Embedded newform 153.4.n.a.106.50

• Label: 153.4.n.a.106.50
• 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			106.50
Character	\(\chi\)	\(=\)	153.106
Dual form			153.4.n.a.13.50

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.50259 + 2.59957i) q^{2} +(-2.94400 + 4.28169i) q^{3} +(9.51555 + 16.4814i) q^{4} +(7.25109 + 1.94292i) q^{5} +(-24.3862 + 11.6256i) q^{6} +(-10.9556 + 2.93555i) q^{7} +57.3523i q^{8} +(-9.66574 - 25.2106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 6 q^{3} + 396 q^{4} - 2 q^{5} - 40 q^{6} - 2 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.50259	+	2.59957i	1.59191	+	0.919087i	0.992980	+	0.118283i	\(0.0377391\pi\)
							0.598926	+	0.800804i	\(0.295594\pi\)
\(3\)	−2.94400	+	4.28169i	−0.566573	+	0.824012i
							\(5\)	7.25109	+	1.94292i	0.648558	+	0.173780i	0.568077	−	0.822976i	\(-0.307688\pi\)
0.0804809	+	0.996756i	\(0.474354\pi\)
\(7\)	−10.9556	+	2.93555i	−0.591548	+	0.158505i	−0.542161	−	0.840275i	\(-0.682394\pi\)
							−0.0493873	+	0.998780i	\(0.515727\pi\)
\(11\)	24.7817	−	6.64024i	0.679269	−	0.182010i	0.0973427	−	0.995251i	\(-0.468966\pi\)
							0.581927	+	0.813241i	\(0.302299\pi\)
\(13\)	−5.15024	−	8.92047i	−0.109878	−	0.190315i	0.805842	−	0.592130i	\(-0.201713\pi\)
							−0.915721	+	0.401815i	\(0.868379\pi\)
\(17\)	41.0874	+	56.7876i	0.586185	+	0.810177i
							\(19\)		−	76.0178i		−	0.917877i	−0.888468	−	0.458939i	\(-0.848230\pi\)
0.888468	−	0.458939i	\(-0.151770\pi\)
\(23\)	12.2215	−	45.6114i	0.110798	−	0.413505i	−0.888140	−	0.459573i	\(-0.848002\pi\)
							0.998938	+	0.0460678i	\(0.0146690\pi\)
\(29\)	58.6265	+	218.797i	0.375402	+	1.40102i	0.852756	+	0.522309i	\(0.174929\pi\)
							−0.477354	+	0.878711i	\(0.658404\pi\)
\(31\)	−14.9697	−	4.01111i	−0.0867301	−	0.0232393i	0.215193	−	0.976572i	\(-0.430962\pi\)
							−0.301923	+	0.953332i	\(0.597629\pi\)
\(37\)	−47.1722	+	47.1722i	−0.209596	+	0.209596i	−0.804096	−	0.594500i	\(-0.797350\pi\)
							0.594500	+	0.804096i	\(0.297350\pi\)
\(41\)	−6.94021	+	25.9012i	−0.0264361	+	0.0986608i	−0.977883	−	0.209151i	\(-0.932930\pi\)
							0.951447	+	0.307812i	\(0.0995967\pi\)
\(43\)	453.019	+	261.551i	1.60662	+	0.927583i	0.990119	+	0.140231i	\(0.0447845\pi\)
							0.616503	+	0.787353i	\(0.288549\pi\)
\(47\)	295.987	−	512.664i	0.918597	−	1.59106i	0.117050	−	0.993126i	\(-0.462656\pi\)
							0.801547	−	0.597931i	\(-0.204010\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.n.a.106.50	yes	208
9.4	even	3	inner	153.4.n.a.4.3	✓	208
17.13	even	4	inner	153.4.n.a.115.3	yes	208
153.13	even	12	inner	153.4.n.a.13.50	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.n.a.4.3	✓	208	9.4	even	3	inner
153.4.n.a.13.50	yes	208	153.13	even	12	inner
153.4.n.a.106.50	yes	208	1.1	even	1	trivial
153.4.n.a.115.3	yes	208	17.13	even	4	inner