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Properties

Label 444.2.ba.a
Level $444$
Weight $2$
Character orbit 444.ba
Analytic conductor $3.545$
Analytic rank $0$
Dimension $432$
Inner twists $8$

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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] = [444,2,Mod(95,444)] 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("444.95"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.ba (of order \(18\), 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: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(432\)
Relative dimension: \(72\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 432 q - 12 q^{4} - 12 q^{9} - 6 q^{10} - 21 q^{12} - 48 q^{13} - 36 q^{16} + 15 q^{18} - 12 q^{21} + 24 q^{22} - 51 q^{24} - 24 q^{25} - 36 q^{28} - 48 q^{30} + 6 q^{33} - 12 q^{34} - 12 q^{36} - 12 q^{37}+ \cdots - 36 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1 −1.41341 + 0.0476073i 0.972840 − 1.43303i 1.99547 − 0.134578i 0.194008 − 1.10027i −1.30680 + 2.07178i 0.736087 + 0.129792i −2.81401 + 0.285212i −1.10716 − 2.78822i −0.221832 + 1.56437i
95.2 −1.40774 − 0.135139i −1.55540 + 0.762062i 1.96347 + 0.380481i −0.173194 + 0.982231i 2.29258 − 0.862592i −1.15780 − 0.204151i −2.71265 − 0.800961i 1.83852 − 2.37062i 0.376550 − 1.35932i
95.3 −1.40187 − 0.186459i −1.73085 − 0.0644872i 1.93047 + 0.522782i 0.706439 − 4.00642i 2.41440 + 0.413135i −2.67985 − 0.472529i −2.60878 − 1.09282i 2.99168 + 0.223235i −1.73737 + 5.48474i
95.4 −1.40113 + 0.191918i 0.580249 + 1.63197i 1.92633 − 0.537804i 0.375185 − 2.12778i −1.12621 − 2.17524i 0.994964 + 0.175439i −2.59583 + 1.12323i −2.32662 + 1.89389i −0.117324 + 3.05331i
95.5 −1.39650 + 0.223126i 1.72980 − 0.0882241i 1.90043 − 0.623191i −0.543983 + 3.08508i −2.39599 + 0.509169i −0.698395 − 0.123146i −2.51490 + 1.29432i 2.98443 − 0.305220i 0.0713109 − 4.42970i
95.6 −1.38033 − 0.307710i −0.926919 − 1.46315i 1.81063 + 0.849483i −0.598538 + 3.39448i 0.829229 + 2.30486i −3.74895 − 0.661040i −2.23787 − 1.72972i −1.28164 + 2.71245i 1.87070 − 4.50133i
95.7 −1.37367 − 0.336197i −0.989335 − 1.42169i 1.77394 + 0.923648i 0.00719629 − 0.0408122i 0.881051 + 2.28555i 4.22835 + 0.745572i −2.12629 − 1.86518i −1.04243 + 2.81307i −0.0236063 + 0.0536432i
95.8 −1.32317 + 0.499217i −0.321693 + 1.70191i 1.50156 − 1.32110i −0.316913 + 1.79730i −0.423970 − 2.41252i −2.52151 − 0.444610i −1.32731 + 2.49765i −2.79303 − 1.09499i −0.477914 − 2.53635i
95.9 −1.26840 − 0.625437i 0.989335 + 1.42169i 1.21766 + 1.58660i 0.00719629 − 0.0408122i −0.365688 − 2.42204i −4.22835 − 0.745572i −0.552151 − 2.77401i −1.04243 + 2.81307i −0.0346532 + 0.0472652i
95.10 −1.25519 − 0.651540i 0.926919 + 1.46315i 1.15099 + 1.63561i −0.598538 + 3.39448i −0.210153 − 2.44046i 3.74895 + 0.661040i −0.379042 − 2.80291i −1.28164 + 2.71245i 2.96292 − 3.87073i
95.11 −1.25441 + 0.653040i −1.67419 − 0.443951i 1.14708 − 1.63836i −0.136712 + 0.775330i 2.39003 − 0.536418i 1.24725 + 0.219924i −0.368988 + 2.80426i 2.60582 + 1.48651i −0.334830 − 1.06186i
95.12 −1.19915 + 0.749687i 1.73157 − 0.0409697i 0.875940 − 1.79798i 0.645741 − 3.66218i −2.04570 + 1.34726i −4.52664 − 0.798168i 0.297534 + 2.81273i 2.99664 − 0.141884i 1.97114 + 4.87562i
95.13 −1.19375 − 0.758267i 1.73085 + 0.0644872i 0.850062 + 1.81036i 0.706439 − 4.00642i −2.01730 − 1.38943i 2.67985 + 0.472529i 0.357977 − 2.80568i 2.99168 + 0.223235i −3.88124 + 4.24698i
95.14 −1.19208 + 0.760890i −1.21678 − 1.23266i 0.842092 − 1.81408i 0.395044 − 2.24041i 2.38841 + 0.543583i 1.02476 + 0.180693i 0.376478 + 2.80326i −0.0388851 + 2.99975i 1.23378 + 2.97132i
95.15 −1.16526 − 0.801357i 1.55540 − 0.762062i 0.715655 + 1.86758i −0.173194 + 0.982231i −2.42312 − 0.358429i 1.15780 + 0.204151i 0.662672 − 2.74970i 1.83852 − 2.37062i 0.988933 − 1.00576i
95.16 −1.13719 + 0.840717i 0.137027 − 1.72662i 0.586389 − 1.91211i −0.165313 + 0.937534i 1.29578 + 2.07869i −3.09139 − 0.545096i 0.940706 + 2.66741i −2.96245 − 0.473187i −0.600210 − 1.20513i
95.17 −1.06855 + 0.926386i 1.55101 + 0.770947i 0.283619 − 1.97979i −0.0496346 + 0.281492i −2.37154 + 0.613037i 3.61541 + 0.637494i 1.53099 + 2.37825i 1.81128 + 2.39150i −0.207733 − 0.346770i
95.18 −1.05214 − 0.944993i −0.972840 + 1.43303i 0.213976 + 1.98852i 0.194008 − 1.10027i 2.37777 − 0.588416i −0.736087 − 0.129792i 1.65401 − 2.29440i −1.10716 − 2.78822i −1.24387 + 0.974299i
95.19 −0.949966 − 1.04765i −0.580249 − 1.63197i −0.195129 + 1.99046i 0.375185 − 2.12778i −1.15851 + 2.15821i −0.994964 − 0.175439i 2.27066 − 1.68644i −2.32662 + 1.89389i −2.58558 + 1.62826i
95.20 −0.926359 − 1.06858i −1.72980 + 0.0882241i −0.283718 + 1.97977i −0.543983 + 3.08508i 1.69669 + 1.76670i 0.698395 + 0.123146i 2.37837 − 1.53081i 2.98443 − 0.305220i 3.80057 − 2.27661i
See next 80 embeddings (of 432 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 95.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
37.h even 18 1 inner
111.n odd 18 1 inner
148.o odd 18 1 inner
444.ba even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 444.2.ba.a ✓ 432
3.b odd 2 1 inner 444.2.ba.a ✓ 432
4.b odd 2 1 inner 444.2.ba.a ✓ 432
12.b even 2 1 inner 444.2.ba.a ✓ 432
37.h even 18 1 inner 444.2.ba.a ✓ 432
111.n odd 18 1 inner 444.2.ba.a ✓ 432
148.o odd 18 1 inner 444.2.ba.a ✓ 432
444.ba even 18 1 inner 444.2.ba.a ✓ 432
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
444.2.ba.a ✓ 432 1.a even 1 1 trivial
444.2.ba.a ✓ 432 3.b odd 2 1 inner
444.2.ba.a ✓ 432 4.b odd 2 1 inner
444.2.ba.a ✓ 432 12.b even 2 1 inner
444.2.ba.a ✓ 432 37.h even 18 1 inner
444.2.ba.a ✓ 432 111.n odd 18 1 inner
444.2.ba.a ✓ 432 148.o odd 18 1 inner
444.2.ba.a ✓ 432 444.ba even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(444, [\chi])\).