Embedded newform 950.2.bc.b.61.14

• Label: 950.2.bc.b.61.14
• 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			61.14
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.b.841.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.961262 + 0.275637i) q^{2} +(0.797908 + 0.112139i) q^{3} +(0.848048 - 0.529919i) q^{4} +(-2.22905 + 0.177042i) q^{5} +(-0.797908 + 0.112139i) q^{6} +(1.52948 - 2.64913i) q^{7} +(-0.669131 + 0.743145i) q^{8} +(-2.25970 - 0.647959i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q + 54 q^{7} - 75 q^{8}+O(q^{10}) \)

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
							\(3\)	0.797908	+	0.112139i	0.460673	+	0.0647433i	0.365692	−	0.930736i	\(-0.380832\pi\)
0.0949804	+	0.995479i	\(0.469721\pi\)
\(5\)	−2.22905	+	0.177042i	−0.996861	+	0.0791755i
							\(7\)	1.52948	−	2.64913i	0.578088	−	1.00128i	−0.417610	−	0.908626i	\(-0.637132\pi\)
0.995699	−	0.0926520i	\(-0.0295344\pi\)
\(11\)	1.63356	+	0.727306i	0.492536	+	0.219291i	0.637947	−	0.770080i	\(-0.279784\pi\)
							−0.145411	+	0.989371i	\(0.546450\pi\)
\(13\)	−1.09689	+	1.05925i	−0.304222	+	0.293784i	−0.831061	−	0.556181i	\(-0.812266\pi\)
							0.526839	+	0.849965i	\(0.323377\pi\)
\(17\)	−0.0564600	+	1.61680i	−0.0136936	+	0.392132i	0.973123	+	0.230288i	\(0.0739667\pi\)
							−0.986816	+	0.161845i	\(0.948256\pi\)
\(19\)	−1.30128	−	4.16013i	−0.298533	−	0.954399i
							\(23\)	−3.29352	−	0.230305i	−0.686746	−	0.0480220i	−0.277879	−	0.960616i	\(-0.589631\pi\)
−0.408867	+	0.912594i	\(0.634076\pi\)
\(29\)	0.290622	+	8.32231i	0.0539671	+	1.54541i	0.660786	+	0.750574i	\(0.270223\pi\)
							−0.606819	+	0.794840i	\(0.707555\pi\)
\(31\)	−9.95212	−	2.11539i	−1.78745	−	0.379935i	−0.809230	−	0.587491i	\(-0.800116\pi\)
							−0.978223	+	0.207556i	\(0.933449\pi\)
\(37\)	−8.41905	−	6.11680i	−1.38408	−	1.00560i	−0.996485	−	0.0837678i	\(-0.973305\pi\)
							−0.387599	−	0.921828i	\(-0.626695\pi\)
\(41\)	1.79798	−	3.68640i	0.280797	−	0.575719i	−0.711052	−	0.703140i	\(-0.751781\pi\)
							0.991849	+	0.127421i	\(0.0406698\pi\)
\(43\)	−4.77999	+	4.01088i	−0.728941	+	0.611654i	−0.929843	−	0.367957i	\(-0.880057\pi\)
							0.200902	+	0.979611i	\(0.435613\pi\)
\(47\)	−0.116246	−	3.32884i	−0.0169562	−	0.485561i	−0.977931	−	0.208927i	\(-0.933003\pi\)
							0.960975	−	0.276635i	\(-0.0892192\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.b.61.14	✓	600
19.5	even	9	inner	950.2.bc.b.461.12	yes	600
25.16	even	5	inner	950.2.bc.b.441.12	yes	600
475.366	even	45	inner	950.2.bc.b.841.14	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.b.61.14	✓	600	1.1	even	1	trivial
950.2.bc.b.441.12	yes	600	25.16	even	5	inner
950.2.bc.b.461.12	yes	600	19.5	even	9	inner
950.2.bc.b.841.14	yes	600	475.366	even	45	inner