Properties

Label 121.2.a.b
Level 121
Weight 2
Character orbit 121.a
Self dual yes
Analytic conductor 0.966
Analytic rank 1
Dimension 1
CM discriminant -11
Inner twists 2

Related objects

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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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 2q^{4} - 3q^{5} - 2q^{9} + O(q^{10}) \) \( q - q^{3} - 2q^{4} - 3q^{5} - 2q^{9} + 2q^{12} + 3q^{15} + 4q^{16} + 6q^{20} - 9q^{23} + 4q^{25} + 5q^{27} - 5q^{31} + 4q^{36} + 7q^{37} + 6q^{45} - 12q^{47} - 4q^{48} - 7q^{49} + 6q^{53} - 15q^{59} - 6q^{60} - 8q^{64} + 13q^{67} + 9q^{69} - 3q^{71} - 4q^{75} - 12q^{80} + q^{81} - 9q^{89} + 18q^{92} + 5q^{93} + 17q^{97} + O(q^{100}) \)

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
0 −1.00000 −2.00000 −3.00000 0 0 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.2.a.b 1
3.b odd 2 1 1089.2.a.g 1
4.b odd 2 1 1936.2.a.h 1
5.b even 2 1 3025.2.a.d 1
7.b odd 2 1 5929.2.a.e 1
8.b even 2 1 7744.2.a.bb 1
8.d odd 2 1 7744.2.a.n 1
11.b odd 2 1 CM 121.2.a.b 1
11.c even 5 4 121.2.c.c 4
11.d odd 10 4 121.2.c.c 4
33.d even 2 1 1089.2.a.g 1
44.c even 2 1 1936.2.a.h 1
55.d odd 2 1 3025.2.a.d 1
77.b even 2 1 5929.2.a.e 1
88.b odd 2 1 7744.2.a.bb 1
88.g even 2 1 7744.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.2.a.b 1 1.a even 1 1 trivial
121.2.a.b 1 11.b odd 2 1 CM
121.2.c.c 4 11.c even 5 4
121.2.c.c 4 11.d odd 10 4
1089.2.a.g 1 3.b odd 2 1
1089.2.a.g 1 33.d even 2 1
1936.2.a.h 1 4.b odd 2 1
1936.2.a.h 1 44.c even 2 1
3025.2.a.d 1 5.b even 2 1
3025.2.a.d 1 55.d odd 2 1
5929.2.a.e 1 7.b odd 2 1
5929.2.a.e 1 77.b even 2 1
7744.2.a.n 1 8.d odd 2 1
7744.2.a.n 1 88.g even 2 1
7744.2.a.bb 1 8.b even 2 1
7744.2.a.bb 1 88.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} \)
$3$ \( 1 + T + 3 T^{2} \)
$5$ \( 1 + 3 T + 5 T^{2} \)
$7$ \( 1 + 7 T^{2} \)
$11$ 1
$13$ \( 1 + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 + 19 T^{2} \)
$23$ \( 1 + 9 T + 23 T^{2} \)
$29$ \( 1 + 29 T^{2} \)
$31$ \( 1 + 5 T + 31 T^{2} \)
$37$ \( 1 - 7 T + 37 T^{2} \)
$41$ \( 1 + 41 T^{2} \)
$43$ \( 1 + 43 T^{2} \)
$47$ \( 1 + 12 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 15 T + 59 T^{2} \)
$61$ \( 1 + 61 T^{2} \)
$67$ \( 1 - 13 T + 67 T^{2} \)
$71$ \( 1 + 3 T + 71 T^{2} \)
$73$ \( 1 + 73 T^{2} \)
$79$ \( 1 + 79 T^{2} \)
$83$ \( 1 + 83 T^{2} \)
$89$ \( 1 + 9 T + 89 T^{2} \)
$97$ \( 1 - 17 T + 97 T^{2} \)
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