Embedded newform 7.10.c.a.2.5

• Label: 7.10.c.a.2.5
• 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			2.5
Root			\(8.80924 + 15.2580i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.2
Dual form			7.10.c.a.4.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(15.6185 - 27.0520i) q^{2} +(-56.7670 - 98.3234i) q^{3} +(-231.874 - 401.617i) q^{4} +(-239.755 + 415.269i) q^{5} -3546.46 q^{6} +(1428.40 - 6189.77i) q^{7} +1507.26 q^{8} +(3396.51 - 5882.92i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 18 q^{2} + 161 q^{3} - 940 q^{4} + 1533 q^{5} - 8708 q^{6} - 1036 q^{7} + 34272 q^{8} - 35734 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	15.6185	−	27.0520i	0.690246	−	1.19554i	−0.281511	−	0.959558i	\(-0.590836\pi\)
							0.971757	−	0.235983i	\(-0.0758309\pi\)
\(3\)	−56.7670	−	98.3234i	−0.404623	−	0.700828i	0.589654	−	0.807656i	\(-0.299264\pi\)
							−0.994278	+	0.106828i	\(0.965931\pi\)
\(5\)	−239.755	+	415.269i	−0.171555	+	0.297142i	−0.938964	−	0.344016i	\(-0.888212\pi\)
							0.767409	+	0.641158i	\(0.221546\pi\)
\(7\)	1428.40	−	6189.77i	0.224859	−	0.974391i
							\(11\)	33637.5	+	58261.9i	0.692719	+	1.19982i	0.970944	+	0.239309i	\(0.0769208\pi\)
−0.278224	+	0.960516i	\(0.589746\pi\)
\(13\)	39620.0			0.384742			0.192371	−	0.981322i	\(-0.438382\pi\)
							0.192371	+	0.981322i	\(0.438382\pi\)
\(17\)	−106607.	−	184649.i	−0.309575	−	0.536199i	0.668695	−	0.743537i	\(-0.266853\pi\)
							−0.978269	+	0.207338i	\(0.933520\pi\)
\(19\)	−566407.	+	981046.i	−0.997097	+	1.72702i	−0.432612	+	0.901580i	\(0.642408\pi\)
							−0.564485	+	0.825443i	\(0.690925\pi\)
\(23\)	365109.	−	632388.i	0.272049	−	0.471203i	−0.697337	−	0.716743i	\(-0.745632\pi\)
							0.969386	+	0.245540i	\(0.0789652\pi\)
\(29\)	383457.			0.100676			0.0503379	−	0.998732i	\(-0.483970\pi\)
							0.0503379	+	0.998732i	\(0.483970\pi\)
\(31\)	2.73393e6	+	4.73530e6i	0.531691	+	0.920915i	0.999316	+	0.0369885i	\(0.0117765\pi\)
							−0.467625	+	0.883927i	\(0.654890\pi\)
\(37\)	3.83476e6	−	6.64200e6i	0.336380	−	0.582627i	−0.647369	−	0.762177i	\(-0.724131\pi\)
							0.983749	+	0.179549i	\(0.0574640\pi\)
\(41\)	1.08865e6			0.0601671			0.0300836	−	0.999547i	\(-0.490423\pi\)
							0.0300836	+	0.999547i	\(0.490423\pi\)
\(43\)	−1.92137e7			−0.857044			−0.428522	−	0.903531i	\(-0.640965\pi\)
							−0.428522	+	0.903531i	\(0.640965\pi\)
\(47\)	2.11223e7	−	3.65850e7i	0.631396	−	1.09361i	−0.355871	−	0.934535i	\(-0.615816\pi\)
							0.987267	−	0.159074i	\(-0.0508510\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.10.c.a.2.5	✓	10
3.2	odd	2		63.10.e.b.37.1		10
4.3	odd	2		112.10.i.c.65.4		10
7.2	even	3		49.10.a.e.1.1		5
7.3	odd	6		49.10.c.g.18.5		10
7.4	even	3	inner	7.10.c.a.4.5	yes	10
7.5	odd	6		49.10.a.f.1.1		5
7.6	odd	2		49.10.c.g.30.5		10
21.11	odd	6		63.10.e.b.46.1		10
28.11	odd	6		112.10.i.c.81.4		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.c.a.2.5	✓	10	1.1	even	1	trivial
7.10.c.a.4.5	yes	10	7.4	even	3	inner
49.10.a.e.1.1		5	7.2	even	3
49.10.a.f.1.1		5	7.5	odd	6
49.10.c.g.18.5		10	7.3	odd	6
49.10.c.g.30.5		10	7.6	odd	2
63.10.e.b.37.1		10	3.2	odd	2
63.10.e.b.46.1		10	21.11	odd	6
112.10.i.c.65.4		10	4.3	odd	2
112.10.i.c.81.4		10	28.11	odd	6