Embedded newform 14.11.d.a.5.2

• Label: 14.11.d.a.5.2
• Level: $14$
• Weight: $11$
• Character: 14.5
• Analytic conductor: $8.895$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 14 = 2 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	14.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.89500153743\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	x^12 + 7168*x^10 - 191104*x^9 + 39872585*x^8 - 837614684*x^7 + 83400850488*x^6 - 1089499860800*x^5 + 117569881085905*x^4 - 972128975671236*x^3 + 70590565447335180*x^2 + 627290085157877232*x + 16876036055394640656
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{24}\cdot 3^{4}\cdot 7^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.2
Root			\(21.7454 - 37.6641i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.5
Dual form			14.11.d.a.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.3137 + 19.5959i) q^{2} +(43.3342 - 25.0190i) q^{3} +(-256.000 - 443.405i) q^{4} +(1336.07 + 771.378i) q^{5} +1132.23i q^{6} +(14974.8 - 7630.84i) q^{7} +11585.2 q^{8} +(-28272.6 + 48969.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 486 q^{3} - 3072 q^{4} - 6666 q^{5} + 30576 q^{7} - 45528 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.3137	+	19.5959i	−0.353553	+	0.612372i
							\(3\)	43.3342	−	25.0190i	0.178330	−	0.102959i	−0.408178	−	0.912902i	\(-0.633836\pi\)
0.586508	+	0.809944i	\(0.300502\pi\)
\(5\)	1336.07	+	771.378i	0.427541	+	0.246841i	0.698299	−	0.715807i	\(-0.253941\pi\)
							−0.270757	+	0.962648i	\(0.587274\pi\)
\(7\)	14974.8	−	7630.84i	0.890988	−	0.454027i
							\(11\)	61649.5	+	106780.i	0.382795	+	0.663020i	0.991461	−	0.130406i	\(-0.0416282\pi\)
−0.608666	+	0.793427i	\(0.708295\pi\)
\(13\)			630844.i			1.69905i	0.527552	+	0.849523i	\(0.323110\pi\)
							−0.527552	+	0.849523i	\(0.676890\pi\)
\(17\)	479470.	−	276822.i	0.337689	−	0.194965i	−0.321561	−	0.946889i	\(-0.604207\pi\)
							0.659249	+	0.751924i	\(0.270874\pi\)
\(19\)	1.91745e6	+	1.10704e6i	0.774383	+	0.447090i	0.834436	−	0.551105i	\(-0.185794\pi\)
							−0.0600532	+	0.998195i	\(0.519127\pi\)
\(23\)	1.93777e6	−	3.35631e6i	0.301066	−	0.521462i	−0.675312	−	0.737533i	\(-0.735991\pi\)
							0.976378	+	0.216071i	\(0.0693241\pi\)
\(29\)	2.46767e7			1.20309			0.601543	−	0.798841i	\(-0.294553\pi\)
							0.601543	+	0.798841i	\(0.294553\pi\)
\(31\)	507444.	−	292973.i	0.0177247	−	0.0102334i	−0.491111	−	0.871097i	\(-0.663409\pi\)
							0.508836	+	0.860863i	\(0.330076\pi\)
\(37\)	−3.87143e7	+	6.70551e7i	−0.558293	+	0.966993i	0.439346	+	0.898318i	\(0.355210\pi\)
							−0.997639	+	0.0686745i	\(0.978123\pi\)
\(41\)			3.18486e7i			0.274898i	0.990509	+	0.137449i	\(0.0438903\pi\)
							−0.990509	+	0.137449i	\(0.956110\pi\)
\(43\)	3.16497e7			0.215292			0.107646	−	0.994189i	\(-0.465669\pi\)
							0.107646	+	0.994189i	\(0.465669\pi\)
\(47\)	−3.33431e8	−	1.92507e8i	−1.45384	−	0.839376i	−0.455145	−	0.890417i	\(-0.650413\pi\)
							−0.998697	+	0.0510417i	\(0.983746\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.11.d.a.5.2	yes	12
3.2	odd	2		126.11.n.b.19.4		12
4.3	odd	2		112.11.s.a.33.4		12
7.2	even	3		98.11.b.c.97.10		12
7.3	odd	6	inner	14.11.d.a.3.2	✓	12
7.4	even	3		98.11.d.a.31.2		12
7.5	odd	6		98.11.b.c.97.9		12
7.6	odd	2		98.11.d.a.19.2		12
21.17	even	6		126.11.n.b.73.4		12
28.3	even	6		112.11.s.a.17.4		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.11.d.a.3.2	✓	12	7.3	odd	6	inner
14.11.d.a.5.2	yes	12	1.1	even	1	trivial
98.11.b.c.97.9		12	7.5	odd	6
98.11.b.c.97.10		12	7.2	even	3
98.11.d.a.19.2		12	7.6	odd	2
98.11.d.a.31.2		12	7.4	even	3
112.11.s.a.17.4		12	28.3	even	6
112.11.s.a.33.4		12	4.3	odd	2
126.11.n.b.19.4		12	3.2	odd	2
126.11.n.b.73.4		12	21.17	even	6