Embedded newform 275.2.bm.b.7.2

• Label: 275.2.bm.b.7.2
• Level: $275$
• Weight: $2$
• Character: 275.7
• Analytic conductor: $2.196$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	275.bm (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.19588605559\)
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			7.2
Character	\(\chi\)	\(=\)	275.7
Dual form			275.2.bm.b.118.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.482327 + 0.0763931i) q^{2} +(-0.517260 - 1.01518i) q^{3} +(-1.67531 + 0.544341i) q^{4} +(0.327041 + 0.450133i) q^{6} +(2.59854 + 1.32402i) q^{7} +(1.63669 - 0.833936i) q^{8} +(1.00032 - 1.37683i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 10 q^{2} + 4 q^{3} - 20 q^{6} + 10 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(177\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.482327	+	0.0763931i	−0.341057	+	0.0540181i	−0.324614	−	0.945847i	\(-0.605234\pi\)
							−0.0164432	+	0.999865i	\(0.505234\pi\)
\(3\)	−0.517260	−	1.01518i	−0.298640	−	0.586114i	0.692114	−	0.721789i	\(-0.256680\pi\)
							−0.990753	+	0.135675i	\(0.956680\pi\)
\(5\)		0			0
							\(7\)	2.59854	+	1.32402i	0.982157	+	0.500434i	0.869891	−	0.493244i	\(-0.164189\pi\)
0.112266	+	0.993678i	\(0.464189\pi\)
\(11\)	−3.15977	−	1.00788i	−0.952708	−	0.303888i
							\(13\)	−0.457808	−	2.89049i	−0.126973	−	0.801677i	−0.966181	−	0.257866i	\(-0.916981\pi\)
0.839208	−	0.543811i	\(-0.183019\pi\)
\(17\)	0.824113	−	5.20325i	0.199877	−	1.26197i	−0.659920	−	0.751336i	\(-0.729410\pi\)
							0.859797	−	0.510636i	\(-0.170590\pi\)
\(19\)	1.26677	−	3.89873i	0.290618	−	0.894429i	−0.694041	−	0.719936i	\(-0.744171\pi\)
							0.984658	−	0.174494i	\(-0.0558288\pi\)
\(23\)	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	0.997818	−	0.0660248i	\(-0.0210316\pi\)
							−0.846060	+	0.533088i	\(0.821032\pi\)
\(31\)	5.45328	+	3.96204i	0.979438	+	0.711603i	0.957583	−	0.288158i	\(-0.0930429\pi\)
							0.0218548	+	0.999761i	\(0.493043\pi\)
\(37\)	0.528116	−	1.03649i	0.0868218	−	0.170397i	−0.843513	−	0.537109i	\(-0.819516\pi\)
							0.930335	+	0.366712i	\(0.119516\pi\)
\(41\)	−3.38136	−	1.09867i	−0.528080	−	0.171584i	0.0328289	−	0.999461i	\(-0.489548\pi\)
							−0.560909	+	0.827877i	\(0.689548\pi\)
\(43\)	5.07292	+	5.07292i	0.773613	+	0.773613i	0.978736	−	0.205123i	\(-0.0657595\pi\)
							−0.205123	+	0.978736i	\(0.565759\pi\)
\(47\)	3.28041	−	1.67145i	0.478497	−	0.243806i	−0.198070	−	0.980188i	\(-0.563467\pi\)
							0.676566	+	0.736382i	\(0.263467\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.2.bm.b.7.2		32
5.2	odd	4		55.2.l.a.18.2	yes	32
5.3	odd	4	inner	275.2.bm.b.18.3		32
5.4	even	2		55.2.l.a.7.3	✓	32
11.8	odd	10	inner	275.2.bm.b.107.3		32
15.2	even	4		495.2.bj.a.73.3		32
15.14	odd	2		495.2.bj.a.172.2		32
20.7	even	4		880.2.cm.a.513.3		32
20.19	odd	2		880.2.cm.a.337.2		32
55.2	even	20		605.2.m.c.578.3		32
55.4	even	10		605.2.m.c.112.3		32
55.7	even	20		605.2.m.d.233.2		32
55.8	even	20	inner	275.2.bm.b.118.2		32
55.9	even	10		605.2.m.d.457.2		32
55.14	even	10		605.2.m.e.602.3		32
55.17	even	20		605.2.e.b.483.8		32
55.19	odd	10		55.2.l.a.52.2	yes	32
55.24	odd	10		605.2.m.c.457.3		32
55.27	odd	20		605.2.e.b.483.9		32
55.29	odd	10		605.2.m.d.112.2		32
55.32	even	4		605.2.m.e.403.3		32
55.37	odd	20		605.2.m.c.233.3		32
55.39	odd	10		605.2.e.b.362.9		32
55.42	odd	20		605.2.m.d.578.2		32
55.47	odd	20		605.2.m.e.118.2		32
55.49	even	10		605.2.e.b.362.8		32
55.52	even	20		55.2.l.a.8.3	yes	32
55.54	odd	2		605.2.m.e.282.2		32
165.74	even	10		495.2.bj.a.217.3		32
165.107	odd	20		495.2.bj.a.118.2		32
220.19	even	10		880.2.cm.a.657.3		32
220.107	odd	20		880.2.cm.a.833.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.l.a.7.3	✓	32	5.4	even	2
55.2.l.a.8.3	yes	32	55.52	even	20
55.2.l.a.18.2	yes	32	5.2	odd	4
55.2.l.a.52.2	yes	32	55.19	odd	10
275.2.bm.b.7.2		32	1.1	even	1	trivial
275.2.bm.b.18.3		32	5.3	odd	4	inner
275.2.bm.b.107.3		32	11.8	odd	10	inner
275.2.bm.b.118.2		32	55.8	even	20	inner
495.2.bj.a.73.3		32	15.2	even	4
495.2.bj.a.118.2		32	165.107	odd	20
495.2.bj.a.172.2		32	15.14	odd	2
495.2.bj.a.217.3		32	165.74	even	10
605.2.e.b.362.8		32	55.49	even	10
605.2.e.b.362.9		32	55.39	odd	10
605.2.e.b.483.8		32	55.17	even	20
605.2.e.b.483.9		32	55.27	odd	20
605.2.m.c.112.3		32	55.4	even	10
605.2.m.c.233.3		32	55.37	odd	20
605.2.m.c.457.3		32	55.24	odd	10
605.2.m.c.578.3		32	55.2	even	20
605.2.m.d.112.2		32	55.29	odd	10
605.2.m.d.233.2		32	55.7	even	20
605.2.m.d.457.2		32	55.9	even	10
605.2.m.d.578.2		32	55.42	odd	20
605.2.m.e.118.2		32	55.47	odd	20
605.2.m.e.282.2		32	55.54	odd	2
605.2.m.e.403.3		32	55.32	even	4
605.2.m.e.602.3		32	55.14	even	10
880.2.cm.a.337.2		32	20.19	odd	2
880.2.cm.a.513.3		32	20.7	even	4
880.2.cm.a.657.3		32	220.19	even	10
880.2.cm.a.833.2		32	220.107	odd	20