Embedded newform 7.10.c.a.4.4

• Label: 7.10.c.a.4.4
• 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			4.4
Root			\(5.89912 - 10.2176i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.4
Dual form			7.10.c.a.2.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.79824 + 16.9710i) q^{2} +(104.977 - 181.826i) q^{3} +(63.9892 - 110.832i) q^{4} +(983.791 + 1703.98i) q^{5} +4114.37 q^{6} +(-5768.52 - 2660.41i) q^{7} +12541.3 q^{8} +(-12198.9 - 21129.2i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 18 q^{2} + 161 q^{3} - 940 q^{4} + 1533 q^{5} - 8708 q^{6} - 1036 q^{7} + 34272 q^{8} - 35734 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	9.79824	+	16.9710i	0.433025	+	0.750021i	0.997132	−	0.0756804i	\(-0.0241129\pi\)
							−0.564107	+	0.825702i	\(0.690780\pi\)
\(3\)	104.977	−	181.826i	0.748255	−	1.29602i	−0.200404	−	0.979713i	\(-0.564225\pi\)
							0.948659	−	0.316302i	\(-0.102441\pi\)
\(5\)	983.791	+	1703.98i	0.703943	+	1.21927i	0.967072	+	0.254505i	\(0.0819124\pi\)
							−0.263128	+	0.964761i	\(0.584754\pi\)
\(7\)	−5768.52	−	2660.41i	−0.908078	−	0.418801i
							\(11\)	−15898.9	+	27537.7i	−0.327416	+	0.567101i	−0.981998	−	0.188890i	\(-0.939511\pi\)
0.654583	+	0.755991i	\(0.272844\pi\)
\(13\)	−100188.			−0.972904			−0.486452	−	0.873707i	\(-0.661709\pi\)
							−0.486452	+	0.873707i	\(0.661709\pi\)
\(17\)	−169129.	+	292940.i	−0.491132	+	0.850666i	−0.999948	−	0.0102097i	\(-0.996750\pi\)
							0.508816	+	0.860875i	\(0.330083\pi\)
\(19\)	−130759.	−	226480.i	−0.230186	−	0.398694i	0.727677	−	0.685920i	\(-0.240600\pi\)
							−0.957863	+	0.287226i	\(0.907267\pi\)
\(23\)	272337.	+	471702.i	0.202923	+	0.351474i	0.949469	−	0.313860i	\(-0.101622\pi\)
							−0.746546	+	0.665334i	\(0.768289\pi\)
\(29\)	2.32128e6			0.609449			0.304724	−	0.952441i	\(-0.401436\pi\)
							0.304724	+	0.952441i	\(0.401436\pi\)
\(31\)	2.53535e6	−	4.39136e6i	0.493073	−	0.854027i	−0.506895	−	0.862008i	\(-0.669207\pi\)
							0.999968	+	0.00798051i	\(0.00254030\pi\)
\(37\)	−2.93235e6	−	5.07897e6i	−0.257222	−	0.445521i	0.708275	−	0.705937i	\(-0.249474\pi\)
							−0.965497	+	0.260416i	\(0.916140\pi\)
\(41\)	−2.81312e7			−1.55475			−0.777376	−	0.629036i	\(-0.783450\pi\)
							−0.777376	+	0.629036i	\(0.783450\pi\)
\(43\)	3.88522e7			1.73304			0.866518	−	0.499147i	\(-0.166353\pi\)
							0.866518	+	0.499147i	\(0.166353\pi\)
\(47\)	−1.26741e7	−	2.19521e7i	−0.378857	−	0.656199i	0.612039	−	0.790827i	\(-0.290349\pi\)
							−0.990896	+	0.134628i	\(0.957016\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.10.c.a.4.4	yes	10
3.2	odd	2		63.10.e.b.46.2		10
4.3	odd	2		112.10.i.c.81.2		10
7.2	even	3	inner	7.10.c.a.2.4	✓	10
7.3	odd	6		49.10.a.f.1.2		5
7.4	even	3		49.10.a.e.1.2		5
7.5	odd	6		49.10.c.g.30.4		10
7.6	odd	2		49.10.c.g.18.4		10
21.2	odd	6		63.10.e.b.37.2		10
28.23	odd	6		112.10.i.c.65.2		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.c.a.2.4	✓	10	7.2	even	3	inner
7.10.c.a.4.4	yes	10	1.1	even	1	trivial
49.10.a.e.1.2		5	7.4	even	3
49.10.a.f.1.2		5	7.3	odd	6
49.10.c.g.18.4		10	7.6	odd	2
49.10.c.g.30.4		10	7.5	odd	6
63.10.e.b.37.2		10	21.2	odd	6
63.10.e.b.46.2		10	3.2	odd	2
112.10.i.c.65.2		10	28.23	odd	6
112.10.i.c.81.2		10	4.3	odd	2