Properties

Label 8.5.d.a
Level 8
Weight 5
Character orbit 8.d
Self dual Yes
Analytic conductor 0.827
Analytic rank 0
Dimension 1
CM disc. -8
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.826959704671\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut +\mathstrut 16q^{4} \) \(\mathstrut -\mathstrut 56q^{6} \) \(\mathstrut +\mathstrut 64q^{8} \) \(\mathstrut +\mathstrut 115q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut +\mathstrut 16q^{4} \) \(\mathstrut -\mathstrut 56q^{6} \) \(\mathstrut +\mathstrut 64q^{8} \) \(\mathstrut +\mathstrut 115q^{9} \) \(\mathstrut -\mathstrut 46q^{11} \) \(\mathstrut -\mathstrut 224q^{12} \) \(\mathstrut +\mathstrut 256q^{16} \) \(\mathstrut -\mathstrut 574q^{17} \) \(\mathstrut +\mathstrut 460q^{18} \) \(\mathstrut +\mathstrut 434q^{19} \) \(\mathstrut -\mathstrut 184q^{22} \) \(\mathstrut -\mathstrut 896q^{24} \) \(\mathstrut +\mathstrut 625q^{25} \) \(\mathstrut -\mathstrut 476q^{27} \) \(\mathstrut +\mathstrut 1024q^{32} \) \(\mathstrut +\mathstrut 644q^{33} \) \(\mathstrut -\mathstrut 2296q^{34} \) \(\mathstrut +\mathstrut 1840q^{36} \) \(\mathstrut +\mathstrut 1736q^{38} \) \(\mathstrut -\mathstrut 1246q^{41} \) \(\mathstrut -\mathstrut 3502q^{43} \) \(\mathstrut -\mathstrut 736q^{44} \) \(\mathstrut -\mathstrut 3584q^{48} \) \(\mathstrut +\mathstrut 2401q^{49} \) \(\mathstrut +\mathstrut 2500q^{50} \) \(\mathstrut +\mathstrut 8036q^{51} \) \(\mathstrut -\mathstrut 1904q^{54} \) \(\mathstrut -\mathstrut 6076q^{57} \) \(\mathstrut -\mathstrut 238q^{59} \) \(\mathstrut +\mathstrut 4096q^{64} \) \(\mathstrut +\mathstrut 2576q^{66} \) \(\mathstrut -\mathstrut 5134q^{67} \) \(\mathstrut -\mathstrut 9184q^{68} \) \(\mathstrut +\mathstrut 7360q^{72} \) \(\mathstrut +\mathstrut 9506q^{73} \) \(\mathstrut -\mathstrut 8750q^{75} \) \(\mathstrut +\mathstrut 6944q^{76} \) \(\mathstrut -\mathstrut 2651q^{81} \) \(\mathstrut -\mathstrut 4984q^{82} \) \(\mathstrut +\mathstrut 11186q^{83} \) \(\mathstrut -\mathstrut 14008q^{86} \) \(\mathstrut -\mathstrut 2944q^{88} \) \(\mathstrut +\mathstrut 5474q^{89} \) \(\mathstrut -\mathstrut 14336q^{96} \) \(\mathstrut -\mathstrut 9982q^{97} \) \(\mathstrut +\mathstrut 9604q^{98} \) \(\mathstrut -\mathstrut 5290q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
4.00000 −14.0000 16.0000 0 −56.0000 0 64.0000 115.000 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 \(T_{3} \) \(\mathstrut +\mathstrut 14 \) acting on \(S_{5}^{\mathrm{new}}(8, [\chi])\).