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Properties

Label 770.2.o.b
Level $770$
Weight $2$
Character orbit 770.o
Analytic conductor $6.148$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [770,2,Mod(439,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.439");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.o (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{2} - 24 q^{4} - 6 q^{5} + 4 q^{7} - 48 q^{8} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{2} - 24 q^{4} - 6 q^{5} + 4 q^{7} - 48 q^{8} - 28 q^{9} - 6 q^{10} + q^{11} + 2 q^{14} + 4 q^{15} - 24 q^{16} + 28 q^{18} + 2 q^{22} - 6 q^{26} - 2 q^{28} + 2 q^{30} + 12 q^{31} + 24 q^{32} - 24 q^{33} + 8 q^{35} + 56 q^{36} + 6 q^{40} - 24 q^{43} + q^{44} + 6 q^{45} - 36 q^{49} - 6 q^{52} - 4 q^{56} + 4 q^{57} - 2 q^{60} + 52 q^{63} + 48 q^{64} - 24 q^{66} - 2 q^{70} - 40 q^{71} + 28 q^{72} - 42 q^{73} + 90 q^{75} - 19 q^{77} + 6 q^{80} - 72 q^{81} + 40 q^{85} - 12 q^{86} - 18 q^{87} - q^{88} + 60 q^{89} - 56 q^{91} - 24 q^{95} - 42 q^{98} - 58 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
439.1 0.500000 − 0.866025i −1.65185 − 2.86109i −0.500000 − 0.866025i −0.232176 − 2.22398i −3.30370 −2.27237 − 1.35511i −1.00000 −3.95721 + 6.85408i −2.04211 − 0.910921i
439.2 0.500000 − 0.866025i −1.64811 − 2.85461i −0.500000 − 0.866025i −0.0163520 + 2.23601i −3.29622 1.01398 − 2.44374i −1.00000 −3.93253 + 6.81133i 1.92826 + 1.13217i
439.3 0.500000 − 0.866025i −1.44024 − 2.49457i −0.500000 − 0.866025i −2.16254 − 0.568686i −2.88048 0.391243 + 2.61666i −1.00000 −2.64857 + 4.58746i −1.57377 + 1.58848i
439.4 0.500000 − 0.866025i −1.23964 − 2.14712i −0.500000 − 0.866025i 2.22778 + 0.192296i −2.47929 2.62866 + 0.300206i −1.00000 −1.57343 + 2.72526i 1.28043 − 1.83317i
439.5 0.500000 − 0.866025i −0.987813 − 1.71094i −0.500000 − 0.866025i 1.00552 + 1.99723i −1.97563 −1.58012 + 2.12208i −1.00000 −0.451550 + 0.782108i 2.23241 + 0.127806i
439.6 0.500000 − 0.866025i −0.781614 − 1.35379i −0.500000 − 0.866025i −0.438635 − 2.19262i −1.56323 0.725302 − 2.54439i −1.00000 0.278160 − 0.481787i −2.11819 − 0.716443i
439.7 0.500000 − 0.866025i −0.732537 − 1.26879i −0.500000 − 0.866025i −0.423475 + 2.19560i −1.46507 1.43527 + 2.22261i −1.00000 0.426780 − 0.739205i 1.68971 + 1.46454i
439.8 0.500000 − 0.866025i −0.730155 − 1.26467i −0.500000 − 0.866025i −2.18303 + 0.484147i −1.46031 1.48697 − 2.18836i −1.00000 0.433747 − 0.751273i −0.672229 + 2.13263i
439.9 0.500000 − 0.866025i −0.617381 − 1.06934i −0.500000 − 0.866025i −1.46901 − 1.68583i −1.23476 −1.62005 + 2.09176i −1.00000 0.737681 − 1.27770i −2.19447 + 0.429285i
439.10 0.500000 − 0.866025i −0.533793 − 0.924557i −0.500000 − 0.866025i 2.23597 + 0.0212957i −1.06759 −1.22107 − 2.34712i −1.00000 0.930129 − 1.61103i 1.13643 − 1.92576i
439.11 0.500000 − 0.866025i −0.340512 − 0.589784i −0.500000 − 0.866025i 1.37482 − 1.76348i −0.681024 −2.52726 + 0.782908i −1.00000 1.26810 − 2.19642i −0.839811 − 2.07237i
439.12 0.500000 − 0.866025i −0.0752187 − 0.130283i −0.500000 − 0.866025i 0.125277 − 2.23256i −0.150437 2.53943 + 0.742498i −1.00000 1.48868 − 2.57848i −1.87081 − 1.22477i
439.13 0.500000 − 0.866025i 0.0752187 + 0.130283i −0.500000 − 0.866025i −1.87081 + 1.22477i 0.150437 2.53943 + 0.742498i −1.00000 1.48868 − 2.57848i 0.125277 + 2.23256i
439.14 0.500000 − 0.866025i 0.340512 + 0.589784i −0.500000 − 0.866025i −0.839811 + 2.07237i 0.681024 −2.52726 + 0.782908i −1.00000 1.26810 − 2.19642i 1.37482 + 1.76348i
439.15 0.500000 − 0.866025i 0.533793 + 0.924557i −0.500000 − 0.866025i 1.13643 + 1.92576i 1.06759 −1.22107 − 2.34712i −1.00000 0.930129 − 1.61103i 2.23597 − 0.0212957i
439.16 0.500000 − 0.866025i 0.617381 + 1.06934i −0.500000 − 0.866025i −2.19447 − 0.429285i 1.23476 −1.62005 + 2.09176i −1.00000 0.737681 − 1.27770i −1.46901 + 1.68583i
439.17 0.500000 − 0.866025i 0.730155 + 1.26467i −0.500000 − 0.866025i −0.672229 − 2.13263i 1.46031 1.48697 − 2.18836i −1.00000 0.433747 − 0.751273i −2.18303 − 0.484147i
439.18 0.500000 − 0.866025i 0.732537 + 1.26879i −0.500000 − 0.866025i 1.68971 − 1.46454i 1.46507 1.43527 + 2.22261i −1.00000 0.426780 − 0.739205i −0.423475 − 2.19560i
439.19 0.500000 − 0.866025i 0.781614 + 1.35379i −0.500000 − 0.866025i −2.11819 + 0.716443i 1.56323 0.725302 − 2.54439i −1.00000 0.278160 − 0.481787i −0.438635 + 2.19262i
439.20 0.500000 − 0.866025i 0.987813 + 1.71094i −0.500000 − 0.866025i 2.23241 − 0.127806i 1.97563 −1.58012 + 2.12208i −1.00000 −0.451550 + 0.782108i 1.00552 − 1.99723i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 439.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
55.d odd 2 1 inner
385.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.o.b yes 48
5.b even 2 1 770.2.o.a ✓ 48
7.d odd 6 1 inner 770.2.o.b yes 48
11.b odd 2 1 770.2.o.a ✓ 48
35.i odd 6 1 770.2.o.a ✓ 48
55.d odd 2 1 inner 770.2.o.b yes 48
77.i even 6 1 770.2.o.a ✓ 48
385.o even 6 1 inner 770.2.o.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.o.a ✓ 48 5.b even 2 1
770.2.o.a ✓ 48 11.b odd 2 1
770.2.o.a ✓ 48 35.i odd 6 1
770.2.o.a ✓ 48 77.i even 6 1
770.2.o.b yes 48 1.a even 1 1 trivial
770.2.o.b yes 48 7.d odd 6 1 inner
770.2.o.b yes 48 55.d odd 2 1 inner
770.2.o.b yes 48 385.o even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{24} - 87 T_{17}^{22} + 5595 T_{17}^{20} - 2505 T_{17}^{19} - 137015 T_{17}^{18} + \cdots + 58982400 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\). Copy content Toggle raw display