Properties

Label 169.4.a.j
Level $169$
Weight $4$
Character orbit 169.a
Self dual yes
Analytic conductor $9.971$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - \beta ) q^{2} + ( 4 - 3 \beta ) q^{3} + ( 5 - 5 \beta ) q^{4} + ( 5 + 5 \beta ) q^{5} + ( 24 - 10 \beta ) q^{6} + ( -8 + \beta ) q^{7} + ( 11 - 7 \beta ) q^{8} + ( 25 - 15 \beta ) q^{9} +O(q^{10})\) \( q + ( 3 - \beta ) q^{2} + ( 4 - 3 \beta ) q^{3} + ( 5 - 5 \beta ) q^{4} + ( 5 + 5 \beta ) q^{5} + ( 24 - 10 \beta ) q^{6} + ( -8 + \beta ) q^{7} + ( 11 - 7 \beta ) q^{8} + ( 25 - 15 \beta ) q^{9} + ( -5 + 5 \beta ) q^{10} + ( -16 + 15 \beta ) q^{11} + ( 80 - 20 \beta ) q^{12} + ( -28 + 10 \beta ) q^{14} + ( -40 - 10 \beta ) q^{15} + ( 21 + 15 \beta ) q^{16} + ( 27 + 16 \beta ) q^{17} + ( 135 - 55 \beta ) q^{18} + ( 68 + 5 \beta ) q^{19} + ( -75 - 25 \beta ) q^{20} + ( -44 + 25 \beta ) q^{21} + ( -108 + 46 \beta ) q^{22} + ( 56 + 33 \beta ) q^{23} + ( 128 - 40 \beta ) q^{24} + 75 \beta q^{25} + ( 172 - 9 \beta ) q^{27} + ( -60 + 40 \beta ) q^{28} + ( 47 - 60 \beta ) q^{29} + ( -80 + 20 \beta ) q^{30} + ( 20 + 100 \beta ) q^{31} + ( -85 + 65 \beta ) q^{32} + ( -244 + 63 \beta ) q^{33} + ( 17 + 5 \beta ) q^{34} + ( -20 - 30 \beta ) q^{35} + ( 425 - 125 \beta ) q^{36} + ( 117 - 44 \beta ) q^{37} + ( 184 - 58 \beta ) q^{38} + ( -85 - 15 \beta ) q^{40} + ( -279 + 20 \beta ) q^{41} + ( -232 + 94 \beta ) q^{42} + ( 276 - 97 \beta ) q^{43} + ( -380 + 80 \beta ) q^{44} + ( -175 - 25 \beta ) q^{45} + ( 36 + 10 \beta ) q^{46} + ( 40 - 140 \beta ) q^{47} + ( -96 - 48 \beta ) q^{48} + ( -275 - 15 \beta ) q^{49} + ( -300 + 150 \beta ) q^{50} + ( -84 - 65 \beta ) q^{51} + ( 355 - 165 \beta ) q^{53} + ( 552 - 190 \beta ) q^{54} + ( 220 + 70 \beta ) q^{55} + ( -116 + 60 \beta ) q^{56} + ( 212 - 199 \beta ) q^{57} + ( 381 - 167 \beta ) q^{58} + ( 432 - 55 \beta ) q^{59} + 200 \beta q^{60} + ( 151 + 200 \beta ) q^{61} + ( -340 + 180 \beta ) q^{62} + ( -260 + 130 \beta ) q^{63} + ( -683 + 95 \beta ) q^{64} + ( -984 + 370 \beta ) q^{66} + ( 192 + 91 \beta ) q^{67} + ( -185 - 135 \beta ) q^{68} + ( -172 - 135 \beta ) q^{69} + ( 60 - 40 \beta ) q^{70} + ( -116 + 105 \beta ) q^{71} + ( 695 - 235 \beta ) q^{72} + ( -335 + 85 \beta ) q^{73} + ( 527 - 205 \beta ) q^{74} + ( -900 + 75 \beta ) q^{75} + ( 240 - 340 \beta ) q^{76} + ( 188 - 121 \beta ) q^{77} + ( 100 + 40 \beta ) q^{79} + ( 405 + 255 \beta ) q^{80} + ( 121 - 120 \beta ) q^{81} + ( -917 + 319 \beta ) q^{82} + ( -80 - 100 \beta ) q^{83} + ( -720 + 220 \beta ) q^{84} + ( 455 + 295 \beta ) q^{85} + ( 1216 - 470 \beta ) q^{86} + ( 908 - 201 \beta ) q^{87} + ( -596 + 172 \beta ) q^{88} + ( -398 - 125 \beta ) q^{89} + ( -425 + 125 \beta ) q^{90} + ( -380 - 280 \beta ) q^{92} + ( -1120 + 40 \beta ) q^{93} + ( 680 - 320 \beta ) q^{94} + ( 440 + 390 \beta ) q^{95} + ( -1120 + 320 \beta ) q^{96} + ( 442 - 469 \beta ) q^{97} + ( -765 + 245 \beta ) q^{98} + ( -1300 + 390 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9} + O(q^{10}) \) \( 2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9} - 5 q^{10} - 17 q^{11} + 140 q^{12} - 46 q^{14} - 90 q^{15} + 57 q^{16} + 70 q^{17} + 215 q^{18} + 141 q^{19} - 175 q^{20} - 63 q^{21} - 170 q^{22} + 145 q^{23} + 216 q^{24} + 75 q^{25} + 335 q^{27} - 80 q^{28} + 34 q^{29} - 140 q^{30} + 140 q^{31} - 105 q^{32} - 425 q^{33} + 39 q^{34} - 70 q^{35} + 725 q^{36} + 190 q^{37} + 310 q^{38} - 185 q^{40} - 538 q^{41} - 370 q^{42} + 455 q^{43} - 680 q^{44} - 375 q^{45} + 82 q^{46} - 60 q^{47} - 240 q^{48} - 565 q^{49} - 450 q^{50} - 233 q^{51} + 545 q^{53} + 914 q^{54} + 510 q^{55} - 172 q^{56} + 225 q^{57} + 595 q^{58} + 809 q^{59} + 200 q^{60} + 502 q^{61} - 500 q^{62} - 390 q^{63} - 1271 q^{64} - 1598 q^{66} + 475 q^{67} - 505 q^{68} - 479 q^{69} + 80 q^{70} - 127 q^{71} + 1155 q^{72} - 585 q^{73} + 849 q^{74} - 1725 q^{75} + 140 q^{76} + 255 q^{77} + 240 q^{79} + 1065 q^{80} + 122 q^{81} - 1515 q^{82} - 260 q^{83} - 1220 q^{84} + 1205 q^{85} + 1962 q^{86} + 1615 q^{87} - 1020 q^{88} - 921 q^{89} - 725 q^{90} - 1040 q^{92} - 2200 q^{93} + 1040 q^{94} + 1270 q^{95} - 1920 q^{96} + 415 q^{97} - 1285 q^{98} - 2210 q^{99} + O(q^{100}) \)

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
2.56155
−1.56155
0.438447 −3.68466 −7.80776 17.8078 −1.61553 −5.43845 −6.93087 −13.4233 7.80776
1.2 4.56155 8.68466 12.8078 −2.80776 39.6155 −9.56155 21.9309 48.4233 −12.8078
\(n\): 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.j 2
3.b odd 2 1 1521.4.a.l 2
13.b even 2 1 169.4.a.f 2
13.c even 3 2 169.4.c.f 4
13.d odd 4 2 169.4.b.e 4
13.e even 6 2 13.4.c.b 4
13.f odd 12 4 169.4.e.g 8
39.d odd 2 1 1521.4.a.t 2
39.h odd 6 2 117.4.g.d 4
52.i odd 6 2 208.4.i.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 13.e even 6 2
117.4.g.d 4 39.h odd 6 2
169.4.a.f 2 13.b even 2 1
169.4.a.j 2 1.a even 1 1 trivial
169.4.b.e 4 13.d odd 4 2
169.4.c.f 4 13.c even 3 2
169.4.e.g 8 13.f odd 12 4
208.4.i.e 4 52.i odd 6 2
1521.4.a.l 2 3.b odd 2 1
1521.4.a.t 2 39.d odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 - 5 T + T^{2} \)
$3$ \( -32 - 5 T + T^{2} \)
$5$ \( -50 - 15 T + T^{2} \)
$7$ \( 52 + 15 T + T^{2} \)
$11$ \( -884 + 17 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 137 - 70 T + T^{2} \)
$19$ \( 4864 - 141 T + T^{2} \)
$23$ \( 628 - 145 T + T^{2} \)
$29$ \( -15011 - 34 T + T^{2} \)
$31$ \( -37600 - 140 T + T^{2} \)
$37$ \( 797 - 190 T + T^{2} \)
$41$ \( 70661 + 538 T + T^{2} \)
$43$ \( 11768 - 455 T + T^{2} \)
$47$ \( -82400 + 60 T + T^{2} \)
$53$ \( -41450 - 545 T + T^{2} \)
$59$ \( 150764 - 809 T + T^{2} \)
$61$ \( -106999 - 502 T + T^{2} \)
$67$ \( 21212 - 475 T + T^{2} \)
$71$ \( -42824 + 127 T + T^{2} \)
$73$ \( 54850 + 585 T + T^{2} \)
$79$ \( 7600 - 240 T + T^{2} \)
$83$ \( -25600 + 260 T + T^{2} \)
$89$ \( 145654 + 921 T + T^{2} \)
$97$ \( -891778 - 415 T + T^{2} \)
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