Properties

Label 8.11.d.a
Level 8
Weight 11
Character orbit 8.d
Self dual Yes
Analytic conductor 5.083
Analytic rank 0
Dimension 1
CM disc. -8
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(5.08285802139\)
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

\(f(q)\) \(=\) \( q - 32q^{2} - 482q^{3} + 1024q^{4} + 15424q^{6} - 32768q^{8} + 173275q^{9} + O(q^{10}) \) \( q - 32q^{2} - 482q^{3} + 1024q^{4} + 15424q^{6} - 32768q^{8} + 173275q^{9} - 97426q^{11} - 493568q^{12} + 1048576q^{16} + 823682q^{17} - 5544800q^{18} + 3353726q^{19} + 3117632q^{22} + 15794176q^{24} + 9765625q^{25} - 55056932q^{27} - 33554432q^{32} + 46959332q^{33} - 26357824q^{34} + 177433600q^{36} - 107319232q^{38} - 37778926q^{41} + 214485614q^{43} - 99764224q^{44} - 505413632q^{48} + 282475249q^{49} - 312500000q^{50} - 397014724q^{51} + 1761821824q^{54} - 1616495932q^{57} + 921043598q^{59} + 1073741824q^{64} - 1502698624q^{66} + 1813708382q^{67} + 843450368q^{68} - 5677875200q^{72} - 1605781582q^{73} - 4707031250q^{75} + 3434215424q^{76} + 16305725749q^{81} + 1208925632q^{82} + 96051518q^{83} - 6863539648q^{86} + 3192455168q^{88} - 11116019374q^{89} + 16173236224q^{96} - 9872978014q^{97} - 9039207968q^{98} - 16881490150q^{99} + O(q^{100}) \)

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
−32.0000 −482.000 1024.00 0 15424.0 0 −32768.0 173275. 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 \( T_{3} + 482 \) acting on \(S_{11}^{\mathrm{new}}(8, [\chi])\).