Newform orbit 336.4.a.h

• Label: 336.4.a.h
• 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 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

$f(q)$

$=$

q + 3 * q^3 - 4 * q^5 + 7 * q^7 + 9 * q^9 - 62 * q^11 - 62 * q^13 - 12 * q^15 + 84 * q^17 - 100 * q^19 + 21 * q^21 + 42 * q^23 - 109 * q^25 + 27 * q^27 - 10 * q^29 + 48 * q^31 - 186 * q^33 - 28 * q^35 - 246 * q^37 - 186 * q^39 - 248 * q^41 - 68 * q^43 - 36 * q^45 - 324 * q^47 + 49 * q^49 + 252 * q^51 + 258 * q^53 + 248 * q^55 - 300 * q^57 - 120 * q^59 + 622 * q^61 + 63 * q^63 + 248 * q^65 - 904 * q^67 + 126 * q^69 + 678 * q^71 - 642 * q^73 - 327 * q^75 - 434 * q^77 - 740 * q^79 + 81 * q^81 - 468 * q^83 - 336 * q^85 - 30 * q^87 + 200 * q^89 - 434 * q^91 + 144 * q^93 + 400 * q^95 - 1266 * q^97 - 558 * 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

−4.00000

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

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.4.a.b	✓	1	4.b	odd	2	1
63.4.a.a		1	12.b	even	2	1
147.4.a.g		1	28.d	even	2	1
147.4.e.b		2	28.f	even	6	2
147.4.e.c		2	28.g	odd	6	2
336.4.a.h		1	1.a	even	1	1	trivial
441.4.a.b		1	84.h	odd	2	1
441.4.e.m		2	84.n	even	6	2
441.4.e.n		2	84.j	odd	6	2
525.4.a.b		1	20.d	odd	2	1
525.4.d.b		2	20.e	even	4	2
1008.4.a.m		1	3.b	odd	2	1
1344.4.a.i		1	8.b	even	2	1
1344.4.a.w		1	8.d	odd	2	1
1575.4.a.k		1	60.h	even	2	1
2352.4.a.l		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 $S_{4}^{\mathrm{new}}(\Gamma_0(336))$:

$T_{5} + 4$

$T_{11} + 62$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$
$3$	$T - 3$
$5$	$T + 4$
$7$	$T - 7$
$11$	$T + 62$