Embedded newform 403.2.be.c.57.15

• Label: 403.2.be.c.57.15
• Level: $403$
• Weight: $2$
• Character: 403.57
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.be (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(34\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			57.15
Character	\(\chi\)	\(=\)	403.57
Dual form			403.2.be.c.99.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.456690 + 0.456690i) q^{2} +(-0.104092 - 0.0600978i) q^{3} +1.58287i q^{4} +(-2.27787 - 0.610354i) q^{5} +(0.0749841 - 0.0200919i) q^{6} +(0.192857 - 0.0516759i) q^{7} +(-1.63626 - 1.63626i) q^{8} +(-1.49278 - 2.58556i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 4 q^{2} - 24 q^{3} + 2 q^{5} - 36 q^{6} - 2 q^{7} - 8 q^{8} + 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.456690	+	0.456690i	−0.322929	+	0.322929i	−0.849890	−	0.526961i	\(-0.823331\pi\)
							0.526961	+	0.849890i	\(0.323331\pi\)
\(3\)	−0.104092	−	0.0600978i	−0.0600978	−	0.0346975i	0.469650	−	0.882853i	\(-0.344380\pi\)
							−0.529748	+	0.848155i	\(0.677713\pi\)
\(5\)	−2.27787	−	0.610354i	−1.01870	−	0.272959i	−0.289437	−	0.957197i	\(-0.593468\pi\)
							−0.729258	+	0.684238i	\(0.760135\pi\)
\(7\)	0.192857	−	0.0516759i	0.0728931	−	0.0195316i	−0.222188	−	0.975004i	\(-0.571320\pi\)
							0.295081	+	0.955472i	\(0.404653\pi\)
\(11\)	1.82540	+	0.489114i	0.550378	+	0.147473i	0.523282	−	0.852159i	\(-0.324707\pi\)
							0.0270960	+	0.999633i	\(0.491374\pi\)
\(13\)	−3.59234	−	0.308363i	−0.996336	−	0.0855246i
							\(17\)	1.39894	−	2.42303i	0.339293	−	0.587672i	−0.645007	−	0.764177i	\(-0.723146\pi\)
0.984300	+	0.176504i	\(0.0564789\pi\)
\(19\)	−1.43002	−	5.33690i	−0.328069	−	1.22437i	−0.911191	−	0.411985i	\(-0.864836\pi\)
							0.583122	−	0.812385i	\(-0.301831\pi\)
\(23\)	3.33646			0.695699			0.347850	−	0.937550i	\(-0.386912\pi\)
							0.347850	+	0.937550i	\(0.386912\pi\)
\(29\)		−	4.13866i		−	0.768531i	−0.923223	−	0.384265i	\(-0.874455\pi\)
							0.923223	−	0.384265i	\(-0.125545\pi\)
\(31\)	−0.888570	+	5.49640i	−0.159592	+	0.987183i
							\(37\)	−1.75787	−	6.56044i	−0.288991	−	1.07853i	−0.945873	−	0.324537i	\(-0.894792\pi\)
0.656882	−	0.753994i	\(-0.271875\pi\)
\(41\)	−9.65163	−	2.58615i	−1.50733	−	0.403888i	−0.591784	−	0.806097i	\(-0.701576\pi\)
							−0.915548	+	0.402208i	\(0.868243\pi\)
\(43\)	−5.53621	+	9.58899i	−0.844264	+	1.46231i	0.0419958	+	0.999118i	\(0.486628\pi\)
							−0.886259	+	0.463189i	\(0.846705\pi\)
\(47\)	−6.37047	−	6.37047i	−0.929229	−	0.929229i	0.0684273	−	0.997656i	\(-0.478202\pi\)
							−0.997656	+	0.0684273i	\(0.978202\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.be.c.57.15	✓	136
13.8	odd	4	inner	403.2.be.c.398.15	yes	136
31.6	odd	6	inner	403.2.be.c.161.15	yes	136
403.99	even	12	inner	403.2.be.c.99.15	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.be.c.57.15	✓	136	1.1	even	1	trivial
403.2.be.c.99.15	yes	136	403.99	even	12	inner
403.2.be.c.161.15	yes	136	31.6	odd	6	inner
403.2.be.c.398.15	yes	136	13.8	odd	4	inner