Properties

Label 276.1.h.b
Level 276
Weight 1
Character orbit 276.h
Self dual Yes
Analytic conductor 0.138
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM disc. -23, -276, 12
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: Yes
Analytic conductor: \(0.137741943487\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{3}, \sqrt{-23})\)
Artin image size \(8\)
Artin image $D_4$
Artin field Galois closure of 4.2.3312.1

$q$-expansion

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

Character Values

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

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

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
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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
12.b Even 1 RM by \(\Q(\sqrt{3}) \) yes
23.b Odd 1 CM by \(\Q(\sqrt{-23}) \) yes
276.h Odd 1 CM by \(\Q(\sqrt{-69}) \) yes

Hecke kernels

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

\( T_{13} + 2 \)
\( T_{47} - 2 \)