Embedded newform 7.10.c.a.2.1

• Label: 7.10.c.a.2.1
• 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			2.1
Root			\(-8.71912 - 15.1020i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.2
Dual form			7.10.c.a.4.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-19.4382 + 33.6680i) q^{2} +(113.728 + 196.982i) q^{3} +(-499.691 - 865.489i) q^{4} +(-162.760 + 281.909i) q^{5} -8842.67 q^{6} +(5234.95 - 3598.46i) q^{7} +18947.7 q^{8} +(-16026.5 + 27758.7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 18 q^{2} + 161 q^{3} - 940 q^{4} + 1533 q^{5} - 8708 q^{6} - 1036 q^{7} + 34272 q^{8} - 35734 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−19.4382	+	33.6680i	−0.859057	+	1.48793i	0.0137724	+	0.999905i	\(0.495616\pi\)
							−0.872829	+	0.488025i	\(0.837717\pi\)
\(3\)	113.728	+	196.982i	0.810626	+	1.40405i	0.912426	+	0.409241i	\(0.134206\pi\)
							−0.101800	+	0.994805i	\(0.532460\pi\)
\(5\)	−162.760	+	281.909i	−0.116462	+	0.201718i	−0.918363	−	0.395739i	\(-0.870489\pi\)
							0.801901	+	0.597456i	\(0.203822\pi\)
\(7\)	5234.95	−	3598.46i	0.824083	−	0.566469i
							\(11\)	24492.8	+	42422.7i	0.504395	+	0.873638i	0.999987	+	0.00508275i	\(0.00161790\pi\)
−0.495592	+	0.868556i	\(0.665049\pi\)
\(13\)	−108737.			−1.05593			−0.527964	−	0.849267i	\(-0.677044\pi\)
							−0.527964	+	0.849267i	\(0.677044\pi\)
\(17\)	114226.	+	197845.i	0.331699	+	0.574520i	0.982845	−	0.184432i	\(-0.0590447\pi\)
							−0.651146	+	0.758953i	\(0.725711\pi\)
\(19\)	28440.4	−	49260.3i	0.0500662	−	0.0867172i	−0.839906	−	0.542732i	\(-0.817390\pi\)
							0.889972	+	0.456014i	\(0.150723\pi\)
\(23\)	721856.	−	1.25029e6i	0.537867	−	0.931614i	−0.461151	−	0.887322i	\(-0.652563\pi\)
							0.999019	−	0.0442921i	\(-0.0141032\pi\)
\(29\)	−266591.			−0.0699929			−0.0349965	−	0.999387i	\(-0.511142\pi\)
							−0.0349965	+	0.999387i	\(0.511142\pi\)
\(31\)	−80792.9	−	139937.i	−0.0157125	−	0.0272149i	0.858062	−	0.513546i	\(-0.171668\pi\)
							−0.873775	+	0.486331i	\(0.838335\pi\)
\(37\)	3.04492e6	−	5.27396e6i	0.267097	−	0.462625i	−0.701014	−	0.713147i	\(-0.747269\pi\)
							0.968111	+	0.250522i	\(0.0806023\pi\)
\(41\)	1.73177e7			0.957113			0.478556	−	0.878057i	\(-0.341160\pi\)
							0.478556	+	0.878057i	\(0.341160\pi\)
\(43\)	1.34512e7			0.600003			0.300001	−	0.953939i	\(-0.403013\pi\)
							0.300001	+	0.953939i	\(0.403013\pi\)
\(47\)	−959805.	+	1.66243e6i	−0.0286908	+	0.0496939i	−0.880014	−	0.474947i	\(-0.842467\pi\)
							0.851323	+	0.524641i	\(0.175800\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.10.c.a.2.1	✓	10
3.2	odd	2		63.10.e.b.37.5		10
4.3	odd	2		112.10.i.c.65.1		10
7.2	even	3		49.10.a.e.1.5		5
7.3	odd	6		49.10.c.g.18.1		10
7.4	even	3	inner	7.10.c.a.4.1	yes	10
7.5	odd	6		49.10.a.f.1.5		5
7.6	odd	2		49.10.c.g.30.1		10
21.11	odd	6		63.10.e.b.46.5		10
28.11	odd	6		112.10.i.c.81.1		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.c.a.2.1	✓	10	1.1	even	1	trivial
7.10.c.a.4.1	yes	10	7.4	even	3	inner
49.10.a.e.1.5		5	7.2	even	3
49.10.a.f.1.5		5	7.5	odd	6
49.10.c.g.18.1		10	7.3	odd	6
49.10.c.g.30.1		10	7.6	odd	2
63.10.e.b.37.5		10	3.2	odd	2
63.10.e.b.46.5		10	21.11	odd	6
112.10.i.c.65.1		10	4.3	odd	2
112.10.i.c.81.1		10	28.11	odd	6