Newform orbit 729.3.b.a

• Label: 729.3.b.a
• Level: $729$
• Weight: $3$
• Character orbit: 729.b
• Analytic conductor: $19.864$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	729.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.8638112719\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 48 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
728.1		−	3.75900i		0		−10.1300				−	0.534675i		0		2.78504					23.0428i		0		−2.00984
728.2		−	3.51633i		0		−8.36459					4.92652i		0		3.39378					15.3473i		0		17.3233
728.3		−	3.39067i		0		−7.49663					3.49531i		0		−10.8678					11.8559i		0		11.8514
728.4		−	3.32411i		0		−7.04970				−	6.87815i		0		6.38137					10.1375i		0		−22.8637
728.5		−	2.62787i		0		−2.90572				−	6.25111i		0		−8.11762				−	2.87564i		0		−16.4271
728.6		−	2.61983i		0		−2.86351				−	0.807612i		0		5.91720				−	2.97742i		0		−2.11581
728.7		−	2.43656i		0		−1.93682					9.03835i		0		−1.73106				−	5.02707i		0		22.0225
728.8		−	2.35177i		0		−1.53083				−	5.78250i		0		−4.87067				−	5.80692i		0		−13.5991
728.9		−	1.94017i		0		0.235738					4.32253i		0		8.88939				−	8.21805i		0		8.38645
728.10		−	1.53349i		0		1.64842				−	0.0504138i		0		9.85801				−	8.66177i		0		−0.0773090
728.11		−	1.30309i		0		2.30195					0.437029i		0		−6.08419				−	8.21202i		0		0.569489
728.12		−	1.16096i		0		2.65217					5.38300i		0		−10.5344				−	7.72291i		0		6.24946
728.13		−	0.581939i		0		3.66135				−	7.66608i		0		1.33395				−	4.45844i		0		−4.46119
728.14		−	0.455989i		0		3.79207				−	2.45401i		0		−4.32645				−	3.55310i		0		−1.11900
728.15		−	0.117696i		0		3.98615				−	6.19791i		0		7.97344				−	0.939939i		0		−0.729470
728.16			0.117696i		0		3.98615					6.19791i		0		7.97344					0.939939i		0		−0.729470
728.17			0.455989i		0		3.79207					2.45401i		0		−4.32645					3.55310i		0		−1.11900
728.18			0.581939i		0		3.66135					7.66608i		0		1.33395					4.45844i		0		−4.46119
728.19			1.16096i		0		2.65217				−	5.38300i		0		−10.5344					7.72291i		0		6.24946
728.20			1.30309i		0		2.30195				−	0.437029i		0		−6.08419					8.21202i		0		0.569489

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	729.3.b.a		30
3.b	odd	2	1	inner	729.3.b.a		30
27.e	even	9	1		27.3.f.a	✓	30
27.e	even	9	1		81.3.f.a		30
27.e	even	9	1		243.3.f.a		30
27.e	even	9	1		243.3.f.b		30
27.e	even	9	1		243.3.f.c		30
27.e	even	9	1		243.3.f.d		30
27.f	odd	18	1		27.3.f.a	✓	30
27.f	odd	18	1		81.3.f.a		30
27.f	odd	18	1		243.3.f.a		30
27.f	odd	18	1		243.3.f.b		30
27.f	odd	18	1		243.3.f.c		30
27.f	odd	18	1		243.3.f.d		30
108.j	odd	18	1		432.3.bc.a		30
108.l	even	18	1		432.3.bc.a		30

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
27.3.f.a	✓	30	27.e	even	9	1
27.3.f.a	✓	30	27.f	odd	18	1
81.3.f.a		30	27.e	even	9	1
81.3.f.a		30	27.f	odd	18	1
243.3.f.a		30	27.e	even	9	1
243.3.f.a		30	27.f	odd	18	1
243.3.f.b		30	27.e	even	9	1
243.3.f.b		30	27.f	odd	18	1
243.3.f.c		30	27.e	even	9	1
243.3.f.c		30	27.f	odd	18	1
243.3.f.d		30	27.e	even	9	1
243.3.f.d		30	27.f	odd	18	1
432.3.bc.a		30	108.j	odd	18	1
432.3.bc.a		30	108.l	even	18	1
729.3.b.a		30	1.a	even	1	1	trivial
729.3.b.a		30	3.b	odd	2	1	inner