Embedded newform 14.13.d.a.3.4

• Label: 14.13.d.a.3.4
• 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			3.4
Root			\(222.636 + 385.618i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.3
Dual form			14.13.d.a.5.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-22.6274 - 39.1918i) q^{2} +(1106.41 + 638.785i) q^{3} +(-1024.00 + 1773.62i) q^{4} +(24805.4 - 14321.4i) q^{5} -57816.2i q^{6} +(-66642.6 + 96953.8i) q^{7} +92681.9 q^{8} +(550372. + 953273. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16384 q^{4} + 18144 q^{5} - 469720 q^{7} + 2362248 q^{9}+O(q^{10}) \)

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
							\(3\)	1106.41	+	638.785i	1.51771	+	0.876248i	0.999783	+	0.0208214i	\(0.00662814\pi\)
0.517923	+	0.855427i	\(0.326705\pi\)
\(5\)	24805.4	−	14321.4i	1.58755	−	0.916571i	0.593838	−	0.804585i	\(-0.297612\pi\)
							0.993710	−	0.111986i	\(-0.0357213\pi\)
\(7\)	−66642.6	+	96953.8i	−0.566453	+	0.824094i
							\(11\)	−318316.	+	551340.i	−0.179681	+	0.311217i	−0.941771	−	0.336254i	\(-0.890840\pi\)
0.762090	+	0.647471i	\(0.224173\pi\)
\(13\)		−	3.67661e6i		−	0.761706i	−0.924636	−	0.380853i	\(-0.875630\pi\)
							0.924636	−	0.380853i	\(-0.124370\pi\)
\(17\)	1.12391e7	+	6.48891e6i	0.465627	+	0.268830i	0.714408	−	0.699730i	\(-0.246696\pi\)
							−0.248780	+	0.968560i	\(0.580030\pi\)
\(19\)	1.90911e7	−	1.10223e7i	0.405798	−	0.234288i	−0.283184	−	0.959065i	\(-0.591391\pi\)
							0.688983	+	0.724778i	\(0.258058\pi\)
\(23\)	5.48310e7	+	9.49701e7i	0.370390	+	0.641534i	0.989625	−	0.143671i	\(-0.0458907\pi\)
							−0.619236	+	0.785205i	\(0.712557\pi\)
\(29\)	−1.71855e8			−0.288917			−0.144459	−	0.989511i	\(-0.546144\pi\)
							−0.144459	+	0.989511i	\(0.546144\pi\)
\(31\)	−1.21966e8	−	7.04171e7i	−0.137426	−	0.0793429i	0.429711	−	0.902967i	\(-0.358616\pi\)
							−0.567137	+	0.823624i	\(0.691949\pi\)
\(37\)	3.10074e8	+	5.37063e8i	0.120852	+	0.209322i	0.920104	−	0.391674i	\(-0.128104\pi\)
							−0.799252	+	0.600996i	\(0.794771\pi\)
\(41\)		−	2.48871e9i		−	0.523927i	−0.965078	−	0.261963i	\(-0.915630\pi\)
							0.965078	−	0.261963i	\(-0.0843700\pi\)
\(43\)	−1.21787e10			−1.92660			−0.963298	−	0.268436i	\(-0.913493\pi\)
							−0.963298	+	0.268436i	\(0.913493\pi\)
\(47\)	−5.32001e9	+	3.07151e9i	−0.493543	+	0.284947i	−0.726043	−	0.687649i	\(-0.758643\pi\)
							0.232500	+	0.972596i	\(0.425309\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.13.d.a.3.4	✓	16
3.2	odd	2		126.13.n.a.73.5		16
4.3	odd	2		112.13.s.c.17.1		16
7.2	even	3		98.13.d.a.19.1		16
7.3	odd	6		98.13.b.c.97.16		16
7.4	even	3		98.13.b.c.97.9		16
7.5	odd	6	inner	14.13.d.a.5.4	yes	16
7.6	odd	2		98.13.d.a.31.1		16
21.5	even	6		126.13.n.a.19.5		16
28.19	even	6		112.13.s.c.33.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.13.d.a.3.4	✓	16	1.1	even	1	trivial
14.13.d.a.5.4	yes	16	7.5	odd	6	inner
98.13.b.c.97.9		16	7.4	even	3
98.13.b.c.97.16		16	7.3	odd	6
98.13.d.a.19.1		16	7.2	even	3
98.13.d.a.31.1		16	7.6	odd	2
112.13.s.c.17.1		16	4.3	odd	2
112.13.s.c.33.1		16	28.19	even	6
126.13.n.a.19.5		16	21.5	even	6
126.13.n.a.73.5		16	3.2	odd	2