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: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - 4 \beta_{2} + 3 \beta_1) q^{3} + (\beta_{3} - 4 \beta_{2} + \beta_1 + 4) q^{4} + (\beta_{3} + 2) q^{5} + ( - \beta_{3} + 12 \beta_{2} - \beta_1 - 12) q^{6} + ( - 11 \beta_{3} + 10 \beta_{2} - 11 \beta_1 - 10) q^{7} + ( - 11 \beta_{3} - 4) q^{8} + ( - 15 \beta_{3} + 25 \beta_{2} - 15 \beta_1 - 25) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - 4 \beta_{2} + 3 \beta_1) q^{3} + (\beta_{3} - 4 \beta_{2} + \beta_1 + 4) q^{4} + (\beta_{3} + 2) q^{5} + ( - \beta_{3} + 12 \beta_{2} - \beta_1 - 12) q^{6} + ( - 11 \beta_{3} + 10 \beta_{2} - 11 \beta_1 - 10) q^{7} + ( - 11 \beta_{3} - 4) q^{8} + ( - 15 \beta_{3} + 25 \beta_{2} - 15 \beta_1 - 25) q^{9} + ( - 4 \beta_{2} + \beta_1) q^{10} + (34 \beta_{2} + 12 \beta_1) q^{11} + ( - 13 \beta_{3} - 28) q^{12} + ( - \beta_{3} + 44) q^{14} + ( - 20 \beta_{2} + 7 \beta_1) q^{15} + (12 \beta_{2} + 15 \beta_1) q^{16} + ( - 17 \beta_{3} + 18 \beta_{2} - 17 \beta_1 - 18) q^{17} + (10 \beta_{3} + 60) q^{18} + (32 \beta_{3} + 26 \beta_{2} + 32 \beta_1 - 26) q^{19} + (5 \beta_{3} - 12 \beta_{2} + 5 \beta_1 + 12) q^{20} + (41 \beta_{3} + 172) q^{21} + (46 \beta_{3} + 48 \beta_{2} + 46 \beta_1 - 48) q^{22} + ( - 104 \beta_{2} + 12 \beta_1) q^{23} + (148 \beta_{2} - 23 \beta_1) q^{24} + (3 \beta_{3} - 117) q^{25} + (9 \beta_{3} + 172) q^{27} + (84 \beta_{2} - 43 \beta_1) q^{28} + (70 \beta_{2} - 96 \beta_1) q^{29} + ( - 13 \beta_{3} + 28 \beta_{2} - 13 \beta_1 - 28) q^{30} + ( - 34 \beta_{3} + 26) q^{31} + ( - 61 \beta_{3} + 92 \beta_{2} - 61 \beta_1 - 92) q^{32} + (90 \beta_{3} + 8 \beta_{2} + 90 \beta_1 - 8) q^{33} + (\beta_{3} + 68) q^{34} + ( - 21 \beta_{3} + 64 \beta_{2} - 21 \beta_1 - 64) q^{35} + (160 \beta_{2} - 70 \beta_1) q^{36} + (102 \beta_{2} + 5 \beta_1) q^{37} + (58 \beta_{3} - 128) q^{38} + ( - 15 \beta_{3} - 52) q^{40} + ( - 126 \beta_{2} + 22 \beta_1) q^{41} + ( - 164 \beta_{2} + 131 \beta_1) q^{42} + (143 \beta_{3} + 72 \beta_{2} + 143 \beta_1 - 72) q^{43} + ( - 2 \beta_{3} + 88) q^{44} + ( - 40 \beta_{3} + 110 \beta_{2} - 40 \beta_1 - 110) q^{45} + ( - 92 \beta_{3} + 48 \beta_{2} - 92 \beta_1 - 48) q^{46} + ( - 121 \beta_{3} - 278) q^{47} + (21 \beta_{3} + 132 \beta_{2} + 21 \beta_1 - 132) q^{48} + ( - 241 \beta_{2} + 99 \beta_1) q^{49} + ( - 12 \beta_{2} - 120 \beta_1) q^{50} + (71 \beta_{3} + 276) q^{51} + ( - 30 \beta_{3} - 74) q^{53} + ( - 36 \beta_{2} + 163 \beta_1) q^{54} + (20 \beta_{2} - 22 \beta_1) q^{55} + (33 \beta_{3} - 524 \beta_{2} + 33 \beta_1 + 524) q^{56} + (46 \beta_{3} - 280) q^{57} + ( - 26 \beta_{3} - 384 \beta_{2} - 26 \beta_1 + 384) q^{58} + ( - 124 \beta_{3} + 246 \beta_{2} - 124 \beta_1 - 246) q^{59} + ( - 41 \beta_{3} - 108) q^{60} + ( - 190 \beta_{3} - 434 \beta_{2} - 190 \beta_1 + 434) q^{61} + (136 \beta_{2} + 60 \beta_1) q^{62} + ( - 910 \beta_{2} + 260 \beta_1) q^{63} + ( - 89 \beta_{3} + 340) q^{64} + (98 \beta_{3} - 360) q^{66} + (150 \beta_{2} - 232 \beta_1) q^{67} + (140 \beta_{2} - 69 \beta_1) q^{68} + ( - 324 \beta_{3} + 560 \beta_{2} - 324 \beta_1 - 560) q^{69} + (43 \beta_{3} + 84) q^{70} + (231 \beta_{3} - 50 \beta_{2} + 231 \beta_1 + 50) q^{71} + (170 \beta_{3} - 760 \beta_{2} + 170 \beta_1 + 760) q^{72} + (260 \beta_{3} - 98) q^{73} + (107 \beta_{3} + 20 \beta_{2} + 107 \beta_1 - 20) q^{74} + (432 \beta_{2} - 348 \beta_1) q^{75} + ( - 24 \beta_{2} + 70 \beta_1) q^{76} + ( - 386 \beta_{3} + 188) q^{77} + ( - 40 \beta_{3} - 524) q^{79} + ( - 36 \beta_{2} + 3 \beta_1) q^{80} + ( - 121 \beta_{2} + 120 \beta_1) q^{81} + ( - 104 \beta_{3} + 88 \beta_{2} - 104 \beta_1 - 88) q^{82} + ( - 182 \beta_{3} - 1070) q^{83} + (295 \beta_{3} - 852 \beta_{2} + 295 \beta_1 + 852) q^{84} + ( - 35 \beta_{3} + 104 \beta_{2} - 35 \beta_1 - 104) q^{85} + (215 \beta_{3} - 572) q^{86} + (306 \beta_{3} - 1432 \beta_{2} + 306 \beta_1 + 1432) q^{87} + (392 \beta_{2} + 458 \beta_1) q^{88} + ( - 166 \beta_{2} - 388 \beta_1) q^{89} + (70 \beta_{3} + 160) q^{90} + ( - 140 \beta_{3} - 464) q^{92} + (304 \beta_{2} + 44 \beta_1) q^{93} + (484 \beta_{2} - 157 \beta_1) q^{94} + (6 \beta_{3} - 76 \beta_{2} + 6 \beta_1 + 76) q^{95} + (337 \beta_{3} + 1100) q^{96} + ( - 508 \beta_{3} + 718 \beta_{2} - 508 \beta_1 - 718) q^{97} + ( - 142 \beta_{3} + 396 \beta_{2} - 142 \beta_1 - 396) q^{98} + ( - 390 \beta_{3} - 130) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 5 q^{3} + 7 q^{4} + 6 q^{5} - 23 q^{6} - 9 q^{7} + 6 q^{8} - 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - 5 q^{3} + 7 q^{4} + 6 q^{5} - 23 q^{6} - 9 q^{7} + 6 q^{8} - 35 q^{9} - 7 q^{10} + 80 q^{11} - 86 q^{12} + 178 q^{14} - 33 q^{15} + 39 q^{16} - 19 q^{17} + 220 q^{18} - 84 q^{19} + 19 q^{20} + 606 q^{21} - 142 q^{22} - 196 q^{23} + 273 q^{24} - 474 q^{25} + 670 q^{27} + 125 q^{28} + 44 q^{29} - 43 q^{30} + 172 q^{31} - 123 q^{32} - 106 q^{33} + 270 q^{34} - 107 q^{35} + 250 q^{36} + 209 q^{37} - 628 q^{38} - 178 q^{40} - 230 q^{41} - 197 q^{42} - 287 q^{43} + 356 q^{44} - 180 q^{45} - 4 q^{46} - 870 q^{47} - 285 q^{48} - 383 q^{49} - 144 q^{50} + 962 q^{51} - 236 q^{53} + 91 q^{54} + 18 q^{55} + 1015 q^{56} - 1212 q^{57} + 794 q^{58} - 368 q^{59} - 350 q^{60} + 1058 q^{61} + 332 q^{62} - 1560 q^{63} + 1538 q^{64} - 1636 q^{66} + 68 q^{67} + 211 q^{68} - 796 q^{69} + 250 q^{70} - 131 q^{71} + 1350 q^{72} - 912 q^{73} - 147 q^{74} + 516 q^{75} + 22 q^{76} + 1524 q^{77} - 2016 q^{79} - 69 q^{80} - 122 q^{81} - 72 q^{82} - 3916 q^{83} + 1409 q^{84} - 173 q^{85} - 2718 q^{86} + 2558 q^{87} + 1242 q^{88} - 720 q^{89} + 500 q^{90} - 1576 q^{92} + 652 q^{93} + 811 q^{94} + 146 q^{95} + 3726 q^{96} - 928 q^{97} - 650 q^{98} + 260 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) Copy content Toggle raw display

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 \( T_{2}^{4} - T_{2}^{3} + 5T_{2}^{2} + 4T_{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + 5 T^{2} + 4 T + 16 \) Copy content Toggle raw display
$3$ \( T^{4} + 5 T^{3} + 57 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$5$ \( (T^{2} - 3 T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 9 T^{3} + 575 T^{2} + \cdots + 244036 \) Copy content Toggle raw display
$11$ \( T^{4} - 80 T^{3} + 5412 T^{2} + \cdots + 976144 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 19 T^{3} + 1499 T^{2} + \cdots + 1295044 \) Copy content Toggle raw display
$19$ \( T^{4} + 84 T^{3} + 9644 T^{2} + \cdots + 6697744 \) Copy content Toggle raw display
$23$ \( T^{4} + 196 T^{3} + \cdots + 80856064 \) Copy content Toggle raw display
$29$ \( T^{4} - 44 T^{3} + \cdots + 1496451856 \) Copy content Toggle raw display
$31$ \( (T^{2} - 86 T - 3064)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 209 T^{3} + \cdots + 116942596 \) Copy content Toggle raw display
$41$ \( T^{4} + 230 T^{3} + \cdots + 124724224 \) Copy content Toggle raw display
$43$ \( T^{4} + 287 T^{3} + \cdots + 4397811856 \) Copy content Toggle raw display
$47$ \( (T^{2} + 435 T - 14918)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 118 T - 344)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 368 T^{3} + \cdots + 991746064 \) Copy content Toggle raw display
$61$ \( T^{4} - 1058 T^{3} + \cdots + 15981005056 \) Copy content Toggle raw display
$67$ \( T^{4} - 68 T^{3} + \cdots + 51799939216 \) Copy content Toggle raw display
$71$ \( T^{4} + 131 T^{3} + \cdots + 49503580036 \) Copy content Toggle raw display
$73$ \( (T^{2} + 456 T - 235316)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 1008 T + 247216)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 1958 T + 817664)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 720 T^{3} + \cdots + 260316284944 \) Copy content Toggle raw display
$97$ \( T^{4} + 928 T^{3} + \cdots + 776999938576 \) Copy content Toggle raw display
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