Properties

Label 276.1.h.d
Level 276
Weight 1
Character orbit 276.h
Analytic conductor 0.138
Analytic rank 0
Dimension 2
Projective image \(D_{6}\)
CM disc. -23
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 276.h (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.137741943487\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.914112.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{2} -\zeta_{6} q^{3} -\zeta_{6} q^{4} - q^{6} - q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{2} -\zeta_{6} q^{3} -\zeta_{6} q^{4} - q^{6} - q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6}^{2} q^{12} + q^{13} + \zeta_{6}^{2} q^{16} + \zeta_{6} q^{18} + q^{23} + \zeta_{6} q^{24} - q^{25} -\zeta_{6}^{2} q^{26} + q^{27} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{29} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{31} + \zeta_{6} q^{32} + q^{36} -\zeta_{6} q^{39} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{41} -\zeta_{6}^{2} q^{46} + q^{47} + q^{48} - q^{49} + \zeta_{6}^{2} q^{50} -\zeta_{6} q^{52} -\zeta_{6}^{2} q^{54} + ( -1 - \zeta_{6} ) q^{58} -2 q^{59} + ( 1 + \zeta_{6} ) q^{62} + q^{64} -\zeta_{6} q^{69} - q^{71} -\zeta_{6}^{2} q^{72} - q^{73} + \zeta_{6} q^{75} - q^{78} -\zeta_{6} q^{81} + ( 1 + \zeta_{6} ) q^{82} + ( -1 + \zeta_{6}^{2} ) q^{87} -\zeta_{6} q^{92} + ( 1 - \zeta_{6}^{2} ) q^{93} -\zeta_{6}^{2} q^{94} -\zeta_{6}^{2} q^{96} + \zeta_{6}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{3} - q^{4} - 2q^{6} - 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + q^{2} - q^{3} - q^{4} - 2q^{6} - 2q^{8} - q^{9} - q^{12} + 2q^{13} - q^{16} + q^{18} + 2q^{23} + q^{24} - 2q^{25} + q^{26} + 2q^{27} + q^{32} + 2q^{36} - q^{39} + q^{46} + 2q^{47} + 2q^{48} - 2q^{49} - q^{50} - q^{52} + q^{54} - 3q^{58} - 4q^{59} + 3q^{62} + 2q^{64} - q^{69} - 2q^{71} + q^{72} - 2q^{73} + q^{75} - 2q^{78} - q^{81} + 3q^{82} - 3q^{87} - q^{92} + 3q^{93} + q^{94} + q^{96} - q^{98} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/276\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(139\) \(185\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
23.b Odd 1 CM by \(\Q(\sqrt{-23}) \) yes
12.b Even 1 yes
276.h Odd 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(276, [\chi])\):

\( T_{13} - 1 \)
\( T_{47} - 1 \)