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

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 9 \zeta_{6} q^{5} - 8 q^{8}+O(q^{10})$$ q + 2*z * q^2 + (4*z - 4) * q^4 + 9*z * q^5 - 8 * q^8 $$q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 9 \zeta_{6} q^{5} - 8 q^{8} + (18 \zeta_{6} - 18) q^{10} + (57 \zeta_{6} - 57) q^{11} + 70 q^{13} - 16 \zeta_{6} q^{16} + (51 \zeta_{6} - 51) q^{17} + 5 \zeta_{6} q^{19} - 36 q^{20} - 114 q^{22} + 69 \zeta_{6} q^{23} + ( - 44 \zeta_{6} + 44) q^{25} + 140 \zeta_{6} q^{26} - 114 q^{29} + ( - 23 \zeta_{6} + 23) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} - 102 q^{34} + 253 \zeta_{6} q^{37} + (10 \zeta_{6} - 10) q^{38} - 72 \zeta_{6} q^{40} - 42 q^{41} - 124 q^{43} - 228 \zeta_{6} q^{44} + (138 \zeta_{6} - 138) q^{46} - 201 \zeta_{6} q^{47} + 88 q^{50} + (280 \zeta_{6} - 280) q^{52} + (393 \zeta_{6} - 393) q^{53} - 513 q^{55} - 228 \zeta_{6} q^{58} + (219 \zeta_{6} - 219) q^{59} - 709 \zeta_{6} q^{61} + 46 q^{62} + 64 q^{64} + 630 \zeta_{6} q^{65} + (419 \zeta_{6} - 419) q^{67} - 204 \zeta_{6} q^{68} + 96 q^{71} + (313 \zeta_{6} - 313) q^{73} + (506 \zeta_{6} - 506) q^{74} - 20 q^{76} - 461 \zeta_{6} q^{79} + ( - 144 \zeta_{6} + 144) q^{80} - 84 \zeta_{6} q^{82} - 588 q^{83} - 459 q^{85} - 248 \zeta_{6} q^{86} + ( - 456 \zeta_{6} + 456) q^{88} + 1017 \zeta_{6} q^{89} - 276 q^{92} + ( - 402 \zeta_{6} + 402) q^{94} + (45 \zeta_{6} - 45) q^{95} + 1834 q^{97} +O(q^{100})$$ q + 2*z * q^2 + (4*z - 4) * q^4 + 9*z * q^5 - 8 * q^8 + (18*z - 18) * q^10 + (57*z - 57) * q^11 + 70 * q^13 - 16*z * q^16 + (51*z - 51) * q^17 + 5*z * q^19 - 36 * q^20 - 114 * q^22 + 69*z * q^23 + (-44*z + 44) * q^25 + 140*z * q^26 - 114 * q^29 + (-23*z + 23) * q^31 + (-32*z + 32) * q^32 - 102 * q^34 + 253*z * q^37 + (10*z - 10) * q^38 - 72*z * q^40 - 42 * q^41 - 124 * q^43 - 228*z * q^44 + (138*z - 138) * q^46 - 201*z * q^47 + 88 * q^50 + (280*z - 280) * q^52 + (393*z - 393) * q^53 - 513 * q^55 - 228*z * q^58 + (219*z - 219) * q^59 - 709*z * q^61 + 46 * q^62 + 64 * q^64 + 630*z * q^65 + (419*z - 419) * q^67 - 204*z * q^68 + 96 * q^71 + (313*z - 313) * q^73 + (506*z - 506) * q^74 - 20 * q^76 - 461*z * q^79 + (-144*z + 144) * q^80 - 84*z * q^82 - 588 * q^83 - 459 * q^85 - 248*z * q^86 + (-456*z + 456) * q^88 + 1017*z * q^89 - 276 * q^92 + (-402*z + 402) * q^94 + (45*z - 45) * q^95 + 1834 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 4 q^{4} + 9 q^{5} - 16 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 4 * q^4 + 9 * q^5 - 16 * q^8 $$2 q + 2 q^{2} - 4 q^{4} + 9 q^{5} - 16 q^{8} - 18 q^{10} - 57 q^{11} + 140 q^{13} - 16 q^{16} - 51 q^{17} + 5 q^{19} - 72 q^{20} - 228 q^{22} + 69 q^{23} + 44 q^{25} + 140 q^{26} - 228 q^{29} + 23 q^{31} + 32 q^{32} - 204 q^{34} + 253 q^{37} - 10 q^{38} - 72 q^{40} - 84 q^{41} - 248 q^{43} - 228 q^{44} - 138 q^{46} - 201 q^{47} + 176 q^{50} - 280 q^{52} - 393 q^{53} - 1026 q^{55} - 228 q^{58} - 219 q^{59} - 709 q^{61} + 92 q^{62} + 128 q^{64} + 630 q^{65} - 419 q^{67} - 204 q^{68} + 192 q^{71} - 313 q^{73} - 506 q^{74} - 40 q^{76} - 461 q^{79} + 144 q^{80} - 84 q^{82} - 1176 q^{83} - 918 q^{85} - 248 q^{86} + 456 q^{88} + 1017 q^{89} - 552 q^{92} + 402 q^{94} - 45 q^{95} + 3668 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 - 4 * q^4 + 9 * q^5 - 16 * q^8 - 18 * q^10 - 57 * q^11 + 140 * q^13 - 16 * q^16 - 51 * q^17 + 5 * q^19 - 72 * q^20 - 228 * q^22 + 69 * q^23 + 44 * q^25 + 140 * q^26 - 228 * q^29 + 23 * q^31 + 32 * q^32 - 204 * q^34 + 253 * q^37 - 10 * q^38 - 72 * q^40 - 84 * q^41 - 248 * q^43 - 228 * q^44 - 138 * q^46 - 201 * q^47 + 176 * q^50 - 280 * q^52 - 393 * q^53 - 1026 * q^55 - 228 * q^58 - 219 * q^59 - 709 * q^61 + 92 * q^62 + 128 * q^64 + 630 * q^65 - 419 * q^67 - 204 * q^68 + 192 * q^71 - 313 * q^73 - 506 * q^74 - 40 * q^76 - 461 * q^79 + 144 * q^80 - 84 * q^82 - 1176 * q^83 - 918 * q^85 - 248 * q^86 + 456 * q^88 + 1017 * q^89 - 552 * q^92 + 402 * q^94 - 45 * q^95 + 3668 * 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 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 $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} - 9T_{5} + 81$$ T5^2 - 9*T5 + 81 $$T_{11}^{2} + 57T_{11} + 3249$$ T11^2 + 57*T11 + 3249 $$T_{13} - 70$$ T13 - 70

## Hecke characteristic polynomials

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