Embedded newform 403.2.ch.a.11.13

• Label: 403.2.ch.a.11.13
• 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			11.13
Character	\(\chi\)	\(=\)	403.11
Dual form			403.2.ch.a.110.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.754685 + 0.0395514i) q^{2} +(2.67759 + 0.870001i) q^{3} +(-1.42106 + 0.149359i) q^{4} +(-0.552551 + 2.06215i) q^{5} +(-2.05515 - 0.550675i) q^{6} +(-3.74165 + 1.43629i) q^{7} +(2.55938 - 0.405366i) q^{8} +(3.98552 + 2.89565i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q - 12 q^{2} - 10 q^{3} - 18 q^{4} - 8 q^{5} - 36 q^{6} - 32 q^{7} + 50 q^{8} + 114 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.754685	+	0.0395514i	−0.533643	+	0.0279670i	−0.317255	−	0.948340i	\(-0.602761\pi\)
							−0.216388	+	0.976307i	\(0.569428\pi\)
\(3\)	2.67759	+	0.870001i	1.54591	+	0.502295i	0.952998	−	0.302975i	\(-0.0979800\pi\)
							0.592907	+	0.805271i	\(0.297980\pi\)
\(5\)	−0.552551	+	2.06215i	−0.247108	+	0.922220i	0.725204	+	0.688534i	\(0.241746\pi\)
							−0.972312	+	0.233686i	\(0.924921\pi\)
\(7\)	−3.74165	+	1.43629i	−1.41421	+	0.542865i	−0.941097	−	0.338136i	\(-0.890204\pi\)
							−0.473114	+	0.881001i	\(0.656870\pi\)
\(11\)	2.84435	−	3.51248i	0.857603	−	1.05905i	−0.140020	−	0.990149i	\(-0.544717\pi\)
							0.997623	−	0.0689033i	\(-0.0219500\pi\)
\(13\)	−1.92095	+	3.05122i	−0.532776	+	0.846256i
							\(17\)	−0.483283	+	4.59813i	−0.117213	+	1.11521i	0.764892	+	0.644159i	\(0.222792\pi\)
−0.882105	+	0.471052i	\(0.843874\pi\)
\(19\)	−7.03895	+	0.368896i	−1.61484	+	0.0846304i	−0.838220	−	0.545333i	\(-0.816403\pi\)
							−0.776625	+	0.629963i	\(0.783070\pi\)
\(23\)	−0.512018	+	4.87153i	−0.106763	+	1.01578i	0.801674	+	0.597762i	\(0.203943\pi\)
							−0.908437	+	0.418022i	\(0.862724\pi\)
\(29\)	−0.108256	−	0.0974739i	−0.0201026	−	0.0181004i	0.659018	−	0.752127i	\(-0.270972\pi\)
							−0.679121	+	0.734027i	\(0.737639\pi\)
\(31\)	0.767662	−	5.51459i	0.137876	−	0.990449i
							\(37\)	6.70481	+	6.70481i	1.10226	+	1.10226i	0.994137	+	0.108127i	\(0.0344855\pi\)
0.108127	+	0.994137i	\(0.465515\pi\)
\(41\)	−0.765551	−	1.17885i	−0.119559	−	0.184105i	0.773824	−	0.633401i	\(-0.218342\pi\)
							−0.893383	+	0.449296i	\(0.851675\pi\)
\(43\)	4.51729	−	5.01696i	0.688881	−	0.765080i	−0.292684	−	0.956209i	\(-0.594548\pi\)
							0.981565	+	0.191129i	\(0.0612150\pi\)
\(47\)	−1.48972	−	2.92375i	−0.217299	−	0.426473i	0.756465	−	0.654034i	\(-0.226925\pi\)
							−0.973764	+	0.227561i	\(0.926925\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ch.a.11.13	yes	560
13.6	odd	12		403.2.cc.a.383.23	yes	560
31.17	odd	30		403.2.cc.a.141.23	✓	560
403.110	even	60	inner	403.2.ch.a.110.13	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.cc.a.141.23	✓	560	31.17	odd	30
403.2.cc.a.383.23	yes	560	13.6	odd	12
403.2.ch.a.11.13	yes	560	1.1	even	1	trivial
403.2.ch.a.110.13	yes	560	403.110	even	60	inner