# Properties

 Label 756.2.k.e Level $756$ Weight $2$ Character orbit 756.k Analytic conductor $6.037$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 756.k (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 Defining polynomial: $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{5} + ( 1 + \beta_{2} + \beta_{4} ) q^{7} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{5} + ( 1 + \beta_{2} + \beta_{4} ) q^{7} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{11} + ( -\beta_{1} + \beta_{3} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{17} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{19} + ( \beta_{2} - \beta_{3} + 5 \beta_{4} ) q^{23} + ( -5 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{25} + ( -2 - \beta_{2} - \beta_{5} ) q^{29} + ( -\beta_{2} - \beta_{3} + \beta_{5} ) q^{31} + ( -6 + \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{35} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{37} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{41} + ( -4 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{43} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{47} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{49} + ( 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} ) q^{53} + ( 5 - 4 \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{5} ) q^{55} + ( 2 - \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{59} + ( 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{61} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 9 \beta_{4} - 2 \beta_{5} ) q^{65} + ( 5 + \beta_{1} + 5 \beta_{4} + \beta_{5} ) q^{67} + ( 2 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{5} ) q^{71} + ( 2 + \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{73} + ( -7 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 7 \beta_{4} + \beta_{5} ) q^{77} + ( -2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{79} + ( 5 + \beta_{1} - \beta_{3} ) q^{83} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{85} + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{89} + ( 4 + 3 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{91} + ( -14 - 5 \beta_{2} - 5 \beta_{3} - 14 \beta_{4} + 5 \beta_{5} ) q^{95} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - q^{5} + 2q^{7} + O(q^{10})$$ $$6q - q^{5} + 2q^{7} - 5q^{11} - 4q^{13} + 4q^{17} - 3q^{19} - 14q^{23} - 10q^{25} - 10q^{29} + 2q^{31} - 26q^{35} + 24q^{41} - 18q^{43} + 9q^{47} - 6q^{53} + 16q^{55} + 5q^{59} - 7q^{61} - 24q^{65} + 16q^{67} + 22q^{71} + q^{73} - 31q^{77} + 8q^{79} + 34q^{83} + 10q^{85} + 3q^{89} + 5q^{91} - 32q^{95} + 28q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 5 \nu^{3} - \nu^{2} + 3 \nu - 6$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 8 \nu^{3} + 8 \nu^{2} - 21 \nu + 12$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu^{4} - 11 \nu^{3} + 20 \nu^{2} - 15 \nu + 9$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 6$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} - 4 \nu^{4} + 14 \nu^{3} - 20 \nu^{2} + 30 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3} + \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} - 5$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} - 5 \beta_{1} - 5$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{5} + 11 \beta_{4} - 9 \beta_{3} - 7 \beta_{2} - 7 \beta_{1} + 16$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-17 \beta_{5} - 16 \beta_{4} + 2 \beta_{3} - 8 \beta_{2} + 12 \beta_{1} + 31$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$379$$ $$\chi(n)$$ $$1$$ $$\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.5 + 0.224437i 0.5 + 2.05195i 0.5 − 1.41036i 0.5 − 0.224437i 0.5 − 2.05195i 0.5 + 1.41036i
0 0 0 −2.14400 3.71351i 0 1.40545 2.24159i 0 0 0
109.2 0 0 0 0.433463 + 0.750780i 0 −2.25729 1.38008i 0 0 0
109.3 0 0 0 1.21053 + 2.09671i 0 1.85185 + 1.88962i 0 0 0
541.1 0 0 0 −2.14400 + 3.71351i 0 1.40545 + 2.24159i 0 0 0
541.2 0 0 0 0.433463 0.750780i 0 −2.25729 + 1.38008i 0 0 0
541.3 0 0 0 1.21053 2.09671i 0 1.85185 1.88962i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 541.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.k.e 6
3.b odd 2 1 756.2.k.f yes 6
7.c even 3 1 inner 756.2.k.e 6
7.c even 3 1 5292.2.a.x 3
7.d odd 6 1 5292.2.a.v 3
9.c even 3 1 2268.2.i.j 6
9.c even 3 1 2268.2.l.k 6
9.d odd 6 1 2268.2.i.k 6
9.d odd 6 1 2268.2.l.j 6
21.g even 6 1 5292.2.a.w 3
21.h odd 6 1 756.2.k.f yes 6
21.h odd 6 1 5292.2.a.u 3
63.g even 3 1 2268.2.i.j 6
63.h even 3 1 2268.2.l.k 6
63.j odd 6 1 2268.2.l.j 6
63.n odd 6 1 2268.2.i.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.k.e 6 1.a even 1 1 trivial
756.2.k.e 6 7.c even 3 1 inner
756.2.k.f yes 6 3.b odd 2 1
756.2.k.f yes 6 21.h odd 6 1
2268.2.i.j 6 9.c even 3 1
2268.2.i.j 6 63.g even 3 1
2268.2.i.k 6 9.d odd 6 1
2268.2.i.k 6 63.n odd 6 1
2268.2.l.j 6 9.d odd 6 1
2268.2.l.j 6 63.j odd 6 1
2268.2.l.k 6 9.c even 3 1
2268.2.l.k 6 63.h even 3 1
5292.2.a.u 3 21.h odd 6 1
5292.2.a.v 3 7.d odd 6 1
5292.2.a.w 3 21.g even 6 1
5292.2.a.x 3 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(756, [\chi])$$:

 $$T_{5}^{6} + T_{5}^{5} + 13 T_{5}^{4} - 30 T_{5}^{3} + 135 T_{5}^{2} - 108 T_{5} + 81$$ $$T_{13}^{3} + 2 T_{13}^{2} - 11 T_{13} - 21$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$81 - 108 T + 135 T^{2} - 30 T^{3} + 13 T^{4} + T^{5} + T^{6}$$
$7$ $$343 - 98 T + 14 T^{2} + 19 T^{3} + 2 T^{4} - 2 T^{5} + T^{6}$$
$11$ $$3969 + 756 T + 459 T^{2} + 66 T^{3} + 37 T^{4} + 5 T^{5} + T^{6}$$
$13$ $$( -21 - 11 T + 2 T^{2} + T^{3} )^{2}$$
$17$ $$81 + 135 T + 189 T^{2} + 78 T^{3} + 31 T^{4} - 4 T^{5} + T^{6}$$
$19$ $$2401 + 1764 T + 1443 T^{2} - 10 T^{3} + 45 T^{4} + 3 T^{5} + T^{6}$$
$23$ $$3969 - 2457 T + 2403 T^{2} + 672 T^{3} + 157 T^{4} + 14 T^{5} + T^{6}$$
$29$ $$( -63 - 12 T + 5 T^{2} + T^{3} )^{2}$$
$31$ $$441 - 231 T + 163 T^{2} - 20 T^{3} + 15 T^{4} - 2 T^{5} + T^{6}$$
$37$ $$18769 - 7809 T + 3249 T^{2} - 274 T^{3} + 57 T^{4} + T^{6}$$
$41$ $$( 81 + 9 T - 12 T^{2} + T^{3} )^{2}$$
$43$ $$( -281 - 30 T + 9 T^{2} + T^{3} )^{2}$$
$47$ $$59049 - 13122 T + 5103 T^{2} + 135 T^{4} - 9 T^{5} + T^{6}$$
$53$ $$321489 + 56133 T + 13203 T^{2} + 540 T^{3} + 135 T^{4} + 6 T^{5} + T^{6}$$
$59$ $$6561 - 1458 T + 729 T^{2} - 72 T^{3} + 43 T^{4} - 5 T^{5} + T^{6}$$
$61$ $$321489 + 50463 T + 11890 T^{2} + 511 T^{3} + 138 T^{4} + 7 T^{5} + T^{6}$$
$67$ $$2809 - 3127 T + 2633 T^{2} - 838 T^{3} + 197 T^{4} - 16 T^{5} + T^{6}$$
$71$ $$( 189 - 72 T - 11 T^{2} + T^{3} )^{2}$$
$73$ $$25921 + 13202 T + 6563 T^{2} + 404 T^{3} + 83 T^{4} - T^{5} + T^{6}$$
$79$ $$762129 - 135315 T + 31009 T^{2} - 506 T^{3} + 219 T^{4} - 8 T^{5} + T^{6}$$
$83$ $$( -99 + 84 T - 17 T^{2} + T^{3} )^{2}$$
$89$ $$1750329 - 297675 T + 54594 T^{2} - 1971 T^{3} + 234 T^{4} - 3 T^{5} + T^{6}$$
$97$ $$( 56 - 40 T - 14 T^{2} + T^{3} )^{2}$$