Properties

Label 8.9.d.a
Level 8
Weight 9
Character orbit 8.d
Self dual Yes
Analytic conductor 3.259
Analytic rank 0
Dimension 1
CM disc. -8
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(3.25902888049\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 34q^{3} \) \(\mathstrut +\mathstrut 256q^{4} \) \(\mathstrut +\mathstrut 544q^{6} \) \(\mathstrut +\mathstrut 4096q^{8} \) \(\mathstrut -\mathstrut 5405q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 34q^{3} \) \(\mathstrut +\mathstrut 256q^{4} \) \(\mathstrut +\mathstrut 544q^{6} \) \(\mathstrut +\mathstrut 4096q^{8} \) \(\mathstrut -\mathstrut 5405q^{9} \) \(\mathstrut -\mathstrut 27166q^{11} \) \(\mathstrut +\mathstrut 8704q^{12} \) \(\mathstrut +\mathstrut 65536q^{16} \) \(\mathstrut +\mathstrut 162434q^{17} \) \(\mathstrut -\mathstrut 86480q^{18} \) \(\mathstrut -\mathstrut 72286q^{19} \) \(\mathstrut -\mathstrut 434656q^{22} \) \(\mathstrut +\mathstrut 139264q^{24} \) \(\mathstrut +\mathstrut 390625q^{25} \) \(\mathstrut -\mathstrut 406844q^{27} \) \(\mathstrut +\mathstrut 1048576q^{32} \) \(\mathstrut -\mathstrut 923644q^{33} \) \(\mathstrut +\mathstrut 2598944q^{34} \) \(\mathstrut -\mathstrut 1383680q^{36} \) \(\mathstrut -\mathstrut 1156576q^{38} \) \(\mathstrut -\mathstrut 4099006q^{41} \) \(\mathstrut +\mathstrut 5426402q^{43} \) \(\mathstrut -\mathstrut 6954496q^{44} \) \(\mathstrut +\mathstrut 2228224q^{48} \) \(\mathstrut +\mathstrut 5764801q^{49} \) \(\mathstrut +\mathstrut 6250000q^{50} \) \(\mathstrut +\mathstrut 5522756q^{51} \) \(\mathstrut -\mathstrut 6509504q^{54} \) \(\mathstrut -\mathstrut 2457724q^{57} \) \(\mathstrut -\mathstrut 24178078q^{59} \) \(\mathstrut +\mathstrut 16777216q^{64} \) \(\mathstrut -\mathstrut 14778304q^{66} \) \(\mathstrut -\mathstrut 13944286q^{67} \) \(\mathstrut +\mathstrut 41583104q^{68} \) \(\mathstrut -\mathstrut 22138880q^{72} \) \(\mathstrut +\mathstrut 33567554q^{73} \) \(\mathstrut +\mathstrut 13281250q^{75} \) \(\mathstrut -\mathstrut 18505216q^{76} \) \(\mathstrut +\mathstrut 21629509q^{81} \) \(\mathstrut -\mathstrut 65584096q^{82} \) \(\mathstrut +\mathstrut 30209954q^{83} \) \(\mathstrut +\mathstrut 86822432q^{86} \) \(\mathstrut -\mathstrut 111271936q^{88} \) \(\mathstrut -\mathstrut 95519806q^{89} \) \(\mathstrut +\mathstrut 35651584q^{96} \) \(\mathstrut -\mathstrut 77418238q^{97} \) \(\mathstrut +\mathstrut 92236816q^{98} \) \(\mathstrut +\mathstrut 146832230q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0
16.0000 34.0000 256.000 0 544.000 0 4096.00 −5405.00 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
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) \(\mathstrut -\mathstrut 34 \) acting on \(S_{9}^{\mathrm{new}}(8, [\chi])\).