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Properties

Label 19.4.e.a
Level $19$
Weight $4$
Character orbit 19.e
Analytic conductor $1.121$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [19,4,Mod(4,19)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("19.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(19, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 19.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.12103629011\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} - 3 q^{3} - 24 q^{4} - 6 q^{5} + 42 q^{6} + 3 q^{7} - 75 q^{8} - 51 q^{9} + 75 q^{10} + 39 q^{11} - 219 q^{12} - 156 q^{13} + 93 q^{14} - 192 q^{15} + 504 q^{16} + 12 q^{17} + 264 q^{18}+ \cdots + 492 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −5.03941 − 1.83419i 0.944789 + 5.35816i 15.9030 + 13.3442i −7.64679 + 6.41642i 5.06673 − 28.7349i −9.43070 + 16.3344i −34.2145 − 59.2612i −2.44557 + 0.890116i 50.3042 − 18.3092i
4.2 −2.16911 − 0.789492i −0.930433 − 5.27675i −2.04660 − 1.71731i 6.28846 − 5.27664i −2.14774 + 12.1804i −1.44586 + 2.50430i 12.3168 + 21.3333i −1.60664 + 0.584768i −17.8062 + 6.48094i
4.3 0.579114 + 0.210780i 1.63522 + 9.27381i −5.83741 − 4.89817i 9.88078 − 8.29096i −1.00776 + 5.71527i 6.05695 − 10.4909i −4.81321 − 8.33672i −57.9579 + 21.0950i 7.46967 − 2.71874i
4.4 2.98397 + 1.08608i −0.341646 − 1.93757i 1.59618 + 1.33936i −6.63193 + 5.56485i 1.08489 − 6.15271i −4.36312 + 7.55715i −9.39359 − 16.2702i 21.7342 − 7.91062i −25.8334 + 9.40257i
5.1 −5.03941 + 1.83419i 0.944789 − 5.35816i 15.9030 − 13.3442i −7.64679 − 6.41642i 5.06673 + 28.7349i −9.43070 − 16.3344i −34.2145 + 59.2612i −2.44557 − 0.890116i 50.3042 + 18.3092i
5.2 −2.16911 + 0.789492i −0.930433 + 5.27675i −2.04660 + 1.71731i 6.28846 + 5.27664i −2.14774 − 12.1804i −1.44586 − 2.50430i 12.3168 − 21.3333i −1.60664 − 0.584768i −17.8062 − 6.48094i
5.3 0.579114 − 0.210780i 1.63522 − 9.27381i −5.83741 + 4.89817i 9.88078 + 8.29096i −1.00776 − 5.71527i 6.05695 + 10.4909i −4.81321 + 8.33672i −57.9579 − 21.0950i 7.46967 + 2.71874i
5.4 2.98397 − 1.08608i −0.341646 + 1.93757i 1.59618 − 1.33936i −6.63193 − 5.56485i 1.08489 + 6.15271i −4.36312 − 7.55715i −9.39359 + 16.2702i 21.7342 + 7.91062i −25.8334 − 9.40257i
6.1 −2.26193 + 1.89799i −8.82799 − 3.21312i 0.124797 − 0.707761i −0.0925462 − 0.524856i 26.0668 − 9.48752i −11.8949 + 20.6025i −10.7499 − 18.6194i 46.9260 + 39.3756i 1.20550 + 1.01154i
6.2 −1.74757 + 1.46638i 4.00345 + 1.45714i −0.485473 + 2.75325i 1.27381 + 7.22413i −9.13303 + 3.32415i 13.1019 − 22.6932i −12.3141 − 21.3286i −6.77881 − 5.68809i −12.8194 − 10.7568i
6.3 1.38101 − 1.15881i 2.58415 + 0.940555i −0.824823 + 4.67780i −3.13553 − 17.7825i 4.65867 − 1.69562i −14.1277 + 24.4699i 11.4927 + 19.9060i −14.8900 − 12.4942i −24.9367 − 20.9244i
6.4 2.98693 − 2.50633i −3.72413 − 1.35547i 1.25086 − 7.09397i 2.58856 + 14.6804i −14.5210 + 5.28519i 5.35146 − 9.26900i 1.55302 + 2.68992i −8.65137 − 7.25936i 44.5258 + 37.3616i
9.1 −0.851642 − 4.82990i 0.458552 − 0.384771i −15.0851 + 5.49052i 6.94640 + 2.52828i −2.24893 − 1.88707i 14.5751 − 25.2448i 19.7481 + 34.2048i −4.62628 + 26.2369i 6.29551 − 35.7036i
9.2 −0.163781 − 0.928850i 1.56059 − 1.30949i 6.68160 − 2.43190i −3.55727 − 1.29474i −1.47192 − 1.23509i −11.7083 + 20.2794i −7.12592 − 12.3424i −3.96782 + 22.5026i −0.620006 + 3.51623i
9.3 0.477474 + 2.70789i −4.62264 + 3.87886i 0.412858 − 0.150268i 5.11043 + 1.86004i −12.7107 − 10.6656i 11.7392 − 20.3328i 11.6027 + 20.0964i 1.63479 − 9.27135i −2.59670 + 14.7266i
9.4 0.824938 + 4.67846i 5.76007 − 4.83328i −13.6899 + 4.98271i −14.0244 − 5.10445i 27.3640 + 22.9611i 3.64598 − 6.31503i −15.6022 − 27.0238i 5.12940 − 29.0903i 12.3117 − 69.8233i
16.1 −2.26193 − 1.89799i −8.82799 + 3.21312i 0.124797 + 0.707761i −0.0925462 + 0.524856i 26.0668 + 9.48752i −11.8949 − 20.6025i −10.7499 + 18.6194i 46.9260 − 39.3756i 1.20550 − 1.01154i
16.2 −1.74757 − 1.46638i 4.00345 − 1.45714i −0.485473 − 2.75325i 1.27381 − 7.22413i −9.13303 − 3.32415i 13.1019 + 22.6932i −12.3141 + 21.3286i −6.77881 + 5.68809i −12.8194 + 10.7568i
16.3 1.38101 + 1.15881i 2.58415 − 0.940555i −0.824823 − 4.67780i −3.13553 + 17.7825i 4.65867 + 1.69562i −14.1277 − 24.4699i 11.4927 − 19.9060i −14.8900 + 12.4942i −24.9367 + 20.9244i
16.4 2.98693 + 2.50633i −3.72413 + 1.35547i 1.25086 + 7.09397i 2.58856 − 14.6804i −14.5210 − 5.28519i 5.35146 + 9.26900i 1.55302 − 2.68992i −8.65137 + 7.25936i 44.5258 − 37.3616i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.4.e.a ✓ 24
3.b odd 2 1 171.4.u.b 24
19.e even 9 1 inner 19.4.e.a ✓ 24
19.e even 9 1 361.4.a.n 12
19.f odd 18 1 361.4.a.m 12
57.l odd 18 1 171.4.u.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.e.a ✓ 24 1.a even 1 1 trivial
19.4.e.a ✓ 24 19.e even 9 1 inner
171.4.u.b 24 3.b odd 2 1
171.4.u.b 24 57.l odd 18 1
361.4.a.m 12 19.f odd 18 1
361.4.a.n 12 19.e even 9 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(19, [\chi])\).