Properties

Label 198.2.a.e
Level $198$
Weight $2$
Character orbit 198.a
Self dual yes
Analytic conductor $1.581$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + 2 q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + 2 q^{7} + q^{8} + q^{11} - 4 q^{13} + 2 q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + q^{22} - 6 q^{23} - 5 q^{25} - 4 q^{26} + 2 q^{28} - 6 q^{29} + 8 q^{31} + q^{32} + 6 q^{34} - 10 q^{37} - 4 q^{38} - 6 q^{41} + 8 q^{43} + q^{44} - 6 q^{46} + 6 q^{47} - 3 q^{49} - 5 q^{50} - 4 q^{52} + 2 q^{56} - 6 q^{58} + 8 q^{61} + 8 q^{62} + q^{64} - 4 q^{67} + 6 q^{68} - 6 q^{71} + 2 q^{73} - 10 q^{74} - 4 q^{76} + 2 q^{77} + 14 q^{79} - 6 q^{82} + 12 q^{83} + 8 q^{86} + q^{88} + 6 q^{89} - 8 q^{91} - 6 q^{92} + 6 q^{94} + 14 q^{97} - 3 q^{98}+O(q^{100}) \) 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 1.00000 0 0 2.00000 1.00000 0 0
\(n\): 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.e 1
3.b odd 2 1 66.2.a.a 1
4.b odd 2 1 1584.2.a.h 1
5.b even 2 1 4950.2.a.g 1
5.c odd 4 2 4950.2.c.r 2
7.b odd 2 1 9702.2.a.bu 1
8.b even 2 1 6336.2.a.bj 1
8.d odd 2 1 6336.2.a.bf 1
9.c even 3 2 1782.2.e.f 2
9.d odd 6 2 1782.2.e.s 2
11.b odd 2 1 2178.2.a.b 1
12.b even 2 1 528.2.a.d 1
15.d odd 2 1 1650.2.a.m 1
15.e even 4 2 1650.2.c.d 2
21.c even 2 1 3234.2.a.d 1
24.f even 2 1 2112.2.a.v 1
24.h odd 2 1 2112.2.a.i 1
33.d even 2 1 726.2.a.i 1
33.f even 10 4 726.2.e.b 4
33.h odd 10 4 726.2.e.k 4
132.d odd 2 1 5808.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.a 1 3.b odd 2 1
198.2.a.e 1 1.a even 1 1 trivial
528.2.a.d 1 12.b even 2 1
726.2.a.i 1 33.d even 2 1
726.2.e.b 4 33.f even 10 4
726.2.e.k 4 33.h odd 10 4
1584.2.a.h 1 4.b odd 2 1
1650.2.a.m 1 15.d odd 2 1
1650.2.c.d 2 15.e even 4 2
1782.2.e.f 2 9.c even 3 2
1782.2.e.s 2 9.d odd 6 2
2112.2.a.i 1 24.h odd 2 1
2112.2.a.v 1 24.f even 2 1
2178.2.a.b 1 11.b odd 2 1
3234.2.a.d 1 21.c even 2 1
4950.2.a.g 1 5.b even 2 1
4950.2.c.r 2 5.c odd 4 2
5808.2.a.l 1 132.d odd 2 1
6336.2.a.bf 1 8.d odd 2 1
6336.2.a.bj 1 8.b even 2 1
9702.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(198))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display
\( T_{17} - 6 \) 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
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T - 6 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T + 6 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T - 14 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
show more
show less