Properties

Label 155.1.c.b
Level 155
Weight 1
Character orbit 155.c
Analytic conductor 0.077
Analytic rank 0
Dimension 2
Projective image \(D_{6}\)
CM disc. -31
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 155 = 5 \cdot 31 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 155.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0773550769581\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.120125.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \zeta_{6} + \zeta_{6}^{2} ) q^{2} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{4} + \zeta_{6} q^{5} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{7} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{8} - q^{9} +O(q^{10})\) \( q + ( \zeta_{6} + \zeta_{6}^{2} ) q^{2} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{4} + \zeta_{6} q^{5} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{7} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{8} - q^{9} + ( -1 + \zeta_{6}^{2} ) q^{10} + ( 2 + \zeta_{6} - \zeta_{6}^{2} ) q^{14} + q^{16} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{18} + q^{19} + ( -1 - \zeta_{6} - \zeta_{6}^{2} ) q^{20} + \zeta_{6}^{2} q^{25} + ( 2 \zeta_{6} + 2 \zeta_{6}^{2} ) q^{28} - q^{31} + ( 1 - \zeta_{6}^{2} ) q^{35} + ( 1 + \zeta_{6} - \zeta_{6}^{2} ) q^{36} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{38} + ( 1 - \zeta_{6}^{2} ) q^{40} - q^{41} -\zeta_{6} q^{45} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{49} + ( -1 - \zeta_{6} ) q^{50} + ( -2 - \zeta_{6} + \zeta_{6}^{2} ) q^{56} - q^{59} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{62} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{63} + q^{64} + ( 1 + 2 \zeta_{6} + \zeta_{6}^{2} ) q^{70} + q^{71} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{72} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{76} + \zeta_{6} q^{80} + q^{81} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{82} + ( 1 - \zeta_{6}^{2} ) q^{90} + \zeta_{6} q^{95} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{97} + ( -2 \zeta_{6} - 2 \zeta_{6}^{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + q^{5} - 2q^{9} + O(q^{10}) \) \( 2q - 4q^{4} + q^{5} - 2q^{9} - 3q^{10} + 6q^{14} + 2q^{16} + 2q^{19} - 2q^{20} - q^{25} - 2q^{31} + 3q^{35} + 4q^{36} + 3q^{40} - 2q^{41} - q^{45} - 4q^{49} - 3q^{50} - 6q^{56} - 2q^{59} + 2q^{64} + 3q^{70} + 2q^{71} - 4q^{76} + q^{80} + 2q^{81} + 3q^{90} + q^{95} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).

\(n\) \(32\) \(96\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
154.1
0.500000 0.866025i
0.500000 + 0.866025i
1.73205i 0 −2.00000 0.500000 0.866025i 0 1.73205i 1.73205i −1.00000 −1.50000 0.866025i
154.2 1.73205i 0 −2.00000 0.500000 + 0.866025i 0 1.73205i 1.73205i −1.00000 −1.50000 + 0.866025i
\(n\): 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
31.b Odd 1 CM by \(\Q(\sqrt{-31}) \) yes
5.b Even 1 yes
155.c Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 \) acting on \(S_{1}^{\mathrm{new}}(155, [\chi])\).