Newform orbit 126.3.r.a

• Label: 126.3.r.a
• Level: $126$
• Weight: $3$
• Character orbit: 126.r
• Analytic conductor: $3.433$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	126.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43325133094\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 64 q^{4} + 8 q^{6} + 2 q^{7} - 20 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1		−	1.41421i	−2.92705	+	0.657548i	−2.00000			2.90307	+	1.67609i	0.929913	+	4.13948i	5.64984	+	4.13271i			2.82843i	8.13526	−	3.84935i	2.37035	−	4.10556i
11.2		−	1.41421i	−2.82827	−	1.00044i	−2.00000			0.857116	+	0.494856i	−1.41483	+	3.99978i	−6.76024	−	1.81637i			2.82843i	6.99824	+	5.65903i	0.699833	−	1.21215i
11.3		−	1.41421i	−0.991189	−	2.83153i	−2.00000			−6.36825	−	3.67671i	−4.00438	+	1.40175i	2.82364	+	6.40524i			2.82843i	−7.03509	+	5.61316i	−5.19965	+	9.00606i
11.4		−	1.41421i	−0.677712	+	2.92245i	−2.00000			−5.54397	−	3.20081i	4.13297	+	0.958429i	4.41544	−	5.43175i			2.82843i	−8.08141	−	3.96116i	−4.52663	+	7.84035i
11.5		−	1.41421i	0.146807	+	2.99641i	−2.00000			2.36201	+	1.36371i	4.23756	−	0.207617i	−2.54299	+	6.52175i			2.82843i	−8.95690	+	0.879787i	1.92857	−	3.34039i
11.6		−	1.41421i	2.09605	−	2.14630i	−2.00000			−2.07383	−	1.19732i	−3.03532	−	2.96426i	−5.45012	−	4.39274i			2.82843i	−0.213187	−	8.99747i	−1.69327	+	2.93283i
11.7		−	1.41421i	2.36405	+	1.84696i	−2.00000			1.22482	+	0.707152i	2.61199	−	3.34328i	3.10205	−	6.27513i			2.82843i	2.17750	+	8.73261i	1.00006	−	1.73216i
11.8		−	1.41421i	2.81732	−	1.03088i	−2.00000			6.63902	+	3.83304i	−1.45789	−	3.98429i	2.93662	+	6.35423i			2.82843i	6.87456	−	5.80865i	5.42073	−	9.38899i
11.9			1.41421i	−2.92348	+	0.673248i	−2.00000			−1.51694	−	0.875808i	−0.952117	−	4.13443i	−1.24611	−	6.88819i		−	2.82843i	8.09347	−	3.93646i	1.23858	−	2.14528i
11.10			1.41421i	−2.24880	−	1.98568i	−2.00000			1.84316	+	1.06415i	2.80817	−	3.18028i	6.16730	+	3.31125i		−	2.82843i	1.11417	+	8.93077i	−1.50493	+	2.60662i
11.11			1.41421i	−1.59142	+	2.54310i	−2.00000			5.46142	+	3.15315i	−3.59649	−	2.25061i	−3.23416	+	6.20807i		−	2.82843i	−3.93476	−	8.09430i	−4.45923	+	7.72361i
11.12			1.41421i	−0.664523	−	2.92548i	−2.00000			−2.12968	−	1.22957i	4.13725	−	0.939777i	−6.56056	+	2.44111i		−	2.82843i	−8.11682	+	3.88809i	1.73888	−	3.01183i
11.13			1.41421i	1.16644	+	2.76395i	−2.00000			−7.41683	−	4.28211i	−3.90882	+	1.64959i	−6.97339	−	0.609798i		−	2.82843i	−6.27884	+	6.44795i	6.05582	−	10.4890i
11.14			1.41421i	1.98625	−	2.24829i	−2.00000			8.39861	+	4.84894i	3.17956	+	2.80898i	−3.70991	−	5.93604i		−	2.82843i	−1.10962	−	8.93133i	−6.85744	+	11.8774i
11.15			1.41421i	2.02240	−	2.21583i	−2.00000			−7.20455	−	4.15955i	3.13365	+	2.86011i	5.54044	−	4.27827i		−	2.82843i	−0.819791	−	8.96259i	5.88249	−	10.1888i
11.16			1.41421i	2.25313	+	1.98076i	−2.00000			2.56482	+	1.48080i	−2.80121	+	3.18641i	6.84216	−	1.47812i		−	2.82843i	1.15321	+	8.92581i	−2.09417	+	3.62721i
23.1		−	1.41421i	−2.92348	−	0.673248i	−2.00000			−1.51694	+	0.875808i	−0.952117	+	4.13443i	−1.24611	+	6.88819i			2.82843i	8.09347	+	3.93646i	1.23858	+	2.14528i
23.2		−	1.41421i	−2.24880	+	1.98568i	−2.00000			1.84316	−	1.06415i	2.80817	+	3.18028i	6.16730	−	3.31125i			2.82843i	1.11417	−	8.93077i	−1.50493	−	2.60662i
23.3		−	1.41421i	−1.59142	−	2.54310i	−2.00000			5.46142	−	3.15315i	−3.59649	+	2.25061i	−3.23416	−	6.20807i			2.82843i	−3.93476	+	8.09430i	−4.45923	−	7.72361i
23.4		−	1.41421i	−0.664523	+	2.92548i	−2.00000			−2.12968	+	1.22957i	4.13725	+	0.939777i	−6.56056	−	2.44111i			2.82843i	−8.11682	−	3.88809i	1.73888	+	3.01183i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.j	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	126.3.r.a	yes	32
3.b	odd	2	1		378.3.r.a		32
7.c	even	3	1		126.3.i.a	✓	32
9.c	even	3	1		378.3.i.a		32
9.d	odd	6	1		126.3.i.a	✓	32
21.h	odd	6	1		378.3.i.a		32
63.h	even	3	1		378.3.r.a		32
63.j	odd	6	1	inner	126.3.r.a	yes	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
126.3.i.a	✓	32	7.c	even	3	1
126.3.i.a	✓	32	9.d	odd	6	1
126.3.r.a	yes	32	1.a	even	1	1	trivial
126.3.r.a	yes	32	63.j	odd	6	1	inner
378.3.i.a		32	9.c	even	3	1
378.3.i.a		32	21.h	odd	6	1
378.3.r.a		32	3.b	odd	2	1
378.3.r.a		32	63.h	even	3	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(126, [\chi])\).