Embedded newform 950.2.bc.b.61.19

• Label: 950.2.bc.b.61.19
• 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			61.19
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.b.841.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.961262 + 0.275637i) q^{2} +(1.24823 + 0.175428i) q^{3} +(0.848048 - 0.529919i) q^{4} +(2.17900 - 0.501953i) q^{5} +(-1.24823 + 0.175428i) q^{6} +(0.830929 - 1.43921i) q^{7} +(-0.669131 + 0.743145i) q^{8} +(-1.35647 - 0.388962i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q + 54 q^{7} - 75 q^{8}+O(q^{10}) \)

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
							\(3\)	1.24823	+	0.175428i	0.720668	+	0.101283i	0.489967	−	0.871741i	\(-0.337009\pi\)
0.230702	+	0.973025i	\(0.425898\pi\)
\(5\)	2.17900	−	0.501953i	0.974479	−	0.224480i
							\(7\)	0.830929	−	1.43921i	0.314062	−	0.543971i	−0.665176	−	0.746687i	\(-0.731643\pi\)
0.979238	+	0.202716i	\(0.0649768\pi\)
\(11\)	−2.06740	−	0.920464i	−0.623344	−	0.277530i	0.0706617	−	0.997500i	\(-0.477489\pi\)
							−0.694005	+	0.719970i	\(0.744156\pi\)
\(13\)	3.64119	−	3.51626i	1.00988	−	0.975234i	0.0101896	−	0.999948i	\(-0.496756\pi\)
							0.999695	+	0.0247143i	\(0.00786761\pi\)
\(17\)	0.118526	−	3.39415i	0.0287469	−	0.823203i	−0.896322	−	0.443404i	\(-0.853771\pi\)
							0.925069	−	0.379799i	\(-0.124007\pi\)
\(19\)	−4.28602	+	0.793728i	−0.983281	+	0.182094i
							\(23\)	1.75623	+	0.122808i	0.366199	+	0.0256072i	0.251671	−	0.967813i	\(-0.419020\pi\)
0.114528	+	0.993420i	\(0.463464\pi\)
\(29\)	−0.167498	−	4.79652i	−0.0311036	−	0.890692i	−0.910230	−	0.414103i	\(-0.864095\pi\)
							0.879126	−	0.476589i	\(-0.158127\pi\)
\(31\)	7.16785	+	1.52357i	1.28738	+	0.273642i	0.800240	−	0.599680i	\(-0.204706\pi\)
							0.487143	+	0.873322i	\(0.338039\pi\)
\(37\)	−1.26161	−	0.916611i	−0.207407	−	0.150690i	0.479233	−	0.877688i	\(-0.340915\pi\)
							−0.686640	+	0.726998i	\(0.740915\pi\)
\(41\)	2.07997	−	4.26457i	0.324837	−	0.666014i	−0.672409	−	0.740180i	\(-0.734740\pi\)
							0.997246	+	0.0741657i	\(0.0236294\pi\)
\(43\)	−2.70054	+	2.26603i	−0.411829	+	0.345566i	−0.825045	−	0.565068i	\(-0.808850\pi\)
							0.413215	+	0.910633i	\(0.364406\pi\)
\(47\)	0.429373	+	12.2956i	0.0626305	+	1.79350i	0.472471	+	0.881346i	\(0.343362\pi\)
							−0.409841	+	0.912157i	\(0.634416\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.b.61.19	✓	600
19.5	even	9	inner	950.2.bc.b.461.7	yes	600
25.16	even	5	inner	950.2.bc.b.441.7	yes	600
475.366	even	45	inner	950.2.bc.b.841.19	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.b.61.19	✓	600	1.1	even	1	trivial
950.2.bc.b.441.7	yes	600	25.16	even	5	inner
950.2.bc.b.461.7	yes	600	19.5	even	9	inner
950.2.bc.b.841.19	yes	600	475.366	even	45	inner