Embedded newform 28.3.g.a.11.5

• Label: 28.3.g.a.11.5
• 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			11.5
Root			\(0.907369 + 0.0534805i\) of defining polynomial
Character	\(\chi\)	\(=\)	28.11
Dual form			28.3.g.a.23.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.19654 - 1.60259i) q^{2} +(-1.86796 - 1.07847i) q^{3} +(-1.13656 - 3.83513i) q^{4} +(3.25304 + 5.63443i) q^{5} +(-3.96343 + 1.70313i) q^{6} +(2.39669 + 6.57692i) q^{7} +(-7.50608 - 2.76746i) q^{8} +(-2.17382 - 3.76517i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} - 4 q^{4} - 2 q^{5} - 12 q^{6} - 8 q^{8} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.19654	−	1.60259i	0.598272	−	0.801293i
							\(3\)	−1.86796	−	1.07847i	−0.622653	−	0.359489i	0.155248	−	0.987875i	\(-0.450382\pi\)
−0.777901	+	0.628387i	\(0.783716\pi\)
\(5\)	3.25304	+	5.63443i	0.650608	+	1.12689i	0.982976	+	0.183736i	\(0.0588190\pi\)
							−0.332368	+	0.943150i	\(0.607848\pi\)
\(7\)	2.39669	+	6.57692i	0.342384	+	0.939560i
							\(11\)	−0.528732	−	0.305264i	−0.0480665	−	0.0277512i	0.475774	−	0.879567i	\(-0.342168\pi\)
−0.523841	+	0.851816i	\(0.675501\pi\)
\(13\)	−10.6645			−0.820347			−0.410173	−	0.912008i	\(-0.634532\pi\)
							−0.410173	+	0.912008i	\(0.634532\pi\)
\(17\)	5.99069	−	10.3762i	0.352393	−	0.610363i	−0.634275	−	0.773108i	\(-0.718701\pi\)
							0.986668	+	0.162744i	\(0.0520346\pi\)
\(19\)	−10.5811	+	6.10898i	−0.556898	+	0.321525i	−0.751900	−	0.659278i	\(-0.770862\pi\)
							0.195001	+	0.980803i	\(0.437529\pi\)
\(23\)	34.7524	−	20.0643i	1.51097	−	0.872361i	0.511055	−	0.859548i	\(-0.329255\pi\)
							0.999918	−	0.0128131i	\(-0.00407865\pi\)
\(29\)	−9.04293			−0.311825			−0.155913	−	0.987771i	\(-0.549832\pi\)
							−0.155913	+	0.987771i	\(0.549832\pi\)
\(31\)	−30.2408	−	17.4595i	−0.975509	−	0.563210i	−0.0745975	−	0.997214i	\(-0.523767\pi\)
							−0.900911	+	0.434004i	\(0.857101\pi\)
\(37\)	25.4082	+	44.0084i	0.686709	+	1.18941i	0.972896	+	0.231241i	\(0.0742787\pi\)
							−0.286187	+	0.958174i	\(0.592388\pi\)
\(41\)	25.7382			0.627761			0.313881	−	0.949462i	\(-0.398371\pi\)
							0.313881	+	0.949462i	\(0.398371\pi\)
\(43\)		−	19.8107i		−	0.460713i	−0.973106	−	0.230357i	\(-0.926011\pi\)
							0.973106	−	0.230357i	\(-0.0739893\pi\)
\(47\)	40.7870	−	23.5484i	0.867809	−	0.501030i	0.00118973	−	0.999999i	\(-0.499621\pi\)
							0.866620	+	0.498969i	\(0.166288\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.3.g.a.11.5	yes	12
3.2	odd	2		252.3.y.c.235.2		12
4.3	odd	2	inner	28.3.g.a.11.1	✓	12
7.2	even	3	inner	28.3.g.a.23.1	yes	12
7.3	odd	6		196.3.c.h.99.4		6
7.4	even	3		196.3.c.i.99.4		6
7.5	odd	6		196.3.g.i.79.1		12
7.6	odd	2		196.3.g.i.67.5		12
8.3	odd	2		448.3.r.h.319.2		12
8.5	even	2		448.3.r.h.319.5		12
12.11	even	2		252.3.y.c.235.6		12
21.2	odd	6		252.3.y.c.163.6		12
28.3	even	6		196.3.c.h.99.3		6
28.11	odd	6		196.3.c.i.99.3		6
28.19	even	6		196.3.g.i.79.5		12
28.23	odd	6	inner	28.3.g.a.23.5	yes	12
28.27	even	2		196.3.g.i.67.1		12
56.37	even	6		448.3.r.h.191.2		12
56.51	odd	6		448.3.r.h.191.5		12
84.23	even	6		252.3.y.c.163.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.3.g.a.11.1	✓	12	4.3	odd	2	inner
28.3.g.a.11.5	yes	12	1.1	even	1	trivial
28.3.g.a.23.1	yes	12	7.2	even	3	inner
28.3.g.a.23.5	yes	12	28.23	odd	6	inner
196.3.c.h.99.3		6	28.3	even	6
196.3.c.h.99.4		6	7.3	odd	6
196.3.c.i.99.3		6	28.11	odd	6
196.3.c.i.99.4		6	7.4	even	3
196.3.g.i.67.1		12	28.27	even	2
196.3.g.i.67.5		12	7.6	odd	2
196.3.g.i.79.1		12	7.5	odd	6
196.3.g.i.79.5		12	28.19	even	6
252.3.y.c.163.2		12	84.23	even	6
252.3.y.c.163.6		12	21.2	odd	6
252.3.y.c.235.2		12	3.2	odd	2
252.3.y.c.235.6		12	12.11	even	2
448.3.r.h.191.2		12	56.37	even	6
448.3.r.h.191.5		12	56.51	odd	6
448.3.r.h.319.2		12	8.3	odd	2
448.3.r.h.319.5		12	8.5	even	2