Embedded newform 63.4.h.a.25.16

• Label: 63.4.h.a.25.16
• Level: $63$
• Weight: $4$
• Character: 63.25
• Analytic conductor: $3.717$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	63.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.71712033036\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			25.16
Character	\(\chi\)	\(=\)	63.25
Dual form			63.4.h.a.58.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.66292 q^{2} +(-4.74469 + 2.11846i) q^{3} -0.908849 q^{4} +(-9.61903 + 16.6607i) q^{5} +(-12.6348 + 5.64129i) q^{6} +(-5.55741 - 17.6668i) q^{7} -23.7236 q^{8} +(18.0243 - 20.1029i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.66292			0.941485			0.470742	−	0.882271i	\(-0.343986\pi\)
							0.470742	+	0.882271i	\(0.343986\pi\)
\(3\)	−4.74469	+	2.11846i	−0.913117	+	0.407698i
							\(5\)	−9.61903	+	16.6607i	−0.860353	+	1.49017i	0.0112364	+	0.999937i	\(0.496423\pi\)
−0.871589	+	0.490237i	\(0.836910\pi\)
\(7\)	−5.55741	−	17.6668i	−0.300072	−	0.953917i
							\(11\)	19.2665	+	33.3705i	0.528097	+	0.914691i	0.999463	+	0.0327533i	\(0.0104276\pi\)
−0.471367	+	0.881937i	\(0.656239\pi\)
\(13\)	38.2897	+	66.3197i	0.816896	+	1.41490i	0.907959	+	0.419060i	\(0.137640\pi\)
							−0.0910628	+	0.995845i	\(0.529026\pi\)
\(17\)	−11.9353	+	20.6725i	−0.170278	+	0.294931i	−0.938517	−	0.345233i	\(-0.887800\pi\)
							0.768239	+	0.640163i	\(0.221133\pi\)
\(19\)	−23.6758	−	41.0076i	−0.285873	−	0.495147i	0.686947	−	0.726707i	\(-0.258950\pi\)
							−0.972821	+	0.231560i	\(0.925617\pi\)
\(23\)	6.76863	−	11.7236i	0.0613634	−	0.106284i	−0.833712	−	0.552200i	\(-0.813788\pi\)
							0.895075	+	0.445916i	\(0.147122\pi\)
\(29\)	−53.5137	+	92.6885i	−0.342664	+	0.593511i	−0.984926	−	0.172974i	\(-0.944662\pi\)
							0.642263	+	0.766485i	\(0.277996\pi\)
\(31\)	−158.726			−0.919616			−0.459808	−	0.888018i	\(-0.652082\pi\)
							−0.459808	+	0.888018i	\(0.652082\pi\)
\(37\)	23.0136	+	39.8608i	0.102254	+	0.177110i	0.912613	−	0.408824i	\(-0.134061\pi\)
							−0.810359	+	0.585934i	\(0.800728\pi\)
\(41\)	101.261	+	175.388i	0.385713	+	0.668075i	0.991868	−	0.127272i	\(-0.0406221\pi\)
							−0.606155	+	0.795347i	\(0.707289\pi\)
\(43\)	−42.1121	+	72.9404i	−0.149350	+	0.258681i	−0.930987	−	0.365052i	\(-0.881051\pi\)
							0.781638	+	0.623733i	\(0.214385\pi\)
\(47\)	473.913			1.47079			0.735396	−	0.677637i	\(-0.236996\pi\)
							0.735396	+	0.677637i	\(0.236996\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.h.a.25.16	yes	44
3.2	odd	2		189.4.h.a.46.7		44
7.2	even	3		63.4.g.a.16.7	yes	44
9.4	even	3		63.4.g.a.4.7	✓	44
9.5	odd	6		189.4.g.a.172.16		44
21.2	odd	6		189.4.g.a.100.16		44
63.23	odd	6		189.4.h.a.37.7		44
63.58	even	3	inner	63.4.h.a.58.16	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.7	✓	44	9.4	even	3
63.4.g.a.16.7	yes	44	7.2	even	3
63.4.h.a.25.16	yes	44	1.1	even	1	trivial
63.4.h.a.58.16	yes	44	63.58	even	3	inner
189.4.g.a.100.16		44	21.2	odd	6
189.4.g.a.172.16		44	9.5	odd	6
189.4.h.a.37.7		44	63.23	odd	6
189.4.h.a.46.7		44	3.2	odd	2