Embedded newform 300.2.n.a.191.18

• Label: 300.2.n.a.191.18
• 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			191.18
Character	\(\chi\)	\(=\)	300.191
Dual form			300.2.n.a.11.18

$q$-expansion

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

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)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.2.n.a.191.18	yes	224
3.2	odd	2	inner	300.2.n.a.191.39	yes	224
4.3	odd	2	inner	300.2.n.a.191.17	yes	224
12.11	even	2	inner	300.2.n.a.191.40	yes	224
25.11	even	5	inner	300.2.n.a.11.40	yes	224
75.11	odd	10	inner	300.2.n.a.11.17	✓	224
100.11	odd	10	inner	300.2.n.a.11.39	yes	224
300.11	even	10	inner	300.2.n.a.11.18	yes	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.n.a.11.17	✓	224	75.11	odd	10	inner
300.2.n.a.11.18	yes	224	300.11	even	10	inner
300.2.n.a.11.39	yes	224	100.11	odd	10	inner
300.2.n.a.11.40	yes	224	25.11	even	5	inner
300.2.n.a.191.17	yes	224	4.3	odd	2	inner
300.2.n.a.191.18	yes	224	1.1	even	1	trivial
300.2.n.a.191.39	yes	224	3.2	odd	2	inner
300.2.n.a.191.40	yes	224	12.11	even	2	inner