Properties

Label 198.6.a.d
Level $198$
Weight $6$
Character orbit 198.a
Self dual yes
Analytic conductor $31.756$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 198.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.7559963230\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4 q^{2} + 16 q^{4} - 81 q^{5} + 98 q^{7} + 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 16 q^{4} - 81 q^{5} + 98 q^{7} + 64 q^{8} - 324 q^{10} - 121 q^{11} + 824 q^{13} + 392 q^{14} + 256 q^{16} - 978 q^{17} - 2140 q^{19} - 1296 q^{20} - 484 q^{22} - 3699 q^{23} + 3436 q^{25} + 3296 q^{26} + 1568 q^{28} - 3480 q^{29} - 7813 q^{31} + 1024 q^{32} - 3912 q^{34} - 7938 q^{35} - 13597 q^{37} - 8560 q^{38} - 5184 q^{40} - 6492 q^{41} + 14234 q^{43} - 1936 q^{44} - 14796 q^{46} + 20352 q^{47} - 7203 q^{49} + 13744 q^{50} + 13184 q^{52} + 366 q^{53} + 9801 q^{55} + 6272 q^{56} - 13920 q^{58} - 9825 q^{59} + 26132 q^{61} - 31252 q^{62} + 4096 q^{64} - 66744 q^{65} + 17093 q^{67} - 15648 q^{68} - 31752 q^{70} + 23583 q^{71} - 35176 q^{73} - 54388 q^{74} - 34240 q^{76} - 11858 q^{77} - 42490 q^{79} - 20736 q^{80} - 25968 q^{82} - 22674 q^{83} + 79218 q^{85} + 56936 q^{86} - 7744 q^{88} + 17145 q^{89} + 80752 q^{91} - 59184 q^{92} + 81408 q^{94} + 173340 q^{95} - 30727 q^{97} - 28812 q^{98}+O(q^{100}) \) 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
4.00000 0 16.0000 −81.0000 0 98.0000 64.0000 0 −324.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(1\)

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 198.6.a.d 1
3.b odd 2 1 22.6.a.a 1
12.b even 2 1 176.6.a.d 1
15.d odd 2 1 550.6.a.g 1
15.e even 4 2 550.6.b.g 2
21.c even 2 1 1078.6.a.b 1
24.f even 2 1 704.6.a.b 1
24.h odd 2 1 704.6.a.i 1
33.d even 2 1 242.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.a 1 3.b odd 2 1
176.6.a.d 1 12.b even 2 1
198.6.a.d 1 1.a even 1 1 trivial
242.6.a.c 1 33.d even 2 1
550.6.a.g 1 15.d odd 2 1
550.6.b.g 2 15.e even 4 2
704.6.a.b 1 24.f even 2 1
704.6.a.i 1 24.h odd 2 1
1078.6.a.b 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(198))\):

\( T_{5} + 81 \) Copy content Toggle raw display
\( T_{7} - 98 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 81 \) Copy content Toggle raw display
$7$ \( T - 98 \) Copy content Toggle raw display
$11$ \( T + 121 \) Copy content Toggle raw display
$13$ \( T - 824 \) Copy content Toggle raw display
$17$ \( T + 978 \) Copy content Toggle raw display
$19$ \( T + 2140 \) Copy content Toggle raw display
$23$ \( T + 3699 \) Copy content Toggle raw display
$29$ \( T + 3480 \) Copy content Toggle raw display
$31$ \( T + 7813 \) Copy content Toggle raw display
$37$ \( T + 13597 \) Copy content Toggle raw display
$41$ \( T + 6492 \) Copy content Toggle raw display
$43$ \( T - 14234 \) Copy content Toggle raw display
$47$ \( T - 20352 \) Copy content Toggle raw display
$53$ \( T - 366 \) Copy content Toggle raw display
$59$ \( T + 9825 \) Copy content Toggle raw display
$61$ \( T - 26132 \) Copy content Toggle raw display
$67$ \( T - 17093 \) Copy content Toggle raw display
$71$ \( T - 23583 \) Copy content Toggle raw display
$73$ \( T + 35176 \) Copy content Toggle raw display
$79$ \( T + 42490 \) Copy content Toggle raw display
$83$ \( T + 22674 \) Copy content Toggle raw display
$89$ \( T - 17145 \) Copy content Toggle raw display
$97$ \( T + 30727 \) Copy content Toggle raw display
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