Newform orbit 980.4.i.e

• Label: 980.4.i.e
• Level: $980$
• Weight: $4$
• Character orbit: 980.i
• Analytic conductor: $57.822$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	980.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(57.8218718056\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 20)
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.

\(f(q)\)	\(=\)	\( q + (4 \zeta_{6} - 4) q^{3} - 5 \zeta_{6} q^{5} + 11 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 5 q^{5} + 11 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)	\(491\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−2.00000

+

3.46410i

0

−2.50000

−

4.33013i

0

0

0

5.50000

+

9.52628i

0

961.1

0

−2.00000

−

3.46410i

0

−2.50000

+

4.33013i

0

0

0

5.50000

−

9.52628i

0

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	980.4.i.e		2
7.b	odd	2	1		980.4.i.n		2
7.c	even	3	1		20.4.a.a	✓	1
7.c	even	3	1	inner	980.4.i.e		2
7.d	odd	6	1		980.4.a.c		1
7.d	odd	6	1		980.4.i.n		2
21.h	odd	6	1		180.4.a.a		1
28.g	odd	6	1		80.4.a.c		1
35.j	even	6	1		100.4.a.a		1
35.l	odd	12	2		100.4.c.a		2
56.k	odd	6	1		320.4.a.k		1
56.p	even	6	1		320.4.a.d		1
63.g	even	3	1		1620.4.i.d		2
63.h	even	3	1		1620.4.i.d		2
63.j	odd	6	1		1620.4.i.j		2
63.n	odd	6	1		1620.4.i.j		2
77.h	odd	6	1		2420.4.a.d		1
84.n	even	6	1		720.4.a.k		1
105.o	odd	6	1		900.4.a.m		1
105.x	even	12	2		900.4.d.k		2
112.u	odd	12	2		1280.4.d.c		2
112.w	even	12	2		1280.4.d.n		2
140.p	odd	6	1		400.4.a.o		1
140.w	even	12	2		400.4.c.j		2
280.bf	even	6	1		1600.4.a.bl		1
280.bi	odd	6	1		1600.4.a.p		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
20.4.a.a	✓	1	7.c	even	3	1
80.4.a.c		1	28.g	odd	6	1
100.4.a.a		1	35.j	even	6	1
100.4.c.a		2	35.l	odd	12	2
180.4.a.a		1	21.h	odd	6	1
320.4.a.d		1	56.p	even	6	1
320.4.a.k		1	56.k	odd	6	1
400.4.a.o		1	140.p	odd	6	1
400.4.c.j		2	140.w	even	12	2
720.4.a.k		1	84.n	even	6	1
900.4.a.m		1	105.o	odd	6	1
900.4.d.k		2	105.x	even	12	2
980.4.a.c		1	7.d	odd	6	1
980.4.i.e		2	1.a	even	1	1	trivial
980.4.i.e		2	7.c	even	3	1	inner
980.4.i.n		2	7.b	odd	2	1
980.4.i.n		2	7.d	odd	6	1
1280.4.d.c		2	112.u	odd	12	2
1280.4.d.n		2	112.w	even	12	2
1600.4.a.p		1	280.bi	odd	6	1
1600.4.a.bl		1	280.bf	even	6	1
1620.4.i.d		2	63.g	even	3	1
1620.4.i.d		2	63.h	even	3	1
1620.4.i.j		2	63.j	odd	6	1
1620.4.i.j		2	63.n	odd	6	1
2420.4.a.d		1	77.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{2} + 4T_{3} + 16 \)

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 4T + 16 \)
$5$	\( T^{2} + 5T + 25 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 60T + 3600 \)