Newform orbit 294.8.e.c

• Label: 294.8.e.c
• Level: $294$
• Weight: $8$
• Character orbit: 294.e
• Analytic conductor: $91.841$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	294.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(91.8411974923\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 6)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - 8 \zeta_{6} q^{2} + (27 \zeta_{6} - 27) q^{3} + (64 \zeta_{6} - 64) q^{4} + 114 \zeta_{6} q^{5} + 216 q^{6} + 512 q^{8} - 729 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} - 27 q^{3} - 64 q^{4} + 114 q^{5} + 432 q^{6} + 1024 q^{8} - 729 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(199\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

67.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−4.00000

−

6.92820i

−13.5000

+

23.3827i

−32.0000

+

55.4256i

57.0000

+

98.7269i

216.000

0

512.000

−364.500

−

631.333i

456.000

−

789.815i

79.1

−4.00000

+

6.92820i

−13.5000

−

23.3827i

−32.0000

−

55.4256i

57.0000

−

98.7269i

216.000

0

512.000

−364.500

+

631.333i

456.000

+

789.815i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	294.8.e.c		2
7.b	odd	2	1		294.8.e.d		2
7.c	even	3	1		6.8.a.a	✓	1
7.c	even	3	1	inner	294.8.e.c		2
7.d	odd	6	1		294.8.a.l		1
7.d	odd	6	1		294.8.e.d		2
21.h	odd	6	1		18.8.a.a		1
28.g	odd	6	1		48.8.a.b		1
35.j	even	6	1		150.8.a.e		1
35.l	odd	12	2		150.8.c.k		2
56.k	odd	6	1		192.8.a.n		1
56.p	even	6	1		192.8.a.f		1
63.g	even	3	1		162.8.c.d		2
63.h	even	3	1		162.8.c.d		2
63.j	odd	6	1		162.8.c.i		2
63.n	odd	6	1		162.8.c.i		2
84.n	even	6	1		144.8.a.h		1
105.o	odd	6	1		450.8.a.ba		1
105.x	even	12	2		450.8.c.a		2
168.s	odd	6	1		576.8.a.h		1
168.v	even	6	1		576.8.a.i		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.8.a.a	✓	1	7.c	even	3	1
18.8.a.a		1	21.h	odd	6	1
48.8.a.b		1	28.g	odd	6	1
144.8.a.h		1	84.n	even	6	1
150.8.a.e		1	35.j	even	6	1
150.8.c.k		2	35.l	odd	12	2
162.8.c.d		2	63.g	even	3	1
162.8.c.d		2	63.h	even	3	1
162.8.c.i		2	63.j	odd	6	1
162.8.c.i		2	63.n	odd	6	1
192.8.a.f		1	56.p	even	6	1
192.8.a.n		1	56.k	odd	6	1
294.8.a.l		1	7.d	odd	6	1
294.8.e.c		2	1.a	even	1	1	trivial
294.8.e.c		2	7.c	even	3	1	inner
294.8.e.d		2	7.b	odd	2	1
294.8.e.d		2	7.d	odd	6	1
450.8.a.ba		1	105.o	odd	6	1
450.8.c.a		2	105.x	even	12	2
576.8.a.h		1	168.s	odd	6	1
576.8.a.i		1	168.v	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 114T_{5} + 12996 \) acting on \(S_{8}^{\mathrm{new}}(294, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 8T + 64 \)
$3$	\( T^{2} + 27T + 729 \)
$5$	\( T^{2} - 114T + 12996 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 7332 T + 53758224 \)