# Properties

 Label 882.4.a.bf Level $882$ Weight $4$ Character orbit 882.a Self dual yes Analytic conductor $52.040$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 882.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$52.0396846251$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{58})$$ Defining polynomial: $$x^{2} - 58$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{58}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} + \beta q^{5} + 8 q^{8} +O(q^{10})$$ $$q + 2 q^{2} + 4 q^{4} + \beta q^{5} + 8 q^{8} + 2 \beta q^{10} + 2 q^{11} -2 \beta q^{13} + 16 q^{16} -3 \beta q^{17} + 10 \beta q^{19} + 4 \beta q^{20} + 4 q^{22} + 30 q^{23} + 107 q^{25} -4 \beta q^{26} + 212 q^{29} + 14 \beta q^{31} + 32 q^{32} -6 \beta q^{34} + 246 q^{37} + 20 \beta q^{38} + 8 \beta q^{40} -21 \beta q^{41} -284 q^{43} + 8 q^{44} + 60 q^{46} + 4 \beta q^{47} + 214 q^{50} -8 \beta q^{52} + 548 q^{53} + 2 \beta q^{55} + 424 q^{58} -44 \beta q^{59} -34 \beta q^{61} + 28 \beta q^{62} + 64 q^{64} -464 q^{65} + 652 q^{67} -12 \beta q^{68} + 770 q^{71} -64 \beta q^{73} + 492 q^{74} + 40 \beta q^{76} + 472 q^{79} + 16 \beta q^{80} -42 \beta q^{82} + 12 \beta q^{83} -696 q^{85} -568 q^{86} + 16 q^{88} + 47 \beta q^{89} + 120 q^{92} + 8 \beta q^{94} + 2320 q^{95} + 20 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} + 8q^{4} + 16q^{8} + O(q^{10})$$ $$2q + 4q^{2} + 8q^{4} + 16q^{8} + 4q^{11} + 32q^{16} + 8q^{22} + 60q^{23} + 214q^{25} + 424q^{29} + 64q^{32} + 492q^{37} - 568q^{43} + 16q^{44} + 120q^{46} + 428q^{50} + 1096q^{53} + 848q^{58} + 128q^{64} - 928q^{65} + 1304q^{67} + 1540q^{71} + 984q^{74} + 944q^{79} - 1392q^{85} - 1136q^{86} + 32q^{88} + 240q^{92} + 4640q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −7.61577 7.61577
2.00000 0 4.00000 −15.2315 0 0 8.00000 0 −30.4631
1.2 2.00000 0 4.00000 15.2315 0 0 8.00000 0 30.4631
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## 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 882.4.a.bf yes 2
3.b odd 2 1 882.4.a.x 2
7.b odd 2 1 inner 882.4.a.bf yes 2
7.c even 3 2 882.4.g.bb 4
7.d odd 6 2 882.4.g.bb 4
21.c even 2 1 882.4.a.x 2
21.g even 6 2 882.4.g.bh 4
21.h odd 6 2 882.4.g.bh 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.4.a.x 2 3.b odd 2 1
882.4.a.x 2 21.c even 2 1
882.4.a.bf yes 2 1.a even 1 1 trivial
882.4.a.bf yes 2 7.b odd 2 1 inner
882.4.g.bb 4 7.c even 3 2
882.4.g.bb 4 7.d odd 6 2
882.4.g.bh 4 21.g even 6 2
882.4.g.bh 4 21.h odd 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5}^{2} - 232$$ $$T_{11} - 2$$ $$T_{13}^{2} - 928$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-232 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$-928 + T^{2}$$
$17$ $$-2088 + T^{2}$$
$19$ $$-23200 + T^{2}$$
$23$ $$( -30 + T )^{2}$$
$29$ $$( -212 + T )^{2}$$
$31$ $$-45472 + T^{2}$$
$37$ $$( -246 + T )^{2}$$
$41$ $$-102312 + T^{2}$$
$43$ $$( 284 + T )^{2}$$
$47$ $$-3712 + T^{2}$$
$53$ $$( -548 + T )^{2}$$
$59$ $$-449152 + T^{2}$$
$61$ $$-268192 + T^{2}$$
$67$ $$( -652 + T )^{2}$$
$71$ $$( -770 + T )^{2}$$
$73$ $$-950272 + T^{2}$$
$79$ $$( -472 + T )^{2}$$
$83$ $$-33408 + T^{2}$$
$89$ $$-512488 + T^{2}$$
$97$ $$-92800 + T^{2}$$