Properties

Label 169.4.a.g
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.

\(f(q)\) \(=\) \( q - \beta q^{2} + ( - 3 \beta + 4) q^{3} + (\beta - 4) q^{4} + ( - \beta + 2) q^{5} + ( - \beta + 12) q^{6} + ( - 11 \beta + 10) q^{7} + (11 \beta - 4) q^{8} + ( - 15 \beta + 25) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + ( - 3 \beta + 4) q^{3} + (\beta - 4) q^{4} + ( - \beta + 2) q^{5} + ( - \beta + 12) q^{6} + ( - 11 \beta + 10) q^{7} + (11 \beta - 4) q^{8} + ( - 15 \beta + 25) q^{9} + ( - \beta + 4) q^{10} + ( - 12 \beta - 34) q^{11} + (13 \beta - 28) q^{12} + (\beta + 44) q^{14} + ( - 7 \beta + 20) q^{15} + ( - 15 \beta - 12) q^{16} + ( - 17 \beta + 18) q^{17} + ( - 10 \beta + 60) q^{18} + (32 \beta + 26) q^{19} + (5 \beta - 12) q^{20} + ( - 41 \beta + 172) q^{21} + (46 \beta + 48) q^{22} + ( - 12 \beta + 104) q^{23} + (23 \beta - 148) q^{24} + ( - 3 \beta - 117) q^{25} + ( - 9 \beta + 172) q^{27} + (43 \beta - 84) q^{28} + (96 \beta - 70) q^{29} + ( - 13 \beta + 28) q^{30} + (34 \beta + 26) q^{31} + ( - 61 \beta + 92) q^{32} + (90 \beta + 8) q^{33} + ( - \beta + 68) q^{34} + ( - 21 \beta + 64) q^{35} + (70 \beta - 160) q^{36} + ( - 5 \beta - 102) q^{37} + ( - 58 \beta - 128) q^{38} + (15 \beta - 52) q^{40} + ( - 22 \beta + 126) q^{41} + ( - 131 \beta + 164) q^{42} + (143 \beta + 72) q^{43} + (2 \beta + 88) q^{44} + ( - 40 \beta + 110) q^{45} + ( - 92 \beta + 48) q^{46} + (121 \beta - 278) q^{47} + (21 \beta + 132) q^{48} + ( - 99 \beta + 241) q^{49} + (120 \beta + 12) q^{50} + ( - 71 \beta + 276) q^{51} + (30 \beta - 74) q^{53} + ( - 163 \beta + 36) q^{54} + (22 \beta - 20) q^{55} + (33 \beta - 524) q^{56} + ( - 46 \beta - 280) q^{57} + ( - 26 \beta - 384) q^{58} + ( - 124 \beta + 246) q^{59} + (41 \beta - 108) q^{60} + ( - 190 \beta - 434) q^{61} + ( - 60 \beta - 136) q^{62} + ( - 260 \beta + 910) q^{63} + (89 \beta + 340) q^{64} + ( - 98 \beta - 360) q^{66} + (232 \beta - 150) q^{67} + (69 \beta - 140) q^{68} + ( - 324 \beta + 560) q^{69} + ( - 43 \beta + 84) q^{70} + (231 \beta - 50) q^{71} + (170 \beta - 760) q^{72} + ( - 260 \beta - 98) q^{73} + (107 \beta + 20) q^{74} + (348 \beta - 432) q^{75} + ( - 70 \beta + 24) q^{76} + (386 \beta + 188) q^{77} + (40 \beta - 524) q^{79} + ( - 3 \beta + 36) q^{80} + ( - 120 \beta + 121) q^{81} + ( - 104 \beta + 88) q^{82} + (182 \beta - 1070) q^{83} + (295 \beta - 852) q^{84} + ( - 35 \beta + 104) q^{85} + ( - 215 \beta - 572) q^{86} + (306 \beta - 1432) q^{87} + ( - 458 \beta - 392) q^{88} + (388 \beta + 166) q^{89} + ( - 70 \beta + 160) q^{90} + (140 \beta - 464) q^{92} + ( - 44 \beta - 304) q^{93} + (157 \beta - 484) q^{94} + (6 \beta - 76) q^{95} + ( - 337 \beta + 1100) q^{96} + ( - 508 \beta + 718) q^{97} + ( - 142 \beta + 396) q^{98} + (390 \beta - 130) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9} + 7 q^{10} - 80 q^{11} - 43 q^{12} + 89 q^{14} + 33 q^{15} - 39 q^{16} + 19 q^{17} + 110 q^{18} + 84 q^{19} - 19 q^{20} + 303 q^{21} + 142 q^{22} + 196 q^{23} - 273 q^{24} - 237 q^{25} + 335 q^{27} - 125 q^{28} - 44 q^{29} + 43 q^{30} + 86 q^{31} + 123 q^{32} + 106 q^{33} + 135 q^{34} + 107 q^{35} - 250 q^{36} - 209 q^{37} - 314 q^{38} - 89 q^{40} + 230 q^{41} + 197 q^{42} + 287 q^{43} + 178 q^{44} + 180 q^{45} + 4 q^{46} - 435 q^{47} + 285 q^{48} + 383 q^{49} + 144 q^{50} + 481 q^{51} - 118 q^{53} - 91 q^{54} - 18 q^{55} - 1015 q^{56} - 606 q^{57} - 794 q^{58} + 368 q^{59} - 175 q^{60} - 1058 q^{61} - 332 q^{62} + 1560 q^{63} + 769 q^{64} - 818 q^{66} - 68 q^{67} - 211 q^{68} + 796 q^{69} + 125 q^{70} + 131 q^{71} - 1350 q^{72} - 456 q^{73} + 147 q^{74} - 516 q^{75} - 22 q^{76} + 762 q^{77} - 1008 q^{79} + 69 q^{80} + 122 q^{81} + 72 q^{82} - 1958 q^{83} - 1409 q^{84} + 173 q^{85} - 1359 q^{86} - 2558 q^{87} - 1242 q^{88} + 720 q^{89} + 250 q^{90} - 788 q^{92} - 652 q^{93} - 811 q^{94} - 146 q^{95} + 1863 q^{96} + 928 q^{97} + 650 q^{98} + 130 q^{99}+O(q^{100}) \) 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
−2.56155 −3.68466 −1.43845 −0.561553 9.43845 −18.1771 24.1771 −13.4233 1.43845
1.2 1.56155 8.68466 −5.56155 3.56155 13.5616 27.1771 −21.1771 48.4233 5.56155
\(n\): 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.g 2
3.b odd 2 1 1521.4.a.r 2
13.b even 2 1 13.4.a.b 2
13.c even 3 2 169.4.c.j 4
13.d odd 4 2 169.4.b.f 4
13.e even 6 2 169.4.c.g 4
13.f odd 12 4 169.4.e.f 8
39.d odd 2 1 117.4.a.d 2
52.b odd 2 1 208.4.a.h 2
65.d even 2 1 325.4.a.f 2
65.h odd 4 2 325.4.b.e 4
91.b odd 2 1 637.4.a.b 2
104.e even 2 1 832.4.a.s 2
104.h odd 2 1 832.4.a.z 2
143.d odd 2 1 1573.4.a.b 2
156.h even 2 1 1872.4.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 13.b even 2 1
117.4.a.d 2 39.d odd 2 1
169.4.a.g 2 1.a even 1 1 trivial
169.4.b.f 4 13.d odd 4 2
169.4.c.g 4 13.e even 6 2
169.4.c.j 4 13.c even 3 2
169.4.e.f 8 13.f odd 12 4
208.4.a.h 2 52.b odd 2 1
325.4.a.f 2 65.d even 2 1
325.4.b.e 4 65.h odd 4 2
637.4.a.b 2 91.b odd 2 1
832.4.a.s 2 104.e even 2 1
832.4.a.z 2 104.h odd 2 1
1521.4.a.r 2 3.b odd 2 1
1573.4.a.b 2 143.d odd 2 1
1872.4.a.bb 2 156.h even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} - 9T - 494 \) Copy content Toggle raw display
$11$ \( T^{2} + 80T + 988 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 19T - 1138 \) Copy content Toggle raw display
$19$ \( T^{2} - 84T - 2588 \) Copy content Toggle raw display
$23$ \( T^{2} - 196T + 8992 \) Copy content Toggle raw display
$29$ \( T^{2} + 44T - 38684 \) Copy content Toggle raw display
$31$ \( T^{2} - 86T - 3064 \) Copy content Toggle raw display
$37$ \( T^{2} + 209T + 10814 \) Copy content Toggle raw display
$41$ \( T^{2} - 230T + 11168 \) Copy content Toggle raw display
$43$ \( T^{2} - 287T - 66316 \) Copy content Toggle raw display
$47$ \( T^{2} + 435T - 14918 \) Copy content Toggle raw display
$53$ \( T^{2} + 118T - 344 \) Copy content Toggle raw display
$59$ \( T^{2} - 368T - 31492 \) Copy content Toggle raw display
$61$ \( T^{2} + 1058 T + 126416 \) Copy content Toggle raw display
$67$ \( T^{2} + 68T - 227596 \) Copy content Toggle raw display
$71$ \( T^{2} - 131T - 222494 \) Copy content Toggle raw display
$73$ \( T^{2} + 456T - 235316 \) Copy content Toggle raw display
$79$ \( T^{2} + 1008 T + 247216 \) Copy content Toggle raw display
$83$ \( T^{2} + 1958 T + 817664 \) Copy content Toggle raw display
$89$ \( T^{2} - 720T - 510212 \) Copy content Toggle raw display
$97$ \( T^{2} - 928T - 881476 \) Copy content Toggle raw display
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