Embedded newform 14.6.c.b.11.1

• Label: 14.6.c.b.11.1
• Level: $14$
• Weight: $6$
• Character: 14.11
• Analytic conductor: $2.245$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 14 = 2 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.24537347738\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{130})\)
Defining polynomial:	\( x^{4} + 130x^{2} + 16900 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			11.1
Root			\(5.70088 - 9.87421i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.11
Dual form			14.6.c.b.9.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.00000 + 3.46410i) q^{2} +(-7.90175 + 13.6862i) q^{3} +(-8.00000 + 13.8564i) q^{4} +(10.5000 + 18.1865i) q^{5} -63.2140 q^{6} +(126.411 - 28.7642i) q^{7} -64.0000 q^{8} +(-3.37544 - 5.84643i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 14 q^{3} - 32 q^{4} + 42 q^{5} + 112 q^{6} + 232 q^{7} - 256 q^{8} - 652 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00000	+	3.46410i	0.353553	+	0.612372i
							\(3\)	−7.90175	+	13.6862i	−0.506898	+	0.877973i	0.493070	+	0.869989i	\(0.335875\pi\)
−0.999968	+	0.00798326i	\(0.997459\pi\)
\(5\)	10.5000	+	18.1865i	0.187830	+	0.325331i	0.944526	−	0.328436i	\(-0.106521\pi\)
							−0.756697	+	0.653766i	\(0.773188\pi\)
\(7\)	126.411	−	28.7642i	0.975075	−	0.221874i
							\(11\)	312.937	−	542.023i	0.779785	−	1.35063i	−0.152280	−	0.988337i	\(-0.548662\pi\)
0.932065	−	0.362290i	\(-0.118005\pi\)
\(13\)	−206.821			−0.339419			−0.169710	−	0.985494i	\(-0.554283\pi\)
							−0.169710	+	0.985494i	\(0.554283\pi\)
\(17\)	−530.732	+	919.254i	−0.445402	+	0.771460i	−0.998080	−	0.0619354i	\(-0.980273\pi\)
							0.552678	+	0.833395i	\(0.313606\pi\)
\(19\)	941.989	+	1631.57i	0.598635	+	1.03687i	0.993023	+	0.117922i	\(0.0376233\pi\)
							−0.394388	+	0.918944i	\(0.629043\pi\)
\(23\)	−1858.68	−	3219.34i	−0.732632	−	1.26896i	−0.955754	−	0.294166i	\(-0.904958\pi\)
							0.223122	−	0.974790i	\(-0.428375\pi\)
\(29\)	−123.747			−0.0273238			−0.0136619	−	0.999907i	\(-0.504349\pi\)
							−0.0136619	+	0.999907i	\(0.504349\pi\)
\(31\)	4554.63	−	7888.85i	0.851234	−	1.47438i	−0.0288611	−	0.999583i	\(-0.509188\pi\)
							0.880095	−	0.474797i	\(-0.157479\pi\)
\(37\)	3014.36	+	5221.03i	0.361986	+	0.626977i	0.988288	−	0.152603i	\(-0.0487656\pi\)
							−0.626302	+	0.779581i	\(0.715432\pi\)
\(41\)	−17201.9			−1.59814			−0.799071	−	0.601236i	\(-0.794675\pi\)
							−0.799071	+	0.601236i	\(0.794675\pi\)
\(43\)	5401.98			0.445535			0.222767	−	0.974872i	\(-0.428491\pi\)
							0.222767	+	0.974872i	\(0.428491\pi\)
\(47\)	−937.621	−	1624.01i	−0.0619131	−	0.107237i	0.833407	−	0.552659i	\(-0.186387\pi\)
							−0.895321	+	0.445422i	\(0.853054\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.6.c.b.11.1	yes	4
3.2	odd	2		126.6.g.e.109.2		4
4.3	odd	2		112.6.i.b.81.2		4
7.2	even	3	inner	14.6.c.b.9.1	✓	4
7.3	odd	6		98.6.a.f.1.1		2
7.4	even	3		98.6.a.c.1.2		2
7.5	odd	6		98.6.c.f.79.2		4
7.6	odd	2		98.6.c.f.67.2		4
21.2	odd	6		126.6.g.e.37.2		4
21.11	odd	6		882.6.a.bt.1.2		2
21.17	even	6		882.6.a.bl.1.2		2
28.3	even	6		784.6.a.r.1.2		2
28.11	odd	6		784.6.a.bc.1.1		2
28.23	odd	6		112.6.i.b.65.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.c.b.9.1	✓	4	7.2	even	3	inner
14.6.c.b.11.1	yes	4	1.1	even	1	trivial
98.6.a.c.1.2		2	7.4	even	3
98.6.a.f.1.1		2	7.3	odd	6
98.6.c.f.67.2		4	7.6	odd	2
98.6.c.f.79.2		4	7.5	odd	6
112.6.i.b.65.2		4	28.23	odd	6
112.6.i.b.81.2		4	4.3	odd	2
126.6.g.e.37.2		4	21.2	odd	6
126.6.g.e.109.2		4	3.2	odd	2
784.6.a.r.1.2		2	28.3	even	6
784.6.a.bc.1.1		2	28.11	odd	6
882.6.a.bl.1.2		2	21.17	even	6
882.6.a.bt.1.2		2	21.11	odd	6