# Properties

 Label 912.2.bo.f 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: $$x^{6} - x^{3} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{18} q^{3} + (2 \zeta_{18}^{5} + \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{5} + (\zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} - 1) q^{7} + \zeta_{18}^{2} q^{9} +O(q^{10})$$ q - z * q^3 + (2*z^5 + z^3 - z^2 + 1) * q^5 + (z^3 - z^2 + z - 1) * q^7 + z^2 * q^9 $$q - \zeta_{18} q^{3} + (2 \zeta_{18}^{5} + \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{5} + (\zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} - 1) q^{7} + \zeta_{18}^{2} q^{9} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{11} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{3} + 2 \zeta_{18}^{2} - \zeta_{18} - 1) q^{13} + ( - \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18} + 2) q^{15} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 2 \zeta_{18} - 2) q^{17} + (2 \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18} + 2) q^{19} + ( - \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{21} + (2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 5 \zeta_{18} + 2) q^{23} + (\zeta_{18}^{5} - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18}^{2}) q^{25} - \zeta_{18}^{3} q^{27} + (2 \zeta_{18}^{4} - 5 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 5 \zeta_{18} + 2) q^{29} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18} - 3) q^{31} + (\zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{33} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18} - 4) q^{35} + ( - 4 \zeta_{18}^{5} + 6 \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18} - 3) q^{37} + ( - \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 2) q^{39} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{3} - 4 \zeta_{18}^{2} + 3 \zeta_{18} - 1) q^{41} + ( - 2 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{43} + (\zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} - 2 \zeta_{18}) q^{45} + (3 \zeta_{18}^{3} - \zeta_{18}^{2} + 3 \zeta_{18}) q^{47} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 2 \zeta_{18}) q^{49} + (2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{51} + ( - \zeta_{18}^{5} + 8 \zeta_{18}^{4} + 8 \zeta_{18}^{3} - 6 \zeta_{18} - 2) q^{53} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{3} + \zeta_{18}^{2} + 2) q^{55} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{2} - 2 \zeta_{18} + 2) q^{57} + (6 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 4 \zeta_{18} - 6) q^{59} + (11 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 1) q^{61} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2}) q^{63} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{2} - 3 \zeta_{18}) q^{65} + ( - 4 \zeta_{18}^{4} + \zeta_{18}^{3} + 4 \zeta_{18}^{2} + \zeta_{18} - 4) q^{67} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - 2 \zeta_{18} + 2) q^{69} + ( - 6 \zeta_{18}^{5} + 5 \zeta_{18}^{4} + \zeta_{18}^{3} + 7 \zeta_{18}^{2} - 7) q^{71} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - \zeta_{18} + 7) q^{73} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 1) q^{75} + q^{77} + ( - 5 \zeta_{18}^{5} + 5 \zeta_{18}^{3} + 10 \zeta_{18}^{2} - 5 \zeta_{18} + 5) q^{79} + \zeta_{18}^{4} q^{81} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 9 \zeta_{18}^{3} - \zeta_{18}^{2} - 2 \zeta_{18} + 9) q^{83} + ( - 7 \zeta_{18}^{4} + 8 \zeta_{18}^{3} - 9 \zeta_{18}^{2} + 8 \zeta_{18} - 7) q^{85} + ( - 2 \zeta_{18}^{5} + 5 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - 2 \zeta_{18}) q^{87} + (5 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 5 \zeta_{18}^{2} - 7 \zeta_{18} - 7) q^{89} + (\zeta_{18}^{5} - 2 \zeta_{18} + 2) q^{91} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{93} + (8 \zeta_{18}^{5} - \zeta_{18}^{4} + 6 \zeta_{18}^{3} - 10 \zeta_{18}^{2} + 2 \zeta_{18} - 3) q^{95} + (\zeta_{18}^{5} - 4 \zeta_{18}^{4} + 4 \zeta_{18} - 1) q^{97} + ( - \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}) q^{99} +O(q^{100})$$ q - z * q^3 + (2*z^5 + z^3 - z^2 + 1) * q^5 + (z^3 - z^2 + z - 1) * q^7 + z^2 * q^9 + (-z^5 - z) * q^11 + (-2*z^5 + z^3 + 2*z^2 - z - 1) * q^13 + (-z^4 - z^3 - z + 2) * q^15 + (2*z^5 - 2*z^4 + 3*z^3 - 3*z^2 + 2*z - 2) * q^17 + (2*z^5 + z^4 + 2*z^3 - 2*z^2 + z + 2) * q^19 + (-z^4 + z^3 - z^2 + z) * q^21 + (2*z^5 + 3*z^4 + 3*z^3 - 5*z + 2) * q^23 + (z^5 - 3*z^4 + 3*z^3 - z^2) * q^25 - z^3 * q^27 + (2*z^4 - 5*z^3 - 2*z^2 - 5*z + 2) * q^29 + (3*z^5 + 3*z^4 + 3*z^3 - 3*z - 3) * q^31 + (z^3 + z^2 - 1) * q^33 + (-2*z^5 + 2*z^3 - 2*z^2 + 3*z - 4) * q^35 + (-4*z^5 + 6*z^4 - 2*z^2 - 2*z - 3) * q^37 + (-z^4 + z^2 + z - 2) * q^39 + (3*z^5 - 3*z^3 - 4*z^2 + 3*z - 1) * q^41 + (-2*z^5 - 5*z^4 - 3*z^3 - z^2 + 1) * q^43 + (z^5 + z^4 + z^2 - 2*z) * q^45 + (3*z^3 - z^2 + 3*z) * q^47 + (-2*z^5 + 3*z^4 + 4*z^3 + 3*z^2 - 2*z) * q^49 + (2*z^5 - 3*z^4 + z^3 - 2*z^2 + 2*z + 2) * q^51 + (-z^5 + 8*z^4 + 8*z^3 - 6*z - 2) * q^53 + (-2*z^5 - z^3 + z^2 + 2) * q^55 + (-z^5 - 2*z^4 - z^2 - 2*z + 2) * q^57 + (6*z^5 + 4*z^4 + 3*z^3 - 3*z^2 - 4*z - 6) * q^59 + (11*z^5 - z^4 - z^3 + 1) * q^61 + (z^5 - z^4 + z^3 - z^2) * q^63 + (-3*z^5 + 3*z^4 + 3*z^2 - 3*z) * q^65 + (-4*z^4 + z^3 + 4*z^2 + z - 4) * q^67 + (-3*z^5 - 3*z^4 - 2*z^3 + 5*z^2 - 2*z + 2) * q^69 + (-6*z^5 + 5*z^4 + z^3 + 7*z^2 - 7) * q^71 + (2*z^5 - 2*z^3 + 5*z^2 - z + 7) * q^73 + (3*z^5 - 3*z^4 + 1) * q^75 + q^77 + (-5*z^5 + 5*z^3 + 10*z^2 - 5*z + 5) * q^79 + z^4 * q^81 + (3*z^5 + 3*z^4 - 9*z^3 - z^2 - 2*z + 9) * q^83 + (-7*z^4 + 8*z^3 - 9*z^2 + 8*z - 7) * q^85 + (-2*z^5 + 5*z^4 + 2*z^3 + 5*z^2 - 2*z) * q^87 + (5*z^5 + 3*z^4 + 4*z^3 - 5*z^2 - 7*z - 7) * q^89 + (z^5 - 2*z + 2) * q^91 + (-3*z^5 - 3*z^4 - 3*z^3 + 3*z^2 + 3*z + 3) * q^93 + (8*z^5 - z^4 + 6*z^3 - 10*z^2 + 2*z - 3) * q^95 + (z^5 - 4*z^4 + 4*z - 1) * q^97 + (-z^4 - z^3 + z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 9 q^{5} - 3 q^{7}+O(q^{10})$$ 6 * q + 9 * q^5 - 3 * q^7 $$6 q + 9 q^{5} - 3 q^{7} - 3 q^{13} + 9 q^{15} - 3 q^{17} + 18 q^{19} + 3 q^{21} + 21 q^{23} + 9 q^{25} - 3 q^{27} - 3 q^{29} - 9 q^{31} - 3 q^{33} - 18 q^{35} - 18 q^{37} - 12 q^{39} - 15 q^{41} - 3 q^{43} + 9 q^{47} + 12 q^{49} + 15 q^{51} + 12 q^{53} + 9 q^{55} + 12 q^{57} - 27 q^{59} + 3 q^{61} + 3 q^{63} - 21 q^{67} + 6 q^{69} - 39 q^{71} + 36 q^{73} + 6 q^{75} + 6 q^{77} + 45 q^{79} + 27 q^{83} - 18 q^{85} + 6 q^{87} - 30 q^{89} + 12 q^{91} + 9 q^{93} - 6 q^{97} - 3 q^{99}+O(q^{100})$$ 6 * q + 9 * q^5 - 3 * q^7 - 3 * q^13 + 9 * q^15 - 3 * q^17 + 18 * q^19 + 3 * q^21 + 21 * q^23 + 9 * q^25 - 3 * q^27 - 3 * q^29 - 9 * q^31 - 3 * q^33 - 18 * q^35 - 18 * q^37 - 12 * q^39 - 15 * q^41 - 3 * q^43 + 9 * q^47 + 12 * q^49 + 15 * q^51 + 12 * q^53 + 9 * q^55 + 12 * q^57 - 27 * q^59 + 3 * q^61 + 3 * q^63 - 21 * q^67 + 6 * q^69 - 39 * q^71 + 36 * q^73 + 6 * q^75 + 6 * q^77 + 45 * q^79 + 27 * q^83 - 18 * q^85 + 6 * q^87 - 30 * q^89 + 12 * q^91 + 9 * q^93 - 6 * q^97 - 3 * q^99

## 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 0.266044 0.460802i 0 −0.939693 + 0.342020i 0
385.1 0 0.173648 0.984808i 0 0.907604 + 0.761570i 0 0.266044 + 0.460802i 0 −0.939693 0.342020i 0
481.1 0 −0.939693 0.342020i 0 0.386659 + 2.19285i 0 −0.326352 + 0.565258i 0 0.766044 + 0.642788i 0
529.1 0 −0.939693 + 0.342020i 0 0.386659 2.19285i 0 −0.326352 0.565258i 0 0.766044 0.642788i 0
625.1 0 0.766044 + 0.642788i 0 3.20574 1.16679i 0 −1.43969 2.49362i 0 0.173648 + 0.984808i 0
769.1 0 0.766044 0.642788i 0 3.20574 + 1.16679i 0 −1.43969 + 2.49362i 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.f 6
4.b odd 2 1 114.2.i.d 6
12.b even 2 1 342.2.u.a 6
19.e even 9 1 inner 912.2.bo.f 6
76.k even 18 1 2166.2.a.u 3
76.l odd 18 1 114.2.i.d 6
76.l odd 18 1 2166.2.a.o 3
228.u odd 18 1 6498.2.a.bn 3
228.v even 18 1 342.2.u.a 6
228.v even 18 1 6498.2.a.bs 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 9T_{5}^{5} + 36T_{5}^{4} - 90T_{5}^{3} + 162T_{5}^{2} - 162T_{5} + 81$$ acting on $$S_{2}^{\mathrm{new}}(912, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + T^{3} + 1$$
$5$ $$T^{6} - 9 T^{5} + 36 T^{4} - 90 T^{3} + \cdots + 81$$
$7$ $$T^{6} + 3 T^{5} + 9 T^{4} + 2 T^{3} + \cdots + 1$$
$11$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$13$ $$T^{6} + 3 T^{5} + 18 T^{4} + 24 T^{3} + \cdots + 9$$
$17$ $$T^{6} + 3 T^{5} + 48 T^{4} + 244 T^{3} + \cdots + 289$$
$19$ $$T^{6} - 18 T^{5} + 162 T^{4} + \cdots + 6859$$
$23$ $$T^{6} - 21 T^{5} + 210 T^{4} + \cdots + 72361$$
$29$ $$T^{6} + 3 T^{5} + 18 T^{4} + \cdots + 45369$$
$31$ $$T^{6} + 9 T^{5} + 81 T^{4} + 54 T^{3} + \cdots + 729$$
$37$ $$(T^{3} + 9 T^{2} - 57 T - 361)^{2}$$
$41$ $$T^{6} + 15 T^{5} + 177 T^{4} + \cdots + 11881$$
$43$ $$T^{6} + 3 T^{5} + 99 T^{4} + \cdots + 3249$$
$47$ $$T^{6} - 9 T^{5} + 63 T^{4} + \cdots + 2809$$
$53$ $$T^{6} - 12 T^{5} + 174 T^{4} + \cdots + 94249$$
$59$ $$T^{6} + 27 T^{5} + 324 T^{4} + \cdots + 289$$
$61$ $$T^{6} - 3 T^{5} - 60 T^{4} + \cdots + 1682209$$
$67$ $$T^{6} + 21 T^{5} + 126 T^{4} + 24 T^{3} + \cdots + 9$$
$71$ $$T^{6} + 39 T^{5} + 561 T^{4} + \cdots + 201601$$
$73$ $$T^{6} - 36 T^{5} + 558 T^{4} + \cdots + 1369$$
$79$ $$T^{6} - 45 T^{5} + 1125 T^{4} + \cdots + 4515625$$
$83$ $$T^{6} - 27 T^{5} + 507 T^{4} + \cdots + 253009$$
$89$ $$T^{6} + 30 T^{5} + 246 T^{4} + \cdots + 11449$$
$97$ $$T^{6} + 6 T^{5} + 3 T^{4} + 35 T^{3} + \cdots + 2809$$