Embedded newform 531.2.m.a.121.53

• Label: 531.2.m.a.121.53
• Level: $531$
• Weight: $2$
• Character: 531.121
• Analytic conductor: $4.240$
• Analytic rank: $0$
• Dimension: $3248$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	531.m (of order \(87\), degree \(56\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.24005634733\)
Analytic rank:	\(0\)
Dimension:	\(3248\)
Relative dimension:	\(58\) over \(\Q(\zeta_{87})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{87}]$

Embedding invariants

Embedding label			121.53
Character	\(\chi\)	\(=\)	531.121
Dual form			531.2.m.a.79.53

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.03999 + 1.43759i) q^{2} +(-1.58171 + 0.705825i) q^{3} +(1.42220 + 3.98187i) q^{4} +(-2.74457 + 1.51918i) q^{5} +(-4.24137 - 0.833970i) q^{6} +(1.63632 - 0.830031i) q^{7} +(-1.48769 + 5.35818i) q^{8} +(2.00362 - 2.23282i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3248 q - 29 q^{2} - 58 q^{3} + 29 q^{4} - 25 q^{5} - 66 q^{6} - 27 q^{7} - 116 q^{8} - 62 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/531\mathbb{Z}\right)^\times\).

\(n\)	\(119\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{25}{29}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.03999	+	1.43759i	1.44249	+	1.01653i	0.992335	+	0.123579i	\(0.0394373\pi\)
							0.450159	+	0.892948i	\(0.351367\pi\)
\(3\)	−1.58171	+	0.705825i	−0.913202	+	0.407508i
							\(5\)	−2.74457	+	1.51918i	−1.22741	+	0.679398i	−0.958863	−	0.283869i	\(-0.908382\pi\)
−0.268545	+	0.963267i	\(0.586543\pi\)
\(7\)	1.63632	−	0.830031i	0.618472	−	0.313722i	−0.115079	−	0.993356i	\(-0.536712\pi\)
							0.733551	+	0.679634i	\(0.237861\pi\)
\(11\)	−6.07319	+	2.29361i	−1.83114	+	0.691551i	−0.843012	+	0.537896i	\(0.819219\pi\)
							−0.988125	+	0.153655i	\(0.950895\pi\)
\(13\)	0.346999	+	2.73091i	0.0962401	+	0.757419i	0.965012	+	0.262206i	\(0.0844501\pi\)
							−0.868772	+	0.495213i	\(0.835090\pi\)
\(17\)	−0.172405	−	3.17982i	−0.0418143	−	0.771220i	−0.941979	−	0.335671i	\(-0.891037\pi\)
							0.900165	−	0.435549i	\(-0.143446\pi\)
\(19\)	4.19585	−	3.97452i	0.962594	−	0.911818i	−0.0335600	−	0.999437i	\(-0.510684\pi\)
							0.996154	+	0.0876192i	\(0.0279259\pi\)
\(23\)	−2.27442	+	7.17586i	−0.474249	+	1.49627i	0.353641	+	0.935381i	\(0.384943\pi\)
							−0.827890	+	0.560890i	\(0.810459\pi\)
\(29\)	0.296693	+	3.27759i	0.0550945	+	0.608633i	0.975842	+	0.218475i	\(0.0701083\pi\)
							−0.920748	+	0.390158i	\(0.872420\pi\)
\(31\)	−2.64078	+	0.784829i	−0.474298	+	0.140959i	−0.513174	−	0.858285i	\(-0.671530\pi\)
							0.0388757	+	0.999244i	\(0.487622\pi\)
\(37\)	2.51381	+	9.05394i	0.413268	+	1.48846i	0.818640	+	0.574306i	\(0.194728\pi\)
							−0.405372	+	0.914152i	\(0.632858\pi\)
\(41\)	5.10804	−	5.59131i	0.797741	−	0.873216i	−0.196486	−	0.980507i	\(-0.562953\pi\)
							0.994228	+	0.107291i	\(0.0342175\pi\)
\(43\)	−1.61507	+	1.32241i	−0.246296	+	0.201665i	−0.748593	−	0.663030i	\(-0.769270\pi\)
							0.502297	+	0.864695i	\(0.332488\pi\)
\(47\)	−1.80978	−	1.00175i	−0.263984	−	0.146121i	0.346749	−	0.937958i	\(-0.387286\pi\)
							−0.610732	+	0.791837i	\(0.709125\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.m.a.121.53	yes	3248
9.7	even	3	inner	531.2.m.a.475.53	yes	3248
59.20	even	29	inner	531.2.m.a.256.53	yes	3248
531.79	even	87	inner	531.2.m.a.79.53	✓	3248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.m.a.79.53	✓	3248	531.79	even	87	inner
531.2.m.a.121.53	yes	3248	1.1	even	1	trivial
531.2.m.a.256.53	yes	3248	59.20	even	29	inner
531.2.m.a.475.53	yes	3248	9.7	even	3	inner