Newform orbit 42.6.a.a

• Label: 42.6.a.a
• Level: $42$
• Weight: $6$
• Character orbit: 42.a
• Self dual: yes
• Analytic conductor: $6.736$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$42 = 2 \cdot 3 \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	42.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$6.73612043215$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

$f(q)$

$=$

q - 4 * q^2 - 9 * q^3 + 16 * q^4 - 54 * q^5 + 36 * q^6 + 49 * q^7 - 64 * q^8 + 81 * q^9 + 216 * q^10 + 216 * q^11 - 144 * q^12 + 998 * q^13 - 196 * q^14 + 486 * q^15 + 256 * q^16 + 1302 * q^17 - 324 * q^18 + 884 * q^19 - 864 * q^20 - 441 * q^21 - 864 * q^22 - 2268 * q^23 + 576 * q^24 - 209 * q^25 - 3992 * q^26 - 729 * q^27 + 784 * q^28 - 1482 * q^29 - 1944 * q^30 + 8360 * q^31 - 1024 * q^32 - 1944 * q^33 - 5208 * q^34 - 2646 * q^35 + 1296 * q^36 - 4714 * q^37 - 3536 * q^38 - 8982 * q^39 + 3456 * q^40 - 9786 * q^41 + 1764 * q^42 + 19436 * q^43 + 3456 * q^44 - 4374 * q^45 + 9072 * q^46 + 22200 * q^47 - 2304 * q^48 + 2401 * q^49 + 836 * q^50 - 11718 * q^51 + 15968 * q^52 + 26790 * q^53 + 2916 * q^54 - 11664 * q^55 - 3136 * q^56 - 7956 * q^57 + 5928 * q^58 + 28092 * q^59 + 7776 * q^60 - 38866 * q^61 - 33440 * q^62 + 3969 * q^63 + 4096 * q^64 - 53892 * q^65 + 7776 * q^66 + 23948 * q^67 + 20832 * q^68 + 20412 * q^69 + 10584 * q^70 - 20628 * q^71 - 5184 * q^72 + 290 * q^73 + 18856 * q^74 + 1881 * q^75 + 14144 * q^76 + 10584 * q^77 + 35928 * q^78 - 99544 * q^79 - 13824 * q^80 + 6561 * q^81 + 39144 * q^82 + 19308 * q^83 - 7056 * q^84 - 70308 * q^85 - 77744 * q^86 + 13338 * q^87 - 13824 * q^88 + 36390 * q^89 + 17496 * q^90 + 48902 * q^91 - 36288 * q^92 - 75240 * q^93 - 88800 * q^94 - 47736 * q^95 + 9216 * q^96 - 79078 * q^97 - 9604 * q^98 + 17496 * 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

−9.00000

16.0000

−54.0000

36.0000

49.0000

−64.0000

81.0000

216.000

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	42.6.a.a	✓	1
3.b	odd	2	1		126.6.a.k		1
4.b	odd	2	1		336.6.a.j		1
5.b	even	2	1		1050.6.a.n		1
5.c	odd	4	2		1050.6.g.o		2
7.b	odd	2	1		294.6.a.h		1
7.c	even	3	2		294.6.e.r		2
7.d	odd	6	2		294.6.e.h		2
12.b	even	2	1		1008.6.a.x		1
21.c	even	2	1		882.6.a.o		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.6.a.a	✓	1	1.a	even	1	1	trivial
126.6.a.k		1	3.b	odd	2	1
294.6.a.h		1	7.b	odd	2	1
294.6.e.h		2	7.d	odd	6	2
294.6.e.r		2	7.c	even	3	2
336.6.a.j		1	4.b	odd	2	1
882.6.a.o		1	21.c	even	2	1
1008.6.a.x		1	12.b	even	2	1
1050.6.a.n		1	5.b	even	2	1
1050.6.g.o		2	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5} + 54$ acting on $S_{6}^{\mathrm{new}}(\Gamma_0(42))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 4$
$3$	$T + 9$
$5$	$T + 54$
$7$	$T - 49$
$11$	$T - 216$