Embedded newform 495.2.bj.a.217.3

• Label: 495.2.bj.a.217.3
• Level: $495$
• Weight: $2$
• Character: 495.217
• Analytic conductor: $3.953$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	495.bj (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.95259490005\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{20})\)
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			217.3
Character	\(\chi\)	\(=\)	495.217
Dual form			495.2.bj.a.73.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0763931 - 0.482327i) q^{2} +(1.67531 + 0.544341i) q^{4} +(-1.47656 - 1.67922i) q^{5} +(-1.32402 - 2.59854i) q^{7} +(0.833936 - 1.63669i) q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 10 q^{2} + 2 q^{5} + 10 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\)	\(46\)	\(56\)	\(397\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0763931	−	0.482327i	0.0540181	−	0.341057i	−0.945847	−	0.324614i	\(-0.894766\pi\)
							0.999865	−	0.0164432i	\(-0.00523428\pi\)
\(3\)		0			0
							\(5\)	−1.47656	−	1.67922i	−0.660336	−	0.750970i
\(7\)	−1.32402	−	2.59854i	−0.500434	−	0.982157i								−0.993678	−	0.112266i	\(-0.964189\pi\)
							0.493244	−	0.869891i	\(-0.335811\pi\)
\(11\)	3.15977	−	1.00788i	0.952708	−	0.303888i
							\(13\)	−2.89049	−	0.457808i	−0.801677	−	0.126973i	−0.257866	−	0.966181i	\(-0.583019\pi\)
−0.543811	+	0.839208i	\(0.683019\pi\)
\(17\)	−5.20325	+	0.824113i	−1.26197	+	0.199877i	−0.751336	−	0.659920i	\(-0.770590\pi\)
							−0.510636	+	0.859797i	\(0.670590\pi\)
\(19\)	−1.26677	−	3.89873i	−0.290618	−	0.894429i	−0.984658	−	0.174494i	\(-0.944171\pi\)
							0.694041	−	0.719936i	\(-0.255829\pi\)
\(23\)	2.12046	−	2.12046i	0.442147	−	0.442147i	−0.450586	−	0.892733i	\(-0.648785\pi\)
							0.892733	+	0.450586i	\(0.148785\pi\)
\(29\)	0.817241	−	2.51521i	0.151758	−	0.467063i	−0.846060	−	0.533088i	\(-0.821032\pi\)
							0.997818	+	0.0660248i	\(0.0210316\pi\)
\(31\)	5.45328	−	3.96204i	0.979438	−	0.711603i	0.0218548	−	0.999761i	\(-0.493043\pi\)
							0.957583	+	0.288158i	\(0.0930429\pi\)
\(37\)	1.03649	−	0.528116i	0.170397	−	0.0868218i	−0.366712	−	0.930335i	\(-0.619516\pi\)
							0.537109	+	0.843513i	\(0.319516\pi\)
\(41\)	3.38136	−	1.09867i	0.528080	−	0.171584i	−0.0328289	−	0.999461i	\(-0.510452\pi\)
							0.560909	+	0.827877i	\(0.310452\pi\)
\(43\)	5.07292	+	5.07292i	0.773613	+	0.773613i	0.978736	−	0.205123i	\(-0.0657595\pi\)
							−0.205123	+	0.978736i	\(0.565759\pi\)
\(47\)	−1.67145	+	3.28041i	−0.243806	+	0.478497i	−0.980188	−	0.198070i	\(-0.936533\pi\)
							0.736382	+	0.676566i	\(0.236533\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.2.bj.a.217.3		32
3.2	odd	2		55.2.l.a.52.2	yes	32
5.3	odd	4	inner	495.2.bj.a.118.2		32
11.7	odd	10	inner	495.2.bj.a.172.2		32
12.11	even	2		880.2.cm.a.657.3		32
15.2	even	4		275.2.bm.b.118.2		32
15.8	even	4		55.2.l.a.8.3	yes	32
15.14	odd	2		275.2.bm.b.107.3		32
33.2	even	10		605.2.e.b.362.8		32
33.5	odd	10		605.2.m.d.112.2		32
33.8	even	10		605.2.m.d.457.2		32
33.14	odd	10		605.2.m.c.457.3		32
33.17	even	10		605.2.m.c.112.3		32
33.20	odd	10		605.2.e.b.362.9		32
33.26	odd	10		605.2.m.e.282.2		32
33.29	even	10		55.2.l.a.7.3	✓	32
33.32	even	2		605.2.m.e.602.3		32
55.18	even	20	inner	495.2.bj.a.73.3		32
60.23	odd	4		880.2.cm.a.833.2		32
132.95	odd	10		880.2.cm.a.337.2		32
165.8	odd	20		605.2.m.d.578.2		32
165.29	even	10		275.2.bm.b.7.2		32
165.38	even	20		605.2.m.d.233.2		32
165.53	even	20		605.2.e.b.483.8		32
165.62	odd	20		275.2.bm.b.18.3		32
165.68	odd	20		605.2.e.b.483.9		32
165.83	odd	20		605.2.m.c.233.3		32
165.98	odd	4		605.2.m.e.118.2		32
165.113	even	20		605.2.m.c.578.3		32
165.128	odd	20		55.2.l.a.18.2	yes	32
165.158	even	20		605.2.m.e.403.3		32
660.623	even	20		880.2.cm.a.513.3		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.l.a.7.3	✓	32	33.29	even	10
55.2.l.a.8.3	yes	32	15.8	even	4
55.2.l.a.18.2	yes	32	165.128	odd	20
55.2.l.a.52.2	yes	32	3.2	odd	2
275.2.bm.b.7.2		32	165.29	even	10
275.2.bm.b.18.3		32	165.62	odd	20
275.2.bm.b.107.3		32	15.14	odd	2
275.2.bm.b.118.2		32	15.2	even	4
495.2.bj.a.73.3		32	55.18	even	20	inner
495.2.bj.a.118.2		32	5.3	odd	4	inner
495.2.bj.a.172.2		32	11.7	odd	10	inner
495.2.bj.a.217.3		32	1.1	even	1	trivial
605.2.e.b.362.8		32	33.2	even	10
605.2.e.b.362.9		32	33.20	odd	10
605.2.e.b.483.8		32	165.53	even	20
605.2.e.b.483.9		32	165.68	odd	20
605.2.m.c.112.3		32	33.17	even	10
605.2.m.c.233.3		32	165.83	odd	20
605.2.m.c.457.3		32	33.14	odd	10
605.2.m.c.578.3		32	165.113	even	20
605.2.m.d.112.2		32	33.5	odd	10
605.2.m.d.233.2		32	165.38	even	20
605.2.m.d.457.2		32	33.8	even	10
605.2.m.d.578.2		32	165.8	odd	20
605.2.m.e.118.2		32	165.98	odd	4
605.2.m.e.282.2		32	33.26	odd	10
605.2.m.e.403.3		32	165.158	even	20
605.2.m.e.602.3		32	33.32	even	2
880.2.cm.a.337.2		32	132.95	odd	10
880.2.cm.a.513.3		32	660.623	even	20
880.2.cm.a.657.3		32	12.11	even	2
880.2.cm.a.833.2		32	60.23	odd	4