# Properties

 Label 912.3.be.a 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 1) q^{3} + (6 \zeta_{6} - 6) q^{5} + 5 q^{7} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z - 1) * q^3 + (6*z - 6) * q^5 + 5 * q^7 + 3*z * q^9 $$q + ( - \zeta_{6} - 1) q^{3} + (6 \zeta_{6} - 6) q^{5} + 5 q^{7} + 3 \zeta_{6} q^{9} + (11 \zeta_{6} - 22) q^{13} + ( - 6 \zeta_{6} + 12) q^{15} + (6 \zeta_{6} - 6) q^{17} + (16 \zeta_{6} + 5) q^{19} + ( - 5 \zeta_{6} - 5) q^{21} - 24 \zeta_{6} q^{23} - 11 \zeta_{6} q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + ( - 18 \zeta_{6} + 36) q^{29} + ( - 34 \zeta_{6} + 17) q^{31} + (30 \zeta_{6} - 30) q^{35} + (70 \zeta_{6} - 35) q^{37} + 33 q^{39} + ( - 24 \zeta_{6} - 24) q^{41} + (25 \zeta_{6} - 25) q^{43} - 18 q^{45} - 42 \zeta_{6} q^{47} - 24 q^{49} + ( - 6 \zeta_{6} + 12) q^{51} + ( - 36 \zeta_{6} + 72) q^{53} + ( - 37 \zeta_{6} + 11) q^{57} + ( - 42 \zeta_{6} - 42) q^{59} - 43 \zeta_{6} q^{61} + 15 \zeta_{6} q^{63} + ( - 132 \zeta_{6} + 66) q^{65} + (33 \zeta_{6} - 66) q^{67} + (48 \zeta_{6} - 24) q^{69} + ( - 36 \zeta_{6} - 36) q^{71} + (11 \zeta_{6} - 11) q^{73} + (22 \zeta_{6} - 11) q^{75} + ( - \zeta_{6} - 1) q^{79} + (9 \zeta_{6} - 9) q^{81} - 126 q^{83} - 36 \zeta_{6} q^{85} - 54 q^{87} + (6 \zeta_{6} - 12) q^{89} + (55 \zeta_{6} - 110) q^{91} + (51 \zeta_{6} - 51) q^{93} + (30 \zeta_{6} - 126) q^{95} + ( - 76 \zeta_{6} - 76) q^{97} +O(q^{100})$$ q + (-z - 1) * q^3 + (6*z - 6) * q^5 + 5 * q^7 + 3*z * q^9 + (11*z - 22) * q^13 + (-6*z + 12) * q^15 + (6*z - 6) * q^17 + (16*z + 5) * q^19 + (-5*z - 5) * q^21 - 24*z * q^23 - 11*z * q^25 + (-6*z + 3) * q^27 + (-18*z + 36) * q^29 + (-34*z + 17) * q^31 + (30*z - 30) * q^35 + (70*z - 35) * q^37 + 33 * q^39 + (-24*z - 24) * q^41 + (25*z - 25) * q^43 - 18 * q^45 - 42*z * q^47 - 24 * q^49 + (-6*z + 12) * q^51 + (-36*z + 72) * q^53 + (-37*z + 11) * q^57 + (-42*z - 42) * q^59 - 43*z * q^61 + 15*z * q^63 + (-132*z + 66) * q^65 + (33*z - 66) * q^67 + (48*z - 24) * q^69 + (-36*z - 36) * q^71 + (11*z - 11) * q^73 + (22*z - 11) * q^75 + (-z - 1) * q^79 + (9*z - 9) * q^81 - 126 * q^83 - 36*z * q^85 - 54 * q^87 + (6*z - 12) * q^89 + (55*z - 110) * q^91 + (51*z - 51) * q^93 + (30*z - 126) * q^95 + (-76*z - 76) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} - 6 q^{5} + 10 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 - 6 * q^5 + 10 * q^7 + 3 * q^9 $$2 q - 3 q^{3} - 6 q^{5} + 10 q^{7} + 3 q^{9} - 33 q^{13} + 18 q^{15} - 6 q^{17} + 26 q^{19} - 15 q^{21} - 24 q^{23} - 11 q^{25} + 54 q^{29} - 30 q^{35} + 66 q^{39} - 72 q^{41} - 25 q^{43} - 36 q^{45} - 42 q^{47} - 48 q^{49} + 18 q^{51} + 108 q^{53} - 15 q^{57} - 126 q^{59} - 43 q^{61} + 15 q^{63} - 99 q^{67} - 108 q^{71} - 11 q^{73} - 3 q^{79} - 9 q^{81} - 252 q^{83} - 36 q^{85} - 108 q^{87} - 18 q^{89} - 165 q^{91} - 51 q^{93} - 222 q^{95} - 228 q^{97}+O(q^{100})$$ 2 * q - 3 * q^3 - 6 * q^5 + 10 * q^7 + 3 * q^9 - 33 * q^13 + 18 * q^15 - 6 * q^17 + 26 * q^19 - 15 * q^21 - 24 * q^23 - 11 * q^25 + 54 * q^29 - 30 * q^35 + 66 * q^39 - 72 * q^41 - 25 * q^43 - 36 * q^45 - 42 * q^47 - 48 * q^49 + 18 * q^51 + 108 * q^53 - 15 * q^57 - 126 * q^59 - 43 * q^61 + 15 * q^63 - 99 * q^67 - 108 * q^71 - 11 * q^73 - 3 * q^79 - 9 * q^81 - 252 * q^83 - 36 * q^85 - 108 * q^87 - 18 * q^89 - 165 * q^91 - 51 * q^93 - 222 * q^95 - 228 * q^97

## 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 −3.00000 5.19615i 0 5.00000 0 1.50000 2.59808i 0
673.1 0 −1.50000 0.866025i 0 −3.00000 + 5.19615i 0 5.00000 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.a 2
4.b odd 2 1 228.3.l.a 2
12.b even 2 1 684.3.y.e 2
19.d odd 6 1 inner 912.3.be.a 2
76.f even 6 1 228.3.l.a 2
228.n odd 6 1 684.3.y.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.l.a 2 4.b odd 2 1
228.3.l.a 2 76.f even 6 1
684.3.y.e 2 12.b even 2 1
684.3.y.e 2 228.n odd 6 1
912.3.be.a 2 1.a even 1 1 trivial
912.3.be.a 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 $$S_{3}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{2} + 6T_{5} + 36$$ T5^2 + 6*T5 + 36 $$T_{7} - 5$$ T7 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 3$$
$5$ $$T^{2} + 6T + 36$$
$7$ $$(T - 5)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 33T + 363$$
$17$ $$T^{2} + 6T + 36$$
$19$ $$T^{2} - 26T + 361$$
$23$ $$T^{2} + 24T + 576$$
$29$ $$T^{2} - 54T + 972$$
$31$ $$T^{2} + 867$$
$37$ $$T^{2} + 3675$$
$41$ $$T^{2} + 72T + 1728$$
$43$ $$T^{2} + 25T + 625$$
$47$ $$T^{2} + 42T + 1764$$
$53$ $$T^{2} - 108T + 3888$$
$59$ $$T^{2} + 126T + 5292$$
$61$ $$T^{2} + 43T + 1849$$
$67$ $$T^{2} + 99T + 3267$$
$71$ $$T^{2} + 108T + 3888$$
$73$ $$T^{2} + 11T + 121$$
$79$ $$T^{2} + 3T + 3$$
$83$ $$(T + 126)^{2}$$
$89$ $$T^{2} + 18T + 108$$
$97$ $$T^{2} + 228T + 17328$$