Embedded newform 29.10.e.a.4.5

• Label: 29.10.e.a.4.5
• Level: $29$
• Weight: $10$
• Character: 29.4
• Analytic conductor: $14.936$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.9360392488\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			4.5
Character	\(\chi\)	\(=\)	29.4
Dual form			29.10.e.a.22.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-23.4413 - 5.35033i) q^{2} +(-66.5806 - 138.256i) q^{3} +(59.5740 + 28.6893i) q^{4} +(-62.4981 + 273.822i) q^{5} +(821.022 + 3597.13i) q^{6} +(9803.33 - 4721.04i) q^{7} +(8381.83 + 6684.29i) q^{8} +(-2409.61 + 3021.55i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 7 q^{2} - 7 q^{3} + 5797 q^{4} + 1367 q^{5} - 7607 q^{6} + 4949 q^{7} + 31766 q^{8} + 112237 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−23.4413	−	5.35033i	−1.03597	−	0.236453i	−0.329470	−	0.944166i	\(-0.606870\pi\)
							−0.706500	+	0.707713i	\(0.749727\pi\)
\(3\)	−66.5806	−	138.256i	−0.474572	−	0.985459i	−0.991582	−	0.129480i	\(-0.958669\pi\)
							0.517010	−	0.855979i	\(-0.327045\pi\)
\(5\)	−62.4981	+	273.822i	−0.0447200	+	0.195931i	−0.992353	−	0.123429i	\(-0.960611\pi\)
							0.947633	+	0.319360i	\(0.103468\pi\)
\(7\)	9803.33	−	4721.04i	1.54324	−	0.743184i	0.547621	−	0.836726i	\(-0.315533\pi\)
							0.995615	+	0.0935429i	\(0.0298192\pi\)
\(11\)	8366.83	−	6672.32i	0.172303	−	0.137407i	−0.533541	−	0.845774i	\(-0.679139\pi\)
							0.705845	+	0.708367i	\(0.250568\pi\)
\(13\)	110123.	+	138089.i	1.06938	+	1.34096i	0.936831	+	0.349781i	\(0.113744\pi\)
							0.132547	+	0.991177i	\(0.457684\pi\)
\(17\)		−	488739.i		−	1.41924i	−0.704583	−	0.709622i	\(-0.748866\pi\)
							0.704583	−	0.709622i	\(-0.251134\pi\)
\(19\)	124958.	−	259478.i	0.219975	−	0.456783i	−0.761553	−	0.648103i	\(-0.775563\pi\)
							0.981528	+	0.191320i	\(0.0612769\pi\)
\(23\)	40897.9	+	179186.i	0.0304738	+	0.133514i	0.987877	−	0.155242i	\(-0.0496156\pi\)
							−0.957403	+	0.288756i	\(0.906758\pi\)
\(29\)	3.77151e6	−	531862.i	0.990202	−	0.139639i
							\(31\)	−5.42298e6	−	1.23776e6i	−1.05465	−	0.240718i	−0.340166	−	0.940365i	\(-0.610483\pi\)
−0.714488	+	0.699647i	\(0.753340\pi\)
\(37\)	1.33741e7	+	1.06655e7i	1.17316	+	0.935561i	0.998793	−	0.0491205i	\(-0.0156418\pi\)
							0.174363	+	0.984681i	\(0.444213\pi\)
\(41\)		−	3.85732e6i		−	0.213186i	−0.994303	−	0.106593i	\(-0.966006\pi\)
							0.994303	−	0.106593i	\(-0.0339941\pi\)
\(43\)	1.37407e7	−	3.13624e6i	0.612918	−	0.139894i	0.0952175	−	0.995456i	\(-0.469645\pi\)
							0.517700	+	0.855562i	\(0.326788\pi\)
\(47\)	−4.65767e7	+	3.71436e7i	−1.39228	+	1.11031i	−0.412342	+	0.911029i	\(0.635289\pi\)
							−0.979942	+	0.199281i	\(0.936139\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.10.e.a.4.5	✓	132
29.22	even	14	inner	29.10.e.a.22.5	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.10.e.a.4.5	✓	132	1.1	even	1	trivial
29.10.e.a.22.5	yes	132	29.22	even	14	inner