# 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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