# Properties

 Label 169.4.b.c Level $169$ Weight $4$ Character orbit 169.b Analytic conductor $9.971$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} + 2 q^{3} - 8 q^{4} + 17 i q^{5} + 8 i q^{6} - 20 i q^{7} - 23 q^{9} +O(q^{10})$$ q + 4*i * q^2 + 2 * q^3 - 8 * q^4 + 17*i * q^5 + 8*i * q^6 - 20*i * q^7 - 23 * q^9 $$q + 4 i q^{2} + 2 q^{3} - 8 q^{4} + 17 i q^{5} + 8 i q^{6} - 20 i q^{7} - 23 q^{9} - 68 q^{10} + 32 i q^{11} - 16 q^{12} + 80 q^{14} + 34 i q^{15} - 64 q^{16} + 13 q^{17} - 92 i q^{18} + 30 i q^{19} - 136 i q^{20} - 40 i q^{21} - 128 q^{22} - 78 q^{23} - 164 q^{25} - 100 q^{27} + 160 i q^{28} + 197 q^{29} - 136 q^{30} - 74 i q^{31} - 256 i q^{32} + 64 i q^{33} + 52 i q^{34} + 340 q^{35} + 184 q^{36} + 227 i q^{37} - 120 q^{38} - 165 i q^{41} + 160 q^{42} + 156 q^{43} - 256 i q^{44} - 391 i q^{45} - 312 i q^{46} + 162 i q^{47} - 128 q^{48} - 57 q^{49} - 656 i q^{50} + 26 q^{51} + 93 q^{53} - 400 i q^{54} - 544 q^{55} + 60 i q^{57} + 788 i q^{58} + 864 i q^{59} - 272 i q^{60} + 145 q^{61} + 296 q^{62} + 460 i q^{63} + 512 q^{64} - 256 q^{66} + 862 i q^{67} - 104 q^{68} - 156 q^{69} + 1360 i q^{70} + 654 i q^{71} - 215 i q^{73} - 908 q^{74} - 328 q^{75} - 240 i q^{76} + 640 q^{77} - 76 q^{79} - 1088 i q^{80} + 421 q^{81} + 660 q^{82} + 628 i q^{83} + 320 i q^{84} + 221 i q^{85} + 624 i q^{86} + 394 q^{87} + 266 i q^{89} + 1564 q^{90} + 624 q^{92} - 148 i q^{93} - 648 q^{94} - 510 q^{95} - 512 i q^{96} + 238 i q^{97} - 228 i q^{98} - 736 i q^{99} +O(q^{100})$$ q + 4*i * q^2 + 2 * q^3 - 8 * q^4 + 17*i * q^5 + 8*i * q^6 - 20*i * q^7 - 23 * q^9 - 68 * q^10 + 32*i * q^11 - 16 * q^12 + 80 * q^14 + 34*i * q^15 - 64 * q^16 + 13 * q^17 - 92*i * q^18 + 30*i * q^19 - 136*i * q^20 - 40*i * q^21 - 128 * q^22 - 78 * q^23 - 164 * q^25 - 100 * q^27 + 160*i * q^28 + 197 * q^29 - 136 * q^30 - 74*i * q^31 - 256*i * q^32 + 64*i * q^33 + 52*i * q^34 + 340 * q^35 + 184 * q^36 + 227*i * q^37 - 120 * q^38 - 165*i * q^41 + 160 * q^42 + 156 * q^43 - 256*i * q^44 - 391*i * q^45 - 312*i * q^46 + 162*i * q^47 - 128 * q^48 - 57 * q^49 - 656*i * q^50 + 26 * q^51 + 93 * q^53 - 400*i * q^54 - 544 * q^55 + 60*i * q^57 + 788*i * q^58 + 864*i * q^59 - 272*i * q^60 + 145 * q^61 + 296 * q^62 + 460*i * q^63 + 512 * q^64 - 256 * q^66 + 862*i * q^67 - 104 * q^68 - 156 * q^69 + 1360*i * q^70 + 654*i * q^71 - 215*i * q^73 - 908 * q^74 - 328 * q^75 - 240*i * q^76 + 640 * q^77 - 76 * q^79 - 1088*i * q^80 + 421 * q^81 + 660 * q^82 + 628*i * q^83 + 320*i * q^84 + 221*i * q^85 + 624*i * q^86 + 394 * q^87 + 266*i * q^89 + 1564 * q^90 + 624 * q^92 - 148*i * q^93 - 648 * q^94 - 510 * q^95 - 512*i * q^96 + 238*i * q^97 - 228*i * q^98 - 736*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} - 16 q^{4} - 46 q^{9}+O(q^{10})$$ 2 * q + 4 * q^3 - 16 * q^4 - 46 * q^9 $$2 q + 4 q^{3} - 16 q^{4} - 46 q^{9} - 136 q^{10} - 32 q^{12} + 160 q^{14} - 128 q^{16} + 26 q^{17} - 256 q^{22} - 156 q^{23} - 328 q^{25} - 200 q^{27} + 394 q^{29} - 272 q^{30} + 680 q^{35} + 368 q^{36} - 240 q^{38} + 320 q^{42} + 312 q^{43} - 256 q^{48} - 114 q^{49} + 52 q^{51} + 186 q^{53} - 1088 q^{55} + 290 q^{61} + 592 q^{62} + 1024 q^{64} - 512 q^{66} - 208 q^{68} - 312 q^{69} - 1816 q^{74} - 656 q^{75} + 1280 q^{77} - 152 q^{79} + 842 q^{81} + 1320 q^{82} + 788 q^{87} + 3128 q^{90} + 1248 q^{92} - 1296 q^{94} - 1020 q^{95}+O(q^{100})$$ 2 * q + 4 * q^3 - 16 * q^4 - 46 * q^9 - 136 * q^10 - 32 * q^12 + 160 * q^14 - 128 * q^16 + 26 * q^17 - 256 * q^22 - 156 * q^23 - 328 * q^25 - 200 * q^27 + 394 * q^29 - 272 * q^30 + 680 * q^35 + 368 * q^36 - 240 * q^38 + 320 * q^42 + 312 * q^43 - 256 * q^48 - 114 * q^49 + 52 * q^51 + 186 * q^53 - 1088 * q^55 + 290 * q^61 + 592 * q^62 + 1024 * q^64 - 512 * q^66 - 208 * q^68 - 312 * q^69 - 1816 * q^74 - 656 * q^75 + 1280 * q^77 - 152 * q^79 + 842 * q^81 + 1320 * q^82 + 788 * q^87 + 3128 * q^90 + 1248 * q^92 - 1296 * q^94 - 1020 * q^95

## 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
 − 1.00000i 1.00000i
4.00000i 2.00000 −8.00000 17.0000i 8.00000i 20.0000i 0 −23.0000 −68.0000
168.2 4.00000i 2.00000 −8.00000 17.0000i 8.00000i 20.0000i 0 −23.0000 −68.0000
 $$n$$: 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.c 2
13.b even 2 1 inner 169.4.b.c 2
13.c even 3 2 169.4.e.c 4
13.d odd 4 1 169.4.a.a 1
13.d odd 4 1 169.4.a.d 1
13.e even 6 2 169.4.e.c 4
13.f odd 12 2 13.4.c.a 2
13.f odd 12 2 169.4.c.d 2
39.f even 4 1 1521.4.a.b 1
39.f even 4 1 1521.4.a.k 1
39.k even 12 2 117.4.g.c 2
52.l even 12 2 208.4.i.b 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$(T - 2)^{2}$$
$5$ $$T^{2} + 289$$
$7$ $$T^{2} + 400$$
$11$ $$T^{2} + 1024$$
$13$ $$T^{2}$$
$17$ $$(T - 13)^{2}$$
$19$ $$T^{2} + 900$$
$23$ $$(T + 78)^{2}$$
$29$ $$(T - 197)^{2}$$
$31$ $$T^{2} + 5476$$
$37$ $$T^{2} + 51529$$
$41$ $$T^{2} + 27225$$
$43$ $$(T - 156)^{2}$$
$47$ $$T^{2} + 26244$$
$53$ $$(T - 93)^{2}$$
$59$ $$T^{2} + 746496$$
$61$ $$(T - 145)^{2}$$
$67$ $$T^{2} + 743044$$
$71$ $$T^{2} + 427716$$
$73$ $$T^{2} + 46225$$
$79$ $$(T + 76)^{2}$$
$83$ $$T^{2} + 394384$$
$89$ $$T^{2} + 70756$$
$97$ $$T^{2} + 56644$$