Embedded newform 153.4.n.a.106.45

• Label: 153.4.n.a.106.45
• 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			106.45
Character	\(\chi\)	\(=\)	153.106
Dual form			153.4.n.a.13.45

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.50903 + 2.02594i) q^{2} +(0.218568 + 5.19155i) q^{3} +(4.20886 + 7.28996i) q^{4} +(-18.5503 - 4.97053i) q^{5} +(-9.75081 + 18.6601i) q^{6} +(-10.4834 + 2.80902i) q^{7} +1.69255i q^{8} +(-26.9045 + 2.26942i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 6 q^{3} + 396 q^{4} - 2 q^{5} - 40 q^{6} - 2 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.50903	+	2.02594i	1.24063	+	0.716278i	0.969222	−	0.246187i	\(-0.0791777\pi\)
							0.271407	+	0.962465i	\(0.412511\pi\)
\(3\)	0.218568	+	5.19155i	0.0420635	+	0.999115i
							\(5\)	−18.5503	−	4.97053i	−1.65919	−	0.444577i	−0.697022	−	0.717050i	\(-0.745492\pi\)
−0.962164	+	0.272472i	\(0.912159\pi\)
\(7\)	−10.4834	+	2.80902i	−0.566050	+	0.151673i	−0.530484	−	0.847695i	\(-0.677990\pi\)
							−0.0355656	+	0.999367i	\(0.511323\pi\)
\(11\)	−22.6073	+	6.05760i	−0.619668	+	0.166039i	−0.554977	−	0.831866i	\(-0.687273\pi\)
							−0.0646909	+	0.997905i	\(0.520606\pi\)
\(13\)	39.3566	+	68.1676i	0.839658	+	1.45433i	0.890181	+	0.455607i	\(0.150578\pi\)
							−0.0505228	+	0.998723i	\(0.516089\pi\)
\(17\)	18.6593	+	67.5635i	0.266209	+	0.963915i
							\(19\)			33.0912i			0.399560i	0.979841	+	0.199780i	\(0.0640228\pi\)
−0.979841	+	0.199780i	\(0.935977\pi\)
\(23\)	−36.6429	+	136.753i	−0.332199	+	1.23978i	0.574676	+	0.818381i	\(0.305128\pi\)
							−0.906875	+	0.421401i	\(0.861539\pi\)
\(29\)	−45.9209	−	171.379i	−0.294045	−	1.09739i	−0.941973	−	0.335689i	\(-0.891031\pi\)
							0.647928	−	0.761702i	\(-0.275636\pi\)
\(31\)	69.6906	+	18.6736i	0.403768	+	0.108189i	0.454987	−	0.890498i	\(-0.349644\pi\)
							−0.0512189	+	0.998687i	\(0.516311\pi\)
\(37\)	−203.311	+	203.311i	−0.903357	+	0.903357i	−0.995725	−	0.0923681i	\(-0.970556\pi\)
							0.0923681	+	0.995725i	\(0.470556\pi\)
\(41\)	46.9839	−	175.346i	0.178967	−	0.667915i	−0.816875	−	0.576815i	\(-0.804295\pi\)
							0.995842	−	0.0910993i	\(-0.0290381\pi\)
\(43\)	−245.898	−	141.969i	−0.872072	−	0.503491i	−0.00403589	−	0.999992i	\(-0.501285\pi\)
							−0.868036	+	0.496501i	\(0.834618\pi\)
\(47\)	92.2805	−	159.834i	0.286393	−	0.496048i	−0.686553	−	0.727080i	\(-0.740877\pi\)
							0.972946	+	0.231032i	\(0.0742103\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.n.a.106.45	yes	208
9.4	even	3	inner	153.4.n.a.4.8	✓	208
17.13	even	4	inner	153.4.n.a.115.8	yes	208
153.13	even	12	inner	153.4.n.a.13.45	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.n.a.4.8	✓	208	9.4	even	3	inner
153.4.n.a.13.45	yes	208	153.13	even	12	inner
153.4.n.a.106.45	yes	208	1.1	even	1	trivial
153.4.n.a.115.8	yes	208	17.13	even	4	inner