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Properties

Label 8.21.d.a
Level 8
Weight 21
Character orbit 8.d
Self dual Yes
Analytic conductor 20.281
Analytic rank 0
Dimension 1
CM disc. -8
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 21 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(20.2811012082\)
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

\(f(q)\) \(=\) \( q + 1024q^{2} + 114226q^{3} + 1048576q^{4} + 116967424q^{6} + 1073741824q^{8} + 9560794675q^{9} + O(q^{10}) \) \( q + 1024q^{2} + 114226q^{3} + 1048576q^{4} + 116967424q^{6} + 1073741824q^{8} + 9560794675q^{9} - 42383023726q^{11} + 119774642176q^{12} + 1099511627776q^{16} - 3353535763774q^{17} + 9790253747200q^{18} - 1014654432526q^{19} - 43400216295424q^{22} + 122649233588224q^{24} + 95367431640625q^{25} + 693809897557924q^{27} + 1125899906842624q^{32} - 4841243268126076q^{33} - 3434020622104576q^{34} + 10025219837132800q^{36} - 1039006138906624q^{38} - 25418071370591326q^{41} + 2781113986388498q^{43} - 44441821486514176q^{44} + 125592815194341376q^{48} + 79792266297612001q^{49} + 97656250000000000q^{50} - 383060976152848924q^{51} + 710461335099314176q^{54} - 115899917209714876q^{57} - 173912197184497198q^{59} + 1152921504606846976q^{64} - 4957433106561101824q^{66} - 356137514166464974q^{67} - 3516437117035085824q^{68} + 10265825113223987200q^{72} - 6016717170316692574q^{73} + 10893440246582031250q^{75} - 1063942286240382976q^{76} + 45914699624497562149q^{81} - 26028105083485517824q^{82} - 31022856480301602574q^{83} + 2847860722061821952q^{86} - 45508425202190516224q^{88} + 61202446863210984674q^{89} + 128607042759005569024q^{96} - 50009130514058267902q^{97} + 81707280688754689024q^{98} - 405215387549939459050q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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
1024.00 114226. 1.04858e6 0 1.16967e8 0 1.07374e9 9.56079e9 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 \( T_{3} - 114226 \) acting on \(S_{21}^{\mathrm{new}}(8, [\chi])\).