Embedded newform 29.6.d.a.7.5

• Label: 29.6.d.a.7.5
• Level: $29$
• Weight: $6$
• Character: 29.7
• Analytic conductor: $4.651$
• Analytic rank: $0$
• Dimension: $66$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	29.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65113077458\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			7.5
Character	\(\chi\)	\(=\)	29.7
Dual form			29.6.d.a.25.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.594851 - 2.60621i) q^{2} +(-6.88516 - 3.31572i) q^{3} +(22.3925 - 10.7837i) q^{4} +(-1.88513 - 8.25929i) q^{5} +(-4.54582 + 19.9165i) q^{6} +(-133.434 - 64.2584i) q^{7} +(-94.7603 - 118.826i) q^{8} +(-115.097 - 144.327i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.594851	−	2.60621i	−0.105156	−	0.460718i	−0.999900	−	0.0141388i	\(-0.995499\pi\)
							0.894744	−	0.446579i	\(-0.147358\pi\)
\(3\)	−6.88516	−	3.31572i	−0.441683	−	0.212703i	0.199806	−	0.979835i	\(-0.435969\pi\)
							−0.641489	+	0.767132i	\(0.721683\pi\)
\(5\)	−1.88513	−	8.25929i	−0.0337222	−	0.147747i	0.955264	−	0.295755i	\(-0.0955710\pi\)
							−0.988986	+	0.148008i	\(0.952714\pi\)
\(7\)	−133.434	−	64.2584i	−1.02925	−	0.495661i	−0.158486	−	0.987361i	\(-0.550661\pi\)
							−0.870764	+	0.491701i	\(0.836375\pi\)
\(11\)	30.2901	−	37.9826i	0.0754777	−	0.0946460i	−0.742659	−	0.669670i	\(-0.766436\pi\)
							0.818137	+	0.575024i	\(0.195007\pi\)
\(13\)	−5.30466	+	6.65183i	−0.00870561	+	0.0109165i	−0.786165	−	0.618016i	\(-0.787937\pi\)
							0.777460	+	0.628933i	\(0.216508\pi\)
\(17\)	847.365			0.711129			0.355564	−	0.934652i	\(-0.384289\pi\)
							0.355564	+	0.934652i	\(0.384289\pi\)
\(19\)	852.670	−	410.624i	0.541872	−	0.260952i	−0.142865	−	0.989742i	\(-0.545631\pi\)
							0.684737	+	0.728790i	\(0.259917\pi\)
\(23\)	−307.429	+	1346.93i	−0.121178	+	0.530917i	0.877503	+	0.479572i	\(0.159208\pi\)
							−0.998681	+	0.0513453i	\(0.983649\pi\)
\(29\)	3783.39	+	2489.40i	0.835385	+	0.549666i
							\(31\)	−857.952	−	3758.93i	−0.160346	−	0.702523i	−0.989623	−	0.143685i	\(-0.954105\pi\)
0.829277	−	0.558837i	\(-0.188752\pi\)
\(37\)	−7255.86	−	9098.56i	−0.871333	−	1.09262i	−0.994958	−	0.100288i	\(-0.968023\pi\)
							0.123625	−	0.992329i	\(-0.460548\pi\)
\(41\)	11592.1			1.07696			0.538482	−	0.842637i	\(-0.318998\pi\)
							0.538482	+	0.842637i	\(0.318998\pi\)
\(43\)	−1172.43	+	5136.75i	−0.0966975	+	0.423660i	−0.999986	−	0.00537513i	\(-0.998289\pi\)
							0.903288	+	0.429035i	\(0.141146\pi\)
\(47\)	−830.220	+	1041.06i	−0.0548212	+	0.0687436i	−0.808486	−	0.588515i	\(-0.799713\pi\)
							0.753665	+	0.657259i	\(0.228284\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.7.5	✓	66
29.5	even	14		841.6.a.h.1.14		33
29.24	even	7		841.6.a.i.1.20		33
29.25	even	7	inner	29.6.d.a.25.5	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.7.5	✓	66	1.1	even	1	trivial
29.6.d.a.25.5	yes	66	29.25	even	7	inner
841.6.a.h.1.14		33	29.5	even	14
841.6.a.i.1.20		33	29.24	even	7