Properties

Label 121.2.a.d
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: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} - q^{3} + 2q^{4} + q^{5} - 2q^{6} + 2q^{7} - 2q^{9} + O(q^{10}) \) \( q + 2q^{2} - q^{3} + 2q^{4} + q^{5} - 2q^{6} + 2q^{7} - 2q^{9} + 2q^{10} - 2q^{12} - 4q^{13} + 4q^{14} - q^{15} - 4q^{16} + 2q^{17} - 4q^{18} + 2q^{20} - 2q^{21} - q^{23} - 4q^{25} - 8q^{26} + 5q^{27} + 4q^{28} - 2q^{30} + 7q^{31} - 8q^{32} + 4q^{34} + 2q^{35} - 4q^{36} + 3q^{37} + 4q^{39} + 8q^{41} - 4q^{42} + 6q^{43} - 2q^{45} - 2q^{46} + 8q^{47} + 4q^{48} - 3q^{49} - 8q^{50} - 2q^{51} - 8q^{52} - 6q^{53} + 10q^{54} + 5q^{59} - 2q^{60} - 12q^{61} + 14q^{62} - 4q^{63} - 8q^{64} - 4q^{65} - 7q^{67} + 4q^{68} + q^{69} + 4q^{70} - 3q^{71} - 4q^{73} + 6q^{74} + 4q^{75} + 8q^{78} + 10q^{79} - 4q^{80} + q^{81} + 16q^{82} + 6q^{83} - 4q^{84} + 2q^{85} + 12q^{86} + 15q^{89} - 4q^{90} - 8q^{91} - 2q^{92} - 7q^{93} + 16q^{94} + 8q^{96} - 7q^{97} - 6q^{98} + O(q^{100}) \)

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
2.00000 −1.00000 2.00000 1.00000 −2.00000 2.00000 0 −2.00000 2.00000
\(n\): 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.d 1
3.b odd 2 1 1089.2.a.b 1
4.b odd 2 1 1936.2.a.i 1
5.b even 2 1 3025.2.a.a 1
7.b odd 2 1 5929.2.a.h 1
8.b even 2 1 7744.2.a.x 1
8.d odd 2 1 7744.2.a.k 1
11.b odd 2 1 11.2.a.a 1
11.c even 5 4 121.2.c.a 4
11.d odd 10 4 121.2.c.e 4
33.d even 2 1 99.2.a.d 1
44.c even 2 1 176.2.a.b 1
55.d odd 2 1 275.2.a.b 1
55.e even 4 2 275.2.b.a 2
77.b even 2 1 539.2.a.a 1
77.h odd 6 2 539.2.e.h 2
77.i even 6 2 539.2.e.g 2
88.b odd 2 1 704.2.a.h 1
88.g even 2 1 704.2.a.c 1
99.g even 6 2 891.2.e.b 2
99.h odd 6 2 891.2.e.k 2
132.d odd 2 1 1584.2.a.g 1
143.d odd 2 1 1859.2.a.b 1
165.d even 2 1 2475.2.a.a 1
165.l odd 4 2 2475.2.c.a 2
176.i even 4 2 2816.2.c.f 2
176.l odd 4 2 2816.2.c.j 2
187.b odd 2 1 3179.2.a.a 1
209.d even 2 1 3971.2.a.b 1
220.g even 2 1 4400.2.a.i 1
220.i odd 4 2 4400.2.b.h 2
231.h odd 2 1 4851.2.a.t 1
253.b even 2 1 5819.2.a.a 1
264.m even 2 1 6336.2.a.br 1
264.p odd 2 1 6336.2.a.bu 1
308.g odd 2 1 8624.2.a.j 1
319.d odd 2 1 9251.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 11.b odd 2 1
99.2.a.d 1 33.d even 2 1
121.2.a.d 1 1.a even 1 1 trivial
121.2.c.a 4 11.c even 5 4
121.2.c.e 4 11.d odd 10 4
176.2.a.b 1 44.c even 2 1
275.2.a.b 1 55.d odd 2 1
275.2.b.a 2 55.e even 4 2
539.2.a.a 1 77.b even 2 1
539.2.e.g 2 77.i even 6 2
539.2.e.h 2 77.h odd 6 2
704.2.a.c 1 88.g even 2 1
704.2.a.h 1 88.b odd 2 1
891.2.e.b 2 99.g even 6 2
891.2.e.k 2 99.h odd 6 2
1089.2.a.b 1 3.b odd 2 1
1584.2.a.g 1 132.d odd 2 1
1859.2.a.b 1 143.d odd 2 1
1936.2.a.i 1 4.b odd 2 1
2475.2.a.a 1 165.d even 2 1
2475.2.c.a 2 165.l odd 4 2
2816.2.c.f 2 176.i even 4 2
2816.2.c.j 2 176.l odd 4 2
3025.2.a.a 1 5.b even 2 1
3179.2.a.a 1 187.b odd 2 1
3971.2.a.b 1 209.d even 2 1
4400.2.a.i 1 220.g even 2 1
4400.2.b.h 2 220.i odd 4 2
4851.2.a.t 1 231.h odd 2 1
5819.2.a.a 1 253.b even 2 1
5929.2.a.h 1 7.b odd 2 1
6336.2.a.br 1 264.m even 2 1
6336.2.a.bu 1 264.p odd 2 1
7744.2.a.k 1 8.d odd 2 1
7744.2.a.x 1 8.b even 2 1
8624.2.a.j 1 308.g odd 2 1
9251.2.a.d 1 319.d odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( 1 + T \)
$5$ \( -1 + T \)
$7$ \( -2 + T \)
$11$ \( T \)
$13$ \( 4 + T \)
$17$ \( -2 + T \)
$19$ \( T \)
$23$ \( 1 + T \)
$29$ \( T \)
$31$ \( -7 + T \)
$37$ \( -3 + T \)
$41$ \( -8 + T \)
$43$ \( -6 + T \)
$47$ \( -8 + T \)
$53$ \( 6 + T \)
$59$ \( -5 + T \)
$61$ \( 12 + T \)
$67$ \( 7 + T \)
$71$ \( 3 + T \)
$73$ \( 4 + T \)
$79$ \( -10 + T \)
$83$ \( -6 + T \)
$89$ \( -15 + T \)
$97$ \( 7 + T \)
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