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

Related objects

Downloads

Learn more about

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

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

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
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 \(T_{3} \) \(\mathstrut -\mathstrut 114226 \) acting on \(S_{21}^{\mathrm{new}}(8, [\chi])\).