# Properties

 Label 912.3.be.d 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: 6.0.6967728.1 Defining polynomial: $$x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1$$ x^6 - x^5 + 8*x^4 + 5*x^3 + 50*x^2 - 7*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - 2) q^{3} + (\beta_{3} - \beta_{2}) q^{5} + (\beta_{5} + \beta_{4} - \beta_1 - 4) q^{7} + (3 \beta_{3} + 3) q^{9}+O(q^{10})$$ q + (-b3 - 2) * q^3 + (b3 - b2) * q^5 + (b5 + b4 - b1 - 4) * q^7 + (3*b3 + 3) * q^9 $$q + ( - \beta_{3} - 2) q^{3} + (\beta_{3} - \beta_{2}) q^{5} + (\beta_{5} + \beta_{4} - \beta_1 - 4) q^{7} + (3 \beta_{3} + 3) q^{9} + (3 \beta_1 - 1) q^{11} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{13} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{15} + ( - 2 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 2) q^{17} + ( - 2 \beta_{5} - 16 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{19} + ( - 3 \beta_{5} + 5 \beta_{3} + \beta_{2} + 2 \beta_1 + 7) q^{21} + (4 \beta_{5} - 2 \beta_{4} + 7 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 11) q^{23} + (4 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 7) q^{25} + ( - 6 \beta_{3} - 3) q^{27} + ( - 4 \beta_{4} - 8 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 8) q^{29} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} - 6 \beta_{2} - 3 \beta_1 - 2) q^{31} + (\beta_{3} - 3 \beta_{2} - 6 \beta_1 + 2) q^{33} + ( - 5 \beta_{3} + 9 \beta_{2}) q^{35} + (9 \beta_{5} - 9 \beta_{4} - 24 \beta_{3} - 3) q^{37} + (\beta_{5} + \beta_{4} - 6 \beta_1 + 7) q^{39} + (8 \beta_{5} + 2 \beta_{3} - 6 \beta_{2} - 12 \beta_1 + 12) q^{41} + (4 \beta_{3} - \beta_{2}) q^{43} + (3 \beta_1 - 3) q^{45} + ( - 12 \beta_{5} + 6 \beta_{4} + 16 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 4) q^{47} + ( - 4 \beta_{5} - 4 \beta_{4} + 6 \beta_1 + 14) q^{49} + ( - 6 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 4) q^{51} + (12 \beta_{4} - 17 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 17) q^{53} + (6 \beta_{5} - 12 \beta_{4} - 64 \beta_{3} - 2 \beta_{2} + 6) q^{55} + (4 \beta_{5} - 2 \beta_{4} + 16 \beta_{3} - 5 \beta_{2} - 4 \beta_1 - 12) q^{57} + (14 \beta_{5} - 21 \beta_{3} - 5 \beta_{2} - 10 \beta_1 - 28) q^{59} + ( - 10 \beta_{5} + 5 \beta_{4} - 38 \beta_{3} - 12 \beta_{2} - 12 \beta_1 - 48) q^{61} + (6 \beta_{5} - 3 \beta_{4} - 15 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 9) q^{63} + (10 \beta_{5} - 10 \beta_{4} - 74 \beta_{3} - 2 \beta_{2} - \beta_1 - 27) q^{65} + (8 \beta_{4} - 4 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{67} + ( - 6 \beta_{5} + 6 \beta_{4} - 14 \beta_{3} + 6 \beta_{2} + 3 \beta_1 - 13) q^{69} + (6 \beta_{5} + 40 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 86) q^{71} + ( - 10 \beta_{5} + 20 \beta_{4} - 31 \beta_{3} + 20 \beta_{2} - 10) q^{73} + ( - 6 \beta_{5} + 6 \beta_{4} - 6 \beta_{3} - 9) q^{75} + (2 \beta_{5} + 2 \beta_{4} - 29 \beta_1 + 7) q^{77} + (7 \beta_{5} + 11 \beta_{3} + \beta_{2} + 2 \beta_1 + 29) q^{79} + 9 \beta_{3} q^{81} + ( - 2 \beta_{5} - 2 \beta_{4} + 42) q^{83} + ( - 8 \beta_{5} + 4 \beta_{4} + 38 \beta_{3} + 14 \beta_{2} + 14 \beta_1 + 30) q^{85} + (4 \beta_{5} + 4 \beta_{4} - 6 \beta_1 - 20) q^{87} + ( - 71 \beta_{3} - \beta_{2} + \beta_1 + 71) q^{89} + ( - 2 \beta_{4} + 4 \beta_{3} + 15 \beta_{2} - 15 \beta_1 - 4) q^{91} + (\beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 9 \beta_{2} + 1) q^{93} + ( - 2 \beta_{5} - 4 \beta_{4} + 9 \beta_{3} - 19 \beta_{2} - 12 \beta_1 + 42) q^{95} + (34 \beta_{5} - 10 \beta_{3} - 8 \beta_{2} - 16 \beta_1 + 14) q^{97} + ( - 3 \beta_{3} + 9 \beta_{2} + 9 \beta_1 - 3) q^{99}+O(q^{100})$$ q + (-b3 - 2) * q^3 + (b3 - b2) * q^5 + (b5 + b4 - b1 - 4) * q^7 + (3*b3 + 3) * q^9 + (3*b1 - 1) * q^11 + (-b4 + 2*b3 - 2*b2 + 2*b1 - 2) * q^13 + (-b3 + b2 - b1 + 1) * q^15 + (-2*b5 + 4*b4 + 4*b3 + 4*b2 - 2) * q^17 + (-2*b5 - 16*b3 + 2*b2 + 3*b1 - 2) * q^19 + (-3*b5 + 5*b3 + b2 + 2*b1 + 7) * q^21 + (4*b5 - 2*b4 + 7*b3 - 3*b2 - 3*b1 + 11) * q^23 + (4*b5 - 2*b4 + 3*b3 + 7) * q^25 + (-6*b3 - 3) * q^27 + (-4*b4 - 8*b3 - 2*b2 + 2*b1 + 8) * q^29 + (-b5 + b4 - 2*b3 - 6*b2 - 3*b1 - 2) * q^31 + (b3 - 3*b2 - 6*b1 + 2) * q^33 + (-5*b3 + 9*b2) * q^35 + (9*b5 - 9*b4 - 24*b3 - 3) * q^37 + (b5 + b4 - 6*b1 + 7) * q^39 + (8*b5 + 2*b3 - 6*b2 - 12*b1 + 12) * q^41 + (4*b3 - b2) * q^43 + (3*b1 - 3) * q^45 + (-12*b5 + 6*b4 + 16*b3 - 4*b2 - 4*b1 + 4) * q^47 + (-4*b5 - 4*b4 + 6*b1 + 14) * q^49 + (-6*b4 - 4*b3 - 4*b2 + 4*b1 + 4) * q^51 + (12*b4 - 17*b3 + 3*b2 - 3*b1 + 17) * q^53 + (6*b5 - 12*b4 - 64*b3 - 2*b2 + 6) * q^55 + (4*b5 - 2*b4 + 16*b3 - 5*b2 - 4*b1 - 12) * q^57 + (14*b5 - 21*b3 - 5*b2 - 10*b1 - 28) * q^59 + (-10*b5 + 5*b4 - 38*b3 - 12*b2 - 12*b1 - 48) * q^61 + (6*b5 - 3*b4 - 15*b3 - 3*b2 - 3*b1 - 9) * q^63 + (10*b5 - 10*b4 - 74*b3 - 2*b2 - b1 - 27) * q^65 + (8*b4 - 4*b3 - b2 + b1 + 4) * q^67 + (-6*b5 + 6*b4 - 14*b3 + 6*b2 + 3*b1 - 13) * q^69 + (6*b5 + 40*b3 - 2*b2 - 4*b1 + 86) * q^71 + (-10*b5 + 20*b4 - 31*b3 + 20*b2 - 10) * q^73 + (-6*b5 + 6*b4 - 6*b3 - 9) * q^75 + (2*b5 + 2*b4 - 29*b1 + 7) * q^77 + (7*b5 + 11*b3 + b2 + 2*b1 + 29) * q^79 + 9*b3 * q^81 + (-2*b5 - 2*b4 + 42) * q^83 + (-8*b5 + 4*b4 + 38*b3 + 14*b2 + 14*b1 + 30) * q^85 + (4*b5 + 4*b4 - 6*b1 - 20) * q^87 + (-71*b3 - b2 + b1 + 71) * q^89 + (-2*b4 + 4*b3 + 15*b2 - 15*b1 - 4) * q^91 + (b5 - 2*b4 + 3*b3 + 9*b2 + 1) * q^93 + (-2*b5 - 4*b4 + 9*b3 - 19*b2 - 12*b1 + 42) * q^95 + (34*b5 - 10*b3 - 8*b2 - 16*b1 + 14) * q^97 + (-3*b3 + 9*b2 + 9*b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 9 q^{3} - 2 q^{5} - 26 q^{7} + 9 q^{9}+O(q^{10})$$ 6 * q - 9 * q^3 - 2 * q^5 - 26 * q^7 + 9 * q^9 $$6 q - 9 q^{3} - 2 q^{5} - 26 q^{7} + 9 q^{9} - 15 q^{13} + 6 q^{15} - 10 q^{17} + 46 q^{19} + 39 q^{21} + 24 q^{23} + 15 q^{25} + 66 q^{29} + 6 q^{35} + 30 q^{39} + 24 q^{41} - 11 q^{43} - 12 q^{45} + 26 q^{47} + 96 q^{49} + 30 q^{51} + 180 q^{53} + 176 q^{55} - 141 q^{57} - 162 q^{59} - 141 q^{61} - 39 q^{63} + 63 q^{67} + 372 q^{71} + 103 q^{73} - 16 q^{77} + 123 q^{79} - 27 q^{81} + 252 q^{83} + 116 q^{85} - 132 q^{87} + 642 q^{89} - 87 q^{91} - 21 q^{93} + 214 q^{95} - 12 q^{97}+O(q^{100})$$ 6 * q - 9 * q^3 - 2 * q^5 - 26 * q^7 + 9 * q^9 - 15 * q^13 + 6 * q^15 - 10 * q^17 + 46 * q^19 + 39 * q^21 + 24 * q^23 + 15 * q^25 + 66 * q^29 + 6 * q^35 + 30 * q^39 + 24 * q^41 - 11 * q^43 - 12 * q^45 + 26 * q^47 + 96 * q^49 + 30 * q^51 + 180 * q^53 + 176 * q^55 - 141 * q^57 - 162 * q^59 - 141 * q^61 - 39 * q^63 + 63 * q^67 + 372 * q^71 + 103 * q^73 - 16 * q^77 + 123 * q^79 - 27 * q^81 + 252 * q^83 + 116 * q^85 - 132 * q^87 + 642 * q^89 - 87 * q^91 - 21 * q^93 + 214 * q^95 - 12 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{5} - 16\nu^{4} + 128\nu^{3} - 100\nu^{2} + 14\nu + 281 ) / 393$$ (2*v^5 - 16*v^4 + 128*v^3 - 100*v^2 + 14*v + 281) / 393 $$\beta_{2}$$ $$=$$ $$( -56\nu^{5} + 55\nu^{4} - 440\nu^{3} - 344\nu^{2} - 1964\nu - 8 ) / 393$$ (-56*v^5 + 55*v^4 - 440*v^3 - 344*v^2 - 1964*v - 8) / 393 $$\beta_{3}$$ $$=$$ $$( -56\nu^{5} + 55\nu^{4} - 440\nu^{3} - 344\nu^{2} - 2750\nu - 8 ) / 393$$ (-56*v^5 + 55*v^4 - 440*v^3 - 344*v^2 - 2750*v - 8) / 393 $$\beta_{4}$$ $$=$$ $$( 196\nu^{5} - 127\nu^{4} + 1540\nu^{3} + 1466\nu^{2} + 10018\nu + 1469 ) / 393$$ (196*v^5 - 127*v^4 + 1540*v^3 + 1466*v^2 + 10018*v + 1469) / 393 $$\beta_{5}$$ $$=$$ $$( -210\nu^{5} + 239\nu^{4} - 1650\nu^{3} - 766\nu^{2} - 10116\nu + 2459 ) / 393$$ (-210*v^5 + 239*v^4 - 1650*v^3 - 766*v^2 - 10116*v + 2459) / 393
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{5} - \beta_{4} - 11\beta_{3} - 9 ) / 2$$ (2*b5 - b4 - 11*b3 - 9) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{5} + \beta_{4} + 7\beta _1 - 15 ) / 2$$ (b5 + b4 + 7*b1 - 15) / 2 $$\nu^{4}$$ $$=$$ $$-4\beta_{5} + 8\beta_{4} + 46\beta_{3} - 3\beta_{2} - 4$$ -4*b5 + 8*b4 + 46*b3 - 3*b2 - 4 $$\nu^{5}$$ $$=$$ $$( -28\beta_{5} + 14\beta_{4} + 193\beta_{3} - 55\beta_{2} - 55\beta _1 + 165 ) / 2$$ (-28*b5 + 14*b4 + 193*b3 - 55*b2 - 55*b1 + 165) / 2

## Character values

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

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

## 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
 1.56632 + 2.71294i 0.0702177 + 0.121621i −1.13654 − 1.96854i 1.56632 − 2.71294i 0.0702177 − 0.121621i −1.13654 + 1.96854i
0 −1.50000 + 0.866025i 0 −3.13264 5.42589i 0 −8.36156 0 1.50000 2.59808i 0
145.2 0 −1.50000 + 0.866025i 0 −0.140435 0.243241i 0 5.24143 0 1.50000 2.59808i 0
145.3 0 −1.50000 + 0.866025i 0 2.27307 + 3.93708i 0 −9.87987 0 1.50000 2.59808i 0
673.1 0 −1.50000 0.866025i 0 −3.13264 + 5.42589i 0 −8.36156 0 1.50000 + 2.59808i 0
673.2 0 −1.50000 0.866025i 0 −0.140435 + 0.243241i 0 5.24143 0 1.50000 + 2.59808i 0
673.3 0 −1.50000 0.866025i 0 2.27307 3.93708i 0 −9.87987 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.d 6
4.b odd 2 1 57.3.g.a 6
12.b even 2 1 171.3.p.e 6
19.d odd 6 1 inner 912.3.be.d 6
76.f even 6 1 57.3.g.a 6
228.n odd 6 1 171.3.p.e 6

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

 $$T_{5}^{6} + 2T_{5}^{5} + 32T_{5}^{4} - 40T_{5}^{3} + 800T_{5}^{2} + 224T_{5} + 64$$ T5^6 + 2*T5^5 + 32*T5^4 - 40*T5^3 + 800*T5^2 + 224*T5 + 64 $$T_{7}^{3} + 13T_{7}^{2} - 13T_{7} - 433$$ T7^3 + 13*T7^2 - 13*T7 - 433

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{2} + 3 T + 3)^{3}$$
$5$ $$T^{6} + 2 T^{5} + 32 T^{4} - 40 T^{3} + \cdots + 64$$
$7$ $$(T^{3} + 13 T^{2} - 13 T - 433)^{2}$$
$11$ $$(T^{3} - 264 T + 304)^{2}$$
$13$ $$T^{6} + 15 T^{5} - 172 T^{4} + \cdots + 3518667$$
$17$ $$T^{6} + 10 T^{5} + 472 T^{4} + \cdots + 5184$$
$19$ $$T^{6} - 46 T^{5} + 1383 T^{4} + \cdots + 47045881$$
$23$ $$T^{6} - 24 T^{5} + 696 T^{4} + \cdots + 1517824$$
$29$ $$T^{6} - 66 T^{5} + 1520 T^{4} + \cdots + 19293888$$
$31$ $$T^{6} + 2033 T^{4} + \cdots + 124768803$$
$37$ $$T^{6} + 5913 T^{4} + \cdots + 1689765867$$
$41$ $$T^{6} - 24 T^{5} + \cdots + 1907539968$$
$43$ $$T^{6} + 11 T^{5} + 110 T^{4} + \cdots + 2209$$
$47$ $$T^{6} - 26 T^{5} + \cdots + 2266521664$$
$53$ $$T^{6} - 180 T^{5} + 11016 T^{4} + \cdots + 2495232$$
$59$ $$T^{6} + 162 T^{5} + \cdots + 48350430912$$
$61$ $$T^{6} + 141 T^{5} + \cdots + 558907255201$$
$67$ $$T^{6} - 63 T^{5} + \cdots + 2349816507$$
$71$ $$T^{6} - 372 T^{5} + \cdots + 100019515392$$
$73$ $$T^{6} - 103 T^{5} + \cdots + 214090364601$$
$79$ $$T^{6} - 123 T^{5} + \cdots + 2549342403$$
$83$ $$(T^{3} - 126 T^{2} + 4908 T - 57704)^{2}$$
$89$ $$T^{6} - 642 T^{5} + \cdots + 3516172210368$$
$97$ $$T^{6} + 12 T^{5} + \cdots + 3000768049152$$