Embedded newform 180.2.x.a.103.32

• Label: 180.2.x.a.103.32
• Level: $180$
• Weight: $2$
• Character: 180.103
• Analytic conductor: $1.437$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	180.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.43730723638\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(32\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			103.32
Character	\(\chi\)	\(=\)	180.103
Dual form			180.2.x.a.7.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41305 + 0.0573585i) q^{2} +(-1.34722 + 1.08857i) q^{3} +(1.99342 + 0.162101i) q^{4} +(-1.42527 + 1.72296i) q^{5} +(-1.96613 + 1.46093i) q^{6} +(-0.891731 + 3.32798i) q^{7} +(2.80750 + 0.343396i) q^{8} +(0.630018 - 2.93310i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 2 q^{2} - 4 q^{5} - 8 q^{6} - 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(91\)	\(101\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41305	+	0.0573585i	0.999177	+	0.0405586i
							\(3\)	−1.34722	+	1.08857i	−0.777819	+	0.628488i
\(5\)	−1.42527	+	1.72296i	−0.637401	+	0.770533i
							\(7\)	−0.891731	+	3.32798i	−0.337042	+	1.25786i	0.564595	+	0.825368i	\(0.309032\pi\)
−0.901638	+	0.432492i	\(0.857634\pi\)
\(11\)	4.34907	−	2.51094i	1.31129	−	0.757076i	0.328983	−	0.944336i	\(-0.393294\pi\)
							0.982310	+	0.187260i	\(0.0599606\pi\)
\(13\)	−3.14262	+	0.842062i	−0.871606	+	0.233546i	−0.666782	−	0.745253i	\(-0.732329\pi\)
							−0.204824	+	0.978799i	\(0.565662\pi\)
\(17\)	3.89332	−	3.89332i	0.944269	−	0.944269i	−0.0542578	−	0.998527i	\(-0.517279\pi\)
							0.998527	+	0.0542578i	\(0.0172793\pi\)
\(19\)	−0.885335			−0.203110			−0.101555	−	0.994830i	\(-0.532382\pi\)
							−0.101555	+	0.994830i	\(0.532382\pi\)
\(23\)	−0.497547	−	1.85687i	−0.103746	−	0.387184i	0.894454	−	0.447160i	\(-0.147564\pi\)
							−0.998200	+	0.0599755i	\(0.980898\pi\)
\(29\)	−2.84378	+	1.64186i	−0.528077	+	0.304885i	−0.740233	−	0.672351i	\(-0.765285\pi\)
							0.212156	+	0.977236i	\(0.431951\pi\)
\(31\)	0.0144367	+	0.00833505i	0.00259291	+	0.00149702i	0.501296	−	0.865276i	\(-0.332857\pi\)
							−0.498703	+	0.866773i	\(0.666190\pi\)
\(37\)	−0.467268	+	0.467268i	−0.0768184	+	0.0768184i	−0.744472	−	0.667654i	\(-0.767299\pi\)
							0.667654	+	0.744472i	\(0.267299\pi\)
\(41\)	1.73731	−	3.00910i	0.271322	−	0.469943i	−0.697879	−	0.716216i	\(-0.745873\pi\)
							0.969201	+	0.246273i	\(0.0792060\pi\)
\(43\)	1.19531	+	0.320283i	0.182283	+	0.0488426i	0.348806	−	0.937195i	\(-0.386587\pi\)
							−0.166523	+	0.986038i	\(0.553254\pi\)
\(47\)	2.43710	−	9.09536i	0.355487	−	1.32670i	−0.524384	−	0.851482i	\(-0.675704\pi\)
							0.879871	−	0.475213i	\(-0.157629\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.2.x.a.103.32	yes	128
3.2	odd	2		540.2.y.a.523.1		128
4.3	odd	2	inner	180.2.x.a.103.28	yes	128
5.2	odd	4	inner	180.2.x.a.67.17	yes	128
5.3	odd	4		900.2.bf.e.607.16		128
5.4	even	2		900.2.bf.e.643.1		128
9.2	odd	6		540.2.y.a.343.22		128
9.7	even	3	inner	180.2.x.a.43.11	yes	128
12.11	even	2		540.2.y.a.523.5		128
15.2	even	4		540.2.y.a.307.16		128
20.3	even	4		900.2.bf.e.607.22		128
20.7	even	4	inner	180.2.x.a.67.11	yes	128
20.19	odd	2		900.2.bf.e.643.5		128
36.7	odd	6	inner	180.2.x.a.43.17	yes	128
36.11	even	6		540.2.y.a.343.16		128
45.2	even	12		540.2.y.a.127.5		128
45.7	odd	12	inner	180.2.x.a.7.28	✓	128
45.34	even	6		900.2.bf.e.43.22		128
45.43	odd	12		900.2.bf.e.7.5		128
60.47	odd	4		540.2.y.a.307.22		128
180.7	even	12	inner	180.2.x.a.7.32	yes	128
180.43	even	12		900.2.bf.e.7.1		128
180.47	odd	12		540.2.y.a.127.1		128
180.79	odd	6		900.2.bf.e.43.16		128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.x.a.7.28	✓	128	45.7	odd	12	inner
180.2.x.a.7.32	yes	128	180.7	even	12	inner
180.2.x.a.43.11	yes	128	9.7	even	3	inner
180.2.x.a.43.17	yes	128	36.7	odd	6	inner
180.2.x.a.67.11	yes	128	20.7	even	4	inner
180.2.x.a.67.17	yes	128	5.2	odd	4	inner
180.2.x.a.103.28	yes	128	4.3	odd	2	inner
180.2.x.a.103.32	yes	128	1.1	even	1	trivial
540.2.y.a.127.1		128	180.47	odd	12
540.2.y.a.127.5		128	45.2	even	12
540.2.y.a.307.16		128	15.2	even	4
540.2.y.a.307.22		128	60.47	odd	4
540.2.y.a.343.16		128	36.11	even	6
540.2.y.a.343.22		128	9.2	odd	6
540.2.y.a.523.1		128	3.2	odd	2
540.2.y.a.523.5		128	12.11	even	2
900.2.bf.e.7.1		128	180.43	even	12
900.2.bf.e.7.5		128	45.43	odd	12
900.2.bf.e.43.16		128	180.79	odd	6
900.2.bf.e.43.22		128	45.34	even	6
900.2.bf.e.607.16		128	5.3	odd	4
900.2.bf.e.607.22		128	20.3	even	4
900.2.bf.e.643.1		128	5.4	even	2
900.2.bf.e.643.5		128	20.19	odd	2