# Properties

 Label 882.4.a.bc Level $882$ Weight $4$ Character orbit 882.a Self dual yes Analytic conductor $52.040$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$7$$ Twist minimal: no (minimal twist has level 294) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 7\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} + ( -6 + \beta ) q^{5} + 8 q^{8} +O(q^{10})$$ $$q + 2 q^{2} + 4 q^{4} + ( -6 + \beta ) q^{5} + 8 q^{8} + ( -12 + 2 \beta ) q^{10} + ( 2 + 6 \beta ) q^{11} + ( 24 + 3 \beta ) q^{13} + 16 q^{16} + ( -42 + \beta ) q^{17} + ( -36 - 2 \beta ) q^{19} + ( -24 + 4 \beta ) q^{20} + ( 4 + 12 \beta ) q^{22} + ( 154 - 6 \beta ) q^{23} + ( 9 - 12 \beta ) q^{25} + ( 48 + 6 \beta ) q^{26} + ( 40 - 18 \beta ) q^{29} + ( 192 - 6 \beta ) q^{31} + 32 q^{32} + ( -84 + 2 \beta ) q^{34} + ( 268 - 12 \beta ) q^{37} + ( -72 - 4 \beta ) q^{38} + ( -48 + 8 \beta ) q^{40} + ( -378 - 5 \beta ) q^{41} + ( 200 + 24 \beta ) q^{43} + ( 8 + 24 \beta ) q^{44} + ( 308 - 12 \beta ) q^{46} + ( -156 + 10 \beta ) q^{47} + ( 18 - 24 \beta ) q^{50} + ( 96 + 12 \beta ) q^{52} + ( 26 + 24 \beta ) q^{53} + ( 576 - 34 \beta ) q^{55} + ( 80 - 36 \beta ) q^{58} + ( -432 - 2 \beta ) q^{59} + ( 708 - 13 \beta ) q^{61} + ( 384 - 12 \beta ) q^{62} + 64 q^{64} + ( 150 + 6 \beta ) q^{65} + ( 72 + 24 \beta ) q^{67} + ( -168 + 4 \beta ) q^{68} + ( 762 + 30 \beta ) q^{71} + ( 372 + 83 \beta ) q^{73} + ( 536 - 24 \beta ) q^{74} + ( -144 - 8 \beta ) q^{76} + ( 488 + 84 \beta ) q^{79} + ( -96 + 16 \beta ) q^{80} + ( -756 - 10 \beta ) q^{82} + ( 156 - 136 \beta ) q^{83} + ( 350 - 48 \beta ) q^{85} + ( 400 + 48 \beta ) q^{86} + ( 16 + 48 \beta ) q^{88} + ( -54 + 29 \beta ) q^{89} + ( 616 - 24 \beta ) q^{92} + ( -312 + 20 \beta ) q^{94} + ( 20 - 24 \beta ) q^{95} + ( -372 - 125 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} + 8q^{4} - 12q^{5} + 16q^{8} + O(q^{10})$$ $$2q + 4q^{2} + 8q^{4} - 12q^{5} + 16q^{8} - 24q^{10} + 4q^{11} + 48q^{13} + 32q^{16} - 84q^{17} - 72q^{19} - 48q^{20} + 8q^{22} + 308q^{23} + 18q^{25} + 96q^{26} + 80q^{29} + 384q^{31} + 64q^{32} - 168q^{34} + 536q^{37} - 144q^{38} - 96q^{40} - 756q^{41} + 400q^{43} + 16q^{44} + 616q^{46} - 312q^{47} + 36q^{50} + 192q^{52} + 52q^{53} + 1152q^{55} + 160q^{58} - 864q^{59} + 1416q^{61} + 768q^{62} + 128q^{64} + 300q^{65} + 144q^{67} - 336q^{68} + 1524q^{71} + 744q^{73} + 1072q^{74} - 288q^{76} + 976q^{79} - 192q^{80} - 1512q^{82} + 312q^{83} + 700q^{85} + 800q^{86} + 32q^{88} - 108q^{89} + 1232q^{92} - 624q^{94} + 40q^{95} - 744q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
2.00000 0 4.00000 −15.8995 0 0 8.00000 0 −31.7990
1.2 2.00000 0 4.00000 3.89949 0 0 8.00000 0 7.79899
 $$n$$: 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.a.bc 2
3.b odd 2 1 294.4.a.j 2
7.b odd 2 1 882.4.a.bi 2
7.c even 3 2 882.4.g.bd 4
7.d odd 6 2 882.4.g.y 4
12.b even 2 1 2352.4.a.cd 2
21.c even 2 1 294.4.a.k yes 2
21.g even 6 2 294.4.e.n 4
21.h odd 6 2 294.4.e.o 4
84.h odd 2 1 2352.4.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.j 2 3.b odd 2 1
294.4.a.k yes 2 21.c even 2 1
294.4.e.n 4 21.g even 6 2
294.4.e.o 4 21.h odd 6 2
882.4.a.bc 2 1.a even 1 1 trivial
882.4.a.bi 2 7.b odd 2 1
882.4.g.y 4 7.d odd 6 2
882.4.g.bd 4 7.c even 3 2
2352.4.a.bn 2 84.h odd 2 1
2352.4.a.cd 2 12.b even 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 12 T_{5} - 62$$ $$T_{11}^{2} - 4 T_{11} - 3524$$ $$T_{13}^{2} - 48 T_{13} - 306$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-62 + 12 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-3524 - 4 T + T^{2}$$
$13$ $$-306 - 48 T + T^{2}$$
$17$ $$1666 + 84 T + T^{2}$$
$19$ $$904 + 72 T + T^{2}$$
$23$ $$20188 - 308 T + T^{2}$$
$29$ $$-30152 - 80 T + T^{2}$$
$31$ $$33336 - 384 T + T^{2}$$
$37$ $$57712 - 536 T + T^{2}$$
$41$ $$140434 + 756 T + T^{2}$$
$43$ $$-16448 - 400 T + T^{2}$$
$47$ $$14536 + 312 T + T^{2}$$
$53$ $$-55772 - 52 T + T^{2}$$
$59$ $$186232 + 864 T + T^{2}$$
$61$ $$484702 - 1416 T + T^{2}$$
$67$ $$-51264 - 144 T + T^{2}$$
$71$ $$492444 - 1524 T + T^{2}$$
$73$ $$-536738 - 744 T + T^{2}$$
$79$ $$-453344 - 976 T + T^{2}$$
$83$ $$-1788272 - 312 T + T^{2}$$
$89$ $$-79502 + 108 T + T^{2}$$
$97$ $$-1392866 + 744 T + T^{2}$$