Embedded newform 7.10.c.a.2.2

• Label: 7.10.c.a.2.2
• Level: $7$
• Weight: $10$
• Character: 7.2
• 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			2.2
Root			\(-5.11725 - 8.86334i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.2
Dual form			7.10.c.a.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.2345 + 21.1908i) q^{2} +(-79.7348 - 138.105i) q^{3} +(-43.3662 - 75.1124i) q^{4} +(1014.15 - 1756.56i) q^{5} +3902.06 q^{6} +(-4235.51 - 4734.35i) q^{7} -10405.9 q^{8} +(-2873.78 + 4977.54i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 18 q^{2} + 161 q^{3} - 940 q^{4} + 1533 q^{5} - 8708 q^{6} - 1036 q^{7} + 34272 q^{8} - 35734 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.2345	+	21.1908i	−0.540694	+	0.936509i	0.458171	+	0.888864i	\(0.348505\pi\)
							−0.998864	+	0.0476448i	\(0.984828\pi\)
\(3\)	−79.7348	−	138.105i	−0.568332	−	0.984381i	−0.996731	−	0.0807907i	\(-0.974255\pi\)
							0.428399	−	0.903590i	\(-0.359078\pi\)
\(5\)	1014.15	−	1756.56i	0.725666	−	1.25689i	−0.233034	−	0.972469i	\(-0.574865\pi\)
							0.958699	−	0.284421i	\(-0.0918014\pi\)
\(7\)	−4235.51	−	4734.35i	−0.666752	−	0.745280i
							\(11\)	−4289.29	−	7429.26i	−0.0883320	−	0.152996i	0.818474	−	0.574543i	\(-0.194820\pi\)
−0.906806	+	0.421548i	\(0.861487\pi\)
\(13\)	−38438.1			−0.373265			−0.186633	−	0.982430i	\(-0.559757\pi\)
							−0.186633	+	0.982430i	\(0.559757\pi\)
\(17\)	150942.	+	261439.i	0.438318	+	0.759190i	0.997560	−	0.0698150i	\(-0.0222409\pi\)
							−0.559242	+	0.829005i	\(0.688908\pi\)
\(19\)	458143.	−	793527.i	0.806510	−	1.39692i	−0.108757	−	0.994068i	\(-0.534687\pi\)
							0.915267	−	0.402848i	\(-0.131980\pi\)
\(23\)	−966607.	+	1.67421e6i	−0.720236	+	1.24749i	0.240669	+	0.970607i	\(0.422633\pi\)
							−0.960905	+	0.276878i	\(0.910700\pi\)
\(29\)	4.52529e6			1.18811			0.594054	−	0.804425i	\(-0.297527\pi\)
							0.594054	+	0.804425i	\(0.297527\pi\)
\(31\)	−147255.	−	255054.i	−0.0286380	−	0.0496025i	0.851351	−	0.524596i	\(-0.175784\pi\)
							−0.879989	+	0.474994i	\(0.842450\pi\)
\(37\)	8.05632e6	−	1.39540e7i	0.706690	−	1.22402i	−0.259388	−	0.965773i	\(-0.583521\pi\)
							0.966078	−	0.258250i	\(-0.0831458\pi\)
\(41\)	1.28783e7			0.711757			0.355878	−	0.934532i	\(-0.384182\pi\)
							0.355878	+	0.934532i	\(0.384182\pi\)
\(43\)	1.15410e7			0.514795			0.257398	−	0.966306i	\(-0.417135\pi\)
							0.257398	+	0.966306i	\(0.417135\pi\)
\(47\)	1.39268e7	−	2.41219e7i	0.416304	−	0.721059i	−0.579261	−	0.815142i	\(-0.696659\pi\)
							0.995564	+	0.0940834i	\(0.0299920\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.10.c.a.2.2	✓	10
3.2	odd	2		63.10.e.b.37.4		10
4.3	odd	2		112.10.i.c.65.5		10
7.2	even	3		49.10.a.e.1.4		5
7.3	odd	6		49.10.c.g.18.2		10
7.4	even	3	inner	7.10.c.a.4.2	yes	10
7.5	odd	6		49.10.a.f.1.4		5
7.6	odd	2		49.10.c.g.30.2		10
21.11	odd	6		63.10.e.b.46.4		10
28.11	odd	6		112.10.i.c.81.5		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.c.a.2.2	✓	10	1.1	even	1	trivial
7.10.c.a.4.2	yes	10	7.4	even	3	inner
49.10.a.e.1.4		5	7.2	even	3
49.10.a.f.1.4		5	7.5	odd	6
49.10.c.g.18.2		10	7.3	odd	6
49.10.c.g.30.2		10	7.6	odd	2
63.10.e.b.37.4		10	3.2	odd	2
63.10.e.b.46.4		10	21.11	odd	6
112.10.i.c.65.5		10	4.3	odd	2
112.10.i.c.81.5		10	28.11	odd	6