# Properties

 Label 882.3.b.f Level $882$ Weight $3$ Character orbit 882.b Analytic conductor $24.033$ 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: $$k$$ $$=$$ $$3$$ 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: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 126) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -2 q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} -2 \beta_{1} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} -2 q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} -2 \beta_{1} q^{8} + ( -2 + \beta_{3} ) q^{10} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{11} + ( 8 + \beta_{3} ) q^{13} + 4 q^{16} + ( -13 \beta_{1} + \beta_{2} ) q^{17} -20 q^{19} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{20} + ( 4 - 2 \beta_{3} ) q^{22} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{23} + ( -33 + 2 \beta_{3} ) q^{25} + ( 8 \beta_{1} + 2 \beta_{2} ) q^{26} + ( -19 \beta_{1} - 2 \beta_{2} ) q^{29} + ( -4 - 2 \beta_{3} ) q^{31} + 4 \beta_{1} q^{32} + ( 26 - \beta_{3} ) q^{34} + 38 q^{37} -20 \beta_{1} q^{38} + ( 4 - 2 \beta_{3} ) q^{40} + ( -27 \beta_{1} - 3 \beta_{2} ) q^{41} + ( 20 + 6 \beta_{3} ) q^{43} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{44} + ( 4 - 2 \beta_{3} ) q^{46} -12 \beta_{1} q^{47} + ( -33 \beta_{1} + 4 \beta_{2} ) q^{50} + ( -16 - 2 \beta_{3} ) q^{52} + ( -3 \beta_{1} - 12 \beta_{2} ) q^{53} + ( 116 - 4 \beta_{3} ) q^{55} + ( 38 + 2 \beta_{3} ) q^{58} + ( -20 \beta_{1} - 4 \beta_{2} ) q^{59} + ( -58 + 4 \beta_{3} ) q^{61} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{62} -8 q^{64} + ( -48 \beta_{1} - 6 \beta_{2} ) q^{65} + ( -48 - 8 \beta_{3} ) q^{67} + ( 26 \beta_{1} - 2 \beta_{2} ) q^{68} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 24 + 5 \beta_{3} ) q^{73} + 38 \beta_{1} q^{74} + 40 q^{76} + ( 76 - 4 \beta_{3} ) q^{79} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{80} + ( 54 + 3 \beta_{3} ) q^{82} + ( -64 \beta_{1} + 4 \beta_{2} ) q^{83} + ( 82 - 14 \beta_{3} ) q^{85} + ( 20 \beta_{1} + 12 \beta_{2} ) q^{86} + ( -8 + 4 \beta_{3} ) q^{88} + ( -51 \beta_{1} + 9 \beta_{2} ) q^{89} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{92} + 24 q^{94} + ( -20 \beta_{1} + 20 \beta_{2} ) q^{95} + ( 72 + 11 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} + O(q^{10})$$ $$4q - 8q^{4} - 8q^{10} + 32q^{13} + 16q^{16} - 80q^{19} + 16q^{22} - 132q^{25} - 16q^{31} + 104q^{34} + 152q^{37} + 16q^{40} + 80q^{43} + 16q^{46} - 64q^{52} + 464q^{55} + 152q^{58} - 232q^{61} - 32q^{64} - 192q^{67} + 96q^{73} + 160q^{76} + 304q^{79} + 216q^{82} + 328q^{85} - 32q^{88} + 96q^{94} + 288q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{3} + 22 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{2} + 16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 2 \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 16$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{2} + 22 \beta_{1}$$$$)/4$$

## Character values

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

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

## 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
 2.57794i − 1.16372i 1.16372i − 2.57794i
1.41421i 0 −2.00000 8.89753i 0 0 2.82843i 0 −12.5830
197.2 1.41421i 0 −2.00000 6.06910i 0 0 2.82843i 0 8.58301
197.3 1.41421i 0 −2.00000 6.06910i 0 0 2.82843i 0 8.58301
197.4 1.41421i 0 −2.00000 8.89753i 0 0 2.82843i 0 −12.5830
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.b.f 4
3.b odd 2 1 inner 882.3.b.f 4
7.b odd 2 1 126.3.b.a 4
7.c even 3 2 882.3.s.i 8
7.d odd 6 2 882.3.s.e 8
21.c even 2 1 126.3.b.a 4
21.g even 6 2 882.3.s.e 8
21.h odd 6 2 882.3.s.i 8
28.d even 2 1 1008.3.d.a 4
35.c odd 2 1 3150.3.e.e 4
35.f even 4 2 3150.3.c.b 8
56.e even 2 1 4032.3.d.j 4
56.h odd 2 1 4032.3.d.i 4
63.l odd 6 2 1134.3.q.c 8
63.o even 6 2 1134.3.q.c 8
84.h odd 2 1 1008.3.d.a 4
105.g even 2 1 3150.3.e.e 4
105.k odd 4 2 3150.3.c.b 8
168.e odd 2 1 4032.3.d.j 4
168.i even 2 1 4032.3.d.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.b.a 4 7.b odd 2 1
126.3.b.a 4 21.c even 2 1
882.3.b.f 4 1.a even 1 1 trivial
882.3.b.f 4 3.b odd 2 1 inner
882.3.s.e 8 7.d odd 6 2
882.3.s.e 8 21.g even 6 2
882.3.s.i 8 7.c even 3 2
882.3.s.i 8 21.h odd 6 2
1008.3.d.a 4 28.d even 2 1
1008.3.d.a 4 84.h odd 2 1
1134.3.q.c 8 63.l odd 6 2
1134.3.q.c 8 63.o even 6 2
3150.3.c.b 8 35.f even 4 2
3150.3.c.b 8 105.k odd 4 2
3150.3.e.e 4 35.c odd 2 1
3150.3.e.e 4 105.g even 2 1
4032.3.d.i 4 56.h odd 2 1
4032.3.d.i 4 168.i even 2 1
4032.3.d.j 4 56.e even 2 1
4032.3.d.j 4 168.e odd 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 116 T_{5}^{2} + 2916$$ $$T_{11}^{4} + 464 T_{11}^{2} + 46656$$ $$T_{13}^{2} - 16 T_{13} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$2916 + 116 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$46656 + 464 T^{2} + T^{4}$$
$13$ $$( -48 - 16 T + T^{2} )^{2}$$
$17$ $$79524 + 788 T^{2} + T^{4}$$
$19$ $$( 20 + T )^{4}$$
$23$ $$46656 + 464 T^{2} + T^{4}$$
$29$ $$248004 + 1892 T^{2} + T^{4}$$
$31$ $$( -432 + 8 T + T^{2} )^{2}$$
$37$ $$( -38 + T )^{4}$$
$41$ $$910116 + 3924 T^{2} + T^{4}$$
$43$ $$( -3632 - 40 T + T^{2} )^{2}$$
$47$ $$( 288 + T^{2} )^{2}$$
$53$ $$64738116 + 16164 T^{2} + T^{4}$$
$59$ $$9216 + 3392 T^{2} + T^{4}$$
$61$ $$( 1572 + 116 T + T^{2} )^{2}$$
$67$ $$( -4864 + 96 T + T^{2} )^{2}$$
$71$ $$46656 + 464 T^{2} + T^{4}$$
$73$ $$( -2224 - 48 T + T^{2} )^{2}$$
$79$ $$( 3984 - 152 T + T^{2} )^{2}$$
$83$ $$53231616 + 18176 T^{2} + T^{4}$$
$89$ $$443556 + 19476 T^{2} + T^{4}$$
$97$ $$( -8368 - 144 T + T^{2} )^{2}$$