Properties

Label 8.13.d.a
Level 8
Weight 13
Character orbit 8.d
Self dual Yes
Analytic conductor 7.312
Analytic rank 0
Dimension 1
CM disc. -8
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 13 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(7.31195053821\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 64q^{2} \) \(\mathstrut +\mathstrut 658q^{3} \) \(\mathstrut +\mathstrut 4096q^{4} \) \(\mathstrut +\mathstrut 42112q^{6} \) \(\mathstrut +\mathstrut 262144q^{8} \) \(\mathstrut -\mathstrut 98477q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 64q^{2} \) \(\mathstrut +\mathstrut 658q^{3} \) \(\mathstrut +\mathstrut 4096q^{4} \) \(\mathstrut +\mathstrut 42112q^{6} \) \(\mathstrut +\mathstrut 262144q^{8} \) \(\mathstrut -\mathstrut 98477q^{9} \) \(\mathstrut +\mathstrut 1923122q^{11} \) \(\mathstrut +\mathstrut 2695168q^{12} \) \(\mathstrut +\mathstrut 16777216q^{16} \) \(\mathstrut -\mathstrut 45296062q^{17} \) \(\mathstrut -\mathstrut 6302528q^{18} \) \(\mathstrut -\mathstrut 87931438q^{19} \) \(\mathstrut +\mathstrut 123079808q^{22} \) \(\mathstrut +\mathstrut 172490752q^{24} \) \(\mathstrut +\mathstrut 244140625q^{25} \) \(\mathstrut -\mathstrut 414486044q^{27} \) \(\mathstrut +\mathstrut 1073741824q^{32} \) \(\mathstrut +\mathstrut 1265414276q^{33} \) \(\mathstrut -\mathstrut 2898947968q^{34} \) \(\mathstrut -\mathstrut 403361792q^{36} \) \(\mathstrut -\mathstrut 5627612032q^{38} \) \(\mathstrut +\mathstrut 8628259682q^{41} \) \(\mathstrut -\mathstrut 7030618702q^{43} \) \(\mathstrut +\mathstrut 7877107712q^{44} \) \(\mathstrut +\mathstrut 11039408128q^{48} \) \(\mathstrut +\mathstrut 13841287201q^{49} \) \(\mathstrut +\mathstrut 15625000000q^{50} \) \(\mathstrut -\mathstrut 29804808796q^{51} \) \(\mathstrut -\mathstrut 26527106816q^{54} \) \(\mathstrut -\mathstrut 57858886204q^{57} \) \(\mathstrut +\mathstrut 8638314482q^{59} \) \(\mathstrut +\mathstrut 68719476736q^{64} \) \(\mathstrut +\mathstrut 80986513664q^{66} \) \(\mathstrut +\mathstrut 175045819538q^{67} \) \(\mathstrut -\mathstrut 185532669952q^{68} \) \(\mathstrut -\mathstrut 25815154688q^{72} \) \(\mathstrut +\mathstrut 49139489378q^{73} \) \(\mathstrut +\mathstrut 160644531250q^{75} \) \(\mathstrut -\mathstrut 360167170048q^{76} \) \(\mathstrut -\mathstrut 220397101595q^{81} \) \(\mathstrut +\mathstrut 552208619648q^{82} \) \(\mathstrut -\mathstrut 192940233262q^{83} \) \(\mathstrut -\mathstrut 449959596928q^{86} \) \(\mathstrut +\mathstrut 504134893568q^{88} \) \(\mathstrut -\mathstrut 866326445278q^{89} \) \(\mathstrut +\mathstrut 706522120192q^{96} \) \(\mathstrut +\mathstrut 1656488134658q^{97} \) \(\mathstrut +\mathstrut 885842380864q^{98} \) \(\mathstrut -\mathstrut 189383285194q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
64.0000 658.000 4096.00 0 42112.0 0 262144. −98477.0 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 \(T_{3} \) \(\mathstrut -\mathstrut 658 \) acting on \(S_{13}^{\mathrm{new}}(8, [\chi])\).