Properties

Label 99.6.a.a
Level $99$
Weight $6$
Character orbit 99.a
Self dual yes
Analytic conductor $15.878$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 99.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.8779981615\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - 31q^{4} + 92q^{5} - 26q^{7} + 63q^{8} + O(q^{10}) \) \( q - q^{2} - 31q^{4} + 92q^{5} - 26q^{7} + 63q^{8} - 92q^{10} - 121q^{11} - 692q^{13} + 26q^{14} + 929q^{16} + 1442q^{17} + 2160q^{19} - 2852q^{20} + 121q^{22} + 1582q^{23} + 5339q^{25} + 692q^{26} + 806q^{28} + 5526q^{29} + 4792q^{31} - 2945q^{32} - 1442q^{34} - 2392q^{35} - 10194q^{37} - 2160q^{38} + 5796q^{40} + 10622q^{41} + 8580q^{43} + 3751q^{44} - 1582q^{46} + 2362q^{47} - 16131q^{49} - 5339q^{50} + 21452q^{52} + 30804q^{53} - 11132q^{55} - 1638q^{56} - 5526q^{58} - 6416q^{59} + 42096q^{61} - 4792q^{62} - 26783q^{64} - 63664q^{65} - 28444q^{67} - 44702q^{68} + 2392q^{70} - 45690q^{71} - 18374q^{73} + 10194q^{74} - 66960q^{76} + 3146q^{77} - 105214q^{79} + 85468q^{80} - 10622q^{82} - 62292q^{83} + 132664q^{85} - 8580q^{86} - 7623q^{88} + 72246q^{89} + 17992q^{91} - 49042q^{92} - 2362q^{94} + 198720q^{95} + 79262q^{97} + 16131q^{98} + O(q^{100}) \)

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
−1.00000 0 −31.0000 92.0000 0 −26.0000 63.0000 0 −92.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.6.a.a 1
3.b odd 2 1 33.6.a.b 1
11.b odd 2 1 1089.6.a.h 1
12.b even 2 1 528.6.a.a 1
15.d odd 2 1 825.6.a.a 1
33.d even 2 1 363.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.b 1 3.b odd 2 1
99.6.a.a 1 1.a even 1 1 trivial
363.6.a.b 1 33.d even 2 1
528.6.a.a 1 12.b even 2 1
825.6.a.a 1 15.d odd 2 1
1089.6.a.h 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(99))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( -92 + T \)
$7$ \( 26 + T \)
$11$ \( 121 + T \)
$13$ \( 692 + T \)
$17$ \( -1442 + T \)
$19$ \( -2160 + T \)
$23$ \( -1582 + T \)
$29$ \( -5526 + T \)
$31$ \( -4792 + T \)
$37$ \( 10194 + T \)
$41$ \( -10622 + T \)
$43$ \( -8580 + T \)
$47$ \( -2362 + T \)
$53$ \( -30804 + T \)
$59$ \( 6416 + T \)
$61$ \( -42096 + T \)
$67$ \( 28444 + T \)
$71$ \( 45690 + T \)
$73$ \( 18374 + T \)
$79$ \( 105214 + T \)
$83$ \( 62292 + T \)
$89$ \( -72246 + T \)
$97$ \( -79262 + T \)
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