Embedded newform 403.2.cd.a.21.14

• Label: 403.2.cd.a.21.14
• Level: $403$
• Weight: $2$
• Character: 403.21
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $576$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(576\)
Relative dimension:	\(36\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			21.14
Character	\(\chi\)	\(=\)	403.21
Dual form			403.2.cd.a.96.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.159860 - 1.00931i) q^{2} +(1.15285 - 0.121170i) q^{3} +(0.908952 - 0.295336i) q^{4} +(0.515241 - 1.92291i) q^{5} +(-0.306593 - 1.14422i) q^{6} +(0.690634 - 0.448503i) q^{7} +(-1.37125 - 2.69124i) q^{8} +(-1.62005 + 0.344353i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 576 q - 12 q^{2} - 28 q^{3} - 8 q^{5} + 6 q^{6} - 16 q^{7} - 46 q^{8} - 104 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{29}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.159860	−	1.00931i	−0.113038	−	0.713693i	−0.977491	−	0.210979i	\(-0.932335\pi\)
							0.864453	−	0.502714i	\(-0.167665\pi\)
\(3\)	1.15285	−	0.121170i	0.665600	−	0.0699574i	0.234298	−	0.972165i	\(-0.424721\pi\)
							0.431302	+	0.902207i	\(0.358054\pi\)
\(5\)	0.515241	−	1.92291i	0.230423	−	0.859950i	−0.749736	−	0.661737i	\(-0.769820\pi\)
							0.980159	−	0.198213i	\(-0.0635137\pi\)
\(7\)	0.690634	−	0.448503i	0.261035	−	0.169518i	−0.407486	−	0.913212i	\(-0.633594\pi\)
							0.668521	+	0.743693i	\(0.266928\pi\)
\(11\)	−2.18180	+	0.114343i	−0.657837	+	0.0344758i	−0.378334	−	0.925669i	\(-0.623503\pi\)
							−0.279503	+	0.960145i	\(0.590170\pi\)
\(13\)	−1.66050	−	3.20043i	−0.460541	−	0.887638i
							\(17\)	3.76309	+	4.17934i	0.912684	+	1.01364i	0.999848	+	0.0174446i	\(0.00555306\pi\)
−0.0871636	+	0.996194i	\(0.527780\pi\)
\(19\)	3.54768	+	1.36183i	0.813893	+	0.312424i	0.729476	−	0.684006i	\(-0.239764\pi\)
							0.0844167	+	0.996431i	\(0.473097\pi\)
\(23\)	−1.86737	+	5.74717i	−0.389373	+	1.19837i	0.543884	+	0.839160i	\(0.316953\pi\)
							−0.933258	+	0.359208i	\(0.883047\pi\)
\(29\)	2.20667	−	3.03722i	0.409769	−	0.563998i	−0.553393	−	0.832920i	\(-0.686667\pi\)
							0.963162	+	0.268922i	\(0.0866674\pi\)
\(31\)	5.52040	−	0.724675i	0.991494	−	0.130155i
							\(37\)	7.12025	−	1.90787i	1.17056	−	0.313651i	0.379385	−	0.925239i	\(-0.376136\pi\)
0.791177	+	0.611588i	\(0.209469\pi\)
\(41\)	−3.54264	+	2.86878i	−0.553268	+	0.448028i	−0.864734	−	0.502230i	\(-0.832513\pi\)
							0.311466	+	0.950257i	\(0.399180\pi\)
\(43\)	4.06056	−	1.80788i	0.619230	−	0.275699i	−0.0730498	−	0.997328i	\(-0.523273\pi\)
							0.692279	+	0.721629i	\(0.256607\pi\)
\(47\)	−0.328086	+	2.07145i	−0.0478562	+	0.302152i	−0.999994	−	0.00351740i	\(-0.998880\pi\)
							0.952138	+	0.305670i	\(0.0988804\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.cd.a.21.14	✓	576
13.5	odd	4	inner	403.2.cd.a.83.14	yes	576
31.3	odd	30	inner	403.2.cd.a.34.14	yes	576
403.96	even	60	inner	403.2.cd.a.96.14	yes	576

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.cd.a.21.14	✓	576	1.1	even	1	trivial
403.2.cd.a.34.14	yes	576	31.3	odd	30	inner
403.2.cd.a.83.14	yes	576	13.5	odd	4	inner
403.2.cd.a.96.14	yes	576	403.96	even	60	inner