# Properties

 Label 912.2.bb.e Level $912$ Weight $2$ Character orbit 912.bb 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.bb (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.954288.1 Defining polynomial: $$x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + ( \beta_{3} + \beta_{4} ) q^{5} + ( 1 + \beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{7} + ( -1 - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + ( \beta_{3} + \beta_{4} ) q^{5} + ( 1 + \beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{7} + ( -1 - \beta_{3} ) q^{9} + ( \beta_{1} - \beta_{5} ) q^{11} + ( -\beta_{1} - \beta_{2} - \beta_{4} ) q^{13} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{15} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{17} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{19} + ( -2 - \beta_{3} + \beta_{5} ) q^{21} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{23} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{25} + q^{27} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{29} + \beta_{2} q^{31} + \beta_{5} q^{33} + ( 4 \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{35} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{37} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{39} + ( -4 + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{41} + ( -8 + 2 \beta_{2} - 4 \beta_{3} - \beta_{4} ) q^{43} + ( 1 - \beta_{2} ) q^{45} + ( -2 + 4 \beta_{1} + 2 \beta_{3} ) q^{47} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} ) q^{49} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{51} + ( 4 + \beta_{1} - 4 \beta_{3} ) q^{53} + ( 2 + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{55} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{57} + ( -\beta_{3} - \beta_{4} ) q^{59} + ( 2 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} ) q^{61} + ( 1 - \beta_{1} - \beta_{3} ) q^{63} + ( 5 + 4 \beta_{1} - \beta_{2} + 10 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{65} + ( 7 + \beta_{1} - 2 \beta_{2} + 7 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{67} + ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{69} + ( 2 \beta_{1} + 5 \beta_{3} - \beta_{4} - \beta_{5} ) q^{71} + ( 4 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{73} + ( 2 - \beta_{1} - \beta_{2} - \beta_{5} ) q^{75} + ( -5 - \beta_{2} ) q^{77} + ( -4 \beta_{3} - 3 \beta_{4} ) q^{79} + \beta_{3} q^{81} + ( 1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{83} + ( -4 - 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{85} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{87} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{89} + ( 2 - 5 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} ) q^{91} -\beta_{4} q^{93} + ( -4 + 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} ) q^{95} + ( -6 + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{97} -\beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 3q^{3} - 2q^{5} - 3q^{9} + O(q^{10})$$ $$6q - 3q^{3} - 2q^{5} - 3q^{9} - 3q^{13} - 2q^{15} - 2q^{17} - 4q^{19} - 9q^{21} + 12q^{23} - 5q^{25} + 6q^{27} + 6q^{29} + 2q^{31} + 6q^{35} - 12q^{41} - 33q^{43} + 4q^{45} - 18q^{47} - 8q^{49} - 2q^{51} + 36q^{53} + 12q^{55} - q^{57} + 2q^{59} + 3q^{61} + 9q^{63} + 19q^{67} - 16q^{71} + 11q^{73} + 10q^{75} - 32q^{77} + 9q^{79} - 3q^{81} - 8q^{85} + 6q^{89} + q^{91} - q^{93} - 26q^{95} - 24q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 5 \nu^{4} + \nu^{3} + 9 \nu^{2} + 21 \nu - 45$$$$)/27$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 4 \nu^{4} + \nu^{3} + 18 \nu^{2} - 33 \nu + 9$$$$)/27$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{5} - \nu^{4} - 2 \nu^{3} + 12 \nu + 9$$$$)/27$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{5} - 8 \nu^{4} + 2 \nu^{3} + 9 \nu^{2} - 12 \nu + 18$$$$)/27$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{5} - \nu^{4} + 4 \nu^{3} - 3 \nu^{2} + 12 \nu + 27$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{5} - 8 \beta_{3} + \beta_{2} + \beta_{1} - 5$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_{1} + 4$$ $$\nu^{5}$$ $$=$$ $$($$$$-3 \beta_{5} + 9 \beta_{4} - 11 \beta_{3} - 8 \beta_{2} + 4 \beta_{1} + 16$$$$)/2$$

## Character values

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

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$1 + \beta_{3}$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.403374 − 1.68443i 1.71903 + 0.211943i −1.62241 + 0.606458i 0.403374 + 1.68443i 1.71903 − 0.211943i −1.62241 − 0.606458i
0 −0.500000 0.866025i 0 −1.66044 2.87597i 0 2.71781i 0 −0.500000 + 0.866025i 0
31.2 0 −0.500000 0.866025i 0 −0.675970 1.17081i 0 1.45735i 0 −0.500000 + 0.866025i 0
31.3 0 −0.500000 0.866025i 0 1.33641 + 2.31473i 0 3.93569i 0 −0.500000 + 0.866025i 0
559.1 0 −0.500000 + 0.866025i 0 −1.66044 + 2.87597i 0 2.71781i 0 −0.500000 0.866025i 0
559.2 0 −0.500000 + 0.866025i 0 −0.675970 + 1.17081i 0 1.45735i 0 −0.500000 0.866025i 0
559.3 0 −0.500000 + 0.866025i 0 1.33641 2.31473i 0 3.93569i 0 −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 559.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bb.e 6
3.b odd 2 1 2736.2.bm.n 6
4.b odd 2 1 912.2.bb.f yes 6
12.b even 2 1 2736.2.bm.o 6
19.d odd 6 1 912.2.bb.f yes 6
57.f even 6 1 2736.2.bm.o 6
76.f even 6 1 inner 912.2.bb.e 6
228.n odd 6 1 2736.2.bm.n 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.bb.e 6 1.a even 1 1 trivial
912.2.bb.e 6 76.f even 6 1 inner
912.2.bb.f yes 6 4.b odd 2 1
912.2.bb.f yes 6 19.d odd 6 1
2736.2.bm.n 6 3.b odd 2 1
2736.2.bm.n 6 228.n odd 6 1
2736.2.bm.o 6 12.b even 2 1
2736.2.bm.o 6 57.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{6} + 2 T_{5}^{5} + 12 T_{5}^{4} + 8 T_{5}^{3} + 88 T_{5}^{2} + 96 T_{5} + 144$$ $$T_{23}^{6} - 12 T_{23}^{5} + 28 T_{23}^{4} + 240 T_{23}^{3} - 416 T_{23}^{2} - 4080 T_{23} + 13872$$ $$T_{31}^{3} - T_{31}^{2} - 9 T_{31} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 1 + T + T^{2} )^{3}$$
$5$ $$144 + 96 T + 88 T^{2} + 8 T^{3} + 12 T^{4} + 2 T^{5} + T^{6}$$
$7$ $$243 + 163 T^{2} + 25 T^{4} + T^{6}$$
$11$ $$48 + 64 T^{2} + 16 T^{4} + T^{6}$$
$13$ $$2883 + 2139 T + 436 T^{2} - 69 T^{3} - 20 T^{4} + 3 T^{5} + T^{6}$$
$17$ $$576 + 480 T + 448 T^{2} + 8 T^{3} + 24 T^{4} + 2 T^{5} + T^{6}$$
$19$ $$6859 + 1444 T + 323 T^{2} + 136 T^{3} + 17 T^{4} + 4 T^{5} + T^{6}$$
$23$ $$13872 - 4080 T - 416 T^{2} + 240 T^{3} + 28 T^{4} - 12 T^{5} + T^{6}$$
$29$ $$192 + 480 T + 448 T^{2} + 120 T^{3} - 8 T^{4} - 6 T^{5} + T^{6}$$
$31$ $$( -3 - 9 T - T^{2} + T^{3} )^{2}$$
$37$ $$328683 + 14971 T^{2} + 217 T^{4} + T^{6}$$
$41$ $$248832 + 82944 T + 5760 T^{2} - 1152 T^{3} - 48 T^{4} + 12 T^{5} + T^{6}$$
$43$ $$2187 + 7533 T + 9540 T^{2} + 3069 T^{3} + 456 T^{4} + 33 T^{5} + T^{6}$$
$47$ $$714432 + 134688 T - 320 T^{2} - 1656 T^{3} + 16 T^{4} + 18 T^{5} + T^{6}$$
$53$ $$80688 - 66912 T + 24400 T^{2} - 4896 T^{3} + 568 T^{4} - 36 T^{5} + T^{6}$$
$59$ $$144 - 96 T + 88 T^{2} - 8 T^{3} + 12 T^{4} - 2 T^{5} + T^{6}$$
$61$ $$638401 - 112659 T + 22278 T^{2} - 1175 T^{3} + 150 T^{4} - 3 T^{5} + T^{6}$$
$67$ $$10201 - 8383 T + 4970 T^{2} - 1375 T^{3} + 278 T^{4} - 19 T^{5} + T^{6}$$
$71$ $$2304 - 1920 T + 2368 T^{2} + 736 T^{3} + 216 T^{4} + 16 T^{5} + T^{6}$$
$73$ $$33489 - 8235 T + 4038 T^{2} + 129 T^{3} + 166 T^{4} - 11 T^{5} + T^{6}$$
$79$ $$151321 - 22173 T + 6750 T^{2} - 265 T^{3} + 138 T^{4} - 9 T^{5} + T^{6}$$
$83$ $$15552 + 4464 T^{2} + 132 T^{4} + T^{6}$$
$89$ $$215472 + 112560 T + 21208 T^{2} + 840 T^{3} - 128 T^{4} - 6 T^{5} + T^{6}$$
$97$ $$62208 - 6912 T - 3200 T^{2} + 384 T^{3} + 208 T^{4} + 24 T^{5} + T^{6}$$