Embedded newform 300.2.w.a.67.12

• Label: 300.2.w.a.67.12
• Level: $300$
• Weight: $2$
• Character: 300.67
• 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			67.12
Character	\(\chi\)	\(=\)	300.67
Dual form			300.2.w.a.103.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.557275 + 1.29979i) q^{2} +(-0.987688 - 0.156434i) q^{3} +(-1.37889 - 1.44868i) q^{4} +(-1.94762 + 1.09854i) q^{5} +(0.753745 - 1.19661i) q^{6} +(2.52433 - 2.52433i) q^{7} +(2.65139 - 0.984953i) q^{8} +(0.951057 + 0.309017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 12 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(151\)	\(277\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{13}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.557275	+	1.29979i	−0.394053	+	0.919088i
							\(3\)	−0.987688	−	0.156434i	−0.570242	−	0.0903175i
\(5\)	−1.94762	+	1.09854i	−0.871001	+	0.491281i
							\(7\)	2.52433	−	2.52433i	0.954108	−	0.954108i	−0.0448845	−	0.998992i	\(-0.514292\pi\)
0.998992	+	0.0448845i	\(0.0142920\pi\)
\(11\)	1.68955	−	0.548967i	0.509417	−	0.165520i	−0.0430199	−	0.999074i	\(-0.513698\pi\)
							0.552437	+	0.833554i	\(0.313698\pi\)
\(13\)	−1.62677	−	3.19271i	−0.451184	−	0.885499i	−0.998810	−	0.0487756i	\(-0.984468\pi\)
							0.547625	−	0.836724i	\(-0.315532\pi\)
\(17\)	0.815187	+	5.14689i	0.197712	+	1.24830i	0.864338	+	0.502911i	\(0.167738\pi\)
							−0.666626	+	0.745392i	\(0.732262\pi\)
\(19\)	6.47859	+	4.70697i	1.48629	+	1.07985i	0.975462	+	0.220170i	\(0.0706611\pi\)
							0.510828	+	0.859683i	\(0.329339\pi\)
\(23\)	3.63061	−	7.12548i	0.757035	−	1.48577i	−0.113435	−	0.993545i	\(-0.536185\pi\)
							0.870471	−	0.492220i	\(-0.163815\pi\)
\(29\)	2.24828	+	3.09449i	0.417494	+	0.574631i	0.965026	−	0.262153i	\(-0.0844326\pi\)
							−0.547532	+	0.836785i	\(0.684433\pi\)
\(31\)	2.52195	−	3.47117i	0.452956	−	0.623440i	−0.520074	−	0.854121i	\(-0.674095\pi\)
							0.973029	+	0.230681i	\(0.0740955\pi\)
\(37\)	4.07246	−	2.07502i	0.669508	−	0.341131i	−0.0859772	−	0.996297i	\(-0.527401\pi\)
							0.755485	+	0.655166i	\(0.227401\pi\)
\(41\)	1.90439	−	5.86111i	0.297416	−	0.915351i	−0.684984	−	0.728558i	\(-0.740191\pi\)
							0.982399	−	0.186793i	\(-0.0598092\pi\)
\(43\)	−6.90605	−	6.90605i	−1.05316	−	1.05316i	−0.998505	−	0.0546581i	\(-0.982593\pi\)
							−0.0546581	−	0.998505i	\(-0.517407\pi\)
\(47\)	0.113466	−	0.716398i	0.0165508	−	0.104497i	−0.978030	−	0.208464i	\(-0.933154\pi\)
							0.994581	+	0.103967i	\(0.0331535\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.2.w.a.67.12	yes	240
3.2	odd	2		900.2.bj.f.667.19		240
4.3	odd	2	inner	300.2.w.a.67.9	✓	240
12.11	even	2		900.2.bj.f.667.22		240
25.3	odd	20	inner	300.2.w.a.103.9	yes	240
75.53	even	20		900.2.bj.f.703.22		240
100.3	even	20	inner	300.2.w.a.103.12	yes	240
300.203	odd	20		900.2.bj.f.703.19		240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.w.a.67.9	✓	240	4.3	odd	2	inner
300.2.w.a.67.12	yes	240	1.1	even	1	trivial
300.2.w.a.103.9	yes	240	25.3	odd	20	inner
300.2.w.a.103.12	yes	240	100.3	even	20	inner
900.2.bj.f.667.19		240	3.2	odd	2
900.2.bj.f.667.22		240	12.11	even	2
900.2.bj.f.703.19		240	300.203	odd	20
900.2.bj.f.703.22		240	75.53	even	20