# Properties

 Label 882.2.l.c Level $882$ Weight $2$ Character orbit 882.l Analytic conductor $7.043$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$7.04280545828$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 48 q^{4} + 16 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48 q - 48 q^{4} + 16 q^{9} - 48 q^{11} + 48 q^{15} + 48 q^{16} + 16 q^{18} - 48 q^{23} - 24 q^{25} - 16 q^{30} - 16 q^{36} + 32 q^{39} + 48 q^{44} - 48 q^{50} - 48 q^{51} + 96 q^{53} - 80 q^{57} - 48 q^{60} - 48 q^{64} - 16 q^{72} + 32 q^{78} - 96 q^{79} + 96 q^{81} + 48 q^{85} - 96 q^{86} + 48 q^{92} + 104 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
227.1 1.00000i −1.70306 + 0.315583i −1.00000 0.220087 0.381202i 0.315583 + 1.70306i 0 1.00000i 2.80082 1.07491i −0.381202 0.220087i
227.2 1.00000i −1.63594 0.568941i −1.00000 1.55389 2.69141i −0.568941 + 1.63594i 0 1.00000i 2.35261 + 1.86151i −2.69141 1.55389i
227.3 1.00000i −1.52614 0.819074i −1.00000 −1.99341 + 3.45268i −0.819074 + 1.52614i 0 1.00000i 1.65823 + 2.50005i 3.45268 + 1.99341i
227.4 1.00000i −0.495324 1.65972i −1.00000 −0.995200 + 1.72374i −1.65972 + 0.495324i 0 1.00000i −2.50931 + 1.64419i 1.72374 + 0.995200i
227.5 1.00000i −0.307440 1.70455i −1.00000 1.55067 2.68584i −1.70455 + 0.307440i 0 1.00000i −2.81096 + 1.04809i −2.68584 1.55067i
227.6 1.00000i −0.149173 + 1.72562i −1.00000 0.948881 1.64351i 1.72562 + 0.149173i 0 1.00000i −2.95549 0.514831i −1.64351 0.948881i
227.7 1.00000i 0.149173 1.72562i −1.00000 −0.948881 + 1.64351i −1.72562 0.149173i 0 1.00000i −2.95549 0.514831i 1.64351 + 0.948881i
227.8 1.00000i 0.307440 + 1.70455i −1.00000 −1.55067 + 2.68584i 1.70455 0.307440i 0 1.00000i −2.81096 + 1.04809i 2.68584 + 1.55067i
227.9 1.00000i 0.495324 + 1.65972i −1.00000 0.995200 1.72374i 1.65972 0.495324i 0 1.00000i −2.50931 + 1.64419i −1.72374 0.995200i
227.10 1.00000i 1.52614 + 0.819074i −1.00000 1.99341 3.45268i 0.819074 1.52614i 0 1.00000i 1.65823 + 2.50005i −3.45268 1.99341i
227.11 1.00000i 1.63594 + 0.568941i −1.00000 −1.55389 + 2.69141i 0.568941 1.63594i 0 1.00000i 2.35261 + 1.86151i 2.69141 + 1.55389i
227.12 1.00000i 1.70306 0.315583i −1.00000 −0.220087 + 0.381202i −0.315583 1.70306i 0 1.00000i 2.80082 1.07491i 0.381202 + 0.220087i
227.13 1.00000i −1.71914 0.211098i −1.00000 0.584859 1.01301i 0.211098 1.71914i 0 1.00000i 2.91088 + 0.725813i 1.01301 + 0.584859i
227.14 1.00000i −1.66366 0.481898i −1.00000 −0.712984 + 1.23492i 0.481898 1.66366i 0 1.00000i 2.53555 + 1.60343i −1.23492 0.712984i
227.15 1.00000i −1.60962 0.639623i −1.00000 −1.35026 + 2.33872i 0.639623 1.60962i 0 1.00000i 2.18177 + 2.05910i −2.33872 1.35026i
227.16 1.00000i −1.45331 + 0.942282i −1.00000 0.724499 1.25487i −0.942282 1.45331i 0 1.00000i 1.22421 2.73885i 1.25487 + 0.724499i
227.17 1.00000i −0.848829 + 1.50980i −1.00000 1.96067 3.39598i −1.50980 0.848829i 0 1.00000i −1.55898 2.56312i 3.39598 + 1.96067i
227.18 1.00000i −0.765075 1.55392i −1.00000 −0.474556 + 0.821956i 1.55392 0.765075i 0 1.00000i −1.82932 + 2.37773i −0.821956 0.474556i
227.19 1.00000i 0.765075 + 1.55392i −1.00000 0.474556 0.821956i −1.55392 + 0.765075i 0 1.00000i −1.82932 + 2.37773i 0.821956 + 0.474556i
227.20 1.00000i 0.848829 1.50980i −1.00000 −1.96067 + 3.39598i 1.50980 + 0.848829i 0 1.00000i −1.55898 2.56312i −3.39598 1.96067i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 509.24 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.b odd 2 1 inner
63.i even 6 1 inner
63.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.l.c 48
3.b odd 2 1 2646.2.l.c 48
7.b odd 2 1 inner 882.2.l.c 48
7.c even 3 1 882.2.m.c 48
7.c even 3 1 882.2.t.c 48
7.d odd 6 1 882.2.m.c 48
7.d odd 6 1 882.2.t.c 48
9.c even 3 1 2646.2.t.c 48
9.d odd 6 1 882.2.t.c 48
21.c even 2 1 2646.2.l.c 48
21.g even 6 1 2646.2.m.c 48
21.g even 6 1 2646.2.t.c 48
21.h odd 6 1 2646.2.m.c 48
21.h odd 6 1 2646.2.t.c 48
63.g even 3 1 2646.2.m.c 48
63.h even 3 1 2646.2.l.c 48
63.i even 6 1 inner 882.2.l.c 48
63.j odd 6 1 inner 882.2.l.c 48
63.k odd 6 1 2646.2.m.c 48
63.l odd 6 1 2646.2.t.c 48
63.n odd 6 1 882.2.m.c 48
63.o even 6 1 882.2.t.c 48
63.s even 6 1 882.2.m.c 48
63.t odd 6 1 2646.2.l.c 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.l.c 48 1.a even 1 1 trivial
882.2.l.c 48 7.b odd 2 1 inner
882.2.l.c 48 63.i even 6 1 inner
882.2.l.c 48 63.j odd 6 1 inner
882.2.m.c 48 7.c even 3 1
882.2.m.c 48 7.d odd 6 1
882.2.m.c 48 63.n odd 6 1
882.2.m.c 48 63.s even 6 1
882.2.t.c 48 7.c even 3 1
882.2.t.c 48 7.d odd 6 1
882.2.t.c 48 9.d odd 6 1
882.2.t.c 48 63.o even 6 1
2646.2.l.c 48 3.b odd 2 1
2646.2.l.c 48 21.c even 2 1
2646.2.l.c 48 63.h even 3 1
2646.2.l.c 48 63.t odd 6 1
2646.2.m.c 48 21.g even 6 1
2646.2.m.c 48 21.h odd 6 1
2646.2.m.c 48 63.g even 3 1
2646.2.m.c 48 63.k odd 6 1
2646.2.t.c 48 9.c even 3 1
2646.2.t.c 48 21.g even 6 1
2646.2.t.c 48 21.h odd 6 1
2646.2.t.c 48 63.l odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$15\!\cdots\!96$$$$T_{5}^{28} +$$$$74\!\cdots\!64$$$$T_{5}^{26} +$$$$30\!\cdots\!78$$$$T_{5}^{24} +$$$$10\!\cdots\!28$$$$T_{5}^{22} +$$$$29\!\cdots\!40$$$$T_{5}^{20} +$$$$68\!\cdots\!88$$$$T_{5}^{18} +$$$$13\!\cdots\!65$$$$T_{5}^{16} +$$$$20\!\cdots\!16$$$$T_{5}^{14} +$$$$24\!\cdots\!28$$$$T_{5}^{12} +$$$$24\!\cdots\!56$$$$T_{5}^{10} +$$$$17\!\cdots\!64$$$$T_{5}^{8} +$$$$91\!\cdots\!12$$$$T_{5}^{6} +$$$$31\!\cdots\!28$$$$T_{5}^{4} +$$$$52\!\cdots\!24$$$$T_{5}^{2} +$$$$58\!\cdots\!16$$">$$T_{5}^{48} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(882, [\chi])$$.