Newform orbit 288.3.t.b

• Label: 288.3.t.b
• Level: $288$
• Weight: $3$
• Character orbit: 288.t
• Analytic conductor: $7.847$
• Analytic rank: $0$
• Dimension: $40$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	288.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.84743161358\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 72)
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.

\(\operatorname{Tr}(f)(q) = \) 40 * q + 6 * q^3 - 18 * q^9 + 16 * q^11 - 4 * q^17 + 76 * q^19 + 118 * q^25 + 144 * q^27 + 156 * q^33 + 108 * q^35 + 20 * q^41 + 16 * q^43 + 166 * q^49 - 330 * q^51 - 258 * q^57 + 64 * q^59 - 102 * q^65 + 64 * q^67 - 292 * q^73 - 138 * q^75 - 42 * q^81 - 554 * q^83 - 688 * q^89 + 204 * q^91 + 92 * q^97 - 90 * 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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
79.1		0		−2.76566	+	1.16239i		0		−0.0166003	+	0.00958419i		0		−4.07208	−	2.35102i		0		6.29772	−	6.42952i		0
79.2		0		−2.76566	+	1.16239i		0		0.0166003	−	0.00958419i		0		4.07208	+	2.35102i		0		6.29772	−	6.42952i		0
79.3		0		−2.66358	−	1.38035i		0		−8.07964	+	4.66478i		0		−4.91220	−	2.83606i		0		5.18928	+	7.35333i		0
79.4		0		−2.66358	−	1.38035i		0		8.07964	−	4.66478i		0		4.91220	+	2.83606i		0		5.18928	+	7.35333i		0
79.5		0		−1.33710	−	2.68555i		0		−1.70411	+	0.983869i		0		8.69613	+	5.02071i		0		−5.42431	+	7.18171i		0
79.6		0		−1.33710	−	2.68555i		0		1.70411	−	0.983869i		0		−8.69613	−	5.02071i		0		−5.42431	+	7.18171i		0
79.7		0		−1.21551	+	2.74272i		0		−5.15803	+	2.97799i		0		−4.09037	−	2.36158i		0		−6.04507	−	6.66762i		0
79.8		0		−1.21551	+	2.74272i		0		5.15803	−	2.97799i		0		4.09037	+	2.36158i		0		−6.04507	−	6.66762i		0
79.9		0		−0.418145	+	2.97072i		0		−4.40783	+	2.54486i		0		10.9609	+	6.32830i		0		−8.65031	−	2.48438i		0
79.10		0		−0.418145	+	2.97072i		0		4.40783	−	2.54486i		0		−10.9609	−	6.32830i		0		−8.65031	−	2.48438i		0
79.11		0		0.136174	−	2.99691i		0		−3.84571	+	2.22032i		0		0.704321	+	0.406640i		0		−8.96291	−	0.816203i		0
79.12		0		0.136174	−	2.99691i		0		3.84571	−	2.22032i		0		−0.704321	−	0.406640i		0		−8.96291	−	0.816203i		0
79.13		0		1.96889	+	2.26351i		0		−5.84790	+	3.37629i		0		−3.50808	−	2.02539i		0		−1.24694	+	8.91320i		0
79.14		0		1.96889	+	2.26351i		0		5.84790	−	3.37629i		0		3.50808	+	2.02539i		0		−1.24694	+	8.91320i		0
79.15		0		2.08749	−	2.15462i		0		−5.42020	+	3.12935i		0		5.96345	+	3.44300i		0		−0.284811	−	8.99549i		0
79.16		0		2.08749	−	2.15462i		0		5.42020	−	3.12935i		0		−5.96345	−	3.44300i		0		−0.284811	−	8.99549i		0
79.17		0		2.73922	−	1.22340i		0		−6.07342	+	3.50649i		0		−8.07247	−	4.66064i		0		6.00661	−	6.70229i		0
79.18		0		2.73922	−	1.22340i		0		6.07342	−	3.50649i		0		8.07247	+	4.66064i		0		6.00661	−	6.70229i		0
79.19		0		2.96823	+	0.435462i		0		−1.50948	+	0.871501i		0		7.93804	+	4.58303i		0		8.62075	+	2.58510i		0
79.20		0		2.96823	+	0.435462i		0		1.50948	−	0.871501i		0		−7.93804	−	4.58303i		0		8.62075	+	2.58510i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
9.c	even	3	1	inner
72.p	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	288.3.t.b		40
3.b	odd	2	1		864.3.t.b		40
4.b	odd	2	1		72.3.p.b	✓	40
8.b	even	2	1		72.3.p.b	✓	40
8.d	odd	2	1	inner	288.3.t.b		40
9.c	even	3	1	inner	288.3.t.b		40
9.c	even	3	1		2592.3.b.e		20
9.d	odd	6	1		864.3.t.b		40
9.d	odd	6	1		2592.3.b.f		20
12.b	even	2	1		216.3.p.b		40
24.f	even	2	1		864.3.t.b		40
24.h	odd	2	1		216.3.p.b		40
36.f	odd	6	1		72.3.p.b	✓	40
36.f	odd	6	1		648.3.b.f		20
36.h	even	6	1		216.3.p.b		40
36.h	even	6	1		648.3.b.e		20
72.j	odd	6	1		216.3.p.b		40
72.j	odd	6	1		648.3.b.e		20
72.l	even	6	1		864.3.t.b		40
72.l	even	6	1		2592.3.b.f		20
72.n	even	6	1		72.3.p.b	✓	40
72.n	even	6	1		648.3.b.f		20
72.p	odd	6	1	inner	288.3.t.b		40
72.p	odd	6	1		2592.3.b.e		20

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.3.p.b	✓	40	4.b	odd	2	1
72.3.p.b	✓	40	8.b	even	2	1
72.3.p.b	✓	40	36.f	odd	6	1
72.3.p.b	✓	40	72.n	even	6	1
216.3.p.b		40	12.b	even	2	1
216.3.p.b		40	24.h	odd	2	1
216.3.p.b		40	36.h	even	6	1
216.3.p.b		40	72.j	odd	6	1
288.3.t.b		40	1.a	even	1	1	trivial
288.3.t.b		40	8.d	odd	2	1	inner
288.3.t.b		40	9.c	even	3	1	inner
288.3.t.b		40	72.p	odd	6	1	inner
648.3.b.e		20	36.h	even	6	1
648.3.b.e		20	72.j	odd	6	1
648.3.b.f		20	36.f	odd	6	1
648.3.b.f		20	72.n	even	6	1
864.3.t.b		40	3.b	odd	2	1
864.3.t.b		40	9.d	odd	6	1
864.3.t.b		40	24.f	even	2	1
864.3.t.b		40	72.l	even	6	1
2592.3.b.e		20	9.c	even	3	1
2592.3.b.e		20	72.p	odd	6	1
2592.3.b.f		20	9.d	odd	6	1
2592.3.b.f		20	72.l	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^40 - 309*T5^38 + 55716*T5^36 - 6712983*T5^34 + 603811641*T5^32 - 41961529638*T5^30 + 2327328612189*T5^28 - 104016298280787*T5^26 + 3784106595265245*T5^24 - 111618882769275558*T5^22 + 2661925178189571849*T5^20 - 50373543727284752943*T5^18 + 743736619369179988308*T5^16 - 8157519137669082514317*T5^14 + 63934766382808694810673*T5^12 - 308579748941050774004376*T5^10 + 1058132405272496522900400*T5^8 - 2083606733974200670567296*T5^6 + 2657225112979175083194624*T5^4 - 976335082686666178560*T5^2 + 358628125468262400