Newform orbit 338.4.a.e

• Label: 338.4.a.e
• Level: $338$
• Weight: $4$
• Character orbit: 338.a
• Self dual: yes
• Analytic conductor: $19.943$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$338 = 2 \cdot 13^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	338.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$19.9426455819$
Analytic rank:	$1$
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 + 2 * q^2 + 3 * q^3 + 4 * q^4 - 11 * q^5 + 6 * q^6 - 19 * q^7 + 8 * q^8 - 18 * q^9 - 22 * q^10 + 38 * q^11 + 12 * q^12 - 38 * q^14 - 33 * q^15 + 16 * q^16 - 51 * q^17 - 36 * q^18 - 90 * q^19 - 44 * q^20 - 57 * q^21 + 76 * q^22 - 52 * q^23 + 24 * q^24 - 4 * q^25 - 135 * q^27 - 76 * q^28 - 190 * q^29 - 66 * q^30 - 292 * q^31 + 32 * q^32 + 114 * q^33 - 102 * q^34 + 209 * q^35 - 72 * q^36 + 441 * q^37 - 180 * q^38 - 88 * q^40 - 312 * q^41 - 114 * q^42 + 373 * q^43 + 152 * q^44 + 198 * q^45 - 104 * q^46 + 41 * q^47 + 48 * q^48 + 18 * q^49 - 8 * q^50 - 153 * q^51 + 468 * q^53 - 270 * q^54 - 418 * q^55 - 152 * q^56 - 270 * q^57 - 380 * q^58 - 530 * q^59 - 132 * q^60 + 592 * q^61 - 584 * q^62 + 342 * q^63 + 64 * q^64 + 228 * q^66 + 206 * q^67 - 204 * q^68 - 156 * q^69 + 418 * q^70 + 863 * q^71 - 144 * q^72 + 322 * q^73 + 882 * q^74 - 12 * q^75 - 360 * q^76 - 722 * q^77 - 460 * q^79 - 176 * q^80 + 81 * q^81 - 624 * q^82 - 528 * q^83 - 228 * q^84 + 561 * q^85 + 746 * q^86 - 570 * q^87 + 304 * q^88 - 870 * q^89 + 396 * q^90 - 208 * q^92 - 876 * q^93 + 82 * q^94 + 990 * q^95 + 96 * q^96 + 346 * q^97 + 36 * q^98 - 684 * 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

2.00000

3.00000

4.00000

−11.0000

6.00000

−19.0000

8.00000

−18.0000

−22.0000

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	338.4.a.e		1
13.b	even	2	1		26.4.a.a	✓	1
13.c	even	3	2		338.4.c.b		2
13.d	odd	4	2		338.4.b.c		2
13.e	even	6	2		338.4.c.f		2
13.f	odd	12	4		338.4.e.b		4
39.d	odd	2	1		234.4.a.g		1
52.b	odd	2	1		208.4.a.c		1
65.d	even	2	1		650.4.a.f		1
65.h	odd	4	2		650.4.b.b		2
91.b	odd	2	1		1274.4.a.b		1
104.e	even	2	1		832.4.a.e		1
104.h	odd	2	1		832.4.a.m		1
156.h	even	2	1		1872.4.a.c		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.4.a.a	✓	1	13.b	even	2	1
208.4.a.c		1	52.b	odd	2	1
234.4.a.g		1	39.d	odd	2	1
338.4.a.e		1	1.a	even	1	1	trivial
338.4.b.c		2	13.d	odd	4	2
338.4.c.b		2	13.c	even	3	2
338.4.c.f		2	13.e	even	6	2
338.4.e.b		4	13.f	odd	12	4
650.4.a.f		1	65.d	even	2	1
650.4.b.b		2	65.h	odd	4	2
832.4.a.e		1	104.e	even	2	1
832.4.a.m		1	104.h	odd	2	1
1274.4.a.b		1	91.b	odd	2	1
1872.4.a.c		1	156.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(338))$:

$T_{3} - 3$

$T_{5} + 11$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 2$
$3$	$T - 3$
$5$	$T + 11$
$7$	$T + 19$
$11$	$T - 38$