Properties

Label 2015.1.h.a
Level 2015
Weight 1
Character orbit 2015.h
Analytic conductor 1.006
Analytic rank 0
Dimension 4
Projective image \(D_{4}\)
CM disc. -155
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 2015 = 5 \cdot 13 \cdot 31 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2015.h (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.00561600046\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.130975.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{3} \) \(- q^{4}\) \( + \zeta_{8}^{2} q^{5} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{3} \) \(- q^{4}\) \( + \zeta_{8}^{2} q^{5} \) \(+ q^{9}\) \( + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{12} \) \( -\zeta_{8}^{3} q^{13} \) \( + ( \zeta_{8} + \zeta_{8}^{3} ) q^{15} \) \(+ q^{16}\) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{17} \) \( + 2 \zeta_{8}^{2} q^{19} \) \( -\zeta_{8}^{2} q^{20} \) \( + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{23} \) \(- q^{25}\) \( -\zeta_{8}^{2} q^{31} \) \(- q^{36}\) \( + ( \zeta_{8} + \zeta_{8}^{3} ) q^{37} \) \( + ( 1 - \zeta_{8}^{2} ) q^{39} \) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{43} \) \( + \zeta_{8}^{2} q^{45} \) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{48} \) \(- q^{49}\) \( + 2 q^{51} \) \( + \zeta_{8}^{3} q^{52} \) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{53} \) \( + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{57} \) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{60} \) \(- q^{64}\) \( + \zeta_{8} q^{65} \) \( + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{68} \) \( -2 q^{69} \) \( -2 \zeta_{8}^{2} q^{71} \) \( + ( \zeta_{8} + \zeta_{8}^{3} ) q^{73} \) \( + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{75} \) \( -2 \zeta_{8}^{2} q^{76} \) \( + \zeta_{8}^{2} q^{80} \) \(- q^{81}\) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{83} \) \( + ( \zeta_{8} + \zeta_{8}^{3} ) q^{85} \) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{92} \) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{93} \) \( -2 q^{95} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut 4q^{16} \) \(\mathstrut -\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 4q^{36} \) \(\mathstrut +\mathstrut 4q^{39} \) \(\mathstrut -\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 8q^{51} \) \(\mathstrut -\mathstrut 4q^{64} \) \(\mathstrut -\mathstrut 8q^{69} \) \(\mathstrut -\mathstrut 4q^{81} \) \(\mathstrut -\mathstrut 8q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2015\mathbb{Z}\right)^\times\).

\(n\) \(716\) \(807\) \(1861\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2014.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 −1.41421 −1.00000 1.00000i 0 0 0 1.00000 0
2014.2 0 −1.41421 −1.00000 1.00000i 0 0 0 1.00000 0
2014.3 0 1.41421 −1.00000 1.00000i 0 0 0 1.00000 0
2014.4 0 1.41421 −1.00000 1.00000i 0 0 0 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
155.c Odd 1 CM by \(\Q(\sqrt{-155}) \) yes
5.b Even 1 yes
13.b Even 1 yes
31.b Odd 1 yes
65.d Even 1 yes
403.b Odd 1 yes
2015.h Odd 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2015, [\chi])\):

\(T_{2} \)
\(T_{3}^{2} \) \(\mathstrut -\mathstrut 2 \)