# Properties

 Label 72.22.a.c Level $72$ Weight $22$ Character orbit 72.a Self dual yes Analytic conductor $201.224$ Analytic rank $1$ 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] = [72,22,Mod(1,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, 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("72.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 72.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: $$201.223687887$$ Analytic rank: $$1$$ 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: no (minimal twist has level 24) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1760166) q^{5} + ( - \beta_{2} - 7 \beta_1 + 284180792) q^{7}+O(q^{10})$$ q + (b1 - 1760166) * q^5 + (-b2 - 7*b1 + 284180792) * q^7 $$q + (\beta_1 - 1760166) q^{5} + ( - \beta_{2} - 7 \beta_1 + 284180792) q^{7} + (56 \beta_{2} + 1042 \beta_1 - 20830252556) q^{11} + ( - 433 \beta_{2} - 11856 \beta_1 + 67921735934) q^{13} + ( - 1730 \beta_{2} + 209590 \beta_1 - 231942606642) q^{17} + (32642 \beta_{2} - 316806 \beta_1 + 1651834707732) q^{19} + (6426 \beta_{2} - 1368418 \beta_1 - 50289135979384) q^{23} + ( - 245830 \beta_{2} + 13033992 \beta_1 + 226064951656623) q^{25} + ( - 430222 \beta_{2} - 61470129 \beta_1 - 10\!\cdots\!82) q^{29}+ \cdots + ( - 463090832696 \beta_{2} + \cdots + 24\!\cdots\!34) q^{97}+O(q^{100})$$ q + (b1 - 1760166) * q^5 + (-b2 - 7*b1 + 284180792) * q^7 + (56*b2 + 1042*b1 - 20830252556) * q^11 + (-433*b2 - 11856*b1 + 67921735934) * q^13 + (-1730*b2 + 209590*b1 - 231942606642) * q^17 + (32642*b2 - 316806*b1 + 1651834707732) * q^19 + (6426*b2 - 1368418*b1 - 50289135979384) * q^23 + (-245830*b2 + 13033992*b1 + 226064951656623) * q^25 + (-430222*b2 - 61470129*b1 - 1080741635075582) * q^29 + (5096903*b2 - 118127179*b1 + 217502867183632) * q^31 + (20035300*b2 + 489193950*b1 - 5147906428613200) * q^35 + (25948165*b2 + 303691330*b1 - 6897757278972074) * q^37 + (2710366*b2 - 3823722438*b1 + 15869533437517334) * q^41 + (90491010*b2 + 818298070*b1 - 92370070410138516) * q^43 + (-131505078*b2 - 14526562046*b1 + 178647502442073776) * q^47 + (-503595526*b2 - 27246071132*b1 + 397607484745538585) * q^49 + (-28257918*b2 - 8579960701*b1 - 949765964116329142) * q^53 + (-1281766300*b2 - 22694786704*b1 + 751808513301585864) * q^55 + (-2103421764*b2 - 140296579248*b1 - 293682125811575516) * q^59 + (3217443397*b2 + 137208913854*b1 + 3540296818538517774) * q^61 + (10844734650*b2 + 26133988998*b1 - 8307777732172287668) * q^65 + (-6734645628*b2 - 420729628496*b1 + 4946693576289238372) * q^67 + (-27636133578*b2 + 854195460654*b1 - 4869648279437313832) * q^71 + (26166463726*b2 + 754279335932*b1 + 4525028754649763818) * q^73 + (44996314760*b2 + 1870160193020*b1 - 57962662430443445920) * q^77 + (30214392091*b2 + 1758709572937*b1 + 92698859172585182720) * q^79 + (66318799972*b2 - 1852553129246*b1 - 195238972761152276820) * q^83 + (-19839442000*b2 + 3402594985298*b1 + 147514264941236199532) * q^85 + (-169189087252*b2 + 7898004310036*b1 - 159272900597554175610) * q^89 + (-324206373306*b2 - 15620051325242*b1 + 439363887754558846352) * q^91 + (-519941163600*b2 - 13107459153104*b1 - 232800325849195135736) * q^95 + (-463090832696*b2 + 5264260583428*b1 + 240803913337907196834) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 5280498 q^{5} + 852542376 q^{7}+O(q^{10})$$ 3 * q - 5280498 * q^5 + 852542376 * q^7 $$3 q - 5280498 q^{5} + 852542376 q^{7} - 62490757668 q^{11} + 203765207802 q^{13} - 695827819926 q^{17} + 4955504123196 q^{19} - 150867407938152 q^{23} + 678194854969869 q^{25} - 32\!\cdots\!46 q^{29}+ \cdots + 72\!\cdots\!02 q^{97}+O(q^{100})$$ 3 * q - 5280498 * q^5 + 852542376 * q^7 - 62490757668 * q^11 + 203765207802 * q^13 - 695827819926 * q^17 + 4955504123196 * q^19 - 150867407938152 * q^23 + 678194854969869 * q^25 - 3242224905226746 * q^29 + 652508601550896 * q^31 - 15443719285839600 * q^35 - 20693271836916222 * q^37 + 47608600312552002 * q^41 - 277110211230415548 * q^43 + 535942507326221328 * q^47 + 1192822454236615755 * q^49 - 2849297892348987426 * q^53 + 2255425539904757592 * q^55 - 881046377434726548 * q^59 + 10620890455615553322 * q^61 - 24923333196516863004 * q^65 + 14840080728867715116 * q^67 - 14608944838311941496 * q^71 + 13575086263949291454 * q^73 - 173887987291330337760 * q^77 + 278096577517755548160 * q^79 - 585716918283456830460 * q^83 + 442542794823708598596 * q^85 - 477818701792662526830 * q^89 + 1318091663263676539056 * q^91 - 698400977547585407208 * q^95 + 722411740013721590502 * q^97

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
 −1949.23 4086.16 −2135.93
0 0 0 −2.53316e7 0 1.44995e9 0 0 0
1.2 0 0 0 −1.51338e7 0 −8.40758e8 0 0 0
1.3 0 0 0 3.51850e7 0 2.43346e8 0 0 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 72.22.a.c 3
3.b odd 2 1 24.22.a.d 3
12.b even 2 1 48.22.a.j 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.22.a.d 3 3.b odd 2 1
48.22.a.j 3 12.b even 2 1
72.22.a.c 3 1.a even 1 1 trivial

## 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 $$S_{22}^{\mathrm{new}}(\Gamma_0(72))$$.

## Hecke characteristic polynomials

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