Embedded newform 403.2.ch.a.11.9

• Label: 403.2.ch.a.11.9
• Level: $403$
• Weight: $2$
• Character: 403.11
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $560$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.ch (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			11.9
Character	\(\chi\)	\(=\)	403.11
Dual form			403.2.ch.a.110.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.62361 + 0.0850899i) q^{2} +(1.10108 + 0.357762i) q^{3} +(0.639831 - 0.0672490i) q^{4} +(-0.314520 + 1.17380i) q^{5} +(-1.81817 - 0.487176i) q^{6} +(-2.76717 + 1.06222i) q^{7} +(2.17853 - 0.345045i) q^{8} +(-1.34267 - 0.975508i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q - 12 q^{2} - 10 q^{3} - 18 q^{4} - 8 q^{5} - 36 q^{6} - 32 q^{7} + 50 q^{8} + 114 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{23}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.62361	+	0.0850899i	−1.14807	+	0.0601676i	−0.616859	−	0.787074i	\(-0.711595\pi\)
							−0.531208	+	0.847241i	\(0.678262\pi\)
\(3\)	1.10108	+	0.357762i	0.635708	+	0.206554i	0.609102	−	0.793092i	\(-0.291530\pi\)
							0.0266060	+	0.999646i	\(0.491530\pi\)
\(5\)	−0.314520	+	1.17380i	−0.140658	+	0.524941i	0.859253	+	0.511551i	\(0.170929\pi\)
							−0.999910	+	0.0133899i	\(0.995738\pi\)
\(7\)	−2.76717	+	1.06222i	−1.04589	+	0.401480i	−0.819814	−	0.572630i	\(-0.805923\pi\)
							−0.226077	+	0.974110i	\(0.572590\pi\)
\(11\)	−1.75835	+	2.17138i	−0.530163	+	0.654697i	−0.969921	−	0.243419i	\(-0.921731\pi\)
							0.439758	+	0.898116i	\(0.355064\pi\)
\(13\)	−0.759635	−	3.52462i	−0.210685	−	0.977554i
							\(17\)	0.588777	−	5.60184i	0.142799	−	1.35865i	−0.654957	−	0.755666i	\(-0.727313\pi\)
0.797757	−	0.602979i	\(-0.206020\pi\)
\(19\)	−2.87595	+	0.150722i	−0.659788	+	0.0345780i	−0.379292	−	0.925277i	\(-0.623832\pi\)
							−0.280496	+	0.959855i	\(0.590499\pi\)
\(23\)	−0.116084	+	1.10446i	−0.0242051	+	0.230296i	0.975731	+	0.218974i	\(0.0702708\pi\)
							−0.999936	+	0.0113226i	\(0.996396\pi\)
\(29\)	−6.53878	−	5.88754i	−1.21422	−	1.09329i	−0.993006	−	0.118064i	\(-0.962331\pi\)
							−0.221215	−	0.975225i	\(-0.571002\pi\)
\(31\)	3.81791	+	4.05260i	0.685718	+	0.727868i
							\(37\)	−1.38504	−	1.38504i	−0.227700	−	0.227700i	0.584031	−	0.811731i	\(-0.301475\pi\)
−0.811731	+	0.584031i	\(0.801475\pi\)
\(41\)	−3.80972	−	5.86645i	−0.594978	−	0.916185i	−0.999998	−	0.00187759i	\(-0.999402\pi\)
							0.405021	−	0.914308i	\(-0.367264\pi\)
\(43\)	−6.38750	+	7.09404i	−0.974085	+	1.08183i	0.0225404	+	0.999746i	\(0.492825\pi\)
							−0.996625	+	0.0820850i	\(0.973842\pi\)
\(47\)	−1.76652	−	3.46699i	−0.257673	−	0.505712i	0.725538	−	0.688182i	\(-0.241591\pi\)
							−0.983211	+	0.182470i	\(0.941591\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ch.a.11.9	yes	560
13.6	odd	12		403.2.cc.a.383.27	yes	560
31.17	odd	30		403.2.cc.a.141.27	✓	560
403.110	even	60	inner	403.2.ch.a.110.9	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.cc.a.141.27	✓	560	31.17	odd	30
403.2.cc.a.383.27	yes	560	13.6	odd	12
403.2.ch.a.11.9	yes	560	1.1	even	1	trivial
403.2.ch.a.110.9	yes	560	403.110	even	60	inner