Embedded newform 99.2.e.d.67.3

• Label: 99.2.e.d.67.3
• Level: $99$
• Weight: $2$
• Character: 99.67
• Analytic conductor: $0.791$
• Analytic rank: $0$
• Dimension: $6$
• 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:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\zeta_{18})\)
Defining polynomial:	\( x^{6} - x^{3} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			67.3
Root			\(-0.766044 + 0.642788i\) of defining polynomial
Character	\(\chi\)	\(=\)	99.67
Dual form			99.2.e.d.34.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.939693 + 1.62760i) q^{2} +(-1.70574 + 0.300767i) q^{3} +(-0.766044 + 1.32683i) q^{4} +(-1.43969 + 2.49362i) q^{5} +(-2.09240 - 2.49362i) q^{6} +(0.326352 + 0.565258i) q^{7} +0.879385 q^{8} +(2.81908 - 1.02606i) q^{9} -5.41147 q^{10} +(0.500000 + 0.866025i) q^{11} +(0.907604 - 2.49362i) q^{12} +(3.37939 - 5.85327i) q^{13} +(-0.613341 + 1.06234i) q^{14} +(1.70574 - 4.68647i) q^{15} +(2.35844 + 4.08494i) q^{16} +0.184793 q^{17} +(4.31908 + 3.62414i) q^{18} -5.22668 q^{19} +(-2.20574 - 3.82045i) q^{20} +(-0.726682 - 0.866025i) q^{21} +(-0.939693 + 1.62760i) q^{22} +(1.59240 - 2.75811i) q^{23} +(-1.50000 + 0.264490i) q^{24} +(-1.64543 - 2.84997i) q^{25} +12.7023 q^{26} +(-4.50000 + 2.59808i) q^{27} -1.00000 q^{28} +(2.01114 + 3.48340i) q^{29} +(9.23055 - 1.62760i) q^{30} +(-0.553033 + 0.957882i) q^{31} +(-3.55303 + 6.15403i) q^{32} +(-1.11334 - 1.32683i) q^{33} +(0.173648 + 0.300767i) q^{34} -1.87939 q^{35} +(-0.798133 + 4.52644i) q^{36} +0.106067 q^{37} +(-4.91147 - 8.50692i) q^{38} +(-4.00387 + 11.0005i) q^{39} +(-1.26604 + 2.19285i) q^{40} +(2.80793 - 4.86348i) q^{41} +(0.726682 - 1.99654i) q^{42} +(-1.92989 - 3.34267i) q^{43} -1.53209 q^{44} +(-1.50000 + 8.50692i) q^{45} +5.98545 q^{46} +(-6.00387 - 10.3990i) q^{47} +(-5.25150 - 6.25849i) q^{48} +(3.28699 - 5.69323i) q^{49} +(3.09240 - 5.35619i) q^{50} +(-0.315207 + 0.0555796i) q^{51} +(5.17752 + 8.96773i) q^{52} -10.0719 q^{53} +(-8.45723 - 4.88279i) q^{54} -2.87939 q^{55} +(0.286989 + 0.497079i) q^{56} +(8.91534 - 1.57202i) q^{57} +(-3.77972 + 6.54666i) q^{58} +(-5.27719 + 9.14036i) q^{59} +(4.91147 + 5.85327i) q^{60} +(3.67365 + 6.36295i) q^{61} -2.07873 q^{62} +(1.50000 + 1.25865i) q^{63} -3.92127 q^{64} +(9.73055 + 16.8538i) q^{65} +(1.11334 - 3.05888i) q^{66} +(5.90420 - 10.2264i) q^{67} +(-0.141559 + 0.245188i) q^{68} +(-1.88666 + 5.18355i) q^{69} +(-1.76604 - 3.05888i) q^{70} -2.47565 q^{71} +(2.47906 - 0.902302i) q^{72} +10.4611 q^{73} +(0.0996702 + 0.172634i) q^{74} +(3.66385 + 4.36640i) q^{75} +(4.00387 - 6.93491i) q^{76} +(-0.326352 + 0.565258i) q^{77} +(-21.6668 + 3.82045i) q^{78} +(0.733956 + 1.27125i) q^{79} -13.5817 q^{80} +(6.89440 - 5.78509i) q^{81} +10.5544 q^{82} +(0.520945 + 0.902302i) q^{83} +(1.70574 - 0.300767i) q^{84} +(-0.266044 + 0.460802i) q^{85} +(3.62701 - 6.28217i) q^{86} +(-4.47818 - 5.33688i) q^{87} +(0.439693 + 0.761570i) q^{88} -3.01960 q^{89} +(-15.2554 + 5.55250i) q^{90} +4.41147 q^{91} +(2.43969 + 4.22567i) q^{92} +(0.655230 - 1.80023i) q^{93} +(11.2836 - 19.5437i) q^{94} +(7.52481 - 13.0334i) q^{95} +(4.20961 - 11.5658i) q^{96} +(2.86959 + 4.97027i) q^{97} +12.3550 q^{98} +(2.29813 + 1.92836i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q - 3 * q^5 - 9 * q^6 + 3 * q^7 - 6 * q^8 - 12 * q^10 + 3 * q^11 + 9 * q^12 + 9 * q^13 + 3 * q^14 + 6 * q^16 - 6 * q^17 + 9 * q^18 - 18 * q^19 - 3 * q^20 + 9 * q^21 + 6 * q^23 - 9 * q^24 + 6 * q^25 + 24 * q^26 - 27 * q^27 - 6 * q^28 + 6 * q^29 + 18 * q^30 + 9 * q^31 - 9 * q^32 + 9 * q^36 - 24 * q^37 - 9 * q^38 - 3 * q^40 + 6 * q^41 - 9 * q^42 - 3 * q^43 - 9 * q^45 - 12 * q^47 + 9 * q^48 + 12 * q^49 + 15 * q^50 - 9 * q^51 + 6 * q^52 + 6 * q^53 - 6 * q^55 - 6 * q^56 + 9 * q^57 + 3 * q^58 - 21 * q^59 + 9 * q^60 + 21 * q^61 - 30 * q^62 + 9 * q^63 - 6 * q^64 + 21 * q^65 - 3 * q^67 - 9 * q^68 - 18 * q^69 - 6 * q^70 + 24 * q^71 + 18 * q^72 - 12 * q^73 + 15 * q^74 + 18 * q^75 - 3 * q^77 - 45 * q^78 + 9 * q^79 - 18 * q^80 + 42 * q^82 + 3 * q^85 - 6 * q^86 + 27 * q^87 - 3 * q^88 - 24 * q^89 - 27 * q^90 + 6 * q^91 + 9 * q^92 - 9 * q^93 + 18 * q^94 + 18 * q^95 - 9 * q^96 + 3 * q^97 + 24 * q^98

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.939693	+	1.62760i	0.664463	+	1.15088i	0.979431	+	0.201781i	\(0.0646730\pi\)
							−0.314968	+	0.949102i	\(0.601994\pi\)
\(3\)	−1.70574	+	0.300767i	−0.984808	+	0.173648i
							\(5\)	−1.43969	+	2.49362i	−0.643850	+	1.11518i	0.340716	+	0.940166i	\(0.389331\pi\)
−0.984566	+	0.175015i	\(0.944003\pi\)
\(7\)	0.326352	+	0.565258i	0.123349	+	0.213647i	0.921087	−	0.389358i	\(-0.127303\pi\)
							−0.797737	+	0.603005i	\(0.793970\pi\)
\(11\)	0.500000	+	0.866025i	0.150756	+	0.261116i
							\(13\)	3.37939	−	5.85327i	0.937273	−	1.62340i	0.166743	−	0.986000i	\(-0.446675\pi\)
0.770530	−	0.637404i	\(-0.219992\pi\)
\(17\)	0.184793			0.0448188			0.0224094	−	0.999749i	\(-0.492866\pi\)
							0.0224094	+	0.999749i	\(0.492866\pi\)
\(19\)	−5.22668			−1.19908			−0.599541	−	0.800344i	\(-0.704650\pi\)
							−0.599541	+	0.800344i	\(0.704650\pi\)
\(23\)	1.59240	−	2.75811i	0.332038	−	0.575106i	−0.650874	−	0.759186i	\(-0.725597\pi\)
							0.982911	+	0.184080i	\(0.0589306\pi\)
\(29\)	2.01114	+	3.48340i	0.373460	+	0.646852i	0.990095	−	0.140397i	\(-0.0448379\pi\)
							−0.616635	+	0.787249i	\(0.711505\pi\)
\(31\)	−0.553033	+	0.957882i	−0.0993277	+	0.172041i	−0.911407	−	0.411507i	\(-0.865003\pi\)
							0.812079	+	0.583548i	\(0.198336\pi\)
\(37\)	0.106067			0.0174373			0.00871864	−	0.999962i	\(-0.497225\pi\)
							0.00871864	+	0.999962i	\(0.497225\pi\)
\(41\)	2.80793	−	4.86348i	0.438526	−	0.759549i	−0.559050	−	0.829134i	\(-0.688834\pi\)
							0.997576	+	0.0695851i	\(0.0221675\pi\)
\(43\)	−1.92989	−	3.34267i	−0.294306	−	0.509753i	0.680517	−	0.732732i	\(-0.261755\pi\)
							−0.974823	+	0.222979i	\(0.928422\pi\)
\(47\)	−6.00387	−	10.3990i	−0.875755	−	1.51685i	−0.855957	−	0.517047i	\(-0.827031\pi\)
							−0.0197977	−	0.999804i	\(-0.506302\pi\)
\(53\)	−10.0719			−1.38348			−0.691742	−	0.722145i	\(-0.743157\pi\)
							−0.691742	+	0.722145i	\(0.743157\pi\)
\(59\)	−5.27719	+	9.14036i	−0.687031	+	1.18997i	0.285762	+	0.958301i	\(0.407753\pi\)
							−0.972794	+	0.231673i	\(0.925580\pi\)
\(61\)	3.67365	+	6.36295i	0.470362	+	0.814692i	0.999426	−	0.0338908i	\(-0.0107899\pi\)
							−0.529063	+	0.848582i	\(0.677457\pi\)
\(67\)	5.90420	−	10.2264i	0.721313	−	1.24935i	−0.239161	−	0.970980i	\(-0.576872\pi\)
							0.960474	−	0.278371i	\(-0.0897943\pi\)
\(71\)	−2.47565			−0.293806			−0.146903	−	0.989151i	\(-0.546930\pi\)
							−0.146903	+	0.989151i	\(0.546930\pi\)
\(73\)	10.4611			1.22438			0.612190	−	0.790711i	\(-0.290289\pi\)
							0.612190	+	0.790711i	\(0.290289\pi\)
\(79\)	0.733956	+	1.27125i	0.0825765	+	0.143027i	0.904356	−	0.426779i	\(-0.140352\pi\)
							−0.821779	+	0.569806i	\(0.807018\pi\)
\(83\)	0.520945	+	0.902302i	0.0571811	+	0.0990406i	0.893199	−	0.449662i	\(-0.148455\pi\)
							−0.836018	+	0.548702i	\(0.815122\pi\)
\(89\)	−3.01960			−0.320077			−0.160038	−	0.987111i	\(-0.551162\pi\)
							−0.160038	+	0.987111i	\(0.551162\pi\)
\(97\)	2.86959	+	4.97027i	0.291362	+	0.504654i	0.974132	−	0.225979i	\(-0.0725582\pi\)
							−0.682770	+	0.730633i	\(0.739225\pi\)

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.d.67.3	yes	6
3.2	odd	2		297.2.e.d.199.1		6
9.2	odd	6		297.2.e.d.100.1		6
9.4	even	3		891.2.a.l.1.1		3
9.5	odd	6		891.2.a.k.1.3		3
9.7	even	3	inner	99.2.e.d.34.3	✓	6
11.10	odd	2		1089.2.e.h.364.1		6
99.32	even	6		9801.2.a.bd.1.1		3
99.43	odd	6		1089.2.e.h.727.1		6
99.76	odd	6		9801.2.a.be.1.3		3

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.e.d.34.3	✓	6	9.7	even	3	inner
99.2.e.d.67.3	yes	6	1.1	even	1	trivial
297.2.e.d.100.1		6	9.2	odd	6
297.2.e.d.199.1		6	3.2	odd	2
891.2.a.k.1.3		3	9.5	odd	6
891.2.a.l.1.1		3	9.4	even	3
1089.2.e.h.364.1		6	11.10	odd	2
1089.2.e.h.727.1		6	99.43	odd	6
9801.2.a.bd.1.1		3	99.32	even	6
9801.2.a.be.1.3		3	99.76	odd	6