Newform orbit 864.2.bk.a

• Label: 864.2.bk.a
• Level: $864$
• Weight: $2$
• Character orbit: 864.bk
• Analytic conductor: $6.899$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.bk (of order \(24\), degree \(8\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89907473464\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{24})\)
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 368 q + 4 q^{2} - 4 q^{4} + 4 q^{5} - 4 q^{7} + 16 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
37.1	−1.41330	+	0.0509169i		0		1.99481	−	0.143921i	−2.15664	+	1.65485i		0		−0.251218	+	0.937559i	−2.81194	+	0.304973i		0		2.96372	−	2.44861i
37.2	−1.39820	−	0.212242i		0		1.90991	+	0.593513i	0.306163	−	0.234927i		0		0.593754	−	2.21592i	−2.54446	−	1.23521i		0		−0.477938	+	0.263494i
37.3	−1.39517	+	0.231288i		0		1.89301	−	0.645374i	2.21674	−	1.70096i		0		−0.400471	+	1.49458i	−2.49181	+	1.33824i		0		−2.69932	+	2.88584i
37.4	−1.34055	+	0.450480i		0		1.59413	−	1.20778i	−1.37652	+	1.05624i		0		1.04771	−	3.91012i	−1.59293	+	2.33721i		0		1.36948	−	2.03604i
37.5	−1.32629	−	0.490865i		0		1.51810	+	1.30206i	−0.489423	+	0.375547i		0		−0.584477	+	2.18130i	−1.37431	−	2.47210i		0		0.833461	−	0.257845i
37.6	−1.32317	−	0.499220i		0		1.50156	+	1.32111i	1.16198	−	0.891617i		0		0.778386	−	2.90498i	−1.32729	−	2.49766i		0		−1.98261	+	0.599678i
37.7	−1.31129	+	0.529637i		0		1.43897	−	1.38902i	0.931920	−	0.715087i		0		−0.366881	+	1.36922i	−1.15123	+	2.58354i		0		−0.843281	+	1.43127i
37.8	−1.16868	−	0.796364i		0		0.731610	+	1.86138i	3.18925	−	2.44720i		0		−0.890421	+	3.32310i	0.627322	−	2.75798i		0		−5.67606	+	0.320180i
37.9	−1.15525	+	0.815721i		0		0.669198	−	1.88472i	−1.28782	+	0.988180i		0		−0.0841003	+	0.313867i	0.764318	+	2.72320i		0		0.681674	−	2.19210i
37.10	−1.04770	+	0.949902i		0		0.195371	−	1.99043i	2.93870	−	2.25494i		0		1.16359	−	4.34257i	1.68603	+	2.27097i		0		−0.936914	+	5.15400i
37.11	−0.930084	−	1.06534i		0		−0.269888	+	1.98171i	−2.36155	+	1.81208i		0		0.338610	−	1.26371i	2.36221	−	1.55563i		0		4.12692	+	0.830461i
37.12	−0.923453	+	1.07109i		0		−0.294470	−	1.97820i	−2.37671	+	1.82371i		0		−0.989396	+	3.69248i	2.39076	+	1.51137i		0		0.241417	−	4.22979i
37.13	−0.917969	+	1.07579i		0		−0.314664	−	1.97509i	2.56852	−	1.97090i		0		−0.846806	+	3.16032i	2.41364	+	1.47456i		0		−0.237547	+	4.57243i
37.14	−0.874211	−	1.11165i		0		−0.471511	+	1.94362i	−1.36482	+	1.04726i		0		1.09705	−	4.09423i	2.57282	−	1.17498i		0		2.35732	+	0.601666i
37.15	−0.830777	−	1.14447i		0		−0.619620	+	1.90160i	1.83769	−	1.41011i		0		0.454899	−	1.69771i	2.69109	−	0.870667i		0		−3.14054	−	0.931695i
37.16	−0.685362	−	1.23704i		0		−1.06056	+	1.69565i	−2.94194	+	2.25743i		0		−1.22999	+	4.59037i	2.82446	+	0.149828i		0		4.80884	+	2.09216i
37.17	−0.683019	+	1.23834i		0		−1.06697	−	1.69162i	0.526137	−	0.403719i		0		0.911495	−	3.40174i	2.82356	−	0.165864i		0		0.140580	+	0.927284i
37.18	−0.624817	−	1.26870i		0		−1.21921	+	1.58541i	1.33786	−	1.02658i		0		−0.437818	+	1.63396i	2.77320	+	0.556221i		0		−2.13834	−	1.05592i
37.19	−0.532138	+	1.31028i		0		−1.43366	−	1.39450i	−0.794880	+	0.609933i		0		0.0316510	−	0.118123i	2.59008	−	1.13642i		0		−0.376196	−	1.36608i
37.20	−0.218918	+	1.39717i		0		−1.90415	−	0.611729i	0.538426	−	0.413148i		0		−0.988129	+	3.68775i	1.27154	−	2.52650i		0		0.459366	+	0.842716i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
32.g	even	8	1	inner
288.bc	even	24	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	864.2.bk.a		368
3.b	odd	2	1		288.2.bc.a	✓	368
9.c	even	3	1	inner	864.2.bk.a		368
9.d	odd	6	1		288.2.bc.a	✓	368
32.g	even	8	1	inner	864.2.bk.a		368
96.p	odd	8	1		288.2.bc.a	✓	368
288.bc	even	24	1	inner	864.2.bk.a		368
288.be	odd	24	1		288.2.bc.a	✓	368

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
288.2.bc.a	✓	368	3.b	odd	2	1
288.2.bc.a	✓	368	9.d	odd	6	1
288.2.bc.a	✓	368	96.p	odd	8	1
288.2.bc.a	✓	368	288.be	odd	24	1
864.2.bk.a		368	1.a	even	1	1	trivial
864.2.bk.a		368	9.c	even	3	1	inner
864.2.bk.a		368	32.g	even	8	1	inner
864.2.bk.a		368	288.bc	even	24	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(864, [\chi])\).