# Properties

 Label 756.2.f.d Level $756$ Weight $2$ Character orbit 756.f 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.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ Defining polynomial: $$x^{4} + 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + ( -1 + \beta_{1} ) q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + ( -1 + \beta_{1} ) q^{7} + \beta_{3} q^{11} -\beta_{1} q^{13} -\beta_{2} q^{17} + \beta_{1} q^{19} + 2 \beta_{3} q^{23} -2 q^{25} + \beta_{3} q^{29} -3 \beta_{1} q^{31} + ( \beta_{2} + \beta_{3} ) q^{35} + q^{37} + \beta_{2} q^{41} + 7 q^{43} -7 \beta_{2} q^{47} + ( -5 - 2 \beta_{1} ) q^{49} + 3 \beta_{1} q^{55} -5 \beta_{2} q^{59} + \beta_{1} q^{61} -\beta_{3} q^{65} -10 q^{67} + 2 \beta_{3} q^{71} + 4 \beta_{1} q^{73} + ( 6 \beta_{2} - \beta_{3} ) q^{77} + 5 q^{79} + 7 \beta_{2} q^{83} + 3 q^{85} -6 \beta_{2} q^{89} + ( 6 + \beta_{1} ) q^{91} + \beta_{3} q^{95} -\beta_{1} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{7} + O(q^{10})$$ $$4 q - 4 q^{7} - 8 q^{25} + 4 q^{37} + 28 q^{43} - 20 q^{49} - 40 q^{67} + 20 q^{79} + 12 q^{85} + 24 q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 5 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$-3 \nu^{3} - 9 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{3} - 9 \beta_{1}$$$$)/6$$

## 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
 − 0.517638i 0.517638i − 1.93185i 1.93185i
0 0 0 −1.73205 0 −1.00000 2.44949i 0 0 0
377.2 0 0 0 −1.73205 0 −1.00000 + 2.44949i 0 0 0
377.3 0 0 0 1.73205 0 −1.00000 2.44949i 0 0 0
377.4 0 0 0 1.73205 0 −1.00000 + 2.44949i 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.b odd 2 1 inner
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.d 4
3.b odd 2 1 inner 756.2.f.d 4
4.b odd 2 1 3024.2.k.h 4
7.b odd 2 1 inner 756.2.f.d 4
9.c even 3 2 2268.2.x.j 8
9.d odd 6 2 2268.2.x.j 8
12.b even 2 1 3024.2.k.h 4
21.c even 2 1 inner 756.2.f.d 4
28.d even 2 1 3024.2.k.h 4
63.l odd 6 2 2268.2.x.j 8
63.o even 6 2 2268.2.x.j 8
84.h odd 2 1 3024.2.k.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.f.d 4 1.a even 1 1 trivial
756.2.f.d 4 3.b odd 2 1 inner
756.2.f.d 4 7.b odd 2 1 inner
756.2.f.d 4 21.c even 2 1 inner
2268.2.x.j 8 9.c even 3 2
2268.2.x.j 8 9.d odd 6 2
2268.2.x.j 8 63.l odd 6 2
2268.2.x.j 8 63.o even 6 2
3024.2.k.h 4 4.b odd 2 1
3024.2.k.h 4 12.b even 2 1
3024.2.k.h 4 28.d even 2 1
3024.2.k.h 4 84.h odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -3 + T^{2} )^{2}$$
$7$ $$( 7 + 2 T + T^{2} )^{2}$$
$11$ $$( 18 + T^{2} )^{2}$$
$13$ $$( 6 + T^{2} )^{2}$$
$17$ $$( -3 + T^{2} )^{2}$$
$19$ $$( 6 + T^{2} )^{2}$$
$23$ $$( 72 + T^{2} )^{2}$$
$29$ $$( 18 + T^{2} )^{2}$$
$31$ $$( 54 + T^{2} )^{2}$$
$37$ $$( -1 + T )^{4}$$
$41$ $$( -3 + T^{2} )^{2}$$
$43$ $$( -7 + T )^{4}$$
$47$ $$( -147 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$( -75 + T^{2} )^{2}$$
$61$ $$( 6 + T^{2} )^{2}$$
$67$ $$( 10 + T )^{4}$$
$71$ $$( 72 + T^{2} )^{2}$$
$73$ $$( 96 + T^{2} )^{2}$$
$79$ $$( -5 + T )^{4}$$
$83$ $$( -147 + T^{2} )^{2}$$
$89$ $$( -108 + T^{2} )^{2}$$
$97$ $$( 6 + T^{2} )^{2}$$