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 disc. -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 \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.40872039654\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
15.d Odd 1 CM by \(\Q(\sqrt{-15}) \) yes

Hecke kernels

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