Embedded newform 11.17.d.a.6.1

• Label: 11.17.d.a.6.1
• Level: $11$
• Weight: $17$
• Character: 11.6
• Analytic conductor: $17.856$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 17 \)
Character orbit:	\([\chi]\)	\(=\)	11.d (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.8556998242\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(15\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			6.1
Character	\(\chi\)	\(=\)	11.6
Dual form			11.17.d.a.2.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-416.677 + 135.386i) q^{2} +(-4354.61 - 3163.81i) q^{3} +(102270. - 74303.7i) q^{4} +(161336. - 496541. i) q^{5} +(2.24280e6 + 728730. i) q^{6} +(1.02203e6 + 1.40670e6i) q^{7} +(-1.56771e7 + 2.15776e7i) q^{8} +(-4.34923e6 - 1.33856e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 5 q^{2} - 150 q^{3} + 542263 q^{4} - 7938 q^{5} - 2437125 q^{6} + 2885320 q^{7} + 30474235 q^{8} - 295234379 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−416.677	+	135.386i	−1.62764	+	0.528853i	−0.973728	−	0.227714i	\(-0.926875\pi\)
							−0.653915	+	0.756568i	\(0.726875\pi\)
\(3\)	−4354.61	−	3163.81i	−0.663711	−	0.482214i	0.204203	−	0.978929i	\(-0.434540\pi\)
							−0.867914	+	0.496714i	\(0.834540\pi\)
\(5\)	161336.	−	496541.i	0.413020	−	1.27115i	−0.500990	−	0.865453i	\(-0.667031\pi\)
							0.914010	−	0.405692i	\(-0.132969\pi\)
\(7\)	1.02203e6	+	1.40670e6i	0.177287	+	0.244015i	0.888408	−	0.459055i	\(-0.151812\pi\)
							−0.711120	+	0.703070i	\(0.751812\pi\)
\(11\)	5.44587e7	+	2.07326e8i	0.254054	+	0.967190i
							\(13\)	9.24565e8	−	3.00409e8i	1.13342	−	0.368270i	0.318545	−	0.947908i	\(-0.396806\pi\)
0.814874	+	0.579638i	\(0.196806\pi\)
\(17\)	2.98073e9	+	9.68497e8i	0.427298	+	0.138838i	0.514768	−	0.857330i	\(-0.327878\pi\)
							−0.0874698	+	0.996167i	\(0.527878\pi\)
\(19\)	1.59793e10	−	2.19937e10i	0.940871	−	1.29500i	−0.0145935	−	0.999894i	\(-0.504645\pi\)
							0.955465	−	0.295105i	\(-0.0953546\pi\)
\(23\)	7.00882e10			0.894999			0.447499	−	0.894284i	\(-0.352315\pi\)
							0.447499	+	0.894284i	\(0.352315\pi\)
\(29\)	−2.51868e11	−	3.46667e11i	−0.503489	−	0.692993i	0.479316	−	0.877643i	\(-0.340885\pi\)
							−0.982804	+	0.184650i	\(0.940885\pi\)
\(31\)	2.78444e11	+	8.56961e11i	0.326470	+	1.00477i	0.970773	+	0.240001i	\(0.0771478\pi\)
							−0.644302	+	0.764771i	\(0.722852\pi\)
\(37\)	2.81012e12	−	2.04167e12i	0.800038	−	0.581262i	−0.110887	−	0.993833i	\(-0.535369\pi\)
							0.910925	+	0.412571i	\(0.135369\pi\)
\(41\)	1.41390e12	−	1.94607e12i	0.177071	−	0.243718i	−0.711251	−	0.702938i	\(-0.751871\pi\)
							0.888322	+	0.459220i	\(0.151871\pi\)
\(43\)		−	4.76661e12i		−	0.407814i	−0.978990	−	0.203907i	\(-0.934636\pi\)
							0.978990	−	0.203907i	\(-0.0653640\pi\)
\(47\)	−3.05840e13	−	2.22206e13i	−1.28443	−	0.933196i	−0.284757	−	0.958600i	\(-0.591913\pi\)
							−0.999677	+	0.0254032i	\(0.991913\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.17.d.a.6.1	yes	60
11.2	odd	10	inner	11.17.d.a.2.1	✓	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.17.d.a.2.1	✓	60	11.2	odd	10	inner
11.17.d.a.6.1	yes	60	1.1	even	1	trivial