Embedded newform 153.4.n.a.13.47

• Label: 153.4.n.a.13.47
• Level: $153$
• Weight: $4$
• Character: 153.13
• Analytic conductor: $9.027$
• Analytic rank: $0$
• Dimension: $208$
• 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			13.47
Character	\(\chi\)	\(=\)	153.13
Dual form			153.4.n.a.106.47

$q$-expansion

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

Character values

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

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

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	0.428030	−	0.903765i	\(-0.359208\pi\)
							0.996698	+	0.0811979i	\(0.0258746\pi\)
\(3\)	1.90134	−	4.83579i	0.365914	−	0.930649i
							\(5\)	1.76280	−	0.472341i	0.157670	−	0.0422474i	−0.179121	−	0.983827i	\(-0.557325\pi\)
0.336790	+	0.941580i	\(0.390659\pi\)
\(7\)	29.0947	+	7.79590i	1.57097	+	0.420939i	0.936115	−	0.351695i	\(-0.114395\pi\)
							0.634851	+	0.772634i	\(0.281061\pi\)
\(11\)	−11.1772	−	2.99492i	−0.306368	−	0.0820911i	0.102360	−	0.994747i	\(-0.467361\pi\)
							−0.408728	+	0.912656i	\(0.634027\pi\)
\(13\)	−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	\(-0.916058\pi\)
0.965429	−	0.260665i	\(-0.0839419\pi\)
\(23\)	−9.67393	−	36.1036i	−0.0877023	−	0.327310i	0.908110	−	0.418732i	\(-0.137525\pi\)
							−0.995812	+	0.0914224i	\(0.970859\pi\)
\(29\)	−25.3112	+	94.4628i	−0.162075	+	0.604872i	0.836320	+	0.548241i	\(0.184703\pi\)
							−0.998395	+	0.0566310i	\(0.981964\pi\)
\(31\)	237.525	−	63.6446i	1.37615	−	0.368739i	0.506430	−	0.862281i	\(-0.330965\pi\)
							0.869722	+	0.493542i	\(0.164298\pi\)
\(37\)	25.1297	+	25.1297i	0.111657	+	0.111657i	0.760728	−	0.649071i	\(-0.224842\pi\)
							−0.649071	+	0.760728i	\(0.724842\pi\)
\(41\)	12.2628	+	45.7654i	0.0467105	+	0.174326i	0.985340	−	0.170600i	\(-0.0545707\pi\)
							−0.938630	+	0.344926i	\(0.887904\pi\)
\(43\)	365.876	−	211.239i	1.29757	−	0.749154i	0.317588	−	0.948229i	\(-0.397127\pi\)
							0.979984	+	0.199075i	\(0.0637938\pi\)
\(47\)	−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.13.47	yes	208
9.7	even	3	inner	153.4.n.a.115.6	yes	208
17.4	even	4	inner	153.4.n.a.4.6	✓	208
153.106	even	12	inner	153.4.n.a.106.47	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.n.a.4.6	✓	208	17.4	even	4	inner
153.4.n.a.13.47	yes	208	1.1	even	1	trivial
153.4.n.a.106.47	yes	208	153.106	even	12	inner
153.4.n.a.115.6	yes	208	9.7	even	3	inner