Embedded newform 7.10.c.a.4.3

• Label: 7.10.c.a.4.3
• Level: $7$
• Weight: $10$
• Character: 7.4
• Analytic conductor: $3.605$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	7.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.60525085315\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Relative dimension:	\(5\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	x^10 - x^9 + 430*x^8 + 61*x^7 + 146753*x^6 + 23608*x^5 + 16136944*x^4 + 30575648*x^3 + 1399072384*x^2 + 1034227200*x + 761760000
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 2^{12}\cdot 3^{3}\cdot 7^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			4.3
Root			\(-0.371984 + 0.644295i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.4
Dual form			7.10.c.a.2.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.74397 - 4.75269i) q^{2} +(-1.70307 + 2.94981i) q^{3} +(240.941 - 417.323i) q^{4} +(-828.924 - 1435.74i) q^{5} +18.6927 q^{6} +(2822.68 - 5690.88i) q^{7} -5454.36 q^{8} +(9835.70 + 17035.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 18 q^{2} + 161 q^{3} - 940 q^{4} + 1533 q^{5} - 8708 q^{6} - 1036 q^{7} + 34272 q^{8} - 35734 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.74397	−	4.75269i	−0.121267	−	0.210041i	0.799000	−	0.601331i	\(-0.205363\pi\)
							−0.920268	+	0.391289i	\(0.872029\pi\)
\(3\)	−1.70307	+	2.94981i	−0.0121391	+	0.0210256i	−0.872031	−	0.489450i	\(-0.837197\pi\)
							0.859892	+	0.510476i	\(0.170531\pi\)
\(5\)	−828.924	−	1435.74i	−0.593129	−	1.02733i	−0.993808	−	0.111112i	\(-0.964559\pi\)
							0.400679	−	0.916219i	\(-0.368774\pi\)
\(7\)	2822.68	−	5690.88i	0.444345	−	0.895856i
							\(11\)	−16835.7	+	29160.2i	−0.346707	+	0.600515i	−0.985662	−	0.168729i	\(-0.946034\pi\)
0.638955	+	0.769244i	\(0.279367\pi\)
\(13\)	47905.5			0.465200			0.232600	−	0.972572i	\(-0.425277\pi\)
							0.232600	+	0.972572i	\(0.425277\pi\)
\(17\)	172909.	−	299487.i	0.502107	−	0.869676i	−0.497890	−	0.867240i	\(-0.665892\pi\)
							0.999997	−	0.00243516i	\(-0.000775137\pi\)
\(19\)	202522.	+	350778.i	0.356518	+	0.617507i	0.987376	−	0.158391i	\(-0.0506306\pi\)
							−0.630859	+	0.775898i	\(0.717297\pi\)
\(23\)	926736.	+	1.60515e6i	0.690527	+	1.19603i	0.971665	+	0.236361i	\(0.0759548\pi\)
							−0.281138	+	0.959667i	\(0.590712\pi\)
\(29\)	682809.			0.179270			0.0896351	−	0.995975i	\(-0.471430\pi\)
							0.0896351	+	0.995975i	\(0.471430\pi\)
\(31\)	4.54839e6	−	7.87804e6i	0.884565	−	1.53211i	0.0383536	−	0.999264i	\(-0.487789\pi\)
							0.846211	−	0.532847i	\(-0.178878\pi\)
\(37\)	7.77984e6	+	1.34751e7i	0.682437	+	1.18202i	0.974235	+	0.225536i	\(0.0724133\pi\)
							−0.291798	+	0.956480i	\(0.594253\pi\)
\(41\)	−2.98719e7			−1.65095			−0.825477	−	0.564435i	\(-0.809094\pi\)
							−0.825477	+	0.564435i	\(0.809094\pi\)
\(43\)	6.28733e6			0.280452			0.140226	−	0.990120i	\(-0.455217\pi\)
							0.140226	+	0.990120i	\(0.455217\pi\)
\(47\)	−5.16042e6	−	8.93811e6i	−0.154257	−	0.267181i	0.778531	−	0.627606i	\(-0.215965\pi\)
							−0.932788	+	0.360425i	\(0.882632\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.10.c.a.4.3	yes	10
3.2	odd	2		63.10.e.b.46.3		10
4.3	odd	2		112.10.i.c.81.3		10
7.2	even	3	inner	7.10.c.a.2.3	✓	10
7.3	odd	6		49.10.a.f.1.3		5
7.4	even	3		49.10.a.e.1.3		5
7.5	odd	6		49.10.c.g.30.3		10
7.6	odd	2		49.10.c.g.18.3		10
21.2	odd	6		63.10.e.b.37.3		10
28.23	odd	6		112.10.i.c.65.3		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.c.a.2.3	✓	10	7.2	even	3	inner
7.10.c.a.4.3	yes	10	1.1	even	1	trivial
49.10.a.e.1.3		5	7.4	even	3
49.10.a.f.1.3		5	7.3	odd	6
49.10.c.g.18.3		10	7.6	odd	2
49.10.c.g.30.3		10	7.5	odd	6
63.10.e.b.37.3		10	21.2	odd	6
63.10.e.b.46.3		10	3.2	odd	2
112.10.i.c.65.3		10	28.23	odd	6
112.10.i.c.81.3		10	4.3	odd	2