# Properties

 Label 169.4.e.b Level $169$ Weight $4$ Character orbit 169.e 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.e (of order $$6$$, 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: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 + 2 \zeta_{6} ) q^{2} + ( 7 - 7 \zeta_{6} ) q^{3} + 4 \zeta_{6} q^{4} + ( 8 - 16 \zeta_{6} ) q^{5} + ( 28 - 14 \zeta_{6} ) q^{6} + ( -26 + 13 \zeta_{6} ) q^{7} + ( 8 - 16 \zeta_{6} ) q^{8} -22 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 2 + 2 \zeta_{6} ) q^{2} + ( 7 - 7 \zeta_{6} ) q^{3} + 4 \zeta_{6} q^{4} + ( 8 - 16 \zeta_{6} ) q^{5} + ( 28 - 14 \zeta_{6} ) q^{6} + ( -26 + 13 \zeta_{6} ) q^{7} + ( 8 - 16 \zeta_{6} ) q^{8} -22 \zeta_{6} q^{9} + ( 48 - 48 \zeta_{6} ) q^{10} + ( 13 + 13 \zeta_{6} ) q^{11} + 28 q^{12} -78 q^{14} + ( -56 - 56 \zeta_{6} ) q^{15} + ( 80 - 80 \zeta_{6} ) q^{16} -27 \zeta_{6} q^{17} + ( 44 - 88 \zeta_{6} ) q^{18} + ( 102 - 51 \zeta_{6} ) q^{19} + ( 64 - 32 \zeta_{6} ) q^{20} + ( -91 + 182 \zeta_{6} ) q^{21} + 78 \zeta_{6} q^{22} + ( -57 + 57 \zeta_{6} ) q^{23} + ( -56 - 56 \zeta_{6} ) q^{24} -67 q^{25} + 35 q^{27} + ( -52 - 52 \zeta_{6} ) q^{28} + ( 69 - 69 \zeta_{6} ) q^{29} -336 \zeta_{6} q^{30} + ( -42 + 84 \zeta_{6} ) q^{31} + ( 192 - 96 \zeta_{6} ) q^{32} + ( 182 - 91 \zeta_{6} ) q^{33} + ( 54 - 108 \zeta_{6} ) q^{34} + 312 \zeta_{6} q^{35} + ( 88 - 88 \zeta_{6} ) q^{36} + ( 23 + 23 \zeta_{6} ) q^{37} + 306 q^{38} -192 q^{40} + ( 227 + 227 \zeta_{6} ) q^{41} + ( -546 + 546 \zeta_{6} ) q^{42} + 85 \zeta_{6} q^{43} + ( -52 + 104 \zeta_{6} ) q^{44} + ( -352 + 176 \zeta_{6} ) q^{45} + ( -228 + 114 \zeta_{6} ) q^{46} + ( -198 + 396 \zeta_{6} ) q^{47} -560 \zeta_{6} q^{48} + ( 164 - 164 \zeta_{6} ) q^{49} + ( -134 - 134 \zeta_{6} ) q^{50} -189 q^{51} + 426 q^{53} + ( 70 + 70 \zeta_{6} ) q^{54} + ( 312 - 312 \zeta_{6} ) q^{55} + 312 \zeta_{6} q^{56} + ( 357 - 714 \zeta_{6} ) q^{57} + ( 276 - 138 \zeta_{6} ) q^{58} + ( 22 - 11 \zeta_{6} ) q^{59} + ( 224 - 448 \zeta_{6} ) q^{60} + 17 \zeta_{6} q^{61} + ( -252 + 252 \zeta_{6} ) q^{62} + ( 286 + 286 \zeta_{6} ) q^{63} -64 q^{64} + 546 q^{66} + ( -95 - 95 \zeta_{6} ) q^{67} + ( 108 - 108 \zeta_{6} ) q^{68} + 399 \zeta_{6} q^{69} + ( -624 + 1248 \zeta_{6} ) q^{70} + ( -674 + 337 \zeta_{6} ) q^{71} + ( -352 + 176 \zeta_{6} ) q^{72} + ( 580 - 1160 \zeta_{6} ) q^{73} + 138 \zeta_{6} q^{74} + ( -469 + 469 \zeta_{6} ) q^{75} + ( 204 + 204 \zeta_{6} ) q^{76} -507 q^{77} -1244 q^{79} + ( -640 - 640 \zeta_{6} ) q^{80} + ( 839 - 839 \zeta_{6} ) q^{81} + 1362 \zeta_{6} q^{82} + ( -246 + 492 \zeta_{6} ) q^{83} + ( -728 + 364 \zeta_{6} ) q^{84} + ( -432 + 216 \zeta_{6} ) q^{85} + ( -170 + 340 \zeta_{6} ) q^{86} -483 \zeta_{6} q^{87} + ( 312 - 312 \zeta_{6} ) q^{88} + ( -177 - 177 \zeta_{6} ) q^{89} -1056 q^{90} -228 q^{92} + ( 294 + 294 \zeta_{6} ) q^{93} + ( -1188 + 1188 \zeta_{6} ) q^{94} -1224 \zeta_{6} q^{95} + ( 672 - 1344 \zeta_{6} ) q^{96} + ( -1426 + 713 \zeta_{6} ) q^{97} + ( 656 - 328 \zeta_{6} ) q^{98} + ( 286 - 572 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{2} + 7 q^{3} + 4 q^{4} + 42 q^{6} - 39 q^{7} - 22 q^{9} + O(q^{10})$$ $$2 q + 6 q^{2} + 7 q^{3} + 4 q^{4} + 42 q^{6} - 39 q^{7} - 22 q^{9} + 48 q^{10} + 39 q^{11} + 56 q^{12} - 156 q^{14} - 168 q^{15} + 80 q^{16} - 27 q^{17} + 153 q^{19} + 96 q^{20} + 78 q^{22} - 57 q^{23} - 168 q^{24} - 134 q^{25} + 70 q^{27} - 156 q^{28} + 69 q^{29} - 336 q^{30} + 288 q^{32} + 273 q^{33} + 312 q^{35} + 88 q^{36} + 69 q^{37} + 612 q^{38} - 384 q^{40} + 681 q^{41} - 546 q^{42} + 85 q^{43} - 528 q^{45} - 342 q^{46} - 560 q^{48} + 164 q^{49} - 402 q^{50} - 378 q^{51} + 852 q^{53} + 210 q^{54} + 312 q^{55} + 312 q^{56} + 414 q^{58} + 33 q^{59} + 17 q^{61} - 252 q^{62} + 858 q^{63} - 128 q^{64} + 1092 q^{66} - 285 q^{67} + 108 q^{68} + 399 q^{69} - 1011 q^{71} - 528 q^{72} + 138 q^{74} - 469 q^{75} + 612 q^{76} - 1014 q^{77} - 2488 q^{79} - 1920 q^{80} + 839 q^{81} + 1362 q^{82} - 1092 q^{84} - 648 q^{85} - 483 q^{87} + 312 q^{88} - 531 q^{89} - 2112 q^{90} - 456 q^{92} + 882 q^{93} - 1188 q^{94} - 1224 q^{95} - 2139 q^{97} + 984 q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 0.5 − 0.866025i 0.5 + 0.866025i
3.00000 1.73205i 3.50000 + 6.06218i 2.00000 3.46410i 13.8564i 21.0000 + 12.1244i −19.5000 11.2583i 13.8564i −11.0000 + 19.0526i 24.0000 + 41.5692i
147.1 3.00000 + 1.73205i 3.50000 6.06218i 2.00000 + 3.46410i 13.8564i 21.0000 12.1244i −19.5000 + 11.2583i 13.8564i −11.0000 19.0526i 24.0000 41.5692i
 $$n$$: 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.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.b 2
13.b even 2 1 13.4.e.a 2
13.c even 3 1 13.4.e.a 2
13.c even 3 1 169.4.b.b 2
13.d odd 4 2 169.4.c.i 4
13.e even 6 1 169.4.b.b 2
13.e even 6 1 inner 169.4.e.b 2
13.f odd 12 2 169.4.a.h 2
13.f odd 12 2 169.4.c.i 4
39.d odd 2 1 117.4.q.c 2
39.i odd 6 1 117.4.q.c 2
39.k even 12 2 1521.4.a.q 2
52.b odd 2 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.b even 2 1
13.4.e.a 2 13.c even 3 1
117.4.q.c 2 39.d odd 2 1
117.4.q.c 2 39.i odd 6 1
169.4.a.h 2 13.f odd 12 2
169.4.b.b 2 13.c even 3 1
169.4.b.b 2 13.e even 6 1
169.4.c.i 4 13.d odd 4 2
169.4.c.i 4 13.f odd 12 2
169.4.e.b 2 1.a even 1 1 trivial
169.4.e.b 2 13.e even 6 1 inner
208.4.w.a 2 52.b odd 2 1
208.4.w.a 2 52.j odd 6 1
1521.4.a.q 2 39.k even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 6 T_{2} + 12$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$12 - 6 T + T^{2}$$
$3$ $$49 - 7 T + T^{2}$$
$5$ $$192 + T^{2}$$
$7$ $$507 + 39 T + T^{2}$$
$11$ $$507 - 39 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$729 + 27 T + T^{2}$$
$19$ $$7803 - 153 T + T^{2}$$
$23$ $$3249 + 57 T + T^{2}$$
$29$ $$4761 - 69 T + T^{2}$$
$31$ $$5292 + T^{2}$$
$37$ $$1587 - 69 T + T^{2}$$
$41$ $$154587 - 681 T + T^{2}$$
$43$ $$7225 - 85 T + T^{2}$$
$47$ $$117612 + T^{2}$$
$53$ $$( -426 + T )^{2}$$
$59$ $$363 - 33 T + T^{2}$$
$61$ $$289 - 17 T + T^{2}$$
$67$ $$27075 + 285 T + T^{2}$$
$71$ $$340707 + 1011 T + T^{2}$$
$73$ $$1009200 + T^{2}$$
$79$ $$( 1244 + T )^{2}$$
$83$ $$181548 + T^{2}$$
$89$ $$93987 + 531 T + T^{2}$$
$97$ $$1525107 + 2139 T + T^{2}$$