Embedded newform 525.2.bc.e.157.1

• Label: 525.2.bc.e.157.1
• Level: $525$
• Weight: $2$
• Character: 525.157
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bc (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			157.1
Character	\(\chi\)	\(=\)	525.157
Dual form			525.2.bc.e.418.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.650518 + 2.42777i) q^{2} +(0.965926 - 0.258819i) q^{3} +(-3.73883 - 2.15861i) q^{4} +2.51341i q^{6} +(-2.09305 - 1.61838i) q^{7} +(4.11829 - 4.11829i) q^{8} +(0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.650518	+	2.42777i	−0.459986	+	1.71669i	0.213015	+	0.977049i	\(0.431672\pi\)
							−0.673001	+	0.739642i	\(0.734995\pi\)
\(3\)	0.965926	−	0.258819i	0.557678	−	0.149429i
							\(5\)		0			0
\(7\)	−2.09305	−	1.61838i	−0.791097	−	0.611691i
							\(11\)	2.73807	−	4.74248i	0.825561	−	1.42991i	−0.0759295	−	0.997113i	\(-0.524192\pi\)
0.901490	−	0.432800i	\(-0.142474\pi\)
\(13\)	−0.579674	−	0.579674i	−0.160773	−	0.160773i	0.622136	−	0.782909i	\(-0.286265\pi\)
							−0.782909	+	0.622136i	\(0.786265\pi\)
\(17\)	−1.22971	−	4.58934i	−0.298248	−	1.11308i	−0.938603	−	0.344999i	\(-0.887879\pi\)
							0.640355	−	0.768079i	\(-0.278787\pi\)
\(19\)	−0.220281	−	0.381538i	−0.0505359	−	0.0875308i	0.839651	−	0.543127i	\(-0.182760\pi\)
							−0.890187	+	0.455596i	\(0.849426\pi\)
\(23\)	−1.70673	−	0.457316i	−0.355877	−	0.0953570i	0.0764510	−	0.997073i	\(-0.475641\pi\)
							−0.432328	+	0.901716i	\(0.642308\pi\)
\(29\)			0.853158i			0.158427i	0.996858	+	0.0792137i	\(0.0252410\pi\)
							−0.996858	+	0.0792137i	\(0.974759\pi\)
\(31\)	2.32463	+	1.34213i	0.417516	+	0.241053i	0.694014	−	0.719962i	\(-0.255841\pi\)
							−0.276498	+	0.961014i	\(0.589174\pi\)
\(37\)	−0.0668745	+	0.249579i	−0.0109941	+	0.0410306i	−0.971205	−	0.238245i	\(-0.923428\pi\)
							0.960211	+	0.279276i	\(0.0900944\pi\)
\(41\)		−	0.321873i		−	0.0502681i	−0.999684	−	0.0251341i	\(-0.991999\pi\)
							0.999684	−	0.0251341i	\(-0.00800127\pi\)
\(43\)	0.631635	−	0.631635i	0.0963234	−	0.0963234i	−0.657303	−	0.753626i	\(-0.728303\pi\)
							0.753626	+	0.657303i	\(0.228303\pi\)
\(47\)	7.91508	+	2.12084i	1.15453	+	0.309356i	0.784780	−	0.619774i	\(-0.212776\pi\)
							0.369752	+	0.929130i	\(0.379443\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bc.e.157.1		32
5.2	odd	4		105.2.u.a.73.1	yes	32
5.3	odd	4	inner	525.2.bc.e.493.8		32
5.4	even	2		105.2.u.a.52.8	✓	32
7.5	odd	6	inner	525.2.bc.e.82.8		32
15.2	even	4		315.2.bz.d.73.8		32
15.14	odd	2		315.2.bz.d.262.1		32
35.2	odd	12		735.2.v.b.313.8		32
35.4	even	6		735.2.m.c.97.16		32
35.9	even	6		735.2.v.b.607.1		32
35.12	even	12		105.2.u.a.103.8	yes	32
35.17	even	12		735.2.m.c.538.16		32
35.19	odd	6		105.2.u.a.82.1	yes	32
35.24	odd	6		735.2.m.c.97.15		32
35.27	even	4		735.2.v.b.178.1		32
35.32	odd	12		735.2.m.c.538.15		32
35.33	even	12	inner	525.2.bc.e.418.1		32
35.34	odd	2		735.2.v.b.472.8		32
105.47	odd	12		315.2.bz.d.208.1		32
105.89	even	6		315.2.bz.d.82.8		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.u.a.52.8	✓	32	5.4	even	2
105.2.u.a.73.1	yes	32	5.2	odd	4
105.2.u.a.82.1	yes	32	35.19	odd	6
105.2.u.a.103.8	yes	32	35.12	even	12
315.2.bz.d.73.8		32	15.2	even	4
315.2.bz.d.82.8		32	105.89	even	6
315.2.bz.d.208.1		32	105.47	odd	12
315.2.bz.d.262.1		32	15.14	odd	2
525.2.bc.e.82.8		32	7.5	odd	6	inner
525.2.bc.e.157.1		32	1.1	even	1	trivial
525.2.bc.e.418.1		32	35.33	even	12	inner
525.2.bc.e.493.8		32	5.3	odd	4	inner
735.2.m.c.97.15		32	35.24	odd	6
735.2.m.c.97.16		32	35.4	even	6
735.2.m.c.538.15		32	35.32	odd	12
735.2.m.c.538.16		32	35.17	even	12
735.2.v.b.178.1		32	35.27	even	4
735.2.v.b.313.8		32	35.2	odd	12
735.2.v.b.472.8		32	35.34	odd	2
735.2.v.b.607.1		32	35.9	even	6