Embedded newform 9.76.c.a.4.30

• Label: 9.76.c.a.4.30
• 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			4.30
Character	\(\chi\)	\(=\)	9.4
Dual form			9.76.c.a.7.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.46451e10 - 9.46481e10i) q^{2} +(-4.22843e17 - 6.55340e17i) q^{3} +(1.29173e22 - 2.23734e22i) q^{4} +(1.01078e26 - 1.75072e26i) q^{5} +(-3.89204e28 + 7.58324e28i) q^{6} +(-2.63190e31 - 4.55859e31i) q^{7} -6.95234e33 q^{8} +(-2.50674e35 + 5.54212e35i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.46451e10	−	9.46481e10i	−0.281143	−	0.486953i	0.690524	−	0.723310i	\(-0.257380\pi\)
							−0.971666	+	0.236356i	\(0.924047\pi\)
\(3\)	−4.22843e17	−	6.55340e17i	−0.542166	−	0.840272i
							\(5\)	1.01078e26	−	1.75072e26i	0.621270	−	1.07607i	−0.367980	−	0.929834i	\(-0.619950\pi\)
0.989250	−	0.146237i	\(-0.0467163\pi\)
\(7\)	−2.63190e31	−	4.55859e31i	−0.535912	−	0.928227i	−0.999119	−	0.0419765i	\(-0.986635\pi\)
							0.463207	−	0.886250i	\(-0.346699\pi\)
\(11\)	−2.96063e38	−	5.12796e38i	−0.262517	−	0.454694i	0.704393	−	0.709810i	\(-0.251219\pi\)
							−0.966910	+	0.255117i	\(0.917886\pi\)
\(13\)	1.85834e41	−	3.21874e41i	0.313508	−	0.543012i	−0.665611	−	0.746299i	\(-0.731829\pi\)
							0.979119	+	0.203287i	\(0.0651623\pi\)
\(17\)	−9.23780e45			−0.666398			−0.333199	−	0.942856i	\(-0.608128\pi\)
							−0.333199	+	0.942856i	\(0.608128\pi\)
\(19\)	1.10101e48			1.22611			0.613057	−	0.790039i	\(-0.289940\pi\)
							0.613057	+	0.790039i	\(0.289940\pi\)
\(23\)	−7.84611e50	+	1.35899e51i	−0.675866	+	1.17063i	0.300349	+	0.953829i	\(0.402897\pi\)
							−0.976215	+	0.216805i	\(0.930436\pi\)
\(29\)	2.87592e54	+	4.98125e54i	0.415769	+	0.720134i	0.995509	−	0.0946682i	\(-0.0301790\pi\)
							−0.579739	+	0.814802i	\(0.696846\pi\)
\(31\)	2.06027e55	−	3.56849e55i	0.244265	−	0.423079i	−0.717660	−	0.696394i	\(-0.754787\pi\)
							0.961925	+	0.273315i	\(0.0881201\pi\)
\(37\)	1.17798e58			0.183473			0.0917365	−	0.995783i	\(-0.470758\pi\)
							0.0917365	+	0.995783i	\(0.470758\pi\)
\(41\)	−2.26563e60	+	3.92418e60i	−0.751265	+	1.30123i	0.195944	+	0.980615i	\(0.437223\pi\)
							−0.947210	+	0.320615i	\(0.896110\pi\)
\(43\)	−7.03909e60	−	1.21921e61i	−0.391246	−	0.677657i	0.601369	−	0.798972i	\(-0.294622\pi\)
							−0.992614	+	0.121315i	\(0.961289\pi\)
\(47\)	3.53127e62	+	6.11634e62i	0.698653	+	1.21010i	0.968934	+	0.247320i	\(0.0795500\pi\)
							−0.270281	+	0.962781i	\(0.587117\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.76.c.a.4.30	✓	148
9.7	even	3	inner	9.76.c.a.7.30	yes	148

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.76.c.a.4.30	✓	148	1.1	even	1	trivial
9.76.c.a.7.30	yes	148	9.7	even	3	inner