Embedded newform 950.2.bg.a.9.3

• Label: 950.2.bg.a.9.3
• 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			9.3
Character	\(\chi\)	\(=\)	950.9
Dual form			950.2.bg.a.739.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.898794 - 0.438371i) q^{2} +(-1.48382 + 2.37460i) q^{3} +(0.615661 + 0.788011i) q^{4} +(1.05677 - 1.97059i) q^{5} +(2.37460 - 1.48382i) q^{6} +(-0.246693 - 0.142428i) q^{7} +(-0.207912 - 0.978148i) q^{8} +(-2.12191 - 4.35057i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q+O(q^{10}) \)

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
							\(3\)	−1.48382	+	2.37460i	−0.856682	+	1.37098i	0.0709491	+	0.997480i	\(0.477397\pi\)
−0.927631	+	0.373498i	\(0.878158\pi\)
\(5\)	1.05677	−	1.97059i	0.472603	−	0.881275i
							\(7\)	−0.246693	−	0.142428i	−0.0932410	−	0.0538327i	0.452654	−	0.891686i	\(-0.350477\pi\)
−0.545896	+	0.837853i	\(0.683810\pi\)
\(11\)	−0.351053	+	3.34005i	−0.105846	+	1.00706i	0.804707	+	0.593672i	\(0.202322\pi\)
							−0.910554	+	0.413390i	\(0.864344\pi\)
\(13\)	0.278518	−	3.98299i	0.0772470	−	1.10468i	−0.792071	−	0.610429i	\(-0.790997\pi\)
							0.869318	−	0.494254i	\(-0.164559\pi\)
\(17\)	−0.648868	+	4.61693i	−0.157374	+	1.11977i	0.738111	+	0.674679i	\(0.235718\pi\)
							−0.895485	+	0.445092i	\(0.853171\pi\)
\(19\)	0.147245	+	4.35641i	0.0337803	+	0.999429i
							\(23\)	2.32039	−	8.09217i	0.483835	−	1.68733i	−0.218788	−	0.975772i	\(-0.570210\pi\)
0.702624	−	0.711562i	\(-0.252012\pi\)
\(29\)	8.78437	−	1.23456i	1.63122	−	0.229253i	0.736679	−	0.676243i	\(-0.236393\pi\)
							0.894539	+	0.446990i	\(0.147504\pi\)
\(31\)	6.53052	+	7.25287i	1.17292	+	1.30265i	0.944280	+	0.329143i	\(0.106760\pi\)
							0.228635	+	0.973512i	\(0.426574\pi\)
\(37\)	1.96344	+	2.70245i	0.322788	+	0.444280i	0.939316	−	0.343053i	\(-0.111461\pi\)
							−0.616528	+	0.787333i	\(0.711461\pi\)
\(41\)	−1.64939	+	6.61532i	−0.257591	+	1.03314i	0.692563	+	0.721358i	\(0.256482\pi\)
							−0.950154	+	0.311782i	\(0.899074\pi\)
\(43\)	0.995139	−	2.73412i	0.151757	−	0.416949i	−0.840397	−	0.541972i	\(-0.817678\pi\)
							0.992154	+	0.125022i	\(0.0399002\pi\)
\(47\)	1.01278	+	7.20633i	0.147730	+	1.05115i	0.913151	+	0.407622i	\(0.133642\pi\)
							−0.765421	+	0.643530i	\(0.777469\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bg.a.9.3	✓	1200
19.17	even	9	inner	950.2.bg.a.359.28	yes	1200
25.14	even	10	inner	950.2.bg.a.389.28	yes	1200
475.264	even	90	inner	950.2.bg.a.739.3	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bg.a.9.3	✓	1200	1.1	even	1	trivial
950.2.bg.a.359.28	yes	1200	19.17	even	9	inner
950.2.bg.a.389.28	yes	1200	25.14	even	10	inner
950.2.bg.a.739.3	yes	1200	475.264	even	90	inner