Embedded newform 7.6.c.a.4.1

• Label: 7.6.c.a.4.1
• Level: $7$
• Weight: $6$
• Character: 7.4
• Analytic conductor: $1.123$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.12268673869\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{37})\)
Defining polynomial:	\( x^{4} - x^{3} + 10x^{2} + 9x + 81 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			4.1
Root			\(1.77069 + 3.06693i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.4
Dual form			7.6.c.a.2.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.54138 - 6.13385i) q^{2} +(5.04138 - 8.73193i) q^{3} +(-9.08276 + 15.7318i) q^{4} +(39.9138 + 69.1328i) q^{5} -71.4138 q^{6} +(43.1587 - 122.247i) q^{7} -97.9863 q^{8} +(70.6689 + 122.402i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 8 q^{3} - 12 q^{4} + 38 q^{5} - 164 q^{6} - 168 q^{7} + 192 q^{8} + 380 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.54138	−	6.13385i	−0.626034	−	1.08432i	−0.988340	−	0.152263i	\(-0.951344\pi\)
							0.362306	−	0.932059i	\(-0.381989\pi\)
\(3\)	5.04138	−	8.73193i	0.323405	−	0.560153i	−0.657783	−	0.753207i	\(-0.728506\pi\)
							0.981188	+	0.193054i	\(0.0618391\pi\)
\(5\)	39.9138	+	69.1328i	0.714000	+	1.23668i	0.963344	+	0.268269i	\(0.0864517\pi\)
							−0.249344	+	0.968415i	\(0.580215\pi\)
\(7\)	43.1587	−	122.247i	0.332907	−	0.942960i
							\(11\)	−175.952	+	304.757i	−0.438442	+	0.759403i	−0.997570	−	0.0696781i	\(-0.977803\pi\)
0.559128	+	0.829082i	\(0.311136\pi\)
\(13\)	−291.683			−0.478688			−0.239344	−	0.970935i	\(-0.576932\pi\)
							−0.239344	+	0.970935i	\(0.576932\pi\)
\(17\)	185.038	−	320.495i	0.155288	−	0.268967i	−0.777876	−	0.628418i	\(-0.783703\pi\)
							0.933164	+	0.359451i	\(0.117036\pi\)
\(19\)	−752.463	−	1303.30i	−0.478190	−	0.828250i	0.521497	−	0.853253i	\(-0.325374\pi\)
							−0.999687	+	0.0250030i	\(0.992040\pi\)
\(23\)	212.855	+	368.676i	0.0839006	+	0.145320i	0.904922	−	0.425577i	\(-0.139929\pi\)
							−0.821022	+	0.570897i	\(0.806596\pi\)
\(29\)	−7783.93			−1.71872			−0.859358	−	0.511374i	\(-0.829137\pi\)
							−0.859358	+	0.511374i	\(0.829137\pi\)
\(31\)	1287.59	−	2230.17i	0.240643	−	0.416805i	−0.720255	−	0.693710i	\(-0.755975\pi\)
							0.960898	+	0.276904i	\(0.0893085\pi\)
\(37\)	−369.809	−	640.528i	−0.0444092	−	0.0769190i	0.842966	−	0.537966i	\(-0.180807\pi\)
							−0.887376	+	0.461047i	\(0.847474\pi\)
\(41\)	7029.84			0.653109			0.326554	−	0.945178i	\(-0.394112\pi\)
							0.326554	+	0.945178i	\(0.394112\pi\)
\(43\)	1835.23			0.151363			0.0756816	−	0.997132i	\(-0.475887\pi\)
							0.0756816	+	0.997132i	\(0.475887\pi\)
\(47\)	766.342	+	1327.34i	0.0506032	+	0.0876473i	0.890217	−	0.455536i	\(-0.150552\pi\)
							−0.839614	+	0.543183i	\(0.817219\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.6.c.a.4.1	yes	4
3.2	odd	2		63.6.e.d.46.2		4
4.3	odd	2		112.6.i.c.81.1		4
7.2	even	3	inner	7.6.c.a.2.1	✓	4
7.3	odd	6		49.6.a.e.1.2		2
7.4	even	3		49.6.a.d.1.2		2
7.5	odd	6		49.6.c.f.30.1		4
7.6	odd	2		49.6.c.f.18.1		4
21.2	odd	6		63.6.e.d.37.2		4
21.11	odd	6		441.6.a.n.1.1		2
21.17	even	6		441.6.a.m.1.1		2
28.3	even	6		784.6.a.t.1.1		2
28.11	odd	6		784.6.a.ba.1.2		2
28.23	odd	6		112.6.i.c.65.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.c.a.2.1	✓	4	7.2	even	3	inner
7.6.c.a.4.1	yes	4	1.1	even	1	trivial
49.6.a.d.1.2		2	7.4	even	3
49.6.a.e.1.2		2	7.3	odd	6
49.6.c.f.18.1		4	7.6	odd	2
49.6.c.f.30.1		4	7.5	odd	6
63.6.e.d.37.2		4	21.2	odd	6
63.6.e.d.46.2		4	3.2	odd	2
112.6.i.c.65.1		4	28.23	odd	6
112.6.i.c.81.1		4	4.3	odd	2
441.6.a.m.1.1		2	21.17	even	6
441.6.a.n.1.1		2	21.11	odd	6
784.6.a.t.1.1		2	28.3	even	6
784.6.a.ba.1.2		2	28.11	odd	6