Embedded newform 570.2.v.c.83.14

• Label: 570.2.v.c.83.14
• Level: $570$
• Weight: $2$
• Character: 570.83
• Analytic conductor: $4.551$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	570.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.55147291521\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			83.14
Character	\(\chi\)	\(=\)	570.83
Dual form			570.2.v.c.467.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.258819 + 0.965926i) q^{2} +(1.25366 - 1.19513i) q^{3} +(-0.866025 - 0.500000i) q^{4} +(1.74044 - 1.40387i) q^{5} +(0.829942 + 1.52026i) q^{6} +(-1.56325 + 1.56325i) q^{7} +(0.707107 - 0.707107i) q^{8} +(0.143305 - 2.99658i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 4 q^{3} + 8 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(211\)	\(457\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.258819	+	0.965926i	−0.183013	+	0.683013i
							\(3\)	1.25366	−	1.19513i	0.723798	−	0.690011i
\(5\)	1.74044	−	1.40387i	0.778350	−	0.627830i
							\(7\)	−1.56325	+	1.56325i	−0.590854	+	0.590854i	−0.937862	−	0.347008i	\(-0.887198\pi\)
0.347008	+	0.937862i	\(0.387198\pi\)
\(11\)		−	0.351777i		−	0.106065i	−0.998593	−	0.0530324i	\(-0.983111\pi\)
							0.998593	−	0.0530324i	\(-0.0168886\pi\)
\(13\)	4.24865	−	1.13842i	1.17836	−	0.315741i	0.384086	−	0.923297i	\(-0.374517\pi\)
							0.794277	+	0.607556i	\(0.207850\pi\)
\(17\)	0.695479	−	2.59556i	0.168678	−	0.629516i	−0.828864	−	0.559450i	\(-0.811012\pi\)
							0.997542	−	0.0700661i	\(-0.0223210\pi\)
\(19\)	0.961041	+	4.25164i	0.220478	+	0.975392i
							\(23\)	−0.278775	−	1.04040i	−0.0581287	−	0.216939i	0.930752	−	0.365652i	\(-0.119154\pi\)
−0.988880	+	0.148712i	\(0.952487\pi\)
\(29\)	4.03278	−	6.98498i	0.748868	−	1.29708i	−0.199497	−	0.979898i	\(-0.563931\pi\)
							0.948366	−	0.317180i	\(-0.102736\pi\)
\(31\)	−2.94483			−0.528907			−0.264453	−	0.964399i	\(-0.585191\pi\)
							−0.264453	+	0.964399i	\(0.585191\pi\)
\(37\)	−4.48320	+	4.48320i	−0.737034	+	0.737034i	−0.972003	−	0.234969i	\(-0.924501\pi\)
							0.234969	+	0.972003i	\(0.424501\pi\)
\(41\)	3.26509	−	1.88510i	0.509922	−	0.294404i	−0.222880	−	0.974846i	\(-0.571546\pi\)
							0.732802	+	0.680442i	\(0.238212\pi\)
\(43\)	1.94063	−	7.24254i	0.295944	−	1.10448i	−0.644521	−	0.764587i	\(-0.722943\pi\)
							0.940465	−	0.339891i	\(-0.110390\pi\)
\(47\)	−1.58789	+	0.425473i	−0.231617	+	0.0620616i	−0.372761	−	0.927928i	\(-0.621589\pi\)
							0.141144	+	0.989989i	\(0.454922\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	570.2.v.c.83.14	✓	144
3.2	odd	2	inner	570.2.v.c.83.25	yes	144
5.2	odd	4	inner	570.2.v.c.197.6	yes	144
15.2	even	4	inner	570.2.v.c.197.20	yes	144
19.11	even	3	inner	570.2.v.c.353.20	yes	144
57.11	odd	6	inner	570.2.v.c.353.6	yes	144
95.87	odd	12	inner	570.2.v.c.467.25	yes	144
285.182	even	12	inner	570.2.v.c.467.14	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.v.c.83.14	✓	144	1.1	even	1	trivial
570.2.v.c.83.25	yes	144	3.2	odd	2	inner
570.2.v.c.197.6	yes	144	5.2	odd	4	inner
570.2.v.c.197.20	yes	144	15.2	even	4	inner
570.2.v.c.353.6	yes	144	57.11	odd	6	inner
570.2.v.c.353.20	yes	144	19.11	even	3	inner
570.2.v.c.467.14	yes	144	285.182	even	12	inner
570.2.v.c.467.25	yes	144	95.87	odd	12	inner