Embedded newform 300.3.u.a.47.69

• Label: 300.3.u.a.47.69
• Level: $300$
• Weight: $3$
• Character: 300.47
• Analytic conductor: $8.174$
• Analytic rank: $0$
• Dimension: $928$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(8.17440793081\)
Analytic rank:	\(0\)
Dimension:	\(928\)
Relative dimension:	\(116\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			47.69
Character	\(\chi\)	\(=\)	300.47
Dual form			300.3.u.a.83.69

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.578549 - 1.91449i) q^{2} +(2.99876 - 0.0861386i) q^{3} +(-3.33056 - 2.21526i) q^{4} +(2.66390 - 4.23127i) q^{5} +(1.57002 - 5.79094i) q^{6} +(6.36797 - 6.36797i) q^{7} +(-6.16798 + 5.09470i) q^{8} +(8.98516 - 0.516619i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 928 q - 20 q^{4} - 6 q^{6} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(151\)	\(277\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{17}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.578549	−	1.91449i	0.289275	−	0.957246i
							\(3\)	2.99876	−	0.0861386i	0.999588	−	0.0287129i
\(5\)	2.66390	−	4.23127i	0.532779	−	0.846254i
							\(7\)	6.36797	−	6.36797i	0.909711	−	0.909711i	−0.0865380	−	0.996249i	\(-0.527580\pi\)
0.996249	+	0.0865380i	\(0.0275804\pi\)
\(11\)	−1.78909	+	5.50626i	−0.162645	+	0.500569i	−0.998855	−	0.0478399i	\(-0.984766\pi\)
							0.836210	+	0.548409i	\(0.184766\pi\)
\(13\)	4.26245	+	2.17183i	0.327881	+	0.167064i	0.610180	−	0.792263i	\(-0.291097\pi\)
							−0.282299	+	0.959326i	\(0.591097\pi\)
\(17\)	−2.00004	+	12.6277i	−0.117649	+	0.742809i	0.856373	+	0.516357i	\(0.172712\pi\)
							−0.974023	+	0.226451i	\(0.927288\pi\)
\(19\)	−18.1431	+	13.1818i	−0.954902	+	0.693777i	−0.951961	−	0.306219i	\(-0.900936\pi\)
							−0.00294068	+	0.999996i	\(0.500936\pi\)
\(23\)	−6.61250	−	12.9778i	−0.287500	−	0.564250i	0.701412	−	0.712757i	\(-0.252554\pi\)
							−0.988911	+	0.148506i	\(0.952554\pi\)
\(29\)	−37.6277	−	27.3381i	−1.29751	−	0.942693i	−0.297578	−	0.954698i	\(-0.596179\pi\)
							−0.999928	+	0.0120043i	\(0.996179\pi\)
\(31\)	19.3368	+	26.6148i	0.623766	+	0.858541i	0.997620	−	0.0689467i	\(-0.0219638\pi\)
							−0.373854	+	0.927488i	\(0.621964\pi\)
\(37\)	29.8119	−	58.5092i	0.805727	−	1.58133i	−0.00791946	−	0.999969i	\(-0.502521\pi\)
							0.813647	−	0.581360i	\(-0.197479\pi\)
\(41\)	−22.2304	+	7.22308i	−0.542204	+	0.176173i	−0.567298	−	0.823512i	\(-0.692011\pi\)
							0.0250945	+	0.999685i	\(0.492011\pi\)
\(43\)	43.9590	+	43.9590i	1.02230	+	1.02230i	0.999746	+	0.0225578i	\(0.00718097\pi\)
							0.0225578	+	0.999746i	\(0.492819\pi\)
\(47\)	11.1120	+	70.1582i	0.236425	+	1.49273i	0.765105	+	0.643906i	\(0.222687\pi\)
							−0.528680	+	0.848821i	\(0.677313\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.3.u.a.47.69	yes	928
3.2	odd	2	inner	300.3.u.a.47.48	yes	928
4.3	odd	2	inner	300.3.u.a.47.12	✓	928
12.11	even	2	inner	300.3.u.a.47.105	yes	928
25.8	odd	20	inner	300.3.u.a.83.105	yes	928
75.8	even	20	inner	300.3.u.a.83.12	yes	928
100.83	even	20	inner	300.3.u.a.83.48	yes	928
300.83	odd	20	inner	300.3.u.a.83.69	yes	928

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.3.u.a.47.12	✓	928	4.3	odd	2	inner
300.3.u.a.47.48	yes	928	3.2	odd	2	inner
300.3.u.a.47.69	yes	928	1.1	even	1	trivial
300.3.u.a.47.105	yes	928	12.11	even	2	inner
300.3.u.a.83.12	yes	928	75.8	even	20	inner
300.3.u.a.83.48	yes	928	100.83	even	20	inner
300.3.u.a.83.69	yes	928	300.83	odd	20	inner
300.3.u.a.83.105	yes	928	25.8	odd	20	inner