Properties

Label 169.4.a.a
Level $169$
Weight $4$
Character orbit 169.a
Self dual yes
Analytic conductor $9.971$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
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

\(f(q)\) \(=\) \( q - 4 q^{2} + 2 q^{3} + 8 q^{4} - 17 q^{5} - 8 q^{6} - 20 q^{7} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 2 q^{3} + 8 q^{4} - 17 q^{5} - 8 q^{6} - 20 q^{7} - 23 q^{9} + 68 q^{10} + 32 q^{11} + 16 q^{12} + 80 q^{14} - 34 q^{15} - 64 q^{16} - 13 q^{17} + 92 q^{18} - 30 q^{19} - 136 q^{20} - 40 q^{21} - 128 q^{22} + 78 q^{23} + 164 q^{25} - 100 q^{27} - 160 q^{28} + 197 q^{29} + 136 q^{30} + 74 q^{31} + 256 q^{32} + 64 q^{33} + 52 q^{34} + 340 q^{35} - 184 q^{36} + 227 q^{37} + 120 q^{38} + 165 q^{41} + 160 q^{42} - 156 q^{43} + 256 q^{44} + 391 q^{45} - 312 q^{46} + 162 q^{47} - 128 q^{48} + 57 q^{49} - 656 q^{50} - 26 q^{51} + 93 q^{53} + 400 q^{54} - 544 q^{55} - 60 q^{57} - 788 q^{58} + 864 q^{59} - 272 q^{60} + 145 q^{61} - 296 q^{62} + 460 q^{63} - 512 q^{64} - 256 q^{66} - 862 q^{67} - 104 q^{68} + 156 q^{69} - 1360 q^{70} - 654 q^{71} - 215 q^{73} - 908 q^{74} + 328 q^{75} - 240 q^{76} - 640 q^{77} - 76 q^{79} + 1088 q^{80} + 421 q^{81} - 660 q^{82} - 628 q^{83} - 320 q^{84} + 221 q^{85} + 624 q^{86} + 394 q^{87} + 266 q^{89} - 1564 q^{90} + 624 q^{92} + 148 q^{93} - 648 q^{94} + 510 q^{95} + 512 q^{96} - 238 q^{97} - 228 q^{98} - 736 q^{99}+O(q^{100}) \) 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
−4.00000 2.00000 8.00000 −17.0000 −8.00000 −20.0000 0 −23.0000 68.0000
\(n\): 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.a 1
3.b odd 2 1 1521.4.a.k 1
13.b even 2 1 169.4.a.d 1
13.c even 3 2 169.4.c.d 2
13.d odd 4 2 169.4.b.c 2
13.e even 6 2 13.4.c.a 2
13.f odd 12 4 169.4.e.c 4
39.d odd 2 1 1521.4.a.b 1
39.h odd 6 2 117.4.g.c 2
52.i odd 6 2 208.4.i.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.a 2 13.e even 6 2
117.4.g.c 2 39.h odd 6 2
169.4.a.a 1 1.a even 1 1 trivial
169.4.a.d 1 13.b even 2 1
169.4.b.c 2 13.d odd 4 2
169.4.c.d 2 13.c even 3 2
169.4.e.c 4 13.f odd 12 4
208.4.i.b 2 52.i odd 6 2
1521.4.a.b 1 39.d odd 2 1
1521.4.a.k 1 3.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T - 2 \) Copy content Toggle raw display
$5$ \( T + 17 \) Copy content Toggle raw display
$7$ \( T + 20 \) Copy content Toggle raw display
$11$ \( T - 32 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 13 \) Copy content Toggle raw display
$19$ \( T + 30 \) Copy content Toggle raw display
$23$ \( T - 78 \) Copy content Toggle raw display
$29$ \( T - 197 \) Copy content Toggle raw display
$31$ \( T - 74 \) Copy content Toggle raw display
$37$ \( T - 227 \) Copy content Toggle raw display
$41$ \( T - 165 \) Copy content Toggle raw display
$43$ \( T + 156 \) Copy content Toggle raw display
$47$ \( T - 162 \) Copy content Toggle raw display
$53$ \( T - 93 \) Copy content Toggle raw display
$59$ \( T - 864 \) Copy content Toggle raw display
$61$ \( T - 145 \) Copy content Toggle raw display
$67$ \( T + 862 \) Copy content Toggle raw display
$71$ \( T + 654 \) Copy content Toggle raw display
$73$ \( T + 215 \) Copy content Toggle raw display
$79$ \( T + 76 \) Copy content Toggle raw display
$83$ \( T + 628 \) Copy content Toggle raw display
$89$ \( T - 266 \) Copy content Toggle raw display
$97$ \( T + 238 \) Copy content Toggle raw display
show more
show less