Embedded newform 315.2.bo.b.4.3

• Label: 315.2.bo.b.4.3
• Level: $315$
• Weight: $2$
• Character: 315.4
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.bo (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(42\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4.3
Character	\(\chi\)	\(=\)	315.4
Dual form			315.2.bo.b.79.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.26367 - 1.30693i) q^{2} +(1.21834 + 1.23112i) q^{3} +(2.41613 + 4.18486i) q^{4} +(-1.53331 - 1.62756i) q^{5} +(-1.14893 - 4.37913i) q^{6} +(-2.64512 + 0.0579738i) q^{7} -7.40313i q^{8} +(-0.0313079 + 2.99984i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q + 44 q^{4} - 6 q^{5} + 6 q^{6} - 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.26367	−	1.30693i	−1.60066	−	0.924139i	−0.991356	−	0.131199i	\(-0.958117\pi\)
							−0.609299	−	0.792940i	\(-0.708549\pi\)
\(3\)	1.21834	+	1.23112i	0.703407	+	0.710787i
							\(5\)	−1.53331	−	1.62756i	−0.685717	−	0.727868i
\(7\)	−2.64512	+	0.0579738i	−0.999760	+	0.0219120i
							\(11\)	−0.0786880			−0.0237253			−0.0118627	−	0.999930i	\(-0.503776\pi\)
−0.0118627	+	0.999930i	\(0.503776\pi\)
\(13\)	−0.00542028	−	0.00312940i	−0.00150332	−	0.000867940i	0.499248	−	0.866459i	\(-0.333610\pi\)
							−0.500751	+	0.865591i	\(0.666943\pi\)
\(17\)	−4.87614	−	2.81524i	−1.18264	−	0.682796i	−0.226013	−	0.974124i	\(-0.572569\pi\)
							−0.956623	+	0.291329i	\(0.905903\pi\)
\(19\)	−3.83039	−	6.63444i	−0.878752	−	1.52204i	−0.852711	−	0.522383i	\(-0.825043\pi\)
							−0.0260415	−	0.999661i	\(-0.508290\pi\)
\(23\)			2.19768i			0.458248i	0.973397	+	0.229124i	\(0.0735861\pi\)
							−0.973397	+	0.229124i	\(0.926414\pi\)
\(29\)	−3.13943	−	5.43765i	−0.582978	−	1.00975i	−0.995124	−	0.0986293i	\(-0.968554\pi\)
							0.412147	−	0.911118i	\(-0.364779\pi\)
\(31\)	−1.85768	−	3.21759i	−0.333649	−	0.577897i	0.649576	−	0.760297i	\(-0.274947\pi\)
							−0.983224	+	0.182400i	\(0.941613\pi\)
\(37\)	2.76672	−	1.59737i	0.454846	−	0.262605i	−0.255029	−	0.966933i	\(-0.582085\pi\)
							0.709875	+	0.704328i	\(0.248752\pi\)
\(41\)	−5.37251	+	9.30547i	−0.839046	+	1.45327i	0.0516475	+	0.998665i	\(0.483553\pi\)
							−0.890693	+	0.454605i	\(0.849781\pi\)
\(43\)	−7.17523	+	4.14262i	−1.09421	+	0.631744i	−0.934695	−	0.355451i	\(-0.884327\pi\)
							−0.159518	+	0.987195i	\(0.550994\pi\)
\(47\)	−2.47765	−	1.43047i	−0.361402	−	0.208656i	0.308293	−	0.951291i	\(-0.400242\pi\)
							−0.669696	+	0.742636i	\(0.733575\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bo.b.4.3	yes	84
3.2	odd	2		945.2.bo.b.739.40		84
5.4	even	2	inner	315.2.bo.b.4.40	yes	84
7.2	even	3		315.2.r.b.184.40	yes	84
9.2	odd	6		945.2.r.b.424.3		84
9.7	even	3		315.2.r.b.214.40	yes	84
15.14	odd	2		945.2.bo.b.739.3		84
21.2	odd	6		945.2.r.b.604.3		84
35.9	even	6		315.2.r.b.184.3	✓	84
45.29	odd	6		945.2.r.b.424.40		84
45.34	even	6		315.2.r.b.214.3	yes	84
63.2	odd	6		945.2.bo.b.289.3		84
63.16	even	3	inner	315.2.bo.b.79.40	yes	84
105.44	odd	6		945.2.r.b.604.40		84
315.79	even	6	inner	315.2.bo.b.79.3	yes	84
315.254	odd	6		945.2.bo.b.289.40		84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.r.b.184.3	✓	84	35.9	even	6
315.2.r.b.184.40	yes	84	7.2	even	3
315.2.r.b.214.3	yes	84	45.34	even	6
315.2.r.b.214.40	yes	84	9.7	even	3
315.2.bo.b.4.3	yes	84	1.1	even	1	trivial
315.2.bo.b.4.40	yes	84	5.4	even	2	inner
315.2.bo.b.79.3	yes	84	315.79	even	6	inner
315.2.bo.b.79.40	yes	84	63.16	even	3	inner
945.2.r.b.424.3		84	9.2	odd	6
945.2.r.b.424.40		84	45.29	odd	6
945.2.r.b.604.3		84	21.2	odd	6
945.2.r.b.604.40		84	105.44	odd	6
945.2.bo.b.289.3		84	63.2	odd	6
945.2.bo.b.289.40		84	315.254	odd	6
945.2.bo.b.739.3		84	15.14	odd	2
945.2.bo.b.739.40		84	3.2	odd	2