Embedded newform 315.2.z.b.104.6

• Label: 315.2.z.b.104.6
• 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			104.6
Character	\(\chi\)	\(=\)	315.104
Dual form			315.2.z.b.209.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18498 - 2.05244i) q^{2} +(0.887757 + 1.48724i) q^{3} +(-1.80833 + 3.13213i) q^{4} +(0.317361 + 2.21343i) q^{5} +(2.00050 - 3.58441i) q^{6} +(-1.79640 - 1.94241i) q^{7} +3.83142 q^{8} +(-1.42378 + 2.64062i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 52 q^{4} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18498	−	2.05244i	−0.837904	−	1.45129i	−0.891644	−	0.452738i	\(-0.850447\pi\)
							0.0537395	−	0.998555i	\(-0.482886\pi\)
\(3\)	0.887757	+	1.48724i	0.512547	+	0.858659i
							\(5\)	0.317361	+	2.21343i	0.141928	+	0.989877i
\(7\)	−1.79640	−	1.94241i	−0.678974	−	0.734162i
							\(11\)	−3.94144	+	2.27559i	−1.18839	+	0.686116i	−0.957940	−	0.286969i	\(-0.907352\pi\)
−0.230448	+	0.973085i	\(0.574019\pi\)
\(13\)	1.57135	−	2.72166i	0.435815	−	0.754854i	−0.561547	−	0.827445i	\(-0.689793\pi\)
							0.997362	+	0.0725912i	\(0.0231268\pi\)
\(17\)			3.56687i			0.865092i	0.901612	+	0.432546i	\(0.142385\pi\)
							−0.901612	+	0.432546i	\(0.857615\pi\)
\(19\)			4.51836i			1.03658i	0.855204	+	0.518291i	\(0.173432\pi\)
							−0.855204	+	0.518291i	\(0.826568\pi\)
\(23\)	−4.14464	+	7.17872i	−0.864217	+	1.49687i	0.00360569	+	0.999993i	\(0.498852\pi\)
							−0.867823	+	0.496874i	\(0.834481\pi\)
\(29\)	2.52016	−	1.45501i	0.467982	−	0.270189i	−0.247413	−	0.968910i	\(-0.579580\pi\)
							0.715394	+	0.698721i	\(0.246247\pi\)
\(31\)	3.60524	+	2.08149i	0.647520	+	0.373846i	0.787506	−	0.616308i	\(-0.211372\pi\)
							−0.139985	+	0.990154i	\(0.544705\pi\)
\(37\)		−	6.75923i		−	1.11121i	−0.831446	−	0.555606i	\(-0.812486\pi\)
							0.831446	−	0.555606i	\(-0.187514\pi\)
\(41\)	1.06535	−	1.84524i	0.166380	−	0.288179i	−0.770764	−	0.637120i	\(-0.780125\pi\)
							0.937144	+	0.348942i	\(0.113459\pi\)
\(43\)	−0.522099	+	0.301434i	−0.0796194	+	0.0459683i	−0.539281	−	0.842126i	\(-0.681304\pi\)
							0.459662	+	0.888094i	\(0.347971\pi\)
\(47\)	4.73507	−	2.73379i	0.690681	−	0.398765i	−0.113186	−	0.993574i	\(-0.536106\pi\)
							0.803867	+	0.594809i	\(0.202772\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.z.b.104.6	yes	80
3.2	odd	2		945.2.z.b.314.35		80
5.4	even	2	inner	315.2.z.b.104.35	yes	80
7.6	odd	2	inner	315.2.z.b.104.5	✓	80
9.2	odd	6	inner	315.2.z.b.209.36	yes	80
9.7	even	3		945.2.z.b.629.5		80
15.14	odd	2		945.2.z.b.314.6		80
21.20	even	2		945.2.z.b.314.36		80
35.34	odd	2	inner	315.2.z.b.104.36	yes	80
45.29	odd	6	inner	315.2.z.b.209.5	yes	80
45.34	even	6		945.2.z.b.629.36		80
63.20	even	6	inner	315.2.z.b.209.35	yes	80
63.34	odd	6		945.2.z.b.629.6		80
105.104	even	2		945.2.z.b.314.5		80
315.34	odd	6		945.2.z.b.629.35		80
315.209	even	6	inner	315.2.z.b.209.6	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.z.b.104.5	✓	80	7.6	odd	2	inner
315.2.z.b.104.6	yes	80	1.1	even	1	trivial
315.2.z.b.104.35	yes	80	5.4	even	2	inner
315.2.z.b.104.36	yes	80	35.34	odd	2	inner
315.2.z.b.209.5	yes	80	45.29	odd	6	inner
315.2.z.b.209.6	yes	80	315.209	even	6	inner
315.2.z.b.209.35	yes	80	63.20	even	6	inner
315.2.z.b.209.36	yes	80	9.2	odd	6	inner
945.2.z.b.314.5		80	105.104	even	2
945.2.z.b.314.6		80	15.14	odd	2
945.2.z.b.314.35		80	3.2	odd	2
945.2.z.b.314.36		80	21.20	even	2
945.2.z.b.629.5		80	9.7	even	3
945.2.z.b.629.6		80	63.34	odd	6
945.2.z.b.629.35		80	315.34	odd	6
945.2.z.b.629.36		80	45.34	even	6