Newform orbit 968.2.g.d

• Label: 968.2.g.d
• Level: $968$
• Weight: $2$
• Character orbit: 968.g
• Analytic conductor: $7.730$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.72951891566\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{3} + 8 q^{4} + 32 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
483.1	−1.40516	−	0.159730i	−3.05739			1.94897	+	0.448895i			3.03830i	4.29614	+	0.488358i	−2.28495			−2.66692	−	0.942081i	6.34764			0.485308	−	4.26931i
483.2	−1.40516	+	0.159730i	−3.05739			1.94897	−	0.448895i		−	3.03830i	4.29614	−	0.488358i	−2.28495			−2.66692	+	0.942081i	6.34764			0.485308	+	4.26931i
483.3	−1.39746	−	0.217060i	1.39971			1.90577	+	0.606665i			0.881650i	−1.95603	−	0.303821i	−0.115822			−2.53155	−	1.26146i	−1.04082			0.191371	−	1.23207i
483.4	−1.39746	+	0.217060i	1.39971			1.90577	−	0.606665i		−	0.881650i	−1.95603	+	0.303821i	−0.115822			−2.53155	+	1.26146i	−1.04082			0.191371	+	1.23207i
483.5	−1.26783	−	0.626593i	2.62943			1.21476	+	1.58882i		−	0.539739i	−3.33366	−	1.64758i	−2.18116			−0.544561	−	2.77551i	3.91392			−0.338197	+	0.684295i
483.6	−1.26783	+	0.626593i	2.62943			1.21476	−	1.58882i			0.539739i	−3.33366	+	1.64758i	−2.18116			−0.544561	+	2.77551i	3.91392			−0.338197	−	0.684295i
483.7	−1.23201	−	0.694372i	−0.544659			1.03569	+	1.71095i			3.64098i	0.671025	+	0.378196i	−1.26159			−0.0879513	−	2.82706i	−2.70335			2.52820	−	4.48573i
483.8	−1.23201	+	0.694372i	−0.544659			1.03569	−	1.71095i		−	3.64098i	0.671025	−	0.378196i	−1.26159			−0.0879513	+	2.82706i	−2.70335			2.52820	+	4.48573i
483.9	−1.09124	−	0.899549i	−3.01105			0.381623	+	1.96325i			1.57847i	3.28579	+	2.70859i	4.30013			1.34960	−	2.48568i	6.06643			1.41991	−	1.72250i
483.10	−1.09124	+	0.899549i	−3.01105			0.381623	−	1.96325i		−	1.57847i	3.28579	−	2.70859i	4.30013			1.34960	+	2.48568i	6.06643			1.41991	+	1.72250i
483.11	−0.623112	−	1.26954i	−1.22454			−1.22346	+	1.58213i		−	2.54666i	0.763027	+	1.55461i	−0.763616			2.77093	+	0.567393i	−1.50049			−3.23309	+	1.58685i
483.12	−0.623112	+	1.26954i	−1.22454			−1.22346	−	1.58213i			2.54666i	0.763027	−	1.55461i	−0.763616			2.77093	−	0.567393i	−1.50049			−3.23309	−	1.58685i
483.13	−0.516053	−	1.31670i	1.70467			−1.46738	+	1.35897i		−	2.47969i	−0.879700	−	2.24453i	−3.93722			2.54660	+	1.23079i	−0.0941080			−3.26500	+	1.27965i
483.14	−0.516053	+	1.31670i	1.70467			−1.46738	−	1.35897i			2.47969i	−0.879700	+	2.24453i	−3.93722			2.54660	−	1.23079i	−0.0941080			−3.26500	−	1.27965i
483.15	−0.319390	−	1.37768i	0.103837			−1.79598	+	0.880033i			2.30596i	−0.0331646	−	0.143054i	−4.67339			1.78602	+	2.19320i	−2.98922			3.17686	−	0.736501i
483.16	−0.319390	+	1.37768i	0.103837			−1.79598	−	0.880033i		−	2.30596i	−0.0331646	+	0.143054i	−4.67339			1.78602	−	2.19320i	−2.98922			3.17686	+	0.736501i
483.17	0.319390	−	1.37768i	0.103837			−1.79598	−	0.880033i		−	2.30596i	0.0331646	−	0.143054i	4.67339			−1.78602	+	2.19320i	−2.98922			−3.17686	−	0.736501i
483.18	0.319390	+	1.37768i	0.103837			−1.79598	+	0.880033i			2.30596i	0.0331646	+	0.143054i	4.67339			−1.78602	−	2.19320i	−2.98922			−3.17686	+	0.736501i
483.19	0.516053	−	1.31670i	1.70467			−1.46738	−	1.35897i			2.47969i	0.879700	−	2.24453i	3.93722			−2.54660	+	1.23079i	−0.0941080			3.26500	+	1.27965i
483.20	0.516053	+	1.31670i	1.70467			−1.46738	+	1.35897i		−	2.47969i	0.879700	+	2.24453i	3.93722			−2.54660	−	1.23079i	−0.0941080			3.26500	−	1.27965i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
11.b	odd	2	1	inner
88.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	968.2.g.d	✓	32
4.b	odd	2	1		3872.2.g.e		32
8.b	even	2	1		3872.2.g.e		32
8.d	odd	2	1	inner	968.2.g.d	✓	32
11.b	odd	2	1	inner	968.2.g.d	✓	32
11.c	even	5	4		968.2.k.k		128
11.d	odd	10	4		968.2.k.k		128
44.c	even	2	1		3872.2.g.e		32
88.b	odd	2	1		3872.2.g.e		32
88.g	even	2	1	inner	968.2.g.d	✓	32
88.k	even	10	4		968.2.k.k		128
88.l	odd	10	4		968.2.k.k		128

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
968.2.g.d	✓	32	1.a	even	1	1	trivial
968.2.g.d	✓	32	8.d	odd	2	1	inner
968.2.g.d	✓	32	11.b	odd	2	1	inner
968.2.g.d	✓	32	88.g	even	2	1	inner
968.2.k.k		128	11.c	even	5	4
968.2.k.k		128	11.d	odd	10	4
968.2.k.k		128	88.k	even	10	4
968.2.k.k		128	88.l	odd	10	4
3872.2.g.e		32	4.b	odd	2	1
3872.2.g.e		32	8.b	even	2	1
3872.2.g.e		32	44.c	even	2	1
3872.2.g.e		32	88.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 2T_{3}^{7} - 14T_{3}^{6} - 20T_{3}^{5} + 60T_{3}^{4} + 44T_{3}^{3} - 68T_{3}^{2} - 32T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\).