# Properties

 Label 342.2.u.a Level $342$ Weight $2$ Character orbit 342.u Analytic conductor $2.731$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 342.u (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.73088374913$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{18} q^{2} + \zeta_{18}^{2} q^{4} + ( -1 + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{5} + ( 1 - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{7} -\zeta_{18}^{3} q^{8} +O(q^{10})$$ $$q -\zeta_{18} q^{2} + \zeta_{18}^{2} q^{4} + ( -1 + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{5} + ( 1 - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{7} -\zeta_{18}^{3} q^{8} + ( 1 + \zeta_{18} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{10} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{11} + ( -1 + \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{13} + ( 1 - \zeta_{18} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{14} + \zeta_{18}^{4} q^{16} + ( 2 + 2 \zeta_{18} - \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{5} ) q^{17} + ( -2 - \zeta_{18} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{19} + ( -\zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{20} + ( -\zeta_{18}^{3} - \zeta_{18}^{5} ) q^{22} + ( 2 + 3 \zeta_{18} - 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{23} + ( -3 \zeta_{18} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{25} + ( 2 + \zeta_{18} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{26} + ( -1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{28} + ( -2 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{29} + ( 3 - 3 \zeta_{18} + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} ) q^{31} -\zeta_{18}^{5} q^{32} + ( 3 - 2 \zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{34} + ( -4 + 4 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{35} + ( -3 + 6 \zeta_{18} + 6 \zeta_{18}^{2} - 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{37} + ( 2 + 2 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{38} + ( -1 + \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{40} + ( 1 - \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{41} + ( -1 + 5 \zeta_{18} - 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 5 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{43} + ( -1 + \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{44} + ( -2 \zeta_{18} - 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{46} + ( \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{47} + ( 3 \zeta_{18} - \zeta_{18}^{2} + 4 \zeta_{18}^{3} - \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{49} + ( -1 + 3 \zeta_{18}^{2} + \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{50} + ( -1 - 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{52} + ( 2 - 8 \zeta_{18} - \zeta_{18}^{2} - 8 \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{53} + ( -2 - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{55} + ( -1 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{4} ) q^{56} + ( 2 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - 5 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{58} + ( -6 + 4 \zeta_{18} - 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{59} + ( 1 - \zeta_{18} - 11 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{61} + ( -3 \zeta_{18} + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{62} + ( -1 + \zeta_{18}^{3} ) q^{64} + ( -3 \zeta_{18} - 3 \zeta_{18}^{5} ) q^{65} + ( 4 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{67} + ( -3 \zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{68} + ( -2 + 4 \zeta_{18} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{70} + ( -7 + 5 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - 5 \zeta_{18}^{4} + 7 \zeta_{18}^{5} ) q^{71} + ( 7 - 7 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{73} + ( -2 + 3 \zeta_{18} - 6 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{5} ) q^{74} + ( -2 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{76} - q^{77} + ( -5 + 5 \zeta_{18}^{2} - 5 \zeta_{18}^{3} - 5 \zeta_{18}^{4} - 10 \zeta_{18}^{5} ) q^{79} + ( -2 + \zeta_{18} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{80} + ( 4 - \zeta_{18} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{82} + ( 9 + 3 \zeta_{18} - 2 \zeta_{18}^{2} - 9 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{83} + ( -7 - 7 \zeta_{18} + 9 \zeta_{18}^{2} + 8 \zeta_{18}^{3} - \zeta_{18}^{4} - 9 \zeta_{18}^{5} ) q^{85} + ( 1 + \zeta_{18} - 5 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{86} + ( \zeta_{18} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{88} + ( 7 - 3 \zeta_{18} - 4 \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{89} + ( -2 + \zeta_{18}^{2} - 2 \zeta_{18}^{4} ) q^{91} + ( -2 + 2 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{92} + ( -1 - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{94} + ( -3 - \zeta_{18} + 2 \zeta_{18}^{2} + 6 \zeta_{18}^{3} - \zeta_{18}^{4} - 10 \zeta_{18}^{5} ) q^{95} + ( -1 - 4 \zeta_{18} - \zeta_{18}^{2} ) q^{97} + ( 3 - 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 4 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 9q^{5} + 3q^{7} - 3q^{8} + O(q^{10})$$ $$6q - 9q^{5} + 3q^{7} - 3q^{8} - 3q^{13} + 6q^{14} + 3q^{17} - 18q^{19} - 3q^{22} + 21q^{23} + 9q^{25} + 6q^{26} - 3q^{28} + 3q^{29} + 9q^{31} + 12q^{34} - 18q^{35} - 18q^{37} + 6q^{38} + 15q^{41} + 3q^{43} - 3q^{44} + 6q^{46} + 9q^{47} + 12q^{49} - 3q^{50} - 3q^{52} - 12q^{53} - 9q^{55} - 6q^{56} + 12q^{58} - 27q^{59} + 3q^{61} - 9q^{62} - 3q^{64} + 21q^{67} + 6q^{68} - 18q^{70} - 39q^{71} + 36q^{73} - 24q^{74} - 9q^{76} - 6q^{77} - 45q^{79} - 9q^{80} + 15q^{82} + 27q^{83} - 18q^{85} + 12q^{86} + 30q^{89} - 12q^{91} + 3q^{92} - 6q^{94} - 6q^{97} + 12q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$325$$ $$\chi(n)$$ $$1$$ $$\zeta_{18}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 −0.173648 + 0.984808i −0.766044 − 0.642788i −0.173648 − 0.984808i −0.766044 + 0.642788i 0.939693 − 0.342020i 0.939693 + 0.342020i
0.173648 0.984808i 0 −0.939693 0.342020i −3.20574 + 1.16679i 0 1.43969 + 2.49362i −0.500000 + 0.866025i 0 0.592396 + 3.35965i
73.1 0.766044 + 0.642788i 0 0.173648 + 0.984808i −0.386659 + 2.19285i 0 0.326352 + 0.565258i −0.500000 + 0.866025i 0 −1.70574 + 1.43128i
199.1 0.173648 + 0.984808i 0 −0.939693 + 0.342020i −3.20574 1.16679i 0 1.43969 2.49362i −0.500000 0.866025i 0 0.592396 3.35965i
253.1 0.766044 0.642788i 0 0.173648 0.984808i −0.386659 2.19285i 0 0.326352 0.565258i −0.500000 0.866025i 0 −1.70574 1.43128i
271.1 −0.939693 + 0.342020i 0 0.766044 0.642788i −0.907604 0.761570i 0 −0.266044 0.460802i −0.500000 + 0.866025i 0 1.11334 + 0.405223i
289.1 −0.939693 0.342020i 0 0.766044 + 0.642788i −0.907604 + 0.761570i 0 −0.266044 + 0.460802i −0.500000 0.866025i 0 1.11334 0.405223i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.u.a 6
3.b odd 2 1 114.2.i.d 6
12.b even 2 1 912.2.bo.f 6
19.e even 9 1 inner 342.2.u.a 6
19.e even 9 1 6498.2.a.bs 3
19.f odd 18 1 6498.2.a.bn 3
57.j even 18 1 2166.2.a.u 3
57.l odd 18 1 114.2.i.d 6
57.l odd 18 1 2166.2.a.o 3
228.v even 18 1 912.2.bo.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.d 6 3.b odd 2 1
114.2.i.d 6 57.l odd 18 1
342.2.u.a 6 1.a even 1 1 trivial
342.2.u.a 6 19.e even 9 1 inner
912.2.bo.f 6 12.b even 2 1
912.2.bo.f 6 228.v even 18 1
2166.2.a.o 3 57.l odd 18 1
2166.2.a.u 3 57.j even 18 1
6498.2.a.bn 3 19.f odd 18 1
6498.2.a.bs 3 19.e even 9 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 9 T_{5}^{5} + 36 T_{5}^{4} + 90 T_{5}^{3} + 162 T_{5}^{2} + 162 T_{5} + 81$$ acting on $$S_{2}^{\mathrm{new}}(342, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{3} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$81 + 162 T + 162 T^{2} + 90 T^{3} + 36 T^{4} + 9 T^{5} + T^{6}$$
$7$ $$1 + 3 T^{2} - 2 T^{3} + 9 T^{4} - 3 T^{5} + T^{6}$$
$11$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$13$ $$9 - 27 T + 9 T^{2} + 24 T^{3} + 18 T^{4} + 3 T^{5} + T^{6}$$
$17$ $$289 - 663 T + 591 T^{2} - 244 T^{3} + 48 T^{4} - 3 T^{5} + T^{6}$$
$19$ $$6859 + 6498 T + 3078 T^{2} + 883 T^{3} + 162 T^{4} + 18 T^{5} + T^{6}$$
$23$ $$72361 - 32280 T + 7674 T^{2} - 1396 T^{3} + 210 T^{4} - 21 T^{5} + T^{6}$$
$29$ $$45369 + 15336 T + 1386 T^{2} + 84 T^{3} + 18 T^{4} - 3 T^{5} + T^{6}$$
$31$ $$729 + 243 T^{2} - 54 T^{3} + 81 T^{4} - 9 T^{5} + T^{6}$$
$37$ $$( -361 - 57 T + 9 T^{2} + T^{3} )^{2}$$
$41$ $$11881 - 12099 T + 5376 T^{2} - 1288 T^{3} + 177 T^{4} - 15 T^{5} + T^{6}$$
$43$ $$3249 + 513 T - 504 T^{2} + 84 T^{3} + 99 T^{4} - 3 T^{5} + T^{6}$$
$47$ $$2809 - 3339 T + 1530 T^{2} - 352 T^{3} + 63 T^{4} - 9 T^{5} + T^{6}$$
$53$ $$94249 - 81048 T + 16962 T^{2} + 1351 T^{3} + 174 T^{4} + 12 T^{5} + T^{6}$$
$59$ $$289 + 459 T + 1701 T^{2} + 1232 T^{3} + 324 T^{4} + 27 T^{5} + T^{6}$$
$61$ $$1682209 + 38910 T + 8979 T^{2} - 1207 T^{3} - 60 T^{4} - 3 T^{5} + T^{6}$$
$67$ $$9 - 54 T + 144 T^{2} - 24 T^{3} + 126 T^{4} - 21 T^{5} + T^{6}$$
$71$ $$201601 + 57921 T + 16386 T^{2} + 3592 T^{3} + 561 T^{4} + 39 T^{5} + T^{6}$$
$73$ $$1369 + 999 T + 12330 T^{2} - 4148 T^{3} + 558 T^{4} - 36 T^{5} + T^{6}$$
$79$ $$4515625 + 1434375 T + 213750 T^{2} + 19000 T^{3} + 1125 T^{4} + 45 T^{5} + T^{6}$$
$83$ $$253009 - 111666 T + 35703 T^{2} - 4988 T^{3} + 507 T^{4} - 27 T^{5} + T^{6}$$
$89$ $$11449 + 7383 T + 2580 T^{2} + 62 T^{3} + 246 T^{4} - 30 T^{5} + T^{6}$$
$97$ $$2809 - 1113 T + 276 T^{2} + 35 T^{3} + 3 T^{4} + 6 T^{5} + T^{6}$$