Embedded newform 11.17.d.a.2.1

• Label: 11.17.d.a.2.1
• Level: $11$
• Weight: $17$
• Character: 11.2
• 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			2.1
Character	\(\chi\)	\(=\)	11.2
Dual form			11.17.d.a.6.1

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.17.d.a.2.1	✓	60	1.1	even	1	trivial
11.17.d.a.6.1	yes	60	11.6	odd	10	inner