Newform orbit 550.2.bb.a

• Label: 550.2.bb.a
• Level: $550$
• Weight: $2$
• Character orbit: 550.bb
• Analytic conductor: $4.392$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	550.bb (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.39177211117\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 10 q^{3} - 120 q^{4} - 2 q^{5} + 4 q^{6} - 20 q^{7} + 36 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
119.1		−	1.00000i	−3.19079	−	1.03675i	−1.00000			0.743607	+	2.10880i	−1.03675	+	3.19079i	0.653908	−	0.900027i			1.00000i	6.67924	+	4.85275i	2.10880	−	0.743607i
119.2		−	1.00000i	−2.56485	−	0.833369i	−1.00000			−0.410172	−	2.19813i	−0.833369	+	2.56485i	−1.55626	+	2.14201i			1.00000i	3.45688	+	2.51157i	−2.19813	+	0.410172i
119.3		−	1.00000i	−2.11706	−	0.687873i	−1.00000			−0.136567	−	2.23189i	−0.687873	+	2.11706i	2.38728	−	3.28581i			1.00000i	1.58171	+	1.14918i	−2.23189	+	0.136567i
119.4		−	1.00000i	−1.70172	−	0.552923i	−1.00000			2.22772	−	0.193009i	−0.552923	+	1.70172i	−1.01423	+	1.39596i			1.00000i	0.163087	+	0.118490i	−0.193009	−	2.22772i
119.5		−	1.00000i	−1.48058	−	0.481068i	−1.00000			−2.22618	+	0.210023i	−0.481068	+	1.48058i	−2.20484	+	3.03470i			1.00000i	−0.466373	−	0.338840i	0.210023	+	2.22618i
119.6		−	1.00000i	−1.34746	−	0.437817i	−1.00000			−0.686067	+	2.12822i	−0.437817	+	1.34746i	1.52296	−	2.09617i			1.00000i	−0.803077	−	0.583470i	2.12822	+	0.686067i
119.7		−	1.00000i	−0.175265	−	0.0569469i	−1.00000			2.23354	−	0.106224i	−0.0569469	+	0.175265i	−1.04501	+	1.43834i			1.00000i	−2.39958	−	1.74339i	−0.106224	−	2.23354i
119.8		−	1.00000i	−0.0496594	−	0.0161353i	−1.00000			−1.98615	−	1.02724i	−0.0161353	+	0.0496594i	1.71999	−	2.36737i			1.00000i	−2.42485	−	1.76175i	−1.02724	+	1.98615i
119.9		−	1.00000i	0.404331	+	0.131375i	−1.00000			−1.35019	+	1.78241i	0.131375	−	0.404331i	−0.0546342	+	0.0751975i			1.00000i	−2.28083	−	1.65712i	1.78241	+	1.35019i
119.10		−	1.00000i	1.09325	+	0.355220i	−1.00000			1.90558	+	1.16995i	0.355220	−	1.09325i	2.57502	−	3.54420i			1.00000i	−1.35803	−	0.986666i	1.16995	−	1.90558i
119.11		−	1.00000i	1.26585	+	0.411301i	−1.00000			−1.84612	−	1.26168i	0.411301	−	1.26585i	−2.09081	+	2.87775i			1.00000i	−0.993835	−	0.722064i	−1.26168	+	1.84612i
119.12		−	1.00000i	1.83495	+	0.596213i	−1.00000			1.35288	+	1.78037i	0.596213	−	1.83495i	−2.15484	+	2.96588i			1.00000i	0.584535	+	0.424689i	1.78037	−	1.35288i
119.13		−	1.00000i	2.41686	+	0.785285i	−1.00000			1.65312	−	1.50572i	0.785285	−	2.41686i	0.793769	−	1.09253i			1.00000i	2.79748	+	2.03249i	−1.50572	−	1.65312i
119.14		−	1.00000i	2.47976	+	0.805722i	−1.00000			−1.00055	−	1.99973i	0.805722	−	2.47976i	0.484600	−	0.666995i			1.00000i	3.07296	+	2.23264i	−1.99973	+	1.00055i
119.15		−	1.00000i	3.03928	+	0.987520i	−1.00000			−1.45046	+	1.70181i	0.987520	−	3.03928i	−1.39886	+	1.92537i			1.00000i	5.83495	+	4.23934i	1.70181	+	1.45046i
119.16			1.00000i	−2.87040	−	0.932650i	−1.00000			−1.66754	+	1.48974i	0.932650	−	2.87040i	−2.46441	+	3.39198i		−	1.00000i	4.94232	+	3.59080i	−1.48974	−	1.66754i
119.17			1.00000i	−2.65147	−	0.861516i	−1.00000			1.24306	−	1.85871i	0.861516	−	2.65147i	−1.77453	+	2.44243i		−	1.00000i	3.86106	+	2.80522i	1.85871	+	1.24306i
119.18			1.00000i	−1.79868	−	0.584426i	−1.00000			1.67483	+	1.48154i	0.584426	−	1.79868i	0.518694	−	0.713921i		−	1.00000i	0.466642	+	0.339035i	−1.48154	+	1.67483i
119.19			1.00000i	−1.44938	−	0.470932i	−1.00000			−2.23360	−	0.104969i	0.470932	−	1.44938i	−0.0783939	+	0.107900i		−	1.00000i	−0.548125	−	0.398236i	0.104969	−	2.23360i
119.20			1.00000i	−1.24110	−	0.403258i	−1.00000			−0.314614	−	2.21382i	0.403258	−	1.24110i	1.32196	−	1.81952i		−	1.00000i	−1.04934	−	0.762390i	2.21382	−	0.314614i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
275.n	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	550.2.bb.a	yes	120
11.c	even	5	1		550.2.n.a	✓	120
25.e	even	10	1		550.2.n.a	✓	120
275.n	even	10	1	inner	550.2.bb.a	yes	120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
550.2.n.a	✓	120	11.c	even	5	1
550.2.n.a	✓	120	25.e	even	10	1
550.2.bb.a	yes	120	1.a	even	1	1	trivial
550.2.bb.a	yes	120	275.n	even	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(550, [\chi])\).