Newform orbit 1225.2.a.t

• Label: 1225.2.a.t
• Level: $1225$
• Weight: $2$
• Character orbit: 1225.a
• Self dual: yes
• Analytic conductor: $9.782$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -7
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(9.78167424761\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{21}) \)
Defining polynomial:	\( x^{2} - x - 5 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b * q^2 + (b + 3) * q^4 + (2*b + 5) * q^8 - 3 * q^9 + (2*b - 3) * q^11 + (5*b + 4) * q^16 - 3*b * q^18 + (-b + 10) * q^22 + (-2*b + 5) * q^23 + (-4*b + 1) * q^29 + (5*b + 15) * q^32 + (-3*b - 9) * q^36 + (4*b - 5) * q^37 + (-2*b - 5) * q^43 + (5*b + 1) * q^44 + (3*b - 10) * q^46 + 10 * q^53 + (-3*b - 20) * q^58 + (10*b + 17) * q^64 + (-6*b + 5) * q^67 + (-2*b - 7) * q^71 + (-6*b - 15) * q^72 + (-b + 20) * q^74 + (-6*b - 1) * q^79 + 9 * q^81 + (-7*b - 10) * q^86 + (8*b + 5) * q^88 + (-3*b + 5) * q^92 + (-6*b + 9) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^2 + 7 * q^4 + 12 * q^8 - 6 * q^9 - 4 * q^11 + 13 * q^16 - 3 * q^18 + 19 * q^22 + 8 * q^23 - 2 * q^29 + 35 * q^32 - 21 * q^36 - 6 * q^37 - 12 * q^43 + 7 * q^44 - 17 * q^46 + 20 * q^53 - 43 * q^58 + 44 * q^64 + 4 * q^67 - 16 * q^71 - 36 * q^72 + 39 * q^74 - 8 * q^79 + 18 * q^81 - 27 * q^86 + 18 * q^88 + 7 * q^92 + 12 * q^99

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

−1.79129
2.79129

−1.79129

0

1.20871

0

0

0

1.41742

−3.00000

0

1.2

2.79129

0

5.79129

0

0

0

10.5826

−3.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\( +1 \)
\(7\)	\( -1 \)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1225.2.a.t	yes	2
5.b	even	2	1		1225.2.a.o	✓	2
5.c	odd	4	2		1225.2.b.e		4
7.b	odd	2	1	CM	1225.2.a.t	yes	2
35.c	odd	2	1		1225.2.a.o	✓	2
35.f	even	4	2		1225.2.b.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1225.2.a.o	✓	2	5.b	even	2	1
1225.2.a.o	✓	2	35.c	odd	2	1
1225.2.a.t	yes	2	1.a	even	1	1	trivial
1225.2.a.t	yes	2	7.b	odd	2	1	CM
1225.2.b.e		4	5.c	odd	4	2
1225.2.b.e		4	35.f	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2}^{2} - T_{2} - 5 \)

\( T_{3} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T - 5 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 4T - 17 \)