Embedded newform 380.2.r.a.49.8

• Label: 380.2.r.a.49.8
• Level: $380$
• Weight: $2$
• Character: 380.49
• Analytic conductor: $3.034$
• Analytic rank: $0$
• Dimension: $20$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	380.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.03431527681\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 20*x^18 + 261*x^16 - 1994*x^14 + 11074*x^12 - 39211*x^10 + 99376*x^8 - 134299*x^6 + 124617*x^4 - 24768*x^2 + 4096
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			49.8
Root			\(1.74361 - 1.00667i\) of defining polynomial
Character	\(\chi\)	\(=\)	380.49
Dual form			380.2.r.a.349.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.74361 + 1.00667i) q^{3} +(0.0408382 + 2.23570i) q^{5} -1.34403i q^{7} +(0.526784 + 0.912416i) q^{9} +5.25594 q^{11} +(-2.10918 + 1.21773i) q^{13} +(-2.17941 + 3.93929i) q^{15} +(1.17765 + 0.679914i) q^{17} +(2.89815 + 3.25587i) q^{19} +(1.35300 - 2.34346i) q^{21} +(-7.05514 + 4.07329i) q^{23} +(-4.99666 + 0.182603i) q^{25} -3.91884i q^{27} +(-1.03597 - 1.79435i) q^{29} -0.513207 q^{31} +(9.16431 + 5.29102i) q^{33} +(3.00483 - 0.0548876i) q^{35} -5.57175i q^{37} -4.90344 q^{39} +(2.70353 - 4.68265i) q^{41} +(-11.0197 - 6.36221i) q^{43} +(-2.01837 + 1.21499i) q^{45} +(2.82785 - 1.63266i) q^{47} +5.19359 q^{49} +(1.36890 + 2.37101i) q^{51} +(10.1892 - 5.88276i) q^{53} +(0.214643 + 11.7507i) q^{55} +(1.77564 + 8.59447i) q^{57} +(0.0175979 - 0.0304805i) q^{59} +(0.518372 + 0.897846i) q^{61} +(1.22631 - 0.708011i) q^{63} +(-2.80861 - 4.66574i) q^{65} +(0.664028 - 0.383377i) q^{67} -16.4019 q^{69} +(5.68450 - 9.84583i) q^{71} +(-1.86429 - 1.07635i) q^{73} +(-8.89606 - 4.71162i) q^{75} -7.06413i q^{77} +(-6.48576 + 11.2337i) q^{79} +(5.52535 - 9.57019i) q^{81} -4.20304i q^{83} +(-1.47199 + 2.66062i) q^{85} -4.17153i q^{87} +(-3.65426 - 6.32937i) q^{89} +(1.63667 + 2.83479i) q^{91} +(-0.894833 - 0.516632i) q^{93} +(-7.16079 + 6.61235i) q^{95} +(0.721716 + 0.416683i) q^{97} +(2.76875 + 4.79561i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	20 * q + q^5 + 10 * q^9 - 5 * q^15 + 14 * q^19 - 8 * q^21 + 9 * q^25 - 16 * q^29 + 8 * q^31 - 2 * q^35 - 8 * q^39 + 26 * q^41 - 32 * q^45 - 44 * q^49 + 26 * q^51 - 12 * q^55 + 4 * q^59 + 2 * q^61 - 18 * q^65 + 48 * q^69 - 2 * q^71 + 46 * q^75 - 16 * q^79 + 26 * q^81 - 39 * q^85 - 40 * q^89 - 4 * q^91 - 43 * q^95 - 20 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(77\)	\(191\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		0			0
							\(3\)	1.74361	+	1.00667i	1.00667	+	0.581203i	0.910216	−	0.414134i	\(-0.135915\pi\)
0.0964577	+	0.995337i	\(0.469249\pi\)
\(5\)	0.0408382	+	2.23570i	0.0182634	+	0.999833i
							\(7\)		−	1.34403i		−	0.507994i	−0.967205	−	0.253997i	\(-0.918255\pi\)
0.967205	−	0.253997i	\(-0.0817454\pi\)
\(11\)	5.25594			1.58473			0.792363	−	0.610050i	\(-0.208851\pi\)
							0.792363	+	0.610050i	\(0.208851\pi\)
\(13\)	−2.10918	+	1.21773i	−0.584980	+	0.337738i	−0.763110	−	0.646269i	\(-0.776329\pi\)
							0.178130	+	0.984007i	\(0.442995\pi\)
\(17\)	1.17765	+	0.679914i	0.285621	+	0.164903i	0.635965	−	0.771718i	\(-0.280602\pi\)
							−0.350344	+	0.936621i	\(0.613935\pi\)
\(19\)	2.89815	+	3.25587i	0.664882	+	0.746949i
							\(23\)	−7.05514	+	4.07329i	−1.47110	+	0.849339i	−0.999473	−	0.0324603i	\(-0.989666\pi\)
−0.471625	+	0.881799i	\(0.656332\pi\)
\(29\)	−1.03597	−	1.79435i	−0.192375	−	0.333203i	0.753662	−	0.657262i	\(-0.228286\pi\)
							−0.946037	+	0.324059i	\(0.894952\pi\)
\(31\)	−0.513207			−0.0921747			−0.0460873	−	0.998937i	\(-0.514675\pi\)
							−0.0460873	+	0.998937i	\(0.514675\pi\)
\(37\)		−	5.57175i		−	0.915991i	−0.888955	−	0.457995i	\(-0.848568\pi\)
							0.888955	−	0.457995i	\(-0.151432\pi\)
\(41\)	2.70353	−	4.68265i	0.422220	−	0.731307i	−0.573936	−	0.818900i	\(-0.694584\pi\)
							0.996156	+	0.0875933i	\(0.0279176\pi\)
\(43\)	−11.0197	−	6.36221i	−1.68048	−	0.970228i	−0.961339	−	0.275369i	\(-0.911200\pi\)
							−0.719146	−	0.694859i	\(-0.755467\pi\)
\(47\)	2.82785	−	1.63266i	0.412485	−	0.238148i	−0.279372	−	0.960183i	\(-0.590126\pi\)
							0.691857	+	0.722035i	\(0.256793\pi\)
\(53\)	10.1892	−	5.88276i	1.39960	−	0.808060i	0.405250	−	0.914206i	\(-0.367185\pi\)
							0.994351	+	0.106146i	\(0.0338512\pi\)
\(59\)	0.0175979	−	0.0304805i	0.00229105	−	0.00396822i	−0.864878	−	0.501983i	\(-0.832604\pi\)
							0.867169	+	0.498015i	\(0.165937\pi\)
\(61\)	0.518372	+	0.897846i	0.0663707	+	0.114957i	0.897301	−	0.441419i	\(-0.145525\pi\)
							−0.830930	+	0.556376i	\(0.812191\pi\)
\(67\)	0.664028	−	0.383377i	0.0811239	−	0.0468369i	−0.458889	−	0.888493i	\(-0.651753\pi\)
							0.540013	+	0.841657i	\(0.318419\pi\)
\(71\)	5.68450	−	9.84583i	0.674625	−	1.16849i	−0.301953	−	0.953323i	\(-0.597638\pi\)
							0.976578	−	0.215163i	\(-0.0690282\pi\)
\(73\)	−1.86429	−	1.07635i	−0.218199	−	0.125977i	0.386917	−	0.922115i	\(-0.373540\pi\)
							−0.605116	+	0.796137i	\(0.706873\pi\)
\(79\)	−6.48576	+	11.2337i	−0.729705	+	1.26389i	0.227302	+	0.973824i	\(0.427009\pi\)
							−0.957008	+	0.290062i	\(0.906324\pi\)
\(83\)		−	4.20304i		−	0.461343i	−0.973032	−	0.230672i	\(-0.925908\pi\)
							0.973032	−	0.230672i	\(-0.0740923\pi\)
\(89\)	−3.65426	−	6.32937i	−0.387351	−	0.670912i	0.604741	−	0.796422i	\(-0.293277\pi\)
							−0.992092	+	0.125510i	\(0.959943\pi\)
\(97\)	0.721716	+	0.416683i	0.0732791	+	0.0423077i	0.536192	−	0.844096i	\(-0.319862\pi\)
							−0.462913	+	0.886404i	\(0.653196\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.2.r.a.49.8	yes	20
3.2	odd	2		3420.2.bj.c.1189.6		20
5.2	odd	4		1900.2.i.g.201.8		20
5.3	odd	4		1900.2.i.g.201.3		20
5.4	even	2	inner	380.2.r.a.49.3	✓	20
15.14	odd	2		3420.2.bj.c.1189.8		20
19.7	even	3	inner	380.2.r.a.349.3	yes	20
57.26	odd	6		3420.2.bj.c.2629.8		20
95.7	odd	12		1900.2.i.g.501.8		20
95.64	even	6	inner	380.2.r.a.349.8	yes	20
95.83	odd	12		1900.2.i.g.501.3		20
285.254	odd	6		3420.2.bj.c.2629.6		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.r.a.49.3	✓	20	5.4	even	2	inner
380.2.r.a.49.8	yes	20	1.1	even	1	trivial
380.2.r.a.349.3	yes	20	19.7	even	3	inner
380.2.r.a.349.8	yes	20	95.64	even	6	inner
1900.2.i.g.201.3		20	5.3	odd	4
1900.2.i.g.201.8		20	5.2	odd	4
1900.2.i.g.501.3		20	95.83	odd	12
1900.2.i.g.501.8		20	95.7	odd	12
3420.2.bj.c.1189.6		20	3.2	odd	2
3420.2.bj.c.1189.8		20	15.14	odd	2
3420.2.bj.c.2629.6		20	285.254	odd	6
3420.2.bj.c.2629.8		20	57.26	odd	6