Embedded newform 10.15.c.a.7.2

• Label: 10.15.c.a.7.2
• Level: $10$
• Weight: $15$
• Character: 10.7
• Analytic conductor: $12.433$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.4328968152\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial:	\( x^{6} - 2x^{5} + 2x^{4} - 11690x^{3} + 819025x^{2} - 12217500x + 91125000 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{7}\cdot 3^{2}\cdot 5^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			7.2
Root			\(8.78753 + 8.78753i\) of defining polynomial
Character	\(\chi\)	\(=\)	10.7
Dual form			10.15.c.a.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-64.0000 - 64.0000i) q^{2} +(1295.91 - 1295.91i) q^{3} +8192.00i q^{4} +(61416.3 - 48286.1i) q^{5} -165877. q^{6} +(905655. + 905655. i) q^{7} +(524288. - 524288. i) q^{8} +1.42420e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 384 q^{2} + 2912 q^{3} + 82500 q^{5} - 372736 q^{6} + 943128 q^{7} + 3145728 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−64.0000	−	64.0000i	−0.500000	−	0.500000i
							\(3\)	1295.91	−	1295.91i	0.592552	−	0.592552i	−0.345768	−	0.938320i	\(-0.612382\pi\)
0.938320	+	0.345768i	\(0.112382\pi\)
\(5\)	61416.3	−	48286.1i	0.786129	−	0.618063i
							\(7\)	905655.	+	905655.i	1.09971	+	1.09971i	0.994445	+	0.105261i	\(0.0335679\pi\)
0.105261	+	0.994445i	\(0.466432\pi\)
\(11\)	−4.06210e6			−0.208450			−0.104225	−	0.994554i	\(-0.533236\pi\)
							−0.104225	+	0.994554i	\(0.533236\pi\)
\(13\)	8.29932e7	−	8.29932e7i	1.32263	−	1.32263i	0.410993	−	0.911638i	\(-0.365182\pi\)
							0.911638	−	0.410993i	\(-0.134818\pi\)
\(17\)	−2.30703e7	−	2.30703e7i	−0.0562226	−	0.0562226i	0.678436	−	0.734659i	\(-0.262658\pi\)
							−0.734659	+	0.678436i	\(0.762658\pi\)
\(19\)		−	1.25725e9i		−	1.40652i	−0.710931	−	0.703261i	\(-0.751726\pi\)
							0.710931	−	0.703261i	\(-0.248274\pi\)
\(23\)	−2.88246e9	+	2.88246e9i	−0.846581	+	0.846581i	−0.989705	−	0.143124i	\(-0.954285\pi\)
							0.143124	+	0.989705i	\(0.454285\pi\)
\(29\)			9.91798e9i			0.574960i	0.957787	+	0.287480i	\(0.0928174\pi\)
							−0.957787	+	0.287480i	\(0.907183\pi\)
\(31\)	−2.55251e9			−0.0927759			−0.0463880	−	0.998923i	\(-0.514771\pi\)
							−0.0463880	+	0.998923i	\(0.514771\pi\)
\(37\)	−4.48451e10	−	4.48451e10i	−0.472392	−	0.472392i	0.430296	−	0.902688i	\(-0.358409\pi\)
							−0.902688	+	0.430296i	\(0.858409\pi\)
\(41\)	1.89208e11			0.971524			0.485762	−	0.874091i	\(-0.338542\pi\)
							0.485762	+	0.874091i	\(0.338542\pi\)
\(43\)	1.27342e11	−	1.27342e11i	0.468482	−	0.468482i	−0.432940	−	0.901423i	\(-0.642524\pi\)
							0.901423	+	0.432940i	\(0.142524\pi\)
\(47\)	−2.61588e11	−	2.61588e11i	−0.516337	−	0.516337i	0.400124	−	0.916461i	\(-0.368967\pi\)
							−0.916461	+	0.400124i	\(0.868967\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10.15.c.a.7.2	yes	6
3.2	odd	2		90.15.g.a.37.1		6
4.3	odd	2		80.15.p.a.17.2		6
5.2	odd	4		50.15.c.b.43.2		6
5.3	odd	4	inner	10.15.c.a.3.2	✓	6
5.4	even	2		50.15.c.b.7.2		6
15.8	even	4		90.15.g.a.73.1		6
20.3	even	4		80.15.p.a.33.2		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.15.c.a.3.2	✓	6	5.3	odd	4	inner
10.15.c.a.7.2	yes	6	1.1	even	1	trivial
50.15.c.b.7.2		6	5.4	even	2
50.15.c.b.43.2		6	5.2	odd	4
80.15.p.a.17.2		6	4.3	odd	2
80.15.p.a.33.2		6	20.3	even	4
90.15.g.a.37.1		6	3.2	odd	2
90.15.g.a.73.1		6	15.8	even	4