Newform orbit 2277.2.a.f

• Label: 2277.2.a.f
• Level: $2277$
• Weight: $2$
• Character orbit: 2277.a
• Self dual: yes
• Analytic conductor: $18.182$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2277 = 3^{2} \cdot 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2277.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(18.1819365402\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 759)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b * q^2 + (b - 1) * q^4 + q^5 + (2*b + 1) * q^7 + (-2*b + 1) * q^8 + b * q^10 - q^11 + 6 * q^13 + (3*b + 2) * q^14 - 3*b * q^16 + 4*b * q^17 + (-4*b + 4) * q^19 + (b - 1) * q^20 - b * q^22 - q^23 - 4 * q^25 + 6*b * q^26 + (b + 1) * q^28 - 6 * q^29 + (-2*b + 1) * q^31 + (b - 5) * q^32 + (4*b + 4) * q^34 + (2*b + 1) * q^35 + (2*b + 6) * q^37 - 4 * q^38 + (-2*b + 1) * q^40 + 3 * q^41 + (10*b - 5) * q^43 + (-b + 1) * q^44 - b * q^46 + (-6*b + 4) * q^47 + (8*b - 2) * q^49 - 4*b * q^50 + (6*b - 6) * q^52 + (-4*b + 1) * q^53 - q^55 + (-4*b - 3) * q^56 - 6*b * q^58 + (6*b - 4) * q^59 + (-12*b + 6) * q^61 + (-b - 2) * q^62 + (2*b + 1) * q^64 + 6 * q^65 + 14 * q^67 + 4 * q^68 + (3*b + 2) * q^70 + (-6*b + 10) * q^71 + (-2*b - 6) * q^73 + (8*b + 2) * q^74 + (4*b - 8) * q^76 + (-2*b - 1) * q^77 + (2*b + 1) * q^79 - 3*b * q^80 + 3*b * q^82 + (4*b + 4) * q^83 + 4*b * q^85 + (5*b + 10) * q^86 + (2*b - 1) * q^88 + 7 * q^89 + (12*b + 6) * q^91 + (-b + 1) * q^92 + (-2*b - 6) * q^94 + (-4*b + 4) * q^95 + (-10*b + 8) * q^97 + (6*b + 8) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^2 - q^4 + 2 * q^5 + 4 * q^7 + q^10 - 2 * q^11 + 12 * q^13 + 7 * q^14 - 3 * q^16 + 4 * q^17 + 4 * q^19 - q^20 - q^22 - 2 * q^23 - 8 * q^25 + 6 * q^26 + 3 * q^28 - 12 * q^29 - 9 * q^32 + 12 * q^34 + 4 * q^35 + 14 * q^37 - 8 * q^38 + 6 * q^41 + q^44 - q^46 + 2 * q^47 + 4 * q^49 - 4 * q^50 - 6 * q^52 - 2 * q^53 - 2 * q^55 - 10 * q^56 - 6 * q^58 - 2 * q^59 - 5 * q^62 + 4 * q^64 + 12 * q^65 + 28 * q^67 + 8 * q^68 + 7 * q^70 + 14 * q^71 - 14 * q^73 + 12 * q^74 - 12 * q^76 - 4 * q^77 + 4 * q^79 - 3 * q^80 + 3 * q^82 + 12 * q^83 + 4 * q^85 + 25 * q^86 + 14 * q^89 + 24 * q^91 + q^92 - 14 * q^94 + 4 * q^95 + 6 * q^97 + 22 * q^98

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

−0.618034
1.61803

−0.618034

0

−1.61803

1.00000

0

−0.236068

2.23607

0

−0.618034

1.2

1.61803

0

0.618034

1.00000

0

4.23607

−2.23607

0

1.61803

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( -1 \)
\(11\)	\( +1 \)
\(23\)	\( +1 \)

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	2277.2.a.f		2
3.b	odd	2	1		759.2.a.c	✓	2
33.d	even	2	1		8349.2.a.j		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
759.2.a.c	✓	2	3.b	odd	2	1
2277.2.a.f		2	1.a	even	1	1	trivial
8349.2.a.j		2	33.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2277))\):

\( T_{2}^{2} - T_{2} - 1 \)

\( T_{5} - 1 \)

\( T_{17}^{2} - 4T_{17} - 16 \)

Hecke characteristic polynomials

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