Embedded newform 29.11.f.a.2.1

• Label: 29.11.f.a.2.1
• 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			2.1
Character	\(\chi\)	\(=\)	29.2
Dual form			29.11.f.a.15.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-60.4165 - 6.80730i) q^{2} +(298.044 + 187.273i) q^{3} +(2605.48 + 594.685i) q^{4} +(-931.971 + 743.222i) q^{5} +(-16731.9 - 13343.3i) q^{6} +(-2936.78 - 12866.9i) q^{7} +(-94601.8 - 33102.6i) q^{8} +(28138.4 + 58429.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 22 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} - 76784 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−60.4165	−	6.80730i	−1.88801	−	0.212728i	−0.909040	−	0.416709i	\(-0.863183\pi\)
							−0.978975	+	0.203981i	\(0.934612\pi\)
\(3\)	298.044	+	187.273i	1.22652	+	0.770672i	0.980102	−	0.198493i	\(-0.0636046\pi\)
							0.246415	+	0.969164i	\(0.420747\pi\)
\(5\)	−931.971	+	743.222i	−0.298231	+	0.237831i	−0.761160	−	0.648564i	\(-0.775370\pi\)
							0.462929	+	0.886395i	\(0.346798\pi\)
\(7\)	−2936.78	−	12866.9i	−0.174736	−	0.765567i	−0.984007	−	0.178132i	\(-0.942995\pi\)
							0.809271	−	0.587436i	\(-0.199862\pi\)
\(11\)	−96335.1	+	33709.1i	−0.598165	+	0.209307i	−0.612343	−	0.790592i	\(-0.709773\pi\)
							0.0141774	+	0.999899i	\(0.495487\pi\)
\(13\)	164540.	−	341671.i	0.443154	−	0.920218i	−0.553049	−	0.833149i	\(-0.686536\pi\)
							0.996202	−	0.0870691i	\(-0.0277501\pi\)
\(17\)	−1.54855e6	+	1.54855e6i	−1.09064	+	1.09064i	−0.0951793	+	0.995460i	\(0.530342\pi\)
							−0.995460	+	0.0951793i	\(0.969658\pi\)
\(19\)	−1.92235e6	−	3.05941e6i	−0.776364	−	1.23558i	−0.967216	−	0.253957i	\(-0.918268\pi\)
							0.190852	−	0.981619i	\(-0.438875\pi\)
\(23\)	1.68011e6	−	2.10679e6i	0.261034	−	0.327327i	−0.633992	−	0.773339i	\(-0.718585\pi\)
							0.895027	+	0.446013i	\(0.147156\pi\)
\(29\)	1.97568e7	−	5.51163e6i	0.963220	−	0.268714i
							\(31\)	−5.32208e7	−	5.99655e6i	−1.85897	−	0.209456i	−0.889888	−	0.456179i	\(-0.849218\pi\)
−0.969086	+	0.246723i	\(0.920646\pi\)
\(37\)	−2.04540e7	−	7.15717e6i	−0.294965	−	0.103213i	0.178740	−	0.983896i	\(-0.442798\pi\)
							−0.473705	+	0.880684i	\(0.657084\pi\)
\(41\)	2.62368e7	+	2.62368e7i	0.226460	+	0.226460i	0.811212	−	0.584752i	\(-0.198808\pi\)
							−0.584752	+	0.811212i	\(0.698808\pi\)
\(43\)	−3.00426e7	−	2.66636e8i	−0.204360	−	1.81374i	−0.508440	−	0.861097i	\(-0.669778\pi\)
							0.304080	−	0.952646i	\(-0.401651\pi\)
\(47\)	−9.58214e7	−	2.73842e8i	−0.417805	−	1.19402i	−0.939995	−	0.341188i	\(-0.889171\pi\)
							0.522190	−	0.852829i	\(-0.325115\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.11.f.a.2.1	✓	288
29.15	odd	28	inner	29.11.f.a.15.1	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.11.f.a.2.1	✓	288	1.1	even	1	trivial
29.11.f.a.15.1	yes	288	29.15	odd	28	inner