Embedded newform 210.3.s.a.11.14

• Label: 210.3.s.a.11.14
• Level: $210$
• Weight: $3$
• Character: 210.11
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.14
Character	\(\chi\)	\(=\)	210.11
Dual form			210.3.s.a.191.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.22474 - 0.707107i) q^{2} +(-1.69912 - 2.47244i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-1.93649 + 1.11803i) q^{5} +(-3.82928 - 1.82665i) q^{6} +(-6.91642 + 1.07847i) q^{7} -2.82843i q^{8} +(-3.22595 + 8.40198i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 40 q^{4} - 8 q^{6} + 20 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.22474	−	0.707107i	0.612372	−	0.353553i
							\(3\)	−1.69912	−	2.47244i	−0.566375	−	0.824148i
\(5\)	−1.93649	+	1.11803i	−0.387298	+	0.223607i
							\(7\)	−6.91642	+	1.07847i	−0.988060	+	0.154067i
\(11\)	−17.8029	−	10.2785i	−1.61844	−	0.934408i								−0.987323	−	0.158723i	\(-0.949262\pi\)
							−0.631119	−	0.775686i	\(-0.717404\pi\)
\(13\)	8.06735			0.620566			0.310283	−	0.950644i	\(-0.399576\pi\)
							0.310283	+	0.950644i	\(0.399576\pi\)
\(17\)	8.43792	+	4.87164i	0.496348	+	0.286567i	0.727204	−	0.686421i	\(-0.240819\pi\)
							−0.230856	+	0.972988i	\(0.574153\pi\)
\(19\)	−2.22102	−	3.84692i	−0.116896	−	0.202470i	0.801640	−	0.597807i	\(-0.203961\pi\)
							−0.918536	+	0.395337i	\(0.870628\pi\)
\(23\)	−36.8634	+	21.2831i	−1.60276	+	0.925352i	−0.611824	+	0.790994i	\(0.709564\pi\)
							−0.990933	+	0.134358i	\(0.957103\pi\)
\(29\)		−	45.5918i		−	1.57213i	−0.618143	−	0.786066i	\(-0.712115\pi\)
							0.618143	−	0.786066i	\(-0.287885\pi\)
\(31\)	1.31018	−	2.26930i	0.0422639	−	0.0732033i	−0.844120	−	0.536155i	\(-0.819876\pi\)
							0.886384	+	0.462952i	\(0.153210\pi\)
\(37\)	−16.4912	−	28.5636i	−0.445708	−	0.771989i	0.552393	−	0.833584i	\(-0.313715\pi\)
							−0.998101	+	0.0615947i	\(0.980381\pi\)
\(41\)		−	27.8799i		−	0.679999i	−0.940426	−	0.339999i	\(-0.889573\pi\)
							0.940426	−	0.339999i	\(-0.110427\pi\)
\(43\)	−35.3520			−0.822140			−0.411070	−	0.911604i	\(-0.634845\pi\)
							−0.411070	+	0.911604i	\(0.634845\pi\)
\(47\)	59.3971	−	34.2929i	1.26377	−	0.729636i	0.289966	−	0.957037i	\(-0.406356\pi\)
							0.973801	+	0.227401i	\(0.0730226\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.s.a.11.14	yes	40
3.2	odd	2	inner	210.3.s.a.11.10	✓	40
7.2	even	3	inner	210.3.s.a.191.10	yes	40
21.2	odd	6	inner	210.3.s.a.191.14	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.s.a.11.10	✓	40	3.2	odd	2	inner
210.3.s.a.11.14	yes	40	1.1	even	1	trivial
210.3.s.a.191.10	yes	40	7.2	even	3	inner
210.3.s.a.191.14	yes	40	21.2	odd	6	inner