Newform orbit 1872.4.a.s

• Label: 1872.4.a.s
• Level: $1872$
• Weight: $4$
• Character orbit: 1872.a
• Self dual: yes
• Analytic conductor: $110.452$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1872.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(110.451575531\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{10}) \)
Defining polynomial:	\( x^{2} - 10 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b - 12) * q^5 + (-3*b - 4) * q^7 + (-8*b + 18) * q^11 - 13 * q^13 + (-18*b + 6) * q^17 + (-9*b - 60) * q^19 + (-16*b + 36) * q^23 + (-24*b + 59) * q^25 + (-8*b - 42) * q^29 + (21*b - 184) * q^31 + (32*b - 72) * q^35 + (30*b + 78) * q^37 + (45*b + 108) * q^41 + (-42*b - 52) * q^43 + (34*b - 330) * q^47 + (24*b + 33) * q^49 + 618 * q^53 + (114*b - 536) * q^55 + (-58*b - 234) * q^59 + (-84*b - 326) * q^61 + (-13*b + 156) * q^65 + (-87*b + 128) * q^67 + (-8*b - 378) * q^71 + (54*b - 562) * q^73 + (-22*b + 888) * q^77 + (12*b + 160) * q^79 + (-42*b + 330) * q^83 + (222*b - 792) * q^85 + (3*b + 1056) * q^89 + (39*b + 52) * q^91 + (48*b + 360) * q^95 + (-102*b - 58) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 24 * q^5 - 8 * q^7 + 36 * q^11 - 26 * q^13 + 12 * q^17 - 120 * q^19 + 72 * q^23 + 118 * q^25 - 84 * q^29 - 368 * q^31 - 144 * q^35 + 156 * q^37 + 216 * q^41 - 104 * q^43 - 660 * q^47 + 66 * q^49 + 1236 * q^53 - 1072 * q^55 - 468 * q^59 - 652 * q^61 + 312 * q^65 + 256 * q^67 - 756 * q^71 - 1124 * q^73 + 1776 * q^77 + 320 * q^79 + 660 * q^83 - 1584 * q^85 + 2112 * q^89 + 104 * q^91 + 720 * q^95 - 116 * 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

−3.16228
3.16228

0

0

0

−18.3246

0

14.9737

0

0

0

1.2

0

0

0

−5.67544

0

−22.9737

0

0

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(3\)	\( -1 \)
\(13\)	\( +1 \)

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	1872.4.a.s		2
3.b	odd	2	1		624.4.a.m		2
4.b	odd	2	1		468.4.a.d		2
12.b	even	2	1		156.4.a.d	✓	2
24.f	even	2	1		2496.4.a.t		2
24.h	odd	2	1		2496.4.a.bb		2
156.h	even	2	1		2028.4.a.e		2
156.l	odd	4	2		2028.4.b.f		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.4.a.d	✓	2	12.b	even	2	1
468.4.a.d		2	4.b	odd	2	1
624.4.a.m		2	3.b	odd	2	1
1872.4.a.s		2	1.a	even	1	1	trivial
2028.4.a.e		2	156.h	even	2	1
2028.4.b.f		4	156.l	odd	4	2
2496.4.a.t		2	24.f	even	2	1
2496.4.a.bb		2	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1872))\):

\( T_{5}^{2} + 24T_{5} + 104 \)

\( T_{7}^{2} + 8T_{7} - 344 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 24T + 104 \)
$7$	\( T^{2} + 8T - 344 \)
$11$	\( T^{2} - 36T - 2236 \)