Newform orbit 441.2.o.e

• Label: 441.2.o.e
• Level: $441$
• Weight: $2$
• Character orbit: 441.o
• 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.o (of order \(6\), degree \(2\), 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{4} + 16 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
146.1	−2.34591	−	1.35441i	−1.70185	+	0.322036i	2.66888	+	4.62263i	0.601464	+	1.04177i	4.42857	+	1.54954i		0			−	9.04141i	2.79259	−	1.09611i		−	3.25853i
146.2	−2.34591	−	1.35441i	1.70185	−	0.322036i	2.66888	+	4.62263i	−0.601464	−	1.04177i	−4.42857	−	1.54954i		0			−	9.04141i	2.79259	−	1.09611i			3.25853i
146.3	−2.05485	−	1.18637i	−1.27902	−	1.16795i	1.81495	+	3.14358i	−1.71774	−	2.97522i	1.24259	+	3.91735i		0			−	3.86732i	0.271802	+	2.98766i			8.15151i
146.4	−2.05485	−	1.18637i	1.27902	+	1.16795i	1.81495	+	3.14358i	1.71774	+	2.97522i	−1.24259	−	3.91735i		0			−	3.86732i	0.271802	+	2.98766i		−	8.15151i
146.5	−1.28562	−	0.742253i	−0.729965	+	1.57072i	0.101880	+	0.176462i	−0.154215	−	0.267109i	2.10433	−	1.47753i		0				2.66653i	−1.93430	−	2.29314i			0.457868i
146.6	−1.28562	−	0.742253i	0.729965	−	1.57072i	0.101880	+	0.176462i	0.154215	+	0.267109i	−2.10433	+	1.47753i		0				2.66653i	−1.93430	−	2.29314i		−	0.457868i
146.7	−1.02035	−	0.589100i	−1.61957	+	0.613991i	−0.305921	−	0.529871i	2.16601	+	3.75164i	2.01424	+	0.327604i		0				3.07728i	2.24603	−	1.98880i		−	5.10399i
146.8	−1.02035	−	0.589100i	1.61957	−	0.613991i	−0.305921	−	0.529871i	−2.16601	−	3.75164i	−2.01424	−	0.327604i		0				3.07728i	2.24603	−	1.98880i			5.10399i
146.9	−0.850109	−	0.490811i	−0.900030	−	1.47985i	−0.518210	−	0.897565i	0.940599	+	1.62916i	0.0387990	+	1.69978i		0				2.98061i	−1.37989	+	2.66381i		−	1.84662i
146.10	−0.850109	−	0.490811i	0.900030	+	1.47985i	−0.518210	−	0.897565i	−0.940599	−	1.62916i	−0.0387990	−	1.69978i		0				2.98061i	−1.37989	+	2.66381i			1.84662i
146.11	−0.367369	−	0.212101i	−1.08640	+	1.34897i	−0.910027	−	1.57621i	−1.80381	−	3.12430i	0.685229	−	0.265143i		0				1.62047i	−0.639450	−	2.93106i			1.53036i
146.12	−0.367369	−	0.212101i	1.08640	−	1.34897i	−0.910027	−	1.57621i	1.80381	+	3.12430i	−0.685229	+	0.265143i		0				1.62047i	−0.639450	−	2.93106i		−	1.53036i
146.13	0.105953	+	0.0611722i	−0.792362	+	1.54018i	−0.992516	−	1.71909i	0.264715	+	0.458500i	−0.178170	+	0.114717i		0			−	0.487547i	−1.74433	−	2.44076i			0.0647728i
146.14	0.105953	+	0.0611722i	0.792362	−	1.54018i	−0.992516	−	1.71909i	−0.264715	−	0.458500i	0.178170	−	0.114717i		0			−	0.487547i	−1.74433	−	2.44076i		−	0.0647728i
146.15	0.575298	+	0.332148i	−1.69462	−	0.358137i	−0.779355	−	1.34988i	0.0141520	+	0.0245119i	−0.855956	−	0.768901i		0			−	2.36404i	2.74348	+	1.21381i			0.0188022i
146.16	0.575298	+	0.332148i	1.69462	+	0.358137i	−0.779355	−	1.34988i	−0.0141520	−	0.0245119i	0.855956	+	0.768901i		0			−	2.36404i	2.74348	+	1.21381i		−	0.0188022i
146.17	1.58658	+	0.916012i	−1.44264	−	0.958541i	0.678156	+	1.17460i	0.322784	+	0.559079i	−1.41082	−	2.84227i		0			−	1.17925i	1.16240	+	2.76565i			1.18270i
146.18	1.58658	+	0.916012i	1.44264	+	0.958541i	0.678156	+	1.17460i	−0.322784	−	0.559079i	1.41082	+	2.84227i		0			−	1.17925i	1.16240	+	2.76565i		−	1.18270i
146.19	1.61855	+	0.934468i	−1.01272	−	1.40514i	0.746462	+	1.29291i	−1.25287	−	2.17003i	−0.326074	−	3.22063i		0			−	0.947692i	−0.948811	+	2.84601i		−	4.68306i
146.20	1.61855	+	0.934468i	1.01272	+	1.40514i	0.746462	+	1.29291i	1.25287	+	2.17003i	0.326074	+	3.22063i		0			−	0.947692i	−0.948811	+	2.84601i			4.68306i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
9.d	odd	6	1	inner
63.o	even	6	1	inner

Twists

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

    

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\):

T2^24 - 18*T2^22 + 207*T2^20 - 12*T2^19 - 1434*T2^18 + 108*T2^17 + 7227*T2^16 - 540*T2^15 - 24510*T2^14 + 61340*T2^12 + 9612*T2^11 - 101154*T2^10 - 34488*T2^9 + 112545*T2^8 + 70116*T2^7 - 42660*T2^6 - 33444*T2^5 + 14130*T2^4 + 10800*T2^3 + 678*T2^2 - 504*T2 + 49

T5^48 + 72*T5^46 + 3024*T5^44 + 85168*T5^42 + 1793667*T5^40 + 29110188*T5^38 + 375560984*T5^36 + 3876266316*T5^34 + 32327781627*T5^32 + 216703289260*T5^30 + 1167814077690*T5^28 + 4979579304444*T5^26 + 16657040868748*T5^24 + 42196954289376*T5^22 + 79686691764444*T5^20 + 103567280273012*T5^18 + 95399108748591*T5^16 + 52794932235564*T5^14 + 20837133890780*T5^12 + 5059172837124*T5^10 + 859511233110*T5^8 + 68412222392*T5^6 + 3795958686*T5^4 + 3040548*T5^2 + 2401