Embedded newform 14.13.d.a.5.5

• Label: 14.13.d.a.5.5
• Level: $14$
• Weight: $13$
• Character: 14.5
• 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			5.5
Root			\(234.838 - 406.752i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.5
Dual form			14.13.d.a.3.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(22.6274 - 39.1918i) q^{2} +(-1161.18 + 670.406i) q^{3} +(-1024.00 - 1773.62i) q^{4} +(14223.7 + 8212.03i) q^{5} +60678.2i q^{6} +(-96002.4 + 68006.1i) q^{7} -92681.9 q^{8} +(633167. - 1.09668e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16384 q^{4} + 18144 q^{5} - 469720 q^{7} + 2362248 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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
							\(3\)	−1161.18	+	670.406i	−1.59283	+	0.919623i	−0.600017	+	0.799987i	\(0.704839\pi\)
−0.992818	+	0.119636i	\(0.961827\pi\)
\(5\)	14223.7	+	8212.03i	0.910314	+	0.525570i	0.880532	−	0.473986i	\(-0.157185\pi\)
							0.0297818	+	0.999556i	\(0.490519\pi\)
\(7\)	−96002.4	+	68006.1i	−0.816007	+	0.578042i
							\(11\)	−464134.	−	803904.i	−0.261992	−	0.453783i	0.704780	−	0.709426i	\(-0.251046\pi\)
−0.966771	+	0.255644i	\(0.917713\pi\)
\(13\)		−	8.24891e6i		−	1.70898i	−0.519470	−	0.854489i	\(-0.673870\pi\)
							0.519470	−	0.854489i	\(-0.326130\pi\)
\(17\)	1.17285e7	−	6.77147e6i	0.485904	−	0.280537i	−0.236970	−	0.971517i	\(-0.576154\pi\)
							0.722874	+	0.690980i	\(0.242821\pi\)
\(19\)	3.88828e7	+	2.24490e7i	0.826487	+	0.477173i	0.852648	−	0.522485i	\(-0.174995\pi\)
							−0.0261613	+	0.999658i	\(0.508328\pi\)
\(23\)	3.53913e7	−	6.12994e7i	0.239072	−	0.414085i	−0.721376	−	0.692543i	\(-0.756490\pi\)
							0.960448	+	0.278458i	\(0.0898234\pi\)
\(29\)	−5.34101e8			−0.897915			−0.448957	−	0.893553i	\(-0.648205\pi\)
							−0.448957	+	0.893553i	\(0.648205\pi\)
\(31\)	1.14672e9	−	6.62060e8i	1.29208	−	0.745980i	0.313054	−	0.949735i	\(-0.398648\pi\)
							0.979022	+	0.203755i	\(0.0653146\pi\)
\(37\)	5.25480e8	−	9.10159e8i	0.204808	−	0.354737i	−0.745264	−	0.666770i	\(-0.767676\pi\)
							0.950071	+	0.312032i	\(0.101010\pi\)
\(41\)		−	1.78011e9i		−	0.374751i	−0.982288	−	0.187375i	\(-0.940002\pi\)
							0.982288	−	0.187375i	\(-0.0599981\pi\)
\(43\)	−4.20662e9			−0.665461			−0.332730	−	0.943022i	\(-0.607970\pi\)
							−0.332730	+	0.943022i	\(0.607970\pi\)
\(47\)	−1.50464e10	−	8.68706e9i	−1.39587	−	0.805908i	−0.401916	−	0.915676i	\(-0.631656\pi\)
							−0.993957	+	0.109768i	\(0.964989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.13.d.a.5.5	yes	16
3.2	odd	2		126.13.n.a.19.1		16
4.3	odd	2		112.13.s.c.33.8		16
7.2	even	3		98.13.b.c.97.1		16
7.3	odd	6	inner	14.13.d.a.3.5	✓	16
7.4	even	3		98.13.d.a.31.8		16
7.5	odd	6		98.13.b.c.97.8		16
7.6	odd	2		98.13.d.a.19.8		16
21.17	even	6		126.13.n.a.73.1		16
28.3	even	6		112.13.s.c.17.8		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.13.d.a.3.5	✓	16	7.3	odd	6	inner
14.13.d.a.5.5	yes	16	1.1	even	1	trivial
98.13.b.c.97.1		16	7.2	even	3
98.13.b.c.97.8		16	7.5	odd	6
98.13.d.a.19.8		16	7.6	odd	2
98.13.d.a.31.8		16	7.4	even	3
112.13.s.c.17.8		16	28.3	even	6
112.13.s.c.33.8		16	4.3	odd	2
126.13.n.a.19.1		16	3.2	odd	2
126.13.n.a.73.1		16	21.17	even	6