Newform orbit 384.4.a.m

• Label: 384.4.a.m
• Level: $384$
• Weight: $4$
• Character orbit: 384.a
• Self dual: yes
• Analytic conductor: $22.657$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$384 = 2^{7} \cdot 3$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	384.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$22.6567334422$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7})$
Defining polynomial:	$x^{2} - 7$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 4\sqrt{7}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 3 * q^3 + (b - 8) * q^5 + (-3*b - 2) * q^7 + 9 * q^9 + (2*b + 12) * q^11 + (4*b - 38) * q^13 + (3*b - 24) * q^15 + (2*b - 10) * q^17 + (-10*b - 16) * q^19 + (-9*b - 6) * q^21 + (2*b - 36) * q^23 + (-16*b + 51) * q^25 + 27 * q^27 + (-5*b - 220) * q^29 + (5*b - 218) * q^31 + (6*b + 36) * q^33 + (22*b - 320) * q^35 + (-6*b + 94) * q^37 + (12*b - 114) * q^39 + (38*b - 2) * q^41 + (2*b - 296) * q^43 + (9*b - 72) * q^45 + (-26*b - 212) * q^47 + (12*b + 669) * q^49 + (6*b - 30) * q^51 + (39*b - 204) * q^53 + (-4*b + 128) * q^55 + (-30*b - 48) * q^57 + (-24*b - 116) * q^59 + (-10*b - 306) * q^61 + (-27*b - 18) * q^63 + (-70*b + 752) * q^65 + (8*b - 492) * q^67 + (6*b - 108) * q^69 + (2*b - 284) * q^71 + (36*b + 182) * q^73 + (-48*b + 153) * q^75 + (-40*b - 696) * q^77 + (61*b + 310) * q^79 + 81 * q^81 + (-78*b + 156) * q^83 + (-26*b + 304) * q^85 + (-15*b - 660) * q^87 + (-60*b - 686) * q^89 + (106*b - 1268) * q^91 + (15*b - 654) * q^93 + (64*b - 992) * q^95 + (20*b + 890) * q^97 + (18*b + 108) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 6 * q^3 - 16 * q^5 - 4 * q^7 + 18 * q^9 + 24 * q^11 - 76 * q^13 - 48 * q^15 - 20 * q^17 - 32 * q^19 - 12 * q^21 - 72 * q^23 + 102 * q^25 + 54 * q^27 - 440 * q^29 - 436 * q^31 + 72 * q^33 - 640 * q^35 + 188 * q^37 - 228 * q^39 - 4 * q^41 - 592 * q^43 - 144 * q^45 - 424 * q^47 + 1338 * q^49 - 60 * q^51 - 408 * q^53 + 256 * q^55 - 96 * q^57 - 232 * q^59 - 612 * q^61 - 36 * q^63 + 1504 * q^65 - 984 * q^67 - 216 * q^69 - 568 * q^71 + 364 * q^73 + 306 * q^75 - 1392 * q^77 + 620 * q^79 + 162 * q^81 + 312 * q^83 + 608 * q^85 - 1320 * q^87 - 1372 * q^89 - 2536 * q^91 - 1308 * q^93 - 1984 * q^95 + 1780 * q^97 + 216 * 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

−2.64575
2.64575

0

3.00000

0

−18.5830

0

29.7490

0

9.00000

0

1.2

0

3.00000

0

2.58301

0

−33.7490

0

9.00000

0

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$

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	384.4.a.m	yes	2
3.b	odd	2	1		1152.4.a.w		2
4.b	odd	2	1		384.4.a.i	✓	2
8.b	even	2	1		384.4.a.l	yes	2
8.d	odd	2	1		384.4.a.p	yes	2
12.b	even	2	1		1152.4.a.x		2
16.e	even	4	2		768.4.d.s		4
16.f	odd	4	2		768.4.d.r		4
24.f	even	2	1		1152.4.a.n		2
24.h	odd	2	1		1152.4.a.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
384.4.a.i	✓	2	4.b	odd	2	1
384.4.a.l	yes	2	8.b	even	2	1
384.4.a.m	yes	2	1.a	even	1	1	trivial
384.4.a.p	yes	2	8.d	odd	2	1
768.4.d.r		4	16.f	odd	4	2
768.4.d.s		4	16.e	even	4	2
1152.4.a.m		2	24.h	odd	2	1
1152.4.a.n		2	24.f	even	2	1
1152.4.a.w		2	3.b	odd	2	1
1152.4.a.x		2	12.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 $S_{4}^{\mathrm{new}}(\Gamma_0(384))$:

$T_{5}^{2} + 16T_{5} - 48$

$T_{7}^{2} + 4T_{7} - 1004$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$(T - 3)^{2}$
$5$	$T^{2} + 16T - 48$
$7$	$T^{2} + 4T - 1004$
$11$	$T^{2} - 24T - 304$