Embedded newform 14.16.c.a.11.2

• Label: 14.16.c.a.11.2
• 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			11.2
Root			\(-879.415 - 1523.19i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.11
Dual form			14.16.c.a.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-64.0000 - 110.851i) q^{2} +(-1561.33 + 2704.30i) q^{3} +(-8192.00 + 14189.0i) q^{4} +(-101823. - 176362. i) q^{5} +399701. q^{6} +(1.89733e6 - 1.07131e6i) q^{7} +2.09715e6 q^{8} +(2.29895e6 + 3.98189e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 640 q^{2} + 1979 q^{3} - 81920 q^{4} + 300953 q^{5} - 506624 q^{6} + 4054316 q^{7} + 20971520 q^{8} - 17367046 q^{9}+O(q^{10}) \)

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
							\(3\)	−1561.33	+	2704.30i	−0.412179	+	0.713914i	−0.995128	−	0.0985943i	\(-0.968565\pi\)
0.582949	+	0.812509i	\(0.301899\pi\)
\(5\)	−101823.	−	176362.i	−0.582868	−	1.00956i	−0.995138	−	0.0984946i	\(-0.968597\pi\)
							0.412270	−	0.911062i	\(-0.364736\pi\)
\(7\)	1.89733e6	−	1.07131e6i	0.870778	−	0.491677i
							\(11\)	4.76832e7	−	8.25896e7i	0.737768	−	1.27785i	−0.215730	−	0.976453i	\(-0.569213\pi\)
0.953498	−	0.301399i	\(-0.0974535\pi\)
\(13\)	−3.63234e8			−1.60550			−0.802752	−	0.596312i	\(-0.796632\pi\)
							−0.802752	+	0.596312i	\(0.796632\pi\)
\(17\)	−1.36763e9	+	2.36880e9i	−0.808354	+	1.40011i	0.105650	+	0.994403i	\(0.466308\pi\)
							−0.914003	+	0.405707i	\(0.867026\pi\)
\(19\)	−4.59140e8	−	7.95255e8i	−0.117840	−	0.204105i	0.801071	−	0.598569i	\(-0.204264\pi\)
							−0.918912	+	0.394464i	\(0.870930\pi\)
\(23\)	1.44643e10	+	2.50529e10i	0.885805	+	1.53426i	0.844789	+	0.535100i	\(0.179726\pi\)
							0.0410159	+	0.999158i	\(0.486941\pi\)
\(29\)	−1.41399e11			−1.52216			−0.761080	−	0.648658i	\(-0.775331\pi\)
							−0.761080	+	0.648658i	\(0.775331\pi\)
\(31\)	−6.39485e9	+	1.10762e10i	−0.0417462	+	0.0723066i	−0.886144	−	0.463411i	\(-0.846625\pi\)
							0.844397	+	0.535717i	\(0.179959\pi\)
\(37\)	2.69720e11	+	4.67169e11i	0.467090	+	0.809024i	0.999293	−	0.0375927i	\(-0.0119689\pi\)
							−0.532203	+	0.846617i	\(0.678636\pi\)
\(41\)	−3.82990e11			−0.307120			−0.153560	−	0.988139i	\(-0.549074\pi\)
							−0.153560	+	0.988139i	\(0.549074\pi\)
\(43\)	−1.55388e12			−0.871773			−0.435887	−	0.900002i	\(-0.643565\pi\)
							−0.435887	+	0.900002i	\(0.643565\pi\)
\(47\)	5.31506e11	+	9.20595e11i	0.153029	+	0.265054i	0.932340	−	0.361584i	\(-0.117764\pi\)
							−0.779311	+	0.626638i	\(0.784431\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.16.c.a.11.2	yes	10
3.2	odd	2		126.16.g.e.109.5		10
7.2	even	3	inner	14.16.c.a.9.2	✓	10
7.3	odd	6		98.16.a.k.1.2		5
7.4	even	3		98.16.a.j.1.4		5
7.5	odd	6		98.16.c.m.79.4		10
7.6	odd	2		98.16.c.m.67.4		10
21.2	odd	6		126.16.g.e.37.5		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.16.c.a.9.2	✓	10	7.2	even	3	inner
14.16.c.a.11.2	yes	10	1.1	even	1	trivial
98.16.a.j.1.4		5	7.4	even	3
98.16.a.k.1.2		5	7.3	odd	6
98.16.c.m.67.4		10	7.6	odd	2
98.16.c.m.79.4		10	7.5	odd	6
126.16.g.e.37.5		10	21.2	odd	6
126.16.g.e.109.5		10	3.2	odd	2