Newform orbit 312.2.bp.a

• Label: 312.2.bp.a
• Level: $312$
• Weight: $2$
• Character orbit: 312.bp
• Analytic conductor: $2.491$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	312.bp (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.49133254306\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
41.1		0		−1.73198	−	0.0152365i		0		−0.477579	−	0.477579i		0		0.988569	−	0.264886i		0		2.99954	+	0.0527789i		0
41.2		0		−1.66291	−	0.484504i		0		1.70770	+	1.70770i		0		−3.27107	+	0.876480i		0		2.53051	+	1.61137i		0
41.3		0		−1.50587	+	0.855777i		0		−1.23702	−	1.23702i		0		0.738861	−	0.197977i		0		1.53529	−	2.57738i		0
41.4		0		−0.922814	−	1.46575i		0		2.19797	+	2.19797i		0		0.457363	−	0.122550i		0		−1.29683	+	2.70522i		0
41.5		0		−0.807967	−	1.53205i		0		−2.19797	−	2.19797i		0		0.457363	−	0.122550i		0		−1.69438	+	2.47570i		0
41.6		0		−0.769874	+	1.55155i		0		0.190181	+	0.190181i		0		3.71203	−	0.994635i		0		−1.81459	−	2.38899i		0
41.7		0		−0.123974	+	1.72761i		0		3.14772	+	3.14772i		0		2.26593	−	0.607154i		0		−2.96926	−	0.428358i		0
41.8		0		−0.0393877	+	1.73160i		0		−0.741654	−	0.741654i		0		−4.39168	+	1.17675i		0		−2.99690	−	0.136408i		0
41.9		0		0.411861	−	1.68237i		0		−1.70770	−	1.70770i		0		−3.27107	+	0.876480i		0		−2.66074	−	1.38580i		0
41.10		0		0.852797	−	1.50756i		0		0.477579	+	0.477579i		0		0.988569	−	0.264886i		0		−1.54548	−	2.57128i		0
41.11		0		1.49406	−	0.876234i		0		1.23702	+	1.23702i		0		0.738861	−	0.197977i		0		1.46443	−	2.61829i		0
41.12		0		1.51931	+	0.831691i		0		0.741654	+	0.741654i		0		−4.39168	+	1.17675i		0		1.61658	+	2.52719i		0
41.13		0		1.55814	+	0.756439i		0		−3.14772	−	3.14772i		0		2.26593	−	0.607154i		0		1.85560	+	2.35728i		0
41.14		0		1.72861	+	0.109042i		0		−0.190181	−	0.190181i		0		3.71203	−	0.994635i		0		2.97622	+	0.376984i		0
89.1		0		−1.68472	+	0.402146i		0		−2.36656	+	2.36656i		0		−0.332282	−	1.24009i		0		2.67656	−	1.35501i		0
89.2		0		−1.43902	+	0.963958i		0		0.504580	−	0.504580i		0		−0.836809	−	3.12301i		0		1.14157	−	2.77431i		0
89.3		0		−1.43567	−	0.968940i		0		2.19967	−	2.19967i		0		1.17763	+	4.39498i		0		1.12231	+	2.78216i		0
89.4		0		−1.26859	−	1.17927i		0		−1.56802	+	1.56802i		0		−0.418596	−	1.56222i		0		0.218643	+	2.99202i		0
89.5		0		−1.19941	+	1.24957i		0		1.44076	−	1.44076i		0		0.918532	+	3.42801i		0		−0.122851	−	2.99748i		0
89.6		0		−0.386983	−	1.68827i		0		1.56802	−	1.56802i		0		−0.418596	−	1.56222i		0		−2.70049	+	1.30666i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
13.f	odd	12	1	inner
39.k	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	312.2.bp.a	✓	56
3.b	odd	2	1	inner	312.2.bp.a	✓	56
4.b	odd	2	1		624.2.cn.f		56
12.b	even	2	1		624.2.cn.f		56
13.f	odd	12	1	inner	312.2.bp.a	✓	56
39.k	even	12	1	inner	312.2.bp.a	✓	56
52.l	even	12	1		624.2.cn.f		56
156.v	odd	12	1		624.2.cn.f		56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
312.2.bp.a	✓	56	1.a	even	1	1	trivial
312.2.bp.a	✓	56	3.b	odd	2	1	inner
312.2.bp.a	✓	56	13.f	odd	12	1	inner
312.2.bp.a	✓	56	39.k	even	12	1	inner
624.2.cn.f		56	4.b	odd	2	1
624.2.cn.f		56	12.b	even	2	1
624.2.cn.f		56	52.l	even	12	1
624.2.cn.f		56	156.v	odd	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(312, [\chi])\).