Embedded newform 11.18.c.a.4.4

• Label: 11.18.c.a.4.4
• Level: $11$
• Weight: $18$
• Character: 11.4
• Analytic conductor: $20.154$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(20.1544296079\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			4.4
Character	\(\chi\)	\(=\)	11.4
Dual form			11.18.c.a.3.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-329.817 - 239.626i) q^{2} +(5133.68 - 15799.8i) q^{3} +(10855.0 + 33408.3i) q^{4} +(530211. - 385221. i) q^{5} +(-5.47922e6 + 3.98089e6i) q^{6} +(2.00658e6 + 6.17563e6i) q^{7} +(-1.20870e7 + 3.71998e7i) q^{8} +(-1.18803e8 - 8.63158e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 795 q^{2} - 6360 q^{3} - 1037785 q^{4} - 91530 q^{5} + 11140861 q^{6} + 49560700 q^{7} + 9544335 q^{8} - 151892774 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−329.817	−	239.626i	−0.910999	−	0.661879i	0.0302687	−	0.999542i	\(-0.490364\pi\)
							−0.941267	+	0.337663i	\(0.890364\pi\)
\(3\)	5133.68	−	15799.8i	0.451750	−	1.39034i	−0.423159	−	0.906055i	\(-0.639079\pi\)
							0.874909	−	0.484287i	\(-0.160921\pi\)
\(5\)	530211.	−	385221.i	0.607021	−	0.441026i	−0.241343	−	0.970440i	\(-0.577588\pi\)
							0.848364	+	0.529413i	\(0.177588\pi\)
\(7\)	2.00658e6	+	6.17563e6i	0.131560	+	0.404900i	0.995039	−	0.0994845i	\(-0.0317194\pi\)
							−0.863479	+	0.504385i	\(0.831719\pi\)
\(11\)	−6.36686e8	+	3.16352e8i	−0.895545	+	0.444971i
							\(13\)	−1.75082e9	−	1.27204e9i	−0.595282	−	0.432498i	0.248919	−	0.968524i	\(-0.419925\pi\)
−0.844201	+	0.536027i	\(0.819925\pi\)
\(17\)	−2.68856e10	+	1.95335e10i	−0.934769	+	0.679149i	−0.947156	−	0.320774i	\(-0.896057\pi\)
							0.0123870	+	0.999923i	\(0.496057\pi\)
\(19\)	−3.73803e10	+	1.15045e11i	−0.504937	+	1.55404i	0.295940	+	0.955207i	\(0.404367\pi\)
							−0.800877	+	0.598829i	\(0.795633\pi\)
\(23\)	1.00155e11			0.266678			0.133339	−	0.991070i	\(-0.457430\pi\)
							0.133339	+	0.991070i	\(0.457430\pi\)
\(29\)	−3.79947e11	−	1.16936e12i	−0.141039	−	0.434074i	0.855441	−	0.517900i	\(-0.173286\pi\)
							−0.996480	+	0.0838259i	\(0.973286\pi\)
\(31\)	−2.18169e12	−	1.58509e12i	−0.459429	−	0.333794i	0.333878	−	0.942616i	\(-0.391642\pi\)
							−0.793307	+	0.608822i	\(0.791642\pi\)
\(37\)	−9.36771e12	−	2.88309e13i	−0.438449	−	1.34941i	−0.889511	−	0.456914i	\(-0.848955\pi\)
							0.451062	−	0.892493i	\(-0.351045\pi\)
\(41\)	9.88456e12	−	3.04216e13i	0.193328	−	0.595002i	−0.806664	−	0.591010i	\(-0.798729\pi\)
							0.999992	−	0.00399206i	\(-0.00127071\pi\)
\(43\)	−8.93947e13			−1.16635			−0.583176	−	0.812346i	\(-0.698190\pi\)
							−0.583176	+	0.812346i	\(0.698190\pi\)
\(47\)	−3.34981e12	+	1.03097e13i	−0.0205205	+	0.0631557i	−0.960792	−	0.277269i	\(-0.910571\pi\)
							0.940272	+	0.340424i	\(0.110571\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.18.c.a.4.4	yes	64
11.3	even	5	inner	11.18.c.a.3.4	✓	64
11.5	even	5		121.18.a.i.1.25		32
11.6	odd	10		121.18.a.j.1.8		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.18.c.a.3.4	✓	64	11.3	even	5	inner
11.18.c.a.4.4	yes	64	1.1	even	1	trivial
121.18.a.i.1.25		32	11.5	even	5
121.18.a.j.1.8		32	11.6	odd	10