# Properties

 Label 912.3.be.b 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{5} - 11 q^{7} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z - 1) * q^3 + (-2*z + 2) * q^5 - 11 * q^7 + 3*z * q^9 $$q + ( - \zeta_{6} - 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{5} - 11 q^{7} + 3 \zeta_{6} q^{9} + 8 q^{11} + (3 \zeta_{6} - 6) q^{13} + (2 \zeta_{6} - 4) q^{15} + ( - 26 \zeta_{6} + 26) q^{17} - 19 q^{19} + (11 \zeta_{6} + 11) q^{21} - 32 \zeta_{6} q^{23} + 21 \zeta_{6} q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + (14 \zeta_{6} - 28) q^{29} + (62 \zeta_{6} - 31) q^{31} + ( - 8 \zeta_{6} - 8) q^{33} + (22 \zeta_{6} - 22) q^{35} + (54 \zeta_{6} - 27) q^{37} + 9 q^{39} + ( - 8 \zeta_{6} - 8) q^{41} + ( - 47 \zeta_{6} + 47) q^{43} + 6 q^{45} + 70 \zeta_{6} q^{47} + 72 q^{49} + (26 \zeta_{6} - 52) q^{51} + (4 \zeta_{6} - 8) q^{53} + ( - 16 \zeta_{6} + 16) q^{55} + (19 \zeta_{6} + 19) q^{57} + (62 \zeta_{6} + 62) q^{59} - 35 \zeta_{6} q^{61} - 33 \zeta_{6} q^{63} + (12 \zeta_{6} - 6) q^{65} + (9 \zeta_{6} - 18) q^{67} + (64 \zeta_{6} - 32) q^{69} + (76 \zeta_{6} + 76) q^{71} + (59 \zeta_{6} - 59) q^{73} + ( - 42 \zeta_{6} + 21) q^{75} - 88 q^{77} + (15 \zeta_{6} + 15) q^{79} + (9 \zeta_{6} - 9) q^{81} + 2 q^{83} - 52 \zeta_{6} q^{85} + 42 q^{87} + (46 \zeta_{6} - 92) q^{89} + ( - 33 \zeta_{6} + 66) q^{91} + ( - 93 \zeta_{6} + 93) q^{93} + (38 \zeta_{6} - 38) q^{95} + ( - 12 \zeta_{6} - 12) q^{97} + 24 \zeta_{6} q^{99} +O(q^{100})$$ q + (-z - 1) * q^3 + (-2*z + 2) * q^5 - 11 * q^7 + 3*z * q^9 + 8 * q^11 + (3*z - 6) * q^13 + (2*z - 4) * q^15 + (-26*z + 26) * q^17 - 19 * q^19 + (11*z + 11) * q^21 - 32*z * q^23 + 21*z * q^25 + (-6*z + 3) * q^27 + (14*z - 28) * q^29 + (62*z - 31) * q^31 + (-8*z - 8) * q^33 + (22*z - 22) * q^35 + (54*z - 27) * q^37 + 9 * q^39 + (-8*z - 8) * q^41 + (-47*z + 47) * q^43 + 6 * q^45 + 70*z * q^47 + 72 * q^49 + (26*z - 52) * q^51 + (4*z - 8) * q^53 + (-16*z + 16) * q^55 + (19*z + 19) * q^57 + (62*z + 62) * q^59 - 35*z * q^61 - 33*z * q^63 + (12*z - 6) * q^65 + (9*z - 18) * q^67 + (64*z - 32) * q^69 + (76*z + 76) * q^71 + (59*z - 59) * q^73 + (-42*z + 21) * q^75 - 88 * q^77 + (15*z + 15) * q^79 + (9*z - 9) * q^81 + 2 * q^83 - 52*z * q^85 + 42 * q^87 + (46*z - 92) * q^89 + (-33*z + 66) * q^91 + (-93*z + 93) * q^93 + (38*z - 38) * q^95 + (-12*z - 12) * q^97 + 24*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + 2 q^{5} - 22 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 + 2 * q^5 - 22 * q^7 + 3 * q^9 $$2 q - 3 q^{3} + 2 q^{5} - 22 q^{7} + 3 q^{9} + 16 q^{11} - 9 q^{13} - 6 q^{15} + 26 q^{17} - 38 q^{19} + 33 q^{21} - 32 q^{23} + 21 q^{25} - 42 q^{29} - 24 q^{33} - 22 q^{35} + 18 q^{39} - 24 q^{41} + 47 q^{43} + 12 q^{45} + 70 q^{47} + 144 q^{49} - 78 q^{51} - 12 q^{53} + 16 q^{55} + 57 q^{57} + 186 q^{59} - 35 q^{61} - 33 q^{63} - 27 q^{67} + 228 q^{71} - 59 q^{73} - 176 q^{77} + 45 q^{79} - 9 q^{81} + 4 q^{83} - 52 q^{85} + 84 q^{87} - 138 q^{89} + 99 q^{91} + 93 q^{93} - 38 q^{95} - 36 q^{97} + 24 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 + 2 * q^5 - 22 * q^7 + 3 * q^9 + 16 * q^11 - 9 * q^13 - 6 * q^15 + 26 * q^17 - 38 * q^19 + 33 * q^21 - 32 * q^23 + 21 * q^25 - 42 * q^29 - 24 * q^33 - 22 * q^35 + 18 * q^39 - 24 * q^41 + 47 * q^43 + 12 * q^45 + 70 * q^47 + 144 * q^49 - 78 * q^51 - 12 * q^53 + 16 * q^55 + 57 * q^57 + 186 * q^59 - 35 * q^61 - 33 * q^63 - 27 * q^67 + 228 * q^71 - 59 * q^73 - 176 * q^77 + 45 * q^79 - 9 * q^81 + 4 * q^83 - 52 * q^85 + 84 * q^87 - 138 * q^89 + 99 * q^91 + 93 * q^93 - 38 * q^95 - 36 * q^97 + 24 * 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 −11.0000 0 1.50000 2.59808i 0
673.1 0 −1.50000 0.866025i 0 1.00000 1.73205i 0 −11.0000 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.b 2
4.b odd 2 1 228.3.l.c 2
12.b even 2 1 684.3.y.b 2
19.d odd 6 1 inner 912.3.be.b 2
76.f even 6 1 228.3.l.c 2
228.n odd 6 1 684.3.y.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.l.c 2 4.b odd 2 1
228.3.l.c 2 76.f even 6 1
684.3.y.b 2 12.b even 2 1
684.3.y.b 2 228.n odd 6 1
912.3.be.b 2 1.a even 1 1 trivial
912.3.be.b 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 $$T_{7} + 11$$ T7 + 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 3$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$(T + 11)^{2}$$
$11$ $$(T - 8)^{2}$$
$13$ $$T^{2} + 9T + 27$$
$17$ $$T^{2} - 26T + 676$$
$19$ $$(T + 19)^{2}$$
$23$ $$T^{2} + 32T + 1024$$
$29$ $$T^{2} + 42T + 588$$
$31$ $$T^{2} + 2883$$
$37$ $$T^{2} + 2187$$
$41$ $$T^{2} + 24T + 192$$
$43$ $$T^{2} - 47T + 2209$$
$47$ $$T^{2} - 70T + 4900$$
$53$ $$T^{2} + 12T + 48$$
$59$ $$T^{2} - 186T + 11532$$
$61$ $$T^{2} + 35T + 1225$$
$67$ $$T^{2} + 27T + 243$$
$71$ $$T^{2} - 228T + 17328$$
$73$ $$T^{2} + 59T + 3481$$
$79$ $$T^{2} - 45T + 675$$
$83$ $$(T - 2)^{2}$$
$89$ $$T^{2} + 138T + 6348$$
$97$ $$T^{2} + 36T + 432$$