Embedded newform 315.2.z.b.104.9

• Label: 315.2.z.b.104.9
• 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			104.9
Character	\(\chi\)	\(=\)	315.104
Dual form			315.2.z.b.209.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.938003 - 1.62467i) q^{2} +(-1.62365 - 0.603134i) q^{3} +(-0.759700 + 1.31584i) q^{4} +(-1.09295 + 1.95076i) q^{5} +(0.543094 + 3.20363i) q^{6} +(2.27915 + 1.34367i) q^{7} -0.901609 q^{8} +(2.27246 + 1.95855i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 52 q^{4} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.938003	−	1.62467i	−0.663268	−	1.14881i	−0.979752	−	0.200217i	\(-0.935835\pi\)
							0.316483	−	0.948598i	\(-0.397498\pi\)
\(3\)	−1.62365	−	0.603134i	−0.937413	−	0.348219i
							\(5\)	−1.09295	+	1.95076i	−0.488781	+	0.872406i
\(7\)	2.27915	+	1.34367i	0.861439	+	0.507861i
							\(11\)	1.72215	−	0.994281i	0.519246	−	0.299787i	−0.217380	−	0.976087i	\(-0.569751\pi\)
0.736626	+	0.676300i	\(0.236418\pi\)
\(13\)	2.28487	−	3.95751i	0.633709	−	1.09762i	−0.353078	−	0.935594i	\(-0.614865\pi\)
							0.986787	−	0.162022i	\(-0.0518016\pi\)
\(17\)			6.36210i			1.54304i	0.636207	+	0.771519i	\(0.280502\pi\)
							−0.636207	+	0.771519i	\(0.719498\pi\)
\(19\)		−	2.43428i		−	0.558463i	−0.960224	−	0.279231i	\(-0.909920\pi\)
							0.960224	−	0.279231i	\(-0.0900796\pi\)
\(23\)	0.00774549	−	0.0134156i	0.00161505	−	0.00279734i	−0.865217	−	0.501398i	\(-0.832819\pi\)
							0.866832	+	0.498601i	\(0.166153\pi\)
\(29\)	4.97858	−	2.87439i	0.924500	−	0.533760i	0.0394318	−	0.999222i	\(-0.487445\pi\)
							0.885068	+	0.465462i	\(0.154112\pi\)
\(31\)	4.68768	+	2.70644i	0.841933	+	0.486090i	0.857921	−	0.513782i	\(-0.171756\pi\)
							−0.0159878	+	0.999872i	\(0.505089\pi\)
\(37\)			11.1208i			1.82824i	0.405442	+	0.914121i	\(0.367118\pi\)
							−0.405442	+	0.914121i	\(0.632882\pi\)
\(41\)	3.35005	−	5.80246i	0.523190	−	0.906192i	−0.476446	−	0.879204i	\(-0.658075\pi\)
							0.999636	−	0.0269879i	\(-0.00859156\pi\)
\(43\)	7.95328	−	4.59183i	1.21286	−	0.700247i	0.249482	−	0.968379i	\(-0.419740\pi\)
							0.963382	+	0.268132i	\(0.0864063\pi\)
\(47\)	4.89285	−	2.82489i	0.713696	−	0.412052i	−0.0987323	−	0.995114i	\(-0.531479\pi\)
							0.812428	+	0.583062i	\(0.198145\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.z.b.104.9	✓	80
3.2	odd	2		945.2.z.b.314.32		80
5.4	even	2	inner	315.2.z.b.104.32	yes	80
7.6	odd	2	inner	315.2.z.b.104.10	yes	80
9.2	odd	6	inner	315.2.z.b.209.31	yes	80
9.7	even	3		945.2.z.b.629.9		80
15.14	odd	2		945.2.z.b.314.10		80
21.20	even	2		945.2.z.b.314.31		80
35.34	odd	2	inner	315.2.z.b.104.31	yes	80
45.29	odd	6	inner	315.2.z.b.209.10	yes	80
45.34	even	6		945.2.z.b.629.31		80
63.20	even	6	inner	315.2.z.b.209.32	yes	80
63.34	odd	6		945.2.z.b.629.10		80
105.104	even	2		945.2.z.b.314.9		80
315.34	odd	6		945.2.z.b.629.32		80
315.209	even	6	inner	315.2.z.b.209.9	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.z.b.104.9	✓	80	1.1	even	1	trivial
315.2.z.b.104.10	yes	80	7.6	odd	2	inner
315.2.z.b.104.31	yes	80	35.34	odd	2	inner
315.2.z.b.104.32	yes	80	5.4	even	2	inner
315.2.z.b.209.9	yes	80	315.209	even	6	inner
315.2.z.b.209.10	yes	80	45.29	odd	6	inner
315.2.z.b.209.31	yes	80	9.2	odd	6	inner
315.2.z.b.209.32	yes	80	63.20	even	6	inner
945.2.z.b.314.9		80	105.104	even	2
945.2.z.b.314.10		80	15.14	odd	2
945.2.z.b.314.31		80	21.20	even	2
945.2.z.b.314.32		80	3.2	odd	2
945.2.z.b.629.9		80	9.7	even	3
945.2.z.b.629.10		80	63.34	odd	6
945.2.z.b.629.31		80	45.34	even	6
945.2.z.b.629.32		80	315.34	odd	6