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Properties

Label 372.2.ba.a
Level $372$
Weight $2$
Character orbit 372.ba
Analytic conductor $2.970$
Analytic rank $0$
Dimension $128$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [372,2,Mod(43,372)] 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("372.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(372, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([15, 0, 19])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 372 = 2^{2} \cdot 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 372.ba (of order \(30\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [128,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.97043495519\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 16 q^{3} + 2 q^{4} + 3 q^{6} - 6 q^{8} + 16 q^{9} - 2 q^{10} + 12 q^{11} - 9 q^{12} - 6 q^{13} + 5 q^{14} + 10 q^{16} + 3 q^{18} + 18 q^{19} - 11 q^{20} - 10 q^{21} + 10 q^{22} + 48 q^{23} - 3 q^{24}+ \cdots + 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.41365 + 0.0399207i −0.913545 − 0.406737i 1.99681 − 0.112868i −0.693805 + 1.20171i 1.30767 + 0.538514i 0.641292 + 0.577421i −2.81829 + 0.239270i 0.669131 + 0.743145i 0.932824 − 1.72649i
43.2 −1.32462 − 0.495371i −0.913545 − 0.406737i 1.50922 + 1.31235i 2.10619 − 3.64802i 1.00861 + 0.991313i 2.54717 + 2.29348i −1.34903 − 2.48598i 0.669131 + 0.743145i −4.59701 + 3.78889i
43.3 −1.13579 − 0.842597i −0.913545 − 0.406737i 0.580059 + 1.91404i −0.616106 + 1.06713i 0.694885 + 1.23172i −3.40982 − 3.07022i 0.953932 − 2.66271i 0.669131 + 0.743145i 1.59893 − 0.692907i
43.4 −0.950246 + 1.04739i −0.913545 − 0.406737i −0.194065 − 1.99056i −1.86380 + 3.22820i 1.29411 − 0.570341i −3.29887 − 2.97031i 2.26931 + 1.68826i 0.669131 + 0.743145i −1.61013 − 5.01973i
43.5 −0.911865 + 1.08097i −0.913545 − 0.406737i −0.337004 − 1.97140i 1.50921 − 2.61403i 1.27270 − 0.616629i −0.591680 − 0.532751i 2.43833 + 1.43336i 0.669131 + 0.743145i 1.44950 + 4.01506i
43.6 −0.737627 − 1.20661i −0.913545 − 0.406737i −0.911812 + 1.78006i −0.0815079 + 0.141176i 0.183084 + 1.40231i 0.360225 + 0.324348i 2.82041 − 0.212818i 0.669131 + 0.743145i 0.230466 − 0.00578697i
43.7 −0.172404 + 1.40367i −0.913545 − 0.406737i −1.94055 − 0.483996i 0.321996 − 0.557713i 0.728421 − 1.21219i 0.791488 + 0.712659i 1.01393 − 2.64045i 0.669131 + 0.743145i 0.727330 + 0.548127i
43.8 0.0211360 − 1.41406i −0.913545 − 0.406737i −1.99911 − 0.0597749i 1.35828 − 2.35261i −0.594457 + 1.28321i −2.24937 − 2.02534i −0.126778 + 2.82558i 0.669131 + 0.743145i −3.29801 − 1.97040i
43.9 0.0752808 − 1.41221i −0.913545 − 0.406737i −1.98867 − 0.212625i −0.409437 + 0.709165i −0.643169 + 1.25950i 3.42813 + 3.08670i −0.449979 + 2.79240i 0.669131 + 0.743145i 0.970666 + 0.631596i
43.10 0.390684 + 1.35918i −0.913545 − 0.406737i −1.69473 + 1.06202i −0.946973 + 1.64020i 0.195920 − 1.40058i −1.24257 − 1.11882i −2.10558 − 1.88853i 0.669131 + 0.743145i −2.59930 − 0.646304i
43.11 0.840373 − 1.13744i −0.913545 − 0.406737i −0.587545 − 1.91175i −1.43833 + 2.49126i −1.23036 + 0.697294i −1.07716 − 0.969876i −2.66826 − 0.938286i 0.669131 + 0.743145i 1.62493 + 3.72960i
43.12 1.07002 + 0.924693i −0.913545 − 0.406737i 0.289884 + 1.97888i 1.52904 − 2.64838i −0.601405 − 1.27997i −0.728432 − 0.655883i −1.51968 + 2.38549i 0.669131 + 0.743145i 4.08504 − 1.41992i
43.13 1.16691 − 0.798945i −0.913545 − 0.406737i 0.723375 − 1.86460i 1.01023 − 1.74977i −1.39099 + 0.255246i 3.39926 + 3.06071i −0.645596 − 2.75376i 0.669131 + 0.743145i −0.219118 − 2.84895i
43.14 1.17786 + 0.782714i −0.913545 − 0.406737i 0.774717 + 1.84386i −0.969598 + 1.67939i −0.757672 − 1.19412i 1.41801 + 1.27678i −0.530704 + 2.77819i 0.669131 + 0.743145i −2.45654 + 1.21917i
43.15 1.38712 − 0.275519i −0.913545 − 0.406737i 1.84818 − 0.764352i 1.01376 − 1.75588i −1.37926 − 0.312492i −2.45297 − 2.20866i 2.35304 − 1.56945i 0.669131 + 0.743145i 0.922421 − 2.71492i
43.16 1.39044 − 0.258226i −0.913545 − 0.406737i 1.86664 − 0.718094i −1.82914 + 3.16817i −1.37526 − 0.329641i 0.718054 + 0.646539i 2.41002 − 1.48048i 0.669131 + 0.743145i −1.72521 + 4.87747i
55.1 −1.40988 − 0.110669i 0.978148 − 0.207912i 1.97550 + 0.312058i −1.54291 + 2.67239i −1.40208 + 0.184880i 1.51565 − 3.40420i −2.75068 − 0.658590i 0.913545 − 0.406737i 2.47106 − 3.59700i
55.2 −1.32483 + 0.494790i 0.978148 − 0.207912i 1.51037 − 1.31103i 0.981892 − 1.70069i −1.19301 + 0.759426i 0.529784 − 1.18991i −1.35230 + 2.48421i 0.913545 − 0.406737i −0.459361 + 2.73896i
55.3 −1.32319 − 0.499154i 0.978148 − 0.207912i 1.50169 + 1.32096i −0.474253 + 0.821430i −1.39806 − 0.213139i −0.788705 + 1.77146i −1.32767 − 2.49746i 0.913545 − 0.406737i 1.03755 − 0.850186i
55.4 −1.03878 + 0.959657i 0.978148 − 0.207912i 0.158117 − 1.99374i 0.276419 − 0.478772i −0.816554 + 1.15466i −1.75660 + 3.94539i 1.74906 + 2.22279i 0.913545 − 0.406737i 0.172319 + 0.762605i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
124.p even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 372.2.ba.a ✓ 128
4.b odd 2 1 372.2.ba.b yes 128
31.h odd 30 1 372.2.ba.b yes 128
124.p even 30 1 inner 372.2.ba.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
372.2.ba.a ✓ 128 1.a even 1 1 trivial
372.2.ba.a ✓ 128 124.p even 30 1 inner
372.2.ba.b yes 128 4.b odd 2 1
372.2.ba.b yes 128 31.h odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{128} + 52 T_{7}^{126} - 160 T_{7}^{125} + 761 T_{7}^{124} - 8224 T_{7}^{123} + \cdots + 35\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(372, [\chi])\). Copy content Toggle raw display