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Properties

Label 552.1.h.a
Level $552$
Weight $1$
Character orbit 552.h
Self dual yes
Analytic conductor $0.275$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -23, -552, 24
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [552,1,Mod(275,552)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(552, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("552.275"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 552.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.275483886973\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{6}, \sqrt{-23})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.12696.2
Stark unit: Root of $x^{4} - 6x^{3} - 13x^{2} - 6x + 1$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} - q^{12} + q^{16} - q^{18} + q^{23} + q^{24} + q^{25} - q^{27} + 2 q^{29} - q^{32} + q^{36} - q^{46} + 2 q^{47} - q^{48} - q^{49} - q^{50}+ \cdots + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/552\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(185\) \(277\) \(415\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
275.1
0
−1.00000 −1.00000 1.00000 0 1.00000 0 −1.00000 1.00000 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
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
24.f even 2 1 RM by \(\Q(\sqrt{6}) \)
552.h odd 2 1 CM by \(\Q(\sqrt{-138}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.1.h.a ✓ 1
3.b odd 2 1 552.1.h.b yes 1
4.b odd 2 1 2208.1.h.a 1
8.b even 2 1 2208.1.h.b 1
8.d odd 2 1 552.1.h.b yes 1
12.b even 2 1 2208.1.h.b 1
23.b odd 2 1 CM 552.1.h.a ✓ 1
24.f even 2 1 RM 552.1.h.a ✓ 1
24.h odd 2 1 2208.1.h.a 1
69.c even 2 1 552.1.h.b yes 1
92.b even 2 1 2208.1.h.a 1
184.e odd 2 1 2208.1.h.b 1
184.h even 2 1 552.1.h.b yes 1
276.h odd 2 1 2208.1.h.b 1
552.b even 2 1 2208.1.h.a 1
552.h odd 2 1 CM 552.1.h.a ✓ 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.1.h.a ✓ 1 1.a even 1 1 trivial
552.1.h.a ✓ 1 23.b odd 2 1 CM
552.1.h.a ✓ 1 24.f even 2 1 RM
552.1.h.a ✓ 1 552.h odd 2 1 CM
552.1.h.b yes 1 3.b odd 2 1
552.1.h.b yes 1 8.d odd 2 1
552.1.h.b yes 1 69.c even 2 1
552.1.h.b yes 1 184.h even 2 1
2208.1.h.a 1 4.b odd 2 1
2208.1.h.a 1 24.h odd 2 1
2208.1.h.a 1 92.b even 2 1
2208.1.h.a 1 552.b even 2 1
2208.1.h.b 1 8.b even 2 1
2208.1.h.b 1 12.b even 2 1
2208.1.h.b 1 184.e odd 2 1
2208.1.h.b 1 276.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(552, [\chi])\):

\( T_{13} \) Copy content Toggle raw display
\( T_{29} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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