Embedded newform 29.11.f.a.2.6

• Label: 29.11.f.a.2.6
• 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			2.6
Character	\(\chi\)	\(=\)	29.2
Dual form			29.11.f.a.15.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-35.2938 - 3.97666i) q^{2} +(164.902 + 103.615i) q^{3} +(231.515 + 52.8418i) q^{4} +(-2509.40 + 2001.18i) q^{5} +(-5408.00 - 4312.74i) q^{6} +(-2684.83 - 11763.0i) q^{7} +(26367.7 + 9226.46i) q^{8} +(-9163.66 - 19028.5i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 22 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} - 76784 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−35.2938	−	3.97666i	−1.10293	−	0.124271i	−0.458325	−	0.888785i	\(-0.651551\pi\)
							−0.644607	+	0.764514i	\(0.722979\pi\)
\(3\)	164.902	+	103.615i	0.678611	+	0.426400i	0.826752	−	0.562567i	\(-0.190186\pi\)
							−0.148141	+	0.988966i	\(0.547329\pi\)
\(5\)	−2509.40	+	2001.18i	−0.803009	+	0.640378i	−0.936498	−	0.350672i	\(-0.885953\pi\)
							0.133489	+	0.991050i	\(0.457382\pi\)
\(7\)	−2684.83	−	11763.0i	−0.159745	−	0.699888i	−0.989830	−	0.142252i	\(-0.954566\pi\)
							0.830085	−	0.557636i	\(-0.188292\pi\)
\(11\)	−180735.	+	63241.9i	−1.12222	+	0.392683i	−0.826694	−	0.562651i	\(-0.809781\pi\)
							−0.295528	+	0.955334i	\(0.595496\pi\)
\(13\)	−168197.	+	349264.i	−0.453003	+	0.940670i	0.541958	+	0.840406i	\(0.317683\pi\)
							−0.994961	+	0.100265i	\(0.968031\pi\)
\(17\)	1.23365e6	−	1.23365e6i	0.868856	−	0.868856i	−0.123490	−	0.992346i	\(-0.539409\pi\)
							0.992346	+	0.123490i	\(0.0394087\pi\)
\(19\)	1.27578e6	+	2.03039e6i	0.515238	+	0.819997i	0.998459	−	0.0555016i	\(-0.0176758\pi\)
							−0.483220	+	0.875499i	\(0.660533\pi\)
\(23\)	6.19426e6	−	7.76736e6i	0.962388	−	1.20680i	−0.0159691	−	0.999872i	\(-0.505083\pi\)
							0.978357	−	0.206924i	\(-0.0663452\pi\)
\(29\)	−4.12700e6	−	2.00917e7i	−0.201207	−	0.979549i
							\(31\)	5.05898e7	+	5.70011e6i	1.76707	+	0.199101i	0.934779	−	0.355230i	\(-0.115598\pi\)
0.832296	+	0.554332i	\(0.187026\pi\)
\(37\)	5.14772e7	+	1.80126e7i	0.742345	+	0.259758i	0.674843	−	0.737961i	\(-0.264211\pi\)
							0.0675023	+	0.997719i	\(0.478497\pi\)
\(41\)	7.09615e7	+	7.09615e7i	0.612496	+	0.612496i	0.943596	−	0.331099i	\(-0.107420\pi\)
							−0.331099	+	0.943596i	\(0.607420\pi\)
\(43\)	−2.17706e7	−	1.93220e8i	−0.148091	−	1.31434i	−0.819424	−	0.573188i	\(-0.805707\pi\)
							0.671333	−	0.741156i	\(-0.265722\pi\)
\(47\)	−2.76349e6	−	7.89760e6i	−0.0120495	−	0.0344355i	0.937702	−	0.347442i	\(-0.112950\pi\)
							−0.949751	+	0.313006i	\(0.898664\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.11.f.a.2.6	✓	288
29.15	odd	28	inner	29.11.f.a.15.6	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.11.f.a.2.6	✓	288	1.1	even	1	trivial
29.11.f.a.15.6	yes	288	29.15	odd	28	inner