Embedded newform 403.2.ch.a.11.17

• Label: 403.2.ch.a.11.17
• 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			11.17
Character	\(\chi\)	\(=\)	403.11
Dual form			403.2.ch.a.110.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.268345 + 0.0140634i) q^{2} +(-1.39144 - 0.452107i) q^{3} +(-1.91723 + 0.201509i) q^{4} +(-1.07119 + 3.99775i) q^{5} +(0.379744 + 0.101752i) q^{6} +(-1.20967 + 0.464347i) q^{7} +(1.04246 - 0.165109i) q^{8} +(-0.695342 - 0.505196i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q - 12 q^{2} - 10 q^{3} - 18 q^{4} - 8 q^{5} - 36 q^{6} - 32 q^{7} + 50 q^{8} + 114 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.268345	+	0.0140634i	−0.189748	+	0.00994429i	−0.146973	−	0.989140i	\(-0.546953\pi\)
							−0.0427750	+	0.999085i	\(0.513620\pi\)
\(3\)	−1.39144	−	0.452107i	−0.803349	−	0.261024i	−0.121571	−	0.992583i	\(-0.538793\pi\)
							−0.681778	+	0.731559i	\(0.738793\pi\)
\(5\)	−1.07119	+	3.99775i	−0.479052	+	1.78785i	0.126416	+	0.991977i	\(0.459652\pi\)
							−0.605468	+	0.795869i	\(0.707014\pi\)
\(7\)	−1.20967	+	0.464347i	−0.457211	+	0.175507i	−0.576063	−	0.817405i	\(-0.695412\pi\)
							0.118852	+	0.992912i	\(0.462078\pi\)
\(11\)	1.24522	−	1.53772i	0.375449	−	0.463640i	−0.554041	−	0.832490i	\(-0.686915\pi\)
							0.929489	+	0.368849i	\(0.120248\pi\)
\(13\)	1.46495	−	3.29453i	0.406305	−	0.913738i
							\(17\)	0.211471	−	2.01201i	0.0512892	−	0.487984i	−0.938483	−	0.345326i	\(-0.887768\pi\)
0.989772	−	0.142658i	\(-0.0455649\pi\)
\(19\)	−2.22067	+	0.116380i	−0.509456	+	0.0266995i	−0.305333	−	0.952246i	\(-0.598768\pi\)
							−0.204123	+	0.978945i	\(0.565434\pi\)
\(23\)	−0.110416	+	1.05054i	−0.0230234	+	0.219053i	0.976961	+	0.213419i	\(0.0684599\pi\)
							−0.999984	+	0.00563363i	\(0.998207\pi\)
\(29\)	3.09211	+	2.78415i	0.574191	+	0.517004i	0.904277	−	0.426947i	\(-0.140411\pi\)
							−0.330085	+	0.943951i	\(0.607078\pi\)
\(31\)	3.95159	−	3.92236i	0.709727	−	0.704477i
							\(37\)	−7.05549	−	7.05549i	−1.15992	−	1.15992i	−0.984493	−	0.175423i	\(-0.943871\pi\)
−0.175423	−	0.984493i	\(-0.556129\pi\)
\(41\)	−6.12012	−	9.42416i	−0.955802	−	1.47181i	−0.880163	−	0.474671i	\(-0.842567\pi\)
							−0.0756387	−	0.997135i	\(-0.524100\pi\)
\(43\)	2.61505	−	2.90431i	0.398791	−	0.442903i	−0.509987	−	0.860182i	\(-0.670350\pi\)
							0.908778	+	0.417280i	\(0.137016\pi\)
\(47\)	4.91173	+	9.63982i	0.716450	+	1.40611i	0.905587	+	0.424161i	\(0.139431\pi\)
							−0.189137	+	0.981951i	\(0.560569\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ch.a.11.17	yes	560
13.6	odd	12		403.2.cc.a.383.19	yes	560
31.17	odd	30		403.2.cc.a.141.19	✓	560
403.110	even	60	inner	403.2.ch.a.110.17	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.cc.a.141.19	✓	560	31.17	odd	30
403.2.cc.a.383.19	yes	560	13.6	odd	12
403.2.ch.a.11.17	yes	560	1.1	even	1	trivial
403.2.ch.a.110.17	yes	560	403.110	even	60	inner