Properties

 Label 882.5.c.c Level $882$ Weight $5$ Character orbit 882.c Analytic conductor $91.172$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 882.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$91.1723074400$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.2048.2 Defining polynomial: $$x^{4} + 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 98) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{2} q^{2} + 8 q^{4} + ( -5 \beta_{1} - 13 \beta_{3} ) q^{5} + 16 \beta_{2} q^{8} +O(q^{10})$$ $$q + 2 \beta_{2} q^{2} + 8 q^{4} + ( -5 \beta_{1} - 13 \beta_{3} ) q^{5} + 16 \beta_{2} q^{8} + ( -16 \beta_{1} - 36 \beta_{3} ) q^{10} + ( -6 + 103 \beta_{2} ) q^{11} + ( -101 \beta_{1} + 97 \beta_{3} ) q^{13} + 64 q^{16} + ( -282 \beta_{1} - 177 \beta_{3} ) q^{17} + ( -207 \beta_{1} + 10 \beta_{3} ) q^{19} + ( -40 \beta_{1} - 104 \beta_{3} ) q^{20} + ( 412 - 12 \beta_{2} ) q^{22} + ( -26 - 144 \beta_{2} ) q^{23} + ( 237 - 274 \beta_{2} ) q^{25} + ( 396 \beta_{1} - 8 \beta_{3} ) q^{26} + ( 352 + 22 \beta_{2} ) q^{29} + ( 698 \beta_{1} - 74 \beta_{3} ) q^{31} + 128 \beta_{2} q^{32} + ( 210 \beta_{1} - 918 \beta_{3} ) q^{34} + ( -848 - 36 \beta_{2} ) q^{37} + ( 434 \beta_{1} - 394 \beta_{3} ) q^{38} + ( -128 \beta_{1} - 288 \beta_{3} ) q^{40} + ( 251 \beta_{1} - 460 \beta_{3} ) q^{41} + ( -506 + 1175 \beta_{2} ) q^{43} + ( -48 + 824 \beta_{2} ) q^{44} + ( -576 - 52 \beta_{2} ) q^{46} + ( -1732 \beta_{1} + 1418 \beta_{3} ) q^{47} + ( -1096 + 474 \beta_{2} ) q^{50} + ( -808 \beta_{1} + 776 \beta_{3} ) q^{52} + ( 4170 + 706 \beta_{2} ) q^{53} + ( -794 \beta_{1} - 1776 \beta_{3} ) q^{55} + ( 88 + 704 \beta_{2} ) q^{58} + ( 1379 \beta_{1} - 2820 \beta_{3} ) q^{59} + ( 1523 \beta_{1} + 365 \beta_{3} ) q^{61} + ( -1544 \beta_{1} + 1248 \beta_{3} ) q^{62} + 512 q^{64} + ( 1512 + 938 \beta_{2} ) q^{65} + ( -5204 - 786 \beta_{2} ) q^{67} + ( -2256 \beta_{1} - 1416 \beta_{3} ) q^{68} + ( 496 - 1244 \beta_{2} ) q^{71} + ( 1385 \beta_{1} - 2736 \beta_{3} ) q^{73} + ( -144 - 1696 \beta_{2} ) q^{74} + ( -1656 \beta_{1} + 80 \beta_{3} ) q^{76} + ( -7404 - 2270 \beta_{2} ) q^{79} + ( -320 \beta_{1} - 832 \beta_{3} ) q^{80} + ( -1422 \beta_{1} - 418 \beta_{3} ) q^{82} + ( -4703 \beta_{1} - 1164 \beta_{3} ) q^{83} + ( -7422 - 5442 \beta_{2} ) q^{85} + ( 4700 - 1012 \beta_{2} ) q^{86} + ( 3296 - 96 \beta_{2} ) q^{88} + ( 3573 \beta_{1} - 3746 \beta_{3} ) q^{89} + ( -208 - 1152 \beta_{2} ) q^{92} + ( 6300 \beta_{1} - 628 \beta_{3} ) q^{94} + ( -1810 - 1476 \beta_{2} ) q^{95} + ( -5324 \beta_{1} + 141 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 32q^{4} + O(q^{10})$$ $$4q + 32q^{4} - 24q^{11} + 256q^{16} + 1648q^{22} - 104q^{23} + 948q^{25} + 1408q^{29} - 3392q^{37} - 2024q^{43} - 192q^{44} - 2304q^{46} - 4384q^{50} + 16680q^{53} + 352q^{58} + 2048q^{64} + 6048q^{65} - 20816q^{67} + 1984q^{71} - 576q^{74} - 29616q^{79} - 29688q^{85} + 18800q^{86} + 13184q^{88} - 832q^{92} - 7240q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{1}$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
685.1
 − 1.84776i 1.84776i 0.765367i − 0.765367i
−2.82843 0 8.00000 0.710974i 0 0 −22.6274 0 2.01094i
685.2 −2.82843 0 8.00000 0.710974i 0 0 −22.6274 0 2.01094i
685.3 2.82843 0 8.00000 27.8477i 0 0 22.6274 0 78.7652i
685.4 2.82843 0 8.00000 27.8477i 0 0 22.6274 0 78.7652i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 882.5.c.c 4
3.b odd 2 1 98.5.b.a 4
7.b odd 2 1 inner 882.5.c.c 4
12.b even 2 1 784.5.c.a 4
21.c even 2 1 98.5.b.a 4
21.g even 6 2 98.5.d.c 8
21.h odd 6 2 98.5.d.c 8
84.h odd 2 1 784.5.c.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.5.b.a 4 3.b odd 2 1
98.5.b.a 4 21.c even 2 1
98.5.d.c 8 21.g even 6 2
98.5.d.c 8 21.h odd 6 2
784.5.c.a 4 12.b even 2 1
784.5.c.a 4 84.h odd 2 1
882.5.c.c 4 1.a even 1 1 trivial
882.5.c.c 4 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} + 776 T_{5}^{2} + 392$$ $$T_{11}^{2} + 12 T_{11} - 21182$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -8 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$392 + 776 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -21182 + 12 T + T^{2} )^{2}$$
$13$ $$707030408 + 78440 T^{2} + T^{4}$$
$17$ $$43821617058 + 443412 T^{2} + T^{4}$$
$19$ $$2981309762 + 171796 T^{2} + T^{4}$$
$23$ $$( -40796 + 52 T + T^{2} )^{2}$$
$29$ $$( 122936 - 704 T + T^{2} )^{2}$$
$31$ $$286409447552 + 1970720 T^{2} + T^{4}$$
$37$ $$( 716512 + 1696 T + T^{2} )^{2}$$
$41$ $$288069342722 + 1098404 T^{2} + T^{4}$$
$43$ $$( -2505214 + 1012 T + T^{2} )^{2}$$
$47$ $$30777535627808 + 20042192 T^{2} + T^{4}$$
$53$ $$( 16392028 - 8340 T + T^{2} )^{2}$$
$59$ $$382444812731522 + 39416164 T^{2} + T^{4}$$
$61$ $$21754848065672 + 9811016 T^{2} + T^{4}$$
$67$ $$( 25846024 + 10408 T + T^{2} )^{2}$$
$71$ $$( -2849056 - 992 T + T^{2} )^{2}$$
$73$ $$345644675616962 + 37615684 T^{2} + T^{4}$$
$79$ $$( 44513416 + 14808 T + T^{2} )^{2}$$
$83$ $$2011288822677218 + 93892420 T^{2} + T^{4}$$
$89$ $$1571934000441218 + 107195380 T^{2} + T^{4}$$
$97$ $$1439024660341058 + 113459428 T^{2} + T^{4}$$