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(235,513)] 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("513.235"); 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([4, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,2,0,34,-3] 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} + 34 q^{4} - 3 q^{5} + q^{7} + 36 q^{8} - 8 q^{10} - 7 q^{11} + 8 q^{13} - q^{14} + 22 q^{16} + 7 q^{17} + 7 q^{19} + 3 q^{20} - 8 q^{22} + 10 q^{23} - 9 q^{25} + 4 q^{26} - 10 q^{28} - 10 q^{29}+ \cdots - 18 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1 −2.46808 0 4.09140 0.957619 − 1.65865i 0 −0.324708 + 0.562412i −5.16173 0 −2.36348 + 4.09366i
235.2 −2.09768 0 2.40028 −1.44796 + 2.50795i 0 0.116480 − 0.201749i −0.839660 0 3.03737 − 5.26088i
235.3 −1.95703 0 1.82996 −0.0981173 + 0.169944i 0 −2.23368 + 3.86885i 0.332766 0 0.192018 − 0.332586i
235.4 −1.77797 0 1.16118 −0.639786 + 1.10814i 0 0.657761 − 1.13928i 1.49140 0 1.13752 − 1.97024i
235.5 −1.60662 0 0.581222 1.87940 − 3.25521i 0 2.27973 − 3.94861i 2.27943 0 −3.01948 + 5.22989i
235.6 −0.791858 0 −1.37296 1.29546 − 2.24381i 0 −0.373088 + 0.646207i 2.67091 0 −1.02582 + 1.77678i
235.7 −0.539090 0 −1.70938 −0.473662 + 0.820407i 0 1.18430 − 2.05126i 1.99969 0 0.255347 − 0.442274i
235.8 −0.370889 0 −1.86244 −1.77761 + 3.07890i 0 −0.124876 + 0.216291i 1.43254 0 0.659294 − 1.14193i
235.9 −0.146534 0 −1.97853 1.28502 − 2.22572i 0 −1.73898 + 3.01201i 0.582990 0 −0.188299 + 0.326143i
235.10 0.194693 0 −1.96209 −0.952817 + 1.65033i 0 1.69446 − 2.93489i −0.771394 0 −0.185507 + 0.321308i
235.11 1.23359 0 −0.478252 −0.275772 + 0.477650i 0 −1.62156 + 2.80862i −3.05715 0 −0.340189 + 0.589225i
235.12 1.69423 0 0.870412 0.0441088 − 0.0763987i 0 1.84695 − 3.19901i −1.91378 0 0.0747304 − 0.129437i
235.13 2.02031 0 2.08166 −2.09369 + 3.62638i 0 −0.976107 + 1.69067i 0.164982 0 −4.22991 + 7.32643i
235.14 2.39943 0 3.75726 0.359839 − 0.623259i 0 1.65862 − 2.87282i 4.21641 0 0.863408 − 1.49547i
235.15 2.60319 0 4.77661 1.43897 − 2.49237i 0 −1.80240 + 3.12185i 7.22804 0 3.74592 − 6.48812i
235.16 2.61030 0 4.81368 −1.00100 + 1.73379i 0 0.257107 − 0.445323i 7.34455 0 −2.61292 + 4.52572i
334.1 −2.46808 0 4.09140 0.957619 + 1.65865i 0 −0.324708 − 0.562412i −5.16173 0 −2.36348 − 4.09366i
334.2 −2.09768 0 2.40028 −1.44796 − 2.50795i 0 0.116480 + 0.201749i −0.839660 0 3.03737 + 5.26088i
334.3 −1.95703 0 1.82996 −0.0981173 − 0.169944i 0 −2.23368 − 3.86885i 0.332766 0 0.192018 + 0.332586i
334.4 −1.77797 0 1.16118 −0.639786 − 1.10814i 0 0.657761 + 1.13928i 1.49140 0 1.13752 + 1.97024i
See all 32 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 235.16
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{2}^{16} - T_{2}^{15} - 24 T_{2}^{14} + 17 T_{2}^{13} + 235 T_{2}^{12} - 96 T_{2}^{11} - 1193 T_{2}^{10} + \cdots - 9 \) Copy content Toggle raw display
\( T_{5}^{32} + 3 T_{5}^{31} + 49 T_{5}^{30} + 110 T_{5}^{29} + 1345 T_{5}^{28} + 2690 T_{5}^{27} + \cdots + 35721 \) Copy content Toggle raw display