# Properties

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

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## 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: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -2 - \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -2 - \zeta_{6} ) q^{7} + ( 1 - 2 \zeta_{6} ) q^{13} + ( 5 - 10 \zeta_{6} ) q^{19} -5 q^{25} + ( 6 - 12 \zeta_{6} ) q^{31} + q^{37} -8 q^{43} + ( 3 + 5 \zeta_{6} ) q^{49} + ( 5 - 10 \zeta_{6} ) q^{61} + 11 q^{67} + ( -1 + 2 \zeta_{6} ) q^{73} -13 q^{79} + ( -4 + 5 \zeta_{6} ) q^{91} + ( -11 + 22 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 5q^{7} + O(q^{10})$$ $$2q - 5q^{7} - 10q^{25} + 2q^{37} - 16q^{43} + 11q^{49} + 22q^{67} - 26q^{79} - 3q^{91} + O(q^{100})$$

## 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 0 0 −2.50000 0.866025i 0 0 0
377.2 0 0 0 0 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 CM by $$\Q(\sqrt{-3})$$
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.b 2
3.b odd 2 1 CM 756.2.f.b 2
4.b odd 2 1 3024.2.k.c 2
7.b odd 2 1 inner 756.2.f.b 2
9.c even 3 1 2268.2.x.d 2
9.c even 3 1 2268.2.x.f 2
9.d odd 6 1 2268.2.x.d 2
9.d odd 6 1 2268.2.x.f 2
12.b even 2 1 3024.2.k.c 2
21.c even 2 1 inner 756.2.f.b 2
28.d even 2 1 3024.2.k.c 2
63.l odd 6 1 2268.2.x.d 2
63.l odd 6 1 2268.2.x.f 2
63.o even 6 1 2268.2.x.d 2
63.o even 6 1 2268.2.x.f 2
84.h odd 2 1 3024.2.k.c 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 5 T^{2} )^{2}$$
$7$ $$1 + 5 T + 7 T^{2}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ $$( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$( 1 - T + 19 T^{2} )( 1 + T + 19 T^{2} )$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$( 1 - 29 T^{2} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$
$37$ $$( 1 - T + 37 T^{2} )^{2}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 + 8 T + 43 T^{2} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$( 1 - 53 T^{2} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 - 13 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$
$67$ $$( 1 - 11 T + 67 T^{2} )^{2}$$
$71$ $$( 1 - 71 T^{2} )^{2}$$
$73$ $$( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$
$79$ $$( 1 + 13 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} )$$
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