Embedded newform 315.3.s.a.11.4

• Label: 315.3.s.a.11.4
• Level: $315$
• Weight: $3$
• Character: 315.11
• Analytic conductor: $8.583$
• Analytic rank: $0$
• Dimension: $128$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(8.58312832735\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.4
Character	\(\chi\)	\(=\)	315.11
Dual form			315.3.s.a.86.61

$q$-expansion

\(f(q)\)	\(=\)	\(q-3.61579i q^{2} +(-0.772202 + 2.89891i) q^{3} -9.07391 q^{4} +(1.93649 + 1.11803i) q^{5} +(10.4819 + 2.79212i) q^{6} +(-2.28768 - 6.61563i) q^{7} +18.3462i q^{8} +(-7.80741 - 4.47710i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 256 q^{4} + 8 q^{6} + 2 q^{7} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	3.61579i		−	1.80789i	−0.427645	−	0.903947i	\(-0.640657\pi\)
							0.427645	−	0.903947i	\(-0.359343\pi\)
\(3\)	−0.772202	+	2.89891i	−0.257401	+	0.966305i
							\(5\)	1.93649	+	1.11803i	0.387298	+	0.223607i
\(7\)	−2.28768	−	6.61563i	−0.326812	−	0.945089i
							\(11\)	−9.19774	+	5.31032i	−0.836158	+	0.482756i	−0.855956	−	0.517048i	\(-0.827031\pi\)
0.0197983	+	0.999804i	\(0.493698\pi\)
\(13\)	10.3447	+	17.9175i	0.795743	+	1.37827i	0.922366	+	0.386317i	\(0.126253\pi\)
							−0.126623	+	0.991951i	\(0.540414\pi\)
\(17\)	12.3742	+	7.14424i	0.727893	+	0.420249i	0.817651	−	0.575714i	\(-0.195276\pi\)
							−0.0897577	+	0.995964i	\(0.528609\pi\)
\(19\)	14.2108	+	24.6137i	0.747934	+	1.29546i	0.948811	+	0.315844i	\(0.102287\pi\)
							−0.200877	+	0.979616i	\(0.564379\pi\)
\(23\)	−12.7719	−	7.37383i	−0.555298	−	0.320601i	0.195958	−	0.980612i	\(-0.437218\pi\)
							−0.751256	+	0.660011i	\(0.770552\pi\)
\(29\)	29.3278	+	16.9324i	1.01130	+	0.583875i	0.911573	−	0.411139i	\(-0.134869\pi\)
							0.0997292	+	0.995015i	\(0.468202\pi\)
\(31\)	−10.2835			−0.331725			−0.165862	−	0.986149i	\(-0.553041\pi\)
							−0.165862	+	0.986149i	\(0.553041\pi\)
\(37\)	3.55169	+	6.15171i	0.0959916	+	0.166262i	0.910022	−	0.414560i	\(-0.136064\pi\)
							−0.814030	+	0.580822i	\(0.802731\pi\)
\(41\)	−3.48373	+	2.01133i	−0.0849691	+	0.0490569i	−0.541883	−	0.840454i	\(-0.682288\pi\)
							0.456913	+	0.889511i	\(0.348955\pi\)
\(43\)	−23.5499	+	40.7896i	−0.547671	+	0.948594i	0.450762	+	0.892644i	\(0.351152\pi\)
							−0.998434	+	0.0559504i	\(0.982181\pi\)
\(47\)			71.8542i			1.52881i	0.644734	+	0.764407i	\(0.276968\pi\)
							−0.644734	+	0.764407i	\(0.723032\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.3.s.a.11.4	✓	128
7.2	even	3		315.3.bd.a.191.4	yes	128
9.5	odd	6		315.3.bd.a.221.4	yes	128
63.23	odd	6	inner	315.3.s.a.86.61	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.3.s.a.11.4	✓	128	1.1	even	1	trivial
315.3.s.a.86.61	yes	128	63.23	odd	6	inner
315.3.bd.a.191.4	yes	128	7.2	even	3
315.3.bd.a.221.4	yes	128	9.5	odd	6