Newform orbit 392.3.h.a

• Label: 392.3.h.a
• Level: $392$
• Weight: $3$
• Character orbit: 392.h
• Analytic conductor: $10.681$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	392.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.6812263629\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{2} + 8 q^{4} - 20 q^{8} + 64 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
293.1	−1.95479	−	0.422821i	−3.86988			3.64244	+	1.65306i	−4.67764			7.56482	+	1.63627i		0		−6.42128	−	4.77149i	5.97596			9.14383	+	1.97781i
293.2	−1.95479	−	0.422821i	3.86988			3.64244	+	1.65306i	4.67764			−7.56482	−	1.63627i		0		−6.42128	−	4.77149i	5.97596			−9.14383	−	1.97781i
293.3	−1.95479	+	0.422821i	−3.86988			3.64244	−	1.65306i	−4.67764			7.56482	−	1.63627i		0		−6.42128	+	4.77149i	5.97596			9.14383	−	1.97781i
293.4	−1.95479	+	0.422821i	3.86988			3.64244	−	1.65306i	4.67764			−7.56482	+	1.63627i		0		−6.42128	+	4.77149i	5.97596			−9.14383	+	1.97781i
293.5	−1.61618	−	1.17812i	−2.33563			1.22408	+	3.80810i	3.10110			3.77480	+	2.75165i		0		2.50807	−	7.59668i	−3.54484			−5.01193	−	3.65346i
293.6	−1.61618	−	1.17812i	2.33563			1.22408	+	3.80810i	−3.10110			−3.77480	−	2.75165i		0		2.50807	−	7.59668i	−3.54484			5.01193	+	3.65346i
293.7	−1.61618	+	1.17812i	−2.33563			1.22408	−	3.80810i	3.10110			3.77480	−	2.75165i		0		2.50807	+	7.59668i	−3.54484			−5.01193	+	3.65346i
293.8	−1.61618	+	1.17812i	2.33563			1.22408	−	3.80810i	−3.10110			−3.77480	+	2.75165i		0		2.50807	+	7.59668i	−3.54484			5.01193	−	3.65346i
293.9	−0.621472	−	1.90099i	−3.40276			−3.22755	+	2.36283i	4.31716			2.11472	+	6.46862i		0		6.49755	+	4.66711i	2.57876			−2.68300	−	8.20689i
293.10	−0.621472	−	1.90099i	3.40276			−3.22755	+	2.36283i	−4.31716			−2.11472	−	6.46862i		0		6.49755	+	4.66711i	2.57876			2.68300	+	8.20689i
293.11	−0.621472	+	1.90099i	−3.40276			−3.22755	−	2.36283i	4.31716			2.11472	−	6.46862i		0		6.49755	−	4.66711i	2.57876			−2.68300	+	8.20689i
293.12	−0.621472	+	1.90099i	3.40276			−3.22755	−	2.36283i	−4.31716			−2.11472	+	6.46862i		0		6.49755	−	4.66711i	2.57876			2.68300	−	8.20689i
293.13	0.324499	−	1.97350i	−0.253256			−3.78940	−	1.28080i	3.57178			−0.0821813	+	0.499801i		0		−3.75731	+	7.06276i	−8.93586			1.15904	−	7.04890i
293.14	0.324499	−	1.97350i	0.253256			−3.78940	−	1.28080i	−3.57178			0.0821813	−	0.499801i		0		−3.75731	+	7.06276i	−8.93586			−1.15904	+	7.04890i
293.15	0.324499	+	1.97350i	−0.253256			−3.78940	+	1.28080i	3.57178			−0.0821813	−	0.499801i		0		−3.75731	−	7.06276i	−8.93586			1.15904	+	7.04890i
293.16	0.324499	+	1.97350i	0.253256			−3.78940	+	1.28080i	−3.57178			0.0821813	+	0.499801i		0		−3.75731	−	7.06276i	−8.93586			−1.15904	−	7.04890i
293.17	1.28255	−	1.53463i	−3.89635			−0.710153	−	3.93646i	−8.85969			−4.99725	+	5.97944i		0		−6.95179	−	3.95886i	6.18155			−11.3630	+	13.5963i
293.18	1.28255	−	1.53463i	3.89635			−0.710153	−	3.93646i	8.85969			4.99725	−	5.97944i		0		−6.95179	−	3.95886i	6.18155			11.3630	−	13.5963i
293.19	1.28255	+	1.53463i	−3.89635			−0.710153	+	3.93646i	−8.85969			−4.99725	−	5.97944i		0		−6.95179	+	3.95886i	6.18155			−11.3630	−	13.5963i
293.20	1.28255	+	1.53463i	3.89635			−0.710153	+	3.93646i	8.85969			4.99725	+	5.97944i		0		−6.95179	+	3.95886i	6.18155			11.3630	+	13.5963i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
8.b	even	2	1	inner
56.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	392.3.h.a		28
4.b	odd	2	1		1568.3.h.a		28
7.b	odd	2	1	inner	392.3.h.a		28
7.c	even	3	1		56.3.j.a	✓	28
7.c	even	3	1		392.3.j.e		28
7.d	odd	6	1		56.3.j.a	✓	28
7.d	odd	6	1		392.3.j.e		28
8.b	even	2	1	inner	392.3.h.a		28
8.d	odd	2	1		1568.3.h.a		28
28.d	even	2	1		1568.3.h.a		28
28.f	even	6	1		224.3.n.a		28
28.g	odd	6	1		224.3.n.a		28
56.e	even	2	1		1568.3.h.a		28
56.h	odd	2	1	inner	392.3.h.a		28
56.j	odd	6	1		56.3.j.a	✓	28
56.j	odd	6	1		392.3.j.e		28
56.k	odd	6	1		224.3.n.a		28
56.m	even	6	1		224.3.n.a		28
56.p	even	6	1		56.3.j.a	✓	28
56.p	even	6	1		392.3.j.e		28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.3.j.a	✓	28	7.c	even	3	1
56.3.j.a	✓	28	7.d	odd	6	1
56.3.j.a	✓	28	56.j	odd	6	1
56.3.j.a	✓	28	56.p	even	6	1
224.3.n.a		28	28.f	even	6	1
224.3.n.a		28	28.g	odd	6	1
224.3.n.a		28	56.k	odd	6	1
224.3.n.a		28	56.m	even	6	1
392.3.h.a		28	1.a	even	1	1	trivial
392.3.h.a		28	7.b	odd	2	1	inner
392.3.h.a		28	8.b	even	2	1	inner
392.3.h.a		28	56.h	odd	2	1	inner
392.3.j.e		28	7.c	even	3	1
392.3.j.e		28	7.d	odd	6	1
392.3.j.e		28	56.j	odd	6	1
392.3.j.e		28	56.p	even	6	1
1568.3.h.a		28	4.b	odd	2	1
1568.3.h.a		28	8.d	odd	2	1
1568.3.h.a		28	28.d	even	2	1
1568.3.h.a		28	56.e	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 79T_{3}^{12} + 2333T_{3}^{10} - 32667T_{3}^{8} + 220483T_{3}^{6} - 618077T_{3}^{4} + 407087T_{3}^{2} - 23625 \) acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\).