Properties

 Label 882.4.a.h 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 42) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 −15.0000 0 0 8.00000 0 −30.0000
 $$n$$: 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.h 1
3.b odd 2 1 294.4.a.g 1
7.b odd 2 1 882.4.a.r 1
7.c even 3 2 882.4.g.l 2
7.d odd 6 2 126.4.g.a 2
12.b even 2 1 2352.4.a.q 1
21.c even 2 1 294.4.a.a 1
21.g even 6 2 42.4.e.b 2
21.h odd 6 2 294.4.e.e 2
84.h odd 2 1 2352.4.a.u 1
84.j odd 6 2 336.4.q.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 21.g even 6 2
126.4.g.a 2 7.d odd 6 2
294.4.a.a 1 21.c even 2 1
294.4.a.g 1 3.b odd 2 1
294.4.e.e 2 21.h odd 6 2
336.4.q.d 2 84.j odd 6 2
882.4.a.h 1 1.a even 1 1 trivial
882.4.a.r 1 7.b odd 2 1
882.4.g.l 2 7.c even 3 2
2352.4.a.q 1 12.b even 2 1
2352.4.a.u 1 84.h odd 2 1

Hecke kernels

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

 $$T_{5} + 15$$ $$T_{11} - 9$$ $$T_{13} - 88$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$T$$
$5$ $$15 + T$$
$7$ $$T$$
$11$ $$-9 + T$$
$13$ $$-88 + T$$
$17$ $$84 + T$$
$19$ $$104 + T$$
$23$ $$-84 + T$$
$29$ $$51 + T$$
$31$ $$185 + T$$
$37$ $$-44 + T$$
$41$ $$168 + T$$
$43$ $$-326 + T$$
$47$ $$138 + T$$
$53$ $$639 + T$$
$59$ $$-159 + T$$
$61$ $$722 + T$$
$67$ $$166 + T$$
$71$ $$1086 + T$$
$73$ $$218 + T$$
$79$ $$583 + T$$
$83$ $$597 + T$$
$89$ $$1038 + T$$
$97$ $$-169 + T$$