Embedded newform 11.11.d.a.6.9

• Label: 11.11.d.a.6.9
• Level: $11$
• Weight: $11$
• Character: 11.6
• Analytic conductor: $6.989$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.98892977941\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(9\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			6.9
Character	\(\chi\)	\(=\)	11.6
Dual form			11.11.d.a.2.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(48.9033 - 15.8897i) q^{2} +(165.281 + 120.084i) q^{3} +(1310.62 - 952.222i) q^{4} +(259.656 - 799.138i) q^{5} +(9990.90 + 3246.24i) q^{6} +(6472.97 + 8909.28i) q^{7} +(18014.0 - 24794.1i) q^{8} +(-5349.36 - 16463.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 5 q^{2} - 78 q^{3} + 2839 q^{4} + 566 q^{5} - 3525 q^{6} - 9740 q^{7} - 56325 q^{8} + 27883 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	48.9033	−	15.8897i	1.52823	−	0.496552i	0.580129	−	0.814524i	\(-0.303002\pi\)
							0.948100	+	0.317973i	\(0.103002\pi\)
\(3\)	165.281	+	120.084i	0.680170	+	0.494173i	0.873414	−	0.486978i	\(-0.161901\pi\)
							−0.193244	+	0.981151i	\(0.561901\pi\)
\(5\)	259.656	−	799.138i	0.0830898	−	0.255724i	−0.900877	−	0.434074i	\(-0.857076\pi\)
							0.983967	+	0.178350i	\(0.0570758\pi\)
\(7\)	6472.97	+	8909.28i	0.385135	+	0.530093i	0.956936	−	0.290299i	\(-0.0937549\pi\)
							−0.571800	+	0.820393i	\(0.693755\pi\)
\(11\)	−160863.	−	7773.63i	−0.998834	−	0.0482681i
							\(13\)	317694.	−	103225.i	0.855643	−	0.278015i	0.151835	−	0.988406i	\(-0.451482\pi\)
0.703808	+	0.710390i	\(0.251482\pi\)
\(17\)	−1.58896e6	−	516284.i	−1.11910	−	0.363617i	−0.309675	−	0.950843i	\(-0.600220\pi\)
							−0.809423	+	0.587225i	\(0.800220\pi\)
\(19\)	−1.68812e6	+	2.32350e6i	−0.681767	+	0.938371i	−0.999953	−	0.00965664i	\(-0.996926\pi\)
							0.318187	+	0.948028i	\(0.396926\pi\)
\(23\)	−6.17837e6			−0.959919			−0.479960	−	0.877291i	\(-0.659349\pi\)
							−0.479960	+	0.877291i	\(0.659349\pi\)
\(29\)	8.83955e6	+	1.21666e7i	0.430963	+	0.593170i	0.968174	−	0.250278i	\(-0.0805220\pi\)
							−0.537211	+	0.843448i	\(0.680522\pi\)
\(31\)	−4.09149e6	−	1.25923e7i	−0.142914	−	0.439843i	0.853823	−	0.520563i	\(-0.174278\pi\)
							−0.996737	+	0.0807203i	\(0.974278\pi\)
\(37\)	8.78268e7	−	6.38099e7i	1.26654	−	0.920194i	0.267479	−	0.963564i	\(-0.413809\pi\)
							0.999059	+	0.0433700i	\(0.0138094\pi\)
\(41\)	5.72523e7	−	7.88010e7i	0.494167	−	0.680162i	−0.486983	−	0.873411i	\(-0.661903\pi\)
							0.981150	+	0.193249i	\(0.0619026\pi\)
\(43\)		−	1.88418e8i		−	1.28168i	−0.767673	−	0.640842i	\(-0.778585\pi\)
							0.767673	−	0.640842i	\(-0.221415\pi\)
\(47\)	5.83203e7	+	4.23722e7i	0.254291	+	0.184753i	0.707626	−	0.706587i	\(-0.249766\pi\)
							−0.453336	+	0.891340i	\(0.649766\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.11.d.a.6.9	yes	36
3.2	odd	2		99.11.k.a.28.1		36
11.2	odd	10	inner	11.11.d.a.2.9	✓	36
11.3	even	5		121.11.b.c.120.4		36
11.8	odd	10		121.11.b.c.120.33		36
33.2	even	10		99.11.k.a.46.1		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.11.d.a.2.9	✓	36	11.2	odd	10	inner
11.11.d.a.6.9	yes	36	1.1	even	1	trivial
99.11.k.a.28.1		36	3.2	odd	2
99.11.k.a.46.1		36	33.2	even	10
121.11.b.c.120.4		36	11.3	even	5
121.11.b.c.120.33		36	11.8	odd	10