Embedded newform 950.2.bg.a.9.9

• Label: 950.2.bg.a.9.9
• 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			9.9
Character	\(\chi\)	\(=\)	950.9
Dual form			950.2.bg.a.739.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.898794 - 0.438371i) q^{2} +(-0.654190 + 1.04692i) q^{3} +(0.615661 + 0.788011i) q^{4} +(2.23588 - 0.0292423i) q^{5} +(1.04692 - 0.654190i) q^{6} +(3.55795 + 2.05418i) q^{7} +(-0.207912 - 0.978148i) q^{8} +(0.647030 + 1.32661i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q+O(q^{10}) \)

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
							\(3\)	−0.654190	+	1.04692i	−0.377697	+	0.604441i	−0.981217	−	0.192906i	\(-0.938209\pi\)
0.603520	+	0.797348i	\(0.293764\pi\)
\(5\)	2.23588	−	0.0292423i	0.999914	−	0.0130776i
							\(7\)	3.55795	+	2.05418i	1.34478	+	0.776408i	0.987504	−	0.157591i	\(-0.0503729\pi\)
0.357274	+	0.934000i	\(0.383706\pi\)
\(11\)	−0.483440	+	4.59962i	−0.145763	+	1.38684i	0.640031	+	0.768349i	\(0.278922\pi\)
							−0.785793	+	0.618489i	\(0.787745\pi\)
\(13\)	−0.0214057	+	0.306115i	−0.00593686	+	0.0849011i	−0.999629	−	0.0272427i	\(-0.991327\pi\)
							0.993692	+	0.112144i	\(0.0357717\pi\)
\(17\)	−0.398033	+	2.83216i	−0.0965373	+	0.686899i	0.880534	+	0.473984i	\(0.157185\pi\)
							−0.977071	+	0.212915i	\(0.931704\pi\)
\(19\)	3.66089	−	2.36599i	0.839866	−	0.542794i
							\(23\)	0.269214	−	0.938860i	0.0561350	−	0.195766i	−0.928198	−	0.372088i	\(-0.878642\pi\)
0.984333	+	0.176322i	\(0.0564199\pi\)
\(29\)	−5.77203	+	0.811206i	−1.07184	+	0.150637i	−0.652951	−	0.757400i	\(-0.726469\pi\)
							−0.418889	+	0.908038i	\(0.637580\pi\)
\(31\)	−2.40294	−	2.66873i	−0.431580	−	0.479318i	0.487650	−	0.873039i	\(-0.337854\pi\)
							−0.919230	+	0.393721i	\(0.871188\pi\)
\(37\)	−6.43810	−	8.86129i	−1.05842	−	1.45679i	−0.881276	−	0.472602i	\(-0.843315\pi\)
							−0.177141	−	0.984185i	\(-0.556685\pi\)
\(41\)	0.844355	−	3.38652i	0.131866	−	0.528886i	−0.867632	−	0.497207i	\(-0.834359\pi\)
							0.999498	−	0.0316793i	\(-0.0100855\pi\)
\(43\)	1.52036	−	4.17716i	0.231853	−	0.637011i	−0.768141	−	0.640280i	\(-0.778818\pi\)
							0.999995	+	0.00326878i	\(0.00104049\pi\)
\(47\)	−0.696440	−	4.95543i	−0.101586	−	0.722823i	−0.972754	−	0.231839i	\(-0.925526\pi\)
							0.871168	−	0.490985i	\(-0.163363\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bg.a.9.9	✓	1200
19.17	even	9	inner	950.2.bg.a.359.34	yes	1200
25.14	even	10	inner	950.2.bg.a.389.34	yes	1200
475.264	even	90	inner	950.2.bg.a.739.9	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bg.a.9.9	✓	1200	1.1	even	1	trivial
950.2.bg.a.359.34	yes	1200	19.17	even	9	inner
950.2.bg.a.389.34	yes	1200	25.14	even	10	inner
950.2.bg.a.739.9	yes	1200	475.264	even	90	inner