Embedded newform 29.8.e.a.4.12

• Label: 29.8.e.a.4.12
• Level: $29$
• Weight: $8$
• Character: 29.4
• 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			4.12
Character	\(\chi\)	\(=\)	29.4
Dual form			29.8.e.a.22.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.92390 + 2.26507i) q^{2} +(1.31301 + 2.72650i) q^{3} +(-21.9707 - 10.5805i) q^{4} +(100.667 - 441.052i) q^{5} +(6.85452 + 30.0316i) q^{6} +(-925.876 + 445.878i) q^{7} +(-1212.74 - 967.127i) q^{8} +(1357.86 - 1702.71i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{2} - 7 q^{3} + 709 q^{4} + 191 q^{5} + 1801 q^{6} - 667 q^{7} + 9086 q^{8} + 9103 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	9.92390	+	2.26507i	0.877158	+	0.200205i	0.637312	−	0.770606i	\(-0.280046\pi\)
							0.239845	+	0.970811i	\(0.422903\pi\)
\(3\)	1.31301	+	2.72650i	0.0280766	+	0.0583018i	0.914535	−	0.404506i	\(-0.132556\pi\)
							−0.886459	+	0.462807i	\(0.846842\pi\)
\(5\)	100.667	−	441.052i	0.360158	−	1.57795i	−0.392634	−	0.919695i	\(-0.628436\pi\)
							0.752791	−	0.658259i	\(-0.228707\pi\)
\(7\)	−925.876	+	445.878i	−1.02026	+	0.491330i	−0.867765	−	0.496975i	\(-0.834444\pi\)
							−0.152492	+	0.988305i	\(0.548730\pi\)
\(11\)	2245.84	−	1791.00i	0.508749	−	0.405714i	−0.335192	−	0.942150i	\(-0.608801\pi\)
							0.843941	+	0.536436i	\(0.180230\pi\)
\(13\)	764.823	+	959.058i	0.0965515	+	0.121072i	0.827760	−	0.561082i	\(-0.189615\pi\)
							−0.731209	+	0.682154i	\(0.761043\pi\)
\(17\)			5677.81i			0.280291i	0.990131	+	0.140146i	\(0.0447571\pi\)
							−0.990131	+	0.140146i	\(0.955243\pi\)
\(19\)	12944.2	−	26879.0i	0.432951	−	0.899032i	−0.564345	−	0.825539i	\(-0.690871\pi\)
							0.997296	−	0.0734926i	\(-0.0234145\pi\)
\(23\)	9093.34	+	39840.5i	0.155839	+	0.682775i	0.991122	+	0.132955i	\(0.0424465\pi\)
							−0.835283	+	0.549820i	\(0.814696\pi\)
\(29\)	−110364.	+	71202.0i	−0.840298	+	0.542125i
							\(31\)	158452.	+	36165.7i	0.955285	+	0.218038i	0.671628	−	0.740888i	\(-0.265595\pi\)
0.283657	+	0.958926i	\(0.408452\pi\)
\(37\)	−82952.2	−	66152.2i	−0.269229	−	0.214703i	0.479564	−	0.877507i	\(-0.340795\pi\)
							−0.748793	+	0.662804i	\(0.769366\pi\)
\(41\)			332756.i			0.754018i	0.926210	+	0.377009i	\(0.123047\pi\)
							−0.926210	+	0.377009i	\(0.876953\pi\)
\(43\)	596523.	−	136153.i	1.14416	−	0.261148i	0.391898	−	0.920009i	\(-0.371819\pi\)
							0.752265	+	0.658861i	\(0.228961\pi\)
\(47\)	1.01581e6	−	810081.i	1.42715	−	1.13811i	0.458794	−	0.888543i	\(-0.348281\pi\)
							0.968356	−	0.249572i	\(-0.0802900\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.8.e.a.4.12	✓	96
29.22	even	14	inner	29.8.e.a.22.12	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.e.a.4.12	✓	96	1.1	even	1	trivial
29.8.e.a.22.12	yes	96	29.22	even	14	inner