Properties

 Label 864.2.y.a Level $864$ Weight $2$ Character orbit 864.y Analytic conductor $6.899$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 864.y (of order $$9$$, degree $$6$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$8$$ over $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 12 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48 q - 12 q^{9} - 12 q^{17} - 48 q^{21} + 24 q^{25} + 6 q^{29} - 6 q^{33} + 30 q^{37} - 12 q^{41} + 30 q^{45} - 6 q^{49} - 36 q^{53} - 6 q^{57} - 12 q^{61} - 60 q^{65} - 78 q^{69} + 48 q^{73} - 12 q^{77} - 36 q^{81} + 102 q^{85} - 66 q^{89} + 36 q^{93} + 6 q^{97} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1 0 −1.72103 0.195105i 0 0.517524 + 0.188363i 0 0.780902 4.42872i 0 2.92387 + 0.671562i 0
97.2 0 −1.11085 + 1.32892i 0 −0.229433 0.0835067i 0 −0.342276 + 1.94115i 0 −0.532044 2.95244i 0
97.3 0 −0.602286 1.62396i 0 −2.89262 1.05283i 0 −0.0578408 + 0.328031i 0 −2.27450 + 1.95618i 0
97.4 0 −0.418682 + 1.68069i 0 3.54422 + 1.28999i 0 0.131496 0.745751i 0 −2.64941 1.40735i 0
97.5 0 0.418682 1.68069i 0 3.54422 + 1.28999i 0 −0.131496 + 0.745751i 0 −2.64941 1.40735i 0
97.6 0 0.602286 + 1.62396i 0 −2.89262 1.05283i 0 0.0578408 0.328031i 0 −2.27450 + 1.95618i 0
97.7 0 1.11085 1.32892i 0 −0.229433 0.0835067i 0 0.342276 1.94115i 0 −0.532044 2.95244i 0
97.8 0 1.72103 + 0.195105i 0 0.517524 + 0.188363i 0 −0.780902 + 4.42872i 0 2.92387 + 0.671562i 0
193.1 0 −1.57833 0.713349i 0 0.394613 2.23796i 0 3.88380 3.25889i 0 1.98227 + 2.25180i 0
193.2 0 −1.46485 0.924244i 0 −0.217354 + 1.23268i 0 −2.00770 + 1.68466i 0 1.29155 + 2.70775i 0
193.3 0 −1.34012 + 1.09730i 0 0.181867 1.03142i 0 −0.246536 + 0.206868i 0 0.591863 2.94104i 0
193.4 0 −0.0827900 + 1.73007i 0 −0.532774 + 3.02151i 0 −2.03971 + 1.71152i 0 −2.98629 0.286465i 0
193.5 0 0.0827900 1.73007i 0 −0.532774 + 3.02151i 0 2.03971 1.71152i 0 −2.98629 0.286465i 0
193.6 0 1.34012 1.09730i 0 0.181867 1.03142i 0 0.246536 0.206868i 0 0.591863 2.94104i 0
193.7 0 1.46485 + 0.924244i 0 −0.217354 + 1.23268i 0 2.00770 1.68466i 0 1.29155 + 2.70775i 0
193.8 0 1.57833 + 0.713349i 0 0.394613 2.23796i 0 −3.88380 + 3.25889i 0 1.98227 + 2.25180i 0
385.1 0 −1.57833 + 0.713349i 0 0.394613 + 2.23796i 0 3.88380 + 3.25889i 0 1.98227 2.25180i 0
385.2 0 −1.46485 + 0.924244i 0 −0.217354 1.23268i 0 −2.00770 1.68466i 0 1.29155 2.70775i 0
385.3 0 −1.34012 1.09730i 0 0.181867 + 1.03142i 0 −0.246536 0.206868i 0 0.591863 + 2.94104i 0
385.4 0 −0.0827900 1.73007i 0 −0.532774 3.02151i 0 −2.03971 1.71152i 0 −2.98629 + 0.286465i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 769.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
27.e even 9 1 inner
108.j odd 18 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.y.a 48
4.b odd 2 1 inner 864.2.y.a 48
27.e even 9 1 inner 864.2.y.a 48
108.j odd 18 1 inner 864.2.y.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.y.a 48 1.a even 1 1 trivial
864.2.y.a 48 4.b odd 2 1 inner
864.2.y.a 48 27.e even 9 1 inner
864.2.y.a 48 108.j odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$:

 $$T_{5}^{24} - \cdots$$ $$19\!\cdots\!67$$$$T_{7}^{24} +$$$$72\!\cdots\!76$$$$T_{7}^{22} +$$$$16\!\cdots\!88$$$$T_{7}^{20} -$$$$14\!\cdots\!38$$$$T_{7}^{18} +$$$$37\!\cdots\!46$$$$T_{7}^{16} +$$$$14\!\cdots\!59$$$$T_{7}^{14} -$$$$13\!\cdots\!07$$$$T_{7}^{12} +$$$$93\!\cdots\!36$$$$T_{7}^{10} +$$$$61\!\cdots\!35$$$$T_{7}^{8} +$$$$10\!\cdots\!34$$$$T_{7}^{6} + 945447002172 T_{7}^{4} + 108970914168 T_{7}^{2} + 7695324729$$">$$T_{7}^{48} + \cdots$$