# Properties

 Label 882.4.g.z Level $882$ Weight $4$ Character orbit 882.g Analytic conductor $52.040$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{193})$$ Defining polynomial: $$x^{4} - x^{3} + 49 x^{2} + 48 x + 2304$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \beta_{2} q^{2} + ( -4 + 4 \beta_{2} ) q^{4} + ( -\beta_{1} - 3 \beta_{2} ) q^{5} + 8 q^{8} +O(q^{10})$$ $$q -2 \beta_{2} q^{2} + ( -4 + 4 \beta_{2} ) q^{4} + ( -\beta_{1} - 3 \beta_{2} ) q^{5} + 8 q^{8} + ( -8 + 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{10} + ( -16 + 7 \beta_{1} + 9 \beta_{2} + 7 \beta_{3} ) q^{11} + ( 27 - 5 \beta_{3} ) q^{13} -16 \beta_{2} q^{16} + ( -44 - 10 \beta_{1} + 54 \beta_{2} - 10 \beta_{3} ) q^{17} + ( -3 \beta_{1} - 58 \beta_{2} ) q^{19} + ( 16 - 4 \beta_{3} ) q^{20} + ( 32 - 14 \beta_{3} ) q^{22} + ( 14 \beta_{1} + 54 \beta_{2} ) q^{23} + ( 61 + 7 \beta_{1} - 68 \beta_{2} + 7 \beta_{3} ) q^{25} + ( -10 \beta_{1} - 44 \beta_{2} ) q^{26} + ( -40 + 7 \beta_{3} ) q^{29} + ( 63 - 28 \beta_{1} - 35 \beta_{2} - 28 \beta_{3} ) q^{31} + ( -32 + 32 \beta_{2} ) q^{32} + ( 88 + 20 \beta_{3} ) q^{34} + ( -21 \beta_{1} - 134 \beta_{2} ) q^{37} + ( -122 + 6 \beta_{1} + 116 \beta_{2} + 6 \beta_{3} ) q^{38} + ( -8 \beta_{1} - 24 \beta_{2} ) q^{40} + 168 q^{41} + ( 143 + 21 \beta_{3} ) q^{43} + ( -28 \beta_{1} - 36 \beta_{2} ) q^{44} + ( 136 - 28 \beta_{1} - 108 \beta_{2} - 28 \beta_{3} ) q^{46} + ( 24 \beta_{1} - 348 \beta_{2} ) q^{47} + ( -122 - 14 \beta_{3} ) q^{50} + ( -108 + 20 \beta_{1} + 88 \beta_{2} + 20 \beta_{3} ) q^{52} + ( -156 - 63 \beta_{1} + 219 \beta_{2} - 63 \beta_{3} ) q^{53} + ( 400 - 37 \beta_{3} ) q^{55} + ( 14 \beta_{1} + 66 \beta_{2} ) q^{58} + ( -412 + 61 \beta_{1} + 351 \beta_{2} + 61 \beta_{3} ) q^{59} + ( 34 \beta_{1} - 220 \beta_{2} ) q^{61} + ( -126 + 56 \beta_{3} ) q^{62} + 64 q^{64} + ( -42 \beta_{1} - 306 \beta_{2} ) q^{65} + ( 503 + 35 \beta_{1} - 538 \beta_{2} + 35 \beta_{3} ) q^{67} + ( 40 \beta_{1} - 216 \beta_{2} ) q^{68} + ( -812 - 28 \beta_{3} ) q^{71} + ( -143 + 97 \beta_{1} + 46 \beta_{2} + 97 \beta_{3} ) q^{73} + ( -310 + 42 \beta_{1} + 268 \beta_{2} + 42 \beta_{3} ) q^{74} + ( 244 - 12 \beta_{3} ) q^{76} + ( 98 \beta_{1} - 311 \beta_{2} ) q^{79} + ( -64 + 16 \beta_{1} + 48 \beta_{2} + 16 \beta_{3} ) q^{80} -336 \beta_{2} q^{82} + ( 188 - 89 \beta_{3} ) q^{83} + ( -304 - 14 \beta_{3} ) q^{85} + ( 42 \beta_{1} - 328 \beta_{2} ) q^{86} + ( -128 + 56 \beta_{1} + 72 \beta_{2} + 56 \beta_{3} ) q^{88} + ( -54 \beta_{1} - 1170 \beta_{2} ) q^{89} + ( -272 + 56 \beta_{3} ) q^{92} + ( -648 - 48 \beta_{1} + 696 \beta_{2} - 48 \beta_{3} ) q^{94} + ( -388 + 70 \beta_{1} + 318 \beta_{2} + 70 \beta_{3} ) q^{95} + ( -46 + 155 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 8q^{4} - 7q^{5} + 32q^{8} + O(q^{10})$$ $$4q - 4q^{2} - 8q^{4} - 7q^{5} + 32q^{8} - 14q^{10} - 25q^{11} + 98q^{13} - 32q^{16} - 98q^{17} - 119q^{19} + 56q^{20} + 100q^{22} + 122q^{23} + 129q^{25} - 98q^{26} - 146q^{29} + 98q^{31} - 64q^{32} + 392q^{34} - 289q^{37} - 238q^{38} - 56q^{40} + 672q^{41} + 614q^{43} - 100q^{44} + 244q^{46} - 672q^{47} - 516q^{50} - 196q^{52} - 375q^{53} + 1526q^{55} + 146q^{58} - 763q^{59} - 406q^{61} - 392q^{62} + 256q^{64} - 654q^{65} + 1041q^{67} - 392q^{68} - 3304q^{71} - 189q^{73} - 578q^{74} + 952q^{76} - 524q^{79} - 112q^{80} - 672q^{82} + 574q^{83} - 1244q^{85} - 614q^{86} - 200q^{88} - 2394q^{89} - 976q^{92} - 1344q^{94} - 706q^{95} + 126q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 49 x^{2} + 48 x + 2304$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 49 \nu^{2} - 49 \nu + 2304$$$$)/2352$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 97$$$$)/49$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 48 \beta_{2} + \beta_{1} - 49$$ $$\nu^{3}$$ $$=$$ $$49 \beta_{3} - 97$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$

## 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
 3.72311 + 6.44862i −3.22311 − 5.58259i 3.72311 − 6.44862i −3.22311 + 5.58259i
−1.00000 1.73205i 0 −2.00000 + 3.46410i −5.22311 9.04669i 0 0 8.00000 0 −10.4462 + 18.0934i
361.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 1.72311 + 2.98452i 0 0 8.00000 0 3.44622 5.96903i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −5.22311 + 9.04669i 0 0 8.00000 0 −10.4462 18.0934i
667.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 1.72311 2.98452i 0 0 8.00000 0 3.44622 + 5.96903i
 $$n$$: 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.z 4
3.b odd 2 1 882.4.g.bj 4
7.b odd 2 1 126.4.g.e 4
7.c even 3 1 882.4.a.bh 2
7.c even 3 1 inner 882.4.g.z 4
7.d odd 6 1 126.4.g.e 4
7.d odd 6 1 882.4.a.bd 2
21.c even 2 1 126.4.g.f yes 4
21.g even 6 1 126.4.g.f yes 4
21.g even 6 1 882.4.a.ba 2
21.h odd 6 1 882.4.a.u 2
21.h odd 6 1 882.4.g.bj 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.g.e 4 7.b odd 2 1
126.4.g.e 4 7.d odd 6 1
126.4.g.f yes 4 21.c even 2 1
126.4.g.f yes 4 21.g even 6 1
882.4.a.u 2 21.h odd 6 1
882.4.a.ba 2 21.g even 6 1
882.4.a.bd 2 7.d odd 6 1
882.4.a.bh 2 7.c even 3 1
882.4.g.z 4 1.a even 1 1 trivial
882.4.g.z 4 7.c even 3 1 inner
882.4.g.bj 4 3.b odd 2 1
882.4.g.bj 4 21.h odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 7 T_{5}^{3} + 85 T_{5}^{2} - 252 T_{5} + 1296$$ $$T_{11}^{4} + 25 T_{11}^{3} + 2833 T_{11}^{2} - 55200 T_{11} + 4875264$$ $$T_{13}^{2} - 49 T_{13} - 606$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$1296 - 252 T + 85 T^{2} + 7 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$4875264 - 55200 T + 2833 T^{2} + 25 T^{3} + T^{4}$$
$13$ $$( -606 - 49 T + T^{2} )^{2}$$
$17$ $$5875776 - 237552 T + 12028 T^{2} + 98 T^{3} + T^{4}$$
$19$ $$9647236 + 369614 T + 11055 T^{2} + 119 T^{3} + T^{4}$$
$23$ $$32901696 + 699792 T + 20620 T^{2} - 122 T^{3} + T^{4}$$
$29$ $$( -1032 + 73 T + T^{2} )^{2}$$
$31$ $$1255072329 + 3471846 T + 45031 T^{2} - 98 T^{3} + T^{4}$$
$37$ $$158404 - 115022 T + 83919 T^{2} + 289 T^{3} + T^{4}$$
$41$ $$( -168 + T )^{4}$$
$43$ $$( 2284 - 307 T + T^{2} )^{2}$$
$47$ $$7242690816 + 57189888 T + 366480 T^{2} + 672 T^{3} + T^{4}$$
$53$ $$24444697104 - 58630500 T + 296973 T^{2} + 375 T^{3} + T^{4}$$
$59$ $$1155728016 - 25938948 T + 616165 T^{2} + 763 T^{3} + T^{4}$$
$61$ $$212226624 - 5914608 T + 179404 T^{2} + 406 T^{3} + T^{4}$$
$67$ $$44865170596 - 220498374 T + 871867 T^{2} - 1041 T^{3} + T^{4}$$
$71$ $$( 644448 + 1652 T + T^{2} )^{2}$$
$73$ $$198073062916 - 84115206 T + 480775 T^{2} + 189 T^{3} + T^{4}$$
$79$ $$155826773001 - 206848476 T + 669325 T^{2} + 524 T^{3} + T^{4}$$
$83$ $$( -361596 - 287 T + T^{2} )^{2}$$
$89$ $$1669553420544 + 3093316128 T + 4439124 T^{2} + 2394 T^{3} + T^{4}$$
$97$ $$( -1158214 - 63 T + T^{2} )^{2}$$