Embedded newform 350.2.q.c.221.10

• Label: 350.2.q.c.221.10
• Level: $350$
• Weight: $2$
• Character: 350.221
• 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			221.10
Character	\(\chi\)	\(=\)	350.221
Dual form			350.2.q.c.331.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.978148 + 0.207912i) q^{2} +(2.85773 - 1.27234i) q^{3} +(0.913545 - 0.406737i) q^{4} +(-0.123564 + 2.23265i) q^{5} +(-2.53074 + 1.83869i) q^{6} +(0.935911 - 2.47469i) q^{7} +(-0.809017 + 0.587785i) q^{8} +(4.54035 - 5.04257i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 10 q^{2} + 2 q^{3} + 10 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{7} - 20 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.978148	+	0.207912i	−0.691655	+	0.147016i
							\(3\)	2.85773	−	1.27234i	1.64991	−	0.734587i	0.650225	−	0.759742i	\(-0.274675\pi\)
0.999684	+	0.0251548i	\(0.00800786\pi\)
\(5\)	−0.123564	+	2.23265i	−0.0552597	+	0.998472i
							\(7\)	0.935911	−	2.47469i	0.353741	−	0.935343i
\(11\)	−2.54604	−	2.82766i	−0.767658	−	0.852571i								0.224895	−	0.974383i	\(-0.427796\pi\)
							−0.992553	+	0.121812i	\(0.961130\pi\)
\(13\)	−0.0562694	−	0.173179i	−0.0156063	−	0.0480313i	0.942950	−	0.332934i	\(-0.108039\pi\)
							−0.958556	+	0.284903i	\(0.908039\pi\)
\(17\)	0.333410	+	3.17219i	0.0808638	+	0.769368i	0.957543	+	0.288292i	\(0.0930873\pi\)
							−0.876679	+	0.481076i	\(0.840246\pi\)
\(19\)	7.48761	+	3.33370i	1.71777	+	0.764802i	0.997466	+	0.0711489i	\(0.0226666\pi\)
							0.720309	+	0.693654i	\(0.244000\pi\)
\(23\)	−5.18548	+	1.10221i	−1.08125	+	0.229826i	−0.713893	−	0.700254i	\(-0.753070\pi\)
							−0.367355	+	0.930081i	\(0.619737\pi\)
\(29\)	−2.37157	−	1.72304i	−0.440389	−	0.319961i	0.345401	−	0.938455i	\(-0.387743\pi\)
							−0.785789	+	0.618494i	\(0.787743\pi\)
\(31\)	0.536163	+	5.10125i	0.0962978	+	0.916212i	0.930879	+	0.365327i	\(0.119043\pi\)
							−0.834581	+	0.550885i	\(0.814290\pi\)
\(37\)	0.748513	−	0.831308i	0.123055	−	0.136666i	−0.678469	−	0.734629i	\(-0.737356\pi\)
							0.801524	+	0.597963i	\(0.204023\pi\)
\(41\)	1.81207	+	5.57697i	0.282997	+	0.870976i	0.986992	+	0.160771i	\(0.0513980\pi\)
							−0.703994	+	0.710206i	\(0.748602\pi\)
\(43\)	3.41084			0.520148			0.260074	−	0.965589i	\(-0.416253\pi\)
							0.260074	+	0.965589i	\(0.416253\pi\)
\(47\)	0.312491	−	2.97316i	0.0455815	−	0.433679i	−0.947804	−	0.318853i	\(-0.896702\pi\)
							0.993386	−	0.114826i	\(-0.0366311\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.c.81.1	✓	80	25.6	even	5	inner
350.2.q.c.121.1	yes	80	7.2	even	3	inner
350.2.q.c.221.10	yes	80	1.1	even	1	trivial
350.2.q.c.331.10	yes	80	175.156	even	15	inner