Embedded newform 165.4.m.d.31.1

• Label: 165.4.m.d.31.1
• Level: $165$
• Weight: $4$
• Character: 165.31
• Analytic conductor: $9.735$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	165.m (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.73531515095\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			31.1
Character	\(\chi\)	\(=\)	165.31
Dual form			165.4.m.d.16.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73360 + 5.33547i) q^{2} +(-2.42705 + 1.76336i) q^{3} +(-18.9898 - 13.7969i) q^{4} +(-1.54508 - 4.75528i) q^{5} +(-5.20080 - 16.0064i) q^{6} +(24.0593 + 17.4801i) q^{7} +(70.2246 - 51.0211i) q^{8} +(2.78115 - 8.55951i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{2} - 18 q^{3} - 16 q^{4} + 30 q^{5} + 6 q^{6} + 23 q^{7} + 139 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).

\(n\)	\(46\)	\(56\)	\(67\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.73360	+	5.33547i	−0.612920	+	1.88637i	−0.184370	+	0.982857i	\(0.559024\pi\)
							−0.428550	+	0.903518i	\(0.640976\pi\)
\(3\)	−2.42705	+	1.76336i	−0.467086	+	0.339358i
							\(5\)	−1.54508	−	4.75528i	−0.138197	−	0.425325i
\(7\)	24.0593	+	17.4801i	1.29908	+	0.943836i								0.999946	−	0.0103753i	\(-0.00330261\pi\)
							0.299133	+	0.954211i	\(0.403303\pi\)
\(11\)	−13.8924	−	33.7343i	−0.380793	−	0.924660i
							\(13\)	13.8054	−	42.4887i	0.294534	−	0.906481i	−0.688844	−	0.724909i	\(-0.741882\pi\)
0.983378	−	0.181572i	\(-0.0581184\pi\)
\(17\)	9.38131	+	28.8727i	0.133841	+	0.411921i	0.995408	−	0.0957232i	\(-0.0305164\pi\)
							−0.861567	+	0.507644i	\(0.830516\pi\)
\(19\)	52.3269	−	38.0177i	0.631822	−	0.459045i	−0.225209	−	0.974310i	\(-0.572307\pi\)
							0.857031	+	0.515265i	\(0.172307\pi\)
\(23\)	−69.8631			−0.633368			−0.316684	−	0.948531i	\(-0.602569\pi\)
							−0.316684	+	0.948531i	\(0.602569\pi\)
\(29\)	27.9329	+	20.2944i	0.178862	+	0.129951i	0.673614	−	0.739083i	\(-0.264741\pi\)
							−0.494752	+	0.869034i	\(0.664741\pi\)
\(31\)	57.4367	−	176.772i	0.332772	−	1.02417i	−0.635037	−	0.772482i	\(-0.719015\pi\)
							0.967809	−	0.251686i	\(-0.0809850\pi\)
\(37\)	217.123	+	157.749i	0.964723	+	0.700912i	0.954243	−	0.299033i	\(-0.0966639\pi\)
							0.0104804	+	0.999945i	\(0.496664\pi\)
\(41\)	95.6041	−	69.4605i	0.364167	−	0.264583i	−0.390621	−	0.920552i	\(-0.627740\pi\)
							0.754788	+	0.655969i	\(0.227740\pi\)
\(43\)	449.777			1.59513			0.797563	−	0.603236i	\(-0.206122\pi\)
							0.797563	+	0.603236i	\(0.206122\pi\)
\(47\)	−0.654345	+	0.475409i	−0.00203077	+	0.00147544i	−0.588800	−	0.808279i	\(-0.700400\pi\)
							0.586769	+	0.809754i	\(0.300400\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.4.m.d.31.1	yes	24
11.4	even	5		1815.4.a.bg.1.1		12
11.5	even	5	inner	165.4.m.d.16.1	✓	24
11.7	odd	10		1815.4.a.bo.1.12		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.m.d.16.1	✓	24	11.5	even	5	inner
165.4.m.d.31.1	yes	24	1.1	even	1	trivial
1815.4.a.bg.1.1		12	11.4	even	5
1815.4.a.bo.1.12		12	11.7	odd	10