Embedded newform 950.2.bc.a.61.4

• Label: 950.2.bc.a.61.4
• 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			61.4
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.a.841.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.961262 - 0.275637i) q^{2} +(-2.47903 - 0.348405i) q^{3} +(0.848048 - 0.529919i) q^{4} +(1.80399 - 1.32121i) q^{5} +(-2.47903 + 0.348405i) q^{6} +(-0.861925 + 1.49290i) q^{7} +(0.669131 - 0.743145i) q^{8} +(3.14043 + 0.900503i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q - 42 q^{7} + 75 q^{8}+O(q^{10}) \)

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
							\(3\)	−2.47903	−	0.348405i	−1.43127	−	0.201152i	−0.619439	−	0.785045i	\(-0.712640\pi\)
−0.811831	+	0.583893i	\(0.801529\pi\)
\(5\)	1.80399	−	1.32121i	0.806770	−	0.590865i
							\(7\)	−0.861925	+	1.49290i	−0.325777	+	0.564262i	−0.981669	−	0.190592i	\(-0.938959\pi\)
0.655892	+	0.754855i	\(0.272293\pi\)
\(11\)	−5.33008	−	2.37310i	−1.60708	−	0.715518i	−0.610037	−	0.792373i	\(-0.708845\pi\)
							−0.997042	+	0.0768556i	\(0.975512\pi\)
\(13\)	2.19871	−	2.12327i	0.609813	−	0.588889i	−0.324262	−	0.945967i	\(-0.605116\pi\)
							0.934075	+	0.357078i	\(0.116227\pi\)
\(17\)	0.0284793	−	0.815539i	0.00690723	−	0.197797i	−0.991032	−	0.133627i	\(-0.957338\pi\)
							0.997939	−	0.0641704i	\(-0.0204401\pi\)
\(19\)	−3.84698	+	2.04957i	−0.882558	+	0.470204i
							\(23\)	−0.0966127	−	0.00675582i	−0.0201451	−	0.00140869i	0.0596804	−	0.998218i	\(-0.480992\pi\)
−0.0798255	+	0.996809i	\(0.525436\pi\)
\(29\)	−0.133710	−	3.82895i	−0.0248293	−	0.711017i	−0.946436	−	0.322892i	\(-0.895345\pi\)
							0.921607	−	0.388125i	\(-0.126877\pi\)
\(31\)	−2.11962	−	0.450538i	−0.380694	−	0.0809191i	0.0135899	−	0.999908i	\(-0.495674\pi\)
							−0.394284	+	0.918989i	\(0.629007\pi\)
\(37\)	−9.69654	−	7.04495i	−1.59410	−	1.15818i	−0.897771	−	0.440462i	\(-0.854815\pi\)
							−0.696331	−	0.717721i	\(-0.745185\pi\)
\(41\)	−4.56915	+	9.36814i	−0.713581	+	1.46306i	0.165413	+	0.986224i	\(0.447104\pi\)
							−0.878994	+	0.476833i	\(0.841785\pi\)
\(43\)	−3.85800	+	3.23725i	−0.588340	+	0.493676i	−0.887674	−	0.460473i	\(-0.847680\pi\)
							0.299334	+	0.954148i	\(0.403236\pi\)
\(47\)	−0.442184	−	12.6625i	−0.0644991	−	1.84701i	−0.414240	−	0.910167i	\(-0.635953\pi\)
							0.349741	−	0.936846i	\(-0.386269\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.a.61.4	✓	600
19.5	even	9	inner	950.2.bc.a.461.22	yes	600
25.16	even	5	inner	950.2.bc.a.441.22	yes	600
475.366	even	45	inner	950.2.bc.a.841.4	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.a.61.4	✓	600	1.1	even	1	trivial
950.2.bc.a.441.22	yes	600	25.16	even	5	inner
950.2.bc.a.461.22	yes	600	19.5	even	9	inner
950.2.bc.a.841.4	yes	600	475.366	even	45	inner