Newform orbit 168.2.v.a

• Label: 168.2.v.a
• Level: $168$
• Weight: $2$
• Character orbit: 168.v
• Analytic conductor: $1.341$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	168.v (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34148675396\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.40790	+	0.133456i	−1.71959	−	0.207430i	1.96438	−	0.375786i	0.692094	+	1.19874i	2.44869	+	0.0625523i	−2.08863	−	1.62408i	−2.71550	+	0.791228i	2.91395	+	0.713387i	−1.13438	−	1.59535i
11.2	−1.40124	+	0.191140i	1.66017	−	0.493786i	1.92693	−	0.535666i	1.09212	+	1.89161i	−2.23191	+	1.00924i	−0.451847	+	2.60688i	−2.59770	+	1.11891i	2.51235	−	1.63954i	−1.89188	−	2.44184i
11.3	−1.32999	+	0.480762i	−0.316681	+	1.70285i	1.53774	−	1.27882i	−1.86088	−	3.22314i	−0.397486	−	2.41702i	−2.46429	+	0.962962i	−1.43036	+	2.44009i	−2.79943	−	1.07852i	4.02451	+	3.39209i
11.4	−1.32990	−	0.481016i	1.28463	+	1.16178i	1.53725	+	1.27940i	−0.646402	−	1.11960i	−1.14959	−	2.16297i	2.42742	−	1.05244i	−1.42896	−	2.44091i	0.300539	+	2.98491i	0.321101	+	1.79988i
11.5	−1.18883	−	0.765954i	0.728136	−	1.57157i	0.826627	+	1.82118i	−1.00081	−	1.73346i	−2.06938	+	1.31060i	−2.50986	−	0.837013i	0.412221	−	2.79823i	−1.93964	−	2.28863i	−0.137955	+	2.82736i
11.6	−1.08135	+	0.911422i	−0.316681	+	1.70285i	0.338619	−	1.97113i	1.86088	+	3.22314i	−1.20958	−	2.13001i	2.46429	−	0.962962i	1.43036	+	2.44009i	−2.79943	−	1.07852i	−4.94989	−	1.78928i
11.7	−0.982844	−	1.01687i	−0.987527	−	1.42295i	−0.0680365	+	1.99884i	1.77783	+	3.07929i	−0.476367	+	2.40272i	0.793456	+	2.52397i	2.09943	−	1.89537i	−1.04958	+	2.81041i	1.38390	−	4.83428i
11.8	−0.935706	−	1.06040i	−1.45135	+	0.945295i	−0.248908	+	1.98445i	−0.187787	−	0.325257i	2.36043	+	0.654497i	0.198565	−	2.63829i	2.33722	−	1.59292i	1.21283	−	2.74391i	−0.169190	+	0.503475i
11.9	−0.866151	+	1.11794i	1.66017	−	0.493786i	−0.499565	−	1.93660i	−1.09212	−	1.89161i	−0.885940	+	2.28366i	0.451847	−	2.60688i	2.59770	+	1.11891i	2.51235	−	1.63954i	3.06064	+	0.417497i
11.10	−0.819527	+	1.15255i	−1.71959	−	0.207430i	−0.656749	−	1.88910i	−0.692094	−	1.19874i	1.64832	−	1.81192i	2.08863	+	1.62408i	2.71550	+	0.791228i	2.91395	+	0.713387i	1.94880	+	0.184728i
11.11	−0.450483	−	1.34055i	−0.0929747	+	1.72955i	−1.59413	+	1.20779i	−0.187787	−	0.325257i	2.36043	−	0.654497i	−0.198565	+	2.63829i	2.33722	+	1.59292i	−2.98271	−	0.321609i	−0.351427	+	0.398260i
11.12	−0.389211	−	1.35960i	1.72608	+	0.143748i	−1.69703	+	1.05834i	1.77783	+	3.07929i	−0.476367	−	2.40272i	−0.793456	−	2.52397i	2.09943	+	1.89537i	2.95867	+	0.496241i	3.49466	−	3.61563i
11.13	−0.248376	+	1.39223i	1.28463	+	1.16178i	−1.87662	−	0.691593i	0.646402	+	1.11960i	−1.93654	+	1.49994i	−2.42742	+	1.05244i	1.42896	−	2.44091i	0.300539	+	2.98491i	−1.71929	+	0.621859i
11.14	−0.0689217	−	1.41253i	0.996948	−	1.41637i	−1.99050	+	0.194708i	−1.00081	−	1.73346i	−2.06938	−	1.31060i	2.50986	+	0.837013i	0.412221	+	2.79823i	−1.01219	−	2.82409i	−2.37959	+	1.53315i
11.15	0.0689217	+	1.41253i	0.728136	−	1.57157i	−1.99050	+	0.194708i	1.00081	+	1.73346i	2.27007	+	0.920201i	2.50986	+	0.837013i	−0.412221	−	2.79823i	−1.93964	−	2.28863i	−2.37959	+	1.53315i
11.16	0.248376	−	1.39223i	−1.64844	−	0.531631i	−1.87662	−	0.691593i	−0.646402	−	1.11960i	−1.14959	+	2.16297i	−2.42742	+	1.05244i	−1.42896	+	2.44091i	2.43474	+	1.75273i	−1.71929	+	0.621859i
11.17	0.389211	+	1.35960i	−0.987527	−	1.42295i	−1.69703	+	1.05834i	−1.77783	−	3.07929i	1.55029	−	1.89647i	−0.793456	−	2.52397i	−2.09943	−	1.89537i	−1.04958	+	2.81041i	3.49466	−	3.61563i
11.18	0.450483	+	1.34055i	−1.45135	+	0.945295i	−1.59413	+	1.20779i	0.187787	+	0.325257i	−1.92102	−	1.51976i	−0.198565	+	2.63829i	−2.33722	−	1.59292i	1.21283	−	2.74391i	−0.351427	+	0.398260i
11.19	0.819527	−	1.15255i	1.03943	+	1.38549i	−0.656749	−	1.88910i	0.692094	+	1.19874i	2.44869	−	0.0625523i	2.08863	+	1.62408i	−2.71550	−	0.791228i	−0.839162	+	2.88024i	1.94880	+	0.184728i
11.20	0.866151	−	1.11794i	−0.402456	−	1.68465i	−0.499565	−	1.93660i	1.09212	+	1.89161i	−2.23191	−	1.00924i	0.451847	−	2.60688i	−2.59770	−	1.11891i	−2.67606	+	1.35599i	3.06064	+	0.417497i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.c	even	3	1	inner
8.d	odd	2	1	inner
21.h	odd	6	1	inner
24.f	even	2	1	inner
56.k	odd	6	1	inner
168.v	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	168.2.v.a	✓	56
3.b	odd	2	1	inner	168.2.v.a	✓	56
4.b	odd	2	1		672.2.bd.a		56
7.c	even	3	1	inner	168.2.v.a	✓	56
8.b	even	2	1		672.2.bd.a		56
8.d	odd	2	1	inner	168.2.v.a	✓	56
12.b	even	2	1		672.2.bd.a		56
21.h	odd	6	1	inner	168.2.v.a	✓	56
24.f	even	2	1	inner	168.2.v.a	✓	56
24.h	odd	2	1		672.2.bd.a		56
28.g	odd	6	1		672.2.bd.a		56
56.k	odd	6	1	inner	168.2.v.a	✓	56
56.p	even	6	1		672.2.bd.a		56
84.n	even	6	1		672.2.bd.a		56
168.s	odd	6	1		672.2.bd.a		56
168.v	even	6	1	inner	168.2.v.a	✓	56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.2.v.a	✓	56	1.a	even	1	1	trivial
168.2.v.a	✓	56	3.b	odd	2	1	inner
168.2.v.a	✓	56	7.c	even	3	1	inner
168.2.v.a	✓	56	8.d	odd	2	1	inner
168.2.v.a	✓	56	21.h	odd	6	1	inner
168.2.v.a	✓	56	24.f	even	2	1	inner
168.2.v.a	✓	56	56.k	odd	6	1	inner
168.2.v.a	✓	56	168.v	even	6	1	inner
672.2.bd.a		56	4.b	odd	2	1
672.2.bd.a		56	8.b	even	2	1
672.2.bd.a		56	12.b	even	2	1
672.2.bd.a		56	24.h	odd	2	1
672.2.bd.a		56	28.g	odd	6	1
672.2.bd.a		56	56.p	even	6	1
672.2.bd.a		56	84.n	even	6	1
672.2.bd.a		56	168.s	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(168, [\chi])\).