# Properties

 Label 756.2.t.b Level $756$ Weight $2$ Character orbit 756.t Analytic conductor $6.037$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 756.t (of order $$6$$, degree $$2$$, 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 1 + 2 \zeta_{6} ) q^{7} + ( -3 + 6 \zeta_{6} ) q^{13} + ( -4 + 2 \zeta_{6} ) q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + ( 5 + 5 \zeta_{6} ) q^{31} + 11 \zeta_{6} q^{37} -5 q^{43} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 10 - 5 \zeta_{6} ) q^{61} + ( -5 + 5 \zeta_{6} ) q^{67} + ( 8 + 8 \zeta_{6} ) q^{73} -17 \zeta_{6} q^{79} + ( -15 + 12 \zeta_{6} ) q^{91} + ( 11 - 22 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{7} + O(q^{10})$$ $$2 q + 4 q^{7} - 6 q^{19} + 5 q^{25} + 15 q^{31} + 11 q^{37} - 10 q^{43} + 2 q^{49} + 15 q^{61} - 5 q^{67} + 24 q^{73} - 17 q^{79} - 18 q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$29$$ $$325$$ $$379$$ $$\chi(n)$$ $$-1$$ $$\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 2.00000 + 1.73205i 0 0 0
593.1 0 0 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.t.b 2
3.b odd 2 1 CM 756.2.t.b 2
7.c even 3 1 5292.2.f.a 2
7.d odd 6 1 inner 756.2.t.b 2
7.d odd 6 1 5292.2.f.a 2
9.c even 3 1 2268.2.w.b 2
9.c even 3 1 2268.2.bm.c 2
9.d odd 6 1 2268.2.w.b 2
9.d odd 6 1 2268.2.bm.c 2
21.g even 6 1 inner 756.2.t.b 2
21.g even 6 1 5292.2.f.a 2
21.h odd 6 1 5292.2.f.a 2
63.i even 6 1 2268.2.bm.c 2
63.k odd 6 1 2268.2.w.b 2
63.s even 6 1 2268.2.w.b 2
63.t odd 6 1 2268.2.bm.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.t.b 2 1.a even 1 1 trivial
756.2.t.b 2 3.b odd 2 1 CM
756.2.t.b 2 7.d odd 6 1 inner
756.2.t.b 2 21.g even 6 1 inner
2268.2.w.b 2 9.c even 3 1
2268.2.w.b 2 9.d odd 6 1
2268.2.w.b 2 63.k odd 6 1
2268.2.w.b 2 63.s even 6 1
2268.2.bm.c 2 9.c even 3 1
2268.2.bm.c 2 9.d odd 6 1
2268.2.bm.c 2 63.i even 6 1
2268.2.bm.c 2 63.t odd 6 1
5292.2.f.a 2 7.c even 3 1
5292.2.f.a 2 7.d odd 6 1
5292.2.f.a 2 21.g even 6 1
5292.2.f.a 2 21.h odd 6 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{13}^{2} + 27$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 - 4 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$27 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$12 + 6 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$75 - 15 T + T^{2}$$
$37$ $$121 - 11 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 5 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$75 - 15 T + T^{2}$$
$67$ $$25 + 5 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$192 - 24 T + T^{2}$$
$79$ $$289 + 17 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$363 + T^{2}$$