Embedded newform 10.11.c.c.3.2

• Label: 10.11.c.c.3.2
• Level: $10$
• Weight: $11$
• Character: 10.3
• Analytic conductor: $6.354$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.35357252674\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial:	\( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{8}\cdot 5^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.2
Root			\(10.1043 - 10.1043i\) of defining polynomial
Character	\(\chi\)	\(=\)	10.3
Dual form			10.11.c.c.7.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.0000 + 16.0000i) q^{2} +(4.29207 + 4.29207i) q^{3} -512.000i q^{4} +(-2978.33 + 946.124i) q^{5} -137.346 q^{6} +(21284.7 - 21284.7i) q^{7} +(8192.00 + 8192.00i) q^{8} -59012.2i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 96 q^{2} + 128 q^{3} + 5460 q^{5} - 4096 q^{6} + 13512 q^{7} + 49152 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.0000	+	16.0000i	−0.500000	+	0.500000i
							\(3\)	4.29207	+	4.29207i	0.0176629	+	0.0176629i	0.715883	−	0.698220i	\(-0.246024\pi\)
−0.698220	+	0.715883i	\(0.746024\pi\)
\(5\)	−2978.33	+	946.124i	−0.953067	+	0.302760i
							\(7\)	21284.7	−	21284.7i	1.26642	−	1.26642i	0.318492	−	0.947926i	\(-0.396824\pi\)
0.947926	−	0.318492i	\(-0.103176\pi\)
\(11\)	155649.			0.966456			0.483228	−	0.875495i	\(-0.339464\pi\)
							0.483228	+	0.875495i	\(0.339464\pi\)
\(13\)	−358614.	−	358614.i	−0.965853	−	0.965853i	0.0335834	−	0.999436i	\(-0.489308\pi\)
							−0.999436	+	0.0335834i	\(0.989308\pi\)
\(17\)	−609397.	+	609397.i	−0.429196	+	0.429196i	−0.888354	−	0.459158i	\(-0.848151\pi\)
							0.459158	+	0.888354i	\(0.348151\pi\)
\(19\)		−	335226.i		−	0.135385i	−0.997706	−	0.0676924i	\(-0.978436\pi\)
							0.997706	−	0.0676924i	\(-0.0215637\pi\)
\(23\)	−5.50132e6	−	5.50132e6i	−0.854727	−	0.854727i	0.135984	−	0.990711i	\(-0.456580\pi\)
							−0.990711	+	0.135984i	\(0.956580\pi\)
\(29\)		−	1.33815e6i		−	0.0652402i	−0.999468	−	0.0326201i	\(-0.989615\pi\)
							0.999468	−	0.0326201i	\(-0.0103851\pi\)
\(31\)	2.59306e7			0.905743			0.452871	−	0.891576i	\(-0.350400\pi\)
							0.452871	+	0.891576i	\(0.350400\pi\)
\(37\)	−5.50890e7	+	5.50890e7i	−0.794431	+	0.794431i	−0.982211	−	0.187780i	\(-0.939871\pi\)
							0.187780	+	0.982211i	\(0.439871\pi\)
\(41\)	1.37064e8			1.18305			0.591525	−	0.806287i	\(-0.298526\pi\)
							0.591525	+	0.806287i	\(0.298526\pi\)
\(43\)	−9.34290e7	−	9.34290e7i	−0.635535	−	0.635535i	0.313916	−	0.949451i	\(-0.398359\pi\)
							−0.949451	+	0.313916i	\(0.898359\pi\)
\(47\)	1.14979e8	−	1.14979e8i	0.501334	−	0.501334i	−0.410518	−	0.911852i	\(-0.634652\pi\)
							0.911852	+	0.410518i	\(0.134652\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10.11.c.c.3.2	✓	6
3.2	odd	2		90.11.g.c.73.3		6
4.3	odd	2		80.11.p.c.33.2		6
5.2	odd	4	inner	10.11.c.c.7.2	yes	6
5.3	odd	4		50.11.c.e.7.2		6
5.4	even	2		50.11.c.e.43.2		6
15.2	even	4		90.11.g.c.37.3		6
20.7	even	4		80.11.p.c.17.2		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.11.c.c.3.2	✓	6	1.1	even	1	trivial
10.11.c.c.7.2	yes	6	5.2	odd	4	inner
50.11.c.e.7.2		6	5.3	odd	4
50.11.c.e.43.2		6	5.4	even	2
80.11.p.c.17.2		6	20.7	even	4
80.11.p.c.33.2		6	4.3	odd	2
90.11.g.c.37.3		6	15.2	even	4
90.11.g.c.73.3		6	3.2	odd	2