Embedded newform 350.2.q.c.81.1

• Label: 350.2.q.c.81.1
• Level: $350$
• Weight: $2$
• Character: 350.81
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $80$
• 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:	\(80\)
Relative dimension:	\(10\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			81.1
Character	\(\chi\)	\(=\)	350.81
Dual form			350.2.q.c.121.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.669131 - 0.743145i) q^{2} +(-0.326983 - 3.11103i) q^{3} +(-0.104528 - 0.994522i) q^{4} +(1.99531 + 1.00932i) q^{5} +(-2.53074 - 1.83869i) q^{6} +(0.935911 - 2.47469i) q^{7} +(-0.809017 - 0.587785i) q^{8} +(-6.63717 + 1.41077i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 10 q^{2} + 2 q^{3} + 10 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{7} - 20 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.669131	−	0.743145i	0.473147	−	0.525483i
							\(3\)	−0.326983	−	3.11103i	−0.188784	−	1.79616i	−0.521626	−	0.853174i	\(-0.674675\pi\)
0.332843	−	0.942982i	\(-0.391992\pi\)
\(5\)	1.99531	+	1.00932i	0.892332	+	0.451380i
							\(7\)	0.935911	−	2.47469i	0.353741	−	0.935343i
\(11\)	3.72184	+	0.791102i	1.12218	+	0.238526i								0.731393	−	0.681956i	\(-0.238870\pi\)
							0.390784	+	0.920482i	\(0.372204\pi\)
\(13\)	−0.0562694	+	0.173179i	−0.0156063	+	0.0480313i	−0.958556	−	0.284903i	\(-0.908039\pi\)
							0.942950	+	0.332934i	\(0.108039\pi\)
\(17\)	−2.91390	+	1.29735i	−0.706724	+	0.314654i	−0.728439	−	0.685110i	\(-0.759754\pi\)
							0.0217151	+	0.999764i	\(0.493087\pi\)
\(19\)	−0.856737	+	8.15130i	−0.196549	+	1.87004i	0.240567	+	0.970632i	\(0.422667\pi\)
							−0.437116	+	0.899405i	\(0.644000\pi\)
\(23\)	3.54728	−	3.93966i	0.739660	−	0.821475i	−0.249491	−	0.968377i	\(-0.580263\pi\)
							0.989151	+	0.146902i	\(0.0469301\pi\)
\(29\)	−2.37157	+	1.72304i	−0.440389	+	0.319961i	−0.785789	−	0.618494i	\(-0.787743\pi\)
							0.345401	+	0.938455i	\(0.387743\pi\)
\(31\)	−4.68590	+	2.08630i	−0.841612	+	0.374710i	−0.781822	−	0.623501i	\(-0.785710\pi\)
							−0.0597895	+	0.998211i	\(0.519043\pi\)
\(37\)	−1.09419	+	0.232577i	−0.179884	+	0.0382355i	−0.296973	−	0.954886i	\(-0.595977\pi\)
							0.117089	+	0.993121i	\(0.462644\pi\)
\(41\)	1.81207	−	5.57697i	0.282997	−	0.870976i	−0.703994	−	0.710206i	\(-0.748602\pi\)
							0.986992	−	0.160771i	\(-0.0513980\pi\)
\(43\)	3.41084			0.520148			0.260074	−	0.965589i	\(-0.416253\pi\)
							0.260074	+	0.965589i	\(0.416253\pi\)
\(47\)	−2.73107	−	1.21595i	−0.398368	−	0.177365i	0.197767	−	0.980249i	\(-0.436631\pi\)
							−0.596135	+	0.802884i	\(0.703298\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.c.81.1	✓	80
7.2	even	3	inner	350.2.q.c.331.10	yes	80
25.21	even	5	inner	350.2.q.c.221.10	yes	80
175.121	even	15	inner	350.2.q.c.121.1	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.c.81.1	✓	80	1.1	even	1	trivial
350.2.q.c.121.1	yes	80	175.121	even	15	inner
350.2.q.c.221.10	yes	80	25.21	even	5	inner
350.2.q.c.331.10	yes	80	7.2	even	3	inner