# Properties

 Label 72.22.a.b Level $72$ Weight $22$ Character orbit 72.a Self dual yes Analytic conductor $201.224$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{358549})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 89637$$ x^2 - x - 89637 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{7}\cdot 3$$ Twist minimal: no (minimal twist has level 8) 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 $$\beta = 192\sqrt{358549}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 20 \beta - 1054070) q^{5} + (4222 \beta + 222385896) q^{7}+O(q^{10})$$ q + (-20*b - 1054070) * q^5 + (4222*b + 222385896) * q^7 $$q + ( - 20 \beta - 1054070) q^{5} + (4222 \beta + 222385896) q^{7} + ( - 527181 \beta - 26903201660) q^{11} + ( - 4314508 \beta - 245183338466) q^{13} + (30718184 \beta + 3296932336046) q^{17} + (366379451 \beta + 9651198962660) q^{19} + (944258822 \beta + 204868932888136) q^{23} + (42162800 \beta - 470439074503825) q^{25} + ( - 23192603812 \beta + 12\!\cdots\!30) q^{29}+ \cdots + (156001529627000 \beta + 39\!\cdots\!94) q^{97}+O(q^{100})$$ q + (-20*b - 1054070) * q^5 + (4222*b + 222385896) * q^7 + (-527181*b - 26903201660) * q^11 + (-4314508*b - 245183338466) * q^13 + (30718184*b + 3296932336046) * q^17 + (366379451*b + 9651198962660) * q^19 + (944258822*b + 204868932888136) * q^23 + (42162800*b - 470439074503825) * q^25 + (-23192603812*b + 1202393761072530) * q^29 + (-18061983176*b - 4344953585279584) * q^31 + (-8898001460*b - 1350500251768560) * q^35 + (-50855746684*b - 1093102048125930) * q^37 + (13526799376*b - 34089019286779338) * q^41 + (-769879839131*b + 132264826133002012) * q^43 + (2319694970380*b - 213247205779311216) * q^47 + (1877826505824*b - 273483788820065767) * q^49 + (-2918184305860*b + 1527990137794759066) * q^53 + (1093749709870*b + 167718685847412520) * q^55 + (7304311733439*b - 391712498761407212) * q^59 + (23337258872596*b - 3588639524539298546) * q^61 + (9451460216880*b + 1398984934878350380) * q^65 + (-77794800439793*b + 8337061587005769044) * q^67 + (231241233933906*b - 4724131574924358184) * q^71 + (-528252850485192*b - 5793070167251503766) * q^73 + (-230822936447696*b - 35401963412776636512) * q^77 + (355065026452268*b - 42640351109357948912) * q^79 + (591846679323509*b + 190907311020043122908) * q^83 + (-98317762929800*b - 11595580332466203700) * q^85 + (1314255056990664*b + 29871466215347989830) * q^89 + (-1994649782382620*b - 295294267388978008272) * q^91 + (-579213567168770*b - 107025816003941936920) * q^95 + (156001529627000*b + 391697330463355975394) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2108140 q^{5} + 444771792 q^{7}+O(q^{10})$$ 2 * q - 2108140 * q^5 + 444771792 * q^7 $$2 q - 2108140 q^{5} + 444771792 q^{7} - 53806403320 q^{11} - 490366676932 q^{13} + 6593864672092 q^{17} + 19302397925320 q^{19} + 409737865776272 q^{23} - 940878149007650 q^{25} + 24\!\cdots\!60 q^{29}+ \cdots + 78\!\cdots\!88 q^{97}+O(q^{100})$$ 2 * q - 2108140 * q^5 + 444771792 * q^7 - 53806403320 * q^11 - 490366676932 * q^13 + 6593864672092 * q^17 + 19302397925320 * q^19 + 409737865776272 * q^23 - 940878149007650 * q^25 + 2404787522145060 * q^29 - 8689907170559168 * q^31 - 2701000503537120 * q^35 - 2186204096251860 * q^37 - 68178038573558676 * q^41 + 264529652266004024 * q^43 - 426494411558622432 * q^47 - 546967577640131534 * q^49 + 3055980275589518132 * q^53 + 335437371694825040 * q^55 - 783424997522814424 * q^59 - 7177279049078597092 * q^61 + 2797969869756700760 * q^65 + 16674123174011538088 * q^67 - 9448263149848716368 * q^71 - 11586140334503007532 * q^73 - 70803926825553273024 * q^77 - 85280702218715897824 * q^79 + 381814622040086245816 * q^83 - 23191160664932407400 * q^85 + 59742932430695979660 * q^89 - 590588534777956016544 * q^91 - 214051632007883873840 * q^95 + 783394660926711950788 * q^97

## 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
 299.895 −298.895
0 0 0 −3.35342e6 0 7.07779e8 0 0 0
1.2 0 0 0 1.24528e6 0 −2.63007e8 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.b 2
3.b odd 2 1 8.22.a.a 2
12.b even 2 1 16.22.a.e 2
24.f even 2 1 64.22.a.h 2
24.h odd 2 1 64.22.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.22.a.a 2 3.b odd 2 1
16.22.a.e 2 12.b even 2 1
64.22.a.h 2 24.f even 2 1
64.22.a.k 2 24.h odd 2 1
72.22.a.b 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 2108140T_{5} - 4175956569500$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(72))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2108140 T - 4175956569500$$
$7$ $$T^{2} - 444771792 T - 18\!\cdots\!08$$
$11$ $$T^{2} + 53806403320 T - 29\!\cdots\!96$$
$13$ $$T^{2} + 490366676932 T - 18\!\cdots\!48$$
$17$ $$T^{2} - 6593864672092 T - 16\!\cdots\!00$$
$19$ $$T^{2} - 19302397925320 T - 16\!\cdots\!36$$
$23$ $$T^{2} - 409737865776272 T + 30\!\cdots\!72$$
$29$ $$T^{2} + \cdots - 56\!\cdots\!84$$
$31$ $$T^{2} + \cdots + 14\!\cdots\!20$$
$37$ $$T^{2} + \cdots - 32\!\cdots\!16$$
$41$ $$T^{2} + \cdots + 11\!\cdots\!08$$
$43$ $$T^{2} + \cdots + 96\!\cdots\!48$$
$47$ $$T^{2} + \cdots - 25\!\cdots\!44$$
$53$ $$T^{2} + \cdots + 22\!\cdots\!56$$
$59$ $$T^{2} + \cdots - 55\!\cdots\!12$$
$61$ $$T^{2} + \cdots + 56\!\cdots\!40$$
$67$ $$T^{2} + \cdots - 10\!\cdots\!28$$
$71$ $$T^{2} + \cdots - 68\!\cdots\!40$$
$73$ $$T^{2} + \cdots - 36\!\cdots\!48$$
$79$ $$T^{2} + \cdots + 15\!\cdots\!80$$
$83$ $$T^{2} + \cdots + 31\!\cdots\!48$$
$89$ $$T^{2} + \cdots - 21\!\cdots\!56$$
$97$ $$T^{2} + \cdots + 15\!\cdots\!36$$