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 $2$

Related objects

Downloads

Learn more

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: \(x^{4} + 9 x^{2} + 16\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + 3 \beta_{3} ) q^{3} + ( 3 + \beta_{3} ) q^{4} + ( \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + 6 \beta_{2} ) q^{6} + ( -11 \beta_{1} - 5 \beta_{2} ) q^{7} + ( 11 \beta_{1} + 2 \beta_{2} ) q^{8} + ( 10 + 15 \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + 3 \beta_{3} ) q^{3} + ( 3 + \beta_{3} ) q^{4} + ( \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + 6 \beta_{2} ) q^{6} + ( -11 \beta_{1} - 5 \beta_{2} ) q^{7} + ( 11 \beta_{1} + 2 \beta_{2} ) q^{8} + ( 10 + 15 \beta_{3} ) q^{9} + ( -3 - \beta_{3} ) q^{10} + ( -12 \beta_{1} + 17 \beta_{2} ) q^{11} + ( 15 + 13 \beta_{3} ) q^{12} + ( 45 - \beta_{3} ) q^{14} + ( 7 \beta_{1} + 10 \beta_{2} ) q^{15} + ( -27 + 15 \beta_{3} ) q^{16} + ( -1 - 17 \beta_{3} ) q^{17} + ( 10 \beta_{1} + 30 \beta_{2} ) q^{18} + ( -32 \beta_{1} + 13 \beta_{2} ) q^{19} + ( 5 \beta_{1} + 6 \beta_{2} ) q^{20} + ( -41 \beta_{1} - 86 \beta_{2} ) q^{21} + ( 94 - 46 \beta_{3} ) q^{22} + ( -92 - 12 \beta_{3} ) q^{23} + ( 23 \beta_{1} + 74 \beta_{2} ) q^{24} + ( 120 - 3 \beta_{3} ) q^{25} + ( 163 + 9 \beta_{3} ) q^{27} + ( -43 \beta_{1} - 42 \beta_{2} ) q^{28} + ( 26 - 96 \beta_{3} ) q^{29} + ( -15 - 13 \beta_{3} ) q^{30} + ( -34 \beta_{1} + 13 \beta_{2} ) q^{31} + ( 61 \beta_{1} + 46 \beta_{2} ) q^{32} + ( 90 \beta_{1} - 4 \beta_{2} ) q^{33} + ( -\beta_{1} - 34 \beta_{2} ) q^{34} + ( 43 + 21 \beta_{3} ) q^{35} + ( 90 + 70 \beta_{3} ) q^{36} + ( -5 \beta_{1} + 51 \beta_{2} ) q^{37} + ( 186 - 58 \beta_{3} ) q^{38} + ( -37 - 15 \beta_{3} ) q^{40} + ( 22 \beta_{1} + 63 \beta_{2} ) q^{41} + ( 33 + 131 \beta_{3} ) q^{42} + ( -215 + 143 \beta_{3} ) q^{43} + ( -2 \beta_{1} + 44 \beta_{2} ) q^{44} + ( 40 \beta_{1} + 55 \beta_{2} ) q^{45} + ( -92 \beta_{1} - 24 \beta_{2} ) q^{46} + ( 121 \beta_{1} + 139 \beta_{2} ) q^{47} + ( 153 - 21 \beta_{3} ) q^{48} + ( -142 - 99 \beta_{3} ) q^{49} + ( 120 \beta_{1} - 6 \beta_{2} ) q^{50} + ( -205 - 71 \beta_{3} ) q^{51} + ( -44 - 30 \beta_{3} ) q^{53} + ( 163 \beta_{1} + 18 \beta_{2} ) q^{54} + ( 2 - 22 \beta_{3} ) q^{55} + ( 491 + 33 \beta_{3} ) q^{56} + ( 46 \beta_{1} - 140 \beta_{2} ) q^{57} + ( 26 \beta_{1} - 192 \beta_{2} ) q^{58} + ( -124 \beta_{1} - 123 \beta_{2} ) q^{59} + ( 41 \beta_{1} + 54 \beta_{2} ) q^{60} + ( -624 + 190 \beta_{3} ) q^{61} + ( 196 - 60 \beta_{3} ) q^{62} + ( -260 \beta_{1} - 455 \beta_{2} ) q^{63} + ( -429 + 89 \beta_{3} ) q^{64} + ( -458 + 98 \beta_{3} ) q^{66} + ( -232 \beta_{1} - 75 \beta_{2} ) q^{67} + ( -71 - 69 \beta_{3} ) q^{68} + ( -236 - 324 \beta_{3} ) q^{69} + ( 43 \beta_{1} + 42 \beta_{2} ) q^{70} + ( -231 \beta_{1} - 25 \beta_{2} ) q^{71} + ( 170 \beta_{1} + 380 \beta_{2} ) q^{72} + ( -260 \beta_{1} + 49 \beta_{2} ) q^{73} + ( 127 - 107 \beta_{3} ) q^{74} + ( 84 + 348 \beta_{3} ) q^{75} + ( -70 \beta_{1} - 12 \beta_{2} ) q^{76} + ( -574 + 386 \beta_{3} ) q^{77} + ( -484 - 40 \beta_{3} ) q^{79} + ( 3 \beta_{1} + 18 \beta_{2} ) q^{80} + ( 1 + 120 \beta_{3} ) q^{81} + ( 16 - 104 \beta_{3} ) q^{82} + ( -182 \beta_{1} - 535 \beta_{2} ) q^{83} + ( -295 \beta_{1} - 426 \beta_{2} ) q^{84} + ( -35 \beta_{1} - 52 \beta_{2} ) q^{85} + ( -215 \beta_{1} + 286 \beta_{2} ) q^{86} + ( -1126 - 306 \beta_{3} ) q^{87} + ( 850 - 458 \beta_{3} ) q^{88} + ( 388 \beta_{1} - 83 \beta_{2} ) q^{89} + ( -90 - 70 \beta_{3} ) q^{90} + ( -324 - 140 \beta_{3} ) q^{92} + ( 44 \beta_{1} - 152 \beta_{2} ) q^{93} + ( -327 - 157 \beta_{3} ) q^{94} + ( 70 + 6 \beta_{3} ) q^{95} + ( 337 \beta_{1} + 550 \beta_{2} ) q^{96} + ( 508 \beta_{1} + 359 \beta_{2} ) q^{97} + ( -142 \beta_{1} - 198 \beta_{2} ) q^{98} + ( 390 \beta_{1} + 65 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{3} + 14 q^{4} + 70 q^{9} + O(q^{10}) \) \( 4 q + 10 q^{3} + 14 q^{4} + 70 q^{9} - 14 q^{10} + 86 q^{12} + 178 q^{14} - 78 q^{16} - 38 q^{17} + 284 q^{22} - 392 q^{23} + 474 q^{25} + 670 q^{27} - 88 q^{29} - 86 q^{30} + 214 q^{35} + 500 q^{36} + 628 q^{38} - 178 q^{40} + 394 q^{42} - 574 q^{43} + 570 q^{48} - 766 q^{49} - 962 q^{51} - 236 q^{53} - 36 q^{55} + 2030 q^{56} - 2116 q^{61} + 664 q^{62} - 1538 q^{64} - 1636 q^{66} - 422 q^{68} - 1592 q^{69} + 294 q^{74} + 1032 q^{75} - 1524 q^{77} - 2016 q^{79} + 244 q^{81} - 144 q^{82} - 5116 q^{87} + 2484 q^{88} - 500 q^{90} - 1576 q^{92} - 1622 q^{94} + 292 q^{95} + O(q^{100}) \)

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

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

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 \( T_{2}^{4} + 9 T_{2}^{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).

Hecke characteristic polynomials

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