# Properties

 Label 169.4.e.c Level $169$ Weight $4$ Character orbit 169.e Analytic conductor $9.971$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.e (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{12} q^{2} + (2 \zeta_{12}^{2} - 2) q^{3} + 8 \zeta_{12}^{2} q^{4} - 17 \zeta_{12}^{3} q^{5} + (8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{6} + ( - 20 \zeta_{12}^{3} + 20 \zeta_{12}) q^{7} + 23 \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + 4*z * q^2 + (2*z^2 - 2) * q^3 + 8*z^2 * q^4 - 17*z^3 * q^5 + (8*z^3 - 8*z) * q^6 + (-20*z^3 + 20*z) * q^7 + 23*z^2 * q^9 $$q + 4 \zeta_{12} q^{2} + (2 \zeta_{12}^{2} - 2) q^{3} + 8 \zeta_{12}^{2} q^{4} - 17 \zeta_{12}^{3} q^{5} + (8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{6} + ( - 20 \zeta_{12}^{3} + 20 \zeta_{12}) q^{7} + 23 \zeta_{12}^{2} q^{9} + ( - 68 \zeta_{12}^{2} + 68) q^{10} + 32 \zeta_{12} q^{11} - 16 q^{12} + 80 q^{14} + 34 \zeta_{12} q^{15} + ( - 64 \zeta_{12}^{2} + 64) q^{16} - 13 \zeta_{12}^{2} q^{17} + 92 \zeta_{12}^{3} q^{18} + (30 \zeta_{12}^{3} - 30 \zeta_{12}) q^{19} + ( - 136 \zeta_{12}^{3} + 136 \zeta_{12}) q^{20} + 40 \zeta_{12}^{3} q^{21} + 128 \zeta_{12}^{2} q^{22} + ( - 78 \zeta_{12}^{2} + 78) q^{23} - 164 q^{25} - 100 q^{27} + 160 \zeta_{12} q^{28} + (197 \zeta_{12}^{2} - 197) q^{29} + 136 \zeta_{12}^{2} q^{30} + 74 \zeta_{12}^{3} q^{31} + ( - 256 \zeta_{12}^{3} + 256 \zeta_{12}) q^{32} + (64 \zeta_{12}^{3} - 64 \zeta_{12}) q^{33} - 52 \zeta_{12}^{3} q^{34} - 340 \zeta_{12}^{2} q^{35} + (184 \zeta_{12}^{2} - 184) q^{36} + 227 \zeta_{12} q^{37} - 120 q^{38} - 165 \zeta_{12} q^{41} + (160 \zeta_{12}^{2} - 160) q^{42} - 156 \zeta_{12}^{2} q^{43} + 256 \zeta_{12}^{3} q^{44} + ( - 391 \zeta_{12}^{3} + 391 \zeta_{12}) q^{45} + ( - 312 \zeta_{12}^{3} + 312 \zeta_{12}) q^{46} - 162 \zeta_{12}^{3} q^{47} + 128 \zeta_{12}^{2} q^{48} + ( - 57 \zeta_{12}^{2} + 57) q^{49} - 656 \zeta_{12} q^{50} + 26 q^{51} + 93 q^{53} - 400 \zeta_{12} q^{54} + ( - 544 \zeta_{12}^{2} + 544) q^{55} - 60 \zeta_{12}^{3} q^{57} + (788 \zeta_{12}^{3} - 788 \zeta_{12}) q^{58} + (864 \zeta_{12}^{3} - 864 \zeta_{12}) q^{59} + 272 \zeta_{12}^{3} q^{60} - 145 \zeta_{12}^{2} q^{61} + (296 \zeta_{12}^{2} - 296) q^{62} + 460 \zeta_{12} q^{63} + 512 q^{64} - 256 q^{66} + 862 \zeta_{12} q^{67} + ( - 104 \zeta_{12}^{2} + 104) q^{68} + 156 \zeta_{12}^{2} q^{69} - 1360 \zeta_{12}^{3} q^{70} + (654 \zeta_{12}^{3} - 654 \zeta_{12}) q^{71} + 215 \zeta_{12}^{3} q^{73} + 908 \zeta_{12}^{2} q^{74} + ( - 328 \zeta_{12}^{2} + 328) q^{75} - 240 \zeta_{12} q^{76} + 640 q^{77} - 76 q^{79} - 1088 \zeta_{12} q^{80} + (421 \zeta_{12}^{2} - 421) q^{81} - 660 \zeta_{12}^{2} q^{82} - 628 \zeta_{12}^{3} q^{83} + (320 \zeta_{12}^{3} - 320 \zeta_{12}) q^{84} + (221 \zeta_{12}^{3} - 221 \zeta_{12}) q^{85} - 624 \zeta_{12}^{3} q^{86} - 394 \zeta_{12}^{2} q^{87} + 266 \zeta_{12} q^{89} + 1564 q^{90} + 624 q^{92} - 148 \zeta_{12} q^{93} + ( - 648 \zeta_{12}^{2} + 648) q^{94} + 510 \zeta_{12}^{2} q^{95} + 512 \zeta_{12}^{3} q^{96} + (238 \zeta_{12}^{3} - 238 \zeta_{12}) q^{97} + ( - 228 \zeta_{12}^{3} + 228 \zeta_{12}) q^{98} + 736 \zeta_{12}^{3} q^{99} +O(q^{100})$$ q + 4*z * q^2 + (2*z^2 - 2) * q^3 + 8*z^2 * q^4 - 17*z^3 * q^5 + (8*z^3 - 8*z) * q^6 + (-20*z^3 + 20*z) * q^7 + 23*z^2 * q^9 + (-68*z^2 + 68) * q^10 + 32*z * q^11 - 16 * q^12 + 80 * q^14 + 34*z * q^15 + (-64*z^2 + 64) * q^16 - 13*z^2 * q^17 + 92*z^3 * q^18 + (30*z^3 - 30*z) * q^19 + (-136*z^3 + 136*z) * q^20 + 40*z^3 * q^21 + 128*z^2 * q^22 + (-78*z^2 + 78) * q^23 - 164 * q^25 - 100 * q^27 + 160*z * q^28 + (197*z^2 - 197) * q^29 + 136*z^2 * q^30 + 74*z^3 * q^31 + (-256*z^3 + 256*z) * q^32 + (64*z^3 - 64*z) * q^33 - 52*z^3 * q^34 - 340*z^2 * q^35 + (184*z^2 - 184) * q^36 + 227*z * q^37 - 120 * q^38 - 165*z * q^41 + (160*z^2 - 160) * q^42 - 156*z^2 * q^43 + 256*z^3 * q^44 + (-391*z^3 + 391*z) * q^45 + (-312*z^3 + 312*z) * q^46 - 162*z^3 * q^47 + 128*z^2 * q^48 + (-57*z^2 + 57) * q^49 - 656*z * q^50 + 26 * q^51 + 93 * q^53 - 400*z * q^54 + (-544*z^2 + 544) * q^55 - 60*z^3 * q^57 + (788*z^3 - 788*z) * q^58 + (864*z^3 - 864*z) * q^59 + 272*z^3 * q^60 - 145*z^2 * q^61 + (296*z^2 - 296) * q^62 + 460*z * q^63 + 512 * q^64 - 256 * q^66 + 862*z * q^67 + (-104*z^2 + 104) * q^68 + 156*z^2 * q^69 - 1360*z^3 * q^70 + (654*z^3 - 654*z) * q^71 + 215*z^3 * q^73 + 908*z^2 * q^74 + (-328*z^2 + 328) * q^75 - 240*z * q^76 + 640 * q^77 - 76 * q^79 - 1088*z * q^80 + (421*z^2 - 421) * q^81 - 660*z^2 * q^82 - 628*z^3 * q^83 + (320*z^3 - 320*z) * q^84 + (221*z^3 - 221*z) * q^85 - 624*z^3 * q^86 - 394*z^2 * q^87 + 266*z * q^89 + 1564 * q^90 + 624 * q^92 - 148*z * q^93 + (-648*z^2 + 648) * q^94 + 510*z^2 * q^95 + 512*z^3 * q^96 + (238*z^3 - 238*z) * q^97 + (-228*z^3 + 228*z) * q^98 + 736*z^3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 16 q^{4} + 46 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 + 16 * q^4 + 46 * q^9 $$4 q - 4 q^{3} + 16 q^{4} + 46 q^{9} + 136 q^{10} - 64 q^{12} + 320 q^{14} + 128 q^{16} - 26 q^{17} + 256 q^{22} + 156 q^{23} - 656 q^{25} - 400 q^{27} - 394 q^{29} + 272 q^{30} - 680 q^{35} - 368 q^{36} - 480 q^{38} - 320 q^{42} - 312 q^{43} + 256 q^{48} + 114 q^{49} + 104 q^{51} + 372 q^{53} + 1088 q^{55} - 290 q^{61} - 592 q^{62} + 2048 q^{64} - 1024 q^{66} + 208 q^{68} + 312 q^{69} + 1816 q^{74} + 656 q^{75} + 2560 q^{77} - 304 q^{79} - 842 q^{81} - 1320 q^{82} - 788 q^{87} + 6256 q^{90} + 2496 q^{92} + 1296 q^{94} + 1020 q^{95}+O(q^{100})$$ 4 * q - 4 * q^3 + 16 * q^4 + 46 * q^9 + 136 * q^10 - 64 * q^12 + 320 * q^14 + 128 * q^16 - 26 * q^17 + 256 * q^22 + 156 * q^23 - 656 * q^25 - 400 * q^27 - 394 * q^29 + 272 * q^30 - 680 * q^35 - 368 * q^36 - 480 * q^38 - 320 * q^42 - 312 * q^43 + 256 * q^48 + 114 * q^49 + 104 * q^51 + 372 * q^53 + 1088 * q^55 - 290 * q^61 - 592 * q^62 + 2048 * q^64 - 1024 * q^66 + 208 * q^68 + 312 * q^69 + 1816 * q^74 + 656 * q^75 + 2560 * q^77 - 304 * q^79 - 842 * q^81 - 1320 * q^82 - 788 * q^87 + 6256 * q^90 + 2496 * q^92 + 1296 * q^94 + 1020 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−3.46410 + 2.00000i −1.00000 1.73205i 4.00000 6.92820i 17.0000i 6.92820 + 4.00000i −17.3205 10.0000i 0 11.5000 19.9186i 34.0000 + 58.8897i
23.2 3.46410 2.00000i −1.00000 1.73205i 4.00000 6.92820i 17.0000i −6.92820 4.00000i 17.3205 + 10.0000i 0 11.5000 19.9186i 34.0000 + 58.8897i
147.1 −3.46410 2.00000i −1.00000 + 1.73205i 4.00000 + 6.92820i 17.0000i 6.92820 4.00000i −17.3205 + 10.0000i 0 11.5000 + 19.9186i 34.0000 58.8897i
147.2 3.46410 + 2.00000i −1.00000 + 1.73205i 4.00000 + 6.92820i 17.0000i −6.92820 + 4.00000i 17.3205 10.0000i 0 11.5000 + 19.9186i 34.0000 58.8897i
 $$n$$: 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
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 16T_{2}^{2} + 256$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 16T^{2} + 256$$
$3$ $$(T^{2} + 2 T + 4)^{2}$$
$5$ $$(T^{2} + 289)^{2}$$
$7$ $$T^{4} - 400 T^{2} + 160000$$
$11$ $$T^{4} - 1024 T^{2} + \cdots + 1048576$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 13 T + 169)^{2}$$
$19$ $$T^{4} - 900 T^{2} + 810000$$
$23$ $$(T^{2} - 78 T + 6084)^{2}$$
$29$ $$(T^{2} + 197 T + 38809)^{2}$$
$31$ $$(T^{2} + 5476)^{2}$$
$37$ $$T^{4} - 51529 T^{2} + \cdots + 2655237841$$
$41$ $$T^{4} - 27225 T^{2} + \cdots + 741200625$$
$43$ $$(T^{2} + 156 T + 24336)^{2}$$
$47$ $$(T^{2} + 26244)^{2}$$
$53$ $$(T - 93)^{4}$$
$59$ $$T^{4} - 746496 T^{2} + \cdots + 557256278016$$
$61$ $$(T^{2} + 145 T + 21025)^{2}$$
$67$ $$T^{4} - 743044 T^{2} + \cdots + 552114385936$$
$71$ $$T^{4} - 427716 T^{2} + \cdots + 182940976656$$
$73$ $$(T^{2} + 46225)^{2}$$
$79$ $$(T + 76)^{4}$$
$83$ $$(T^{2} + 394384)^{2}$$
$89$ $$T^{4} - 70756 T^{2} + \cdots + 5006411536$$
$97$ $$T^{4} - 56644 T^{2} + \cdots + 3208542736$$
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