# Properties

 Label 882.4.g.u Level $882$ Weight $4$ Character orbit 882.g Analytic conductor $52.040$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 882.g (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$52.0396846251$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 9 \zeta_{6} q^{5} -8 q^{8} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 9 \zeta_{6} q^{5} -8 q^{8} + ( -18 + 18 \zeta_{6} ) q^{10} + ( -57 + 57 \zeta_{6} ) q^{11} + 70 q^{13} -16 \zeta_{6} q^{16} + ( -51 + 51 \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} -36 q^{20} -114 q^{22} + 69 \zeta_{6} q^{23} + ( 44 - 44 \zeta_{6} ) q^{25} + 140 \zeta_{6} q^{26} -114 q^{29} + ( 23 - 23 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} -102 q^{34} + 253 \zeta_{6} q^{37} + ( -10 + 10 \zeta_{6} ) q^{38} -72 \zeta_{6} q^{40} -42 q^{41} -124 q^{43} -228 \zeta_{6} q^{44} + ( -138 + 138 \zeta_{6} ) q^{46} -201 \zeta_{6} q^{47} + 88 q^{50} + ( -280 + 280 \zeta_{6} ) q^{52} + ( -393 + 393 \zeta_{6} ) q^{53} -513 q^{55} -228 \zeta_{6} q^{58} + ( -219 + 219 \zeta_{6} ) q^{59} -709 \zeta_{6} q^{61} + 46 q^{62} + 64 q^{64} + 630 \zeta_{6} q^{65} + ( -419 + 419 \zeta_{6} ) q^{67} -204 \zeta_{6} q^{68} + 96 q^{71} + ( -313 + 313 \zeta_{6} ) q^{73} + ( -506 + 506 \zeta_{6} ) q^{74} -20 q^{76} -461 \zeta_{6} q^{79} + ( 144 - 144 \zeta_{6} ) q^{80} -84 \zeta_{6} q^{82} -588 q^{83} -459 q^{85} -248 \zeta_{6} q^{86} + ( 456 - 456 \zeta_{6} ) q^{88} + 1017 \zeta_{6} q^{89} -276 q^{92} + ( 402 - 402 \zeta_{6} ) q^{94} + ( -45 + 45 \zeta_{6} ) q^{95} + 1834 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 4q^{4} + 9q^{5} - 16q^{8} + O(q^{10})$$ $$2q + 2q^{2} - 4q^{4} + 9q^{5} - 16q^{8} - 18q^{10} - 57q^{11} + 140q^{13} - 16q^{16} - 51q^{17} + 5q^{19} - 72q^{20} - 228q^{22} + 69q^{23} + 44q^{25} + 140q^{26} - 228q^{29} + 23q^{31} + 32q^{32} - 204q^{34} + 253q^{37} - 10q^{38} - 72q^{40} - 84q^{41} - 248q^{43} - 228q^{44} - 138q^{46} - 201q^{47} + 176q^{50} - 280q^{52} - 393q^{53} - 1026q^{55} - 228q^{58} - 219q^{59} - 709q^{61} + 92q^{62} + 128q^{64} + 630q^{65} - 419q^{67} - 204q^{68} + 192q^{71} - 313q^{73} - 506q^{74} - 40q^{76} - 461q^{79} + 144q^{80} - 84q^{82} - 1176q^{83} - 918q^{85} - 248q^{86} + 456q^{88} + 1017q^{89} - 552q^{92} + 402q^{94} - 45q^{95} + 3668q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i 4.50000 + 7.79423i 0 0 −8.00000 0 −9.00000 + 15.5885i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i 4.50000 7.79423i 0 0 −8.00000 0 −9.00000 15.5885i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.g.u 2
3.b odd 2 1 98.4.c.a 2
7.b odd 2 1 126.4.g.d 2
7.c even 3 1 882.4.a.c 1
7.c even 3 1 inner 882.4.g.u 2
7.d odd 6 1 126.4.g.d 2
7.d odd 6 1 882.4.a.f 1
21.c even 2 1 14.4.c.a 2
21.g even 6 1 14.4.c.a 2
21.g even 6 1 98.4.a.d 1
21.h odd 6 1 98.4.a.f 1
21.h odd 6 1 98.4.c.a 2
84.h odd 2 1 112.4.i.a 2
84.j odd 6 1 112.4.i.a 2
84.j odd 6 1 784.4.a.p 1
84.n even 6 1 784.4.a.c 1
105.g even 2 1 350.4.e.e 2
105.k odd 4 2 350.4.j.b 4
105.o odd 6 1 2450.4.a.d 1
105.p even 6 1 350.4.e.e 2
105.p even 6 1 2450.4.a.q 1
105.w odd 12 2 350.4.j.b 4
168.e odd 2 1 448.4.i.e 2
168.i even 2 1 448.4.i.b 2
168.ba even 6 1 448.4.i.b 2
168.be odd 6 1 448.4.i.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 21.c even 2 1
14.4.c.a 2 21.g even 6 1
98.4.a.d 1 21.g even 6 1
98.4.a.f 1 21.h odd 6 1
98.4.c.a 2 3.b odd 2 1
98.4.c.a 2 21.h odd 6 1
112.4.i.a 2 84.h odd 2 1
112.4.i.a 2 84.j odd 6 1
126.4.g.d 2 7.b odd 2 1
126.4.g.d 2 7.d odd 6 1
350.4.e.e 2 105.g even 2 1
350.4.e.e 2 105.p even 6 1
350.4.j.b 4 105.k odd 4 2
350.4.j.b 4 105.w odd 12 2
448.4.i.b 2 168.i even 2 1
448.4.i.b 2 168.ba even 6 1
448.4.i.e 2 168.e odd 2 1
448.4.i.e 2 168.be odd 6 1
784.4.a.c 1 84.n even 6 1
784.4.a.p 1 84.j odd 6 1
882.4.a.c 1 7.c even 3 1
882.4.a.f 1 7.d odd 6 1
882.4.g.u 2 1.a even 1 1 trivial
882.4.g.u 2 7.c even 3 1 inner
2450.4.a.d 1 105.o odd 6 1
2450.4.a.q 1 105.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} - 9 T_{5} + 81$$ $$T_{11}^{2} + 57 T_{11} + 3249$$ $$T_{13} - 70$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$81 - 9 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$3249 + 57 T + T^{2}$$
$13$ $$( -70 + T )^{2}$$
$17$ $$2601 + 51 T + T^{2}$$
$19$ $$25 - 5 T + T^{2}$$
$23$ $$4761 - 69 T + T^{2}$$
$29$ $$( 114 + T )^{2}$$
$31$ $$529 - 23 T + T^{2}$$
$37$ $$64009 - 253 T + T^{2}$$
$41$ $$( 42 + T )^{2}$$
$43$ $$( 124 + T )^{2}$$
$47$ $$40401 + 201 T + T^{2}$$
$53$ $$154449 + 393 T + T^{2}$$
$59$ $$47961 + 219 T + T^{2}$$
$61$ $$502681 + 709 T + T^{2}$$
$67$ $$175561 + 419 T + T^{2}$$
$71$ $$( -96 + T )^{2}$$
$73$ $$97969 + 313 T + T^{2}$$
$79$ $$212521 + 461 T + T^{2}$$
$83$ $$( 588 + T )^{2}$$
$89$ $$1034289 - 1017 T + T^{2}$$
$97$ $$( -1834 + T )^{2}$$