# Properties

 Label 99.8.a.g Level $99$ Weight $8$ Character orbit 99.a Self dual yes Analytic conductor $30.926$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,8,Mod(1,99)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("99.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 99.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$30.9261175229$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 341x^{2} + 1417x - 1412$$ x^4 - x^3 - 341*x^2 + 1417*x - 1412 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 11) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 + 151) q^{4} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 135) q^{5} + ( - 6 \beta_{3} - 4 \beta_{2} + 24 \beta_1 + 42) q^{7} + ( - 14 \beta_{3} + 14 \beta_{2} + 120 \beta_1 + 98) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 + b2 + b1 + 151) * q^4 + (-b3 + 2*b2 + 2*b1 - 135) * q^5 + (-6*b3 - 4*b2 + 24*b1 + 42) * q^7 + (-14*b3 + 14*b2 + 120*b1 + 98) * q^8 $$q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 + 151) q^{4} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 135) q^{5} + ( - 6 \beta_{3} - 4 \beta_{2} + 24 \beta_1 + 42) q^{7} + ( - 14 \beta_{3} + 14 \beta_{2} + 120 \beta_1 + 98) q^{8} + (14 \beta_{3} + 28 \beta_{2} - 61 \beta_1 + 364) q^{10} + 1331 q^{11} + (50 \beta_{3} - 20 \beta_{2} + 280 \beta_1 + 1080) q^{13} + (112 \beta_{3} - 28 \beta_{2} - 398 \beta_1 + 7532) q^{14} + (174 \beta_{3} + 174 \beta_{2} + 314 \beta_1 + 13186) q^{16} + ( - 116 \beta_{3} - 24 \beta_{2} + 332 \beta_1 - 13598) q^{17} + ( - 220 \beta_{3} + 40 \beta_{2} + 500 \beta_1 + 16896) q^{19} + ( - 157 \beta_{3} + 47 \beta_{2} + 2175 \beta_1 - 4023) q^{20} + 1331 \beta_1 q^{22} + (263 \beta_{3} - 158 \beta_{2} + 3134 \beta_1 + 2359) q^{23} + (255 \beta_{3} - 822 \beta_{2} + 1586 \beta_1 + 5902) q^{25} + ( - 400 \beta_{3} + 20 \beta_{2} + 2240 \beta_1 + 77820) q^{26} + ( - 1170 \beta_{3} - 250 \beta_{2} + 6974 \beta_1 - 119190) q^{28} + ( - 22 \beta_{3} - 1252 \beta_{2} + 388 \beta_1 - 58212) q^{29} + ( - 511 \beta_{3} - 66 \beta_{2} - 3966 \beta_1 + 47353) q^{31} + ( - 504 \beta_{3} + 784 \beta_{2} + 14844 \beta_1 + 43568) q^{32} + (1980 \beta_{3} + 20 \beta_{2} - 19250 \beta_1 + 102124) q^{34} + (462 \beta_{3} + 1860 \beta_{2} - 10376 \beta_1 - 38874) q^{35} + ( - 1185 \beta_{3} + 1890 \beta_{2} + 15458 \beta_1 + 31205) q^{37} + (3540 \beta_{3} + 1020 \beta_{2} + 10836 \beta_1 + 146820) q^{38} + (2534 \beta_{3} - 798 \beta_{2} + 2312 \beta_1 + 563150) q^{40} + ( - 2458 \beta_{3} + 292 \beta_{2} - 10328 \beta_1 - 73148) q^{41} + ( - 4094 \beta_{3} + 1004 \beta_{2} + 5036 \beta_1 + 174958) q^{43} + (1331 \beta_{3} + 1331 \beta_{2} + 1331 \beta_1 + 200981) q^{44} + ( - 390 \beta_{3} + 1080 \beta_{2} + 7165 \beta_1 + 879408) q^{46} + ( - 2104 \beta_{3} - 1536 \beta_{2} + 6520 \beta_1 + 432128) q^{47} + (4060 \beta_{3} - 1960 \beta_{2} - 32760 \beta_1 + 36485) q^{49} + ( - 1162 \beta_{3} - 9100 \beta_{2} - 28344 \beta_1 + 530964) q^{50} + (1420 \beta_{3} + 5060 \beta_{2} + 29340 \beta_1 + 506620) q^{52} + (740 \beta_{3} + 2280 \beta_{2} - 47296 \beta_1 - 275050) q^{53} + ( - 1331 \beta_{3} + 2662 \beta_{2} + 2662 \beta_1 - 179685) q^{55} + (9268 \beta_{3} + 7308 \beta_{2} - 122072 \beta_1 + 1078420) q^{56} + (1948 \beta_{3} - 15888 \beta_{2} - 128816 \beta_1 + 265984) q^{58} + ( - 8599 \beta_{3} - 9434 \beta_{2} - 13054 \beta_1 + 1166653) q^{59} + (3882 \beta_{3} - 2468 \beta_{2} - 55548 \beta_1 + 79240) q^{61} + (3254 \beta_{3} - 4824 \beta_{2} + 19251 \beta_1 - 1069648) q^{62} + ( - 1156 \beta_{3} + 2764 \beta_{2} + 41964 \beta_1 + 2383892) q^{64} + (160 \beta_{3} + 16640 \beta_{2} - 9500 \beta_1 - 1135660) q^{65} + (7747 \beta_{3} - 3662 \beta_{2} - 31930 \beta_1 - 839209) q^{67} + ( - 32142 \beta_{3} - 15918 \beta_{2} + 120698 \beta_1 - 3743586) q^{68} + ( - 18704 \beta_{3} + 13804 \beta_{2} + 73390 \beta_1 - 3153276) q^{70} + ( - 9283 \beta_{3} - 1178 \beta_{2} - 189438 \beta_1 + 852109) q^{71} + ( - 9602 \beta_{3} - 1868 \beta_{2} + 116928 \beta_1 + 2864624) q^{73} + (30158 \beta_{3} + 40028 \beta_{2} + 105103 \beta_1 + 4142892) q^{74} + ( - 11584 \beta_{3} + 18976 \beta_{2} + 292376 \beta_1 + 534816) q^{76} + ( - 7986 \beta_{3} - 5324 \beta_{2} + 31944 \beta_1 + 55902) q^{77} + ( - 14902 \beta_{3} - 5972 \beta_{2} - 174612 \beta_1 + 139338) q^{79} + ( - 12270 \beta_{3} - 14078 \beta_{2} + 343734 \beta_1 + 1117838) q^{80} + (23792 \beta_{3} - 6532 \beta_{2} - 165444 \beta_1 - 2780364) q^{82} + (28086 \beta_{3} - 34396 \beta_{2} - 159940 \beta_1 + 1070818) q^{83} + (22774 \beta_{3} - 12396 \beta_{2} - 151824 \beta_1 + 2587018) q^{85} + (61348 \beta_{3} + 18088 \beta_{2} + 72458 \beta_1 + 1508808) q^{86} + ( - 18634 \beta_{3} + 18634 \beta_{2} + 159720 \beta_1 + 130438) q^{88} + (26089 \beta_{3} - 20186 \beta_{2} + 147254 \beta_1 - 4554729) q^{89} + (28200 \beta_{3} - 34080 \beta_{2} + 162360 \beta_1 - 267800) q^{91} + ( - 22119 \beta_{3} + 41429 \beta_{2} + 530301 \beta_1 + 1583923) q^{92} + (37512 \beta_{3} - 13448 \beta_{2} + 268472 \beta_1 + 2128904) q^{94} + (744 \beta_{3} + 32832 \beta_{2} - 83148 \beta_1 + 1407720) q^{95} + (49773 \beta_{3} + 61502 \beta_{2} - 14314 \beta_1 + 2853399) q^{97} + ( - 87640 \beta_{3} - 58240 \beta_{2} + 56365 \beta_1 - 9122400) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 + b2 + b1 + 151) * q^4 + (-b3 + 2*b2 + 2*b1 - 135) * q^5 + (-6*b3 - 4*b2 + 24*b1 + 42) * q^7 + (-14*b3 + 14*b2 + 120*b1 + 98) * q^8 + (14*b3 + 28*b2 - 61*b1 + 364) * q^10 + 1331 * q^11 + (50*b3 - 20*b2 + 280*b1 + 1080) * q^13 + (112*b3 - 28*b2 - 398*b1 + 7532) * q^14 + (174*b3 + 174*b2 + 314*b1 + 13186) * q^16 + (-116*b3 - 24*b2 + 332*b1 - 13598) * q^17 + (-220*b3 + 40*b2 + 500*b1 + 16896) * q^19 + (-157*b3 + 47*b2 + 2175*b1 - 4023) * q^20 + 1331*b1 * q^22 + (263*b3 - 158*b2 + 3134*b1 + 2359) * q^23 + (255*b3 - 822*b2 + 1586*b1 + 5902) * q^25 + (-400*b3 + 20*b2 + 2240*b1 + 77820) * q^26 + (-1170*b3 - 250*b2 + 6974*b1 - 119190) * q^28 + (-22*b3 - 1252*b2 + 388*b1 - 58212) * q^29 + (-511*b3 - 66*b2 - 3966*b1 + 47353) * q^31 + (-504*b3 + 784*b2 + 14844*b1 + 43568) * q^32 + (1980*b3 + 20*b2 - 19250*b1 + 102124) * q^34 + (462*b3 + 1860*b2 - 10376*b1 - 38874) * q^35 + (-1185*b3 + 1890*b2 + 15458*b1 + 31205) * q^37 + (3540*b3 + 1020*b2 + 10836*b1 + 146820) * q^38 + (2534*b3 - 798*b2 + 2312*b1 + 563150) * q^40 + (-2458*b3 + 292*b2 - 10328*b1 - 73148) * q^41 + (-4094*b3 + 1004*b2 + 5036*b1 + 174958) * q^43 + (1331*b3 + 1331*b2 + 1331*b1 + 200981) * q^44 + (-390*b3 + 1080*b2 + 7165*b1 + 879408) * q^46 + (-2104*b3 - 1536*b2 + 6520*b1 + 432128) * q^47 + (4060*b3 - 1960*b2 - 32760*b1 + 36485) * q^49 + (-1162*b3 - 9100*b2 - 28344*b1 + 530964) * q^50 + (1420*b3 + 5060*b2 + 29340*b1 + 506620) * q^52 + (740*b3 + 2280*b2 - 47296*b1 - 275050) * q^53 + (-1331*b3 + 2662*b2 + 2662*b1 - 179685) * q^55 + (9268*b3 + 7308*b2 - 122072*b1 + 1078420) * q^56 + (1948*b3 - 15888*b2 - 128816*b1 + 265984) * q^58 + (-8599*b3 - 9434*b2 - 13054*b1 + 1166653) * q^59 + (3882*b3 - 2468*b2 - 55548*b1 + 79240) * q^61 + (3254*b3 - 4824*b2 + 19251*b1 - 1069648) * q^62 + (-1156*b3 + 2764*b2 + 41964*b1 + 2383892) * q^64 + (160*b3 + 16640*b2 - 9500*b1 - 1135660) * q^65 + (7747*b3 - 3662*b2 - 31930*b1 - 839209) * q^67 + (-32142*b3 - 15918*b2 + 120698*b1 - 3743586) * q^68 + (-18704*b3 + 13804*b2 + 73390*b1 - 3153276) * q^70 + (-9283*b3 - 1178*b2 - 189438*b1 + 852109) * q^71 + (-9602*b3 - 1868*b2 + 116928*b1 + 2864624) * q^73 + (30158*b3 + 40028*b2 + 105103*b1 + 4142892) * q^74 + (-11584*b3 + 18976*b2 + 292376*b1 + 534816) * q^76 + (-7986*b3 - 5324*b2 + 31944*b1 + 55902) * q^77 + (-14902*b3 - 5972*b2 - 174612*b1 + 139338) * q^79 + (-12270*b3 - 14078*b2 + 343734*b1 + 1117838) * q^80 + (23792*b3 - 6532*b2 - 165444*b1 - 2780364) * q^82 + (28086*b3 - 34396*b2 - 159940*b1 + 1070818) * q^83 + (22774*b3 - 12396*b2 - 151824*b1 + 2587018) * q^85 + (61348*b3 + 18088*b2 + 72458*b1 + 1508808) * q^86 + (-18634*b3 + 18634*b2 + 159720*b1 + 130438) * q^88 + (26089*b3 - 20186*b2 + 147254*b1 - 4554729) * q^89 + (28200*b3 - 34080*b2 + 162360*b1 - 267800) * q^91 + (-22119*b3 + 41429*b2 + 530301*b1 + 1583923) * q^92 + (37512*b3 - 13448*b2 + 268472*b1 + 2128904) * q^94 + (744*b3 + 32832*b2 - 83148*b1 + 1407720) * q^95 + (49773*b3 + 61502*b2 - 14314*b1 + 2853399) * q^97 + (-87640*b3 - 58240*b2 + 56365*b1 - 9122400) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 604 q^{4} - 537 q^{5} + 170 q^{7} + 420 q^{8}+O(q^{10})$$ 4 * q + 604 * q^4 - 537 * q^5 + 170 * q^7 + 420 * q^8 $$4 q + 604 q^{4} - 537 q^{5} + 170 q^{7} + 420 q^{8} + 1470 q^{10} + 5324 q^{11} + 4250 q^{13} + 29988 q^{14} + 52744 q^{16} - 54300 q^{17} + 67844 q^{19} - 15888 q^{20} + 9015 q^{23} + 22531 q^{25} + 311700 q^{26} - 475840 q^{28} - 234078 q^{29} + 189857 q^{31} + 175560 q^{32} + 406536 q^{34} - 154098 q^{35} + 127895 q^{37} + 584760 q^{38} + 2249268 q^{40} - 289842 q^{41} + 704930 q^{43} + 803924 q^{44} + 3519102 q^{46} + 1729080 q^{47} + 139920 q^{49} + 2115918 q^{50} + 2030120 q^{52} - 1098660 q^{53} - 714747 q^{55} + 4311720 q^{56} + 1046100 q^{58} + 4665777 q^{59} + 310610 q^{61} - 4286670 q^{62} + 9539488 q^{64} - 4526160 q^{65} - 3368245 q^{67} - 14958120 q^{68} - 12580596 q^{70} + 3416541 q^{71} + 11466230 q^{73} + 16581438 q^{74} + 2169824 q^{76} + 226270 q^{77} + 566282 q^{79} + 4469544 q^{80} - 11151780 q^{82} + 4220790 q^{83} + 10312902 q^{85} + 5991972 q^{86} + 559020 q^{88} - 18265191 q^{89} - 1133480 q^{91} + 6399240 q^{92} + 8464656 q^{94} + 5662968 q^{95} + 11425325 q^{97} - 36460200 q^{98}+O(q^{100})$$ 4 * q + 604 * q^4 - 537 * q^5 + 170 * q^7 + 420 * q^8 + 1470 * q^10 + 5324 * q^11 + 4250 * q^13 + 29988 * q^14 + 52744 * q^16 - 54300 * q^17 + 67844 * q^19 - 15888 * q^20 + 9015 * q^23 + 22531 * q^25 + 311700 * q^26 - 475840 * q^28 - 234078 * q^29 + 189857 * q^31 + 175560 * q^32 + 406536 * q^34 - 154098 * q^35 + 127895 * q^37 + 584760 * q^38 + 2249268 * q^40 - 289842 * q^41 + 704930 * q^43 + 803924 * q^44 + 3519102 * q^46 + 1729080 * q^47 + 139920 * q^49 + 2115918 * q^50 + 2030120 * q^52 - 1098660 * q^53 - 714747 * q^55 + 4311720 * q^56 + 1046100 * q^58 + 4665777 * q^59 + 310610 * q^61 - 4286670 * q^62 + 9539488 * q^64 - 4526160 * q^65 - 3368245 * q^67 - 14958120 * q^68 - 12580596 * q^70 + 3416541 * q^71 + 11466230 * q^73 + 16581438 * q^74 + 2169824 * q^76 + 226270 * q^77 + 566282 * q^79 + 4469544 * q^80 - 11151780 * q^82 + 4220790 * q^83 + 10312902 * q^85 + 5991972 * q^86 + 559020 * q^88 - 18265191 * q^89 - 1133480 * q^91 + 6399240 * q^92 + 8464656 * q^94 + 5662968 * q^95 + 11425325 * q^97 - 36460200 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 341x^{2} + 1417x - 1412$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu^{2} - 321\nu + 204 ) / 28$$ (v^3 + 4*v^2 - 321*v + 204) / 28 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} - 4\nu^{2} + 573\nu - 260 ) / 28$$ (-v^3 - 4*v^2 + 573*v - 260) / 28 $$\beta_{3}$$ $$=$$ $$-\nu^{3} - \nu^{2} + 339\nu - 721$$ -v^3 - v^2 + 339*v - 721
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 2 ) / 9$$ (b2 + b1 + 2) / 9 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 2\beta_{2} + 26\beta _1 + 513 ) / 3$$ (b3 - 2*b2 + 26*b1 + 513) / 3 $$\nu^{3}$$ $$=$$ $$( -4\beta_{3} + 115\beta_{2} + 87\beta _1 - 2450 ) / 3$$ (-4*b3 + 115*b2 + 87*b1 - 2450) / 3

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.58394 1.64802 −19.8969 16.6649
−20.7673 0 303.281 −223.515 0 −1410.66 −3640.12 0 4641.80
1.2 −11.0598 0 −5.68000 60.1766 0 698.069 1478.48 0 −665.544
1.3 10.6261 0 −15.0863 −512.130 0 973.904 −1520.45 0 −5441.94
1.4 21.2011 0 321.485 138.468 0 −91.3115 4102.09 0 2935.68
 $$n$$: 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 99.8.a.g 4
3.b odd 2 1 11.8.a.b 4
12.b even 2 1 176.8.a.j 4
15.d odd 2 1 275.8.a.b 4
21.c even 2 1 539.8.a.b 4
33.d even 2 1 121.8.a.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.8.a.b 4 3.b odd 2 1
99.8.a.g 4 1.a even 1 1 trivial
121.8.a.c 4 33.d even 2 1
176.8.a.j 4 12.b even 2 1
275.8.a.b 4 15.d odd 2 1
539.8.a.b 4 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 558T_{2}^{2} - 140T_{2} + 51744$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(99))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 558 T^{2} - 140 T + 51744$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 537 T^{3} + \cdots + 953818350$$
$7$ $$T^{4} - 170 T^{3} + \cdots + 87571440704$$
$11$ $$(T - 1331)^{4}$$
$13$ $$T^{4} + \cdots - 518474088880000$$
$17$ $$T^{4} + 54300 T^{3} + \cdots - 58\!\cdots\!56$$
$19$ $$T^{4} - 67844 T^{3} + \cdots - 31\!\cdots\!84$$
$23$ $$T^{4} - 9015 T^{3} + \cdots + 10\!\cdots\!44$$
$29$ $$T^{4} + 234078 T^{3} + \cdots - 41\!\cdots\!64$$
$31$ $$T^{4} - 189857 T^{3} + \cdots - 61\!\cdots\!36$$
$37$ $$T^{4} - 127895 T^{3} + \cdots + 41\!\cdots\!94$$
$41$ $$T^{4} + 289842 T^{3} + \cdots - 66\!\cdots\!84$$
$43$ $$T^{4} - 704930 T^{3} + \cdots + 28\!\cdots\!64$$
$47$ $$T^{4} - 1729080 T^{3} + \cdots - 74\!\cdots\!04$$
$53$ $$T^{4} + 1098660 T^{3} + \cdots + 18\!\cdots\!84$$
$59$ $$T^{4} - 4665777 T^{3} + \cdots + 29\!\cdots\!64$$
$61$ $$T^{4} - 310610 T^{3} + \cdots + 10\!\cdots\!84$$
$67$ $$T^{4} + 3368245 T^{3} + \cdots - 84\!\cdots\!64$$
$71$ $$T^{4} - 3416541 T^{3} + \cdots + 59\!\cdots\!44$$
$73$ $$T^{4} - 11466230 T^{3} + \cdots - 60\!\cdots\!76$$
$79$ $$T^{4} - 566282 T^{3} + \cdots + 25\!\cdots\!16$$
$83$ $$T^{4} - 4220790 T^{3} + \cdots + 70\!\cdots\!96$$
$89$ $$T^{4} + 18265191 T^{3} + \cdots - 24\!\cdots\!34$$
$97$ $$T^{4} - 11425325 T^{3} + \cdots + 46\!\cdots\!46$$