Embedded newform 300.2.n.a.191.3

• Label: 300.2.n.a.191.3
• 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			191.3
Character	\(\chi\)	\(=\)	300.191
Dual form			300.2.n.a.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39445 - 0.235585i) q^{2} +(-1.04145 + 1.38397i) q^{3} +(1.88900 + 0.657024i) q^{4} +(1.89530 + 1.18653i) q^{5} +(1.77830 - 1.68454i) q^{6} -4.72716i q^{7} +(-2.47934 - 1.36121i) q^{8} +(-0.830762 - 2.88268i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 6 q^{4} + q^{6} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39445	−	0.235585i	−0.986027	−	0.166584i
							\(3\)	−1.04145	+	1.38397i	−0.601282	+	0.799037i
\(5\)	1.89530	+	1.18653i	0.847603	+	0.530631i
							\(7\)		−	4.72716i		−	1.78670i	−0.449361	−	0.893350i	\(-0.648348\pi\)
0.449361	−	0.893350i	\(-0.351652\pi\)
\(11\)	2.39258	+	1.73831i	0.721391	+	0.524122i	0.886828	−	0.462099i	\(-0.152903\pi\)
							−0.165437	+	0.986220i	\(0.552903\pi\)
\(13\)	3.16282	−	2.29792i	0.877207	−	0.637328i	−0.0553039	−	0.998470i	\(-0.517613\pi\)
							0.932511	+	0.361141i	\(0.117613\pi\)
\(17\)	−0.528610	−	0.171756i	−0.128207	−	0.0416569i	0.244211	−	0.969722i	\(-0.421471\pi\)
							−0.372418	+	0.928065i	\(0.621471\pi\)
\(19\)	−1.14860	−	0.373201i	−0.263506	−	0.0856183i	0.174284	−	0.984695i	\(-0.444239\pi\)
							−0.437790	+	0.899077i	\(0.644239\pi\)
\(23\)	−0.664017	−	0.482436i	−0.138457	−	0.100595i	0.516401	−	0.856347i	\(-0.327272\pi\)
							−0.654858	+	0.755752i	\(0.727272\pi\)
\(29\)	6.07319	−	1.97330i	1.12776	−	0.366432i	0.315037	−	0.949079i	\(-0.397983\pi\)
							0.812725	+	0.582647i	\(0.197983\pi\)
\(31\)	4.42801	+	1.43875i	0.795293	+	0.258406i	0.678356	−	0.734733i	\(-0.262693\pi\)
							0.116937	+	0.993139i	\(0.462693\pi\)
\(37\)	3.51289	−	2.55226i	0.577515	−	0.419589i	−0.260312	−	0.965524i	\(-0.583826\pi\)
							0.837827	+	0.545935i	\(0.183826\pi\)
\(41\)	−1.90569	−	2.62296i	−0.297619	−	0.409637i	0.633851	−	0.773455i	\(-0.281473\pi\)
							−0.931470	+	0.363818i	\(0.881473\pi\)
\(43\)			9.83201i			1.49937i	0.661796	+	0.749684i	\(0.269794\pi\)
							−0.661796	+	0.749684i	\(0.730206\pi\)
\(47\)	1.36526	+	4.20183i	0.199143	+	0.612901i	0.999903	+	0.0139148i	\(0.00442937\pi\)
							−0.800760	+	0.598986i	\(0.795571\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.2.n.a.191.3	yes	224
3.2	odd	2	inner	300.2.n.a.191.54	yes	224
4.3	odd	2	inner	300.2.n.a.191.37	yes	224
12.11	even	2	inner	300.2.n.a.191.20	yes	224
25.11	even	5	inner	300.2.n.a.11.20	yes	224
75.11	odd	10	inner	300.2.n.a.11.37	yes	224
100.11	odd	10	inner	300.2.n.a.11.54	yes	224
300.11	even	10	inner	300.2.n.a.11.3	✓	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.n.a.11.3	✓	224	300.11	even	10	inner
300.2.n.a.11.20	yes	224	25.11	even	5	inner
300.2.n.a.11.37	yes	224	75.11	odd	10	inner
300.2.n.a.11.54	yes	224	100.11	odd	10	inner
300.2.n.a.191.3	yes	224	1.1	even	1	trivial
300.2.n.a.191.20	yes	224	12.11	even	2	inner
300.2.n.a.191.37	yes	224	4.3	odd	2	inner
300.2.n.a.191.54	yes	224	3.2	odd	2	inner