Newform orbit 900.2.bg.a

• Label: 900.2.bg.a
• Level: $900$
• Weight: $2$
• Character orbit: 900.bg
• Analytic conductor: $7.187$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.bg (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.18653618192\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(30\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 2 q^{3} - 4 q^{5} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
61.1		0		−1.67673	+	0.434241i		0		−1.00789	+	1.99604i		0		0.516228	−	0.894133i		0		2.62287	−	1.45621i		0
61.2		0		−1.64289	+	0.548549i		0		1.37194	−	1.76573i		0		−2.37185	+	4.10816i		0		2.39819	−	1.80241i		0
61.3		0		−1.62661	−	0.595101i		0		2.22506	+	0.221639i		0		1.44975	−	2.51104i		0		2.29171	+	1.93599i		0
61.4		0		−1.60007	−	0.663151i		0		1.06657	+	1.96530i		0		−2.15417	+	3.73113i		0		2.12046	+	2.12218i		0
61.5		0		−1.59842	+	0.667132i		0		−0.502718	−	2.17882i		0		1.58299	−	2.74182i		0		2.10987	−	2.13271i		0
61.6		0		−1.58608	−	0.695944i		0		−2.23484	+	0.0742255i		0		−0.867218	+	1.50207i		0		2.03132	+	2.20765i		0
61.7		0		−1.27651	+	1.17069i		0		2.23269	−	0.122868i		0		0.279123	−	0.483455i		0		0.258969	−	2.98880i		0
61.8		0		−1.26899	+	1.17885i		0		−1.96299	−	1.07083i		0		−0.910263	+	1.57662i		0		0.220648	−	2.99187i		0
61.9		0		−0.893308	−	1.48391i		0		−2.05489	−	0.881726i		0		2.49276	−	4.31759i		0		−1.40400	+	2.65118i		0
61.10		0		−0.878097	−	1.49297i		0		0.0507636	−	2.23549i		0		−0.545715	+	0.945206i		0		−1.45789	+	2.62194i		0
61.11		0		−0.874214	+	1.49524i		0		0.535572	+	2.17098i		0		1.54954	−	2.68389i		0		−1.47150	−	2.61432i		0
61.12		0		−0.671690	−	1.59651i		0		0.0838407	+	2.23450i		0		−0.905890	+	1.56905i		0		−2.09766	+	2.14471i		0
61.13		0		−0.637479	+	1.61047i		0		−1.19376	+	1.89075i		0		−2.07969	+	3.60213i		0		−2.18724	−	2.05328i		0
61.14		0		−0.631875	−	1.61268i		0		0.308404	+	2.21470i		0		1.91374	−	3.31469i		0		−2.20147	+	2.03802i		0
61.15		0		−0.312229	−	1.70368i		0		2.01459	−	0.970277i		0		−0.856597	+	1.48367i		0		−2.80503	+	1.06387i		0
61.16		0		−0.175370	+	1.72315i		0		−1.95088	−	1.09274i		0		1.38079	−	2.39159i		0		−2.93849	−	0.604379i		0
61.17		0		0.0513649	+	1.73129i		0		2.13415	+	0.667390i		0		−0.801301	+	1.38789i		0		−2.99472	+	0.177855i		0
61.18		0		0.179429	+	1.72273i		0		0.694082	−	2.12562i		0		−0.0881730	+	0.152720i		0		−2.93561	+	0.618215i		0
61.19		0		0.579526	−	1.63222i		0		−2.01757	−	0.964056i		0		−1.83528	+	3.17879i		0		−2.32830	−	1.89183i		0
61.20		0		0.626630	−	1.61472i		0		−1.66045	+	1.49764i		0		0.0756354	−	0.131004i		0		−2.21467	−	2.02367i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
25.d	even	5	1	inner
225.q	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.2.bg.a	✓	240
9.c	even	3	1	inner	900.2.bg.a	✓	240
25.d	even	5	1	inner	900.2.bg.a	✓	240
225.q	even	15	1	inner	900.2.bg.a	✓	240

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
900.2.bg.a	✓	240	1.a	even	1	1	trivial
900.2.bg.a	✓	240	9.c	even	3	1	inner
900.2.bg.a	✓	240	25.d	even	5	1	inner
900.2.bg.a	✓	240	225.q	even	15	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(900, [\chi])\).