Embedded newform 14.16.c.b.11.3

• Label: 14.16.c.b.11.3
• 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 - x^9 + 7979102*x^8 - 3342530557*x^7 + 48610066963550*x^6 - 20531463882664153*x^5 + 122945882607722292337*x^4 - 89586690298994071428648*x^3 + 238785864099387585455090736*x^2 - 108333121326673799746890760320*x + 51761574501460447109413728819456
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{18}\cdot 3^{3}\cdot 5^{4}\cdot 7^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			11.3
Root			\(259.792 + 449.973i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.11
Dual form			14.16.c.b.9.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(64.0000 + 110.851i) q^{2} +(350.525 - 607.127i) q^{3} +(-8192.00 + 14189.0i) q^{4} +(-6451.18 - 11173.8i) q^{5} +89734.4 q^{6} +(-117963. - 2.17569e6i) q^{7} -2.09715e6 q^{8} +(6.92872e6 + 1.20009e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 640 q^{2} + 2395 q^{3} - 81920 q^{4} + 10969 q^{5} + 613120 q^{6} + 2375276 q^{7} - 20971520 q^{8} - 45505414 q^{9}+O(q^{10}) \)

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
							\(3\)	350.525	−	607.127i	0.0925357	−	0.160277i	−0.816042	−	0.577993i	\(-0.803836\pi\)
0.908577	+	0.417716i	\(0.137169\pi\)
\(5\)	−6451.18	−	11173.8i	−0.0369287	−	0.0639624i	0.846970	−	0.531640i	\(-0.178424\pi\)
							−0.883899	+	0.467678i	\(0.845091\pi\)
\(7\)	−117963.	−	2.17569e6i	−0.0541388	−	0.998533i
							\(11\)	4.09267e7	−	7.08871e7i	0.633230	−	1.09679i	−0.353657	−	0.935375i	\(-0.615062\pi\)
0.986887	−	0.161411i	\(-0.0516046\pi\)
\(13\)	−3.93201e7			−0.173796			−0.0868979	−	0.996217i	\(-0.527695\pi\)
							−0.0868979	+	0.996217i	\(0.527695\pi\)
\(17\)	1.47773e9	−	2.55950e9i	0.873428	−	1.51282i	0.0149997	−	0.999887i	\(-0.495225\pi\)
							0.858428	−	0.512934i	\(-0.171441\pi\)
\(19\)	2.67100e9	+	4.62630e9i	0.685522	+	1.18736i	0.973273	+	0.229653i	\(0.0737592\pi\)
							−0.287751	+	0.957705i	\(0.592907\pi\)
\(23\)	8.95531e9	+	1.55111e10i	0.548431	+	0.949910i	0.998382	+	0.0568570i	\(0.0181079\pi\)
							−0.449952	+	0.893053i	\(0.648559\pi\)
\(29\)	1.47317e11			1.58587			0.792936	−	0.609305i	\(-0.208552\pi\)
							0.792936	+	0.609305i	\(0.208552\pi\)
\(31\)	2.97885e10	−	5.15951e10i	0.194462	−	0.336819i	−0.752262	−	0.658864i	\(-0.771037\pi\)
							0.946724	+	0.322046i	\(0.104370\pi\)
\(37\)	−1.06016e11	−	1.83624e11i	−0.183593	−	0.317993i	0.759508	−	0.650498i	\(-0.225440\pi\)
							−0.943102	+	0.332505i	\(0.892106\pi\)
\(41\)	−1.79271e12			−1.43758			−0.718788	−	0.695230i	\(-0.755303\pi\)
							−0.718788	+	0.695230i	\(0.755303\pi\)
\(43\)	−9.09110e11			−0.510038			−0.255019	−	0.966936i	\(-0.582082\pi\)
							−0.255019	+	0.966936i	\(0.582082\pi\)
\(47\)	−2.19025e12	−	3.79363e12i	−0.630609	−	1.09225i	−0.987427	−	0.158073i	\(-0.949472\pi\)
							0.356819	−	0.934174i	\(-0.383861\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.16.c.b.11.3	yes	10
3.2	odd	2		126.16.g.c.109.3		10
7.2	even	3	inner	14.16.c.b.9.3	✓	10
7.3	odd	6		98.16.a.i.1.3		5
7.4	even	3		98.16.a.h.1.3		5
7.5	odd	6		98.16.c.n.79.3		10
7.6	odd	2		98.16.c.n.67.3		10
21.2	odd	6		126.16.g.c.37.3		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.16.c.b.9.3	✓	10	7.2	even	3	inner
14.16.c.b.11.3	yes	10	1.1	even	1	trivial
98.16.a.h.1.3		5	7.4	even	3
98.16.a.i.1.3		5	7.3	odd	6
98.16.c.n.67.3		10	7.6	odd	2
98.16.c.n.79.3		10	7.5	odd	6
126.16.g.c.37.3		10	21.2	odd	6
126.16.g.c.109.3		10	3.2	odd	2