Embedded newform 14.11.d.a.3.4

• Label: 14.11.d.a.3.4
• Level: $14$
• Weight: $11$
• Character: 14.3
• Analytic conductor: $8.895$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 14 = 2 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	14.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.89500153743\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	x^12 + 7168*x^10 - 191104*x^9 + 39872585*x^8 - 837614684*x^7 + 83400850488*x^6 - 1089499860800*x^5 + 117569881085905*x^4 - 972128975671236*x^3 + 70590565447335180*x^2 + 627290085157877232*x + 16876036055394640656
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{24}\cdot 3^{4}\cdot 7^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			3.4
Root			\(26.6851 + 46.2199i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.3
Dual form			14.11.d.a.5.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.3137 + 19.5959i) q^{2} +(-102.202 - 59.0063i) q^{3} +(-256.000 + 443.405i) q^{4} +(32.1973 - 18.5891i) q^{5} -2670.32i q^{6} +(-11036.6 - 12675.5i) q^{7} -11585.2 q^{8} +(-22561.0 - 39076.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 486 q^{3} - 3072 q^{4} - 6666 q^{5} + 30576 q^{7} - 45528 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.3137	+	19.5959i	0.353553	+	0.612372i
							\(3\)	−102.202	−	59.0063i	−0.420584	−	0.242824i	0.274743	−	0.961518i	\(-0.411407\pi\)
−0.695327	+	0.718694i	\(0.744740\pi\)
\(5\)	32.1973	−	18.5891i	0.0103031	−	0.00594852i	−0.494840	−	0.868984i	\(-0.664773\pi\)
							0.505143	+	0.863036i	\(0.331440\pi\)
\(7\)	−11036.6	−	12675.5i	−0.656667	−	0.754181i
							\(11\)	67040.0	−	116117.i	0.416266	−	0.720993i	−0.579295	−	0.815118i	\(-0.696672\pi\)
0.995560	+	0.0941250i	\(0.0300053\pi\)
\(13\)		−	182276.i		−	0.490923i	−0.969406	−	0.245462i	\(-0.921060\pi\)
							0.969406	−	0.245462i	\(-0.0789396\pi\)
\(17\)	−1.39715e6	−	806648.i	−0.984011	−	0.568119i	−0.0805323	−	0.996752i	\(-0.525662\pi\)
							−0.903479	+	0.428633i	\(0.858995\pi\)
\(19\)	−1.40947e6	+	813758.i	−0.569230	+	0.328645i	−0.756842	−	0.653598i	\(-0.773259\pi\)
							0.187612	+	0.982243i	\(0.439925\pi\)
\(23\)	2.93644e6	+	5.08607e6i	0.456228	+	0.790211i	0.998758	−	0.0498260i	\(-0.0158667\pi\)
							−0.542530	+	0.840037i	\(0.682533\pi\)
\(29\)	4.34134e6			0.211657			0.105829	−	0.994384i	\(-0.466250\pi\)
							0.105829	+	0.994384i	\(0.466250\pi\)
\(31\)	−2.35447e7	−	1.35935e7i	−0.822403	−	0.474815i	0.0288413	−	0.999584i	\(-0.490818\pi\)
							−0.851244	+	0.524769i	\(0.824152\pi\)
\(37\)	−1.98766e7	−	3.44273e7i	−0.286638	−	0.496472i	0.686367	−	0.727255i	\(-0.259204\pi\)
							−0.973005	+	0.230784i	\(0.925871\pi\)
\(41\)		−	1.69868e8i		−	1.46619i	−0.680124	−	0.733097i	\(-0.738074\pi\)
							0.680124	−	0.733097i	\(-0.261926\pi\)
\(43\)	1.87794e8			1.27744			0.638719	−	0.769440i	\(-0.279465\pi\)
							0.638719	+	0.769440i	\(0.279465\pi\)
\(47\)	1.73904e7	−	1.00404e7i	0.0758264	−	0.0437784i	−0.461607	−	0.887084i	\(-0.652727\pi\)
							0.537434	+	0.843306i	\(0.319394\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.11.d.a.3.4	✓	12
3.2	odd	2		126.11.n.b.73.2		12
4.3	odd	2		112.11.s.a.17.5		12
7.2	even	3		98.11.d.a.19.6		12
7.3	odd	6		98.11.b.c.97.3		12
7.4	even	3		98.11.b.c.97.4		12
7.5	odd	6	inner	14.11.d.a.5.4	yes	12
7.6	odd	2		98.11.d.a.31.6		12
21.5	even	6		126.11.n.b.19.2		12
28.19	even	6		112.11.s.a.33.5		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.11.d.a.3.4	✓	12	1.1	even	1	trivial
14.11.d.a.5.4	yes	12	7.5	odd	6	inner
98.11.b.c.97.3		12	7.3	odd	6
98.11.b.c.97.4		12	7.4	even	3
98.11.d.a.19.6		12	7.2	even	3
98.11.d.a.31.6		12	7.6	odd	2
112.11.s.a.17.5		12	4.3	odd	2
112.11.s.a.33.5		12	28.19	even	6
126.11.n.b.19.2		12	21.5	even	6
126.11.n.b.73.2		12	3.2	odd	2