Embedded newform 315.2.z.b.104.1

• Label: 315.2.z.b.104.1
• Level: $315$
• Weight: $2$
• Character: 315.104
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			104.1
Character	\(\chi\)	\(=\)	315.104
Dual form			315.2.z.b.209.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37508 - 2.38171i) q^{2} +(-1.72249 - 0.181752i) q^{3} +(-2.78170 + 4.81804i) q^{4} +(2.23561 + 0.0452693i) q^{5} +(1.93568 + 4.35239i) q^{6} +(0.187702 - 2.63908i) q^{7} +9.79992 q^{8} +(2.93393 + 0.626130i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 52 q^{4} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37508	−	2.38171i	−0.972329	−	1.68412i	−0.688481	−	0.725255i	\(-0.741722\pi\)
							−0.283849	−	0.958869i	\(-0.591611\pi\)
\(3\)	−1.72249	−	0.181752i	−0.994479	−	0.104934i
							\(5\)	2.23561	+	0.0452693i	0.999795	+	0.0202451i
\(7\)	0.187702	−	2.63908i	0.0709448	−	0.997480i
							\(11\)	2.46877	−	1.42534i	0.744361	−	0.429757i	−0.0792919	−	0.996851i	\(-0.525266\pi\)
0.823653	+	0.567095i	\(0.191933\pi\)
\(13\)	−0.851765	+	1.47530i	−0.236237	+	0.409175i	−0.959631	−	0.281260i	\(-0.909248\pi\)
							0.723394	+	0.690435i	\(0.242581\pi\)
\(17\)			1.34695i			0.326682i	0.986570	+	0.163341i	\(0.0522271\pi\)
							−0.986570	+	0.163341i	\(0.947773\pi\)
\(19\)		−	2.61005i		−	0.598786i	−0.954130	−	0.299393i	\(-0.903216\pi\)
							0.954130	−	0.299393i	\(-0.0967842\pi\)
\(23\)	0.725086	−	1.25589i	0.151191	−	0.261870i	−0.780475	−	0.625187i	\(-0.785023\pi\)
							0.931666	+	0.363317i	\(0.118356\pi\)
\(29\)	2.81784	−	1.62688i	0.523259	−	0.302104i	−0.215008	−	0.976612i	\(-0.568978\pi\)
							0.738267	+	0.674509i	\(0.235644\pi\)
\(31\)	−2.82771	−	1.63258i	−0.507872	−	0.293220i	0.224086	−	0.974569i	\(-0.428060\pi\)
							−0.731959	+	0.681349i	\(0.761394\pi\)
\(37\)		−	8.65571i		−	1.42299i	−0.702691	−	0.711495i	\(-0.748018\pi\)
							0.702691	−	0.711495i	\(-0.251982\pi\)
\(41\)	2.53591	−	4.39232i	0.396042	−	0.685966i	−0.597191	−	0.802099i	\(-0.703717\pi\)
							0.993234	+	0.116133i	\(0.0370500\pi\)
\(43\)	−5.87241	+	3.39044i	−0.895534	+	0.517037i	−0.875749	−	0.482768i	\(-0.839632\pi\)
							−0.0197853	+	0.999804i	\(0.506298\pi\)
\(47\)	0.359622	−	0.207628i	0.0524562	−	0.0302856i	−0.473543	−	0.880771i	\(-0.657025\pi\)
							0.525999	+	0.850485i	\(0.323692\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.z.b.104.1	✓	80
3.2	odd	2		945.2.z.b.314.39		80
5.4	even	2	inner	315.2.z.b.104.40	yes	80
7.6	odd	2	inner	315.2.z.b.104.2	yes	80
9.2	odd	6	inner	315.2.z.b.209.39	yes	80
9.7	even	3		945.2.z.b.629.1		80
15.14	odd	2		945.2.z.b.314.2		80
21.20	even	2		945.2.z.b.314.40		80
35.34	odd	2	inner	315.2.z.b.104.39	yes	80
45.29	odd	6	inner	315.2.z.b.209.2	yes	80
45.34	even	6		945.2.z.b.629.40		80
63.20	even	6	inner	315.2.z.b.209.40	yes	80
63.34	odd	6		945.2.z.b.629.2		80
105.104	even	2		945.2.z.b.314.1		80
315.34	odd	6		945.2.z.b.629.39		80
315.209	even	6	inner	315.2.z.b.209.1	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.z.b.104.1	✓	80	1.1	even	1	trivial
315.2.z.b.104.2	yes	80	7.6	odd	2	inner
315.2.z.b.104.39	yes	80	35.34	odd	2	inner
315.2.z.b.104.40	yes	80	5.4	even	2	inner
315.2.z.b.209.1	yes	80	315.209	even	6	inner
315.2.z.b.209.2	yes	80	45.29	odd	6	inner
315.2.z.b.209.39	yes	80	9.2	odd	6	inner
315.2.z.b.209.40	yes	80	63.20	even	6	inner
945.2.z.b.314.1		80	105.104	even	2
945.2.z.b.314.2		80	15.14	odd	2
945.2.z.b.314.39		80	3.2	odd	2
945.2.z.b.314.40		80	21.20	even	2
945.2.z.b.629.1		80	9.7	even	3
945.2.z.b.629.2		80	63.34	odd	6
945.2.z.b.629.39		80	315.34	odd	6
945.2.z.b.629.40		80	45.34	even	6