# Properties

 Label 882.3.s.h Level $882$ Weight $3$ Character orbit 882.s Analytic conductor $24.033$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 882.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.0327593166$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.621801639936.1 Defining polynomial: $$x^{8} - 4 x^{7} - 34 x^{6} + 116 x^{5} + 413 x^{4} - 1024 x^{3} - 1664 x^{2} + 2196 x + 4467$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{2} + 2 \beta_{1} q^{4} -\beta_{2} q^{5} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{8} +O(q^{10})$$ $$q -\beta_{4} q^{2} + 2 \beta_{1} q^{4} -\beta_{2} q^{5} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{8} + ( -\beta_{6} - \beta_{7} ) q^{10} + 2 \beta_{5} q^{11} -\beta_{6} q^{13} + ( -4 + 4 \beta_{1} ) q^{16} + 3 \beta_{3} q^{17} + 2 \beta_{7} q^{19} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{20} + 4 q^{22} -30 \beta_{4} q^{23} + 49 \beta_{1} q^{25} -2 \beta_{2} q^{26} + ( -11 \beta_{4} - 11 \beta_{5} ) q^{29} + ( 2 \beta_{6} + 2 \beta_{7} ) q^{31} -4 \beta_{5} q^{32} + 3 \beta_{6} q^{34} + ( 6 - 6 \beta_{1} ) q^{37} -4 \beta_{3} q^{38} -2 \beta_{7} q^{40} + ( 3 \beta_{2} - 3 \beta_{3} ) q^{41} -68 q^{43} -4 \beta_{4} q^{44} + 60 \beta_{1} q^{46} + 8 \beta_{2} q^{47} + ( -49 \beta_{4} - 49 \beta_{5} ) q^{50} + ( -2 \beta_{6} - 2 \beta_{7} ) q^{52} + 29 \beta_{5} q^{53} -2 \beta_{6} q^{55} + ( -22 + 22 \beta_{1} ) q^{58} -8 \beta_{3} q^{59} -8 \beta_{7} q^{61} + ( 4 \beta_{2} - 4 \beta_{3} ) q^{62} -8 q^{64} -74 \beta_{4} q^{65} + 104 \beta_{1} q^{67} + 6 \beta_{2} q^{68} + ( -50 \beta_{4} - 50 \beta_{5} ) q^{71} + ( 5 \beta_{6} + 5 \beta_{7} ) q^{73} + 6 \beta_{5} q^{74} -4 \beta_{6} q^{76} + ( 20 - 20 \beta_{1} ) q^{79} + 4 \beta_{3} q^{80} + 3 \beta_{7} q^{82} -222 q^{85} + 68 \beta_{4} q^{86} + 8 \beta_{1} q^{88} -11 \beta_{2} q^{89} + ( -60 \beta_{4} - 60 \beta_{5} ) q^{92} + ( 8 \beta_{6} + 8 \beta_{7} ) q^{94} + 148 \beta_{5} q^{95} + 13 \beta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{4} + O(q^{10})$$ $$8 q + 8 q^{4} - 16 q^{16} + 32 q^{22} + 196 q^{25} + 24 q^{37} - 544 q^{43} + 240 q^{46} - 88 q^{58} - 64 q^{64} + 416 q^{67} + 80 q^{79} - 1776 q^{85} + 32 q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 34 x^{6} + 116 x^{5} + 413 x^{4} - 1024 x^{3} - 1664 x^{2} + 2196 x + 4467$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{6} + 6 \nu^{5} + 91 \nu^{4} - 192 \nu^{3} - 1187 \nu^{2} + 1284 \nu + 4425$$$$)/2418$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{6} - 6 \nu^{5} - 91 \nu^{4} + 192 \nu^{3} + 2396 \nu^{2} - 2493 \nu - 15306$$$$)/1209$$ $$\beta_{3}$$ $$=$$ $$($$$$-35 \nu^{6} + 105 \nu^{5} + 988 \nu^{4} - 2151 \nu^{3} - 8078 \nu^{2} + 9171 \nu + 4293$$$$)/2418$$ $$\beta_{4}$$ $$=$$ $$($$$$-724 \nu^{7} + 2534 \nu^{6} + 27020 \nu^{5} - 73885 \nu^{4} - 387688 \nu^{3} + 656684 \nu^{2} + 1520571 \nu - 872256$$$$)/1374633$$ $$\beta_{5}$$ $$=$$ $$($$$$-1838 \nu^{7} + 6433 \nu^{6} + 45811 \nu^{5} - 130610 \nu^{4} - 262721 \nu^{3} + 527908 \nu^{2} - 1269957 \nu + 542487$$$$)/2749266$$ $$\beta_{6}$$ $$=$$ $$($$$$-2896 \nu^{7} + 10136 \nu^{6} + 108080 \nu^{5} - 295540 \nu^{4} - 1550752 \nu^{3} + 2626736 \nu^{2} + 11580816 \nu - 6238290$$$$)/1374633$$ $$\beta_{7}$$ $$=$$ $$($$$$-8224 \nu^{7} + 28784 \nu^{6} + 291734 \nu^{5} - 801295 \nu^{4} - 3006376 \nu^{3} + 5325251 \nu^{2} + 564096 \nu - 1196985$$$$)/1374633$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - 4 \beta_{4} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - 4 \beta_{4} + 4 \beta_{2} + 8 \beta_{1} + 38$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{7} + 8 \beta_{6} - 4 \beta_{5} - 61 \beta_{4} + 3 \beta_{2} + 6 \beta_{1} + 28$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$12 \beta_{7} + 31 \beta_{6} - 16 \beta_{5} - 240 \beta_{4} - 16 \beta_{3} + 96 \beta_{2} + 472 \beta_{1} + 382$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$220 \beta_{7} + 309 \beta_{6} - 776 \beta_{5} - 2750 \beta_{4} - 40 \beta_{3} + 230 \beta_{2} + 1160 \beta_{1} + 862$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$315 \beta_{7} + 425 \beta_{6} - 1144 \beta_{5} - 3826 \beta_{4} - 424 \beta_{3} + 1054 \beta_{2} + 7110 \beta_{1} + 1086$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$5978 \beta_{7} + 5783 \beta_{6} - 31052 \beta_{5} - 59216 \beta_{4} - 2828 \beta_{3} + 6580 \beta_{2} + 45724 \beta_{1} + 4650$$$$)/4$$

## Character values

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

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-1 + \beta_{1}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
557.1
 −3.76613 + 0.707107i 2.31664 + 0.707107i 4.76613 − 0.707107i −1.31664 − 0.707107i −3.76613 − 0.707107i 2.31664 − 0.707107i 4.76613 + 0.707107i −1.31664 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i −7.44983 + 4.30116i 0 0 2.82843i 0 6.08276 10.5357i
557.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 7.44983 4.30116i 0 0 2.82843i 0 −6.08276 + 10.5357i
557.3 1.22474 0.707107i 0 1.00000 1.73205i −7.44983 + 4.30116i 0 0 2.82843i 0 −6.08276 + 10.5357i
557.4 1.22474 0.707107i 0 1.00000 1.73205i 7.44983 4.30116i 0 0 2.82843i 0 6.08276 10.5357i
863.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −7.44983 4.30116i 0 0 2.82843i 0 6.08276 + 10.5357i
863.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 7.44983 + 4.30116i 0 0 2.82843i 0 −6.08276 10.5357i
863.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −7.44983 4.30116i 0 0 2.82843i 0 −6.08276 10.5357i
863.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 7.44983 + 4.30116i 0 0 2.82843i 0 6.08276 + 10.5357i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 863.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.s.h 8
3.b odd 2 1 inner 882.3.s.h 8
7.b odd 2 1 inner 882.3.s.h 8
7.c even 3 1 882.3.b.g 4
7.c even 3 1 inner 882.3.s.h 8
7.d odd 6 1 882.3.b.g 4
7.d odd 6 1 inner 882.3.s.h 8
21.c even 2 1 inner 882.3.s.h 8
21.g even 6 1 882.3.b.g 4
21.g even 6 1 inner 882.3.s.h 8
21.h odd 6 1 882.3.b.g 4
21.h odd 6 1 inner 882.3.s.h 8

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} - 74 T_{5}^{2} + 5476$$ $$T_{11}^{4} - 8 T_{11}^{2} + 64$$ $$T_{13}^{2} - 148$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 5476 - 74 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 64 - 8 T^{2} + T^{4} )^{2}$$
$13$ $$( -148 + T^{2} )^{4}$$
$17$ $$( 443556 - 666 T^{2} + T^{4} )^{2}$$
$19$ $$( 350464 + 592 T^{2} + T^{4} )^{2}$$
$23$ $$( 3240000 - 1800 T^{2} + T^{4} )^{2}$$
$29$ $$( 242 + T^{2} )^{4}$$
$31$ $$( 350464 + 592 T^{2} + T^{4} )^{2}$$
$37$ $$( 36 - 6 T + T^{2} )^{4}$$
$41$ $$( 666 + T^{2} )^{4}$$
$43$ $$( 68 + T )^{8}$$
$47$ $$( 22429696 - 4736 T^{2} + T^{4} )^{2}$$
$53$ $$( 2829124 - 1682 T^{2} + T^{4} )^{2}$$
$59$ $$( 22429696 - 4736 T^{2} + T^{4} )^{2}$$
$61$ $$( 89718784 + 9472 T^{2} + T^{4} )^{2}$$
$67$ $$( 10816 - 104 T + T^{2} )^{4}$$
$71$ $$( 5000 + T^{2} )^{4}$$
$73$ $$( 13690000 + 3700 T^{2} + T^{4} )^{2}$$
$79$ $$( 400 - 20 T + T^{2} )^{4}$$
$83$ $$T^{8}$$
$89$ $$( 80174116 - 8954 T^{2} + T^{4} )^{2}$$
$97$ $$( -25012 + T^{2} )^{4}$$