Embedded newform 29.11.f.a.2.9

• Label: 29.11.f.a.2.9
• 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			2.9
Character	\(\chi\)	\(=\)	29.2
Dual form			29.11.f.a.15.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-23.5047 - 2.64834i) q^{2} +(-49.6686 - 31.2088i) q^{3} +(-452.869 - 103.365i) q^{4} +(494.026 - 393.973i) q^{5} +(1084.79 + 865.093i) q^{6} +(-3650.42 - 15993.5i) q^{7} +(33232.7 + 11628.6i) q^{8} +(-24127.4 - 50101.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 22 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} - 76784 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−23.5047	−	2.64834i	−0.734522	−	0.0827607i	−0.263224	−	0.964735i	\(-0.584786\pi\)
							−0.471298	+	0.881974i	\(0.656214\pi\)
\(3\)	−49.6686	−	31.2088i	−0.204397	−	0.128431i	0.425944	−	0.904750i	\(-0.359942\pi\)
							−0.630341	+	0.776318i	\(0.717085\pi\)
\(5\)	494.026	−	393.973i	0.158088	−	0.126071i	−0.541251	−	0.840861i	\(-0.682049\pi\)
							0.699339	+	0.714790i	\(0.253478\pi\)
\(7\)	−3650.42	−	15993.5i	−0.217196	−	0.951599i	−0.959538	−	0.281578i	\(-0.909142\pi\)
							0.742342	−	0.670021i	\(-0.233715\pi\)
\(11\)	−37188.2	+	13012.7i	−0.230909	+	0.0807987i	−0.443254	−	0.896396i	\(-0.646176\pi\)
							0.212344	+	0.977195i	\(0.431890\pi\)
\(13\)	205896.	−	427546.i	0.554536	−	1.15151i	−0.415735	−	0.909486i	\(-0.636475\pi\)
							0.970271	−	0.242021i	\(-0.0778102\pi\)
\(17\)	−404500.	+	404500.i	−0.284888	+	0.284888i	−0.835055	−	0.550167i	\(-0.814564\pi\)
							0.550167	+	0.835055i	\(0.314564\pi\)
\(19\)	−499955.	−	795674.i	−0.201912	−	0.321342i	0.730384	−	0.683037i	\(-0.239341\pi\)
							−0.932297	+	0.361695i	\(0.882198\pi\)
\(23\)	−5.43885e6	+	6.82010e6i	−0.845022	+	1.05962i	0.152433	+	0.988314i	\(0.451289\pi\)
							−0.997454	+	0.0713097i	\(0.977282\pi\)
\(29\)	−5.37218e6	+	1.97951e7i	−0.261915	+	0.965091i
							\(31\)	−1.52063e7	−	1.71333e6i	−0.531146	−	0.0598457i	−0.157681	−	0.987490i	\(-0.550402\pi\)
−0.373465	+	0.927644i	\(0.621830\pi\)
\(37\)	7.33963e7	+	2.56825e7i	1.05844	+	0.370364i	0.802700	−	0.596383i	\(-0.203396\pi\)
							0.255738	+	0.966746i	\(0.417682\pi\)
\(41\)	−7.53798e7	−	7.53798e7i	−0.650632	−	0.650632i	0.302513	−	0.953145i	\(-0.402174\pi\)
							−0.953145	+	0.302513i	\(0.902174\pi\)
\(43\)	−1.70417e7	−	1.51249e8i	−0.115923	−	1.02885i	−0.908158	−	0.418629i	\(-0.862511\pi\)
							0.792234	−	0.610217i	\(-0.208918\pi\)
\(47\)	2.94361e7	+	8.41236e7i	0.128349	+	0.366799i	0.990069	−	0.140585i	\(-0.0448982\pi\)
							−0.861720	+	0.507384i	\(0.830613\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.11.f.a.2.9	✓	288
29.15	odd	28	inner	29.11.f.a.15.9	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.11.f.a.2.9	✓	288	1.1	even	1	trivial
29.11.f.a.15.9	yes	288	29.15	odd	28	inner