Embedded newform 211.3.k.a.10.11

• Label: 211.3.k.a.10.11
• Level: $211$
• Weight: $3$
• Character: 211.10
• Analytic conductor: $5.749$
• Analytic rank: $0$
• Dimension: $272$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 211 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	211.k (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.74933357800\)
Analytic rank:	\(0\)
Dimension:	\(272\)
Relative dimension:	\(34\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			10.11
Character	\(\chi\)	\(=\)	211.10
Dual form			211.3.k.a.190.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.627268 + 1.40887i) q^{2} +(-3.69211 - 3.32439i) q^{3} +(1.08508 + 1.20511i) q^{4} +(2.81134 - 8.65241i) q^{5} +(6.99956 - 3.11640i) q^{6} +(0.0595191 + 0.00625571i) q^{7} +(-8.24533 + 2.67907i) q^{8} +(1.63934 + 15.5973i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 7 q^{2} - 20 q^{3} - 65 q^{4} - 6 q^{5} - 9 q^{6} - 36 q^{7} - 95 q^{8} - 102 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/211\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{19}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.627268	+	1.40887i	−0.313634	+	0.704433i	−0.999734	−	0.0230742i	\(-0.992655\pi\)
							0.686100	+	0.727507i	\(0.259321\pi\)
\(3\)	−3.69211	−	3.32439i	−1.23070	−	1.10813i	−0.990515	−	0.137403i	\(-0.956125\pi\)
							−0.240187	−	0.970727i	\(-0.577209\pi\)
\(5\)	2.81134	−	8.65241i	0.562268	−	1.73048i	−0.113665	−	0.993519i	\(-0.536259\pi\)
							0.675933	−	0.736963i	\(-0.263741\pi\)
\(7\)	0.0595191	+	0.00625571i	0.00850273	+	0.000893673i	0.108779	−	0.994066i	\(-0.465306\pi\)
							−0.100276	+	0.994960i	\(0.531973\pi\)
\(11\)	1.39725	−	4.30031i	0.127023	−	0.390937i	−0.867241	−	0.497888i	\(-0.834109\pi\)
							0.994264	+	0.106951i	\(0.0341089\pi\)
\(13\)	−4.57749	−	3.32574i	−0.352115	−	0.255826i	0.397641	−	0.917541i	\(-0.369829\pi\)
							−0.749756	+	0.661715i	\(0.769829\pi\)
\(17\)	−11.9000	−	1.25074i	−0.699998	−	0.0735727i	−0.252157	−	0.967686i	\(-0.581140\pi\)
							−0.447840	+	0.894114i	\(0.647807\pi\)
\(19\)	−2.68219	+	25.5194i	−0.141168	+	1.34312i	0.662954	+	0.748660i	\(0.269302\pi\)
							−0.804122	+	0.594464i	\(0.797364\pi\)
\(23\)	−24.3048	−	33.4527i	−1.05673	−	1.45446i	−0.882826	−	0.469701i	\(-0.844362\pi\)
							−0.173904	−	0.984763i	\(-0.555638\pi\)
\(29\)	−3.81861	−	8.57673i	−0.131676	−	0.295749i	0.835660	−	0.549247i	\(-0.185086\pi\)
							−0.967336	+	0.253498i	\(0.918419\pi\)
\(31\)	−5.92278	+	3.41952i	−0.191057	+	0.110307i	−0.592477	−	0.805587i	\(-0.701850\pi\)
							0.401420	+	0.915894i	\(0.368517\pi\)
\(37\)	33.2115	−	36.8851i	0.897608	−	0.996895i	−0.102390	−	0.994744i	\(-0.532649\pi\)
							0.999998	−	0.00215042i	\(-0.000684499\pi\)
\(41\)	12.8178	−	11.5412i	0.312629	−	0.281492i	−0.497846	−	0.867265i	\(-0.665876\pi\)
							0.810475	+	0.585773i	\(0.199209\pi\)
\(43\)	−35.2325	+	61.0245i	−0.819361	+	1.41917i	0.0867927	+	0.996226i	\(0.472338\pi\)
							−0.906154	+	0.422948i	\(0.860995\pi\)
\(47\)	−8.66111	+	82.4050i	−0.184279	+	1.75330i	0.377499	+	0.926010i	\(0.376784\pi\)
							−0.561778	+	0.827288i	\(0.689883\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	211.3.k.a.10.11	✓	272
211.190	odd	30	inner	211.3.k.a.190.11	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
211.3.k.a.10.11	✓	272	1.1	even	1	trivial
211.3.k.a.190.11	yes	272	211.190	odd	30	inner