Properties

Label 169.4.a.f
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: \( x^{2} - x - 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta - 2) q^{2} + (3 \beta + 1) q^{3} + 5 \beta q^{4} + (5 \beta - 10) q^{5} + ( - 10 \beta - 14) q^{6} + (\beta + 7) q^{7} + ( - 7 \beta - 4) q^{8} + (15 \beta + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 2) q^{2} + (3 \beta + 1) q^{3} + 5 \beta q^{4} + (5 \beta - 10) q^{5} + ( - 10 \beta - 14) q^{6} + (\beta + 7) q^{7} + ( - 7 \beta - 4) q^{8} + (15 \beta + 10) q^{9} - 5 \beta q^{10} + (15 \beta + 1) q^{11} + (20 \beta + 60) q^{12} + ( - 10 \beta - 18) q^{14} + ( - 10 \beta + 50) q^{15} + ( - 15 \beta + 36) q^{16} + ( - 16 \beta + 43) q^{17} + ( - 55 \beta - 80) q^{18} + (5 \beta - 73) q^{19} + ( - 25 \beta + 100) q^{20} + (25 \beta + 19) q^{21} + ( - 46 \beta - 62) q^{22} + ( - 33 \beta + 89) q^{23} + ( - 40 \beta - 88) q^{24} + ( - 75 \beta + 75) q^{25} + (9 \beta + 163) q^{27} + (40 \beta + 20) q^{28} + (60 \beta - 13) q^{29} + ( - 20 \beta - 60) q^{30} + (100 \beta - 120) q^{31} + (65 \beta + 20) q^{32} + (63 \beta + 181) q^{33} + (5 \beta - 22) q^{34} + (30 \beta - 50) q^{35} + (125 \beta + 300) q^{36} + ( - 44 \beta - 73) q^{37} + (58 \beta + 126) q^{38} + (15 \beta - 100) q^{40} + (20 \beta + 259) q^{41} + ( - 94 \beta - 138) q^{42} + (97 \beta + 179) q^{43} + (80 \beta + 300) q^{44} + ( - 25 \beta + 200) q^{45} + (10 \beta - 46) q^{46} + ( - 140 \beta + 100) q^{47} + (48 \beta - 144) q^{48} + (15 \beta - 290) q^{49} + (150 \beta + 150) q^{50} + (65 \beta - 149) q^{51} + (165 \beta + 190) q^{53} + ( - 190 \beta - 362) q^{54} + ( - 70 \beta + 290) q^{55} + ( - 60 \beta - 56) q^{56} + ( - 199 \beta - 13) q^{57} + ( - 167 \beta - 214) q^{58} + ( - 55 \beta - 377) q^{59} + (200 \beta - 200) q^{60} + ( - 200 \beta + 351) q^{61} + ( - 180 \beta - 160) q^{62} + (130 \beta + 130) q^{63} + ( - 95 \beta - 588) q^{64} + ( - 370 \beta - 614) q^{66} + (91 \beta - 283) q^{67} + (135 \beta - 320) q^{68} + (135 \beta - 307) q^{69} + ( - 40 \beta - 20) q^{70} + (105 \beta + 11) q^{71} + ( - 235 \beta - 460) q^{72} + (85 \beta + 250) q^{73} + (205 \beta + 322) q^{74} + ( - 75 \beta - 825) q^{75} + ( - 340 \beta + 100) q^{76} + (121 \beta + 67) q^{77} + ( - 40 \beta + 140) q^{79} + (255 \beta - 660) q^{80} + (120 \beta + 1) q^{81} + ( - 319 \beta - 598) q^{82} + ( - 100 \beta + 180) q^{83} + (220 \beta + 500) q^{84} + (295 \beta - 750) q^{85} + ( - 470 \beta - 746) q^{86} + (201 \beta + 707) q^{87} + ( - 172 \beta - 424) q^{88} + ( - 125 \beta + 523) q^{89} + ( - 125 \beta - 300) q^{90} + (280 \beta - 660) q^{92} + (40 \beta + 1080) q^{93} + (320 \beta + 360) q^{94} + ( - 390 \beta + 830) q^{95} + (320 \beta + 800) q^{96} + ( - 469 \beta + 27) q^{97} + (245 \beta + 520) q^{98} + (390 \beta + 910) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{2} + 5 q^{3} + 5 q^{4} - 15 q^{5} - 38 q^{6} + 15 q^{7} - 15 q^{8} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 5 q^{2} + 5 q^{3} + 5 q^{4} - 15 q^{5} - 38 q^{6} + 15 q^{7} - 15 q^{8} + 35 q^{9} - 5 q^{10} + 17 q^{11} + 140 q^{12} - 46 q^{14} + 90 q^{15} + 57 q^{16} + 70 q^{17} - 215 q^{18} - 141 q^{19} + 175 q^{20} + 63 q^{21} - 170 q^{22} + 145 q^{23} - 216 q^{24} + 75 q^{25} + 335 q^{27} + 80 q^{28} + 34 q^{29} - 140 q^{30} - 140 q^{31} + 105 q^{32} + 425 q^{33} - 39 q^{34} - 70 q^{35} + 725 q^{36} - 190 q^{37} + 310 q^{38} - 185 q^{40} + 538 q^{41} - 370 q^{42} + 455 q^{43} + 680 q^{44} + 375 q^{45} - 82 q^{46} + 60 q^{47} - 240 q^{48} - 565 q^{49} + 450 q^{50} - 233 q^{51} + 545 q^{53} - 914 q^{54} + 510 q^{55} - 172 q^{56} - 225 q^{57} - 595 q^{58} - 809 q^{59} - 200 q^{60} + 502 q^{61} - 500 q^{62} + 390 q^{63} - 1271 q^{64} - 1598 q^{66} - 475 q^{67} - 505 q^{68} - 479 q^{69} - 80 q^{70} + 127 q^{71} - 1155 q^{72} + 585 q^{73} + 849 q^{74} - 1725 q^{75} - 140 q^{76} + 255 q^{77} + 240 q^{79} - 1065 q^{80} + 122 q^{81} - 1515 q^{82} + 260 q^{83} + 1220 q^{84} - 1205 q^{85} - 1962 q^{86} + 1615 q^{87} - 1020 q^{88} + 921 q^{89} - 725 q^{90} - 1040 q^{92} + 2200 q^{93} + 1040 q^{94} + 1270 q^{95} + 1920 q^{96} - 415 q^{97} + 1285 q^{98} + 2210 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
−4.56155 8.68466 12.8078 2.80776 −39.6155 9.56155 −21.9309 48.4233 −12.8078
1.2 −0.438447 −3.68466 −7.80776 −17.8078 1.61553 5.43845 6.93087 −13.4233 7.80776
\(n\): 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.f 2
3.b odd 2 1 1521.4.a.t 2
13.b even 2 1 169.4.a.j 2
13.c even 3 2 13.4.c.b 4
13.d odd 4 2 169.4.b.e 4
13.e even 6 2 169.4.c.f 4
13.f odd 12 4 169.4.e.g 8
39.d odd 2 1 1521.4.a.l 2
39.i odd 6 2 117.4.g.d 4
52.j 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.c even 3 2
117.4.g.d 4 39.i odd 6 2
169.4.a.f 2 1.a even 1 1 trivial
169.4.a.j 2 13.b even 2 1
169.4.b.e 4 13.d odd 4 2
169.4.c.f 4 13.e even 6 2
169.4.e.g 8 13.f odd 12 4
208.4.i.e 4 52.j odd 6 2
1521.4.a.l 2 39.d odd 2 1
1521.4.a.t 2 3.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

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