# Properties

 Label 99.4.e.a Level $99$ Weight $4$ Character orbit 99.e Analytic conductor $5.841$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.84118909057$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 4 \zeta_{6} + 4) q^{2} + (3 \zeta_{6} - 6) q^{3} - 8 \zeta_{6} q^{4} + 19 \zeta_{6} q^{5} + (24 \zeta_{6} - 12) q^{6} + ( - 26 \zeta_{6} + 26) q^{7} + ( - 27 \zeta_{6} + 27) q^{9} +O(q^{10})$$ q + (-4*z + 4) * q^2 + (3*z - 6) * q^3 - 8*z * q^4 + 19*z * q^5 + (24*z - 12) * q^6 + (-26*z + 26) * q^7 + (-27*z + 27) * q^9 $$q + ( - 4 \zeta_{6} + 4) q^{2} + (3 \zeta_{6} - 6) q^{3} - 8 \zeta_{6} q^{4} + 19 \zeta_{6} q^{5} + (24 \zeta_{6} - 12) q^{6} + ( - 26 \zeta_{6} + 26) q^{7} + ( - 27 \zeta_{6} + 27) q^{9} + 76 q^{10} + (11 \zeta_{6} - 11) q^{11} + (24 \zeta_{6} + 24) q^{12} + 56 \zeta_{6} q^{13} - 104 \zeta_{6} q^{14} + ( - 57 \zeta_{6} - 57) q^{15} + ( - 64 \zeta_{6} + 64) q^{16} + 104 q^{17} - 108 \zeta_{6} q^{18} - 96 q^{19} + ( - 152 \zeta_{6} + 152) q^{20} + (156 \zeta_{6} - 78) q^{21} + 44 \zeta_{6} q^{22} - 40 \zeta_{6} q^{23} + (236 \zeta_{6} - 236) q^{25} + 224 q^{26} + (162 \zeta_{6} - 81) q^{27} - 208 q^{28} + (18 \zeta_{6} - 18) q^{29} + (228 \zeta_{6} - 456) q^{30} - 49 \zeta_{6} q^{31} - 256 \zeta_{6} q^{32} + ( - 66 \zeta_{6} + 33) q^{33} + ( - 416 \zeta_{6} + 416) q^{34} + 494 q^{35} - 216 q^{36} + 75 q^{37} + (384 \zeta_{6} - 384) q^{38} + ( - 168 \zeta_{6} - 168) q^{39} - 296 \zeta_{6} q^{41} + (312 \zeta_{6} + 312) q^{42} + (372 \zeta_{6} - 372) q^{43} + 88 q^{44} + 513 q^{45} - 160 q^{46} + ( - 149 \zeta_{6} + 149) q^{47} + (384 \zeta_{6} - 192) q^{48} - 333 \zeta_{6} q^{49} + 944 \zeta_{6} q^{50} + (312 \zeta_{6} - 624) q^{51} + ( - 448 \zeta_{6} + 448) q^{52} - 417 q^{53} + (324 \zeta_{6} + 324) q^{54} - 209 q^{55} + ( - 288 \zeta_{6} + 576) q^{57} + 72 \zeta_{6} q^{58} + 17 \zeta_{6} q^{59} + (912 \zeta_{6} - 456) q^{60} + (90 \zeta_{6} - 90) q^{61} - 196 q^{62} - 702 \zeta_{6} q^{63} - 512 q^{64} + (1064 \zeta_{6} - 1064) q^{65} + ( - 132 \zeta_{6} - 132) q^{66} - 1073 \zeta_{6} q^{67} - 832 \zeta_{6} q^{68} + (120 \zeta_{6} + 120) q^{69} + ( - 1976 \zeta_{6} + 1976) q^{70} - 285 q^{71} - 962 q^{73} + ( - 300 \zeta_{6} + 300) q^{74} + ( - 1416 \zeta_{6} + 708) q^{75} + 768 \zeta_{6} q^{76} + 286 \zeta_{6} q^{77} + (672 \zeta_{6} - 1344) q^{78} + (596 \zeta_{6} - 596) q^{79} + 1216 q^{80} - 729 \zeta_{6} q^{81} - 1184 q^{82} + ( - 498 \zeta_{6} + 498) q^{83} + ( - 624 \zeta_{6} + 1248) q^{84} + 1976 \zeta_{6} q^{85} + 1488 \zeta_{6} q^{86} + ( - 108 \zeta_{6} + 54) q^{87} + 1230 q^{89} + ( - 2052 \zeta_{6} + 2052) q^{90} + 1456 q^{91} + (320 \zeta_{6} - 320) q^{92} + (147 \zeta_{6} + 147) q^{93} - 596 \zeta_{6} q^{94} - 1824 \zeta_{6} q^{95} + (768 \zeta_{6} + 768) q^{96} + ( - 331 \zeta_{6} + 331) q^{97} - 1332 q^{98} + 297 \zeta_{6} q^{99} +O(q^{100})$$ q + (-4*z + 4) * q^2 + (3*z - 6) * q^3 - 8*z * q^4 + 19*z * q^5 + (24*z - 12) * q^6 + (-26*z + 26) * q^7 + (-27*z + 27) * q^9 + 76 * q^10 + (11*z - 11) * q^11 + (24*z + 24) * q^12 + 56*z * q^13 - 104*z * q^14 + (-57*z - 57) * q^15 + (-64*z + 64) * q^16 + 104 * q^17 - 108*z * q^18 - 96 * q^19 + (-152*z + 152) * q^20 + (156*z - 78) * q^21 + 44*z * q^22 - 40*z * q^23 + (236*z - 236) * q^25 + 224 * q^26 + (162*z - 81) * q^27 - 208 * q^28 + (18*z - 18) * q^29 + (228*z - 456) * q^30 - 49*z * q^31 - 256*z * q^32 + (-66*z + 33) * q^33 + (-416*z + 416) * q^34 + 494 * q^35 - 216 * q^36 + 75 * q^37 + (384*z - 384) * q^38 + (-168*z - 168) * q^39 - 296*z * q^41 + (312*z + 312) * q^42 + (372*z - 372) * q^43 + 88 * q^44 + 513 * q^45 - 160 * q^46 + (-149*z + 149) * q^47 + (384*z - 192) * q^48 - 333*z * q^49 + 944*z * q^50 + (312*z - 624) * q^51 + (-448*z + 448) * q^52 - 417 * q^53 + (324*z + 324) * q^54 - 209 * q^55 + (-288*z + 576) * q^57 + 72*z * q^58 + 17*z * q^59 + (912*z - 456) * q^60 + (90*z - 90) * q^61 - 196 * q^62 - 702*z * q^63 - 512 * q^64 + (1064*z - 1064) * q^65 + (-132*z - 132) * q^66 - 1073*z * q^67 - 832*z * q^68 + (120*z + 120) * q^69 + (-1976*z + 1976) * q^70 - 285 * q^71 - 962 * q^73 + (-300*z + 300) * q^74 + (-1416*z + 708) * q^75 + 768*z * q^76 + 286*z * q^77 + (672*z - 1344) * q^78 + (596*z - 596) * q^79 + 1216 * q^80 - 729*z * q^81 - 1184 * q^82 + (-498*z + 498) * q^83 + (-624*z + 1248) * q^84 + 1976*z * q^85 + 1488*z * q^86 + (-108*z + 54) * q^87 + 1230 * q^89 + (-2052*z + 2052) * q^90 + 1456 * q^91 + (320*z - 320) * q^92 + (147*z + 147) * q^93 - 596*z * q^94 - 1824*z * q^95 + (768*z + 768) * q^96 + (-331*z + 331) * q^97 - 1332 * q^98 + 297*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - 9 q^{3} - 8 q^{4} + 19 q^{5} + 26 q^{7} + 27 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 - 9 * q^3 - 8 * q^4 + 19 * q^5 + 26 * q^7 + 27 * q^9 $$2 q + 4 q^{2} - 9 q^{3} - 8 q^{4} + 19 q^{5} + 26 q^{7} + 27 q^{9} + 152 q^{10} - 11 q^{11} + 72 q^{12} + 56 q^{13} - 104 q^{14} - 171 q^{15} + 64 q^{16} + 208 q^{17} - 108 q^{18} - 192 q^{19} + 152 q^{20} + 44 q^{22} - 40 q^{23} - 236 q^{25} + 448 q^{26} - 416 q^{28} - 18 q^{29} - 684 q^{30} - 49 q^{31} - 256 q^{32} + 416 q^{34} + 988 q^{35} - 432 q^{36} + 150 q^{37} - 384 q^{38} - 504 q^{39} - 296 q^{41} + 936 q^{42} - 372 q^{43} + 176 q^{44} + 1026 q^{45} - 320 q^{46} + 149 q^{47} - 333 q^{49} + 944 q^{50} - 936 q^{51} + 448 q^{52} - 834 q^{53} + 972 q^{54} - 418 q^{55} + 864 q^{57} + 72 q^{58} + 17 q^{59} - 90 q^{61} - 392 q^{62} - 702 q^{63} - 1024 q^{64} - 1064 q^{65} - 396 q^{66} - 1073 q^{67} - 832 q^{68} + 360 q^{69} + 1976 q^{70} - 570 q^{71} - 1924 q^{73} + 300 q^{74} + 768 q^{76} + 286 q^{77} - 2016 q^{78} - 596 q^{79} + 2432 q^{80} - 729 q^{81} - 2368 q^{82} + 498 q^{83} + 1872 q^{84} + 1976 q^{85} + 1488 q^{86} + 2460 q^{89} + 2052 q^{90} + 2912 q^{91} - 320 q^{92} + 441 q^{93} - 596 q^{94} - 1824 q^{95} + 2304 q^{96} + 331 q^{97} - 2664 q^{98} + 297 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 - 9 * q^3 - 8 * q^4 + 19 * q^5 + 26 * q^7 + 27 * q^9 + 152 * q^10 - 11 * q^11 + 72 * q^12 + 56 * q^13 - 104 * q^14 - 171 * q^15 + 64 * q^16 + 208 * q^17 - 108 * q^18 - 192 * q^19 + 152 * q^20 + 44 * q^22 - 40 * q^23 - 236 * q^25 + 448 * q^26 - 416 * q^28 - 18 * q^29 - 684 * q^30 - 49 * q^31 - 256 * q^32 + 416 * q^34 + 988 * q^35 - 432 * q^36 + 150 * q^37 - 384 * q^38 - 504 * q^39 - 296 * q^41 + 936 * q^42 - 372 * q^43 + 176 * q^44 + 1026 * q^45 - 320 * q^46 + 149 * q^47 - 333 * q^49 + 944 * q^50 - 936 * q^51 + 448 * q^52 - 834 * q^53 + 972 * q^54 - 418 * q^55 + 864 * q^57 + 72 * q^58 + 17 * q^59 - 90 * q^61 - 392 * q^62 - 702 * q^63 - 1024 * q^64 - 1064 * q^65 - 396 * q^66 - 1073 * q^67 - 832 * q^68 + 360 * q^69 + 1976 * q^70 - 570 * q^71 - 1924 * q^73 + 300 * q^74 + 768 * q^76 + 286 * q^77 - 2016 * q^78 - 596 * q^79 + 2432 * q^80 - 729 * q^81 - 2368 * q^82 + 498 * q^83 + 1872 * q^84 + 1976 * q^85 + 1488 * q^86 + 2460 * q^89 + 2052 * q^90 + 2912 * q^91 - 320 * q^92 + 441 * q^93 - 596 * q^94 - 1824 * q^95 + 2304 * q^96 + 331 * q^97 - 2664 * q^98 + 297 * q^99

## Character values

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

 $$n$$ $$46$$ $$56$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1
 0.5 + 0.866025i 0.5 − 0.866025i
2.00000 3.46410i −4.50000 + 2.59808i −4.00000 6.92820i 9.50000 + 16.4545i 20.7846i 13.0000 22.5167i 0 13.5000 23.3827i 76.0000
67.1 2.00000 + 3.46410i −4.50000 2.59808i −4.00000 + 6.92820i 9.50000 16.4545i 20.7846i 13.0000 + 22.5167i 0 13.5000 + 23.3827i 76.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.4.e.a 2
3.b odd 2 1 297.4.e.a 2
9.c even 3 1 inner 99.4.e.a 2
9.c even 3 1 891.4.a.a 1
9.d odd 6 1 297.4.e.a 2
9.d odd 6 1 891.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.4.e.a 2 1.a even 1 1 trivial
99.4.e.a 2 9.c even 3 1 inner
297.4.e.a 2 3.b odd 2 1
297.4.e.a 2 9.d odd 6 1
891.4.a.a 1 9.c even 3 1
891.4.a.d 1 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 4T_{2} + 16$$ acting on $$S_{4}^{\mathrm{new}}(99, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 4T + 16$$
$3$ $$T^{2} + 9T + 27$$
$5$ $$T^{2} - 19T + 361$$
$7$ $$T^{2} - 26T + 676$$
$11$ $$T^{2} + 11T + 121$$
$13$ $$T^{2} - 56T + 3136$$
$17$ $$(T - 104)^{2}$$
$19$ $$(T + 96)^{2}$$
$23$ $$T^{2} + 40T + 1600$$
$29$ $$T^{2} + 18T + 324$$
$31$ $$T^{2} + 49T + 2401$$
$37$ $$(T - 75)^{2}$$
$41$ $$T^{2} + 296T + 87616$$
$43$ $$T^{2} + 372T + 138384$$
$47$ $$T^{2} - 149T + 22201$$
$53$ $$(T + 417)^{2}$$
$59$ $$T^{2} - 17T + 289$$
$61$ $$T^{2} + 90T + 8100$$
$67$ $$T^{2} + 1073 T + 1151329$$
$71$ $$(T + 285)^{2}$$
$73$ $$(T + 962)^{2}$$
$79$ $$T^{2} + 596T + 355216$$
$83$ $$T^{2} - 498T + 248004$$
$89$ $$(T - 1230)^{2}$$
$97$ $$T^{2} - 331T + 109561$$