Embedded newform 735.2.y.j.128.7

• Label: 735.2.y.j.128.7
• Level: $735$
• Weight: $2$
• Character: 735.128
• 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			128.7
Character	\(\chi\)	\(=\)	735.128
Dual form			735.2.y.j.557.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.355526 - 0.0952630i) q^{2} +(1.47708 + 0.904561i) q^{3} +(-1.61473 + 0.932263i) q^{4} +(1.32690 + 1.79982i) q^{5} +(0.611312 + 0.180884i) q^{6} +(-1.00579 + 1.00579i) q^{8} +(1.36354 + 2.67222i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{3}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.355526	−	0.0952630i	0.251395	−	0.0673611i	−0.130921	−	0.991393i	\(-0.541793\pi\)
							0.382316	+	0.924032i	\(0.375127\pi\)
\(3\)	1.47708	+	0.904561i	0.852793	+	0.522249i
							\(5\)	1.32690	+	1.79982i	0.593407	+	0.804903i
\(7\)		0			0
							\(11\)	−2.93366	+	1.69375i	−0.884531	+	0.510684i	−0.872150	−	0.489239i	\(-0.837275\pi\)
−0.0123815	+	0.999923i	\(0.503941\pi\)
\(13\)	1.59420	+	1.59420i	0.442151	+	0.442151i	0.892734	−	0.450583i	\(-0.148784\pi\)
							−0.450583	+	0.892734i	\(0.648784\pi\)
\(17\)	−0.0514941	+	0.192179i	−0.0124891	+	0.0466101i	−0.971889	−	0.235439i	\(-0.924347\pi\)
							0.959400	+	0.282049i	\(0.0910140\pi\)
\(19\)	−6.36261	−	3.67345i	−1.45968	−	0.842748i	−0.460687	−	0.887563i	\(-0.652397\pi\)
							−0.998995	+	0.0448148i	\(0.985730\pi\)
\(23\)	−0.810540	−	3.02498i	−0.169009	−	0.630752i	−0.997495	−	0.0707408i	\(-0.977464\pi\)
							0.828485	−	0.560011i	\(-0.189203\pi\)
\(29\)	9.49165			1.76256			0.881278	−	0.472598i	\(-0.156684\pi\)
							0.881278	+	0.472598i	\(0.156684\pi\)
\(31\)	−0.461291	−	0.798980i	−0.0828503	−	0.143501i	0.821623	−	0.570031i	\(-0.193069\pi\)
							−0.904473	+	0.426531i	\(0.859736\pi\)
\(37\)	2.16525	+	8.08083i	0.355965	+	1.32848i	0.879266	+	0.476332i	\(0.158034\pi\)
							−0.523300	+	0.852148i	\(0.675299\pi\)
\(41\)			1.39256i			0.217481i	0.994070	+	0.108741i	\(0.0346818\pi\)
							−0.994070	+	0.108741i	\(0.965318\pi\)
\(43\)	0.864526	+	0.864526i	0.131839	+	0.131839i	0.769947	−	0.638108i	\(-0.220283\pi\)
							−0.638108	+	0.769947i	\(0.720283\pi\)
\(47\)	0.889755	−	0.238409i	0.129784	−	0.0347755i	−0.193342	−	0.981131i	\(-0.561933\pi\)
							0.323126	+	0.946356i	\(0.395266\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.y.j.128.7		48
3.2	odd	2	inner	735.2.y.j.128.6		48
5.2	odd	4	inner	735.2.y.j.422.7		48
7.2	even	3		105.2.j.a.8.6	✓	24
7.3	odd	6		735.2.y.g.263.6		48
7.4	even	3	inner	735.2.y.j.263.6		48
7.5	odd	6		735.2.j.h.638.6		24
7.6	odd	2		735.2.y.g.128.7		48
15.2	even	4	inner	735.2.y.j.422.6		48
21.2	odd	6		105.2.j.a.8.7	yes	24
21.5	even	6		735.2.j.h.638.7		24
21.11	odd	6	inner	735.2.y.j.263.7		48
21.17	even	6		735.2.y.g.263.7		48
21.20	even	2		735.2.y.g.128.6		48
35.2	odd	12		105.2.j.a.92.7	yes	24
35.9	even	6		525.2.j.b.218.7		24
35.12	even	12		735.2.j.h.197.7		24
35.17	even	12		735.2.y.g.557.6		48
35.23	odd	12		525.2.j.b.407.6		24
35.27	even	4		735.2.y.g.422.7		48
35.32	odd	12	inner	735.2.y.j.557.6		48
105.2	even	12		105.2.j.a.92.6	yes	24
105.17	odd	12		735.2.y.g.557.7		48
105.23	even	12		525.2.j.b.407.7		24
105.32	even	12	inner	735.2.y.j.557.7		48
105.44	odd	6		525.2.j.b.218.6		24
105.47	odd	12		735.2.j.h.197.6		24
105.62	odd	4		735.2.y.g.422.6		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.j.a.8.6	✓	24	7.2	even	3
105.2.j.a.8.7	yes	24	21.2	odd	6
105.2.j.a.92.6	yes	24	105.2	even	12
105.2.j.a.92.7	yes	24	35.2	odd	12
525.2.j.b.218.6		24	105.44	odd	6
525.2.j.b.218.7		24	35.9	even	6
525.2.j.b.407.6		24	35.23	odd	12
525.2.j.b.407.7		24	105.23	even	12
735.2.j.h.197.6		24	105.47	odd	12
735.2.j.h.197.7		24	35.12	even	12
735.2.j.h.638.6		24	7.5	odd	6
735.2.j.h.638.7		24	21.5	even	6
735.2.y.g.128.6		48	21.20	even	2
735.2.y.g.128.7		48	7.6	odd	2
735.2.y.g.263.6		48	7.3	odd	6
735.2.y.g.263.7		48	21.17	even	6
735.2.y.g.422.6		48	105.62	odd	4
735.2.y.g.422.7		48	35.27	even	4
735.2.y.g.557.6		48	35.17	even	12
735.2.y.g.557.7		48	105.17	odd	12
735.2.y.j.128.6		48	3.2	odd	2	inner
735.2.y.j.128.7		48	1.1	even	1	trivial
735.2.y.j.263.6		48	7.4	even	3	inner
735.2.y.j.263.7		48	21.11	odd	6	inner
735.2.y.j.422.6		48	15.2	even	4	inner
735.2.y.j.422.7		48	5.2	odd	4	inner
735.2.y.j.557.6		48	35.32	odd	12	inner
735.2.y.j.557.7		48	105.32	even	12	inner