Properties

 Label 24.22.a.d Level $24$ Weight $22$ Character orbit 24.a Self dual yes Analytic conductor $67.075$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [24,22,Mod(1,24)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(24, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("24.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 24.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: $$67.0745626289$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 12529199x - 17012391021$$ x^3 - x^2 - 12529199*x - 17012391021 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{21}\cdot 3^{4}\cdot 7$$ Twist minimal: yes 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 59049 q^{3} + ( - \beta_1 + 1760166) q^{5} + ( - \beta_{2} - 7 \beta_1 + 284180792) q^{7} + 3486784401 q^{9}+O(q^{10})$$ q + 59049 * q^3 + (-b1 + 1760166) * q^5 + (-b2 - 7*b1 + 284180792) * q^7 + 3486784401 * q^9 $$q + 59049 q^{3} + ( - \beta_1 + 1760166) q^{5} + ( - \beta_{2} - 7 \beta_1 + 284180792) q^{7} + 3486784401 q^{9} + ( - 56 \beta_{2} - 1042 \beta_1 + 20830252556) q^{11} + ( - 433 \beta_{2} - 11856 \beta_1 + 67921735934) q^{13} + ( - 59049 \beta_1 + 103936042134) q^{15} + (1730 \beta_{2} - 209590 \beta_1 + 231942606642) q^{17} + (32642 \beta_{2} - 316806 \beta_1 + 1651834707732) q^{19} + ( - 59049 \beta_{2} - 413343 \beta_1 + 16780591586808) q^{21} + ( - 6426 \beta_{2} + 1368418 \beta_1 + 50289135979384) q^{23} + ( - 245830 \beta_{2} + 13033992 \beta_1 + 226064951656623) q^{25} + 205891132094649 q^{27} + (430222 \beta_{2} + 61470129 \beta_1 + 10\!\cdots\!82) q^{29}+ \cdots + ( - 195259926456 \beta_{2} + \cdots + 72\!\cdots\!56) q^{99}+O(q^{100})$$ q + 59049 * q^3 + (-b1 + 1760166) * q^5 + (-b2 - 7*b1 + 284180792) * q^7 + 3486784401 * q^9 + (-56*b2 - 1042*b1 + 20830252556) * q^11 + (-433*b2 - 11856*b1 + 67921735934) * q^13 + (-59049*b1 + 103936042134) * q^15 + (1730*b2 - 209590*b1 + 231942606642) * q^17 + (32642*b2 - 316806*b1 + 1651834707732) * q^19 + (-59049*b2 - 413343*b1 + 16780591586808) * q^21 + (-6426*b2 + 1368418*b1 + 50289135979384) * q^23 + (-245830*b2 + 13033992*b1 + 226064951656623) * q^25 + 205891132094649 * q^27 + (430222*b2 + 61470129*b1 + 1080741635075582) * q^29 + (5096903*b2 - 118127179*b1 + 217502867183632) * q^31 + (-3306744*b2 - 61529058*b1 + 1230005583179244) * q^33 + (-20035300*b2 - 489193950*b1 + 5147906428613200) * q^35 + (25948165*b2 + 303691330*b1 - 6897757278972074) * q^37 + (-25568217*b2 - 700084944*b1 + 4010710585166766) * q^39 + (-2710366*b2 + 3823722438*b1 - 15869533437517334) * q^41 + (90491010*b2 + 818298070*b1 - 92370070410138516) * q^43 + (-3486784401*b1 + 6137319351970566) * q^45 + (131505078*b2 + 14526562046*b1 - 178647502442073776) * q^47 + (-503595526*b2 - 27246071132*b1 + 397607484745538585) * q^49 + (102154770*b2 - 12376079910*b1 + 13695978979603458) * q^51 + (28257918*b2 + 8579960701*b1 + 949765964116329142) * q^53 + (-1281766300*b2 - 22694786704*b1 + 751808513301585864) * q^55 + (1927477458*b2 - 18707077494*b1 + 97539187656866868) * q^57 + (2103421764*b2 + 140296579248*b1 + 293682125811575516) * q^59 + (3217443397*b2 + 137208913854*b1 + 3540296818538517774) * q^61 + (-3486784401*b2 - 24407490807*b1 + 990877152609425592) * q^63 + (-10844734650*b2 - 26133988998*b1 + 8307777732172287668) * q^65 + (-6734645628*b2 - 420729628496*b1 + 4946693576289238372) * q^67 + (-379448874*b2 + 80803714482*b1 + 2969523190446645816) * q^69 + (27636133578*b2 - 854195460654*b1 + 4869648279437313832) * q^71 + (26166463726*b2 + 754279335932*b1 + 4525028754649763818) * q^73 + (-14516015670*b2 + 769644193608*b1 + 13348909330371931527) * q^75 + (-44996314760*b2 - 1870160193020*b1 + 57962662430443445920) * q^77 + (30214392091*b2 + 1758709572937*b1 + 92698859172585182720) * q^79 + 12157665459056928801 * q^81 + (-66318799972*b2 + 1852553129246*b1 + 195238972761152276820) * q^83 + (-19839442000*b2 + 3402594985298*b1 + 147514264941236199532) * q^85 + (25404178878*b2 + 3629749647321*b1 + 63816712809578041518) * q^87 + (169189087252*b2 - 7898004310036*b1 + 159272900597554175610) * q^89 + (-324206373306*b2 - 15620051325242*b1 + 439363887754558846352) * q^91 + (300967025247*b2 - 6975291792771*b1 + 12843326804326285968) * q^93 + (519941163600*b2 + 13107459153104*b1 + 232800325849195135736) * q^95 + (-463090832696*b2 + 5264260583428*b1 + 240803913337907196834) * q^97 + (-195259926456*b2 - 3633229345842*b1 + 72630599681151178956) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 177147 q^{3} + 5280498 q^{5} + 852542376 q^{7} + 10460353203 q^{9}+O(q^{10})$$ 3 * q + 177147 * q^3 + 5280498 * q^5 + 852542376 * q^7 + 10460353203 * q^9 $$3 q + 177147 q^{3} + 5280498 q^{5} + 852542376 q^{7} + 10460353203 q^{9} + 62490757668 q^{11} + 203765207802 q^{13} + 311808126402 q^{15} + 695827819926 q^{17} + 4955504123196 q^{19} + 50341774760424 q^{21} + 150867407938152 q^{23} + 678194854969869 q^{25} + 617673396283947 q^{27} + 32\!\cdots\!46 q^{29}+ \cdots + 21\!\cdots\!68 q^{99}+O(q^{100})$$ 3 * q + 177147 * q^3 + 5280498 * q^5 + 852542376 * q^7 + 10460353203 * q^9 + 62490757668 * q^11 + 203765207802 * q^13 + 311808126402 * q^15 + 695827819926 * q^17 + 4955504123196 * q^19 + 50341774760424 * q^21 + 150867407938152 * q^23 + 678194854969869 * q^25 + 617673396283947 * q^27 + 3242224905226746 * q^29 + 652508601550896 * q^31 + 3690016749537732 * q^33 + 15443719285839600 * q^35 - 20693271836916222 * q^37 + 12032131755500298 * q^39 - 47608600312552002 * q^41 - 277110211230415548 * q^43 + 18411958055911698 * q^45 - 535942507326221328 * q^47 + 1192822454236615755 * q^49 + 41087936938810374 * q^51 + 2849297892348987426 * q^53 + 2255425539904757592 * q^55 + 292617562970600604 * q^57 + 881046377434726548 * q^59 + 10620890455615553322 * q^61 + 2972631457828276776 * q^63 + 24923333196516863004 * q^65 + 14840080728867715116 * q^67 + 8908569571339937448 * q^69 + 14608944838311941496 * q^71 + 13575086263949291454 * q^73 + 40046727991115794581 * q^75 + 173887987291330337760 * q^77 + 278096577517755548160 * q^79 + 36472996377170786403 * q^81 + 585716918283456830460 * q^83 + 442542794823708598596 * q^85 + 191450138428734124554 * q^87 + 477818701792662526830 * q^89 + 1318091663263676539056 * q^91 + 38529980412978857904 * q^93 + 698400977547585407208 * q^95 + 722411740013721590502 * q^97 + 217891799043453536868 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 12529199x - 17012391021$$ :

 $$\beta_{1}$$ $$=$$ $$( 576\nu^{2} - 1212288\nu - 4810808512 ) / 11$$ (576*v^2 - 1212288*v - 4810808512) / 11 $$\beta_{2}$$ $$=$$ $$( 8064\nu^{2} - 13187328\nu - 67352580736 ) / 11$$ (8064*v^2 - 13187328*v - 67352580736) / 11
 $$\nu$$ $$=$$ $$( \beta_{2} - 14\beta _1 + 114688 ) / 344064$$ (b2 - 14*b1 + 114688) / 344064 $$\nu^{2}$$ $$=$$ $$( 451\beta_{2} - 4906\beta _1 + 615835213824 ) / 73728$$ (451*b2 - 4906*b1 + 615835213824) / 73728

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
 −2135.93 4086.16 −1949.23
0 59049.0 0 −3.51850e7 0 2.43346e8 0 3.48678e9 0
1.2 0 59049.0 0 1.51338e7 0 −8.40758e8 0 3.48678e9 0
1.3 0 59049.0 0 2.53316e7 0 1.44995e9 0 3.48678e9 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$

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 24.22.a.d 3
3.b odd 2 1 72.22.a.c 3
4.b odd 2 1 48.22.a.j 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.22.a.d 3 1.a even 1 1 trivial
48.22.a.j 3 4.b odd 2 1
72.22.a.c 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} - 5280498T_{5}^{2} - 1040411335225620T_{5} + 13488669456493277401000$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(24))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 59049)^{3}$$
$5$ $$T^{3} - 5280498 T^{2} + \cdots + 13\!\cdots\!00$$
$7$ $$T^{3} - 852542376 T^{2} + \cdots + 29\!\cdots\!00$$
$11$ $$T^{3} - 62490757668 T^{2} + \cdots - 18\!\cdots\!60$$
$13$ $$T^{3} - 203765207802 T^{2} + \cdots - 64\!\cdots\!24$$
$17$ $$T^{3} - 695827819926 T^{2} + \cdots + 13\!\cdots\!12$$
$19$ $$T^{3} - 4955504123196 T^{2} + \cdots - 18\!\cdots\!64$$
$23$ $$T^{3} - 150867407938152 T^{2} + \cdots - 60\!\cdots\!52$$
$29$ $$T^{3} + \cdots + 20\!\cdots\!76$$
$31$ $$T^{3} - 652508601550896 T^{2} + \cdots - 88\!\cdots\!20$$
$37$ $$T^{3} + \cdots - 10\!\cdots\!76$$
$41$ $$T^{3} + \cdots - 91\!\cdots\!60$$
$43$ $$T^{3} + \cdots - 11\!\cdots\!04$$
$47$ $$T^{3} + \cdots - 45\!\cdots\!00$$
$53$ $$T^{3} + \cdots - 78\!\cdots\!20$$
$59$ $$T^{3} + \cdots + 25\!\cdots\!60$$
$61$ $$T^{3} + \cdots + 12\!\cdots\!96$$
$67$ $$T^{3} + \cdots + 46\!\cdots\!16$$
$71$ $$T^{3} + \cdots - 43\!\cdots\!20$$
$73$ $$T^{3} + \cdots + 27\!\cdots\!00$$
$79$ $$T^{3} + \cdots - 33\!\cdots\!64$$
$83$ $$T^{3} + \cdots - 54\!\cdots\!88$$
$89$ $$T^{3} + \cdots + 14\!\cdots\!88$$
$97$ $$T^{3} + \cdots + 12\!\cdots\!08$$