# 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 $2$

# Related objects

## 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: $$x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} - 4 \nu^{4} + \nu^{3} - 9 \nu^{2} + 21 \nu + 9$$$$)/27$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{5} - \nu^{4} - 2 \nu^{3} + 12 \nu + 36$$$$)/27$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{5} + 8 \nu^{4} - 2 \nu^{3} - 9 \nu^{2} + 12 \nu - 18$$$$)/27$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{5} - 2 \nu^{4} + 14 \nu^{3} + 18 \nu^{2} + 24 \nu + 45$$$$)/27$$ $$\beta_{5}$$ $$=$$ $$($$$$10 \nu^{5} + 5 \nu^{4} - 8 \nu^{3} + 36 \nu^{2} - 6 \nu - 153$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 4 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} - 2 \beta_{1}$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{5} + 7 \beta_{4} + 2 \beta_{3} - 24 \beta_{2} + 4 \beta_{1} + 9$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + 8 \beta_{3} - 6 \beta_{2} - 2 \beta_{1} + 18$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$7 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 33 \beta_{2} + 22 \beta_{1} + 81$$$$)/6$$

## 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 $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 1 - T + T^{2} )^{3}$$
$5$ $$144 + 96 T + 88 T^{2} + 8 T^{3} + 12 T^{4} + 2 T^{5} + T^{6}$$
$7$ $$( -3 - 9 T - T^{2} + T^{3} )^{2}$$
$11$ $$( 36 - 24 T + T^{3} )^{2}$$
$13$ $$9 + 63 T + 438 T^{2} + 27 T^{3} + 22 T^{4} - T^{5} + T^{6}$$
$17$ $$T^{6}$$
$19$ $$6859 + 1444 T + 323 T^{2} + 136 T^{3} + 17 T^{4} + 4 T^{5} + T^{6}$$
$23$ $$24336 + 4368 T + 2968 T^{2} - 704 T^{3} + 168 T^{4} - 14 T^{5} + T^{6}$$
$29$ $$9216 + 3072 T + 1408 T^{2} + 64 T^{3} + 48 T^{4} + 4 T^{5} + T^{6}$$
$31$ $$( -31 + 51 T + 15 T^{2} + T^{3} )^{2}$$
$37$ $$( 1 + T )^{6}$$
$41$ $$9216 - 3072 T + 1408 T^{2} - 64 T^{3} + 48 T^{4} - 4 T^{5} + T^{6}$$
$43$ $$169 - 273 T + 402 T^{2} - 89 T^{3} + 30 T^{4} + 3 T^{5} + T^{6}$$
$47$ $$( 36 + 6 T + T^{2} )^{3}$$
$53$ $$104976 - 23328 T + 7128 T^{2} - 216 T^{3} + 108 T^{4} - 6 T^{5} + T^{6}$$
$59$ $$1296 - 864 T + 576 T^{2} - 72 T^{3} + 24 T^{4} + T^{6}$$
$61$ $$5329 - 803 T + 1070 T^{2} + 289 T^{3} + 158 T^{4} + 13 T^{5} + T^{6}$$
$67$ $$292681 - 43821 T + 11430 T^{2} - 353 T^{3} + 162 T^{4} - 9 T^{5} + T^{6}$$
$71$ $$419904 - 7776 T + 11808 T^{2} + 1512 T^{3} + 312 T^{4} + 18 T^{5} + T^{6}$$
$73$ $$961 - 2573 T + 7478 T^{2} + 1639 T^{3} + 278 T^{4} + 19 T^{5} + T^{6}$$
$79$ $$29241 + 513 T + 1890 T^{2} - 375 T^{3} + 118 T^{4} - 11 T^{5} + T^{6}$$
$83$ $$( 192 - 80 T - 4 T^{2} + T^{3} )^{2}$$
$89$ $$11664 - 8208 T + 4048 T^{2} - 1000 T^{3} + 180 T^{4} - 16 T^{5} + T^{6}$$
$97$ $$2096704 - 388064 T + 74720 T^{2} - 2360 T^{3} + 272 T^{4} - 2 T^{5} + T^{6}$$