Newform orbit 288.3.u.a

• Label: 288.3.u.a
• Level: $288$
• Weight: $3$
• Character orbit: 288.u
• Analytic conductor: $7.847$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(7.84743161358\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{8})\)
Twist minimal:	no (minimal twist has level 32)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{2} - 4 q^{4} + 4 q^{5} - 4 q^{7} + 4 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−1.82416	+	0.820030i		0		2.65510	−	2.99173i	7.60625	−	3.15061i		0		6.84161	+	6.84161i	−2.39002	+	7.63465i		0		−11.2914	+	11.9846i
19.2	−1.61758	−	1.17620i		0		1.23313	+	3.80518i	−2.28872	+	0.948019i		0		−6.37744	−	6.37744i	2.48095	−	7.60558i		0		4.81725	+	1.15849i
19.3	−0.345994	+	1.96984i		0		−3.76058	−	1.36311i	−7.20074	+	2.98264i		0		4.26150	+	4.26150i	3.98625	−	6.93612i		0		−3.38393	−	15.2163i
19.4	0.108191	+	1.99707i		0		−3.97659	+	0.432130i	2.81639	−	1.16659i		0		−6.23443	−	6.23443i	−1.29322	−	7.89478i		0		2.63447	+	5.49832i
19.5	0.360897	−	1.96717i		0		−3.73951	−	1.41989i	0.452310	−	0.187353i		0		0.429965	+	0.429965i	−4.14274	+	6.84381i		0		−0.205317	−	0.957385i
19.6	1.62478	−	1.16623i		0		1.27980	−	3.78974i	4.51028	−	1.86822i		0		−3.85317	−	3.85317i	−2.34032	−	7.65003i		0		5.14942	−	8.29547i
19.7	1.98676	+	0.229757i		0		3.89442	+	0.912943i	−4.18866	+	1.73500i		0		3.93197	+	3.93197i	7.52753	+	2.70857i		0		−8.72048	+	2.48465i
91.1	−1.82416	−	0.820030i		0		2.65510	+	2.99173i	7.60625	+	3.15061i		0		6.84161	−	6.84161i	−2.39002	−	7.63465i		0		−11.2914	−	11.9846i
91.2	−1.61758	+	1.17620i		0		1.23313	−	3.80518i	−2.28872	−	0.948019i		0		−6.37744	+	6.37744i	2.48095	+	7.60558i		0		4.81725	−	1.15849i
91.3	−0.345994	−	1.96984i		0		−3.76058	+	1.36311i	−7.20074	−	2.98264i		0		4.26150	−	4.26150i	3.98625	+	6.93612i		0		−3.38393	+	15.2163i
91.4	0.108191	−	1.99707i		0		−3.97659	−	0.432130i	2.81639	+	1.16659i		0		−6.23443	+	6.23443i	−1.29322	+	7.89478i		0		2.63447	−	5.49832i
91.5	0.360897	+	1.96717i		0		−3.73951	+	1.41989i	0.452310	+	0.187353i		0		0.429965	−	0.429965i	−4.14274	−	6.84381i		0		−0.205317	+	0.957385i
91.6	1.62478	+	1.16623i		0		1.27980	+	3.78974i	4.51028	+	1.86822i		0		−3.85317	+	3.85317i	−2.34032	+	7.65003i		0		5.14942	+	8.29547i
91.7	1.98676	−	0.229757i		0		3.89442	−	0.912943i	−4.18866	−	1.73500i		0		3.93197	−	3.93197i	7.52753	−	2.70857i		0		−8.72048	−	2.48465i
163.1	−1.93931	+	0.488972i		0		3.52181	−	1.89653i	1.85856	+	4.48696i		0		−5.27676	−	5.27676i	−5.90252	+	5.40002i		0		−5.79831	−	7.79280i
163.2	−1.20513	−	1.59614i		0		−1.09531	+	3.84712i	0.642823	+	1.55191i		0		−4.95044	−	4.95044i	7.46052	−	2.88803i		0		1.70238	−	2.89629i
163.3	−0.682385	+	1.87999i		0		−3.06870	−	2.56575i	−1.34740	−	3.25291i		0		0.583225	+	0.583225i	6.91761	−	4.01829i		0		7.03487	−	0.313357i
163.4	0.658450	−	1.88850i		0		−3.13289	−	2.48697i	0.659338	+	1.59178i		0		9.54718	+	9.54718i	−6.75950	+	4.27892i		0		3.44023	−	0.197052i
163.5	1.44490	+	1.38284i		0		0.175499	+	3.99615i	3.18221	+	7.68254i		0		3.67370	+	3.67370i	−5.27246	+	6.01674i		0		−6.02574	+	15.5010i
163.6	1.46783	−	1.35848i		0		0.309042	−	3.98804i	−1.74699	−	4.21761i		0		−0.392379	−	0.392379i	−4.96407	−	6.27359i		0		−8.29385	−	3.81747i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
32.h	odd	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	288.3.u.a		28
3.b	odd	2	1		32.3.h.a	✓	28
12.b	even	2	1		128.3.h.a		28
24.f	even	2	1		256.3.h.a		28
24.h	odd	2	1		256.3.h.b		28
32.h	odd	8	1	inner	288.3.u.a		28
96.o	even	8	1		32.3.h.a	✓	28
96.o	even	8	1		256.3.h.b		28
96.p	odd	8	1		128.3.h.a		28
96.p	odd	8	1		256.3.h.a		28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.3.h.a	✓	28	3.b	odd	2	1
32.3.h.a	✓	28	96.o	even	8	1
128.3.h.a		28	12.b	even	2	1
128.3.h.a		28	96.p	odd	8	1
256.3.h.a		28	24.f	even	2	1
256.3.h.a		28	96.p	odd	8	1
256.3.h.b		28	24.h	odd	2	1
256.3.h.b		28	96.o	even	8	1
288.3.u.a		28	1.a	even	1	1	trivial
288.3.u.a		28	32.h	odd	8	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^28 - 4*T5^27 + 10*T5^26 + 12*T5^25 - 94*T5^24 - 2800*T5^23 + 10288*T5^22 + 322176*T5^21 + 16382956*T5^20 - 45609360*T5^19 + 87548504*T5^18 - 426735408*T5^17 + 971078936*T5^16 - 1194053504*T5^15 + 3277586816*T5^14 + 238475981824*T5^13 + 3790610308400*T5^12 - 7309145658304*T5^11 + 7846018792672*T5^10 - 103487214400448*T5^9 + 233266507183712*T5^8 + 454867372429568*T5^7 - 785966344559872*T5^6 + 1672149101848576*T5^5 + 10194457565394496*T5^4 - 21742919801861888*T5^3 + 37559374313885824*T5^2 - 24719026719545600*T5 + 5805158826700928