# Properties

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

# Related objects

## 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 5 \zeta_{12} q^{2} + ( - 7 \zeta_{12}^{2} + 7) q^{3} + 17 \zeta_{12}^{2} q^{4} - 7 \zeta_{12}^{3} q^{5} + ( - 35 \zeta_{12}^{3} + 35 \zeta_{12}) q^{6} + ( - 13 \zeta_{12}^{3} + 13 \zeta_{12}) q^{7} + 45 \zeta_{12}^{3} q^{8} - 22 \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + 5*z * q^2 + (-7*z^2 + 7) * q^3 + 17*z^2 * q^4 - 7*z^3 * q^5 + (-35*z^3 + 35*z) * q^6 + (-13*z^3 + 13*z) * q^7 + 45*z^3 * q^8 - 22*z^2 * q^9 $$q + 5 \zeta_{12} q^{2} + ( - 7 \zeta_{12}^{2} + 7) q^{3} + 17 \zeta_{12}^{2} q^{4} - 7 \zeta_{12}^{3} q^{5} + ( - 35 \zeta_{12}^{3} + 35 \zeta_{12}) q^{6} + ( - 13 \zeta_{12}^{3} + 13 \zeta_{12}) q^{7} + 45 \zeta_{12}^{3} q^{8} - 22 \zeta_{12}^{2} q^{9} + ( - 35 \zeta_{12}^{2} + 35) q^{10} - 26 \zeta_{12} q^{11} + 119 q^{12} + 65 q^{14} - 49 \zeta_{12} q^{15} + (89 \zeta_{12}^{2} - 89) q^{16} + 77 \zeta_{12}^{2} q^{17} - 110 \zeta_{12}^{3} q^{18} + (126 \zeta_{12}^{3} - 126 \zeta_{12}) q^{19} + ( - 119 \zeta_{12}^{3} + 119 \zeta_{12}) q^{20} - 91 \zeta_{12}^{3} q^{21} - 130 \zeta_{12}^{2} q^{22} + (96 \zeta_{12}^{2} - 96) q^{23} + 315 \zeta_{12} q^{24} + 76 q^{25} + 35 q^{27} + 221 \zeta_{12} q^{28} + ( - 82 \zeta_{12}^{2} + 82) q^{29} - 245 \zeta_{12}^{2} q^{30} + 196 \zeta_{12}^{3} q^{31} + (85 \zeta_{12}^{3} - 85 \zeta_{12}) q^{32} + (182 \zeta_{12}^{3} - 182 \zeta_{12}) q^{33} + 385 \zeta_{12}^{3} q^{34} - 91 \zeta_{12}^{2} q^{35} + ( - 374 \zeta_{12}^{2} + 374) q^{36} - 131 \zeta_{12} q^{37} - 630 q^{38} + 315 q^{40} - 336 \zeta_{12} q^{41} + ( - 455 \zeta_{12}^{2} + 455) q^{42} - 201 \zeta_{12}^{2} q^{43} - 442 \zeta_{12}^{3} q^{44} + (154 \zeta_{12}^{3} - 154 \zeta_{12}) q^{45} + (480 \zeta_{12}^{3} - 480 \zeta_{12}) q^{46} + 105 \zeta_{12}^{3} q^{47} + 623 \zeta_{12}^{2} q^{48} + (174 \zeta_{12}^{2} - 174) q^{49} + 380 \zeta_{12} q^{50} + 539 q^{51} - 432 q^{53} + 175 \zeta_{12} q^{54} + (182 \zeta_{12}^{2} - 182) q^{55} + 585 \zeta_{12}^{2} q^{56} + 882 \zeta_{12}^{3} q^{57} + ( - 410 \zeta_{12}^{3} + 410 \zeta_{12}) q^{58} + ( - 294 \zeta_{12}^{3} + 294 \zeta_{12}) q^{59} - 833 \zeta_{12}^{3} q^{60} + 56 \zeta_{12}^{2} q^{61} + (980 \zeta_{12}^{2} - 980) q^{62} - 286 \zeta_{12} q^{63} + 287 q^{64} - 910 q^{66} - 478 \zeta_{12} q^{67} + (1309 \zeta_{12}^{2} - 1309) q^{68} + 672 \zeta_{12}^{2} q^{69} - 455 \zeta_{12}^{3} q^{70} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{71} + ( - 990 \zeta_{12}^{3} + 990 \zeta_{12}) q^{72} - 98 \zeta_{12}^{3} q^{73} - 655 \zeta_{12}^{2} q^{74} + ( - 532 \zeta_{12}^{2} + 532) q^{75} - 2142 \zeta_{12} q^{76} - 338 q^{77} + 1304 q^{79} + 623 \zeta_{12} q^{80} + ( - 839 \zeta_{12}^{2} + 839) q^{81} - 1680 \zeta_{12}^{2} q^{82} - 308 \zeta_{12}^{3} q^{83} + ( - 1547 \zeta_{12}^{3} + 1547 \zeta_{12}) q^{84} + ( - 539 \zeta_{12}^{3} + 539 \zeta_{12}) q^{85} - 1005 \zeta_{12}^{3} q^{86} - 574 \zeta_{12}^{2} q^{87} + ( - 1170 \zeta_{12}^{2} + 1170) q^{88} - 1190 \zeta_{12} q^{89} - 770 q^{90} - 1632 q^{92} + 1372 \zeta_{12} q^{93} + (525 \zeta_{12}^{2} - 525) q^{94} + 882 \zeta_{12}^{2} q^{95} + 595 \zeta_{12}^{3} q^{96} + ( - 70 \zeta_{12}^{3} + 70 \zeta_{12}) q^{97} + (870 \zeta_{12}^{3} - 870 \zeta_{12}) q^{98} + 572 \zeta_{12}^{3} q^{99} +O(q^{100})$$ q + 5*z * q^2 + (-7*z^2 + 7) * q^3 + 17*z^2 * q^4 - 7*z^3 * q^5 + (-35*z^3 + 35*z) * q^6 + (-13*z^3 + 13*z) * q^7 + 45*z^3 * q^8 - 22*z^2 * q^9 + (-35*z^2 + 35) * q^10 - 26*z * q^11 + 119 * q^12 + 65 * q^14 - 49*z * q^15 + (89*z^2 - 89) * q^16 + 77*z^2 * q^17 - 110*z^3 * q^18 + (126*z^3 - 126*z) * q^19 + (-119*z^3 + 119*z) * q^20 - 91*z^3 * q^21 - 130*z^2 * q^22 + (96*z^2 - 96) * q^23 + 315*z * q^24 + 76 * q^25 + 35 * q^27 + 221*z * q^28 + (-82*z^2 + 82) * q^29 - 245*z^2 * q^30 + 196*z^3 * q^31 + (85*z^3 - 85*z) * q^32 + (182*z^3 - 182*z) * q^33 + 385*z^3 * q^34 - 91*z^2 * q^35 + (-374*z^2 + 374) * q^36 - 131*z * q^37 - 630 * q^38 + 315 * q^40 - 336*z * q^41 + (-455*z^2 + 455) * q^42 - 201*z^2 * q^43 - 442*z^3 * q^44 + (154*z^3 - 154*z) * q^45 + (480*z^3 - 480*z) * q^46 + 105*z^3 * q^47 + 623*z^2 * q^48 + (174*z^2 - 174) * q^49 + 380*z * q^50 + 539 * q^51 - 432 * q^53 + 175*z * q^54 + (182*z^2 - 182) * q^55 + 585*z^2 * q^56 + 882*z^3 * q^57 + (-410*z^3 + 410*z) * q^58 + (-294*z^3 + 294*z) * q^59 - 833*z^3 * q^60 + 56*z^2 * q^61 + (980*z^2 - 980) * q^62 - 286*z * q^63 + 287 * q^64 - 910 * q^66 - 478*z * q^67 + (1309*z^2 - 1309) * q^68 + 672*z^2 * q^69 - 455*z^3 * q^70 + (-9*z^3 + 9*z) * q^71 + (-990*z^3 + 990*z) * q^72 - 98*z^3 * q^73 - 655*z^2 * q^74 + (-532*z^2 + 532) * q^75 - 2142*z * q^76 - 338 * q^77 + 1304 * q^79 + 623*z * q^80 + (-839*z^2 + 839) * q^81 - 1680*z^2 * q^82 - 308*z^3 * q^83 + (-1547*z^3 + 1547*z) * q^84 + (-539*z^3 + 539*z) * q^85 - 1005*z^3 * q^86 - 574*z^2 * q^87 + (-1170*z^2 + 1170) * q^88 - 1190*z * q^89 - 770 * q^90 - 1632 * q^92 + 1372*z * q^93 + (525*z^2 - 525) * q^94 + 882*z^2 * q^95 + 595*z^3 * q^96 + (-70*z^3 + 70*z) * q^97 + (870*z^3 - 870*z) * q^98 + 572*z^3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 14 q^{3} + 34 q^{4} - 44 q^{9}+O(q^{10})$$ 4 * q + 14 * q^3 + 34 * q^4 - 44 * q^9 $$4 q + 14 q^{3} + 34 q^{4} - 44 q^{9} + 70 q^{10} + 476 q^{12} + 260 q^{14} - 178 q^{16} + 154 q^{17} - 260 q^{22} - 192 q^{23} + 304 q^{25} + 140 q^{27} + 164 q^{29} - 490 q^{30} - 182 q^{35} + 748 q^{36} - 2520 q^{38} + 1260 q^{40} + 910 q^{42} - 402 q^{43} + 1246 q^{48} - 348 q^{49} + 2156 q^{51} - 1728 q^{53} - 364 q^{55} + 1170 q^{56} + 112 q^{61} - 1960 q^{62} + 1148 q^{64} - 3640 q^{66} - 2618 q^{68} + 1344 q^{69} - 1310 q^{74} + 1064 q^{75} - 1352 q^{77} + 5216 q^{79} + 1678 q^{81} - 3360 q^{82} - 1148 q^{87} + 2340 q^{88} - 3080 q^{90} - 6528 q^{92} - 1050 q^{94} + 1764 q^{95}+O(q^{100})$$ 4 * q + 14 * q^3 + 34 * q^4 - 44 * q^9 + 70 * q^10 + 476 * q^12 + 260 * q^14 - 178 * q^16 + 154 * q^17 - 260 * q^22 - 192 * q^23 + 304 * q^25 + 140 * q^27 + 164 * q^29 - 490 * q^30 - 182 * q^35 + 748 * q^36 - 2520 * q^38 + 1260 * q^40 + 910 * q^42 - 402 * q^43 + 1246 * q^48 - 348 * q^49 + 2156 * q^51 - 1728 * q^53 - 364 * q^55 + 1170 * q^56 + 112 * q^61 - 1960 * q^62 + 1148 * q^64 - 3640 * q^66 - 2618 * q^68 + 1344 * q^69 - 1310 * q^74 + 1064 * q^75 - 1352 * q^77 + 5216 * q^79 + 1678 * q^81 - 3360 * q^82 - 1148 * q^87 + 2340 * q^88 - 3080 * q^90 - 6528 * q^92 - 1050 * q^94 + 1764 * 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
−4.33013 + 2.50000i 3.50000 + 6.06218i 8.50000 14.7224i 7.00000i −30.3109 17.5000i −11.2583 6.50000i 45.0000i −11.0000 + 19.0526i 17.5000 + 30.3109i
23.2 4.33013 2.50000i 3.50000 + 6.06218i 8.50000 14.7224i 7.00000i 30.3109 + 17.5000i 11.2583 + 6.50000i 45.0000i −11.0000 + 19.0526i 17.5000 + 30.3109i
147.1 −4.33013 2.50000i 3.50000 6.06218i 8.50000 + 14.7224i 7.00000i −30.3109 + 17.5000i −11.2583 + 6.50000i 45.0000i −11.0000 19.0526i 17.5000 30.3109i
147.2 4.33013 + 2.50000i 3.50000 6.06218i 8.50000 + 14.7224i 7.00000i 30.3109 17.5000i 11.2583 6.50000i 45.0000i −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.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.e 4
13.b even 2 1 inner 169.4.e.e 4
13.c even 3 1 169.4.b.a 2
13.c even 3 1 inner 169.4.e.e 4
13.d odd 4 1 169.4.c.a 2
13.d odd 4 1 169.4.c.e 2
13.e even 6 1 169.4.b.a 2
13.e even 6 1 inner 169.4.e.e 4
13.f odd 12 1 13.4.a.a 1
13.f odd 12 1 169.4.a.e 1
13.f odd 12 1 169.4.c.a 2
13.f odd 12 1 169.4.c.e 2
39.k even 12 1 117.4.a.b 1
39.k even 12 1 1521.4.a.a 1
52.l even 12 1 208.4.a.g 1
65.o even 12 1 325.4.b.b 2
65.s odd 12 1 325.4.a.d 1
65.t even 12 1 325.4.b.b 2
91.bc even 12 1 637.4.a.a 1
104.u even 12 1 832.4.a.a 1
104.x odd 12 1 832.4.a.r 1
143.o even 12 1 1573.4.a.a 1
156.v odd 12 1 1872.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 13.f odd 12 1
117.4.a.b 1 39.k even 12 1
169.4.a.e 1 13.f odd 12 1
169.4.b.a 2 13.c even 3 1
169.4.b.a 2 13.e even 6 1
169.4.c.a 2 13.d odd 4 1
169.4.c.a 2 13.f odd 12 1
169.4.c.e 2 13.d odd 4 1
169.4.c.e 2 13.f odd 12 1
169.4.e.e 4 1.a even 1 1 trivial
169.4.e.e 4 13.b even 2 1 inner
169.4.e.e 4 13.c even 3 1 inner
169.4.e.e 4 13.e even 6 1 inner
208.4.a.g 1 52.l even 12 1
325.4.a.d 1 65.s odd 12 1
325.4.b.b 2 65.o even 12 1
325.4.b.b 2 65.t even 12 1
637.4.a.a 1 91.bc even 12 1
832.4.a.a 1 104.u even 12 1
832.4.a.r 1 104.x odd 12 1
1521.4.a.a 1 39.k even 12 1
1573.4.a.a 1 143.o even 12 1
1872.4.a.k 1 156.v odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 25T_{2}^{2} + 625$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 25T^{2} + 625$$
$3$ $$(T^{2} - 7 T + 49)^{2}$$
$5$ $$(T^{2} + 49)^{2}$$
$7$ $$T^{4} - 169 T^{2} + 28561$$
$11$ $$T^{4} - 676 T^{2} + 456976$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 77 T + 5929)^{2}$$
$19$ $$T^{4} - 15876 T^{2} + \cdots + 252047376$$
$23$ $$(T^{2} + 96 T + 9216)^{2}$$
$29$ $$(T^{2} - 82 T + 6724)^{2}$$
$31$ $$(T^{2} + 38416)^{2}$$
$37$ $$T^{4} - 17161 T^{2} + \cdots + 294499921$$
$41$ $$T^{4} - 112896 T^{2} + \cdots + 12745506816$$
$43$ $$(T^{2} + 201 T + 40401)^{2}$$
$47$ $$(T^{2} + 11025)^{2}$$
$53$ $$(T + 432)^{4}$$
$59$ $$T^{4} - 86436 T^{2} + \cdots + 7471182096$$
$61$ $$(T^{2} - 56 T + 3136)^{2}$$
$67$ $$T^{4} - 228484 T^{2} + \cdots + 52204938256$$
$71$ $$T^{4} - 81T^{2} + 6561$$
$73$ $$(T^{2} + 9604)^{2}$$
$79$ $$(T - 1304)^{4}$$
$83$ $$(T^{2} + 94864)^{2}$$
$89$ $$T^{4} - 1416100 T^{2} + \cdots + 2005339210000$$
$97$ $$T^{4} - 4900 T^{2} + \cdots + 24010000$$