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 discriminant -4
Inner twists 2

Related objects

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Newspace parameters

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

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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.9.b.a 1
3.b odd 2 1 36.9.d.a 1
4.b odd 2 1 CM 4.9.b.a 1
5.b even 2 1 100.9.b.a 1
5.c odd 4 2 100.9.d.a 2
8.b even 2 1 64.9.c.a 1
8.d odd 2 1 64.9.c.a 1
12.b even 2 1 36.9.d.a 1
16.e even 4 2 256.9.d.a 2
16.f odd 4 2 256.9.d.a 2
20.d odd 2 1 100.9.b.a 1
20.e even 4 2 100.9.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.9.b.a 1 1.a even 1 1 trivial
4.9.b.a 1 4.b odd 2 1 CM
36.9.d.a 1 3.b odd 2 1
36.9.d.a 1 12.b even 2 1
64.9.c.a 1 8.b even 2 1
64.9.c.a 1 8.d odd 2 1
100.9.b.a 1 5.b even 2 1
100.9.b.a 1 20.d odd 2 1
100.9.d.a 2 5.c odd 4 2
100.9.d.a 2 20.e even 4 2
256.9.d.a 2 16.e even 4 2
256.9.d.a 2 16.f odd 4 2

Hecke kernels

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