Embedded newform 10.11.c.b.7.1

• Label: 10.11.c.b.7.1
• Level: $10$
• Weight: $11$
• Character: 10.7
• Analytic conductor: $6.354$
• Analytic rank: $0$
• Dimension: $2$
• 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:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			7.1
Root			\(1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	10.7
Dual form			10.11.c.b.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(16.0000 + 16.0000i) q^{2} +(57.0000 - 57.0000i) q^{3} +512.000i q^{4} +(2925.00 + 1100.00i) q^{5} +1824.00 q^{6} +(6953.00 + 6953.00i) q^{7} +(-8192.00 + 8192.00i) q^{8} +52551.0i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 32 q^{2} + 114 q^{3} + 5850 q^{5} + 3648 q^{6} + 13906 q^{7} - 16384 q^{8}+O(q^{10}) \)

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
							\(3\)	57.0000	−	57.0000i	0.234568	−	0.234568i	−0.580028	−	0.814596i	\(-0.696959\pi\)
0.814596	+	0.580028i	\(0.196959\pi\)
\(5\)	2925.00	+	1100.00i	0.936000	+	0.352000i
							\(7\)	6953.00	+	6953.00i	0.413697	+	0.413697i	0.883024	−	0.469328i	\(-0.155504\pi\)
−0.469328	+	0.883024i	\(0.655504\pi\)
\(11\)	75242.0			0.467194			0.233597	−	0.972334i	\(-0.424950\pi\)
							0.233597	+	0.972334i	\(0.424950\pi\)
\(13\)	109857.	−	109857.i	0.295877	−	0.295877i	−0.543520	−	0.839396i	\(-0.682909\pi\)
							0.839396	+	0.543520i	\(0.182909\pi\)
\(17\)	−1.52893e6	−	1.52893e6i	−1.07682	−	1.07682i	−0.996793	−	0.0800247i	\(-0.974500\pi\)
							−0.0800247	−	0.996793i	\(-0.525500\pi\)
\(19\)		−	4.03868e6i		−	1.63107i	−0.578711	−	0.815533i	\(-0.696444\pi\)
							0.578711	−	0.815533i	\(-0.303556\pi\)
\(23\)	−712423.	+	712423.i	−0.110688	+	0.110688i	−0.760281	−	0.649594i	\(-0.774939\pi\)
							0.649594	+	0.760281i	\(0.274939\pi\)
\(29\)			446120.i			0.0217501i	0.999941	+	0.0108751i	\(0.00346171\pi\)
							−0.999941	+	0.0108751i	\(0.996538\pi\)
\(31\)	−2.90807e7			−1.01577			−0.507886	−	0.861424i	\(-0.669573\pi\)
							−0.507886	+	0.861424i	\(0.669573\pi\)
\(37\)	−911847.	−	911847.i	−0.0131496	−	0.0131496i	0.700501	−	0.713651i	\(-0.252960\pi\)
							−0.713651	+	0.700501i	\(0.752960\pi\)
\(41\)	−1.63946e8			−1.41508			−0.707540	−	0.706674i	\(-0.750195\pi\)
							−0.707540	+	0.706674i	\(0.750195\pi\)
\(43\)	1.18423e8	−	1.18423e8i	0.805551	−	0.805551i	−0.178406	−	0.983957i	\(-0.557094\pi\)
							0.983957	+	0.178406i	\(0.0570941\pi\)
\(47\)	2.76320e8	+	2.76320e8i	1.20482	+	1.20482i	0.972682	+	0.232142i	\(0.0745733\pi\)
							0.232142	+	0.972682i	\(0.425427\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10.11.c.b.7.1	yes	2
3.2	odd	2		90.11.g.a.37.1		2
4.3	odd	2		80.11.p.a.17.1		2
5.2	odd	4		50.11.c.b.43.1		2
5.3	odd	4	inner	10.11.c.b.3.1	✓	2
5.4	even	2		50.11.c.b.7.1		2
15.8	even	4		90.11.g.a.73.1		2
20.3	even	4		80.11.p.a.33.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.11.c.b.3.1	✓	2	5.3	odd	4	inner
10.11.c.b.7.1	yes	2	1.1	even	1	trivial
50.11.c.b.7.1		2	5.4	even	2
50.11.c.b.43.1		2	5.2	odd	4
80.11.p.a.17.1		2	4.3	odd	2
80.11.p.a.33.1		2	20.3	even	4
90.11.g.a.37.1		2	3.2	odd	2
90.11.g.a.73.1		2	15.8	even	4