Embedded newform 11.4.c.a.9.2

• Label: 11.4.c.a.9.2
• Level: $11$
• Weight: $4$
• Character: 11.9
• 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			9.2
Root			\(0.785330 - 2.41700i\) of defining polynomial
Character	\(\chi\)	\(=\)	11.9
Dual form			11.4.c.a.5.2

$q$-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 \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.c.a.5.2	✓	8	11.5	even	5	inner
11.4.c.a.9.2	yes	8	1.1	even	1	trivial
99.4.f.c.64.1		8	3.2	odd	2
99.4.f.c.82.1		8	33.5	odd	10
121.4.a.f.1.2		4	11.7	odd	10
121.4.a.g.1.3		4	11.4	even	5
121.4.c.b.3.2		8	11.2	odd	10
121.4.c.b.81.2		8	11.8	odd	10
121.4.c.h.3.1		8	11.9	even	5
121.4.c.h.81.1		8	11.3	even	5
121.4.c.i.9.1		8	11.10	odd	2
121.4.c.i.27.1		8	11.6	odd	10
176.4.m.c.49.2		8	44.27	odd	10
176.4.m.c.97.2		8	4.3	odd	2
1089.4.a.y.1.2		4	33.26	odd	10
1089.4.a.bh.1.3		4	33.29	even	10
1936.4.a.bk.1.2		4	44.15	odd	10
1936.4.a.bl.1.2		4	44.7	even	10