Newform orbit 864.2.y.a

• 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: $4$

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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{9}+O(q^{10}) \)

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

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 \(S_{2}^{\mathrm{new}}(864, [\chi])\):

T5^24 - 6*T5^22 - 19*T5^21 + 9*T5^20 + 165*T5^19 + 1126*T5^18 + 774*T5^17 + 2340*T5^16 + 9506*T5^15 + 48069*T5^14 + 16197*T5^13 + 371293*T5^12 + 181422*T5^11 + 667518*T5^10 - 894823*T5^9 + 5052582*T5^8 - 4843890*T5^7 + 11539936*T5^6 - 8086680*T5^5 + 7315518*T5^4 - 3075967*T5^3 - 799254*T5^2 + 514425*T5 + 130321

T7^48 + 3*T7^46 + 369*T7^44 + 10095*T7^42 - 84699*T7^40 - 676188*T7^38 + 41027850*T7^36 - 539353080*T7^34 + 5008322259*T7^32 - 26170413714*T7^30 + 79492328694*T7^28 + 132553183536*T7^26 - 1902843390267*T7^24 + 7243087569876*T7^22 + 16140232315188*T7^20 - 147298700960838*T7^18 + 374679689350446*T7^16 + 149525943983559*T7^14 - 131416309846107*T7^12 + 9310779240336*T7^10 + 61896955807035*T7^8 + 10314008090334*T7^6 + 945447002172*T7^4 + 108970914168*T7^2 + 7695324729