Embedded newform 350.2.q.b.11.3

• Label: 350.2.q.b.11.3
• Level: $350$
• Weight: $2$
• Character: 350.11
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79476407074\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(9\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			11.3
Character	\(\chi\)	\(=\)	350.11
Dual form			350.2.q.b.191.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.913545 - 0.406737i) q^{2} +(-1.27054 - 1.41107i) q^{3} +(0.669131 + 0.743145i) q^{4} +(-0.881031 + 2.05518i) q^{5} +(0.586758 + 1.80585i) q^{6} +(1.87755 - 1.86408i) q^{7} +(-0.309017 - 0.951057i) q^{8} +(-0.0632808 + 0.602076i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + q^{3} + 9 q^{4} + 2 q^{6} + 8 q^{7} + 18 q^{8} + 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.913545	−	0.406737i	−0.645974	−	0.287606i
							\(3\)	−1.27054	−	1.41107i	−0.733545	−	0.814684i	0.254788	−	0.966997i	\(-0.417994\pi\)
−0.988332	+	0.152313i	\(0.951328\pi\)
\(5\)	−0.881031	+	2.05518i	−0.394009	+	0.919107i
							\(7\)	1.87755	−	1.86408i	0.709649	−	0.704556i
\(11\)	0.00668509	+	0.0636044i	0.00201563	+	0.0191775i								0.995483	−	0.0949349i	\(-0.0302643\pi\)
							−0.993468	+	0.114112i	\(0.963598\pi\)
\(13\)	−4.74458	−	3.44714i	−1.31591	−	0.956065i	−0.999973	−	0.00728446i	\(-0.997681\pi\)
							−0.315937	−	0.948780i	\(-0.602319\pi\)
\(17\)	−1.90932	−	0.405839i	−0.463079	−	0.0984305i	−0.0295368	−	0.999564i	\(-0.509403\pi\)
							−0.433542	+	0.901133i	\(0.642737\pi\)
\(19\)	0.411342	−	0.456841i	0.0943683	−	0.104807i	−0.694111	−	0.719868i	\(-0.744202\pi\)
							0.788479	+	0.615062i	\(0.210869\pi\)
\(23\)	−2.42341	−	1.07897i	−0.505316	−	0.224981i	0.138213	−	0.990402i	\(-0.455864\pi\)
							−0.643530	+	0.765421i	\(0.722531\pi\)
\(29\)	−1.95541	+	6.01813i	−0.363111	+	1.11754i	0.588045	+	0.808828i	\(0.299898\pi\)
							−0.951156	+	0.308711i	\(0.900102\pi\)
\(31\)	−5.51123	−	1.17145i	−0.989847	−	0.210398i	−0.315590	−	0.948896i	\(-0.602203\pi\)
							−0.674257	+	0.738497i	\(0.735536\pi\)
\(37\)	0.499343	−	4.75093i	0.0820914	−	0.781048i	−0.873593	−	0.486656i	\(-0.838216\pi\)
							0.955685	−	0.294392i	\(-0.0951170\pi\)
\(41\)	−7.05282	−	5.12417i	−1.10146	−	0.800261i	−0.120166	−	0.992754i	\(-0.538343\pi\)
							−0.981298	+	0.192493i	\(0.938343\pi\)
\(43\)	1.38793			0.211658			0.105829	−	0.994384i	\(-0.466250\pi\)
							0.105829	+	0.994384i	\(0.466250\pi\)
\(47\)	8.72816	−	1.85523i	1.27313	−	0.270613i	0.478708	−	0.877974i	\(-0.341105\pi\)
							0.794426	+	0.607362i	\(0.207772\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.b.11.3	✓	72
7.2	even	3	inner	350.2.q.b.261.7	yes	72
25.16	even	5	inner	350.2.q.b.291.7	yes	72
175.16	even	15	inner	350.2.q.b.191.3	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.b.11.3	✓	72	1.1	even	1	trivial
350.2.q.b.191.3	yes	72	175.16	even	15	inner
350.2.q.b.261.7	yes	72	7.2	even	3	inner
350.2.q.b.291.7	yes	72	25.16	even	5	inner