Embedded newform 950.2.bg.a.9.15

• Label: 950.2.bg.a.9.15
• Level: $950$
• Weight: $2$
• Character: 950.9
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $1200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.bg (of order \(90\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(1200\)
Relative dimension:	\(50\) over \(\Q(\zeta_{90})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{90}]$

Embedding invariants

Embedding label			9.15
Character	\(\chi\)	\(=\)	950.9
Dual form			950.2.bg.a.739.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.898794 - 0.438371i) q^{2} +(0.281867 - 0.451082i) q^{3} +(0.615661 + 0.788011i) q^{4} +(1.08365 + 1.95594i) q^{5} +(-0.451082 + 0.281867i) q^{6} +(-1.73157 - 0.999724i) q^{7} +(-0.207912 - 0.978148i) q^{8} +(1.19109 + 2.44209i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.898794	−	0.438371i	−0.635543	−	0.309975i
							\(3\)	0.281867	−	0.451082i	0.162736	−	0.260432i	−0.756623	−	0.653851i	\(-0.773152\pi\)
0.919359	+	0.393419i	\(0.128708\pi\)
\(5\)	1.08365	+	1.95594i	0.484624	+	0.874722i
							\(7\)	−1.73157	−	0.999724i	−0.654473	−	0.377860i	0.135695	−	0.990751i	\(-0.456673\pi\)
−0.790168	+	0.612891i	\(0.790007\pi\)
\(11\)	−0.120711	+	1.14848i	−0.0363956	+	0.346281i	0.961137	+	0.276073i	\(0.0890331\pi\)
							−0.997532	+	0.0702083i	\(0.977634\pi\)
\(13\)	−0.205645	+	2.94086i	−0.0570357	+	0.815648i	0.881874	+	0.471485i	\(0.156282\pi\)
							−0.938910	+	0.344163i	\(0.888163\pi\)
\(17\)	0.277145	−	1.97199i	0.0672174	−	0.478277i	−0.927351	−	0.374192i	\(-0.877920\pi\)
							0.994568	−	0.104084i	\(-0.0331912\pi\)
\(19\)	−4.27774	−	0.837236i	−0.981380	−	0.192075i
							\(23\)	−0.0780770	+	0.272287i	−0.0162802	+	0.0567757i	−0.968983	−	0.247129i	\(-0.920513\pi\)
0.952702	+	0.303905i	\(0.0982905\pi\)
\(29\)	−9.40528	+	1.32183i	−1.74652	+	0.245457i	−0.938885	−	0.344230i	\(-0.888140\pi\)
							−0.807632	+	0.589687i	\(0.799251\pi\)
\(31\)	5.83821	+	6.48399i	1.04857	+	1.16456i	0.986039	+	0.166516i	\(0.0532519\pi\)
							0.0625352	+	0.998043i	\(0.480081\pi\)
\(37\)	−0.122647	−	0.168809i	−0.0201631	−	0.0277521i	0.798816	−	0.601575i	\(-0.205460\pi\)
							−0.818980	+	0.573823i	\(0.805460\pi\)
\(41\)	−2.30822	+	9.25777i	−0.360484	+	1.44582i	0.466963	+	0.884277i	\(0.345348\pi\)
							−0.827447	+	0.561544i	\(0.810207\pi\)
\(43\)	−0.0890432	+	0.244644i	−0.0135790	+	0.0373079i	−0.946297	−	0.323299i	\(-0.895208\pi\)
							0.932718	+	0.360607i	\(0.117430\pi\)
\(47\)	−0.972318	−	6.91840i	−0.141827	−	1.00915i	−0.922990	−	0.384825i	\(-0.874262\pi\)
							0.781162	−	0.624328i	\(-0.214627\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bg.a.9.15	✓	1200
19.17	even	9	inner	950.2.bg.a.359.40	yes	1200
25.14	even	10	inner	950.2.bg.a.389.40	yes	1200
475.264	even	90	inner	950.2.bg.a.739.15	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bg.a.9.15	✓	1200	1.1	even	1	trivial
950.2.bg.a.359.40	yes	1200	19.17	even	9	inner
950.2.bg.a.389.40	yes	1200	25.14	even	10	inner
950.2.bg.a.739.15	yes	1200	475.264	even	90	inner