# Properties

 Label 912.2.q.d Level $912$ Weight $2$ Character orbit 912.q Analytic conductor $7.282$ Analytic rank $0$ Dimension $2$ 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.q (of order $$3$$, 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 114) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{3} + ( 4 - 4 \zeta_{6} ) q^{5} + 3 q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{3} + ( 4 - 4 \zeta_{6} ) q^{5} + 3 q^{7} -\zeta_{6} q^{9} -2 q^{11} + 7 \zeta_{6} q^{13} + 4 \zeta_{6} q^{15} + ( 5 - 2 \zeta_{6} ) q^{19} + ( -3 + 3 \zeta_{6} ) q^{21} -4 \zeta_{6} q^{23} -11 \zeta_{6} q^{25} + q^{27} -4 \zeta_{6} q^{29} - q^{31} + ( 2 - 2 \zeta_{6} ) q^{33} + ( 12 - 12 \zeta_{6} ) q^{35} + 7 q^{37} -7 q^{39} + ( -4 + 4 \zeta_{6} ) q^{41} + ( 7 - 7 \zeta_{6} ) q^{43} -4 q^{45} + 2 \zeta_{6} q^{47} + 2 q^{49} + 4 \zeta_{6} q^{53} + ( -8 + 8 \zeta_{6} ) q^{55} + ( -3 + 5 \zeta_{6} ) q^{57} + ( -6 + 6 \zeta_{6} ) q^{59} + \zeta_{6} q^{61} -3 \zeta_{6} q^{63} + 28 q^{65} + 3 \zeta_{6} q^{67} + 4 q^{69} + ( 2 - 2 \zeta_{6} ) q^{71} + ( 3 - 3 \zeta_{6} ) q^{73} + 11 q^{75} -6 q^{77} + ( 5 - 5 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} + 4 q^{87} -18 \zeta_{6} q^{89} + 21 \zeta_{6} q^{91} + ( 1 - \zeta_{6} ) q^{93} + ( 12 - 20 \zeta_{6} ) q^{95} + ( -10 + 10 \zeta_{6} ) q^{97} + 2 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} + 4q^{5} + 6q^{7} - q^{9} + O(q^{10})$$ $$2q - q^{3} + 4q^{5} + 6q^{7} - q^{9} - 4q^{11} + 7q^{13} + 4q^{15} + 8q^{19} - 3q^{21} - 4q^{23} - 11q^{25} + 2q^{27} - 4q^{29} - 2q^{31} + 2q^{33} + 12q^{35} + 14q^{37} - 14q^{39} - 4q^{41} + 7q^{43} - 8q^{45} + 2q^{47} + 4q^{49} + 4q^{53} - 8q^{55} - q^{57} - 6q^{59} + q^{61} - 3q^{63} + 56q^{65} + 3q^{67} + 8q^{69} + 2q^{71} + 3q^{73} + 22q^{75} - 12q^{77} + 5q^{79} - q^{81} + 24q^{83} + 8q^{87} - 18q^{89} + 21q^{91} + q^{93} + 4q^{95} - 10q^{97} + 2q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 + 0.866025i 0 2.00000 3.46410i 0 3.00000 0 −0.500000 0.866025i 0
577.1 0 −0.500000 0.866025i 0 2.00000 + 3.46410i 0 3.00000 0 −0.500000 + 0.866025i 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.q.d 2
3.b odd 2 1 2736.2.s.c 2
4.b odd 2 1 114.2.e.a 2
12.b even 2 1 342.2.g.d 2
19.c even 3 1 inner 912.2.q.d 2
57.h odd 6 1 2736.2.s.c 2
76.f even 6 1 2166.2.a.c 1
76.g odd 6 1 114.2.e.a 2
76.g odd 6 1 2166.2.a.f 1
228.m even 6 1 342.2.g.d 2
228.m even 6 1 6498.2.a.l 1
228.n odd 6 1 6498.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.e.a 2 4.b odd 2 1
114.2.e.a 2 76.g odd 6 1
342.2.g.d 2 12.b even 2 1
342.2.g.d 2 228.m even 6 1
912.2.q.d 2 1.a even 1 1 trivial
912.2.q.d 2 19.c even 3 1 inner
2166.2.a.c 1 76.f even 6 1
2166.2.a.f 1 76.g odd 6 1
2736.2.s.c 2 3.b odd 2 1
2736.2.s.c 2 57.h odd 6 1
6498.2.a.l 1 228.m even 6 1
6498.2.a.x 1 228.n odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 4 T_{5} + 16$$ $$T_{7} - 3$$ $$T_{13}^{2} - 7 T_{13} + 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$16 - 4 T + T^{2}$$
$7$ $$( -3 + T )^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$49 - 7 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$19 - 8 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$16 + 4 T + T^{2}$$
$31$ $$( 1 + T )^{2}$$
$37$ $$( -7 + T )^{2}$$
$41$ $$16 + 4 T + T^{2}$$
$43$ $$49 - 7 T + T^{2}$$
$47$ $$4 - 2 T + T^{2}$$
$53$ $$16 - 4 T + T^{2}$$
$59$ $$36 + 6 T + T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$9 - 3 T + T^{2}$$
$71$ $$4 - 2 T + T^{2}$$
$73$ $$9 - 3 T + T^{2}$$
$79$ $$25 - 5 T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$324 + 18 T + T^{2}$$
$97$ $$100 + 10 T + T^{2}$$