Properties

Label 8.3.d.a
Level 8
Weight 3
Character orbit 8.d
Self dual Yes
Analytic conductor 0.218
Analytic rank 0
Dimension 1
CM disc. -8
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(0.217984211488\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 8q^{8} \) \(\mathstrut -\mathstrut 5q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 8q^{8} \) \(\mathstrut -\mathstrut 5q^{9} \) \(\mathstrut +\mathstrut 14q^{11} \) \(\mathstrut -\mathstrut 8q^{12} \) \(\mathstrut +\mathstrut 16q^{16} \) \(\mathstrut +\mathstrut 2q^{17} \) \(\mathstrut +\mathstrut 10q^{18} \) \(\mathstrut -\mathstrut 34q^{19} \) \(\mathstrut -\mathstrut 28q^{22} \) \(\mathstrut +\mathstrut 16q^{24} \) \(\mathstrut +\mathstrut 25q^{25} \) \(\mathstrut +\mathstrut 28q^{27} \) \(\mathstrut -\mathstrut 32q^{32} \) \(\mathstrut -\mathstrut 28q^{33} \) \(\mathstrut -\mathstrut 4q^{34} \) \(\mathstrut -\mathstrut 20q^{36} \) \(\mathstrut +\mathstrut 68q^{38} \) \(\mathstrut -\mathstrut 46q^{41} \) \(\mathstrut +\mathstrut 14q^{43} \) \(\mathstrut +\mathstrut 56q^{44} \) \(\mathstrut -\mathstrut 32q^{48} \) \(\mathstrut +\mathstrut 49q^{49} \) \(\mathstrut -\mathstrut 50q^{50} \) \(\mathstrut -\mathstrut 4q^{51} \) \(\mathstrut -\mathstrut 56q^{54} \) \(\mathstrut +\mathstrut 68q^{57} \) \(\mathstrut -\mathstrut 82q^{59} \) \(\mathstrut +\mathstrut 64q^{64} \) \(\mathstrut +\mathstrut 56q^{66} \) \(\mathstrut +\mathstrut 62q^{67} \) \(\mathstrut +\mathstrut 8q^{68} \) \(\mathstrut +\mathstrut 40q^{72} \) \(\mathstrut -\mathstrut 142q^{73} \) \(\mathstrut -\mathstrut 50q^{75} \) \(\mathstrut -\mathstrut 136q^{76} \) \(\mathstrut -\mathstrut 11q^{81} \) \(\mathstrut +\mathstrut 92q^{82} \) \(\mathstrut +\mathstrut 158q^{83} \) \(\mathstrut -\mathstrut 28q^{86} \) \(\mathstrut -\mathstrut 112q^{88} \) \(\mathstrut +\mathstrut 146q^{89} \) \(\mathstrut +\mathstrut 64q^{96} \) \(\mathstrut -\mathstrut 94q^{97} \) \(\mathstrut -\mathstrut 98q^{98} \) \(\mathstrut -\mathstrut 70q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−2.00000 −2.00000 4.00000 0 4.00000 0 −8.00000 −5.00000 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

There are no other newforms in \(S_{3}^{\mathrm{new}}(8, [\chi])\).