Properties

Label 169.4.b.e
Level $169$
Weight $4$
Character orbit 169.b
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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (3 \beta_{2} + \beta_1) q^{2} + (3 \beta_{3} + 1) q^{3} - 5 \beta_{3} q^{4} + (5 \beta_{2} - 5 \beta_1) q^{5} + (24 \beta_{2} + 10 \beta_1) q^{6} + (8 \beta_{2} + \beta_1) q^{7} + ( - 11 \beta_{2} - 7 \beta_1) q^{8} + (15 \beta_{3} + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta_{2} + \beta_1) q^{2} + (3 \beta_{3} + 1) q^{3} - 5 \beta_{3} q^{4} + (5 \beta_{2} - 5 \beta_1) q^{5} + (24 \beta_{2} + 10 \beta_1) q^{6} + (8 \beta_{2} + \beta_1) q^{7} + ( - 11 \beta_{2} - 7 \beta_1) q^{8} + (15 \beta_{3} + 10) q^{9} + 5 \beta_{3} q^{10} + (16 \beta_{2} + 15 \beta_1) q^{11} + ( - 20 \beta_{3} - 60) q^{12} + ( - 10 \beta_{3} - 18) q^{14} + ( - 40 \beta_{2} + 10 \beta_1) q^{15} + ( - 15 \beta_{3} + 36) q^{16} + (16 \beta_{3} - 43) q^{17} + (135 \beta_{2} + 55 \beta_1) q^{18} + (68 \beta_{2} - 5 \beta_1) q^{19} + (75 \beta_{2} - 25 \beta_1) q^{20} + (44 \beta_{2} + 25 \beta_1) q^{21} + ( - 46 \beta_{3} - 62) q^{22} + (33 \beta_{3} - 89) q^{23} + ( - 128 \beta_{2} - 40 \beta_1) q^{24} + (75 \beta_{3} - 75) q^{25} + (9 \beta_{3} + 163) q^{27} + ( - 60 \beta_{2} - 40 \beta_1) q^{28} + (60 \beta_{3} - 13) q^{29} + (20 \beta_{3} + 60) q^{30} + (20 \beta_{2} - 100 \beta_1) q^{31} + ( - 85 \beta_{2} - 65 \beta_1) q^{32} + (244 \beta_{2} + 63 \beta_1) q^{33} + ( - 17 \beta_{2} + 5 \beta_1) q^{34} + (30 \beta_{3} - 50) q^{35} + ( - 125 \beta_{3} - 300) q^{36} + ( - 117 \beta_{2} - 44 \beta_1) q^{37} + ( - 58 \beta_{3} - 126) q^{38} + (15 \beta_{3} - 100) q^{40} + ( - 279 \beta_{2} - 20 \beta_1) q^{41} + ( - 94 \beta_{3} - 138) q^{42} + ( - 97 \beta_{3} - 179) q^{43} + ( - 380 \beta_{2} - 80 \beta_1) q^{44} + ( - 175 \beta_{2} + 25 \beta_1) q^{45} + ( - 36 \beta_{2} + 10 \beta_1) q^{46} + ( - 40 \beta_{2} - 140 \beta_1) q^{47} + (48 \beta_{3} - 144) q^{48} + ( - 15 \beta_{3} + 290) q^{49} + (300 \beta_{2} + 150 \beta_1) q^{50} + ( - 65 \beta_{3} + 149) q^{51} + (165 \beta_{3} + 190) q^{53} + (552 \beta_{2} + 190 \beta_1) q^{54} + ( - 70 \beta_{3} + 290) q^{55} + (60 \beta_{3} + 56) q^{56} + (212 \beta_{2} + 199 \beta_1) q^{57} + (381 \beta_{2} + 167 \beta_1) q^{58} + ( - 432 \beta_{2} - 55 \beta_1) q^{59} + 200 \beta_1 q^{60} + ( - 200 \beta_{3} + 351) q^{61} + (180 \beta_{3} + 160) q^{62} + (260 \beta_{2} + 130 \beta_1) q^{63} + (95 \beta_{3} + 588) q^{64} + ( - 370 \beta_{3} - 614) q^{66} + (192 \beta_{2} - 91 \beta_1) q^{67} + (135 \beta_{3} - 320) q^{68} + ( - 135 \beta_{3} + 307) q^{69} + (60 \beta_{2} + 40 \beta_1) q^{70} + ( - 116 \beta_{2} - 105 \beta_1) q^{71} + ( - 695 \beta_{2} - 235 \beta_1) q^{72} + (335 \beta_{2} + 85 \beta_1) q^{73} + (205 \beta_{3} + 322) q^{74} + (75 \beta_{3} + 825) q^{75} + ( - 240 \beta_{2} - 340 \beta_1) q^{76} + ( - 121 \beta_{3} - 67) q^{77} + ( - 40 \beta_{3} + 140) q^{79} + (405 \beta_{2} - 255 \beta_1) q^{80} + (120 \beta_{3} + 1) q^{81} + (319 \beta_{3} + 598) q^{82} + ( - 80 \beta_{2} + 100 \beta_1) q^{83} + ( - 720 \beta_{2} - 220 \beta_1) q^{84} + ( - 455 \beta_{2} + 295 \beta_1) q^{85} + ( - 1216 \beta_{2} - 470 \beta_1) q^{86} + (201 \beta_{3} + 707) q^{87} + (172 \beta_{3} + 424) q^{88} + (398 \beta_{2} - 125 \beta_1) q^{89} + (125 \beta_{3} + 300) q^{90} + (280 \beta_{3} - 660) q^{92} + ( - 1120 \beta_{2} - 40 \beta_1) q^{93} + (320 \beta_{3} + 360) q^{94} + (390 \beta_{3} - 830) q^{95} + ( - 1120 \beta_{2} - 320 \beta_1) q^{96} + (442 \beta_{2} + 469 \beta_1) q^{97} + (765 \beta_{2} + 245 \beta_1) q^{98} + (1300 \beta_{2} + 390 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{3} - 10 q^{4} + 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{3} - 10 q^{4} + 70 q^{9} + 10 q^{10} - 280 q^{12} - 92 q^{14} + 114 q^{16} - 140 q^{17} - 340 q^{22} - 290 q^{23} - 150 q^{25} + 670 q^{27} + 68 q^{29} + 280 q^{30} - 140 q^{35} - 1450 q^{36} - 620 q^{38} - 370 q^{40} - 740 q^{42} - 910 q^{43} - 480 q^{48} + 1130 q^{49} + 466 q^{51} + 1090 q^{53} + 1020 q^{55} + 344 q^{56} + 1004 q^{61} + 1000 q^{62} + 2542 q^{64} - 3196 q^{66} - 1010 q^{68} + 958 q^{69} + 1698 q^{74} + 3450 q^{75} - 510 q^{77} + 480 q^{79} + 244 q^{81} + 3030 q^{82} + 3230 q^{87} + 2040 q^{88} + 1450 q^{90} - 2080 q^{92} + 2080 q^{94} - 2540 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
168.1
1.56155i
2.56155i
2.56155i
1.56155i
4.56155i 8.68466 −12.8078 2.80776i 39.6155i 9.56155i 21.9309i 48.4233 12.8078
168.2 0.438447i −3.68466 7.80776 17.8078i 1.61553i 5.43845i 6.93087i −13.4233 −7.80776
168.3 0.438447i −3.68466 7.80776 17.8078i 1.61553i 5.43845i 6.93087i −13.4233 −7.80776
168.4 4.56155i 8.68466 −12.8078 2.80776i 39.6155i 9.56155i 21.9309i 48.4233 12.8078
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.b.e 4
13.b even 2 1 inner 169.4.b.e 4
13.c even 3 2 169.4.e.g 8
13.d odd 4 1 169.4.a.f 2
13.d odd 4 1 169.4.a.j 2
13.e even 6 2 169.4.e.g 8
13.f odd 12 2 13.4.c.b 4
13.f odd 12 2 169.4.c.f 4
39.f even 4 1 1521.4.a.l 2
39.f even 4 1 1521.4.a.t 2
39.k even 12 2 117.4.g.d 4
52.l even 12 2 208.4.i.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 13.f odd 12 2
117.4.g.d 4 39.k even 12 2
169.4.a.f 2 13.d odd 4 1
169.4.a.j 2 13.d odd 4 1
169.4.b.e 4 1.a even 1 1 trivial
169.4.b.e 4 13.b even 2 1 inner
169.4.c.f 4 13.f odd 12 2
169.4.e.g 8 13.c even 3 2
169.4.e.g 8 13.e even 6 2
208.4.i.e 4 52.l even 12 2
1521.4.a.l 2 39.f even 4 1
1521.4.a.t 2 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 21T_{2}^{2} + 4 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 21T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} - 5 T - 32)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 325T^{2} + 2500 \) Copy content Toggle raw display
$7$ \( T^{4} + 121T^{2} + 2704 \) Copy content Toggle raw display
$11$ \( T^{4} + 2057 T^{2} + 781456 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 70 T + 137)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 10153 T^{2} + \cdots + 23658496 \) Copy content Toggle raw display
$23$ \( (T^{2} + 145 T + 628)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 34 T - 15011)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 94800 T^{2} + \cdots + 1413760000 \) Copy content Toggle raw display
$37$ \( T^{4} + 34506 T^{2} + \cdots + 635209 \) Copy content Toggle raw display
$41$ \( T^{4} + 148122 T^{2} + \cdots + 4992976921 \) Copy content Toggle raw display
$43$ \( (T^{2} + 455 T + 11768)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 168400 T^{2} + \cdots + 6789760000 \) Copy content Toggle raw display
$53$ \( (T^{2} - 545 T - 41450)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 352953 T^{2} + \cdots + 22729783696 \) Copy content Toggle raw display
$61$ \( (T^{2} - 502 T - 106999)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 183201 T^{2} + \cdots + 449948944 \) Copy content Toggle raw display
$71$ \( T^{4} + 101777 T^{2} + \cdots + 1833894976 \) Copy content Toggle raw display
$73$ \( T^{4} + 232525 T^{2} + \cdots + 3008522500 \) Copy content Toggle raw display
$79$ \( (T^{2} - 240 T + 7600)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 118800 T^{2} + \cdots + 655360000 \) Copy content Toggle raw display
$89$ \( T^{4} + 556933 T^{2} + \cdots + 21215087716 \) Copy content Toggle raw display
$97$ \( T^{4} + 1955781 T^{2} + \cdots + 795268001284 \) Copy content Toggle raw display
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