Properties

Label 99.4.a.a
Level $99$
Weight $4$
Character orbit 99.a
Self dual yes
Analytic conductor $5.841$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 99.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.84118909057\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + q^{2} - 7 q^{4} + 4 q^{5} - 26 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 7 q^{4} + 4 q^{5} - 26 q^{7} - 15 q^{8} + 4 q^{10} - 11 q^{11} - 32 q^{13} - 26 q^{14} + 41 q^{16} - 74 q^{17} - 60 q^{19} - 28 q^{20} - 11 q^{22} + 182 q^{23} - 109 q^{25} - 32 q^{26} + 182 q^{28} + 90 q^{29} - 8 q^{31} + 161 q^{32} - 74 q^{34} - 104 q^{35} - 66 q^{37} - 60 q^{38} - 60 q^{40} - 422 q^{41} + 408 q^{43} + 77 q^{44} + 182 q^{46} + 506 q^{47} + 333 q^{49} - 109 q^{50} + 224 q^{52} - 348 q^{53} - 44 q^{55} + 390 q^{56} + 90 q^{58} + 200 q^{59} + 132 q^{61} - 8 q^{62} - 167 q^{64} - 128 q^{65} - 1036 q^{67} + 518 q^{68} - 104 q^{70} - 762 q^{71} - 542 q^{73} - 66 q^{74} + 420 q^{76} + 286 q^{77} - 550 q^{79} + 164 q^{80} - 422 q^{82} + 132 q^{83} - 296 q^{85} + 408 q^{86} + 165 q^{88} - 570 q^{89} + 832 q^{91} - 1274 q^{92} + 506 q^{94} - 240 q^{95} + 14 q^{97} + 333 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 −7.00000 4.00000 0 −26.0000 −15.0000 0 4.00000
\(n\): 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 99.4.a.a 1
3.b odd 2 1 33.4.a.b 1
4.b odd 2 1 1584.4.a.l 1
5.b even 2 1 2475.4.a.e 1
11.b odd 2 1 1089.4.a.e 1
12.b even 2 1 528.4.a.h 1
15.d odd 2 1 825.4.a.f 1
15.e even 4 2 825.4.c.f 2
21.c even 2 1 1617.4.a.d 1
24.f even 2 1 2112.4.a.h 1
24.h odd 2 1 2112.4.a.u 1
33.d even 2 1 363.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.b 1 3.b odd 2 1
99.4.a.a 1 1.a even 1 1 trivial
363.4.a.d 1 33.d even 2 1
528.4.a.h 1 12.b even 2 1
825.4.a.f 1 15.d odd 2 1
825.4.c.f 2 15.e even 4 2
1089.4.a.e 1 11.b odd 2 1
1584.4.a.l 1 4.b odd 2 1
1617.4.a.d 1 21.c even 2 1
2112.4.a.h 1 24.f even 2 1
2112.4.a.u 1 24.h odd 2 1
2475.4.a.e 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 4 \) Copy content Toggle raw display
$7$ \( T + 26 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T + 32 \) Copy content Toggle raw display
$17$ \( T + 74 \) Copy content Toggle raw display
$19$ \( T + 60 \) Copy content Toggle raw display
$23$ \( T - 182 \) Copy content Toggle raw display
$29$ \( T - 90 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T + 66 \) Copy content Toggle raw display
$41$ \( T + 422 \) Copy content Toggle raw display
$43$ \( T - 408 \) Copy content Toggle raw display
$47$ \( T - 506 \) Copy content Toggle raw display
$53$ \( T + 348 \) Copy content Toggle raw display
$59$ \( T - 200 \) Copy content Toggle raw display
$61$ \( T - 132 \) Copy content Toggle raw display
$67$ \( T + 1036 \) Copy content Toggle raw display
$71$ \( T + 762 \) Copy content Toggle raw display
$73$ \( T + 542 \) Copy content Toggle raw display
$79$ \( T + 550 \) Copy content Toggle raw display
$83$ \( T - 132 \) Copy content Toggle raw display
$89$ \( T + 570 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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