Newform orbit 253.2.b.a

• Label: 253.2.b.a
• Level: $253$
• Weight: $2$
• Character orbit: 253.b
• Analytic conductor: $2.020$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -11
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 253 = 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	253.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.02021517114\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-11}) \)
Defining polynomial:	\( x^{2} - x + 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
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 - q^3 + 2 * q^4 + (2*b - 1) * q^5 - 2 * q^9 + (2*b - 1) * q^11 - 2 * q^12 + (-2*b + 1) * q^15 + 4 * q^16 + (4*b - 2) * q^20 + (b + 4) * q^23 - 6 * q^25 + 5 * q^27 + 5 * q^31 + (-2*b + 1) * q^33 - 4 * q^36 + (-6*b + 3) * q^37 + (4*b - 2) * q^44 + (-4*b + 2) * q^45 - 12 * q^47 - 4 * q^48 - 7 * q^49 + (-8*b + 4) * q^53 - 11 * q^55 + 15 * q^59 + (-4*b + 2) * q^60 + 8 * q^64 + (-6*b + 3) * q^67 + (-b - 4) * q^69 + 3 * q^71 + 6 * q^75 + (8*b - 4) * q^80 + q^81 + (10*b - 5) * q^89 + (2*b + 8) * q^92 - 5 * q^93 + (-6*b + 3) * q^97 + (-4*b + 2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 2 * q^3 + 4 * q^4 - 4 * q^9 - 4 * q^12 + 8 * q^16 + 9 * q^23 - 12 * q^25 + 10 * q^27 + 10 * q^31 - 8 * q^36 - 24 * q^47 - 8 * q^48 - 14 * q^49 - 22 * q^55 + 30 * q^59 + 16 * q^64 - 9 * q^69 + 6 * q^71 + 12 * q^75 + 2 * q^81 + 18 * q^92 - 10 * q^93

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/253\mathbb{Z}\right)^\times\).

\(n\)	\(24\)	\(166\)
\(\chi(n)\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

252.1

0.500000	−	1.65831i
0.500000	+	1.65831i

0

−1.00000

2.00000

−

3.31662i

0

0

0

−2.00000

0

252.2

0

−1.00000

2.00000

3.31662i

0

0

0

−2.00000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	253.2.b.a	✓	2
11.b	odd	2	1	CM	253.2.b.a	✓	2
23.b	odd	2	1	inner	253.2.b.a	✓	2
253.b	even	2	1	inner	253.2.b.a	✓	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
253.2.b.a	✓	2	1.a	even	1	1	trivial
253.2.b.a	✓	2	11.b	odd	2	1	CM
253.2.b.a	✓	2	23.b	odd	2	1	inner
253.2.b.a	✓	2	253.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(253, [\chi])\).

Hecke characteristic polynomials

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