Embedded newform 210.3.k.a.83.3

• Label: 210.3.k.a.83.3
• Level: $210$
• Weight: $3$
• Character: 210.83
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			83.3
Character	\(\chi\)	\(=\)	210.83
Dual form			210.3.k.a.167.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} +(-2.35150 + 1.86291i) q^{3} +2.00000i q^{4} +(-1.91622 - 4.61824i) q^{5} +(4.21441 + 0.488596i) q^{6} +(6.26242 + 3.12763i) q^{7} +(2.00000 - 2.00000i) q^{8} +(2.05914 - 8.76127i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{2} - 4 q^{7} + 64 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−	1.00000i	−0.500000	−	0.500000i
							\(3\)	−2.35150	+	1.86291i	−0.783835	+	0.620970i
\(5\)	−1.91622	−	4.61824i	−0.383244	−	0.923647i
							\(7\)	6.26242	+	3.12763i	0.894631	+	0.446805i
\(11\)		−	0.117411i		−	0.0106738i								−0.999986	−	0.00533688i	\(-0.998301\pi\)
							0.999986	−	0.00533688i	\(-0.00169879\pi\)
\(13\)	−9.72258	+	9.72258i	−0.747890	+	0.747890i	−0.974083	−	0.226192i	\(-0.927372\pi\)
							0.226192	+	0.974083i	\(0.427372\pi\)
\(17\)	−13.8691	+	13.8691i	−0.815828	+	0.815828i	−0.985500	−	0.169673i	\(-0.945729\pi\)
							0.169673	+	0.985500i	\(0.445729\pi\)
\(19\)	−29.7218			−1.56431			−0.782153	−	0.623086i	\(-0.785879\pi\)
							−0.782153	+	0.623086i	\(0.785879\pi\)
\(23\)	2.25150	−	2.25150i	0.0978912	−	0.0978912i	−0.656465	−	0.754356i	\(-0.727949\pi\)
							0.754356	+	0.656465i	\(0.227949\pi\)
\(29\)	−46.0711			−1.58866			−0.794329	−	0.607488i	\(-0.792177\pi\)
							−0.794329	+	0.607488i	\(0.792177\pi\)
\(31\)		−	1.50946i		−	0.0486921i	−0.999704	−	0.0243461i	\(-0.992250\pi\)
							0.999704	−	0.0243461i	\(-0.00775036\pi\)
\(37\)	5.32611	−	5.32611i	0.143949	−	0.143949i	−0.631460	−	0.775409i	\(-0.717544\pi\)
							0.775409	+	0.631460i	\(0.217544\pi\)
\(41\)	13.4956			0.329160			0.164580	−	0.986364i	\(-0.447373\pi\)
							0.164580	+	0.986364i	\(0.447373\pi\)
\(43\)	36.8754	+	36.8754i	0.857568	+	0.857568i	0.991051	−	0.133483i	\(-0.0426162\pi\)
							−0.133483	+	0.991051i	\(0.542616\pi\)
\(47\)	−29.7803	+	29.7803i	−0.633624	+	0.633624i	−0.948975	−	0.315351i	\(-0.897878\pi\)
							0.315351	+	0.948975i	\(0.397878\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.k.a.83.3	✓	32
3.2	odd	2		210.3.k.b.83.6	yes	32
5.2	odd	4		210.3.k.b.167.11	yes	32
7.6	odd	2	inner	210.3.k.a.83.14	yes	32
15.2	even	4	inner	210.3.k.a.167.14	yes	32
21.20	even	2		210.3.k.b.83.11	yes	32
35.27	even	4		210.3.k.b.167.6	yes	32
105.62	odd	4	inner	210.3.k.a.167.3	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.k.a.83.3	✓	32	1.1	even	1	trivial
210.3.k.a.83.14	yes	32	7.6	odd	2	inner
210.3.k.a.167.3	yes	32	105.62	odd	4	inner
210.3.k.a.167.14	yes	32	15.2	even	4	inner
210.3.k.b.83.6	yes	32	3.2	odd	2
210.3.k.b.83.11	yes	32	21.20	even	2
210.3.k.b.167.6	yes	32	35.27	even	4
210.3.k.b.167.11	yes	32	5.2	odd	4