Newform orbit 198.2.f.a

• Label: 198.2.f.a
• Level: $198$
• Weight: $2$
• Character orbit: 198.f
• Analytic conductor: $1.581$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	198.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.58103796002\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{10})\)
Defining polynomial:	\( x^{4} - x^{3} + x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 66)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (z^3 - z^2 + z - 1) * q^2 - z^3 * q^4 + (-z^2 + 3*z - 1) * q^5 + (z^3 + z - 1) * q^7 + z^2 * q^8
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - q^{4} - 2 q^{7} - q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/198\mathbb{Z}\right)^\times\).

\(n\)	\(145\)	\(155\)
\(\chi(n)\)	\(-\zeta_{10}^{3}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

37.1

0.809017	−	0.587785i
0.809017	+	0.587785i
−0.309017	−	0.951057i
−0.309017	+	0.951057i

−0.809017

−

0.587785i

0

0.309017

+

0.951057i

1.11803

−

0.812299i

0

−0.500000

−

1.53884i

0.309017

−

0.951057i

0

−1.38197

91.1

−0.809017

+

0.587785i

0

0.309017

−

0.951057i

1.11803

+

0.812299i

0

−0.500000

+

1.53884i

0.309017

+

0.951057i

0

−1.38197

163.1

0.309017

−

0.951057i

0

−0.809017

−

0.587785i

−1.11803

−

3.44095i

0

−0.500000

−

0.363271i

−0.809017

+

0.587785i

0

−3.61803

181.1

0.309017

+

0.951057i

0

−0.809017

+

0.587785i

−1.11803

+

3.44095i

0

−0.500000

+

0.363271i

−0.809017

−

0.587785i

0

−3.61803

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	198.2.f.a		4
3.b	odd	2	1		66.2.e.b	✓	4
11.c	even	5	1	inner	198.2.f.a		4
11.c	even	5	1		2178.2.a.v		2
11.d	odd	10	1		2178.2.a.o		2
12.b	even	2	1		528.2.y.g		4
33.d	even	2	1		726.2.e.c		4
33.f	even	10	1		726.2.a.m		2
33.f	even	10	2		726.2.e.a		4
33.f	even	10	1		726.2.e.c		4
33.h	odd	10	1		66.2.e.b	✓	4
33.h	odd	10	1		726.2.a.k		2
33.h	odd	10	2		726.2.e.j		4
132.n	odd	10	1		5808.2.a.by		2
132.o	even	10	1		528.2.y.g		4
132.o	even	10	1		5808.2.a.bz		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
66.2.e.b	✓	4	3.b	odd	2	1
66.2.e.b	✓	4	33.h	odd	10	1
198.2.f.a		4	1.a	even	1	1	trivial
198.2.f.a		4	11.c	even	5	1	inner
528.2.y.g		4	12.b	even	2	1
528.2.y.g		4	132.o	even	10	1
726.2.a.k		2	33.h	odd	10	1
726.2.a.m		2	33.f	even	10	1
726.2.e.a		4	33.f	even	10	2
726.2.e.c		4	33.d	even	2	1
726.2.e.c		4	33.f	even	10	1
726.2.e.j		4	33.h	odd	10	2
2178.2.a.o		2	11.d	odd	10	1
2178.2.a.v		2	11.c	even	5	1
5808.2.a.by		2	132.n	odd	10	1
5808.2.a.bz		2	132.o	even	10	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(198, [\chi])\):

\( T_{5}^{4} + 10T_{5}^{2} - 25T_{5} + 25 \)

\( T_{7}^{4} + 2T_{7}^{3} + 4T_{7}^{2} + 3T_{7} + 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^4 + T^3 + T^2 + T + 1
$3$	\( T^{4} \)
$5$	T^4 + 10*T^2 - 25*T + 25
$7$	T^4 + 2*T^3 + 4*T^2 + 3*T + 1
$11$	T^4 - 4*T^3 + 6*T^2 - 44*T + 121