Embedded newform 29.6.d.a.7.7

• Label: 29.6.d.a.7.7
• 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			7.7
Character	\(\chi\)	\(=\)	29.7
Dual form			29.6.d.a.25.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.522777 + 2.29044i) q^{2} +(-22.0166 - 10.6026i) q^{3} +(23.8582 - 11.4895i) q^{4} +(18.8051 + 82.3907i) q^{5} +(12.7749 - 55.9704i) q^{6} +(123.108 + 59.2858i) q^{7} +(85.6617 + 107.416i) q^{8} +(220.807 + 276.883i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.522777	+	2.29044i	0.0924148	+	0.404896i	0.999884	−	0.0152227i	\(-0.00484572\pi\)
							−0.907469	+	0.420118i	\(0.861989\pi\)
\(3\)	−22.0166	−	10.6026i	−1.41237	−	0.680160i	−0.436738	−	0.899589i	\(-0.643866\pi\)
							−0.975628	+	0.219429i	\(0.929581\pi\)
\(5\)	18.8051	+	82.3907i	0.336396	+	1.47385i	0.806498	+	0.591237i	\(0.201360\pi\)
							−0.470102	+	0.882612i	\(0.655783\pi\)
\(7\)	123.108	+	59.2858i	0.949603	+	0.457305i	0.843547	−	0.537055i	\(-0.180463\pi\)
							0.106056	+	0.994360i	\(0.466178\pi\)
\(11\)	198.633	−	249.078i	0.494959	−	0.620659i	−0.470125	−	0.882600i	\(-0.655791\pi\)
							0.965084	+	0.261941i	\(0.0843626\pi\)
\(13\)	−604.705	+	758.276i	−0.992396	+	1.24443i	−0.0227934	+	0.999740i	\(0.507256\pi\)
							−0.969603	+	0.244685i	\(0.921315\pi\)
\(17\)	1998.65			1.67732			0.838659	−	0.544657i	\(-0.183340\pi\)
							0.838659	+	0.544657i	\(0.183340\pi\)
\(19\)	−1292.05	+	622.217i	−0.821097	+	0.395419i	−0.796769	−	0.604285i	\(-0.793459\pi\)
							−0.0243283	+	0.999704i	\(0.507745\pi\)
\(23\)	134.131	−	587.665i	0.0528700	−	0.231638i	−0.941589	−	0.336764i	\(-0.890667\pi\)
							0.994459	+	0.105126i	\(0.0335246\pi\)
\(29\)	−4197.73	−	1700.07i	−0.926870	−	0.375381i
							\(31\)	−53.2013	−	233.090i	−0.00994302	−	0.0435632i	0.969713	−	0.244247i	\(-0.0785408\pi\)
−0.979656	+	0.200684i	\(0.935684\pi\)
\(37\)	1368.94	+	1716.60i	0.164392	+	0.206141i	0.857204	−	0.514978i	\(-0.172200\pi\)
							−0.692812	+	0.721118i	\(0.743628\pi\)
\(41\)	1050.10			0.0975596			0.0487798	−	0.998810i	\(-0.484467\pi\)
							0.0487798	+	0.998810i	\(0.484467\pi\)
\(43\)	2684.74	−	11762.6i	0.221427	−	0.970135i	−0.734978	−	0.678091i	\(-0.762807\pi\)
							0.956405	−	0.292044i	\(-0.0943354\pi\)
\(47\)	8328.32	−	10443.4i	0.549937	−	0.689599i	−0.426725	−	0.904381i	\(-0.640333\pi\)
							0.976662	+	0.214782i	\(0.0689042\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.7.7	✓	66
29.5	even	14		841.6.a.h.1.21		33
29.24	even	7		841.6.a.i.1.13		33
29.25	even	7	inner	29.6.d.a.25.7	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.7.7	✓	66	1.1	even	1	trivial
29.6.d.a.25.7	yes	66	29.25	even	7	inner
841.6.a.h.1.21		33	29.5	even	14
841.6.a.i.1.13		33	29.24	even	7