Newform orbit 338.6.a.a

• Label: 338.6.a.a
• Level: $338$
• Weight: $6$
• Character orbit: 338.a
• Self dual: yes
• Analytic conductor: $54.210$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$338 = 2 \cdot 13^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	338.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$54.2097310968$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

$f(q)$

$=$

q - 4 * q^2 - 13 * q^3 + 16 * q^4 + 51 * q^5 + 52 * q^6 - 105 * q^7 - 64 * q^8 - 74 * q^9 - 204 * q^10 - 120 * q^11 - 208 * q^12 + 420 * q^14 - 663 * q^15 + 256 * q^16 + 1101 * q^17 + 296 * q^18 - 1170 * q^19 + 816 * q^20 + 1365 * q^21 + 480 * q^22 - 1050 * q^23 + 832 * q^24 - 524 * q^25 + 4121 * q^27 - 1680 * q^28 - 4104 * q^29 + 2652 * q^30 + 9624 * q^31 - 1024 * q^32 + 1560 * q^33 - 4404 * q^34 - 5355 * q^35 - 1184 * q^36 - 8709 * q^37 + 4680 * q^38 - 3264 * q^40 - 9480 * q^41 - 5460 * q^42 - 9995 * q^43 - 1920 * q^44 - 3774 * q^45 + 4200 * q^46 + 2943 * q^47 - 3328 * q^48 - 5782 * q^49 + 2096 * q^50 - 14313 * q^51 - 750 * q^53 - 16484 * q^54 - 6120 * q^55 + 6720 * q^56 + 15210 * q^57 + 16416 * q^58 + 40938 * q^59 - 10608 * q^60 - 57920 * q^61 - 38496 * q^62 + 7770 * q^63 + 4096 * q^64 - 6240 * q^66 + 22812 * q^67 + 17616 * q^68 + 13650 * q^69 + 21420 * q^70 + 63741 * q^71 + 4736 * q^72 - 58866 * q^73 + 34836 * q^74 + 6812 * q^75 - 18720 * q^76 + 12600 * q^77 + 63202 * q^79 + 13056 * q^80 - 35591 * q^81 + 37920 * q^82 + 55458 * q^83 + 21840 * q^84 + 56151 * q^85 + 39980 * q^86 + 53352 * q^87 + 7680 * q^88 + 104778 * q^89 + 15096 * q^90 - 16800 * q^92 - 125112 * q^93 - 11772 * q^94 - 59670 * q^95 + 13312 * q^96 + 160452 * q^97 + 23128 * q^98 + 8880 * 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

−4.00000

−13.0000

16.0000

51.0000

52.0000

−105.000

−64.0000

−74.0000

−204.000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$13$	$-1$

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	338.6.a.a		1
13.b	even	2	1		338.6.a.c		1
13.d	odd	4	2		26.6.b.a	✓	2
39.f	even	4	2		234.6.b.b		2
52.f	even	4	2		208.6.f.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.6.b.a	✓	2	13.d	odd	4	2
208.6.f.b		2	52.f	even	4	2
234.6.b.b		2	39.f	even	4	2
338.6.a.a		1	1.a	even	1	1	trivial
338.6.a.c		1	13.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(\Gamma_0(338))$:

$T_{3} + 13$

$T_{5} - 51$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 4$
$3$	$T + 13$
$5$	$T - 51$
$7$	$T + 105$
$11$	$T + 120$