Newform orbit 195.2.i.a

• Label: 195.2.i.a
• Level: $195$
• Weight: $2$
• Character orbit: 195.i
• Analytic conductor: $1.557$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 195 = 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	195.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.55708283941\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
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.

\(f(q)\)	\(=\)	\( q + (2 \zeta_{6} - 2) q^{2} + (\zeta_{6} - 1) q^{3} - 2 \zeta_{6} q^{4} - q^{5} - 2 \zeta_{6} q^{6} - 5 \zeta_{6} q^{7} - \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} - 5 q^{7} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/195\mathbb{Z}\right)^\times\).

\(n\)	\(106\)	\(131\)	\(157\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

16.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−1.00000

−

1.73205i

−0.500000

−

0.866025i

−1.00000

+

1.73205i

−1.00000

−1.00000

+

1.73205i

−2.50000

+

4.33013i

0

−0.500000

+

0.866025i

1.00000

+

1.73205i

61.1

−1.00000

+

1.73205i

−0.500000

+

0.866025i

−1.00000

−

1.73205i

−1.00000

−1.00000

−

1.73205i

−2.50000

−

4.33013i

0

−0.500000

−

0.866025i

1.00000

−

1.73205i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	195.2.i.a	✓	2
3.b	odd	2	1		585.2.j.b		2
5.b	even	2	1		975.2.i.i		2
5.c	odd	4	2		975.2.bb.f		4
13.c	even	3	1	inner	195.2.i.a	✓	2
13.c	even	3	1		2535.2.a.m		1
13.e	even	6	1		2535.2.a.c		1
39.h	odd	6	1		7605.2.a.s		1
39.i	odd	6	1		585.2.j.b		2
39.i	odd	6	1		7605.2.a.a		1
65.n	even	6	1		975.2.i.i		2
65.q	odd	12	2		975.2.bb.f		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
195.2.i.a	✓	2	1.a	even	1	1	trivial
195.2.i.a	✓	2	13.c	even	3	1	inner
585.2.j.b		2	3.b	odd	2	1
585.2.j.b		2	39.i	odd	6	1
975.2.i.i		2	5.b	even	2	1
975.2.i.i		2	65.n	even	6	1
975.2.bb.f		4	5.c	odd	4	2
975.2.bb.f		4	65.q	odd	12	2
2535.2.a.c		1	13.e	even	6	1
2535.2.a.m		1	13.c	even	3	1
7605.2.a.a		1	39.i	odd	6	1
7605.2.a.s		1	39.h	odd	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 \(S_{2}^{\mathrm{new}}(195, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T + 4 \)
$3$	\( T^{2} + T + 1 \)
$5$	\( (T + 1)^{2} \)
$7$	\( T^{2} + 5T + 25 \)
$11$	\( T^{2} + 2T + 4 \)