Embedded newform 403.2.bi.b.40.1

• Label: 403.2.bi.b.40.1
• Level: $403$
• Weight: $2$
• Character: 403.40
• 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			40.1
Character	\(\chi\)	\(=\)	403.40
Dual form			403.2.bi.b.131.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.779971 + 2.40050i) q^{2} +(2.95884 + 0.628922i) q^{3} +(-3.53603 - 2.56908i) q^{4} +(-2.10392 - 3.64409i) q^{5} +(-3.81754 + 6.61218i) q^{6} +(3.44993 - 1.53601i) q^{7} +(4.84111 - 3.51727i) q^{8} +(5.61858 + 2.50155i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 6 q^{2} + 2 q^{3} - 34 q^{4} - 15 q^{5} - 10 q^{6} - 8 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.779971	+	2.40050i	−0.551523	+	1.69741i	0.153430	+	0.988159i	\(0.450968\pi\)
							−0.704953	+	0.709254i	\(0.749032\pi\)
\(3\)	2.95884	+	0.628922i	1.70829	+	0.363108i	0.955463	−	0.295111i	\(-0.0953565\pi\)
							0.752827	+	0.658219i	\(0.228690\pi\)
\(5\)	−2.10392	−	3.64409i	−0.940901	−	1.62969i	−0.763758	−	0.645502i	\(-0.776648\pi\)
							−0.177142	−	0.984185i	\(-0.556685\pi\)
\(7\)	3.44993	−	1.53601i	1.30395	−	0.580556i	0.367066	−	0.930195i	\(-0.380362\pi\)
							0.936884	+	0.349639i	\(0.113696\pi\)
\(11\)	0.135243	−	1.28675i	0.0407774	−	0.387971i	−0.955031	−	0.296506i	\(-0.904178\pi\)
							0.995808	−	0.0914647i	\(-0.0291549\pi\)
\(13\)	0.669131	−	0.743145i	0.185583	−	0.206111i
							\(17\)	0.0842004	+	0.801113i	0.0204216	+	0.194298i	0.999976	−	0.00687935i	\(-0.00218978\pi\)
−0.979555	+	0.201178i	\(0.935523\pi\)
\(19\)	1.75616	+	1.95041i	0.402891	+	0.447456i	0.910113	−	0.414360i	\(-0.135994\pi\)
							−0.507222	+	0.861815i	\(0.669328\pi\)
\(23\)	−4.08606	+	2.96870i	−0.852003	+	0.619016i	−0.925698	−	0.378264i	\(-0.876521\pi\)
							0.0736946	+	0.997281i	\(0.476521\pi\)
\(29\)	0.738687	−	2.27344i	0.137171	−	0.422168i	−0.858751	−	0.512394i	\(-0.828759\pi\)
							0.995921	+	0.0902258i	\(0.0287589\pi\)
\(31\)	−2.02186	−	5.18769i	−0.363137	−	0.931736i
							\(37\)	0.244018	−	0.422651i	0.0401162	−	0.0694833i	−0.845270	−	0.534339i	\(-0.820561\pi\)
0.885386	+	0.464856i	\(0.153894\pi\)
\(41\)	0.965894	−	0.205307i	0.150847	−	0.0320636i	−0.131869	−	0.991267i	\(-0.542098\pi\)
							0.282716	+	0.959204i	\(0.408764\pi\)
\(43\)	5.99166	+	6.65442i	0.913720	+	1.01479i	0.999829	+	0.0185031i	\(0.00589007\pi\)
							−0.0861088	+	0.996286i	\(0.527443\pi\)
\(47\)	2.12261	+	6.53273i	0.309615	+	0.952897i	0.977915	+	0.209004i	\(0.0670223\pi\)
							−0.668300	+	0.743892i	\(0.732978\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bi.b.40.1	✓	136
31.7	even	15	inner	403.2.bi.b.131.1	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bi.b.40.1	✓	136	1.1	even	1	trivial
403.2.bi.b.131.1	yes	136	31.7	even	15	inner