Embedded newform 211.4.o.a.4.13

• Label: 211.4.o.a.4.13
• Level: $211$
• Weight: $4$
• Character: 211.4
• Analytic conductor: $12.449$
• Analytic rank: $0$
• Dimension: $2496$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 211 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	211.o (of order \(105\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.4494030112\)
Analytic rank:	\(0\)
Dimension:	\(2496\)
Relative dimension:	\(52\) over \(\Q(\zeta_{105})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{105}]$

Embedding invariants

Embedding label			4.13
Character	\(\chi\)	\(=\)	211.4
Dual form			211.4.o.a.53.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.97205 - 0.0889500i) q^{2} +(1.11700 + 3.82337i) q^{3} +(0.839479 + 0.0502943i) q^{4} +(9.58614 + 10.0263i) q^{5} +(-2.97968 - 11.4626i) q^{6} +(-5.32177 - 8.61188i) q^{7} +(21.2008 + 1.90810i) q^{8} +(9.38297 - 5.99407i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2496 q - 44 q^{2} - 56 q^{3} - 270 q^{4} - 50 q^{5} - 89 q^{6} - 55 q^{7} + 211 q^{8} - 596 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{105}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.97205	−	0.0889500i	−1.05078	−	0.0314486i	−0.499939	−	0.866060i	\(-0.666644\pi\)
							−0.550838	+	0.834612i	\(0.685692\pi\)
\(3\)	1.11700	+	3.82337i	0.214966	+	0.735809i	0.994053	+	0.108893i	\(0.0347306\pi\)
							−0.779087	+	0.626916i	\(0.784317\pi\)
\(5\)	9.58614	+	10.0263i	0.857411	+	0.896781i	0.995608	−	0.0936234i	\(-0.0298450\pi\)
							−0.138197	+	0.990405i	\(0.544131\pi\)
\(7\)	−5.32177	−	8.61188i	−0.287349	−	0.464998i	0.674617	−	0.738168i	\(-0.264309\pi\)
							−0.961966	+	0.273170i	\(0.911928\pi\)
\(11\)	7.03762	−	51.9538i	0.192902	−	1.42406i	−0.592747	−	0.805389i	\(-0.701957\pi\)
							0.785649	−	0.618672i	\(-0.212329\pi\)
\(13\)	2.60554	+	6.09597i	0.0555883	+	0.130055i	0.945020	−	0.327013i	\(-0.106042\pi\)
							−0.889431	+	0.457069i	\(0.848899\pi\)
\(17\)	74.3546	−	25.3952i	1.06080	−	0.362308i	0.262730	−	0.964869i	\(-0.415377\pi\)
							0.798072	+	0.602562i	\(0.205853\pi\)
\(19\)	−2.70945	−	25.7787i	−0.0327153	−	0.311265i	−0.998627	−	0.0523921i	\(-0.983315\pi\)
							0.965911	−	0.258873i	\(-0.0833512\pi\)
\(23\)	−49.0310	−	35.6231i	−0.444507	−	0.322953i	0.342916	−	0.939366i	\(-0.388585\pi\)
							−0.787423	+	0.616413i	\(0.788585\pi\)
\(29\)	61.9052	−	82.5785i	0.396397	−	0.528774i	−0.556993	−	0.830517i	\(-0.688045\pi\)
							0.953389	+	0.301744i	\(0.0975686\pi\)
\(31\)	−4.65272	−	1.43517i	−0.0269566	−	0.00831500i	0.281248	−	0.959635i	\(-0.409252\pi\)
							−0.308204	+	0.951320i	\(0.599728\pi\)
\(37\)	288.985	+	118.437i	1.28402	+	0.526241i	0.915103	−	0.403221i	\(-0.132109\pi\)
							0.368918	+	0.929462i	\(0.379728\pi\)
\(41\)	131.179	+	73.1419i	0.499676	+	0.278606i	0.716478	−	0.697610i	\(-0.245753\pi\)
							−0.216802	+	0.976216i	\(0.569563\pi\)
\(43\)	−104.073	+	96.5661i	−0.369094	+	0.342470i	−0.842813	−	0.538207i	\(-0.819102\pi\)
							0.473718	+	0.880676i	\(0.342911\pi\)
\(47\)	330.380	+	211.055i	1.02534	+	0.655010i	0.939941	−	0.341337i	\(-0.110880\pi\)
							0.0853960	+	0.996347i	\(0.472784\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	211.4.o.a.4.13	✓	2496
211.53	even	105	inner	211.4.o.a.53.13	yes	2496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
211.4.o.a.4.13	✓	2496	1.1	even	1	trivial
211.4.o.a.53.13	yes	2496	211.53	even	105	inner