# Properties

 Label 756.2.k.d Level 756 Weight 2 Character orbit 756.k 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.k (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{10})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + ( 2 + 3 \beta_{2} ) q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + ( 2 + 3 \beta_{2} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{3} ) q^{17} + ( 7 + 7 \beta_{2} ) q^{19} -\beta_{1} q^{23} + 5 \beta_{2} q^{25} + \beta_{3} q^{29} + 3 \beta_{2} q^{31} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{35} + ( 4 + 4 \beta_{2} ) q^{37} + 3 \beta_{3} q^{41} + 5 q^{43} + 3 \beta_{1} q^{47} + ( -5 + 3 \beta_{2} ) q^{49} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{53} + 20 q^{55} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{59} + ( 3 + 3 \beta_{2} ) q^{61} + 10 \beta_{2} q^{67} -4 \beta_{3} q^{71} -5 \beta_{2} q^{73} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{77} + ( -12 - 12 \beta_{2} ) q^{79} + 2 \beta_{3} q^{83} -10 q^{85} -3 \beta_{1} q^{89} + ( 7 \beta_{1} + 7 \beta_{3} ) q^{95} -5 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{7} + O(q^{10})$$ $$4q + 2q^{7} + 14q^{19} - 10q^{25} - 6q^{31} + 8q^{37} + 20q^{43} - 26q^{49} + 80q^{55} + 6q^{61} - 20q^{67} + 10q^{73} - 24q^{79} - 40q^{85} - 20q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 10 x^{2} + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/10$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$10 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$10 \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −1.58114 − 2.73861i 1.58114 + 2.73861i −1.58114 + 2.73861i 1.58114 − 2.73861i
0 0 0 −1.58114 2.73861i 0 0.500000 + 2.59808i 0 0 0
109.2 0 0 0 1.58114 + 2.73861i 0 0.500000 + 2.59808i 0 0 0
541.1 0 0 0 −1.58114 + 2.73861i 0 0.500000 2.59808i 0 0 0
541.2 0 0 0 1.58114 2.73861i 0 0.500000 2.59808i 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.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.k.d 4
3.b odd 2 1 inner 756.2.k.d 4
7.c even 3 1 inner 756.2.k.d 4
7.c even 3 1 5292.2.a.q 2
7.d odd 6 1 5292.2.a.r 2
9.c even 3 1 2268.2.i.i 4
9.c even 3 1 2268.2.l.i 4
9.d odd 6 1 2268.2.i.i 4
9.d odd 6 1 2268.2.l.i 4
21.g even 6 1 5292.2.a.r 2
21.h odd 6 1 inner 756.2.k.d 4
21.h odd 6 1 5292.2.a.q 2
63.g even 3 1 2268.2.i.i 4
63.h even 3 1 2268.2.l.i 4
63.j odd 6 1 2268.2.l.i 4
63.n odd 6 1 2268.2.i.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.k.d 4 1.a even 1 1 trivial
756.2.k.d 4 3.b odd 2 1 inner
756.2.k.d 4 7.c even 3 1 inner
756.2.k.d 4 21.h odd 6 1 inner
2268.2.i.i 4 9.c even 3 1
2268.2.i.i 4 9.d odd 6 1
2268.2.i.i 4 63.g even 3 1
2268.2.i.i 4 63.n odd 6 1
2268.2.l.i 4 9.c even 3 1
2268.2.l.i 4 9.d odd 6 1
2268.2.l.i 4 63.h even 3 1
2268.2.l.i 4 63.j odd 6 1
5292.2.a.q 2 7.c even 3 1
5292.2.a.q 2 21.h odd 6 1
5292.2.a.r 2 7.d odd 6 1
5292.2.a.r 2 21.g even 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 10 T_{5}^{2} + 100$$ $$T_{13}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 25 T^{4} + 625 T^{8}$$
$7$ $$( 1 - T + 7 T^{2} )^{2}$$
$11$ $$1 + 18 T^{2} + 203 T^{4} + 2178 T^{6} + 14641 T^{8}$$
$13$ $$( 1 + 13 T^{2} )^{4}$$
$17$ $$1 - 24 T^{2} + 287 T^{4} - 6936 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2}$$
$23$ $$1 - 36 T^{2} + 767 T^{4} - 19044 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 48 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 8 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 5 T + 43 T^{2} )^{4}$$
$47$ $$1 - 4 T^{2} - 2193 T^{4} - 8836 T^{6} + 4879681 T^{8}$$
$53$ $$1 - 16 T^{2} - 2553 T^{4} - 44944 T^{6} + 7890481 T^{8}$$
$59$ $$1 + 42 T^{2} - 1717 T^{4} + 146202 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 - 3 T - 52 T^{2} - 183 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 18 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 5 T - 48 T^{2} - 365 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 126 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$1 - 88 T^{2} - 177 T^{4} - 697048 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 + 5 T + 97 T^{2} )^{4}$$