Embedded newform 56.3.k.d.11.2

• Label: 56.3.k.d.11.2
• Level: $56$
• Weight: $3$
• Character: 56.11
• Analytic conductor: $1.526$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	56.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.52588948042\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - 3*x^11 - 4*x^10 + 3*x^9 + 86*x^8 - 163*x^7 + 155*x^6 - 166*x^5 + 164*x^4 - 116*x^3 + 60*x^2 - 20*x + 4
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.2
Root			\(-0.407369 - 0.812545i\) of defining polynomial
Character	\(\chi\)	\(=\)	56.11
Dual form			56.3.k.d.51.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19654 + 1.60259i) q^{2} +(-2.66613 + 4.61787i) q^{3} +(-1.13656 - 3.83513i) q^{4} +(1.86796 - 1.07847i) q^{5} +(-4.21039 - 9.79818i) q^{6} +(-6.91861 + 1.06433i) q^{7} +(7.50608 + 2.76746i) q^{8} +(-9.71647 - 16.8294i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{2} - 6 q^{3} - 4 q^{4} - 56 q^{6} + 8 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(17\)	\(29\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19654	+	1.60259i	−0.598272	+	0.801293i
							\(3\)	−2.66613	+	4.61787i	−0.888709	+	1.53929i	−0.0473064	+	0.998880i	\(0.515064\pi\)
−0.841403	+	0.540409i	\(0.818270\pi\)
\(5\)	1.86796	−	1.07847i	0.373592	−	0.215693i	−0.301435	−	0.953487i	\(-0.597466\pi\)
							0.675026	+	0.737794i	\(0.264132\pi\)
\(7\)	−6.91861	+	1.06433i	−0.988373	+	0.152047i
							\(11\)	−2.62956	+	4.55453i	−0.239051	+	0.414048i	−0.960442	−	0.278480i	\(-0.910170\pi\)
0.721392	+	0.692527i	\(0.243503\pi\)
\(13\)			21.4116i			1.64704i	0.567285	+	0.823522i	\(0.307994\pi\)
							−0.567285	+	0.823522i	\(0.692006\pi\)
\(17\)	−0.463429	+	0.802683i	−0.0272606	+	0.0472167i	−0.879334	−	0.476206i	\(-0.842012\pi\)
							0.852073	+	0.523423i	\(0.175345\pi\)
\(19\)	2.96505	+	5.13561i	0.156055	+	0.270295i	0.933443	−	0.358726i	\(-0.116789\pi\)
							−0.777388	+	0.629022i	\(0.783456\pi\)
\(23\)	7.52507	−	4.34460i	0.327177	−	0.188896i	−0.327410	−	0.944882i	\(-0.606176\pi\)
							0.654587	+	0.755987i	\(0.272843\pi\)
\(29\)		−	9.42223i		−	0.324904i	−0.986716	−	0.162452i	\(-0.948060\pi\)
							0.986716	−	0.162452i	\(-0.0519403\pi\)
\(31\)	29.8813	+	17.2520i	0.963912	+	0.556515i	0.897375	−	0.441269i	\(-0.145471\pi\)
							0.0665375	+	0.997784i	\(0.478805\pi\)
\(37\)	−11.0853	+	6.40011i	−0.299603	+	0.172976i	−0.642265	−	0.766483i	\(-0.722005\pi\)
							0.342662	+	0.939459i	\(0.388672\pi\)
\(41\)	43.1339			1.05205			0.526023	−	0.850470i	\(-0.323683\pi\)
							0.526023	+	0.850470i	\(0.323683\pi\)
\(43\)	−41.7382			−0.970656			−0.485328	−	0.874332i	\(-0.661300\pi\)
							−0.485328	+	0.874332i	\(0.661300\pi\)
\(47\)	39.8357	−	22.9991i	0.847567	−	0.489343i	−0.0122620	−	0.999925i	\(-0.503903\pi\)
							0.859829	+	0.510582i	\(0.170570\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	56.3.k.d.11.2	✓	12
4.3	odd	2		224.3.o.d.207.6		12
7.2	even	3	inner	56.3.k.d.51.6	yes	12
7.3	odd	6		392.3.g.i.99.3		6
7.4	even	3		392.3.g.j.99.3		6
7.5	odd	6		392.3.k.l.275.6		12
7.6	odd	2		392.3.k.l.67.2		12
8.3	odd	2	inner	56.3.k.d.11.6	yes	12
8.5	even	2		224.3.o.d.207.5		12
28.3	even	6		1568.3.g.l.687.5		6
28.11	odd	6		1568.3.g.j.687.2		6
28.23	odd	6		224.3.o.d.79.5		12
56.3	even	6		392.3.g.i.99.4		6
56.11	odd	6		392.3.g.j.99.4		6
56.19	even	6		392.3.k.l.275.2		12
56.27	even	2		392.3.k.l.67.6		12
56.37	even	6		224.3.o.d.79.6		12
56.45	odd	6		1568.3.g.l.687.6		6
56.51	odd	6	inner	56.3.k.d.51.2	yes	12
56.53	even	6		1568.3.g.j.687.1		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.k.d.11.2	✓	12	1.1	even	1	trivial
56.3.k.d.11.6	yes	12	8.3	odd	2	inner
56.3.k.d.51.2	yes	12	56.51	odd	6	inner
56.3.k.d.51.6	yes	12	7.2	even	3	inner
224.3.o.d.79.5		12	28.23	odd	6
224.3.o.d.79.6		12	56.37	even	6
224.3.o.d.207.5		12	8.5	even	2
224.3.o.d.207.6		12	4.3	odd	2
392.3.g.i.99.3		6	7.3	odd	6
392.3.g.i.99.4		6	56.3	even	6
392.3.g.j.99.3		6	7.4	even	3
392.3.g.j.99.4		6	56.11	odd	6
392.3.k.l.67.2		12	7.6	odd	2
392.3.k.l.67.6		12	56.27	even	2
392.3.k.l.275.2		12	56.19	even	6
392.3.k.l.275.6		12	7.5	odd	6
1568.3.g.j.687.1		6	56.53	even	6
1568.3.g.j.687.2		6	28.11	odd	6
1568.3.g.l.687.5		6	28.3	even	6
1568.3.g.l.687.6		6	56.45	odd	6