Embedded newform 403.2.ch.a.11.3

• Label: 403.2.ch.a.11.3
• 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			11.3
Character	\(\chi\)	\(=\)	403.11
Dual form			403.2.ch.a.110.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.38428 + 0.124955i) q^{2} +(-0.874415 - 0.284115i) q^{3} +(3.68013 - 0.386797i) q^{4} +(0.207198 - 0.773272i) q^{5} +(2.12035 + 0.568146i) q^{6} +(-0.517083 + 0.198489i) q^{7} +(-4.00980 + 0.635090i) q^{8} +(-1.74317 - 1.26649i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q - 12 q^{2} - 10 q^{3} - 18 q^{4} - 8 q^{5} - 36 q^{6} - 32 q^{7} + 50 q^{8} + 114 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.38428	+	0.124955i	−1.68594	+	0.0883563i	−0.871029	−	0.491231i	\(-0.836547\pi\)
							−0.814910	+	0.579587i	\(0.803214\pi\)
\(3\)	−0.874415	−	0.284115i	−0.504844	−	0.164034i	0.0455122	−	0.998964i	\(-0.485508\pi\)
							−0.550356	+	0.834930i	\(0.685508\pi\)
\(5\)	0.207198	−	0.773272i	0.0926616	−	0.345818i	−0.903993	−	0.427547i	\(-0.859378\pi\)
							0.996655	+	0.0817297i	\(0.0260444\pi\)
\(7\)	−0.517083	+	0.198489i	−0.195439	+	0.0750220i	−0.454119	−	0.890941i	\(-0.650046\pi\)
							0.258680	+	0.965963i	\(0.416713\pi\)
\(11\)	2.54692	−	3.14519i	0.767927	−	0.948310i	−0.231774	−	0.972770i	\(-0.574453\pi\)
							0.999701	+	0.0244592i	\(0.00778639\pi\)
\(13\)	1.20575	+	3.39796i	0.334415	+	0.942426i
							\(17\)	0.139942	−	1.33146i	0.0339410	−	0.322927i	−0.964357	−	0.264605i	\(-0.914758\pi\)
0.998298	−	0.0583220i	\(-0.0185750\pi\)
\(19\)	−1.04434	+	0.0547316i	−0.239588	+	0.0125563i	−0.171752	−	0.985140i	\(-0.554943\pi\)
							−0.0678362	+	0.997696i	\(0.521610\pi\)
\(23\)	−0.215166	+	2.04717i	−0.0448652	+	0.426864i	0.948917	+	0.315525i	\(0.102181\pi\)
							−0.993783	+	0.111339i	\(0.964486\pi\)
\(29\)	−6.76445	−	6.09074i	−1.25613	−	1.13102i	−0.985741	−	0.168270i	\(-0.946182\pi\)
							−0.270386	−	0.962752i	\(-0.587151\pi\)
\(31\)	−3.84929	−	4.02281i	−0.691353	−	0.722518i
							\(37\)	−6.86854	−	6.86854i	−1.12918	−	1.12918i	−0.990310	−	0.138871i	\(-0.955653\pi\)
−0.138871	−	0.990310i	\(-0.544347\pi\)
\(41\)	−5.90533	−	9.09341i	−0.922258	−	1.42015i	−0.907402	−	0.420263i	\(-0.861938\pi\)
							−0.0148553	−	0.999890i	\(-0.504729\pi\)
\(43\)	−0.409780	+	0.455107i	−0.0624909	+	0.0694032i	−0.773580	−	0.633699i	\(-0.781536\pi\)
							0.711089	+	0.703102i	\(0.248202\pi\)
\(47\)	−3.43197	−	6.73563i	−0.500605	−	0.982493i	−0.993653	−	0.112490i	\(-0.964117\pi\)
							0.493048	−	0.870002i	\(-0.335883\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ch.a.11.3	yes	560
13.6	odd	12		403.2.cc.a.383.33	yes	560
31.17	odd	30		403.2.cc.a.141.33	✓	560
403.110	even	60	inner	403.2.ch.a.110.3	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.cc.a.141.33	✓	560	31.17	odd	30
403.2.cc.a.383.33	yes	560	13.6	odd	12
403.2.ch.a.11.3	yes	560	1.1	even	1	trivial
403.2.ch.a.110.3	yes	560	403.110	even	60	inner