Properties

Label 552.1.h.b
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

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

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 4.0.12696.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} - q^{12} + q^{16} + q^{18} - q^{23} - q^{24} + q^{25} - q^{27} - 2q^{29} + q^{32} + q^{36} - q^{46} - 2q^{47} - q^{48} - q^{49} + q^{50} - q^{54} - 2q^{58} + q^{64} + q^{69} + 2q^{71} + q^{72} - 2q^{73} - q^{75} + q^{81} + 2q^{87} - q^{92} - 2q^{94} - q^{96} - q^{98} + O(q^{100}) \)

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.

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

Hecke kernels

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

\( T_{13} \)
\( T_{29} + 2 \)

Hecke characteristic polynomials

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