# Properties

 Label 169.4.a.f Level $169$ Weight $4$ Character orbit 169.a Self dual yes Analytic conductor $9.971$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 2) q^{2} + (3 \beta + 1) q^{3} + 5 \beta q^{4} + (5 \beta - 10) q^{5} + ( - 10 \beta - 14) q^{6} + (\beta + 7) q^{7} + ( - 7 \beta - 4) q^{8} + (15 \beta + 10) q^{9}+O(q^{10})$$ q + (-b - 2) * q^2 + (3*b + 1) * q^3 + 5*b * q^4 + (5*b - 10) * q^5 + (-10*b - 14) * q^6 + (b + 7) * q^7 + (-7*b - 4) * q^8 + (15*b + 10) * q^9 $$q + ( - \beta - 2) q^{2} + (3 \beta + 1) q^{3} + 5 \beta q^{4} + (5 \beta - 10) q^{5} + ( - 10 \beta - 14) q^{6} + (\beta + 7) q^{7} + ( - 7 \beta - 4) q^{8} + (15 \beta + 10) q^{9} - 5 \beta q^{10} + (15 \beta + 1) q^{11} + (20 \beta + 60) q^{12} + ( - 10 \beta - 18) q^{14} + ( - 10 \beta + 50) q^{15} + ( - 15 \beta + 36) q^{16} + ( - 16 \beta + 43) q^{17} + ( - 55 \beta - 80) q^{18} + (5 \beta - 73) q^{19} + ( - 25 \beta + 100) q^{20} + (25 \beta + 19) q^{21} + ( - 46 \beta - 62) q^{22} + ( - 33 \beta + 89) q^{23} + ( - 40 \beta - 88) q^{24} + ( - 75 \beta + 75) q^{25} + (9 \beta + 163) q^{27} + (40 \beta + 20) q^{28} + (60 \beta - 13) q^{29} + ( - 20 \beta - 60) q^{30} + (100 \beta - 120) q^{31} + (65 \beta + 20) q^{32} + (63 \beta + 181) q^{33} + (5 \beta - 22) q^{34} + (30 \beta - 50) q^{35} + (125 \beta + 300) q^{36} + ( - 44 \beta - 73) q^{37} + (58 \beta + 126) q^{38} + (15 \beta - 100) q^{40} + (20 \beta + 259) q^{41} + ( - 94 \beta - 138) q^{42} + (97 \beta + 179) q^{43} + (80 \beta + 300) q^{44} + ( - 25 \beta + 200) q^{45} + (10 \beta - 46) q^{46} + ( - 140 \beta + 100) q^{47} + (48 \beta - 144) q^{48} + (15 \beta - 290) q^{49} + (150 \beta + 150) q^{50} + (65 \beta - 149) q^{51} + (165 \beta + 190) q^{53} + ( - 190 \beta - 362) q^{54} + ( - 70 \beta + 290) q^{55} + ( - 60 \beta - 56) q^{56} + ( - 199 \beta - 13) q^{57} + ( - 167 \beta - 214) q^{58} + ( - 55 \beta - 377) q^{59} + (200 \beta - 200) q^{60} + ( - 200 \beta + 351) q^{61} + ( - 180 \beta - 160) q^{62} + (130 \beta + 130) q^{63} + ( - 95 \beta - 588) q^{64} + ( - 370 \beta - 614) q^{66} + (91 \beta - 283) q^{67} + (135 \beta - 320) q^{68} + (135 \beta - 307) q^{69} + ( - 40 \beta - 20) q^{70} + (105 \beta + 11) q^{71} + ( - 235 \beta - 460) q^{72} + (85 \beta + 250) q^{73} + (205 \beta + 322) q^{74} + ( - 75 \beta - 825) q^{75} + ( - 340 \beta + 100) q^{76} + (121 \beta + 67) q^{77} + ( - 40 \beta + 140) q^{79} + (255 \beta - 660) q^{80} + (120 \beta + 1) q^{81} + ( - 319 \beta - 598) q^{82} + ( - 100 \beta + 180) q^{83} + (220 \beta + 500) q^{84} + (295 \beta - 750) q^{85} + ( - 470 \beta - 746) q^{86} + (201 \beta + 707) q^{87} + ( - 172 \beta - 424) q^{88} + ( - 125 \beta + 523) q^{89} + ( - 125 \beta - 300) q^{90} + (280 \beta - 660) q^{92} + (40 \beta + 1080) q^{93} + (320 \beta + 360) q^{94} + ( - 390 \beta + 830) q^{95} + (320 \beta + 800) q^{96} + ( - 469 \beta + 27) q^{97} + (245 \beta + 520) q^{98} + (390 \beta + 910) q^{99} +O(q^{100})$$ q + (-b - 2) * q^2 + (3*b + 1) * q^3 + 5*b * q^4 + (5*b - 10) * q^5 + (-10*b - 14) * q^6 + (b + 7) * q^7 + (-7*b - 4) * q^8 + (15*b + 10) * q^9 - 5*b * q^10 + (15*b + 1) * q^11 + (20*b + 60) * q^12 + (-10*b - 18) * q^14 + (-10*b + 50) * q^15 + (-15*b + 36) * q^16 + (-16*b + 43) * q^17 + (-55*b - 80) * q^18 + (5*b - 73) * q^19 + (-25*b + 100) * q^20 + (25*b + 19) * q^21 + (-46*b - 62) * q^22 + (-33*b + 89) * q^23 + (-40*b - 88) * q^24 + (-75*b + 75) * q^25 + (9*b + 163) * q^27 + (40*b + 20) * q^28 + (60*b - 13) * q^29 + (-20*b - 60) * q^30 + (100*b - 120) * q^31 + (65*b + 20) * q^32 + (63*b + 181) * q^33 + (5*b - 22) * q^34 + (30*b - 50) * q^35 + (125*b + 300) * q^36 + (-44*b - 73) * q^37 + (58*b + 126) * q^38 + (15*b - 100) * q^40 + (20*b + 259) * q^41 + (-94*b - 138) * q^42 + (97*b + 179) * q^43 + (80*b + 300) * q^44 + (-25*b + 200) * q^45 + (10*b - 46) * q^46 + (-140*b + 100) * q^47 + (48*b - 144) * q^48 + (15*b - 290) * q^49 + (150*b + 150) * q^50 + (65*b - 149) * q^51 + (165*b + 190) * q^53 + (-190*b - 362) * q^54 + (-70*b + 290) * q^55 + (-60*b - 56) * q^56 + (-199*b - 13) * q^57 + (-167*b - 214) * q^58 + (-55*b - 377) * q^59 + (200*b - 200) * q^60 + (-200*b + 351) * q^61 + (-180*b - 160) * q^62 + (130*b + 130) * q^63 + (-95*b - 588) * q^64 + (-370*b - 614) * q^66 + (91*b - 283) * q^67 + (135*b - 320) * q^68 + (135*b - 307) * q^69 + (-40*b - 20) * q^70 + (105*b + 11) * q^71 + (-235*b - 460) * q^72 + (85*b + 250) * q^73 + (205*b + 322) * q^74 + (-75*b - 825) * q^75 + (-340*b + 100) * q^76 + (121*b + 67) * q^77 + (-40*b + 140) * q^79 + (255*b - 660) * q^80 + (120*b + 1) * q^81 + (-319*b - 598) * q^82 + (-100*b + 180) * q^83 + (220*b + 500) * q^84 + (295*b - 750) * q^85 + (-470*b - 746) * q^86 + (201*b + 707) * q^87 + (-172*b - 424) * q^88 + (-125*b + 523) * q^89 + (-125*b - 300) * q^90 + (280*b - 660) * q^92 + (40*b + 1080) * q^93 + (320*b + 360) * q^94 + (-390*b + 830) * q^95 + (320*b + 800) * q^96 + (-469*b + 27) * q^97 + (245*b + 520) * q^98 + (390*b + 910) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 5 q^{2} + 5 q^{3} + 5 q^{4} - 15 q^{5} - 38 q^{6} + 15 q^{7} - 15 q^{8} + 35 q^{9}+O(q^{10})$$ 2 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 - 15 * q^5 - 38 * q^6 + 15 * q^7 - 15 * q^8 + 35 * q^9 $$2 q - 5 q^{2} + 5 q^{3} + 5 q^{4} - 15 q^{5} - 38 q^{6} + 15 q^{7} - 15 q^{8} + 35 q^{9} - 5 q^{10} + 17 q^{11} + 140 q^{12} - 46 q^{14} + 90 q^{15} + 57 q^{16} + 70 q^{17} - 215 q^{18} - 141 q^{19} + 175 q^{20} + 63 q^{21} - 170 q^{22} + 145 q^{23} - 216 q^{24} + 75 q^{25} + 335 q^{27} + 80 q^{28} + 34 q^{29} - 140 q^{30} - 140 q^{31} + 105 q^{32} + 425 q^{33} - 39 q^{34} - 70 q^{35} + 725 q^{36} - 190 q^{37} + 310 q^{38} - 185 q^{40} + 538 q^{41} - 370 q^{42} + 455 q^{43} + 680 q^{44} + 375 q^{45} - 82 q^{46} + 60 q^{47} - 240 q^{48} - 565 q^{49} + 450 q^{50} - 233 q^{51} + 545 q^{53} - 914 q^{54} + 510 q^{55} - 172 q^{56} - 225 q^{57} - 595 q^{58} - 809 q^{59} - 200 q^{60} + 502 q^{61} - 500 q^{62} + 390 q^{63} - 1271 q^{64} - 1598 q^{66} - 475 q^{67} - 505 q^{68} - 479 q^{69} - 80 q^{70} + 127 q^{71} - 1155 q^{72} + 585 q^{73} + 849 q^{74} - 1725 q^{75} - 140 q^{76} + 255 q^{77} + 240 q^{79} - 1065 q^{80} + 122 q^{81} - 1515 q^{82} + 260 q^{83} + 1220 q^{84} - 1205 q^{85} - 1962 q^{86} + 1615 q^{87} - 1020 q^{88} + 921 q^{89} - 725 q^{90} - 1040 q^{92} + 2200 q^{93} + 1040 q^{94} + 1270 q^{95} + 1920 q^{96} - 415 q^{97} + 1285 q^{98} + 2210 q^{99}+O(q^{100})$$ 2 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 - 15 * q^5 - 38 * q^6 + 15 * q^7 - 15 * q^8 + 35 * q^9 - 5 * q^10 + 17 * q^11 + 140 * q^12 - 46 * q^14 + 90 * q^15 + 57 * q^16 + 70 * q^17 - 215 * q^18 - 141 * q^19 + 175 * q^20 + 63 * q^21 - 170 * q^22 + 145 * q^23 - 216 * q^24 + 75 * q^25 + 335 * q^27 + 80 * q^28 + 34 * q^29 - 140 * q^30 - 140 * q^31 + 105 * q^32 + 425 * q^33 - 39 * q^34 - 70 * q^35 + 725 * q^36 - 190 * q^37 + 310 * q^38 - 185 * q^40 + 538 * q^41 - 370 * q^42 + 455 * q^43 + 680 * q^44 + 375 * q^45 - 82 * q^46 + 60 * q^47 - 240 * q^48 - 565 * q^49 + 450 * q^50 - 233 * q^51 + 545 * q^53 - 914 * q^54 + 510 * q^55 - 172 * q^56 - 225 * q^57 - 595 * q^58 - 809 * q^59 - 200 * q^60 + 502 * q^61 - 500 * q^62 + 390 * q^63 - 1271 * q^64 - 1598 * q^66 - 475 * q^67 - 505 * q^68 - 479 * q^69 - 80 * q^70 + 127 * q^71 - 1155 * q^72 + 585 * q^73 + 849 * q^74 - 1725 * q^75 - 140 * q^76 + 255 * q^77 + 240 * q^79 - 1065 * q^80 + 122 * q^81 - 1515 * q^82 + 260 * q^83 + 1220 * q^84 - 1205 * q^85 - 1962 * q^86 + 1615 * q^87 - 1020 * q^88 + 921 * q^89 - 725 * q^90 - 1040 * q^92 + 2200 * q^93 + 1040 * q^94 + 1270 * q^95 + 1920 * q^96 - 415 * q^97 + 1285 * q^98 + 2210 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.56155 −1.56155
−4.56155 8.68466 12.8078 2.80776 −39.6155 9.56155 −21.9309 48.4233 −12.8078
1.2 −0.438447 −3.68466 −7.80776 −17.8078 1.61553 5.43845 6.93087 −13.4233 7.80776
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.a.f 2
3.b odd 2 1 1521.4.a.t 2
13.b even 2 1 169.4.a.j 2
13.c even 3 2 13.4.c.b 4
13.d odd 4 2 169.4.b.e 4
13.e even 6 2 169.4.c.f 4
13.f odd 12 4 169.4.e.g 8
39.d odd 2 1 1521.4.a.l 2
39.i odd 6 2 117.4.g.d 4
52.j odd 6 2 208.4.i.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 13.c even 3 2
117.4.g.d 4 39.i odd 6 2
169.4.a.f 2 1.a even 1 1 trivial
169.4.a.j 2 13.b even 2 1
169.4.b.e 4 13.d odd 4 2
169.4.c.f 4 13.e even 6 2
169.4.e.g 8 13.f odd 12 4
208.4.i.e 4 52.j odd 6 2
1521.4.a.l 2 39.d odd 2 1
1521.4.a.t 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 5T_{2} + 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(169))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 5T + 2$$
$3$ $$T^{2} - 5T - 32$$
$5$ $$T^{2} + 15T - 50$$
$7$ $$T^{2} - 15T + 52$$
$11$ $$T^{2} - 17T - 884$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 70T + 137$$
$19$ $$T^{2} + 141T + 4864$$
$23$ $$T^{2} - 145T + 628$$
$29$ $$T^{2} - 34T - 15011$$
$31$ $$T^{2} + 140T - 37600$$
$37$ $$T^{2} + 190T + 797$$
$41$ $$T^{2} - 538T + 70661$$
$43$ $$T^{2} - 455T + 11768$$
$47$ $$T^{2} - 60T - 82400$$
$53$ $$T^{2} - 545T - 41450$$
$59$ $$T^{2} + 809T + 150764$$
$61$ $$T^{2} - 502T - 106999$$
$67$ $$T^{2} + 475T + 21212$$
$71$ $$T^{2} - 127T - 42824$$
$73$ $$T^{2} - 585T + 54850$$
$79$ $$T^{2} - 240T + 7600$$
$83$ $$T^{2} - 260T - 25600$$
$89$ $$T^{2} - 921T + 145654$$
$97$ $$T^{2} + 415T - 891778$$