Embedded newform 300.2.w.a.103.3

• Label: 300.2.w.a.103.3
• 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			103.3
Character	\(\chi\)	\(=\)	300.103
Dual form			300.2.w.a.67.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37682 + 0.323077i) q^{2} +(0.987688 - 0.156434i) q^{3} +(1.79124 - 0.889636i) q^{4} +(-2.11409 - 0.728446i) q^{5} +(-1.30932 + 0.534481i) q^{6} +(0.284106 + 0.284106i) q^{7} +(-2.17879 + 1.80357i) q^{8} +(0.951057 - 0.309017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37682	+	0.323077i	−0.973556	+	0.228450i
							\(3\)	0.987688	−	0.156434i	0.570242	−	0.0903175i
\(5\)	−2.11409	−	0.728446i	−0.945449	−	0.325771i
							\(7\)	0.284106	+	0.284106i	0.107382	+	0.107382i	0.758756	−	0.651374i	\(-0.225807\pi\)
−0.651374	+	0.758756i	\(0.725807\pi\)
\(11\)	4.58935	+	1.49117i	1.38374	+	0.449605i	0.903897	−	0.427750i	\(-0.140694\pi\)
							0.479843	+	0.877354i	\(0.340694\pi\)
\(13\)	2.80753	−	5.51008i	0.778668	−	1.52822i	−0.0689536	−	0.997620i	\(-0.521966\pi\)
							0.847621	−	0.530602i	\(-0.178034\pi\)
\(17\)	−0.127955	+	0.807879i	−0.0310338	+	0.195939i	−0.998334	−	0.0576933i	\(-0.981625\pi\)
							0.967301	+	0.253633i	\(0.0816255\pi\)
\(19\)	1.06893	−	0.776624i	0.245230	−	0.178170i	−0.458380	−	0.888756i	\(-0.651570\pi\)
							0.703610	+	0.710586i	\(0.251570\pi\)
\(23\)	−2.81503	−	5.52481i	−0.586974	−	1.15200i	−0.973278	−	0.229630i	\(-0.926248\pi\)
							0.386303	−	0.922372i	\(-0.373752\pi\)
\(29\)	4.62807	−	6.37000i	0.859412	−	1.18288i	−0.122298	−	0.992493i	\(-0.539026\pi\)
							0.981709	−	0.190385i	\(-0.0609737\pi\)
\(31\)	1.92954	+	2.65579i	0.346556	+	0.476994i	0.946342	−	0.323167i	\(-0.104748\pi\)
							−0.599786	+	0.800161i	\(0.704748\pi\)
\(37\)	4.49965	+	2.29269i	0.739738	+	0.376915i	0.782896	−	0.622153i	\(-0.213742\pi\)
							−0.0431579	+	0.999068i	\(0.513742\pi\)
\(41\)	0.177400	+	0.545980i	0.0277051	+	0.0852677i	0.963953	−	0.266073i	\(-0.0857261\pi\)
							−0.936248	+	0.351340i	\(0.885726\pi\)
\(43\)	−8.22849	+	8.22849i	−1.25483	+	1.25483i	−0.301304	+	0.953528i	\(0.597422\pi\)
							−0.953528	+	0.301304i	\(0.902578\pi\)
\(47\)	−0.342542	−	2.16273i	−0.0499649	−	0.315466i	−0.999995	−	0.00330631i	\(-0.998948\pi\)
							0.950030	−	0.312160i	\(-0.101052\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.2.w.a.103.3	yes	240
3.2	odd	2		900.2.bj.f.703.28		240
4.3	odd	2	inner	300.2.w.a.103.24	yes	240
12.11	even	2		900.2.bj.f.703.7		240
25.17	odd	20	inner	300.2.w.a.67.24	yes	240
75.17	even	20		900.2.bj.f.667.7		240
100.67	even	20	inner	300.2.w.a.67.3	✓	240
300.167	odd	20		900.2.bj.f.667.28		240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.w.a.67.3	✓	240	100.67	even	20	inner
300.2.w.a.67.24	yes	240	25.17	odd	20	inner
300.2.w.a.103.3	yes	240	1.1	even	1	trivial
300.2.w.a.103.24	yes	240	4.3	odd	2	inner
900.2.bj.f.667.7		240	75.17	even	20
900.2.bj.f.667.28		240	300.167	odd	20
900.2.bj.f.703.7		240	12.11	even	2
900.2.bj.f.703.28		240	3.2	odd	2