Embedded newform 300.2.n.a.191.6

• Label: 300.2.n.a.191.6
• Level: $300$
• Weight: $2$
• Character: 300.191
• Analytic conductor: $2.396$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	300.n (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.39551206064\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(56\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			191.6
Character	\(\chi\)	\(=\)	300.191
Dual form			300.2.n.a.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31876 + 0.510760i) q^{2} +(-1.38129 + 1.04500i) q^{3} +(1.47825 - 1.34714i) q^{4} +(-1.62369 - 1.53741i) q^{5} +(1.28785 - 2.08361i) q^{6} +1.57078i q^{7} +(-1.26139 + 2.53158i) q^{8} +(0.815942 - 2.88691i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 6 q^{4} + q^{6} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(151\)	\(277\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31876	+	0.510760i	−0.932503	+	0.361162i
							\(3\)	−1.38129	+	1.04500i	−0.797490	+	0.603332i
\(5\)	−1.62369	−	1.53741i	−0.726136	−	0.687551i
							\(7\)			1.57078i			0.593698i	0.954924	+	0.296849i	\(0.0959358\pi\)
−0.954924	+	0.296849i	\(0.904064\pi\)
\(11\)	−0.805168	−	0.584989i	−0.242767	−	0.176381i	0.459748	−	0.888049i	\(-0.347940\pi\)
							−0.702515	+	0.711669i	\(0.747940\pi\)
\(13\)	3.32120	−	2.41299i	0.921135	−	0.669244i	−0.0226710	−	0.999743i	\(-0.507217\pi\)
							0.943806	+	0.330499i	\(0.107217\pi\)
\(17\)	2.67618	+	0.869543i	0.649068	+	0.210895i	0.615003	−	0.788524i	\(-0.289155\pi\)
							0.0340651	+	0.999420i	\(0.489155\pi\)
\(19\)	2.16804	+	0.704440i	0.497383	+	0.161610i	0.546957	−	0.837161i	\(-0.315786\pi\)
							−0.0495736	+	0.998770i	\(0.515786\pi\)
\(23\)	3.32508	+	2.41581i	0.693327	+	0.503731i	0.877752	−	0.479115i	\(-0.159042\pi\)
							−0.184425	+	0.982847i	\(0.559042\pi\)
\(29\)	5.85151	−	1.90127i	1.08660	−	0.353057i	0.289668	−	0.957127i	\(-0.406455\pi\)
							0.796931	+	0.604070i	\(0.206455\pi\)
\(31\)	5.79822	+	1.88396i	1.04139	+	0.338368i	0.779284	−	0.626670i	\(-0.215583\pi\)
							0.262107	+	0.965039i	\(0.415583\pi\)
\(37\)	−6.24168	+	4.53485i	−1.02613	+	0.745524i	−0.967530	−	0.252758i	\(-0.918662\pi\)
							−0.0585964	+	0.998282i	\(0.518662\pi\)
\(41\)	0.677043	+	0.931869i	0.105736	+	0.145534i	0.858606	−	0.512636i	\(-0.171331\pi\)
							−0.752870	+	0.658170i	\(0.771331\pi\)
\(43\)		−	10.1022i		−	1.54057i	−0.637697	−	0.770287i	\(-0.720113\pi\)
							0.637697	−	0.770287i	\(-0.279887\pi\)
\(47\)	−2.73594	−	8.42037i	−0.399078	−	1.22824i	−0.925740	−	0.378162i	\(-0.876556\pi\)
							0.526661	−	0.850075i	\(-0.323444\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.2.n.a.191.6	yes	224
3.2	odd	2	inner	300.2.n.a.191.51	yes	224
4.3	odd	2	inner	300.2.n.a.191.28	yes	224
12.11	even	2	inner	300.2.n.a.191.29	yes	224
25.11	even	5	inner	300.2.n.a.11.29	yes	224
75.11	odd	10	inner	300.2.n.a.11.28	yes	224
100.11	odd	10	inner	300.2.n.a.11.51	yes	224
300.11	even	10	inner	300.2.n.a.11.6	✓	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.n.a.11.6	✓	224	300.11	even	10	inner
300.2.n.a.11.28	yes	224	75.11	odd	10	inner
300.2.n.a.11.29	yes	224	25.11	even	5	inner
300.2.n.a.11.51	yes	224	100.11	odd	10	inner
300.2.n.a.191.6	yes	224	1.1	even	1	trivial
300.2.n.a.191.28	yes	224	4.3	odd	2	inner
300.2.n.a.191.29	yes	224	12.11	even	2	inner
300.2.n.a.191.51	yes	224	3.2	odd	2	inner