# Properties

 Label 24.28.a.d Level $24$ Weight $28$ Character orbit 24.a Self dual yes Analytic conductor $110.845$ 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] = [24,28,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, 28, names="a")

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

chi := DirichletCharacter("24.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ $$=$$ $$28$$ 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: $$110.845337961$$ 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: $$x^{4} - 2x^{3} - 360714331909x^{2} - 43287560841177118x + 8819337660421091919513$$ x^4 - 2*x^3 - 360714331909*x^2 - 43287560841177118*x + 8819337660421091919513 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{42}\cdot 3^{8}\cdot 5\cdot 7$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 1594323 q^{3} + ( - \beta_1 + 1237262606) q^{5} + ( - \beta_{2} - 10 \beta_1 - 22062432888) q^{7} + 2541865828329 q^{9}+O(q^{10})$$ q + 1594323 * q^3 + (-b1 + 1237262606) * q^5 + (-b2 - 10*b1 - 22062432888) * q^7 + 2541865828329 * q^9 $$q + 1594323 q^{3} + ( - \beta_1 + 1237262606) q^{5} + ( - \beta_{2} - 10 \beta_1 - 22062432888) q^{7} + 2541865828329 q^{9} + ( - \beta_{3} + 133 \beta_{2} + \cdots - 27797408429212) q^{11}+ \cdots + ( - 2541865828329 \beta_{3} + \cdots - 70\!\cdots\!48) q^{99}+O(q^{100})$$ q + 1594323 * q^3 + (-b1 + 1237262606) * q^5 + (-b2 - 10*b1 - 22062432888) * q^7 + 2541865828329 * q^9 + (-b3 + 133*b2 + 1666*b1 - 27797408429212) * q^11 + (-2*b3 - 1225*b2 + 152257*b1 - 1834914410666) * q^13 + (-1594323*b1 + 1972596229785738) * q^15 + (161*b3 - 120463*b2 + 830704*b1 + 8121184312766866) * q^17 + (-69*b3 - 379477*b2 - 19114656*b1 - 8220430924248100) * q^19 + (-1594323*b2 - 15943230*b1 - 35174644189294824) * q^21 + (-10580*b3 + 1938762*b2 - 653088220*b1 - 80375299946703544) * q^23 + (21850*b3 + 10188500*b2 - 2037184874*b1 + 4448971884818228519) * q^25 + 4052555153018976267 * q^27 + (208486*b3 + 75186076*b2 - 9379101221*b1 + 35839965461989539702) * q^29 + (-482386*b3 + 82635161*b2 - 32558419694*b1 - 3815763038373088912) * q^31 + (-1594323*b3 + 212044959*b2 + 2656142118*b1 - 44318047599086563476) * q^33 + (4757225*b3 - 3317690425*b2 + 133009596194*b1 + 75200739403001402736) * q^35 + (-1169410*b3 + 1636240081*b2 + 232101192885*b1 + 278893180487639416254) * q^37 + (-3188646*b3 - 1953045675*b2 + 242746837011*b1 - 2925446247956249118) * q^39 + (10951543*b3 + 13159846595*b2 - 10713351188*b1 + 2675146334922496697370) * q^41 + (-36549451*b3 + 14469559881*b2 + 2359224973236*b1 + 2471847763898014050052) * q^43 + (-2541865828329*b1 + 3144955538860687165374) * q^45 + (-9783096*b3 - 40146783810*b2 - 1332336136024*b1 + 12548063986343726522544) * q^47 + (223859476*b3 + 6332809010*b2 - 22672666058686*b1 + 57099574619301143463209) * q^49 + (256686003*b3 - 192056931549*b2 + 1324410493392*b1 + 12947790937083408101718) * q^51 + (-1003703750*b3 - 411293962632*b2 - 20392326716785*b1 + 25906778093526786370622) * q^53 + (-132342650*b3 + 1331265192950*b2 + 63765455728788*b1 - 51507563897552219118728) * q^55 + (-110008287*b3 - 605008909071*b2 - 30474935697888*b1 - 13106022092440003536300) * q^57 + (898198072*b3 + 227057783484*b2 + 270068033700148*b1 - 499180336322451236021356) * q^59 + (3584569938*b3 + 2908087614565*b2 - 66000599566023*b1 - 271580667288359985347450) * q^61 + (-2541865828329*b2 - 25418658283290*b1 - 56079744247809091684152) * q^63 + (3248542425*b3 - 3953378935275*b2 + 370248494276844*b1 - 1582437336661951771423564) * q^65 + (-25921634448*b3 - 2673346829244*b2 - 460049762786732*b1 - 44219581749077454612340) * q^67 + (-16867937340*b3 + 3091012848126*b2 - 1041233570175060*b1 - 128144189336928234380712) * q^69 + (48667489172*b3 - 11883861067386*b2 - 1041085410522712*b1 + 2719370306845073501541784) * q^71 + (35494031292*b3 + 6309787741222*b2 - 1494792536027962*b1 + 810786335349112696070362) * q^73 + (34835957550*b3 + 16243759885500*b2 - 3247930699870302*b1 + 7093098202319052547097637) * q^75 + (-115641315810*b3 + 65419771642498*b2 + 14232220240982140*b1 - 15669488746977631055806176) * q^77 + (-24601325720*b3 - 15607436423285*b2 - 7983463162573630*b1 + 30193943637441553384236928) * q^79 + 6461081889226673298932241 * q^81 + (-294162533301*b3 + 48132209258861*b2 + 27076626619722326*b1 + 16697538792647804747924380) * q^83 + (446856257400*b3 - 564239090216200*b2 - 1251341818112062*b1 + 1291191799867050483787772) * q^85 + (332394024978*b3 + 119870890246548*b2 - 14953316795968383*b1 + 57140481255255548906311746) * q^87 + (188092661170*b3 + 245453181114946*b2 + 308413204801800*b1 - 46657475909906854786318902) * q^89 + (-676908110779*b3 + 394473663361005*b2 - 30025750893358296*b1 + 133069079711078369294861232) * q^91 + (-769079094678*b3 + 131747137791003*b2 - 51908637361797162*b1 - 6083558774628098233446576) * q^93 + (2175028961800*b3 - 1041253487429400*b2 + 41549021642966208*b1 + 187572614981561087964537352) * q^95 + (-2926923552810*b3 + 1633489885827898*b2 - 117469384385908100*b1 + 27369206681811914113422818) * q^97 + (-2541865828329*b3 + 338068155167757*b2 + 4234748469996114*b1 - 70657282602318487140746748) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6377292 q^{3} + 4949050424 q^{5} - 88249731552 q^{7} + 10167463313316 q^{9}+O(q^{10})$$ 4 * q + 6377292 * q^3 + 4949050424 * q^5 - 88249731552 * q^7 + 10167463313316 * q^9 $$4 q + 6377292 q^{3} + 4949050424 q^{5} - 88249731552 q^{7} + 10167463313316 q^{9} - 111189633716848 q^{11} - 7339657642664 q^{13} + 78\!\cdots\!52 q^{15}+ \cdots - 28\!\cdots\!92 q^{99}+O(q^{100})$$ 4 * q + 6377292 * q^3 + 4949050424 * q^5 - 88249731552 * q^7 + 10167463313316 * q^9 - 111189633716848 * q^11 - 7339657642664 * q^13 + 7890384919142952 * q^15 + 32484737251067464 * q^17 - 32881723696992400 * q^19 - 140698576757179296 * q^21 - 321501199786814176 * q^23 + 17795887539272914076 * q^25 + 16210220612075905068 * q^27 + 143359861847958158808 * q^29 - 15263052153492355648 * q^31 - 177272190396346253904 * q^33 + 300802957612005610944 * q^35 + 1115572721950557665016 * q^37 - 11701784991824996472 * q^39 + 10700585339689986789480 * q^41 + 9887391055592056200208 * q^43 + 12579822155442748661496 * q^45 + 50192255945374906090176 * q^47 + 228398298477204573852836 * q^49 + 51791163748333632406872 * q^51 + 103627112374107145482488 * q^53 - 206030255590208876474912 * q^55 - 52424088369760014145200 * q^57 - 1996721345289804944085424 * q^59 - 1086322669153439941389800 * q^61 - 224318976991236366736608 * q^63 - 6329749346647807085694256 * q^65 - 176878326996309818449360 * q^67 - 512576757347712937522848 * q^69 + 10877481227380294006167136 * q^71 + 3243145341396450784281448 * q^73 + 28372392809276210188390548 * q^75 - 62677954987910524223224704 * q^77 + 120775774549766213536947712 * q^79 + 25844327556906693195728964 * q^81 + 66790155170591218991697520 * q^83 + 5164767199468201935151088 * q^85 + 228561925021022195625246984 * q^87 - 186629903639627419145275608 * q^89 + 532276318844313477179444928 * q^91 - 24334235098512392933786304 * q^93 + 750290459926244351858149408 * q^95 + 109476826727247656453691272 * q^97 - 282629130409273948562986992 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 360714331909x^{2} - 43287560841177118x + 8819337660421091919513$$ :

 $$\beta_{1}$$ $$=$$ $$( 45950336 \nu^{3} - 24972080483328 \nu^{2} + \cdots + 30\!\cdots\!28 ) / 784193203840095$$ (45950336*v^3 - 24972080483328*v^2 - 10799286174332356736*v + 3012065727257252564401728) / 784193203840095 $$\beta_{2}$$ $$=$$ $$( 39230848 \nu^{3} + 75313256469504 \nu^{2} + \cdots - 14\!\cdots\!80 ) / 156838640768019$$ (39230848*v^3 + 75313256469504*v^2 - 151705309522009763968*v - 14856886659496624584646080) / 156838640768019 $$\beta_{3}$$ $$=$$ $$( 3643222147456 \nu^{3} + \cdots - 70\!\cdots\!92 ) / 784193203840095$$ (3643222147456*v^3 - 265175617827376128*v^2 - 1053452435345671119101056*v - 70454771880046552470162528192) / 784193203840095
 $$\nu$$ $$=$$ $$( \beta_{3} - 3549\beta_{2} - 64136\beta _1 + 1486356480 ) / 2972712960$$ (b3 - 3549*b2 - 64136*b1 + 1486356480) / 2972712960 $$\nu^{2}$$ $$=$$ $$( 147449\beta_{3} - 40817973\beta_{2} - 11516408584\beta _1 + 53615008466478563328 ) / 297271296$$ (147449*b3 - 40817973*b2 - 11516408584*b1 + 53615008466478563328) / 297271296 $$\nu^{3}$$ $$=$$ $$( 1036344307171 \beta_{3} + \cdots + 96\!\cdots\!40 ) / 2972712960$$ (1036344307171*b3 - 1055917495400439*b2 - 26927518838229656*b1 + 96512728289770781634723840) / 2972712960

## 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
 −263517. 108627. 637889. −482996.
0 1.59432e6 0 −2.94911e9 0 −2.52860e11 0 2.54187e12 0
1.2 0 1.59432e6 0 −8.07135e8 0 1.51306e11 0 2.54187e12 0
1.3 0 1.59432e6 0 3.92930e9 0 4.56279e11 0 2.54187e12 0
1.4 0 1.59432e6 0 4.77599e9 0 −4.42974e11 0 2.54187e12 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 24.28.a.d 4
4.b odd 2 1 48.28.a.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.28.a.d 4 1.a even 1 1 trivial
48.28.a.l 4 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 4949050424 T_{5}^{3} + \cdots + 44\!\cdots\!00$$ acting on $$S_{28}^{\mathrm{new}}(\Gamma_0(24))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T - 1594323)^{4}$$
$5$ $$T^{4} + \cdots + 44\!\cdots\!00$$
$7$ $$T^{4} + \cdots + 77\!\cdots\!80$$
$11$ $$T^{4} + \cdots + 41\!\cdots\!48$$
$13$ $$T^{4} + \cdots + 97\!\cdots\!48$$
$17$ $$T^{4} + \cdots + 28\!\cdots\!44$$
$19$ $$T^{4} + \cdots + 16\!\cdots\!32$$
$23$ $$T^{4} + \cdots + 48\!\cdots\!60$$
$29$ $$T^{4} + \cdots - 99\!\cdots\!84$$
$31$ $$T^{4} + \cdots + 10\!\cdots\!00$$
$37$ $$T^{4} + \cdots - 30\!\cdots\!00$$
$41$ $$T^{4} + \cdots + 26\!\cdots\!92$$
$43$ $$T^{4} + \cdots - 42\!\cdots\!76$$
$47$ $$T^{4} + \cdots + 88\!\cdots\!00$$
$53$ $$T^{4} + \cdots + 58\!\cdots\!00$$
$59$ $$T^{4} + \cdots - 27\!\cdots\!52$$
$61$ $$T^{4} + \cdots + 11\!\cdots\!92$$
$67$ $$T^{4} + \cdots + 40\!\cdots\!52$$
$71$ $$T^{4} + \cdots + 59\!\cdots\!48$$
$73$ $$T^{4} + \cdots - 70\!\cdots\!64$$
$79$ $$T^{4} + \cdots - 66\!\cdots\!44$$
$83$ $$T^{4} + \cdots + 61\!\cdots\!52$$
$89$ $$T^{4} + \cdots + 31\!\cdots\!60$$
$97$ $$T^{4} + \cdots + 93\!\cdots\!24$$