# 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: $$x^{2} - x + 1$$ 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.

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

## 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 $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} + 7T_{5} + 49$$ T5^2 + 7*T5 + 49 $$T_{11}^{2} - 35T_{11} + 1225$$ T11^2 - 35*T11 + 1225 $$T_{13} + 66$$ T13 + 66

## Hecke characteristic polynomials

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