Properties

Label 169.4.c.j
Level $169$
Weight $4$
Character orbit 169.c
Analytic conductor $9.971$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{3} + ( 4 + \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{4} + ( 2 + \beta_{3} ) q^{5} + ( -12 - \beta_{1} + 12 \beta_{2} - \beta_{3} ) q^{6} + ( -10 - 11 \beta_{1} + 10 \beta_{2} - 11 \beta_{3} ) q^{7} + ( -4 - 11 \beta_{3} ) q^{8} + ( -25 - 15 \beta_{1} + 25 \beta_{2} - 15 \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{3} + ( 4 + \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{4} + ( 2 + \beta_{3} ) q^{5} + ( -12 - \beta_{1} + 12 \beta_{2} - \beta_{3} ) q^{6} + ( -10 - 11 \beta_{1} + 10 \beta_{2} - 11 \beta_{3} ) q^{7} + ( -4 - 11 \beta_{3} ) q^{8} + ( -25 - 15 \beta_{1} + 25 \beta_{2} - 15 \beta_{3} ) q^{9} + ( \beta_{1} - 4 \beta_{2} ) q^{10} + ( 12 \beta_{1} + 34 \beta_{2} ) q^{11} + ( -28 - 13 \beta_{3} ) q^{12} + ( 44 - \beta_{3} ) q^{14} + ( 7 \beta_{1} - 20 \beta_{2} ) q^{15} + ( 15 \beta_{1} + 12 \beta_{2} ) q^{16} + ( -18 - 17 \beta_{1} + 18 \beta_{2} - 17 \beta_{3} ) q^{17} + ( 60 + 10 \beta_{3} ) q^{18} + ( -26 + 32 \beta_{1} + 26 \beta_{2} + 32 \beta_{3} ) q^{19} + ( 12 + 5 \beta_{1} - 12 \beta_{2} + 5 \beta_{3} ) q^{20} + ( 172 + 41 \beta_{3} ) q^{21} + ( -48 + 46 \beta_{1} + 48 \beta_{2} + 46 \beta_{3} ) q^{22} + ( 12 \beta_{1} - 104 \beta_{2} ) q^{23} + ( -23 \beta_{1} + 148 \beta_{2} ) q^{24} + ( -117 + 3 \beta_{3} ) q^{25} + ( 172 + 9 \beta_{3} ) q^{27} + ( -43 \beta_{1} + 84 \beta_{2} ) q^{28} + ( -96 \beta_{1} + 70 \beta_{2} ) q^{29} + ( -28 - 13 \beta_{1} + 28 \beta_{2} - 13 \beta_{3} ) q^{30} + ( 26 - 34 \beta_{3} ) q^{31} + ( -92 - 61 \beta_{1} + 92 \beta_{2} - 61 \beta_{3} ) q^{32} + ( -8 + 90 \beta_{1} + 8 \beta_{2} + 90 \beta_{3} ) q^{33} + ( 68 + \beta_{3} ) q^{34} + ( -64 - 21 \beta_{1} + 64 \beta_{2} - 21 \beta_{3} ) q^{35} + ( -70 \beta_{1} + 160 \beta_{2} ) q^{36} + ( 5 \beta_{1} + 102 \beta_{2} ) q^{37} + ( -128 + 58 \beta_{3} ) q^{38} + ( -52 - 15 \beta_{3} ) q^{40} + ( 22 \beta_{1} - 126 \beta_{2} ) q^{41} + ( 131 \beta_{1} - 164 \beta_{2} ) q^{42} + ( -72 + 143 \beta_{1} + 72 \beta_{2} + 143 \beta_{3} ) q^{43} + ( 88 - 2 \beta_{3} ) q^{44} + ( -110 - 40 \beta_{1} + 110 \beta_{2} - 40 \beta_{3} ) q^{45} + ( -48 - 92 \beta_{1} + 48 \beta_{2} - 92 \beta_{3} ) q^{46} + ( -278 - 121 \beta_{3} ) q^{47} + ( -132 + 21 \beta_{1} + 132 \beta_{2} + 21 \beta_{3} ) q^{48} + ( 99 \beta_{1} - 241 \beta_{2} ) q^{49} + ( -120 \beta_{1} - 12 \beta_{2} ) q^{50} + ( 276 + 71 \beta_{3} ) q^{51} + ( -74 - 30 \beta_{3} ) q^{53} + ( 163 \beta_{1} - 36 \beta_{2} ) q^{54} + ( -22 \beta_{1} + 20 \beta_{2} ) q^{55} + ( 524 + 33 \beta_{1} - 524 \beta_{2} + 33 \beta_{3} ) q^{56} + ( -280 + 46 \beta_{3} ) q^{57} + ( 384 - 26 \beta_{1} - 384 \beta_{2} - 26 \beta_{3} ) q^{58} + ( -246 - 124 \beta_{1} + 246 \beta_{2} - 124 \beta_{3} ) q^{59} + ( -108 - 41 \beta_{3} ) q^{60} + ( 434 - 190 \beta_{1} - 434 \beta_{2} - 190 \beta_{3} ) q^{61} + ( 60 \beta_{1} + 136 \beta_{2} ) q^{62} + ( 260 \beta_{1} - 910 \beta_{2} ) q^{63} + ( 340 - 89 \beta_{3} ) q^{64} + ( -360 + 98 \beta_{3} ) q^{66} + ( -232 \beta_{1} + 150 \beta_{2} ) q^{67} + ( -69 \beta_{1} + 140 \beta_{2} ) q^{68} + ( -560 - 324 \beta_{1} + 560 \beta_{2} - 324 \beta_{3} ) q^{69} + ( 84 + 43 \beta_{3} ) q^{70} + ( 50 + 231 \beta_{1} - 50 \beta_{2} + 231 \beta_{3} ) q^{71} + ( 760 + 170 \beta_{1} - 760 \beta_{2} + 170 \beta_{3} ) q^{72} + ( -98 + 260 \beta_{3} ) q^{73} + ( -20 + 107 \beta_{1} + 20 \beta_{2} + 107 \beta_{3} ) q^{74} + ( -348 \beta_{1} + 432 \beta_{2} ) q^{75} + ( 70 \beta_{1} - 24 \beta_{2} ) q^{76} + ( 188 - 386 \beta_{3} ) q^{77} + ( -524 - 40 \beta_{3} ) q^{79} + ( 3 \beta_{1} - 36 \beta_{2} ) q^{80} + ( 120 \beta_{1} - 121 \beta_{2} ) q^{81} + ( -88 - 104 \beta_{1} + 88 \beta_{2} - 104 \beta_{3} ) q^{82} + ( -1070 - 182 \beta_{3} ) q^{83} + ( 852 + 295 \beta_{1} - 852 \beta_{2} + 295 \beta_{3} ) q^{84} + ( -104 - 35 \beta_{1} + 104 \beta_{2} - 35 \beta_{3} ) q^{85} + ( -572 + 215 \beta_{3} ) q^{86} + ( 1432 + 306 \beta_{1} - 1432 \beta_{2} + 306 \beta_{3} ) q^{87} + ( 458 \beta_{1} + 392 \beta_{2} ) q^{88} + ( -388 \beta_{1} - 166 \beta_{2} ) q^{89} + ( 160 + 70 \beta_{3} ) q^{90} + ( -464 - 140 \beta_{3} ) q^{92} + ( 44 \beta_{1} + 304 \beta_{2} ) q^{93} + ( -157 \beta_{1} + 484 \beta_{2} ) q^{94} + ( 76 + 6 \beta_{1} - 76 \beta_{2} + 6 \beta_{3} ) q^{95} + ( 1100 + 337 \beta_{3} ) q^{96} + ( -718 - 508 \beta_{1} + 718 \beta_{2} - 508 \beta_{3} ) q^{97} + ( -396 - 142 \beta_{1} + 396 \beta_{2} - 142 \beta_{3} ) q^{98} + ( -130 - 390 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 5 q^{3} + 7 q^{4} + 6 q^{5} - 23 q^{6} - 9 q^{7} + 6 q^{8} - 35 q^{9} + O(q^{10}) \) \( 4 q + q^{2} - 5 q^{3} + 7 q^{4} + 6 q^{5} - 23 q^{6} - 9 q^{7} + 6 q^{8} - 35 q^{9} - 7 q^{10} + 80 q^{11} - 86 q^{12} + 178 q^{14} - 33 q^{15} + 39 q^{16} - 19 q^{17} + 220 q^{18} - 84 q^{19} + 19 q^{20} + 606 q^{21} - 142 q^{22} - 196 q^{23} + 273 q^{24} - 474 q^{25} + 670 q^{27} + 125 q^{28} + 44 q^{29} - 43 q^{30} + 172 q^{31} - 123 q^{32} - 106 q^{33} + 270 q^{34} - 107 q^{35} + 250 q^{36} + 209 q^{37} - 628 q^{38} - 178 q^{40} - 230 q^{41} - 197 q^{42} - 287 q^{43} + 356 q^{44} - 180 q^{45} - 4 q^{46} - 870 q^{47} - 285 q^{48} - 383 q^{49} - 144 q^{50} + 962 q^{51} - 236 q^{53} + 91 q^{54} + 18 q^{55} + 1015 q^{56} - 1212 q^{57} + 794 q^{58} - 368 q^{59} - 350 q^{60} + 1058 q^{61} + 332 q^{62} - 1560 q^{63} + 1538 q^{64} - 1636 q^{66} + 68 q^{67} + 211 q^{68} - 796 q^{69} + 250 q^{70} - 131 q^{71} + 1350 q^{72} - 912 q^{73} - 147 q^{74} + 516 q^{75} + 22 q^{76} + 1524 q^{77} - 2016 q^{79} - 69 q^{80} - 122 q^{81} - 72 q^{82} - 3916 q^{83} + 1409 q^{84} - 173 q^{85} - 2718 q^{86} + 2558 q^{87} + 1242 q^{88} - 720 q^{89} + 500 q^{90} - 1576 q^{92} + 652 q^{93} + 811 q^{94} + 146 q^{95} + 3726 q^{96} - 928 q^{97} - 650 q^{98} + 260 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 5 \nu^{2} - 5 \nu + 16 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4 \beta_{2} + \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(5 \beta_{3} - 4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
−0.780776 + 1.35234i
1.28078 2.21837i
−0.780776 1.35234i
1.28078 + 2.21837i
−0.780776 + 1.35234i −4.34233 + 7.52113i 2.78078 + 4.81645i 3.56155 −6.78078 11.7446i −13.5885 23.5360i −21.1771 −24.2116 41.9358i −2.78078 + 4.81645i
22.2 1.28078 2.21837i 1.84233 3.19101i 0.719224 + 1.24573i −0.561553 −4.71922 8.17394i 9.08854 + 15.7418i 24.1771 6.71165 + 11.6249i −0.719224 + 1.24573i
146.1 −0.780776 1.35234i −4.34233 7.52113i 2.78078 4.81645i 3.56155 −6.78078 + 11.7446i −13.5885 + 23.5360i −21.1771 −24.2116 + 41.9358i −2.78078 4.81645i
146.2 1.28078 + 2.21837i 1.84233 + 3.19101i 0.719224 1.24573i −0.561553 −4.71922 + 8.17394i 9.08854 15.7418i 24.1771 6.71165 11.6249i −0.719224 1.24573i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.c.j 4
13.b even 2 1 169.4.c.g 4
13.c even 3 1 169.4.a.g 2
13.c even 3 1 inner 169.4.c.j 4
13.d odd 4 2 169.4.e.f 8
13.e even 6 1 13.4.a.b 2
13.e even 6 1 169.4.c.g 4
13.f odd 12 2 169.4.b.f 4
13.f odd 12 2 169.4.e.f 8
39.h odd 6 1 117.4.a.d 2
39.i odd 6 1 1521.4.a.r 2
52.i odd 6 1 208.4.a.h 2
65.l even 6 1 325.4.a.f 2
65.r odd 12 2 325.4.b.e 4
91.t odd 6 1 637.4.a.b 2
104.p odd 6 1 832.4.a.z 2
104.s even 6 1 832.4.a.s 2
143.i odd 6 1 1573.4.a.b 2
156.r even 6 1 1872.4.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 13.e even 6 1
117.4.a.d 2 39.h odd 6 1
169.4.a.g 2 13.c even 3 1
169.4.b.f 4 13.f odd 12 2
169.4.c.g 4 13.b even 2 1
169.4.c.g 4 13.e even 6 1
169.4.c.j 4 1.a even 1 1 trivial
169.4.c.j 4 13.c even 3 1 inner
169.4.e.f 8 13.d odd 4 2
169.4.e.f 8 13.f odd 12 2
208.4.a.h 2 52.i odd 6 1
325.4.a.f 2 65.l even 6 1
325.4.b.e 4 65.r odd 12 2
637.4.a.b 2 91.t odd 6 1
832.4.a.s 2 104.s even 6 1
832.4.a.z 2 104.p odd 6 1
1521.4.a.r 2 39.i odd 6 1
1573.4.a.b 2 143.i odd 6 1
1872.4.a.bb 2 156.r even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} + 5 T_{2}^{2} + 4 T_{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 4 T + 5 T^{2} - T^{3} + T^{4} \)
$3$ \( 1024 - 160 T + 57 T^{2} + 5 T^{3} + T^{4} \)
$5$ \( ( -2 - 3 T + T^{2} )^{2} \)
$7$ \( 244036 - 4446 T + 575 T^{2} + 9 T^{3} + T^{4} \)
$11$ \( 976144 - 79040 T + 5412 T^{2} - 80 T^{3} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 1295044 - 21622 T + 1499 T^{2} + 19 T^{3} + T^{4} \)
$19$ \( 6697744 - 217392 T + 9644 T^{2} + 84 T^{3} + T^{4} \)
$23$ \( 80856064 + 1762432 T + 29424 T^{2} + 196 T^{3} + T^{4} \)
$29$ \( 1496451856 + 1702096 T + 40620 T^{2} - 44 T^{3} + T^{4} \)
$31$ \( ( -3064 - 86 T + T^{2} )^{2} \)
$37$ \( 116942596 - 2260126 T + 32867 T^{2} - 209 T^{3} + T^{4} \)
$41$ \( 124724224 + 2568640 T + 41732 T^{2} + 230 T^{3} + T^{4} \)
$43$ \( 4397811856 - 19032692 T + 148685 T^{2} + 287 T^{3} + T^{4} \)
$47$ \( ( -14918 + 435 T + T^{2} )^{2} \)
$53$ \( ( -344 + 118 T + T^{2} )^{2} \)
$59$ \( 991746064 - 11589056 T + 166916 T^{2} + 368 T^{3} + T^{4} \)
$61$ \( 15981005056 - 133748128 T + 992948 T^{2} - 1058 T^{3} + T^{4} \)
$67$ \( 51799939216 + 15476528 T + 232220 T^{2} - 68 T^{3} + T^{4} \)
$71$ \( 49503580036 - 29146714 T + 239655 T^{2} + 131 T^{3} + T^{4} \)
$73$ \( ( -235316 + 456 T + T^{2} )^{2} \)
$79$ \( ( 247216 + 1008 T + T^{2} )^{2} \)
$83$ \( ( 817664 + 1958 T + T^{2} )^{2} \)
$89$ \( 260316284944 - 367352640 T + 1028612 T^{2} + 720 T^{3} + T^{4} \)
$97$ \( 776999938576 - 818009728 T + 1742660 T^{2} + 928 T^{3} + T^{4} \)
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