Embedded newform 603.2.l.a.401.13

• Label: 603.2.l.a.401.13
• Level: $603$
• Weight: $2$
• Character: 603.401
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $132$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.81497924188\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(66\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			401.13
Character	\(\chi\)	\(=\)	603.401
Dual form			603.2.l.a.200.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.948352 - 1.64259i) q^{2} +(0.853494 - 1.50717i) q^{3} +(-0.798743 + 1.38346i) q^{4} +(1.60208 - 2.77488i) q^{5} +(-3.28507 + 0.0273793i) q^{6} +(0.395721 - 0.228469i) q^{7} -0.763450 q^{8} +(-1.54310 - 2.57271i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 66 q^{4} + 10 q^{6} + 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/603\mathbb{Z}\right)^\times\).

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.948352	−	1.64259i	−0.670586	−	1.16149i	−0.977738	−	0.209829i	\(-0.932709\pi\)
							0.307152	−	0.951660i	\(-0.400624\pi\)
\(3\)	0.853494	−	1.50717i	0.492765	−	0.870162i
							\(5\)	1.60208	−	2.77488i	0.716472	−	1.24097i	−0.245917	−	0.969291i	\(-0.579089\pi\)
0.962389	−	0.271675i	\(-0.0875776\pi\)
\(7\)	0.395721	−	0.228469i	0.149568	−	0.0863533i	−0.423348	−	0.905967i	\(-0.639145\pi\)
							0.572916	+	0.819614i	\(0.305812\pi\)
\(11\)	−1.42798	−	2.47334i	−0.430553	−	0.745740i	0.566368	−	0.824153i	\(-0.308348\pi\)
							−0.996921	+	0.0784126i	\(0.975015\pi\)
\(13\)	3.82550	+	2.20865i	1.06100	+	0.612570i	0.925709	−	0.378236i	\(-0.123469\pi\)
							0.135293	+	0.990806i	\(0.456802\pi\)
\(17\)			1.61213i			0.390999i	0.980704	+	0.195499i	\(0.0626328\pi\)
							−0.980704	+	0.195499i	\(0.937367\pi\)
\(19\)	−1.90625			−0.437323			−0.218662	−	0.975801i	\(-0.570169\pi\)
							−0.218662	+	0.975801i	\(0.570169\pi\)
\(23\)	−0.758829	−	0.438110i	−0.158227	−	0.0913523i	0.418796	−	0.908080i	\(-0.362452\pi\)
							−0.577023	+	0.816728i	\(0.695786\pi\)
\(29\)	5.93678	−	3.42760i	1.10243	−	0.636490i	0.165574	−	0.986197i	\(-0.447052\pi\)
							0.936859	+	0.349707i	\(0.113719\pi\)
\(31\)	0.150077	+	0.0866472i	0.0269547	+	0.0155623i	0.513417	−	0.858139i	\(-0.328379\pi\)
							−0.486462	+	0.873702i	\(0.661713\pi\)
\(37\)	5.70239			0.937468			0.468734	−	0.883339i	\(-0.344710\pi\)
							0.468734	+	0.883339i	\(0.344710\pi\)
\(41\)	−0.559635	+	0.969316i	−0.0874003	+	0.151382i	−0.906412	−	0.422396i	\(-0.861189\pi\)
							0.819011	+	0.573778i	\(0.194523\pi\)
\(43\)	−0.317624	+	0.183380i	−0.0484372	+	0.0279652i	−0.524023	−	0.851704i	\(-0.675569\pi\)
							0.475586	+	0.879669i	\(0.342236\pi\)
\(47\)	−4.40822	+	2.54508i	−0.643004	+	0.371239i	−0.785771	−	0.618518i	\(-0.787734\pi\)
							0.142767	+	0.989756i	\(0.454400\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.l.a.401.13	yes	132
9.2	odd	6	inner	603.2.l.a.200.54	yes	132
67.66	odd	2	inner	603.2.l.a.401.54	yes	132
603.200	even	6	inner	603.2.l.a.200.13	✓	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.l.a.200.13	✓	132	603.200	even	6	inner
603.2.l.a.200.54	yes	132	9.2	odd	6	inner
603.2.l.a.401.13	yes	132	1.1	even	1	trivial
603.2.l.a.401.54	yes	132	67.66	odd	2	inner