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Properties

Label 147.2.o.b
Level $147$
Weight $2$
Character orbit 147.o
Analytic conductor $1.174$
Analytic rank $0$
Dimension $192$
Inner twists $4$

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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] = [147,2,Mod(5,147)] 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("147.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([21, 29])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.o (of order \(42\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [192] 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: \(1.17380090971\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(16\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 14 q^{3} - 44 q^{4} - 28 q^{6} - 28 q^{7} - 26 q^{9} - 28 q^{10} + 21 q^{12} - 28 q^{13} - 10 q^{15} - 20 q^{16} - 3 q^{18} - 42 q^{19} - 14 q^{21} - 34 q^{22} - 14 q^{24} - 16 q^{25} + 7 q^{27}+ \cdots - 136 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.409128 − 2.71438i 0.647849 − 1.60633i −5.28935 + 1.63155i 0.153380 − 2.04671i −4.62525 − 1.10132i 2.11947 + 1.58362i 4.21061 + 8.74343i −2.16058 − 2.08132i −5.61832 + 0.421035i
5.2 −0.364280 − 2.41684i −1.72403 + 0.166545i −3.79727 + 1.17130i −0.194601 + 2.59677i 1.03054 + 4.10602i −2.07858 + 1.63692i 2.09318 + 4.34653i 2.94453 − 0.574255i 6.34687 − 0.475632i
5.3 −0.322503 − 2.13967i −0.521326 + 1.65173i −2.56303 + 0.790591i 0.187651 − 2.50402i 3.70229 + 0.582777i 0.453192 − 2.60665i 0.640481 + 1.32997i −2.45644 − 1.72218i −5.41830 + 0.406046i
5.4 −0.262096 − 1.73889i 1.69470 − 0.357748i −1.04390 + 0.322001i −0.214075 + 2.85663i −1.06626 − 2.85314i 1.71512 − 2.01454i −0.692471 − 1.43793i 2.74403 − 1.21255i 5.02347 − 0.376457i
5.5 −0.240046 − 1.59260i −0.767290 − 1.55283i −0.567611 + 0.175085i −0.00908860 + 0.121279i −2.28885 + 1.59474i −1.90644 − 1.83452i −0.982525 − 2.04023i −1.82253 + 2.38293i 0.195331 − 0.0146380i
5.6 −0.230083 − 1.52650i 1.48236 + 0.895883i −0.366123 + 0.112934i 0.148232 − 1.97802i 1.02650 − 2.46895i −0.513287 + 2.59548i −1.08298 − 2.24883i 1.39479 + 2.65604i −3.05356 + 0.228832i
5.7 −0.105834 − 0.702164i −1.63171 + 0.580973i 1.42931 − 0.440884i −0.123919 + 1.65358i 0.580629 + 1.08424i 2.61893 + 0.375748i −1.07704 − 2.23650i 2.32494 − 1.89596i 1.17420 − 0.0879943i
5.8 −0.0569582 − 0.377893i 0.421048 + 1.68009i 1.77159 − 0.546462i −0.273303 + 3.64698i 0.610914 − 0.254806i −2.49430 − 0.882305i −0.639038 − 1.32698i −2.64544 + 1.41480i 1.39374 − 0.104446i
5.9 0.0569582 + 0.377893i −1.71778 + 0.221866i 1.77159 − 0.546462i 0.273303 − 3.64698i −0.181683 − 0.636501i −2.49430 − 0.882305i 0.639038 + 1.32698i 2.90155 − 0.762233i 1.39374 − 0.104446i
5.10 0.105834 + 0.702164i 0.0553172 + 1.73117i 1.42931 − 0.440884i 0.123919 − 1.65358i −1.20971 + 0.222059i 2.61893 + 0.375748i 1.07704 + 2.23650i −2.99388 + 0.191527i 1.17420 − 0.0879943i
5.11 0.230083 + 1.52650i −1.37552 − 1.05259i −0.366123 + 0.112934i −0.148232 + 1.97802i 1.29029 − 2.34192i −0.513287 + 2.59548i 1.08298 + 2.24883i 0.784115 + 2.89571i −3.05356 + 0.228832i
5.12 0.240046 + 1.59260i 1.72581 + 0.146939i −0.567611 + 0.175085i 0.00908860 − 0.121279i 0.180258 + 2.78379i −1.90644 − 1.83452i 0.982525 + 2.04023i 2.95682 + 0.507176i 0.195331 − 0.0146380i
5.13 0.262096 + 1.73889i −0.286126 − 1.70825i −1.04390 + 0.322001i 0.214075 − 2.85663i 2.89547 − 0.945268i 1.71512 − 2.01454i 0.692471 + 1.43793i −2.83626 + 0.977553i 5.02347 − 0.376457i
5.14 0.322503 + 2.13967i −1.34709 + 1.08873i −2.56303 + 0.790591i −0.187651 + 2.50402i −2.76397 − 2.53121i 0.453192 − 2.60665i −0.640481 − 1.32997i 0.629315 − 2.93325i −5.41830 + 0.406046i
5.15 0.364280 + 2.41684i 0.474825 + 1.66570i −3.79727 + 1.17130i 0.194601 − 2.59677i −3.85275 + 1.75436i −2.07858 + 1.63692i −2.09318 − 4.34653i −2.54908 + 1.58183i 6.34687 − 0.475632i
5.16 0.409128 + 2.71438i 1.25860 − 1.18992i −5.28935 + 1.63155i −0.153380 + 2.04671i 3.74484 + 2.92950i 2.11947 + 1.58362i −4.21061 − 8.74343i 0.168163 − 2.99528i −5.61832 + 0.421035i
17.1 −2.54105 + 0.190425i 1.00837 − 1.40826i 4.44299 − 0.669673i 2.54958 − 2.36567i −2.29414 + 3.77047i −1.65898 + 2.06101i −6.19375 + 1.41368i −0.966394 − 2.84008i −6.02812 + 6.49677i
17.2 −2.24468 + 0.168215i 0.109375 + 1.72859i 3.03261 − 0.457093i 0.910664 − 0.844973i −0.536288 − 3.86174i 2.61371 − 0.410513i −2.34128 + 0.534381i −2.97607 + 0.378131i −1.90201 + 2.04988i
17.3 −2.16211 + 0.162028i 1.58630 + 0.695456i 2.67080 − 0.402559i −2.50568 + 2.32493i −3.54243 − 1.24663i −2.58708 + 0.554071i −1.48172 + 0.338193i 2.03268 + 2.20640i 5.04085 − 5.43274i
17.4 −1.78385 + 0.133681i −1.52980 + 0.812227i 1.18659 − 0.178850i −0.675229 + 0.626521i 2.62036 − 1.65340i −2.37457 − 1.16680i 1.39521 − 0.318447i 1.68057 − 2.48509i 1.12075 − 1.20789i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 5.16
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
49.h odd 42 1 inner
147.o even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.2.o.b ✓ 192
3.b odd 2 1 inner 147.2.o.b ✓ 192
49.h odd 42 1 inner 147.2.o.b ✓ 192
147.o even 42 1 inner 147.2.o.b ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.o.b ✓ 192 1.a even 1 1 trivial
147.2.o.b ✓ 192 3.b odd 2 1 inner
147.2.o.b ✓ 192 49.h odd 42 1 inner
147.2.o.b ✓ 192 147.o even 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{192} + 38 T_{2}^{190} + 671 T_{2}^{188} + 6776 T_{2}^{186} + 33956 T_{2}^{184} + \cdots + 18\!\cdots\!29 \) acting on \(S_{2}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display