Properties

Label 15.3.d.b
Level $15$
Weight $3$
Character orbit 15.d
Self dual yes
Analytic conductor $0.409$
Analytic rank $0$
Dimension $1$
CM discriminant -15
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 15.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.408720396540\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - 3 q^{3} - 3 q^{4} + 5 q^{5} - 3 q^{6} - 7 q^{8} + 9 q^{9} + O(q^{10}) \) \( q + q^{2} - 3 q^{3} - 3 q^{4} + 5 q^{5} - 3 q^{6} - 7 q^{8} + 9 q^{9} + 5 q^{10} + 9 q^{12} - 15 q^{15} + 5 q^{16} - 14 q^{17} + 9 q^{18} - 22 q^{19} - 15 q^{20} + 34 q^{23} + 21 q^{24} + 25 q^{25} - 27 q^{27} - 15 q^{30} + 2 q^{31} + 33 q^{32} - 14 q^{34} - 27 q^{36} - 22 q^{38} - 35 q^{40} + 45 q^{45} + 34 q^{46} - 14 q^{47} - 15 q^{48} + 49 q^{49} + 25 q^{50} + 42 q^{51} - 86 q^{53} - 27 q^{54} + 66 q^{57} + 45 q^{60} - 118 q^{61} + 2 q^{62} + 13 q^{64} + 42 q^{68} - 102 q^{69} - 63 q^{72} - 75 q^{75} + 66 q^{76} + 98 q^{79} + 25 q^{80} + 81 q^{81} + 154 q^{83} - 70 q^{85} + 45 q^{90} - 102 q^{92} - 6 q^{93} - 14 q^{94} - 110 q^{95} - 99 q^{96} + 49 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
0
1.00000 −3.00000 −3.00000 5.00000 −3.00000 0 −7.00000 9.00000 5.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.3.d.b yes 1
3.b odd 2 1 15.3.d.a 1
4.b odd 2 1 240.3.c.b 1
5.b even 2 1 15.3.d.a 1
5.c odd 4 2 75.3.c.d 2
8.b even 2 1 960.3.c.c 1
8.d odd 2 1 960.3.c.a 1
9.c even 3 2 405.3.h.a 2
9.d odd 6 2 405.3.h.b 2
12.b even 2 1 240.3.c.a 1
15.d odd 2 1 CM 15.3.d.b yes 1
15.e even 4 2 75.3.c.d 2
20.d odd 2 1 240.3.c.a 1
20.e even 4 2 1200.3.l.l 2
24.f even 2 1 960.3.c.d 1
24.h odd 2 1 960.3.c.b 1
40.e odd 2 1 960.3.c.d 1
40.f even 2 1 960.3.c.b 1
45.h odd 6 2 405.3.h.a 2
45.j even 6 2 405.3.h.b 2
60.h even 2 1 240.3.c.b 1
60.l odd 4 2 1200.3.l.l 2
120.i odd 2 1 960.3.c.c 1
120.m even 2 1 960.3.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.d.a 1 3.b odd 2 1
15.3.d.a 1 5.b even 2 1
15.3.d.b yes 1 1.a even 1 1 trivial
15.3.d.b yes 1 15.d odd 2 1 CM
75.3.c.d 2 5.c odd 4 2
75.3.c.d 2 15.e even 4 2
240.3.c.a 1 12.b even 2 1
240.3.c.a 1 20.d odd 2 1
240.3.c.b 1 4.b odd 2 1
240.3.c.b 1 60.h even 2 1
405.3.h.a 2 9.c even 3 2
405.3.h.a 2 45.h odd 6 2
405.3.h.b 2 9.d odd 6 2
405.3.h.b 2 45.j even 6 2
960.3.c.a 1 8.d odd 2 1
960.3.c.a 1 120.m even 2 1
960.3.c.b 1 24.h odd 2 1
960.3.c.b 1 40.f even 2 1
960.3.c.c 1 8.b even 2 1
960.3.c.c 1 120.i odd 2 1
960.3.c.d 1 24.f even 2 1
960.3.c.d 1 40.e odd 2 1
1200.3.l.l 2 20.e even 4 2
1200.3.l.l 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{3}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 3 + T \)
$5$ \( -5 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( T \)
$17$ \( 14 + T \)
$19$ \( 22 + T \)
$23$ \( -34 + T \)
$29$ \( T \)
$31$ \( -2 + T \)
$37$ \( T \)
$41$ \( T \)
$43$ \( T \)
$47$ \( 14 + T \)
$53$ \( 86 + T \)
$59$ \( T \)
$61$ \( 118 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( T \)
$79$ \( -98 + T \)
$83$ \( -154 + T \)
$89$ \( T \)
$97$ \( T \)
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