Properties

Label 198.2.a.a
Level $198$
Weight $2$
Character orbit 198.a
Self dual yes
Analytic conductor $1.581$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 198.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.58103796002\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - 2q^{5} - 4q^{7} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} - 2q^{5} - 4q^{7} - q^{8} + 2q^{10} + q^{11} - 6q^{13} + 4q^{14} + q^{16} - 2q^{17} + 4q^{19} - 2q^{20} - q^{22} - 4q^{23} - q^{25} + 6q^{26} - 4q^{28} - 6q^{29} - q^{32} + 2q^{34} + 8q^{35} + 6q^{37} - 4q^{38} + 2q^{40} + 6q^{41} + 4q^{43} + q^{44} + 4q^{46} + 12q^{47} + 9q^{49} + q^{50} - 6q^{52} - 2q^{53} - 2q^{55} + 4q^{56} + 6q^{58} - 12q^{59} - 14q^{61} + q^{64} + 12q^{65} + 4q^{67} - 2q^{68} - 8q^{70} + 12q^{71} - 6q^{73} - 6q^{74} + 4q^{76} - 4q^{77} - 4q^{79} - 2q^{80} - 6q^{82} - 4q^{83} + 4q^{85} - 4q^{86} - q^{88} - 10q^{89} + 24q^{91} - 4q^{92} - 12q^{94} - 8q^{95} - 14q^{97} - 9q^{98} + O(q^{100}) \)

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 1.00000 −2.00000 0 −4.00000 −1.00000 0 2.00000
\(n\): 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 198.2.a.a 1
3.b odd 2 1 66.2.a.b 1
4.b odd 2 1 1584.2.a.f 1
5.b even 2 1 4950.2.a.bu 1
5.c odd 4 2 4950.2.c.p 2
7.b odd 2 1 9702.2.a.x 1
8.b even 2 1 6336.2.a.bw 1
8.d odd 2 1 6336.2.a.cj 1
9.c even 3 2 1782.2.e.v 2
9.d odd 6 2 1782.2.e.e 2
11.b odd 2 1 2178.2.a.g 1
12.b even 2 1 528.2.a.j 1
15.d odd 2 1 1650.2.a.k 1
15.e even 4 2 1650.2.c.e 2
21.c even 2 1 3234.2.a.t 1
24.f even 2 1 2112.2.a.e 1
24.h odd 2 1 2112.2.a.r 1
33.d even 2 1 726.2.a.c 1
33.f even 10 4 726.2.e.o 4
33.h odd 10 4 726.2.e.g 4
132.d odd 2 1 5808.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.b 1 3.b odd 2 1
198.2.a.a 1 1.a even 1 1 trivial
528.2.a.j 1 12.b even 2 1
726.2.a.c 1 33.d even 2 1
726.2.e.g 4 33.h odd 10 4
726.2.e.o 4 33.f even 10 4
1584.2.a.f 1 4.b odd 2 1
1650.2.a.k 1 15.d odd 2 1
1650.2.c.e 2 15.e even 4 2
1782.2.e.e 2 9.d odd 6 2
1782.2.e.v 2 9.c even 3 2
2112.2.a.e 1 24.f even 2 1
2112.2.a.r 1 24.h odd 2 1
2178.2.a.g 1 11.b odd 2 1
3234.2.a.t 1 21.c even 2 1
4950.2.a.bu 1 5.b even 2 1
4950.2.c.p 2 5.c odd 4 2
5808.2.a.bc 1 132.d odd 2 1
6336.2.a.bw 1 8.b even 2 1
6336.2.a.cj 1 8.d odd 2 1
9702.2.a.x 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(198))\):

\( T_{5} + 2 \)
\( T_{13} + 6 \)
\( T_{17} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( 2 + T \)
$7$ \( 4 + T \)
$11$ \( -1 + T \)
$13$ \( 6 + T \)
$17$ \( 2 + T \)
$19$ \( -4 + T \)
$23$ \( 4 + T \)
$29$ \( 6 + T \)
$31$ \( T \)
$37$ \( -6 + T \)
$41$ \( -6 + T \)
$43$ \( -4 + T \)
$47$ \( -12 + T \)
$53$ \( 2 + T \)
$59$ \( 12 + T \)
$61$ \( 14 + T \)
$67$ \( -4 + T \)
$71$ \( -12 + T \)
$73$ \( 6 + T \)
$79$ \( 4 + T \)
$83$ \( 4 + T \)
$89$ \( 10 + T \)
$97$ \( 14 + T \)
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