# Properties

 Label 99.4.a.g Level $99$ Weight $4$ Character orbit 99.a Self dual yes Analytic conductor $5.841$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 99.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.84118909057$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (2 \beta + 5) q^{4} + ( - \beta + 10) q^{5} + (5 \beta - 8) q^{7} + ( - \beta + 21) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + (2*b + 5) * q^4 + (-b + 10) * q^5 + (5*b - 8) * q^7 + (-b + 21) * q^8 $$q + (\beta + 1) q^{2} + (2 \beta + 5) q^{4} + ( - \beta + 10) q^{5} + (5 \beta - 8) q^{7} + ( - \beta + 21) q^{8} + (9 \beta - 2) q^{10} - 11 q^{11} + ( - 11 \beta + 40) q^{13} + ( - 3 \beta + 52) q^{14} + (4 \beta - 31) q^{16} + ( - 6 \beta + 82) q^{17} + ( - 30 \beta - 18) q^{19} + (15 \beta + 26) q^{20} + ( - 11 \beta - 11) q^{22} + (9 \beta + 86) q^{23} + ( - 20 \beta - 13) q^{25} + (29 \beta - 92) q^{26} + (9 \beta + 80) q^{28} + ( - 56 \beta + 54) q^{29} + ( - 26 \beta - 224) q^{31} + ( - 19 \beta - 151) q^{32} + (76 \beta + 10) q^{34} + (58 \beta - 140) q^{35} + (10 \beta + 54) q^{37} + ( - 48 \beta - 378) q^{38} + ( - 31 \beta + 222) q^{40} + ( - 4 \beta + 106) q^{41} + (92 \beta + 78) q^{43} + ( - 22 \beta - 55) q^{44} + (95 \beta + 194) q^{46} + ( - 21 \beta - 10) q^{47} + ( - 80 \beta + 21) q^{49} + ( - 33 \beta - 253) q^{50} + (25 \beta - 64) q^{52} + (187 \beta - 66) q^{53} + (11 \beta - 110) q^{55} + (113 \beta - 228) q^{56} + ( - 2 \beta - 618) q^{58} + (102 \beta + 344) q^{59} + (67 \beta - 48) q^{61} + ( - 250 \beta - 536) q^{62} + ( - 202 \beta - 131) q^{64} + ( - 150 \beta + 532) q^{65} + ( - 128 \beta + 224) q^{67} + (134 \beta + 266) q^{68} + ( - 82 \beta + 556) q^{70} + ( - 275 \beta + 66) q^{71} + (36 \beta + 214) q^{73} + (64 \beta + 174) q^{74} + ( - 186 \beta - 810) q^{76} + ( - 55 \beta + 88) q^{77} + ( - \beta + 212) q^{79} + (71 \beta - 358) q^{80} + (102 \beta + 58) q^{82} + (30 \beta + 360) q^{83} + ( - 142 \beta + 892) q^{85} + (170 \beta + 1182) q^{86} + (11 \beta - 231) q^{88} + (176 \beta + 528) q^{89} + (288 \beta - 980) q^{91} + (217 \beta + 646) q^{92} + ( - 31 \beta - 262) q^{94} + ( - 282 \beta + 180) q^{95} + (432 \beta + 26) q^{97} + ( - 59 \beta - 939) q^{98}+O(q^{100})$$ q + (b + 1) * q^2 + (2*b + 5) * q^4 + (-b + 10) * q^5 + (5*b - 8) * q^7 + (-b + 21) * q^8 + (9*b - 2) * q^10 - 11 * q^11 + (-11*b + 40) * q^13 + (-3*b + 52) * q^14 + (4*b - 31) * q^16 + (-6*b + 82) * q^17 + (-30*b - 18) * q^19 + (15*b + 26) * q^20 + (-11*b - 11) * q^22 + (9*b + 86) * q^23 + (-20*b - 13) * q^25 + (29*b - 92) * q^26 + (9*b + 80) * q^28 + (-56*b + 54) * q^29 + (-26*b - 224) * q^31 + (-19*b - 151) * q^32 + (76*b + 10) * q^34 + (58*b - 140) * q^35 + (10*b + 54) * q^37 + (-48*b - 378) * q^38 + (-31*b + 222) * q^40 + (-4*b + 106) * q^41 + (92*b + 78) * q^43 + (-22*b - 55) * q^44 + (95*b + 194) * q^46 + (-21*b - 10) * q^47 + (-80*b + 21) * q^49 + (-33*b - 253) * q^50 + (25*b - 64) * q^52 + (187*b - 66) * q^53 + (11*b - 110) * q^55 + (113*b - 228) * q^56 + (-2*b - 618) * q^58 + (102*b + 344) * q^59 + (67*b - 48) * q^61 + (-250*b - 536) * q^62 + (-202*b - 131) * q^64 + (-150*b + 532) * q^65 + (-128*b + 224) * q^67 + (134*b + 266) * q^68 + (-82*b + 556) * q^70 + (-275*b + 66) * q^71 + (36*b + 214) * q^73 + (64*b + 174) * q^74 + (-186*b - 810) * q^76 + (-55*b + 88) * q^77 + (-b + 212) * q^79 + (71*b - 358) * q^80 + (102*b + 58) * q^82 + (30*b + 360) * q^83 + (-142*b + 892) * q^85 + (170*b + 1182) * q^86 + (11*b - 231) * q^88 + (176*b + 528) * q^89 + (288*b - 980) * q^91 + (217*b + 646) * q^92 + (-31*b - 262) * q^94 + (-282*b + 180) * q^95 + (432*b + 26) * q^97 + (-59*b - 939) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 10 q^{4} + 20 q^{5} - 16 q^{7} + 42 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 10 * q^4 + 20 * q^5 - 16 * q^7 + 42 * q^8 $$2 q + 2 q^{2} + 10 q^{4} + 20 q^{5} - 16 q^{7} + 42 q^{8} - 4 q^{10} - 22 q^{11} + 80 q^{13} + 104 q^{14} - 62 q^{16} + 164 q^{17} - 36 q^{19} + 52 q^{20} - 22 q^{22} + 172 q^{23} - 26 q^{25} - 184 q^{26} + 160 q^{28} + 108 q^{29} - 448 q^{31} - 302 q^{32} + 20 q^{34} - 280 q^{35} + 108 q^{37} - 756 q^{38} + 444 q^{40} + 212 q^{41} + 156 q^{43} - 110 q^{44} + 388 q^{46} - 20 q^{47} + 42 q^{49} - 506 q^{50} - 128 q^{52} - 132 q^{53} - 220 q^{55} - 456 q^{56} - 1236 q^{58} + 688 q^{59} - 96 q^{61} - 1072 q^{62} - 262 q^{64} + 1064 q^{65} + 448 q^{67} + 532 q^{68} + 1112 q^{70} + 132 q^{71} + 428 q^{73} + 348 q^{74} - 1620 q^{76} + 176 q^{77} + 424 q^{79} - 716 q^{80} + 116 q^{82} + 720 q^{83} + 1784 q^{85} + 2364 q^{86} - 462 q^{88} + 1056 q^{89} - 1960 q^{91} + 1292 q^{92} - 524 q^{94} + 360 q^{95} + 52 q^{97} - 1878 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 10 * q^4 + 20 * q^5 - 16 * q^7 + 42 * q^8 - 4 * q^10 - 22 * q^11 + 80 * q^13 + 104 * q^14 - 62 * q^16 + 164 * q^17 - 36 * q^19 + 52 * q^20 - 22 * q^22 + 172 * q^23 - 26 * q^25 - 184 * q^26 + 160 * q^28 + 108 * q^29 - 448 * q^31 - 302 * q^32 + 20 * q^34 - 280 * q^35 + 108 * q^37 - 756 * q^38 + 444 * q^40 + 212 * q^41 + 156 * q^43 - 110 * q^44 + 388 * q^46 - 20 * q^47 + 42 * q^49 - 506 * q^50 - 128 * q^52 - 132 * q^53 - 220 * q^55 - 456 * q^56 - 1236 * q^58 + 688 * q^59 - 96 * q^61 - 1072 * q^62 - 262 * q^64 + 1064 * q^65 + 448 * q^67 + 532 * q^68 + 1112 * q^70 + 132 * q^71 + 428 * q^73 + 348 * q^74 - 1620 * q^76 + 176 * q^77 + 424 * q^79 - 716 * q^80 + 116 * q^82 + 720 * q^83 + 1784 * q^85 + 2364 * q^86 - 462 * q^88 + 1056 * q^89 - 1960 * q^91 + 1292 * q^92 - 524 * q^94 + 360 * q^95 + 52 * q^97 - 1878 * q^98

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
−2.46410 0 −1.92820 13.4641 0 −25.3205 24.4641 0 −33.1769
1.2 4.46410 0 11.9282 6.53590 0 9.32051 17.5359 0 29.1769
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$11$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.4.a.g yes 2
3.b odd 2 1 99.4.a.d 2
4.b odd 2 1 1584.4.a.bk 2
5.b even 2 1 2475.4.a.m 2
11.b odd 2 1 1089.4.a.l 2
12.b even 2 1 1584.4.a.w 2
15.d odd 2 1 2475.4.a.r 2
33.d even 2 1 1089.4.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.4.a.d 2 3.b odd 2 1
99.4.a.g yes 2 1.a even 1 1 trivial
1089.4.a.l 2 11.b odd 2 1
1089.4.a.w 2 33.d even 2 1
1584.4.a.w 2 12.b even 2 1
1584.4.a.bk 2 4.b odd 2 1
2475.4.a.m 2 5.b even 2 1
2475.4.a.r 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2T_{2} - 11$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(99))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 11$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 20T + 88$$
$7$ $$T^{2} + 16T - 236$$
$11$ $$(T + 11)^{2}$$
$13$ $$T^{2} - 80T + 148$$
$17$ $$T^{2} - 164T + 6292$$
$19$ $$T^{2} + 36T - 10476$$
$23$ $$T^{2} - 172T + 6424$$
$29$ $$T^{2} - 108T - 34716$$
$31$ $$T^{2} + 448T + 42064$$
$37$ $$T^{2} - 108T + 1716$$
$41$ $$T^{2} - 212T + 11044$$
$43$ $$T^{2} - 156T - 95484$$
$47$ $$T^{2} + 20T - 5192$$
$53$ $$T^{2} + 132T - 415272$$
$59$ $$T^{2} - 688T - 6512$$
$61$ $$T^{2} + 96T - 51564$$
$67$ $$T^{2} - 448T - 146432$$
$71$ $$T^{2} - 132T - 903144$$
$73$ $$T^{2} - 428T + 30244$$
$79$ $$T^{2} - 424T + 44932$$
$83$ $$T^{2} - 720T + 118800$$
$89$ $$T^{2} - 1056T - 92928$$
$97$ $$T^{2} - 52T - 2238812$$