Properties

Label 121.2.a.a
Level $121$
Weight $2$
Character orbit 121.a
Self dual yes
Analytic conductor $0.966$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 121.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.966189864457\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + 2q^{3} - q^{4} + q^{5} - 2q^{6} + 2q^{7} + 3q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + 2q^{3} - q^{4} + q^{5} - 2q^{6} + 2q^{7} + 3q^{8} + q^{9} - q^{10} - 2q^{12} - q^{13} - 2q^{14} + 2q^{15} - q^{16} + 5q^{17} - q^{18} - 6q^{19} - q^{20} + 4q^{21} + 2q^{23} + 6q^{24} - 4q^{25} + q^{26} - 4q^{27} - 2q^{28} - 9q^{29} - 2q^{30} - 2q^{31} - 5q^{32} - 5q^{34} + 2q^{35} - q^{36} - 3q^{37} + 6q^{38} - 2q^{39} + 3q^{40} + 5q^{41} - 4q^{42} + q^{45} - 2q^{46} + 2q^{47} - 2q^{48} - 3q^{49} + 4q^{50} + 10q^{51} + q^{52} + 9q^{53} + 4q^{54} + 6q^{56} - 12q^{57} + 9q^{58} + 8q^{59} - 2q^{60} - 6q^{61} + 2q^{62} + 2q^{63} + 7q^{64} - q^{65} + 2q^{67} - 5q^{68} + 4q^{69} - 2q^{70} + 12q^{71} + 3q^{72} + 2q^{73} + 3q^{74} - 8q^{75} + 6q^{76} + 2q^{78} + 10q^{79} - q^{80} - 11q^{81} - 5q^{82} - 6q^{83} - 4q^{84} + 5q^{85} - 18q^{87} - 9q^{89} - q^{90} - 2q^{91} - 2q^{92} - 4q^{93} - 2q^{94} - 6q^{95} - 10q^{96} - 13q^{97} + 3q^{98} + O(q^{100}) \)

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 2.00000 −1.00000 1.00000 −2.00000 2.00000 3.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 121.2.a.a 1
3.b odd 2 1 1089.2.a.i 1
4.b odd 2 1 1936.2.a.a 1
5.b even 2 1 3025.2.a.e 1
7.b odd 2 1 5929.2.a.a 1
8.b even 2 1 7744.2.a.f 1
8.d odd 2 1 7744.2.a.be 1
11.b odd 2 1 121.2.a.c yes 1
11.c even 5 4 121.2.c.d 4
11.d odd 10 4 121.2.c.b 4
33.d even 2 1 1089.2.a.c 1
44.c even 2 1 1936.2.a.b 1
55.d odd 2 1 3025.2.a.b 1
77.b even 2 1 5929.2.a.g 1
88.b odd 2 1 7744.2.a.c 1
88.g even 2 1 7744.2.a.bf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.2.a.a 1 1.a even 1 1 trivial
121.2.a.c yes 1 11.b odd 2 1
121.2.c.b 4 11.d odd 10 4
121.2.c.d 4 11.c even 5 4
1089.2.a.c 1 33.d even 2 1
1089.2.a.i 1 3.b odd 2 1
1936.2.a.a 1 4.b odd 2 1
1936.2.a.b 1 44.c even 2 1
3025.2.a.b 1 55.d odd 2 1
3025.2.a.e 1 5.b even 2 1
5929.2.a.a 1 7.b odd 2 1
5929.2.a.g 1 77.b even 2 1
7744.2.a.c 1 88.b odd 2 1
7744.2.a.f 1 8.b even 2 1
7744.2.a.be 1 8.d odd 2 1
7744.2.a.bf 1 88.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(121))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -2 + T \)
$5$ \( -1 + T \)
$7$ \( -2 + T \)
$11$ \( T \)
$13$ \( 1 + T \)
$17$ \( -5 + T \)
$19$ \( 6 + T \)
$23$ \( -2 + T \)
$29$ \( 9 + T \)
$31$ \( 2 + T \)
$37$ \( 3 + T \)
$41$ \( -5 + T \)
$43$ \( T \)
$47$ \( -2 + T \)
$53$ \( -9 + T \)
$59$ \( -8 + T \)
$61$ \( 6 + T \)
$67$ \( -2 + T \)
$71$ \( -12 + T \)
$73$ \( -2 + T \)
$79$ \( -10 + T \)
$83$ \( 6 + T \)
$89$ \( 9 + T \)
$97$ \( 13 + T \)
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