Newform orbit 441.4.e.e

• Label: 441.4.e.e
• Level: $441$
• Weight: $4$
• Character orbit: 441.e
• Analytic conductor: $26.020$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(26.0198423125\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \zeta_{6} q^{2} + ( - 7 \zeta_{6} + 7) q^{4} - 16 \zeta_{6} q^{5} - 15 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 7 q^{4} - 16 q^{5} - 30 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

226.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−0.500000

+

0.866025i

0

3.50000

+

6.06218i

−8.00000

+

13.8564i

0

0

−15.0000

0

−8.00000

−

13.8564i

361.1

−0.500000

−

0.866025i

0

3.50000

−

6.06218i

−8.00000

−

13.8564i

0

0

−15.0000

0

−8.00000

+

13.8564i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	441.4.e.e		2
3.b	odd	2	1		49.4.c.b		2
7.b	odd	2	1		441.4.e.h		2
7.c	even	3	1		441.4.a.i		1
7.c	even	3	1	inner	441.4.e.e		2
7.d	odd	6	1		63.4.a.b		1
7.d	odd	6	1		441.4.e.h		2
21.c	even	2	1		49.4.c.c		2
21.g	even	6	1		7.4.a.a	✓	1
21.g	even	6	1		49.4.c.c		2
21.h	odd	6	1		49.4.a.b		1
21.h	odd	6	1		49.4.c.b		2
28.f	even	6	1		1008.4.a.c		1
35.i	odd	6	1		1575.4.a.e		1
84.j	odd	6	1		112.4.a.f		1
84.n	even	6	1		784.4.a.g		1
105.o	odd	6	1		1225.4.a.j		1
105.p	even	6	1		175.4.a.b		1
105.w	odd	12	2		175.4.b.b		2
168.ba	even	6	1		448.4.a.i		1
168.be	odd	6	1		448.4.a.e		1
231.k	odd	6	1		847.4.a.b		1
273.ba	even	6	1		1183.4.a.b		1
357.s	even	6	1		2023.4.a.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.4.a.a	✓	1	21.g	even	6	1
49.4.a.b		1	21.h	odd	6	1
49.4.c.b		2	3.b	odd	2	1
49.4.c.b		2	21.h	odd	6	1
49.4.c.c		2	21.c	even	2	1
49.4.c.c		2	21.g	even	6	1
63.4.a.b		1	7.d	odd	6	1
112.4.a.f		1	84.j	odd	6	1
175.4.a.b		1	105.p	even	6	1
175.4.b.b		2	105.w	odd	12	2
441.4.a.i		1	7.c	even	3	1
441.4.e.e		2	1.a	even	1	1	trivial
441.4.e.e		2	7.c	even	3	1	inner
441.4.e.h		2	7.b	odd	2	1
441.4.e.h		2	7.d	odd	6	1
448.4.a.e		1	168.be	odd	6	1
448.4.a.i		1	168.ba	even	6	1
784.4.a.g		1	84.n	even	6	1
847.4.a.b		1	231.k	odd	6	1
1008.4.a.c		1	28.f	even	6	1
1183.4.a.b		1	273.ba	even	6	1
1225.4.a.j		1	105.o	odd	6	1
1575.4.a.e		1	35.i	odd	6	1
2023.4.a.a		1	357.s	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{2} + T_{2} + 1 \)

\( T_{5}^{2} + 16T_{5} + 256 \)

\( T_{13} + 28 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 16T + 256 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 8T + 64 \)