# Properties

 Label 24.22.d.a Level $24$ Weight $22$ Character orbit 24.d Analytic conductor $67.075$ Analytic rank $0$ Dimension $42$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [24,22,Mod(13,24)]

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([0, 1, 0]))

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

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

chi := DirichletCharacter("24.13");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 24.d (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: $$42$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$42 q - 574 q^{2} + 392288 q^{4} - 44759142 q^{6} - 1129900996 q^{7} - 2157413692 q^{8} - 146444944842 q^{9}+O(q^{10})$$ 42 * q - 574 * q^2 + 392288 * q^4 - 44759142 * q^6 - 1129900996 * q^7 - 2157413692 * q^8 - 146444944842 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$42 q - 574 q^{2} + 392288 q^{4} - 44759142 q^{6} - 1129900996 q^{7} - 2157413692 q^{8} - 146444944842 q^{9} + 31553345052 q^{10} - 274707993996 q^{12} + 1565450468972 q^{14} + 2306601562500 q^{15} - 3863799224840 q^{16} - 3481428994004 q^{17} + 2001414246174 q^{18} + 72162740641304 q^{20} - 372001656147968 q^{22} - 174381110476664 q^{23} - 137569089593988 q^{24} - 37\!\cdots\!46 q^{25}+ \cdots + 13\!\cdots\!98 q^{98}+O(q^{100})$$ 42 * q - 574 * q^2 + 392288 * q^4 - 44759142 * q^6 - 1129900996 * q^7 - 2157413692 * q^8 - 146444944842 * q^9 + 31553345052 * q^10 - 274707993996 * q^12 + 1565450468972 * q^14 + 2306601562500 * q^15 - 3863799224840 * q^16 - 3481428994004 * q^17 + 2001414246174 * q^18 + 72162740641304 * q^20 - 372001656147968 * q^22 - 174381110476664 * q^23 - 137569089593988 * q^24 - 3726960758912246 * q^25 + 1821164548445192 * q^26 + 6111351944921536 * q^28 - 3695078269676844 * q^30 + 13907033543056644 * q^31 + 20291511717782536 * q^32 - 26951524530024580 * q^34 - 1367823679099488 * q^36 + 152044668976276888 * q^38 - 65123248681532808 * q^39 + 48530287268994440 * q^40 + 85901867185267604 * q^41 + 164202494061337308 * q^42 - 583729140892614912 * q^44 - 425500141029899224 * q^46 - 1351800616447161624 * q^47 - 375977886927896448 * q^48 + 3990677511431283522 * q^49 + 3155333405061952034 * q^50 + 1393522695028139216 * q^52 + 156065478127743942 * q^54 - 89167753774449664 * q^55 - 1787796305200178632 * q^56 - 2098638974607729624 * q^57 - 7254382453095841940 * q^58 - 6972548942016787968 * q^60 - 15146072229522685484 * q^62 + 3939721167527163396 * q^63 + 34964359458467535488 * q^64 - 12603807792794356144 * q^65 - 43379940017819256192 * q^66 + 41027785784298520512 * q^68 + 17400423406741356136 * q^70 - 101755763943373821288 * q^71 + 7522436407769418492 * q^72 + 7031925768882255172 * q^73 + 156212595524575515440 * q^74 + 120422011169859891760 * q^76 + 29908411169744718936 * q^78 + 260653560486670585748 * q^79 - 168480498883332591744 * q^80 + 510621949280391009642 * q^81 - 354029829364220187068 * q^82 - 220644844170169622760 * q^84 + 49699114310265976040 * q^86 - 298108097029722826188 * q^87 + 184453501508512377472 * q^88 - 493168983092689682020 * q^89 - 110019711326684133852 * q^90 + 461231995073255535744 * q^92 + 690723142138680165384 * q^94 + 2751494378446039662064 * q^95 + 1069806011881439468808 * q^96 - 227026944571268558932 * q^97 + 1371753640376248773498 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −1432.06 215.300i 59049.0i 2.00444e6 + 616646.i 3.89605e7i −1.27133e7 + 8.45618e7i −5.76846e8 −2.73772e9 1.31463e9i −3.48678e9 8.38821e9 5.57938e10i
13.2 −1432.06 + 215.300i 59049.0i 2.00444e6 616646.i 3.89605e7i −1.27133e7 8.45618e7i −5.76846e8 −2.73772e9 + 1.31463e9i −3.48678e9 8.38821e9 + 5.57938e10i
13.3 −1425.38 255.822i 59049.0i 1.96626e6 + 729286.i 6.52910e6i −1.51060e7 + 8.41672e7i −1.01290e8 −2.61610e9 1.54252e9i −3.48678e9 −1.67029e9 + 9.30645e9i
13.4 −1425.38 + 255.822i 59049.0i 1.96626e6 729286.i 6.52910e6i −1.51060e7 8.41672e7i −1.01290e8 −2.61610e9 + 1.54252e9i −3.48678e9 −1.67029e9 9.30645e9i
13.5 −1327.20 579.385i 59049.0i 1.42578e6 + 1.53792e6i 3.76014e7i 3.42121e7 7.83699e7i 3.92911e8 −1.00125e9 2.86721e9i −3.48678e9 2.17857e10 4.99046e10i
13.6 −1327.20 + 579.385i 59049.0i 1.42578e6 1.53792e6i 3.76014e7i 3.42121e7 + 7.83699e7i 3.92911e8 −1.00125e9 + 2.86721e9i −3.48678e9 2.17857e10 + 4.99046e10i
13.7 −1304.11 629.633i 59049.0i 1.30428e6 + 1.64223e6i 1.97665e7i 3.71792e7 7.70066e7i 1.00011e9 −6.66923e8 2.96287e9i −3.48678e9 −1.24457e10 + 2.57778e10i
13.8 −1304.11 + 629.633i 59049.0i 1.30428e6 1.64223e6i 1.97665e7i 3.71792e7 + 7.70066e7i 1.00011e9 −6.66923e8 + 2.96287e9i −3.48678e9 −1.24457e10 2.57778e10i
13.9 −1040.85 1006.87i 59049.0i 69588.1 + 2.09600e6i 3.31530e7i −5.94545e7 + 6.14612e7i −9.65383e8 2.03796e9 2.25169e9i −3.48678e9 −3.33807e10 + 3.45073e10i
13.10 −1040.85 + 1006.87i 59049.0i 69588.1 2.09600e6i 3.31530e7i −5.94545e7 6.14612e7i −9.65383e8 2.03796e9 + 2.25169e9i −3.48678e9 −3.33807e10 3.45073e10i
13.11 −1000.56 1046.91i 59049.0i −94895.0 + 2.09500e6i 1.17189e7i −6.18191e7 + 5.90823e7i 8.98360e8 2.28823e9 1.99684e9i −3.48678e9 1.22686e10 1.17255e10i
13.12 −1000.56 + 1046.91i 59049.0i −94895.0 2.09500e6i 1.17189e7i −6.18191e7 5.90823e7i 8.98360e8 2.28823e9 + 1.99684e9i −3.48678e9 1.22686e10 + 1.17255e10i
13.13 −975.110 1070.66i 59049.0i −195474. + 2.08802e6i 1.04371e6i 6.32214e7 5.75793e7i −9.20837e8 2.42617e9 1.82676e9i −3.48678e9 1.11746e9 1.01773e9i
13.14 −975.110 + 1070.66i 59049.0i −195474. 2.08802e6i 1.04371e6i 6.32214e7 + 5.75793e7i −9.20837e8 2.42617e9 + 1.82676e9i −3.48678e9 1.11746e9 + 1.01773e9i
13.15 −745.078 1241.78i 59049.0i −986870. + 1.85044e6i 1.87294e7i 7.33257e7 4.39961e7i 4.95293e8 3.03313e9 1.53250e8i −3.48678e9 −2.32578e10 + 1.39549e10i
13.16 −745.078 + 1241.78i 59049.0i −986870. 1.85044e6i 1.87294e7i 7.33257e7 + 4.39961e7i 4.95293e8 3.03313e9 + 1.53250e8i −3.48678e9 −2.32578e10 1.39549e10i
13.17 −299.073 1416.94i 59049.0i −1.91826e6 + 847535.i 2.45935e7i −8.36686e7 + 1.76600e7i −1.36613e9 1.77460e9 + 2.46458e9i −3.48678e9 3.48475e10 7.35527e9i
13.18 −299.073 + 1416.94i 59049.0i −1.91826e6 847535.i 2.45935e7i −8.36686e7 1.76600e7i −1.36613e9 1.77460e9 2.46458e9i −3.48678e9 3.48475e10 + 7.35527e9i
13.19 −225.622 1430.47i 59049.0i −1.99534e6 + 645491.i 1.24189e7i −8.44679e7 + 1.33227e7i 4.04688e8 1.37355e9 + 2.70864e9i −3.48678e9 −1.77648e10 + 2.80197e9i
13.20 −225.622 + 1430.47i 59049.0i −1.99534e6 645491.i 1.24189e7i −8.44679e7 1.33227e7i 4.04688e8 1.37355e9 2.70864e9i −3.48678e9 −1.77648e10 2.80197e9i
See all 42 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.42 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.b 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.d.a 42
8.b even 2 1 inner 24.22.d.a 42

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.22.d.a 42 1.a even 1 1 trivial
24.22.d.a 42 8.b even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{22}^{\mathrm{new}}(24, [\chi])$$.