Newform orbit 180.2.n.d

• Label: 180.2.n.d
• Level: $180$
• Weight: $2$
• Character orbit: 180.n
• Analytic conductor: $1.437$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	180.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.43730723638\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{4} - 18 q^{5} - 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−1.41350	+	0.0449209i	1.62879	+	0.589100i	1.99596	−	0.126991i	−1.91817	+	1.14918i	−2.32876	−	0.759526i	1.44473	+	2.50234i	−2.81559	+	0.269163i	2.30592	+	1.91904i	2.65972	−	1.71053i
59.2	−1.38574	+	0.282343i	−0.959839	−	1.44177i	1.84056	−	0.782509i	0.520093	−	2.17474i	1.73716	+	1.72692i	0.550457	+	0.953419i	−2.32961	+	1.60403i	−1.15742	+	2.76774i	−0.106691	+	3.16048i
59.3	−1.38444	−	0.288666i	0.112475	+	1.72840i	1.83334	+	0.799280i	2.23134	−	0.145382i	0.343213	−	2.42533i	0.573658	+	0.993605i	−2.30743	−	1.63578i	−2.97470	+	0.388803i	−3.13112	−	0.442837i
59.4	−1.25828	−	0.645549i	0.424796	−	1.67915i	1.16653	+	1.62456i	−1.98302	+	1.03326i	−1.61849	+	1.83861i	−1.60868	−	2.78632i	−0.419091	−	2.79721i	−2.63910	−	1.42659i	3.16221	−	0.0199985i
59.5	−1.24631	−	0.668359i	−1.66946	+	0.461412i	1.10659	+	1.66597i	−1.69235	−	1.46149i	2.38906	+	0.540736i	0.667441	+	1.15604i	−0.265693	−	2.81592i	2.57420	−	1.54062i	1.13240	+	2.95257i
59.6	−1.00727	−	0.992675i	1.30213	−	1.14213i	0.0291929	+	1.99979i	1.64183	−	1.51803i	−2.44536	−	0.142161i	1.65751	+	2.87089i	1.95573	−	2.04331i	0.391092	−	2.97440i	−3.16067	−	0.100737i
59.7	−0.937387	+	1.05892i	−0.959839	−	1.44177i	−0.242609	−	1.98523i	−1.62334	+	1.53778i	2.42646	+	0.335110i	0.550457	+	0.953419i	2.32961	+	1.60403i	−1.15742	+	2.76774i	−0.106691	−	3.16048i
59.8	−0.745653	+	1.20167i	1.62879	+	0.589100i	−0.888004	−	1.79205i	0.0361315	−	2.23578i	−1.92241	+	1.51800i	1.44473	+	2.50234i	2.81559	+	0.269163i	2.30592	+	1.91904i	2.65972	+	1.71053i
59.9	−0.442228	+	1.34329i	0.112475	+	1.72840i	−1.60887	−	1.18808i	0.989764	+	2.00509i	−2.37148	−	0.613258i	0.573658	+	0.993605i	2.30743	−	1.63578i	−2.97470	+	0.388803i	−3.13112	+	0.442837i
59.10	−0.356046	−	1.36866i	−1.30213	+	1.14213i	−1.74646	+	0.974612i	1.64183	−	1.51803i	2.02680	+	1.37553i	−1.65751	−	2.87089i	1.95573	+	2.04331i	0.391092	−	2.97440i	−2.66223	−	1.70662i
59.11	−0.0700780	+	1.41248i	0.424796	−	1.67915i	−1.99018	−	0.197967i	−0.0966770	−	2.23398i	2.34199	+	0.717686i	−1.60868	−	2.78632i	0.419091	−	2.79721i	−2.63910	−	1.42659i	3.16221	+	0.0199985i
59.12	−0.0443404	+	1.41352i	−1.66946	+	0.461412i	−1.99607	−	0.125352i	−2.11186	−	0.734875i	−0.578190	−	2.38027i	0.667441	+	1.15604i	0.265693	−	2.81592i	2.57420	−	1.54062i	1.13240	−	2.95257i
59.13	0.0443404	−	1.41352i	1.66946	−	0.461412i	−1.99607	−	0.125352i	−1.69235	−	1.46149i	−0.578190	−	2.38027i	−0.667441	−	1.15604i	−0.265693	+	2.81592i	2.57420	−	1.54062i	−2.14088	+	2.32737i
59.14	0.0700780	−	1.41248i	−0.424796	+	1.67915i	−1.99018	−	0.197967i	−1.98302	+	1.03326i	2.34199	+	0.717686i	1.60868	+	2.78632i	−0.419091	+	2.79721i	−2.63910	−	1.42659i	1.32049	+	2.87338i
59.15	0.356046	+	1.36866i	1.30213	−	1.14213i	−1.74646	+	0.974612i	−0.493735	+	2.18088i	2.02680	+	1.37553i	1.65751	+	2.87089i	−1.95573	−	2.04331i	0.391092	−	2.97440i	−3.16067	+	0.100737i
59.16	0.442228	−	1.34329i	−0.112475	−	1.72840i	−1.60887	−	1.18808i	2.23134	−	0.145382i	−2.37148	−	0.613258i	−0.573658	−	0.993605i	−2.30743	+	1.63578i	−2.97470	+	0.388803i	0.791468	−	3.06163i
59.17	0.745653	−	1.20167i	−1.62879	−	0.589100i	−0.888004	−	1.79205i	−1.91817	+	1.14918i	−1.92241	+	1.51800i	−1.44473	−	2.50234i	−2.81559	−	0.269163i	2.30592	+	1.91904i	−0.0493612	+	3.16189i
59.18	0.937387	−	1.05892i	0.959839	+	1.44177i	−0.242609	−	1.98523i	0.520093	−	2.17474i	2.42646	+	0.335110i	−0.550457	−	0.953419i	−2.32961	−	1.60403i	−1.15742	+	2.76774i	−1.81534	−	2.58931i
59.19	1.00727	+	0.992675i	−1.30213	+	1.14213i	0.0291929	+	1.99979i	−0.493735	+	2.18088i	−2.44536	−	0.142161i	−1.65751	−	2.87089i	−1.95573	+	2.04331i	0.391092	−	2.97440i	−2.66223	+	1.70662i
59.20	1.24631	+	0.668359i	1.66946	−	0.461412i	1.10659	+	1.66597i	−2.11186	−	0.734875i	2.38906	+	0.540736i	−0.667441	−	1.15604i	0.265693	+	2.81592i	2.57420	−	1.54062i	−2.14088	−	2.32737i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
9.d	odd	6	1	inner
20.d	odd	2	1	inner
36.h	even	6	1	inner
45.h	odd	6	1	inner
180.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	180.2.n.d	✓	48
3.b	odd	2	1		540.2.n.d		48
4.b	odd	2	1	inner	180.2.n.d	✓	48
5.b	even	2	1	inner	180.2.n.d	✓	48
5.c	odd	4	2		900.2.r.g		48
9.c	even	3	1		540.2.n.d		48
9.d	odd	6	1	inner	180.2.n.d	✓	48
12.b	even	2	1		540.2.n.d		48
15.d	odd	2	1		540.2.n.d		48
20.d	odd	2	1	inner	180.2.n.d	✓	48
20.e	even	4	2		900.2.r.g		48
36.f	odd	6	1		540.2.n.d		48
36.h	even	6	1	inner	180.2.n.d	✓	48
45.h	odd	6	1	inner	180.2.n.d	✓	48
45.j	even	6	1		540.2.n.d		48
45.l	even	12	2		900.2.r.g		48
60.h	even	2	1		540.2.n.d		48
180.n	even	6	1	inner	180.2.n.d	✓	48
180.p	odd	6	1		540.2.n.d		48
180.v	odd	12	2		900.2.r.g		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
180.2.n.d	✓	48	1.a	even	1	1	trivial
180.2.n.d	✓	48	4.b	odd	2	1	inner
180.2.n.d	✓	48	5.b	even	2	1	inner
180.2.n.d	✓	48	9.d	odd	6	1	inner
180.2.n.d	✓	48	20.d	odd	2	1	inner
180.2.n.d	✓	48	36.h	even	6	1	inner
180.2.n.d	✓	48	45.h	odd	6	1	inner
180.2.n.d	✓	48	180.n	even	6	1	inner
540.2.n.d		48	3.b	odd	2	1
540.2.n.d		48	9.c	even	3	1
540.2.n.d		48	12.b	even	2	1
540.2.n.d		48	15.d	odd	2	1
540.2.n.d		48	36.f	odd	6	1
540.2.n.d		48	45.j	even	6	1
540.2.n.d		48	60.h	even	2	1
540.2.n.d		48	180.p	odd	6	1
900.2.r.g		48	5.c	odd	4	2
900.2.r.g		48	20.e	even	4	2
900.2.r.g		48	45.l	even	12	2
900.2.r.g		48	180.v	odd	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^24 + 34*T7^22 + 730*T7^20 + 9700*T7^18 + 94189*T7^16 + 620404*T7^14 + 2963314*T7^12 + 8702080*T7^10 + 18514873*T7^8 + 26555616*T7^6 + 27788076*T7^4 + 17884800*T7^2 + 7290000