# Properties

 Label 882.3.b.a Level $882$ Weight $3$ Character orbit 882.b Analytic conductor $24.033$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -2 q^{4} + 3 \beta q^{5} -2 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} -2 q^{4} + 3 \beta q^{5} -2 \beta q^{8} -6 q^{10} + 12 \beta q^{11} -8 q^{13} + 4 q^{16} + 9 \beta q^{17} + 16 q^{19} -6 \beta q^{20} -24 q^{22} -12 \beta q^{23} + 7 q^{25} -8 \beta q^{26} + 3 \beta q^{29} -44 q^{31} + 4 \beta q^{32} -18 q^{34} -34 q^{37} + 16 \beta q^{38} + 12 q^{40} -33 \beta q^{41} -40 q^{43} -24 \beta q^{44} + 24 q^{46} + 60 \beta q^{47} + 7 \beta q^{50} + 16 q^{52} + 27 \beta q^{53} -72 q^{55} -6 q^{58} -24 \beta q^{59} -50 q^{61} -44 \beta q^{62} -8 q^{64} -24 \beta q^{65} + 8 q^{67} -18 \beta q^{68} -36 \beta q^{71} + 16 q^{73} -34 \beta q^{74} -32 q^{76} -76 q^{79} + 12 \beta q^{80} + 66 q^{82} -84 \beta q^{83} -54 q^{85} -40 \beta q^{86} + 48 q^{88} -9 \beta q^{89} + 24 \beta q^{92} -120 q^{94} + 48 \beta q^{95} -176 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} + O(q^{10})$$ $$2q - 4q^{4} - 12q^{10} - 16q^{13} + 8q^{16} + 32q^{19} - 48q^{22} + 14q^{25} - 88q^{31} - 36q^{34} - 68q^{37} + 24q^{40} - 80q^{43} + 48q^{46} + 32q^{52} - 144q^{55} - 12q^{58} - 100q^{61} - 16q^{64} + 16q^{67} + 32q^{73} - 64q^{76} - 152q^{79} + 132q^{82} - 108q^{85} + 96q^{88} - 240q^{94} - 352q^{97} + O(q^{100})$$

## 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
 − 1.41421i 1.41421i
1.41421i 0 −2.00000 4.24264i 0 0 2.82843i 0 −6.00000
197.2 1.41421i 0 −2.00000 4.24264i 0 0 2.82843i 0 −6.00000
 $$n$$: 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.a 2
3.b odd 2 1 inner 882.3.b.a 2
7.b odd 2 1 18.3.b.a 2
7.c even 3 2 882.3.s.d 4
7.d odd 6 2 882.3.s.b 4
21.c even 2 1 18.3.b.a 2
21.g even 6 2 882.3.s.b 4
21.h odd 6 2 882.3.s.d 4
28.d even 2 1 144.3.e.b 2
35.c odd 2 1 450.3.d.f 2
35.f even 4 2 450.3.b.b 4
56.e even 2 1 576.3.e.f 2
56.h odd 2 1 576.3.e.c 2
63.l odd 6 2 162.3.d.b 4
63.o even 6 2 162.3.d.b 4
77.b even 2 1 2178.3.c.d 2
84.h odd 2 1 144.3.e.b 2
91.b odd 2 1 3042.3.c.e 2
91.i even 4 2 3042.3.d.a 4
105.g even 2 1 450.3.d.f 2
105.k odd 4 2 450.3.b.b 4
112.j even 4 2 2304.3.h.c 4
112.l odd 4 2 2304.3.h.f 4
140.c even 2 1 3600.3.l.d 2
140.j odd 4 2 3600.3.c.b 4
168.e odd 2 1 576.3.e.f 2
168.i even 2 1 576.3.e.c 2
231.h odd 2 1 2178.3.c.d 2
252.s odd 6 2 1296.3.q.f 4
252.bi even 6 2 1296.3.q.f 4
273.g even 2 1 3042.3.c.e 2
273.o odd 4 2 3042.3.d.a 4
336.v odd 4 2 2304.3.h.c 4
336.y even 4 2 2304.3.h.f 4
420.o odd 2 1 3600.3.l.d 2
420.w even 4 2 3600.3.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 7.b odd 2 1
18.3.b.a 2 21.c even 2 1
144.3.e.b 2 28.d even 2 1
144.3.e.b 2 84.h odd 2 1
162.3.d.b 4 63.l odd 6 2
162.3.d.b 4 63.o even 6 2
450.3.b.b 4 35.f even 4 2
450.3.b.b 4 105.k odd 4 2
450.3.d.f 2 35.c odd 2 1
450.3.d.f 2 105.g even 2 1
576.3.e.c 2 56.h odd 2 1
576.3.e.c 2 168.i even 2 1
576.3.e.f 2 56.e even 2 1
576.3.e.f 2 168.e odd 2 1
882.3.b.a 2 1.a even 1 1 trivial
882.3.b.a 2 3.b odd 2 1 inner
882.3.s.b 4 7.d odd 6 2
882.3.s.b 4 21.g even 6 2
882.3.s.d 4 7.c even 3 2
882.3.s.d 4 21.h odd 6 2
1296.3.q.f 4 252.s odd 6 2
1296.3.q.f 4 252.bi even 6 2
2178.3.c.d 2 77.b even 2 1
2178.3.c.d 2 231.h odd 2 1
2304.3.h.c 4 112.j even 4 2
2304.3.h.c 4 336.v odd 4 2
2304.3.h.f 4 112.l odd 4 2
2304.3.h.f 4 336.y even 4 2
3042.3.c.e 2 91.b odd 2 1
3042.3.c.e 2 273.g even 2 1
3042.3.d.a 4 91.i even 4 2
3042.3.d.a 4 273.o odd 4 2
3600.3.c.b 4 140.j odd 4 2
3600.3.c.b 4 420.w even 4 2
3600.3.l.d 2 140.c even 2 1
3600.3.l.d 2 420.o odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 18$$ $$T_{11}^{2} + 288$$ $$T_{13} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$18 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$288 + T^{2}$$
$13$ $$( 8 + T )^{2}$$
$17$ $$162 + T^{2}$$
$19$ $$( -16 + T )^{2}$$
$23$ $$288 + T^{2}$$
$29$ $$18 + T^{2}$$
$31$ $$( 44 + T )^{2}$$
$37$ $$( 34 + T )^{2}$$
$41$ $$2178 + T^{2}$$
$43$ $$( 40 + T )^{2}$$
$47$ $$7200 + T^{2}$$
$53$ $$1458 + T^{2}$$
$59$ $$1152 + T^{2}$$
$61$ $$( 50 + T )^{2}$$
$67$ $$( -8 + T )^{2}$$
$71$ $$2592 + T^{2}$$
$73$ $$( -16 + T )^{2}$$
$79$ $$( 76 + T )^{2}$$
$83$ $$14112 + T^{2}$$
$89$ $$162 + T^{2}$$
$97$ $$( 176 + T )^{2}$$