Embedded newform 432.2.bg.a.157.46

• Label: 432.2.bg.a.157.46
• Level: $432$
• Weight: $2$
• Character: 432.157
• 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			157.46
Character	\(\chi\)	\(=\)	432.157
Dual form			432.2.bg.a.421.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.663640 - 1.24883i) q^{2} +(-1.73025 + 0.0789784i) q^{3} +(-1.11916 - 1.65755i) q^{4} +(1.18369 - 2.53844i) q^{5} +(-1.04963 + 2.21320i) q^{6} +(-1.66871 + 0.294239i) q^{7} +(-2.81272 + 0.297632i) q^{8} +(2.98752 - 0.273305i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.663640	−	1.24883i	0.469264	−	0.883058i
							\(3\)	−1.73025	+	0.0789784i	−0.998960	+	0.0455982i
\(5\)	1.18369	−	2.53844i	0.529363	−	1.13522i								−0.441867	−	0.897081i	\(-0.645684\pi\)
							0.971230	−	0.238142i	\(-0.0765385\pi\)
\(7\)	−1.66871	+	0.294239i	−0.630714	+	0.111212i	−0.479861	−	0.877344i	\(-0.659313\pi\)
							−0.150853	+	0.988556i	\(0.548202\pi\)
\(11\)	−3.10862	+	1.44957i	−0.937285	+	0.437063i	−0.830375	−	0.557206i	\(-0.811874\pi\)
							−0.106910	+	0.994269i	\(0.534096\pi\)
\(13\)	−0.388861	−	4.44470i	−0.107851	−	1.23274i	−0.836545	−	0.547899i	\(-0.815428\pi\)
							0.728694	−	0.684839i	\(-0.240128\pi\)
\(17\)	−1.31452	+	2.27681i	−0.318817	+	0.552208i	−0.980242	−	0.197804i	\(-0.936619\pi\)
							0.661424	+	0.750012i	\(0.269952\pi\)
\(19\)	−0.0505563	+	0.188679i	−0.0115984	+	0.0432858i	−0.971483	−	0.237111i	\(-0.923799\pi\)
							0.959884	+	0.280397i	\(0.0904661\pi\)
\(23\)	1.44265	+	0.254379i	0.300814	+	0.0530417i	0.322018	−	0.946734i	\(-0.395639\pi\)
							−0.0212039	+	0.999775i	\(0.506750\pi\)
\(29\)	−9.05837	−	0.792505i	−1.68210	−	0.147164i	−0.794339	−	0.607474i	\(-0.792183\pi\)
							−0.887758	+	0.460310i	\(0.847738\pi\)
\(31\)	0.660193	−	3.74414i	0.118574	−	0.672468i	−0.866344	−	0.499448i	\(-0.833536\pi\)
							0.984918	−	0.173020i	\(-0.0553525\pi\)
\(37\)	0.404048	+	1.50793i	0.0664252	+	0.247902i	0.991152	−	0.132728i	\(-0.0423737\pi\)
							−0.924727	+	0.380630i	\(0.875707\pi\)
\(41\)	−3.67611	+	4.38102i	−0.574112	+	0.684200i	−0.972469	−	0.233030i	\(-0.925136\pi\)
							0.398358	+	0.917230i	\(0.369580\pi\)
\(43\)	4.29311	−	2.00191i	0.654693	−	0.305288i	−0.0667422	−	0.997770i	\(-0.521261\pi\)
							0.721435	+	0.692482i	\(0.243483\pi\)
\(47\)	−1.85132	−	10.4994i	−0.270044	−	1.53149i	−0.754278	−	0.656555i	\(-0.772013\pi\)
							0.484235	−	0.874938i	\(-0.339098\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.157.46	✓	840
16.5	even	4	inner	432.2.bg.a.373.64	yes	840
27.16	even	9	inner	432.2.bg.a.205.64	yes	840
432.421	even	36	inner	432.2.bg.a.421.46	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.157.46	✓	840	1.1	even	1	trivial
432.2.bg.a.205.64	yes	840	27.16	even	9	inner
432.2.bg.a.373.64	yes	840	16.5	even	4	inner
432.2.bg.a.421.46	yes	840	432.421	even	36	inner