Embedded newform 84.3.m.a.73.1

• Label: 84.3.m.a.73.1
• Level: $84$
• Weight: $3$
• Character: 84.73
• Analytic conductor: $2.289$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	84.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.28883422063\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			73.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	84.73
Dual form			84.3.m.a.61.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.50000 + 0.866025i) q^{3} +(1.50000 - 0.866025i) q^{5} +(6.50000 + 2.59808i) q^{7} +(1.50000 + 2.59808i) q^{9} +(1.50000 - 2.59808i) q^{11} +6.92820i q^{13} +3.00000 q^{15} +(-15.0000 - 8.66025i) q^{17} +(9.00000 - 5.19615i) q^{19} +(7.50000 + 9.52628i) q^{21} +(-18.0000 - 31.1769i) q^{23} +(-11.0000 + 19.0526i) q^{25} +5.19615i q^{27} -51.0000 q^{29} +(-10.5000 - 6.06218i) q^{31} +(4.50000 - 2.59808i) q^{33} +(12.0000 - 1.73205i) q^{35} +(-11.0000 - 19.0526i) q^{37} +(-6.00000 + 10.3923i) q^{39} +24.2487i q^{41} +10.0000 q^{43} +(4.50000 + 2.59808i) q^{45} +(78.0000 - 45.0333i) q^{47} +(35.5000 + 33.7750i) q^{49} +(-15.0000 - 25.9808i) q^{51} +(-25.5000 + 44.1673i) q^{53} -5.19615i q^{55} +18.0000 q^{57} +(64.5000 + 37.2391i) q^{59} +(60.0000 - 34.6410i) q^{61} +(3.00000 + 20.7846i) q^{63} +(6.00000 + 10.3923i) q^{65} +(-34.0000 + 58.8897i) q^{67} -62.3538i q^{69} +(-18.0000 - 10.3923i) q^{73} +(-33.0000 + 19.0526i) q^{75} +(16.5000 - 12.9904i) q^{77} +(-62.5000 - 108.253i) q^{79} +(-4.50000 + 7.79423i) q^{81} +154.153i q^{83} -30.0000 q^{85} +(-76.5000 - 44.1673i) q^{87} +(63.0000 - 36.3731i) q^{89} +(-18.0000 + 45.0333i) q^{91} +(-10.5000 - 18.1865i) q^{93} +(9.00000 - 15.5885i) q^{95} -147.224i q^{97} +9.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 3 * q^3 + 3 * q^5 + 13 * q^7 + 3 * q^9 + 3 * q^11 + 6 * q^15 - 30 * q^17 + 18 * q^19 + 15 * q^21 - 36 * q^23 - 22 * q^25 - 102 * q^29 - 21 * q^31 + 9 * q^33 + 24 * q^35 - 22 * q^37 - 12 * q^39 + 20 * q^43 + 9 * q^45 + 156 * q^47 + 71 * q^49 - 30 * q^51 - 51 * q^53 + 36 * q^57 + 129 * q^59 + 120 * q^61 + 6 * q^63 + 12 * q^65 - 68 * q^67 - 36 * q^73 - 66 * q^75 + 33 * q^77 - 125 * q^79 - 9 * q^81 - 60 * q^85 - 153 * q^87 + 126 * q^89 - 36 * q^91 - 21 * q^93 + 18 * q^95 + 18 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(43\)	\(73\)
\(\chi(n)\)	\(1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)

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			0
							\(3\)	1.50000	+	0.866025i	0.500000	+	0.288675i
\(5\)	1.50000	−	0.866025i	0.300000	−	0.173205i								−0.342443	−	0.939539i	\(-0.611254\pi\)
							0.642443	+	0.766334i	\(0.277921\pi\)
\(7\)	6.50000	+	2.59808i	0.928571	+	0.371154i
							\(11\)	1.50000	−	2.59808i	0.136364	−	0.236189i	−0.789754	−	0.613424i	\(-0.789792\pi\)
0.926118	+	0.377235i	\(0.123125\pi\)
\(13\)			6.92820i			0.532939i	0.963843	+	0.266469i	\(0.0858571\pi\)
							−0.963843	+	0.266469i	\(0.914143\pi\)
\(17\)	−15.0000	−	8.66025i	−0.882353	−	0.509427i	−0.0109194	−	0.999940i	\(-0.503476\pi\)
							−0.871433	+	0.490514i	\(0.836809\pi\)
\(19\)	9.00000	−	5.19615i	0.473684	−	0.273482i	−0.244096	−	0.969751i	\(-0.578491\pi\)
							0.717781	+	0.696269i	\(0.245158\pi\)
\(23\)	−18.0000	−	31.1769i	−0.782609	−	1.35552i	−0.930417	−	0.366502i	\(-0.880555\pi\)
							0.147809	−	0.989016i	\(-0.452778\pi\)
\(29\)	−51.0000			−1.75862			−0.879310	−	0.476249i	\(-0.841996\pi\)
							−0.879310	+	0.476249i	\(0.841996\pi\)
\(31\)	−10.5000	−	6.06218i	−0.338710	−	0.195554i	0.320992	−	0.947082i	\(-0.395984\pi\)
							−0.659701	+	0.751528i	\(0.729317\pi\)
\(37\)	−11.0000	−	19.0526i	−0.297297	−	0.514934i	0.678219	−	0.734859i	\(-0.262752\pi\)
							−0.975517	+	0.219925i	\(0.929419\pi\)
\(41\)			24.2487i			0.591432i	0.955276	+	0.295716i	\(0.0955582\pi\)
							−0.955276	+	0.295716i	\(0.904442\pi\)
\(43\)	10.0000			0.232558			0.116279	−	0.993217i	\(-0.462903\pi\)
							0.116279	+	0.993217i	\(0.462903\pi\)
\(47\)	78.0000	−	45.0333i	1.65957	−	0.958156i	0.686664	−	0.726975i	\(-0.259075\pi\)
							0.972911	−	0.231180i	\(-0.0742588\pi\)
\(53\)	−25.5000	+	44.1673i	−0.481132	+	0.833345i	−0.999766	−	0.0216515i	\(-0.993108\pi\)
							0.518634	+	0.854997i	\(0.326441\pi\)
\(59\)	64.5000	+	37.2391i	1.09322	+	0.631171i	0.934432	−	0.356142i	\(-0.115908\pi\)
							0.158788	+	0.987313i	\(0.449241\pi\)
\(61\)	60.0000	−	34.6410i	0.983607	−	0.567886i	0.0802495	−	0.996775i	\(-0.474428\pi\)
							0.903357	+	0.428889i	\(0.141095\pi\)
\(67\)	−34.0000	+	58.8897i	−0.507463	+	0.878951i	0.492500	+	0.870312i	\(0.336083\pi\)
							−0.999963	+	0.00863871i	\(0.997250\pi\)
\(71\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(73\)	−18.0000	−	10.3923i	−0.246575	−	0.142360i	0.371620	−	0.928385i	\(-0.378803\pi\)
							−0.618195	+	0.786025i	\(0.712136\pi\)
\(79\)	−62.5000	−	108.253i	−0.791139	−	1.37029i	−0.925262	−	0.379329i	\(-0.876155\pi\)
							0.134123	−	0.990965i	\(-0.457178\pi\)
\(83\)			154.153i			1.85726i	0.371008	+	0.928630i	\(0.379012\pi\)
							−0.371008	+	0.928630i	\(0.620988\pi\)
\(89\)	63.0000	−	36.3731i	0.707865	−	0.408686i	−0.102405	−	0.994743i	\(-0.532654\pi\)
							0.810270	+	0.586057i	\(0.199320\pi\)
\(97\)		−	147.224i		−	1.51778i	−0.651221	−	0.758888i	\(-0.725743\pi\)
							0.651221	−	0.758888i	\(-0.274257\pi\)

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	84.3.m.a.73.1	yes	2
3.2	odd	2		252.3.z.b.73.1		2
4.3	odd	2		336.3.bh.b.241.1		2
5.2	odd	4		2100.3.be.c.1249.2		4
5.3	odd	4		2100.3.be.c.1249.1		4
5.4	even	2		2100.3.bd.b.1501.1		2
7.2	even	3		588.3.m.a.313.1		2
7.3	odd	6		588.3.d.a.97.2		2
7.4	even	3		588.3.d.a.97.1		2
7.5	odd	6	inner	84.3.m.a.61.1	✓	2
7.6	odd	2		588.3.m.a.325.1		2
12.11	even	2		1008.3.cg.b.577.1		2
21.2	odd	6		1764.3.z.e.901.1		2
21.5	even	6		252.3.z.b.145.1		2
21.11	odd	6		1764.3.d.c.685.1		2
21.17	even	6		1764.3.d.c.685.2		2
21.20	even	2		1764.3.z.e.325.1		2
28.3	even	6		2352.3.f.c.97.1		2
28.11	odd	6		2352.3.f.c.97.2		2
28.19	even	6		336.3.bh.b.145.1		2
35.12	even	12		2100.3.be.c.649.1		4
35.19	odd	6		2100.3.bd.b.901.1		2
35.33	even	12		2100.3.be.c.649.2		4
84.47	odd	6		1008.3.cg.b.145.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.3.m.a.61.1	✓	2	7.5	odd	6	inner
84.3.m.a.73.1	yes	2	1.1	even	1	trivial
252.3.z.b.73.1		2	3.2	odd	2
252.3.z.b.145.1		2	21.5	even	6
336.3.bh.b.145.1		2	28.19	even	6
336.3.bh.b.241.1		2	4.3	odd	2
588.3.d.a.97.1		2	7.4	even	3
588.3.d.a.97.2		2	7.3	odd	6
588.3.m.a.313.1		2	7.2	even	3
588.3.m.a.325.1		2	7.6	odd	2
1008.3.cg.b.145.1		2	84.47	odd	6
1008.3.cg.b.577.1		2	12.11	even	2
1764.3.d.c.685.1		2	21.11	odd	6
1764.3.d.c.685.2		2	21.17	even	6
1764.3.z.e.325.1		2	21.20	even	2
1764.3.z.e.901.1		2	21.2	odd	6
2100.3.bd.b.901.1		2	35.19	odd	6
2100.3.bd.b.1501.1		2	5.4	even	2
2100.3.be.c.649.1		4	35.12	even	12
2100.3.be.c.649.2		4	35.33	even	12
2100.3.be.c.1249.1		4	5.3	odd	4
2100.3.be.c.1249.2		4	5.2	odd	4
2352.3.f.c.97.1		2	28.3	even	6
2352.3.f.c.97.2		2	28.11	odd	6