# Properties

 Label 912.3.be.c Level $912$ Weight $3$ Character orbit 912.be Analytic conductor $24.850$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 912.be (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.8502001097$$ 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 228) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{5} + q^{7} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z - 1) * q^3 + (-2*z + 2) * q^5 + q^7 + 3*z * q^9 $$q + ( - \zeta_{6} - 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{5} + q^{7} + 3 \zeta_{6} q^{9} - 16 q^{11} + ( - 9 \zeta_{6} + 18) q^{13} + (2 \zeta_{6} - 4) q^{15} + (22 \zeta_{6} - 22) q^{17} - 19 q^{19} + ( - \zeta_{6} - 1) q^{21} + 40 \zeta_{6} q^{23} + 21 \zeta_{6} q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + ( - 10 \zeta_{6} + 20) q^{29} + ( - 58 \zeta_{6} + 29) q^{31} + (16 \zeta_{6} + 16) q^{33} + ( - 2 \zeta_{6} + 2) q^{35} + ( - 18 \zeta_{6} + 9) q^{37} - 27 q^{39} + ( - 8 \zeta_{6} - 8) q^{41} + (49 \zeta_{6} - 49) q^{43} + 6 q^{45} + 46 \zeta_{6} q^{47} - 48 q^{49} + ( - 22 \zeta_{6} + 44) q^{51} + (28 \zeta_{6} - 56) q^{53} + (32 \zeta_{6} - 32) q^{55} + (19 \zeta_{6} + 19) q^{57} + (38 \zeta_{6} + 38) q^{59} + 97 \zeta_{6} q^{61} + 3 \zeta_{6} q^{63} + ( - 36 \zeta_{6} + 18) q^{65} + ( - 15 \zeta_{6} + 30) q^{67} + ( - 80 \zeta_{6} + 40) q^{69} + (28 \zeta_{6} + 28) q^{71} + (35 \zeta_{6} - 35) q^{73} + ( - 42 \zeta_{6} + 21) q^{75} - 16 q^{77} + (51 \zeta_{6} + 51) q^{79} + (9 \zeta_{6} - 9) q^{81} + 146 q^{83} + 44 \zeta_{6} q^{85} - 30 q^{87} + (22 \zeta_{6} - 44) q^{89} + ( - 9 \zeta_{6} + 18) q^{91} + (87 \zeta_{6} - 87) q^{93} + (38 \zeta_{6} - 38) q^{95} + (36 \zeta_{6} + 36) q^{97} - 48 \zeta_{6} q^{99} +O(q^{100})$$ q + (-z - 1) * q^3 + (-2*z + 2) * q^5 + q^7 + 3*z * q^9 - 16 * q^11 + (-9*z + 18) * q^13 + (2*z - 4) * q^15 + (22*z - 22) * q^17 - 19 * q^19 + (-z - 1) * q^21 + 40*z * q^23 + 21*z * q^25 + (-6*z + 3) * q^27 + (-10*z + 20) * q^29 + (-58*z + 29) * q^31 + (16*z + 16) * q^33 + (-2*z + 2) * q^35 + (-18*z + 9) * q^37 - 27 * q^39 + (-8*z - 8) * q^41 + (49*z - 49) * q^43 + 6 * q^45 + 46*z * q^47 - 48 * q^49 + (-22*z + 44) * q^51 + (28*z - 56) * q^53 + (32*z - 32) * q^55 + (19*z + 19) * q^57 + (38*z + 38) * q^59 + 97*z * q^61 + 3*z * q^63 + (-36*z + 18) * q^65 + (-15*z + 30) * q^67 + (-80*z + 40) * q^69 + (28*z + 28) * q^71 + (35*z - 35) * q^73 + (-42*z + 21) * q^75 - 16 * q^77 + (51*z + 51) * q^79 + (9*z - 9) * q^81 + 146 * q^83 + 44*z * q^85 - 30 * q^87 + (22*z - 44) * q^89 + (-9*z + 18) * q^91 + (87*z - 87) * q^93 + (38*z - 38) * q^95 + (36*z + 36) * q^97 - 48*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9 $$2 q - 3 q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} - 32 q^{11} + 27 q^{13} - 6 q^{15} - 22 q^{17} - 38 q^{19} - 3 q^{21} + 40 q^{23} + 21 q^{25} + 30 q^{29} + 48 q^{33} + 2 q^{35} - 54 q^{39} - 24 q^{41} - 49 q^{43} + 12 q^{45} + 46 q^{47} - 96 q^{49} + 66 q^{51} - 84 q^{53} - 32 q^{55} + 57 q^{57} + 114 q^{59} + 97 q^{61} + 3 q^{63} + 45 q^{67} + 84 q^{71} - 35 q^{73} - 32 q^{77} + 153 q^{79} - 9 q^{81} + 292 q^{83} + 44 q^{85} - 60 q^{87} - 66 q^{89} + 27 q^{91} - 87 q^{93} - 38 q^{95} + 108 q^{97} - 48 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9 - 32 * q^11 + 27 * q^13 - 6 * q^15 - 22 * q^17 - 38 * q^19 - 3 * q^21 + 40 * q^23 + 21 * q^25 + 30 * q^29 + 48 * q^33 + 2 * q^35 - 54 * q^39 - 24 * q^41 - 49 * q^43 + 12 * q^45 + 46 * q^47 - 96 * q^49 + 66 * q^51 - 84 * q^53 - 32 * q^55 + 57 * q^57 + 114 * q^59 + 97 * q^61 + 3 * q^63 + 45 * q^67 + 84 * q^71 - 35 * q^73 - 32 * q^77 + 153 * q^79 - 9 * q^81 + 292 * q^83 + 44 * q^85 - 60 * q^87 - 66 * q^89 + 27 * q^91 - 87 * q^93 - 38 * q^95 + 108 * q^97 - 48 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.50000 + 0.866025i 0 1.00000 + 1.73205i 0 1.00000 0 1.50000 2.59808i 0
673.1 0 −1.50000 0.866025i 0 1.00000 1.73205i 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
19.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.be.c 2
4.b odd 2 1 228.3.l.b 2
12.b even 2 1 684.3.y.a 2
19.d odd 6 1 inner 912.3.be.c 2
76.f even 6 1 228.3.l.b 2
228.n odd 6 1 684.3.y.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.l.b 2 4.b odd 2 1
228.3.l.b 2 76.f even 6 1
684.3.y.a 2 12.b even 2 1
684.3.y.a 2 228.n odd 6 1
912.3.be.c 2 1.a even 1 1 trivial
912.3.be.c 2 19.d odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{2} - 2T_{5} + 4$$ T5^2 - 2*T5 + 4 $$T_{7} - 1$$ T7 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 3$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T + 16)^{2}$$
$13$ $$T^{2} - 27T + 243$$
$17$ $$T^{2} + 22T + 484$$
$19$ $$(T + 19)^{2}$$
$23$ $$T^{2} - 40T + 1600$$
$29$ $$T^{2} - 30T + 300$$
$31$ $$T^{2} + 2523$$
$37$ $$T^{2} + 243$$
$41$ $$T^{2} + 24T + 192$$
$43$ $$T^{2} + 49T + 2401$$
$47$ $$T^{2} - 46T + 2116$$
$53$ $$T^{2} + 84T + 2352$$
$59$ $$T^{2} - 114T + 4332$$
$61$ $$T^{2} - 97T + 9409$$
$67$ $$T^{2} - 45T + 675$$
$71$ $$T^{2} - 84T + 2352$$
$73$ $$T^{2} + 35T + 1225$$
$79$ $$T^{2} - 153T + 7803$$
$83$ $$(T - 146)^{2}$$
$89$ $$T^{2} + 66T + 1452$$
$97$ $$T^{2} - 108T + 3888$$