# Properties

 Label 24.22.f.a Level $24$ Weight $22$ Character orbit 24.f Analytic conductor $67.075$ Analytic rank $0$ Dimension $2$ CM discriminant -8 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [24,22,Mod(11,24)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(24, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

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

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

chi := DirichletCharacter("24.11");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 24.f (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$67.0745626289$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 1024 \beta q^{2} + (67717 \beta - 35905) q^{3} - 2097152 q^{4} + (36766720 \beta + 138684416) q^{6} + 2147483648 \beta q^{8} + ( - 4862757770 \beta - 7882015153) q^{9} +O(q^{10})$$ q - 1024*b * q^2 + (67717*b - 35905) * q^3 - 2097152 * q^4 + (36766720*b + 138684416) * q^6 + 2147483648*b * q^8 + (-4862757770*b - 7882015153) * q^9 $$q - 1024 \beta q^{2} + (67717 \beta - 35905) q^{3} - 2097152 q^{4} + (36766720 \beta + 138684416) q^{6} + 2147483648 \beta q^{8} + ( - 4862757770 \beta - 7882015153) q^{9} - 21068191850 \beta q^{11} + ( - 142012841984 \beta + 75298242560) q^{12} + 4398046511104 q^{16} - 1958571155372 \beta q^{17} + (8071183516672 \beta - 9958927912960) q^{18} - 52860065645374 q^{19} - 43147656908800 q^{22} + ( - 77105400381440 \beta - 290842300383232) q^{24} - 476837158203125 q^{25} + ( - 359149102383851 \beta + 941586489890645) q^{27} - 45\!\cdots\!96 \beta q^{32} + \cdots + (16\!\cdots\!50 \beta - 20\!\cdots\!00) q^{99} +O(q^{100})$$ q - 1024*b * q^2 + (67717*b - 35905) * q^3 - 2097152 * q^4 + (36766720*b + 138684416) * q^6 + 2147483648*b * q^8 + (-4862757770*b - 7882015153) * q^9 - 21068191850*b * q^11 + (-142012841984*b + 75298242560) * q^12 + 4398046511104 * q^16 - 1958571155372*b * q^17 + (8071183516672*b - 9958927912960) * q^18 - 52860065645374 * q^19 - 43147656908800 * q^22 + (-77105400381440*b - 290842300383232) * q^24 - 476837158203125 * q^25 + (-359149102383851*b + 941586489890645) * q^27 - 4503599627370496*b * q^32 + (756453428374250*b + 2853349495012900) * q^33 - 4011153726201856 * q^34 + (10197942182871040*b + 16529783842144256) * q^36 + 54128707220862976*b * q^38 + 83351054401637080*b * q^41 + 169093942006962710 * q^43 + 44183200674611200*b * q^44 + (297822515592429568*b - 157911859981189120) * q^48 + 558545864083284007 * q^49 + 488281250000000000*b * q^50 + (70322497333631660*b + 265257125856651448) * q^51 + (-964184565648020480*b - 735537361682126848) * q^54 + (-3579525065307791158*b + 1897940656997153470) * q^57 - 3006432882784735790*b * q^59 - 9223372036854775808 * q^64 + (-2921829882893209600*b + 1549216621310464000) * q^66 + 12900350914197686030 * q^67 + 4107421415630700544*b * q^68 + (-16926498654355718144*b + 20885385590519889920) * q^72 + 46067613696354796130 * q^73 + (-32289981842041015625*b + 17120838165283203125) * q^75 + 110855592388327374848 * q^76 + (76656660857016977620*b + 14833336612730867609) * q^81 + 170702959414552739840 * q^82 + 26207594499604744246*b * q^83 - 173152196615129815040*b * q^86 + 90487194981603737600 * q^88 + 46180916335255203220*b * q^89 + (161701744620737658880*b + 609940511933295755264) * q^96 + 1427356980098163174710 * q^97 - 571950964821282823168*b * q^98 + (166059807408011103050*b - 204899027236876349000) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 71810 q^{3} - 4194304 q^{4} + 277368832 q^{6} - 15764030306 q^{9}+O(q^{10})$$ 2 * q - 71810 * q^3 - 4194304 * q^4 + 277368832 * q^6 - 15764030306 * q^9 $$2 q - 71810 q^{3} - 4194304 q^{4} + 277368832 q^{6} - 15764030306 q^{9} + 150596485120 q^{12} + 8796093022208 q^{16} - 19917855825920 q^{18} - 105720131290748 q^{19} - 86295313817600 q^{22} - 581684600766464 q^{24} - 953674316406250 q^{25} + 18\!\cdots\!90 q^{27}+ \cdots - 40\!\cdots\!00 q^{99}+O(q^{100})$$ 2 * q - 71810 * q^3 - 4194304 * q^4 + 277368832 * q^6 - 15764030306 * q^9 + 150596485120 * q^12 + 8796093022208 * q^16 - 19917855825920 * q^18 - 105720131290748 * q^19 - 86295313817600 * q^22 - 581684600766464 * q^24 - 953674316406250 * q^25 + 1883172979781290 * q^27 + 5706698990025800 * q^33 - 8022307452403712 * q^34 + 33059567684288512 * q^36 + 338187884013925420 * q^43 - 315823719962378240 * q^48 + 1117091728166568014 * q^49 + 530514251713302896 * q^51 - 1471074723364253696 * q^54 + 3795881313994306940 * q^57 - 18446744073709551616 * q^64 + 3098433242620928000 * q^66 + 25800701828395372060 * q^67 + 41770771181039779840 * q^72 + 92135227392709592260 * q^73 + 34241676330566406250 * q^75 + 221711184776654749696 * q^76 + 29666673225461735218 * q^81 + 341405918829105479680 * q^82 + 180974389963207475200 * q^88 + 1219881023866591510528 * q^96 + 2854713960196326349420 * q^97 - 409798054473752698000 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$13$$ $$17$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 1.41421i − 1.41421i
1448.15i −35905.0 + 95766.3i −2.09715e6 0 1.38684e8 + 5.19960e7i 0 3.03700e9i −7.88202e9 6.87698e9i 0
11.2 1448.15i −35905.0 95766.3i −2.09715e6 0 1.38684e8 5.19960e7i 0 3.03700e9i −7.88202e9 + 6.87698e9i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
3.b odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.22.f.a 2
3.b odd 2 1 inner 24.22.f.a 2
8.d odd 2 1 CM 24.22.f.a 2
24.f even 2 1 inner 24.22.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.22.f.a 2 1.a even 1 1 trivial
24.22.f.a 2 3.b odd 2 1 inner
24.22.f.a 2 8.d odd 2 1 CM
24.22.f.a 2 24.f even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{22}^{\mathrm{new}}(24, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2097152$$
$3$ $$T^{2} + 71810 T + 10460353203$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 88\!\cdots\!00$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 76\!\cdots\!68$$
$19$ $$(T + 52860065645374)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 13\!\cdots\!00$$
$43$ $$(T - 16\!\cdots\!10)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 18\!\cdots\!00$$
$61$ $$T^{2}$$
$67$ $$(T - 12\!\cdots\!30)^{2}$$
$71$ $$T^{2}$$
$73$ $$(T - 46\!\cdots\!30)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 13\!\cdots\!32$$
$89$ $$T^{2} + 42\!\cdots\!00$$
$97$ $$(T - 14\!\cdots\!10)^{2}$$