Embedded newform 735.2.p.g.509.11

• Label: 735.2.p.g.509.11
• Level: $735$
• Weight: $2$
• Character: 735.509
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $64$
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			509.11
Character	\(\chi\)	\(=\)	735.509
Dual form			735.2.p.g.374.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451669 + 0.782314i) q^{2} +(0.697946 + 1.58520i) q^{3} +(0.591990 + 1.02536i) q^{4} +(-0.0199888 + 2.23598i) q^{5} +(-1.55537 - 0.169975i) q^{6} -2.87621 q^{8} +(-2.02574 + 2.21277i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 16 q^{4} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(346\)	\(442\)	\(491\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.451669	+	0.782314i	−0.319378	+	0.553180i	−0.980358	−	0.197224i	\(-0.936807\pi\)
							0.660980	+	0.750403i	\(0.270141\pi\)
\(3\)	0.697946	+	1.58520i	0.402959	+	0.915218i
							\(5\)	−0.0199888	+	2.23598i	−0.00893927	+	0.999960i
\(7\)		0			0
							\(11\)	3.51985	−	2.03219i	1.06127	−	0.612727i	0.135490	−	0.990779i	\(-0.456739\pi\)
0.925785	+	0.378051i	\(0.123406\pi\)
\(13\)	2.88298			0.799595			0.399798	−	0.916603i	\(-0.369080\pi\)
							0.399798	+	0.916603i	\(0.369080\pi\)
\(17\)	−5.77521	+	3.33432i	−1.40069	+	0.808691i	−0.994464	−	0.105080i	\(-0.966490\pi\)
							−0.406230	+	0.913771i	\(0.633157\pi\)
\(19\)	4.94242	+	2.85351i	1.13387	+	0.654639i	0.944905	−	0.327345i	\(-0.106154\pi\)
							0.188964	+	0.981984i	\(0.439487\pi\)
\(23\)	−0.815927	+	1.41323i	−0.170133	+	0.294678i	−0.938466	−	0.345371i	\(-0.887753\pi\)
							0.768333	+	0.640050i	\(0.221086\pi\)
\(29\)		−	5.89707i		−	1.09506i	−0.836786	−	0.547530i	\(-0.815568\pi\)
							0.836786	−	0.547530i	\(-0.184432\pi\)
\(31\)	1.62006	−	0.935342i	0.290971	−	0.167992i	−0.347409	−	0.937714i	\(-0.612938\pi\)
							0.638380	+	0.769722i	\(0.279605\pi\)
\(37\)	−1.38541	−	0.799864i	−0.227759	−	0.131497i	0.381779	−	0.924254i	\(-0.375312\pi\)
							−0.609538	+	0.792757i	\(0.708645\pi\)
\(41\)	9.12244			1.42469			0.712343	−	0.701832i	\(-0.247634\pi\)
							0.712343	+	0.701832i	\(0.247634\pi\)
\(43\)		−	7.53359i		−	1.14886i	−0.818553	−	0.574431i	\(-0.805223\pi\)
							0.818553	−	0.574431i	\(-0.194777\pi\)
\(47\)	5.98810	+	3.45723i	0.873454	+	0.504289i	0.868495	−	0.495699i	\(-0.165088\pi\)
							0.00495956	+	0.999988i	\(0.498421\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.p.g.509.11		64
3.2	odd	2	inner	735.2.p.g.509.24		64
5.4	even	2	inner	735.2.p.g.509.22		64
7.2	even	3		735.2.g.c.734.21	yes	32
7.3	odd	6	inner	735.2.p.g.374.9		64
7.4	even	3	inner	735.2.p.g.374.12		64
7.5	odd	6		735.2.g.c.734.24	yes	32
7.6	odd	2	inner	735.2.p.g.509.10		64
15.14	odd	2	inner	735.2.p.g.509.9		64
21.2	odd	6		735.2.g.c.734.10	yes	32
21.5	even	6		735.2.g.c.734.11	yes	32
21.11	odd	6	inner	735.2.p.g.374.23		64
21.17	even	6	inner	735.2.p.g.374.22		64
21.20	even	2	inner	735.2.p.g.509.21		64
35.4	even	6	inner	735.2.p.g.374.21		64
35.9	even	6		735.2.g.c.734.12	yes	32
35.19	odd	6		735.2.g.c.734.9	✓	32
35.24	odd	6	inner	735.2.p.g.374.24		64
35.34	odd	2	inner	735.2.p.g.509.23		64
105.44	odd	6		735.2.g.c.734.23	yes	32
105.59	even	6	inner	735.2.p.g.374.11		64
105.74	odd	6	inner	735.2.p.g.374.10		64
105.89	even	6		735.2.g.c.734.22	yes	32
105.104	even	2	inner	735.2.p.g.509.12		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.2.g.c.734.9	✓	32	35.19	odd	6
735.2.g.c.734.10	yes	32	21.2	odd	6
735.2.g.c.734.11	yes	32	21.5	even	6
735.2.g.c.734.12	yes	32	35.9	even	6
735.2.g.c.734.21	yes	32	7.2	even	3
735.2.g.c.734.22	yes	32	105.89	even	6
735.2.g.c.734.23	yes	32	105.44	odd	6
735.2.g.c.734.24	yes	32	7.5	odd	6
735.2.p.g.374.9		64	7.3	odd	6	inner
735.2.p.g.374.10		64	105.74	odd	6	inner
735.2.p.g.374.11		64	105.59	even	6	inner
735.2.p.g.374.12		64	7.4	even	3	inner
735.2.p.g.374.21		64	35.4	even	6	inner
735.2.p.g.374.22		64	21.17	even	6	inner
735.2.p.g.374.23		64	21.11	odd	6	inner
735.2.p.g.374.24		64	35.24	odd	6	inner
735.2.p.g.509.9		64	15.14	odd	2	inner
735.2.p.g.509.10		64	7.6	odd	2	inner
735.2.p.g.509.11		64	1.1	even	1	trivial
735.2.p.g.509.12		64	105.104	even	2	inner
735.2.p.g.509.21		64	21.20	even	2	inner
735.2.p.g.509.22		64	5.4	even	2	inner
735.2.p.g.509.23		64	35.34	odd	2	inner
735.2.p.g.509.24		64	3.2	odd	2	inner