Embedded newform 378.2.g.b.163.1

• Label: 378.2.g.b.163.1
• Level: $378$
• Weight: $2$
• Character: 378.163
• Analytic conductor: $3.018$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.01834519640\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	378.163
Dual form			378.2.g.b.109.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-2.00000 - 1.73205i) q^{7} +1.00000 q^{8} +(3.00000 + 5.19615i) q^{11} +5.00000 q^{13} +(2.50000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(3.00000 + 5.19615i) q^{17} +(2.00000 - 3.46410i) q^{19} -6.00000 q^{22} +(3.00000 - 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-2.50000 + 4.33013i) q^{26} +(-0.500000 + 2.59808i) q^{28} -6.00000 q^{29} +(0.500000 + 0.866025i) q^{31} +(-0.500000 - 0.866025i) q^{32} -6.00000 q^{34} +(0.500000 - 0.866025i) q^{37} +(2.00000 + 3.46410i) q^{38} +6.00000 q^{41} -1.00000 q^{43} +(3.00000 - 5.19615i) q^{44} +(3.00000 + 5.19615i) q^{46} +(-3.00000 + 5.19615i) q^{47} +(1.00000 + 6.92820i) q^{49} -5.00000 q^{50} +(-2.50000 - 4.33013i) q^{52} +(-3.00000 - 5.19615i) q^{53} +(-2.00000 - 1.73205i) q^{56} +(3.00000 - 5.19615i) q^{58} +(-3.00000 - 5.19615i) q^{59} +(0.500000 - 0.866025i) q^{61} -1.00000 q^{62} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{67} +(3.00000 - 5.19615i) q^{68} -12.0000 q^{71} +(-1.00000 - 1.73205i) q^{73} +(0.500000 + 0.866025i) q^{74} -4.00000 q^{76} +(3.00000 - 15.5885i) q^{77} +(0.500000 - 0.866025i) q^{79} +(-3.00000 + 5.19615i) q^{82} -6.00000 q^{83} +(0.500000 - 0.866025i) q^{86} +(3.00000 + 5.19615i) q^{88} +(-10.0000 - 8.66025i) q^{91} -6.00000 q^{92} +(-3.00000 - 5.19615i) q^{94} +17.0000 q^{97} +(-6.50000 - 2.59808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^2 - q^4 - 4 * q^7 + 2 * q^8 + 6 * q^11 + 10 * q^13 + 5 * q^14 - q^16 + 6 * q^17 + 4 * q^19 - 12 * q^22 + 6 * q^23 + 5 * q^25 - 5 * q^26 - q^28 - 12 * q^29 + q^31 - q^32 - 12 * q^34 + q^37 + 4 * q^38 + 12 * q^41 - 2 * q^43 + 6 * q^44 + 6 * q^46 - 6 * q^47 + 2 * q^49 - 10 * q^50 - 5 * q^52 - 6 * q^53 - 4 * q^56 + 6 * q^58 - 6 * q^59 + q^61 - 2 * q^62 + 2 * q^64 + q^67 + 6 * q^68 - 24 * q^71 - 2 * q^73 + q^74 - 8 * q^76 + 6 * q^77 + q^79 - 6 * q^82 - 12 * q^83 + q^86 + 6 * q^88 - 20 * q^91 - 12 * q^92 - 6 * q^94 + 34 * q^97 - 13 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(325\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

Coefficient data

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

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

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.500000	+	0.866025i	−0.353553	+	0.612372i
							\(3\)		0			0
\(5\)		0			0									−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(7\)	−2.00000	−	1.73205i	−0.755929	−	0.654654i
							\(11\)	3.00000	+	5.19615i	0.904534	+	1.56670i	0.821541	+	0.570149i	\(0.193114\pi\)
0.0829925	+	0.996550i	\(0.473552\pi\)
\(13\)	5.00000			1.38675			0.693375	−	0.720577i	\(-0.256123\pi\)
							0.693375	+	0.720577i	\(0.256123\pi\)
\(17\)	3.00000	+	5.19615i	0.727607	+	1.26025i	0.957892	+	0.287129i	\(0.0927008\pi\)
							−0.230285	+	0.973123i	\(0.573966\pi\)
\(19\)	2.00000	−	3.46410i	0.458831	−	0.794719i	−0.540068	−	0.841621i	\(-0.681602\pi\)
							0.998899	+	0.0469020i	\(0.0149348\pi\)
\(23\)	3.00000	−	5.19615i	0.625543	−	1.08347i	−0.362892	−	0.931831i	\(-0.618211\pi\)
							0.988436	−	0.151642i	\(-0.0484560\pi\)
\(29\)	−6.00000			−1.11417			−0.557086	−	0.830455i	\(-0.688081\pi\)
							−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	0.500000	+	0.866025i	0.0898027	+	0.155543i	0.907428	−	0.420208i	\(-0.138043\pi\)
							−0.817625	+	0.575751i	\(0.804710\pi\)
\(37\)	0.500000	−	0.866025i	0.0821995	−	0.142374i	−0.821995	−	0.569495i	\(-0.807139\pi\)
							0.904194	+	0.427121i	\(0.140472\pi\)
\(41\)	6.00000			0.937043			0.468521	−	0.883452i	\(-0.344787\pi\)
							0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	−1.00000			−0.152499			−0.0762493	−	0.997089i	\(-0.524294\pi\)
							−0.0762493	+	0.997089i	\(0.524294\pi\)
\(47\)	−3.00000	+	5.19615i	−0.437595	+	0.757937i	−0.997503	−	0.0706177i	\(-0.977503\pi\)
							0.559908	+	0.828554i	\(0.310836\pi\)
\(53\)	−3.00000	−	5.19615i	−0.412082	−	0.713746i	0.583036	−	0.812447i	\(-0.301865\pi\)
							−0.995117	+	0.0987002i	\(0.968532\pi\)
\(59\)	−3.00000	−	5.19615i	−0.390567	−	0.676481i	0.601958	−	0.798528i	\(-0.294388\pi\)
							−0.992524	+	0.122047i	\(0.961054\pi\)
\(61\)	0.500000	−	0.866025i	0.0640184	−	0.110883i	−0.832240	−	0.554416i	\(-0.812942\pi\)
							0.896258	+	0.443533i	\(0.146275\pi\)
\(67\)	0.500000	+	0.866025i	0.0610847	+	0.105802i	0.894951	−	0.446165i	\(-0.147211\pi\)
							−0.833866	+	0.551967i	\(0.813877\pi\)
\(71\)	−12.0000			−1.42414			−0.712069	−	0.702109i	\(-0.752242\pi\)
							−0.712069	+	0.702109i	\(0.752242\pi\)
\(73\)	−1.00000	−	1.73205i	−0.117041	−	0.202721i	0.801553	−	0.597924i	\(-0.204008\pi\)
							−0.918594	+	0.395203i	\(0.870674\pi\)
\(79\)	0.500000	−	0.866025i	0.0562544	−	0.0974355i	−0.836527	−	0.547926i	\(-0.815418\pi\)
							0.892781	+	0.450490i	\(0.148751\pi\)
\(83\)	−6.00000			−0.658586			−0.329293	−	0.944228i	\(-0.606810\pi\)
							−0.329293	+	0.944228i	\(0.606810\pi\)
\(89\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(97\)	17.0000			1.72609			0.863044	−	0.505128i	\(-0.168555\pi\)
							0.863044	+	0.505128i	\(0.168555\pi\)

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	378.2.g.b.163.1	yes	2
3.2	odd	2		378.2.g.e.163.1	yes	2
7.2	even	3		2646.2.a.x.1.1		1
7.4	even	3	inner	378.2.g.b.109.1	✓	2
7.5	odd	6		2646.2.a.w.1.1		1
9.2	odd	6		1134.2.e.c.919.1		2
9.4	even	3		1134.2.h.d.541.1		2
9.5	odd	6		1134.2.h.n.541.1		2
9.7	even	3		1134.2.e.m.919.1		2
21.2	odd	6		2646.2.a.h.1.1		1
21.5	even	6		2646.2.a.g.1.1		1
21.11	odd	6		378.2.g.e.109.1	yes	2
63.4	even	3		1134.2.e.m.865.1		2
63.11	odd	6		1134.2.h.n.109.1		2
63.25	even	3		1134.2.h.d.109.1		2
63.32	odd	6		1134.2.e.c.865.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.g.b.109.1	✓	2	7.4	even	3	inner
378.2.g.b.163.1	yes	2	1.1	even	1	trivial
378.2.g.e.109.1	yes	2	21.11	odd	6
378.2.g.e.163.1	yes	2	3.2	odd	2
1134.2.e.c.865.1		2	63.32	odd	6
1134.2.e.c.919.1		2	9.2	odd	6
1134.2.e.m.865.1		2	63.4	even	3
1134.2.e.m.919.1		2	9.7	even	3
1134.2.h.d.109.1		2	63.25	even	3
1134.2.h.d.541.1		2	9.4	even	3
1134.2.h.n.109.1		2	63.11	odd	6
1134.2.h.n.541.1		2	9.5	odd	6
2646.2.a.g.1.1		1	21.5	even	6
2646.2.a.h.1.1		1	21.2	odd	6
2646.2.a.w.1.1		1	7.5	odd	6
2646.2.a.x.1.1		1	7.2	even	3