Embedded newform 350.2.q.b.11.2

• Label: 350.2.q.b.11.2
• 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			11.2
Character	\(\chi\)	\(=\)	350.11
Dual form			350.2.q.b.191.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.913545 - 0.406737i) q^{2} +(-2.03944 - 2.26502i) q^{3} +(0.669131 + 0.743145i) q^{4} +(2.23454 - 0.0827180i) q^{5} +(0.941850 + 2.89872i) q^{6} +(2.10598 + 1.60151i) q^{7} +(-0.309017 - 0.951057i) q^{8} +(-0.657446 + 6.25519i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + q^{3} + 9 q^{4} + 2 q^{6} + 8 q^{7} + 18 q^{8} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.03944	−	2.26502i	−1.17747	−	1.30771i	−0.941915	−	0.335851i	\(-0.890976\pi\)
−0.235555	−	0.971861i	\(-0.575691\pi\)
\(5\)	2.23454	−	0.0827180i	0.999316	−	0.0369926i
							\(7\)	2.10598	+	1.60151i	0.795986	+	0.605314i
\(11\)	0.561604	+	5.34331i	0.169330	+	1.61107i								0.667923	+	0.744231i	\(0.267184\pi\)
							−0.498593	+	0.866836i	\(0.666150\pi\)
\(13\)	2.50842	+	1.82247i	0.695709	+	0.505462i	0.878532	−	0.477684i	\(-0.158523\pi\)
							−0.182823	+	0.983146i	\(0.558523\pi\)
\(17\)	4.98199	+	1.05896i	1.20831	+	0.256834i	0.767668	−	0.640848i	\(-0.221417\pi\)
							0.440643	+	0.897682i	\(0.354750\pi\)
\(19\)	−2.23022	+	2.47691i	−0.511648	+	0.568243i	−0.942511	−	0.334174i	\(-0.891543\pi\)
							0.430863	+	0.902417i	\(0.358209\pi\)
\(23\)	−7.22753	−	3.21790i	−1.50704	−	0.670979i	−0.523563	−	0.851987i	\(-0.675397\pi\)
							−0.983482	+	0.181008i	\(0.942064\pi\)
\(29\)	0.580062	−	1.78525i	0.107715	−	0.331512i	−0.882643	−	0.470044i	\(-0.844238\pi\)
							0.990358	+	0.138531i	\(0.0442382\pi\)
\(31\)	5.77457	+	1.22742i	1.03714	+	0.220452i	0.694860	−	0.719145i	\(-0.255466\pi\)
							0.342284	+	0.939597i	\(0.388800\pi\)
\(37\)	−0.0567874	+	0.540296i	−0.00933580	+	0.0888242i	−0.998198	−	0.0600121i	\(-0.980886\pi\)
							0.988862	+	0.148836i	\(0.0475527\pi\)
\(41\)	−3.40344	−	2.47275i	−0.531529	−	0.386178i	0.289401	−	0.957208i	\(-0.406544\pi\)
							−0.820929	+	0.571030i	\(0.806544\pi\)
\(43\)	1.15448			0.176056			0.0880282	−	0.996118i	\(-0.471943\pi\)
							0.0880282	+	0.996118i	\(0.471943\pi\)
\(47\)	−4.29218	+	0.912330i	−0.626078	+	0.133077i	−0.510016	−	0.860165i	\(-0.670361\pi\)
							−0.116062	+	0.993242i	\(0.537027\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.b.11.2	✓	72
7.2	even	3	inner	350.2.q.b.261.8	yes	72
25.16	even	5	inner	350.2.q.b.291.8	yes	72
175.16	even	15	inner	350.2.q.b.191.2	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.b.11.2	✓	72	1.1	even	1	trivial
350.2.q.b.191.2	yes	72	175.16	even	15	inner
350.2.q.b.261.8	yes	72	7.2	even	3	inner
350.2.q.b.291.8	yes	72	25.16	even	5	inner