Embedded newform 14.14.c.a.11.1

• Label: 14.14.c.a.11.1
• Level: $14$
• Weight: $14$
• Character: 14.11
• Analytic conductor: $15.012$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(15.0123300533\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	x^8 - 2*x^7 + 692094*x^6 + 445371928*x^5 + 480078817147*x^4 + 153633603309480*x^3 + 48219450150076326*x^2 + 340591169755393830*x + 2353882543131872025
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{14}\cdot 3\cdot 7^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			11.1
Root			\(-3.53342 - 6.12007i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.11
Dual form			14.14.c.a.9.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-32.0000 - 55.4256i) q^{2} +(-855.669 + 1482.06i) q^{3} +(-2048.00 + 3547.24i) q^{4} +(12545.8 + 21729.9i) q^{5} +109526. q^{6} +(48813.9 + 307419. i) q^{7} +262144. q^{8} +(-667176. - 1.15558e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 256 q^{2} - 182 q^{3} - 16384 q^{4} - 64400 q^{5} + 23296 q^{6} - 113736 q^{7} + 2097152 q^{8} - 3983752 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−32.0000	−	55.4256i	−0.353553	−	0.612372i
							\(3\)	−855.669	+	1482.06i	−0.677669	+	1.17376i	0.298012	+	0.954562i	\(0.403676\pi\)
−0.975681	+	0.219195i	\(0.929657\pi\)
\(5\)	12545.8	+	21729.9i	0.359080	+	0.621945i	0.987807	−	0.155681i	\(-0.0497571\pi\)
							−0.628727	+	0.777626i	\(0.716424\pi\)
\(7\)	48813.9	+	307419.i	0.156822	+	0.987627i
							\(11\)	850293.	−	1.47275e6i	0.144716	−	0.250655i	−0.784551	−	0.620064i	\(-0.787107\pi\)
0.929267	+	0.369409i	\(0.120440\pi\)
\(13\)	−9.03160e6			−0.518959			−0.259479	−	0.965749i	\(-0.583551\pi\)
							−0.259479	+	0.965749i	\(0.583551\pi\)
\(17\)	−9.01266e7	+	1.56104e8i	−0.905597	+	1.56854i	−0.0854829	+	0.996340i	\(0.527243\pi\)
							−0.820114	+	0.572200i	\(0.806090\pi\)
\(19\)	−1.70281e7	−	2.94935e7i	−0.0830360	−	0.143823i	0.821517	−	0.570185i	\(-0.193128\pi\)
							−0.904553	+	0.426362i	\(0.859795\pi\)
\(23\)	−5.27016e8	−	9.12819e8i	−0.742323	−	1.28574i	−0.951435	−	0.307850i	\(-0.900390\pi\)
							0.209111	−	0.977892i	\(-0.432943\pi\)
\(29\)	2.18869e9			0.683277			0.341638	−	0.939832i	\(-0.389018\pi\)
							0.341638	+	0.939832i	\(0.389018\pi\)
\(31\)	3.02690e8	−	5.24275e8i	0.0612559	−	0.106098i	−0.833771	−	0.552110i	\(-0.813823\pi\)
							0.895027	+	0.446012i	\(0.147156\pi\)
\(37\)	−8.94467e9	−	1.54926e10i	−0.573130	−	0.992690i	−0.996242	−	0.0866130i	\(-0.972396\pi\)
							0.423112	−	0.906077i	\(-0.360938\pi\)
\(41\)	4.48250e10			1.47376			0.736878	−	0.676026i	\(-0.236299\pi\)
							0.736878	+	0.676026i	\(0.236299\pi\)
\(43\)	−3.24463e9			−0.0782744			−0.0391372	−	0.999234i	\(-0.512461\pi\)
							−0.0391372	+	0.999234i	\(0.512461\pi\)
\(47\)	−3.67969e10	−	6.37340e10i	−0.497937	−	0.862453i	0.502060	−	0.864833i	\(-0.332576\pi\)
							−0.999997	+	0.00238018i	\(0.999242\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.14.c.a.11.1	yes	8
3.2	odd	2		126.14.g.d.109.1		8
7.2	even	3	inner	14.14.c.a.9.1	✓	8
7.3	odd	6		98.14.a.k.1.1		4
7.4	even	3		98.14.a.l.1.4		4
7.5	odd	6		98.14.c.n.79.4		8
7.6	odd	2		98.14.c.n.67.4		8
21.2	odd	6		126.14.g.d.37.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.14.c.a.9.1	✓	8	7.2	even	3	inner
14.14.c.a.11.1	yes	8	1.1	even	1	trivial
98.14.a.k.1.1		4	7.3	odd	6
98.14.a.l.1.4		4	7.4	even	3
98.14.c.n.67.4		8	7.6	odd	2
98.14.c.n.79.4		8	7.5	odd	6
126.14.g.d.37.1		8	21.2	odd	6
126.14.g.d.109.1		8	3.2	odd	2