Newform orbit 441.2.a.i

• Label: 441.2.a.i
• Level: $441$
• Weight: $2$
• Character orbit: 441.a
• Self dual: yes
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} + (\beta - 2) q^{5} + (\beta + 3) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 6 q^{8}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−1.41421
1.41421

−0.414214

0

−1.82843

−3.41421

0

0

1.58579

0

1.41421

1.2

2.41421

0

3.82843

−0.585786

0

0

4.41421

0

−1.41421

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(7\)	\(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	441.2.a.i		2
3.b	odd	2	1		147.2.a.e	yes	2
4.b	odd	2	1		7056.2.a.cf		2
7.b	odd	2	1		441.2.a.j		2
7.c	even	3	2		441.2.e.g		4
7.d	odd	6	2		441.2.e.f		4
12.b	even	2	1		2352.2.a.bc		2
15.d	odd	2	1		3675.2.a.bd		2
21.c	even	2	1		147.2.a.d	✓	2
21.g	even	6	2		147.2.e.e		4
21.h	odd	6	2		147.2.e.d		4
24.f	even	2	1		9408.2.a.dt		2
24.h	odd	2	1		9408.2.a.di		2
28.d	even	2	1		7056.2.a.cv		2
84.h	odd	2	1		2352.2.a.be		2
84.j	odd	6	2		2352.2.q.bb		4
84.n	even	6	2		2352.2.q.bd		4
105.g	even	2	1		3675.2.a.bf		2
168.e	odd	2	1		9408.2.a.dq		2
168.i	even	2	1		9408.2.a.ef		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
147.2.a.d	✓	2	21.c	even	2	1
147.2.a.e	yes	2	3.b	odd	2	1
147.2.e.d		4	21.h	odd	6	2
147.2.e.e		4	21.g	even	6	2
441.2.a.i		2	1.a	even	1	1	trivial
441.2.a.j		2	7.b	odd	2	1
441.2.e.f		4	7.d	odd	6	2
441.2.e.g		4	7.c	even	3	2
2352.2.a.bc		2	12.b	even	2	1
2352.2.a.be		2	84.h	odd	2	1
2352.2.q.bb		4	84.j	odd	6	2
2352.2.q.bd		4	84.n	even	6	2
3675.2.a.bd		2	15.d	odd	2	1
3675.2.a.bf		2	105.g	even	2	1
7056.2.a.cf		2	4.b	odd	2	1
7056.2.a.cv		2	28.d	even	2	1
9408.2.a.di		2	24.h	odd	2	1
9408.2.a.dq		2	168.e	odd	2	1
9408.2.a.dt		2	24.f	even	2	1
9408.2.a.ef		2	168.i	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(441))\):

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

\( T_{5}^{2} + 4T_{5} + 2 \)

\( T_{13}^{2} - 8T_{13} + 14 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 2T - 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 4T + 2 \)
$7$	\( T^{2} \)
$11$	\( (T - 2)^{2} \)