Properties

Label 4.9.b.a
Level 4
Weight 9
Character orbit 4.b
Self dual Yes
Analytic conductor 1.630
Analytic rank 0
Dimension 1
CM disc. -4
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 4.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(1.62951444024\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 256q^{4} \) \(\mathstrut -\mathstrut 1054q^{5} \) \(\mathstrut +\mathstrut 4096q^{8} \) \(\mathstrut +\mathstrut 6561q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 256q^{4} \) \(\mathstrut -\mathstrut 1054q^{5} \) \(\mathstrut +\mathstrut 4096q^{8} \) \(\mathstrut +\mathstrut 6561q^{9} \) \(\mathstrut -\mathstrut 16864q^{10} \) \(\mathstrut -\mathstrut 478q^{13} \) \(\mathstrut +\mathstrut 65536q^{16} \) \(\mathstrut -\mathstrut 63358q^{17} \) \(\mathstrut +\mathstrut 104976q^{18} \) \(\mathstrut -\mathstrut 269824q^{20} \) \(\mathstrut +\mathstrut 720291q^{25} \) \(\mathstrut -\mathstrut 7648q^{26} \) \(\mathstrut -\mathstrut 1407838q^{29} \) \(\mathstrut +\mathstrut 1048576q^{32} \) \(\mathstrut -\mathstrut 1013728q^{34} \) \(\mathstrut +\mathstrut 1679616q^{36} \) \(\mathstrut +\mathstrut 925922q^{37} \) \(\mathstrut -\mathstrut 4317184q^{40} \) \(\mathstrut +\mathstrut 3577922q^{41} \) \(\mathstrut -\mathstrut 6915294q^{45} \) \(\mathstrut +\mathstrut 5764801q^{49} \) \(\mathstrut +\mathstrut 11524656q^{50} \) \(\mathstrut -\mathstrut 122368q^{52} \) \(\mathstrut -\mathstrut 9620638q^{53} \) \(\mathstrut -\mathstrut 22525408q^{58} \) \(\mathstrut +\mathstrut 20722082q^{61} \) \(\mathstrut +\mathstrut 16777216q^{64} \) \(\mathstrut +\mathstrut 503812q^{65} \) \(\mathstrut -\mathstrut 16219648q^{68} \) \(\mathstrut +\mathstrut 26873856q^{72} \) \(\mathstrut -\mathstrut 54717118q^{73} \) \(\mathstrut +\mathstrut 14814752q^{74} \) \(\mathstrut -\mathstrut 69074944q^{80} \) \(\mathstrut +\mathstrut 43046721q^{81} \) \(\mathstrut +\mathstrut 57246752q^{82} \) \(\mathstrut +\mathstrut 66779332q^{85} \) \(\mathstrut -\mathstrut 30265918q^{89} \) \(\mathstrut -\mathstrut 110644704q^{90} \) \(\mathstrut -\mathstrut 173379838q^{97} \) \(\mathstrut +\mathstrut 92236816q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0
16.0000 0 256.000 −1054.00 0 0 4096.00 6561.00 −16864.0
\(n\): 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
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) acting on \(S_{9}^{\mathrm{new}}(4, [\chi])\).