Newform orbit 350.6.a.e

• Label: 350.6.a.e
• Level: $350$
• Weight: $6$
• Character orbit: 350.a
• Self dual: yes
• Analytic conductor: $56.134$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(56.1343369345\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

q - 4 * q^2 + 17 * q^3 + 16 * q^4 - 68 * q^6 + 49 * q^7 - 64 * q^8 + 46 * q^9 - 715 * q^11 + 272 * q^12 - 331 * q^13 - 196 * q^14 + 256 * q^16 + 1699 * q^17 - 184 * q^18 - 1718 * q^19 + 833 * q^21 + 2860 * q^22 + 3950 * q^23 - 1088 * q^24 + 1324 * q^26 - 3349 * q^27 + 784 * q^28 + 4579 * q^29 + 6756 * q^31 - 1024 * q^32 - 12155 * q^33 - 6796 * q^34 + 736 * q^36 + 16518 * q^37 + 6872 * q^38 - 5627 * q^39 + 18876 * q^41 - 3332 * q^42 - 2258 * q^43 - 11440 * q^44 - 15800 * q^46 + 537 * q^47 + 4352 * q^48 + 2401 * q^49 + 28883 * q^51 - 5296 * q^52 + 10984 * q^53 + 13396 * q^54 - 3136 * q^56 - 29206 * q^57 - 18316 * q^58 - 25956 * q^59 + 39188 * q^61 - 27024 * q^62 + 2254 * q^63 + 4096 * q^64 + 48620 * q^66 - 4416 * q^67 + 27184 * q^68 + 67150 * q^69 - 31880 * q^71 - 2944 * q^72 + 5018 * q^73 - 66072 * q^74 - 27488 * q^76 - 35035 * q^77 + 22508 * q^78 - 27977 * q^79 - 68111 * q^81 - 75504 * q^82 - 37644 * q^83 + 13328 * q^84 + 9032 * q^86 + 77843 * q^87 + 45760 * q^88 - 17216 * q^89 - 16219 * q^91 + 63200 * q^92 + 114852 * q^93 - 2148 * q^94 - 17408 * q^96 + 63175 * q^97 - 9604 * q^98 - 32890 * 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

17.0000

16.0000

0

−68.0000

49.0000

−64.0000

46.0000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(5\)	\( +1 \)
\(7\)	\( -1 \)

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	350.6.a.e		1
5.b	even	2	1		70.6.a.e	✓	1
5.c	odd	4	2		350.6.c.a		2
15.d	odd	2	1		630.6.a.b		1
20.d	odd	2	1		560.6.a.h		1
35.c	odd	2	1		490.6.a.m		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.6.a.e	✓	1	5.b	even	2	1
350.6.a.e		1	1.a	even	1	1	trivial
350.6.c.a		2	5.c	odd	4	2
490.6.a.m		1	35.c	odd	2	1
560.6.a.h		1	20.d	odd	2	1
630.6.a.b		1	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(350))\):

\( T_{3} - 17 \)

\( T_{13} + 331 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T + 4 \)
$3$	\( T - 17 \)
$5$	\( T \)
$7$	\( T - 49 \)
$11$	\( T + 715 \)