Embedded newform 11.13.d.a.2.2

• Label: 11.13.d.a.2.2
• Level: $11$
• Weight: $13$
• Character: 11.2
• Analytic conductor: $10.054$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	11.d (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.0539319900\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			2.2
Character	\(\chi\)	\(=\)	11.2
Dual form			11.13.d.a.6.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-92.8176 - 30.1583i) q^{2} +(691.311 - 502.267i) q^{3} +(4391.85 + 3190.87i) q^{4} +(1083.89 + 3335.88i) q^{5} +(-79313.3 + 25770.5i) q^{6} +(-69230.1 + 95287.1i) q^{7} +(-76445.3 - 105218. i) q^{8} +(61414.8 - 189015. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 5 q^{2} - 1083 q^{3} + 16247 q^{4} - 723 q^{5} + 147195 q^{6} - 219605 q^{7} + 921595 q^{8} - 3695582 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−92.8176	−	30.1583i	−1.45027	−	0.471223i	−0.525190	−	0.850985i	\(-0.676006\pi\)
							−0.925084	+	0.379762i	\(0.876006\pi\)
\(3\)	691.311	−	502.267i	0.948301	−	0.688981i	−0.00210356	−	0.999998i	\(-0.500670\pi\)
							0.950404	+	0.311017i	\(0.100670\pi\)
\(5\)	1083.89	+	3335.88i	0.0693693	+	0.213497i	0.979731	−	0.200316i	\(-0.0641969\pi\)
							−0.910362	+	0.413813i	\(0.864197\pi\)
\(7\)	−69230.1	+	95287.1i	−0.588446	+	0.809927i	−0.994590	−	0.103883i	\(-0.966873\pi\)
							0.406143	+	0.913809i	\(0.366873\pi\)
\(11\)	1.64518e6	+	657119.i	0.928662	+	0.370926i
							\(13\)	4.52595e6	+	1.47057e6i	0.937668	+	0.304667i	0.737695	−	0.675134i	\(-0.235914\pi\)
0.199973	+	0.979801i	\(0.435914\pi\)
\(17\)	−7.78023e6	+	2.52795e6i	−0.322328	+	0.104731i	−0.465712	−	0.884936i	\(-0.654202\pi\)
							0.143384	+	0.989667i	\(0.454202\pi\)
\(19\)	−1.08616e6	−	1.49497e6i	−0.0230872	−	0.0317768i	0.797317	−	0.603560i	\(-0.206252\pi\)
							−0.820405	+	0.571783i	\(0.806252\pi\)
\(23\)	2.35715e8			1.59228			0.796141	−	0.605112i	\(-0.206871\pi\)
							0.796141	+	0.605112i	\(0.206871\pi\)
\(29\)	−3.92506e8	+	5.40238e8i	−0.659869	+	0.908232i	−0.999477	−	0.0323350i	\(-0.989706\pi\)
							0.339608	+	0.940567i	\(0.389706\pi\)
\(31\)	2.64449e8	−	8.13891e8i	0.297970	−	0.917056i	−0.684238	−	0.729259i	\(-0.739865\pi\)
							0.982208	−	0.187797i	\(-0.0601349\pi\)
\(37\)	−1.37581e9	−	9.99587e8i	−0.536227	−	0.389592i	0.286455	−	0.958094i	\(-0.407523\pi\)
							−0.822682	+	0.568502i	\(0.807523\pi\)
\(41\)	3.78919e9	+	5.21538e9i	0.797708	+	1.09795i	0.993105	+	0.117226i	\(0.0374003\pi\)
							−0.195398	+	0.980724i	\(0.562600\pi\)
\(43\)			2.12549e9i			0.336240i	0.985767	+	0.168120i	\(0.0537695\pi\)
							−0.985767	+	0.168120i	\(0.946230\pi\)
\(47\)	−1.36670e10	+	9.92965e9i	−1.26790	+	0.921185i	−0.999117	−	0.0420154i	\(-0.986622\pi\)
							−0.268785	+	0.963200i	\(0.586622\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.13.d.a.2.2	✓	44
11.6	odd	10	inner	11.13.d.a.6.2	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.13.d.a.2.2	✓	44	1.1	even	1	trivial
11.13.d.a.6.2	yes	44	11.6	odd	10	inner