# Properties

 Label 756.2.t.d Level $756$ Weight $2$ Character orbit 756.t Analytic conductor $6.037$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + ( -2 + \beta_{2} ) q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + ( -2 + \beta_{2} ) q^{7} + ( 2 + 4 \beta_{2} ) q^{13} + ( -\beta_{1} - \beta_{3} ) q^{17} + ( -1 + \beta_{2} ) q^{19} + ( \beta_{1} + 2 \beta_{3} ) q^{23} + 13 \beta_{2} q^{25} + ( -2 \beta_{1} - \beta_{3} ) q^{29} + ( -2 - \beta_{2} ) q^{31} + ( -2 \beta_{1} + \beta_{3} ) q^{35} + ( 8 + 8 \beta_{2} ) q^{37} -\beta_{3} q^{41} -5 q^{43} + \beta_{1} q^{47} + ( 3 - 5 \beta_{2} ) q^{49} + ( \beta_{1} - \beta_{3} ) q^{53} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{59} + ( -3 + 3 \beta_{2} ) q^{61} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{65} + 2 \beta_{2} q^{67} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{71} + ( -10 - 5 \beta_{2} ) q^{73} + ( 4 + 4 \beta_{2} ) q^{79} -4 \beta_{3} q^{83} + 18 q^{85} + 3 \beta_{1} q^{89} + ( -8 - 10 \beta_{2} ) q^{91} + ( -\beta_{1} + \beta_{3} ) q^{95} + ( 9 + 18 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{7} + O(q^{10})$$ $$4 q - 10 q^{7} - 6 q^{19} - 26 q^{25} - 6 q^{31} + 16 q^{37} - 20 q^{43} + 22 q^{49} - 18 q^{61} - 4 q^{67} - 30 q^{73} + 8 q^{79} + 72 q^{85} - 12 q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$3 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$3 \nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$$$/3$$

## Character values

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

 $$n$$ $$29$$ $$325$$ $$379$$ $$\chi(n)$$ $$-1$$ $$1 + \beta_{2}$$ $$1$$

## 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
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 0 0 −2.12132 3.67423i 0 −2.50000 + 0.866025i 0 0 0
269.2 0 0 0 2.12132 + 3.67423i 0 −2.50000 + 0.866025i 0 0 0
593.1 0 0 0 −2.12132 + 3.67423i 0 −2.50000 0.866025i 0 0 0
593.2 0 0 0 2.12132 3.67423i 0 −2.50000 0.866025i 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 inner
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.d 4
3.b odd 2 1 inner 756.2.t.d 4
7.c even 3 1 5292.2.f.d 4
7.d odd 6 1 inner 756.2.t.d 4
7.d odd 6 1 5292.2.f.d 4
9.c even 3 1 2268.2.w.g 4
9.c even 3 1 2268.2.bm.h 4
9.d odd 6 1 2268.2.w.g 4
9.d odd 6 1 2268.2.bm.h 4
21.g even 6 1 inner 756.2.t.d 4
21.g even 6 1 5292.2.f.d 4
21.h odd 6 1 5292.2.f.d 4
63.i even 6 1 2268.2.bm.h 4
63.k odd 6 1 2268.2.w.g 4
63.s even 6 1 2268.2.w.g 4
63.t odd 6 1 2268.2.bm.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.t.d 4 1.a even 1 1 trivial
756.2.t.d 4 3.b odd 2 1 inner
756.2.t.d 4 7.d odd 6 1 inner
756.2.t.d 4 21.g even 6 1 inner
2268.2.w.g 4 9.c even 3 1
2268.2.w.g 4 9.d odd 6 1
2268.2.w.g 4 63.k odd 6 1
2268.2.w.g 4 63.s even 6 1
2268.2.bm.h 4 9.c even 3 1
2268.2.bm.h 4 9.d odd 6 1
2268.2.bm.h 4 63.i even 6 1
2268.2.bm.h 4 63.t odd 6 1
5292.2.f.d 4 7.c even 3 1
5292.2.f.d 4 7.d odd 6 1
5292.2.f.d 4 21.g even 6 1
5292.2.f.d 4 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 $$S_{2}^{\mathrm{new}}(756, [\chi])$$:

 $$T_{5}^{4} + 18 T_{5}^{2} + 324$$ $$T_{13}^{2} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$324 + 18 T^{2} + T^{4}$$
$7$ $$( 7 + 5 T + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$( 12 + T^{2} )^{2}$$
$17$ $$324 + 18 T^{2} + T^{4}$$
$19$ $$( 3 + 3 T + T^{2} )^{2}$$
$23$ $$2916 - 54 T^{2} + T^{4}$$
$29$ $$( 54 + T^{2} )^{2}$$
$31$ $$( 3 + 3 T + T^{2} )^{2}$$
$37$ $$( 64 - 8 T + T^{2} )^{2}$$
$41$ $$( -18 + T^{2} )^{2}$$
$43$ $$( 5 + T )^{4}$$
$47$ $$324 + 18 T^{2} + T^{4}$$
$53$ $$2916 - 54 T^{2} + T^{4}$$
$59$ $$5184 + 72 T^{2} + T^{4}$$
$61$ $$( 27 + 9 T + T^{2} )^{2}$$
$67$ $$( 4 + 2 T + T^{2} )^{2}$$
$71$ $$( 216 + T^{2} )^{2}$$
$73$ $$( 75 + 15 T + T^{2} )^{2}$$
$79$ $$( 16 - 4 T + T^{2} )^{2}$$
$83$ $$( -288 + T^{2} )^{2}$$
$89$ $$26244 + 162 T^{2} + T^{4}$$
$97$ $$( 243 + T^{2} )^{2}$$