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 discriminant -3
Inner twists 2

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Newspace parameters

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

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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.43.b.a 1
3.b odd 2 1 CM 3.43.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.43.b.a 1 1.a even 1 1 trivial
3.43.b.a 1 3.b odd 2 1 CM

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2097152 T )( 1 + 2097152 T ) \)
$3$ \( 1 + 10460353203 T \)
$5$ \( ( 1 - 476837158203125 T )( 1 + 476837158203125 T ) \)
$7$ \( 1 - 146246101081752386 T + \)\(31\!\cdots\!49\)\( T^{2} \)
$11$ \( ( 1 - \)\(74\!\cdots\!11\)\( T )( 1 + \)\(74\!\cdots\!11\)\( T ) \)
$13$ \( 1 + \)\(35\!\cdots\!26\)\( T + \)\(61\!\cdots\!69\)\( T^{2} \)
$17$ \( ( 1 - \)\(69\!\cdots\!17\)\( T )( 1 + \)\(69\!\cdots\!17\)\( T ) \)
$19$ \( 1 + \)\(16\!\cdots\!62\)\( T + \)\(51\!\cdots\!61\)\( T^{2} \)
$23$ \( ( 1 - \)\(39\!\cdots\!23\)\( T )( 1 + \)\(39\!\cdots\!23\)\( T ) \)
$29$ \( ( 1 - \)\(51\!\cdots\!29\)\( T )( 1 + \)\(51\!\cdots\!29\)\( T ) \)
$31$ \( 1 - \)\(40\!\cdots\!62\)\( T + \)\(43\!\cdots\!61\)\( T^{2} \)
$37$ \( 1 - \)\(16\!\cdots\!26\)\( T + \)\(73\!\cdots\!69\)\( T^{2} \)
$41$ \( ( 1 - \)\(73\!\cdots\!41\)\( T )( 1 + \)\(73\!\cdots\!41\)\( T ) \)
$43$ \( 1 - \)\(30\!\cdots\!14\)\( T + \)\(40\!\cdots\!49\)\( T^{2} \)
$47$ \( ( 1 - \)\(13\!\cdots\!47\)\( T )( 1 + \)\(13\!\cdots\!47\)\( T ) \)
$53$ \( ( 1 - \)\(16\!\cdots\!53\)\( T )( 1 + \)\(16\!\cdots\!53\)\( T ) \)
$59$ \( ( 1 - \)\(15\!\cdots\!59\)\( T )( 1 + \)\(15\!\cdots\!59\)\( T ) \)
$61$ \( 1 - \)\(55\!\cdots\!22\)\( T + \)\(96\!\cdots\!21\)\( T^{2} \)
$67$ \( 1 + \)\(39\!\cdots\!34\)\( T + \)\(49\!\cdots\!89\)\( T^{2} \)
$71$ \( ( 1 - \)\(75\!\cdots\!71\)\( T )( 1 + \)\(75\!\cdots\!71\)\( T ) \)
$73$ \( 1 + \)\(11\!\cdots\!46\)\( T + \)\(18\!\cdots\!29\)\( T^{2} \)
$79$ \( 1 - \)\(14\!\cdots\!58\)\( T + \)\(50\!\cdots\!41\)\( T^{2} \)
$83$ \( ( 1 - \)\(19\!\cdots\!83\)\( T )( 1 + \)\(19\!\cdots\!83\)\( T ) \)
$89$ \( ( 1 - \)\(86\!\cdots\!89\)\( T )( 1 + \)\(86\!\cdots\!89\)\( T ) \)
$97$ \( 1 - \)\(22\!\cdots\!06\)\( T + \)\(27\!\cdots\!09\)\( T^{2} \)
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