Embedded newform 1183.2.a.k.1.3

• Label: 1183.2.a.k.1.3
• Level: $1183$
• Weight: $2$
• Character: 1183.1
• Self dual: yes
• Analytic conductor: $9.446$
• Analytic rank: $1$
• Dimension: $4$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1183 = 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1183.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(9.44630255912\)
Analytic rank:	\(1\)
Dimension:	\(4\)
Coefficient field:	4.4.27004.1
Defining polynomial:	\( x^{4} - x^{3} - 6x^{2} + 3x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			\(-0.231361\) of defining polynomial
Character	\(\chi\)	\(=\)	1183.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+0.231361 q^{2} -3.32225 q^{3} -1.94647 q^{4} -2.23136 q^{5} -0.768639 q^{6} -1.00000 q^{7} -0.913059 q^{8} +8.03736 q^{9} -0.516249 q^{10} +3.32225 q^{11} +6.46667 q^{12} -0.231361 q^{14} +7.41314 q^{15} +3.68170 q^{16} -1.37578 q^{17} +1.85953 q^{18} +3.23531 q^{19} +4.34328 q^{20} +3.32225 q^{21} +0.768639 q^{22} +0.838502 q^{23} +3.03341 q^{24} -0.0210289 q^{25} -16.7354 q^{27} +1.94647 q^{28} -0.607142 q^{29} +1.71511 q^{30} +1.71511 q^{31} +2.67792 q^{32} -11.0374 q^{33} -0.318302 q^{34} +2.23136 q^{35} -15.6445 q^{36} +1.55361 q^{37} +0.748524 q^{38} +2.03736 q^{40} -9.17783 q^{41} +0.768639 q^{42} +1.23136 q^{43} -6.46667 q^{44} -17.9343 q^{45} +0.193997 q^{46} -1.62817 q^{47} -12.2315 q^{48} +1.00000 q^{49} -0.00486525 q^{50} +4.57069 q^{51} +8.39607 q^{53} -3.87192 q^{54} -7.41314 q^{55} +0.913059 q^{56} -10.7485 q^{57} -0.140469 q^{58} +8.82234 q^{59} -14.4295 q^{60} +5.46667 q^{61} +0.396810 q^{62} -8.03736 q^{63} -6.74383 q^{64} -2.55361 q^{66} -10.1857 q^{67} +2.67792 q^{68} -2.78572 q^{69} +0.516249 q^{70} -5.21428 q^{71} -7.33859 q^{72} -3.96355 q^{73} +0.359445 q^{74} +0.0698632 q^{75} -6.29744 q^{76} -3.32225 q^{77} +6.45051 q^{79} -8.21520 q^{80} +31.4871 q^{81} -2.12339 q^{82} -4.64055 q^{83} -6.46667 q^{84} +3.06986 q^{85} +0.284889 q^{86} +2.01708 q^{87} -3.03341 q^{88} +9.12826 q^{89} -4.14929 q^{90} -1.63212 q^{92} -5.69803 q^{93} -0.376695 q^{94} -7.21915 q^{95} -8.89672 q^{96} -15.3589 q^{97} +0.231361 q^{98} +26.7022 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q - q^2 + q^3 + 5 * q^4 - 7 * q^5 - 5 * q^6 - 4 * q^7 - 6 * q^8 + 7 * q^9 - 11 * q^10 - q^11 + 12 * q^12 + q^14 + 3 * q^15 + 19 * q^16 - 4 * q^17 - 3 * q^18 + q^19 - 2 * q^20 - q^21 + 5 * q^22 - 2 * q^23 - 3 * q^24 + 5 * q^25 - 26 * q^27 - 5 * q^28 + q^29 - 4 * q^30 - 4 * q^31 - 33 * q^32 - 19 * q^33 + 3 * q^34 + 7 * q^35 - 34 * q^36 - 10 * q^37 - 23 * q^38 - 17 * q^40 - 22 * q^41 + 5 * q^42 + 3 * q^43 - 12 * q^44 - 11 * q^45 + 24 * q^46 + 2 * q^47 + 11 * q^48 + 4 * q^49 + 43 * q^50 + 7 * q^51 + 2 * q^53 + 5 * q^54 - 3 * q^55 + 6 * q^56 - 17 * q^57 - 11 * q^58 - 8 * q^59 - 11 * q^60 + 8 * q^61 - 5 * q^62 - 7 * q^63 + 14 * q^64 + 6 * q^66 - 6 * q^67 - 33 * q^68 - 18 * q^69 + 11 * q^70 - 14 * q^71 + 5 * q^72 - 8 * q^73 + 20 * q^74 - 7 * q^75 + 32 * q^76 + q^77 - 26 * q^79 + 7 * q^80 + 24 * q^81 - 14 * q^82 - 12 * q^84 + 5 * q^85 + 12 * q^86 + 13 * q^87 + 3 * q^88 - q^89 + 26 * q^90 + 12 * q^92 - 7 * q^93 + 33 * q^94 + 21 * q^95 - 58 * q^96 + 3 * q^97 - q^98 + 23 * q^99

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0.231361	0.163597	0.0817984	−	0.996649i	\(-0.473934\pi\)
			0.0817984	+	0.996649i	\(0.473934\pi\)
\(3\)	−3.32225	−1.91810	−0.959052	−	0.283231i	\(-0.908594\pi\)
			−0.959052	+	0.283231i	\(0.908594\pi\)
\(5\)	−2.23136	−0.997895	−0.498947	−	0.866632i	\(-0.666280\pi\)
			−0.498947	+	0.866632i	\(0.666280\pi\)
\(7\)	−1.00000	−0.377964
			\(11\)	3.32225	1.00170	0.500848	−	0.865535i	\(-0.333021\pi\)
0.500848	+	0.865535i				\(0.333021\pi\)
\(13\)	0	0
			\(17\)	−1.37578	−0.333676	−0.166838	−	0.985984i	\(-0.553356\pi\)
−0.166838	+	0.985984i				\(0.553356\pi\)
\(19\)	3.23531	0.742231	0.371116	−	0.928587i	\(-0.378975\pi\)
			0.371116	+	0.928587i	\(0.378975\pi\)
\(23\)	0.838502	0.174840	0.0874199	−	0.996172i	\(-0.472138\pi\)
			0.0874199	+	0.996172i	\(0.472138\pi\)
\(29\)	−0.607142	−0.112743	−0.0563717	−	0.998410i	\(-0.517953\pi\)
			−0.0563717	+	0.998410i	\(0.517953\pi\)
\(31\)	1.71511	0.308043	0.154022	−	0.988067i	\(-0.450777\pi\)
			0.154022	+	0.988067i	\(0.450777\pi\)
\(37\)	1.55361	0.255413	0.127706	−	0.991812i	\(-0.459239\pi\)
			0.127706	+	0.991812i	\(0.459239\pi\)
\(41\)	−9.17783	−1.43334	−0.716668	−	0.697414i	\(-0.754334\pi\)
			−0.716668	+	0.697414i	\(0.754334\pi\)
\(43\)	1.23136	0.187781	0.0938904	−	0.995583i	\(-0.470070\pi\)
			0.0938904	+	0.995583i	\(0.470070\pi\)
\(47\)	−1.62817	−0.237493	−0.118747	−	0.992925i	\(-0.537888\pi\)
			−0.118747	+	0.992925i	\(0.537888\pi\)
\(53\)	8.39607	1.15329	0.576644	−	0.816995i	\(-0.304362\pi\)
			0.576644	+	0.816995i	\(0.304362\pi\)
\(59\)	8.82234	1.14857	0.574285	−	0.818655i	\(-0.305280\pi\)
			0.574285	+	0.818655i	\(0.305280\pi\)
\(61\)	5.46667	0.699936	0.349968	−	0.936762i	\(-0.386193\pi\)
			0.349968	+	0.936762i	\(0.386193\pi\)
\(67\)	−10.1857	−1.24439	−0.622193	−	0.782864i	\(-0.713758\pi\)
			−0.622193	+	0.782864i	\(0.713758\pi\)
\(71\)	−5.21428	−0.618822	−0.309411	−	0.950928i	\(-0.600132\pi\)
			−0.309411	+	0.950928i	\(0.600132\pi\)
\(73\)	−3.96355	−0.463898	−0.231949	−	0.972728i	\(-0.574510\pi\)
			−0.231949	+	0.972728i	\(0.574510\pi\)
\(79\)	6.45051	0.725739	0.362869	−	0.931840i	\(-0.381797\pi\)
			0.362869	+	0.931840i	\(0.381797\pi\)
\(83\)	−4.64055	−0.509367	−0.254684	−	0.967024i	\(-0.581971\pi\)
			−0.254684	+	0.967024i	\(0.581971\pi\)
\(89\)	9.12826	0.967593	0.483797	−	0.875180i	\(-0.339257\pi\)
			0.483797	+	0.875180i	\(0.339257\pi\)
\(97\)	−15.3589	−1.55946	−0.779729	−	0.626117i	\(-0.784643\pi\)
			−0.779729	+	0.626117i	\(0.784643\pi\)

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	1183.2.a.k.1.3		4
7.6	odd	2		8281.2.a.bp.1.3		4
13.3	even	3		91.2.f.c.22.2	✓	8
13.5	odd	4		1183.2.c.g.337.4		8
13.8	odd	4		1183.2.c.g.337.5		8
13.9	even	3		91.2.f.c.29.2	yes	8
13.12	even	2		1183.2.a.l.1.2		4
39.29	odd	6		819.2.o.h.568.3		8
39.35	odd	6		819.2.o.h.757.3		8
52.3	odd	6		1456.2.s.q.113.1		8
52.35	odd	6		1456.2.s.q.1121.1		8
91.3	odd	6		637.2.g.j.373.2		8
91.9	even	3		637.2.g.k.263.2		8
91.16	even	3		637.2.h.h.165.3		8
91.48	odd	6		637.2.f.i.393.2		8
91.55	odd	6		637.2.f.i.295.2		8
91.61	odd	6		637.2.g.j.263.2		8
91.68	odd	6		637.2.h.i.165.3		8
91.74	even	3		637.2.h.h.471.3		8
91.81	even	3		637.2.g.k.373.2		8
91.87	odd	6		637.2.h.i.471.3		8
91.90	odd	2		8281.2.a.bt.1.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.f.c.22.2	✓	8	13.3	even	3
91.2.f.c.29.2	yes	8	13.9	even	3
637.2.f.i.295.2		8	91.55	odd	6
637.2.f.i.393.2		8	91.48	odd	6
637.2.g.j.263.2		8	91.61	odd	6
637.2.g.j.373.2		8	91.3	odd	6
637.2.g.k.263.2		8	91.9	even	3
637.2.g.k.373.2		8	91.81	even	3
637.2.h.h.165.3		8	91.16	even	3
637.2.h.h.471.3		8	91.74	even	3
637.2.h.i.165.3		8	91.68	odd	6
637.2.h.i.471.3		8	91.87	odd	6
819.2.o.h.568.3		8	39.29	odd	6
819.2.o.h.757.3		8	39.35	odd	6
1183.2.a.k.1.3		4	1.1	even	1	trivial
1183.2.a.l.1.2		4	13.12	even	2
1183.2.c.g.337.4		8	13.5	odd	4
1183.2.c.g.337.5		8	13.8	odd	4
1456.2.s.q.113.1		8	52.3	odd	6
1456.2.s.q.1121.1		8	52.35	odd	6
8281.2.a.bp.1.3		4	7.6	odd	2
8281.2.a.bt.1.2		4	91.90	odd	2