Properties

 Label 882.4.a.f Level $882$ Weight $4$ Character orbit 882.a Self dual yes Analytic conductor $52.040$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 882.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$52.0396846251$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−2.00000 0 4.00000 9.00000 0 0 −8.00000 0 −18.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 21.h odd 6 2
98.4.a.d 1 3.b odd 2 1
98.4.a.f 1 21.c even 2 1
98.4.c.a 2 21.g even 6 2
112.4.i.a 2 84.n even 6 2
126.4.g.d 2 7.c even 3 2
350.4.e.e 2 105.o odd 6 2
350.4.j.b 4 105.x even 12 4
448.4.i.b 2 168.s odd 6 2
448.4.i.e 2 168.v even 6 2
784.4.a.c 1 84.h odd 2 1
784.4.a.p 1 12.b even 2 1
882.4.a.c 1 7.b odd 2 1
882.4.a.f 1 1.a even 1 1 trivial
882.4.g.u 2 7.d odd 6 2
2450.4.a.d 1 105.g even 2 1
2450.4.a.q 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5} - 9$$ T5 - 9 $$T_{11} - 57$$ T11 - 57 $$T_{13} + 70$$ T13 + 70

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T - 9$$
$7$ $$T$$
$11$ $$T - 57$$
$13$ $$T + 70$$
$17$ $$T + 51$$
$19$ $$T - 5$$
$23$ $$T + 69$$
$29$ $$T + 114$$
$31$ $$T - 23$$
$37$ $$T + 253$$
$41$ $$T - 42$$
$43$ $$T + 124$$
$47$ $$T + 201$$
$53$ $$T - 393$$
$59$ $$T + 219$$
$61$ $$T + 709$$
$67$ $$T - 419$$
$71$ $$T - 96$$
$73$ $$T + 313$$
$79$ $$T - 461$$
$83$ $$T - 588$$
$89$ $$T - 1017$$
$97$ $$T + 1834$$