Newform orbit 1275.2.a.n

• Label: 1275.2.a.n
• Level: $1275$
• Weight: $2$
• Character orbit: 1275.a
• Self dual: yes
• Analytic conductor: $10.181$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1275.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(10.1809262577\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{17}) \)
Defining polynomial:	\( x^{2} - x - 4 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} + q^{3} + (\beta + 2) q^{4} + \beta q^{6} + (\beta + 4) q^{8} + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} + 5 q^{4} + q^{6} + 9 q^{8} + 2 q^{9}+O(q^{10}) \)

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

−1.56155
2.56155

−1.56155

1.00000

0.438447

0

−1.56155

0

2.43845

1.00000

0

1.2

2.56155

1.00000

4.56155

0

2.56155

0

6.56155

1.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(5\)	\(1\)
\(17\)	\(1\)

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	1275.2.a.n		2
3.b	odd	2	1		3825.2.a.s		2
5.b	even	2	1		51.2.a.b	✓	2
5.c	odd	4	2		1275.2.b.d		4
15.d	odd	2	1		153.2.a.e		2
20.d	odd	2	1		816.2.a.m		2
35.c	odd	2	1		2499.2.a.o		2
40.e	odd	2	1		3264.2.a.bg		2
40.f	even	2	1		3264.2.a.bl		2
55.d	odd	2	1		6171.2.a.p		2
60.h	even	2	1		2448.2.a.v		2
65.d	even	2	1		8619.2.a.q		2
85.c	even	2	1		867.2.a.f		2
85.j	even	4	2		867.2.d.c		4
85.m	even	8	4		867.2.e.f		8
85.p	odd	16	8		867.2.h.j		16
105.g	even	2	1		7497.2.a.v		2
120.i	odd	2	1		9792.2.a.cy		2
120.m	even	2	1		9792.2.a.cz		2
255.h	odd	2	1		2601.2.a.t		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
51.2.a.b	✓	2	5.b	even	2	1
153.2.a.e		2	15.d	odd	2	1
816.2.a.m		2	20.d	odd	2	1
867.2.a.f		2	85.c	even	2	1
867.2.d.c		4	85.j	even	4	2
867.2.e.f		8	85.m	even	8	4
867.2.h.j		16	85.p	odd	16	8
1275.2.a.n		2	1.a	even	1	1	trivial
1275.2.b.d		4	5.c	odd	4	2
2448.2.a.v		2	60.h	even	2	1
2499.2.a.o		2	35.c	odd	2	1
2601.2.a.t		2	255.h	odd	2	1
3264.2.a.bg		2	40.e	odd	2	1
3264.2.a.bl		2	40.f	even	2	1
3825.2.a.s		2	3.b	odd	2	1
6171.2.a.p		2	55.d	odd	2	1
7497.2.a.v		2	105.g	even	2	1
8619.2.a.q		2	65.d	even	2	1
9792.2.a.cy		2	120.i	odd	2	1
9792.2.a.cz		2	120.m	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1275))\):

\( T_{2}^{2} - T_{2} - 4 \)

\( T_{7} \)

Hecke characteristic polynomials

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