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Properties

Label 245.2.p.b
Level $245$
Weight $2$
Character orbit 245.p
Analytic conductor $1.956$
Analytic rank $0$
Dimension $144$
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] = [245,2,Mod(29,245)] 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("245.29"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([7, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.p (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [144] 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: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(24\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 16 q^{4} - q^{5} - 34 q^{6} + 26 q^{9} - 7 q^{10} + 32 q^{11} - 4 q^{14} - 39 q^{15} - 32 q^{16} - 68 q^{19} + 15 q^{20} - 12 q^{21} + 22 q^{24} - 21 q^{25} - 28 q^{26} + 2 q^{29} - 4 q^{30} + 28 q^{31}+ \cdots + 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.17076 + 2.43110i 2.76955 − 0.632131i −3.29260 − 4.12879i 0.816587 + 2.08163i −1.70569 + 7.47311i 2.28728 − 1.32979i 8.63099 − 1.96997i 4.56789 − 2.19978i −6.01668 − 0.451878i
29.2 −1.11708 + 2.31965i −2.48024 + 0.566099i −2.88591 − 3.61882i 2.19467 − 0.428287i 1.45749 − 6.38567i −2.15133 − 1.54006i 6.59807 − 1.50597i 3.12822 − 1.50647i −1.45815 + 5.56929i
29.3 −1.07760 + 2.23765i 1.15159 − 0.262843i −2.59890 − 3.25892i −1.78699 − 1.34413i −0.652799 + 2.86010i −2.58340 + 0.571000i 5.25023 − 1.19833i −1.44583 + 0.696276i 4.93335 − 2.55023i
29.4 −0.999626 + 2.07574i 0.298086 − 0.0680361i −2.06248 − 2.58627i 0.860589 − 2.06383i −0.156748 + 0.686760i 2.25670 + 1.38106i 2.93788 − 0.670551i −2.61868 + 1.26109i 3.42371 + 3.84942i
29.5 −0.902155 + 1.87334i −1.48830 + 0.339694i −1.44855 − 1.81643i 0.0467901 + 2.23558i 0.706310 − 3.09455i 0.998185 + 2.45023i 0.655370 − 0.149584i −0.603274 + 0.290522i −4.23022 − 1.92918i
29.6 −0.689327 + 1.43140i 0.0543958 − 0.0124155i −0.326759 − 0.409743i −1.54631 + 1.61522i −0.0197249 + 0.0864206i −1.15602 − 2.37984i −2.28605 + 0.521777i −2.70010 + 1.30030i −1.24611 − 3.32680i
29.7 −0.655500 + 1.36116i 1.90587 − 0.435002i −0.176096 − 0.220818i 1.94446 + 1.10412i −0.657190 + 2.87934i −2.62444 + 0.335138i −2.52979 + 0.577408i 0.740208 − 0.356465i −2.77747 + 1.92298i
29.8 −0.613663 + 1.27428i −1.95484 + 0.446179i −0.000237008 0 0.000297199i −1.21687 − 1.87596i 0.631052 − 2.76482i 1.75956 − 1.97584i −2.75725 + 0.629324i 0.919404 − 0.442762i 3.13725 − 0.399428i
29.9 −0.468810 + 0.973493i 3.06012 − 0.698453i 0.519073 + 0.650897i −1.40221 − 1.74178i −0.754675 + 3.30645i 0.788721 − 2.52545i −2.98380 + 0.681033i 6.17360 − 2.97305i 2.35298 − 0.548481i
29.10 −0.291861 + 0.606056i −2.72598 + 0.622187i 0.964858 + 1.20989i 2.19115 + 0.445950i 0.418528 − 1.83369i 2.59609 − 0.510203i −2.32648 + 0.531004i 4.34094 − 2.09049i −0.909782 + 1.19780i
29.11 −0.139647 + 0.289980i −0.237528 + 0.0542142i 1.18239 + 1.48267i −2.15876 − 0.582879i 0.0174490 − 0.0764491i 0.373253 + 2.61929i −1.22263 + 0.279057i −2.64943 + 1.27590i 0.470487 − 0.544600i
29.12 −0.0373027 + 0.0774598i 1.33177 − 0.303967i 1.24237 + 1.55788i −0.624880 + 2.14698i −0.0261332 + 0.114497i 2.45766 − 0.979757i −0.334654 + 0.0763825i −1.02170 + 0.492026i −0.142995 − 0.128491i
29.13 0.0373027 − 0.0774598i −1.33177 + 0.303967i 1.24237 + 1.55788i 1.49454 + 1.66324i −0.0261332 + 0.114497i −2.45766 + 0.979757i 0.334654 − 0.0763825i −1.02170 + 0.492026i 0.184584 − 0.0537234i
29.14 0.139647 − 0.289980i 0.237528 − 0.0542142i 1.18239 + 1.48267i 1.69208 − 1.46181i 0.0174490 − 0.0764491i −0.373253 − 2.61929i 1.22263 − 0.279057i −2.64943 + 1.27590i −0.187601 − 0.694804i
29.15 0.291861 − 0.606056i 2.72598 − 0.622187i 0.964858 + 1.20989i −1.78067 + 1.35249i 0.418528 − 1.83369i −2.59609 + 0.510203i 2.32648 − 0.531004i 4.34094 − 2.09049i 0.299978 + 1.47392i
29.16 0.468810 − 0.973493i −3.06012 + 0.698453i 0.519073 + 0.650897i 0.507620 − 2.17769i −0.754675 + 3.30645i −0.788721 + 2.52545i 2.98380 − 0.681033i 6.17360 − 2.97305i −1.88199 − 1.51509i
29.17 0.613663 − 1.27428i 1.95484 − 0.446179i −0.000237008 0 0.000297199i 0.282411 − 2.21816i 0.631052 − 2.76482i −1.75956 + 1.97584i 2.75725 − 0.629324i 0.919404 − 0.442762i −2.65326 − 1.72108i
29.18 0.655500 − 1.36116i −1.90587 + 0.435002i −0.176096 − 0.220818i −1.27284 + 1.83844i −0.657190 + 2.87934i 2.62444 − 0.335138i 2.52979 − 0.577408i 0.740208 − 0.356465i 1.66807 + 2.93764i
29.19 0.689327 − 1.43140i −0.0543958 + 0.0124155i −0.326759 − 0.409743i 2.09399 + 0.784343i −0.0197249 + 0.0864206i 1.15602 + 2.37984i 2.28605 − 0.521777i −2.70010 + 1.30030i 2.56615 − 2.45668i
29.20 0.902155 − 1.87334i 1.48830 − 0.339694i −1.44855 − 1.81643i 0.927825 + 2.03449i 0.706310 − 3.09455i −0.998185 − 2.45023i −0.655370 + 0.149584i −0.603274 + 0.290522i 4.64834 + 0.0972885i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
49.e even 7 1 inner
245.p even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.p.b ✓ 144
5.b even 2 1 inner 245.2.p.b ✓ 144
49.e even 7 1 inner 245.2.p.b ✓ 144
245.p even 14 1 inner 245.2.p.b ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.p.b ✓ 144 1.a even 1 1 trivial
245.2.p.b ✓ 144 5.b even 2 1 inner
245.2.p.b ✓ 144 49.e even 7 1 inner
245.2.p.b ✓ 144 245.p even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{144} - 32 T_{2}^{142} + 613 T_{2}^{140} - 9114 T_{2}^{138} + 115945 T_{2}^{136} + \cdots + 8208541201 \) acting on \(S_{2}^{\mathrm{new}}(245, [\chi])\). Copy content Toggle raw display