Embedded newform 10.9.c.a.7.1

• Label: 10.9.c.a.7.1
• Level: $10$
• Weight: $9$
• Character: 10.7
• Analytic conductor: $4.074$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.07378610061\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(i)\)
Coefficient field:	\(\Q(i, \sqrt{249})\)
Defining polynomial:	\( x^{4} + 125x^{2} + 3844 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2\cdot 5^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			7.1
Root			\(8.38987i\) of defining polynomial
Character	\(\chi\)	\(=\)	10.7
Dual form			10.9.c.a.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.00000 - 8.00000i) q^{2} +(-25.9493 + 25.9493i) q^{3} +128.000i q^{4} +(535.341 + 322.544i) q^{5} +415.189 q^{6} +(2031.01 + 2031.01i) q^{7} +(1024.00 - 1024.00i) q^{8} +5214.26i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} + 54 q^{3} + 90 q^{5} - 864 q^{6} - 1186 q^{7} + 4096 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.00000	−	8.00000i	−0.500000	−	0.500000i
							\(3\)	−25.9493	+	25.9493i	−0.320362	+	0.320362i	−0.848906	−	0.528544i	\(-0.822738\pi\)
0.528544	+	0.848906i	\(0.322738\pi\)
\(5\)	535.341	+	322.544i	0.856546	+	0.516070i
							\(7\)	2031.01	+	2031.01i	0.845902	+	0.845902i	0.989619	−	0.143717i	\(-0.0459055\pi\)
−0.143717	+	0.989619i	\(0.545905\pi\)
\(11\)	−15298.7			−1.04492			−0.522462	−	0.852663i	\(-0.674986\pi\)
							−0.522462	+	0.852663i	\(0.674986\pi\)
\(13\)	8879.26	−	8879.26i	0.310888	−	0.310888i	−0.534366	−	0.845253i	\(-0.679449\pi\)
							0.845253	+	0.534366i	\(0.179449\pi\)
\(17\)	46027.8	+	46027.8i	0.551092	+	0.551092i	0.926756	−	0.375664i	\(-0.122585\pi\)
							−0.375664	+	0.926756i	\(0.622585\pi\)
\(19\)			73288.0i			0.562365i	0.959654	+	0.281182i	\(0.0907266\pi\)
							−0.959654	+	0.281182i	\(0.909273\pi\)
\(23\)	263518.	−	263518.i	0.941671	−	0.941671i	−0.0567188	−	0.998390i	\(-0.518064\pi\)
							0.998390	+	0.0567188i	\(0.0180638\pi\)
\(29\)		−	1.20564e6i		−	1.70461i	−0.523047	−	0.852304i	\(-0.675205\pi\)
							0.523047	−	0.852304i	\(-0.324795\pi\)
\(31\)	−1.40909e6			−1.52578			−0.762892	−	0.646525i	\(-0.776221\pi\)
							−0.762892	+	0.646525i	\(0.776221\pi\)
\(37\)	1.51835e6	+	1.51835e6i	0.810149	+	0.810149i	0.984656	−	0.174507i	\(-0.0558332\pi\)
							−0.174507	+	0.984656i	\(0.555833\pi\)
\(41\)	2.03090e6			0.718708			0.359354	−	0.933201i	\(-0.382997\pi\)
							0.359354	+	0.933201i	\(0.382997\pi\)
\(43\)	−441189.	+	441189.i	−0.129048	+	0.129048i	−0.768681	−	0.639633i	\(-0.779086\pi\)
							0.639633	+	0.768681i	\(0.279086\pi\)
\(47\)	−3.91949e6	−	3.91949e6i	−0.803227	−	0.803227i	0.180372	−	0.983599i	\(-0.442270\pi\)
							−0.983599	+	0.180372i	\(0.942270\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10.9.c.a.7.1	yes	4
3.2	odd	2		90.9.g.b.37.1		4
4.3	odd	2		80.9.p.b.17.2		4
5.2	odd	4		50.9.c.e.43.2		4
5.3	odd	4	inner	10.9.c.a.3.1	✓	4
5.4	even	2		50.9.c.e.7.2		4
15.8	even	4		90.9.g.b.73.1		4
20.3	even	4		80.9.p.b.33.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.9.c.a.3.1	✓	4	5.3	odd	4	inner
10.9.c.a.7.1	yes	4	1.1	even	1	trivial
50.9.c.e.7.2		4	5.4	even	2
50.9.c.e.43.2		4	5.2	odd	4
80.9.p.b.17.2		4	4.3	odd	2
80.9.p.b.33.2		4	20.3	even	4
90.9.g.b.37.1		4	3.2	odd	2
90.9.g.b.73.1		4	15.8	even	4