Newform orbit 136.2.i.b

• Label: 136.2.i.b
• Level: $136$
• Weight: $2$
• Character orbit: 136.i
• Analytic conductor: $1.086$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	136.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.08596546749\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{4} - 4 q^{6} - 8 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	−1.40037	+	0.197414i	1.80863	−	1.80863i	1.92206	−	0.552904i	0.797964	−	0.797964i	−2.17570	+	2.88980i	−0.453490	+	0.453490i	−2.58243	+	1.15371i		−	3.54230i	−0.959914	+	1.27497i
13.2	−1.38301	−	0.295438i	−1.22371	+	1.22371i	1.82543	+	0.817188i	1.56683	−	1.56683i	2.05393	−	1.33087i	−2.62341	+	2.62341i	−2.28316	−	1.66948i			0.00507325i	−2.62985	+	1.70404i
13.3	−1.06549	+	0.929913i	0.504638	−	0.504638i	0.270523	−	1.98162i	−2.91725	+	2.91725i	−0.0684155	+	1.00695i	−1.08595	+	1.08595i	1.55450	+	2.36295i			2.49068i	0.395502	−	5.82108i
13.4	−0.668019	−	1.24650i	−0.600469	+	0.600469i	−1.10750	+	1.66536i	−1.35840	+	1.35840i	1.14961	+	0.347358i	−1.77843	+	1.77843i	2.81570	+	0.268003i			2.27887i	2.60069	+	0.785806i
13.5	−0.513367	+	1.31775i	0.739113	−	0.739113i	−1.47291	−	1.35297i	1.54400	−	1.54400i	0.594527	+	1.35340i	1.33694	−	1.33694i	2.53902	−	1.24635i			1.90743i	1.24196	+	2.82723i
13.6	−0.176636	−	1.40314i	−2.01739	+	2.01739i	−1.93760	+	0.495689i	2.48369	−	2.48369i	3.18703	+	2.47434i	2.60434	−	2.60434i	1.03777	+	2.63117i		−	5.13975i	−3.92368	−	3.04626i
13.7	0.176636	−	1.40314i	2.01739	−	2.01739i	−1.93760	−	0.495689i	−2.48369	+	2.48369i	−2.47434	−	3.18703i	2.60434	−	2.60434i	−1.03777	+	2.63117i		−	5.13975i	3.04626	+	3.92368i
13.8	0.513367	+	1.31775i	−0.739113	+	0.739113i	−1.47291	+	1.35297i	−1.54400	+	1.54400i	−1.35340	−	0.594527i	1.33694	−	1.33694i	−2.53902	−	1.24635i			1.90743i	−2.82723	−	1.24196i
13.9	0.668019	−	1.24650i	0.600469	−	0.600469i	−1.10750	−	1.66536i	1.35840	−	1.35840i	−0.347358	−	1.14961i	−1.77843	+	1.77843i	−2.81570	+	0.268003i			2.27887i	−0.785806	−	2.60069i
13.10	1.06549	+	0.929913i	−0.504638	+	0.504638i	0.270523	+	1.98162i	2.91725	−	2.91725i	−1.00695	+	0.0684155i	−1.08595	+	1.08595i	−1.55450	+	2.36295i			2.49068i	5.82108	−	0.395502i
13.11	1.38301	−	0.295438i	1.22371	−	1.22371i	1.82543	−	0.817188i	−1.56683	+	1.56683i	1.33087	−	2.05393i	−2.62341	+	2.62341i	2.28316	−	1.66948i			0.00507325i	−1.70404	+	2.62985i
13.12	1.40037	+	0.197414i	−1.80863	+	1.80863i	1.92206	+	0.552904i	−0.797964	+	0.797964i	−2.88980	+	2.17570i	−0.453490	+	0.453490i	2.58243	+	1.15371i		−	3.54230i	−1.27497	+	0.959914i
21.1	−1.40037	−	0.197414i	1.80863	+	1.80863i	1.92206	+	0.552904i	0.797964	+	0.797964i	−2.17570	−	2.88980i	−0.453490	−	0.453490i	−2.58243	−	1.15371i			3.54230i	−0.959914	−	1.27497i
21.2	−1.38301	+	0.295438i	−1.22371	−	1.22371i	1.82543	−	0.817188i	1.56683	+	1.56683i	2.05393	+	1.33087i	−2.62341	−	2.62341i	−2.28316	+	1.66948i		−	0.00507325i	−2.62985	−	1.70404i
21.3	−1.06549	−	0.929913i	0.504638	+	0.504638i	0.270523	+	1.98162i	−2.91725	−	2.91725i	−0.0684155	−	1.00695i	−1.08595	−	1.08595i	1.55450	−	2.36295i		−	2.49068i	0.395502	+	5.82108i
21.4	−0.668019	+	1.24650i	−0.600469	−	0.600469i	−1.10750	−	1.66536i	−1.35840	−	1.35840i	1.14961	−	0.347358i	−1.77843	−	1.77843i	2.81570	−	0.268003i		−	2.27887i	2.60069	−	0.785806i
21.5	−0.513367	−	1.31775i	0.739113	+	0.739113i	−1.47291	+	1.35297i	1.54400	+	1.54400i	0.594527	−	1.35340i	1.33694	+	1.33694i	2.53902	+	1.24635i		−	1.90743i	1.24196	−	2.82723i
21.6	−0.176636	+	1.40314i	−2.01739	−	2.01739i	−1.93760	−	0.495689i	2.48369	+	2.48369i	3.18703	−	2.47434i	2.60434	+	2.60434i	1.03777	−	2.63117i			5.13975i	−3.92368	+	3.04626i
21.7	0.176636	+	1.40314i	2.01739	+	2.01739i	−1.93760	+	0.495689i	−2.48369	−	2.48369i	−2.47434	+	3.18703i	2.60434	+	2.60434i	−1.03777	−	2.63117i			5.13975i	3.04626	−	3.92368i
21.8	0.513367	−	1.31775i	−0.739113	−	0.739113i	−1.47291	−	1.35297i	−1.54400	−	1.54400i	−1.35340	+	0.594527i	1.33694	+	1.33694i	−2.53902	+	1.24635i		−	1.90743i	−2.82723	+	1.24196i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
17.c	even	4	1	inner
136.i	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	136.2.i.b	✓	24
4.b	odd	2	1		544.2.m.b		24
8.b	even	2	1	inner	136.2.i.b	✓	24
8.d	odd	2	1		544.2.m.b		24
17.c	even	4	1	inner	136.2.i.b	✓	24
68.f	odd	4	1		544.2.m.b		24
136.i	even	4	1	inner	136.2.i.b	✓	24
136.j	odd	4	1		544.2.m.b		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
136.2.i.b	✓	24	1.a	even	1	1	trivial
136.2.i.b	✓	24	8.b	even	2	1	inner
136.2.i.b	✓	24	17.c	even	4	1	inner
136.2.i.b	✓	24	136.i	even	4	1	inner
544.2.m.b		24	4.b	odd	2	1
544.2.m.b		24	8.d	odd	2	1
544.2.m.b		24	68.f	odd	4	1
544.2.m.b		24	136.j	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 120T_{3}^{20} + 4048T_{3}^{16} + 33088T_{3}^{12} + 54272T_{3}^{8} + 27712T_{3}^{4} + 4096 \) acting on \(S_{2}^{\mathrm{new}}(136, [\chi])\).