Embedded newform 735.2.y.g.422.7

• Label: 735.2.y.g.422.7
• Level: $735$
• Weight: $2$
• Character: 735.422
• 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			422.7
Character	\(\chi\)	\(=\)	735.422
Dual form			735.2.y.g.263.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0952630 + 0.355526i) q^{2} +(-0.904561 + 1.47708i) q^{3} +(1.61473 - 0.932263i) q^{4} +(-2.22214 - 0.249219i) q^{5} +(-0.611312 - 0.180884i) q^{6} +(1.00579 + 1.00579i) q^{8} +(-1.36354 - 2.67222i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{3}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0952630	+	0.355526i	0.0673611	+	0.251395i	0.991393	−	0.130921i	\(-0.0417933\pi\)
							−0.924032	+	0.382316i	\(0.875127\pi\)
\(3\)	−0.904561	+	1.47708i	−0.522249	+	0.852793i
							\(5\)	−2.22214	−	0.249219i	−0.993770	−	0.111454i
\(7\)		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	\(-0.851216\pi\)
							0.450583	+	0.892734i	\(0.351216\pi\)
\(17\)	0.192179	+	0.0514941i	0.0466101	+	0.0124891i	0.282049	−	0.959400i	\(-0.408986\pi\)
							−0.235439	+	0.971889i	\(0.575653\pi\)
\(19\)	−6.36261	−	3.67345i	−1.45968	−	0.842748i	−0.460687	−	0.887563i	\(-0.652397\pi\)
							−0.998995	+	0.0448148i	\(0.985730\pi\)
\(23\)	−3.02498	+	0.810540i	−0.630752	+	0.169009i	−0.560011	−	0.828485i	\(-0.689203\pi\)
							−0.0707408	+	0.997495i	\(0.522536\pi\)
\(29\)	−9.49165			−1.76256			−0.881278	−	0.472598i	\(-0.843316\pi\)
							−0.881278	+	0.472598i	\(0.843316\pi\)
\(31\)	0.461291	+	0.798980i	0.0828503	+	0.143501i	0.904473	−	0.426531i	\(-0.140264\pi\)
							−0.821623	+	0.570031i	\(0.806931\pi\)
\(37\)	−8.08083	+	2.16525i	−1.32848	+	0.355965i	−0.852148	−	0.523300i	\(-0.824701\pi\)
							−0.476332	+	0.879266i	\(0.658034\pi\)
\(41\)		−	1.39256i		−	0.217481i	−0.994070	−	0.108741i	\(-0.965318\pi\)
							0.994070	−	0.108741i	\(-0.0346818\pi\)
\(43\)	0.864526	−	0.864526i	0.131839	−	0.131839i	−0.638108	−	0.769947i	\(-0.720283\pi\)
							0.769947	+	0.638108i	\(0.220283\pi\)
\(47\)	−0.238409	−	0.889755i	−0.0347755	−	0.129784i	0.946356	−	0.323126i	\(-0.104734\pi\)
							−0.981131	+	0.193342i	\(0.938067\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.y.g.422.7		48
3.2	odd	2	inner	735.2.y.g.422.6		48
5.3	odd	4	inner	735.2.y.g.128.7		48
7.2	even	3		735.2.j.h.197.7		24
7.3	odd	6		735.2.y.j.557.6		48
7.4	even	3	inner	735.2.y.g.557.6		48
7.5	odd	6		105.2.j.a.92.7	yes	24
7.6	odd	2		735.2.y.j.422.7		48
15.8	even	4	inner	735.2.y.g.128.6		48
21.2	odd	6		735.2.j.h.197.6		24
21.5	even	6		105.2.j.a.92.6	yes	24
21.11	odd	6	inner	735.2.y.g.557.7		48
21.17	even	6		735.2.y.j.557.7		48
21.20	even	2		735.2.y.j.422.6		48
35.3	even	12		735.2.y.j.263.6		48
35.12	even	12		525.2.j.b.218.7		24
35.13	even	4		735.2.y.j.128.7		48
35.18	odd	12	inner	735.2.y.g.263.6		48
35.19	odd	6		525.2.j.b.407.6		24
35.23	odd	12		735.2.j.h.638.6		24
35.33	even	12		105.2.j.a.8.6	✓	24
105.23	even	12		735.2.j.h.638.7		24
105.38	odd	12		735.2.y.j.263.7		48
105.47	odd	12		525.2.j.b.218.6		24
105.53	even	12	inner	735.2.y.g.263.7		48
105.68	odd	12		105.2.j.a.8.7	yes	24
105.83	odd	4		735.2.y.j.128.6		48
105.89	even	6		525.2.j.b.407.7		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.j.a.8.6	✓	24	35.33	even	12
105.2.j.a.8.7	yes	24	105.68	odd	12
105.2.j.a.92.6	yes	24	21.5	even	6
105.2.j.a.92.7	yes	24	7.5	odd	6
525.2.j.b.218.6		24	105.47	odd	12
525.2.j.b.218.7		24	35.12	even	12
525.2.j.b.407.6		24	35.19	odd	6
525.2.j.b.407.7		24	105.89	even	6
735.2.j.h.197.6		24	21.2	odd	6
735.2.j.h.197.7		24	7.2	even	3
735.2.j.h.638.6		24	35.23	odd	12
735.2.j.h.638.7		24	105.23	even	12
735.2.y.g.128.6		48	15.8	even	4	inner
735.2.y.g.128.7		48	5.3	odd	4	inner
735.2.y.g.263.6		48	35.18	odd	12	inner
735.2.y.g.263.7		48	105.53	even	12	inner
735.2.y.g.422.6		48	3.2	odd	2	inner
735.2.y.g.422.7		48	1.1	even	1	trivial
735.2.y.g.557.6		48	7.4	even	3	inner
735.2.y.g.557.7		48	21.11	odd	6	inner
735.2.y.j.128.6		48	105.83	odd	4
735.2.y.j.128.7		48	35.13	even	4
735.2.y.j.263.6		48	35.3	even	12
735.2.y.j.263.7		48	105.38	odd	12
735.2.y.j.422.6		48	21.20	even	2
735.2.y.j.422.7		48	7.6	odd	2
735.2.y.j.557.6		48	7.3	odd	6
735.2.y.j.557.7		48	21.17	even	6