Embedded newform 10.11.c.c.7.1

• Label: 10.11.c.c.7.1
• Level: $10$
• Weight: $11$
• Character: 10.7
• 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			7.1
Root			\(2.29143 + 2.29143i\) of defining polynomial
Character	\(\chi\)	\(=\)	10.7
Dual form			10.11.c.c.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.0000 - 16.0000i) q^{2} +(-266.441 + 266.441i) q^{3} +512.000i q^{4} +(2583.39 - 1758.32i) q^{5} +8526.10 q^{6} +(-13021.6 - 13021.6i) q^{7} +(8192.00 - 8192.00i) q^{8} -82932.3i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 96 q^{2} + 128 q^{3} + 5460 q^{5} - 4096 q^{6} + 13512 q^{7} + 49152 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.0000	−	16.0000i	−0.500000	−	0.500000i
							\(3\)	−266.441	+	266.441i	−1.09646	+	1.09646i	−0.101643	+	0.994821i	\(0.532410\pi\)
−0.994821	+	0.101643i	\(0.967590\pi\)
\(5\)	2583.39	−	1758.32i	0.826686	−	0.562663i
							\(7\)	−13021.6	−	13021.6i	−0.774769	−	0.774769i	0.204167	−	0.978936i	\(-0.434552\pi\)
−0.978936	+	0.204167i	\(0.934552\pi\)
\(11\)	239729.			1.48853			0.744266	−	0.667884i	\(-0.232800\pi\)
							0.744266	+	0.667884i	\(0.232800\pi\)
\(13\)	287980.	−	287980.i	0.775613	−	0.775613i	−0.203468	−	0.979082i	\(-0.565221\pi\)
							0.979082	+	0.203468i	\(0.0652214\pi\)
\(17\)	−884527.	−	884527.i	−0.622969	−	0.622969i	0.323320	−	0.946290i	\(-0.395201\pi\)
							−0.946290	+	0.323320i	\(0.895201\pi\)
\(19\)		−	197744.i		−	0.0798611i	−0.999202	−	0.0399305i	\(-0.987286\pi\)
							0.999202	−	0.0399305i	\(-0.0127137\pi\)
\(23\)	−3.55115e6	+	3.55115e6i	−0.551734	+	0.551734i	−0.926941	−	0.375207i	\(-0.877572\pi\)
							0.375207	+	0.926941i	\(0.377572\pi\)
\(29\)		−	3.57499e7i		−	1.74295i	−0.490439	−	0.871476i	\(-0.663163\pi\)
							0.490439	−	0.871476i	\(-0.336837\pi\)
\(31\)	9.65516e6			0.337249			0.168625	−	0.985680i	\(-0.446067\pi\)
							0.168625	+	0.985680i	\(0.446067\pi\)
\(37\)	1.34842e7	+	1.34842e7i	0.194453	+	0.194453i	0.797617	−	0.603164i	\(-0.206094\pi\)
							−0.603164	+	0.797617i	\(0.706094\pi\)
\(41\)	1.08475e8			0.936286			0.468143	−	0.883653i	\(-0.344923\pi\)
							0.468143	+	0.883653i	\(0.344923\pi\)
\(43\)	8.62678e7	−	8.62678e7i	0.586822	−	0.586822i	−0.349947	−	0.936769i	\(-0.613801\pi\)
							0.936769	+	0.349947i	\(0.113801\pi\)
\(47\)	−1.76759e8	−	1.76759e8i	−0.770713	−	0.770713i	0.207518	−	0.978231i	\(-0.433461\pi\)
							−0.978231	+	0.207518i	\(0.933461\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10.11.c.c.7.1	yes	6
3.2	odd	2		90.11.g.c.37.2		6
4.3	odd	2		80.11.p.c.17.3		6
5.2	odd	4		50.11.c.e.43.3		6
5.3	odd	4	inner	10.11.c.c.3.1	✓	6
5.4	even	2		50.11.c.e.7.3		6
15.8	even	4		90.11.g.c.73.2		6
20.3	even	4		80.11.p.c.33.3		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.11.c.c.3.1	✓	6	5.3	odd	4	inner
10.11.c.c.7.1	yes	6	1.1	even	1	trivial
50.11.c.e.7.3		6	5.4	even	2
50.11.c.e.43.3		6	5.2	odd	4
80.11.p.c.17.3		6	4.3	odd	2
80.11.p.c.33.3		6	20.3	even	4
90.11.g.c.37.2		6	3.2	odd	2
90.11.g.c.73.2		6	15.8	even	4