Embedded newform 950.2.bc.b.61.11

• Label: 950.2.bc.b.61.11
• 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			61.11
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.b.841.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.961262 + 0.275637i) q^{2} +(-0.890453 - 0.125145i) q^{3} +(0.848048 - 0.529919i) q^{4} +(0.325758 + 2.21221i) q^{5} +(0.890453 - 0.125145i) q^{6} +(-0.413882 + 0.716865i) q^{7} +(-0.669131 + 0.743145i) q^{8} +(-2.10654 - 0.604041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q + 54 q^{7} - 75 q^{8}+O(q^{10}) \)

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
							\(3\)	−0.890453	−	0.125145i	−0.514103	−	0.0722525i	−0.122649	−	0.992450i	\(-0.539139\pi\)
−0.391454	+	0.920198i	\(0.628028\pi\)
\(5\)	0.325758	+	2.21221i	0.145684	+	0.989331i
							\(7\)	−0.413882	+	0.716865i	−0.156433	+	0.270950i	−0.933580	−	0.358369i	\(-0.883333\pi\)
0.777147	+	0.629319i	\(0.216666\pi\)
\(11\)	3.24961	+	1.44682i	0.979794	+	0.436232i	0.833204	−	0.552965i	\(-0.186504\pi\)
							0.146589	+	0.989197i	\(0.453170\pi\)
\(13\)	4.54413	−	4.38822i	1.26032	−	1.21707i	0.295144	−	0.955453i	\(-0.404632\pi\)
							0.965171	−	0.261619i	\(-0.0842565\pi\)
\(17\)	−0.244358	+	6.99749i	−0.0592654	+	1.69714i	0.498298	+	0.867006i	\(0.333959\pi\)
							−0.557563	+	0.830134i	\(0.688264\pi\)
\(19\)	−4.35027	+	0.274112i	−0.998021	+	0.0628856i
							\(23\)	7.69317	+	0.537959i	1.60414	+	0.112172i	0.843545	−	0.537058i	\(-0.180465\pi\)
0.760592	+	0.649231i	\(0.224909\pi\)
\(29\)	−0.138084	−	3.95420i	−0.0256415	−	0.734276i	−0.942332	−	0.334679i	\(-0.891372\pi\)
							0.916691	−	0.399597i	\(-0.130850\pi\)
\(31\)	−6.72859	−	1.43021i	−1.20849	−	0.256873i	−0.440748	−	0.897631i	\(-0.645287\pi\)
							−0.767743	+	0.640758i	\(0.778620\pi\)
\(37\)	4.93766	+	3.58742i	0.811747	+	0.589769i	0.914337	−	0.404955i	\(-0.132713\pi\)
							−0.102590	+	0.994724i	\(0.532713\pi\)
\(41\)	−2.15831	+	4.42519i	−0.337071	+	0.691099i	−0.998250	−	0.0591294i	\(-0.981168\pi\)
							0.661179	+	0.750228i	\(0.270056\pi\)
\(43\)	−8.05692	+	6.76055i	−1.22867	+	1.03097i	−0.230344	+	0.973109i	\(0.573985\pi\)
							−0.998324	+	0.0578656i	\(0.981571\pi\)
\(47\)	−0.136299	−	3.90311i	−0.0198813	−	0.569327i	−0.967933	−	0.251209i	\(-0.919172\pi\)
							0.948051	−	0.318117i	\(-0.103051\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.b.61.11	✓	600
19.5	even	9	inner	950.2.bc.b.461.15	yes	600
25.16	even	5	inner	950.2.bc.b.441.15	yes	600
475.366	even	45	inner	950.2.bc.b.841.11	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.b.61.11	✓	600	1.1	even	1	trivial
950.2.bc.b.441.15	yes	600	25.16	even	5	inner
950.2.bc.b.461.15	yes	600	19.5	even	9	inner
950.2.bc.b.841.11	yes	600	475.366	even	45	inner