Embedded newform 153.4.n.a.4.6

• Label: 153.4.n.a.4.6
• Level: $153$
• Weight: $4$
• Character: 153.4
• 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			4.6
Character	\(\chi\)	\(=\)	153.4
Dual form			153.4.n.a.115.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.02974 + 2.32657i) q^{2} +(-4.83579 - 1.90134i) q^{3} +(6.82586 - 11.8227i) q^{4} +(-0.472341 - 1.76280i) q^{5} +(23.9106 - 3.58890i) q^{6} +(-7.79590 + 29.0947i) q^{7} +26.2982i q^{8} +(19.7698 + 18.3890i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 6 q^{3} + 396 q^{4} - 2 q^{5} - 40 q^{6} - 2 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(37\)	\(137\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.02974	+	2.32657i	−1.42473	+	0.822567i	−0.996698	−	0.0811979i	\(-0.974125\pi\)
							−0.428030	+	0.903765i	\(0.640792\pi\)
\(3\)	−4.83579	−	1.90134i	−0.930649	−	0.365914i
							\(5\)	−0.472341	−	1.76280i	−0.0422474	−	0.157670i	0.941580	−	0.336790i	\(-0.109341\pi\)
−0.983827	+	0.179121i	\(0.942675\pi\)
\(7\)	−7.79590	+	29.0947i	−0.420939	+	1.57097i	0.351695	+	0.936115i	\(0.385605\pi\)
							−0.772634	+	0.634851i	\(0.781061\pi\)
\(11\)	2.99492	−	11.1772i	0.0820911	−	0.306368i	−0.912656	−	0.408728i	\(-0.865973\pi\)
							0.994747	+	0.102360i	\(0.0326392\pi\)
\(13\)	−31.0423	+	53.7668i	−0.662275	+	1.14709i	0.317741	+	0.948177i	\(0.397076\pi\)
							−0.980016	+	0.198917i	\(0.936258\pi\)
\(17\)	−63.5651	−	29.5377i	−0.906871	−	0.421409i
							\(19\)			43.1761i			0.521331i	0.965429	+	0.260665i	\(0.0839419\pi\)
−0.965429	+	0.260665i	\(0.916058\pi\)
\(23\)	36.1036	−	9.67393i	0.327310	−	0.0877023i	−0.0914224	−	0.995812i	\(-0.529141\pi\)
							0.418732	+	0.908110i	\(0.362475\pi\)
\(29\)	94.4628	+	25.3112i	0.604872	+	0.162075i	0.548241	−	0.836320i	\(-0.315297\pi\)
							0.0566310	+	0.998395i	\(0.481964\pi\)
\(31\)	−63.6446	−	237.525i	−0.368739	−	1.37615i	−0.862281	−	0.506430i	\(-0.830965\pi\)
							0.493542	−	0.869722i	\(-0.335702\pi\)
\(37\)	25.1297	−	25.1297i	0.111657	−	0.111657i	−0.649071	−	0.760728i	\(-0.724842\pi\)
							0.760728	+	0.649071i	\(0.224842\pi\)
\(41\)	−45.7654	+	12.2628i	−0.174326	+	0.0467105i	−0.344926	−	0.938630i	\(-0.612096\pi\)
							0.170600	+	0.985340i	\(0.445429\pi\)
\(43\)	−365.876	+	211.239i	−1.29757	+	0.749154i	−0.979984	−	0.199075i	\(-0.936206\pi\)
							−0.317588	+	0.948229i	\(0.602873\pi\)
\(47\)	−254.717	−	441.182i	−0.790516	−	1.36921i	−0.925648	−	0.378386i	\(-0.876479\pi\)
							0.135132	−	0.990828i	\(-0.456854\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.n.a.4.6	✓	208
9.7	even	3	inner	153.4.n.a.106.47	yes	208
17.13	even	4	inner	153.4.n.a.13.47	yes	208
153.115	even	12	inner	153.4.n.a.115.6	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.n.a.4.6	✓	208	1.1	even	1	trivial
153.4.n.a.13.47	yes	208	17.13	even	4	inner
153.4.n.a.106.47	yes	208	9.7	even	3	inner
153.4.n.a.115.6	yes	208	153.115	even	12	inner