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Properties

Label 93.3.n.b
Level $93$
Weight $3$
Character orbit 93.n
Analytic conductor $2.534$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [93,3,Mod(13,93)] 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("93.13"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(93, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([0, 11])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 93.n (of order \(30\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48] 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: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{2} + 18 q^{3} - 30 q^{4} + 9 q^{6} + q^{7} + 30 q^{8} - 18 q^{9} + 6 q^{10} - 68 q^{11} + 45 q^{12} + 18 q^{13} - 5 q^{14} - 146 q^{16} + 31 q^{17} - 9 q^{18} + 55 q^{19} + 191 q^{20} - 108 q^{21}+ \cdots + 81 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −3.14762 + 2.28688i 0.704489 + 1.58231i 3.44162 − 10.5922i 3.39584 + 5.88177i −5.83601 − 3.36942i 5.72390 + 6.35704i 8.58109 + 26.4099i −2.00739 + 2.22943i −24.1397 − 10.7477i
13.2 −2.19842 + 1.59725i 0.704489 + 1.58231i 1.04579 − 3.21862i −4.22402 − 7.31621i −4.07610 − 2.35334i −1.04976 − 1.16587i −0.517049 − 1.59131i −2.00739 + 2.22943i 20.9720 + 9.33732i
13.3 −1.31388 + 0.954589i 0.704489 + 1.58231i −0.421031 + 1.29580i 2.68233 + 4.64594i −2.43606 − 1.40646i −3.70180 − 4.11127i −2.69120 − 8.28266i −2.00739 + 2.22943i −7.95922 − 3.54367i
13.4 0.438624 − 0.318679i 0.704489 + 1.58231i −1.14523 + 3.52467i −1.56109 − 2.70388i 0.813253 + 0.469532i 7.06741 + 7.84916i 1.29107 + 3.97350i −2.00739 + 2.22943i −1.54640 − 0.688501i
13.5 1.42893 − 1.03818i 0.704489 + 1.58231i −0.272043 + 0.837261i 3.69711 + 6.40358i 2.64938 + 1.52962i −4.88801 − 5.42868i 2.66371 + 8.19804i −2.00739 + 2.22943i 11.9310 + 5.31200i
13.6 2.70969 − 1.96871i 0.704489 + 1.58231i 2.23056 − 6.86495i −0.695623 − 1.20485i 5.02405 + 2.90063i 0.652428 + 0.724595i −3.33091 − 10.2515i −2.00739 + 2.22943i −4.25693 − 1.89531i
22.1 −1.06480 + 3.27712i 0.360114 − 1.69420i −6.36964 − 4.62782i −3.91137 − 6.77469i 5.16865 + 2.98412i 9.66339 − 4.30242i 10.7976 − 7.84490i −2.74064 − 1.22021i 26.3663 − 5.60433i
22.2 −1.00524 + 3.09380i 0.360114 − 1.69420i −5.32503 − 3.86886i 3.52360 + 6.10306i 4.87952 + 2.81719i −10.3732 + 4.61844i 6.79541 − 4.93716i −2.74064 − 1.22021i −22.4237 + 4.76631i
22.3 −0.379628 + 1.16837i 0.360114 − 1.69420i 2.01509 + 1.46405i 0.344159 + 0.596100i 1.84275 + 1.06391i 6.16016 − 2.74268i −6.45105 + 4.68696i −2.74064 − 1.22021i −0.827121 + 0.175810i
22.4 0.354096 − 1.08979i 0.360114 − 1.69420i 2.17380 + 1.57936i 4.18594 + 7.25026i −1.71882 − 0.992360i 0.00755191 − 0.00336233i 6.19905 − 4.50387i −2.74064 − 1.22021i 9.38353 − 1.99453i
22.5 0.551190 − 1.69639i 0.360114 − 1.69420i 0.662149 + 0.481079i −3.20115 − 5.54456i −2.67553 − 1.54472i −2.32756 + 1.03630i 6.95320 − 5.05179i −2.74064 − 1.22021i −11.1702 + 2.37429i
22.6 1.10898 − 3.41310i 0.360114 − 1.69420i −7.18331 − 5.21898i 1.09496 + 1.89653i −5.38311 − 3.10794i 4.68036 − 2.08383i −14.1656 + 10.2919i −2.74064 − 1.22021i 7.68734 − 1.63399i
34.1 −2.93428 + 2.13188i 1.72256 + 0.181049i 2.82902 − 8.70683i −1.33804 + 2.31756i −5.44045 + 3.14105i −11.6250 + 2.47096i 5.77759 + 17.7816i 2.93444 + 0.623735i −1.01456 − 9.65292i
34.2 −2.00288 + 1.45518i 1.72256 + 0.181049i 0.657913 − 2.02485i 3.52838 − 6.11133i −3.71354 + 2.14401i 5.21989 − 1.10952i −1.43133 − 4.40519i 2.93444 + 0.623735i 1.82615 + 17.3747i
34.3 −0.429658 + 0.312165i 1.72256 + 0.181049i −1.14891 + 3.53598i −2.43322 + 4.21446i −0.796629 + 0.459934i −8.91470 + 1.89488i −1.26663 − 3.89828i 2.93444 + 0.623735i −0.270154 − 2.57034i
34.4 0.773018 − 0.561630i 1.72256 + 0.181049i −0.953940 + 2.93593i −0.901489 + 1.56142i 1.43325 − 0.827489i 10.0656 − 2.13952i 2.09256 + 6.44023i 2.93444 + 0.623735i 0.180076 + 1.71331i
34.5 2.05224 − 1.49104i 1.72256 + 0.181049i 0.752428 − 2.31573i 2.48305 − 4.30077i 3.80507 − 2.19686i −7.51879 + 1.59817i 1.22686 + 3.77588i 2.93444 + 0.623735i −1.31680 − 12.5286i
34.6 3.12423 − 2.26989i 1.72256 + 0.181049i 3.37236 − 10.3791i −4.63323 + 8.02500i 5.79264 − 3.34438i −0.284519 + 0.0604764i −8.24985 − 25.3904i 2.93444 + 0.623735i 3.74054 + 35.5888i
43.1 −3.14762 − 2.28688i 0.704489 − 1.58231i 3.44162 + 10.5922i 3.39584 − 5.88177i −5.83601 + 3.36942i 5.72390 − 6.35704i 8.58109 − 26.4099i −2.00739 − 2.22943i −24.1397 + 10.7477i
43.2 −2.19842 − 1.59725i 0.704489 − 1.58231i 1.04579 + 3.21862i −4.22402 + 7.31621i −4.07610 + 2.35334i −1.04976 + 1.16587i −0.517049 + 1.59131i −2.00739 − 2.22943i 20.9720 − 9.33732i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.h odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.3.n.b ✓ 48
3.b odd 2 1 279.3.bc.d 48
31.h odd 30 1 inner 93.3.n.b ✓ 48
93.p even 30 1 279.3.bc.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.3.n.b ✓ 48 1.a even 1 1 trivial
93.3.n.b ✓ 48 31.h odd 30 1 inner
279.3.bc.d 48 3.b odd 2 1
279.3.bc.d 48 93.p even 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 6 T_{2}^{47} + 57 T_{2}^{46} + 236 T_{2}^{45} + 1550 T_{2}^{44} + 5582 T_{2}^{43} + \cdots + 13347506923776 \) acting on \(S_{3}^{\mathrm{new}}(93, [\chi])\). Copy content Toggle raw display