Embedded newform 11.13.d.a.6.6

• Label: 11.13.d.a.6.6
• Level: $11$
• Weight: $13$
• Character: 11.6
• Analytic conductor: $10.054$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	11.d (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.0539319900\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			6.6
Character	\(\chi\)	\(=\)	11.6
Dual form			11.13.d.a.2.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.21322 + 2.66864i) q^{2} +(-83.6942 - 60.8074i) q^{3} +(-3253.40 + 2363.73i) q^{4} +(3579.35 - 11016.1i) q^{5} +(849.672 + 276.075i) q^{6} +(61235.8 + 84283.8i) q^{7} +(41204.4 - 56713.0i) q^{8} +(-160917. - 495252. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 5 q^{2} - 1083 q^{3} + 16247 q^{4} - 723 q^{5} + 147195 q^{6} - 219605 q^{7} + 921595 q^{8} - 3695582 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.21322	+	2.66864i	−0.128332	+	0.0416975i	−0.372479	−	0.928041i	\(-0.621492\pi\)
							0.244147	+	0.969738i	\(0.421492\pi\)
\(3\)	−83.6942	−	60.8074i	−0.114807	−	0.0834121i	0.528900	−	0.848684i	\(-0.322605\pi\)
							−0.643707	+	0.765272i	\(0.722605\pi\)
\(5\)	3579.35	−	11016.1i	0.229078	−	0.705031i	−0.768773	−	0.639521i	\(-0.779133\pi\)
							0.997852	−	0.0655099i	\(-0.0208674\pi\)
\(7\)	61235.8	+	84283.8i	0.520496	+	0.716401i	0.985645	−	0.168831i	\(-0.0539994\pi\)
							−0.465149	+	0.885232i	\(0.653999\pi\)
\(11\)	1.75888e6	+	211618.i	0.992840	+	0.119453i
							\(13\)	8.42006e6	−	2.73584e6i	1.74444	−	0.566802i	0.749029	−	0.662537i	\(-0.230520\pi\)
0.995407	+	0.0957355i	\(0.0305203\pi\)
\(17\)	1.10446e7	+	3.58859e6i	0.457567	+	0.148673i	0.528726	−	0.848792i	\(-0.322670\pi\)
							−0.0711592	+	0.997465i	\(0.522670\pi\)
\(19\)	1.20274e7	−	1.65543e7i	0.255653	−	0.351876i	−0.661828	−	0.749656i	\(-0.730219\pi\)
							0.917481	+	0.397780i	\(0.130219\pi\)
\(23\)	−2.79184e8			−1.88592			−0.942962	−	0.332902i	\(-0.891972\pi\)
							−0.942962	+	0.332902i	\(0.891972\pi\)
\(29\)	3.86363e8	+	5.31783e8i	0.649543	+	0.894019i	0.999079	−	0.0429039i	\(-0.0136609\pi\)
							−0.349536	+	0.936923i	\(0.613661\pi\)
\(31\)	−3.45867e8	−	1.06447e9i	−0.389707	−	1.19940i	−0.933007	−	0.359857i	\(-0.882825\pi\)
							0.543300	−	0.839539i	\(-0.317175\pi\)
\(37\)	2.63099e9	−	1.91153e9i	1.02544	−	0.745024i	0.0580474	−	0.998314i	\(-0.481513\pi\)
							0.967391	+	0.253290i	\(0.0815125\pi\)
\(41\)	3.21175e9	−	4.42059e9i	0.676143	−	0.930631i	−0.323737	−	0.946147i	\(-0.604939\pi\)
							0.999880	+	0.0155162i	\(0.00493917\pi\)
\(43\)			6.76887e9i			1.07079i	0.844601	+	0.535396i	\(0.179838\pi\)
							−0.844601	+	0.535396i	\(0.820162\pi\)
\(47\)	−3.26255e9	−	2.37038e9i	−0.302671	−	0.219903i	0.426075	−	0.904688i	\(-0.359896\pi\)
							−0.728745	+	0.684785i	\(0.759896\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.13.d.a.6.6	yes	44
11.2	odd	10	inner	11.13.d.a.2.6	✓	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.13.d.a.2.6	✓	44	11.2	odd	10	inner
11.13.d.a.6.6	yes	44	1.1	even	1	trivial