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

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

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:

Atkin-Lehner signs

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

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 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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 24 \) Copy content Toggle raw display
$3$ \( T - 252 \) Copy content Toggle raw display
$5$ \( T + 4830 \) Copy content Toggle raw display
$7$ \( T - 16744 \) Copy content Toggle raw display
$11$ \( T + 534612 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 6905934 \) Copy content Toggle raw display
$19$ \( T + 10661420 \) Copy content Toggle raw display
$23$ \( T - 18643272 \) Copy content Toggle raw display
$29$ \( T - 128406630 \) Copy content Toggle raw display
$31$ \( T - 52843168 \) Copy content Toggle raw display
$37$ \( T - 182213314 \) Copy content Toggle raw display
$41$ \( T + 308120442 \) Copy content Toggle raw display
$43$ \( T + 17125708 \) Copy content Toggle raw display
$47$ \( T + 2687348496 \) Copy content Toggle raw display
$53$ \( T + 1596055698 \) Copy content Toggle raw display
$59$ \( T - 5189203740 \) Copy content Toggle raw display
$61$ \( T - 6956478662 \) Copy content Toggle raw display
$67$ \( T - 15481826884 \) Copy content Toggle raw display
$71$ \( T + 9791485272 \) Copy content Toggle raw display
$73$ \( T + 1463791322 \) Copy content Toggle raw display
$79$ \( T - 38116845680 \) Copy content Toggle raw display
$83$ \( T - 29335099668 \) Copy content Toggle raw display
$89$ \( T - 24992917110 \) Copy content Toggle raw display
$97$ \( T + 75013568546 \) Copy content Toggle raw display
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