Newform orbit 195.3.u.c

• Label: 195.3.u.c
• Level: $195$
• Weight: $3$
• Character orbit: 195.u
• Analytic conductor: $5.313$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 195 = 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	195.u (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.31336515503\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(i)\)
Coefficient field:	\(\Q(\zeta_{8})\)
Defining polynomial:	\( x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{2} + ( - 2 \zeta_{8}^{2} - \zeta_{8} + 2) q^{3} + 4 q^{4} + (3 \zeta_{8}^{3} + 4 \zeta_{8}) q^{5} + (8 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 2) q^{6} - 5 q^{7} + (4 \zeta_{8}^{3} - 7 \zeta_{8}^{2} - 4 \zeta_{8}) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{3} + 16 q^{4} + 8 q^{6} - 20 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(106\)	\(131\)	\(157\)
\(\chi(n)\)	\(\zeta_{8}^{2}\)	\(-1\)	\(\zeta_{8}^{2}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

47.1

0.707107	+	0.707107i
−0.707107	−	0.707107i
0.707107	−	0.707107i
−0.707107	+	0.707107i

−2.82843

1.29289

−

2.70711i

4.00000

0.707107

+

4.94975i

−3.65685

+

7.65685i

−5.00000

0

−5.65685

−

7.00000i

−2.00000

−

14.0000i

47.2

2.82843

2.70711

−

1.29289i

4.00000

−0.707107

−

4.94975i

7.65685

−

3.65685i

−5.00000

0

5.65685

−

7.00000i

−2.00000

−

14.0000i

83.1

−2.82843

1.29289

+

2.70711i

4.00000

0.707107

−

4.94975i

−3.65685

−

7.65685i

−5.00000

0

−5.65685

+

7.00000i

−2.00000

+

14.0000i

83.2

2.82843

2.70711

+

1.29289i

4.00000

−0.707107

+

4.94975i

7.65685

+

3.65685i

−5.00000

0

5.65685

+

7.00000i

−2.00000

+

14.0000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
65.f	even	4	1	inner
195.u	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	195.3.u.c	yes	4
3.b	odd	2	1	inner	195.3.u.c	yes	4
5.c	odd	4	1		195.3.j.c	✓	4
13.d	odd	4	1		195.3.j.c	✓	4
15.e	even	4	1		195.3.j.c	✓	4
39.f	even	4	1		195.3.j.c	✓	4
65.f	even	4	1	inner	195.3.u.c	yes	4
195.u	odd	4	1	inner	195.3.u.c	yes	4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
195.3.j.c	✓	4	5.c	odd	4	1
195.3.j.c	✓	4	13.d	odd	4	1
195.3.j.c	✓	4	15.e	even	4	1
195.3.j.c	✓	4	39.f	even	4	1
195.3.u.c	yes	4	1.a	even	1	1	trivial
195.3.u.c	yes	4	3.b	odd	2	1	inner
195.3.u.c	yes	4	65.f	even	4	1	inner
195.3.u.c	yes	4	195.u	odd	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 8 \) acting on \(S_{3}^{\mathrm{new}}(195, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - 8)^{2} \)
$3$	\( T^{4} - 8 T^{3} + 32 T^{2} - 72 T + 81 \)
$5$	\( T^{4} + 48T^{2} + 625 \)
$7$	\( (T + 5)^{4} \)
$11$	\( T^{4} + 28561 \)