Embedded newform 29.11.f.a.2.17

• Label: 29.11.f.a.2.17
• Level: $29$
• Weight: $11$
• Character: 29.2
• Analytic conductor: $18.425$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	29.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(18.4253603275\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(24\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			2.17
Character	\(\chi\)	\(=\)	29.2
Dual form			29.11.f.a.15.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(23.5437 + 2.65273i) q^{2} +(243.217 + 152.824i) q^{3} +(-451.060 - 102.951i) q^{4} +(3023.75 - 2411.36i) q^{5} +(5320.82 + 4243.21i) q^{6} +(-7168.45 - 31407.0i) q^{7} +(-33246.3 - 11633.4i) q^{8} +(10179.2 + 21137.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 22 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} - 76784 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	23.5437	+	2.65273i	0.735739	+	0.0828979i	0.471879	−	0.881663i	\(-0.343576\pi\)
							0.263860	+	0.964561i	\(0.415004\pi\)
\(3\)	243.217	+	152.824i	1.00089	+	0.628903i	0.929607	−	0.368552i	\(-0.120147\pi\)
							0.0712870	+	0.997456i	\(0.477289\pi\)
\(5\)	3023.75	−	2411.36i	0.967599	−	0.771635i	−0.00598044	−	0.999982i	\(-0.501904\pi\)
							0.973580	+	0.228347i	\(0.0733322\pi\)
\(7\)	−7168.45	−	31407.0i	−0.426516	−	1.86869i	−0.491622	−	0.870809i	\(-0.663596\pi\)
							0.0651065	−	0.997878i	\(-0.479261\pi\)
\(11\)	−31649.8	+	11074.7i	−0.196520	+	0.0687655i	−0.426745	−	0.904372i	\(-0.640340\pi\)
							0.230224	+	0.973138i	\(0.426054\pi\)
\(13\)	167228.	−	347252.i	0.450393	−	0.935251i	−0.544915	−	0.838491i	\(-0.683438\pi\)
							0.995308	−	0.0967594i	\(-0.0308477\pi\)
\(17\)	−616161.	+	616161.i	−0.433960	+	0.433960i	−0.889973	−	0.456013i	\(-0.849277\pi\)
							0.456013	+	0.889973i	\(0.349277\pi\)
\(19\)	2.40566e6	+	3.82859e6i	0.971553	+	1.54622i	0.831095	+	0.556130i	\(0.187715\pi\)
							0.140458	+	0.990087i	\(0.455143\pi\)
\(23\)	−578240.	+	725090.i	−0.0898398	+	0.112656i	−0.824720	−	0.565542i	\(-0.808667\pi\)
							0.734880	+	0.678197i	\(0.237239\pi\)
\(29\)	6.51911e6	−	1.94476e7i	0.317832	−	0.948147i
							\(31\)	1.51411e7	+	1.70599e6i	0.528870	+	0.0595893i	0.372362	−	0.928087i	\(-0.378548\pi\)
0.156507	+	0.987677i	\(0.449976\pi\)
\(37\)	9.22538e6	+	3.22810e6i	0.133038	+	0.0465520i	0.395977	−	0.918260i	\(-0.370406\pi\)
							−0.262939	+	0.964813i	\(0.584692\pi\)
\(41\)	7.76709e7	+	7.76709e7i	0.670408	+	0.670408i	0.957810	−	0.287402i	\(-0.0927917\pi\)
							−0.287402	+	0.957810i	\(0.592792\pi\)
\(43\)	−8.63610e6	−	7.66475e7i	−0.0587456	−	0.521382i	−0.988281	−	0.152648i	\(-0.951220\pi\)
							0.929535	−	0.368734i	\(-0.120209\pi\)
\(47\)	4.56381e7	+	1.30426e8i	0.198993	+	0.568690i	0.999532	−	0.0305908i	\(-0.00973887\pi\)
							−0.800539	+	0.599281i	\(0.795453\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.11.f.a.2.17	✓	288
29.15	odd	28	inner	29.11.f.a.15.17	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.11.f.a.2.17	✓	288	1.1	even	1	trivial
29.11.f.a.15.17	yes	288	29.15	odd	28	inner