Newform orbit 680.2.h.c

• Label: 680.2.h.c
• Level: $680$
• Weight: $2$
• Character orbit: 680.h
• Analytic conductor: $5.430$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	680.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.42982733745\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Twist minimal:	yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 96 q^{9} - 48 q^{15} + 24 q^{34} - 96 q^{36} + 96 q^{49} + 48 q^{60} + 192 q^{64} - 144 q^{66} - 96 q^{70} + 96 q^{76} - 192 q^{84} - 192 q^{86} - 48 q^{89} - 144 q^{94}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
509.1	−1.39273	−	0.245576i		−	3.26492i	1.87939	+	0.684040i	−1.77664	−	1.35778i	−0.801785	+	4.54715i	−3.95202			−2.44949	−	1.41421i	−7.65971			2.14094	+	2.32731i
509.2	−1.39273	−	0.245576i		−	1.76041i	1.87939	+	0.684040i	1.35778	+	1.77664i	−0.432315	+	2.45178i	0.109289			−2.44949	−	1.41421i	−0.0990599			−1.45471	−	2.80781i
509.3	−1.39273	−	0.245576i			1.76041i	1.87939	+	0.684040i	−1.35778	−	1.77664i	0.432315	−	2.45178i	−0.109289			−2.44949	−	1.41421i	−0.0990599			1.45471	+	2.80781i
509.4	−1.39273	−	0.245576i			3.26492i	1.87939	+	0.684040i	1.77664	+	1.35778i	0.801785	−	4.54715i	3.95202			−2.44949	−	1.41421i	−7.65971			−2.14094	−	2.32731i
509.5	−1.39273	+	0.245576i		−	3.26492i	1.87939	−	0.684040i	1.77664	−	1.35778i	0.801785	+	4.54715i	3.95202			−2.44949	+	1.41421i	−7.65971			−2.14094	+	2.32731i
509.6	−1.39273	+	0.245576i		−	1.76041i	1.87939	−	0.684040i	−1.35778	+	1.77664i	0.432315	+	2.45178i	−0.109289			−2.44949	+	1.41421i	−0.0990599			1.45471	−	2.80781i
509.7	−1.39273	+	0.245576i			1.76041i	1.87939	−	0.684040i	1.35778	−	1.77664i	−0.432315	−	2.45178i	0.109289			−2.44949	+	1.41421i	−0.0990599			−1.45471	+	2.80781i
509.8	−1.39273	+	0.245576i			3.26492i	1.87939	−	0.684040i	−1.77664	+	1.35778i	−0.801785	−	4.54715i	−3.95202			−2.44949	+	1.41421i	−7.65971			2.14094	−	2.32731i
509.9	−0.909039	−	1.08335i		−	2.68098i	−0.347296	+	1.96962i	2.10758	−	0.747066i	−2.90444	+	2.43712i	0.590069			2.44949	−	1.41421i	−4.18766			−2.72521	−	1.60414i
509.10	−0.909039	−	1.08335i		−	1.45525i	−0.347296	+	1.96962i	−0.747066	+	2.10758i	−1.57654	+	1.32288i	−4.94720			2.44949	−	1.41421i	0.882256			2.96236	−	1.10654i
509.11	−0.909039	−	1.08335i			1.45525i	−0.347296	+	1.96962i	0.747066	−	2.10758i	1.57654	−	1.32288i	4.94720			2.44949	−	1.41421i	0.882256			−2.96236	+	1.10654i
509.12	−0.909039	−	1.08335i			2.68098i	−0.347296	+	1.96962i	−2.10758	+	0.747066i	2.90444	−	2.43712i	−0.590069			2.44949	−	1.41421i	−4.18766			2.72521	+	1.60414i
509.13	−0.909039	+	1.08335i		−	2.68098i	−0.347296	−	1.96962i	−2.10758	−	0.747066i	2.90444	+	2.43712i	−0.590069			2.44949	+	1.41421i	−4.18766			2.72521	−	1.60414i
509.14	−0.909039	+	1.08335i		−	1.45525i	−0.347296	−	1.96962i	0.747066	+	2.10758i	1.57654	+	1.32288i	4.94720			2.44949	+	1.41421i	0.882256			−2.96236	−	1.10654i
509.15	−0.909039	+	1.08335i			1.45525i	−0.347296	−	1.96962i	−0.747066	−	2.10758i	−1.57654	−	1.32288i	−4.94720			2.44949	+	1.41421i	0.882256			2.96236	+	1.10654i
509.16	−0.909039	+	1.08335i			2.68098i	−0.347296	−	1.96962i	2.10758	+	0.747066i	−2.90444	−	2.43712i	0.590069			2.44949	+	1.41421i	−4.18766			−2.72521	+	1.60414i
509.17	−0.483690	−	1.32893i		−	2.44428i	−1.53209	+	1.28558i	−0.380200	−	2.20351i	−3.24827	+	1.18227i	−2.45108			2.44949	+	1.41421i	−2.97453			−2.74440	+	1.57107i
509.18	−0.483690	−	1.32893i		−	0.980457i	−1.53209	+	1.28558i	−2.20351	−	0.380200i	−1.30295	+	0.474237i	2.74570			2.44949	+	1.41421i	2.03870			0.560555	+	3.11220i
509.19	−0.483690	−	1.32893i			0.980457i	−1.53209	+	1.28558i	2.20351	+	0.380200i	1.30295	−	0.474237i	−2.74570			2.44949	+	1.41421i	2.03870			−0.560555	−	3.11220i
509.20	−0.483690	−	1.32893i			2.44428i	−1.53209	+	1.28558i	0.380200	+	2.20351i	3.24827	−	1.18227i	2.45108			2.44949	+	1.41421i	−2.97453			2.74440	−	1.57107i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
8.b	even	2	1	inner
17.b	even	2	1	inner
40.f	even	2	1	inner
85.c	even	2	1	inner
136.h	even	2	1	inner
680.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	680.2.h.c	✓	48
5.b	even	2	1	inner	680.2.h.c	✓	48
8.b	even	2	1	inner	680.2.h.c	✓	48
17.b	even	2	1	inner	680.2.h.c	✓	48
40.f	even	2	1	inner	680.2.h.c	✓	48
85.c	even	2	1	inner	680.2.h.c	✓	48
136.h	even	2	1	inner	680.2.h.c	✓	48
680.h	even	2	1	inner	680.2.h.c	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
680.2.h.c	✓	48	1.a	even	1	1	trivial
680.2.h.c	✓	48	5.b	even	2	1	inner
680.2.h.c	✓	48	8.b	even	2	1	inner
680.2.h.c	✓	48	17.b	even	2	1	inner
680.2.h.c	✓	48	40.f	even	2	1	inner
680.2.h.c	✓	48	85.c	even	2	1	inner
680.2.h.c	✓	48	136.h	even	2	1	inner
680.2.h.c	✓	48	680.h	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 30T_{3}^{10} + 342T_{3}^{8} + 1872T_{3}^{6} + 5100T_{3}^{4} + 6456T_{3}^{2} + 2888 \) acting on \(S_{2}^{\mathrm{new}}(680, [\chi])\).