Embedded newform 301.2.ba.a.100.24

• Label: 301.2.ba.a.100.24
• Level: $301$
• Weight: $2$
• Character: 301.100
• Analytic conductor: $2.403$
• Analytic rank: $0$
• Dimension: $324$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 301 = 7 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	301.ba (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.40349710084\)
Analytic rank:	\(0\)
Dimension:	\(324\)
Relative dimension:	\(27\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			100.24
Character	\(\chi\)	\(=\)	301.100
Dual form			301.2.ba.a.298.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.160916 - 2.14727i) q^{2} +(0.474797 - 0.323711i) q^{3} +(-2.60723 - 0.392976i) q^{4} +(0.364428 - 1.59666i) q^{5} +(-0.618694 - 1.07161i) q^{6} +(-2.04739 - 1.67577i) q^{7} +(-0.305066 + 1.33658i) q^{8} +(-0.975380 + 2.48523i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q - 4 q^{2} - 11 q^{3} + 20 q^{4} - q^{5} - 16 q^{6} - 7 q^{7} - 14 q^{8} + 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/301\mathbb{Z}\right)^\times\).

\(n\)	\(87\)	\(218\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{10}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.160916	−	2.14727i	0.113785	−	1.51835i	−0.589689	−	0.807630i	\(-0.700750\pi\)
							0.703474	−	0.710721i	\(-0.251631\pi\)
\(3\)	0.474797	−	0.323711i	0.274124	−	0.186895i	−0.418453	−	0.908238i	\(-0.637428\pi\)
							0.692577	+	0.721344i	\(0.256475\pi\)
\(5\)	0.364428	−	1.59666i	0.162977	−	0.714049i	−0.825715	−	0.564088i	\(-0.809228\pi\)
							0.988692	−	0.149961i	\(-0.0479149\pi\)
\(7\)	−2.04739	−	1.67577i	−0.773839	−	0.633382i
							\(11\)	0.998889	−	2.54513i	0.301176	−	0.767385i	−0.697531	−	0.716555i	\(-0.745718\pi\)
0.998707	−	0.0508300i	\(-0.0161867\pi\)
\(13\)	2.67838	+	2.48517i	0.742848	+	0.689262i	0.957823	−	0.287357i	\(-0.0927767\pi\)
							−0.214976	+	0.976619i	\(0.568967\pi\)
\(17\)	−1.16305	−	5.09567i	−0.282082	−	1.23588i	−0.895119	−	0.445827i	\(-0.852910\pi\)
							0.613038	−	0.790054i	\(-0.289947\pi\)
\(19\)	−1.66356	+	2.08604i	−0.381647	+	0.478570i	−0.935137	−	0.354286i	\(-0.884724\pi\)
							0.553490	+	0.832856i	\(0.313296\pi\)
\(23\)	1.82684	+	2.29078i	0.380922	+	0.477661i	0.934921	−	0.354857i	\(-0.115470\pi\)
							−0.553999	+	0.832517i	\(0.686899\pi\)
\(29\)	0.652703	−	8.70972i	0.121204	−	1.61735i	−0.522885	−	0.852403i	\(-0.675144\pi\)
							0.644089	−	0.764950i	\(-0.277237\pi\)
\(31\)	−0.167880	+	2.24021i	−0.0301522	+	0.402353i	0.961624	+	0.274370i	\(0.0884694\pi\)
							−0.991776	+	0.127983i	\(0.959150\pi\)
\(37\)	5.61684			0.923404			0.461702	−	0.887035i	\(-0.347239\pi\)
							0.461702	+	0.887035i	\(0.347239\pi\)
\(41\)	7.13290	−	3.43502i	1.11397	−	0.536461i	0.215947	−	0.976405i	\(-0.430716\pi\)
							0.898025	+	0.439944i	\(0.145002\pi\)
\(43\)	−0.452897	−	6.54178i	−0.0690661	−	0.997612i
							\(47\)	4.66686	+	11.8910i	0.680732	+	1.73448i	0.678632	+	0.734479i	\(0.262573\pi\)
0.00210006	+	0.999998i	\(0.499332\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	301.2.ba.a.100.24	yes	324
7.4	even	3		301.2.z.a.186.24	✓	324
43.40	even	21		301.2.z.a.212.24	yes	324
301.298	even	21	inner	301.2.ba.a.298.24	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
301.2.z.a.186.24	✓	324	7.4	even	3
301.2.z.a.212.24	yes	324	43.40	even	21
301.2.ba.a.100.24	yes	324	1.1	even	1	trivial
301.2.ba.a.298.24	yes	324	301.298	even	21	inner