Embedded newform 153.4.n.a.106.13

• Label: 153.4.n.a.106.13
• 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			106.13
Character	\(\chi\)	\(=\)	153.106
Dual form			153.4.n.a.13.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.01007 - 1.73786i) q^{2} +(-1.82835 + 4.86386i) q^{3} +(2.04034 + 3.53398i) q^{4} +(2.60647 + 0.698403i) q^{5} +(13.9562 - 11.4631i) q^{6} +(7.74620 - 2.07559i) q^{7} +13.6225i q^{8} +(-20.3143 - 17.7857i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 6 q^{3} + 396 q^{4} - 2 q^{5} - 40 q^{6} - 2 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.01007	−	1.73786i	−1.06422	−	0.614428i	−0.137623	−	0.990485i	\(-0.543946\pi\)
							−0.926597	+	0.376057i	\(0.877280\pi\)
\(3\)	−1.82835	+	4.86386i	−0.351866	+	0.936050i
							\(5\)	2.60647	+	0.698403i	0.233130	+	0.0624670i	0.373493	−	0.927633i	\(-0.378160\pi\)
−0.140362	+	0.990100i	\(0.544827\pi\)
\(7\)	7.74620	−	2.07559i	0.418255	−	0.112071i	−0.0435526	−	0.999051i	\(-0.513868\pi\)
							0.461808	+	0.886980i	\(0.347201\pi\)
\(11\)	37.8060	−	10.1301i	1.03627	−	0.277667i	0.299701	−	0.954033i	\(-0.403113\pi\)
							0.736565	+	0.676366i	\(0.236446\pi\)
\(13\)	−19.2637	−	33.3657i	−0.410984	−	0.711845i	0.584014	−	0.811744i	\(-0.301481\pi\)
							−0.994998	+	0.0998989i	\(0.968148\pi\)
\(17\)	−47.6145	+	51.4379i	−0.679307	+	0.733854i
							\(19\)			81.9790i			0.989856i	0.868934	+	0.494928i	\(0.164806\pi\)
−0.868934	+	0.494928i	\(0.835194\pi\)
\(23\)	−41.8183	+	156.068i	−0.379118	+	1.41489i	0.468115	+	0.883668i	\(0.344933\pi\)
							−0.847233	+	0.531222i	\(0.821733\pi\)
\(29\)	78.9638	+	294.697i	0.505628	+	1.88703i	0.459677	+	0.888086i	\(0.347965\pi\)
							0.0459509	+	0.998944i	\(0.485368\pi\)
\(31\)	148.968	+	39.9159i	0.863079	+	0.231261i	0.663093	−	0.748537i	\(-0.269244\pi\)
							0.199986	+	0.979799i	\(0.435910\pi\)
\(37\)	−156.738	+	156.738i	−0.696419	+	0.696419i	−0.963636	−	0.267218i	\(-0.913896\pi\)
							0.267218	+	0.963636i	\(0.413896\pi\)
\(41\)	10.1270	−	37.7944i	0.0385748	−	0.143963i	−0.943953	−	0.330081i	\(-0.892924\pi\)
							0.982527	+	0.186118i	\(0.0595906\pi\)
\(43\)	−169.705	−	97.9793i	−0.601855	−	0.347481i	0.167916	−	0.985801i	\(-0.446296\pi\)
							−0.769771	+	0.638320i	\(0.779630\pi\)
\(47\)	−218.266	+	378.048i	−0.677391	+	1.17327i	0.298373	+	0.954449i	\(0.403556\pi\)
							−0.975764	+	0.218826i	\(0.929777\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.n.a.106.13	yes	208
9.4	even	3	inner	153.4.n.a.4.40	✓	208
17.13	even	4	inner	153.4.n.a.115.40	yes	208
153.13	even	12	inner	153.4.n.a.13.13	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.n.a.4.40	✓	208	9.4	even	3	inner
153.4.n.a.13.13	yes	208	153.13	even	12	inner
153.4.n.a.106.13	yes	208	1.1	even	1	trivial
153.4.n.a.115.40	yes	208	17.13	even	4	inner