Embedded newform 11.14.c.a.3.5

• Label: 11.14.c.a.3.5
• Level: $11$
• Weight: $14$
• Character: 11.3
• Analytic conductor: $11.795$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.7954021847\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			3.5
Character	\(\chi\)	\(=\)	11.3
Dual form			11.14.c.a.4.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-38.1571 + 27.7228i) q^{2} +(-502.353 - 1546.08i) q^{3} +(-1844.05 + 5675.41i) q^{4} +(29618.8 + 21519.3i) q^{5} +(62030.1 + 45067.5i) q^{6} +(-28662.8 + 88215.0i) q^{7} +(-206371. - 635143. i) q^{8} +(-848181. + 616240. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 59 q^{2} - 864 q^{3} - 44697 q^{4} - 52240 q^{5} - 130115 q^{6} + 390696 q^{7} + 876527 q^{8} - 3520244 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−38.1571	+	27.7228i	−0.421581	+	0.306296i	−0.778274	−	0.627925i	\(-0.783904\pi\)
							0.356693	+	0.934222i	\(0.383904\pi\)
\(3\)	−502.353	−	1546.08i	−0.397851	−	1.22446i	−0.926719	−	0.375756i	\(-0.877383\pi\)
							0.528867	−	0.848705i	\(-0.322617\pi\)
\(5\)	29618.8	+	21519.3i	0.847739	+	0.615918i	0.924522	−	0.381130i	\(-0.124465\pi\)
							−0.0767828	+	0.997048i	\(0.524465\pi\)
\(7\)	−28662.8	+	88215.0i	−0.0920834	+	0.283403i	−0.986483	−	0.163866i	\(-0.947604\pi\)
							0.894399	+	0.447269i	\(0.147604\pi\)
\(11\)	−4.12239e6	−	4.18671e6i	−0.701612	−	0.712559i
							\(13\)	1.95148e7	−	1.41784e7i	1.12133	−	0.814693i	0.136919	−	0.990582i	\(-0.456280\pi\)
0.984410	+	0.175889i	\(0.0562801\pi\)
\(17\)	−4.30059e7	−	3.12456e7i	−0.432126	−	0.313958i	0.350373	−	0.936610i	\(-0.386055\pi\)
							−0.782498	+	0.622653i	\(0.786055\pi\)
\(19\)	−8.45952e7	−	2.60357e8i	−0.412522	−	1.26961i	−0.914448	−	0.404703i	\(-0.867375\pi\)
							0.501926	−	0.864911i	\(-0.332625\pi\)
\(23\)	8.98669e8			1.26581			0.632906	−	0.774229i	\(-0.281862\pi\)
							0.632906	+	0.774229i	\(0.281862\pi\)
\(29\)	9.11558e7	−	2.80549e8i	0.0284575	−	0.0875833i	−0.935819	−	0.352481i	\(-0.885338\pi\)
							0.964277	+	0.264898i	\(0.0853382\pi\)
\(31\)	−2.22932e9	+	1.61970e9i	−0.451151	+	0.327780i	−0.790050	−	0.613042i	\(-0.789945\pi\)
							0.338899	+	0.940823i	\(0.389945\pi\)
\(37\)	4.64129e9	−	1.42844e10i	0.297391	−	0.915274i	−0.685017	−	0.728527i	\(-0.740205\pi\)
							0.982408	−	0.186747i	\(-0.0597946\pi\)
\(41\)	−2.46605e9	−	7.58973e9i	−0.0810788	−	0.249535i	0.902298	−	0.431114i	\(-0.141879\pi\)
							−0.983376	+	0.181579i	\(0.941879\pi\)
\(43\)	6.16804e10			1.48800			0.743999	−	0.668181i	\(-0.232927\pi\)
							0.743999	+	0.668181i	\(0.232927\pi\)
\(47\)	−1.88772e10	−	5.80979e10i	−0.255447	−	0.786184i	−0.993741	−	0.111706i	\(-0.964369\pi\)
							0.738294	−	0.674479i	\(-0.235631\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.14.c.a.3.5	✓	48
11.2	odd	10		121.14.a.i.1.10		24
11.4	even	5	inner	11.14.c.a.4.5	yes	48
11.9	even	5		121.14.a.h.1.15		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.14.c.a.3.5	✓	48	1.1	even	1	trivial
11.14.c.a.4.5	yes	48	11.4	even	5	inner
121.14.a.h.1.15		24	11.9	even	5
121.14.a.i.1.10		24	11.2	odd	10