Embedded newform 14.16.c.b.11.5

• Label: 14.16.c.b.11.5
• 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			11.5
Root			\(-942.293 - 1632.10i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.11
Dual form			14.16.c.b.9.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(64.0000 + 110.851i) q^{2} +(3194.86 - 5533.66i) q^{3} +(-8192.00 + 14189.0i) q^{4} +(-99492.8 - 172327. i) q^{5} +817884. q^{6} +(2.03263e6 + 784833. i) q^{7} -2.09715e6 q^{8} +(-1.32398e7 - 2.29320e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 640 q^{2} + 2395 q^{3} - 81920 q^{4} + 10969 q^{5} + 613120 q^{6} + 2375276 q^{7} - 20971520 q^{8} - 45505414 q^{9}+O(q^{10}) \)

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
							\(3\)	3194.86	−	5533.66i	0.843417	−	1.46084i	−0.0435710	−	0.999050i	\(-0.513873\pi\)
0.886988	−	0.461792i	\(-0.152793\pi\)
\(5\)	−99492.8	−	172327.i	−0.569530	−	0.986455i	−0.996612	−	0.0822422i	\(-0.973792\pi\)
							0.427082	−	0.904213i	\(-0.359541\pi\)
\(7\)	2.03263e6	+	784833.i	0.932876	+	0.360198i
							\(11\)	2.41555e7	−	4.18386e7i	0.373741	−	0.647339i	−0.616396	−	0.787436i	\(-0.711408\pi\)
0.990138	+	0.140097i	\(0.0447414\pi\)
\(13\)	−2.49387e8			−1.10230			−0.551149	−	0.834407i	\(-0.685810\pi\)
							−0.551149	+	0.834407i	\(0.685810\pi\)
\(17\)	−9.16603e8	+	1.58760e9i	−0.541769	+	0.938372i	0.457033	+	0.889450i	\(0.348912\pi\)
							−0.998803	+	0.0489224i	\(0.984421\pi\)
\(19\)	−1.64208e9	−	2.84417e9i	−0.421446	−	0.729967i	0.574635	−	0.818410i	\(-0.305144\pi\)
							−0.996081	+	0.0884433i	\(0.971811\pi\)
\(23\)	−1.55962e10	−	2.70135e10i	−0.955127	−	1.65433i	−0.734077	−	0.679067i	\(-0.762385\pi\)
							−0.221050	−	0.975262i	\(-0.570949\pi\)
\(29\)	1.10574e11			1.19033			0.595166	−	0.803603i	\(-0.297086\pi\)
							0.595166	+	0.803603i	\(0.297086\pi\)
\(31\)	1.78686e10	−	3.09493e10i	0.116648	−	0.202040i	−0.801789	−	0.597607i	\(-0.796118\pi\)
							0.918437	+	0.395567i	\(0.129452\pi\)
\(37\)	4.58620e11	+	7.94354e11i	0.794219	+	1.37563i	0.923334	+	0.383998i	\(0.125453\pi\)
							−0.129115	+	0.991630i	\(0.541214\pi\)
\(41\)	−2.82110e11			−0.226225			−0.113112	−	0.993582i	\(-0.536082\pi\)
							−0.113112	+	0.993582i	\(0.536082\pi\)
\(43\)	1.12963e12			0.633758			0.316879	−	0.948466i	\(-0.397365\pi\)
							0.316879	+	0.948466i	\(0.397365\pi\)
\(47\)	−1.96516e12	−	3.40375e12i	−0.565800	−	0.979995i	−0.996975	−	0.0777263i	\(-0.975234\pi\)
							0.431174	−	0.902269i	\(-0.358099\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.16.c.b.11.5	yes	10
3.2	odd	2		126.16.g.c.109.5		10
7.2	even	3	inner	14.16.c.b.9.5	✓	10
7.3	odd	6		98.16.a.i.1.5		5
7.4	even	3		98.16.a.h.1.1		5
7.5	odd	6		98.16.c.n.79.1		10
7.6	odd	2		98.16.c.n.67.1		10
21.2	odd	6		126.16.g.c.37.5		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.16.c.b.9.5	✓	10	7.2	even	3	inner
14.16.c.b.11.5	yes	10	1.1	even	1	trivial
98.16.a.h.1.1		5	7.4	even	3
98.16.a.i.1.5		5	7.3	odd	6
98.16.c.n.67.1		10	7.6	odd	2
98.16.c.n.79.1		10	7.5	odd	6
126.16.g.c.37.5		10	21.2	odd	6
126.16.g.c.109.5		10	3.2	odd	2