Embedded newform 300.2.n.a.131.15

• Label: 300.2.n.a.131.15
• 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			131.15
Character	\(\chi\)	\(=\)	300.131
Dual form			300.2.n.a.71.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.936272 + 1.05990i) q^{2} +(1.00821 + 1.40837i) q^{3} +(-0.246789 - 1.98472i) q^{4} +(-1.49656 - 1.66142i) q^{5} +(-2.43670 - 0.250011i) q^{6} -4.44836i q^{7} +(2.33467 + 1.59666i) q^{8} +(-0.967016 + 2.83987i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 6 q^{4} + q^{6} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.936272	+	1.05990i	−0.662044	+	0.749465i
							\(3\)	1.00821	+	1.40837i	0.582092	+	0.813123i
\(5\)	−1.49656	−	1.66142i	−0.669282	−	0.743009i
							\(7\)		−	4.44836i		−	1.68132i	−0.541562	−	0.840661i	\(-0.682167\pi\)
0.541562	−	0.840661i	\(-0.317833\pi\)
\(11\)	−1.38440	−	4.26075i	−0.417413	−	1.28466i	−0.910075	−	0.414444i	\(-0.863976\pi\)
							0.492662	−	0.870221i	\(-0.336024\pi\)
\(13\)	1.09944	−	3.38374i	0.304931	−	0.938481i	−0.674772	−	0.738026i	\(-0.735758\pi\)
							0.979703	−	0.200454i	\(-0.0642419\pi\)
\(17\)	0.229964	−	0.316518i	0.0557744	−	0.0767669i	−0.780219	−	0.625507i	\(-0.784892\pi\)
							0.835993	+	0.548740i	\(0.184892\pi\)
\(19\)	−1.18868	+	1.63607i	−0.272701	+	0.375341i	−0.923299	−	0.384081i	\(-0.874518\pi\)
							0.650598	+	0.759422i	\(0.274518\pi\)
\(23\)	−0.927671	−	2.85508i	−0.193433	−	0.595325i	−0.999991	−	0.00417066i	\(-0.998672\pi\)
							0.806559	−	0.591154i	\(-0.201328\pi\)
\(29\)	4.65476	+	6.40673i	0.864368	+	1.18970i	0.980510	+	0.196468i	\(0.0629471\pi\)
							−0.116142	+	0.993233i	\(0.537053\pi\)
\(31\)	−0.713204	+	0.981641i	−0.128095	+	0.176308i	−0.868247	−	0.496131i	\(-0.834753\pi\)
							0.740152	+	0.672439i	\(0.234753\pi\)
\(37\)	−0.145467	+	0.447702i	−0.0239146	+	0.0736017i	−0.962302	−	0.271985i	\(-0.912320\pi\)
							0.938387	+	0.345586i	\(0.112320\pi\)
\(41\)	7.88511	+	2.56203i	1.23145	+	0.400121i	0.851238	−	0.524779i	\(-0.175852\pi\)
							0.380209	+	0.924901i	\(0.375852\pi\)
\(43\)		−	5.44861i		−	0.830905i	−0.909615	−	0.415453i	\(-0.863623\pi\)
							0.909615	−	0.415453i	\(-0.136377\pi\)
\(47\)	5.11114	−	3.71346i	0.745537	−	0.541664i	−0.148903	−	0.988852i	\(-0.547574\pi\)
							0.894440	+	0.447187i	\(0.147574\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.2.n.a.131.15	yes	224
3.2	odd	2	inner	300.2.n.a.131.42	yes	224
4.3	odd	2	inner	300.2.n.a.131.4	yes	224
12.11	even	2	inner	300.2.n.a.131.53	yes	224
25.21	even	5	inner	300.2.n.a.71.53	yes	224
75.71	odd	10	inner	300.2.n.a.71.4	✓	224
100.71	odd	10	inner	300.2.n.a.71.42	yes	224
300.71	even	10	inner	300.2.n.a.71.15	yes	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.n.a.71.4	✓	224	75.71	odd	10	inner
300.2.n.a.71.15	yes	224	300.71	even	10	inner
300.2.n.a.71.42	yes	224	100.71	odd	10	inner
300.2.n.a.71.53	yes	224	25.21	even	5	inner
300.2.n.a.131.4	yes	224	4.3	odd	2	inner
300.2.n.a.131.15	yes	224	1.1	even	1	trivial
300.2.n.a.131.42	yes	224	3.2	odd	2	inner
300.2.n.a.131.53	yes	224	12.11	even	2	inner