Embedded newform 29.8.e.a.5.2

• Label: 29.8.e.a.5.2
• Level: $29$
• Weight: $8$
• Character: 29.5
• Analytic conductor: $9.059$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.05916573904\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			5.2
Character	\(\chi\)	\(=\)	29.5
Dual form			29.8.e.a.6.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-15.7422 + 12.5540i) q^{2} +(39.5051 - 9.01678i) q^{3} +(61.7319 - 270.465i) q^{4} +(-50.2510 - 63.0128i) q^{5} +(-508.702 + 637.892i) q^{6} +(190.445 + 834.393i) q^{7} +(1305.38 + 2710.65i) q^{8} +(-491.067 + 236.485i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{2} - 7 q^{3} + 709 q^{4} + 191 q^{5} + 1801 q^{6} - 667 q^{7} + 9086 q^{8} + 9103 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{11}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−15.7422	+	12.5540i	−1.39143	+	1.10963i	−0.411248	+	0.911524i	\(0.634907\pi\)
							−0.980181	+	0.198104i	\(0.936522\pi\)
\(3\)	39.5051	−	9.01678i	0.844751	−	0.192809i	0.221822	−	0.975087i	\(-0.428800\pi\)
							0.622929	+	0.782278i	\(0.285942\pi\)
\(5\)	−50.2510	−	63.0128i	−0.179784	−	0.225441i	0.683771	−	0.729696i	\(-0.260339\pi\)
							−0.863555	+	0.504255i	\(0.831767\pi\)
\(7\)	190.445	+	834.393i	0.209858	+	0.919449i	0.964660	+	0.263496i	\(0.0848756\pi\)
							−0.754802	+	0.655952i	\(0.772267\pi\)
\(11\)	−1159.21	+	2407.13i	−0.262596	+	0.545286i	−0.990024	−	0.140898i	\(-0.955001\pi\)
							0.727428	+	0.686184i	\(0.240715\pi\)
\(13\)	−6972.47	−	3357.76i	−0.880207	−	0.423885i	−0.0615068	−	0.998107i	\(-0.519591\pi\)
							−0.818700	+	0.574221i	\(0.805305\pi\)
\(17\)			10975.7i			0.541826i	0.962604	+	0.270913i	\(0.0873255\pi\)
							−0.962604	+	0.270913i	\(0.912674\pi\)
\(19\)	−29804.4	−	6802.65i	−0.996878	−	0.227531i	−0.307199	−	0.951645i	\(-0.599392\pi\)
							−0.689680	+	0.724114i	\(0.742249\pi\)
\(23\)	−17012.2	+	21332.6i	−0.291549	+	0.365591i	−0.905937	−	0.423413i	\(-0.860832\pi\)
							0.614387	+	0.789005i	\(0.289403\pi\)
\(29\)	−7936.02	+	131099.i	−0.0604240	+	0.998173i
							\(31\)	−201423.	+	160629.i	−1.21435	+	0.968409i	−0.999965	−	0.00834145i	\(-0.997345\pi\)
−0.214381	+	0.976750i	\(0.568773\pi\)
\(37\)	−244040.	−	506755.i	−0.792056	−	1.64472i	−0.764088	−	0.645112i	\(-0.776811\pi\)
							−0.0279680	−	0.999609i	\(-0.508904\pi\)
\(41\)			502775.i			1.13928i	0.821894	+	0.569640i	\(0.192917\pi\)
							−0.821894	+	0.569640i	\(0.807083\pi\)
\(43\)	64038.3	+	51068.9i	0.122829	+	0.0979528i	0.682978	−	0.730439i	\(-0.260685\pi\)
							−0.560149	+	0.828392i	\(0.689256\pi\)
\(47\)	156875.	−	325754.i	0.220400	−	0.457665i	−0.761225	−	0.648488i	\(-0.775402\pi\)
							0.981624	+	0.190824i	\(0.0611158\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.8.e.a.5.2	✓	96
29.6	even	14	inner	29.8.e.a.6.2	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.e.a.5.2	✓	96	1.1	even	1	trivial
29.8.e.a.6.2	yes	96	29.6	even	14	inner