Embedded newform 14.16.c.a.11.1

• Label: 14.16.c.a.11.1
• Level: $14$
• Weight: $16$
• Character: 14.11
• Analytic conductor: $19.977$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 14 = 2 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.9770907140\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Relative dimension:	\(5\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	x^10 - 2*x^9 + 11041039*x^8 + 10251836788*x^7 + 100086056086567*x^6 + 67517626050179350*x^5 + 266968630608932668831*x^4 + 130403899669863282233290*x^3 + 531639154391012701500584575*x^2 + 238948900825518555440992768500*x + 120052637231856930389048520650625
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{18}\cdot 3^{3}\cdot 5^{2}\cdot 7^{7} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			11.1
Root			\(-1239.01 - 2146.02i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.11
Dual form			14.16.c.a.9.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-64.0000 - 110.851i) q^{2} +(-2280.51 + 3949.96i) q^{3} +(-8192.00 + 14189.0i) q^{4} +(151908. + 263112. i) q^{5} +583811. q^{6} +(1.80034e6 - 1.22733e6i) q^{7} +2.09715e6 q^{8} +(-3.22701e6 - 5.58934e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 640 q^{2} + 1979 q^{3} - 81920 q^{4} + 300953 q^{5} - 506624 q^{6} + 4054316 q^{7} + 20971520 q^{8} - 17367046 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−64.0000	−	110.851i	−0.353553	−	0.612372i
							\(3\)	−2280.51	+	3949.96i	−0.602036	+	1.04276i	0.390476	+	0.920613i	\(0.372310\pi\)
−0.992512	+	0.122144i	\(0.961023\pi\)
\(5\)	151908.	+	263112.i	0.869570	+	1.50614i	0.862437	+	0.506164i	\(0.168937\pi\)
							0.00713256	+	0.999975i	\(0.497730\pi\)
\(7\)	1.80034e6	−	1.22733e6i	0.826264	−	0.563284i
							\(11\)	−4.22225e7	+	7.31316e7i	−0.653280	+	1.13151i	0.329042	+	0.944315i	\(0.393274\pi\)
−0.982322	+	0.187199i	\(0.940059\pi\)
\(13\)	2.50950e8			1.10920			0.554602	−	0.832115i	\(-0.312870\pi\)
							0.554602	+	0.832115i	\(0.312870\pi\)
\(17\)	7.61920e8	−	1.31968e9i	0.450342	−	0.780015i	−0.548065	−	0.836436i	\(-0.684635\pi\)
							0.998407	+	0.0564203i	\(0.0179687\pi\)
\(19\)	3.41200e9	+	5.90976e9i	0.875704	+	1.51676i	0.856011	+	0.516957i	\(0.172935\pi\)
							0.0196926	+	0.999806i	\(0.493731\pi\)
\(23\)	−2.60957e8	−	4.51991e8i	−0.0159812	−	0.0276803i	0.857924	−	0.513776i	\(-0.171754\pi\)
							−0.873905	+	0.486096i	\(0.838421\pi\)
\(29\)	−9.79433e10			−1.05436			−0.527181	−	0.849753i	\(-0.676751\pi\)
							−0.527181	+	0.849753i	\(0.676751\pi\)
\(31\)	−1.45644e10	+	2.52262e10i	−0.0950777	+	0.164679i	−0.909641	−	0.415395i	\(-0.863643\pi\)
							0.814563	+	0.580075i	\(0.196977\pi\)
\(37\)	−1.03107e11	−	1.78586e11i	−0.178556	−	0.309268i	0.762830	−	0.646599i	\(-0.223809\pi\)
							−0.941386	+	0.337331i	\(0.890476\pi\)
\(41\)	−6.30546e11			−0.505635			−0.252818	−	0.967514i	\(-0.581357\pi\)
							−0.252818	+	0.967514i	\(0.581357\pi\)
\(43\)	2.95173e11			0.165601			0.0828006	−	0.996566i	\(-0.473614\pi\)
							0.0828006	+	0.996566i	\(0.473614\pi\)
\(47\)	−2.25742e12	−	3.90997e12i	−0.649948	−	1.12574i	−0.983135	−	0.182882i	\(-0.941457\pi\)
							0.333187	−	0.942861i	\(-0.391876\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.16.c.a.11.1	yes	10
3.2	odd	2		126.16.g.e.109.1		10
7.2	even	3	inner	14.16.c.a.9.1	✓	10
7.3	odd	6		98.16.a.k.1.1		5
7.4	even	3		98.16.a.j.1.5		5
7.5	odd	6		98.16.c.m.79.5		10
7.6	odd	2		98.16.c.m.67.5		10
21.2	odd	6		126.16.g.e.37.1		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.16.c.a.9.1	✓	10	7.2	even	3	inner
14.16.c.a.11.1	yes	10	1.1	even	1	trivial
98.16.a.j.1.5		5	7.4	even	3
98.16.a.k.1.1		5	7.3	odd	6
98.16.c.m.67.5		10	7.6	odd	2
98.16.c.m.79.5		10	7.5	odd	6
126.16.g.e.37.1		10	21.2	odd	6
126.16.g.e.109.1		10	3.2	odd	2