Embedded newform 308.2.v.a.127.6

• Label: 308.2.v.a.127.6
• Level: $308$
• Weight: $2$
• Character: 308.127
• Analytic conductor: $2.459$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	308.v (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.45939238226\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			127.6
Character	\(\chi\)	\(=\)	308.127
Dual form			308.2.v.a.211.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.732938 + 1.20946i) q^{2} +(-1.80272 + 2.48123i) q^{3} +(-0.925604 - 1.77292i) q^{4} +(0.791257 - 2.43524i) q^{5} +(-1.67968 - 3.99891i) q^{6} +(0.809017 - 0.587785i) q^{7} +(2.82270 + 0.179958i) q^{8} +(-1.97966 - 6.09276i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - q^{2} - q^{4} + 18 q^{7} - 7 q^{8} + 22 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/308\mathbb{Z}\right)^\times\).

\(n\)	\(45\)	\(57\)	\(155\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{10}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.732938	+	1.20946i	−0.518265	+	0.855220i
							\(3\)	−1.80272	+	2.48123i	−1.04080	+	1.43254i	−0.144281	+	0.989537i	\(0.546087\pi\)
−0.896520	+	0.443003i	\(0.853913\pi\)
\(5\)	0.791257	−	2.43524i	0.353861	−	1.08907i	−0.602806	−	0.797888i	\(-0.705951\pi\)
							0.956667	−	0.291185i	\(-0.0940494\pi\)
\(7\)	0.809017	−	0.587785i	0.305780	−	0.222162i
							\(11\)	2.69637	−	1.93121i	0.812986	−	0.582283i
\(13\)	2.73482	−	0.888596i	0.758502	−	0.246452i								0.0958665	−	0.995394i	\(-0.469438\pi\)
							0.662636	+	0.748942i	\(0.269438\pi\)
\(17\)	−1.73219	−	0.562823i	−0.420118	−	0.136505i	0.0913262	−	0.995821i	\(-0.470889\pi\)
							−0.511444	+	0.859316i	\(0.670889\pi\)
\(19\)	−6.35256	−	4.61541i	−1.45738	−	1.05885i	−0.984036	−	0.177968i	\(-0.943048\pi\)
							−0.473341	−	0.880879i	\(-0.656952\pi\)
\(23\)			0.805676i			0.167995i	0.996466	+	0.0839976i	\(0.0267688\pi\)
							−0.996466	+	0.0839976i	\(0.973231\pi\)
\(29\)	1.12872	+	1.55356i	0.209599	+	0.288488i	0.900854	−	0.434123i	\(-0.142942\pi\)
							−0.691255	+	0.722611i	\(0.742942\pi\)
\(31\)	6.48103	−	2.10581i	1.16403	−	0.378215i	0.337616	−	0.941284i	\(-0.390379\pi\)
							0.826410	+	0.563069i	\(0.190379\pi\)
\(37\)	8.06311	−	5.85819i	1.32557	−	0.963081i	0.325723	−	0.945465i	\(-0.394392\pi\)
							0.999845	−	0.0176159i	\(-0.00560762\pi\)
\(41\)	−1.30199	+	1.79204i	−0.203337	+	0.279869i	−0.898491	−	0.438991i	\(-0.855336\pi\)
							0.695155	+	0.718860i	\(0.255336\pi\)
\(43\)	9.84817			1.50183			0.750916	−	0.660398i	\(-0.229612\pi\)
							0.750916	+	0.660398i	\(0.229612\pi\)
\(47\)	−4.72684	+	6.50594i	−0.689481	+	0.948988i	−0.999999	−	0.00157892i	\(-0.999497\pi\)
							0.310518	+	0.950567i	\(0.399497\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	308.2.v.a.127.6	✓	72
4.3	odd	2		308.2.v.b.127.3	yes	72
11.2	odd	10		308.2.v.b.211.3	yes	72
44.35	even	10	inner	308.2.v.a.211.6	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.2.v.a.127.6	✓	72	1.1	even	1	trivial
308.2.v.a.211.6	yes	72	44.35	even	10	inner
308.2.v.b.127.3	yes	72	4.3	odd	2
308.2.v.b.211.3	yes	72	11.2	odd	10