Newform orbit 336.4.a.b

• Label: 336.4.a.b
• Level: $336$
• Weight: $4$
• Character orbit: 336.a
• Self dual: yes
• Analytic conductor: $19.825$
• Analytic rank: $1$
• 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:	$1$
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 - 10 * q^5 + 7 * q^7 + 9 * q^9 + 52 * q^11 - 10 * q^13 + 30 * q^15 - 54 * q^17 + 52 * q^19 - 21 * q^21 - 48 * q^23 - 25 * q^25 - 27 * q^27 - 186 * q^29 - 224 * q^31 - 156 * q^33 - 70 * q^35 + 94 * q^37 + 30 * q^39 - 478 * q^41 + 316 * q^43 - 90 * q^45 - 256 * q^47 + 49 * q^49 + 162 * q^51 - 66 * q^53 - 520 * q^55 - 156 * q^57 - 420 * q^59 + 342 * q^61 + 63 * q^63 + 100 * q^65 - 668 * q^67 + 144 * q^69 + 272 * q^71 - 86 * q^73 + 75 * q^75 + 364 * q^77 - 1360 * q^79 + 81 * q^81 - 188 * q^83 + 540 * q^85 + 558 * q^87 - 366 * q^89 - 70 * q^91 + 672 * q^93 - 520 * q^95 + 1554 * q^97 + 468 * 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

−10.0000

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.b		1
3.b	odd	2	1		1008.4.a.q		1
4.b	odd	2	1		168.4.a.e	✓	1
7.b	odd	2	1		2352.4.a.bh		1
8.b	even	2	1		1344.4.a.x		1
8.d	odd	2	1		1344.4.a.k		1
12.b	even	2	1		504.4.a.e		1
28.d	even	2	1		1176.4.a.g		1

    

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

$T_{5} + 10$

$T_{11} - 52$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$
$3$	$T + 3$
$5$	$T + 10$
$7$	$T - 7$
$11$	$T - 52$