# Properties

 Label 882.4.a.ba 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: $$\Q(\sqrt{193})$$ Defining polynomial: $$x^{2} - x - 48$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{193})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 4 q^{4} + ( 4 - \beta ) q^{5} -8 q^{8} +O(q^{10})$$ $$q -2 q^{2} + 4 q^{4} + ( 4 - \beta ) q^{5} -8 q^{8} + ( -8 + 2 \beta ) q^{10} + ( -16 + 7 \beta ) q^{11} + ( -27 + 5 \beta ) q^{13} + 16 q^{16} + ( 44 + 10 \beta ) q^{17} + ( -61 + 3 \beta ) q^{19} + ( 16 - 4 \beta ) q^{20} + ( 32 - 14 \beta ) q^{22} + ( 68 - 14 \beta ) q^{23} + ( -61 - 7 \beta ) q^{25} + ( 54 - 10 \beta ) q^{26} + ( 40 - 7 \beta ) q^{29} + ( 63 - 28 \beta ) q^{31} -32 q^{32} + ( -88 - 20 \beta ) q^{34} + ( 155 - 21 \beta ) q^{37} + ( 122 - 6 \beta ) q^{38} + ( -32 + 8 \beta ) q^{40} + 168 q^{41} + ( 143 + 21 \beta ) q^{43} + ( -64 + 28 \beta ) q^{44} + ( -136 + 28 \beta ) q^{46} + ( 324 + 24 \beta ) q^{47} + ( 122 + 14 \beta ) q^{50} + ( -108 + 20 \beta ) q^{52} + ( -156 - 63 \beta ) q^{53} + ( -400 + 37 \beta ) q^{55} + ( -80 + 14 \beta ) q^{58} + ( 412 - 61 \beta ) q^{59} + ( -186 - 34 \beta ) q^{61} + ( -126 + 56 \beta ) q^{62} + 64 q^{64} + ( -348 + 42 \beta ) q^{65} + ( -503 - 35 \beta ) q^{67} + ( 176 + 40 \beta ) q^{68} + ( 812 + 28 \beta ) q^{71} + ( -143 + 97 \beta ) q^{73} + ( -310 + 42 \beta ) q^{74} + ( -244 + 12 \beta ) q^{76} + ( 213 + 98 \beta ) q^{79} + ( 64 - 16 \beta ) q^{80} -336 q^{82} + ( 188 - 89 \beta ) q^{83} + ( -304 - 14 \beta ) q^{85} + ( -286 - 42 \beta ) q^{86} + ( 128 - 56 \beta ) q^{88} + ( 1224 - 54 \beta ) q^{89} + ( 272 - 56 \beta ) q^{92} + ( -648 - 48 \beta ) q^{94} + ( -388 + 70 \beta ) q^{95} + ( 46 - 155 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 8q^{4} + 7q^{5} - 16q^{8} + O(q^{10})$$ $$2q - 4q^{2} + 8q^{4} + 7q^{5} - 16q^{8} - 14q^{10} - 25q^{11} - 49q^{13} + 32q^{16} + 98q^{17} - 119q^{19} + 28q^{20} + 50q^{22} + 122q^{23} - 129q^{25} + 98q^{26} + 73q^{29} + 98q^{31} - 64q^{32} - 196q^{34} + 289q^{37} + 238q^{38} - 56q^{40} + 336q^{41} + 307q^{43} - 100q^{44} - 244q^{46} + 672q^{47} + 258q^{50} - 196q^{52} - 375q^{53} - 763q^{55} - 146q^{58} + 763q^{59} - 406q^{61} - 196q^{62} + 128q^{64} - 654q^{65} - 1041q^{67} + 392q^{68} + 1652q^{71} - 189q^{73} - 578q^{74} - 476q^{76} + 524q^{79} + 112q^{80} - 672q^{82} + 287q^{83} - 622q^{85} - 614q^{86} + 200q^{88} + 2394q^{89} + 488q^{92} - 1344q^{94} - 706q^{95} - 63q^{97} + O(q^{100})$$

## 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
 7.44622 −6.44622
−2.00000 0 4.00000 −3.44622 0 0 −8.00000 0 6.89244
1.2 −2.00000 0 4.00000 10.4462 0 0 −8.00000 0 −20.8924
 $$n$$: 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.ba 2
3.b odd 2 1 882.4.a.bd 2
7.b odd 2 1 882.4.a.u 2
7.c even 3 2 126.4.g.f yes 4
7.d odd 6 2 882.4.g.bj 4
21.c even 2 1 882.4.a.bh 2
21.g even 6 2 882.4.g.z 4
21.h odd 6 2 126.4.g.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.g.e 4 21.h odd 6 2
126.4.g.f yes 4 7.c even 3 2
882.4.a.u 2 7.b odd 2 1
882.4.a.ba 2 1.a even 1 1 trivial
882.4.a.bd 2 3.b odd 2 1
882.4.a.bh 2 21.c even 2 1
882.4.g.z 4 21.g even 6 2
882.4.g.bj 4 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5}^{2} - 7 T_{5} - 36$$ $$T_{11}^{2} + 25 T_{11} - 2208$$ $$T_{13}^{2} + 49 T_{13} - 606$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-36 - 7 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-2208 + 25 T + T^{2}$$
$13$ $$-606 + 49 T + T^{2}$$
$17$ $$-2424 - 98 T + T^{2}$$
$19$ $$3106 + 119 T + T^{2}$$
$23$ $$-5736 - 122 T + T^{2}$$
$29$ $$-1032 - 73 T + T^{2}$$
$31$ $$-35427 - 98 T + T^{2}$$
$37$ $$-398 - 289 T + T^{2}$$
$41$ $$( -168 + T )^{2}$$
$43$ $$2284 - 307 T + T^{2}$$
$47$ $$85104 - 672 T + T^{2}$$
$53$ $$-156348 + 375 T + T^{2}$$
$59$ $$-33996 - 763 T + T^{2}$$
$61$ $$-14568 + 406 T + T^{2}$$
$67$ $$211814 + 1041 T + T^{2}$$
$71$ $$644448 - 1652 T + T^{2}$$
$73$ $$-445054 + 189 T + T^{2}$$
$79$ $$-394749 - 524 T + T^{2}$$
$83$ $$-361596 - 287 T + T^{2}$$
$89$ $$1292112 - 2394 T + T^{2}$$
$97$ $$-1158214 + 63 T + T^{2}$$