Embedded newform 29.10.d.a.7.8

• Label: 29.10.d.a.7.8
• Level: $29$
• Weight: $10$
• Character: 29.7
• Analytic conductor: $14.936$
• Analytic rank: $0$
• Dimension: $126$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	29.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.9360392488\)
Analytic rank:	\(0\)
Dimension:	\(126\)
Relative dimension:	\(21\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			7.8
Character	\(\chi\)	\(=\)	29.7
Dual form			29.10.d.a.25.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.55763 - 15.5870i) q^{2} +(-34.2508 - 16.4943i) q^{3} +(230.999 - 111.243i) q^{4} +(376.920 + 1651.40i) q^{5} +(-135.245 + 592.548i) q^{6} +(-3361.64 - 1618.88i) q^{7} +(-7659.50 - 9604.70i) q^{8} +(-11371.1 - 14258.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 23 q^{2} - 5 q^{3} - 4699 q^{4} - 1031 q^{5} + 12781 q^{6} - 4959 q^{7} - 43354 q^{8} - 76050 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.55763	−	15.5870i	−0.157226	−	0.688854i	−0.990674	−	0.136255i	\(-0.956493\pi\)
							0.833447	−	0.552599i	\(-0.186364\pi\)
\(3\)	−34.2508	−	16.4943i	−0.244132	−	0.117568i	0.307817	−	0.951446i	\(-0.400402\pi\)
							−0.551949	+	0.833878i	\(0.686116\pi\)
\(5\)	376.920	+	1651.40i	0.269702	+	1.18164i	0.910360	+	0.413816i	\(0.135804\pi\)
							−0.640658	+	0.767826i	\(0.721338\pi\)
\(7\)	−3361.64	−	1618.88i	−0.529188	−	0.254843i	0.150158	−	0.988662i	\(-0.452022\pi\)
							−0.679345	+	0.733819i	\(0.737736\pi\)
\(11\)	−11612.5	+	14561.6i	−0.239143	+	0.299875i	−0.886891	−	0.461979i	\(-0.847139\pi\)
							0.647748	+	0.761855i	\(0.275711\pi\)
\(13\)	−57820.9	+	72505.1i	−0.561487	+	0.704082i	−0.978832	−	0.204666i	\(-0.934389\pi\)
							0.417345	+	0.908748i	\(0.362961\pi\)
\(17\)	−88481.3			−0.256940			−0.128470	−	0.991713i	\(-0.541007\pi\)
							−0.128470	+	0.991713i	\(0.541007\pi\)
\(19\)	−642878.	+	309594.i	−1.13172	+	0.545006i	−0.903493	−	0.428603i	\(-0.859006\pi\)
							−0.228223	+	0.973609i	\(0.573292\pi\)
\(23\)	−212011.	+	928881.i	−0.157973	+	0.692125i	0.832455	+	0.554093i	\(0.186935\pi\)
							−0.990428	+	0.138032i	\(0.955922\pi\)
\(29\)	−2.48419e6	+	2.88720e6i	−0.652220	+	0.758030i
							\(31\)	−624360.	−	2.73550e6i	−0.121425	−	0.531997i	−0.998651	−	0.0519201i	\(-0.983466\pi\)
0.877226	−	0.480077i	\(-0.159391\pi\)
\(37\)	−582616.	−	730577.i	−0.0511063	−	0.0640853i	0.755621	−	0.655009i	\(-0.227335\pi\)
							−0.806728	+	0.590923i	\(0.798764\pi\)
\(41\)	1.96065e6			0.108361			0.0541804	−	0.998531i	\(-0.482745\pi\)
							0.0541804	+	0.998531i	\(0.482745\pi\)
\(43\)	2.09959e6	−	9.19891e6i	0.0936541	−	0.410325i	−0.906269	−	0.422701i	\(-0.861082\pi\)
							0.999923	+	0.0123754i	\(0.00393930\pi\)
\(47\)	−7.18336e6	+	9.00765e6i	−0.214727	+	0.269260i	−0.877516	−	0.479547i	\(-0.840801\pi\)
							0.662789	+	0.748806i	\(0.269373\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.10.d.a.7.8	✓	126
29.25	even	7	inner	29.10.d.a.25.8	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.10.d.a.7.8	✓	126	1.1	even	1	trivial
29.10.d.a.25.8	yes	126	29.25	even	7	inner