Embedded newform 735.2.y.g.557.5

• Label: 735.2.y.g.557.5
• Level: $735$
• Weight: $2$
• Character: 735.557
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.y (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			557.5
Character	\(\chi\)	\(=\)	735.557
Dual form			735.2.y.g.128.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.474084 - 0.127030i) q^{2} +(-1.40400 - 1.01429i) q^{3} +(-1.52343 - 0.879554i) q^{4} +(2.23532 + 0.0580193i) q^{5} +(0.536770 + 0.659208i) q^{6} +(1.30461 + 1.30461i) q^{8} +(0.942448 + 2.84812i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{3}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\)	\(346\)	\(442\)	\(491\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.474084	−	0.127030i	−0.335228	−	0.0898241i	0.0872785	−	0.996184i	\(-0.472183\pi\)
							−0.422507	+	0.906360i	\(0.638850\pi\)
\(3\)	−1.40400	−	1.01429i	−0.810601	−	0.585598i
							\(5\)	2.23532	+	0.0580193i	0.999663	+	0.0259470i
\(7\)		0			0
							\(11\)	−2.31347	−	1.33568i	−0.697538	−	0.402724i	0.108892	−	0.994054i	\(-0.465270\pi\)
−0.806430	+	0.591330i	\(0.798603\pi\)
\(13\)	−2.14945	+	2.14945i	−0.596149	+	0.596149i	−0.939285	−	0.343137i	\(-0.888511\pi\)
							0.343137	+	0.939285i	\(0.388511\pi\)
\(17\)	1.19588	+	4.46307i	0.290042	+	1.08245i	0.945075	+	0.326852i	\(0.105988\pi\)
							−0.655033	+	0.755600i	\(0.727345\pi\)
\(19\)	−4.54082	+	2.62164i	−1.04174	+	0.601446i	−0.920324	−	0.391156i	\(-0.872075\pi\)
							−0.121411	+	0.992602i	\(0.538742\pi\)
\(23\)	0.932688	−	3.48084i	0.194479	−	0.725805i	−0.797922	−	0.602760i	\(-0.794068\pi\)
							0.992401	−	0.123045i	\(-0.0392658\pi\)
\(29\)	2.86924			0.532804			0.266402	−	0.963862i	\(-0.414165\pi\)
							0.266402	+	0.963862i	\(0.414165\pi\)
\(31\)	−2.64299	+	4.57780i	−0.474696	+	0.822197i	−0.999580	−	0.0289764i	\(-0.990775\pi\)
							0.524884	+	0.851174i	\(0.324109\pi\)
\(37\)	−0.784341	+	2.92720i	−0.128945	+	0.481229i	−0.999950	−	0.0100459i	\(-0.996802\pi\)
							0.871005	+	0.491275i	\(0.163469\pi\)
\(41\)			11.5768i			1.80800i	0.427537	+	0.903998i	\(0.359381\pi\)
							−0.427537	+	0.903998i	\(0.640619\pi\)
\(43\)	0.759108	−	0.759108i	0.115763	−	0.115763i	−0.646852	−	0.762615i	\(-0.723915\pi\)
							0.762615	+	0.646852i	\(0.223915\pi\)
\(47\)	10.4642	+	2.80388i	1.52636	+	0.408988i	0.921831	−	0.387593i	\(-0.126693\pi\)
							0.604533	+	0.796580i	\(0.293360\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.y.g.557.5		48
3.2	odd	2	inner	735.2.y.g.557.8		48
5.3	odd	4	inner	735.2.y.g.263.5		48
7.2	even	3	inner	735.2.y.g.422.8		48
7.3	odd	6		105.2.j.a.92.8	yes	24
7.4	even	3		735.2.j.h.197.8		24
7.5	odd	6		735.2.y.j.422.8		48
7.6	odd	2		735.2.y.j.557.5		48
15.8	even	4	inner	735.2.y.g.263.8		48
21.2	odd	6	inner	735.2.y.g.422.5		48
21.5	even	6		735.2.y.j.422.5		48
21.11	odd	6		735.2.j.h.197.5		24
21.17	even	6		105.2.j.a.92.5	yes	24
21.20	even	2		735.2.y.j.557.8		48
35.3	even	12		105.2.j.a.8.5	✓	24
35.13	even	4		735.2.y.j.263.5		48
35.17	even	12		525.2.j.b.218.8		24
35.18	odd	12		735.2.j.h.638.5		24
35.23	odd	12	inner	735.2.y.g.128.8		48
35.24	odd	6		525.2.j.b.407.5		24
35.33	even	12		735.2.y.j.128.8		48
105.17	odd	12		525.2.j.b.218.5		24
105.23	even	12	inner	735.2.y.g.128.5		48
105.38	odd	12		105.2.j.a.8.8	yes	24
105.53	even	12		735.2.j.h.638.8		24
105.59	even	6		525.2.j.b.407.8		24
105.68	odd	12		735.2.y.j.128.5		48
105.83	odd	4		735.2.y.j.263.8		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.j.a.8.5	✓	24	35.3	even	12
105.2.j.a.8.8	yes	24	105.38	odd	12
105.2.j.a.92.5	yes	24	21.17	even	6
105.2.j.a.92.8	yes	24	7.3	odd	6
525.2.j.b.218.5		24	105.17	odd	12
525.2.j.b.218.8		24	35.17	even	12
525.2.j.b.407.5		24	35.24	odd	6
525.2.j.b.407.8		24	105.59	even	6
735.2.j.h.197.5		24	21.11	odd	6
735.2.j.h.197.8		24	7.4	even	3
735.2.j.h.638.5		24	35.18	odd	12
735.2.j.h.638.8		24	105.53	even	12
735.2.y.g.128.5		48	105.23	even	12	inner
735.2.y.g.128.8		48	35.23	odd	12	inner
735.2.y.g.263.5		48	5.3	odd	4	inner
735.2.y.g.263.8		48	15.8	even	4	inner
735.2.y.g.422.5		48	21.2	odd	6	inner
735.2.y.g.422.8		48	7.2	even	3	inner
735.2.y.g.557.5		48	1.1	even	1	trivial
735.2.y.g.557.8		48	3.2	odd	2	inner
735.2.y.j.128.5		48	105.68	odd	12
735.2.y.j.128.8		48	35.33	even	12
735.2.y.j.263.5		48	35.13	even	4
735.2.y.j.263.8		48	105.83	odd	4
735.2.y.j.422.5		48	21.5	even	6
735.2.y.j.422.8		48	7.5	odd	6
735.2.y.j.557.5		48	7.6	odd	2
735.2.y.j.557.8		48	21.20	even	2