# 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: $$\Q(\sqrt{2}, \sqrt{-3})$$ 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.

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

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

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 8 T^{2} + 39 T^{4} + 200 T^{6} + 625 T^{8}$$
$7$ $$( 1 + 5 T + 7 T^{2} )^{2}$$
$11$ $$( 1 + 11 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$
$17$ $$1 - 16 T^{2} - 33 T^{4} - 4624 T^{6} + 83521 T^{8}$$
$19$ $$( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 - 8 T^{2} - 465 T^{4} - 4232 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 4 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 64 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 5 T + 43 T^{2} )^{4}$$
$47$ $$1 - 76 T^{2} + 3567 T^{4} - 167884 T^{6} + 4879681 T^{8}$$
$53$ $$1 + 52 T^{2} - 105 T^{4} + 146068 T^{6} + 7890481 T^{8}$$
$59$ $$1 - 46 T^{2} - 1365 T^{4} - 160126 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 74 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 15 T + 148 T^{2} + 1095 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2}$$
$83$ $$( 1 - 122 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$1 - 16 T^{2} - 7665 T^{4} - 126736 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 + 49 T^{2} + 9409 T^{4} )^{2}$$