Properties

Label 169.4.c.f
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{2} + (3 \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{3} - 5 \beta_1 q^{4} + (5 \beta_{3} + 10) q^{5} + ( - 14 \beta_{2} - 10 \beta_1) q^{6} + (7 \beta_{2} + \beta_1) q^{7} + ( - 7 \beta_{3} + 4) q^{8} + ( - 10 \beta_{2} - 15 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{2} + (3 \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{3} - 5 \beta_1 q^{4} + (5 \beta_{3} + 10) q^{5} + ( - 14 \beta_{2} - 10 \beta_1) q^{6} + (7 \beta_{2} + \beta_1) q^{7} + ( - 7 \beta_{3} + 4) q^{8} + ( - 10 \beta_{2} - 15 \beta_1) q^{9} + ( - 5 \beta_{3} - 5 \beta_1) q^{10} + ( - 15 \beta_{3} - \beta_{2} - 15 \beta_1 + 1) q^{11} + ( - 20 \beta_{3} + 60) q^{12} + (10 \beta_{3} - 18) q^{14} + (10 \beta_{3} - 50 \beta_{2} + 10 \beta_1 + 50) q^{15} + ( - 15 \beta_{3} + 36 \beta_{2} - 15 \beta_1 - 36) q^{16} + ( - 43 \beta_{2} + 16 \beta_1) q^{17} + ( - 55 \beta_{3} + 80) q^{18} + ( - 73 \beta_{2} + 5 \beta_1) q^{19} + (100 \beta_{2} - 25 \beta_1) q^{20} + (25 \beta_{3} - 19) q^{21} + (62 \beta_{2} + 46 \beta_1) q^{22} + ( - 33 \beta_{3} + 89 \beta_{2} - 33 \beta_1 - 89) q^{23} + (40 \beta_{3} + 88 \beta_{2} + 40 \beta_1 - 88) q^{24} + (75 \beta_{3} + 75) q^{25} + ( - 9 \beta_{3} + 163) q^{27} + ( - 40 \beta_{3} - 20 \beta_{2} - 40 \beta_1 + 20) q^{28} + (60 \beta_{3} - 13 \beta_{2} + 60 \beta_1 + 13) q^{29} + (60 \beta_{2} + 20 \beta_1) q^{30} + (100 \beta_{3} + 120) q^{31} + (20 \beta_{2} + 65 \beta_1) q^{32} + (181 \beta_{2} + 63 \beta_1) q^{33} + (5 \beta_{3} + 22) q^{34} + (50 \beta_{2} - 30 \beta_1) q^{35} + (125 \beta_{3} + 300 \beta_{2} + 125 \beta_1 - 300) q^{36} + (44 \beta_{3} + 73 \beta_{2} + 44 \beta_1 - 73) q^{37} + ( - 58 \beta_{3} + 126) q^{38} + ( - 15 \beta_{3} - 100) q^{40} + ( - 20 \beta_{3} - 259 \beta_{2} - 20 \beta_1 + 259) q^{41} + ( - 94 \beta_{3} - 138 \beta_{2} - 94 \beta_1 + 138) q^{42} + ( - 179 \beta_{2} - 97 \beta_1) q^{43} + (80 \beta_{3} - 300) q^{44} + (200 \beta_{2} - 25 \beta_1) q^{45} + ( - 46 \beta_{2} + 10 \beta_1) q^{46} + ( - 140 \beta_{3} - 100) q^{47} + (144 \beta_{2} - 48 \beta_1) q^{48} + (15 \beta_{3} - 290 \beta_{2} + 15 \beta_1 + 290) q^{49} + ( - 150 \beta_{3} - 150 \beta_{2} - 150 \beta_1 + 150) q^{50} + ( - 65 \beta_{3} - 149) q^{51} + ( - 165 \beta_{3} + 190) q^{53} + (190 \beta_{3} + 362 \beta_{2} + 190 \beta_1 - 362) q^{54} + ( - 70 \beta_{3} + 290 \beta_{2} - 70 \beta_1 - 290) q^{55} + (56 \beta_{2} + 60 \beta_1) q^{56} + ( - 199 \beta_{3} + 13) q^{57} + ( - 214 \beta_{2} - 167 \beta_1) q^{58} + ( - 377 \beta_{2} - 55 \beta_1) q^{59} + (200 \beta_{3} + 200) q^{60} + ( - 351 \beta_{2} + 200 \beta_1) q^{61} + ( - 180 \beta_{3} - 160 \beta_{2} - 180 \beta_1 + 160) q^{62} + ( - 130 \beta_{3} - 130 \beta_{2} - 130 \beta_1 + 130) q^{63} + (95 \beta_{3} - 588) q^{64} + (370 \beta_{3} - 614) q^{66} + ( - 91 \beta_{3} + 283 \beta_{2} - 91 \beta_1 - 283) q^{67} + (135 \beta_{3} - 320 \beta_{2} + 135 \beta_1 + 320) q^{68} + (307 \beta_{2} - 135 \beta_1) q^{69} + ( - 40 \beta_{3} + 20) q^{70} + (11 \beta_{2} + 105 \beta_1) q^{71} + ( - 460 \beta_{2} - 235 \beta_1) q^{72} + (85 \beta_{3} - 250) q^{73} + ( - 322 \beta_{2} - 205 \beta_1) q^{74} + ( - 75 \beta_{3} - 825 \beta_{2} - 75 \beta_1 + 825) q^{75} + (340 \beta_{3} - 100 \beta_{2} + 340 \beta_1 + 100) q^{76} + ( - 121 \beta_{3} + 67) q^{77} + (40 \beta_{3} + 140) q^{79} + ( - 255 \beta_{3} + 660 \beta_{2} - 255 \beta_1 - 660) q^{80} + (120 \beta_{3} + \beta_{2} + 120 \beta_1 - 1) q^{81} + (598 \beta_{2} + 319 \beta_1) q^{82} + ( - 100 \beta_{3} - 180) q^{83} + (500 \beta_{2} + 220 \beta_1) q^{84} + ( - 750 \beta_{2} + 295 \beta_1) q^{85} + ( - 470 \beta_{3} + 746) q^{86} + ( - 707 \beta_{2} - 201 \beta_1) q^{87} + ( - 172 \beta_{3} - 424 \beta_{2} - 172 \beta_1 + 424) q^{88} + (125 \beta_{3} - 523 \beta_{2} + 125 \beta_1 + 523) q^{89} + (125 \beta_{3} - 300) q^{90} + ( - 280 \beta_{3} - 660) q^{92} + ( - 40 \beta_{3} - 1080 \beta_{2} - 40 \beta_1 + 1080) q^{93} + (320 \beta_{3} + 360 \beta_{2} + 320 \beta_1 - 360) q^{94} + ( - 830 \beta_{2} + 390 \beta_1) q^{95} + (320 \beta_{3} - 800) q^{96} + (27 \beta_{2} - 469 \beta_1) q^{97} + (520 \beta_{2} + 245 \beta_1) q^{98} + (390 \beta_{3} - 910) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 5 q^{2} - 5 q^{3} - 5 q^{4} + 30 q^{5} - 38 q^{6} + 15 q^{7} + 30 q^{8} - 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 5 q^{2} - 5 q^{3} - 5 q^{4} + 30 q^{5} - 38 q^{6} + 15 q^{7} + 30 q^{8} - 35 q^{9} + 5 q^{10} + 17 q^{11} + 280 q^{12} - 92 q^{14} + 90 q^{15} - 57 q^{16} - 70 q^{17} + 430 q^{18} - 141 q^{19} + 175 q^{20} - 126 q^{21} + 170 q^{22} - 145 q^{23} - 216 q^{24} + 150 q^{25} + 670 q^{27} + 80 q^{28} - 34 q^{29} + 140 q^{30} + 280 q^{31} + 105 q^{32} + 425 q^{33} + 78 q^{34} + 70 q^{35} - 725 q^{36} - 190 q^{37} + 620 q^{38} - 370 q^{40} + 538 q^{41} + 370 q^{42} - 455 q^{43} - 1360 q^{44} + 375 q^{45} - 82 q^{46} - 120 q^{47} + 240 q^{48} + 565 q^{49} + 450 q^{50} - 466 q^{51} + 1090 q^{53} - 914 q^{54} - 510 q^{55} + 172 q^{56} + 450 q^{57} - 595 q^{58} - 809 q^{59} + 400 q^{60} - 502 q^{61} + 500 q^{62} + 390 q^{63} - 2542 q^{64} - 3196 q^{66} - 475 q^{67} + 505 q^{68} + 479 q^{69} + 160 q^{70} + 127 q^{71} - 1155 q^{72} - 1170 q^{73} - 849 q^{74} + 1725 q^{75} - 140 q^{76} + 510 q^{77} + 480 q^{79} - 1065 q^{80} - 122 q^{81} + 1515 q^{82} - 520 q^{83} + 1220 q^{84} - 1205 q^{85} + 3924 q^{86} - 1615 q^{87} + 1020 q^{88} + 921 q^{89} - 1450 q^{90} - 2080 q^{92} + 2200 q^{93} - 1040 q^{94} - 1270 q^{95} - 3840 q^{96} - 415 q^{97} + 1285 q^{98} - 4420 q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

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

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
1.28078 + 2.21837i
−0.780776 1.35234i
1.28078 2.21837i
−0.780776 + 1.35234i
−2.28078 + 3.95042i −4.34233 + 7.52113i −6.40388 11.0918i −2.80776 −19.8078 34.3081i 4.78078 + 8.28055i 21.9309 −24.2116 41.9358i 6.40388 11.0918i
22.2 −0.219224 + 0.379706i 1.84233 3.19101i 3.90388 + 6.76172i 17.8078 0.807764 + 1.39909i 2.71922 + 4.70983i −6.93087 6.71165 + 11.6249i −3.90388 + 6.76172i
146.1 −2.28078 3.95042i −4.34233 7.52113i −6.40388 + 11.0918i −2.80776 −19.8078 + 34.3081i 4.78078 8.28055i 21.9309 −24.2116 + 41.9358i 6.40388 + 11.0918i
146.2 −0.219224 0.379706i 1.84233 + 3.19101i 3.90388 6.76172i 17.8078 0.807764 1.39909i 2.71922 4.70983i −6.93087 6.71165 11.6249i −3.90388 6.76172i
\(n\): 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.f 4
13.b even 2 1 13.4.c.b 4
13.c even 3 1 169.4.a.j 2
13.c even 3 1 inner 169.4.c.f 4
13.d odd 4 2 169.4.e.g 8
13.e even 6 1 13.4.c.b 4
13.e even 6 1 169.4.a.f 2
13.f odd 12 2 169.4.b.e 4
13.f odd 12 2 169.4.e.g 8
39.d odd 2 1 117.4.g.d 4
39.h odd 6 1 117.4.g.d 4
39.h odd 6 1 1521.4.a.t 2
39.i odd 6 1 1521.4.a.l 2
52.b odd 2 1 208.4.i.e 4
52.i odd 6 1 208.4.i.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 13.b even 2 1
13.4.c.b 4 13.e even 6 1
117.4.g.d 4 39.d odd 2 1
117.4.g.d 4 39.h odd 6 1
169.4.a.f 2 13.e even 6 1
169.4.a.j 2 13.c even 3 1
169.4.b.e 4 13.f odd 12 2
169.4.c.f 4 1.a even 1 1 trivial
169.4.c.f 4 13.c even 3 1 inner
169.4.e.g 8 13.d odd 4 2
169.4.e.g 8 13.f odd 12 2
208.4.i.e 4 52.b odd 2 1
208.4.i.e 4 52.i odd 6 1
1521.4.a.l 2 39.i odd 6 1
1521.4.a.t 2 39.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 5T_{2}^{3} + 23T_{2}^{2} + 10T_{2} + 4 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} + 5 T^{3} + 57 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$5$ \( (T^{2} - 15 T - 50)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 15 T^{3} + 173 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$11$ \( T^{4} - 17 T^{3} + 1173 T^{2} + \cdots + 781456 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 70 T^{3} + 4763 T^{2} + \cdots + 18769 \) Copy content Toggle raw display
$19$ \( T^{4} + 141 T^{3} + \cdots + 23658496 \) Copy content Toggle raw display
$23$ \( T^{4} + 145 T^{3} + 20397 T^{2} + \cdots + 394384 \) Copy content Toggle raw display
$29$ \( T^{4} + 34 T^{3} + \cdots + 225330121 \) Copy content Toggle raw display
$31$ \( (T^{2} - 140 T - 37600)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 190 T^{3} + 35303 T^{2} + \cdots + 635209 \) Copy content Toggle raw display
$41$ \( T^{4} - 538 T^{3} + \cdots + 4992976921 \) Copy content Toggle raw display
$43$ \( T^{4} + 455 T^{3} + \cdots + 138485824 \) Copy content Toggle raw display
$47$ \( (T^{2} + 60 T - 82400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 545 T - 41450)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 809 T^{3} + \cdots + 22729783696 \) Copy content Toggle raw display
$61$ \( T^{4} + 502 T^{3} + \cdots + 11448786001 \) Copy content Toggle raw display
$67$ \( T^{4} + 475 T^{3} + \cdots + 449948944 \) Copy content Toggle raw display
$71$ \( T^{4} - 127 T^{3} + \cdots + 1833894976 \) Copy content Toggle raw display
$73$ \( (T^{2} + 585 T + 54850)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 240 T + 7600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 260 T - 25600)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 921 T^{3} + \cdots + 21215087716 \) Copy content Toggle raw display
$97$ \( T^{4} + 415 T^{3} + \cdots + 795268001284 \) Copy content Toggle raw display
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