Newform orbit 297.2.e.a

• Label: 297.2.e.a
• Level: $297$
• Weight: $2$
• Character orbit: 297.e
• Analytic conductor: $2.372$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	297.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.37155694003\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 99)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/297\mathbb{Z}\right)^\times\).

\(n\)	\(56\)	\(244\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

100.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.00000

+

1.73205i

0

−1.00000

−

1.73205i

−1.00000

−

1.73205i

0

−2.00000

+

3.46410i

0

0

4.00000

199.1

−1.00000

−

1.73205i

0

−1.00000

+

1.73205i

−1.00000

+

1.73205i

0

−2.00000

−

3.46410i

0

0

4.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	297.2.e.a		2
3.b	odd	2	1		99.2.e.c	✓	2
9.c	even	3	1	inner	297.2.e.a		2
9.c	even	3	1		891.2.a.h		1
9.d	odd	6	1		99.2.e.c	✓	2
9.d	odd	6	1		891.2.a.a		1
33.d	even	2	1		1089.2.e.a		2
99.g	even	6	1		1089.2.e.a		2
99.g	even	6	1		9801.2.a.l		1
99.h	odd	6	1		9801.2.a.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
99.2.e.c	✓	2	3.b	odd	2	1
99.2.e.c	✓	2	9.d	odd	6	1
297.2.e.a		2	1.a	even	1	1	trivial
297.2.e.a		2	9.c	even	3	1	inner
891.2.a.a		1	9.d	odd	6	1
891.2.a.h		1	9.c	even	3	1
1089.2.e.a		2	33.d	even	2	1
1089.2.e.a		2	99.g	even	6	1
9801.2.a.a		1	99.h	odd	6	1
9801.2.a.l		1	99.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(297, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T + 4 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 2T + 4 \)
$7$	\( T^{2} + 4T + 16 \)
$11$	\( T^{2} - T + 1 \)