Embedded newform 11.4.c.a.9.1

• Label: 11.4.c.a.9.1
• 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			9.1
Root			\(-1.09435 + 3.36805i\) of defining polynomial
Character	\(\chi\)	\(=\)	11.9
Dual form			11.4.c.a.5.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.976313 + 3.00478i) q^{2} +(1.24700 - 0.906001i) q^{3} +(-1.60339 - 1.16493i) q^{4} +(-5.33576 - 16.4218i) q^{5} +(1.50487 + 4.63152i) q^{6} +(7.73807 + 5.62204i) q^{7} +(-15.3824 + 11.1760i) q^{8} +(-7.60928 + 23.4190i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 7 q^{2} - 3 q^{3} + 3 q^{4} - 7 q^{5} - 29 q^{6} - 35 q^{7} + 47 q^{8} + 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.976313	+	3.00478i	−0.345179	+	1.06235i	0.616309	+	0.787504i	\(0.288627\pi\)
							−0.961488	+	0.274847i	\(0.911373\pi\)
\(3\)	1.24700	−	0.906001i	0.239986	−	0.174360i	−0.461291	−	0.887249i	\(-0.652614\pi\)
							0.701277	+	0.712889i	\(0.252614\pi\)
\(5\)	−5.33576	−	16.4218i	−0.477245	−	1.46881i	−0.842906	−	0.538060i	\(-0.819157\pi\)
							0.365661	−	0.930748i	\(-0.380843\pi\)
\(7\)	7.73807	+	5.62204i	0.417817	+	0.303562i	0.776759	−	0.629798i	\(-0.216862\pi\)
							−0.358942	+	0.933360i	\(0.616862\pi\)
\(11\)	−8.01537	−	35.5915i	−0.219702	−	0.975567i
							\(13\)	−2.00292	+	6.16434i	−0.0427314	+	0.131514i	−0.970146	−	0.242521i	\(-0.922026\pi\)
0.927415	+	0.374034i	\(0.122026\pi\)
\(17\)	29.2135	+	89.9099i	0.416783	+	1.28273i	0.910646	+	0.413187i	\(0.135585\pi\)
							−0.493863	+	0.869540i	\(0.664415\pi\)
\(19\)	42.5672	−	30.9269i	0.513978	−	0.373427i	−0.300353	−	0.953828i	\(-0.597104\pi\)
							0.814331	+	0.580401i	\(0.197104\pi\)
\(23\)	82.2579			0.745737			0.372868	−	0.927884i	\(-0.378374\pi\)
							0.372868	+	0.927884i	\(0.378374\pi\)
\(29\)	−121.265	−	88.1041i	−0.776494	−	0.564156i	0.127430	−	0.991848i	\(-0.459327\pi\)
							−0.903925	+	0.427691i	\(0.859327\pi\)
\(31\)	40.6007	−	124.956i	0.235229	−	0.723961i	−0.761862	−	0.647740i	\(-0.775714\pi\)
							0.997091	−	0.0762213i	\(-0.0242855\pi\)
\(37\)	99.3085	+	72.1518i	0.441249	+	0.320586i	0.786131	−	0.618060i	\(-0.212081\pi\)
							−0.344882	+	0.938646i	\(0.612081\pi\)
\(41\)	−153.793	+	111.737i	−0.585815	+	0.425619i	−0.840816	−	0.541321i	\(-0.817924\pi\)
							0.255001	+	0.966941i	\(0.417924\pi\)
\(43\)	−320.348			−1.13611			−0.568054	−	0.822991i	\(-0.692304\pi\)
							−0.568054	+	0.822991i	\(0.692304\pi\)
\(47\)	356.916	−	259.314i	1.10769	−	0.804785i	0.125393	−	0.992107i	\(-0.459981\pi\)
							0.982299	+	0.187322i	\(0.0599808\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.4.c.a.9.1	yes	8
3.2	odd	2		99.4.f.c.64.2		8
4.3	odd	2		176.4.m.c.97.1		8
11.2	odd	10		121.4.c.b.3.1		8
11.3	even	5		121.4.c.h.81.2		8
11.4	even	5		121.4.a.g.1.1		4
11.5	even	5	inner	11.4.c.a.5.1	✓	8
11.6	odd	10		121.4.c.i.27.2		8
11.7	odd	10		121.4.a.f.1.4		4
11.8	odd	10		121.4.c.b.81.1		8
11.9	even	5		121.4.c.h.3.2		8
11.10	odd	2		121.4.c.i.9.2		8
33.5	odd	10		99.4.f.c.82.2		8
33.26	odd	10		1089.4.a.y.1.4		4
33.29	even	10		1089.4.a.bh.1.1		4
44.7	even	10		1936.4.a.bl.1.3		4
44.15	odd	10		1936.4.a.bk.1.3		4
44.27	odd	10		176.4.m.c.49.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.c.a.5.1	✓	8	11.5	even	5	inner
11.4.c.a.9.1	yes	8	1.1	even	1	trivial
99.4.f.c.64.2		8	3.2	odd	2
99.4.f.c.82.2		8	33.5	odd	10
121.4.a.f.1.4		4	11.7	odd	10
121.4.a.g.1.1		4	11.4	even	5
121.4.c.b.3.1		8	11.2	odd	10
121.4.c.b.81.1		8	11.8	odd	10
121.4.c.h.3.2		8	11.9	even	5
121.4.c.h.81.2		8	11.3	even	5
121.4.c.i.9.2		8	11.10	odd	2
121.4.c.i.27.2		8	11.6	odd	10
176.4.m.c.49.1		8	44.27	odd	10
176.4.m.c.97.1		8	4.3	odd	2
1089.4.a.y.1.4		4	33.26	odd	10
1089.4.a.bh.1.1		4	33.29	even	10
1936.4.a.bk.1.3		4	44.15	odd	10
1936.4.a.bl.1.3		4	44.7	even	10