Embedded newform 10.15.c.a.7.3

• Label: 10.15.c.a.7.3
• 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			7.3
Root			\(-23.8745 - 23.8745i\) of defining polynomial
Character	\(\chi\)	\(=\)	10.7
Dual form			10.15.c.a.3.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-64.0000 - 64.0000i) q^{2} +(1920.76 - 1920.76i) q^{3} +8192.00i q^{4} +(-66391.4 + 41178.9i) q^{5} -245857. q^{6} +(-374464. - 374464. i) q^{7} +(524288. - 524288. i) q^{8} -2.59566e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 384 q^{2} + 2912 q^{3} + 82500 q^{5} - 372736 q^{6} + 943128 q^{7} + 3145728 q^{8}+O(q^{10}) \)

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
							\(3\)	1920.76	−	1920.76i	0.878262	−	0.878262i	−0.115093	−	0.993355i	\(-0.536717\pi\)
0.993355	+	0.115093i	\(0.0367165\pi\)
\(5\)	−66391.4	+	41178.9i	−0.849809	+	0.527090i
							\(7\)	−374464.	−	374464.i	−0.454699	−	0.454699i	0.442212	−	0.896911i	\(-0.354194\pi\)
−0.896911	+	0.442212i	\(0.854194\pi\)
\(11\)	−1.40802e7			−0.722536			−0.361268	−	0.932462i	\(-0.617656\pi\)
							−0.361268	+	0.932462i	\(0.617656\pi\)
\(13\)	−6.05534e7	+	6.05534e7i	−0.965017	+	0.965017i	−0.999408	−	0.0343912i	\(-0.989051\pi\)
							0.0343912	+	0.999408i	\(0.489051\pi\)
\(17\)	3.46706e8	+	3.46706e8i	0.844928	+	0.844928i	0.989495	−	0.144567i	\(-0.0461790\pi\)
							−0.144567	+	0.989495i	\(0.546179\pi\)
\(19\)			1.12580e8i			0.125946i	0.998015	+	0.0629730i	\(0.0200582\pi\)
							−0.998015	+	0.0629730i	\(0.979942\pi\)
\(23\)	−3.26997e9	+	3.26997e9i	−0.960394	+	0.960394i	−0.999245	−	0.0388509i	\(-0.987630\pi\)
							0.0388509	+	0.999245i	\(0.487630\pi\)
\(29\)		−	2.41211e10i		−	1.39834i	−0.714958	−	0.699168i	\(-0.753554\pi\)
							0.714958	−	0.699168i	\(-0.246446\pi\)
\(31\)	−3.63836e10			−1.32243			−0.661216	−	0.750196i	\(-0.729959\pi\)
							−0.661216	+	0.750196i	\(0.729959\pi\)
\(37\)	−4.43100e10	−	4.43100e10i	−0.466756	−	0.466756i	0.434106	−	0.900862i	\(-0.357064\pi\)
							−0.900862	+	0.434106i	\(0.857064\pi\)
\(41\)	2.75199e10			0.141306			0.0706528	−	0.997501i	\(-0.477492\pi\)
							0.0706528	+	0.997501i	\(0.477492\pi\)
\(43\)	−2.95811e11	+	2.95811e11i	−1.08827	+	1.08827i	−0.0925605	+	0.995707i	\(0.529505\pi\)
							−0.995707	+	0.0925605i	\(0.970495\pi\)
\(47\)	−6.71176e11	−	6.71176e11i	−1.32480	−	1.32480i	−0.909837	−	0.414965i	\(-0.863794\pi\)
							−0.414965	−	0.909837i	\(-0.636206\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10.15.c.a.7.3	yes	6
3.2	odd	2		90.15.g.a.37.3		6
4.3	odd	2		80.15.p.a.17.1		6
5.2	odd	4		50.15.c.b.43.1		6
5.3	odd	4	inner	10.15.c.a.3.3	✓	6
5.4	even	2		50.15.c.b.7.1		6
15.8	even	4		90.15.g.a.73.3		6
20.3	even	4		80.15.p.a.33.1		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.15.c.a.3.3	✓	6	5.3	odd	4	inner
10.15.c.a.7.3	yes	6	1.1	even	1	trivial
50.15.c.b.7.1		6	5.4	even	2
50.15.c.b.43.1		6	5.2	odd	4
80.15.p.a.17.1		6	4.3	odd	2
80.15.p.a.33.1		6	20.3	even	4
90.15.g.a.37.3		6	3.2	odd	2
90.15.g.a.73.3		6	15.8	even	4