Embedded newform 152.4.t.a.5.10

• Label: 152.4.t.a.5.10
• Level: $152$
• Weight: $4$
• Character: 152.5
• Analytic conductor: $8.968$
• Analytic rank: $0$
• Dimension: $348$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 152 = 2^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	152.t (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.96829032087\)
Analytic rank:	\(0\)
Dimension:	\(348\)
Relative dimension:	\(58\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			5.10
Character	\(\chi\)	\(=\)	152.5
Dual form			152.4.t.a.61.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.49413 - 1.33392i) q^{2} +(4.10931 + 0.724583i) q^{3} +(4.44133 + 6.65392i) q^{4} +(9.57274 - 11.4084i) q^{5} +(-9.28261 - 7.28868i) q^{6} +(-5.68735 - 9.85077i) q^{7} +(-2.20147 - 22.5201i) q^{8} +(-9.01028 - 3.27947i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 6 q^{2} + 12 q^{4} - 24 q^{6} - 6 q^{7} - 3 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/152\mathbb{Z}\right)^\times\).

\(n\)	\(39\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.49413	−	1.33392i	−0.881807	−	0.471611i
							\(3\)	4.10931	+	0.724583i	0.790837	+	0.139446i	0.554455	−	0.832214i	\(-0.312927\pi\)
0.236383	+	0.971660i	\(0.424038\pi\)
\(5\)	9.57274	−	11.4084i	0.856212	−	1.02039i	−0.143316	−	0.989677i	\(-0.545776\pi\)
							0.999528	−	0.0307170i	\(-0.00977906\pi\)
\(7\)	−5.68735	−	9.85077i	−0.307088	−	0.531892i	0.670636	−	0.741786i	\(-0.266021\pi\)
							−0.977724	+	0.209895i	\(0.932688\pi\)
\(11\)	−7.59327	−	4.38398i	−0.208132	−	0.120165i	0.392311	−	0.919833i	\(-0.371676\pi\)
							−0.600443	+	0.799667i	\(0.705009\pi\)
\(13\)	28.0243	−	4.94144i	0.597888	−	0.105424i	0.133490	−	0.991050i	\(-0.457382\pi\)
							0.464398	+	0.885626i	\(0.346271\pi\)
\(17\)	61.2208	−	22.2825i	0.873425	−	0.317901i	0.133871	−	0.990999i	\(-0.457259\pi\)
							0.739553	+	0.673098i	\(0.235037\pi\)
\(19\)	−72.9262	−	39.2526i	−0.880548	−	0.473956i
							\(23\)	57.3873	−	48.1536i	0.520264	−	0.436553i	−0.344460	−	0.938801i	\(-0.611938\pi\)
0.864724	+	0.502248i	\(0.167493\pi\)
\(29\)	−34.5775	+	95.0009i	−0.221410	+	0.608318i	−0.999811	−	0.0194489i	\(-0.993809\pi\)
							0.778401	+	0.627767i	\(0.216031\pi\)
\(31\)	−26.3768	−	45.6860i	−0.152820	−	0.264692i	0.779443	−	0.626473i	\(-0.215502\pi\)
							−0.932263	+	0.361781i	\(0.882169\pi\)
\(37\)		−	113.127i		−	0.502646i	−0.967903	−	0.251323i	\(-0.919134\pi\)
							0.967903	−	0.251323i	\(-0.0808657\pi\)
\(41\)	52.0756	−	295.335i	0.198362	−	1.12497i	−0.709187	−	0.705020i	\(-0.750938\pi\)
							0.907549	−	0.419946i	\(-0.137951\pi\)
\(43\)	115.810	−	138.016i	0.410716	−	0.489472i	−0.520540	−	0.853837i	\(-0.674269\pi\)
							0.931256	+	0.364365i	\(0.118714\pi\)
\(47\)	147.696	+	53.7568i	0.458375	+	0.166835i	0.560879	−	0.827898i	\(-0.310463\pi\)
							−0.102504	+	0.994733i	\(0.532686\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.4.t.a.5.10	✓	348
8.5	even	2	inner	152.4.t.a.5.24	yes	348
19.4	even	9	inner	152.4.t.a.61.24	yes	348
152.61	even	18	inner	152.4.t.a.61.10	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.t.a.5.10	✓	348	1.1	even	1	trivial
152.4.t.a.5.24	yes	348	8.5	even	2	inner
152.4.t.a.61.10	yes	348	152.61	even	18	inner
152.4.t.a.61.24	yes	348	19.4	even	9	inner