# Properties

 Label 882.2.g.m Level $882$ Weight $2$ Character orbit 882.g Analytic conductor $7.043$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 882.g (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.04280545828$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

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

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 $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.707107 1.22474i 0 0 −1.00000 0 0.707107 1.22474i
361.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.707107 + 1.22474i 0 0 −1.00000 0 −0.707107 + 1.22474i
667.1 0.500000 0.866025i 0 −0.500000 0.866025i −0.707107 + 1.22474i 0 0 −1.00000 0 0.707107 + 1.22474i
667.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.707107 1.22474i 0 0 −1.00000 0 −0.707107 1.22474i
 $$n$$: 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
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.g.m 4
3.b odd 2 1 882.2.g.k 4
7.b odd 2 1 inner 882.2.g.m 4
7.c even 3 1 882.2.a.m 2
7.c even 3 1 inner 882.2.g.m 4
7.d odd 6 1 882.2.a.m 2
7.d odd 6 1 inner 882.2.g.m 4
21.c even 2 1 882.2.g.k 4
21.g even 6 1 882.2.a.o yes 2
21.g even 6 1 882.2.g.k 4
21.h odd 6 1 882.2.a.o yes 2
21.h odd 6 1 882.2.g.k 4
28.f even 6 1 7056.2.a.cs 2
28.g odd 6 1 7056.2.a.cs 2
84.j odd 6 1 7056.2.a.ci 2
84.n even 6 1 7056.2.a.ci 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.a.m 2 7.c even 3 1
882.2.a.m 2 7.d odd 6 1
882.2.a.o yes 2 21.g even 6 1
882.2.a.o yes 2 21.h odd 6 1
882.2.g.k 4 3.b odd 2 1
882.2.g.k 4 21.c even 2 1
882.2.g.k 4 21.g even 6 1
882.2.g.k 4 21.h odd 6 1
882.2.g.m 4 1.a even 1 1 trivial
882.2.g.m 4 7.b odd 2 1 inner
882.2.g.m 4 7.c even 3 1 inner
882.2.g.m 4 7.d odd 6 1 inner
7056.2.a.ci 2 84.j odd 6 1
7056.2.a.ci 2 84.n even 6 1
7056.2.a.cs 2 28.f even 6 1
7056.2.a.cs 2 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} + 2 T_{5}^{2} + 4$$ $$T_{11}^{2} - 4 T_{11} + 16$$ $$T_{13}^{2} - 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ 1
$5$ $$1 - 8 T^{2} + 39 T^{4} - 200 T^{6} + 625 T^{8}$$
$7$ 1
$11$ $$( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 8 T^{2} + 169 T^{4} )^{2}$$
$17$ $$1 + 16 T^{2} - 33 T^{4} + 4624 T^{6} + 83521 T^{8}$$
$19$ $$1 - 6 T^{2} - 325 T^{4} - 2166 T^{6} + 130321 T^{8}$$
$23$ $$( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 2 T + 29 T^{2} )^{4}$$
$31$ $$( 1 - 31 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 16 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{4}$$
$47$ $$1 - 62 T^{2} + 1635 T^{4} - 136958 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 18 T + 167 T^{2} - 1062 T^{3} + 3481 T^{4} )( 1 + 18 T + 167 T^{2} + 1062 T^{3} + 3481 T^{4} )$$
$61$ $$1 - 120 T^{2} + 10679 T^{4} - 446520 T^{6} + 13845841 T^{8}$$
$67$ $$( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 71 T^{2} )^{4}$$
$73$ $$1 + 96 T^{2} + 3887 T^{4} + 511584 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 134 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$1 - 128 T^{2} + 8463 T^{4} - 1013888 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 + 144 T^{2} + 9409 T^{4} )^{2}$$