# Properties

 Label 169.4.b.b 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: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \beta q^{2} - 7 q^{3} - 4 q^{4} - 8 \beta q^{5} + 14 \beta q^{6} - 13 \beta q^{7} - 8 \beta q^{8} + 22 q^{9} +O(q^{10})$$ q - 2*b * q^2 - 7 * q^3 - 4 * q^4 - 8*b * q^5 + 14*b * q^6 - 13*b * q^7 - 8*b * q^8 + 22 * q^9 $$q - 2 \beta q^{2} - 7 q^{3} - 4 q^{4} - 8 \beta q^{5} + 14 \beta q^{6} - 13 \beta q^{7} - 8 \beta q^{8} + 22 q^{9} - 48 q^{10} - 13 \beta q^{11} + 28 q^{12} - 78 q^{14} + 56 \beta q^{15} - 80 q^{16} + 27 q^{17} - 44 \beta q^{18} + 51 \beta q^{19} + 32 \beta q^{20} + 91 \beta q^{21} - 78 q^{22} + 57 q^{23} + 56 \beta q^{24} - 67 q^{25} + 35 q^{27} + 52 \beta q^{28} - 69 q^{29} + 336 q^{30} + 42 \beta q^{31} + 96 \beta q^{32} + 91 \beta q^{33} - 54 \beta q^{34} - 312 q^{35} - 88 q^{36} - 23 \beta q^{37} + 306 q^{38} - 192 q^{40} - 227 \beta q^{41} + 546 q^{42} - 85 q^{43} + 52 \beta q^{44} - 176 \beta q^{45} - 114 \beta q^{46} + 198 \beta q^{47} + 560 q^{48} - 164 q^{49} + 134 \beta q^{50} - 189 q^{51} + 426 q^{53} - 70 \beta q^{54} - 312 q^{55} - 312 q^{56} - 357 \beta q^{57} + 138 \beta q^{58} + 11 \beta q^{59} - 224 \beta q^{60} - 17 q^{61} + 252 q^{62} - 286 \beta q^{63} - 64 q^{64} + 546 q^{66} + 95 \beta q^{67} - 108 q^{68} - 399 q^{69} + 624 \beta q^{70} - 337 \beta q^{71} - 176 \beta q^{72} - 580 \beta q^{73} - 138 q^{74} + 469 q^{75} - 204 \beta q^{76} - 507 q^{77} - 1244 q^{79} + 640 \beta q^{80} - 839 q^{81} - 1362 q^{82} + 246 \beta q^{83} - 364 \beta q^{84} - 216 \beta q^{85} + 170 \beta q^{86} + 483 q^{87} - 312 q^{88} + 177 \beta q^{89} - 1056 q^{90} - 228 q^{92} - 294 \beta q^{93} + 1188 q^{94} + 1224 q^{95} - 672 \beta q^{96} - 713 \beta q^{97} + 328 \beta q^{98} - 286 \beta q^{99} +O(q^{100})$$ q - 2*b * q^2 - 7 * q^3 - 4 * q^4 - 8*b * q^5 + 14*b * q^6 - 13*b * q^7 - 8*b * q^8 + 22 * q^9 - 48 * q^10 - 13*b * q^11 + 28 * q^12 - 78 * q^14 + 56*b * q^15 - 80 * q^16 + 27 * q^17 - 44*b * q^18 + 51*b * q^19 + 32*b * q^20 + 91*b * q^21 - 78 * q^22 + 57 * q^23 + 56*b * q^24 - 67 * q^25 + 35 * q^27 + 52*b * q^28 - 69 * q^29 + 336 * q^30 + 42*b * q^31 + 96*b * q^32 + 91*b * q^33 - 54*b * q^34 - 312 * q^35 - 88 * q^36 - 23*b * q^37 + 306 * q^38 - 192 * q^40 - 227*b * q^41 + 546 * q^42 - 85 * q^43 + 52*b * q^44 - 176*b * q^45 - 114*b * q^46 + 198*b * q^47 + 560 * q^48 - 164 * q^49 + 134*b * q^50 - 189 * q^51 + 426 * q^53 - 70*b * q^54 - 312 * q^55 - 312 * q^56 - 357*b * q^57 + 138*b * q^58 + 11*b * q^59 - 224*b * q^60 - 17 * q^61 + 252 * q^62 - 286*b * q^63 - 64 * q^64 + 546 * q^66 + 95*b * q^67 - 108 * q^68 - 399 * q^69 + 624*b * q^70 - 337*b * q^71 - 176*b * q^72 - 580*b * q^73 - 138 * q^74 + 469 * q^75 - 204*b * q^76 - 507 * q^77 - 1244 * q^79 + 640*b * q^80 - 839 * q^81 - 1362 * q^82 + 246*b * q^83 - 364*b * q^84 - 216*b * q^85 + 170*b * q^86 + 483 * q^87 - 312 * q^88 + 177*b * q^89 - 1056 * q^90 - 228 * q^92 - 294*b * q^93 + 1188 * q^94 + 1224 * q^95 - 672*b * q^96 - 713*b * q^97 + 328*b * q^98 - 286*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 14 q^{3} - 8 q^{4} + 44 q^{9}+O(q^{10})$$ 2 * q - 14 * q^3 - 8 * q^4 + 44 * q^9 $$2 q - 14 q^{3} - 8 q^{4} + 44 q^{9} - 96 q^{10} + 56 q^{12} - 156 q^{14} - 160 q^{16} + 54 q^{17} - 156 q^{22} + 114 q^{23} - 134 q^{25} + 70 q^{27} - 138 q^{29} + 672 q^{30} - 624 q^{35} - 176 q^{36} + 612 q^{38} - 384 q^{40} + 1092 q^{42} - 170 q^{43} + 1120 q^{48} - 328 q^{49} - 378 q^{51} + 852 q^{53} - 624 q^{55} - 624 q^{56} - 34 q^{61} + 504 q^{62} - 128 q^{64} + 1092 q^{66} - 216 q^{68} - 798 q^{69} - 276 q^{74} + 938 q^{75} - 1014 q^{77} - 2488 q^{79} - 1678 q^{81} - 2724 q^{82} + 966 q^{87} - 624 q^{88} - 2112 q^{90} - 456 q^{92} + 2376 q^{94} + 2448 q^{95}+O(q^{100})$$ 2 * q - 14 * q^3 - 8 * q^4 + 44 * q^9 - 96 * q^10 + 56 * q^12 - 156 * q^14 - 160 * q^16 + 54 * q^17 - 156 * q^22 + 114 * q^23 - 134 * q^25 + 70 * q^27 - 138 * q^29 + 672 * q^30 - 624 * q^35 - 176 * q^36 + 612 * q^38 - 384 * q^40 + 1092 * q^42 - 170 * q^43 + 1120 * q^48 - 328 * q^49 - 378 * q^51 + 852 * q^53 - 624 * q^55 - 624 * q^56 - 34 * q^61 + 504 * q^62 - 128 * q^64 + 1092 * q^66 - 216 * q^68 - 798 * q^69 - 276 * q^74 + 938 * q^75 - 1014 * q^77 - 2488 * q^79 - 1678 * q^81 - 2724 * q^82 + 966 * q^87 - 624 * q^88 - 2112 * q^90 - 456 * q^92 + 2376 * q^94 + 2448 * 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
 0.5 + 0.866025i 0.5 − 0.866025i
3.46410i −7.00000 −4.00000 13.8564i 24.2487i 22.5167i 13.8564i 22.0000 −48.0000
168.2 3.46410i −7.00000 −4.00000 13.8564i 24.2487i 22.5167i 13.8564i 22.0000 −48.0000
 $$n$$: 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.b 2
13.b even 2 1 inner 169.4.b.b 2
13.c even 3 1 13.4.e.a 2
13.c even 3 1 169.4.e.b 2
13.d odd 4 2 169.4.a.h 2
13.e even 6 1 13.4.e.a 2
13.e even 6 1 169.4.e.b 2
13.f odd 12 4 169.4.c.i 4
39.f even 4 2 1521.4.a.q 2
39.h odd 6 1 117.4.q.c 2
39.i odd 6 1 117.4.q.c 2
52.i odd 6 1 208.4.w.a 2
52.j odd 6 1 208.4.w.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 12$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 12$$
$3$ $$(T + 7)^{2}$$
$5$ $$T^{2} + 192$$
$7$ $$T^{2} + 507$$
$11$ $$T^{2} + 507$$
$13$ $$T^{2}$$
$17$ $$(T - 27)^{2}$$
$19$ $$T^{2} + 7803$$
$23$ $$(T - 57)^{2}$$
$29$ $$(T + 69)^{2}$$
$31$ $$T^{2} + 5292$$
$37$ $$T^{2} + 1587$$
$41$ $$T^{2} + 154587$$
$43$ $$(T + 85)^{2}$$
$47$ $$T^{2} + 117612$$
$53$ $$(T - 426)^{2}$$
$59$ $$T^{2} + 363$$
$61$ $$(T + 17)^{2}$$
$67$ $$T^{2} + 27075$$
$71$ $$T^{2} + 340707$$
$73$ $$T^{2} + 1009200$$
$79$ $$(T + 1244)^{2}$$
$83$ $$T^{2} + 181548$$
$89$ $$T^{2} + 93987$$
$97$ $$T^{2} + 1525107$$