Newform orbit 1197.2.db.a

• Label: 1197.2.db.a
• Level: $1197$
• Weight: $2$
• Character orbit: 1197.db
• Analytic conductor: $9.558$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1197 = 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1197.db (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.55809312195\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 48 q^{4} + 8 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
647.1	−2.42896	+	1.40236i		0		2.93323	−	5.08050i	−0.761918	−	1.31968i		0		1.46953	+	2.20011i			10.8443i		0		3.70134	+	2.13697i
647.2	−2.35524	+	1.35980i		0		2.69812	−	4.67327i	0.153495	+	0.265861i		0		−2.56747	−	0.638832i			9.23639i		0		−0.723035	−	0.417444i
647.3	−2.18268	+	1.26017i		0		2.17605	−	3.76903i	1.50551	+	2.60762i		0		2.61298	−	0.415128i			5.92809i		0		−6.57208	−	3.79439i
647.4	−2.15278	+	1.24291i		0		2.08963	−	3.61935i	−1.15891	−	2.00729i		0		−1.57780	−	2.12380i			5.41726i		0		4.98974	+	2.88083i
647.5	−2.10194	+	1.21356i		0		1.94545	−	3.36961i	1.33568	+	2.31346i		0		1.49746	−	2.18120i			4.58942i		0		−5.61505	−	3.24185i
647.6	−1.99261	+	1.15043i		0		1.64700	−	2.85268i	−0.963221	−	1.66835i		0		1.46823	+	2.20098i			2.97731i		0		3.83865	+	2.21624i
647.7	−1.97511	+	1.14033i		0		1.60071	−	2.77251i	0.970205	+	1.68044i		0		−2.34380	+	1.22744i			2.74002i		0		−3.83252	−	2.21271i
647.8	−1.88715	+	1.08955i		0		1.37423	−	2.38023i	−0.464876	−	0.805189i		0		1.91409	−	1.82654i			1.63096i		0		1.75458	+	1.01301i
647.9	−1.69639	+	0.979413i		0		0.918501	−	1.59089i	0.263928	+	0.457136i		0		−1.75471	+	1.98015i		−	0.319283i		0		−0.895450	−	0.516989i
647.10	−1.58089	+	0.912729i		0		0.666150	−	1.15381i	−1.39335	−	2.41335i		0		−2.63134	+	0.275746i		−	1.21886i		0		4.40546	+	2.54350i
647.11	−1.46489	+	0.845756i		0		0.430607	−	0.745834i	−1.32536	−	2.29559i		0		1.85706	−	1.88450i		−	1.92627i		0		3.88302	+	2.24186i
647.12	−1.42438	+	0.822364i		0		0.352564	−	0.610659i	−1.92638	−	3.33659i		0		0.236940	+	2.63512i		−	2.12971i		0		5.48778	+	3.16837i
647.13	−1.31077	+	0.756772i		0		0.145408	−	0.251853i	0.416154	+	0.720799i		0		−0.851166	−	2.50510i		−	2.58693i		0		−1.09096	−	0.629867i
647.14	−1.25928	+	0.727048i		0		0.0571968	−	0.0990677i	1.78585	+	3.09319i		0		2.17606	+	1.50491i		−	2.74185i		0		−4.49779	−	2.59680i
647.15	−0.959209	+	0.553800i		0		−0.386612	+	0.669632i	1.12128	+	1.94211i		0		2.59227	+	0.529289i		−	3.07162i		0		−2.15108	−	1.24192i
647.16	−0.895160	+	0.516821i		0		−0.465792	+	0.806775i	1.42456	+	2.46740i		0		−1.84133	−	1.89987i		−	3.03021i		0		−2.55041	−	1.47248i
647.17	−0.850513	+	0.491044i		0		−0.517752	+	0.896772i	0.359764	+	0.623130i		0		−2.63950	−	0.181813i		−	2.98113i		0		−0.611969	−	0.353320i
647.18	−0.839260	+	0.484547i		0		−0.530428	+	0.918729i	−0.237468	−	0.411306i		0		−1.39585	−	2.24758i		−	2.96626i		0		0.398594	+	0.230129i
647.19	−0.818824	+	0.472749i		0		−0.553018	+	0.957855i	−2.19579	−	3.80322i		0		2.57960	−	0.587948i		−	2.93675i		0		3.59593	+	2.07611i
647.20	−0.509165	+	0.293966i		0		−0.827168	+	1.43270i	−1.37766	−	2.38618i		0		−0.00278646	−	2.64575i		−	2.14850i		0		1.40292	+	0.809974i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1197.2.db.a	✓	96
3.b	odd	2	1	inner	1197.2.db.a	✓	96
7.d	odd	6	1	inner	1197.2.db.a	✓	96
21.g	even	6	1	inner	1197.2.db.a	✓	96

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1197.2.db.a	✓	96	1.a	even	1	1	trivial
1197.2.db.a	✓	96	3.b	odd	2	1	inner
1197.2.db.a	✓	96	7.d	odd	6	1	inner
1197.2.db.a	✓	96	21.g	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1197, [\chi])\).