# Properties

 Label 64.22.a.l Level $64$ Weight $22$ Character orbit 64.a Self dual yes Analytic conductor $178.866$ 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] = [64,22,Mod(1,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("64.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 64.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: $$178.865500344$$ 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: $$x^{3} - x^{2} - 4963x + 96223$$ x^3 - x^2 - 4963*x + 96223 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{21}\cdot 3\cdot 5\cdot 7$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 32255) q^{3} + ( - \beta_{2} - 9 \beta_1 + 8037261) q^{5} + (4 \beta_{2} - 4014 \beta_1 + 98664098) q^{7} + ( - 306 \beta_{2} - 120578 \beta_1 + 6281605939) q^{9}+O(q^{10})$$ q + (b1 - 32255) * q^3 + (-b2 - 9*b1 + 8037261) * q^5 + (4*b2 - 4014*b1 + 98664098) * q^7 + (-306*b2 - 120578*b1 + 6281605939) * q^9 $$q + (\beta_1 - 32255) q^{3} + ( - \beta_{2} - 9 \beta_1 + 8037261) q^{5} + (4 \beta_{2} - 4014 \beta_1 + 98664098) q^{7} + ( - 306 \beta_{2} - 120578 \beta_1 + 6281605939) q^{9} + ( - 2856 \beta_{2} + 241515 \beta_1 + 13444955723) q^{11} + (8551 \beta_{2} - 1219041 \beta_1 - 44577735635) q^{13} + ( - 21060 \beta_{2} + 24214790 \beta_1 - 407720224330) q^{15} + (206750 \beta_{2} - 78703794 \beta_1 + 2599270326072) q^{17} + (319144 \beta_{2} - 10179315 \beta_1 - 11929396534227) q^{19} + (1323540 \beta_{2} + 391662132 \beta_1 - 66179872171740) q^{21} + (5214236 \beta_{2} + 1679798214 \beta_1 + 64589693893622) q^{23} + (292556 \beta_{2} + 2369077164 \beta_1 + 375854023120779) q^{25} + (29609784 \beta_{2} + 11178145762 \beta_1 - 17\!\cdots\!38) q^{27}+ \cdots + (15464975405472 \beta_{2} + \cdots + 31\!\cdots\!83) q^{99}+O(q^{100})$$ q + (b1 - 32255) * q^3 + (-b2 - 9*b1 + 8037261) * q^5 + (4*b2 - 4014*b1 + 98664098) * q^7 + (-306*b2 - 120578*b1 + 6281605939) * q^9 + (-2856*b2 + 241515*b1 + 13444955723) * q^11 + (8551*b2 - 1219041*b1 - 44577735635) * q^13 + (-21060*b2 + 24214790*b1 - 407720224330) * q^15 + (206750*b2 - 78703794*b1 + 2599270326072) * q^17 + (319144*b2 - 10179315*b1 - 11929396534227) * q^19 + (1323540*b2 + 391662132*b1 - 66179872171740) * q^21 + (5214236*b2 + 1679798214*b1 + 64589693893622) * q^23 + (292556*b2 + 2369077164*b1 + 375854023120779) * q^25 + (29609784*b2 + 11178145762*b1 - 1760671157218238) * q^27 + (-33737045*b2 - 1660970493*b1 - 1869113920498543) * q^29 + (74881888*b2 - 5920500456*b1 + 3748921264001176) * q^31 + (-141916374*b2 + 36046394810*b1 + 3338037660191858) * q^33 + (83942384*b2 - 107547831684*b1 - 1758047959172804) * q^35 + (-10140941*b2 + 208805051403*b1 + 8090763662016401) * q^37 + (576660060*b2 - 68445178114*b1 - 17641746644228050) * q^39 + (1361738284*b2 - 272353907700*b1 - 99386278879320634) * q^41 + (5014813136*b2 - 85889919573*b1 + 11111339343224811) * q^43 + (2549104623*b2 - 2128341923353*b1 + 309137868773272277) * q^45 + (18460661896*b2 - 276404718564*b1 - 40291335714295108) * q^47 + (-5695384968*b2 - 1156764401352*b1 - 283467391737079375) * q^49 + (29006905464*b2 + 6370268425034*b1 - 1318131720028028758) * q^51 + (-21744109661*b2 - 43814693061*b1 + 379481145606515761) * q^53 + (-3901779356*b2 + 13244749165026*b1 + 2319179746504802226) * q^55 + (10714965606*b2 - 15939600411050*b1 + 227237828911355998) * q^57 + (-79875012000*b2 - 30637171373409*b1 - 3075197953846188001) * q^59 + (115270423775*b2 - 28283296240809*b1 + 2184976859120054997) * q^61 + (-130171243644*b2 - 79144304421414*b1 + 7261764567978521130) * q^63 + (42445710468*b2 - 51481464297948*b1 - 6904843446032282168) * q^65 + (-323347301080*b2 + 29752893747513*b1 + 5264341440545960537) * q^67 + (-389846437380*b2 - 163983745168292*b1 + 24329491431145035820) * q^69 + (-934927894380*b2 + 137472678497106*b1 - 13713239988715498270) * q^71 + (-378453267642*b2 - 93484221354954*b1 - 6474024821894165672) * q^73 + (-717970683600*b2 + 162109742402975*b1 + 25077165094461530975) * q^75 + (589853232572*b2 - 198923328266724*b1 - 22793346147343450388) * q^77 + (2322314505128*b2 + 279671030545140*b1 - 43911866990278864876) * q^79 + (485482873122*b2 - 1942146171617678*b1 + 166809246069484652275) * q^81 + (4513980072832*b2 - 529310118462219*b1 - 21337821507520406283) * q^83 + (-26161327462*b2 - 2455621500102678*b1 - 130085249047167064858) * q^85 + (-295157018772*b2 - 1203447813013314*b1 + 33966719140332693294) * q^87 + (4599855411654*b2 + 1887497379621558*b1 + 143296519800052541736) * q^89 + (-2503713552880*b2 + 482908923309420*b1 + 99060406226366062252) * q^91 + (3594910420368*b2 + 3119957848026128*b1 - 213346162989254242096) * q^93 + (9966832058044*b2 - 984050788352274*b1 - 345457140866077157474) * q^95 + (-21789767623018*b2 - 751754024242938*b1 - 108019528253374798368) * q^97 + (15464975405472*b2 - 188974335581689*b1 + 316661031554270978183) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 96764 q^{3} + 24111774 q^{5} + 295988280 q^{7} + 18844697239 q^{9}+O(q^{10})$$ 3 * q - 96764 * q^3 + 24111774 * q^5 + 295988280 * q^7 + 18844697239 * q^9 $$3 q - 96764 q^{3} + 24111774 q^{5} + 295988280 q^{7} + 18844697239 q^{9} + 40335108684 q^{11} - 133734425946 q^{13} - 1223136458200 q^{15} + 7797732274422 q^{17} - 35788199781996 q^{19} - 198539224853088 q^{21} + 193770761479080 q^{23} + 11\!\cdots\!01 q^{25}+ \cdots + 94\!\cdots\!60 q^{99}+O(q^{100})$$ 3 * q - 96764 * q^3 + 24111774 * q^5 + 295988280 * q^7 + 18844697239 * q^9 + 40335108684 * q^11 - 133734425946 * q^13 - 1223136458200 * q^15 + 7797732274422 * q^17 - 35788199781996 * q^19 - 198539224853088 * q^21 + 193770761479080 * q^23 + 1127564438439501 * q^25 - 5282002293508952 * q^27 - 5607343422466122 * q^29 + 11246757871503072 * q^31 + 10014149026970384 * q^33 - 5274251425350096 * q^35 + 24272499791100606 * q^37 - 52925308377862264 * q^39 - 298159108991869602 * q^41 + 33333932139754860 * q^43 + 927411477977893478 * q^45 - 120874283547603888 * q^47 - 850403331975639477 * q^49 - 3954388789815661240 * q^51 + 1138443393004854222 * q^53 + 6957552484263571704 * q^55 + 681697547133656944 * q^57 - 9225624498709937412 * q^59 + 6554902294063924182 * q^61 + 21785214559631141976 * q^63 - 20714581819561144452 * q^65 + 15793054074531629124 * q^67 + 72988310309689939168 * q^69 - 41139582493467997704 * q^71 - 19422167949903851970 * q^73 + 75231657393126995900 * q^75 - 68380237365358617888 * q^77 - 131735321299806049488 * q^79 + 500425796062282339147 * q^81 - 64013993832679681068 * q^83 - 390258202763001297252 * q^85 + 101898953973185066568 * q^87 + 429891446897537246766 * q^89 + 297181701588021496176 * q^91 - 640035369009914700160 * q^93 - 1036372406649019824696 * q^95 - 324059336514148638042 * q^97 + 949982905688477352860 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4963x + 96223$$ :

 $$\beta_{1}$$ $$=$$ $$64\nu^{2} + 640\nu - 211989$$ 64*v^2 + 640*v - 211989 $$\beta_{2}$$ $$=$$ $$8896\nu^{2} + 662400\nu - 29657664$$ 8896*v^2 + 662400*v - 29657664
 $$\nu$$ $$=$$ $$( \beta_{2} - 139\beta _1 + 191193 ) / 573440$$ (b2 - 139*b1 + 191193) / 573440 $$\nu^{2}$$ $$=$$ $$( -\beta_{2} + 1035\beta _1 + 189750951 ) / 57344$$ (-b2 + 1035*b1 + 189750951) / 57344

## 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
 21.2235 57.9766 −78.2002
0 −201833. 0 2.11555e7 0 7.32981e8 0 3.02761e10 0
1.2 0 7983.67 0 −3.09730e7 0 9.17385e7 0 −1.03966e10 0
1.3 0 97085.2 0 3.39292e7 0 −5.28731e8 0 −1.03483e9 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
$$2$$ $$1$$

## 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 64.22.a.l 3
4.b odd 2 1 64.22.a.m 3
8.b even 2 1 8.22.a.b 3
8.d odd 2 1 16.22.a.f 3
24.h odd 2 1 72.22.a.f 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.22.a.b 3 8.b even 2 1
16.22.a.f 3 8.d odd 2 1
64.22.a.l 3 1.a even 1 1 trivial
64.22.a.m 3 4.b odd 2 1
72.22.a.f 3 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 96764T_{3}^{2} - 20431242576T_{3} + 156439878638400$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(64))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + \cdots + 156439878638400$$
$5$ $$T^{3} - 24111774 T^{2} + \cdots + 22\!\cdots\!00$$
$7$ $$T^{3} - 295988280 T^{2} + \cdots + 35\!\cdots\!56$$
$11$ $$T^{3} - 40335108684 T^{2} + \cdots + 59\!\cdots\!44$$
$13$ $$T^{3} + 133734425946 T^{2} + \cdots + 64\!\cdots\!48$$
$17$ $$T^{3} - 7797732274422 T^{2} + \cdots + 13\!\cdots\!00$$
$19$ $$T^{3} + 35788199781996 T^{2} + \cdots + 14\!\cdots\!64$$
$23$ $$T^{3} - 193770761479080 T^{2} + \cdots + 13\!\cdots\!24$$
$29$ $$T^{3} + \cdots + 45\!\cdots\!12$$
$31$ $$T^{3} + \cdots - 23\!\cdots\!00$$
$37$ $$T^{3} + \cdots + 15\!\cdots\!16$$
$41$ $$T^{3} + \cdots + 68\!\cdots\!24$$
$43$ $$T^{3} + \cdots - 88\!\cdots\!08$$
$47$ $$T^{3} + \cdots - 79\!\cdots\!64$$
$53$ $$T^{3} + \cdots + 28\!\cdots\!76$$
$59$ $$T^{3} + \cdots - 10\!\cdots\!56$$
$61$ $$T^{3} + \cdots + 14\!\cdots\!00$$
$67$ $$T^{3} + \cdots + 42\!\cdots\!48$$
$71$ $$T^{3} + \cdots - 34\!\cdots\!00$$
$73$ $$T^{3} + \cdots - 28\!\cdots\!56$$
$79$ $$T^{3} + \cdots - 47\!\cdots\!00$$
$83$ $$T^{3} + \cdots + 44\!\cdots\!76$$
$89$ $$T^{3} + \cdots + 24\!\cdots\!76$$
$97$ $$T^{3} + \cdots + 10\!\cdots\!44$$