Embedded newform 300.2.n.a.131.23

• Label: 300.2.n.a.131.23
• Level: $300$
• Weight: $2$
• Character: 300.131
• 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			131.23
Character	\(\chi\)	\(=\)	300.131
Dual form			300.2.n.a.71.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.446933 + 1.34173i) q^{2} +(1.72317 + 0.175164i) q^{3} +(-1.60050 - 1.19933i) q^{4} +(-2.16063 - 0.575913i) q^{5} +(-1.00517 + 2.23375i) q^{6} +4.70514i q^{7} +(2.32450 - 1.61143i) q^{8} +(2.93863 + 0.603677i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 6 q^{4} + q^{6} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(151\)	\(277\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.446933	+	1.34173i	−0.316030	+	0.948749i
							\(3\)	1.72317	+	0.175164i	0.994873	+	0.101131i
\(5\)	−2.16063	−	0.575913i	−0.966263	−	0.257556i
							\(7\)			4.70514i			1.77837i	0.457543	+	0.889187i	\(0.348729\pi\)
−0.457543	+	0.889187i	\(0.651271\pi\)
\(11\)	0.0957905	+	0.294813i	0.0288819	+	0.0888895i	0.964458	−	0.264235i	\(-0.0851193\pi\)
							−0.935576	+	0.353124i	\(0.885119\pi\)
\(13\)	−1.61206	+	4.96142i	−0.447106	+	1.37605i	0.433051	+	0.901369i	\(0.357437\pi\)
							−0.880158	+	0.474682i	\(0.842563\pi\)
\(17\)	−2.69217	+	3.70546i	−0.652947	+	0.898705i	−0.999222	−	0.0394280i	\(-0.987446\pi\)
							0.346275	+	0.938133i	\(0.387446\pi\)
\(19\)	2.88355	−	3.96886i	0.661531	−	0.910519i	−0.338000	−	0.941146i	\(-0.609750\pi\)
							0.999531	+	0.0306269i	\(0.00975038\pi\)
\(23\)	−0.222611	−	0.685126i	−0.0464176	−	0.142859i	0.925162	−	0.379573i	\(-0.123929\pi\)
							−0.971579	+	0.236715i	\(0.923929\pi\)
\(29\)	−2.34652	−	3.22971i	−0.435738	−	0.599742i	0.533520	−	0.845787i	\(-0.320869\pi\)
							−0.969258	+	0.246045i	\(0.920869\pi\)
\(31\)	2.15258	−	2.96277i	0.386614	−	0.532129i	−0.570707	−	0.821154i	\(-0.693331\pi\)
							0.957322	+	0.289024i	\(0.0933309\pi\)
\(37\)	1.08397	−	3.33612i	0.178204	−	0.548454i	−0.821562	−	0.570120i	\(-0.806897\pi\)
							0.999765	+	0.0216654i	\(0.00689684\pi\)
\(41\)	6.26489	+	2.03559i	0.978411	+	0.317905i	0.754207	−	0.656637i	\(-0.228022\pi\)
							0.224204	+	0.974542i	\(0.428022\pi\)
\(43\)		−	2.91650i		−	0.444762i	−0.974960	−	0.222381i	\(-0.928617\pi\)
							0.974960	−	0.222381i	\(-0.0713829\pi\)
\(47\)	3.23343	−	2.34923i	0.471645	−	0.342670i	−0.326437	−	0.945219i	\(-0.605848\pi\)
							0.798082	+	0.602549i	\(0.205848\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.2.n.a.131.23	yes	224
3.2	odd	2	inner	300.2.n.a.131.34	yes	224
4.3	odd	2	inner	300.2.n.a.131.12	yes	224
12.11	even	2	inner	300.2.n.a.131.45	yes	224
25.21	even	5	inner	300.2.n.a.71.45	yes	224
75.71	odd	10	inner	300.2.n.a.71.12	✓	224
100.71	odd	10	inner	300.2.n.a.71.34	yes	224
300.71	even	10	inner	300.2.n.a.71.23	yes	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.n.a.71.12	✓	224	75.71	odd	10	inner
300.2.n.a.71.23	yes	224	300.71	even	10	inner
300.2.n.a.71.34	yes	224	100.71	odd	10	inner
300.2.n.a.71.45	yes	224	25.21	even	5	inner
300.2.n.a.131.12	yes	224	4.3	odd	2	inner
300.2.n.a.131.23	yes	224	1.1	even	1	trivial
300.2.n.a.131.34	yes	224	3.2	odd	2	inner
300.2.n.a.131.45	yes	224	12.11	even	2	inner