# Properties

 Label 912.2.f.c Level $912$ Weight $2$ Character orbit 912.f 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.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 228) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - 4 q^{7} - 3 q^{9}+O(q^{10})$$ q + b * q^3 - 4 * q^7 - 3 * q^9 $$q + \beta q^{3} - 4 q^{7} - 3 q^{9} - 4 \beta q^{13} + ( - \beta + 4) q^{19} - 4 \beta q^{21} + 5 q^{25} - 3 \beta q^{27} - 6 \beta q^{31} + 4 \beta q^{37} + 12 q^{39} - 8 q^{43} + 9 q^{49} + (4 \beta + 3) q^{57} - 14 q^{61} + 12 q^{63} - 2 \beta q^{67} - 10 q^{73} + 5 \beta q^{75} - 10 \beta q^{79} + 9 q^{81} + 16 \beta q^{91} + 18 q^{93} - 8 \beta q^{97} +O(q^{100})$$ q + b * q^3 - 4 * q^7 - 3 * q^9 - 4*b * q^13 + (-b + 4) * q^19 - 4*b * q^21 + 5 * q^25 - 3*b * q^27 - 6*b * q^31 + 4*b * q^37 + 12 * q^39 - 8 * q^43 + 9 * q^49 + (4*b + 3) * q^57 - 14 * q^61 + 12 * q^63 - 2*b * q^67 - 10 * q^73 + 5*b * q^75 - 10*b * q^79 + 9 * q^81 + 16*b * q^91 + 18 * q^93 - 8*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q - 8 * q^7 - 6 * q^9 $$2 q - 8 q^{7} - 6 q^{9} + 8 q^{19} + 10 q^{25} + 24 q^{39} - 16 q^{43} + 18 q^{49} + 6 q^{57} - 28 q^{61} + 24 q^{63} - 20 q^{73} + 18 q^{81} + 36 q^{93}+O(q^{100})$$ 2 * q - 8 * q^7 - 6 * q^9 + 8 * q^19 + 10 * q^25 + 24 * q^39 - 16 * q^43 + 18 * q^49 + 6 * q^57 - 28 * q^61 + 24 * q^63 - 20 * q^73 + 18 * q^81 + 36 * q^93

## Character values

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

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.73205i 0 0 0 −4.00000 0 −3.00000 0
113.2 0 1.73205i 0 0 0 −4.00000 0 −3.00000 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.b odd 2 1 inner
57.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.f.c 2
3.b odd 2 1 CM 912.2.f.c 2
4.b odd 2 1 228.2.d.a 2
12.b even 2 1 228.2.d.a 2
19.b odd 2 1 inner 912.2.f.c 2
57.d even 2 1 inner 912.2.f.c 2
76.d even 2 1 228.2.d.a 2
228.b odd 2 1 228.2.d.a 2

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

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{7} + 4$$ T7 + 4 $$T_{29}$$ T29

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3$$
$5$ $$T^{2}$$
$7$ $$(T + 4)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 48$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 8T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 108$$
$37$ $$T^{2} + 48$$
$41$ $$T^{2}$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 14)^{2}$$
$67$ $$T^{2} + 12$$
$71$ $$T^{2}$$
$73$ $$(T + 10)^{2}$$
$79$ $$T^{2} + 300$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 192$$