Newform orbit 1380.2.r.c

• Label: 1380.2.r.c
• Level: $1380$
• Weight: $2$
• Character orbit: 1380.r
• Analytic conductor: $11.019$
• Analytic rank: $0$
• Dimension: $80$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.0193554789\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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 + 8 * q^3 + 32 * q^13 + 24 * q^21 - 32 * q^25 - 28 * q^27 - 32 * q^31 - 44 * q^33 + 24 * q^37 - 32 * q^43 + 88 * q^45 + 16 * q^51 + 8 * q^55 + 16 * q^57 - 32 * q^61 - 12 * q^63 - 16 * q^67 - 32 * q^73 + 4 * q^75 - 64 * q^81 - 32 * q^85 + 64 * q^91 + 8 * q^93 - 96 * q^97

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
737.1		0		−1.72892	−	0.104128i		0		1.51193	+	1.64744i		0		3.27368	+	3.27368i		0		2.97831	+	0.360057i		0
737.2		0		−1.71368	−	0.251606i		0		0.257544	−	2.22119i		0		−2.77975	−	2.77975i		0		2.87339	+	0.862345i		0
737.3		0		−1.67967	−	0.422758i		0		2.05360	−	0.884710i		0		0.924264	+	0.924264i		0		2.64255	+	1.42018i		0
737.4		0		−1.57061	−	0.730184i		0		−0.115647	+	2.23308i		0		−0.487335	−	0.487335i		0		1.93366	+	2.29368i		0
737.5		0		−1.53876	+	0.795128i		0		1.56652	−	1.59562i		0		0.769646	+	0.769646i		0		1.73554	−	2.44702i		0
737.6		0		−1.50371	−	0.859570i		0		−0.917834	−	2.03901i		0		0.160200	+	0.160200i		0		1.52228	+	2.58509i		0
737.7		0		−1.43700	+	0.966965i		0		1.81165	+	1.31070i		0		−2.64771	−	2.64771i		0		1.12996	−	2.77906i		0
737.8		0		−1.43142	+	0.975211i		0		0.832554	+	2.07530i		0		0.0455472	+	0.0455472i		0		1.09793	−	2.79187i		0
737.9		0		−1.31796	−	1.12382i		0		2.16179	+	0.571544i		0		−3.03605	−	3.03605i		0		0.474038	+	2.96231i		0
737.10		0		−1.29281	−	1.15267i		0		−0.532114	+	2.17183i		0		−0.268444	−	0.268444i		0		0.342726	+	2.98036i		0
737.11		0		−1.21045	−	1.23887i		0		−2.23574	+	0.0384947i		0		1.22712	+	1.22712i		0		−0.0696199	+	2.99919i		0
737.12		0		−0.975211	+	1.43142i		0		−0.832554	−	2.07530i		0		0.0455472	+	0.0455472i		0		−1.09793	−	2.79187i		0
737.13		0		−0.966965	+	1.43700i		0		−1.81165	−	1.31070i		0		−2.64771	−	2.64771i		0		−1.12996	−	2.77906i		0
737.14		0		−0.795128	+	1.53876i		0		−1.56652	+	1.59562i		0		0.769646	+	0.769646i		0		−1.73554	−	2.44702i		0
737.15		0		−0.727046	−	1.57207i		0		1.08694	−	1.95412i		0		2.39152	+	2.39152i		0		−1.94281	+	2.28593i		0
737.16		0		−0.136960	−	1.72663i		0		−2.01176	−	0.976134i		0		−0.818985	−	0.818985i		0		−2.96248	+	0.472958i		0
737.17		0		−0.118851	−	1.72797i		0		0.238514	+	2.22331i		0		2.86460	+	2.86460i		0		−2.97175	+	0.410741i		0
737.18		0		−0.0819537	−	1.73011i		0		−1.50202	+	1.65649i		0		−2.00070	−	2.00070i		0		−2.98657	+	0.283578i		0
737.19		0		0.104128	+	1.72892i		0		−1.51193	−	1.64744i		0		3.27368	+	3.27368i		0		−2.97831	+	0.360057i		0
737.20		0		0.251606	+	1.71368i		0		−0.257544	+	2.22119i		0		−2.77975	−	2.77975i		0		−2.87339	+	0.862345i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1380.2.r.c	✓	80
3.b	odd	2	1	inner	1380.2.r.c	✓	80
5.c	odd	4	1	inner	1380.2.r.c	✓	80
15.e	even	4	1	inner	1380.2.r.c	✓	80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1380.2.r.c	✓	80	1.a	even	1	1	trivial
1380.2.r.c	✓	80	3.b	odd	2	1	inner
1380.2.r.c	✓	80	5.c	odd	4	1	inner
1380.2.r.c	✓	80	15.e	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^40 - 20*T7^37 + 1084*T7^36 - 172*T7^35 + 200*T7^34 - 20844*T7^33 + 435334*T7^32 - 201084*T7^31 + 214872*T7^30 - 7124348*T7^29 + 82449308*T7^28 - 72439852*T7^27 + 74321528*T7^26 - 979986500*T7^25 + 7619403577*T7^24 - 10117013092*T7^23 + 10181196584*T7^22 - 52138093536*T7^21 + 303672701728*T7^20 - 519513817848*T7^19 + 523355549280*T7^18 - 841916345792*T7^17 + 3931236498712*T7^16 - 7514326340560*T7^15 + 7633498070816*T7^14 - 3010140157248*T7^13 + 6250082676720*T7^12 - 13130926266720*T7^11 + 14597439704064*T7^10 - 2659852019520*T7^9 + 293682355920*T7^8 - 2283198800256*T7^7 + 4449737410560*T7^6 + 480708836352*T7^5 + 78849339840*T7^4 - 105410460672*T7^3 + 42138206208*T7^2 - 3386105856*T7 + 136048896