Newform orbit 825.2.f.a

• Label: 825.2.f.a
• Level: $825$
• Weight: $2$
• Character orbit: 825.f
• Analytic conductor: $6.588$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -11
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.58765816676\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-11}) \)
Defining polynomial:	\( x^{2} - x + 3 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\)	\(376\)	\(551\)	\(727\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

626.1

0.500000	+	1.65831i
0.500000	−	1.65831i

0

−0.500000

−

1.65831i

−2.00000

0

0

0

0

−2.50000

+

1.65831i

0

626.2

0

−0.500000

+

1.65831i

−2.00000

0

0

0

0

−2.50000

−

1.65831i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by \(\Q(\sqrt{-11}) \)
3.b	odd	2	1	inner
33.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	825.2.f.a		2
3.b	odd	2	1	inner	825.2.f.a		2
5.b	even	2	1		33.2.d.a	✓	2
5.c	odd	4	2		825.2.d.a		4
11.b	odd	2	1	CM	825.2.f.a		2
15.d	odd	2	1		33.2.d.a	✓	2
15.e	even	4	2		825.2.d.a		4
20.d	odd	2	1		528.2.b.a		2
33.d	even	2	1	inner	825.2.f.a		2
40.e	odd	2	1		2112.2.b.f		2
40.f	even	2	1		2112.2.b.e		2
45.h	odd	6	2		891.2.g.a		4
45.j	even	6	2		891.2.g.a		4
55.d	odd	2	1		33.2.d.a	✓	2
55.e	even	4	2		825.2.d.a		4
55.h	odd	10	4		363.2.f.c		8
55.j	even	10	4		363.2.f.c		8
60.h	even	2	1		528.2.b.a		2
120.i	odd	2	1		2112.2.b.e		2
120.m	even	2	1		2112.2.b.f		2
165.d	even	2	1		33.2.d.a	✓	2
165.l	odd	4	2		825.2.d.a		4
165.o	odd	10	4		363.2.f.c		8
165.r	even	10	4		363.2.f.c		8
220.g	even	2	1		528.2.b.a		2
440.c	even	2	1		2112.2.b.f		2
440.o	odd	2	1		2112.2.b.e		2
495.o	odd	6	2		891.2.g.a		4
495.r	even	6	2		891.2.g.a		4
660.g	odd	2	1		528.2.b.a		2
1320.b	odd	2	1		2112.2.b.f		2
1320.u	even	2	1		2112.2.b.e		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.2.d.a	✓	2	5.b	even	2	1
33.2.d.a	✓	2	15.d	odd	2	1
33.2.d.a	✓	2	55.d	odd	2	1
33.2.d.a	✓	2	165.d	even	2	1
363.2.f.c		8	55.h	odd	10	4
363.2.f.c		8	55.j	even	10	4
363.2.f.c		8	165.o	odd	10	4
363.2.f.c		8	165.r	even	10	4
528.2.b.a		2	20.d	odd	2	1
528.2.b.a		2	60.h	even	2	1
528.2.b.a		2	220.g	even	2	1
528.2.b.a		2	660.g	odd	2	1
825.2.d.a		4	5.c	odd	4	2
825.2.d.a		4	15.e	even	4	2
825.2.d.a		4	55.e	even	4	2
825.2.d.a		4	165.l	odd	4	2
825.2.f.a		2	1.a	even	1	1	trivial
825.2.f.a		2	3.b	odd	2	1	inner
825.2.f.a		2	11.b	odd	2	1	CM
825.2.f.a		2	33.d	even	2	1	inner
891.2.g.a		4	45.h	odd	6	2
891.2.g.a		4	45.j	even	6	2
891.2.g.a		4	495.o	odd	6	2
891.2.g.a		4	495.r	even	6	2
2112.2.b.e		2	40.f	even	2	1
2112.2.b.e		2	120.i	odd	2	1
2112.2.b.e		2	440.o	odd	2	1
2112.2.b.e		2	1320.u	even	2	1
2112.2.b.f		2	40.e	odd	2	1
2112.2.b.f		2	120.m	even	2	1
2112.2.b.f		2	440.c	even	2	1
2112.2.b.f		2	1320.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2} \)

\( T_{23}^{2} + 11 \)

\( T_{37} - 7 \)

Hecke characteristic polynomials

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