Embedded newform 950.2.bc.b.61.17

• Label: 950.2.bc.b.61.17
• 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			61.17
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.b.841.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.961262 + 0.275637i) q^{2} +(1.01884 + 0.143188i) q^{3} +(0.848048 - 0.529919i) q^{4} +(1.62816 + 1.53268i) q^{5} +(-1.01884 + 0.143188i) q^{6} +(-0.935339 + 1.62005i) q^{7} +(-0.669131 + 0.743145i) q^{8} +(-1.86626 - 0.535142i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q + 54 q^{7} - 75 q^{8}+O(q^{10}) \)

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
							\(3\)	1.01884	+	0.143188i	0.588225	+	0.0826697i	0.427006	−	0.904249i	\(-0.359568\pi\)
0.161220	+	0.986919i	\(0.448457\pi\)
\(5\)	1.62816	+	1.53268i	0.728135	+	0.685434i
							\(7\)	−0.935339	+	1.62005i	−0.353525	+	0.612323i	−0.986864	−	0.161551i	\(-0.948350\pi\)
0.633340	+	0.773874i	\(0.281684\pi\)
\(11\)	−1.40262	−	0.624487i	−0.422906	−	0.188290i	0.184238	−	0.982882i	\(-0.441018\pi\)
							−0.607144	+	0.794592i	\(0.707685\pi\)
\(13\)	−2.65664	+	2.56549i	−0.736820	+	0.711539i	−0.965049	−	0.262069i	\(-0.915595\pi\)
							0.228229	+	0.973607i	\(0.426706\pi\)
\(17\)	−0.163874	+	4.69273i	−0.0397452	+	1.13815i	0.801671	+	0.597766i	\(0.203945\pi\)
							−0.841416	+	0.540388i	\(0.818277\pi\)
\(19\)	−3.60210	−	2.45456i	−0.826378	−	0.563116i
							\(23\)	−7.21843	−	0.504762i	−1.50515	−	0.105250i	−0.706785	−	0.707428i	\(-0.749855\pi\)
−0.798362	+	0.602178i	\(0.794300\pi\)
\(29\)	0.126816	+	3.63155i	0.0235492	+	0.674361i	0.952568	+	0.304327i	\(0.0984317\pi\)
							−0.929018	+	0.370034i	\(0.879346\pi\)
\(31\)	−0.535259	−	0.113773i	−0.0961354	−	0.0204342i	0.159593	−	0.987183i	\(-0.448982\pi\)
							−0.255728	+	0.966749i	\(0.582315\pi\)
\(37\)	0.783351	+	0.569138i	0.128782	+	0.0935657i	0.650312	−	0.759668i	\(-0.274638\pi\)
							−0.521529	+	0.853233i	\(0.674638\pi\)
\(41\)	3.13100	−	6.41949i	0.488979	−	1.00256i	−0.501534	−	0.865138i	\(-0.667231\pi\)
							0.990513	−	0.137418i	\(-0.0438805\pi\)
\(43\)	8.31502	−	6.97713i	1.26803	−	1.06400i	0.273251	−	0.961943i	\(-0.411901\pi\)
							0.994778	−	0.102060i	\(-0.0325434\pi\)
\(47\)	0.239464	+	6.85736i	0.0349294	+	1.00025i	0.882715	+	0.469909i	\(0.155713\pi\)
							−0.847785	+	0.530339i	\(0.822064\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.b.61.17	✓	600
19.5	even	9	inner	950.2.bc.b.461.9	yes	600
25.16	even	5	inner	950.2.bc.b.441.9	yes	600
475.366	even	45	inner	950.2.bc.b.841.17	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.b.61.17	✓	600	1.1	even	1	trivial
950.2.bc.b.441.9	yes	600	25.16	even	5	inner
950.2.bc.b.461.9	yes	600	19.5	even	9	inner
950.2.bc.b.841.17	yes	600	475.366	even	45	inner