Embedded newform 10.15.c.a.7.1

• Label: 10.15.c.a.7.1
• 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			7.1
Root			\(16.0869 + 16.0869i\) of defining polynomial
Character	\(\chi\)	\(=\)	10.7
Dual form			10.15.c.a.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-64.0000 - 64.0000i) q^{2} +(-1760.67 + 1760.67i) q^{3} +8192.00i q^{4} +(46225.0 + 62982.2i) q^{5} +225366. q^{6} +(-59627.2 - 59627.2i) q^{7} +(524288. - 524288. i) q^{8} -1.41695e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 384 q^{2} + 2912 q^{3} + 82500 q^{5} - 372736 q^{6} + 943128 q^{7} + 3145728 q^{8}+O(q^{10}) \)

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
							\(3\)	−1760.67	+	1760.67i	−0.805062	+	0.805062i	−0.983882	−	0.178820i	\(-0.942772\pi\)
0.178820	+	0.983882i	\(0.442772\pi\)
\(5\)	46225.0	+	62982.2i	0.591681	+	0.806173i
							\(7\)	−59627.2	−	59627.2i	−0.0724033	−	0.0724033i	0.669978	−	0.742381i	\(-0.266304\pi\)
−0.742381	+	0.669978i	\(0.766304\pi\)
\(11\)	−4.64100e6			−0.238157			−0.119078	−	0.992885i	\(-0.537994\pi\)
							−0.119078	+	0.992885i	\(0.537994\pi\)
\(13\)	−4.85144e7	+	4.85144e7i	−0.773157	+	0.773157i	−0.978657	−	0.205500i	\(-0.934118\pi\)
							0.205500	+	0.978657i	\(0.434118\pi\)
\(17\)	−4.70811e8	−	4.70811e8i	−1.14737	−	1.14737i	−0.987068	−	0.160303i	\(-0.948753\pi\)
							−0.160303	−	0.987068i	\(-0.551247\pi\)
\(19\)		−	7.64295e8i		−	0.855038i	−0.904006	−	0.427519i	\(-0.859388\pi\)
							0.904006	−	0.427519i	\(-0.140612\pi\)
\(23\)	1.43685e9	−	1.43685e9i	0.422005	−	0.422005i	−0.463889	−	0.885893i	\(-0.653546\pi\)
							0.885893	+	0.463889i	\(0.153546\pi\)
\(29\)		−	3.55005e9i		−	0.205802i	−0.994692	−	0.102901i	\(-0.967188\pi\)
							0.994692	−	0.102901i	\(-0.0328124\pi\)
\(31\)	4.07968e10			1.48284			0.741419	−	0.671042i	\(-0.234153\pi\)
							0.741419	+	0.671042i	\(0.234153\pi\)
\(37\)	−1.25794e11	−	1.25794e11i	−1.32510	−	1.32510i	−0.909590	−	0.415507i	\(-0.863604\pi\)
							−0.415507	−	0.909590i	\(-0.636396\pi\)
\(41\)	−1.93888e11			−0.995551			−0.497776	−	0.867306i	\(-0.665850\pi\)
							−0.497776	+	0.867306i	\(0.665850\pi\)
\(43\)	−2.99264e11	+	2.99264e11i	−1.10097	+	1.10097i	−0.106674	+	0.994294i	\(0.534020\pi\)
							−0.994294	+	0.106674i	\(0.965980\pi\)
\(47\)	4.49650e11	+	4.49650e11i	0.887544	+	0.887544i	0.994287	−	0.106743i	\(-0.0340422\pi\)
							−0.106743	+	0.994287i	\(0.534042\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10.15.c.a.7.1	yes	6
3.2	odd	2		90.15.g.a.37.2		6
4.3	odd	2		80.15.p.a.17.3		6
5.2	odd	4		50.15.c.b.43.3		6
5.3	odd	4	inner	10.15.c.a.3.1	✓	6
5.4	even	2		50.15.c.b.7.3		6
15.8	even	4		90.15.g.a.73.2		6
20.3	even	4		80.15.p.a.33.3		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.15.c.a.3.1	✓	6	5.3	odd	4	inner
10.15.c.a.7.1	yes	6	1.1	even	1	trivial
50.15.c.b.7.3		6	5.4	even	2
50.15.c.b.43.3		6	5.2	odd	4
80.15.p.a.17.3		6	4.3	odd	2
80.15.p.a.33.3		6	20.3	even	4
90.15.g.a.37.2		6	3.2	odd	2
90.15.g.a.73.2		6	15.8	even	4