Newform orbit 294.8.a.c

• Label: 294.8.a.c
• Level: $294$
• Weight: $8$
• Character orbit: 294.a
• Self dual: yes
• Analytic conductor: $91.841$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$294 = 2 \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	294.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$91.8411974923$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

$f(q)$

$=$

q - 8 * q^2 - 27 * q^3 + 64 * q^4 + 122 * q^5 + 216 * q^6 - 512 * q^8 + 729 * q^9 - 976 * q^10 - 1012 * q^11 - 1728 * q^12 - 3126 * q^13 - 3294 * q^15 + 4096 * q^16 + 28294 * q^17 - 5832 * q^18 + 22228 * q^19 + 7808 * q^20 + 8096 * q^22 - 108640 * q^23 + 13824 * q^24 - 63241 * q^25 + 25008 * q^26 - 19683 * q^27 - 41354 * q^29 + 26352 * q^30 - 46656 * q^31 - 32768 * q^32 + 27324 * q^33 - 226352 * q^34 + 46656 * q^36 - 85714 * q^37 - 177824 * q^38 + 84402 * q^39 - 62464 * q^40 + 155694 * q^41 + 926804 * q^43 - 64768 * q^44 + 88938 * q^45 + 869120 * q^46 + 529152 * q^47 - 110592 * q^48 + 505928 * q^50 - 763938 * q^51 - 200064 * q^52 - 294066 * q^53 + 157464 * q^54 - 123464 * q^55 - 600156 * q^57 + 330832 * q^58 + 667292 * q^59 - 210816 * q^60 - 833430 * q^61 + 373248 * q^62 + 262144 * q^64 - 381372 * q^65 - 218592 * q^66 + 1153996 * q^67 + 1810816 * q^68 + 2933280 * q^69 + 3842336 * q^71 - 373248 * q^72 - 1483690 * q^73 + 685712 * q^74 + 1707507 * q^75 + 1422592 * q^76 - 675216 * q^78 - 3763824 * q^79 + 499712 * q^80 + 531441 * q^81 - 1245552 * q^82 + 5393092 * q^83 + 3451868 * q^85 - 7414432 * q^86 + 1116558 * q^87 + 518144 * q^88 - 2502690 * q^89 - 711504 * q^90 - 6952960 * q^92 + 1259712 * q^93 - 4233216 * q^94 + 2711816 * q^95 + 884736 * q^96 + 4597550 * q^97 - 737748 * 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

−8.00000

−27.0000

64.0000

122.000

216.000

0

−512.000

729.000

−976.000

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	294.8.a.c		1
7.b	odd	2	1		42.8.a.c	✓	1
7.c	even	3	2		294.8.e.n		2
7.d	odd	6	2		294.8.e.j		2
21.c	even	2	1		126.8.a.g		1
28.d	even	2	1		336.8.a.d		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.8.a.c	✓	1	7.b	odd	2	1
126.8.a.g		1	21.c	even	2	1
294.8.a.c		1	1.a	even	1	1	trivial
294.8.e.j		2	7.d	odd	6	2
294.8.e.n		2	7.c	even	3	2
336.8.a.d		1	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5} - 122$ acting on $S_{8}^{\mathrm{new}}(\Gamma_0(294))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 8$
$3$	$T + 27$
$5$	$T - 122$
$7$	$T$
$11$	$T + 1012$