Newform orbit 88.4.g.b

• Label: 88.4.g.b
• Level: $88$
• Weight: $4$
• Character orbit: 88.g
• Analytic conductor: $5.192$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	88.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.19216808051\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{3} + 4 q^{4} + 120 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
43.1	−2.79244	−	0.449748i	2.78486			7.59545	+	2.51179i			12.2050i	−7.77656	−	1.25249i	0.410438			−20.0802	−	10.4301i	−19.2446			5.48915	−	34.0816i
43.2	−2.79244	+	0.449748i	2.78486			7.59545	−	2.51179i		−	12.2050i	−7.77656	+	1.25249i	0.410438			−20.0802	+	10.4301i	−19.2446			5.48915	+	34.0816i
43.3	−2.60314	−	1.10619i	−7.48903			5.55268	+	5.75915i		−	19.6372i	19.4950	+	8.28430i	−30.5072			−8.08367	−	21.1342i	29.0855			−21.7226	+	51.1185i
43.4	−2.60314	+	1.10619i	−7.48903			5.55268	−	5.75915i			19.6372i	19.4950	−	8.28430i	−30.5072			−8.08367	+	21.1342i	29.0855			−21.7226	−	51.1185i
43.5	−2.54722	−	1.22949i	−5.79901			4.97668	+	6.26359i			6.95743i	14.7714	+	7.12985i	18.5713			−4.97567	−	22.0736i	6.62848			8.55412	−	17.7221i
43.6	−2.54722	+	1.22949i	−5.79901			4.97668	−	6.26359i		−	6.95743i	14.7714	−	7.12985i	18.5713			−4.97567	+	22.0736i	6.62848			8.55412	+	17.7221i
43.7	−2.35973	−	1.55938i	8.46261			3.13667	+	7.35944i		−	10.4798i	−19.9695	−	13.1964i	9.39496			4.07446	−	22.2576i	44.6158			−16.3420	+	24.7295i
43.8	−2.35973	+	1.55938i	8.46261			3.13667	−	7.35944i			10.4798i	−19.9695	+	13.1964i	9.39496			4.07446	+	22.2576i	44.6158			−16.3420	−	24.7295i
43.9	−1.76049	−	2.21375i	2.74386			−1.80136	+	7.79456i			2.48060i	−4.83054	−	6.07422i	−31.7835			20.4264	−	9.73449i	−19.4712			5.49143	−	4.36708i
43.10	−1.76049	+	2.21375i	2.74386			−1.80136	−	7.79456i		−	2.48060i	−4.83054	+	6.07422i	−31.7835			20.4264	+	9.73449i	−19.4712			5.49143	+	4.36708i
43.11	−1.38546	−	2.46587i	−3.47705			−4.16099	+	6.83273i			2.75693i	4.81732	+	8.57393i	10.5613			22.6135	+	0.793945i	−14.9101			6.79822	−	3.81963i
43.12	−1.38546	+	2.46587i	−3.47705			−4.16099	−	6.83273i		−	2.75693i	4.81732	−	8.57393i	10.5613			22.6135	−	0.793945i	−14.9101			6.79822	+	3.81963i
43.13	−0.724448	−	2.73408i	7.53140			−6.95035	+	3.96139i			13.2419i	−5.45611	−	20.5914i	22.8914			15.8659	+	16.1330i	29.7220			36.2045	−	9.59310i
43.14	−0.724448	+	2.73408i	7.53140			−6.95035	−	3.96139i		−	13.2419i	−5.45611	+	20.5914i	22.8914			15.8659	−	16.1330i	29.7220			36.2045	+	9.59310i
43.15	−0.570618	−	2.77027i	−0.757662			−7.34879	+	3.16153i		−	18.5495i	0.432335	+	2.09893i	13.1999			12.9516	+	18.5541i	−26.4259			−51.3871	+	10.5847i
43.16	−0.570618	+	2.77027i	−0.757662			−7.34879	−	3.16153i			18.5495i	0.432335	−	2.09893i	13.1999			12.9516	−	18.5541i	−26.4259			−51.3871	−	10.5847i
43.17	0.570618	−	2.77027i	−0.757662			−7.34879	−	3.16153i			18.5495i	−0.432335	+	2.09893i	−13.1999			−12.9516	+	18.5541i	−26.4259			51.3871	+	10.5847i
43.18	0.570618	+	2.77027i	−0.757662			−7.34879	+	3.16153i		−	18.5495i	−0.432335	−	2.09893i	−13.1999			−12.9516	−	18.5541i	−26.4259			51.3871	−	10.5847i
43.19	0.724448	−	2.73408i	7.53140			−6.95035	−	3.96139i		−	13.2419i	5.45611	−	20.5914i	−22.8914			−15.8659	+	16.1330i	29.7220			−36.2045	−	9.59310i
43.20	0.724448	+	2.73408i	7.53140			−6.95035	+	3.96139i			13.2419i	5.45611	+	20.5914i	−22.8914			−15.8659	−	16.1330i	29.7220			−36.2045	+	9.59310i

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	88.4.g.b	✓	32
4.b	odd	2	1		352.4.g.b		32
8.b	even	2	1		352.4.g.b		32
8.d	odd	2	1	inner	88.4.g.b	✓	32
11.b	odd	2	1	inner	88.4.g.b	✓	32
44.c	even	2	1		352.4.g.b		32
88.b	odd	2	1		352.4.g.b		32
88.g	even	2	1	inner	88.4.g.b	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
88.4.g.b	✓	32	1.a	even	1	1	trivial
88.4.g.b	✓	32	8.d	odd	2	1	inner
88.4.g.b	✓	32	11.b	odd	2	1	inner
88.4.g.b	✓	32	88.g	even	2	1	inner
352.4.g.b		32	4.b	odd	2	1
352.4.g.b		32	8.b	even	2	1
352.4.g.b		32	44.c	even	2	1
352.4.g.b		32	88.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 4T_{3}^{7} - 115T_{3}^{6} + 342T_{3}^{5} + 3927T_{3}^{4} - 7512T_{3}^{3} - 35773T_{3}^{2} + 52318T_{3} + 55720 \) acting on \(S_{4}^{\mathrm{new}}(88, [\chi])\).