Newform orbit 950.4.a.h

• Label: 950.4.a.h
• Level: $950$
• Weight: $4$
• Character orbit: 950.a
• Self dual: yes
• Analytic conductor: $56.052$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	950.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(56.0518145055\)
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 + 2 * q^2 - b * q^3 + 4 * q^4 - 2*b * q^6 + (-b - 28) * q^7 + 8 * q^8 + (b + 17) * q^9 + (2*b + 4) * q^11 - 4*b * q^12 + (7*b - 10) * q^13 + (-2*b - 56) * q^14 + 16 * q^16 + (15*b + 18) * q^17 + (2*b + 34) * q^18 - 19 * q^19 + (29*b + 44) * q^21 + (4*b + 8) * q^22 + (-13*b + 84) * q^23 - 8*b * q^24 + (14*b - 20) * q^26 + (9*b - 44) * q^27 + (-4*b - 112) * q^28 + (29*b - 54) * q^29 - 16*b * q^31 + 32 * q^32 + (-6*b - 88) * q^33 + (30*b + 36) * q^34 + (4*b + 68) * q^36 + (16*b - 198) * q^37 - 38 * q^38 + (3*b - 308) * q^39 + (6*b - 398) * q^41 + (58*b + 88) * q^42 + (-48*b - 124) * q^43 + (8*b + 16) * q^44 + (-26*b + 168) * q^46 + (-40*b + 120) * q^47 - 16*b * q^48 + (57*b + 485) * q^49 + (-33*b - 660) * q^51 + (28*b - 40) * q^52 + (-9*b - 194) * q^53 + (18*b - 88) * q^54 + (-8*b - 224) * q^56 + 19*b * q^57 + (58*b - 108) * q^58 + (-71*b + 136) * q^59 + (44*b - 362) * q^61 - 32*b * q^62 + (-46*b - 520) * q^63 + 64 * q^64 + (-12*b - 176) * q^66 + (43*b + 448) * q^67 + (60*b + 72) * q^68 + (-71*b + 572) * q^69 + (22*b + 192) * q^71 + (8*b + 136) * q^72 + (b - 62) * q^73 + (32*b - 396) * q^74 - 76 * q^76 + (-62*b - 200) * q^77 + (6*b - 616) * q^78 + (58*b + 24) * q^79 + (8*b - 855) * q^81 + (12*b - 796) * q^82 + (6*b - 1116) * q^83 + (116*b + 176) * q^84 + (-96*b - 248) * q^86 + (25*b - 1276) * q^87 + (16*b + 32) * q^88 + (-10*b - 430) * q^89 + (-193*b - 28) * q^91 + (-52*b + 336) * q^92 + (16*b + 704) * q^93 + (-80*b + 240) * q^94 - 32*b * q^96 + (76*b + 894) * q^97 + (114*b + 970) * q^98 + (40*b + 156) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^2 - q^3 + 8 * q^4 - 2 * q^6 - 57 * q^7 + 16 * q^8 + 35 * q^9 + 10 * q^11 - 4 * q^12 - 13 * q^13 - 114 * q^14 + 32 * q^16 + 51 * q^17 + 70 * q^18 - 38 * q^19 + 117 * q^21 + 20 * q^22 + 155 * q^23 - 8 * q^24 - 26 * q^26 - 79 * q^27 - 228 * q^28 - 79 * q^29 - 16 * q^31 + 64 * q^32 - 182 * q^33 + 102 * q^34 + 140 * q^36 - 380 * q^37 - 76 * q^38 - 613 * q^39 - 790 * q^41 + 234 * q^42 - 296 * q^43 + 40 * q^44 + 310 * q^46 + 200 * q^47 - 16 * q^48 + 1027 * q^49 - 1353 * q^51 - 52 * q^52 - 397 * q^53 - 158 * q^54 - 456 * q^56 + 19 * q^57 - 158 * q^58 + 201 * q^59 - 680 * q^61 - 32 * q^62 - 1086 * q^63 + 128 * q^64 - 364 * q^66 + 939 * q^67 + 204 * q^68 + 1073 * q^69 + 406 * q^71 + 280 * q^72 - 123 * q^73 - 760 * q^74 - 152 * q^76 - 462 * q^77 - 1226 * q^78 + 106 * q^79 - 1702 * q^81 - 1580 * q^82 - 2226 * q^83 + 468 * q^84 - 592 * q^86 - 2527 * q^87 + 80 * q^88 - 870 * q^89 - 249 * q^91 + 620 * q^92 + 1424 * q^93 + 400 * q^94 - 32 * q^96 + 1864 * q^97 + 2054 * q^98 + 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

7.15207
−6.15207

2.00000

−7.15207

4.00000

0

−14.3041

−35.1521

8.00000

24.1521

0

1.2

2.00000

6.15207

4.00000

0

12.3041

−21.8479

8.00000

10.8479

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(5\)	\( +1 \)
\(19\)	\( +1 \)

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	950.4.a.h		2
5.b	even	2	1		38.4.a.b	✓	2
5.c	odd	4	2		950.4.b.g		4
15.d	odd	2	1		342.4.a.k		2
20.d	odd	2	1		304.4.a.d		2
35.c	odd	2	1		1862.4.a.b		2
40.e	odd	2	1		1216.4.a.l		2
40.f	even	2	1		1216.4.a.j		2
95.d	odd	2	1		722.4.a.i		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
38.4.a.b	✓	2	5.b	even	2	1
304.4.a.d		2	20.d	odd	2	1
342.4.a.k		2	15.d	odd	2	1
722.4.a.i		2	95.d	odd	2	1
950.4.a.h		2	1.a	even	1	1	trivial
950.4.b.g		4	5.c	odd	4	2
1216.4.a.j		2	40.f	even	2	1
1216.4.a.l		2	40.e	odd	2	1
1862.4.a.b		2	35.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(950))\):

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

\( T_{7}^{2} + 57T_{7} + 768 \)

Hecke characteristic polynomials

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