Embedded newform 950.2.bg.a.9.8

• Label: 950.2.bg.a.9.8
• 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			9.8
Character	\(\chi\)	\(=\)	950.9
Dual form			950.2.bg.a.739.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.898794 - 0.438371i) q^{2} +(-0.716423 + 1.14652i) q^{3} +(0.615661 + 0.788011i) q^{4} +(-2.00197 - 0.996049i) q^{5} +(1.14652 - 0.716423i) q^{6} +(-1.22683 - 0.708310i) q^{7} +(-0.207912 - 0.978148i) q^{8} +(0.513875 + 1.05360i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q+O(q^{10}) \)

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
							\(3\)	−0.716423	+	1.14652i	−0.413627	+	0.661942i	−0.987610	−	0.156925i	\(-0.949842\pi\)
0.573983	+	0.818867i	\(0.305397\pi\)
\(5\)	−2.00197	−	0.996049i	−0.895309	−	0.445446i
							\(7\)	−1.22683	−	0.708310i	−0.463698	−	0.267716i	0.249900	−	0.968272i	\(-0.419602\pi\)
−0.713598	+	0.700556i	\(0.752936\pi\)
\(11\)	−0.318534	+	3.03065i	−0.0960417	+	0.913775i	0.835341	+	0.549733i	\(0.185270\pi\)
							−0.931382	+	0.364043i	\(0.881396\pi\)
\(13\)	−0.0648613	+	0.927560i	−0.0179893	+	0.257259i	0.980242	+	0.197804i	\(0.0633808\pi\)
							−0.998231	+	0.0594552i	\(0.981064\pi\)
\(17\)	0.284391	−	2.02355i	0.0689750	−	0.490782i	−0.924895	−	0.380224i	\(-0.875847\pi\)
							0.993870	−	0.110559i	\(-0.0352640\pi\)
\(19\)	−0.516092	+	4.32824i	−0.118400	+	0.992966i
							\(23\)	2.25179	−	7.85293i	0.469531	−	1.63745i	−0.268935	−	0.963158i	\(-0.586672\pi\)
0.738466	−	0.674291i	\(-0.235550\pi\)
\(29\)	−2.54013	+	0.356991i	−0.471690	+	0.0662916i	−0.371013	−	0.928628i	\(-0.620990\pi\)
							−0.100676	+	0.994919i	\(0.532101\pi\)
\(31\)	−5.67133	−	6.29865i	−1.01860	−	1.13127i	−0.991297	−	0.131644i	\(-0.957975\pi\)
							−0.0273037	−	0.999627i	\(-0.508692\pi\)
\(37\)	1.36736	+	1.88201i	0.224792	+	0.309400i	0.906485	−	0.422239i	\(-0.138756\pi\)
							−0.681692	+	0.731639i	\(0.738756\pi\)
\(41\)	0.00364587	−	0.0146228i	0.000569389	−	0.00228369i	−0.970010	−	0.243064i	\(-0.921848\pi\)
							0.970580	+	0.240780i	\(0.0774032\pi\)
\(43\)	−0.902727	+	2.48022i	−0.137665	+	0.378230i	−0.989298	−	0.145907i	\(-0.953390\pi\)
							0.851634	+	0.524137i	\(0.175612\pi\)
\(47\)	−0.585350	−	4.16498i	−0.0853821	−	0.607525i	−0.985219	−	0.171299i	\(-0.945204\pi\)
							0.899837	−	0.436226i	\(-0.143685\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bg.a.9.8	✓	1200
19.17	even	9	inner	950.2.bg.a.359.33	yes	1200
25.14	even	10	inner	950.2.bg.a.389.33	yes	1200
475.264	even	90	inner	950.2.bg.a.739.8	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bg.a.9.8	✓	1200	1.1	even	1	trivial
950.2.bg.a.359.33	yes	1200	19.17	even	9	inner
950.2.bg.a.389.33	yes	1200	25.14	even	10	inner
950.2.bg.a.739.8	yes	1200	475.264	even	90	inner