Properties

Label 8.16.a.a
Level 8
Weight 16
Character orbit 8.a
Self dual Yes
Analytic conductor 11.415
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 16 \)
Character orbit: \([\chi]\) = 8.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(11.415480408\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 3444q^{3} \) \(\mathstrut +\mathstrut 313358q^{5} \) \(\mathstrut -\mathstrut 2324616q^{7} \) \(\mathstrut -\mathstrut 2487771q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 3444q^{3} \) \(\mathstrut +\mathstrut 313358q^{5} \) \(\mathstrut -\mathstrut 2324616q^{7} \) \(\mathstrut -\mathstrut 2487771q^{9} \) \(\mathstrut -\mathstrut 55249084q^{11} \) \(\mathstrut -\mathstrut 110259578q^{13} \) \(\mathstrut -\mathstrut 1079204952q^{15} \) \(\mathstrut -\mathstrut 2601428750q^{17} \) \(\mathstrut +\mathstrut 1952124284q^{19} \) \(\mathstrut +\mathstrut 8005977504q^{21} \) \(\mathstrut -\mathstrut 25430340376q^{23} \) \(\mathstrut +\mathstrut 67675658039q^{25} \) \(\mathstrut +\mathstrut 57985519032q^{27} \) \(\mathstrut -\mathstrut 2277224202q^{29} \) \(\mathstrut -\mathstrut 190667257120q^{31} \) \(\mathstrut +\mathstrut 190277845296q^{33} \) \(\mathstrut -\mathstrut 728437020528q^{35} \) \(\mathstrut -\mathstrut 288229450002q^{37} \) \(\mathstrut +\mathstrut 379733986632q^{39} \) \(\mathstrut +\mathstrut 756412456602q^{41} \) \(\mathstrut -\mathstrut 354186592988q^{43} \) \(\mathstrut -\mathstrut 779562945018q^{45} \) \(\mathstrut +\mathstrut 6035922573648q^{47} \) \(\mathstrut +\mathstrut 656278037513q^{49} \) \(\mathstrut +\mathstrut 8959320615000q^{51} \) \(\mathstrut -\mathstrut 12198920684962q^{53} \) \(\mathstrut -\mathstrut 17312742464072q^{55} \) \(\mathstrut -\mathstrut 6723116034096q^{57} \) \(\mathstrut -\mathstrut 4090911936748q^{59} \) \(\mathstrut +\mathstrut 17565907389910q^{61} \) \(\mathstrut +\mathstrut 5783112270936q^{63} \) \(\mathstrut -\mathstrut 34550720842924q^{65} \) \(\mathstrut -\mathstrut 3931246965172q^{67} \) \(\mathstrut +\mathstrut 87582092254944q^{69} \) \(\mathstrut +\mathstrut 58825436072248q^{71} \) \(\mathstrut +\mathstrut 107571519617914q^{73} \) \(\mathstrut -\mathstrut 233074966286316q^{75} \) \(\mathstrut +\mathstrut 128432904651744q^{77} \) \(\mathstrut +\mathstrut 61543860115504q^{79} \) \(\mathstrut -\mathstrut 164005332829911q^{81} \) \(\mathstrut +\mathstrut 13432070277436q^{83} \) \(\mathstrut -\mathstrut 815178510242500q^{85} \) \(\mathstrut +\mathstrut 7842760151688q^{87} \) \(\mathstrut +\mathstrut 269696339030634q^{89} \) \(\mathstrut +\mathstrut 256311179172048q^{91} \) \(\mathstrut +\mathstrut 656658033521280q^{93} \) \(\mathstrut +\mathstrut 611713761385672q^{95} \) \(\mathstrut -\mathstrut 793796744596318q^{97} \) \(\mathstrut +\mathstrut 137447068951764q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −3444.00 0 313358. 0 −2.32462e6 0 −2.48777e6 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) \(\mathstrut +\mathstrut 3444 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(8))\).