Newform orbit 1216.4.a.l

• Label: 1216.4.a.l
• Level: $1216$
• Weight: $4$
• Character orbit: 1216.a
• Self dual: yes
• Analytic conductor: $71.746$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(71.7463225670\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{177}) \)
Defining polynomial:	\( x^{2} - x - 44 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b * q^3 + (2*b - 6) * q^5 + (-b - 28) * q^7 + (b + 17) * q^9 + (2*b + 4) * q^11 + (7*b - 10) * q^13 + (-4*b + 88) * q^15 + (-15*b - 18) * q^17 - 19 * q^19 + (-29*b - 44) * q^21 + (-13*b + 84) * q^23 + (-20*b + 87) * q^25 + (-9*b + 44) * q^27 + (-29*b + 54) * q^29 + 16*b * q^31 + (6*b + 88) * q^33 + (-52*b + 80) * q^35 + (16*b - 198) * q^37 + (-3*b + 308) * q^39 + (6*b - 398) * q^41 + (48*b + 124) * q^43 + (30*b - 14) * q^45 + (-40*b + 120) * q^47 + (57*b + 485) * q^49 + (-33*b - 660) * q^51 + (-9*b - 194) * q^53 + 152 * q^55 - 19*b * q^57 + (-71*b + 136) * q^59 + (-44*b + 362) * q^61 + (-46*b - 520) * q^63 + (-48*b + 676) * q^65 + (-43*b - 448) * q^67 + (71*b - 572) * q^69 + (-22*b - 192) * q^71 + (-b + 62) * q^73 + (67*b - 880) * q^75 + (-62*b - 200) * q^77 + (-58*b - 24) * q^79 + (8*b - 855) * q^81 + (-6*b + 1116) * q^83 + (24*b - 1212) * q^85 + (25*b - 1276) * q^87 + (-10*b - 430) * q^89 + (-193*b - 28) * q^91 + (16*b + 704) * q^93 + (-38*b + 114) * q^95 + (-76*b - 894) * q^97 + (40*b + 156) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^3 - 10 * q^5 - 57 * q^7 + 35 * q^9 + 10 * q^11 - 13 * q^13 + 172 * q^15 - 51 * q^17 - 38 * q^19 - 117 * q^21 + 155 * q^23 + 154 * q^25 + 79 * q^27 + 79 * q^29 + 16 * q^31 + 182 * q^33 + 108 * q^35 - 380 * q^37 + 613 * q^39 - 790 * q^41 + 296 * q^43 + 2 * q^45 + 200 * q^47 + 1027 * q^49 - 1353 * q^51 - 397 * q^53 + 304 * q^55 - 19 * q^57 + 201 * q^59 + 680 * q^61 - 1086 * q^63 + 1304 * q^65 - 939 * q^67 - 1073 * q^69 - 406 * q^71 + 123 * q^73 - 1693 * q^75 - 462 * q^77 - 106 * q^79 - 1702 * q^81 + 2226 * q^83 - 2400 * q^85 - 2527 * q^87 - 870 * q^89 - 249 * q^91 + 1424 * q^93 + 190 * q^95 - 1864 * q^97 + 352 * 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

−6.15207
7.15207

0

−6.15207

0

−18.3041

0

−21.8479

0

10.8479

0

1.2

0

7.15207

0

8.30413

0

−35.1521

0

24.1521

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(19\)	\( +1 \)

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	1216.4.a.l		2
4.b	odd	2	1		1216.4.a.j		2
8.b	even	2	1		304.4.a.d		2
8.d	odd	2	1		38.4.a.b	✓	2
24.f	even	2	1		342.4.a.k		2
40.e	odd	2	1		950.4.a.h		2
40.k	even	4	2		950.4.b.g		4
56.e	even	2	1		1862.4.a.b		2
152.b	even	2	1		722.4.a.i		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
38.4.a.b	✓	2	8.d	odd	2	1
304.4.a.d		2	8.b	even	2	1
342.4.a.k		2	24.f	even	2	1
722.4.a.i		2	152.b	even	2	1
950.4.a.h		2	40.e	odd	2	1
950.4.b.g		4	40.k	even	4	2
1216.4.a.j		2	4.b	odd	2	1
1216.4.a.l		2	1.a	even	1	1	trivial
1862.4.a.b		2	56.e	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1216))\):

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

\( T_{5}^{2} + 10T_{5} - 152 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - T - 44 \)
$5$	\( T^{2} + 10T - 152 \)
$7$	\( T^{2} + 57T + 768 \)
$11$	\( T^{2} - 10T - 152 \)