Properties

Label 12.9.c.a
Level 12
Weight 9
Character orbit 12.c
Self dual Yes
Analytic conductor 4.889
Analytic rank 0
Dimension 1
CM disc. -3
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 12.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(4.88854332073\)
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

\(f(q)\) \(=\) \( q + 81q^{3} + 4034q^{7} + 6561q^{9} + O(q^{10}) \) \( q + 81q^{3} + 4034q^{7} + 6561q^{9} - 35806q^{13} - 258526q^{19} + 326754q^{21} + 390625q^{25} + 531441q^{27} - 1809406q^{31} + 503522q^{37} - 2900286q^{39} + 3492194q^{43} + 10508355q^{49} - 20940606q^{57} - 23826526q^{61} + 26467074q^{63} - 5421406q^{67} + 16169282q^{73} + 31640625q^{75} - 18887038q^{79} + 43046721q^{81} - 144441404q^{91} - 146561886q^{93} + 176908034q^{97} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0
0 81.0000 0 0 0 4034.00 0 6561.00 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
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{9}^{\mathrm{new}}(12, [\chi])\).