Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [513,2,Mod(64,513)] 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("513.64"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,-1,0,-17,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.09632562369\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{2} - 17 q^{4} + 6 q^{5} + q^{7} + 36 q^{8} - 8 q^{10} - 7 q^{11} - 4 q^{13} + 2 q^{14} - 11 q^{16} + 7 q^{17} + 7 q^{19} + 3 q^{20} + 16 q^{22} - 5 q^{23} + 18 q^{25} + 4 q^{26} - 10 q^{28}+ \cdots - 18 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −1.30515 + 2.26059i 0 −2.40684 − 4.16877i 2.00201 0 0.257107 + 0.445323i 7.34455 0 −2.61292 + 4.52572i
64.2 −1.30160 + 2.25443i 0 −2.38830 − 4.13666i −2.87794 0 −1.80240 − 3.12185i 7.22804 0 3.74592 − 6.48812i
64.3 −1.19971 + 2.07797i 0 −1.87863 − 3.25388i −0.719678 0 1.65862 + 2.87282i 4.21641 0 0.863408 − 1.49547i
64.4 −1.01016 + 1.74964i 0 −1.04083 − 1.80277i 4.18739 0 −0.976107 − 1.69067i 0.164982 0 −4.22991 + 7.32643i
64.5 −0.847114 + 1.46725i 0 −0.435206 − 0.753799i −0.0882176 0 1.84695 + 3.19901i −1.91378 0 0.0747304 − 0.129437i
64.6 −0.616796 + 1.06832i 0 0.239126 + 0.414178i 0.551543 0 −1.62156 − 2.80862i −3.05715 0 −0.340189 + 0.589225i
64.7 −0.0973467 + 0.168609i 0 0.981047 + 1.69922i 1.90563 0 1.69446 + 2.93489i −0.771394 0 −0.185507 + 0.321308i
64.8 0.0732670 − 0.126902i 0 0.989264 + 1.71346i −2.57004 0 −1.73898 − 3.01201i 0.582990 0 −0.188299 + 0.326143i
64.9 0.185445 − 0.321199i 0 0.931221 + 1.61292i 3.55521 0 −0.124876 − 0.216291i 1.43254 0 0.659294 − 1.14193i
64.10 0.269545 − 0.466866i 0 0.854691 + 1.48037i 0.947325 0 1.18430 + 2.05126i 1.99969 0 0.255347 − 0.442274i
64.11 0.395929 − 0.685769i 0 0.686481 + 1.18902i −2.59093 0 −0.373088 − 0.646207i 2.67091 0 −1.02582 + 1.77678i
64.12 0.803309 − 1.39137i 0 −0.290611 − 0.503353i −3.75880 0 2.27973 + 3.94861i 2.27943 0 −3.01948 + 5.22989i
64.13 0.888985 − 1.53977i 0 −0.580589 − 1.00561i 1.27957 0 0.657761 + 1.13928i 1.49140 0 1.13752 − 1.97024i
64.14 0.978515 − 1.69484i 0 −0.914982 − 1.58480i 0.196235 0 −2.23368 − 3.86885i 0.332766 0 0.192018 − 0.332586i
64.15 1.04884 − 1.81665i 0 −1.20014 − 2.07870i 2.89593 0 0.116480 + 0.201749i −0.839660 0 3.03737 − 5.26088i
64.16 1.23404 − 2.13742i 0 −2.04570 − 3.54325i −1.91524 0 −0.324708 − 0.562412i −5.16173 0 −2.36348 + 4.09366i
505.1 −1.30515 − 2.26059i 0 −2.40684 + 4.16877i 2.00201 0 0.257107 − 0.445323i 7.34455 0 −2.61292 − 4.52572i
505.2 −1.30160 − 2.25443i 0 −2.38830 + 4.13666i −2.87794 0 −1.80240 + 3.12185i 7.22804 0 3.74592 + 6.48812i
505.3 −1.19971 − 2.07797i 0 −1.87863 + 3.25388i −0.719678 0 1.65862 − 2.87282i 4.21641 0 0.863408 + 1.49547i
505.4 −1.01016 − 1.74964i 0 −1.04083 + 1.80277i 4.18739 0 −0.976107 + 1.69067i 0.164982 0 −4.22991 − 7.32643i
See all 32 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 64.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 513.2.g.c 32
3.b odd 2 1 171.2.g.c ✓ 32
9.c even 3 1 513.2.h.c 32
9.d odd 6 1 171.2.h.c yes 32
19.c even 3 1 513.2.h.c 32
57.h odd 6 1 171.2.h.c yes 32
171.g even 3 1 inner 513.2.g.c 32
171.n odd 6 1 171.2.g.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.g.c ✓ 32 3.b odd 2 1
171.2.g.c ✓ 32 171.n odd 6 1
171.2.h.c yes 32 9.d odd 6 1
171.2.h.c yes 32 57.h odd 6 1
513.2.g.c 32 1.a even 1 1 trivial
513.2.g.c 32 171.g even 3 1 inner
513.2.h.c 32 9.c even 3 1
513.2.h.c 32 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(513, [\chi])\):

\( T_{2}^{32} + T_{2}^{31} + 25 T_{2}^{30} + 10 T_{2}^{29} + 358 T_{2}^{28} + 34 T_{2}^{27} + 3447 T_{2}^{26} + \cdots + 81 \) Copy content Toggle raw display
\( T_{5}^{16} - 3 T_{5}^{15} - 40 T_{5}^{14} + 115 T_{5}^{13} + 600 T_{5}^{12} - 1690 T_{5}^{11} + \cdots - 189 \) Copy content Toggle raw display