Embedded newform 671.2.ct.a.97.19

• Label: 671.2.ct.a.97.19
• Level: $671$
• Weight: $2$
• Character: 671.97
• Analytic conductor: $5.358$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 671 = 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	671.ct (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.35796197563\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(60\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			97.19
Character	\(\chi\)	\(=\)	671.97
Dual form			671.2.ct.a.588.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08694 + 0.627543i) q^{2} +(2.31883 + 1.68473i) q^{3} +(-0.212380 + 0.367854i) q^{4} +(0.405510 - 3.85817i) q^{5} +(-3.57766 - 0.376027i) q^{6} +(-1.83445 + 4.12023i) q^{7} -3.04328i q^{8} +(1.61161 + 4.96003i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 9 q^{2} - 2 q^{3} + 229 q^{4} - 22 q^{6} - 13 q^{7} - 114 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/671\mathbb{Z}\right)^\times\).

\(n\)	\(123\)	\(551\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{7}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08694	+	0.627543i	−0.768580	+	0.443740i	−0.832368	−	0.554224i	\(-0.813015\pi\)
							0.0637881	+	0.997963i	\(0.479682\pi\)
\(3\)	2.31883	+	1.68473i	1.33878	+	0.972679i	0.999488	+	0.0319964i	\(0.0101865\pi\)
							0.339289	+	0.940682i	\(0.389813\pi\)
\(5\)	0.405510	−	3.85817i	0.181350	−	1.72543i	−0.404097	−	0.914716i	\(-0.632414\pi\)
							0.585446	−	0.810711i	\(-0.300919\pi\)
\(7\)	−1.83445	+	4.12023i	−0.693355	+	1.55730i	0.131097	+	0.991370i	\(0.458150\pi\)
							−0.824452	+	0.565931i	\(0.808517\pi\)
\(11\)	−0.188094	+	3.31129i	−0.0567125	+	0.998391i
							\(13\)	−4.31724	+	0.917659i	−1.19739	+	0.254513i	−0.763111	−	0.646267i	\(-0.776329\pi\)
−0.434277	+	0.900780i	\(0.642996\pi\)
\(17\)	−0.419827	−	0.942947i	−0.101823	−	0.228698i	0.855449	−	0.517887i	\(-0.173281\pi\)
							−0.957272	+	0.289189i	\(0.906615\pi\)
\(19\)	−2.75705	+	1.22752i	−0.632511	+	0.281612i	−0.697842	−	0.716252i	\(-0.745856\pi\)
							0.0653311	+	0.997864i	\(0.479190\pi\)
\(23\)	1.16986	+	1.61018i	0.243934	+	0.335746i	0.913375	−	0.407119i	\(-0.133467\pi\)
							−0.669442	+	0.742865i	\(0.733467\pi\)
\(29\)	2.09260	+	4.70005i	0.388585	+	0.872777i	0.996870	+	0.0790580i	\(0.0251912\pi\)
							−0.608285	+	0.793719i	\(0.708142\pi\)
\(31\)	3.40239	+	7.64189i	0.611087	+	1.37252i	0.908544	+	0.417789i	\(0.137195\pi\)
							−0.297457	+	0.954735i	\(0.596139\pi\)
\(37\)	5.29547	+	7.28859i	0.870570	+	1.19824i	0.978944	+	0.204127i	\(0.0654355\pi\)
							−0.108374	+	0.994110i	\(0.534564\pi\)
\(41\)	−1.61429	−	4.96827i	−0.252110	−	0.775914i	−0.994385	−	0.105819i	\(-0.966254\pi\)
							0.742276	−	0.670095i	\(-0.233746\pi\)
\(43\)	−2.67648	−	0.281310i	−0.408160	−	0.0428993i	−0.101776	−	0.994807i	\(-0.532452\pi\)
							−0.306384	+	0.951908i	\(0.599119\pi\)
\(47\)	0.821219	+	7.81338i	0.119787	+	1.13970i	0.874968	+	0.484181i	\(0.160882\pi\)
							−0.755181	+	0.655517i	\(0.772451\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.ct.a.97.19	yes	480
11.5	even	5		671.2.ch.a.280.42	✓	480
61.39	even	30		671.2.ch.a.405.42	yes	480
671.588	even	30	inner	671.2.ct.a.588.19	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.ch.a.280.42	✓	480	11.5	even	5
671.2.ch.a.405.42	yes	480	61.39	even	30
671.2.ct.a.97.19	yes	480	1.1	even	1	trivial
671.2.ct.a.588.19	yes	480	671.588	even	30	inner