# Properties

 Label 756.2.b.b Level $756$ Weight $2$ Character orbit 756.b Analytic conductor $6.037$ Analytic rank $0$ Dimension $4$ CM discriminant -84 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} -2 q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} + \beta_{3} q^{7} -2 \beta_{2} q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} -2 q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} + \beta_{3} q^{7} -2 \beta_{2} q^{8} + ( 3 - \beta_{3} ) q^{10} + ( -3 \beta_{1} - \beta_{2} ) q^{11} + ( 2 \beta_{1} + \beta_{2} ) q^{14} + 4 q^{16} + ( -2 \beta_{1} - \beta_{2} ) q^{17} + ( -6 - \beta_{3} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{20} + ( -1 + 3 \beta_{3} ) q^{22} + ( -3 \beta_{1} - 4 \beta_{2} ) q^{23} + ( -3 + 3 \beta_{3} ) q^{25} -2 \beta_{3} q^{28} + ( 3 + 2 \beta_{3} ) q^{31} + 4 \beta_{2} q^{32} + 2 \beta_{3} q^{34} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{35} + ( -4 + 3 \beta_{3} ) q^{37} + ( -2 \beta_{1} - 7 \beta_{2} ) q^{38} + ( -6 + 2 \beta_{3} ) q^{40} + ( \beta_{1} + 8 \beta_{2} ) q^{41} + ( 6 \beta_{1} + 2 \beta_{2} ) q^{44} + ( 5 + 3 \beta_{3} ) q^{46} + 7 q^{49} + 6 \beta_{1} q^{50} + ( 12 - 5 \beta_{3} ) q^{55} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{56} + ( 4 \beta_{1} + 5 \beta_{2} ) q^{62} -8 q^{64} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{68} + ( -7 + 3 \beta_{3} ) q^{70} + ( -3 \beta_{1} - 7 \beta_{2} ) q^{71} + ( 6 \beta_{1} - \beta_{2} ) q^{74} + ( 12 + 2 \beta_{3} ) q^{76} + ( \beta_{1} - 10 \beta_{2} ) q^{77} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{80} + ( -15 - \beta_{3} ) q^{82} + ( 7 - 3 \beta_{3} ) q^{85} + ( 2 - 6 \beta_{3} ) q^{88} + ( -5 \beta_{1} - 4 \beta_{2} ) q^{89} + ( 6 \beta_{1} + 8 \beta_{2} ) q^{92} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{95} + 7 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} + O(q^{10})$$ $$4 q - 8 q^{4} + 12 q^{10} + 16 q^{16} - 24 q^{19} - 4 q^{22} - 12 q^{25} + 12 q^{31} - 16 q^{37} - 24 q^{40} + 20 q^{46} + 28 q^{49} + 48 q^{55} - 32 q^{64} - 28 q^{70} + 48 q^{76} - 60 q^{82} + 28 q^{85} + 8 q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{2} - 5 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 − 1.16372i 2.57794i − 2.57794i 1.16372i
1.41421i 0 −2.00000 0.250492i 0 2.64575 2.82843i 0 0.354249
55.2 1.41421i 0 −2.00000 3.99215i 0 −2.64575 2.82843i 0 5.64575
55.3 1.41421i 0 −2.00000 3.99215i 0 −2.64575 2.82843i 0 5.64575
55.4 1.41421i 0 −2.00000 0.250492i 0 2.64575 2.82843i 0 0.354249
 $$n$$: 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
84.h odd 2 1 CM by $$\Q(\sqrt{-21})$$
3.b odd 2 1 inner
28.d 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.b.b yes 4
3.b odd 2 1 inner 756.2.b.b yes 4
4.b odd 2 1 756.2.b.a 4
7.b odd 2 1 756.2.b.a 4
12.b even 2 1 756.2.b.a 4
21.c even 2 1 756.2.b.a 4
28.d even 2 1 inner 756.2.b.b yes 4
84.h odd 2 1 CM 756.2.b.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.b.a 4 4.b odd 2 1
756.2.b.a 4 7.b odd 2 1
756.2.b.a 4 12.b even 2 1
756.2.b.a 4 21.c even 2 1
756.2.b.b yes 4 1.a even 1 1 trivial
756.2.b.b yes 4 3.b odd 2 1 inner
756.2.b.b yes 4 28.d even 2 1 inner
756.2.b.b yes 4 84.h odd 2 1 CM

## Hecke kernels

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

 $$T_{5}^{4} + 16 T_{5}^{2} + 1$$ $$T_{19}^{2} + 12 T_{19} + 29$$ $$T_{47}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$1 + 16 T^{2} + T^{4}$$
$7$ $$( -7 + T^{2} )^{2}$$
$11$ $$961 + 64 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( 14 + T^{2} )^{2}$$
$19$ $$( 29 + 12 T + T^{2} )^{2}$$
$23$ $$361 + 88 T^{2} + T^{4}$$
$29$ $$T^{4}$$
$31$ $$( -19 - 6 T + T^{2} )^{2}$$
$37$ $$( -47 + 8 T + T^{2} )^{2}$$
$41$ $$11881 + 232 T^{2} + T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$841 + 184 T^{2} + T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$6889 + 184 T^{2} + T^{4}$$
$97$ $$T^{4}$$
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