Embedded newform 99.2.f.c.91.1

• Label: 99.2.f.c.91.1
• Level: $99$
• Weight: $2$
• Character: 99.91
• Analytic conductor: $0.791$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	99.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.790518980011\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	8.0.484000000.9
Defining polynomial:	\( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			91.1
Root			\(-1.73855 + 1.26313i\) of defining polynomial
Character	\(\chi\)	\(=\)	99.91
Dual form			99.2.f.c.37.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73855 + 1.26313i) q^{2} +(0.809017 - 2.48990i) q^{4} +(1.73855 + 1.26313i) q^{5} +(-1.30902 + 4.02874i) q^{7} +(0.410415 + 1.26313i) q^{8} -4.61803 q^{10} +(-3.22344 + 0.780656i) q^{11} +(-0.190983 + 0.138757i) q^{13} +(-2.81303 - 8.65761i) q^{14} +(1.92705 + 1.40008i) q^{16} +(4.96199 + 3.60510i) q^{17} +(-0.736068 - 2.26538i) q^{19} +(4.55157 - 3.30691i) q^{20} +(4.61803 - 5.42882i) q^{22} +3.98439 q^{23} +(-0.118034 - 0.363271i) q^{25} +(0.156765 - 0.482472i) q^{26} +(8.97214 + 6.51864i) q^{28} +(2.14896 - 6.61382i) q^{29} +(3.73607 - 2.71441i) q^{31} -7.77501 q^{32} -13.1803 q^{34} +(-7.36460 + 5.35069i) q^{35} +(0.545085 - 1.67760i) q^{37} +(4.14116 + 3.00873i) q^{38} +(-0.881966 + 2.71441i) q^{40} +(-0.917716 - 2.82444i) q^{41} +2.70820 q^{43} +(-0.664066 + 8.65761i) q^{44} +(-6.92705 + 5.03280i) q^{46} +(1.07448 + 3.30691i) q^{47} +(-8.85410 - 6.43288i) q^{49} +(0.664066 + 0.482472i) q^{50} +(0.190983 + 0.587785i) q^{52} +(1.48490 - 1.07884i) q^{53} +(-6.59017 - 2.71441i) q^{55} -5.62605 q^{56} +(4.61803 + 14.2128i) q^{58} +(-2.30573 + 7.09629i) q^{59} +(9.16312 + 6.65740i) q^{61} +(-3.06668 + 9.43826i) q^{62} +(9.66312 - 7.02067i) q^{64} -0.507301 q^{65} -2.85410 q^{67} +(12.9907 - 9.43826i) q^{68} +(6.04508 - 18.6049i) q^{70} +(-8.69273 - 6.31564i) q^{71} +(0.763932 - 2.35114i) q^{73} +(1.17137 + 3.60510i) q^{74} -6.23607 q^{76} +(1.07448 - 14.0083i) q^{77} +(7.66312 - 5.56758i) q^{79} +(1.58178 + 4.86822i) q^{80} +(5.16312 + 3.75123i) q^{82} +(-6.70053 - 4.86822i) q^{83} +(4.07295 + 12.5352i) q^{85} +(-4.70834 + 3.42081i) q^{86} +(-2.30902 - 3.75123i) q^{88} +8.90937 q^{89} +(-0.309017 - 0.951057i) q^{91} +(3.22344 - 9.92073i) q^{92} +(-6.04508 - 4.39201i) q^{94} +(1.58178 - 4.86822i) q^{95} +(-9.54508 + 6.93491i) q^{97} +23.5188 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	8 * q + 2 * q^4 - 6 * q^7 - 28 * q^10 - 6 * q^13 + 2 * q^16 + 12 * q^19 + 28 * q^22 + 8 * q^25 + 36 * q^28 + 12 * q^31 - 16 * q^34 - 18 * q^37 - 16 * q^40 - 32 * q^43 - 42 * q^46 - 44 * q^49 + 6 * q^52 - 8 * q^55 + 28 * q^58 + 42 * q^61 + 46 * q^64 + 4 * q^67 + 26 * q^70 + 24 * q^73 - 32 * q^76 + 30 * q^79 + 10 * q^82 + 46 * q^85 - 14 * q^88 + 2 * q^91 - 26 * q^94 - 54 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\)	\(46\)	\(56\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.73855	+	1.26313i	−1.22934	+	0.893166i	−0.996840	−	0.0794309i	\(-0.974690\pi\)
							−0.232497	+	0.972597i	\(0.574690\pi\)
\(3\)		0			0
							\(5\)	1.73855	+	1.26313i	0.777501	+	0.564888i	0.904228	−	0.427050i	\(-0.140447\pi\)
−0.126727	+	0.991938i	\(0.540447\pi\)
\(7\)	−1.30902	+	4.02874i	−0.494762	+	1.52272i	0.322566	+	0.946547i	\(0.395455\pi\)
							−0.817327	+	0.576173i	\(0.804545\pi\)
\(11\)	−3.22344	+	0.780656i	−0.971904	+	0.235377i
							\(13\)	−0.190983	+	0.138757i	−0.0529692	+	0.0384843i	−0.613955	−	0.789341i	\(-0.710422\pi\)
0.560986	+	0.827826i	\(0.310422\pi\)
\(17\)	4.96199	+	3.60510i	1.20346	+	0.874364i	0.994620	−	0.103586i	\(-0.0330318\pi\)
							0.208838	+	0.977950i	\(0.433032\pi\)
\(19\)	−0.736068	−	2.26538i	−0.168866	−	0.519715i	0.830435	−	0.557116i	\(-0.188092\pi\)
							−0.999300	+	0.0374011i	\(0.988092\pi\)
\(23\)	3.98439			0.830803			0.415402	−	0.909638i	\(-0.363641\pi\)
							0.415402	+	0.909638i	\(0.363641\pi\)
\(29\)	2.14896	−	6.61382i	0.399052	−	1.22816i	−0.526709	−	0.850046i	\(-0.676574\pi\)
							0.925761	−	0.378110i	\(-0.123426\pi\)
\(31\)	3.73607	−	2.71441i	0.671018	−	0.487523i	−0.199348	−	0.979929i	\(-0.563882\pi\)
							0.870366	+	0.492406i	\(0.163882\pi\)
\(37\)	0.545085	−	1.67760i	0.0896114	−	0.275796i	−0.896201	−	0.443649i	\(-0.853684\pi\)
							0.985812	+	0.167854i	\(0.0536836\pi\)
\(41\)	−0.917716	−	2.82444i	−0.143323	−	0.441103i	0.853468	−	0.521145i	\(-0.174495\pi\)
							−0.996792	+	0.0800413i	\(0.974495\pi\)
\(43\)	2.70820			0.412997			0.206499	−	0.978447i	\(-0.433793\pi\)
							0.206499	+	0.978447i	\(0.433793\pi\)
\(47\)	1.07448	+	3.30691i	0.156729	+	0.482363i	0.998332	−	0.0577343i	\(-0.0183876\pi\)
							−0.841603	+	0.540097i	\(0.818388\pi\)
\(53\)	1.48490	−	1.07884i	0.203966	−	0.148190i	−0.481113	−	0.876658i	\(-0.659767\pi\)
							0.685079	+	0.728468i	\(0.259767\pi\)
\(59\)	−2.30573	+	7.09629i	−0.300180	+	0.923859i	0.681252	+	0.732049i	\(0.261436\pi\)
							−0.981432	+	0.191810i	\(0.938564\pi\)
\(61\)	9.16312	+	6.65740i	1.17322	+	0.852392i	0.991391	−	0.130938i	\(-0.0417990\pi\)
							0.181827	+	0.983331i	\(0.441799\pi\)
\(67\)	−2.85410			−0.348684			−0.174342	−	0.984685i	\(-0.555780\pi\)
							−0.174342	+	0.984685i	\(0.555780\pi\)
\(71\)	−8.69273	−	6.31564i	−1.03164	−	0.749528i	−0.0630016	−	0.998013i	\(-0.520067\pi\)
							−0.968636	+	0.248485i	\(0.920067\pi\)
\(73\)	0.763932	−	2.35114i	0.0894115	−	0.275180i	−0.896346	−	0.443356i	\(-0.853788\pi\)
							0.985757	+	0.168176i	\(0.0537877\pi\)
\(79\)	7.66312	−	5.56758i	0.862168	−	0.626402i	−0.0663057	−	0.997799i	\(-0.521121\pi\)
							0.928474	+	0.371397i	\(0.121121\pi\)
\(83\)	−6.70053	−	4.86822i	−0.735479	−	0.534357i	0.155813	−	0.987787i	\(-0.450200\pi\)
							−0.891292	+	0.453430i	\(0.850200\pi\)
\(89\)	8.90937			0.944392			0.472196	−	0.881494i	\(-0.343462\pi\)
							0.472196	+	0.881494i	\(0.343462\pi\)
\(97\)	−9.54508	+	6.93491i	−0.969157	+	0.704133i	−0.955259	−	0.295770i	\(-0.904424\pi\)
							−0.0138974	+	0.999903i	\(0.504424\pi\)

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	99.2.f.c.91.1	yes	8
3.2	odd	2	inner	99.2.f.c.91.2	yes	8
9.2	odd	6		891.2.n.e.190.1		16
9.4	even	3		891.2.n.e.784.1		16
9.5	odd	6		891.2.n.e.784.2		16
9.7	even	3		891.2.n.e.190.2		16
11.2	odd	10		1089.2.a.w.1.1		4
11.4	even	5	inner	99.2.f.c.37.1	✓	8
11.9	even	5		1089.2.a.v.1.4		4
33.2	even	10		1089.2.a.w.1.4		4
33.20	odd	10		1089.2.a.v.1.1		4
33.26	odd	10	inner	99.2.f.c.37.2	yes	8
99.4	even	15		891.2.n.e.136.2		16
99.59	odd	30		891.2.n.e.136.1		16
99.70	even	15		891.2.n.e.433.1		16
99.92	odd	30		891.2.n.e.433.2		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.f.c.37.1	✓	8	11.4	even	5	inner
99.2.f.c.37.2	yes	8	33.26	odd	10	inner
99.2.f.c.91.1	yes	8	1.1	even	1	trivial
99.2.f.c.91.2	yes	8	3.2	odd	2	inner
891.2.n.e.136.1		16	99.59	odd	30
891.2.n.e.136.2		16	99.4	even	15
891.2.n.e.190.1		16	9.2	odd	6
891.2.n.e.190.2		16	9.7	even	3
891.2.n.e.433.1		16	99.70	even	15
891.2.n.e.433.2		16	99.92	odd	30
891.2.n.e.784.1		16	9.4	even	3
891.2.n.e.784.2		16	9.5	odd	6
1089.2.a.v.1.1		4	33.20	odd	10
1089.2.a.v.1.4		4	11.9	even	5
1089.2.a.w.1.1		4	11.2	odd	10
1089.2.a.w.1.4		4	33.2	even	10