Newform orbit 1656.4.a.h

• Label: 1656.4.a.h
• Level: $1656$
• Weight: $4$
• Character orbit: 1656.a
• Self dual: yes
• Analytic conductor: $97.707$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1656.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(97.7071629695\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2\cdot 3 \)
Twist minimal:	no (minimal twist has level 552)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b + 7) * q^5 + (b + 5) * q^7 + (-2*b + 10) * q^11 + (2*b - 20) * q^13 + (-b - 7) * q^17 + (-9*b - 57) * q^19 + 23 * q^23 + (14*b - 31) * q^25 + (-34*b - 16) * q^29 + (6*b - 90) * q^31 + (12*b + 80) * q^35 + (-64*b + 14) * q^37 + (8*b - 66) * q^41 + (7*b - 457) * q^43 + (44*b - 64) * q^47 + (10*b - 273) * q^49 + (-21*b + 201) * q^53 + (-4*b - 20) * q^55 + (-60*b + 196) * q^59 + (36*b - 430) * q^61 + (-6*b - 50) * q^65 + (-83*b - 187) * q^67 + (-4*b + 68) * q^71 + (-60*b - 258) * q^73 - 40 * q^77 + (19*b + 399) * q^79 + (34*b + 318) * q^83 + (-14*b - 94) * q^85 + (141*b + 215) * q^89 + (-10*b - 10) * q^91 + (-120*b - 804) * q^95 + (222*b - 200) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 14 * q^5 + 10 * q^7 + 20 * q^11 - 40 * q^13 - 14 * q^17 - 114 * q^19 + 46 * q^23 - 62 * q^25 - 32 * q^29 - 180 * q^31 + 160 * q^35 + 28 * q^37 - 132 * q^41 - 914 * q^43 - 128 * q^47 - 546 * q^49 + 402 * q^53 - 40 * q^55 + 392 * q^59 - 860 * q^61 - 100 * q^65 - 374 * q^67 + 136 * q^71 - 516 * q^73 - 80 * q^77 + 798 * q^79 + 636 * q^83 - 188 * q^85 + 430 * q^89 - 20 * q^91 - 1608 * q^95 - 400 * q^97

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

0

0

0.291796

0

−1.70820

0

0

0

1.2

0

0

0

13.7082

0

11.7082

0

0

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(3\)	\( -1 \)
\(23\)	\( -1 \)

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	1656.4.a.h		2
3.b	odd	2	1		552.4.a.e	✓	2
12.b	even	2	1		1104.4.a.i		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
552.4.a.e	✓	2	3.b	odd	2	1
1104.4.a.i		2	12.b	even	2	1
1656.4.a.h		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 14T_{5} + 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1656))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 14T + 4 \)
$7$	\( T^{2} - 10T - 20 \)
$11$	\( T^{2} - 20T - 80 \)