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 discriminant -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 \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00561600046\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.130975.1

$q$-expansion

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
155.c odd 2 1 CM by \(\Q(\sqrt{-155}) \)
5.b even 2 1 inner
13.b even 2 1 inner
31.b odd 2 1 inner
65.d even 2 1 inner
403.b odd 2 1 inner
2015.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2015.1.h.a 4
5.b even 2 1 inner 2015.1.h.a 4
13.b even 2 1 inner 2015.1.h.a 4
31.b odd 2 1 inner 2015.1.h.a 4
65.d even 2 1 inner 2015.1.h.a 4
155.c odd 2 1 CM 2015.1.h.a 4
403.b odd 2 1 inner 2015.1.h.a 4
2015.h odd 2 1 inner 2015.1.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.1.h.a 4 1.a even 1 1 trivial
2015.1.h.a 4 5.b even 2 1 inner
2015.1.h.a 4 13.b even 2 1 inner
2015.1.h.a 4 31.b odd 2 1 inner
2015.1.h.a 4 65.d even 2 1 inner
2015.1.h.a 4 155.c odd 2 1 CM
2015.1.h.a 4 403.b odd 2 1 inner
2015.1.h.a 4 2015.h odd 2 1 inner

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( ( 1 + T^{4} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( 1 + T^{2} )^{4} \)
$11$ \( ( 1 + T^{2} )^{4} \)
$13$ \( 1 + T^{4} \)
$17$ \( ( 1 + T^{4} )^{2} \)
$19$ \( ( 1 + T^{2} )^{4} \)
$23$ \( ( 1 + T^{4} )^{2} \)
$29$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$31$ \( ( 1 + T^{2} )^{2} \)
$37$ \( ( 1 + T^{4} )^{2} \)
$41$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$43$ \( ( 1 + T^{4} )^{2} \)
$47$ \( ( 1 + T^{2} )^{4} \)
$53$ \( ( 1 + T^{4} )^{2} \)
$59$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$61$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$67$ \( ( 1 + T^{2} )^{4} \)
$71$ \( ( 1 + T^{2} )^{4} \)
$73$ \( ( 1 + T^{4} )^{2} \)
$79$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$83$ \( ( 1 + T^{4} )^{2} \)
$89$ \( ( 1 + T^{2} )^{4} \)
$97$ \( ( 1 + T^{2} )^{4} \)
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