Properties

Label 552.1.h.c
Level $552$
Weight $1$
Character orbit 552.h
Analytic conductor $0.275$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -23
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: no
Analytic conductor: \(0.275483886973\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.7312896.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} + \zeta_{6} q^{3} -\zeta_{6} q^{4} - q^{6} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{2} + \zeta_{6} q^{3} -\zeta_{6} q^{4} - q^{6} + q^{8} + \zeta_{6}^{2} q^{9} -\zeta_{6}^{2} q^{12} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{13} + \zeta_{6}^{2} q^{16} -\zeta_{6} q^{18} - q^{23} + \zeta_{6} q^{24} + q^{25} + ( -1 - \zeta_{6} ) q^{26} - q^{27} + q^{29} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{31} -\zeta_{6} q^{32} + q^{36} + ( -1 + \zeta_{6}^{2} ) q^{39} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{41} -\zeta_{6}^{2} q^{46} + q^{47} - q^{48} - q^{49} + \zeta_{6}^{2} q^{50} + ( 1 - \zeta_{6}^{2} ) q^{52} -\zeta_{6}^{2} q^{54} + \zeta_{6}^{2} q^{58} + ( 1 + \zeta_{6} ) q^{62} + q^{64} -\zeta_{6} q^{69} - q^{71} + \zeta_{6}^{2} q^{72} + q^{73} + \zeta_{6} q^{75} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{78} -\zeta_{6} q^{81} + ( 1 + \zeta_{6} ) q^{82} + \zeta_{6} q^{87} + \zeta_{6} q^{92} + ( 1 - \zeta_{6}^{2} ) q^{93} + \zeta_{6}^{2} q^{94} -\zeta_{6}^{2} q^{96} -\zeta_{6}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{3} - q^{4} - 2q^{6} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{3} - q^{4} - 2q^{6} + 2q^{8} - q^{9} + q^{12} - q^{16} - q^{18} - 2q^{23} + q^{24} + 2q^{25} - 3q^{26} - 2q^{27} + 2q^{29} - q^{32} + 2q^{36} - 3q^{39} + q^{46} + 2q^{47} - 2q^{48} - 2q^{49} - q^{50} + 3q^{52} + q^{54} - q^{58} + 3q^{62} + 2q^{64} - q^{69} - 2q^{71} - q^{72} + 2q^{73} + q^{75} - q^{81} + 3q^{82} + q^{87} + q^{92} + 3q^{93} - q^{94} + q^{96} + q^{98} + O(q^{100}) \)

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
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 0 1.00000 −0.500000 0.866025i 0
275.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 0 1.00000 −0.500000 + 0.866025i 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 inner
552.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.1.h.c 2
3.b odd 2 1 552.1.h.d yes 2
4.b odd 2 1 2208.1.h.d 2
8.b even 2 1 2208.1.h.c 2
8.d odd 2 1 552.1.h.d yes 2
12.b even 2 1 2208.1.h.c 2
23.b odd 2 1 CM 552.1.h.c 2
24.f even 2 1 inner 552.1.h.c 2
24.h odd 2 1 2208.1.h.d 2
69.c even 2 1 552.1.h.d yes 2
92.b even 2 1 2208.1.h.d 2
184.e odd 2 1 2208.1.h.c 2
184.h even 2 1 552.1.h.d yes 2
276.h odd 2 1 2208.1.h.c 2
552.b even 2 1 2208.1.h.d 2
552.h odd 2 1 inner 552.1.h.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.1.h.c 2 1.a even 1 1 trivial
552.1.h.c 2 23.b odd 2 1 CM
552.1.h.c 2 24.f even 2 1 inner
552.1.h.c 2 552.h odd 2 1 inner
552.1.h.d yes 2 3.b odd 2 1
552.1.h.d yes 2 8.d odd 2 1
552.1.h.d yes 2 69.c even 2 1
552.1.h.d yes 2 184.h even 2 1
2208.1.h.c 2 8.b even 2 1
2208.1.h.c 2 12.b even 2 1
2208.1.h.c 2 184.e odd 2 1
2208.1.h.c 2 276.h odd 2 1
2208.1.h.d 2 4.b odd 2 1
2208.1.h.d 2 24.h odd 2 1
2208.1.h.d 2 92.b even 2 1
2208.1.h.d 2 552.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 \(S_{1}^{\mathrm{new}}(552, [\chi])\):

\( T_{13}^{2} + 3 \)
\( T_{29} - 1 \)

Hecke characteristic polynomials

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