Embedded newform 315.2.bz.d.208.1

• Label: 315.2.bz.d.208.1
• Level: $315$
• Weight: $2$
• Character: 315.208
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.bz (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
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			208.1
Character	\(\chi\)	\(=\)	315.208
Dual form			315.2.bz.d.262.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.650518 - 2.42777i) q^{2} +(-3.73883 + 2.15861i) q^{4} +(1.42436 - 1.72371i) q^{5} +(2.09305 - 1.61838i) q^{7} +(4.11829 + 4.11829i) q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 12 q^{5} + 8 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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	−0.673001	−	0.739642i	\(-0.734995\pi\)
							0.213015	−	0.977049i	\(-0.431672\pi\)
\(3\)		0			0
							\(5\)	1.42436	−	1.72371i	0.636995	−	0.770868i
\(7\)	2.09305	−	1.61838i	0.791097	−	0.611691i
							\(11\)	−2.73807	−	4.74248i	−0.825561	−	1.42991i	−0.901490	−	0.432800i	\(-0.857526\pi\)
0.0759295	−	0.997113i	\(-0.475808\pi\)
\(13\)	0.579674	−	0.579674i	0.160773	−	0.160773i	−0.622136	−	0.782909i	\(-0.713735\pi\)
							0.782909	+	0.622136i	\(0.213735\pi\)
\(17\)	−1.22971	+	4.58934i	−0.298248	+	1.11308i	0.640355	+	0.768079i	\(0.278787\pi\)
							−0.938603	+	0.344999i	\(0.887879\pi\)
\(19\)	−0.220281	+	0.381538i	−0.0505359	+	0.0875308i	−0.890187	−	0.455596i	\(-0.849426\pi\)
							0.839651	+	0.543127i	\(0.182760\pi\)
\(23\)	−1.70673	+	0.457316i	−0.355877	+	0.0953570i	−0.432328	−	0.901716i	\(-0.642308\pi\)
							0.0764510	+	0.997073i	\(0.475641\pi\)
\(29\)			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	−0.276498	−	0.961014i	\(-0.589174\pi\)
							0.694014	+	0.719962i	\(0.255841\pi\)
\(37\)	0.0668745	+	0.249579i	0.0109941	+	0.0410306i	0.971205	−	0.238245i	\(-0.0765722\pi\)
							−0.960211	+	0.279276i	\(0.909906\pi\)
\(41\)		−	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	\(-0.271697\pi\)
							−0.753626	+	0.657303i	\(0.771697\pi\)
\(47\)	7.91508	−	2.12084i	1.15453	−	0.309356i	0.369752	−	0.929130i	\(-0.379443\pi\)
							0.784780	+	0.619774i	\(0.212776\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bz.d.208.1		32
3.2	odd	2		105.2.u.a.103.8	yes	32
5.2	odd	4	inner	315.2.bz.d.82.8		32
7.3	odd	6	inner	315.2.bz.d.73.8		32
15.2	even	4		105.2.u.a.82.1	yes	32
15.8	even	4		525.2.bc.e.82.8		32
15.14	odd	2		525.2.bc.e.418.1		32
21.2	odd	6		735.2.m.c.538.16		32
21.5	even	6		735.2.m.c.538.15		32
21.11	odd	6		735.2.v.b.178.1		32
21.17	even	6		105.2.u.a.73.1	yes	32
21.20	even	2		735.2.v.b.313.8		32
35.17	even	12	inner	315.2.bz.d.262.1		32
105.2	even	12		735.2.m.c.97.15		32
105.17	odd	12		105.2.u.a.52.8	✓	32
105.32	even	12		735.2.v.b.472.8		32
105.38	odd	12		525.2.bc.e.157.1		32
105.47	odd	12		735.2.m.c.97.16		32
105.59	even	6		525.2.bc.e.493.8		32
105.62	odd	4		735.2.v.b.607.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.u.a.52.8	✓	32	105.17	odd	12
105.2.u.a.73.1	yes	32	21.17	even	6
105.2.u.a.82.1	yes	32	15.2	even	4
105.2.u.a.103.8	yes	32	3.2	odd	2
315.2.bz.d.73.8		32	7.3	odd	6	inner
315.2.bz.d.82.8		32	5.2	odd	4	inner
315.2.bz.d.208.1		32	1.1	even	1	trivial
315.2.bz.d.262.1		32	35.17	even	12	inner
525.2.bc.e.82.8		32	15.8	even	4
525.2.bc.e.157.1		32	105.38	odd	12
525.2.bc.e.418.1		32	15.14	odd	2
525.2.bc.e.493.8		32	105.59	even	6
735.2.m.c.97.15		32	105.2	even	12
735.2.m.c.97.16		32	105.47	odd	12
735.2.m.c.538.15		32	21.5	even	6
735.2.m.c.538.16		32	21.2	odd	6
735.2.v.b.178.1		32	21.11	odd	6
735.2.v.b.313.8		32	21.20	even	2
735.2.v.b.472.8		32	105.32	even	12
735.2.v.b.607.1		32	105.62	odd	4