Newform orbit 294.8.a.b

• Label: 294.8.a.b
• 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 - 165 * q^5 + 216 * q^6 - 512 * q^8 + 729 * q^9 + 1320 * q^10 - 2055 * q^11 - 1728 * q^12 - 2720 * q^13 + 4455 * q^15 + 4096 * q^16 + 15708 * q^17 - 5832 * q^18 - 6752 * q^19 - 10560 * q^20 + 16440 * q^22 + 30828 * q^23 + 13824 * q^24 - 50900 * q^25 + 21760 * q^26 - 19683 * q^27 - 118305 * q^29 - 35640 * q^30 + 147517 * q^31 - 32768 * q^32 + 55485 * q^33 - 125664 * q^34 + 46656 * q^36 + 311732 * q^37 + 54016 * q^38 + 73440 * q^39 + 84480 * q^40 + 491400 * q^41 - 577174 * q^43 - 131520 * q^44 - 120285 * q^45 - 246624 * q^46 + 854862 * q^47 - 110592 * q^48 + 407200 * q^50 - 424116 * q^51 - 174080 * q^52 + 1166883 * q^53 + 157464 * q^54 + 339075 * q^55 + 182304 * q^57 + 946440 * q^58 + 167079 * q^59 + 285120 * q^60 - 1027274 * q^61 - 1180136 * q^62 + 262144 * q^64 + 448800 * q^65 - 443880 * q^66 - 3268114 * q^67 + 1005312 * q^68 - 832356 * q^69 + 3046842 * q^71 - 373248 * q^72 + 3209110 * q^73 - 2493856 * q^74 + 1374300 * q^75 - 432128 * q^76 - 587520 * q^78 + 7127987 * q^79 - 675840 * q^80 + 531441 * q^81 - 3931200 * q^82 - 4365909 * q^83 - 2591820 * q^85 + 4617392 * q^86 + 3194235 * q^87 + 1052160 * q^88 + 11996214 * q^89 + 962280 * q^90 + 1972992 * q^92 - 3982959 * q^93 - 6838896 * q^94 + 1114080 * q^95 + 884736 * q^96 - 13343639 * q^97 - 1498095 * 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

−165.000

216.000

0

−512.000

729.000

1320.00

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.b		1
7.b	odd	2	1		294.8.a.i		1
7.c	even	3	2		294.8.e.p		2
7.d	odd	6	2		42.8.e.a	✓	2
21.g	even	6	2		126.8.g.b		2

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5} + 165$ acting on $S_{8}^{\mathrm{new}}(\Gamma_0(294))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 8$
$3$	$T + 27$
$5$	$T + 165$
$7$	$T$
$11$	$T + 2055$