Newform orbit 1008.2.df.e

• Label: 1008.2.df.e
• Level: $1008$
• Weight: $2$
• Character orbit: 1008.df
• Analytic conductor: $8.049$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(8.04892052375\)
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 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^9 - 8 * q^15 - 10 * q^21 + 48 * q^25 - 18 * q^27 + 18 * q^29 - 18 * q^31 + 12 * q^33 + 4 * q^39 - 6 * q^41 + 6 * q^43 - 18 * q^45 - 18 * q^47 - 12 * q^49 - 6 * q^51 - 12 * q^53 + 4 * q^57 + 18 * q^61 + 32 * q^63 - 36 * q^65 - 12 * q^77 - 6 * q^79 + 6 * q^81 + 54 * q^87 - 18 * q^89 - 6 * q^91 + 4 * q^93 + 54 * q^95 + 64 * q^99

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
689.1		0		−1.73204	+	0.00732041i		0		2.59337				0		2.61256	−	0.417759i		0		2.99989	−	0.0253584i		0
689.2		0		−1.72872	−	0.107304i		0		−3.60777				0		−0.415579	−	2.61291i		0		2.97697	+	0.370999i		0
689.3		0		−1.70521	+	0.303722i		0		2.22830				0		−2.51733	−	0.814280i		0		2.81551	−	1.03582i		0
689.4		0		−1.64232	−	0.550252i		0		−1.28783				0		−1.10056	+	2.40599i		0		2.39445	+	1.80738i		0
689.5		0		−1.51239	+	0.844203i		0		−2.84763				0		2.64481	−	0.0704652i		0		1.57464	−	2.55353i		0
689.6		0		−1.24547	+	1.20367i		0		2.04899				0		−1.41312	+	2.23676i		0		0.102371	−	2.99825i		0
689.7		0		−0.994080	−	1.41838i		0		−1.58600				0		1.06431	−	2.42224i		0		−1.02361	+	2.81997i		0
689.8		0		−0.958980	−	1.44234i		0		4.11484				0		2.54819	−	0.711830i		0		−1.16071	+	2.76636i		0
689.9		0		−0.930597	−	1.46082i		0		1.36828				0		−2.64451	−	0.0810554i		0		−1.26798	+	2.71887i		0
689.10		0		−0.549450	+	1.64259i		0		−1.05582				0		−1.79851	−	1.94045i		0		−2.39621	−	1.80504i		0
689.11		0		−0.310783	+	1.70394i		0		2.76937				0		1.21939	+	2.34800i		0		−2.80683	−	1.05911i		0
689.12		0		−0.149059	+	1.72562i		0		−2.22094				0		2.45091	+	0.996507i		0		−2.95556	−	0.514438i		0
689.13		0		0.210634	−	1.71920i		0		−1.07485				0		1.26701	+	2.32265i		0		−2.91127	−	0.724244i		0
689.14		0		0.498607	+	1.65873i		0		−0.623597				0		0.996837	−	2.45078i		0		−2.50278	+	1.65411i		0
689.15		0		0.601162	−	1.62438i		0		−0.207028				0		1.37075	−	2.26297i		0		−2.27721	−	1.95303i		0
689.16		0		0.859090	−	1.50398i		0		−4.19811				0		−1.67151	+	2.05087i		0		−1.52393	−	2.58411i		0
689.17		0		1.03050	−	1.39215i		0		2.20884				0		−2.16520	−	1.52049i		0		−0.876153	−	2.86921i		0
689.18		0		1.11231	+	1.32769i		0		3.53401				0		−1.30083	−	2.30388i		0		−0.525517	+	2.95361i		0
689.19		0		1.14647	+	1.29831i		0		−0.0525740				0		−2.44149	+	1.01937i		0		−0.371197	+	2.97695i		0
689.20		0		1.45594	+	0.938214i		0		−1.81173				0		−1.69266	+	2.03345i		0		1.23951	+	2.73196i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.s	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1008.2.df.e		48
3.b	odd	2	1		3024.2.df.e		48
4.b	odd	2	1		504.2.cx.a	yes	48
7.d	odd	6	1		1008.2.ca.e		48
9.c	even	3	1		3024.2.ca.e		48
9.d	odd	6	1		1008.2.ca.e		48
12.b	even	2	1		1512.2.cx.a		48
21.g	even	6	1		3024.2.ca.e		48
28.f	even	6	1		504.2.bs.a	✓	48
36.f	odd	6	1		1512.2.bs.a		48
36.h	even	6	1		504.2.bs.a	✓	48
63.k	odd	6	1		3024.2.df.e		48
63.s	even	6	1	inner	1008.2.df.e		48
84.j	odd	6	1		1512.2.bs.a		48
252.n	even	6	1		1512.2.cx.a		48
252.bn	odd	6	1		504.2.cx.a	yes	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
504.2.bs.a	✓	48	28.f	even	6	1
504.2.bs.a	✓	48	36.h	even	6	1
504.2.cx.a	yes	48	4.b	odd	2	1
504.2.cx.a	yes	48	252.bn	odd	6	1
1008.2.ca.e		48	7.d	odd	6	1
1008.2.ca.e		48	9.d	odd	6	1
1008.2.df.e		48	1.a	even	1	1	trivial
1008.2.df.e		48	63.s	even	6	1	inner
1512.2.bs.a		48	36.f	odd	6	1
1512.2.bs.a		48	84.j	odd	6	1
1512.2.cx.a		48	12.b	even	2	1
1512.2.cx.a		48	252.n	even	6	1
3024.2.ca.e		48	9.c	even	3	1
3024.2.ca.e		48	21.g	even	6	1
3024.2.df.e		48	3.b	odd	2	1
3024.2.df.e		48	63.k	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^24 - 72*T5^22 - 12*T5^21 + 2175*T5^20 + 768*T5^19 - 36005*T5^18 - 20280*T5^17 + 357756*T5^16 + 287724*T5^15 - 2188401*T5^14 - 2387970*T5^13 + 8008817*T5^12 + 11791020*T5^11 - 15323967*T5^10 - 33359910*T5^9 + 6433746*T5^8 + 47326098*T5^7 + 23713832*T5^6 - 19294080*T5^5 - 25983819*T5^4 - 11529738*T5^3 - 2383875*T5^2 - 220260*T5 - 6476