# Properties

 Label 882.4.a.t Level $882$ Weight $4$ Character orbit 882.a Self dual yes Analytic conductor $52.040$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 294) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 4 q^{4} + ( -6 + \beta ) q^{5} -8 q^{8} +O(q^{10})$$ $$q -2 q^{2} + 4 q^{4} + ( -6 + \beta ) q^{5} -8 q^{8} + ( 12 - 2 \beta ) q^{10} + ( -2 + 6 \beta ) q^{11} + ( 24 + 15 \beta ) q^{13} + 16 q^{16} + ( -66 - 11 \beta ) q^{17} + ( 60 - 46 \beta ) q^{19} + ( -24 + 4 \beta ) q^{20} + ( 4 - 12 \beta ) q^{22} + ( 38 - 102 \beta ) q^{23} + ( -87 - 12 \beta ) q^{25} + ( -48 - 30 \beta ) q^{26} + ( 56 + 150 \beta ) q^{29} + ( 216 + 54 \beta ) q^{31} -32 q^{32} + ( 132 + 22 \beta ) q^{34} + ( -140 + 180 \beta ) q^{37} + ( -120 + 92 \beta ) q^{38} + ( 48 - 8 \beta ) q^{40} + ( -18 + 127 \beta ) q^{41} + ( -64 - 288 \beta ) q^{43} + ( -8 + 24 \beta ) q^{44} + ( -76 + 204 \beta ) q^{46} + ( 132 - 338 \beta ) q^{47} + ( 174 + 24 \beta ) q^{50} + ( 96 + 60 \beta ) q^{52} + ( -134 - 192 \beta ) q^{53} + ( 24 - 38 \beta ) q^{55} + ( -112 - 300 \beta ) q^{58} + ( -168 + 298 \beta ) q^{59} + ( -252 - 353 \beta ) q^{61} + ( -432 - 108 \beta ) q^{62} + 64 q^{64} + ( -114 - 66 \beta ) q^{65} + ( -192 + 144 \beta ) q^{67} + ( -264 - 44 \beta ) q^{68} + ( 198 + 342 \beta ) q^{71} + ( -156 + 595 \beta ) q^{73} + ( 280 - 360 \beta ) q^{74} + ( 240 - 184 \beta ) q^{76} + ( -424 + 300 \beta ) q^{79} + ( -96 + 16 \beta ) q^{80} + ( 36 - 254 \beta ) q^{82} + ( 324 + 80 \beta ) q^{83} + 374 q^{85} + ( 128 + 576 \beta ) q^{86} + ( 16 - 48 \beta ) q^{88} + ( 306 - 175 \beta ) q^{89} + ( 152 - 408 \beta ) q^{92} + ( -264 + 676 \beta ) q^{94} + ( -452 + 336 \beta ) q^{95} + ( -1092 - 133 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 8q^{4} - 12q^{5} - 16q^{8} + O(q^{10})$$ $$2q - 4q^{2} + 8q^{4} - 12q^{5} - 16q^{8} + 24q^{10} - 4q^{11} + 48q^{13} + 32q^{16} - 132q^{17} + 120q^{19} - 48q^{20} + 8q^{22} + 76q^{23} - 174q^{25} - 96q^{26} + 112q^{29} + 432q^{31} - 64q^{32} + 264q^{34} - 280q^{37} - 240q^{38} + 96q^{40} - 36q^{41} - 128q^{43} - 16q^{44} - 152q^{46} + 264q^{47} + 348q^{50} + 192q^{52} - 268q^{53} + 48q^{55} - 224q^{58} - 336q^{59} - 504q^{61} - 864q^{62} + 128q^{64} - 228q^{65} - 384q^{67} - 528q^{68} + 396q^{71} - 312q^{73} + 560q^{74} + 480q^{76} - 848q^{79} - 192q^{80} + 72q^{82} + 648q^{83} + 748q^{85} + 256q^{86} + 32q^{88} + 612q^{89} + 304q^{92} - 528q^{94} - 904q^{95} - 2184q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
−2.00000 0 4.00000 −7.41421 0 0 −8.00000 0 14.8284
1.2 −2.00000 0 4.00000 −4.58579 0 0 −8.00000 0 9.17157
 $$n$$: 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.t 2
3.b odd 2 1 294.4.a.o yes 2
7.b odd 2 1 882.4.a.bb 2
7.c even 3 2 882.4.g.bk 4
7.d odd 6 2 882.4.g.be 4
12.b even 2 1 2352.4.a.bu 2
21.c even 2 1 294.4.a.l 2
21.g even 6 2 294.4.e.m 4
21.h odd 6 2 294.4.e.k 4
84.h odd 2 1 2352.4.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.l 2 21.c even 2 1
294.4.a.o yes 2 3.b odd 2 1
294.4.e.k 4 21.h odd 6 2
294.4.e.m 4 21.g even 6 2
882.4.a.t 2 1.a even 1 1 trivial
882.4.a.bb 2 7.b odd 2 1
882.4.g.be 4 7.d odd 6 2
882.4.g.bk 4 7.c even 3 2
2352.4.a.bu 2 12.b even 2 1
2352.4.a.bw 2 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5}^{2} + 12 T_{5} + 34$$ $$T_{11}^{2} + 4 T_{11} - 68$$ $$T_{13}^{2} - 48 T_{13} + 126$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$34 + 12 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-68 + 4 T + T^{2}$$
$13$ $$126 - 48 T + T^{2}$$
$17$ $$4114 + 132 T + T^{2}$$
$19$ $$-632 - 120 T + T^{2}$$
$23$ $$-19364 - 76 T + T^{2}$$
$29$ $$-41864 - 112 T + T^{2}$$
$31$ $$40824 - 432 T + T^{2}$$
$37$ $$-45200 + 280 T + T^{2}$$
$41$ $$-31934 + 36 T + T^{2}$$
$43$ $$-161792 + 128 T + T^{2}$$
$47$ $$-211064 - 264 T + T^{2}$$
$53$ $$-55772 + 268 T + T^{2}$$
$59$ $$-149384 + 336 T + T^{2}$$
$61$ $$-185714 + 504 T + T^{2}$$
$67$ $$-4608 + 384 T + T^{2}$$
$71$ $$-194724 - 396 T + T^{2}$$
$73$ $$-683714 + 312 T + T^{2}$$
$79$ $$-224 + 848 T + T^{2}$$
$83$ $$92176 - 648 T + T^{2}$$
$89$ $$32386 - 612 T + T^{2}$$
$97$ $$1157086 + 2184 T + T^{2}$$