# Properties

 Label 882.4.a.z 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: $$\Q(\sqrt{1345})$$ Defining polynomial: $$x^{2} - x - 336$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{1345})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 4 q^{4} + ( 3 - \beta ) q^{5} -8 q^{8} +O(q^{10})$$ $$q -2 q^{2} + 4 q^{4} + ( 3 - \beta ) q^{5} -8 q^{8} + ( -6 + 2 \beta ) q^{10} + ( -33 - \beta ) q^{11} + ( -20 - \beta ) q^{13} + 16 q^{16} + ( -48 + 4 \beta ) q^{17} + ( -20 - 3 \beta ) q^{19} + ( 12 - 4 \beta ) q^{20} + ( 66 + 2 \beta ) q^{22} + ( -72 - 4 \beta ) q^{23} + ( 220 - 5 \beta ) q^{25} + ( 40 + 2 \beta ) q^{26} + ( -33 - 11 \beta ) q^{29} + ( 259 + 2 \beta ) q^{31} -32 q^{32} + ( 96 - 8 \beta ) q^{34} + ( 8 - 9 \beta ) q^{37} + ( 40 + 6 \beta ) q^{38} + ( -24 + 8 \beta ) q^{40} + ( -216 + 6 \beta ) q^{41} + ( -46 - 15 \beta ) q^{43} + ( -132 - 4 \beta ) q^{44} + ( 144 + 8 \beta ) q^{46} + ( 294 - 12 \beta ) q^{47} + ( -440 + 10 \beta ) q^{50} + ( -80 - 4 \beta ) q^{52} + ( 123 - 3 \beta ) q^{53} + ( 237 + 31 \beta ) q^{55} + ( 66 + 22 \beta ) q^{58} + ( 9 - 25 \beta ) q^{59} + ( -110 - 4 \beta ) q^{61} + ( -518 - 4 \beta ) q^{62} + 64 q^{64} + ( 276 + 18 \beta ) q^{65} + ( 338 + 11 \beta ) q^{67} + ( -192 + 16 \beta ) q^{68} + ( -246 + 20 \beta ) q^{71} + ( 478 - 35 \beta ) q^{73} + ( -16 + 18 \beta ) q^{74} + ( -80 - 12 \beta ) q^{76} + ( -259 - 8 \beta ) q^{79} + ( 48 - 16 \beta ) q^{80} + ( 432 - 12 \beta ) q^{82} + ( -123 + 25 \beta ) q^{83} + ( -1488 + 56 \beta ) q^{85} + ( 92 + 30 \beta ) q^{86} + ( 264 + 8 \beta ) q^{88} + ( 366 + 42 \beta ) q^{89} + ( -288 - 16 \beta ) q^{92} + ( -588 + 24 \beta ) q^{94} + ( 948 + 14 \beta ) q^{95} + ( -959 - 35 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 8q^{4} + 5q^{5} - 16q^{8} + O(q^{10})$$ $$2q - 4q^{2} + 8q^{4} + 5q^{5} - 16q^{8} - 10q^{10} - 67q^{11} - 41q^{13} + 32q^{16} - 92q^{17} - 43q^{19} + 20q^{20} + 134q^{22} - 148q^{23} + 435q^{25} + 82q^{26} - 77q^{29} + 520q^{31} - 64q^{32} + 184q^{34} + 7q^{37} + 86q^{38} - 40q^{40} - 426q^{41} - 107q^{43} - 268q^{44} + 296q^{46} + 576q^{47} - 870q^{50} - 164q^{52} + 243q^{53} + 505q^{55} + 154q^{58} - 7q^{59} - 224q^{61} - 1040q^{62} + 128q^{64} + 570q^{65} + 687q^{67} - 368q^{68} - 472q^{71} + 921q^{73} - 14q^{74} - 172q^{76} - 526q^{79} + 80q^{80} + 852q^{82} - 221q^{83} - 2920q^{85} + 214q^{86} + 536q^{88} + 774q^{89} - 592q^{92} - 1152q^{94} + 1910q^{95} - 1953q^{97} + O(q^{100})$$

## 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
 18.8371 −17.8371
−2.00000 0 4.00000 −15.8371 0 0 −8.00000 0 31.6742
1.2 −2.00000 0 4.00000 20.8371 0 0 −8.00000 0 −41.6742
 $$n$$: 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.z 2
3.b odd 2 1 294.4.a.m 2
7.b odd 2 1 882.4.a.v 2
7.c even 3 2 882.4.g.bf 4
7.d odd 6 2 126.4.g.g 4
12.b even 2 1 2352.4.a.ca 2
21.c even 2 1 294.4.a.n 2
21.g even 6 2 42.4.e.c 4
21.h odd 6 2 294.4.e.l 4
84.h odd 2 1 2352.4.a.bq 2
84.j odd 6 2 336.4.q.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.c 4 21.g even 6 2
126.4.g.g 4 7.d odd 6 2
294.4.a.m 2 3.b odd 2 1
294.4.a.n 2 21.c even 2 1
294.4.e.l 4 21.h odd 6 2
336.4.q.j 4 84.j odd 6 2
882.4.a.v 2 7.b odd 2 1
882.4.a.z 2 1.a even 1 1 trivial
882.4.g.bf 4 7.c even 3 2
2352.4.a.bq 2 84.h odd 2 1
2352.4.a.ca 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5}^{2} - 5 T_{5} - 330$$ $$T_{11}^{2} + 67 T_{11} + 786$$ $$T_{13}^{2} + 41 T_{13} + 84$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-330 - 5 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$786 + 67 T + T^{2}$$
$13$ $$84 + 41 T + T^{2}$$
$17$ $$-3264 + 92 T + T^{2}$$
$19$ $$-2564 + 43 T + T^{2}$$
$23$ $$96 + 148 T + T^{2}$$
$29$ $$-39204 + 77 T + T^{2}$$
$31$ $$66255 - 520 T + T^{2}$$
$37$ $$-27224 - 7 T + T^{2}$$
$41$ $$33264 + 426 T + T^{2}$$
$43$ $$-72794 + 107 T + T^{2}$$
$47$ $$34524 - 576 T + T^{2}$$
$53$ $$11736 - 243 T + T^{2}$$
$59$ $$-210144 + 7 T + T^{2}$$
$61$ $$7164 + 224 T + T^{2}$$
$67$ $$77306 - 687 T + T^{2}$$
$71$ $$-78804 + 472 T + T^{2}$$
$73$ $$-199846 - 921 T + T^{2}$$
$79$ $$47649 + 526 T + T^{2}$$
$83$ $$-197946 + 221 T + T^{2}$$
$89$ $$-443376 - 774 T + T^{2}$$
$97$ $$541646 + 1953 T + T^{2}$$