Embedded newform 950.2.bc.a.61.11

• Label: 950.2.bc.a.61.11
• Level: $950$
• Weight: $2$
• Character: 950.61
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $600$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.bc (of order \(45\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(600\)
Relative dimension:	\(25\) over \(\Q(\zeta_{45})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{45}]$

Embedding invariants

Embedding label			61.11
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.a.841.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.961262 - 0.275637i) q^{2} +(-0.363561 - 0.0510952i) q^{3} +(0.848048 - 0.529919i) q^{4} +(-1.52899 + 1.63162i) q^{5} +(-0.363561 + 0.0510952i) q^{6} +(1.89241 - 3.27775i) q^{7} +(0.669131 - 0.743145i) q^{8} +(-2.75422 - 0.789760i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q - 42 q^{7} + 75 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.961262	−	0.275637i	0.679715	−	0.194905i
							\(3\)	−0.363561	−	0.0510952i	−0.209902	−	0.0294998i	0.0334383	−	0.999441i	\(-0.489354\pi\)
−0.243340	+	0.969941i	\(0.578243\pi\)
\(5\)	−1.52899	+	1.63162i	−0.683786	+	0.729683i
							\(7\)	1.89241	−	3.27775i	0.715263	−	1.23887i	−0.247595	−	0.968864i	\(-0.579640\pi\)
0.962858	−	0.270008i	\(-0.0870265\pi\)
\(11\)	−4.79956	−	2.13690i	−1.44712	−	0.644300i	−0.475258	−	0.879847i	\(-0.657645\pi\)
							−0.971863	+	0.235547i	\(0.924312\pi\)
\(13\)	3.39639	−	3.27986i	0.941989	−	0.909668i	−0.0540540	−	0.998538i	\(-0.517214\pi\)
							0.996043	+	0.0888696i	\(0.0283254\pi\)
\(17\)	−0.139818	+	4.00385i	−0.0339107	+	0.971076i	0.856563	+	0.516042i	\(0.172595\pi\)
							−0.890474	+	0.455034i	\(0.849627\pi\)
\(19\)	−3.31886	+	2.82580i	−0.761399	+	0.648284i
							\(23\)	−1.79253	−	0.125346i	−0.373767	−	0.0261364i	−0.118362	−	0.992970i	\(-0.537764\pi\)
−0.255405	+	0.966834i	\(0.582209\pi\)
\(29\)	−0.208652	−	5.97500i	−0.0387456	−	1.10953i	−0.850648	−	0.525736i	\(-0.823790\pi\)
							0.811902	−	0.583794i	\(-0.198432\pi\)
\(31\)	−7.63023	−	1.62185i	−1.37043	−	0.291294i	−0.536843	−	0.843682i	\(-0.680383\pi\)
							−0.833587	+	0.552389i	\(0.813716\pi\)
\(37\)	−7.09134	−	5.15216i	−1.16581	−	0.847010i	−0.175308	−	0.984514i	\(-0.556092\pi\)
							−0.990501	+	0.137504i	\(0.956092\pi\)
\(41\)	4.99401	−	10.2392i	0.779933	−	1.59910i	−0.0205818	−	0.999788i	\(-0.506552\pi\)
							0.800515	−	0.599312i	\(-0.204559\pi\)
\(43\)	7.51284	−	6.30402i	1.14570	−	0.961355i	0.146088	−	0.989272i	\(-0.453332\pi\)
							0.999610	+	0.0279169i	\(0.00888739\pi\)
\(47\)	0.258120	+	7.39158i	0.0376506	+	1.07817i	0.860353	+	0.509699i	\(0.170243\pi\)
							−0.822702	+	0.568473i	\(0.807534\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.a.61.11	✓	600
19.5	even	9	inner	950.2.bc.a.461.15	yes	600
25.16	even	5	inner	950.2.bc.a.441.15	yes	600
475.366	even	45	inner	950.2.bc.a.841.11	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.a.61.11	✓	600	1.1	even	1	trivial
950.2.bc.a.441.15	yes	600	25.16	even	5	inner
950.2.bc.a.461.15	yes	600	19.5	even	9	inner
950.2.bc.a.841.11	yes	600	475.366	even	45	inner