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Properties

Label 325.3.w.e
Level $325$
Weight $3$
Character orbit 325.w
Analytic conductor $8.856$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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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] = [325,3,Mod(24,325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 7])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("325.24"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 325.w (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,0,-12] 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: \(8.85560859171\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 12 q^{3} - 12 q^{6} - 44 q^{7} + 36 q^{8} + 72 q^{9} - 12 q^{11} + 120 q^{12} - 36 q^{13} - 48 q^{14} + 128 q^{16} - 32 q^{17} + 136 q^{18} - 68 q^{19} - 48 q^{21} - 72 q^{22} - 28 q^{23} + 56 q^{24}+ \cdots + 76 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 −3.30742 − 0.886220i 0.0285599 − 0.0164891i 6.68953 + 3.86220i 0 −0.109072 + 0.0292259i 0.692695 − 0.185607i −9.01754 − 9.01754i −4.49946 + 7.79329i 0
24.2 −2.72224 − 0.729421i −4.15192 + 2.39711i 3.41442 + 1.97132i 0 13.0510 − 3.49701i −4.75472 + 1.27402i 0.114321 + 0.114321i 6.99232 − 12.1111i 0
24.3 −2.66986 − 0.715386i 3.65949 − 2.11281i 3.15226 + 1.81996i 0 −11.2818 + 3.02294i 3.52221 − 0.943772i 0.703773 + 0.703773i 4.42790 − 7.66935i 0
24.4 −1.16577 − 0.312366i −2.06426 + 1.19180i −2.20266 − 1.27171i 0 2.77872 − 0.744556i 7.49531 − 2.00836i 5.58415 + 5.58415i −1.65922 + 2.87385i 0
24.5 −0.401493 − 0.107580i 3.61165 − 2.08519i −3.31448 − 1.91361i 0 −1.67438 + 0.448649i −10.0361 + 2.68917i 2.30053 + 2.30053i 4.19603 − 7.26773i 0
24.6 −0.346703 − 0.0928988i 0.511202 − 0.295143i −3.35253 − 1.93558i 0 −0.204654 + 0.0548368i −0.327396 + 0.0877254i 1.99774 + 1.99774i −4.32578 + 7.49247i 0
24.7 1.14195 + 0.305985i −3.98555 + 2.30106i −2.25367 − 1.30116i 0 −5.25539 + 1.40818i −11.4086 + 3.05692i −5.51932 − 5.51932i 6.08971 − 10.5477i 0
24.8 2.20214 + 0.590063i −1.83281 + 1.05817i 1.03716 + 0.598805i 0 −4.66050 + 1.24878i −1.17338 + 0.314408i −4.51768 − 4.51768i −2.26053 + 3.91536i 0
24.9 3.53665 + 0.947644i 4.05506 − 2.34119i 8.14580 + 4.70298i 0 16.5600 − 4.43723i −10.5506 + 2.82701i 13.9961 + 13.9961i 6.46235 − 11.1931i 0
24.10 3.73273 + 1.00018i −2.83142 + 1.63472i 9.46878 + 5.46680i 0 −12.2039 + 3.27004i 8.61234 − 2.30767i 18.9464 + 18.9464i 0.844632 − 1.46295i 0
124.1 −0.974678 − 3.63755i −4.47773 − 2.58522i −8.81765 + 5.09087i 0 −5.03951 + 18.8077i −0.239364 + 0.893318i 16.4612 + 16.4612i 8.86671 + 15.3576i 0
124.2 −0.749686 − 2.79787i 0.461903 + 0.266680i −3.80193 + 2.19504i 0 0.399852 − 1.49227i −2.69020 + 10.0400i 0.798969 + 0.798969i −4.35776 − 7.54787i 0
124.3 −0.624525 − 2.33076i 1.82226 + 1.05208i −1.57831 + 0.911238i 0 1.31411 − 4.90432i 0.565655 − 2.11105i −3.71537 − 3.71537i −2.28624 − 3.95988i 0
124.4 −0.340684 − 1.27145i −3.38958 − 1.95698i 1.96358 − 1.13368i 0 −1.33342 + 4.97639i 2.05866 − 7.68303i −5.83343 − 5.83343i 3.15951 + 5.47243i 0
124.5 −0.206966 − 0.772409i 5.13291 + 2.96349i 2.91032 − 1.68027i 0 1.22668 − 4.57805i 0.592215 − 2.21018i −4.16197 − 4.16197i 13.0645 + 22.6284i 0
124.6 −0.0113525 − 0.0423683i −2.77843 − 1.60413i 3.46244 − 1.99904i 0 −0.0364219 + 0.135928i −2.16799 + 8.09104i −0.248066 − 0.248066i 0.646457 + 1.11970i 0
124.7 0.598846 + 2.23492i 3.51738 + 2.03076i −1.17216 + 0.676749i 0 −2.43222 + 9.07719i −2.56698 + 9.58012i 4.32988 + 4.32988i 3.74797 + 6.49168i 0
124.8 0.614128 + 2.29196i 1.05591 + 0.609633i −1.41181 + 0.815107i 0 −0.748785 + 2.79450i 1.33475 − 4.98135i 3.97609 + 3.97609i −3.75670 − 6.50679i 0
124.9 0.683209 + 2.54977i −3.75874 − 2.17011i −2.57045 + 1.48405i 0 2.96528 − 11.0666i −2.17385 + 8.11293i 1.92611 + 1.92611i 4.91876 + 8.51954i 0
124.10 1.01171 + 3.77575i −0.585883 − 0.338260i −9.76865 + 5.63993i 0 0.684442 − 2.55437i 1.21531 − 4.53560i −20.1218 − 20.1218i −4.27116 − 7.39787i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.10
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.s odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.3.w.e 40
5.b even 2 1 325.3.w.f 40
5.c odd 4 1 65.3.p.a ✓ 40
5.c odd 4 1 325.3.t.d 40
13.f odd 12 1 325.3.w.f 40
65.o even 12 1 65.3.p.a ✓ 40
65.s odd 12 1 inner 325.3.w.e 40
65.t even 12 1 325.3.t.d 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.3.p.a ✓ 40 5.c odd 4 1
65.3.p.a ✓ 40 65.o even 12 1
325.3.t.d 40 5.c odd 4 1
325.3.t.d 40 65.t even 12 1
325.3.w.e 40 1.a even 1 1 trivial
325.3.w.e 40 65.s odd 12 1 inner
325.3.w.f 40 5.b even 2 1
325.3.w.f 40 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 12 T_{2}^{37} - 336 T_{2}^{36} - 132 T_{2}^{35} + 72 T_{2}^{34} + 5796 T_{2}^{33} + \cdots + 158986881 \) acting on \(S_{3}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display