# Properties

 Label 9.22.a.f Level $9$ Weight $22$ Character orbit 9.a Self dual yes Analytic conductor $25.153$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,22,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 9.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: $$25.1529609858$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3085})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 771$$ x^2 - x - 771 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ 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 $$\beta = 36\sqrt{3085}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + 1901008 q^{4} + 40 \beta q^{5} - 1019030740 q^{7} + 196144 \beta q^{8} +O(q^{10})$$ q - b * q^2 + 1901008 * q^4 + 40*b * q^5 - 1019030740 * q^7 + 196144*b * q^8 $$q - \beta q^{2} + 1901008 q^{4} + 40 \beta q^{5} - 1019030740 q^{7} + 196144 \beta q^{8} - 159926400 q^{10} + 35120480 \beta q^{11} + 315303754610 q^{13} + 1019030740 \beta q^{14} - 4770917824256 q^{16} + 7440440592 \beta q^{17} + 18274662257384 q^{19} + 76040320 \beta q^{20} - 140417298316800 q^{22} - 148975726016 \beta q^{23} - 476830761147125 q^{25} - 315303754610 \beta q^{26} - 19\!\cdots\!20 q^{28} + \cdots - 47\!\cdots\!93 \beta q^{98} +O(q^{100})$$ q - b * q^2 + 1901008 * q^4 + 40*b * q^5 - 1019030740 * q^7 + 196144*b * q^8 - 159926400 * q^10 + 35120480*b * q^11 + 315303754610 * q^13 + 1019030740*b * q^14 - 4770917824256 * q^16 + 7440440592*b * q^17 + 18274662257384 * q^19 + 76040320*b * q^20 - 140417298316800 * q^22 - 148975726016*b * q^23 - 476830761147125 * q^25 - 315303754610*b * q^26 - 1937185588985920 * q^28 + 875115102680*b * q^29 - 4895670932287252 * q^31 + 4359574042368*b * q^32 - 29748071957310720 * q^34 - 40761229600*b * q^35 + 10829421922473830 * q^37 - 18274662257384*b * q^38 + 31368603801600 * q^40 - 52643783782160*b * q^41 - 55629090972850840 * q^43 + 66764313443840*b * q^44 + 595628788728130560 * q^46 + 109544410508096*b * q^47 + 479877784981663593 * q^49 + 476830761147125*b * q^50 + 599394959943646880 * q^52 - 824250418858152*b * q^53 + 5616691932672000 * q^55 - 199876765466560*b * q^56 - 3498850198931068800 * q^58 - 3222944112793280*b * q^59 - 516377057302468258 * q^61 + 4895670932287252*b * q^62 - 7424934696259923968 * q^64 + 12612150184400*b * q^65 - 25495243630423487440 * q^67 + 14144337088916736*b * q^68 + 162969917737536000 * q^70 - 12065177363585280*b * q^71 + 59948603086371356630 * q^73 - 10829421922473830*b * q^74 + 34740279148585043072 * q^76 - 35788848723555200*b * q^77 - 35815879987560028564 * q^79 - 190836712970240*b * q^80 + 210478270566480825600 * q^82 + 52172010553013792*b * q^83 + 1189922878292428800 * q^85 + 55629090972850840*b * q^86 + 27542010561050419200 * q^88 + 210035268343744800*b * q^89 - 321304218385006711400 * q^91 - 283204046962224128*b * q^92 - 437976080317049103360 * q^94 + 730986490295360*b * q^95 - 678228467968157723890 * q^97 - 479877784981663593*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3802016 q^{4} - 2038061480 q^{7}+O(q^{10})$$ 2 * q + 3802016 * q^4 - 2038061480 * q^7 $$2 q + 3802016 q^{4} - 2038061480 q^{7} - 319852800 q^{10} + 630607509220 q^{13} - 9541835648512 q^{16} + 36549324514768 q^{19} - 280834596633600 q^{22} - 953661522294250 q^{25} - 38\!\cdots\!40 q^{28}+ \cdots - 13\!\cdots\!80 q^{97}+O(q^{100})$$ 2 * q + 3802016 * q^4 - 2038061480 * q^7 - 319852800 * q^10 + 630607509220 * q^13 - 9541835648512 * q^16 + 36549324514768 * q^19 - 280834596633600 * q^22 - 953661522294250 * q^25 - 3874371177971840 * q^28 - 9791341864574504 * q^31 - 59496143914621440 * q^34 + 21658843844947660 * q^37 + 62737207603200 * q^40 - 111258181945701680 * q^43 + 1191257577456261120 * q^46 + 959755569963327186 * q^49 + 1198789919887293760 * q^52 + 11233383865344000 * q^55 - 6997700397862137600 * q^58 - 1032754114604936516 * q^61 - 14849869392519847936 * q^64 - 50990487260846974880 * q^67 + 325939835475072000 * q^70 + 119897206172742713260 * q^73 + 69480558297170086144 * q^76 - 71631759975120057128 * q^79 + 420956541132961651200 * q^82 + 2379845756584857600 * q^85 + 55084021122100838400 * q^88 - 642608436770013422800 * q^91 - 875952160634098206720 * q^94 - 1356456935936315447780 * 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
 28.2714 −27.2714
−1999.54 0 1.90101e6 79981.6 0 −1.01903e9 3.92198e8 0 −1.59926e8
1.2 1999.54 0 1.90101e6 −79981.6 0 −1.01903e9 −3.92198e8 0 −1.59926e8
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.22.a.f 2
3.b odd 2 1 inner 9.22.a.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.22.a.f 2 1.a even 1 1 trivial
9.22.a.f 2 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3998160$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3998160$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 6397056000$$
$7$ $$(T + 1019030740)^{2}$$
$11$ $$T^{2} - 49\!\cdots\!00$$
$13$ $$(T - 315303754610)^{2}$$
$17$ $$T^{2} - 22\!\cdots\!40$$
$19$ $$(T - 18274662257384)^{2}$$
$23$ $$T^{2} - 88\!\cdots\!60$$
$29$ $$T^{2} - 30\!\cdots\!00$$
$31$ $$(T + 48\!\cdots\!52)^{2}$$
$37$ $$(T - 10\!\cdots\!30)^{2}$$
$41$ $$T^{2} - 11\!\cdots\!00$$
$43$ $$(T + 55\!\cdots\!40)^{2}$$
$47$ $$T^{2} - 47\!\cdots\!60$$
$53$ $$T^{2} - 27\!\cdots\!40$$
$59$ $$T^{2} - 41\!\cdots\!00$$
$61$ $$(T + 51\!\cdots\!58)^{2}$$
$67$ $$(T + 25\!\cdots\!40)^{2}$$
$71$ $$T^{2} - 58\!\cdots\!00$$
$73$ $$(T - 59\!\cdots\!30)^{2}$$
$79$ $$(T + 35\!\cdots\!64)^{2}$$
$83$ $$T^{2} - 10\!\cdots\!40$$
$89$ $$T^{2} - 17\!\cdots\!00$$
$97$ $$(T + 67\!\cdots\!90)^{2}$$
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