Embedded newform 11.13.d.a.2.11

• Label: 11.13.d.a.2.11
• 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			2.11
Character	\(\chi\)	\(=\)	11.2
Dual form			11.13.d.a.6.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(116.921 + 37.9900i) q^{2} +(-222.926 + 161.965i) q^{3} +(8913.56 + 6476.08i) q^{4} +(698.010 + 2148.25i) q^{5} +(-32217.8 + 10468.2i) q^{6} +(9655.01 - 13289.0i) q^{7} +(500175. + 688432. i) q^{8} +(-140761. + 433218. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 5 q^{2} - 1083 q^{3} + 16247 q^{4} - 723 q^{5} + 147195 q^{6} - 219605 q^{7} + 921595 q^{8} - 3695582 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	116.921	+	37.9900i	1.82689	+	0.593593i	0.999488	+	0.0319847i	\(0.0101828\pi\)
							0.827403	+	0.561608i	\(0.189817\pi\)
\(3\)	−222.926	+	161.965i	−0.305797	+	0.222174i	−0.730091	−	0.683350i	\(-0.760522\pi\)
							0.424294	+	0.905524i	\(0.360522\pi\)
\(5\)	698.010	+	2148.25i	0.0446727	+	0.137488i	0.970905	−	0.239465i	\(-0.0769719\pi\)
							−0.926232	+	0.376953i	\(0.876972\pi\)
\(7\)	9655.01	−	13289.0i	0.0820662	−	0.112955i	−0.766010	−	0.642828i	\(-0.777761\pi\)
							0.848076	+	0.529874i	\(0.177761\pi\)
\(11\)	856125.	−	1.55096e6i	0.483260	−	0.875477i
							\(13\)	−3.90111e6	−	1.26755e6i	−0.808217	−	0.262606i	−0.124375	−	0.992235i	\(-0.539693\pi\)
−0.683843	+	0.729630i	\(0.739693\pi\)
\(17\)	3.80432e7	−	1.23610e7i	1.57610	−	0.512106i	0.615051	−	0.788487i	\(-0.289135\pi\)
							0.961048	+	0.276382i	\(0.0891353\pi\)
\(19\)	−3.13996e7	−	4.32179e7i	−0.667426	−	0.918633i	0.332273	−	0.943183i	\(-0.392185\pi\)
							−0.999699	+	0.0245504i	\(0.992185\pi\)
\(23\)	−7.06799e7			−0.477451			−0.238726	−	0.971087i	\(-0.576730\pi\)
							−0.238726	+	0.971087i	\(0.576730\pi\)
\(29\)	−5.75463e7	+	7.92057e7i	−0.0967452	+	0.133158i	−0.854645	−	0.519212i	\(-0.826225\pi\)
							0.757900	+	0.652371i	\(0.226225\pi\)
\(31\)	2.87933e8	−	8.86168e8i	0.324431	−	0.998495i	−0.647266	−	0.762264i	\(-0.724088\pi\)
							0.971697	−	0.236231i	\(-0.0759121\pi\)
\(37\)	−3.20345e9	−	2.32744e9i	−1.24855	−	0.907128i	−0.250417	−	0.968138i	\(-0.580568\pi\)
							−0.998137	+	0.0610104i	\(0.980568\pi\)
\(41\)	1.52522e9	+	2.09929e9i	0.321093	+	0.441946i	0.938800	−	0.344462i	\(-0.111939\pi\)
							−0.617708	+	0.786408i	\(0.711939\pi\)
\(43\)			9.88951e9i			1.56446i	0.622991	+	0.782229i	\(0.285917\pi\)
							−0.622991	+	0.782229i	\(0.714083\pi\)
\(47\)	−4.66236e9	+	3.38740e9i	−0.432533	+	0.314253i	−0.782661	−	0.622449i	\(-0.786138\pi\)
							0.350128	+	0.936702i	\(0.386138\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.13.d.a.2.11	✓	44
11.6	odd	10	inner	11.13.d.a.6.11	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.13.d.a.2.11	✓	44	1.1	even	1	trivial
11.13.d.a.6.11	yes	44	11.6	odd	10	inner