Embedded newform 300.2.n.a.191.5

• Label: 300.2.n.a.191.5
• 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			191.5
Character	\(\chi\)	\(=\)	300.191
Dual form			300.2.n.a.11.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38528 + 0.284622i) q^{2} +(1.64016 - 0.556662i) q^{3} +(1.83798 - 0.788561i) q^{4} +(1.45398 - 1.69881i) q^{5} +(-2.11364 + 1.23796i) q^{6} +2.78321i q^{7} +(-2.32167 + 1.61550i) q^{8} +(2.38025 - 1.82603i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 6 q^{4} + q^{6} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38528	+	0.284622i	−0.979538	+	0.201258i
							\(3\)	1.64016	−	0.556662i	0.946947	−	0.321389i
\(5\)	1.45398	−	1.69881i	0.650237	−	0.759731i
							\(7\)			2.78321i			1.05196i	0.850498	+	0.525978i	\(0.176301\pi\)
−0.850498	+	0.525978i	\(0.823699\pi\)
\(11\)	3.44703	+	2.50441i	1.03932	+	0.755108i	0.970152	−	0.242497i	\(-0.0779664\pi\)
							0.0691653	+	0.997605i	\(0.477966\pi\)
\(13\)	0.188992	−	0.137311i	0.0524170	−	0.0380832i	−0.561268	−	0.827634i	\(-0.689686\pi\)
							0.613685	+	0.789551i	\(0.289686\pi\)
\(17\)	−2.43212	−	0.790243i	−0.589875	−	0.191662i	−0.00115566	−	0.999999i	\(-0.500368\pi\)
							−0.588720	+	0.808337i	\(0.700368\pi\)
\(19\)	−7.47793	−	2.42973i	−1.71555	−	0.557417i	−0.724312	−	0.689473i	\(-0.757842\pi\)
							−0.991242	+	0.132055i	\(0.957842\pi\)
\(23\)	2.43562	+	1.76958i	0.507861	+	0.368983i	0.812012	−	0.583641i	\(-0.198373\pi\)
							−0.304151	+	0.952624i	\(0.598373\pi\)
\(29\)	5.68764	−	1.84803i	1.05617	−	0.343170i	0.271082	−	0.962556i	\(-0.412619\pi\)
							0.785086	+	0.619386i	\(0.212619\pi\)
\(31\)	2.25520	+	0.732759i	0.405046	+	0.131607i	0.504452	−	0.863440i	\(-0.331694\pi\)
							−0.0994066	+	0.995047i	\(0.531694\pi\)
\(37\)	−3.01910	+	2.19350i	−0.496337	+	0.360610i	−0.807616	−	0.589709i	\(-0.799243\pi\)
							0.311279	+	0.950318i	\(0.399243\pi\)
\(41\)	−6.28532	−	8.65100i	−0.981602	−	1.35106i	−0.935962	−	0.352102i	\(-0.885467\pi\)
							−0.0456408	−	0.998958i	\(-0.514533\pi\)
\(43\)			6.67397i			1.01777i	0.860834	+	0.508885i	\(0.169942\pi\)
							−0.860834	+	0.508885i	\(0.830058\pi\)
\(47\)	0.254574	+	0.783497i	0.0371334	+	0.114285i	0.967905	−	0.251316i	\(-0.0808634\pi\)
							−0.930772	+	0.365601i	\(0.880863\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.2.n.a.191.5	yes	224
3.2	odd	2	inner	300.2.n.a.191.52	yes	224
4.3	odd	2	inner	300.2.n.a.191.31	yes	224
12.11	even	2	inner	300.2.n.a.191.26	yes	224
25.11	even	5	inner	300.2.n.a.11.26	yes	224
75.11	odd	10	inner	300.2.n.a.11.31	yes	224
100.11	odd	10	inner	300.2.n.a.11.52	yes	224
300.11	even	10	inner	300.2.n.a.11.5	✓	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.n.a.11.5	✓	224	300.11	even	10	inner
300.2.n.a.11.26	yes	224	25.11	even	5	inner
300.2.n.a.11.31	yes	224	75.11	odd	10	inner
300.2.n.a.11.52	yes	224	100.11	odd	10	inner
300.2.n.a.191.5	yes	224	1.1	even	1	trivial
300.2.n.a.191.26	yes	224	12.11	even	2	inner
300.2.n.a.191.31	yes	224	4.3	odd	2	inner
300.2.n.a.191.52	yes	224	3.2	odd	2	inner