Embedded newform 210.3.k.b.167.6

• Label: 210.3.k.b.167.6
• Level: $210$
• Weight: $3$
• Character: 210.167
• 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			167.6
Character	\(\chi\)	\(=\)	210.167
Dual form			210.3.k.b.83.6

$q$-expansion

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

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	−	1.00000i	0.500000	−	0.500000i
							\(3\)	−1.86291	−	2.35150i	−0.620970	−	0.783835i
\(5\)	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	0.226192	−	0.974083i	\(-0.427372\pi\)
							−0.974083	+	0.226192i	\(0.927372\pi\)
\(17\)	13.8691	+	13.8691i	0.815828	+	0.815828i	0.985500	−	0.169673i	\(-0.0542710\pi\)
							−0.169673	+	0.985500i	\(0.554271\pi\)
\(19\)	−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	\(-0.272051\pi\)
							−0.754356	+	0.656465i	\(0.772051\pi\)
\(29\)	46.0711			1.58866			0.794329	−	0.607488i	\(-0.207823\pi\)
							0.794329	+	0.607488i	\(0.207823\pi\)
\(31\)			1.50946i			0.0486921i	0.999704	+	0.0243461i	\(0.00775036\pi\)
							−0.999704	+	0.0243461i	\(0.992250\pi\)
\(37\)	5.32611	+	5.32611i	0.143949	+	0.143949i	0.775409	−	0.631460i	\(-0.217544\pi\)
							−0.631460	+	0.775409i	\(0.717544\pi\)
\(41\)	−13.4956			−0.329160			−0.164580	−	0.986364i	\(-0.552627\pi\)
							−0.164580	+	0.986364i	\(0.552627\pi\)
\(43\)	36.8754	−	36.8754i	0.857568	−	0.857568i	−0.133483	−	0.991051i	\(-0.542616\pi\)
							0.991051	+	0.133483i	\(0.0426162\pi\)
\(47\)	29.7803	+	29.7803i	0.633624	+	0.633624i	0.948975	−	0.315351i	\(-0.102122\pi\)
							−0.315351	+	0.948975i	\(0.602122\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.k.a.83.3	✓	32	35.13	even	4
210.3.k.a.83.14	yes	32	5.3	odd	4
210.3.k.a.167.3	yes	32	3.2	odd	2
210.3.k.a.167.14	yes	32	21.20	even	2
210.3.k.b.83.6	yes	32	105.83	odd	4	inner
210.3.k.b.83.11	yes	32	15.8	even	4	inner
210.3.k.b.167.6	yes	32	1.1	even	1	trivial
210.3.k.b.167.11	yes	32	7.6	odd	2	inner