Properties

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

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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: \(0\)
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

\(f(q)\) \(=\) \( q + 5 q^{2} - 7 q^{3} + 17 q^{4} + 7 q^{5} - 35 q^{6} + 13 q^{7} + 45 q^{8} + 22 q^{9} + O(q^{10}) \) \( q + 5 q^{2} - 7 q^{3} + 17 q^{4} + 7 q^{5} - 35 q^{6} + 13 q^{7} + 45 q^{8} + 22 q^{9} + 35 q^{10} + 26 q^{11} - 119 q^{12} + 65 q^{14} - 49 q^{15} + 89 q^{16} + 77 q^{17} + 110 q^{18} + 126 q^{19} + 119 q^{20} - 91 q^{21} + 130 q^{22} - 96 q^{23} - 315 q^{24} - 76 q^{25} + 35 q^{27} + 221 q^{28} - 82 q^{29} - 245 q^{30} - 196 q^{31} + 85 q^{32} - 182 q^{33} + 385 q^{34} + 91 q^{35} + 374 q^{36} + 131 q^{37} + 630 q^{38} + 315 q^{40} - 336 q^{41} - 455 q^{42} - 201 q^{43} + 442 q^{44} + 154 q^{45} - 480 q^{46} + 105 q^{47} - 623 q^{48} - 174 q^{49} - 380 q^{50} - 539 q^{51} - 432 q^{53} + 175 q^{54} + 182 q^{55} + 585 q^{56} - 882 q^{57} - 410 q^{58} + 294 q^{59} - 833 q^{60} - 56 q^{61} - 980 q^{62} + 286 q^{63} - 287 q^{64} - 910 q^{66} - 478 q^{67} + 1309 q^{68} + 672 q^{69} + 455 q^{70} - 9 q^{71} + 990 q^{72} - 98 q^{73} + 655 q^{74} + 532 q^{75} + 2142 q^{76} + 338 q^{77} + 1304 q^{79} + 623 q^{80} - 839 q^{81} - 1680 q^{82} + 308 q^{83} - 1547 q^{84} + 539 q^{85} - 1005 q^{86} + 574 q^{87} + 1170 q^{88} + 1190 q^{89} + 770 q^{90} - 1632 q^{92} + 1372 q^{93} + 525 q^{94} + 882 q^{95} - 595 q^{96} - 70 q^{97} - 870 q^{98} + 572 q^{99} + O(q^{100}) \)

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
5.00000 −7.00000 17.0000 7.00000 −35.0000 13.0000 45.0000 22.0000 35.0000
\(n\): 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.e 1
3.b odd 2 1 1521.4.a.a 1
13.b even 2 1 13.4.a.a 1
13.c even 3 2 169.4.c.a 2
13.d odd 4 2 169.4.b.a 2
13.e even 6 2 169.4.c.e 2
13.f odd 12 4 169.4.e.e 4
39.d odd 2 1 117.4.a.b 1
52.b odd 2 1 208.4.a.g 1
65.d even 2 1 325.4.a.d 1
65.h odd 4 2 325.4.b.b 2
91.b odd 2 1 637.4.a.a 1
104.e even 2 1 832.4.a.r 1
104.h odd 2 1 832.4.a.a 1
143.d odd 2 1 1573.4.a.a 1
156.h even 2 1 1872.4.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 13.b even 2 1
117.4.a.b 1 39.d odd 2 1
169.4.a.e 1 1.a even 1 1 trivial
169.4.b.a 2 13.d odd 4 2
169.4.c.a 2 13.c even 3 2
169.4.c.e 2 13.e even 6 2
169.4.e.e 4 13.f odd 12 4
208.4.a.g 1 52.b odd 2 1
325.4.a.d 1 65.d even 2 1
325.4.b.b 2 65.h odd 4 2
637.4.a.a 1 91.b odd 2 1
832.4.a.a 1 104.h odd 2 1
832.4.a.r 1 104.e even 2 1
1521.4.a.a 1 3.b odd 2 1
1573.4.a.a 1 143.d odd 2 1
1872.4.a.k 1 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(169))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -5 + T \)
$3$ \( 7 + T \)
$5$ \( -7 + T \)
$7$ \( -13 + T \)
$11$ \( -26 + T \)
$13$ \( T \)
$17$ \( -77 + T \)
$19$ \( -126 + T \)
$23$ \( 96 + T \)
$29$ \( 82 + T \)
$31$ \( 196 + T \)
$37$ \( -131 + T \)
$41$ \( 336 + T \)
$43$ \( 201 + T \)
$47$ \( -105 + T \)
$53$ \( 432 + T \)
$59$ \( -294 + T \)
$61$ \( 56 + T \)
$67$ \( 478 + T \)
$71$ \( 9 + T \)
$73$ \( 98 + T \)
$79$ \( -1304 + T \)
$83$ \( -308 + T \)
$89$ \( -1190 + T \)
$97$ \( 70 + T \)
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