# Properties

 Label 882.3.b.h Level $882$ Weight $3$ Character orbit 882.b Analytic conductor $24.033$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$24.0327593166$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} -2 q^{4} + 4 \zeta_{8}^{2} q^{5} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} -2 q^{4} + 4 \zeta_{8}^{2} q^{5} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{10} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{11} + ( 9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{13} + 4 q^{16} + 4 \zeta_{8}^{2} q^{17} + ( 16 \zeta_{8} - 16 \zeta_{8}^{3} ) q^{19} -8 \zeta_{8}^{2} q^{20} + 4 q^{22} + ( 26 \zeta_{8} + 26 \zeta_{8}^{3} ) q^{23} + 9 q^{25} + 18 \zeta_{8}^{2} q^{26} + ( 23 \zeta_{8} + 23 \zeta_{8}^{3} ) q^{29} + ( -36 \zeta_{8} + 36 \zeta_{8}^{3} ) q^{31} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{32} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{34} -32 q^{37} + 32 \zeta_{8}^{2} q^{38} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{40} -38 \zeta_{8}^{2} q^{41} + 20 q^{43} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{44} -52 q^{46} -20 \zeta_{8}^{2} q^{47} + ( 9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{50} + ( -18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{52} + ( 67 \zeta_{8} + 67 \zeta_{8}^{3} ) q^{53} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{55} -46 q^{58} + 4 \zeta_{8}^{2} q^{59} + ( 59 \zeta_{8} - 59 \zeta_{8}^{3} ) q^{61} -72 \zeta_{8}^{2} q^{62} -8 q^{64} + ( 36 \zeta_{8} + 36 \zeta_{8}^{3} ) q^{65} -48 q^{67} -8 \zeta_{8}^{2} q^{68} + ( 54 \zeta_{8} + 54 \zeta_{8}^{3} ) q^{71} + ( -85 \zeta_{8} + 85 \zeta_{8}^{3} ) q^{73} + ( -32 \zeta_{8} - 32 \zeta_{8}^{3} ) q^{74} + ( -32 \zeta_{8} + 32 \zeta_{8}^{3} ) q^{76} -148 q^{79} + 16 \zeta_{8}^{2} q^{80} + ( 38 \zeta_{8} - 38 \zeta_{8}^{3} ) q^{82} + 80 \zeta_{8}^{2} q^{83} -16 q^{85} + ( 20 \zeta_{8} + 20 \zeta_{8}^{3} ) q^{86} -8 q^{88} -106 \zeta_{8}^{2} q^{89} + ( -52 \zeta_{8} - 52 \zeta_{8}^{3} ) q^{92} + ( 20 \zeta_{8} - 20 \zeta_{8}^{3} ) q^{94} + ( 64 \zeta_{8} + 64 \zeta_{8}^{3} ) q^{95} + ( 109 \zeta_{8} - 109 \zeta_{8}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} + O(q^{10})$$ $$4 q - 8 q^{4} + 16 q^{16} + 16 q^{22} + 36 q^{25} - 128 q^{37} + 80 q^{43} - 208 q^{46} - 184 q^{58} - 32 q^{64} - 192 q^{67} - 592 q^{79} - 64 q^{85} - 32 q^{88} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
1.41421i 0 −2.00000 4.00000i 0 0 2.82843i 0 −5.65685
197.2 1.41421i 0 −2.00000 4.00000i 0 0 2.82843i 0 5.65685
197.3 1.41421i 0 −2.00000 4.00000i 0 0 2.82843i 0 5.65685
197.4 1.41421i 0 −2.00000 4.00000i 0 0 2.82843i 0 −5.65685
 $$n$$: 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
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{5}^{2} + 16$$ $$T_{11}^{2} + 8$$ $$T_{13}^{2} - 162$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( 16 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( 8 + T^{2} )^{2}$$
$13$ $$( -162 + T^{2} )^{2}$$
$17$ $$( 16 + T^{2} )^{2}$$
$19$ $$( -512 + T^{2} )^{2}$$
$23$ $$( 1352 + T^{2} )^{2}$$
$29$ $$( 1058 + T^{2} )^{2}$$
$31$ $$( -2592 + T^{2} )^{2}$$
$37$ $$( 32 + T )^{4}$$
$41$ $$( 1444 + T^{2} )^{2}$$
$43$ $$( -20 + T )^{4}$$
$47$ $$( 400 + T^{2} )^{2}$$
$53$ $$( 8978 + T^{2} )^{2}$$
$59$ $$( 16 + T^{2} )^{2}$$
$61$ $$( -6962 + T^{2} )^{2}$$
$67$ $$( 48 + T )^{4}$$
$71$ $$( 5832 + T^{2} )^{2}$$
$73$ $$( -14450 + T^{2} )^{2}$$
$79$ $$( 148 + T )^{4}$$
$83$ $$( 6400 + T^{2} )^{2}$$
$89$ $$( 11236 + T^{2} )^{2}$$
$97$ $$( -23762 + T^{2} )^{2}$$