# Properties

 Label 912.2.bo.a Level $912$ Weight $2$ Character orbit 912.bo Analytic conductor $7.282$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.bo (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{18} q^{3} + ( -1 + \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{5} + ( -3 + \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{3} ) q^{7} + \zeta_{18}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{18} q^{3} + ( -1 + \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{5} + ( -3 + \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{3} ) q^{7} + \zeta_{18}^{2} q^{9} + ( -\zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{11} + ( 3 + 3 \zeta_{18} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{13} + ( 2 - \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{15} + ( -2 - \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{17} + ( 2 - \zeta_{18} + 4 \zeta_{18}^{2} - \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{19} + ( -3 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{21} + ( -2 + \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{23} + ( -\zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{25} + \zeta_{18}^{3} q^{27} + ( -2 + \zeta_{18} + 6 \zeta_{18}^{2} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{29} + ( 3 - \zeta_{18} + 6 \zeta_{18}^{2} - 3 \zeta_{18}^{3} - 5 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{31} + ( 1 - \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{33} + ( 8 - 3 \zeta_{18} + 4 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{5} ) q^{35} + ( 3 - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{37} + ( 3 \zeta_{18} + 3 \zeta_{18}^{2} - \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{39} + ( 3 - 3 \zeta_{18} + 6 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{41} + ( 3 - 3 \zeta_{18}^{2} + \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{43} + ( 2 \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{45} + ( -4 - \zeta_{18} - 3 \zeta_{18}^{2} - \zeta_{18}^{3} - 4 \zeta_{18}^{4} ) q^{47} + ( -6 \zeta_{18} + 7 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 7 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{49} + ( -2 - 2 \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{51} + ( 2 + 2 \zeta_{18} - 4 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{53} + ( 6 \zeta_{18} - 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 6 \zeta_{18}^{4} ) q^{55} + ( 2 + 2 \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{57} + ( -6 \zeta_{18} + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 6 \zeta_{18}^{4} ) q^{59} + ( -3 - 2 \zeta_{18} + 5 \zeta_{18}^{3} + 5 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{61} + ( -3 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{63} + ( -5 \zeta_{18} + \zeta_{18}^{2} - 6 \zeta_{18}^{3} + \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{65} + ( 2 - 3 \zeta_{18} - 6 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{67} + ( -2 \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{69} + ( -3 + 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 9 \zeta_{18}^{4} ) q^{71} + ( -5 + 3 \zeta_{18} + \zeta_{18}^{2} + 6 \zeta_{18}^{3} - 6 \zeta_{18}^{5} ) q^{73} + ( -1 + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{75} + ( 3 - 4 \zeta_{18} - 4 \zeta_{18}^{2} - 4 \zeta_{18}^{4} + 8 \zeta_{18}^{5} ) q^{77} + ( 5 + 9 \zeta_{18} + 2 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{5} ) q^{79} + \zeta_{18}^{4} q^{81} + ( 3 - 8 \zeta_{18} + \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 7 \zeta_{18}^{4} + 7 \zeta_{18}^{5} ) q^{83} + ( 3 + 3 \zeta_{18}^{2} + 3 \zeta_{18}^{4} ) q^{85} + ( -2 \zeta_{18} + \zeta_{18}^{2} + 6 \zeta_{18}^{3} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{87} + ( 1 + \zeta_{18} - 9 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} + 9 \zeta_{18}^{5} ) q^{89} + ( -8 + 8 \zeta_{18}^{3} + 8 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{91} + ( 5 + 3 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{93} + ( -5 + 6 \zeta_{18} - 4 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{95} + ( -9 - 4 \zeta_{18} + 4 \zeta_{18}^{4} + 9 \zeta_{18}^{5} ) q^{97} + ( -2 + \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 9q^{5} - 9q^{7} + O(q^{10})$$ $$6q - 9q^{5} - 9q^{7} + 15q^{13} + 9q^{15} - 9q^{17} + 12q^{19} - 3q^{21} - 9q^{23} + 9q^{25} + 3q^{27} - 9q^{29} + 9q^{31} + 9q^{33} + 36q^{35} + 18q^{37} + 27q^{41} + 21q^{43} - 27q^{47} - 12q^{49} - 9q^{51} + 9q^{55} + 18q^{57} - 9q^{59} - 3q^{61} + 3q^{63} - 18q^{65} + 3q^{67} - 9q^{71} - 12q^{73} - 6q^{75} + 18q^{77} + 21q^{79} + 9q^{83} + 18q^{85} + 18q^{87} - 24q^{91} + 33q^{93} - 36q^{95} - 54q^{97} - 9q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$-\zeta_{18}^{5}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −0.173648 − 0.984808i −0.173648 + 0.984808i 0.939693 + 0.342020i 0.939693 − 0.342020i −0.766044 − 0.642788i −0.766044 + 0.642788i
0 −0.173648 0.984808i 0 −0.907604 + 0.761570i 0 −0.733956 + 1.27125i 0 −0.939693 + 0.342020i 0
385.1 0 −0.173648 + 0.984808i 0 −0.907604 0.761570i 0 −0.733956 1.27125i 0 −0.939693 0.342020i 0
481.1 0 0.939693 + 0.342020i 0 −0.386659 2.19285i 0 −1.32635 + 2.29731i 0 0.766044 + 0.642788i 0
529.1 0 0.939693 0.342020i 0 −0.386659 + 2.19285i 0 −1.32635 2.29731i 0 0.766044 0.642788i 0
625.1 0 −0.766044 0.642788i 0 −3.20574 + 1.16679i 0 −2.43969 4.22567i 0 0.173648 + 0.984808i 0
769.1 0 −0.766044 + 0.642788i 0 −3.20574 1.16679i 0 −2.43969 + 4.22567i 0 0.173648 0.984808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 769.1 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 912.2.bo.a 6
4.b odd 2 1 114.2.i.a 6
12.b even 2 1 342.2.u.e 6
19.e even 9 1 inner 912.2.bo.a 6
76.k even 18 1 2166.2.a.s 3
76.l odd 18 1 114.2.i.a 6
76.l odd 18 1 2166.2.a.q 3
228.u odd 18 1 6498.2.a.bm 3
228.v even 18 1 342.2.u.e 6
228.v even 18 1 6498.2.a.br 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.a 6 4.b odd 2 1
114.2.i.a 6 76.l odd 18 1
342.2.u.e 6 12.b even 2 1
342.2.u.e 6 228.v even 18 1
912.2.bo.a 6 1.a even 1 1 trivial
912.2.bo.a 6 19.e even 9 1 inner
2166.2.a.q 3 76.l odd 18 1
2166.2.a.s 3 76.k even 18 1
6498.2.a.bm 3 228.u odd 18 1
6498.2.a.br 3 228.v even 18 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 $$S_{2}^{\mathrm{new}}(912, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$1 - T^{3} + T^{6}$$
$5$ $$81 + 162 T + 162 T^{2} + 90 T^{3} + 36 T^{4} + 9 T^{5} + T^{6}$$
$7$ $$361 + 456 T + 405 T^{2} + 178 T^{3} + 57 T^{4} + 9 T^{5} + T^{6}$$
$11$ $$81 - 81 T + 81 T^{2} - 18 T^{3} + 9 T^{4} + T^{6}$$
$13$ $$1369 - 1443 T + 885 T^{2} - 352 T^{3} + 96 T^{4} - 15 T^{5} + T^{6}$$
$17$ $$81 + 81 T + 81 T^{2} + 72 T^{3} + 36 T^{4} + 9 T^{5} + T^{6}$$
$19$ $$6859 - 4332 T + 1482 T^{2} - 385 T^{3} + 78 T^{4} - 12 T^{5} + T^{6}$$
$23$ $$81 + 162 T + 162 T^{2} + 90 T^{3} + 36 T^{4} + 9 T^{5} + T^{6}$$
$29$ $$29241 - 6156 T + 2106 T^{2} - 72 T^{3} - 18 T^{4} + 9 T^{5} + T^{6}$$
$31$ $$292681 - 35706 T + 9225 T^{2} - 488 T^{3} + 147 T^{4} - 9 T^{5} + T^{6}$$
$37$ $$( 1 + 15 T - 9 T^{2} + T^{3} )^{2}$$
$41$ $$210681 - 111537 T + 27702 T^{2} - 4104 T^{3} + 405 T^{4} - 27 T^{5} + T^{6}$$
$43$ $$32041 - 23091 T + 7662 T^{2} - 1592 T^{3} + 231 T^{4} - 21 T^{5} + T^{6}$$
$47$ $$23409 + 15147 T + 6804 T^{2} + 2016 T^{3} + 333 T^{4} + 27 T^{5} + T^{6}$$
$53$ $$81 + 324 T + 486 T^{2} + 315 T^{3} + 90 T^{4} + T^{6}$$
$59$ $$6561 - 6561 T + 729 T^{2} + 648 T^{3} + 162 T^{4} + 9 T^{5} + T^{6}$$
$61$ $$94249 + 12894 T + 1821 T^{2} + 161 T^{3} - 12 T^{4} + 3 T^{5} + T^{6}$$
$67$ $$2809 + 1272 T + 5502 T^{2} - 224 T^{3} - 66 T^{4} - 3 T^{5} + T^{6}$$
$71$ $$263169 - 13851 T + 9720 T^{2} - 1728 T^{3} - 27 T^{4} + 9 T^{5} + T^{6}$$
$73$ $$54289 - 32853 T + 5754 T^{2} - 244 T^{3} + 78 T^{4} + 12 T^{5} + T^{6}$$
$79$ $$2809 + 14151 T + 30144 T^{2} - 3356 T^{3} + 231 T^{4} - 21 T^{5} + T^{6}$$
$83$ $$408321 - 92016 T + 26487 T^{2} + 18 T^{3} + 225 T^{4} - 9 T^{5} + T^{6}$$
$89$ $$431649 - 5913 T - 4374 T^{2} + 900 T^{3} + 90 T^{4} + T^{6}$$
$97$ $$2679769 + 751383 T + 141156 T^{2} + 17675 T^{3} + 1323 T^{4} + 54 T^{5} + T^{6}$$