Newform orbit 387.2.t.a

• Label: 387.2.t.a
• Level: $387$
• Weight: $2$
• Character orbit: 387.t
• Analytic conductor: $3.090$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	387.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.09021055822\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 24 q^{4} + 6 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
80.1	−2.57549				0		4.63314			−0.649717	+	1.12534i		0		−2.31737	+	1.33793i	−6.78161				0		1.67334	−	2.89831i
80.2	−2.21275				0		2.89628			−0.826031	+	1.43073i		0		0.490111	−	0.282966i	−1.98324				0		1.82780	−	3.16585i
80.3	−2.13263				0		2.54810			2.00714	−	3.47647i		0		4.48706	−	2.59060i	−1.16889				0		−4.28049	+	7.41402i
80.4	−1.39585				0		−0.0516143			1.64095	−	2.84221i		0		−2.17205	+	1.25403i	2.86374				0		−2.29052	+	3.96729i
80.5	−0.947726				0		−1.10182			−1.48568	+	2.57328i		0		1.45658	−	0.840957i	2.93967				0		1.40802	−	2.43876i
80.6	−0.906465				0		−1.17832			0.576829	−	0.999097i		0		−2.46580	+	1.42363i	2.88104				0		−0.522875	+	0.905646i
80.7	−0.504218				0		−1.74576			−0.366350	+	0.634537i		0		2.02147	−	1.16709i	1.88868				0		0.184720	−	0.319945i
80.8	0.504218				0		−1.74576			0.366350	−	0.634537i		0		2.02147	−	1.16709i	−1.88868				0		0.184720	−	0.319945i
80.9	0.906465				0		−1.17832			−0.576829	+	0.999097i		0		−2.46580	+	1.42363i	−2.88104				0		−0.522875	+	0.905646i
80.10	0.947726				0		−1.10182			1.48568	−	2.57328i		0		1.45658	−	0.840957i	−2.93967				0		1.40802	−	2.43876i
80.11	1.39585				0		−0.0516143			−1.64095	+	2.84221i		0		−2.17205	+	1.25403i	−2.86374				0		−2.29052	+	3.96729i
80.12	2.13263				0		2.54810			−2.00714	+	3.47647i		0		4.48706	−	2.59060i	1.16889				0		−4.28049	+	7.41402i
80.13	2.21275				0		2.89628			0.826031	−	1.43073i		0		0.490111	−	0.282966i	1.98324				0		1.82780	−	3.16585i
80.14	2.57549				0		4.63314			0.649717	−	1.12534i		0		−2.31737	+	1.33793i	6.78161				0		1.67334	−	2.89831i
179.1	−2.57549				0		4.63314			−0.649717	−	1.12534i		0		−2.31737	−	1.33793i	−6.78161				0		1.67334	+	2.89831i
179.2	−2.21275				0		2.89628			−0.826031	−	1.43073i		0		0.490111	+	0.282966i	−1.98324				0		1.82780	+	3.16585i
179.3	−2.13263				0		2.54810			2.00714	+	3.47647i		0		4.48706	+	2.59060i	−1.16889				0		−4.28049	−	7.41402i
179.4	−1.39585				0		−0.0516143			1.64095	+	2.84221i		0		−2.17205	−	1.25403i	2.86374				0		−2.29052	−	3.96729i
179.5	−0.947726				0		−1.10182			−1.48568	−	2.57328i		0		1.45658	+	0.840957i	2.93967				0		1.40802	+	2.43876i
179.6	−0.906465				0		−1.17832			0.576829	+	0.999097i		0		−2.46580	−	1.42363i	2.88104				0		−0.522875	−	0.905646i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
43.d	odd	6	1	inner
129.h	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	387.2.t.a	✓	28
3.b	odd	2	1	inner	387.2.t.a	✓	28
43.d	odd	6	1	inner	387.2.t.a	✓	28
129.h	even	6	1	inner	387.2.t.a	✓	28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
387.2.t.a	✓	28	1.a	even	1	1	trivial
387.2.t.a	✓	28	3.b	odd	2	1	inner
387.2.t.a	✓	28	43.d	odd	6	1	inner
387.2.t.a	✓	28	129.h	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(387, [\chi])\).