Embedded newform 380.3.g.a.189.1

• Label: 380.3.g.a.189.1
• Level: $380$
• Weight: $3$
• Character: 380.189
• Analytic conductor: $10.354$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	380.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3542500457\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{-6}, \sqrt{14})\)
Defining polynomial:	\( x^{4} - 4x^{2} + 25 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{5} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			189.1
Root			\(1.87083 - 1.22474i\) of defining polynomial
Character	\(\chi\)	\(=\)	380.189
Dual form			380.3.g.a.189.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-3.74166 q^{3} -5.00000 q^{5} -9.79796i q^{7} +5.00000 q^{9} +4.00000 q^{11} -11.2250 q^{13} +18.7083 q^{15} -19.5959i q^{17} +(-5.00000 + 18.3303i) q^{19} +36.6606i q^{21} +9.79796i q^{23} +25.0000 q^{25} +14.9666 q^{27} +36.6606i q^{29} +36.6606i q^{31} -14.9666 q^{33} +48.9898i q^{35} -33.6749 q^{37} +42.0000 q^{39} -36.6606i q^{41} +68.5857i q^{43} -25.0000 q^{45} +9.79796i q^{47} -47.0000 q^{49} +73.3212i q^{51} +56.1249 q^{53} -20.0000 q^{55} +(18.7083 - 68.5857i) q^{57} -73.3212i q^{59} +100.000 q^{61} -48.9898i q^{63} +56.1249 q^{65} -11.2250 q^{67} -36.6606i q^{69} +36.6606i q^{71} +19.5959i q^{73} -93.5414 q^{75} -39.1918i q^{77} +109.982i q^{79} -101.000 q^{81} +29.3939i q^{83} +97.9796i q^{85} -137.171i q^{87} -146.642i q^{89} +109.982i q^{91} -137.171i q^{93} +(25.0000 - 91.6515i) q^{95} -123.475 q^{97} +20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{5} + 20 q^{9} + 16 q^{11} - 20 q^{19} + 100 q^{25} + 168 q^{39} - 100 q^{45} - 188 q^{49} - 80 q^{55} + 400 q^{61} - 404 q^{81} + 100 q^{95} + 80 q^{99}+O(q^{100}) \)

Character values

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

\(n\)	\(21\)	\(77\)	\(191\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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
							\(3\)	−3.74166			−1.24722			−0.623610	−	0.781736i	\(-0.714334\pi\)
−0.623610	+	0.781736i	\(0.714334\pi\)
\(5\)	−5.00000			−1.00000
							\(7\)		−	9.79796i		−	1.39971i	−0.714286	−	0.699854i	\(-0.753248\pi\)
0.714286	−	0.699854i	\(-0.246752\pi\)
\(11\)	4.00000			0.363636			0.181818	−	0.983332i	\(-0.441802\pi\)
							0.181818	+	0.983332i	\(0.441802\pi\)
\(13\)	−11.2250			−0.863459			−0.431730	−	0.902003i	\(-0.642097\pi\)
							−0.431730	+	0.902003i	\(0.642097\pi\)
\(17\)		−	19.5959i		−	1.15270i	−0.817203	−	0.576351i	\(-0.804476\pi\)
							0.817203	−	0.576351i	\(-0.195524\pi\)
\(19\)	−5.00000	+	18.3303i	−0.263158	+	0.964753i
							\(23\)			9.79796i			0.425998i	0.977052	+	0.212999i	\(0.0683231\pi\)
−0.977052	+	0.212999i	\(0.931677\pi\)
\(29\)			36.6606i			1.26416i	0.774904	+	0.632079i	\(0.217798\pi\)
							−0.774904	+	0.632079i	\(0.782202\pi\)
\(31\)			36.6606i			1.18260i	0.806452	+	0.591300i	\(0.201385\pi\)
							−0.806452	+	0.591300i	\(0.798615\pi\)
\(37\)	−33.6749			−0.910133			−0.455066	−	0.890457i	\(-0.650384\pi\)
							−0.455066	+	0.890457i	\(0.650384\pi\)
\(41\)		−	36.6606i		−	0.894161i	−0.894494	−	0.447081i	\(-0.852464\pi\)
							0.894494	−	0.447081i	\(-0.147536\pi\)
\(43\)			68.5857i			1.59502i	0.603308	+	0.797508i	\(0.293849\pi\)
							−0.603308	+	0.797508i	\(0.706151\pi\)
\(47\)			9.79796i			0.208467i	0.994553	+	0.104234i	\(0.0332390\pi\)
							−0.994553	+	0.104234i	\(0.966761\pi\)
\(53\)	56.1249			1.05896			0.529480	−	0.848323i	\(-0.322387\pi\)
							0.529480	+	0.848323i	\(0.322387\pi\)
\(59\)		−	73.3212i		−	1.24273i	−0.783520	−	0.621366i	\(-0.786578\pi\)
							0.783520	−	0.621366i	\(-0.213422\pi\)
\(61\)	100.000			1.63934			0.819672	−	0.572833i	\(-0.194156\pi\)
							0.819672	+	0.572833i	\(0.194156\pi\)
\(67\)	−11.2250			−0.167537			−0.0837684	−	0.996485i	\(-0.526696\pi\)
							−0.0837684	+	0.996485i	\(0.526696\pi\)
\(71\)			36.6606i			0.516347i	0.966099	+	0.258173i	\(0.0831205\pi\)
							−0.966099	+	0.258173i	\(0.916879\pi\)
\(73\)			19.5959i			0.268437i	0.990952	+	0.134219i	\(0.0428524\pi\)
							−0.990952	+	0.134219i	\(0.957148\pi\)
\(79\)			109.982i			1.39217i	0.717957	+	0.696087i	\(0.245077\pi\)
							−0.717957	+	0.696087i	\(0.754923\pi\)
\(83\)			29.3939i			0.354143i	0.984198	+	0.177072i	\(0.0566624\pi\)
							−0.984198	+	0.177072i	\(0.943338\pi\)
\(89\)		−	146.642i		−	1.64767i	−0.566831	−	0.823834i	\(-0.691831\pi\)
							0.566831	−	0.823834i	\(-0.308169\pi\)
\(97\)	−123.475			−1.27293			−0.636467	−	0.771304i	\(-0.719605\pi\)
							−0.636467	+	0.771304i	\(0.719605\pi\)

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	380.3.g.a.189.1	✓	4
3.2	odd	2		3420.3.h.c.2089.1		4
5.2	odd	4		1900.3.e.c.1101.4		4
5.3	odd	4		1900.3.e.c.1101.1		4
5.4	even	2	inner	380.3.g.a.189.4	yes	4
15.14	odd	2		3420.3.h.c.2089.4		4
19.18	odd	2	inner	380.3.g.a.189.3	yes	4
57.56	even	2		3420.3.h.c.2089.2		4
95.18	even	4		1900.3.e.c.1101.3		4
95.37	even	4		1900.3.e.c.1101.2		4
95.94	odd	2	inner	380.3.g.a.189.2	yes	4
285.284	even	2		3420.3.h.c.2089.3		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.g.a.189.1	✓	4	1.1	even	1	trivial
380.3.g.a.189.2	yes	4	95.94	odd	2	inner
380.3.g.a.189.3	yes	4	19.18	odd	2	inner
380.3.g.a.189.4	yes	4	5.4	even	2	inner
1900.3.e.c.1101.1		4	5.3	odd	4
1900.3.e.c.1101.2		4	95.37	even	4
1900.3.e.c.1101.3		4	95.18	even	4
1900.3.e.c.1101.4		4	5.2	odd	4
3420.3.h.c.2089.1		4	3.2	odd	2
3420.3.h.c.2089.2		4	57.56	even	2
3420.3.h.c.2089.3		4	285.284	even	2
3420.3.h.c.2089.4		4	15.14	odd	2