Embedded newform 300.2.w.a.103.2

• Label: 300.2.w.a.103.2
• Level: $300$
• Weight: $2$
• Character: 300.103
• Analytic conductor: $2.396$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	300.w (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.39551206064\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(30\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			103.2
Character	\(\chi\)	\(=\)	300.103
Dual form			300.2.w.a.67.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37859 + 0.315398i) q^{2} +(0.987688 - 0.156434i) q^{3} +(1.80105 - 0.869612i) q^{4} +(2.22681 - 0.203301i) q^{5} +(-1.31228 + 0.527175i) q^{6} +(2.91192 + 2.91192i) q^{7} +(-2.20864 + 1.76689i) q^{8} +(0.951057 - 0.309017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 12 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(151\)	\(277\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{7}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37859	+	0.315398i	−0.974814	+	0.223020i
							\(3\)	0.987688	−	0.156434i	0.570242	−	0.0903175i
\(5\)	2.22681	−	0.203301i	0.995858	−	0.0909190i
							\(7\)	2.91192	+	2.91192i	1.10060	+	1.10060i	0.994338	+	0.106266i	\(0.0338896\pi\)
0.106266	+	0.994338i	\(0.466110\pi\)
\(11\)	−2.79799	−	0.909123i	−0.843627	−	0.274111i	−0.144853	−	0.989453i	\(-0.546271\pi\)
							−0.698774	+	0.715342i	\(0.746271\pi\)
\(13\)	−0.922241	+	1.81000i	−0.255784	+	0.502004i	−0.982813	−	0.184602i	\(-0.940900\pi\)
							0.727030	+	0.686606i	\(0.240900\pi\)
\(17\)	0.653515	−	4.12613i	0.158501	−	1.00073i	−0.772314	−	0.635241i	\(-0.780901\pi\)
							0.930814	−	0.365492i	\(-0.119099\pi\)
\(19\)	−2.04108	+	1.48293i	−0.468256	+	0.340208i	−0.796761	−	0.604294i	\(-0.793455\pi\)
							0.328505	+	0.944502i	\(0.393455\pi\)
\(23\)	0.577981	+	1.13435i	0.120517	+	0.236529i	0.943382	−	0.331709i	\(-0.107625\pi\)
							−0.822864	+	0.568238i	\(0.807625\pi\)
\(29\)	−4.10725	+	5.65314i	−0.762697	+	1.04976i	0.234288	+	0.972167i	\(0.424724\pi\)
							−0.996985	+	0.0775947i	\(0.975276\pi\)
\(31\)	−4.56619	−	6.28482i	−0.820111	−	1.12879i	−0.989684	−	0.143269i	\(-0.954239\pi\)
							0.169572	−	0.985518i	\(-0.445761\pi\)
\(37\)	9.09435	+	4.63380i	1.49510	+	0.761792i	0.994584	−	0.103934i	\(-0.0331431\pi\)
							0.500517	+	0.865726i	\(0.333143\pi\)
\(41\)	−1.48215	−	4.56160i	−0.231474	−	0.712402i	−0.997570	−	0.0696763i	\(-0.977803\pi\)
							0.766096	−	0.642726i	\(-0.222197\pi\)
\(43\)	4.56222	−	4.56222i	0.695733	−	0.695733i	−0.267755	−	0.963487i	\(-0.586282\pi\)
							0.963487	+	0.267755i	\(0.0862817\pi\)
\(47\)	−0.753158	−	4.75525i	−0.109859	−	0.693625i	−0.979727	−	0.200339i	\(-0.935796\pi\)
							0.869867	−	0.493286i	\(-0.164204\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.2.w.a.103.2	yes	240
3.2	odd	2		900.2.bj.f.703.29		240
4.3	odd	2	inner	300.2.w.a.103.23	yes	240
12.11	even	2		900.2.bj.f.703.8		240
25.17	odd	20	inner	300.2.w.a.67.23	yes	240
75.17	even	20		900.2.bj.f.667.8		240
100.67	even	20	inner	300.2.w.a.67.2	✓	240
300.167	odd	20		900.2.bj.f.667.29		240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.w.a.67.2	✓	240	100.67	even	20	inner
300.2.w.a.67.23	yes	240	25.17	odd	20	inner
300.2.w.a.103.2	yes	240	1.1	even	1	trivial
300.2.w.a.103.23	yes	240	4.3	odd	2	inner
900.2.bj.f.667.8		240	75.17	even	20
900.2.bj.f.667.29		240	300.167	odd	20
900.2.bj.f.703.8		240	12.11	even	2
900.2.bj.f.703.29		240	3.2	odd	2