# Properties

 Label 24.22.f.b Level $24$ Weight $22$ Character orbit 24.f Analytic conductor $67.075$ Analytic rank $0$ Dimension $80$ CM no 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: $$80$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 71808 q^{3} + 2982260 q^{4} - 346652136 q^{6} + 15764030304 q^{9}+O(q^{10})$$ 80 * q + 71808 * q^3 + 2982260 * q^4 - 346652136 * q^6 + 15764030304 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 71808 q^{3} + 2982260 q^{4} - 346652136 q^{6} + 15764030304 q^{9} - 8532785112 q^{10} - 563812835292 q^{12} - 12550257321688 q^{16} + 23322230265720 q^{18} + 59712367669312 q^{19} + 67957211435872 q^{22} + 488194083606216 q^{24} + 80\!\cdots\!96 q^{25}+ \cdots + 19\!\cdots\!44 q^{99}+O(q^{100})$$ 80 * q + 71808 * q^3 + 2982260 * q^4 - 346652136 * q^6 + 15764030304 * q^9 - 8532785112 * q^10 - 563812835292 * q^12 - 12550257321688 * q^16 + 23322230265720 * q^18 + 59712367669312 * q^19 + 67957211435872 * q^22 + 488194083606216 * q^24 + 8010864257812496 * q^25 - 3030912775631808 * q^27 - 513917837541840 * q^28 - 4895417474723568 * q^30 - 11065770169999248 * q^33 - 34905338787330896 * q^34 - 127617605449326660 * q^36 - 124593160320877344 * q^40 + 160682116306893864 * q^42 - 692082307682152640 * q^43 + 538923820601694240 * q^46 + 1483584891764811192 * q^48 - 7341952695930987568 * q^49 - 2882918345847326592 * q^51 - 260212440736095456 * q^52 + 5258554078654363176 * q^54 - 5485164827868902352 * q^57 + 17620328507958022728 * q^58 + 26442181372193439216 * q^60 + 74447721991404798704 * q^64 - 60948251998564042584 * q^66 + 1506920368457616640 * q^67 - 84796009333194281808 * q^70 - 138074428186734782256 * q^72 - 56244572703277703072 * q^73 - 4985308401855910080 * q^75 - 484517825624256411272 * q^76 + 284064012590825039376 * q^78 - 38379817742148850608 * q^81 - 163069147218078286208 * q^82 + 306566734178567191200 * q^84 + 327139332144232703536 * q^88 - 1128285545319952095672 * q^90 + 263337661282428114624 * q^91 - 1918272831310264652736 * q^94 - 2782570738011293928240 * q^96 - 2011688553228787103168 * q^97 + 1946221033902479530944 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1447.69 36.5854i 83218.8 59455.7i 2.09448e6 + 105929.i 2.15453e7 −1.22650e8 + 8.30290e7i 7.78742e8i −3.02828e9 2.29979e8i 3.39039e9 9.89567e9i −3.11909e10 7.88241e8i
11.2 −1447.69 + 36.5854i 83218.8 + 59455.7i 2.09448e6 105929.i 2.15453e7 −1.22650e8 8.30290e7i 7.78742e8i −3.02828e9 + 2.29979e8i 3.39039e9 + 9.89567e9i −3.11909e10 + 7.88241e8i
11.3 −1447.32 49.2747i 27542.1 98497.6i 2.09230e6 + 142632.i −3.60058e7 −4.47156e7 + 1.41200e8i 5.30004e8i −3.02119e9 3.09531e8i −8.94322e9 5.42566e9i 5.21117e10 + 1.77417e9i
11.4 −1447.32 + 49.2747i 27542.1 + 98497.6i 2.09230e6 142632.i −3.60058e7 −4.47156e7 1.41200e8i 5.30004e8i −3.02119e9 + 3.09531e8i −8.94322e9 + 5.42566e9i 5.21117e10 1.77417e9i
11.5 −1432.54 212.110i −18617.4 100567.i 2.00717e6 + 607710.i 1.24708e7 5.33885e6 + 1.48015e8i 1.44834e9i −2.74644e9 1.29631e9i −9.76714e9 + 3.74460e9i −1.78649e10 2.64519e9i
11.6 −1432.54 + 212.110i −18617.4 + 100567.i 2.00717e6 607710.i 1.24708e7 5.33885e6 1.48015e8i 1.44834e9i −2.74644e9 + 1.29631e9i −9.76714e9 3.74460e9i −1.78649e10 + 2.64519e9i
11.7 −1407.10 342.382i −94251.2 39712.2i 1.86270e6 + 963530.i −1.78035e7 1.19024e8 + 8.81488e7i 2.30554e8i −2.29111e9 1.99354e9i 7.30624e9 + 7.48584e9i 2.50513e10 + 6.09560e9i
11.8 −1407.10 + 342.382i −94251.2 + 39712.2i 1.86270e6 963530.i −1.78035e7 1.19024e8 8.81488e7i 2.30554e8i −2.29111e9 + 1.99354e9i 7.30624e9 7.48584e9i 2.50513e10 6.09560e9i
11.9 −1356.69 506.515i −91907.9 + 44869.7i 1.58404e6 + 1.37436e6i 3.41502e7 1.47417e8 1.43212e7i 1.23766e8i −1.45290e9 2.66692e9i 6.43378e9 8.24775e9i −4.63311e10 1.72976e10i
11.10 −1356.69 + 506.515i −91907.9 44869.7i 1.58404e6 1.37436e6i 3.41502e7 1.47417e8 + 1.43212e7i 1.23766e8i −1.45290e9 + 2.66692e9i 6.43378e9 + 8.24775e9i −4.63311e10 + 1.72976e10i
11.11 −1278.89 679.406i −43276.9 + 92668.5i 1.17397e6 + 1.73777e6i −3.88324e6 1.18306e8 8.91103e7i 7.66700e8i −3.20723e8 3.02002e9i −6.71457e9 8.02082e9i 4.96624e9 + 2.63830e9i
11.12 −1278.89 + 679.406i −43276.9 92668.5i 1.17397e6 1.73777e6i −3.88324e6 1.18306e8 + 8.91103e7i 7.66700e8i −3.20723e8 + 3.02002e9i −6.71457e9 + 8.02082e9i 4.96624e9 2.63830e9i
11.13 −1259.59 714.555i 99544.4 23479.2i 1.07597e6 + 1.80009e6i −1.95847e7 −1.42162e8 4.15557e7i 3.09866e8i −6.90211e7 3.03622e9i 9.35780e9 4.67445e9i 2.46687e10 + 1.39943e10i
11.14 −1259.59 + 714.555i 99544.4 + 23479.2i 1.07597e6 1.80009e6i −1.95847e7 −1.42162e8 + 4.15557e7i 3.09866e8i −6.90211e7 + 3.03622e9i 9.35780e9 + 4.67445e9i 2.46687e10 1.39943e10i
11.15 −1239.13 749.479i 71611.8 + 73021.3i 973714. + 1.85740e6i −183918. −3.40080e7 1.44154e8i 8.83191e8i 1.85527e8 3.03133e9i −2.03868e8 + 1.04584e10i 2.27898e8 + 1.37843e8i
11.16 −1239.13 + 749.479i 71611.8 73021.3i 973714. 1.85740e6i −183918. −3.40080e7 + 1.44154e8i 8.83191e8i 1.85527e8 + 3.03133e9i −2.03868e8 1.04584e10i 2.27898e8 1.37843e8i
11.17 −1162.26 863.883i 2559.62 102244.i 604564. + 2.00812e6i 2.80198e7 −9.13017e7 + 1.16623e8i 6.76763e8i 1.03212e9 2.85624e9i −1.04472e10 5.23411e8i −3.25664e10 2.42058e10i
11.18 −1162.26 + 863.883i 2559.62 + 102244.i 604564. 2.00812e6i 2.80198e7 −9.13017e7 1.16623e8i 6.76763e8i 1.03212e9 + 2.85624e9i −1.04472e10 + 5.23411e8i −3.25664e10 + 2.42058e10i
11.19 −1052.87 994.292i −76264.6 68147.4i 119920. + 2.09372e6i 1.46521e6 1.25383e7 + 1.47580e8i 5.38101e8i 1.95551e9 2.32365e9i 1.17221e9 + 1.03945e10i −1.54268e9 1.45685e9i
11.20 −1052.87 + 994.292i −76264.6 + 68147.4i 119920. 2.09372e6i 1.46521e6 1.25383e7 1.47580e8i 5.38101e8i 1.95551e9 + 2.32365e9i 1.17221e9 1.03945e10i −1.54268e9 + 1.45685e9i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.80 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d 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.b 80
3.b odd 2 1 inner 24.22.f.b 80
8.d odd 2 1 inner 24.22.f.b 80
24.f even 2 1 inner 24.22.f.b 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.22.f.b 80 1.a even 1 1 trivial
24.22.f.b 80 3.b odd 2 1 inner
24.22.f.b 80 8.d odd 2 1 inner
24.22.f.b 80 24.f even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{40} + \cdots + 15\!\cdots\!00$$ acting on $$S_{22}^{\mathrm{new}}(24, [\chi])$$.