Newform orbit 468.2.bz.a

• Label: 468.2.bz.a
• Level: $468$
• Weight: $2$
• Character orbit: 468.bz
• Analytic conductor: $3.737$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	468.bz (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.73699881460\)
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		−1.73205	+	0.00164220i		0		−3.31021	+	0.886969i		0		−1.61187	−	1.61187i		0		2.99999	−	0.00568874i		0
41.2		0		−1.70662	−	0.295698i		0		−0.625781	+	0.167678i		0		−0.0920834	−	0.0920834i		0		2.82513	+	1.00929i		0
41.3		0		−1.44462	−	0.955549i		0		3.78167	−	1.01330i		0		1.85325	+	1.85325i		0		1.17385	+	2.76081i		0
41.4		0		−1.35995	+	1.07263i		0		2.74150	−	0.734582i		0		−0.574219	−	0.574219i		0		0.698942	−	2.91744i		0
41.5		0		−0.799158	+	1.53667i		0		−1.48330	+	0.397449i		0		3.49497	+	3.49497i		0		−1.72269	−	2.45608i		0
41.6		0		−0.458507	+	1.67026i		0		−1.27901	+	0.342710i		0		−1.51170	−	1.51170i		0		−2.57954	−	1.53165i		0
41.7		0		−0.354861	−	1.69531i		0		−3.02915	+	0.811657i		0		2.25148	+	2.25148i		0		−2.74815	+	1.20320i		0
41.8		0		0.00961860	−	1.73202i		0		0.133404	−	0.0357456i		0		−0.944253	−	0.944253i		0		−2.99981	−	0.0333193i		0
41.9		0		0.690572	+	1.58843i		0		−0.407567	+	0.109207i		0		−1.26050	−	1.26050i		0		−2.04622	+	2.19385i		0
41.10		0		1.06006	−	1.36977i		0		2.36232	−	0.632983i		0		1.46097	+	1.46097i		0		−0.752556	−	2.90408i		0
41.11		0		1.20133	+	1.24772i		0		2.65533	−	0.711493i		0		2.08103	+	2.08103i		0		−0.113592	+	2.99785i		0
41.12		0		1.54375	−	0.785394i		0		−3.50693	+	0.939679i		0		−3.18836	−	3.18836i		0		1.76631	−	2.42490i		0
41.13		0		1.64009	−	0.556883i		0		−0.620128	+	0.166163i		0		1.81075	+	1.81075i		0		2.37976	−	1.82667i		0
41.14		0		1.71036	+	0.273286i		0		2.58785	−	0.693413i		0		−2.76946	−	2.76946i		0		2.85063	+	0.934833i		0
137.1		0		−1.73205	−	0.00164220i		0		−3.31021	−	0.886969i		0		−1.61187	+	1.61187i		0		2.99999	+	0.00568874i		0
137.2		0		−1.70662	+	0.295698i		0		−0.625781	−	0.167678i		0		−0.0920834	+	0.0920834i		0		2.82513	−	1.00929i		0
137.3		0		−1.44462	+	0.955549i		0		3.78167	+	1.01330i		0		1.85325	−	1.85325i		0		1.17385	−	2.76081i		0
137.4		0		−1.35995	−	1.07263i		0		2.74150	+	0.734582i		0		−0.574219	+	0.574219i		0		0.698942	+	2.91744i		0
137.5		0		−0.799158	−	1.53667i		0		−1.48330	−	0.397449i		0		3.49497	−	3.49497i		0		−1.72269	+	2.45608i		0
137.6		0		−0.458507	−	1.67026i		0		−1.27901	−	0.342710i		0		−1.51170	+	1.51170i		0		−2.57954	+	1.53165i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
117.bc	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	468.2.bz.a	yes	56
3.b	odd	2	1		1404.2.cc.a		56
9.c	even	3	1		1404.2.bz.a		56
9.d	odd	6	1		468.2.bw.a	✓	56
13.f	odd	12	1		468.2.bw.a	✓	56
39.k	even	12	1		1404.2.bz.a		56
117.bb	odd	12	1		1404.2.cc.a		56
117.bc	even	12	1	inner	468.2.bz.a	yes	56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
468.2.bw.a	✓	56	9.d	odd	6	1
468.2.bw.a	✓	56	13.f	odd	12	1
468.2.bz.a	yes	56	1.a	even	1	1	trivial
468.2.bz.a	yes	56	117.bc	even	12	1	inner
1404.2.bz.a		56	9.c	even	3	1
1404.2.bz.a		56	39.k	even	12	1
1404.2.cc.a		56	3.b	odd	2	1
1404.2.cc.a		56	117.bb	odd	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(468, [\chi])\).