Newform orbit 540.2.n.d

• Label: 540.2.n.d
• Level: $540$
• Weight: $2$
• Character orbit: 540.n
• Analytic conductor: $4.312$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	540.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.31192170915\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 48 * q - 2 * q^4 + 18 * q^5 - 16 * q^10 + 42 * q^14 + 30 * q^16 - 26 * q^25 - 4 * q^34 + 10 * q^40 - 96 * q^41 + 4 * q^46 + 32 * q^49 + 90 * q^50 - 6 * q^56 + 8 * q^61 - 20 * q^64 + 6 * q^65 + 4 * q^70 - 72 * q^74 - 4 * q^85 - 108 * q^86 - 62 * q^94

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
179.1	−1.41350	+	0.0449209i		0		1.99596	−	0.126991i	−0.0361315	+	2.23578i		0		−1.44473	−	2.50234i	−2.81559	+	0.269163i		0		−0.0493612	−	3.16189i
179.2	−1.38574	+	0.282343i		0		1.84056	−	0.782509i	1.62334	−	1.53778i		0		−0.550457	−	0.953419i	−2.32961	+	1.60403i		0		−1.81534	+	2.58931i
179.3	−1.38444	−	0.288666i		0		1.83334	+	0.799280i	−0.989764	−	2.00509i		0		−0.573658	−	0.993605i	−2.30743	−	1.63578i		0		0.791468	+	3.06163i
179.4	−1.25828	−	0.645549i		0		1.16653	+	1.62456i	0.0966770	+	2.23398i		0		1.60868	+	2.78632i	−0.419091	−	2.79721i		0		1.32049	−	2.87338i
179.5	−1.24631	−	0.668359i		0		1.10659	+	1.66597i	2.11186	+	0.734875i		0		−0.667441	−	1.15604i	−0.265693	−	2.81592i		0		−2.14088	−	2.32737i
179.6	−1.00727	−	0.992675i		0		0.0291929	+	1.99979i	0.493735	−	2.18088i		0		−1.65751	−	2.87089i	1.95573	−	2.04331i		0		−2.66223	+	1.70662i
179.7	−0.937387	+	1.05892i		0		−0.242609	−	1.98523i	−0.520093	+	2.17474i		0		−0.550457	−	0.953419i	2.32961	+	1.60403i		0		−1.81534	−	2.58931i
179.8	−0.745653	+	1.20167i		0		−0.888004	−	1.79205i	1.91817	−	1.14918i		0		−1.44473	−	2.50234i	2.81559	+	0.269163i		0		−0.0493612	+	3.16189i
179.9	−0.442228	+	1.34329i		0		−1.60887	−	1.18808i	−2.23134	+	0.145382i		0		−0.573658	−	0.993605i	2.30743	−	1.63578i		0		0.791468	−	3.06163i
179.10	−0.356046	−	1.36866i		0		−1.74646	+	0.974612i	0.493735	−	2.18088i		0		1.65751	+	2.87089i	1.95573	+	2.04331i		0		−3.16067	+	0.100737i
179.11	−0.0700780	+	1.41248i		0		−1.99018	−	0.197967i	1.98302	−	1.03326i		0		1.60868	+	2.78632i	0.419091	−	2.79721i		0		1.32049	+	2.87338i
179.12	−0.0443404	+	1.41352i		0		−1.99607	−	0.125352i	1.69235	+	1.46149i		0		−0.667441	−	1.15604i	0.265693	−	2.81592i		0		−2.14088	+	2.32737i
179.13	0.0443404	−	1.41352i		0		−1.99607	−	0.125352i	2.11186	+	0.734875i		0		0.667441	+	1.15604i	−0.265693	+	2.81592i		0		1.13240	−	2.95257i
179.14	0.0700780	−	1.41248i		0		−1.99018	−	0.197967i	0.0966770	+	2.23398i		0		−1.60868	−	2.78632i	−0.419091	+	2.79721i		0		3.16221	+	0.0199985i
179.15	0.356046	+	1.36866i		0		−1.74646	+	0.974612i	−1.64183	+	1.51803i		0		−1.65751	−	2.87089i	−1.95573	−	2.04331i		0		−2.66223	−	1.70662i
179.16	0.442228	−	1.34329i		0		−1.60887	−	1.18808i	−0.989764	−	2.00509i		0		0.573658	+	0.993605i	−2.30743	+	1.63578i		0		−3.13112	+	0.442837i
179.17	0.745653	−	1.20167i		0		−0.888004	−	1.79205i	−0.0361315	+	2.23578i		0		1.44473	+	2.50234i	−2.81559	−	0.269163i		0		2.65972	+	1.71053i
179.18	0.937387	−	1.05892i		0		−0.242609	−	1.98523i	1.62334	−	1.53778i		0		0.550457	+	0.953419i	−2.32961	−	1.60403i		0		−0.106691	−	3.16048i
179.19	1.00727	+	0.992675i		0		0.0291929	+	1.99979i	−1.64183	+	1.51803i		0		1.65751	+	2.87089i	−1.95573	+	2.04331i		0		−3.16067	−	0.100737i
179.20	1.24631	+	0.668359i		0		1.10659	+	1.66597i	1.69235	+	1.46149i		0		0.667441	+	1.15604i	0.265693	+	2.81592i		0		1.13240	+	2.95257i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
9.d	odd	6	1	inner
20.d	odd	2	1	inner
36.h	even	6	1	inner
45.h	odd	6	1	inner
180.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	540.2.n.d		48
3.b	odd	2	1		180.2.n.d	✓	48
4.b	odd	2	1	inner	540.2.n.d		48
5.b	even	2	1	inner	540.2.n.d		48
9.c	even	3	1		180.2.n.d	✓	48
9.d	odd	6	1	inner	540.2.n.d		48
12.b	even	2	1		180.2.n.d	✓	48
15.d	odd	2	1		180.2.n.d	✓	48
15.e	even	4	2		900.2.r.g		48
20.d	odd	2	1	inner	540.2.n.d		48
36.f	odd	6	1		180.2.n.d	✓	48
36.h	even	6	1	inner	540.2.n.d		48
45.h	odd	6	1	inner	540.2.n.d		48
45.j	even	6	1		180.2.n.d	✓	48
45.k	odd	12	2		900.2.r.g		48
60.h	even	2	1		180.2.n.d	✓	48
60.l	odd	4	2		900.2.r.g		48
180.n	even	6	1	inner	540.2.n.d		48
180.p	odd	6	1		180.2.n.d	✓	48
180.x	even	12	2		900.2.r.g		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
180.2.n.d	✓	48	3.b	odd	2	1
180.2.n.d	✓	48	9.c	even	3	1
180.2.n.d	✓	48	12.b	even	2	1
180.2.n.d	✓	48	15.d	odd	2	1
180.2.n.d	✓	48	36.f	odd	6	1
180.2.n.d	✓	48	45.j	even	6	1
180.2.n.d	✓	48	60.h	even	2	1
180.2.n.d	✓	48	180.p	odd	6	1
540.2.n.d		48	1.a	even	1	1	trivial
540.2.n.d		48	4.b	odd	2	1	inner
540.2.n.d		48	5.b	even	2	1	inner
540.2.n.d		48	9.d	odd	6	1	inner
540.2.n.d		48	20.d	odd	2	1	inner
540.2.n.d		48	36.h	even	6	1	inner
540.2.n.d		48	45.h	odd	6	1	inner
540.2.n.d		48	180.n	even	6	1	inner
900.2.r.g		48	15.e	even	4	2
900.2.r.g		48	45.k	odd	12	2
900.2.r.g		48	60.l	odd	4	2
900.2.r.g		48	180.x	even	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^24 + 34*T7^22 + 730*T7^20 + 9700*T7^18 + 94189*T7^16 + 620404*T7^14 + 2963314*T7^12 + 8702080*T7^10 + 18514873*T7^8 + 26555616*T7^6 + 27788076*T7^4 + 17884800*T7^2 + 7290000