Embedded newform 9.76.c.a.7.63

• Label: 9.76.c.a.7.63
• Level: $9$
• Weight: $76$
• Character: 9.7
• Analytic conductor: $320.606$
• Analytic rank: $0$
• Dimension: $148$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 76 \)
Character orbit:	\([\chi]\)	\(=\)	9.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(320.605553540\)
Analytic rank:	\(0\)
Dimension:	\(148\)
Relative dimension:	\(74\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			7.63
Character	\(\chi\)	\(=\)	9.7
Dual form			9.76.c.a.4.63

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.42120e11 - 2.46159e11i) q^{2} +(5.28418e17 + 5.73621e17i) q^{3} +(-2.15068e22 - 3.72508e22i) q^{4} +(4.18108e25 + 7.24184e25i) q^{5} +(2.16301e29 - 4.85521e28i) q^{6} +(-1.54233e31 + 2.67140e31i) q^{7} -1.48789e33 q^{8} +(-4.98146e34 + 6.06224e35i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	148 * q - 137438953473 * q^2 + 785092363818710037 * q^3 - 1378931012997936402399233 * q^4 - 128851949435224459885424655 * q^5 + 538792206380571785448697304391 * q^6 + 2305169592807634547412733660793 * q^7 + 24179496392511120352908521695703106 * q^8 + 330787849704966093938607389501153829 * q^9

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.42120e11	−	2.46159e11i	0.731191	−	1.26646i	−0.225184	−	0.974316i	\(-0.572298\pi\)
							0.956375	−	0.292143i	\(-0.0943684\pi\)
\(3\)	5.28418e17	+	5.73621e17i	0.677534	+	0.735492i
							\(5\)	4.18108e25	+	7.24184e25i	0.256988	+	0.445116i	0.965434	−	0.260649i	\(-0.0839365\pi\)
−0.708446	+	0.705766i	\(0.750603\pi\)
\(7\)	−1.54233e31	+	2.67140e31i	−0.314052	+	0.543953i	−0.979235	−	0.202726i	\(-0.935020\pi\)
							0.665184	+	0.746680i	\(0.268353\pi\)
\(11\)	−3.70902e38	+	6.42421e38i	−0.328877	+	0.569632i	−0.982289	−	0.187371i	\(-0.940003\pi\)
							0.653412	+	0.757002i	\(0.273337\pi\)
\(13\)	−1.48457e41	−	2.57135e41i	−0.250452	−	0.433796i	0.713198	−	0.700962i	\(-0.247246\pi\)
							−0.963650	+	0.267167i	\(0.913913\pi\)
\(17\)	−2.13448e46			−1.53978			−0.769889	−	0.638178i	\(-0.779688\pi\)
							−0.769889	+	0.638178i	\(0.779688\pi\)
\(19\)	−3.60644e47			−0.401623			−0.200811	−	0.979630i	\(-0.564358\pi\)
							−0.200811	+	0.979630i	\(0.564358\pi\)
\(23\)	−1.05304e50	−	1.82392e50i	−0.0907091	−	0.157113i	0.817101	−	0.576495i	\(-0.195580\pi\)
							−0.907810	+	0.419382i	\(0.862247\pi\)
\(29\)	6.46170e54	−	1.11920e55i	0.934162	−	1.61802i	0.158039	−	0.987433i	\(-0.449483\pi\)
							0.776122	−	0.630583i	\(-0.217184\pi\)
\(31\)	1.46028e55	+	2.52928e55i	0.173130	+	0.299870i	0.939513	−	0.342514i	\(-0.111279\pi\)
							−0.766383	+	0.642385i	\(0.777945\pi\)
\(37\)	−7.59616e58			−1.18312			−0.591561	−	0.806260i	\(-0.701488\pi\)
							−0.591561	+	0.806260i	\(0.701488\pi\)
\(41\)	6.61970e59	+	1.14656e60i	0.219504	+	0.380193i	0.954657	−	0.297709i	\(-0.0962226\pi\)
							−0.735152	+	0.677902i	\(0.762889\pi\)
\(43\)	−4.32187e60	+	7.48570e60i	−0.240218	+	0.416069i	−0.960776	−	0.277325i	\(-0.910552\pi\)
							0.720559	+	0.693394i	\(0.243885\pi\)
\(47\)	8.16740e60	−	1.41464e61i	0.0161590	−	0.0279882i	−0.857833	−	0.513929i	\(-0.828190\pi\)
							0.873992	+	0.485941i	\(0.161523\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.76.c.a.7.63	yes	148
9.4	even	3	inner	9.76.c.a.4.63	✓	148

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.76.c.a.4.63	✓	148	9.4	even	3	inner
9.76.c.a.7.63	yes	148	1.1	even	1	trivial