Embedded newform 315.3.bd.a.221.4

• Label: 315.3.bd.a.221.4
• Level: $315$
• Weight: $3$
• Character: 315.221
• Analytic conductor: $8.583$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	315.bd (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			221.4
Character	\(\chi\)	\(=\)	315.221
Dual form			315.3.bd.a.191.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.13136 - 1.80789i) q^{2} +(-2.12443 + 2.11820i) q^{3} +(4.53696 + 7.85824i) q^{4} +2.23607i q^{5} +(10.4819 - 2.79212i) q^{6} +(6.87314 + 1.32662i) q^{7} -18.3462i q^{8} +(0.0264256 - 8.99996i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 128 q^{4} + 8 q^{6} + 2 q^{7} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.13136	−	1.80789i	−1.56568	−	0.903947i	−0.996663	−	0.0816216i	\(-0.973990\pi\)
							−0.569018	−	0.822325i	\(-0.692677\pi\)
\(3\)	−2.12443	+	2.11820i	−0.708144	+	0.706068i
							\(5\)			2.23607i			0.447214i
\(7\)	6.87314	+	1.32662i	0.981877	+	0.189517i
							\(11\)			10.6206i			0.965512i	0.875755	+	0.482756i	\(0.160364\pi\)
−0.875755	+	0.482756i	\(0.839636\pi\)
\(13\)	10.3447	−	17.9175i	0.795743	−	1.37827i	−0.126623	−	0.991951i	\(-0.540414\pi\)
							0.922366	−	0.386317i	\(-0.126253\pi\)
\(17\)	−12.3742	−	7.14424i	−0.727893	−	0.420249i	0.0897577	−	0.995964i	\(-0.471391\pi\)
							−0.817651	+	0.575714i	\(0.804724\pi\)
\(19\)	14.2108	+	24.6137i	0.747934	+	1.29546i	0.948811	+	0.315844i	\(0.102287\pi\)
							−0.200877	+	0.979616i	\(0.564379\pi\)
\(23\)		−	14.7477i		−	0.641203i	−0.947214	−	0.320601i	\(-0.896115\pi\)
							0.947214	−	0.320601i	\(-0.103885\pi\)
\(29\)	29.3278	−	16.9324i	1.01130	−	0.583875i	0.0997292	−	0.995015i	\(-0.468202\pi\)
							0.911573	+	0.411139i	\(0.134869\pi\)
\(31\)	5.14174	+	8.90575i	0.165862	+	0.287282i	0.936961	−	0.349433i	\(-0.113626\pi\)
							−0.771099	+	0.636716i	\(0.780293\pi\)
\(37\)	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	0.456913	−	0.889511i	\(-0.348955\pi\)
							−0.541883	+	0.840454i	\(0.682288\pi\)
\(43\)	−23.5499	−	40.7896i	−0.547671	−	0.948594i	−0.998434	−	0.0559504i	\(-0.982181\pi\)
							0.450762	−	0.892644i	\(-0.351152\pi\)
\(47\)	62.2276	+	35.9271i	1.32399	+	0.764407i	0.984363	−	0.176153i	\(-0.0563652\pi\)
							0.339629	+	0.940560i	\(0.389699\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.3.s.a.11.4	✓	128	9.2	odd	6
315.3.s.a.86.61	yes	128	7.2	even	3
315.3.bd.a.191.4	yes	128	63.2	odd	6	inner
315.3.bd.a.221.4	yes	128	1.1	even	1	trivial