# Properties

 Label 756.2.j.a Level $756$ Weight $2$ Character orbit 756.j 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.j (of order $$3$$, degree $$2$$, not 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: no (minimal twist has level 252) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} + \beta_{3} ) q^{5} -\beta_{3} q^{7} +O(q^{10})$$ $$q + ( -1 + \beta_{1} + \beta_{3} ) q^{5} -\beta_{3} q^{7} + ( -\beta_{2} - 2 \beta_{3} ) q^{11} + ( 1 - \beta_{3} ) q^{13} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{19} + ( -2 + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{23} + ( 2 \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{25} + ( -5 \beta_{3} + \beta_{5} ) q^{29} + ( 1 + 3 \beta_{1} - \beta_{3} ) q^{31} + ( 1 - \beta_{1} - \beta_{2} ) q^{35} + ( -1 + 3 \beta_{1} + 3 \beta_{2} ) q^{37} + ( -2 - \beta_{1} + 2 \beta_{3} ) q^{41} + ( \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{43} + ( \beta_{2} - 5 \beta_{3} - \beta_{5} ) q^{47} + ( -1 + \beta_{3} ) q^{49} + ( 6 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{53} + ( -4 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{55} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{59} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{61} + ( -\beta_{2} + \beta_{3} ) q^{65} + ( 2 + \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{67} + ( 5 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{71} + ( 3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{73} + ( -2 - \beta_{1} + 2 \beta_{3} ) q^{77} + ( -5 \beta_{2} - \beta_{3} - \beta_{5} ) q^{79} + ( -\beta_{2} - 6 \beta_{3} - \beta_{5} ) q^{83} + ( 5 - 4 \beta_{1} - 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{85} + ( -2 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} ) q^{89} - q^{91} + ( -8 - \beta_{1} + 8 \beta_{3} ) q^{95} + ( -2 \beta_{2} + 5 \beta_{3} - \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{5} - 3 q^{7} + O(q^{10})$$ $$6 q - 3 q^{5} - 3 q^{7} - 6 q^{11} + 3 q^{13} + 6 q^{19} - 6 q^{23} - 6 q^{25} - 15 q^{29} + 3 q^{31} + 6 q^{35} - 6 q^{37} - 6 q^{41} - 3 q^{43} - 15 q^{47} - 3 q^{49} + 36 q^{53} - 24 q^{55} - 3 q^{59} + 6 q^{61} + 3 q^{65} + 6 q^{67} + 30 q^{71} + 18 q^{73} - 6 q^{77} - 3 q^{79} - 18 q^{83} + 15 q^{85} - 12 q^{89} - 6 q^{91} - 24 q^{95} + 15 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} + 5 \nu^{3} - 2 \nu^{2}$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu^{4} - 11 \nu^{3} + 17 \nu^{2} - 12 \nu + 3$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 16 \nu^{3} - 19 \nu^{2} + 21 \nu - 6$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} + 8 \nu^{2} - 7 \nu + 8$$ $$\beta_{5}$$ $$=$$ $$($$$$7 \nu^{5} - 16 \nu^{4} + 62 \nu^{3} - 68 \nu^{2} + 99 \nu - 30$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - 2 \beta_{1} - 6$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - 6 \beta_{3} - 4 \beta_{2} + \beta_{1} - 6$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{5} - 3 \beta_{4} - 13 \beta_{3} - \beta_{2} + 11 \beta_{1} + 19$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-3 \beta_{5} - 6 \beta_{4} + 19 \beta_{3} + 19 \beta_{2} + 11 \beta_{1} + 37$$$$)/3$$

## Character values

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

 $$n$$ $$29$$ $$325$$ $$379$$ $$\chi(n)$$ $$-1 + \beta_{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
253.1
 0.5 − 1.41036i 0.5 + 0.224437i 0.5 + 2.05195i 0.5 + 1.41036i 0.5 − 0.224437i 0.5 − 2.05195i
0 0 0 −1.97141 + 3.41458i 0 −0.500000 0.866025i 0 0 0
253.2 0 0 0 −0.555632 + 0.962383i 0 −0.500000 0.866025i 0 0 0
253.3 0 0 0 1.02704 1.77889i 0 −0.500000 0.866025i 0 0 0
505.1 0 0 0 −1.97141 3.41458i 0 −0.500000 + 0.866025i 0 0 0
505.2 0 0 0 −0.555632 0.962383i 0 −0.500000 + 0.866025i 0 0 0
505.3 0 0 0 1.02704 + 1.77889i 0 −0.500000 + 0.866025i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 505.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.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.j.a 6
3.b odd 2 1 252.2.j.b 6
4.b odd 2 1 3024.2.r.i 6
7.b odd 2 1 5292.2.j.e 6
7.c even 3 1 5292.2.i.d 6
7.c even 3 1 5292.2.l.g 6
7.d odd 6 1 5292.2.i.g 6
7.d odd 6 1 5292.2.l.d 6
9.c even 3 1 inner 756.2.j.a 6
9.c even 3 1 2268.2.a.j 3
9.d odd 6 1 252.2.j.b 6
9.d odd 6 1 2268.2.a.g 3
12.b even 2 1 1008.2.r.g 6
21.c even 2 1 1764.2.j.d 6
21.g even 6 1 1764.2.i.e 6
21.g even 6 1 1764.2.l.g 6
21.h odd 6 1 1764.2.i.f 6
21.h odd 6 1 1764.2.l.d 6
36.f odd 6 1 3024.2.r.i 6
36.f odd 6 1 9072.2.a.bz 3
36.h even 6 1 1008.2.r.g 6
36.h even 6 1 9072.2.a.bt 3
63.g even 3 1 5292.2.i.d 6
63.h even 3 1 5292.2.l.g 6
63.i even 6 1 1764.2.l.g 6
63.j odd 6 1 1764.2.l.d 6
63.k odd 6 1 5292.2.i.g 6
63.l odd 6 1 5292.2.j.e 6
63.n odd 6 1 1764.2.i.f 6
63.o even 6 1 1764.2.j.d 6
63.s even 6 1 1764.2.i.e 6
63.t odd 6 1 5292.2.l.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.j.b 6 3.b odd 2 1
252.2.j.b 6 9.d odd 6 1
756.2.j.a 6 1.a even 1 1 trivial
756.2.j.a 6 9.c even 3 1 inner
1008.2.r.g 6 12.b even 2 1
1008.2.r.g 6 36.h even 6 1
1764.2.i.e 6 21.g even 6 1
1764.2.i.e 6 63.s even 6 1
1764.2.i.f 6 21.h odd 6 1
1764.2.i.f 6 63.n odd 6 1
1764.2.j.d 6 21.c even 2 1
1764.2.j.d 6 63.o even 6 1
1764.2.l.d 6 21.h odd 6 1
1764.2.l.d 6 63.j odd 6 1
1764.2.l.g 6 21.g even 6 1
1764.2.l.g 6 63.i even 6 1
2268.2.a.g 3 9.d odd 6 1
2268.2.a.j 3 9.c even 3 1
3024.2.r.i 6 4.b odd 2 1
3024.2.r.i 6 36.f odd 6 1
5292.2.i.d 6 7.c even 3 1
5292.2.i.d 6 63.g even 3 1
5292.2.i.g 6 7.d odd 6 1
5292.2.i.g 6 63.k odd 6 1
5292.2.j.e 6 7.b odd 2 1
5292.2.j.e 6 63.l odd 6 1
5292.2.l.d 6 7.d odd 6 1
5292.2.l.d 6 63.t odd 6 1
5292.2.l.g 6 7.c even 3 1
5292.2.l.g 6 63.h even 3 1
9072.2.a.bt 3 36.h even 6 1
9072.2.a.bz 3 36.f odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 3 T_{5}^{5} + 15 T_{5}^{4} + 63 T_{5}^{2} + 54 T_{5} + 81$$ acting on $$S_{2}^{\mathrm{new}}(756, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$81 + 54 T + 63 T^{2} + 15 T^{4} + 3 T^{5} + T^{6}$$
$7$ $$( 1 + T + T^{2} )^{3}$$
$11$ $$81 - 27 T + 63 T^{2} + 36 T^{3} + 33 T^{4} + 6 T^{5} + T^{6}$$
$13$ $$( 1 - T + T^{2} )^{3}$$
$17$ $$( -9 - 33 T + T^{3} )^{2}$$
$19$ $$( 49 - 36 T - 3 T^{2} + T^{3} )^{2}$$
$23$ $$9801 + 1485 T + 819 T^{2} + 108 T^{3} + 51 T^{4} + 6 T^{5} + T^{6}$$
$29$ $$3969 - 3024 T + 3249 T^{2} + 846 T^{3} + 177 T^{4} + 15 T^{5} + T^{6}$$
$31$ $$2809 - 4134 T + 6243 T^{2} + 128 T^{3} + 87 T^{4} - 3 T^{5} + T^{6}$$
$37$ $$( -107 - 78 T + 3 T^{2} + T^{3} )^{2}$$
$41$ $$81 - 27 T + 63 T^{2} + 36 T^{3} + 33 T^{4} + 6 T^{5} + T^{6}$$
$43$ $$214369 + 50004 T + 13053 T^{2} + 602 T^{3} + 117 T^{4} + 3 T^{5} + T^{6}$$
$47$ $$6561 - 2916 T + 2511 T^{2} + 702 T^{3} + 189 T^{4} + 15 T^{5} + T^{6}$$
$53$ $$( 387 + 39 T - 18 T^{2} + T^{3} )^{2}$$
$59$ $$6561 - 4374 T + 2673 T^{2} - 324 T^{3} + 63 T^{4} + 3 T^{5} + T^{6}$$
$61$ $$961 + 1395 T + 1839 T^{2} + 332 T^{3} + 81 T^{4} - 6 T^{5} + T^{6}$$
$67$ $$3481 - 1593 T + 1083 T^{2} + 44 T^{3} + 63 T^{4} - 6 T^{5} + T^{6}$$
$71$ $$( 297 + 18 T - 15 T^{2} + T^{3} )^{2}$$
$73$ $$( 79 - 12 T - 9 T^{2} + T^{3} )^{2}$$
$79$ $$1857769 + 318942 T + 58845 T^{2} + 2024 T^{3} + 243 T^{4} + 3 T^{5} + T^{6}$$
$83$ $$729 + 2025 T + 5139 T^{2} + 1296 T^{3} + 249 T^{4} + 18 T^{5} + T^{6}$$
$89$ $$( -1089 - 195 T + 6 T^{2} + T^{3} )^{2}$$
$97$ $$529 + 414 T + 669 T^{2} - 316 T^{3} + 207 T^{4} - 15 T^{5} + T^{6}$$