Embedded newform 152.4.p.a.45.9

• Label: 152.4.p.a.45.9
• 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			45.9
Character	\(\chi\)	\(=\)	152.45
Dual form			152.4.p.a.125.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.55889 + 1.20502i) q^{2} +(-4.74696 + 2.74066i) q^{3} +(5.09583 - 6.16705i) q^{4} +(2.75286 - 1.58936i) q^{5} +(8.84438 - 12.7332i) q^{6} +6.39182 q^{7} +(-5.60823 + 21.9214i) q^{8} +(1.52239 - 2.63686i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - q^{2} - 7 q^{4} - 11 q^{6} - 8 q^{7} - 46 q^{8} + 484 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.55889	+	1.20502i	−0.904704	+	0.426040i
							\(3\)	−4.74696	+	2.74066i	−0.913552	+	0.527440i	−0.881572	−	0.472049i	\(-0.843515\pi\)
−0.0319798	+	0.999489i	\(0.510181\pi\)
\(5\)	2.75286	−	1.58936i	0.246223	−	0.142157i	−0.371811	−	0.928309i	\(-0.621263\pi\)
							0.618034	+	0.786152i	\(0.287930\pi\)
\(7\)	6.39182			0.345126			0.172563	−	0.984999i	\(-0.444795\pi\)
							0.172563	+	0.984999i	\(0.444795\pi\)
\(11\)			61.8197i			1.69448i	0.531206	+	0.847242i	\(0.321739\pi\)
							−0.531206	+	0.847242i	\(0.678261\pi\)
\(13\)	−48.9939	−	28.2866i	−1.04527	−	0.603485i	−0.123946	−	0.992289i	\(-0.539555\pi\)
							−0.921320	+	0.388804i	\(0.872888\pi\)
\(17\)	−3.16069	−	5.47448i	−0.0450929	−	0.0781033i	0.842598	−	0.538543i	\(-0.181025\pi\)
							−0.887691	+	0.460440i	\(0.847692\pi\)
\(19\)	50.2899	−	65.8022i	0.607226	−	0.794529i
							\(23\)	36.5338	−	63.2785i	0.331210	−	0.573673i	−0.651539	−	0.758615i	\(-0.725876\pi\)
0.982749	+	0.184942i	\(0.0592098\pi\)
\(29\)	−226.243	−	130.622i	−1.44870	−	0.836408i	−0.450297	−	0.892879i	\(-0.648682\pi\)
							−0.998404	+	0.0564711i	\(0.982015\pi\)
\(31\)	−112.649			−0.652654			−0.326327	−	0.945257i	\(-0.605811\pi\)
							−0.326327	+	0.945257i	\(0.605811\pi\)
\(37\)		−	227.883i		−	1.01253i	−0.862378	−	0.506266i	\(-0.831026\pi\)
							0.862378	−	0.506266i	\(-0.168974\pi\)
\(41\)	−25.3562	−	43.9183i	−0.0965848	−	0.167290i	0.813684	−	0.581307i	\(-0.197459\pi\)
							−0.910269	+	0.414017i	\(0.864125\pi\)
\(43\)	−407.779	+	235.431i	−1.44618	+	0.834953i	−0.998251	−	0.0591165i	\(-0.981172\pi\)
							−0.447929	+	0.894069i	\(0.647838\pi\)
\(47\)	67.6767	−	117.219i	0.210035	−	0.363792i	−0.741690	−	0.670743i	\(-0.765975\pi\)
							0.951725	+	0.306951i	\(0.0993088\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.4.p.a.45.9	✓	116
8.5	even	2	inner	152.4.p.a.45.48	yes	116
19.11	even	3	inner	152.4.p.a.125.48	yes	116
152.125	even	6	inner	152.4.p.a.125.9	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.p.a.45.9	✓	116	1.1	even	1	trivial
152.4.p.a.45.48	yes	116	8.5	even	2	inner
152.4.p.a.125.9	yes	116	152.125	even	6	inner
152.4.p.a.125.48	yes	116	19.11	even	3	inner