Newform orbit 140.2.s.b

• Label: 140.2.s.b
• Level: $140$
• Weight: $2$
• Character orbit: 140.s
• Analytic conductor: $1.118$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	140.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.11790562830\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 6 q^{4} - 6 q^{5} + 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−1.41371	+	0.0377920i	0.634715	−	0.366453i	1.99714	−	0.106854i	−0.661137	+	2.13609i	−0.883452	+	0.542044i	2.56107	−	0.664037i	−2.81934	+	0.226536i	−1.23142	+	2.13289i	0.853928	−	3.04480i
19.2	−1.38503	+	0.285823i	2.24836	−	1.29809i	1.83661	−	0.791746i	2.07525	−	0.832668i	−2.74302	+	2.44053i	−2.57589	−	0.603960i	−2.31746	+	1.62154i	1.87009	−	3.23909i	−2.63629	+	1.74642i
19.3	−1.36614	−	0.365598i	−1.28100	+	0.739583i	1.73268	+	0.998916i	−1.26649	−	1.84283i	2.02041	−	0.542044i	0.664037	+	2.56107i	−2.00188	−	1.99812i	−0.406034	+	0.703271i	1.05646	+	2.98058i
19.4	−1.13507	+	0.843578i	−1.62820	+	0.940044i	0.576753	−	1.91503i	0.430625	−	2.19421i	1.05512	−	2.44053i	0.603960	−	2.57589i	0.960828	+	2.66023i	0.267367	−	0.463092i	1.36220	+	2.85384i
19.5	−0.999687	−	1.00031i	1.28100	−	0.739583i	−0.00125109	+	2.00000i	−1.26649	−	1.84283i	−2.02041	−	0.542044i	−0.664037	−	2.56107i	2.00188	−	1.99812i	−0.406034	+	0.703271i	−0.577314	+	3.10913i
19.6	−0.674125	−	1.24320i	−0.634715	+	0.366453i	−1.09111	+	1.67615i	−0.661137	+	2.13609i	0.883452	+	0.542044i	−2.56107	+	0.664037i	2.81934	+	0.226536i	−1.23142	+	2.13289i	3.10129	−	0.618067i
19.7	−0.444985	−	1.34238i	−2.24836	+	1.29809i	−1.60398	+	1.19468i	2.07525	−	0.832668i	2.74302	+	2.44053i	2.57589	+	0.603960i	2.31746	+	1.62154i	1.87009	−	3.23909i	−2.04121	−	2.41525i
19.8	−0.163027	+	1.40479i	−1.62820	+	0.940044i	−1.94684	−	0.458035i	−1.68493	+	1.47004i	−1.05512	−	2.44053i	0.603960	−	2.57589i	0.960828	−	2.66023i	0.267367	−	0.463092i	−1.79040	−	2.60662i
19.9	0.163027	−	1.40479i	1.62820	−	0.940044i	−1.94684	−	0.458035i	0.430625	−	2.19421i	−1.05512	−	2.44053i	−0.603960	+	2.57589i	−0.960828	+	2.66023i	0.267367	−	0.463092i	−3.01219	−	0.962651i
19.10	0.444985	+	1.34238i	2.24836	−	1.29809i	−1.60398	+	1.19468i	0.316513	+	2.21355i	2.74302	+	2.44053i	−2.57589	−	0.603960i	−2.31746	−	1.62154i	1.87009	−	3.23909i	−2.83059	+	1.40988i
19.11	0.674125	+	1.24320i	0.634715	−	0.366453i	−1.09111	+	1.67615i	1.51934	−	1.64061i	0.883452	+	0.542044i	2.56107	−	0.664037i	−2.81934	−	0.226536i	−1.23142	+	2.13289i	3.06384	+	0.782877i
19.12	0.999687	+	1.00031i	−1.28100	+	0.739583i	−0.00125109	+	2.00000i	−2.22918	−	0.175395i	−2.02041	−	0.542044i	0.664037	+	2.56107i	−2.00188	+	1.99812i	−0.406034	+	0.703271i	−2.05303	−	2.40522i
19.13	1.13507	−	0.843578i	1.62820	−	0.940044i	0.576753	−	1.91503i	−1.68493	+	1.47004i	1.05512	−	2.44053i	−0.603960	+	2.57589i	−0.960828	−	2.66023i	0.267367	−	0.463092i	−0.672416	+	3.08996i
19.14	1.36614	+	0.365598i	1.28100	−	0.739583i	1.73268	+	0.998916i	−2.22918	−	0.175395i	2.02041	−	0.542044i	−0.664037	−	2.56107i	2.00188	+	1.99812i	−0.406034	+	0.703271i	−2.98125	−	1.05460i
19.15	1.38503	−	0.285823i	−2.24836	+	1.29809i	1.83661	−	0.791746i	0.316513	+	2.21355i	−2.74302	+	2.44053i	2.57589	+	0.603960i	2.31746	−	1.62154i	1.87009	−	3.23909i	1.07106	+	2.97537i
19.16	1.41371	−	0.0377920i	−0.634715	+	0.366453i	1.99714	−	0.106854i	1.51934	−	1.64061i	−0.883452	+	0.542044i	−2.56107	+	0.664037i	2.81934	−	0.226536i	−1.23142	+	2.13289i	2.08591	−	2.37676i
59.1	−1.41371	−	0.0377920i	0.634715	+	0.366453i	1.99714	+	0.106854i	−0.661137	−	2.13609i	−0.883452	−	0.542044i	2.56107	+	0.664037i	−2.81934	−	0.226536i	−1.23142	−	2.13289i	0.853928	+	3.04480i
59.2	−1.38503	−	0.285823i	2.24836	+	1.29809i	1.83661	+	0.791746i	2.07525	+	0.832668i	−2.74302	−	2.44053i	−2.57589	+	0.603960i	−2.31746	−	1.62154i	1.87009	+	3.23909i	−2.63629	−	1.74642i
59.3	−1.36614	+	0.365598i	−1.28100	−	0.739583i	1.73268	−	0.998916i	−1.26649	+	1.84283i	2.02041	+	0.542044i	0.664037	−	2.56107i	−2.00188	+	1.99812i	−0.406034	−	0.703271i	1.05646	−	2.98058i
59.4	−1.13507	−	0.843578i	−1.62820	−	0.940044i	0.576753	+	1.91503i	0.430625	+	2.19421i	1.05512	+	2.44053i	0.603960	+	2.57589i	0.960828	−	2.66023i	0.267367	+	0.463092i	1.36220	−	2.85384i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
7.d	odd	6	1	inner
20.d	odd	2	1	inner
28.f	even	6	1	inner
35.i	odd	6	1	inner
140.s	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	140.2.s.b	✓	32
4.b	odd	2	1	inner	140.2.s.b	✓	32
5.b	even	2	1	inner	140.2.s.b	✓	32
5.c	odd	4	2		700.2.p.e		32
7.b	odd	2	1		980.2.s.e		32
7.c	even	3	1		980.2.c.d		32
7.c	even	3	1		980.2.s.e		32
7.d	odd	6	1	inner	140.2.s.b	✓	32
7.d	odd	6	1		980.2.c.d		32
20.d	odd	2	1	inner	140.2.s.b	✓	32
20.e	even	4	2		700.2.p.e		32
28.d	even	2	1		980.2.s.e		32
28.f	even	6	1	inner	140.2.s.b	✓	32
28.f	even	6	1		980.2.c.d		32
28.g	odd	6	1		980.2.c.d		32
28.g	odd	6	1		980.2.s.e		32
35.c	odd	2	1		980.2.s.e		32
35.i	odd	6	1	inner	140.2.s.b	✓	32
35.i	odd	6	1		980.2.c.d		32
35.j	even	6	1		980.2.c.d		32
35.j	even	6	1		980.2.s.e		32
35.k	even	12	2		700.2.p.e		32
140.c	even	2	1		980.2.s.e		32
140.p	odd	6	1		980.2.c.d		32
140.p	odd	6	1		980.2.s.e		32
140.s	even	6	1	inner	140.2.s.b	✓	32
140.s	even	6	1		980.2.c.d		32
140.x	odd	12	2		700.2.p.e		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.2.s.b	✓	32	1.a	even	1	1	trivial
140.2.s.b	✓	32	4.b	odd	2	1	inner
140.2.s.b	✓	32	5.b	even	2	1	inner
140.2.s.b	✓	32	7.d	odd	6	1	inner
140.2.s.b	✓	32	20.d	odd	2	1	inner
140.2.s.b	✓	32	28.f	even	6	1	inner
140.2.s.b	✓	32	35.i	odd	6	1	inner
140.2.s.b	✓	32	140.s	even	6	1	inner
700.2.p.e		32	5.c	odd	4	2
700.2.p.e		32	20.e	even	4	2
700.2.p.e		32	35.k	even	12	2
700.2.p.e		32	140.x	odd	12	2
980.2.c.d		32	7.c	even	3	1
980.2.c.d		32	7.d	odd	6	1
980.2.c.d		32	28.f	even	6	1
980.2.c.d		32	28.g	odd	6	1
980.2.c.d		32	35.i	odd	6	1
980.2.c.d		32	35.j	even	6	1
980.2.c.d		32	140.p	odd	6	1
980.2.c.d		32	140.s	even	6	1
980.2.s.e		32	7.b	odd	2	1
980.2.s.e		32	7.c	even	3	1
980.2.s.e		32	28.d	even	2	1
980.2.s.e		32	28.g	odd	6	1
980.2.s.e		32	35.c	odd	2	1
980.2.s.e		32	35.j	even	6	1
980.2.s.e		32	140.c	even	2	1
980.2.s.e		32	140.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 13T_{3}^{14} + 116T_{3}^{12} - 535T_{3}^{10} + 1780T_{3}^{8} - 3353T_{3}^{6} + 4445T_{3}^{4} - 2156T_{3}^{2} + 784 \) acting on \(S_{2}^{\mathrm{new}}(140, [\chi])\).