Newform orbit 336.4.a.a

• Label: 336.4.a.a
• 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 168)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

$f(q)$

$=$

q - 3 * q^3 - 16 * q^5 - 7 * q^7 + 9 * q^9 + 18 * q^11 - 54 * q^13 + 48 * q^15 - 128 * q^17 - 52 * q^19 + 21 * q^21 + 202 * q^23 + 131 * q^25 - 27 * q^27 + 302 * q^29 + 200 * q^31 - 54 * q^33 + 112 * q^35 - 150 * q^37 + 162 * q^39 + 172 * q^41 - 164 * q^43 - 144 * q^45 + 460 * q^47 + 49 * q^49 + 384 * q^51 - 190 * q^53 - 288 * q^55 + 156 * q^57 - 96 * q^59 + 622 * q^61 - 63 * q^63 + 864 * q^65 - 744 * q^67 - 606 * q^69 + 54 * q^71 + 742 * q^73 - 393 * q^75 - 126 * q^77 + 92 * q^79 + 81 * q^81 + 228 * q^83 + 2048 * q^85 - 906 * q^87 - 116 * q^89 + 378 * q^91 - 600 * q^93 + 832 * q^95 - 554 * q^97 + 162 * 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

−16.0000

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.a		1
3.b	odd	2	1		1008.4.a.t		1
4.b	odd	2	1		168.4.a.d	✓	1
7.b	odd	2	1		2352.4.a.bj		1
8.b	even	2	1		1344.4.a.z		1
8.d	odd	2	1		1344.4.a.l		1
12.b	even	2	1		504.4.a.h		1
28.d	even	2	1		1176.4.a.h		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.4.a.d	✓	1	4.b	odd	2	1
336.4.a.a		1	1.a	even	1	1	trivial
504.4.a.h		1	12.b	even	2	1
1008.4.a.t		1	3.b	odd	2	1
1176.4.a.h		1	28.d	even	2	1
1344.4.a.l		1	8.d	odd	2	1
1344.4.a.z		1	8.b	even	2	1
2352.4.a.bj		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 $S_{4}^{\mathrm{new}}(\Gamma_0(336))$:

$T_{5} + 16$

$T_{11} - 18$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$
$3$	$T + 3$
$5$	$T + 16$
$7$	$T + 7$
$11$	$T - 18$