Newform orbit 288.2.w.a

• Label: 288.2.w.a
• Level: $288$
• Weight: $2$
• Character orbit: 288.w
• Analytic conductor: $2.300$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	288.w (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29969157821\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{2} - 4 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
35.1	−1.39224	−	0.248345i		0		1.87665	+	0.691511i	4.01952	−	1.66494i		0		2.72280	+	2.72280i	−2.44101	−	1.42881i		0		−6.00960	+	1.31976i
35.2	−1.33776	−	0.458699i		0		1.57919	+	1.22726i	−2.70076	+	1.11869i		0		−1.06647	−	1.06647i	−1.54963	−	2.36614i		0		4.12611	−	0.257702i
35.3	−1.19734	+	0.752583i		0		0.867238	−	1.80219i	−0.0913223	+	0.0378270i		0		−3.05457	−	3.05457i	0.317923	+	2.81050i		0		0.0808758	−	0.114019i
35.4	−0.345448	−	1.37137i		0		−1.76133	+	0.947475i	−1.64859	+	0.682869i		0		2.51270	+	2.51270i	1.90779	+	2.08814i		0		1.50597	+	2.02494i
35.5	0.349793	+	1.37027i		0		−1.75529	+	0.958624i	−2.97412	+	1.23192i		0		0.237717	+	0.237717i	−1.92756	−	2.06990i		0		−2.72839	−	3.64443i
35.6	0.430512	−	1.34709i		0		−1.62932	−	1.15988i	1.78959	−	0.741273i		0		−2.33709	−	2.33709i	−2.26391	+	1.69550i		0		−0.228123	−	2.72987i
35.7	1.16775	+	0.797726i		0		0.727265	+	1.86308i	1.65625	−	0.686041i		0		−0.456585	−	0.456585i	−0.636970	+	2.75577i		0		2.48135	+	0.520112i
35.8	1.32473	−	0.495070i		0		1.50981	−	1.31167i	−0.0505677	+	0.0209458i		0		1.44150	+	1.44150i	1.35072	−	2.48506i		0		−0.0566189	+	0.0527821i
107.1	−1.39224	+	0.248345i		0		1.87665	−	0.691511i	4.01952	+	1.66494i		0		2.72280	−	2.72280i	−2.44101	+	1.42881i		0		−6.00960	−	1.31976i
107.2	−1.33776	+	0.458699i		0		1.57919	−	1.22726i	−2.70076	−	1.11869i		0		−1.06647	+	1.06647i	−1.54963	+	2.36614i		0		4.12611	+	0.257702i
107.3	−1.19734	−	0.752583i		0		0.867238	+	1.80219i	−0.0913223	−	0.0378270i		0		−3.05457	+	3.05457i	0.317923	−	2.81050i		0		0.0808758	+	0.114019i
107.4	−0.345448	+	1.37137i		0		−1.76133	−	0.947475i	−1.64859	−	0.682869i		0		2.51270	−	2.51270i	1.90779	−	2.08814i		0		1.50597	−	2.02494i
107.5	0.349793	−	1.37027i		0		−1.75529	−	0.958624i	−2.97412	−	1.23192i		0		0.237717	−	0.237717i	−1.92756	+	2.06990i		0		−2.72839	+	3.64443i
107.6	0.430512	+	1.34709i		0		−1.62932	+	1.15988i	1.78959	+	0.741273i		0		−2.33709	+	2.33709i	−2.26391	−	1.69550i		0		−0.228123	+	2.72987i
107.7	1.16775	−	0.797726i		0		0.727265	−	1.86308i	1.65625	+	0.686041i		0		−0.456585	+	0.456585i	−0.636970	−	2.75577i		0		2.48135	−	0.520112i
107.8	1.32473	+	0.495070i		0		1.50981	+	1.31167i	−0.0505677	−	0.0209458i		0		1.44150	−	1.44150i	1.35072	+	2.48506i		0		−0.0566189	−	0.0527821i
179.1	−1.40684	+	0.144194i		0		1.95842	−	0.405715i	0.366958	+	0.885915i		0		1.21471	+	1.21471i	−2.69668	+	0.853169i		0		−0.643996	−	1.19343i
179.2	−1.04926	−	0.948179i		0		0.201912	+	1.98978i	−1.39555	−	3.36915i		0		−1.05755	−	1.05755i	1.67481	−	2.27926i		0		−1.73026	+	4.85836i
179.3	−0.850468	+	1.12991i		0		−0.553410	−	1.92191i	0.352978	+	0.852163i		0		−3.43393	−	3.43393i	2.64225	+	1.00922i		0		−1.26307	−	0.325903i
179.4	−0.337906	−	1.37325i		0		−1.77164	+	0.928061i	1.32268	+	3.19322i		0		−2.32913	−	2.32913i	1.87311	+	2.11931i		0		3.93816	−	2.89538i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
96.o	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	288.2.w.a	✓	32
3.b	odd	2	1		288.2.w.b	yes	32
4.b	odd	2	1		1152.2.w.b		32
12.b	even	2	1		1152.2.w.a		32
32.g	even	8	1		1152.2.w.a		32
32.h	odd	8	1		288.2.w.b	yes	32
96.o	even	8	1	inner	288.2.w.a	✓	32
96.p	odd	8	1		1152.2.w.b		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
288.2.w.a	✓	32	1.a	even	1	1	trivial
288.2.w.a	✓	32	96.o	even	8	1	inner
288.2.w.b	yes	32	3.b	odd	2	1
288.2.w.b	yes	32	32.h	odd	8	1
1152.2.w.a		32	12.b	even	2	1
1152.2.w.a		32	32.g	even	8	1
1152.2.w.b		32	4.b	odd	2	1
1152.2.w.b		32	96.p	odd	8	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^32 - 16*T5^29 + 224*T5^27 + 896*T5^26 + 6080*T5^25 + 48512*T5^24 + 87168*T5^23 + 188416*T5^22 + 181376*T5^21 + 155648*T5^20 - 380672*T5^19 - 6281728*T5^18 - 13038592*T5^17 + 83787904*T5^16 + 47824896*T5^15 + 17272832*T5^14 - 315368448*T5^13 - 209092608*T5^12 + 803289088*T5^11 + 588259328*T5^10 - 1798410240*T5^9 + 4815478784*T5^8 - 2732515328*T5^7 + 2096988160*T5^6 + 140255232*T5^5 + 54394880*T5^4 + 24920064*T5^3 + 3506176*T5^2 + 196608*T5 + 4096