# Properties

 Label 756.2.f.a Level $756$ Weight $2$ Character orbit 756.f Analytic conductor $6.037$ Analytic rank $0$ Dimension $2$ 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.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -3 q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} + ( -3 + 6 \zeta_{6} ) q^{11} + ( 2 - 4 \zeta_{6} ) q^{13} + 6 q^{17} + ( 1 - 2 \zeta_{6} ) q^{19} + ( 3 - 6 \zeta_{6} ) q^{23} + 4 q^{25} + ( 6 - 12 \zeta_{6} ) q^{29} + ( 3 - 6 \zeta_{6} ) q^{31} + ( -9 + 6 \zeta_{6} ) q^{35} + q^{37} + 3 q^{41} + 10 q^{43} + 6 q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 9 - 18 \zeta_{6} ) q^{55} -6 q^{59} + ( -8 + 16 \zeta_{6} ) q^{61} + ( -6 + 12 \zeta_{6} ) q^{65} + 2 q^{67} + ( 3 - 6 \zeta_{6} ) q^{71} + ( -2 + 4 \zeta_{6} ) q^{73} + ( 3 + 12 \zeta_{6} ) q^{77} + 14 q^{79} -6 q^{83} -18 q^{85} -9 q^{89} + ( -2 - 8 \zeta_{6} ) q^{91} + ( -3 + 6 \zeta_{6} ) q^{95} + ( -4 + 8 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{5} + 4q^{7} + O(q^{10})$$ $$2q - 6q^{5} + 4q^{7} + 12q^{17} + 8q^{25} - 12q^{35} + 2q^{37} + 6q^{41} + 20q^{43} + 12q^{47} + 2q^{49} - 12q^{59} + 4q^{67} + 18q^{77} + 28q^{79} - 12q^{83} - 36q^{85} - 18q^{89} - 12q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$29$$ $$325$$ $$379$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
377.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −3.00000 0 2.00000 1.73205i 0 0 0
377.2 0 0 0 −3.00000 0 2.00000 + 1.73205i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.f.a 2
3.b odd 2 1 756.2.f.c yes 2
4.b odd 2 1 3024.2.k.a 2
7.b odd 2 1 756.2.f.c yes 2
9.c even 3 1 2268.2.x.g 2
9.c even 3 1 2268.2.x.h 2
9.d odd 6 1 2268.2.x.a 2
9.d odd 6 1 2268.2.x.b 2
12.b even 2 1 3024.2.k.d 2
21.c even 2 1 inner 756.2.f.a 2
28.d even 2 1 3024.2.k.d 2
63.l odd 6 1 2268.2.x.a 2
63.l odd 6 1 2268.2.x.b 2
63.o even 6 1 2268.2.x.g 2
63.o even 6 1 2268.2.x.h 2
84.h odd 2 1 3024.2.k.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.f.a 2 1.a even 1 1 trivial
756.2.f.a 2 21.c even 2 1 inner
756.2.f.c yes 2 3.b odd 2 1
756.2.f.c yes 2 7.b odd 2 1
2268.2.x.a 2 9.d odd 6 1
2268.2.x.a 2 63.l odd 6 1
2268.2.x.b 2 9.d odd 6 1
2268.2.x.b 2 63.l odd 6 1
2268.2.x.g 2 9.c even 3 1
2268.2.x.g 2 63.o even 6 1
2268.2.x.h 2 9.c even 3 1
2268.2.x.h 2 63.o even 6 1
3024.2.k.a 2 4.b odd 2 1
3024.2.k.a 2 84.h odd 2 1
3024.2.k.d 2 12.b even 2 1
3024.2.k.d 2 28.d even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 3 T + 5 T^{2} )^{2}$$
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 + 5 T^{2} + 121 T^{4}$$
$13$ $$1 - 14 T^{2} + 169 T^{4}$$
$17$ $$( 1 - 6 T + 17 T^{2} )^{2}$$
$19$ $$1 - 35 T^{2} + 361 T^{4}$$
$23$ $$1 - 19 T^{2} + 529 T^{4}$$
$29$ $$1 + 50 T^{2} + 841 T^{4}$$
$31$ $$1 - 35 T^{2} + 961 T^{4}$$
$37$ $$( 1 - T + 37 T^{2} )^{2}$$
$41$ $$( 1 - 3 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 10 T + 43 T^{2} )^{2}$$
$47$ $$( 1 - 6 T + 47 T^{2} )^{2}$$
$53$ $$( 1 - 53 T^{2} )^{2}$$
$59$ $$( 1 + 6 T + 59 T^{2} )^{2}$$
$61$ $$1 + 70 T^{2} + 3721 T^{4}$$
$67$ $$( 1 - 2 T + 67 T^{2} )^{2}$$
$71$ $$1 - 115 T^{2} + 5041 T^{4}$$
$73$ $$1 - 134 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 14 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 6 T + 83 T^{2} )^{2}$$
$89$ $$( 1 + 9 T + 89 T^{2} )^{2}$$
$97$ $$1 - 146 T^{2} + 9409 T^{4}$$