# Properties

 Label 169.4.c.e Level $169$ Weight $4$ Character orbit 169.c 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.c (of order $$3$$, degree $$2$$, 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 5 \zeta_{6} + 5) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} - 17 \zeta_{6} q^{4} - 7 q^{5} - 35 \zeta_{6} q^{6} + 13 \zeta_{6} q^{7} - 45 q^{8} - 22 \zeta_{6} q^{9} +O(q^{10})$$ q + (-5*z + 5) * q^2 + (-7*z + 7) * q^3 - 17*z * q^4 - 7 * q^5 - 35*z * q^6 + 13*z * q^7 - 45 * q^8 - 22*z * q^9 $$q + ( - 5 \zeta_{6} + 5) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} - 17 \zeta_{6} q^{4} - 7 q^{5} - 35 \zeta_{6} q^{6} + 13 \zeta_{6} q^{7} - 45 q^{8} - 22 \zeta_{6} q^{9} + (35 \zeta_{6} - 35) q^{10} + ( - 26 \zeta_{6} + 26) q^{11} - 119 q^{12} + 65 q^{14} + (49 \zeta_{6} - 49) q^{15} + (89 \zeta_{6} - 89) q^{16} - 77 \zeta_{6} q^{17} - 110 q^{18} + 126 \zeta_{6} q^{19} + 119 \zeta_{6} q^{20} + 91 q^{21} - 130 \zeta_{6} q^{22} + ( - 96 \zeta_{6} + 96) q^{23} + (315 \zeta_{6} - 315) q^{24} - 76 q^{25} + 35 q^{27} + ( - 221 \zeta_{6} + 221) q^{28} + ( - 82 \zeta_{6} + 82) q^{29} + 245 \zeta_{6} q^{30} + 196 q^{31} + 85 \zeta_{6} q^{32} - 182 \zeta_{6} q^{33} - 385 q^{34} - 91 \zeta_{6} q^{35} + (374 \zeta_{6} - 374) q^{36} + ( - 131 \zeta_{6} + 131) q^{37} + 630 q^{38} + 315 q^{40} + (336 \zeta_{6} - 336) q^{41} + ( - 455 \zeta_{6} + 455) q^{42} + 201 \zeta_{6} q^{43} - 442 q^{44} + 154 \zeta_{6} q^{45} - 480 \zeta_{6} q^{46} - 105 q^{47} + 623 \zeta_{6} q^{48} + ( - 174 \zeta_{6} + 174) q^{49} + (380 \zeta_{6} - 380) q^{50} - 539 q^{51} - 432 q^{53} + ( - 175 \zeta_{6} + 175) q^{54} + (182 \zeta_{6} - 182) q^{55} - 585 \zeta_{6} q^{56} + 882 q^{57} - 410 \zeta_{6} q^{58} + 294 \zeta_{6} q^{59} + 833 q^{60} + 56 \zeta_{6} q^{61} + ( - 980 \zeta_{6} + 980) q^{62} + ( - 286 \zeta_{6} + 286) q^{63} - 287 q^{64} - 910 q^{66} + (478 \zeta_{6} - 478) q^{67} + (1309 \zeta_{6} - 1309) q^{68} - 672 \zeta_{6} q^{69} - 455 q^{70} - 9 \zeta_{6} q^{71} + 990 \zeta_{6} q^{72} + 98 q^{73} - 655 \zeta_{6} q^{74} + (532 \zeta_{6} - 532) q^{75} + ( - 2142 \zeta_{6} + 2142) q^{76} + 338 q^{77} + 1304 q^{79} + ( - 623 \zeta_{6} + 623) q^{80} + ( - 839 \zeta_{6} + 839) q^{81} + 1680 \zeta_{6} q^{82} - 308 q^{83} - 1547 \zeta_{6} q^{84} + 539 \zeta_{6} q^{85} + 1005 q^{86} - 574 \zeta_{6} q^{87} + (1170 \zeta_{6} - 1170) q^{88} + ( - 1190 \zeta_{6} + 1190) q^{89} + 770 q^{90} - 1632 q^{92} + ( - 1372 \zeta_{6} + 1372) q^{93} + (525 \zeta_{6} - 525) q^{94} - 882 \zeta_{6} q^{95} + 595 q^{96} - 70 \zeta_{6} q^{97} - 870 \zeta_{6} q^{98} - 572 q^{99} +O(q^{100})$$ q + (-5*z + 5) * q^2 + (-7*z + 7) * q^3 - 17*z * q^4 - 7 * q^5 - 35*z * q^6 + 13*z * q^7 - 45 * q^8 - 22*z * q^9 + (35*z - 35) * q^10 + (-26*z + 26) * q^11 - 119 * q^12 + 65 * q^14 + (49*z - 49) * q^15 + (89*z - 89) * q^16 - 77*z * q^17 - 110 * q^18 + 126*z * q^19 + 119*z * q^20 + 91 * q^21 - 130*z * q^22 + (-96*z + 96) * q^23 + (315*z - 315) * q^24 - 76 * q^25 + 35 * q^27 + (-221*z + 221) * q^28 + (-82*z + 82) * q^29 + 245*z * q^30 + 196 * q^31 + 85*z * q^32 - 182*z * q^33 - 385 * q^34 - 91*z * q^35 + (374*z - 374) * q^36 + (-131*z + 131) * q^37 + 630 * q^38 + 315 * q^40 + (336*z - 336) * q^41 + (-455*z + 455) * q^42 + 201*z * q^43 - 442 * q^44 + 154*z * q^45 - 480*z * q^46 - 105 * q^47 + 623*z * q^48 + (-174*z + 174) * q^49 + (380*z - 380) * q^50 - 539 * q^51 - 432 * q^53 + (-175*z + 175) * q^54 + (182*z - 182) * q^55 - 585*z * q^56 + 882 * q^57 - 410*z * q^58 + 294*z * q^59 + 833 * q^60 + 56*z * q^61 + (-980*z + 980) * q^62 + (-286*z + 286) * q^63 - 287 * q^64 - 910 * q^66 + (478*z - 478) * q^67 + (1309*z - 1309) * q^68 - 672*z * q^69 - 455 * q^70 - 9*z * q^71 + 990*z * q^72 + 98 * q^73 - 655*z * q^74 + (532*z - 532) * q^75 + (-2142*z + 2142) * q^76 + 338 * q^77 + 1304 * q^79 + (-623*z + 623) * q^80 + (-839*z + 839) * q^81 + 1680*z * q^82 - 308 * q^83 - 1547*z * q^84 + 539*z * q^85 + 1005 * q^86 - 574*z * q^87 + (1170*z - 1170) * q^88 + (-1190*z + 1190) * q^89 + 770 * q^90 - 1632 * q^92 + (-1372*z + 1372) * q^93 + (525*z - 525) * q^94 - 882*z * q^95 + 595 * q^96 - 70*z * q^97 - 870*z * q^98 - 572 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{2} + 7 q^{3} - 17 q^{4} - 14 q^{5} - 35 q^{6} + 13 q^{7} - 90 q^{8} - 22 q^{9}+O(q^{10})$$ 2 * q + 5 * q^2 + 7 * q^3 - 17 * q^4 - 14 * q^5 - 35 * q^6 + 13 * q^7 - 90 * q^8 - 22 * q^9 $$2 q + 5 q^{2} + 7 q^{3} - 17 q^{4} - 14 q^{5} - 35 q^{6} + 13 q^{7} - 90 q^{8} - 22 q^{9} - 35 q^{10} + 26 q^{11} - 238 q^{12} + 130 q^{14} - 49 q^{15} - 89 q^{16} - 77 q^{17} - 220 q^{18} + 126 q^{19} + 119 q^{20} + 182 q^{21} - 130 q^{22} + 96 q^{23} - 315 q^{24} - 152 q^{25} + 70 q^{27} + 221 q^{28} + 82 q^{29} + 245 q^{30} + 392 q^{31} + 85 q^{32} - 182 q^{33} - 770 q^{34} - 91 q^{35} - 374 q^{36} + 131 q^{37} + 1260 q^{38} + 630 q^{40} - 336 q^{41} + 455 q^{42} + 201 q^{43} - 884 q^{44} + 154 q^{45} - 480 q^{46} - 210 q^{47} + 623 q^{48} + 174 q^{49} - 380 q^{50} - 1078 q^{51} - 864 q^{53} + 175 q^{54} - 182 q^{55} - 585 q^{56} + 1764 q^{57} - 410 q^{58} + 294 q^{59} + 1666 q^{60} + 56 q^{61} + 980 q^{62} + 286 q^{63} - 574 q^{64} - 1820 q^{66} - 478 q^{67} - 1309 q^{68} - 672 q^{69} - 910 q^{70} - 9 q^{71} + 990 q^{72} + 196 q^{73} - 655 q^{74} - 532 q^{75} + 2142 q^{76} + 676 q^{77} + 2608 q^{79} + 623 q^{80} + 839 q^{81} + 1680 q^{82} - 616 q^{83} - 1547 q^{84} + 539 q^{85} + 2010 q^{86} - 574 q^{87} - 1170 q^{88} + 1190 q^{89} + 1540 q^{90} - 3264 q^{92} + 1372 q^{93} - 525 q^{94} - 882 q^{95} + 1190 q^{96} - 70 q^{97} - 870 q^{98} - 1144 q^{99}+O(q^{100})$$ 2 * q + 5 * q^2 + 7 * q^3 - 17 * q^4 - 14 * q^5 - 35 * q^6 + 13 * q^7 - 90 * q^8 - 22 * q^9 - 35 * q^10 + 26 * q^11 - 238 * q^12 + 130 * q^14 - 49 * q^15 - 89 * q^16 - 77 * q^17 - 220 * q^18 + 126 * q^19 + 119 * q^20 + 182 * q^21 - 130 * q^22 + 96 * q^23 - 315 * q^24 - 152 * q^25 + 70 * q^27 + 221 * q^28 + 82 * q^29 + 245 * q^30 + 392 * q^31 + 85 * q^32 - 182 * q^33 - 770 * q^34 - 91 * q^35 - 374 * q^36 + 131 * q^37 + 1260 * q^38 + 630 * q^40 - 336 * q^41 + 455 * q^42 + 201 * q^43 - 884 * q^44 + 154 * q^45 - 480 * q^46 - 210 * q^47 + 623 * q^48 + 174 * q^49 - 380 * q^50 - 1078 * q^51 - 864 * q^53 + 175 * q^54 - 182 * q^55 - 585 * q^56 + 1764 * q^57 - 410 * q^58 + 294 * q^59 + 1666 * q^60 + 56 * q^61 + 980 * q^62 + 286 * q^63 - 574 * q^64 - 1820 * q^66 - 478 * q^67 - 1309 * q^68 - 672 * q^69 - 910 * q^70 - 9 * q^71 + 990 * q^72 + 196 * q^73 - 655 * q^74 - 532 * q^75 + 2142 * q^76 + 676 * q^77 + 2608 * q^79 + 623 * q^80 + 839 * q^81 + 1680 * q^82 - 616 * q^83 - 1547 * q^84 + 539 * q^85 + 2010 * q^86 - 574 * q^87 - 1170 * q^88 + 1190 * q^89 + 1540 * q^90 - 3264 * q^92 + 1372 * q^93 - 525 * q^94 - 882 * q^95 + 1190 * q^96 - 70 * q^97 - 870 * q^98 - 1144 * q^99

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 0.5 + 0.866025i 0.5 − 0.866025i
2.50000 4.33013i 3.50000 6.06218i −8.50000 14.7224i −7.00000 −17.5000 30.3109i 6.50000 + 11.2583i −45.0000 −11.0000 19.0526i −17.5000 + 30.3109i
146.1 2.50000 + 4.33013i 3.50000 + 6.06218i −8.50000 + 14.7224i −7.00000 −17.5000 + 30.3109i 6.50000 11.2583i −45.0000 −11.0000 + 19.0526i −17.5000 30.3109i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.c.e 2
13.b even 2 1 169.4.c.a 2
13.c even 3 1 13.4.a.a 1
13.c even 3 1 inner 169.4.c.e 2
13.d odd 4 2 169.4.e.e 4
13.e even 6 1 169.4.a.e 1
13.e even 6 1 169.4.c.a 2
13.f odd 12 2 169.4.b.a 2
13.f odd 12 2 169.4.e.e 4
39.h odd 6 1 1521.4.a.a 1
39.i odd 6 1 117.4.a.b 1
52.j odd 6 1 208.4.a.g 1
65.n even 6 1 325.4.a.d 1
65.q odd 12 2 325.4.b.b 2
91.n odd 6 1 637.4.a.a 1
104.n odd 6 1 832.4.a.a 1
104.r even 6 1 832.4.a.r 1
143.k odd 6 1 1573.4.a.a 1
156.p even 6 1 1872.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 13.c even 3 1
117.4.a.b 1 39.i odd 6 1
169.4.a.e 1 13.e even 6 1
169.4.b.a 2 13.f odd 12 2
169.4.c.a 2 13.b even 2 1
169.4.c.a 2 13.e even 6 1
169.4.c.e 2 1.a even 1 1 trivial
169.4.c.e 2 13.c even 3 1 inner
169.4.e.e 4 13.d odd 4 2
169.4.e.e 4 13.f odd 12 2
208.4.a.g 1 52.j odd 6 1
325.4.a.d 1 65.n even 6 1
325.4.b.b 2 65.q odd 12 2
637.4.a.a 1 91.n odd 6 1
832.4.a.a 1 104.n odd 6 1
832.4.a.r 1 104.r even 6 1
1521.4.a.a 1 39.h odd 6 1
1573.4.a.a 1 143.k odd 6 1
1872.4.a.k 1 156.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 5T_{2} + 25$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5T + 25$$
$3$ $$T^{2} - 7T + 49$$
$5$ $$(T + 7)^{2}$$
$7$ $$T^{2} - 13T + 169$$
$11$ $$T^{2} - 26T + 676$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 77T + 5929$$
$19$ $$T^{2} - 126T + 15876$$
$23$ $$T^{2} - 96T + 9216$$
$29$ $$T^{2} - 82T + 6724$$
$31$ $$(T - 196)^{2}$$
$37$ $$T^{2} - 131T + 17161$$
$41$ $$T^{2} + 336T + 112896$$
$43$ $$T^{2} - 201T + 40401$$
$47$ $$(T + 105)^{2}$$
$53$ $$(T + 432)^{2}$$
$59$ $$T^{2} - 294T + 86436$$
$61$ $$T^{2} - 56T + 3136$$
$67$ $$T^{2} + 478T + 228484$$
$71$ $$T^{2} + 9T + 81$$
$73$ $$(T - 98)^{2}$$
$79$ $$(T - 1304)^{2}$$
$83$ $$(T + 308)^{2}$$
$89$ $$T^{2} - 1190 T + 1416100$$
$97$ $$T^{2} + 70T + 4900$$