \zeta_{8}^{3} q^{52} - \zeta_{8} q^{63} - \zeta_{8}^{2} q^{64} - 2 \zeta_{8} q^{73} - 2 \zeta_{8}^{2} q^{79} - q^{81} + \zeta_{8}^{2} q^{84} - 2 q^{91} - 2 \zeta_{8}^{3} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{16} + 4 q^{21} - 4 q^{36} - 4 q^{81} - 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(\zeta_{8}^{2}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
0.707107 − 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
−0.707107 − 0.707107i
0 −0.707107 + 0.707107i − 1.00000i 0 0 −0.707107 − 0.707107i 0 − 1.00000i 0
293.2 0 0.707107 − 0.707107i − 1.00000i 0 0 0.707107 + 0.707107i 0 − 1.00000i 0
482.1 0 −0.707107 − 0.707107i 1.00000i 0 0 −0.707107 + 0.707107i 0 1.00000i 0
482.2 0 0.707107 + 0.707107i 1.00000i 0 0 0.707107 − 0.707107i 0 1.00000i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
105.g even 2 1 RM by \(\Q(\sqrt{105}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
21.c even 2 1 inner
35.f even 4 2 inner
105.k odd 4 2 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 1 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 16 \) Copy content Toggle raw display
$79$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 16 \) Copy content Toggle raw display
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