Embedded newform 29.10.d.a.7.7

• Label: 29.10.d.a.7.7
• 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			7.7
Character	\(\chi\)	\(=\)	29.7
Dual form			29.10.d.a.25.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.60417 - 15.7909i) q^{2} +(194.806 + 93.8135i) q^{3} +(224.933 - 108.322i) q^{4} +(-195.891 - 858.256i) q^{5} +(779.288 - 3414.28i) q^{6} +(-8317.05 - 4005.28i) q^{7} +(-7691.72 - 9645.11i) q^{8} +(16876.1 + 21162.0i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 23 q^{2} - 5 q^{3} - 4699 q^{4} - 1031 q^{5} + 12781 q^{6} - 4959 q^{7} - 43354 q^{8} - 76050 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.60417	−	15.7909i	−0.159284	−	0.697867i	−0.989988	−	0.141153i	\(-0.954919\pi\)
							0.830704	−	0.556714i	\(-0.187938\pi\)
\(3\)	194.806	+	93.8135i	1.38853	+	0.668682i	0.970801	−	0.239888i	\(-0.0771108\pi\)
							0.417732	+	0.908570i	\(0.362825\pi\)
\(5\)	−195.891	−	858.256i	−0.140168	−	0.614118i	−0.995395	−	0.0958629i	\(-0.969439\pi\)
							0.855226	−	0.518255i	\(-0.173418\pi\)
\(7\)	−8317.05	−	4005.28i	−1.30927	−	0.630509i	−0.356523	−	0.934286i	\(-0.616038\pi\)
							−0.952743	+	0.303777i	\(0.901752\pi\)
\(11\)	49033.1	−	61485.5i	1.00977	−	1.26621i	0.0461485	−	0.998935i	\(-0.485305\pi\)
							0.963620	−	0.267275i	\(-0.0861233\pi\)
\(13\)	−5731.42	+	7186.97i	−0.0556567	+	0.0697912i	−0.808881	−	0.587973i	\(-0.799926\pi\)
							0.753224	+	0.657764i	\(0.228498\pi\)
\(17\)	239884.			0.696597			0.348299	−	0.937384i	\(-0.386760\pi\)
							0.348299	+	0.937384i	\(0.386760\pi\)
\(19\)	−361145.	+	173919.i	−0.635757	+	0.306164i	−0.723860	−	0.689947i	\(-0.757634\pi\)
							0.0881031	+	0.996111i	\(0.471920\pi\)
\(23\)	−16956.9	+	74293.2i	−0.0126349	+	0.0553572i	−0.980852	−	0.194753i	\(-0.937610\pi\)
							0.968217	+	0.250110i	\(0.0804667\pi\)
\(29\)	3.80764e6	−	95118.6i	0.999688	−	0.0249732i
							\(31\)	814097.	+	3.56679e6i	0.158325	+	0.693666i	0.990311	+	0.138869i	\(0.0443468\pi\)
−0.831986	+	0.554797i	\(0.812796\pi\)
\(37\)	4.38837e6	+	5.50284e6i	0.384942	+	0.482702i	0.936118	−	0.351687i	\(-0.114392\pi\)
							−0.551176	+	0.834389i	\(0.685821\pi\)
\(41\)	−1.37468e7			−0.759754			−0.379877	−	0.925037i	\(-0.624034\pi\)
							−0.379877	+	0.925037i	\(0.624034\pi\)
\(43\)	4.12491e6	−	1.80724e7i	0.183995	−	0.806135i	−0.795708	−	0.605681i	\(-0.792901\pi\)
							0.979703	−	0.200455i	\(-0.0642420\pi\)
\(47\)	−2.96824e7	+	3.72206e7i	−0.887277	+	1.11261i	0.105712	+	0.994397i	\(0.466288\pi\)
							−0.992988	+	0.118213i	\(0.962284\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.10.d.a.7.7	✓	126
29.25	even	7	inner	29.10.d.a.25.7	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.10.d.a.7.7	✓	126	1.1	even	1	trivial
29.10.d.a.25.7	yes	126	29.25	even	7	inner