# Properties

 Label 882.3.c.f Level $882$ Weight $3$ Character orbit 882.c 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.0327593166$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 14) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + 2 q^{4} + ( \beta_{2} + 2 \beta_{3} ) q^{5} + 2 \beta_{1} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + 2 q^{4} + ( \beta_{2} + 2 \beta_{3} ) q^{5} + 2 \beta_{1} q^{8} + ( -4 \beta_{2} - \beta_{3} ) q^{10} + ( 9 + 3 \beta_{1} ) q^{11} + ( 6 \beta_{2} + 2 \beta_{3} ) q^{13} + 4 q^{16} + ( -5 \beta_{2} + 2 \beta_{3} ) q^{17} + ( \beta_{2} + \beta_{3} ) q^{19} + ( 2 \beta_{2} + 4 \beta_{3} ) q^{20} + ( 6 + 9 \beta_{1} ) q^{22} + ( 15 - 9 \beta_{1} ) q^{23} + ( -2 + 12 \beta_{1} ) q^{25} + ( -4 \beta_{2} - 6 \beta_{3} ) q^{26} + ( -12 - 6 \beta_{1} ) q^{29} + ( 7 \beta_{2} + 15 \beta_{3} ) q^{31} + 4 \beta_{1} q^{32} + ( -4 \beta_{2} + 5 \beta_{3} ) q^{34} + ( 31 + 24 \beta_{1} ) q^{37} + ( -2 \beta_{2} - \beta_{3} ) q^{38} + ( -8 \beta_{2} - 2 \beta_{3} ) q^{40} + ( 2 \beta_{2} + 10 \beta_{3} ) q^{41} + ( -2 + 6 \beta_{1} ) q^{43} + ( 18 + 6 \beta_{1} ) q^{44} + ( -18 + 15 \beta_{1} ) q^{46} + ( -29 \beta_{2} - \beta_{3} ) q^{47} + ( 24 - 2 \beta_{1} ) q^{50} + ( 12 \beta_{2} + 4 \beta_{3} ) q^{52} + ( -39 + 12 \beta_{1} ) q^{53} + ( -3 \beta_{2} + 15 \beta_{3} ) q^{55} + ( -12 - 12 \beta_{1} ) q^{58} + ( -13 \beta_{2} + 25 \beta_{3} ) q^{59} + ( -7 \beta_{2} - 32 \beta_{3} ) q^{61} + ( -30 \beta_{2} - 7 \beta_{3} ) q^{62} + 8 q^{64} + ( -42 + 42 \beta_{1} ) q^{65} + ( 29 + 45 \beta_{1} ) q^{67} + ( -10 \beta_{2} + 4 \beta_{3} ) q^{68} + ( 6 + 30 \beta_{1} ) q^{71} + ( -53 \beta_{2} + 16 \beta_{3} ) q^{73} + ( 48 + 31 \beta_{1} ) q^{74} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{76} + ( -55 - 15 \beta_{1} ) q^{79} + ( 4 \beta_{2} + 8 \beta_{3} ) q^{80} + ( -20 \beta_{2} - 2 \beta_{3} ) q^{82} + ( 68 \beta_{2} + 4 \beta_{3} ) q^{83} + ( -9 - 24 \beta_{1} ) q^{85} + ( 12 - 2 \beta_{1} ) q^{86} + ( 12 + 18 \beta_{1} ) q^{88} + ( 63 \beta_{2} - 24 \beta_{3} ) q^{89} + ( 30 - 18 \beta_{1} ) q^{92} + ( 2 \beta_{2} + 29 \beta_{3} ) q^{94} + ( -15 + 9 \beta_{1} ) q^{95} + ( 22 \beta_{2} + 26 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} + O(q^{10})$$ $$4q + 8q^{4} + 36q^{11} + 16q^{16} + 24q^{22} + 60q^{23} - 8q^{25} - 48q^{29} + 124q^{37} - 8q^{43} + 72q^{44} - 72q^{46} + 96q^{50} - 156q^{53} - 48q^{58} + 32q^{64} - 168q^{65} + 116q^{67} + 24q^{71} + 192q^{74} - 220q^{79} - 36q^{85} + 48q^{86} + 48q^{88} + 120q^{92} - 60q^{95} + O(q^{100})$$

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

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

## 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
 0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 − 1.22474i −0.707107 + 1.22474i
−1.41421 0 2.00000 6.63103i 0 0 −2.82843 0 9.37769i
685.2 −1.41421 0 2.00000 6.63103i 0 0 −2.82843 0 9.37769i
685.3 1.41421 0 2.00000 3.16693i 0 0 2.82843 0 4.47871i
685.4 1.41421 0 2.00000 3.16693i 0 0 2.82843 0 4.47871i
 $$n$$: 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 882.3.c.f 4
3.b odd 2 1 98.3.b.b 4
7.b odd 2 1 inner 882.3.c.f 4
7.c even 3 1 126.3.n.c 4
7.c even 3 1 882.3.n.b 4
7.d odd 6 1 126.3.n.c 4
7.d odd 6 1 882.3.n.b 4
12.b even 2 1 784.3.c.e 4
21.c even 2 1 98.3.b.b 4
21.g even 6 1 14.3.d.a 4
21.g even 6 1 98.3.d.a 4
21.h odd 6 1 14.3.d.a 4
21.h odd 6 1 98.3.d.a 4
28.f even 6 1 1008.3.cg.l 4
28.g odd 6 1 1008.3.cg.l 4
84.h odd 2 1 784.3.c.e 4
84.j odd 6 1 112.3.s.b 4
84.j odd 6 1 784.3.s.c 4
84.n even 6 1 112.3.s.b 4
84.n even 6 1 784.3.s.c 4
105.o odd 6 1 350.3.k.a 4
105.p even 6 1 350.3.k.a 4
105.w odd 12 2 350.3.i.a 8
105.x even 12 2 350.3.i.a 8
168.s odd 6 1 448.3.s.d 4
168.v even 6 1 448.3.s.c 4
168.ba even 6 1 448.3.s.d 4
168.be odd 6 1 448.3.s.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a 4 21.g even 6 1
14.3.d.a 4 21.h odd 6 1
98.3.b.b 4 3.b odd 2 1
98.3.b.b 4 21.c even 2 1
98.3.d.a 4 21.g even 6 1
98.3.d.a 4 21.h odd 6 1
112.3.s.b 4 84.j odd 6 1
112.3.s.b 4 84.n even 6 1
126.3.n.c 4 7.c even 3 1
126.3.n.c 4 7.d odd 6 1
350.3.i.a 8 105.w odd 12 2
350.3.i.a 8 105.x even 12 2
350.3.k.a 4 105.o odd 6 1
350.3.k.a 4 105.p even 6 1
448.3.s.c 4 168.v even 6 1
448.3.s.c 4 168.be odd 6 1
448.3.s.d 4 168.s odd 6 1
448.3.s.d 4 168.ba even 6 1
784.3.c.e 4 12.b even 2 1
784.3.c.e 4 84.h odd 2 1
784.3.s.c 4 84.j odd 6 1
784.3.s.c 4 84.n even 6 1
882.3.c.f 4 1.a even 1 1 trivial
882.3.c.f 4 7.b odd 2 1 inner
882.3.n.b 4 7.c even 3 1
882.3.n.b 4 7.d odd 6 1
1008.3.cg.l 4 28.f even 6 1
1008.3.cg.l 4 28.g odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 54 T_{5}^{2} + 441$$ $$T_{23}^{2} - 30 T_{23} + 63$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$441 + 54 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 63 - 18 T + T^{2} )^{2}$$
$13$ $$7056 + 264 T^{2} + T^{4}$$
$17$ $$2601 + 198 T^{2} + T^{4}$$
$19$ $$9 + 18 T^{2} + T^{4}$$
$23$ $$( 63 - 30 T + T^{2} )^{2}$$
$29$ $$( 72 + 24 T + T^{2} )^{2}$$
$31$ $$1447209 + 2994 T^{2} + T^{4}$$
$37$ $$( -191 - 62 T + T^{2} )^{2}$$
$41$ $$345744 + 1224 T^{2} + T^{4}$$
$43$ $$( -68 + 4 T + T^{2} )^{2}$$
$47$ $$6335289 + 5058 T^{2} + T^{4}$$
$53$ $$( 1233 + 78 T + T^{2} )^{2}$$
$59$ $$10517049 + 8514 T^{2} + T^{4}$$
$61$ $$35964009 + 12582 T^{2} + T^{4}$$
$67$ $$( -3209 - 58 T + T^{2} )^{2}$$
$71$ $$( -1764 - 12 T + T^{2} )^{2}$$
$73$ $$47485881 + 19926 T^{2} + T^{4}$$
$79$ $$( 2575 + 110 T + T^{2} )^{2}$$
$83$ $$189778176 + 27936 T^{2} + T^{4}$$
$89$ $$71419401 + 30726 T^{2} + T^{4}$$
$97$ $$6780816 + 11016 T^{2} + T^{4}$$