Embedded newform 14.13.d.a.3.1

• Label: 14.13.d.a.3.1
• Level: $14$
• Weight: $13$
• Character: 14.3
• Analytic conductor: $12.796$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 14 = 2 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	14.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.7959134419\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial:	x^16 + 527434*x^14 - 31307480*x^13 + 193554267483*x^12 - 12267558721140*x^11 + 36740634631350658*x^10 - 2684653740993140180*x^9 + 5051027688394458110177*x^8 - 329042050982173184619740*x^7 + 328571000441888143595366884*x^6 - 20799163177686535050419379920*x^5 + 15169931128056380497155715556632*x^4 - 685746033739471361446313307383040*x^3 + 159607415711477686151635843287775328*x^2 + 6015487435743293961505549572273215360*x + 735121207672317095821497912841120438336
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{48}\cdot 3^{8}\cdot 7^{7} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			3.1
Root			\(-213.975 - 370.616i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.3
Dual form			14.13.d.a.5.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-22.6274 - 39.1918i) q^{2} +(-745.979 - 430.691i) q^{3} +(-1024.00 + 1773.62i) q^{4} +(-7532.28 + 4348.76i) q^{5} +38981.7i q^{6} +(-93774.2 + 71047.1i) q^{7} +92681.9 q^{8} +(105269. + 182331. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16384 q^{4} + 18144 q^{5} - 469720 q^{7} + 2362248 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−22.6274	−	39.1918i	−0.353553	−	0.612372i
							\(3\)	−745.979	−	430.691i	−1.02329	−	0.590797i	−0.108235	−	0.994125i	\(-0.534520\pi\)
−0.915055	+	0.403328i	\(0.867853\pi\)
\(5\)	−7532.28	+	4348.76i	−0.482066	+	0.278321i	−0.721277	−	0.692647i	\(-0.756445\pi\)
							0.239211	+	0.970968i	\(0.423111\pi\)
\(7\)	−93774.2	+	71047.1i	−0.797067	+	0.603891i
							\(11\)	257922.	−	446735.i	0.145590	−	0.252170i	−0.784003	−	0.620758i	\(-0.786825\pi\)
0.929593	+	0.368587i	\(0.120158\pi\)
\(13\)		−	2.74786e6i		−	0.569292i	−0.958633	−	0.284646i	\(-0.908124\pi\)
							0.958633	−	0.284646i	\(-0.0918760\pi\)
\(17\)	5.50579e6	+	3.17877e6i	0.228100	+	0.131694i	0.609695	−	0.792636i	\(-0.291292\pi\)
							−0.381595	+	0.924330i	\(0.624625\pi\)
\(19\)	6.71736e7	−	3.87827e7i	1.42783	−	0.824359i	0.430882	−	0.902408i	\(-0.358203\pi\)
							0.996949	+	0.0780497i	\(0.0248693\pi\)
\(23\)	5.20289e7	+	9.01168e7i	0.351462	+	0.608750i	0.986506	−	0.163726i	\(-0.0523514\pi\)
							−0.635044	+	0.772476i	\(0.719018\pi\)
\(29\)	−1.41564e8			−0.237994			−0.118997	−	0.992895i	\(-0.537968\pi\)
							−0.118997	+	0.992895i	\(0.537968\pi\)
\(31\)	−6.99677e8	−	4.03959e8i	−0.788365	−	0.455163i	0.0510215	−	0.998698i	\(-0.483752\pi\)
							−0.839387	+	0.543535i	\(0.817086\pi\)
\(37\)	1.86759e9	+	3.23475e9i	0.727898	+	1.26076i	0.957770	+	0.287535i	\(0.0928357\pi\)
							−0.229873	+	0.973221i	\(0.573831\pi\)
\(41\)			4.75235e9i			1.00047i	0.865889	+	0.500236i	\(0.166754\pi\)
							−0.865889	+	0.500236i	\(0.833246\pi\)
\(43\)	8.94118e9			1.41444			0.707219	−	0.706994i	\(-0.249949\pi\)
							0.707219	+	0.706994i	\(0.249949\pi\)
\(47\)	1.79163e10	−	1.03440e10i	1.66211	−	0.959621i	0.690407	−	0.723421i	\(-0.257432\pi\)
							0.971705	−	0.236199i	\(-0.0759018\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.13.d.a.3.1	✓	16
3.2	odd	2		126.13.n.a.73.7		16
4.3	odd	2		112.13.s.c.17.7		16
7.2	even	3		98.13.d.a.19.4		16
7.3	odd	6		98.13.b.c.97.10		16
7.4	even	3		98.13.b.c.97.15		16
7.5	odd	6	inner	14.13.d.a.5.1	yes	16
7.6	odd	2		98.13.d.a.31.4		16
21.5	even	6		126.13.n.a.19.7		16
28.19	even	6		112.13.s.c.33.7		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.13.d.a.3.1	✓	16	1.1	even	1	trivial
14.13.d.a.5.1	yes	16	7.5	odd	6	inner
98.13.b.c.97.10		16	7.3	odd	6
98.13.b.c.97.15		16	7.4	even	3
98.13.d.a.19.4		16	7.2	even	3
98.13.d.a.31.4		16	7.6	odd	2
112.13.s.c.17.7		16	4.3	odd	2
112.13.s.c.33.7		16	28.19	even	6
126.13.n.a.19.7		16	21.5	even	6
126.13.n.a.73.7		16	3.2	odd	2