Embedded newform 28.3.g.a.11.2

• Label: 28.3.g.a.11.2
• Level: $28$
• Weight: $3$
• Character: 28.11
• Analytic conductor: $0.763$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	28.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.762944740209\)
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:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.2
Root			\(-2.29733 + 1.90372i\) of defining polynomial
Character	\(\chi\)	\(=\)	28.11
Dual form			28.3.g.a.23.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.51615 - 1.30434i) q^{2} +(-3.95004 - 2.28056i) q^{3} +(0.597396 + 3.95514i) q^{4} +(-2.62655 - 4.54932i) q^{5} +(3.01422 + 8.60985i) q^{6} +(5.86799 - 3.81663i) q^{7} +(4.25310 - 6.77577i) q^{8} +(5.90188 + 10.2224i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} - 4 q^{4} - 2 q^{5} - 12 q^{6} - 8 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.51615	−	1.30434i	−0.758073	−	0.652170i
							\(3\)	−3.95004	−	2.28056i	−1.31668	−	0.760186i	−0.333487	−	0.942755i	\(-0.608225\pi\)
−0.983193	+	0.182569i	\(0.941559\pi\)
\(5\)	−2.62655	−	4.54932i	−0.525310	−	0.909864i	−0.999565	−	0.0294768i	\(-0.990616\pi\)
							0.474255	−	0.880388i	\(-0.342717\pi\)
\(7\)	5.86799	−	3.81663i	0.838285	−	0.545233i
							\(11\)	−1.91795	−	1.10733i	−0.174359	−	0.100666i	0.410281	−	0.911959i	\(-0.365431\pi\)
−0.584640	+	0.811293i	\(0.698764\pi\)
\(13\)	−3.29755			−0.253658			−0.126829	−	0.991925i	\(-0.540480\pi\)
							−0.126829	+	0.991925i	\(0.540480\pi\)
\(17\)	6.69943	−	11.6038i	0.394084	−	0.682574i	−0.598900	−	0.800824i	\(-0.704395\pi\)
							0.992984	+	0.118250i	\(0.0377285\pi\)
\(19\)	6.72474	−	3.88253i	0.353934	−	0.204344i	−0.312483	−	0.949923i	\(-0.601161\pi\)
							0.666416	+	0.745580i	\(0.267827\pi\)
\(23\)	−3.66033	+	2.11329i	−0.159145	+	0.0918823i	−0.577457	−	0.816421i	\(-0.695955\pi\)
							0.418313	+	0.908303i	\(0.362622\pi\)
\(29\)	39.4113			1.35901			0.679505	−	0.733671i	\(-0.262195\pi\)
							0.679505	+	0.733671i	\(0.262195\pi\)
\(31\)	17.2001	+	9.93050i	0.554843	+	0.320339i	0.751073	−	0.660219i	\(-0.229537\pi\)
							−0.196230	+	0.980558i	\(0.562870\pi\)
\(37\)	12.8704	+	22.2922i	0.347850	+	0.602493i	0.985867	−	0.167529i	\(-0.0535787\pi\)
							−0.638018	+	0.770022i	\(0.720245\pi\)
\(41\)	−55.0188			−1.34192			−0.670961	−	0.741493i	\(-0.734118\pi\)
							−0.670961	+	0.741493i	\(0.734118\pi\)
\(43\)		−	78.2646i		−	1.82011i	−0.414490	−	0.910054i	\(-0.636040\pi\)
							0.414490	−	0.910054i	\(-0.363960\pi\)
\(47\)	−18.3993	+	10.6228i	−0.391474	+	0.226018i	−0.682799	−	0.730607i	\(-0.739237\pi\)
							0.291324	+	0.956624i	\(0.405904\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.3.g.a.11.2	✓	12
3.2	odd	2		252.3.y.c.235.5		12
4.3	odd	2	inner	28.3.g.a.11.3	yes	12
7.2	even	3	inner	28.3.g.a.23.3	yes	12
7.3	odd	6		196.3.c.h.99.5		6
7.4	even	3		196.3.c.i.99.5		6
7.5	odd	6		196.3.g.i.79.3		12
7.6	odd	2		196.3.g.i.67.2		12
8.3	odd	2		448.3.r.h.319.1		12
8.5	even	2		448.3.r.h.319.6		12
12.11	even	2		252.3.y.c.235.4		12
21.2	odd	6		252.3.y.c.163.4		12
28.3	even	6		196.3.c.h.99.6		6
28.11	odd	6		196.3.c.i.99.6		6
28.19	even	6		196.3.g.i.79.2		12
28.23	odd	6	inner	28.3.g.a.23.2	yes	12
28.27	even	2		196.3.g.i.67.3		12
56.37	even	6		448.3.r.h.191.1		12
56.51	odd	6		448.3.r.h.191.6		12
84.23	even	6		252.3.y.c.163.5		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.3.g.a.11.2	✓	12	1.1	even	1	trivial
28.3.g.a.11.3	yes	12	4.3	odd	2	inner
28.3.g.a.23.2	yes	12	28.23	odd	6	inner
28.3.g.a.23.3	yes	12	7.2	even	3	inner
196.3.c.h.99.5		6	7.3	odd	6
196.3.c.h.99.6		6	28.3	even	6
196.3.c.i.99.5		6	7.4	even	3
196.3.c.i.99.6		6	28.11	odd	6
196.3.g.i.67.2		12	7.6	odd	2
196.3.g.i.67.3		12	28.27	even	2
196.3.g.i.79.2		12	28.19	even	6
196.3.g.i.79.3		12	7.5	odd	6
252.3.y.c.163.4		12	21.2	odd	6
252.3.y.c.163.5		12	84.23	even	6
252.3.y.c.235.4		12	12.11	even	2
252.3.y.c.235.5		12	3.2	odd	2
448.3.r.h.191.1		12	56.37	even	6
448.3.r.h.191.6		12	56.51	odd	6
448.3.r.h.319.1		12	8.3	odd	2
448.3.r.h.319.6		12	8.5	even	2