Embedded newform 301.2.ba.a.100.14

• Label: 301.2.ba.a.100.14
• 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			100.14
Character	\(\chi\)	\(=\)	301.100
Dual form			301.2.ba.a.298.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.00583219 + 0.0778252i) q^{2} +(1.15971 - 0.790674i) q^{3} +(1.97164 + 0.297177i) q^{4} +(0.549945 - 2.40947i) q^{5} +(0.0547707 + 0.0948657i) q^{6} +(2.19863 + 1.47174i) q^{7} +(-0.0693594 + 0.303883i) q^{8} +(-0.376271 + 0.958724i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q - 4 q^{2} - 11 q^{3} + 20 q^{4} - q^{5} - 16 q^{6} - 7 q^{7} - 14 q^{8} + 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.00583219	+	0.0778252i	−0.00412398	+	0.0550307i	−0.998886	−	0.0471863i	\(-0.984975\pi\)
							0.994762	+	0.102217i	\(0.0325936\pi\)
\(3\)	1.15971	−	0.790674i	0.669556	−	0.456496i	−0.180229	−	0.983625i	\(-0.557684\pi\)
							0.849785	+	0.527129i	\(0.176732\pi\)
\(5\)	0.549945	−	2.40947i	0.245943	−	1.07755i	−0.689560	−	0.724228i	\(-0.742196\pi\)
							0.935503	−	0.353318i	\(-0.114947\pi\)
\(7\)	2.19863	+	1.47174i	0.831004	+	0.556266i
							\(11\)	−1.69897	+	4.32890i	−0.512258	+	1.30521i	0.407672	+	0.913129i	\(0.366341\pi\)
−0.919930	+	0.392083i	\(0.871754\pi\)
\(13\)	−4.33612	−	4.02333i	−1.20262	−	1.11587i	−0.990339	−	0.138664i	\(-0.955719\pi\)
							−0.212285	−	0.977208i	\(-0.568090\pi\)
\(17\)	−1.76436	−	7.73018i	−0.427921	−	1.87484i	−0.481775	−	0.876295i	\(-0.660008\pi\)
							0.0538546	−	0.998549i	\(-0.482849\pi\)
\(19\)	−5.12204	+	6.42284i	−1.17508	+	1.47350i	−0.325890	+	0.945408i	\(0.605664\pi\)
							−0.849187	+	0.528093i	\(0.822907\pi\)
\(23\)	0.197631	+	0.247821i	0.0412089	+	0.0516743i	0.802009	−	0.597312i	\(-0.203765\pi\)
							−0.760800	+	0.648986i	\(0.775193\pi\)
\(29\)	0.283234	−	3.77949i	0.0525952	−	0.701835i	−0.907122	−	0.420867i	\(-0.861726\pi\)
							0.959717	−	0.280967i	\(-0.0906552\pi\)
\(31\)	−0.140512	+	1.87500i	−0.0252366	+	0.336760i	0.970267	+	0.242037i	\(0.0778155\pi\)
							−0.995504	+	0.0947227i	\(0.969804\pi\)
\(37\)	−1.29187			−0.212383			−0.106191	−	0.994346i	\(-0.533866\pi\)
							−0.106191	+	0.994346i	\(0.533866\pi\)
\(41\)	1.57050	−	0.756311i	0.245270	−	0.118116i	−0.307210	−	0.951642i	\(-0.599396\pi\)
							0.552481	+	0.833526i	\(0.313681\pi\)
\(43\)	2.73961	−	5.95773i	0.417787	−	0.908545i
							\(47\)	−2.54200	−	6.47691i	−0.370789	−	0.944755i	−0.987508	−	0.157571i	\(-0.949634\pi\)
0.616719	−	0.787184i	\(-0.288462\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	301.2.ba.a.100.14	yes	324
7.4	even	3		301.2.z.a.186.14	✓	324
43.40	even	21		301.2.z.a.212.14	yes	324
301.298	even	21	inner	301.2.ba.a.298.14	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
301.2.z.a.186.14	✓	324	7.4	even	3
301.2.z.a.212.14	yes	324	43.40	even	21
301.2.ba.a.100.14	yes	324	1.1	even	1	trivial
301.2.ba.a.298.14	yes	324	301.298	even	21	inner