Embedded newform 671.2.bp.a.137.2

• Label: 671.2.bp.a.137.2
• Level: $671$
• Weight: $2$
• Character: 671.137
• Analytic conductor: $5.358$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 671 = 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	671.bp (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.35796197563\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(60\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			137.2
Character	\(\chi\)	\(=\)	671.137
Dual form			671.2.bp.a.240.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.43783 + 1.08539i) q^{2} -2.39080 q^{3} +(3.42669 - 3.80572i) q^{4} +(-2.00788 + 0.893966i) q^{5} +(5.82838 - 2.59496i) q^{6} +(-0.136549 + 0.236510i) q^{7} +(-2.57374 + 7.92116i) q^{8} +2.71594 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 8 q^{2} - 12 q^{3} + 50 q^{4} - 6 q^{5} - 14 q^{6} - q^{7} + 436 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/671\mathbb{Z}\right)^\times\).

\(n\)	\(123\)	\(551\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{7}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.43783	+	1.08539i	−1.72381	+	0.767489i	−0.727090	+	0.686543i	\(0.759128\pi\)
							−0.996718	+	0.0809462i	\(0.974206\pi\)
\(3\)	−2.39080			−1.38033			−0.690166	−	0.723651i	\(-0.742462\pi\)
							−0.690166	+	0.723651i	\(0.742462\pi\)
\(5\)	−2.00788	+	0.893966i	−0.897952	+	0.399794i	−0.803202	−	0.595707i	\(-0.796872\pi\)
							−0.0947500	+	0.995501i	\(0.530205\pi\)
\(7\)	−0.136549	+	0.236510i	−0.0516108	+	0.0893925i	−0.890677	−	0.454637i	\(-0.849769\pi\)
							0.839066	+	0.544030i	\(0.183102\pi\)
\(11\)	−0.989217	−	3.16567i	−0.298260	−	0.954485i
							\(13\)	2.52804	+	0.537352i	0.701153	+	0.149035i	0.544676	−	0.838647i	\(-0.316653\pi\)
0.156478	+	0.987682i	\(0.449986\pi\)
\(17\)	−0.661861	+	6.29718i	−0.160525	+	1.52729i	0.556855	+	0.830610i	\(0.312008\pi\)
							−0.717380	+	0.696682i	\(0.754659\pi\)
\(19\)	2.39793	+	4.15334i	0.550123	+	0.952840i	0.998265	+	0.0588783i	\(0.0187524\pi\)
							−0.448142	+	0.893962i	\(0.647914\pi\)
\(23\)	−1.06951	+	3.29162i	−0.223009	+	0.686351i	0.775479	+	0.631374i	\(0.217509\pi\)
							−0.998488	+	0.0549773i	\(0.982491\pi\)
\(29\)	−2.88367	−	1.28389i	−0.535485	−	0.238413i	0.121134	−	0.992636i	\(-0.461347\pi\)
							−0.656618	+	0.754223i	\(0.728014\pi\)
\(31\)	2.49021	−	0.529310i	0.447255	−	0.0950669i	0.0212221	−	0.999775i	\(-0.493244\pi\)
							0.426032	+	0.904708i	\(0.359911\pi\)
\(37\)	−8.56021			−1.40729			−0.703645	−	0.710552i	\(-0.748446\pi\)
							−0.703645	+	0.710552i	\(0.748446\pi\)
\(41\)	1.91438	+	5.89186i	0.298976	+	0.920153i	0.981857	+	0.189624i	\(0.0607267\pi\)
							−0.682881	+	0.730530i	\(0.739273\pi\)
\(43\)	−0.617922	−	0.131343i	−0.0942322	−	0.0200297i	0.160554	−	0.987027i	\(-0.448672\pi\)
							−0.254786	+	0.966997i	\(0.582005\pi\)
\(47\)	0.490907	−	4.67067i	0.0716062	−	0.681288i	−0.898561	−	0.438849i	\(-0.855386\pi\)
							0.970167	−	0.242438i	\(-0.0779472\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.bp.a.137.2	yes	480
11.9	even	5		671.2.bl.a.625.59	yes	480
61.57	even	15		671.2.bl.a.423.59	✓	480
671.240	even	15	inner	671.2.bp.a.240.2	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.bl.a.423.59	✓	480	61.57	even	15
671.2.bl.a.625.59	yes	480	11.9	even	5
671.2.bp.a.137.2	yes	480	1.1	even	1	trivial
671.2.bp.a.240.2	yes	480	671.240	even	15	inner