# Properties

 Label 6.22.a.b Level $6$ Weight $22$ Character orbit 6.a Self dual yes Analytic conductor $16.769$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6,22,Mod(1,6)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6, 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("6.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6 = 2 \cdot 3$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 6.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: $$16.7686406572$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 1024 q^{2} - 59049 q^{3} + 1048576 q^{4} + 12954174 q^{5} - 60466176 q^{6} - 479513104 q^{7} + 1073741824 q^{8} + 3486784401 q^{9}+O(q^{10})$$ q + 1024 * q^2 - 59049 * q^3 + 1048576 * q^4 + 12954174 * q^5 - 60466176 * q^6 - 479513104 * q^7 + 1073741824 * q^8 + 3486784401 * q^9 $$q + 1024 q^{2} - 59049 q^{3} + 1048576 q^{4} + 12954174 q^{5} - 60466176 q^{6} - 479513104 q^{7} + 1073741824 q^{8} + 3486784401 q^{9} + 13265074176 q^{10} + 115657781700 q^{11} - 61917364224 q^{12} + 295658246702 q^{13} - 491021418496 q^{14} - 764931020526 q^{15} + 1099511627776 q^{16} + 6626983431906 q^{17} + 3570467226624 q^{18} + 28576184164796 q^{19} + 13583435956224 q^{20} + 28314769278096 q^{21} + 118433568460800 q^{22} + 335385196791000 q^{23} - 63403380965376 q^{24} - 309026534180849 q^{25} + 302754044622848 q^{26} - 205891132094649 q^{27} - 502805932539904 q^{28} - 699224214482106 q^{29} - 783289365018624 q^{30} - 34\!\cdots\!92 q^{31}+ \cdots + 40\!\cdots\!00 q^{99}+O(q^{100})$$ q + 1024 * q^2 - 59049 * q^3 + 1048576 * q^4 + 12954174 * q^5 - 60466176 * q^6 - 479513104 * q^7 + 1073741824 * q^8 + 3486784401 * q^9 + 13265074176 * q^10 + 115657781700 * q^11 - 61917364224 * q^12 + 295658246702 * q^13 - 491021418496 * q^14 - 764931020526 * q^15 + 1099511627776 * q^16 + 6626983431906 * q^17 + 3570467226624 * q^18 + 28576184164796 * q^19 + 13583435956224 * q^20 + 28314769278096 * q^21 + 118433568460800 * q^22 + 335385196791000 * q^23 - 63403380965376 * q^24 - 309026534180849 * q^25 + 302754044622848 * q^26 - 205891132094649 * q^27 - 502805932539904 * q^28 - 699224214482106 * q^29 - 783289365018624 * q^30 - 3484957262657992 * q^31 + 1125899906842624 * q^32 - 6829476351603300 * q^33 + 6786031034271744 * q^34 - 6211696184496096 * q^35 + 3656158440062976 * q^36 - 35181531093012298 * q^37 + 29262012584751104 * q^38 - 17458323809506398 * q^39 + 13909438419173376 * q^40 + 6132056240639994 * q^41 + 28994323740770304 * q^42 + 233260850096910596 * q^43 + 121275974103859200 * q^44 + 45168411831039774 * q^45 + 343434441513984000 * q^46 - 580205712121346400 * q^47 - 64925062108545024 * q^48 - 328613047175569191 * q^49 - 316443171001189376 * q^50 - 391316744670617394 * q^51 + 310020141693796352 * q^52 - 1394471665941750306 * q^53 - 210832519264920576 * q^54 + 1498251028595815800 * q^55 - 514873274920861696 * q^56 - 1687395098747039004 * q^57 - 716005595629676544 * q^58 + 2352476807159705700 * q^59 - 802088309779070976 * q^60 + 9920628300330384590 * q^61 - 3568596236961783808 * q^62 - 1671958811102290704 * q^63 + 1152921504606846976 * q^64 + 3830008372312634148 * q^65 - 6993383784041779200 * q^66 + 26068981808996843996 * q^67 + 6948895779094265856 * q^68 - 19804160485311759000 * q^69 - 6360776892924002304 * q^70 - 13336955952504341400 * q^71 + 3743906242624487424 * q^72 + 9037529597968684202 * q^73 - 36025887839244593152 * q^74 + 18247707816844952601 * q^75 + 29964300886785130496 * q^76 - 55459421904721396800 * q^77 - 17877323580934551552 * q^78 - 77283864571811027992 * q^79 + 14243264941233537024 * q^80 + 12157665459056928801 * q^81 + 6279225590415353856 * q^82 - 155698418876868248388 * q^83 + 29690187510548791296 * q^84 + 85847096472027475644 * q^85 + 238859110499236450304 * q^86 + 41288490640953877194 * q^87 + 124186597482351820800 * q^88 + 253837312813381912218 * q^89 + 46252453714984728576 * q^90 - 141772003599273783008 * q^91 + 351676868110319616000 * q^92 + 205783241402691769608 * q^93 - 594130649212258713600 * q^94 + 370180861926812058504 * q^95 - 66483263599150104576 * q^96 - 1030722092535365121598 * q^97 - 336499760307782851584 * q^98 + 403273749085823261700 * q^99

## 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
 0
1024.00 −59049.0 1.04858e6 1.29542e7 −6.04662e7 −4.79513e8 1.07374e9 3.48678e9 1.32651e10
 $$n$$: 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 6.22.a.b 1
3.b odd 2 1 18.22.a.a 1
4.b odd 2 1 48.22.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.22.a.b 1 1.a even 1 1 trivial
18.22.a.a 1 3.b odd 2 1
48.22.a.f 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 12954174$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(6))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1024$$
$3$ $$T + 59049$$
$5$ $$T - 12954174$$
$7$ $$T + 479513104$$
$11$ $$T - 115657781700$$
$13$ $$T - 295658246702$$
$17$ $$T - 6626983431906$$
$19$ $$T - 28576184164796$$
$23$ $$T - 335385196791000$$
$29$ $$T + 699224214482106$$
$31$ $$T + 3484957262657992$$
$37$ $$T + 35\!\cdots\!98$$
$41$ $$T - 6132056240639994$$
$43$ $$T - 23\!\cdots\!96$$
$47$ $$T + 58\!\cdots\!00$$
$53$ $$T + 13\!\cdots\!06$$
$59$ $$T - 23\!\cdots\!00$$
$61$ $$T - 99\!\cdots\!90$$
$67$ $$T - 26\!\cdots\!96$$
$71$ $$T + 13\!\cdots\!00$$
$73$ $$T - 90\!\cdots\!02$$
$79$ $$T + 77\!\cdots\!92$$
$83$ $$T + 15\!\cdots\!88$$
$89$ $$T - 25\!\cdots\!18$$
$97$ $$T + 10\!\cdots\!98$$