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Properties

Label 3525.1.l.b
Level $3525$
Weight $1$
Character orbit 3525.l
Analytic conductor $1.759$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -47
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3525,1,Mod(1268,3525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3525, 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("3525.1268");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3525.l (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: \(1.75920416953\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
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 6.0.186384375.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{24}^{9} q^{2} - \zeta_{24}^{7} q^{3} + \zeta_{24}^{4} q^{6} + ( - \zeta_{24}^{5} - \zeta_{24}) q^{7} - \zeta_{24}^{3} q^{8} - \zeta_{24}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{9} q^{2} - \zeta_{24}^{7} q^{3} + \zeta_{24}^{4} q^{6} + ( - \zeta_{24}^{5} - \zeta_{24}) q^{7} - \zeta_{24}^{3} q^{8} - \zeta_{24}^{2} q^{9} + ( - \zeta_{24}^{10} + \zeta_{24}^{2}) q^{14} + q^{16} + \zeta_{24}^{9} q^{17} - \zeta_{24}^{11} q^{18} + (\zeta_{24}^{8} - 1) q^{21} + \zeta_{24}^{10} q^{24} + \zeta_{24}^{9} q^{27} - \zeta_{24}^{6} q^{34} + ( - \zeta_{24}^{9} - \zeta_{24}^{5}) q^{42} + \zeta_{24}^{9} q^{47} - \zeta_{24}^{7} q^{48} + (\zeta_{24}^{10} + \cdots + \zeta_{24}^{2}) q^{49} + \cdots + (\zeta_{24}^{11} + \cdots - \zeta_{24}^{3}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{6} + 8 q^{16} - 12 q^{21} + 4 q^{51} - 16 q^{61} + 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3525\mathbb{Z}\right)^\times\).

\(n\) \(1552\) \(2026\) \(2351\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1268.1
−0.258819 + 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.258819 + 0.965926i
−0.707107 − 0.707107i −0.965926 − 0.258819i 0 0 0.500000 + 0.866025i 1.22474 − 1.22474i −0.707107 + 0.707107i 0.866025 + 0.500000i 0
1268.2 −0.707107 − 0.707107i 0.258819 + 0.965926i 0 0 0.500000 − 0.866025i −1.22474 + 1.22474i −0.707107 + 0.707107i −0.866025 + 0.500000i 0
1268.3 0.707107 + 0.707107i −0.258819 − 0.965926i 0 0 0.500000 − 0.866025i 1.22474 − 1.22474i 0.707107 − 0.707107i −0.866025 + 0.500000i 0
1268.4 0.707107 + 0.707107i 0.965926 + 0.258819i 0 0 0.500000 + 0.866025i −1.22474 + 1.22474i 0.707107 − 0.707107i 0.866025 + 0.500000i 0
1832.1 −0.707107 + 0.707107i −0.965926 + 0.258819i 0 0 0.500000 − 0.866025i 1.22474 + 1.22474i −0.707107 − 0.707107i 0.866025 − 0.500000i 0
1832.2 −0.707107 + 0.707107i 0.258819 − 0.965926i 0 0 0.500000 + 0.866025i −1.22474 − 1.22474i −0.707107 − 0.707107i −0.866025 − 0.500000i 0
1832.3 0.707107 − 0.707107i −0.258819 + 0.965926i 0 0 0.500000 + 0.866025i 1.22474 + 1.22474i 0.707107 + 0.707107i −0.866025 − 0.500000i 0
1832.4 0.707107 − 0.707107i 0.965926 − 0.258819i 0 0 0.500000 − 0.866025i −1.22474 − 1.22474i 0.707107 + 0.707107i 0.866025 − 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1268.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.b odd 2 1 CM by \(\Q(\sqrt{-47}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner
141.c even 2 1 inner
235.b odd 2 1 inner
235.e even 4 2 inner
705.g even 2 1 inner
705.l odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3525.1.l.b ✓ 8
3.b odd 2 1 inner 3525.1.l.b ✓ 8
5.b even 2 1 inner 3525.1.l.b ✓ 8
5.c odd 4 2 inner 3525.1.l.b ✓ 8
15.d odd 2 1 inner 3525.1.l.b ✓ 8
15.e even 4 2 inner 3525.1.l.b ✓ 8
47.b odd 2 1 CM 3525.1.l.b ✓ 8
141.c even 2 1 inner 3525.1.l.b ✓ 8
235.b odd 2 1 inner 3525.1.l.b ✓ 8
235.e even 4 2 inner 3525.1.l.b ✓ 8
705.g even 2 1 inner 3525.1.l.b ✓ 8
705.l odd 4 2 inner 3525.1.l.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3525.1.l.b ✓ 8 1.a even 1 1 trivial
3525.1.l.b ✓ 8 3.b odd 2 1 inner
3525.1.l.b ✓ 8 5.b even 2 1 inner
3525.1.l.b ✓ 8 5.c odd 4 2 inner
3525.1.l.b ✓ 8 15.d odd 2 1 inner
3525.1.l.b ✓ 8 15.e even 4 2 inner
3525.1.l.b ✓ 8 47.b odd 2 1 CM
3525.1.l.b ✓ 8 141.c even 2 1 inner
3525.1.l.b ✓ 8 235.b odd 2 1 inner
3525.1.l.b ✓ 8 235.e even 4 2 inner
3525.1.l.b ✓ 8 705.g even 2 1 inner
3525.1.l.b ✓ 8 705.l odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} \) 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
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$61$ \( (T + 2)^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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