Newform orbit 1216.2.m.a

• Label: 1216.2.m.a
• Level: $1216$
• Weight: $2$
• Character orbit: 1216.m
• Analytic conductor: $9.710$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.m (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.70980888579\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(i)\)
Twist minimal:	no (minimal twist has level 304)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 4 q^{5} + 8 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
303.1		0		−2.33030	−	2.33030i		0		1.33252	−	1.33252i		0		−2.39899				0				7.86060i		0
303.2		0		−2.19409	−	2.19409i		0		−1.29836	+	1.29836i		0		4.71096				0				6.62806i		0
303.3		0		−2.03444	−	2.03444i		0		2.23466	−	2.23466i		0		0.725567				0				5.27790i		0
303.4		0		−1.96469	−	1.96469i		0		−0.884107	+	0.884107i		0		0.721104				0				4.72004i		0
303.5		0		−1.96187	−	1.96187i		0		−3.06749	+	3.06749i		0		−1.97168				0				4.69788i		0
303.6		0		−1.65943	−	1.65943i		0		−0.668279	+	0.668279i		0		−1.43332				0				2.50744i		0
303.7		0		−1.63122	−	1.63122i		0		0.837183	−	0.837183i		0		−1.95985				0				2.32176i		0
303.8		0		−1.42780	−	1.42780i		0		0.391081	−	0.391081i		0		3.36082				0				1.07723i		0
303.9		0		−1.34292	−	1.34292i		0		2.53450	−	2.53450i		0		2.70706				0				0.606865i		0
303.10		0		−1.08919	−	1.08919i		0		−0.770023	+	0.770023i		0		−1.47744				0			−	0.627346i		0
303.11		0		−1.07779	−	1.07779i		0		−1.37400	+	1.37400i		0		4.00321				0			−	0.676721i		0
303.12		0		−1.03801	−	1.03801i		0		2.13321	−	2.13321i		0		−3.87204				0			−	0.845075i		0
303.13		0		−1.02920	−	1.02920i		0		−2.13462	+	2.13462i		0		−4.26829				0			−	0.881504i		0
303.14		0		−0.623611	−	0.623611i		0		2.23509	−	2.23509i		0		0.609161				0			−	2.22222i		0
303.15		0		−0.599104	−	0.599104i		0		−1.69205	+	1.69205i		0		1.26000				0			−	2.28215i		0
303.16		0		−0.549728	−	0.549728i		0		1.49999	−	1.49999i		0		3.48044				0			−	2.39560i		0
303.17		0		−0.308215	−	0.308215i		0		0.114298	−	0.114298i		0		−4.05189				0			−	2.81001i		0
303.18		0		−0.105438	−	0.105438i		0		0.199404	−	0.199404i		0		−0.290770				0			−	2.97777i		0
303.19		0		−0.101486	−	0.101486i		0		−2.62302	+	2.62302i		0		2.14594				0			−	2.97940i		0
303.20		0		0.101486	+	0.101486i		0		−2.62302	+	2.62302i		0		2.14594				0			−	2.97940i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.f	odd	4	1	inner
19.b	odd	2	1	inner
304.m	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1216.2.m.a		76
4.b	odd	2	1		304.2.m.a	✓	76
16.e	even	4	1		304.2.m.a	✓	76
16.f	odd	4	1	inner	1216.2.m.a		76
19.b	odd	2	1	inner	1216.2.m.a		76
76.d	even	2	1		304.2.m.a	✓	76
304.j	odd	4	1		304.2.m.a	✓	76
304.m	even	4	1	inner	1216.2.m.a		76

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
304.2.m.a	✓	76	4.b	odd	2	1
304.2.m.a	✓	76	16.e	even	4	1
304.2.m.a	✓	76	76.d	even	2	1
304.2.m.a	✓	76	304.j	odd	4	1
1216.2.m.a		76	1.a	even	1	1	trivial
1216.2.m.a		76	16.f	odd	4	1	inner
1216.2.m.a		76	19.b	odd	2	1	inner
1216.2.m.a		76	304.m	even	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1216, [\chi])\).