# Properties

 Label 756.2.be.b Level 756 Weight 2 Character orbit 756.be 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.be (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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{1} q^{5} + ( 1 + 2 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{1} q^{5} + ( 1 + 2 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} + 4 \beta_{2} q^{10} + ( -\beta_{1} - \beta_{3} ) q^{11} - q^{13} + ( \beta_{1} + 2 \beta_{3} ) q^{14} + ( -4 + 4 \beta_{2} ) q^{16} + ( \beta_{1} - \beta_{3} ) q^{17} + ( -4 - 4 \beta_{2} ) q^{19} + 4 \beta_{3} q^{20} + ( 2 - 4 \beta_{2} ) q^{22} + ( \beta_{1} - 2 \beta_{3} ) q^{23} + 3 \beta_{2} q^{25} -\beta_{1} q^{26} + ( -4 + 6 \beta_{2} ) q^{28} -5 \beta_{3} q^{29} + ( 6 - 3 \beta_{2} ) q^{31} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} + 2 q^{34} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{35} + ( -5 + 5 \beta_{2} ) q^{37} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{38} + ( -8 + 8 \beta_{2} ) q^{40} -5 \beta_{3} q^{41} + ( -5 + 10 \beta_{2} ) q^{43} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{44} + ( 4 - 2 \beta_{2} ) q^{46} + ( -3 \beta_{1} + 6 \beta_{3} ) q^{47} + ( -3 + 8 \beta_{2} ) q^{49} + 3 \beta_{3} q^{50} -2 \beta_{2} q^{52} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{53} + ( 4 - 8 \beta_{2} ) q^{55} + ( -4 \beta_{1} + 6 \beta_{3} ) q^{56} + ( 10 - 10 \beta_{2} ) q^{58} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{59} + ( -5 + 5 \beta_{2} ) q^{61} + ( 6 \beta_{1} - 3 \beta_{3} ) q^{62} -8 q^{64} -2 \beta_{1} q^{65} + ( -6 + 3 \beta_{2} ) q^{67} + 2 \beta_{1} q^{68} + ( -8 + 12 \beta_{2} ) q^{70} + ( 12 \beta_{1} - 6 \beta_{3} ) q^{71} + 4 \beta_{2} q^{73} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{74} + ( 8 - 16 \beta_{2} ) q^{76} + ( \beta_{1} - 5 \beta_{3} ) q^{77} + ( 9 + 9 \beta_{2} ) q^{79} + ( -8 \beta_{1} + 8 \beta_{3} ) q^{80} + ( 10 - 10 \beta_{2} ) q^{82} + ( -2 \beta_{1} + \beta_{3} ) q^{83} + 4 q^{85} + ( -5 \beta_{1} + 10 \beta_{3} ) q^{86} + ( 8 - 4 \beta_{2} ) q^{88} -10 \beta_{1} q^{89} + ( -1 - 2 \beta_{2} ) q^{91} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{92} + ( -12 + 6 \beta_{2} ) q^{94} + ( -8 \beta_{1} - 8 \beta_{3} ) q^{95} + 5 q^{97} + ( -3 \beta_{1} + 8 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} + 8q^{7} + O(q^{10})$$ $$4q + 4q^{4} + 8q^{7} + 8q^{10} - 4q^{13} - 8q^{16} - 24q^{19} + 6q^{25} - 4q^{28} + 18q^{31} + 8q^{34} - 10q^{37} - 16q^{40} + 12q^{46} + 4q^{49} - 4q^{52} + 20q^{58} - 10q^{61} - 32q^{64} - 18q^{67} - 8q^{70} + 8q^{73} + 54q^{79} + 20q^{82} + 16q^{85} + 24q^{88} - 8q^{91} - 36q^{94} + 20q^{97} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i −2.44949 1.41421i 0 2.00000 + 1.73205i 2.82843i 0 2.00000 + 3.46410i
107.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 2.44949 + 1.41421i 0 2.00000 + 1.73205i 2.82843i 0 2.00000 + 3.46410i
431.1 −1.22474 + 0.707107i 0 1.00000 1.73205i −2.44949 + 1.41421i 0 2.00000 1.73205i 2.82843i 0 2.00000 3.46410i
431.2 1.22474 0.707107i 0 1.00000 1.73205i 2.44949 1.41421i 0 2.00000 1.73205i 2.82843i 0 2.00000 3.46410i
 $$n$$: 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
28.g odd 6 1 inner
84.n 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.be.b yes 4
3.b odd 2 1 inner 756.2.be.b yes 4
4.b odd 2 1 756.2.be.a 4
7.c even 3 1 756.2.be.a 4
12.b even 2 1 756.2.be.a 4
21.h odd 6 1 756.2.be.a 4
28.g odd 6 1 inner 756.2.be.b yes 4
84.n even 6 1 inner 756.2.be.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.be.a 4 4.b odd 2 1
756.2.be.a 4 7.c even 3 1
756.2.be.a 4 12.b even 2 1
756.2.be.a 4 21.h odd 6 1
756.2.be.b yes 4 1.a even 1 1 trivial
756.2.be.b yes 4 3.b odd 2 1 inner
756.2.be.b yes 4 28.g odd 6 1 inner
756.2.be.b yes 4 84.n even 6 1 inner

## Hecke kernels

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

 $$T_{5}^{4} - 8 T_{5}^{2} + 64$$ $$T_{19}^{2} + 12 T_{19} + 48$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 4 T^{4}$$
$3$ 1
$5$ $$1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8}$$
$7$ $$( 1 - 4 T + 7 T^{2} )^{2}$$
$11$ $$1 - 16 T^{2} + 135 T^{4} - 1936 T^{6} + 14641 T^{8}$$
$13$ $$( 1 + T + 13 T^{2} )^{4}$$
$17$ $$1 + 32 T^{2} + 735 T^{4} + 9248 T^{6} + 83521 T^{8}$$
$19$ $$( 1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 - 40 T^{2} + 1071 T^{4} - 21160 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 8 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 32 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 11 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 - 40 T^{2} - 609 T^{4} - 88360 T^{6} + 4879681 T^{8}$$
$53$ $$1 + 8 T^{2} - 2745 T^{4} + 22472 T^{6} + 7890481 T^{8}$$
$59$ $$1 - 64 T^{2} + 615 T^{4} - 222784 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 9 T + 94 T^{2} + 603 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 74 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 27 T + 322 T^{2} - 2133 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 160 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$1 - 22 T^{2} - 7437 T^{4} - 174262 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 - 5 T + 97 T^{2} )^{4}$$