Newform orbit 336.4.a.f

• Label: 336.4.a.f
• Level: $336$
• Weight: $4$
• Character orbit: 336.a
• Self dual: yes
• Analytic conductor: $19.825$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

$f(q)$

$=$

q + 3 * q^3 - 18 * q^5 - 7 * q^7 + 9 * q^9 + 36 * q^11 - 34 * q^13 - 54 * q^15 + 42 * q^17 + 124 * q^19 - 21 * q^21 + 199 * q^25 + 27 * q^27 + 102 * q^29 + 160 * q^31 + 108 * q^33 + 126 * q^35 + 398 * q^37 - 102 * q^39 - 318 * q^41 + 268 * q^43 - 162 * q^45 - 240 * q^47 + 49 * q^49 + 126 * q^51 - 498 * q^53 - 648 * q^55 + 372 * q^57 + 132 * q^59 + 398 * q^61 - 63 * q^63 + 612 * q^65 - 92 * q^67 + 720 * q^71 - 502 * q^73 + 597 * q^75 - 252 * q^77 + 1024 * q^79 + 81 * q^81 + 204 * q^83 - 756 * q^85 + 306 * q^87 + 354 * q^89 + 238 * q^91 + 480 * q^93 - 2232 * q^95 - 286 * q^97 + 324 * 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

3.00000

0

−18.0000

0

−7.00000

0

9.00000

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$7$	$+1$

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	336.4.a.f		1
3.b	odd	2	1		1008.4.a.v		1
4.b	odd	2	1		21.4.a.a	✓	1
7.b	odd	2	1		2352.4.a.r		1
8.b	even	2	1		1344.4.a.n		1
8.d	odd	2	1		1344.4.a.ba		1
12.b	even	2	1		63.4.a.c		1
20.d	odd	2	1		525.4.a.g		1
20.e	even	4	2		525.4.d.c		2
28.d	even	2	1		147.4.a.c		1
28.f	even	6	2		147.4.e.g		2
28.g	odd	6	2		147.4.e.i		2
60.h	even	2	1		1575.4.a.b		1
84.h	odd	2	1		441.4.a.j		1
84.j	odd	6	2		441.4.e.d		2
84.n	even	6	2		441.4.e.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.4.a.a	✓	1	4.b	odd	2	1
63.4.a.c		1	12.b	even	2	1
147.4.a.c		1	28.d	even	2	1
147.4.e.g		2	28.f	even	6	2
147.4.e.i		2	28.g	odd	6	2
336.4.a.f		1	1.a	even	1	1	trivial
441.4.a.j		1	84.h	odd	2	1
441.4.e.b		2	84.n	even	6	2
441.4.e.d		2	84.j	odd	6	2
525.4.a.g		1	20.d	odd	2	1
525.4.d.c		2	20.e	even	4	2
1008.4.a.v		1	3.b	odd	2	1
1344.4.a.n		1	8.b	even	2	1
1344.4.a.ba		1	8.d	odd	2	1
1575.4.a.b		1	60.h	even	2	1
2352.4.a.r		1	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(336))$:

$T_{5} + 18$

$T_{11} - 36$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$
$3$	$T - 3$
$5$	$T + 18$
$7$	$T + 7$
$11$	$T - 36$