Embedded newform 403.2.bi.b.14.10

• Label: 403.2.bi.b.14.10
• Level: $403$
• Weight: $2$
• Character: 403.14
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bi (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(17\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			14.10
Character	\(\chi\)	\(=\)	403.14
Dual form			403.2.bi.b.144.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.201597 - 0.620451i) q^{2} +(-0.733986 + 0.815174i) q^{3} +(1.27372 + 0.925409i) q^{4} +(0.554991 - 0.961272i) q^{5} +(0.357806 + 0.619739i) q^{6} +(-0.401317 + 3.81828i) q^{7} +(1.88652 - 1.37064i) q^{8} +(0.187812 + 1.78691i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 6 q^{2} + 2 q^{3} - 34 q^{4} - 15 q^{5} - 10 q^{6} - 8 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{11}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.201597	−	0.620451i	0.142550	−	0.438725i	−0.854137	−	0.520047i	\(-0.825914\pi\)
							0.996688	+	0.0813222i	\(0.0259143\pi\)
\(3\)	−0.733986	+	0.815174i	−0.423767	+	0.470641i	−0.916787	−	0.399377i	\(-0.869226\pi\)
							0.493020	+	0.870018i	\(0.335893\pi\)
\(5\)	0.554991	−	0.961272i	0.248199	−	0.429894i	−0.714827	−	0.699301i	\(-0.753495\pi\)
							0.963026	+	0.269407i	\(0.0868278\pi\)
\(7\)	−0.401317	+	3.81828i	−0.151684	+	1.44317i	0.608546	+	0.793519i	\(0.291753\pi\)
							−0.760230	+	0.649655i	\(0.774914\pi\)
\(11\)	−3.37044	+	1.50062i	−1.01623	+	0.452453i	−0.846131	−	0.532975i	\(-0.821074\pi\)
							−0.170095	+	0.985428i	\(0.554407\pi\)
\(13\)	−0.978148	−	0.207912i	−0.271289	−	0.0576643i
							\(17\)	−4.94022	−	2.19953i	−1.19818	−	0.533464i	−0.292025	−	0.956411i	\(-0.594329\pi\)
−0.906154	+	0.422947i	\(0.860996\pi\)
\(19\)	4.83877	−	1.02851i	1.11009	−	0.235957i	0.383848	−	0.923396i	\(-0.374599\pi\)
							0.726242	+	0.687439i	\(0.241265\pi\)
\(23\)	4.24248	−	3.08234i	0.884618	−	0.642713i	−0.0498510	−	0.998757i	\(-0.515875\pi\)
							0.934469	+	0.356044i	\(0.115875\pi\)
\(29\)	−2.22442	+	6.84606i	−0.413065	+	1.27128i	0.500906	+	0.865502i	\(0.333000\pi\)
							−0.913971	+	0.405781i	\(0.867000\pi\)
\(31\)	5.55305	+	0.404533i	0.997357	+	0.0726562i
							\(37\)	2.67380	+	4.63116i	0.439570	+	0.761358i	0.997656	−	0.0684254i	\(-0.0217975\pi\)
−0.558086	+	0.829783i	\(0.688464\pi\)
\(41\)	−3.52991	−	3.92036i	−0.551279	−	0.612257i	0.401523	−	0.915849i	\(-0.368481\pi\)
							−0.952802	+	0.303591i	\(0.901814\pi\)
\(43\)	9.31829	−	1.98066i	1.42103	−	0.302048i	0.567618	−	0.823292i	\(-0.307865\pi\)
							0.853408	+	0.521244i	\(0.174532\pi\)
\(47\)	0.430826	+	1.32595i	0.0628424	+	0.193409i	0.977548	−	0.210711i	\(-0.0675778\pi\)
							−0.914706	+	0.404120i	\(0.867578\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bi.b.14.10	✓	136
31.20	even	15	inner	403.2.bi.b.144.10	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bi.b.14.10	✓	136	1.1	even	1	trivial
403.2.bi.b.144.10	yes	136	31.20	even	15	inner