Properties

 Label 882.4.g.d 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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -7 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -7 \zeta_{6} q^{5} + 8 q^{8} + ( -14 + 14 \zeta_{6} ) q^{10} + ( 35 - 35 \zeta_{6} ) q^{11} -66 q^{13} -16 \zeta_{6} q^{16} + ( -59 + 59 \zeta_{6} ) q^{17} + 137 \zeta_{6} q^{19} + 28 q^{20} -70 q^{22} -7 \zeta_{6} q^{23} + ( 76 - 76 \zeta_{6} ) q^{25} + 132 \zeta_{6} q^{26} -106 q^{29} + ( 75 - 75 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} + 118 q^{34} -11 \zeta_{6} q^{37} + ( 274 - 274 \zeta_{6} ) q^{38} -56 \zeta_{6} q^{40} -498 q^{41} + 260 q^{43} + 140 \zeta_{6} q^{44} + ( -14 + 14 \zeta_{6} ) q^{46} + 171 \zeta_{6} q^{47} -152 q^{50} + ( 264 - 264 \zeta_{6} ) q^{52} + ( -417 + 417 \zeta_{6} ) q^{53} -245 q^{55} + 212 \zeta_{6} q^{58} + ( 17 - 17 \zeta_{6} ) q^{59} + 51 \zeta_{6} q^{61} -150 q^{62} + 64 q^{64} + 462 \zeta_{6} q^{65} + ( -439 + 439 \zeta_{6} ) q^{67} -236 \zeta_{6} q^{68} + 784 q^{71} + ( 295 - 295 \zeta_{6} ) q^{73} + ( -22 + 22 \zeta_{6} ) q^{74} -548 q^{76} + 495 \zeta_{6} q^{79} + ( -112 + 112 \zeta_{6} ) q^{80} + 996 \zeta_{6} q^{82} + 932 q^{83} + 413 q^{85} -520 \zeta_{6} q^{86} + ( 280 - 280 \zeta_{6} ) q^{88} + 873 \zeta_{6} q^{89} + 28 q^{92} + ( 342 - 342 \zeta_{6} ) q^{94} + ( 959 - 959 \zeta_{6} ) q^{95} + 290 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{4} - 7q^{5} + 16q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{4} - 7q^{5} + 16q^{8} - 14q^{10} + 35q^{11} - 132q^{13} - 16q^{16} - 59q^{17} + 137q^{19} + 56q^{20} - 140q^{22} - 7q^{23} + 76q^{25} + 132q^{26} - 212q^{29} + 75q^{31} - 32q^{32} + 236q^{34} - 11q^{37} + 274q^{38} - 56q^{40} - 996q^{41} + 520q^{43} + 140q^{44} - 14q^{46} + 171q^{47} - 304q^{50} + 264q^{52} - 417q^{53} - 490q^{55} + 212q^{58} + 17q^{59} + 51q^{61} - 300q^{62} + 128q^{64} + 462q^{65} - 439q^{67} - 236q^{68} + 1568q^{71} + 295q^{73} - 22q^{74} - 1096q^{76} + 495q^{79} - 112q^{80} + 996q^{82} + 1864q^{83} + 826q^{85} - 520q^{86} + 280q^{88} + 873q^{89} + 56q^{92} + 342q^{94} + 959q^{95} + 580q^{97} + O(q^{100})$$

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 −3.50000 6.06218i 0 0 8.00000 0 −7.00000 + 12.1244i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −3.50000 + 6.06218i 0 0 8.00000 0 −7.00000 12.1244i
 $$n$$: 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.d 2
3.b odd 2 1 98.4.c.e 2
7.b odd 2 1 126.4.g.c 2
7.c even 3 1 882.4.a.p 1
7.c even 3 1 inner 882.4.g.d 2
7.d odd 6 1 126.4.g.c 2
7.d odd 6 1 882.4.a.k 1
21.c even 2 1 14.4.c.b 2
21.g even 6 1 14.4.c.b 2
21.g even 6 1 98.4.a.b 1
21.h odd 6 1 98.4.a.c 1
21.h odd 6 1 98.4.c.e 2
84.h odd 2 1 112.4.i.b 2
84.j odd 6 1 112.4.i.b 2
84.j odd 6 1 784.4.a.l 1
84.n even 6 1 784.4.a.j 1
105.g even 2 1 350.4.e.b 2
105.k odd 4 2 350.4.j.d 4
105.o odd 6 1 2450.4.a.bf 1
105.p even 6 1 350.4.e.b 2
105.p even 6 1 2450.4.a.bh 1
105.w odd 12 2 350.4.j.d 4
168.e odd 2 1 448.4.i.d 2
168.i even 2 1 448.4.i.c 2
168.ba even 6 1 448.4.i.c 2
168.be odd 6 1 448.4.i.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 21.c even 2 1
14.4.c.b 2 21.g even 6 1
98.4.a.b 1 21.g even 6 1
98.4.a.c 1 21.h odd 6 1
98.4.c.e 2 3.b odd 2 1
98.4.c.e 2 21.h odd 6 1
112.4.i.b 2 84.h odd 2 1
112.4.i.b 2 84.j odd 6 1
126.4.g.c 2 7.b odd 2 1
126.4.g.c 2 7.d odd 6 1
350.4.e.b 2 105.g even 2 1
350.4.e.b 2 105.p even 6 1
350.4.j.d 4 105.k odd 4 2
350.4.j.d 4 105.w odd 12 2
448.4.i.c 2 168.i even 2 1
448.4.i.c 2 168.ba even 6 1
448.4.i.d 2 168.e odd 2 1
448.4.i.d 2 168.be odd 6 1
784.4.a.j 1 84.n even 6 1
784.4.a.l 1 84.j odd 6 1
882.4.a.k 1 7.d odd 6 1
882.4.a.p 1 7.c even 3 1
882.4.g.d 2 1.a even 1 1 trivial
882.4.g.d 2 7.c even 3 1 inner
2450.4.a.bf 1 105.o odd 6 1
2450.4.a.bh 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 $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} + 7 T_{5} + 49$$ $$T_{11}^{2} - 35 T_{11} + 1225$$ $$T_{13} + 66$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$49 + 7 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$1225 - 35 T + T^{2}$$
$13$ $$( 66 + T )^{2}$$
$17$ $$3481 + 59 T + T^{2}$$
$19$ $$18769 - 137 T + T^{2}$$
$23$ $$49 + 7 T + T^{2}$$
$29$ $$( 106 + T )^{2}$$
$31$ $$5625 - 75 T + T^{2}$$
$37$ $$121 + 11 T + T^{2}$$
$41$ $$( 498 + T )^{2}$$
$43$ $$( -260 + T )^{2}$$
$47$ $$29241 - 171 T + T^{2}$$
$53$ $$173889 + 417 T + T^{2}$$
$59$ $$289 - 17 T + T^{2}$$
$61$ $$2601 - 51 T + T^{2}$$
$67$ $$192721 + 439 T + T^{2}$$
$71$ $$( -784 + T )^{2}$$
$73$ $$87025 - 295 T + T^{2}$$
$79$ $$245025 - 495 T + T^{2}$$
$83$ $$( -932 + T )^{2}$$
$89$ $$762129 - 873 T + T^{2}$$
$97$ $$( -290 + T )^{2}$$