Newform orbit 1008.4.bt.d

• Label: 1008.4.bt.d
• Level: $1008$
• Weight: $4$
• Character orbit: 1008.bt
• Analytic conductor: $59.474$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.bt (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(59.4739252858\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1		0			0			0		−10.9253	−	18.9231i		0		12.1367	+	13.9893i		0			0			0
17.2		0			0			0		−9.74197	−	16.8736i		0		18.4446	−	1.67197i		0			0			0
17.3		0			0			0		−8.48442	−	14.6954i		0		−3.52711	+	18.1813i		0			0			0
17.4		0			0			0		−7.89184	−	13.6691i		0		−10.6185	−	15.1739i		0			0			0
17.5		0			0			0		−7.29543	−	12.6361i		0		−18.4933	−	0.998739i		0			0			0
17.6		0			0			0		−5.74088	−	9.94350i		0		−17.2542	−	6.73007i		0			0			0
17.7		0			0			0		−5.00025	−	8.66069i		0		−1.56940	+	18.4536i		0			0			0
17.8		0			0			0		−3.34783	−	5.79860i		0		12.7404	−	13.4418i		0			0			0
17.9		0			0			0		−3.30930	−	5.73188i		0		4.31556	−	18.0104i		0			0			0
17.10		0			0			0		−2.08296	−	3.60779i		0		18.2790	+	2.97950i		0			0			0
17.11		0			0			0		−1.56246	−	2.70625i		0		−17.1949	+	6.88015i		0			0			0
17.12		0			0			0		−1.35769	−	2.35159i		0		8.74105	+	16.3277i		0			0			0
17.13		0			0			0		1.35769	+	2.35159i		0		8.74105	+	16.3277i		0			0			0
17.14		0			0			0		1.56246	+	2.70625i		0		−17.1949	+	6.88015i		0			0			0
17.15		0			0			0		2.08296	+	3.60779i		0		18.2790	+	2.97950i		0			0			0
17.16		0			0			0		3.30930	+	5.73188i		0		4.31556	−	18.0104i		0			0			0
17.17		0			0			0		3.34783	+	5.79860i		0		12.7404	−	13.4418i		0			0			0
17.18		0			0			0		5.00025	+	8.66069i		0		−1.56940	+	18.4536i		0			0			0
17.19		0			0			0		5.74088	+	9.94350i		0		−17.2542	−	6.73007i		0			0			0
17.20		0			0			0		7.29543	+	12.6361i		0		−18.4933	−	0.998739i		0			0			0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1008.4.bt.d		48
3.b	odd	2	1	inner	1008.4.bt.d		48
4.b	odd	2	1		504.4.bl.a	✓	48
7.d	odd	6	1	inner	1008.4.bt.d		48
12.b	even	2	1		504.4.bl.a	✓	48
21.g	even	6	1	inner	1008.4.bt.d		48
28.f	even	6	1		504.4.bl.a	✓	48
84.j	odd	6	1		504.4.bl.a	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
504.4.bl.a	✓	48	4.b	odd	2	1
504.4.bl.a	✓	48	12.b	even	2	1
504.4.bl.a	✓	48	28.f	even	6	1
504.4.bl.a	✓	48	84.j	odd	6	1
1008.4.bt.d		48	1.a	even	1	1	trivial
1008.4.bt.d		48	3.b	odd	2	1	inner
1008.4.bt.d		48	7.d	odd	6	1	inner
1008.4.bt.d		48	21.g	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^48 + 1962*T5^46 + 2221785*T5^44 + 1697605406*T5^42 + 972264420678*T5^40 + 432211934571006*T5^38 + 153759105859962197*T5^36 + 44257649461949153178*T5^34 + 10419718425869166122070*T5^32 + 2007801423882459620521154*T5^30 + 317299420997063659541665341*T5^28 + 40880579943218673138744733362*T5^26 + 4283322559921515419832348643501*T5^24 + 360916908067903646494619364212268*T5^22 + 24343045857052591203616069407393204*T5^20 + 1292982646345789189539019382213521456*T5^18 + 54018106929713231906196337524140074848*T5^16 + 1735886013851174487469346127021612235200*T5^14 + 42752720590477032724376332051226160292160*T5^12 + 774892835427645502253354036328599188733184*T5^10 + 10520293200060378987270596685473591963493888*T5^8 + 100739330172879909289227162248352080465474560*T5^6 + 698882626964643343748540662138995422813365248*T5^4 + 3009932888363881819521955439793514784011382784*T5^2 + 8020472652070673187683207018321520344367763456