Embedded newform 99.2.f.c.82.2

• Label: 99.2.f.c.82.2
• Level: $99$
• Weight: $2$
• Character: 99.82
• 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			82.2
Root			\(0.476925 + 1.46782i\) of defining polynomial
Character	\(\chi\)	\(=\)	99.82
Dual form			99.2.f.c.64.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.476925 + 1.46782i) q^{2} +(-0.309017 + 0.224514i) q^{4} +(-0.476925 + 1.46782i) q^{5} +(-0.190983 + 0.138757i) q^{7} +(2.02029 + 1.46782i) q^{8} -2.38197 q^{10} +(-2.31504 - 2.37499i) q^{11} +(-1.30902 - 4.02874i) q^{13} +(-0.294756 - 0.214153i) q^{14} +(-1.42705 + 4.39201i) q^{16} +(1.83812 - 5.65714i) q^{17} +(3.73607 + 2.71441i) q^{19} +(-0.182169 - 0.560659i) q^{20} +(2.38197 - 4.53077i) q^{22} -7.49164 q^{23} +(2.11803 + 1.53884i) q^{25} +(5.28918 - 3.84281i) q^{26} +(0.0278640 - 0.0857567i) q^{28} +(1.54336 - 1.12132i) q^{29} +(-0.736068 - 2.26538i) q^{31} -2.13287 q^{32} +9.18034 q^{34} +(-0.112587 - 0.346506i) q^{35} +(-5.04508 + 3.66547i) q^{37} +(-2.20246 + 6.77846i) q^{38} +(-3.11803 + 2.26538i) q^{40} +(4.51750 + 3.28216i) q^{41} -10.7082 q^{43} +(1.24861 + 0.214153i) q^{44} +(-3.57295 - 10.9964i) q^{46} +(0.771681 + 0.560659i) q^{47} +(-2.14590 + 6.60440i) q^{49} +(-1.24861 + 3.84281i) q^{50} +(1.30902 + 0.951057i) q^{52} +(2.79197 + 8.59279i) q^{53} +(4.59017 - 2.26538i) q^{55} -0.589512 q^{56} +(2.38197 + 1.73060i) q^{58} +(-6.83254 + 4.96413i) q^{59} +(1.33688 - 4.11450i) q^{61} +(2.97414 - 2.16084i) q^{62} +(1.83688 + 5.65334i) q^{64} +6.53779 q^{65} +3.85410 q^{67} +(0.702099 + 2.16084i) q^{68} +(0.454915 - 0.330515i) q^{70} +(2.38463 - 7.33912i) q^{71} +(5.23607 - 3.80423i) q^{73} +(-7.78639 - 5.65714i) q^{74} -1.76393 q^{76} +(0.771681 + 0.132354i) q^{77} +(-0.163119 - 0.502029i) q^{79} +(-5.76611 - 4.18932i) q^{80} +(-2.66312 + 8.19624i) q^{82} +(-1.36119 + 4.18932i) q^{83} +(7.42705 + 5.39607i) q^{85} +(-5.10701 - 15.7178i) q^{86} +(-1.19098 - 8.19624i) q^{88} +16.7518 q^{89} +(0.809017 + 0.587785i) q^{91} +(2.31504 - 1.68198i) q^{92} +(-0.454915 + 1.40008i) q^{94} +(-5.76611 + 4.18932i) q^{95} +(-3.95492 - 12.1720i) q^{97} -10.7175 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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 \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\)	\(46\)	\(56\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.476925	+	1.46782i	0.337237	+	1.03791i	0.965610	+	0.259996i	\(0.0837213\pi\)
							−0.628373	+	0.777912i	\(0.716279\pi\)
\(3\)		0			0
							\(5\)	−0.476925	+	1.46782i	−0.213287	+	0.656431i	0.785983	+	0.618248i	\(0.212157\pi\)
−0.999271	+	0.0381834i	\(0.987843\pi\)
\(7\)	−0.190983	+	0.138757i	−0.0721848	+	0.0524453i	−0.623292	−	0.781989i	\(-0.714205\pi\)
							0.551108	+	0.834434i	\(0.314205\pi\)
\(11\)	−2.31504	−	2.37499i	−0.698012	−	0.716086i
							\(13\)	−1.30902	−	4.02874i	−0.363056	−	1.11737i	−0.951189	−	0.308608i	\(-0.900137\pi\)
0.588133	−	0.808764i	\(-0.299863\pi\)
\(17\)	1.83812	−	5.65714i	0.445809	−	1.37206i	−0.435785	−	0.900051i	\(-0.643529\pi\)
							0.881594	−	0.472008i	\(-0.156471\pi\)
\(19\)	3.73607	+	2.71441i	0.857113	+	0.622729i	0.927098	−	0.374819i	\(-0.122295\pi\)
							−0.0699852	+	0.997548i	\(0.522295\pi\)
\(23\)	−7.49164			−1.56211			−0.781057	−	0.624460i	\(-0.785319\pi\)
							−0.781057	+	0.624460i	\(0.785319\pi\)
\(29\)	1.54336	−	1.12132i	0.286595	−	0.208224i	−0.435194	−	0.900337i	\(-0.643320\pi\)
							0.721789	+	0.692113i	\(0.243320\pi\)
\(31\)	−0.736068	−	2.26538i	−0.132202	−	0.406875i	0.862943	−	0.505302i	\(-0.168619\pi\)
							−0.995144	+	0.0984270i	\(0.968619\pi\)
\(37\)	−5.04508	+	3.66547i	−0.829407	+	0.602599i	−0.919391	−	0.393344i	\(-0.871318\pi\)
							0.0899846	+	0.995943i	\(0.471318\pi\)
\(41\)	4.51750	+	3.28216i	0.705515	+	0.512587i	0.881724	−	0.471766i	\(-0.156383\pi\)
							−0.176209	+	0.984353i	\(0.556383\pi\)
\(43\)	−10.7082			−1.63299			−0.816493	−	0.577355i	\(-0.804085\pi\)
							−0.816493	+	0.577355i	\(0.804085\pi\)
\(47\)	0.771681	+	0.560659i	0.112561	+	0.0817805i	0.642642	−	0.766167i	\(-0.277838\pi\)
							−0.530081	+	0.847947i	\(0.677838\pi\)
\(53\)	2.79197	+	8.59279i	0.383506	+	1.18031i	0.937558	+	0.347829i	\(0.113081\pi\)
							−0.554052	+	0.832482i	\(0.686919\pi\)
\(59\)	−6.83254	+	4.96413i	−0.889521	+	0.646275i	−0.935753	−	0.352656i	\(-0.885279\pi\)
							0.0462319	+	0.998931i	\(0.485279\pi\)
\(61\)	1.33688	−	4.11450i	0.171170	−	0.526807i	−0.828268	−	0.560332i	\(-0.810673\pi\)
							0.999438	+	0.0335251i	\(0.0106734\pi\)
\(67\)	3.85410			0.470853			0.235427	−	0.971892i	\(-0.424351\pi\)
							0.235427	+	0.971892i	\(0.424351\pi\)
\(71\)	2.38463	−	7.33912i	0.283003	−	0.870994i	−0.703987	−	0.710213i	\(-0.748599\pi\)
							0.986990	−	0.160781i	\(-0.0514012\pi\)
\(73\)	5.23607	−	3.80423i	0.612835	−	0.445251i	−0.237576	−	0.971369i	\(-0.576353\pi\)
							0.850412	+	0.526118i	\(0.176353\pi\)
\(79\)	−0.163119	−	0.502029i	−0.0183523	−	0.0564826i	0.941461	−	0.337122i	\(-0.109453\pi\)
							−0.959813	+	0.280639i	\(0.909453\pi\)
\(83\)	−1.36119	+	4.18932i	−0.149410	+	0.459838i	−0.997552	−	0.0699322i	\(-0.977722\pi\)
							0.848141	+	0.529770i	\(0.177722\pi\)
\(89\)	16.7518			1.77569			0.887844	−	0.460145i	\(-0.152202\pi\)
							0.887844	+	0.460145i	\(0.152202\pi\)
\(97\)	−3.95492	−	12.1720i	−0.401561	−	1.23588i	−0.923733	−	0.383037i	\(-0.874878\pi\)
							0.522172	−	0.852840i	\(-0.325122\pi\)

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.82.2	yes	8
3.2	odd	2	inner	99.2.f.c.82.1	yes	8
9.2	odd	6		891.2.n.e.676.1		16
9.4	even	3		891.2.n.e.379.1		16
9.5	odd	6		891.2.n.e.379.2		16
9.7	even	3		891.2.n.e.676.2		16
11.3	even	5		1089.2.a.v.1.3		4
11.8	odd	10		1089.2.a.w.1.2		4
11.9	even	5	inner	99.2.f.c.64.2	yes	8
33.8	even	10		1089.2.a.w.1.3		4
33.14	odd	10		1089.2.a.v.1.2		4
33.20	odd	10	inner	99.2.f.c.64.1	✓	8
99.20	odd	30		891.2.n.e.757.2		16
99.31	even	15		891.2.n.e.460.2		16
99.86	odd	30		891.2.n.e.460.1		16
99.97	even	15		891.2.n.e.757.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.f.c.64.1	✓	8	33.20	odd	10	inner
99.2.f.c.64.2	yes	8	11.9	even	5	inner
99.2.f.c.82.1	yes	8	3.2	odd	2	inner
99.2.f.c.82.2	yes	8	1.1	even	1	trivial
891.2.n.e.379.1		16	9.4	even	3
891.2.n.e.379.2		16	9.5	odd	6
891.2.n.e.460.1		16	99.86	odd	30
891.2.n.e.460.2		16	99.31	even	15
891.2.n.e.676.1		16	9.2	odd	6
891.2.n.e.676.2		16	9.7	even	3
891.2.n.e.757.1		16	99.97	even	15
891.2.n.e.757.2		16	99.20	odd	30
1089.2.a.v.1.2		4	33.14	odd	10
1089.2.a.v.1.3		4	11.3	even	5
1089.2.a.w.1.2		4	11.8	odd	10
1089.2.a.w.1.3		4	33.8	even	10