Properties

Label 15.3.d.a
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

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

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

$p$ $F_p(T)$
$2$ \( 1 + T + 4 T^{2} \)
$3$ \( 1 - 3 T \)
$5$ \( 1 + 5 T \)
$7$ \( ( 1 - 7 T )( 1 + 7 T ) \)
$11$ \( ( 1 - 11 T )( 1 + 11 T ) \)
$13$ \( ( 1 - 13 T )( 1 + 13 T ) \)
$17$ \( 1 - 14 T + 289 T^{2} \)
$19$ \( 1 + 22 T + 361 T^{2} \)
$23$ \( 1 + 34 T + 529 T^{2} \)
$29$ \( ( 1 - 29 T )( 1 + 29 T ) \)
$31$ \( 1 - 2 T + 961 T^{2} \)
$37$ \( ( 1 - 37 T )( 1 + 37 T ) \)
$41$ \( ( 1 - 41 T )( 1 + 41 T ) \)
$43$ \( ( 1 - 43 T )( 1 + 43 T ) \)
$47$ \( 1 - 14 T + 2209 T^{2} \)
$53$ \( 1 - 86 T + 2809 T^{2} \)
$59$ \( ( 1 - 59 T )( 1 + 59 T ) \)
$61$ \( 1 + 118 T + 3721 T^{2} \)
$67$ \( ( 1 - 67 T )( 1 + 67 T ) \)
$71$ \( ( 1 - 71 T )( 1 + 71 T ) \)
$73$ \( ( 1 - 73 T )( 1 + 73 T ) \)
$79$ \( 1 - 98 T + 6241 T^{2} \)
$83$ \( 1 + 154 T + 6889 T^{2} \)
$89$ \( ( 1 - 89 T )( 1 + 89 T ) \)
$97$ \( ( 1 - 97 T )( 1 + 97 T ) \)
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