# Properties

 Label 882.3.s.d Level $882$ Weight $3$ Character orbit 882.s Analytic conductor $24.033$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 882.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.0327593166$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 3 \beta_{1} q^{5} + 2 \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 3 \beta_{1} q^{5} + 2 \beta_{3} q^{8} + 6 \beta_{2} q^{10} + ( -12 \beta_{1} + 12 \beta_{3} ) q^{11} -8 q^{13} + ( -4 + 4 \beta_{2} ) q^{16} + ( -9 \beta_{1} + 9 \beta_{3} ) q^{17} + ( -16 + 16 \beta_{2} ) q^{19} + 6 \beta_{3} q^{20} -24 q^{22} -12 \beta_{1} q^{23} -7 \beta_{2} q^{25} -8 \beta_{1} q^{26} -3 \beta_{3} q^{29} + 44 \beta_{2} q^{31} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} -18 q^{34} + ( 34 - 34 \beta_{2} ) q^{37} + ( -16 \beta_{1} + 16 \beta_{3} ) q^{38} + ( -12 + 12 \beta_{2} ) q^{40} + 33 \beta_{3} q^{41} -40 q^{43} -24 \beta_{1} q^{44} -24 \beta_{2} q^{46} + 60 \beta_{1} q^{47} -7 \beta_{3} q^{50} -16 \beta_{2} q^{52} + ( -27 \beta_{1} + 27 \beta_{3} ) q^{53} -72 q^{55} + ( 6 - 6 \beta_{2} ) q^{58} + ( 24 \beta_{1} - 24 \beta_{3} ) q^{59} + ( 50 - 50 \beta_{2} ) q^{61} + 44 \beta_{3} q^{62} -8 q^{64} -24 \beta_{1} q^{65} -8 \beta_{2} q^{67} -18 \beta_{1} q^{68} + 36 \beta_{3} q^{71} -16 \beta_{2} q^{73} + ( 34 \beta_{1} - 34 \beta_{3} ) q^{74} -32 q^{76} + ( 76 - 76 \beta_{2} ) q^{79} + ( -12 \beta_{1} + 12 \beta_{3} ) q^{80} + ( -66 + 66 \beta_{2} ) q^{82} + 84 \beta_{3} q^{83} -54 q^{85} -40 \beta_{1} q^{86} -48 \beta_{2} q^{88} -9 \beta_{1} q^{89} -24 \beta_{3} q^{92} + 120 \beta_{2} q^{94} + ( -48 \beta_{1} + 48 \beta_{3} ) q^{95} -176 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} + O(q^{10})$$ $$4 q + 4 q^{4} + 12 q^{10} - 32 q^{13} - 8 q^{16} - 32 q^{19} - 96 q^{22} - 14 q^{25} + 88 q^{31} - 72 q^{34} + 68 q^{37} - 24 q^{40} - 160 q^{43} - 48 q^{46} - 32 q^{52} - 288 q^{55} + 12 q^{58} + 100 q^{61} - 32 q^{64} - 16 q^{67} - 32 q^{73} - 128 q^{76} + 152 q^{79} - 132 q^{82} - 216 q^{85} - 96 q^{88} + 240 q^{94} - 704 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
557.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i −3.67423 + 2.12132i 0 0 2.82843i 0 3.00000 5.19615i
557.2 1.22474 0.707107i 0 1.00000 1.73205i 3.67423 2.12132i 0 0 2.82843i 0 3.00000 5.19615i
863.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −3.67423 2.12132i 0 0 2.82843i 0 3.00000 + 5.19615i
863.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 3.67423 + 2.12132i 0 0 2.82843i 0 3.00000 + 5.19615i
 $$n$$: 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
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.s.d 4
3.b odd 2 1 inner 882.3.s.d 4
7.b odd 2 1 882.3.s.b 4
7.c even 3 1 882.3.b.a 2
7.c even 3 1 inner 882.3.s.d 4
7.d odd 6 1 18.3.b.a 2
7.d odd 6 1 882.3.s.b 4
21.c even 2 1 882.3.s.b 4
21.g even 6 1 18.3.b.a 2
21.g even 6 1 882.3.s.b 4
21.h odd 6 1 882.3.b.a 2
21.h odd 6 1 inner 882.3.s.d 4
28.f even 6 1 144.3.e.b 2
35.i odd 6 1 450.3.d.f 2
35.k even 12 2 450.3.b.b 4
56.j odd 6 1 576.3.e.c 2
56.m even 6 1 576.3.e.f 2
63.i even 6 1 162.3.d.b 4
63.k odd 6 1 162.3.d.b 4
63.s even 6 1 162.3.d.b 4
63.t odd 6 1 162.3.d.b 4
77.i even 6 1 2178.3.c.d 2
84.j odd 6 1 144.3.e.b 2
91.s odd 6 1 3042.3.c.e 2
91.bb even 12 2 3042.3.d.a 4
105.p even 6 1 450.3.d.f 2
105.w odd 12 2 450.3.b.b 4
112.v even 12 2 2304.3.h.c 4
112.x odd 12 2 2304.3.h.f 4
140.s even 6 1 3600.3.l.d 2
140.x odd 12 2 3600.3.c.b 4
168.ba even 6 1 576.3.e.c 2
168.be odd 6 1 576.3.e.f 2
231.k odd 6 1 2178.3.c.d 2
252.n even 6 1 1296.3.q.f 4
252.r odd 6 1 1296.3.q.f 4
252.bj even 6 1 1296.3.q.f 4
252.bn odd 6 1 1296.3.q.f 4
273.ba even 6 1 3042.3.c.e 2
273.cb odd 12 2 3042.3.d.a 4
336.bo even 12 2 2304.3.h.f 4
336.br odd 12 2 2304.3.h.c 4
420.be odd 6 1 3600.3.l.d 2
420.br even 12 2 3600.3.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 7.d odd 6 1
18.3.b.a 2 21.g even 6 1
144.3.e.b 2 28.f even 6 1
144.3.e.b 2 84.j odd 6 1
162.3.d.b 4 63.i even 6 1
162.3.d.b 4 63.k odd 6 1
162.3.d.b 4 63.s even 6 1
162.3.d.b 4 63.t odd 6 1
450.3.b.b 4 35.k even 12 2
450.3.b.b 4 105.w odd 12 2
450.3.d.f 2 35.i odd 6 1
450.3.d.f 2 105.p even 6 1
576.3.e.c 2 56.j odd 6 1
576.3.e.c 2 168.ba even 6 1
576.3.e.f 2 56.m even 6 1
576.3.e.f 2 168.be odd 6 1
882.3.b.a 2 7.c even 3 1
882.3.b.a 2 21.h odd 6 1
882.3.s.b 4 7.b odd 2 1
882.3.s.b 4 7.d odd 6 1
882.3.s.b 4 21.c even 2 1
882.3.s.b 4 21.g even 6 1
882.3.s.d 4 1.a even 1 1 trivial
882.3.s.d 4 3.b odd 2 1 inner
882.3.s.d 4 7.c even 3 1 inner
882.3.s.d 4 21.h odd 6 1 inner
1296.3.q.f 4 252.n even 6 1
1296.3.q.f 4 252.r odd 6 1
1296.3.q.f 4 252.bj even 6 1
1296.3.q.f 4 252.bn odd 6 1
2178.3.c.d 2 77.i even 6 1
2178.3.c.d 2 231.k odd 6 1
2304.3.h.c 4 112.v even 12 2
2304.3.h.c 4 336.br odd 12 2
2304.3.h.f 4 112.x odd 12 2
2304.3.h.f 4 336.bo even 12 2
3042.3.c.e 2 91.s odd 6 1
3042.3.c.e 2 273.ba even 6 1
3042.3.d.a 4 91.bb even 12 2
3042.3.d.a 4 273.cb odd 12 2
3600.3.c.b 4 140.x odd 12 2
3600.3.c.b 4 420.br even 12 2
3600.3.l.d 2 140.s even 6 1
3600.3.l.d 2 420.be odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} - 18 T_{5}^{2} + 324$$ $$T_{11}^{4} - 288 T_{11}^{2} + 82944$$ $$T_{13} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$324 - 18 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$82944 - 288 T^{2} + T^{4}$$
$13$ $$( 8 + T )^{4}$$
$17$ $$26244 - 162 T^{2} + T^{4}$$
$19$ $$( 256 + 16 T + T^{2} )^{2}$$
$23$ $$82944 - 288 T^{2} + T^{4}$$
$29$ $$( 18 + T^{2} )^{2}$$
$31$ $$( 1936 - 44 T + T^{2} )^{2}$$
$37$ $$( 1156 - 34 T + T^{2} )^{2}$$
$41$ $$( 2178 + T^{2} )^{2}$$
$43$ $$( 40 + T )^{4}$$
$47$ $$51840000 - 7200 T^{2} + T^{4}$$
$53$ $$2125764 - 1458 T^{2} + T^{4}$$
$59$ $$1327104 - 1152 T^{2} + T^{4}$$
$61$ $$( 2500 - 50 T + T^{2} )^{2}$$
$67$ $$( 64 + 8 T + T^{2} )^{2}$$
$71$ $$( 2592 + T^{2} )^{2}$$
$73$ $$( 256 + 16 T + T^{2} )^{2}$$
$79$ $$( 5776 - 76 T + T^{2} )^{2}$$
$83$ $$( 14112 + T^{2} )^{2}$$
$89$ $$26244 - 162 T^{2} + T^{4}$$
$97$ $$( 176 + T )^{4}$$