Embedded newform 2366.2.d.r.337.12

• Label: 2366.2.d.r.337.12
• Level: $2366$
• Weight: $2$
• Character: 2366.337
• Analytic conductor: $18.893$
• Analytic rank: $0$
• Dimension: $12$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2366.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(18.8926051182\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - 6*x^11 + 39*x^10 - 140*x^9 + 460*x^8 - 1066*x^7 + 2127*x^6 - 3172*x^5 + 3842*x^4 - 3394*x^3 + 2141*x^2 - 832*x + 169
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	no (minimal twist has level 182)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.12
Root			\(0.500000 + 2.47866i\) of defining polynomial
Character	\(\chi\)	\(=\)	2366.337
Dual form			2366.2.d.r.337.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +3.34469 q^{3} -1.00000 q^{4} +1.56356i q^{5} +3.34469i q^{6} -1.00000i q^{7} -1.00000i q^{8} +8.18694 q^{9} -1.56356 q^{10} +2.86614i q^{11} -3.34469 q^{12} +1.00000 q^{14} +5.22961i q^{15} +1.00000 q^{16} +2.22961 q^{17} +8.18694i q^{18} -7.23602i q^{19} -1.56356i q^{20} -3.34469i q^{21} -2.86614 q^{22} +1.66735 q^{23} -3.34469i q^{24} +2.55529 q^{25} +17.3487 q^{27} +1.00000i q^{28} +4.82757 q^{29} -5.22961 q^{30} -0.597963i q^{31} +1.00000i q^{32} +9.58634i q^{33} +2.22961i q^{34} +1.56356 q^{35} -8.18694 q^{36} -0.0385636i q^{37} +7.23602 q^{38} +1.56356 q^{40} +7.95469i q^{41} +3.34469 q^{42} -10.0914 q^{43} -2.86614i q^{44} +12.8007i q^{45} +1.66735i q^{46} +7.02636i q^{47} +3.34469 q^{48} -1.00000 q^{49} +2.55529i q^{50} +7.45736 q^{51} -5.98404 q^{53} +17.3487i q^{54} -4.48137 q^{55} -1.00000 q^{56} -24.2022i q^{57} +4.82757i q^{58} +0.896206i q^{59} -5.22961i q^{60} -14.2569 q^{61} +0.597963 q^{62} -8.18694i q^{63} -1.00000 q^{64} -9.58634 q^{66} +1.64086i q^{67} -2.22961 q^{68} +5.57677 q^{69} +1.56356i q^{70} -2.29466i q^{71} -8.18694i q^{72} +11.2277i q^{73} +0.0385636 q^{74} +8.54664 q^{75} +7.23602i q^{76} +2.86614 q^{77} +4.26098 q^{79} +1.56356i q^{80} +33.4651 q^{81} -7.95469 q^{82} -4.94829i q^{83} +3.34469i q^{84} +3.48613i q^{85} -10.0914i q^{86} +16.1467 q^{87} +2.86614 q^{88} -2.42120i q^{89} -12.8007 q^{90} -1.66735 q^{92} -2.00000i q^{93} -7.02636 q^{94} +11.3139 q^{95} +3.34469i q^{96} -4.88829i q^{97} -1.00000i q^{98} +23.4649i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	12 * q + 4 * q^3 - 12 * q^4 + 12 * q^9 + 4 * q^10 - 4 * q^12 + 12 * q^14 + 12 * q^16 - 8 * q^17 + 4 * q^22 + 12 * q^23 - 24 * q^25 + 40 * q^27 + 20 * q^29 - 28 * q^30 - 4 * q^35 - 12 * q^36 + 8 * q^38 - 4 * q^40 + 4 * q^42 - 52 * q^43 + 4 * q^48 - 12 * q^49 - 36 * q^51 + 36 * q^53 + 12 * q^55 - 12 * q^56 + 56 * q^61 + 4 * q^62 - 12 * q^64 + 8 * q^68 - 64 * q^69 + 96 * q^75 - 4 * q^77 + 44 * q^79 + 68 * q^81 - 12 * q^82 - 4 * q^87 - 4 * q^88 + 12 * q^90 - 12 * q^92 - 16 * q^94 - 64 * q^95

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times\).

\(n\)	\(339\)	\(2199\)
\(\chi(n)\)	\(1\)	\(-1\)

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
			\(3\)	3.34469	1.93106	0.965528	−	0.260298i	\(-0.0838210\pi\)
0.965528	+	0.260298i				\(0.0838210\pi\)
\(5\)	1.56356i	0.699244i	0.936891	+	0.349622i	\(0.113690\pi\)
			−0.936891	+	0.349622i	\(0.886310\pi\)
\(7\)	− 1.00000i	− 0.377964i
			\(11\)	2.86614i	0.864173i	0.901832	+	0.432087i	\(0.142223\pi\)
−0.901832	+	0.432087i				\(0.857777\pi\)
\(13\)	0	0
			\(17\)	2.22961	0.540760	0.270380	−	0.962754i	\(-0.412851\pi\)
0.270380	+	0.962754i				\(0.412851\pi\)
\(19\)	− 7.23602i	− 1.66006i	−0.557722	−	0.830028i	\(-0.688324\pi\)
			0.557722	−	0.830028i	\(-0.311676\pi\)
\(23\)	1.66735	0.347667	0.173833	−	0.984775i	\(-0.444385\pi\)
			0.173833	+	0.984775i	\(0.444385\pi\)
\(29\)	4.82757	0.896458	0.448229	−	0.893919i	\(-0.352055\pi\)
			0.448229	+	0.893919i	\(0.352055\pi\)
\(31\)	− 0.597963i	− 0.107397i	−0.998557	−	0.0536987i	\(-0.982899\pi\)
			0.998557	−	0.0536987i	\(-0.0171010\pi\)
\(37\)	− 0.0385636i	− 0.00633982i	−0.999995	−	0.00316991i	\(-0.998991\pi\)
			0.999995	−	0.00316991i	\(-0.00100902\pi\)
\(41\)	7.95469i	1.24231i	0.783686	+	0.621157i	\(0.213337\pi\)
			−0.783686	+	0.621157i	\(0.786663\pi\)
\(43\)	−10.0914	−1.53893	−0.769463	−	0.638691i	\(-0.779476\pi\)
			−0.769463	+	0.638691i	\(0.779476\pi\)
\(47\)	7.02636i	1.02490i	0.858717	+	0.512450i	\(0.171262\pi\)
			−0.858717	+	0.512450i	\(0.828738\pi\)
\(53\)	−5.98404	−0.821971	−0.410985	−	0.911642i	\(-0.634815\pi\)
			−0.410985	+	0.911642i	\(0.634815\pi\)
\(59\)	0.896206i	0.116676i	0.998297	+	0.0583381i	\(0.0185801\pi\)
			−0.998297	+	0.0583381i	\(0.981420\pi\)
\(61\)	−14.2569	−1.82541	−0.912706	−	0.408616i	\(-0.866011\pi\)
			−0.912706	+	0.408616i	\(0.866011\pi\)
\(67\)	1.64086i	0.200464i	0.994964	+	0.100232i	\(0.0319584\pi\)
			−0.994964	+	0.100232i	\(0.968042\pi\)
\(71\)	− 2.29466i	− 0.272326i	−0.990686	−	0.136163i	\(-0.956523\pi\)
			0.990686	−	0.136163i	\(-0.0434771\pi\)
\(73\)	11.2277i	1.31411i	0.753844	+	0.657054i	\(0.228198\pi\)
			−0.753844	+	0.657054i	\(0.771802\pi\)
\(79\)	4.26098	0.479397	0.239699	−	0.970847i	\(-0.422951\pi\)
			0.239699	+	0.970847i	\(0.422951\pi\)
\(83\)	− 4.94829i	− 0.543145i	−0.962418	−	0.271572i	\(-0.912456\pi\)
			0.962418	−	0.271572i	\(-0.0875437\pi\)
\(89\)	− 2.42120i	− 0.256647i	−0.991732	−	0.128323i	\(-0.959040\pi\)
			0.991732	−	0.128323i	\(-0.0409595\pi\)
\(97\)	− 4.88829i	− 0.496330i	−0.968718	−	0.248165i	\(-0.920172\pi\)
			0.968718	−	0.248165i	\(-0.0798276\pi\)

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	2366.2.d.r.337.12		12
13.3	even	3		182.2.m.b.43.1	✓	12
13.4	even	6		182.2.m.b.127.1	yes	12
13.5	odd	4		2366.2.a.bh.1.6		6
13.8	odd	4		2366.2.a.bf.1.6		6
13.12	even	2	inner	2366.2.d.r.337.6		12
39.17	odd	6		1638.2.bj.g.127.6		12
39.29	odd	6		1638.2.bj.g.1135.4		12
52.3	odd	6		1456.2.cc.d.225.6		12
52.43	odd	6		1456.2.cc.d.673.6		12
91.3	odd	6		1274.2.v.d.667.4		12
91.4	even	6		1274.2.o.d.569.1		12
91.16	even	3		1274.2.o.d.459.4		12
91.17	odd	6		1274.2.o.e.569.3		12
91.30	even	6		1274.2.v.e.361.6		12
91.55	odd	6		1274.2.m.c.589.3		12
91.68	odd	6		1274.2.o.e.459.6		12
91.69	odd	6		1274.2.m.c.491.3		12
91.81	even	3		1274.2.v.e.667.6		12
91.82	odd	6		1274.2.v.d.361.4		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.m.b.43.1	✓	12	13.3	even	3
182.2.m.b.127.1	yes	12	13.4	even	6
1274.2.m.c.491.3		12	91.69	odd	6
1274.2.m.c.589.3		12	91.55	odd	6
1274.2.o.d.459.4		12	91.16	even	3
1274.2.o.d.569.1		12	91.4	even	6
1274.2.o.e.459.6		12	91.68	odd	6
1274.2.o.e.569.3		12	91.17	odd	6
1274.2.v.d.361.4		12	91.82	odd	6
1274.2.v.d.667.4		12	91.3	odd	6
1274.2.v.e.361.6		12	91.30	even	6
1274.2.v.e.667.6		12	91.81	even	3
1456.2.cc.d.225.6		12	52.3	odd	6
1456.2.cc.d.673.6		12	52.43	odd	6
1638.2.bj.g.127.6		12	39.17	odd	6
1638.2.bj.g.1135.4		12	39.29	odd	6
2366.2.a.bf.1.6		6	13.8	odd	4
2366.2.a.bh.1.6		6	13.5	odd	4
2366.2.d.r.337.6		12	13.12	even	2	inner
2366.2.d.r.337.12		12	1.1	even	1	trivial