Embedded newform 11.10.c.a.9.5

• Label: 11.10.c.a.9.5
• Level: $11$
• Weight: $10$
• Character: 11.9
• Analytic conductor: $5.665$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.66539419780\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			9.5
Character	\(\chi\)	\(=\)	11.9
Dual form			11.10.c.a.5.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.04500 + 3.21619i) q^{2} +(-135.164 + 98.2024i) q^{3} +(404.965 + 294.224i) q^{4} +(-656.025 - 2019.04i) q^{5} +(-174.591 - 537.335i) q^{6} +(-6768.61 - 4917.68i) q^{7} +(-2770.23 + 2012.69i) q^{8} +(2543.22 - 7827.23i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 21 q^{2} + 69 q^{3} - 1625 q^{4} + 225 q^{5} + 8173 q^{6} - 10675 q^{7} + 22863 q^{8} - 93587 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.04500	+	3.21619i	−0.0461830	+	0.142137i	−0.971489	−	0.237085i	\(-0.923808\pi\)
							0.925306	+	0.379221i	\(0.123808\pi\)
\(3\)	−135.164	+	98.2024i	−0.963420	+	0.699965i	−0.953942	−	0.299990i	\(-0.903017\pi\)
							−0.00947724	+	0.999955i	\(0.503017\pi\)
\(5\)	−656.025	−	2019.04i	−0.469414	−	1.44471i	−0.853346	−	0.521345i	\(-0.825430\pi\)
							0.383932	−	0.923361i	\(-0.374570\pi\)
\(7\)	−6768.61	−	4917.68i	−1.06551	−	0.774140i	−0.0904115	−	0.995904i	\(-0.528818\pi\)
							−0.975100	+	0.221765i	\(0.928818\pi\)
\(11\)	−11590.6	−	47155.1i	−0.238693	−	0.971095i
							\(13\)	2988.51	−	9197.69i	0.0290208	−	0.0893170i	−0.935497	−	0.353335i	\(-0.885048\pi\)
0.964518	+	0.264018i	\(0.0850477\pi\)
\(17\)	−11885.1	−	36578.5i	−0.0345129	−	0.106220i	0.932316	−	0.361645i	\(-0.117785\pi\)
							−0.966829	+	0.255425i	\(0.917785\pi\)
\(19\)	−778393.	+	565536.i	−1.37028	+	0.995563i	−0.372560	+	0.928008i	\(0.621520\pi\)
							−0.997716	+	0.0675548i	\(0.978480\pi\)
\(23\)	−1.41652e6			−1.05547			−0.527736	−	0.849409i	\(-0.676959\pi\)
							−0.527736	+	0.849409i	\(0.676959\pi\)
\(29\)	2.76241e6	+	2.00701e6i	0.725266	+	0.526936i	0.888062	−	0.459723i	\(-0.152051\pi\)
							−0.162796	+	0.986660i	\(0.552051\pi\)
\(31\)	1.83290e6	−	5.64110e6i	0.356461	−	1.09707i	−0.598696	−	0.800976i	\(-0.704314\pi\)
							0.955157	−	0.296098i	\(-0.0956856\pi\)
\(37\)	4.20237e6	+	3.05320e6i	0.368626	+	0.267823i	0.756641	−	0.653831i	\(-0.226839\pi\)
							−0.388015	+	0.921653i	\(0.626839\pi\)
\(41\)	1.36819e7	−	9.94051e6i	0.756171	−	0.549391i	−0.141562	−	0.989929i	\(-0.545213\pi\)
							0.897734	+	0.440539i	\(0.145213\pi\)
\(43\)	−2.68909e7			−1.19949			−0.599746	−	0.800190i	\(-0.704732\pi\)
							−0.599746	+	0.800190i	\(0.704732\pi\)
\(47\)	8.77183e6	−	6.37311e6i	0.262210	−	0.190507i	−0.448911	−	0.893577i	\(-0.648188\pi\)
							0.711121	+	0.703070i	\(0.248188\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.10.c.a.9.5	yes	32
3.2	odd	2		99.10.f.a.64.4		32
11.4	even	5		121.10.a.h.1.9		16
11.5	even	5	inner	11.10.c.a.5.5	✓	32
11.7	odd	10		121.10.a.i.1.8		16
33.5	odd	10		99.10.f.a.82.4		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.10.c.a.5.5	✓	32	11.5	even	5	inner
11.10.c.a.9.5	yes	32	1.1	even	1	trivial
99.10.f.a.64.4		32	3.2	odd	2
99.10.f.a.82.4		32	33.5	odd	10
121.10.a.h.1.9		16	11.4	even	5
121.10.a.i.1.8		16	11.7	odd	10