Newform orbit 722.4.a.c

• Label: 722.4.a.c
• Level: $722$
• Weight: $4$
• Character orbit: 722.a
• Self dual: yes
• Analytic conductor: $42.599$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	722.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(42.5993790241\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

q + 2 * q^2 - 5 * q^3 + 4 * q^4 - 12 * q^5 - 10 * q^6 + 8 * q^7 + 8 * q^8 - 2 * q^9 - 24 * q^10 + 9 * q^11 - 20 * q^12 + 26 * q^13 + 16 * q^14 + 60 * q^15 + 16 * q^16 + 114 * q^17 - 4 * q^18 - 48 * q^20 - 40 * q^21 + 18 * q^22 - 78 * q^23 - 40 * q^24 + 19 * q^25 + 52 * q^26 + 145 * q^27 + 32 * q^28 - 204 * q^29 + 120 * q^30 + 98 * q^31 + 32 * q^32 - 45 * q^33 + 228 * q^34 - 96 * q^35 - 8 * q^36 - 334 * q^37 - 130 * q^39 - 96 * q^40 + 177 * q^41 - 80 * q^42 - 316 * q^43 + 36 * q^44 + 24 * q^45 - 156 * q^46 - 492 * q^47 - 80 * q^48 - 279 * q^49 + 38 * q^50 - 570 * q^51 + 104 * q^52 + 678 * q^53 + 290 * q^54 - 108 * q^55 + 64 * q^56 - 408 * q^58 - 579 * q^59 + 240 * q^60 - 352 * q^61 + 196 * q^62 - 16 * q^63 + 64 * q^64 - 312 * q^65 - 90 * q^66 + 755 * q^67 + 456 * q^68 + 390 * q^69 - 192 * q^70 + 6 * q^71 - 16 * q^72 - 145 * q^73 - 668 * q^74 - 95 * q^75 + 72 * q^77 - 260 * q^78 - 316 * q^79 - 192 * q^80 - 671 * q^81 + 354 * q^82 - 567 * q^83 - 160 * q^84 - 1368 * q^85 - 632 * q^86 + 1020 * q^87 + 72 * q^88 - 114 * q^89 + 48 * q^90 + 208 * q^91 - 312 * q^92 - 490 * q^93 - 984 * q^94 - 160 * q^96 - 943 * q^97 - 558 * q^98 - 18 * 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

−5.00000

4.00000

−12.0000

−10.0000

8.00000

8.00000

−2.00000

−24.0000

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(19\)	\( +1 \)

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	722.4.a.c		1
19.b	odd	2	1		722.4.a.b		1
19.c	even	3	2		38.4.c.b	✓	2
57.h	odd	6	2		342.4.g.c		2
76.g	odd	6	2		304.4.i.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
38.4.c.b	✓	2	19.c	even	3	2
304.4.i.a		2	76.g	odd	6	2
342.4.g.c		2	57.h	odd	6	2
722.4.a.b		1	19.b	odd	2	1
722.4.a.c		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(722))\):

\( T_{3} + 5 \)

\( T_{5} + 12 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T - 2 \)
$3$	\( T + 5 \)
$5$	\( T + 12 \)
$7$	\( T - 8 \)
$11$	\( T - 9 \)