Embedded newform 11.9.d.a.6.5

• Label: 11.9.d.a.6.5
• Level: $11$
• Weight: $9$
• Character: 11.6
• Analytic conductor: $4.481$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	11.d (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.48116471067\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			6.5
Character	\(\chi\)	\(=\)	11.6
Dual form			11.9.d.a.2.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.67744 - 2.16963i) q^{2} +(-62.7933 - 45.6220i) q^{3} +(-167.227 + 121.498i) q^{4} +(-217.294 + 668.763i) q^{5} +(-518.281 - 168.400i) q^{6} +(-790.545 - 1088.09i) q^{7} +(-1909.53 + 2628.24i) q^{8} +(-165.830 - 510.373i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 5 q^{2} + 144 q^{3} + 951 q^{4} - 708 q^{5} - 4485 q^{6} + 5470 q^{7} - 3845 q^{8} - 14225 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.67744	−	2.16963i	0.417340	−	0.135602i	−0.0928173	−	0.995683i	\(-0.529587\pi\)
							0.510157	+	0.860081i	\(0.329587\pi\)
\(3\)	−62.7933	−	45.6220i	−0.775226	−	0.563234i	0.128317	−	0.991733i	\(-0.459043\pi\)
							−0.903542	+	0.428499i	\(0.859043\pi\)
\(5\)	−217.294	+	668.763i	−0.347671	+	1.07002i	0.612467	+	0.790496i	\(0.290177\pi\)
							−0.960138	+	0.279525i	\(0.909823\pi\)
\(7\)	−790.545	−	1088.09i	−0.329256	−	0.453183i	0.612009	−	0.790851i	\(-0.290362\pi\)
							−0.941265	+	0.337668i	\(0.890362\pi\)
\(11\)	−14415.0	−	2562.63i	−0.984563	−	0.175031i
							\(13\)	18414.6	−	5983.26i	0.644746	−	0.209491i	0.0316498	−	0.999499i	\(-0.489924\pi\)
0.613096	+	0.790008i	\(0.289924\pi\)
\(17\)	65096.6	+	21151.2i	0.779404	+	0.253244i	0.671586	−	0.740927i	\(-0.265613\pi\)
							0.107819	+	0.994171i	\(0.465613\pi\)
\(19\)	−117001.	+	161038.i	−0.897793	+	1.23571i	0.0733740	+	0.997304i	\(0.476623\pi\)
							−0.971167	+	0.238401i	\(0.923377\pi\)
\(23\)	−350509.			−1.25253			−0.626265	−	0.779610i	\(-0.715417\pi\)
							−0.626265	+	0.779610i	\(0.715417\pi\)
\(29\)	224769.	+	309368.i	0.317793	+	0.437405i	0.937792	−	0.347198i	\(-0.112867\pi\)
							−0.619998	+	0.784603i	\(0.712867\pi\)
\(31\)	−312538.	−	961894.i	−0.338420	−	1.04155i	−0.965013	−	0.262203i	\(-0.915551\pi\)
							0.626592	−	0.779347i	\(-0.284449\pi\)
\(37\)	−310990.	+	225947.i	−0.165936	+	0.120559i	−0.667654	−	0.744472i	\(-0.732701\pi\)
							0.501718	+	0.865031i	\(0.332701\pi\)
\(41\)	−1.87467e6	+	2.58026e6i	−0.663420	+	0.913120i	−0.999589	−	0.0286824i	\(-0.990869\pi\)
							0.336168	+	0.941802i	\(0.390869\pi\)
\(43\)			823502.i			0.240875i	0.992721	+	0.120437i	\(0.0384297\pi\)
							−0.992721	+	0.120437i	\(0.961570\pi\)
\(47\)	410370.	+	298151.i	0.0840978	+	0.0611006i	0.629040	−	0.777373i	\(-0.283448\pi\)
							−0.544942	+	0.838474i	\(0.683448\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.9.d.a.6.5	yes	28
3.2	odd	2		99.9.k.a.28.3		28
11.2	odd	10	inner	11.9.d.a.2.5	✓	28
11.3	even	5		121.9.b.b.120.12		28
11.8	odd	10		121.9.b.b.120.17		28
33.2	even	10		99.9.k.a.46.3		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.9.d.a.2.5	✓	28	11.2	odd	10	inner
11.9.d.a.6.5	yes	28	1.1	even	1	trivial
99.9.k.a.28.3		28	3.2	odd	2
99.9.k.a.46.3		28	33.2	even	10
121.9.b.b.120.12		28	11.3	even	5
121.9.b.b.120.17		28	11.8	odd	10