Properties

Label 169.4.b.f
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (3 \beta_{3} + 1) q^{3} + (\beta_{3} + 3) q^{4} + (\beta_{2} + \beta_1) q^{5} + (6 \beta_{2} + \beta_1) q^{6} + ( - 5 \beta_{2} - 11 \beta_1) q^{7} + (2 \beta_{2} + 11 \beta_1) q^{8} + (15 \beta_{3} + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (3 \beta_{3} + 1) q^{3} + (\beta_{3} + 3) q^{4} + (\beta_{2} + \beta_1) q^{5} + (6 \beta_{2} + \beta_1) q^{6} + ( - 5 \beta_{2} - 11 \beta_1) q^{7} + (2 \beta_{2} + 11 \beta_1) q^{8} + (15 \beta_{3} + 10) q^{9} + ( - \beta_{3} - 3) q^{10} + (17 \beta_{2} - 12 \beta_1) q^{11} + (13 \beta_{3} + 15) q^{12} + ( - \beta_{3} + 45) q^{14} + (10 \beta_{2} + 7 \beta_1) q^{15} + (15 \beta_{3} - 27) q^{16} + ( - 17 \beta_{3} - 1) q^{17} + (30 \beta_{2} + 10 \beta_1) q^{18} + (13 \beta_{2} - 32 \beta_1) q^{19} + (6 \beta_{2} + 5 \beta_1) q^{20} + ( - 86 \beta_{2} - 41 \beta_1) q^{21} + ( - 46 \beta_{3} + 94) q^{22} + ( - 12 \beta_{3} - 92) q^{23} + (74 \beta_{2} + 23 \beta_1) q^{24} + ( - 3 \beta_{3} + 120) q^{25} + (9 \beta_{3} + 163) q^{27} + ( - 42 \beta_{2} - 43 \beta_1) q^{28} + ( - 96 \beta_{3} + 26) q^{29} + ( - 13 \beta_{3} - 15) q^{30} + (13 \beta_{2} - 34 \beta_1) q^{31} + (46 \beta_{2} + 61 \beta_1) q^{32} + ( - 4 \beta_{2} + 90 \beta_1) q^{33} + ( - 34 \beta_{2} - \beta_1) q^{34} + (21 \beta_{3} + 43) q^{35} + (70 \beta_{3} + 90) q^{36} + (51 \beta_{2} - 5 \beta_1) q^{37} + ( - 58 \beta_{3} + 186) q^{38} + ( - 15 \beta_{3} - 37) q^{40} + (63 \beta_{2} + 22 \beta_1) q^{41} + (131 \beta_{3} + 33) q^{42} + (143 \beta_{3} - 215) q^{43} + (44 \beta_{2} - 2 \beta_1) q^{44} + (55 \beta_{2} + 40 \beta_1) q^{45} + ( - 24 \beta_{2} - 92 \beta_1) q^{46} + (139 \beta_{2} + 121 \beta_1) q^{47} + ( - 21 \beta_{3} + 153) q^{48} + ( - 99 \beta_{3} - 142) q^{49} + ( - 6 \beta_{2} + 120 \beta_1) q^{50} + ( - 71 \beta_{3} - 205) q^{51} + ( - 30 \beta_{3} - 44) q^{53} + (18 \beta_{2} + 163 \beta_1) q^{54} + ( - 22 \beta_{3} + 2) q^{55} + (33 \beta_{3} + 491) q^{56} + ( - 140 \beta_{2} + 46 \beta_1) q^{57} + ( - 192 \beta_{2} + 26 \beta_1) q^{58} + ( - 123 \beta_{2} - 124 \beta_1) q^{59} + (54 \beta_{2} + 41 \beta_1) q^{60} + (190 \beta_{3} - 624) q^{61} + ( - 60 \beta_{3} + 196) q^{62} + ( - 455 \beta_{2} - 260 \beta_1) q^{63} + (89 \beta_{3} - 429) q^{64} + (98 \beta_{3} - 458) q^{66} + ( - 75 \beta_{2} - 232 \beta_1) q^{67} + ( - 69 \beta_{3} - 71) q^{68} + ( - 324 \beta_{3} - 236) q^{69} + (42 \beta_{2} + 43 \beta_1) q^{70} + ( - 25 \beta_{2} - 231 \beta_1) q^{71} + (380 \beta_{2} + 170 \beta_1) q^{72} + (49 \beta_{2} - 260 \beta_1) q^{73} + ( - 107 \beta_{3} + 127) q^{74} + (348 \beta_{3} + 84) q^{75} + ( - 12 \beta_{2} - 70 \beta_1) q^{76} + (386 \beta_{3} - 574) q^{77} + ( - 40 \beta_{3} - 484) q^{79} + (18 \beta_{2} + 3 \beta_1) q^{80} + (120 \beta_{3} + 1) q^{81} + ( - 104 \beta_{3} + 16) q^{82} + ( - 535 \beta_{2} - 182 \beta_1) q^{83} + ( - 426 \beta_{2} - 295 \beta_1) q^{84} + ( - 52 \beta_{2} - 35 \beta_1) q^{85} + (286 \beta_{2} - 215 \beta_1) q^{86} + ( - 306 \beta_{3} - 1126) q^{87} + ( - 458 \beta_{3} + 850) q^{88} + ( - 83 \beta_{2} + 388 \beta_1) q^{89} + ( - 70 \beta_{3} - 90) q^{90} + ( - 140 \beta_{3} - 324) q^{92} + ( - 152 \beta_{2} + 44 \beta_1) q^{93} + ( - 157 \beta_{3} - 327) q^{94} + (6 \beta_{3} + 70) q^{95} + (550 \beta_{2} + 337 \beta_1) q^{96} + (359 \beta_{2} + 508 \beta_1) q^{97} + ( - 198 \beta_{2} - 142 \beta_1) q^{98} + (65 \beta_{2} + 390 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{3} + 14 q^{4} + 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{3} + 14 q^{4} + 70 q^{9} - 14 q^{10} + 86 q^{12} + 178 q^{14} - 78 q^{16} - 38 q^{17} + 284 q^{22} - 392 q^{23} + 474 q^{25} + 670 q^{27} - 88 q^{29} - 86 q^{30} + 214 q^{35} + 500 q^{36} + 628 q^{38} - 178 q^{40} + 394 q^{42} - 574 q^{43} + 570 q^{48} - 766 q^{49} - 962 q^{51} - 236 q^{53} - 36 q^{55} + 2030 q^{56} - 2116 q^{61} + 664 q^{62} - 1538 q^{64} - 1636 q^{66} - 422 q^{68} - 1592 q^{69} + 294 q^{74} + 1032 q^{75} - 1524 q^{77} - 2016 q^{79} + 244 q^{81} - 144 q^{82} - 5116 q^{87} + 2484 q^{88} - 500 q^{90} - 1576 q^{92} - 1622 q^{94} + 292 q^{95}+O(q^{100}) \) 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
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 2 \) 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
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 5\beta_1 \) 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
2.56155i
1.56155i
1.56155i
2.56155i
2.56155i −3.68466 1.43845 0.561553i 9.43845i 18.1771i 24.1771i −13.4233 −1.43845
168.2 1.56155i 8.68466 5.56155 3.56155i 13.5616i 27.1771i 21.1771i 48.4233 −5.56155
168.3 1.56155i 8.68466 5.56155 3.56155i 13.5616i 27.1771i 21.1771i 48.4233 −5.56155
168.4 2.56155i −3.68466 1.43845 0.561553i 9.43845i 18.1771i 24.1771i −13.4233 −1.43845
\(n\): 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.f 4
13.b even 2 1 inner 169.4.b.f 4
13.c even 3 2 169.4.e.f 8
13.d odd 4 1 13.4.a.b 2
13.d odd 4 1 169.4.a.g 2
13.e even 6 2 169.4.e.f 8
13.f odd 12 2 169.4.c.g 4
13.f odd 12 2 169.4.c.j 4
39.f even 4 1 117.4.a.d 2
39.f even 4 1 1521.4.a.r 2
52.f even 4 1 208.4.a.h 2
65.f even 4 1 325.4.b.e 4
65.g odd 4 1 325.4.a.f 2
65.k even 4 1 325.4.b.e 4
91.i even 4 1 637.4.a.b 2
104.j odd 4 1 832.4.a.s 2
104.m even 4 1 832.4.a.z 2
143.g even 4 1 1573.4.a.b 2
156.l odd 4 1 1872.4.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 13.d odd 4 1
117.4.a.d 2 39.f even 4 1
169.4.a.g 2 13.d odd 4 1
169.4.b.f 4 1.a even 1 1 trivial
169.4.b.f 4 13.b even 2 1 inner
169.4.c.g 4 13.f odd 12 2
169.4.c.j 4 13.f odd 12 2
169.4.e.f 8 13.c even 3 2
169.4.e.f 8 13.e even 6 2
208.4.a.h 2 52.f even 4 1
325.4.a.f 2 65.g odd 4 1
325.4.b.e 4 65.f even 4 1
325.4.b.e 4 65.k even 4 1
637.4.a.b 2 91.i even 4 1
832.4.a.s 2 104.j odd 4 1
832.4.a.z 2 104.m even 4 1
1521.4.a.r 2 39.f even 4 1
1573.4.a.b 2 143.g even 4 1
1872.4.a.bb 2 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 9T_{2}^{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} - 5 T - 32)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 13T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} + 1069 T^{2} + 244036 \) Copy content Toggle raw display
$11$ \( T^{4} + 4424 T^{2} + 976144 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 19 T - 1138)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 12232 T^{2} + \cdots + 6697744 \) Copy content Toggle raw display
$23$ \( (T^{2} + 196 T + 8992)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 44 T - 38684)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 13524 T^{2} + \cdots + 9388096 \) Copy content Toggle raw display
$37$ \( T^{4} + 22053 T^{2} + \cdots + 116942596 \) Copy content Toggle raw display
$41$ \( T^{4} + 30564 T^{2} + \cdots + 124724224 \) Copy content Toggle raw display
$43$ \( (T^{2} + 287 T - 66316)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 219061 T^{2} + \cdots + 222546724 \) Copy content Toggle raw display
$53$ \( (T^{2} + 118 T - 344)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 198408 T^{2} + \cdots + 991746064 \) Copy content Toggle raw display
$61$ \( (T^{2} + 1058 T + 126416)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 459816 T^{2} + \cdots + 51799939216 \) Copy content Toggle raw display
$71$ \( T^{4} + 462149 T^{2} + \cdots + 49503580036 \) Copy content Toggle raw display
$73$ \( T^{4} + 678568 T^{2} + \cdots + 55373619856 \) Copy content Toggle raw display
$79$ \( (T^{2} + 1008 T + 247216)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 2198436 T^{2} + \cdots + 668574416896 \) Copy content Toggle raw display
$89$ \( T^{4} + 1538824 T^{2} + \cdots + 260316284944 \) Copy content Toggle raw display
$97$ \( T^{4} + 2624136 T^{2} + \cdots + 776999938576 \) Copy content Toggle raw display
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