# Properties

 Label 882.3.c.a Level $882$ Weight $3$ Character orbit 882.c Analytic conductor $24.033$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 882.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.0327593166$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.2048.2 Defining polynomial: $$x^{4} + 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 98) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + 2 q^{4} + ( \beta_{1} + 4 \beta_{3} ) q^{5} + 2 \beta_{2} q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + 2 q^{4} + ( \beta_{1} + 4 \beta_{3} ) q^{5} + 2 \beta_{2} q^{8} + ( 3 \beta_{1} + 5 \beta_{3} ) q^{10} + ( -12 - 4 \beta_{2} ) q^{11} + ( -4 \beta_{1} + \beta_{3} ) q^{13} + 4 q^{16} + 9 \beta_{3} q^{17} + ( 14 \beta_{1} - 6 \beta_{3} ) q^{19} + ( 2 \beta_{1} + 8 \beta_{3} ) q^{20} + ( -8 - 12 \beta_{2} ) q^{22} + ( -20 + 12 \beta_{2} ) q^{23} + ( -9 - 23 \beta_{2} ) q^{25} + ( 5 \beta_{1} - 3 \beta_{3} ) q^{26} + ( 16 - 13 \beta_{2} ) q^{29} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{31} + 4 \beta_{2} q^{32} + ( 9 \beta_{1} + 9 \beta_{3} ) q^{34} + ( -32 + 3 \beta_{2} ) q^{37} + ( -20 \beta_{1} + 8 \beta_{3} ) q^{38} + ( 6 \beta_{1} + 10 \beta_{3} ) q^{40} + ( \beta_{1} - 16 \beta_{3} ) q^{41} + ( -44 - 8 \beta_{2} ) q^{43} + ( -24 - 8 \beta_{2} ) q^{44} + ( 24 - 20 \beta_{2} ) q^{46} + ( 28 \beta_{1} + 20 \beta_{3} ) q^{47} + ( -46 - 9 \beta_{2} ) q^{50} + ( -8 \beta_{1} + 2 \beta_{3} ) q^{52} + ( 24 - 16 \beta_{2} ) q^{53} + ( -24 \beta_{1} - 68 \beta_{3} ) q^{55} + ( -26 + 16 \beta_{2} ) q^{58} + ( -18 \beta_{1} - 46 \beta_{3} ) q^{59} + ( 52 \beta_{1} + 23 \beta_{3} ) q^{61} + ( -12 \beta_{1} + 4 \beta_{3} ) q^{62} + 8 q^{64} + 7 \beta_{2} q^{65} + ( 64 + 24 \beta_{2} ) q^{67} + 18 \beta_{3} q^{68} + ( -8 - 40 \beta_{2} ) q^{71} + ( -15 \beta_{1} + 56 \beta_{3} ) q^{73} + ( 6 - 32 \beta_{2} ) q^{74} + ( 28 \beta_{1} - 12 \beta_{3} ) q^{76} + ( -48 + 20 \beta_{2} ) q^{79} + ( 4 \beta_{1} + 16 \beta_{3} ) q^{80} + ( -17 \beta_{1} - 15 \beta_{3} ) q^{82} + ( 6 \beta_{1} - 26 \beta_{3} ) q^{83} + ( -72 - 45 \beta_{2} ) q^{85} + ( -16 - 44 \beta_{2} ) q^{86} + ( -16 - 24 \beta_{2} ) q^{88} + ( 32 \beta_{1} - 9 \beta_{3} ) q^{89} + ( -40 + 24 \beta_{2} ) q^{92} + ( -8 \beta_{1} + 48 \beta_{3} ) q^{94} + ( 20 - 12 \beta_{2} ) q^{95} + ( 81 \beta_{1} - 8 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} + O(q^{10})$$ $$4 q + 8 q^{4} - 48 q^{11} + 16 q^{16} - 32 q^{22} - 80 q^{23} - 36 q^{25} + 64 q^{29} - 128 q^{37} - 176 q^{43} - 96 q^{44} + 96 q^{46} - 184 q^{50} + 96 q^{53} - 104 q^{58} + 32 q^{64} + 256 q^{67} - 32 q^{71} + 24 q^{74} - 192 q^{79} - 288 q^{85} - 64 q^{86} - 64 q^{88} - 160 q^{92} + 80 q^{95} + O(q^{100})$$

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
685.1
 1.84776i − 1.84776i − 0.765367i 0.765367i
−1.41421 0 2.00000 1.21371i 0 0 −2.82843 0 1.71644i
685.2 −1.41421 0 2.00000 1.21371i 0 0 −2.82843 0 1.71644i
685.3 1.41421 0 2.00000 8.15640i 0 0 2.82843 0 11.5349i
685.4 1.41421 0 2.00000 8.15640i 0 0 2.82843 0 11.5349i
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.c.a 4
3.b odd 2 1 98.3.b.a 4
7.b odd 2 1 inner 882.3.c.a 4
7.c even 3 2 882.3.n.j 8
7.d odd 6 2 882.3.n.j 8
12.b even 2 1 784.3.c.b 4
21.c even 2 1 98.3.b.a 4
21.g even 6 2 98.3.d.b 8
21.h odd 6 2 98.3.d.b 8
84.h odd 2 1 784.3.c.b 4
84.j odd 6 2 784.3.s.j 8
84.n even 6 2 784.3.s.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.3.b.a 4 3.b odd 2 1
98.3.b.a 4 21.c even 2 1
98.3.d.b 8 21.g even 6 2
98.3.d.b 8 21.h odd 6 2
784.3.c.b 4 12.b even 2 1
784.3.c.b 4 84.h odd 2 1
784.3.s.j 8 84.j odd 6 2
784.3.s.j 8 84.n even 6 2
882.3.c.a 4 1.a even 1 1 trivial
882.3.c.a 4 7.b odd 2 1 inner
882.3.n.j 8 7.c even 3 2
882.3.n.j 8 7.d odd 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} + 68 T_{5}^{2} + 98$$ $$T_{23}^{2} + 40 T_{23} + 112$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$98 + 68 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 112 + 24 T + T^{2} )^{2}$$
$13$ $$98 + 68 T^{2} + T^{4}$$
$17$ $$13122 + 324 T^{2} + T^{4}$$
$19$ $$128 + 928 T^{2} + T^{4}$$
$23$ $$( 112 + 40 T + T^{2} )^{2}$$
$29$ $$( -82 - 32 T + T^{2} )^{2}$$
$31$ $$512 + 320 T^{2} + T^{4}$$
$37$ $$( 1006 + 64 T + T^{2} )^{2}$$
$41$ $$164738 + 1028 T^{2} + T^{4}$$
$43$ $$( 1808 + 88 T + T^{2} )^{2}$$
$47$ $$4524032 + 4736 T^{2} + T^{4}$$
$53$ $$( 64 - 48 T + T^{2} )^{2}$$
$59$ $$36992 + 9760 T^{2} + T^{4}$$
$61$ $$41714978 + 12932 T^{2} + T^{4}$$
$67$ $$( 2944 - 128 T + T^{2} )^{2}$$
$71$ $$( -3136 + 16 T + T^{2} )^{2}$$
$73$ $$42154562 + 13444 T^{2} + T^{4}$$
$79$ $$( 1504 + 96 T + T^{2} )^{2}$$
$83$ $$1812608 + 2848 T^{2} + T^{4}$$
$89$ $$269378 + 4420 T^{2} + T^{4}$$
$97$ $$54100802 + 26500 T^{2} + T^{4}$$