Embedded newform 950.2.bc.b.61.6

• Label: 950.2.bc.b.61.6
• 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			61.6
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.b.841.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.961262 + 0.275637i) q^{2} +(-2.13154 - 0.299568i) q^{3} +(0.848048 - 0.529919i) q^{4} +(1.98945 + 1.02082i) q^{5} +(2.13154 - 0.299568i) q^{6} +(2.50721 - 4.34262i) q^{7} +(-0.669131 + 0.743145i) q^{8} +(1.56993 + 0.450170i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q + 54 q^{7} - 75 q^{8}+O(q^{10}) \)

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
							\(3\)	−2.13154	−	0.299568i	−1.23064	−	0.172956i	−0.506277	−	0.862371i	\(-0.668979\pi\)
−0.724367	+	0.689415i	\(0.757868\pi\)
\(5\)	1.98945	+	1.02082i	0.889711	+	0.456525i
							\(7\)	2.50721	−	4.34262i	0.947637	−	1.64135i	0.197252	−	0.980353i	\(-0.436798\pi\)
0.750384	−	0.661002i	\(-0.229868\pi\)
\(11\)	−6.03719	−	2.68793i	−1.82028	−	0.810441i	−0.945227	−	0.326414i	\(-0.894160\pi\)
							−0.875053	−	0.484027i	\(-0.839174\pi\)
\(13\)	−2.66272	+	2.57136i	−0.738506	+	0.713167i	−0.965406	−	0.260751i	\(-0.916030\pi\)
							0.226900	+	0.973918i	\(0.427141\pi\)
\(17\)	0.0185393	−	0.530897i	0.00449644	−	0.128761i	−0.995067	−	0.0992077i	\(-0.968369\pi\)
							0.999563	−	0.0295537i	\(-0.00940859\pi\)
\(19\)	−3.02115	−	3.14208i	−0.693099	−	0.720842i
							\(23\)	4.83849	+	0.338340i	1.00890	+	0.0705488i	0.564628	−	0.825346i	\(-0.309020\pi\)
0.444267	+	0.895894i	\(0.353464\pi\)
\(29\)	0.0775369	+	2.22037i	0.0143982	+	0.412312i	0.985096	+	0.172007i	\(0.0550251\pi\)
							−0.970698	+	0.240305i	\(0.922753\pi\)
\(31\)	−8.99713	−	1.91240i	−1.61593	−	0.343477i	−0.690778	−	0.723067i	\(-0.742732\pi\)
							−0.925155	+	0.379590i	\(0.876065\pi\)
\(37\)	2.13702	+	1.55264i	0.351325	+	0.255252i	0.749425	−	0.662090i	\(-0.230330\pi\)
							−0.398100	+	0.917342i	\(0.630330\pi\)
\(41\)	−2.08989	+	4.28492i	−0.326387	+	0.669192i	−0.997385	−	0.0722667i	\(-0.976977\pi\)
							0.670999	+	0.741459i	\(0.265866\pi\)
\(43\)	−7.41856	+	6.22491i	−1.13132	+	0.949289i	−0.999120	−	0.0419337i	\(-0.986648\pi\)
							−0.132199	+	0.991223i	\(0.542204\pi\)
\(47\)	0.261492	+	7.48816i	0.0381426	+	1.09226i	0.856044	+	0.516902i	\(0.172915\pi\)
							−0.817902	+	0.575358i	\(0.804863\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.b.61.6	✓	600
19.5	even	9	inner	950.2.bc.b.461.20	yes	600
25.16	even	5	inner	950.2.bc.b.441.20	yes	600
475.366	even	45	inner	950.2.bc.b.841.6	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.b.61.6	✓	600	1.1	even	1	trivial
950.2.bc.b.441.20	yes	600	25.16	even	5	inner
950.2.bc.b.461.20	yes	600	19.5	even	9	inner
950.2.bc.b.841.6	yes	600	475.366	even	45	inner