Embedded newform 312.2.q.c.217.1

• Label: 312.2.q.c.217.1
• Level: $312$
• Weight: $2$
• Character: 312.217
• Analytic conductor: $2.491$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	312.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.49133254306\)
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:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	312.217
Dual form			312.2.q.c.289.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +3.00000 q^{5} +(2.00000 + 3.46410i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-2.00000 + 3.46410i) q^{11} +(-2.50000 - 2.59808i) q^{13} +(1.50000 - 2.59808i) q^{15} +(-1.50000 - 2.59808i) q^{17} +(2.00000 + 3.46410i) q^{19} +4.00000 q^{21} +(4.00000 - 6.92820i) q^{23} +4.00000 q^{25} -1.00000 q^{27} +(2.50000 - 4.33013i) q^{29} -8.00000 q^{31} +(2.00000 + 3.46410i) q^{33} +(6.00000 + 10.3923i) q^{35} +(-3.50000 + 6.06218i) q^{37} +(-3.50000 + 0.866025i) q^{39} +(4.50000 - 7.79423i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(-1.50000 - 2.59808i) q^{45} -4.00000 q^{47} +(-4.50000 + 7.79423i) q^{49} -3.00000 q^{51} -5.00000 q^{53} +(-6.00000 + 10.3923i) q^{55} +4.00000 q^{57} +(-2.00000 - 3.46410i) q^{59} +(2.50000 + 4.33013i) q^{61} +(2.00000 - 3.46410i) q^{63} +(-7.50000 - 7.79423i) q^{65} +(-4.00000 + 6.92820i) q^{67} +(-4.00000 - 6.92820i) q^{69} +(2.00000 + 3.46410i) q^{71} +11.0000 q^{73} +(2.00000 - 3.46410i) q^{75} -16.0000 q^{77} -4.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} +(-4.50000 - 7.79423i) q^{85} +(-2.50000 - 4.33013i) q^{87} +(3.00000 - 5.19615i) q^{89} +(4.00000 - 13.8564i) q^{91} +(-4.00000 + 6.92820i) q^{93} +(6.00000 + 10.3923i) q^{95} +(7.00000 + 12.1244i) q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^3 + 6 * q^5 + 4 * q^7 - q^9 - 4 * q^11 - 5 * q^13 + 3 * q^15 - 3 * q^17 + 4 * q^19 + 8 * q^21 + 8 * q^23 + 8 * q^25 - 2 * q^27 + 5 * q^29 - 16 * q^31 + 4 * q^33 + 12 * q^35 - 7 * q^37 - 7 * q^39 + 9 * q^41 - 8 * q^43 - 3 * q^45 - 8 * q^47 - 9 * q^49 - 6 * q^51 - 10 * q^53 - 12 * q^55 + 8 * q^57 - 4 * q^59 + 5 * q^61 + 4 * q^63 - 15 * q^65 - 8 * q^67 - 8 * q^69 + 4 * q^71 + 22 * q^73 + 4 * q^75 - 32 * q^77 - 8 * q^79 - q^81 - 9 * q^85 - 5 * q^87 + 6 * q^89 + 8 * q^91 - 8 * q^93 + 12 * q^95 + 14 * q^97 + 8 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).

\(n\)	\(79\)	\(145\)	\(157\)	\(209\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)

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
							\(3\)	0.500000	−	0.866025i	0.288675	−	0.500000i
\(5\)	3.00000			1.34164										0.670820	−	0.741620i	\(-0.265942\pi\)
							0.670820	+	0.741620i	\(0.265942\pi\)
\(7\)	2.00000	+	3.46410i	0.755929	+	1.30931i	0.944911	+	0.327327i	\(0.106148\pi\)
							−0.188982	+	0.981981i	\(0.560519\pi\)
\(11\)	−2.00000	+	3.46410i	−0.603023	+	1.04447i	0.389338	+	0.921095i	\(0.372704\pi\)
							−0.992361	+	0.123371i	\(0.960630\pi\)
\(13\)	−2.50000	−	2.59808i	−0.693375	−	0.720577i
							\(17\)	−1.50000	−	2.59808i	−0.363803	−	0.630126i	0.624780	−	0.780801i	\(-0.285189\pi\)
−0.988583	+	0.150675i	\(0.951855\pi\)
\(19\)	2.00000	+	3.46410i	0.458831	+	0.794719i	0.998899	−	0.0469020i	\(-0.0149348\pi\)
							−0.540068	+	0.841621i	\(0.681602\pi\)
\(23\)	4.00000	−	6.92820i	0.834058	−	1.44463i	−0.0607377	−	0.998154i	\(-0.519345\pi\)
							0.894795	−	0.446476i	\(-0.147321\pi\)
\(29\)	2.50000	−	4.33013i	0.464238	−	0.804084i	−0.534928	−	0.844897i	\(-0.679661\pi\)
							0.999167	+	0.0408130i	\(0.0129948\pi\)
\(31\)	−8.00000			−1.43684			−0.718421	−	0.695608i	\(-0.755135\pi\)
							−0.718421	+	0.695608i	\(0.755135\pi\)
\(37\)	−3.50000	+	6.06218i	−0.575396	+	0.996616i	0.420602	+	0.907245i	\(0.361819\pi\)
							−0.995998	+	0.0893706i	\(0.971514\pi\)
\(41\)	4.50000	−	7.79423i	0.702782	−	1.21725i	−0.264704	−	0.964330i	\(-0.585274\pi\)
							0.967486	−	0.252924i	\(-0.0813924\pi\)
\(43\)	−4.00000	−	6.92820i	−0.609994	−	1.05654i	−0.991241	−	0.132068i	\(-0.957838\pi\)
							0.381246	−	0.924473i	\(-0.375495\pi\)
\(47\)	−4.00000			−0.583460			−0.291730	−	0.956501i	\(-0.594231\pi\)
							−0.291730	+	0.956501i	\(0.594231\pi\)
\(53\)	−5.00000			−0.686803			−0.343401	−	0.939189i	\(-0.611579\pi\)
							−0.343401	+	0.939189i	\(0.611579\pi\)
\(59\)	−2.00000	−	3.46410i	−0.260378	−	0.450988i	0.705965	−	0.708247i	\(-0.250514\pi\)
							−0.966342	+	0.257260i	\(0.917180\pi\)
\(61\)	2.50000	+	4.33013i	0.320092	+	0.554416i	0.980507	−	0.196485i	\(-0.0629528\pi\)
							−0.660415	+	0.750901i	\(0.729619\pi\)
\(67\)	−4.00000	+	6.92820i	−0.488678	+	0.846415i	−0.999915	−	0.0130248i	\(-0.995854\pi\)
							0.511237	+	0.859440i	\(0.329187\pi\)
\(71\)	2.00000	+	3.46410i	0.237356	+	0.411113i	0.959955	−	0.280155i	\(-0.0903858\pi\)
							−0.722599	+	0.691268i	\(0.757052\pi\)
\(73\)	11.0000			1.28745			0.643726	−	0.765256i	\(-0.277388\pi\)
							0.643726	+	0.765256i	\(0.277388\pi\)
\(79\)	−4.00000			−0.450035			−0.225018	−	0.974355i	\(-0.572244\pi\)
							−0.225018	+	0.974355i	\(0.572244\pi\)
\(83\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(89\)	3.00000	−	5.19615i	0.317999	−	0.550791i	−0.662071	−	0.749441i	\(-0.730322\pi\)
							0.980071	+	0.198650i	\(0.0636557\pi\)
\(97\)	7.00000	+	12.1244i	0.710742	+	1.23104i	0.964579	+	0.263795i	\(0.0849741\pi\)
							−0.253837	+	0.967247i	\(0.581693\pi\)

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	312.2.q.c.217.1	✓	2
3.2	odd	2		936.2.t.b.217.1		2
4.3	odd	2		624.2.q.e.529.1		2
12.11	even	2		1872.2.t.a.1153.1		2
13.3	even	3	inner	312.2.q.c.289.1	yes	2
13.4	even	6		4056.2.a.b.1.1		1
13.6	odd	12		4056.2.c.b.337.1		2
13.7	odd	12		4056.2.c.b.337.2		2
13.9	even	3		4056.2.a.j.1.1		1
39.29	odd	6		936.2.t.b.289.1		2
52.3	odd	6		624.2.q.e.289.1		2
52.35	odd	6		8112.2.a.bh.1.1		1
52.43	odd	6		8112.2.a.r.1.1		1
156.107	even	6		1872.2.t.a.289.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.q.c.217.1	✓	2	1.1	even	1	trivial
312.2.q.c.289.1	yes	2	13.3	even	3	inner
624.2.q.e.289.1		2	52.3	odd	6
624.2.q.e.529.1		2	4.3	odd	2
936.2.t.b.217.1		2	3.2	odd	2
936.2.t.b.289.1		2	39.29	odd	6
1872.2.t.a.289.1		2	156.107	even	6
1872.2.t.a.1153.1		2	12.11	even	2
4056.2.a.b.1.1		1	13.4	even	6
4056.2.a.j.1.1		1	13.9	even	3
4056.2.c.b.337.1		2	13.6	odd	12
4056.2.c.b.337.2		2	13.7	odd	12
8112.2.a.r.1.1		1	52.43	odd	6
8112.2.a.bh.1.1		1	52.35	odd	6