Embedded newform 11.4.c.a.5.2

• Label: 11.4.c.a.5.2
• Level: $11$
• Weight: $4$
• Character: 11.5
• Analytic conductor: $0.649$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.649021010063\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	8.0.29283765625.1
Defining polynomial:	\( x^{8} - x^{7} + 10x^{6} - 19x^{5} + 109x^{4} + 171x^{3} + 810x^{2} + 729x + 6561 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			5.2
Root			\(0.785330 + 2.41700i\) of defining polynomial
Character	\(\chi\)	\(=\)	11.5
Dual form			11.4.c.a.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.903364 + 2.78027i) q^{2} +(-3.67405 - 2.66936i) q^{3} +(-0.441690 + 0.320907i) q^{4} +(1.90871 - 5.87440i) q^{5} +(4.10252 - 12.6263i) q^{6} +(-25.9914 + 18.8838i) q^{7} +(17.6291 + 12.8083i) q^{8} +(-1.97025 - 6.06380i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 7 q^{2} - 3 q^{3} + 3 q^{4} - 7 q^{5} - 29 q^{6} - 35 q^{7} + 47 q^{8} + 31 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.903364	+	2.78027i	0.319387	+	0.982973i	0.973911	+	0.226932i	\(0.0728694\pi\)
							−0.654523	+	0.756042i	\(0.727131\pi\)
\(3\)	−3.67405	−	2.66936i	−0.707072	−	0.513718i	0.175156	−	0.984541i	\(-0.443957\pi\)
							−0.882228	+	0.470823i	\(0.843957\pi\)
\(5\)	1.90871	−	5.87440i	0.170720	−	0.525422i	−0.828692	−	0.559705i	\(-0.810915\pi\)
							0.999412	+	0.0342824i	\(0.0109146\pi\)
\(7\)	−25.9914	+	18.8838i	−1.40340	+	1.01963i	−0.409161	+	0.912462i	\(0.634178\pi\)
							−0.994241	+	0.107169i	\(0.965822\pi\)
\(11\)	35.3867	+	8.87592i	0.969954	+	0.243290i
							\(13\)	−1.38969	−	4.27704i	−0.0296486	−	0.0912490i	0.935137	−	0.354286i	\(-0.115276\pi\)
−0.964786	+	0.263037i	\(0.915276\pi\)
\(17\)	18.3792	−	56.5653i	0.262212	−	0.807006i	−0.730111	−	0.683329i	\(-0.760531\pi\)
							0.992323	−	0.123677i	\(-0.0394686\pi\)
\(19\)	−23.4557	−	17.0416i	−0.283217	−	0.205769i	0.437103	−	0.899412i	\(-0.356005\pi\)
							−0.720319	+	0.693643i	\(0.756005\pi\)
\(23\)	−38.3004			−0.347226			−0.173613	−	0.984814i	\(-0.555544\pi\)
							−0.173613	+	0.984814i	\(0.555544\pi\)
\(29\)	−31.9032	+	23.1790i	−0.204285	+	0.148422i	−0.685224	−	0.728332i	\(-0.740296\pi\)
							0.480939	+	0.876754i	\(0.340296\pi\)
\(31\)	−82.2162	−	253.036i	−0.476338	−	1.46602i	−0.844145	−	0.536115i	\(-0.819891\pi\)
							0.367807	−	0.929902i	\(-0.380109\pi\)
\(37\)	−90.7978	+	65.9685i	−0.403434	+	0.293112i	−0.770938	−	0.636910i	\(-0.780212\pi\)
							0.367504	+	0.930022i	\(0.380212\pi\)
\(41\)	108.589	+	78.8943i	0.413627	+	0.300518i	0.775069	−	0.631877i	\(-0.217715\pi\)
							−0.361442	+	0.932395i	\(0.617715\pi\)
\(43\)	−252.470			−0.895380			−0.447690	−	0.894189i	\(-0.647753\pi\)
							−0.447690	+	0.894189i	\(0.647753\pi\)
\(47\)	147.464	+	107.139i	0.457656	+	0.332507i	0.792611	−	0.609727i	\(-0.208721\pi\)
							−0.334955	+	0.942234i	\(0.608721\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.4.c.a.5.2	✓	8
3.2	odd	2		99.4.f.c.82.1		8
4.3	odd	2		176.4.m.c.49.2		8
11.2	odd	10		121.4.c.i.9.1		8
11.3	even	5		121.4.a.g.1.3		4
11.4	even	5		121.4.c.h.3.1		8
11.5	even	5		121.4.c.h.81.1		8
11.6	odd	10		121.4.c.b.81.2		8
11.7	odd	10		121.4.c.b.3.2		8
11.8	odd	10		121.4.a.f.1.2		4
11.9	even	5	inner	11.4.c.a.9.2	yes	8
11.10	odd	2		121.4.c.i.27.1		8
33.8	even	10		1089.4.a.bh.1.3		4
33.14	odd	10		1089.4.a.y.1.2		4
33.20	odd	10		99.4.f.c.64.1		8
44.3	odd	10		1936.4.a.bk.1.2		4
44.19	even	10		1936.4.a.bl.1.2		4
44.31	odd	10		176.4.m.c.97.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.c.a.5.2	✓	8	1.1	even	1	trivial
11.4.c.a.9.2	yes	8	11.9	even	5	inner
99.4.f.c.64.1		8	33.20	odd	10
99.4.f.c.82.1		8	3.2	odd	2
121.4.a.f.1.2		4	11.8	odd	10
121.4.a.g.1.3		4	11.3	even	5
121.4.c.b.3.2		8	11.7	odd	10
121.4.c.b.81.2		8	11.6	odd	10
121.4.c.h.3.1		8	11.4	even	5
121.4.c.h.81.1		8	11.5	even	5
121.4.c.i.9.1		8	11.2	odd	10
121.4.c.i.27.1		8	11.10	odd	2
176.4.m.c.49.2		8	4.3	odd	2
176.4.m.c.97.2		8	44.31	odd	10
1089.4.a.y.1.2		4	33.14	odd	10
1089.4.a.bh.1.3		4	33.8	even	10
1936.4.a.bk.1.2		4	44.3	odd	10
1936.4.a.bl.1.2		4	44.19	even	10