Newform orbit 1040.4.a.j

• Label: 1040.4.a.j
• Level: $1040$
• Weight: $4$
• Character orbit: 1040.a
• Self dual: yes
• Analytic conductor: $61.362$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(61.3619864060\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\)	\(=\)	q + (-3*b - 1) * q^3 - 5 * q^5 - 8*b * q^7 + (6*b + 19) * q^9 + (-5*b - 27) * q^11 - 13 * q^13 + (15*b + 5) * q^15 + (16*b + 34) * q^17 + (59*b + 21) * q^19 + (8*b + 120) * q^21 + (-9*b - 147) * q^23 + 25 * q^25 + (18*b - 82) * q^27 + (14*b + 156) * q^29 + (-3*b - 85) * q^31 + (86*b + 102) * q^33 + 40*b * q^35 + (34*b - 204) * q^37 + (39*b + 13) * q^39 + (76*b + 70) * q^41 + (19*b + 161) * q^43 + (-30*b - 95) * q^45 + (108*b + 92) * q^47 - 23 * q^49 + (-118*b - 274) * q^51 + (-236*b - 58) * q^53 + (25*b + 135) * q^55 + (-122*b - 906) * q^57 + (169*b + 191) * q^59 + (114*b + 24) * q^61 + (-152*b - 240) * q^63 + 65 * q^65 + (-86*b + 810) * q^67 + (450*b + 282) * q^69 + (65*b - 465) * q^71 + (-210*b + 228) * q^73 + (-75*b - 25) * q^75 + (216*b + 200) * q^77 + (146*b - 266) * q^79 + (66*b - 701) * q^81 + (512*b + 60) * q^83 + (-80*b - 170) * q^85 + (-482*b - 366) * q^87 + (184*b + 146) * q^89 + 104*b * q^91 + (258*b + 130) * q^93 + (-295*b - 105) * q^95 + (136*b - 546) * q^97 + (-257*b - 663) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 2 * q^3 - 10 * q^5 + 38 * q^9 - 54 * q^11 - 26 * q^13 + 10 * q^15 + 68 * q^17 + 42 * q^19 + 240 * q^21 - 294 * q^23 + 50 * q^25 - 164 * q^27 + 312 * q^29 - 170 * q^31 + 204 * q^33 - 408 * q^37 + 26 * q^39 + 140 * q^41 + 322 * q^43 - 190 * q^45 + 184 * q^47 - 46 * q^49 - 548 * q^51 - 116 * q^53 + 270 * q^55 - 1812 * q^57 + 382 * q^59 + 48 * q^61 - 480 * q^63 + 130 * q^65 + 1620 * q^67 + 564 * q^69 - 930 * q^71 + 456 * q^73 - 50 * q^75 + 400 * q^77 - 532 * q^79 - 1402 * q^81 + 120 * q^83 - 340 * q^85 - 732 * q^87 + 292 * q^89 + 260 * q^93 - 210 * q^95 - 1092 * q^97 - 1326 * 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

1.61803
−0.618034

0

−7.70820

0

−5.00000

0

−17.8885

0

32.4164

0

1.2

0

5.70820

0

−5.00000

0

17.8885

0

5.58359

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(5\)	\( +1 \)
\(13\)	\( +1 \)

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	1040.4.a.j		2
4.b	odd	2	1		130.4.a.e	✓	2
12.b	even	2	1		1170.4.a.z		2
20.d	odd	2	1		650.4.a.s		2
20.e	even	4	2		650.4.b.k		4
52.b	odd	2	1		1690.4.a.t		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
130.4.a.e	✓	2	4.b	odd	2	1
650.4.a.s		2	20.d	odd	2	1
650.4.b.k		4	20.e	even	4	2
1040.4.a.j		2	1.a	even	1	1	trivial
1170.4.a.z		2	12.b	even	2	1
1690.4.a.t		2	52.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1040))\):

\( T_{3}^{2} + 2T_{3} - 44 \)

\( T_{7}^{2} - 320 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 2T - 44 \)
$5$	\( (T + 5)^{2} \)
$7$	\( T^{2} - 320 \)
$11$	\( T^{2} + 54T + 604 \)