Embedded newform 11.9.d.a.2.5

• Label: 11.9.d.a.2.5
• Level: $11$
• Weight: $9$
• Character: 11.2
• Analytic conductor: $4.481$
• Analytic rank: $0$
• Dimension: $28$
• 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			2.5
Character	\(\chi\)	\(=\)	11.2
Dual form			11.9.d.a.6.5

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

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.9.d.a.2.5	✓	28	1.1	even	1	trivial
11.9.d.a.6.5	yes	28	11.6	odd	10	inner
99.9.k.a.28.3		28	33.17	even	10
99.9.k.a.46.3		28	3.2	odd	2
121.9.b.b.120.12		28	11.7	odd	10
121.9.b.b.120.17		28	11.4	even	5