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: \(x^{2} - x - 4\)
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.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( 4 - 3 \beta ) q^{3} + ( -4 + \beta ) q^{4} + ( 2 - \beta ) q^{5} + ( 12 - \beta ) q^{6} + ( 10 - 11 \beta ) q^{7} + ( -4 + 11 \beta ) q^{8} + ( 25 - 15 \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{2} + ( 4 - 3 \beta ) q^{3} + ( -4 + \beta ) q^{4} + ( 2 - \beta ) q^{5} + ( 12 - \beta ) q^{6} + ( 10 - 11 \beta ) q^{7} + ( -4 + 11 \beta ) q^{8} + ( 25 - 15 \beta ) q^{9} + ( 4 - \beta ) q^{10} + ( -34 - 12 \beta ) q^{11} + ( -28 + 13 \beta ) q^{12} + ( 44 + \beta ) q^{14} + ( 20 - 7 \beta ) q^{15} + ( -12 - 15 \beta ) q^{16} + ( 18 - 17 \beta ) q^{17} + ( 60 - 10 \beta ) q^{18} + ( 26 + 32 \beta ) q^{19} + ( -12 + 5 \beta ) q^{20} + ( 172 - 41 \beta ) q^{21} + ( 48 + 46 \beta ) q^{22} + ( 104 - 12 \beta ) q^{23} + ( -148 + 23 \beta ) q^{24} + ( -117 - 3 \beta ) q^{25} + ( 172 - 9 \beta ) q^{27} + ( -84 + 43 \beta ) q^{28} + ( -70 + 96 \beta ) q^{29} + ( 28 - 13 \beta ) q^{30} + ( 26 + 34 \beta ) q^{31} + ( 92 - 61 \beta ) q^{32} + ( 8 + 90 \beta ) q^{33} + ( 68 - \beta ) q^{34} + ( 64 - 21 \beta ) q^{35} + ( -160 + 70 \beta ) q^{36} + ( -102 - 5 \beta ) q^{37} + ( -128 - 58 \beta ) q^{38} + ( -52 + 15 \beta ) q^{40} + ( 126 - 22 \beta ) q^{41} + ( 164 - 131 \beta ) q^{42} + ( 72 + 143 \beta ) q^{43} + ( 88 + 2 \beta ) q^{44} + ( 110 - 40 \beta ) q^{45} + ( 48 - 92 \beta ) q^{46} + ( -278 + 121 \beta ) q^{47} + ( 132 + 21 \beta ) q^{48} + ( 241 - 99 \beta ) q^{49} + ( 12 + 120 \beta ) q^{50} + ( 276 - 71 \beta ) q^{51} + ( -74 + 30 \beta ) q^{53} + ( 36 - 163 \beta ) q^{54} + ( -20 + 22 \beta ) q^{55} + ( -524 + 33 \beta ) q^{56} + ( -280 - 46 \beta ) q^{57} + ( -384 - 26 \beta ) q^{58} + ( 246 - 124 \beta ) q^{59} + ( -108 + 41 \beta ) q^{60} + ( -434 - 190 \beta ) q^{61} + ( -136 - 60 \beta ) q^{62} + ( 910 - 260 \beta ) q^{63} + ( 340 + 89 \beta ) q^{64} + ( -360 - 98 \beta ) q^{66} + ( -150 + 232 \beta ) q^{67} + ( -140 + 69 \beta ) q^{68} + ( 560 - 324 \beta ) q^{69} + ( 84 - 43 \beta ) q^{70} + ( -50 + 231 \beta ) q^{71} + ( -760 + 170 \beta ) q^{72} + ( -98 - 260 \beta ) q^{73} + ( 20 + 107 \beta ) q^{74} + ( -432 + 348 \beta ) q^{75} + ( 24 - 70 \beta ) q^{76} + ( 188 + 386 \beta ) q^{77} + ( -524 + 40 \beta ) q^{79} + ( 36 - 3 \beta ) q^{80} + ( 121 - 120 \beta ) q^{81} + ( 88 - 104 \beta ) q^{82} + ( -1070 + 182 \beta ) q^{83} + ( -852 + 295 \beta ) q^{84} + ( 104 - 35 \beta ) q^{85} + ( -572 - 215 \beta ) q^{86} + ( -1432 + 306 \beta ) q^{87} + ( -392 - 458 \beta ) q^{88} + ( 166 + 388 \beta ) q^{89} + ( 160 - 70 \beta ) q^{90} + ( -464 + 140 \beta ) q^{92} + ( -304 - 44 \beta ) q^{93} + ( -484 + 157 \beta ) q^{94} + ( -76 + 6 \beta ) q^{95} + ( 1100 - 337 \beta ) q^{96} + ( 718 - 508 \beta ) q^{97} + ( 396 - 142 \beta ) q^{98} + ( -130 + 390 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9} + O(q^{10}) \) \( 2 q - q^{2} + 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9} + 7 q^{10} - 80 q^{11} - 43 q^{12} + 89 q^{14} + 33 q^{15} - 39 q^{16} + 19 q^{17} + 110 q^{18} + 84 q^{19} - 19 q^{20} + 303 q^{21} + 142 q^{22} + 196 q^{23} - 273 q^{24} - 237 q^{25} + 335 q^{27} - 125 q^{28} - 44 q^{29} + 43 q^{30} + 86 q^{31} + 123 q^{32} + 106 q^{33} + 135 q^{34} + 107 q^{35} - 250 q^{36} - 209 q^{37} - 314 q^{38} - 89 q^{40} + 230 q^{41} + 197 q^{42} + 287 q^{43} + 178 q^{44} + 180 q^{45} + 4 q^{46} - 435 q^{47} + 285 q^{48} + 383 q^{49} + 144 q^{50} + 481 q^{51} - 118 q^{53} - 91 q^{54} - 18 q^{55} - 1015 q^{56} - 606 q^{57} - 794 q^{58} + 368 q^{59} - 175 q^{60} - 1058 q^{61} - 332 q^{62} + 1560 q^{63} + 769 q^{64} - 818 q^{66} - 68 q^{67} - 211 q^{68} + 796 q^{69} + 125 q^{70} + 131 q^{71} - 1350 q^{72} - 456 q^{73} + 147 q^{74} - 516 q^{75} - 22 q^{76} + 762 q^{77} - 1008 q^{79} + 69 q^{80} + 122 q^{81} + 72 q^{82} - 1958 q^{83} - 1409 q^{84} + 173 q^{85} - 1359 q^{86} - 2558 q^{87} - 1242 q^{88} + 720 q^{89} + 250 q^{90} - 788 q^{92} - 652 q^{93} - 811 q^{94} - 146 q^{95} + 1863 q^{96} + 928 q^{97} + 650 q^{98} + 130 q^{99} + O(q^{100}) \)

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(169))\).

Hecke characteristic polynomials

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