Embedded newform 403.2.cc.a.383.17

• Label: 403.2.cc.a.383.17
• Level: $403$
• Weight: $2$
• Character: 403.383
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $560$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.cc (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(560\)
Relative dimension:	\(35\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			383.17
Character	\(\chi\)	\(=\)	403.383
Dual form			403.2.cc.a.141.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.104943 - 0.0681505i) q^{2} +(-0.195678 - 0.920593i) q^{3} +(-0.807105 + 1.81279i) q^{4} +(-0.806301 - 0.216048i) q^{5} +(-0.0832739 - 0.0832739i) q^{6} +(-1.51385 - 0.239771i) q^{7} +(0.0779919 + 0.492421i) q^{8} +(1.93143 - 0.859930i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q - 12 q^{2} - 4 q^{3} - 18 q^{4} - 8 q^{5} - 42 q^{6} - 8 q^{7} - 40 q^{8} - 72 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{23}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.104943	−	0.0681505i	0.0742056	−	0.0481897i	−0.507003	−	0.861944i	\(-0.669247\pi\)
							0.581209	+	0.813755i	\(0.302580\pi\)
\(3\)	−0.195678	−	0.920593i	−0.112975	−	0.531505i	−0.997842	−	0.0656613i	\(-0.979084\pi\)
							0.884867	−	0.465843i	\(-0.154249\pi\)
\(5\)	−0.806301	−	0.216048i	−0.360589	−	0.0966195i	0.0739761	−	0.997260i	\(-0.476431\pi\)
							−0.434565	+	0.900641i	\(0.643098\pi\)
\(7\)	−1.51385	−	0.239771i	−0.572182	−	0.0906247i	−0.136363	−	0.990659i	\(-0.543541\pi\)
							−0.435819	+	0.900034i	\(0.643541\pi\)
\(11\)	0.941302	−	5.94314i	0.283813	−	1.79193i	−0.273760	−	0.961798i	\(-0.588268\pi\)
							0.557573	−	0.830128i	\(-0.311732\pi\)
\(13\)	1.96438	−	3.02345i	0.544820	−	0.838553i
							\(17\)	4.47874	+	3.25399i	1.08625	+	0.789210i	0.978763	−	0.204996i	\(-0.0657183\pi\)
0.107491	+	0.994206i	\(0.465718\pi\)
\(19\)	5.56828	+	2.83718i	1.27745	+	0.650893i	0.955256	−	0.295780i	\(-0.0955796\pi\)
							0.322194	+	0.946674i	\(0.395580\pi\)
\(23\)	5.60053	−	2.49352i	1.16779	−	0.519934i	0.271083	−	0.962556i	\(-0.412618\pi\)
							0.896708	+	0.442622i	\(0.145952\pi\)
\(29\)	−1.15167	+	5.41818i	−0.213860	+	1.00613i	0.731937	+	0.681372i	\(0.238616\pi\)
							−0.945797	+	0.324759i	\(0.894717\pi\)
\(31\)	0.155875	−	5.56558i	0.0279959	−	0.999608i
							\(37\)	−7.56923	−	2.02817i	−1.24437	−	0.333429i	−0.424213	−	0.905562i	\(-0.639449\pi\)
−0.820161	+	0.572133i	\(0.806116\pi\)
\(41\)	−5.02110	−	2.55838i	−0.784164	−	0.399552i	0.0155906	−	0.999878i	\(-0.495037\pi\)
							−0.799755	+	0.600327i	\(0.795037\pi\)
\(43\)	1.93087	+	5.94262i	0.294455	+	0.906240i	0.983404	+	0.181430i	\(0.0580726\pi\)
							−0.688948	+	0.724810i	\(0.741927\pi\)
\(47\)	0.781558	−	0.398223i	0.114002	−	0.0580869i	−0.396060	−	0.918225i	\(-0.629623\pi\)
							0.510061	+	0.860138i	\(0.329623\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.cc.a.383.17	yes	560
13.11	odd	12		403.2.ch.a.11.19	yes	560
31.17	odd	30		403.2.ch.a.110.19	yes	560
403.141	even	60	inner	403.2.cc.a.141.17	✓	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.cc.a.141.17	✓	560	403.141	even	60	inner
403.2.cc.a.383.17	yes	560	1.1	even	1	trivial
403.2.ch.a.11.19	yes	560	13.11	odd	12
403.2.ch.a.110.19	yes	560	31.17	odd	30