Embedded newform 300.2.n.a.131.31

• Label: 300.2.n.a.131.31
• Level: $300$
• Weight: $2$
• Character: 300.131
• 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			131.31
Character	\(\chi\)	\(=\)	300.131
Dual form			300.2.n.a.71.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.322989 + 1.37684i) q^{2} +(0.0437168 + 1.73150i) q^{3} +(-1.79136 + 0.889407i) q^{4} +(-1.79390 + 1.33489i) q^{5} +(-2.36987 + 0.619447i) q^{6} -3.04159i q^{7} +(-1.80316 - 2.17913i) q^{8} +(-2.99618 + 0.151391i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 6 q^{4} + q^{6} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(151\)	\(277\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.322989	+	1.37684i	0.228388	+	0.973570i
							\(3\)	0.0437168	+	1.73150i	0.0252399	+	0.999681i
\(5\)	−1.79390	+	1.33489i	−0.802257	+	0.596979i
							\(7\)		−	3.04159i		−	1.14961i	−0.818290	−	0.574806i	\(-0.805077\pi\)
0.818290	−	0.574806i	\(-0.194923\pi\)
\(11\)	0.519850	+	1.59994i	0.156741	+	0.482399i	0.998333	−	0.0577147i	\(-0.0183814\pi\)
							−0.841592	+	0.540113i	\(0.818381\pi\)
\(13\)	−1.88691	+	5.80732i	−0.523336	+	1.61066i	0.244248	+	0.969713i	\(0.421459\pi\)
							−0.767584	+	0.640949i	\(0.778541\pi\)
\(17\)	−0.841506	+	1.15823i	−0.204095	+	0.280913i	−0.898779	−	0.438403i	\(-0.855544\pi\)
							0.694684	+	0.719316i	\(0.255544\pi\)
\(19\)	2.14597	−	2.95368i	0.492320	−	0.677621i	−0.488494	−	0.872568i	\(-0.662453\pi\)
							0.980814	+	0.194947i	\(0.0624534\pi\)
\(23\)	1.33200	+	4.09948i	0.277742	+	0.854801i	0.988481	+	0.151346i	\(0.0483606\pi\)
							−0.710739	+	0.703456i	\(0.751639\pi\)
\(29\)	1.32412	+	1.82249i	0.245883	+	0.338428i	0.914064	−	0.405570i	\(-0.132927\pi\)
							−0.668181	+	0.743998i	\(0.732927\pi\)
\(31\)	−3.80357	+	5.23517i	−0.683142	+	0.940264i	−0.999966	−	0.00822000i	\(-0.997383\pi\)
							0.316824	+	0.948484i	\(0.397383\pi\)
\(37\)	−3.25448	+	10.0162i	−0.535033	+	1.64666i	0.208546	+	0.978012i	\(0.433127\pi\)
							−0.743579	+	0.668648i	\(0.766873\pi\)
\(41\)	−8.40760	−	2.73179i	−1.31305	−	0.426635i	−0.432945	−	0.901420i	\(-0.642526\pi\)
							−0.880101	+	0.474786i	\(0.842526\pi\)
\(43\)			2.84174i			0.433361i	0.976243	+	0.216681i	\(0.0695230\pi\)
							−0.976243	+	0.216681i	\(0.930477\pi\)
\(47\)	5.06530	−	3.68016i	0.738850	−	0.536806i	−0.153501	−	0.988149i	\(-0.549055\pi\)
							0.892351	+	0.451342i	\(0.149055\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.2.n.a.131.31	yes	224
3.2	odd	2	inner	300.2.n.a.131.26	yes	224
4.3	odd	2	inner	300.2.n.a.131.20	yes	224
12.11	even	2	inner	300.2.n.a.131.37	yes	224
25.21	even	5	inner	300.2.n.a.71.37	yes	224
75.71	odd	10	inner	300.2.n.a.71.20	✓	224
100.71	odd	10	inner	300.2.n.a.71.26	yes	224
300.71	even	10	inner	300.2.n.a.71.31	yes	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.n.a.71.20	✓	224	75.71	odd	10	inner
300.2.n.a.71.26	yes	224	100.71	odd	10	inner
300.2.n.a.71.31	yes	224	300.71	even	10	inner
300.2.n.a.71.37	yes	224	25.21	even	5	inner
300.2.n.a.131.20	yes	224	4.3	odd	2	inner
300.2.n.a.131.26	yes	224	3.2	odd	2	inner
300.2.n.a.131.31	yes	224	1.1	even	1	trivial
300.2.n.a.131.37	yes	224	12.11	even	2	inner