Embedded newform 99.2.e.e.67.3

• Label: 99.2.e.e.67.3
• Level: $99$
• Weight: $2$
• Character: 99.67
• Analytic conductor: $0.791$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.790518980011\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	8.0.508277025.1
Defining polynomial:	\( x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			67.3
Root			\(0.947217 - 0.807294i\) of defining polynomial
Character	\(\chi\)	\(=\)	99.67
Dual form			99.2.e.e.34.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.447217 + 0.774602i) q^{2} +(1.22553 - 1.22396i) q^{3} +(0.599994 - 1.03922i) q^{4} +(-1.87447 + 3.24667i) q^{5} +(1.49616 + 0.401921i) q^{6} +(-0.725528 - 1.25665i) q^{7} +2.86218 q^{8} +(0.00384004 - 3.00000i) q^{9} -3.35317 q^{10} +(-0.500000 - 0.866025i) q^{11} +(-0.536655 - 2.00796i) q^{12} +(-2.87831 + 4.98537i) q^{13} +(0.648937 - 1.12399i) q^{14} +(1.67659 + 6.27316i) q^{15} +(0.0800260 + 0.138609i) q^{16} -4.79655 q^{17} +(2.32552 - 1.33868i) q^{18} +0.702126 q^{19} +(2.24934 + 3.89597i) q^{20} +(-2.42725 - 0.652045i) q^{21} +(0.447217 - 0.774602i) q^{22} +(0.825523 - 1.42985i) q^{23} +(3.50768 - 3.50319i) q^{24} +(-4.52724 - 7.84141i) q^{25} -5.14891 q^{26} +(-3.66717 - 3.68128i) q^{27} -1.74125 q^{28} +(2.15278 + 3.72872i) q^{29} +(-4.10941 + 4.10415i) q^{30} +(1.65278 - 2.86269i) q^{31} +(2.79060 - 4.83346i) q^{32} +(-1.67275 - 0.449358i) q^{33} +(-2.14510 - 3.71542i) q^{34} +5.43991 q^{35} +(-3.11535 - 1.80397i) q^{36} +9.73779 q^{37} +(0.314002 + 0.543868i) q^{38} +(2.57445 + 9.63265i) q^{39} +(-5.36505 + 9.29255i) q^{40} +(-2.12380 + 3.67853i) q^{41} +(-0.580431 - 2.17176i) q^{42} +(2.05278 + 3.55552i) q^{43} -1.19999 q^{44} +(9.73280 + 5.63586i) q^{45} +1.47675 q^{46} +(-0.898274 - 1.55586i) q^{47} +(0.267726 + 0.0719207i) q^{48} +(2.44722 - 4.23870i) q^{49} +(4.04932 - 7.01363i) q^{50} +(-5.87831 + 5.87079i) q^{51} +(3.45393 + 5.98239i) q^{52} -1.15318 q^{53} +(1.21151 - 4.48693i) q^{54} +3.74893 q^{55} +(-2.07659 - 3.59676i) q^{56} +(0.860475 - 0.859374i) q^{57} +(-1.92552 + 3.33509i) q^{58} +(2.32552 - 4.02792i) q^{59} +(7.52513 + 2.02152i) q^{60} +(-1.27447 - 2.20745i) q^{61} +2.95660 q^{62} +(-3.77274 + 2.17176i) q^{63} +5.31212 q^{64} +(-10.7906 - 18.6898i) q^{65} +(-0.400006 - 1.49667i) q^{66} +(-4.47062 + 7.74334i) q^{67} +(-2.87790 + 4.98467i) q^{68} +(-0.738375 - 2.76273i) q^{69} +(2.43282 + 4.21377i) q^{70} +5.14204 q^{71} +(0.0109909 - 8.58653i) q^{72} +10.5378 q^{73} +(4.35490 + 7.54291i) q^{74} +(-15.1458 - 4.06871i) q^{75} +(0.421271 - 0.729663i) q^{76} +(-0.725528 + 1.25665i) q^{77} +(-6.31013 + 6.30206i) q^{78} +(0.543371 + 0.941146i) q^{79} -0.600024 q^{80} +(-8.99997 - 0.0230402i) q^{81} -3.79920 q^{82} +(1.90171 + 3.29386i) q^{83} +(-2.13395 + 2.13122i) q^{84} +(8.99096 - 15.5728i) q^{85} +(-1.83608 + 3.18018i) q^{86} +(7.20210 + 1.93474i) q^{87} +(-1.43109 - 2.47872i) q^{88} -4.01536 q^{89} +(-0.0128763 + 10.0595i) q^{90} +8.35317 q^{91} +(-0.990617 - 1.71580i) q^{92} +(-1.47830 - 5.53125i) q^{93} +(0.803446 - 1.39161i) q^{94} +(-1.31611 + 2.27957i) q^{95} +(-2.49601 - 9.33913i) q^{96} +(-1.64550 - 2.85009i) q^{97} +4.37775 q^{98} +(-2.59999 + 1.49667i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	8 * q - q^2 + 5 * q^3 - 11 * q^4 - 4 * q^5 + 17 * q^6 - q^7 - 5 * q^9 + 2 * q^10 - 4 * q^11 - 2 * q^12 - 7 * q^13 - q^14 - q^15 - 17 * q^16 - 10 * q^17 - 2 * q^18 + 18 * q^19 + 10 * q^20 - 13 * q^21 - q^22 - 14 * q^23 + 18 * q^24 - 14 * q^25 + 44 * q^26 + 5 * q^27 - 2 * q^28 + 6 * q^29 - 37 * q^30 + 2 * q^31 + 34 * q^32 - 4 * q^33 - 16 * q^34 - 16 * q^35 + 11 * q^36 + 6 * q^37 - 3 * q^38 - 22 * q^39 - 12 * q^40 + 2 * q^41 - q^42 + 21 * q^43 + 22 * q^44 + 49 * q^45 + 4 * q^46 + 7 * q^47 - 59 * q^48 + 15 * q^49 - 23 * q^50 - 31 * q^51 + 10 * q^52 - 12 * q^53 - 37 * q^54 + 8 * q^55 - 18 * q^56 + 33 * q^57 + 21 * q^58 - 2 * q^59 + 73 * q^60 - 15 * q^61 - 40 * q^62 - 5 * q^63 + 32 * q^64 - 19 * q^65 - 19 * q^66 - 14 * q^67 + 7 * q^68 - 2 * q^69 + 38 * q^70 - 6 * q^71 + 75 * q^72 + 44 * q^73 + 36 * q^74 + 10 * q^75 - 42 * q^76 - q^77 + 29 * q^78 - 11 * q^79 - 68 * q^80 + 7 * q^81 - 34 * q^82 - 18 * q^83 + 34 * q^84 - 13 * q^85 + 24 * q^86 - 9 * q^87 - 12 * q^89 - 80 * q^90 + 38 * q^91 - 67 * q^92 + 20 * q^93 + 19 * q^94 + 30 * q^95 - 50 * q^96 - 26 * q^97 + 30 * q^98 - 5 * q^99

Character values

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

\(n\)	\(46\)	\(56\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.447217	+	0.774602i	0.316230	+	0.547727i	0.979698	−	0.200478i	\(-0.0642495\pi\)
							−0.663468	+	0.748205i	\(0.730916\pi\)
\(3\)	1.22553	−	1.22396i	0.707559	−	0.706654i
							\(5\)	−1.87447	+	3.24667i	−0.838287	+	1.45195i	0.0530397	+	0.998592i	\(0.483109\pi\)
−0.891326	+	0.453362i	\(0.850224\pi\)
\(7\)	−0.725528	−	1.25665i	−0.274224	−	0.474970i	0.695715	−	0.718318i	\(-0.255088\pi\)
							−0.969939	+	0.243348i	\(0.921754\pi\)
\(11\)	−0.500000	−	0.866025i	−0.150756	−	0.261116i
							\(13\)	−2.87831	+	4.98537i	−0.798298	+	1.38269i	0.122425	+	0.992478i	\(0.460933\pi\)
−0.920724	+	0.390216i	\(0.872400\pi\)
\(17\)	−4.79655			−1.16333			−0.581667	−	0.813427i	\(-0.697599\pi\)
							−0.581667	+	0.813427i	\(0.697599\pi\)
\(19\)	0.702126			0.161079			0.0805393	−	0.996751i	\(-0.474336\pi\)
							0.0805393	+	0.996751i	\(0.474336\pi\)
\(23\)	0.825523	−	1.42985i	0.172133	−	0.298144i	−0.767032	−	0.641609i	\(-0.778267\pi\)
							0.939165	+	0.343465i	\(0.111601\pi\)
\(29\)	2.15278	+	3.72872i	0.399761	+	0.692406i	0.993696	−	0.112107i	\(-0.0357599\pi\)
							−0.593935	+	0.804513i	\(0.702427\pi\)
\(31\)	1.65278	−	2.86269i	0.296848	−	0.514155i	−0.678565	−	0.734540i	\(-0.737398\pi\)
							0.975413	+	0.220385i	\(0.0707313\pi\)
\(37\)	9.73779			1.60088			0.800441	−	0.599411i	\(-0.204599\pi\)
							0.800441	+	0.599411i	\(0.204599\pi\)
\(41\)	−2.12380	+	3.67853i	−0.331682	+	0.574490i	−0.982842	−	0.184450i	\(-0.940949\pi\)
							0.651160	+	0.758941i	\(0.274283\pi\)
\(43\)	2.05278	+	3.55552i	0.313046	+	0.542212i	0.979020	−	0.203763i	\(-0.0653171\pi\)
							−0.665974	+	0.745975i	\(0.731984\pi\)
\(47\)	−0.898274	−	1.55586i	−0.131027	−	0.226945i	0.793046	−	0.609162i	\(-0.208494\pi\)
							−0.924073	+	0.382217i	\(0.875161\pi\)
\(53\)	−1.15318			−0.158402			−0.0792009	−	0.996859i	\(-0.525237\pi\)
							−0.0792009	+	0.996859i	\(0.525237\pi\)
\(59\)	2.32552	−	4.02792i	0.302757	−	0.524391i	−0.674002	−	0.738729i	\(-0.735426\pi\)
							0.976759	+	0.214338i	\(0.0687595\pi\)
\(61\)	−1.27447	−	2.20745i	−0.163179	−	0.282635i	0.772828	−	0.634616i	\(-0.218842\pi\)
							−0.936007	+	0.351981i	\(0.885508\pi\)
\(67\)	−4.47062	+	7.74334i	−0.546173	+	0.946000i	0.452359	+	0.891836i	\(0.350583\pi\)
							−0.998532	+	0.0541636i	\(0.982751\pi\)
\(71\)	5.14204			0.610248			0.305124	−	0.952313i	\(-0.401302\pi\)
							0.305124	+	0.952313i	\(0.401302\pi\)
\(73\)	10.5378			1.23336			0.616678	−	0.787215i	\(-0.288478\pi\)
							0.616678	+	0.787215i	\(0.288478\pi\)
\(79\)	0.543371	+	0.941146i	0.0611340	+	0.105887i	0.894973	−	0.446121i	\(-0.147195\pi\)
							−0.833839	+	0.552008i	\(0.813862\pi\)
\(83\)	1.90171	+	3.29386i	0.208740	+	0.361548i	0.951318	−	0.308212i	\(-0.0997305\pi\)
							−0.742578	+	0.669759i	\(0.766397\pi\)
\(89\)	−4.01536			−0.425627			−0.212814	−	0.977093i	\(-0.568263\pi\)
							−0.212814	+	0.977093i	\(0.568263\pi\)
\(97\)	−1.64550	−	2.85009i	−0.167075	−	0.289383i	0.770315	−	0.637664i	\(-0.220099\pi\)
							−0.937390	+	0.348280i	\(0.886766\pi\)

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.e.e.67.3	yes	8
3.2	odd	2		297.2.e.e.199.2		8
9.2	odd	6		297.2.e.e.100.2		8
9.4	even	3		891.2.a.q.1.2		4
9.5	odd	6		891.2.a.p.1.3		4
9.7	even	3	inner	99.2.e.e.34.3	✓	8
11.10	odd	2		1089.2.e.i.364.2		8
99.32	even	6		9801.2.a.bl.1.2		4
99.43	odd	6		1089.2.e.i.727.2		8
99.76	odd	6		9801.2.a.bi.1.3		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.e.e.34.3	✓	8	9.7	even	3	inner
99.2.e.e.67.3	yes	8	1.1	even	1	trivial
297.2.e.e.100.2		8	9.2	odd	6
297.2.e.e.199.2		8	3.2	odd	2
891.2.a.p.1.3		4	9.5	odd	6
891.2.a.q.1.2		4	9.4	even	3
1089.2.e.i.364.2		8	11.10	odd	2
1089.2.e.i.727.2		8	99.43	odd	6
9801.2.a.bi.1.3		4	99.76	odd	6
9801.2.a.bl.1.2		4	99.32	even	6