Embedded newform 300.2.n.a.11.18

• Label: 300.2.n.a.11.18
• Level: $300$
• Weight: $2$
• Character: 300.11
• 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			11.18
Character	\(\chi\)	\(=\)	300.11
Dual form			300.2.n.a.191.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.776415 - 1.18202i) q^{2} +(-0.846655 + 1.51102i) q^{3} +(-0.794359 + 1.83548i) q^{4} +(0.448288 + 2.19067i) q^{5} +(2.44341 - 0.172411i) q^{6} -2.41810i q^{7} +(2.78634 - 0.486145i) q^{8} +(-1.56635 - 2.55862i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224q - 6q^{4} + q^{6} - 6q^{9} + O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(151\)	\(277\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.776415	−	1.18202i	−0.549008	−	0.835817i
							\(3\)	−0.846655	+	1.51102i	−0.488816	+	0.872387i
\(5\)	0.448288	+	2.19067i	0.200481	+	0.979698i
							\(7\)		−	2.41810i		−	0.913955i	−0.889478	−	0.456978i	\(-0.848932\pi\)
0.889478	−	0.456978i	\(-0.151068\pi\)
\(11\)	−4.39698	+	3.19459i	−1.32574	+	0.963206i	−0.325897	+	0.945405i	\(0.605666\pi\)
							−0.999842	+	0.0178004i	\(0.994334\pi\)
\(13\)	3.38184	+	2.45705i	0.937955	+	0.681464i	0.947928	−	0.318486i	\(-0.103174\pi\)
							−0.00997257	+	0.999950i	\(0.503174\pi\)
\(17\)	−3.42396	+	1.11251i	−0.830433	+	0.269824i	−0.693228	−	0.720719i	\(-0.743812\pi\)
							−0.137205	+	0.990543i	\(0.543812\pi\)
\(19\)	−7.02945	+	2.28401i	−1.61267	+	0.523987i	−0.970196	−	0.242323i	\(-0.922091\pi\)
							−0.642471	+	0.766310i	\(0.722091\pi\)
\(23\)	−0.603374	+	0.438377i	−0.125812	+	0.0914079i	−0.648912	−	0.760864i	\(-0.724776\pi\)
							0.523100	+	0.852272i	\(0.324776\pi\)
\(29\)	−3.66968	−	1.19235i	−0.681443	−	0.221414i	−0.0522158	−	0.998636i	\(-0.516628\pi\)
							−0.629227	+	0.777222i	\(0.716628\pi\)
\(31\)	0.927208	−	0.301268i	0.166531	−	0.0541093i	−0.224565	−	0.974459i	\(-0.572096\pi\)
							0.391096	+	0.920350i	\(0.372096\pi\)
\(37\)	4.42936	+	3.21812i	0.728183	+	0.529056i	0.888988	−	0.457930i	\(-0.151409\pi\)
							−0.160805	+	0.986986i	\(0.551409\pi\)
\(41\)	1.05987	−	1.45879i	0.165524	−	0.227824i	−0.718195	−	0.695842i	\(-0.755032\pi\)
							0.883719	+	0.468017i	\(0.155032\pi\)
\(43\)			4.88416i			0.744827i	0.928067	+	0.372414i	\(0.121470\pi\)
							−0.928067	+	0.372414i	\(0.878530\pi\)
\(47\)	2.13830	−	6.58101i	0.311903	−	0.959940i	−0.665107	−	0.746748i	\(-0.731614\pi\)
							0.977010	−	0.213192i	\(-0.0683859\pi\)