# Properties

 Label 912.2.f.e Level $912$ Weight $2$ Character orbit 912.f 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.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: $$1$$ Twist minimal: no (minimal twist has level 114) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} + 1) q^{3} + ( - 4 \zeta_{6} + 2) q^{5} - q^{7} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (z + 1) * q^3 + (-4*z + 2) * q^5 - q^7 + 3*z * q^9 $$q + (\zeta_{6} + 1) q^{3} + ( - 4 \zeta_{6} + 2) q^{5} - q^{7} + 3 \zeta_{6} q^{9} + ( - 4 \zeta_{6} + 2) q^{11} + (2 \zeta_{6} - 1) q^{13} + ( - 6 \zeta_{6} + 6) q^{15} + ( - 2 \zeta_{6} + 1) q^{17} + (2 \zeta_{6} + 3) q^{19} + ( - \zeta_{6} - 1) q^{21} + ( - 6 \zeta_{6} + 3) q^{23} - 7 q^{25} + (6 \zeta_{6} - 3) q^{27} + 9 q^{29} + ( - 12 \zeta_{6} + 6) q^{31} + ( - 6 \zeta_{6} + 6) q^{33} + (4 \zeta_{6} - 2) q^{35} + ( - 8 \zeta_{6} + 4) q^{37} + (3 \zeta_{6} - 3) q^{39} - 2 q^{43} + ( - 6 \zeta_{6} + 12) q^{45} + (4 \zeta_{6} - 2) q^{47} - 6 q^{49} + ( - 3 \zeta_{6} + 3) q^{51} - 9 q^{53} - 12 q^{55} + (7 \zeta_{6} + 1) q^{57} + 3 q^{59} - 8 q^{61} - 3 \zeta_{6} q^{63} + 6 q^{65} + (10 \zeta_{6} - 5) q^{67} + ( - 9 \zeta_{6} + 9) q^{69} + 12 q^{71} + 11 q^{73} + ( - 7 \zeta_{6} - 7) q^{75} + (4 \zeta_{6} - 2) q^{77} + (8 \zeta_{6} - 4) q^{79} + (9 \zeta_{6} - 9) q^{81} + (12 \zeta_{6} - 6) q^{83} - 6 q^{85} + (9 \zeta_{6} + 9) q^{87} + 6 q^{89} + ( - 2 \zeta_{6} + 1) q^{91} + ( - 18 \zeta_{6} + 18) q^{93} + ( - 16 \zeta_{6} + 14) q^{95} + (16 \zeta_{6} - 8) q^{97} + ( - 6 \zeta_{6} + 12) q^{99} +O(q^{100})$$ q + (z + 1) * q^3 + (-4*z + 2) * q^5 - q^7 + 3*z * q^9 + (-4*z + 2) * q^11 + (2*z - 1) * q^13 + (-6*z + 6) * q^15 + (-2*z + 1) * q^17 + (2*z + 3) * q^19 + (-z - 1) * q^21 + (-6*z + 3) * q^23 - 7 * q^25 + (6*z - 3) * q^27 + 9 * q^29 + (-12*z + 6) * q^31 + (-6*z + 6) * q^33 + (4*z - 2) * q^35 + (-8*z + 4) * q^37 + (3*z - 3) * q^39 - 2 * q^43 + (-6*z + 12) * q^45 + (4*z - 2) * q^47 - 6 * q^49 + (-3*z + 3) * q^51 - 9 * q^53 - 12 * q^55 + (7*z + 1) * q^57 + 3 * q^59 - 8 * q^61 - 3*z * q^63 + 6 * q^65 + (10*z - 5) * q^67 + (-9*z + 9) * q^69 + 12 * q^71 + 11 * q^73 + (-7*z - 7) * q^75 + (4*z - 2) * q^77 + (8*z - 4) * q^79 + (9*z - 9) * q^81 + (12*z - 6) * q^83 - 6 * q^85 + (9*z + 9) * q^87 + 6 * q^89 + (-2*z + 1) * q^91 + (-18*z + 18) * q^93 + (-16*z + 14) * q^95 + (16*z - 8) * q^97 + (-6*z + 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 - 2 * q^7 + 3 * q^9 $$2 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 6 q^{15} + 8 q^{19} - 3 q^{21} - 14 q^{25} + 18 q^{29} + 6 q^{33} - 3 q^{39} - 4 q^{43} + 18 q^{45} - 12 q^{49} + 3 q^{51} - 18 q^{53} - 24 q^{55} + 9 q^{57} + 6 q^{59} - 16 q^{61} - 3 q^{63} + 12 q^{65} + 9 q^{69} + 24 q^{71} + 22 q^{73} - 21 q^{75} - 9 q^{81} - 12 q^{85} + 27 q^{87} + 12 q^{89} + 18 q^{93} + 12 q^{95} + 18 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 - 2 * q^7 + 3 * q^9 + 6 * q^15 + 8 * q^19 - 3 * q^21 - 14 * q^25 + 18 * q^29 + 6 * q^33 - 3 * q^39 - 4 * q^43 + 18 * q^45 - 12 * q^49 + 3 * q^51 - 18 * q^53 - 24 * q^55 + 9 * q^57 + 6 * q^59 - 16 * q^61 - 3 * q^63 + 12 * q^65 + 9 * q^69 + 24 * q^71 + 22 * q^73 - 21 * q^75 - 9 * q^81 - 12 * q^85 + 27 * q^87 + 12 * q^89 + 18 * q^93 + 12 * q^95 + 18 * q^99

## 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 1.50000 0.866025i 0 3.46410i 0 −1.00000 0 1.50000 2.59808i 0
113.2 0 1.50000 + 0.866025i 0 3.46410i 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
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.e 2
3.b odd 2 1 912.2.f.a 2
4.b odd 2 1 114.2.b.c yes 2
12.b even 2 1 114.2.b.b 2
19.b odd 2 1 912.2.f.a 2
57.d even 2 1 inner 912.2.f.e 2
76.d even 2 1 114.2.b.b 2
228.b odd 2 1 114.2.b.c yes 2

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

 $$T_{5}^{2} + 12$$ T5^2 + 12 $$T_{7} + 1$$ T7 + 1 $$T_{29} - 9$$ T29 - 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$T^{2} + 12$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + 12$$
$13$ $$T^{2} + 3$$
$17$ $$T^{2} + 3$$
$19$ $$T^{2} - 8T + 19$$
$23$ $$T^{2} + 27$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2} + 108$$
$37$ $$T^{2} + 48$$
$41$ $$T^{2}$$
$43$ $$(T + 2)^{2}$$
$47$ $$T^{2} + 12$$
$53$ $$(T + 9)^{2}$$
$59$ $$(T - 3)^{2}$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 75$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T - 11)^{2}$$
$79$ $$T^{2} + 48$$
$83$ $$T^{2} + 108$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 192$$