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Properties

Label 93.3.l.b
Level $93$
Weight $3$
Character orbit 93.l
Analytic conductor $2.534$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [93,3,Mod(2,93)] 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("93.2"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(93, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 8])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 93.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.53406645855\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 5 q^{3} + 36 q^{4} - 10 q^{6} - 54 q^{7} + 5 q^{9} + 114 q^{10} + 14 q^{12} + 26 q^{13} + 35 q^{15} - 184 q^{16} + 15 q^{18} - 90 q^{19} + 71 q^{21} - 114 q^{22} - 101 q^{24} - 188 q^{25} + 116 q^{27}+ \cdots - 616 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.30253 + 3.16915i −1.34522 − 2.68149i −3.50585 − 10.7899i 0.369845i 11.5955 + 1.91099i 2.37994 + 7.32470i 27.3649 + 8.89139i −5.38077 + 7.21439i −1.17210 − 0.851578i
2.2 −2.17320 + 2.99115i 0.124604 + 2.99741i −2.98814 − 9.19654i − 5.36452i −9.23651 − 6.14127i −3.32687 − 10.2391i 19.9368 + 6.47787i −8.96895 + 0.746980i 16.0461 + 11.6582i
2.3 −1.72290 + 2.37137i 2.91360 + 0.714793i −1.41895 − 4.36707i 1.18691i −6.71489 + 5.67772i 2.57067 + 7.91172i 1.64979 + 0.536048i 7.97814 + 4.16524i −2.81460 − 2.04493i
2.4 −1.51008 + 2.07845i 1.83238 − 2.37537i −0.803542 − 2.47305i − 5.23090i 2.17003 + 7.39553i −2.44289 − 7.51843i −3.41994 − 1.11121i −2.28474 − 8.70517i 10.8722 + 7.89911i
2.5 −1.27198 + 1.75074i −2.99978 + 0.0359714i −0.211065 − 0.649592i − 6.70333i 3.75270 − 5.29759i −0.186362 − 0.573564i −6.82672 − 2.21814i 8.99741 − 0.215813i 11.7358 + 8.52653i
2.6 −1.06274 + 1.46274i −1.77299 − 2.42002i 0.225888 + 0.695211i 3.73375i 5.42408 − 0.0215715i −0.683387 − 2.10325i −8.13517 − 2.64328i −2.71298 + 8.58136i −5.46149 − 3.96801i
2.7 −0.556283 + 0.765658i −0.525348 + 2.95364i 0.959286 + 2.95238i − 2.58079i −1.96924 − 2.04530i 1.96270 + 6.04057i −6.39449 − 2.07770i −8.44802 − 3.10338i 1.97600 + 1.43565i
2.8 −0.545338 + 0.750594i 2.29503 + 1.93206i 0.970071 + 2.98557i 4.88462i −2.70176 + 0.669006i −3.26048 − 10.0347i −6.29947 − 2.04682i 1.53429 + 8.86826i −3.66636 − 2.66377i
2.9 −0.170762 + 0.235034i 1.53643 − 2.57670i 1.20999 + 3.72396i 6.49329i 0.343247 + 0.801117i 1.26783 + 3.90197i −2.18707 − 0.710624i −4.27874 − 7.91785i −1.52614 − 1.10881i
2.10 0.170762 − 0.235034i 2.92537 − 0.664992i 1.20999 + 3.72396i − 6.49329i 0.343247 − 0.801117i 1.26783 + 3.90197i 2.18707 + 0.710624i 8.11557 − 3.89069i −1.52614 − 1.10881i
2.11 0.545338 − 0.750594i −1.12830 − 2.77974i 0.970071 + 2.98557i − 4.88462i −2.70176 − 0.669006i −3.26048 − 10.0347i 6.29947 + 2.04682i −6.45390 + 6.27274i −3.66636 − 2.66377i
2.12 0.556283 − 0.765658i −2.97142 − 0.413090i 0.959286 + 2.95238i 2.58079i −1.96924 + 2.04530i 1.96270 + 6.04057i 6.39449 + 2.07770i 8.65871 + 2.45493i 1.97600 + 1.43565i
2.13 1.06274 − 1.46274i 1.75369 + 2.43404i 0.225888 + 0.695211i − 3.73375i 5.42408 + 0.0215715i −0.683387 − 2.10325i 8.13517 + 2.64328i −2.84915 + 8.53712i −5.46149 − 3.96801i
2.14 1.27198 − 1.75074i −0.961195 + 2.84185i −0.211065 − 0.649592i 6.70333i 3.75270 + 5.29759i −0.186362 − 0.573564i 6.82672 + 2.21814i −7.15221 − 5.46314i 11.7358 + 8.52653i
2.15 1.51008 − 2.07845i 2.82535 − 1.00867i −0.803542 − 2.47305i 5.23090i 2.17003 − 7.39553i −2.44289 − 7.51843i 3.41994 + 1.11121i 6.96516 − 5.69969i 10.8722 + 7.89911i
2.16 1.72290 − 2.37137i 0.220544 − 2.99188i −1.41895 − 4.36707i − 1.18691i −6.71489 − 5.67772i 2.57067 + 7.91172i −1.64979 − 0.536048i −8.90272 − 1.31968i −2.81460 − 2.04493i
2.17 2.17320 − 2.99115i −2.81220 − 1.04476i −2.98814 − 9.19654i 5.36452i −9.23651 + 6.14127i −3.32687 − 10.2391i −19.9368 − 6.47787i 6.81697 + 5.87613i 16.0461 + 11.6582i
2.18 2.30253 − 3.16915i 2.13455 + 2.10801i −3.50585 − 10.7899i − 0.369845i 11.5955 − 1.91099i 2.37994 + 7.32470i −27.3649 − 8.89139i 0.112621 + 8.99930i −1.17210 − 0.851578i
8.1 −3.46751 − 1.12666i 2.05322 + 2.18730i 7.51820 + 5.46229i 5.25837i −4.65521 − 9.89779i −8.69344 − 6.31616i −11.3431 − 15.6125i −0.568590 + 8.98202i 5.92441 − 18.2335i
8.2 −3.20669 − 1.04192i −0.616051 − 2.93607i 5.96120 + 4.33106i 6.02940i −1.08365 + 10.0569i 4.41911 + 3.21067i −6.67573 − 9.18835i −8.24096 + 3.61753i 6.28213 − 19.3344i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.d even 5 1 inner
93.l odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.3.l.b ✓ 72
3.b odd 2 1 inner 93.3.l.b ✓ 72
31.d even 5 1 inner 93.3.l.b ✓ 72
93.l odd 10 1 inner 93.3.l.b ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.3.l.b ✓ 72 1.a even 1 1 trivial
93.3.l.b ✓ 72 3.b odd 2 1 inner
93.3.l.b ✓ 72 31.d even 5 1 inner
93.3.l.b ✓ 72 93.l odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - 54 T_{2}^{70} + 1756 T_{2}^{68} - 45015 T_{2}^{66} + 990028 T_{2}^{64} + \cdots + 25\!\cdots\!01 \) acting on \(S_{3}^{\mathrm{new}}(93, [\chi])\). Copy content Toggle raw display