Embedded newform 432.2.bg.a.13.2

• Label: 432.2.bg.a.13.2
• Level: $432$
• Weight: $2$
• Character: 432.13
• Analytic conductor: $3.450$
• Analytic rank: $0$
• Dimension: $840$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.bg (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.44953736732\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(70\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			13.2
Character	\(\chi\)	\(=\)	432.13
Dual form			432.2.bg.a.133.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41151 - 0.0874761i) q^{2} +(0.0727719 + 1.73052i) q^{3} +(1.98470 + 0.246946i) q^{4} +(-0.723668 + 0.0633127i) q^{5} +(0.0486614 - 2.44901i) q^{6} +(0.398828 - 1.09577i) q^{7} +(-2.77981 - 0.522179i) q^{8} +(-2.98941 + 0.251867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41151	−	0.0874761i	−0.998085	−	0.0618550i
							\(3\)	0.0727719	+	1.73052i	0.0420149	+	0.999117i
\(5\)	−0.723668	+	0.0633127i	−0.323634	+	0.0283143i								−0.247815	−	0.968807i	\(-0.579713\pi\)
							−0.0758188	+	0.997122i	\(0.524157\pi\)
\(7\)	0.398828	−	1.09577i	0.150743	−	0.414163i	−0.841220	−	0.540693i	\(-0.818162\pi\)
							0.991963	+	0.126531i	\(0.0403842\pi\)
\(11\)	−0.304496	+	3.48040i	−0.0918089	+	1.04938i	0.800916	+	0.598777i	\(0.204347\pi\)
							−0.892725	+	0.450603i	\(0.851209\pi\)
\(13\)	−0.440114	+	0.308171i	−0.122066	+	0.0854713i	−0.633015	−	0.774140i	\(-0.718183\pi\)
							0.510949	+	0.859611i	\(0.329294\pi\)
\(17\)	−3.29464	+	5.70648i	−0.799067	+	1.38402i	0.121158	+	0.992633i	\(0.461339\pi\)
							−0.920224	+	0.391391i	\(0.871994\pi\)
\(19\)	0.0169360	−	0.0632060i	0.00388538	−	0.0145004i	−0.963956	−	0.266062i	\(-0.914277\pi\)
							0.967841	+	0.251562i	\(0.0809441\pi\)
\(23\)	−0.415820	−	1.14246i	−0.0867045	−	0.238219i	0.888761	−	0.458371i	\(-0.151567\pi\)
							−0.975466	+	0.220152i	\(0.929345\pi\)
\(29\)	1.23837	−	1.76857i	0.229959	−	0.328415i	−0.687545	−	0.726141i	\(-0.741312\pi\)
							0.917504	+	0.397726i	\(0.130201\pi\)
\(31\)	−3.93790	+	1.43328i	−0.707267	+	0.257424i	−0.670511	−	0.741900i	\(-0.733925\pi\)
							−0.0367568	+	0.999324i	\(0.511703\pi\)
\(37\)	1.92956	+	7.20122i	0.317218	+	1.18387i	0.921907	+	0.387411i	\(0.126631\pi\)
							−0.604689	+	0.796461i	\(0.706703\pi\)
\(41\)	−6.52466	−	1.15047i	−1.01898	−	0.179674i	−0.360887	−	0.932610i	\(-0.617526\pi\)
							−0.658094	+	0.752936i	\(0.728637\pi\)
\(43\)	0.375213	−	4.28870i	0.0572194	−	0.654021i	−0.912383	−	0.409337i	\(-0.865760\pi\)
							0.969603	−	0.244684i	\(-0.0786843\pi\)
\(47\)	−6.70108	−	2.43900i	−0.977454	−	0.355764i	−0.196604	−	0.980483i	\(-0.562991\pi\)
							−0.780850	+	0.624719i	\(0.785214\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.13.2	✓	840
16.5	even	4	inner	432.2.bg.a.229.62	yes	840
27.25	even	9	inner	432.2.bg.a.349.62	yes	840
432.133	even	36	inner	432.2.bg.a.133.2	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.13.2	✓	840	1.1	even	1	trivial
432.2.bg.a.133.2	yes	840	432.133	even	36	inner
432.2.bg.a.229.62	yes	840	16.5	even	4	inner
432.2.bg.a.349.62	yes	840	27.25	even	9	inner