Newform orbit 360.2.w.e

• Label: 360.2.w.e
• Level: $360$
• Weight: $2$
• Character orbit: 360.w
• Analytic conductor: $2.875$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	360.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.87461447277\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	no (minimal twist has level 120)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
163.1	−1.31581	+	0.518298i		0		1.46273	−	1.36397i	2.22965	+	0.169312i		0		0.645414	+	0.645414i	−1.21775	+	2.55286i		0		−3.02156	+	0.932839i
163.2	−1.25738	+	0.647304i		0		1.16200	−	1.62781i	−1.28903	−	1.82713i		0		−1.45533	−	1.45533i	−0.407381	+	2.79894i		0		2.80350	+	1.46300i
163.3	−1.08304	−	0.909406i		0		0.345961	+	1.96985i	−0.780766	+	2.09533i		0		−2.10796	−	2.10796i	1.41670	−	2.44805i		0		2.75111	−	1.55930i
163.4	−0.647304	+	1.25738i		0		−1.16200	−	1.62781i	1.28903	+	1.82713i		0		1.45533	+	1.45533i	2.79894	−	0.407381i		0		−3.13178	+	0.438090i
163.5	−0.624608	−	1.26880i		0		−1.21973	+	1.58501i	−2.11218	−	0.733965i		0		1.93078	+	1.93078i	2.77292	+	0.557590i		0		0.388025	+	3.13838i
163.6	−0.518298	+	1.31581i		0		−1.46273	−	1.36397i	−2.22965	−	0.169312i		0		−0.645414	−	0.645414i	2.55286	−	1.21775i		0		1.37841	−	2.84605i
163.7	−0.109339	−	1.40998i		0		−1.97609	+	0.308331i	0.0696909	−	2.23498i		0		−1.21782	−	1.21782i	0.650804	+	2.75254i		0		−3.15890	+	0.146107i
163.8	0.804501	−	1.16309i		0		−0.705556	−	1.87141i	−1.51371	+	1.64581i		0		−3.43671	−	3.43671i	−2.74424	−	0.684930i		0		0.696440	+	3.08463i
163.9	0.909406	+	1.08304i		0		−0.345961	+	1.96985i	0.780766	−	2.09533i		0		2.10796	+	2.10796i	−2.44805	+	1.41670i		0		2.97936	−	1.05990i
163.10	1.16309	−	0.804501i		0		0.705556	−	1.87141i	1.51371	−	1.64581i		0		3.43671	+	3.43671i	−0.684930	−	2.74424i		0		0.436527	−	3.13200i
163.11	1.26880	+	0.624608i		0		1.21973	+	1.58501i	2.11218	+	0.733965i		0		−1.93078	−	1.93078i	0.557590	+	2.77292i		0		2.22150	+	2.25054i
163.12	1.40998	+	0.109339i		0		1.97609	+	0.308331i	−0.0696909	+	2.23498i		0		1.21782	+	1.21782i	2.75254	+	0.650804i		0		−0.342633	+	3.14366i
307.1	−1.31581	−	0.518298i		0		1.46273	+	1.36397i	2.22965	−	0.169312i		0		0.645414	−	0.645414i	−1.21775	−	2.55286i		0		−3.02156	−	0.932839i
307.2	−1.25738	−	0.647304i		0		1.16200	+	1.62781i	−1.28903	+	1.82713i		0		−1.45533	+	1.45533i	−0.407381	−	2.79894i		0		2.80350	−	1.46300i
307.3	−1.08304	+	0.909406i		0		0.345961	−	1.96985i	−0.780766	−	2.09533i		0		−2.10796	+	2.10796i	1.41670	+	2.44805i		0		2.75111	+	1.55930i
307.4	−0.647304	−	1.25738i		0		−1.16200	+	1.62781i	1.28903	−	1.82713i		0		1.45533	−	1.45533i	2.79894	+	0.407381i		0		−3.13178	−	0.438090i
307.5	−0.624608	+	1.26880i		0		−1.21973	−	1.58501i	−2.11218	+	0.733965i		0		1.93078	−	1.93078i	2.77292	−	0.557590i		0		0.388025	−	3.13838i
307.6	−0.518298	−	1.31581i		0		−1.46273	+	1.36397i	−2.22965	+	0.169312i		0		−0.645414	+	0.645414i	2.55286	+	1.21775i		0		1.37841	+	2.84605i
307.7	−0.109339	+	1.40998i		0		−1.97609	−	0.308331i	0.0696909	+	2.23498i		0		−1.21782	+	1.21782i	0.650804	−	2.75254i		0		−3.15890	−	0.146107i
307.8	0.804501	+	1.16309i		0		−0.705556	+	1.87141i	−1.51371	−	1.64581i		0		−3.43671	+	3.43671i	−2.74424	+	0.684930i		0		0.696440	−	3.08463i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
8.d	odd	2	1	inner
40.k	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	360.2.w.e		24
3.b	odd	2	1		120.2.v.a	✓	24
4.b	odd	2	1		1440.2.bi.e		24
5.c	odd	4	1	inner	360.2.w.e		24
8.b	even	2	1		1440.2.bi.e		24
8.d	odd	2	1	inner	360.2.w.e		24
12.b	even	2	1		480.2.bh.a		24
15.d	odd	2	1		600.2.v.b		24
15.e	even	4	1		120.2.v.a	✓	24
15.e	even	4	1		600.2.v.b		24
20.e	even	4	1		1440.2.bi.e		24
24.f	even	2	1		120.2.v.a	✓	24
24.h	odd	2	1		480.2.bh.a		24
40.i	odd	4	1		1440.2.bi.e		24
40.k	even	4	1	inner	360.2.w.e		24
60.h	even	2	1		2400.2.bh.b		24
60.l	odd	4	1		480.2.bh.a		24
60.l	odd	4	1		2400.2.bh.b		24
120.i	odd	2	1		2400.2.bh.b		24
120.m	even	2	1		600.2.v.b		24
120.q	odd	4	1		120.2.v.a	✓	24
120.q	odd	4	1		600.2.v.b		24
120.w	even	4	1		480.2.bh.a		24
120.w	even	4	1		2400.2.bh.b		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.2.v.a	✓	24	3.b	odd	2	1
120.2.v.a	✓	24	15.e	even	4	1
120.2.v.a	✓	24	24.f	even	2	1
120.2.v.a	✓	24	120.q	odd	4	1
360.2.w.e		24	1.a	even	1	1	trivial
360.2.w.e		24	5.c	odd	4	1	inner
360.2.w.e		24	8.d	odd	2	1	inner
360.2.w.e		24	40.k	even	4	1	inner
480.2.bh.a		24	12.b	even	2	1
480.2.bh.a		24	24.h	odd	2	1
480.2.bh.a		24	60.l	odd	4	1
480.2.bh.a		24	120.w	even	4	1
600.2.v.b		24	15.d	odd	2	1
600.2.v.b		24	15.e	even	4	1
600.2.v.b		24	120.m	even	2	1
600.2.v.b		24	120.q	odd	4	1
1440.2.bi.e		24	4.b	odd	2	1
1440.2.bi.e		24	8.b	even	2	1
1440.2.bi.e		24	20.e	even	4	1
1440.2.bi.e		24	40.i	odd	4	1
2400.2.bh.b		24	60.h	even	2	1
2400.2.bh.b		24	60.l	odd	4	1
2400.2.bh.b		24	120.i	odd	2	1
2400.2.bh.b		24	120.w	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 720T_{7}^{20} + 98656T_{7}^{16} + 4752640T_{7}^{12} + 81309952T_{7}^{8} + 440926208T_{7}^{4} + 268435456 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).