Newform orbit 192.8.a.d

• Label: 192.8.a.d
• Level: $192$
• Weight: $8$
• Character orbit: 192.a
• Self dual: yes
• Analytic conductor: $59.978$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$192 = 2^{6} \cdot 3$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	192.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$59.9779248930$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

$f(q)$

$=$

q - 27 * q^3 + 26 * q^5 - 1056 * q^7 + 729 * q^9 + 6412 * q^11 - 5206 * q^13 - 702 * q^15 - 6238 * q^17 + 41492 * q^19 + 28512 * q^21 + 29432 * q^23 - 77449 * q^25 - 19683 * q^27 + 210498 * q^29 - 185240 * q^31 - 173124 * q^33 - 27456 * q^35 - 507630 * q^37 + 140562 * q^39 + 360042 * q^41 + 620044 * q^43 + 18954 * q^45 + 847680 * q^47 + 291593 * q^49 + 168426 * q^51 - 1423750 * q^53 + 166712 * q^55 - 1120284 * q^57 - 2548724 * q^59 + 706058 * q^61 - 769824 * q^63 - 135356 * q^65 - 2418796 * q^67 - 794664 * q^69 - 265976 * q^71 - 5791238 * q^73 + 2091123 * q^75 - 6771072 * q^77 - 2955688 * q^79 + 531441 * q^81 + 3462932 * q^83 - 162188 * q^85 - 5683446 * q^87 - 2211126 * q^89 + 5497536 * q^91 + 5001480 * q^93 + 1078792 * q^95 - 15594814 * q^97 + 4674348 * 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

0

0

−27.0000

0

26.0000

0

−1056.00

0

729.000

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	192.8.a.d		1
3.b	odd	2	1		576.8.a.o		1
4.b	odd	2	1		192.8.a.l		1
8.b	even	2	1		48.8.a.f		1
8.d	odd	2	1		24.8.a.a	✓	1
12.b	even	2	1		576.8.a.p		1
24.f	even	2	1		72.8.a.c		1
24.h	odd	2	1		144.8.a.f		1
40.e	odd	2	1		600.8.a.e		1
40.k	even	4	2		600.8.f.e		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
24.8.a.a	✓	1	8.d	odd	2	1
48.8.a.f		1	8.b	even	2	1
72.8.a.c		1	24.f	even	2	1
144.8.a.f		1	24.h	odd	2	1
192.8.a.d		1	1.a	even	1	1	trivial
192.8.a.l		1	4.b	odd	2	1
576.8.a.o		1	3.b	odd	2	1
576.8.a.p		1	12.b	even	2	1
600.8.a.e		1	40.e	odd	2	1
600.8.f.e		2	40.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{8}^{\mathrm{new}}(\Gamma_0(192))$:

$T_{5} - 26$

$T_{7} + 1056$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$
$3$	$T + 27$
$5$	$T - 26$
$7$	$T + 1056$
$11$	$T - 6412$