Embedded newform 300.2.n.a.191.55

• Label: 300.2.n.a.191.55
• 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			191.55
Character	\(\chi\)	\(=\)	300.191
Dual form			300.2.n.a.11.55

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40679 + 0.144680i) q^{2} +(-0.926171 + 1.46363i) q^{3} +(1.95814 + 0.407070i) q^{4} +(-2.17904 + 0.501802i) q^{5} +(-1.51469 + 1.92502i) q^{6} +2.67645i q^{7} +(2.69580 + 0.855967i) q^{8} +(-1.28441 - 2.71114i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 6 q^{4} + q^{6} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40679	+	0.144680i	0.994753	+	0.102304i
							\(3\)	−0.926171	+	1.46363i	−0.534725	+	0.845026i
\(5\)	−2.17904	+	0.501802i	−0.974494	+	0.224413i
							\(7\)			2.67645i			1.01160i	0.862650	+	0.505801i	\(0.168803\pi\)
−0.862650	+	0.505801i	\(0.831197\pi\)
\(11\)	1.85172	+	1.34535i	0.558315	+	0.405640i	0.830842	−	0.556509i	\(-0.187859\pi\)
							−0.272527	+	0.962148i	\(0.587859\pi\)
\(13\)	−2.84737	+	2.06874i	−0.789719	+	0.573765i	−0.907880	−	0.419230i	\(-0.862300\pi\)
							0.118161	+	0.992994i	\(0.462300\pi\)
\(17\)	−1.64171	−	0.533423i	−0.398172	−	0.129374i	0.103084	−	0.994673i	\(-0.467129\pi\)
							−0.501257	+	0.865299i	\(0.667129\pi\)
\(19\)	6.93479	+	2.25325i	1.59095	+	0.516931i	0.964848	−	0.262809i	\(-0.0846490\pi\)
							0.626103	+	0.779740i	\(0.284649\pi\)
\(23\)	−6.63409	−	4.81995i	−1.38330	−	1.00503i	−0.996563	−	0.0828400i	\(-0.973601\pi\)
							−0.386740	−	0.922189i	\(-0.626399\pi\)
\(29\)	5.87446	−	1.90873i	1.09086	−	0.354442i	0.292281	−	0.956332i	\(-0.405586\pi\)
							0.798579	+	0.601891i	\(0.205586\pi\)
\(31\)	1.24970	+	0.406051i	0.224452	+	0.0729288i	0.419084	−	0.907947i	\(-0.362351\pi\)
							−0.194632	+	0.980876i	\(0.562351\pi\)
\(37\)	7.58270	−	5.50915i	1.24659	−	0.905699i	0.248569	−	0.968614i	\(-0.420040\pi\)
							0.998019	+	0.0629150i	\(0.0200397\pi\)
\(41\)	−4.57985	−	6.30362i	−0.715252	−	0.984460i	−0.999668	−	0.0257592i	\(-0.991800\pi\)
							0.284416	−	0.958701i	\(-0.408200\pi\)
\(43\)			2.46655i			0.376145i	0.982155	+	0.188073i	\(0.0602240\pi\)
							−0.982155	+	0.188073i	\(0.939776\pi\)
\(47\)	1.80384	+	5.55165i	0.263117	+	0.809792i	0.992121	+	0.125282i	\(0.0399836\pi\)
							−0.729004	+	0.684510i	\(0.760016\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.2.n.a.191.55	yes	224
3.2	odd	2	inner	300.2.n.a.191.2	yes	224
4.3	odd	2	inner	300.2.n.a.191.22	yes	224
12.11	even	2	inner	300.2.n.a.191.35	yes	224
25.11	even	5	inner	300.2.n.a.11.35	yes	224
75.11	odd	10	inner	300.2.n.a.11.22	yes	224
100.11	odd	10	inner	300.2.n.a.11.2	✓	224
300.11	even	10	inner	300.2.n.a.11.55	yes	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.n.a.11.2	✓	224	100.11	odd	10	inner
300.2.n.a.11.22	yes	224	75.11	odd	10	inner
300.2.n.a.11.35	yes	224	25.11	even	5	inner
300.2.n.a.11.55	yes	224	300.11	even	10	inner
300.2.n.a.191.2	yes	224	3.2	odd	2	inner
300.2.n.a.191.22	yes	224	4.3	odd	2	inner
300.2.n.a.191.35	yes	224	12.11	even	2	inner
300.2.n.a.191.55	yes	224	1.1	even	1	trivial