# Properties

 Label 912.2.q.l Level $912$ Weight $2$ Character orbit 912.q Analytic conductor $7.282$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.954288.1 Defining polynomial: $$x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27$$ x^6 - x^5 - 2*x^4 + 3*x^3 - 6*x^2 - 9*x + 27 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 57) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{3} + \beta_{2} - 1) q^{5} + ( - \beta_{3} - \beta_1) q^{7} - \beta_{2} q^{9}+O(q^{10})$$ q + (-b2 + 1) * q^3 + (-b3 + b2 - 1) * q^5 + (-b3 - b1) * q^7 - b2 * q^9 $$q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{3} + \beta_{2} - 1) q^{5} + ( - \beta_{3} - \beta_1) q^{7} - \beta_{2} q^{9} + ( - \beta_{4} - \beta_{3} - \beta_1) q^{11} + ( - \beta_{5} + \beta_{4}) q^{13} + (\beta_{2} + \beta_1) q^{15} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{19} - \beta_{3} q^{21} + ( - \beta_{5} + \beta_{4} + 4 \beta_{2} - \beta_1) q^{23} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{25} - q^{27} + ( - 2 \beta_{2} - 2 \beta_1) q^{29} + (\beta_{4} + \beta_{3} + \beta_1 - 5) q^{31} + ( - \beta_{5} - \beta_{3}) q^{33} + ( - \beta_{5} - \beta_{3} - 6 \beta_{2} + 6) q^{35} - q^{37} + \beta_{4} q^{39} + (2 \beta_{3} - 2 \beta_{2} + 2) q^{41} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{43} + (\beta_{3} + \beta_1 + 1) q^{45} - 6 \beta_{2} q^{47} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{49} + (3 \beta_{2} + 3 \beta_1) q^{53} + (\beta_{5} - 2 \beta_{3} - 3 \beta_{2} + 3) q^{55} + (\beta_{4} + \beta_{3} - 1) q^{57} + ( - \beta_{5} - \beta_{3}) q^{59} + ( - \beta_{5} + \beta_{4} - 4 \beta_{2} + 2 \beta_1) q^{61} + \beta_1 q^{63} + ( - 2 \beta_{4} + \beta_{3} + \beta_1 + 3) q^{65} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - \beta_1) q^{67} + (\beta_{4} - \beta_{3} - \beta_1 + 4) q^{69} + (2 \beta_{5} + 2 \beta_{3} + 6 \beta_{2} - 6) q^{71} + ( - 2 \beta_{3} + 7 \beta_{2} - 7) q^{73} + (\beta_{4} - 2) q^{75} + ( - 3 \beta_{3} - 3 \beta_1 + 3) q^{77} + (\beta_{5} - \beta_{3} - 3 \beta_{2} + 3) q^{79} + (\beta_{2} - 1) q^{81} + ( - 2 \beta_{4} + 2) q^{83} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{87} + (5 \beta_{2} - \beta_1) q^{89} + (\beta_{5} - \beta_{4} + 3 \beta_{2} + \beta_1) q^{91} + (\beta_{5} + \beta_{3} + 5 \beta_{2} - 5) q^{93} + (\beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 6 \beta_{2} + 2 \beta_1 - 2) q^{95} + (3 \beta_{5} - 2 \beta_{3} + \beta_{2} - 1) q^{97} + ( - \beta_{5} + \beta_{4} + \beta_1) q^{99}+O(q^{100})$$ q + (-b2 + 1) * q^3 + (-b3 + b2 - 1) * q^5 + (-b3 - b1) * q^7 - b2 * q^9 + (-b4 - b3 - b1) * q^11 + (-b5 + b4) * q^13 + (b2 + b1) * q^15 + (-b5 + b4 + b3 - b2 + b1) * q^19 - b3 * q^21 + (-b5 + b4 + 4*b2 - b1) * q^23 + (-b5 + b4 - 2*b2) * q^25 - q^27 + (-2*b2 - 2*b1) * q^29 + (b4 + b3 + b1 - 5) * q^31 + (-b5 - b3) * q^33 + (-b5 - b3 - 6*b2 + 6) * q^35 - q^37 + b4 * q^39 + (2*b3 - 2*b2 + 2) * q^41 + (-b5 - b3 + b2 - 1) * q^43 + (b3 + b1 + 1) * q^45 - 6*b2 * q^47 + (-b4 - 2*b3 - 2*b1 - 1) * q^49 + (3*b2 + 3*b1) * q^53 + (b5 - 2*b3 - 3*b2 + 3) * q^55 + (b4 + b3 - 1) * q^57 + (-b5 - b3) * q^59 + (-b5 + b4 - 4*b2 + 2*b1) * q^61 + b1 * q^63 + (-2*b4 + b3 + b1 + 3) * q^65 + (-2*b5 + 2*b4 + 2*b2 - b1) * q^67 + (b4 - b3 - b1 + 4) * q^69 + (2*b5 + 2*b3 + 6*b2 - 6) * q^71 + (-2*b3 + 7*b2 - 7) * q^73 + (b4 - 2) * q^75 + (-3*b3 - 3*b1 + 3) * q^77 + (b5 - b3 - 3*b2 + 3) * q^79 + (b2 - 1) * q^81 + (-2*b4 + 2) * q^83 + (-2*b3 - 2*b1 - 2) * q^87 + (5*b2 - b1) * q^89 + (b5 - b4 + 3*b2 + b1) * q^91 + (b5 + b3 + 5*b2 - 5) * q^93 + (b5 - 2*b4 + 3*b3 + 6*b2 + 2*b1 - 2) * q^95 + (3*b5 - 2*b3 + b2 - 1) * q^97 + (-b5 + b4 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{3} - 2 q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10})$$ 6 * q + 3 * q^3 - 2 * q^5 + 2 * q^7 - 3 * q^9 $$6 q + 3 q^{3} - 2 q^{5} + 2 q^{7} - 3 q^{9} + q^{13} + 2 q^{15} - 4 q^{19} + q^{21} + 14 q^{23} - 5 q^{25} - 6 q^{27} - 4 q^{29} - 30 q^{31} + 18 q^{35} - 6 q^{37} + 2 q^{39} + 4 q^{41} - 3 q^{43} + 4 q^{45} - 18 q^{47} - 4 q^{49} + 6 q^{53} + 12 q^{55} - 5 q^{57} - 13 q^{61} - q^{63} + 12 q^{65} + 9 q^{67} + 28 q^{69} - 18 q^{71} - 19 q^{73} - 10 q^{75} + 24 q^{77} + 11 q^{79} - 3 q^{81} + 8 q^{83} - 8 q^{87} + 16 q^{89} + 7 q^{91} - 15 q^{93} - 2 q^{95} + 2 q^{97}+O(q^{100})$$ 6 * q + 3 * q^3 - 2 * q^5 + 2 * q^7 - 3 * q^9 + q^13 + 2 * q^15 - 4 * q^19 + q^21 + 14 * q^23 - 5 * q^25 - 6 * q^27 - 4 * q^29 - 30 * q^31 + 18 * q^35 - 6 * q^37 + 2 * q^39 + 4 * q^41 - 3 * q^43 + 4 * q^45 - 18 * q^47 - 4 * q^49 + 6 * q^53 + 12 * q^55 - 5 * q^57 - 13 * q^61 - q^63 + 12 * q^65 + 9 * q^67 + 28 * q^69 - 18 * q^71 - 19 * q^73 - 10 * q^75 + 24 * q^77 + 11 * q^79 - 3 * q^81 + 8 * q^83 - 8 * q^87 + 16 * q^89 + 7 * q^91 - 15 * q^93 - 2 * q^95 + 2 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} - 4\nu^{4} + \nu^{3} - 9\nu^{2} + 21\nu + 9 ) / 27$$ (v^5 - 4*v^4 + v^3 - 9*v^2 + 21*v + 9) / 27 $$\beta_{2}$$ $$=$$ $$( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27$$ (-2*v^5 - v^4 - 2*v^3 + 12*v + 36) / 27 $$\beta_{3}$$ $$=$$ $$( -2\nu^{5} + 8\nu^{4} - 2\nu^{3} - 9\nu^{2} + 12\nu - 18 ) / 27$$ (-2*v^5 + 8*v^4 - 2*v^3 - 9*v^2 + 12*v - 18) / 27 $$\beta_{4}$$ $$=$$ $$( -4\nu^{5} - 2\nu^{4} + 14\nu^{3} + 18\nu^{2} + 24\nu + 45 ) / 27$$ (-4*v^5 - 2*v^4 + 14*v^3 + 18*v^2 + 24*v + 45) / 27 $$\beta_{5}$$ $$=$$ $$( 10\nu^{5} + 5\nu^{4} - 8\nu^{3} + 36\nu^{2} - 6\nu - 153 ) / 27$$ (10*v^5 + 5*v^4 - 8*v^3 + 36*v^2 - 6*v - 153) / 27
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} + 2\beta_{3} + 3\beta_{2} + 4\beta_1 ) / 6$$ (b5 + b4 + 2*b3 + 3*b2 + 4*b1) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + \beta_{4} - \beta_{3} + 3\beta_{2} - 2\beta_1 ) / 3$$ (b5 + b4 - b3 + 3*b2 - 2*b1) / 3 $$\nu^{3}$$ $$=$$ $$( -2\beta_{5} + 7\beta_{4} + 2\beta_{3} - 24\beta_{2} + 4\beta _1 + 9 ) / 6$$ (-2*b5 + 7*b4 + 2*b3 - 24*b2 + 4*b1 + 9) / 6 $$\nu^{4}$$ $$=$$ $$( \beta_{5} + \beta_{4} + 8\beta_{3} - 6\beta_{2} - 2\beta _1 + 18 ) / 3$$ (b5 + b4 + 8*b3 - 6*b2 - 2*b1 + 18) / 3 $$\nu^{5}$$ $$=$$ $$( 7\beta_{5} - 2\beta_{4} + 2\beta_{3} - 33\beta_{2} + 22\beta _1 + 81 ) / 6$$ (7*b5 - 2*b4 + 2*b3 - 33*b2 + 22*b1 + 81) / 6

## Character values

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

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.403374 + 1.68443i 1.71903 − 0.211943i −1.62241 − 0.606458i 0.403374 − 1.68443i 1.71903 + 0.211943i −1.62241 + 0.606458i
0 0.500000 0.866025i 0 −1.66044 + 2.87597i 0 −2.32088 0 −0.500000 0.866025i 0
49.2 0 0.500000 0.866025i 0 −0.675970 + 1.17081i 0 −0.351939 0 −0.500000 0.866025i 0
49.3 0 0.500000 0.866025i 0 1.33641 2.31473i 0 3.67282 0 −0.500000 0.866025i 0
577.1 0 0.500000 + 0.866025i 0 −1.66044 2.87597i 0 −2.32088 0 −0.500000 + 0.866025i 0
577.2 0 0.500000 + 0.866025i 0 −0.675970 1.17081i 0 −0.351939 0 −0.500000 + 0.866025i 0
577.3 0 0.500000 + 0.866025i 0 1.33641 + 2.31473i 0 3.67282 0 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 577.3 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.q.l 6
3.b odd 2 1 2736.2.s.z 6
4.b odd 2 1 57.2.e.b 6
12.b even 2 1 171.2.f.b 6
19.c even 3 1 inner 912.2.q.l 6
57.h odd 6 1 2736.2.s.z 6
76.f even 6 1 1083.2.a.o 3
76.g odd 6 1 57.2.e.b 6
76.g odd 6 1 1083.2.a.l 3
228.m even 6 1 171.2.f.b 6
228.m even 6 1 3249.2.a.y 3
228.n odd 6 1 3249.2.a.t 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.e.b 6 4.b odd 2 1
57.2.e.b 6 76.g odd 6 1
171.2.f.b 6 12.b even 2 1
171.2.f.b 6 228.m even 6 1
912.2.q.l 6 1.a even 1 1 trivial
912.2.q.l 6 19.c even 3 1 inner
1083.2.a.l 3 76.g odd 6 1
1083.2.a.o 3 76.f even 6 1
2736.2.s.z 6 3.b odd 2 1
2736.2.s.z 6 57.h odd 6 1
3249.2.a.t 3 228.n odd 6 1
3249.2.a.y 3 228.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{6} + 2T_{5}^{5} + 12T_{5}^{4} + 8T_{5}^{3} + 88T_{5}^{2} + 96T_{5} + 144$$ T5^6 + 2*T5^5 + 12*T5^4 + 8*T5^3 + 88*T5^2 + 96*T5 + 144 $$T_{7}^{3} - T_{7}^{2} - 9T_{7} - 3$$ T7^3 - T7^2 - 9*T7 - 3 $$T_{13}^{6} - T_{13}^{5} + 22T_{13}^{4} + 27T_{13}^{3} + 438T_{13}^{2} + 63T_{13} + 9$$ T13^6 - T13^5 + 22*T13^4 + 27*T13^3 + 438*T13^2 + 63*T13 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{2} - T + 1)^{3}$$
$5$ $$T^{6} + 2 T^{5} + 12 T^{4} + 8 T^{3} + \cdots + 144$$
$7$ $$(T^{3} - T^{2} - 9 T - 3)^{2}$$
$11$ $$(T^{3} - 24 T + 36)^{2}$$
$13$ $$T^{6} - T^{5} + 22 T^{4} + 27 T^{3} + \cdots + 9$$
$17$ $$T^{6}$$
$19$ $$T^{6} + 4 T^{5} + 17 T^{4} + \cdots + 6859$$
$23$ $$T^{6} - 14 T^{5} + 168 T^{4} + \cdots + 24336$$
$29$ $$T^{6} + 4 T^{5} + 48 T^{4} + \cdots + 9216$$
$31$ $$(T^{3} + 15 T^{2} + 51 T - 31)^{2}$$
$37$ $$(T + 1)^{6}$$
$41$ $$T^{6} - 4 T^{5} + 48 T^{4} + \cdots + 9216$$
$43$ $$T^{6} + 3 T^{5} + 30 T^{4} - 89 T^{3} + \cdots + 169$$
$47$ $$(T^{2} + 6 T + 36)^{3}$$
$53$ $$T^{6} - 6 T^{5} + 108 T^{4} + \cdots + 104976$$
$59$ $$T^{6} + 24 T^{4} - 72 T^{3} + \cdots + 1296$$
$61$ $$T^{6} + 13 T^{5} + 158 T^{4} + \cdots + 5329$$
$67$ $$T^{6} - 9 T^{5} + 162 T^{4} + \cdots + 292681$$
$71$ $$T^{6} + 18 T^{5} + 312 T^{4} + \cdots + 419904$$
$73$ $$T^{6} + 19 T^{5} + 278 T^{4} + \cdots + 961$$
$79$ $$T^{6} - 11 T^{5} + 118 T^{4} + \cdots + 29241$$
$83$ $$(T^{3} - 4 T^{2} - 80 T + 192)^{2}$$
$89$ $$T^{6} - 16 T^{5} + 180 T^{4} + \cdots + 11664$$
$97$ $$T^{6} - 2 T^{5} + 272 T^{4} + \cdots + 2096704$$
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