Properties

Label 169.4.a.b
Level $169$
Weight $4$
Character orbit 169.a
Self dual yes
Analytic conductor $9.971$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.97132279097\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3 q^{2} - q^{3} + q^{4} + 9 q^{5} + 3 q^{6} - 15 q^{7} + 21 q^{8} - 26 q^{9} + O(q^{10}) \) \( q - 3 q^{2} - q^{3} + q^{4} + 9 q^{5} + 3 q^{6} - 15 q^{7} + 21 q^{8} - 26 q^{9} - 27 q^{10} + 48 q^{11} - q^{12} + 45 q^{14} - 9 q^{15} - 71 q^{16} + 45 q^{17} + 78 q^{18} - 6 q^{19} + 9 q^{20} + 15 q^{21} - 144 q^{22} - 162 q^{23} - 21 q^{24} - 44 q^{25} + 53 q^{27} - 15 q^{28} - 144 q^{29} + 27 q^{30} - 264 q^{31} + 45 q^{32} - 48 q^{33} - 135 q^{34} - 135 q^{35} - 26 q^{36} - 303 q^{37} + 18 q^{38} + 189 q^{40} + 192 q^{41} - 45 q^{42} + 97 q^{43} + 48 q^{44} - 234 q^{45} + 486 q^{46} - 111 q^{47} + 71 q^{48} - 118 q^{49} + 132 q^{50} - 45 q^{51} - 414 q^{53} - 159 q^{54} + 432 q^{55} - 315 q^{56} + 6 q^{57} + 432 q^{58} - 522 q^{59} - 9 q^{60} + 376 q^{61} + 792 q^{62} + 390 q^{63} + 433 q^{64} + 144 q^{66} + 36 q^{67} + 45 q^{68} + 162 q^{69} + 405 q^{70} - 357 q^{71} - 546 q^{72} + 1098 q^{73} + 909 q^{74} + 44 q^{75} - 6 q^{76} - 720 q^{77} - 830 q^{79} - 639 q^{80} + 649 q^{81} - 576 q^{82} + 438 q^{83} + 15 q^{84} + 405 q^{85} - 291 q^{86} + 144 q^{87} + 1008 q^{88} + 438 q^{89} + 702 q^{90} - 162 q^{92} + 264 q^{93} + 333 q^{94} - 54 q^{95} - 45 q^{96} + 852 q^{97} + 354 q^{98} - 1248 q^{99} + O(q^{100}) \)

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
−3.00000 −1.00000 1.00000 9.00000 3.00000 −15.0000 21.0000 −26.0000 −27.0000
\(n\): 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 169.4.a.b 1
3.b odd 2 1 1521.4.a.i 1
13.b even 2 1 169.4.a.c 1
13.c even 3 2 169.4.c.c 2
13.d odd 4 2 13.4.b.a 2
13.e even 6 2 169.4.c.b 2
13.f odd 12 4 169.4.e.d 4
39.d odd 2 1 1521.4.a.d 1
39.f even 4 2 117.4.b.a 2
52.f even 4 2 208.4.f.b 2
65.f even 4 2 325.4.d.b 2
65.g odd 4 2 325.4.c.b 2
65.k even 4 2 325.4.d.a 2
104.j odd 4 2 832.4.f.e 2
104.m even 4 2 832.4.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 13.d odd 4 2
117.4.b.a 2 39.f even 4 2
169.4.a.b 1 1.a even 1 1 trivial
169.4.a.c 1 13.b even 2 1
169.4.c.b 2 13.e even 6 2
169.4.c.c 2 13.c even 3 2
169.4.e.d 4 13.f odd 12 4
208.4.f.b 2 52.f even 4 2
325.4.c.b 2 65.g odd 4 2
325.4.d.a 2 65.k even 4 2
325.4.d.b 2 65.f even 4 2
832.4.f.c 2 104.m even 4 2
832.4.f.e 2 104.j odd 4 2
1521.4.a.d 1 39.d odd 2 1
1521.4.a.i 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(169))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 + T \)
$3$ \( 1 + T \)
$5$ \( -9 + T \)
$7$ \( 15 + T \)
$11$ \( -48 + T \)
$13$ \( T \)
$17$ \( -45 + T \)
$19$ \( 6 + T \)
$23$ \( 162 + T \)
$29$ \( 144 + T \)
$31$ \( 264 + T \)
$37$ \( 303 + T \)
$41$ \( -192 + T \)
$43$ \( -97 + T \)
$47$ \( 111 + T \)
$53$ \( 414 + T \)
$59$ \( 522 + T \)
$61$ \( -376 + T \)
$67$ \( -36 + T \)
$71$ \( 357 + T \)
$73$ \( -1098 + T \)
$79$ \( 830 + T \)
$83$ \( -438 + T \)
$89$ \( -438 + T \)
$97$ \( -852 + T \)
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