Embedded newform 300.2.r.a.59.6

• Label: 300.2.r.a.59.6
• Level: $300$
• Weight: $2$
• Character: 300.59
• 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.r (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			59.6
Character	\(\chi\)	\(=\)	300.59
Dual form			300.2.r.a.239.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32465 + 0.495274i) q^{2} +(1.60045 + 0.662243i) q^{3} +(1.50941 - 1.31213i) q^{4} +(0.462356 + 2.18774i) q^{5} +(-2.44803 - 0.0845826i) q^{6} +2.28606 q^{7} +(-1.34958 + 2.48569i) q^{8} +(2.12287 + 2.11977i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 6 q^{4} - 7 q^{6} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(151\)	\(277\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32465	+	0.495274i	−0.936671	+	0.350211i
							\(3\)	1.60045	+	0.662243i	0.924019	+	0.382346i
\(5\)	0.462356	+	2.18774i	0.206772	+	0.978389i
							\(7\)	2.28606			0.864051			0.432025	−	0.901861i	\(-0.357799\pi\)
0.432025	+	0.901861i	\(0.357799\pi\)
\(11\)	−2.72636	−	1.98082i	−0.822029	−	0.597239i	0.0952639	−	0.995452i	\(-0.469631\pi\)
							−0.917293	+	0.398213i	\(0.869631\pi\)
\(13\)	0.498773	+	0.686502i	0.138335	+	0.190401i	0.872563	−	0.488501i	\(-0.162456\pi\)
							−0.734229	+	0.678902i	\(0.762456\pi\)
\(17\)	−0.267534	+	0.823384i	−0.0648864	+	0.199700i	−0.978244	−	0.207459i	\(-0.933481\pi\)
							0.913357	+	0.407159i	\(0.133481\pi\)
\(19\)	0.177992	+	0.0578330i	0.0408341	+	0.0132678i	0.329363	−	0.944203i	\(-0.393166\pi\)
							−0.288529	+	0.957471i	\(0.593166\pi\)
\(23\)	3.95361	−	5.44167i	0.824384	−	1.13467i	−0.164558	−	0.986367i	\(-0.552620\pi\)
							0.988942	−	0.148300i	\(-0.0473801\pi\)
\(29\)	−3.23281	+	1.05040i	−0.600318	+	0.195055i	−0.593382	−	0.804921i	\(-0.702208\pi\)
							−0.00693593	+	0.999976i	\(0.502208\pi\)
\(31\)	−6.19921	−	2.01424i	−1.11341	−	0.361769i	−0.306161	−	0.951980i	\(-0.599045\pi\)
							−0.807249	+	0.590211i	\(0.799045\pi\)
\(37\)	5.08496	+	6.99884i	0.835962	+	1.15060i	0.986784	+	0.162042i	\(0.0518079\pi\)
							−0.150822	+	0.988561i	\(0.548192\pi\)
\(41\)	5.24265	+	7.21588i	0.818764	+	1.12693i	0.989912	+	0.141686i	\(0.0452524\pi\)
							−0.171148	+	0.985245i	\(0.554748\pi\)
\(43\)	−2.72313			−0.415273			−0.207636	−	0.978206i	\(-0.566577\pi\)
							−0.207636	+	0.978206i	\(0.566577\pi\)
\(47\)	9.30378	−	3.02298i	1.35710	−	0.440947i	0.462023	−	0.886868i	\(-0.347124\pi\)
							0.895072	+	0.445921i	\(0.147124\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.2.r.a.59.6	✓	224
3.2	odd	2	inner	300.2.r.a.59.51	yes	224
4.3	odd	2	inner	300.2.r.a.59.29	yes	224
12.11	even	2	inner	300.2.r.a.59.28	yes	224
25.14	even	10	inner	300.2.r.a.239.28	yes	224
75.14	odd	10	inner	300.2.r.a.239.29	yes	224
100.39	odd	10	inner	300.2.r.a.239.51	yes	224
300.239	even	10	inner	300.2.r.a.239.6	yes	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.r.a.59.6	✓	224	1.1	even	1	trivial
300.2.r.a.59.28	yes	224	12.11	even	2	inner
300.2.r.a.59.29	yes	224	4.3	odd	2	inner
300.2.r.a.59.51	yes	224	3.2	odd	2	inner
300.2.r.a.239.6	yes	224	300.239	even	10	inner
300.2.r.a.239.28	yes	224	25.14	even	10	inner
300.2.r.a.239.29	yes	224	75.14	odd	10	inner
300.2.r.a.239.51	yes	224	100.39	odd	10	inner