Embedded newform 153.4.n.a.13.45

• Label: 153.4.n.a.13.45
• 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			13.45
Character	\(\chi\)	\(=\)	153.13
Dual form			153.4.n.a.106.45

$q$-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 \(\left(\mathbb{Z}/153\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.n.a.4.8	✓	208	17.4	even	4	inner
153.4.n.a.13.45	yes	208	1.1	even	1	trivial
153.4.n.a.106.45	yes	208	153.106	even	12	inner
153.4.n.a.115.8	yes	208	9.7	even	3	inner