Embedded newform 29.10.d.a.7.3

• Label: 29.10.d.a.7.3
• Level: $29$
• Weight: $10$
• Character: 29.7
• Analytic conductor: $14.936$
• Analytic rank: $0$
• Dimension: $126$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	29.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.9360392488\)
Analytic rank:	\(0\)
Dimension:	\(126\)
Relative dimension:	\(21\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			7.3
Character	\(\chi\)	\(=\)	29.7
Dual form			29.10.d.a.25.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.97654 - 34.9475i) q^{2} +(73.3338 + 35.3157i) q^{3} +(-696.407 + 335.372i) q^{4} +(440.975 + 1932.04i) q^{5} +(649.245 - 2844.53i) q^{6} +(-672.333 - 323.778i) q^{7} +(5832.25 + 7313.41i) q^{8} +(-8141.50 - 10209.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 23 q^{2} - 5 q^{3} - 4699 q^{4} - 1031 q^{5} + 12781 q^{6} - 4959 q^{7} - 43354 q^{8} - 76050 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.97654	−	34.9475i	−0.352517	−	1.54448i	−0.771349	−	0.636413i	\(-0.780418\pi\)
							0.418832	−	0.908064i	\(-0.362440\pi\)
\(3\)	73.3338	+	35.3157i	0.522707	+	0.251723i	0.676581	−	0.736368i	\(-0.263461\pi\)
							−0.153874	+	0.988090i	\(0.549175\pi\)
\(5\)	440.975	+	1932.04i	0.315536	+	1.38245i	0.845292	+	0.534305i	\(0.179426\pi\)
							−0.529756	+	0.848150i	\(0.677716\pi\)
\(7\)	−672.333	−	323.778i	−0.105838	−	0.0509691i	0.380215	−	0.924898i	\(-0.375850\pi\)
							−0.486054	+	0.873929i	\(0.661564\pi\)
\(11\)	36593.4	−	45886.6i	0.753590	−	0.944972i	−0.246115	−	0.969241i	\(-0.579154\pi\)
							0.999705	+	0.0242685i	\(0.00772567\pi\)
\(13\)	114554.	−	143646.i	1.11241	−	1.39492i	0.202917	−	0.979196i	\(-0.434958\pi\)
							0.909492	−	0.415721i	\(-0.136471\pi\)
\(17\)	−109313.			−0.317433			−0.158717	−	0.987324i	\(-0.550736\pi\)
							−0.158717	+	0.987324i	\(0.550736\pi\)
\(19\)	851732.	−	410172.i	1.49938	−	0.722063i	0.509042	−	0.860742i	\(-0.330000\pi\)
							0.990337	+	0.138678i	\(0.0442854\pi\)
\(23\)	63801.9	−	279535.i	0.0475399	−	0.208286i	−0.945580	−	0.325391i	\(-0.894504\pi\)
							0.993120	+	0.117105i	\(0.0373613\pi\)
\(29\)	3.65504e6	+	1.07136e6i	0.959625	+	0.281284i
							\(31\)	270111.	+	1.18344e6i	0.0525310	+	0.230153i	0.994378	−	0.105889i	\(-0.0337687\pi\)
−0.941847	+	0.336042i	\(0.890912\pi\)
\(37\)	2.05091e6	+	2.57176e6i	0.179903	+	0.225591i	0.863604	−	0.504171i	\(-0.168202\pi\)
							−0.683701	+	0.729763i	\(0.739631\pi\)
\(41\)	6.56385e6			0.362770			0.181385	−	0.983412i	\(-0.441942\pi\)
							0.181385	+	0.983412i	\(0.441942\pi\)
\(43\)	−7.43168e6	+	3.25603e7i	−0.331497	+	1.45238i	0.484737	+	0.874660i	\(0.338915\pi\)
							−0.816234	+	0.577722i	\(0.803942\pi\)
\(47\)	1.50747e7	−	1.89030e7i	0.450617	−	0.565056i	−0.503690	−	0.863885i	\(-0.668025\pi\)
							0.954307	+	0.298829i	\(0.0965960\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.10.d.a.7.3	✓	126
29.25	even	7	inner	29.10.d.a.25.3	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.10.d.a.7.3	✓	126	1.1	even	1	trivial
29.10.d.a.25.3	yes	126	29.25	even	7	inner