Newform orbit 407.2.bb.a

• Label: 407.2.bb.a
• Level: $407$
• Weight: $2$
• Character orbit: 407.bb
• Analytic conductor: $3.250$
• Analytic rank: $0$
• Dimension: $24$
• CM discriminant: -11
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 407 = 11 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	407.bb (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.24991136227\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(2\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{36}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 30 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
32.1		0		−1.05773	+	2.90608i	1.28558	+	1.53209i	−3.26028	−	2.28287i		0			0			0		−5.02840	−	4.21933i		0
32.2		0		1.17651	−	3.23244i	1.28558	+	1.53209i	1.98821	+	1.39216i		0			0			0		−6.76633	−	5.67762i		0
54.1		0		−2.85390	−	0.503220i	0.684040	+	1.87939i	−0.346886	−	3.96493i		0			0			0		5.07246	+	1.84622i		0
54.2		0		1.34509	+	0.237176i	0.684040	+	1.87939i	0.211541	+	2.41792i		0			0			0		−1.06606	−	0.388015i		0
76.1		0		−0.125125	+	0.149118i	1.96962	+	0.347296i	−0.861876	+	1.84830i		0			0			0		0.514365	+	2.91711i		0
76.2		0		1.33317	−	1.58881i	1.96962	+	0.347296i	−1.58743	+	3.40426i		0			0			0		−0.226031	−	1.28189i		0
87.1		0		−1.34509	+	0.237176i	−0.684040	+	1.87939i	3.74189	+	0.327373i		0			0			0		−1.06606	+	0.388015i		0
87.2		0		2.85390	−	0.503220i	−0.684040	+	1.87939i	2.03161	+	0.177743i		0			0			0		5.07246	−	1.84622i		0
98.1		0		−2.85390	+	0.503220i	0.684040	−	1.87939i	−0.346886	+	3.96493i		0			0			0		5.07246	−	1.84622i		0
98.2		0		1.34509	−	0.237176i	0.684040	−	1.87939i	0.211541	−	2.41792i		0			0			0		−1.06606	+	0.388015i		0
109.1		0		−1.33317	+	1.58881i	−1.96962	−	0.347296i	−2.19975	−	1.02576i		0			0			0		−0.226031	−	1.28189i		0
109.2		0		0.125125	−	0.149118i	−1.96962	−	0.347296i	3.60717	+	1.68205i		0			0			0		0.514365	+	2.91711i		0
131.1		0		−1.34509	−	0.237176i	−0.684040	−	1.87939i	3.74189	−	0.327373i		0			0			0		−1.06606	−	0.388015i		0
131.2		0		2.85390	+	0.503220i	−0.684040	−	1.87939i	2.03161	−	0.177743i		0			0			0		5.07246	+	1.84622i		0
153.1		0		−1.17651	+	3.23244i	−1.28558	−	1.53209i	−2.15446	+	3.07689i		0			0			0		−6.76633	−	5.67762i		0
153.2		0		1.05773	−	2.90608i	−1.28558	−	1.53209i	−1.16974	+	1.67056i		0			0			0		−5.02840	−	4.21933i		0
241.1		0		−0.125125	−	0.149118i	1.96962	−	0.347296i	−0.861876	−	1.84830i		0			0			0		0.514365	−	2.91711i		0
241.2		0		1.33317	+	1.58881i	1.96962	−	0.347296i	−1.58743	−	3.40426i		0			0			0		−0.226031	+	1.28189i		0
274.1		0		−1.17651	−	3.23244i	−1.28558	+	1.53209i	−2.15446	−	3.07689i		0			0			0		−6.76633	+	5.67762i		0
274.2		0		1.05773	+	2.90608i	−1.28558	+	1.53209i	−1.16974	−	1.67056i		0			0			0		−5.02840	+	4.21933i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by \(\Q(\sqrt{-11}) \)
37.i	odd	36	1	inner
407.bb	even	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	407.2.bb.a	✓	24
11.b	odd	2	1	CM	407.2.bb.a	✓	24
37.i	odd	36	1	inner	407.2.bb.a	✓	24
407.bb	even	36	1	inner	407.2.bb.a	✓	24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
407.2.bb.a	✓	24	1.a	even	1	1	trivial
407.2.bb.a	✓	24	11.b	odd	2	1	CM
407.2.bb.a	✓	24	37.i	odd	36	1	inner
407.2.bb.a	✓	24	407.bb	even	36	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(407, [\chi])\).