Embedded newform 9.76.c.a.4.17

• Label: 9.76.c.a.4.17
• Level: $9$
• Weight: $76$
• Character: 9.4
• Analytic conductor: $320.606$
• Analytic rank: $0$
• Dimension: $148$
• CM: no
• 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			4.17
Character	\(\chi\)	\(=\)	9.4
Dual form			9.76.c.a.7.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17057e11 - 2.02749e11i) q^{2} +(5.96990e17 + 5.01866e17i) q^{3} +(-8.51530e21 + 1.47489e22i) q^{4} +(1.05568e26 - 1.82849e26i) q^{5} +(3.18709e28 - 1.79786e29i) q^{6} +(-3.87881e31 - 6.71830e31i) q^{7} -4.85748e33 q^{8} +(1.04527e35 + 5.99218e35i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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 \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17057e11	−	2.02749e11i	−0.602245	−	1.04312i	−0.992480	−	0.122404i	\(-0.960940\pi\)
							0.390235	−	0.920715i	\(-0.372394\pi\)
\(3\)	5.96990e17	+	5.01866e17i	0.765456	+	0.643489i
							\(5\)	1.05568e26	−	1.82849e26i	0.648870	−	1.12388i	−0.334523	−	0.942388i	\(-0.608575\pi\)
0.983393	−	0.181488i	\(-0.0580915\pi\)
\(7\)	−3.87881e31	−	6.71830e31i	−0.789810	−	1.36799i	−0.926083	−	0.377319i	\(-0.876846\pi\)
							0.136274	−	0.990671i	\(-0.456487\pi\)
\(11\)	4.26736e38	+	7.39129e38i	0.378385	+	0.655382i	0.990827	−	0.135133i	\(-0.0431462\pi\)
							−0.612442	+	0.790515i	\(0.709813\pi\)
\(13\)	1.75598e41	−	3.04145e41i	0.296240	−	0.513103i	−0.679032	−	0.734108i	\(-0.737600\pi\)
							0.975273	+	0.221005i	\(0.0709337\pi\)
\(17\)	3.11771e45			0.224906			0.112453	−	0.993657i	\(-0.464129\pi\)
							0.112453	+	0.993657i	\(0.464129\pi\)
\(19\)	−7.30236e47			−0.813211			−0.406606	−	0.913604i	\(-0.633288\pi\)
							−0.406606	+	0.913604i	\(0.633288\pi\)
\(23\)	−5.76492e49	+	9.98514e49i	−0.0496592	+	0.0860123i	−0.889787	−	0.456377i	\(-0.849147\pi\)
							0.840127	+	0.542389i	\(0.182480\pi\)
\(29\)	1.48583e54	+	2.57353e54i	0.214805	+	0.372052i	0.953212	−	0.302302i	\(-0.0977552\pi\)
							−0.738408	+	0.674355i	\(0.764422\pi\)
\(31\)	−2.74227e55	+	4.74974e55i	−0.325122	+	0.563127i	−0.981537	−	0.191272i	\(-0.938739\pi\)
							0.656415	+	0.754400i	\(0.272072\pi\)
\(37\)	−5.59266e58			−0.871072			−0.435536	−	0.900171i	\(-0.643441\pi\)
							−0.435536	+	0.900171i	\(0.643441\pi\)
\(41\)	2.63912e60	−	4.57109e60i	0.875113	−	1.51574i	0.0184708	−	0.999829i	\(-0.494120\pi\)
							0.856642	−	0.515911i	\(-0.172546\pi\)
\(43\)	−2.92178e60	−	5.06068e60i	−0.162398	−	0.281282i	0.773330	−	0.634004i	\(-0.218590\pi\)
							−0.935728	+	0.352722i	\(0.885256\pi\)
\(47\)	4.85250e62	+	8.40477e62i	0.960054	+	1.66286i	0.722353	+	0.691524i	\(0.243060\pi\)
							0.237701	+	0.971338i	\(0.423606\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.76.c.a.4.17	✓	148
9.7	even	3	inner	9.76.c.a.7.17	yes	148

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.76.c.a.4.17	✓	148	1.1	even	1	trivial
9.76.c.a.7.17	yes	148	9.7	even	3	inner