Embedded newform 1183.2.e.d.508.2

• Label: 1183.2.e.d.508.2
• Level: $1183$
• Weight: $2$
• Character: 1183.508
• Analytic conductor: $9.446$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1183 = 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1183.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.44630255912\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial:	\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			508.2
Root			\(-0.309017 - 0.535233i\) of defining polynomial
Character	\(\chi\)	\(=\)	1183.508
Dual form			1183.2.e.d.170.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.190983 - 0.330792i) q^{2} +(-1.11803 + 1.93649i) q^{3} +(0.927051 - 1.60570i) q^{4} +(-1.11803 - 1.93649i) q^{5} +0.854102 q^{6} +(2.00000 - 1.73205i) q^{7} -1.47214 q^{8} +(-1.00000 - 1.73205i) q^{9} +(-0.427051 + 0.739674i) q^{10} +(-1.50000 + 2.59808i) q^{11} +(2.07295 + 3.59045i) q^{12} +(-0.954915 - 0.330792i) q^{14} +5.00000 q^{15} +(-1.57295 - 2.72443i) q^{16} +(3.73607 - 6.47106i) q^{17} +(-0.381966 + 0.661585i) q^{18} +(1.50000 + 2.59808i) q^{19} -4.14590 q^{20} +(1.11803 + 5.80948i) q^{21} +1.14590 q^{22} +(1.88197 + 3.25966i) q^{23} +(1.64590 - 2.85078i) q^{24} -2.23607 q^{27} +(-0.927051 - 4.81710i) q^{28} -4.47214 q^{29} +(-0.954915 - 1.65396i) q^{30} +(2.50000 - 4.33013i) q^{31} +(-2.07295 + 3.59045i) q^{32} +(-3.35410 - 5.80948i) q^{33} -2.85410 q^{34} +(-5.59017 - 1.93649i) q^{35} -3.70820 q^{36} +(-4.35410 - 7.54153i) q^{37} +(0.572949 - 0.992377i) q^{38} +(1.64590 + 2.85078i) q^{40} -4.47214 q^{41} +(1.70820 - 1.47935i) q^{42} -8.00000 q^{43} +(2.78115 + 4.81710i) q^{44} +(-2.23607 + 3.87298i) q^{45} +(0.718847 - 1.24508i) q^{46} +(0.736068 + 1.27491i) q^{47} +7.03444 q^{48} +(1.00000 - 6.92820i) q^{49} +(8.35410 + 14.4697i) q^{51} +(-0.736068 + 1.27491i) q^{53} +(0.427051 + 0.739674i) q^{54} +6.70820 q^{55} +(-2.94427 + 2.54981i) q^{56} -6.70820 q^{57} +(0.854102 + 1.47935i) q^{58} +(3.73607 - 6.47106i) q^{59} +(4.63525 - 8.02850i) q^{60} +(-1.50000 - 2.59808i) q^{61} -1.90983 q^{62} +(-5.00000 - 1.73205i) q^{63} -4.70820 q^{64} +(-1.28115 + 2.21902i) q^{66} +(-1.50000 + 2.59808i) q^{67} +(-6.92705 - 11.9980i) q^{68} -8.41641 q^{69} +(0.427051 + 2.21902i) q^{70} -8.94427 q^{71} +(1.47214 + 2.54981i) q^{72} +(5.35410 - 9.27358i) q^{73} +(-1.66312 + 2.88061i) q^{74} +5.56231 q^{76} +(1.50000 + 7.79423i) q^{77} +(-5.35410 - 9.27358i) q^{79} +(-3.51722 + 6.09201i) q^{80} +(5.50000 - 9.52628i) q^{81} +(0.854102 + 1.47935i) q^{82} +(10.3647 + 3.59045i) q^{84} -16.7082 q^{85} +(1.52786 + 2.64634i) q^{86} +(5.00000 - 8.66025i) q^{87} +(2.20820 - 3.82472i) q^{88} +(-1.11803 - 1.93649i) q^{89} +1.70820 q^{90} +6.97871 q^{92} +(5.59017 + 9.68246i) q^{93} +(0.281153 - 0.486971i) q^{94} +(3.35410 - 5.80948i) q^{95} +(-4.63525 - 8.02850i) q^{96} +17.4164 q^{97} +(-2.48278 + 0.992377i) q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q - 3 * q^2 - 3 * q^4 - 10 * q^6 + 8 * q^7 + 12 * q^8 - 4 * q^9 + 5 * q^10 - 6 * q^11 + 15 * q^12 - 15 * q^14 + 20 * q^15 - 13 * q^16 + 6 * q^17 - 6 * q^18 + 6 * q^19 - 30 * q^20 + 18 * q^22 + 12 * q^23 + 20 * q^24 + 3 * q^28 - 15 * q^30 + 10 * q^31 - 15 * q^32 + 2 * q^34 + 12 * q^36 - 4 * q^37 + 9 * q^38 + 20 * q^40 - 20 * q^42 - 32 * q^43 - 9 * q^44 + 23 * q^46 - 6 * q^47 - 30 * q^48 + 4 * q^49 + 20 * q^51 + 6 * q^53 - 5 * q^54 + 24 * q^56 - 10 * q^58 + 6 * q^59 - 15 * q^60 - 6 * q^61 - 30 * q^62 - 20 * q^63 + 8 * q^64 + 15 * q^66 - 6 * q^67 - 21 * q^68 + 20 * q^69 - 5 * q^70 - 12 * q^72 + 8 * q^73 + 9 * q^74 - 18 * q^76 + 6 * q^77 - 8 * q^79 + 15 * q^80 + 22 * q^81 - 10 * q^82 + 75 * q^84 - 40 * q^85 + 24 * q^86 + 20 * q^87 - 18 * q^88 - 20 * q^90 - 66 * q^92 - 19 * q^94 + 15 * q^96 + 16 * q^97 - 39 * q^98 + 24 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\)	\(339\)	\(1016\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.190983	−	0.330792i	−0.135045	−	0.233905i	0.790569	−	0.612372i	\(-0.209785\pi\)
							−0.925615	+	0.378467i	\(0.876451\pi\)
\(3\)	−1.11803	+	1.93649i	−0.645497	+	1.11803i	0.338689	+	0.940898i	\(0.390016\pi\)
							−0.984186	+	0.177136i	\(0.943317\pi\)
\(5\)	−1.11803	−	1.93649i	−0.500000	−	0.866025i	0.500000	−	0.866025i	\(-0.333333\pi\)
							−1.00000			\(\pi\)
\(7\)	2.00000	−	1.73205i	0.755929	−	0.654654i
							\(11\)	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	\(-0.982718\pi\)
0.546259	+	0.837616i	\(0.316051\pi\)
\(13\)		0			0
							\(17\)	3.73607	−	6.47106i	0.906130	−	1.56946i	0.0867359	−	0.996231i	\(-0.472356\pi\)
0.819394	−	0.573231i	\(-0.194310\pi\)
\(19\)	1.50000	+	2.59808i	0.344124	+	0.596040i	0.985194	−	0.171442i	\(-0.0548427\pi\)
							−0.641071	+	0.767482i	\(0.721509\pi\)
\(23\)	1.88197	+	3.25966i	0.392417	+	0.679686i	0.992768	−	0.120051i	\(-0.0383057\pi\)
							−0.600351	+	0.799737i	\(0.704972\pi\)
\(29\)	−4.47214			−0.830455			−0.415227	−	0.909718i	\(-0.636298\pi\)
							−0.415227	+	0.909718i	\(0.636298\pi\)
\(31\)	2.50000	−	4.33013i	0.449013	−	0.777714i	−0.549309	−	0.835619i	\(-0.685109\pi\)
							0.998322	+	0.0579057i	\(0.0184423\pi\)
\(37\)	−4.35410	−	7.54153i	−0.715810	−	1.23982i	−0.962646	−	0.270762i	\(-0.912724\pi\)
							0.246836	−	0.969057i	\(-0.420609\pi\)
\(41\)	−4.47214			−0.698430			−0.349215	−	0.937043i	\(-0.613552\pi\)
							−0.349215	+	0.937043i	\(0.613552\pi\)
\(43\)	−8.00000			−1.21999			−0.609994	−	0.792406i	\(-0.708828\pi\)
							−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	0.736068	+	1.27491i	0.107367	+	0.185964i	0.914703	−	0.404128i	\(-0.132425\pi\)
							−0.807336	+	0.590092i	\(0.799091\pi\)
\(53\)	−0.736068	+	1.27491i	−0.101107	+	0.175122i	−0.912141	−	0.409877i	\(-0.865572\pi\)
							0.811034	+	0.584999i	\(0.198905\pi\)
\(59\)	3.73607	−	6.47106i	0.486395	−	0.842460i	−0.513483	−	0.858100i	\(-0.671645\pi\)
							0.999878	+	0.0156395i	\(0.00497842\pi\)
\(61\)	−1.50000	−	2.59808i	−0.192055	−	0.332650i	0.753876	−	0.657017i	\(-0.228182\pi\)
							−0.945931	+	0.324367i	\(0.894849\pi\)
\(67\)	−1.50000	+	2.59808i	−0.183254	+	0.317406i	−0.942987	−	0.332830i	\(-0.891996\pi\)
							0.759733	+	0.650236i	\(0.225330\pi\)
\(71\)	−8.94427			−1.06149			−0.530745	−	0.847532i	\(-0.678088\pi\)
							−0.530745	+	0.847532i	\(0.678088\pi\)
\(73\)	5.35410	−	9.27358i	0.626650	−	1.08539i	−0.361569	−	0.932345i	\(-0.617759\pi\)
							0.988219	−	0.153045i	\(-0.0489079\pi\)
\(79\)	−5.35410	−	9.27358i	−0.602384	−	1.04336i	−0.992459	−	0.122576i	\(-0.960884\pi\)
							0.390076	−	0.920783i	\(-0.372449\pi\)
\(83\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(89\)	−1.11803	−	1.93649i	−0.118511	−	0.205268i	0.800667	−	0.599110i	\(-0.204479\pi\)
							−0.919178	+	0.393842i	\(0.871146\pi\)
\(97\)	17.4164			1.76837			0.884184	−	0.467139i	\(-0.154715\pi\)
							0.884184	+	0.467139i	\(0.154715\pi\)

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.e.d.508.2		4
7.2	even	3	inner	1183.2.e.d.170.2		4
7.3	odd	6		8281.2.a.ba.1.1		2
7.4	even	3		8281.2.a.z.1.1		2
13.12	even	2		91.2.e.b.53.1	✓	4
39.38	odd	2		819.2.j.c.235.2		4
52.51	odd	2		1456.2.r.j.417.2		4
91.12	odd	6		637.2.e.h.79.1		4
91.25	even	6		637.2.a.f.1.2		2
91.38	odd	6		637.2.a.e.1.2		2
91.51	even	6		91.2.e.b.79.1	yes	4
91.90	odd	2		637.2.e.h.508.1		4
273.38	even	6		5733.2.a.w.1.1		2
273.116	odd	6		5733.2.a.v.1.1		2
273.233	odd	6		819.2.j.c.352.2		4
364.51	odd	6		1456.2.r.j.625.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.b.53.1	✓	4	13.12	even	2
91.2.e.b.79.1	yes	4	91.51	even	6
637.2.a.e.1.2		2	91.38	odd	6
637.2.a.f.1.2		2	91.25	even	6
637.2.e.h.79.1		4	91.12	odd	6
637.2.e.h.508.1		4	91.90	odd	2
819.2.j.c.235.2		4	39.38	odd	2
819.2.j.c.352.2		4	273.233	odd	6
1183.2.e.d.170.2		4	7.2	even	3	inner
1183.2.e.d.508.2		4	1.1	even	1	trivial
1456.2.r.j.417.2		4	52.51	odd	2
1456.2.r.j.625.2		4	364.51	odd	6
5733.2.a.v.1.1		2	273.116	odd	6
5733.2.a.w.1.1		2	273.38	even	6
8281.2.a.z.1.1		2	7.4	even	3
8281.2.a.ba.1.1		2	7.3	odd	6