Newform orbit 294.6.e.g

• Label: 294.6.e.g
• Level: $294$
• Weight: $6$
• Character orbit: 294.e
• Analytic conductor: $47.153$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	294.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(47.1528430250\)
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.

\(f(q)\)	\(=\)	\( q - 4 \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + (16 \zeta_{6} - 16) q^{4} + 66 \zeta_{6} q^{5} - 36 q^{6} + 64 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 9 q^{3} - 16 q^{4} + 66 q^{5} - 72 q^{6} + 128 q^{8} - 81 q^{9}+O(q^{10}) \)

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

−2.00000

−

3.46410i

4.50000

−

7.79423i

−8.00000

+

13.8564i

33.0000

+

57.1577i

−36.0000

0

64.0000

−40.5000

−

70.1481i

132.000

−

228.631i

79.1

−2.00000

+

3.46410i

4.50000

+

7.79423i

−8.00000

−

13.8564i

33.0000

−

57.1577i

−36.0000

0

64.0000

−40.5000

+

70.1481i

132.000

+

228.631i

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	294.6.e.g		2
7.b	odd	2	1		294.6.e.a		2
7.c	even	3	1		6.6.a.a	✓	1
7.c	even	3	1	inner	294.6.e.g		2
7.d	odd	6	1		294.6.a.m		1
7.d	odd	6	1		294.6.e.a		2
21.g	even	6	1		882.6.a.a		1
21.h	odd	6	1		18.6.a.b		1
28.g	odd	6	1		48.6.a.c		1
35.j	even	6	1		150.6.a.d		1
35.l	odd	12	2		150.6.c.b		2
56.k	odd	6	1		192.6.a.g		1
56.p	even	6	1		192.6.a.o		1
63.g	even	3	1		162.6.c.e		2
63.h	even	3	1		162.6.c.e		2
63.j	odd	6	1		162.6.c.h		2
63.n	odd	6	1		162.6.c.h		2
77.h	odd	6	1		726.6.a.a		1
84.n	even	6	1		144.6.a.j		1
91.r	even	6	1		1014.6.a.c		1
105.o	odd	6	1		450.6.a.m		1
105.x	even	12	2		450.6.c.j		2
112.u	odd	12	2		768.6.d.p		2
112.w	even	12	2		768.6.d.c		2
168.s	odd	6	1		576.6.a.j		1
168.v	even	6	1		576.6.a.i		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.6.a.a	✓	1	7.c	even	3	1
18.6.a.b		1	21.h	odd	6	1
48.6.a.c		1	28.g	odd	6	1
144.6.a.j		1	84.n	even	6	1
150.6.a.d		1	35.j	even	6	1
150.6.c.b		2	35.l	odd	12	2
162.6.c.e		2	63.g	even	3	1
162.6.c.e		2	63.h	even	3	1
162.6.c.h		2	63.j	odd	6	1
162.6.c.h		2	63.n	odd	6	1
192.6.a.g		1	56.k	odd	6	1
192.6.a.o		1	56.p	even	6	1
294.6.a.m		1	7.d	odd	6	1
294.6.e.a		2	7.b	odd	2	1
294.6.e.a		2	7.d	odd	6	1
294.6.e.g		2	1.a	even	1	1	trivial
294.6.e.g		2	7.c	even	3	1	inner
450.6.a.m		1	105.o	odd	6	1
450.6.c.j		2	105.x	even	12	2
576.6.a.i		1	168.v	even	6	1
576.6.a.j		1	168.s	odd	6	1
726.6.a.a		1	77.h	odd	6	1
768.6.d.c		2	112.w	even	12	2
768.6.d.p		2	112.u	odd	12	2
882.6.a.a		1	21.g	even	6	1
1014.6.a.c		1	91.r	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(294, [\chi])\):

\( T_{5}^{2} - 66T_{5} + 4356 \)

\( T_{11}^{2} - 60T_{11} + 3600 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 4T + 16 \)
$3$	\( T^{2} - 9T + 81 \)
$5$	\( T^{2} - 66T + 4356 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 60T + 3600 \)