Newform orbit 560.4.a.r

• Label: 560.4.a.r
• Level: $560$
• Weight: $4$
• Character orbit: 560.a
• Self dual: yes
• Analytic conductor: $33.041$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(33.0410696032\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b - 1) * q^3 - 5 * q^5 + 7 * q^7 + (-2*b + 6) * q^9 + (8*b + 7) * q^11 + (b + 25) * q^13 + (-5*b + 5) * q^15 + (11*b - 25) * q^17 + (-11*b - 18) * q^19 + (7*b - 7) * q^21 + (17*b - 122) * q^23 + 25 * q^25 + (-19*b - 43) * q^27 + (6*b - 13) * q^29 + (45*b + 60) * q^31 + (-b + 249) * q^33 - 35 * q^35 + (-15*b + 282) * q^37 + (24*b + 7) * q^39 + (31*b - 164) * q^41 + (-17*b + 130) * q^43 + (10*b - 30) * q^45 + (33*b + 175) * q^47 + 49 * q^49 + (-36*b + 377) * q^51 + (32*b - 28) * q^53 + (-40*b - 35) * q^55 + (-7*b - 334) * q^57 + 616 * q^59 + (-27*b + 168) * q^61 + (-14*b + 42) * q^63 + (-5*b - 125) * q^65 + (16*b + 76) * q^67 + (-139*b + 666) * q^69 + 952 * q^71 + (-86*b + 338) * q^73 + (25*b - 25) * q^75 + (56*b + 49) * q^77 + (-62*b - 507) * q^79 + (30*b - 727) * q^81 + (-150*b + 188) * q^83 + (-55*b + 125) * q^85 + (-19*b + 205) * q^87 + (11*b - 108) * q^89 + (7*b + 175) * q^91 + (15*b + 1380) * q^93 + (55*b + 90) * q^95 + (55*b + 1371) * q^97 + (34*b - 470) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 2 * q^3 - 10 * q^5 + 14 * q^7 + 12 * q^9 + 14 * q^11 + 50 * q^13 + 10 * q^15 - 50 * q^17 - 36 * q^19 - 14 * q^21 - 244 * q^23 + 50 * q^25 - 86 * q^27 - 26 * q^29 + 120 * q^31 + 498 * q^33 - 70 * q^35 + 564 * q^37 + 14 * q^39 - 328 * q^41 + 260 * q^43 - 60 * q^45 + 350 * q^47 + 98 * q^49 + 754 * q^51 - 56 * q^53 - 70 * q^55 - 668 * q^57 + 1232 * q^59 + 336 * q^61 + 84 * q^63 - 250 * q^65 + 152 * q^67 + 1332 * q^69 + 1904 * q^71 + 676 * q^73 - 50 * q^75 + 98 * q^77 - 1014 * q^79 - 1454 * q^81 + 376 * q^83 + 250 * q^85 + 410 * q^87 - 216 * q^89 + 350 * q^91 + 2760 * q^93 + 180 * q^95 + 2742 * q^97 - 940 * 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

−1.41421
1.41421

0

−6.65685

0

−5.00000

0

7.00000

0

17.3137

0

1.2

0

4.65685

0

−5.00000

0

7.00000

0

−5.31371

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(5\)	\( +1 \)
\(7\)	\( -1 \)

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	560.4.a.r		2
4.b	odd	2	1		35.4.a.b	✓	2
8.b	even	2	1		2240.4.a.bo		2
8.d	odd	2	1		2240.4.a.bn		2
12.b	even	2	1		315.4.a.f		2
20.d	odd	2	1		175.4.a.c		2
20.e	even	4	2		175.4.b.c		4
28.d	even	2	1		245.4.a.k		2
28.f	even	6	2		245.4.e.i		4
28.g	odd	6	2		245.4.e.h		4
60.h	even	2	1		1575.4.a.z		2
84.h	odd	2	1		2205.4.a.u		2
140.c	even	2	1		1225.4.a.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.4.a.b	✓	2	4.b	odd	2	1
175.4.a.c		2	20.d	odd	2	1
175.4.b.c		4	20.e	even	4	2
245.4.a.k		2	28.d	even	2	1
245.4.e.h		4	28.g	odd	6	2
245.4.e.i		4	28.f	even	6	2
315.4.a.f		2	12.b	even	2	1
560.4.a.r		2	1.a	even	1	1	trivial
1225.4.a.m		2	140.c	even	2	1
1575.4.a.z		2	60.h	even	2	1
2205.4.a.u		2	84.h	odd	2	1
2240.4.a.bn		2	8.d	odd	2	1
2240.4.a.bo		2	8.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(560))\):

\( T_{3}^{2} + 2T_{3} - 31 \)

\( T_{11}^{2} - 14T_{11} - 1999 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 2T - 31 \)
$5$	\( (T + 5)^{2} \)
$7$	\( (T - 7)^{2} \)
$11$	\( T^{2} - 14T - 1999 \)