Embedded newform 152.4.p.a.45.15

• Label: 152.4.p.a.45.15
• Level: $152$
• Weight: $4$
• Character: 152.45
• Analytic conductor: $8.968$
• Analytic rank: $0$
• Dimension: $116$
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(8.96829032087\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			45.15
Character	\(\chi\)	\(=\)	152.45
Dual form			152.4.p.a.125.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.95037 - 2.04843i) q^{2} +(-0.706252 + 0.407755i) q^{3} +(-0.392134 + 7.99038i) q^{4} +(-12.8180 + 7.40046i) q^{5} +(2.21271 + 0.651437i) q^{6} +26.8247 q^{7} +(17.1326 - 14.7809i) q^{8} +(-13.1675 + 22.8067i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - q^{2} - 7 q^{4} - 11 q^{6} - 8 q^{7} - 46 q^{8} + 484 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.95037	−	2.04843i	−0.689559	−	0.724230i
							\(3\)	−0.706252	+	0.407755i	−0.135918	+	0.0784725i	−0.566417	−	0.824119i	\(-0.691671\pi\)
0.430499	+	0.902591i	\(0.358338\pi\)
\(5\)	−12.8180	+	7.40046i	−1.14647	+	0.661917i	−0.948026	−	0.318194i	\(-0.896924\pi\)
							−0.198449	+	0.980111i	\(0.563590\pi\)
\(7\)	26.8247			1.44840			0.724198	−	0.689592i	\(-0.242210\pi\)
							0.724198	+	0.689592i	\(0.242210\pi\)
\(11\)		−	49.8833i		−	1.36731i	−0.729807	−	0.683654i	\(-0.760390\pi\)
							0.729807	−	0.683654i	\(-0.239610\pi\)
\(13\)	−38.4561	−	22.2026i	−0.820446	−	0.473685i	0.0301244	−	0.999546i	\(-0.490410\pi\)
							−0.850570	+	0.525862i	\(0.823743\pi\)
\(17\)	−17.0158	−	29.4722i	−0.242761	−	0.420474i	0.718739	−	0.695280i	\(-0.244720\pi\)
							−0.961500	+	0.274806i	\(0.911386\pi\)
\(19\)	−47.2126	+	68.0439i	−0.570069	+	0.821596i
							\(23\)	92.5634	−	160.324i	0.839165	−	1.45348i	−0.0514285	−	0.998677i	\(-0.516377\pi\)
0.890594	−	0.454800i	\(-0.150289\pi\)
\(29\)	−18.4731	−	10.6654i	−0.118289	−	0.0682939i	0.439688	−	0.898150i	\(-0.355089\pi\)
							−0.557977	+	0.829856i	\(0.688422\pi\)
\(31\)	−336.831			−1.95151			−0.975753	−	0.218876i	\(-0.929761\pi\)
							−0.975753	+	0.218876i	\(0.929761\pi\)
\(37\)		−	330.438i		−	1.46821i	−0.679036	−	0.734105i	\(-0.737602\pi\)
							0.679036	−	0.734105i	\(-0.262398\pi\)
\(41\)	−86.7691	−	150.289i	−0.330514	−	0.572467i	0.652099	−	0.758134i	\(-0.273889\pi\)
							−0.982613	+	0.185667i	\(0.940555\pi\)
\(43\)	−84.5716	+	48.8274i	−0.299931	+	0.173165i	−0.642412	−	0.766359i	\(-0.722066\pi\)
							0.342481	+	0.939525i	\(0.388733\pi\)
\(47\)	156.165	−	270.486i	0.484660	−	0.839456i	−0.515184	−	0.857079i	\(-0.672277\pi\)
							0.999845	+	0.0176230i	\(0.00560988\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.4.p.a.45.15	✓	116
8.5	even	2	inner	152.4.p.a.45.24	yes	116
19.11	even	3	inner	152.4.p.a.125.24	yes	116
152.125	even	6	inner	152.4.p.a.125.15	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.p.a.45.15	✓	116	1.1	even	1	trivial
152.4.p.a.45.24	yes	116	8.5	even	2	inner
152.4.p.a.125.15	yes	116	152.125	even	6	inner
152.4.p.a.125.24	yes	116	19.11	even	3	inner