Newform orbit 588.3.c.d

• Label: 588.3.c.d
• Level: $588$
• Weight: $3$
• Character orbit: 588.c
• Analytic conductor: $16.022$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	588.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.0218395444\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-5}) \)
Defining polynomial:	\( x^{2} + 5 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b - 2) * q^3 + 3*b * q^5 + (-4*b - 1) * q^9 + 3*b * q^11 - 14 * q^13 + (-6*b - 15) * q^15 + 12*b * q^17 - 8 * q^19 + 6*b * q^23 - 20 * q^25 + (7*b + 22) * q^27 - 21*b * q^29 + 31 * q^31 + (-6*b - 15) * q^33 - 28 * q^37 + (-14*b + 28) * q^39 - 30*b * q^41 - 52 * q^43 + (-3*b + 60) * q^45 - 18*b * q^47 + (-24*b - 60) * q^51 - 3*b * q^53 - 45 * q^55 + (-8*b + 16) * q^57 + 9*b * q^59 - 14 * q^61 - 42*b * q^65 - 4 * q^67 + (-12*b - 30) * q^69 + 18*b * q^71 - 98 * q^73 + (-20*b + 40) * q^75 + 101 * q^79 + (8*b - 79) * q^81 + 39*b * q^83 - 180 * q^85 + (42*b + 105) * q^87 - 30*b * q^89 + (31*b - 62) * q^93 - 24*b * q^95 + 13 * q^97 + (-3*b + 60) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^3 - 2 * q^9 - 28 * q^13 - 30 * q^15 - 16 * q^19 - 40 * q^25 + 44 * q^27 + 62 * q^31 - 30 * q^33 - 56 * q^37 + 56 * q^39 - 104 * q^43 + 120 * q^45 - 120 * q^51 - 90 * q^55 + 32 * q^57 - 28 * q^61 - 8 * q^67 - 60 * q^69 - 196 * q^73 + 80 * q^75 + 202 * q^79 - 158 * q^81 - 360 * q^85 + 210 * q^87 - 124 * q^93 + 26 * q^97 + 120 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(493\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

197.1

	−	2.23607i
		2.23607i

0

−2.00000

−

2.23607i

0

−

6.70820i

0

0

0

−1.00000

+

8.94427i

0

197.2

0

−2.00000

+

2.23607i

0

6.70820i

0

0

0

−1.00000

−

8.94427i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	588.3.c.d		2
3.b	odd	2	1	inner	588.3.c.d		2
7.b	odd	2	1		588.3.c.g		2
7.c	even	3	2		588.3.p.e		4
7.d	odd	6	2		84.3.p.b	✓	4
21.c	even	2	1		588.3.c.g		2
21.g	even	6	2		84.3.p.b	✓	4
21.h	odd	6	2		588.3.p.e		4
28.f	even	6	2		336.3.bn.e		4
84.j	odd	6	2		336.3.bn.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.3.p.b	✓	4	7.d	odd	6	2
84.3.p.b	✓	4	21.g	even	6	2
336.3.bn.e		4	28.f	even	6	2
336.3.bn.e		4	84.j	odd	6	2
588.3.c.d		2	1.a	even	1	1	trivial
588.3.c.d		2	3.b	odd	2	1	inner
588.3.c.g		2	7.b	odd	2	1
588.3.c.g		2	21.c	even	2	1
588.3.p.e		4	7.c	even	3	2
588.3.p.e		4	21.h	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(588, [\chi])\):

\( T_{5}^{2} + 45 \)

\( T_{13} + 14 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 4T + 9 \)
$5$	\( T^{2} + 45 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 45 \)