# Properties

 Label 912.2.bn.a Level $912$ Weight $2$ Character orbit 912.bn Analytic conductor $7.282$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 228) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{6} ) q^{3} - q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{6} ) q^{3} - q^{7} + 3 \zeta_{6} q^{9} + ( 2 - \zeta_{6} ) q^{13} + ( 5 - 2 \zeta_{6} ) q^{19} + ( 1 + \zeta_{6} ) q^{21} -5 \zeta_{6} q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -1 + 2 \zeta_{6} ) q^{31} + ( 7 - 14 \zeta_{6} ) q^{37} -3 q^{39} + ( 5 - 5 \zeta_{6} ) q^{43} -6 q^{49} + ( -7 - \zeta_{6} ) q^{57} -13 \zeta_{6} q^{61} -3 \zeta_{6} q^{63} + ( 18 - 9 \zeta_{6} ) q^{67} + ( 7 - 7 \zeta_{6} ) q^{73} + ( -5 + 10 \zeta_{6} ) q^{75} + ( -7 - 7 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -2 + \zeta_{6} ) q^{91} + ( 3 - 3 \zeta_{6} ) q^{93} + ( 8 + 8 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} - 2 q^{7} + 3 q^{9} + O(q^{10})$$ $$2 q - 3 q^{3} - 2 q^{7} + 3 q^{9} + 3 q^{13} + 8 q^{19} + 3 q^{21} - 5 q^{25} - 6 q^{39} + 5 q^{43} - 12 q^{49} - 15 q^{57} - 13 q^{61} - 3 q^{63} + 27 q^{67} + 7 q^{73} - 21 q^{79} - 9 q^{81} - 3 q^{91} + 3 q^{93} + 24 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$\zeta_{6}$$ $$1$$ $$-1$$ $$1$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.50000 0.866025i 0 0 0 −1.00000 0 1.50000 + 2.59808i 0
449.1 0 −1.50000 + 0.866025i 0 0 0 −1.00000 0 1.50000 2.59808i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
19.d odd 6 1 inner
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.a 2
3.b odd 2 1 CM 912.2.bn.a 2
4.b odd 2 1 228.2.p.b 2
12.b even 2 1 228.2.p.b 2
19.d odd 6 1 inner 912.2.bn.a 2
57.f even 6 1 inner 912.2.bn.a 2
76.f even 6 1 228.2.p.b 2
228.n odd 6 1 228.2.p.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.p.b 2 4.b odd 2 1
228.2.p.b 2 12.b even 2 1
228.2.p.b 2 76.f even 6 1
228.2.p.b 2 228.n odd 6 1
912.2.bn.a 2 1.a even 1 1 trivial
912.2.bn.a 2 3.b odd 2 1 CM
912.2.bn.a 2 19.d odd 6 1 inner
912.2.bn.a 2 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 $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}$$ $$T_{7} + 1$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$3 - 3 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$19 - 8 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$3 + T^{2}$$
$37$ $$147 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$25 - 5 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$169 + 13 T + T^{2}$$
$67$ $$243 - 27 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$49 - 7 T + T^{2}$$
$79$ $$147 + 21 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$192 - 24 T + T^{2}$$