Properties

Label 121.12.a.b
Level 121
Weight 12
Character orbit 121.a
Self dual yes
Analytic conductor 92.970
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 121 = 11^{2} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 121.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9695248477\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 24q^{2} + 252q^{3} - 1472q^{4} + 4830q^{5} + 6048q^{6} + 16744q^{7} - 84480q^{8} - 113643q^{9} + O(q^{10}) \) \( q + 24q^{2} + 252q^{3} - 1472q^{4} + 4830q^{5} + 6048q^{6} + 16744q^{7} - 84480q^{8} - 113643q^{9} + 115920q^{10} - 370944q^{12} + 577738q^{13} + 401856q^{14} + 1217160q^{15} + 987136q^{16} + 6905934q^{17} - 2727432q^{18} - 10661420q^{19} - 7109760q^{20} + 4219488q^{21} + 18643272q^{23} - 21288960q^{24} - 25499225q^{25} + 13865712q^{26} - 73279080q^{27} - 24647168q^{28} - 128406630q^{29} + 29211840q^{30} - 52843168q^{31} + 196706304q^{32} + 165742416q^{34} + 80873520q^{35} + 167282496q^{36} - 182213314q^{37} - 255874080q^{38} + 145589976q^{39} - 408038400q^{40} - 308120442q^{41} + 101267712q^{42} + 17125708q^{43} - 548895690q^{45} + 447438528q^{46} + 2687348496q^{47} + 248758272q^{48} - 1696965207q^{49} - 611981400q^{50} + 1740295368q^{51} - 850430336q^{52} - 1596055698q^{53} - 1758697920q^{54} - 1414533120q^{56} - 2686677840q^{57} - 3081759120q^{58} - 5189203740q^{59} - 1791659520q^{60} - 6956478662q^{61} - 1268236032q^{62} - 1902838392q^{63} + 2699296768q^{64} + 2790474540q^{65} - 15481826884q^{67} - 10165534848q^{68} + 4698104544q^{69} + 1940964480q^{70} + 9791485272q^{71} + 9600560640q^{72} - 1463791322q^{73} - 4373119536q^{74} - 6425804700q^{75} + 15693610240q^{76} + 3494159424q^{78} - 38116845680q^{79} + 4767866880q^{80} + 1665188361q^{81} - 7394890608q^{82} + 29335099668q^{83} - 6211086336q^{84} + 33355661220q^{85} + 411016992q^{86} - 32358470760q^{87} - 24992917110q^{89} - 13173496560q^{90} + 9673645072q^{91} - 27442896384q^{92} - 13316478336q^{93} + 64496363904q^{94} - 51494658600q^{95} + 49569988608q^{96} + 75013568546q^{97} - 40727164968q^{98} + O(q^{100}) \)

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
24.0000 252.000 −1472.00 4830.00 6048.00 16744.0 −84480.0 −113643. 115920.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 121.12.a.b 1
11.b odd 2 1 1.12.a.a 1
33.d even 2 1 9.12.a.b 1
44.c even 2 1 16.12.a.a 1
55.d odd 2 1 25.12.a.b 1
55.e even 4 2 25.12.b.b 2
77.b even 2 1 49.12.a.a 1
77.h odd 6 2 49.12.c.b 2
77.i even 6 2 49.12.c.c 2
88.b odd 2 1 64.12.a.b 1
88.g even 2 1 64.12.a.f 1
99.g even 6 2 81.12.c.b 2
99.h odd 6 2 81.12.c.d 2
132.d odd 2 1 144.12.a.d 1
143.d odd 2 1 169.12.a.a 1
165.d even 2 1 225.12.a.b 1
165.l odd 4 2 225.12.b.d 2
176.i even 4 2 256.12.b.c 2
176.l odd 4 2 256.12.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 11.b odd 2 1
9.12.a.b 1 33.d even 2 1
16.12.a.a 1 44.c even 2 1
25.12.a.b 1 55.d odd 2 1
25.12.b.b 2 55.e even 4 2
49.12.a.a 1 77.b even 2 1
49.12.c.b 2 77.h odd 6 2
49.12.c.c 2 77.i even 6 2
64.12.a.b 1 88.b odd 2 1
64.12.a.f 1 88.g even 2 1
81.12.c.b 2 99.g even 6 2
81.12.c.d 2 99.h odd 6 2
121.12.a.b 1 1.a even 1 1 trivial
144.12.a.d 1 132.d odd 2 1
169.12.a.a 1 143.d odd 2 1
225.12.a.b 1 165.d even 2 1
225.12.b.d 2 165.l odd 4 2
256.12.b.c 2 176.i even 4 2
256.12.b.e 2 176.l odd 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 24 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(121))\).