# Properties

 Label 912.2.bn.k Level $912$ Weight $2$ Character orbit 912.bn Analytic conductor $7.282$ Analytic rank $0$ Dimension $4$ 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.bn (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 228) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{3} ) q^{3} + ( -2 + \beta_{1} - \beta_{2} ) q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{3} ) q^{3} + ( -2 + \beta_{1} - \beta_{2} ) q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{9} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{11} + ( 2 - \beta_{2} ) q^{13} + ( 4 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{15} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{17} + ( -3 - 2 \beta_{2} ) q^{19} + ( -5 - \beta_{1} + 2 \beta_{2} ) q^{21} + ( -3 + 2 \beta_{2} + \beta_{3} ) q^{23} + ( 3 - 6 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{25} + ( 5 - 2 \beta_{1} + 2 \beta_{3} ) q^{27} + ( 1 - 2 \beta_{1} - 8 \beta_{2} + \beta_{3} ) q^{29} + ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{31} + ( 2 - 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{33} + ( -4 + 4 \beta_{1} ) q^{35} + ( -2 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{37} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{39} + ( -3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{41} + ( -3 - 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{43} + ( -6 - 3 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{45} + ( 5 - 2 \beta_{2} + \beta_{3} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{49} + ( 3 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{51} + ( 1 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{53} + ( 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{55} + ( 2 - 2 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} ) q^{57} + ( -4 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{59} + ( 2 - 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{61} + ( -5 + 2 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} ) q^{63} + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{65} + ( 2 - \beta_{2} ) q^{67} + ( 1 + 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{69} + ( 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} ) q^{71} + ( -9 + 2 \beta_{1} + 7 \beta_{2} - 4 \beta_{3} ) q^{73} + ( -12 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{75} + ( -8 + 4 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} ) q^{77} + ( -3 + 6 \beta_{1} + 3 \beta_{2} ) q^{79} + ( -\beta_{2} + 5 \beta_{3} ) q^{81} + ( -2 - 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -1 + 2 \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{85} + ( 5 - 8 \beta_{1} - 10 \beta_{2} + \beta_{3} ) q^{87} + ( 1 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{89} + ( -1 - \beta_{2} - 3 \beta_{3} ) q^{91} + ( -3 + 3 \beta_{1} - 6 \beta_{2} ) q^{93} + ( 4 - 5 \beta_{1} + 9 \beta_{2} + 2 \beta_{3} ) q^{95} + ( -6 + 9 \beta_{1} + 3 \beta_{2} ) q^{97} + ( 6 - 6 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{3} - 9 q^{5} - 2 q^{7} + 5 q^{9} + O(q^{10})$$ $$4 q + q^{3} - 9 q^{5} - 2 q^{7} + 5 q^{9} + 6 q^{13} + 11 q^{15} + 15 q^{17} - 16 q^{19} - 17 q^{21} - 9 q^{23} + 9 q^{25} + 16 q^{27} - 15 q^{29} - 8 q^{33} - 12 q^{35} + 3 q^{39} - 3 q^{41} - 8 q^{43} - 17 q^{45} + 15 q^{47} + 6 q^{49} + 11 q^{51} + 9 q^{53} + 2 q^{55} - q^{57} - 9 q^{59} + 8 q^{61} - 19 q^{63} - 18 q^{65} + 6 q^{67} + q^{69} + 3 q^{71} - 16 q^{73} - 54 q^{75} - 7 q^{81} - 17 q^{85} - 9 q^{87} + 9 q^{89} - 3 q^{91} - 21 q^{93} + 27 q^{95} - 9 q^{97} - 4 q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + 2 \beta_{1} + 3$$

## Character values

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

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 −1.18614 − 1.26217i 1.68614 + 0.396143i −1.18614 + 1.26217i 1.68614 − 0.396143i
0 −1.18614 + 1.26217i 0 −3.68614 2.12819i 0 2.37228 0 −0.186141 2.99422i 0
65.2 0 1.68614 0.396143i 0 −0.813859 0.469882i 0 −3.37228 0 2.68614 1.33591i 0
449.1 0 −1.18614 1.26217i 0 −3.68614 + 2.12819i 0 2.37228 0 −0.186141 + 2.99422i 0
449.2 0 1.68614 + 0.396143i 0 −0.813859 + 0.469882i 0 −3.37228 0 2.68614 + 1.33591i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.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.bn.k 4
3.b odd 2 1 912.2.bn.l 4
4.b odd 2 1 228.2.p.d yes 4
12.b even 2 1 228.2.p.c 4
19.d odd 6 1 912.2.bn.l 4
57.f even 6 1 inner 912.2.bn.k 4
76.f even 6 1 228.2.p.c 4
228.n odd 6 1 228.2.p.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.p.c 4 12.b even 2 1
228.2.p.c 4 76.f even 6 1
228.2.p.d yes 4 4.b odd 2 1
228.2.p.d yes 4 228.n odd 6 1
912.2.bn.k 4 1.a even 1 1 trivial
912.2.bn.k 4 57.f even 6 1 inner
912.2.bn.l 4 3.b odd 2 1
912.2.bn.l 4 19.d odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 9 T_{5}^{3} + 31 T_{5}^{2} + 36 T_{5} + 16$$ $$T_{7}^{2} + T_{7} - 8$$ $$T_{17}^{4} - 15 T_{17}^{3} + 91 T_{17}^{2} - 240 T_{17} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 - 3 T - 2 T^{2} - T^{3} + T^{4}$$
$5$ $$16 + 36 T + 31 T^{2} + 9 T^{3} + T^{4}$$
$7$ $$( -8 + T + T^{2} )^{2}$$
$11$ $$64 + 28 T^{2} + T^{4}$$
$13$ $$( 3 - 3 T + T^{2} )^{2}$$
$17$ $$256 - 240 T + 91 T^{2} - 15 T^{3} + T^{4}$$
$19$ $$( 19 + 8 T + T^{2} )^{2}$$
$23$ $$16 + 36 T + 31 T^{2} + 9 T^{3} + T^{4}$$
$29$ $$2304 + 720 T + 177 T^{2} + 15 T^{3} + T^{4}$$
$31$ $$324 + 63 T^{2} + T^{4}$$
$37$ $$576 + 51 T^{2} + T^{4}$$
$41$ $$5184 - 216 T + 81 T^{2} + 3 T^{3} + T^{4}$$
$43$ $$289 - 136 T + 81 T^{2} + 8 T^{3} + T^{4}$$
$47$ $$256 - 240 T + 91 T^{2} - 15 T^{3} + T^{4}$$
$53$ $$144 - 108 T + 69 T^{2} - 9 T^{3} + T^{4}$$
$59$ $$144 + 108 T + 69 T^{2} + 9 T^{3} + T^{4}$$
$61$ $$289 + 136 T + 81 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$( 3 - 3 T + T^{2} )^{2}$$
$71$ $$5184 + 216 T + 81 T^{2} - 3 T^{3} + T^{4}$$
$73$ $$961 + 496 T + 225 T^{2} + 16 T^{3} + T^{4}$$
$79$ $$9801 - 99 T^{2} + T^{4}$$
$83$ $$256 + 76 T^{2} + T^{4}$$
$89$ $$144 - 108 T + 69 T^{2} - 9 T^{3} + T^{4}$$
$97$ $$46656 - 1944 T - 189 T^{2} + 9 T^{3} + T^{4}$$