Embedded newform 1014.2.b.e.337.1

• Label: 1014.2.b.e.337.1
• Level: $1014$
• Weight: $2$
• Character: 1014.337
• Analytic conductor: $8.097$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1014.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.09683076496\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			\(0.866025 - 0.500000i\) of defining polynomial
Character	\(\chi\)	\(=\)	1014.337
Dual form			1014.2.b.e.337.4

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -3.73205i q^{5} -1.00000i q^{6} -2.73205i q^{7} +1.00000i q^{8} +1.00000 q^{9} -3.73205 q^{10} +1.26795i q^{11} -1.00000 q^{12} -2.73205 q^{14} -3.73205i q^{15} +1.00000 q^{16} +5.73205 q^{17} -1.00000i q^{18} -4.73205i q^{19} +3.73205i q^{20} -2.73205i q^{21} +1.26795 q^{22} -4.19615 q^{23} +1.00000i q^{24} -8.92820 q^{25} +1.00000 q^{27} +2.73205i q^{28} -4.46410 q^{29} -3.73205 q^{30} -1.46410i q^{31} -1.00000i q^{32} +1.26795i q^{33} -5.73205i q^{34} -10.1962 q^{35} -1.00000 q^{36} +3.53590i q^{37} -4.73205 q^{38} +3.73205 q^{40} +9.39230i q^{41} -2.73205 q^{42} +9.66025 q^{43} -1.26795i q^{44} -3.73205i q^{45} +4.19615i q^{46} +2.19615i q^{47} +1.00000 q^{48} -0.464102 q^{49} +8.92820i q^{50} +5.73205 q^{51} -6.46410 q^{53} -1.00000i q^{54} +4.73205 q^{55} +2.73205 q^{56} -4.73205i q^{57} +4.46410i q^{58} +8.00000i q^{59} +3.73205i q^{60} -9.19615 q^{61} -1.46410 q^{62} -2.73205i q^{63} -1.00000 q^{64} +1.26795 q^{66} -13.1244i q^{67} -5.73205 q^{68} -4.19615 q^{69} +10.1962i q^{70} -4.73205i q^{71} +1.00000i q^{72} -6.26795i q^{73} +3.53590 q^{74} -8.92820 q^{75} +4.73205i q^{76} +3.46410 q^{77} -2.53590 q^{79} -3.73205i q^{80} +1.00000 q^{81} +9.39230 q^{82} +0.196152i q^{83} +2.73205i q^{84} -21.3923i q^{85} -9.66025i q^{86} -4.46410 q^{87} -1.26795 q^{88} -9.46410i q^{89} -3.73205 q^{90} +4.19615 q^{92} -1.46410i q^{93} +2.19615 q^{94} -17.6603 q^{95} -1.00000i q^{96} +6.00000i q^{97} +0.464102i q^{98} +1.26795i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 8 * q^10 - 4 * q^12 - 4 * q^14 + 4 * q^16 + 16 * q^17 + 12 * q^22 + 4 * q^23 - 8 * q^25 + 4 * q^27 - 4 * q^29 - 8 * q^30 - 20 * q^35 - 4 * q^36 - 12 * q^38 + 8 * q^40 - 4 * q^42 + 4 * q^43 + 4 * q^48 + 12 * q^49 + 16 * q^51 - 12 * q^53 + 12 * q^55 + 4 * q^56 - 16 * q^61 + 8 * q^62 - 4 * q^64 + 12 * q^66 - 16 * q^68 + 4 * q^69 + 28 * q^74 - 8 * q^75 - 24 * q^79 + 4 * q^81 - 4 * q^82 - 4 * q^87 - 12 * q^88 - 8 * q^90 - 4 * q^92 - 12 * q^94 - 36 * q^95

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\)	\(677\)	\(847\)
\(\chi(n)\)	\(1\)	\(-1\)

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	− 1.00000i	− 0.707107i
			\(3\)	1.00000	0.577350
\(5\)	− 3.73205i	− 1.66902i				−0.550990	−	0.834512i	\(-0.685750\pi\)
			0.550990	−	0.834512i	\(-0.314250\pi\)
\(7\)	− 2.73205i	− 1.03262i	−0.856402	−	0.516309i	\(-0.827306\pi\)
			0.856402	−	0.516309i	\(-0.172694\pi\)
\(11\)	1.26795i	0.382301i	0.981561	+	0.191151i	\(0.0612219\pi\)
			−0.981561	+	0.191151i	\(0.938778\pi\)
\(13\)	0	0
			\(17\)	5.73205	1.39023	0.695113	−	0.718900i	\(-0.255354\pi\)
0.695113	+	0.718900i				\(0.255354\pi\)
\(19\)	− 4.73205i	− 1.08561i	−0.839860	−	0.542803i	\(-0.817363\pi\)
			0.839860	−	0.542803i	\(-0.182637\pi\)
\(23\)	−4.19615	−0.874958	−0.437479	−	0.899229i	\(-0.644129\pi\)
			−0.437479	+	0.899229i	\(0.644129\pi\)
\(29\)	−4.46410	−0.828963	−0.414481	−	0.910058i	\(-0.636037\pi\)
			−0.414481	+	0.910058i	\(0.636037\pi\)
\(31\)	− 1.46410i	− 0.262960i	−0.991319	−	0.131480i	\(-0.958027\pi\)
			0.991319	−	0.131480i	\(-0.0419730\pi\)
\(37\)	3.53590i	0.581298i	0.956830	+	0.290649i	\(0.0938712\pi\)
			−0.956830	+	0.290649i	\(0.906129\pi\)
\(41\)	9.39230i	1.46683i	0.679780	+	0.733416i	\(0.262075\pi\)
			−0.679780	+	0.733416i	\(0.737925\pi\)
\(43\)	9.66025	1.47317	0.736587	−	0.676342i	\(-0.236436\pi\)
			0.736587	+	0.676342i	\(0.236436\pi\)
\(47\)	2.19615i	0.320342i	0.987089	+	0.160171i	\(0.0512045\pi\)
			−0.987089	+	0.160171i	\(0.948795\pi\)
\(53\)	−6.46410	−0.887913	−0.443956	−	0.896048i	\(-0.646425\pi\)
			−0.443956	+	0.896048i	\(0.646425\pi\)
\(59\)	8.00000i	1.04151i	0.853706	+	0.520756i	\(0.174350\pi\)
			−0.853706	+	0.520756i	\(0.825650\pi\)
\(61\)	−9.19615	−1.17745	−0.588723	−	0.808335i	\(-0.700369\pi\)
			−0.588723	+	0.808335i	\(0.700369\pi\)
\(67\)	− 13.1244i	− 1.60340i	−0.597730	−	0.801698i	\(-0.703930\pi\)
			0.597730	−	0.801698i	\(-0.296070\pi\)
\(71\)	− 4.73205i	− 0.561591i	−0.959768	−	0.280796i	\(-0.909402\pi\)
			0.959768	−	0.280796i	\(-0.0905983\pi\)
\(73\)	− 6.26795i	− 0.733608i	−0.930298	−	0.366804i	\(-0.880452\pi\)
			0.930298	−	0.366804i	\(-0.119548\pi\)
\(79\)	−2.53590	−0.285311	−0.142655	−	0.989772i	\(-0.545564\pi\)
			−0.142655	+	0.989772i	\(0.545564\pi\)
\(83\)	0.196152i	0.0215305i	0.999942	+	0.0107653i	\(0.00342676\pi\)
			−0.999942	+	0.0107653i	\(0.996573\pi\)
\(89\)	− 9.46410i	− 1.00319i	−0.865102	−	0.501596i	\(-0.832746\pi\)
			0.865102	−	0.501596i	\(-0.167254\pi\)
\(97\)	6.00000i	0.609208i	0.952479	+	0.304604i	\(0.0985241\pi\)
			−0.952479	+	0.304604i	\(0.901476\pi\)

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	1014.2.b.e.337.1		4
3.2	odd	2		3042.2.b.i.1351.4		4
13.2	odd	12		1014.2.e.i.529.1		4
13.3	even	3		1014.2.i.a.823.2		4
13.4	even	6		1014.2.i.a.361.2		4
13.5	odd	4		1014.2.a.i.1.1		2
13.6	odd	12		1014.2.e.i.991.1		4
13.7	odd	12		1014.2.e.g.991.2		4
13.8	odd	4		1014.2.a.k.1.2		2
13.9	even	3		78.2.i.a.49.1	yes	4
13.10	even	6		78.2.i.a.43.1	✓	4
13.11	odd	12		1014.2.e.g.529.2		4
13.12	even	2	inner	1014.2.b.e.337.4		4
39.5	even	4		3042.2.a.y.1.2		2
39.8	even	4		3042.2.a.p.1.1		2
39.23	odd	6		234.2.l.c.199.2		4
39.35	odd	6		234.2.l.c.127.2		4
39.38	odd	2		3042.2.b.i.1351.1		4
52.23	odd	6		624.2.bv.e.433.2		4
52.31	even	4		8112.2.a.bj.1.1		2
52.35	odd	6		624.2.bv.e.49.1		4
52.47	even	4		8112.2.a.bp.1.2		2
65.9	even	6		1950.2.bc.d.751.2		4
65.22	odd	12		1950.2.y.b.49.1		4
65.23	odd	12		1950.2.y.b.199.1		4
65.48	odd	12		1950.2.y.g.49.2		4
65.49	even	6		1950.2.bc.d.901.2		4
65.62	odd	12		1950.2.y.g.199.2		4
156.23	even	6		1872.2.by.h.433.1		4
156.35	even	6		1872.2.by.h.1297.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.i.a.43.1	✓	4	13.10	even	6
78.2.i.a.49.1	yes	4	13.9	even	3
234.2.l.c.127.2		4	39.35	odd	6
234.2.l.c.199.2		4	39.23	odd	6
624.2.bv.e.49.1		4	52.35	odd	6
624.2.bv.e.433.2		4	52.23	odd	6
1014.2.a.i.1.1		2	13.5	odd	4
1014.2.a.k.1.2		2	13.8	odd	4
1014.2.b.e.337.1		4	1.1	even	1	trivial
1014.2.b.e.337.4		4	13.12	even	2	inner
1014.2.e.g.529.2		4	13.11	odd	12
1014.2.e.g.991.2		4	13.7	odd	12
1014.2.e.i.529.1		4	13.2	odd	12
1014.2.e.i.991.1		4	13.6	odd	12
1014.2.i.a.361.2		4	13.4	even	6
1014.2.i.a.823.2		4	13.3	even	3
1872.2.by.h.433.1		4	156.23	even	6
1872.2.by.h.1297.2		4	156.35	even	6
1950.2.y.b.49.1		4	65.22	odd	12
1950.2.y.b.199.1		4	65.23	odd	12
1950.2.y.g.49.2		4	65.48	odd	12
1950.2.y.g.199.2		4	65.62	odd	12
1950.2.bc.d.751.2		4	65.9	even	6
1950.2.bc.d.901.2		4	65.49	even	6
3042.2.a.p.1.1		2	39.8	even	4
3042.2.a.y.1.2		2	39.5	even	4
3042.2.b.i.1351.1		4	39.38	odd	2
3042.2.b.i.1351.4		4	3.2	odd	2
8112.2.a.bj.1.1		2	52.31	even	4
8112.2.a.bp.1.2		2	52.47	even	4