# Properties

 Label 912.2.bo.d 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{18} q^{3} + ( 1 - \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{5} + ( 1 - \zeta_{18} + 3 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{7} + \zeta_{18}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{18} q^{3} + ( 1 - \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{5} + ( 1 - \zeta_{18} + 3 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{7} + \zeta_{18}^{2} q^{9} + ( -\zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{11} + ( 1 + \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{13} + ( -\zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{15} + ( -2 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{17} + ( 5 \zeta_{18} - 3 \zeta_{18}^{4} ) q^{19} + ( -2 - \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{21} + ( -6 + 3 \zeta_{18} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{23} + ( -4 - 4 \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{25} -\zeta_{18}^{3} q^{27} + ( -\zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{29} + ( 5 - \zeta_{18} - 5 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{31} + ( -1 + \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{33} + ( -7 \zeta_{18} - 4 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{5} ) q^{35} + ( -1 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - 4 \zeta_{18}^{4} ) q^{37} + ( 2 - \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{39} + ( -3 - 3 \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{41} + ( -3 + 3 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{43} + ( 2 + \zeta_{18}^{2} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{45} + ( -4 + 3 \zeta_{18} + 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 4 \zeta_{18}^{4} ) q^{47} + ( 2 \zeta_{18} - \zeta_{18}^{2} - 8 \zeta_{18}^{3} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{49} + ( 2 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{51} + ( -2 \zeta_{18} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 9 \zeta_{18}^{5} ) q^{53} + ( -8 - 4 \zeta_{18} - 7 \zeta_{18}^{2} + 7 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 8 \zeta_{18}^{5} ) q^{55} + ( -5 \zeta_{18}^{2} + 3 \zeta_{18}^{5} ) q^{57} + ( 4 - 6 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} + 6 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{59} + ( -3 + 2 \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{61} + ( 2 + 2 \zeta_{18} + \zeta_{18}^{2} - 3 \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{63} + ( 3 \zeta_{18} - 5 \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 5 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{65} + ( 6 - 11 \zeta_{18} - 4 \zeta_{18}^{2} - 11 \zeta_{18}^{3} + 6 \zeta_{18}^{4} ) q^{67} + ( 2 + 6 \zeta_{18} - 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{69} + ( -3 + 3 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 7 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{71} + ( 3 - 5 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{73} + ( 1 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - 3 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{75} + ( -7 - 6 \zeta_{18} - 6 \zeta_{18}^{2} - 4 \zeta_{18}^{4} + 10 \zeta_{18}^{5} ) q^{77} + ( -5 + 5 \zeta_{18} - 4 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{79} + \zeta_{18}^{4} q^{81} + ( -1 + 4 \zeta_{18} - 3 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{83} + ( 3 - 2 \zeta_{18} + 5 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{85} + ( \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{87} + ( 7 + 7 \zeta_{18} - \zeta_{18}^{2} - 7 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{89} + ( 8 - 6 \zeta_{18} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{91} + ( 1 - 5 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + 5 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{93} + ( -3 + 2 \zeta_{18} - 6 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 5 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{95} + ( 7 + 4 \zeta_{18} + 8 \zeta_{18}^{2} - 8 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 7 \zeta_{18}^{5} ) q^{97} + ( 2 + \zeta_{18} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 3q^{5} + 3q^{7} + O(q^{10})$$ $$6q + 3q^{5} + 3q^{7} + 9q^{13} + 3q^{15} - 3q^{17} - 15q^{21} - 27q^{23} - 15q^{25} - 3q^{27} - 3q^{29} + 15q^{31} + 3q^{33} - 12q^{35} - 6q^{37} + 12q^{39} - 15q^{41} - 3q^{43} + 6q^{45} - 15q^{47} - 24q^{49} - 3q^{51} + 6q^{53} - 27q^{55} + 27q^{59} - 15q^{61} + 3q^{63} + 12q^{65} + 3q^{67} + 6q^{69} - 3q^{71} + 12q^{73} + 6q^{75} - 42q^{77} - 27q^{79} - 3q^{83} + 12q^{85} - 6q^{87} + 42q^{89} + 42q^{91} + 3q^{93} - 24q^{95} + 18q^{97} + 3q^{99} + O(q^{100})$$

## 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 −0.0923963 + 0.0775297i 0 −2.14543 + 3.71599i 0 −0.939693 + 0.342020i 0
385.1 0 0.173648 0.984808i 0 −0.0923963 0.0775297i 0 −2.14543 3.71599i 0 −0.939693 0.342020i 0
481.1 0 −0.939693 0.342020i 0 −0.613341 3.47843i 0 1.85844 3.21891i 0 0.766044 + 0.642788i 0
529.1 0 −0.939693 + 0.342020i 0 −0.613341 + 3.47843i 0 1.85844 + 3.21891i 0 0.766044 0.642788i 0
625.1 0 0.766044 + 0.642788i 0 2.20574 0.802823i 0 1.78699 + 3.09516i 0 0.173648 + 0.984808i 0
769.1 0 0.766044 0.642788i 0 2.20574 + 0.802823i 0 1.78699 3.09516i 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.d 6
4.b odd 2 1 114.2.i.c 6
12.b even 2 1 342.2.u.b 6
19.e even 9 1 inner 912.2.bo.d 6
76.k even 18 1 2166.2.a.p 3
76.l odd 18 1 114.2.i.c 6
76.l odd 18 1 2166.2.a.r 3
228.u odd 18 1 6498.2.a.bu 3
228.v even 18 1 342.2.u.b 6
228.v even 18 1 6498.2.a.bp 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.c 6 4.b odd 2 1
114.2.i.c 6 76.l odd 18 1
342.2.u.b 6 12.b even 2 1
342.2.u.b 6 228.v even 18 1
912.2.bo.d 6 1.a even 1 1 trivial
912.2.bo.d 6 19.e even 9 1 inner
2166.2.a.p 3 76.k even 18 1
2166.2.a.r 3 76.l odd 18 1
6498.2.a.bp 3 228.v even 18 1
6498.2.a.bu 3 228.u odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 3 T_{5}^{5} + 12 T_{5}^{4} - 46 T_{5}^{3} + 60 T_{5}^{2} + 12 T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(912, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$1 + T^{3} + T^{6}$$
$5$ $$1 + 12 T + 60 T^{2} - 46 T^{3} + 12 T^{4} - 3 T^{5} + T^{6}$$
$7$ $$3249 - 1026 T + 495 T^{2} - 60 T^{3} + 27 T^{4} - 3 T^{5} + T^{6}$$
$11$ $$1369 - 777 T + 441 T^{2} - 74 T^{3} + 21 T^{4} + T^{6}$$
$13$ $$1 - 9 T + 27 T^{2} - 28 T^{3} + 36 T^{4} - 9 T^{5} + T^{6}$$
$17$ $$9 - 27 T + 9 T^{2} + 24 T^{3} + 18 T^{4} + 3 T^{5} + T^{6}$$
$19$ $$6859 - 107 T^{3} + T^{6}$$
$23$ $$157609 + 85752 T + 22626 T^{2} + 3556 T^{3} + 378 T^{4} + 27 T^{5} + T^{6}$$
$29$ $$1 - 12 T + 60 T^{2} + 46 T^{3} + 12 T^{4} + 3 T^{5} + T^{6}$$
$31$ $$11881 - 7848 T + 3549 T^{2} - 862 T^{3} + 153 T^{4} - 15 T^{5} + T^{6}$$
$37$ $$( 17 - 45 T + 3 T^{2} + T^{3} )^{2}$$
$41$ $$2809 + 2703 T + 1308 T^{2} + 316 T^{3} + 87 T^{4} + 15 T^{5} + T^{6}$$
$43$ $$63001 - 17319 T + 2406 T^{2} - 280 T^{3} - 3 T^{4} + 3 T^{5} + T^{6}$$
$47$ $$5041 - 1491 T + 570 T^{2} - 80 T^{3} + 51 T^{4} + 15 T^{5} + T^{6}$$
$53$ $$395641 + 18870 T + 6186 T^{2} + 1205 T^{3} - 30 T^{4} - 6 T^{5} + T^{6}$$
$59$ $$81 + 567 T + 1701 T^{2} + 72 T^{3} + 198 T^{4} - 27 T^{5} + T^{6}$$
$61$ $$289 + 408 T + 375 T^{2} + 215 T^{3} + 84 T^{4} + 15 T^{5} + T^{6}$$
$67$ $$7017201 - 619866 T + 10890 T^{2} - 834 T^{3} + 72 T^{4} - 3 T^{5} + T^{6}$$
$71$ $$26569 - 9291 T + 14946 T^{2} - 1304 T^{3} - 87 T^{4} + 3 T^{5} + T^{6}$$
$73$ $$3249 + 1539 T + 522 T^{2} + 300 T^{3} + 54 T^{4} - 12 T^{5} + T^{6}$$
$79$ $$32761 + 27693 T + 9918 T^{2} + 2152 T^{3} + 351 T^{4} + 27 T^{5} + T^{6}$$
$83$ $$2601 - 1836 T + 1143 T^{2} - 210 T^{3} + 45 T^{4} + 3 T^{5} + T^{6}$$
$89$ $$239121 - 92421 T + 35406 T^{2} - 7104 T^{3} + 756 T^{4} - 42 T^{5} + T^{6}$$
$97$ $$218089 + 121887 T + 19332 T^{2} + 451 T^{3} + 171 T^{4} - 18 T^{5} + T^{6}$$