Embedded newform 950.2.bc.a.61.3

• Label: 950.2.bc.a.61.3
• 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			61.3
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.a.841.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.961262 - 0.275637i) q^{2} +(-2.52861 - 0.355373i) q^{3} +(0.848048 - 0.529919i) q^{4} +(2.12329 + 0.701180i) q^{5} +(-2.52861 + 0.355373i) q^{6} +(1.00675 - 1.74375i) q^{7} +(0.669131 - 0.743145i) q^{8} +(3.38381 + 0.970291i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q - 42 q^{7} + 75 q^{8}+O(q^{10}) \)

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
							\(3\)	−2.52861	−	0.355373i	−1.45990	−	0.205175i	−0.635900	−	0.771771i	\(-0.719371\pi\)
−0.823995	+	0.566597i	\(0.808260\pi\)
\(5\)	2.12329	+	0.701180i	0.949563	+	0.313577i
							\(7\)	1.00675	−	1.74375i	0.380517	−	0.659075i	−0.610619	−	0.791924i	\(-0.709079\pi\)
0.991136	+	0.132850i	\(0.0424128\pi\)
\(11\)	1.49972	+	0.667718i	0.452182	+	0.201324i	0.620175	−	0.784464i	\(-0.287062\pi\)
							−0.167992	+	0.985788i	\(0.553728\pi\)
\(13\)	0.651350	−	0.629001i	0.180652	−	0.174453i	−0.598834	−	0.800873i	\(-0.704369\pi\)
							0.779486	+	0.626420i	\(0.215480\pi\)
\(17\)	−0.253883	+	7.27025i	−0.0615756	+	1.76329i	0.439830	+	0.898081i	\(0.355039\pi\)
							−0.501405	+	0.865213i	\(0.667183\pi\)
\(19\)	0.863350	−	4.27254i	0.198066	−	0.980189i
							\(23\)	−2.73986	−	0.191590i	−0.571300	−	0.0399492i	−0.218815	−	0.975766i	\(-0.570219\pi\)
−0.352486	+	0.935817i	\(0.614663\pi\)
\(29\)	−0.337915	−	9.67663i	−0.0627493	−	1.79691i	−0.469056	−	0.883169i	\(-0.655406\pi\)
							0.406306	−	0.913737i	\(-0.366817\pi\)
\(31\)	3.10264	+	0.659486i	0.557250	+	0.118447i	0.477923	−	0.878402i	\(-0.341390\pi\)
							0.0793266	+	0.996849i	\(0.474723\pi\)
\(37\)	5.54338	+	4.02750i	0.911325	+	0.662117i	0.941350	−	0.337433i	\(-0.109558\pi\)
							−0.0300244	+	0.999549i	\(0.509558\pi\)
\(41\)	3.56403	−	7.30735i	0.556609	−	1.14122i	−0.416128	−	0.909306i	\(-0.636613\pi\)
							0.972737	−	0.231911i	\(-0.0744978\pi\)
\(43\)	0.227207	−	0.190650i	0.0346488	−	0.0290738i	−0.625299	−	0.780385i	\(-0.715023\pi\)
							0.659948	+	0.751312i	\(0.270578\pi\)
\(47\)	−0.175941	−	5.03828i	−0.0256636	−	0.734909i	−0.942218	−	0.335000i	\(-0.891264\pi\)
							0.916555	−	0.399910i	\(-0.130958\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.a.61.3	✓	600
19.5	even	9	inner	950.2.bc.a.461.23	yes	600
25.16	even	5	inner	950.2.bc.a.441.23	yes	600
475.366	even	45	inner	950.2.bc.a.841.3	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.a.61.3	✓	600	1.1	even	1	trivial
950.2.bc.a.441.23	yes	600	25.16	even	5	inner
950.2.bc.a.461.23	yes	600	19.5	even	9	inner
950.2.bc.a.841.3	yes	600	475.366	even	45	inner