Newform orbit 585.2.ba.a

• Label: 585.2.ba.a
• Level: $585$
• Weight: $2$
• Character orbit: 585.ba
• Analytic conductor: $4.671$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	585.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.67124851824\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(56\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 112 q^{4} + 18 q^{6} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
121.1		−	2.77680i	−1.50399	−	0.859080i	−5.71062			−0.866025	−	0.500000i	−2.38549	+	4.17628i	−3.75002	−	2.16508i			10.3037i	1.52396	+	2.58409i	−1.38840	+	2.40478i
121.2		−	2.76598i	−0.979416	+	1.42855i	−5.65062			0.866025	+	0.500000i	3.95132	+	2.70904i	1.09992	+	0.635041i			10.0975i	−1.08149	−	2.79828i	1.38299	−	2.39541i
121.3		−	2.65635i	0.750872	+	1.56083i	−5.05622			−0.866025	−	0.500000i	4.14612	−	1.99458i	1.89927	+	1.09654i			8.11839i	−1.87238	+	2.34397i	−1.32818	+	2.30047i
121.4		−	2.50837i	1.70856	−	0.284280i	−4.29192			0.866025	+	0.500000i	−0.713080	−	4.28571i	4.08889	+	2.36072i			5.74899i	2.83837	−	0.971420i	1.25419	−	2.17231i
121.5		−	2.43934i	−1.56128	−	0.749940i	−3.95037			0.866025	+	0.500000i	−1.82936	+	3.80848i	1.33395	+	0.770158i			4.75760i	1.87518	+	2.34173i	1.21967	−	2.11253i
121.6		−	2.33968i	0.527435	−	1.64979i	−3.47410			0.866025	+	0.500000i	−3.85998	−	1.23403i	−0.902915	−	0.521298i			3.44893i	−2.44363	−	1.74031i	1.16984	−	2.02622i
121.7		−	2.26374i	0.586458	+	1.62974i	−3.12453			0.866025	+	0.500000i	3.68932	−	1.32759i	−4.50002	−	2.59809i			2.54566i	−2.31213	+	1.91155i	1.13187	−	1.96046i
121.8		−	2.25334i	−1.06935	+	1.36253i	−3.07755			−0.866025	−	0.500000i	3.07024	+	2.40962i	−2.39223	−	1.38116i			2.42809i	−0.712960	−	2.91405i	−1.12667	+	1.95145i
121.9		−	2.12050i	1.07836	−	1.35541i	−2.49653			−0.866025	−	0.500000i	−2.87416	−	2.28666i	−1.04503	−	0.603347i			1.05290i	−0.674289	−	2.92324i	−1.06025	+	1.83641i
121.10		−	2.08617i	1.50529	+	0.856804i	−2.35211			−0.866025	−	0.500000i	1.78744	−	3.14029i	0.0572545	+	0.0330559i			0.734567i	1.53178	+	2.57947i	−1.04309	+	1.80668i
121.11		−	2.08465i	−1.49480	+	0.874972i	−2.34576			−0.866025	−	0.500000i	1.82401	+	3.11613i	2.30189	+	1.32900i			0.720788i	1.46885	−	2.61581i	−1.04232	+	1.80536i
121.12		−	2.07056i	−0.654636	−	1.60357i	−2.28723			−0.866025	−	0.500000i	−3.32030	+	1.35547i	−0.776370	−	0.448238i			0.594732i	−2.14290	+	2.09952i	−1.03528	+	1.79316i
121.13		−	1.85442i	−1.71713	−	0.226835i	−1.43889			0.866025	+	0.500000i	−0.420648	+	3.18429i	−0.980446	−	0.566061i		−	1.04054i	2.89709	+	0.779010i	0.927212	−	1.60598i
121.14		−	1.71421i	−0.685514	+	1.59062i	−0.938508			0.866025	+	0.500000i	2.72665	+	1.17511i	1.17697	+	0.679524i		−	1.81962i	−2.06014	−	2.18078i	0.857104	−	1.48455i
121.15		−	1.42082i	1.53777	−	0.797039i	−0.0187380			−0.866025	−	0.500000i	−1.13245	−	2.18490i	4.00328	+	2.31129i		−	2.81502i	1.72946	−	2.45132i	−0.710412	+	1.23047i
121.16		−	1.41330i	1.16595	−	1.28084i	0.00257576			0.866025	+	0.500000i	−1.81022	−	1.64784i	−1.36630	−	0.788834i		−	2.83025i	−0.281110	−	2.98680i	0.706651	−	1.22396i
121.17		−	1.34514i	1.71577	+	0.236930i	0.190612			0.866025	+	0.500000i	0.318703	−	2.30794i	0.0652890	+	0.0376946i		−	2.94667i	2.88773	+	0.813035i	0.672568	−	1.16492i
121.18		−	1.21945i	0.420152	+	1.68032i	0.512939			−0.866025	−	0.500000i	2.04907	−	0.512355i	−3.34948	−	1.93382i		−	3.06441i	−2.64694	+	1.41198i	−0.609725	+	1.05608i
121.19		−	0.952617i	−0.144224	−	1.72604i	1.09252			0.866025	+	0.500000i	−1.64425	+	0.137390i	3.01390	+	1.74008i		−	2.94599i	−2.95840	+	0.497872i	0.476308	−	0.824990i
121.20		−	0.930301i	−1.61216	−	0.633196i	1.13454			−0.866025	−	0.500000i	−0.589062	+	1.49979i	2.52214	+	1.45616i		−	2.91607i	2.19813	+	2.04163i	−0.465150	+	0.805664i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
117.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	585.2.ba.a	✓	112
9.c	even	3	1		585.2.bm.a	yes	112
13.e	even	6	1		585.2.bm.a	yes	112
117.l	even	6	1	inner	585.2.ba.a	✓	112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
585.2.ba.a	✓	112	1.a	even	1	1	trivial
585.2.ba.a	✓	112	117.l	even	6	1	inner
585.2.bm.a	yes	112	9.c	even	3	1
585.2.bm.a	yes	112	13.e	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(585, [\chi])\).