Embedded newform 315.2.z.b.104.20

• Label: 315.2.z.b.104.20
• 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			104.20
Character	\(\chi\)	\(=\)	315.104
Dual form			315.2.z.b.209.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.139034 - 0.240814i) q^{2} +(1.62845 + 0.590047i) q^{3} +(0.961339 - 1.66509i) q^{4} +(-0.326583 + 2.21209i) q^{5} +(-0.0843182 - 0.474190i) q^{6} +(2.60030 - 0.488328i) q^{7} -1.09077 q^{8} +(2.30369 + 1.92172i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 52 q^{4} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.139034	−	0.240814i	−0.0983120	−	0.170281i	0.812674	−	0.582719i	\(-0.198011\pi\)
							−0.910986	+	0.412437i	\(0.864678\pi\)
\(3\)	1.62845	+	0.590047i	0.940185	+	0.340664i
							\(5\)	−0.326583	+	2.21209i	−0.146052	+	0.989277i
\(7\)	2.60030	−	0.488328i	0.982819	−	0.184570i
							\(11\)	−3.37792	+	1.95024i	−1.01848	+	0.588020i	−0.913663	−	0.406472i	\(-0.866759\pi\)
−0.104817	+	0.994492i	\(0.533426\pi\)
\(13\)	0.562755	−	0.974721i	0.156080	−	0.270339i	−0.777372	−	0.629042i	\(-0.783447\pi\)
							0.933452	+	0.358703i	\(0.116781\pi\)
\(17\)		−	2.20034i		−	0.533661i	−0.963743	−	0.266831i	\(-0.914024\pi\)
							0.963743	−	0.266831i	\(-0.0859764\pi\)
\(19\)			3.75853i			0.862266i	0.902288	+	0.431133i	\(0.141886\pi\)
							−0.902288	+	0.431133i	\(0.858114\pi\)
\(23\)	3.74981	−	6.49487i	0.781890	−	1.35427i	−0.148949	−	0.988845i	\(-0.547589\pi\)
							0.930839	−	0.365429i	\(-0.119078\pi\)
\(29\)	−3.65123	+	2.10804i	−0.678016	+	0.391453i	−0.799107	−	0.601189i	\(-0.794694\pi\)
							0.121091	+	0.992641i	\(0.461361\pi\)
\(31\)	−5.45664	−	3.15039i	−0.980042	−	0.565827i	−0.0777590	−	0.996972i	\(-0.524776\pi\)
							−0.902283	+	0.431145i	\(0.858110\pi\)
\(37\)			4.76824i			0.783894i	0.919988	+	0.391947i	\(0.128198\pi\)
							−0.919988	+	0.391947i	\(0.871802\pi\)
\(41\)	4.00455	−	6.93608i	0.625405	−	1.08323i	−0.363057	−	0.931767i	\(-0.618267\pi\)
							0.988462	−	0.151467i	\(-0.0483996\pi\)
\(43\)	5.00066	−	2.88713i	0.762593	−	0.440283i	−0.0676328	−	0.997710i	\(-0.521545\pi\)
							0.830226	+	0.557427i	\(0.188211\pi\)
\(47\)	−3.16020	+	1.82454i	−0.460962	+	0.266137i	−0.712449	−	0.701724i	\(-0.752414\pi\)
							0.251487	+	0.967861i	\(0.419081\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.z.b.104.20	yes	80
3.2	odd	2		945.2.z.b.314.22		80
5.4	even	2	inner	315.2.z.b.104.21	yes	80
7.6	odd	2	inner	315.2.z.b.104.19	✓	80
9.2	odd	6	inner	315.2.z.b.209.22	yes	80
9.7	even	3		945.2.z.b.629.19		80
15.14	odd	2		945.2.z.b.314.20		80
21.20	even	2		945.2.z.b.314.21		80
35.34	odd	2	inner	315.2.z.b.104.22	yes	80
45.29	odd	6	inner	315.2.z.b.209.19	yes	80
45.34	even	6		945.2.z.b.629.21		80
63.20	even	6	inner	315.2.z.b.209.21	yes	80
63.34	odd	6		945.2.z.b.629.20		80
105.104	even	2		945.2.z.b.314.19		80
315.34	odd	6		945.2.z.b.629.22		80
315.209	even	6	inner	315.2.z.b.209.20	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.z.b.104.19	✓	80	7.6	odd	2	inner
315.2.z.b.104.20	yes	80	1.1	even	1	trivial
315.2.z.b.104.21	yes	80	5.4	even	2	inner
315.2.z.b.104.22	yes	80	35.34	odd	2	inner
315.2.z.b.209.19	yes	80	45.29	odd	6	inner
315.2.z.b.209.20	yes	80	315.209	even	6	inner
315.2.z.b.209.21	yes	80	63.20	even	6	inner
315.2.z.b.209.22	yes	80	9.2	odd	6	inner
945.2.z.b.314.19		80	105.104	even	2
945.2.z.b.314.20		80	15.14	odd	2
945.2.z.b.314.21		80	21.20	even	2
945.2.z.b.314.22		80	3.2	odd	2
945.2.z.b.629.19		80	9.7	even	3
945.2.z.b.629.20		80	63.34	odd	6
945.2.z.b.629.21		80	45.34	even	6
945.2.z.b.629.22		80	315.34	odd	6