# Properties

 Label 912.2.bn.h 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + \beta_{1} q^{5} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + \beta_{1} q^{5} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{9} + ( 1 - 2 \beta_{2} + \beta_{3} ) q^{11} + ( -4 + 2 \beta_{2} ) q^{13} + ( 2 - \beta_{3} ) q^{15} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{17} + ( 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{19} + ( -4 - \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{21} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{23} -3 \beta_{2} q^{25} + ( -5 + \beta_{3} ) q^{27} + ( -\beta_{1} - \beta_{3} ) q^{29} -3 \beta_{3} q^{31} + ( -2 + 3 \beta_{2} - 2 \beta_{3} ) q^{33} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{35} + ( -2 + 4 \beta_{2} - 3 \beta_{3} ) q^{37} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{39} + ( -3 + 3 \beta_{2} ) q^{41} + ( -4 + 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{43} + ( \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{45} + ( -4 + 7 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} ) q^{47} + ( 3 + 8 \beta_{1} - 4 \beta_{3} ) q^{49} + ( -6 - 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{51} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{55} + ( 7 - 2 \beta_{1} - 7 \beta_{2} ) q^{57} + ( -9 + \beta_{1} + 9 \beta_{2} - 2 \beta_{3} ) q^{59} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{61} + ( 2 + 3 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{63} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{65} + ( 2 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{67} + ( -10 + 5 \beta_{1} + 10 \beta_{2} ) q^{69} + ( -6 + 6 \beta_{2} ) q^{71} + ( 1 + 4 \beta_{1} - \beta_{2} - 8 \beta_{3} ) q^{73} + ( -3 - 3 \beta_{1} + 3 \beta_{2} ) q^{75} + \beta_{3} q^{77} -6 \beta_{1} q^{79} + ( -4 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{81} + ( 9 - 18 \beta_{2} + \beta_{3} ) q^{83} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{85} + ( -2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{87} + ( 2 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 8 + 6 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{91} + ( -3 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{93} + ( 6 + \beta_{3} ) q^{95} + ( -3 - 6 \beta_{1} - 3 \beta_{2} ) q^{97} + ( 3 - \beta_{1} - 5 \beta_{2} + 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} - 8q^{7} + 2q^{9} + O(q^{10})$$ $$4q - 2q^{3} - 8q^{7} + 2q^{9} - 12q^{13} + 8q^{15} - 12q^{17} + 2q^{19} - 8q^{21} - 6q^{25} - 20q^{27} - 2q^{33} - 12q^{35} + 12q^{39} - 6q^{41} - 8q^{43} - 8q^{45} - 12q^{47} + 12q^{49} - 16q^{51} - 12q^{53} - 4q^{55} + 14q^{57} - 18q^{59} + 8q^{61} + 20q^{63} + 6q^{67} - 20q^{69} - 12q^{71} + 2q^{73} - 6q^{75} + 14q^{81} - 8q^{85} - 12q^{87} + 24q^{89} + 24q^{91} - 12q^{93} + 24q^{95} - 18q^{97} + 2q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## 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
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
0 −1.72474 0.158919i 0 −1.22474 0.707107i 0 0.449490 0 2.94949 + 0.548188i 0
65.2 0 0.724745 1.57313i 0 1.22474 + 0.707107i 0 −4.44949 0 −1.94949 2.28024i 0
449.1 0 −1.72474 + 0.158919i 0 −1.22474 + 0.707107i 0 0.449490 0 2.94949 0.548188i 0
449.2 0 0.724745 + 1.57313i 0 1.22474 0.707107i 0 −4.44949 0 −1.94949 + 2.28024i 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.h 4
3.b odd 2 1 912.2.bn.g 4
4.b odd 2 1 114.2.h.f yes 4
12.b even 2 1 114.2.h.e 4
19.d odd 6 1 912.2.bn.g 4
57.f even 6 1 inner 912.2.bn.h 4
76.f even 6 1 114.2.h.e 4
228.n odd 6 1 114.2.h.f yes 4

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

## Hecke kernels

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

 $$T_{5}^{4} - 2 T_{5}^{2} + 4$$ $$T_{7}^{2} + 4 T_{7} - 2$$ $$T_{17}^{4} + 12 T_{17}^{3} + 52 T_{17}^{2} + 48 T_{17} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 6 T + T^{2} + 2 T^{3} + T^{4}$$
$5$ $$4 - 2 T^{2} + T^{4}$$
$7$ $$( -2 + 4 T + T^{2} )^{2}$$
$11$ $$1 + 10 T^{2} + T^{4}$$
$13$ $$( 12 + 6 T + T^{2} )^{2}$$
$17$ $$16 + 48 T + 52 T^{2} + 12 T^{3} + T^{4}$$
$19$ $$361 - 38 T - 15 T^{2} - 2 T^{3} + T^{4}$$
$23$ $$2500 - 50 T^{2} + T^{4}$$
$29$ $$36 + 6 T^{2} + T^{4}$$
$31$ $$( 18 + T^{2} )^{2}$$
$37$ $$36 + 60 T^{2} + T^{4}$$
$41$ $$( 9 + 3 T + T^{2} )^{2}$$
$43$ $$64 - 64 T + 72 T^{2} + 8 T^{3} + T^{4}$$
$47$ $$7396 - 1032 T - 38 T^{2} + 12 T^{3} + T^{4}$$
$53$ $$144 + 144 T + 132 T^{2} + 12 T^{3} + T^{4}$$
$59$ $$5625 + 1350 T + 249 T^{2} + 18 T^{3} + T^{4}$$
$61$ $$100 - 80 T + 54 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$225 + 90 T - 3 T^{2} - 6 T^{3} + T^{4}$$
$71$ $$( 36 + 6 T + T^{2} )^{2}$$
$73$ $$9025 + 190 T + 99 T^{2} - 2 T^{3} + T^{4}$$
$79$ $$5184 - 72 T^{2} + T^{4}$$
$83$ $$58081 + 490 T^{2} + T^{4}$$
$89$ $$14400 - 2880 T + 456 T^{2} - 24 T^{3} + T^{4}$$
$97$ $$2025 - 810 T + 63 T^{2} + 18 T^{3} + T^{4}$$