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Properties

Label 765.2.n.e
Level $765$
Weight $2$
Character orbit 765.n
Analytic conductor $6.109$
Analytic rank $0$
Dimension $24$
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] = [765,2,Mod(188,765)] 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("765.188"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(765, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 765.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.10855575463\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{7} - 20 q^{10} - 20 q^{13} - 8 q^{16} - 40 q^{22} - 24 q^{28} + 72 q^{31} + 16 q^{37} - 88 q^{40} + 36 q^{43} + 56 q^{46} + 32 q^{52} - 88 q^{55} - 80 q^{58} - 8 q^{61} + 104 q^{67} - 136 q^{70}+ \cdots + 88 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1 −1.73886 − 1.73886i 0 4.04726i −2.10859 − 0.744208i 0 −0.649310 + 0.649310i 3.55991 − 3.55991i 0 2.37247 + 4.96062i
188.2 −1.63882 − 1.63882i 0 3.37143i −0.913805 − 2.04082i 0 −2.58679 + 2.58679i 2.24752 − 2.24752i 0 −1.84698 + 4.84209i
188.3 −1.34783 − 1.34783i 0 1.63327i 1.47114 − 1.68397i 0 2.83235 − 2.83235i −0.494282 + 0.494282i 0 −4.25254 + 0.286850i
188.4 −1.16413 − 1.16413i 0 0.710382i 2.23590 − 0.0275136i 0 2.04131 − 2.04131i −1.50128 + 1.50128i 0 −2.63490 − 2.57084i
188.5 −0.281958 − 0.281958i 0 − 1.84100i −0.408357 + 2.19846i 0 −0.380936 + 0.380936i −1.08300 + 1.08300i 0 0.735014 − 0.504734i
188.6 −0.198302 − 0.198302i 0 − 1.92135i −1.54549 + 1.61600i 0 0.743380 − 0.743380i −0.777612 + 0.777612i 0 0.626930 − 0.0139813i
188.7 0.198302 + 0.198302i 0 − 1.92135i 1.54549 − 1.61600i 0 0.743380 − 0.743380i 0.777612 − 0.777612i 0 0.626930 − 0.0139813i
188.8 0.281958 + 0.281958i 0 − 1.84100i 0.408357 − 2.19846i 0 −0.380936 + 0.380936i 1.08300 − 1.08300i 0 0.735014 − 0.504734i
188.9 1.16413 + 1.16413i 0 0.710382i −2.23590 + 0.0275136i 0 2.04131 − 2.04131i 1.50128 − 1.50128i 0 −2.63490 − 2.57084i
188.10 1.34783 + 1.34783i 0 1.63327i −1.47114 + 1.68397i 0 2.83235 − 2.83235i 0.494282 − 0.494282i 0 −4.25254 + 0.286850i
188.11 1.63882 + 1.63882i 0 3.37143i 0.913805 + 2.04082i 0 −2.58679 + 2.58679i −2.24752 + 2.24752i 0 −1.84698 + 4.84209i
188.12 1.73886 + 1.73886i 0 4.04726i 2.10859 + 0.744208i 0 −0.649310 + 0.649310i −3.55991 + 3.55991i 0 2.37247 + 4.96062i
647.1 −1.73886 + 1.73886i 0 − 4.04726i −2.10859 + 0.744208i 0 −0.649310 − 0.649310i 3.55991 + 3.55991i 0 2.37247 − 4.96062i
647.2 −1.63882 + 1.63882i 0 − 3.37143i −0.913805 + 2.04082i 0 −2.58679 − 2.58679i 2.24752 + 2.24752i 0 −1.84698 − 4.84209i
647.3 −1.34783 + 1.34783i 0 − 1.63327i 1.47114 + 1.68397i 0 2.83235 + 2.83235i −0.494282 − 0.494282i 0 −4.25254 − 0.286850i
647.4 −1.16413 + 1.16413i 0 − 0.710382i 2.23590 + 0.0275136i 0 2.04131 + 2.04131i −1.50128 − 1.50128i 0 −2.63490 + 2.57084i
647.5 −0.281958 + 0.281958i 0 1.84100i −0.408357 − 2.19846i 0 −0.380936 − 0.380936i −1.08300 − 1.08300i 0 0.735014 + 0.504734i
647.6 −0.198302 + 0.198302i 0 1.92135i −1.54549 − 1.61600i 0 0.743380 + 0.743380i −0.777612 − 0.777612i 0 0.626930 + 0.0139813i
647.7 0.198302 − 0.198302i 0 1.92135i 1.54549 + 1.61600i 0 0.743380 + 0.743380i 0.777612 + 0.777612i 0 0.626930 + 0.0139813i
647.8 0.281958 − 0.281958i 0 1.84100i 0.408357 + 2.19846i 0 −0.380936 − 0.380936i 1.08300 + 1.08300i 0 0.735014 + 0.504734i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 188.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 765.2.n.e ✓ 24
3.b odd 2 1 inner 765.2.n.e ✓ 24
5.c odd 4 1 inner 765.2.n.e ✓ 24
15.e even 4 1 inner 765.2.n.e ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
765.2.n.e ✓ 24 1.a even 1 1 trivial
765.2.n.e ✓ 24 3.b odd 2 1 inner
765.2.n.e ✓ 24 5.c odd 4 1 inner
765.2.n.e ✓ 24 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 86T_{2}^{20} + 2499T_{2}^{16} + 28102T_{2}^{12} + 103201T_{2}^{8} + 3224T_{2}^{4} + 16 \) acting on \(S_{2}^{\mathrm{new}}(765, [\chi])\). Copy content Toggle raw display