Embedded newform 153.4.n.a.106.18

• Label: 153.4.n.a.106.18
• Level: $153$
• Weight: $4$
• Character: 153.106
• Analytic conductor: $9.027$
• Analytic rank: $0$
• Dimension: $208$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	153.n (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.02729223088\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(52\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			106.18
Character	\(\chi\)	\(=\)	153.106
Dual form			153.4.n.a.13.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.93856 - 1.11923i) q^{2} +(4.53363 + 2.53894i) q^{3} +(-1.49466 - 2.58882i) q^{4} +(-12.3493 - 3.30900i) q^{5} +(-5.94705 - 9.99606i) q^{6} +(-13.8994 + 3.72434i) q^{7} +24.5991i q^{8} +(14.1075 + 23.0212i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 6 q^{3} + 396 q^{4} - 2 q^{5} - 40 q^{6} - 2 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/153\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(137\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.93856	−	1.11923i	−0.685385	−	0.395707i	0.116496	−	0.993191i	\(-0.462834\pi\)
							−0.801881	+	0.597484i	\(0.796167\pi\)
\(3\)	4.53363	+	2.53894i	0.872497	+	0.488620i
							\(5\)	−12.3493	−	3.30900i	−1.10456	−	0.295966i	−0.339939	−	0.940447i	\(-0.610406\pi\)
−0.764620	+	0.644482i	\(0.777073\pi\)
\(7\)	−13.8994	+	3.72434i	−0.750499	+	0.201096i	−0.613739	−	0.789509i	\(-0.710335\pi\)
							−0.136760	+	0.990604i	\(0.543669\pi\)
\(11\)	62.9315	−	16.8624i	1.72496	−	0.462202i	0.745948	−	0.666004i	\(-0.231997\pi\)
							0.979012	+	0.203802i	\(0.0653300\pi\)
\(13\)	16.9356	+	29.3333i	0.361314	+	0.625815i	0.988177	−	0.153315i	\(-0.0489948\pi\)
							−0.626863	+	0.779129i	\(0.715661\pi\)
\(17\)	70.0800	−	1.33802i	0.999818	−	0.0190892i
							\(19\)			124.540i			1.50376i	0.659298	+	0.751881i	\(0.270853\pi\)
−0.659298	+	0.751881i	\(0.729147\pi\)
\(23\)	3.41612	−	12.7491i	0.0309700	−	0.115582i	−0.948710	−	0.316146i	\(-0.897611\pi\)
							0.979680	+	0.200565i	\(0.0642776\pi\)
\(29\)	−22.0161	−	82.1651i	−0.140975	−	0.526127i	−0.999902	−	0.0140276i	\(-0.995535\pi\)
							0.858926	−	0.512099i	\(-0.171132\pi\)
\(31\)	203.420	+	54.5063i	1.17856	+	0.315794i	0.794355	−	0.607453i	\(-0.207809\pi\)
							0.384205	+	0.923248i	\(0.374476\pi\)
\(37\)	−195.968	+	195.968i	−0.870728	+	0.870728i	−0.992552	−	0.121824i	\(-0.961126\pi\)
							0.121824	+	0.992552i	\(0.461126\pi\)
\(41\)	18.3502	−	68.4838i	0.0698980	−	0.260863i	−0.922130	−	0.386880i	\(-0.873553\pi\)
							0.992028	+	0.126017i	\(0.0402195\pi\)
\(43\)	197.347	+	113.939i	0.699888	+	0.404081i	0.807306	−	0.590133i	\(-0.200925\pi\)
							−0.107417	+	0.994214i	\(0.534258\pi\)
\(47\)	−75.4210	+	130.633i	−0.234070	+	0.405421i	−0.959002	−	0.283400i	\(-0.908538\pi\)
							0.724932	+	0.688820i	\(0.241871\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.n.a.106.18	yes	208
9.4	even	3	inner	153.4.n.a.4.35	✓	208
17.13	even	4	inner	153.4.n.a.115.35	yes	208
153.13	even	12	inner	153.4.n.a.13.18	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.n.a.4.35	✓	208	9.4	even	3	inner
153.4.n.a.13.18	yes	208	153.13	even	12	inner
153.4.n.a.106.18	yes	208	1.1	even	1	trivial
153.4.n.a.115.35	yes	208	17.13	even	4	inner