# Properties

 Label 882.4.g.l 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 42) 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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 15 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 15 \zeta_{6} q^{5} + 8 q^{8} + ( 30 - 30 \zeta_{6} ) q^{10} + ( -9 + 9 \zeta_{6} ) q^{11} + 88 q^{13} -16 \zeta_{6} q^{16} + ( 84 - 84 \zeta_{6} ) q^{17} + 104 \zeta_{6} q^{19} -60 q^{20} + 18 q^{22} -84 \zeta_{6} q^{23} + ( -100 + 100 \zeta_{6} ) q^{25} -176 \zeta_{6} q^{26} -51 q^{29} + ( 185 - 185 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} -168 q^{34} -44 \zeta_{6} q^{37} + ( 208 - 208 \zeta_{6} ) q^{38} + 120 \zeta_{6} q^{40} -168 q^{41} + 326 q^{43} -36 \zeta_{6} q^{44} + ( -168 + 168 \zeta_{6} ) q^{46} + 138 \zeta_{6} q^{47} + 200 q^{50} + ( -352 + 352 \zeta_{6} ) q^{52} + ( 639 - 639 \zeta_{6} ) q^{53} -135 q^{55} + 102 \zeta_{6} q^{58} + ( -159 + 159 \zeta_{6} ) q^{59} + 722 \zeta_{6} q^{61} -370 q^{62} + 64 q^{64} + 1320 \zeta_{6} q^{65} + ( 166 - 166 \zeta_{6} ) q^{67} + 336 \zeta_{6} q^{68} -1086 q^{71} + ( 218 - 218 \zeta_{6} ) q^{73} + ( -88 + 88 \zeta_{6} ) q^{74} -416 q^{76} + 583 \zeta_{6} q^{79} + ( 240 - 240 \zeta_{6} ) q^{80} + 336 \zeta_{6} q^{82} -597 q^{83} + 1260 q^{85} -652 \zeta_{6} q^{86} + ( -72 + 72 \zeta_{6} ) q^{88} + 1038 \zeta_{6} q^{89} + 336 q^{92} + ( 276 - 276 \zeta_{6} ) q^{94} + ( -1560 + 1560 \zeta_{6} ) q^{95} + 169 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{4} + 15q^{5} + 16q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{4} + 15q^{5} + 16q^{8} + 30q^{10} - 9q^{11} + 176q^{13} - 16q^{16} + 84q^{17} + 104q^{19} - 120q^{20} + 36q^{22} - 84q^{23} - 100q^{25} - 176q^{26} - 102q^{29} + 185q^{31} - 32q^{32} - 336q^{34} - 44q^{37} + 208q^{38} + 120q^{40} - 336q^{41} + 652q^{43} - 36q^{44} - 168q^{46} + 138q^{47} + 400q^{50} - 352q^{52} + 639q^{53} - 270q^{55} + 102q^{58} - 159q^{59} + 722q^{61} - 740q^{62} + 128q^{64} + 1320q^{65} + 166q^{67} + 336q^{68} - 2172q^{71} + 218q^{73} - 88q^{74} - 832q^{76} + 583q^{79} + 240q^{80} + 336q^{82} - 1194q^{83} + 2520q^{85} - 652q^{86} - 72q^{88} + 1038q^{89} + 672q^{92} + 276q^{94} - 1560q^{95} + 338q^{97} + O(q^{100})$$

## 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 7.50000 + 12.9904i 0 0 8.00000 0 15.0000 25.9808i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i 7.50000 12.9904i 0 0 8.00000 0 15.0000 + 25.9808i
 $$n$$: 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.l 2
3.b odd 2 1 294.4.e.e 2
7.b odd 2 1 126.4.g.a 2
7.c even 3 1 882.4.a.h 1
7.c even 3 1 inner 882.4.g.l 2
7.d odd 6 1 126.4.g.a 2
7.d odd 6 1 882.4.a.r 1
21.c even 2 1 42.4.e.b 2
21.g even 6 1 42.4.e.b 2
21.g even 6 1 294.4.a.a 1
21.h odd 6 1 294.4.a.g 1
21.h odd 6 1 294.4.e.e 2
84.h odd 2 1 336.4.q.d 2
84.j odd 6 1 336.4.q.d 2
84.j odd 6 1 2352.4.a.u 1
84.n even 6 1 2352.4.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 21.c even 2 1
42.4.e.b 2 21.g even 6 1
126.4.g.a 2 7.b odd 2 1
126.4.g.a 2 7.d odd 6 1
294.4.a.a 1 21.g even 6 1
294.4.a.g 1 21.h odd 6 1
294.4.e.e 2 3.b odd 2 1
294.4.e.e 2 21.h odd 6 1
336.4.q.d 2 84.h odd 2 1
336.4.q.d 2 84.j odd 6 1
882.4.a.h 1 7.c even 3 1
882.4.a.r 1 7.d odd 6 1
882.4.g.l 2 1.a even 1 1 trivial
882.4.g.l 2 7.c even 3 1 inner
2352.4.a.q 1 84.n even 6 1
2352.4.a.u 1 84.j odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 15 T_{5} + 225$$ $$T_{11}^{2} + 9 T_{11} + 81$$ $$T_{13} - 88$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$225 - 15 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$81 + 9 T + T^{2}$$
$13$ $$( -88 + T )^{2}$$
$17$ $$7056 - 84 T + T^{2}$$
$19$ $$10816 - 104 T + T^{2}$$
$23$ $$7056 + 84 T + T^{2}$$
$29$ $$( 51 + T )^{2}$$
$31$ $$34225 - 185 T + T^{2}$$
$37$ $$1936 + 44 T + T^{2}$$
$41$ $$( 168 + T )^{2}$$
$43$ $$( -326 + T )^{2}$$
$47$ $$19044 - 138 T + T^{2}$$
$53$ $$408321 - 639 T + T^{2}$$
$59$ $$25281 + 159 T + T^{2}$$
$61$ $$521284 - 722 T + T^{2}$$
$67$ $$27556 - 166 T + T^{2}$$
$71$ $$( 1086 + T )^{2}$$
$73$ $$47524 - 218 T + T^{2}$$
$79$ $$339889 - 583 T + T^{2}$$
$83$ $$( 597 + T )^{2}$$
$89$ $$1077444 - 1038 T + T^{2}$$
$97$ $$( -169 + T )^{2}$$