Newform orbit 864.2.bn.a

• Label: 864.2.bn.a
• Level: $864$
• Weight: $2$
• Character orbit: 864.bn
• 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.bn (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.

\(\operatorname{Tr}(f)(q) = \) \( 368 q + 12 q^{2} - 4 q^{4} + 12 q^{5} - 4 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
35.1	−1.41409	−	0.0185843i		0		1.99931	+	0.0525599i	−0.473408	−	0.363259i		0		−2.79519	+	0.748968i	−2.82623	−	0.111480i		0		0.662692	+	0.522479i
35.2	−1.40263	−	0.180610i		0		1.93476	+	0.506659i	0.0140517	+	0.0107822i		0		0.714379	−	0.191417i	−2.62225	−	1.06009i		0		−0.0177620	−	0.0176614i
35.3	−1.36587	+	0.366606i		0		1.73120	−	1.00147i	−0.924826	−	0.709644i		0		1.85343	−	0.496625i	−1.99745	+	2.00255i		0		1.52335	+	0.630235i
35.4	−1.34272	+	0.443967i		0		1.60579	−	1.19224i	3.31478	+	2.54352i		0		−0.681945	+	0.182727i	−1.62680	+	2.31376i		0		−5.58006	−	1.94358i
35.5	−1.33268	+	0.473259i		0		1.55205	−	1.26140i	2.25126	+	1.72746i		0		−0.665173	+	0.178233i	−1.47141	+	2.41556i		0		−3.81774	−	1.23671i
35.6	−1.31542	−	0.519298i		0		1.46066	+	1.36619i	−2.54801	−	1.95516i		0		−3.63481	+	0.973946i	−1.21192	−	2.55563i		0		2.33640	+	3.89503i
35.7	−1.24919	−	0.662973i		0		1.12093	+	1.65635i	1.36141	+	1.04465i		0		4.23968	−	1.13602i	−0.302134	−	2.81224i		0		−1.00808	−	2.20753i
35.8	−1.16912	−	0.795718i		0		0.733665	+	1.86057i	0.274775	+	0.210843i		0		−1.33748	+	0.358377i	0.622752	−	2.75902i		0		−0.153473	−	0.465143i
35.9	−1.12313	+	0.859413i		0		0.522820	−	1.93046i	−3.20104	−	2.45624i		0		2.32357	−	0.622598i	1.07187	+	2.61746i		0		5.70609	+	0.00765608i
35.10	−1.07440	+	0.919596i		0		0.308687	−	1.97603i	−1.05601	−	0.810304i		0		−0.165787	+	0.0444224i	1.48550	+	2.40693i		0		1.87973	−	0.100507i
35.11	−1.00355	−	0.996435i		0		0.0142344	+	1.99995i	−2.06818	−	1.58697i		0		3.83711	−	1.02815i	1.97853	−	2.02124i		0		0.494213	+	3.65341i
35.12	−0.908095	+	1.08414i		0		−0.350728	−	1.96901i	−0.210277	−	0.161351i		0		−2.90937	+	0.779563i	2.45318	+	1.40781i		0		0.365879	−	0.0814479i
35.13	−0.886889	−	1.10156i		0		−0.426855	+	1.95392i	1.69953	+	1.30410i		0		−0.980387	+	0.262694i	2.53092	−	1.26270i		0		−0.0707598	−	3.02872i
35.14	−0.883288	+	1.10445i		0		−0.439606	−	1.95109i	2.71277	+	2.08158i		0		3.12273	−	0.836733i	2.54317	+	1.23785i		0		−4.69515	+	1.15747i
35.15	−0.840941	+	1.13702i		0		−0.585638	−	1.91234i	0.802533	+	0.615805i		0		−4.10026	+	1.09866i	2.66685	+	0.942277i		0		−1.37507	+	0.394642i
35.16	−0.728560	−	1.21211i		0		−0.938401	+	1.76618i	−2.86669	−	2.19969i		0		−0.326320	+	0.0874372i	2.82448	−	0.149329i		0		−0.577700	+	5.07734i
35.17	−0.488984	−	1.32699i		0		−1.52179	+	1.29775i	1.11620	+	0.856491i		0		−1.82515	+	0.489046i	2.46623	+	1.38481i		0		0.590748	−	1.90000i
35.18	−0.445719	+	1.34214i		0		−1.60267	−	1.19643i	−0.218659	−	0.167783i		0		4.22533	−	1.13217i	2.32012	−	1.61773i		0		0.322648	−	0.218686i
35.19	−0.426698	−	1.34831i		0		−1.63586	+	1.15064i	3.24976	+	2.49363i		0		4.30845	−	1.15445i	2.24943	+	1.71466i		0		1.97551	−	5.44570i
35.20	−0.195262	−	1.40067i		0		−1.92375	+	0.546996i	−0.729399	−	0.559687i		0		1.92647	−	0.516196i	1.14179	+	2.58772i		0		−0.641512	+	1.13093i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
32.h	odd	8	1	inner
288.bf	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.bn.a		368
3.b	odd	2	1		288.2.bf.a	✓	368
9.c	even	3	1		288.2.bf.a	✓	368
9.d	odd	6	1	inner	864.2.bn.a		368
32.h	odd	8	1	inner	864.2.bn.a		368
96.o	even	8	1		288.2.bf.a	✓	368
288.bd	odd	24	1		288.2.bf.a	✓	368
288.bf	even	24	1	inner	864.2.bn.a		368

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
288.2.bf.a	✓	368	3.b	odd	2	1
288.2.bf.a	✓	368	9.c	even	3	1
288.2.bf.a	✓	368	96.o	even	8	1
288.2.bf.a	✓	368	288.bd	odd	24	1
864.2.bn.a		368	1.a	even	1	1	trivial
864.2.bn.a		368	9.d	odd	6	1	inner
864.2.bn.a		368	32.h	odd	8	1	inner
864.2.bn.a		368	288.bf	even	24	1	inner

Hecke kernels

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