Embedded newform 11.12.c.a.3.7

• Label: 11.12.c.a.3.7
• Level: $11$
• Weight: $12$
• Character: 11.3
• Analytic conductor: $8.452$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.45177498616\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			3.7
Character	\(\chi\)	\(=\)	11.3
Dual form			11.12.c.a.4.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(21.2132 - 15.4123i) q^{2} +(-143.663 - 442.150i) q^{3} +(-420.407 + 1293.88i) q^{4} +(-5027.95 - 3653.02i) q^{5} +(-9862.09 - 7165.22i) q^{6} +(-13499.1 + 41546.0i) q^{7} +(27617.8 + 84998.8i) q^{8} +(-31542.8 + 22917.2i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} - 276 q^{3} - 6137 q^{4} + 6038 q^{5} - 30371 q^{6} + 55440 q^{7} - 347809 q^{8} - 240812 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	21.2132	−	15.4123i	0.468749	−	0.340566i	−0.328205	−	0.944607i	\(-0.606444\pi\)
							0.796953	+	0.604041i	\(0.206444\pi\)
\(3\)	−143.663	−	442.150i	−0.341334	−	1.05052i	−0.963517	−	0.267646i	\(-0.913754\pi\)
							0.622184	−	0.782871i	\(-0.286246\pi\)
\(5\)	−5027.95	−	3653.02i	−0.719541	−	0.522777i	0.166697	−	0.986008i	\(-0.446690\pi\)
							−0.886238	+	0.463231i	\(0.846690\pi\)
\(7\)	−13499.1	+	41546.0i	−0.303575	+	0.934309i	0.676630	+	0.736324i	\(0.263440\pi\)
							−0.980205	+	0.197985i	\(0.936560\pi\)
\(11\)	−530273.	−	64208.3i	−0.992749	−	0.120208i
							\(13\)	−1.32258e6	+	960914.i	−0.987950	+	0.717788i	−0.959471	−	0.281806i	\(-0.909066\pi\)
−0.0284790	+	0.999594i	\(0.509066\pi\)
\(17\)	−2.07776e6	−	1.50958e6i	−0.354916	−	0.257862i	0.396012	−	0.918245i	\(-0.370394\pi\)
							−0.750929	+	0.660383i	\(0.770394\pi\)
\(19\)	−4.25898e6	−	1.31078e7i	−0.394603	−	1.21446i	−0.929270	−	0.369401i	\(-0.879563\pi\)
							0.534667	−	0.845063i	\(-0.320437\pi\)
\(23\)	3.68903e7			1.19511			0.597557	−	0.801826i	\(-0.296138\pi\)
							0.597557	+	0.801826i	\(0.296138\pi\)
\(29\)	−5.11328e6	+	1.57370e7i	−0.0462925	+	0.142474i	−0.971531	−	0.236912i	\(-0.923865\pi\)
							0.925239	+	0.379386i	\(0.123865\pi\)
\(31\)	−1.86432e7	+	1.35451e7i	−0.116958	+	0.0849750i	−0.644727	−	0.764413i	\(-0.723029\pi\)
							0.527769	+	0.849388i	\(0.323029\pi\)
\(37\)	−1.88451e8	+	5.79993e8i	−0.446776	+	1.37503i	0.433749	+	0.901034i	\(0.357191\pi\)
							−0.880525	+	0.474001i	\(0.842809\pi\)
\(41\)	−2.46174e8	−	7.57646e8i	−0.331842	−	1.02130i	−0.968257	−	0.249957i	\(-0.919583\pi\)
							0.636415	−	0.771347i	\(-0.280417\pi\)
\(43\)	−8.98553e8			−0.932110			−0.466055	−	0.884756i	\(-0.654325\pi\)
							−0.466055	+	0.884756i	\(0.654325\pi\)
\(47\)	5.90561e8	+	1.81756e9i	0.375601	+	1.15598i	0.943073	+	0.332587i	\(0.107922\pi\)
							−0.567472	+	0.823393i	\(0.692078\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.12.c.a.3.7	✓	40
3.2	odd	2		99.12.f.a.91.4		40
11.2	odd	10		121.12.a.h.1.14		20
11.4	even	5	inner	11.12.c.a.4.7	yes	40
11.9	even	5		121.12.a.j.1.7		20
33.26	odd	10		99.12.f.a.37.4		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.12.c.a.3.7	✓	40	1.1	even	1	trivial
11.12.c.a.4.7	yes	40	11.4	even	5	inner
99.12.f.a.37.4		40	33.26	odd	10
99.12.f.a.91.4		40	3.2	odd	2
121.12.a.h.1.14		20	11.2	odd	10
121.12.a.j.1.7		20	11.9	even	5