Properties

Label 169.12.a.a
Level 169
Weight 12
Character orbit 169.a
Self dual yes
Analytic conductor 129.850
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 169 = 13^{2} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 169.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(129.849997515\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + 24q^{2} + 252q^{3} - 1472q^{4} - 4830q^{5} + 6048q^{6} + 16744q^{7} - 84480q^{8} - 113643q^{9} + O(q^{10}) \) \( q + 24q^{2} + 252q^{3} - 1472q^{4} - 4830q^{5} + 6048q^{6} + 16744q^{7} - 84480q^{8} - 113643q^{9} - 115920q^{10} - 534612q^{11} - 370944q^{12} + 401856q^{14} - 1217160q^{15} + 987136q^{16} - 6905934q^{17} - 2727432q^{18} - 10661420q^{19} + 7109760q^{20} + 4219488q^{21} - 12830688q^{22} + 18643272q^{23} - 21288960q^{24} - 25499225q^{25} - 73279080q^{27} - 24647168q^{28} + 128406630q^{29} - 29211840q^{30} + 52843168q^{31} + 196706304q^{32} - 134722224q^{33} - 165742416q^{34} - 80873520q^{35} + 167282496q^{36} + 182213314q^{37} - 255874080q^{38} + 408038400q^{40} - 308120442q^{41} + 101267712q^{42} - 17125708q^{43} + 786948864q^{44} + 548895690q^{45} + 447438528q^{46} - 2687348496q^{47} + 248758272q^{48} - 1696965207q^{49} - 611981400q^{50} - 1740295368q^{51} - 1596055698q^{53} - 1758697920q^{54} + 2582175960q^{55} - 1414533120q^{56} - 2686677840q^{57} + 3081759120q^{58} + 5189203740q^{59} + 1791659520q^{60} + 6956478662q^{61} + 1268236032q^{62} - 1902838392q^{63} + 2699296768q^{64} - 3233333376q^{66} + 15481826884q^{67} + 10165534848q^{68} + 4698104544q^{69} - 1940964480q^{70} - 9791485272q^{71} + 9600560640q^{72} - 1463791322q^{73} + 4373119536q^{74} - 6425804700q^{75} + 15693610240q^{76} - 8951543328q^{77} + 38116845680q^{79} - 4767866880q^{80} + 1665188361q^{81} - 7394890608q^{82} + 29335099668q^{83} - 6211086336q^{84} + 33355661220q^{85} - 411016992q^{86} + 32358470760q^{87} + 45164021760q^{88} + 24992917110q^{89} + 13173496560q^{90} - 27442896384q^{92} + 13316478336q^{93} - 64496363904q^{94} + 51494658600q^{95} + 49569988608q^{96} - 75013568546q^{97} - 40727164968q^{98} + 60754911516q^{99} + O(q^{100}) \)

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 169.12.a.a 1
13.b even 2 1 1.12.a.a 1
39.d odd 2 1 9.12.a.b 1
52.b odd 2 1 16.12.a.a 1
65.d even 2 1 25.12.a.b 1
65.h odd 4 2 25.12.b.b 2
91.b odd 2 1 49.12.a.a 1
91.r even 6 2 49.12.c.b 2
91.s odd 6 2 49.12.c.c 2
104.e even 2 1 64.12.a.b 1
104.h odd 2 1 64.12.a.f 1
117.n odd 6 2 81.12.c.b 2
117.t even 6 2 81.12.c.d 2
143.d odd 2 1 121.12.a.b 1
156.h even 2 1 144.12.a.d 1
195.e odd 2 1 225.12.a.b 1
195.s even 4 2 225.12.b.d 2
208.o odd 4 2 256.12.b.c 2
208.p even 4 2 256.12.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 13.b even 2 1
9.12.a.b 1 39.d odd 2 1
16.12.a.a 1 52.b odd 2 1
25.12.a.b 1 65.d even 2 1
25.12.b.b 2 65.h odd 4 2
49.12.a.a 1 91.b odd 2 1
49.12.c.b 2 91.r even 6 2
49.12.c.c 2 91.s odd 6 2
64.12.a.b 1 104.e even 2 1
64.12.a.f 1 104.h odd 2 1
81.12.c.b 2 117.n odd 6 2
81.12.c.d 2 117.t even 6 2
121.12.a.b 1 143.d odd 2 1
144.12.a.d 1 156.h even 2 1
169.12.a.a 1 1.a even 1 1 trivial
225.12.a.b 1 195.e odd 2 1
225.12.b.d 2 195.s even 4 2
256.12.b.c 2 208.o odd 4 2
256.12.b.e 2 208.p even 4 2

Atkin-Lehner signs

\( p \) Sign
\(13\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 24 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(169))\).