Newform orbit 672.4.i.c

• Label: 672.4.i.c
• Level: $672$
• Weight: $4$
• Character orbit: 672.i
• Analytic conductor: $39.649$
• Analytic rank: $0$
• Dimension: $80$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	672.i (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(39.6492835239\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 64 q^{7} + 104 q^{9} + 8 q^{15} - 976 q^{25} - 568 q^{39} - 4048 q^{49} - 1448 q^{57} + 2152 q^{63} - 4992 q^{79} + 1568 q^{81}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
209.1		0		−5.19433	−	0.137767i		0			−	6.83655i		0		−13.6972	+	12.4654i		0		26.9620	+	1.43121i		0
209.2		0		−5.19433	−	0.137767i		0			−	6.83655i		0		−13.6972	−	12.4654i		0		26.9620	+	1.43121i		0
209.3		0		−5.19433	+	0.137767i		0				6.83655i		0		−13.6972	−	12.4654i		0		26.9620	−	1.43121i		0
209.4		0		−5.19433	+	0.137767i		0				6.83655i		0		−13.6972	+	12.4654i		0		26.9620	−	1.43121i		0
209.5		0		−5.10794	−	0.953394i		0			−	17.0023i		0		5.04990	−	17.8185i		0		25.1821	+	9.73975i		0
209.6		0		−5.10794	−	0.953394i		0			−	17.0023i		0		5.04990	+	17.8185i		0		25.1821	+	9.73975i		0
209.7		0		−5.10794	+	0.953394i		0				17.0023i		0		5.04990	+	17.8185i		0		25.1821	−	9.73975i		0
209.8		0		−5.10794	+	0.953394i		0				17.0023i		0		5.04990	−	17.8185i		0		25.1821	−	9.73975i		0
209.9		0		−4.74822	−	2.11055i		0				15.5059i		0		14.6158	+	11.3745i		0		18.0912	+	20.0427i		0
209.10		0		−4.74822	−	2.11055i		0				15.5059i		0		14.6158	−	11.3745i		0		18.0912	+	20.0427i		0
209.11		0		−4.74822	+	2.11055i		0			−	15.5059i		0		14.6158	−	11.3745i		0		18.0912	−	20.0427i		0
209.12		0		−4.74822	+	2.11055i		0			−	15.5059i		0		14.6158	+	11.3745i		0		18.0912	−	20.0427i		0
209.13		0		−4.72061	−	2.17161i		0			−	2.27823i		0		13.7109	−	12.4503i		0		17.5682	+	20.5026i		0
209.14		0		−4.72061	−	2.17161i		0			−	2.27823i		0		13.7109	+	12.4503i		0		17.5682	+	20.5026i		0
209.15		0		−4.72061	+	2.17161i		0				2.27823i		0		13.7109	+	12.4503i		0		17.5682	−	20.5026i		0
209.16		0		−4.72061	+	2.17161i		0				2.27823i		0		13.7109	−	12.4503i		0		17.5682	−	20.5026i		0
209.17		0		−4.05635	−	3.24747i		0				13.2397i		0		−18.4748	−	1.29729i		0		5.90788	+	26.3457i		0
209.18		0		−4.05635	−	3.24747i		0				13.2397i		0		−18.4748	+	1.29729i		0		5.90788	+	26.3457i		0
209.19		0		−4.05635	+	3.24747i		0			−	13.2397i		0		−18.4748	+	1.29729i		0		5.90788	−	26.3457i		0
209.20		0		−4.05635	+	3.24747i		0			−	13.2397i		0		−18.4748	−	1.29729i		0		5.90788	−	26.3457i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
8.b	even	2	1	inner
21.c	even	2	1	inner
24.h	odd	2	1	inner
56.h	odd	2	1	inner
168.i	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	672.4.i.c		80
3.b	odd	2	1	inner	672.4.i.c		80
4.b	odd	2	1		168.4.i.c	✓	80
7.b	odd	2	1	inner	672.4.i.c		80
8.b	even	2	1	inner	672.4.i.c		80
8.d	odd	2	1		168.4.i.c	✓	80
12.b	even	2	1		168.4.i.c	✓	80
21.c	even	2	1	inner	672.4.i.c		80
24.f	even	2	1		168.4.i.c	✓	80
24.h	odd	2	1	inner	672.4.i.c		80
28.d	even	2	1		168.4.i.c	✓	80
56.e	even	2	1		168.4.i.c	✓	80
56.h	odd	2	1	inner	672.4.i.c		80
84.h	odd	2	1		168.4.i.c	✓	80
168.e	odd	2	1		168.4.i.c	✓	80
168.i	even	2	1	inner	672.4.i.c		80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.4.i.c	✓	80	4.b	odd	2	1
168.4.i.c	✓	80	8.d	odd	2	1
168.4.i.c	✓	80	12.b	even	2	1
168.4.i.c	✓	80	24.f	even	2	1
168.4.i.c	✓	80	28.d	even	2	1
168.4.i.c	✓	80	56.e	even	2	1
168.4.i.c	✓	80	84.h	odd	2	1
168.4.i.c	✓	80	168.e	odd	2	1
672.4.i.c		80	1.a	even	1	1	trivial
672.4.i.c		80	3.b	odd	2	1	inner
672.4.i.c		80	7.b	odd	2	1	inner
672.4.i.c		80	8.b	even	2	1	inner
672.4.i.c		80	21.c	even	2	1	inner
672.4.i.c		80	24.h	odd	2	1	inner
672.4.i.c		80	56.h	odd	2	1	inner
672.4.i.c		80	168.i	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^20 + 1372*T5^18 + 782932*T5^16 + 239880544*T5^14 + 42479046160*T5^12 + 4358828758336*T5^10 + 246895591977664*T5^8 + 6978739579684864*T5^6 + 82397884120202752*T5^4 + 347349051281676288*T5^2 + 395329100828147712