Embedded newform 9.76.c.a.4.2

• Label: 9.76.c.a.4.2
• 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			4.2
Character	\(\chi\)	\(=\)	9.4
Dual form			9.76.c.a.7.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.89250e11 - 3.27791e11i) q^{2} +(6.80741e17 + 3.80602e17i) q^{3} +(-5.27418e22 + 9.13515e22i) q^{4} +(1.46942e26 - 2.54510e26i) q^{5} +(-4.07234e27 - 2.95170e29i) q^{6} +(2.97431e31 + 5.15165e31i) q^{7} +2.56262e34 q^{8} +(3.18550e35 + 5.18184e35i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.89250e11	−	3.27791e11i	−0.973669	−	1.68644i	−0.684258	−	0.729240i	\(-0.739874\pi\)
							−0.289411	−	0.957205i	\(-0.593459\pi\)
\(3\)	6.80741e17	+	3.80602e17i	0.872841	+	0.488005i
							\(5\)	1.46942e26	−	2.54510e26i	0.903170	−	1.56434i	0.0798146	−	0.996810i	\(-0.474567\pi\)
0.823355	−	0.567526i	\(-0.192100\pi\)
\(7\)	2.97431e31	+	5.15165e31i	0.605632	+	1.04899i	0.991951	+	0.126621i	\(0.0404131\pi\)
							−0.386319	+	0.922365i	\(0.626254\pi\)
\(11\)	−7.86726e38	−	1.36265e39i	−0.697586	−	1.20825i	−0.969301	−	0.245876i	\(-0.920924\pi\)
							0.271716	−	0.962378i	\(-0.412409\pi\)
\(13\)	−1.36738e41	+	2.36837e41i	−0.230682	+	0.399552i	−0.958009	−	0.286739i	\(-0.907429\pi\)
							0.727327	+	0.686291i	\(0.240762\pi\)
\(17\)	−3.93216e45			−0.283659			−0.141829	−	0.989891i	\(-0.545298\pi\)
							−0.141829	+	0.989891i	\(0.545298\pi\)
\(19\)	−3.96560e47			−0.441620			−0.220810	−	0.975317i	\(-0.570870\pi\)
							−0.220810	+	0.975317i	\(0.570870\pi\)
\(23\)	−6.63007e49	+	1.14836e50i	−0.0571116	+	0.0989201i	−0.893168	−	0.449724i	\(-0.851522\pi\)
							0.836056	+	0.548644i	\(0.184856\pi\)
\(29\)	−8.05810e53	−	1.39570e54i	−0.116495	−	0.201776i	0.801881	−	0.597483i	\(-0.203833\pi\)
							−0.918376	+	0.395708i	\(0.870499\pi\)
\(31\)	−4.55115e55	+	7.88283e55i	−0.539583	+	0.934584i	0.459344	+	0.888259i	\(0.348085\pi\)
							−0.998926	+	0.0463259i	\(0.985249\pi\)
\(37\)	−6.83721e58			−1.06491			−0.532457	−	0.846457i	\(-0.678731\pi\)
							−0.532457	+	0.846457i	\(0.678731\pi\)
\(41\)	−1.09017e60	+	1.88823e60i	−0.361492	+	0.626123i	−0.988207	−	0.153126i	\(-0.951066\pi\)
							0.626715	+	0.779249i	\(0.284399\pi\)
\(43\)	7.23369e60	+	1.25291e61i	0.402062	+	0.696392i	0.993975	−	0.109610i	\(-0.0349603\pi\)
							−0.591913	+	0.806002i	\(0.701627\pi\)
\(47\)	6.74567e61	+	1.16838e62i	0.133461	+	0.231162i	0.925009	−	0.379946i	\(-0.124057\pi\)
							−0.791547	+	0.611108i	\(0.790724\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.76.c.a.4.2	✓	148
9.7	even	3	inner	9.76.c.a.7.2	yes	148

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.76.c.a.4.2	✓	148	1.1	even	1	trivial
9.76.c.a.7.2	yes	148	9.7	even	3	inner