Embedded newform 29.6.d.a.7.6

• Label: 29.6.d.a.7.6
• 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			7.6
Character	\(\chi\)	\(=\)	29.7
Dual form			29.6.d.a.25.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.402095 + 1.76169i) q^{2} +(22.1010 + 10.6433i) q^{3} +(25.8891 - 12.4675i) q^{4} +(-5.52960 - 24.2268i) q^{5} +(-9.86349 + 43.2148i) q^{6} +(-77.8191 - 37.4757i) q^{7} +(68.4265 + 85.8041i) q^{8} +(223.667 + 280.469i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.402095	+	1.76169i	0.0710810	+	0.311426i	0.997953	−	0.0639513i	\(-0.0203702\pi\)
							−0.926872	+	0.375377i	\(0.877513\pi\)
\(3\)	22.1010	+	10.6433i	1.41778	+	0.682767i	0.976682	−	0.214692i	\(-0.0688747\pi\)
							0.441098	+	0.897459i	\(0.354589\pi\)
\(5\)	−5.52960	−	24.2268i	−0.0989166	−	0.433382i	0.901083	−	0.433646i	\(-0.142773\pi\)
							−1.00000		0.000264016i	\(0.999916\pi\)
\(7\)	−77.8191	−	37.4757i	−0.600263	−	0.289071i	0.108971	−	0.994045i	\(-0.465244\pi\)
							−0.709233	+	0.704974i	\(0.750959\pi\)
\(11\)	30.7776	−	38.5938i	0.0766924	−	0.0961693i	−0.742005	−	0.670394i	\(-0.766125\pi\)
							0.818698	+	0.574225i	\(0.194696\pi\)
\(13\)	−374.380	+	469.457i	−0.614404	+	0.770438i	−0.987545	−	0.157337i	\(-0.949709\pi\)
							0.373141	+	0.927775i	\(0.378281\pi\)
\(17\)	−772.414			−0.648228			−0.324114	−	0.946018i	\(-0.605066\pi\)
							−0.324114	+	0.946018i	\(0.605066\pi\)
\(19\)	−2052.80	+	988.576i	−1.30456	+	0.628241i	−0.951583	−	0.307393i	\(-0.900543\pi\)
							−0.352973	+	0.935634i	\(0.614829\pi\)
\(23\)	723.606	−	3170.32i	0.285222	−	1.24964i	−0.605778	−	0.795634i	\(-0.707138\pi\)
							0.891000	−	0.454004i	\(-0.150005\pi\)
\(29\)	−3188.01	−	3216.79i	−0.703921	−	0.710278i
							\(31\)	888.608	+	3893.24i	0.166076	+	0.727625i	0.987540	+	0.157366i	\(0.0503001\pi\)
−0.821465	+	0.570259i	\(0.806843\pi\)
\(37\)	−2481.53	−	3111.74i	−0.297999	−	0.373679i	0.610179	−	0.792264i	\(-0.291098\pi\)
							−0.908178	+	0.418585i	\(0.862526\pi\)
\(41\)	18429.7			1.71222			0.856108	−	0.516796i	\(-0.172875\pi\)
							0.856108	+	0.516796i	\(0.172875\pi\)
\(43\)	−2666.31	+	11681.9i	−0.219907	+	0.963476i	0.737639	+	0.675195i	\(0.235941\pi\)
							−0.957546	+	0.288281i	\(0.906916\pi\)
\(47\)	−3396.07	+	4258.54i	−0.224250	+	0.281201i	−0.881210	−	0.472725i	\(-0.843271\pi\)
							0.656960	+	0.753925i	\(0.271842\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.7.6	✓	66
29.5	even	14		841.6.a.h.1.19		33
29.24	even	7		841.6.a.i.1.15		33
29.25	even	7	inner	29.6.d.a.25.6	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.7.6	✓	66	1.1	even	1	trivial
29.6.d.a.25.6	yes	66	29.25	even	7	inner
841.6.a.h.1.19		33	29.5	even	14
841.6.a.i.1.15		33	29.24	even	7