# Properties

 Label 882.4.a.v Level $882$ Weight $4$ Character orbit 882.a Self dual yes Analytic conductor $52.040$ Analytic rank $1$ 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 4 q^{4} + ( -2 - \beta ) q^{5} -8 q^{8} +O(q^{10})$$ $$q -2 q^{2} + 4 q^{4} + ( -2 - \beta ) q^{5} -8 q^{8} + ( 4 + 2 \beta ) q^{10} + ( -34 + \beta ) q^{11} + ( 21 - \beta ) q^{13} + 16 q^{16} + ( 44 + 4 \beta ) q^{17} + ( 23 - 3 \beta ) q^{19} + ( -8 - 4 \beta ) q^{20} + ( 68 - 2 \beta ) q^{22} + ( -76 + 4 \beta ) q^{23} + ( 215 + 5 \beta ) q^{25} + ( -42 + 2 \beta ) q^{26} + ( -44 + 11 \beta ) q^{29} + ( -261 + 2 \beta ) q^{31} -32 q^{32} + ( -88 - 8 \beta ) q^{34} + ( -1 + 9 \beta ) q^{37} + ( -46 + 6 \beta ) q^{38} + ( 16 + 8 \beta ) q^{40} + ( 210 + 6 \beta ) q^{41} + ( -61 + 15 \beta ) q^{43} + ( -136 + 4 \beta ) q^{44} + ( 152 - 8 \beta ) q^{46} + ( -282 - 12 \beta ) q^{47} + ( -430 - 10 \beta ) q^{50} + ( 84 - 4 \beta ) q^{52} + ( 120 + 3 \beta ) q^{53} + ( -268 + 31 \beta ) q^{55} + ( 88 - 22 \beta ) q^{58} + ( 16 - 25 \beta ) q^{59} + ( 114 - 4 \beta ) q^{61} + ( 522 - 4 \beta ) q^{62} + 64 q^{64} + ( 294 - 18 \beta ) q^{65} + ( 349 - 11 \beta ) q^{67} + ( 176 + 16 \beta ) q^{68} + ( -226 - 20 \beta ) q^{71} + ( -443 - 35 \beta ) q^{73} + ( 2 - 18 \beta ) q^{74} + ( 92 - 12 \beta ) q^{76} + ( -267 + 8 \beta ) q^{79} + ( -32 - 16 \beta ) q^{80} + ( -420 - 12 \beta ) q^{82} + ( 98 + 25 \beta ) q^{83} + ( -1432 - 56 \beta ) q^{85} + ( 122 - 30 \beta ) q^{86} + ( 272 - 8 \beta ) q^{88} + ( -408 + 42 \beta ) q^{89} + ( -304 + 16 \beta ) q^{92} + ( 564 + 24 \beta ) q^{94} + ( 962 - 14 \beta ) q^{95} + ( 994 - 35 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 8q^{4} - 5q^{5} - 16q^{8} + O(q^{10})$$ $$2q - 4q^{2} + 8q^{4} - 5q^{5} - 16q^{8} + 10q^{10} - 67q^{11} + 41q^{13} + 32q^{16} + 92q^{17} + 43q^{19} - 20q^{20} + 134q^{22} - 148q^{23} + 435q^{25} - 82q^{26} - 77q^{29} - 520q^{31} - 64q^{32} - 184q^{34} + 7q^{37} - 86q^{38} + 40q^{40} + 426q^{41} - 107q^{43} - 268q^{44} + 296q^{46} - 576q^{47} - 870q^{50} + 164q^{52} + 243q^{53} - 505q^{55} + 154q^{58} + 7q^{59} + 224q^{61} + 1040q^{62} + 128q^{64} + 570q^{65} + 687q^{67} + 368q^{68} - 472q^{71} - 921q^{73} - 14q^{74} + 172q^{76} - 526q^{79} - 80q^{80} - 852q^{82} + 221q^{83} - 2920q^{85} + 214q^{86} + 536q^{88} - 774q^{89} - 592q^{92} + 1152q^{94} + 1910q^{95} + 1953q^{97} + O(q^{100})$$

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

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