# Properties

 Label 912.3.be.f Level $912$ Weight $3$ Character orbit 912.be Analytic conductor $24.850$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.92607408.1 Defining polynomial: $$x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 94x^{2} - 77x + 43$$ x^6 - 3*x^5 + 20*x^4 - 35*x^3 + 94*x^2 - 77*x + 43 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 57) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 1) q^{3} + (\beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{5} + (\beta_{5} - \beta_{4} - \beta_1 + 4) q^{7} + 3 \beta_{3} q^{9}+O(q^{10})$$ q + (b3 + 1) * q^3 + (b5 - 2*b3 - b2 - b1 + 2) * q^5 + (b5 - b4 - b1 + 4) * q^7 + 3*b3 * q^9 $$q + (\beta_{3} + 1) q^{3} + (\beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{5} + (\beta_{5} - \beta_{4} - \beta_1 + 4) q^{7} + 3 \beta_{3} q^{9} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} - \beta_1 + 5) q^{11} + (\beta_{5} - 3 \beta_{4} - \beta_{2} - 3 \beta_1) q^{13} + (2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 4) q^{15} + ( - \beta_{5} + 3 \beta_{4} - 11 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 11) q^{17} + (2 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} - 2) q^{19} + (2 \beta_{5} - \beta_{4} + 4 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{21} + (3 \beta_{4} - 20 \beta_{3} - 3 \beta_1) q^{23} + (\beta_{5} - \beta_{4} - 6 \beta_{3} + \beta_{2} + \beta_1) q^{25} + (6 \beta_{3} - 3) q^{27} + (\beta_{5} + 3 \beta_{4} + 11 \beta_{3} - \beta_{2} + 3 \beta_1 - 22) q^{29} + ( - \beta_{5} - \beta_{4} - 30 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 15) q^{31} + (3 \beta_{4} + 5 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 5) q^{33} + (2 \beta_{5} - 5 \beta_{4} - 11 \beta_{3} - 7 \beta_{2} - 7 \beta_1 + 11) q^{35} + (4 \beta_{5} + 4 \beta_{4} - 18 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 9) q^{37} + (4 \beta_{5} - 4 \beta_{4} + \beta_{2} - 5 \beta_1) q^{39} + (2 \beta_{4} + 10 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 10) q^{41} + ( - \beta_{5} + 2 \beta_{4} + 37 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 37) q^{43} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{2} + 6) q^{45} + (2 \beta_{5} + 2 \beta_{4} + 16 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{47} + (12 \beta_{5} - 12 \beta_{4} - 3 \beta_{2} - 9 \beta_1 - 3) q^{49} + ( - 5 \beta_{5} + 7 \beta_{4} - 11 \beta_{3} + 5 \beta_{2} + 7 \beta_1 + 22) q^{51} + ( - 9 \beta_{5} + 13 \beta_{3} + 9 \beta_{2} - 26) q^{53} + (9 \beta_{5} - 7 \beta_{4} + 41 \beta_{3} - 16 \beta_{2} - 16 \beta_1 - 41) q^{55} + (7 \beta_{4} + 8 \beta_{3} + 8 \beta_{2} - 7) q^{57} + ( - 5 \beta_{5} + 6 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{59} + ( - 5 \beta_{5} + 5 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} - 5 \beta_1) q^{61} + (3 \beta_{5} + 12 \beta_{3} + 3 \beta_{2}) q^{63} + ( - 11 \beta_{5} - 11 \beta_{4} + 8 \beta_{3} - 15 \beta_{2} - 4 \beta_1 - 4) q^{65} + (11 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} - 11 \beta_{2} - 2 \beta_1 + 16) q^{67} + (3 \beta_{5} + 3 \beta_{4} - 40 \beta_{3} - 3 \beta_{2} - 6 \beta_1 + 20) q^{69} + (17 \beta_{5} - 13 \beta_{4} - 5 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 5) q^{71} + (6 \beta_{5} - 14 \beta_{4} + 17 \beta_{3} - 20 \beta_{2} - 20 \beta_1 - 17) q^{73} + ( - 12 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 6) q^{75} + (3 \beta_{5} - 3 \beta_{4} + \beta_{2} - 4 \beta_1 + 44) q^{77} + (5 \beta_{5} + 6 \beta_{4} - 3 \beta_{3} + 11 \beta_{2} - 11 \beta_1 - 3) q^{79} + (9 \beta_{3} - 9) q^{81} + ( - 24 \beta_{5} + 24 \beta_{4} + 9 \beta_{2} + 15 \beta_1 + 47) q^{83} + (14 \beta_{5} + 20 \beta_{4} + 14 \beta_{3} + 14 \beta_{2} - 20 \beta_1) q^{85} + ( - 2 \beta_{5} + 2 \beta_{4} - 5 \beta_{2} + 7 \beta_1 - 33) q^{87} + (3 \beta_{5} + 12 \beta_{4} + 7 \beta_{3} - 3 \beta_{2} + 12 \beta_1 - 14) q^{89} + (16 \beta_{5} - 19 \beta_{4} - 49 \beta_{3} - 16 \beta_{2} - 19 \beta_1 + 98) q^{91} + (\beta_{5} - 4 \beta_{4} - 45 \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 45) q^{93} + (16 \beta_{5} - 15 \beta_{4} + 75 \beta_{3} - 9 \beta_{2} - 21 \beta_1 - 49) q^{95} + (7 \beta_{5} - 7 \beta_{4} - 23 \beta_{3} - 23) q^{97} + (3 \beta_{5} + 6 \beta_{4} + 15 \beta_{3} + 3 \beta_{2} - 6 \beta_1) q^{99}+O(q^{100})$$ q + (b3 + 1) * q^3 + (b5 - 2*b3 - b2 - b1 + 2) * q^5 + (b5 - b4 - b1 + 4) * q^7 + 3*b3 * q^9 + (-b5 + b4 + 2*b2 - b1 + 5) * q^11 + (b5 - 3*b4 - b2 - 3*b1) * q^13 + (2*b5 - b4 - 2*b3 - 2*b2 - b1 + 4) * q^15 + (-b5 + 3*b4 - 11*b3 + 4*b2 + 4*b1 + 11) * q^17 + (2*b4 + 5*b3 + 5*b2 - 2) * q^19 + (2*b5 - b4 + 4*b3 + b2 - b1 + 4) * q^21 + (3*b4 - 20*b3 - 3*b1) * q^23 + (b5 - b4 - 6*b3 + b2 + b1) * q^25 + (6*b3 - 3) * q^27 + (b5 + 3*b4 + 11*b3 - b2 + 3*b1 - 22) * q^29 + (-b5 - b4 - 30*b3 - 3*b2 - 2*b1 + 15) * q^31 + (3*b4 + 5*b3 + 3*b2 - 3*b1 + 5) * q^33 + (2*b5 - 5*b4 - 11*b3 - 7*b2 - 7*b1 + 11) * q^35 + (4*b5 + 4*b4 - 18*b3 - 4*b2 - 8*b1 + 9) * q^37 + (4*b5 - 4*b4 + b2 - 5*b1) * q^39 + (2*b4 + 10*b3 + 2*b2 - 2*b1 + 10) * q^41 + (-b5 + 2*b4 + 37*b3 + 3*b2 + 3*b1 - 37) * q^43 + (3*b5 - 3*b4 - 3*b2 + 6) * q^45 + (2*b5 + 2*b4 + 16*b3 + 2*b2 - 2*b1) * q^47 + (12*b5 - 12*b4 - 3*b2 - 9*b1 - 3) * q^49 + (-5*b5 + 7*b4 - 11*b3 + 5*b2 + 7*b1 + 22) * q^51 + (-9*b5 + 13*b3 + 9*b2 - 26) * q^53 + (9*b5 - 7*b4 + 41*b3 - 16*b2 - 16*b1 - 41) * q^55 + (7*b4 + 8*b3 + 8*b2 - 7) * q^57 + (-5*b5 + 6*b4 - 2*b3 + b2 - b1 - 2) * q^59 + (-5*b5 + 5*b4 + 4*b3 - 5*b2 - 5*b1) * q^61 + (3*b5 + 12*b3 + 3*b2) * q^63 + (-11*b5 - 11*b4 + 8*b3 - 15*b2 - 4*b1 - 4) * q^65 + (11*b5 - 2*b4 - 8*b3 - 11*b2 - 2*b1 + 16) * q^67 + (3*b5 + 3*b4 - 40*b3 - 3*b2 - 6*b1 + 20) * q^69 + (17*b5 - 13*b4 - 5*b3 + 4*b2 - 4*b1 - 5) * q^71 + (6*b5 - 14*b4 + 17*b3 - 20*b2 - 20*b1 - 17) * q^73 + (-12*b3 + 3*b2 + 3*b1 + 6) * q^75 + (3*b5 - 3*b4 + b2 - 4*b1 + 44) * q^77 + (5*b5 + 6*b4 - 3*b3 + 11*b2 - 11*b1 - 3) * q^79 + (9*b3 - 9) * q^81 + (-24*b5 + 24*b4 + 9*b2 + 15*b1 + 47) * q^83 + (14*b5 + 20*b4 + 14*b3 + 14*b2 - 20*b1) * q^85 + (-2*b5 + 2*b4 - 5*b2 + 7*b1 - 33) * q^87 + (3*b5 + 12*b4 + 7*b3 - 3*b2 + 12*b1 - 14) * q^89 + (16*b5 - 19*b4 - 49*b3 - 16*b2 - 19*b1 + 98) * q^91 + (b5 - 4*b4 - 45*b3 - 5*b2 - 5*b1 + 45) * q^93 + (16*b5 - 15*b4 + 75*b3 - 9*b2 - 21*b1 - 49) * q^95 + (7*b5 - 7*b4 - 23*b3 - 23) * q^97 + (3*b5 + 6*b4 + 15*b3 + 3*b2 - 6*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 9 q^{3} + 4 q^{5} + 22 q^{7} + 9 q^{9}+O(q^{10})$$ 6 * q + 9 * q^3 + 4 * q^5 + 22 * q^7 + 9 * q^9 $$6 q + 9 q^{3} + 4 q^{5} + 22 q^{7} + 9 q^{9} + 36 q^{11} - 3 q^{13} + 12 q^{15} + 38 q^{17} + 10 q^{19} + 33 q^{21} - 54 q^{23} - 21 q^{25} - 102 q^{29} + 54 q^{33} + 24 q^{35} - 6 q^{39} + 96 q^{41} - 107 q^{43} + 24 q^{45} + 50 q^{47} - 48 q^{49} + 114 q^{51} - 90 q^{53} - 148 q^{55} - 3 q^{57} + 27 q^{61} + 33 q^{63} + 39 q^{67} - 84 q^{71} - 77 q^{73} + 260 q^{77} - 9 q^{79} - 27 q^{81} + 348 q^{83} + 68 q^{85} - 204 q^{87} - 72 q^{89} + 393 q^{91} + 129 q^{93} - 104 q^{95} - 228 q^{97} + 54 q^{99}+O(q^{100})$$ 6 * q + 9 * q^3 + 4 * q^5 + 22 * q^7 + 9 * q^9 + 36 * q^11 - 3 * q^13 + 12 * q^15 + 38 * q^17 + 10 * q^19 + 33 * q^21 - 54 * q^23 - 21 * q^25 - 102 * q^29 + 54 * q^33 + 24 * q^35 - 6 * q^39 + 96 * q^41 - 107 * q^43 + 24 * q^45 + 50 * q^47 - 48 * q^49 + 114 * q^51 - 90 * q^53 - 148 * q^55 - 3 * q^57 + 27 * q^61 + 33 * q^63 + 39 * q^67 - 84 * q^71 - 77 * q^73 + 260 * q^77 - 9 * q^79 - 27 * q^81 + 348 * q^83 + 68 * q^85 - 204 * q^87 - 72 * q^89 + 393 * q^91 + 129 * q^93 - 104 * q^95 - 228 * q^97 + 54 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 94x^{2} - 77x + 43$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + 5$$ v^2 + 5 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2\nu - 6$$ -v^2 + 2*v - 6 $$\beta_{3}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 26\nu^{3} + 34\nu^{2} - 56\nu + 34 ) / 23$$ (-2*v^5 + 5*v^4 - 26*v^3 + 34*v^2 - 56*v + 34) / 23 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + 14\nu^{4} - 13\nu^{3} + 109\nu^{2} + 41\nu + 63 ) / 23$$ (-v^5 + 14*v^4 - 13*v^3 + 109*v^2 + 41*v + 63) / 23 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} - 9\nu^{4} + 33\nu^{3} - 121\nu^{2} + 271\nu - 98 ) / 23$$ (-v^5 - 9*v^4 + 33*v^3 - 121*v^2 + 271*v - 98) / 23
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 1 ) / 2$$ (b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 - 5$$ b1 - 5 $$\nu^{3}$$ $$=$$ $$( \beta_{5} + \beta_{4} - \beta_{3} - 8\beta_{2} - 6\beta _1 - 15 ) / 2$$ (b5 + b4 - b3 - 8*b2 - 6*b1 - 15) / 2 $$\nu^{4}$$ $$=$$ $$2\beta_{4} - \beta_{3} - 3\beta_{2} - 11\beta _1 + 33$$ 2*b4 - b3 - 3*b2 - 11*b1 + 33 $$\nu^{5}$$ $$=$$ $$( -13\beta_{5} - 3\beta_{4} - 15\beta_{3} + 61\beta_{2} + 29\beta _1 + 196 ) / 2$$ (-13*b5 - 3*b4 - 15*b3 + 61*b2 + 29*b1 + 196) / 2

## Character values

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

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$ $$1$$ $$1$$

## 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 + 2.93068i 0.5 − 2.69511i 0.5 + 0.630453i 0.5 − 2.93068i 0.5 + 2.69511i 0.5 − 0.630453i
0 1.50000 0.866025i 0 −3.20750 5.55555i 0 2.26281 0 1.50000 2.59808i 0
145.2 0 1.50000 0.866025i 0 2.32722 + 4.03087i 0 10.6817 0 1.50000 2.59808i 0
145.3 0 1.50000 0.866025i 0 2.88028 + 4.98878i 0 −1.94451 0 1.50000 2.59808i 0
673.1 0 1.50000 + 0.866025i 0 −3.20750 + 5.55555i 0 2.26281 0 1.50000 + 2.59808i 0
673.2 0 1.50000 + 0.866025i 0 2.32722 4.03087i 0 10.6817 0 1.50000 + 2.59808i 0
673.3 0 1.50000 + 0.866025i 0 2.88028 4.98878i 0 −1.94451 0 1.50000 + 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 673.3 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.f 6
4.b odd 2 1 57.3.g.b 6
12.b even 2 1 171.3.p.c 6
19.d odd 6 1 inner 912.3.be.f 6
76.f even 6 1 57.3.g.b 6
228.n odd 6 1 171.3.p.c 6

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

 $$T_{5}^{6} - 4T_{5}^{5} + 56T_{5}^{4} - 184T_{5}^{3} + 2288T_{5}^{2} - 6880T_{5} + 29584$$ T5^6 - 4*T5^5 + 56*T5^4 - 184*T5^3 + 2288*T5^2 - 6880*T5 + 29584 $$T_{7}^{3} - 11T_{7}^{2} - T_{7} + 47$$ T7^3 - 11*T7^2 - T7 + 47

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{2} - 3 T + 3)^{3}$$
$5$ $$T^{6} - 4 T^{5} + 56 T^{4} + \cdots + 29584$$
$7$ $$(T^{3} - 11 T^{2} - T + 47)^{2}$$
$11$ $$(T^{3} - 18 T^{2} - 96 T + 1084)^{2}$$
$13$ $$T^{6} + 3 T^{5} - 340 T^{4} + \cdots + 2883$$
$17$ $$T^{6} - 38 T^{5} + 1408 T^{4} + \cdots + 52186176$$
$19$ $$T^{6} - 10 T^{5} - 249 T^{4} + \cdots + 47045881$$
$23$ $$T^{6} + 54 T^{5} + 2352 T^{4} + \cdots + 21049744$$
$29$ $$T^{6} + 102 T^{5} + \cdots + 487228608$$
$31$ $$T^{6} + 2345 T^{4} + \cdots + 112326483$$
$37$ $$T^{6} + 4713 T^{4} + \cdots + 3652564347$$
$41$ $$T^{6} - 96 T^{5} + 3824 T^{4} + \cdots + 1354752$$
$43$ $$T^{6} + 107 T^{5} + \cdots + 1477402969$$
$47$ $$T^{6} - 50 T^{5} + 1976 T^{4} + \cdots + 5798464$$
$53$ $$T^{6} + 90 T^{5} + \cdots + 6327041328$$
$59$ $$T^{6} - 1048 T^{4} + \cdots + 25509168$$
$61$ $$T^{6} - 27 T^{5} + \cdots + 1215986641$$
$67$ $$T^{6} - 39 T^{5} + \cdots + 11787475467$$
$71$ $$T^{6} + 84 T^{5} + \cdots + 149905029888$$
$73$ $$T^{6} + 77 T^{5} + \cdots + 434693631969$$
$79$ $$T^{6} + 9 T^{5} + \cdots + 32898206883$$
$83$ $$(T^{3} - 174 T^{2} - 4140 T + 1174072)^{2}$$
$89$ $$T^{6} + 72 T^{5} + \cdots + 591631797168$$
$97$ $$T^{6} + 228 T^{5} + \cdots + 68659968$$