Properties

Label 3.43.b.a
Level 3
Weight 43
Character orbit 3.b
Self dual Yes
Analytic conductor 33.518
Analytic rank 0
Dimension 1
CM disc. -3
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 43 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(33.5183121516\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 10460353203q^{3} \) \(\mathstrut +\mathstrut 4398046511104q^{4} \) \(\mathstrut +\mathstrut 146246101081752386q^{7} \) \(\mathstrut +\mathstrut 109418989131512359209q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 10460353203q^{3} \) \(\mathstrut +\mathstrut 4398046511104q^{4} \) \(\mathstrut +\mathstrut 146246101081752386q^{7} \) \(\mathstrut +\mathstrut 109418989131512359209q^{9} \) \(\mathstrut -\mathstrut 46005119909369701466112q^{12} \) \(\mathstrut -\mathstrut 357172201574584820855926q^{13} \) \(\mathstrut +\mathstrut 19342813113834066795298816q^{16} \) \(\mathstrut -\mathstrut 165282232922573499748305862q^{19} \) \(\mathstrut -\mathstrut 1529785871876770335747992358q^{21} \) \(\mathstrut +\mathstrut 227373675443232059478759765625q^{25} \) \(\mathstrut -\mathstrut 1144561273430837494885949696427q^{27} \) \(\mathstrut +\mathstrut 643197154625164001525727494144q^{28} \) \(\mathstrut +\mathstrut 40071891552502334872877231380562q^{31} \) \(\mathstrut +\mathstrut 481229803398374426442198455156736q^{36} \) \(\mathstrut +\mathstrut 1627039758795395209154445850600826q^{37} \) \(\mathstrut +\mathstrut 3736147382763269974235466735630978q^{39} \) \(\mathstrut +\mathstrut 30195211903983689990258309024586314q^{43} \) \(\mathstrut -\mathstrut 202332657110324584212719915287707648q^{48} \) \(\mathstrut -\mathstrut 290585560202928234894752897205283053q^{49} \) \(\mathstrut -\mathstrut 1570859954998437386602722199343202304q^{52} \) \(\mathstrut +\mathstrut 1728910534550633759095110917395375986q^{57} \) \(\mathstrut +\mathstrut 55873936229113631301915176656255273322q^{61} \) \(\mathstrut +\mathstrut 16002100544790322202836897262824822674q^{63} \) \(\mathstrut +\mathstrut 85070591730234615865843651857942052864q^{64} \) \(\mathstrut -\mathstrut 397023094599476479692793424374906556134q^{67} \) \(\mathstrut -\mathstrut 1167857467376379700865578403060403787246q^{73} \) \(\mathstrut -\mathstrut 2378408954200494918040931224822998046875q^{75} \) \(\mathstrut -\mathstrut 726918947852603065933043618503771291648q^{76} \) \(\mathstrut +\mathstrut 14082525970872707793274472783233500883058q^{79} \) \(\mathstrut +\mathstrut 11972515182562019788602740026717047105681q^{81} \) \(\mathstrut -\mathstrut 6728069416543820527759940480274354143232q^{84} \) \(\mathstrut -\mathstrut 52235041895068770435557926308880512739436q^{91} \) \(\mathstrut -\mathstrut 419166139151486441252479945097433828640086q^{93} \) \(\mathstrut +\mathstrut 226617616347969910436414292537604801615106q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0
0 −1.04604e10 4.39805e12 0 0 1.46246e17 0 1.09419e20 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
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) acting on \(S_{43}^{\mathrm{new}}(3, [\chi])\).