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Properties

Label 368.2.i.b
Level $368$
Weight $2$
Character orbit 368.i
Analytic conductor $2.938$
Analytic rank $0$
Dimension $80$
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] = [368,2,Mod(91,368)] 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("368.91"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(368, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 368.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [80] 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: \(2.93849479438\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{2} - 4 q^{3} - 16 q^{6} - 4 q^{8} + 20 q^{12} - 4 q^{13} - 16 q^{16} + 24 q^{18} + 4 q^{23} + 4 q^{24} - 24 q^{26} + 44 q^{27} - 20 q^{29} - 24 q^{32} + 16 q^{35} + 104 q^{36} - 128 q^{39} - 16 q^{46}+ \cdots - 88 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −1.41406 + 0.0206295i −0.728611 − 0.728611i 1.99915 − 0.0583429i −2.62365 + 2.62365i 1.04533 + 1.01527i 1.75252i −2.82572 + 0.123742i − 1.93825i 3.65588 − 3.76413i
91.2 −1.41406 + 0.0206295i −0.728611 − 0.728611i 1.99915 − 0.0583429i 2.62365 − 2.62365i 1.04533 + 1.01527i − 1.75252i −2.82572 + 0.123742i − 1.93825i −3.65588 + 3.76413i
91.3 −1.39222 + 0.248418i 2.02263 + 2.02263i 1.87658 − 0.691707i −2.65107 + 2.65107i −3.31841 − 2.31349i − 3.33656i −2.44078 + 1.42919i 5.18204i 3.03230 − 4.34945i
91.4 −1.39222 + 0.248418i 2.02263 + 2.02263i 1.87658 − 0.691707i 2.65107 − 2.65107i −3.31841 − 2.31349i 3.33656i −2.44078 + 1.42919i 5.18204i −3.03230 + 4.34945i
91.5 −1.33867 − 0.456028i 0.860099 + 0.860099i 1.58408 + 1.22094i −0.0592628 + 0.0592628i −0.759160 − 1.54362i − 2.65917i −1.56377 − 2.35682i − 1.52046i 0.106359 − 0.0523078i
91.6 −1.33867 − 0.456028i 0.860099 + 0.860099i 1.58408 + 1.22094i 0.0592628 − 0.0592628i −0.759160 − 1.54362i 2.65917i −1.56377 − 2.35682i − 1.52046i −0.106359 + 0.0523078i
91.7 −1.17696 + 0.784064i 0.290642 + 0.290642i 0.770488 − 1.84563i −0.682402 + 0.682402i −0.569956 − 0.114193i 1.60196i 0.540255 + 2.77635i − 2.83105i 0.268116 − 1.33821i
91.8 −1.17696 + 0.784064i 0.290642 + 0.290642i 0.770488 − 1.84563i 0.682402 − 0.682402i −0.569956 − 0.114193i − 1.60196i 0.540255 + 2.77635i − 2.83105i −0.268116 + 1.33821i
91.9 −1.11619 + 0.868407i −1.89356 − 1.89356i 0.491738 − 1.93861i −1.65650 + 1.65650i 3.75794 + 0.469183i − 3.05009i 1.13463 + 2.59087i 4.17114i 0.410443 − 3.28747i
91.10 −1.11619 + 0.868407i −1.89356 − 1.89356i 0.491738 − 1.93861i 1.65650 − 1.65650i 3.75794 + 0.469183i 3.05009i 1.13463 + 2.59087i 4.17114i −0.410443 + 3.28747i
91.11 −0.949316 − 1.04824i −1.48829 − 1.48829i −0.197597 + 1.99021i −1.49485 + 1.49485i −0.147221 + 2.97294i 1.97830i 2.27380 − 1.68222i 1.43002i 2.98604 + 0.147870i
91.12 −0.949316 − 1.04824i −1.48829 − 1.48829i −0.197597 + 1.99021i 1.49485 − 1.49485i −0.147221 + 2.97294i − 1.97830i 2.27380 − 1.68222i 1.43002i −2.98604 − 0.147870i
91.13 −0.786933 + 1.17505i 1.81358 + 1.81358i −0.761472 − 1.84937i −1.82931 + 1.82931i −3.55821 + 0.703875i 3.62156i 2.77232 + 0.560564i 3.57815i −0.709980 − 3.58907i
91.14 −0.786933 + 1.17505i 1.81358 + 1.81358i −0.761472 − 1.84937i 1.82931 − 1.82931i −3.55821 + 0.703875i − 3.62156i 2.77232 + 0.560564i 3.57815i 0.709980 + 3.58907i
91.15 −0.750696 − 1.19852i 0.670749 + 0.670749i −0.872910 + 1.79945i −1.44943 + 1.44943i 0.300379 − 1.30744i 2.27607i 2.81197 − 0.304639i − 2.10019i 2.82526 + 0.649093i
91.16 −0.750696 − 1.19852i 0.670749 + 0.670749i −0.872910 + 1.79945i 1.44943 − 1.44943i 0.300379 − 1.30744i − 2.27607i 2.81197 − 0.304639i − 2.10019i −2.82526 − 0.649093i
91.17 −0.399394 + 1.35664i −0.289574 − 0.289574i −1.68097 − 1.08367i −1.68071 + 1.68071i 0.508503 − 0.277195i − 3.16447i 2.14153 − 1.84766i − 2.83229i −1.60886 − 2.95139i
91.18 −0.399394 + 1.35664i −0.289574 − 0.289574i −1.68097 − 1.08367i 1.68071 − 1.68071i 0.508503 − 0.277195i 3.16447i 2.14153 − 1.84766i − 2.83229i 1.60886 + 2.95139i
91.19 −0.158965 − 1.40525i 0.802883 + 0.802883i −1.94946 + 0.446771i −2.06308 + 2.06308i 1.00062 − 1.25588i − 4.69475i 0.937721 + 2.66846i − 1.71076i 3.22711 + 2.57119i
91.20 −0.158965 − 1.40525i 0.802883 + 0.802883i −1.94946 + 0.446771i 2.06308 − 2.06308i 1.00062 − 1.25588i 4.69475i 0.937721 + 2.66846i − 1.71076i −3.22711 − 2.57119i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.40
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner
23.b odd 2 1 inner
368.i even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.2.i.b ✓ 80
4.b odd 2 1 1472.2.i.b 80
16.e even 4 1 1472.2.i.b 80
16.f odd 4 1 inner 368.2.i.b ✓ 80
23.b odd 2 1 inner 368.2.i.b ✓ 80
92.b even 2 1 1472.2.i.b 80
368.i even 4 1 inner 368.2.i.b ✓ 80
368.k odd 4 1 1472.2.i.b 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
368.2.i.b ✓ 80 1.a even 1 1 trivial
368.2.i.b ✓ 80 16.f odd 4 1 inner
368.2.i.b ✓ 80 23.b odd 2 1 inner
368.2.i.b ✓ 80 368.i even 4 1 inner
1472.2.i.b 80 4.b odd 2 1
1472.2.i.b 80 16.e even 4 1
1472.2.i.b 80 92.b even 2 1
1472.2.i.b 80 368.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 2 T_{3}^{39} + 2 T_{3}^{38} - 10 T_{3}^{37} + 207 T_{3}^{36} + 360 T_{3}^{35} + 356 T_{3}^{34} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display