Embedded newform 211.3.k.a.10.23

• Label: 211.3.k.a.10.23
• 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			10.23
Character	\(\chi\)	\(=\)	211.10
Dual form			211.3.k.a.190.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.594653 - 1.33561i) q^{2} +(-2.60043 - 2.34144i) q^{3} +(1.24627 + 1.38413i) q^{4} +(0.688900 - 2.12022i) q^{5} +(-4.67360 + 2.08082i) q^{6} +(10.4989 + 1.10348i) q^{7} +(8.15157 - 2.64861i) q^{8} +(0.339147 + 3.22677i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 7 q^{2} - 20 q^{3} - 65 q^{4} - 6 q^{5} - 9 q^{6} - 36 q^{7} - 95 q^{8} - 102 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.594653	−	1.33561i	0.297326	−	0.667806i	−0.701674	−	0.712498i	\(-0.747564\pi\)
							0.999001	+	0.0446917i	\(0.0142305\pi\)
\(3\)	−2.60043	−	2.34144i	−0.866809	−	0.780478i	0.110148	−	0.993915i	\(-0.464868\pi\)
							−0.976957	+	0.213437i	\(0.931534\pi\)
\(5\)	0.688900	−	2.12022i	0.137780	−	0.424044i	−0.858232	−	0.513262i	\(-0.828437\pi\)
							0.996012	+	0.0892184i	\(0.0284369\pi\)
\(7\)	10.4989	+	1.10348i	1.49984	+	0.157640i	0.818576	−	0.574398i	\(-0.194764\pi\)
							0.681264	+	0.732038i	\(0.261431\pi\)
\(11\)	0.627573	−	1.93147i	0.0570521	−	0.175588i	−0.918470	−	0.395492i	\(-0.870574\pi\)
							0.975522	+	0.219903i	\(0.0705742\pi\)
\(13\)	−6.37512	−	4.63180i	−0.490394	−	0.356292i	0.314942	−	0.949111i	\(-0.398015\pi\)
							−0.805336	+	0.592819i	\(0.798015\pi\)
\(17\)	−8.53704	−	0.897279i	−0.502179	−	0.0527811i	−0.149947	−	0.988694i	\(-0.547910\pi\)
							−0.352232	+	0.935913i	\(0.614577\pi\)
\(19\)	1.14392	−	10.8837i	0.0602063	−	0.572824i	−0.922284	−	0.386512i	\(-0.873680\pi\)
							0.982491	−	0.186312i	\(-0.0596536\pi\)
\(23\)	−15.9856	−	22.0023i	−0.695026	−	0.956621i	−0.999991	−	0.00425759i	\(-0.998645\pi\)
							0.304965	−	0.952364i	\(-0.401355\pi\)
\(29\)	−2.33461	−	5.24363i	−0.0805039	−	0.180815i	0.868809	−	0.495147i	\(-0.164886\pi\)
							−0.949313	+	0.314333i	\(0.898219\pi\)
\(31\)	22.4731	−	12.9748i	0.724937	−	0.418543i	−0.0916299	−	0.995793i	\(-0.529208\pi\)
							0.816567	+	0.577250i	\(0.195874\pi\)
\(37\)	−27.6547	+	30.7136i	−0.747423	+	0.830098i	−0.990152	−	0.139997i	\(-0.955291\pi\)
							0.242729	+	0.970094i	\(0.421958\pi\)
\(41\)	37.5772	−	33.8347i	0.916518	−	0.825236i	−0.0685069	−	0.997651i	\(-0.521824\pi\)
							0.985024	+	0.172415i	\(0.0551569\pi\)
\(43\)	−5.61271	+	9.72151i	−0.130528	+	0.226082i	−0.923880	−	0.382682i	\(-0.875001\pi\)
							0.793352	+	0.608763i	\(0.208334\pi\)
\(47\)	−1.41677	+	13.4797i	−0.0301441	+	0.286802i	0.969057	+	0.246837i	\(0.0793913\pi\)
							−0.999201	+	0.0399649i	\(0.987275\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	211.3.k.a.10.23	✓	272
211.190	odd	30	inner	211.3.k.a.190.23	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
211.3.k.a.10.23	✓	272	1.1	even	1	trivial
211.3.k.a.190.23	yes	272	211.190	odd	30	inner