Newform orbit 348.2.u.a

• Label: 348.2.u.a
• Level: $348$
• Weight: $2$
• Character orbit: 348.u
• Analytic conductor: $2.779$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	348.u (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.77879399034\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(10\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
77.1		0		−1.61032	−	0.637874i		0		−0.257187	+	1.12681i		0		1.28537	−	0.619002i		0		2.18623	+	2.05436i		0
77.2		0		−1.43419	+	0.971134i		0		−0.315574	+	1.38262i		0		−4.35058	+	2.09513i		0		1.11380	−	2.78558i		0
77.3		0		−0.910998	+	1.47312i		0		−0.668669	+	2.92963i		0		2.92243	−	1.40737i		0		−1.34017	−	2.68402i		0
77.4		0		−0.564571	−	1.63746i		0		0.446237	−	1.95509i		0		1.32620	−	0.638664i		0		−2.36252	+	1.84892i		0
77.5		0		−0.206229	+	1.71973i		0		0.668669	−	2.92963i		0		2.92243	−	1.40737i		0		−2.91494	−	0.709316i		0
77.6		0		0.0554564	−	1.73116i		0		−0.717531	+	3.14371i		0		−2.94019	+	1.41592i		0		−2.99385	−	0.192008i		0
77.7		0		0.515802	+	1.65347i		0		0.315574	−	1.38262i		0		−4.35058	+	2.09513i		0		−2.46790	+	1.70572i		0
77.8		0		1.03600	−	1.38805i		0		0.717531	−	3.14371i		0		−2.94019	+	1.41592i		0		−0.853388	−	2.87606i		0
77.9		0		1.46234	−	0.928210i		0		−0.446237	+	1.95509i		0		1.32620	−	0.638664i		0		1.27685	−	2.71471i		0
77.10		0		1.65670	+	0.505306i		0		0.257187	−	1.12681i		0		1.28537	−	0.619002i		0		2.48933	+	1.67428i		0
89.1		0		−1.73198	−	0.0158657i		0		1.59205	+	1.99637i		0		−0.403835	−	1.76932i		0		2.99950	+	0.0549580i		0
89.2		0		−1.73192	−	0.0214617i		0		−0.983380	−	1.23312i		0		0.714491	+	3.13039i		0		2.99908	+	0.0743397i		0
89.3		0		−0.826331	+	1.52223i		0		−0.603757	−	0.757088i		0		−0.729341	−	3.19545i		0		−1.63435	−	2.51573i		0
89.4		0		−0.770787	−	1.55109i		0		0.983380	+	1.23312i		0		0.714491	+	3.13039i		0		−1.81177	+	2.39112i		0
89.5		0		−0.765772	−	1.55357i		0		−1.59205	−	1.99637i		0		−0.403835	−	1.76932i		0		−1.82719	+	2.37937i		0
89.6		0		0.360287	+	1.69416i		0		−0.868530	−	1.08910i		0		0.633399	+	2.77510i		0		−2.74039	+	1.22077i		0
89.7		0		1.01295	−	1.40497i		0		0.603757	+	0.757088i		0		−0.729341	−	3.19545i		0		−0.947872	−	2.84632i		0
89.8		0		1.08314	+	1.35160i		0		2.49932	+	3.13404i		0		−0.562661	−	2.46518i		0		−0.653631	+	2.92793i		0
89.9		0		1.68271	−	0.410463i		0		0.868530	+	1.08910i		0		0.633399	+	2.77510i		0		2.66304	−	1.38138i		0
89.10		0		1.68770	+	0.389436i		0		−2.49932	−	3.13404i		0		−0.562661	−	2.46518i		0		2.69668	+	1.31450i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
29.f	odd	28	1	inner
87.k	even	28	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	348.2.u.a	✓	120
3.b	odd	2	1	inner	348.2.u.a	✓	120
29.f	odd	28	1	inner	348.2.u.a	✓	120
87.k	even	28	1	inner	348.2.u.a	✓	120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
348.2.u.a	✓	120	1.a	even	1	1	trivial
348.2.u.a	✓	120	3.b	odd	2	1	inner
348.2.u.a	✓	120	29.f	odd	28	1	inner
348.2.u.a	✓	120	87.k	even	28	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(348, [\chi])\).