Newform orbit 441.2.i.d

• Label: 441.2.i.d
• Level: $441$
• Weight: $2$
• Character orbit: 441.i
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{4} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
68.1		−	2.70883i	−0.572034	−	1.63486i	−5.33776			−0.601464	−	1.04177i	−4.42857	+	1.54954i		0				9.04141i	−2.34556	+	1.87039i	−2.82197	+	1.62926i
68.2		−	2.70883i	0.572034	+	1.63486i	−5.33776			0.601464	+	1.04177i	4.42857	−	1.54954i		0				9.04141i	−2.34556	+	1.87039i	2.82197	−	1.62926i
68.3		−	2.37274i	−1.65098	−	0.523694i	−3.62990			1.71774	+	2.97522i	−1.24259	+	3.91735i		0				3.86732i	2.45149	+	1.72922i	7.05942	−	4.07576i
68.4		−	2.37274i	1.65098	+	0.523694i	−3.62990			−1.71774	−	2.97522i	1.24259	−	3.91735i		0				3.86732i	2.45149	+	1.72922i	−7.05942	+	4.07576i
68.5		−	1.48451i	−0.995298	+	1.41753i	−0.203760			−0.154215	−	0.267109i	2.10433	+	1.47753i		0			−	2.66653i	−1.01876	−	2.82172i	−0.396525	+	0.228934i
68.6		−	1.48451i	0.995298	−	1.41753i	−0.203760			0.154215	+	0.267109i	−2.10433	−	1.47753i		0			−	2.66653i	−1.01876	−	2.82172i	0.396525	−	0.228934i
68.7		−	1.17820i	−0.278055	−	1.70959i	0.611843			−2.16601	−	3.75164i	−2.01424	+	0.327604i		0			−	3.07728i	−2.84537	+	0.950717i	−4.42019	+	2.55200i
68.8		−	1.17820i	0.278055	+	1.70959i	0.611843			2.16601	+	3.75164i	2.01424	−	0.327604i		0			−	3.07728i	−2.84537	+	0.950717i	4.42019	−	2.55200i
68.9		−	0.981621i	−1.73160	−	0.0395255i	1.03642			−0.940599	−	1.62916i	−0.0387990	+	1.69978i		0			−	2.98061i	2.99688	+	0.136885i	−1.59922	+	0.923312i
68.10		−	0.981621i	1.73160	+	0.0395255i	1.03642			0.940599	+	1.62916i	0.0387990	−	1.69978i		0			−	2.98061i	2.99688	+	0.136885i	1.59922	−	0.923312i
68.11		−	0.424201i	−0.625041	+	1.61534i	1.82005			−1.80381	−	3.12430i	0.685229	+	0.265143i		0			−	1.62047i	−2.21865	−	2.01931i	−1.32533	+	0.765180i
68.12		−	0.424201i	0.625041	−	1.61534i	1.82005			1.80381	+	3.12430i	−0.685229	−	0.265143i		0			−	1.62047i	−2.21865	−	2.01931i	1.32533	−	0.765180i
68.13			0.122344i	−0.937657	+	1.45630i	1.98503			0.264715	+	0.458500i	−0.178170	−	0.114717i		0				0.487547i	−1.24160	−	2.73101i	−0.0560949	+	0.0323864i
68.14			0.122344i	0.937657	−	1.45630i	1.98503			−0.264715	−	0.458500i	0.178170	+	0.114717i		0				0.487547i	−1.24160	−	2.73101i	0.0560949	−	0.0323864i
68.15			0.664297i	−1.15747	−	1.28852i	1.55871			−0.0141520	−	0.0245119i	0.855956	−	0.768901i		0				2.36404i	−0.320544	+	2.98283i	0.0162832	−	0.00940110i
68.16			0.664297i	1.15747	+	1.28852i	1.55871			0.0141520	+	0.0245119i	−0.855956	+	0.768901i		0				2.36404i	−0.320544	+	2.98283i	−0.0162832	+	0.00940110i
68.17			1.83202i	−1.55144	−	0.770089i	−1.35631			−0.322784	−	0.559079i	1.41082	−	2.84227i		0				1.17925i	1.81393	+	2.38949i	1.02425	−	0.591348i
68.18			1.83202i	1.55144	+	0.770089i	−1.35631			0.322784	+	0.559079i	−1.41082	+	2.84227i		0				1.17925i	1.81393	+	2.38949i	−1.02425	+	0.591348i
68.19			1.86894i	−1.72324	−	0.174470i	−1.49292			1.25287	+	2.17003i	0.326074	−	3.22063i		0				0.947692i	2.93912	+	0.601309i	−4.05565	+	2.34153i
68.20			1.86894i	1.72324	+	0.174470i	−1.49292			−1.25287	−	2.17003i	−0.326074	+	3.22063i		0				0.947692i	2.93912	+	0.601309i	4.05565	−	2.34153i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
63.i	even	6	1	inner
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	441.2.i.d		48
3.b	odd	2	1		1323.2.i.d		48
7.b	odd	2	1	inner	441.2.i.d		48
7.c	even	3	1		441.2.o.e	✓	48
7.c	even	3	1		441.2.s.d		48
7.d	odd	6	1		441.2.o.e	✓	48
7.d	odd	6	1		441.2.s.d		48
9.c	even	3	1		1323.2.s.d		48
9.d	odd	6	1		441.2.s.d		48
21.c	even	2	1		1323.2.i.d		48
21.g	even	6	1		1323.2.o.e		48
21.g	even	6	1		1323.2.s.d		48
21.h	odd	6	1		1323.2.o.e		48
21.h	odd	6	1		1323.2.s.d		48
63.g	even	3	1		1323.2.o.e		48
63.h	even	3	1		1323.2.i.d		48
63.i	even	6	1	inner	441.2.i.d		48
63.j	odd	6	1	inner	441.2.i.d		48
63.k	odd	6	1		1323.2.o.e		48
63.l	odd	6	1		1323.2.s.d		48
63.n	odd	6	1		441.2.o.e	✓	48
63.o	even	6	1		441.2.s.d		48
63.s	even	6	1		441.2.o.e	✓	48
63.t	odd	6	1		1323.2.i.d		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
441.2.i.d		48	1.a	even	1	1	trivial
441.2.i.d		48	7.b	odd	2	1	inner
441.2.i.d		48	63.i	even	6	1	inner
441.2.i.d		48	63.j	odd	6	1	inner
441.2.o.e	✓	48	7.c	even	3	1
441.2.o.e	✓	48	7.d	odd	6	1
441.2.o.e	✓	48	63.n	odd	6	1
441.2.o.e	✓	48	63.s	even	6	1
441.2.s.d		48	7.c	even	3	1
441.2.s.d		48	7.d	odd	6	1
441.2.s.d		48	9.d	odd	6	1
441.2.s.d		48	63.o	even	6	1
1323.2.i.d		48	3.b	odd	2	1
1323.2.i.d		48	21.c	even	2	1
1323.2.i.d		48	63.h	even	3	1
1323.2.i.d		48	63.t	odd	6	1
1323.2.o.e		48	21.g	even	6	1
1323.2.o.e		48	21.h	odd	6	1
1323.2.o.e		48	63.g	even	3	1
1323.2.o.e		48	63.k	odd	6	1
1323.2.s.d		48	9.c	even	3	1
1323.2.s.d		48	21.g	even	6	1
1323.2.s.d		48	21.h	odd	6	1
1323.2.s.d		48	63.l	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^24 + 36*T2^22 + 558*T2^20 + 4884*T2^18 + 26613*T2^16 + 93876*T2^14 + 216050*T2^12 + 319272*T2^10 + 290826*T2^8 + 151416*T2^6 + 39240*T2^4 + 3828*T2^2 + 49